LMFDB - Embedded newform 384.4.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -2.58301 q^{5} +33.7490 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -2.58301 q^{5} +33.7490 q^{7} +9.00000 q^{9} +33.1660 q^{11} -4.33202 q^{13} -7.74902 q^{15} +11.1660 q^{17} -121.830 q^{19} +101.247 q^{21} +14.8340 q^{23} -118.328 q^{25} +27.0000 q^{27} +272.915 q^{29} +165.085 q^{31} +99.4980 q^{33} -87.1739 q^{35} -30.5020 q^{37} -12.9961 q^{39} +400.154 q^{41} -274.834 q^{43} -23.2470 q^{45} +487.158 q^{47} +795.996 q^{49} +33.4980 q^{51} -208.737 q^{53} -85.6680 q^{55} -365.490 q^{57} -369.992 q^{59} +411.830 q^{61} +303.741 q^{63} +11.1896 q^{65} -407.336 q^{67} +44.5020 q^{69} +262.834 q^{71} +562.988 q^{73} -354.984 q^{75} +1119.32 q^{77} -955.563 q^{79} +81.0000 q^{81} -669.474 q^{83} -28.8419 q^{85} +818.745 q^{87} -1320.98 q^{89} -146.201 q^{91} +495.255 q^{93} +314.688 q^{95} +1101.66 q^{97} +298.494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 16 q^{5} + 4 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 16 * q^5 + 4 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} + 16 q^{5} + 4 q^{7} + 18 q^{9} + 24 q^{11} + 76 q^{13} + 48 q^{15} - 20 q^{17} - 32 q^{19} + 12 q^{21} + 72 q^{23} + 102 q^{25} + 54 q^{27} + 440 q^{29} + 436 q^{31} + 72 q^{33} - 640 q^{35} - 188 q^{37} + 228 q^{39} - 4 q^{41} - 592 q^{43} + 144 q^{45} + 424 q^{47} + 1338 q^{49} - 60 q^{51} + 408 q^{53} - 256 q^{55} - 96 q^{57} - 232 q^{59} + 612 q^{61} + 36 q^{63} + 1504 q^{65} - 984 q^{67} + 216 q^{69} + 568 q^{71} + 364 q^{73} + 306 q^{75} + 1392 q^{77} - 620 q^{79} + 162 q^{81} + 312 q^{83} - 608 q^{85} + 1320 q^{87} - 1372 q^{89} - 2536 q^{91} + 1308 q^{93} + 1984 q^{95} + 1780 q^{97} + 216 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 16 * q^5 + 4 * q^7 + 18 * q^9 + 24 * q^11 + 76 * q^13 + 48 * q^15 - 20 * q^17 - 32 * q^19 + 12 * q^21 + 72 * q^23 + 102 * q^25 + 54 * q^27 + 440 * q^29 + 436 * q^31 + 72 * q^33 - 640 * q^35 - 188 * q^37 + 228 * q^39 - 4 * q^41 - 592 * q^43 + 144 * q^45 + 424 * q^47 + 1338 * q^49 - 60 * q^51 + 408 * q^53 - 256 * q^55 - 96 * q^57 - 232 * q^59 + 612 * q^61 + 36 * q^63 + 1504 * q^65 - 984 * q^67 + 216 * q^69 + 568 * q^71 + 364 * q^73 + 306 * q^75 + 1392 * q^77 - 620 * q^79 + 162 * q^81 + 312 * q^83 - 608 * q^85 + 1320 * q^87 - 1372 * q^89 - 2536 * q^91 + 1308 * q^93 + 1984 * q^95 + 1780 * q^97 + 216 * 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
$5$	−2.58301	−0.231031	−0.115516	−	0.993306i	$-0.536852\pi$
$5$	−2.58301	−0.231031	−0.115516	+	0.993306i	$0.536852\pi$
$6$	0	0
$7$	33.7490	1.82228	0.911138	−	0.412102i	$-0.135205\pi$
$7$	33.7490	1.82228	0.911138	+	0.412102i	$0.135205\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	33.1660	0.909084	0.454542	−	0.890725i	$-0.349803\pi$
$11$	33.1660	0.909084	0.454542	+	0.890725i	$0.349803\pi$
$12$	0	0
$13$	−4.33202	−0.0924220	−0.0462110	−	0.998932i	$-0.514715\pi$
$13$	−4.33202	−0.0924220	−0.0462110	+	0.998932i	$0.514715\pi$
$14$	0	0
$15$	−7.74902	−0.133386
$16$	0	0
$17$	11.1660	0.159303	0.0796516	−	0.996823i	$-0.474619\pi$
$17$	11.1660	0.159303	0.0796516	+	0.996823i	$0.474619\pi$
$18$	0	0
$19$	−121.830	−1.47104	−0.735519	−	0.677504i	$-0.763062\pi$
$19$	−121.830	−1.47104	−0.735519	+	0.677504i	$0.763062\pi$
$20$	0	0
$21$	101.247	1.05209
$22$	0	0
$23$	14.8340	0.134483	0.0672413	−	0.997737i	$-0.478580\pi$
$23$	14.8340	0.134483	0.0672413	+	0.997737i	$0.478580\pi$
$24$	0	0
$25$	−118.328	−0.946625
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	272.915	1.74755	0.873777	−	0.486327i	$-0.161664\pi$
$29$	272.915	1.74755	0.873777	+	0.486327i	$0.161664\pi$
$30$	0	0
$31$	165.085	0.956456	0.478228	−	0.878236i	$-0.341279\pi$
$31$	165.085	0.956456	0.478228	+	0.878236i	$0.341279\pi$
$32$	0	0
$33$	99.4980	0.524860
$34$	0	0
$35$	−87.1739	−0.421002
$36$	0	0
$37$	−30.5020	−0.135527	−0.0677634	−	0.997701i	$-0.521586\pi$
$37$	−30.5020	−0.135527	−0.0677634	+	0.997701i	$0.521586\pi$
$38$	0	0
$39$	−12.9961	−0.0533599
$40$	0	0
$41$	400.154	1.52423	0.762117	−	0.647439i	$-0.224160\pi$
$41$	400.154	1.52423	0.762117	+	0.647439i	$0.224160\pi$
$42$	0	0
$43$	−274.834	−0.974693	−0.487346	−	0.873209i	$-0.662035\pi$
$43$	−274.834	−0.974693	−0.487346	+	0.873209i	$0.662035\pi$
$44$	0	0
$45$	−23.2470	−0.0770103
$46$	0	0
$47$	487.158	1.51190	0.755950	−	0.654629i	$-0.227175\pi$
$47$	487.158	1.51190	0.755950	+	0.654629i	$0.227175\pi$
$48$	0	0
$49$	795.996	2.32069
$50$	0	0
$51$	33.4980	0.0919738
$52$	0	0
$53$	−208.737	−0.540986	−0.270493	−	0.962722i	$-0.587187\pi$
$53$	−208.737	−0.540986	−0.270493	+	0.962722i	$0.587187\pi$
$54$	0	0
$55$	−85.6680	−0.210027
$56$	0	0
$57$	−365.490	−0.849304
$58$	0	0
$59$	−369.992	−0.816422	−0.408211	−	0.912888i	$-0.633847\pi$
$59$	−369.992	−0.816422	−0.408211	+	0.912888i	$0.633847\pi$
$60$	0	0
$61$	411.830	0.864417	0.432208	−	0.901774i	$-0.357734\pi$
$61$	411.830	0.864417	0.432208	+	0.901774i	$0.357734\pi$
$62$	0	0
$63$	303.741	0.607425
$64$	0	0
$65$	11.1896	0.0213524
$66$	0	0
$67$	−407.336	−0.742746	−0.371373	−	0.928484i	$-0.621113\pi$
$67$	−407.336	−0.742746	−0.371373	+	0.928484i	$0.621113\pi$
$68$	0	0
$69$	44.5020	0.0776436
$70$	0	0
$71$	262.834	0.439333	0.219667	−	0.975575i	$-0.429503\pi$
$71$	262.834	0.439333	0.219667	+	0.975575i	$0.429503\pi$
$72$	0	0
$73$	562.988	0.902641	0.451320	−	0.892362i	$-0.350953\pi$
$73$	562.988	0.902641	0.451320	+	0.892362i	$0.350953\pi$
$74$	0	0
$75$	−354.984	−0.546534
$76$	0	0
$77$	1119.32	1.65660
$78$	0	0
$79$	−955.563	−1.36088	−0.680438	−	0.732805i	$-0.738211\pi$
$79$	−955.563	−1.36088	−0.680438	+	0.732805i	$0.738211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−669.474	−0.885354	−0.442677	−	0.896681i	$-0.645971\pi$
$83$	−669.474	−0.885354	−0.442677	+	0.896681i	$0.645971\pi$
$84$	0	0
$85$	−28.8419	−0.0368040
$86$	0	0
$87$	818.745	1.00895
$88$	0	0
$89$	−1320.98	−1.57330	−0.786650	−	0.617400i	$-0.788186\pi$
$89$	−1320.98	−1.57330	−0.786650	+	0.617400i	$0.788186\pi$
$90$	0	0
$91$	−146.201	−0.168418
$92$	0	0
$93$	495.255	0.552210
$94$	0	0
$95$	314.688	0.339856
$96$	0	0
$97$	1101.66	1.15316	0.576581	−	0.817040i	$-0.304387\pi$
$97$	1101.66	1.15316	0.576581	+	0.817040i	$0.304387\pi$
$98$	0	0
$99$	298.494	0.303028
$100$	0	0
$101$	−721.093	−0.710410	−0.355205	−	0.934788i	$-0.615589\pi$
$101$	−721.093	−0.710410	−0.355205	+	0.934788i	$0.615589\pi$
$102$	0	0
$103$	305.231	0.291994	0.145997	−	0.989285i	$-0.453361\pi$
$103$	305.231	0.291994	0.145997	+	0.989285i	$0.453361\pi$
$104$	0	0
$105$	−261.522	−0.243066
$106$	0	0
$107$	1706.32	1.54164	0.770822	−	0.637051i	$-0.219846\pi$
$107$	1706.32	1.54164	0.770822	+	0.637051i	$0.219846\pi$
$108$	0	0
$109$	−276.656	−0.243109	−0.121554	−	0.992585i	$-0.538788\pi$
$109$	−276.656	−0.243109	−0.121554	+	0.992585i	$0.538788\pi$
$110$	0	0
$111$	−91.5059	−0.0782465
$112$	0	0
$113$	1350.96	1.12467	0.562333	−	0.826911i	$-0.309904\pi$
$113$	1350.96	1.12467	0.562333	+	0.826911i	$0.309904\pi$
$114$	0	0
$115$	−38.3163	−0.0310697
$116$	0	0
$117$	−38.9882	−0.0308073
$118$	0	0
$119$	376.842	0.290294
$120$	0	0
$121$	−231.016	−0.173566
$122$	0	0
$123$	1200.46	0.880017
$124$	0	0
$125$	628.518	0.449731
$126$	0	0
$127$	−383.448	−0.267918	−0.133959	−	0.990987i	$-0.542769\pi$
$127$	−383.448	−0.267918	−0.133959	+	0.990987i	$0.542769\pi$
$128$	0	0
$129$	−824.502	−0.562739
$130$	0	0
$131$	2063.95	1.37655	0.688276	−	0.725449i	$-0.258368\pi$
$131$	2063.95	1.37655	0.688276	+	0.725449i	$0.258368\pi$
$132$	0	0
$133$	−4111.64	−2.68064
$134$	0	0
$135$	−69.7411	−0.0444619
$136$	0	0
$137$	−3130.79	−1.95242	−0.976211	−	0.216822i	$-0.930431\pi$
$137$	−3130.79	−1.95242	−0.976211	+	0.216822i	$0.930431\pi$
$138$	0	0
$139$	−2872.60	−1.75288	−0.876442	−	0.481508i	$-0.840089\pi$
$139$	−2872.60	−1.75288	−0.876442	+	0.481508i	$0.840089\pi$
$140$	0	0
$141$	1461.47	0.872896
$142$	0	0
$143$	−143.676	−0.0840194
$144$	0	0
$145$	−704.941	−0.403739
$146$	0	0
$147$	2387.99	1.33985
$148$	0	0
$149$	2487.33	1.36758	0.683792	−	0.729677i	$-0.260330\pi$
$149$	2487.33	1.36758	0.683792	+	0.729677i	$0.260330\pi$
$150$	0	0
$151$	−1745.23	−0.940562	−0.470281	−	0.882517i	$-0.655847\pi$
$151$	−1745.23	−0.940562	−0.470281	+	0.882517i	$0.655847\pi$
$152$	0	0
$153$	100.494	0.0531011
$154$	0	0
$155$	−426.415	−0.220971
$156$	0	0
$157$	−1137.16	−0.578058	−0.289029	−	0.957320i	$-0.593332\pi$
$157$	−1137.16	−0.578058	−0.289029	+	0.957320i	$0.593332\pi$
$158$	0	0
$159$	−626.212	−0.312338
$160$	0	0
$161$	500.633	0.245064
$162$	0	0
$163$	−374.170	−0.179799	−0.0898995	−	0.995951i	$-0.528655\pi$
$163$	−374.170	−0.179799	−0.0898995	+	0.995951i	$0.528655\pi$
$164$	0	0
$165$	−257.004	−0.121259
$166$	0	0
$167$	−4062.64	−1.88249	−0.941247	−	0.337718i	$-0.890345\pi$
$167$	−4062.64	−1.88249	−0.941247	+	0.337718i	$0.890345\pi$
$168$	0	0
$169$	−2178.23	−0.991458
$170$	0	0
$171$	−1096.47	−0.490346
$172$	0	0
$173$	−2601.84	−1.14343	−0.571717	−	0.820451i	$-0.693723\pi$
$173$	−2601.84	−1.14343	−0.571717	+	0.820451i	$0.693723\pi$
$174$	0	0
$175$	−3993.46	−1.72501
$176$	0	0
$177$	−1109.98	−0.471361
$178$	0	0
$179$	−1205.98	−0.503569	−0.251785	−	0.967783i	$-0.581017\pi$
$179$	−1205.98	−0.503569	−0.251785	+	0.967783i	$0.581017\pi$
$180$	0	0
$181$	−4390.60	−1.80304	−0.901522	−	0.432734i	$-0.857549\pi$
$181$	−4390.60	−1.80304	−0.901522	+	0.432734i	$0.857549\pi$
$182$	0	0
$183$	1235.49	0.499071
$184$	0	0
$185$	78.7867	0.0313109
$186$	0	0
$187$	370.332	0.144820
$188$	0	0
$189$	911.223	0.350697
$190$	0	0
$191$	4531.27	1.71660	0.858302	−	0.513145i	$-0.171520\pi$
$191$	4531.27	1.71660	0.858302	+	0.513145i	$0.171520\pi$
$192$	0	0
$193$	2972.93	1.10879	0.554395	−	0.832254i	$-0.312950\pi$
$193$	2972.93	1.10879	0.554395	+	0.832254i	$0.312950\pi$
$194$	0	0
$195$	33.5689	0.0123278
$196$	0	0
$197$	−2212.37	−0.800125	−0.400062	−	0.916488i	$-0.631012\pi$
$197$	−2212.37	−0.800125	−0.400062	+	0.916488i	$0.631012\pi$
$198$	0	0
$199$	−2786.40	−0.992575	−0.496288	−	0.868158i	$-0.665304\pi$
$199$	−2786.40	−0.992575	−0.496288	+	0.868158i	$0.665304\pi$
$200$	0	0
$201$	−1222.01	−0.428825
$202$	0	0
$203$	9210.61	3.18452
$204$	0	0
$205$	−1033.60	−0.352145
$206$	0	0
$207$	133.506	0.0448275
$208$	0	0
$209$	−4040.62	−1.33730
$210$	0	0
$211$	1150.56	0.375393	0.187697	−	0.982227i	$-0.439898\pi$
$211$	1150.56	0.375393	0.187697	+	0.982227i	$0.439898\pi$
$212$	0	0
$213$	788.502	0.253649
$214$	0	0
$215$	709.898	0.225184
$216$	0	0
$217$	5571.46	1.74293
$218$	0	0
$219$	1688.96	0.521140
$220$	0	0
$221$	−48.3714	−0.0147231
$222$	0	0
$223$	−1487.32	−0.446630	−0.223315	−	0.974746i	$-0.571688\pi$
$223$	−1487.32	−0.446630	−0.223315	+	0.974746i	$0.571688\pi$
$224$	0	0
$225$	−1064.95	−0.315542
$226$	0	0
$227$	−3071.13	−0.897964	−0.448982	−	0.893541i	$-0.648213\pi$
$227$	−3071.13	−0.897964	−0.448982	+	0.893541i	$0.648213\pi$
$228$	0	0
$229$	3048.89	0.879808	0.439904	−	0.898045i	$-0.355012\pi$
$229$	3048.89	0.879808	0.439904	+	0.898045i	$0.355012\pi$
$230$	0	0
$231$	3357.96	0.956440
$232$	0	0
$233$	−1628.38	−0.457848	−0.228924	−	0.973444i	$-0.573521\pi$
$233$	−1628.38	−0.457848	−0.228924	+	0.973444i	$0.573521\pi$
$234$	0	0
$235$	−1258.33	−0.349296
$236$	0	0
$237$	−2866.69	−0.785703
$238$	0	0
$239$	−2870.19	−0.776808	−0.388404	−	0.921489i	$-0.626974\pi$
$239$	−2870.19	−0.776808	−0.388404	+	0.921489i	$0.626974\pi$
$240$	0	0
$241$	1933.35	0.516756	0.258378	−	0.966044i	$-0.416812\pi$
$241$	1933.35	0.516756	0.258378	+	0.966044i	$0.416812\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−2056.06	−0.536151
$246$	0	0
$247$	527.770	0.135956
$248$	0	0
$249$	−2008.42	−0.511159
$250$	0	0
$251$	−1230.71	−0.309488	−0.154744	−	0.987955i	$-0.549455\pi$
$251$	−1230.71	−0.309488	−0.154744	+	0.987955i	$0.549455\pi$
$252$	0	0
$253$	491.984	0.122256
$254$	0	0
$255$	−86.5256	−0.0212488
$256$	0	0
$257$	−2097.30	−0.509052	−0.254526	−	0.967066i	$-0.581919\pi$
$257$	−2097.30	−0.509052	−0.254526	+	0.967066i	$0.581919\pi$
$258$	0	0
$259$	−1029.41	−0.246967
$260$	0	0
$261$	2456.24	0.582518
$262$	0	0
$263$	−105.423	−0.0247172	−0.0123586	−	0.999924i	$-0.503934\pi$
$263$	−105.423	−0.0247172	−0.0123586	+	0.999924i	$0.503934\pi$
$264$	0	0
$265$	539.169	0.124985
$266$	0	0
$267$	−3962.94	−0.908345
$268$	0	0
$269$	7087.79	1.60651	0.803253	−	0.595639i	$-0.203101\pi$
$269$	7087.79	1.60651	0.803253	+	0.595639i	$0.203101\pi$
$270$	0	0
$271$	−4012.14	−0.899337	−0.449669	−	0.893195i	$-0.648458\pi$
$271$	−4012.14	−0.899337	−0.449669	+	0.893195i	$0.648458\pi$
$272$	0	0
$273$	−438.604	−0.0972364
$274$	0	0
$275$	−3924.47	−0.860562
$276$	0	0
$277$	7765.19	1.68435	0.842176	−	0.539203i	$-0.181275\pi$
$277$	7765.19	1.68435	0.842176	+	0.539203i	$0.181275\pi$
$278$	0	0
$279$	1485.76	0.318819
$280$	0	0
$281$	3564.75	0.756780	0.378390	−	0.925646i	$-0.376478\pi$
$281$	3564.75	0.756780	0.378390	+	0.925646i	$0.376478\pi$
$282$	0	0
$283$	−843.352	−0.177145	−0.0885725	−	0.996070i	$-0.528231\pi$
$283$	−843.352	−0.177145	−0.0885725	+	0.996070i	$0.528231\pi$
$284$	0	0
$285$	944.063	0.196216
$286$	0	0
$287$	13504.8	2.77757
$288$	0	0
$289$	−4788.32	−0.974622
$290$	0	0
$291$	3304.98	0.665778
$292$	0	0
$293$	4777.18	0.952511	0.476255	−	0.879307i	$-0.341994\pi$
$293$	4777.18	0.952511	0.476255	+	0.879307i	$0.341994\pi$
$294$	0	0
$295$	955.692	0.188619
$296$	0	0
$297$	895.482	0.174953
$298$	0	0
$299$	−64.2612	−0.0124292
$300$	0	0
$301$	−9275.38	−1.77616
$302$	0	0
$303$	−2163.28	−0.410155
$304$	0	0
$305$	−1063.76	−0.199707
$306$	0	0
$307$	2021.88	0.375879	0.187940	−	0.982181i	$-0.439819\pi$
$307$	2021.88	0.375879	0.187940	+	0.982181i	$0.439819\pi$
$308$	0	0
$309$	915.694	0.168583
$310$	0	0
$311$	−3979.50	−0.725585	−0.362792	−	0.931870i	$-0.618177\pi$
$311$	−3979.50	−0.725585	−0.362792	+	0.931870i	$0.618177\pi$
$312$	0	0
$313$	1209.07	0.218342	0.109171	−	0.994023i	$-0.465180\pi$
$313$	1209.07	0.218342	0.109171	+	0.994023i	$0.465180\pi$
$314$	0	0
$315$	−784.565	−0.140334
$316$	0	0
$317$	−1505.12	−0.266676	−0.133338	−	0.991071i	$-0.542570\pi$
$317$	−1505.12	−0.266676	−0.133338	+	0.991071i	$0.542570\pi$
$318$	0	0
$319$	9051.50	1.58867
$320$	0	0
$321$	5118.95	0.890068
$322$	0	0
$323$	−1360.36	−0.234341
$324$	0	0
$325$	512.600	0.0874890
$326$	0	0
$327$	−829.969	−0.140359
$328$	0	0
$329$	16441.1	2.75510
$330$	0	0
$331$	−9935.60	−1.64988	−0.824939	−	0.565221i	$-0.808791\pi$
$331$	−9935.60	−1.64988	−0.824939	+	0.565221i	$0.808791\pi$
$332$	0	0
$333$	−274.518	−0.0451756
$334$	0	0
$335$	1052.15	0.171597
$336$	0	0
$337$	1474.83	0.238395	0.119198	−	0.992871i	$-0.461968\pi$
$337$	1474.83	0.238395	0.119198	+	0.992871i	$0.461968\pi$
$338$	0	0
$339$	4052.87	0.649327
$340$	0	0
$341$	5475.21	0.869499
$342$	0	0
$343$	15288.2	2.40666
$344$	0	0
$345$	−114.949	−0.0179381
$346$	0	0
$347$	494.478	0.0764985	0.0382493	−	0.999268i	$-0.487822\pi$
$347$	494.478	0.0764985	0.0382493	+	0.999268i	$0.487822\pi$
$348$	0	0
$349$	−2740.76	−0.420370	−0.210185	−	0.977662i	$-0.567407\pi$
$349$	−2740.76	−0.420370	−0.210185	+	0.977662i	$0.567407\pi$
$350$	0	0
$351$	−116.965	−0.0177866
$352$	0	0
$353$	−2800.30	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$353$	−2800.30	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$354$	0	0
$355$	−678.902	−0.101500
$356$	0	0
$357$	1130.53	0.167602
$358$	0	0
$359$	−6704.23	−0.985615	−0.492808	−	0.870138i	$-0.664029\pi$
$359$	−6704.23	−0.985615	−0.492808	+	0.870138i	$0.664029\pi$
$360$	0	0
$361$	7983.56	1.16395
$362$	0	0
$363$	−693.047	−0.100208
$364$	0	0
$365$	−1454.20	−0.208538
$366$	0	0
$367$	−3834.79	−0.545434	−0.272717	−	0.962094i	$-0.587922\pi$
$367$	−3834.79	−0.545434	−0.272717	+	0.962094i	$0.587922\pi$
$368$	0	0
$369$	3601.39	0.508078
$370$	0	0
$371$	−7044.68	−0.985826
$372$	0	0
$373$	3257.11	0.452136	0.226068	−	0.974111i	$-0.427413\pi$
$373$	3257.11	0.452136	0.226068	+	0.974111i	$0.427413\pi$
$374$	0	0
$375$	1885.55	0.259652
$376$	0	0
$377$	−1182.27	−0.161512
$378$	0	0
$379$	−7539.90	−1.02190	−0.510948	−	0.859612i	$-0.670706\pi$
$379$	−7539.90	−1.02190	−0.510948	+	0.859612i	$0.670706\pi$
$380$	0	0
$381$	−1150.35	−0.154682
$382$	0	0
$383$	−1207.10	−0.161044	−0.0805221	−	0.996753i	$-0.525659\pi$
$383$	−1207.10	−0.161044	−0.0805221	+	0.996753i	$0.525659\pi$
$384$	0	0
$385$	−2891.21	−0.382727
$386$	0	0
$387$	−2473.51	−0.324898
$388$	0	0
$389$	4639.01	0.604645	0.302323	−	0.953206i	$-0.402238\pi$
$389$	4639.01	0.604645	0.302323	+	0.953206i	$0.402238\pi$
$390$	0	0
$391$	165.636	0.0214235
$392$	0	0
$393$	6191.86	0.794753
$394$	0	0
$395$	2468.23	0.314405
$396$	0	0
$397$	12815.5	1.62013	0.810065	−	0.586340i	$-0.199432\pi$
$397$	12815.5	1.62013	0.810065	+	0.586340i	$0.199432\pi$
$398$	0	0
$399$	−12334.9	−1.54767
$400$	0	0
$401$	4573.44	0.569542	0.284771	−	0.958596i	$-0.408082\pi$
$401$	4573.44	0.569542	0.284771	+	0.958596i	$0.408082\pi$
$402$	0	0
$403$	−715.152	−0.0883976
$404$	0	0
$405$	−209.223	−0.0256701
$406$	0	0
$407$	−1011.63	−0.123205
$408$	0	0
$409$	−9921.53	−1.19948	−0.599741	−	0.800194i	$-0.704730\pi$
$409$	−9921.53	−1.19948	−0.599741	+	0.800194i	$0.704730\pi$
$410$	0	0
$411$	−9392.38	−1.12723
$412$	0	0
$413$	−12486.9	−1.48775
$414$	0	0
$415$	1729.26	0.204544
$416$	0	0
$417$	−8617.80	−1.01203
$418$	0	0
$419$	−10898.4	−1.27069	−0.635346	−	0.772227i	$-0.719143\pi$
$419$	−10898.4	−1.27069	−0.635346	+	0.772227i	$0.719143\pi$
$420$	0	0
$421$	10708.5	1.23967	0.619833	−	0.784734i	$-0.287200\pi$
$421$	10708.5	1.23967	0.619833	+	0.784734i	$0.287200\pi$
$422$	0	0
$423$	4384.42	0.503967
$424$	0	0
$425$	−1321.25	−0.150800
$426$	0	0
$427$	13898.9	1.57521
$428$	0	0
$429$	−431.028	−0.0485086
$430$	0	0
$431$	−5263.90	−0.588290	−0.294145	−	0.955761i	$-0.595035\pi$
$431$	−5263.90	−0.588290	−0.294145	+	0.955761i	$0.595035\pi$
$432$	0	0
$433$	−11624.4	−1.29015	−0.645075	−	0.764119i	$-0.723174\pi$
$433$	−11624.4	−1.29015	−0.645075	+	0.764119i	$0.723174\pi$
$434$	0	0
$435$	−2114.82	−0.233099
$436$	0	0
$437$	−1807.23	−0.197829
$438$	0	0
$439$	10859.3	1.18060	0.590301	−	0.807183i	$-0.299009\pi$
$439$	10859.3	1.18060	0.590301	+	0.807183i	$0.299009\pi$
$440$	0	0
$441$	7163.96	0.773563
$442$	0	0
$443$	8610.37	0.923456	0.461728	−	0.887022i	$-0.347230\pi$
$443$	8610.37	0.923456	0.461728	+	0.887022i	$0.347230\pi$
$444$	0	0
$445$	3412.10	0.363481
$446$	0	0
$447$	7461.99	0.789575
$448$	0	0
$449$	1412.37	0.148449	0.0742247	−	0.997242i	$-0.476352\pi$
$449$	1412.37	0.148449	0.0742247	+	0.997242i	$0.476352\pi$
$450$	0	0
$451$	13271.5	1.38566
$452$	0	0
$453$	−5235.69	−0.543034
$454$	0	0
$455$	377.639	0.0389099
$456$	0	0
$457$	5237.47	0.536102	0.268051	−	0.963405i	$-0.413620\pi$
$457$	5237.47	0.536102	0.268051	+	0.963405i	$0.413620\pi$
$458$	0	0
$459$	301.482	0.0306579
$460$	0	0
$461$	−2987.66	−0.301842	−0.150921	−	0.988546i	$-0.548224\pi$
$461$	−2987.66	−0.301842	−0.150921	+	0.988546i	$0.548224\pi$
$462$	0	0
$463$	−15939.5	−1.59994	−0.799971	−	0.600039i	$-0.795152\pi$
$463$	−15939.5	−1.59994	−0.799971	+	0.600039i	$0.795152\pi$
$464$	0	0
$465$	−1279.25	−0.127578
$466$	0	0
$467$	3893.47	0.385800	0.192900	−	0.981218i	$-0.438211\pi$
$467$	3893.47	0.385800	0.192900	+	0.981218i	$0.438211\pi$
$468$	0	0
$469$	−13747.2	−1.35349
$470$	0	0
$471$	−3411.47	−0.333742
$472$	0	0
$473$	−9115.15	−0.886078
$474$	0	0
$475$	14415.9	1.39252
$476$	0	0
$477$	−1878.63	−0.180329
$478$	0	0
$479$	15411.9	1.47012	0.735060	−	0.678002i	$-0.237154\pi$
$479$	15411.9	1.47012	0.735060	+	0.678002i	$0.237154\pi$
$480$	0	0
$481$	132.135	0.0125257
$482$	0	0
$483$	1501.90	0.141488
$484$	0	0
$485$	−2845.59	−0.266416
$486$	0	0
$487$	−6539.41	−0.608478	−0.304239	−	0.952596i	$-0.598402\pi$
$487$	−6539.41	−0.608478	−0.304239	+	0.952596i	$0.598402\pi$
$488$	0	0
$489$	−1122.51	−0.103807
$490$	0	0
$491$	−14024.1	−1.28900	−0.644498	−	0.764606i	$-0.722934\pi$
$491$	−14024.1	−1.28900	−0.644498	+	0.764606i	$0.722934\pi$
$492$	0	0
$493$	3047.37	0.278391
$494$	0	0
$495$	−771.012	−0.0700089
$496$	0	0
$497$	8870.39	0.800586
$498$	0	0
$499$	12664.6	1.13617	0.568083	−	0.822971i	$-0.307685\pi$
$499$	12664.6	1.13617	0.568083	+	0.822971i	$0.307685\pi$
$500$	0	0
$501$	−12187.9	−1.08686
$502$	0	0
$503$	−6345.10	−0.562454	−0.281227	−	0.959641i	$-0.590741\pi$
$503$	−6345.10	−0.562454	−0.281227	+	0.959641i	$0.590741\pi$
$504$	0	0
$505$	1862.59	0.164127
$506$	0	0
$507$	−6534.70	−0.572419
$508$	0	0
$509$	−18974.6	−1.65232	−0.826162	−	0.563432i	$-0.809481\pi$
$509$	−18974.6	−1.65232	−0.826162	+	0.563432i	$0.809481\pi$
$510$	0	0
$511$	19000.3	1.64486
$512$	0	0
$513$	−3289.41	−0.283101
$514$	0	0
$515$	−788.414	−0.0674596
$516$	0	0
$517$	16157.1	1.37445
$518$	0	0
$519$	−7805.52	−0.660162
$520$	0	0
$521$	−12203.6	−1.02620	−0.513100	−	0.858329i	$-0.671503\pi$
$521$	−12203.6	−1.02620	−0.513100	+	0.858329i	$0.671503\pi$
$522$	0	0
$523$	10491.5	0.877174	0.438587	−	0.898689i	$-0.355479\pi$
$523$	10491.5	0.877174	0.438587	+	0.898689i	$0.355479\pi$
$524$	0	0
$525$	−11980.4	−0.995936
$526$	0	0
$527$	1843.34	0.152367
$528$	0	0
$529$	−11947.0	−0.981914
$530$	0	0
$531$	−3329.93	−0.272141
$532$	0	0
$533$	−1733.48	−0.140873
$534$	0	0
$535$	−4407.42	−0.356167
$536$	0	0
$537$	−3617.93	−0.290736
$538$	0	0
$539$	26400.0	2.10970
$540$	0	0
$541$	20107.6	1.59796	0.798979	−	0.601359i	$-0.205374\pi$
$541$	20107.6	1.59796	0.798979	+	0.601359i	$0.205374\pi$
$542$	0	0
$543$	−13171.8	−1.04099
$544$	0	0
$545$	714.604	0.0561657
$546$	0	0
$547$	11513.0	0.899928	0.449964	−	0.893047i	$-0.351437\pi$
$547$	11513.0	0.899928	0.449964	+	0.893047i	$0.351437\pi$
$548$	0	0
$549$	3706.47	0.288139
$550$	0	0
$551$	−33249.3	−2.57072
$552$	0	0
$553$	−32249.3	−2.47989
$554$	0	0
$555$	236.360	0.0180774
$556$	0	0
$557$	13498.7	1.02686	0.513429	−	0.858132i	$-0.328375\pi$
$557$	13498.7	1.02686	0.513429	+	0.858132i	$0.328375\pi$
$558$	0	0
$559$	1190.59	0.0900831
$560$	0	0
$561$	1111.00	0.0836119
$562$	0	0
$563$	−8463.07	−0.633528	−0.316764	−	0.948504i	$-0.602596\pi$
$563$	−8463.07	−0.633528	−0.316764	+	0.948504i	$0.602596\pi$
$564$	0	0
$565$	−3489.53	−0.259833
$566$	0	0
$567$	2733.67	0.202475
$568$	0	0
$569$	12590.4	0.927619	0.463809	−	0.885935i	$-0.346482\pi$
$569$	12590.4	0.927619	0.463809	+	0.885935i	$0.346482\pi$
$570$	0	0
$571$	−15848.6	−1.16155	−0.580775	−	0.814064i	$-0.697250\pi$
$571$	−15848.6	−1.16155	−0.580775	+	0.814064i	$0.697250\pi$
$572$	0	0
$573$	13593.8	0.991082
$574$	0	0
$575$	−1755.28	−0.127305
$576$	0	0
$577$	−8452.27	−0.609831	−0.304916	−	0.952379i	$-0.598628\pi$
$577$	−8452.27	−0.609831	−0.304916	+	0.952379i	$0.598628\pi$
$578$	0	0
$579$	8918.80	0.640160
$580$	0	0
$581$	−22594.1	−1.61336
$582$	0	0
$583$	−6922.98	−0.491802
$584$	0	0
$585$	100.707	0.00711745
$586$	0	0
$587$	−8795.17	−0.618425	−0.309213	−	0.950993i	$-0.600065\pi$
$587$	−8795.17	−0.618425	−0.309213	+	0.950993i	$0.600065\pi$
$588$	0	0
$589$	−20112.3	−1.40698
$590$	0	0
$591$	−6637.10	−0.461952
$592$	0	0
$593$	−11229.1	−0.777609	−0.388805	−	0.921320i	$-0.627112\pi$
$593$	−11229.1	−0.777609	−0.388805	+	0.921320i	$0.627112\pi$
$594$	0	0
$595$	−973.385	−0.0670670
$596$	0	0
$597$	−8359.19	−0.573064
$598$	0	0
$599$	1749.39	0.119329	0.0596645	−	0.998218i	$-0.480997\pi$
$599$	1749.39	0.119329	0.0596645	+	0.998218i	$0.480997\pi$
$600$	0	0
$601$	−9355.37	−0.634964	−0.317482	−	0.948264i	$-0.602837\pi$
$601$	−9355.37	−0.634964	−0.317482	+	0.948264i	$0.602837\pi$
$602$	0	0
$603$	−3666.02	−0.247582
$604$	0	0
$605$	596.715	0.0400990
$606$	0	0
$607$	7370.07	0.492820	0.246410	−	0.969166i	$-0.420749\pi$
$607$	7370.07	0.492820	0.246410	+	0.969166i	$0.420749\pi$
$608$	0	0
$609$	27631.8	1.83859
$610$	0	0
$611$	−2110.38	−0.139733
$612$	0	0
$613$	−22685.2	−1.49470	−0.747348	−	0.664433i	$-0.768673\pi$
$613$	−22685.2	−1.49470	−0.747348	+	0.664433i	$0.768673\pi$
$614$	0	0
$615$	−3100.80	−0.203311
$616$	0	0
$617$	−12502.8	−0.815790	−0.407895	−	0.913029i	$-0.633737\pi$
$617$	−12502.8	−0.815790	−0.407895	+	0.913029i	$0.633737\pi$
$618$	0	0
$619$	9362.12	0.607908	0.303954	−	0.952687i	$-0.401693\pi$
$619$	9362.12	0.607908	0.303954	+	0.952687i	$0.401693\pi$
$620$	0	0
$621$	400.518	0.0258812
$622$	0	0
$623$	−44581.8	−2.86698
$624$	0	0
$625$	13167.5	0.842723
$626$	0	0
$627$	−12121.9	−0.772089
$628$	0	0
$629$	−340.585	−0.0215899
$630$	0	0
$631$	12813.0	0.808361	0.404181	−	0.914679i	$-0.367557\pi$
$631$	12813.0	0.808361	0.404181	+	0.914679i	$0.367557\pi$
$632$	0	0
$633$	3451.69	0.216733
$634$	0	0
$635$	990.449	0.0618973
$636$	0	0
$637$	−3448.27	−0.214483
$638$	0	0
$639$	2365.51	0.146444
$640$	0	0
$641$	24298.6	1.49725	0.748625	−	0.662993i	$-0.230714\pi$
$641$	24298.6	1.49725	0.748625	+	0.662993i	$0.230714\pi$
$642$	0	0
$643$	−836.946	−0.0513311	−0.0256656	−	0.999671i	$-0.508170\pi$
$643$	−836.946	−0.0513311	−0.0256656	+	0.999671i	$0.508170\pi$
$644$	0	0
$645$	2129.69	0.130010
$646$	0	0
$647$	17491.2	1.06283	0.531416	−	0.847111i	$-0.321660\pi$
$647$	17491.2	1.06283	0.531416	+	0.847111i	$0.321660\pi$
$648$	0	0
$649$	−12271.2	−0.742196
$650$	0	0
$651$	16714.4	1.00628
$652$	0	0
$653$	5732.48	0.343536	0.171768	−	0.985137i	$-0.445052\pi$
$653$	5732.48	0.343536	0.171768	+	0.985137i	$0.445052\pi$
$654$	0	0
$655$	−5331.20	−0.318026
$656$	0	0
$657$	5066.89	0.300880
$658$	0	0
$659$	24847.8	1.46879	0.734396	−	0.678721i	$-0.237465\pi$
$659$	24847.8	1.46879	0.734396	+	0.678721i	$0.237465\pi$
$660$	0	0
$661$	−113.775	−0.00669490	−0.00334745	−	0.999994i	$-0.501066\pi$
$661$	−113.775	−0.00669490	−0.00334745	+	0.999994i	$0.501066\pi$
$662$	0	0
$663$	−145.114	−0.00850040
$664$	0	0
$665$	10620.4	0.619310
$666$	0	0
$667$	4048.42	0.235016
$668$	0	0
$669$	−4461.97	−0.257862
$670$	0	0
$671$	13658.8	0.785828
$672$	0	0
$673$	1734.60	0.0993521	0.0496760	−	0.998765i	$-0.484181\pi$
$673$	1734.60	0.0993521	0.0496760	+	0.998765i	$0.484181\pi$
$674$	0	0
$675$	−3194.86	−0.182178
$676$	0	0
$677$	−16665.4	−0.946090	−0.473045	−	0.881038i	$-0.656845\pi$
$677$	−16665.4	−0.946090	−0.473045	+	0.881038i	$0.656845\pi$
$678$	0	0
$679$	37179.9	2.10138
$680$	0	0
$681$	−9213.38	−0.518440
$682$	0	0
$683$	−1374.13	−0.0769834	−0.0384917	−	0.999259i	$-0.512255\pi$
$683$	−1374.13	−0.0769834	−0.0384917	+	0.999259i	$0.512255\pi$
$684$	0	0
$685$	8086.86	0.451070
$686$	0	0
$687$	9146.66	0.507957
$688$	0	0
$689$	904.254	0.0499990
$690$	0	0
$691$	−20910.3	−1.15118	−0.575589	−	0.817739i	$-0.695227\pi$
$691$	−20910.3	−1.15118	−0.575589	+	0.817739i	$0.695227\pi$
$692$	0	0
$693$	10073.9	0.552201
$694$	0	0
$695$	7419.94	0.404971
$696$	0	0
$697$	4468.13	0.242815
$698$	0	0
$699$	−4885.14	−0.264339
$700$	0	0
$701$	22052.5	1.18817	0.594087	−	0.804401i	$-0.297514\pi$
$701$	22052.5	1.18817	0.594087	+	0.804401i	$0.297514\pi$
$702$	0	0
$703$	3716.06	0.199365
$704$	0	0
$705$	−3775.00	−0.201666
$706$	0	0
$707$	−24336.2	−1.29456
$708$	0	0
$709$	5578.04	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$709$	5578.04	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$710$	0	0
$711$	−8600.07	−0.453626
$712$	0	0
$713$	2448.87	0.128627
$714$	0	0
$715$	371.115	0.0194111
$716$	0	0
$717$	−8610.57	−0.448490
$718$	0	0
$719$	6503.71	0.337340	0.168670	−	0.985673i	$-0.446053\pi$
$719$	6503.71	0.337340	0.168670	+	0.985673i	$0.446053\pi$
$720$	0	0
$721$	10301.3	0.532093
$722$	0	0
$723$	5800.06	0.298349
$724$	0	0
$725$	−32293.5	−1.65428
$726$	0	0
$727$	−20080.7	−1.02442	−0.512210	−	0.858860i	$-0.671173\pi$
$727$	−20080.7	−1.02442	−0.512210	+	0.858860i	$0.671173\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3068.80	−0.155272
$732$	0	0
$733$	21791.2	1.09806	0.549029	−	0.835803i	$-0.314998\pi$
$733$	21791.2	1.09806	0.549029	+	0.835803i	$0.314998\pi$
$734$	0	0
$735$	−6168.19	−0.309547
$736$	0	0
$737$	−13509.7	−0.675219
$738$	0	0
$739$	−2598.27	−0.129335	−0.0646677	−	0.997907i	$-0.520599\pi$
$739$	−2598.27	−0.129335	−0.0646677	+	0.997907i	$0.520599\pi$
$740$	0	0
$741$	1583.31	0.0784944
$742$	0	0
$743$	−7715.44	−0.380958	−0.190479	−	0.981691i	$-0.561004\pi$
$743$	−7715.44	−0.380958	−0.190479	+	0.981691i	$0.561004\pi$
$744$	0	0
$745$	−6424.79	−0.315954
$746$	0	0
$747$	−6025.27	−0.295118
$748$	0	0
$749$	57586.5	2.80930
$750$	0	0
$751$	7545.94	0.366651	0.183326	−	0.983052i	$-0.441314\pi$
$751$	7545.94	0.366651	0.183326	+	0.983052i	$0.441314\pi$
$752$	0	0
$753$	−3692.12	−0.178683
$754$	0	0
$755$	4507.94	0.217299
$756$	0	0
$757$	25801.6	1.23881	0.619403	−	0.785073i	$-0.287375\pi$
$757$	25801.6	1.23881	0.619403	+	0.785073i	$0.287375\pi$
$758$	0	0
$759$	1475.95	0.0705846
$760$	0	0
$761$	−8363.87	−0.398410	−0.199205	−	0.979958i	$-0.563836\pi$
$761$	−8363.87	−0.398410	−0.199205	+	0.979958i	$0.563836\pi$
$762$	0	0
$763$	−9336.87	−0.443011
$764$	0	0
$765$	−259.577	−0.0122680
$766$	0	0
$767$	1602.81	0.0754553
$768$	0	0
$769$	26200.1	1.22861	0.614305	−	0.789069i	$-0.289436\pi$
$769$	26200.1	1.22861	0.614305	+	0.789069i	$0.289436\pi$
$770$	0	0
$771$	−6291.91	−0.293901
$772$	0	0
$773$	22712.6	1.05681	0.528407	−	0.848991i	$-0.322790\pi$
$773$	22712.6	1.05681	0.528407	+	0.848991i	$0.322790\pi$
$774$	0	0
$775$	−19534.2	−0.905405
$776$	0	0
$777$	−3088.23	−0.142587
$778$	0	0
$779$	−48750.8	−2.24221
$780$	0	0
$781$	8717.15	0.399391
$782$	0	0
$783$	7368.71	0.336317
$784$	0	0
$785$	2937.29	0.133549
$786$	0	0
$787$	19483.8	0.882492	0.441246	−	0.897386i	$-0.354537\pi$
$787$	19483.8	0.882492	0.441246	+	0.897386i	$0.354537\pi$
$788$	0	0
$789$	−316.268	−0.0142705
$790$	0	0
$791$	45593.5	2.04945
$792$	0	0
$793$	−1784.06	−0.0798912
$794$	0	0
$795$	1617.51	0.0721599
$796$	0	0
$797$	−25878.6	−1.15015	−0.575075	−	0.818101i	$-0.695027\pi$
$797$	−25878.6	−1.15015	−0.575075	+	0.818101i	$0.695027\pi$
$798$	0	0
$799$	5439.61	0.240851
$800$	0	0
$801$	−11888.8	−0.524433
$802$	0	0
$803$	18672.1	0.820577
$804$	0	0
$805$	−1293.14	−0.0566175
$806$	0	0
$807$	21263.4	0.927516
$808$	0	0
$809$	10331.9	0.449011	0.224506	−	0.974473i	$-0.427923\pi$
$809$	10331.9	0.449011	0.224506	+	0.974473i	$0.427923\pi$
$810$	0	0
$811$	−16025.0	−0.693850	−0.346925	−	0.937893i	$-0.612774\pi$
$811$	−16025.0	−0.693850	−0.346925	+	0.937893i	$0.612774\pi$
$812$	0	0
$813$	−12036.4	−0.519233
$814$	0	0
$815$	966.483	0.0415392
$816$	0	0
$817$	33483.0	1.43381
$818$	0	0
$819$	−1315.81	−0.0561395
$820$	0	0
$821$	41867.7	1.77977	0.889885	−	0.456184i	$-0.150784\pi$
$821$	41867.7	1.77977	0.889885	+	0.456184i	$0.150784\pi$
$822$	0	0
$823$	−23743.2	−1.00563	−0.502816	−	0.864394i	$-0.667703\pi$
$823$	−23743.2	−1.00563	−0.502816	+	0.864394i	$0.667703\pi$
$824$	0	0
$825$	−11773.4	−0.496846
$826$	0	0
$827$	−21250.1	−0.893518	−0.446759	−	0.894654i	$-0.647422\pi$
$827$	−21250.1	−0.893518	−0.446759	+	0.894654i	$0.647422\pi$
$828$	0	0
$829$	9622.35	0.403134	0.201567	−	0.979475i	$-0.435397\pi$
$829$	9622.35	0.403134	0.201567	+	0.979475i	$0.435397\pi$
$830$	0	0
$831$	23295.6	0.972461
$832$	0	0
$833$	8888.10	0.369693
$834$	0	0
$835$	10493.8	0.434915
$836$	0	0
$837$	4457.29	0.184070
$838$	0	0
$839$	28899.6	1.18918	0.594592	−	0.804028i	$-0.297314\pi$
$839$	28899.6	1.18918	0.594592	+	0.804028i	$0.297314\pi$
$840$	0	0
$841$	50093.6	2.05394
$842$	0	0
$843$	10694.3	0.436927
$844$	0	0
$845$	5626.39	0.229058
$846$	0	0
$847$	−7796.55	−0.316284
$848$	0	0
$849$	−2530.06	−0.102275
$850$	0	0
$851$	−452.466	−0.0182260
$852$	0	0
$853$	−37772.1	−1.51617	−0.758085	−	0.652156i	$-0.773865\pi$
$853$	−37772.1	−1.51617	−0.758085	+	0.652156i	$0.773865\pi$
$854$	0	0
$855$	2832.19	0.113285
$856$	0	0
$857$	−312.274	−0.0124470	−0.00622349	−	0.999981i	$-0.501981\pi$
$857$	−312.274	−0.0124470	−0.00622349	+	0.999981i	$0.501981\pi$
$858$	0	0
$859$	162.343	0.00644829	0.00322414	−	0.999995i	$-0.498974\pi$
$859$	162.343	0.00644829	0.00322414	+	0.999995i	$0.498974\pi$
$860$	0	0
$861$	40514.4	1.60363
$862$	0	0
$863$	23147.8	0.913050	0.456525	−	0.889711i	$-0.349094\pi$
$863$	23147.8	0.913050	0.456525	+	0.889711i	$0.349094\pi$
$864$	0	0
$865$	6720.57	0.264169
$866$	0	0
$867$	−14365.0	−0.562699
$868$	0	0
$869$	−31692.2	−1.23715
$870$	0	0
$871$	1764.59	0.0686461
$872$	0	0
$873$	9914.94	0.384387
$874$	0	0
$875$	21211.9	0.819533
$876$	0	0
$877$	6345.78	0.244335	0.122167	−	0.992510i	$-0.461016\pi$
$877$	6345.78	0.244335	0.122167	+	0.992510i	$0.461016\pi$
$878$	0	0
$879$	14331.5	0.549932
$880$	0	0
$881$	14383.2	0.550035	0.275017	−	0.961439i	$-0.411316\pi$
$881$	14383.2	0.550035	0.275017	+	0.961439i	$0.411316\pi$
$882$	0	0
$883$	−22457.8	−0.855906	−0.427953	−	0.903801i	$-0.640765\pi$
$883$	−22457.8	−0.855906	−0.427953	+	0.903801i	$0.640765\pi$
$884$	0	0
$885$	2867.07	0.108899
$886$	0	0
$887$	4401.93	0.166632	0.0833159	−	0.996523i	$-0.473449\pi$
$887$	4401.93	0.166632	0.0833159	+	0.996523i	$0.473449\pi$
$888$	0	0
$889$	−12941.0	−0.488220
$890$	0	0
$891$	2686.45	0.101009
$892$	0	0
$893$	−59350.5	−2.22406
$894$	0	0
$895$	3115.04	0.116340
$896$	0	0
$897$	−192.783	−0.00717598
$898$	0	0
$899$	45054.2	1.67146
$900$	0	0
$901$	−2330.76	−0.0861808
$902$	0	0
$903$	−27826.1	−1.02547
$904$	0	0
$905$	11340.9	0.416559
$906$	0	0
$907$	7507.85	0.274856	0.137428	−	0.990512i	$-0.456117\pi$
$907$	7507.85	0.274856	0.137428	+	0.990512i	$0.456117\pi$
$908$	0	0
$909$	−6489.84	−0.236803
$910$	0	0
$911$	9565.13	0.347867	0.173934	−	0.984757i	$-0.444352\pi$
$911$	9565.13	0.347867	0.173934	+	0.984757i	$0.444352\pi$
$912$	0	0
$913$	−22203.8	−0.804861
$914$	0	0
$915$	−3191.28	−0.115301
$916$	0	0
$917$	69656.4	2.50846
$918$	0	0
$919$	13117.8	0.470856	0.235428	−	0.971892i	$-0.424351\pi$
$919$	13117.8	0.470856	0.235428	+	0.971892i	$0.424351\pi$
$920$	0	0
$921$	6065.65	0.217014
$922$	0	0
$923$	−1138.60	−0.0406041
$924$	0	0
$925$	3609.24	0.128293
$926$	0	0
$927$	2747.08	0.0973312
$928$	0	0
$929$	−11126.8	−0.392960	−0.196480	−	0.980508i	$-0.562951\pi$
$929$	−11126.8	−0.392960	−0.196480	+	0.980508i	$0.562951\pi$
$930$	0	0
$931$	−96976.2	−3.41382
$932$	0	0
$933$	−11938.5	−0.418917
$934$	0	0
$935$	−956.570	−0.0334579
$936$	0	0
$937$	43331.3	1.51075	0.755374	−	0.655294i	$-0.227455\pi$
$937$	43331.3	1.51075	0.755374	+	0.655294i	$0.227455\pi$
$938$	0	0
$939$	3627.22	0.126060
$940$	0	0
$941$	6592.51	0.228385	0.114192	−	0.993459i	$-0.463572\pi$
$941$	6592.51	0.228385	0.114192	+	0.993459i	$0.463572\pi$
$942$	0	0
$943$	5935.88	0.204983
$944$	0	0
$945$	−2353.69	−0.0810219
$946$	0	0
$947$	−27300.7	−0.936806	−0.468403	−	0.883515i	$-0.655170\pi$
$947$	−27300.7	−0.936806	−0.468403	+	0.883515i	$0.655170\pi$
$948$	0	0
$949$	−2438.88	−0.0834239
$950$	0	0
$951$	−4515.37	−0.153965
$952$	0	0
$953$	−55888.0	−1.89967	−0.949837	−	0.312746i	$-0.898751\pi$
$953$	−55888.0	−1.89967	−0.949837	+	0.312746i	$0.898751\pi$
$954$	0	0
$955$	−11704.3	−0.396589
$956$	0	0
$957$	27154.5	0.917221
$958$	0	0
$959$	−105661.	−3.55785
$960$	0	0
$961$	−2537.95	−0.0851919
$962$	0	0
$963$	15356.8	0.513881
$964$	0	0
$965$	−7679.10	−0.256165
$966$	0	0
$967$	−22882.7	−0.760970	−0.380485	−	0.924787i	$-0.624243\pi$
$967$	−22882.7	−0.760970	−0.380485	+	0.924787i	$0.624243\pi$
$968$	0	0
$969$	−4081.07	−0.135297
$970$	0	0
$971$	28928.3	0.956081	0.478040	−	0.878338i	$-0.341347\pi$
$971$	28928.3	0.956081	0.478040	+	0.878338i	$0.341347\pi$
$972$	0	0
$973$	−96947.5	−3.19424
$974$	0	0
$975$	1537.80	0.0505118
$976$	0	0
$977$	16672.9	0.545972	0.272986	−	0.962018i	$-0.411989\pi$
$977$	16672.9	0.545972	0.272986	+	0.962018i	$0.411989\pi$
$978$	0	0
$979$	−43811.6	−1.43026
$980$	0	0
$981$	−2489.91	−0.0810363
$982$	0	0
$983$	−17762.7	−0.576340	−0.288170	−	0.957579i	$-0.593047\pi$
$983$	−17762.7	−0.576340	−0.288170	+	0.957579i	$0.593047\pi$
$984$	0	0
$985$	5714.55	0.184854
$986$	0	0
$987$	49323.3	1.59066
$988$	0	0
$989$	−4076.88	−0.131079
$990$	0	0
$991$	−34238.2	−1.09749	−0.548745	−	0.835989i	$-0.684894\pi$
$991$	−34238.2	−1.09749	−0.548745	+	0.835989i	$0.684894\pi$
$992$	0	0
$993$	−29806.8	−0.952558
$994$	0	0
$995$	7197.28	0.229316
$996$	0	0
$997$	−10081.1	−0.320232	−0.160116	−	0.987098i	$-0.551187\pi$
$997$	−10081.1	−0.320232	−0.160116	+	0.987098i	$0.551187\pi$
$998$	0	0
$999$	−823.553	−0.0260822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.a.p.1.1	yes	2
3.2	odd	2		1152.4.a.n.1.2		2
4.3	odd	2		384.4.a.l.1.1	yes	2
8.3	odd	2		384.4.a.m.1.2	yes	2
8.5	even	2		384.4.a.i.1.2	✓	2
12.11	even	2		1152.4.a.m.1.2		2
16.3	odd	4		768.4.d.s.385.2		4
16.5	even	4		768.4.d.r.385.1		4
16.11	odd	4		768.4.d.s.385.3		4
16.13	even	4		768.4.d.r.385.4		4
24.5	odd	2		1152.4.a.x.1.1		2
24.11	even	2		1152.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.i.1.2	✓	2	8.5	even	2
384.4.a.l.1.1	yes	2	4.3	odd	2
384.4.a.m.1.2	yes	2	8.3	odd	2
384.4.a.p.1.1	yes	2	1.1	even	1	trivial
768.4.d.r.385.1		4	16.5	even	4
768.4.d.r.385.4		4	16.13	even	4
768.4.d.s.385.2		4	16.3	odd	4
768.4.d.s.385.3		4	16.11	odd	4
1152.4.a.m.1.2		2	12.11	even	2
1152.4.a.n.1.2		2	3.2	odd	2
1152.4.a.w.1.1		2	24.11	even	2
1152.4.a.x.1.1		2	24.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -2.58301 q^{5} +33.7490 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -2.58301 q^{5} +33.7490 q^{7} +9.00000 q^{9} +33.1660 q^{11} -4.33202 q^{13} -7.74902 q^{15} +11.1660 q^{17} -121.830 q^{19} +101.247 q^{21} +14.8340 q^{23} -118.328 q^{25} +27.0000 q^{27} +272.915 q^{29} +165.085 q^{31} +99.4980 q^{33} -87.1739 q^{35} -30.5020 q^{37} -12.9961 q^{39} +400.154 q^{41} -274.834 q^{43} -23.2470 q^{45} +487.158 q^{47} +795.996 q^{49} +33.4980 q^{51} -208.737 q^{53} -85.6680 q^{55} -365.490 q^{57} -369.992 q^{59} +411.830 q^{61} +303.741 q^{63} +11.1896 q^{65} -407.336 q^{67} +44.5020 q^{69} +262.834 q^{71} +562.988 q^{73} -354.984 q^{75} +1119.32 q^{77} -955.563 q^{79} +81.0000 q^{81} -669.474 q^{83} -28.8419 q^{85} +818.745 q^{87} -1320.98 q^{89} -146.201 q^{91} +495.255 q^{93} +314.688 q^{95} +1101.66 q^{97} +298.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 16 q^{5} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 16 * q^5 + 4 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} + 16 q^{5} + 4 q^{7} + 18 q^{9} + 24 q^{11} + 76 q^{13} + 48 q^{15} - 20 q^{17} - 32 q^{19} + 12 q^{21} + 72 q^{23} + 102 q^{25} + 54 q^{27} + 440 q^{29} + 436 q^{31} + 72 q^{33} - 640 q^{35} - 188 q^{37} + 228 q^{39} - 4 q^{41} - 592 q^{43} + 144 q^{45} + 424 q^{47} + 1338 q^{49} - 60 q^{51} + 408 q^{53} - 256 q^{55} - 96 q^{57} - 232 q^{59} + 612 q^{61} + 36 q^{63} + 1504 q^{65} - 984 q^{67} + 216 q^{69} + 568 q^{71} + 364 q^{73} + 306 q^{75} + 1392 q^{77} - 620 q^{79} + 162 q^{81} + 312 q^{83} - 608 q^{85} + 1320 q^{87} - 1372 q^{89} - 2536 q^{91} + 1308 q^{93} + 1984 q^{95} + 1780 q^{97} + 216 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 16 * q^5 + 4 * q^7 + 18 * q^9 + 24 * q^11 + 76 * q^13 + 48 * q^15 - 20 * q^17 - 32 * q^19 + 12 * q^21 + 72 * q^23 + 102 * q^25 + 54 * q^27 + 440 * q^29 + 436 * q^31 + 72 * q^33 - 640 * q^35 - 188 * q^37 + 228 * q^39 - 4 * q^41 - 592 * q^43 + 144 * q^45 + 424 * q^47 + 1338 * q^49 - 60 * q^51 + 408 * q^53 - 256 * q^55 - 96 * q^57 - 232 * q^59 + 612 * q^61 + 36 * q^63 + 1504 * q^65 - 984 * q^67 + 216 * q^69 + 568 * q^71 + 364 * q^73 + 306 * q^75 + 1392 * q^77 - 620 * q^79 + 162 * q^81 + 312 * q^83 - 608 * q^85 + 1320 * q^87 - 1372 * q^89 - 2536 * q^91 + 1308 * q^93 + 1984 * q^95 + 1780 * q^97 + 216 * q^99

Introduction

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