LMFDB - Embedded newform 384.8.a.t.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,8,Mod(1,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,108,0,336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.955849786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 620x^{2} - 700x + 83625 $ x^4 - 620x^2 - 700x + 83625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$12.7912$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} +514.363 q^{5} -1600.84 q^{7} +729.000 q^{9} +6913.89 q^{11} +1238.70 q^{13} +13887.8 q^{15} +27118.0 q^{17} -44175.4 q^{19} -43222.7 q^{21} +15503.8 q^{23} +186445. q^{25} +19683.0 q^{27} -94577.1 q^{29} -97999.9 q^{31} +186675. q^{33} -823415. q^{35} +556688. q^{37} +33444.8 q^{39} -203369. q^{41} +59526.7 q^{43} +374971. q^{45} +698902. q^{47} +1.73915e6 q^{49} +732187. q^{51} -91646.2 q^{53} +3.55625e6 q^{55} -1.19274e6 q^{57} +1.47653e6 q^{59} +42411.3 q^{61} -1.16701e6 q^{63} +637140. q^{65} +722597. q^{67} +418603. q^{69} -941020. q^{71} -2.12119e6 q^{73} +5.03401e6 q^{75} -1.10680e7 q^{77} -2.91670e6 q^{79} +531441. q^{81} +5.42934e6 q^{83} +1.39485e7 q^{85} -2.55358e6 q^{87} +3.26957e6 q^{89} -1.98296e6 q^{91} -2.64600e6 q^{93} -2.27222e7 q^{95} +3.55685e6 q^{97} +5.04023e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} + 336 q^{5} - 680 q^{7} + 2916 q^{9} + 3856 q^{11} + 10680 q^{13} + 9072 q^{15} + 26232 q^{17} - 15456 q^{19} - 18360 q^{21} + 11312 q^{23} + 159052 q^{25} + 78732 q^{27} - 1856 q^{29} - 71752 q^{31}+ \cdots + 2811024 q^{99}+O(q^{100}) $ 4 * q + 108 * q^3 + 336 * q^5 - 680 * q^7 + 2916 * q^9 + 3856 * q^11 + 10680 * q^13 + 9072 * q^15 + 26232 * q^17 - 15456 * q^19 - 18360 * q^21 + 11312 * q^23 + 159052 * q^25 + 78732 * q^27 - 1856 * q^29 - 71752 * q^31 + 104112 * q^33 - 179040 * q^35 - 180088 * q^37 + 288360 * q^39 + 11224 * q^41 + 66688 * q^43 + 244944 * q^45 + 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 - 864576 * q^53 + 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 - 1901176 * q^61 - 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 + 305424 * q^69 + 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 - 8979488 * q^77 - 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 + 12122592 * q^85 - 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 - 1937304 * q^93 - 22523904 * q^95 + 2180520 * q^97 + 2811024 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	514.363	1.84024	0.920121	−	0.391634i	$-0.128090\pi$
$5$	514.363	1.84024	0.920121	+	0.391634i	$0.128090\pi$
$6$	0	0
$7$	−1600.84	−1.76403	−0.882014	−	0.471223i	$-0.843813\pi$
$7$	−1600.84	−1.76403	−0.882014	+	0.471223i	$0.843813\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	6913.89	1.56620	0.783101	−	0.621894i	$-0.213637\pi$
$11$	6913.89	1.56620	0.783101	+	0.621894i	$0.213637\pi$
$12$	0	0
$13$	1238.70	0.156373	0.0781867	−	0.996939i	$-0.475087\pi$
$13$	1238.70	0.156373	0.0781867	+	0.996939i	$0.475087\pi$
$14$	0	0
$15$	13887.8	1.06246
$16$	0	0
$17$	27118.0	1.33871	0.669356	−	0.742942i	$-0.266570\pi$
$17$	27118.0	1.33871	0.669356	+	0.742942i	$0.266570\pi$
$18$	0	0
$19$	−44175.4	−1.47755	−0.738776	−	0.673951i	$-0.764596\pi$
$19$	−44175.4	−1.47755	−0.738776	+	0.673951i	$0.764596\pi$
$20$	0	0
$21$	−43222.7	−1.01846
$22$	0	0
$23$	15503.8	0.265700	0.132850	−	0.991136i	$-0.457587\pi$
$23$	15503.8	0.265700	0.132850	+	0.991136i	$0.457587\pi$
$24$	0	0
$25$	186445.	2.38649
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−94577.1	−0.720100	−0.360050	−	0.932933i	$-0.617240\pi$
$29$	−94577.1	−0.720100	−0.360050	+	0.932933i	$0.617240\pi$
$30$	0	0
$31$	−97999.9	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$31$	−97999.9	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$32$	0	0
$33$	186675.	0.904248
$34$	0	0
$35$	−823415.	−3.24624
$36$	0	0
$37$	556688.	1.80678	0.903390	−	0.428819i	$-0.141070\pi$
$37$	556688.	1.80678	0.903390	+	0.428819i	$0.141070\pi$
$38$	0	0
$39$	33444.8	0.0902822
$40$	0	0
$41$	−203369.	−0.460831	−0.230416	−	0.973092i	$-0.574009\pi$
$41$	−203369.	−0.460831	−0.230416	+	0.973092i	$0.574009\pi$
$42$	0	0
$43$	59526.7	0.114175	0.0570876	−	0.998369i	$-0.481819\pi$
$43$	59526.7	0.114175	0.0570876	+	0.998369i	$0.481819\pi$
$44$	0	0
$45$	374971.	0.613414
$46$	0	0
$47$	698902.	0.981914	0.490957	−	0.871184i	$-0.336647\pi$
$47$	698902.	0.981914	0.490957	+	0.871184i	$0.336647\pi$
$48$	0	0
$49$	1.73915e6	2.11179
$50$	0	0
$51$	732187.	0.772905
$52$	0	0
$53$	−91646.2	−0.0845569	−0.0422784	−	0.999106i	$-0.513462\pi$
$53$	−91646.2	−0.0845569	−0.0422784	+	0.999106i	$0.513462\pi$
$54$	0	0
$55$	3.55625e6	2.88219
$56$	0	0
$57$	−1.19274e6	−0.853065
$58$	0	0
$59$	1.47653e6	0.935965	0.467982	−	0.883738i	$-0.344981\pi$
$59$	1.47653e6	0.935965	0.467982	+	0.883738i	$0.344981\pi$
$60$	0	0
$61$	42411.3	0.0239236	0.0119618	−	0.999928i	$-0.496192\pi$
$61$	42411.3	0.0239236	0.0119618	+	0.999928i	$0.496192\pi$
$62$	0	0
$63$	−1.16701e6	−0.588009
$64$	0	0
$65$	637140.	0.287765
$66$	0	0
$67$	722597.	0.293518	0.146759	−	0.989172i	$-0.453116\pi$
$67$	722597.	0.293518	0.146759	+	0.989172i	$0.453116\pi$
$68$	0	0
$69$	418603.	0.153402
$70$	0	0
$71$	−941020.	−0.312029	−0.156014	−	0.987755i	$-0.549865\pi$
$71$	−941020.	−0.312029	−0.156014	+	0.987755i	$0.549865\pi$
$72$	0	0
$73$	−2.12119e6	−0.638190	−0.319095	−	0.947723i	$-0.603379\pi$
$73$	−2.12119e6	−0.638190	−0.319095	+	0.947723i	$0.603379\pi$
$74$	0	0
$75$	5.03401e6	1.37784
$76$	0	0
$77$	−1.10680e7	−2.76283
$78$	0	0
$79$	−2.91670e6	−0.665576	−0.332788	−	0.943002i	$-0.607989\pi$
$79$	−2.91670e6	−0.665576	−0.332788	+	0.943002i	$0.607989\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.42934e6	1.04225	0.521127	−	0.853479i	$-0.325512\pi$
$83$	5.42934e6	1.04225	0.521127	+	0.853479i	$0.325512\pi$
$84$	0	0
$85$	1.39485e7	2.46355
$86$	0	0
$87$	−2.55358e6	−0.415750
$88$	0	0
$89$	3.26957e6	0.491615	0.245807	−	0.969319i	$-0.420947\pi$
$89$	3.26957e6	0.491615	0.245807	+	0.969319i	$0.420947\pi$
$90$	0	0
$91$	−1.98296e6	−0.275847
$92$	0	0
$93$	−2.64600e6	−0.341114
$94$	0	0
$95$	−2.27222e7	−2.71905
$96$	0	0
$97$	3.55685e6	0.395698	0.197849	−	0.980232i	$-0.436604\pi$
$97$	3.55685e6	0.395698	0.197849	+	0.980232i	$0.436604\pi$
$98$	0	0
$99$	5.04023e6	0.522068
$100$	0	0
$101$	−1.61387e7	−1.55864	−0.779318	−	0.626629i	$-0.784434\pi$
$101$	−1.61387e7	−1.55864	−0.779318	+	0.626629i	$0.784434\pi$
$102$	0	0
$103$	6.41643e6	0.578579	0.289290	−	0.957242i	$-0.406581\pi$
$103$	6.41643e6	0.578579	0.289290	+	0.957242i	$0.406581\pi$
$104$	0	0
$105$	−2.22322e7	−1.87422
$106$	0	0
$107$	−1.80279e7	−1.42266	−0.711332	−	0.702857i	$-0.751908\pi$
$107$	−1.80279e7	−1.42266	−0.711332	+	0.702857i	$0.751908\pi$
$108$	0	0
$109$	−1.37005e7	−1.01331	−0.506657	−	0.862148i	$-0.669119\pi$
$109$	−1.37005e7	−1.01331	−0.506657	+	0.862148i	$0.669119\pi$
$110$	0	0
$111$	1.50306e7	1.04315
$112$	0	0
$113$	2.57644e7	1.67975	0.839877	−	0.542777i	$-0.182627\pi$
$113$	2.57644e7	1.67975	0.839877	+	0.542777i	$0.182627\pi$
$114$	0	0
$115$	7.97460e6	0.488952
$116$	0	0
$117$	903009.	0.0521245
$118$	0	0
$119$	−4.34117e7	−2.36152
$120$	0	0
$121$	2.83147e7	1.45299
$122$	0	0
$123$	−5.49097e6	−0.266061
$124$	0	0
$125$	5.57157e7	2.55148
$126$	0	0
$127$	2.92424e7	1.26678	0.633388	−	0.773834i	$-0.281664\pi$
$127$	2.92424e7	1.26678	0.633388	+	0.773834i	$0.281664\pi$
$128$	0	0
$129$	1.60722e6	0.0659191
$130$	0	0
$131$	8.70400e6	0.338275	0.169137	−	0.985592i	$-0.445902\pi$
$131$	8.70400e6	0.338275	0.169137	+	0.985592i	$0.445902\pi$
$132$	0	0
$133$	7.07179e7	2.60644
$134$	0	0
$135$	1.01242e7	0.354155
$136$	0	0
$137$	5.05997e7	1.68123	0.840613	−	0.541636i	$-0.182195\pi$
$137$	5.05997e7	1.68123	0.840613	+	0.541636i	$0.182195\pi$
$138$	0	0
$139$	3.36806e7	1.06372	0.531861	−	0.846831i	$-0.321493\pi$
$139$	3.36806e7	1.06372	0.531861	+	0.846831i	$0.321493\pi$
$140$	0	0
$141$	1.88703e7	0.566909
$142$	0	0
$143$	8.56421e6	0.244913
$144$	0	0
$145$	−4.86470e7	−1.32516
$146$	0	0
$147$	4.69572e7	1.21925
$148$	0	0
$149$	−1.74744e7	−0.432763	−0.216381	−	0.976309i	$-0.569425\pi$
$149$	−1.74744e7	−0.432763	−0.216381	+	0.976309i	$0.569425\pi$
$150$	0	0
$151$	4.95302e6	0.117071	0.0585357	−	0.998285i	$-0.481357\pi$
$151$	4.95302e6	0.117071	0.0585357	+	0.998285i	$0.481357\pi$
$152$	0	0
$153$	1.97690e7	0.446237
$154$	0	0
$155$	−5.04076e7	−1.08726
$156$	0	0
$157$	6.63928e7	1.36922	0.684608	−	0.728911i	$-0.259973\pi$
$157$	6.63928e7	1.36922	0.684608	+	0.728911i	$0.259973\pi$
$158$	0	0
$159$	−2.47445e6	−0.0488189
$160$	0	0
$161$	−2.48192e7	−0.468702
$162$	0	0
$163$	−2.86225e7	−0.517668	−0.258834	−	0.965922i	$-0.583338\pi$
$163$	−2.86225e7	−0.517668	−0.258834	+	0.965922i	$0.583338\pi$
$164$	0	0
$165$	9.60188e7	1.66404
$166$	0	0
$167$	−6.60559e6	−0.109750	−0.0548749	−	0.998493i	$-0.517476\pi$
$167$	−6.60559e6	−0.109750	−0.0548749	+	0.998493i	$0.517476\pi$
$168$	0	0
$169$	−6.12141e7	−0.975547
$170$	0	0
$171$	−3.22039e7	−0.492517
$172$	0	0
$173$	6.19023e7	0.908961	0.454481	−	0.890757i	$-0.349825\pi$
$173$	6.19023e7	0.908961	0.454481	+	0.890757i	$0.349825\pi$
$174$	0	0
$175$	−2.98469e8	−4.20984
$176$	0	0
$177$	3.98662e7	0.540379
$178$	0	0
$179$	5.07847e7	0.661832	0.330916	−	0.943660i	$-0.392642\pi$
$179$	5.07847e7	0.661832	0.330916	+	0.943660i	$0.392642\pi$
$180$	0	0
$181$	−1.01132e8	−1.26770	−0.633848	−	0.773458i	$-0.718525\pi$
$181$	−1.01132e8	−1.26770	−0.633848	+	0.773458i	$0.718525\pi$
$182$	0	0
$183$	1.14510e6	0.0138123
$184$	0	0
$185$	2.86340e8	3.32492
$186$	0	0
$187$	1.87491e8	2.09669
$188$	0	0
$189$	−3.15094e7	−0.339487
$190$	0	0
$191$	−8.98727e7	−0.933277	−0.466639	−	0.884448i	$-0.654535\pi$
$191$	−8.98727e7	−0.933277	−0.466639	+	0.884448i	$0.654535\pi$
$192$	0	0
$193$	1.09530e8	1.09669	0.548343	−	0.836253i	$-0.315259\pi$
$193$	1.09530e8	1.09669	0.548343	+	0.836253i	$0.315259\pi$
$194$	0	0
$195$	1.72028e7	0.166141
$196$	0	0
$197$	−1.38205e8	−1.28793	−0.643967	−	0.765054i	$-0.722712\pi$
$197$	−1.38205e8	−1.28793	−0.643967	+	0.765054i	$0.722712\pi$
$198$	0	0
$199$	−1.13605e7	−0.102191	−0.0510955	−	0.998694i	$-0.516271\pi$
$199$	−1.13605e7	−0.102191	−0.0510955	+	0.998694i	$0.516271\pi$
$200$	0	0
$201$	1.95101e7	0.169463
$202$	0	0
$203$	1.51403e8	1.27028
$204$	0	0
$205$	−1.04606e8	−0.848041
$206$	0	0
$207$	1.13023e7	0.0885666
$208$	0	0
$209$	−3.05424e8	−2.31415
$210$	0	0
$211$	1.70994e8	1.25312	0.626560	−	0.779373i	$-0.284462\pi$
$211$	1.70994e8	1.25312	0.626560	+	0.779373i	$0.284462\pi$
$212$	0	0
$213$	−2.54075e7	−0.180150
$214$	0	0
$215$	3.06183e7	0.210110
$216$	0	0
$217$	1.56882e8	1.04223
$218$	0	0
$219$	−5.72722e7	−0.368459
$220$	0	0
$221$	3.35910e7	0.209339
$222$	0	0
$223$	−1.91458e8	−1.15613	−0.578064	−	0.815991i	$-0.696192\pi$
$223$	−1.91458e8	−1.15613	−0.578064	+	0.815991i	$0.696192\pi$
$224$	0	0
$225$	1.35918e8	0.795498
$226$	0	0
$227$	6.12196e7	0.347376	0.173688	−	0.984801i	$-0.444432\pi$
$227$	6.12196e7	0.347376	0.173688	+	0.984801i	$0.444432\pi$
$228$	0	0
$229$	2.19050e8	1.20537	0.602683	−	0.797980i	$-0.294098\pi$
$229$	2.19050e8	1.20537	0.602683	+	0.797980i	$0.294098\pi$
$230$	0	0
$231$	−2.98837e8	−1.59512
$232$	0	0
$233$	−9.49591e7	−0.491803	−0.245901	−	0.969295i	$-0.579084\pi$
$233$	−9.49591e7	−0.491803	−0.245901	+	0.969295i	$0.579084\pi$
$234$	0	0
$235$	3.59489e8	1.80696
$236$	0	0
$237$	−7.87510e7	−0.384270
$238$	0	0
$239$	3.17475e8	1.50424	0.752120	−	0.659026i	$-0.229031\pi$
$239$	3.17475e8	1.50424	0.752120	+	0.659026i	$0.229031\pi$
$240$	0	0
$241$	1.25861e8	0.579202	0.289601	−	0.957147i	$-0.406477\pi$
$241$	1.25861e8	0.579202	0.289601	+	0.957147i	$0.406477\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	8.94557e8	3.88621
$246$	0	0
$247$	−5.47199e7	−0.231050
$248$	0	0
$249$	1.46592e8	0.601746
$250$	0	0
$251$	1.46376e8	0.584268	0.292134	−	0.956377i	$-0.405635\pi$
$251$	1.46376e8	0.584268	0.292134	+	0.956377i	$0.405635\pi$
$252$	0	0
$253$	1.07192e8	0.416140
$254$	0	0
$255$	3.76610e8	1.42233
$256$	0	0
$257$	2.66104e8	0.977880	0.488940	−	0.872317i	$-0.337384\pi$
$257$	2.66104e8	0.977880	0.488940	+	0.872317i	$0.337384\pi$
$258$	0	0
$259$	−8.91169e8	−3.18721
$260$	0	0
$261$	−6.89467e7	−0.240033
$262$	0	0
$263$	−2.53538e8	−0.859404	−0.429702	−	0.902971i	$-0.641381\pi$
$263$	−2.53538e8	−0.859404	−0.429702	+	0.902971i	$0.641381\pi$
$264$	0	0
$265$	−4.71395e7	−0.155605
$266$	0	0
$267$	8.82783e7	0.283834
$268$	0	0
$269$	3.85972e7	0.120899	0.0604496	−	0.998171i	$-0.480747\pi$
$269$	3.85972e7	0.120899	0.0604496	+	0.998171i	$0.480747\pi$
$270$	0	0
$271$	1.07533e8	0.328208	0.164104	−	0.986443i	$-0.447527\pi$
$271$	1.07533e8	0.328208	0.164104	+	0.986443i	$0.447527\pi$
$272$	0	0
$273$	−5.35398e7	−0.159260
$274$	0	0
$275$	1.28906e9	3.73773
$276$	0	0
$277$	4.32477e8	1.22260	0.611300	−	0.791399i	$-0.290647\pi$
$277$	4.32477e8	1.22260	0.611300	+	0.791399i	$0.290647\pi$
$278$	0	0
$279$	−7.14420e7	−0.196942
$280$	0	0
$281$	3.52335e8	0.947290	0.473645	−	0.880716i	$-0.342938\pi$
$281$	3.52335e8	0.947290	0.473645	+	0.880716i	$0.342938\pi$
$282$	0	0
$283$	−7.31369e8	−1.91816	−0.959078	−	0.283140i	$-0.908624\pi$
$283$	−7.31369e8	−1.91816	−0.959078	+	0.283140i	$0.908624\pi$
$284$	0	0
$285$	−6.13500e8	−1.56985
$286$	0	0
$287$	3.25562e8	0.812919
$288$	0	0
$289$	3.25049e8	0.792148
$290$	0	0
$291$	9.60349e7	0.228457
$292$	0	0
$293$	2.95868e8	0.687165	0.343582	−	0.939123i	$-0.388360\pi$
$293$	2.95868e8	0.687165	0.343582	+	0.939123i	$0.388360\pi$
$294$	0	0
$295$	7.59472e8	1.72240
$296$	0	0
$297$	1.36086e8	0.301416
$298$	0	0
$299$	1.92045e7	0.0415484
$300$	0	0
$301$	−9.52928e7	−0.201408
$302$	0	0
$303$	−4.35746e8	−0.899879
$304$	0	0
$305$	2.18148e7	0.0440252
$306$	0	0
$307$	−2.83413e8	−0.559031	−0.279516	−	0.960141i	$-0.590174\pi$
$307$	−2.83413e8	−0.559031	−0.279516	+	0.960141i	$0.590174\pi$
$308$	0	0
$309$	1.73243e8	0.334043
$310$	0	0
$311$	−4.72364e8	−0.890461	−0.445231	−	0.895416i	$-0.646878\pi$
$311$	−4.72364e8	−0.890461	−0.445231	+	0.895416i	$0.646878\pi$
$312$	0	0
$313$	4.16123e8	0.767037	0.383518	−	0.923533i	$-0.374712\pi$
$313$	4.16123e8	0.767037	0.383518	+	0.923533i	$0.374712\pi$
$314$	0	0
$315$	−6.00269e8	−1.08208
$316$	0	0
$317$	−1.07692e9	−1.89878	−0.949391	−	0.314098i	$-0.898298\pi$
$317$	−1.07692e9	−1.89878	−0.949391	+	0.314098i	$0.898298\pi$
$318$	0	0
$319$	−6.53895e8	−1.12782
$320$	0	0
$321$	−4.86754e8	−0.821375
$322$	0	0
$323$	−1.19795e9	−1.97802
$324$	0	0
$325$	2.30948e8	0.373184
$326$	0	0
$327$	−3.69914e8	−0.585037
$328$	0	0
$329$	−1.11883e9	−1.73212
$330$	0	0
$331$	2.78980e8	0.422839	0.211419	−	0.977395i	$-0.432191\pi$
$331$	2.78980e8	0.422839	0.211419	+	0.977395i	$0.432191\pi$
$332$	0	0
$333$	4.05825e8	0.602260
$334$	0	0
$335$	3.71678e8	0.540144
$336$	0	0
$337$	−1.04948e9	−1.49372	−0.746860	−	0.664981i	$-0.768440\pi$
$337$	−1.04948e9	−1.49372	−0.746860	+	0.664981i	$0.768440\pi$
$338$	0	0
$339$	6.95639e8	0.969806
$340$	0	0
$341$	−6.77561e8	−0.925354
$342$	0	0
$343$	−1.46575e9	−1.96124
$344$	0	0
$345$	2.15314e8	0.282297
$346$	0	0
$347$	9.01438e7	0.115820	0.0579099	−	0.998322i	$-0.481556\pi$
$347$	9.01438e7	0.115820	0.0579099	+	0.998322i	$0.481556\pi$
$348$	0	0
$349$	−2.29227e8	−0.288654	−0.144327	−	0.989530i	$-0.546102\pi$
$349$	−2.29227e8	−0.288654	−0.144327	+	0.989530i	$0.546102\pi$
$350$	0	0
$351$	2.43813e7	0.0300941
$352$	0	0
$353$	−5.11895e8	−0.619398	−0.309699	−	0.950835i	$-0.600228\pi$
$353$	−5.11895e8	−0.619398	−0.309699	+	0.950835i	$0.600228\pi$
$354$	0	0
$355$	−4.84026e8	−0.574208
$356$	0	0
$357$	−1.17212e9	−1.36343
$358$	0	0
$359$	2.60775e8	0.297464	0.148732	−	0.988878i	$-0.452481\pi$
$359$	2.60775e8	0.297464	0.148732	+	0.988878i	$0.452481\pi$
$360$	0	0
$361$	1.05759e9	1.18316
$362$	0	0
$363$	7.64497e8	0.838885
$364$	0	0
$365$	−1.09106e9	−1.17443
$366$	0	0
$367$	−3.30282e8	−0.348782	−0.174391	−	0.984677i	$-0.555796\pi$
$367$	−3.30282e8	−0.348782	−0.174391	+	0.984677i	$0.555796\pi$
$368$	0	0
$369$	−1.48256e8	−0.153610
$370$	0	0
$371$	1.46711e8	0.149161
$372$	0	0
$373$	1.96509e9	1.96066	0.980331	−	0.197362i	$-0.0632374\pi$
$373$	1.96509e9	1.96066	0.980331	+	0.197362i	$0.0632374\pi$
$374$	0	0
$375$	1.50432e9	1.47310
$376$	0	0
$377$	−1.17152e8	−0.112605
$378$	0	0
$379$	5.56639e8	0.525214	0.262607	−	0.964903i	$-0.415418\pi$
$379$	5.56639e8	0.525214	0.262607	+	0.964903i	$0.415418\pi$
$380$	0	0
$381$	7.89545e8	0.731374
$382$	0	0
$383$	−1.33569e9	−1.21482	−0.607409	−	0.794389i	$-0.707791\pi$
$383$	−1.33569e9	−1.21482	−0.607409	+	0.794389i	$0.707791\pi$
$384$	0	0
$385$	−5.69300e9	−5.08427
$386$	0	0
$387$	4.33949e7	0.0380584
$388$	0	0
$389$	2.32242e8	0.200041	0.100020	−	0.994985i	$-0.468109\pi$
$389$	2.32242e8	0.200041	0.100020	+	0.994985i	$0.468109\pi$
$390$	0	0
$391$	4.20433e8	0.355695
$392$	0	0
$393$	2.35008e8	0.195303
$394$	0	0
$395$	−1.50025e9	−1.22482
$396$	0	0
$397$	−1.01386e9	−0.813228	−0.406614	−	0.913600i	$-0.633291\pi$
$397$	−1.01386e9	−0.813228	−0.406614	+	0.913600i	$0.633291\pi$
$398$	0	0
$399$	1.90938e9	1.50483
$400$	0	0
$401$	−3.08859e8	−0.239197	−0.119598	−	0.992822i	$-0.538161\pi$
$401$	−3.08859e8	−0.239197	−0.119598	+	0.992822i	$0.538161\pi$
$402$	0	0
$403$	−1.21392e8	−0.0923895
$404$	0	0
$405$	2.73354e8	0.204471
$406$	0	0
$407$	3.84888e9	2.82979
$408$	0	0
$409$	−6.47968e8	−0.468298	−0.234149	−	0.972201i	$-0.575230\pi$
$409$	−6.47968e8	−0.468298	−0.234149	+	0.972201i	$0.575230\pi$
$410$	0	0
$411$	1.36619e9	0.970656
$412$	0	0
$413$	−2.36369e9	−1.65107
$414$	0	0
$415$	2.79265e9	1.91800
$416$	0	0
$417$	9.09377e8	0.614141
$418$	0	0
$419$	−2.39227e9	−1.58877	−0.794386	−	0.607413i	$-0.792207\pi$
$419$	−2.39227e9	−1.58877	−0.794386	+	0.607413i	$0.792207\pi$
$420$	0	0
$421$	−1.55058e9	−1.01276	−0.506381	−	0.862310i	$-0.669017\pi$
$421$	−1.55058e9	−1.01276	−0.506381	+	0.862310i	$0.669017\pi$
$422$	0	0
$423$	5.09499e8	0.327305
$424$	0	0
$425$	5.05601e9	3.19482
$426$	0	0
$427$	−6.78938e7	−0.0422019
$428$	0	0
$429$	2.31234e8	0.141400
$430$	0	0
$431$	1.74806e8	0.105168	0.0525842	−	0.998616i	$-0.483254\pi$
$431$	1.74806e8	0.105168	0.0525842	+	0.998616i	$0.483254\pi$
$432$	0	0
$433$	−1.27805e9	−0.756553	−0.378277	−	0.925693i	$-0.623483\pi$
$433$	−1.27805e9	−0.756553	−0.378277	+	0.925693i	$0.623483\pi$
$434$	0	0
$435$	−1.31347e9	−0.765081
$436$	0	0
$437$	−6.84888e8	−0.392586
$438$	0	0
$439$	4.91676e8	0.277366	0.138683	−	0.990337i	$-0.455713\pi$
$439$	4.91676e8	0.277366	0.138683	+	0.990337i	$0.455713\pi$
$440$	0	0
$441$	1.26784e9	0.703932
$442$	0	0
$443$	−1.84094e9	−1.00607	−0.503034	−	0.864267i	$-0.667783\pi$
$443$	−1.84094e9	−1.00607	−0.503034	+	0.864267i	$0.667783\pi$
$444$	0	0
$445$	1.68174e9	0.904691
$446$	0	0
$447$	−4.71808e8	−0.249856
$448$	0	0
$449$	9.00232e8	0.469345	0.234672	−	0.972075i	$-0.424598\pi$
$449$	9.00232e8	0.469345	0.234672	+	0.972075i	$0.424598\pi$
$450$	0	0
$451$	−1.40607e9	−0.721755
$452$	0	0
$453$	1.33732e8	0.0675912
$454$	0	0
$455$	−1.01996e9	−0.507626
$456$	0	0
$457$	4.88220e7	0.0239281	0.0119641	−	0.999928i	$-0.496192\pi$
$457$	4.88220e7	0.0239281	0.0119641	+	0.999928i	$0.496192\pi$
$458$	0	0
$459$	5.33764e8	0.257635
$460$	0	0
$461$	−2.72400e9	−1.29495	−0.647476	−	0.762086i	$-0.724175\pi$
$461$	−2.72400e9	−1.29495	−0.647476	+	0.762086i	$0.724175\pi$
$462$	0	0
$463$	−3.09504e9	−1.44921	−0.724607	−	0.689162i	$-0.757979\pi$
$463$	−3.09504e9	−1.44921	−0.724607	+	0.689162i	$0.757979\pi$
$464$	0	0
$465$	−1.36100e9	−0.627732
$466$	0	0
$467$	8.17657e8	0.371503	0.185751	−	0.982597i	$-0.440528\pi$
$467$	8.17657e8	0.371503	0.185751	+	0.982597i	$0.440528\pi$
$468$	0	0
$469$	−1.15676e9	−0.517774
$470$	0	0
$471$	1.79261e9	0.790518
$472$	0	0
$473$	4.11561e8	0.178822
$474$	0	0
$475$	−8.23627e9	−3.52617
$476$	0	0
$477$	−6.68101e7	−0.0281856
$478$	0	0
$479$	1.81239e9	0.753491	0.376746	−	0.926317i	$-0.377043\pi$
$479$	1.81239e9	0.753491	0.376746	+	0.926317i	$0.377043\pi$
$480$	0	0
$481$	6.89567e8	0.282533
$482$	0	0
$483$	−6.70118e8	−0.270605
$484$	0	0
$485$	1.82951e9	0.728181
$486$	0	0
$487$	−1.61893e9	−0.635150	−0.317575	−	0.948233i	$-0.602869\pi$
$487$	−1.61893e9	−0.635150	−0.317575	+	0.948233i	$0.602869\pi$
$488$	0	0
$489$	−7.72808e8	−0.298876
$490$	0	0
$491$	−8.08518e8	−0.308251	−0.154125	−	0.988051i	$-0.549256\pi$
$491$	−8.08518e8	−0.308251	−0.154125	+	0.988051i	$0.549256\pi$
$492$	0	0
$493$	−2.56474e9	−0.964006
$494$	0	0
$495$	2.59251e9	0.960731
$496$	0	0
$497$	1.50642e9	0.550427
$498$	0	0
$499$	5.29358e9	1.90721	0.953603	−	0.301066i	$-0.0973426\pi$
$499$	5.29358e9	1.90721	0.953603	+	0.301066i	$0.0973426\pi$
$500$	0	0
$501$	−1.78351e8	−0.0633641
$502$	0	0
$503$	1.78698e9	0.626083	0.313042	−	0.949739i	$-0.398652\pi$
$503$	1.78698e9	0.626083	0.313042	+	0.949739i	$0.398652\pi$
$504$	0	0
$505$	−8.30117e9	−2.86827
$506$	0	0
$507$	−1.65278e9	−0.563233
$508$	0	0
$509$	−1.41019e9	−0.473987	−0.236994	−	0.971511i	$-0.576162\pi$
$509$	−1.41019e9	−0.473987	−0.236994	+	0.971511i	$0.576162\pi$
$510$	0	0
$511$	3.39570e9	1.12579
$512$	0	0
$513$	−8.69504e8	−0.284355
$514$	0	0
$515$	3.30037e9	1.06473
$516$	0	0
$517$	4.83213e9	1.53788
$518$	0	0
$519$	1.67136e9	0.524789
$520$	0	0
$521$	1.59145e9	0.493015	0.246507	−	0.969141i	$-0.420717\pi$
$521$	1.59145e9	0.493015	0.246507	+	0.969141i	$0.420717\pi$
$522$	0	0
$523$	−5.05199e9	−1.54421	−0.772105	−	0.635495i	$-0.780796\pi$
$523$	−5.05199e9	−1.54421	−0.772105	+	0.635495i	$0.780796\pi$
$524$	0	0
$525$	−8.05865e9	−2.43055
$526$	0	0
$527$	−2.65757e9	−0.790946
$528$	0	0
$529$	−3.16446e9	−0.929404
$530$	0	0
$531$	1.07639e9	0.311988
$532$	0	0
$533$	−2.51913e8	−0.0720618
$534$	0	0
$535$	−9.27290e9	−2.61805
$536$	0	0
$537$	1.37119e9	0.382109
$538$	0	0
$539$	1.20243e10	3.30750
$540$	0	0
$541$	−7.88845e8	−0.214191	−0.107095	−	0.994249i	$-0.534155\pi$
$541$	−7.88845e8	−0.214191	−0.107095	+	0.994249i	$0.534155\pi$
$542$	0	0
$543$	−2.73057e9	−0.731904
$544$	0	0
$545$	−7.04704e9	−1.86474
$546$	0	0
$547$	3.89546e9	1.01766	0.508830	−	0.860867i	$-0.330078\pi$
$547$	3.89546e9	1.01766	0.508830	+	0.860867i	$0.330078\pi$
$548$	0	0
$549$	3.09178e7	0.00797454
$550$	0	0
$551$	4.17798e9	1.06399
$552$	0	0
$553$	4.66918e9	1.17409
$554$	0	0
$555$	7.73117e9	1.91964
$556$	0	0
$557$	−2.61110e9	−0.640223	−0.320111	−	0.947380i	$-0.603720\pi$
$557$	−2.61110e9	−0.640223	−0.320111	+	0.947380i	$0.603720\pi$
$558$	0	0
$559$	7.37354e7	0.0178540
$560$	0	0
$561$	5.06226e9	1.21053
$562$	0	0
$563$	1.68140e9	0.397093	0.198546	−	0.980091i	$-0.436378\pi$
$563$	1.68140e9	0.397093	0.198546	+	0.980091i	$0.436378\pi$
$564$	0	0
$565$	1.32523e10	3.09115
$566$	0	0
$567$	−8.50753e8	−0.196003
$568$	0	0
$569$	−4.99371e9	−1.13640	−0.568199	−	0.822891i	$-0.692360\pi$
$569$	−4.99371e9	−1.13640	−0.568199	+	0.822891i	$0.692360\pi$
$570$	0	0
$571$	−2.41336e9	−0.542495	−0.271248	−	0.962510i	$-0.587436\pi$
$571$	−2.41336e9	−0.542495	−0.271248	+	0.962510i	$0.587436\pi$
$572$	0	0
$573$	−2.42656e9	−0.538828
$574$	0	0
$575$	2.89061e9	0.634091
$576$	0	0
$577$	1.88306e9	0.408082	0.204041	−	0.978962i	$-0.434592\pi$
$577$	1.88306e9	0.408082	0.204041	+	0.978962i	$0.434592\pi$
$578$	0	0
$579$	2.95731e9	0.633173
$580$	0	0
$581$	−8.69151e9	−1.83857
$582$	0	0
$583$	−6.33632e8	−0.132433
$584$	0	0
$585$	4.64475e8	0.0959217
$586$	0	0
$587$	−7.88029e9	−1.60808	−0.804042	−	0.594573i	$-0.797321\pi$
$587$	−7.88029e9	−1.60808	−0.804042	+	0.594573i	$0.797321\pi$
$588$	0	0
$589$	4.32919e9	0.872977
$590$	0	0
$591$	−3.73155e9	−0.743589
$592$	0	0
$593$	−7.58062e9	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$593$	−7.58062e9	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$594$	0	0
$595$	−2.23294e10	−4.34578
$596$	0	0
$597$	−3.06734e8	−0.0590000
$598$	0	0
$599$	2.83376e9	0.538727	0.269363	−	0.963039i	$-0.413187\pi$
$599$	2.83376e9	0.538727	0.269363	+	0.963039i	$0.413187\pi$
$600$	0	0
$601$	−7.75307e9	−1.45684	−0.728422	−	0.685129i	$-0.759746\pi$
$601$	−7.75307e9	−1.45684	−0.728422	+	0.685129i	$0.759746\pi$
$602$	0	0
$603$	5.26773e8	0.0978393
$604$	0	0
$605$	1.45640e10	2.67386
$606$	0	0
$607$	−3.33242e9	−0.604782	−0.302391	−	0.953184i	$-0.597785\pi$
$607$	−3.33242e9	−0.604782	−0.302391	+	0.953184i	$0.597785\pi$
$608$	0	0
$609$	4.08788e9	0.733395
$610$	0	0
$611$	8.65727e8	0.153545
$612$	0	0
$613$	−3.90408e9	−0.684553	−0.342276	−	0.939599i	$-0.611198\pi$
$613$	−3.90408e9	−0.684553	−0.342276	+	0.939599i	$0.611198\pi$
$614$	0	0
$615$	−2.82435e9	−0.489617
$616$	0	0
$617$	−7.25879e9	−1.24413	−0.622066	−	0.782965i	$-0.713706\pi$
$617$	−7.25879e9	−1.24413	−0.622066	+	0.782965i	$0.713706\pi$
$618$	0	0
$619$	5.14839e9	0.872477	0.436238	−	0.899831i	$-0.356310\pi$
$619$	5.14839e9	0.872477	0.436238	+	0.899831i	$0.356310\pi$
$620$	0	0
$621$	3.05162e8	0.0511340
$622$	0	0
$623$	−5.23406e9	−0.867222
$624$	0	0
$625$	1.40921e10	2.30885
$626$	0	0
$627$	−8.24644e9	−1.33607
$628$	0	0
$629$	1.50963e10	2.41876
$630$	0	0
$631$	−1.08002e10	−1.71131	−0.855656	−	0.517544i	$-0.826846\pi$
$631$	−1.08002e10	−1.71131	−0.855656	+	0.517544i	$0.826846\pi$
$632$	0	0
$633$	4.61684e9	0.723489
$634$	0	0
$635$	1.50412e10	2.33118
$636$	0	0
$637$	2.15428e9	0.330229
$638$	0	0
$639$	−6.86003e8	−0.104010
$640$	0	0
$641$	1.02491e10	1.53703	0.768515	−	0.639832i	$-0.220996\pi$
$641$	1.02491e10	1.53703	0.768515	+	0.639832i	$0.220996\pi$
$642$	0	0
$643$	−2.34758e9	−0.348243	−0.174122	−	0.984724i	$-0.555709\pi$
$643$	−2.34758e9	−0.348243	−0.174122	+	0.984724i	$0.555709\pi$
$644$	0	0
$645$	8.26695e8	0.121307
$646$	0	0
$647$	2.17174e9	0.315241	0.157620	−	0.987500i	$-0.449618\pi$
$647$	2.17174e9	0.315241	0.157620	+	0.987500i	$0.449618\pi$
$648$	0	0
$649$	1.02085e10	1.46591
$650$	0	0
$651$	4.23583e9	0.601734
$652$	0	0
$653$	−5.34174e9	−0.750734	−0.375367	−	0.926876i	$-0.622483\pi$
$653$	−5.34174e9	−0.750734	−0.375367	+	0.926876i	$0.622483\pi$
$654$	0	0
$655$	4.47702e9	0.622507
$656$	0	0
$657$	−1.54635e9	−0.212730
$658$	0	0
$659$	4.65528e9	0.633646	0.316823	−	0.948485i	$-0.397384\pi$
$659$	4.65528e9	0.633646	0.316823	+	0.948485i	$0.397384\pi$
$660$	0	0
$661$	2.32068e8	0.0312544	0.0156272	−	0.999878i	$-0.495026\pi$
$661$	2.32068e8	0.0312544	0.0156272	+	0.999878i	$0.495026\pi$
$662$	0	0
$663$	9.06957e8	0.120862
$664$	0	0
$665$	3.63747e10	4.79649
$666$	0	0
$667$	−1.46631e9	−0.191331
$668$	0	0
$669$	−5.16936e9	−0.667491
$670$	0	0
$671$	2.93227e8	0.0374692
$672$	0	0
$673$	−8.04519e9	−1.01738	−0.508691	−	0.860949i	$-0.669870\pi$
$673$	−8.04519e9	−1.01738	−0.508691	+	0.860949i	$0.669870\pi$
$674$	0	0
$675$	3.66979e9	0.459281
$676$	0	0
$677$	−9.36064e9	−1.15943	−0.579716	−	0.814819i	$-0.696836\pi$
$677$	−9.36064e9	−1.15943	−0.579716	+	0.814819i	$0.696836\pi$
$678$	0	0
$679$	−5.69395e9	−0.698023
$680$	0	0
$681$	1.65293e9	0.200558
$682$	0	0
$683$	−1.27346e10	−1.52938	−0.764688	−	0.644401i	$-0.777107\pi$
$683$	−1.27346e10	−1.52938	−0.764688	+	0.644401i	$0.777107\pi$
$684$	0	0
$685$	2.60266e10	3.09386
$686$	0	0
$687$	5.91435e9	0.695919
$688$	0	0
$689$	−1.13522e8	−0.0132224
$690$	0	0
$691$	−5.00174e9	−0.576698	−0.288349	−	0.957525i	$-0.593106\pi$
$691$	−5.00174e9	−0.576698	−0.288349	+	0.957525i	$0.593106\pi$
$692$	0	0
$693$	−8.06861e9	−0.920942
$694$	0	0
$695$	1.73241e10	1.95751
$696$	0	0
$697$	−5.51497e9	−0.616920
$698$	0	0
$699$	−2.56390e9	−0.283942
$700$	0	0
$701$	3.22052e9	0.353112	0.176556	−	0.984291i	$-0.443504\pi$
$701$	3.22052e9	0.353112	0.176556	+	0.984291i	$0.443504\pi$
$702$	0	0
$703$	−2.45919e10	−2.66961
$704$	0	0
$705$	9.70622e9	1.04325
$706$	0	0
$707$	2.58356e10	2.74948
$708$	0	0
$709$	1.13062e10	1.19140	0.595698	−	0.803209i	$-0.296876\pi$
$709$	1.13062e10	1.19140	0.595698	+	0.803209i	$0.296876\pi$
$710$	0	0
$711$	−2.12628e9	−0.221859
$712$	0	0
$713$	−1.51937e9	−0.156983
$714$	0	0
$715$	4.40511e9	0.450698
$716$	0	0
$717$	8.57183e9	0.868473
$718$	0	0
$719$	−1.69246e9	−0.169811	−0.0849056	−	0.996389i	$-0.527059\pi$
$719$	−1.69246e9	−0.169811	−0.0849056	+	0.996389i	$0.527059\pi$
$720$	0	0
$721$	−1.02717e10	−1.02063
$722$	0	0
$723$	3.39824e9	0.334403
$724$	0	0
$725$	−1.76334e10	−1.71851
$726$	0	0
$727$	−5.86766e8	−0.0566362	−0.0283181	−	0.999599i	$-0.509015\pi$
$727$	−5.86766e8	−0.0566362	−0.0283181	+	0.999599i	$0.509015\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.61425e9	0.152848
$732$	0	0
$733$	−6.03168e9	−0.565685	−0.282842	−	0.959166i	$-0.591277\pi$
$733$	−6.03168e9	−0.565685	−0.282842	+	0.959166i	$0.591277\pi$
$734$	0	0
$735$	2.41530e10	2.24371
$736$	0	0
$737$	4.99596e9	0.459708
$738$	0	0
$739$	1.95957e10	1.78610	0.893050	−	0.449958i	$-0.148561\pi$
$739$	1.95957e10	1.78610	0.893050	+	0.449958i	$0.148561\pi$
$740$	0	0
$741$	−1.47744e9	−0.133397
$742$	0	0
$743$	−1.08121e10	−0.967051	−0.483525	−	0.875330i	$-0.660644\pi$
$743$	−1.08121e10	−0.967051	−0.483525	+	0.875330i	$0.660644\pi$
$744$	0	0
$745$	−8.98818e9	−0.796388
$746$	0	0
$747$	3.95799e9	0.347418
$748$	0	0
$749$	2.88598e10	2.50962
$750$	0	0
$751$	−6.78087e9	−0.584179	−0.292090	−	0.956391i	$-0.594351\pi$
$751$	−6.78087e9	−0.584179	−0.292090	+	0.956391i	$0.594351\pi$
$752$	0	0
$753$	3.95215e9	0.337327
$754$	0	0
$755$	2.54765e9	0.215440
$756$	0	0
$757$	−1.21107e10	−1.01469	−0.507346	−	0.861742i	$-0.669373\pi$
$757$	−1.21107e10	−1.01469	−0.507346	+	0.861742i	$0.669373\pi$
$758$	0	0
$759$	2.89418e9	0.240259
$760$	0	0
$761$	−1.02887e10	−0.846282	−0.423141	−	0.906064i	$-0.639072\pi$
$761$	−1.02887e10	−0.846282	−0.423141	+	0.906064i	$0.639072\pi$
$762$	0	0
$763$	2.19324e10	1.78751
$764$	0	0
$765$	1.01685e10	0.821185
$766$	0	0
$767$	1.82897e9	0.146360
$768$	0	0
$769$	−2.16478e10	−1.71661	−0.858304	−	0.513142i	$-0.828482\pi$
$769$	−2.16478e10	−1.71661	−0.858304	+	0.513142i	$0.828482\pi$
$770$	0	0
$771$	7.18481e9	0.564579
$772$	0	0
$773$	1.75000e9	0.136273	0.0681363	−	0.997676i	$-0.478295\pi$
$773$	1.75000e9	0.136273	0.0681363	+	0.997676i	$0.478295\pi$
$774$	0	0
$775$	−1.82716e10	−1.41000
$776$	0	0
$777$	−2.40616e10	−1.84014
$778$	0	0
$779$	8.98392e9	0.680902
$780$	0	0
$781$	−6.50611e9	−0.488700
$782$	0	0
$783$	−1.86156e9	−0.138583
$784$	0	0
$785$	3.41500e10	2.51969
$786$	0	0
$787$	−6.02838e9	−0.440848	−0.220424	−	0.975404i	$-0.570744\pi$
$787$	−6.02838e9	−0.440848	−0.220424	+	0.975404i	$0.570744\pi$
$788$	0	0
$789$	−6.84552e9	−0.496177
$790$	0	0
$791$	−4.12448e10	−2.96313
$792$	0	0
$793$	5.25347e7	0.00374102
$794$	0	0
$795$	−1.27277e9	−0.0898387
$796$	0	0
$797$	−1.39431e10	−0.975561	−0.487781	−	0.872966i	$-0.662193\pi$
$797$	−1.39431e10	−0.975561	−0.487781	+	0.872966i	$0.662193\pi$
$798$	0	0
$799$	1.89528e10	1.31450
$800$	0	0
$801$	2.38351e9	0.163872
$802$	0	0
$803$	−1.46657e10	−0.999536
$804$	0	0
$805$	−1.27661e10	−0.862526
$806$	0	0
$807$	1.04213e9	0.0698012
$808$	0	0
$809$	−2.16864e10	−1.44002	−0.720010	−	0.693964i	$-0.755863\pi$
$809$	−2.16864e10	−1.44002	−0.720010	+	0.693964i	$0.755863\pi$
$810$	0	0
$811$	−2.69392e8	−0.0177342	−0.00886712	−	0.999961i	$-0.502823\pi$
$811$	−2.69392e8	−0.0177342	−0.00886712	+	0.999961i	$0.502823\pi$
$812$	0	0
$813$	2.90339e9	0.189491
$814$	0	0
$815$	−1.47224e10	−0.952634
$816$	0	0
$817$	−2.62961e9	−0.168700
$818$	0	0
$819$	−1.44558e9	−0.0919490
$820$	0	0
$821$	7.91181e9	0.498971	0.249485	−	0.968379i	$-0.419739\pi$
$821$	7.91181e9	0.498971	0.249485	+	0.968379i	$0.419739\pi$
$822$	0	0
$823$	2.93009e10	1.83224	0.916119	−	0.400907i	$-0.131305\pi$
$823$	2.93009e10	1.83224	0.916119	+	0.400907i	$0.131305\pi$
$824$	0	0
$825$	3.48046e10	2.15798
$826$	0	0
$827$	−1.07918e10	−0.663475	−0.331738	−	0.943372i	$-0.607635\pi$
$827$	−1.07918e10	−0.663475	−0.331738	+	0.943372i	$0.607635\pi$
$828$	0	0
$829$	6.46317e9	0.394008	0.197004	−	0.980403i	$-0.436879\pi$
$829$	6.46317e9	0.394008	0.197004	+	0.980403i	$0.436879\pi$
$830$	0	0
$831$	1.16769e10	0.705868
$832$	0	0
$833$	4.71624e10	2.82708
$834$	0	0
$835$	−3.39767e9	−0.201966
$836$	0	0
$837$	−1.92893e9	−0.113705
$838$	0	0
$839$	1.06936e9	0.0625111	0.0312556	−	0.999511i	$-0.490049\pi$
$839$	1.06936e9	0.0625111	0.0312556	+	0.999511i	$0.490049\pi$
$840$	0	0
$841$	−8.30506e9	−0.481456
$842$	0	0
$843$	9.51303e9	0.546918
$844$	0	0
$845$	−3.14863e10	−1.79524
$846$	0	0
$847$	−4.53274e10	−2.56312
$848$	0	0
$849$	−1.97470e10	−1.10745
$850$	0	0
$851$	8.63079e9	0.480062
$852$	0	0
$853$	1.95676e10	1.07948	0.539742	−	0.841830i	$-0.318522\pi$
$853$	1.95676e10	1.07948	0.539742	+	0.841830i	$0.318522\pi$
$854$	0	0
$855$	−1.65645e10	−0.906352
$856$	0	0
$857$	1.19700e10	0.649621	0.324811	−	0.945779i	$-0.394699\pi$
$857$	1.19700e10	0.649621	0.324811	+	0.945779i	$0.394699\pi$
$858$	0	0
$859$	−5.21229e9	−0.280577	−0.140289	−	0.990111i	$-0.544803\pi$
$859$	−5.21229e9	−0.280577	−0.140289	+	0.990111i	$0.544803\pi$
$860$	0	0
$861$	8.79018e9	0.469339
$862$	0	0
$863$	1.00199e10	0.530670	0.265335	−	0.964156i	$-0.414517\pi$
$863$	1.00199e10	0.530670	0.265335	+	0.964156i	$0.414517\pi$
$864$	0	0
$865$	3.18403e10	1.67271
$866$	0	0
$867$	8.77632e9	0.457347
$868$	0	0
$869$	−2.01658e10	−1.04243
$870$	0	0
$871$	8.95078e8	0.0458984
$872$	0	0
$873$	2.59294e9	0.131899
$874$	0	0
$875$	−8.91921e10	−4.50089
$876$	0	0
$877$	9.64156e9	0.482668	0.241334	−	0.970442i	$-0.422415\pi$
$877$	9.64156e9	0.482668	0.241334	+	0.970442i	$0.422415\pi$
$878$	0	0
$879$	7.98843e9	0.396735
$880$	0	0
$881$	−8.98936e9	−0.442908	−0.221454	−	0.975171i	$-0.571080\pi$
$881$	−8.98936e9	−0.442908	−0.221454	+	0.975171i	$0.571080\pi$
$882$	0	0
$883$	−3.63832e10	−1.77844	−0.889218	−	0.457484i	$-0.848751\pi$
$883$	−3.63832e10	−1.77844	−0.889218	+	0.457484i	$0.848751\pi$
$884$	0	0
$885$	2.05057e10	0.994429
$886$	0	0
$887$	3.75420e10	1.80628	0.903140	−	0.429345i	$-0.141256\pi$
$887$	3.75420e10	1.80628	0.903140	+	0.429345i	$0.141256\pi$
$888$	0	0
$889$	−4.68125e10	−2.23463
$890$	0	0
$891$	3.67432e9	0.174023
$892$	0	0
$893$	−3.08743e10	−1.45083
$894$	0	0
$895$	2.61218e10	1.21793
$896$	0	0
$897$	5.18522e8	0.0239880
$898$	0	0
$899$	9.26855e9	0.425454
$900$	0	0
$901$	−2.48526e9	−0.113197
$902$	0	0
$903$	−2.57291e9	−0.116283
$904$	0	0
$905$	−5.20188e10	−2.33287
$906$	0	0
$907$	3.50289e10	1.55884	0.779419	−	0.626502i	$-0.215514\pi$
$907$	3.50289e10	1.55884	0.779419	+	0.626502i	$0.215514\pi$
$908$	0	0
$909$	−1.17651e10	−0.519545
$910$	0	0
$911$	−4.82258e9	−0.211332	−0.105666	−	0.994402i	$-0.533697\pi$
$911$	−4.82258e9	−0.211332	−0.105666	+	0.994402i	$0.533697\pi$
$912$	0	0
$913$	3.75378e10	1.63238
$914$	0	0
$915$	5.89000e8	0.0254180
$916$	0	0
$917$	−1.39337e10	−0.596726
$918$	0	0
$919$	3.67472e10	1.56178	0.780890	−	0.624668i	$-0.214766\pi$
$919$	3.67472e10	1.56178	0.780890	+	0.624668i	$0.214766\pi$
$920$	0	0
$921$	−7.65216e9	−0.322757
$922$	0	0
$923$	−1.16564e9	−0.0487930
$924$	0	0
$925$	1.03791e11	4.31187
$926$	0	0
$927$	4.67757e9	0.192860
$928$	0	0
$929$	4.07264e8	0.0166656	0.00833280	−	0.999965i	$-0.497348\pi$
$929$	4.07264e8	0.0166656	0.00833280	+	0.999965i	$0.497348\pi$
$930$	0	0
$931$	−7.68278e10	−3.12029
$932$	0	0
$933$	−1.27538e10	−0.514108
$934$	0	0
$935$	9.64386e10	3.85842
$936$	0	0
$937$	3.52666e10	1.40047	0.700237	−	0.713910i	$-0.253077\pi$
$937$	3.52666e10	1.40047	0.700237	+	0.713910i	$0.253077\pi$
$938$	0	0
$939$	1.12353e10	0.442849
$940$	0	0
$941$	2.39832e10	0.938305	0.469153	−	0.883117i	$-0.344559\pi$
$941$	2.39832e10	0.938305	0.469153	+	0.883117i	$0.344559\pi$
$942$	0	0
$943$	−3.15300e9	−0.122443
$944$	0	0
$945$	−1.62073e10	−0.624739
$946$	0	0
$947$	4.78013e10	1.82901	0.914503	−	0.404580i	$-0.132582\pi$
$947$	4.78013e10	1.82901	0.914503	+	0.404580i	$0.132582\pi$
$948$	0	0
$949$	−2.62751e9	−0.0997960
$950$	0	0
$951$	−2.90768e10	−1.09626
$952$	0	0
$953$	−2.95772e10	−1.10696	−0.553480	−	0.832863i	$-0.686700\pi$
$953$	−2.95772e10	−1.10696	−0.553480	+	0.832863i	$0.686700\pi$
$954$	0	0
$955$	−4.62272e10	−1.71746
$956$	0	0
$957$	−1.76552e10	−0.651149
$958$	0	0
$959$	−8.10022e10	−2.96573
$960$	0	0
$961$	−1.79086e10	−0.650924
$962$	0	0
$963$	−1.31423e10	−0.474221
$964$	0	0
$965$	5.63382e10	2.01817
$966$	0	0
$967$	−4.15404e10	−1.47733	−0.738665	−	0.674072i	$-0.764544\pi$
$967$	−4.15404e10	−1.47733	−0.738665	+	0.674072i	$0.764544\pi$
$968$	0	0
$969$	−3.23446e10	−1.14201
$970$	0	0
$971$	−1.82050e8	−0.00638151	−0.00319076	−	0.999995i	$-0.501016\pi$
$971$	−1.82050e8	−0.00638151	−0.00319076	+	0.999995i	$0.501016\pi$
$972$	0	0
$973$	−5.39174e10	−1.87644
$974$	0	0
$975$	6.23560e9	0.215458
$976$	0	0
$977$	−2.91815e10	−1.00110	−0.500549	−	0.865708i	$-0.666869\pi$
$977$	−2.91815e10	−1.00110	−0.500549	+	0.865708i	$0.666869\pi$
$978$	0	0
$979$	2.26054e10	0.769969
$980$	0	0
$981$	−9.98767e9	−0.337771
$982$	0	0
$983$	1.38398e10	0.464722	0.232361	−	0.972630i	$-0.425355\pi$
$983$	1.38398e10	0.464722	0.232361	+	0.972630i	$0.425355\pi$
$984$	0	0
$985$	−7.10878e10	−2.37011
$986$	0	0
$987$	−3.02085e10	−1.00004
$988$	0	0
$989$	9.22891e8	0.0303364
$990$	0	0
$991$	4.71569e10	1.53917	0.769586	−	0.638543i	$-0.220463\pi$
$991$	4.71569e10	1.53917	0.769586	+	0.638543i	$0.220463\pi$
$992$	0	0
$993$	7.53246e9	0.244126
$994$	0	0
$995$	−5.84344e9	−0.188056
$996$	0	0
$997$	−2.57126e10	−0.821698	−0.410849	−	0.911703i	$-0.634768\pi$
$997$	−2.57126e10	−0.821698	−0.410849	+	0.911703i	$0.634768\pi$
$998$	0	0
$999$	1.09573e10	0.347715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.t.1.4	yes	4
4.3	odd	2		384.8.a.p.1.4	yes	4
8.3	odd	2		384.8.a.q.1.1	yes	4
8.5	even	2		384.8.a.m.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.m.1.1	✓	4	8.5	even	2
384.8.a.p.1.4	yes	4	4.3	odd	2
384.8.a.q.1.1	yes	4	8.3	odd	2
384.8.a.t.1.4	yes	4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +514.363 q^{5} -1600.84 q^{7} +729.000 q^{9} +6913.89 q^{11} +1238.70 q^{13} +13887.8 q^{15} +27118.0 q^{17} -44175.4 q^{19} -43222.7 q^{21} +15503.8 q^{23} +186445. q^{25} +19683.0 q^{27} -94577.1 q^{29} -97999.9 q^{31} +186675. q^{33} -823415. q^{35} +556688. q^{37} +33444.8 q^{39} -203369. q^{41} +59526.7 q^{43} +374971. q^{45} +698902. q^{47} +1.73915e6 q^{49} +732187. q^{51} -91646.2 q^{53} +3.55625e6 q^{55} -1.19274e6 q^{57} +1.47653e6 q^{59} +42411.3 q^{61} -1.16701e6 q^{63} +637140. q^{65} +722597. q^{67} +418603. q^{69} -941020. q^{71} -2.12119e6 q^{73} +5.03401e6 q^{75} -1.10680e7 q^{77} -2.91670e6 q^{79} +531441. q^{81} +5.42934e6 q^{83} +1.39485e7 q^{85} -2.55358e6 q^{87} +3.26957e6 q^{89} -1.98296e6 q^{91} -2.64600e6 q^{93} -2.27222e7 q^{95} +3.55685e6 q^{97} +5.04023e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} + 336 q^{5} - 680 q^{7} + 2916 q^{9} + 3856 q^{11} + 10680 q^{13} + 9072 q^{15} + 26232 q^{17} - 15456 q^{19} - 18360 q^{21} + 11312 q^{23} + 159052 q^{25} + 78732 q^{27} - 1856 q^{29} - 71752 q^{31}+ \cdots + 2811024 q^{99}+O(q^{100}) \) 4 * q + 108 * q^3 + 336 * q^5 - 680 * q^7 + 2916 * q^9 + 3856 * q^11 + 10680 * q^13 + 9072 * q^15 + 26232 * q^17 - 15456 * q^19 - 18360 * q^21 + 11312 * q^23 + 159052 * q^25 + 78732 * q^27 - 1856 * q^29 - 71752 * q^31 + 104112 * q^33 - 179040 * q^35 - 180088 * q^37 + 288360 * q^39 + 11224 * q^41 + 66688 * q^43 + 244944 * q^45 + 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 - 864576 * q^53 + 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 - 1901176 * q^61 - 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 + 305424 * q^69 + 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 - 8979488 * q^77 - 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 + 12122592 * q^85 - 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 - 1937304 * q^93 - 22523904 * q^95 + 2180520 * q^97 + 2811024 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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