LMFDB - Embedded newform 1856.4.a.bi.1.4

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} + \cdots + 128592 $ x^10 - 4x^9 - 135x^8 + 788x^7 + 3323x^6 - 26136x^5 + 2315x^4 + 188664x^3 - 265632x^2 - 18800*x + 128592
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.47788$ 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)

$f(q)$	$=$	$q-3.91971 q^{3} -7.82384 q^{5} +17.9594 q^{7} -11.6359 q^{9} +O(q^{10})$	$q-3.91971 q^{3} -7.82384 q^{5} +17.9594 q^{7} -11.6359 q^{9} +43.9600 q^{11} -16.3449 q^{13} +30.6672 q^{15} -103.343 q^{17} +131.247 q^{19} -70.3957 q^{21} +19.3205 q^{23} -63.7875 q^{25} +151.441 q^{27} +29.0000 q^{29} -164.702 q^{31} -172.311 q^{33} -140.512 q^{35} +299.840 q^{37} +64.0672 q^{39} +309.015 q^{41} +6.49592 q^{43} +91.0373 q^{45} -197.683 q^{47} -20.4589 q^{49} +405.074 q^{51} +402.061 q^{53} -343.936 q^{55} -514.449 q^{57} -512.698 q^{59} -796.465 q^{61} -208.974 q^{63} +127.880 q^{65} -976.161 q^{67} -75.7309 q^{69} +327.169 q^{71} -335.380 q^{73} +250.029 q^{75} +789.497 q^{77} +430.636 q^{79} -279.438 q^{81} +665.667 q^{83} +808.539 q^{85} -113.672 q^{87} -721.405 q^{89} -293.545 q^{91} +645.583 q^{93} -1026.85 q^{95} +1504.43 q^{97} -511.513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) $ 10 * q + 4 * q^5 + 50 * q^9	$ 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) $ 10 * q + 4 * q^5 + 50 * q^9 + 220 * q^13 + 76 * q^17 + 104 * q^21 + 446 * q^25 + 290 * q^29 - 1120 * q^33 + 1708 * q^37 - 980 * q^41 - 348 * q^45 + 1146 * q^49 + 44 * q^53 - 40 * q^57 + 1492 * q^61 - 2016 * q^65 + 2328 * q^69 - 3100 * q^73 + 3016 * q^77 - 2174 * q^81 + 9440 * q^85 + 636 * q^89 + 2536 * q^93 + 620 * 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
$3$	−3.91971	−0.754348	−0.377174	−	0.926142i	$-0.623104\pi$
$3$	−3.91971	−0.754348	−0.377174	+	0.926142i	$0.623104\pi$
$4$	0	0
$5$	−7.82384	−0.699786	−0.349893	−	0.936790i	$-0.613782\pi$
$5$	−7.82384	−0.699786	−0.349893	+	0.936790i	$0.613782\pi$
$6$	0	0
$7$	17.9594	0.969718	0.484859	−	0.874592i	$-0.338871\pi$
$7$	17.9594	0.969718	0.484859	+	0.874592i	$0.338871\pi$
$8$	0	0
$9$	−11.6359	−0.430958
$10$	0	0
$11$	43.9600	1.20495	0.602475	−	0.798138i	$-0.294181\pi$
$11$	43.9600	1.20495	0.602475	+	0.798138i	$0.294181\pi$
$12$	0	0
$13$	−16.3449	−0.348712	−0.174356	−	0.984683i	$-0.555784\pi$
$13$	−16.3449	−0.348712	−0.174356	+	0.984683i	$0.555784\pi$
$14$	0	0
$15$	30.6672	0.527882
$16$	0	0
$17$	−103.343	−1.47437	−0.737187	−	0.675689i	$-0.763846\pi$
$17$	−103.343	−1.47437	−0.737187	+	0.675689i	$0.763846\pi$
$18$	0	0
$19$	131.247	1.58474	0.792370	−	0.610041i	$-0.208847\pi$
$19$	131.247	1.58474	0.792370	+	0.610041i	$0.208847\pi$
$20$	0	0
$21$	−70.3957	−0.731505
$22$	0	0
$23$	19.3205	0.175157	0.0875785	−	0.996158i	$-0.472087\pi$
$23$	19.3205	0.175157	0.0875785	+	0.996158i	$0.472087\pi$
$24$	0	0
$25$	−63.7875	−0.510300
$26$	0	0
$27$	151.441	1.07944
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−164.702	−0.954236	−0.477118	−	0.878839i	$-0.658319\pi$
$31$	−164.702	−0.954236	−0.477118	+	0.878839i	$0.658319\pi$
$32$	0	0
$33$	−172.311	−0.908952
$34$	0	0
$35$	−140.512	−0.678595
$36$	0	0
$37$	299.840	1.33225	0.666127	−	0.745838i	$-0.267951\pi$
$37$	299.840	1.33225	0.666127	+	0.745838i	$0.267951\pi$
$38$	0	0
$39$	64.0672	0.263050
$40$	0	0
$41$	309.015	1.17707	0.588537	−	0.808471i	$-0.299704\pi$
$41$	309.015	1.17707	0.588537	+	0.808471i	$0.299704\pi$
$42$	0	0
$43$	6.49592	0.0230376	0.0115188	−	0.999934i	$-0.496333\pi$
$43$	6.49592	0.0230376	0.0115188	+	0.999934i	$0.496333\pi$
$44$	0	0
$45$	91.0373	0.301579
$46$	0	0
$47$	−197.683	−0.613511	−0.306756	−	0.951788i	$-0.599243\pi$
$47$	−197.683	−0.613511	−0.306756	+	0.951788i	$0.599243\pi$
$48$	0	0
$49$	−20.4589	−0.0596470
$50$	0	0
$51$	405.074	1.11219
$52$	0	0
$53$	402.061	1.04203	0.521013	−	0.853549i	$-0.325554\pi$
$53$	402.061	1.04203	0.521013	+	0.853549i	$0.325554\pi$
$54$	0	0
$55$	−343.936	−0.843206
$56$	0	0
$57$	−514.449	−1.19545
$58$	0	0
$59$	−512.698	−1.13132	−0.565658	−	0.824640i	$-0.691378\pi$
$59$	−512.698	−1.13132	−0.565658	+	0.824640i	$0.691378\pi$
$60$	0	0
$61$	−796.465	−1.67175	−0.835876	−	0.548918i	$-0.815040\pi$
$61$	−796.465	−1.67175	−0.835876	+	0.548918i	$0.815040\pi$
$62$	0	0
$63$	−208.974	−0.417908
$64$	0	0
$65$	127.880	0.244024
$66$	0	0
$67$	−976.161	−1.77996	−0.889978	−	0.456003i	$-0.849280\pi$
$67$	−976.161	−1.77996	−0.889978	+	0.456003i	$0.849280\pi$
$68$	0	0
$69$	−75.7309	−0.132129
$70$	0	0
$71$	327.169	0.546871	0.273435	−	0.961890i	$-0.411840\pi$
$71$	327.169	0.546871	0.273435	+	0.961890i	$0.411840\pi$
$72$	0	0
$73$	−335.380	−0.537716	−0.268858	−	0.963180i	$-0.586646\pi$
$73$	−335.380	−0.537716	−0.268858	+	0.963180i	$0.586646\pi$
$74$	0	0
$75$	250.029	0.384944
$76$	0	0
$77$	789.497	1.16846
$78$	0	0
$79$	430.636	0.613296	0.306648	−	0.951823i	$-0.400793\pi$
$79$	430.636	0.613296	0.306648	+	0.951823i	$0.400793\pi$
$80$	0	0
$81$	−279.438	−0.383316
$82$	0	0
$83$	665.667	0.880319	0.440159	−	0.897920i	$-0.354922\pi$
$83$	665.667	0.880319	0.440159	+	0.897920i	$0.354922\pi$
$84$	0	0
$85$	808.539	1.03175
$86$	0	0
$87$	−113.672	−0.140079
$88$	0	0
$89$	−721.405	−0.859200	−0.429600	−	0.903019i	$-0.641345\pi$
$89$	−721.405	−0.859200	−0.429600	+	0.903019i	$0.641345\pi$
$90$	0	0
$91$	−293.545	−0.338152
$92$	0	0
$93$	645.583	0.719826
$94$	0	0
$95$	−1026.85	−1.10898
$96$	0	0
$97$	1504.43	1.57476	0.787378	−	0.616470i	$-0.211438\pi$
$97$	1504.43	1.57476	0.787378	+	0.616470i	$0.211438\pi$
$98$	0	0
$99$	−511.513	−0.519283
$100$	0	0
$101$	−991.587	−0.976897	−0.488449	−	0.872593i	$-0.662437\pi$
$101$	−991.587	−0.976897	−0.488449	+	0.872593i	$0.662437\pi$
$102$	0	0
$103$	596.603	0.570729	0.285364	−	0.958419i	$-0.407885\pi$
$103$	596.603	0.570729	0.285364	+	0.958419i	$0.407885\pi$
$104$	0	0
$105$	550.765	0.511897
$106$	0	0
$107$	−1055.86	−0.953961	−0.476980	−	0.878914i	$-0.658269\pi$
$107$	−1055.86	−0.953961	−0.476980	+	0.878914i	$0.658269\pi$
$108$	0	0
$109$	311.926	0.274102	0.137051	−	0.990564i	$-0.456238\pi$
$109$	311.926	0.274102	0.137051	+	0.990564i	$0.456238\pi$
$110$	0	0
$111$	−1175.29	−1.00498
$112$	0	0
$113$	−1625.66	−1.35335	−0.676676	−	0.736281i	$-0.736580\pi$
$113$	−1625.66	−1.35335	−0.676676	+	0.736281i	$0.736580\pi$
$114$	0	0
$115$	−151.161	−0.122572
$116$	0	0
$117$	190.187	0.150280
$118$	0	0
$119$	−1855.98	−1.42973
$120$	0	0
$121$	601.484	0.451904
$122$	0	0
$123$	−1211.25	−0.887923
$124$	0	0
$125$	1477.04	1.05689
$126$	0	0
$127$	1116.94	0.780415	0.390207	−	0.920727i	$-0.372403\pi$
$127$	1116.94	0.780415	0.390207	+	0.920727i	$0.372403\pi$
$128$	0	0
$129$	−25.4621	−0.0173784
$130$	0	0
$131$	420.709	0.280592	0.140296	−	0.990110i	$-0.455195\pi$
$131$	420.709	0.280592	0.140296	+	0.990110i	$0.455195\pi$
$132$	0	0
$133$	2357.12	1.53675
$134$	0	0
$135$	−1184.85	−0.755377
$136$	0	0
$137$	−1631.76	−1.01759	−0.508797	−	0.860887i	$-0.669910\pi$
$137$	−1631.76	−1.01759	−0.508797	+	0.860887i	$0.669910\pi$
$138$	0	0
$139$	1718.61	1.04871	0.524355	−	0.851500i	$-0.324306\pi$
$139$	1718.61	1.04871	0.524355	+	0.851500i	$0.324306\pi$
$140$	0	0
$141$	774.860	0.462801
$142$	0	0
$143$	−718.521	−0.420180
$144$	0	0
$145$	−226.891	−0.129947
$146$	0	0
$147$	80.1931	0.0449947
$148$	0	0
$149$	1687.69	0.927926	0.463963	−	0.885855i	$-0.346427\pi$
$149$	1687.69	0.927926	0.463963	+	0.885855i	$0.346427\pi$
$150$	0	0
$151$	3229.87	1.74068	0.870340	−	0.492451i	$-0.163899\pi$
$151$	3229.87	1.74068	0.870340	+	0.492451i	$0.163899\pi$
$152$	0	0
$153$	1202.49	0.635394
$154$	0	0
$155$	1288.60	0.667761
$156$	0	0
$157$	3482.28	1.77017	0.885084	−	0.465430i	$-0.154100\pi$
$157$	3482.28	1.77017	0.885084	+	0.465430i	$0.154100\pi$
$158$	0	0
$159$	−1575.96	−0.786050
$160$	0	0
$161$	346.986	0.169853
$162$	0	0
$163$	−3032.67	−1.45728	−0.728640	−	0.684896i	$-0.759847\pi$
$163$	−3032.67	−1.45728	−0.728640	+	0.684896i	$0.759847\pi$
$164$	0	0
$165$	1348.13	0.636071
$166$	0	0
$167$	3633.07	1.68344	0.841722	−	0.539911i	$-0.181542\pi$
$167$	3633.07	1.68344	0.841722	+	0.539911i	$0.181542\pi$
$168$	0	0
$169$	−1929.84	−0.878400
$170$	0	0
$171$	−1527.17	−0.682957
$172$	0	0
$173$	413.122	0.181555	0.0907776	−	0.995871i	$-0.471065\pi$
$173$	413.122	0.181555	0.0907776	+	0.995871i	$0.471065\pi$
$174$	0	0
$175$	−1145.59	−0.494847
$176$	0	0
$177$	2009.63	0.853406
$178$	0	0
$179$	1116.67	0.466279	0.233140	−	0.972443i	$-0.425100\pi$
$179$	1116.67	0.466279	0.233140	+	0.972443i	$0.425100\pi$
$180$	0	0
$181$	4702.48	1.93112	0.965559	−	0.260184i	$-0.0837832\pi$
$181$	4702.48	1.93112	0.965559	+	0.260184i	$0.0837832\pi$
$182$	0	0
$183$	3121.91	1.26108
$184$	0	0
$185$	−2345.90	−0.932292
$186$	0	0
$187$	−4542.96	−1.77655
$188$	0	0
$189$	2719.80	1.04675
$190$	0	0
$191$	−130.901	−0.0495900	−0.0247950	−	0.999693i	$-0.507893\pi$
$191$	−130.901	−0.0495900	−0.0247950	+	0.999693i	$0.507893\pi$
$192$	0	0
$193$	−2208.52	−0.823694	−0.411847	−	0.911253i	$-0.635116\pi$
$193$	−2208.52	−0.823694	−0.411847	+	0.911253i	$0.635116\pi$
$194$	0	0
$195$	−501.251	−0.184079
$196$	0	0
$197$	−3577.67	−1.29390	−0.646950	−	0.762532i	$-0.723956\pi$
$197$	−3577.67	−1.29390	−0.646950	+	0.762532i	$0.723956\pi$
$198$	0	0
$199$	4776.19	1.70138	0.850692	−	0.525665i	$-0.176183\pi$
$199$	4776.19	1.70138	0.850692	+	0.525665i	$0.176183\pi$
$200$	0	0
$201$	3826.27	1.34271
$202$	0	0
$203$	520.823	0.180072
$204$	0	0
$205$	−2417.68	−0.823699
$206$	0	0
$207$	−224.811	−0.0754854
$208$	0	0
$209$	5769.61	1.90953
$210$	0	0
$211$	3369.91	1.09950	0.549749	−	0.835330i	$-0.314723\pi$
$211$	3369.91	1.09950	0.549749	+	0.835330i	$0.314723\pi$
$212$	0	0
$213$	−1282.41	−0.412531
$214$	0	0
$215$	−50.8230	−0.0161214
$216$	0	0
$217$	−2957.95	−0.925340
$218$	0	0
$219$	1314.59	0.405625
$220$	0	0
$221$	1689.13	0.514131
$222$	0	0
$223$	−2022.88	−0.607453	−0.303727	−	0.952759i	$-0.598231\pi$
$223$	−2022.88	−0.607453	−0.303727	+	0.952759i	$0.598231\pi$
$224$	0	0
$225$	742.224	0.219918
$226$	0	0
$227$	5457.88	1.59583	0.797913	−	0.602773i	$-0.205937\pi$
$227$	5457.88	1.59583	0.797913	+	0.602773i	$0.205937\pi$
$228$	0	0
$229$	3052.35	0.880807	0.440404	−	0.897800i	$-0.354835\pi$
$229$	3052.35	0.880807	0.440404	+	0.897800i	$0.354835\pi$
$230$	0	0
$231$	−3094.60	−0.881427
$232$	0	0
$233$	3292.17	0.925654	0.462827	−	0.886449i	$-0.346835\pi$
$233$	3292.17	0.925654	0.462827	+	0.886449i	$0.346835\pi$
$234$	0	0
$235$	1546.64	0.429326
$236$	0	0
$237$	−1687.97	−0.462639
$238$	0	0
$239$	−2251.88	−0.609464	−0.304732	−	0.952438i	$-0.598567\pi$
$239$	−2251.88	−0.609464	−0.304732	+	0.952438i	$0.598567\pi$
$240$	0	0
$241$	2660.33	0.711067	0.355533	−	0.934664i	$-0.384299\pi$
$241$	2660.33	0.711067	0.355533	+	0.934664i	$0.384299\pi$
$242$	0	0
$243$	−2993.60	−0.790287
$244$	0	0
$245$	160.067	0.0417401
$246$	0	0
$247$	−2145.21	−0.552618
$248$	0	0
$249$	−2609.22	−0.664067
$250$	0	0
$251$	5118.30	1.28711	0.643555	−	0.765400i	$-0.277459\pi$
$251$	5118.30	1.28711	0.643555	+	0.765400i	$0.277459\pi$
$252$	0	0
$253$	849.331	0.211055
$254$	0	0
$255$	−3169.24	−0.778295
$256$	0	0
$257$	1672.75	0.406006	0.203003	−	0.979178i	$-0.434930\pi$
$257$	1672.75	0.406006	0.203003	+	0.979178i	$0.434930\pi$
$258$	0	0
$259$	5384.95	1.29191
$260$	0	0
$261$	−337.440	−0.0800270
$262$	0	0
$263$	797.134	0.186895	0.0934475	−	0.995624i	$-0.470211\pi$
$263$	797.134	0.186895	0.0934475	+	0.995624i	$0.470211\pi$
$264$	0	0
$265$	−3145.66	−0.729194
$266$	0	0
$267$	2827.70	0.648136
$268$	0	0
$269$	4574.34	1.03681	0.518406	−	0.855135i	$-0.326526\pi$
$269$	4574.34	1.03681	0.518406	+	0.855135i	$0.326526\pi$
$270$	0	0
$271$	−1367.52	−0.306535	−0.153268	−	0.988185i	$-0.548980\pi$
$271$	−1367.52	−0.306535	−0.153268	+	0.988185i	$0.548980\pi$
$272$	0	0
$273$	1150.61	0.255085
$274$	0	0
$275$	−2804.10	−0.614886
$276$	0	0
$277$	−728.979	−0.158123	−0.0790615	−	0.996870i	$-0.525192\pi$
$277$	−728.979	−0.158123	−0.0790615	+	0.996870i	$0.525192\pi$
$278$	0	0
$279$	1916.45	0.411236
$280$	0	0
$281$	−3272.55	−0.694746	−0.347373	−	0.937727i	$-0.612926\pi$
$281$	−3272.55	−0.694746	−0.347373	+	0.937727i	$0.612926\pi$
$282$	0	0
$283$	2232.69	0.468973	0.234487	−	0.972119i	$-0.424659\pi$
$283$	2232.69	0.468973	0.234487	+	0.972119i	$0.424659\pi$
$284$	0	0
$285$	4024.97	0.836556
$286$	0	0
$287$	5549.73	1.14143
$288$	0	0
$289$	5766.76	1.17378
$290$	0	0
$291$	−5896.92	−1.18792
$292$	0	0
$293$	5177.52	1.03233	0.516167	−	0.856488i	$-0.327358\pi$
$293$	5177.52	1.03233	0.516167	+	0.856488i	$0.327358\pi$
$294$	0	0
$295$	4011.27	0.791678
$296$	0	0
$297$	6657.37	1.30067
$298$	0	0
$299$	−315.792	−0.0610793
$300$	0	0
$301$	116.663	0.0223400
$302$	0	0
$303$	3886.73	0.736921
$304$	0	0
$305$	6231.42	1.16987
$306$	0	0
$307$	8919.66	1.65821	0.829107	−	0.559090i	$-0.188849\pi$
$307$	8919.66	1.65821	0.829107	+	0.559090i	$0.188849\pi$
$308$	0	0
$309$	−2338.51	−0.430528
$310$	0	0
$311$	−9867.55	−1.79916	−0.899578	−	0.436761i	$-0.856126\pi$
$311$	−9867.55	−1.79916	−0.899578	+	0.436761i	$0.856126\pi$
$312$	0	0
$313$	3016.90	0.544809	0.272404	−	0.962183i	$-0.412181\pi$
$313$	3016.90	0.544809	0.272404	+	0.962183i	$0.412181\pi$
$314$	0	0
$315$	1634.98	0.292446
$316$	0	0
$317$	−7285.42	−1.29082	−0.645410	−	0.763837i	$-0.723313\pi$
$317$	−7285.42	−1.29082	−0.645410	+	0.763837i	$0.723313\pi$
$318$	0	0
$319$	1274.84	0.223754
$320$	0	0
$321$	4138.66	0.719619
$322$	0	0
$323$	−13563.4	−2.33650
$324$	0	0
$325$	1042.60	0.177948
$326$	0	0
$327$	−1222.66	−0.206768
$328$	0	0
$329$	−3550.27	−0.594933
$330$	0	0
$331$	−1092.68	−0.181448	−0.0907239	−	0.995876i	$-0.528918\pi$
$331$	−1092.68	−0.181448	−0.0907239	+	0.995876i	$0.528918\pi$
$332$	0	0
$333$	−3488.90	−0.574146
$334$	0	0
$335$	7637.33	1.24559
$336$	0	0
$337$	6323.23	1.02210	0.511051	−	0.859550i	$-0.329256\pi$
$337$	6323.23	1.02210	0.511051	+	0.859550i	$0.329256\pi$
$338$	0	0
$339$	6372.10	1.02090
$340$	0	0
$341$	−7240.29	−1.14981
$342$	0	0
$343$	−6527.51	−1.02756
$344$	0	0
$345$	592.506	0.0924622
$346$	0	0
$347$	1576.44	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$347$	1576.44	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$348$	0	0
$349$	−4452.86	−0.682969	−0.341484	−	0.939887i	$-0.610930\pi$
$349$	−4452.86	−0.682969	−0.341484	+	0.939887i	$0.610930\pi$
$350$	0	0
$351$	−2475.29	−0.376414
$352$	0	0
$353$	−6455.03	−0.973277	−0.486638	−	0.873603i	$-0.661777\pi$
$353$	−6455.03	−0.973277	−0.486638	+	0.873603i	$0.661777\pi$
$354$	0	0
$355$	−2559.72	−0.382692
$356$	0	0
$357$	7274.90	1.07851
$358$	0	0
$359$	1824.52	0.268230	0.134115	−	0.990966i	$-0.457181\pi$
$359$	1824.52	0.268230	0.134115	+	0.990966i	$0.457181\pi$
$360$	0	0
$361$	10366.7	1.51140
$362$	0	0
$363$	−2357.64	−0.340893
$364$	0	0
$365$	2623.96	0.376286
$366$	0	0
$367$	7840.96	1.11524	0.557622	−	0.830095i	$-0.311714\pi$
$367$	7840.96	1.11524	0.557622	+	0.830095i	$0.311714\pi$
$368$	0	0
$369$	−3595.66	−0.507270
$370$	0	0
$371$	7220.79	1.01047
$372$	0	0
$373$	3736.89	0.518737	0.259368	−	0.965778i	$-0.416486\pi$
$373$	3736.89	0.518737	0.259368	+	0.965778i	$0.416486\pi$
$374$	0	0
$375$	−5789.58	−0.797261
$376$	0	0
$377$	−474.002	−0.0647542
$378$	0	0
$379$	−5590.74	−0.757723	−0.378861	−	0.925453i	$-0.623684\pi$
$379$	−5590.74	−0.757723	−0.378861	+	0.925453i	$0.623684\pi$
$380$	0	0
$381$	−4378.09	−0.588705
$382$	0	0
$383$	7768.46	1.03642	0.518211	−	0.855253i	$-0.326598\pi$
$383$	7768.46	1.03642	0.518211	+	0.855253i	$0.326598\pi$
$384$	0	0
$385$	−6176.90	−0.817672
$386$	0	0
$387$	−75.5857	−0.00992826
$388$	0	0
$389$	768.019	0.100103	0.0500516	−	0.998747i	$-0.484061\pi$
$389$	768.019	0.100103	0.0500516	+	0.998747i	$0.484061\pi$
$390$	0	0
$391$	−1996.64	−0.258247
$392$	0	0
$393$	−1649.06	−0.211664
$394$	0	0
$395$	−3369.23	−0.429176
$396$	0	0
$397$	−1885.77	−0.238398	−0.119199	−	0.992870i	$-0.538033\pi$
$397$	−1885.77	−0.238398	−0.119199	+	0.992870i	$0.538033\pi$
$398$	0	0
$399$	−9239.21	−1.15925
$400$	0	0
$401$	6644.32	0.827436	0.413718	−	0.910405i	$-0.364230\pi$
$401$	6644.32	0.827436	0.413718	+	0.910405i	$0.364230\pi$
$402$	0	0
$403$	2692.03	0.332753
$404$	0	0
$405$	2186.28	0.268239
$406$	0	0
$407$	13181.0	1.60530
$408$	0	0
$409$	10830.3	1.30935	0.654673	−	0.755912i	$-0.272806\pi$
$409$	10830.3	1.30935	0.654673	+	0.755912i	$0.272806\pi$
$410$	0	0
$411$	6396.01	0.767620
$412$	0	0
$413$	−9207.77	−1.09706
$414$	0	0
$415$	−5208.07	−0.616034
$416$	0	0
$417$	−6736.46	−0.791093
$418$	0	0
$419$	−6355.74	−0.741046	−0.370523	−	0.928823i	$-0.620822\pi$
$419$	−6355.74	−0.741046	−0.370523	+	0.928823i	$0.620822\pi$
$420$	0	0
$421$	4899.22	0.567158	0.283579	−	0.958949i	$-0.408478\pi$
$421$	4899.22	0.567158	0.283579	+	0.958949i	$0.408478\pi$
$422$	0	0
$423$	2300.22	0.264398
$424$	0	0
$425$	6591.99	0.752373
$426$	0	0
$427$	−14304.1	−1.62113
$428$	0	0
$429$	2816.39	0.316962
$430$	0	0
$431$	2076.82	0.232105	0.116052	−	0.993243i	$-0.462976\pi$
$431$	2076.82	0.232105	0.116052	+	0.993243i	$0.462976\pi$
$432$	0	0
$433$	13370.3	1.48392	0.741960	−	0.670444i	$-0.233896\pi$
$433$	13370.3	1.48392	0.741960	+	0.670444i	$0.233896\pi$
$434$	0	0
$435$	889.348	0.0980253
$436$	0	0
$437$	2535.76	0.277578
$438$	0	0
$439$	17679.2	1.92205	0.961027	−	0.276453i	$-0.0891590\pi$
$439$	17679.2	1.92205	0.961027	+	0.276453i	$0.0891590\pi$
$440$	0	0
$441$	238.058	0.0257054
$442$	0	0
$443$	2773.29	0.297433	0.148716	−	0.988880i	$-0.452486\pi$
$443$	2773.29	0.297433	0.148716	+	0.988880i	$0.452486\pi$
$444$	0	0
$445$	5644.16	0.601256
$446$	0	0
$447$	−6615.25	−0.699979
$448$	0	0
$449$	1827.52	0.192085	0.0960423	−	0.995377i	$-0.469382\pi$
$449$	1827.52	0.192085	0.0960423	+	0.995377i	$0.469382\pi$
$450$	0	0
$451$	13584.3	1.41831
$452$	0	0
$453$	−12660.1	−1.31308
$454$	0	0
$455$	2296.65	0.236634
$456$	0	0
$457$	509.464	0.0521481	0.0260741	−	0.999660i	$-0.491699\pi$
$457$	509.464	0.0521481	0.0260741	+	0.999660i	$0.491699\pi$
$458$	0	0
$459$	−15650.4	−1.59150
$460$	0	0
$461$	10720.1	1.08304	0.541522	−	0.840687i	$-0.317848\pi$
$461$	10720.1	1.08304	0.541522	+	0.840687i	$0.317848\pi$
$462$	0	0
$463$	−2487.38	−0.249673	−0.124836	−	0.992177i	$-0.539841\pi$
$463$	−2487.38	−0.249673	−0.124836	+	0.992177i	$0.539841\pi$
$464$	0	0
$465$	−5050.94	−0.503724
$466$	0	0
$467$	−15190.5	−1.50521	−0.752603	−	0.658475i	$-0.771202\pi$
$467$	−15190.5	−1.50521	−0.752603	+	0.658475i	$0.771202\pi$
$468$	0	0
$469$	−17531.3	−1.72606
$470$	0	0
$471$	−13649.5	−1.33532
$472$	0	0
$473$	285.561	0.0277592
$474$	0	0
$475$	−8371.90	−0.808693
$476$	0	0
$477$	−4678.33	−0.449070
$478$	0	0
$479$	−7231.04	−0.689759	−0.344880	−	0.938647i	$-0.612080\pi$
$479$	−7231.04	−0.689759	−0.344880	+	0.938647i	$0.612080\pi$
$480$	0	0
$481$	−4900.85	−0.464573
$482$	0	0
$483$	−1360.08	−0.128128
$484$	0	0
$485$	−11770.4	−1.10199
$486$	0	0
$487$	−15795.6	−1.46975	−0.734876	−	0.678202i	$-0.762760\pi$
$487$	−15795.6	−1.46975	−0.734876	+	0.678202i	$0.762760\pi$
$488$	0	0
$489$	11887.2	1.09930
$490$	0	0
$491$	−6556.62	−0.602640	−0.301320	−	0.953523i	$-0.597427\pi$
$491$	−6556.62	−0.602640	−0.301320	+	0.953523i	$0.597427\pi$
$492$	0	0
$493$	−2996.95	−0.273784
$494$	0	0
$495$	4002.00	0.363387
$496$	0	0
$497$	5875.77	0.530310
$498$	0	0
$499$	15972.1	1.43288	0.716442	−	0.697647i	$-0.245769\pi$
$499$	15972.1	1.43288	0.716442	+	0.697647i	$0.245769\pi$
$500$	0	0
$501$	−14240.6	−1.26990
$502$	0	0
$503$	−9876.92	−0.875527	−0.437764	−	0.899090i	$-0.644229\pi$
$503$	−9876.92	−0.875527	−0.437764	+	0.899090i	$0.644229\pi$
$504$	0	0
$505$	7758.02	0.683619
$506$	0	0
$507$	7564.43	0.662620
$508$	0	0
$509$	20333.0	1.77062	0.885309	−	0.465004i	$-0.153947\pi$
$509$	20333.0	1.77062	0.885309	+	0.465004i	$0.153947\pi$
$510$	0	0
$511$	−6023.23	−0.521433
$512$	0	0
$513$	19876.2	1.71063
$514$	0	0
$515$	−4667.73	−0.399388
$516$	0	0
$517$	−8690.15	−0.739250
$518$	0	0
$519$	−1619.32	−0.136956
$520$	0	0
$521$	−12391.4	−1.04199	−0.520996	−	0.853559i	$-0.674439\pi$
$521$	−12391.4	−1.04199	−0.520996	+	0.853559i	$0.674439\pi$
$522$	0	0
$523$	−22155.6	−1.85239	−0.926193	−	0.377049i	$-0.876939\pi$
$523$	−22155.6	−1.85239	−0.926193	+	0.377049i	$0.876939\pi$
$524$	0	0
$525$	4490.37	0.373287
$526$	0	0
$527$	17020.8	1.40690
$528$	0	0
$529$	−11793.7	−0.969320
$530$	0	0
$531$	5965.69	0.487550
$532$	0	0
$533$	−5050.81	−0.410459
$534$	0	0
$535$	8260.88	0.667568
$536$	0	0
$537$	−4377.03	−0.351737
$538$	0	0
$539$	−899.375	−0.0718717
$540$	0	0
$541$	1227.50	0.0975495	0.0487747	−	0.998810i	$-0.484468\pi$
$541$	1227.50	0.0975495	0.0487747	+	0.998810i	$0.484468\pi$
$542$	0	0
$543$	−18432.3	−1.45674
$544$	0	0
$545$	−2440.46	−0.191812
$546$	0	0
$547$	5744.16	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$547$	5744.16	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$548$	0	0
$549$	9267.57	0.720456
$550$	0	0
$551$	3806.15	0.294279
$552$	0	0
$553$	7733.98	0.594724
$554$	0	0
$555$	9195.25	0.703273
$556$	0	0
$557$	20336.5	1.54701	0.773506	−	0.633789i	$-0.218501\pi$
$557$	20336.5	1.54701	0.773506	+	0.633789i	$0.218501\pi$
$558$	0	0
$559$	−106.175	−0.00803349
$560$	0	0
$561$	17807.1	1.34013
$562$	0	0
$563$	9206.34	0.689167	0.344584	−	0.938756i	$-0.388020\pi$
$563$	9206.34	0.689167	0.344584	+	0.938756i	$0.388020\pi$
$564$	0	0
$565$	12718.9	0.947056
$566$	0	0
$567$	−5018.54	−0.371709
$568$	0	0
$569$	−11456.0	−0.844043	−0.422022	−	0.906586i	$-0.638679\pi$
$569$	−11456.0	−0.844043	−0.422022	+	0.906586i	$0.638679\pi$
$570$	0	0
$571$	−6973.91	−0.511119	−0.255560	−	0.966793i	$-0.582260\pi$
$571$	−6973.91	−0.511119	−0.255560	+	0.966793i	$0.582260\pi$
$572$	0	0
$573$	513.095	0.0374081
$574$	0	0
$575$	−1232.41	−0.0893826
$576$	0	0
$577$	24733.0	1.78449	0.892244	−	0.451553i	$-0.149130\pi$
$577$	24733.0	1.78449	0.892244	+	0.451553i	$0.149130\pi$
$578$	0	0
$579$	8656.76	0.621352
$580$	0	0
$581$	11955.0	0.853661
$582$	0	0
$583$	17674.6	1.25559
$584$	0	0
$585$	−1487.99	−0.105164
$586$	0	0
$587$	−975.516	−0.0685926	−0.0342963	−	0.999412i	$-0.510919\pi$
$587$	−975.516	−0.0685926	−0.0342963	+	0.999412i	$0.510919\pi$
$588$	0	0
$589$	−21616.6	−1.51222
$590$	0	0
$591$	14023.4	0.976052
$592$	0	0
$593$	−6055.96	−0.419374	−0.209687	−	0.977769i	$-0.567244\pi$
$593$	−6055.96	−0.419374	−0.209687	+	0.977769i	$0.567244\pi$
$594$	0	0
$595$	14520.9	1.00050
$596$	0	0
$597$	−18721.3	−1.28344
$598$	0	0
$599$	18960.0	1.29330	0.646648	−	0.762789i	$-0.276170\pi$
$599$	18960.0	1.29330	0.646648	+	0.762789i	$0.276170\pi$
$600$	0	0
$601$	7851.43	0.532889	0.266445	−	0.963850i	$-0.414151\pi$
$601$	7851.43	0.532889	0.266445	+	0.963850i	$0.414151\pi$
$602$	0	0
$603$	11358.5	0.767087
$604$	0	0
$605$	−4705.91	−0.316236
$606$	0	0
$607$	−16556.1	−1.10707	−0.553536	−	0.832825i	$-0.686722\pi$
$607$	−16556.1	−1.10707	−0.553536	+	0.832825i	$0.686722\pi$
$608$	0	0
$609$	−2041.48	−0.135837
$610$	0	0
$611$	3231.11	0.213939
$612$	0	0
$613$	4367.15	0.287745	0.143873	−	0.989596i	$-0.454044\pi$
$613$	4367.15	0.287745	0.143873	+	0.989596i	$0.454044\pi$
$614$	0	0
$615$	9476.61	0.621356
$616$	0	0
$617$	−15641.2	−1.02057	−0.510286	−	0.860005i	$-0.670460\pi$
$617$	−15641.2	−1.02057	−0.510286	+	0.860005i	$0.670460\pi$
$618$	0	0
$619$	−4387.43	−0.284888	−0.142444	−	0.989803i	$-0.545496\pi$
$619$	−4387.43	−0.284888	−0.142444	+	0.989803i	$0.545496\pi$
$620$	0	0
$621$	2925.93	0.189072
$622$	0	0
$623$	−12956.0	−0.833181
$624$	0	0
$625$	−3582.71	−0.229294
$626$	0	0
$627$	−22615.2	−1.44045
$628$	0	0
$629$	−30986.3	−1.96424
$630$	0	0
$631$	11562.9	0.729496	0.364748	−	0.931106i	$-0.381155\pi$
$631$	11562.9	0.729496	0.364748	+	0.931106i	$0.381155\pi$
$632$	0	0
$633$	−13209.1	−0.829405
$634$	0	0
$635$	−8738.79	−0.546123
$636$	0	0
$637$	334.399	0.0207996
$638$	0	0
$639$	−3806.90	−0.235679
$640$	0	0
$641$	−4969.64	−0.306223	−0.153111	−	0.988209i	$-0.548929\pi$
$641$	−4969.64	−0.306223	−0.153111	+	0.988209i	$0.548929\pi$
$642$	0	0
$643$	19774.3	1.21279	0.606394	−	0.795164i	$-0.292615\pi$
$643$	19774.3	1.21279	0.606394	+	0.795164i	$0.292615\pi$
$644$	0	0
$645$	199.211	0.0121612
$646$	0	0
$647$	−20098.1	−1.22123	−0.610616	−	0.791927i	$-0.709078\pi$
$647$	−20098.1	−1.22123	−0.610616	+	0.791927i	$0.709078\pi$
$648$	0	0
$649$	−22538.2	−1.36318
$650$	0	0
$651$	11594.3	0.698029
$652$	0	0
$653$	16447.7	0.985678	0.492839	−	0.870121i	$-0.335959\pi$
$653$	16447.7	0.985678	0.492839	+	0.870121i	$0.335959\pi$
$654$	0	0
$655$	−3291.56	−0.196354
$656$	0	0
$657$	3902.44	0.231733
$658$	0	0
$659$	30950.8	1.82955	0.914773	−	0.403967i	$-0.132369\pi$
$659$	30950.8	1.82955	0.914773	+	0.403967i	$0.132369\pi$
$660$	0	0
$661$	−15743.1	−0.926380	−0.463190	−	0.886259i	$-0.653295\pi$
$661$	−15743.1	−0.926380	−0.463190	+	0.886259i	$0.653295\pi$
$662$	0	0
$663$	−6620.89	−0.387834
$664$	0	0
$665$	−18441.7	−1.07540
$666$	0	0
$667$	560.296	0.0325258
$668$	0	0
$669$	7929.10	0.458232
$670$	0	0
$671$	−35012.6	−2.01438
$672$	0	0
$673$	14464.1	0.828454	0.414227	−	0.910174i	$-0.364052\pi$
$673$	14464.1	0.828454	0.414227	+	0.910174i	$0.364052\pi$
$674$	0	0
$675$	−9660.07	−0.550839
$676$	0	0
$677$	−11382.6	−0.646188	−0.323094	−	0.946367i	$-0.604723\pi$
$677$	−11382.6	−0.646188	−0.323094	+	0.946367i	$0.604723\pi$
$678$	0	0
$679$	27018.6	1.52707
$680$	0	0
$681$	−21393.3	−1.20381
$682$	0	0
$683$	−5724.70	−0.320717	−0.160358	−	0.987059i	$-0.551265\pi$
$683$	−5724.70	−0.320717	−0.160358	+	0.987059i	$0.551265\pi$
$684$	0	0
$685$	12766.6	0.712097
$686$	0	0
$687$	−11964.3	−0.664435
$688$	0	0
$689$	−6571.64	−0.363367
$690$	0	0
$691$	−20221.2	−1.11324	−0.556622	−	0.830766i	$-0.687903\pi$
$691$	−20221.2	−1.11324	−0.556622	+	0.830766i	$0.687903\pi$
$692$	0	0
$693$	−9186.49	−0.503558
$694$	0	0
$695$	−13446.1	−0.733873
$696$	0	0
$697$	−31934.5	−1.73545
$698$	0	0
$699$	−12904.4	−0.698266
$700$	0	0
$701$	24050.0	1.29580	0.647900	−	0.761725i	$-0.275647\pi$
$701$	24050.0	1.29580	0.647900	+	0.761725i	$0.275647\pi$
$702$	0	0
$703$	39353.0	2.11128
$704$	0	0
$705$	−6062.38	−0.323862
$706$	0	0
$707$	−17808.3	−0.947315
$708$	0	0
$709$	−8186.65	−0.433647	−0.216824	−	0.976211i	$-0.569570\pi$
$709$	−8186.65	−0.433647	−0.216824	+	0.976211i	$0.569570\pi$
$710$	0	0
$711$	−5010.83	−0.264305
$712$	0	0
$713$	−3182.13	−0.167141
$714$	0	0
$715$	5621.60	0.294036
$716$	0	0
$717$	8826.72	0.459749
$718$	0	0
$719$	5534.51	0.287069	0.143534	−	0.989645i	$-0.454153\pi$
$719$	5534.51	0.287069	0.143534	+	0.989645i	$0.454153\pi$
$720$	0	0
$721$	10714.6	0.553446
$722$	0	0
$723$	−10427.7	−0.536392
$724$	0	0
$725$	−1849.84	−0.0947604
$726$	0	0
$727$	−9167.05	−0.467658	−0.233829	−	0.972278i	$-0.575126\pi$
$727$	−9167.05	−0.467658	−0.233829	+	0.972278i	$0.575126\pi$
$728$	0	0
$729$	19278.9	0.979468
$730$	0	0
$731$	−671.307	−0.0339661
$732$	0	0
$733$	10378.4	0.522967	0.261484	−	0.965208i	$-0.415788\pi$
$733$	10378.4	0.522967	0.261484	+	0.965208i	$0.415788\pi$
$734$	0	0
$735$	−627.418	−0.0314866
$736$	0	0
$737$	−42912.1	−2.14476
$738$	0	0
$739$	29962.1	1.49144	0.745718	−	0.666261i	$-0.232106\pi$
$739$	29962.1	1.49144	0.745718	+	0.666261i	$0.232106\pi$
$740$	0	0
$741$	8408.61	0.416866
$742$	0	0
$743$	−2325.42	−0.114820	−0.0574100	−	0.998351i	$-0.518284\pi$
$743$	−2325.42	−0.114820	−0.0574100	+	0.998351i	$0.518284\pi$
$744$	0	0
$745$	−13204.2	−0.649349
$746$	0	0
$747$	−7745.62	−0.379381
$748$	0	0
$749$	−18962.6	−0.925073
$750$	0	0
$751$	−11777.6	−0.572263	−0.286132	−	0.958190i	$-0.592369\pi$
$751$	−11777.6	−0.572263	−0.286132	+	0.958190i	$0.592369\pi$
$752$	0	0
$753$	−20062.3	−0.970929
$754$	0	0
$755$	−25270.0	−1.21810
$756$	0	0
$757$	18967.8	0.910698	0.455349	−	0.890313i	$-0.349515\pi$
$757$	18967.8	0.910698	0.455349	+	0.890313i	$0.349515\pi$
$758$	0	0
$759$	−3329.13	−0.159209
$760$	0	0
$761$	403.225	0.0192075	0.00960374	−	0.999954i	$-0.496943\pi$
$761$	403.225	0.0192075	0.00960374	+	0.999954i	$0.496943\pi$
$762$	0	0
$763$	5602.01	0.265801
$764$	0	0
$765$	−9408.06	−0.444639
$766$	0	0
$767$	8379.99	0.394503
$768$	0	0
$769$	−16714.5	−0.783797	−0.391898	−	0.920008i	$-0.628182\pi$
$769$	−16714.5	−0.783797	−0.391898	+	0.920008i	$0.628182\pi$
$770$	0	0
$771$	−6556.71	−0.306270
$772$	0	0
$773$	−25335.4	−1.17885	−0.589424	−	0.807824i	$-0.700645\pi$
$773$	−25335.4	−1.17885	−0.589424	+	0.807824i	$0.700645\pi$
$774$	0	0
$775$	10505.9	0.486947
$776$	0	0
$777$	−21107.5	−0.974551
$778$	0	0
$779$	40557.2	1.86535
$780$	0	0
$781$	14382.4	0.658952
$782$	0	0
$783$	4391.80	0.200447
$784$	0	0
$785$	−27244.8	−1.23874
$786$	0	0
$787$	−21378.3	−0.968300	−0.484150	−	0.874985i	$-0.660871\pi$
$787$	−21378.3	−0.968300	−0.484150	+	0.874985i	$0.660871\pi$
$788$	0	0
$789$	−3124.53	−0.140984
$790$	0	0
$791$	−29195.8	−1.31237
$792$	0	0
$793$	13018.1	0.582960
$794$	0	0
$795$	12330.1	0.550067
$796$	0	0
$797$	6639.05	0.295065	0.147533	−	0.989057i	$-0.452867\pi$
$797$	6639.05	0.295065	0.147533	+	0.989057i	$0.452867\pi$
$798$	0	0
$799$	20429.1	0.904545
$800$	0	0
$801$	8394.18	0.370279
$802$	0	0
$803$	−14743.3	−0.647920
$804$	0	0
$805$	−2714.76	−0.118861
$806$	0	0
$807$	−17930.1	−0.782117
$808$	0	0
$809$	−37636.6	−1.63564	−0.817820	−	0.575474i	$-0.804817\pi$
$809$	−37636.6	−1.63564	−0.817820	+	0.575474i	$0.804817\pi$
$810$	0	0
$811$	−2388.15	−0.103402	−0.0517011	−	0.998663i	$-0.516464\pi$
$811$	−2388.15	−0.103402	−0.0517011	+	0.998663i	$0.516464\pi$
$812$	0	0
$813$	5360.29	0.231234
$814$	0	0
$815$	23727.1	1.01978
$816$	0	0
$817$	852.568	0.0365086
$818$	0	0
$819$	3415.65	0.145730
$820$	0	0
$821$	4200.67	0.178568	0.0892841	−	0.996006i	$-0.471542\pi$
$821$	4200.67	0.178568	0.0892841	+	0.996006i	$0.471542\pi$
$822$	0	0
$823$	−26576.6	−1.12564	−0.562821	−	0.826579i	$-0.690284\pi$
$823$	−26576.6	−1.12564	−0.562821	+	0.826579i	$0.690284\pi$
$824$	0	0
$825$	10991.3	0.463838
$826$	0	0
$827$	2869.25	0.120645	0.0603227	−	0.998179i	$-0.480787\pi$
$827$	2869.25	0.120645	0.0603227	+	0.998179i	$0.480787\pi$
$828$	0	0
$829$	25083.9	1.05090	0.525452	−	0.850823i	$-0.323896\pi$
$829$	25083.9	1.05090	0.525452	+	0.850823i	$0.323896\pi$
$830$	0	0
$831$	2857.39	0.119280
$832$	0	0
$833$	2114.29	0.0879420
$834$	0	0
$835$	−28424.5	−1.17805
$836$	0	0
$837$	−24942.7	−1.03004
$838$	0	0
$839$	−23659.9	−0.973577	−0.486789	−	0.873520i	$-0.661832\pi$
$839$	−23659.9	−0.973577	−0.486789	+	0.873520i	$0.661832\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	12827.4	0.524081
$844$	0	0
$845$	15098.8	0.614692
$846$	0	0
$847$	10802.3	0.438219
$848$	0	0
$849$	−8751.48	−0.353769
$850$	0	0
$851$	5793.07	0.233354
$852$	0	0
$853$	−12955.2	−0.520022	−0.260011	−	0.965606i	$-0.583726\pi$
$853$	−12955.2	−0.520022	−0.260011	+	0.965606i	$0.583726\pi$
$854$	0	0
$855$	11948.3	0.477923
$856$	0	0
$857$	−20262.8	−0.807658	−0.403829	−	0.914834i	$-0.632321\pi$
$857$	−20262.8	−0.807658	−0.403829	+	0.914834i	$0.632321\pi$
$858$	0	0
$859$	29328.0	1.16491	0.582455	−	0.812863i	$-0.302092\pi$
$859$	29328.0	1.16491	0.582455	+	0.812863i	$0.302092\pi$
$860$	0	0
$861$	−21753.3	−0.861035
$862$	0	0
$863$	7441.90	0.293540	0.146770	−	0.989171i	$-0.453112\pi$
$863$	7441.90	0.293540	0.146770	+	0.989171i	$0.453112\pi$
$864$	0	0
$865$	−3232.20	−0.127050
$866$	0	0
$867$	−22604.0	−0.885437
$868$	0	0
$869$	18930.8	0.738991
$870$	0	0
$871$	15955.2	0.620692
$872$	0	0
$873$	−17505.3	−0.678655
$874$	0	0
$875$	26526.9	1.02488
$876$	0	0
$877$	−27923.6	−1.07516	−0.537578	−	0.843214i	$-0.680661\pi$
$877$	−27923.6	−1.07516	−0.537578	+	0.843214i	$0.680661\pi$
$878$	0	0
$879$	−20294.4	−0.778740
$880$	0	0
$881$	4060.07	0.155264	0.0776318	−	0.996982i	$-0.475264\pi$
$881$	4060.07	0.155264	0.0776318	+	0.996982i	$0.475264\pi$
$882$	0	0
$883$	33800.0	1.28818	0.644088	−	0.764951i	$-0.277237\pi$
$883$	33800.0	1.28818	0.644088	+	0.764951i	$0.277237\pi$
$884$	0	0
$885$	−15723.0	−0.597201
$886$	0	0
$887$	42193.1	1.59719	0.798594	−	0.601870i	$-0.205577\pi$
$887$	42193.1	1.59719	0.798594	+	0.601870i	$0.205577\pi$
$888$	0	0
$889$	20059.7	0.756782
$890$	0	0
$891$	−12284.1	−0.461877
$892$	0	0
$893$	−25945.2	−0.972256
$894$	0	0
$895$	−8736.66	−0.326295
$896$	0	0
$897$	1237.81	0.0460751
$898$	0	0
$899$	−4776.35	−0.177197
$900$	0	0
$901$	−41550.2	−1.53633
$902$	0	0
$903$	−457.285	−0.0168521
$904$	0	0
$905$	−36791.4	−1.35137
$906$	0	0
$907$	27271.6	0.998389	0.499194	−	0.866490i	$-0.333629\pi$
$907$	27271.6	0.998389	0.499194	+	0.866490i	$0.333629\pi$
$908$	0	0
$909$	11538.0	0.421002
$910$	0	0
$911$	46628.7	1.69580	0.847901	−	0.530154i	$-0.177866\pi$
$911$	46628.7	1.69580	0.847901	+	0.530154i	$0.177866\pi$
$912$	0	0
$913$	29262.7	1.06074
$914$	0	0
$915$	−24425.3	−0.882488
$916$	0	0
$917$	7555.69	0.272095
$918$	0	0
$919$	−13563.8	−0.486863	−0.243432	−	0.969918i	$-0.578273\pi$
$919$	−13563.8	−0.486863	−0.243432	+	0.969918i	$0.578273\pi$
$920$	0	0
$921$	−34962.5	−1.25087
$922$	0	0
$923$	−5347.54	−0.190700
$924$	0	0
$925$	−19126.0	−0.679849
$926$	0	0
$927$	−6942.00	−0.245960
$928$	0	0
$929$	8134.85	0.287293	0.143647	−	0.989629i	$-0.454117\pi$
$929$	8134.85	0.287293	0.143647	+	0.989629i	$0.454117\pi$
$930$	0	0
$931$	−2685.17	−0.0945250
$932$	0	0
$933$	38677.9	1.35719
$934$	0	0
$935$	35543.4	1.24320
$936$	0	0
$937$	−36312.5	−1.26604	−0.633020	−	0.774136i	$-0.718185\pi$
$937$	−36312.5	−1.26604	−0.633020	+	0.774136i	$0.718185\pi$
$938$	0	0
$939$	−11825.4	−0.410976
$940$	0	0
$941$	−13095.9	−0.453682	−0.226841	−	0.973932i	$-0.572840\pi$
$941$	−13095.9	−0.453682	−0.226841	+	0.973932i	$0.572840\pi$
$942$	0	0
$943$	5970.33	0.206173
$944$	0	0
$945$	−21279.3	−0.732503
$946$	0	0
$947$	−6114.86	−0.209827	−0.104914	−	0.994481i	$-0.533457\pi$
$947$	−6114.86	−0.209827	−0.104914	+	0.994481i	$0.533457\pi$
$948$	0	0
$949$	5481.74	0.187508
$950$	0	0
$951$	28556.7	0.973728
$952$	0	0
$953$	−39834.5	−1.35400	−0.677002	−	0.735981i	$-0.736721\pi$
$953$	−39834.5	−1.35400	−0.677002	+	0.735981i	$0.736721\pi$
$954$	0	0
$955$	1024.15	0.0347024
$956$	0	0
$957$	−4997.00	−0.168788
$958$	0	0
$959$	−29305.4	−0.986779
$960$	0	0
$961$	−2664.32	−0.0894337
$962$	0	0
$963$	12285.9	0.411117
$964$	0	0
$965$	17279.1	0.576409
$966$	0	0
$967$	−34378.5	−1.14327	−0.571633	−	0.820510i	$-0.693690\pi$
$967$	−34378.5	−1.14327	−0.571633	+	0.820510i	$0.693690\pi$
$968$	0	0
$969$	53164.7	1.76253
$970$	0	0
$971$	−42229.8	−1.39569	−0.697847	−	0.716247i	$-0.745858\pi$
$971$	−42229.8	−1.39569	−0.697847	+	0.716247i	$0.745858\pi$
$972$	0	0
$973$	30865.3	1.01695
$974$	0	0
$975$	−4086.69	−0.134235
$976$	0	0
$977$	−20253.1	−0.663207	−0.331604	−	0.943419i	$-0.607590\pi$
$977$	−20253.1	−0.663207	−0.331604	+	0.943419i	$0.607590\pi$
$978$	0	0
$979$	−31713.0	−1.03529
$980$	0	0
$981$	−3629.53	−0.118126
$982$	0	0
$983$	−1439.49	−0.0467067	−0.0233533	−	0.999727i	$-0.507434\pi$
$983$	−1439.49	−0.0467067	−0.0233533	+	0.999727i	$0.507434\pi$
$984$	0	0
$985$	27991.1	0.905453
$986$	0	0
$987$	13916.0	0.448787
$988$	0	0
$989$	125.505	0.00403520
$990$	0	0
$991$	3058.11	0.0980264	0.0490132	−	0.998798i	$-0.484392\pi$
$991$	3058.11	0.0980264	0.0490132	+	0.998798i	$0.484392\pi$
$992$	0	0
$993$	4282.99	0.136875
$994$	0	0
$995$	−37368.2	−1.19060
$996$	0	0
$997$	−5863.96	−0.186272	−0.0931361	−	0.995653i	$-0.529689\pi$
$997$	−5863.96	−0.186272	−0.0931361	+	0.995653i	$0.529689\pi$
$998$	0	0
$999$	45408.2	1.43809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bi.1.4		10
4.3	odd	2	inner	1856.4.a.bi.1.7		10
8.3	odd	2		928.4.a.f.1.4	✓	10
8.5	even	2		928.4.a.f.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.f.1.4	✓	10	8.3	odd	2
928.4.a.f.1.7	yes	10	8.5	even	2
1856.4.a.bi.1.4		10	1.1	even	1	trivial
1856.4.a.bi.1.7		10	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)

\(f(q)\)	\(=\)	\(q-3.91971 q^{3} -7.82384 q^{5} +17.9594 q^{7} -11.6359 q^{9} +O(q^{10})\)	\(q-3.91971 q^{3} -7.82384 q^{5} +17.9594 q^{7} -11.6359 q^{9} +43.9600 q^{11} -16.3449 q^{13} +30.6672 q^{15} -103.343 q^{17} +131.247 q^{19} -70.3957 q^{21} +19.3205 q^{23} -63.7875 q^{25} +151.441 q^{27} +29.0000 q^{29} -164.702 q^{31} -172.311 q^{33} -140.512 q^{35} +299.840 q^{37} +64.0672 q^{39} +309.015 q^{41} +6.49592 q^{43} +91.0373 q^{45} -197.683 q^{47} -20.4589 q^{49} +405.074 q^{51} +402.061 q^{53} -343.936 q^{55} -514.449 q^{57} -512.698 q^{59} -796.465 q^{61} -208.974 q^{63} +127.880 q^{65} -976.161 q^{67} -75.7309 q^{69} +327.169 q^{71} -335.380 q^{73} +250.029 q^{75} +789.497 q^{77} +430.636 q^{79} -279.438 q^{81} +665.667 q^{83} +808.539 q^{85} -113.672 q^{87} -721.405 q^{89} -293.545 q^{91} +645.583 q^{93} -1026.85 q^{95} +1504.43 q^{97} -511.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) \) 10 * q + 4 * q^5 + 50 * q^9	\( 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) \) 10 * q + 4 * q^5 + 50 * q^9 + 220 * q^13 + 76 * q^17 + 104 * q^21 + 446 * q^25 + 290 * q^29 - 1120 * q^33 + 1708 * q^37 - 980 * q^41 - 348 * q^45 + 1146 * q^49 + 44 * q^53 - 40 * q^57 + 1492 * q^61 - 2016 * q^65 + 2328 * q^69 - 3100 * q^73 + 3016 * q^77 - 2174 * q^81 + 9440 * q^85 + 636 * q^89 + 2536 * q^93 + 620 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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