LMFDB - Embedded newform 1840.3.k.d.321.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.14
Root			$6.02373i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.79379 q^{3} +2.23607i q^{5} +7.10180i q^{7} +5.39287 q^{9} +O(q^{10})$	$q+3.79379 q^{3} +2.23607i q^{5} +7.10180i q^{7} +5.39287 q^{9} +11.2644i q^{11} +20.0597 q^{13} +8.48318i q^{15} -1.63128i q^{17} +29.4164i q^{19} +26.9428i q^{21} +(-20.0280 + 11.3084i) q^{23} -5.00000 q^{25} -13.6847 q^{27} -50.3233 q^{29} -11.1316 q^{31} +42.7347i q^{33} -15.8801 q^{35} -40.5429i q^{37} +76.1025 q^{39} -7.24039 q^{41} -71.7020i q^{43} +12.0588i q^{45} +6.40666 q^{47} -1.43550 q^{49} -6.18873i q^{51} +20.4148i q^{53} -25.1879 q^{55} +111.600i q^{57} +65.8889 q^{59} +37.7281i q^{61} +38.2991i q^{63} +44.8549i q^{65} -124.242i q^{67} +(-75.9821 + 42.9016i) q^{69} -43.5656 q^{71} +48.1194 q^{73} -18.9690 q^{75} -79.9972 q^{77} +101.026i q^{79} -100.453 q^{81} +102.409i q^{83} +3.64765 q^{85} -190.916 q^{87} +9.63875i q^{89} +142.460i q^{91} -42.2310 q^{93} -65.7771 q^{95} -143.631i q^{97} +60.7473i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	3.79379			1.26460			0.632299	−	0.774724i	$-0.282111\pi$
$3$	3.79379			1.26460			0.632299	+	0.774724i	$0.282111\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			7.10180i			1.01454i	0.861787	+	0.507271i	$0.169346\pi$
$7$			7.10180i			1.01454i	−0.861787	+	0.507271i	$0.830654\pi$
$8$		0			0
$9$	5.39287			0.599208
$10$		0			0
$11$			11.2644i			1.02403i	0.858975	+	0.512017i	$0.171101\pi$
$11$			11.2644i			1.02403i	−0.858975	+	0.512017i	$0.828899\pi$
$12$		0			0
$13$	20.0597			1.54306			0.771528	−	0.636195i	$-0.219493\pi$
$13$	20.0597			1.54306			0.771528	+	0.636195i	$0.219493\pi$
$14$		0			0
$15$			8.48318i			0.565545i
$16$		0			0
$17$		−	1.63128i		−	0.0959575i	−0.998848	−	0.0479788i	$-0.984722\pi$
$17$		−	1.63128i		−	0.0959575i	0.998848	−	0.0479788i	$-0.0152780\pi$
$18$		0			0
$19$			29.4164i			1.54823i	0.633044	+	0.774116i	$0.281805\pi$
$19$			29.4164i			1.54823i	−0.633044	+	0.774116i	$0.718195\pi$
$20$		0			0
$21$			26.9428i			1.28299i
$22$		0			0
$23$	−20.0280	+	11.3084i	−0.870783	+	0.491668i
$23$	−20.0280	+	11.3084i	−0.870783	+	0.491668i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−13.6847			−0.506841
$28$		0			0
$29$	−50.3233			−1.73529			−0.867644	−	0.497187i	$-0.834366\pi$
$29$	−50.3233			−1.73529			−0.867644	+	0.497187i	$0.834366\pi$
$30$		0			0
$31$	−11.1316			−0.359084			−0.179542	−	0.983750i	$-0.557462\pi$
$31$	−11.1316			−0.359084			−0.179542	+	0.983750i	$0.557462\pi$
$32$		0			0
$33$			42.7347i			1.29499i
$34$		0			0
$35$	−15.8801			−0.453717
$36$		0			0
$37$		−	40.5429i		−	1.09575i	−0.836559	−	0.547877i	$-0.815436\pi$
$37$		−	40.5429i		−	1.09575i	0.836559	−	0.547877i	$-0.184564\pi$
$38$		0			0
$39$	76.1025			1.95135
$40$		0			0
$41$	−7.24039			−0.176595			−0.0882975	−	0.996094i	$-0.528143\pi$
$41$	−7.24039			−0.176595			−0.0882975	+	0.996094i	$0.528143\pi$
$42$		0			0
$43$		−	71.7020i		−	1.66749i	−0.552150	−	0.833745i	$-0.686192\pi$
$43$		−	71.7020i		−	1.66749i	0.552150	−	0.833745i	$-0.313808\pi$
$44$		0			0
$45$			12.0588i			0.267974i
$46$		0			0
$47$	6.40666			0.136312			0.0681560	−	0.997675i	$-0.478288\pi$
$47$	6.40666			0.136312			0.0681560	+	0.997675i	$0.478288\pi$
$48$		0			0
$49$	−1.43550			−0.0292959
$50$		0			0
$51$		−	6.18873i		−	0.121348i
$52$		0			0
$53$			20.4148i			0.385184i	0.981279	+	0.192592i	$0.0616894\pi$
$53$			20.4148i			0.385184i	−0.981279	+	0.192592i	$0.938311\pi$
$54$		0			0
$55$	−25.1879			−0.457962
$56$		0			0
$57$			111.600i			1.95789i
$58$		0			0
$59$	65.8889			1.11676			0.558381	−	0.829585i	$-0.311423\pi$
$59$	65.8889			1.11676			0.558381	+	0.829585i	$0.311423\pi$
$60$		0			0
$61$			37.7281i			0.618493i	0.950982	+	0.309247i	$0.100077\pi$
$61$			37.7281i			0.618493i	−0.950982	+	0.309247i	$0.899923\pi$
$62$		0			0
$63$			38.2991i			0.607922i
$64$		0			0
$65$			44.8549i			0.690076i
$66$		0			0
$67$		−	124.242i		−	1.85437i	−0.374609	−	0.927183i	$-0.622223\pi$
$67$		−	124.242i		−	1.85437i	0.374609	−	0.927183i	$-0.377777\pi$
$68$		0			0
$69$	−75.9821	+	42.9016i	−1.10119	+	0.621763i
$70$		0			0
$71$	−43.5656			−0.613600			−0.306800	−	0.951774i	$-0.599258\pi$
$71$	−43.5656			−0.613600			−0.306800	+	0.951774i	$0.599258\pi$
$72$		0			0
$73$	48.1194			0.659169			0.329585	−	0.944126i	$-0.393091\pi$
$73$	48.1194			0.659169			0.329585	+	0.944126i	$0.393091\pi$
$74$		0			0
$75$	−18.9690			−0.252920
$76$		0			0
$77$	−79.9972			−1.03893
$78$		0			0
$79$			101.026i			1.27882i	0.768868	+	0.639408i	$0.220821\pi$
$79$			101.026i			1.27882i	−0.768868	+	0.639408i	$0.779179\pi$
$80$		0			0
$81$	−100.453			−1.24016
$82$		0			0
$83$			102.409i			1.23384i	0.787025	+	0.616921i	$0.211620\pi$
$83$			102.409i			1.23384i	−0.787025	+	0.616921i	$0.788380\pi$
$84$		0			0
$85$	3.64765			0.0429135
$86$		0			0
$87$	−190.916			−2.19444
$88$		0			0
$89$			9.63875i			0.108301i	0.998533	+	0.0541503i	$0.0172450\pi$
$89$			9.63875i			0.108301i	−0.998533	+	0.0541503i	$0.982755\pi$
$90$		0			0
$91$			142.460i			1.56550i
$92$		0			0
$93$	−42.2310			−0.454097
$94$		0			0
$95$	−65.7771			−0.692390
$96$		0			0
$97$		−	143.631i		−	1.48074i	−0.672202	−	0.740368i	$-0.734651\pi$
$97$		−	143.631i		−	1.48074i	0.672202	−	0.740368i	$-0.265349\pi$
$98$		0			0
$99$			60.7473i			0.613609i
$100$		0			0
$101$	103.099			1.02078			0.510391	−	0.859943i	$-0.329501\pi$
$101$	103.099			1.02078			0.510391	+	0.859943i	$0.329501\pi$
$102$		0			0
$103$			98.8637i			0.959841i	0.877312	+	0.479921i	$0.159335\pi$
$103$			98.8637i			0.959841i	−0.877312	+	0.479921i	$0.840665\pi$
$104$		0			0
$105$	−60.2458			−0.573770
$106$		0			0
$107$			22.5494i			0.210742i	0.994433	+	0.105371i	$0.0336029\pi$
$107$			22.5494i			0.210742i	−0.994433	+	0.105371i	$0.966397\pi$
$108$		0			0
$109$			30.2389i			0.277421i	0.990333	+	0.138711i	$0.0442958\pi$
$109$			30.2389i			0.277421i	−0.990333	+	0.138711i	$0.955704\pi$
$110$		0			0
$111$		−	153.811i		−	1.38569i
$112$		0			0
$113$			213.437i			1.88882i	0.328771	+	0.944410i	$0.393365\pi$
$113$			213.437i			1.88882i	−0.328771	+	0.944410i	$0.606635\pi$
$114$		0			0
$115$	−25.2863	−	44.7840i	−0.219881	−	0.389426i
$116$		0			0
$117$	108.180			0.924612
$118$		0			0
$119$	11.5850			0.0973530
$120$		0			0
$121$	−5.88598			−0.0486445
$122$		0			0
$123$	−27.4686			−0.223322
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	29.9509			0.235834			0.117917	−	0.993023i	$-0.462378\pi$
$127$	29.9509			0.235834			0.117917	+	0.993023i	$0.462378\pi$
$128$		0			0
$129$		−	272.023i		−	2.10870i
$130$		0			0
$131$	116.486			0.889208			0.444604	−	0.895727i	$-0.353344\pi$
$131$	116.486			0.889208			0.444604	+	0.895727i	$0.353344\pi$
$132$		0			0
$133$	−208.909			−1.57075
$134$		0			0
$135$		−	30.5999i		−	0.226666i
$136$		0			0
$137$			94.8211i			0.692125i	0.938212	+	0.346062i	$0.112481\pi$
$137$			94.8211i			0.692125i	−0.938212	+	0.346062i	$0.887519\pi$
$138$		0			0
$139$	−89.2774			−0.642284			−0.321142	−	0.947031i	$-0.604067\pi$
$139$	−89.2774			−0.642284			−0.321142	+	0.947031i	$0.604067\pi$
$140$		0			0
$141$	24.3056			0.172380
$142$		0			0
$143$			225.960i			1.58014i
$144$		0			0
$145$		−	112.526i		−	0.776044i
$146$		0			0
$147$	−5.44599			−0.0370476
$148$		0			0
$149$			182.441i			1.22443i	0.790690	+	0.612217i	$0.209722\pi$
$149$			182.441i			1.22443i	−0.790690	+	0.612217i	$0.790278\pi$
$150$		0			0
$151$	−29.7608			−0.197092			−0.0985458	−	0.995133i	$-0.531419\pi$
$151$	−29.7608			−0.197092			−0.0985458	+	0.995133i	$0.531419\pi$
$152$		0			0
$153$		−	8.79728i		−	0.0574985i
$154$		0			0
$155$		−	24.8910i		−	0.160587i
$156$		0			0
$157$			64.1093i			0.408340i	0.978935	+	0.204170i	$0.0654495\pi$
$157$			64.1093i			0.408340i	−0.978935	+	0.204170i	$0.934551\pi$
$158$		0			0
$159$			77.4494i			0.487103i
$160$		0			0
$161$	−80.3097	−	142.235i	−0.498818	−	0.883446i
$162$		0			0
$163$	75.3328			0.462164			0.231082	−	0.972934i	$-0.425773\pi$
$163$	75.3328			0.462164			0.231082	+	0.972934i	$0.425773\pi$
$164$		0			0
$165$	−95.5577			−0.579137
$166$		0			0
$167$	−272.459			−1.63149			−0.815745	−	0.578412i	$-0.803673\pi$
$167$	−272.459			−1.63149			−0.815745	+	0.578412i	$0.803673\pi$
$168$		0			0
$169$	233.393			1.38102
$170$		0			0
$171$			158.639i			0.927713i
$172$		0			0
$173$	261.815			1.51338			0.756691	−	0.653773i	$-0.226815\pi$
$173$	261.815			1.51338			0.756691	+	0.653773i	$0.226815\pi$
$174$		0			0
$175$		−	35.5090i		−	0.202908i
$176$		0			0
$177$	249.969			1.41225
$178$		0			0
$179$	184.406			1.03020			0.515101	−	0.857130i	$-0.327754\pi$
$179$	184.406			1.03020			0.515101	+	0.857130i	$0.327754\pi$
$180$		0			0
$181$			191.867i			1.06004i	0.847986	+	0.530019i	$0.177815\pi$
$181$			191.867i			1.06004i	−0.847986	+	0.530019i	$0.822185\pi$
$182$		0			0
$183$			143.133i			0.782145i
$184$		0			0
$185$	90.6566			0.490036
$186$		0			0
$187$	18.3753			0.0982637
$188$		0			0
$189$		−	97.1859i		−	0.514211i
$190$		0			0
$191$			32.6185i			0.170778i	0.996348	+	0.0853888i	$0.0272132\pi$
$191$			32.6185i			0.170778i	−0.996348	+	0.0853888i	$0.972787\pi$
$192$		0			0
$193$	316.748			1.64118			0.820592	−	0.571515i	$-0.193644\pi$
$193$	316.748			1.64118			0.820592	+	0.571515i	$0.193644\pi$
$194$		0			0
$195$			170.170i			0.872669i
$196$		0			0
$197$	−194.946			−0.989573			−0.494787	−	0.869015i	$-0.664754\pi$
$197$	−194.946			−0.989573			−0.494787	+	0.869015i	$0.664754\pi$
$198$		0			0
$199$			74.0815i			0.372269i	0.982524	+	0.186134i	$0.0595960\pi$
$199$			74.0815i			0.372269i	−0.982524	+	0.186134i	$0.940404\pi$
$200$		0			0
$201$		−	471.350i		−	2.34503i
$202$		0			0
$203$		−	357.386i		−	1.76052i
$204$		0			0
$205$		−	16.1900i		−	0.0789757i
$206$		0			0
$207$	−108.008	+	60.9846i	−0.521780	+	0.294612i
$208$		0			0
$209$	−331.357			−1.58544
$210$		0			0
$211$	4.75017			0.0225126			0.0112563	−	0.999937i	$-0.496417\pi$
$211$	4.75017			0.0225126			0.0112563	+	0.999937i	$0.496417\pi$
$212$		0			0
$213$	−165.279			−0.775958
$214$		0			0
$215$	160.331			0.745724
$216$		0			0
$217$		−	79.0544i		−	0.364306i
$218$		0			0
$219$	182.555			0.833584
$220$		0			0
$221$		−	32.7230i		−	0.148068i
$222$		0			0
$223$	−211.977			−0.950571			−0.475286	−	0.879832i	$-0.657655\pi$
$223$	−211.977			−0.950571			−0.475286	+	0.879832i	$0.657655\pi$
$224$		0			0
$225$	−26.9644			−0.119842
$226$		0			0
$227$		−	389.941i		−	1.71780i	−0.512141	−	0.858901i	$-0.671148\pi$
$227$		−	389.941i		−	1.71780i	0.512141	−	0.858901i	$-0.328852\pi$
$228$		0			0
$229$		−	156.000i		−	0.681224i	−0.940204	−	0.340612i	$-0.889366\pi$
$229$		−	156.000i		−	0.681224i	0.940204	−	0.340612i	$-0.110634\pi$
$230$		0			0
$231$	−303.493			−1.31382
$232$		0			0
$233$	−46.4968			−0.199557			−0.0997785	−	0.995010i	$-0.531813\pi$
$233$	−46.4968			−0.199557			−0.0997785	+	0.995010i	$0.531813\pi$
$234$		0			0
$235$			14.3257i			0.0609606i
$236$		0			0
$237$			383.274i			1.61719i
$238$		0			0
$239$	454.735			1.90266			0.951328	−	0.308181i	$-0.0997201\pi$
$239$	454.735			1.90266			0.951328	+	0.308181i	$0.0997201\pi$
$240$		0			0
$241$			149.357i			0.619739i	0.950779	+	0.309870i	$0.100285\pi$
$241$			149.357i			0.619739i	−0.950779	+	0.309870i	$0.899715\pi$
$242$		0			0
$243$	−257.935			−1.06146
$244$		0			0
$245$		−	3.20988i		−	0.0131015i
$246$		0			0
$247$			590.085i			2.38901i
$248$		0			0
$249$			388.518i			1.56031i
$250$		0			0
$251$		−	364.513i		−	1.45224i	−0.687567	−	0.726121i	$-0.741321\pi$
$251$		−	364.513i		−	1.45224i	0.687567	−	0.726121i	$-0.258679\pi$
$252$		0			0
$253$	−127.382	−	225.603i	−0.503485	−	0.891711i
$254$		0			0
$255$	13.8384			0.0542683
$256$		0			0
$257$	59.7088			0.232330			0.116165	−	0.993230i	$-0.462940\pi$
$257$	59.7088			0.232330			0.116165	+	0.993230i	$0.462940\pi$
$258$		0			0
$259$	287.927			1.11169
$260$		0			0
$261$	−271.387			−1.03980
$262$		0			0
$263$		−	282.085i		−	1.07257i	−0.844038	−	0.536284i	$-0.819828\pi$
$263$		−	282.085i		−	1.07257i	0.844038	−	0.536284i	$-0.180172\pi$
$264$		0			0
$265$	−45.6488			−0.172260
$266$		0			0
$267$			36.5674i			0.136957i
$268$		0			0
$269$	14.7823			0.0549529			0.0274764	−	0.999622i	$-0.491253\pi$
$269$	14.7823			0.0549529			0.0274764	+	0.999622i	$0.491253\pi$
$270$		0			0
$271$	34.7150			0.128100			0.0640499	−	0.997947i	$-0.479598\pi$
$271$	34.7150			0.128100			0.0640499	+	0.997947i	$0.479598\pi$
$272$		0			0
$273$			540.465i			1.97972i
$274$		0			0
$275$		−	56.3218i		−	0.204807i
$276$		0			0
$277$	−191.042			−0.689682			−0.344841	−	0.938661i	$-0.612067\pi$
$277$	−191.042			−0.689682			−0.344841	+	0.938661i	$0.612067\pi$
$278$		0			0
$279$	−60.0314			−0.215166
$280$		0			0
$281$			471.349i			1.67740i	0.544596	+	0.838698i	$0.316683\pi$
$281$			471.349i			1.67740i	−0.544596	+	0.838698i	$0.683317\pi$
$282$		0			0
$283$		−	23.4603i		−	0.0828984i	−0.999141	−	0.0414492i	$-0.986803\pi$
$283$		−	23.4603i		−	0.0828984i	0.999141	−	0.0414492i	$-0.0131975\pi$
$284$		0			0
$285$	−249.545			−0.875595
$286$		0			0
$287$		−	51.4198i		−	0.179163i
$288$		0			0
$289$	286.339			0.990792
$290$		0			0
$291$		−	544.908i		−	1.87254i
$292$		0			0
$293$			289.288i			0.987331i	0.869652	+	0.493666i	$0.164343\pi$
$293$			289.288i			0.987331i	−0.869652	+	0.493666i	$0.835657\pi$
$294$		0			0
$295$			147.332i			0.499431i
$296$		0			0
$297$		−	154.149i		−	0.519022i
$298$		0			0
$299$	−401.756	+	226.843i	−1.34367	+	0.758672i
$300$		0			0
$301$	509.213			1.69174
$302$		0			0
$303$	391.136			1.29088
$304$		0			0
$305$	−84.3625			−0.276599
$306$		0			0
$307$	563.775			1.83640			0.918200	−	0.396117i	$-0.129642\pi$
$307$	563.775			1.83640			0.918200	+	0.396117i	$0.129642\pi$
$308$		0			0
$309$			375.068i			1.21381i
$310$		0			0
$311$	−76.9428			−0.247404			−0.123702	−	0.992319i	$-0.539477\pi$
$311$	−76.9428			−0.247404			−0.123702	+	0.992319i	$0.539477\pi$
$312$		0			0
$313$		−	436.773i		−	1.39544i	−0.716370	−	0.697721i	$-0.754198\pi$
$313$		−	436.773i		−	1.39544i	0.716370	−	0.697721i	$-0.245802\pi$
$314$		0			0
$315$	−85.6394			−0.271871
$316$		0			0
$317$	95.6774			0.301822			0.150911	−	0.988547i	$-0.451779\pi$
$317$	95.6774			0.301822			0.150911	+	0.988547i	$0.451779\pi$
$318$		0			0
$319$		−	566.860i		−	1.77699i
$320$		0			0
$321$			85.5476i			0.266503i
$322$		0			0
$323$	47.9863			0.148565
$324$		0			0
$325$	−100.299			−0.308611
$326$		0			0
$327$			114.720i			0.350827i
$328$		0			0
$329$			45.4988i			0.138294i
$330$		0			0
$331$	−515.137			−1.55631			−0.778153	−	0.628074i	$-0.783843\pi$
$331$	−515.137			−1.55631			−0.778153	+	0.628074i	$0.783843\pi$
$332$		0			0
$333$		−	218.643i		−	0.656584i
$334$		0			0
$335$	277.815			0.829297
$336$		0			0
$337$		−	251.793i		−	0.747161i	−0.927598	−	0.373580i	$-0.878130\pi$
$337$		−	251.793i		−	0.747161i	0.927598	−	0.373580i	$-0.121870\pi$
$338$		0			0
$339$			809.734i			2.38860i
$340$		0			0
$341$		−	125.391i		−	0.367714i
$342$		0			0
$343$			337.793i			0.984820i
$344$		0			0
$345$	−95.9309	−	169.901i	−0.278061	−	0.492467i
$346$		0			0
$347$	167.899			0.483858			0.241929	−	0.970294i	$-0.422220\pi$
$347$	167.899			0.483858			0.241929	+	0.970294i	$0.422220\pi$
$348$		0			0
$349$	131.699			0.377360			0.188680	−	0.982039i	$-0.439579\pi$
$349$	131.699			0.377360			0.188680	+	0.982039i	$0.439579\pi$
$350$		0			0
$351$	−274.511			−0.782084
$352$		0			0
$353$	232.683			0.659159			0.329580	−	0.944128i	$-0.393093\pi$
$353$	232.683			0.659159			0.329580	+	0.944128i	$0.393093\pi$
$354$		0			0
$355$		−	97.4157i		−	0.274410i
$356$		0			0
$357$	43.9511			0.123112
$358$		0			0
$359$			205.862i			0.573432i	0.958016	+	0.286716i	$0.0925636\pi$
$359$			205.862i			0.573432i	−0.958016	+	0.286716i	$0.907436\pi$
$360$		0			0
$361$	−504.325			−1.39702
$362$		0			0
$363$	−22.3302			−0.0615157
$364$		0			0
$365$			107.598i			0.294789i
$366$		0			0
$367$		−	158.406i		−	0.431623i	−0.976435	−	0.215811i	$-0.930760\pi$
$367$		−	158.406i		−	0.431623i	0.976435	−	0.215811i	$-0.0692396\pi$
$368$		0			0
$369$	−39.0465			−0.105817
$370$		0			0
$371$	−144.981			−0.390786
$372$		0			0
$373$			67.1037i			0.179903i	0.995946	+	0.0899513i	$0.0286712\pi$
$373$			67.1037i			0.179903i	−0.995946	+	0.0899513i	$0.971329\pi$
$374$		0			0
$375$		−	42.4159i		−	0.113109i
$376$		0			0
$377$	−1009.47			−2.67765
$378$		0			0
$379$			10.5032i			0.0277128i	0.999904	+	0.0138564i	$0.00441078\pi$
$379$			10.5032i			0.0277128i	−0.999904	+	0.0138564i	$0.995589\pi$
$380$		0			0
$381$	113.628			0.298235
$382$		0			0
$383$		−	360.978i		−	0.942501i	−0.882000	−	0.471250i	$-0.843803\pi$
$383$		−	360.978i		−	0.942501i	0.882000	−	0.471250i	$-0.156197\pi$
$384$		0			0
$385$		−	178.879i		−	0.464621i
$386$		0			0
$387$		−	386.680i		−	0.999173i
$388$		0			0
$389$		−	47.2280i		−	0.121409i	−0.998156	−	0.0607044i	$-0.980665\pi$
$389$		−	47.2280i		−	0.121409i	0.998156	−	0.0607044i	$-0.0193347\pi$
$390$		0			0
$391$	18.4471	+	32.6712i	0.0471793	+	0.0835582i
$392$		0			0
$393$	441.925			1.12449
$394$		0			0
$395$	−225.902			−0.571904
$396$		0			0
$397$	4.85826			0.0122374			0.00611872	−	0.999981i	$-0.498052\pi$
$397$	4.85826			0.0122374			0.00611872	+	0.999981i	$0.498052\pi$
$398$		0			0
$399$	−792.559			−1.98636
$400$		0			0
$401$		−	297.502i		−	0.741900i	−0.928653	−	0.370950i	$-0.879032\pi$
$401$		−	297.502i		−	0.741900i	0.928653	−	0.370950i	$-0.120968\pi$
$402$		0			0
$403$	−223.297			−0.554087
$404$		0			0
$405$		−	224.619i		−	0.554615i
$406$		0			0
$407$	456.690			1.12209
$408$		0			0
$409$	238.943			0.584213			0.292106	−	0.956386i	$-0.405644\pi$
$409$	238.943			0.584213			0.292106	+	0.956386i	$0.405644\pi$
$410$		0			0
$411$			359.732i			0.875259i
$412$		0			0
$413$			467.930i			1.13300i
$414$		0			0
$415$	−228.993			−0.551791
$416$		0			0
$417$	−338.700			−0.812231
$418$		0			0
$419$		−	197.316i		−	0.470920i	−0.971884	−	0.235460i	$-0.924340\pi$
$419$		−	197.316i		−	0.470920i	0.971884	−	0.235460i	$-0.0756597\pi$
$420$		0			0
$421$			459.256i			1.09087i	0.838153	+	0.545435i	$0.183635\pi$
$421$			459.256i			1.09087i	−0.838153	+	0.545435i	$0.816365\pi$
$422$		0			0
$423$	34.5503			0.0816793
$424$		0			0
$425$			8.15639i			0.0191915i
$426$		0			0
$427$	−267.937			−0.627487
$428$		0			0
$429$			857.247i			1.99824i
$430$		0			0
$431$			475.283i			1.10275i	0.834259	+	0.551373i	$0.185896\pi$
$431$			475.283i			1.10275i	−0.834259	+	0.551373i	$0.814104\pi$
$432$		0			0
$433$		−	694.309i		−	1.60349i	−0.597669	−	0.801743i	$-0.703906\pi$
$433$		−	694.309i		−	1.60349i	0.597669	−	0.801743i	$-0.296094\pi$
$434$		0			0
$435$		−	426.902i		−	0.981384i
$436$		0			0
$437$	−332.651	−	589.152i	−0.761216	−	1.34817i
$438$		0			0
$439$	692.132			1.57661			0.788305	−	0.615284i	$-0.210959\pi$
$439$	692.132			1.57661			0.788305	+	0.615284i	$0.210959\pi$
$440$		0			0
$441$	−7.74147			−0.0175544
$442$		0			0
$443$	282.065			0.636716			0.318358	−	0.947971i	$-0.396869\pi$
$443$	282.065			0.636716			0.318358	+	0.947971i	$0.396869\pi$
$444$		0			0
$445$	−21.5529			−0.0484335
$446$		0			0
$447$			692.142i			1.54842i
$448$		0			0
$449$	−4.51574			−0.0100573			−0.00502866	−	0.999987i	$-0.501601\pi$
$449$	−4.51574			−0.0100573			−0.00502866	+	0.999987i	$0.501601\pi$
$450$		0			0
$451$		−	81.5584i		−	0.180839i
$452$		0			0
$453$	−112.907			−0.249242
$454$		0			0
$455$	−318.551			−0.700111
$456$		0			0
$457$			399.040i			0.873174i	0.899662	+	0.436587i	$0.143813\pi$
$457$			399.040i			0.873174i	−0.899662	+	0.436587i	$0.856187\pi$
$458$		0			0
$459$			22.3235i			0.0486352i
$460$		0			0
$461$	44.2537			0.0959950			0.0479975	−	0.998847i	$-0.484716\pi$
$461$	44.2537			0.0959950			0.0479975	+	0.998847i	$0.484716\pi$
$462$		0			0
$463$	668.258			1.44332			0.721661	−	0.692247i	$-0.243379\pi$
$463$	668.258			1.44332			0.721661	+	0.692247i	$0.243379\pi$
$464$		0			0
$465$		−	94.4315i		−	0.203078i
$466$		0			0
$467$			670.150i			1.43501i	0.696554	+	0.717505i	$0.254716\pi$
$467$			670.150i			1.43501i	−0.696554	+	0.717505i	$0.745284\pi$
$468$		0			0
$469$	882.345			1.88133
$470$		0			0
$471$			243.218i			0.516385i
$472$		0			0
$473$	807.678			1.70756
$474$		0			0
$475$		−	147.082i		−	0.309646i
$476$		0			0
$477$			110.094i			0.230805i
$478$		0			0
$479$			310.492i			0.648209i	0.946021	+	0.324105i	$0.105063\pi$
$479$			310.492i			0.648209i	−0.946021	+	0.324105i	$0.894937\pi$
$480$		0			0
$481$		−	813.279i		−	1.69081i
$482$		0			0
$483$	−304.679	−	539.609i	−0.630804	−	1.11720i
$484$		0			0
$485$	321.169			0.662205
$486$		0			0
$487$	−829.644			−1.70358			−0.851790	−	0.523883i	$-0.824483\pi$
$487$	−829.644			−1.70358			−0.851790	+	0.523883i	$0.824483\pi$
$488$		0			0
$489$	285.797			0.584452
$490$		0			0
$491$	123.794			0.252126			0.126063	−	0.992022i	$-0.459766\pi$
$491$	123.794			0.252126			0.126063	+	0.992022i	$0.459766\pi$
$492$		0			0
$493$			82.0913i			0.166514i
$494$		0			0
$495$	−135.835			−0.274414
$496$		0			0
$497$		−	309.394i		−	0.622523i
$498$		0			0
$499$	−757.919			−1.51887			−0.759437	−	0.650580i	$-0.774526\pi$
$499$	−757.919			−1.51887			−0.759437	+	0.650580i	$0.774526\pi$
$500$		0			0
$501$	−1033.65			−2.06318
$502$		0			0
$503$		−	242.915i		−	0.482933i	−0.970409	−	0.241467i	$-0.922372\pi$
$503$		−	242.915i		−	0.482933i	0.970409	−	0.241467i	$-0.0776284\pi$
$504$		0			0
$505$			230.536i			0.456507i
$506$		0			0
$507$	885.445			1.74644
$508$		0			0
$509$	822.585			1.61608			0.808040	−	0.589127i	$-0.200528\pi$
$509$	822.585			1.61608			0.808040	+	0.589127i	$0.200528\pi$
$510$		0			0
$511$			341.734i			0.668755i
$512$		0			0
$513$		−	402.555i		−	0.784707i
$514$		0			0
$515$	−221.066			−0.429254
$516$		0			0
$517$			72.1670i			0.139588i
$518$		0			0
$519$	993.273			1.91382
$520$		0			0
$521$			95.3538i			0.183021i	0.995804	+	0.0915103i	$0.0291694\pi$
$521$			95.3538i			0.183021i	−0.995804	+	0.0915103i	$0.970831\pi$
$522$		0			0
$523$			277.463i			0.530522i	0.964177	+	0.265261i	$0.0854581\pi$
$523$			277.463i			0.530522i	−0.964177	+	0.265261i	$0.914542\pi$
$524$		0			0
$525$		−	134.714i		−	0.256598i
$526$		0			0
$527$			18.1588i			0.0344568i
$528$		0			0
$529$	273.242	−	452.968i	0.516525	−	0.856272i
$530$		0			0
$531$	355.331			0.669173
$532$		0			0
$533$	−145.240			−0.272496
$534$		0			0
$535$	−50.4219			−0.0942465
$536$		0			0
$537$	699.599			1.30279
$538$		0			0
$539$		−	16.1700i		−	0.0300000i
$540$		0			0
$541$	−666.108			−1.23125			−0.615627	−	0.788038i	$-0.711097\pi$
$541$	−666.108			−1.23125			−0.615627	+	0.788038i	$0.711097\pi$
$542$		0			0
$543$			727.903i			1.34052i
$544$		0			0
$545$	−67.6163			−0.124067
$546$		0			0
$547$	−349.611			−0.639143			−0.319571	−	0.947562i	$-0.603539\pi$
$547$	−349.611			−0.639143			−0.319571	+	0.947562i	$0.603539\pi$
$548$		0			0
$549$			203.463i			0.370606i
$550$		0			0
$551$		−	1480.33i		−	2.68663i
$552$		0			0
$553$	−717.469			−1.29741
$554$		0			0
$555$	343.933			0.619698
$556$		0			0
$557$			8.96150i			0.0160889i	0.999968	+	0.00804443i	$0.00256065\pi$
$557$			8.96150i			0.0160889i	−0.999968	+	0.00804443i	$0.997439\pi$
$558$		0			0
$559$		−	1438.32i		−	2.57303i
$560$		0			0
$561$	69.7122			0.124264
$562$		0			0
$563$			732.683i			1.30139i	0.759339	+	0.650696i	$0.225523\pi$
$563$			732.683i			1.30139i	−0.759339	+	0.650696i	$0.774477\pi$
$564$		0			0
$565$	−477.259			−0.844706
$566$		0			0
$567$		−	713.395i		−	1.25819i
$568$		0			0
$569$			42.5363i			0.0747563i	0.999301	+	0.0373781i	$0.0119006\pi$
$569$			42.5363i			0.0747563i	−0.999301	+	0.0373781i	$0.988099\pi$
$570$		0			0
$571$			459.356i			0.804476i	0.915535	+	0.402238i	$0.131768\pi$
$571$			459.356i			0.804476i	−0.915535	+	0.402238i	$0.868232\pi$
$572$		0			0
$573$			123.748i			0.215965i
$574$		0			0
$575$	100.140	−	56.5418i	0.174157	−	0.0983336i
$576$		0			0
$577$	831.608			1.44126			0.720630	−	0.693319i	$-0.243852\pi$
$577$	831.608			1.44126			0.720630	+	0.693319i	$0.243852\pi$
$578$		0			0
$579$	1201.68			2.07544
$580$		0			0
$581$	−727.287			−1.25179
$582$		0			0
$583$	−229.959			−0.394441
$584$		0			0
$585$			241.897i			0.413499i
$586$		0			0
$587$	166.970			0.284447			0.142223	−	0.989835i	$-0.454575\pi$
$587$	166.970			0.284447			0.142223	+	0.989835i	$0.454575\pi$
$588$		0			0
$589$		−	327.452i		−	0.555945i
$590$		0			0
$591$	−739.585			−1.25141
$592$		0			0
$593$	900.895			1.51922			0.759608	−	0.650381i	$-0.225391\pi$
$593$	900.895			1.51922			0.759608	+	0.650381i	$0.225391\pi$
$594$		0			0
$595$			25.9049i			0.0435376i
$596$		0			0
$597$			281.050i			0.470771i
$598$		0			0
$599$	129.221			0.215727			0.107864	−	0.994166i	$-0.465599\pi$
$599$	129.221			0.215727			0.107864	+	0.994166i	$0.465599\pi$
$600$		0			0
$601$	580.916			0.966582			0.483291	−	0.875460i	$-0.339441\pi$
$601$	580.916			0.966582			0.483291	+	0.875460i	$0.339441\pi$
$602$		0			0
$603$		−	670.024i		−	1.11115i
$604$		0			0
$605$		−	13.1615i		−	0.0217545i
$606$		0			0
$607$	1063.36			1.75183			0.875915	−	0.482466i	$-0.160259\pi$
$607$	1063.36			1.75183			0.875915	+	0.482466i	$0.160259\pi$
$608$		0			0
$609$		−	1355.85i		−	2.22635i
$610$		0			0
$611$	128.516			0.210337
$612$		0			0
$613$			442.412i			0.721715i	0.932621	+	0.360858i	$0.117516\pi$
$613$			442.412i			0.721715i	−0.932621	+	0.360858i	$0.882484\pi$
$614$		0			0
$615$		−	61.4216i		−	0.0998725i
$616$		0			0
$617$		−	936.724i		−	1.51819i	−0.650979	−	0.759095i	$-0.725642\pi$
$617$		−	936.724i		−	1.51819i	0.650979	−	0.759095i	$-0.274358\pi$
$618$		0			0
$619$		−	401.856i		−	0.649202i	−0.945851	−	0.324601i	$-0.894770\pi$
$619$		−	401.856i		−	0.649202i	0.945851	−	0.324601i	$-0.105230\pi$
$620$		0			0
$621$	274.077	−	154.752i	0.441348	−	0.249197i
$622$		0			0
$623$	−68.4524			−0.109875
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−1257.10			−2.00495
$628$		0			0
$629$	−66.1367			−0.105146
$630$		0			0
$631$		−	1033.92i		−	1.63854i	−0.573411	−	0.819268i	$-0.694380\pi$
$631$		−	1033.92i		−	1.63854i	0.573411	−	0.819268i	$-0.305620\pi$
$632$		0			0
$633$	18.0212			0.0284694
$634$		0			0
$635$			66.9722i			0.105468i
$636$		0			0
$637$	−28.7958			−0.0452053
$638$		0			0
$639$	−234.944			−0.367674
$640$		0			0
$641$		−	188.175i		−	0.293564i	−0.989169	−	0.146782i	$-0.953108\pi$
$641$		−	188.175i		−	0.293564i	0.989169	−	0.146782i	$-0.0468916\pi$
$642$		0			0
$643$		−	1055.82i		−	1.64203i	−0.570909	−	0.821013i	$-0.693409\pi$
$643$		−	1055.82i		−	1.64203i	0.570909	−	0.821013i	$-0.306591\pi$
$644$		0			0
$645$	608.261			0.943041
$646$		0			0
$647$	443.636			0.685681			0.342841	−	0.939394i	$-0.388611\pi$
$647$	443.636			0.685681			0.342841	+	0.939394i	$0.388611\pi$
$648$		0			0
$649$			742.197i			1.14360i
$650$		0			0
$651$		−	299.916i		−	0.460701i
$652$		0			0
$653$	117.465			0.179886			0.0899429	−	0.995947i	$-0.471332\pi$
$653$	117.465			0.179886			0.0899429	+	0.995947i	$0.471332\pi$
$654$		0			0
$655$			260.471i			0.397666i
$656$		0			0
$657$	259.502			0.394980
$658$		0			0
$659$			664.743i			1.00871i	0.863495	+	0.504357i	$0.168270\pi$
$659$			664.743i			1.00871i	−0.863495	+	0.504357i	$0.831730\pi$
$660$		0			0
$661$		−	426.231i		−	0.644828i	−0.946599	−	0.322414i	$-0.895506\pi$
$661$		−	426.231i		−	0.644828i	0.946599	−	0.322414i	$-0.104494\pi$
$662$		0			0
$663$		−	124.144i		−	0.187246i
$664$		0			0
$665$		−	467.135i		−	0.702459i
$666$		0			0
$667$	1007.88	−	569.075i	1.51106	−	0.853185i
$668$		0			0
$669$	−804.198			−1.20209
$670$		0			0
$671$	−424.983			−0.633358
$672$		0			0
$673$	−824.444			−1.22503			−0.612514	−	0.790460i	$-0.709842\pi$
$673$	−824.444			−1.22503			−0.612514	+	0.790460i	$0.709842\pi$
$674$		0			0
$675$	68.4235			0.101368
$676$		0			0
$677$			530.039i			0.782924i	0.920194	+	0.391462i	$0.128031\pi$
$677$			530.039i			0.782924i	−0.920194	+	0.391462i	$0.871969\pi$
$678$		0			0
$679$	1020.04			1.50227
$680$		0			0
$681$		−	1479.36i		−	2.17233i
$682$		0			0
$683$	−414.954			−0.607546			−0.303773	−	0.952744i	$-0.598246\pi$
$683$	−414.954			−0.607546			−0.303773	+	0.952744i	$0.598246\pi$
$684$		0			0
$685$	−212.026			−0.309527
$686$		0			0
$687$		−	591.833i		−	0.861475i
$688$		0			0
$689$			409.515i			0.594361i
$690$		0			0
$691$	363.154			0.525548			0.262774	−	0.964857i	$-0.415363\pi$
$691$	363.154			0.525548			0.262774	+	0.964857i	$0.415363\pi$
$692$		0			0
$693$	−431.415			−0.622532
$694$		0			0
$695$		−	199.630i		−	0.287238i
$696$		0			0
$697$			11.8111i			0.0169456i
$698$		0			0
$699$	−176.399			−0.252360
$700$		0			0
$701$		−	928.839i		−	1.32502i	−0.749053	−	0.662510i	$-0.769491\pi$
$701$		−	928.839i		−	1.32502i	0.749053	−	0.662510i	$-0.230509\pi$
$702$		0			0
$703$	1192.63			1.69648
$704$		0			0
$705$			54.3489i			0.0770906i
$706$		0			0
$707$			732.188i			1.03563i
$708$		0			0
$709$			44.8088i			0.0632000i	0.999501	+	0.0316000i	$0.0100603\pi$
$709$			44.8088i			0.0632000i	−0.999501	+	0.0316000i	$0.989940\pi$
$710$		0			0
$711$			544.823i			0.766277i
$712$		0			0
$713$	222.944	−	125.880i	0.312684	−	0.176550i
$714$		0			0
$715$	−505.263			−0.706661
$716$		0			0
$717$	1725.17			2.40609
$718$		0			0
$719$	−1308.36			−1.81969			−0.909844	−	0.414950i	$-0.863799\pi$
$719$	−1308.36			−1.81969			−0.909844	+	0.414950i	$0.863799\pi$
$720$		0			0
$721$	−702.110			−0.973800
$722$		0			0
$723$			566.630i			0.783721i
$724$		0			0
$725$	251.617			0.347057
$726$		0			0
$727$		−	515.858i		−	0.709570i	−0.934948	−	0.354785i	$-0.884554\pi$
$727$		−	515.858i		−	0.709570i	0.934948	−	0.354785i	$-0.115446\pi$
$728$		0			0
$729$	−74.4769			−0.102163
$730$		0			0
$731$	−116.966			−0.160008
$732$		0			0
$733$		−	405.638i		−	0.553394i	−0.960957	−	0.276697i	$-0.910760\pi$
$733$		−	405.638i		−	0.553394i	0.960957	−	0.276697i	$-0.0892398\pi$
$734$		0			0
$735$		−	12.1776i		−	0.0165682i
$736$		0			0
$737$	1399.51			1.89893
$738$		0			0
$739$	−601.397			−0.813798			−0.406899	−	0.913473i	$-0.633390\pi$
$739$	−601.397			−0.813798			−0.406899	+	0.913473i	$0.633390\pi$
$740$		0			0
$741$			2238.66i			3.02114i
$742$		0			0
$743$		−	775.124i		−	1.04324i	−0.853179	−	0.521618i	$-0.825329\pi$
$743$		−	775.124i		−	1.04324i	0.853179	−	0.521618i	$-0.174671\pi$
$744$		0			0
$745$	−407.950			−0.547583
$746$		0			0
$747$			552.278i			0.739328i
$748$		0			0
$749$	−160.141			−0.213806
$750$		0			0
$751$			533.250i			0.710054i	0.934856	+	0.355027i	$0.115528\pi$
$751$			533.250i			0.710054i	−0.934856	+	0.355027i	$0.884472\pi$
$752$		0			0
$753$		−	1382.89i		−	1.83650i
$754$		0			0
$755$		−	66.5473i		−	0.0881421i
$756$		0			0
$757$			794.660i			1.04975i	0.851180	+	0.524875i	$0.175888\pi$
$757$			794.660i			1.04975i	−0.851180	+	0.524875i	$0.824112\pi$
$758$		0			0
$759$	−483.260	−	855.890i	−0.636706	−	1.12766i
$760$		0			0
$761$	−1322.01			−1.73720			−0.868599	−	0.495515i	$-0.834979\pi$
$761$	−1322.01			−1.73720			−0.868599	+	0.495515i	$0.834979\pi$
$762$		0			0
$763$	−214.751			−0.281456
$764$		0			0
$765$	19.6713			0.0257141
$766$		0			0
$767$	1321.71			1.72323
$768$		0			0
$769$		−	1443.85i		−	1.87757i	−0.344500	−	0.938786i	$-0.611952\pi$
$769$		−	1443.85i		−	1.87757i	0.344500	−	0.938786i	$-0.388048\pi$
$770$		0			0
$771$	226.523			0.293804
$772$		0			0
$773$		−	64.8298i		−	0.0838678i	−0.999120	−	0.0419339i	$-0.986648\pi$
$773$		−	64.8298i		−	0.0838678i	0.999120	−	0.0419339i	$-0.0133519\pi$
$774$		0			0
$775$	55.6580			0.0718168
$776$		0			0
$777$	1092.34			1.40584
$778$		0			0
$779$		−	212.986i		−	0.273410i
$780$		0			0
$781$		−	490.739i		−	0.628347i
$782$		0			0
$783$	688.659			0.879514
$784$		0			0
$785$	−143.353			−0.182615
$786$		0			0
$787$		−	1247.04i		−	1.58455i	−0.610163	−	0.792276i	$-0.708896\pi$
$787$		−	1247.04i		−	1.58455i	0.610163	−	0.792276i	$-0.291104\pi$
$788$		0			0
$789$		−	1070.17i		−	1.35637i
$790$		0			0
$791$	−1515.78			−1.91629
$792$		0			0
$793$			756.815i			0.954370i
$794$		0			0
$795$	−173.182			−0.217839
$796$		0			0
$797$			965.218i			1.21106i	0.795821	+	0.605532i	$0.207040\pi$
$797$			965.218i			1.21106i	−0.795821	+	0.605532i	$0.792960\pi$
$798$		0			0
$799$		−	10.4511i		−	0.0130802i
$800$		0			0
$801$			51.9805i			0.0648946i
$802$		0			0
$803$			542.034i			0.675011i
$804$		0			0
$805$	318.047	−	179.578i	0.395089	−	0.223078i
$806$		0			0
$807$	56.0811			0.0694933
$808$		0			0
$809$	−1310.48			−1.61988			−0.809939	−	0.586514i	$-0.800500\pi$
$809$	−1310.48			−1.61988			−0.809939	+	0.586514i	$0.800500\pi$
$810$		0			0
$811$	1174.00			1.44760			0.723799	−	0.690010i	$-0.242394\pi$
$811$	1174.00			1.44760			0.723799	+	0.690010i	$0.242394\pi$
$812$		0			0
$813$	131.702			0.161995
$814$		0			0
$815$			168.449i			0.206686i
$816$		0			0
$817$	2109.22			2.58166
$818$		0			0
$819$			768.270i			0.938058i
$820$		0			0
$821$	−1238.00			−1.50791			−0.753956	−	0.656925i	$-0.771857\pi$
$821$	−1238.00			−1.50791			−0.753956	+	0.656925i	$0.771857\pi$
$822$		0			0
$823$	937.653			1.13931			0.569656	−	0.821883i	$-0.307077\pi$
$823$	937.653			1.13931			0.569656	+	0.821883i	$0.307077\pi$
$824$		0			0
$825$		−	213.673i		−	0.258998i
$826$		0			0
$827$			199.863i			0.241672i	0.992672	+	0.120836i	$0.0385575\pi$
$827$			199.863i			0.241672i	−0.992672	+	0.120836i	$0.961443\pi$
$828$		0			0
$829$	−892.787			−1.07695			−0.538473	−	0.842643i	$-0.680998\pi$
$829$	−892.787			−1.07695			−0.538473	+	0.842643i	$0.680998\pi$
$830$		0			0
$831$	−724.774			−0.872171
$832$		0			0
$833$			2.34170i			0.00281117i
$834$		0			0
$835$		−	609.236i		−	0.729624i
$836$		0			0
$837$	152.333			0.181998
$838$		0			0
$839$		−	515.108i		−	0.613955i	−0.951717	−	0.306977i	$-0.900682\pi$
$839$		−	515.108i		−	0.613955i	0.951717	−	0.306977i	$-0.0993176\pi$
$840$		0			0
$841$	1691.44			2.01122
$842$		0			0
$843$			1788.20i			2.12123i
$844$		0			0
$845$			521.883i			0.617613i
$846$		0			0
$847$		−	41.8010i		−	0.0493519i
$848$		0			0
$849$		−	89.0034i		−	0.104833i
$850$		0			0
$851$	458.474	+	811.993i	0.538747	+	0.954163i
$852$		0			0
$853$	−483.197			−0.566467			−0.283234	−	0.959051i	$-0.591407\pi$
$853$	−483.197			−0.566467			−0.283234	+	0.959051i	$0.591407\pi$
$854$		0			0
$855$	−354.727			−0.414886
$856$		0			0
$857$	920.413			1.07399			0.536997	−	0.843584i	$-0.319559\pi$
$857$	920.413			1.07399			0.536997	+	0.843584i	$0.319559\pi$
$858$		0			0
$859$	−1173.30			−1.36589			−0.682947	−	0.730468i	$-0.739302\pi$
$859$	−1173.30			−1.36589			−0.682947	+	0.730468i	$0.739302\pi$
$860$		0			0
$861$		−	195.076i		−	0.226569i
$862$		0			0
$863$	−866.353			−1.00388			−0.501942	−	0.864901i	$-0.667381\pi$
$863$	−866.353			−1.00388			−0.501942	+	0.864901i	$0.667381\pi$
$864$		0			0
$865$			585.436i			0.676805i
$866$		0			0
$867$	1086.31			1.25295
$868$		0			0
$869$	−1138.00			−1.30955
$870$		0			0
$871$		−	2492.27i		−	2.86139i
$872$		0			0
$873$		−	774.586i		−	0.887269i
$874$		0			0
$875$	79.4005			0.0907434
$876$		0			0
$877$	281.590			0.321084			0.160542	−	0.987029i	$-0.448676\pi$
$877$	281.590			0.321084			0.160542	+	0.987029i	$0.448676\pi$
$878$		0			0
$879$			1097.50i			1.24858i
$880$		0			0
$881$		−	919.123i		−	1.04327i	−0.853168	−	0.521636i	$-0.825322\pi$
$881$		−	919.123i		−	1.04327i	0.853168	−	0.521636i	$-0.174678\pi$
$882$		0			0
$883$	821.800			0.930691			0.465346	−	0.885129i	$-0.345930\pi$
$883$	821.800			0.930691			0.465346	+	0.885129i	$0.345930\pi$
$884$		0			0
$885$			558.948i			0.631579i
$886$		0			0
$887$	−1280.91			−1.44409			−0.722047	−	0.691844i	$-0.756799\pi$
$887$	−1280.91			−1.44409			−0.722047	+	0.691844i	$0.756799\pi$
$888$		0			0
$889$			212.705i			0.239263i
$890$		0			0
$891$		−	1131.54i		−	1.26996i
$892$		0			0
$893$			188.461i			0.211043i
$894$		0			0
$895$			412.345i			0.460720i
$896$		0			0
$897$	−1524.18	+	860.595i	−1.69920	+	0.959415i
$898$		0			0
$899$	560.180			0.623114
$900$		0			0
$901$	33.3022			0.0369613
$902$		0			0
$903$	1931.85			2.13937
$904$		0			0
$905$	−429.027			−0.474063
$906$		0			0
$907$			1726.39i			1.90341i	0.307021	+	0.951703i	$0.400668\pi$
$907$			1726.39i			1.90341i	−0.307021	+	0.951703i	$0.599332\pi$
$908$		0			0
$909$	556.000			0.611661
$910$		0			0
$911$		−	791.171i		−	0.868464i	−0.900801	−	0.434232i	$-0.857020\pi$
$911$		−	791.171i		−	0.868464i	0.900801	−	0.434232i	$-0.142980\pi$
$912$		0			0
$913$	−1153.57			−1.26350
$914$		0			0
$915$	−320.054			−0.349786
$916$		0			0
$917$			827.262i			0.902139i
$918$		0			0
$919$		−	1272.36i		−	1.38451i	−0.721655	−	0.692253i	$-0.756618\pi$
$919$		−	1272.36i		−	1.38451i	0.721655	−	0.692253i	$-0.243382\pi$
$920$		0			0
$921$	2138.85			2.32231
$922$		0			0
$923$	−873.915			−0.946820
$924$		0			0
$925$			202.714i			0.219151i
$926$		0			0
$927$			533.159i			0.575145i
$928$		0			0
$929$	672.653			0.724062			0.362031	−	0.932166i	$-0.382083\pi$
$929$	672.653			0.724062			0.362031	+	0.932166i	$0.382083\pi$
$930$		0			0
$931$		−	42.2273i		−	0.0453569i
$932$		0			0
$933$	−291.905			−0.312867
$934$		0			0
$935$			41.0885i			0.0439449i
$936$		0			0
$937$			1599.57i			1.70711i	0.520999	+	0.853557i	$0.325560\pi$
$937$			1599.57i			1.70711i	−0.520999	+	0.853557i	$0.674440\pi$
$938$		0			0
$939$		−	1657.03i		−	1.76467i
$940$		0			0
$941$		−	1628.06i		−	1.73014i	−0.501651	−	0.865070i	$-0.667274\pi$
$941$		−	1628.06i		−	1.73014i	0.501651	−	0.865070i	$-0.332726\pi$
$942$		0			0
$943$	145.011	−	81.8770i	0.153776	−	0.0868261i
$944$		0			0
$945$	217.314			0.229962
$946$		0			0
$947$	1829.88			1.93229			0.966147	−	0.257992i	$-0.0830608\pi$
$947$	1829.88			1.93229			0.966147	+	0.257992i	$0.0830608\pi$
$948$		0			0
$949$	965.262			1.01714
$950$		0			0
$951$	362.980			0.381683
$952$		0			0
$953$		−	655.714i		−	0.688053i	−0.938960	−	0.344026i	$-0.888209\pi$
$953$		−	655.714i		−	0.688053i	0.938960	−	0.344026i	$-0.111791\pi$
$954$		0			0
$955$	−72.9373			−0.0763741
$956$		0			0
$957$		−	2150.55i		−	2.24718i
$958$		0			0
$959$	−673.400			−0.702190
$960$		0			0
$961$	−837.087			−0.871059
$962$		0			0
$963$			121.606i			0.126278i
$964$		0			0
$965$			708.271i			0.733960i
$966$		0			0
$967$	−143.641			−0.148543			−0.0742716	−	0.997238i	$-0.523663\pi$
$967$	−143.641			−0.148543			−0.0742716	+	0.997238i	$0.523663\pi$
$968$		0			0
$969$	182.050			0.187874
$970$		0			0
$971$			1107.60i			1.14068i	0.821408	+	0.570341i	$0.193189\pi$
$971$			1107.60i			1.14068i	−0.821408	+	0.570341i	$0.806811\pi$
$972$		0			0
$973$		−	634.030i		−	0.651624i
$974$		0			0
$975$	−380.513			−0.390269
$976$		0			0
$977$			626.322i			0.641066i	0.947237	+	0.320533i	$0.103862\pi$
$977$			626.322i			0.641066i	−0.947237	+	0.320533i	$0.896138\pi$
$978$		0			0
$979$	−108.574			−0.110903
$980$		0			0
$981$			163.075i			0.166233i
$982$		0			0
$983$			202.538i			0.206041i	0.994679	+	0.103020i	$0.0328507\pi$
$983$			202.538i			0.206041i	−0.994679	+	0.103020i	$0.967149\pi$
$984$		0			0
$985$		−	435.912i		−	0.442551i
$986$		0			0
$987$			172.613i			0.174887i
$988$		0			0
$989$	810.833	+	1436.05i	0.819851	+	1.45202i
$990$		0			0
$991$	−354.302			−0.357520			−0.178760	−	0.983893i	$-0.557209\pi$
$991$	−354.302			−0.357520			−0.178760	+	0.983893i	$0.557209\pi$
$992$		0			0
$993$	−1954.33			−1.96810
$994$		0			0
$995$	−165.651			−0.166484
$996$		0			0
$997$	184.596			0.185152			0.0925758	−	0.995706i	$-0.470490\pi$
$997$	184.596			0.185152			0.0925758	+	0.995706i	$0.470490\pi$
$998$		0			0
$999$			554.817i			0.555372i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.14		16
4.3	odd	2		230.3.d.a.91.10	yes	16
12.11	even	2		2070.3.c.a.91.1		16
20.3	even	4		1150.3.c.c.1149.9		32
20.7	even	4		1150.3.c.c.1149.24		32
20.19	odd	2		1150.3.d.b.551.8		16
23.22	odd	2	inner	1840.3.k.d.321.13		16
92.91	even	2		230.3.d.a.91.9	✓	16
276.275	odd	2		2070.3.c.a.91.8		16
460.183	odd	4		1150.3.c.c.1149.23		32
460.367	odd	4		1150.3.c.c.1149.10		32
460.459	even	2		1150.3.d.b.551.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.9	✓	16	92.91	even	2
230.3.d.a.91.10	yes	16	4.3	odd	2
1150.3.c.c.1149.9		32	20.3	even	4
1150.3.c.c.1149.10		32	460.367	odd	4
1150.3.c.c.1149.23		32	460.183	odd	4
1150.3.c.c.1149.24		32	20.7	even	4
1150.3.d.b.551.7		16	460.459	even	2
1150.3.d.b.551.8		16	20.19	odd	2
1840.3.k.d.321.13		16	23.22	odd	2	inner
1840.3.k.d.321.14		16	1.1	even	1	trivial
2070.3.c.a.91.1		16	12.11	even	2
2070.3.c.a.91.8		16	276.275	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.79379 q^{3} +2.23607i q^{5} +7.10180i q^{7} +5.39287 q^{9} +O(q^{10})\)	\(q+3.79379 q^{3} +2.23607i q^{5} +7.10180i q^{7} +5.39287 q^{9} +11.2644i q^{11} +20.0597 q^{13} +8.48318i q^{15} -1.63128i q^{17} +29.4164i q^{19} +26.9428i q^{21} +(-20.0280 + 11.3084i) q^{23} -5.00000 q^{25} -13.6847 q^{27} -50.3233 q^{29} -11.1316 q^{31} +42.7347i q^{33} -15.8801 q^{35} -40.5429i q^{37} +76.1025 q^{39} -7.24039 q^{41} -71.7020i q^{43} +12.0588i q^{45} +6.40666 q^{47} -1.43550 q^{49} -6.18873i q^{51} +20.4148i q^{53} -25.1879 q^{55} +111.600i q^{57} +65.8889 q^{59} +37.7281i q^{61} +38.2991i q^{63} +44.8549i q^{65} -124.242i q^{67} +(-75.9821 + 42.9016i) q^{69} -43.5656 q^{71} +48.1194 q^{73} -18.9690 q^{75} -79.9972 q^{77} +101.026i q^{79} -100.453 q^{81} +102.409i q^{83} +3.64765 q^{85} -190.916 q^{87} +9.63875i q^{89} +142.460i q^{91} -42.2310 q^{93} -65.7771 q^{95} -143.631i q^{97} +60.7473i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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