LMFDB - Embedded newform 1840.4.a.k.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} - 2x^{3} - 60x^{2} - 45x + 108 $ x^4 - 2x^3 - 60x^2 - 45*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.12571$ 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)

$f(q)$	$=$	$q-10.1257 q^{3} +5.00000 q^{5} +24.6431 q^{7} +75.5301 q^{9} +O(q^{10})$	$q-10.1257 q^{3} +5.00000 q^{5} +24.6431 q^{7} +75.5301 q^{9} +17.3867 q^{11} +4.00699 q^{13} -50.6286 q^{15} +48.2740 q^{17} -79.3172 q^{19} -249.529 q^{21} +23.0000 q^{23} +25.0000 q^{25} -491.402 q^{27} -254.267 q^{29} +220.696 q^{31} -176.053 q^{33} +123.215 q^{35} -422.904 q^{37} -40.5736 q^{39} -170.251 q^{41} +228.920 q^{43} +377.650 q^{45} -580.087 q^{47} +264.282 q^{49} -488.809 q^{51} -260.354 q^{53} +86.9335 q^{55} +803.144 q^{57} -353.130 q^{59} -80.6108 q^{61} +1861.29 q^{63} +20.0349 q^{65} +820.011 q^{67} -232.891 q^{69} -614.845 q^{71} +511.586 q^{73} -253.143 q^{75} +428.462 q^{77} -160.464 q^{79} +2936.48 q^{81} +32.5646 q^{83} +241.370 q^{85} +2574.64 q^{87} -25.0375 q^{89} +98.7445 q^{91} -2234.70 q^{93} -396.586 q^{95} -249.798 q^{97} +1313.22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9	$ 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) $ 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9 - 21 * q^11 + 70 * q^13 - 70 * q^15 + 56 * q^17 - 173 * q^19 - 120 * q^21 + 92 * q^23 + 100 * q^25 - 389 * q^27 - 118 * q^29 - 17 * q^31 - 89 * q^33 - 40 * q^35 - 343 * q^37 + 221 * q^39 + 139 * q^41 + 50 * q^43 + 320 * q^45 - 367 * q^47 - 124 * q^49 - 439 * q^51 - 353 * q^53 - 105 * q^55 - 238 * q^57 + 453 * q^59 - 327 * q^61 + 1723 * q^63 + 350 * q^65 + 455 * q^67 - 322 * q^69 - 195 * q^71 - 633 * q^73 - 350 * q^75 - 2 * q^77 + 1140 * q^79 + 1456 * q^81 + 1199 * q^83 + 280 * q^85 + 1775 * q^87 - 2170 * q^89 - 557 * q^91 - 3241 * q^93 - 865 * q^95 - 703 * q^97 + 2745 * 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
$3$	−10.1257	−1.94869	−0.974347	−	0.225050i	$-0.927745\pi$
$3$	−10.1257	−1.94869	−0.974347	+	0.225050i	$0.927745\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	24.6431	1.33060	0.665301	−	0.746575i	$-0.268303\pi$
$7$	24.6431	1.33060	0.665301	+	0.746575i	$0.268303\pi$
$8$	0	0
$9$	75.5301	2.79741
$10$	0	0
$11$	17.3867	0.476572	0.238286	−	0.971195i	$-0.423414\pi$
$11$	17.3867	0.476572	0.238286	+	0.971195i	$0.423414\pi$
$12$	0	0
$13$	4.00699	0.0854876	0.0427438	−	0.999086i	$-0.486390\pi$
$13$	4.00699	0.0854876	0.0427438	+	0.999086i	$0.486390\pi$
$14$	0	0
$15$	−50.6286	−0.871483
$16$	0	0
$17$	48.2740	0.688716	0.344358	−	0.938838i	$-0.388097\pi$
$17$	48.2740	0.688716	0.344358	+	0.938838i	$0.388097\pi$
$18$	0	0
$19$	−79.3172	−0.957717	−0.478859	−	0.877892i	$-0.658949\pi$
$19$	−79.3172	−0.957717	−0.478859	+	0.877892i	$0.658949\pi$
$20$	0	0
$21$	−249.529	−2.59294
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−491.402	−3.50260
$28$	0	0
$29$	−254.267	−1.62815	−0.814074	−	0.580761i	$-0.802755\pi$
$29$	−254.267	−1.62815	−0.814074	+	0.580761i	$0.802755\pi$
$30$	0	0
$31$	220.696	1.27865	0.639325	−	0.768937i	$-0.279214\pi$
$31$	220.696	1.27865	0.639325	+	0.768937i	$0.279214\pi$
$32$	0	0
$33$	−176.053	−0.928693
$34$	0	0
$35$	123.215	0.595063
$36$	0	0
$37$	−422.904	−1.87905	−0.939527	−	0.342476i	$-0.888735\pi$
$37$	−422.904	−1.87905	−0.939527	+	0.342476i	$0.888735\pi$
$38$	0	0
$39$	−40.5736	−0.166589
$40$	0	0
$41$	−170.251	−0.648505	−0.324252	−	0.945971i	$-0.605113\pi$
$41$	−170.251	−0.648505	−0.324252	+	0.945971i	$0.605113\pi$
$42$	0	0
$43$	228.920	0.811860	0.405930	−	0.913904i	$-0.366948\pi$
$43$	228.920	0.811860	0.405930	+	0.913904i	$0.366948\pi$
$44$	0	0
$45$	377.650	1.25104
$46$	0	0
$47$	−580.087	−1.80031	−0.900154	−	0.435572i	$-0.856546\pi$
$47$	−580.087	−1.80031	−0.900154	+	0.435572i	$0.856546\pi$
$48$	0	0
$49$	264.282	0.770500
$50$	0	0
$51$	−488.809	−1.34210
$52$	0	0
$53$	−260.354	−0.674762	−0.337381	−	0.941368i	$-0.609541\pi$
$53$	−260.354	−0.674762	−0.337381	+	0.941368i	$0.609541\pi$
$54$	0	0
$55$	86.9335	0.213129
$56$	0	0
$57$	803.144	1.86630
$58$	0	0
$59$	−353.130	−0.779215	−0.389607	−	0.920981i	$-0.627389\pi$
$59$	−353.130	−0.779215	−0.389607	+	0.920981i	$0.627389\pi$
$60$	0	0
$61$	−80.6108	−0.169199	−0.0845996	−	0.996415i	$-0.526961\pi$
$61$	−80.6108	−0.169199	−0.0845996	+	0.996415i	$0.526961\pi$
$62$	0	0
$63$	1861.29	3.72224
$64$	0	0
$65$	20.0349	0.0382312
$66$	0	0
$67$	820.011	1.49523	0.747614	−	0.664133i	$-0.231199\pi$
$67$	820.011	1.49523	0.747614	+	0.664133i	$0.231199\pi$
$68$	0	0
$69$	−232.891	−0.406331
$70$	0	0
$71$	−614.845	−1.02773	−0.513864	−	0.857872i	$-0.671786\pi$
$71$	−614.845	−1.02773	−0.513864	+	0.857872i	$0.671786\pi$
$72$	0	0
$73$	511.586	0.820228	0.410114	−	0.912034i	$-0.365489\pi$
$73$	511.586	0.820228	0.410114	+	0.912034i	$0.365489\pi$
$74$	0	0
$75$	−253.143	−0.389739
$76$	0	0
$77$	428.462	0.634127
$78$	0	0
$79$	−160.464	−0.228526	−0.114263	−	0.993451i	$-0.536451\pi$
$79$	−160.464	−0.228526	−0.114263	+	0.993451i	$0.536451\pi$
$80$	0	0
$81$	2936.48	4.02809
$82$	0	0
$83$	32.5646	0.0430654	0.0215327	−	0.999768i	$-0.493145\pi$
$83$	32.5646	0.0430654	0.0215327	+	0.999768i	$0.493145\pi$
$84$	0	0
$85$	241.370	0.308003
$86$	0	0
$87$	2574.64	3.17276
$88$	0	0
$89$	−25.0375	−0.0298199	−0.0149100	−	0.999889i	$-0.504746\pi$
$89$	−25.0375	−0.0298199	−0.0149100	+	0.999889i	$0.504746\pi$
$90$	0	0
$91$	98.7445	0.113750
$92$	0	0
$93$	−2234.70	−2.49170
$94$	0	0
$95$	−396.586	−0.428304
$96$	0	0
$97$	−249.798	−0.261476	−0.130738	−	0.991417i	$-0.541735\pi$
$97$	−249.798	−0.261476	−0.130738	+	0.991417i	$0.541735\pi$
$98$	0	0
$99$	1313.22	1.33317
$100$	0	0
$101$	620.493	0.611300	0.305650	−	0.952144i	$-0.401126\pi$
$101$	620.493	0.611300	0.305650	+	0.952144i	$0.401126\pi$
$102$	0	0
$103$	−1473.24	−1.40935	−0.704673	−	0.709532i	$-0.748906\pi$
$103$	−1473.24	−1.40935	−0.704673	+	0.709532i	$0.748906\pi$
$104$	0	0
$105$	−1247.64	−1.15960
$106$	0	0
$107$	940.141	0.849410	0.424705	−	0.905332i	$-0.360378\pi$
$107$	940.141	0.849410	0.424705	+	0.905332i	$0.360378\pi$
$108$	0	0
$109$	−636.264	−0.559111	−0.279555	−	0.960130i	$-0.590187\pi$
$109$	−636.264	−0.559111	−0.279555	+	0.960130i	$0.590187\pi$
$110$	0	0
$111$	4282.20	3.66170
$112$	0	0
$113$	832.451	0.693013	0.346506	−	0.938048i	$-0.387368\pi$
$113$	832.451	0.693013	0.346506	+	0.938048i	$0.387368\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	302.648	0.239144
$118$	0	0
$119$	1189.62	0.916406
$120$	0	0
$121$	−1028.70	−0.772879
$122$	0	0
$123$	1723.91	1.26374
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1614.60	−1.12813	−0.564067	−	0.825729i	$-0.690764\pi$
$127$	−1614.60	−1.12813	−0.564067	+	0.825729i	$0.690764\pi$
$128$	0	0
$129$	−2317.98	−1.58207
$130$	0	0
$131$	1974.56	1.31693	0.658464	−	0.752612i	$-0.271206\pi$
$131$	1974.56	1.31693	0.658464	+	0.752612i	$0.271206\pi$
$132$	0	0
$133$	−1954.62	−1.27434
$134$	0	0
$135$	−2457.01	−1.56641
$136$	0	0
$137$	753.805	0.470087	0.235044	−	0.971985i	$-0.424477\pi$
$137$	753.805	0.470087	0.235044	+	0.971985i	$0.424477\pi$
$138$	0	0
$139$	−1014.10	−0.618810	−0.309405	−	0.950930i	$-0.600130\pi$
$139$	−1014.10	−0.618810	−0.309405	+	0.950930i	$0.600130\pi$
$140$	0	0
$141$	5873.80	3.50825
$142$	0	0
$143$	69.6683	0.0407410
$144$	0	0
$145$	−1271.34	−0.728130
$146$	0	0
$147$	−2676.04	−1.50147
$148$	0	0
$149$	−2771.80	−1.52399	−0.761995	−	0.647583i	$-0.775780\pi$
$149$	−2771.80	−1.52399	−0.761995	+	0.647583i	$0.775780\pi$
$150$	0	0
$151$	−3108.62	−1.67534	−0.837668	−	0.546180i	$-0.816082\pi$
$151$	−3108.62	−1.67534	−0.837668	+	0.546180i	$0.816082\pi$
$152$	0	0
$153$	3646.14	1.92662
$154$	0	0
$155$	1103.48	0.571830
$156$	0	0
$157$	3712.87	1.88739	0.943693	−	0.330822i	$-0.107326\pi$
$157$	3712.87	1.88739	0.943693	+	0.330822i	$0.107326\pi$
$158$	0	0
$159$	2636.27	1.31490
$160$	0	0
$161$	566.791	0.277450
$162$	0	0
$163$	−915.791	−0.440063	−0.220032	−	0.975493i	$-0.570616\pi$
$163$	−915.791	−0.440063	−0.220032	+	0.975493i	$0.570616\pi$
$164$	0	0
$165$	−880.264	−0.415324
$166$	0	0
$167$	−1432.32	−0.663688	−0.331844	−	0.943334i	$-0.607671\pi$
$167$	−1432.32	−0.663688	−0.331844	+	0.943334i	$0.607671\pi$
$168$	0	0
$169$	−2180.94	−0.992692
$170$	0	0
$171$	−5990.84	−2.67913
$172$	0	0
$173$	−3479.54	−1.52916	−0.764580	−	0.644529i	$-0.777054\pi$
$173$	−3479.54	−1.52916	−0.764580	+	0.644529i	$0.777054\pi$
$174$	0	0
$175$	616.077	0.266120
$176$	0	0
$177$	3575.70	1.51845
$178$	0	0
$179$	−3642.71	−1.52105	−0.760527	−	0.649307i	$-0.775059\pi$
$179$	−3642.71	−1.52105	−0.760527	+	0.649307i	$0.775059\pi$
$180$	0	0
$181$	2409.15	0.989342	0.494671	−	0.869080i	$-0.335289\pi$
$181$	2409.15	0.989342	0.494671	+	0.869080i	$0.335289\pi$
$182$	0	0
$183$	816.241	0.329717
$184$	0	0
$185$	−2114.52	−0.840338
$186$	0	0
$187$	839.326	0.328223
$188$	0	0
$189$	−12109.6	−4.66057
$190$	0	0
$191$	189.608	0.0718299	0.0359150	−	0.999355i	$-0.488565\pi$
$191$	189.608	0.0718299	0.0359150	+	0.999355i	$0.488565\pi$
$192$	0	0
$193$	−1855.45	−0.692012	−0.346006	−	0.938232i	$-0.612462\pi$
$193$	−1855.45	−0.692012	−0.346006	+	0.938232i	$0.612462\pi$
$194$	0	0
$195$	−202.868	−0.0745009
$196$	0	0
$197$	2429.85	0.878779	0.439390	−	0.898297i	$-0.355195\pi$
$197$	2429.85	0.878779	0.439390	+	0.898297i	$0.355195\pi$
$198$	0	0
$199$	−4333.49	−1.54368	−0.771842	−	0.635815i	$-0.780664\pi$
$199$	−4333.49	−1.54368	−0.771842	+	0.635815i	$0.780664\pi$
$200$	0	0
$201$	−8303.20	−2.91374
$202$	0	0
$203$	−6265.94	−2.16642
$204$	0	0
$205$	−851.254	−0.290020
$206$	0	0
$207$	1737.19	0.583300
$208$	0	0
$209$	−1379.07	−0.456421
$210$	0	0
$211$	816.788	0.266493	0.133247	−	0.991083i	$-0.457460\pi$
$211$	816.788	0.266493	0.133247	+	0.991083i	$0.457460\pi$
$212$	0	0
$213$	6225.75	2.00273
$214$	0	0
$215$	1144.60	0.363075
$216$	0	0
$217$	5438.63	1.70137
$218$	0	0
$219$	−5180.18	−1.59837
$220$	0	0
$221$	193.433	0.0588767
$222$	0	0
$223$	4513.80	1.35546	0.677728	−	0.735313i	$-0.262965\pi$
$223$	4513.80	1.35546	0.677728	+	0.735313i	$0.262965\pi$
$224$	0	0
$225$	1888.25	0.559482
$226$	0	0
$227$	−2792.85	−0.816599	−0.408300	−	0.912848i	$-0.633878\pi$
$227$	−2792.85	−0.816599	−0.408300	+	0.912848i	$0.633878\pi$
$228$	0	0
$229$	1404.50	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$229$	1404.50	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$230$	0	0
$231$	−4338.48	−1.23572
$232$	0	0
$233$	1073.79	0.301916	0.150958	−	0.988540i	$-0.451764\pi$
$233$	1073.79	0.301916	0.150958	+	0.988540i	$0.451764\pi$
$234$	0	0
$235$	−2900.44	−0.805122
$236$	0	0
$237$	1624.81	0.445328
$238$	0	0
$239$	2573.18	0.696423	0.348212	−	0.937416i	$-0.386789\pi$
$239$	2573.18	0.696423	0.348212	+	0.937416i	$0.386789\pi$
$240$	0	0
$241$	−696.127	−0.186064	−0.0930321	−	0.995663i	$-0.529656\pi$
$241$	−696.127	−0.186064	−0.0930321	+	0.995663i	$0.529656\pi$
$242$	0	0
$243$	−16466.1	−4.34692
$244$	0	0
$245$	1321.41	0.344578
$246$	0	0
$247$	−317.823	−0.0818729
$248$	0	0
$249$	−329.740	−0.0839213
$250$	0	0
$251$	−6467.81	−1.62647	−0.813236	−	0.581934i	$-0.802296\pi$
$251$	−6467.81	−1.62647	−0.813236	+	0.581934i	$0.802296\pi$
$252$	0	0
$253$	399.894	0.0993721
$254$	0	0
$255$	−2444.04	−0.600204
$256$	0	0
$257$	5332.72	1.29434	0.647172	−	0.762344i	$-0.275952\pi$
$257$	5332.72	1.29434	0.647172	+	0.762344i	$0.275952\pi$
$258$	0	0
$259$	−10421.7	−2.50027
$260$	0	0
$261$	−19204.8	−4.55460
$262$	0	0
$263$	6872.85	1.61140	0.805699	−	0.592325i	$-0.201790\pi$
$263$	6872.85	1.61140	0.805699	+	0.592325i	$0.201790\pi$
$264$	0	0
$265$	−1301.77	−0.301763
$266$	0	0
$267$	253.523	0.0581099
$268$	0	0
$269$	−1926.13	−0.436572	−0.218286	−	0.975885i	$-0.570047\pi$
$269$	−1926.13	−0.436572	−0.218286	+	0.975885i	$0.570047\pi$
$270$	0	0
$271$	3653.06	0.818847	0.409423	−	0.912344i	$-0.365730\pi$
$271$	3653.06	0.818847	0.409423	+	0.912344i	$0.365730\pi$
$272$	0	0
$273$	−999.859	−0.221664
$274$	0	0
$275$	434.668	0.0953144
$276$	0	0
$277$	−1047.37	−0.227185	−0.113592	−	0.993527i	$-0.536236\pi$
$277$	−1047.37	−0.227185	−0.113592	+	0.993527i	$0.536236\pi$
$278$	0	0
$279$	16669.2	3.57691
$280$	0	0
$281$	−2758.90	−0.585701	−0.292851	−	0.956158i	$-0.594604\pi$
$281$	−2758.90	−0.585701	−0.292851	+	0.956158i	$0.594604\pi$
$282$	0	0
$283$	−3355.58	−0.704837	−0.352418	−	0.935843i	$-0.614641\pi$
$283$	−3355.58	−0.704837	−0.352418	+	0.935843i	$0.614641\pi$
$284$	0	0
$285$	4015.72	0.834634
$286$	0	0
$287$	−4195.50	−0.862902
$288$	0	0
$289$	−2582.62	−0.525670
$290$	0	0
$291$	2529.39	0.509537
$292$	0	0
$293$	419.625	0.0836681	0.0418341	−	0.999125i	$-0.486680\pi$
$293$	419.625	0.0836681	0.0418341	+	0.999125i	$0.486680\pi$
$294$	0	0
$295$	−1765.65	−0.348475
$296$	0	0
$297$	−8543.85	−1.66924
$298$	0	0
$299$	92.1607	0.0178254
$300$	0	0
$301$	5641.30	1.08026
$302$	0	0
$303$	−6282.93	−1.19124
$304$	0	0
$305$	−403.054	−0.0756682
$306$	0	0
$307$	4133.41	0.768425	0.384212	−	0.923245i	$-0.374473\pi$
$307$	4133.41	0.768425	0.384212	+	0.923245i	$0.374473\pi$
$308$	0	0
$309$	14917.6	2.74639
$310$	0	0
$311$	6991.03	1.27468	0.637339	−	0.770584i	$-0.280035\pi$
$311$	6991.03	1.27468	0.637339	+	0.770584i	$0.280035\pi$
$312$	0	0
$313$	9380.53	1.69399	0.846996	−	0.531600i	$-0.178409\pi$
$313$	9380.53	1.69399	0.846996	+	0.531600i	$0.178409\pi$
$314$	0	0
$315$	9306.47	1.66464
$316$	0	0
$317$	−7995.35	−1.41660	−0.708302	−	0.705910i	$-0.750538\pi$
$317$	−7995.35	−1.41660	−0.708302	+	0.705910i	$0.750538\pi$
$318$	0	0
$319$	−4420.87	−0.775929
$320$	0	0
$321$	−9519.60	−1.65524
$322$	0	0
$323$	−3828.96	−0.659595
$324$	0	0
$325$	100.175	0.0170975
$326$	0	0
$327$	6442.63	1.08954
$328$	0	0
$329$	−14295.1	−2.39549
$330$	0	0
$331$	4798.86	0.796886	0.398443	−	0.917193i	$-0.369551\pi$
$331$	4798.86	0.796886	0.398443	+	0.917193i	$0.369551\pi$
$332$	0	0
$333$	−31942.0	−5.25648
$334$	0	0
$335$	4100.06	0.668687
$336$	0	0
$337$	6895.51	1.11461	0.557303	−	0.830309i	$-0.311836\pi$
$337$	6895.51	1.11461	0.557303	+	0.830309i	$0.311836\pi$
$338$	0	0
$339$	−8429.16	−1.35047
$340$	0	0
$341$	3837.17	0.609368
$342$	0	0
$343$	−1939.86	−0.305372
$344$	0	0
$345$	−1164.46	−0.181717
$346$	0	0
$347$	8744.17	1.35277	0.676386	−	0.736548i	$-0.263545\pi$
$347$	8744.17	1.35277	0.676386	+	0.736548i	$0.263545\pi$
$348$	0	0
$349$	−6387.08	−0.979635	−0.489818	−	0.871825i	$-0.662937\pi$
$349$	−6387.08	−0.979635	−0.489818	+	0.871825i	$0.662937\pi$
$350$	0	0
$351$	−1969.04	−0.299429
$352$	0	0
$353$	−589.384	−0.0888661	−0.0444331	−	0.999012i	$-0.514148\pi$
$353$	−589.384	−0.0888661	−0.0444331	+	0.999012i	$0.514148\pi$
$354$	0	0
$355$	−3074.23	−0.459614
$356$	0	0
$357$	−12045.8	−1.78580
$358$	0	0
$359$	−7214.13	−1.06058	−0.530289	−	0.847817i	$-0.677917\pi$
$359$	−7214.13	−1.06058	−0.530289	+	0.847817i	$0.677917\pi$
$360$	0	0
$361$	−567.774	−0.0827780
$362$	0	0
$363$	10416.3	1.50611
$364$	0	0
$365$	2557.93	0.366817
$366$	0	0
$367$	−12356.4	−1.75750	−0.878748	−	0.477286i	$-0.841621\pi$
$367$	−12356.4	−1.75750	−0.878748	+	0.477286i	$0.841621\pi$
$368$	0	0
$369$	−12859.1	−1.81413
$370$	0	0
$371$	−6415.92	−0.897839
$372$	0	0
$373$	7145.98	0.991969	0.495985	−	0.868331i	$-0.334807\pi$
$373$	7145.98	0.991969	0.495985	+	0.868331i	$0.334807\pi$
$374$	0	0
$375$	−1265.71	−0.174297
$376$	0	0
$377$	−1018.85	−0.139186
$378$	0	0
$379$	−2170.69	−0.294197	−0.147099	−	0.989122i	$-0.546993\pi$
$379$	−2170.69	−0.294197	−0.147099	+	0.989122i	$0.546993\pi$
$380$	0	0
$381$	16349.0	2.19839
$382$	0	0
$383$	7967.21	1.06294	0.531469	−	0.847078i	$-0.321640\pi$
$383$	7967.21	1.06294	0.531469	+	0.847078i	$0.321640\pi$
$384$	0	0
$385$	2142.31	0.283590
$386$	0	0
$387$	17290.3	2.27111
$388$	0	0
$389$	−568.951	−0.0741567	−0.0370783	−	0.999312i	$-0.511805\pi$
$389$	−568.951	−0.0741567	−0.0370783	+	0.999312i	$0.511805\pi$
$390$	0	0
$391$	1110.30	0.143607
$392$	0	0
$393$	−19993.8	−2.56629
$394$	0	0
$395$	−802.319	−0.102200
$396$	0	0
$397$	8564.88	1.08277	0.541384	−	0.840775i	$-0.317900\pi$
$397$	8564.88	1.08277	0.541384	+	0.840775i	$0.317900\pi$
$398$	0	0
$399$	19791.9	2.48330
$400$	0	0
$401$	−12455.6	−1.55113	−0.775563	−	0.631270i	$-0.782534\pi$
$401$	−12455.6	−1.55113	−0.775563	+	0.631270i	$0.782534\pi$
$402$	0	0
$403$	884.325	0.109309
$404$	0	0
$405$	14682.4	1.80142
$406$	0	0
$407$	−7352.91	−0.895504
$408$	0	0
$409$	11838.9	1.43129	0.715645	−	0.698465i	$-0.246133\pi$
$409$	11838.9	1.43129	0.715645	+	0.698465i	$0.246133\pi$
$410$	0	0
$411$	−7632.82	−0.916056
$412$	0	0
$413$	−8702.22	−1.03682
$414$	0	0
$415$	162.823	0.0192594
$416$	0	0
$417$	10268.5	1.20587
$418$	0	0
$419$	1531.77	0.178596	0.0892981	−	0.996005i	$-0.471538\pi$
$419$	1531.77	0.178596	0.0892981	+	0.996005i	$0.471538\pi$
$420$	0	0
$421$	−9985.06	−1.15592	−0.577959	−	0.816066i	$-0.696151\pi$
$421$	−9985.06	−1.15592	−0.577959	+	0.816066i	$0.696151\pi$
$422$	0	0
$423$	−43814.0	−5.03620
$424$	0	0
$425$	1206.85	0.137743
$426$	0	0
$427$	−1986.50	−0.225137
$428$	0	0
$429$	−705.441	−0.0793917
$430$	0	0
$431$	13786.7	1.54079	0.770396	−	0.637566i	$-0.220059\pi$
$431$	13786.7	1.54079	0.770396	+	0.637566i	$0.220059\pi$
$432$	0	0
$433$	−2621.92	−0.290996	−0.145498	−	0.989359i	$-0.546478\pi$
$433$	−2621.92	−0.290996	−0.145498	+	0.989359i	$0.546478\pi$
$434$	0	0
$435$	12873.2	1.41890
$436$	0	0
$437$	−1824.30	−0.199698
$438$	0	0
$439$	−12062.7	−1.31143	−0.655717	−	0.755007i	$-0.727633\pi$
$439$	−12062.7	−1.31143	−0.655717	+	0.755007i	$0.727633\pi$
$440$	0	0
$441$	19961.2	2.15541
$442$	0	0
$443$	−3659.26	−0.392453	−0.196227	−	0.980559i	$-0.562869\pi$
$443$	−3659.26	−0.392453	−0.196227	+	0.980559i	$0.562869\pi$
$444$	0	0
$445$	−125.188	−0.0133359
$446$	0	0
$447$	28066.4	2.96979
$448$	0	0
$449$	−10529.6	−1.10674	−0.553368	−	0.832937i	$-0.686658\pi$
$449$	−10529.6	−1.10674	−0.553368	+	0.832937i	$0.686658\pi$
$450$	0	0
$451$	−2960.10	−0.309059
$452$	0	0
$453$	31477.0	3.26472
$454$	0	0
$455$	493.723	0.0508705
$456$	0	0
$457$	6443.23	0.659522	0.329761	−	0.944064i	$-0.393032\pi$
$457$	6443.23	0.659522	0.329761	+	0.944064i	$0.393032\pi$
$458$	0	0
$459$	−23721.9	−2.41230
$460$	0	0
$461$	−3263.86	−0.329747	−0.164873	−	0.986315i	$-0.552722\pi$
$461$	−3263.86	−0.329747	−0.164873	+	0.986315i	$0.552722\pi$
$462$	0	0
$463$	−9518.12	−0.955388	−0.477694	−	0.878526i	$-0.658527\pi$
$463$	−9518.12	−0.955388	−0.477694	+	0.878526i	$0.658527\pi$
$464$	0	0
$465$	−11173.5	−1.11432
$466$	0	0
$467$	−19092.6	−1.89187	−0.945934	−	0.324360i	$-0.894851\pi$
$467$	−19092.6	−1.89187	−0.945934	+	0.324360i	$0.894851\pi$
$468$	0	0
$469$	20207.6	1.98955
$470$	0	0
$471$	−37595.5	−3.67794
$472$	0	0
$473$	3980.17	0.386910
$474$	0	0
$475$	−1982.93	−0.191543
$476$	0	0
$477$	−19664.6	−1.88758
$478$	0	0
$479$	6324.81	0.603315	0.301658	−	0.953416i	$-0.402460\pi$
$479$	6324.81	0.603315	0.301658	+	0.953416i	$0.402460\pi$
$480$	0	0
$481$	−1694.57	−0.160636
$482$	0	0
$483$	−5739.16	−0.540664
$484$	0	0
$485$	−1248.99	−0.116936
$486$	0	0
$487$	7873.07	0.732573	0.366286	−	0.930502i	$-0.380629\pi$
$487$	7873.07	0.732573	0.366286	+	0.930502i	$0.380629\pi$
$488$	0	0
$489$	9273.04	0.857549
$490$	0	0
$491$	3556.82	0.326918	0.163459	−	0.986550i	$-0.447735\pi$
$491$	3556.82	0.326918	0.163459	+	0.986550i	$0.447735\pi$
$492$	0	0
$493$	−12274.5	−1.12133
$494$	0	0
$495$	6566.10	0.596210
$496$	0	0
$497$	−15151.7	−1.36750
$498$	0	0
$499$	1933.37	0.173446	0.0867231	−	0.996232i	$-0.472360\pi$
$499$	1933.37	0.173446	0.0867231	+	0.996232i	$0.472360\pi$
$500$	0	0
$501$	14503.2	1.29332
$502$	0	0
$503$	−2114.36	−0.187425	−0.0937123	−	0.995599i	$-0.529873\pi$
$503$	−2114.36	−0.187425	−0.0937123	+	0.995599i	$0.529873\pi$
$504$	0	0
$505$	3102.46	0.273382
$506$	0	0
$507$	22083.6	1.93445
$508$	0	0
$509$	−316.452	−0.0275570	−0.0137785	−	0.999905i	$-0.504386\pi$
$509$	−316.452	−0.0275570	−0.0137785	+	0.999905i	$0.504386\pi$
$510$	0	0
$511$	12607.1	1.09140
$512$	0	0
$513$	38976.6	3.35450
$514$	0	0
$515$	−7366.20	−0.630279
$516$	0	0
$517$	−10085.8	−0.857976
$518$	0	0
$519$	35232.8	2.97987
$520$	0	0
$521$	−309.041	−0.0259872	−0.0129936	−	0.999916i	$-0.504136\pi$
$521$	−309.041	−0.0259872	−0.0129936	+	0.999916i	$0.504136\pi$
$522$	0	0
$523$	5892.46	0.492656	0.246328	−	0.969186i	$-0.420776\pi$
$523$	5892.46	0.492656	0.246328	+	0.969186i	$0.420776\pi$
$524$	0	0
$525$	−6238.22	−0.518587
$526$	0	0
$527$	10653.9	0.880626
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−26672.0	−2.17978
$532$	0	0
$533$	−682.193	−0.0554391
$534$	0	0
$535$	4700.71	0.379868
$536$	0	0
$537$	36885.0	2.96407
$538$	0	0
$539$	4594.99	0.367199
$540$	0	0
$541$	1560.55	0.124017	0.0620085	−	0.998076i	$-0.480249\pi$
$541$	1560.55	0.124017	0.0620085	+	0.998076i	$0.480249\pi$
$542$	0	0
$543$	−24394.4	−1.92792
$544$	0	0
$545$	−3181.32	−0.250042
$546$	0	0
$547$	−17756.3	−1.38795	−0.693973	−	0.720001i	$-0.744141\pi$
$547$	−17756.3	−1.38795	−0.693973	+	0.720001i	$0.744141\pi$
$548$	0	0
$549$	−6088.54	−0.473319
$550$	0	0
$551$	20167.8	1.55930
$552$	0	0
$553$	−3954.32	−0.304078
$554$	0	0
$555$	21411.0	1.63756
$556$	0	0
$557$	1212.77	0.0922559	0.0461280	−	0.998936i	$-0.485312\pi$
$557$	1212.77	0.0922559	0.0461280	+	0.998936i	$0.485312\pi$
$558$	0	0
$559$	917.280	0.0694040
$560$	0	0
$561$	−8498.78	−0.639605
$562$	0	0
$563$	−12558.2	−0.940078	−0.470039	−	0.882646i	$-0.655760\pi$
$563$	−12558.2	−0.940078	−0.470039	+	0.882646i	$0.655760\pi$
$564$	0	0
$565$	4162.26	0.309925
$566$	0	0
$567$	72363.9	5.35978
$568$	0	0
$569$	9776.72	0.720319	0.360159	−	0.932891i	$-0.382722\pi$
$569$	9776.72	0.720319	0.360159	+	0.932891i	$0.382722\pi$
$570$	0	0
$571$	18733.9	1.37301	0.686505	−	0.727125i	$-0.259144\pi$
$571$	18733.9	1.37301	0.686505	+	0.727125i	$0.259144\pi$
$572$	0	0
$573$	−1919.91	−0.139975
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	5113.58	0.368944	0.184472	−	0.982838i	$-0.440942\pi$
$577$	5113.58	0.368944	0.184472	+	0.982838i	$0.440942\pi$
$578$	0	0
$579$	18787.8	1.34852
$580$	0	0
$581$	802.492	0.0573029
$582$	0	0
$583$	−4526.70	−0.321572
$584$	0	0
$585$	1513.24	0.106948
$586$	0	0
$587$	5379.95	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$587$	5379.95	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$588$	0	0
$589$	−17505.0	−1.22458
$590$	0	0
$591$	−24603.9	−1.71247
$592$	0	0
$593$	−15060.3	−1.04292	−0.521461	−	0.853275i	$-0.674613\pi$
$593$	−15060.3	−1.04292	−0.521461	+	0.853275i	$0.674613\pi$
$594$	0	0
$595$	5948.10	0.409829
$596$	0	0
$597$	43879.7	3.00817
$598$	0	0
$599$	3772.04	0.257298	0.128649	−	0.991690i	$-0.458936\pi$
$599$	3772.04	0.257298	0.128649	+	0.991690i	$0.458936\pi$
$600$	0	0
$601$	−14663.9	−0.995261	−0.497631	−	0.867389i	$-0.665796\pi$
$601$	−14663.9	−0.995261	−0.497631	+	0.867389i	$0.665796\pi$
$602$	0	0
$603$	61935.5	4.18277
$604$	0	0
$605$	−5143.51	−0.345642
$606$	0	0
$607$	24514.5	1.63923	0.819614	−	0.572915i	$-0.194188\pi$
$607$	24514.5	1.63923	0.819614	+	0.572915i	$0.194188\pi$
$608$	0	0
$609$	63447.1	4.22168
$610$	0	0
$611$	−2324.40	−0.153904
$612$	0	0
$613$	14451.2	0.952167	0.476084	−	0.879400i	$-0.342056\pi$
$613$	14451.2	0.952167	0.476084	+	0.879400i	$0.342056\pi$
$614$	0	0
$615$	8619.55	0.565161
$616$	0	0
$617$	3292.19	0.214811	0.107406	−	0.994215i	$-0.465746\pi$
$617$	3292.19	0.214811	0.107406	+	0.994215i	$0.465746\pi$
$618$	0	0
$619$	12595.5	0.817858	0.408929	−	0.912566i	$-0.365902\pi$
$619$	12595.5	0.817858	0.408929	+	0.912566i	$0.365902\pi$
$620$	0	0
$621$	−11302.2	−0.730343
$622$	0	0
$623$	−617.002	−0.0396785
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	13964.0	0.889425
$628$	0	0
$629$	−20415.3	−1.29413
$630$	0	0
$631$	−19889.3	−1.25480	−0.627401	−	0.778697i	$-0.715881\pi$
$631$	−19889.3	−1.25480	−0.627401	+	0.778697i	$0.715881\pi$
$632$	0	0
$633$	−8270.56	−0.519313
$634$	0	0
$635$	−8073.02	−0.504517
$636$	0	0
$637$	1058.97	0.0658682
$638$	0	0
$639$	−46439.3	−2.87498
$640$	0	0
$641$	−19276.0	−1.18776	−0.593880	−	0.804553i	$-0.702405\pi$
$641$	−19276.0	−1.18776	−0.593880	+	0.804553i	$0.702405\pi$
$642$	0	0
$643$	10219.2	0.626758	0.313379	−	0.949628i	$-0.398539\pi$
$643$	10219.2	0.626758	0.313379	+	0.949628i	$0.398539\pi$
$644$	0	0
$645$	−11589.9	−0.707522
$646$	0	0
$647$	−20818.4	−1.26500	−0.632500	−	0.774560i	$-0.717971\pi$
$647$	−20818.4	−1.26500	−0.632500	+	0.774560i	$0.717971\pi$
$648$	0	0
$649$	−6139.77	−0.371352
$650$	0	0
$651$	−55070.0	−3.31546
$652$	0	0
$653$	−15135.3	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$653$	−15135.3	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$654$	0	0
$655$	9872.78	0.588949
$656$	0	0
$657$	38640.2	2.29451
$658$	0	0
$659$	13207.9	0.780737	0.390369	−	0.920659i	$-0.372348\pi$
$659$	13207.9	0.780737	0.390369	+	0.920659i	$0.372348\pi$
$660$	0	0
$661$	6671.34	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$661$	6671.34	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$662$	0	0
$663$	−1958.65	−0.114733
$664$	0	0
$665$	−9773.11	−0.569902
$666$	0	0
$667$	−5848.15	−0.339492
$668$	0	0
$669$	−45705.5	−2.64137
$670$	0	0
$671$	−1401.56	−0.0806355
$672$	0	0
$673$	−9534.41	−0.546099	−0.273049	−	0.962000i	$-0.588032\pi$
$673$	−9534.41	−0.546099	−0.273049	+	0.962000i	$0.588032\pi$
$674$	0	0
$675$	−12285.0	−0.700520
$676$	0	0
$677$	17748.4	1.00757	0.503787	−	0.863828i	$-0.331940\pi$
$677$	17748.4	1.00757	0.503787	+	0.863828i	$0.331940\pi$
$678$	0	0
$679$	−6155.80	−0.347920
$680$	0	0
$681$	28279.6	1.59130
$682$	0	0
$683$	−9583.57	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$683$	−9583.57	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$684$	0	0
$685$	3769.03	0.210229
$686$	0	0
$687$	−14221.5	−0.789790
$688$	0	0
$689$	−1043.24	−0.0576837
$690$	0	0
$691$	−9410.74	−0.518092	−0.259046	−	0.965865i	$-0.583408\pi$
$691$	−9410.74	−0.518092	−0.259046	+	0.965865i	$0.583408\pi$
$692$	0	0
$693$	32361.8	1.77391
$694$	0	0
$695$	−5070.48	−0.276740
$696$	0	0
$697$	−8218.69	−0.446636
$698$	0	0
$699$	−10872.9	−0.588343
$700$	0	0
$701$	−19517.9	−1.05161	−0.525806	−	0.850605i	$-0.676236\pi$
$701$	−19517.9	−1.05161	−0.525806	+	0.850605i	$0.676236\pi$
$702$	0	0
$703$	33543.6	1.79960
$704$	0	0
$705$	29369.0	1.56894
$706$	0	0
$707$	15290.9	0.813397
$708$	0	0
$709$	−28430.7	−1.50598	−0.752990	−	0.658033i	$-0.771389\pi$
$709$	−28430.7	−1.50598	−0.752990	+	0.658033i	$0.771389\pi$
$710$	0	0
$711$	−12119.8	−0.639282
$712$	0	0
$713$	5076.00	0.266617
$714$	0	0
$715$	348.342	0.0182199
$716$	0	0
$717$	−26055.3	−1.35712
$718$	0	0
$719$	−18716.0	−0.970779	−0.485390	−	0.874298i	$-0.661322\pi$
$719$	−18716.0	−0.970779	−0.485390	+	0.874298i	$0.661322\pi$
$720$	0	0
$721$	−36305.2	−1.87528
$722$	0	0
$723$	7048.78	0.362582
$724$	0	0
$725$	−6356.69	−0.325630
$726$	0	0
$727$	−18419.5	−0.939670	−0.469835	−	0.882754i	$-0.655687\pi$
$727$	−18419.5	−0.939670	−0.469835	+	0.882754i	$0.655687\pi$
$728$	0	0
$729$	87446.1	4.44272
$730$	0	0
$731$	11050.9	0.559141
$732$	0	0
$733$	−21548.4	−1.08582	−0.542912	−	0.839790i	$-0.682678\pi$
$733$	−21548.4	−1.08582	−0.542912	+	0.839790i	$0.682678\pi$
$734$	0	0
$735$	−13380.2	−0.671478
$736$	0	0
$737$	14257.3	0.712584
$738$	0	0
$739$	−12066.4	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$739$	−12066.4	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$740$	0	0
$741$	3218.19	0.159545
$742$	0	0
$743$	21951.0	1.08385	0.541926	−	0.840426i	$-0.317695\pi$
$743$	21951.0	1.08385	0.541926	+	0.840426i	$0.317695\pi$
$744$	0	0
$745$	−13859.0	−0.681549
$746$	0	0
$747$	2459.61	0.120472
$748$	0	0
$749$	23168.0	1.13023
$750$	0	0
$751$	3112.43	0.151230	0.0756152	−	0.997137i	$-0.475908\pi$
$751$	3112.43	0.151230	0.0756152	+	0.997137i	$0.475908\pi$
$752$	0	0
$753$	65491.2	3.16950
$754$	0	0
$755$	−15543.1	−0.749233
$756$	0	0
$757$	−7684.64	−0.368960	−0.184480	−	0.982836i	$-0.559060\pi$
$757$	−7684.64	−0.368960	−0.184480	+	0.982836i	$0.559060\pi$
$758$	0	0
$759$	−4049.21	−0.193646
$760$	0	0
$761$	−36484.6	−1.73793	−0.868965	−	0.494874i	$-0.835214\pi$
$761$	−36484.6	−1.73793	−0.868965	+	0.494874i	$0.835214\pi$
$762$	0	0
$763$	−15679.5	−0.743954
$764$	0	0
$765$	18230.7	0.861611
$766$	0	0
$767$	−1414.99	−0.0666132
$768$	0	0
$769$	2004.39	0.0939925	0.0469962	−	0.998895i	$-0.485035\pi$
$769$	2004.39	0.0939925	0.0469962	+	0.998895i	$0.485035\pi$
$770$	0	0
$771$	−53997.6	−2.52228
$772$	0	0
$773$	7716.17	0.359031	0.179516	−	0.983755i	$-0.442547\pi$
$773$	7716.17	0.359031	0.179516	+	0.983755i	$0.442547\pi$
$774$	0	0
$775$	5517.40	0.255730
$776$	0	0
$777$	105527.	4.87226
$778$	0	0
$779$	13503.8	0.621084
$780$	0	0
$781$	−10690.1	−0.489786
$782$	0	0
$783$	124947.	5.70275
$784$	0	0
$785$	18564.4	0.844065
$786$	0	0
$787$	57.0149	0.00258241	0.00129121	−	0.999999i	$-0.499589\pi$
$787$	57.0149	0.00258241	0.00129121	+	0.999999i	$0.499589\pi$
$788$	0	0
$789$	−69592.5	−3.14012
$790$	0	0
$791$	20514.2	0.922124
$792$	0	0
$793$	−323.006	−0.0144644
$794$	0	0
$795$	13181.3	0.588043
$796$	0	0
$797$	−15184.7	−0.674870	−0.337435	−	0.941349i	$-0.609559\pi$
$797$	−15184.7	−0.674870	−0.337435	+	0.941349i	$0.609559\pi$
$798$	0	0
$799$	−28003.2	−1.23990
$800$	0	0
$801$	−1891.09	−0.0834186
$802$	0	0
$803$	8894.80	0.390898
$804$	0	0
$805$	2833.95	0.124079
$806$	0	0
$807$	19503.4	0.850746
$808$	0	0
$809$	−30286.4	−1.31621	−0.658105	−	0.752926i	$-0.728642\pi$
$809$	−30286.4	−1.31621	−0.658105	+	0.752926i	$0.728642\pi$
$810$	0	0
$811$	43936.2	1.90235	0.951176	−	0.308650i	$-0.0998771\pi$
$811$	43936.2	1.90235	0.951176	+	0.308650i	$0.0998771\pi$
$812$	0	0
$813$	−36989.8	−1.59568
$814$	0	0
$815$	−4578.96	−0.196802
$816$	0	0
$817$	−18157.3	−0.777532
$818$	0	0
$819$	7458.18	0.318205
$820$	0	0
$821$	−5245.69	−0.222991	−0.111496	−	0.993765i	$-0.535564\pi$
$821$	−5245.69	−0.222991	−0.111496	+	0.993765i	$0.535564\pi$
$822$	0	0
$823$	−10678.0	−0.452260	−0.226130	−	0.974097i	$-0.572607\pi$
$823$	−10678.0	−0.452260	−0.226130	+	0.974097i	$0.572607\pi$
$824$	0	0
$825$	−4401.32	−0.185739
$826$	0	0
$827$	−3393.69	−0.142697	−0.0713484	−	0.997451i	$-0.522730\pi$
$827$	−3393.69	−0.142697	−0.0713484	+	0.997451i	$0.522730\pi$
$828$	0	0
$829$	9601.74	0.402270	0.201135	−	0.979564i	$-0.435537\pi$
$829$	9601.74	0.402270	0.201135	+	0.979564i	$0.435537\pi$
$830$	0	0
$831$	10605.3	0.442713
$832$	0	0
$833$	12757.9	0.530656
$834$	0	0
$835$	−7161.58	−0.296810
$836$	0	0
$837$	−108450.	−4.47860
$838$	0	0
$839$	−11992.8	−0.493489	−0.246744	−	0.969081i	$-0.579361\pi$
$839$	−11992.8	−0.493489	−0.246744	+	0.969081i	$0.579361\pi$
$840$	0	0
$841$	40263.0	1.65087
$842$	0	0
$843$	27935.8	1.14135
$844$	0	0
$845$	−10904.7	−0.443945
$846$	0	0
$847$	−25350.4	−1.02839
$848$	0	0
$849$	33977.7	1.37351
$850$	0	0
$851$	−9726.79	−0.391810
$852$	0	0
$853$	−15441.5	−0.619821	−0.309911	−	0.950766i	$-0.600299\pi$
$853$	−15441.5	−0.619821	−0.309911	+	0.950766i	$0.600299\pi$
$854$	0	0
$855$	−29954.2	−1.19814
$856$	0	0
$857$	−44572.4	−1.77662	−0.888310	−	0.459244i	$-0.848120\pi$
$857$	−44572.4	−1.77662	−0.888310	+	0.459244i	$0.848120\pi$
$858$	0	0
$859$	−2519.56	−0.100077	−0.0500386	−	0.998747i	$-0.515934\pi$
$859$	−2519.56	−0.100077	−0.0500386	+	0.998747i	$0.515934\pi$
$860$	0	0
$861$	42482.5	1.68153
$862$	0	0
$863$	−28980.1	−1.14310	−0.571548	−	0.820568i	$-0.693657\pi$
$863$	−28980.1	−1.14310	−0.571548	+	0.820568i	$0.693657\pi$
$864$	0	0
$865$	−17397.7	−0.683861
$866$	0	0
$867$	26150.9	1.02437
$868$	0	0
$869$	−2789.94	−0.108909
$870$	0	0
$871$	3285.77	0.127823
$872$	0	0
$873$	−18867.3	−0.731455
$874$	0	0
$875$	3080.39	0.119013
$876$	0	0
$877$	−7955.28	−0.306307	−0.153153	−	0.988202i	$-0.548943\pi$
$877$	−7955.28	−0.306307	−0.153153	+	0.988202i	$0.548943\pi$
$878$	0	0
$879$	−4249.00	−0.163044
$880$	0	0
$881$	29722.0	1.13662	0.568309	−	0.822815i	$-0.307598\pi$
$881$	29722.0	1.13662	0.568309	+	0.822815i	$0.307598\pi$
$882$	0	0
$883$	19379.7	0.738596	0.369298	−	0.929311i	$-0.379598\pi$
$883$	19379.7	0.738596	0.369298	+	0.929311i	$0.379598\pi$
$884$	0	0
$885$	17878.5	0.679072
$886$	0	0
$887$	−22901.7	−0.866925	−0.433463	−	0.901172i	$-0.642708\pi$
$887$	−22901.7	−0.866925	−0.433463	+	0.901172i	$0.642708\pi$
$888$	0	0
$889$	−39788.8	−1.50110
$890$	0	0
$891$	51055.7	1.91967
$892$	0	0
$893$	46010.9	1.72419
$894$	0	0
$895$	−18213.5	−0.680236
$896$	0	0
$897$	−933.193	−0.0347362
$898$	0	0
$899$	−56115.8	−2.08183
$900$	0	0
$901$	−12568.3	−0.464719
$902$	0	0
$903$	−57122.2	−2.10510
$904$	0	0
$905$	12045.8	0.442447
$906$	0	0
$907$	−42542.2	−1.55743	−0.778716	−	0.627376i	$-0.784129\pi$
$907$	−42542.2	−1.55743	−0.778716	+	0.627376i	$0.784129\pi$
$908$	0	0
$909$	46865.9	1.71006
$910$	0	0
$911$	−220.864	−0.00803243	−0.00401622	−	0.999992i	$-0.501278\pi$
$911$	−220.864	−0.00803243	−0.00401622	+	0.999992i	$0.501278\pi$
$912$	0	0
$913$	566.191	0.0205237
$914$	0	0
$915$	4081.21	0.147454
$916$	0	0
$917$	48659.1	1.75231
$918$	0	0
$919$	−27835.4	−0.999135	−0.499567	−	0.866275i	$-0.666508\pi$
$919$	−27835.4	−0.999135	−0.499567	+	0.866275i	$0.666508\pi$
$920$	0	0
$921$	−41853.8	−1.49743
$922$	0	0
$923$	−2463.68	−0.0878580
$924$	0	0
$925$	−10572.6	−0.375811
$926$	0	0
$927$	−111274.	−3.94252
$928$	0	0
$929$	−2172.71	−0.0767323	−0.0383661	−	0.999264i	$-0.512215\pi$
$929$	−2172.71	−0.0767323	−0.0383661	+	0.999264i	$0.512215\pi$
$930$	0	0
$931$	−20962.1	−0.737921
$932$	0	0
$933$	−70789.1	−2.48396
$934$	0	0
$935$	4196.63	0.146786
$936$	0	0
$937$	54906.5	1.91432	0.957160	−	0.289560i	$-0.0935090\pi$
$937$	54906.5	1.91432	0.957160	+	0.289560i	$0.0935090\pi$
$938$	0	0
$939$	−94984.6	−3.30107
$940$	0	0
$941$	5980.06	0.207167	0.103584	−	0.994621i	$-0.466969\pi$
$941$	5980.06	0.207167	0.103584	+	0.994621i	$0.466969\pi$
$942$	0	0
$943$	−3915.77	−0.135223
$944$	0	0
$945$	−60548.2	−2.08427
$946$	0	0
$947$	20268.6	0.695504	0.347752	−	0.937587i	$-0.386945\pi$
$947$	20268.6	0.695504	0.347752	+	0.937587i	$0.386945\pi$
$948$	0	0
$949$	2049.92	0.0701193
$950$	0	0
$951$	80958.6	2.76053
$952$	0	0
$953$	−21797.9	−0.740925	−0.370463	−	0.928847i	$-0.620801\pi$
$953$	−21797.9	−0.740925	−0.370463	+	0.928847i	$0.620801\pi$
$954$	0	0
$955$	948.038	0.0321233
$956$	0	0
$957$	44764.5	1.51205
$958$	0	0
$959$	18576.1	0.625499
$960$	0	0
$961$	18915.6	0.634945
$962$	0	0
$963$	71008.9	2.37615
$964$	0	0
$965$	−9277.26	−0.309477
$966$	0	0
$967$	−48950.6	−1.62786	−0.813932	−	0.580960i	$-0.802677\pi$
$967$	−48950.6	−1.62786	−0.813932	+	0.580960i	$0.802677\pi$
$968$	0	0
$969$	38771.0	1.28535
$970$	0	0
$971$	26426.2	0.873386	0.436693	−	0.899611i	$-0.356150\pi$
$971$	26426.2	0.873386	0.436693	+	0.899611i	$0.356150\pi$
$972$	0	0
$973$	−24990.5	−0.823389
$974$	0	0
$975$	−1014.34	−0.0333178
$976$	0	0
$977$	5770.09	0.188947	0.0944736	−	0.995527i	$-0.469883\pi$
$977$	5770.09	0.188947	0.0944736	+	0.995527i	$0.469883\pi$
$978$	0	0
$979$	−435.320	−0.0142113
$980$	0	0
$981$	−48057.1	−1.56406
$982$	0	0
$983$	484.532	0.0157214	0.00786071	−	0.999969i	$-0.497498\pi$
$983$	484.532	0.0157214	0.00786071	+	0.999969i	$0.497498\pi$
$984$	0	0
$985$	12149.2	0.393002
$986$	0	0
$987$	144749.	4.66808
$988$	0	0
$989$	5265.16	0.169285
$990$	0	0
$991$	26511.5	0.849813	0.424907	−	0.905237i	$-0.360307\pi$
$991$	26511.5	0.849813	0.424907	+	0.905237i	$0.360307\pi$
$992$	0	0
$993$	−48591.9	−1.55289
$994$	0	0
$995$	−21667.4	−0.690356
$996$	0	0
$997$	−4628.20	−0.147018	−0.0735088	−	0.997295i	$-0.523420\pi$
$997$	−4628.20	−0.147018	−0.0735088	+	0.997295i	$0.523420\pi$
$998$	0	0
$999$	207816.	6.58158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.k.1.1		4
4.3	odd	2		230.4.a.j.1.4	✓	4
12.11	even	2		2070.4.a.bg.1.1		4
20.3	even	4		1150.4.b.o.599.4		8
20.7	even	4		1150.4.b.o.599.5		8
20.19	odd	2		1150.4.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.4	✓	4	4.3	odd	2
1150.4.a.n.1.1		4	20.19	odd	2
1150.4.b.o.599.4		8	20.3	even	4
1150.4.b.o.599.5		8	20.7	even	4
1840.4.a.k.1.1		4	1.1	even	1	trivial
2070.4.a.bg.1.1		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)

\(f(q)\)	\(=\)	\(q-10.1257 q^{3} +5.00000 q^{5} +24.6431 q^{7} +75.5301 q^{9} +O(q^{10})\)	\(q-10.1257 q^{3} +5.00000 q^{5} +24.6431 q^{7} +75.5301 q^{9} +17.3867 q^{11} +4.00699 q^{13} -50.6286 q^{15} +48.2740 q^{17} -79.3172 q^{19} -249.529 q^{21} +23.0000 q^{23} +25.0000 q^{25} -491.402 q^{27} -254.267 q^{29} +220.696 q^{31} -176.053 q^{33} +123.215 q^{35} -422.904 q^{37} -40.5736 q^{39} -170.251 q^{41} +228.920 q^{43} +377.650 q^{45} -580.087 q^{47} +264.282 q^{49} -488.809 q^{51} -260.354 q^{53} +86.9335 q^{55} +803.144 q^{57} -353.130 q^{59} -80.6108 q^{61} +1861.29 q^{63} +20.0349 q^{65} +820.011 q^{67} -232.891 q^{69} -614.845 q^{71} +511.586 q^{73} -253.143 q^{75} +428.462 q^{77} -160.464 q^{79} +2936.48 q^{81} +32.5646 q^{83} +241.370 q^{85} +2574.64 q^{87} -25.0375 q^{89} +98.7445 q^{91} -2234.70 q^{93} -396.586 q^{95} -249.798 q^{97} +1313.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9	\( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) \) 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9 - 21 * q^11 + 70 * q^13 - 70 * q^15 + 56 * q^17 - 173 * q^19 - 120 * q^21 + 92 * q^23 + 100 * q^25 - 389 * q^27 - 118 * q^29 - 17 * q^31 - 89 * q^33 - 40 * q^35 - 343 * q^37 + 221 * q^39 + 139 * q^41 + 50 * q^43 + 320 * q^45 - 367 * q^47 - 124 * q^49 - 439 * q^51 - 353 * q^53 - 105 * q^55 - 238 * q^57 + 453 * q^59 - 327 * q^61 + 1723 * q^63 + 350 * q^65 + 455 * q^67 - 322 * q^69 - 195 * q^71 - 633 * q^73 - 350 * q^75 - 2 * q^77 + 1140 * q^79 + 1456 * q^81 + 1199 * q^83 + 280 * q^85 + 1775 * q^87 - 2170 * q^89 - 557 * q^91 - 3241 * q^93 - 865 * q^95 - 703 * q^97 + 2745 * q^99

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