LMFDB - Embedded newform 384.10.a.m.1.1

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			1.1
Root			$43.8567$ 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)

$f(q)$	$=$	$q+81.0000 q^{3} -2126.01 q^{5} +1445.12 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} -2126.01 q^{5} +1445.12 q^{7} +6561.00 q^{9} -20159.9 q^{11} -150057. q^{13} -172206. q^{15} +351448. q^{17} +584303. q^{19} +117055. q^{21} +289317. q^{23} +2.56678e6 q^{25} +531441. q^{27} -2.55200e6 q^{29} +9.90613e6 q^{31} -1.63295e6 q^{33} -3.07234e6 q^{35} -3.78890e6 q^{37} -1.21546e7 q^{39} +1.02647e7 q^{41} +3.60796e6 q^{43} -1.39487e7 q^{45} +1.06168e7 q^{47} -3.82652e7 q^{49} +2.84673e7 q^{51} -4.14129e7 q^{53} +4.28601e7 q^{55} +4.73285e7 q^{57} +9.14005e7 q^{59} -1.51926e8 q^{61} +9.48144e6 q^{63} +3.19022e8 q^{65} +1.97423e8 q^{67} +2.34347e7 q^{69} +1.32283e8 q^{71} -1.50306e8 q^{73} +2.07909e8 q^{75} -2.91335e7 q^{77} +3.92882e8 q^{79} +4.30467e7 q^{81} +4.76638e7 q^{83} -7.47181e8 q^{85} -2.06712e8 q^{87} -1.00694e9 q^{89} -2.16850e8 q^{91} +8.02397e8 q^{93} -1.24223e9 q^{95} +8.13065e8 q^{97} -1.32269e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2126.01	−1.52125	−0.760623	−	0.649194i	$-0.775106\pi$
$5$	−2126.01	−1.52125	−0.760623	+	0.649194i	$0.775106\pi$
$6$	0	0
$7$	1445.12	0.227490	0.113745	−	0.993510i	$-0.463715\pi$
$7$	1445.12	0.227490	0.113745	+	0.993510i	$0.463715\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−20159.9	−0.415166	−0.207583	−	0.978217i	$-0.566560\pi$
$11$	−20159.9	−0.415166	−0.207583	+	0.978217i	$0.566560\pi$
$12$	0	0
$13$	−150057.	−1.45717	−0.728586	−	0.684955i	$-0.759822\pi$
$13$	−150057.	−1.45717	−0.728586	+	0.684955i	$0.759822\pi$
$14$	0	0
$15$	−172206.	−0.878292
$16$	0	0
$17$	351448.	1.02057	0.510283	−	0.860006i	$-0.329541\pi$
$17$	351448.	1.02057	0.510283	+	0.860006i	$0.329541\pi$
$18$	0	0
$19$	584303.	1.02860	0.514300	−	0.857610i	$-0.328052\pi$
$19$	584303.	1.02860	0.514300	+	0.857610i	$0.328052\pi$
$20$	0	0
$21$	117055.	0.131342
$22$	0	0
$23$	289317.	0.215575	0.107787	−	0.994174i	$-0.465623\pi$
$23$	289317.	0.215575	0.107787	+	0.994174i	$0.465623\pi$
$24$	0	0
$25$	2.56678e6	1.31419
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	−2.55200e6	−0.670022	−0.335011	−	0.942214i	$-0.608740\pi$
$29$	−2.55200e6	−0.670022	−0.335011	+	0.942214i	$0.608740\pi$
$30$	0	0
$31$	9.90613e6	1.92653	0.963267	−	0.268546i	$-0.0865432\pi$
$31$	9.90613e6	1.92653	0.963267	+	0.268546i	$0.0865432\pi$
$32$	0	0
$33$	−1.63295e6	−0.239696
$34$	0	0
$35$	−3.07234e6	−0.346069
$36$	0	0
$37$	−3.78890e6	−0.332358	−0.166179	−	0.986096i	$-0.553143\pi$
$37$	−3.78890e6	−0.332358	−0.166179	+	0.986096i	$0.553143\pi$
$38$	0	0
$39$	−1.21546e7	−0.841298
$40$	0	0
$41$	1.02647e7	0.567308	0.283654	−	0.958927i	$-0.408453\pi$
$41$	1.02647e7	0.567308	0.283654	+	0.958927i	$0.408453\pi$
$42$	0	0
$43$	3.60796e6	0.160936	0.0804680	−	0.996757i	$-0.474359\pi$
$43$	3.60796e6	0.160936	0.0804680	+	0.996757i	$0.474359\pi$
$44$	0	0
$45$	−1.39487e7	−0.507082
$46$	0	0
$47$	1.06168e7	0.317362	0.158681	−	0.987330i	$-0.449276\pi$
$47$	1.06168e7	0.317362	0.158681	+	0.987330i	$0.449276\pi$
$48$	0	0
$49$	−3.82652e7	−0.948248
$50$	0	0
$51$	2.84673e7	0.589224
$52$	0	0
$53$	−4.14129e7	−0.720933	−0.360466	−	0.932772i	$-0.617382\pi$
$53$	−4.14129e7	−0.720933	−0.360466	+	0.932772i	$0.617382\pi$
$54$	0	0
$55$	4.28601e7	0.631569
$56$	0	0
$57$	4.73285e7	0.593863
$58$	0	0
$59$	9.14005e7	0.982007	0.491003	−	0.871158i	$-0.336630\pi$
$59$	9.14005e7	0.982007	0.491003	+	0.871158i	$0.336630\pi$
$60$	0	0
$61$	−1.51926e8	−1.40491	−0.702454	−	0.711729i	$-0.747913\pi$
$61$	−1.51926e8	−1.40491	−0.702454	+	0.711729i	$0.747913\pi$
$62$	0	0
$63$	9.48144e6	0.0758302
$64$	0	0
$65$	3.19022e8	2.21672
$66$	0	0
$67$	1.97423e8	1.19691	0.598456	−	0.801156i	$-0.295781\pi$
$67$	1.97423e8	1.19691	0.598456	+	0.801156i	$0.295781\pi$
$68$	0	0
$69$	2.34347e7	0.124462
$70$	0	0
$71$	1.32283e8	0.617791	0.308896	−	0.951096i	$-0.400041\pi$
$71$	1.32283e8	0.617791	0.308896	+	0.951096i	$0.400041\pi$
$72$	0	0
$73$	−1.50306e8	−0.619474	−0.309737	−	0.950822i	$-0.600241\pi$
$73$	−1.50306e8	−0.619474	−0.309737	+	0.950822i	$0.600241\pi$
$74$	0	0
$75$	2.07909e8	0.758748
$76$	0	0
$77$	−2.91335e7	−0.0944463
$78$	0	0
$79$	3.92882e8	1.13485	0.567427	−	0.823423i	$-0.307939\pi$
$79$	3.92882e8	1.13485	0.567427	+	0.823423i	$0.307939\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	4.76638e7	0.110240	0.0551198	−	0.998480i	$-0.482446\pi$
$83$	4.76638e7	0.110240	0.0551198	+	0.998480i	$0.482446\pi$
$84$	0	0
$85$	−7.47181e8	−1.55253
$86$	0	0
$87$	−2.06712e8	−0.386837
$88$	0	0
$89$	−1.00694e9	−1.70118	−0.850588	−	0.525832i	$-0.823754\pi$
$89$	−1.00694e9	−1.70118	−0.850588	+	0.525832i	$0.823754\pi$
$90$	0	0
$91$	−2.16850e8	−0.331493
$92$	0	0
$93$	8.02397e8	1.11228
$94$	0	0
$95$	−1.24223e9	−1.56475
$96$	0	0
$97$	8.13065e8	0.932508	0.466254	−	0.884651i	$-0.345603\pi$
$97$	8.13065e8	0.932508	0.466254	+	0.884651i	$0.345603\pi$
$98$	0	0
$99$	−1.32269e8	−0.138389
$100$	0	0
$101$	−1.06164e9	−1.01515	−0.507575	−	0.861607i	$-0.669458\pi$
$101$	−1.06164e9	−1.01515	−0.507575	+	0.861607i	$0.669458\pi$
$102$	0	0
$103$	−2.08863e9	−1.82849	−0.914246	−	0.405160i	$-0.867216\pi$
$103$	−2.08863e9	−1.82849	−0.914246	+	0.405160i	$0.867216\pi$
$104$	0	0
$105$	−2.48859e8	−0.199803
$106$	0	0
$107$	−1.01080e9	−0.745487	−0.372744	−	0.927934i	$-0.621583\pi$
$107$	−1.01080e9	−0.745487	−0.372744	+	0.927934i	$0.621583\pi$
$108$	0	0
$109$	2.42877e9	1.64804	0.824019	−	0.566562i	$-0.191727\pi$
$109$	2.42877e9	1.64804	0.824019	+	0.566562i	$0.191727\pi$
$110$	0	0
$111$	−3.06901e8	−0.191887
$112$	0	0
$113$	−1.64111e9	−0.946858	−0.473429	−	0.880832i	$-0.656984\pi$
$113$	−1.64111e9	−0.946858	−0.473429	+	0.880832i	$0.656984\pi$
$114$	0	0
$115$	−6.15089e8	−0.327943
$116$	0	0
$117$	−9.84523e8	−0.485724
$118$	0	0
$119$	5.07886e8	0.232169
$120$	0	0
$121$	−1.95153e9	−0.827637
$122$	0	0
$123$	8.31441e8	0.327535
$124$	0	0
$125$	−1.30463e9	−0.477960
$126$	0	0
$127$	−8.28568e8	−0.282626	−0.141313	−	0.989965i	$-0.545132\pi$
$127$	−8.28568e8	−0.282626	−0.141313	+	0.989965i	$0.545132\pi$
$128$	0	0
$129$	2.92245e8	0.0929165
$130$	0	0
$131$	−4.06490e8	−0.120595	−0.0602975	−	0.998180i	$-0.519205\pi$
$131$	−4.06490e8	−0.120595	−0.0602975	+	0.998180i	$0.519205\pi$
$132$	0	0
$133$	8.44389e8	0.233997
$134$	0	0
$135$	−1.12985e9	−0.292764
$136$	0	0
$137$	−4.35544e9	−1.05631	−0.528153	−	0.849149i	$-0.677115\pi$
$137$	−4.35544e9	−1.05631	−0.528153	+	0.849149i	$0.677115\pi$
$138$	0	0
$139$	4.87849e9	1.10846	0.554228	−	0.832365i	$-0.313014\pi$
$139$	4.87849e9	1.10846	0.554228	+	0.832365i	$0.313014\pi$
$140$	0	0
$141$	8.59963e8	0.183229
$142$	0	0
$143$	3.02513e9	0.604968
$144$	0	0
$145$	5.42556e9	1.01927
$146$	0	0
$147$	−3.09948e9	−0.547471
$148$	0	0
$149$	−6.24917e9	−1.03869	−0.519343	−	0.854566i	$-0.673823\pi$
$149$	−6.24917e9	−1.03869	−0.519343	+	0.854566i	$0.673823\pi$
$150$	0	0
$151$	−7.69710e9	−1.20484	−0.602422	−	0.798178i	$-0.705798\pi$
$151$	−7.69710e9	−1.20484	−0.602422	+	0.798178i	$0.705798\pi$
$152$	0	0
$153$	2.30585e9	0.340189
$154$	0	0
$155$	−2.10605e10	−2.93073
$156$	0	0
$157$	−7.53416e8	−0.0989661	−0.0494831	−	0.998775i	$-0.515757\pi$
$157$	−7.53416e8	−0.0989661	−0.0494831	+	0.998775i	$0.515757\pi$
$158$	0	0
$159$	−3.35445e9	−0.416231
$160$	0	0
$161$	4.18098e8	0.0490413
$162$	0	0
$163$	9.24666e9	1.02598	0.512992	−	0.858393i	$-0.328537\pi$
$163$	9.24666e9	1.02598	0.512992	+	0.858393i	$0.328537\pi$
$164$	0	0
$165$	3.47167e9	0.364637
$166$	0	0
$167$	1.32098e10	1.31423	0.657114	−	0.753791i	$-0.271777\pi$
$167$	1.32098e10	1.31423	0.657114	+	0.753791i	$0.271777\pi$
$168$	0	0
$169$	1.19125e10	1.12335
$170$	0	0
$171$	3.83361e9	0.342867
$172$	0	0
$173$	6.06746e8	0.0514991	0.0257495	−	0.999668i	$-0.491803\pi$
$173$	6.06746e8	0.0514991	0.0257495	+	0.999668i	$0.491803\pi$
$174$	0	0
$175$	3.70930e9	0.298966
$176$	0	0
$177$	7.40344e9	0.566962
$178$	0	0
$179$	6.46950e9	0.471012	0.235506	−	0.971873i	$-0.424325\pi$
$179$	6.46950e9	0.471012	0.235506	+	0.971873i	$0.424325\pi$
$180$	0	0
$181$	−4.92300e9	−0.340938	−0.170469	−	0.985363i	$-0.554528\pi$
$181$	−4.92300e9	−0.340938	−0.170469	+	0.985363i	$0.554528\pi$
$182$	0	0
$183$	−1.23060e10	−0.811125
$184$	0	0
$185$	8.05523e9	0.505598
$186$	0	0
$187$	−7.08517e9	−0.423704
$188$	0	0
$189$	7.67997e8	0.0437806
$190$	0	0
$191$	1.45071e10	0.788735	0.394367	−	0.918953i	$-0.370964\pi$
$191$	1.45071e10	0.788735	0.394367	+	0.918953i	$0.370964\pi$
$192$	0	0
$193$	3.02370e10	1.56867	0.784333	−	0.620340i	$-0.213005\pi$
$193$	3.02370e10	1.56867	0.784333	+	0.620340i	$0.213005\pi$
$194$	0	0
$195$	2.58408e10	1.27982
$196$	0	0
$197$	−3.70014e10	−1.75033	−0.875166	−	0.483823i	$-0.839248\pi$
$197$	−3.70014e10	−1.75033	−0.875166	+	0.483823i	$0.839248\pi$
$198$	0	0
$199$	−5.46975e9	−0.247246	−0.123623	−	0.992329i	$-0.539451\pi$
$199$	−5.46975e9	−0.247246	−0.123623	+	0.992329i	$0.539451\pi$
$200$	0	0
$201$	1.59913e10	0.691037
$202$	0	0
$203$	−3.68795e9	−0.152424
$204$	0	0
$205$	−2.18228e10	−0.863015
$206$	0	0
$207$	1.89821e9	0.0718583
$208$	0	0
$209$	−1.17795e10	−0.427040
$210$	0	0
$211$	−5.46026e10	−1.89645	−0.948226	−	0.317595i	$-0.897125\pi$
$211$	−5.46026e10	−1.89645	−0.948226	+	0.317595i	$0.897125\pi$
$212$	0	0
$213$	1.07149e10	0.356682
$214$	0	0
$215$	−7.67054e9	−0.244823
$216$	0	0
$217$	1.43156e10	0.438268
$218$	0	0
$219$	−1.21748e10	−0.357654
$220$	0	0
$221$	−5.27372e10	−1.48714
$222$	0	0
$223$	5.33790e9	0.144543	0.0722717	−	0.997385i	$-0.476975\pi$
$223$	5.33790e9	0.144543	0.0722717	+	0.997385i	$0.476975\pi$
$224$	0	0
$225$	1.68406e10	0.438063
$226$	0	0
$227$	−5.32409e10	−1.33085	−0.665425	−	0.746465i	$-0.731750\pi$
$227$	−5.32409e10	−1.33085	−0.665425	+	0.746465i	$0.731750\pi$
$228$	0	0
$229$	3.47725e10	0.835558	0.417779	−	0.908549i	$-0.362809\pi$
$229$	3.47725e10	0.835558	0.417779	+	0.908549i	$0.362809\pi$
$230$	0	0
$231$	−2.35982e9	−0.0545286
$232$	0	0
$233$	4.88898e10	1.08672	0.543358	−	0.839501i	$-0.317152\pi$
$233$	4.88898e10	1.08672	0.543358	+	0.839501i	$0.317152\pi$
$234$	0	0
$235$	−2.25714e10	−0.482785
$236$	0	0
$237$	3.18234e10	0.655209
$238$	0	0
$239$	−4.43988e10	−0.880198	−0.440099	−	0.897949i	$-0.645057\pi$
$239$	−4.43988e10	−0.880198	−0.440099	+	0.897949i	$0.645057\pi$
$240$	0	0
$241$	−5.37387e10	−1.02615	−0.513075	−	0.858344i	$-0.671494\pi$
$241$	−5.37387e10	−1.02615	−0.513075	+	0.858344i	$0.671494\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	8.13521e10	1.44252
$246$	0	0
$247$	−8.76786e10	−1.49885
$248$	0	0
$249$	3.86077e9	0.0636468
$250$	0	0
$251$	−9.51803e10	−1.51362	−0.756808	−	0.653638i	$-0.773242\pi$
$251$	−9.51803e10	−1.51362	−0.756808	+	0.653638i	$0.773242\pi$
$252$	0	0
$253$	−5.83260e9	−0.0894994
$254$	0	0
$255$	−6.05217e10	−0.896355
$256$	0	0
$257$	−8.76361e10	−1.25310	−0.626548	−	0.779383i	$-0.715533\pi$
$257$	−8.76361e10	−1.25310	−0.626548	+	0.779383i	$0.715533\pi$
$258$	0	0
$259$	−5.47543e9	−0.0756082
$260$	0	0
$261$	−1.67436e10	−0.223341
$262$	0	0
$263$	−1.12390e11	−1.44853	−0.724265	−	0.689522i	$-0.757821\pi$
$263$	−1.12390e11	−1.44853	−0.724265	+	0.689522i	$0.757821\pi$
$264$	0	0
$265$	8.80442e10	1.09672
$266$	0	0
$267$	−8.15623e10	−0.982175
$268$	0	0
$269$	−1.30872e11	−1.52392	−0.761960	−	0.647624i	$-0.775763\pi$
$269$	−1.30872e11	−1.52392	−0.761960	+	0.647624i	$0.775763\pi$
$270$	0	0
$271$	−5.12023e10	−0.576670	−0.288335	−	0.957530i	$-0.593102\pi$
$271$	−5.12023e10	−0.576670	−0.288335	+	0.957530i	$0.593102\pi$
$272$	0	0
$273$	−1.75649e10	−0.191387
$274$	0	0
$275$	−5.17460e10	−0.545607
$276$	0	0
$277$	−7.35027e10	−0.750143	−0.375072	−	0.926996i	$-0.622382\pi$
$277$	−7.35027e10	−0.750143	−0.375072	+	0.926996i	$0.622382\pi$
$278$	0	0
$279$	6.49941e10	0.642178
$280$	0	0
$281$	8.63145e10	0.825857	0.412929	−	0.910763i	$-0.364506\pi$
$281$	8.63145e10	0.825857	0.412929	+	0.910763i	$0.364506\pi$
$282$	0	0
$283$	6.02580e10	0.558439	0.279219	−	0.960227i	$-0.409924\pi$
$283$	6.02580e10	0.558439	0.279219	+	0.960227i	$0.409924\pi$
$284$	0	0
$285$	−1.00621e11	−0.903412
$286$	0	0
$287$	1.48337e10	0.129057
$288$	0	0
$289$	4.92803e9	0.0415559
$290$	0	0
$291$	6.58583e10	0.538384
$292$	0	0
$293$	−1.73424e11	−1.37469	−0.687344	−	0.726332i	$-0.741223\pi$
$293$	−1.73424e11	−1.37469	−0.687344	+	0.726332i	$0.741223\pi$
$294$	0	0
$295$	−1.94318e11	−1.49387
$296$	0	0
$297$	−1.07138e10	−0.0798987
$298$	0	0
$299$	−4.34139e10	−0.314130
$300$	0	0
$301$	5.21394e9	0.0366114
$302$	0	0
$303$	−8.59927e10	−0.586098
$304$	0	0
$305$	3.22996e11	2.13721
$306$	0	0
$307$	−1.83367e11	−1.17814	−0.589071	−	0.808081i	$-0.700506\pi$
$307$	−1.83367e11	−1.17814	−0.589071	+	0.808081i	$0.700506\pi$
$308$	0	0
$309$	−1.69179e11	−1.05568
$310$	0	0
$311$	−6.67223e10	−0.404436	−0.202218	−	0.979341i	$-0.564815\pi$
$311$	−6.67223e10	−0.404436	−0.202218	+	0.979341i	$0.564815\pi$
$312$	0	0
$313$	1.40731e11	0.828784	0.414392	−	0.910098i	$-0.363994\pi$
$313$	1.40731e11	0.828784	0.414392	+	0.910098i	$0.363994\pi$
$314$	0	0
$315$	−2.01576e10	−0.115356
$316$	0	0
$317$	1.34125e11	0.746007	0.373004	−	0.927830i	$-0.378328\pi$
$317$	1.34125e11	0.746007	0.373004	+	0.927830i	$0.378328\pi$
$318$	0	0
$319$	5.14480e10	0.278170
$320$	0	0
$321$	−8.18751e10	−0.430407
$322$	0	0
$323$	2.05352e11	1.04976
$324$	0	0
$325$	−3.85162e11	−1.91500
$326$	0	0
$327$	1.96730e11	0.951496
$328$	0	0
$329$	1.53426e10	0.0721967
$330$	0	0
$331$	−1.72093e11	−0.788020	−0.394010	−	0.919106i	$-0.628912\pi$
$331$	−1.72093e11	−0.788020	−0.394010	+	0.919106i	$0.628912\pi$
$332$	0	0
$333$	−2.48590e10	−0.110786
$334$	0	0
$335$	−4.19723e11	−1.82080
$336$	0	0
$337$	2.52423e11	1.06609	0.533046	−	0.846086i	$-0.321047\pi$
$337$	2.52423e11	1.06609	0.533046	+	0.846086i	$0.321047\pi$
$338$	0	0
$339$	−1.32930e11	−0.546669
$340$	0	0
$341$	−1.99707e11	−0.799831
$342$	0	0
$343$	−1.13614e11	−0.443208
$344$	0	0
$345$	−4.98222e10	−0.189338
$346$	0	0
$347$	3.95517e11	1.46448	0.732238	−	0.681049i	$-0.238476\pi$
$347$	3.95517e11	1.46448	0.732238	+	0.681049i	$0.238476\pi$
$348$	0	0
$349$	−4.50336e10	−0.162488	−0.0812441	−	0.996694i	$-0.525889\pi$
$349$	−4.50336e10	−0.162488	−0.0812441	+	0.996694i	$0.525889\pi$
$350$	0	0
$351$	−7.97463e10	−0.280433
$352$	0	0
$353$	1.82955e11	0.627130	0.313565	−	0.949567i	$-0.398477\pi$
$353$	1.82955e11	0.627130	0.313565	+	0.949567i	$0.398477\pi$
$354$	0	0
$355$	−2.81235e11	−0.939812
$356$	0	0
$357$	4.11387e10	0.134043
$358$	0	0
$359$	5.89461e10	0.187297	0.0936483	−	0.995605i	$-0.470147\pi$
$359$	5.89461e10	0.187297	0.0936483	+	0.995605i	$0.470147\pi$
$360$	0	0
$361$	1.87222e10	0.0580195
$362$	0	0
$363$	−1.58074e11	−0.477837
$364$	0	0
$365$	3.19551e11	0.942373
$366$	0	0
$367$	5.40036e11	1.55391	0.776955	−	0.629556i	$-0.216763\pi$
$367$	5.40036e11	1.55391	0.776955	+	0.629556i	$0.216763\pi$
$368$	0	0
$369$	6.73467e10	0.189103
$370$	0	0
$371$	−5.98467e10	−0.164005
$372$	0	0
$373$	−6.12254e11	−1.63773	−0.818865	−	0.573986i	$-0.805396\pi$
$373$	−6.12254e11	−1.63773	−0.818865	+	0.573986i	$0.805396\pi$
$374$	0	0
$375$	−1.05675e11	−0.275950
$376$	0	0
$377$	3.82944e11	0.976337
$378$	0	0
$379$	4.08851e11	1.01786	0.508931	−	0.860808i	$-0.330041\pi$
$379$	4.08851e11	1.01786	0.508931	+	0.860808i	$0.330041\pi$
$380$	0	0
$381$	−6.71140e10	−0.163174
$382$	0	0
$383$	−6.10228e11	−1.44910	−0.724549	−	0.689224i	$-0.757952\pi$
$383$	−6.10228e11	−1.44910	−0.724549	+	0.689224i	$0.757952\pi$
$384$	0	0
$385$	6.19380e10	0.143676
$386$	0	0
$387$	2.36718e10	0.0536454
$388$	0	0
$389$	−9.17629e10	−0.203186	−0.101593	−	0.994826i	$-0.532394\pi$
$389$	−9.17629e10	−0.203186	−0.101593	+	0.994826i	$0.532394\pi$
$390$	0	0
$391$	1.01680e11	0.220009
$392$	0	0
$393$	−3.29257e10	−0.0696255
$394$	0	0
$395$	−8.35270e11	−1.72639
$396$	0	0
$397$	−2.90677e11	−0.587291	−0.293646	−	0.955914i	$-0.594869\pi$
$397$	−2.90677e11	−0.587291	−0.293646	+	0.955914i	$0.594869\pi$
$398$	0	0
$399$	6.83955e10	0.135098
$400$	0	0
$401$	−4.22231e10	−0.0815456	−0.0407728	−	0.999168i	$-0.512982\pi$
$401$	−4.22231e10	−0.0815456	−0.0407728	+	0.999168i	$0.512982\pi$
$402$	0	0
$403$	−1.48648e12	−2.80729
$404$	0	0
$405$	−9.15176e10	−0.169027
$406$	0	0
$407$	7.63840e10	0.137984
$408$	0	0
$409$	−2.59030e11	−0.457715	−0.228857	−	0.973460i	$-0.573499\pi$
$409$	−2.59030e11	−0.457715	−0.228857	+	0.973460i	$0.573499\pi$
$410$	0	0
$411$	−3.52791e11	−0.609859
$412$	0	0
$413$	1.32085e11	0.223397
$414$	0	0
$415$	−1.01334e11	−0.167701
$416$	0	0
$417$	3.95157e11	0.639968
$418$	0	0
$419$	8.54680e11	1.35469	0.677346	−	0.735665i	$-0.263130\pi$
$419$	8.54680e11	1.35469	0.677346	+	0.735665i	$0.263130\pi$
$420$	0	0
$421$	5.85059e11	0.907675	0.453837	−	0.891085i	$-0.350055\pi$
$421$	5.85059e11	0.907675	0.453837	+	0.891085i	$0.350055\pi$
$422$	0	0
$423$	6.96570e10	0.105787
$424$	0	0
$425$	9.02089e11	1.34122
$426$	0	0
$427$	−2.19552e11	−0.319603
$428$	0	0
$429$	2.45036e11	0.349278
$430$	0	0
$431$	−1.32751e12	−1.85306	−0.926532	−	0.376215i	$-0.877225\pi$
$431$	−1.32751e12	−1.85306	−0.926532	+	0.376215i	$0.877225\pi$
$432$	0	0
$433$	−9.21842e11	−1.26026	−0.630131	−	0.776489i	$-0.716999\pi$
$433$	−9.21842e11	−1.26026	−0.630131	+	0.776489i	$0.716999\pi$
$434$	0	0
$435$	4.39470e11	0.588475
$436$	0	0
$437$	1.69049e11	0.221741
$438$	0	0
$439$	−4.11660e11	−0.528992	−0.264496	−	0.964387i	$-0.585206\pi$
$439$	−4.11660e11	−0.528992	−0.264496	+	0.964387i	$0.585206\pi$
$440$	0	0
$441$	−2.51058e11	−0.316083
$442$	0	0
$443$	−9.98623e10	−0.123193	−0.0615964	−	0.998101i	$-0.519619\pi$
$443$	−9.98623e10	−0.123193	−0.0615964	+	0.998101i	$0.519619\pi$
$444$	0	0
$445$	2.14077e12	2.58791
$446$	0	0
$447$	−5.06183e11	−0.599686
$448$	0	0
$449$	−1.41938e12	−1.64812	−0.824060	−	0.566503i	$-0.808296\pi$
$449$	−1.41938e12	−1.64812	−0.824060	+	0.566503i	$0.808296\pi$
$450$	0	0
$451$	−2.06935e11	−0.235527
$452$	0	0
$453$	−6.23465e11	−0.695617
$454$	0	0
$455$	4.61025e11	0.504282
$456$	0	0
$457$	1.45069e12	1.55579	0.777897	−	0.628392i	$-0.216287\pi$
$457$	1.45069e12	1.55579	0.777897	+	0.628392i	$0.216287\pi$
$458$	0	0
$459$	1.86774e11	0.196408
$460$	0	0
$461$	1.12468e12	1.15978	0.579891	−	0.814694i	$-0.303095\pi$
$461$	1.12468e12	1.15978	0.579891	+	0.814694i	$0.303095\pi$
$462$	0	0
$463$	1.05071e12	1.06260	0.531299	−	0.847184i	$-0.321704\pi$
$463$	1.05071e12	1.06260	0.531299	+	0.847184i	$0.321704\pi$
$464$	0	0
$465$	−1.70590e12	−1.69206
$466$	0	0
$467$	−1.95982e12	−1.90673	−0.953366	−	0.301817i	$-0.902407\pi$
$467$	−1.95982e12	−1.90673	−0.953366	+	0.301817i	$0.902407\pi$
$468$	0	0
$469$	2.85301e11	0.272286
$470$	0	0
$471$	−6.10267e10	−0.0571381
$472$	0	0
$473$	−7.27361e10	−0.0668152
$474$	0	0
$475$	1.49977e12	1.35178
$476$	0	0
$477$	−2.71710e11	−0.240311
$478$	0	0
$479$	1.53456e12	1.33191	0.665954	−	0.745993i	$-0.268025\pi$
$479$	1.53456e12	1.33191	0.665954	+	0.745993i	$0.268025\pi$
$480$	0	0
$481$	5.68551e11	0.484302
$482$	0	0
$483$	3.38659e10	0.0283140
$484$	0	0
$485$	−1.72858e12	−1.41857
$486$	0	0
$487$	−4.84071e11	−0.389968	−0.194984	−	0.980806i	$-0.562465\pi$
$487$	−4.84071e11	−0.389968	−0.194984	+	0.980806i	$0.562465\pi$
$488$	0	0
$489$	7.48979e11	0.592352
$490$	0	0
$491$	−6.19993e11	−0.481415	−0.240708	−	0.970598i	$-0.577380\pi$
$491$	−6.19993e11	−0.481415	−0.240708	+	0.970598i	$0.577380\pi$
$492$	0	0
$493$	−8.96895e11	−0.683802
$494$	0	0
$495$	2.81205e11	0.210523
$496$	0	0
$497$	1.91165e11	0.140542
$498$	0	0
$499$	1.55499e10	0.0112273	0.00561364	−	0.999984i	$-0.498213\pi$
$499$	1.55499e10	0.0112273	0.00561364	+	0.999984i	$0.498213\pi$
$500$	0	0
$501$	1.06999e12	0.758770
$502$	0	0
$503$	1.35355e12	0.942799	0.471400	−	0.881920i	$-0.343749\pi$
$503$	1.35355e12	0.942799	0.471400	+	0.881920i	$0.343749\pi$
$504$	0	0
$505$	2.25705e12	1.54429
$506$	0	0
$507$	9.64916e11	0.648565
$508$	0	0
$509$	−8.85998e11	−0.585063	−0.292532	−	0.956256i	$-0.594498\pi$
$509$	−8.85998e11	−0.585063	−0.292532	+	0.956256i	$0.594498\pi$
$510$	0	0
$511$	−2.17210e11	−0.140925
$512$	0	0
$513$	3.10523e11	0.197954
$514$	0	0
$515$	4.44043e12	2.78159
$516$	0	0
$517$	−2.14034e11	−0.131758
$518$	0	0
$519$	4.91464e10	0.0297330
$520$	0	0
$521$	−2.02158e12	−1.20205	−0.601025	−	0.799231i	$-0.705241\pi$
$521$	−2.02158e12	−1.20205	−0.601025	+	0.799231i	$0.705241\pi$
$522$	0	0
$523$	1.78264e12	1.04185	0.520927	−	0.853601i	$-0.325586\pi$
$523$	1.78264e12	1.04185	0.520927	+	0.853601i	$0.325586\pi$
$524$	0	0
$525$	3.00454e11	0.172608
$526$	0	0
$527$	3.48149e12	1.96616
$528$	0	0
$529$	−1.71745e12	−0.953527
$530$	0	0
$531$	5.99679e11	0.327336
$532$	0	0
$533$	−1.54029e12	−0.826665
$534$	0	0
$535$	2.14898e12	1.13407
$536$	0	0
$537$	5.24029e11	0.271939
$538$	0	0
$539$	7.71424e11	0.393680
$540$	0	0
$541$	1.53295e12	0.769380	0.384690	−	0.923046i	$-0.374308\pi$
$541$	1.53295e12	0.769380	0.384690	+	0.923046i	$0.374308\pi$
$542$	0	0
$543$	−3.98763e11	−0.196841
$544$	0	0
$545$	−5.16358e12	−2.50707
$546$	0	0
$547$	3.82914e12	1.82877	0.914384	−	0.404848i	$-0.132676\pi$
$547$	3.82914e12	1.82877	0.914384	+	0.404848i	$0.132676\pi$
$548$	0	0
$549$	−9.96787e11	−0.468303
$550$	0	0
$551$	−1.49114e12	−0.689185
$552$	0	0
$553$	5.67762e11	0.258169
$554$	0	0
$555$	6.52474e11	0.291907
$556$	0	0
$557$	−1.93062e12	−0.849861	−0.424930	−	0.905226i	$-0.639701\pi$
$557$	−1.93062e12	−0.849861	−0.424930	+	0.905226i	$0.639701\pi$
$558$	0	0
$559$	−5.41399e11	−0.234511
$560$	0	0
$561$	−5.73899e11	−0.244626
$562$	0	0
$563$	−2.31015e12	−0.969064	−0.484532	−	0.874774i	$-0.661010\pi$
$563$	−2.31015e12	−0.969064	−0.484532	+	0.874774i	$0.661010\pi$
$564$	0	0
$565$	3.48901e12	1.44040
$566$	0	0
$567$	6.22077e10	0.0252767
$568$	0	0
$569$	1.54228e12	0.616818	0.308409	−	0.951254i	$-0.400203\pi$
$569$	1.54228e12	0.616818	0.308409	+	0.951254i	$0.400203\pi$
$570$	0	0
$571$	−4.11331e12	−1.61931	−0.809653	−	0.586909i	$-0.800345\pi$
$571$	−4.11331e12	−1.61931	−0.809653	+	0.586909i	$0.800345\pi$
$572$	0	0
$573$	1.17508e12	0.455376
$574$	0	0
$575$	7.42611e11	0.283306
$576$	0	0
$577$	−2.75851e12	−1.03606	−0.518029	−	0.855363i	$-0.673334\pi$
$577$	−2.75851e12	−1.03606	−0.518029	+	0.855363i	$0.673334\pi$
$578$	0	0
$579$	2.44920e12	0.905670
$580$	0	0
$581$	6.88800e10	0.0250784
$582$	0	0
$583$	8.34881e11	0.299307
$584$	0	0
$585$	2.09310e12	0.738905
$586$	0	0
$587$	4.68695e12	1.62937	0.814684	−	0.579905i	$-0.196910\pi$
$587$	4.68695e12	1.62937	0.814684	+	0.579905i	$0.196910\pi$
$588$	0	0
$589$	5.78818e12	1.98163
$590$	0	0
$591$	−2.99712e12	−1.01055
$592$	0	0
$593$	−3.97181e12	−1.31899	−0.659496	−	0.751708i	$-0.729230\pi$
$593$	−3.97181e12	−1.31899	−0.659496	+	0.751708i	$0.729230\pi$
$594$	0	0
$595$	−1.07977e12	−0.353186
$596$	0	0
$597$	−4.43050e11	−0.142747
$598$	0	0
$599$	−8.21210e11	−0.260635	−0.130318	−	0.991472i	$-0.541600\pi$
$599$	−8.21210e11	−0.260635	−0.130318	+	0.991472i	$0.541600\pi$
$600$	0	0
$601$	9.05981e11	0.283259	0.141630	−	0.989920i	$-0.454766\pi$
$601$	9.05981e11	0.283259	0.141630	+	0.989920i	$0.454766\pi$
$602$	0	0
$603$	1.29529e12	0.398970
$604$	0	0
$605$	4.14896e12	1.25904
$606$	0	0
$607$	4.71983e12	1.41116	0.705582	−	0.708628i	$-0.250686\pi$
$607$	4.71983e12	1.41116	0.705582	+	0.708628i	$0.250686\pi$
$608$	0	0
$609$	−2.98724e11	−0.0880018
$610$	0	0
$611$	−1.59313e12	−0.462450
$612$	0	0
$613$	−3.50974e12	−1.00393	−0.501964	−	0.864889i	$-0.667389\pi$
$613$	−3.50974e12	−1.00393	−0.501964	+	0.864889i	$0.667389\pi$
$614$	0	0
$615$	−1.76765e12	−0.498262
$616$	0	0
$617$	−1.71750e12	−0.477104	−0.238552	−	0.971130i	$-0.576673\pi$
$617$	−1.71750e12	−0.477104	−0.238552	+	0.971130i	$0.576673\pi$
$618$	0	0
$619$	−5.13938e12	−1.40703	−0.703514	−	0.710681i	$-0.748387\pi$
$619$	−5.13938e12	−1.40703	−0.703514	+	0.710681i	$0.748387\pi$
$620$	0	0
$621$	1.53755e11	0.0414874
$622$	0	0
$623$	−1.45515e12	−0.387002
$624$	0	0
$625$	−2.23959e12	−0.587095
$626$	0	0
$627$	−9.54139e11	−0.246552
$628$	0	0
$629$	−1.33160e12	−0.339193
$630$	0	0
$631$	5.82848e12	1.46360	0.731802	−	0.681518i	$-0.238680\pi$
$631$	5.82848e12	1.46360	0.731802	+	0.681518i	$0.238680\pi$
$632$	0	0
$633$	−4.42281e12	−1.09492
$634$	0	0
$635$	1.76154e12	0.429943
$636$	0	0
$637$	5.74196e12	1.38176
$638$	0	0
$639$	8.67909e11	0.205930
$640$	0	0
$641$	−4.81980e12	−1.12763	−0.563817	−	0.825900i	$-0.690668\pi$
$641$	−4.81980e12	−1.12763	−0.563817	+	0.825900i	$0.690668\pi$
$642$	0	0
$643$	6.81149e12	1.57142	0.785711	−	0.618594i	$-0.212297\pi$
$643$	6.81149e12	1.57142	0.785711	+	0.618594i	$0.212297\pi$
$644$	0	0
$645$	−6.21314e11	−0.141349
$646$	0	0
$647$	2.47142e12	0.554468	0.277234	−	0.960802i	$-0.410582\pi$
$647$	2.47142e12	0.554468	0.277234	+	0.960802i	$0.410582\pi$
$648$	0	0
$649$	−1.84263e12	−0.407696
$650$	0	0
$651$	1.15956e12	0.253034
$652$	0	0
$653$	8.66067e12	1.86398	0.931992	−	0.362478i	$-0.118069\pi$
$653$	8.66067e12	1.86398	0.931992	+	0.362478i	$0.118069\pi$
$654$	0	0
$655$	8.64200e11	0.183455
$656$	0	0
$657$	−9.86158e11	−0.206491
$658$	0	0
$659$	4.22504e12	0.872663	0.436332	−	0.899786i	$-0.356277\pi$
$659$	4.22504e12	0.872663	0.436332	+	0.899786i	$0.356277\pi$
$660$	0	0
$661$	−3.23326e11	−0.0658770	−0.0329385	−	0.999457i	$-0.510487\pi$
$661$	−3.23326e11	−0.0658770	−0.0329385	+	0.999457i	$0.510487\pi$
$662$	0	0
$663$	−4.27171e12	−0.858601
$664$	0	0
$665$	−1.79518e12	−0.355967
$666$	0	0
$667$	−7.38335e11	−0.144440
$668$	0	0
$669$	4.32370e11	0.0834522
$670$	0	0
$671$	3.06282e12	0.583270
$672$	0	0
$673$	5.71913e12	1.07464	0.537319	−	0.843379i	$-0.319437\pi$
$673$	5.71913e12	1.07464	0.537319	+	0.843379i	$0.319437\pi$
$674$	0	0
$675$	1.36409e12	0.252916
$676$	0	0
$677$	−3.54197e12	−0.648031	−0.324016	−	0.946052i	$-0.605033\pi$
$677$	−3.54197e12	−0.648031	−0.324016	+	0.946052i	$0.605033\pi$
$678$	0	0
$679$	1.17498e12	0.212137
$680$	0	0
$681$	−4.31251e12	−0.768366
$682$	0	0
$683$	−1.15530e12	−0.203143	−0.101571	−	0.994828i	$-0.532387\pi$
$683$	−1.15530e12	−0.203143	−0.101571	+	0.994828i	$0.532387\pi$
$684$	0	0
$685$	9.25970e12	1.60690
$686$	0	0
$687$	2.81658e12	0.482410
$688$	0	0
$689$	6.21429e12	1.05052
$690$	0	0
$691$	−3.67044e12	−0.612445	−0.306222	−	0.951960i	$-0.599065\pi$
$691$	−3.67044e12	−0.612445	−0.306222	+	0.951960i	$0.599065\pi$
$692$	0	0
$693$	−1.91145e11	−0.0314821
$694$	0	0
$695$	−1.03717e13	−1.68623
$696$	0	0
$697$	3.60751e12	0.578975
$698$	0	0
$699$	3.96007e12	0.627416
$700$	0	0
$701$	6.79947e12	1.06352	0.531758	−	0.846896i	$-0.321532\pi$
$701$	6.79947e12	1.06352	0.531758	+	0.846896i	$0.321532\pi$
$702$	0	0
$703$	−2.21387e12	−0.341863
$704$	0	0
$705$	−1.82829e12	−0.278736
$706$	0	0
$707$	−1.53420e12	−0.230937
$708$	0	0
$709$	−2.40116e12	−0.356873	−0.178437	−	0.983951i	$-0.557104\pi$
$709$	−2.40116e12	−0.356873	−0.178437	+	0.983951i	$0.557104\pi$
$710$	0	0
$711$	2.57770e12	0.378285
$712$	0	0
$713$	2.86601e12	0.415312
$714$	0	0
$715$	−6.43145e12	−0.920305
$716$	0	0
$717$	−3.59630e12	−0.508183
$718$	0	0
$719$	8.47504e12	1.18266	0.591332	−	0.806428i	$-0.298602\pi$
$719$	8.47504e12	1.18266	0.591332	+	0.806428i	$0.298602\pi$
$720$	0	0
$721$	−3.01832e12	−0.415964
$722$	0	0
$723$	−4.35283e12	−0.592447
$724$	0	0
$725$	−6.55040e12	−0.880536
$726$	0	0
$727$	8.63217e12	1.14608	0.573040	−	0.819527i	$-0.305764\pi$
$727$	8.63217e12	1.14608	0.573040	+	0.819527i	$0.305764\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	1.26801e12	0.164246
$732$	0	0
$733$	4.13998e12	0.529700	0.264850	−	0.964290i	$-0.414677\pi$
$733$	4.13998e12	0.529700	0.264850	+	0.964290i	$0.414677\pi$
$734$	0	0
$735$	6.58952e12	0.832839
$736$	0	0
$737$	−3.98004e12	−0.496917
$738$	0	0
$739$	−1.04729e13	−1.29171	−0.645856	−	0.763459i	$-0.723499\pi$
$739$	−1.04729e13	−1.29171	−0.645856	+	0.763459i	$0.723499\pi$
$740$	0	0
$741$	−7.10197e12	−0.865360
$742$	0	0
$743$	−2.13032e12	−0.256446	−0.128223	−	0.991745i	$-0.540927\pi$
$743$	−2.13032e12	−0.256446	−0.128223	+	0.991745i	$0.540927\pi$
$744$	0	0
$745$	1.32858e13	1.58010
$746$	0	0
$747$	3.12722e11	0.0367465
$748$	0	0
$749$	−1.46074e12	−0.169591
$750$	0	0
$751$	9.83639e11	0.112838	0.0564191	−	0.998407i	$-0.482032\pi$
$751$	9.83639e11	0.112838	0.0564191	+	0.998407i	$0.482032\pi$
$752$	0	0
$753$	−7.70961e12	−0.873886
$754$	0	0
$755$	1.63641e13	1.83286
$756$	0	0
$757$	2.86648e12	0.317261	0.158631	−	0.987338i	$-0.449292\pi$
$757$	2.86648e12	0.317261	0.158631	+	0.987338i	$0.449292\pi$
$758$	0	0
$759$	−4.72441e11	−0.0516725
$760$	0	0
$761$	1.06618e13	1.15240	0.576198	−	0.817310i	$-0.304536\pi$
$761$	1.06618e13	1.15240	0.576198	+	0.817310i	$0.304536\pi$
$762$	0	0
$763$	3.50987e12	0.374913
$764$	0	0
$765$	−4.90226e12	−0.517511
$766$	0	0
$767$	−1.37153e13	−1.43095
$768$	0	0
$769$	4.89791e12	0.505059	0.252529	−	0.967589i	$-0.418738\pi$
$769$	4.89791e12	0.505059	0.252529	+	0.967589i	$0.418738\pi$
$770$	0	0
$771$	−7.09853e12	−0.723475
$772$	0	0
$773$	−1.00025e13	−1.00763	−0.503816	−	0.863811i	$-0.668071\pi$
$773$	−1.00025e13	−1.00763	−0.503816	+	0.863811i	$0.668071\pi$
$774$	0	0
$775$	2.54268e13	2.53183
$776$	0	0
$777$	−4.43510e11	−0.0436524
$778$	0	0
$779$	5.99769e12	0.583533
$780$	0	0
$781$	−2.66681e12	−0.256486
$782$	0	0
$783$	−1.35624e12	−0.128946
$784$	0	0
$785$	1.60177e12	0.150552
$786$	0	0
$787$	6.62330e12	0.615443	0.307721	−	0.951476i	$-0.400433\pi$
$787$	6.62330e12	0.615443	0.307721	+	0.951476i	$0.400433\pi$
$788$	0	0
$789$	−9.10360e12	−0.836309
$790$	0	0
$791$	−2.37161e12	−0.215401
$792$	0	0
$793$	2.27975e13	2.04719
$794$	0	0
$795$	7.13158e12	0.633189
$796$	0	0
$797$	−9.63840e12	−0.846140	−0.423070	−	0.906097i	$-0.639048\pi$
$797$	−9.63840e12	−0.846140	−0.423070	+	0.906097i	$0.639048\pi$
$798$	0	0
$799$	3.73127e12	0.323889
$800$	0	0
$801$	−6.60655e12	−0.567059
$802$	0	0
$803$	3.03016e12	0.257185
$804$	0	0
$805$	−8.88879e11	−0.0746038
$806$	0	0
$807$	−1.06007e13	−0.879836
$808$	0	0
$809$	8.06927e12	0.662317	0.331159	−	0.943575i	$-0.392560\pi$
$809$	8.06927e12	0.662317	0.331159	+	0.943575i	$0.392560\pi$
$810$	0	0
$811$	−1.14952e13	−0.933089	−0.466544	−	0.884498i	$-0.654501\pi$
$811$	−1.14952e13	−0.933089	−0.466544	+	0.884498i	$0.654501\pi$
$812$	0	0
$813$	−4.14738e12	−0.332941
$814$	0	0
$815$	−1.96585e13	−1.56077
$816$	0	0
$817$	2.10814e12	0.165539
$818$	0	0
$819$	−1.42275e12	−0.110498
$820$	0	0
$821$	2.53548e13	1.94767	0.973837	−	0.227248i	$-0.0729727\pi$
$821$	2.53548e13	1.94767	0.973837	+	0.227248i	$0.0729727\pi$
$822$	0	0
$823$	1.35033e13	1.02598	0.512992	−	0.858394i	$-0.328537\pi$
$823$	1.35033e13	1.02598	0.512992	+	0.858394i	$0.328537\pi$
$824$	0	0
$825$	−4.19143e12	−0.315006
$826$	0	0
$827$	7.32566e12	0.544593	0.272296	−	0.962213i	$-0.412217\pi$
$827$	7.32566e12	0.544593	0.272296	+	0.962213i	$0.412217\pi$
$828$	0	0
$829$	1.72332e12	0.126727	0.0633635	−	0.997991i	$-0.479817\pi$
$829$	1.72332e12	0.126727	0.0633635	+	0.997991i	$0.479817\pi$
$830$	0	0
$831$	−5.95372e12	−0.433095
$832$	0	0
$833$	−1.34482e13	−0.967750
$834$	0	0
$835$	−2.80840e13	−1.99927
$836$	0	0
$837$	5.26453e12	0.370762
$838$	0	0
$839$	−1.89755e13	−1.32210	−0.661051	−	0.750341i	$-0.729889\pi$
$839$	−1.89755e13	−1.32210	−0.661051	+	0.750341i	$0.729889\pi$
$840$	0	0
$841$	−7.99446e12	−0.551071
$842$	0	0
$843$	6.99147e12	0.476809
$844$	0	0
$845$	−2.53261e13	−1.70889
$846$	0	0
$847$	−2.82019e12	−0.188280
$848$	0	0
$849$	4.88090e12	0.322415
$850$	0	0
$851$	−1.09619e12	−0.0716480
$852$	0	0
$853$	1.14364e13	0.739638	0.369819	−	0.929104i	$-0.379420\pi$
$853$	1.14364e13	0.739638	0.369819	+	0.929104i	$0.379420\pi$
$854$	0	0
$855$	−8.15028e12	−0.521585
$856$	0	0
$857$	−3.91509e12	−0.247929	−0.123965	−	0.992287i	$-0.539561\pi$
$857$	−3.91509e12	−0.247929	−0.123965	+	0.992287i	$0.539561\pi$
$858$	0	0
$859$	−2.45009e13	−1.53537	−0.767685	−	0.640828i	$-0.778591\pi$
$859$	−2.45009e13	−1.53537	−0.767685	+	0.640828i	$0.778591\pi$
$860$	0	0
$861$	1.20153e12	0.0745112
$862$	0	0
$863$	−9.55511e12	−0.586391	−0.293195	−	0.956053i	$-0.594719\pi$
$863$	−9.55511e12	−0.586391	−0.293195	+	0.956053i	$0.594719\pi$
$864$	0	0
$865$	−1.28994e12	−0.0783427
$866$	0	0
$867$	3.99170e11	0.0239923
$868$	0	0
$869$	−7.92047e12	−0.471153
$870$	0	0
$871$	−2.96247e13	−1.74410
$872$	0	0
$873$	5.33452e12	0.310836
$874$	0	0
$875$	−1.88534e12	−0.108731
$876$	0	0
$877$	1.71224e12	0.0977385	0.0488692	−	0.998805i	$-0.484438\pi$
$877$	1.71224e12	0.0977385	0.0488692	+	0.998805i	$0.484438\pi$
$878$	0	0
$879$	−1.40473e13	−0.793676
$880$	0	0
$881$	−9.35035e12	−0.522921	−0.261461	−	0.965214i	$-0.584204\pi$
$881$	−9.35035e12	−0.522921	−0.261461	+	0.965214i	$0.584204\pi$
$882$	0	0
$883$	−1.69552e13	−0.938598	−0.469299	−	0.883039i	$-0.655493\pi$
$883$	−1.69552e13	−0.938598	−0.469299	+	0.883039i	$0.655493\pi$
$884$	0	0
$885$	−1.57398e13	−0.862489
$886$	0	0
$887$	−1.72616e13	−0.936322	−0.468161	−	0.883643i	$-0.655083\pi$
$887$	−1.72616e13	−0.936322	−0.468161	+	0.883643i	$0.655083\pi$
$888$	0	0
$889$	−1.19738e12	−0.0642946
$890$	0	0
$891$	−8.67818e11	−0.0461295
$892$	0	0
$893$	6.20344e12	0.326438
$894$	0	0
$895$	−1.37542e13	−0.716525
$896$	0	0
$897$	−3.51653e12	−0.181363
$898$	0	0
$899$	−2.52804e13	−1.29082
$900$	0	0
$901$	−1.45545e13	−0.735760
$902$	0	0
$903$	4.22329e11	0.0211376
$904$	0	0
$905$	1.04663e13	0.518651
$906$	0	0
$907$	−7.85379e12	−0.385342	−0.192671	−	0.981263i	$-0.561715\pi$
$907$	−7.85379e12	−0.385342	−0.192671	+	0.981263i	$0.561715\pi$
$908$	0	0
$909$	−6.96541e12	−0.338384
$910$	0	0
$911$	8.85344e12	0.425872	0.212936	−	0.977066i	$-0.431697\pi$
$911$	8.85344e12	0.425872	0.212936	+	0.977066i	$0.431697\pi$
$912$	0	0
$913$	−9.60898e11	−0.0457677
$914$	0	0
$915$	2.61627e13	1.23392
$916$	0	0
$917$	−5.87428e11	−0.0274342
$918$	0	0
$919$	−1.44239e13	−0.667055	−0.333527	−	0.942740i	$-0.608239\pi$
$919$	−1.44239e13	−0.667055	−0.333527	+	0.942740i	$0.608239\pi$
$920$	0	0
$921$	−1.48527e13	−0.680201
$922$	0	0
$923$	−1.98500e13	−0.900227
$924$	0	0
$925$	−9.72527e12	−0.436781
$926$	0	0
$927$	−1.37035e13	−0.609497
$928$	0	0
$929$	2.42057e13	1.06622	0.533111	−	0.846045i	$-0.321023\pi$
$929$	2.42057e13	1.06622	0.533111	+	0.846045i	$0.321023\pi$
$930$	0	0
$931$	−2.23585e13	−0.975369
$932$	0	0
$933$	−5.40451e12	−0.233501
$934$	0	0
$935$	1.50631e13	0.644558
$936$	0	0
$937$	8.32006e12	0.352613	0.176306	−	0.984335i	$-0.443585\pi$
$937$	8.32006e12	0.352613	0.176306	+	0.984335i	$0.443585\pi$
$938$	0	0
$939$	1.13992e13	0.478499
$940$	0	0
$941$	−2.32963e13	−0.968576	−0.484288	−	0.874909i	$-0.660921\pi$
$941$	−2.32963e13	−0.968576	−0.484288	+	0.874909i	$0.660921\pi$
$942$	0	0
$943$	2.96975e12	0.122297
$944$	0	0
$945$	−1.63277e12	−0.0666010
$946$	0	0
$947$	3.61981e12	0.146255	0.0731276	−	0.997323i	$-0.476702\pi$
$947$	3.61981e12	0.146255	0.0731276	+	0.997323i	$0.476702\pi$
$948$	0	0
$949$	2.25544e13	0.902680
$950$	0	0
$951$	1.08641e13	0.430708
$952$	0	0
$953$	−3.39005e13	−1.33134	−0.665669	−	0.746248i	$-0.731854\pi$
$953$	−3.39005e13	−1.33134	−0.665669	+	0.746248i	$0.731854\pi$
$954$	0	0
$955$	−3.08422e13	−1.19986
$956$	0	0
$957$	4.16729e12	0.160602
$958$	0	0
$959$	−6.29415e12	−0.240300
$960$	0	0
$961$	7.16919e13	2.71153
$962$	0	0
$963$	−6.63189e12	−0.248496
$964$	0	0
$965$	−6.42840e13	−2.38633
$966$	0	0
$967$	2.32172e13	0.853870	0.426935	−	0.904282i	$-0.359593\pi$
$967$	2.32172e13	0.853870	0.426935	+	0.904282i	$0.359593\pi$
$968$	0	0
$969$	1.66335e13	0.606077
$970$	0	0
$971$	2.45225e13	0.885274	0.442637	−	0.896701i	$-0.354043\pi$
$971$	2.45225e13	0.885274	0.442637	+	0.896701i	$0.354043\pi$
$972$	0	0
$973$	7.05001e12	0.252163
$974$	0	0
$975$	−3.11981e13	−1.10563
$976$	0	0
$977$	−3.10059e13	−1.08872	−0.544362	−	0.838850i	$-0.683228\pi$
$977$	−3.10059e13	−1.08872	−0.544362	+	0.838850i	$0.683228\pi$
$978$	0	0
$979$	2.02999e13	0.706270
$980$	0	0
$981$	1.59352e13	0.549346
$982$	0	0
$983$	6.65150e11	0.0227211	0.0113605	−	0.999935i	$-0.496384\pi$
$983$	6.65150e11	0.0227211	0.0113605	+	0.999935i	$0.496384\pi$
$984$	0	0
$985$	7.86653e13	2.66269
$986$	0	0
$987$	1.24275e12	0.0416828
$988$	0	0
$989$	1.04384e12	0.0346938
$990$	0	0
$991$	5.23636e13	1.72464	0.862320	−	0.506364i	$-0.169011\pi$
$991$	5.23636e13	1.72464	0.862320	+	0.506364i	$0.169011\pi$
$992$	0	0
$993$	−1.39395e13	−0.454964
$994$	0	0
$995$	1.16287e13	0.376122
$996$	0	0
$997$	4.66861e13	1.49644	0.748219	−	0.663451i	$-0.230909\pi$
$997$	4.66861e13	1.49644	0.748219	+	0.663451i	$0.230909\pi$
$998$	0	0
$999$	−2.01358e12	−0.0639623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.m.1.1	yes	5
4.3	odd	2		384.10.a.i.1.1	✓	5
8.3	odd	2		384.10.a.p.1.5	yes	5
8.5	even	2		384.10.a.l.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.i.1.1	✓	5	4.3	odd	2
384.10.a.l.1.5	yes	5	8.5	even	2
384.10.a.m.1.1	yes	5	1.1	even	1	trivial
384.10.a.p.1.5	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)

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} -2126.01 q^{5} +1445.12 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} -2126.01 q^{5} +1445.12 q^{7} +6561.00 q^{9} -20159.9 q^{11} -150057. q^{13} -172206. q^{15} +351448. q^{17} +584303. q^{19} +117055. q^{21} +289317. q^{23} +2.56678e6 q^{25} +531441. q^{27} -2.55200e6 q^{29} +9.90613e6 q^{31} -1.63295e6 q^{33} -3.07234e6 q^{35} -3.78890e6 q^{37} -1.21546e7 q^{39} +1.02647e7 q^{41} +3.60796e6 q^{43} -1.39487e7 q^{45} +1.06168e7 q^{47} -3.82652e7 q^{49} +2.84673e7 q^{51} -4.14129e7 q^{53} +4.28601e7 q^{55} +4.73285e7 q^{57} +9.14005e7 q^{59} -1.51926e8 q^{61} +9.48144e6 q^{63} +3.19022e8 q^{65} +1.97423e8 q^{67} +2.34347e7 q^{69} +1.32283e8 q^{71} -1.50306e8 q^{73} +2.07909e8 q^{75} -2.91335e7 q^{77} +3.92882e8 q^{79} +4.30467e7 q^{81} +4.76638e7 q^{83} -7.47181e8 q^{85} -2.06712e8 q^{87} -1.00694e9 q^{89} -2.16850e8 q^{91} +8.02397e8 q^{93} -1.24223e9 q^{95} +8.13065e8 q^{97} -1.32269e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 405 q^{3} - 772 q^{5} + 38 q^{7} + 32805 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more