LMFDB - Embedded newform 392.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,4,Mod(1,392)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("392.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	392.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:	$23.1287487223$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685 q^{3} -5.65685 q^{5} +5.00000 q^{9} +O(q^{10})$	$q-5.65685 q^{3} -5.65685 q^{5} +5.00000 q^{9} -4.00000 q^{11} +5.65685 q^{13} +32.0000 q^{15} -124.451 q^{17} -130.108 q^{19} +120.000 q^{23} -93.0000 q^{25} +124.451 q^{27} +218.000 q^{29} +147.078 q^{31} +22.6274 q^{33} +130.000 q^{37} -32.0000 q^{39} -147.078 q^{41} +332.000 q^{43} -28.2843 q^{45} +124.451 q^{47} +704.000 q^{51} -498.000 q^{53} +22.6274 q^{55} +736.000 q^{57} +548.715 q^{59} +650.538 q^{61} -32.0000 q^{65} +156.000 q^{67} -678.823 q^{69} +240.000 q^{71} +526.087 q^{75} +1112.00 q^{79} -839.000 q^{81} +28.2843 q^{83} +704.000 q^{85} -1233.19 q^{87} -1357.65 q^{89} -832.000 q^{93} +736.000 q^{95} +1482.10 q^{97} -20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9}+O(q^{10}) $ 2 * q + 10 * q^9	$ 2 q + 10 q^{9} - 8 q^{11} + 64 q^{15} + 240 q^{23} - 186 q^{25} + 436 q^{29} + 260 q^{37} - 64 q^{39} + 664 q^{43} + 1408 q^{51} - 996 q^{53} + 1472 q^{57} - 64 q^{65} + 312 q^{67} + 480 q^{71} + 2224 q^{79} - 1678 q^{81} + 1408 q^{85} - 1664 q^{93} + 1472 q^{95} - 40 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 - 8 * q^11 + 64 * q^15 + 240 * q^23 - 186 * q^25 + 436 * q^29 + 260 * q^37 - 64 * q^39 + 664 * q^43 + 1408 * q^51 - 996 * q^53 + 1472 * q^57 - 64 * q^65 + 312 * q^67 + 480 * q^71 + 2224 * q^79 - 1678 * q^81 + 1408 * q^85 - 1664 * q^93 + 1472 * q^95 - 40 * 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
$3$	−5.65685	−1.08866	−0.544331	−	0.838870i	$-0.683216\pi$
$3$	−5.65685	−1.08866	−0.544331	+	0.838870i	$0.683216\pi$
$4$	0	0
$5$	−5.65685	−0.505964	−0.252982	−	0.967471i	$-0.581411\pi$
$5$	−5.65685	−0.505964	−0.252982	+	0.967471i	$0.581411\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.00000	0.185185
$10$	0	0
$11$	−4.00000	−0.109640	−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000	−0.109640	−0.0548202	+	0.998496i	$0.517459\pi$
$12$	0	0
$13$	5.65685	0.120687	0.0603434	−	0.998178i	$-0.480780\pi$
$13$	5.65685	0.120687	0.0603434	+	0.998178i	$0.480780\pi$
$14$	0	0
$15$	32.0000	0.550824
$16$	0	0
$17$	−124.451	−1.77551	−0.887757	−	0.460312i	$-0.847738\pi$
$17$	−124.451	−1.77551	−0.887757	+	0.460312i	$0.847738\pi$
$18$	0	0
$19$	−130.108	−1.57099	−0.785493	−	0.618870i	$-0.787591\pi$
$19$	−130.108	−1.57099	−0.785493	+	0.618870i	$0.787591\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	0	0
$25$	−93.0000	−0.744000
$26$	0	0
$27$	124.451	0.887058
$28$	0	0
$29$	218.000	1.39592	0.697958	−	0.716138i	$-0.254092\pi$
$29$	218.000	1.39592	0.697958	+	0.716138i	$0.254092\pi$
$30$	0	0
$31$	147.078	0.852130	0.426065	−	0.904693i	$-0.359900\pi$
$31$	147.078	0.852130	0.426065	+	0.904693i	$0.359900\pi$
$32$	0	0
$33$	22.6274	0.119361
$34$	0	0
$35$	0	0
$36$	0	0
$37$	130.000	0.577618	0.288809	−	0.957387i	$-0.406741\pi$
$37$	130.000	0.577618	0.288809	+	0.957387i	$0.406741\pi$
$38$	0	0
$39$	−32.0000	−0.131387
$40$	0	0
$41$	−147.078	−0.560238	−0.280119	−	0.959965i	$-0.590374\pi$
$41$	−147.078	−0.560238	−0.280119	+	0.959965i	$0.590374\pi$
$42$	0	0
$43$	332.000	1.17743	0.588715	−	0.808340i	$-0.299634\pi$
$43$	332.000	1.17743	0.588715	+	0.808340i	$0.299634\pi$
$44$	0	0
$45$	−28.2843	−0.0936971
$46$	0	0
$47$	124.451	0.386234	0.193117	−	0.981176i	$-0.438140\pi$
$47$	124.451	0.386234	0.193117	+	0.981176i	$0.438140\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	704.000	1.93294
$52$	0	0
$53$	−498.000	−1.29067	−0.645335	−	0.763899i	$-0.723282\pi$
$53$	−498.000	−1.29067	−0.645335	+	0.763899i	$0.723282\pi$
$54$	0	0
$55$	22.6274	0.0554742
$56$	0	0
$57$	736.000	1.71027
$58$	0	0
$59$	548.715	1.21079	0.605395	−	0.795925i	$-0.293015\pi$
$59$	548.715	1.21079	0.605395	+	0.795925i	$0.293015\pi$
$60$	0	0
$61$	650.538	1.36546	0.682729	−	0.730672i	$-0.260793\pi$
$61$	650.538	1.36546	0.682729	+	0.730672i	$0.260793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−32.0000	−0.0610633
$66$	0	0
$67$	156.000	0.284454	0.142227	−	0.989834i	$-0.454574\pi$
$67$	156.000	0.284454	0.142227	+	0.989834i	$0.454574\pi$
$68$	0	0
$69$	−678.823	−1.18436
$70$	0	0
$71$	240.000	0.401166	0.200583	−	0.979677i	$-0.435716\pi$
$71$	240.000	0.401166	0.200583	+	0.979677i	$0.435716\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	526.087	0.809965
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1112.00	1.58367	0.791834	−	0.610736i	$-0.209126\pi$
$79$	1112.00	1.58367	0.791834	+	0.610736i	$0.209126\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	28.2843	0.0374048	0.0187024	−	0.999825i	$-0.494046\pi$
$83$	28.2843	0.0374048	0.0187024	+	0.999825i	$0.494046\pi$
$84$	0	0
$85$	704.000	0.898347
$86$	0	0
$87$	−1233.19	−1.51968
$88$	0	0
$89$	−1357.65	−1.61697	−0.808484	−	0.588519i	$-0.799711\pi$
$89$	−1357.65	−1.61697	−0.808484	+	0.588519i	$0.799711\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−832.000	−0.927682
$94$	0	0
$95$	736.000	0.794863
$96$	0	0
$97$	1482.10	1.55138	0.775691	−	0.631113i	$-0.217402\pi$
$97$	1482.10	1.55138	0.775691	+	0.631113i	$0.217402\pi$
$98$	0	0
$99$	−20.0000	−0.0203038
$100$	0	0
$101$	1238.85	1.22050	0.610249	−	0.792210i	$-0.291069\pi$
$101$	1238.85	1.22050	0.610249	+	0.792210i	$0.291069\pi$
$102$	0	0
$103$	−644.881	−0.616913	−0.308457	−	0.951238i	$-0.599812\pi$
$103$	−644.881	−0.616913	−0.308457	+	0.951238i	$0.599812\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1044.00	0.943246	0.471623	−	0.881800i	$-0.343668\pi$
$107$	1044.00	0.943246	0.471623	+	0.881800i	$0.343668\pi$
$108$	0	0
$109$	−502.000	−0.441127	−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000	−0.441127	−0.220564	+	0.975373i	$0.570790\pi$
$110$	0	0
$111$	−735.391	−0.628831
$112$	0	0
$113$	130.000	0.108225	0.0541123	−	0.998535i	$-0.482767\pi$
$113$	130.000	0.108225	0.0541123	+	0.998535i	$0.482767\pi$
$114$	0	0
$115$	−678.823	−0.550439
$116$	0	0
$117$	28.2843	0.0223494
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	832.000	0.609910
$124$	0	0
$125$	1233.19	0.882402
$126$	0	0
$127$	−2416.00	−1.68807	−0.844037	−	0.536285i	$-0.819827\pi$
$127$	−2416.00	−1.68807	−0.844037	+	0.536285i	$0.819827\pi$
$128$	0	0
$129$	−1878.08	−1.28182
$130$	0	0
$131$	−1329.36	−0.886617	−0.443308	−	0.896369i	$-0.646195\pi$
$131$	−1329.36	−0.886617	−0.443308	+	0.896369i	$0.646195\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−704.000	−0.448820
$136$	0	0
$137$	−730.000	−0.455242	−0.227621	−	0.973750i	$-0.573095\pi$
$137$	−730.000	−0.455242	−0.227621	+	0.973750i	$0.573095\pi$
$138$	0	0
$139$	−28.2843	−0.0172593	−0.00862964	−	0.999963i	$-0.502747\pi$
$139$	−28.2843	−0.0172593	−0.00862964	+	0.999963i	$0.502747\pi$
$140$	0	0
$141$	−704.000	−0.420479
$142$	0	0
$143$	−22.6274	−0.0132322
$144$	0	0
$145$	−1233.19	−0.706284
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2770.00	−1.52300	−0.761501	−	0.648164i	$-0.775537\pi$
$149$	−2770.00	−1.52300	−0.761501	+	0.648164i	$0.775537\pi$
$150$	0	0
$151$	2776.00	1.49608	0.748039	−	0.663655i	$-0.230996\pi$
$151$	2776.00	1.49608	0.748039	+	0.663655i	$0.230996\pi$
$152$	0	0
$153$	−622.254	−0.328799
$154$	0	0
$155$	−832.000	−0.431147
$156$	0	0
$157$	2743.57	1.39466	0.697328	−	0.716752i	$-0.254372\pi$
$157$	2743.57	1.39466	0.697328	+	0.716752i	$0.254372\pi$
$158$	0	0
$159$	2817.11	1.40510
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1460.00	0.701571	0.350785	−	0.936456i	$-0.385915\pi$
$163$	1460.00	0.701571	0.350785	+	0.936456i	$0.385915\pi$
$164$	0	0
$165$	−128.000	−0.0603926
$166$	0	0
$167$	−1482.10	−0.686755	−0.343377	−	0.939198i	$-0.611571\pi$
$167$	−1482.10	−0.686755	−0.343377	+	0.939198i	$0.611571\pi$
$168$	0	0
$169$	−2165.00	−0.985435
$170$	0	0
$171$	−650.538	−0.290923
$172$	0	0
$173$	4078.59	1.79243	0.896213	−	0.443625i	$-0.146308\pi$
$173$	4078.59	1.79243	0.896213	+	0.443625i	$0.146308\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3104.00	−1.31814
$178$	0	0
$179$	220.000	0.0918635	0.0459318	−	0.998945i	$-0.485374\pi$
$179$	220.000	0.0918635	0.0459318	+	0.998945i	$0.485374\pi$
$180$	0	0
$181$	−1917.67	−0.787511	−0.393756	−	0.919215i	$-0.628824\pi$
$181$	−1917.67	−0.787511	−0.393756	+	0.919215i	$0.628824\pi$
$182$	0	0
$183$	−3680.00	−1.48652
$184$	0	0
$185$	−735.391	−0.292254
$186$	0	0
$187$	497.803	0.194668
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3816.00	−1.44563	−0.722817	−	0.691040i	$-0.757153\pi$
$191$	−3816.00	−1.44563	−0.722817	+	0.691040i	$0.757153\pi$
$192$	0	0
$193$	2290.00	0.854082	0.427041	−	0.904232i	$-0.359556\pi$
$193$	2290.00	0.854082	0.427041	+	0.904232i	$0.359556\pi$
$194$	0	0
$195$	181.019	0.0664773
$196$	0	0
$197$	−1634.00	−0.590953	−0.295476	−	0.955350i	$-0.595478\pi$
$197$	−1634.00	−0.590953	−0.295476	+	0.955350i	$0.595478\pi$
$198$	0	0
$199$	4174.76	1.48714	0.743570	−	0.668658i	$-0.233131\pi$
$199$	4174.76	1.48714	0.743570	+	0.668658i	$0.233131\pi$
$200$	0	0
$201$	−882.469	−0.309675
$202$	0	0
$203$	0	0
$204$	0	0
$205$	832.000	0.283460
$206$	0	0
$207$	600.000	0.201463
$208$	0	0
$209$	520.431	0.172244
$210$	0	0
$211$	236.000	0.0769996	0.0384998	−	0.999259i	$-0.487742\pi$
$211$	236.000	0.0769996	0.0384998	+	0.999259i	$0.487742\pi$
$212$	0	0
$213$	−1357.65	−0.436734
$214$	0	0
$215$	−1878.08	−0.595738
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−704.000	−0.214281
$222$	0	0
$223$	5453.21	1.63755	0.818775	−	0.574114i	$-0.194653\pi$
$223$	5453.21	1.63755	0.818775	+	0.574114i	$0.194653\pi$
$224$	0	0
$225$	−465.000	−0.137778
$226$	0	0
$227$	3920.20	1.14622	0.573112	−	0.819477i	$-0.305736\pi$
$227$	3920.20	1.14622	0.573112	+	0.819477i	$0.305736\pi$
$228$	0	0
$229$	1544.32	0.445640	0.222820	−	0.974860i	$-0.428474\pi$
$229$	1544.32	0.445640	0.222820	+	0.974860i	$0.428474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−250.000	−0.0702920	−0.0351460	−	0.999382i	$-0.511190\pi$
$233$	−250.000	−0.0702920	−0.0351460	+	0.999382i	$0.511190\pi$
$234$	0	0
$235$	−704.000	−0.195421
$236$	0	0
$237$	−6290.42	−1.72408
$238$	0	0
$239$	−1712.00	−0.463348	−0.231674	−	0.972794i	$-0.574420\pi$
$239$	−1712.00	−0.463348	−0.231674	+	0.972794i	$0.574420\pi$
$240$	0	0
$241$	−2093.04	−0.559437	−0.279718	−	0.960082i	$-0.590241\pi$
$241$	−2093.04	−0.559437	−0.279718	+	0.960082i	$0.590241\pi$
$242$	0	0
$243$	1385.93	0.365874
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−736.000	−0.189597
$248$	0	0
$249$	−160.000	−0.0407212
$250$	0	0
$251$	2008.18	0.505002	0.252501	−	0.967597i	$-0.418747\pi$
$251$	2008.18	0.505002	0.252501	+	0.967597i	$0.418747\pi$
$252$	0	0
$253$	−480.000	−0.119278
$254$	0	0
$255$	−3982.43	−0.977997
$256$	0	0
$257$	−4072.94	−0.988571	−0.494285	−	0.869300i	$-0.664570\pi$
$257$	−4072.94	−0.988571	−0.494285	+	0.869300i	$0.664570\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1090.00	0.258503
$262$	0	0
$263$	−352.000	−0.0825295	−0.0412647	−	0.999148i	$-0.513139\pi$
$263$	−352.000	−0.0825295	−0.0412647	+	0.999148i	$0.513139\pi$
$264$	0	0
$265$	2817.11	0.653034
$266$	0	0
$267$	7680.00	1.76033
$268$	0	0
$269$	−2687.01	−0.609032	−0.304516	−	0.952507i	$-0.598495\pi$
$269$	−2687.01	−0.609032	−0.304516	+	0.952507i	$0.598495\pi$
$270$	0	0
$271$	−6788.23	−1.52161	−0.760803	−	0.648983i	$-0.775195\pi$
$271$	−6788.23	−1.52161	−0.760803	+	0.648983i	$0.775195\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	372.000	0.0815725
$276$	0	0
$277$	5390.00	1.16915	0.584573	−	0.811341i	$-0.301262\pi$
$277$	5390.00	1.16915	0.584573	+	0.811341i	$0.301262\pi$
$278$	0	0
$279$	735.391	0.157802
$280$	0	0
$281$	2166.00	0.459832	0.229916	−	0.973211i	$-0.426155\pi$
$281$	2166.00	0.459832	0.229916	+	0.973211i	$0.426155\pi$
$282$	0	0
$283$	−4746.10	−0.996913	−0.498457	−	0.866915i	$-0.666100\pi$
$283$	−4746.10	−0.996913	−0.498457	+	0.866915i	$0.666100\pi$
$284$	0	0
$285$	−4163.44	−0.865337
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10575.0	2.15245
$290$	0	0
$291$	−8384.00	−1.68893
$292$	0	0
$293$	7461.39	1.48771	0.743855	−	0.668341i	$-0.232995\pi$
$293$	7461.39	1.48771	0.743855	+	0.668341i	$0.232995\pi$
$294$	0	0
$295$	−3104.00	−0.612616
$296$	0	0
$297$	−497.803	−0.0972575
$298$	0	0
$299$	678.823	0.131295
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7008.00	−1.32871
$304$	0	0
$305$	−3680.00	−0.690873
$306$	0	0
$307$	−8977.43	−1.66895	−0.834477	−	0.551043i	$-0.814230\pi$
$307$	−8977.43	−1.66895	−0.834477	+	0.551043i	$0.814230\pi$
$308$	0	0
$309$	3648.00	0.671610
$310$	0	0
$311$	1945.96	0.354808	0.177404	−	0.984138i	$-0.443230\pi$
$311$	1945.96	0.354808	0.177404	+	0.984138i	$0.443230\pi$
$312$	0	0
$313$	7500.99	1.35457	0.677286	−	0.735720i	$-0.263156\pi$
$313$	7500.99	1.35457	0.677286	+	0.735720i	$0.263156\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5766.00	1.02161	0.510806	−	0.859696i	$-0.329347\pi$
$317$	5766.00	1.02161	0.510806	+	0.859696i	$0.329347\pi$
$318$	0	0
$319$	−872.000	−0.153049
$320$	0	0
$321$	−5905.76	−1.02688
$322$	0	0
$323$	16192.0	2.78931
$324$	0	0
$325$	−526.087	−0.0897910
$326$	0	0
$327$	2839.74	0.480239
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10780.0	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$331$	10780.0	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$332$	0	0
$333$	650.000	0.106966
$334$	0	0
$335$	−882.469	−0.143924
$336$	0	0
$337$	350.000	0.0565748	0.0282874	−	0.999600i	$-0.490995\pi$
$337$	350.000	0.0565748	0.0282874	+	0.999600i	$0.490995\pi$
$338$	0	0
$339$	−735.391	−0.117820
$340$	0	0
$341$	−588.313	−0.0934279
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3840.00	0.599242
$346$	0	0
$347$	−7940.00	−1.22836	−0.614181	−	0.789165i	$-0.710513\pi$
$347$	−7940.00	−1.22836	−0.614181	+	0.789165i	$0.710513\pi$
$348$	0	0
$349$	4259.61	0.653329	0.326664	−	0.945140i	$-0.394075\pi$
$349$	4259.61	0.653329	0.326664	+	0.945140i	$0.394075\pi$
$350$	0	0
$351$	704.000	0.107056
$352$	0	0
$353$	5407.95	0.815400	0.407700	−	0.913116i	$-0.366331\pi$
$353$	5407.95	0.815400	0.407700	+	0.913116i	$0.366331\pi$
$354$	0	0
$355$	−1357.65	−0.202976
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9480.00	−1.39369	−0.696846	−	0.717221i	$-0.745414\pi$
$359$	−9480.00	−1.39369	−0.696846	+	0.717221i	$0.745414\pi$
$360$	0	0
$361$	10069.0	1.46800
$362$	0	0
$363$	7438.76	1.07558
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3461.99	0.492411	0.246205	−	0.969218i	$-0.420816\pi$
$367$	3461.99	0.492411	0.246205	+	0.969218i	$0.420816\pi$
$368$	0	0
$369$	−735.391	−0.103748
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11630.0	1.61442	0.807210	−	0.590265i	$-0.200977\pi$
$373$	11630.0	1.61442	0.807210	+	0.590265i	$0.200977\pi$
$374$	0	0
$375$	−6976.00	−0.960638
$376$	0	0
$377$	1233.19	0.168469
$378$	0	0
$379$	6284.00	0.851682	0.425841	−	0.904798i	$-0.359978\pi$
$379$	6284.00	0.851682	0.425841	+	0.904798i	$0.359978\pi$
$380$	0	0
$381$	13667.0	1.83774
$382$	0	0
$383$	12841.1	1.71318	0.856589	−	0.515999i	$-0.172579\pi$
$383$	12841.1	1.71318	0.856589	+	0.515999i	$0.172579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1660.00	0.218043
$388$	0	0
$389$	−6238.00	−0.813057	−0.406528	−	0.913638i	$-0.633261\pi$
$389$	−6238.00	−0.813057	−0.406528	+	0.913638i	$0.633261\pi$
$390$	0	0
$391$	−14934.1	−1.93158
$392$	0	0
$393$	7520.00	0.965226
$394$	0	0
$395$	−6290.42	−0.801280
$396$	0	0
$397$	−2868.03	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$397$	−2868.03	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2526.00	0.314570	0.157285	−	0.987553i	$-0.449726\pi$
$401$	2526.00	0.314570	0.157285	+	0.987553i	$0.449726\pi$
$402$	0	0
$403$	832.000	0.102841
$404$	0	0
$405$	4746.10	0.582310
$406$	0	0
$407$	−520.000	−0.0633303
$408$	0	0
$409$	9401.69	1.13664	0.568318	−	0.822809i	$-0.307595\pi$
$409$	9401.69	1.13664	0.568318	+	0.822809i	$0.307595\pi$
$410$	0	0
$411$	4129.50	0.495604
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−160.000	−0.0189255
$416$	0	0
$417$	160.000	0.0187895
$418$	0	0
$419$	−8694.58	−1.01374	−0.506871	−	0.862022i	$-0.669198\pi$
$419$	−8694.58	−1.01374	−0.506871	+	0.862022i	$0.669198\pi$
$420$	0	0
$421$	−354.000	−0.0409808	−0.0204904	−	0.999790i	$-0.506523\pi$
$421$	−354.000	−0.0409808	−0.0204904	+	0.999790i	$0.506523\pi$
$422$	0	0
$423$	622.254	0.0715249
$424$	0	0
$425$	11573.9	1.32098
$426$	0	0
$427$	0	0
$428$	0	0
$429$	128.000	0.0144054
$430$	0	0
$431$	5184.00	0.579361	0.289680	−	0.957123i	$-0.406451\pi$
$431$	5184.00	0.579361	0.289680	+	0.957123i	$0.406451\pi$
$432$	0	0
$433$	2025.15	0.224764	0.112382	−	0.993665i	$-0.464152\pi$
$433$	2025.15	0.224764	0.112382	+	0.993665i	$0.464152\pi$
$434$	0	0
$435$	6976.00	0.768905
$436$	0	0
$437$	−15612.9	−1.70908
$438$	0	0
$439$	−5430.58	−0.590404	−0.295202	−	0.955435i	$-0.595387\pi$
$439$	−5430.58	−0.590404	−0.295202	+	0.955435i	$0.595387\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5612.00	−0.601883	−0.300941	−	0.953643i	$-0.597301\pi$
$443$	−5612.00	−0.601883	−0.300941	+	0.953643i	$0.597301\pi$
$444$	0	0
$445$	7680.00	0.818128
$446$	0	0
$447$	15669.5	1.65803
$448$	0	0
$449$	−11902.0	−1.25098	−0.625490	−	0.780232i	$-0.715101\pi$
$449$	−11902.0	−1.25098	−0.625490	+	0.780232i	$0.715101\pi$
$450$	0	0
$451$	588.313	0.0614248
$452$	0	0
$453$	−15703.4	−1.62872
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5034.00	0.515275	0.257637	−	0.966242i	$-0.417056\pi$
$457$	5034.00	0.515275	0.257637	+	0.966242i	$0.417056\pi$
$458$	0	0
$459$	−15488.0	−1.57498
$460$	0	0
$461$	−10154.1	−1.02586	−0.512930	−	0.858430i	$-0.671440\pi$
$461$	−10154.1	−1.02586	−0.512930	+	0.858430i	$0.671440\pi$
$462$	0	0
$463$	−16760.0	−1.68230	−0.841148	−	0.540805i	$-0.818120\pi$
$463$	−16760.0	−1.68230	−0.841148	+	0.540805i	$0.818120\pi$
$464$	0	0
$465$	4706.50	0.469374
$466$	0	0
$467$	−12371.5	−1.22588	−0.612941	−	0.790129i	$-0.710014\pi$
$467$	−12371.5	−1.22588	−0.612941	+	0.790129i	$0.710014\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15520.0	−1.51831
$472$	0	0
$473$	−1328.00	−0.129094
$474$	0	0
$475$	12100.0	1.16881
$476$	0	0
$477$	−2490.00	−0.239013
$478$	0	0
$479$	−11483.4	−1.09539	−0.547694	−	0.836679i	$-0.684494\pi$
$479$	−11483.4	−1.09539	−0.547694	+	0.836679i	$0.684494\pi$
$480$	0	0
$481$	735.391	0.0697109
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8384.00	−0.784944
$486$	0	0
$487$	13080.0	1.21707	0.608533	−	0.793528i	$-0.291758\pi$
$487$	13080.0	1.21707	0.608533	+	0.793528i	$0.291758\pi$
$488$	0	0
$489$	−8259.01	−0.763773
$490$	0	0
$491$	11284.0	1.03715	0.518574	−	0.855033i	$-0.326463\pi$
$491$	11284.0	1.03715	0.518574	+	0.855033i	$0.326463\pi$
$492$	0	0
$493$	−27130.3	−2.47847
$494$	0	0
$495$	113.137	0.0102730
$496$	0	0
$497$	0	0
$498$	0	0
$499$	18172.0	1.63024	0.815121	−	0.579291i	$-0.196671\pi$
$499$	18172.0	1.63024	0.815121	+	0.579291i	$0.196671\pi$
$500$	0	0
$501$	8384.00	0.747644
$502$	0	0
$503$	−1380.27	−0.122352	−0.0611762	−	0.998127i	$-0.519485\pi$
$503$	−1380.27	−0.122352	−0.0611762	+	0.998127i	$0.519485\pi$
$504$	0	0
$505$	−7008.00	−0.617529
$506$	0	0
$507$	12247.1	1.07281
$508$	0	0
$509$	−3365.83	−0.293100	−0.146550	−	0.989203i	$-0.546817\pi$
$509$	−3365.83	−0.293100	−0.146550	+	0.989203i	$0.546817\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−16192.0	−1.39356
$514$	0	0
$515$	3648.00	0.312136
$516$	0	0
$517$	−497.803	−0.0423469
$518$	0	0
$519$	−23072.0	−1.95135
$520$	0	0
$521$	2862.37	0.240696	0.120348	−	0.992732i	$-0.461599\pi$
$521$	2862.37	0.240696	0.120348	+	0.992732i	$0.461599\pi$
$522$	0	0
$523$	13536.9	1.13179	0.565894	−	0.824478i	$-0.308531\pi$
$523$	13536.9	1.13179	0.565894	+	0.824478i	$0.308531\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18304.0	−1.51297
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	2743.57	0.224220
$532$	0	0
$533$	−832.000	−0.0676134
$534$	0	0
$535$	−5905.76	−0.477249
$536$	0	0
$537$	−1244.51	−0.100008
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4474.00	−0.355550	−0.177775	−	0.984071i	$-0.556890\pi$
$541$	−4474.00	−0.355550	−0.177775	+	0.984071i	$0.556890\pi$
$542$	0	0
$543$	10848.0	0.857334
$544$	0	0
$545$	2839.74	0.223195
$546$	0	0
$547$	−9100.00	−0.711312	−0.355656	−	0.934617i	$-0.615743\pi$
$547$	−9100.00	−0.711312	−0.355656	+	0.934617i	$0.615743\pi$
$548$	0	0
$549$	3252.69	0.252862
$550$	0	0
$551$	−28363.5	−2.19297
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4160.00	0.318166
$556$	0	0
$557$	−17610.0	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$557$	−17610.0	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$558$	0	0
$559$	1878.08	0.142100
$560$	0	0
$561$	−2816.00	−0.211928
$562$	0	0
$563$	14340.1	1.07347	0.536736	−	0.843751i	$-0.319657\pi$
$563$	14340.1	1.07347	0.536736	+	0.843751i	$0.319657\pi$
$564$	0	0
$565$	−735.391	−0.0547578
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13050.0	−0.961484	−0.480742	−	0.876862i	$-0.659633\pi$
$569$	−13050.0	−0.961484	−0.480742	+	0.876862i	$0.659633\pi$
$570$	0	0
$571$	9340.00	0.684530	0.342265	−	0.939603i	$-0.388806\pi$
$571$	9340.00	0.684530	0.342265	+	0.939603i	$0.388806\pi$
$572$	0	0
$573$	21586.6	1.57381
$574$	0	0
$575$	−11160.0	−0.809399
$576$	0	0
$577$	−8756.81	−0.631804	−0.315902	−	0.948792i	$-0.602307\pi$
$577$	−8756.81	−0.631804	−0.315902	+	0.948792i	$0.602307\pi$
$578$	0	0
$579$	−12954.2	−0.929807
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1992.00	0.141510
$584$	0	0
$585$	−160.000	−0.0113080
$586$	0	0
$587$	−1272.79	−0.0894953	−0.0447477	−	0.998998i	$-0.514248\pi$
$587$	−1272.79	−0.0894953	−0.0447477	+	0.998998i	$0.514248\pi$
$588$	0	0
$589$	−19136.0	−1.33868
$590$	0	0
$591$	9243.30	0.643348
$592$	0	0
$593$	21744.9	1.50583	0.752916	−	0.658117i	$-0.228647\pi$
$593$	21744.9	1.50583	0.752916	+	0.658117i	$0.228647\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−23616.0	−1.61899
$598$	0	0
$599$	8288.00	0.565340	0.282670	−	0.959217i	$-0.408780\pi$
$599$	8288.00	0.565340	0.282670	+	0.959217i	$0.408780\pi$
$600$	0	0
$601$	−19776.4	−1.34225	−0.671127	−	0.741342i	$-0.734189\pi$
$601$	−19776.4	−1.34225	−0.671127	+	0.741342i	$0.734189\pi$
$602$	0	0
$603$	780.000	0.0526767
$604$	0	0
$605$	7438.76	0.499882
$606$	0	0
$607$	26293.1	1.75816	0.879079	−	0.476675i	$-0.158158\pi$
$607$	26293.1	1.75816	0.879079	+	0.476675i	$0.158158\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	704.000	0.0466134
$612$	0	0
$613$	24738.0	1.62995	0.814974	−	0.579497i	$-0.196751\pi$
$613$	24738.0	1.62995	0.814974	+	0.579497i	$0.196751\pi$
$614$	0	0
$615$	−4706.50	−0.308593
$616$	0	0
$617$	1590.00	0.103746	0.0518728	−	0.998654i	$-0.483481\pi$
$617$	1590.00	0.103746	0.0518728	+	0.998654i	$0.483481\pi$
$618$	0	0
$619$	8852.98	0.574848	0.287424	−	0.957803i	$-0.407201\pi$
$619$	8852.98	0.574848	0.287424	+	0.957803i	$0.407201\pi$
$620$	0	0
$621$	14934.1	0.965032
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4649.00	0.297536
$626$	0	0
$627$	−2944.00	−0.187515
$628$	0	0
$629$	−16178.6	−1.02557
$630$	0	0
$631$	−18256.0	−1.15176	−0.575879	−	0.817535i	$-0.695340\pi$
$631$	−18256.0	−1.15176	−0.575879	+	0.817535i	$0.695340\pi$
$632$	0	0
$633$	−1335.02	−0.0838265
$634$	0	0
$635$	13667.0	0.854105
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1200.00	0.0742899
$640$	0	0
$641$	−24370.0	−1.50165	−0.750825	−	0.660502i	$-0.770344\pi$
$641$	−24370.0	−1.50165	−0.750825	+	0.660502i	$0.770344\pi$
$642$	0	0
$643$	627.911	0.0385107	0.0192554	−	0.999815i	$-0.493870\pi$
$643$	627.911	0.0385107	0.0192554	+	0.999815i	$0.493870\pi$
$644$	0	0
$645$	10624.0	0.648558
$646$	0	0
$647$	13949.8	0.847640	0.423820	−	0.905746i	$-0.360689\pi$
$647$	13949.8	0.847640	0.423820	+	0.905746i	$0.360689\pi$
$648$	0	0
$649$	−2194.86	−0.132752
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5078.00	−0.304315	−0.152157	−	0.988356i	$-0.548622\pi$
$653$	−5078.00	−0.304315	−0.152157	+	0.988356i	$0.548622\pi$
$654$	0	0
$655$	7520.00	0.448597
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5172.00	0.305725	0.152862	−	0.988247i	$-0.451151\pi$
$659$	5172.00	0.305725	0.152862	+	0.988247i	$0.451151\pi$
$660$	0	0
$661$	11511.7	0.677388	0.338694	−	0.940897i	$-0.390015\pi$
$661$	11511.7	0.677388	0.338694	+	0.940897i	$0.390015\pi$
$662$	0	0
$663$	3982.43	0.233280
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26160.0	1.51862
$668$	0	0
$669$	−30848.0	−1.78274
$670$	0	0
$671$	−2602.15	−0.149709
$672$	0	0
$673$	16350.0	0.936473	0.468236	−	0.883603i	$-0.344890\pi$
$673$	16350.0	0.936473	0.468236	+	0.883603i	$0.344890\pi$
$674$	0	0
$675$	−11573.9	−0.659971
$676$	0	0
$677$	−11692.7	−0.663793	−0.331896	−	0.943316i	$-0.607688\pi$
$677$	−11692.7	−0.663793	−0.331896	+	0.943316i	$0.607688\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−22176.0	−1.24785
$682$	0	0
$683$	26420.0	1.48014	0.740068	−	0.672532i	$-0.234793\pi$
$683$	26420.0	1.48014	0.740068	+	0.672532i	$0.234793\pi$
$684$	0	0
$685$	4129.50	0.230336
$686$	0	0
$687$	−8736.00	−0.485152
$688$	0	0
$689$	−2817.11	−0.155767
$690$	0	0
$691$	−16263.5	−0.895356	−0.447678	−	0.894195i	$-0.647749\pi$
$691$	−16263.5	−0.895356	−0.447678	+	0.894195i	$0.647749\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	160.000	0.00873258
$696$	0	0
$697$	18304.0	0.994711
$698$	0	0
$699$	1414.21	0.0765243
$700$	0	0
$701$	−3686.00	−0.198600	−0.0992998	−	0.995058i	$-0.531660\pi$
$701$	−3686.00	−0.198600	−0.0992998	+	0.995058i	$0.531660\pi$
$702$	0	0
$703$	−16914.0	−0.907430
$704$	0	0
$705$	3982.43	0.212747
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2882.00	0.152660	0.0763299	−	0.997083i	$-0.475680\pi$
$709$	2882.00	0.152660	0.0763299	+	0.997083i	$0.475680\pi$
$710$	0	0
$711$	5560.00	0.293272
$712$	0	0
$713$	17649.4	0.927033
$714$	0	0
$715$	128.000	0.00669501
$716$	0	0
$717$	9684.53	0.504429
$718$	0	0
$719$	22457.7	1.16486	0.582428	−	0.812882i	$-0.302103\pi$
$719$	22457.7	1.16486	0.582428	+	0.812882i	$0.302103\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11840.0	0.609038
$724$	0	0
$725$	−20274.0	−1.03856
$726$	0	0
$727$	−30603.6	−1.56124	−0.780622	−	0.625004i	$-0.785097\pi$
$727$	−30603.6	−1.56124	−0.780622	+	0.625004i	$0.785097\pi$
$728$	0	0
$729$	14813.0	0.752578
$730$	0	0
$731$	−41317.7	−2.09055
$732$	0	0
$733$	−729.734	−0.0367713	−0.0183856	−	0.999831i	$-0.505853\pi$
$733$	−729.734	−0.0367713	−0.0183856	+	0.999831i	$0.505853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−624.000	−0.0311877
$738$	0	0
$739$	−20268.0	−1.00889	−0.504445	−	0.863444i	$-0.668303\pi$
$739$	−20268.0	−1.00889	−0.504445	+	0.863444i	$0.668303\pi$
$740$	0	0
$741$	4163.44	0.206408
$742$	0	0
$743$	19688.0	0.972117	0.486058	−	0.873926i	$-0.338434\pi$
$743$	19688.0	0.972117	0.486058	+	0.873926i	$0.338434\pi$
$744$	0	0
$745$	15669.5	0.770585
$746$	0	0
$747$	141.421	0.00692682
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31056.0	1.50899	0.754494	−	0.656307i	$-0.227883\pi$
$751$	31056.0	1.50899	0.754494	+	0.656307i	$0.227883\pi$
$752$	0	0
$753$	−11360.0	−0.549776
$754$	0	0
$755$	−15703.4	−0.756962
$756$	0	0
$757$	27250.0	1.30835	0.654173	−	0.756345i	$-0.273017\pi$
$757$	27250.0	1.30835	0.654173	+	0.756345i	$0.273017\pi$
$758$	0	0
$759$	2715.29	0.129853
$760$	0	0
$761$	−35921.0	−1.71109	−0.855543	−	0.517732i	$-0.826776\pi$
$761$	−35921.0	−1.71109	−0.855543	+	0.517732i	$0.826776\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3520.00	0.166361
$766$	0	0
$767$	3104.00	0.146126
$768$	0	0
$769$	25897.1	1.21440	0.607199	−	0.794550i	$-0.292293\pi$
$769$	25897.1	1.21440	0.607199	+	0.794550i	$0.292293\pi$
$770$	0	0
$771$	23040.0	1.07622
$772$	0	0
$773$	1465.13	0.0681719	0.0340860	−	0.999419i	$-0.489148\pi$
$773$	1465.13	0.0681719	0.0340860	+	0.999419i	$0.489148\pi$
$774$	0	0
$775$	−13678.3	−0.633985
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19136.0	0.880126
$780$	0	0
$781$	−960.000	−0.0439840
$782$	0	0
$783$	27130.3	1.23826
$784$	0	0
$785$	−15520.0	−0.705647
$786$	0	0
$787$	5277.85	0.239053	0.119527	−	0.992831i	$-0.461862\pi$
$787$	5277.85	0.239053	0.119527	+	0.992831i	$0.461862\pi$
$788$	0	0
$789$	1991.21	0.0898467
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3680.00	0.164793
$794$	0	0
$795$	−15936.0	−0.710933
$796$	0	0
$797$	−10335.1	−0.459331	−0.229666	−	0.973270i	$-0.573763\pi$
$797$	−10335.1	−0.459331	−0.229666	+	0.973270i	$0.573763\pi$
$798$	0	0
$799$	−15488.0	−0.685765
$800$	0	0
$801$	−6788.23	−0.299438
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15200.0	0.663030
$808$	0	0
$809$	−25398.0	−1.10377	−0.551883	−	0.833922i	$-0.686090\pi$
$809$	−25398.0	−1.10377	−0.551883	+	0.833922i	$0.686090\pi$
$810$	0	0
$811$	37363.5	1.61777	0.808885	−	0.587968i	$-0.200072\pi$
$811$	37363.5	1.61777	0.808885	+	0.587968i	$0.200072\pi$
$812$	0	0
$813$	38400.0	1.65652
$814$	0	0
$815$	−8259.01	−0.354970
$816$	0	0
$817$	−43195.7	−1.84973
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6894.00	0.293060	0.146530	−	0.989206i	$-0.453190\pi$
$821$	6894.00	0.293060	0.146530	+	0.989206i	$0.453190\pi$
$822$	0	0
$823$	15008.0	0.635657	0.317829	−	0.948148i	$-0.397046\pi$
$823$	15008.0	0.635657	0.317829	+	0.948148i	$0.397046\pi$
$824$	0	0
$825$	−2104.35	−0.0888049
$826$	0	0
$827$	30740.0	1.29254	0.646272	−	0.763107i	$-0.276327\pi$
$827$	30740.0	1.29254	0.646272	+	0.763107i	$0.276327\pi$
$828$	0	0
$829$	−8219.41	−0.344357	−0.172178	−	0.985066i	$-0.555081\pi$
$829$	−8219.41	−0.344357	−0.172178	+	0.985066i	$0.555081\pi$
$830$	0	0
$831$	−30490.4	−1.27281
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8384.00	0.347473
$836$	0	0
$837$	18304.0	0.755889
$838$	0	0
$839$	22344.6	0.919452	0.459726	−	0.888061i	$-0.347948\pi$
$839$	22344.6	0.919452	0.459726	+	0.888061i	$0.347948\pi$
$840$	0	0
$841$	23135.0	0.948583
$842$	0	0
$843$	−12252.7	−0.500601
$844$	0	0
$845$	12247.1	0.498595
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26848.0	1.08530
$850$	0	0
$851$	15600.0	0.628391
$852$	0	0
$853$	−11624.8	−0.466620	−0.233310	−	0.972402i	$-0.574956\pi$
$853$	−11624.8	−0.466620	−0.233310	+	0.972402i	$0.574956\pi$
$854$	0	0
$855$	3680.00	0.147197
$856$	0	0
$857$	−8881.26	−0.354000	−0.177000	−	0.984211i	$-0.556639\pi$
$857$	−8881.26	−0.354000	−0.177000	+	0.984211i	$0.556639\pi$
$858$	0	0
$859$	17055.4	0.677443	0.338721	−	0.940887i	$-0.390006\pi$
$859$	17055.4	0.677443	0.338721	+	0.940887i	$0.390006\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24552.0	−0.968435	−0.484218	−	0.874948i	$-0.660896\pi$
$863$	−24552.0	−0.968435	−0.484218	+	0.874948i	$0.660896\pi$
$864$	0	0
$865$	−23072.0	−0.906904
$866$	0	0
$867$	−59821.2	−2.34329
$868$	0	0
$869$	−4448.00	−0.173634
$870$	0	0
$871$	882.469	0.0343299
$872$	0	0
$873$	7410.48	0.287293
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11306.0	0.435321	0.217660	−	0.976025i	$-0.430157\pi$
$877$	11306.0	0.435321	0.217660	+	0.976025i	$0.430157\pi$
$878$	0	0
$879$	−42208.0	−1.61961
$880$	0	0
$881$	1357.65	0.0519185	0.0259593	−	0.999663i	$-0.491736\pi$
$881$	1357.65	0.0519185	0.0259593	+	0.999663i	$0.491736\pi$
$882$	0	0
$883$	−980.000	−0.0373495	−0.0186748	−	0.999826i	$-0.505945\pi$
$883$	−980.000	−0.0373495	−0.0186748	+	0.999826i	$0.505945\pi$
$884$	0	0
$885$	17558.9	0.666932
$886$	0	0
$887$	7410.48	0.280518	0.140259	−	0.990115i	$-0.455206\pi$
$887$	7410.48	0.280518	0.140259	+	0.990115i	$0.455206\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3356.00	0.126184
$892$	0	0
$893$	−16192.0	−0.606769
$894$	0	0
$895$	−1244.51	−0.0464797
$896$	0	0
$897$	−3840.00	−0.142936
$898$	0	0
$899$	32063.0	1.18950
$900$	0	0
$901$	61976.5	2.29161
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10848.0	0.398453
$906$	0	0
$907$	−3580.00	−0.131061	−0.0655303	−	0.997851i	$-0.520874\pi$
$907$	−3580.00	−0.131061	−0.0655303	+	0.997851i	$0.520874\pi$
$908$	0	0
$909$	6194.26	0.226018
$910$	0	0
$911$	51920.0	1.88824	0.944120	−	0.329602i	$-0.106915\pi$
$911$	51920.0	1.88824	0.944120	+	0.329602i	$0.106915\pi$
$912$	0	0
$913$	−113.137	−0.00410109
$914$	0	0
$915$	20817.2	0.752127
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26032.0	0.934403	0.467202	−	0.884151i	$-0.345262\pi$
$919$	26032.0	0.934403	0.467202	+	0.884151i	$0.345262\pi$
$920$	0	0
$921$	50784.0	1.81693
$922$	0	0
$923$	1357.65	0.0484154
$924$	0	0
$925$	−12090.0	−0.429748
$926$	0	0
$927$	−3224.41	−0.114243
$928$	0	0
$929$	−7003.19	−0.247327	−0.123664	−	0.992324i	$-0.539464\pi$
$929$	−7003.19	−0.247327	−0.123664	+	0.992324i	$0.539464\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−11008.0	−0.386266
$934$	0	0
$935$	−2816.00	−0.0984952
$936$	0	0
$937$	22582.2	0.787329	0.393664	−	0.919254i	$-0.371207\pi$
$937$	22582.2	0.787329	0.393664	+	0.919254i	$0.371207\pi$
$938$	0	0
$939$	−42432.0	−1.47467
$940$	0	0
$941$	32555.2	1.12781	0.563905	−	0.825840i	$-0.309298\pi$
$941$	32555.2	1.12781	0.563905	+	0.825840i	$0.309298\pi$
$942$	0	0
$943$	−17649.4	−0.609484
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9996.00	−0.343006	−0.171503	−	0.985184i	$-0.554862\pi$
$947$	−9996.00	−0.343006	−0.171503	+	0.985184i	$0.554862\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−32617.4	−1.11219
$952$	0	0
$953$	27498.0	0.934677	0.467339	−	0.884078i	$-0.345213\pi$
$953$	27498.0	0.934677	0.467339	+	0.884078i	$0.345213\pi$
$954$	0	0
$955$	21586.6	0.731439
$956$	0	0
$957$	4932.78	0.166619
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8159.00	−0.273875
$962$	0	0
$963$	5220.00	0.174675
$964$	0	0
$965$	−12954.2	−0.432135
$966$	0	0
$967$	16104.0	0.535543	0.267771	−	0.963482i	$-0.413713\pi$
$967$	16104.0	0.535543	0.267771	+	0.963482i	$0.413713\pi$
$968$	0	0
$969$	−91595.8	−3.03662
$970$	0	0
$971$	−2211.83	−0.0731009	−0.0365505	−	0.999332i	$-0.511637\pi$
$971$	−2211.83	−0.0731009	−0.0365505	+	0.999332i	$0.511637\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2976.00	0.0977521
$976$	0	0
$977$	−19314.0	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$977$	−19314.0	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$978$	0	0
$979$	5430.58	0.177285
$980$	0	0
$981$	−2510.00	−0.0816902
$982$	0	0
$983$	36056.8	1.16992	0.584961	−	0.811062i	$-0.301110\pi$
$983$	36056.8	1.16992	0.584961	+	0.811062i	$0.301110\pi$
$984$	0	0
$985$	9243.30	0.299001
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39840.0	1.28093
$990$	0	0
$991$	−11272.0	−0.361319	−0.180659	−	0.983546i	$-0.557823\pi$
$991$	−11272.0	−0.361319	−0.180659	+	0.983546i	$0.557823\pi$
$992$	0	0
$993$	−60980.9	−1.94881
$994$	0	0
$995$	−23616.0	−0.752440
$996$	0	0
$997$	9973.03	0.316800	0.158400	−	0.987375i	$-0.449367\pi$
$997$	9973.03	0.316800	0.158400	+	0.987375i	$0.449367\pi$
$998$	0	0
$999$	16178.6	0.512381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.4.a.g.1.1	✓	2
4.3	odd	2		784.4.a.x.1.2		2
7.2	even	3		392.4.i.j.361.2		4
7.3	odd	6		392.4.i.j.177.1		4
7.4	even	3		392.4.i.j.177.2		4
7.5	odd	6		392.4.i.j.361.1		4
7.6	odd	2	inner	392.4.a.g.1.2	yes	2
28.27	even	2		784.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.g.1.1	✓	2	1.1	even	1	trivial
392.4.a.g.1.2	yes	2	7.6	odd	2	inner
392.4.i.j.177.1		4	7.3	odd	6
392.4.i.j.177.2		4	7.4	even	3
392.4.i.j.361.1		4	7.5	odd	6
392.4.i.j.361.2		4	7.2	even	3
784.4.a.x.1.1		2	28.27	even	2
784.4.a.x.1.2		2	4.3	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q-5.65685 q^{3} -5.65685 q^{5} +5.00000 q^{9} +O(q^{10})\)	\(q-5.65685 q^{3} -5.65685 q^{5} +5.00000 q^{9} -4.00000 q^{11} +5.65685 q^{13} +32.0000 q^{15} -124.451 q^{17} -130.108 q^{19} +120.000 q^{23} -93.0000 q^{25} +124.451 q^{27} +218.000 q^{29} +147.078 q^{31} +22.6274 q^{33} +130.000 q^{37} -32.0000 q^{39} -147.078 q^{41} +332.000 q^{43} -28.2843 q^{45} +124.451 q^{47} +704.000 q^{51} -498.000 q^{53} +22.6274 q^{55} +736.000 q^{57} +548.715 q^{59} +650.538 q^{61} -32.0000 q^{65} +156.000 q^{67} -678.823 q^{69} +240.000 q^{71} +526.087 q^{75} +1112.00 q^{79} -839.000 q^{81} +28.2843 q^{83} +704.000 q^{85} -1233.19 q^{87} -1357.65 q^{89} -832.000 q^{93} +736.000 q^{95} +1482.10 q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9}+O(q^{10}) \) 2 * q + 10 * q^9	\( 2 q + 10 q^{9} - 8 q^{11} + 64 q^{15} + 240 q^{23} - 186 q^{25} + 436 q^{29} + 260 q^{37} - 64 q^{39} + 664 q^{43} + 1408 q^{51} - 996 q^{53} + 1472 q^{57} - 64 q^{65} + 312 q^{67} + 480 q^{71} + 2224 q^{79} - 1678 q^{81} + 1408 q^{85} - 1664 q^{93} + 1472 q^{95} - 40 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 - 8 * q^11 + 64 * q^15 + 240 * q^23 - 186 * q^25 + 436 * q^29 + 260 * q^37 - 64 * q^39 + 664 * q^43 + 1408 * q^51 - 996 * q^53 + 1472 * q^57 - 64 * q^65 + 312 * q^67 + 480 * q^71 + 2224 * q^79 - 1678 * q^81 + 1408 * q^85 - 1664 * q^93 + 1472 * q^95 - 40 * q^99

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