LMFDB - Embedded newform 1840.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 $ x^6 - 2x^5 - 80x^4 + 105x^3 + 1328x^2 - 816*x - 4032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.08353$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.08353 q^{3} +5.00000 q^{5} +9.74988 q^{7} +23.1764 q^{9} +O(q^{10})$	$q-7.08353 q^{3} +5.00000 q^{5} +9.74988 q^{7} +23.1764 q^{9} +53.3636 q^{11} -20.8824 q^{13} -35.4177 q^{15} -36.6370 q^{17} -12.4419 q^{19} -69.0635 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0845 q^{27} -124.071 q^{29} -34.7845 q^{31} -378.002 q^{33} +48.7494 q^{35} -296.700 q^{37} +147.921 q^{39} -130.454 q^{41} -127.970 q^{43} +115.882 q^{45} +471.440 q^{47} -247.940 q^{49} +259.519 q^{51} +741.340 q^{53} +266.818 q^{55} +88.1325 q^{57} -694.335 q^{59} -882.394 q^{61} +225.967 q^{63} -104.412 q^{65} +688.620 q^{67} +162.921 q^{69} +514.508 q^{71} +148.812 q^{73} -177.088 q^{75} +520.288 q^{77} -456.735 q^{79} -817.617 q^{81} +496.304 q^{83} -183.185 q^{85} +878.861 q^{87} +59.4468 q^{89} -203.601 q^{91} +246.397 q^{93} -62.2095 q^{95} +1198.10 q^{97} +1236.78 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * 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
$3$	−7.08353	−1.36323	−0.681613	−	0.731713i	$-0.738721\pi$
$3$	−7.08353	−1.36323	−0.681613	+	0.731713i	$0.738721\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	9.74988	0.526444	0.263222	−	0.964735i	$-0.415215\pi$
$7$	9.74988	0.526444	0.263222	+	0.964735i	$0.415215\pi$
$8$	0	0
$9$	23.1764	0.858386
$10$	0	0
$11$	53.3636	1.46270	0.731351	−	0.682001i	$-0.238890\pi$
$11$	53.3636	1.46270	0.731351	+	0.682001i	$0.238890\pi$
$12$	0	0
$13$	−20.8824	−0.445518	−0.222759	−	0.974874i	$-0.571506\pi$
$13$	−20.8824	−0.445518	−0.222759	+	0.974874i	$0.571506\pi$
$14$	0	0
$15$	−35.4177	−0.609653
$16$	0	0
$17$	−36.6370	−0.522693	−0.261347	−	0.965245i	$-0.584167\pi$
$17$	−36.6370	−0.522693	−0.261347	+	0.965245i	$0.584167\pi$
$18$	0	0
$19$	−12.4419	−0.150230	−0.0751149	−	0.997175i	$-0.523932\pi$
$19$	−12.4419	−0.150230	−0.0751149	+	0.997175i	$0.523932\pi$
$20$	0	0
$21$	−69.0635	−0.717662
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0845	0.193052
$28$	0	0
$29$	−124.071	−0.794462	−0.397231	−	0.917719i	$-0.630029\pi$
$29$	−124.071	−0.794462	−0.397231	+	0.917719i	$0.630029\pi$
$30$	0	0
$31$	−34.7845	−0.201531	−0.100766	−	0.994910i	$-0.532129\pi$
$31$	−34.7845	−0.201531	−0.100766	+	0.994910i	$0.532129\pi$
$32$	0	0
$33$	−378.002	−1.99399
$34$	0	0
$35$	48.7494	0.235433
$36$	0	0
$37$	−296.700	−1.31830	−0.659150	−	0.752011i	$-0.729084\pi$
$37$	−296.700	−1.31830	−0.659150	+	0.752011i	$0.729084\pi$
$38$	0	0
$39$	147.921	0.607342
$40$	0	0
$41$	−130.454	−0.496913	−0.248456	−	0.968643i	$-0.579923\pi$
$41$	−130.454	−0.496913	−0.248456	+	0.968643i	$0.579923\pi$
$42$	0	0
$43$	−127.970	−0.453844	−0.226922	−	0.973913i	$-0.572866\pi$
$43$	−127.970	−0.453844	−0.226922	+	0.973913i	$0.572866\pi$
$44$	0	0
$45$	115.882	0.383882
$46$	0	0
$47$	471.440	1.46312	0.731559	−	0.681778i	$-0.238793\pi$
$47$	471.440	1.46312	0.731559	+	0.681778i	$0.238793\pi$
$48$	0	0
$49$	−247.940	−0.722857
$50$	0	0
$51$	259.519	0.712549
$52$	0	0
$53$	741.340	1.92134	0.960669	−	0.277696i	$-0.0895707\pi$
$53$	741.340	1.92134	0.960669	+	0.277696i	$0.0895707\pi$
$54$	0	0
$55$	266.818	0.654140
$56$	0	0
$57$	88.1325	0.204797
$58$	0	0
$59$	−694.335	−1.53211	−0.766057	−	0.642772i	$-0.777784\pi$
$59$	−694.335	−1.53211	−0.766057	+	0.642772i	$0.777784\pi$
$60$	0	0
$61$	−882.394	−1.85211	−0.926057	−	0.377383i	$-0.876824\pi$
$61$	−882.394	−1.85211	−0.926057	+	0.377383i	$0.876824\pi$
$62$	0	0
$63$	225.967	0.451892
$64$	0	0
$65$	−104.412	−0.199242
$66$	0	0
$67$	688.620	1.25565	0.627824	−	0.778356i	$-0.283946\pi$
$67$	688.620	1.25565	0.627824	+	0.778356i	$0.283946\pi$
$68$	0	0
$69$	162.921	0.284252
$70$	0	0
$71$	514.508	0.860012	0.430006	−	0.902826i	$-0.358511\pi$
$71$	514.508	0.860012	0.430006	+	0.902826i	$0.358511\pi$
$72$	0	0
$73$	148.812	0.238592	0.119296	−	0.992859i	$-0.461936\pi$
$73$	148.812	0.238592	0.119296	+	0.992859i	$0.461936\pi$
$74$	0	0
$75$	−177.088	−0.272645
$76$	0	0
$77$	520.288	0.770030
$78$	0	0
$79$	−456.735	−0.650465	−0.325232	−	0.945634i	$-0.605443\pi$
$79$	−456.735	−0.650465	−0.325232	+	0.945634i	$0.605443\pi$
$80$	0	0
$81$	−817.617	−1.12156
$82$	0	0
$83$	496.304	0.656343	0.328172	−	0.944618i	$-0.393568\pi$
$83$	496.304	0.656343	0.328172	+	0.944618i	$0.393568\pi$
$84$	0	0
$85$	−183.185	−0.233755
$86$	0	0
$87$	878.861	1.08303
$88$	0	0
$89$	59.4468	0.0708017	0.0354008	−	0.999373i	$-0.488729\pi$
$89$	59.4468	0.0708017	0.0354008	+	0.999373i	$0.488729\pi$
$90$	0	0
$91$	−203.601	−0.234540
$92$	0	0
$93$	246.397	0.274733
$94$	0	0
$95$	−62.2095	−0.0671848
$96$	0	0
$97$	1198.10	1.25411	0.627054	−	0.778976i	$-0.284261\pi$
$97$	1198.10	1.25411	0.627054	+	0.778976i	$0.284261\pi$
$98$	0	0
$99$	1236.78	1.25556
$100$	0	0
$101$	1131.78	1.11502	0.557508	−	0.830172i	$-0.311758\pi$
$101$	1131.78	1.11502	0.557508	+	0.830172i	$0.311758\pi$
$102$	0	0
$103$	447.760	0.428341	0.214170	−	0.976796i	$-0.431295\pi$
$103$	447.760	0.428341	0.214170	+	0.976796i	$0.431295\pi$
$104$	0	0
$105$	−345.318	−0.320948
$106$	0	0
$107$	−245.608	−0.221904	−0.110952	−	0.993826i	$-0.535390\pi$
$107$	−245.608	−0.221904	−0.110952	+	0.993826i	$0.535390\pi$
$108$	0	0
$109$	923.365	0.811398	0.405699	−	0.914007i	$-0.367028\pi$
$109$	923.365	0.811398	0.405699	+	0.914007i	$0.367028\pi$
$110$	0	0
$111$	2101.68	1.79714
$112$	0	0
$113$	−1682.09	−1.40034	−0.700168	−	0.713979i	$-0.746891\pi$
$113$	−1682.09	−1.40034	−0.700168	+	0.713979i	$0.746891\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−483.980	−0.382427
$118$	0	0
$119$	−357.206	−0.275169
$120$	0	0
$121$	1516.67	1.13950
$122$	0	0
$123$	924.072	0.677405
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1391.71	−0.972395	−0.486198	−	0.873849i	$-0.661616\pi$
$127$	−1391.71	−0.972395	−0.486198	+	0.873849i	$0.661616\pi$
$128$	0	0
$129$	906.482	0.618692
$130$	0	0
$131$	−863.103	−0.575646	−0.287823	−	0.957684i	$-0.592932\pi$
$131$	−863.103	−0.575646	−0.287823	+	0.957684i	$0.592932\pi$
$132$	0	0
$133$	−121.307	−0.0790875
$134$	0	0
$135$	135.422	0.0863355
$136$	0	0
$137$	−916.993	−0.571854	−0.285927	−	0.958251i	$-0.592301\pi$
$137$	−916.993	−0.571854	−0.285927	+	0.958251i	$0.592301\pi$
$138$	0	0
$139$	1907.38	1.16390	0.581949	−	0.813225i	$-0.302290\pi$
$139$	1907.38	1.16390	0.581949	+	0.813225i	$0.302290\pi$
$140$	0	0
$141$	−3339.46	−1.99456
$142$	0	0
$143$	−1114.36	−0.651661
$144$	0	0
$145$	−620.355	−0.355294
$146$	0	0
$147$	1756.29	0.985418
$148$	0	0
$149$	−1684.17	−0.925989	−0.462994	−	0.886361i	$-0.653225\pi$
$149$	−1684.17	−0.925989	−0.462994	+	0.886361i	$0.653225\pi$
$150$	0	0
$151$	−2559.58	−1.37944	−0.689720	−	0.724077i	$-0.742266\pi$
$151$	−2559.58	−1.37944	−0.689720	+	0.724077i	$0.742266\pi$
$152$	0	0
$153$	−849.115	−0.448672
$154$	0	0
$155$	−173.922	−0.0901276
$156$	0	0
$157$	1838.50	0.934573	0.467286	−	0.884106i	$-0.345232\pi$
$157$	1838.50	0.934573	0.467286	+	0.884106i	$0.345232\pi$
$158$	0	0
$159$	−5251.31	−2.61922
$160$	0	0
$161$	−224.247	−0.109771
$162$	0	0
$163$	−2158.12	−1.03704	−0.518519	−	0.855066i	$-0.673517\pi$
$163$	−2158.12	−1.03704	−0.518519	+	0.855066i	$0.673517\pi$
$164$	0	0
$165$	−1890.01	−0.891741
$166$	0	0
$167$	−1835.31	−0.850422	−0.425211	−	0.905094i	$-0.639800\pi$
$167$	−1835.31	−0.850422	−0.425211	+	0.905094i	$0.639800\pi$
$168$	0	0
$169$	−1760.92	−0.801513
$170$	0	0
$171$	−288.359	−0.128955
$172$	0	0
$173$	1741.58	0.765377	0.382688	−	0.923877i	$-0.374998\pi$
$173$	1741.58	0.765377	0.382688	+	0.923877i	$0.374998\pi$
$174$	0	0
$175$	243.747	0.105289
$176$	0	0
$177$	4918.35	2.08862
$178$	0	0
$179$	−137.457	−0.0573969	−0.0286984	−	0.999588i	$-0.509136\pi$
$179$	−137.457	−0.0573969	−0.0286984	+	0.999588i	$0.509136\pi$
$180$	0	0
$181$	−3711.82	−1.52430	−0.762148	−	0.647403i	$-0.775855\pi$
$181$	−3711.82	−1.52430	−0.762148	+	0.647403i	$0.775855\pi$
$182$	0	0
$183$	6250.47	2.52485
$184$	0	0
$185$	−1483.50	−0.589562
$186$	0	0
$187$	−1955.08	−0.764544
$188$	0	0
$189$	264.070	0.101631
$190$	0	0
$191$	2372.28	0.898703	0.449352	−	0.893355i	$-0.351655\pi$
$191$	2372.28	0.898703	0.449352	+	0.893355i	$0.351655\pi$
$192$	0	0
$193$	−2913.02	−1.08644	−0.543222	−	0.839589i	$-0.682796\pi$
$193$	−2913.02	−1.08644	−0.543222	+	0.839589i	$0.682796\pi$
$194$	0	0
$195$	739.606	0.271612
$196$	0	0
$197$	2349.11	0.849579	0.424790	−	0.905292i	$-0.360348\pi$
$197$	2349.11	0.849579	0.424790	+	0.905292i	$0.360348\pi$
$198$	0	0
$199$	−118.357	−0.0421613	−0.0210807	−	0.999778i	$-0.506711\pi$
$199$	−118.357	−0.0421613	−0.0210807	+	0.999778i	$0.506711\pi$
$200$	0	0
$201$	−4877.86	−1.71173
$202$	0	0
$203$	−1209.68	−0.418240
$204$	0	0
$205$	−652.268	−0.222226
$206$	0	0
$207$	−533.058	−0.178986
$208$	0	0
$209$	−663.944	−0.219741
$210$	0	0
$211$	3428.92	1.11875	0.559376	−	0.828914i	$-0.311041\pi$
$211$	3428.92	1.11875	0.559376	+	0.828914i	$0.311041\pi$
$212$	0	0
$213$	−3644.53	−1.17239
$214$	0	0
$215$	−639.852	−0.202965
$216$	0	0
$217$	−339.144	−0.106095
$218$	0	0
$219$	−1054.12	−0.325254
$220$	0	0
$221$	765.069	0.232869
$222$	0	0
$223$	−4835.66	−1.45211	−0.726053	−	0.687639i	$-0.758647\pi$
$223$	−4835.66	−1.45211	−0.726053	+	0.687639i	$0.758647\pi$
$224$	0	0
$225$	579.410	0.171677
$226$	0	0
$227$	2498.43	0.730515	0.365257	−	0.930907i	$-0.380981\pi$
$227$	2498.43	0.730515	0.365257	+	0.930907i	$0.380981\pi$
$228$	0	0
$229$	−6091.07	−1.75768	−0.878841	−	0.477115i	$-0.841683\pi$
$229$	−6091.07	−1.75768	−0.878841	+	0.477115i	$0.841683\pi$
$230$	0	0
$231$	−3685.48	−1.04973
$232$	0	0
$233$	−6561.92	−1.84500	−0.922501	−	0.385994i	$-0.873859\pi$
$233$	−6561.92	−1.84500	−0.922501	+	0.385994i	$0.873859\pi$
$234$	0	0
$235$	2357.20	0.654327
$236$	0	0
$237$	3235.30	0.886731
$238$	0	0
$239$	−4770.26	−1.29106	−0.645528	−	0.763737i	$-0.723363\pi$
$239$	−4770.26	−1.29106	−0.645528	+	0.763737i	$0.723363\pi$
$240$	0	0
$241$	1186.98	0.317263	0.158632	−	0.987338i	$-0.449292\pi$
$241$	1186.98	0.317263	0.158632	+	0.987338i	$0.449292\pi$
$242$	0	0
$243$	5060.33	1.33589
$244$	0	0
$245$	−1239.70	−0.323271
$246$	0	0
$247$	259.817	0.0669301
$248$	0	0
$249$	−3515.59	−0.894744
$250$	0	0
$251$	5011.26	1.26019	0.630095	−	0.776518i	$-0.283016\pi$
$251$	5011.26	1.26019	0.630095	+	0.776518i	$0.283016\pi$
$252$	0	0
$253$	−1227.36	−0.304994
$254$	0	0
$255$	1297.60	0.318662
$256$	0	0
$257$	−3009.66	−0.730496	−0.365248	−	0.930910i	$-0.619016\pi$
$257$	−3009.66	−0.730496	−0.365248	+	0.930910i	$0.619016\pi$
$258$	0	0
$259$	−2892.78	−0.694011
$260$	0	0
$261$	−2875.52	−0.681955
$262$	0	0
$263$	−1214.04	−0.284643	−0.142322	−	0.989820i	$-0.545457\pi$
$263$	−1214.04	−0.284643	−0.142322	+	0.989820i	$0.545457\pi$
$264$	0	0
$265$	3706.70	0.859249
$266$	0	0
$267$	−421.093	−0.0965187
$268$	0	0
$269$	386.914	0.0876972	0.0438486	−	0.999038i	$-0.486038\pi$
$269$	386.914	0.0876972	0.0438486	+	0.999038i	$0.486038\pi$
$270$	0	0
$271$	−3023.10	−0.677640	−0.338820	−	0.940851i	$-0.610028\pi$
$271$	−3023.10	−0.677640	−0.338820	+	0.940851i	$0.610028\pi$
$272$	0	0
$273$	1442.21	0.319732
$274$	0	0
$275$	1334.09	0.292540
$276$	0	0
$277$	3120.52	0.676874	0.338437	−	0.940989i	$-0.390102\pi$
$277$	3120.52	0.676874	0.338437	+	0.940989i	$0.390102\pi$
$278$	0	0
$279$	−806.179	−0.172992
$280$	0	0
$281$	2778.40	0.589841	0.294920	−	0.955522i	$-0.404707\pi$
$281$	2778.40	0.589841	0.294920	+	0.955522i	$0.404707\pi$
$282$	0	0
$283$	−100.476	−0.0211048	−0.0105524	−	0.999944i	$-0.503359\pi$
$283$	−100.476	−0.0211048	−0.0105524	+	0.999944i	$0.503359\pi$
$284$	0	0
$285$	440.663	0.0915881
$286$	0	0
$287$	−1271.91	−0.261597
$288$	0	0
$289$	−3570.73	−0.726792
$290$	0	0
$291$	−8486.77	−1.70963
$292$	0	0
$293$	−2710.50	−0.540441	−0.270220	−	0.962799i	$-0.587097\pi$
$293$	−2710.50	−0.540441	−0.270220	+	0.962799i	$0.587097\pi$
$294$	0	0
$295$	−3471.68	−0.685182
$296$	0	0
$297$	1445.32	0.282378
$298$	0	0
$299$	480.295	0.0928970
$300$	0	0
$301$	−1247.69	−0.238923
$302$	0	0
$303$	−8017.02	−1.52002
$304$	0	0
$305$	−4411.97	−0.828291
$306$	0	0
$307$	−5421.57	−1.00790	−0.503950	−	0.863733i	$-0.668121\pi$
$307$	−5421.57	−1.00790	−0.503950	+	0.863733i	$0.668121\pi$
$308$	0	0
$309$	−3171.72	−0.583925
$310$	0	0
$311$	−3284.38	−0.598843	−0.299422	−	0.954121i	$-0.596794\pi$
$311$	−3284.38	−0.598843	−0.299422	+	0.954121i	$0.596794\pi$
$312$	0	0
$313$	−743.560	−0.134276	−0.0671381	−	0.997744i	$-0.521387\pi$
$313$	−743.560	−0.134276	−0.0671381	+	0.997744i	$0.521387\pi$
$314$	0	0
$315$	1129.84	0.202092
$316$	0	0
$317$	394.714	0.0699348	0.0349674	−	0.999388i	$-0.488867\pi$
$317$	394.714	0.0699348	0.0349674	+	0.999388i	$0.488867\pi$
$318$	0	0
$319$	−6620.87	−1.16206
$320$	0	0
$321$	1739.77	0.302506
$322$	0	0
$323$	455.834	0.0785241
$324$	0	0
$325$	−522.060	−0.0891037
$326$	0	0
$327$	−6540.69	−1.10612
$328$	0	0
$329$	4596.48	0.770250
$330$	0	0
$331$	−746.521	−0.123965	−0.0619826	−	0.998077i	$-0.519742\pi$
$331$	−746.521	−0.123965	−0.0619826	+	0.998077i	$0.519742\pi$
$332$	0	0
$333$	−6876.44	−1.13161
$334$	0	0
$335$	3443.10	0.561542
$336$	0	0
$337$	−2946.64	−0.476303	−0.238151	−	0.971228i	$-0.576541\pi$
$337$	−2946.64	−0.476303	−0.238151	+	0.971228i	$0.576541\pi$
$338$	0	0
$339$	11915.1	1.90897
$340$	0	0
$341$	−1856.22	−0.294780
$342$	0	0
$343$	−5761.59	−0.906987
$344$	0	0
$345$	814.606	0.127122
$346$	0	0
$347$	2656.97	0.411049	0.205524	−	0.978652i	$-0.434110\pi$
$347$	2656.97	0.411049	0.205524	+	0.978652i	$0.434110\pi$
$348$	0	0
$349$	6115.81	0.938029	0.469014	−	0.883191i	$-0.344609\pi$
$349$	6115.81	0.938029	0.469014	+	0.883191i	$0.344609\pi$
$350$	0	0
$351$	−565.589	−0.0860083
$352$	0	0
$353$	3270.11	0.493061	0.246531	−	0.969135i	$-0.420709\pi$
$353$	3270.11	0.493061	0.246531	+	0.969135i	$0.420709\pi$
$354$	0	0
$355$	2572.54	0.384609
$356$	0	0
$357$	2530.28	0.375117
$358$	0	0
$359$	−5636.87	−0.828697	−0.414349	−	0.910118i	$-0.635991\pi$
$359$	−5636.87	−0.828697	−0.414349	+	0.910118i	$0.635991\pi$
$360$	0	0
$361$	−6704.20	−0.977431
$362$	0	0
$363$	−10743.4	−1.55339
$364$	0	0
$365$	744.062	0.106701
$366$	0	0
$367$	1213.41	0.172587	0.0862935	−	0.996270i	$-0.472498\pi$
$367$	1213.41	0.172587	0.0862935	+	0.996270i	$0.472498\pi$
$368$	0	0
$369$	−3023.45	−0.426543
$370$	0	0
$371$	7227.98	1.01148
$372$	0	0
$373$	−12072.5	−1.67585	−0.837925	−	0.545786i	$-0.816231\pi$
$373$	−12072.5	−1.67585	−0.837925	+	0.545786i	$0.816231\pi$
$374$	0	0
$375$	−885.441	−0.121931
$376$	0	0
$377$	2590.90	0.353948
$378$	0	0
$379$	−3783.90	−0.512838	−0.256419	−	0.966566i	$-0.582543\pi$
$379$	−3783.90	−0.512838	−0.256419	+	0.966566i	$0.582543\pi$
$380$	0	0
$381$	9858.21	1.32559
$382$	0	0
$383$	−2905.46	−0.387630	−0.193815	−	0.981038i	$-0.562086\pi$
$383$	−2905.46	−0.387630	−0.193815	+	0.981038i	$0.562086\pi$
$384$	0	0
$385$	2601.44	0.344368
$386$	0	0
$387$	−2965.89	−0.389573
$388$	0	0
$389$	−4190.97	−0.546249	−0.273124	−	0.961979i	$-0.588057\pi$
$389$	−4190.97	−0.546249	−0.273124	+	0.961979i	$0.588057\pi$
$390$	0	0
$391$	842.651	0.108989
$392$	0	0
$393$	6113.82	0.784736
$394$	0	0
$395$	−2283.68	−0.290897
$396$	0	0
$397$	6627.84	0.837889	0.418945	−	0.908012i	$-0.362400\pi$
$397$	6627.84	0.837889	0.418945	+	0.908012i	$0.362400\pi$
$398$	0	0
$399$	859.281	0.107814
$400$	0	0
$401$	7203.09	0.897021	0.448510	−	0.893778i	$-0.351955\pi$
$401$	7203.09	0.897021	0.448510	+	0.893778i	$0.351955\pi$
$402$	0	0
$403$	726.383	0.0897860
$404$	0	0
$405$	−4088.08	−0.501577
$406$	0	0
$407$	−15833.0	−1.92828
$408$	0	0
$409$	−9723.67	−1.17556	−0.587781	−	0.809020i	$-0.699998\pi$
$409$	−9723.67	−1.17556	−0.587781	+	0.809020i	$0.699998\pi$
$410$	0	0
$411$	6495.55	0.779566
$412$	0	0
$413$	−6769.68	−0.806572
$414$	0	0
$415$	2481.52	0.293526
$416$	0	0
$417$	−13511.0	−1.58666
$418$	0	0
$419$	−9591.61	−1.11833	−0.559165	−	0.829056i	$-0.688878\pi$
$419$	−9591.61	−1.11833	−0.559165	+	0.829056i	$0.688878\pi$
$420$	0	0
$421$	−7907.04	−0.915358	−0.457679	−	0.889117i	$-0.651319\pi$
$421$	−7907.04	−0.915358	−0.457679	+	0.889117i	$0.651319\pi$
$422$	0	0
$423$	10926.3	1.25592
$424$	0	0
$425$	−915.925	−0.104539
$426$	0	0
$427$	−8603.23	−0.975034
$428$	0	0
$429$	7893.60	0.888361
$430$	0	0
$431$	−12217.2	−1.36539	−0.682695	−	0.730703i	$-0.739192\pi$
$431$	−12217.2	−1.36539	−0.682695	+	0.730703i	$0.739192\pi$
$432$	0	0
$433$	6320.07	0.701439	0.350719	−	0.936481i	$-0.385937\pi$
$433$	6320.07	0.701439	0.350719	+	0.936481i	$0.385937\pi$
$434$	0	0
$435$	4394.30	0.484347
$436$	0	0
$437$	286.164	0.0313251
$438$	0	0
$439$	−8937.53	−0.971674	−0.485837	−	0.874049i	$-0.661485\pi$
$439$	−8937.53	−0.971674	−0.485837	+	0.874049i	$0.661485\pi$
$440$	0	0
$441$	−5746.36	−0.620490
$442$	0	0
$443$	−8445.19	−0.905741	−0.452870	−	0.891576i	$-0.649600\pi$
$443$	−8445.19	−0.905741	−0.452870	+	0.891576i	$0.649600\pi$
$444$	0	0
$445$	297.234	0.0316635
$446$	0	0
$447$	11929.9	1.26233
$448$	0	0
$449$	15483.3	1.62740	0.813701	−	0.581284i	$-0.197450\pi$
$449$	15483.3	1.62740	0.813701	+	0.581284i	$0.197450\pi$
$450$	0	0
$451$	−6961.47	−0.726835
$452$	0	0
$453$	18130.8	1.88049
$454$	0	0
$455$	−1018.00	−0.104890
$456$	0	0
$457$	5505.28	0.563515	0.281757	−	0.959486i	$-0.409083\pi$
$457$	5505.28	0.563515	0.281757	+	0.959486i	$0.409083\pi$
$458$	0	0
$459$	−992.294	−0.100907
$460$	0	0
$461$	15252.3	1.54094	0.770468	−	0.637479i	$-0.220023\pi$
$461$	15252.3	1.54094	0.770468	+	0.637479i	$0.220023\pi$
$462$	0	0
$463$	2794.72	0.280522	0.140261	−	0.990115i	$-0.455206\pi$
$463$	2794.72	0.280522	0.140261	+	0.990115i	$0.455206\pi$
$464$	0	0
$465$	1231.98	0.122864
$466$	0	0
$467$	15124.6	1.49868	0.749340	−	0.662185i	$-0.230371\pi$
$467$	15124.6	1.49868	0.749340	+	0.662185i	$0.230371\pi$
$468$	0	0
$469$	6713.96	0.661028
$470$	0	0
$471$	−13023.0	−1.27403
$472$	0	0
$473$	−6828.95	−0.663839
$474$	0	0
$475$	−311.047	−0.0300460
$476$	0	0
$477$	17181.6	1.64925
$478$	0	0
$479$	−8437.47	−0.804839	−0.402419	−	0.915455i	$-0.631831\pi$
$479$	−8437.47	−0.804839	−0.402419	+	0.915455i	$0.631831\pi$
$480$	0	0
$481$	6195.80	0.587327
$482$	0	0
$483$	1588.46	0.149643
$484$	0	0
$485$	5990.49	0.560854
$486$	0	0
$487$	16061.3	1.49447	0.747233	−	0.664562i	$-0.231382\pi$
$487$	16061.3	1.49447	0.747233	+	0.664562i	$0.231382\pi$
$488$	0	0
$489$	15287.1	1.41372
$490$	0	0
$491$	−14763.6	−1.35697	−0.678486	−	0.734614i	$-0.737363\pi$
$491$	−14763.6	−1.35697	−0.678486	+	0.734614i	$0.737363\pi$
$492$	0	0
$493$	4545.59	0.415260
$494$	0	0
$495$	6183.88	0.561505
$496$	0	0
$497$	5016.39	0.452748
$498$	0	0
$499$	5407.58	0.485123	0.242562	−	0.970136i	$-0.422012\pi$
$499$	5407.58	0.485123	0.242562	+	0.970136i	$0.422012\pi$
$500$	0	0
$501$	13000.5	1.15932
$502$	0	0
$503$	956.117	0.0847538	0.0423769	−	0.999102i	$-0.486507\pi$
$503$	956.117	0.0847538	0.0423769	+	0.999102i	$0.486507\pi$
$504$	0	0
$505$	5658.92	0.498650
$506$	0	0
$507$	12473.6	1.09264
$508$	0	0
$509$	−13313.3	−1.15933	−0.579667	−	0.814853i	$-0.696818\pi$
$509$	−13313.3	−1.15933	−0.579667	+	0.814853i	$0.696818\pi$
$510$	0	0
$511$	1450.90	0.125605
$512$	0	0
$513$	−336.982	−0.0290022
$514$	0	0
$515$	2238.80	0.191560
$516$	0	0
$517$	25157.7	2.14011
$518$	0	0
$519$	−12336.6	−1.04338
$520$	0	0
$521$	−14092.8	−1.18506	−0.592531	−	0.805548i	$-0.701871\pi$
$521$	−14092.8	−1.18506	−0.592531	+	0.805548i	$0.701871\pi$
$522$	0	0
$523$	−4042.31	−0.337970	−0.168985	−	0.985619i	$-0.554049\pi$
$523$	−4042.31	−0.337970	−0.168985	+	0.985619i	$0.554049\pi$
$524$	0	0
$525$	−1726.59	−0.143532
$526$	0	0
$527$	1274.40	0.105339
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−16092.2	−1.31515
$532$	0	0
$533$	2724.18	0.221384
$534$	0	0
$535$	−1228.04	−0.0992387
$536$	0	0
$537$	973.683	0.0782449
$538$	0	0
$539$	−13231.0	−1.05732
$540$	0	0
$541$	9518.12	0.756407	0.378203	−	0.925722i	$-0.376542\pi$
$541$	9518.12	0.756407	0.378203	+	0.925722i	$0.376542\pi$
$542$	0	0
$543$	26292.8	2.07796
$544$	0	0
$545$	4616.83	0.362868
$546$	0	0
$547$	−210.407	−0.0164467	−0.00822335	−	0.999966i	$-0.502618\pi$
$547$	−210.407	−0.0164467	−0.00822335	+	0.999966i	$0.502618\pi$
$548$	0	0
$549$	−20450.7	−1.58983
$550$	0	0
$551$	1543.68	0.119352
$552$	0	0
$553$	−4453.11	−0.342433
$554$	0	0
$555$	10508.4	0.803706
$556$	0	0
$557$	10799.1	0.821493	0.410747	−	0.911750i	$-0.365268\pi$
$557$	10799.1	0.821493	0.410747	+	0.911750i	$0.365268\pi$
$558$	0	0
$559$	2672.33	0.202196
$560$	0	0
$561$	13848.9	1.04225
$562$	0	0
$563$	10335.4	0.773685	0.386843	−	0.922146i	$-0.373566\pi$
$563$	10335.4	0.773685	0.386843	+	0.922146i	$0.373566\pi$
$564$	0	0
$565$	−8410.46	−0.626249
$566$	0	0
$567$	−7971.66	−0.590438
$568$	0	0
$569$	2786.04	0.205267	0.102633	−	0.994719i	$-0.467273\pi$
$569$	2786.04	0.205267	0.102633	+	0.994719i	$0.467273\pi$
$570$	0	0
$571$	−17929.4	−1.31405	−0.657026	−	0.753868i	$-0.728186\pi$
$571$	−17929.4	−1.31405	−0.657026	+	0.753868i	$0.728186\pi$
$572$	0	0
$573$	−16804.1	−1.22514
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	879.238	0.0634370	0.0317185	−	0.999497i	$-0.489902\pi$
$577$	879.238	0.0634370	0.0317185	+	0.999497i	$0.489902\pi$
$578$	0	0
$579$	20634.5	1.48107
$580$	0	0
$581$	4838.90	0.345528
$582$	0	0
$583$	39560.6	2.81034
$584$	0	0
$585$	−2419.90	−0.171026
$586$	0	0
$587$	11291.2	0.793929	0.396964	−	0.917834i	$-0.370064\pi$
$587$	11291.2	0.793929	0.396964	+	0.917834i	$0.370064\pi$
$588$	0	0
$589$	432.785	0.0302760
$590$	0	0
$591$	−16640.0	−1.15817
$592$	0	0
$593$	−18648.4	−1.29140	−0.645699	−	0.763592i	$-0.723434\pi$
$593$	−18648.4	−1.29140	−0.645699	+	0.763592i	$0.723434\pi$
$594$	0	0
$595$	−1786.03	−0.123059
$596$	0	0
$597$	838.386	0.0574754
$598$	0	0
$599$	−8385.44	−0.571986	−0.285993	−	0.958232i	$-0.592323\pi$
$599$	−8385.44	−0.571986	−0.285993	+	0.958232i	$0.592323\pi$
$600$	0	0
$601$	12270.2	0.832796	0.416398	−	0.909182i	$-0.363292\pi$
$601$	12270.2	0.832796	0.416398	+	0.909182i	$0.363292\pi$
$602$	0	0
$603$	15959.8	1.07783
$604$	0	0
$605$	7583.35	0.509598
$606$	0	0
$607$	21513.1	1.43854	0.719268	−	0.694733i	$-0.244477\pi$
$607$	21513.1	1.43854	0.719268	+	0.694733i	$0.244477\pi$
$608$	0	0
$609$	8568.78	0.570155
$610$	0	0
$611$	−9844.80	−0.651846
$612$	0	0
$613$	−25145.0	−1.65677	−0.828384	−	0.560161i	$-0.810739\pi$
$613$	−25145.0	−1.65677	−0.828384	+	0.560161i	$0.810739\pi$
$614$	0	0
$615$	4620.36	0.302945
$616$	0	0
$617$	−18040.3	−1.17710	−0.588552	−	0.808459i	$-0.700302\pi$
$617$	−18040.3	−1.17710	−0.588552	+	0.808459i	$0.700302\pi$
$618$	0	0
$619$	21792.3	1.41504	0.707519	−	0.706695i	$-0.249814\pi$
$619$	21792.3	1.41504	0.707519	+	0.706695i	$0.249814\pi$
$620$	0	0
$621$	−622.943	−0.0402541
$622$	0	0
$623$	579.599	0.0372731
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4703.07	0.299557
$628$	0	0
$629$	10870.2	0.689067
$630$	0	0
$631$	23936.3	1.51012	0.755062	−	0.655653i	$-0.227607\pi$
$631$	23936.3	1.51012	0.755062	+	0.655653i	$0.227607\pi$
$632$	0	0
$633$	−24288.9	−1.52511
$634$	0	0
$635$	−6958.54	−0.434868
$636$	0	0
$637$	5177.58	0.322046
$638$	0	0
$639$	11924.5	0.738222
$640$	0	0
$641$	−21552.3	−1.32803	−0.664014	−	0.747720i	$-0.731148\pi$
$641$	−21552.3	−1.32803	−0.664014	+	0.747720i	$0.731148\pi$
$642$	0	0
$643$	14596.1	0.895203	0.447602	−	0.894233i	$-0.352278\pi$
$643$	14596.1	0.895203	0.447602	+	0.894233i	$0.352278\pi$
$644$	0	0
$645$	4532.41	0.276688
$646$	0	0
$647$	−1689.29	−0.102647	−0.0513235	−	0.998682i	$-0.516344\pi$
$647$	−1689.29	−0.102647	−0.0513235	+	0.998682i	$0.516344\pi$
$648$	0	0
$649$	−37052.2	−2.24103
$650$	0	0
$651$	2402.34	0.144631
$652$	0	0
$653$	22611.4	1.35506	0.677529	−	0.735496i	$-0.263051\pi$
$653$	22611.4	1.35506	0.677529	+	0.735496i	$0.263051\pi$
$654$	0	0
$655$	−4315.51	−0.257437
$656$	0	0
$657$	3448.94	0.204804
$658$	0	0
$659$	−12332.2	−0.728976	−0.364488	−	0.931208i	$-0.618756\pi$
$659$	−12332.2	−0.728976	−0.364488	+	0.931208i	$0.618756\pi$
$660$	0	0
$661$	−4928.54	−0.290012	−0.145006	−	0.989431i	$-0.546320\pi$
$661$	−4928.54	−0.290012	−0.145006	+	0.989431i	$0.546320\pi$
$662$	0	0
$663$	−5419.39	−0.317454
$664$	0	0
$665$	−606.535	−0.0353690
$666$	0	0
$667$	2853.63	0.165657
$668$	0	0
$669$	34253.5	1.97955
$670$	0	0
$671$	−47087.7	−2.70909
$672$	0	0
$673$	−32249.0	−1.84711	−0.923556	−	0.383464i	$-0.874731\pi$
$673$	−32249.0	−1.84711	−0.923556	+	0.383464i	$0.874731\pi$
$674$	0	0
$675$	677.111	0.0386104
$676$	0	0
$677$	3209.18	0.182184	0.0910922	−	0.995842i	$-0.470964\pi$
$677$	3209.18	0.182184	0.0910922	+	0.995842i	$0.470964\pi$
$678$	0	0
$679$	11681.3	0.660217
$680$	0	0
$681$	−17697.7	−0.995857
$682$	0	0
$683$	−7822.62	−0.438249	−0.219125	−	0.975697i	$-0.570320\pi$
$683$	−7822.62	−0.438249	−0.219125	+	0.975697i	$0.570320\pi$
$684$	0	0
$685$	−4584.96	−0.255741
$686$	0	0
$687$	43146.3	2.39612
$688$	0	0
$689$	−15481.0	−0.855991
$690$	0	0
$691$	−26788.8	−1.47481	−0.737406	−	0.675450i	$-0.763950\pi$
$691$	−26788.8	−1.47481	−0.737406	+	0.675450i	$0.763950\pi$
$692$	0	0
$693$	12058.4	0.660983
$694$	0	0
$695$	9536.89	0.520511
$696$	0	0
$697$	4779.43	0.259733
$698$	0	0
$699$	46481.6	2.51516
$700$	0	0
$701$	4678.98	0.252101	0.126050	−	0.992024i	$-0.459770\pi$
$701$	4678.98	0.252101	0.126050	+	0.992024i	$0.459770\pi$
$702$	0	0
$703$	3691.51	0.198048
$704$	0	0
$705$	−16697.3	−0.891995
$706$	0	0
$707$	11034.7	0.586993
$708$	0	0
$709$	20765.8	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$709$	20765.8	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$710$	0	0
$711$	−10585.5	−0.558350
$712$	0	0
$713$	800.043	0.0420222
$714$	0	0
$715$	−5571.80	−0.291431
$716$	0	0
$717$	33790.3	1.76000
$718$	0	0
$719$	−10950.9	−0.568013	−0.284007	−	0.958822i	$-0.591664\pi$
$719$	−10950.9	−0.568013	−0.284007	+	0.958822i	$0.591664\pi$
$720$	0	0
$721$	4365.60	0.225497
$722$	0	0
$723$	−8408.04	−0.432501
$724$	0	0
$725$	−3101.77	−0.158892
$726$	0	0
$727$	−324.297	−0.0165440	−0.00827202	−	0.999966i	$-0.502633\pi$
$727$	−324.297	−0.0165440	−0.00827202	+	0.999966i	$0.502633\pi$
$728$	0	0
$729$	−13769.4	−0.699557
$730$	0	0
$731$	4688.45	0.237221
$732$	0	0
$733$	−13731.9	−0.691951	−0.345976	−	0.938243i	$-0.612452\pi$
$733$	−13731.9	−0.691951	−0.345976	+	0.938243i	$0.612452\pi$
$734$	0	0
$735$	8781.45	0.440692
$736$	0	0
$737$	36747.2	1.83664
$738$	0	0
$739$	20578.6	1.02435	0.512175	−	0.858881i	$-0.328840\pi$
$739$	20578.6	1.02435	0.512175	+	0.858881i	$0.328840\pi$
$740$	0	0
$741$	−1840.42	−0.0912409
$742$	0	0
$743$	28523.6	1.40839	0.704193	−	0.710009i	$-0.251309\pi$
$743$	28523.6	1.40839	0.704193	+	0.710009i	$0.251309\pi$
$744$	0	0
$745$	−8420.84	−0.414115
$746$	0	0
$747$	11502.6	0.563396
$748$	0	0
$749$	−2394.64	−0.116820
$750$	0	0
$751$	−11944.5	−0.580376	−0.290188	−	0.956970i	$-0.593718\pi$
$751$	−11944.5	−0.580376	−0.290188	+	0.956970i	$0.593718\pi$
$752$	0	0
$753$	−35497.4	−1.71793
$754$	0	0
$755$	−12797.9	−0.616904
$756$	0	0
$757$	−31224.2	−1.49916	−0.749578	−	0.661916i	$-0.769744\pi$
$757$	−31224.2	−1.49916	−0.749578	+	0.661916i	$0.769744\pi$
$758$	0	0
$759$	8694.06	0.415776
$760$	0	0
$761$	6874.77	0.327477	0.163739	−	0.986504i	$-0.447645\pi$
$761$	6874.77	0.327477	0.163739	+	0.986504i	$0.447645\pi$
$762$	0	0
$763$	9002.70	0.427155
$764$	0	0
$765$	−4245.57	−0.200652
$766$	0	0
$767$	14499.4	0.682585
$768$	0	0
$769$	−26845.2	−1.25886	−0.629431	−	0.777057i	$-0.716712\pi$
$769$	−26845.2	−1.25886	−0.629431	+	0.777057i	$0.716712\pi$
$770$	0	0
$771$	21319.0	0.995832
$772$	0	0
$773$	5403.89	0.251442	0.125721	−	0.992066i	$-0.459876\pi$
$773$	5403.89	0.251442	0.125721	+	0.992066i	$0.459876\pi$
$774$	0	0
$775$	−869.612	−0.0403063
$776$	0	0
$777$	20491.1	0.946094
$778$	0	0
$779$	1623.09	0.0746511
$780$	0	0
$781$	27456.0	1.25794
$782$	0	0
$783$	−3360.40	−0.153373
$784$	0	0
$785$	9192.48	0.417954
$786$	0	0
$787$	−3186.99	−0.144351	−0.0721754	−	0.997392i	$-0.522994\pi$
$787$	−3186.99	−0.144351	−0.0721754	+	0.997392i	$0.522994\pi$
$788$	0	0
$789$	8599.72	0.388033
$790$	0	0
$791$	−16400.2	−0.737198
$792$	0	0
$793$	18426.5	0.825151
$794$	0	0
$795$	−26256.5	−1.17135
$796$	0	0
$797$	1971.38	0.0876160	0.0438080	−	0.999040i	$-0.486051\pi$
$797$	1971.38	0.0876160	0.0438080	+	0.999040i	$0.486051\pi$
$798$	0	0
$799$	−17272.2	−0.764762
$800$	0	0
$801$	1377.76	0.0607752
$802$	0	0
$803$	7941.16	0.348988
$804$	0	0
$805$	−1121.24	−0.0490911
$806$	0	0
$807$	−2740.72	−0.119551
$808$	0	0
$809$	−17872.0	−0.776693	−0.388347	−	0.921513i	$-0.626954\pi$
$809$	−17872.0	−0.776693	−0.388347	+	0.921513i	$0.626954\pi$
$810$	0	0
$811$	13371.5	0.578959	0.289480	−	0.957184i	$-0.406518\pi$
$811$	13371.5	0.578959	0.289480	+	0.957184i	$0.406518\pi$
$812$	0	0
$813$	21414.3	0.923777
$814$	0	0
$815$	−10790.6	−0.463778
$816$	0	0
$817$	1592.19	0.0681809
$818$	0	0
$819$	−4718.74	−0.201326
$820$	0	0
$821$	−10641.2	−0.452350	−0.226175	−	0.974087i	$-0.572622\pi$
$821$	−10641.2	−0.452350	−0.226175	+	0.974087i	$0.572622\pi$
$822$	0	0
$823$	19507.2	0.826220	0.413110	−	0.910681i	$-0.364443\pi$
$823$	19507.2	0.826220	0.413110	+	0.910681i	$0.364443\pi$
$824$	0	0
$825$	−9450.06	−0.398799
$826$	0	0
$827$	−2183.64	−0.0918171	−0.0459085	−	0.998946i	$-0.514618\pi$
$827$	−2183.64	−0.0918171	−0.0459085	+	0.998946i	$0.514618\pi$
$828$	0	0
$829$	19725.3	0.826402	0.413201	−	0.910640i	$-0.364411\pi$
$829$	19725.3	0.826402	0.413201	+	0.910640i	$0.364411\pi$
$830$	0	0
$831$	−22104.3	−0.922733
$832$	0	0
$833$	9083.78	0.377832
$834$	0	0
$835$	−9176.54	−0.380320
$836$	0	0
$837$	−942.118	−0.0389061
$838$	0	0
$839$	−36445.7	−1.49970	−0.749849	−	0.661609i	$-0.769874\pi$
$839$	−36445.7	−1.49970	−0.749849	+	0.661609i	$0.769874\pi$
$840$	0	0
$841$	−8995.39	−0.368830
$842$	0	0
$843$	−19680.9	−0.804087
$844$	0	0
$845$	−8804.62	−0.358448
$846$	0	0
$847$	14787.3	0.599881
$848$	0	0
$849$	711.722	0.0287706
$850$	0	0
$851$	6824.09	0.274885
$852$	0	0
$853$	−23757.7	−0.953634	−0.476817	−	0.879003i	$-0.658209\pi$
$853$	−23757.7	−0.953634	−0.476817	+	0.879003i	$0.658209\pi$
$854$	0	0
$855$	−1441.79	−0.0576705
$856$	0	0
$857$	−20155.1	−0.803367	−0.401683	−	0.915779i	$-0.631575\pi$
$857$	−20155.1	−0.803367	−0.401683	+	0.915779i	$0.631575\pi$
$858$	0	0
$859$	13473.5	0.535170	0.267585	−	0.963534i	$-0.413774\pi$
$859$	13473.5	0.535170	0.267585	+	0.963534i	$0.413774\pi$
$860$	0	0
$861$	9009.59	0.356615
$862$	0	0
$863$	23495.9	0.926777	0.463389	−	0.886155i	$-0.346633\pi$
$863$	23495.9	0.926777	0.463389	+	0.886155i	$0.346633\pi$
$864$	0	0
$865$	8707.92	0.342287
$866$	0	0
$867$	25293.4	0.990782
$868$	0	0
$869$	−24373.0	−0.951436
$870$	0	0
$871$	−14380.1	−0.559414
$872$	0	0
$873$	27767.6	1.07651
$874$	0	0
$875$	1218.73	0.0470866
$876$	0	0
$877$	−34995.1	−1.34744	−0.673718	−	0.738988i	$-0.735304\pi$
$877$	−34995.1	−1.34744	−0.673718	+	0.738988i	$0.735304\pi$
$878$	0	0
$879$	19199.9	0.736743
$880$	0	0
$881$	45327.7	1.73340	0.866702	−	0.498826i	$-0.166235\pi$
$881$	45327.7	1.73340	0.866702	+	0.498826i	$0.166235\pi$
$882$	0	0
$883$	−4482.87	−0.170850	−0.0854251	−	0.996345i	$-0.527225\pi$
$883$	−4482.87	−0.170850	−0.0854251	+	0.996345i	$0.527225\pi$
$884$	0	0
$885$	24591.7	0.934059
$886$	0	0
$887$	42709.2	1.61672	0.808362	−	0.588686i	$-0.200355\pi$
$887$	42709.2	1.61672	0.808362	+	0.588686i	$0.200355\pi$
$888$	0	0
$889$	−13569.0	−0.511911
$890$	0	0
$891$	−43631.0	−1.64051
$892$	0	0
$893$	−5865.60	−0.219804
$894$	0	0
$895$	−687.286	−0.0256687
$896$	0	0
$897$	−3402.19	−0.126640
$898$	0	0
$899$	4315.74	0.160109
$900$	0	0
$901$	−27160.5	−1.00427
$902$	0	0
$903$	8838.09	0.325707
$904$	0	0
$905$	−18559.1	−0.681686
$906$	0	0
$907$	−37998.3	−1.39108	−0.695541	−	0.718486i	$-0.744835\pi$
$907$	−37998.3	−1.39108	−0.695541	+	0.718486i	$0.744835\pi$
$908$	0	0
$909$	26230.7	0.957114
$910$	0	0
$911$	−12028.8	−0.437467	−0.218734	−	0.975785i	$-0.570193\pi$
$911$	−12028.8	−0.437467	−0.218734	+	0.975785i	$0.570193\pi$
$912$	0	0
$913$	26484.6	0.960034
$914$	0	0
$915$	31252.3	1.12915
$916$	0	0
$917$	−8415.15	−0.303045
$918$	0	0
$919$	39289.5	1.41027	0.705136	−	0.709072i	$-0.250886\pi$
$919$	39289.5	1.41027	0.705136	+	0.709072i	$0.250886\pi$
$920$	0	0
$921$	38403.9	1.37400
$922$	0	0
$923$	−10744.2	−0.383151
$924$	0	0
$925$	−7417.49	−0.263660
$926$	0	0
$927$	10377.5	0.367681
$928$	0	0
$929$	−29766.0	−1.05123	−0.525615	−	0.850723i	$-0.676165\pi$
$929$	−29766.0	−1.05123	−0.525615	+	0.850723i	$0.676165\pi$
$930$	0	0
$931$	3084.84	0.108595
$932$	0	0
$933$	23265.0	0.816359
$934$	0	0
$935$	−9775.41	−0.341915
$936$	0	0
$937$	4561.05	0.159021	0.0795107	−	0.996834i	$-0.474664\pi$
$937$	4561.05	0.159021	0.0795107	+	0.996834i	$0.474664\pi$
$938$	0	0
$939$	5267.03	0.183049
$940$	0	0
$941$	−30191.6	−1.04593	−0.522963	−	0.852355i	$-0.675174\pi$
$941$	−30191.6	−1.04593	−0.522963	+	0.852355i	$0.675174\pi$
$942$	0	0
$943$	3000.43	0.103613
$944$	0	0
$945$	1320.35	0.0454508
$946$	0	0
$947$	−20719.5	−0.710975	−0.355487	−	0.934681i	$-0.615685\pi$
$947$	−20719.5	−0.710975	−0.355487	+	0.934681i	$0.615685\pi$
$948$	0	0
$949$	−3107.56	−0.106297
$950$	0	0
$951$	−2795.97	−0.0953369
$952$	0	0
$953$	−35366.0	−1.20212	−0.601059	−	0.799205i	$-0.705254\pi$
$953$	−35366.0	−1.20212	−0.601059	+	0.799205i	$0.705254\pi$
$954$	0	0
$955$	11861.4	0.401912
$956$	0	0
$957$	46899.1	1.58415
$958$	0	0
$959$	−8940.57	−0.301049
$960$	0	0
$961$	−28581.0	−0.959385
$962$	0	0
$963$	−5692.30	−0.190480
$964$	0	0
$965$	−14565.1	−0.485873
$966$	0	0
$967$	−27910.5	−0.928171	−0.464086	−	0.885790i	$-0.653617\pi$
$967$	−27910.5	−0.928171	−0.464086	+	0.885790i	$0.653617\pi$
$968$	0	0
$969$	−3228.91	−0.107046
$970$	0	0
$971$	30430.1	1.00571	0.502857	−	0.864370i	$-0.332282\pi$
$971$	30430.1	1.00571	0.502857	+	0.864370i	$0.332282\pi$
$972$	0	0
$973$	18596.7	0.612727
$974$	0	0
$975$	3698.03	0.121468
$976$	0	0
$977$	−28854.8	−0.944878	−0.472439	−	0.881363i	$-0.656626\pi$
$977$	−28854.8	−0.944878	−0.472439	+	0.881363i	$0.656626\pi$
$978$	0	0
$979$	3172.29	0.103562
$980$	0	0
$981$	21400.3	0.696492
$982$	0	0
$983$	−14350.6	−0.465629	−0.232814	−	0.972521i	$-0.574793\pi$
$983$	−14350.6	−0.465629	−0.232814	+	0.972521i	$0.574793\pi$
$984$	0	0
$985$	11745.5	0.379943
$986$	0	0
$987$	−32559.3	−1.05002
$988$	0	0
$989$	2943.32	0.0946330
$990$	0	0
$991$	33658.5	1.07891	0.539454	−	0.842015i	$-0.318631\pi$
$991$	33658.5	1.07891	0.539454	+	0.842015i	$0.318631\pi$
$992$	0	0
$993$	5288.00	0.168993
$994$	0	0
$995$	−591.785	−0.0188551
$996$	0	0
$997$	23620.7	0.750326	0.375163	−	0.926959i	$-0.377587\pi$
$997$	23620.7	0.750326	0.375163	+	0.926959i	$0.377587\pi$
$998$	0	0
$999$	−8035.95	−0.254501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.u.1.1		6
4.3	odd	2		920.4.a.a.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.a.1.6	✓	6	4.3	odd	2
1840.4.a.u.1.1		6	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-7.08353 q^{3} +5.00000 q^{5} +9.74988 q^{7} +23.1764 q^{9} +O(q^{10})\)	\(q-7.08353 q^{3} +5.00000 q^{5} +9.74988 q^{7} +23.1764 q^{9} +53.3636 q^{11} -20.8824 q^{13} -35.4177 q^{15} -36.6370 q^{17} -12.4419 q^{19} -69.0635 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0845 q^{27} -124.071 q^{29} -34.7845 q^{31} -378.002 q^{33} +48.7494 q^{35} -296.700 q^{37} +147.921 q^{39} -130.454 q^{41} -127.970 q^{43} +115.882 q^{45} +471.440 q^{47} -247.940 q^{49} +259.519 q^{51} +741.340 q^{53} +266.818 q^{55} +88.1325 q^{57} -694.335 q^{59} -882.394 q^{61} +225.967 q^{63} -104.412 q^{65} +688.620 q^{67} +162.921 q^{69} +514.508 q^{71} +148.812 q^{73} -177.088 q^{75} +520.288 q^{77} -456.735 q^{79} -817.617 q^{81} +496.304 q^{83} -183.185 q^{85} +878.861 q^{87} +59.4468 q^{89} -203.601 q^{91} +246.397 q^{93} -62.2095 q^{95} +1198.10 q^{97} +1236.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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