LMFDB - Embedded newform 768.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.48361$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -18.5422 q^{5} -9.32669 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -18.5422 q^{5} -9.32669 q^{7} +9.00000 q^{9} +39.7378 q^{11} +32.9533 q^{13} -55.6266 q^{15} +90.5998 q^{17} -72.5998 q^{19} -27.9801 q^{21} -45.3466 q^{23} +218.813 q^{25} +27.0000 q^{27} -143.364 q^{29} +90.4865 q^{31} +119.213 q^{33} +172.937 q^{35} -1.77977 q^{37} +98.8599 q^{39} -195.827 q^{41} -407.027 q^{43} -166.880 q^{45} -278.467 q^{47} -256.013 q^{49} +271.799 q^{51} -241.303 q^{53} -736.826 q^{55} -217.799 q^{57} +149.724 q^{59} -508.314 q^{61} -83.9402 q^{63} -611.027 q^{65} -950.026 q^{67} -136.040 q^{69} +803.559 q^{71} -449.786 q^{73} +656.440 q^{75} -370.622 q^{77} -157.220 q^{79} +81.0000 q^{81} +175.063 q^{83} -1679.92 q^{85} -430.093 q^{87} -127.200 q^{89} -307.345 q^{91} +271.459 q^{93} +1346.16 q^{95} +158.826 q^{97} +357.640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9	$ 3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9} - 52 q^{13} - 30 q^{15} + 26 q^{17} + 28 q^{19} - 42 q^{21} - 164 q^{23} + 53 q^{25} + 81 q^{27} - 174 q^{29} - 318 q^{31} - 92 q^{35} - 296 q^{37} - 156 q^{39} - 118 q^{41} - 260 q^{43} - 90 q^{45} - 204 q^{47} + 327 q^{49} + 78 q^{51} - 1086 q^{53} - 512 q^{55} + 84 q^{57} + 196 q^{59} - 1536 q^{61} - 126 q^{63} - 872 q^{65} - 660 q^{67} - 492 q^{69} + 852 q^{71} - 478 q^{73} + 159 q^{75} - 2304 q^{77} - 22 q^{79} + 243 q^{81} + 1136 q^{83} - 2732 q^{85} - 522 q^{87} + 110 q^{89} - 632 q^{91} - 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100}) $ 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9 - 52 * q^13 - 30 * q^15 + 26 * q^17 + 28 * q^19 - 42 * q^21 - 164 * q^23 + 53 * q^25 + 81 * q^27 - 174 * q^29 - 318 * q^31 - 92 * q^35 - 296 * q^37 - 156 * q^39 - 118 * q^41 - 260 * q^43 - 90 * q^45 - 204 * q^47 + 327 * q^49 + 78 * q^51 - 1086 * q^53 - 512 * q^55 + 84 * q^57 + 196 * q^59 - 1536 * q^61 - 126 * q^63 - 872 * q^65 - 660 * q^67 - 492 * q^69 + 852 * q^71 - 478 * q^73 + 159 * q^75 - 2304 * q^77 - 22 * q^79 + 243 * q^81 + 1136 * q^83 - 2732 * q^85 - 522 * q^87 + 110 * q^89 - 632 * q^91 - 954 * q^93 + 2552 * q^95 - 1222 * q^97

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−18.5422	−1.65846	−0.829232	−	0.558904i	$-0.811222\pi$
$5$	−18.5422	−1.65846	−0.829232	+	0.558904i	$0.811222\pi$
$6$	0	0
$7$	−9.32669	−0.503594	−0.251797	−	0.967780i	$-0.581022\pi$
$7$	−9.32669	−0.503594	−0.251797	+	0.967780i	$0.581022\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	39.7378	1.08922	0.544609	−	0.838690i	$-0.316678\pi$
$11$	39.7378	1.08922	0.544609	+	0.838690i	$0.316678\pi$
$12$	0	0
$13$	32.9533	0.703046	0.351523	−	0.936179i	$-0.385664\pi$
$13$	32.9533	0.703046	0.351523	+	0.936179i	$0.385664\pi$
$14$	0	0
$15$	−55.6266	−0.957515
$16$	0	0
$17$	90.5998	1.29257	0.646285	−	0.763096i	$-0.276322\pi$
$17$	90.5998	1.29257	0.646285	+	0.763096i	$0.276322\pi$
$18$	0	0
$19$	−72.5998	−0.876607	−0.438304	−	0.898827i	$-0.644421\pi$
$19$	−72.5998	−0.876607	−0.438304	+	0.898827i	$0.644421\pi$
$20$	0	0
$21$	−27.9801	−0.290750
$22$	0	0
$23$	−45.3466	−0.411105	−0.205553	−	0.978646i	$-0.565899\pi$
$23$	−45.3466	−0.411105	−0.205553	+	0.978646i	$0.565899\pi$
$24$	0	0
$25$	218.813	1.75051
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−143.364	−0.918003	−0.459002	−	0.888435i	$-0.651793\pi$
$29$	−143.364	−0.918003	−0.459002	+	0.888435i	$0.651793\pi$
$30$	0	0
$31$	90.4865	0.524253	0.262127	−	0.965033i	$-0.415576\pi$
$31$	90.4865	0.524253	0.262127	+	0.965033i	$0.415576\pi$
$32$	0	0
$33$	119.213	0.628860
$34$	0	0
$35$	172.937	0.835193
$36$	0	0
$37$	−1.77977	−0.00790792	−0.00395396	−	0.999992i	$-0.501259\pi$
$37$	−1.77977	−0.00790792	−0.00395396	+	0.999992i	$0.501259\pi$
$38$	0	0
$39$	98.8599	0.405904
$40$	0	0
$41$	−195.827	−0.745927	−0.372964	−	0.927846i	$-0.621658\pi$
$41$	−195.827	−0.745927	−0.372964	+	0.927846i	$0.621658\pi$
$42$	0	0
$43$	−407.027	−1.44351	−0.721755	−	0.692148i	$-0.756664\pi$
$43$	−407.027	−1.44351	−0.721755	+	0.692148i	$0.756664\pi$
$44$	0	0
$45$	−166.880	−0.552822
$46$	0	0
$47$	−278.467	−0.864224	−0.432112	−	0.901820i	$-0.642231\pi$
$47$	−278.467	−0.864224	−0.432112	+	0.901820i	$0.642231\pi$
$48$	0	0
$49$	−256.013	−0.746393
$50$	0	0
$51$	271.799	0.746265
$52$	0	0
$53$	−241.303	−0.625386	−0.312693	−	0.949854i	$-0.601231\pi$
$53$	−241.303	−0.625386	−0.312693	+	0.949854i	$0.601231\pi$
$54$	0	0
$55$	−736.826	−1.80643
$56$	0	0
$57$	−217.799	−0.506109
$58$	0	0
$59$	149.724	0.330380	0.165190	−	0.986262i	$-0.447176\pi$
$59$	149.724	0.330380	0.165190	+	0.986262i	$0.447176\pi$
$60$	0	0
$61$	−508.314	−1.06693	−0.533466	−	0.845821i	$-0.679111\pi$
$61$	−508.314	−1.06693	−0.533466	+	0.845821i	$0.679111\pi$
$62$	0	0
$63$	−83.9402	−0.167865
$64$	0	0
$65$	−611.027	−1.16598
$66$	0	0
$67$	−950.026	−1.73230	−0.866150	−	0.499784i	$-0.833413\pi$
$67$	−950.026	−1.73230	−0.866150	+	0.499784i	$0.833413\pi$
$68$	0	0
$69$	−136.040	−0.237352
$70$	0	0
$71$	803.559	1.34317	0.671584	−	0.740929i	$-0.265614\pi$
$71$	803.559	1.34317	0.671584	+	0.740929i	$0.265614\pi$
$72$	0	0
$73$	−449.786	−0.721143	−0.360571	−	0.932732i	$-0.617418\pi$
$73$	−449.786	−0.721143	−0.360571	+	0.932732i	$0.617418\pi$
$74$	0	0
$75$	656.440	1.01065
$76$	0	0
$77$	−370.622	−0.548524
$78$	0	0
$79$	−157.220	−0.223906	−0.111953	−	0.993713i	$-0.535711\pi$
$79$	−157.220	−0.223906	−0.111953	+	0.993713i	$0.535711\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	175.063	0.231513	0.115757	−	0.993278i	$-0.463071\pi$
$83$	175.063	0.231513	0.115757	+	0.993278i	$0.463071\pi$
$84$	0	0
$85$	−1679.92	−2.14368
$86$	0	0
$87$	−430.093	−0.530009
$88$	0	0
$89$	−127.200	−0.151496	−0.0757479	−	0.997127i	$-0.524134\pi$
$89$	−127.200	−0.151496	−0.0757479	+	0.997127i	$0.524134\pi$
$90$	0	0
$91$	−307.345	−0.354050
$92$	0	0
$93$	271.459	0.302678
$94$	0	0
$95$	1346.16	1.45382
$96$	0	0
$97$	158.826	0.166251	0.0831254	−	0.996539i	$-0.473510\pi$
$97$	158.826	0.166251	0.0831254	+	0.996539i	$0.473510\pi$
$98$	0	0
$99$	357.640	0.363073
$100$	0	0
$101$	−1366.26	−1.34602	−0.673010	−	0.739633i	$-0.734999\pi$
$101$	−1366.26	−1.34602	−0.673010	+	0.739633i	$0.734999\pi$
$102$	0	0
$103$	1741.30	1.66578	0.832889	−	0.553440i	$-0.186685\pi$
$103$	1741.30	1.66578	0.832889	+	0.553440i	$0.186685\pi$
$104$	0	0
$105$	518.812	0.482199
$106$	0	0
$107$	649.378	0.586708	0.293354	−	0.956004i	$-0.405229\pi$
$107$	649.378	0.586708	0.293354	+	0.956004i	$0.405229\pi$
$108$	0	0
$109$	−1413.18	−1.24182	−0.620908	−	0.783883i	$-0.713236\pi$
$109$	−1413.18	−1.24182	−0.620908	+	0.783883i	$0.713236\pi$
$110$	0	0
$111$	−5.33932	−0.00456564
$112$	0	0
$113$	1096.43	0.912771	0.456386	−	0.889782i	$-0.349144\pi$
$113$	1096.43	0.912771	0.456386	+	0.889782i	$0.349144\pi$
$114$	0	0
$115$	840.826	0.681804
$116$	0	0
$117$	296.580	0.234349
$118$	0	0
$119$	−844.997	−0.650930
$120$	0	0
$121$	248.092	0.186395
$122$	0	0
$123$	−587.481	−0.430661
$124$	0	0
$125$	−1739.50	−1.24469
$126$	0	0
$127$	−737.794	−0.515501	−0.257751	−	0.966211i	$-0.582981\pi$
$127$	−737.794	−0.515501	−0.257751	+	0.966211i	$0.582981\pi$
$128$	0	0
$129$	−1221.08	−0.833411
$130$	0	0
$131$	−147.698	−0.0985074	−0.0492537	−	0.998786i	$-0.515684\pi$
$131$	−147.698	−0.0985074	−0.0492537	+	0.998786i	$0.515684\pi$
$132$	0	0
$133$	677.116	0.441454
$134$	0	0
$135$	−500.639	−0.319172
$136$	0	0
$137$	1880.79	1.17289	0.586447	−	0.809988i	$-0.300526\pi$
$137$	1880.79	1.17289	0.586447	+	0.809988i	$0.300526\pi$
$138$	0	0
$139$	−629.333	−0.384024	−0.192012	−	0.981393i	$-0.561501\pi$
$139$	−629.333	−0.384024	−0.192012	+	0.981393i	$0.561501\pi$
$140$	0	0
$141$	−835.400	−0.498960
$142$	0	0
$143$	1309.49	0.765770
$144$	0	0
$145$	2658.29	1.52248
$146$	0	0
$147$	−768.038	−0.430930
$148$	0	0
$149$	429.457	0.236124	0.118062	−	0.993006i	$-0.462332\pi$
$149$	429.457	0.236124	0.118062	+	0.993006i	$0.462332\pi$
$150$	0	0
$151$	27.3124	0.0147196	0.00735978	−	0.999973i	$-0.497657\pi$
$151$	27.3124	0.0147196	0.00735978	+	0.999973i	$0.497657\pi$
$152$	0	0
$153$	815.398	0.430857
$154$	0	0
$155$	−1677.82	−0.869456
$156$	0	0
$157$	−1251.08	−0.635970	−0.317985	−	0.948096i	$-0.603006\pi$
$157$	−1251.08	−0.635970	−0.317985	+	0.948096i	$0.603006\pi$
$158$	0	0
$159$	−723.908	−0.361067
$160$	0	0
$161$	422.934	0.207030
$162$	0	0
$163$	−127.884	−0.0614517	−0.0307258	−	0.999528i	$-0.509782\pi$
$163$	−127.884	−0.0614517	−0.0307258	+	0.999528i	$0.509782\pi$
$164$	0	0
$165$	−2210.48	−1.04294
$166$	0	0
$167$	2079.65	0.963642	0.481821	−	0.876270i	$-0.339976\pi$
$167$	2079.65	0.963642	0.481821	+	0.876270i	$0.339976\pi$
$168$	0	0
$169$	−1111.08	−0.505726
$170$	0	0
$171$	−653.398	−0.292202
$172$	0	0
$173$	−685.140	−0.301099	−0.150550	−	0.988602i	$-0.548104\pi$
$173$	−685.140	−0.301099	−0.150550	+	0.988602i	$0.548104\pi$
$174$	0	0
$175$	−2040.80	−0.881544
$176$	0	0
$177$	449.172	0.190745
$178$	0	0
$179$	429.423	0.179310	0.0896552	−	0.995973i	$-0.471424\pi$
$179$	429.423	0.179310	0.0896552	+	0.995973i	$0.471424\pi$
$180$	0	0
$181$	−2842.85	−1.16745	−0.583723	−	0.811953i	$-0.698404\pi$
$181$	−2842.85	−1.16745	−0.583723	+	0.811953i	$0.698404\pi$
$182$	0	0
$183$	−1524.94	−0.615994
$184$	0	0
$185$	33.0009	0.0131150
$186$	0	0
$187$	3600.24	1.40789
$188$	0	0
$189$	−251.821	−0.0969167
$190$	0	0
$191$	2546.78	0.964808	0.482404	−	0.875949i	$-0.339764\pi$
$191$	2546.78	0.964808	0.482404	+	0.875949i	$0.339764\pi$
$192$	0	0
$193$	−3579.97	−1.33519	−0.667596	−	0.744524i	$-0.732677\pi$
$193$	−3579.97	−1.33519	−0.667596	+	0.744524i	$0.732677\pi$
$194$	0	0
$195$	−1833.08	−0.673177
$196$	0	0
$197$	−3872.58	−1.40056	−0.700280	−	0.713869i	$-0.746941\pi$
$197$	−3872.58	−1.40056	−0.700280	+	0.713869i	$0.746941\pi$
$198$	0	0
$199$	−5558.64	−1.98011	−0.990054	−	0.140686i	$-0.955069\pi$
$199$	−5558.64	−1.98011	−0.990054	+	0.140686i	$0.955069\pi$
$200$	0	0
$201$	−2850.08	−1.00014
$202$	0	0
$203$	1337.12	0.462301
$204$	0	0
$205$	3631.06	1.23709
$206$	0	0
$207$	−408.120	−0.137035
$208$	0	0
$209$	−2884.96	−0.954816
$210$	0	0
$211$	4658.93	1.52007	0.760034	−	0.649884i	$-0.225182\pi$
$211$	4658.93	1.52007	0.760034	+	0.649884i	$0.225182\pi$
$212$	0	0
$213$	2410.68	0.775478
$214$	0	0
$215$	7547.17	2.39401
$216$	0	0
$217$	−843.940	−0.264011
$218$	0	0
$219$	−1349.36	−0.416352
$220$	0	0
$221$	2985.56	0.908736
$222$	0	0
$223$	−1545.42	−0.464077	−0.232038	−	0.972707i	$-0.574540\pi$
$223$	−1545.42	−0.464077	−0.232038	+	0.972707i	$0.574540\pi$
$224$	0	0
$225$	1969.32	0.583502
$226$	0	0
$227$	−6545.76	−1.91391	−0.956954	−	0.290240i	$-0.906265\pi$
$227$	−6545.76	−1.91391	−0.956954	+	0.290240i	$0.906265\pi$
$228$	0	0
$229$	5463.48	1.57658	0.788291	−	0.615303i	$-0.210966\pi$
$229$	5463.48	1.57658	0.788291	+	0.615303i	$0.210966\pi$
$230$	0	0
$231$	−1111.87	−0.316690
$232$	0	0
$233$	−4722.40	−1.32779	−0.663894	−	0.747827i	$-0.731097\pi$
$233$	−4722.40	−1.32779	−0.663894	+	0.747827i	$0.731097\pi$
$234$	0	0
$235$	5163.38	1.43328
$236$	0	0
$237$	−471.659	−0.129272
$238$	0	0
$239$	−1054.38	−0.285363	−0.142682	−	0.989769i	$-0.545573\pi$
$239$	−1054.38	−0.285363	−0.142682	+	0.989769i	$0.545573\pi$
$240$	0	0
$241$	−3134.40	−0.837777	−0.418888	−	0.908038i	$-0.637580\pi$
$241$	−3134.40	−0.837777	−0.418888	+	0.908038i	$0.637580\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	4747.04	1.23787
$246$	0	0
$247$	−2392.40	−0.616295
$248$	0	0
$249$	525.188	0.133664
$250$	0	0
$251$	−4881.91	−1.22766	−0.613831	−	0.789437i	$-0.710372\pi$
$251$	−4881.91	−1.22766	−0.613831	+	0.789437i	$0.710372\pi$
$252$	0	0
$253$	−1801.97	−0.447783
$254$	0	0
$255$	−5039.76	−1.23765
$256$	0	0
$257$	−540.458	−0.131178	−0.0655892	−	0.997847i	$-0.520893\pi$
$257$	−540.458	−0.131178	−0.0655892	+	0.997847i	$0.520893\pi$
$258$	0	0
$259$	16.5994	0.00398238
$260$	0	0
$261$	−1290.28	−0.306001
$262$	0	0
$263$	−4800.12	−1.12543	−0.562715	−	0.826651i	$-0.690243\pi$
$263$	−4800.12	−1.12543	−0.562715	+	0.826651i	$0.690243\pi$
$264$	0	0
$265$	4474.28	1.03718
$266$	0	0
$267$	−381.599	−0.0874662
$268$	0	0
$269$	−3321.28	−0.752795	−0.376397	−	0.926458i	$-0.622837\pi$
$269$	−3321.28	−0.752795	−0.376397	+	0.926458i	$0.622837\pi$
$270$	0	0
$271$	5274.04	1.18220	0.591098	−	0.806600i	$-0.298695\pi$
$271$	5274.04	1.18220	0.591098	+	0.806600i	$0.298695\pi$
$272$	0	0
$273$	−922.036	−0.204411
$274$	0	0
$275$	8695.15	1.90668
$276$	0	0
$277$	−3190.24	−0.691996	−0.345998	−	0.938235i	$-0.612460\pi$
$277$	−3190.24	−0.691996	−0.345998	+	0.938235i	$0.612460\pi$
$278$	0	0
$279$	814.378	0.174751
$280$	0	0
$281$	545.619	0.115832	0.0579162	−	0.998321i	$-0.481554\pi$
$281$	545.619	0.115832	0.0579162	+	0.998321i	$0.481554\pi$
$282$	0	0
$283$	−5927.74	−1.24511	−0.622557	−	0.782574i	$-0.713906\pi$
$283$	−5927.74	−1.24511	−0.622557	+	0.782574i	$0.713906\pi$
$284$	0	0
$285$	4038.48	0.839365
$286$	0	0
$287$	1826.42	0.375645
$288$	0	0
$289$	3295.33	0.670736
$290$	0	0
$291$	476.478	0.0959850
$292$	0	0
$293$	5406.01	1.07789	0.538946	−	0.842340i	$-0.318823\pi$
$293$	5406.01	1.07789	0.538946	+	0.842340i	$0.318823\pi$
$294$	0	0
$295$	−2776.21	−0.547923
$296$	0	0
$297$	1072.92	0.209620
$298$	0	0
$299$	−1494.32	−0.289026
$300$	0	0
$301$	3796.21	0.726944
$302$	0	0
$303$	−4098.78	−0.777125
$304$	0	0
$305$	9425.25	1.76947
$306$	0	0
$307$	1558.56	0.289745	0.144873	−	0.989450i	$-0.453723\pi$
$307$	1558.56	0.289745	0.144873	+	0.989450i	$0.453723\pi$
$308$	0	0
$309$	5223.89	0.961738
$310$	0	0
$311$	8348.21	1.52213	0.761067	−	0.648673i	$-0.224676\pi$
$311$	8348.21	1.52213	0.761067	+	0.648673i	$0.224676\pi$
$312$	0	0
$313$	5213.09	0.941410	0.470705	−	0.882291i	$-0.344000\pi$
$313$	5213.09	0.941410	0.470705	+	0.882291i	$0.344000\pi$
$314$	0	0
$315$	1556.44	0.278398
$316$	0	0
$317$	9070.57	1.60711	0.803555	−	0.595230i	$-0.202939\pi$
$317$	9070.57	1.60711	0.803555	+	0.595230i	$0.202939\pi$
$318$	0	0
$319$	−5696.98	−0.999905
$320$	0	0
$321$	1948.13	0.338736
$322$	0	0
$323$	−6577.53	−1.13308
$324$	0	0
$325$	7210.61	1.23069
$326$	0	0
$327$	−4239.54	−0.716963
$328$	0	0
$329$	2597.17	0.435218
$330$	0	0
$331$	186.537	0.0309758	0.0154879	−	0.999880i	$-0.495070\pi$
$331$	186.537	0.0309758	0.0154879	+	0.999880i	$0.495070\pi$
$332$	0	0
$333$	−16.0180	−0.00263597
$334$	0	0
$335$	17615.6	2.87296
$336$	0	0
$337$	829.350	0.134058	0.0670290	−	0.997751i	$-0.478648\pi$
$337$	829.350	0.134058	0.0670290	+	0.997751i	$0.478648\pi$
$338$	0	0
$339$	3289.28	0.526989
$340$	0	0
$341$	3595.73	0.571026
$342$	0	0
$343$	5586.81	0.879473
$344$	0	0
$345$	2522.48	0.393640
$346$	0	0
$347$	12005.6	1.85734	0.928669	−	0.370908i	$-0.120954\pi$
$347$	12005.6	1.85734	0.928669	+	0.370908i	$0.120954\pi$
$348$	0	0
$349$	77.8551	0.0119412	0.00597062	−	0.999982i	$-0.498099\pi$
$349$	77.8551	0.0119412	0.00597062	+	0.999982i	$0.498099\pi$
$350$	0	0
$351$	889.739	0.135301
$352$	0	0
$353$	−4925.06	−0.742591	−0.371296	−	0.928515i	$-0.621086\pi$
$353$	−4925.06	−0.742591	−0.371296	+	0.928515i	$0.621086\pi$
$354$	0	0
$355$	−14899.8	−2.22760
$356$	0	0
$357$	−2534.99	−0.375815
$358$	0	0
$359$	12260.2	1.80242	0.901209	−	0.433384i	$-0.142681\pi$
$359$	12260.2	1.80242	0.901209	+	0.433384i	$0.142681\pi$
$360$	0	0
$361$	−1588.27	−0.231560
$362$	0	0
$363$	744.275	0.107615
$364$	0	0
$365$	8340.02	1.19599
$366$	0	0
$367$	−8600.86	−1.22333	−0.611664	−	0.791118i	$-0.709500\pi$
$367$	−8600.86	−1.22333	−0.611664	+	0.791118i	$0.709500\pi$
$368$	0	0
$369$	−1762.44	−0.248642
$370$	0	0
$371$	2250.56	0.314941
$372$	0	0
$373$	−3996.47	−0.554770	−0.277385	−	0.960759i	$-0.589468\pi$
$373$	−3996.47	−0.554770	−0.277385	+	0.960759i	$0.589468\pi$
$374$	0	0
$375$	−5218.51	−0.718620
$376$	0	0
$377$	−4724.33	−0.645399
$378$	0	0
$379$	−10404.7	−1.41017	−0.705087	−	0.709121i	$-0.749092\pi$
$379$	−10404.7	−1.41017	−0.705087	+	0.709121i	$0.749092\pi$
$380$	0	0
$381$	−2213.38	−0.297625
$382$	0	0
$383$	6814.19	0.909109	0.454554	−	0.890719i	$-0.349798\pi$
$383$	6814.19	0.909109	0.454554	+	0.890719i	$0.349798\pi$
$384$	0	0
$385$	6872.15	0.909707
$386$	0	0
$387$	−3663.24	−0.481170
$388$	0	0
$389$	−779.329	−0.101577	−0.0507886	−	0.998709i	$-0.516173\pi$
$389$	−779.329	−0.101577	−0.0507886	+	0.998709i	$0.516173\pi$
$390$	0	0
$391$	−4108.39	−0.531382
$392$	0	0
$393$	−443.095	−0.0568733
$394$	0	0
$395$	2915.20	0.371340
$396$	0	0
$397$	12514.1	1.58202	0.791011	−	0.611802i	$-0.209555\pi$
$397$	12514.1	1.58202	0.791011	+	0.611802i	$0.209555\pi$
$398$	0	0
$399$	2031.35	0.254874
$400$	0	0
$401$	−7949.68	−0.989995	−0.494998	−	0.868894i	$-0.664831\pi$
$401$	−7949.68	−0.989995	−0.494998	+	0.868894i	$0.664831\pi$
$402$	0	0
$403$	2981.83	0.368574
$404$	0	0
$405$	−1501.92	−0.184274
$406$	0	0
$407$	−70.7243	−0.00861345
$408$	0	0
$409$	−15183.9	−1.83569	−0.917846	−	0.396937i	$-0.870073\pi$
$409$	−15183.9	−1.83569	−0.917846	+	0.396937i	$0.870073\pi$
$410$	0	0
$411$	5642.36	0.677170
$412$	0	0
$413$	−1396.43	−0.166377
$414$	0	0
$415$	−3246.05	−0.383957
$416$	0	0
$417$	−1888.00	−0.221716
$418$	0	0
$419$	11767.2	1.37199	0.685996	−	0.727606i	$-0.259367\pi$
$419$	11767.2	1.37199	0.685996	+	0.727606i	$0.259367\pi$
$420$	0	0
$421$	14459.8	1.67394	0.836970	−	0.547248i	$-0.184325\pi$
$421$	14459.8	1.67394	0.836970	+	0.547248i	$0.184325\pi$
$422$	0	0
$423$	−2506.20	−0.288075
$424$	0	0
$425$	19824.4	2.26265
$426$	0	0
$427$	4740.89	0.537301
$428$	0	0
$429$	3928.47	0.442118
$430$	0	0
$431$	−9248.50	−1.03361	−0.516804	−	0.856104i	$-0.672878\pi$
$431$	−9248.50	−1.03361	−0.516804	+	0.856104i	$0.672878\pi$
$432$	0	0
$433$	−3456.12	−0.383581	−0.191790	−	0.981436i	$-0.561429\pi$
$433$	−3456.12	−0.383581	−0.191790	+	0.981436i	$0.561429\pi$
$434$	0	0
$435$	7974.87	0.879002
$436$	0	0
$437$	3292.16	0.360378
$438$	0	0
$439$	−8075.68	−0.877975	−0.438988	−	0.898493i	$-0.644663\pi$
$439$	−8075.68	−0.877975	−0.438988	+	0.898493i	$0.644663\pi$
$440$	0	0
$441$	−2304.12	−0.248798
$442$	0	0
$443$	−11447.7	−1.22776	−0.613878	−	0.789401i	$-0.710391\pi$
$443$	−11447.7	−1.22776	−0.613878	+	0.789401i	$0.710391\pi$
$444$	0	0
$445$	2358.56	0.251251
$446$	0	0
$447$	1288.37	0.136326
$448$	0	0
$449$	−1010.18	−0.106177	−0.0530886	−	0.998590i	$-0.516907\pi$
$449$	−1010.18	−0.106177	−0.0530886	+	0.998590i	$0.516907\pi$
$450$	0	0
$451$	−7781.73	−0.812477
$452$	0	0
$453$	81.9373	0.00849835
$454$	0	0
$455$	5698.86	0.587179
$456$	0	0
$457$	−15949.2	−1.63254	−0.816272	−	0.577667i	$-0.803963\pi$
$457$	−15949.2	−1.63254	−0.816272	+	0.577667i	$0.803963\pi$
$458$	0	0
$459$	2446.19	0.248755
$460$	0	0
$461$	−6737.61	−0.680698	−0.340349	−	0.940299i	$-0.610545\pi$
$461$	−6737.61	−0.680698	−0.340349	+	0.940299i	$0.610545\pi$
$462$	0	0
$463$	−2602.69	−0.261246	−0.130623	−	0.991432i	$-0.541698\pi$
$463$	−2602.69	−0.261246	−0.130623	+	0.991432i	$0.541698\pi$
$464$	0	0
$465$	−5033.45	−0.501980
$466$	0	0
$467$	9326.18	0.924120	0.462060	−	0.886849i	$-0.347110\pi$
$467$	9326.18	0.924120	0.462060	+	0.886849i	$0.347110\pi$
$468$	0	0
$469$	8860.60	0.872376
$470$	0	0
$471$	−3753.25	−0.367178
$472$	0	0
$473$	−16174.3	−1.57230
$474$	0	0
$475$	−15885.8	−1.53451
$476$	0	0
$477$	−2171.72	−0.208462
$478$	0	0
$479$	11472.0	1.09430	0.547149	−	0.837035i	$-0.315713\pi$
$479$	11472.0	1.09430	0.547149	+	0.837035i	$0.315713\pi$
$480$	0	0
$481$	−58.6494	−0.00555963
$482$	0	0
$483$	1268.80	0.119529
$484$	0	0
$485$	−2944.98	−0.275721
$486$	0	0
$487$	15048.0	1.40018	0.700090	−	0.714054i	$-0.253143\pi$
$487$	15048.0	1.40018	0.700090	+	0.714054i	$0.253143\pi$
$488$	0	0
$489$	−383.651	−0.0354792
$490$	0	0
$491$	14373.5	1.32112	0.660558	−	0.750775i	$-0.270320\pi$
$491$	14373.5	1.32112	0.660558	+	0.750775i	$0.270320\pi$
$492$	0	0
$493$	−12988.8	−1.18658
$494$	0	0
$495$	−6631.43	−0.602143
$496$	0	0
$497$	−7494.55	−0.676411
$498$	0	0
$499$	9167.08	0.822395	0.411197	−	0.911546i	$-0.365111\pi$
$499$	9167.08	0.822395	0.411197	+	0.911546i	$0.365111\pi$
$500$	0	0
$501$	6238.95	0.556359
$502$	0	0
$503$	−7690.02	−0.681672	−0.340836	−	0.940123i	$-0.610710\pi$
$503$	−7690.02	−0.681672	−0.340836	+	0.940123i	$0.610710\pi$
$504$	0	0
$505$	25333.5	2.23233
$506$	0	0
$507$	−3333.24	−0.291981
$508$	0	0
$509$	−6854.21	−0.596872	−0.298436	−	0.954430i	$-0.596465\pi$
$509$	−6854.21	−0.596872	−0.298436	+	0.954430i	$0.596465\pi$
$510$	0	0
$511$	4195.01	0.363163
$512$	0	0
$513$	−1960.19	−0.168703
$514$	0	0
$515$	−32287.5	−2.76264
$516$	0	0
$517$	−11065.6	−0.941328
$518$	0	0
$519$	−2055.42	−0.173840
$520$	0	0
$521$	−1641.63	−0.138044	−0.0690221	−	0.997615i	$-0.521988\pi$
$521$	−1641.63	−0.138044	−0.0690221	+	0.997615i	$0.521988\pi$
$522$	0	0
$523$	1976.34	0.165238	0.0826188	−	0.996581i	$-0.473672\pi$
$523$	1976.34	0.165238	0.0826188	+	0.996581i	$0.473672\pi$
$524$	0	0
$525$	−6122.41	−0.508960
$526$	0	0
$527$	8198.06	0.677634
$528$	0	0
$529$	−10110.7	−0.830992
$530$	0	0
$531$	1347.52	0.110127
$532$	0	0
$533$	−6453.14	−0.524421
$534$	0	0
$535$	−12040.9	−0.973034
$536$	0	0
$537$	1288.27	0.103525
$538$	0	0
$539$	−10173.4	−0.812984
$540$	0	0
$541$	2892.17	0.229841	0.114921	−	0.993375i	$-0.463339\pi$
$541$	2892.17	0.229841	0.114921	+	0.993375i	$0.463339\pi$
$542$	0	0
$543$	−8528.56	−0.674025
$544$	0	0
$545$	26203.5	2.05951
$546$	0	0
$547$	−7033.62	−0.549791	−0.274896	−	0.961474i	$-0.588643\pi$
$547$	−7033.62	−0.549791	−0.274896	+	0.961474i	$0.588643\pi$
$548$	0	0
$549$	−4574.82	−0.355644
$550$	0	0
$551$	10408.2	0.804728
$552$	0	0
$553$	1466.34	0.112758
$554$	0	0
$555$	99.0028	0.00757196
$556$	0	0
$557$	8702.92	0.662037	0.331019	−	0.943624i	$-0.392608\pi$
$557$	8702.92	0.662037	0.331019	+	0.943624i	$0.392608\pi$
$558$	0	0
$559$	−13412.9	−1.01485
$560$	0	0
$561$	10800.7	0.812845
$562$	0	0
$563$	14119.7	1.05697	0.528484	−	0.848943i	$-0.322761\pi$
$563$	14119.7	1.05697	0.528484	+	0.848943i	$0.322761\pi$
$564$	0	0
$565$	−20330.2	−1.51380
$566$	0	0
$567$	−755.462	−0.0559549
$568$	0	0
$569$	−383.132	−0.0282280	−0.0141140	−	0.999900i	$-0.504493\pi$
$569$	−383.132	−0.0282280	−0.0141140	+	0.999900i	$0.504493\pi$
$570$	0	0
$571$	21917.8	1.60636	0.803178	−	0.595739i	$-0.203141\pi$
$571$	21917.8	1.60636	0.803178	+	0.595739i	$0.203141\pi$
$572$	0	0
$573$	7640.33	0.557032
$574$	0	0
$575$	−9922.44	−0.719642
$576$	0	0
$577$	1570.50	0.113311	0.0566556	−	0.998394i	$-0.481956\pi$
$577$	1570.50	0.113311	0.0566556	+	0.998394i	$0.481956\pi$
$578$	0	0
$579$	−10739.9	−0.770873
$580$	0	0
$581$	−1632.76	−0.116589
$582$	0	0
$583$	−9588.84	−0.681182
$584$	0	0
$585$	−5499.24	−0.388659
$586$	0	0
$587$	−14387.7	−1.01166	−0.505830	−	0.862633i	$-0.668814\pi$
$587$	−14387.7	−1.01166	−0.505830	+	0.862633i	$0.668814\pi$
$588$	0	0
$589$	−6569.30	−0.459564
$590$	0	0
$591$	−11617.8	−0.808613
$592$	0	0
$593$	17903.0	1.23978	0.619889	−	0.784690i	$-0.287178\pi$
$593$	17903.0	1.23978	0.619889	+	0.784690i	$0.287178\pi$
$594$	0	0
$595$	15668.1	1.07955
$596$	0	0
$597$	−16675.9	−1.14322
$598$	0	0
$599$	−9474.88	−0.646299	−0.323150	−	0.946348i	$-0.604742\pi$
$599$	−9474.88	−0.646299	−0.323150	+	0.946348i	$0.604742\pi$
$600$	0	0
$601$	16945.6	1.15012	0.575061	−	0.818110i	$-0.304978\pi$
$601$	16945.6	1.15012	0.575061	+	0.818110i	$0.304978\pi$
$602$	0	0
$603$	−8550.23	−0.577433
$604$	0	0
$605$	−4600.16	−0.309129
$606$	0	0
$607$	18736.2	1.25285	0.626423	−	0.779483i	$-0.284518\pi$
$607$	18736.2	1.25285	0.626423	+	0.779483i	$0.284518\pi$
$608$	0	0
$609$	4011.35	0.266910
$610$	0	0
$611$	−9176.39	−0.607589
$612$	0	0
$613$	−27850.0	−1.83499	−0.917497	−	0.397742i	$-0.869794\pi$
$613$	−27850.0	−1.83499	−0.917497	+	0.397742i	$0.869794\pi$
$614$	0	0
$615$	10893.2	0.714237
$616$	0	0
$617$	12836.3	0.837551	0.418776	−	0.908090i	$-0.362459\pi$
$617$	12836.3	0.837551	0.418776	+	0.908090i	$0.362459\pi$
$618$	0	0
$619$	18030.3	1.17076	0.585378	−	0.810760i	$-0.300946\pi$
$619$	18030.3	1.17076	0.585378	+	0.810760i	$0.300946\pi$
$620$	0	0
$621$	−1224.36	−0.0791173
$622$	0	0
$623$	1186.35	0.0762924
$624$	0	0
$625$	4902.56	0.313764
$626$	0	0
$627$	−8654.87	−0.551263
$628$	0	0
$629$	−161.247	−0.0102215
$630$	0	0
$631$	−16460.1	−1.03846	−0.519229	−	0.854635i	$-0.673781\pi$
$631$	−16460.1	−1.03846	−0.519229	+	0.854635i	$0.673781\pi$
$632$	0	0
$633$	13976.8	0.877611
$634$	0	0
$635$	13680.3	0.854940
$636$	0	0
$637$	−8436.46	−0.524749
$638$	0	0
$639$	7232.03	0.447723
$640$	0	0
$641$	−19443.3	−1.19807	−0.599035	−	0.800723i	$-0.704449\pi$
$641$	−19443.3	−1.19807	−0.599035	+	0.800723i	$0.704449\pi$
$642$	0	0
$643$	7368.87	0.451944	0.225972	−	0.974134i	$-0.427444\pi$
$643$	7368.87	0.451944	0.225972	+	0.974134i	$0.427444\pi$
$644$	0	0
$645$	22641.5	1.38218
$646$	0	0
$647$	11042.3	0.670969	0.335484	−	0.942046i	$-0.391100\pi$
$647$	11042.3	0.670969	0.335484	+	0.942046i	$0.391100\pi$
$648$	0	0
$649$	5949.70	0.359856
$650$	0	0
$651$	−2531.82	−0.152427
$652$	0	0
$653$	−14174.3	−0.849440	−0.424720	−	0.905325i	$-0.639627\pi$
$653$	−14174.3	−0.849440	−0.424720	+	0.905325i	$0.639627\pi$
$654$	0	0
$655$	2738.65	0.163371
$656$	0	0
$657$	−4048.07	−0.240381
$658$	0	0
$659$	−1917.19	−0.113328	−0.0566640	−	0.998393i	$-0.518046\pi$
$659$	−1917.19	−0.113328	−0.0566640	+	0.998393i	$0.518046\pi$
$660$	0	0
$661$	28084.2	1.65257	0.826284	−	0.563254i	$-0.190451\pi$
$661$	28084.2	1.65257	0.826284	+	0.563254i	$0.190451\pi$
$662$	0	0
$663$	8956.69	0.524659
$664$	0	0
$665$	−12555.2	−0.732136
$666$	0	0
$667$	6501.09	0.377396
$668$	0	0
$669$	−4636.27	−0.267935
$670$	0	0
$671$	−20199.3	−1.16212
$672$	0	0
$673$	1756.20	0.100589	0.0502947	−	0.998734i	$-0.483984\pi$
$673$	1756.20	0.100589	0.0502947	+	0.998734i	$0.483984\pi$
$674$	0	0
$675$	5907.96	0.336885
$676$	0	0
$677$	−10424.0	−0.591766	−0.295883	−	0.955224i	$-0.595614\pi$
$677$	−10424.0	−0.591766	−0.295883	+	0.955224i	$0.595614\pi$
$678$	0	0
$679$	−1481.32	−0.0837230
$680$	0	0
$681$	−19637.3	−1.10500
$682$	0	0
$683$	5828.41	0.326527	0.163264	−	0.986582i	$-0.447798\pi$
$683$	5828.41	0.326527	0.163264	+	0.986582i	$0.447798\pi$
$684$	0	0
$685$	−34873.9	−1.94520
$686$	0	0
$687$	16390.4	0.910240
$688$	0	0
$689$	−7951.72	−0.439675
$690$	0	0
$691$	10673.5	0.587611	0.293806	−	0.955865i	$-0.405078\pi$
$691$	10673.5	0.587611	0.293806	+	0.955865i	$0.405078\pi$
$692$	0	0
$693$	−3335.60	−0.182841
$694$	0	0
$695$	11669.2	0.636890
$696$	0	0
$697$	−17741.9	−0.964163
$698$	0	0
$699$	−14167.2	−0.766599
$700$	0	0
$701$	−14367.0	−0.774084	−0.387042	−	0.922062i	$-0.626503\pi$
$701$	−14367.0	−0.774084	−0.387042	+	0.922062i	$0.626503\pi$
$702$	0	0
$703$	129.211	0.00693214
$704$	0	0
$705$	15490.1	0.827507
$706$	0	0
$707$	12742.7	0.677848
$708$	0	0
$709$	−25026.4	−1.32565	−0.662825	−	0.748774i	$-0.730643\pi$
$709$	−25026.4	−1.32565	−0.662825	+	0.748774i	$0.730643\pi$
$710$	0	0
$711$	−1414.98	−0.0746354
$712$	0	0
$713$	−4103.26	−0.215523
$714$	0	0
$715$	−24280.8	−1.27000
$716$	0	0
$717$	−3163.13	−0.164755
$718$	0	0
$719$	692.065	0.0358966	0.0179483	−	0.999839i	$-0.494287\pi$
$719$	692.065	0.0358966	0.0179483	+	0.999839i	$0.494287\pi$
$720$	0	0
$721$	−16240.6	−0.838876
$722$	0	0
$723$	−9403.19	−0.483691
$724$	0	0
$725$	−31370.0	−1.60697
$726$	0	0
$727$	23929.5	1.22076	0.610382	−	0.792107i	$-0.291016\pi$
$727$	23929.5	1.22076	0.610382	+	0.792107i	$0.291016\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−36876.5	−1.86584
$732$	0	0
$733$	−5613.22	−0.282850	−0.141425	−	0.989949i	$-0.545168\pi$
$733$	−5613.22	−0.282850	−0.141425	+	0.989949i	$0.545168\pi$
$734$	0	0
$735$	14241.1	0.714683
$736$	0	0
$737$	−37751.9	−1.88685
$738$	0	0
$739$	4790.30	0.238449	0.119225	−	0.992867i	$-0.461959\pi$
$739$	4790.30	0.238449	0.119225	+	0.992867i	$0.461959\pi$
$740$	0	0
$741$	−7177.21	−0.355818
$742$	0	0
$743$	−16695.0	−0.824333	−0.412166	−	0.911109i	$-0.635228\pi$
$743$	−16695.0	−0.824333	−0.412166	+	0.911109i	$0.635228\pi$
$744$	0	0
$745$	−7963.07	−0.391603
$746$	0	0
$747$	1575.56	0.0771711
$748$	0	0
$749$	−6056.55	−0.295463
$750$	0	0
$751$	27366.8	1.32973	0.664866	−	0.746963i	$-0.268489\pi$
$751$	27366.8	1.32973	0.664866	+	0.746963i	$0.268489\pi$
$752$	0	0
$753$	−14645.7	−0.708791
$754$	0	0
$755$	−506.433	−0.0244119
$756$	0	0
$757$	5712.23	0.274260	0.137130	−	0.990553i	$-0.456212\pi$
$757$	5712.23	0.274260	0.137130	+	0.990553i	$0.456212\pi$
$758$	0	0
$759$	−5405.92	−0.258528
$760$	0	0
$761$	−14015.9	−0.667643	−0.333822	−	0.942636i	$-0.608338\pi$
$761$	−14015.9	−0.667643	−0.333822	+	0.942636i	$0.608338\pi$
$762$	0	0
$763$	13180.3	0.625372
$764$	0	0
$765$	−15119.3	−0.714560
$766$	0	0
$767$	4933.90	0.232272
$768$	0	0
$769$	−3430.70	−0.160877	−0.0804384	−	0.996760i	$-0.525632\pi$
$769$	−3430.70	−0.160877	−0.0804384	+	0.996760i	$0.525632\pi$
$770$	0	0
$771$	−1621.37	−0.0757359
$772$	0	0
$773$	14821.8	0.689654	0.344827	−	0.938666i	$-0.387938\pi$
$773$	14821.8	0.689654	0.344827	+	0.938666i	$0.387938\pi$
$774$	0	0
$775$	19799.6	0.917708
$776$	0	0
$777$	49.7982	0.00229923
$778$	0	0
$779$	14217.0	0.653885
$780$	0	0
$781$	31931.7	1.46300
$782$	0	0
$783$	−3870.84	−0.176670
$784$	0	0
$785$	23197.8	1.05473
$786$	0	0
$787$	16917.4	0.766253	0.383126	−	0.923696i	$-0.374847\pi$
$787$	16917.4	0.766253	0.383126	+	0.923696i	$0.374847\pi$
$788$	0	0
$789$	−14400.4	−0.649768
$790$	0	0
$791$	−10226.0	−0.459666
$792$	0	0
$793$	−16750.6	−0.750103
$794$	0	0
$795$	13422.9	0.598817
$796$	0	0
$797$	−23546.7	−1.04651	−0.523254	−	0.852177i	$-0.675282\pi$
$797$	−23546.7	−1.04651	−0.523254	+	0.852177i	$0.675282\pi$
$798$	0	0
$799$	−25229.0	−1.11707
$800$	0	0
$801$	−1144.80	−0.0504986
$802$	0	0
$803$	−17873.5	−0.785482
$804$	0	0
$805$	−7842.13	−0.343352
$806$	0	0
$807$	−9963.83	−0.434626
$808$	0	0
$809$	17647.6	0.766942	0.383471	−	0.923553i	$-0.374729\pi$
$809$	17647.6	0.766942	0.383471	+	0.923553i	$0.374729\pi$
$810$	0	0
$811$	11690.3	0.506169	0.253085	−	0.967444i	$-0.418555\pi$
$811$	11690.3	0.506169	0.253085	+	0.967444i	$0.418555\pi$
$812$	0	0
$813$	15822.1	0.682541
$814$	0	0
$815$	2371.25	0.101915
$816$	0	0
$817$	29550.0	1.26539
$818$	0	0
$819$	−2766.11	−0.118017
$820$	0	0
$821$	−10576.3	−0.449594	−0.224797	−	0.974406i	$-0.572172\pi$
$821$	−10576.3	−0.449594	−0.224797	+	0.974406i	$0.572172\pi$
$822$	0	0
$823$	44624.4	1.89005	0.945023	−	0.327005i	$-0.106039\pi$
$823$	44624.4	1.89005	0.945023	+	0.327005i	$0.106039\pi$
$824$	0	0
$825$	26085.5	1.10082
$826$	0	0
$827$	2532.98	0.106506	0.0532528	−	0.998581i	$-0.483041\pi$
$827$	2532.98	0.106506	0.0532528	+	0.998581i	$0.483041\pi$
$828$	0	0
$829$	24029.6	1.00673	0.503366	−	0.864073i	$-0.332095\pi$
$829$	24029.6	1.00673	0.503366	+	0.864073i	$0.332095\pi$
$830$	0	0
$831$	−9570.72	−0.399524
$832$	0	0
$833$	−23194.7	−0.964765
$834$	0	0
$835$	−38561.3	−1.59817
$836$	0	0
$837$	2443.13	0.100893
$838$	0	0
$839$	−39117.5	−1.60964	−0.804819	−	0.593520i	$-0.797738\pi$
$839$	−39117.5	−1.60964	−0.804819	+	0.593520i	$0.797738\pi$
$840$	0	0
$841$	−3835.65	−0.157270
$842$	0	0
$843$	1636.86	0.0668758
$844$	0	0
$845$	20601.9	0.838729
$846$	0	0
$847$	−2313.87	−0.0938674
$848$	0	0
$849$	−17783.2	−0.718867
$850$	0	0
$851$	80.7068	0.00325099
$852$	0	0
$853$	35436.1	1.42240	0.711201	−	0.702988i	$-0.248151\pi$
$853$	35436.1	1.42240	0.711201	+	0.702988i	$0.248151\pi$
$854$	0	0
$855$	12115.4	0.484607
$856$	0	0
$857$	−20451.8	−0.815191	−0.407596	−	0.913163i	$-0.633633\pi$
$857$	−20451.8	−0.815191	−0.407596	+	0.913163i	$0.633633\pi$
$858$	0	0
$859$	6477.74	0.257297	0.128648	−	0.991690i	$-0.458936\pi$
$859$	6477.74	0.257297	0.128648	+	0.991690i	$0.458936\pi$
$860$	0	0
$861$	5479.25	0.216879
$862$	0	0
$863$	−2068.34	−0.0815843	−0.0407922	−	0.999168i	$-0.512988\pi$
$863$	−2068.34	−0.0815843	−0.0407922	+	0.999168i	$0.512988\pi$
$864$	0	0
$865$	12704.0	0.499363
$866$	0	0
$867$	9885.98	0.387250
$868$	0	0
$869$	−6247.56	−0.243882
$870$	0	0
$871$	−31306.5	−1.21789
$872$	0	0
$873$	1429.43	0.0554170
$874$	0	0
$875$	16223.8	0.626817
$876$	0	0
$877$	−33095.0	−1.27427	−0.637137	−	0.770750i	$-0.719882\pi$
$877$	−33095.0	−1.27427	−0.637137	+	0.770750i	$0.719882\pi$
$878$	0	0
$879$	16218.0	0.622321
$880$	0	0
$881$	33169.3	1.26845	0.634225	−	0.773149i	$-0.281319\pi$
$881$	33169.3	1.26845	0.634225	+	0.773149i	$0.281319\pi$
$882$	0	0
$883$	−28990.4	−1.10488	−0.552438	−	0.833554i	$-0.686302\pi$
$883$	−28990.4	−1.10488	−0.552438	+	0.833554i	$0.686302\pi$
$884$	0	0
$885$	−8328.64	−0.316344
$886$	0	0
$887$	−458.565	−0.0173586	−0.00867931	−	0.999962i	$-0.502763\pi$
$887$	−458.565	−0.0173586	−0.00867931	+	0.999962i	$0.502763\pi$
$888$	0	0
$889$	6881.18	0.259603
$890$	0	0
$891$	3218.76	0.121024
$892$	0	0
$893$	20216.6	0.757585
$894$	0	0
$895$	−7962.44	−0.297380
$896$	0	0
$897$	−4482.96	−0.166869
$898$	0	0
$899$	−12972.5	−0.481266
$900$	0	0
$901$	−21862.0	−0.808355
$902$	0	0
$903$	11388.6	0.419701
$904$	0	0
$905$	52712.8	1.93617
$906$	0	0
$907$	−45653.8	−1.67134	−0.835672	−	0.549229i	$-0.814922\pi$
$907$	−45653.8	−1.67134	−0.835672	+	0.549229i	$0.814922\pi$
$908$	0	0
$909$	−12296.3	−0.448673
$910$	0	0
$911$	−50760.6	−1.84608	−0.923038	−	0.384710i	$-0.874302\pi$
$911$	−50760.6	−1.84608	−0.923038	+	0.384710i	$0.874302\pi$
$912$	0	0
$913$	6956.60	0.252169
$914$	0	0
$915$	28275.8	1.02160
$916$	0	0
$917$	1377.54	0.0496078
$918$	0	0
$919$	25939.0	0.931065	0.465532	−	0.885031i	$-0.345863\pi$
$919$	25939.0	0.931065	0.465532	+	0.885031i	$0.345863\pi$
$920$	0	0
$921$	4675.68	0.167284
$922$	0	0
$923$	26479.9	0.944309
$924$	0	0
$925$	−389.438	−0.0138429
$926$	0	0
$927$	15671.7	0.555260
$928$	0	0
$929$	41850.4	1.47801	0.739003	−	0.673702i	$-0.235297\pi$
$929$	41850.4	1.47801	0.739003	+	0.673702i	$0.235297\pi$
$930$	0	0
$931$	18586.5	0.654294
$932$	0	0
$933$	25044.6	0.878805
$934$	0	0
$935$	−66756.3	−2.33494
$936$	0	0
$937$	−18888.1	−0.658534	−0.329267	−	0.944237i	$-0.606802\pi$
$937$	−18888.1	−0.658534	−0.329267	+	0.944237i	$0.606802\pi$
$938$	0	0
$939$	15639.3	0.543523
$940$	0	0
$941$	21571.8	0.747313	0.373656	−	0.927567i	$-0.378104\pi$
$941$	21571.8	0.747313	0.373656	+	0.927567i	$0.378104\pi$
$942$	0	0
$943$	8880.09	0.306655
$944$	0	0
$945$	4669.31	0.160733
$946$	0	0
$947$	−2981.14	−0.102296	−0.0511479	−	0.998691i	$-0.516288\pi$
$947$	−2981.14	−0.102296	−0.0511479	+	0.998691i	$0.516288\pi$
$948$	0	0
$949$	−14821.9	−0.506997
$950$	0	0
$951$	27211.7	0.927865
$952$	0	0
$953$	8353.84	0.283953	0.141977	−	0.989870i	$-0.454654\pi$
$953$	8353.84	0.283953	0.141977	+	0.989870i	$0.454654\pi$
$954$	0	0
$955$	−47222.9	−1.60010
$956$	0	0
$957$	−17090.9	−0.577296
$958$	0	0
$959$	−17541.5	−0.590662
$960$	0	0
$961$	−21603.2	−0.725159
$962$	0	0
$963$	5844.40	0.195569
$964$	0	0
$965$	66380.6	2.21437
$966$	0	0
$967$	20156.9	0.670323	0.335162	−	0.942161i	$-0.391209\pi$
$967$	20156.9	0.670323	0.335162	+	0.942161i	$0.391209\pi$
$968$	0	0
$969$	−19732.6	−0.654182
$970$	0	0
$971$	−32976.8	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$971$	−32976.8	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$972$	0	0
$973$	5869.60	0.193392
$974$	0	0
$975$	21631.8	0.710537
$976$	0	0
$977$	2934.18	0.0960827	0.0480413	−	0.998845i	$-0.484702\pi$
$977$	2934.18	0.0960827	0.0480413	+	0.998845i	$0.484702\pi$
$978$	0	0
$979$	−5054.63	−0.165012
$980$	0	0
$981$	−12718.6	−0.413939
$982$	0	0
$983$	−11965.0	−0.388223	−0.194111	−	0.980979i	$-0.562182\pi$
$983$	−11965.0	−0.388223	−0.194111	+	0.980979i	$0.562182\pi$
$984$	0	0
$985$	71806.2	2.32278
$986$	0	0
$987$	7791.52	0.251273
$988$	0	0
$989$	18457.3	0.593435
$990$	0	0
$991$	43262.0	1.38674	0.693371	−	0.720581i	$-0.256125\pi$
$991$	43262.0	1.38674	0.693371	+	0.720581i	$0.256125\pi$
$992$	0	0
$993$	559.610	0.0178839
$994$	0	0
$995$	103069.	3.28394
$996$	0	0
$997$	35664.7	1.13291	0.566456	−	0.824092i	$-0.308314\pi$
$997$	35664.7	1.13291	0.566456	+	0.824092i	$0.308314\pi$
$998$	0	0
$999$	−48.0539	−0.00152188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.s.1.1		3
3.2	odd	2		2304.4.a.bv.1.3		3
4.3	odd	2		768.4.a.q.1.1		3
8.3	odd	2		768.4.a.t.1.3		3
8.5	even	2		768.4.a.r.1.3		3
12.11	even	2		2304.4.a.bw.1.3		3
16.3	odd	4		96.4.d.a.49.3		6
16.5	even	4		24.4.d.a.13.2	yes	6
16.11	odd	4		96.4.d.a.49.4		6
16.13	even	4		24.4.d.a.13.1	✓	6
24.5	odd	2		2304.4.a.bt.1.1		3
24.11	even	2		2304.4.a.bu.1.1		3
48.5	odd	4		72.4.d.d.37.5		6
48.11	even	4		288.4.d.d.145.6		6
48.29	odd	4		72.4.d.d.37.6		6
48.35	even	4		288.4.d.d.145.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.d.a.13.1	✓	6	16.13	even	4
24.4.d.a.13.2	yes	6	16.5	even	4
72.4.d.d.37.5		6	48.5	odd	4
72.4.d.d.37.6		6	48.29	odd	4
96.4.d.a.49.3		6	16.3	odd	4
96.4.d.a.49.4		6	16.11	odd	4
288.4.d.d.145.1		6	48.35	even	4
288.4.d.d.145.6		6	48.11	even	4
768.4.a.q.1.1		3	4.3	odd	2
768.4.a.r.1.3		3	8.5	even	2
768.4.a.s.1.1		3	1.1	even	1	trivial
768.4.a.t.1.3		3	8.3	odd	2
2304.4.a.bt.1.1		3	24.5	odd	2
2304.4.a.bu.1.1		3	24.11	even	2
2304.4.a.bv.1.3		3	3.2	odd	2
2304.4.a.bw.1.3		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -18.5422 q^{5} -9.32669 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -18.5422 q^{5} -9.32669 q^{7} +9.00000 q^{9} +39.7378 q^{11} +32.9533 q^{13} -55.6266 q^{15} +90.5998 q^{17} -72.5998 q^{19} -27.9801 q^{21} -45.3466 q^{23} +218.813 q^{25} +27.0000 q^{27} -143.364 q^{29} +90.4865 q^{31} +119.213 q^{33} +172.937 q^{35} -1.77977 q^{37} +98.8599 q^{39} -195.827 q^{41} -407.027 q^{43} -166.880 q^{45} -278.467 q^{47} -256.013 q^{49} +271.799 q^{51} -241.303 q^{53} -736.826 q^{55} -217.799 q^{57} +149.724 q^{59} -508.314 q^{61} -83.9402 q^{63} -611.027 q^{65} -950.026 q^{67} -136.040 q^{69} +803.559 q^{71} -449.786 q^{73} +656.440 q^{75} -370.622 q^{77} -157.220 q^{79} +81.0000 q^{81} +175.063 q^{83} -1679.92 q^{85} -430.093 q^{87} -127.200 q^{89} -307.345 q^{91} +271.459 q^{93} +1346.16 q^{95} +158.826 q^{97} +357.640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9	\( 3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9} - 52 q^{13} - 30 q^{15} + 26 q^{17} + 28 q^{19} - 42 q^{21} - 164 q^{23} + 53 q^{25} + 81 q^{27} - 174 q^{29} - 318 q^{31} - 92 q^{35} - 296 q^{37} - 156 q^{39} - 118 q^{41} - 260 q^{43} - 90 q^{45} - 204 q^{47} + 327 q^{49} + 78 q^{51} - 1086 q^{53} - 512 q^{55} + 84 q^{57} + 196 q^{59} - 1536 q^{61} - 126 q^{63} - 872 q^{65} - 660 q^{67} - 492 q^{69} + 852 q^{71} - 478 q^{73} + 159 q^{75} - 2304 q^{77} - 22 q^{79} + 243 q^{81} + 1136 q^{83} - 2732 q^{85} - 522 q^{87} + 110 q^{89} - 632 q^{91} - 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9 - 52 * q^13 - 30 * q^15 + 26 * q^17 + 28 * q^19 - 42 * q^21 - 164 * q^23 + 53 * q^25 + 81 * q^27 - 174 * q^29 - 318 * q^31 - 92 * q^35 - 296 * q^37 - 156 * q^39 - 118 * q^41 - 260 * q^43 - 90 * q^45 - 204 * q^47 + 327 * q^49 + 78 * q^51 - 1086 * q^53 - 512 * q^55 + 84 * q^57 + 196 * q^59 - 1536 * q^61 - 126 * q^63 - 872 * q^65 - 660 * q^67 - 492 * q^69 + 852 * q^71 - 478 * q^73 + 159 * q^75 - 2304 * q^77 - 22 * q^79 + 243 * q^81 + 1136 * q^83 - 2732 * q^85 - 522 * q^87 + 110 * q^89 - 632 * q^91 - 954 * q^93 + 2552 * q^95 - 1222 * q^97

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