LMFDB - Embedded newform 384.8.a.q.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:	$4$
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			1.1
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)

$f(q)$	$=$	$q+27.0000 q^{3} -514.363 q^{5} +1600.84 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -514.363 q^{5} +1600.84 q^{7} +729.000 q^{9} +6913.89 q^{11} -1238.70 q^{13} -13887.8 q^{15} +27118.0 q^{17} -44175.4 q^{19} +43222.7 q^{21} -15503.8 q^{23} +186445. q^{25} +19683.0 q^{27} +94577.1 q^{29} +97999.9 q^{31} +186675. q^{33} -823415. q^{35} -556688. q^{37} -33444.8 q^{39} -203369. q^{41} +59526.7 q^{43} -374971. q^{45} -698902. q^{47} +1.73915e6 q^{49} +732187. q^{51} +91646.2 q^{53} -3.55625e6 q^{55} -1.19274e6 q^{57} +1.47653e6 q^{59} -42411.3 q^{61} +1.16701e6 q^{63} +637140. q^{65} +722597. q^{67} -418603. q^{69} +941020. q^{71} -2.12119e6 q^{73} +5.03401e6 q^{75} +1.10680e7 q^{77} +2.91670e6 q^{79} +531441. q^{81} +5.42934e6 q^{83} -1.39485e7 q^{85} +2.55358e6 q^{87} +3.26957e6 q^{89} -1.98296e6 q^{91} +2.64600e6 q^{93} +2.27222e7 q^{95} +3.55685e6 q^{97} +5.04023e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	$ 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}+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

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
$5$	−514.363	−1.84024	−0.920121	−	0.391634i	$-0.871910\pi$
$5$	−514.363	−1.84024	−0.920121	+	0.391634i	$0.871910\pi$
$6$	0	0
$7$	1600.84	1.76403	0.882014	−	0.471223i	$-0.156187\pi$
$7$	1600.84	1.76403	0.882014	+	0.471223i	$0.156187\pi$
$8$	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	$-0.524913\pi$
$13$	−1238.70	−0.156373	−0.0781867	+	0.996939i	$0.524913\pi$
$14$	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	$-0.542413\pi$
$23$	−15503.8	−0.265700	−0.132850	+	0.991136i	$0.542413\pi$
$24$	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	$-0.382760\pi$
$29$	94577.1	0.720100	0.360050	+	0.932933i	$0.382760\pi$
$30$	0	0
$31$	97999.9	0.590826	0.295413	−	0.955370i	$-0.404543\pi$
$31$	97999.9	0.590826	0.295413	+	0.955370i	$0.404543\pi$
$32$	0	0
$33$	186675.	0.904248
$34$	0	0
$35$	−823415.	−3.24624
$36$	0	0
$37$	−556688.	−1.80678	−0.903390	−	0.428819i	$-0.858930\pi$
$37$	−556688.	−1.80678	−0.903390	+	0.428819i	$0.858930\pi$
$38$	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	$-0.663353\pi$
$47$	−698902.	−0.981914	−0.490957	+	0.871184i	$0.663353\pi$
$48$	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	$-0.486538\pi$
$53$	91646.2	0.0845569	0.0422784	+	0.999106i	$0.486538\pi$
$54$	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	$-0.503808\pi$
$61$	−42411.3	−0.0239236	−0.0119618	+	0.999928i	$0.503808\pi$
$62$	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	$-0.450135\pi$
$71$	941020.	0.312029	0.156014	+	0.987755i	$0.450135\pi$
$72$	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	$-0.392011\pi$
$79$	2.91670e6	0.665576	0.332788	+	0.943002i	$0.392011\pi$
$80$	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	$-0.215566\pi$
$101$	1.61387e7	1.55864	0.779318	+	0.626629i	$0.215566\pi$
$102$	0	0
$103$	−6.41643e6	−0.578579	−0.289290	−	0.957242i	$-0.593419\pi$
$103$	−6.41643e6	−0.578579	−0.289290	+	0.957242i	$0.593419\pi$
$104$	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	$-0.330881\pi$
$109$	1.37005e7	1.01331	0.506657	+	0.862148i	$0.330881\pi$
$110$	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	$-0.718336\pi$
$127$	−2.92424e7	−1.26678	−0.633388	+	0.773834i	$0.718336\pi$
$128$	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	$-0.430575\pi$
$149$	1.74744e7	0.432763	0.216381	+	0.976309i	$0.430575\pi$
$150$	0	0
$151$	−4.95302e6	−0.117071	−0.0585357	−	0.998285i	$-0.518643\pi$
$151$	−4.95302e6	−0.117071	−0.0585357	+	0.998285i	$0.518643\pi$
$152$	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	$-0.740027\pi$
$157$	−6.63928e7	−1.36922	−0.684608	+	0.728911i	$0.740027\pi$
$158$	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	$-0.482524\pi$
$167$	6.60559e6	0.109750	0.0548749	+	0.998493i	$0.482524\pi$
$168$	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	$-0.650175\pi$
$173$	−6.19023e7	−0.908961	−0.454481	+	0.890757i	$0.650175\pi$
$174$	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	$-0.281475\pi$
$181$	1.01132e8	1.26770	0.633848	+	0.773458i	$0.281475\pi$
$182$	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	$-0.345465\pi$
$191$	8.98727e7	0.933277	0.466639	+	0.884448i	$0.345465\pi$
$192$	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	$-0.277288\pi$
$197$	1.38205e8	1.28793	0.643967	+	0.765054i	$0.277288\pi$
$198$	0	0
$199$	1.13605e7	0.102191	0.0510955	−	0.998694i	$-0.483729\pi$
$199$	1.13605e7	0.102191	0.0510955	+	0.998694i	$0.483729\pi$
$200$	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	$-0.303808\pi$
$223$	1.91458e8	1.15613	0.578064	+	0.815991i	$0.303808\pi$
$224$	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	$-0.705902\pi$
$229$	−2.19050e8	−1.20537	−0.602683	+	0.797980i	$0.705902\pi$
$230$	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	$-0.770969\pi$
$239$	−3.17475e8	−1.50424	−0.752120	+	0.659026i	$0.770969\pi$
$240$	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	$-0.358619\pi$
$263$	2.53538e8	0.859404	0.429702	+	0.902971i	$0.358619\pi$
$264$	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	$-0.519253\pi$
$269$	−3.85972e7	−0.120899	−0.0604496	+	0.998171i	$0.519253\pi$
$270$	0	0
$271$	−1.07533e8	−0.328208	−0.164104	−	0.986443i	$-0.552473\pi$
$271$	−1.07533e8	−0.328208	−0.164104	+	0.986443i	$0.552473\pi$
$272$	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	$-0.709353\pi$
$277$	−4.32477e8	−1.22260	−0.611300	+	0.791399i	$0.709353\pi$
$278$	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	$-0.611640\pi$
$293$	−2.95868e8	−0.687165	−0.343582	+	0.939123i	$0.611640\pi$
$294$	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	$-0.353122\pi$
$311$	4.72364e8	0.890461	0.445231	+	0.895416i	$0.353122\pi$
$312$	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	$-0.101702\pi$
$317$	1.07692e9	1.89878	0.949391	+	0.314098i	$0.101702\pi$
$318$	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	$-0.453898\pi$
$349$	2.29227e8	0.288654	0.144327	+	0.989530i	$0.453898\pi$
$350$	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	$-0.547519\pi$
$359$	−2.60775e8	−0.297464	−0.148732	+	0.988878i	$0.547519\pi$
$360$	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	$-0.444204\pi$
$367$	3.30282e8	0.348782	0.174391	+	0.984677i	$0.444204\pi$
$368$	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	$-0.936763\pi$
$373$	−1.96509e9	−1.96066	−0.980331	+	0.197362i	$0.936763\pi$
$374$	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	$-0.292209\pi$
$383$	1.33569e9	1.21482	0.607409	+	0.794389i	$0.292209\pi$
$384$	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	$-0.531891\pi$
$389$	−2.32242e8	−0.200041	−0.100020	+	0.994985i	$0.531891\pi$
$390$	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	$-0.366709\pi$
$397$	1.01386e9	0.813228	0.406614	+	0.913600i	$0.366709\pi$
$398$	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	$-0.330983\pi$
$421$	1.55058e9	1.01276	0.506381	+	0.862310i	$0.330983\pi$
$422$	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	$-0.516746\pi$
$431$	−1.74806e8	−0.105168	−0.0525842	+	0.998616i	$0.516746\pi$
$432$	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	$-0.544287\pi$
$439$	−4.91676e8	−0.277366	−0.138683	+	0.990337i	$0.544287\pi$
$440$	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	$-0.275825\pi$
$461$	2.72400e9	1.29495	0.647476	+	0.762086i	$0.275825\pi$
$462$	0	0
$463$	3.09504e9	1.44921	0.724607	−	0.689162i	$-0.242021\pi$
$463$	3.09504e9	1.44921	0.724607	+	0.689162i	$0.242021\pi$
$464$	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	$-0.622957\pi$
$479$	−1.81239e9	−0.753491	−0.376746	+	0.926317i	$0.622957\pi$
$480$	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	$-0.397131\pi$
$487$	1.61893e9	0.635150	0.317575	+	0.948233i	$0.397131\pi$
$488$	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	$-0.601348\pi$
$503$	−1.78698e9	−0.626083	−0.313042	+	0.949739i	$0.601348\pi$
$504$	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	$-0.423838\pi$
$509$	1.41019e9	0.473987	0.236994	+	0.971511i	$0.423838\pi$
$510$	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	$-0.465845\pi$
$541$	7.88845e8	0.214191	0.107095	+	0.994249i	$0.465845\pi$
$542$	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	$-0.396280\pi$
$557$	2.61110e9	0.640223	0.320111	+	0.947380i	$0.396280\pi$
$558$	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	$-0.586813\pi$
$599$	−2.83376e9	−0.538727	−0.269363	+	0.963039i	$0.586813\pi$
$600$	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	$-0.402215\pi$
$607$	3.33242e9	0.604782	0.302391	+	0.953184i	$0.402215\pi$
$608$	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	$-0.388802\pi$
$613$	3.90408e9	0.684553	0.342276	+	0.939599i	$0.388802\pi$
$614$	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	$-0.173154\pi$
$631$	1.08002e10	1.71131	0.855656	+	0.517544i	$0.173154\pi$
$632$	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	$-0.550382\pi$
$647$	−2.17174e9	−0.315241	−0.157620	+	0.987500i	$0.550382\pi$
$648$	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	$-0.377517\pi$
$653$	5.34174e9	0.750734	0.375367	+	0.926876i	$0.377517\pi$
$654$	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	$-0.504974\pi$
$661$	−2.32068e8	−0.0312544	−0.0156272	+	0.999878i	$0.504974\pi$
$662$	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	$-0.303164\pi$
$677$	9.36064e9	1.15943	0.579716	+	0.814819i	$0.303164\pi$
$678$	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	$-0.556496\pi$
$701$	−3.22052e9	−0.353112	−0.176556	+	0.984291i	$0.556496\pi$
$702$	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	$-0.703124\pi$
$709$	−1.13062e10	−1.19140	−0.595698	+	0.803209i	$0.703124\pi$
$710$	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	$-0.472941\pi$
$719$	1.69246e9	0.169811	0.0849056	+	0.996389i	$0.472941\pi$
$720$	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	$-0.490985\pi$
$727$	5.86766e8	0.0566362	0.0283181	+	0.999599i	$0.490985\pi$
$728$	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	$-0.408723\pi$
$733$	6.03168e9	0.565685	0.282842	+	0.959166i	$0.408723\pi$
$734$	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	$-0.339356\pi$
$743$	1.08121e10	0.967051	0.483525	+	0.875330i	$0.339356\pi$
$744$	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	$-0.405649\pi$
$751$	6.78087e9	0.584179	0.292090	+	0.956391i	$0.405649\pi$
$752$	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	$-0.330627\pi$
$757$	1.21107e10	1.01469	0.507346	+	0.861742i	$0.330627\pi$
$758$	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	$-0.521705\pi$
$773$	−1.75000e9	−0.136273	−0.0681363	+	0.997676i	$0.521705\pi$
$774$	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	$-0.337807\pi$
$797$	1.39431e10	0.975561	0.487781	+	0.872966i	$0.337807\pi$
$798$	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	$-0.580261\pi$
$821$	−7.91181e9	−0.498971	−0.249485	+	0.968379i	$0.580261\pi$
$822$	0	0
$823$	−2.93009e10	−1.83224	−0.916119	−	0.400907i	$-0.868695\pi$
$823$	−2.93009e10	−1.83224	−0.916119	+	0.400907i	$0.868695\pi$
$824$	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	$-0.563121\pi$
$829$	−6.46317e9	−0.394008	−0.197004	+	0.980403i	$0.563121\pi$
$830$	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	$-0.509951\pi$
$839$	−1.06936e9	−0.0625111	−0.0312556	+	0.999511i	$0.509951\pi$
$840$	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	$-0.681478\pi$
$853$	−1.95676e10	−1.07948	−0.539742	+	0.841830i	$0.681478\pi$
$854$	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	$-0.585483\pi$
$863$	−1.00199e10	−0.530670	−0.265335	+	0.964156i	$0.585483\pi$
$864$	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	$-0.577585\pi$
$877$	−9.64156e9	−0.482668	−0.241334	+	0.970442i	$0.577585\pi$
$878$	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	$-0.858744\pi$
$887$	−3.75420e10	−1.80628	−0.903140	+	0.429345i	$0.858744\pi$
$888$	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	$-0.466303\pi$
$911$	4.82258e9	0.211332	0.105666	+	0.994402i	$0.466303\pi$
$912$	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	$-0.785234\pi$
$919$	−3.67472e10	−1.56178	−0.780890	+	0.624668i	$0.785234\pi$
$920$	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	$-0.655441\pi$
$941$	−2.39832e10	−0.938305	−0.469153	+	0.883117i	$0.655441\pi$
$942$	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	$-0.235456\pi$
$967$	4.15404e10	1.47733	0.738665	+	0.674072i	$0.235456\pi$
$968$	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	$-0.574645\pi$
$983$	−1.38398e10	−0.464722	−0.232361	+	0.972630i	$0.574645\pi$
$984$	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	$-0.779537\pi$
$991$	−4.71569e10	−1.53917	−0.769586	+	0.638543i	$0.779537\pi$
$992$	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	$-0.365232\pi$
$997$	2.57126e10	0.821698	0.410849	+	0.911703i	$0.365232\pi$
$998$	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.q.1.1	yes	4
4.3	odd	2		384.8.a.m.1.1	✓	4
8.3	odd	2		384.8.a.t.1.4	yes	4
8.5	even	2		384.8.a.p.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.m.1.1	✓	4	4.3	odd	2
384.8.a.p.1.4	yes	4	8.5	even	2
384.8.a.q.1.1	yes	4	1.1	even	1	trivial
384.8.a.t.1.4	yes	4	8.3	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -514.363 q^{5} +1600.84 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -514.363 q^{5} +1600.84 q^{7} +729.000 q^{9} +6913.89 q^{11} -1238.70 q^{13} -13887.8 q^{15} +27118.0 q^{17} -44175.4 q^{19} +43222.7 q^{21} -15503.8 q^{23} +186445. q^{25} +19683.0 q^{27} +94577.1 q^{29} +97999.9 q^{31} +186675. q^{33} -823415. q^{35} -556688. q^{37} -33444.8 q^{39} -203369. q^{41} +59526.7 q^{43} -374971. q^{45} -698902. q^{47} +1.73915e6 q^{49} +732187. q^{51} +91646.2 q^{53} -3.55625e6 q^{55} -1.19274e6 q^{57} +1.47653e6 q^{59} -42411.3 q^{61} +1.16701e6 q^{63} +637140. q^{65} +722597. q^{67} -418603. q^{69} +941020. q^{71} -2.12119e6 q^{73} +5.03401e6 q^{75} +1.10680e7 q^{77} +2.91670e6 q^{79} +531441. q^{81} +5.42934e6 q^{83} -1.39485e7 q^{85} +2.55358e6 q^{87} +3.26957e6 q^{89} -1.98296e6 q^{91} +2.64600e6 q^{93} +2.27222e7 q^{95} +3.55685e6 q^{97} +5.04023e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9} + 3856 q^{11} - 10680 q^{13} - 9072 q^{15} + 26232 q^{17} - 15456 q^{19} + 18360 q^{21} - 11312 q^{23} + 159052 q^{25} + 78732 q^{27} + 1856 q^{29} + 71752 q^{31} + 104112 q^{33} - 179040 q^{35} + 180088 q^{37} - 288360 q^{39} + 11224 q^{41} + 66688 q^{43} - 244944 q^{45} - 1334448 q^{47} + 2401140 q^{49} + 708264 q^{51} + 864576 q^{53} - 3304896 q^{55} - 417312 q^{57} + 1878448 q^{59} + 1901176 q^{61} + 495720 q^{63} + 4366944 q^{65} + 5505488 q^{67} - 305424 q^{69} - 967696 q^{71} + 3244760 q^{73} + 4294404 q^{75} + 8979488 q^{77} + 6471816 q^{79} + 2125764 q^{81} + 17019600 q^{83} - 12122592 q^{85} + 50112 q^{87} + 13559816 q^{89} + 6692304 q^{91} + 1937304 q^{93} + 22523904 q^{95} + 2180520 q^{97} + 2811024 q^{99}+O(q^{100}) \) 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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