LMFDB - Embedded newform 1856.4.a.m.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:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.69042$ 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)

$f(q)$	$=$	$q+9.69042 q^{3} -15.0000 q^{5} +9.38083 q^{7} +66.9042 q^{9} +O(q^{10})$	$q+9.69042 q^{3} -15.0000 q^{5} +9.38083 q^{7} +66.9042 q^{9} -46.4521 q^{11} +51.9042 q^{13} -145.356 q^{15} +4.33418 q^{17} -123.808 q^{19} +90.9042 q^{21} -86.2850 q^{23} +100.000 q^{25} +386.688 q^{27} -29.0000 q^{29} -275.877 q^{31} -450.140 q^{33} -140.712 q^{35} +97.5233 q^{37} +502.973 q^{39} +374.521 q^{41} -436.162 q^{43} -1003.56 q^{45} +101.455 q^{47} -255.000 q^{49} +42.0000 q^{51} -85.5259 q^{53} +696.781 q^{55} -1199.75 q^{57} -731.808 q^{59} -526.329 q^{61} +627.617 q^{63} -778.562 q^{65} +466.565 q^{67} -836.137 q^{69} +908.904 q^{71} +250.093 q^{73} +969.042 q^{75} -435.759 q^{77} -897.356 q^{79} +1940.75 q^{81} -468.575 q^{83} -65.0127 q^{85} -281.022 q^{87} +314.137 q^{89} +486.904 q^{91} -2673.36 q^{93} +1857.12 q^{95} +555.386 q^{97} -3107.84 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9}+O(q^{10}) $ 2 * q + 10 * q^3 - 30 * q^5 + 40 * q^9	$ 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} - 46 q^{11} + 10 q^{13} - 150 q^{15} + 140 q^{17} - 60 q^{19} + 88 q^{21} - 60 q^{23} + 200 q^{25} + 370 q^{27} - 58 q^{29} + 58 q^{31} - 450 q^{33} + 120 q^{37} + 490 q^{39} + 280 q^{41} - 150 q^{43} - 600 q^{45} + 550 q^{47} - 510 q^{49} + 84 q^{51} - 490 q^{53} + 690 q^{55} - 1180 q^{57} - 1276 q^{59} - 396 q^{61} + 880 q^{63} - 150 q^{65} - 80 q^{67} - 828 q^{69} + 1724 q^{71} + 200 q^{73} + 1000 q^{75} - 440 q^{77} - 1654 q^{79} + 2662 q^{81} - 1500 q^{83} - 2100 q^{85} - 290 q^{87} - 216 q^{89} + 880 q^{91} - 2570 q^{93} + 900 q^{95} + 1880 q^{97} - 3120 q^{99}+O(q^{100}) $ 2 * q + 10 * q^3 - 30 * q^5 + 40 * q^9 - 46 * q^11 + 10 * q^13 - 150 * q^15 + 140 * q^17 - 60 * q^19 + 88 * q^21 - 60 * q^23 + 200 * q^25 + 370 * q^27 - 58 * q^29 + 58 * q^31 - 450 * q^33 + 120 * q^37 + 490 * q^39 + 280 * q^41 - 150 * q^43 - 600 * q^45 + 550 * q^47 - 510 * q^49 + 84 * q^51 - 490 * q^53 + 690 * q^55 - 1180 * q^57 - 1276 * q^59 - 396 * q^61 + 880 * q^63 - 150 * q^65 - 80 * q^67 - 828 * q^69 + 1724 * q^71 + 200 * q^73 + 1000 * q^75 - 440 * q^77 - 1654 * q^79 + 2662 * q^81 - 1500 * q^83 - 2100 * q^85 - 290 * q^87 - 216 * q^89 + 880 * q^91 - 2570 * q^93 + 900 * q^95 + 1880 * q^97 - 3120 * 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
$3$	9.69042	1.86492	0.932461	−	0.361271i	$-0.117657\pi$
$3$	9.69042	1.86492	0.932461	+	0.361271i	$0.117657\pi$
$4$	0	0
$5$	−15.0000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−15.0000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	9.38083	0.506517	0.253259	−	0.967399i	$-0.418498\pi$
$7$	9.38083	0.506517	0.253259	+	0.967399i	$0.418498\pi$
$8$	0	0
$9$	66.9042	2.47793
$10$	0	0
$11$	−46.4521	−1.27326	−0.636629	−	0.771171i	$-0.719672\pi$
$11$	−46.4521	−1.27326	−0.636629	+	0.771171i	$0.719672\pi$
$12$	0	0
$13$	51.9042	1.10736	0.553678	−	0.832731i	$-0.313224\pi$
$13$	51.9042	1.10736	0.553678	+	0.832731i	$0.313224\pi$
$14$	0	0
$15$	−145.356	−2.50205
$16$	0	0
$17$	4.33418	0.0618349	0.0309174	−	0.999522i	$-0.490157\pi$
$17$	4.33418	0.0618349	0.0309174	+	0.999522i	$0.490157\pi$
$18$	0	0
$19$	−123.808	−1.49493	−0.747463	−	0.664304i	$-0.768728\pi$
$19$	−123.808	−1.49493	−0.747463	+	0.664304i	$0.768728\pi$
$20$	0	0
$21$	90.9042	0.944615
$22$	0	0
$23$	−86.2850	−0.782246	−0.391123	−	0.920338i	$-0.627913\pi$
$23$	−86.2850	−0.782246	−0.391123	+	0.920338i	$0.627913\pi$
$24$	0	0
$25$	100.000	0.800000
$26$	0	0
$27$	386.688	2.75623
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−275.877	−1.59835	−0.799177	−	0.601096i	$-0.794731\pi$
$31$	−275.877	−1.59835	−0.799177	+	0.601096i	$0.794731\pi$
$32$	0	0
$33$	−450.140	−2.37452
$34$	0	0
$35$	−140.712	−0.679564
$36$	0	0
$37$	97.5233	0.433317	0.216659	−	0.976247i	$-0.430484\pi$
$37$	97.5233	0.433317	0.216659	+	0.976247i	$0.430484\pi$
$38$	0	0
$39$	502.973	2.06513
$40$	0	0
$41$	374.521	1.42659	0.713297	−	0.700862i	$-0.247201\pi$
$41$	374.521	1.42659	0.713297	+	0.700862i	$0.247201\pi$
$42$	0	0
$43$	−436.162	−1.54684	−0.773420	−	0.633894i	$-0.781455\pi$
$43$	−436.162	−1.54684	−0.773420	+	0.633894i	$0.781455\pi$
$44$	0	0
$45$	−1003.56	−3.32449
$46$	0	0
$47$	101.455	0.314865	0.157433	−	0.987530i	$-0.449678\pi$
$47$	101.455	0.314865	0.157433	+	0.987530i	$0.449678\pi$
$48$	0	0
$49$	−255.000	−0.743440
$50$	0	0
$51$	42.0000	0.115317
$52$	0	0
$53$	−85.5259	−0.221658	−0.110829	−	0.993839i	$-0.535351\pi$
$53$	−85.5259	−0.221658	−0.110829	+	0.993839i	$0.535351\pi$
$54$	0	0
$55$	696.781	1.70825
$56$	0	0
$57$	−1199.75	−2.78792
$58$	0	0
$59$	−731.808	−1.61480	−0.807401	−	0.590003i	$-0.799127\pi$
$59$	−731.808	−1.61480	−0.807401	+	0.590003i	$0.799127\pi$
$60$	0	0
$61$	−526.329	−1.10475	−0.552373	−	0.833597i	$-0.686278\pi$
$61$	−526.329	−1.10475	−0.552373	+	0.833597i	$0.686278\pi$
$62$	0	0
$63$	627.617	1.25512
$64$	0	0
$65$	−778.562	−1.48567
$66$	0	0
$67$	466.565	0.850746	0.425373	−	0.905018i	$-0.360143\pi$
$67$	466.565	0.850746	0.425373	+	0.905018i	$0.360143\pi$
$68$	0	0
$69$	−836.137	−1.45883
$70$	0	0
$71$	908.904	1.51925	0.759627	−	0.650359i	$-0.225381\pi$
$71$	908.904	1.51925	0.759627	+	0.650359i	$0.225381\pi$
$72$	0	0
$73$	250.093	0.400975	0.200488	−	0.979696i	$-0.435747\pi$
$73$	250.093	0.400975	0.200488	+	0.979696i	$0.435747\pi$
$74$	0	0
$75$	969.042	1.49194
$76$	0	0
$77$	−435.759	−0.644927
$78$	0	0
$79$	−897.356	−1.27798	−0.638990	−	0.769215i	$-0.720648\pi$
$79$	−897.356	−1.27798	−0.638990	+	0.769215i	$0.720648\pi$
$80$	0	0
$81$	1940.75	2.66221
$82$	0	0
$83$	−468.575	−0.619672	−0.309836	−	0.950790i	$-0.600274\pi$
$83$	−468.575	−0.619672	−0.309836	+	0.950790i	$0.600274\pi$
$84$	0	0
$85$	−65.0127	−0.0829602
$86$	0	0
$87$	−281.022	−0.346307
$88$	0	0
$89$	314.137	0.374140	0.187070	−	0.982347i	$-0.440101\pi$
$89$	314.137	0.374140	0.187070	+	0.982347i	$0.440101\pi$
$90$	0	0
$91$	486.904	0.560895
$92$	0	0
$93$	−2673.36	−2.98080
$94$	0	0
$95$	1857.12	2.00565
$96$	0	0
$97$	555.386	0.581349	0.290675	−	0.956822i	$-0.406120\pi$
$97$	555.386	0.581349	0.290675	+	0.956822i	$0.406120\pi$
$98$	0	0
$99$	−3107.84	−3.15504
$100$	0	0
$101$	197.808	0.194878	0.0974389	−	0.995242i	$-0.468935\pi$
$101$	197.808	0.194878	0.0974389	+	0.995242i	$0.468935\pi$
$102$	0	0
$103$	−1328.24	−1.27063	−0.635315	−	0.772253i	$-0.719130\pi$
$103$	−1328.24	−1.27063	−0.635315	+	0.772253i	$0.719130\pi$
$104$	0	0
$105$	−1363.56	−1.26733
$106$	0	0
$107$	1323.94	1.19617	0.598083	−	0.801434i	$-0.295929\pi$
$107$	1323.94	1.19617	0.598083	+	0.801434i	$0.295929\pi$
$108$	0	0
$109$	377.658	0.331863	0.165932	−	0.986137i	$-0.446937\pi$
$109$	377.658	0.331863	0.165932	+	0.986137i	$0.446937\pi$
$110$	0	0
$111$	945.042	0.808102
$112$	0	0
$113$	1170.56	0.974487	0.487243	−	0.873266i	$-0.338002\pi$
$113$	1170.56	0.974487	0.487243	+	0.873266i	$0.338002\pi$
$114$	0	0
$115$	1294.27	1.04949
$116$	0	0
$117$	3472.60	2.74395
$118$	0	0
$119$	40.6582	0.0313204
$120$	0	0
$121$	826.796	0.621184
$122$	0	0
$123$	3629.26	2.66048
$124$	0	0
$125$	375.000	0.268328
$126$	0	0
$127$	194.953	0.136215	0.0681075	−	0.997678i	$-0.478304\pi$
$127$	194.953	0.136215	0.0681075	+	0.997678i	$0.478304\pi$
$128$	0	0
$129$	−4226.59	−2.88473
$130$	0	0
$131$	−2275.97	−1.51796	−0.758980	−	0.651114i	$-0.774302\pi$
$131$	−2275.97	−1.51796	−0.758980	+	0.651114i	$0.774302\pi$
$132$	0	0
$133$	−1161.42	−0.757205
$134$	0	0
$135$	−5800.32	−3.69787
$136$	0	0
$137$	−1885.89	−1.17607	−0.588037	−	0.808834i	$-0.700099\pi$
$137$	−1885.89	−1.17607	−0.588037	+	0.808834i	$0.700099\pi$
$138$	0	0
$139$	−1974.17	−1.20465	−0.602326	−	0.798250i	$-0.705759\pi$
$139$	−1974.17	−1.20465	−0.602326	+	0.798250i	$0.705759\pi$
$140$	0	0
$141$	983.137	0.587199
$142$	0	0
$143$	−2411.06	−1.40995
$144$	0	0
$145$	435.000	0.249136
$146$	0	0
$147$	−2471.06	−1.38646
$148$	0	0
$149$	−1844.26	−1.01401	−0.507006	−	0.861942i	$-0.669248\pi$
$149$	−1844.26	−1.01401	−0.507006	+	0.861942i	$0.669248\pi$
$150$	0	0
$151$	−2608.33	−1.40571	−0.702857	−	0.711331i	$-0.748093\pi$
$151$	−2608.33	−1.40571	−0.702857	+	0.711331i	$0.748093\pi$
$152$	0	0
$153$	289.975	0.153223
$154$	0	0
$155$	4138.16	2.14442
$156$	0	0
$157$	−3344.31	−1.70003	−0.850016	−	0.526758i	$-0.823408\pi$
$157$	−3344.31	−1.70003	−0.850016	+	0.526758i	$0.823408\pi$
$158$	0	0
$159$	−828.781	−0.413375
$160$	0	0
$161$	−809.425	−0.396221
$162$	0	0
$163$	−3391.86	−1.62988	−0.814942	−	0.579542i	$-0.803231\pi$
$163$	−3391.86	−1.62988	−0.814942	+	0.579542i	$0.803231\pi$
$164$	0	0
$165$	6752.10	3.18576
$166$	0	0
$167$	1938.52	0.898245	0.449122	−	0.893470i	$-0.351737\pi$
$167$	1938.52	0.898245	0.449122	+	0.893470i	$0.351737\pi$
$168$	0	0
$169$	497.042	0.226236
$170$	0	0
$171$	−8283.29	−3.70432
$172$	0	0
$173$	−886.098	−0.389415	−0.194708	−	0.980861i	$-0.562376\pi$
$173$	−886.098	−0.389415	−0.194708	+	0.980861i	$0.562376\pi$
$174$	0	0
$175$	938.083	0.405214
$176$	0	0
$177$	−7091.53	−3.01148
$178$	0	0
$179$	−136.629	−0.0570511	−0.0285256	−	0.999593i	$-0.509081\pi$
$179$	−136.629	−0.0570511	−0.0285256	+	0.999593i	$0.509081\pi$
$180$	0	0
$181$	−1763.25	−0.724096	−0.362048	−	0.932160i	$-0.617922\pi$
$181$	−1763.25	−0.724096	−0.362048	+	0.932160i	$0.617922\pi$
$182$	0	0
$183$	−5100.35	−2.06027
$184$	0	0
$185$	−1462.85	−0.581356
$186$	0	0
$187$	−201.332	−0.0787317
$188$	0	0
$189$	3627.45	1.39608
$190$	0	0
$191$	−1488.58	−0.563924	−0.281962	−	0.959426i	$-0.590985\pi$
$191$	−1488.58	−0.563924	−0.281962	+	0.959426i	$0.590985\pi$
$192$	0	0
$193$	3166.90	1.18113	0.590567	−	0.806989i	$-0.298904\pi$
$193$	3166.90	1.18113	0.590567	+	0.806989i	$0.298904\pi$
$194$	0	0
$195$	−7544.59	−2.77066
$196$	0	0
$197$	409.813	0.148213	0.0741066	−	0.997250i	$-0.476389\pi$
$197$	409.813	0.148213	0.0741066	+	0.997250i	$0.476389\pi$
$198$	0	0
$199$	−1774.90	−0.632259	−0.316130	−	0.948716i	$-0.602383\pi$
$199$	−1774.90	−0.632259	−0.316130	+	0.948716i	$0.602383\pi$
$200$	0	0
$201$	4521.21	1.58657
$202$	0	0
$203$	−272.044	−0.0940579
$204$	0	0
$205$	−5617.81	−1.91398
$206$	0	0
$207$	−5772.82	−1.93835
$208$	0	0
$209$	5751.15	1.90342
$210$	0	0
$211$	1133.08	0.369690	0.184845	−	0.982768i	$-0.440822\pi$
$211$	1133.08	0.369690	0.184845	+	0.982768i	$0.440822\pi$
$212$	0	0
$213$	8807.66	2.83329
$214$	0	0
$215$	6542.43	2.07530
$216$	0	0
$217$	−2587.96	−0.809594
$218$	0	0
$219$	2423.51	0.747788
$220$	0	0
$221$	224.962	0.0684732
$222$	0	0
$223$	3587.81	1.07739	0.538694	−	0.842501i	$-0.318918\pi$
$223$	3587.81	1.07739	0.538694	+	0.842501i	$0.318918\pi$
$224$	0	0
$225$	6690.42	1.98235
$226$	0	0
$227$	2932.95	0.857563	0.428782	−	0.903408i	$-0.358943\pi$
$227$	2932.95	0.857563	0.428782	+	0.903408i	$0.358943\pi$
$228$	0	0
$229$	−3400.71	−0.981334	−0.490667	−	0.871347i	$-0.663247\pi$
$229$	−3400.71	−0.981334	−0.490667	+	0.871347i	$0.663247\pi$
$230$	0	0
$231$	−4222.69	−1.20274
$232$	0	0
$233$	2657.48	0.747198	0.373599	−	0.927590i	$-0.378124\pi$
$233$	2657.48	0.747198	0.373599	+	0.927590i	$0.378124\pi$
$234$	0	0
$235$	−1521.82	−0.422436
$236$	0	0
$237$	−8695.76	−2.38333
$238$	0	0
$239$	2887.21	0.781414	0.390707	−	0.920515i	$-0.372231\pi$
$239$	2887.21	0.781414	0.390707	+	0.920515i	$0.372231\pi$
$240$	0	0
$241$	−543.550	−0.145283	−0.0726413	−	0.997358i	$-0.523143\pi$
$241$	−543.550	−0.145283	−0.0726413	+	0.997358i	$0.523143\pi$
$242$	0	0
$243$	8366.14	2.20859
$244$	0	0
$245$	3825.00	0.997430
$246$	0	0
$247$	−6426.17	−1.65541
$248$	0	0
$249$	−4540.69	−1.15564
$250$	0	0
$251$	−625.414	−0.157274	−0.0786370	−	0.996903i	$-0.525057\pi$
$251$	−625.414	−0.157274	−0.0786370	+	0.996903i	$0.525057\pi$
$252$	0	0
$253$	4008.12	0.996001
$254$	0	0
$255$	−630.000	−0.154714
$256$	0	0
$257$	−1586.88	−0.385164	−0.192582	−	0.981281i	$-0.561686\pi$
$257$	−1586.88	−0.385164	−0.192582	+	0.981281i	$0.561686\pi$
$258$	0	0
$259$	914.850	0.219483
$260$	0	0
$261$	−1940.22	−0.460140
$262$	0	0
$263$	2138.34	0.501353	0.250676	−	0.968071i	$-0.419347\pi$
$263$	2138.34	0.501353	0.250676	+	0.968071i	$0.419347\pi$
$264$	0	0
$265$	1282.89	0.297386
$266$	0	0
$267$	3044.12	0.697743
$268$	0	0
$269$	8073.35	1.82989	0.914945	−	0.403578i	$-0.132234\pi$
$269$	8073.35	1.82989	0.914945	+	0.403578i	$0.132234\pi$
$270$	0	0
$271$	501.027	0.112307	0.0561536	−	0.998422i	$-0.482116\pi$
$271$	501.027	0.112307	0.0561536	+	0.998422i	$0.482116\pi$
$272$	0	0
$273$	4718.30	1.04602
$274$	0	0
$275$	−4645.21	−1.01861
$276$	0	0
$277$	−733.970	−0.159206	−0.0796028	−	0.996827i	$-0.525365\pi$
$277$	−733.970	−0.159206	−0.0796028	+	0.996827i	$0.525365\pi$
$278$	0	0
$279$	−18457.3	−3.96061
$280$	0	0
$281$	5301.69	1.12552	0.562762	−	0.826619i	$-0.309739\pi$
$281$	5301.69	1.12552	0.562762	+	0.826619i	$0.309739\pi$
$282$	0	0
$283$	−2764.13	−0.580603	−0.290301	−	0.956935i	$-0.593756\pi$
$283$	−2764.13	−0.580603	−0.290301	+	0.956935i	$0.593756\pi$
$284$	0	0
$285$	17996.3	3.74038
$286$	0	0
$287$	3513.32	0.722594
$288$	0	0
$289$	−4894.21	−0.996176
$290$	0	0
$291$	5381.92	1.08417
$292$	0	0
$293$	−6385.81	−1.27325	−0.636626	−	0.771172i	$-0.719671\pi$
$293$	−6385.81	−1.27325	−0.636626	+	0.771172i	$0.719671\pi$
$294$	0	0
$295$	10977.1	2.16648
$296$	0	0
$297$	−17962.5	−3.50939
$298$	0	0
$299$	−4478.55	−0.866225
$300$	0	0
$301$	−4091.56	−0.783501
$302$	0	0
$303$	1916.84	0.363432
$304$	0	0
$305$	7894.94	1.48217
$306$	0	0
$307$	−2061.35	−0.383217	−0.191608	−	0.981471i	$-0.561370\pi$
$307$	−2061.35	−0.383217	−0.191608	+	0.981471i	$0.561370\pi$
$308$	0	0
$309$	−12871.2	−2.36963
$310$	0	0
$311$	−253.074	−0.0461431	−0.0230716	−	0.999734i	$-0.507345\pi$
$311$	−253.074	−0.0461431	−0.0230716	+	0.999734i	$0.507345\pi$
$312$	0	0
$313$	−3176.42	−0.573616	−0.286808	−	0.957988i	$-0.592594\pi$
$313$	−3176.42	−0.573616	−0.286808	+	0.957988i	$0.592594\pi$
$314$	0	0
$315$	−9414.25	−1.68391
$316$	0	0
$317$	7399.52	1.31104	0.655518	−	0.755180i	$-0.272450\pi$
$317$	7399.52	1.31104	0.655518	+	0.755180i	$0.272450\pi$
$318$	0	0
$319$	1347.11	0.236438
$320$	0	0
$321$	12829.5	2.23075
$322$	0	0
$323$	−536.607	−0.0924385
$324$	0	0
$325$	5190.42	0.885885
$326$	0	0
$327$	3659.67	0.618899
$328$	0	0
$329$	951.729	0.159485
$330$	0	0
$331$	7797.09	1.29476	0.647382	−	0.762166i	$-0.275864\pi$
$331$	7797.09	1.29476	0.647382	+	0.762166i	$0.275864\pi$
$332$	0	0
$333$	6524.72	1.07373
$334$	0	0
$335$	−6998.47	−1.14140
$336$	0	0
$337$	−1840.50	−0.297503	−0.148751	−	0.988875i	$-0.547525\pi$
$337$	−1840.50	−0.297503	−0.148751	+	0.988875i	$0.547525\pi$
$338$	0	0
$339$	11343.2	1.81734
$340$	0	0
$341$	12815.1	2.03512
$342$	0	0
$343$	−5609.74	−0.883083
$344$	0	0
$345$	12542.1	1.95722
$346$	0	0
$347$	7110.10	1.09997	0.549986	−	0.835174i	$-0.314633\pi$
$347$	7110.10	1.09997	0.549986	+	0.835174i	$0.314633\pi$
$348$	0	0
$349$	7585.59	1.16346	0.581729	−	0.813382i	$-0.302376\pi$
$349$	7585.59	1.16346	0.581729	+	0.813382i	$0.302376\pi$
$350$	0	0
$351$	20070.7	3.05212
$352$	0	0
$353$	−11358.4	−1.71259	−0.856297	−	0.516484i	$-0.827241\pi$
$353$	−11358.4	−1.71259	−0.856297	+	0.516484i	$0.827241\pi$
$354$	0	0
$355$	−13633.6	−2.03829
$356$	0	0
$357$	393.995	0.0584101
$358$	0	0
$359$	−1010.18	−0.148511	−0.0742553	−	0.997239i	$-0.523658\pi$
$359$	−1010.18	−0.148511	−0.0742553	+	0.997239i	$0.523658\pi$
$360$	0	0
$361$	8469.50	1.23480
$362$	0	0
$363$	8011.99	1.15846
$364$	0	0
$365$	−3751.40	−0.537965
$366$	0	0
$367$	917.074	0.130438	0.0652192	−	0.997871i	$-0.479225\pi$
$367$	917.074	0.130438	0.0652192	+	0.997871i	$0.479225\pi$
$368$	0	0
$369$	25057.0	3.53500
$370$	0	0
$371$	−802.304	−0.112274
$372$	0	0
$373$	−4529.53	−0.628767	−0.314384	−	0.949296i	$-0.601798\pi$
$373$	−4529.53	−0.628767	−0.314384	+	0.949296i	$0.601798\pi$
$374$	0	0
$375$	3633.91	0.500411
$376$	0	0
$377$	−1505.22	−0.205631
$378$	0	0
$379$	2018.50	0.273571	0.136785	−	0.990601i	$-0.456323\pi$
$379$	2018.50	0.273571	0.136785	+	0.990601i	$0.456323\pi$
$380$	0	0
$381$	1889.18	0.254030
$382$	0	0
$383$	−777.540	−0.103735	−0.0518674	−	0.998654i	$-0.516517\pi$
$383$	−777.540	−0.103735	−0.0518674	+	0.998654i	$0.516517\pi$
$384$	0	0
$385$	6536.39	0.865260
$386$	0	0
$387$	−29181.1	−3.83296
$388$	0	0
$389$	−535.610	−0.0698110	−0.0349055	−	0.999391i	$-0.511113\pi$
$389$	−535.610	−0.0698110	−0.0349055	+	0.999391i	$0.511113\pi$
$390$	0	0
$391$	−373.975	−0.0483701
$392$	0	0
$393$	−22055.1	−2.83088
$394$	0	0
$395$	13460.3	1.71459
$396$	0	0
$397$	9301.99	1.17595	0.587977	−	0.808878i	$-0.299925\pi$
$397$	9301.99	1.17595	0.587977	+	0.808878i	$0.299925\pi$
$398$	0	0
$399$	−11254.7	−1.41213
$400$	0	0
$401$	−10293.4	−1.28187	−0.640935	−	0.767595i	$-0.721453\pi$
$401$	−10293.4	−1.28187	−0.640935	+	0.767595i	$0.721453\pi$
$402$	0	0
$403$	−14319.2	−1.76995
$404$	0	0
$405$	−29111.3	−3.57173
$406$	0	0
$407$	−4530.16	−0.551724
$408$	0	0
$409$	6548.69	0.791716	0.395858	−	0.918312i	$-0.370447\pi$
$409$	6548.69	0.791716	0.395858	+	0.918312i	$0.370447\pi$
$410$	0	0
$411$	−18275.0	−2.19329
$412$	0	0
$413$	−6864.97	−0.817925
$414$	0	0
$415$	7028.63	0.831378
$416$	0	0
$417$	−19130.5	−2.24658
$418$	0	0
$419$	−2920.80	−0.340550	−0.170275	−	0.985397i	$-0.554466\pi$
$419$	−2920.80	−0.340550	−0.170275	+	0.985397i	$0.554466\pi$
$420$	0	0
$421$	1151.49	0.133302	0.0666508	−	0.997776i	$-0.478769\pi$
$421$	1151.49	0.133302	0.0666508	+	0.997776i	$0.478769\pi$
$422$	0	0
$423$	6787.74	0.780215
$424$	0	0
$425$	433.418	0.0494679
$426$	0	0
$427$	−4937.40	−0.559573
$428$	0	0
$429$	−23364.1	−2.62944
$430$	0	0
$431$	−6172.91	−0.689881	−0.344940	−	0.938625i	$-0.612101\pi$
$431$	−6172.91	−0.689881	−0.344940	+	0.938625i	$0.612101\pi$
$432$	0	0
$433$	9273.18	1.02919	0.514597	−	0.857432i	$-0.327942\pi$
$433$	9273.18	1.02919	0.514597	+	0.857432i	$0.327942\pi$
$434$	0	0
$435$	4215.33	0.464620
$436$	0	0
$437$	10682.8	1.16940
$438$	0	0
$439$	−5494.49	−0.597353	−0.298676	−	0.954354i	$-0.596545\pi$
$439$	−5494.49	−0.597353	−0.298676	+	0.954354i	$0.596545\pi$
$440$	0	0
$441$	−17060.6	−1.84219
$442$	0	0
$443$	16755.0	1.79696	0.898480	−	0.439014i	$-0.144672\pi$
$443$	16755.0	1.79696	0.898480	+	0.439014i	$0.144672\pi$
$444$	0	0
$445$	−4712.06	−0.501962
$446$	0	0
$447$	−17871.7	−1.89105
$448$	0	0
$449$	12805.5	1.34594	0.672970	−	0.739670i	$-0.265018\pi$
$449$	12805.5	1.34594	0.672970	+	0.739670i	$0.265018\pi$
$450$	0	0
$451$	−17397.3	−1.81642
$452$	0	0
$453$	−25275.8	−2.62155
$454$	0	0
$455$	−7303.56	−0.752519
$456$	0	0
$457$	1421.09	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$457$	1421.09	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$458$	0	0
$459$	1675.97	0.170431
$460$	0	0
$461$	−4998.00	−0.504946	−0.252473	−	0.967604i	$-0.581244\pi$
$461$	−4998.00	−0.504946	−0.252473	+	0.967604i	$0.581244\pi$
$462$	0	0
$463$	5460.18	0.548069	0.274035	−	0.961720i	$-0.411642\pi$
$463$	5460.18	0.548069	0.274035	+	0.961720i	$0.411642\pi$
$464$	0	0
$465$	40100.4	3.99917
$466$	0	0
$467$	−4249.90	−0.421118	−0.210559	−	0.977581i	$-0.567528\pi$
$467$	−4249.90	−0.421118	−0.210559	+	0.977581i	$0.567528\pi$
$468$	0	0
$469$	4376.77	0.430918
$470$	0	0
$471$	−32407.7	−3.17042
$472$	0	0
$473$	20260.6	1.96952
$474$	0	0
$475$	−12380.8	−1.19594
$476$	0	0
$477$	−5722.04	−0.549254
$478$	0	0
$479$	−8962.37	−0.854908	−0.427454	−	0.904037i	$-0.640589\pi$
$479$	−8962.37	−0.854908	−0.427454	+	0.904037i	$0.640589\pi$
$480$	0	0
$481$	5061.87	0.479836
$482$	0	0
$483$	−7843.66	−0.738921
$484$	0	0
$485$	−8330.79	−0.779962
$486$	0	0
$487$	17592.3	1.63692	0.818462	−	0.574561i	$-0.194827\pi$
$487$	17592.3	1.63692	0.818462	+	0.574561i	$0.194827\pi$
$488$	0	0
$489$	−32868.6	−3.03961
$490$	0	0
$491$	12314.0	1.13182	0.565908	−	0.824469i	$-0.308526\pi$
$491$	12314.0	1.13182	0.565908	+	0.824469i	$0.308526\pi$
$492$	0	0
$493$	−125.691	−0.0114824
$494$	0	0
$495$	46617.6	4.23294
$496$	0	0
$497$	8526.28	0.769529
$498$	0	0
$499$	11696.0	1.04927	0.524636	−	0.851327i	$-0.324201\pi$
$499$	11696.0	1.04927	0.524636	+	0.851327i	$0.324201\pi$
$500$	0	0
$501$	18785.0	1.67516
$502$	0	0
$503$	12239.0	1.08491	0.542456	−	0.840084i	$-0.317495\pi$
$503$	12239.0	1.08491	0.542456	+	0.840084i	$0.317495\pi$
$504$	0	0
$505$	−2967.12	−0.261456
$506$	0	0
$507$	4816.54	0.421913
$508$	0	0
$509$	15875.4	1.38244	0.691222	−	0.722642i	$-0.257073\pi$
$509$	15875.4	1.38244	0.691222	+	0.722642i	$0.257073\pi$
$510$	0	0
$511$	2346.08	0.203101
$512$	0	0
$513$	−47875.2	−4.12035
$514$	0	0
$515$	19923.5	1.70473
$516$	0	0
$517$	−4712.78	−0.400905
$518$	0	0
$519$	−8586.66	−0.726229
$520$	0	0
$521$	5240.51	0.440674	0.220337	−	0.975424i	$-0.429284\pi$
$521$	5240.51	0.440674	0.220337	+	0.975424i	$0.429284\pi$
$522$	0	0
$523$	5333.16	0.445895	0.222947	−	0.974830i	$-0.428432\pi$
$523$	5333.16	0.445895	0.222947	+	0.974830i	$0.428432\pi$
$524$	0	0
$525$	9090.42	0.755692
$526$	0	0
$527$	−1195.70	−0.0988340
$528$	0	0
$529$	−4721.90	−0.388091
$530$	0	0
$531$	−48961.0	−4.00137
$532$	0	0
$533$	19439.2	1.57975
$534$	0	0
$535$	−19859.0	−1.60482
$536$	0	0
$537$	−1323.99	−0.106396
$538$	0	0
$539$	11845.3	0.946591
$540$	0	0
$541$	−5954.19	−0.473181	−0.236590	−	0.971610i	$-0.576030\pi$
$541$	−5954.19	−0.473181	−0.236590	+	0.971610i	$0.576030\pi$
$542$	0	0
$543$	−17086.6	−1.35038
$544$	0	0
$545$	−5664.87	−0.445241
$546$	0	0
$547$	−20027.1	−1.56544	−0.782721	−	0.622373i	$-0.786169\pi$
$547$	−20027.1	−1.56544	−0.782721	+	0.622373i	$0.786169\pi$
$548$	0	0
$549$	−35213.6	−2.73749
$550$	0	0
$551$	3590.44	0.277601
$552$	0	0
$553$	−8417.95	−0.647319
$554$	0	0
$555$	−14175.6	−1.08418
$556$	0	0
$557$	12043.0	0.916121	0.458061	−	0.888921i	$-0.348544\pi$
$557$	12043.0	0.916121	0.458061	+	0.888921i	$0.348544\pi$
$558$	0	0
$559$	−22638.6	−1.71290
$560$	0	0
$561$	−1950.99	−0.146828
$562$	0	0
$563$	−3490.32	−0.261278	−0.130639	−	0.991430i	$-0.541703\pi$
$563$	−3490.32	−0.261278	−0.130639	+	0.991430i	$0.541703\pi$
$564$	0	0
$565$	−17558.4	−1.30741
$566$	0	0
$567$	18205.9	1.34846
$568$	0	0
$569$	−10053.0	−0.740672	−0.370336	−	0.928898i	$-0.620757\pi$
$569$	−10053.0	−0.740672	−0.370336	+	0.928898i	$0.620757\pi$
$570$	0	0
$571$	−12115.1	−0.887916	−0.443958	−	0.896048i	$-0.646426\pi$
$571$	−12115.1	−0.887916	−0.443958	+	0.896048i	$0.646426\pi$
$572$	0	0
$573$	−14424.9	−1.05167
$574$	0	0
$575$	−8628.50	−0.625797
$576$	0	0
$577$	−5076.79	−0.366290	−0.183145	−	0.983086i	$-0.558628\pi$
$577$	−5076.79	−0.366290	−0.183145	+	0.983086i	$0.558628\pi$
$578$	0	0
$579$	30688.6	2.20272
$580$	0	0
$581$	−4395.62	−0.313875
$582$	0	0
$583$	3972.85	0.282228
$584$	0	0
$585$	−52089.1	−3.68140
$586$	0	0
$587$	−5463.28	−0.384146	−0.192073	−	0.981381i	$-0.561521\pi$
$587$	−5463.28	−0.384146	−0.192073	+	0.981381i	$0.561521\pi$
$588$	0	0
$589$	34155.9	2.38942
$590$	0	0
$591$	3971.26	0.276406
$592$	0	0
$593$	−17533.7	−1.21420	−0.607100	−	0.794625i	$-0.707667\pi$
$593$	−17533.7	−1.21420	−0.607100	+	0.794625i	$0.707667\pi$
$594$	0	0
$595$	−609.873	−0.0420208
$596$	0	0
$597$	−17199.6	−1.17911
$598$	0	0
$599$	19594.9	1.33661	0.668303	−	0.743889i	$-0.267021\pi$
$599$	19594.9	1.33661	0.668303	+	0.743889i	$0.267021\pi$
$600$	0	0
$601$	18610.1	1.26310	0.631550	−	0.775335i	$-0.282419\pi$
$601$	18610.1	1.26310	0.631550	+	0.775335i	$0.282419\pi$
$602$	0	0
$603$	31215.1	2.10809
$604$	0	0
$605$	−12401.9	−0.833406
$606$	0	0
$607$	19450.3	1.30060	0.650301	−	0.759677i	$-0.274643\pi$
$607$	19450.3	1.30060	0.650301	+	0.759677i	$0.274643\pi$
$608$	0	0
$609$	−2636.22	−0.175411
$610$	0	0
$611$	5265.92	0.348668
$612$	0	0
$613$	−13569.7	−0.894085	−0.447042	−	0.894513i	$-0.647523\pi$
$613$	−13569.7	−0.894085	−0.447042	+	0.894513i	$0.647523\pi$
$614$	0	0
$615$	−54438.9	−3.56941
$616$	0	0
$617$	−25585.1	−1.66940	−0.834698	−	0.550708i	$-0.814358\pi$
$617$	−25585.1	−1.66940	−0.834698	+	0.550708i	$0.814358\pi$
$618$	0	0
$619$	25071.6	1.62797	0.813983	−	0.580888i	$-0.197295\pi$
$619$	25071.6	1.62797	0.813983	+	0.580888i	$0.197295\pi$
$620$	0	0
$621$	−33365.4	−2.15605
$622$	0	0
$623$	2946.87	0.189509
$624$	0	0
$625$	−18125.0	−1.16000
$626$	0	0
$627$	55731.1	3.54974
$628$	0	0
$629$	422.684	0.0267941
$630$	0	0
$631$	−9298.18	−0.586616	−0.293308	−	0.956018i	$-0.594756\pi$
$631$	−9298.18	−0.586616	−0.293308	+	0.956018i	$0.594756\pi$
$632$	0	0
$633$	10980.0	0.689442
$634$	0	0
$635$	−2924.30	−0.182752
$636$	0	0
$637$	−13235.6	−0.823253
$638$	0	0
$639$	60809.5	3.76461
$640$	0	0
$641$	−4145.34	−0.255431	−0.127715	−	0.991811i	$-0.540764\pi$
$641$	−4145.34	−0.255431	−0.127715	+	0.991811i	$0.540764\pi$
$642$	0	0
$643$	−1461.53	−0.0896376	−0.0448188	−	0.998995i	$-0.514271\pi$
$643$	−1461.53	−0.0896376	−0.0448188	+	0.998995i	$0.514271\pi$
$644$	0	0
$645$	63398.9	3.87028
$646$	0	0
$647$	−11331.7	−0.688555	−0.344277	−	0.938868i	$-0.611876\pi$
$647$	−11331.7	−0.688555	−0.344277	+	0.938868i	$0.611876\pi$
$648$	0	0
$649$	33994.0	2.05606
$650$	0	0
$651$	−25078.4	−1.50983
$652$	0	0
$653$	−3510.14	−0.210356	−0.105178	−	0.994453i	$-0.533541\pi$
$653$	−3510.14	−0.210356	−0.105178	+	0.994453i	$0.533541\pi$
$654$	0	0
$655$	34139.6	2.03656
$656$	0	0
$657$	16732.3	0.993590
$658$	0	0
$659$	17364.9	1.02647	0.513234	−	0.858249i	$-0.328447\pi$
$659$	17364.9	1.02647	0.513234	+	0.858249i	$0.328447\pi$
$660$	0	0
$661$	2526.28	0.148655	0.0743275	−	0.997234i	$-0.476319\pi$
$661$	2526.28	0.148655	0.0743275	+	0.997234i	$0.476319\pi$
$662$	0	0
$663$	2179.97	0.127697
$664$	0	0
$665$	17421.4	1.01590
$666$	0	0
$667$	2502.26	0.145259
$668$	0	0
$669$	34767.4	2.00924
$670$	0	0
$671$	24449.1	1.40663
$672$	0	0
$673$	−29160.2	−1.67020	−0.835099	−	0.550099i	$-0.814590\pi$
$673$	−29160.2	−1.67020	−0.835099	+	0.550099i	$0.814590\pi$
$674$	0	0
$675$	38668.8	2.20498
$676$	0	0
$677$	11657.4	0.661787	0.330893	−	0.943668i	$-0.392650\pi$
$677$	11657.4	0.661787	0.330893	+	0.943668i	$0.392650\pi$
$678$	0	0
$679$	5209.98	0.294464
$680$	0	0
$681$	28421.5	1.59929
$682$	0	0
$683$	−1990.76	−0.111529	−0.0557644	−	0.998444i	$-0.517760\pi$
$683$	−1990.76	−0.111529	−0.0557644	+	0.998444i	$0.517760\pi$
$684$	0	0
$685$	28288.3	1.57787
$686$	0	0
$687$	−32954.3	−1.83011
$688$	0	0
$689$	−4439.15	−0.245454
$690$	0	0
$691$	−1300.38	−0.0715903	−0.0357952	−	0.999359i	$-0.511396\pi$
$691$	−1300.38	−0.0715903	−0.0357952	+	0.999359i	$0.511396\pi$
$692$	0	0
$693$	−29154.1	−1.59808
$694$	0	0
$695$	29612.5	1.61621
$696$	0	0
$697$	1623.24	0.0882132
$698$	0	0
$699$	25752.1	1.39347
$700$	0	0
$701$	−7431.72	−0.400417	−0.200208	−	0.979753i	$-0.564162\pi$
$701$	−7431.72	−0.400417	−0.200208	+	0.979753i	$0.564162\pi$
$702$	0	0
$703$	−12074.2	−0.647777
$704$	0	0
$705$	−14747.1	−0.787811
$706$	0	0
$707$	1855.61	0.0987090
$708$	0	0
$709$	14169.4	0.750555	0.375277	−	0.926913i	$-0.377547\pi$
$709$	14169.4	0.750555	0.375277	+	0.926913i	$0.377547\pi$
$710$	0	0
$711$	−60036.9	−3.16675
$712$	0	0
$713$	23804.0	1.25031
$714$	0	0
$715$	36165.8	1.89164
$716$	0	0
$717$	27978.2	1.45728
$718$	0	0
$719$	−18065.4	−0.937030	−0.468515	−	0.883456i	$-0.655211\pi$
$719$	−18065.4	−0.937030	−0.468515	+	0.883456i	$0.655211\pi$
$720$	0	0
$721$	−12460.0	−0.643596
$722$	0	0
$723$	−5267.22	−0.270941
$724$	0	0
$725$	−2900.00	−0.148556
$726$	0	0
$727$	−13699.8	−0.698895	−0.349447	−	0.936956i	$-0.613631\pi$
$727$	−13699.8	−0.698895	−0.349447	+	0.936956i	$0.613631\pi$
$728$	0	0
$729$	28671.0	1.45664
$730$	0	0
$731$	−1890.40	−0.0956486
$732$	0	0
$733$	32434.9	1.63439	0.817197	−	0.576359i	$-0.195527\pi$
$733$	32434.9	1.63439	0.817197	+	0.576359i	$0.195527\pi$
$734$	0	0
$735$	37065.8	1.86013
$736$	0	0
$737$	−21672.9	−1.08322
$738$	0	0
$739$	−13377.2	−0.665885	−0.332942	−	0.942947i	$-0.608041\pi$
$739$	−13377.2	−0.665885	−0.332942	+	0.942947i	$0.608041\pi$
$740$	0	0
$741$	−62272.2	−3.08722
$742$	0	0
$743$	4826.96	0.238337	0.119168	−	0.992874i	$-0.461977\pi$
$743$	4826.96	0.238337	0.119168	+	0.992874i	$0.461977\pi$
$744$	0	0
$745$	27663.9	1.36044
$746$	0	0
$747$	−31349.6	−1.53551
$748$	0	0
$749$	12419.6	0.605878
$750$	0	0
$751$	19252.6	0.935468	0.467734	−	0.883869i	$-0.345071\pi$
$751$	19252.6	0.935468	0.467734	+	0.883869i	$0.345071\pi$
$752$	0	0
$753$	−6060.52	−0.293304
$754$	0	0
$755$	39124.9	1.88596
$756$	0	0
$757$	17362.6	0.833624	0.416812	−	0.908993i	$-0.363147\pi$
$757$	17362.6	0.833624	0.416812	+	0.908993i	$0.363147\pi$
$758$	0	0
$759$	38840.3	1.85746
$760$	0	0
$761$	820.256	0.0390726	0.0195363	−	0.999809i	$-0.493781\pi$
$761$	820.256	0.0390726	0.0195363	+	0.999809i	$0.493781\pi$
$762$	0	0
$763$	3542.75	0.168094
$764$	0	0
$765$	−4349.62	−0.205570
$766$	0	0
$767$	−37983.9	−1.78816
$768$	0	0
$769$	39151.3	1.83593	0.917967	−	0.396658i	$-0.129830\pi$
$769$	39151.3	1.83593	0.917967	+	0.396658i	$0.129830\pi$
$770$	0	0
$771$	−15377.6	−0.718300
$772$	0	0
$773$	1212.21	0.0564040	0.0282020	−	0.999602i	$-0.491022\pi$
$773$	1212.21	0.0564040	0.0282020	+	0.999602i	$0.491022\pi$
$774$	0	0
$775$	−27587.7	−1.27868
$776$	0	0
$777$	8865.28	0.409318
$778$	0	0
$779$	−46368.8	−2.13265
$780$	0	0
$781$	−42220.5	−1.93440
$782$	0	0
$783$	−11213.9	−0.511818
$784$	0	0
$785$	50164.6	2.28083
$786$	0	0
$787$	−13735.8	−0.622144	−0.311072	−	0.950386i	$-0.600688\pi$
$787$	−13735.8	−0.622144	−0.311072	+	0.950386i	$0.600688\pi$
$788$	0	0
$789$	20721.4	0.934984
$790$	0	0
$791$	10980.8	0.493594
$792$	0	0
$793$	−27318.7	−1.22335
$794$	0	0
$795$	12431.7	0.554601
$796$	0	0
$797$	33087.2	1.47053	0.735263	−	0.677782i	$-0.237059\pi$
$797$	33087.2	1.47053	0.735263	+	0.677782i	$0.237059\pi$
$798$	0	0
$799$	439.723	0.0194697
$800$	0	0
$801$	21017.1	0.927094
$802$	0	0
$803$	−11617.4	−0.510545
$804$	0	0
$805$	12141.4	0.531587
$806$	0	0
$807$	78234.1	3.41260
$808$	0	0
$809$	−13428.2	−0.583572	−0.291786	−	0.956484i	$-0.594249\pi$
$809$	−13428.2	−0.583572	−0.291786	+	0.956484i	$0.594249\pi$
$810$	0	0
$811$	3078.22	0.133281	0.0666406	−	0.997777i	$-0.478772\pi$
$811$	3078.22	0.133281	0.0666406	+	0.997777i	$0.478772\pi$
$812$	0	0
$813$	4855.16	0.209444
$814$	0	0
$815$	50877.9	2.18672
$816$	0	0
$817$	54000.5	2.31241
$818$	0	0
$819$	32575.9	1.38986
$820$	0	0
$821$	−41468.6	−1.76280	−0.881402	−	0.472367i	$-0.843400\pi$
$821$	−41468.6	−1.76280	−0.881402	+	0.472367i	$0.843400\pi$
$822$	0	0
$823$	−37436.9	−1.58562	−0.792811	−	0.609467i	$-0.791383\pi$
$823$	−37436.9	−1.58562	−0.792811	+	0.609467i	$0.791383\pi$
$824$	0	0
$825$	−45014.0	−1.89962
$826$	0	0
$827$	−17889.3	−0.752204	−0.376102	−	0.926578i	$-0.622736\pi$
$827$	−17889.3	−0.752204	−0.376102	+	0.926578i	$0.622736\pi$
$828$	0	0
$829$	−17587.3	−0.736829	−0.368415	−	0.929662i	$-0.620099\pi$
$829$	−17587.3	−0.736829	−0.368415	+	0.929662i	$0.620099\pi$
$830$	0	0
$831$	−7112.47	−0.296906
$832$	0	0
$833$	−1105.22	−0.0459705
$834$	0	0
$835$	−29077.7	−1.20512
$836$	0	0
$837$	−106678.	−4.40543
$838$	0	0
$839$	24163.4	0.994296	0.497148	−	0.867666i	$-0.334381\pi$
$839$	24163.4	0.994296	0.497148	+	0.867666i	$0.334381\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	51375.6	2.09901
$844$	0	0
$845$	−7455.62	−0.303528
$846$	0	0
$847$	7756.03	0.314640
$848$	0	0
$849$	−26785.6	−1.08278
$850$	0	0
$851$	−8414.80	−0.338961
$852$	0	0
$853$	29895.5	1.20000	0.600001	−	0.799999i	$-0.295167\pi$
$853$	29895.5	1.20000	0.600001	+	0.799999i	$0.295167\pi$
$854$	0	0
$855$	124249.	4.96987
$856$	0	0
$857$	−29755.8	−1.18604	−0.593022	−	0.805186i	$-0.702065\pi$
$857$	−29755.8	−1.18604	−0.593022	+	0.805186i	$0.702065\pi$
$858$	0	0
$859$	−26496.8	−1.05246	−0.526228	−	0.850344i	$-0.676394\pi$
$859$	−26496.8	−1.05246	−0.526228	+	0.850344i	$0.676394\pi$
$860$	0	0
$861$	34045.5	1.34758
$862$	0	0
$863$	−1177.38	−0.0464409	−0.0232204	−	0.999730i	$-0.507392\pi$
$863$	−1177.38	−0.0464409	−0.0232204	+	0.999730i	$0.507392\pi$
$864$	0	0
$865$	13291.5	0.522455
$866$	0	0
$867$	−47427.0	−1.85779
$868$	0	0
$869$	41684.1	1.62720
$870$	0	0
$871$	24216.7	0.942078
$872$	0	0
$873$	37157.6	1.44054
$874$	0	0
$875$	3517.81	0.135913
$876$	0	0
$877$	45838.3	1.76494	0.882468	−	0.470373i	$-0.155881\pi$
$877$	45838.3	1.76494	0.882468	+	0.470373i	$0.155881\pi$
$878$	0	0
$879$	−61881.2	−2.37452
$880$	0	0
$881$	−18368.1	−0.702427	−0.351214	−	0.936295i	$-0.614231\pi$
$881$	−18368.1	−0.702427	−0.351214	+	0.936295i	$0.614231\pi$
$882$	0	0
$883$	37241.6	1.41934	0.709671	−	0.704533i	$-0.248843\pi$
$883$	37241.6	1.41934	0.709671	+	0.704533i	$0.248843\pi$
$884$	0	0
$885$	106373.	4.04032
$886$	0	0
$887$	−13717.0	−0.519248	−0.259624	−	0.965710i	$-0.583599\pi$
$887$	−13717.0	−0.519248	−0.259624	+	0.965710i	$0.583599\pi$
$888$	0	0
$889$	1828.82	0.0689953
$890$	0	0
$891$	−90152.1	−3.38968
$892$	0	0
$893$	−12560.9	−0.470700
$894$	0	0
$895$	2049.44	0.0765421
$896$	0	0
$897$	−43399.0	−1.61544
$898$	0	0
$899$	8000.43	0.296807
$900$	0	0
$901$	−370.684	−0.0137062
$902$	0	0
$903$	−39648.9	−1.46117
$904$	0	0
$905$	26448.7	0.971476
$906$	0	0
$907$	24889.8	0.911195	0.455597	−	0.890186i	$-0.349426\pi$
$907$	24889.8	0.911195	0.455597	+	0.890186i	$0.349426\pi$
$908$	0	0
$909$	13234.2	0.482894
$910$	0	0
$911$	−32962.0	−1.19877	−0.599385	−	0.800461i	$-0.704588\pi$
$911$	−32962.0	−1.19877	−0.599385	+	0.800461i	$0.704588\pi$
$912$	0	0
$913$	21766.3	0.789002
$914$	0	0
$915$	76505.2	2.76414
$916$	0	0
$917$	−21350.5	−0.768873
$918$	0	0
$919$	43961.9	1.57799	0.788993	−	0.614403i	$-0.210603\pi$
$919$	43961.9	1.57799	0.788993	+	0.614403i	$0.210603\pi$
$920$	0	0
$921$	−19975.4	−0.714670
$922$	0	0
$923$	47175.9	1.68236
$924$	0	0
$925$	9752.33	0.346654
$926$	0	0
$927$	−88864.5	−3.14854
$928$	0	0
$929$	−22666.1	−0.800487	−0.400243	−	0.916409i	$-0.631074\pi$
$929$	−22666.1	−0.800487	−0.400243	+	0.916409i	$0.631074\pi$
$930$	0	0
$931$	31571.1	1.11139
$932$	0	0
$933$	−2452.39	−0.0860533
$934$	0	0
$935$	3019.97	0.105630
$936$	0	0
$937$	52377.0	1.82613	0.913064	−	0.407817i	$-0.133710\pi$
$937$	52377.0	1.82613	0.913064	+	0.407817i	$0.133710\pi$
$938$	0	0
$939$	−30780.8	−1.06975
$940$	0	0
$941$	10124.8	0.350754	0.175377	−	0.984501i	$-0.443885\pi$
$941$	10124.8	0.350754	0.175377	+	0.984501i	$0.443885\pi$
$942$	0	0
$943$	−32315.5	−1.11595
$944$	0	0
$945$	−54411.8	−1.87303
$946$	0	0
$947$	28852.3	0.990045	0.495022	−	0.868880i	$-0.335160\pi$
$947$	28852.3	0.990045	0.495022	+	0.868880i	$0.335160\pi$
$948$	0	0
$949$	12980.9	0.444022
$950$	0	0
$951$	71704.4	2.44498
$952$	0	0
$953$	−13047.0	−0.443478	−0.221739	−	0.975106i	$-0.571173\pi$
$953$	−13047.0	−0.443478	−0.221739	+	0.975106i	$0.571173\pi$
$954$	0	0
$955$	22328.6	0.756584
$956$	0	0
$957$	13054.1	0.440938
$958$	0	0
$959$	−17691.2	−0.595702
$960$	0	0
$961$	46317.1	1.55474
$962$	0	0
$963$	88576.8	2.96402
$964$	0	0
$965$	−47503.6	−1.58466
$966$	0	0
$967$	−43405.7	−1.44347	−0.721734	−	0.692171i	$-0.756654\pi$
$967$	−43405.7	−1.44347	−0.721734	+	0.692171i	$0.756654\pi$
$968$	0	0
$969$	−5199.95	−0.172391
$970$	0	0
$971$	12205.2	0.403382	0.201691	−	0.979449i	$-0.435356\pi$
$971$	12205.2	0.403382	0.201691	+	0.979449i	$0.435356\pi$
$972$	0	0
$973$	−18519.3	−0.610177
$974$	0	0
$975$	50297.3	1.65210
$976$	0	0
$977$	−4221.98	−0.138253	−0.0691265	−	0.997608i	$-0.522021\pi$
$977$	−4221.98	−0.138253	−0.0691265	+	0.997608i	$0.522021\pi$
$978$	0	0
$979$	−14592.3	−0.476377
$980$	0	0
$981$	25266.9	0.822335
$982$	0	0
$983$	29129.7	0.945162	0.472581	−	0.881287i	$-0.343322\pi$
$983$	29129.7	0.945162	0.472581	+	0.881287i	$0.343322\pi$
$984$	0	0
$985$	−6147.20	−0.198849
$986$	0	0
$987$	9222.65	0.297427
$988$	0	0
$989$	37634.2	1.21001
$990$	0	0
$991$	−37778.8	−1.21098	−0.605491	−	0.795852i	$-0.707023\pi$
$991$	−37778.8	−1.21098	−0.605491	+	0.795852i	$0.707023\pi$
$992$	0	0
$993$	75557.0	2.41463
$994$	0	0
$995$	26623.6	0.848265
$996$	0	0
$997$	−28534.9	−0.906428	−0.453214	−	0.891402i	$-0.649723\pi$
$997$	−28534.9	−0.906428	−0.453214	+	0.891402i	$0.649723\pi$
$998$	0	0
$999$	37711.1	1.19432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.m.1.2		2
4.3	odd	2		1856.4.a.g.1.1		2
8.3	odd	2		116.4.a.b.1.2	✓	2
8.5	even	2		464.4.a.c.1.1		2
24.11	even	2		1044.4.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.4.a.b.1.2	✓	2	8.3	odd	2
464.4.a.c.1.1		2	8.5	even	2
1044.4.a.b.1.1		2	24.11	even	2
1856.4.a.g.1.1		2	4.3	odd	2
1856.4.a.m.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+9.69042 q^{3} -15.0000 q^{5} +9.38083 q^{7} +66.9042 q^{9} +O(q^{10})\)	\(q+9.69042 q^{3} -15.0000 q^{5} +9.38083 q^{7} +66.9042 q^{9} -46.4521 q^{11} +51.9042 q^{13} -145.356 q^{15} +4.33418 q^{17} -123.808 q^{19} +90.9042 q^{21} -86.2850 q^{23} +100.000 q^{25} +386.688 q^{27} -29.0000 q^{29} -275.877 q^{31} -450.140 q^{33} -140.712 q^{35} +97.5233 q^{37} +502.973 q^{39} +374.521 q^{41} -436.162 q^{43} -1003.56 q^{45} +101.455 q^{47} -255.000 q^{49} +42.0000 q^{51} -85.5259 q^{53} +696.781 q^{55} -1199.75 q^{57} -731.808 q^{59} -526.329 q^{61} +627.617 q^{63} -778.562 q^{65} +466.565 q^{67} -836.137 q^{69} +908.904 q^{71} +250.093 q^{73} +969.042 q^{75} -435.759 q^{77} -897.356 q^{79} +1940.75 q^{81} -468.575 q^{83} -65.0127 q^{85} -281.022 q^{87} +314.137 q^{89} +486.904 q^{91} -2673.36 q^{93} +1857.12 q^{95} +555.386 q^{97} -3107.84 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9}+O(q^{10}) \) 2 * q + 10 * q^3 - 30 * q^5 + 40 * q^9	\( 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} - 46 q^{11} + 10 q^{13} - 150 q^{15} + 140 q^{17} - 60 q^{19} + 88 q^{21} - 60 q^{23} + 200 q^{25} + 370 q^{27} - 58 q^{29} + 58 q^{31} - 450 q^{33} + 120 q^{37} + 490 q^{39} + 280 q^{41} - 150 q^{43} - 600 q^{45} + 550 q^{47} - 510 q^{49} + 84 q^{51} - 490 q^{53} + 690 q^{55} - 1180 q^{57} - 1276 q^{59} - 396 q^{61} + 880 q^{63} - 150 q^{65} - 80 q^{67} - 828 q^{69} + 1724 q^{71} + 200 q^{73} + 1000 q^{75} - 440 q^{77} - 1654 q^{79} + 2662 q^{81} - 1500 q^{83} - 2100 q^{85} - 290 q^{87} - 216 q^{89} + 880 q^{91} - 2570 q^{93} + 900 q^{95} + 1880 q^{97} - 3120 q^{99}+O(q^{100}) \) 2 * q + 10 * q^3 - 30 * q^5 + 40 * q^9 - 46 * q^11 + 10 * q^13 - 150 * q^15 + 140 * q^17 - 60 * q^19 + 88 * q^21 - 60 * q^23 + 200 * q^25 + 370 * q^27 - 58 * q^29 + 58 * q^31 - 450 * q^33 + 120 * q^37 + 490 * q^39 + 280 * q^41 - 150 * q^43 - 600 * q^45 + 550 * q^47 - 510 * q^49 + 84 * q^51 - 490 * q^53 + 690 * q^55 - 1180 * q^57 - 1276 * q^59 - 396 * q^61 + 880 * q^63 - 150 * q^65 - 80 * q^67 - 828 * q^69 + 1724 * q^71 + 200 * q^73 + 1000 * q^75 - 440 * q^77 - 1654 * q^79 + 2662 * q^81 - 1500 * q^83 - 2100 * q^85 - 290 * q^87 - 216 * q^89 + 880 * q^91 - 2570 * q^93 + 900 * q^95 + 1880 * q^97 - 3120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more