LMFDB - Embedded newform 1856.4.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 19x^{6} + 92x^{4} - 51x^{2} + 5 $ x^8 - 19x^6 + 92x^4 - 51*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.702141$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.67525 q^{3} -6.01400 q^{5} -26.8310 q^{7} +17.5590 q^{9} +O(q^{10})$	$q-6.67525 q^{3} -6.01400 q^{5} -26.8310 q^{7} +17.5590 q^{9} -13.6469 q^{11} +1.20286 q^{13} +40.1450 q^{15} -100.089 q^{17} +61.7320 q^{19} +179.104 q^{21} +149.816 q^{23} -88.8319 q^{25} +63.0209 q^{27} -29.0000 q^{29} -79.7358 q^{31} +91.0969 q^{33} +161.361 q^{35} -186.987 q^{37} -8.02937 q^{39} -115.600 q^{41} +549.253 q^{43} -105.600 q^{45} +322.835 q^{47} +376.901 q^{49} +668.121 q^{51} +531.641 q^{53} +82.0727 q^{55} -412.077 q^{57} +161.629 q^{59} +489.592 q^{61} -471.126 q^{63} -7.23397 q^{65} +526.205 q^{67} -1000.06 q^{69} -175.324 q^{71} -1096.16 q^{73} +592.975 q^{75} +366.161 q^{77} -290.565 q^{79} -894.774 q^{81} -1209.79 q^{83} +601.936 q^{85} +193.582 q^{87} +914.177 q^{89} -32.2738 q^{91} +532.257 q^{93} -371.256 q^{95} +1362.12 q^{97} -239.627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) $ 8 * q + 20 * q^5 + 40 * q^9	$ 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) $ 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

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
$3$	−6.67525	−1.28465	−0.642327	−	0.766431i	$-0.722031\pi$
$3$	−6.67525	−1.28465	−0.642327	+	0.766431i	$0.722031\pi$
$4$	0	0
$5$	−6.01400	−0.537908	−0.268954	−	0.963153i	$-0.586678\pi$
$5$	−6.01400	−0.537908	−0.268954	+	0.963153i	$0.586678\pi$
$6$	0	0
$7$	−26.8310	−1.44874	−0.724368	−	0.689413i	$-0.757868\pi$
$7$	−26.8310	−1.44874	−0.724368	+	0.689413i	$0.757868\pi$
$8$	0	0
$9$	17.5590	0.650334
$10$	0	0
$11$	−13.6469	−0.374065	−0.187032	−	0.982354i	$-0.559887\pi$
$11$	−13.6469	−0.374065	−0.187032	+	0.982354i	$0.559887\pi$
$12$	0	0
$13$	1.20286	0.0256625	0.0128312	−	0.999918i	$-0.495916\pi$
$13$	1.20286	0.0256625	0.0128312	+	0.999918i	$0.495916\pi$
$14$	0	0
$15$	40.1450	0.691025
$16$	0	0
$17$	−100.089	−1.42795	−0.713977	−	0.700170i	$-0.753108\pi$
$17$	−100.089	−1.42795	−0.713977	+	0.700170i	$0.753108\pi$
$18$	0	0
$19$	61.7320	0.745384	0.372692	−	0.927955i	$-0.378435\pi$
$19$	61.7320	0.745384	0.372692	+	0.927955i	$0.378435\pi$
$20$	0	0
$21$	179.104	1.86112
$22$	0	0
$23$	149.816	1.35821	0.679104	−	0.734042i	$-0.262369\pi$
$23$	149.816	1.35821	0.679104	+	0.734042i	$0.262369\pi$
$24$	0	0
$25$	−88.8319	−0.710655
$26$	0	0
$27$	63.0209	0.449199
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−79.7358	−0.461967	−0.230984	−	0.972958i	$-0.574194\pi$
$31$	−79.7358	−0.461967	−0.230984	+	0.972958i	$0.574194\pi$
$32$	0	0
$33$	91.0969	0.480543
$34$	0	0
$35$	161.361	0.779287
$36$	0	0
$37$	−186.987	−0.830823	−0.415412	−	0.909634i	$-0.636362\pi$
$37$	−186.987	−0.830823	−0.415412	+	0.909634i	$0.636362\pi$
$38$	0	0
$39$	−8.02937	−0.0329674
$40$	0	0
$41$	−115.600	−0.440332	−0.220166	−	0.975462i	$-0.570660\pi$
$41$	−115.600	−0.440332	−0.220166	+	0.975462i	$0.570660\pi$
$42$	0	0
$43$	549.253	1.94791	0.973957	−	0.226732i	$-0.0728041\pi$
$43$	549.253	1.94791	0.973957	+	0.226732i	$0.0728041\pi$
$44$	0	0
$45$	−105.600	−0.349820
$46$	0	0
$47$	322.835	1.00192	0.500961	−	0.865470i	$-0.332980\pi$
$47$	322.835	1.00192	0.500961	+	0.865470i	$0.332980\pi$
$48$	0	0
$49$	376.901	1.09884
$50$	0	0
$51$	668.121	1.83442
$52$	0	0
$53$	531.641	1.37786	0.688929	−	0.724829i	$-0.258081\pi$
$53$	531.641	1.37786	0.688929	+	0.724829i	$0.258081\pi$
$54$	0	0
$55$	82.0727	0.201212
$56$	0	0
$57$	−412.077	−0.957560
$58$	0	0
$59$	161.629	0.356649	0.178324	−	0.983972i	$-0.442932\pi$
$59$	161.629	0.356649	0.178324	+	0.983972i	$0.442932\pi$
$60$	0	0
$61$	489.592	1.02764	0.513818	−	0.857899i	$-0.328231\pi$
$61$	489.592	1.02764	0.513818	+	0.857899i	$0.328231\pi$
$62$	0	0
$63$	−471.126	−0.942163
$64$	0	0
$65$	−7.23397	−0.0138040
$66$	0	0
$67$	526.205	0.959494	0.479747	−	0.877407i	$-0.340728\pi$
$67$	526.205	0.959494	0.479747	+	0.877407i	$0.340728\pi$
$68$	0	0
$69$	−1000.06	−1.74483
$70$	0	0
$71$	−175.324	−0.293058	−0.146529	−	0.989206i	$-0.546810\pi$
$71$	−175.324	−0.293058	−0.146529	+	0.989206i	$0.546810\pi$
$72$	0	0
$73$	−1096.16	−1.75747	−0.878735	−	0.477310i	$-0.841612\pi$
$73$	−1096.16	−1.75747	−0.878735	+	0.477310i	$0.841612\pi$
$74$	0	0
$75$	592.975	0.912945
$76$	0	0
$77$	366.161	0.541921
$78$	0	0
$79$	−290.565	−0.413812	−0.206906	−	0.978361i	$-0.566339\pi$
$79$	−290.565	−0.413812	−0.206906	+	0.978361i	$0.566339\pi$
$80$	0	0
$81$	−894.774	−1.22740
$82$	0	0
$83$	−1209.79	−1.59990	−0.799952	−	0.600064i	$-0.795142\pi$
$83$	−1209.79	−1.59990	−0.799952	+	0.600064i	$0.795142\pi$
$84$	0	0
$85$	601.936	0.768108
$86$	0	0
$87$	193.582	0.238554
$88$	0	0
$89$	914.177	1.08879	0.544397	−	0.838828i	$-0.316759\pi$
$89$	914.177	1.08879	0.544397	+	0.838828i	$0.316759\pi$
$90$	0	0
$91$	−32.2738	−0.0371781
$92$	0	0
$93$	532.257	0.593468
$94$	0	0
$95$	−371.256	−0.400948
$96$	0	0
$97$	1362.12	1.42579	0.712897	−	0.701269i	$-0.247383\pi$
$97$	1362.12	1.42579	0.712897	+	0.701269i	$0.247383\pi$
$98$	0	0
$99$	−239.627	−0.243267
$100$	0	0
$101$	92.3886	0.0910199	0.0455100	−	0.998964i	$-0.485509\pi$
$101$	92.3886	0.0910199	0.0455100	+	0.998964i	$0.485509\pi$
$102$	0	0
$103$	1306.76	1.25008	0.625041	−	0.780592i	$-0.285082\pi$
$103$	1306.76	1.25008	0.625041	+	0.780592i	$0.285082\pi$
$104$	0	0
$105$	−1077.13	−1.00111
$106$	0	0
$107$	−908.747	−0.821046	−0.410523	−	0.911850i	$-0.634654\pi$
$107$	−908.747	−0.821046	−0.410523	+	0.911850i	$0.634654\pi$
$108$	0	0
$109$	875.932	0.769716	0.384858	−	0.922976i	$-0.374250\pi$
$109$	875.932	0.769716	0.384858	+	0.922976i	$0.374250\pi$
$110$	0	0
$111$	1248.19	1.06732
$112$	0	0
$113$	−437.446	−0.364172	−0.182086	−	0.983283i	$-0.558285\pi$
$113$	−437.446	−0.364172	−0.182086	+	0.983283i	$0.558285\pi$
$114$	0	0
$115$	−900.992	−0.730591
$116$	0	0
$117$	21.1210	0.0166892
$118$	0	0
$119$	2685.49	2.06873
$120$	0	0
$121$	−1144.76	−0.860076
$122$	0	0
$123$	771.656	0.565674
$124$	0	0
$125$	1285.98	0.920175
$126$	0	0
$127$	55.9680	0.0391052	0.0195526	−	0.999809i	$-0.493776\pi$
$127$	55.9680	0.0391052	0.0195526	+	0.999809i	$0.493776\pi$
$128$	0	0
$129$	−3666.40	−2.50239
$130$	0	0
$131$	−1406.15	−0.937832	−0.468916	−	0.883243i	$-0.655355\pi$
$131$	−1406.15	−0.937832	−0.468916	+	0.883243i	$0.655355\pi$
$132$	0	0
$133$	−1656.33	−1.07986
$134$	0	0
$135$	−379.007	−0.241628
$136$	0	0
$137$	1131.92	0.705886	0.352943	−	0.935645i	$-0.385181\pi$
$137$	1131.92	0.705886	0.352943	+	0.935645i	$0.385181\pi$
$138$	0	0
$139$	1666.80	1.01709	0.508546	−	0.861035i	$-0.330183\pi$
$139$	1666.80	1.01709	0.508546	+	0.861035i	$0.330183\pi$
$140$	0	0
$141$	−2155.01	−1.28712
$142$	0	0
$143$	−16.4153	−0.00959942
$144$	0	0
$145$	174.406	0.0998870
$146$	0	0
$147$	−2515.91	−1.41162
$148$	0	0
$149$	1915.09	1.05296	0.526478	−	0.850189i	$-0.323512\pi$
$149$	1915.09	1.05296	0.526478	+	0.850189i	$0.323512\pi$
$150$	0	0
$151$	880.626	0.474598	0.237299	−	0.971437i	$-0.423738\pi$
$151$	880.626	0.474598	0.237299	+	0.971437i	$0.423738\pi$
$152$	0	0
$153$	−1757.47	−0.928647
$154$	0	0
$155$	479.531	0.248496
$156$	0	0
$157$	902.086	0.458562	0.229281	−	0.973360i	$-0.426362\pi$
$157$	902.086	0.458562	0.229281	+	0.973360i	$0.426362\pi$
$158$	0	0
$159$	−3548.84	−1.77007
$160$	0	0
$161$	−4019.71	−1.96768
$162$	0	0
$163$	−311.858	−0.149857	−0.0749283	−	0.997189i	$-0.523873\pi$
$163$	−311.858	−0.149857	−0.0749283	+	0.997189i	$0.523873\pi$
$164$	0	0
$165$	−547.856	−0.258488
$166$	0	0
$167$	847.806	0.392845	0.196423	−	0.980519i	$-0.437068\pi$
$167$	847.806	0.392845	0.196423	+	0.980519i	$0.437068\pi$
$168$	0	0
$169$	−2195.55	−0.999341
$170$	0	0
$171$	1083.95	0.484748
$172$	0	0
$173$	590.459	0.259490	0.129745	−	0.991547i	$-0.458584\pi$
$173$	590.459	0.259490	0.129745	+	0.991547i	$0.458584\pi$
$174$	0	0
$175$	2383.45	1.02955
$176$	0	0
$177$	−1078.91	−0.458170
$178$	0	0
$179$	−890.897	−0.372004	−0.186002	−	0.982549i	$-0.559553\pi$
$179$	−890.897	−0.372004	−0.186002	+	0.982549i	$0.559553\pi$
$180$	0	0
$181$	−2434.50	−0.999749	−0.499875	−	0.866098i	$-0.666621\pi$
$181$	−2434.50	−0.999749	−0.499875	+	0.866098i	$0.666621\pi$
$182$	0	0
$183$	−3268.15	−1.32016
$184$	0	0
$185$	1124.54	0.446907
$186$	0	0
$187$	1365.91	0.534147
$188$	0	0
$189$	−1690.91	−0.650771
$190$	0	0
$191$	3475.46	1.31663	0.658313	−	0.752744i	$-0.271270\pi$
$191$	3475.46	1.31663	0.658313	+	0.752744i	$0.271270\pi$
$192$	0	0
$193$	−4428.52	−1.65167	−0.825834	−	0.563913i	$-0.809295\pi$
$193$	−4428.52	−1.65167	−0.825834	+	0.563913i	$0.809295\pi$
$194$	0	0
$195$	48.2886	0.0177334
$196$	0	0
$197$	−3024.38	−1.09380	−0.546899	−	0.837198i	$-0.684192\pi$
$197$	−3024.38	−1.09380	−0.546899	+	0.837198i	$0.684192\pi$
$198$	0	0
$199$	−681.016	−0.242593	−0.121296	−	0.992616i	$-0.538705\pi$
$199$	−681.016	−0.242593	−0.121296	+	0.992616i	$0.538705\pi$
$200$	0	0
$201$	−3512.55	−1.23262
$202$	0	0
$203$	778.098	0.269024
$204$	0	0
$205$	695.215	0.236858
$206$	0	0
$207$	2630.62	0.883289
$208$	0	0
$209$	−842.453	−0.278822
$210$	0	0
$211$	−2622.80	−0.855738	−0.427869	−	0.903841i	$-0.640736\pi$
$211$	−2622.80	−0.855738	−0.427869	+	0.903841i	$0.640736\pi$
$212$	0	0
$213$	1170.33	0.376478
$214$	0	0
$215$	−3303.21	−1.04780
$216$	0	0
$217$	2139.39	0.669268
$218$	0	0
$219$	7317.12	2.25774
$220$	0	0
$221$	−120.393	−0.0366448
$222$	0	0
$223$	−4758.60	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$223$	−4758.60	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$224$	0	0
$225$	−1559.80	−0.462163
$226$	0	0
$227$	−1784.04	−0.521633	−0.260816	−	0.965388i	$-0.583992\pi$
$227$	−1784.04	−0.521633	−0.260816	+	0.965388i	$0.583992\pi$
$228$	0	0
$229$	1088.82	0.314196	0.157098	−	0.987583i	$-0.449786\pi$
$229$	1088.82	0.314196	0.157098	+	0.987583i	$0.449786\pi$
$230$	0	0
$231$	−2444.22	−0.696180
$232$	0	0
$233$	−562.244	−0.158085	−0.0790425	−	0.996871i	$-0.525186\pi$
$233$	−562.244	−0.158085	−0.0790425	+	0.996871i	$0.525186\pi$
$234$	0	0
$235$	−1941.53	−0.538942
$236$	0	0
$237$	1939.60	0.531605
$238$	0	0
$239$	1180.82	0.319586	0.159793	−	0.987151i	$-0.448917\pi$
$239$	1180.82	0.319586	0.159793	+	0.987151i	$0.448917\pi$
$240$	0	0
$241$	−6564.26	−1.75453	−0.877264	−	0.480009i	$-0.840634\pi$
$241$	−6564.26	−1.75453	−0.877264	+	0.480009i	$0.840634\pi$
$242$	0	0
$243$	4271.28	1.12758
$244$	0	0
$245$	−2266.68	−0.591073
$246$	0	0
$247$	74.2546	0.0191284
$248$	0	0
$249$	8075.67	2.05532
$250$	0	0
$251$	−2693.32	−0.677293	−0.338647	−	0.940914i	$-0.609969\pi$
$251$	−2693.32	−0.677293	−0.338647	+	0.940914i	$0.609969\pi$
$252$	0	0
$253$	−2044.53	−0.508057
$254$	0	0
$255$	−4018.08	−0.986752
$256$	0	0
$257$	3834.84	0.930781	0.465391	−	0.885105i	$-0.345914\pi$
$257$	3834.84	0.930781	0.465391	+	0.885105i	$0.345914\pi$
$258$	0	0
$259$	5017.04	1.20364
$260$	0	0
$261$	−509.212	−0.120764
$262$	0	0
$263$	−1556.96	−0.365043	−0.182522	−	0.983202i	$-0.558426\pi$
$263$	−1556.96	−0.365043	−0.182522	+	0.983202i	$0.558426\pi$
$264$	0	0
$265$	−3197.29	−0.741161
$266$	0	0
$267$	−6102.37	−1.39872
$268$	0	0
$269$	−7321.11	−1.65939	−0.829695	−	0.558217i	$-0.811486\pi$
$269$	−7321.11	−1.65939	−0.829695	+	0.558217i	$0.811486\pi$
$270$	0	0
$271$	4667.48	1.04623	0.523116	−	0.852261i	$-0.324769\pi$
$271$	4667.48	1.04623	0.523116	+	0.852261i	$0.324769\pi$
$272$	0	0
$273$	215.436	0.0477610
$274$	0	0
$275$	1212.28	0.265831
$276$	0	0
$277$	8072.85	1.75109	0.875543	−	0.483140i	$-0.160504\pi$
$277$	8072.85	1.75109	0.875543	+	0.483140i	$0.160504\pi$
$278$	0	0
$279$	−1400.08	−0.300433
$280$	0	0
$281$	5327.62	1.13103	0.565515	−	0.824738i	$-0.308678\pi$
$281$	5327.62	1.13103	0.565515	+	0.824738i	$0.308678\pi$
$282$	0	0
$283$	422.493	0.0887442	0.0443721	−	0.999015i	$-0.485871\pi$
$283$	422.493	0.0887442	0.0443721	+	0.999015i	$0.485871\pi$
$284$	0	0
$285$	2478.23	0.515079
$286$	0	0
$287$	3101.65	0.637925
$288$	0	0
$289$	5104.85	1.03905
$290$	0	0
$291$	−9092.48	−1.83165
$292$	0	0
$293$	5421.56	1.08099	0.540496	−	0.841346i	$-0.318236\pi$
$293$	5421.56	1.08099	0.540496	+	0.841346i	$0.318236\pi$
$294$	0	0
$295$	−972.034	−0.191844
$296$	0	0
$297$	−860.043	−0.168030
$298$	0	0
$299$	180.207	0.0348549
$300$	0	0
$301$	−14737.0	−2.82201
$302$	0	0
$303$	−616.718	−0.116929
$304$	0	0
$305$	−2944.40	−0.552774
$306$	0	0
$307$	−3691.49	−0.686269	−0.343134	−	0.939286i	$-0.611489\pi$
$307$	−3691.49	−0.686269	−0.343134	+	0.939286i	$0.611489\pi$
$308$	0	0
$309$	−8722.93	−1.60592
$310$	0	0
$311$	−9190.89	−1.67578	−0.837890	−	0.545839i	$-0.816211\pi$
$311$	−9190.89	−1.67578	−0.837890	+	0.545839i	$0.816211\pi$
$312$	0	0
$313$	−9667.91	−1.74589	−0.872944	−	0.487821i	$-0.837792\pi$
$313$	−9667.91	−1.74589	−0.872944	+	0.487821i	$0.837792\pi$
$314$	0	0
$315$	2833.35	0.506797
$316$	0	0
$317$	−2156.34	−0.382058	−0.191029	−	0.981584i	$-0.561182\pi$
$317$	−2156.34	−0.382058	−0.191029	+	0.981584i	$0.561182\pi$
$318$	0	0
$319$	395.761	0.0694620
$320$	0	0
$321$	6066.12	1.05476
$322$	0	0
$323$	−6178.71	−1.06437
$324$	0	0
$325$	−106.852	−0.0182372
$326$	0	0
$327$	−5847.07	−0.988818
$328$	0	0
$329$	−8661.98	−1.45152
$330$	0	0
$331$	114.473	0.0190090	0.00950451	−	0.999955i	$-0.496975\pi$
$331$	114.473	0.0190090	0.00950451	+	0.999955i	$0.496975\pi$
$332$	0	0
$333$	−3283.31	−0.540313
$334$	0	0
$335$	−3164.59	−0.516120
$336$	0	0
$337$	2375.44	0.383972	0.191986	−	0.981398i	$-0.438507\pi$
$337$	2375.44	0.383972	0.191986	+	0.981398i	$0.438507\pi$
$338$	0	0
$339$	2920.06	0.467835
$340$	0	0
$341$	1088.15	0.172805
$342$	0	0
$343$	−909.600	−0.143189
$344$	0	0
$345$	6014.35	0.938556
$346$	0	0
$347$	4814.75	0.744868	0.372434	−	0.928059i	$-0.378523\pi$
$347$	4814.75	0.744868	0.372434	+	0.928059i	$0.378523\pi$
$348$	0	0
$349$	−6814.33	−1.04517	−0.522583	−	0.852589i	$-0.675031\pi$
$349$	−6814.33	−1.04517	−0.522583	+	0.852589i	$0.675031\pi$
$350$	0	0
$351$	75.8050	0.0115276
$352$	0	0
$353$	−2869.29	−0.432626	−0.216313	−	0.976324i	$-0.569403\pi$
$353$	−2869.29	−0.432626	−0.216313	+	0.976324i	$0.569403\pi$
$354$	0	0
$355$	1054.40	0.157638
$356$	0	0
$357$	−17926.3	−2.65760
$358$	0	0
$359$	6994.09	1.02823	0.514114	−	0.857722i	$-0.328121\pi$
$359$	6994.09	1.02823	0.514114	+	0.857722i	$0.328121\pi$
$360$	0	0
$361$	−3048.16	−0.444403
$362$	0	0
$363$	7641.57	1.10490
$364$	0	0
$365$	6592.27	0.945357
$366$	0	0
$367$	−1376.68	−0.195810	−0.0979051	−	0.995196i	$-0.531214\pi$
$367$	−1376.68	−0.195810	−0.0979051	+	0.995196i	$0.531214\pi$
$368$	0	0
$369$	−2029.82	−0.286363
$370$	0	0
$371$	−14264.4	−1.99615
$372$	0	0
$373$	−1470.10	−0.204072	−0.102036	−	0.994781i	$-0.532536\pi$
$373$	−1470.10	−0.204072	−0.102036	+	0.994781i	$0.532536\pi$
$374$	0	0
$375$	−8584.27	−1.18211
$376$	0	0
$377$	−34.8828	−0.00476540
$378$	0	0
$379$	2387.32	0.323557	0.161779	−	0.986827i	$-0.448277\pi$
$379$	2387.32	0.323557	0.161779	+	0.986827i	$0.448277\pi$
$380$	0	0
$381$	−373.601	−0.0502366
$382$	0	0
$383$	−2802.81	−0.373934	−0.186967	−	0.982366i	$-0.559866\pi$
$383$	−2802.81	−0.373934	−0.186967	+	0.982366i	$0.559866\pi$
$384$	0	0
$385$	−2202.09	−0.291504
$386$	0	0
$387$	9644.35	1.26680
$388$	0	0
$389$	3964.33	0.516708	0.258354	−	0.966050i	$-0.416820\pi$
$389$	3964.33	0.516708	0.258354	+	0.966050i	$0.416820\pi$
$390$	0	0
$391$	−14995.0	−1.93946
$392$	0	0
$393$	9386.41	1.20479
$394$	0	0
$395$	1747.46	0.222593
$396$	0	0
$397$	8763.56	1.10789	0.553943	−	0.832555i	$-0.313123\pi$
$397$	8763.56	1.10789	0.553943	+	0.832555i	$0.313123\pi$
$398$	0	0
$399$	11056.4	1.38725
$400$	0	0
$401$	−1199.79	−0.149413	−0.0747067	−	0.997206i	$-0.523802\pi$
$401$	−1199.79	−0.149413	−0.0747067	+	0.997206i	$0.523802\pi$
$402$	0	0
$403$	−95.9107	−0.0118552
$404$	0	0
$405$	5381.17	0.660228
$406$	0	0
$407$	2551.80	0.310782
$408$	0	0
$409$	5855.65	0.707930	0.353965	−	0.935259i	$-0.384833\pi$
$409$	5855.65	0.707930	0.353965	+	0.935259i	$0.384833\pi$
$410$	0	0
$411$	−7555.85	−0.906819
$412$	0	0
$413$	−4336.66	−0.516690
$414$	0	0
$415$	7275.69	0.860601
$416$	0	0
$417$	−11126.3	−1.30661
$418$	0	0
$419$	−10653.8	−1.24218	−0.621089	−	0.783740i	$-0.713310\pi$
$419$	−10653.8	−1.24218	−0.621089	+	0.783740i	$0.713310\pi$
$420$	0	0
$421$	−5502.49	−0.636995	−0.318497	−	0.947924i	$-0.603178\pi$
$421$	−5502.49	−0.636995	−0.318497	+	0.947924i	$0.603178\pi$
$422$	0	0
$423$	5668.67	0.651584
$424$	0	0
$425$	8891.11	1.01478
$426$	0	0
$427$	−13136.2	−1.48877
$428$	0	0
$429$	109.576	0.0123319
$430$	0	0
$431$	12530.7	1.40043	0.700215	−	0.713932i	$-0.253088\pi$
$431$	12530.7	1.40043	0.700215	+	0.713932i	$0.253088\pi$
$432$	0	0
$433$	15274.9	1.69530	0.847651	−	0.530555i	$-0.178016\pi$
$433$	15274.9	1.69530	0.847651	+	0.530555i	$0.178016\pi$
$434$	0	0
$435$	−1164.20	−0.128320
$436$	0	0
$437$	9248.43	1.01239
$438$	0	0
$439$	14935.0	1.62371	0.811854	−	0.583861i	$-0.198459\pi$
$439$	14935.0	1.62371	0.811854	+	0.583861i	$0.198459\pi$
$440$	0	0
$441$	6618.02	0.714611
$442$	0	0
$443$	2841.36	0.304734	0.152367	−	0.988324i	$-0.451310\pi$
$443$	2841.36	0.304734	0.152367	+	0.988324i	$0.451310\pi$
$444$	0	0
$445$	−5497.86	−0.585671
$446$	0	0
$447$	−12783.7	−1.35268
$448$	0	0
$449$	−12922.6	−1.35825	−0.679125	−	0.734023i	$-0.737641\pi$
$449$	−12922.6	−1.35825	−0.679125	+	0.734023i	$0.737641\pi$
$450$	0	0
$451$	1577.58	0.164713
$452$	0	0
$453$	−5878.41	−0.609694
$454$	0	0
$455$	194.094	0.0199984
$456$	0	0
$457$	8496.75	0.869718	0.434859	−	0.900498i	$-0.356798\pi$
$457$	8496.75	0.869718	0.434859	+	0.900498i	$0.356798\pi$
$458$	0	0
$459$	−6307.71	−0.641435
$460$	0	0
$461$	−13723.4	−1.38647	−0.693236	−	0.720711i	$-0.743815\pi$
$461$	−13723.4	−1.38647	−0.693236	+	0.720711i	$0.743815\pi$
$462$	0	0
$463$	15885.1	1.59448	0.797241	−	0.603661i	$-0.206292\pi$
$463$	15885.1	1.59448	0.797241	+	0.603661i	$0.206292\pi$
$464$	0	0
$465$	−3200.99	−0.319231
$466$	0	0
$467$	−10524.3	−1.04284	−0.521419	−	0.853301i	$-0.674597\pi$
$467$	−10524.3	−1.04284	−0.521419	+	0.853301i	$0.674597\pi$
$468$	0	0
$469$	−14118.6	−1.39005
$470$	0	0
$471$	−6021.65	−0.589094
$472$	0	0
$473$	−7495.63	−0.728646
$474$	0	0
$475$	−5483.77	−0.529710
$476$	0	0
$477$	9335.10	0.896069
$478$	0	0
$479$	−15346.6	−1.46390	−0.731948	−	0.681361i	$-0.761388\pi$
$479$	−15346.6	−1.46390	−0.731948	+	0.681361i	$0.761388\pi$
$480$	0	0
$481$	−224.918	−0.0213210
$482$	0	0
$483$	26832.6	2.52779
$484$	0	0
$485$	−8191.77	−0.766946
$486$	0	0
$487$	13338.0	1.24107	0.620537	−	0.784177i	$-0.286915\pi$
$487$	13338.0	1.24107	0.620537	+	0.784177i	$0.286915\pi$
$488$	0	0
$489$	2081.73	0.192514
$490$	0	0
$491$	10422.3	0.957949	0.478975	−	0.877829i	$-0.341009\pi$
$491$	10422.3	0.957949	0.478975	+	0.877829i	$0.341009\pi$
$492$	0	0
$493$	2902.59	0.265164
$494$	0	0
$495$	1441.12	0.130855
$496$	0	0
$497$	4704.11	0.424564
$498$	0	0
$499$	−9871.95	−0.885630	−0.442815	−	0.896613i	$-0.646020\pi$
$499$	−9871.95	−0.885630	−0.442815	+	0.896613i	$0.646020\pi$
$500$	0	0
$501$	−5659.32	−0.504670
$502$	0	0
$503$	−11313.8	−1.00290	−0.501449	−	0.865187i	$-0.667200\pi$
$503$	−11313.8	−1.00290	−0.501449	+	0.865187i	$0.667200\pi$
$504$	0	0
$505$	−555.625	−0.0489603
$506$	0	0
$507$	14655.9	1.28381
$508$	0	0
$509$	−12854.8	−1.11941	−0.559705	−	0.828692i	$-0.689086\pi$
$509$	−12854.8	−1.11941	−0.559705	+	0.828692i	$0.689086\pi$
$510$	0	0
$511$	29410.9	2.54611
$512$	0	0
$513$	3890.41	0.334826
$514$	0	0
$515$	−7858.83	−0.672430
$516$	0	0
$517$	−4405.71	−0.374783
$518$	0	0
$519$	−3941.47	−0.333355
$520$	0	0
$521$	−3470.70	−0.291850	−0.145925	−	0.989296i	$-0.546616\pi$
$521$	−3470.70	−0.291850	−0.145925	+	0.989296i	$0.546616\pi$
$522$	0	0
$523$	−6660.80	−0.556896	−0.278448	−	0.960451i	$-0.589820\pi$
$523$	−6660.80	−0.556896	−0.278448	+	0.960451i	$0.589820\pi$
$524$	0	0
$525$	−15910.1	−1.32262
$526$	0	0
$527$	7980.70	0.659667
$528$	0	0
$529$	10277.8	0.844727
$530$	0	0
$531$	2838.04	0.231941
$532$	0	0
$533$	−139.050	−0.0113000
$534$	0	0
$535$	5465.20	0.441647
$536$	0	0
$537$	5946.96	0.477896
$538$	0	0
$539$	−5143.55	−0.411036
$540$	0	0
$541$	−9270.08	−0.736695	−0.368347	−	0.929688i	$-0.620076\pi$
$541$	−9270.08	−0.736695	−0.368347	+	0.929688i	$0.620076\pi$
$542$	0	0
$543$	16250.9	1.28433
$544$	0	0
$545$	−5267.85	−0.414036
$546$	0	0
$547$	3281.59	0.256510	0.128255	−	0.991741i	$-0.459062\pi$
$547$	3281.59	0.256510	0.128255	+	0.991741i	$0.459062\pi$
$548$	0	0
$549$	8596.75	0.668307
$550$	0	0
$551$	−1790.23	−0.138414
$552$	0	0
$553$	7796.15	0.599505
$554$	0	0
$555$	−7506.58	−0.574120
$556$	0	0
$557$	3047.61	0.231834	0.115917	−	0.993259i	$-0.463019\pi$
$557$	3047.61	0.231834	0.115917	+	0.993259i	$0.463019\pi$
$558$	0	0
$559$	660.672	0.0499883
$560$	0	0
$561$	−9117.81	−0.686193
$562$	0	0
$563$	16418.7	1.22907	0.614536	−	0.788889i	$-0.289343\pi$
$563$	16418.7	1.22907	0.614536	+	0.788889i	$0.289343\pi$
$564$	0	0
$565$	2630.80	0.195891
$566$	0	0
$567$	24007.7	1.77818
$568$	0	0
$569$	−10017.9	−0.738090	−0.369045	−	0.929411i	$-0.620315\pi$
$569$	−10017.9	−0.738090	−0.369045	+	0.929411i	$0.620315\pi$
$570$	0	0
$571$	21821.2	1.59928	0.799639	−	0.600481i	$-0.205024\pi$
$571$	21821.2	1.59928	0.799639	+	0.600481i	$0.205024\pi$
$572$	0	0
$573$	−23199.6	−1.69141
$574$	0	0
$575$	−13308.4	−0.965217
$576$	0	0
$577$	−15664.3	−1.13018	−0.565088	−	0.825030i	$-0.691158\pi$
$577$	−15664.3	−1.13018	−0.565088	+	0.825030i	$0.691158\pi$
$578$	0	0
$579$	29561.5	2.12182
$580$	0	0
$581$	32459.9	2.31784
$582$	0	0
$583$	−7255.28	−0.515408
$584$	0	0
$585$	−127.021	−0.00897725
$586$	0	0
$587$	15834.4	1.11339	0.556693	−	0.830718i	$-0.312070\pi$
$587$	15834.4	1.11339	0.556693	+	0.830718i	$0.312070\pi$
$588$	0	0
$589$	−4922.25	−0.344343
$590$	0	0
$591$	20188.5	1.40515
$592$	0	0
$593$	−929.744	−0.0643845	−0.0321922	−	0.999482i	$-0.510249\pi$
$593$	−929.744	−0.0643845	−0.0321922	+	0.999482i	$0.510249\pi$
$594$	0	0
$595$	−16150.5	−1.11279
$596$	0	0
$597$	4545.96	0.311648
$598$	0	0
$599$	−15058.6	−1.02718	−0.513588	−	0.858037i	$-0.671684\pi$
$599$	−15058.6	−1.02718	−0.513588	+	0.858037i	$0.671684\pi$
$600$	0	0
$601$	1537.76	0.104370	0.0521852	−	0.998637i	$-0.483381\pi$
$601$	1537.76	0.104370	0.0521852	+	0.998637i	$0.483381\pi$
$602$	0	0
$603$	9239.64	0.623992
$604$	0	0
$605$	6884.59	0.462642
$606$	0	0
$607$	10039.9	0.671344	0.335672	−	0.941979i	$-0.391037\pi$
$607$	10039.9	0.671344	0.335672	+	0.941979i	$0.391037\pi$
$608$	0	0
$609$	−5194.00	−0.345602
$610$	0	0
$611$	388.324	0.0257118
$612$	0	0
$613$	13124.4	0.864748	0.432374	−	0.901694i	$-0.357676\pi$
$613$	13124.4	0.864748	0.432374	+	0.901694i	$0.357676\pi$
$614$	0	0
$615$	−4640.74	−0.304281
$616$	0	0
$617$	21273.2	1.38805	0.694025	−	0.719951i	$-0.255836\pi$
$617$	21273.2	1.38805	0.694025	+	0.719951i	$0.255836\pi$
$618$	0	0
$619$	20457.2	1.32834	0.664172	−	0.747580i	$-0.268784\pi$
$619$	20457.2	1.32834	0.664172	+	0.747580i	$0.268784\pi$
$620$	0	0
$621$	9441.53	0.610106
$622$	0	0
$623$	−24528.3	−1.57737
$624$	0	0
$625$	3370.08	0.215685
$626$	0	0
$627$	5623.59	0.358189
$628$	0	0
$629$	18715.4	1.18638
$630$	0	0
$631$	−20186.9	−1.27358	−0.636790	−	0.771038i	$-0.719738\pi$
$631$	−20186.9	−1.27358	−0.636790	+	0.771038i	$0.719738\pi$
$632$	0	0
$633$	17507.8	1.09933
$634$	0	0
$635$	−336.591	−0.0210350
$636$	0	0
$637$	453.358	0.0281989
$638$	0	0
$639$	−3078.52	−0.190586
$640$	0	0
$641$	25797.0	1.58958	0.794789	−	0.606885i	$-0.207581\pi$
$641$	25797.0	1.58958	0.794789	+	0.606885i	$0.207581\pi$
$642$	0	0
$643$	17174.5	1.05334	0.526668	−	0.850071i	$-0.323441\pi$
$643$	17174.5	1.05334	0.526668	+	0.850071i	$0.323441\pi$
$644$	0	0
$645$	22049.7	1.34606
$646$	0	0
$647$	21899.4	1.33069	0.665343	−	0.746538i	$-0.268286\pi$
$647$	21899.4	1.33069	0.665343	+	0.746538i	$0.268286\pi$
$648$	0	0
$649$	−2205.74	−0.133410
$650$	0	0
$651$	−14281.0	−0.859778
$652$	0	0
$653$	6017.79	0.360634	0.180317	−	0.983609i	$-0.442288\pi$
$653$	6017.79	0.360634	0.180317	+	0.983609i	$0.442288\pi$
$654$	0	0
$655$	8456.58	0.504467
$656$	0	0
$657$	−19247.4	−1.14294
$658$	0	0
$659$	−8860.50	−0.523757	−0.261879	−	0.965101i	$-0.584342\pi$
$659$	−8860.50	−0.523757	−0.261879	+	0.965101i	$0.584342\pi$
$660$	0	0
$661$	2592.78	0.152568	0.0762840	−	0.997086i	$-0.475694\pi$
$661$	2592.78	0.152568	0.0762840	+	0.997086i	$0.475694\pi$
$662$	0	0
$663$	803.653	0.0470759
$664$	0	0
$665$	9961.16	0.580868
$666$	0	0
$667$	−4344.66	−0.252213
$668$	0	0
$669$	31764.9	1.83573
$670$	0	0
$671$	−6681.43	−0.384402
$672$	0	0
$673$	4289.21	0.245672	0.122836	−	0.992427i	$-0.460801\pi$
$673$	4289.21	0.245672	0.122836	+	0.992427i	$0.460801\pi$
$674$	0	0
$675$	−5598.26	−0.319226
$676$	0	0
$677$	33650.0	1.91030	0.955151	−	0.296118i	$-0.0956922\pi$
$677$	33650.0	1.91030	0.955151	+	0.296118i	$0.0956922\pi$
$678$	0	0
$679$	−36546.9	−2.06560
$680$	0	0
$681$	11908.9	0.670118
$682$	0	0
$683$	33232.1	1.86177	0.930885	−	0.365312i	$-0.119038\pi$
$683$	33232.1	1.86177	0.930885	+	0.365312i	$0.119038\pi$
$684$	0	0
$685$	−6807.36	−0.379702
$686$	0	0
$687$	−7268.12	−0.403633
$688$	0	0
$689$	639.487	0.0353592
$690$	0	0
$691$	4839.06	0.266406	0.133203	−	0.991089i	$-0.457474\pi$
$691$	4839.06	0.266406	0.133203	+	0.991089i	$0.457474\pi$
$692$	0	0
$693$	6429.43	0.352430
$694$	0	0
$695$	−10024.1	−0.547102
$696$	0	0
$697$	11570.3	0.628774
$698$	0	0
$699$	3753.12	0.203084
$700$	0	0
$701$	−21422.0	−1.15420	−0.577102	−	0.816672i	$-0.695816\pi$
$701$	−21422.0	−1.15420	−0.577102	+	0.816672i	$0.695816\pi$
$702$	0	0
$703$	−11543.1	−0.619282
$704$	0	0
$705$	12960.2	0.692354
$706$	0	0
$707$	−2478.88	−0.131864
$708$	0	0
$709$	−28554.5	−1.51254	−0.756268	−	0.654262i	$-0.772979\pi$
$709$	−28554.5	−1.51254	−0.756268	+	0.654262i	$0.772979\pi$
$710$	0	0
$711$	−5102.04	−0.269116
$712$	0	0
$713$	−11945.7	−0.627447
$714$	0	0
$715$	98.7216	0.00516360
$716$	0	0
$717$	−7882.28	−0.410557
$718$	0	0
$719$	5643.80	0.292737	0.146369	−	0.989230i	$-0.453241\pi$
$719$	5643.80	0.292737	0.146369	+	0.989230i	$0.453241\pi$
$720$	0	0
$721$	−35061.5	−1.81104
$722$	0	0
$723$	43818.1	2.25396
$724$	0	0
$725$	2576.12	0.131965
$726$	0	0
$727$	−18702.5	−0.954109	−0.477054	−	0.878874i	$-0.658296\pi$
$727$	−18702.5	−0.954109	−0.477054	+	0.878874i	$0.658296\pi$
$728$	0	0
$729$	−4352.99	−0.221155
$730$	0	0
$731$	−54974.3	−2.78153
$732$	0	0
$733$	21502.0	1.08349	0.541744	−	0.840544i	$-0.317764\pi$
$733$	21502.0	1.08349	0.541744	+	0.840544i	$0.317764\pi$
$734$	0	0
$735$	15130.7	0.759324
$736$	0	0
$737$	−7181.09	−0.358913
$738$	0	0
$739$	−12906.2	−0.642440	−0.321220	−	0.947005i	$-0.604093\pi$
$739$	−12906.2	−0.642440	−0.321220	+	0.947005i	$0.604093\pi$
$740$	0	0
$741$	−495.669	−0.0245733
$742$	0	0
$743$	−7233.93	−0.357183	−0.178592	−	0.983923i	$-0.557154\pi$
$743$	−7233.93	−0.357183	−0.178592	+	0.983923i	$0.557154\pi$
$744$	0	0
$745$	−11517.3	−0.566393
$746$	0	0
$747$	−21242.8	−1.04047
$748$	0	0
$749$	24382.6	1.18948
$750$	0	0
$751$	−13042.1	−0.633708	−0.316854	−	0.948474i	$-0.602626\pi$
$751$	−13042.1	−0.633708	−0.316854	+	0.948474i	$0.602626\pi$
$752$	0	0
$753$	17978.6	0.870087
$754$	0	0
$755$	−5296.08	−0.255290
$756$	0	0
$757$	10658.2	0.511731	0.255865	−	0.966712i	$-0.417640\pi$
$757$	10658.2	0.511731	0.255865	+	0.966712i	$0.417640\pi$
$758$	0	0
$759$	13647.8	0.652677
$760$	0	0
$761$	−34805.8	−1.65796	−0.828981	−	0.559276i	$-0.811079\pi$
$761$	−34805.8	−1.65796	−0.828981	+	0.559276i	$0.811079\pi$
$762$	0	0
$763$	−23502.1	−1.11512
$764$	0	0
$765$	10569.4	0.499527
$766$	0	0
$767$	194.416	0.00915248
$768$	0	0
$769$	−208.091	−0.00975809	−0.00487904	−	0.999988i	$-0.501553\pi$
$769$	−208.091	−0.00975809	−0.00487904	+	0.999988i	$0.501553\pi$
$770$	0	0
$771$	−25598.5	−1.19573
$772$	0	0
$773$	25816.1	1.20122	0.600609	−	0.799543i	$-0.294925\pi$
$773$	25816.1	1.20122	0.600609	+	0.799543i	$0.294925\pi$
$774$	0	0
$775$	7083.08	0.328299
$776$	0	0
$777$	−33490.0	−1.54627
$778$	0	0
$779$	−7136.19	−0.328216
$780$	0	0
$781$	2392.64	0.109623
$782$	0	0
$783$	−1827.61	−0.0834142
$784$	0	0
$785$	−5425.14	−0.246664
$786$	0	0
$787$	−34090.5	−1.54409	−0.772043	−	0.635570i	$-0.780765\pi$
$787$	−34090.5	−1.54409	−0.772043	+	0.635570i	$0.780765\pi$
$788$	0	0
$789$	10393.1	0.468954
$790$	0	0
$791$	11737.1	0.527589
$792$	0	0
$793$	588.908	0.0263717
$794$	0	0
$795$	21342.7	0.952135
$796$	0	0
$797$	−15555.4	−0.691344	−0.345672	−	0.938355i	$-0.612349\pi$
$797$	−15555.4	−0.691344	−0.345672	+	0.938355i	$0.612349\pi$
$798$	0	0
$799$	−32312.3	−1.43070
$800$	0	0
$801$	16052.1	0.708080
$802$	0	0
$803$	14959.2	0.657407
$804$	0	0
$805$	24174.5	1.05843
$806$	0	0
$807$	48870.3	2.13174
$808$	0	0
$809$	24602.1	1.06918	0.534589	−	0.845112i	$-0.320467\pi$
$809$	24602.1	1.06918	0.534589	+	0.845112i	$0.320467\pi$
$810$	0	0
$811$	2656.73	0.115031	0.0575157	−	0.998345i	$-0.481682\pi$
$811$	2656.73	0.115031	0.0575157	+	0.998345i	$0.481682\pi$
$812$	0	0
$813$	−31156.6	−1.34405
$814$	0	0
$815$	1875.51	0.0806090
$816$	0	0
$817$	33906.5	1.45194
$818$	0	0
$819$	−566.696	−0.0241782
$820$	0	0
$821$	41970.4	1.78414	0.892068	−	0.451901i	$-0.149254\pi$
$821$	41970.4	1.78414	0.892068	+	0.451901i	$0.149254\pi$
$822$	0	0
$823$	−32434.5	−1.37375	−0.686875	−	0.726776i	$-0.741018\pi$
$823$	−32434.5	−1.37375	−0.686875	+	0.726776i	$0.741018\pi$
$824$	0	0
$825$	−8092.30	−0.341500
$826$	0	0
$827$	−12389.6	−0.520955	−0.260477	−	0.965480i	$-0.583880\pi$
$827$	−12389.6	−0.520955	−0.260477	+	0.965480i	$0.583880\pi$
$828$	0	0
$829$	9167.56	0.384080	0.192040	−	0.981387i	$-0.438490\pi$
$829$	9167.56	0.384080	0.192040	+	0.981387i	$0.438490\pi$
$830$	0	0
$831$	−53888.4	−2.24954
$832$	0	0
$833$	−37723.7	−1.56909
$834$	0	0
$835$	−5098.70	−0.211315
$836$	0	0
$837$	−5025.02	−0.207515
$838$	0	0
$839$	22146.4	0.911296	0.455648	−	0.890160i	$-0.349408\pi$
$839$	22146.4	0.911296	0.455648	+	0.890160i	$0.349408\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−35563.2	−1.45298
$844$	0	0
$845$	13204.0	0.537554
$846$	0	0
$847$	30715.0	1.24602
$848$	0	0
$849$	−2820.25	−0.114006
$850$	0	0
$851$	−28013.6	−1.12843
$852$	0	0
$853$	3153.19	0.126569	0.0632844	−	0.997996i	$-0.479842\pi$
$853$	3153.19	0.126569	0.0632844	+	0.997996i	$0.479842\pi$
$854$	0	0
$855$	−6518.89	−0.260750
$856$	0	0
$857$	15765.5	0.628400	0.314200	−	0.949357i	$-0.398264\pi$
$857$	15765.5	0.628400	0.314200	+	0.949357i	$0.398264\pi$
$858$	0	0
$859$	−15872.5	−0.630455	−0.315228	−	0.949016i	$-0.602081\pi$
$859$	−15872.5	−0.630455	−0.315228	+	0.949016i	$0.602081\pi$
$860$	0	0
$861$	−20704.3	−0.819513
$862$	0	0
$863$	−29406.5	−1.15992	−0.579959	−	0.814645i	$-0.696932\pi$
$863$	−29406.5	−1.15992	−0.579959	+	0.814645i	$0.696932\pi$
$864$	0	0
$865$	−3551.02	−0.139582
$866$	0	0
$867$	−34076.2	−1.33482
$868$	0	0
$869$	3965.33	0.154792
$870$	0	0
$871$	632.948	0.0246230
$872$	0	0
$873$	23917.4	0.927243
$874$	0	0
$875$	−34504.2	−1.33309
$876$	0	0
$877$	4454.90	0.171529	0.0857647	−	0.996315i	$-0.472667\pi$
$877$	4454.90	0.171529	0.0857647	+	0.996315i	$0.472667\pi$
$878$	0	0
$879$	−36190.3	−1.38870
$880$	0	0
$881$	−3307.10	−0.126469	−0.0632343	−	0.997999i	$-0.520142\pi$
$881$	−3307.10	−0.126469	−0.0632343	+	0.997999i	$0.520142\pi$
$882$	0	0
$883$	−10956.9	−0.417587	−0.208793	−	0.977960i	$-0.566954\pi$
$883$	−10956.9	−0.417587	−0.208793	+	0.977960i	$0.566954\pi$
$884$	0	0
$885$	6488.58	0.246453
$886$	0	0
$887$	49594.1	1.87735	0.938674	−	0.344805i	$-0.112055\pi$
$887$	49594.1	1.87735	0.938674	+	0.344805i	$0.112055\pi$
$888$	0	0
$889$	−1501.68	−0.0566531
$890$	0	0
$891$	12210.9	0.459127
$892$	0	0
$893$	19929.2	0.746816
$894$	0	0
$895$	5357.85	0.200104
$896$	0	0
$897$	−1202.93	−0.0447765
$898$	0	0
$899$	2312.34	0.0857851
$900$	0	0
$901$	−53211.5	−1.96752
$902$	0	0
$903$	98373.2	3.62531
$904$	0	0
$905$	14641.0	0.537773
$906$	0	0
$907$	−36367.9	−1.33140	−0.665699	−	0.746220i	$-0.731866\pi$
$907$	−36367.9	−1.33140	−0.665699	+	0.746220i	$0.731866\pi$
$908$	0	0
$909$	1622.25	0.0591934
$910$	0	0
$911$	−18435.7	−0.670473	−0.335236	−	0.942134i	$-0.608816\pi$
$911$	−18435.7	−0.670473	−0.335236	+	0.942134i	$0.608816\pi$
$912$	0	0
$913$	16510.0	0.598467
$914$	0	0
$915$	19654.6	0.710122
$916$	0	0
$917$	37728.4	1.35867
$918$	0	0
$919$	−27486.9	−0.986625	−0.493312	−	0.869852i	$-0.664214\pi$
$919$	−27486.9	−0.986625	−0.493312	+	0.869852i	$0.664214\pi$
$920$	0	0
$921$	24641.6	0.881617
$922$	0	0
$923$	−210.889	−0.00752060
$924$	0	0
$925$	16610.4	0.590429
$926$	0	0
$927$	22945.4	0.812972
$928$	0	0
$929$	6757.44	0.238648	0.119324	−	0.992855i	$-0.461927\pi$
$929$	6757.44	0.238648	0.119324	+	0.992855i	$0.461927\pi$
$930$	0	0
$931$	23266.9	0.819055
$932$	0	0
$933$	61351.5	2.15280
$934$	0	0
$935$	−8214.59	−0.287322
$936$	0	0
$937$	−11255.9	−0.392439	−0.196219	−	0.980560i	$-0.562866\pi$
$937$	−11255.9	−0.392439	−0.196219	+	0.980560i	$0.562866\pi$
$938$	0	0
$939$	64535.8	2.24286
$940$	0	0
$941$	53727.2	1.86127	0.930636	−	0.365946i	$-0.119254\pi$
$941$	53727.2	1.86127	0.930636	+	0.365946i	$0.119254\pi$
$942$	0	0
$943$	−17318.6	−0.598062
$944$	0	0
$945$	10169.1	0.350055
$946$	0	0
$947$	28284.1	0.970549	0.485274	−	0.874362i	$-0.338720\pi$
$947$	28284.1	0.970549	0.485274	+	0.874362i	$0.338720\pi$
$948$	0	0
$949$	−1318.52	−0.0451010
$950$	0	0
$951$	14394.2	0.490812
$952$	0	0
$953$	14711.2	0.500045	0.250023	−	0.968240i	$-0.419562\pi$
$953$	14711.2	0.500045	0.250023	+	0.968240i	$0.419562\pi$
$954$	0	0
$955$	−20901.4	−0.708224
$956$	0	0
$957$	−2641.81	−0.0892346
$958$	0	0
$959$	−30370.5	−1.02264
$960$	0	0
$961$	−23433.2	−0.786586
$962$	0	0
$963$	−15956.7	−0.533954
$964$	0	0
$965$	26633.1	0.888446
$966$	0	0
$967$	2907.91	0.0967032	0.0483516	−	0.998830i	$-0.484603\pi$
$967$	2907.91	0.0967032	0.0483516	+	0.998830i	$0.484603\pi$
$968$	0	0
$969$	41244.4	1.36735
$970$	0	0
$971$	18395.7	0.607978	0.303989	−	0.952676i	$-0.401681\pi$
$971$	18395.7	0.607978	0.303989	+	0.952676i	$0.401681\pi$
$972$	0	0
$973$	−44721.8	−1.47350
$974$	0	0
$975$	713.263	0.0234284
$976$	0	0
$977$	−56026.4	−1.83464	−0.917319	−	0.398153i	$-0.869651\pi$
$977$	−56026.4	−1.83464	−0.917319	+	0.398153i	$0.869651\pi$
$978$	0	0
$979$	−12475.7	−0.407279
$980$	0	0
$981$	15380.5	0.500573
$982$	0	0
$983$	−4069.08	−0.132028	−0.0660141	−	0.997819i	$-0.521028\pi$
$983$	−4069.08	−0.132028	−0.0660141	+	0.997819i	$0.521028\pi$
$984$	0	0
$985$	18188.6	0.588363
$986$	0	0
$987$	57820.9	1.86470
$988$	0	0
$989$	82286.8	2.64567
$990$	0	0
$991$	−36893.1	−1.18259	−0.591295	−	0.806455i	$-0.701383\pi$
$991$	−36893.1	−1.18259	−0.591295	+	0.806455i	$0.701383\pi$
$992$	0	0
$993$	−764.134	−0.0244200
$994$	0	0
$995$	4095.63	0.130493
$996$	0	0
$997$	−32184.0	−1.02234	−0.511172	−	0.859479i	$-0.670788\pi$
$997$	−32184.0	−1.02234	−0.511172	+	0.859479i	$0.670788\pi$
$998$	0	0
$999$	−11784.1	−0.373205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.be.1.2		8
4.3	odd	2	inner	1856.4.a.be.1.7		8
8.3	odd	2		928.4.a.c.1.2	✓	8
8.5	even	2		928.4.a.c.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.c.1.2	✓	8	8.3	odd	2
928.4.a.c.1.7	yes	8	8.5	even	2
1856.4.a.be.1.2		8	1.1	even	1	trivial
1856.4.a.be.1.7		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-6.67525 q^{3} -6.01400 q^{5} -26.8310 q^{7} +17.5590 q^{9} +O(q^{10})\)	\(q-6.67525 q^{3} -6.01400 q^{5} -26.8310 q^{7} +17.5590 q^{9} -13.6469 q^{11} +1.20286 q^{13} +40.1450 q^{15} -100.089 q^{17} +61.7320 q^{19} +179.104 q^{21} +149.816 q^{23} -88.8319 q^{25} +63.0209 q^{27} -29.0000 q^{29} -79.7358 q^{31} +91.0969 q^{33} +161.361 q^{35} -186.987 q^{37} -8.02937 q^{39} -115.600 q^{41} +549.253 q^{43} -105.600 q^{45} +322.835 q^{47} +376.901 q^{49} +668.121 q^{51} +531.641 q^{53} +82.0727 q^{55} -412.077 q^{57} +161.629 q^{59} +489.592 q^{61} -471.126 q^{63} -7.23397 q^{65} +526.205 q^{67} -1000.06 q^{69} -175.324 q^{71} -1096.16 q^{73} +592.975 q^{75} +366.161 q^{77} -290.565 q^{79} -894.774 q^{81} -1209.79 q^{83} +601.936 q^{85} +193.582 q^{87} +914.177 q^{89} -32.2738 q^{91} +532.257 q^{93} -371.256 q^{95} +1362.12 q^{97} -239.627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) \) 8 * q + 20 * q^5 + 40 * q^9	\( 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) \) 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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