LMFDB - Embedded newform 384.10.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,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, 10, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$197.773761087$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 3854x^{3} + 12258x^{2} + 2877633x + 16772643 $ x^5 - x^4 - 3854x^3 + 12258x^2 + 2877633*x + 16772643
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{29}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$44.2229$ 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)

$f(q)$	$=$	$q+81.0000 q^{3} +1251.33 q^{5} -561.535 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} +1251.33 q^{5} -561.535 q^{7} +6561.00 q^{9} +80532.1 q^{11} -119077. q^{13} +101357. q^{15} -603707. q^{17} +722380. q^{19} -45484.3 q^{21} -229161. q^{23} -387309. q^{25} +531441. q^{27} -7.34286e6 q^{29} -4.26391e6 q^{31} +6.52310e6 q^{33} -702663. q^{35} +1.01156e7 q^{37} -9.64521e6 q^{39} -1.68312e7 q^{41} -1.63523e7 q^{43} +8.20995e6 q^{45} -4.92462e7 q^{47} -4.00383e7 q^{49} -4.89003e7 q^{51} +8.72617e7 q^{53} +1.00772e8 q^{55} +5.85128e7 q^{57} +1.14769e8 q^{59} -1.21503e7 q^{61} -3.68423e6 q^{63} -1.49004e8 q^{65} -2.43537e8 q^{67} -1.85620e7 q^{69} -1.05574e8 q^{71} -3.19551e7 q^{73} -3.13721e7 q^{75} -4.52216e7 q^{77} -5.34767e8 q^{79} +4.30467e7 q^{81} +7.54300e8 q^{83} -7.55434e8 q^{85} -5.94772e8 q^{87} -5.93697e8 q^{89} +6.68658e7 q^{91} -3.45376e8 q^{93} +9.03932e8 q^{95} -7.63848e8 q^{97} +5.28371e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 405 q^{3} - 772 q^{5} + 38 q^{7} + 32805 q^{9}+O(q^{10}) $ 5 * q + 405 * q^3 - 772 * q^5 + 38 * q^7 + 32805 * q^9	$ 5 q + 405 q^{3} - 772 q^{5} + 38 q^{7} + 32805 q^{9} + 41100 q^{11} - 22486 q^{13} - 62532 q^{15} - 1130 q^{17} - 664712 q^{19} + 3078 q^{21} - 369972 q^{23} + 823383 q^{25} + 2657205 q^{27} - 7390736 q^{29} - 9149938 q^{31} + 3329100 q^{33} - 3167800 q^{35} - 11922058 q^{37} - 1821366 q^{39} - 8471746 q^{41} - 8948896 q^{43} - 5065092 q^{45} + 5051660 q^{47} + 39616113 q^{49} - 91530 q^{51} + 31431984 q^{53} + 67216 q^{55} - 53841672 q^{57} + 204260948 q^{59} - 190850874 q^{61} + 249318 q^{63} - 165466760 q^{65} + 274483500 q^{67} - 29967732 q^{69} - 162722908 q^{71} - 508927538 q^{73} + 66694023 q^{75} - 428895960 q^{77} - 491411266 q^{79} + 215233605 q^{81} + 766279260 q^{83} - 713985400 q^{85} - 598649616 q^{87} - 954097990 q^{89} + 503505932 q^{91} - 741144978 q^{93} - 968680288 q^{95} - 677085326 q^{97} + 269657100 q^{99}+O(q^{100}) $ 5 * q + 405 * q^3 - 772 * q^5 + 38 * q^7 + 32805 * q^9 + 41100 * q^11 - 22486 * q^13 - 62532 * q^15 - 1130 * q^17 - 664712 * q^19 + 3078 * q^21 - 369972 * q^23 + 823383 * q^25 + 2657205 * q^27 - 7390736 * q^29 - 9149938 * q^31 + 3329100 * q^33 - 3167800 * q^35 - 11922058 * q^37 - 1821366 * q^39 - 8471746 * q^41 - 8948896 * q^43 - 5065092 * q^45 + 5051660 * q^47 + 39616113 * q^49 - 91530 * q^51 + 31431984 * q^53 + 67216 * q^55 - 53841672 * q^57 + 204260948 * q^59 - 190850874 * q^61 + 249318 * q^63 - 165466760 * q^65 + 274483500 * q^67 - 29967732 * q^69 - 162722908 * q^71 - 508927538 * q^73 + 66694023 * q^75 - 428895960 * q^77 - 491411266 * q^79 + 215233605 * q^81 + 766279260 * q^83 - 713985400 * q^85 - 598649616 * q^87 - 954097990 * q^89 + 503505932 * q^91 - 741144978 * q^93 - 968680288 * q^95 - 677085326 * q^97 + 269657100 * 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
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	0	0
$5$	1251.33	0.895376	0.447688	−	0.894190i	$-0.352248\pi$
$5$	1251.33	0.895376	0.447688	+	0.894190i	$0.352248\pi$
$6$	0	0
$7$	−561.535	−0.0883966	−0.0441983	−	0.999023i	$-0.514073\pi$
$7$	−561.535	−0.0883966	−0.0441983	+	0.999023i	$0.514073\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	80532.1	1.65845	0.829224	−	0.558917i	$-0.188783\pi$
$11$	80532.1	1.65845	0.829224	+	0.558917i	$0.188783\pi$
$12$	0	0
$13$	−119077.	−1.15633	−0.578165	−	0.815920i	$-0.696231\pi$
$13$	−119077.	−1.15633	−0.578165	+	0.815920i	$0.696231\pi$
$14$	0	0
$15$	101357.	0.516945
$16$	0	0
$17$	−603707.	−1.75310	−0.876549	−	0.481313i	$-0.840160\pi$
$17$	−603707.	−1.75310	−0.876549	+	0.481313i	$0.840160\pi$
$18$	0	0
$19$	722380.	1.27167	0.635835	−	0.771825i	$-0.280656\pi$
$19$	722380.	1.27167	0.635835	+	0.771825i	$0.280656\pi$
$20$	0	0
$21$	−45484.3	−0.0510358
$22$	0	0
$23$	−229161.	−0.170752	−0.0853760	−	0.996349i	$-0.527209\pi$
$23$	−229161.	−0.170752	−0.0853760	+	0.996349i	$0.527209\pi$
$24$	0	0
$25$	−387309.	−0.198302
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	−7.34286e6	−1.92786	−0.963928	−	0.266165i	$-0.914244\pi$
$29$	−7.34286e6	−1.92786	−0.963928	+	0.266165i	$0.914244\pi$
$30$	0	0
$31$	−4.26391e6	−0.829239	−0.414620	−	0.909995i	$-0.636085\pi$
$31$	−4.26391e6	−0.829239	−0.414620	+	0.909995i	$0.636085\pi$
$32$	0	0
$33$	6.52310e6	0.957505
$34$	0	0
$35$	−702663.	−0.0791482
$36$	0	0
$37$	1.01156e7	0.887323	0.443662	−	0.896194i	$-0.353679\pi$
$37$	1.01156e7	0.887323	0.443662	+	0.896194i	$0.353679\pi$
$38$	0	0
$39$	−9.64521e6	−0.667607
$40$	0	0
$41$	−1.68312e7	−0.930224	−0.465112	−	0.885252i	$-0.653986\pi$
$41$	−1.68312e7	−0.930224	−0.465112	+	0.885252i	$0.653986\pi$
$42$	0	0
$43$	−1.63523e7	−0.729410	−0.364705	−	0.931123i	$-0.618830\pi$
$43$	−1.63523e7	−0.729410	−0.364705	+	0.931123i	$0.618830\pi$
$44$	0	0
$45$	8.20995e6	0.298459
$46$	0	0
$47$	−4.92462e7	−1.47208	−0.736041	−	0.676936i	$-0.763307\pi$
$47$	−4.92462e7	−1.47208	−0.736041	+	0.676936i	$0.763307\pi$
$48$	0	0
$49$	−4.00383e7	−0.992186
$50$	0	0
$51$	−4.89003e7	−1.01215
$52$	0	0
$53$	8.72617e7	1.51909	0.759543	−	0.650457i	$-0.225423\pi$
$53$	8.72617e7	1.51909	0.759543	+	0.650457i	$0.225423\pi$
$54$	0	0
$55$	1.00772e8	1.48493
$56$	0	0
$57$	5.85128e7	0.734199
$58$	0	0
$59$	1.14769e8	1.23308	0.616541	−	0.787323i	$-0.288533\pi$
$59$	1.14769e8	1.23308	0.616541	+	0.787323i	$0.288533\pi$
$60$	0	0
$61$	−1.21503e7	−0.112358	−0.0561790	−	0.998421i	$-0.517892\pi$
$61$	−1.21503e7	−0.112358	−0.0561790	+	0.998421i	$0.517892\pi$
$62$	0	0
$63$	−3.68423e6	−0.0294655
$64$	0	0
$65$	−1.49004e8	−1.03535
$66$	0	0
$67$	−2.43537e8	−1.47648	−0.738241	−	0.674537i	$-0.764343\pi$
$67$	−2.43537e8	−1.47648	−0.738241	+	0.674537i	$0.764343\pi$
$68$	0	0
$69$	−1.85620e7	−0.0985837
$70$	0	0
$71$	−1.05574e8	−0.493056	−0.246528	−	0.969136i	$-0.579290\pi$
$71$	−1.05574e8	−0.493056	−0.246528	+	0.969136i	$0.579290\pi$
$72$	0	0
$73$	−3.19551e7	−0.131700	−0.0658502	−	0.997830i	$-0.520976\pi$
$73$	−3.19551e7	−0.131700	−0.0658502	+	0.997830i	$0.520976\pi$
$74$	0	0
$75$	−3.13721e7	−0.114490
$76$	0	0
$77$	−4.52216e7	−0.146601
$78$	0	0
$79$	−5.34767e8	−1.54470	−0.772348	−	0.635200i	$-0.780918\pi$
$79$	−5.34767e8	−1.54470	−0.772348	+	0.635200i	$0.780918\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	7.54300e8	1.74459	0.872294	−	0.488983i	$-0.162632\pi$
$83$	7.54300e8	1.74459	0.872294	+	0.488983i	$0.162632\pi$
$84$	0	0
$85$	−7.55434e8	−1.56968
$86$	0	0
$87$	−5.94772e8	−1.11305
$88$	0	0
$89$	−5.93697e8	−1.00302	−0.501510	−	0.865152i	$-0.667222\pi$
$89$	−5.93697e8	−1.00302	−0.501510	+	0.865152i	$0.667222\pi$
$90$	0	0
$91$	6.68658e7	0.102216
$92$	0	0
$93$	−3.45376e8	−0.478762
$94$	0	0
$95$	9.03932e8	1.13862
$96$	0	0
$97$	−7.63848e8	−0.876060	−0.438030	−	0.898960i	$-0.644324\pi$
$97$	−7.63848e8	−0.876060	−0.438030	+	0.898960i	$0.644324\pi$
$98$	0	0
$99$	5.28371e8	0.552816
$100$	0	0
$101$	1.64929e9	1.57707	0.788536	−	0.614989i	$-0.210840\pi$
$101$	1.64929e9	1.57707	0.788536	+	0.614989i	$0.210840\pi$
$102$	0	0
$103$	−7.77578e8	−0.680732	−0.340366	−	0.940293i	$-0.610551\pi$
$103$	−7.77578e8	−0.680732	−0.340366	+	0.940293i	$0.610551\pi$
$104$	0	0
$105$	−5.69157e7	−0.0456962
$106$	0	0
$107$	−1.08863e9	−0.802887	−0.401443	−	0.915884i	$-0.631491\pi$
$107$	−1.08863e9	−0.802887	−0.401443	+	0.915884i	$0.631491\pi$
$108$	0	0
$109$	−1.28952e9	−0.875002	−0.437501	−	0.899218i	$-0.644136\pi$
$109$	−1.28952e9	−0.875002	−0.437501	+	0.899218i	$0.644136\pi$
$110$	0	0
$111$	8.19360e8	0.512296
$112$	0	0
$113$	1.76262e9	1.01697	0.508483	−	0.861072i	$-0.330207\pi$
$113$	1.76262e9	1.01697	0.508483	+	0.861072i	$0.330207\pi$
$114$	0	0
$115$	−2.86755e8	−0.152887
$116$	0	0
$117$	−7.81262e8	−0.385443
$118$	0	0
$119$	3.39003e8	0.154968
$120$	0	0
$121$	4.12747e9	1.75045
$122$	0	0
$123$	−1.36333e9	−0.537065
$124$	0	0
$125$	−2.92865e9	−1.07293
$126$	0	0
$127$	1.81850e9	0.620293	0.310147	−	0.950689i	$-0.399622\pi$
$127$	1.81850e9	0.620293	0.310147	+	0.950689i	$0.399622\pi$
$128$	0	0
$129$	−1.32454e9	−0.421125
$130$	0	0
$131$	4.95102e9	1.46884	0.734418	−	0.678697i	$-0.237455\pi$
$131$	4.95102e9	1.46884	0.734418	+	0.678697i	$0.237455\pi$
$132$	0	0
$133$	−4.05642e8	−0.112411
$134$	0	0
$135$	6.65006e8	0.172315
$136$	0	0
$137$	4.04562e9	0.981165	0.490583	−	0.871395i	$-0.336784\pi$
$137$	4.04562e9	0.981165	0.490583	+	0.871395i	$0.336784\pi$
$138$	0	0
$139$	6.05001e9	1.37464	0.687321	−	0.726354i	$-0.258787\pi$
$139$	6.05001e9	1.37464	0.687321	+	0.726354i	$0.258787\pi$
$140$	0	0
$141$	−3.98894e9	−0.849907
$142$	0	0
$143$	−9.58950e9	−1.91771
$144$	0	0
$145$	−9.18831e9	−1.72615
$146$	0	0
$147$	−3.24310e9	−0.572839
$148$	0	0
$149$	−3.15234e9	−0.523955	−0.261978	−	0.965074i	$-0.584375\pi$
$149$	−3.15234e9	−0.523955	−0.261978	+	0.965074i	$0.584375\pi$
$150$	0	0
$151$	−3.09017e8	−0.0483711	−0.0241855	−	0.999707i	$-0.507699\pi$
$151$	−3.09017e8	−0.0483711	−0.0241855	+	0.999707i	$0.507699\pi$
$152$	0	0
$153$	−3.96092e9	−0.584366
$154$	0	0
$155$	−5.33553e9	−0.742481
$156$	0	0
$157$	1.07723e10	1.41501	0.707506	−	0.706707i	$-0.249820\pi$
$157$	1.07723e10	1.41501	0.707506	+	0.706707i	$0.249820\pi$
$158$	0	0
$159$	7.06820e9	0.877045
$160$	0	0
$161$	1.28682e8	0.0150939
$162$	0	0
$163$	−5.36455e9	−0.595236	−0.297618	−	0.954685i	$-0.596192\pi$
$163$	−5.36455e9	−0.595236	−0.297618	+	0.954685i	$0.596192\pi$
$164$	0	0
$165$	8.16252e9	0.857327
$166$	0	0
$167$	−1.33915e10	−1.33231	−0.666154	−	0.745814i	$-0.732061\pi$
$167$	−1.33915e10	−1.33231	−0.666154	+	0.745814i	$0.732061\pi$
$168$	0	0
$169$	3.57476e9	0.337099
$170$	0	0
$171$	4.73953e9	0.423890
$172$	0	0
$173$	−6.95779e9	−0.590559	−0.295280	−	0.955411i	$-0.595413\pi$
$173$	−6.95779e9	−0.590559	−0.295280	+	0.955411i	$0.595413\pi$
$174$	0	0
$175$	2.17488e8	0.0175293
$176$	0	0
$177$	9.29632e9	0.711920
$178$	0	0
$179$	−1.35492e10	−0.986451	−0.493225	−	0.869902i	$-0.664182\pi$
$179$	−1.35492e10	−0.986451	−0.493225	+	0.869902i	$0.664182\pi$
$180$	0	0
$181$	−1.02270e10	−0.708261	−0.354130	−	0.935196i	$-0.615223\pi$
$181$	−1.02270e10	−0.708261	−0.354130	+	0.935196i	$0.615223\pi$
$182$	0	0
$183$	−9.84177e8	−0.0648699
$184$	0	0
$185$	1.26578e10	0.794488
$186$	0	0
$187$	−4.86178e10	−2.90742
$188$	0	0
$189$	−2.98423e8	−0.0170119
$190$	0	0
$191$	−3.80601e9	−0.206928	−0.103464	−	0.994633i	$-0.532993\pi$
$191$	−3.80601e9	−0.206928	−0.103464	+	0.994633i	$0.532993\pi$
$192$	0	0
$193$	1.58965e10	0.824695	0.412348	−	0.911027i	$-0.364709\pi$
$193$	1.58965e10	0.824695	0.412348	+	0.911027i	$0.364709\pi$
$194$	0	0
$195$	−1.20693e10	−0.597759
$196$	0	0
$197$	1.49591e10	0.707630	0.353815	−	0.935315i	$-0.384884\pi$
$197$	1.49591e10	0.707630	0.353815	+	0.935315i	$0.384884\pi$
$198$	0	0
$199$	−6.22732e8	−0.0281489	−0.0140745	−	0.999901i	$-0.504480\pi$
$199$	−6.22732e8	−0.0281489	−0.0140745	+	0.999901i	$0.504480\pi$
$200$	0	0
$201$	−1.97265e10	−0.852448
$202$	0	0
$203$	4.12328e9	0.170416
$204$	0	0
$205$	−2.10613e10	−0.832900
$206$	0	0
$207$	−1.50353e9	−0.0569173
$208$	0	0
$209$	5.81747e10	2.10900
$210$	0	0
$211$	2.01982e10	0.701523	0.350761	−	0.936465i	$-0.385923\pi$
$211$	2.01982e10	0.701523	0.350761	+	0.936465i	$0.385923\pi$
$212$	0	0
$213$	−8.55153e9	−0.284666
$214$	0	0
$215$	−2.04621e10	−0.653096
$216$	0	0
$217$	2.39433e9	0.0733020
$218$	0	0
$219$	−2.58836e9	−0.0760372
$220$	0	0
$221$	7.18874e10	2.02716
$222$	0	0
$223$	1.61462e10	0.437218	0.218609	−	0.975813i	$-0.429848\pi$
$223$	1.61462e10	0.437218	0.218609	+	0.975813i	$0.429848\pi$
$224$	0	0
$225$	−2.54114e9	−0.0661008
$226$	0	0
$227$	1.36800e10	0.341955	0.170978	−	0.985275i	$-0.445307\pi$
$227$	1.36800e10	0.341955	0.170978	+	0.985275i	$0.445307\pi$
$228$	0	0
$229$	−7.10441e10	−1.70714	−0.853569	−	0.520980i	$-0.825567\pi$
$229$	−7.10441e10	−1.70714	−0.853569	+	0.520980i	$0.825567\pi$
$230$	0	0
$231$	−3.66295e9	−0.0846403
$232$	0	0
$233$	9.55113e9	0.212302	0.106151	−	0.994350i	$-0.466147\pi$
$233$	9.55113e9	0.212302	0.106151	+	0.994350i	$0.466147\pi$
$234$	0	0
$235$	−6.16230e10	−1.31807
$236$	0	0
$237$	−4.33162e10	−0.891831
$238$	0	0
$239$	2.30873e10	0.457702	0.228851	−	0.973461i	$-0.426503\pi$
$239$	2.30873e10	0.457702	0.228851	+	0.973461i	$0.426503\pi$
$240$	0	0
$241$	−1.68503e10	−0.321759	−0.160880	−	0.986974i	$-0.551433\pi$
$241$	−1.68503e10	−0.321759	−0.160880	+	0.986974i	$0.551433\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	−5.01009e10	−0.888379
$246$	0	0
$247$	−8.60186e10	−1.47047
$248$	0	0
$249$	6.10983e10	1.00724
$250$	0	0
$251$	−1.76046e10	−0.279959	−0.139980	−	0.990154i	$-0.544704\pi$
$251$	−1.76046e10	−0.279959	−0.139980	+	0.990154i	$0.544704\pi$
$252$	0	0
$253$	−1.84548e10	−0.283183
$254$	0	0
$255$	−6.11901e10	−0.906256
$256$	0	0
$257$	−1.74568e10	−0.249613	−0.124806	−	0.992181i	$-0.539831\pi$
$257$	−1.74568e10	−0.249613	−0.124806	+	0.992181i	$0.539831\pi$
$258$	0	0
$259$	−5.68024e9	−0.0784364
$260$	0	0
$261$	−4.81765e10	−0.642618
$262$	0	0
$263$	−5.18217e10	−0.667899	−0.333950	−	0.942591i	$-0.608382\pi$
$263$	−5.18217e10	−0.667899	−0.333950	+	0.942591i	$0.608382\pi$
$264$	0	0
$265$	1.09193e11	1.36015
$266$	0	0
$267$	−4.80894e10	−0.579094
$268$	0	0
$269$	−1.56192e11	−1.81876	−0.909379	−	0.415970i	$-0.863442\pi$
$269$	−1.56192e11	−1.81876	−0.909379	+	0.415970i	$0.863442\pi$
$270$	0	0
$271$	−5.59149e10	−0.629746	−0.314873	−	0.949134i	$-0.601962\pi$
$271$	−5.59149e10	−0.629746	−0.314873	+	0.949134i	$0.601962\pi$
$272$	0	0
$273$	5.41613e9	0.0590143
$274$	0	0
$275$	−3.11908e10	−0.328874
$276$	0	0
$277$	−1.68080e10	−0.171536	−0.0857682	−	0.996315i	$-0.527334\pi$
$277$	−1.68080e10	−0.171536	−0.0857682	+	0.996315i	$0.527334\pi$
$278$	0	0
$279$	−2.79755e10	−0.276413
$280$	0	0
$281$	2.76924e10	0.264961	0.132481	−	0.991186i	$-0.457706\pi$
$281$	2.76924e10	0.264961	0.132481	+	0.991186i	$0.457706\pi$
$282$	0	0
$283$	9.56295e9	0.0886243	0.0443122	−	0.999018i	$-0.485890\pi$
$283$	9.56295e9	0.0886243	0.0443122	+	0.999018i	$0.485890\pi$
$284$	0	0
$285$	7.32185e10	0.657384
$286$	0	0
$287$	9.45131e9	0.0822287
$288$	0	0
$289$	2.45874e11	2.07335
$290$	0	0
$291$	−6.18716e10	−0.505793
$292$	0	0
$293$	−1.50049e11	−1.18940	−0.594700	−	0.803947i	$-0.702729\pi$
$293$	−1.50049e11	−1.18940	−0.594700	+	0.803947i	$0.702729\pi$
$294$	0	0
$295$	1.43614e11	1.10407
$296$	0	0
$297$	4.27980e10	0.319168
$298$	0	0
$299$	2.72877e10	0.197446
$300$	0	0
$301$	9.18241e9	0.0644774
$302$	0	0
$303$	1.33593e11	0.910522
$304$	0	0
$305$	−1.52040e10	−0.100603
$306$	0	0
$307$	−5.83624e10	−0.374982	−0.187491	−	0.982266i	$-0.560036\pi$
$307$	−5.83624e10	−0.374982	−0.187491	+	0.982266i	$0.560036\pi$
$308$	0	0
$309$	−6.29838e10	−0.393021
$310$	0	0
$311$	−2.08426e11	−1.26337	−0.631685	−	0.775225i	$-0.717636\pi$
$311$	−2.08426e11	−1.26337	−0.631685	+	0.775225i	$0.717636\pi$
$312$	0	0
$313$	−4.10734e10	−0.241886	−0.120943	−	0.992659i	$-0.538592\pi$
$313$	−4.10734e10	−0.241886	−0.120943	+	0.992659i	$0.538592\pi$
$314$	0	0
$315$	−4.61017e9	−0.0263827
$316$	0	0
$317$	−5.35043e10	−0.297593	−0.148796	−	0.988868i	$-0.547540\pi$
$317$	−5.35043e10	−0.297593	−0.148796	+	0.988868i	$0.547540\pi$
$318$	0	0
$319$	−5.91336e11	−3.19725
$320$	0	0
$321$	−8.81793e10	−0.463547
$322$	0	0
$323$	−4.36106e11	−2.22936
$324$	0	0
$325$	4.61195e10	0.229303
$326$	0	0
$327$	−1.04451e11	−0.505183
$328$	0	0
$329$	2.76535e10	0.130127
$330$	0	0
$331$	−1.06052e11	−0.485615	−0.242808	−	0.970074i	$-0.578068\pi$
$331$	−1.06052e11	−0.485615	−0.242808	+	0.970074i	$0.578068\pi$
$332$	0	0
$333$	6.63681e10	0.295774
$334$	0	0
$335$	−3.04744e11	−1.32201
$336$	0	0
$337$	−1.09163e11	−0.461042	−0.230521	−	0.973067i	$-0.574043\pi$
$337$	−1.09163e11	−0.461042	−0.230521	+	0.973067i	$0.574043\pi$
$338$	0	0
$339$	1.42772e11	0.587145
$340$	0	0
$341$	−3.43381e11	−1.37525
$342$	0	0
$343$	4.51429e10	0.176103
$344$	0	0
$345$	−2.32272e10	−0.0882694
$346$	0	0
$347$	−1.61488e11	−0.597939	−0.298970	−	0.954263i	$-0.596643\pi$
$347$	−1.61488e11	−0.597939	−0.298970	+	0.954263i	$0.596643\pi$
$348$	0	0
$349$	−5.05572e11	−1.82418	−0.912091	−	0.409987i	$-0.865533\pi$
$349$	−5.05572e11	−1.82418	−0.912091	+	0.409987i	$0.865533\pi$
$350$	0	0
$351$	−6.32822e10	−0.222536
$352$	0	0
$353$	−1.61940e11	−0.555096	−0.277548	−	0.960712i	$-0.589522\pi$
$353$	−1.61940e11	−0.555096	−0.277548	+	0.960712i	$0.589522\pi$
$354$	0	0
$355$	−1.32108e11	−0.441470
$356$	0	0
$357$	2.74592e10	0.0894708
$358$	0	0
$359$	−2.90648e11	−0.923513	−0.461756	−	0.887007i	$-0.652781\pi$
$359$	−2.90648e11	−0.923513	−0.461756	+	0.887007i	$0.652781\pi$
$360$	0	0
$361$	1.99145e11	0.617144
$362$	0	0
$363$	3.34325e11	1.01062
$364$	0	0
$365$	−3.99862e10	−0.117921
$366$	0	0
$367$	−1.37994e11	−0.397065	−0.198533	−	0.980094i	$-0.563618\pi$
$367$	−1.37994e11	−0.397065	−0.198533	+	0.980094i	$0.563618\pi$
$368$	0	0
$369$	−1.10430e11	−0.310075
$370$	0	0
$371$	−4.90005e10	−0.134282
$372$	0	0
$373$	1.34309e11	0.359265	0.179632	−	0.983734i	$-0.442509\pi$
$373$	1.34309e11	0.359265	0.179632	+	0.983734i	$0.442509\pi$
$374$	0	0
$375$	−2.37220e11	−0.619457
$376$	0	0
$377$	8.74364e11	2.22924
$378$	0	0
$379$	−4.43415e11	−1.10391	−0.551955	−	0.833874i	$-0.686118\pi$
$379$	−4.43415e11	−1.10391	−0.551955	+	0.833874i	$0.686118\pi$
$380$	0	0
$381$	1.47299e11	0.358126
$382$	0	0
$383$	6.08821e11	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$383$	6.08821e11	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$384$	0	0
$385$	−5.65869e10	−0.131263
$386$	0	0
$387$	−1.07288e11	−0.243137
$388$	0	0
$389$	−2.16458e11	−0.479293	−0.239646	−	0.970860i	$-0.577032\pi$
$389$	−2.16458e11	−0.479293	−0.239646	+	0.970860i	$0.577032\pi$
$390$	0	0
$391$	1.38346e11	0.299345
$392$	0	0
$393$	4.01032e11	0.848033
$394$	0	0
$395$	−6.69168e11	−1.38308
$396$	0	0
$397$	8.65679e11	1.74904	0.874520	−	0.484989i	$-0.161176\pi$
$397$	8.65679e11	1.74904	0.874520	+	0.484989i	$0.161176\pi$
$398$	0	0
$399$	−3.28570e10	−0.0649007
$400$	0	0
$401$	−4.77629e10	−0.0922445	−0.0461223	−	0.998936i	$-0.514686\pi$
$401$	−4.77629e10	−0.0922445	−0.0461223	+	0.998936i	$0.514686\pi$
$402$	0	0
$403$	5.07732e11	0.958874
$404$	0	0
$405$	5.38655e10	0.0994862
$406$	0	0
$407$	8.14626e11	1.47158
$408$	0	0
$409$	6.58925e11	1.16434	0.582172	−	0.813066i	$-0.302203\pi$
$409$	6.58925e11	1.16434	0.582172	+	0.813066i	$0.302203\pi$
$410$	0	0
$411$	3.27695e11	0.566476
$412$	0	0
$413$	−6.44470e10	−0.109000
$414$	0	0
$415$	9.43875e11	1.56206
$416$	0	0
$417$	4.90051e11	0.793650
$418$	0	0
$419$	−3.54261e11	−0.561514	−0.280757	−	0.959779i	$-0.590585\pi$
$419$	−3.54261e11	−0.561514	−0.280757	+	0.959779i	$0.590585\pi$
$420$	0	0
$421$	−5.97395e11	−0.926812	−0.463406	−	0.886146i	$-0.653373\pi$
$421$	−5.97395e11	−0.926812	−0.463406	+	0.886146i	$0.653373\pi$
$422$	0	0
$423$	−3.23104e11	−0.490694
$424$	0	0
$425$	2.33821e11	0.347643
$426$	0	0
$427$	6.82284e9	0.00993207
$428$	0	0
$429$	−7.76749e11	−1.10719
$430$	0	0
$431$	−9.12658e11	−1.27397	−0.636987	−	0.770875i	$-0.719819\pi$
$431$	−9.12658e11	−1.27397	−0.636987	+	0.770875i	$0.719819\pi$
$432$	0	0
$433$	4.10826e11	0.561645	0.280823	−	0.959760i	$-0.409393\pi$
$433$	4.10826e11	0.561645	0.280823	+	0.959760i	$0.409393\pi$
$434$	0	0
$435$	−7.44253e11	−0.996596
$436$	0	0
$437$	−1.65541e11	−0.217140
$438$	0	0
$439$	−3.04423e11	−0.391190	−0.195595	−	0.980685i	$-0.562664\pi$
$439$	−3.04423e11	−0.391190	−0.195595	+	0.980685i	$0.562664\pi$
$440$	0	0
$441$	−2.62691e11	−0.330729
$442$	0	0
$443$	1.08842e12	1.34270	0.671352	−	0.741139i	$-0.265714\pi$
$443$	1.08842e12	1.34270	0.671352	+	0.741139i	$0.265714\pi$
$444$	0	0
$445$	−7.42908e11	−0.898080
$446$	0	0
$447$	−2.55339e11	−0.302506
$448$	0	0
$449$	−9.78713e11	−1.13644	−0.568220	−	0.822876i	$-0.692368\pi$
$449$	−9.78713e11	−1.13644	−0.568220	+	0.822876i	$0.692368\pi$
$450$	0	0
$451$	−1.35545e12	−1.54273
$452$	0	0
$453$	−2.50304e10	−0.0279271
$454$	0	0
$455$	8.36708e10	0.0915214
$456$	0	0
$457$	5.55228e11	0.595454	0.297727	−	0.954651i	$-0.403771\pi$
$457$	5.55228e11	0.595454	0.297727	+	0.954651i	$0.403771\pi$
$458$	0	0
$459$	−3.20835e11	−0.337384
$460$	0	0
$461$	1.08518e12	1.11905	0.559524	−	0.828814i	$-0.310984\pi$
$461$	1.08518e12	1.11905	0.559524	+	0.828814i	$0.310984\pi$
$462$	0	0
$463$	−6.82120e11	−0.689837	−0.344918	−	0.938633i	$-0.612093\pi$
$463$	−6.82120e11	−0.689837	−0.344918	+	0.938633i	$0.612093\pi$
$464$	0	0
$465$	−4.32178e11	−0.428671
$466$	0	0
$467$	1.55597e12	1.51382	0.756911	−	0.653518i	$-0.226708\pi$
$467$	1.55597e12	1.51382	0.756911	+	0.653518i	$0.226708\pi$
$468$	0	0
$469$	1.36755e11	0.130516
$470$	0	0
$471$	8.72556e11	0.816957
$472$	0	0
$473$	−1.31689e12	−1.20969
$474$	0	0
$475$	−2.79784e11	−0.252175
$476$	0	0
$477$	5.72524e11	0.506362
$478$	0	0
$479$	−7.60423e11	−0.660002	−0.330001	−	0.943981i	$-0.607049\pi$
$479$	−7.60423e11	−0.660002	−0.330001	+	0.943981i	$0.607049\pi$
$480$	0	0
$481$	−1.20453e12	−1.02604
$482$	0	0
$483$	1.04232e10	0.00871446
$484$	0	0
$485$	−9.55822e11	−0.784403
$486$	0	0
$487$	1.67749e12	1.35139	0.675693	−	0.737183i	$-0.263845\pi$
$487$	1.67749e12	1.35139	0.675693	+	0.737183i	$0.263845\pi$
$488$	0	0
$489$	−4.34528e11	−0.343659
$490$	0	0
$491$	2.47048e12	1.91829	0.959145	−	0.282914i	$-0.0913010\pi$
$491$	2.47048e12	1.91829	0.959145	+	0.282914i	$0.0913010\pi$
$492$	0	0
$493$	4.43294e12	3.37972
$494$	0	0
$495$	6.61164e11	0.494978
$496$	0	0
$497$	5.92838e10	0.0435845
$498$	0	0
$499$	7.96398e11	0.575013	0.287507	−	0.957779i	$-0.407174\pi$
$499$	7.96398e11	0.575013	0.287507	+	0.957779i	$0.407174\pi$
$500$	0	0
$501$	−1.08471e12	−0.769209
$502$	0	0
$503$	1.33174e12	0.927605	0.463803	−	0.885939i	$-0.346485\pi$
$503$	1.33174e12	0.927605	0.463803	+	0.885939i	$0.346485\pi$
$504$	0	0
$505$	2.06380e12	1.41207
$506$	0	0
$507$	2.89556e11	0.194624
$508$	0	0
$509$	1.57766e12	1.04180	0.520898	−	0.853619i	$-0.325597\pi$
$509$	1.57766e12	1.04180	0.520898	+	0.853619i	$0.325597\pi$
$510$	0	0
$511$	1.79439e10	0.0116419
$512$	0	0
$513$	3.83902e11	0.244733
$514$	0	0
$515$	−9.73003e11	−0.609511
$516$	0	0
$517$	−3.96590e12	−2.44137
$518$	0	0
$519$	−5.63581e11	−0.340960
$520$	0	0
$521$	−1.00242e12	−0.596044	−0.298022	−	0.954559i	$-0.596327\pi$
$521$	−1.00242e12	−0.596044	−0.298022	+	0.954559i	$0.596327\pi$
$522$	0	0
$523$	−4.90824e11	−0.286859	−0.143429	−	0.989661i	$-0.545813\pi$
$523$	−4.90824e11	−0.286859	−0.143429	+	0.989661i	$0.545813\pi$
$524$	0	0
$525$	1.76165e10	0.0101205
$526$	0	0
$527$	2.57415e12	1.45374
$528$	0	0
$529$	−1.74864e12	−0.970844
$530$	0	0
$531$	7.53002e11	0.411027
$532$	0	0
$533$	2.00420e12	1.07565
$534$	0	0
$535$	−1.36223e12	−0.718885
$536$	0	0
$537$	−1.09749e12	−0.569528
$538$	0	0
$539$	−3.22437e12	−1.64549
$540$	0	0
$541$	3.88983e12	1.95228	0.976141	−	0.217139i	$-0.0696725\pi$
$541$	3.88983e12	1.95228	0.976141	+	0.217139i	$0.0696725\pi$
$542$	0	0
$543$	−8.28384e11	−0.408915
$544$	0	0
$545$	−1.61361e12	−0.783456
$546$	0	0
$547$	2.08571e12	0.996117	0.498059	−	0.867143i	$-0.334046\pi$
$547$	2.08571e12	0.996117	0.498059	+	0.867143i	$0.334046\pi$
$548$	0	0
$549$	−7.97183e10	−0.0374526
$550$	0	0
$551$	−5.30433e12	−2.45160
$552$	0	0
$553$	3.00291e11	0.136546
$554$	0	0
$555$	1.02529e12	0.458698
$556$	0	0
$557$	−1.11503e12	−0.490840	−0.245420	−	0.969417i	$-0.578926\pi$
$557$	−1.11503e12	−0.490840	−0.245420	+	0.969417i	$0.578926\pi$
$558$	0	0
$559$	1.94718e12	0.843438
$560$	0	0
$561$	−3.93804e12	−1.67860
$562$	0	0
$563$	1.53114e12	0.642283	0.321142	−	0.947031i	$-0.395933\pi$
$563$	1.53114e12	0.642283	0.321142	+	0.947031i	$0.395933\pi$
$564$	0	0
$565$	2.20561e12	0.910566
$566$	0	0
$567$	−2.41722e10	−0.00982185
$568$	0	0
$569$	3.50044e12	1.39997	0.699983	−	0.714159i	$-0.253191\pi$
$569$	3.50044e12	1.39997	0.699983	+	0.714159i	$0.253191\pi$
$570$	0	0
$571$	−1.49783e12	−0.589658	−0.294829	−	0.955550i	$-0.595263\pi$
$571$	−1.49783e12	−0.589658	−0.294829	+	0.955550i	$0.595263\pi$
$572$	0	0
$573$	−3.08287e11	−0.119470
$574$	0	0
$575$	8.87562e10	0.0338605
$576$	0	0
$577$	−3.78055e12	−1.41992	−0.709961	−	0.704241i	$-0.751287\pi$
$577$	−3.78055e12	−1.41992	−0.709961	+	0.704241i	$0.751287\pi$
$578$	0	0
$579$	1.28762e12	0.476138
$580$	0	0
$581$	−4.23566e11	−0.154216
$582$	0	0
$583$	7.02737e12	2.51932
$584$	0	0
$585$	−9.77614e11	−0.345117
$586$	0	0
$587$	4.05765e12	1.41060	0.705298	−	0.708911i	$-0.250813\pi$
$587$	4.05765e12	1.41060	0.705298	+	0.708911i	$0.250813\pi$
$588$	0	0
$589$	−3.08016e12	−1.05452
$590$	0	0
$591$	1.21168e12	0.408550
$592$	0	0
$593$	1.66492e11	0.0552900	0.0276450	−	0.999618i	$-0.491199\pi$
$593$	1.66492e11	0.0552900	0.0276450	+	0.999618i	$0.491199\pi$
$594$	0	0
$595$	4.24203e11	0.138754
$596$	0	0
$597$	−5.04413e10	−0.0162518
$598$	0	0
$599$	−2.82896e12	−0.897855	−0.448928	−	0.893568i	$-0.648194\pi$
$599$	−2.82896e12	−0.897855	−0.448928	+	0.893568i	$0.648194\pi$
$600$	0	0
$601$	−5.28549e12	−1.65253	−0.826266	−	0.563280i	$-0.809539\pi$
$601$	−5.28549e12	−1.65253	−0.826266	+	0.563280i	$0.809539\pi$
$602$	0	0
$603$	−1.59785e12	−0.492161
$604$	0	0
$605$	5.16481e12	1.56731
$606$	0	0
$607$	4.17198e12	1.24736	0.623682	−	0.781678i	$-0.285636\pi$
$607$	4.17198e12	1.24736	0.623682	+	0.781678i	$0.285636\pi$
$608$	0	0
$609$	3.33985e11	0.0983897
$610$	0	0
$611$	5.86407e12	1.70221
$612$	0	0
$613$	2.86672e12	0.819999	0.410000	−	0.912086i	$-0.365529\pi$
$613$	2.86672e12	0.819999	0.410000	+	0.912086i	$0.365529\pi$
$614$	0	0
$615$	−1.70597e12	−0.480875
$616$	0	0
$617$	1.49703e12	0.415859	0.207930	−	0.978144i	$-0.433328\pi$
$617$	1.49703e12	0.415859	0.207930	+	0.978144i	$0.433328\pi$
$618$	0	0
$619$	5.86995e12	1.60704	0.803520	−	0.595277i	$-0.202958\pi$
$619$	5.86995e12	1.60704	0.803520	+	0.595277i	$0.202958\pi$
$620$	0	0
$621$	−1.21786e11	−0.0328612
$622$	0	0
$623$	3.33382e11	0.0886636
$624$	0	0
$625$	−2.90823e12	−0.762374
$626$	0	0
$627$	4.71215e12	1.21763
$628$	0	0
$629$	−6.10683e12	−1.55556
$630$	0	0
$631$	4.13958e11	0.103950	0.0519749	−	0.998648i	$-0.483448\pi$
$631$	4.13958e11	0.103950	0.0519749	+	0.998648i	$0.483448\pi$
$632$	0	0
$633$	1.63605e12	0.405024
$634$	0	0
$635$	2.27554e12	0.555396
$636$	0	0
$637$	4.76763e12	1.14729
$638$	0	0
$639$	−6.92674e11	−0.164352
$640$	0	0
$641$	−7.29223e12	−1.70608	−0.853040	−	0.521845i	$-0.825244\pi$
$641$	−7.29223e12	−1.70608	−0.853040	+	0.521845i	$0.825244\pi$
$642$	0	0
$643$	−7.25513e12	−1.67377	−0.836885	−	0.547379i	$-0.815626\pi$
$643$	−7.25513e12	−1.67377	−0.836885	+	0.547379i	$0.815626\pi$
$644$	0	0
$645$	−1.65743e12	−0.377065
$646$	0	0
$647$	4.45684e12	0.999902	0.499951	−	0.866054i	$-0.333351\pi$
$647$	4.45684e12	0.999902	0.499951	+	0.866054i	$0.333351\pi$
$648$	0	0
$649$	9.24262e12	2.04500
$650$	0	0
$651$	1.93941e11	0.0423209
$652$	0	0
$653$	4.47800e11	0.0963773	0.0481886	−	0.998838i	$-0.484655\pi$
$653$	4.47800e11	0.0963773	0.0481886	+	0.998838i	$0.484655\pi$
$654$	0	0
$655$	6.19533e12	1.31516
$656$	0	0
$657$	−2.09657e11	−0.0439001
$658$	0	0
$659$	4.30424e12	0.889021	0.444510	−	0.895774i	$-0.353378\pi$
$659$	4.30424e12	0.889021	0.444510	+	0.895774i	$0.353378\pi$
$660$	0	0
$661$	5.41518e11	0.110333	0.0551666	−	0.998477i	$-0.482431\pi$
$661$	5.41518e11	0.110333	0.0551666	+	0.998477i	$0.482431\pi$
$662$	0	0
$663$	5.82288e12	1.17038
$664$	0	0
$665$	−5.07590e11	−0.100650
$666$	0	0
$667$	1.68270e12	0.329185
$668$	0	0
$669$	1.30784e12	0.252428
$670$	0	0
$671$	−9.78491e11	−0.186340
$672$	0	0
$673$	8.18075e12	1.53718	0.768591	−	0.639740i	$-0.220958\pi$
$673$	8.18075e12	1.53718	0.768591	+	0.639740i	$0.220958\pi$
$674$	0	0
$675$	−2.05832e11	−0.0381633
$676$	0	0
$677$	−4.69000e12	−0.858073	−0.429036	−	0.903287i	$-0.641147\pi$
$677$	−4.69000e12	−0.858073	−0.429036	+	0.903287i	$0.641147\pi$
$678$	0	0
$679$	4.28927e11	0.0774408
$680$	0	0
$681$	1.10808e12	0.197428
$682$	0	0
$683$	−6.17085e12	−1.08506	−0.542528	−	0.840038i	$-0.682533\pi$
$683$	−6.17085e12	−1.08506	−0.542528	+	0.840038i	$0.682533\pi$
$684$	0	0
$685$	5.06238e12	0.878512
$686$	0	0
$687$	−5.75457e12	−0.985616
$688$	0	0
$689$	−1.03908e13	−1.75656
$690$	0	0
$691$	−5.51124e12	−0.919597	−0.459799	−	0.888023i	$-0.652078\pi$
$691$	−5.51124e12	−0.919597	−0.459799	+	0.888023i	$0.652078\pi$
$692$	0	0
$693$	−2.96699e11	−0.0488671
$694$	0	0
$695$	7.57053e12	1.23082
$696$	0	0
$697$	1.01611e13	1.63077
$698$	0	0
$699$	7.73642e11	0.122572
$700$	0	0
$701$	7.87462e12	1.23168	0.615841	−	0.787871i	$-0.288817\pi$
$701$	7.87462e12	1.23168	0.615841	+	0.787871i	$0.288817\pi$
$702$	0	0
$703$	7.30727e12	1.12838
$704$	0	0
$705$	−4.99146e12	−0.760986
$706$	0	0
$707$	−9.26135e11	−0.139408
$708$	0	0
$709$	8.87402e12	1.31890	0.659451	−	0.751747i	$-0.270789\pi$
$709$	8.87402e12	1.31890	0.659451	+	0.751747i	$0.270789\pi$
$710$	0	0
$711$	−3.50861e12	−0.514899
$712$	0	0
$713$	9.77121e11	0.141594
$714$	0	0
$715$	−1.19996e13	−1.71707
$716$	0	0
$717$	1.87007e12	0.264255
$718$	0	0
$719$	7.24721e12	1.01132	0.505662	−	0.862731i	$-0.331248\pi$
$719$	7.24721e12	1.01132	0.505662	+	0.862731i	$0.331248\pi$
$720$	0	0
$721$	4.36637e11	0.0601745
$722$	0	0
$723$	−1.36487e12	−0.185768
$724$	0	0
$725$	2.84396e12	0.382298
$726$	0	0
$727$	−7.22078e12	−0.958693	−0.479346	−	0.877626i	$-0.659126\pi$
$727$	−7.22078e12	−0.958693	−0.479346	+	0.877626i	$0.659126\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	9.87202e12	1.27873
$732$	0	0
$733$	9.93345e12	1.27096	0.635480	−	0.772117i	$-0.280802\pi$
$733$	9.93345e12	1.27096	0.635480	+	0.772117i	$0.280802\pi$
$734$	0	0
$735$	−4.05818e12	−0.512906
$736$	0	0
$737$	−1.96125e13	−2.44867
$738$	0	0
$739$	2.09416e12	0.258292	0.129146	−	0.991626i	$-0.458776\pi$
$739$	2.09416e12	0.258292	0.129146	+	0.991626i	$0.458776\pi$
$740$	0	0
$741$	−6.96751e12	−0.848976
$742$	0	0
$743$	2.98254e10	0.00359035	0.00179518	−	0.999998i	$-0.499429\pi$
$743$	2.98254e10	0.00359035	0.00179518	+	0.999998i	$0.499429\pi$
$744$	0	0
$745$	−3.94460e12	−0.469137
$746$	0	0
$747$	4.94896e12	0.581529
$748$	0	0
$749$	6.11306e11	0.0709725
$750$	0	0
$751$	−1.54090e13	−1.76765	−0.883823	−	0.467821i	$-0.845039\pi$
$751$	−1.54090e13	−1.76765	−0.883823	+	0.467821i	$0.845039\pi$
$752$	0	0
$753$	−1.42597e12	−0.161635
$754$	0	0
$755$	−3.86681e11	−0.0433103
$756$	0	0
$757$	−5.80995e11	−0.0643044	−0.0321522	−	0.999483i	$-0.510236\pi$
$757$	−5.80995e11	−0.0643044	−0.0321522	+	0.999483i	$0.510236\pi$
$758$	0	0
$759$	−1.49484e12	−0.163496
$760$	0	0
$761$	4.96002e12	0.536108	0.268054	−	0.963404i	$-0.413619\pi$
$761$	4.96002e12	0.536108	0.268054	+	0.963404i	$0.413619\pi$
$762$	0	0
$763$	7.24111e11	0.0773472
$764$	0	0
$765$	−4.95640e12	−0.523227
$766$	0	0
$767$	−1.36664e13	−1.42585
$768$	0	0
$769$	1.42174e13	1.46606	0.733032	−	0.680194i	$-0.238104\pi$
$769$	1.42174e13	1.46606	0.733032	+	0.680194i	$0.238104\pi$
$770$	0	0
$771$	−1.41400e12	−0.144114
$772$	0	0
$773$	−9.34947e12	−0.941844	−0.470922	−	0.882175i	$-0.656079\pi$
$773$	−9.34947e12	−0.941844	−0.470922	+	0.882175i	$0.656079\pi$
$774$	0	0
$775$	1.65145e12	0.164440
$776$	0	0
$777$	−4.60099e11	−0.0452853
$778$	0	0
$779$	−1.21585e13	−1.18294
$780$	0	0
$781$	−8.50213e12	−0.817708
$782$	0	0
$783$	−3.90230e12	−0.371016
$784$	0	0
$785$	1.34797e13	1.26697
$786$	0	0
$787$	−3.37342e12	−0.313461	−0.156731	−	0.987641i	$-0.550095\pi$
$787$	−3.37342e12	−0.313461	−0.156731	+	0.987641i	$0.550095\pi$
$788$	0	0
$789$	−4.19756e12	−0.385612
$790$	0	0
$791$	−9.89775e11	−0.0898964
$792$	0	0
$793$	1.44682e12	0.129923
$794$	0	0
$795$	8.84462e12	0.785284
$796$	0	0
$797$	−1.69600e13	−1.48890	−0.744448	−	0.667680i	$-0.767287\pi$
$797$	−1.69600e13	−1.48890	−0.744448	+	0.667680i	$0.767287\pi$
$798$	0	0
$799$	2.97303e13	2.58070
$800$	0	0
$801$	−3.89524e12	−0.334340
$802$	0	0
$803$	−2.57341e12	−0.218418
$804$	0	0
$805$	1.61023e11	0.0135147
$806$	0	0
$807$	−1.26516e13	−1.05006
$808$	0	0
$809$	−1.04004e13	−0.853652	−0.426826	−	0.904334i	$-0.640368\pi$
$809$	−1.04004e13	−0.853652	−0.426826	+	0.904334i	$0.640368\pi$
$810$	0	0
$811$	−1.07761e12	−0.0874717	−0.0437359	−	0.999043i	$-0.513926\pi$
$811$	−1.07761e12	−0.0874717	−0.0437359	+	0.999043i	$0.513926\pi$
$812$	0	0
$813$	−4.52911e12	−0.363584
$814$	0	0
$815$	−6.71280e12	−0.532959
$816$	0	0
$817$	−1.18126e13	−0.927568
$818$	0	0
$819$	4.38706e11	0.0340719
$820$	0	0
$821$	−1.51220e12	−0.116162	−0.0580812	−	0.998312i	$-0.518498\pi$
$821$	−1.51220e12	−0.116162	−0.0580812	+	0.998312i	$0.518498\pi$
$822$	0	0
$823$	−4.40763e12	−0.334893	−0.167446	−	0.985881i	$-0.553552\pi$
$823$	−4.40763e12	−0.334893	−0.167446	+	0.985881i	$0.553552\pi$
$824$	0	0
$825$	−2.52646e12	−0.189876
$826$	0	0
$827$	−1.34950e13	−1.00323	−0.501613	−	0.865092i	$-0.667260\pi$
$827$	−1.34950e13	−1.00323	−0.501613	+	0.865092i	$0.667260\pi$
$828$	0	0
$829$	2.57710e13	1.89512	0.947558	−	0.319585i	$-0.103544\pi$
$829$	2.57710e13	1.89512	0.947558	+	0.319585i	$0.103544\pi$
$830$	0	0
$831$	−1.36145e12	−0.0990366
$832$	0	0
$833$	2.41714e13	1.73940
$834$	0	0
$835$	−1.67571e13	−1.19292
$836$	0	0
$837$	−2.26601e12	−0.159587
$838$	0	0
$839$	−3.79371e12	−0.264323	−0.132161	−	0.991228i	$-0.542192\pi$
$839$	−3.79371e12	−0.264323	−0.132161	+	0.991228i	$0.542192\pi$
$840$	0	0
$841$	3.94105e13	2.71663
$842$	0	0
$843$	2.24308e12	0.152975
$844$	0	0
$845$	4.47319e12	0.301830
$846$	0	0
$847$	−2.31772e12	−0.154734
$848$	0	0
$849$	7.74599e11	0.0511673
$850$	0	0
$851$	−2.31809e12	−0.151512
$852$	0	0
$853$	9.12640e12	0.590240	0.295120	−	0.955460i	$-0.404640\pi$
$853$	9.12640e12	0.590240	0.295120	+	0.955460i	$0.404640\pi$
$854$	0	0
$855$	5.93070e12	0.379541
$856$	0	0
$857$	−4.91982e12	−0.311555	−0.155778	−	0.987792i	$-0.549788\pi$
$857$	−4.91982e12	−0.311555	−0.155778	+	0.987792i	$0.549788\pi$
$858$	0	0
$859$	−7.92238e12	−0.496462	−0.248231	−	0.968701i	$-0.579849\pi$
$859$	−7.92238e12	−0.496462	−0.248231	+	0.968701i	$0.579849\pi$
$860$	0	0
$861$	7.65556e11	0.0474748
$862$	0	0
$863$	−2.91668e12	−0.178995	−0.0894975	−	0.995987i	$-0.528526\pi$
$863$	−2.91668e12	−0.178995	−0.0894975	+	0.995987i	$0.528526\pi$
$864$	0	0
$865$	−8.70646e12	−0.528773
$866$	0	0
$867$	1.99158e13	1.19705
$868$	0	0
$869$	−4.30659e13	−2.56180
$870$	0	0
$871$	2.89996e13	1.70730
$872$	0	0
$873$	−5.01160e12	−0.292020
$874$	0	0
$875$	1.64454e12	0.0948435
$876$	0	0
$877$	6.56703e12	0.374861	0.187431	−	0.982278i	$-0.439984\pi$
$877$	6.56703e12	0.374861	0.187431	+	0.982278i	$0.439984\pi$
$878$	0	0
$879$	−1.21540e13	−0.686701
$880$	0	0
$881$	−1.07579e13	−0.601640	−0.300820	−	0.953681i	$-0.597260\pi$
$881$	−1.07579e13	−0.601640	−0.300820	+	0.953681i	$0.597260\pi$
$882$	0	0
$883$	−2.76747e13	−1.53201	−0.766003	−	0.642837i	$-0.777757\pi$
$883$	−2.76747e13	−1.53201	−0.766003	+	0.642837i	$0.777757\pi$
$884$	0	0
$885$	1.16327e13	0.637436
$886$	0	0
$887$	−2.30611e13	−1.25090	−0.625451	−	0.780264i	$-0.715085\pi$
$887$	−2.30611e13	−1.25090	−0.625451	+	0.780264i	$0.715085\pi$
$888$	0	0
$889$	−1.02115e12	−0.0548318
$890$	0	0
$891$	3.46664e12	0.184272
$892$	0	0
$893$	−3.55744e13	−1.87200
$894$	0	0
$895$	−1.69545e13	−0.883244
$896$	0	0
$897$	2.21031e12	0.113995
$898$	0	0
$899$	3.13093e13	1.59865
$900$	0	0
$901$	−5.26805e13	−2.66311
$902$	0	0
$903$	7.43775e11	0.0372260
$904$	0	0
$905$	−1.27973e13	−0.634160
$906$	0	0
$907$	6.86843e12	0.336996	0.168498	−	0.985702i	$-0.446108\pi$
$907$	6.86843e12	0.336996	0.168498	+	0.985702i	$0.446108\pi$
$908$	0	0
$909$	1.08210e13	0.525690
$910$	0	0
$911$	−3.69252e13	−1.77619	−0.888096	−	0.459659i	$-0.847972\pi$
$911$	−3.69252e13	−1.77619	−0.888096	+	0.459659i	$0.847972\pi$
$912$	0	0
$913$	6.07453e13	2.89331
$914$	0	0
$915$	−1.23153e12	−0.0580829
$916$	0	0
$917$	−2.78017e12	−0.129840
$918$	0	0
$919$	1.23010e13	0.568880	0.284440	−	0.958694i	$-0.408192\pi$
$919$	1.23010e13	0.568880	0.284440	+	0.958694i	$0.408192\pi$
$920$	0	0
$921$	−4.72735e12	−0.216496
$922$	0	0
$923$	1.25715e13	0.570135
$924$	0	0
$925$	−3.91785e12	−0.175958
$926$	0	0
$927$	−5.10169e12	−0.226911
$928$	0	0
$929$	1.34100e13	0.590689	0.295345	−	0.955391i	$-0.404566\pi$
$929$	1.34100e13	0.590689	0.295345	+	0.955391i	$0.404566\pi$
$930$	0	0
$931$	−2.89228e13	−1.26173
$932$	0	0
$933$	−1.68825e13	−0.729407
$934$	0	0
$935$	−6.08367e13	−2.60323
$936$	0	0
$937$	1.18685e13	0.502998	0.251499	−	0.967858i	$-0.419077\pi$
$937$	1.18685e13	0.502998	0.251499	+	0.967858i	$0.419077\pi$
$938$	0	0
$939$	−3.32695e12	−0.139653
$940$	0	0
$941$	−1.42000e13	−0.590385	−0.295193	−	0.955438i	$-0.595384\pi$
$941$	−1.42000e13	−0.590385	−0.295193	+	0.955438i	$0.595384\pi$
$942$	0	0
$943$	3.85706e12	0.158838
$944$	0	0
$945$	−3.73424e11	−0.0152321
$946$	0	0
$947$	−2.04481e13	−0.826185	−0.413092	−	0.910689i	$-0.635551\pi$
$947$	−2.04481e13	−0.826185	−0.413092	+	0.910689i	$0.635551\pi$
$948$	0	0
$949$	3.80510e12	0.152289
$950$	0	0
$951$	−4.33385e12	−0.171815
$952$	0	0
$953$	2.34108e13	0.919387	0.459694	−	0.888078i	$-0.347959\pi$
$953$	2.34108e13	0.919387	0.459694	+	0.888078i	$0.347959\pi$
$954$	0	0
$955$	−4.76256e12	−0.185279
$956$	0	0
$957$	−4.78982e13	−1.84593
$958$	0	0
$959$	−2.27176e12	−0.0867317
$960$	0	0
$961$	−8.25873e12	−0.312362
$962$	0	0
$963$	−7.14252e12	−0.267629
$964$	0	0
$965$	1.98917e13	0.738412
$966$	0	0
$967$	3.10773e13	1.14294	0.571471	−	0.820622i	$-0.306373\pi$
$967$	3.10773e13	1.14294	0.571471	+	0.820622i	$0.306373\pi$
$968$	0	0
$969$	−3.53246e13	−1.28712
$970$	0	0
$971$	3.30119e13	1.19175	0.595874	−	0.803078i	$-0.296806\pi$
$971$	3.30119e13	1.19175	0.595874	+	0.803078i	$0.296806\pi$
$972$	0	0
$973$	−3.39729e12	−0.121514
$974$	0	0
$975$	3.73568e12	0.132388
$976$	0	0
$977$	2.09262e12	0.0734791	0.0367396	−	0.999325i	$-0.488303\pi$
$977$	2.09262e12	0.0734791	0.0367396	+	0.999325i	$0.488303\pi$
$978$	0	0
$979$	−4.78116e13	−1.66346
$980$	0	0
$981$	−8.46054e12	−0.291667
$982$	0	0
$983$	1.01778e13	0.347666	0.173833	−	0.984775i	$-0.444385\pi$
$983$	1.01778e13	0.347666	0.173833	+	0.984775i	$0.444385\pi$
$984$	0	0
$985$	1.87186e13	0.633595
$986$	0	0
$987$	2.23993e12	0.0751290
$988$	0	0
$989$	3.74732e12	0.124548
$990$	0	0
$991$	−4.55312e13	−1.49961	−0.749803	−	0.661661i	$-0.769852\pi$
$991$	−4.55312e13	−1.49961	−0.749803	+	0.661661i	$0.769852\pi$
$992$	0	0
$993$	−8.59020e12	−0.280370
$994$	0	0
$995$	−7.79240e11	−0.0252039
$996$	0	0
$997$	5.73400e13	1.83793	0.918966	−	0.394337i	$-0.129026\pi$
$997$	5.73400e13	1.83793	0.918966	+	0.394337i	$0.129026\pi$
$998$	0	0
$999$	5.37582e12	0.170765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.m.1.4	yes	5
4.3	odd	2		384.10.a.i.1.4	✓	5
8.3	odd	2		384.10.a.p.1.2	yes	5
8.5	even	2		384.10.a.l.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.i.1.4	✓	5	4.3	odd	2
384.10.a.l.1.2	yes	5	8.5	even	2
384.10.a.m.1.4	yes	5	1.1	even	1	trivial
384.10.a.p.1.2	yes	5	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} +1251.33 q^{5} -561.535 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} +1251.33 q^{5} -561.535 q^{7} +6561.00 q^{9} +80532.1 q^{11} -119077. q^{13} +101357. q^{15} -603707. q^{17} +722380. q^{19} -45484.3 q^{21} -229161. q^{23} -387309. q^{25} +531441. q^{27} -7.34286e6 q^{29} -4.26391e6 q^{31} +6.52310e6 q^{33} -702663. q^{35} +1.01156e7 q^{37} -9.64521e6 q^{39} -1.68312e7 q^{41} -1.63523e7 q^{43} +8.20995e6 q^{45} -4.92462e7 q^{47} -4.00383e7 q^{49} -4.89003e7 q^{51} +8.72617e7 q^{53} +1.00772e8 q^{55} +5.85128e7 q^{57} +1.14769e8 q^{59} -1.21503e7 q^{61} -3.68423e6 q^{63} -1.49004e8 q^{65} -2.43537e8 q^{67} -1.85620e7 q^{69} -1.05574e8 q^{71} -3.19551e7 q^{73} -3.13721e7 q^{75} -4.52216e7 q^{77} -5.34767e8 q^{79} +4.30467e7 q^{81} +7.54300e8 q^{83} -7.55434e8 q^{85} -5.94772e8 q^{87} -5.93697e8 q^{89} +6.68658e7 q^{91} -3.45376e8 q^{93} +9.03932e8 q^{95} -7.63848e8 q^{97} +5.28371e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 405 q^{3} - 772 q^{5} + 38 q^{7} + 32805 q^{9}+O(q^{10}) \) 5 * q + 405 * q^3 - 772 * q^5 + 38 * q^7 + 32805 * q^9	\( 5 q + 405 q^{3} - 772 q^{5} + 38 q^{7} + 32805 q^{9} + 41100 q^{11} - 22486 q^{13} - 62532 q^{15} - 1130 q^{17} - 664712 q^{19} + 3078 q^{21} - 369972 q^{23} + 823383 q^{25} + 2657205 q^{27} - 7390736 q^{29} - 9149938 q^{31} + 3329100 q^{33} - 3167800 q^{35} - 11922058 q^{37} - 1821366 q^{39} - 8471746 q^{41} - 8948896 q^{43} - 5065092 q^{45} + 5051660 q^{47} + 39616113 q^{49} - 91530 q^{51} + 31431984 q^{53} + 67216 q^{55} - 53841672 q^{57} + 204260948 q^{59} - 190850874 q^{61} + 249318 q^{63} - 165466760 q^{65} + 274483500 q^{67} - 29967732 q^{69} - 162722908 q^{71} - 508927538 q^{73} + 66694023 q^{75} - 428895960 q^{77} - 491411266 q^{79} + 215233605 q^{81} + 766279260 q^{83} - 713985400 q^{85} - 598649616 q^{87} - 954097990 q^{89} + 503505932 q^{91} - 741144978 q^{93} - 968680288 q^{95} - 677085326 q^{97} + 269657100 q^{99}+O(q^{100}) \) 5 * q + 405 * q^3 - 772 * q^5 + 38 * q^7 + 32805 * q^9 + 41100 * q^11 - 22486 * q^13 - 62532 * q^15 - 1130 * q^17 - 664712 * q^19 + 3078 * q^21 - 369972 * q^23 + 823383 * q^25 + 2657205 * q^27 - 7390736 * q^29 - 9149938 * q^31 + 3329100 * q^33 - 3167800 * q^35 - 11922058 * q^37 - 1821366 * q^39 - 8471746 * q^41 - 8948896 * q^43 - 5065092 * q^45 + 5051660 * q^47 + 39616113 * q^49 - 91530 * q^51 + 31431984 * q^53 + 67216 * q^55 - 53841672 * q^57 + 204260948 * q^59 - 190850874 * q^61 + 249318 * q^63 - 165466760 * q^65 + 274483500 * q^67 - 29967732 * q^69 - 162722908 * q^71 - 508927538 * q^73 + 66694023 * q^75 - 428895960 * q^77 - 491411266 * q^79 + 215233605 * q^81 + 766279260 * q^83 - 713985400 * q^85 - 598649616 * q^87 - 954097990 * q^89 + 503505932 * q^91 - 741144978 * q^93 - 968680288 * q^95 - 677085326 * q^97 + 269657100 * q^99

Introduction

L-functions

Modular forms

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Database

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