LMFDB - Embedded newform 384.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,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, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{366}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 366 $ x^2 - 366
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-19.1311$ 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)

$f(q)$	$=$	$q-27.0000 q^{3} -241.049 q^{5} +1255.25 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} -241.049 q^{5} +1255.25 q^{7} +729.000 q^{9} +2415.27 q^{11} +13408.9 q^{13} +6508.32 q^{15} +25571.4 q^{17} +38780.3 q^{19} -33891.6 q^{21} -49641.1 q^{23} -20020.4 q^{25} -19683.0 q^{27} +231355. q^{29} -246856. q^{31} -65212.4 q^{33} -302576. q^{35} +297314. q^{37} -362041. q^{39} -616533. q^{41} +968744. q^{43} -175725. q^{45} +359217. q^{47} +752097. q^{49} -690426. q^{51} -1.39933e6 q^{53} -582199. q^{55} -1.04707e6 q^{57} -214279. q^{59} -1.75876e6 q^{61} +915074. q^{63} -3.23221e6 q^{65} -1.08452e6 q^{67} +1.34031e6 q^{69} +1.02995e6 q^{71} +822221. q^{73} +540550. q^{75} +3.03176e6 q^{77} +4.25755e6 q^{79} +531441. q^{81} +5.69750e6 q^{83} -6.16395e6 q^{85} -6.24659e6 q^{87} -6.57345e6 q^{89} +1.68315e7 q^{91} +6.66512e6 q^{93} -9.34796e6 q^{95} +7.55287e6 q^{97} +1.76073e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} - 176 q^{5} + 980 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q - 54 * q^3 - 176 * q^5 + 980 * q^7 + 1458 * q^9	$ 2 q - 54 q^{3} - 176 q^{5} + 980 q^{7} + 1458 q^{9} - 3128 q^{11} + 8452 q^{13} + 4752 q^{15} + 5228 q^{17} + 87968 q^{19} - 26460 q^{21} - 104792 q^{23} - 93914 q^{25} - 39366 q^{27} + 98760 q^{29} + 1860 q^{31} + 84456 q^{33} - 320480 q^{35} + 455660 q^{37} - 228204 q^{39} - 205188 q^{41} + 803088 q^{43} - 128304 q^{45} - 522488 q^{47} + 4314 q^{49} - 141156 q^{51} - 2468376 q^{53} - 942784 q^{55} - 2375136 q^{57} + 989288 q^{59} - 2482292 q^{61} + 714420 q^{63} - 3554656 q^{65} + 434008 q^{67} + 2829384 q^{69} + 1554840 q^{71} + 279244 q^{73} + 2535678 q^{75} + 4557520 q^{77} + 1015396 q^{79} + 1062882 q^{81} + 2899560 q^{83} - 7487264 q^{85} - 2666520 q^{87} - 8115868 q^{89} + 18195880 q^{91} - 50220 q^{93} - 6148352 q^{95} - 2553676 q^{97} - 2280312 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 - 176 * q^5 + 980 * q^7 + 1458 * q^9 - 3128 * q^11 + 8452 * q^13 + 4752 * q^15 + 5228 * q^17 + 87968 * q^19 - 26460 * q^21 - 104792 * q^23 - 93914 * q^25 - 39366 * q^27 + 98760 * q^29 + 1860 * q^31 + 84456 * q^33 - 320480 * q^35 + 455660 * q^37 - 228204 * q^39 - 205188 * q^41 + 803088 * q^43 - 128304 * q^45 - 522488 * q^47 + 4314 * q^49 - 141156 * q^51 - 2468376 * q^53 - 942784 * q^55 - 2375136 * q^57 + 989288 * q^59 - 2482292 * q^61 + 714420 * q^63 - 3554656 * q^65 + 434008 * q^67 + 2829384 * q^69 + 1554840 * q^71 + 279244 * q^73 + 2535678 * q^75 + 4557520 * q^77 + 1015396 * q^79 + 1062882 * q^81 + 2899560 * q^83 - 7487264 * q^85 - 2666520 * q^87 - 8115868 * q^89 + 18195880 * q^91 - 50220 * q^93 - 6148352 * q^95 - 2553676 * q^97 - 2280312 * 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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−241.049	−0.862403	−0.431202	−	0.902256i	$-0.641910\pi$
$5$	−241.049	−0.862403	−0.431202	+	0.902256i	$0.641910\pi$
$6$	0	0
$7$	1255.25	1.38320	0.691601	−	0.722280i	$-0.256906\pi$
$7$	1255.25	1.38320	0.691601	+	0.722280i	$0.256906\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	2415.27	0.547132	0.273566	−	0.961853i	$-0.411797\pi$
$11$	2415.27	0.547132	0.273566	+	0.961853i	$0.411797\pi$
$12$	0	0
$13$	13408.9	1.69275	0.846375	−	0.532588i	$-0.178780\pi$
$13$	13408.9	1.69275	0.846375	+	0.532588i	$0.178780\pi$
$14$	0	0
$15$	6508.32	0.497909
$16$	0	0
$17$	25571.4	1.26236	0.631179	−	0.775637i	$-0.282571\pi$
$17$	25571.4	1.26236	0.631179	+	0.775637i	$0.282571\pi$
$18$	0	0
$19$	38780.3	1.29710	0.648551	−	0.761171i	$-0.275375\pi$
$19$	38780.3	1.29710	0.648551	+	0.761171i	$0.275375\pi$
$20$	0	0
$21$	−33891.6	−0.798592
$22$	0	0
$23$	−49641.1	−0.850735	−0.425367	−	0.905021i	$-0.639855\pi$
$23$	−49641.1	−0.850735	−0.425367	+	0.905021i	$0.639855\pi$
$24$	0	0
$25$	−20020.4	−0.256261
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	231355.	1.76152	0.880758	−	0.473567i	$-0.157034\pi$
$29$	231355.	1.76152	0.880758	+	0.473567i	$0.157034\pi$
$30$	0	0
$31$	−246856.	−1.48826	−0.744129	−	0.668036i	$-0.767135\pi$
$31$	−246856.	−1.48826	−0.744129	+	0.668036i	$0.767135\pi$
$32$	0	0
$33$	−65212.4	−0.315887
$34$	0	0
$35$	−302576.	−1.19288
$36$	0	0
$37$	297314.	0.964961	0.482480	−	0.875907i	$-0.339736\pi$
$37$	297314.	0.964961	0.482480	+	0.875907i	$0.339736\pi$
$38$	0	0
$39$	−362041.	−0.977309
$40$	0	0
$41$	−616533.	−1.39705	−0.698526	−	0.715585i	$-0.746160\pi$
$41$	−616533.	−1.39705	−0.698526	+	0.715585i	$0.746160\pi$
$42$	0	0
$43$	968744.	1.85810	0.929050	−	0.369953i	$-0.120626\pi$
$43$	968744.	1.85810	0.929050	+	0.369953i	$0.120626\pi$
$44$	0	0
$45$	−175725.	−0.287468
$46$	0	0
$47$	359217.	0.504678	0.252339	−	0.967639i	$-0.418800\pi$
$47$	359217.	0.504678	0.252339	+	0.967639i	$0.418800\pi$
$48$	0	0
$49$	752097.	0.913246
$50$	0	0
$51$	−690426.	−0.728823
$52$	0	0
$53$	−1.39933e6	−1.29108	−0.645541	−	0.763725i	$-0.723368\pi$
$53$	−1.39933e6	−1.29108	−0.645541	+	0.763725i	$0.723368\pi$
$54$	0	0
$55$	−582199.	−0.471848
$56$	0	0
$57$	−1.04707e6	−0.748882
$58$	0	0
$59$	−214279.	−0.135831	−0.0679153	−	0.997691i	$-0.521635\pi$
$59$	−214279.	−0.135831	−0.0679153	+	0.997691i	$0.521635\pi$
$60$	0	0
$61$	−1.75876e6	−0.992091	−0.496046	−	0.868296i	$-0.665215\pi$
$61$	−1.75876e6	−0.992091	−0.496046	+	0.868296i	$0.665215\pi$
$62$	0	0
$63$	915074.	0.461067
$64$	0	0
$65$	−3.23221e6	−1.45983
$66$	0	0
$67$	−1.08452e6	−0.440532	−0.220266	−	0.975440i	$-0.570693\pi$
$67$	−1.08452e6	−0.440532	−0.220266	+	0.975440i	$0.570693\pi$
$68$	0	0
$69$	1.34031e6	0.491172
$70$	0	0
$71$	1.02995e6	0.341517	0.170758	−	0.985313i	$-0.445378\pi$
$71$	1.02995e6	0.341517	0.170758	+	0.985313i	$0.445378\pi$
$72$	0	0
$73$	822221.	0.247376	0.123688	−	0.992321i	$-0.460528\pi$
$73$	822221.	0.247376	0.123688	+	0.992321i	$0.460528\pi$
$74$	0	0
$75$	540550.	0.147952
$76$	0	0
$77$	3.03176e6	0.756794
$78$	0	0
$79$	4.25755e6	0.971550	0.485775	−	0.874084i	$-0.338537\pi$
$79$	4.25755e6	0.971550	0.485775	+	0.874084i	$0.338537\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.69750e6	1.09373	0.546866	−	0.837220i	$-0.315821\pi$
$83$	5.69750e6	1.09373	0.546866	+	0.837220i	$0.315821\pi$
$84$	0	0
$85$	−6.16395e6	−1.08866
$86$	0	0
$87$	−6.24659e6	−1.01701
$88$	0	0
$89$	−6.57345e6	−0.988390	−0.494195	−	0.869351i	$-0.664537\pi$
$89$	−6.57345e6	−0.988390	−0.494195	+	0.869351i	$0.664537\pi$
$90$	0	0
$91$	1.68315e7	2.34141
$92$	0	0
$93$	6.66512e6	0.859246
$94$	0	0
$95$	−9.34796e6	−1.11862
$96$	0	0
$97$	7.55287e6	0.840254	0.420127	−	0.907465i	$-0.361985\pi$
$97$	7.55287e6	0.840254	0.420127	+	0.907465i	$0.361985\pi$
$98$	0	0
$99$	1.76073e6	0.182377
$100$	0	0
$101$	−7.02082e6	−0.678052	−0.339026	−	0.940777i	$-0.610098\pi$
$101$	−7.02082e6	−0.678052	−0.339026	+	0.940777i	$0.610098\pi$
$102$	0	0
$103$	1.83883e7	1.65810	0.829051	−	0.559172i	$-0.188881\pi$
$103$	1.83883e7	1.65810	0.829051	+	0.559172i	$0.188881\pi$
$104$	0	0
$105$	8.16954e6	0.688708
$106$	0	0
$107$	−1.41357e7	−1.11551	−0.557755	−	0.830005i	$-0.688337\pi$
$107$	−1.41357e7	−1.11551	−0.557755	+	0.830005i	$0.688337\pi$
$108$	0	0
$109$	4.63280e6	0.342650	0.171325	−	0.985215i	$-0.445195\pi$
$109$	4.63280e6	0.342650	0.171325	+	0.985215i	$0.445195\pi$
$110$	0	0
$111$	−8.02748e6	−0.557120
$112$	0	0
$113$	1.05522e7	0.687970	0.343985	−	0.938975i	$-0.388223\pi$
$113$	1.05522e7	0.687970	0.343985	+	0.938975i	$0.388223\pi$
$114$	0	0
$115$	1.19659e7	0.733676
$116$	0	0
$117$	9.77512e6	0.564250
$118$	0	0
$119$	3.20983e7	1.74609
$120$	0	0
$121$	−1.36536e7	−0.700647
$122$	0	0
$123$	1.66464e7	0.806588
$124$	0	0
$125$	2.36578e7	1.08340
$126$	0	0
$127$	−9.59436e6	−0.415626	−0.207813	−	0.978169i	$-0.566635\pi$
$127$	−9.59436e6	−0.415626	−0.207813	+	0.978169i	$0.566635\pi$
$128$	0	0
$129$	−2.61561e7	−1.07277
$130$	0	0
$131$	−2.51622e7	−0.977910	−0.488955	−	0.872309i	$-0.662622\pi$
$131$	−2.51622e7	−0.977910	−0.488955	+	0.872309i	$0.662622\pi$
$132$	0	0
$133$	4.86788e7	1.79415
$134$	0	0
$135$	4.74457e6	0.165970
$136$	0	0
$137$	−4.08875e7	−1.35853	−0.679263	−	0.733895i	$-0.737701\pi$
$137$	−4.08875e7	−1.35853	−0.679263	+	0.733895i	$0.737701\pi$
$138$	0	0
$139$	−3.93884e7	−1.24399	−0.621994	−	0.783022i	$-0.713677\pi$
$139$	−3.93884e7	−1.24399	−0.621994	+	0.783022i	$0.713677\pi$
$140$	0	0
$141$	−9.69885e6	−0.291376
$142$	0	0
$143$	3.23863e7	0.926157
$144$	0	0
$145$	−5.57680e7	−1.51914
$146$	0	0
$147$	−2.03066e7	−0.527263
$148$	0	0
$149$	−3.70043e6	−0.0916431	−0.0458216	−	0.998950i	$-0.514591\pi$
$149$	−3.70043e6	−0.0916431	−0.0458216	+	0.998950i	$0.514591\pi$
$150$	0	0
$151$	1.24292e7	0.293781	0.146891	−	0.989153i	$-0.453073\pi$
$151$	1.24292e7	0.293781	0.146891	+	0.989153i	$0.453073\pi$
$152$	0	0
$153$	1.86415e7	0.420786
$154$	0	0
$155$	5.95045e7	1.28348
$156$	0	0
$157$	2.82244e7	0.582072	0.291036	−	0.956712i	$-0.406000\pi$
$157$	2.82244e7	0.582072	0.291036	+	0.956712i	$0.406000\pi$
$158$	0	0
$159$	3.77819e7	0.745407
$160$	0	0
$161$	−6.23118e7	−1.17674
$162$	0	0
$163$	6.47825e7	1.17166	0.585830	−	0.810434i	$-0.300769\pi$
$163$	6.47825e7	1.17166	0.585830	+	0.810434i	$0.300769\pi$
$164$	0	0
$165$	1.57194e7	0.272422
$166$	0	0
$167$	5.65214e7	0.939086	0.469543	−	0.882910i	$-0.344419\pi$
$167$	5.65214e7	0.939086	0.469543	+	0.882910i	$0.344419\pi$
$168$	0	0
$169$	1.17051e8	1.86540
$170$	0	0
$171$	2.82709e7	0.432367
$172$	0	0
$173$	1.07852e7	0.158367	0.0791836	−	0.996860i	$-0.474769\pi$
$173$	1.07852e7	0.158367	0.0791836	+	0.996860i	$0.474769\pi$
$174$	0	0
$175$	−2.51305e7	−0.354460
$176$	0	0
$177$	5.78553e6	0.0784218
$178$	0	0
$179$	1.53198e6	0.0199649	0.00998247	−	0.999950i	$-0.496822\pi$
$179$	1.53198e6	0.0199649	0.00998247	+	0.999950i	$0.496822\pi$
$180$	0	0
$181$	1.26121e8	1.58093	0.790467	−	0.612505i	$-0.209838\pi$
$181$	1.26121e8	1.58093	0.790467	+	0.612505i	$0.209838\pi$
$182$	0	0
$183$	4.74865e7	0.572784
$184$	0	0
$185$	−7.16673e7	−0.832185
$186$	0	0
$187$	6.17618e7	0.690676
$188$	0	0
$189$	−2.47070e7	−0.266197
$190$	0	0
$191$	1.41139e8	1.46565	0.732826	−	0.680417i	$-0.238201\pi$
$191$	1.41139e8	1.46565	0.732826	+	0.680417i	$0.238201\pi$
$192$	0	0
$193$	−1.15274e8	−1.15420	−0.577100	−	0.816674i	$-0.695816\pi$
$193$	−1.15274e8	−1.15420	−0.577100	+	0.816674i	$0.695816\pi$
$194$	0	0
$195$	8.72697e7	0.842835
$196$	0	0
$197$	−3.22603e7	−0.300633	−0.150316	−	0.988638i	$-0.548029\pi$
$197$	−3.22603e7	−0.300633	−0.150316	+	0.988638i	$0.548029\pi$
$198$	0	0
$199$	5.49688e7	0.494459	0.247230	−	0.968957i	$-0.420480\pi$
$199$	5.49688e7	0.494459	0.247230	+	0.968957i	$0.420480\pi$
$200$	0	0
$201$	2.92822e7	0.254341
$202$	0	0
$203$	2.90408e8	2.43653
$204$	0	0
$205$	1.48615e8	1.20482
$206$	0	0
$207$	−3.61884e7	−0.283578
$208$	0	0
$209$	9.36651e7	0.709686
$210$	0	0
$211$	−5.55364e7	−0.406995	−0.203498	−	0.979075i	$-0.565231\pi$
$211$	−5.55364e7	−0.406995	−0.203498	+	0.979075i	$0.565231\pi$
$212$	0	0
$213$	−2.78087e7	−0.197175
$214$	0	0
$215$	−2.33515e8	−1.60243
$216$	0	0
$217$	−3.09865e8	−2.05856
$218$	0	0
$219$	−2.22000e7	−0.142823
$220$	0	0
$221$	3.42885e8	2.13686
$222$	0	0
$223$	1.26355e8	0.763004	0.381502	−	0.924368i	$-0.375407\pi$
$223$	1.26355e8	0.763004	0.381502	+	0.924368i	$0.375407\pi$
$224$	0	0
$225$	−1.45949e7	−0.0854203
$226$	0	0
$227$	1.81110e8	1.02767	0.513834	−	0.857890i	$-0.328225\pi$
$227$	1.81110e8	1.02767	0.513834	+	0.857890i	$0.328225\pi$
$228$	0	0
$229$	−1.46997e8	−0.808880	−0.404440	−	0.914564i	$-0.632534\pi$
$229$	−1.46997e8	−0.808880	−0.404440	+	0.914564i	$0.632534\pi$
$230$	0	0
$231$	−8.18576e7	−0.436935
$232$	0	0
$233$	1.68176e8	0.870998	0.435499	−	0.900189i	$-0.356572\pi$
$233$	1.68176e8	0.870998	0.435499	+	0.900189i	$0.356572\pi$
$234$	0	0
$235$	−8.65888e7	−0.435236
$236$	0	0
$237$	−1.14954e8	−0.560925
$238$	0	0
$239$	2.06268e8	0.977326	0.488663	−	0.872473i	$-0.337485\pi$
$239$	2.06268e8	0.977326	0.488663	+	0.872473i	$0.337485\pi$
$240$	0	0
$241$	−3.39136e8	−1.56068	−0.780341	−	0.625354i	$-0.784954\pi$
$241$	−3.39136e8	−1.56068	−0.780341	+	0.625354i	$0.784954\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−1.81292e8	−0.787586
$246$	0	0
$247$	5.20003e8	2.19567
$248$	0	0
$249$	−1.53833e8	−0.631467
$250$	0	0
$251$	3.62781e7	0.144806	0.0724030	−	0.997375i	$-0.476933\pi$
$251$	3.62781e7	0.144806	0.0724030	+	0.997375i	$0.476933\pi$
$252$	0	0
$253$	−1.19897e8	−0.465464
$254$	0	0
$255$	1.66427e8	0.628539
$256$	0	0
$257$	−5.14951e8	−1.89234	−0.946172	−	0.323665i	$-0.895085\pi$
$257$	−5.14951e8	−1.89234	−0.946172	+	0.323665i	$0.895085\pi$
$258$	0	0
$259$	3.73202e8	1.33474
$260$	0	0
$261$	1.68658e8	0.587172
$262$	0	0
$263$	9.03571e7	0.306279	0.153139	−	0.988205i	$-0.451062\pi$
$263$	9.03571e7	0.306279	0.153139	+	0.988205i	$0.451062\pi$
$264$	0	0
$265$	3.37307e8	1.11343
$266$	0	0
$267$	1.77483e8	0.570647
$268$	0	0
$269$	3.05304e8	0.956311	0.478156	−	0.878275i	$-0.341306\pi$
$269$	3.05304e8	0.956311	0.478156	+	0.878275i	$0.341306\pi$
$270$	0	0
$271$	−1.67428e8	−0.511019	−0.255509	−	0.966807i	$-0.582243\pi$
$271$	−1.67428e8	−0.511019	−0.255509	+	0.966807i	$0.582243\pi$
$272$	0	0
$273$	−4.54451e8	−1.35182
$274$	0	0
$275$	−4.83547e7	−0.140208
$276$	0	0
$277$	−2.46391e8	−0.696538	−0.348269	−	0.937395i	$-0.613230\pi$
$277$	−2.46391e8	−0.696538	−0.348269	+	0.937395i	$0.613230\pi$
$278$	0	0
$279$	−1.79958e8	−0.496086
$280$	0	0
$281$	−3.35835e8	−0.902929	−0.451465	−	0.892289i	$-0.649098\pi$
$281$	−3.35835e8	−0.902929	−0.451465	+	0.892289i	$0.649098\pi$
$282$	0	0
$283$	−3.73786e8	−0.980325	−0.490162	−	0.871631i	$-0.663063\pi$
$283$	−3.73786e8	−0.980325	−0.490162	+	0.871631i	$0.663063\pi$
$284$	0	0
$285$	2.52395e8	0.645838
$286$	0	0
$287$	−7.73899e8	−1.93240
$288$	0	0
$289$	2.43555e8	0.593547
$290$	0	0
$291$	−2.03927e8	−0.485121
$292$	0	0
$293$	4.15803e8	0.965720	0.482860	−	0.875697i	$-0.339598\pi$
$293$	4.15803e8	0.965720	0.482860	+	0.875697i	$0.339598\pi$
$294$	0	0
$295$	5.16517e7	0.117141
$296$	0	0
$297$	−4.75398e7	−0.105296
$298$	0	0
$299$	−6.65635e8	−1.44008
$300$	0	0
$301$	1.21601e9	2.57013
$302$	0	0
$303$	1.89562e8	0.391474
$304$	0	0
$305$	4.23947e8	0.855583
$306$	0	0
$307$	2.19257e8	0.432484	0.216242	−	0.976340i	$-0.430620\pi$
$307$	2.19257e8	0.432484	0.216242	+	0.976340i	$0.430620\pi$
$308$	0	0
$309$	−4.96484e8	−0.957306
$310$	0	0
$311$	−2.13624e8	−0.402707	−0.201354	−	0.979519i	$-0.564534\pi$
$311$	−2.13624e8	−0.402707	−0.201354	+	0.979519i	$0.564534\pi$
$312$	0	0
$313$	8.19563e8	1.51070	0.755348	−	0.655324i	$-0.227468\pi$
$313$	8.19563e8	1.51070	0.755348	+	0.655324i	$0.227468\pi$
$314$	0	0
$315$	−2.20578e8	−0.397626
$316$	0	0
$317$	−4.31470e8	−0.760753	−0.380376	−	0.924832i	$-0.624206\pi$
$317$	−4.31470e8	−0.760753	−0.380376	+	0.924832i	$0.624206\pi$
$318$	0	0
$319$	5.58786e8	0.963781
$320$	0	0
$321$	3.81664e8	0.644040
$322$	0	0
$323$	9.91666e8	1.63741
$324$	0	0
$325$	−2.68452e8	−0.433785
$326$	0	0
$327$	−1.25086e8	−0.197829
$328$	0	0
$329$	4.50905e8	0.698071
$330$	0	0
$331$	8.22158e7	0.124611	0.0623056	−	0.998057i	$-0.480155\pi$
$331$	8.22158e7	0.124611	0.0623056	+	0.998057i	$0.480155\pi$
$332$	0	0
$333$	2.16742e8	0.321654
$334$	0	0
$335$	2.61424e8	0.379916
$336$	0	0
$337$	8.93277e8	1.27140	0.635699	−	0.771937i	$-0.280712\pi$
$337$	8.93277e8	1.27140	0.635699	+	0.771937i	$0.280712\pi$
$338$	0	0
$339$	−2.84910e8	−0.397199
$340$	0	0
$341$	−5.96226e8	−0.814274
$342$	0	0
$343$	−8.96820e7	−0.119999
$344$	0	0
$345$	−3.23080e8	−0.423588
$346$	0	0
$347$	7.08416e8	0.910197	0.455099	−	0.890441i	$-0.349604\pi$
$347$	7.08416e8	0.910197	0.455099	+	0.890441i	$0.349604\pi$
$348$	0	0
$349$	1.85138e8	0.233134	0.116567	−	0.993183i	$-0.462811\pi$
$349$	1.85138e8	0.233134	0.116567	+	0.993183i	$0.462811\pi$
$350$	0	0
$351$	−2.63928e8	−0.325770
$352$	0	0
$353$	−1.62041e9	−1.96071	−0.980355	−	0.197240i	$-0.936802\pi$
$353$	−1.62041e9	−1.96071	−0.980355	+	0.197240i	$0.936802\pi$
$354$	0	0
$355$	−2.48269e8	−0.294525
$356$	0	0
$357$	−8.66654e8	−1.00811
$358$	0	0
$359$	2.30456e8	0.262880	0.131440	−	0.991324i	$-0.458040\pi$
$359$	2.30456e8	0.262880	0.131440	+	0.991324i	$0.458040\pi$
$360$	0	0
$361$	6.10043e8	0.682472
$362$	0	0
$363$	3.68648e8	0.404519
$364$	0	0
$365$	−1.98195e8	−0.213338
$366$	0	0
$367$	1.30643e9	1.37960	0.689801	−	0.723999i	$-0.257698\pi$
$367$	1.30643e9	1.37960	0.689801	+	0.723999i	$0.257698\pi$
$368$	0	0
$369$	−4.49452e8	−0.465684
$370$	0	0
$371$	−1.75650e9	−1.78583
$372$	0	0
$373$	−6.36011e8	−0.634576	−0.317288	−	0.948329i	$-0.602772\pi$
$373$	−6.36011e8	−0.634576	−0.317288	+	0.948329i	$0.602772\pi$
$374$	0	0
$375$	−6.38762e8	−0.625503
$376$	0	0
$377$	3.10223e9	2.98180
$378$	0	0
$379$	−7.80291e8	−0.736240	−0.368120	−	0.929778i	$-0.619998\pi$
$379$	−7.80291e8	−0.736240	−0.368120	+	0.929778i	$0.619998\pi$
$380$	0	0
$381$	2.59048e8	0.239962
$382$	0	0
$383$	3.06788e6	0.00279024	0.00139512	−	0.999999i	$-0.499556\pi$
$383$	3.06788e6	0.00279024	0.00139512	+	0.999999i	$0.499556\pi$
$384$	0	0
$385$	−7.30803e8	−0.652661
$386$	0	0
$387$	7.06214e8	0.619367
$388$	0	0
$389$	−1.50816e9	−1.29905	−0.649524	−	0.760341i	$-0.725032\pi$
$389$	−1.50816e9	−1.29905	−0.649524	+	0.760341i	$0.725032\pi$
$390$	0	0
$391$	−1.26939e9	−1.07393
$392$	0	0
$393$	6.79379e8	0.564597
$394$	0	0
$395$	−1.02628e9	−0.837868
$396$	0	0
$397$	−2.76390e8	−0.221694	−0.110847	−	0.993837i	$-0.535356\pi$
$397$	−2.76390e8	−0.221694	−0.110847	+	0.993837i	$0.535356\pi$
$398$	0	0
$399$	−1.31433e9	−1.03585
$400$	0	0
$401$	−1.86311e9	−1.44289	−0.721444	−	0.692473i	$-0.756521\pi$
$401$	−1.86311e9	−1.44289	−0.721444	+	0.692473i	$0.756521\pi$
$402$	0	0
$403$	−3.31008e9	−2.51925
$404$	0	0
$405$	−1.28103e8	−0.0958226
$406$	0	0
$407$	7.18095e8	0.527961
$408$	0	0
$409$	1.11458e9	0.805524	0.402762	−	0.915305i	$-0.368050\pi$
$409$	1.11458e9	0.805524	0.402762	+	0.915305i	$0.368050\pi$
$410$	0	0
$411$	1.10396e9	0.784346
$412$	0	0
$413$	−2.68973e8	−0.187881
$414$	0	0
$415$	−1.37338e9	−0.943239
$416$	0	0
$417$	1.06349e9	0.718217
$418$	0	0
$419$	−1.75638e9	−1.16646	−0.583228	−	0.812308i	$-0.698211\pi$
$419$	−1.75638e9	−1.16646	−0.583228	+	0.812308i	$0.698211\pi$
$420$	0	0
$421$	8.24327e8	0.538409	0.269204	−	0.963083i	$-0.413239\pi$
$421$	8.24327e8	0.538409	0.269204	+	0.963083i	$0.413239\pi$
$422$	0	0
$423$	2.61869e8	0.168226
$424$	0	0
$425$	−5.11948e8	−0.323493
$426$	0	0
$427$	−2.20767e9	−1.37226
$428$	0	0
$429$	−8.74429e8	−0.534717
$430$	0	0
$431$	−2.21141e9	−1.33045	−0.665225	−	0.746643i	$-0.731665\pi$
$431$	−2.21141e9	−1.33045	−0.665225	+	0.746643i	$0.731665\pi$
$432$	0	0
$433$	−8.91107e8	−0.527499	−0.263750	−	0.964591i	$-0.584959\pi$
$433$	−8.91107e8	−0.527499	−0.263750	+	0.964591i	$0.584959\pi$
$434$	0	0
$435$	1.50573e9	0.877074
$436$	0	0
$437$	−1.92510e9	−1.10349
$438$	0	0
$439$	1.22630e8	0.0691785	0.0345893	−	0.999402i	$-0.488988\pi$
$439$	1.22630e8	0.0691785	0.0345893	+	0.999402i	$0.488988\pi$
$440$	0	0
$441$	5.48279e8	0.304415
$442$	0	0
$443$	3.32516e9	1.81719	0.908594	−	0.417681i	$-0.137157\pi$
$443$	3.32516e9	1.81719	0.908594	+	0.417681i	$0.137157\pi$
$444$	0	0
$445$	1.58452e9	0.852390
$446$	0	0
$447$	9.99115e7	0.0529102
$448$	0	0
$449$	−9.94013e8	−0.518239	−0.259119	−	0.965845i	$-0.583432\pi$
$449$	−9.94013e8	−0.518239	−0.259119	+	0.965845i	$0.583432\pi$
$450$	0	0
$451$	−1.48910e9	−0.764372
$452$	0	0
$453$	−3.35589e8	−0.169615
$454$	0	0
$455$	−4.05722e9	−2.01924
$456$	0	0
$457$	2.67555e9	1.31131	0.655657	−	0.755059i	$-0.272392\pi$
$457$	2.67555e9	1.31131	0.655657	+	0.755059i	$0.272392\pi$
$458$	0	0
$459$	−5.03321e8	−0.242941
$460$	0	0
$461$	−2.02753e9	−0.963861	−0.481931	−	0.876209i	$-0.660064\pi$
$461$	−2.02753e9	−0.963861	−0.481931	+	0.876209i	$0.660064\pi$
$462$	0	0
$463$	−3.02693e9	−1.41732	−0.708661	−	0.705549i	$-0.750700\pi$
$463$	−3.02693e9	−1.41732	−0.708661	+	0.705549i	$0.750700\pi$
$464$	0	0
$465$	−1.60662e9	−0.741017
$466$	0	0
$467$	3.30446e9	1.50138	0.750690	−	0.660654i	$-0.229721\pi$
$467$	3.30446e9	1.50138	0.750690	+	0.660654i	$0.229721\pi$
$468$	0	0
$469$	−1.36134e9	−0.609345
$470$	0	0
$471$	−7.62060e8	−0.336059
$472$	0	0
$473$	2.33978e9	1.01663
$474$	0	0
$475$	−7.76397e8	−0.332396
$476$	0	0
$477$	−1.02011e9	−0.430361
$478$	0	0
$479$	6.88113e8	0.286079	0.143039	−	0.989717i	$-0.454312\pi$
$479$	6.88113e8	0.286079	0.143039	+	0.989717i	$0.454312\pi$
$480$	0	0
$481$	3.98667e9	1.63344
$482$	0	0
$483$	1.68242e9	0.679390
$484$	0	0
$485$	−1.82061e9	−0.724638
$486$	0	0
$487$	1.85856e9	0.729163	0.364581	−	0.931172i	$-0.381212\pi$
$487$	1.85856e9	0.729163	0.364581	+	0.931172i	$0.381212\pi$
$488$	0	0
$489$	−1.74913e9	−0.676458
$490$	0	0
$491$	−3.10855e9	−1.18515	−0.592574	−	0.805516i	$-0.701888\pi$
$491$	−3.10855e9	−1.18515	−0.592574	+	0.805516i	$0.701888\pi$
$492$	0	0
$493$	5.91607e9	2.22366
$494$	0	0
$495$	−4.24423e8	−0.157283
$496$	0	0
$497$	1.29284e9	0.472387
$498$	0	0
$499$	−2.95742e6	−0.00106552	−0.000532759	−	1.00000i	$-0.500170\pi$
$499$	−2.95742e6	−0.00106552	−0.000532759		1.00000i	$0.500170\pi$
$500$	0	0
$501$	−1.52608e9	−0.542182
$502$	0	0
$503$	−7.89330e8	−0.276548	−0.138274	−	0.990394i	$-0.544155\pi$
$503$	−7.89330e8	−0.276548	−0.138274	+	0.990394i	$0.544155\pi$
$504$	0	0
$505$	1.69236e9	0.584755
$506$	0	0
$507$	−3.16038e9	−1.07699
$508$	0	0
$509$	−4.84341e8	−0.162794	−0.0813971	−	0.996682i	$-0.525938\pi$
$509$	−4.84341e8	−0.162794	−0.0813971	+	0.996682i	$0.525938\pi$
$510$	0	0
$511$	1.03209e9	0.342171
$512$	0	0
$513$	−7.63313e8	−0.249627
$514$	0	0
$515$	−4.43248e9	−1.42995
$516$	0	0
$517$	8.67607e8	0.276125
$518$	0	0
$519$	−2.91199e8	−0.0914334
$520$	0	0
$521$	−4.48726e9	−1.39011	−0.695056	−	0.718956i	$-0.744620\pi$
$521$	−4.48726e9	−1.39011	−0.695056	+	0.718956i	$0.744620\pi$
$522$	0	0
$523$	2.86000e9	0.874200	0.437100	−	0.899413i	$-0.356006\pi$
$523$	2.86000e9	0.874200	0.437100	+	0.899413i	$0.356006\pi$
$524$	0	0
$525$	6.78523e8	0.204648
$526$	0	0
$527$	−6.31245e9	−1.87871
$528$	0	0
$529$	−9.40585e8	−0.276251
$530$	0	0
$531$	−1.56209e8	−0.0452769
$532$	0	0
$533$	−8.26705e9	−2.36486
$534$	0	0
$535$	3.40739e9	0.962020
$536$	0	0
$537$	−4.13635e7	−0.0115268
$538$	0	0
$539$	1.81652e9	0.499666
$540$	0	0
$541$	2.29923e9	0.624297	0.312148	−	0.950033i	$-0.398951\pi$
$541$	2.29923e9	0.624297	0.312148	+	0.950033i	$0.398951\pi$
$542$	0	0
$543$	−3.40528e9	−0.912752
$544$	0	0
$545$	−1.11673e9	−0.295503
$546$	0	0
$547$	−8.06117e8	−0.210592	−0.105296	−	0.994441i	$-0.533579\pi$
$547$	−8.06117e8	−0.210592	−0.105296	+	0.994441i	$0.533579\pi$
$548$	0	0
$549$	−1.28213e9	−0.330697
$550$	0	0
$551$	8.97203e9	2.28486
$552$	0	0
$553$	5.34427e9	1.34385
$554$	0	0
$555$	1.93502e9	0.480462
$556$	0	0
$557$	5.67124e9	1.39054	0.695272	−	0.718747i	$-0.255284\pi$
$557$	5.67124e9	1.39054	0.695272	+	0.718747i	$0.255284\pi$
$558$	0	0
$559$	1.29898e10	3.14530
$560$	0	0
$561$	−1.66757e9	−0.398762
$562$	0	0
$563$	2.83103e9	0.668598	0.334299	−	0.942467i	$-0.391500\pi$
$563$	2.83103e9	0.668598	0.334299	+	0.942467i	$0.391500\pi$
$564$	0	0
$565$	−2.54360e9	−0.593307
$566$	0	0
$567$	6.67089e8	0.153689
$568$	0	0
$569$	1.35162e8	0.0307583	0.0153792	−	0.999882i	$-0.495104\pi$
$569$	1.35162e8	0.0307583	0.0153792	+	0.999882i	$0.495104\pi$
$570$	0	0
$571$	−6.87175e9	−1.54469	−0.772344	−	0.635204i	$-0.780916\pi$
$571$	−6.87175e9	−1.54469	−0.772344	+	0.635204i	$0.780916\pi$
$572$	0	0
$573$	−3.81076e9	−0.846194
$574$	0	0
$575$	9.93834e8	0.218010
$576$	0	0
$577$	2.78800e9	0.604195	0.302098	−	0.953277i	$-0.402313\pi$
$577$	2.78800e9	0.604195	0.302098	+	0.953277i	$0.402313\pi$
$578$	0	0
$579$	3.11240e9	0.666377
$580$	0	0
$581$	7.15176e9	1.51285
$582$	0	0
$583$	−3.37976e9	−0.706392
$584$	0	0
$585$	−2.35628e9	−0.486611
$586$	0	0
$587$	3.87741e9	0.791240	0.395620	−	0.918414i	$-0.370530\pi$
$587$	3.87741e9	0.791240	0.395620	+	0.918414i	$0.370530\pi$
$588$	0	0
$589$	−9.57317e9	−1.93042
$590$	0	0
$591$	8.71027e8	0.173570
$592$	0	0
$593$	3.58123e9	0.705245	0.352623	−	0.935766i	$-0.385290\pi$
$593$	3.58123e9	0.705245	0.352623	+	0.935766i	$0.385290\pi$
$594$	0	0
$595$	−7.73727e9	−1.50584
$596$	0	0
$597$	−1.48416e9	−0.285476
$598$	0	0
$599$	−2.81275e9	−0.534734	−0.267367	−	0.963595i	$-0.586154\pi$
$599$	−2.81275e9	−0.534734	−0.267367	+	0.963595i	$0.586154\pi$
$600$	0	0
$601$	7.21845e9	1.35639	0.678193	−	0.734884i	$-0.262764\pi$
$601$	7.21845e9	1.35639	0.678193	+	0.734884i	$0.262764\pi$
$602$	0	0
$603$	−7.90619e8	−0.146844
$604$	0	0
$605$	3.29119e9	0.604240
$606$	0	0
$607$	9.43874e9	1.71298	0.856492	−	0.516160i	$-0.172639\pi$
$607$	9.43874e9	1.71298	0.856492	+	0.516160i	$0.172639\pi$
$608$	0	0
$609$	−7.84100e9	−1.40673
$610$	0	0
$611$	4.81672e9	0.854293
$612$	0	0
$613$	6.64596e9	1.16532	0.582661	−	0.812715i	$-0.302011\pi$
$613$	6.64596e9	1.16532	0.582661	+	0.812715i	$0.302011\pi$
$614$	0	0
$615$	−4.01259e9	−0.695604
$616$	0	0
$617$	−6.57521e9	−1.12697	−0.563484	−	0.826127i	$-0.690539\pi$
$617$	−6.57521e9	−1.12697	−0.563484	+	0.826127i	$0.690539\pi$
$618$	0	0
$619$	5.34093e9	0.905107	0.452553	−	0.891737i	$-0.350513\pi$
$619$	5.34093e9	0.905107	0.452553	+	0.891737i	$0.350513\pi$
$620$	0	0
$621$	9.77086e8	0.163724
$622$	0	0
$623$	−8.25129e9	−1.36714
$624$	0	0
$625$	−4.13861e9	−0.678070
$626$	0	0
$627$	−2.52896e9	−0.409737
$628$	0	0
$629$	7.60273e9	1.21813
$630$	0	0
$631$	2.30915e9	0.365889	0.182945	−	0.983123i	$-0.441437\pi$
$631$	2.30915e9	0.365889	0.182945	+	0.983123i	$0.441437\pi$
$632$	0	0
$633$	1.49948e9	0.234979
$634$	0	0
$635$	2.31271e9	0.358437
$636$	0	0
$637$	1.00848e10	1.54590
$638$	0	0
$639$	7.50834e8	0.113839
$640$	0	0
$641$	4.73538e9	0.710152	0.355076	−	0.934837i	$-0.384455\pi$
$641$	4.73538e9	0.710152	0.355076	+	0.934837i	$0.384455\pi$
$642$	0	0
$643$	−4.22576e8	−0.0626854	−0.0313427	−	0.999509i	$-0.509978\pi$
$643$	−4.22576e8	−0.0626854	−0.0313427	+	0.999509i	$0.509978\pi$
$644$	0	0
$645$	6.30490e9	0.925165
$646$	0	0
$647$	−7.01310e9	−1.01799	−0.508997	−	0.860769i	$-0.669983\pi$
$647$	−7.01310e9	−1.01799	−0.508997	+	0.860769i	$0.669983\pi$
$648$	0	0
$649$	−5.17543e8	−0.0743173
$650$	0	0
$651$	8.36636e9	1.18851
$652$	0	0
$653$	1.04168e10	1.46399	0.731995	−	0.681310i	$-0.238590\pi$
$653$	1.04168e10	1.46399	0.731995	+	0.681310i	$0.238590\pi$
$654$	0	0
$655$	6.06532e9	0.843353
$656$	0	0
$657$	5.99399e8	0.0824588
$658$	0	0
$659$	1.07019e10	1.45667	0.728337	−	0.685219i	$-0.240294\pi$
$659$	1.07019e10	1.45667	0.728337	+	0.685219i	$0.240294\pi$
$660$	0	0
$661$	7.65225e9	1.03059	0.515293	−	0.857014i	$-0.327683\pi$
$661$	7.65225e9	1.03059	0.515293	+	0.857014i	$0.327683\pi$
$662$	0	0
$663$	−9.25789e9	−1.23371
$664$	0	0
$665$	−1.17340e10	−1.54728
$666$	0	0
$667$	−1.14847e10	−1.49858
$668$	0	0
$669$	−3.41159e9	−0.440520
$670$	0	0
$671$	−4.24788e9	−0.542805
$672$	0	0
$673$	−3.47157e9	−0.439009	−0.219505	−	0.975611i	$-0.570444\pi$
$673$	−3.47157e9	−0.439009	−0.219505	+	0.975611i	$0.570444\pi$
$674$	0	0
$675$	3.94061e8	0.0493174
$676$	0	0
$677$	−1.01392e10	−1.25587	−0.627936	−	0.778265i	$-0.716100\pi$
$677$	−1.01392e10	−1.25587	−0.627936	+	0.778265i	$0.716100\pi$
$678$	0	0
$679$	9.48070e9	1.16224
$680$	0	0
$681$	−4.88998e9	−0.593324
$682$	0	0
$683$	6.29238e9	0.755689	0.377844	−	0.925869i	$-0.376665\pi$
$683$	6.29238e9	0.755689	0.377844	+	0.925869i	$0.376665\pi$
$684$	0	0
$685$	9.85588e9	1.17160
$686$	0	0
$687$	3.96892e9	0.467007
$688$	0	0
$689$	−1.87635e10	−2.18548
$690$	0	0
$691$	1.09543e10	1.26303	0.631514	−	0.775365i	$-0.282434\pi$
$691$	1.09543e10	1.26303	0.631514	+	0.775365i	$0.282434\pi$
$692$	0	0
$693$	2.21015e9	0.252265
$694$	0	0
$695$	9.49453e9	1.07282
$696$	0	0
$697$	−1.57656e10	−1.76358
$698$	0	0
$699$	−4.54074e9	−0.502871
$700$	0	0
$701$	−1.41208e10	−1.54827	−0.774137	−	0.633018i	$-0.781816\pi$
$701$	−1.41208e10	−1.54827	−0.774137	+	0.633018i	$0.781816\pi$
$702$	0	0
$703$	1.15299e10	1.25165
$704$	0	0
$705$	2.33790e9	0.251283
$706$	0	0
$707$	−8.81285e9	−0.937883
$708$	0	0
$709$	1.66122e10	1.75052	0.875258	−	0.483656i	$-0.160691\pi$
$709$	1.66122e10	1.75052	0.875258	+	0.483656i	$0.160691\pi$
$710$	0	0
$711$	3.10376e9	0.323850
$712$	0	0
$713$	1.22542e10	1.26611
$714$	0	0
$715$	−7.80668e9	−0.798721
$716$	0	0
$717$	−5.56924e9	−0.564259
$718$	0	0
$719$	1.53237e10	1.53749	0.768747	−	0.639553i	$-0.220881\pi$
$719$	1.53237e10	1.53749	0.768747	+	0.639553i	$0.220881\pi$
$720$	0	0
$721$	2.30818e10	2.29349
$722$	0	0
$723$	9.15667e9	0.901060
$724$	0	0
$725$	−4.63182e9	−0.451407
$726$	0	0
$727$	−3.65903e9	−0.353180	−0.176590	−	0.984285i	$-0.556507\pi$
$727$	−3.65903e9	−0.353180	−0.176590	+	0.984285i	$0.556507\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.47721e10	2.34559
$732$	0	0
$733$	−5.09932e9	−0.478243	−0.239121	−	0.970990i	$-0.576859\pi$
$733$	−5.09932e9	−0.478243	−0.239121	+	0.970990i	$0.576859\pi$
$734$	0	0
$735$	4.89489e9	0.454713
$736$	0	0
$737$	−2.61942e9	−0.241029
$738$	0	0
$739$	−1.78005e9	−0.162247	−0.0811236	−	0.996704i	$-0.525851\pi$
$739$	−1.78005e9	−0.162247	−0.0811236	+	0.996704i	$0.525851\pi$
$740$	0	0
$741$	−1.40401e10	−1.26767
$742$	0	0
$743$	3.36627e9	0.301084	0.150542	−	0.988604i	$-0.451898\pi$
$743$	3.36627e9	0.301084	0.150542	+	0.988604i	$0.451898\pi$
$744$	0	0
$745$	8.91984e8	0.0790333
$746$	0	0
$747$	4.15348e9	0.364578
$748$	0	0
$749$	−1.77438e10	−1.54298
$750$	0	0
$751$	−2.21469e10	−1.90798	−0.953989	−	0.299842i	$-0.903066\pi$
$751$	−2.21469e10	−1.90798	−0.953989	+	0.299842i	$0.903066\pi$
$752$	0	0
$753$	−9.79509e8	−0.0836038
$754$	0	0
$755$	−2.99605e9	−0.253358
$756$	0	0
$757$	1.43073e10	1.19873	0.599365	−	0.800476i	$-0.295420\pi$
$757$	1.43073e10	1.19873	0.599365	+	0.800476i	$0.295420\pi$
$758$	0	0
$759$	3.23722e9	0.268736
$760$	0	0
$761$	−1.23905e10	−1.01916	−0.509579	−	0.860424i	$-0.670199\pi$
$761$	−1.23905e10	−1.01916	−0.509579	+	0.860424i	$0.670199\pi$
$762$	0	0
$763$	5.81530e9	0.473954
$764$	0	0
$765$	−4.49352e9	−0.362887
$766$	0	0
$767$	−2.87325e9	−0.229927
$768$	0	0
$769$	−1.76200e10	−1.39722	−0.698608	−	0.715505i	$-0.746197\pi$
$769$	−1.76200e10	−1.39722	−0.698608	+	0.715505i	$0.746197\pi$
$770$	0	0
$771$	1.39037e10	1.09255
$772$	0	0
$773$	9.38497e9	0.730811	0.365405	−	0.930849i	$-0.380930\pi$
$773$	9.38497e9	0.730811	0.365405	+	0.930849i	$0.380930\pi$
$774$	0	0
$775$	4.94216e9	0.381382
$776$	0	0
$777$	−1.00765e10	−0.770610
$778$	0	0
$779$	−2.39093e10	−1.81212
$780$	0	0
$781$	2.48761e9	0.186855
$782$	0	0
$783$	−4.55377e9	−0.339004
$784$	0	0
$785$	−6.80347e9	−0.501980
$786$	0	0
$787$	2.16968e10	1.58666	0.793330	−	0.608791i	$-0.208345\pi$
$787$	2.16968e10	1.58666	0.793330	+	0.608791i	$0.208345\pi$
$788$	0	0
$789$	−2.43964e9	−0.176830
$790$	0	0
$791$	1.32456e10	0.951600
$792$	0	0
$793$	−2.35831e10	−1.67936
$794$	0	0
$795$	−9.10728e9	−0.642841
$796$	0	0
$797$	−6.39497e9	−0.447439	−0.223720	−	0.974654i	$-0.571820\pi$
$797$	−6.39497e9	−0.447439	−0.223720	+	0.974654i	$0.571820\pi$
$798$	0	0
$799$	9.18566e9	0.637084
$800$	0	0
$801$	−4.79204e9	−0.329463
$802$	0	0
$803$	1.98589e9	0.135348
$804$	0	0
$805$	1.50202e10	1.01482
$806$	0	0
$807$	−8.24320e9	−0.552126
$808$	0	0
$809$	−4.80487e9	−0.319052	−0.159526	−	0.987194i	$-0.550997\pi$
$809$	−4.80487e9	−0.319052	−0.159526	+	0.987194i	$0.550997\pi$
$810$	0	0
$811$	−1.20422e10	−0.792745	−0.396372	−	0.918090i	$-0.629731\pi$
$811$	−1.20422e10	−0.792745	−0.396372	+	0.918090i	$0.629731\pi$
$812$	0	0
$813$	4.52057e9	0.295037
$814$	0	0
$815$	−1.56158e10	−1.01044
$816$	0	0
$817$	3.75682e10	2.41015
$818$	0	0
$819$	1.22702e10	0.780471
$820$	0	0
$821$	2.71353e10	1.71133	0.855666	−	0.517528i	$-0.173148\pi$
$821$	2.71353e10	1.71133	0.855666	+	0.517528i	$0.173148\pi$
$822$	0	0
$823$	2.01740e10	1.26152	0.630758	−	0.775980i	$-0.282744\pi$
$823$	2.01740e10	1.26152	0.630758	+	0.775980i	$0.282744\pi$
$824$	0	0
$825$	1.30558e9	0.0809494
$826$	0	0
$827$	2.66064e10	1.63575	0.817875	−	0.575396i	$-0.195152\pi$
$827$	2.66064e10	1.63575	0.817875	+	0.575396i	$0.195152\pi$
$828$	0	0
$829$	−8.75290e9	−0.533594	−0.266797	−	0.963753i	$-0.585965\pi$
$829$	−8.75290e9	−0.533594	−0.266797	+	0.963753i	$0.585965\pi$
$830$	0	0
$831$	6.65255e9	0.402147
$832$	0	0
$833$	1.92321e10	1.15284
$834$	0	0
$835$	−1.36244e10	−0.809871
$836$	0	0
$837$	4.85887e9	0.286415
$838$	0	0
$839$	1.75179e9	0.102404	0.0512018	−	0.998688i	$-0.483695\pi$
$839$	1.75179e9	0.102404	0.0512018	+	0.998688i	$0.483695\pi$
$840$	0	0
$841$	3.62754e10	2.10294
$842$	0	0
$843$	9.06754e9	0.521306
$844$	0	0
$845$	−2.82151e10	−1.60873
$846$	0	0
$847$	−1.71386e10	−0.969135
$848$	0	0
$849$	1.00922e10	0.565991
$850$	0	0
$851$	−1.47590e10	−0.820926
$852$	0	0
$853$	−5.65193e9	−0.311799	−0.155900	−	0.987773i	$-0.549828\pi$
$853$	−5.65193e9	−0.311799	−0.155900	+	0.987773i	$0.549828\pi$
$854$	0	0
$855$	−6.81466e9	−0.372875
$856$	0	0
$857$	−7.32497e9	−0.397533	−0.198766	−	0.980047i	$-0.563694\pi$
$857$	−7.32497e9	−0.397533	−0.198766	+	0.980047i	$0.563694\pi$
$858$	0	0
$859$	−1.93541e10	−1.04183	−0.520914	−	0.853609i	$-0.674409\pi$
$859$	−1.93541e10	−1.04183	−0.520914	+	0.853609i	$0.674409\pi$
$860$	0	0
$861$	2.08953e10	1.11567
$862$	0	0
$863$	1.55088e10	0.821371	0.410685	−	0.911777i	$-0.365289\pi$
$863$	1.55088e10	0.821371	0.410685	+	0.911777i	$0.365289\pi$
$864$	0	0
$865$	−2.59975e9	−0.136576
$866$	0	0
$867$	−6.57599e9	−0.342685
$868$	0	0
$869$	1.02832e10	0.531566
$870$	0	0
$871$	−1.45423e10	−0.745711
$872$	0	0
$873$	5.50604e9	0.280085
$874$	0	0
$875$	2.96964e10	1.49856
$876$	0	0
$877$	2.13613e10	1.06937	0.534686	−	0.845051i	$-0.320430\pi$
$877$	2.13613e10	1.06937	0.534686	+	0.845051i	$0.320430\pi$
$878$	0	0
$879$	−1.12267e10	−0.557559
$880$	0	0
$881$	4.16334e9	0.205129	0.102564	−	0.994726i	$-0.467295\pi$
$881$	4.16334e9	0.205129	0.102564	+	0.994726i	$0.467295\pi$
$882$	0	0
$883$	−1.68226e10	−0.822301	−0.411151	−	0.911567i	$-0.634873\pi$
$883$	−1.68226e10	−0.822301	−0.411151	+	0.911567i	$0.634873\pi$
$884$	0	0
$885$	−1.39460e9	−0.0676312
$886$	0	0
$887$	2.68120e9	0.129002	0.0645010	−	0.997918i	$-0.479454\pi$
$887$	2.68120e9	0.129002	0.0645010	+	0.997918i	$0.479454\pi$
$888$	0	0
$889$	−1.20433e10	−0.574895
$890$	0	0
$891$	1.28358e9	0.0607924
$892$	0	0
$893$	1.39305e10	0.654618
$894$	0	0
$895$	−3.69283e8	−0.0172178
$896$	0	0
$897$	1.79721e10	0.831431
$898$	0	0
$899$	−5.71115e10	−2.62159
$900$	0	0
$901$	−3.57827e10	−1.62981
$902$	0	0
$903$	−3.28323e10	−1.48386
$904$	0	0
$905$	−3.04014e10	−1.36340
$906$	0	0
$907$	2.89904e9	0.129011	0.0645057	−	0.997917i	$-0.479453\pi$
$907$	2.89904e9	0.129011	0.0645057	+	0.997917i	$0.479453\pi$
$908$	0	0
$909$	−5.11818e9	−0.226017
$910$	0	0
$911$	−1.65643e10	−0.725868	−0.362934	−	0.931815i	$-0.618225\pi$
$911$	−1.65643e10	−0.725868	−0.362934	+	0.931815i	$0.618225\pi$
$912$	0	0
$913$	1.37610e10	0.598416
$914$	0	0
$915$	−1.14466e10	−0.493971
$916$	0	0
$917$	−3.15847e10	−1.35265
$918$	0	0
$919$	6.44040e9	0.273721	0.136861	−	0.990590i	$-0.456299\pi$
$919$	6.44040e9	0.273721	0.136861	+	0.990590i	$0.456299\pi$
$920$	0	0
$921$	−5.91995e9	−0.249695
$922$	0	0
$923$	1.38106e10	0.578103
$924$	0	0
$925$	−5.95234e9	−0.247282
$926$	0	0
$927$	1.34051e10	0.552701
$928$	0	0
$929$	3.45986e10	1.41580	0.707902	−	0.706311i	$-0.249642\pi$
$929$	3.45986e10	1.41580	0.707902	+	0.706311i	$0.249642\pi$
$930$	0	0
$931$	2.91666e10	1.18457
$932$	0	0
$933$	5.76786e9	0.232503
$934$	0	0
$935$	−1.48876e10	−0.595641
$936$	0	0
$937$	−3.81853e10	−1.51638	−0.758190	−	0.652034i	$-0.773916\pi$
$937$	−3.81853e10	−1.51638	−0.758190	+	0.652034i	$0.773916\pi$
$938$	0	0
$939$	−2.21282e10	−0.872201
$940$	0	0
$941$	6.28907e9	0.246049	0.123025	−	0.992404i	$-0.460741\pi$
$941$	6.28907e9	0.246049	0.123025	+	0.992404i	$0.460741\pi$
$942$	0	0
$943$	3.06054e10	1.18852
$944$	0	0
$945$	5.95560e9	0.229569
$946$	0	0
$947$	−2.30506e10	−0.881977	−0.440988	−	0.897513i	$-0.645372\pi$
$947$	−2.30506e10	−0.881977	−0.440988	+	0.897513i	$0.645372\pi$
$948$	0	0
$949$	1.10251e10	0.418746
$950$	0	0
$951$	1.16497e10	0.439221
$952$	0	0
$953$	1.46553e10	0.548493	0.274246	−	0.961659i	$-0.411572\pi$
$953$	1.46553e10	0.548493	0.274246	+	0.961659i	$0.411572\pi$
$954$	0	0
$955$	−3.40214e10	−1.26398
$956$	0	0
$957$	−1.50872e10	−0.556439
$958$	0	0
$959$	−5.13238e10	−1.87912
$960$	0	0
$961$	3.34254e10	1.21491
$962$	0	0
$963$	−1.03049e10	−0.371837
$964$	0	0
$965$	2.77867e10	0.995385
$966$	0	0
$967$	2.23044e9	0.0793226	0.0396613	−	0.999213i	$-0.487372\pi$
$967$	2.23044e9	0.0793226	0.0396613	+	0.999213i	$0.487372\pi$
$968$	0	0
$969$	−2.67750e10	−0.945357
$970$	0	0
$971$	−3.25778e10	−1.14197	−0.570984	−	0.820961i	$-0.693438\pi$
$971$	−3.25778e10	−1.14197	−0.570984	+	0.820961i	$0.693438\pi$
$972$	0	0
$973$	−4.94421e10	−1.72069
$974$	0	0
$975$	7.24820e9	0.250446
$976$	0	0
$977$	−3.10973e10	−1.06682	−0.533411	−	0.845856i	$-0.679090\pi$
$977$	−3.10973e10	−1.06682	−0.533411	+	0.845856i	$0.679090\pi$
$978$	0	0
$979$	−1.58767e10	−0.540780
$980$	0	0
$981$	3.37731e9	0.114217
$982$	0	0
$983$	−3.12824e9	−0.105042	−0.0525210	−	0.998620i	$-0.516726\pi$
$983$	−3.12824e9	−0.105042	−0.0525210	+	0.998620i	$0.516726\pi$
$984$	0	0
$985$	7.77630e9	0.259266
$986$	0	0
$987$	−1.21744e10	−0.403031
$988$	0	0
$989$	−4.80895e10	−1.58075
$990$	0	0
$991$	4.75891e10	1.55328	0.776639	−	0.629946i	$-0.216923\pi$
$991$	4.75891e10	1.55328	0.776639	+	0.629946i	$0.216923\pi$
$992$	0	0
$993$	−2.21983e9	−0.0719444
$994$	0	0
$995$	−1.32502e10	−0.426423
$996$	0	0
$997$	−1.10819e10	−0.354144	−0.177072	−	0.984198i	$-0.556662\pi$
$997$	−1.10819e10	−0.354144	−0.177072	+	0.984198i	$0.556662\pi$
$998$	0	0
$999$	−5.85204e9	−0.185707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.e.1.1	✓	2
4.3	odd	2		384.8.a.g.1.1	yes	2
8.3	odd	2		384.8.a.f.1.2	yes	2
8.5	even	2		384.8.a.h.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.e.1.1	✓	2	1.1	even	1	trivial
384.8.a.f.1.2	yes	2	8.3	odd	2
384.8.a.g.1.1	yes	2	4.3	odd	2
384.8.a.h.1.2	yes	2	8.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -241.049 q^{5} +1255.25 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} -241.049 q^{5} +1255.25 q^{7} +729.000 q^{9} +2415.27 q^{11} +13408.9 q^{13} +6508.32 q^{15} +25571.4 q^{17} +38780.3 q^{19} -33891.6 q^{21} -49641.1 q^{23} -20020.4 q^{25} -19683.0 q^{27} +231355. q^{29} -246856. q^{31} -65212.4 q^{33} -302576. q^{35} +297314. q^{37} -362041. q^{39} -616533. q^{41} +968744. q^{43} -175725. q^{45} +359217. q^{47} +752097. q^{49} -690426. q^{51} -1.39933e6 q^{53} -582199. q^{55} -1.04707e6 q^{57} -214279. q^{59} -1.75876e6 q^{61} +915074. q^{63} -3.23221e6 q^{65} -1.08452e6 q^{67} +1.34031e6 q^{69} +1.02995e6 q^{71} +822221. q^{73} +540550. q^{75} +3.03176e6 q^{77} +4.25755e6 q^{79} +531441. q^{81} +5.69750e6 q^{83} -6.16395e6 q^{85} -6.24659e6 q^{87} -6.57345e6 q^{89} +1.68315e7 q^{91} +6.66512e6 q^{93} -9.34796e6 q^{95} +7.55287e6 q^{97} +1.76073e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} - 176 q^{5} + 980 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q - 54 * q^3 - 176 * q^5 + 980 * q^7 + 1458 * q^9	\( 2 q - 54 q^{3} - 176 q^{5} + 980 q^{7} + 1458 q^{9} - 3128 q^{11} + 8452 q^{13} + 4752 q^{15} + 5228 q^{17} + 87968 q^{19} - 26460 q^{21} - 104792 q^{23} - 93914 q^{25} - 39366 q^{27} + 98760 q^{29} + 1860 q^{31} + 84456 q^{33} - 320480 q^{35} + 455660 q^{37} - 228204 q^{39} - 205188 q^{41} + 803088 q^{43} - 128304 q^{45} - 522488 q^{47} + 4314 q^{49} - 141156 q^{51} - 2468376 q^{53} - 942784 q^{55} - 2375136 q^{57} + 989288 q^{59} - 2482292 q^{61} + 714420 q^{63} - 3554656 q^{65} + 434008 q^{67} + 2829384 q^{69} + 1554840 q^{71} + 279244 q^{73} + 2535678 q^{75} + 4557520 q^{77} + 1015396 q^{79} + 1062882 q^{81} + 2899560 q^{83} - 7487264 q^{85} - 2666520 q^{87} - 8115868 q^{89} + 18195880 q^{91} - 50220 q^{93} - 6148352 q^{95} - 2553676 q^{97} - 2280312 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 - 176 * q^5 + 980 * q^7 + 1458 * q^9 - 3128 * q^11 + 8452 * q^13 + 4752 * q^15 + 5228 * q^17 + 87968 * q^19 - 26460 * q^21 - 104792 * q^23 - 93914 * q^25 - 39366 * q^27 + 98760 * q^29 + 1860 * q^31 + 84456 * q^33 - 320480 * q^35 + 455660 * q^37 - 228204 * q^39 - 205188 * q^41 + 803088 * q^43 - 128304 * q^45 - 522488 * q^47 + 4314 * q^49 - 141156 * q^51 - 2468376 * q^53 - 942784 * q^55 - 2375136 * q^57 + 989288 * q^59 - 2482292 * q^61 + 714420 * q^63 - 3554656 * q^65 + 434008 * q^67 + 2829384 * q^69 + 1554840 * q^71 + 279244 * q^73 + 2535678 * q^75 + 4557520 * q^77 + 1015396 * q^79 + 1062882 * q^81 + 2899560 * q^83 - 7487264 * q^85 - 2666520 * q^87 - 8115868 * q^89 + 18195880 * q^91 - 50220 * q^93 - 6148352 * q^95 - 2553676 * q^97 - 2280312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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