LMFDB - Embedded newform 1472.4.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 $ x^8 - 4x^7 - 137x^6 + 344x^5 + 6175x^4 - 7924x^3 - 89643x^2 + 45072*x + 51084
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$5.24371$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24371 q^{3} +5.01937 q^{5} -22.4881 q^{7} -16.4784 q^{9} +50.5296 q^{11} +89.2035 q^{13} +16.2813 q^{15} -66.9007 q^{17} -97.7214 q^{19} -72.9448 q^{21} -23.0000 q^{23} -99.8060 q^{25} -141.031 q^{27} -29.9936 q^{29} -73.5761 q^{31} +163.903 q^{33} -112.876 q^{35} -1.12672 q^{37} +289.350 q^{39} +34.3225 q^{41} +160.562 q^{43} -82.7110 q^{45} -289.634 q^{47} +162.715 q^{49} -217.006 q^{51} +392.969 q^{53} +253.627 q^{55} -316.980 q^{57} +174.989 q^{59} +474.108 q^{61} +370.567 q^{63} +447.745 q^{65} -671.911 q^{67} -74.6052 q^{69} +481.540 q^{71} -778.702 q^{73} -323.741 q^{75} -1136.32 q^{77} +132.340 q^{79} -12.5472 q^{81} -808.059 q^{83} -335.799 q^{85} -97.2903 q^{87} -1055.47 q^{89} -2006.02 q^{91} -238.659 q^{93} -490.500 q^{95} +824.410 q^{97} -832.646 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

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
$3$	3.24371	0.624251	0.312126	−	0.950041i	$-0.398959\pi$
$3$	3.24371	0.624251	0.312126	+	0.950041i	$0.398959\pi$
$4$	0	0
$5$	5.01937	0.448946	0.224473	−	0.974480i	$-0.427934\pi$
$5$	5.01937	0.448946	0.224473	+	0.974480i	$0.427934\pi$
$6$	0	0
$7$	−22.4881	−1.21424	−0.607122	−	0.794609i	$-0.707676\pi$
$7$	−22.4881	−1.21424	−0.607122	+	0.794609i	$0.707676\pi$
$8$	0	0
$9$	−16.4784	−0.610310
$10$	0	0
$11$	50.5296	1.38502	0.692512	−	0.721406i	$-0.256504\pi$
$11$	50.5296	1.38502	0.692512	+	0.721406i	$0.256504\pi$
$12$	0	0
$13$	89.2035	1.90312	0.951561	−	0.307460i	$-0.0994790\pi$
$13$	89.2035	1.90312	0.951561	+	0.307460i	$0.0994790\pi$
$14$	0	0
$15$	16.2813	0.280255
$16$	0	0
$17$	−66.9007	−0.954459	−0.477230	−	0.878779i	$-0.658359\pi$
$17$	−66.9007	−0.954459	−0.477230	+	0.878779i	$0.658359\pi$
$18$	0	0
$19$	−97.7214	−1.17994	−0.589969	−	0.807426i	$-0.700860\pi$
$19$	−97.7214	−1.17994	−0.589969	+	0.807426i	$0.700860\pi$
$20$	0	0
$21$	−72.9448	−0.757993
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−99.8060	−0.798448
$26$	0	0
$27$	−141.031	−1.00524
$28$	0	0
$29$	−29.9936	−0.192057	−0.0960287	−	0.995379i	$-0.530614\pi$
$29$	−29.9936	−0.192057	−0.0960287	+	0.995379i	$0.530614\pi$
$30$	0	0
$31$	−73.5761	−0.426279	−0.213140	−	0.977022i	$-0.568369\pi$
$31$	−73.5761	−0.426279	−0.213140	+	0.977022i	$0.568369\pi$
$32$	0	0
$33$	163.903	0.864603
$34$	0	0
$35$	−112.876	−0.545130
$36$	0	0
$37$	−1.12672	−0.00500628	−0.00250314	−	0.999997i	$-0.500797\pi$
$37$	−1.12672	−0.00500628	−0.00250314	+	0.999997i	$0.500797\pi$
$38$	0	0
$39$	289.350	1.18803
$40$	0	0
$41$	34.3225	0.130738	0.0653692	−	0.997861i	$-0.479177\pi$
$41$	34.3225	0.130738	0.0653692	+	0.997861i	$0.479177\pi$
$42$	0	0
$43$	160.562	0.569429	0.284714	−	0.958612i	$-0.408101\pi$
$43$	160.562	0.569429	0.284714	+	0.958612i	$0.408101\pi$
$44$	0	0
$45$	−82.7110	−0.273996
$46$	0	0
$47$	−289.634	−0.898883	−0.449442	−	0.893310i	$-0.648377\pi$
$47$	−289.634	−0.898883	−0.449442	+	0.893310i	$0.648377\pi$
$48$	0	0
$49$	162.715	0.474388
$50$	0	0
$51$	−217.006	−0.595823
$52$	0	0
$53$	392.969	1.01846	0.509230	−	0.860630i	$-0.329930\pi$
$53$	392.969	1.01846	0.509230	+	0.860630i	$0.329930\pi$
$54$	0	0
$55$	253.627	0.621801
$56$	0	0
$57$	−316.980	−0.736579
$58$	0	0
$59$	174.989	0.386129	0.193065	−	0.981186i	$-0.438157\pi$
$59$	174.989	0.386129	0.193065	+	0.981186i	$0.438157\pi$
$60$	0	0
$61$	474.108	0.995136	0.497568	−	0.867425i	$-0.334226\pi$
$61$	474.108	0.995136	0.497568	+	0.867425i	$0.334226\pi$
$62$	0	0
$63$	370.567	0.741065
$64$	0	0
$65$	447.745	0.854399
$66$	0	0
$67$	−671.911	−1.22518	−0.612589	−	0.790401i	$-0.709872\pi$
$67$	−671.911	−1.22518	−0.612589	+	0.790401i	$0.709872\pi$
$68$	0	0
$69$	−74.6052	−0.130165
$70$	0	0
$71$	481.540	0.804906	0.402453	−	0.915441i	$-0.368158\pi$
$71$	481.540	0.804906	0.402453	+	0.915441i	$0.368158\pi$
$72$	0	0
$73$	−778.702	−1.24850	−0.624248	−	0.781226i	$-0.714595\pi$
$73$	−778.702	−1.24850	−0.624248	+	0.781226i	$0.714595\pi$
$74$	0	0
$75$	−323.741	−0.498432
$76$	0	0
$77$	−1136.32	−1.68176
$78$	0	0
$79$	132.340	0.188474	0.0942369	−	0.995550i	$-0.469959\pi$
$79$	132.340	0.188474	0.0942369	+	0.995550i	$0.469959\pi$
$80$	0	0
$81$	−12.5472	−0.0172115
$82$	0	0
$83$	−808.059	−1.06863	−0.534313	−	0.845287i	$-0.679430\pi$
$83$	−808.059	−1.06863	−0.534313	+	0.845287i	$0.679430\pi$
$84$	0	0
$85$	−335.799	−0.428500
$86$	0	0
$87$	−97.2903	−0.119892
$88$	0	0
$89$	−1055.47	−1.25707	−0.628534	−	0.777782i	$-0.716345\pi$
$89$	−1055.47	−1.25707	−0.628534	+	0.777782i	$0.716345\pi$
$90$	0	0
$91$	−2006.02	−2.31085
$92$	0	0
$93$	−238.659	−0.266106
$94$	0	0
$95$	−490.500	−0.529729
$96$	0	0
$97$	824.410	0.862950	0.431475	−	0.902125i	$-0.357993\pi$
$97$	824.410	0.862950	0.431475	+	0.902125i	$0.357993\pi$
$98$	0	0
$99$	−832.646	−0.845294
$100$	0	0
$101$	−1026.60	−1.01139	−0.505694	−	0.862713i	$-0.668763\pi$
$101$	−1026.60	−1.01139	−0.505694	+	0.862713i	$0.668763\pi$
$102$	0	0
$103$	−1258.22	−1.20365	−0.601825	−	0.798628i	$-0.705560\pi$
$103$	−1258.22	−1.20365	−0.601825	+	0.798628i	$0.705560\pi$
$104$	0	0
$105$	−366.137	−0.340298
$106$	0	0
$107$	507.430	0.458459	0.229229	−	0.973372i	$-0.426379\pi$
$107$	507.430	0.458459	0.229229	+	0.973372i	$0.426379\pi$
$108$	0	0
$109$	479.750	0.421575	0.210787	−	0.977532i	$-0.432397\pi$
$109$	479.750	0.421575	0.210787	+	0.977532i	$0.432397\pi$
$110$	0	0
$111$	−3.65476	−0.00312518
$112$	0	0
$113$	−1634.89	−1.36104	−0.680519	−	0.732731i	$-0.738245\pi$
$113$	−1634.89	−1.36104	−0.680519	+	0.732731i	$0.738245\pi$
$114$	0	0
$115$	−115.445	−0.0936117
$116$	0	0
$117$	−1469.93	−1.16149
$118$	0	0
$119$	1504.47	1.15895
$120$	0	0
$121$	1222.25	0.918291
$122$	0	0
$123$	111.332	0.0816136
$124$	0	0
$125$	−1128.38	−0.807405
$126$	0	0
$127$	−433.550	−0.302924	−0.151462	−	0.988463i	$-0.548398\pi$
$127$	−433.550	−0.302924	−0.151462	+	0.988463i	$0.548398\pi$
$128$	0	0
$129$	520.815	0.355467
$130$	0	0
$131$	−1097.96	−0.732281	−0.366140	−	0.930560i	$-0.619321\pi$
$131$	−1097.96	−0.732281	−0.366140	+	0.930560i	$0.619321\pi$
$132$	0	0
$133$	2197.57	1.43273
$134$	0	0
$135$	−707.886	−0.451298
$136$	0	0
$137$	−1578.08	−0.984122	−0.492061	−	0.870561i	$-0.663756\pi$
$137$	−1578.08	−0.984122	−0.492061	+	0.870561i	$0.663756\pi$
$138$	0	0
$139$	−1969.53	−1.20182	−0.600911	−	0.799316i	$-0.705195\pi$
$139$	−1969.53	−1.20182	−0.600911	+	0.799316i	$0.705195\pi$
$140$	0	0
$141$	−939.489	−0.561129
$142$	0	0
$143$	4507.42	2.63587
$144$	0	0
$145$	−150.549	−0.0862233
$146$	0	0
$147$	527.800	0.296137
$148$	0	0
$149$	1291.44	0.710060	0.355030	−	0.934855i	$-0.384471\pi$
$149$	1291.44	0.710060	0.355030	+	0.934855i	$0.384471\pi$
$150$	0	0
$151$	−1388.00	−0.748038	−0.374019	−	0.927421i	$-0.622021\pi$
$151$	−1388.00	−0.748038	−0.374019	+	0.927421i	$0.622021\pi$
$152$	0	0
$153$	1102.42	0.582516
$154$	0	0
$155$	−369.305	−0.191376
$156$	0	0
$157$	−2602.69	−1.32304	−0.661519	−	0.749929i	$-0.730088\pi$
$157$	−2602.69	−1.32304	−0.661519	+	0.749929i	$0.730088\pi$
$158$	0	0
$159$	1274.68	0.635776
$160$	0	0
$161$	517.227	0.253187
$162$	0	0
$163$	1483.18	0.712711	0.356356	−	0.934350i	$-0.384019\pi$
$163$	1483.18	0.712711	0.356356	+	0.934350i	$0.384019\pi$
$164$	0	0
$165$	822.691	0.388160
$166$	0	0
$167$	−3685.95	−1.70795	−0.853975	−	0.520314i	$-0.825815\pi$
$167$	−3685.95	−1.70795	−0.853975	+	0.520314i	$0.825815\pi$
$168$	0	0
$169$	5760.26	2.62187
$170$	0	0
$171$	1610.29	0.720129
$172$	0	0
$173$	1816.77	0.798417	0.399208	−	0.916860i	$-0.369285\pi$
$173$	1816.77	0.798417	0.399208	+	0.916860i	$0.369285\pi$
$174$	0	0
$175$	2244.45	0.969510
$176$	0	0
$177$	567.613	0.241042
$178$	0	0
$179$	−3595.87	−1.50150	−0.750749	−	0.660588i	$-0.770307\pi$
$179$	−3595.87	−1.50150	−0.750749	+	0.660588i	$0.770307\pi$
$180$	0	0
$181$	2646.44	1.08679	0.543394	−	0.839478i	$-0.317139\pi$
$181$	2646.44	1.08679	0.543394	+	0.839478i	$0.317139\pi$
$182$	0	0
$183$	1537.87	0.621215
$184$	0	0
$185$	−5.65544	−0.00224755
$186$	0	0
$187$	−3380.47	−1.32195
$188$	0	0
$189$	3171.52	1.22060
$190$	0	0
$191$	−3057.51	−1.15829	−0.579146	−	0.815224i	$-0.696614\pi$
$191$	−3057.51	−1.15829	−0.579146	+	0.815224i	$0.696614\pi$
$192$	0	0
$193$	2810.64	1.04826	0.524130	−	0.851638i	$-0.324390\pi$
$193$	2810.64	1.04826	0.524130	+	0.851638i	$0.324390\pi$
$194$	0	0
$195$	1452.35	0.533360
$196$	0	0
$197$	3369.29	1.21854	0.609269	−	0.792963i	$-0.291463\pi$
$197$	3369.29	1.21854	0.609269	+	0.792963i	$0.291463\pi$
$198$	0	0
$199$	−3191.01	−1.13671	−0.568353	−	0.822785i	$-0.692419\pi$
$199$	−3191.01	−1.13671	−0.568353	+	0.822785i	$0.692419\pi$
$200$	0	0
$201$	−2179.48	−0.764819
$202$	0	0
$203$	674.498	0.233204
$204$	0	0
$205$	172.277	0.0586944
$206$	0	0
$207$	379.003	0.127258
$208$	0	0
$209$	−4937.83	−1.63424
$210$	0	0
$211$	−3367.08	−1.09857	−0.549287	−	0.835634i	$-0.685101\pi$
$211$	−3367.08	−1.09857	−0.549287	+	0.835634i	$0.685101\pi$
$212$	0	0
$213$	1561.98	0.502464
$214$	0	0
$215$	805.918	0.255643
$216$	0	0
$217$	1654.59	0.517607
$218$	0	0
$219$	−2525.88	−0.779376
$220$	0	0
$221$	−5967.78	−1.81645
$222$	0	0
$223$	−4129.95	−1.24019	−0.620094	−	0.784528i	$-0.712906\pi$
$223$	−4129.95	−1.24019	−0.620094	+	0.784528i	$0.712906\pi$
$224$	0	0
$225$	1644.64	0.487301
$226$	0	0
$227$	−2844.36	−0.831661	−0.415830	−	0.909442i	$-0.636509\pi$
$227$	−2844.36	−0.831661	−0.415830	+	0.909442i	$0.636509\pi$
$228$	0	0
$229$	3004.28	0.866936	0.433468	−	0.901169i	$-0.357290\pi$
$229$	3004.28	0.866936	0.433468	+	0.901169i	$0.357290\pi$
$230$	0	0
$231$	−3685.88	−1.04984
$232$	0	0
$233$	−457.644	−0.128675	−0.0643374	−	0.997928i	$-0.520493\pi$
$233$	−457.644	−0.128675	−0.0643374	+	0.997928i	$0.520493\pi$
$234$	0	0
$235$	−1453.78	−0.403550
$236$	0	0
$237$	429.273	0.117655
$238$	0	0
$239$	1302.50	0.352518	0.176259	−	0.984344i	$-0.443600\pi$
$239$	1302.50	0.352518	0.176259	+	0.984344i	$0.443600\pi$
$240$	0	0
$241$	−4474.43	−1.19595	−0.597973	−	0.801516i	$-0.704027\pi$
$241$	−4474.43	−1.19595	−0.597973	+	0.801516i	$0.704027\pi$
$242$	0	0
$243$	3767.14	0.994494
$244$	0	0
$245$	816.727	0.212975
$246$	0	0
$247$	−8717.09	−2.24557
$248$	0	0
$249$	−2621.10	−0.667091
$250$	0	0
$251$	−3834.68	−0.964314	−0.482157	−	0.876085i	$-0.660146\pi$
$251$	−3834.68	−0.964314	−0.482157	+	0.876085i	$0.660146\pi$
$252$	0	0
$253$	−1162.18	−0.288797
$254$	0	0
$255$	−1089.23	−0.267492
$256$	0	0
$257$	385.508	0.0935693	0.0467846	−	0.998905i	$-0.485103\pi$
$257$	385.508	0.0935693	0.0467846	+	0.998905i	$0.485103\pi$
$258$	0	0
$259$	25.3379	0.00607884
$260$	0	0
$261$	494.245	0.117215
$262$	0	0
$263$	−3080.35	−0.722215	−0.361108	−	0.932524i	$-0.617601\pi$
$263$	−3080.35	−0.722215	−0.361108	+	0.932524i	$0.617601\pi$
$264$	0	0
$265$	1972.45	0.457234
$266$	0	0
$267$	−3423.62	−0.784727
$268$	0	0
$269$	5963.04	1.35157	0.675786	−	0.737098i	$-0.263804\pi$
$269$	5963.04	1.35157	0.675786	+	0.737098i	$0.263804\pi$
$270$	0	0
$271$	−6227.62	−1.39594	−0.697972	−	0.716125i	$-0.745914\pi$
$271$	−6227.62	−1.39594	−0.697972	+	0.716125i	$0.745914\pi$
$272$	0	0
$273$	−6506.93	−1.44255
$274$	0	0
$275$	−5043.16	−1.10587
$276$	0	0
$277$	615.008	0.133402	0.0667009	−	0.997773i	$-0.478753\pi$
$277$	615.008	0.133402	0.0667009	+	0.997773i	$0.478753\pi$
$278$	0	0
$279$	1212.41	0.260163
$280$	0	0
$281$	8285.01	1.75887	0.879435	−	0.476019i	$-0.157921\pi$
$281$	8285.01	1.75887	0.879435	+	0.476019i	$0.157921\pi$
$282$	0	0
$283$	3165.51	0.664911	0.332456	−	0.943119i	$-0.392123\pi$
$283$	3165.51	0.664911	0.332456	+	0.943119i	$0.392123\pi$
$284$	0	0
$285$	−1591.04	−0.330684
$286$	0	0
$287$	−771.848	−0.158748
$288$	0	0
$289$	−437.293	−0.0890073
$290$	0	0
$291$	2674.14	0.538698
$292$	0	0
$293$	−8523.21	−1.69942	−0.849712	−	0.527248i	$-0.823224\pi$
$293$	−8523.21	−1.69942	−0.849712	+	0.527248i	$0.823224\pi$
$294$	0	0
$295$	878.334	0.173351
$296$	0	0
$297$	−7126.25	−1.39228
$298$	0	0
$299$	−2051.68	−0.396828
$300$	0	0
$301$	−3610.73	−0.691425
$302$	0	0
$303$	−3329.98	−0.631360
$304$	0	0
$305$	2379.72	0.446762
$306$	0	0
$307$	10084.1	1.87469	0.937343	−	0.348407i	$-0.113277\pi$
$307$	10084.1	1.87469	0.937343	+	0.348407i	$0.113277\pi$
$308$	0	0
$309$	−4081.29	−0.751381
$310$	0	0
$311$	3592.20	0.654967	0.327484	−	0.944857i	$-0.393799\pi$
$311$	3592.20	0.654967	0.327484	+	0.944857i	$0.393799\pi$
$312$	0	0
$313$	1308.03	0.236212	0.118106	−	0.993001i	$-0.462318\pi$
$313$	1308.03	0.236212	0.118106	+	0.993001i	$0.462318\pi$
$314$	0	0
$315$	1860.01	0.332698
$316$	0	0
$317$	−1877.87	−0.332718	−0.166359	−	0.986065i	$-0.553201\pi$
$317$	−1877.87	−0.332718	−0.166359	+	0.986065i	$0.553201\pi$
$318$	0	0
$319$	−1515.56	−0.266004
$320$	0	0
$321$	1645.95	0.286194
$322$	0	0
$323$	6537.64	1.12620
$324$	0	0
$325$	−8903.04	−1.51954
$326$	0	0
$327$	1556.17	0.263169
$328$	0	0
$329$	6513.33	1.09146
$330$	0	0
$331$	8312.07	1.38028	0.690140	−	0.723675i	$-0.257549\pi$
$331$	8312.07	1.38028	0.690140	+	0.723675i	$0.257549\pi$
$332$	0	0
$333$	18.5666	0.00305538
$334$	0	0
$335$	−3372.57	−0.550039
$336$	0	0
$337$	8346.55	1.34916	0.674578	−	0.738204i	$-0.264326\pi$
$337$	8346.55	1.34916	0.674578	+	0.738204i	$0.264326\pi$
$338$	0	0
$339$	−5303.09	−0.849629
$340$	0	0
$341$	−3717.78	−0.590407
$342$	0	0
$343$	4054.27	0.638221
$344$	0	0
$345$	−374.471	−0.0584372
$346$	0	0
$347$	10694.4	1.65449	0.827245	−	0.561842i	$-0.189907\pi$
$347$	10694.4	1.65449	0.827245	+	0.561842i	$0.189907\pi$
$348$	0	0
$349$	−8.08786	−0.00124050	−0.000620248	−	1.00000i	$-0.500197\pi$
$349$	−8.08786	−0.00124050	−0.000620248		1.00000i	$0.500197\pi$
$350$	0	0
$351$	−12580.5	−1.91309
$352$	0	0
$353$	−7139.25	−1.07644	−0.538221	−	0.842804i	$-0.680903\pi$
$353$	−7139.25	−1.07644	−0.538221	+	0.842804i	$0.680903\pi$
$354$	0	0
$355$	2417.03	0.361359
$356$	0	0
$357$	4880.06	0.723474
$358$	0	0
$359$	−11540.4	−1.69660	−0.848301	−	0.529514i	$-0.822374\pi$
$359$	−11540.4	−1.69660	−0.848301	+	0.529514i	$0.822374\pi$
$360$	0	0
$361$	2690.48	0.392256
$362$	0	0
$363$	3964.60	0.573244
$364$	0	0
$365$	−3908.59	−0.560507
$366$	0	0
$367$	4341.92	0.617565	0.308783	−	0.951133i	$-0.400078\pi$
$367$	4341.92	0.617565	0.308783	+	0.951133i	$0.400078\pi$
$368$	0	0
$369$	−565.579	−0.0797910
$370$	0	0
$371$	−8837.13	−1.23666
$372$	0	0
$373$	10170.6	1.41183	0.705915	−	0.708297i	$-0.250536\pi$
$373$	10170.6	1.41183	0.705915	+	0.708297i	$0.250536\pi$
$374$	0	0
$375$	−3660.14	−0.504024
$376$	0	0
$377$	−2675.53	−0.365509
$378$	0	0
$379$	11102.5	1.50474	0.752370	−	0.658740i	$-0.228910\pi$
$379$	11102.5	1.50474	0.752370	+	0.658740i	$0.228910\pi$
$380$	0	0
$381$	−1406.31	−0.189101
$382$	0	0
$383$	7798.68	1.04045	0.520227	−	0.854028i	$-0.325847\pi$
$383$	7798.68	1.04045	0.520227	+	0.854028i	$0.325847\pi$
$384$	0	0
$385$	−5703.59	−0.755017
$386$	0	0
$387$	−2645.80	−0.347528
$388$	0	0
$389$	4144.11	0.540141	0.270070	−	0.962841i	$-0.412953\pi$
$389$	4144.11	0.540141	0.270070	+	0.962841i	$0.412953\pi$
$390$	0	0
$391$	1538.72	0.199019
$392$	0	0
$393$	−3561.44	−0.457127
$394$	0	0
$395$	664.264	0.0846145
$396$	0	0
$397$	13652.5	1.72595	0.862974	−	0.505248i	$-0.168599\pi$
$397$	13652.5	1.72595	0.862974	+	0.505248i	$0.168599\pi$
$398$	0	0
$399$	7128.27	0.894386
$400$	0	0
$401$	−4312.25	−0.537016	−0.268508	−	0.963277i	$-0.586531\pi$
$401$	−4312.25	−0.537016	−0.268508	+	0.963277i	$0.586531\pi$
$402$	0	0
$403$	−6563.24	−0.811262
$404$	0	0
$405$	−62.9788	−0.00772702
$406$	0	0
$407$	−56.9329	−0.00693381
$408$	0	0
$409$	4307.52	0.520765	0.260383	−	0.965505i	$-0.416151\pi$
$409$	4307.52	0.520765	0.260383	+	0.965505i	$0.416151\pi$
$410$	0	0
$411$	−5118.84	−0.614340
$412$	0	0
$413$	−3935.17	−0.468855
$414$	0	0
$415$	−4055.94	−0.479755
$416$	0	0
$417$	−6388.57	−0.750239
$418$	0	0
$419$	−3968.03	−0.462651	−0.231326	−	0.972876i	$-0.574306\pi$
$419$	−3968.03	−0.462651	−0.231326	+	0.972876i	$0.574306\pi$
$420$	0	0
$421$	−10423.7	−1.20670	−0.603350	−	0.797477i	$-0.706168\pi$
$421$	−10423.7	−1.20670	−0.603350	+	0.797477i	$0.706168\pi$
$422$	0	0
$423$	4772.70	0.548598
$424$	0	0
$425$	6677.09	0.762086
$426$	0	0
$427$	−10661.8	−1.20834
$428$	0	0
$429$	14620.7	1.64545
$430$	0	0
$431$	−7390.02	−0.825905	−0.412952	−	0.910753i	$-0.635502\pi$
$431$	−7390.02	−0.825905	−0.412952	+	0.910753i	$0.635502\pi$
$432$	0	0
$433$	−9821.24	−1.09002	−0.545010	−	0.838430i	$-0.683474\pi$
$433$	−9821.24	−1.09002	−0.545010	+	0.838430i	$0.683474\pi$
$434$	0	0
$435$	−488.336	−0.0538250
$436$	0	0
$437$	2247.59	0.246034
$438$	0	0
$439$	4275.53	0.464829	0.232414	−	0.972617i	$-0.425337\pi$
$439$	4275.53	0.464829	0.232414	+	0.972617i	$0.425337\pi$
$440$	0	0
$441$	−2681.28	−0.289524
$442$	0	0
$443$	−5079.47	−0.544770	−0.272385	−	0.962188i	$-0.587812\pi$
$443$	−5079.47	−0.544770	−0.272385	+	0.962188i	$0.587812\pi$
$444$	0	0
$445$	−5297.77	−0.564356
$446$	0	0
$447$	4189.06	0.443256
$448$	0	0
$449$	17452.0	1.83433	0.917163	−	0.398513i	$-0.130474\pi$
$449$	17452.0	1.83433	0.917163	+	0.398513i	$0.130474\pi$
$450$	0	0
$451$	1734.30	0.181076
$452$	0	0
$453$	−4502.26	−0.466964
$454$	0	0
$455$	−10068.9	−1.03745
$456$	0	0
$457$	3153.12	0.322750	0.161375	−	0.986893i	$-0.448407\pi$
$457$	3153.12	0.322750	0.161375	+	0.986893i	$0.448407\pi$
$458$	0	0
$459$	9435.08	0.959459
$460$	0	0
$461$	−5406.53	−0.546220	−0.273110	−	0.961983i	$-0.588052\pi$
$461$	−5406.53	−0.546220	−0.273110	+	0.961983i	$0.588052\pi$
$462$	0	0
$463$	9702.07	0.973851	0.486926	−	0.873443i	$-0.338118\pi$
$463$	9702.07	0.973851	0.486926	+	0.873443i	$0.338118\pi$
$464$	0	0
$465$	−1197.92	−0.119467
$466$	0	0
$467$	−19192.0	−1.90172	−0.950858	−	0.309627i	$-0.899796\pi$
$467$	−19192.0	−1.90172	−0.950858	+	0.309627i	$0.899796\pi$
$468$	0	0
$469$	15110.0	1.48767
$470$	0	0
$471$	−8442.35	−0.825908
$472$	0	0
$473$	8113.13	0.788672
$474$	0	0
$475$	9753.18	0.942119
$476$	0	0
$477$	−6475.49	−0.621577
$478$	0	0
$479$	2427.60	0.231565	0.115783	−	0.993275i	$-0.463062\pi$
$479$	2427.60	0.231565	0.115783	+	0.993275i	$0.463062\pi$
$480$	0	0
$481$	−100.508	−0.00952756
$482$	0	0
$483$	1677.73	0.158053
$484$	0	0
$485$	4138.02	0.387418
$486$	0	0
$487$	−13649.6	−1.27006	−0.635031	−	0.772487i	$-0.719013\pi$
$487$	−13649.6	−1.27006	−0.635031	+	0.772487i	$0.719013\pi$
$488$	0	0
$489$	4811.01	0.444911
$490$	0	0
$491$	18411.1	1.69222	0.846109	−	0.533010i	$-0.178939\pi$
$491$	18411.1	1.69222	0.846109	+	0.533010i	$0.178939\pi$
$492$	0	0
$493$	2006.59	0.183311
$494$	0	0
$495$	−4179.36	−0.379491
$496$	0	0
$497$	−10828.9	−0.977352
$498$	0	0
$499$	18746.4	1.68177	0.840884	−	0.541215i	$-0.182035\pi$
$499$	18746.4	1.68177	0.840884	+	0.541215i	$0.182035\pi$
$500$	0	0
$501$	−11956.2	−1.06619
$502$	0	0
$503$	6877.91	0.609683	0.304842	−	0.952403i	$-0.401396\pi$
$503$	6877.91	0.609683	0.304842	+	0.952403i	$0.401396\pi$
$504$	0	0
$505$	−5152.86	−0.454058
$506$	0	0
$507$	18684.6	1.63671
$508$	0	0
$509$	−2182.76	−0.190077	−0.0950385	−	0.995474i	$-0.530297\pi$
$509$	−2182.76	−0.190077	−0.0950385	+	0.995474i	$0.530297\pi$
$510$	0	0
$511$	17511.5	1.51598
$512$	0	0
$513$	13781.8	1.18612
$514$	0	0
$515$	−6315.46	−0.540374
$516$	0	0
$517$	−14635.1	−1.24497
$518$	0	0
$519$	5893.05	0.498413
$520$	0	0
$521$	−10786.2	−0.907012	−0.453506	−	0.891253i	$-0.649827\pi$
$521$	−10786.2	−0.907012	−0.453506	+	0.891253i	$0.649827\pi$
$522$	0	0
$523$	−2309.04	−0.193054	−0.0965269	−	0.995330i	$-0.530773\pi$
$523$	−2309.04	−0.193054	−0.0965269	+	0.995330i	$0.530773\pi$
$524$	0	0
$525$	7280.33	0.605218
$526$	0	0
$527$	4922.30	0.406866
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−2883.53	−0.235659
$532$	0	0
$533$	3061.69	0.248811
$534$	0	0
$535$	2546.98	0.205823
$536$	0	0
$537$	−11664.0	−0.937312
$538$	0	0
$539$	8221.94	0.657039
$540$	0	0
$541$	10591.1	0.841673	0.420836	−	0.907137i	$-0.361737\pi$
$541$	10591.1	0.841673	0.420836	+	0.907137i	$0.361737\pi$
$542$	0	0
$543$	8584.28	0.678429
$544$	0	0
$545$	2408.04	0.189264
$546$	0	0
$547$	−22197.1	−1.73506	−0.867530	−	0.497384i	$-0.834294\pi$
$547$	−22197.1	−1.73506	−0.867530	+	0.497384i	$0.834294\pi$
$548$	0	0
$549$	−7812.53	−0.607342
$550$	0	0
$551$	2931.01	0.226616
$552$	0	0
$553$	−2976.08	−0.228853
$554$	0	0
$555$	−18.3446	−0.00140303
$556$	0	0
$557$	−14996.6	−1.14080	−0.570400	−	0.821367i	$-0.693212\pi$
$557$	−14996.6	−1.14080	−0.570400	+	0.821367i	$0.693212\pi$
$558$	0	0
$559$	14322.7	1.08369
$560$	0	0
$561$	−10965.2	−0.825229
$562$	0	0
$563$	−6781.84	−0.507674	−0.253837	−	0.967247i	$-0.581693\pi$
$563$	−6781.84	−0.507674	−0.253837	+	0.967247i	$0.581693\pi$
$564$	0	0
$565$	−8206.09	−0.611032
$566$	0	0
$567$	282.162	0.0208989
$568$	0	0
$569$	2121.99	0.156341	0.0781707	−	0.996940i	$-0.475092\pi$
$569$	2121.99	0.156341	0.0781707	+	0.996940i	$0.475092\pi$
$570$	0	0
$571$	−11539.3	−0.845720	−0.422860	−	0.906195i	$-0.638974\pi$
$571$	−11539.3	−0.845720	−0.422860	+	0.906195i	$0.638974\pi$
$572$	0	0
$573$	−9917.66	−0.723065
$574$	0	0
$575$	2295.54	0.166488
$576$	0	0
$577$	−17547.9	−1.26608	−0.633042	−	0.774118i	$-0.718194\pi$
$577$	−17547.9	−1.26608	−0.633042	+	0.774118i	$0.718194\pi$
$578$	0	0
$579$	9116.89	0.654378
$580$	0	0
$581$	18171.7	1.29757
$582$	0	0
$583$	19856.6	1.41059
$584$	0	0
$585$	−7378.11	−0.521448
$586$	0	0
$587$	16611.1	1.16800	0.583999	−	0.811754i	$-0.301487\pi$
$587$	16611.1	1.16800	0.583999	+	0.811754i	$0.301487\pi$
$588$	0	0
$589$	7189.96	0.502984
$590$	0	0
$591$	10929.0	0.760675
$592$	0	0
$593$	6065.78	0.420054	0.210027	−	0.977696i	$-0.432645\pi$
$593$	6065.78	0.420054	0.210027	+	0.977696i	$0.432645\pi$
$594$	0	0
$595$	7551.49	0.520304
$596$	0	0
$597$	−10350.7	−0.709590
$598$	0	0
$599$	12573.5	0.857659	0.428829	−	0.903385i	$-0.358926\pi$
$599$	12573.5	0.857659	0.428829	+	0.903385i	$0.358926\pi$
$600$	0	0
$601$	−18446.2	−1.25198	−0.625988	−	0.779832i	$-0.715304\pi$
$601$	−18446.2	−1.25198	−0.625988	+	0.779832i	$0.715304\pi$
$602$	0	0
$603$	11072.0	0.747739
$604$	0	0
$605$	6134.90	0.412263
$606$	0	0
$607$	13715.7	0.917135	0.458568	−	0.888659i	$-0.348363\pi$
$607$	13715.7	0.917135	0.458568	+	0.888659i	$0.348363\pi$
$608$	0	0
$609$	2187.87	0.145578
$610$	0	0
$611$	−25836.4	−1.71068
$612$	0	0
$613$	9497.86	0.625800	0.312900	−	0.949786i	$-0.398700\pi$
$613$	9497.86	0.625800	0.312900	+	0.949786i	$0.398700\pi$
$614$	0	0
$615$	558.817	0.0366401
$616$	0	0
$617$	4749.94	0.309928	0.154964	−	0.987920i	$-0.450474\pi$
$617$	4749.94	0.309928	0.154964	+	0.987920i	$0.450474\pi$
$618$	0	0
$619$	−17961.0	−1.16626	−0.583130	−	0.812379i	$-0.698172\pi$
$619$	−17961.0	−1.16626	−0.583130	+	0.812379i	$0.698172\pi$
$620$	0	0
$621$	3243.71	0.209607
$622$	0	0
$623$	23735.4	1.52639
$624$	0	0
$625$	6811.98	0.435966
$626$	0	0
$627$	−16016.9	−1.02018
$628$	0	0
$629$	75.3786	0.00477829
$630$	0	0
$631$	−20618.4	−1.30080	−0.650401	−	0.759591i	$-0.725399\pi$
$631$	−20618.4	−1.30080	−0.650401	+	0.759591i	$0.725399\pi$
$632$	0	0
$633$	−10921.8	−0.685786
$634$	0	0
$635$	−2176.14	−0.135996
$636$	0	0
$637$	14514.7	0.902818
$638$	0	0
$639$	−7935.00	−0.491242
$640$	0	0
$641$	18608.2	1.14661	0.573306	−	0.819341i	$-0.305661\pi$
$641$	18608.2	1.14661	0.573306	+	0.819341i	$0.305661\pi$
$642$	0	0
$643$	24685.2	1.51398	0.756989	−	0.653428i	$-0.226670\pi$
$643$	24685.2	1.51398	0.756989	+	0.653428i	$0.226670\pi$
$644$	0	0
$645$	2614.16	0.159585
$646$	0	0
$647$	26900.9	1.63460	0.817298	−	0.576215i	$-0.195471\pi$
$647$	26900.9	1.63460	0.817298	+	0.576215i	$0.195471\pi$
$648$	0	0
$649$	8842.13	0.534798
$650$	0	0
$651$	5367.00	0.323117
$652$	0	0
$653$	11712.3	0.701898	0.350949	−	0.936395i	$-0.385859\pi$
$653$	11712.3	0.701898	0.350949	+	0.936395i	$0.385859\pi$
$654$	0	0
$655$	−5511.04	−0.328754
$656$	0	0
$657$	12831.8	0.761970
$658$	0	0
$659$	−324.827	−0.0192010	−0.00960051	−	0.999954i	$-0.503056\pi$
$659$	−324.827	−0.0192010	−0.00960051	+	0.999954i	$0.503056\pi$
$660$	0	0
$661$	−11829.6	−0.696095	−0.348047	−	0.937477i	$-0.613155\pi$
$661$	−11829.6	−0.696095	−0.348047	+	0.937477i	$0.613155\pi$
$662$	0	0
$663$	−19357.7	−1.13392
$664$	0	0
$665$	11030.4	0.643220
$666$	0	0
$667$	689.852	0.0400467
$668$	0	0
$669$	−13396.3	−0.774189
$670$	0	0
$671$	23956.5	1.37829
$672$	0	0
$673$	2893.24	0.165715	0.0828576	−	0.996561i	$-0.473595\pi$
$673$	2893.24	0.165715	0.0828576	+	0.996561i	$0.473595\pi$
$674$	0	0
$675$	14075.7	0.802630
$676$	0	0
$677$	15638.7	0.887805	0.443903	−	0.896075i	$-0.353594\pi$
$677$	15638.7	0.887805	0.443903	+	0.896075i	$0.353594\pi$
$678$	0	0
$679$	−18539.4	−1.04783
$680$	0	0
$681$	−9226.28	−0.519166
$682$	0	0
$683$	−27288.8	−1.52881	−0.764404	−	0.644738i	$-0.776967\pi$
$683$	−27288.8	−1.52881	−0.764404	+	0.644738i	$0.776967\pi$
$684$	0	0
$685$	−7920.98	−0.441818
$686$	0	0
$687$	9744.99	0.541186
$688$	0	0
$689$	35054.2	1.93825
$690$	0	0
$691$	−25680.2	−1.41378	−0.706889	−	0.707325i	$-0.749902\pi$
$691$	−25680.2	−1.41378	−0.706889	+	0.707325i	$0.749902\pi$
$692$	0	0
$693$	18724.6	1.02639
$694$	0	0
$695$	−9885.78	−0.539553
$696$	0	0
$697$	−2296.20	−0.124784
$698$	0	0
$699$	−1484.46	−0.0803254
$700$	0	0
$701$	17178.5	0.925568	0.462784	−	0.886471i	$-0.346851\pi$
$701$	17178.5	0.925568	0.462784	+	0.886471i	$0.346851\pi$
$702$	0	0
$703$	110.105	0.00590710
$704$	0	0
$705$	−4715.64	−0.251917
$706$	0	0
$707$	23086.2	1.22807
$708$	0	0
$709$	−6738.71	−0.356950	−0.178475	−	0.983944i	$-0.557116\pi$
$709$	−6738.71	−0.356950	−0.178475	+	0.983944i	$0.557116\pi$
$710$	0	0
$711$	−2180.75	−0.115028
$712$	0	0
$713$	1692.25	0.0888854
$714$	0	0
$715$	22624.4	1.18336
$716$	0	0
$717$	4224.93	0.220060
$718$	0	0
$719$	24261.2	1.25840	0.629200	−	0.777243i	$-0.283383\pi$
$719$	24261.2	1.25840	0.629200	+	0.777243i	$0.283383\pi$
$720$	0	0
$721$	28295.0	1.46153
$722$	0	0
$723$	−14513.7	−0.746572
$724$	0	0
$725$	2993.54	0.153348
$726$	0	0
$727$	−33964.0	−1.73268	−0.866338	−	0.499458i	$-0.833533\pi$
$727$	−33964.0	−1.73268	−0.866338	+	0.499458i	$0.833533\pi$
$728$	0	0
$729$	12558.3	0.638026
$730$	0	0
$731$	−10741.7	−0.543496
$732$	0	0
$733$	−37420.1	−1.88560	−0.942798	−	0.333365i	$-0.891816\pi$
$733$	−37420.1	−1.88560	−0.942798	+	0.333365i	$0.891816\pi$
$734$	0	0
$735$	2649.22	0.132950
$736$	0	0
$737$	−33951.4	−1.69690
$738$	0	0
$739$	−5606.82	−0.279094	−0.139547	−	0.990215i	$-0.544565\pi$
$739$	−5606.82	−0.279094	−0.139547	+	0.990215i	$0.544565\pi$
$740$	0	0
$741$	−28275.7	−1.40180
$742$	0	0
$743$	38168.4	1.88461	0.942303	−	0.334760i	$-0.108655\pi$
$743$	38168.4	1.88461	0.942303	+	0.334760i	$0.108655\pi$
$744$	0	0
$745$	6482.22	0.318779
$746$	0	0
$747$	13315.5	0.652193
$748$	0	0
$749$	−11411.1	−0.556681
$750$	0	0
$751$	17551.3	0.852804	0.426402	−	0.904534i	$-0.359781\pi$
$751$	17551.3	0.852804	0.426402	+	0.904534i	$0.359781\pi$
$752$	0	0
$753$	−12438.6	−0.601974
$754$	0	0
$755$	−6966.87	−0.335828
$756$	0	0
$757$	17994.6	0.863967	0.431984	−	0.901881i	$-0.357814\pi$
$757$	17994.6	0.863967	0.431984	+	0.901881i	$0.357814\pi$
$758$	0	0
$759$	−3769.78	−0.180282
$760$	0	0
$761$	13231.7	0.630286	0.315143	−	0.949044i	$-0.397947\pi$
$761$	13231.7	0.630286	0.315143	+	0.949044i	$0.397947\pi$
$762$	0	0
$763$	−10788.7	−0.511895
$764$	0	0
$765$	5533.43	0.261518
$766$	0	0
$767$	15609.6	0.734851
$768$	0	0
$769$	19279.4	0.904074	0.452037	−	0.891999i	$-0.350697\pi$
$769$	19279.4	0.904074	0.452037	+	0.891999i	$0.350697\pi$
$770$	0	0
$771$	1250.47	0.0584107
$772$	0	0
$773$	8171.25	0.380206	0.190103	−	0.981764i	$-0.439118\pi$
$773$	8171.25	0.380206	0.190103	+	0.981764i	$0.439118\pi$
$774$	0	0
$775$	7343.34	0.340362
$776$	0	0
$777$	82.1886	0.00379472
$778$	0	0
$779$	−3354.04	−0.154263
$780$	0	0
$781$	24332.1	1.11481
$782$	0	0
$783$	4230.02	0.193063
$784$	0	0
$785$	−13063.8	−0.593972
$786$	0	0
$787$	24615.9	1.11495	0.557473	−	0.830195i	$-0.311771\pi$
$787$	24615.9	1.11495	0.557473	+	0.830195i	$0.311771\pi$
$788$	0	0
$789$	−9991.76	−0.450844
$790$	0	0
$791$	36765.5	1.65263
$792$	0	0
$793$	42292.1	1.89387
$794$	0	0
$795$	6398.06	0.285429
$796$	0	0
$797$	−37349.4	−1.65995	−0.829976	−	0.557798i	$-0.811646\pi$
$797$	−37349.4	−1.65995	−0.829976	+	0.557798i	$0.811646\pi$
$798$	0	0
$799$	19376.7	0.857948
$800$	0	0
$801$	17392.4	0.767202
$802$	0	0
$803$	−39347.6	−1.72920
$804$	0	0
$805$	2596.15	0.113667
$806$	0	0
$807$	19342.3	0.843720
$808$	0	0
$809$	−16964.4	−0.737254	−0.368627	−	0.929577i	$-0.620172\pi$
$809$	−16964.4	−0.737254	−0.368627	+	0.929577i	$0.620172\pi$
$810$	0	0
$811$	19028.5	0.823899	0.411950	−	0.911207i	$-0.364848\pi$
$811$	19028.5	0.823899	0.411950	+	0.911207i	$0.364848\pi$
$812$	0	0
$813$	−20200.6	−0.871420
$814$	0	0
$815$	7444.64	0.319969
$816$	0	0
$817$	−15690.3	−0.671891
$818$	0	0
$819$	33055.9	1.41034
$820$	0	0
$821$	16919.1	0.719223	0.359612	−	0.933102i	$-0.382909\pi$
$821$	16919.1	0.719223	0.359612	+	0.933102i	$0.382909\pi$
$822$	0	0
$823$	−23348.3	−0.988908	−0.494454	−	0.869204i	$-0.664632\pi$
$823$	−23348.3	−0.988908	−0.494454	+	0.869204i	$0.664632\pi$
$824$	0	0
$825$	−16358.5	−0.690340
$826$	0	0
$827$	27804.4	1.16911	0.584555	−	0.811354i	$-0.301269\pi$
$827$	27804.4	1.16911	0.584555	+	0.811354i	$0.301269\pi$
$828$	0	0
$829$	−14429.4	−0.604528	−0.302264	−	0.953224i	$-0.597742\pi$
$829$	−14429.4	−0.604528	−0.302264	+	0.953224i	$0.597742\pi$
$830$	0	0
$831$	1994.91	0.0832762
$832$	0	0
$833$	−10885.8	−0.452784
$834$	0	0
$835$	−18501.2	−0.766777
$836$	0	0
$837$	10376.5	0.428512
$838$	0	0
$839$	30557.7	1.25741	0.628707	−	0.777643i	$-0.283585\pi$
$839$	30557.7	1.25741	0.628707	+	0.777643i	$0.283585\pi$
$840$	0	0
$841$	−23489.4	−0.963114
$842$	0	0
$843$	26874.1	1.09798
$844$	0	0
$845$	28912.8	1.17708
$846$	0	0
$847$	−27486.0	−1.11503
$848$	0	0
$849$	10268.0	0.415072
$850$	0	0
$851$	25.9146	0.00104388
$852$	0	0
$853$	46453.7	1.86465	0.932325	−	0.361622i	$-0.117777\pi$
$853$	46453.7	1.86465	0.932325	+	0.361622i	$0.117777\pi$
$854$	0	0
$855$	8082.64	0.323299
$856$	0	0
$857$	−43077.2	−1.71702	−0.858512	−	0.512794i	$-0.828610\pi$
$857$	−43077.2	−1.71702	−0.858512	+	0.512794i	$0.828610\pi$
$858$	0	0
$859$	−27121.5	−1.07727	−0.538634	−	0.842540i	$-0.681059\pi$
$859$	−27121.5	−1.07727	−0.538634	+	0.842540i	$0.681059\pi$
$860$	0	0
$861$	−2503.65	−0.0990989
$862$	0	0
$863$	−12067.2	−0.475983	−0.237992	−	0.971267i	$-0.576489\pi$
$863$	−12067.2	−0.475983	−0.237992	+	0.971267i	$0.576489\pi$
$864$	0	0
$865$	9119.01	0.358446
$866$	0	0
$867$	−1418.45	−0.0555630
$868$	0	0
$869$	6687.10	0.261041
$870$	0	0
$871$	−59936.8	−2.33166
$872$	0	0
$873$	−13584.9	−0.526667
$874$	0	0
$875$	25375.2	0.980387
$876$	0	0
$877$	−33239.2	−1.27983	−0.639913	−	0.768447i	$-0.721030\pi$
$877$	−33239.2	−1.27983	−0.639913	+	0.768447i	$0.721030\pi$
$878$	0	0
$879$	−27646.8	−1.06087
$880$	0	0
$881$	29990.7	1.14689	0.573445	−	0.819244i	$-0.305606\pi$
$881$	29990.7	1.14689	0.573445	+	0.819244i	$0.305606\pi$
$882$	0	0
$883$	31116.9	1.18592	0.592960	−	0.805232i	$-0.297959\pi$
$883$	31116.9	1.18592	0.592960	+	0.805232i	$0.297959\pi$
$884$	0	0
$885$	2849.06	0.108215
$886$	0	0
$887$	2202.57	0.0833765	0.0416882	−	0.999131i	$-0.486726\pi$
$887$	2202.57	0.0833765	0.0416882	+	0.999131i	$0.486726\pi$
$888$	0	0
$889$	9749.71	0.367823
$890$	0	0
$891$	−634.004	−0.0238383
$892$	0	0
$893$	28303.5	1.06063
$894$	0	0
$895$	−18049.0	−0.674091
$896$	0	0
$897$	−6655.04	−0.247721
$898$	0	0
$899$	2206.81	0.0818701
$900$	0	0
$901$	−26289.9	−0.972079
$902$	0	0
$903$	−11712.1	−0.431623
$904$	0	0
$905$	13283.5	0.487909
$906$	0	0
$907$	25884.4	0.947606	0.473803	−	0.880631i	$-0.342881\pi$
$907$	25884.4	0.947606	0.473803	+	0.880631i	$0.342881\pi$
$908$	0	0
$909$	16916.6	0.617260
$910$	0	0
$911$	40217.9	1.46265	0.731326	−	0.682028i	$-0.238902\pi$
$911$	40217.9	1.46265	0.731326	+	0.682028i	$0.238902\pi$
$912$	0	0
$913$	−40830.9	−1.48007
$914$	0	0
$915$	7719.12	0.278892
$916$	0	0
$917$	24690.9	0.889168
$918$	0	0
$919$	−26633.7	−0.956000	−0.478000	−	0.878360i	$-0.658638\pi$
$919$	−26633.7	−0.956000	−0.478000	+	0.878360i	$0.658638\pi$
$920$	0	0
$921$	32709.8	1.17028
$922$	0	0
$923$	42955.1	1.53183
$924$	0	0
$925$	112.454	0.00399725
$926$	0	0
$927$	20733.4	0.734600
$928$	0	0
$929$	−11575.8	−0.408815	−0.204408	−	0.978886i	$-0.565527\pi$
$929$	−11575.8	−0.408815	−0.204408	+	0.978886i	$0.565527\pi$
$930$	0	0
$931$	−15900.8	−0.559749
$932$	0	0
$933$	11652.0	0.408864
$934$	0	0
$935$	−16967.8	−0.593483
$936$	0	0
$937$	−11824.3	−0.412254	−0.206127	−	0.978525i	$-0.566086\pi$
$937$	−11824.3	−0.412254	−0.206127	+	0.978525i	$0.566086\pi$
$938$	0	0
$939$	4242.88	0.147456
$940$	0	0
$941$	−106.265	−0.00368135	−0.00184068	−	0.999998i	$-0.500586\pi$
$941$	−106.265	−0.00368135	−0.00184068	+	0.999998i	$0.500586\pi$
$942$	0	0
$943$	−789.418	−0.0272608
$944$	0	0
$945$	15919.0	0.547985
$946$	0	0
$947$	−7880.25	−0.270405	−0.135203	−	0.990818i	$-0.543169\pi$
$947$	−7880.25	−0.270405	−0.135203	+	0.990818i	$0.543169\pi$
$948$	0	0
$949$	−69463.0	−2.37604
$950$	0	0
$951$	−6091.26	−0.207700
$952$	0	0
$953$	10290.6	0.349784	0.174892	−	0.984588i	$-0.444042\pi$
$953$	10290.6	0.349784	0.174892	+	0.984588i	$0.444042\pi$
$954$	0	0
$955$	−15346.8	−0.520010
$956$	0	0
$957$	−4916.04	−0.166053
$958$	0	0
$959$	35488.1	1.19496
$960$	0	0
$961$	−24377.6	−0.818286
$962$	0	0
$963$	−8361.62	−0.279802
$964$	0	0
$965$	14107.6	0.470612
$966$	0	0
$967$	−4662.32	−0.155047	−0.0775234	−	0.996991i	$-0.524701\pi$
$967$	−4662.32	−0.155047	−0.0775234	+	0.996991i	$0.524701\pi$
$968$	0	0
$969$	21206.2	0.703034
$970$	0	0
$971$	6268.19	0.207164	0.103582	−	0.994621i	$-0.466970\pi$
$971$	6268.19	0.207164	0.103582	+	0.994621i	$0.466970\pi$
$972$	0	0
$973$	44291.0	1.45930
$974$	0	0
$975$	−28878.8	−0.948577
$976$	0	0
$977$	−26085.0	−0.854180	−0.427090	−	0.904209i	$-0.640461\pi$
$977$	−26085.0	−0.854180	−0.427090	+	0.904209i	$0.640461\pi$
$978$	0	0
$979$	−53332.3	−1.74107
$980$	0	0
$981$	−7905.49	−0.257291
$982$	0	0
$983$	−54148.4	−1.75693	−0.878467	−	0.477802i	$-0.841434\pi$
$983$	−54148.4	−1.75693	−0.878467	+	0.477802i	$0.841434\pi$
$984$	0	0
$985$	16911.7	0.547058
$986$	0	0
$987$	21127.3	0.681348
$988$	0	0
$989$	−3692.92	−0.118734
$990$	0	0
$991$	−40567.7	−1.30038	−0.650189	−	0.759772i	$-0.725310\pi$
$991$	−40567.7	−1.30038	−0.650189	+	0.759772i	$0.725310\pi$
$992$	0	0
$993$	26961.9	0.861642
$994$	0	0
$995$	−16016.8	−0.510319
$996$	0	0
$997$	−49354.5	−1.56778	−0.783888	−	0.620902i	$-0.786766\pi$
$997$	−49354.5	−1.56778	−0.783888	+	0.620902i	$0.786766\pi$
$998$	0	0
$999$	158.903	0.00503250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.6		8
4.3	odd	2		1472.4.a.bh.1.3		8
8.3	odd	2		736.4.a.e.1.6	✓	8
8.5	even	2		736.4.a.f.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.6	✓	8	8.3	odd	2
736.4.a.f.1.3	yes	8	8.5	even	2
1472.4.a.bg.1.6		8	1.1	even	1	trivial
1472.4.a.bh.1.3		8	4.3	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24371 q^{3} +5.01937 q^{5} -22.4881 q^{7} -16.4784 q^{9} +50.5296 q^{11} +89.2035 q^{13} +16.2813 q^{15} -66.9007 q^{17} -97.7214 q^{19} -72.9448 q^{21} -23.0000 q^{23} -99.8060 q^{25} -141.031 q^{27} -29.9936 q^{29} -73.5761 q^{31} +163.903 q^{33} -112.876 q^{35} -1.12672 q^{37} +289.350 q^{39} +34.3225 q^{41} +160.562 q^{43} -82.7110 q^{45} -289.634 q^{47} +162.715 q^{49} -217.006 q^{51} +392.969 q^{53} +253.627 q^{55} -316.980 q^{57} +174.989 q^{59} +474.108 q^{61} +370.567 q^{63} +447.745 q^{65} -671.911 q^{67} -74.6052 q^{69} +481.540 q^{71} -778.702 q^{73} -323.741 q^{75} -1136.32 q^{77} +132.340 q^{79} -12.5472 q^{81} -808.059 q^{83} -335.799 q^{85} -97.2903 q^{87} -1055.47 q^{89} -2006.02 q^{91} -238.659 q^{93} -490.500 q^{95} +824.410 q^{97} -832.646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more