LMFDB - Embedded newform 1840.4.a.m.1.2

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:	$0$
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} - 68x^{2} - 111x + 342 $ x^4 - 68x^2 - 111x + 342
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			1.2
Root			$1.58997$ 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)

$f(q)$	$=$	$q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} +O(q^{10})$	$q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} -47.9296 q^{11} +42.3717 q^{13} +2.94985 q^{15} +1.70534 q^{17} -21.4208 q^{19} -10.9190 q^{21} +23.0000 q^{23} +25.0000 q^{25} +31.6530 q^{27} +57.6332 q^{29} -295.699 q^{31} +28.2770 q^{33} -92.5387 q^{35} -7.85184 q^{37} -24.9980 q^{39} +465.929 q^{41} -182.374 q^{43} +133.260 q^{45} -449.193 q^{47} -0.463605 q^{49} -1.00610 q^{51} -368.316 q^{53} +239.648 q^{55} +12.6376 q^{57} +377.032 q^{59} +849.042 q^{61} -493.267 q^{63} -211.858 q^{65} -92.3424 q^{67} -13.5693 q^{69} +626.854 q^{71} +439.227 q^{73} -14.7492 q^{75} -887.068 q^{77} -641.707 q^{79} +700.928 q^{81} +609.932 q^{83} -8.52672 q^{85} -34.0018 q^{87} +1122.87 q^{89} +784.204 q^{91} +174.453 q^{93} +107.104 q^{95} -1428.80 q^{97} +1277.42 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9	$ 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9 + 39 * q^11 - 20 * q^13 - 20 * q^15 - 23 * q^17 - 53 * q^19 + 300 * q^21 + 92 * q^23 + 100 * q^25 - 137 * q^27 + 161 * q^29 - 388 * q^31 + 87 * q^33 - 5 * q^35 + 466 * q^37 - 1047 * q^39 + 484 * q^41 - 894 * q^43 - 160 * q^45 + 265 * q^47 + 1643 * q^49 - 1825 * q^51 + 576 * q^53 - 195 * q^55 + 178 * q^57 + 94 * q^59 + 1153 * q^61 - 60 * q^63 + 100 * q^65 + 1472 * q^67 + 92 * q^69 - 200 * q^71 + 1147 * q^73 + 100 * q^75 - 2176 * q^77 + 908 * q^79 - 1056 * q^81 + 1048 * q^83 + 115 * q^85 + 2167 * q^87 - 1784 * q^89 - 2329 * q^91 + 1483 * q^93 + 265 * q^95 - 2047 * q^97 + 2665 * 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
$3$	−0.589969	−0.113540	−0.0567698	−	0.998387i	$-0.518080\pi$
$3$	−0.589969	−0.113540	−0.0567698	+	0.998387i	$0.518080\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	18.5077	0.999324	0.499662	−	0.866220i	$-0.333458\pi$
$7$	18.5077	0.999324	0.499662	+	0.866220i	$0.333458\pi$
$8$	0	0
$9$	−26.6519	−0.987109
$10$	0	0
$11$	−47.9296	−1.31376	−0.656878	−	0.753997i	$-0.728123\pi$
$11$	−47.9296	−1.31376	−0.656878	+	0.753997i	$0.728123\pi$
$12$	0	0
$13$	42.3717	0.903984	0.451992	−	0.892022i	$-0.350714\pi$
$13$	42.3717	0.903984	0.451992	+	0.892022i	$0.350714\pi$
$14$	0	0
$15$	2.94985	0.0507765
$16$	0	0
$17$	1.70534	0.0243298	0.0121649	−	0.999926i	$-0.496128\pi$
$17$	1.70534	0.0243298	0.0121649	+	0.999926i	$0.496128\pi$
$18$	0	0
$19$	−21.4208	−0.258645	−0.129323	−	0.991603i	$-0.541280\pi$
$19$	−21.4208	−0.258645	−0.129323	+	0.991603i	$0.541280\pi$
$20$	0	0
$21$	−10.9190	−0.113463
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	31.6530	0.225616
$28$	0	0
$29$	57.6332	0.369042	0.184521	−	0.982829i	$-0.440927\pi$
$29$	57.6332	0.369042	0.184521	+	0.982829i	$0.440927\pi$
$30$	0	0
$31$	−295.699	−1.71320	−0.856599	−	0.515983i	$-0.827427\pi$
$31$	−295.699	−1.71320	−0.856599	+	0.515983i	$0.827427\pi$
$32$	0	0
$33$	28.2770	0.149163
$34$	0	0
$35$	−92.5387	−0.446911
$36$	0	0
$37$	−7.85184	−0.0348874	−0.0174437	−	0.999848i	$-0.505553\pi$
$37$	−7.85184	−0.0348874	−0.0174437	+	0.999848i	$0.505553\pi$
$38$	0	0
$39$	−24.9980	−0.102638
$40$	0	0
$41$	465.929	1.77478	0.887390	−	0.461020i	$-0.152516\pi$
$41$	465.929	1.77478	0.887390	+	0.461020i	$0.152516\pi$
$42$	0	0
$43$	−182.374	−0.646784	−0.323392	−	0.946265i	$-0.604823\pi$
$43$	−182.374	−0.646784	−0.323392	+	0.946265i	$0.604823\pi$
$44$	0	0
$45$	133.260	0.441448
$46$	0	0
$47$	−449.193	−1.39408	−0.697038	−	0.717035i	$-0.745499\pi$
$47$	−449.193	−1.39408	−0.697038	+	0.717035i	$0.745499\pi$
$48$	0	0
$49$	−0.463605	−0.00135162
$50$	0	0
$51$	−1.00610	−0.00276240
$52$	0	0
$53$	−368.316	−0.954567	−0.477283	−	0.878749i	$-0.658379\pi$
$53$	−368.316	−0.954567	−0.477283	+	0.878749i	$0.658379\pi$
$54$	0	0
$55$	239.648	0.587529
$56$	0	0
$57$	12.6376	0.0293665
$58$	0	0
$59$	377.032	0.831955	0.415977	−	0.909375i	$-0.363440\pi$
$59$	377.032	0.831955	0.415977	+	0.909375i	$0.363440\pi$
$60$	0	0
$61$	849.042	1.78211	0.891055	−	0.453896i	$-0.149966\pi$
$61$	849.042	1.78211	0.891055	+	0.453896i	$0.149966\pi$
$62$	0	0
$63$	−493.267	−0.986441
$64$	0	0
$65$	−211.858	−0.404274
$66$	0	0
$67$	−92.3424	−0.168379	−0.0841897	−	0.996450i	$-0.526830\pi$
$67$	−92.3424	−0.168379	−0.0841897	+	0.996450i	$0.526830\pi$
$68$	0	0
$69$	−13.5693	−0.0236747
$70$	0	0
$71$	626.854	1.04780	0.523901	−	0.851779i	$-0.324476\pi$
$71$	626.854	1.04780	0.523901	+	0.851779i	$0.324476\pi$
$72$	0	0
$73$	439.227	0.704214	0.352107	−	0.935960i	$-0.385465\pi$
$73$	439.227	0.704214	0.352107	+	0.935960i	$0.385465\pi$
$74$	0	0
$75$	−14.7492	−0.0227079
$76$	0	0
$77$	−887.068	−1.31287
$78$	0	0
$79$	−641.707	−0.913894	−0.456947	−	0.889494i	$-0.651057\pi$
$79$	−641.707	−0.913894	−0.456947	+	0.889494i	$0.651057\pi$
$80$	0	0
$81$	700.928	0.961492
$82$	0	0
$83$	609.932	0.806611	0.403306	−	0.915065i	$-0.367861\pi$
$83$	609.932	0.806611	0.403306	+	0.915065i	$0.367861\pi$
$84$	0	0
$85$	−8.52672	−0.0108806
$86$	0	0
$87$	−34.0018	−0.0419009
$88$	0	0
$89$	1122.87	1.33735	0.668673	−	0.743557i	$-0.266863\pi$
$89$	1122.87	1.33735	0.668673	+	0.743557i	$0.266863\pi$
$90$	0	0
$91$	784.204	0.903373
$92$	0	0
$93$	174.453	0.194516
$94$	0	0
$95$	107.104	0.115670
$96$	0	0
$97$	−1428.80	−1.49559	−0.747795	−	0.663930i	$-0.768887\pi$
$97$	−1428.80	−1.49559	−0.747795	+	0.663930i	$0.768887\pi$
$98$	0	0
$99$	1277.42	1.29682
$100$	0	0
$101$	−1512.15	−1.48975	−0.744875	−	0.667204i	$-0.767491\pi$
$101$	−1512.15	−1.48975	−0.744875	+	0.667204i	$0.767491\pi$
$102$	0	0
$103$	−957.279	−0.915762	−0.457881	−	0.889013i	$-0.651391\pi$
$103$	−957.279	−0.915762	−0.457881	+	0.889013i	$0.651391\pi$
$104$	0	0
$105$	54.5950	0.0507422
$106$	0	0
$107$	1742.16	1.57403	0.787013	−	0.616936i	$-0.211626\pi$
$107$	1742.16	1.57403	0.787013	+	0.616936i	$0.211626\pi$
$108$	0	0
$109$	1166.77	1.02529	0.512644	−	0.858601i	$-0.328666\pi$
$109$	1166.77	1.02529	0.512644	+	0.858601i	$0.328666\pi$
$110$	0	0
$111$	4.63234	0.00396111
$112$	0	0
$113$	−393.287	−0.327410	−0.163705	−	0.986509i	$-0.552345\pi$
$113$	−393.287	−0.327410	−0.163705	+	0.986509i	$0.552345\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−1129.29	−0.892331
$118$	0	0
$119$	31.5621	0.0243134
$120$	0	0
$121$	966.245	0.725954
$122$	0	0
$123$	−274.884	−0.201508
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1067.87	0.746127	0.373063	−	0.927806i	$-0.378307\pi$
$127$	1067.87	0.746127	0.373063	+	0.927806i	$0.378307\pi$
$128$	0	0
$129$	107.595	0.0734357
$130$	0	0
$131$	175.497	0.117047	0.0585237	−	0.998286i	$-0.481361\pi$
$131$	175.497	0.117047	0.0585237	+	0.998286i	$0.481361\pi$
$132$	0	0
$133$	−396.450	−0.258471
$134$	0	0
$135$	−158.265	−0.100898
$136$	0	0
$137$	475.898	0.296779	0.148389	−	0.988929i	$-0.452591\pi$
$137$	475.898	0.296779	0.148389	+	0.988929i	$0.452591\pi$
$138$	0	0
$139$	−153.167	−0.0934638	−0.0467319	−	0.998907i	$-0.514881\pi$
$139$	−153.167	−0.0934638	−0.0467319	+	0.998907i	$0.514881\pi$
$140$	0	0
$141$	265.010	0.158283
$142$	0	0
$143$	−2030.86	−1.18761
$144$	0	0
$145$	−288.166	−0.165041
$146$	0	0
$147$	0.273513	0.000153462 0
$148$	0	0
$149$	506.906	0.278707	0.139353	−	0.990243i	$-0.455498\pi$
$149$	506.906	0.278707	0.139353	+	0.990243i	$0.455498\pi$
$150$	0	0
$151$	−2437.84	−1.31383	−0.656916	−	0.753964i	$-0.728139\pi$
$151$	−2437.84	−1.31383	−0.656916	+	0.753964i	$0.728139\pi$
$152$	0	0
$153$	−45.4507	−0.0240162
$154$	0	0
$155$	1478.50	0.766166
$156$	0	0
$157$	255.966	0.130117	0.0650584	−	0.997881i	$-0.479277\pi$
$157$	255.966	0.130117	0.0650584	+	0.997881i	$0.479277\pi$
$158$	0	0
$159$	217.295	0.108381
$160$	0	0
$161$	425.678	0.208373
$162$	0	0
$163$	−321.632	−0.154553	−0.0772767	−	0.997010i	$-0.524622\pi$
$163$	−321.632	−0.154553	−0.0772767	+	0.997010i	$0.524622\pi$
$164$	0	0
$165$	−141.385	−0.0667079
$166$	0	0
$167$	2926.22	1.35591	0.677957	−	0.735102i	$-0.262866\pi$
$167$	2926.22	1.35591	0.677957	+	0.735102i	$0.262866\pi$
$168$	0	0
$169$	−401.640	−0.182813
$170$	0	0
$171$	570.905	0.255311
$172$	0	0
$173$	1811.84	0.796254	0.398127	−	0.917330i	$-0.369660\pi$
$173$	1811.84	0.796254	0.398127	+	0.917330i	$0.369660\pi$
$174$	0	0
$175$	462.693	0.199865
$176$	0	0
$177$	−222.437	−0.0944599
$178$	0	0
$179$	−912.664	−0.381093	−0.190547	−	0.981678i	$-0.561026\pi$
$179$	−912.664	−0.381093	−0.190547	+	0.981678i	$0.561026\pi$
$180$	0	0
$181$	3670.55	1.50735	0.753673	−	0.657249i	$-0.228280\pi$
$181$	3670.55	1.50735	0.753673	+	0.657249i	$0.228280\pi$
$182$	0	0
$183$	−500.909	−0.202340
$184$	0	0
$185$	39.2592	0.0156021
$186$	0	0
$187$	−81.7364	−0.0319634
$188$	0	0
$189$	585.826	0.225463
$190$	0	0
$191$	1840.96	0.697419	0.348710	−	0.937231i	$-0.386620\pi$
$191$	1840.96	0.697419	0.348710	+	0.937231i	$0.386620\pi$
$192$	0	0
$193$	−611.817	−0.228184	−0.114092	−	0.993470i	$-0.536396\pi$
$193$	−611.817	−0.228184	−0.114092	+	0.993470i	$0.536396\pi$
$194$	0	0
$195$	124.990	0.0459011
$196$	0	0
$197$	2830.26	1.02359	0.511796	−	0.859107i	$-0.328980\pi$
$197$	2830.26	1.02359	0.511796	+	0.859107i	$0.328980\pi$
$198$	0	0
$199$	1162.74	0.414195	0.207097	−	0.978320i	$-0.433598\pi$
$199$	1162.74	0.414195	0.207097	+	0.978320i	$0.433598\pi$
$200$	0	0
$201$	54.4792	0.0191177
$202$	0	0
$203$	1066.66	0.368792
$204$	0	0
$205$	−2329.65	−0.793705
$206$	0	0
$207$	−612.995	−0.205826
$208$	0	0
$209$	1026.69	0.339797
$210$	0	0
$211$	−1399.58	−0.456641	−0.228320	−	0.973586i	$-0.573323\pi$
$211$	−1399.58	−0.456641	−0.228320	+	0.973586i	$0.573323\pi$
$212$	0	0
$213$	−369.825	−0.118967
$214$	0	0
$215$	911.869	0.289251
$216$	0	0
$217$	−5472.72	−1.71204
$218$	0	0
$219$	−259.130	−0.0799562
$220$	0	0
$221$	72.2583	0.0219938
$222$	0	0
$223$	−4257.98	−1.27863	−0.639317	−	0.768943i	$-0.720783\pi$
$223$	−4257.98	−1.27863	−0.639317	+	0.768943i	$0.720783\pi$
$224$	0	0
$225$	−666.298	−0.197422
$226$	0	0
$227$	4025.03	1.17688	0.588438	−	0.808542i	$-0.299743\pi$
$227$	4025.03	1.17688	0.588438	+	0.808542i	$0.299743\pi$
$228$	0	0
$229$	3623.04	1.04549	0.522745	−	0.852489i	$-0.324908\pi$
$229$	3623.04	1.04549	0.522745	+	0.852489i	$0.324908\pi$
$230$	0	0
$231$	523.343	0.149063
$232$	0	0
$233$	6502.81	1.82838	0.914192	−	0.405282i	$-0.132827\pi$
$233$	6502.81	1.82838	0.914192	+	0.405282i	$0.132827\pi$
$234$	0	0
$235$	2245.96	0.623449
$236$	0	0
$237$	378.587	0.103763
$238$	0	0
$239$	2690.07	0.728059	0.364030	−	0.931387i	$-0.381401\pi$
$239$	2690.07	0.728059	0.364030	+	0.931387i	$0.381401\pi$
$240$	0	0
$241$	−44.6958	−0.0119465	−0.00597326	−	0.999982i	$-0.501901\pi$
$241$	−44.6958	−0.0119465	−0.00597326	+	0.999982i	$0.501901\pi$
$242$	0	0
$243$	−1268.16	−0.334783
$244$	0	0
$245$	2.31802	0.000604461 0
$246$	0	0
$247$	−907.635	−0.233811
$248$	0	0
$249$	−359.841	−0.0915824
$250$	0	0
$251$	6801.41	1.71036	0.855181	−	0.518329i	$-0.173446\pi$
$251$	6801.41	1.71036	0.855181	+	0.518329i	$0.173446\pi$
$252$	0	0
$253$	−1102.38	−0.273937
$254$	0	0
$255$	5.03050	0.00123538
$256$	0	0
$257$	5576.00	1.35339	0.676695	−	0.736263i	$-0.263411\pi$
$257$	5576.00	1.35339	0.676695	+	0.736263i	$0.263411\pi$
$258$	0	0
$259$	−145.320	−0.0348638
$260$	0	0
$261$	−1536.04	−0.364285
$262$	0	0
$263$	5669.58	1.32928	0.664641	−	0.747163i	$-0.268585\pi$
$263$	5669.58	1.32928	0.664641	+	0.747163i	$0.268585\pi$
$264$	0	0
$265$	1841.58	0.426895
$266$	0	0
$267$	−662.458	−0.151842
$268$	0	0
$269$	6040.21	1.36906	0.684532	−	0.728983i	$-0.260006\pi$
$269$	6040.21	1.36906	0.684532	+	0.728983i	$0.260006\pi$
$270$	0	0
$271$	6899.26	1.54650	0.773248	−	0.634104i	$-0.218631\pi$
$271$	6899.26	1.54650	0.773248	+	0.634104i	$0.218631\pi$
$272$	0	0
$273$	−462.657	−0.102569
$274$	0	0
$275$	−1198.24	−0.262751
$276$	0	0
$277$	5617.68	1.21853	0.609267	−	0.792965i	$-0.291464\pi$
$277$	5617.68	1.21853	0.609267	+	0.792965i	$0.291464\pi$
$278$	0	0
$279$	7880.96	1.69111
$280$	0	0
$281$	3069.89	0.651723	0.325861	−	0.945418i	$-0.394346\pi$
$281$	3069.89	0.651723	0.325861	+	0.945418i	$0.394346\pi$
$282$	0	0
$283$	−1404.86	−0.295089	−0.147544	−	0.989055i	$-0.547137\pi$
$283$	−1404.86	−0.295089	−0.147544	+	0.989055i	$0.547137\pi$
$284$	0	0
$285$	−63.1880	−0.0131331
$286$	0	0
$287$	8623.30	1.77358
$288$	0	0
$289$	−4910.09	−0.999408
$290$	0	0
$291$	842.946	0.169809
$292$	0	0
$293$	−2407.05	−0.479936	−0.239968	−	0.970781i	$-0.577137\pi$
$293$	−2407.05	−0.479936	−0.239968	+	0.970781i	$0.577137\pi$
$294$	0	0
$295$	−1885.16	−0.372062
$296$	0	0
$297$	−1517.12	−0.296404
$298$	0	0
$299$	974.549	0.188494
$300$	0	0
$301$	−3375.33	−0.646347
$302$	0	0
$303$	892.124	0.169146
$304$	0	0
$305$	−4245.21	−0.796984
$306$	0	0
$307$	459.743	0.0854689	0.0427344	−	0.999086i	$-0.486393\pi$
$307$	459.743	0.0854689	0.0427344	+	0.999086i	$0.486393\pi$
$308$	0	0
$309$	564.765	0.103975
$310$	0	0
$311$	−4119.48	−0.751107	−0.375553	−	0.926801i	$-0.622547\pi$
$311$	−4119.48	−0.751107	−0.375553	+	0.926801i	$0.622547\pi$
$312$	0	0
$313$	1684.15	0.304133	0.152066	−	0.988370i	$-0.451407\pi$
$313$	1684.15	0.304133	0.152066	+	0.988370i	$0.451407\pi$
$314$	0	0
$315$	2466.34	0.441150
$316$	0	0
$317$	8686.47	1.53906	0.769528	−	0.638613i	$-0.220492\pi$
$317$	8686.47	1.53906	0.769528	+	0.638613i	$0.220492\pi$
$318$	0	0
$319$	−2762.34	−0.484831
$320$	0	0
$321$	−1027.82	−0.178714
$322$	0	0
$323$	−36.5298	−0.00629279
$324$	0	0
$325$	1059.29	0.180797
$326$	0	0
$327$	−688.360	−0.116411
$328$	0	0
$329$	−8313.55	−1.39313
$330$	0	0
$331$	−4307.91	−0.715359	−0.357680	−	0.933844i	$-0.616432\pi$
$331$	−4307.91	−0.715359	−0.357680	+	0.933844i	$0.616432\pi$
$332$	0	0
$333$	209.267	0.0344377
$334$	0	0
$335$	461.712	0.0753015
$336$	0	0
$337$	−290.156	−0.0469015	−0.0234507	−	0.999725i	$-0.507465\pi$
$337$	−290.156	−0.0469015	−0.0234507	+	0.999725i	$0.507465\pi$
$338$	0	0
$339$	232.027	0.0371740
$340$	0	0
$341$	14172.7	2.25072
$342$	0	0
$343$	−6356.73	−1.00067
$344$	0	0
$345$	67.8465	0.0105876
$346$	0	0
$347$	−1042.94	−0.161349	−0.0806743	−	0.996741i	$-0.525707\pi$
$347$	−1042.94	−0.161349	−0.0806743	+	0.996741i	$0.525707\pi$
$348$	0	0
$349$	−1819.56	−0.279080	−0.139540	−	0.990216i	$-0.544562\pi$
$349$	−1819.56	−0.279080	−0.139540	+	0.990216i	$0.544562\pi$
$350$	0	0
$351$	1341.19	0.203953
$352$	0	0
$353$	−4514.29	−0.680656	−0.340328	−	0.940307i	$-0.610538\pi$
$353$	−4514.29	−0.680656	−0.340328	+	0.940307i	$0.610538\pi$
$354$	0	0
$355$	−3134.27	−0.468591
$356$	0	0
$357$	−18.6207	−0.00276053
$358$	0	0
$359$	−11527.9	−1.69476	−0.847379	−	0.530988i	$-0.821821\pi$
$359$	−11527.9	−1.69476	−0.847379	+	0.530988i	$0.821821\pi$
$360$	0	0
$361$	−6400.15	−0.933103
$362$	0	0
$363$	−570.055	−0.0824246
$364$	0	0
$365$	−2196.13	−0.314934
$366$	0	0
$367$	−6894.09	−0.980568	−0.490284	−	0.871563i	$-0.663107\pi$
$367$	−6894.09	−0.980568	−0.490284	+	0.871563i	$0.663107\pi$
$368$	0	0
$369$	−12417.9	−1.75190
$370$	0	0
$371$	−6816.69	−0.953922
$372$	0	0
$373$	7733.25	1.07349	0.536746	−	0.843744i	$-0.319653\pi$
$373$	7733.25	1.07349	0.536746	+	0.843744i	$0.319653\pi$
$374$	0	0
$375$	73.7462	0.0101553
$376$	0	0
$377$	2442.02	0.333608
$378$	0	0
$379$	9495.72	1.28697	0.643486	−	0.765458i	$-0.277488\pi$
$379$	9495.72	1.28697	0.643486	+	0.765458i	$0.277488\pi$
$380$	0	0
$381$	−630.011	−0.0847150
$382$	0	0
$383$	12877.0	1.71797	0.858987	−	0.511998i	$-0.171094\pi$
$383$	12877.0	1.71797	0.858987	+	0.511998i	$0.171094\pi$
$384$	0	0
$385$	4435.34	0.587132
$386$	0	0
$387$	4860.61	0.638447
$388$	0	0
$389$	−8200.40	−1.06884	−0.534418	−	0.845221i	$-0.679469\pi$
$389$	−8200.40	−1.06884	−0.534418	+	0.845221i	$0.679469\pi$
$390$	0	0
$391$	39.2229	0.00507312
$392$	0	0
$393$	−103.538	−0.0132895
$394$	0	0
$395$	3208.53	0.408706
$396$	0	0
$397$	7842.68	0.991468	0.495734	−	0.868475i	$-0.334899\pi$
$397$	7842.68	0.991468	0.495734	+	0.868475i	$0.334899\pi$
$398$	0	0
$399$	233.893	0.0293467
$400$	0	0
$401$	−14534.2	−1.80999	−0.904993	−	0.425427i	$-0.860124\pi$
$401$	−14534.2	−1.80999	−0.904993	+	0.425427i	$0.860124\pi$
$402$	0	0
$403$	−12529.3	−1.54870
$404$	0	0
$405$	−3504.64	−0.429992
$406$	0	0
$407$	376.335	0.0458335
$408$	0	0
$409$	13388.0	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$409$	13388.0	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$410$	0	0
$411$	−280.765	−0.0336962
$412$	0	0
$413$	6978.00	0.831393
$414$	0	0
$415$	−3049.66	−0.360727
$416$	0	0
$417$	90.3640	0.0106119
$418$	0	0
$419$	3103.10	0.361805	0.180903	−	0.983501i	$-0.442098\pi$
$419$	3103.10	0.361805	0.180903	+	0.983501i	$0.442098\pi$
$420$	0	0
$421$	−6075.82	−0.703367	−0.351684	−	0.936119i	$-0.614391\pi$
$421$	−6075.82	−0.703367	−0.351684	+	0.936119i	$0.614391\pi$
$422$	0	0
$423$	11971.9	1.37610
$424$	0	0
$425$	42.6336	0.00486596
$426$	0	0
$427$	15713.8	1.78090
$428$	0	0
$429$	1198.14	0.134841
$430$	0	0
$431$	−14120.3	−1.57808	−0.789040	−	0.614341i	$-0.789422\pi$
$431$	−14120.3	−1.57808	−0.789040	+	0.614341i	$0.789422\pi$
$432$	0	0
$433$	8555.96	0.949592	0.474796	−	0.880096i	$-0.342522\pi$
$433$	8555.96	0.949592	0.474796	+	0.880096i	$0.342522\pi$
$434$	0	0
$435$	170.009	0.0187387
$436$	0	0
$437$	−492.678	−0.0539313
$438$	0	0
$439$	10894.1	1.18439	0.592194	−	0.805796i	$-0.298262\pi$
$439$	10894.1	1.18439	0.592194	+	0.805796i	$0.298262\pi$
$440$	0	0
$441$	12.3560	0.00133419
$442$	0	0
$443$	−16120.5	−1.72891	−0.864456	−	0.502708i	$-0.832337\pi$
$443$	−16120.5	−1.72891	−0.864456	+	0.502708i	$0.832337\pi$
$444$	0	0
$445$	−5614.34	−0.598079
$446$	0	0
$447$	−299.059	−0.0316443
$448$	0	0
$449$	4811.29	0.505699	0.252849	−	0.967506i	$-0.418632\pi$
$449$	4811.29	0.505699	0.252849	+	0.967506i	$0.418632\pi$
$450$	0	0
$451$	−22331.8	−2.33163
$452$	0	0
$453$	1438.25	0.149172
$454$	0	0
$455$	−3921.02	−0.404001
$456$	0	0
$457$	−226.329	−0.0231667	−0.0115834	−	0.999933i	$-0.503687\pi$
$457$	−226.329	−0.0231667	−0.0115834	+	0.999933i	$0.503687\pi$
$458$	0	0
$459$	53.9793	0.00548919
$460$	0	0
$461$	−4349.17	−0.439395	−0.219697	−	0.975568i	$-0.570507\pi$
$461$	−4349.17	−0.439395	−0.219697	+	0.975568i	$0.570507\pi$
$462$	0	0
$463$	989.313	0.0993030	0.0496515	−	0.998767i	$-0.484189\pi$
$463$	989.313	0.0993030	0.0496515	+	0.998767i	$0.484189\pi$
$464$	0	0
$465$	−872.267	−0.0869902
$466$	0	0
$467$	−8512.91	−0.843535	−0.421767	−	0.906704i	$-0.638590\pi$
$467$	−8512.91	−0.843535	−0.421767	+	0.906704i	$0.638590\pi$
$468$	0	0
$469$	−1709.05	−0.168266
$470$	0	0
$471$	−151.012	−0.0147734
$472$	0	0
$473$	8741.10	0.849717
$474$	0	0
$475$	−535.519	−0.0517291
$476$	0	0
$477$	9816.33	0.942261
$478$	0	0
$479$	12058.6	1.15025	0.575125	−	0.818065i	$-0.304953\pi$
$479$	12058.6	1.15025	0.575125	+	0.818065i	$0.304953\pi$
$480$	0	0
$481$	−332.696	−0.0315377
$482$	0	0
$483$	−251.137	−0.0236587
$484$	0	0
$485$	7143.98	0.668848
$486$	0	0
$487$	−11605.3	−1.07985	−0.539924	−	0.841714i	$-0.681547\pi$
$487$	−11605.3	−1.07985	−0.539924	+	0.841714i	$0.681547\pi$
$488$	0	0
$489$	189.753	0.0175479
$490$	0	0
$491$	4651.88	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$491$	4651.88	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$492$	0	0
$493$	98.2845	0.00897872
$494$	0	0
$495$	−6387.08	−0.579955
$496$	0	0
$497$	11601.6	1.04709
$498$	0	0
$499$	7953.03	0.713480	0.356740	−	0.934204i	$-0.383888\pi$
$499$	7953.03	0.713480	0.356740	+	0.934204i	$0.383888\pi$
$500$	0	0
$501$	−1726.38	−0.153950
$502$	0	0
$503$	11805.6	1.04649	0.523245	−	0.852182i	$-0.324721\pi$
$503$	11805.6	1.04649	0.523245	+	0.852182i	$0.324721\pi$
$504$	0	0
$505$	7560.76	0.666237
$506$	0	0
$507$	236.955	0.0207565
$508$	0	0
$509$	−2732.89	−0.237983	−0.118991	−	0.992895i	$-0.537966\pi$
$509$	−2732.89	−0.237983	−0.118991	+	0.992895i	$0.537966\pi$
$510$	0	0
$511$	8129.09	0.703738
$512$	0	0
$513$	−678.032	−0.0583545
$514$	0	0
$515$	4786.40	0.409541
$516$	0	0
$517$	21529.6	1.83147
$518$	0	0
$519$	−1068.93	−0.0904065
$520$	0	0
$521$	14710.9	1.23704	0.618518	−	0.785771i	$-0.287733\pi$
$521$	14710.9	1.23704	0.618518	+	0.785771i	$0.287733\pi$
$522$	0	0
$523$	7034.35	0.588127	0.294064	−	0.955786i	$-0.404992\pi$
$523$	7034.35	0.588127	0.294064	+	0.955786i	$0.404992\pi$
$524$	0	0
$525$	−272.975	−0.0226926
$526$	0	0
$527$	−504.269	−0.0416818
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−10048.6	−0.821230
$532$	0	0
$533$	19742.2	1.60437
$534$	0	0
$535$	−8710.79	−0.703926
$536$	0	0
$537$	538.444	0.0432692
$538$	0	0
$539$	22.2204	0.00177569
$540$	0	0
$541$	1552.90	0.123409	0.0617045	−	0.998094i	$-0.480346\pi$
$541$	1552.90	0.123409	0.0617045	+	0.998094i	$0.480346\pi$
$542$	0	0
$543$	−2165.51	−0.171144
$544$	0	0
$545$	−5833.86	−0.458523
$546$	0	0
$547$	−174.657	−0.0136523	−0.00682614	−	0.999977i	$-0.502173\pi$
$547$	−174.657	−0.0136523	−0.00682614	+	0.999977i	$0.502173\pi$
$548$	0	0
$549$	−22628.6	−1.75914
$550$	0	0
$551$	−1234.55	−0.0954510
$552$	0	0
$553$	−11876.5	−0.913276
$554$	0	0
$555$	−23.1617	−0.00177146
$556$	0	0
$557$	−1990.63	−0.151429	−0.0757143	−	0.997130i	$-0.524124\pi$
$557$	−1990.63	−0.151429	−0.0757143	+	0.997130i	$0.524124\pi$
$558$	0	0
$559$	−7727.48	−0.584683
$560$	0	0
$561$	48.2220	0.00362912
$562$	0	0
$563$	−208.006	−0.0155709	−0.00778543	−	0.999970i	$-0.502478\pi$
$563$	−208.006	−0.0155709	−0.00778543	+	0.999970i	$0.502478\pi$
$564$	0	0
$565$	1966.44	0.146422
$566$	0	0
$567$	12972.6	0.960842
$568$	0	0
$569$	−3003.05	−0.221256	−0.110628	−	0.993862i	$-0.535286\pi$
$569$	−3003.05	−0.221256	−0.110628	+	0.993862i	$0.535286\pi$
$570$	0	0
$571$	10796.1	0.791249	0.395624	−	0.918412i	$-0.370528\pi$
$571$	10796.1	0.791249	0.395624	+	0.918412i	$0.370528\pi$
$572$	0	0
$573$	−1086.11	−0.0791847
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−26011.3	−1.87671	−0.938357	−	0.345667i	$-0.887653\pi$
$577$	−26011.3	−1.87671	−0.938357	+	0.345667i	$0.887653\pi$
$578$	0	0
$579$	360.953	0.0259079
$580$	0	0
$581$	11288.5	0.806066
$582$	0	0
$583$	17653.2	1.25407
$584$	0	0
$585$	5646.44	0.399062
$586$	0	0
$587$	2774.97	0.195120	0.0975598	−	0.995230i	$-0.468896\pi$
$587$	2774.97	0.195120	0.0975598	+	0.995230i	$0.468896\pi$
$588$	0	0
$589$	6334.11	0.443111
$590$	0	0
$591$	−1669.77	−0.116218
$592$	0	0
$593$	−2384.07	−0.165096	−0.0825481	−	0.996587i	$-0.526306\pi$
$593$	−2384.07	−0.165096	−0.0825481	+	0.996587i	$0.526306\pi$
$594$	0	0
$595$	−157.810	−0.0108733
$596$	0	0
$597$	−685.984	−0.0470276
$598$	0	0
$599$	1090.87	0.0744102	0.0372051	−	0.999308i	$-0.488155\pi$
$599$	1090.87	0.0744102	0.0372051	+	0.999308i	$0.488155\pi$
$600$	0	0
$601$	−5192.37	−0.352414	−0.176207	−	0.984353i	$-0.556383\pi$
$601$	−5192.37	−0.352414	−0.176207	+	0.984353i	$0.556383\pi$
$602$	0	0
$603$	2461.10	0.166209
$604$	0	0
$605$	−4831.22	−0.324657
$606$	0	0
$607$	−1205.44	−0.0806054	−0.0403027	−	0.999188i	$-0.512832\pi$
$607$	−1205.44	−0.0806054	−0.0403027	+	0.999188i	$0.512832\pi$
$608$	0	0
$609$	−629.297	−0.0418726
$610$	0	0
$611$	−19033.1	−1.26022
$612$	0	0
$613$	−14243.2	−0.938466	−0.469233	−	0.883075i	$-0.655469\pi$
$613$	−14243.2	−0.938466	−0.469233	+	0.883075i	$0.655469\pi$
$614$	0	0
$615$	1374.42	0.0901170
$616$	0	0
$617$	−422.492	−0.0275671	−0.0137835	−	0.999905i	$-0.504388\pi$
$617$	−422.492	−0.0275671	−0.0137835	+	0.999905i	$0.504388\pi$
$618$	0	0
$619$	8826.35	0.573119	0.286560	−	0.958062i	$-0.407488\pi$
$619$	8826.35	0.573119	0.286560	+	0.958062i	$0.407488\pi$
$620$	0	0
$621$	728.019	0.0470441
$622$	0	0
$623$	20781.7	1.33644
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−605.715	−0.0385804
$628$	0	0
$629$	−13.3901	−0.000848804 0
$630$	0	0
$631$	19684.1	1.24186	0.620929	−	0.783867i	$-0.286756\pi$
$631$	19684.1	1.24186	0.620929	+	0.783867i	$0.286756\pi$
$632$	0	0
$633$	825.710	0.0518468
$634$	0	0
$635$	−5339.35	−0.333678
$636$	0	0
$637$	−19.6437	−0.00122184
$638$	0	0
$639$	−16706.9	−1.03429
$640$	0	0
$641$	15113.1	0.931253	0.465626	−	0.884981i	$-0.345829\pi$
$641$	15113.1	0.931253	0.465626	+	0.884981i	$0.345829\pi$
$642$	0	0
$643$	16917.8	1.03760	0.518798	−	0.854897i	$-0.326380\pi$
$643$	16917.8	1.03760	0.518798	+	0.854897i	$0.326380\pi$
$644$	0	0
$645$	−537.975	−0.0328414
$646$	0	0
$647$	−19564.5	−1.18881	−0.594405	−	0.804166i	$-0.702612\pi$
$647$	−19564.5	−1.18881	−0.594405	+	0.804166i	$0.702612\pi$
$648$	0	0
$649$	−18071.0	−1.09299
$650$	0	0
$651$	3228.74	0.194384
$652$	0	0
$653$	−22735.0	−1.36247	−0.681233	−	0.732067i	$-0.738556\pi$
$653$	−22735.0	−1.36247	−0.681233	+	0.732067i	$0.738556\pi$
$654$	0	0
$655$	−877.483	−0.0523452
$656$	0	0
$657$	−11706.2	−0.695136
$658$	0	0
$659$	30730.7	1.81654	0.908269	−	0.418388i	$-0.137405\pi$
$659$	30730.7	1.81654	0.908269	+	0.418388i	$0.137405\pi$
$660$	0	0
$661$	−29204.3	−1.71848	−0.859240	−	0.511573i	$-0.829063\pi$
$661$	−29204.3	−1.71848	−0.859240	+	0.511573i	$0.829063\pi$
$662$	0	0
$663$	−42.6302	−0.00249716
$664$	0	0
$665$	1982.25	0.115592
$666$	0	0
$667$	1325.56	0.0769506
$668$	0	0
$669$	2512.08	0.145176
$670$	0	0
$671$	−40694.2	−2.34126
$672$	0	0
$673$	−17530.1	−1.00406	−0.502032	−	0.864849i	$-0.667414\pi$
$673$	−17530.1	−1.00406	−0.502032	+	0.864849i	$0.667414\pi$
$674$	0	0
$675$	791.325	0.0451231
$676$	0	0
$677$	−20553.4	−1.16681	−0.583407	−	0.812180i	$-0.698281\pi$
$677$	−20553.4	−1.16681	−0.583407	+	0.812180i	$0.698281\pi$
$678$	0	0
$679$	−26443.8	−1.49458
$680$	0	0
$681$	−2374.65	−0.133622
$682$	0	0
$683$	−13174.8	−0.738098	−0.369049	−	0.929410i	$-0.620316\pi$
$683$	−13174.8	−0.738098	−0.369049	+	0.929410i	$0.620316\pi$
$684$	0	0
$685$	−2379.49	−0.132723
$686$	0	0
$687$	−2137.48	−0.118705
$688$	0	0
$689$	−15606.2	−0.862913
$690$	0	0
$691$	14800.8	0.814831	0.407415	−	0.913243i	$-0.366430\pi$
$691$	14800.8	0.814831	0.407415	+	0.913243i	$0.366430\pi$
$692$	0	0
$693$	23642.1	1.29594
$694$	0	0
$695$	765.836	0.0417983
$696$	0	0
$697$	794.570	0.0431800
$698$	0	0
$699$	−3836.46	−0.207594
$700$	0	0
$701$	−12311.5	−0.663336	−0.331668	−	0.943396i	$-0.607611\pi$
$701$	−12311.5	−0.663336	−0.331668	+	0.943396i	$0.607611\pi$
$702$	0	0
$703$	168.192	0.00902347
$704$	0	0
$705$	−1325.05	−0.0707862
$706$	0	0
$707$	−27986.5	−1.48874
$708$	0	0
$709$	3893.89	0.206260	0.103130	−	0.994668i	$-0.467114\pi$
$709$	3893.89	0.206260	0.103130	+	0.994668i	$0.467114\pi$
$710$	0	0
$711$	17102.7	0.902113
$712$	0	0
$713$	−6801.08	−0.357227
$714$	0	0
$715$	10154.3	0.531117
$716$	0	0
$717$	−1587.06	−0.0826636
$718$	0	0
$719$	19942.5	1.03440	0.517198	−	0.855866i	$-0.326975\pi$
$719$	19942.5	1.03440	0.517198	+	0.855866i	$0.326975\pi$
$720$	0	0
$721$	−17717.1	−0.915143
$722$	0	0
$723$	26.3692	0.00135640
$724$	0	0
$725$	1440.83	0.0738084
$726$	0	0
$727$	−36167.0	−1.84506	−0.922530	−	0.385926i	$-0.873882\pi$
$727$	−36167.0	−1.84506	−0.922530	+	0.385926i	$0.873882\pi$
$728$	0	0
$729$	−18176.9	−0.923481
$730$	0	0
$731$	−311.010	−0.0157361
$732$	0	0
$733$	−11669.5	−0.588025	−0.294013	−	0.955802i	$-0.594991\pi$
$733$	−11669.5	−0.588025	−0.294013	+	0.955802i	$0.594991\pi$
$734$	0	0
$735$	−1.36756	−6.86304e−5 0
$736$	0	0
$737$	4425.93	0.221209
$738$	0	0
$739$	−75.0536	−0.00373598	−0.00186799	−	0.999998i	$-0.500595\pi$
$739$	−75.0536	−0.00373598	−0.00186799	+	0.999998i	$0.500595\pi$
$740$	0	0
$741$	535.477	0.0265469
$742$	0	0
$743$	−28279.8	−1.39635	−0.698173	−	0.715929i	$-0.746003\pi$
$743$	−28279.8	−1.39635	−0.698173	+	0.715929i	$0.746003\pi$
$744$	0	0
$745$	−2534.53	−0.124641
$746$	0	0
$747$	−16255.9	−0.796213
$748$	0	0
$749$	32243.4	1.57296
$750$	0	0
$751$	17268.1	0.839044	0.419522	−	0.907745i	$-0.362198\pi$
$751$	17268.1	0.839044	0.419522	+	0.907745i	$0.362198\pi$
$752$	0	0
$753$	−4012.62	−0.194194
$754$	0	0
$755$	12189.2	0.587564
$756$	0	0
$757$	−12525.8	−0.601400	−0.300700	−	0.953719i	$-0.597220\pi$
$757$	−12525.8	−0.601400	−0.300700	+	0.953719i	$0.597220\pi$
$758$	0	0
$759$	650.371	0.0311027
$760$	0	0
$761$	−18670.1	−0.889343	−0.444672	−	0.895694i	$-0.646680\pi$
$761$	−18670.1	−0.889343	−0.444672	+	0.895694i	$0.646680\pi$
$762$	0	0
$763$	21594.3	1.02460
$764$	0	0
$765$	227.254	0.0107404
$766$	0	0
$767$	15975.5	0.752074
$768$	0	0
$769$	−32969.6	−1.54605	−0.773027	−	0.634373i	$-0.781258\pi$
$769$	−32969.6	−1.54605	−0.773027	+	0.634373i	$0.781258\pi$
$770$	0	0
$771$	−3289.67	−0.153664
$772$	0	0
$773$	−23251.3	−1.08188	−0.540938	−	0.841062i	$-0.681931\pi$
$773$	−23251.3	−1.08188	−0.540938	+	0.841062i	$0.681931\pi$
$774$	0	0
$775$	−7392.48	−0.342640
$776$	0	0
$777$	85.7342	0.00395843
$778$	0	0
$779$	−9980.57	−0.459039
$780$	0	0
$781$	−30044.8	−1.37655
$782$	0	0
$783$	1824.26	0.0832617
$784$	0	0
$785$	−1279.83	−0.0581900
$786$	0	0
$787$	29782.8	1.34897	0.674486	−	0.738288i	$-0.264365\pi$
$787$	29782.8	1.34897	0.674486	+	0.738288i	$0.264365\pi$
$788$	0	0
$789$	−3344.88	−0.150926
$790$	0	0
$791$	−7278.86	−0.327189
$792$	0	0
$793$	35975.3	1.61100
$794$	0	0
$795$	−1086.47	−0.0484696
$796$	0	0
$797$	35307.2	1.56919	0.784596	−	0.620008i	$-0.212871\pi$
$797$	35307.2	1.56919	0.784596	+	0.620008i	$0.212871\pi$
$798$	0	0
$799$	−766.029	−0.0339176
$800$	0	0
$801$	−29926.6	−1.32011
$802$	0	0
$803$	−21052.0	−0.925165
$804$	0	0
$805$	−2128.39	−0.0931874
$806$	0	0
$807$	−3563.54	−0.155443
$808$	0	0
$809$	7752.46	0.336912	0.168456	−	0.985709i	$-0.446122\pi$
$809$	7752.46	0.336912	0.168456	+	0.985709i	$0.446122\pi$
$810$	0	0
$811$	−23471.6	−1.01627	−0.508137	−	0.861276i	$-0.669666\pi$
$811$	−23471.6	−1.01627	−0.508137	+	0.861276i	$0.669666\pi$
$812$	0	0
$813$	−4070.35	−0.175589
$814$	0	0
$815$	1608.16	0.0691184
$816$	0	0
$817$	3906.59	0.167288
$818$	0	0
$819$	−20900.6	−0.891727
$820$	0	0
$821$	2467.55	0.104894	0.0524470	−	0.998624i	$-0.483298\pi$
$821$	2467.55	0.104894	0.0524470	+	0.998624i	$0.483298\pi$
$822$	0	0
$823$	−21984.5	−0.931144	−0.465572	−	0.885010i	$-0.654151\pi$
$823$	−21984.5	−0.931144	−0.465572	+	0.885010i	$0.654151\pi$
$824$	0	0
$825$	706.925	0.0298327
$826$	0	0
$827$	587.293	0.0246943	0.0123472	−	0.999924i	$-0.496070\pi$
$827$	587.293	0.0246943	0.0123472	+	0.999924i	$0.496070\pi$
$828$	0	0
$829$	41822.8	1.75219	0.876096	−	0.482137i	$-0.160139\pi$
$829$	41822.8	1.75219	0.876096	+	0.482137i	$0.160139\pi$
$830$	0	0
$831$	−3314.26	−0.138352
$832$	0	0
$833$	−0.790605	−3.28846e−5 0
$834$	0	0
$835$	−14631.1	−0.606383
$836$	0	0
$837$	−9359.77	−0.386524
$838$	0	0
$839$	10126.5	0.416693	0.208346	−	0.978055i	$-0.433192\pi$
$839$	10126.5	0.416693	0.208346	+	0.978055i	$0.433192\pi$
$840$	0	0
$841$	−21067.4	−0.863808
$842$	0	0
$843$	−1811.14	−0.0739964
$844$	0	0
$845$	2008.20	0.0817564
$846$	0	0
$847$	17883.0	0.725463
$848$	0	0
$849$	828.822	0.0335043
$850$	0	0
$851$	−180.592	−0.00727453
$852$	0	0
$853$	11584.4	0.464995	0.232498	−	0.972597i	$-0.425310\pi$
$853$	11584.4	0.464995	0.232498	+	0.972597i	$0.425310\pi$
$854$	0	0
$855$	−2854.53	−0.114179
$856$	0	0
$857$	−9221.47	−0.367561	−0.183780	−	0.982967i	$-0.558834\pi$
$857$	−9221.47	−0.367561	−0.183780	+	0.982967i	$0.558834\pi$
$858$	0	0
$859$	−34654.2	−1.37647	−0.688235	−	0.725488i	$-0.741614\pi$
$859$	−34654.2	−1.37647	−0.688235	+	0.725488i	$0.741614\pi$
$860$	0	0
$861$	−5087.48	−0.201372
$862$	0	0
$863$	−42887.7	−1.69167	−0.845837	−	0.533441i	$-0.820899\pi$
$863$	−42887.7	−1.69167	−0.845837	+	0.533441i	$0.820899\pi$
$864$	0	0
$865$	−9059.22	−0.356096
$866$	0	0
$867$	2896.80	0.113472
$868$	0	0
$869$	30756.7	1.20063
$870$	0	0
$871$	−3912.70	−0.152212
$872$	0	0
$873$	38080.2	1.47631
$874$	0	0
$875$	−2313.47	−0.0893823
$876$	0	0
$877$	45935.5	1.76868	0.884339	−	0.466846i	$-0.154610\pi$
$877$	45935.5	1.76868	0.884339	+	0.466846i	$0.154610\pi$
$878$	0	0
$879$	1420.08	0.0544917
$880$	0	0
$881$	−71.9309	−0.00275075	−0.00137538	−	0.999999i	$-0.500438\pi$
$881$	−71.9309	−0.00275075	−0.00137538	+	0.999999i	$0.500438\pi$
$882$	0	0
$883$	−20003.8	−0.762379	−0.381189	−	0.924497i	$-0.624485\pi$
$883$	−20003.8	−0.762379	−0.381189	+	0.924497i	$0.624485\pi$
$884$	0	0
$885$	1112.19	0.0422437
$886$	0	0
$887$	−26357.3	−0.997737	−0.498869	−	0.866678i	$-0.666251\pi$
$887$	−26357.3	−0.997737	−0.498869	+	0.866678i	$0.666251\pi$
$888$	0	0
$889$	19763.9	0.745623
$890$	0	0
$891$	−33595.2	−1.26317
$892$	0	0
$893$	9622.06	0.360571
$894$	0	0
$895$	4563.32	0.170430
$896$	0	0
$897$	−574.954	−0.0214015
$898$	0	0
$899$	−17042.1	−0.632242
$900$	0	0
$901$	−628.105	−0.0232244
$902$	0	0
$903$	1991.34	0.0733860
$904$	0	0
$905$	−18352.7	−0.674106
$906$	0	0
$907$	39300.4	1.43875	0.719377	−	0.694620i	$-0.244428\pi$
$907$	39300.4	1.43875	0.719377	+	0.694620i	$0.244428\pi$
$908$	0	0
$909$	40301.8	1.47055
$910$	0	0
$911$	−31099.2	−1.13103	−0.565513	−	0.824740i	$-0.691322\pi$
$911$	−31099.2	−1.13103	−0.565513	+	0.824740i	$0.691322\pi$
$912$	0	0
$913$	−29233.8	−1.05969
$914$	0	0
$915$	2504.54	0.0904893
$916$	0	0
$917$	3248.05	0.116968
$918$	0	0
$919$	−18935.8	−0.679690	−0.339845	−	0.940481i	$-0.610375\pi$
$919$	−18935.8	−0.679690	−0.339845	+	0.940481i	$0.610375\pi$
$920$	0	0
$921$	−271.235	−0.00970411
$922$	0	0
$923$	26560.9	0.947195
$924$	0	0
$925$	−196.296	−0.00697748
$926$	0	0
$927$	25513.3	0.903957
$928$	0	0
$929$	40156.1	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$929$	40156.1	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$930$	0	0
$931$	9.93077	0.000349590 0
$932$	0	0
$933$	2430.37	0.0852804
$934$	0	0
$935$	408.682	0.0142945
$936$	0	0
$937$	126.898	0.00442432	0.00221216	−	0.999998i	$-0.499296\pi$
$937$	126.898	0.00442432	0.00221216	+	0.999998i	$0.499296\pi$
$938$	0	0
$939$	−993.594	−0.0345311
$940$	0	0
$941$	−19266.6	−0.667452	−0.333726	−	0.942670i	$-0.608306\pi$
$941$	−19266.6	−0.667452	−0.333726	+	0.942670i	$0.608306\pi$
$942$	0	0
$943$	10716.4	0.370067
$944$	0	0
$945$	−2929.13	−0.100830
$946$	0	0
$947$	54560.4	1.87220	0.936101	−	0.351732i	$-0.114407\pi$
$947$	54560.4	1.87220	0.936101	+	0.351732i	$0.114407\pi$
$948$	0	0
$949$	18610.8	0.636598
$950$	0	0
$951$	−5124.75	−0.174744
$952$	0	0
$953$	−44976.2	−1.52877	−0.764387	−	0.644758i	$-0.776958\pi$
$953$	−44976.2	−1.52877	−0.764387	+	0.644758i	$0.776958\pi$
$954$	0	0
$955$	−9204.79	−0.311895
$956$	0	0
$957$	1629.69	0.0550476
$958$	0	0
$959$	8807.79	0.296578
$960$	0	0
$961$	57647.0	1.93505
$962$	0	0
$963$	−46431.9	−1.55374
$964$	0	0
$965$	3059.08	0.102047
$966$	0	0
$967$	39431.7	1.31131	0.655656	−	0.755059i	$-0.272392\pi$
$967$	39431.7	1.31131	0.655656	+	0.755059i	$0.272392\pi$
$968$	0	0
$969$	21.5515	0.000714482 0
$970$	0	0
$971$	−20249.9	−0.669258	−0.334629	−	0.942350i	$-0.608611\pi$
$971$	−20249.9	−0.669258	−0.334629	+	0.942350i	$0.608611\pi$
$972$	0	0
$973$	−2834.78	−0.0934006
$974$	0	0
$975$	−624.950	−0.0205276
$976$	0	0
$977$	−9673.55	−0.316770	−0.158385	−	0.987377i	$-0.550629\pi$
$977$	−9673.55	−0.316770	−0.158385	+	0.987377i	$0.550629\pi$
$978$	0	0
$979$	−53818.6	−1.75695
$980$	0	0
$981$	−31096.7	−1.01207
$982$	0	0
$983$	17264.3	0.560168	0.280084	−	0.959975i	$-0.409638\pi$
$983$	17264.3	0.560168	0.280084	+	0.959975i	$0.409638\pi$
$984$	0	0
$985$	−14151.3	−0.457764
$986$	0	0
$987$	4904.74	0.158176
$988$	0	0
$989$	−4194.60	−0.134864
$990$	0	0
$991$	−23892.9	−0.765876	−0.382938	−	0.923774i	$-0.625088\pi$
$991$	−23892.9	−0.765876	−0.382938	+	0.923774i	$0.625088\pi$
$992$	0	0
$993$	2541.53	0.0812217
$994$	0	0
$995$	−5813.72	−0.185234
$996$	0	0
$997$	−35830.2	−1.13817	−0.569083	−	0.822280i	$-0.692702\pi$
$997$	−35830.2	−1.13817	−0.569083	+	0.822280i	$0.692702\pi$
$998$	0	0
$999$	−248.534	−0.00787115

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.m.1.2		4
4.3	odd	2		230.4.a.h.1.3	✓	4
12.11	even	2		2070.4.a.bj.1.2		4
20.3	even	4		1150.4.b.n.599.7		8
20.7	even	4		1150.4.b.n.599.2		8
20.19	odd	2		1150.4.a.p.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.3	✓	4	4.3	odd	2
1150.4.a.p.1.2		4	20.19	odd	2
1150.4.b.n.599.2		8	20.7	even	4
1150.4.b.n.599.7		8	20.3	even	4
1840.4.a.m.1.2		4	1.1	even	1	trivial
2070.4.a.bj.1.2		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)

\(f(q)\)	\(=\)	\(q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} +O(q^{10})\)	\(q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} -47.9296 q^{11} +42.3717 q^{13} +2.94985 q^{15} +1.70534 q^{17} -21.4208 q^{19} -10.9190 q^{21} +23.0000 q^{23} +25.0000 q^{25} +31.6530 q^{27} +57.6332 q^{29} -295.699 q^{31} +28.2770 q^{33} -92.5387 q^{35} -7.85184 q^{37} -24.9980 q^{39} +465.929 q^{41} -182.374 q^{43} +133.260 q^{45} -449.193 q^{47} -0.463605 q^{49} -1.00610 q^{51} -368.316 q^{53} +239.648 q^{55} +12.6376 q^{57} +377.032 q^{59} +849.042 q^{61} -493.267 q^{63} -211.858 q^{65} -92.3424 q^{67} -13.5693 q^{69} +626.854 q^{71} +439.227 q^{73} -14.7492 q^{75} -887.068 q^{77} -641.707 q^{79} +700.928 q^{81} +609.932 q^{83} -8.52672 q^{85} -34.0018 q^{87} +1122.87 q^{89} +784.204 q^{91} +174.453 q^{93} +107.104 q^{95} -1428.80 q^{97} +1277.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9	\( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9 + 39 * q^11 - 20 * q^13 - 20 * q^15 - 23 * q^17 - 53 * q^19 + 300 * q^21 + 92 * q^23 + 100 * q^25 - 137 * q^27 + 161 * q^29 - 388 * q^31 + 87 * q^33 - 5 * q^35 + 466 * q^37 - 1047 * q^39 + 484 * q^41 - 894 * q^43 - 160 * q^45 + 265 * q^47 + 1643 * q^49 - 1825 * q^51 + 576 * q^53 - 195 * q^55 + 178 * q^57 + 94 * q^59 + 1153 * q^61 - 60 * q^63 + 100 * q^65 + 1472 * q^67 + 92 * q^69 - 200 * q^71 + 1147 * q^73 + 100 * q^75 - 2176 * q^77 + 908 * q^79 - 1056 * q^81 + 1048 * q^83 + 115 * q^85 + 2167 * q^87 - 1784 * q^89 - 2329 * q^91 + 1483 * q^93 + 265 * q^95 - 2047 * q^97 + 2665 * q^99

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