LMFDB - Embedded newform 768.4.a.u.1.3

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:	$4$
Coefficient field:	4.4.9792.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 2x + 7 $ x^4 - 2x^3 - 7x^2 + 2*x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.21597$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +5.67763 q^{5} -33.0917 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.67763 q^{5} -33.0917 q^{7} +9.00000 q^{9} -34.6274 q^{11} +82.2421 q^{13} -17.0329 q^{15} +97.8823 q^{17} +55.8823 q^{19} +99.2750 q^{21} +130.418 q^{23} -92.7645 q^{25} -27.0000 q^{27} -147.451 q^{29} -101.223 q^{31} +103.882 q^{33} -187.882 q^{35} -184.439 q^{37} -246.726 q^{39} -237.411 q^{41} -199.882 q^{43} +51.0987 q^{45} +334.813 q^{47} +752.058 q^{49} -293.647 q^{51} -102.030 q^{53} -196.602 q^{55} -167.647 q^{57} -105.961 q^{59} -717.803 q^{61} -297.825 q^{63} +466.940 q^{65} -316.471 q^{67} -391.255 q^{69} +800.045 q^{71} -301.058 q^{73} +278.294 q^{75} +1145.88 q^{77} +42.8329 q^{79} +81.0000 q^{81} -1236.67 q^{83} +555.739 q^{85} +442.354 q^{87} -1325.29 q^{89} -2721.53 q^{91} +303.670 q^{93} +317.279 q^{95} -505.765 q^{97} -311.647 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.67763	0.507823	0.253911	−	0.967227i	$-0.418283\pi$
$5$	5.67763	0.507823	0.253911	+	0.967227i	$0.418283\pi$
$6$	0	0
$7$	−33.0917	−1.78678	−0.893391	−	0.449280i	$-0.851680\pi$
$7$	−33.0917	−1.78678	−0.893391	+	0.449280i	$0.851680\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−34.6274	−0.949142	−0.474571	−	0.880217i	$-0.657397\pi$
$11$	−34.6274	−0.949142	−0.474571	+	0.880217i	$0.657397\pi$
$12$	0	0
$13$	82.2421	1.75460	0.877302	−	0.479939i	$-0.159341\pi$
$13$	82.2421	1.75460	0.877302	+	0.479939i	$0.159341\pi$
$14$	0	0
$15$	−17.0329	−0.293192
$16$	0	0
$17$	97.8823	1.39647	0.698233	−	0.715870i	$-0.253970\pi$
$17$	97.8823	1.39647	0.698233	+	0.715870i	$0.253970\pi$
$18$	0	0
$19$	55.8823	0.674751	0.337375	−	0.941370i	$-0.390461\pi$
$19$	55.8823	0.674751	0.337375	+	0.941370i	$0.390461\pi$
$20$	0	0
$21$	99.2750	1.03160
$22$	0	0
$23$	130.418	1.18235	0.591176	−	0.806542i	$-0.298664\pi$
$23$	130.418	1.18235	0.591176	+	0.806542i	$0.298664\pi$
$24$	0	0
$25$	−92.7645	−0.742116
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−147.451	−0.944173	−0.472086	−	0.881552i	$-0.656499\pi$
$29$	−147.451	−0.944173	−0.472086	+	0.881552i	$0.656499\pi$
$30$	0	0
$31$	−101.223	−0.586459	−0.293230	−	0.956042i	$-0.594730\pi$
$31$	−101.223	−0.586459	−0.293230	+	0.956042i	$0.594730\pi$
$32$	0	0
$33$	103.882	0.547987
$34$	0	0
$35$	−187.882	−0.907368
$36$	0	0
$37$	−184.439	−0.819504	−0.409752	−	0.912197i	$-0.634385\pi$
$37$	−184.439	−0.819504	−0.409752	+	0.912197i	$0.634385\pi$
$38$	0	0
$39$	−246.726	−1.01302
$40$	0	0
$41$	−237.411	−0.904327	−0.452164	−	0.891935i	$-0.649348\pi$
$41$	−237.411	−0.904327	−0.452164	+	0.891935i	$0.649348\pi$
$42$	0	0
$43$	−199.882	−0.708878	−0.354439	−	0.935079i	$-0.615328\pi$
$43$	−199.882	−0.708878	−0.354439	+	0.935079i	$0.615328\pi$
$44$	0	0
$45$	51.0987	0.169274
$46$	0	0
$47$	334.813	1.03910	0.519548	−	0.854441i	$-0.326100\pi$
$47$	334.813	1.03910	0.519548	+	0.854441i	$0.326100\pi$
$48$	0	0
$49$	752.058	2.19259
$50$	0	0
$51$	−293.647	−0.806250
$52$	0	0
$53$	−102.030	−0.264433	−0.132216	−	0.991221i	$-0.542209\pi$
$53$	−102.030	−0.264433	−0.132216	+	0.991221i	$0.542209\pi$
$54$	0	0
$55$	−196.602	−0.481996
$56$	0	0
$57$	−167.647	−0.389568
$58$	0	0
$59$	−105.961	−0.233813	−0.116907	−	0.993143i	$-0.537298\pi$
$59$	−105.961	−0.233813	−0.116907	+	0.993143i	$0.537298\pi$
$60$	0	0
$61$	−717.803	−1.50664	−0.753321	−	0.657653i	$-0.771549\pi$
$61$	−717.803	−1.50664	−0.753321	+	0.657653i	$0.771549\pi$
$62$	0	0
$63$	−297.825	−0.595594
$64$	0	0
$65$	466.940	0.891028
$66$	0	0
$67$	−316.471	−0.577061	−0.288530	−	0.957471i	$-0.593167\pi$
$67$	−316.471	−0.577061	−0.288530	+	0.957471i	$0.593167\pi$
$68$	0	0
$69$	−391.255	−0.682632
$70$	0	0
$71$	800.045	1.33729	0.668647	−	0.743580i	$-0.266874\pi$
$71$	800.045	1.33729	0.668647	+	0.743580i	$0.266874\pi$
$72$	0	0
$73$	−301.058	−0.482687	−0.241344	−	0.970440i	$-0.577588\pi$
$73$	−301.058	−0.482687	−0.241344	+	0.970440i	$0.577588\pi$
$74$	0	0
$75$	278.294	0.428461
$76$	0	0
$77$	1145.88	1.69591
$78$	0	0
$79$	42.8329	0.0610010	0.0305005	−	0.999535i	$-0.490290\pi$
$79$	42.8329	0.0610010	0.0305005	+	0.999535i	$0.490290\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1236.67	−1.63544	−0.817721	−	0.575614i	$-0.804763\pi$
$83$	−1236.67	−1.63544	−0.817721	+	0.575614i	$0.804763\pi$
$84$	0	0
$85$	555.739	0.709158
$86$	0	0
$87$	442.354	0.545119
$88$	0	0
$89$	−1325.29	−1.57844	−0.789218	−	0.614113i	$-0.789514\pi$
$89$	−1325.29	−1.57844	−0.789218	+	0.614113i	$0.789514\pi$
$90$	0	0
$91$	−2721.53	−3.13509
$92$	0	0
$93$	303.670	0.338592
$94$	0	0
$95$	317.279	0.342654
$96$	0	0
$97$	−505.765	−0.529408	−0.264704	−	0.964330i	$-0.585274\pi$
$97$	−505.765	−0.529408	−0.264704	+	0.964330i	$0.585274\pi$
$98$	0	0
$99$	−311.647	−0.316381
$100$	0	0
$101$	1281.97	1.26298	0.631491	−	0.775383i	$-0.282443\pi$
$101$	1281.97	1.26298	0.631491	+	0.775383i	$0.282443\pi$
$102$	0	0
$103$	161.562	0.154555	0.0772775	−	0.997010i	$-0.475377\pi$
$103$	161.562	0.154555	0.0772775	+	0.997010i	$0.475377\pi$
$104$	0	0
$105$	563.647	0.523869
$106$	0	0
$107$	−481.726	−0.435235	−0.217618	−	0.976034i	$-0.569829\pi$
$107$	−481.726	−0.435235	−0.217618	+	0.976034i	$0.569829\pi$
$108$	0	0
$109$	286.637	0.251879	0.125940	−	0.992038i	$-0.459805\pi$
$109$	286.637	0.251879	0.125940	+	0.992038i	$0.459805\pi$
$110$	0	0
$111$	553.318	0.473141
$112$	0	0
$113$	−1397.29	−1.16324	−0.581621	−	0.813460i	$-0.697581\pi$
$113$	−1397.29	−1.16324	−0.581621	+	0.813460i	$0.697581\pi$
$114$	0	0
$115$	740.468	0.600426
$116$	0	0
$117$	740.179	0.584868
$118$	0	0
$119$	−3239.09	−2.49518
$120$	0	0
$121$	−131.942	−0.0991300
$122$	0	0
$123$	712.234	0.522113
$124$	0	0
$125$	−1236.39	−0.884686
$126$	0	0
$127$	−443.829	−0.310106	−0.155053	−	0.987906i	$-0.549555\pi$
$127$	−443.829	−0.310106	−0.155053	+	0.987906i	$0.549555\pi$
$128$	0	0
$129$	599.647	0.409271
$130$	0	0
$131$	−1365.57	−0.910765	−0.455382	−	0.890296i	$-0.650497\pi$
$131$	−1365.57	−0.910765	−0.455382	+	0.890296i	$0.650497\pi$
$132$	0	0
$133$	−1849.24	−1.20563
$134$	0	0
$135$	−153.296	−0.0977305
$136$	0	0
$137$	1223.06	0.762721	0.381360	−	0.924426i	$-0.375456\pi$
$137$	1223.06	0.762721	0.381360	+	0.924426i	$0.375456\pi$
$138$	0	0
$139$	−2176.70	−1.32824	−0.664121	−	0.747625i	$-0.731194\pi$
$139$	−2176.70	−1.32824	−0.664121	+	0.747625i	$0.731194\pi$
$140$	0	0
$141$	−1004.44	−0.599922
$142$	0	0
$143$	−2847.83	−1.66537
$144$	0	0
$145$	−837.174	−0.479473
$146$	0	0
$147$	−2256.17	−1.26589
$148$	0	0
$149$	−2875.88	−1.58122	−0.790610	−	0.612320i	$-0.790236\pi$
$149$	−2875.88	−1.58122	−0.790610	+	0.612320i	$0.790236\pi$
$150$	0	0
$151$	2330.03	1.25573	0.627863	−	0.778323i	$-0.283930\pi$
$151$	2330.03	1.25573	0.627863	+	0.778323i	$0.283930\pi$
$152$	0	0
$153$	880.940	0.465489
$154$	0	0
$155$	−574.708	−0.297817
$156$	0	0
$157$	2447.31	1.24405	0.622027	−	0.782996i	$-0.286310\pi$
$157$	2447.31	1.24405	0.622027	+	0.782996i	$0.286310\pi$
$158$	0	0
$159$	306.091	0.152670
$160$	0	0
$161$	−4315.76	−2.11261
$162$	0	0
$163$	1436.82	0.690432	0.345216	−	0.938523i	$-0.387806\pi$
$163$	1436.82	0.690432	0.345216	+	0.938523i	$0.387806\pi$
$164$	0	0
$165$	589.805	0.278280
$166$	0	0
$167$	1769.42	0.819889	0.409945	−	0.912110i	$-0.365548\pi$
$167$	1769.42	0.819889	0.409945	+	0.912110i	$0.365548\pi$
$168$	0	0
$169$	4566.76	2.07863
$170$	0	0
$171$	502.940	0.224917
$172$	0	0
$173$	−3897.86	−1.71300	−0.856499	−	0.516148i	$-0.827365\pi$
$173$	−3897.86	−1.71300	−0.856499	+	0.516148i	$0.827365\pi$
$174$	0	0
$175$	3069.73	1.32600
$176$	0	0
$177$	317.884	0.134992
$178$	0	0
$179$	−4146.74	−1.73152	−0.865760	−	0.500460i	$-0.833164\pi$
$179$	−4146.74	−1.73152	−0.865760	+	0.500460i	$0.833164\pi$
$180$	0	0
$181$	1104.22	0.453457	0.226728	−	0.973958i	$-0.427197\pi$
$181$	1104.22	0.453457	0.226728	+	0.973958i	$0.427197\pi$
$182$	0	0
$183$	2153.41	0.869860
$184$	0	0
$185$	−1047.18	−0.416163
$186$	0	0
$187$	−3389.41	−1.32544
$188$	0	0
$189$	893.475	0.343866
$190$	0	0
$191$	35.0686	0.0132852	0.00664260	−	0.999978i	$-0.497886\pi$
$191$	35.0686	0.0132852	0.00664260	+	0.999978i	$0.497886\pi$
$192$	0	0
$193$	615.884	0.229701	0.114851	−	0.993383i	$-0.463361\pi$
$193$	615.884	0.229701	0.114851	+	0.993383i	$0.463361\pi$
$194$	0	0
$195$	−1400.82	−0.514435
$196$	0	0
$197$	589.638	0.213249	0.106624	−	0.994299i	$-0.465996\pi$
$197$	589.638	0.213249	0.106624	+	0.994299i	$0.465996\pi$
$198$	0	0
$199$	−4813.82	−1.71479	−0.857394	−	0.514661i	$-0.827918\pi$
$199$	−4813.82	−1.71479	−0.857394	+	0.514661i	$0.827918\pi$
$200$	0	0
$201$	949.413	0.333166
$202$	0	0
$203$	4879.41	1.68703
$204$	0	0
$205$	−1347.93	−0.459238
$206$	0	0
$207$	1173.77	0.394118
$208$	0	0
$209$	−1935.06	−0.640434
$210$	0	0
$211$	3294.35	1.07484	0.537422	−	0.843313i	$-0.319398\pi$
$211$	3294.35	1.07484	0.537422	+	0.843313i	$0.319398\pi$
$212$	0	0
$213$	−2400.13	−0.772087
$214$	0	0
$215$	−1134.86	−0.359984
$216$	0	0
$217$	3349.65	1.04787
$218$	0	0
$219$	903.174	0.278680
$220$	0	0
$221$	8050.04	2.45025
$222$	0	0
$223$	−1572.78	−0.472294	−0.236147	−	0.971717i	$-0.575885\pi$
$223$	−1572.78	−0.472294	−0.236147	+	0.971717i	$0.575885\pi$
$224$	0	0
$225$	−834.881	−0.247372
$226$	0	0
$227$	1492.51	0.436394	0.218197	−	0.975905i	$-0.429982\pi$
$227$	1492.51	0.436394	0.218197	+	0.975905i	$0.429982\pi$
$228$	0	0
$229$	6163.31	1.77853	0.889265	−	0.457393i	$-0.151217\pi$
$229$	6163.31	1.77853	0.889265	+	0.457393i	$0.151217\pi$
$230$	0	0
$231$	−3437.64	−0.979134
$232$	0	0
$233$	−2540.35	−0.714266	−0.357133	−	0.934054i	$-0.616246\pi$
$233$	−2540.35	−0.714266	−0.357133	+	0.934054i	$0.616246\pi$
$234$	0	0
$235$	1900.95	0.527677
$236$	0	0
$237$	−128.499	−0.0352190
$238$	0	0
$239$	1339.25	0.362465	0.181232	−	0.983440i	$-0.441991\pi$
$239$	1339.25	0.362465	0.181232	+	0.983440i	$0.441991\pi$
$240$	0	0
$241$	2542.71	0.679627	0.339814	−	0.940493i	$-0.389636\pi$
$241$	2542.71	0.679627	0.339814	+	0.940493i	$0.389636\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	4269.91	1.11345
$246$	0	0
$247$	4595.87	1.18392
$248$	0	0
$249$	3710.00	0.944223
$250$	0	0
$251$	−4701.45	−1.18228	−0.591141	−	0.806568i	$-0.701322\pi$
$251$	−4701.45	−1.18228	−0.591141	+	0.806568i	$0.701322\pi$
$252$	0	0
$253$	−4516.05	−1.12222
$254$	0	0
$255$	−1667.22	−0.409432
$256$	0	0
$257$	−863.884	−0.209679	−0.104840	−	0.994489i	$-0.533433\pi$
$257$	−863.884	−0.209679	−0.104840	+	0.994489i	$0.533433\pi$
$258$	0	0
$259$	6103.41	1.46428
$260$	0	0
$261$	−1327.06	−0.314724
$262$	0	0
$263$	7091.36	1.66263	0.831315	−	0.555801i	$-0.187588\pi$
$263$	7091.36	1.66263	0.831315	+	0.555801i	$0.187588\pi$
$264$	0	0
$265$	−579.290	−0.134285
$266$	0	0
$267$	3975.88	0.911311
$268$	0	0
$269$	395.596	0.0896650	0.0448325	−	0.998995i	$-0.485725\pi$
$269$	395.596	0.0896650	0.0448325	+	0.998995i	$0.485725\pi$
$270$	0	0
$271$	−5532.21	−1.24007	−0.620033	−	0.784576i	$-0.712881\pi$
$271$	−5532.21	−1.24007	−0.620033	+	0.784576i	$0.712881\pi$
$272$	0	0
$273$	8164.58	1.81005
$274$	0	0
$275$	3212.20	0.704373
$276$	0	0
$277$	−3830.31	−0.830834	−0.415417	−	0.909631i	$-0.636364\pi$
$277$	−3830.31	−0.830834	−0.415417	+	0.909631i	$0.636364\pi$
$278$	0	0
$279$	−911.009	−0.195486
$280$	0	0
$281$	−5283.17	−1.12159	−0.560797	−	0.827954i	$-0.689505\pi$
$281$	−5283.17	−1.12159	−0.560797	+	0.827954i	$0.689505\pi$
$282$	0	0
$283$	−4639.76	−0.974577	−0.487288	−	0.873241i	$-0.662014\pi$
$283$	−4639.76	−0.974577	−0.487288	+	0.873241i	$0.662014\pi$
$284$	0	0
$285$	−951.836	−0.197831
$286$	0	0
$287$	7856.33	1.61584
$288$	0	0
$289$	4667.94	0.950119
$290$	0	0
$291$	1517.29	0.305654
$292$	0	0
$293$	−6022.12	−1.20074	−0.600369	−	0.799723i	$-0.704980\pi$
$293$	−6022.12	−1.20074	−0.600369	+	0.799723i	$0.704980\pi$
$294$	0	0
$295$	−601.609	−0.118736
$296$	0	0
$297$	934.940	0.182662
$298$	0	0
$299$	10725.9	2.07456
$300$	0	0
$301$	6614.44	1.26661
$302$	0	0
$303$	−3845.92	−0.729183
$304$	0	0
$305$	−4075.42	−0.765107
$306$	0	0
$307$	−2998.82	−0.557497	−0.278749	−	0.960364i	$-0.589920\pi$
$307$	−2998.82	−0.557497	−0.278749	+	0.960364i	$0.589920\pi$
$308$	0	0
$309$	−484.685	−0.0892323
$310$	0	0
$311$	−2403.97	−0.438318	−0.219159	−	0.975689i	$-0.570331\pi$
$311$	−2403.97	−0.438318	−0.219159	+	0.975689i	$0.570331\pi$
$312$	0	0
$313$	−5845.75	−1.05566	−0.527830	−	0.849350i	$-0.676994\pi$
$313$	−5845.75	−1.05566	−0.527830	+	0.849350i	$0.676994\pi$
$314$	0	0
$315$	−1690.94	−0.302456
$316$	0	0
$317$	−8917.38	−1.57997	−0.789984	−	0.613127i	$-0.789911\pi$
$317$	−8917.38	−1.57997	−0.789984	+	0.613127i	$0.789911\pi$
$318$	0	0
$319$	5105.86	0.896154
$320$	0	0
$321$	1445.18	0.251283
$322$	0	0
$323$	5469.88	0.942267
$324$	0	0
$325$	−7629.15	−1.30212
$326$	0	0
$327$	−859.910	−0.145422
$328$	0	0
$329$	−11079.5	−1.85664
$330$	0	0
$331$	9011.29	1.49639	0.748196	−	0.663478i	$-0.230920\pi$
$331$	9011.29	1.49639	0.748196	+	0.663478i	$0.230920\pi$
$332$	0	0
$333$	−1659.96	−0.273168
$334$	0	0
$335$	−1796.81	−0.293045
$336$	0	0
$337$	4516.35	0.730033	0.365017	−	0.931001i	$-0.381063\pi$
$337$	4516.35	0.730033	0.365017	+	0.931001i	$0.381063\pi$
$338$	0	0
$339$	4191.88	0.671598
$340$	0	0
$341$	3505.10	0.556633
$342$	0	0
$343$	−13536.4	−2.13090
$344$	0	0
$345$	−2221.40	−0.346656
$346$	0	0
$347$	3651.48	0.564904	0.282452	−	0.959281i	$-0.408852\pi$
$347$	3651.48	0.564904	0.282452	+	0.959281i	$0.408852\pi$
$348$	0	0
$349$	−3250.36	−0.498532	−0.249266	−	0.968435i	$-0.580189\pi$
$349$	−3250.36	−0.498532	−0.249266	+	0.968435i	$0.580189\pi$
$350$	0	0
$351$	−2220.54	−0.337674
$352$	0	0
$353$	592.345	0.0893125	0.0446563	−	0.999002i	$-0.485781\pi$
$353$	592.345	0.0893125	0.0446563	+	0.999002i	$0.485781\pi$
$354$	0	0
$355$	4542.36	0.679108
$356$	0	0
$357$	9717.26	1.44059
$358$	0	0
$359$	−3443.48	−0.506239	−0.253120	−	0.967435i	$-0.581457\pi$
$359$	−3443.48	−0.506239	−0.253120	+	0.967435i	$0.581457\pi$
$360$	0	0
$361$	−3736.17	−0.544711
$362$	0	0
$363$	395.826	0.0572327
$364$	0	0
$365$	−1709.30	−0.245120
$366$	0	0
$367$	−3297.39	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$367$	−3297.39	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$368$	0	0
$369$	−2136.70	−0.301442
$370$	0	0
$371$	3376.35	0.472483
$372$	0	0
$373$	−50.1819	−0.00696601	−0.00348300	−	0.999994i	$-0.501109\pi$
$373$	−50.1819	−0.00696601	−0.00348300	+	0.999994i	$0.501109\pi$
$374$	0	0
$375$	3709.16	0.510774
$376$	0	0
$377$	−12126.7	−1.65665
$378$	0	0
$379$	6769.28	0.917453	0.458726	−	0.888578i	$-0.348306\pi$
$379$	6769.28	0.917453	0.458726	+	0.888578i	$0.348306\pi$
$380$	0	0
$381$	1331.49	0.179040
$382$	0	0
$383$	−4208.46	−0.561468	−0.280734	−	0.959786i	$-0.590578\pi$
$383$	−4208.46	−0.561468	−0.280734	+	0.959786i	$0.590578\pi$
$384$	0	0
$385$	6505.88	0.861221
$386$	0	0
$387$	−1798.94	−0.236293
$388$	0	0
$389$	−2490.47	−0.324607	−0.162303	−	0.986741i	$-0.551892\pi$
$389$	−2490.47	−0.324607	−0.162303	+	0.986741i	$0.551892\pi$
$390$	0	0
$391$	12765.6	1.65112
$392$	0	0
$393$	4096.70	0.525830
$394$	0	0
$395$	243.190	0.0309777
$396$	0	0
$397$	7905.51	0.999411	0.499706	−	0.866195i	$-0.333442\pi$
$397$	7905.51	0.999411	0.499706	+	0.866195i	$0.333442\pi$
$398$	0	0
$399$	5547.71	0.696072
$400$	0	0
$401$	−10389.4	−1.29382	−0.646911	−	0.762566i	$-0.723939\pi$
$401$	−10389.4	−1.29382	−0.646911	+	0.762566i	$0.723939\pi$
$402$	0	0
$403$	−8324.81	−1.02900
$404$	0	0
$405$	459.888	0.0564248
$406$	0	0
$407$	6386.66	0.777826
$408$	0	0
$409$	13448.1	1.62583	0.812917	−	0.582379i	$-0.197878\pi$
$409$	13448.1	1.62583	0.812917	+	0.582379i	$0.197878\pi$
$410$	0	0
$411$	−3669.17	−0.440357
$412$	0	0
$413$	3506.44	0.417773
$414$	0	0
$415$	−7021.33	−0.830515
$416$	0	0
$417$	6530.11	0.766860
$418$	0	0
$419$	5191.34	0.605283	0.302641	−	0.953105i	$-0.402132\pi$
$419$	5191.34	0.605283	0.302641	+	0.953105i	$0.402132\pi$
$420$	0	0
$421$	5061.52	0.585946	0.292973	−	0.956121i	$-0.405355\pi$
$421$	5061.52	0.585946	0.292973	+	0.956121i	$0.405355\pi$
$422$	0	0
$423$	3013.32	0.346365
$424$	0	0
$425$	−9080.00	−1.03634
$426$	0	0
$427$	23753.3	2.69204
$428$	0	0
$429$	8543.49	0.961501
$430$	0	0
$431$	3005.64	0.335909	0.167954	−	0.985795i	$-0.446284\pi$
$431$	3005.64	0.335909	0.167954	+	0.985795i	$0.446284\pi$
$432$	0	0
$433$	5895.88	0.654360	0.327180	−	0.944962i	$-0.393902\pi$
$433$	5895.88	0.654360	0.327180	+	0.944962i	$0.393902\pi$
$434$	0	0
$435$	2511.52	0.276824
$436$	0	0
$437$	7288.07	0.797794
$438$	0	0
$439$	11556.8	1.25644	0.628218	−	0.778038i	$-0.283785\pi$
$439$	11556.8	1.25644	0.628218	+	0.778038i	$0.283785\pi$
$440$	0	0
$441$	6768.52	0.730863
$442$	0	0
$443$	−14007.8	−1.50233	−0.751163	−	0.660117i	$-0.770507\pi$
$443$	−14007.8	−1.50233	−0.751163	+	0.660117i	$0.770507\pi$
$444$	0	0
$445$	−7524.53	−0.801566
$446$	0	0
$447$	8627.65	0.912917
$448$	0	0
$449$	3783.06	0.397625	0.198813	−	0.980038i	$-0.436291\pi$
$449$	3783.06	0.397625	0.198813	+	0.980038i	$0.436291\pi$
$450$	0	0
$451$	8220.94	0.858335
$452$	0	0
$453$	−6990.08	−0.724994
$454$	0	0
$455$	−15451.8	−1.59207
$456$	0	0
$457$	1545.53	0.158198	0.0790992	−	0.996867i	$-0.474796\pi$
$457$	1545.53	0.158198	0.0790992	+	0.996867i	$0.474796\pi$
$458$	0	0
$459$	−2642.82	−0.268750
$460$	0	0
$461$	12730.2	1.28613	0.643065	−	0.765811i	$-0.277662\pi$
$461$	12730.2	1.28613	0.643065	+	0.765811i	$0.277662\pi$
$462$	0	0
$463$	19656.4	1.97303	0.986513	−	0.163682i	$-0.0523370\pi$
$463$	19656.4	1.97303	0.986513	+	0.163682i	$0.0523370\pi$
$464$	0	0
$465$	1724.12	0.171945
$466$	0	0
$467$	6016.26	0.596144	0.298072	−	0.954543i	$-0.403656\pi$
$467$	6016.26	0.596144	0.298072	+	0.954543i	$0.403656\pi$
$468$	0	0
$469$	10472.6	1.03108
$470$	0	0
$471$	−7341.92	−0.718255
$472$	0	0
$473$	6921.41	0.672826
$474$	0	0
$475$	−5183.89	−0.500743
$476$	0	0
$477$	−918.272	−0.0881442
$478$	0	0
$479$	−13890.6	−1.32501	−0.662505	−	0.749057i	$-0.730507\pi$
$479$	−13890.6	−1.32501	−0.662505	+	0.749057i	$0.730507\pi$
$480$	0	0
$481$	−15168.7	−1.43791
$482$	0	0
$483$	12947.3	1.21971
$484$	0	0
$485$	−2871.54	−0.268846
$486$	0	0
$487$	−9314.23	−0.866669	−0.433335	−	0.901233i	$-0.642663\pi$
$487$	−9314.23	−0.866669	−0.433335	+	0.901233i	$0.642663\pi$
$488$	0	0
$489$	−4310.46	−0.398621
$490$	0	0
$491$	7253.34	0.666677	0.333339	−	0.942807i	$-0.391825\pi$
$491$	7253.34	0.666677	0.333339	+	0.942807i	$0.391825\pi$
$492$	0	0
$493$	−14432.9	−1.31851
$494$	0	0
$495$	−1769.42	−0.160665
$496$	0	0
$497$	−26474.8	−2.38945
$498$	0	0
$499$	−16208.2	−1.45407	−0.727034	−	0.686602i	$-0.759102\pi$
$499$	−16208.2	−1.45407	−0.727034	+	0.686602i	$0.759102\pi$
$500$	0	0
$501$	−5308.25	−0.473363
$502$	0	0
$503$	2182.04	0.193425	0.0967123	−	0.995312i	$-0.469167\pi$
$503$	2182.04	0.193425	0.0967123	+	0.995312i	$0.469167\pi$
$504$	0	0
$505$	7278.58	0.641371
$506$	0	0
$507$	−13700.3	−1.20010
$508$	0	0
$509$	1795.97	0.156395	0.0781974	−	0.996938i	$-0.475084\pi$
$509$	1795.97	0.156395	0.0781974	+	0.996938i	$0.475084\pi$
$510$	0	0
$511$	9962.51	0.862457
$512$	0	0
$513$	−1508.82	−0.129856
$514$	0	0
$515$	917.288	0.0784865
$516$	0	0
$517$	−11593.7	−0.986249
$518$	0	0
$519$	11693.6	0.989000
$520$	0	0
$521$	20946.1	1.76135	0.880676	−	0.473718i	$-0.157088\pi$
$521$	20946.1	1.76135	0.880676	+	0.473718i	$0.157088\pi$
$522$	0	0
$523$	5363.64	0.448443	0.224221	−	0.974538i	$-0.428016\pi$
$523$	5363.64	0.448443	0.224221	+	0.974538i	$0.428016\pi$
$524$	0	0
$525$	−9209.19	−0.765566
$526$	0	0
$527$	−9907.96	−0.818970
$528$	0	0
$529$	4841.96	0.397958
$530$	0	0
$531$	−953.652	−0.0779378
$532$	0	0
$533$	−19525.2	−1.58674
$534$	0	0
$535$	−2735.06	−0.221022
$536$	0	0
$537$	12440.2	0.999693
$538$	0	0
$539$	−26041.8	−2.08108
$540$	0	0
$541$	−3431.20	−0.272678	−0.136339	−	0.990662i	$-0.543534\pi$
$541$	−3431.20	−0.272678	−0.136339	+	0.990662i	$0.543534\pi$
$542$	0	0
$543$	−3312.65	−0.261803
$544$	0	0
$545$	1627.42	0.127910
$546$	0	0
$547$	−19449.0	−1.52026	−0.760128	−	0.649773i	$-0.774864\pi$
$547$	−19449.0	−1.52026	−0.760128	+	0.649773i	$0.774864\pi$
$548$	0	0
$549$	−6460.22	−0.502214
$550$	0	0
$551$	−8239.91	−0.637082
$552$	0	0
$553$	−1417.41	−0.108996
$554$	0	0
$555$	3141.54	0.240272
$556$	0	0
$557$	−9898.44	−0.752981	−0.376491	−	0.926420i	$-0.622869\pi$
$557$	−9898.44	−0.752981	−0.376491	+	0.926420i	$0.622869\pi$
$558$	0	0
$559$	−16438.7	−1.24380
$560$	0	0
$561$	10168.2	0.765246
$562$	0	0
$563$	−13809.8	−1.03378	−0.516888	−	0.856053i	$-0.672909\pi$
$563$	−13809.8	−1.03378	−0.516888	+	0.856053i	$0.672909\pi$
$564$	0	0
$565$	−7933.32	−0.590721
$566$	0	0
$567$	−2680.42	−0.198531
$568$	0	0
$569$	123.053	0.00906615	0.00453308	−	0.999990i	$-0.498557\pi$
$569$	123.053	0.00906615	0.00453308	+	0.999990i	$0.498557\pi$
$570$	0	0
$571$	−2540.72	−0.186210	−0.0931049	−	0.995656i	$-0.529679\pi$
$571$	−2540.72	−0.186210	−0.0931049	+	0.995656i	$0.529679\pi$
$572$	0	0
$573$	−105.206	−0.00767022
$574$	0	0
$575$	−12098.2	−0.877443
$576$	0	0
$577$	−15618.2	−1.12685	−0.563427	−	0.826166i	$-0.690517\pi$
$577$	−15618.2	−1.12685	−0.563427	+	0.826166i	$0.690517\pi$
$578$	0	0
$579$	−1847.65	−0.132618
$580$	0	0
$581$	40923.3	2.92218
$582$	0	0
$583$	3533.04	0.250984
$584$	0	0
$585$	4202.46	0.297009
$586$	0	0
$587$	1809.56	0.127238	0.0636188	−	0.997974i	$-0.479736\pi$
$587$	1809.56	0.127238	0.0636188	+	0.997974i	$0.479736\pi$
$588$	0	0
$589$	−5656.58	−0.395714
$590$	0	0
$591$	−1768.91	−0.123119
$592$	0	0
$593$	898.229	0.0622021	0.0311010	−	0.999516i	$-0.490099\pi$
$593$	898.229	0.0622021	0.0311010	+	0.999516i	$0.490099\pi$
$594$	0	0
$595$	−18390.3	−1.26711
$596$	0	0
$597$	14441.5	0.990033
$598$	0	0
$599$	22990.9	1.56825	0.784125	−	0.620603i	$-0.213112\pi$
$599$	22990.9	1.56825	0.784125	+	0.620603i	$0.213112\pi$
$600$	0	0
$601$	26893.7	1.82532	0.912661	−	0.408717i	$-0.134024\pi$
$601$	26893.7	1.82532	0.912661	+	0.408717i	$0.134024\pi$
$602$	0	0
$603$	−2848.24	−0.192354
$604$	0	0
$605$	−749.118	−0.0503405
$606$	0	0
$607$	−6306.93	−0.421730	−0.210865	−	0.977515i	$-0.567628\pi$
$607$	−6306.93	−0.421730	−0.210865	+	0.977515i	$0.567628\pi$
$608$	0	0
$609$	−14638.2	−0.974008
$610$	0	0
$611$	27535.7	1.82320
$612$	0	0
$613$	2612.97	0.172164	0.0860822	−	0.996288i	$-0.472565\pi$
$613$	2612.97	0.172164	0.0860822	+	0.996288i	$0.472565\pi$
$614$	0	0
$615$	4043.80	0.265141
$616$	0	0
$617$	2803.88	0.182949	0.0914747	−	0.995807i	$-0.470842\pi$
$617$	2803.88	0.182949	0.0914747	+	0.995807i	$0.470842\pi$
$618$	0	0
$619$	10547.1	0.684849	0.342425	−	0.939545i	$-0.388752\pi$
$619$	10547.1	0.684849	0.342425	+	0.939545i	$0.388752\pi$
$620$	0	0
$621$	−3521.30	−0.227544
$622$	0	0
$623$	43856.2	2.82032
$624$	0	0
$625$	4575.82	0.292852
$626$	0	0
$627$	5805.17	0.369755
$628$	0	0
$629$	−18053.3	−1.14441
$630$	0	0
$631$	−14161.4	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$631$	−14161.4	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$632$	0	0
$633$	−9883.04	−0.620562
$634$	0	0
$635$	−2519.90	−0.157479
$636$	0	0
$637$	61850.8	3.84713
$638$	0	0
$639$	7200.40	0.445764
$640$	0	0
$641$	5622.61	0.346458	0.173229	−	0.984882i	$-0.444580\pi$
$641$	5622.61	0.346458	0.173229	+	0.984882i	$0.444580\pi$
$642$	0	0
$643$	−29438.7	−1.80552	−0.902759	−	0.430146i	$-0.858462\pi$
$643$	−29438.7	−1.80552	−0.902759	+	0.430146i	$0.858462\pi$
$644$	0	0
$645$	3404.57	0.207837
$646$	0	0
$647$	−4607.39	−0.279962	−0.139981	−	0.990154i	$-0.544704\pi$
$647$	−4607.39	−0.279962	−0.139981	+	0.990154i	$0.544704\pi$
$648$	0	0
$649$	3669.17	0.221922
$650$	0	0
$651$	−10048.9	−0.604991
$652$	0	0
$653$	−16634.1	−0.996850	−0.498425	−	0.866933i	$-0.666088\pi$
$653$	−16634.1	−0.996850	−0.498425	+	0.866933i	$0.666088\pi$
$654$	0	0
$655$	−7753.19	−0.462507
$656$	0	0
$657$	−2709.52	−0.160896
$658$	0	0
$659$	−20619.9	−1.21887	−0.609437	−	0.792835i	$-0.708604\pi$
$659$	−20619.9	−1.21887	−0.609437	+	0.792835i	$0.708604\pi$
$660$	0	0
$661$	−881.112	−0.0518476	−0.0259238	−	0.999664i	$-0.508253\pi$
$661$	−881.112	−0.0518476	−0.0259238	+	0.999664i	$0.508253\pi$
$662$	0	0
$663$	−24150.1	−1.41465
$664$	0	0
$665$	−10499.3	−0.612248
$666$	0	0
$667$	−19230.4	−1.11635
$668$	0	0
$669$	4718.35	0.272679
$670$	0	0
$671$	24855.6	1.43002
$672$	0	0
$673$	8943.86	0.512274	0.256137	−	0.966641i	$-0.417550\pi$
$673$	8943.86	0.512274	0.256137	+	0.966641i	$0.417550\pi$
$674$	0	0
$675$	2504.64	0.142820
$676$	0	0
$677$	−21748.7	−1.23467	−0.617333	−	0.786702i	$-0.711787\pi$
$677$	−21748.7	−1.23467	−0.617333	+	0.786702i	$0.711787\pi$
$678$	0	0
$679$	16736.6	0.945937
$680$	0	0
$681$	−4477.53	−0.251952
$682$	0	0
$683$	−7922.37	−0.443838	−0.221919	−	0.975065i	$-0.571232\pi$
$683$	−7922.37	−0.443838	−0.221919	+	0.975065i	$0.571232\pi$
$684$	0	0
$685$	6944.06	0.387327
$686$	0	0
$687$	−18489.9	−1.02683
$688$	0	0
$689$	−8391.18	−0.463975
$690$	0	0
$691$	177.517	0.00977288	0.00488644	−	0.999988i	$-0.498445\pi$
$691$	177.517	0.00977288	0.00488644	+	0.999988i	$0.498445\pi$
$692$	0	0
$693$	10312.9	0.565303
$694$	0	0
$695$	−12358.5	−0.674511
$696$	0	0
$697$	−23238.3	−1.26286
$698$	0	0
$699$	7621.05	0.412382
$700$	0	0
$701$	767.980	0.0413783	0.0206891	−	0.999786i	$-0.493414\pi$
$701$	767.980	0.0413783	0.0206891	+	0.999786i	$0.493414\pi$
$702$	0	0
$703$	−10306.9	−0.552961
$704$	0	0
$705$	−5702.84	−0.304654
$706$	0	0
$707$	−42422.7	−2.25667
$708$	0	0
$709$	31634.2	1.67566	0.837832	−	0.545928i	$-0.183823\pi$
$709$	31634.2	1.67566	0.837832	+	0.545928i	$0.183823\pi$
$710$	0	0
$711$	385.496	0.0203337
$712$	0	0
$713$	−13201.4	−0.693401
$714$	0	0
$715$	−16168.9	−0.845712
$716$	0	0
$717$	−4017.76	−0.209269
$718$	0	0
$719$	−17351.7	−0.900011	−0.450005	−	0.893026i	$-0.648578\pi$
$719$	−17351.7	−0.900011	−0.450005	+	0.893026i	$0.648578\pi$
$720$	0	0
$721$	−5346.35	−0.276156
$722$	0	0
$723$	−7628.12	−0.392383
$724$	0	0
$725$	13678.2	0.700686
$726$	0	0
$727$	−16364.6	−0.834842	−0.417421	−	0.908713i	$-0.637066\pi$
$727$	−16364.6	−0.834842	−0.417421	+	0.908713i	$0.637066\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−19564.9	−0.989925
$732$	0	0
$733$	−23552.7	−1.18682	−0.593410	−	0.804900i	$-0.702219\pi$
$733$	−23552.7	−1.18682	−0.593410	+	0.804900i	$0.702219\pi$
$734$	0	0
$735$	−12809.7	−0.642849
$736$	0	0
$737$	10958.6	0.547713
$738$	0	0
$739$	3519.05	0.175169	0.0875847	−	0.996157i	$-0.472085\pi$
$739$	3519.05	0.175169	0.0875847	+	0.996157i	$0.472085\pi$
$740$	0	0
$741$	−13787.6	−0.683537
$742$	0	0
$743$	20752.4	1.02467	0.512336	−	0.858785i	$-0.328780\pi$
$743$	20752.4	1.02467	0.512336	+	0.858785i	$0.328780\pi$
$744$	0	0
$745$	−16328.2	−0.802979
$746$	0	0
$747$	−11130.0	−0.545148
$748$	0	0
$749$	15941.1	0.777671
$750$	0	0
$751$	3679.08	0.178764	0.0893818	−	0.995997i	$-0.471511\pi$
$751$	3679.08	0.178764	0.0893818	+	0.995997i	$0.471511\pi$
$752$	0	0
$753$	14104.3	0.682591
$754$	0	0
$755$	13229.0	0.637687
$756$	0	0
$757$	−527.275	−0.0253159	−0.0126579	−	0.999920i	$-0.504029\pi$
$757$	−527.275	−0.0253159	−0.0126579	+	0.999920i	$0.504029\pi$
$758$	0	0
$759$	13548.2	0.647914
$760$	0	0
$761$	−3231.54	−0.153933	−0.0769666	−	0.997034i	$-0.524523\pi$
$761$	−3231.54	−0.153933	−0.0769666	+	0.997034i	$0.524523\pi$
$762$	0	0
$763$	−9485.29	−0.450053
$764$	0	0
$765$	5001.65	0.236386
$766$	0	0
$767$	−8714.48	−0.410250
$768$	0	0
$769$	13681.7	0.641581	0.320791	−	0.947150i	$-0.396051\pi$
$769$	13681.7	0.641581	0.320791	+	0.947150i	$0.396051\pi$
$770$	0	0
$771$	2591.65	0.121058
$772$	0	0
$773$	−9216.63	−0.428847	−0.214424	−	0.976741i	$-0.568787\pi$
$773$	−9216.63	−0.428847	−0.214424	+	0.976741i	$0.568787\pi$
$774$	0	0
$775$	9389.92	0.435221
$776$	0	0
$777$	−18310.2	−0.845400
$778$	0	0
$779$	−13267.1	−0.610196
$780$	0	0
$781$	−27703.5	−1.26928
$782$	0	0
$783$	3981.18	0.181706
$784$	0	0
$785$	13894.9	0.631759
$786$	0	0
$787$	33169.0	1.50235	0.751174	−	0.660104i	$-0.229488\pi$
$787$	33169.0	1.50235	0.751174	+	0.660104i	$0.229488\pi$
$788$	0	0
$789$	−21274.1	−0.959920
$790$	0	0
$791$	46238.8	2.07846
$792$	0	0
$793$	−59033.6	−2.64356
$794$	0	0
$795$	1737.87	0.0775294
$796$	0	0
$797$	−23781.9	−1.05696	−0.528481	−	0.848945i	$-0.677238\pi$
$797$	−23781.9	−1.05696	−0.528481	+	0.848945i	$0.677238\pi$
$798$	0	0
$799$	32772.3	1.45106
$800$	0	0
$801$	−11927.6	−0.526145
$802$	0	0
$803$	10424.9	0.458139
$804$	0	0
$805$	−24503.3	−1.07283
$806$	0	0
$807$	−1186.79	−0.0517681
$808$	0	0
$809$	29842.4	1.29691	0.648456	−	0.761252i	$-0.275415\pi$
$809$	29842.4	1.29691	0.648456	+	0.761252i	$0.275415\pi$
$810$	0	0
$811$	19074.5	0.825888	0.412944	−	0.910756i	$-0.364500\pi$
$811$	19074.5	0.825888	0.412944	+	0.910756i	$0.364500\pi$
$812$	0	0
$813$	16596.6	0.715952
$814$	0	0
$815$	8157.74	0.350617
$816$	0	0
$817$	−11169.9	−0.478316
$818$	0	0
$819$	−24493.7	−1.04503
$820$	0	0
$821$	−7493.97	−0.318564	−0.159282	−	0.987233i	$-0.550918\pi$
$821$	−7493.97	−0.318564	−0.159282	+	0.987233i	$0.550918\pi$
$822$	0	0
$823$	−1780.90	−0.0754294	−0.0377147	−	0.999289i	$-0.512008\pi$
$823$	−1780.90	−0.0754294	−0.0377147	+	0.999289i	$0.512008\pi$
$824$	0	0
$825$	−9636.59	−0.406670
$826$	0	0
$827$	−10860.1	−0.456640	−0.228320	−	0.973586i	$-0.573323\pi$
$827$	−10860.1	−0.456640	−0.228320	+	0.973586i	$0.573323\pi$
$828$	0	0
$829$	−34105.1	−1.42885	−0.714426	−	0.699711i	$-0.753312\pi$
$829$	−34105.1	−1.42885	−0.714426	+	0.699711i	$0.753312\pi$
$830$	0	0
$831$	11490.9	0.479682
$832$	0	0
$833$	73613.1	3.06188
$834$	0	0
$835$	10046.1	0.416358
$836$	0	0
$837$	2733.03	0.112864
$838$	0	0
$839$	−32688.9	−1.34511	−0.672555	−	0.740047i	$-0.734803\pi$
$839$	−32688.9	−1.34511	−0.672555	+	0.740047i	$0.734803\pi$
$840$	0	0
$841$	−2647.12	−0.108537
$842$	0	0
$843$	15849.5	0.647552
$844$	0	0
$845$	25928.4	1.05558
$846$	0	0
$847$	4366.18	0.177124
$848$	0	0
$849$	13919.3	0.562672
$850$	0	0
$851$	−24054.3	−0.968943
$852$	0	0
$853$	17391.2	0.698080	0.349040	−	0.937108i	$-0.386508\pi$
$853$	17391.2	0.698080	0.349040	+	0.937108i	$0.386508\pi$
$854$	0	0
$855$	2855.51	0.114218
$856$	0	0
$857$	4182.34	0.166705	0.0833525	−	0.996520i	$-0.473437\pi$
$857$	4182.34	0.166705	0.0833525	+	0.996520i	$0.473437\pi$
$858$	0	0
$859$	−20585.0	−0.817639	−0.408820	−	0.912615i	$-0.634059\pi$
$859$	−20585.0	−0.817639	−0.408820	+	0.912615i	$0.634059\pi$
$860$	0	0
$861$	−23569.0	−0.932903
$862$	0	0
$863$	14159.7	0.558518	0.279259	−	0.960216i	$-0.409911\pi$
$863$	14159.7	0.558518	0.279259	+	0.960216i	$0.409911\pi$
$864$	0	0
$865$	−22130.6	−0.869900
$866$	0	0
$867$	−14003.8	−0.548552
$868$	0	0
$869$	−1483.19	−0.0578986
$870$	0	0
$871$	−26027.2	−1.01251
$872$	0	0
$873$	−4551.88	−0.176469
$874$	0	0
$875$	40914.1	1.58074
$876$	0	0
$877$	−5156.38	−0.198539	−0.0992695	−	0.995061i	$-0.531651\pi$
$877$	−5156.38	−0.198539	−0.0992695	+	0.995061i	$0.531651\pi$
$878$	0	0
$879$	18066.4	0.693246
$880$	0	0
$881$	841.065	0.0321637	0.0160818	−	0.999871i	$-0.494881\pi$
$881$	841.065	0.0321637	0.0160818	+	0.999871i	$0.494881\pi$
$882$	0	0
$883$	−7845.99	−0.299024	−0.149512	−	0.988760i	$-0.547770\pi$
$883$	−7845.99	−0.299024	−0.149512	+	0.988760i	$0.547770\pi$
$884$	0	0
$885$	1804.83	0.0685521
$886$	0	0
$887$	−38781.9	−1.46806	−0.734029	−	0.679118i	$-0.762363\pi$
$887$	−38781.9	−1.46806	−0.734029	+	0.679118i	$0.762363\pi$
$888$	0	0
$889$	14687.1	0.554092
$890$	0	0
$891$	−2804.82	−0.105460
$892$	0	0
$893$	18710.1	0.701131
$894$	0	0
$895$	−23543.7	−0.879305
$896$	0	0
$897$	−32177.6	−1.19775
$898$	0	0
$899$	14925.5	0.553719
$900$	0	0
$901$	−9986.95	−0.369271
$902$	0	0
$903$	−19843.3	−0.731278
$904$	0	0
$905$	6269.33	0.230276
$906$	0	0
$907$	−36431.2	−1.33371	−0.666856	−	0.745187i	$-0.732360\pi$
$907$	−36431.2	−1.33371	−0.666856	+	0.745187i	$0.732360\pi$
$908$	0	0
$909$	11537.8	0.420994
$910$	0	0
$911$	−51077.9	−1.85762	−0.928808	−	0.370562i	$-0.879165\pi$
$911$	−51077.9	−1.85762	−0.928808	+	0.370562i	$0.879165\pi$
$912$	0	0
$913$	42822.6	1.55227
$914$	0	0
$915$	12226.3	0.441735
$916$	0	0
$917$	45188.9	1.62734
$918$	0	0
$919$	37080.1	1.33097	0.665484	−	0.746412i	$-0.268225\pi$
$919$	37080.1	1.33097	0.665484	+	0.746412i	$0.268225\pi$
$920$	0	0
$921$	8996.46	0.321871
$922$	0	0
$923$	65797.3	2.34642
$924$	0	0
$925$	17109.4	0.608167
$926$	0	0
$927$	1454.06	0.0515183
$928$	0	0
$929$	38259.5	1.35119	0.675595	−	0.737273i	$-0.263887\pi$
$929$	38259.5	1.35119	0.675595	+	0.737273i	$0.263887\pi$
$930$	0	0
$931$	42026.7	1.47945
$932$	0	0
$933$	7211.92	0.253063
$934$	0	0
$935$	−19243.8	−0.673091
$936$	0	0
$937$	−4413.55	−0.153879	−0.0769394	−	0.997036i	$-0.524515\pi$
$937$	−4413.55	−0.153879	−0.0769394	+	0.997036i	$0.524515\pi$
$938$	0	0
$939$	17537.3	0.609486
$940$	0	0
$941$	31511.2	1.09164	0.545821	−	0.837902i	$-0.316218\pi$
$941$	31511.2	1.09164	0.545821	+	0.837902i	$0.316218\pi$
$942$	0	0
$943$	−30962.8	−1.06923
$944$	0	0
$945$	5072.82	0.174623
$946$	0	0
$947$	−23561.3	−0.808490	−0.404245	−	0.914651i	$-0.632466\pi$
$947$	−23561.3	−0.808490	−0.404245	+	0.914651i	$0.632466\pi$
$948$	0	0
$949$	−24759.6	−0.846925
$950$	0	0
$951$	26752.1	0.912195
$952$	0	0
$953$	−35039.7	−1.19103	−0.595513	−	0.803346i	$-0.703051\pi$
$953$	−35039.7	−1.19103	−0.595513	+	0.803346i	$0.703051\pi$
$954$	0	0
$955$	199.107	0.00674653
$956$	0	0
$957$	−15317.6	−0.517395
$958$	0	0
$959$	−40473.0	−1.36282
$960$	0	0
$961$	−19544.9	−0.656066
$962$	0	0
$963$	−4335.53	−0.145078
$964$	0	0
$965$	3496.76	0.116647
$966$	0	0
$967$	−7975.38	−0.265223	−0.132612	−	0.991168i	$-0.542336\pi$
$967$	−7975.38	−0.265223	−0.132612	+	0.991168i	$0.542336\pi$
$968$	0	0
$969$	−16409.6	−0.544018
$970$	0	0
$971$	−21386.8	−0.706834	−0.353417	−	0.935466i	$-0.614980\pi$
$971$	−21386.8	−0.706834	−0.353417	+	0.935466i	$0.614980\pi$
$972$	0	0
$973$	72030.7	2.37328
$974$	0	0
$975$	22887.4	0.751779
$976$	0	0
$977$	−40142.3	−1.31450	−0.657250	−	0.753673i	$-0.728280\pi$
$977$	−40142.3	−1.31450	−0.657250	+	0.753673i	$0.728280\pi$
$978$	0	0
$979$	45891.5	1.49816
$980$	0	0
$981$	2579.73	0.0839597
$982$	0	0
$983$	9205.44	0.298686	0.149343	−	0.988785i	$-0.452284\pi$
$983$	9205.44	0.298686	0.149343	+	0.988785i	$0.452284\pi$
$984$	0	0
$985$	3347.75	0.108292
$986$	0	0
$987$	33238.6	1.07193
$988$	0	0
$989$	−26068.3	−0.838144
$990$	0	0
$991$	22082.2	0.707834	0.353917	−	0.935277i	$-0.384850\pi$
$991$	22082.2	0.707834	0.353917	+	0.935277i	$0.384850\pi$
$992$	0	0
$993$	−27033.9	−0.863942
$994$	0	0
$995$	−27331.1	−0.870808
$996$	0	0
$997$	−7207.66	−0.228956	−0.114478	−	0.993426i	$-0.536520\pi$
$997$	−7207.66	−0.228956	−0.114478	+	0.993426i	$0.536520\pi$
$998$	0	0
$999$	4979.87	0.157714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.u.1.3		4
3.2	odd	2		2304.4.a.cb.1.2		4
4.3	odd	2		768.4.a.v.1.3		4
8.3	odd	2	inner	768.4.a.u.1.2		4
8.5	even	2		768.4.a.v.1.2		4
12.11	even	2		2304.4.a.by.1.2		4
16.3	odd	4		384.4.d.f.193.6	yes	8
16.5	even	4		384.4.d.f.193.7	yes	8
16.11	odd	4		384.4.d.f.193.3	yes	8
16.13	even	4		384.4.d.f.193.2	✓	8
24.5	odd	2		2304.4.a.by.1.3		4
24.11	even	2		2304.4.a.cb.1.3		4
48.5	odd	4		1152.4.d.p.577.4		8
48.11	even	4		1152.4.d.p.577.3		8
48.29	odd	4		1152.4.d.p.577.6		8
48.35	even	4		1152.4.d.p.577.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.f.193.2	✓	8	16.13	even	4
384.4.d.f.193.3	yes	8	16.11	odd	4
384.4.d.f.193.6	yes	8	16.3	odd	4
384.4.d.f.193.7	yes	8	16.5	even	4
768.4.a.u.1.2		4	8.3	odd	2	inner
768.4.a.u.1.3		4	1.1	even	1	trivial
768.4.a.v.1.2		4	8.5	even	2
768.4.a.v.1.3		4	4.3	odd	2
1152.4.d.p.577.3		8	48.11	even	4
1152.4.d.p.577.4		8	48.5	odd	4
1152.4.d.p.577.5		8	48.35	even	4
1152.4.d.p.577.6		8	48.29	odd	4
2304.4.a.by.1.2		4	12.11	even	2
2304.4.a.by.1.3		4	24.5	odd	2
2304.4.a.cb.1.2		4	3.2	odd	2
2304.4.a.cb.1.3		4	24.11	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.67763 q^{5} -33.0917 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.67763 q^{5} -33.0917 q^{7} +9.00000 q^{9} -34.6274 q^{11} +82.2421 q^{13} -17.0329 q^{15} +97.8823 q^{17} +55.8823 q^{19} +99.2750 q^{21} +130.418 q^{23} -92.7645 q^{25} -27.0000 q^{27} -147.451 q^{29} -101.223 q^{31} +103.882 q^{33} -187.882 q^{35} -184.439 q^{37} -246.726 q^{39} -237.411 q^{41} -199.882 q^{43} +51.0987 q^{45} +334.813 q^{47} +752.058 q^{49} -293.647 q^{51} -102.030 q^{53} -196.602 q^{55} -167.647 q^{57} -105.961 q^{59} -717.803 q^{61} -297.825 q^{63} +466.940 q^{65} -316.471 q^{67} -391.255 q^{69} +800.045 q^{71} -301.058 q^{73} +278.294 q^{75} +1145.88 q^{77} +42.8329 q^{79} +81.0000 q^{81} -1236.67 q^{83} +555.739 q^{85} +442.354 q^{87} -1325.29 q^{89} -2721.53 q^{91} +303.670 q^{93} +317.279 q^{95} -505.765 q^{97} -311.647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * q^99

Introduction

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