LMFDB - Embedded newform 1840.4.a.l.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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.92791$ 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)

$f(q)$	$=$	$q-8.92791 q^{3} -5.00000 q^{5} -23.5524 q^{7} +52.7075 q^{9} +O(q^{10})$	$q-8.92791 q^{3} -5.00000 q^{5} -23.5524 q^{7} +52.7075 q^{9} -1.63537 q^{11} -88.6599 q^{13} +44.6395 q^{15} -7.31435 q^{17} +6.53738 q^{19} +210.274 q^{21} -23.0000 q^{23} +25.0000 q^{25} -229.514 q^{27} +119.240 q^{29} +156.456 q^{31} +14.6004 q^{33} +117.762 q^{35} -293.173 q^{37} +791.547 q^{39} +74.3404 q^{41} -468.081 q^{43} -263.538 q^{45} +393.971 q^{47} +211.715 q^{49} +65.3018 q^{51} +233.171 q^{53} +8.17685 q^{55} -58.3651 q^{57} +766.648 q^{59} +178.365 q^{61} -1241.39 q^{63} +443.299 q^{65} -246.904 q^{67} +205.342 q^{69} +650.678 q^{71} +695.444 q^{73} -223.198 q^{75} +38.5169 q^{77} -660.717 q^{79} +625.978 q^{81} +1328.39 q^{83} +36.5718 q^{85} -1064.57 q^{87} -824.702 q^{89} +2088.15 q^{91} -1396.82 q^{93} -32.6869 q^{95} +383.833 q^{97} -86.1963 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9	$ 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 - 93 * q^11 + 32 * q^13 + 20 * q^15 + 108 * q^17 - 185 * q^19 + 302 * q^21 - 92 * q^23 + 100 * q^25 - 259 * q^27 + 294 * q^29 + 211 * q^31 - 237 * q^33 + 130 * q^35 + 5 * q^37 + 421 * q^39 - 369 * q^41 + 100 * q^43 - 320 * q^45 + 363 * q^47 + 600 * q^49 + 315 * q^51 + 21 * q^53 + 465 * q^55 - 160 * q^57 + 33 * q^59 - 307 * q^61 - 167 * q^63 - 160 * q^65 - 725 * q^67 + 92 * q^69 + 1257 * q^71 + 509 * q^73 - 100 * q^75 - 1962 * q^77 - 1202 * q^79 - 992 * q^81 + 1377 * q^83 - 540 * q^85 + 2829 * q^87 - 984 * q^89 + 995 * q^91 - 1843 * q^93 + 925 * q^95 + 137 * q^97 - 2013 * 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
$3$	−8.92791	−1.71818	−0.859088	−	0.511828i	$-0.828969\pi$
$3$	−8.92791	−1.71818	−0.859088	+	0.511828i	$0.828969\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−23.5524	−1.27171	−0.635855	−	0.771809i	$-0.719352\pi$
$7$	−23.5524	−1.27171	−0.635855	+	0.771809i	$0.719352\pi$
$8$	0	0
$9$	52.7075	1.95213
$10$	0	0
$11$	−1.63537	−0.0448257	−0.0224129	−	0.999749i	$-0.507135\pi$
$11$	−1.63537	−0.0448257	−0.0224129	+	0.999749i	$0.507135\pi$
$12$	0	0
$13$	−88.6599	−1.89152	−0.945762	−	0.324860i	$-0.894683\pi$
$13$	−88.6599	−1.89152	−0.945762	+	0.324860i	$0.894683\pi$
$14$	0	0
$15$	44.6395	0.768392
$16$	0	0
$17$	−7.31435	−0.104352	−0.0521762	−	0.998638i	$-0.516616\pi$
$17$	−7.31435	−0.104352	−0.0521762	+	0.998638i	$0.516616\pi$
$18$	0	0
$19$	6.53738	0.0789357	0.0394678	−	0.999221i	$-0.487434\pi$
$19$	6.53738	0.0789357	0.0394678	+	0.999221i	$0.487434\pi$
$20$	0	0
$21$	210.274	2.18502
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−229.514	−1.63593
$28$	0	0
$29$	119.240	0.763530	0.381765	−	0.924259i	$-0.375316\pi$
$29$	119.240	0.763530	0.381765	+	0.924259i	$0.375316\pi$
$30$	0	0
$31$	156.456	0.906462	0.453231	−	0.891393i	$-0.350271\pi$
$31$	156.456	0.906462	0.453231	+	0.891393i	$0.350271\pi$
$32$	0	0
$33$	14.6004	0.0770185
$34$	0	0
$35$	117.762	0.568726
$36$	0	0
$37$	−293.173	−1.30263	−0.651315	−	0.758807i	$-0.725782\pi$
$37$	−293.173	−1.30263	−0.651315	+	0.758807i	$0.725782\pi$
$38$	0	0
$39$	791.547	3.24997
$40$	0	0
$41$	74.3404	0.283171	0.141586	−	0.989926i	$-0.454780\pi$
$41$	74.3404	0.283171	0.141586	+	0.989926i	$0.454780\pi$
$42$	0	0
$43$	−468.081	−1.66004	−0.830020	−	0.557733i	$-0.811671\pi$
$43$	−468.081	−1.66004	−0.830020	+	0.557733i	$0.811671\pi$
$44$	0	0
$45$	−263.538	−0.873019
$46$	0	0
$47$	393.971	1.22269	0.611346	−	0.791363i	$-0.290628\pi$
$47$	393.971	1.22269	0.611346	+	0.791363i	$0.290628\pi$
$48$	0	0
$49$	211.715	0.617245
$50$	0	0
$51$	65.3018	0.179296
$52$	0	0
$53$	233.171	0.604311	0.302155	−	0.953259i	$-0.402294\pi$
$53$	233.171	0.604311	0.302155	+	0.953259i	$0.402294\pi$
$54$	0	0
$55$	8.17685	0.0200467
$56$	0	0
$57$	−58.3651	−0.135625
$58$	0	0
$59$	766.648	1.69168	0.845840	−	0.533437i	$-0.179100\pi$
$59$	766.648	1.69168	0.845840	+	0.533437i	$0.179100\pi$
$60$	0	0
$61$	178.365	0.374383	0.187191	−	0.982323i	$-0.440062\pi$
$61$	178.365	0.374383	0.187191	+	0.982323i	$0.440062\pi$
$62$	0	0
$63$	−1241.39	−2.48254
$64$	0	0
$65$	443.299	0.845915
$66$	0	0
$67$	−246.904	−0.450211	−0.225105	−	0.974334i	$-0.572273\pi$
$67$	−246.904	−0.450211	−0.225105	+	0.974334i	$0.572273\pi$
$68$	0	0
$69$	205.342	0.358265
$70$	0	0
$71$	650.678	1.08762	0.543812	−	0.839207i	$-0.316980\pi$
$71$	650.678	1.08762	0.543812	+	0.839207i	$0.316980\pi$
$72$	0	0
$73$	695.444	1.11501	0.557504	−	0.830174i	$-0.311759\pi$
$73$	695.444	1.11501	0.557504	+	0.830174i	$0.311759\pi$
$74$	0	0
$75$	−223.198	−0.343635
$76$	0	0
$77$	38.5169	0.0570053
$78$	0	0
$79$	−660.717	−0.940967	−0.470484	−	0.882409i	$-0.655921\pi$
$79$	−660.717	−0.940967	−0.470484	+	0.882409i	$0.655921\pi$
$80$	0	0
$81$	625.978	0.858681
$82$	0	0
$83$	1328.39	1.75674	0.878372	−	0.477977i	$-0.158630\pi$
$83$	1328.39	1.75674	0.878372	+	0.477977i	$0.158630\pi$
$84$	0	0
$85$	36.5718	0.0466678
$86$	0	0
$87$	−1064.57	−1.31188
$88$	0	0
$89$	−824.702	−0.982227	−0.491113	−	0.871096i	$-0.663410\pi$
$89$	−824.702	−0.982227	−0.491113	+	0.871096i	$0.663410\pi$
$90$	0	0
$91$	2088.15	2.40547
$92$	0	0
$93$	−1396.82	−1.55746
$94$	0	0
$95$	−32.6869	−0.0353011
$96$	0	0
$97$	383.833	0.401776	0.200888	−	0.979614i	$-0.435617\pi$
$97$	383.833	0.401776	0.200888	+	0.979614i	$0.435617\pi$
$98$	0	0
$99$	−86.1963	−0.0875056
$100$	0	0
$101$	−674.579	−0.664586	−0.332293	−	0.943176i	$-0.607822\pi$
$101$	−674.579	−0.664586	−0.332293	+	0.943176i	$0.607822\pi$
$102$	0	0
$103$	−169.297	−0.161955	−0.0809773	−	0.996716i	$-0.525804\pi$
$103$	−169.297	−0.161955	−0.0809773	+	0.996716i	$0.525804\pi$
$104$	0	0
$105$	−1051.37	−0.977171
$106$	0	0
$107$	1533.31	1.38533	0.692665	−	0.721259i	$-0.256436\pi$
$107$	1533.31	1.38533	0.692665	+	0.721259i	$0.256436\pi$
$108$	0	0
$109$	1239.75	1.08941	0.544707	−	0.838626i	$-0.316641\pi$
$109$	1239.75	1.08941	0.544707	+	0.838626i	$0.316641\pi$
$110$	0	0
$111$	2617.42	2.23815
$112$	0	0
$113$	2299.49	1.91432	0.957159	−	0.289561i	$-0.0935094\pi$
$113$	2299.49	1.91432	0.957159	+	0.289561i	$0.0935094\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−4673.04	−3.69250
$118$	0	0
$119$	172.270	0.132706
$120$	0	0
$121$	−1328.33	−0.997991
$122$	0	0
$123$	−663.704	−0.486538
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1177.82	0.822948	0.411474	−	0.911422i	$-0.365014\pi$
$127$	1177.82	0.822948	0.411474	+	0.911422i	$0.365014\pi$
$128$	0	0
$129$	4178.99	2.85224
$130$	0	0
$131$	−2385.56	−1.59105	−0.795525	−	0.605921i	$-0.792805\pi$
$131$	−2385.56	−1.59105	−0.795525	+	0.605921i	$0.792805\pi$
$132$	0	0
$133$	−153.971	−0.100383
$134$	0	0
$135$	1147.57	0.731609
$136$	0	0
$137$	1123.22	0.700460	0.350230	−	0.936664i	$-0.386103\pi$
$137$	1123.22	0.700460	0.350230	+	0.936664i	$0.386103\pi$
$138$	0	0
$139$	1801.75	1.09944	0.549721	−	0.835348i	$-0.314734\pi$
$139$	1801.75	1.09944	0.549721	+	0.835348i	$0.314734\pi$
$140$	0	0
$141$	−3517.33	−2.10080
$142$	0	0
$143$	144.992	0.0847889
$144$	0	0
$145$	−596.202	−0.341461
$146$	0	0
$147$	−1890.17	−1.06054
$148$	0	0
$149$	663.138	0.364607	0.182303	−	0.983242i	$-0.441645\pi$
$149$	663.138	0.364607	0.182303	+	0.983242i	$0.441645\pi$
$150$	0	0
$151$	−1617.66	−0.871808	−0.435904	−	0.899993i	$-0.643571\pi$
$151$	−1617.66	−0.871808	−0.435904	+	0.899993i	$0.643571\pi$
$152$	0	0
$153$	−385.521	−0.203709
$154$	0	0
$155$	−782.280	−0.405382
$156$	0	0
$157$	875.080	0.444834	0.222417	−	0.974952i	$-0.428605\pi$
$157$	875.080	0.444834	0.222417	+	0.974952i	$0.428605\pi$
$158$	0	0
$159$	−2081.73	−1.03831
$160$	0	0
$161$	541.705	0.265170
$162$	0	0
$163$	−2532.39	−1.21689	−0.608443	−	0.793597i	$-0.708206\pi$
$163$	−2532.39	−1.21689	−0.608443	+	0.793597i	$0.708206\pi$
$164$	0	0
$165$	−73.0022	−0.0344437
$166$	0	0
$167$	1131.65	0.524371	0.262186	−	0.965017i	$-0.415557\pi$
$167$	1131.65	0.524371	0.262186	+	0.965017i	$0.415557\pi$
$168$	0	0
$169$	5663.57	2.57786
$170$	0	0
$171$	344.569	0.154093
$172$	0	0
$173$	−3186.23	−1.40026	−0.700128	−	0.714017i	$-0.746874\pi$
$173$	−3186.23	−1.40026	−0.700128	+	0.714017i	$0.746874\pi$
$174$	0	0
$175$	−588.810	−0.254342
$176$	0	0
$177$	−6844.56	−2.90660
$178$	0	0
$179$	−1663.48	−0.694606	−0.347303	−	0.937753i	$-0.612902\pi$
$179$	−1663.48	−0.694606	−0.347303	+	0.937753i	$0.612902\pi$
$180$	0	0
$181$	−3747.35	−1.53889	−0.769443	−	0.638716i	$-0.779466\pi$
$181$	−3747.35	−1.53889	−0.769443	+	0.638716i	$0.779466\pi$
$182$	0	0
$183$	−1592.43	−0.643256
$184$	0	0
$185$	1465.86	0.582554
$186$	0	0
$187$	11.9617	0.00467767
$188$	0	0
$189$	5405.61	2.08042
$190$	0	0
$191$	−2262.97	−0.857291	−0.428645	−	0.903473i	$-0.641009\pi$
$191$	−2262.97	−0.857291	−0.428645	+	0.903473i	$0.641009\pi$
$192$	0	0
$193$	−2001.25	−0.746390	−0.373195	−	0.927753i	$-0.621738\pi$
$193$	−2001.25	−0.746390	−0.373195	+	0.927753i	$0.621738\pi$
$194$	0	0
$195$	−3957.73	−1.45343
$196$	0	0
$197$	−4804.08	−1.73744	−0.868722	−	0.495300i	$-0.835058\pi$
$197$	−4804.08	−1.73744	−0.868722	+	0.495300i	$0.835058\pi$
$198$	0	0
$199$	−3885.03	−1.38393	−0.691967	−	0.721929i	$-0.743256\pi$
$199$	−3885.03	−1.38393	−0.691967	+	0.721929i	$0.743256\pi$
$200$	0	0
$201$	2204.34	0.773542
$202$	0	0
$203$	−2808.39	−0.970989
$204$	0	0
$205$	−371.702	−0.126638
$206$	0	0
$207$	−1212.27	−0.407047
$208$	0	0
$209$	−10.6910	−0.00353835
$210$	0	0
$211$	−4568.39	−1.49052	−0.745262	−	0.666772i	$-0.767676\pi$
$211$	−4568.39	−1.49052	−0.745262	+	0.666772i	$0.767676\pi$
$212$	0	0
$213$	−5809.20	−1.86873
$214$	0	0
$215$	2340.41	0.742393
$216$	0	0
$217$	−3684.91	−1.15276
$218$	0	0
$219$	−6208.86	−1.91578
$220$	0	0
$221$	648.489	0.197385
$222$	0	0
$223$	−3081.07	−0.925218	−0.462609	−	0.886563i	$-0.653087\pi$
$223$	−3081.07	−0.925218	−0.462609	+	0.886563i	$0.653087\pi$
$224$	0	0
$225$	1317.69	0.390426
$226$	0	0
$227$	−678.971	−0.198524	−0.0992618	−	0.995061i	$-0.531648\pi$
$227$	−678.971	−0.198524	−0.0992618	+	0.995061i	$0.531648\pi$
$228$	0	0
$229$	6399.27	1.84662	0.923310	−	0.384056i	$-0.125473\pi$
$229$	6399.27	1.84662	0.923310	+	0.384056i	$0.125473\pi$
$230$	0	0
$231$	−343.875	−0.0979451
$232$	0	0
$233$	6083.30	1.71043	0.855215	−	0.518274i	$-0.173425\pi$
$233$	6083.30	1.71043	0.855215	+	0.518274i	$0.173425\pi$
$234$	0	0
$235$	−1969.85	−0.546805
$236$	0	0
$237$	5898.82	1.61675
$238$	0	0
$239$	−438.313	−0.118628	−0.0593140	−	0.998239i	$-0.518891\pi$
$239$	−438.313	−0.118628	−0.0593140	+	0.998239i	$0.518891\pi$
$240$	0	0
$241$	2854.41	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$241$	2854.41	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$242$	0	0
$243$	608.207	0.160562
$244$	0	0
$245$	−1058.57	−0.276040
$246$	0	0
$247$	−579.603	−0.149309
$248$	0	0
$249$	−11859.7	−3.01840
$250$	0	0
$251$	2292.93	0.576607	0.288303	−	0.957539i	$-0.406909\pi$
$251$	2292.93	0.576607	0.288303	+	0.957539i	$0.406909\pi$
$252$	0	0
$253$	37.6135	0.00934681
$254$	0	0
$255$	−326.509	−0.0801835
$256$	0	0
$257$	3547.59	0.861060	0.430530	−	0.902576i	$-0.358327\pi$
$257$	3547.59	0.861060	0.430530	+	0.902576i	$0.358327\pi$
$258$	0	0
$259$	6904.92	1.65657
$260$	0	0
$261$	6284.86	1.49051
$262$	0	0
$263$	2278.35	0.534178	0.267089	−	0.963672i	$-0.413938\pi$
$263$	2278.35	0.534178	0.267089	+	0.963672i	$0.413938\pi$
$264$	0	0
$265$	−1165.85	−0.270256
$266$	0	0
$267$	7362.86	1.68764
$268$	0	0
$269$	3915.36	0.887450	0.443725	−	0.896163i	$-0.353657\pi$
$269$	3915.36	0.887450	0.443725	+	0.896163i	$0.353657\pi$
$270$	0	0
$271$	−6252.71	−1.40157	−0.700784	−	0.713374i	$-0.747166\pi$
$271$	−6252.71	−1.40157	−0.700784	+	0.713374i	$0.747166\pi$
$272$	0	0
$273$	−18642.8	−4.13302
$274$	0	0
$275$	−40.8843	−0.00896514
$276$	0	0
$277$	−4700.53	−1.01959	−0.509797	−	0.860295i	$-0.670279\pi$
$277$	−4700.53	−1.01959	−0.509797	+	0.860295i	$0.670279\pi$
$278$	0	0
$279$	8246.40	1.76953
$280$	0	0
$281$	5508.07	1.16934	0.584669	−	0.811272i	$-0.301224\pi$
$281$	5508.07	1.16934	0.584669	+	0.811272i	$0.301224\pi$
$282$	0	0
$283$	−6639.20	−1.39456	−0.697278	−	0.716801i	$-0.745606\pi$
$283$	−6639.20	−1.39456	−0.697278	+	0.716801i	$0.745606\pi$
$284$	0	0
$285$	291.826	0.0606535
$286$	0	0
$287$	−1750.89	−0.360111
$288$	0	0
$289$	−4859.50	−0.989111
$290$	0	0
$291$	−3426.82	−0.690323
$292$	0	0
$293$	−2120.85	−0.422872	−0.211436	−	0.977392i	$-0.567814\pi$
$293$	−2120.85	−0.422872	−0.211436	+	0.977392i	$0.567814\pi$
$294$	0	0
$295$	−3833.24	−0.756542
$296$	0	0
$297$	375.341	0.0733316
$298$	0	0
$299$	2039.18	0.394410
$300$	0	0
$301$	11024.4	2.11109
$302$	0	0
$303$	6022.58	1.14188
$304$	0	0
$305$	−891.827	−0.167429
$306$	0	0
$307$	−7422.87	−1.37995	−0.689977	−	0.723832i	$-0.742379\pi$
$307$	−7422.87	−1.37995	−0.689977	+	0.723832i	$0.742379\pi$
$308$	0	0
$309$	1511.47	0.278266
$310$	0	0
$311$	2563.97	0.467490	0.233745	−	0.972298i	$-0.424902\pi$
$311$	2563.97	0.467490	0.233745	+	0.972298i	$0.424902\pi$
$312$	0	0
$313$	9403.53	1.69814	0.849072	−	0.528277i	$-0.177162\pi$
$313$	9403.53	1.69814	0.849072	+	0.528277i	$0.177162\pi$
$314$	0	0
$315$	6206.94	1.11023
$316$	0	0
$317$	407.110	0.0721312	0.0360656	−	0.999349i	$-0.488517\pi$
$317$	407.110	0.0721312	0.0360656	+	0.999349i	$0.488517\pi$
$318$	0	0
$319$	−195.002	−0.0342258
$320$	0	0
$321$	−13689.2	−2.38024
$322$	0	0
$323$	−47.8167	−0.00823713
$324$	0	0
$325$	−2216.50	−0.378305
$326$	0	0
$327$	−11068.3	−1.87180
$328$	0	0
$329$	−9278.95	−1.55491
$330$	0	0
$331$	7122.33	1.18272	0.591358	−	0.806409i	$-0.298592\pi$
$331$	7122.33	1.18272	0.591358	+	0.806409i	$0.298592\pi$
$332$	0	0
$333$	−15452.4	−2.54290
$334$	0	0
$335$	1234.52	0.201340
$336$	0	0
$337$	−8267.78	−1.33642	−0.668212	−	0.743971i	$-0.732940\pi$
$337$	−8267.78	−1.33642	−0.668212	+	0.743971i	$0.732940\pi$
$338$	0	0
$339$	−20529.7	−3.28914
$340$	0	0
$341$	−255.864	−0.0406328
$342$	0	0
$343$	3092.08	0.486753
$344$	0	0
$345$	−1026.71	−0.160221
$346$	0	0
$347$	−1048.83	−0.162260	−0.0811302	−	0.996704i	$-0.525853\pi$
$347$	−1048.83	−0.162260	−0.0811302	+	0.996704i	$0.525853\pi$
$348$	0	0
$349$	−9135.87	−1.40124	−0.700619	−	0.713535i	$-0.747093\pi$
$349$	−9135.87	−1.40124	−0.700619	+	0.713535i	$0.747093\pi$
$350$	0	0
$351$	20348.7	3.09440
$352$	0	0
$353$	5253.06	0.792046	0.396023	−	0.918240i	$-0.370390\pi$
$353$	5253.06	0.792046	0.396023	+	0.918240i	$0.370390\pi$
$354$	0	0
$355$	−3253.39	−0.486400
$356$	0	0
$357$	−1538.01	−0.228012
$358$	0	0
$359$	11269.7	1.65680	0.828400	−	0.560137i	$-0.189252\pi$
$359$	11269.7	1.65680	0.828400	+	0.560137i	$0.189252\pi$
$360$	0	0
$361$	−6816.26	−0.993769
$362$	0	0
$363$	11859.2	1.71472
$364$	0	0
$365$	−3477.22	−0.498646
$366$	0	0
$367$	5675.46	0.807238	0.403619	−	0.914927i	$-0.367752\pi$
$367$	5675.46	0.807238	0.403619	+	0.914927i	$0.367752\pi$
$368$	0	0
$369$	3918.29	0.552787
$370$	0	0
$371$	−5491.73	−0.768508
$372$	0	0
$373$	3662.20	0.508369	0.254185	−	0.967156i	$-0.418193\pi$
$373$	3662.20	0.508369	0.254185	+	0.967156i	$0.418193\pi$
$374$	0	0
$375$	1115.99	0.153678
$376$	0	0
$377$	−10571.8	−1.44424
$378$	0	0
$379$	−5520.89	−0.748256	−0.374128	−	0.927377i	$-0.622058\pi$
$379$	−5520.89	−0.748256	−0.374128	+	0.927377i	$0.622058\pi$
$380$	0	0
$381$	−10515.4	−1.41397
$382$	0	0
$383$	2309.13	0.308071	0.154036	−	0.988065i	$-0.450773\pi$
$383$	2309.13	0.308071	0.154036	+	0.988065i	$0.450773\pi$
$384$	0	0
$385$	−192.584	−0.0254935
$386$	0	0
$387$	−24671.4	−3.24061
$388$	0	0
$389$	−8194.51	−1.06807	−0.534034	−	0.845463i	$-0.679324\pi$
$389$	−8194.51	−1.06807	−0.534034	+	0.845463i	$0.679324\pi$
$390$	0	0
$391$	168.230	0.0217590
$392$	0	0
$393$	21298.1	2.73370
$394$	0	0
$395$	3303.58	0.420813
$396$	0	0
$397$	7273.90	0.919564	0.459782	−	0.888032i	$-0.347928\pi$
$397$	7273.90	0.919564	0.459782	+	0.888032i	$0.347928\pi$
$398$	0	0
$399$	1374.64	0.172476
$400$	0	0
$401$	−1835.81	−0.228618	−0.114309	−	0.993445i	$-0.536465\pi$
$401$	−1835.81	−0.228618	−0.114309	+	0.993445i	$0.536465\pi$
$402$	0	0
$403$	−13871.4	−1.71460
$404$	0	0
$405$	−3129.89	−0.384014
$406$	0	0
$407$	479.446	0.0583914
$408$	0	0
$409$	−2865.80	−0.346466	−0.173233	−	0.984881i	$-0.555421\pi$
$409$	−2865.80	−0.346466	−0.173233	+	0.984881i	$0.555421\pi$
$410$	0	0
$411$	−10028.0	−1.20351
$412$	0	0
$413$	−18056.4	−2.15132
$414$	0	0
$415$	−6641.95	−0.785640
$416$	0	0
$417$	−16085.9	−1.88904
$418$	0	0
$419$	−11044.0	−1.28767	−0.643836	−	0.765164i	$-0.722658\pi$
$419$	−11044.0	−1.28767	−0.643836	+	0.765164i	$0.722658\pi$
$420$	0	0
$421$	6343.24	0.734324	0.367162	−	0.930157i	$-0.380329\pi$
$421$	6343.24	0.734324	0.367162	+	0.930157i	$0.380329\pi$
$422$	0	0
$423$	20765.2	2.38685
$424$	0	0
$425$	−182.859	−0.0208705
$426$	0	0
$427$	−4200.93	−0.476106
$428$	0	0
$429$	−1294.47	−0.145682
$430$	0	0
$431$	4117.01	0.460115	0.230058	−	0.973177i	$-0.426109\pi$
$431$	4117.01	0.460115	0.230058	+	0.973177i	$0.426109\pi$
$432$	0	0
$433$	−758.791	−0.0842151	−0.0421076	−	0.999113i	$-0.513407\pi$
$433$	−758.791	−0.0842151	−0.0421076	+	0.999113i	$0.513407\pi$
$434$	0	0
$435$	5322.83	0.586690
$436$	0	0
$437$	−150.360	−0.0164592
$438$	0	0
$439$	−4150.10	−0.451193	−0.225596	−	0.974221i	$-0.572433\pi$
$439$	−4150.10	−0.451193	−0.225596	+	0.974221i	$0.572433\pi$
$440$	0	0
$441$	11159.0	1.20494
$442$	0	0
$443$	4597.24	0.493051	0.246525	−	0.969136i	$-0.420711\pi$
$443$	4597.24	0.493051	0.246525	+	0.969136i	$0.420711\pi$
$444$	0	0
$445$	4123.51	0.439265
$446$	0	0
$447$	−5920.43	−0.626458
$448$	0	0
$449$	6402.58	0.672954	0.336477	−	0.941692i	$-0.390765\pi$
$449$	6402.58	0.672954	0.336477	+	0.941692i	$0.390765\pi$
$450$	0	0
$451$	−121.574	−0.0126933
$452$	0	0
$453$	14442.3	1.49792
$454$	0	0
$455$	−10440.8	−1.07576
$456$	0	0
$457$	551.499	0.0564508	0.0282254	−	0.999602i	$-0.491014\pi$
$457$	551.499	0.0564508	0.0282254	+	0.999602i	$0.491014\pi$
$458$	0	0
$459$	1678.75	0.170713
$460$	0	0
$461$	−3901.96	−0.394213	−0.197107	−	0.980382i	$-0.563154\pi$
$461$	−3901.96	−0.394213	−0.197107	+	0.980382i	$0.563154\pi$
$462$	0	0
$463$	−4828.15	−0.484628	−0.242314	−	0.970198i	$-0.577906\pi$
$463$	−4828.15	−0.484628	−0.242314	+	0.970198i	$0.577906\pi$
$464$	0	0
$465$	6984.12	0.696518
$466$	0	0
$467$	14136.3	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$467$	14136.3	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$468$	0	0
$469$	5815.18	0.572538
$470$	0	0
$471$	−7812.63	−0.764303
$472$	0	0
$473$	765.487	0.0744125
$474$	0	0
$475$	163.434	0.0157871
$476$	0	0
$477$	12289.8	1.17969
$478$	0	0
$479$	−15246.1	−1.45430	−0.727152	−	0.686476i	$-0.759157\pi$
$479$	−15246.1	−1.45430	−0.727152	+	0.686476i	$0.759157\pi$
$480$	0	0
$481$	25992.7	2.46396
$482$	0	0
$483$	−4836.29	−0.455608
$484$	0	0
$485$	−1919.16	−0.179680
$486$	0	0
$487$	−3876.39	−0.360690	−0.180345	−	0.983603i	$-0.557721\pi$
$487$	−3876.39	−0.360690	−0.180345	+	0.983603i	$0.557721\pi$
$488$	0	0
$489$	22609.0	2.09083
$490$	0	0
$491$	5027.14	0.462061	0.231030	−	0.972947i	$-0.425790\pi$
$491$	5027.14	0.462061	0.231030	+	0.972947i	$0.425790\pi$
$492$	0	0
$493$	−872.166	−0.0796762
$494$	0	0
$495$	430.982	0.0391337
$496$	0	0
$497$	−15325.0	−1.38314
$498$	0	0
$499$	−7178.10	−0.643960	−0.321980	−	0.946747i	$-0.604348\pi$
$499$	−7178.10	−0.643960	−0.321980	+	0.946747i	$0.604348\pi$
$500$	0	0
$501$	−10103.3	−0.900962
$502$	0	0
$503$	18169.6	1.61063	0.805313	−	0.592850i	$-0.201997\pi$
$503$	18169.6	1.61063	0.805313	+	0.592850i	$0.201997\pi$
$504$	0	0
$505$	3372.90	0.297212
$506$	0	0
$507$	−50563.8	−4.42923
$508$	0	0
$509$	2296.14	0.199950	0.0999750	−	0.994990i	$-0.468124\pi$
$509$	2296.14	0.199950	0.0999750	+	0.994990i	$0.468124\pi$
$510$	0	0
$511$	−16379.4	−1.41797
$512$	0	0
$513$	−1500.42	−0.129133
$514$	0	0
$515$	846.484	0.0724283
$516$	0	0
$517$	−644.288	−0.0548081
$518$	0	0
$519$	28446.3	2.40589
$520$	0	0
$521$	12945.5	1.08859	0.544293	−	0.838895i	$-0.316798\pi$
$521$	12945.5	1.08859	0.544293	+	0.838895i	$0.316798\pi$
$522$	0	0
$523$	16093.6	1.34555	0.672775	−	0.739847i	$-0.265102\pi$
$523$	16093.6	1.34555	0.672775	+	0.739847i	$0.265102\pi$
$524$	0	0
$525$	5256.84	0.437004
$526$	0	0
$527$	−1144.37	−0.0945915
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	40408.1	3.30238
$532$	0	0
$533$	−6591.01	−0.535625
$534$	0	0
$535$	−7666.53	−0.619538
$536$	0	0
$537$	14851.4	1.19346
$538$	0	0
$539$	−346.232	−0.0276684
$540$	0	0
$541$	−15424.7	−1.22580	−0.612900	−	0.790161i	$-0.709997\pi$
$541$	−15424.7	−1.22580	−0.612900	+	0.790161i	$0.709997\pi$
$542$	0	0
$543$	33456.0	2.64408
$544$	0	0
$545$	−6198.73	−0.487201
$546$	0	0
$547$	−5307.37	−0.414857	−0.207428	−	0.978250i	$-0.566509\pi$
$547$	−5307.37	−0.414857	−0.207428	+	0.978250i	$0.566509\pi$
$548$	0	0
$549$	9401.20	0.730844
$550$	0	0
$551$	779.519	0.0602698
$552$	0	0
$553$	15561.5	1.19664
$554$	0	0
$555$	−13087.1	−1.00093
$556$	0	0
$557$	−13530.9	−1.02930	−0.514651	−	0.857400i	$-0.672079\pi$
$557$	−13530.9	−1.02930	−0.514651	+	0.857400i	$0.672079\pi$
$558$	0	0
$559$	41500.0	3.14001
$560$	0	0
$561$	−106.793	−0.00803706
$562$	0	0
$563$	18109.0	1.35560	0.677799	−	0.735248i	$-0.262934\pi$
$563$	18109.0	1.35560	0.677799	+	0.735248i	$0.262934\pi$
$564$	0	0
$565$	−11497.5	−0.856109
$566$	0	0
$567$	−14743.3	−1.09199
$568$	0	0
$569$	−908.991	−0.0669717	−0.0334858	−	0.999439i	$-0.510661\pi$
$569$	−908.991	−0.0669717	−0.0334858	+	0.999439i	$0.510661\pi$
$570$	0	0
$571$	8894.71	0.651895	0.325948	−	0.945388i	$-0.394317\pi$
$571$	8894.71	0.651895	0.325948	+	0.945388i	$0.394317\pi$
$572$	0	0
$573$	20203.6	1.47298
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−12947.3	−0.934148	−0.467074	−	0.884218i	$-0.654692\pi$
$577$	−12947.3	−0.934148	−0.467074	+	0.884218i	$0.654692\pi$
$578$	0	0
$579$	17867.0	1.28243
$580$	0	0
$581$	−31286.8	−2.23407
$582$	0	0
$583$	−381.321	−0.0270887
$584$	0	0
$585$	23365.2	1.65134
$586$	0	0
$587$	19404.1	1.36438	0.682190	−	0.731175i	$-0.261028\pi$
$587$	19404.1	1.36438	0.682190	+	0.731175i	$0.261028\pi$
$588$	0	0
$589$	1022.81	0.0715522
$590$	0	0
$591$	42890.4	2.98524
$592$	0	0
$593$	−13633.9	−0.944141	−0.472070	−	0.881561i	$-0.656493\pi$
$593$	−13633.9	−0.944141	−0.472070	+	0.881561i	$0.656493\pi$
$594$	0	0
$595$	−861.352	−0.0593479
$596$	0	0
$597$	34685.2	2.37784
$598$	0	0
$599$	−11276.5	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$599$	−11276.5	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$600$	0	0
$601$	−380.766	−0.0258432	−0.0129216	−	0.999917i	$-0.504113\pi$
$601$	−380.766	−0.0258432	−0.0129216	+	0.999917i	$0.504113\pi$
$602$	0	0
$603$	−13013.7	−0.878870
$604$	0	0
$605$	6641.63	0.446315
$606$	0	0
$607$	21669.2	1.44897	0.724485	−	0.689290i	$-0.242077\pi$
$607$	21669.2	1.44897	0.724485	+	0.689290i	$0.242077\pi$
$608$	0	0
$609$	25073.1	1.66833
$610$	0	0
$611$	−34929.4	−2.31275
$612$	0	0
$613$	−936.980	−0.0617362	−0.0308681	−	0.999523i	$-0.509827\pi$
$613$	−936.980	−0.0617362	−0.0308681	+	0.999523i	$0.509827\pi$
$614$	0	0
$615$	3318.52	0.217586
$616$	0	0
$617$	−17539.2	−1.14441	−0.572205	−	0.820111i	$-0.693912\pi$
$617$	−17539.2	−1.14441	−0.572205	+	0.820111i	$0.693912\pi$
$618$	0	0
$619$	4740.20	0.307794	0.153897	−	0.988087i	$-0.450818\pi$
$619$	4740.20	0.307794	0.153897	+	0.988087i	$0.450818\pi$
$620$	0	0
$621$	5278.83	0.341114
$622$	0	0
$623$	19423.7	1.24911
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	95.4486	0.00607950
$628$	0	0
$629$	2144.37	0.135933
$630$	0	0
$631$	18505.6	1.16751	0.583754	−	0.811930i	$-0.301583\pi$
$631$	18505.6	1.16751	0.583754	+	0.811930i	$0.301583\pi$
$632$	0	0
$633$	40786.1	2.56098
$634$	0	0
$635$	−5889.09	−0.368033
$636$	0	0
$637$	−18770.6	−1.16753
$638$	0	0
$639$	34295.6	2.12318
$640$	0	0
$641$	10163.6	0.626266	0.313133	−	0.949709i	$-0.398621\pi$
$641$	10163.6	0.626266	0.313133	+	0.949709i	$0.398621\pi$
$642$	0	0
$643$	−7132.33	−0.437436	−0.218718	−	0.975788i	$-0.570188\pi$
$643$	−7132.33	−0.437436	−0.218718	+	0.975788i	$0.570188\pi$
$644$	0	0
$645$	−20894.9	−1.27556
$646$	0	0
$647$	−14536.9	−0.883315	−0.441657	−	0.897184i	$-0.645609\pi$
$647$	−14536.9	−0.883315	−0.441657	+	0.897184i	$0.645609\pi$
$648$	0	0
$649$	−1253.75	−0.0758307
$650$	0	0
$651$	32898.5	1.98064
$652$	0	0
$653$	3318.33	0.198861	0.0994305	−	0.995045i	$-0.468298\pi$
$653$	3318.33	0.198861	0.0994305	+	0.995045i	$0.468298\pi$
$654$	0	0
$655$	11927.8	0.711539
$656$	0	0
$657$	36655.1	2.17664
$658$	0	0
$659$	−3088.17	−0.182546	−0.0912730	−	0.995826i	$-0.529094\pi$
$659$	−3088.17	−0.182546	−0.0912730	+	0.995826i	$0.529094\pi$
$660$	0	0
$661$	27602.0	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$661$	27602.0	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$662$	0	0
$663$	−5789.65	−0.339142
$664$	0	0
$665$	769.854	0.0448927
$666$	0	0
$667$	−2742.53	−0.159207
$668$	0	0
$669$	27507.5	1.58969
$670$	0	0
$671$	−291.694	−0.0167820
$672$	0	0
$673$	8091.33	0.463444	0.231722	−	0.972782i	$-0.425564\pi$
$673$	8091.33	0.463444	0.231722	+	0.972782i	$0.425564\pi$
$674$	0	0
$675$	−5737.85	−0.327185
$676$	0	0
$677$	−9515.37	−0.540185	−0.270093	−	0.962834i	$-0.587054\pi$
$677$	−9515.37	−0.540185	−0.270093	+	0.962834i	$0.587054\pi$
$678$	0	0
$679$	−9040.18	−0.510943
$680$	0	0
$681$	6061.79	0.341099
$682$	0	0
$683$	630.346	0.0353141	0.0176570	−	0.999844i	$-0.494379\pi$
$683$	630.346	0.0353141	0.0176570	+	0.999844i	$0.494379\pi$
$684$	0	0
$685$	−5616.09	−0.313255
$686$	0	0
$687$	−57132.1	−3.17282
$688$	0	0
$689$	−20672.9	−1.14307
$690$	0	0
$691$	6515.92	0.358723	0.179361	−	0.983783i	$-0.442597\pi$
$691$	6515.92	0.358723	0.179361	+	0.983783i	$0.442597\pi$
$692$	0	0
$693$	2030.13	0.111282
$694$	0	0
$695$	−9008.76	−0.491686
$696$	0	0
$697$	−543.751	−0.0295496
$698$	0	0
$699$	−54311.1	−2.93882
$700$	0	0
$701$	−32850.2	−1.76995	−0.884975	−	0.465638i	$-0.845825\pi$
$701$	−32850.2	−1.76995	−0.884975	+	0.465638i	$0.845825\pi$
$702$	0	0
$703$	−1916.58	−0.102824
$704$	0	0
$705$	17586.7	0.939507
$706$	0	0
$707$	15888.0	0.845160
$708$	0	0
$709$	−21841.2	−1.15693	−0.578466	−	0.815707i	$-0.696348\pi$
$709$	−21841.2	−1.15693	−0.578466	+	0.815707i	$0.696348\pi$
$710$	0	0
$711$	−34824.7	−1.83689
$712$	0	0
$713$	−3598.49	−0.189010
$714$	0	0
$715$	−724.959	−0.0379188
$716$	0	0
$717$	3913.22	0.203824
$718$	0	0
$719$	−25807.3	−1.33859	−0.669296	−	0.742996i	$-0.733404\pi$
$719$	−25807.3	−1.33859	−0.669296	+	0.742996i	$0.733404\pi$
$720$	0	0
$721$	3987.34	0.205959
$722$	0	0
$723$	−25483.9	−1.31087
$724$	0	0
$725$	2981.01	0.152706
$726$	0	0
$727$	−14264.0	−0.727679	−0.363839	−	0.931462i	$-0.618534\pi$
$727$	−14264.0	−0.727679	−0.363839	+	0.931462i	$0.618534\pi$
$728$	0	0
$729$	−22331.4	−1.13455
$730$	0	0
$731$	3423.71	0.173229
$732$	0	0
$733$	811.706	0.0409018	0.0204509	−	0.999791i	$-0.493490\pi$
$733$	811.706	0.0409018	0.0204509	+	0.999791i	$0.493490\pi$
$734$	0	0
$735$	9450.86	0.474286
$736$	0	0
$737$	403.780	0.0201810
$738$	0	0
$739$	30696.9	1.52802	0.764008	−	0.645207i	$-0.223229\pi$
$739$	30696.9	1.52802	0.764008	+	0.645207i	$0.223229\pi$
$740$	0	0
$741$	5174.64	0.256539
$742$	0	0
$743$	26935.6	1.32998	0.664988	−	0.746854i	$-0.268437\pi$
$743$	26935.6	1.32998	0.664988	+	0.746854i	$0.268437\pi$
$744$	0	0
$745$	−3315.69	−0.163057
$746$	0	0
$747$	70016.2	3.42939
$748$	0	0
$749$	−36113.0	−1.76174
$750$	0	0
$751$	−22371.2	−1.08700	−0.543501	−	0.839409i	$-0.682902\pi$
$751$	−22371.2	−1.08700	−0.543501	+	0.839409i	$0.682902\pi$
$752$	0	0
$753$	−20471.0	−0.990712
$754$	0	0
$755$	8088.28	0.389884
$756$	0	0
$757$	697.182	0.0334736	0.0167368	−	0.999860i	$-0.494672\pi$
$757$	697.182	0.0334736	0.0167368	+	0.999860i	$0.494672\pi$
$758$	0	0
$759$	−335.810	−0.0160595
$760$	0	0
$761$	39756.9	1.89380	0.946902	−	0.321522i	$-0.104194\pi$
$761$	39756.9	1.89380	0.946902	+	0.321522i	$0.104194\pi$
$762$	0	0
$763$	−29199.0	−1.38542
$764$	0	0
$765$	1927.61	0.0911016
$766$	0	0
$767$	−67970.9	−3.19985
$768$	0	0
$769$	−12574.8	−0.589673	−0.294836	−	0.955548i	$-0.595265\pi$
$769$	−12574.8	−0.589673	−0.294836	+	0.955548i	$0.595265\pi$
$770$	0	0
$771$	−31672.5	−1.47945
$772$	0	0
$773$	2478.65	0.115331	0.0576654	−	0.998336i	$-0.481634\pi$
$773$	2478.65	0.115331	0.0576654	+	0.998336i	$0.481634\pi$
$774$	0	0
$775$	3911.40	0.181292
$776$	0	0
$777$	−61646.5	−2.84628
$778$	0	0
$779$	485.991	0.0223523
$780$	0	0
$781$	−1064.10	−0.0487535
$782$	0	0
$783$	−27367.3	−1.24908
$784$	0	0
$785$	−4375.40	−0.198936
$786$	0	0
$787$	7716.40	0.349504	0.174752	−	0.984612i	$-0.444088\pi$
$787$	7716.40	0.349504	0.174752	+	0.984612i	$0.444088\pi$
$788$	0	0
$789$	−20340.9	−0.917812
$790$	0	0
$791$	−54158.5	−2.43446
$792$	0	0
$793$	−15813.9	−0.708154
$794$	0	0
$795$	10408.6	0.464347
$796$	0	0
$797$	−26798.1	−1.19101	−0.595507	−	0.803350i	$-0.703049\pi$
$797$	−26798.1	−1.19101	−0.595507	+	0.803350i	$0.703049\pi$
$798$	0	0
$799$	−2881.64	−0.127591
$800$	0	0
$801$	−43468.0	−1.91743
$802$	0	0
$803$	−1137.31	−0.0499810
$804$	0	0
$805$	−2708.52	−0.118588
$806$	0	0
$807$	−34956.0	−1.52479
$808$	0	0
$809$	45278.0	1.96772	0.983862	−	0.178927i	$-0.0572625\pi$
$809$	45278.0	1.96772	0.983862	+	0.178927i	$0.0572625\pi$
$810$	0	0
$811$	24889.6	1.07767	0.538836	−	0.842411i	$-0.318864\pi$
$811$	24889.6	1.07767	0.538836	+	0.842411i	$0.318864\pi$
$812$	0	0
$813$	55823.6	2.40814
$814$	0	0
$815$	12662.0	0.544208
$816$	0	0
$817$	−3060.03	−0.131036
$818$	0	0
$819$	110061.	4.69579
$820$	0	0
$821$	19274.0	0.819326	0.409663	−	0.912237i	$-0.365646\pi$
$821$	19274.0	0.819326	0.409663	+	0.912237i	$0.365646\pi$
$822$	0	0
$823$	21620.8	0.915740	0.457870	−	0.889019i	$-0.348613\pi$
$823$	21620.8	0.915740	0.457870	+	0.889019i	$0.348613\pi$
$824$	0	0
$825$	365.011	0.0154037
$826$	0	0
$827$	9012.81	0.378967	0.189484	−	0.981884i	$-0.439319\pi$
$827$	9012.81	0.378967	0.189484	+	0.981884i	$0.439319\pi$
$828$	0	0
$829$	26574.7	1.11336	0.556682	−	0.830726i	$-0.312074\pi$
$829$	26574.7	1.11336	0.556682	+	0.830726i	$0.312074\pi$
$830$	0	0
$831$	41965.9	1.75184
$832$	0	0
$833$	−1548.56	−0.0644110
$834$	0	0
$835$	−5658.27	−0.234506
$836$	0	0
$837$	−35908.9	−1.48291
$838$	0	0
$839$	1261.30	0.0519009	0.0259504	−	0.999663i	$-0.491739\pi$
$839$	1261.30	0.0519009	0.0259504	+	0.999663i	$0.491739\pi$
$840$	0	0
$841$	−10170.7	−0.417022
$842$	0	0
$843$	−49175.6	−2.00913
$844$	0	0
$845$	−28317.8	−1.15286
$846$	0	0
$847$	31285.2	1.26915
$848$	0	0
$849$	59274.1	2.39609
$850$	0	0
$851$	6742.98	0.271617
$852$	0	0
$853$	32765.1	1.31519	0.657594	−	0.753373i	$-0.271574\pi$
$853$	32765.1	1.31519	0.657594	+	0.753373i	$0.271574\pi$
$854$	0	0
$855$	−1722.84	−0.0689123
$856$	0	0
$857$	722.029	0.0287795	0.0143898	−	0.999896i	$-0.495419\pi$
$857$	722.029	0.0287795	0.0143898	+	0.999896i	$0.495419\pi$
$858$	0	0
$859$	36626.9	1.45482	0.727411	−	0.686202i	$-0.240723\pi$
$859$	36626.9	1.45482	0.727411	+	0.686202i	$0.240723\pi$
$860$	0	0
$861$	15631.8	0.618735
$862$	0	0
$863$	33207.9	1.30986	0.654931	−	0.755688i	$-0.272698\pi$
$863$	33207.9	1.30986	0.654931	+	0.755688i	$0.272698\pi$
$864$	0	0
$865$	15931.1	0.626214
$866$	0	0
$867$	43385.2	1.69947
$868$	0	0
$869$	1080.52	0.0421795
$870$	0	0
$871$	21890.5	0.851585
$872$	0	0
$873$	20230.9	0.784320
$874$	0	0
$875$	2944.05	0.113745
$876$	0	0
$877$	−39507.6	−1.52118	−0.760592	−	0.649230i	$-0.775091\pi$
$877$	−39507.6	−1.52118	−0.760592	+	0.649230i	$0.775091\pi$
$878$	0	0
$879$	18934.8	0.726569
$880$	0	0
$881$	47928.6	1.83287	0.916434	−	0.400187i	$-0.131055\pi$
$881$	47928.6	1.83287	0.916434	+	0.400187i	$0.131055\pi$
$882$	0	0
$883$	−48176.2	−1.83608	−0.918040	−	0.396488i	$-0.870229\pi$
$883$	−48176.2	−1.83608	−0.918040	+	0.396488i	$0.870229\pi$
$884$	0	0
$885$	34222.8	1.29987
$886$	0	0
$887$	−31421.7	−1.18944	−0.594722	−	0.803932i	$-0.702738\pi$
$887$	−31421.7	−1.18944	−0.594722	+	0.803932i	$0.702738\pi$
$888$	0	0
$889$	−27740.4	−1.04655
$890$	0	0
$891$	−1023.71	−0.0384910
$892$	0	0
$893$	2575.54	0.0965141
$894$	0	0
$895$	8317.41	0.310637
$896$	0	0
$897$	−18205.6	−0.677666
$898$	0	0
$899$	18655.9	0.692111
$900$	0	0
$901$	−1705.49	−0.0630613
$902$	0	0
$903$	−98425.1	−3.62722
$904$	0	0
$905$	18736.7	0.688211
$906$	0	0
$907$	−27694.1	−1.01385	−0.506927	−	0.861989i	$-0.669219\pi$
$907$	−27694.1	−1.01385	−0.506927	+	0.861989i	$0.669219\pi$
$908$	0	0
$909$	−35555.4	−1.29736
$910$	0	0
$911$	27325.1	0.993764	0.496882	−	0.867818i	$-0.334478\pi$
$911$	27325.1	0.993764	0.496882	+	0.867818i	$0.334478\pi$
$912$	0	0
$913$	−2172.41	−0.0787473
$914$	0	0
$915$	7962.15	0.287673
$916$	0	0
$917$	56185.7	2.02335
$918$	0	0
$919$	−28135.6	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$919$	−28135.6	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$920$	0	0
$921$	66270.7	2.37100
$922$	0	0
$923$	−57689.1	−2.05727
$924$	0	0
$925$	−7329.32	−0.260526
$926$	0	0
$927$	−8923.21	−0.316156
$928$	0	0
$929$	32497.6	1.14770	0.573850	−	0.818961i	$-0.305449\pi$
$929$	32497.6	1.14770	0.573850	+	0.818961i	$0.305449\pi$
$930$	0	0
$931$	1384.06	0.0487226
$932$	0	0
$933$	−22890.9	−0.803230
$934$	0	0
$935$	−59.8084	−0.00209192
$936$	0	0
$937$	−12209.3	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$937$	−12209.3	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$938$	0	0
$939$	−83953.8	−2.91771
$940$	0	0
$941$	−1317.78	−0.0456518	−0.0228259	−	0.999739i	$-0.507266\pi$
$941$	−1317.78	−0.0456518	−0.0228259	+	0.999739i	$0.507266\pi$
$942$	0	0
$943$	−1709.83	−0.0590453
$944$	0	0
$945$	−27028.0	−0.930394
$946$	0	0
$947$	11954.8	0.410219	0.205110	−	0.978739i	$-0.434245\pi$
$947$	11954.8	0.410219	0.205110	+	0.978739i	$0.434245\pi$
$948$	0	0
$949$	−61657.9	−2.10906
$950$	0	0
$951$	−3634.64	−0.123934
$952$	0	0
$953$	−19658.8	−0.668216	−0.334108	−	0.942535i	$-0.608435\pi$
$953$	−19658.8	−0.668216	−0.334108	+	0.942535i	$0.608435\pi$
$954$	0	0
$955$	11314.8	0.383392
$956$	0	0
$957$	1740.96	0.0588059
$958$	0	0
$959$	−26454.5	−0.890782
$960$	0	0
$961$	−5312.52	−0.178326
$962$	0	0
$963$	80816.8	2.70434
$964$	0	0
$965$	10006.3	0.333796
$966$	0	0
$967$	7604.58	0.252892	0.126446	−	0.991973i	$-0.459643\pi$
$967$	7604.58	0.252892	0.126446	+	0.991973i	$0.459643\pi$
$968$	0	0
$969$	426.903	0.0141528
$970$	0	0
$971$	14731.2	0.486866	0.243433	−	0.969918i	$-0.421726\pi$
$971$	14731.2	0.486866	0.243433	+	0.969918i	$0.421726\pi$
$972$	0	0
$973$	−42435.5	−1.39817
$974$	0	0
$975$	19788.7	0.649995
$976$	0	0
$977$	−50584.6	−1.65644	−0.828222	−	0.560400i	$-0.810647\pi$
$977$	−50584.6	−1.65644	−0.828222	+	0.560400i	$0.810647\pi$
$978$	0	0
$979$	1348.69	0.0440290
$980$	0	0
$981$	65343.9	2.12668
$982$	0	0
$983$	−15625.9	−0.507007	−0.253503	−	0.967334i	$-0.581583\pi$
$983$	−15625.9	−0.507007	−0.253503	+	0.967334i	$0.581583\pi$
$984$	0	0
$985$	24020.4	0.777009
$986$	0	0
$987$	82841.6	2.67161
$988$	0	0
$989$	10765.9	0.346142
$990$	0	0
$991$	−18267.8	−0.585567	−0.292783	−	0.956179i	$-0.594581\pi$
$991$	−18267.8	−0.585567	−0.292783	+	0.956179i	$0.594581\pi$
$992$	0	0
$993$	−63587.5	−2.03211
$994$	0	0
$995$	19425.2	0.618914
$996$	0	0
$997$	21580.5	0.685519	0.342760	−	0.939423i	$-0.388638\pi$
$997$	21580.5	0.685519	0.342760	+	0.939423i	$0.388638\pi$
$998$	0	0
$999$	67287.3	2.13101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.l.1.1		4
4.3	odd	2		230.4.a.i.1.4	✓	4
12.11	even	2		2070.4.a.bi.1.3		4
20.3	even	4		1150.4.b.m.599.4		8
20.7	even	4		1150.4.b.m.599.5		8
20.19	odd	2		1150.4.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.4	✓	4	4.3	odd	2
1150.4.a.o.1.1		4	20.19	odd	2
1150.4.b.m.599.4		8	20.3	even	4
1150.4.b.m.599.5		8	20.7	even	4
1840.4.a.l.1.1		4	1.1	even	1	trivial
2070.4.a.bi.1.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-8.92791 q^{3} -5.00000 q^{5} -23.5524 q^{7} +52.7075 q^{9} +O(q^{10})\)	\(q-8.92791 q^{3} -5.00000 q^{5} -23.5524 q^{7} +52.7075 q^{9} -1.63537 q^{11} -88.6599 q^{13} +44.6395 q^{15} -7.31435 q^{17} +6.53738 q^{19} +210.274 q^{21} -23.0000 q^{23} +25.0000 q^{25} -229.514 q^{27} +119.240 q^{29} +156.456 q^{31} +14.6004 q^{33} +117.762 q^{35} -293.173 q^{37} +791.547 q^{39} +74.3404 q^{41} -468.081 q^{43} -263.538 q^{45} +393.971 q^{47} +211.715 q^{49} +65.3018 q^{51} +233.171 q^{53} +8.17685 q^{55} -58.3651 q^{57} +766.648 q^{59} +178.365 q^{61} -1241.39 q^{63} +443.299 q^{65} -246.904 q^{67} +205.342 q^{69} +650.678 q^{71} +695.444 q^{73} -223.198 q^{75} +38.5169 q^{77} -660.717 q^{79} +625.978 q^{81} +1328.39 q^{83} +36.5718 q^{85} -1064.57 q^{87} -824.702 q^{89} +2088.15 q^{91} -1396.82 q^{93} -32.6869 q^{95} +383.833 q^{97} -86.1963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9	\( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 - 93 * q^11 + 32 * q^13 + 20 * q^15 + 108 * q^17 - 185 * q^19 + 302 * q^21 - 92 * q^23 + 100 * q^25 - 259 * q^27 + 294 * q^29 + 211 * q^31 - 237 * q^33 + 130 * q^35 + 5 * q^37 + 421 * q^39 - 369 * q^41 + 100 * q^43 - 320 * q^45 + 363 * q^47 + 600 * q^49 + 315 * q^51 + 21 * q^53 + 465 * q^55 - 160 * q^57 + 33 * q^59 - 307 * q^61 - 167 * q^63 - 160 * q^65 - 725 * q^67 + 92 * q^69 + 1257 * q^71 + 509 * q^73 - 100 * q^75 - 1962 * q^77 - 1202 * q^79 - 992 * q^81 + 1377 * q^83 - 540 * q^85 + 2829 * q^87 - 984 * q^89 + 995 * q^91 - 1843 * q^93 + 925 * q^95 + 137 * q^97 - 2013 * q^99

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