LMFDB - Embedded newform 1840.3.k.c.321.2

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} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + \cdots + 1535848276 $ x^16 - 64x^14 - 16x^13 + 2252x^12 + 648x^11 - 30106x^10 + 12360x^9 + 374528x^8 + 196544x^7 + 1261236x^6 - 4237944x^5 + 38013345x^4 - 8913096x^3 + 327235800x^2 + 72566336x + 1535848276
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.2
Root			$5.89296 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.c.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.89296 q^{3} +2.23607i q^{5} -1.21480i q^{7} +25.7269 q^{9} +O(q^{10})$	$q-5.89296 q^{3} +2.23607i q^{5} -1.21480i q^{7} +25.7269 q^{9} -19.1790i q^{11} -6.79358 q^{13} -13.1771i q^{15} -21.3706i q^{17} -20.5552i q^{19} +7.15878i q^{21} +(13.0815 + 18.9176i) q^{23} -5.00000 q^{25} -98.5711 q^{27} +15.1176 q^{29} -7.27990 q^{31} +113.021i q^{33} +2.71638 q^{35} -51.4644i q^{37} +40.0343 q^{39} +14.1873 q^{41} +37.4927i q^{43} +57.5272i q^{45} +58.7126 q^{47} +47.5243 q^{49} +125.936i q^{51} +24.6357i q^{53} +42.8856 q^{55} +121.131i q^{57} +68.9831 q^{59} -98.9407i q^{61} -31.2532i q^{63} -15.1909i q^{65} +52.1637i q^{67} +(-77.0886 - 111.480i) q^{69} -82.9700 q^{71} -68.0710 q^{73} +29.4648 q^{75} -23.2987 q^{77} -3.71508i q^{79} +349.333 q^{81} -90.1400i q^{83} +47.7860 q^{85} -89.0872 q^{87} +103.447i q^{89} +8.25286i q^{91} +42.9001 q^{93} +45.9629 q^{95} +73.2599i q^{97} -493.418i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * 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
$3$	−5.89296			−1.96432			−0.982159	−	0.188050i	$-0.939783\pi$
$3$	−5.89296			−1.96432			−0.982159	+	0.188050i	$0.939783\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	1.21480i		−	0.173543i	−0.996228	−	0.0867716i	$-0.972345\pi$
$7$		−	1.21480i		−	0.173543i	0.996228	−	0.0867716i	$-0.0276550\pi$
$8$		0			0
$9$	25.7269			2.85855
$10$		0			0
$11$		−	19.1790i		−	1.74355i	−0.489908	−	0.871774i	$-0.662970\pi$
$11$		−	19.1790i		−	1.74355i	0.489908	−	0.871774i	$-0.337030\pi$
$12$		0			0
$13$	−6.79358			−0.522583			−0.261291	−	0.965260i	$-0.584148\pi$
$13$	−6.79358			−0.522583			−0.261291	+	0.965260i	$0.584148\pi$
$14$		0			0
$15$		−	13.1771i		−	0.878470i
$16$		0			0
$17$		−	21.3706i		−	1.25709i	−0.777772	−	0.628546i	$-0.783650\pi$
$17$		−	21.3706i		−	1.25709i	0.777772	−	0.628546i	$-0.216350\pi$
$18$		0			0
$19$		−	20.5552i		−	1.08185i	−0.841070	−	0.540927i	$-0.818074\pi$
$19$		−	20.5552i		−	1.08185i	0.841070	−	0.540927i	$-0.181926\pi$
$20$		0			0
$21$			7.15878i			0.340894i
$22$		0			0
$23$	13.0815	+	18.9176i	0.568760	+	0.822503i
$23$	13.0815	+	18.9176i	0.568760	+	0.822503i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−98.5711			−3.65078
$28$		0			0
$29$	15.1176			0.521296			0.260648	−	0.965434i	$-0.416064\pi$
$29$	15.1176			0.521296			0.260648	+	0.965434i	$0.416064\pi$
$30$		0			0
$31$	−7.27990			−0.234836			−0.117418	−	0.993083i	$-0.537462\pi$
$31$	−7.27990			−0.234836			−0.117418	+	0.993083i	$0.537462\pi$
$32$		0			0
$33$			113.021i			3.42488i
$34$		0			0
$35$	2.71638			0.0776109
$36$		0			0
$37$		−	51.4644i		−	1.39093i	−0.718560	−	0.695465i	$-0.755199\pi$
$37$		−	51.4644i		−	1.39093i	0.718560	−	0.695465i	$-0.244801\pi$
$38$		0			0
$39$	40.0343			1.02652
$40$		0			0
$41$	14.1873			0.346032			0.173016	−	0.984919i	$-0.444649\pi$
$41$	14.1873			0.346032			0.173016	+	0.984919i	$0.444649\pi$
$42$		0			0
$43$			37.4927i			0.871924i	0.899965	+	0.435962i	$0.143592\pi$
$43$			37.4927i			0.871924i	−0.899965	+	0.435962i	$0.856408\pi$
$44$		0			0
$45$			57.5272i			1.27838i
$46$		0			0
$47$	58.7126			1.24920			0.624602	−	0.780943i	$-0.285261\pi$
$47$	58.7126			1.24920			0.624602	+	0.780943i	$0.285261\pi$
$48$		0			0
$49$	47.5243			0.969883
$50$		0			0
$51$			125.936i			2.46933i
$52$		0			0
$53$			24.6357i			0.464824i	0.972617	+	0.232412i	$0.0746618\pi$
$53$			24.6357i			0.464824i	−0.972617	+	0.232412i	$0.925338\pi$
$54$		0			0
$55$	42.8856			0.779739
$56$		0			0
$57$			121.131i			2.12511i
$58$		0			0
$59$	68.9831			1.16920			0.584602	−	0.811320i	$-0.301251\pi$
$59$	68.9831			1.16920			0.584602	+	0.811320i	$0.301251\pi$
$60$		0			0
$61$		−	98.9407i		−	1.62198i	−0.585061	−	0.810989i	$-0.698930\pi$
$61$		−	98.9407i		−	1.62198i	0.585061	−	0.810989i	$-0.301070\pi$
$62$		0			0
$63$		−	31.2532i		−	0.496082i
$64$		0			0
$65$		−	15.1909i		−	0.233706i
$66$		0			0
$67$			52.1637i			0.778563i	0.921119	+	0.389281i	$0.127277\pi$
$67$			52.1637i			0.778563i	−0.921119	+	0.389281i	$0.872723\pi$
$68$		0			0
$69$	−77.0886	−	111.480i	−1.11723	−	1.61566i
$70$		0			0
$71$	−82.9700			−1.16859			−0.584296	−	0.811541i	$-0.698629\pi$
$71$	−82.9700			−1.16859			−0.584296	+	0.811541i	$0.698629\pi$
$72$		0			0
$73$	−68.0710			−0.932480			−0.466240	−	0.884658i	$-0.654392\pi$
$73$	−68.0710			−0.932480			−0.466240	+	0.884658i	$0.654392\pi$
$74$		0			0
$75$	29.4648			0.392864
$76$		0			0
$77$	−23.2987			−0.302581
$78$		0			0
$79$		−	3.71508i		−	0.0470264i	−0.999724	−	0.0235132i	$-0.992515\pi$
$79$		−	3.71508i		−	0.0470264i	0.999724	−	0.0235132i	$-0.00748517\pi$
$80$		0			0
$81$	349.333			4.31275
$82$		0			0
$83$		−	90.1400i		−	1.08602i	−0.839725	−	0.543012i	$-0.817284\pi$
$83$		−	90.1400i		−	1.08602i	0.839725	−	0.543012i	$-0.182716\pi$
$84$		0			0
$85$	47.7860			0.562189
$86$		0			0
$87$	−89.0872			−1.02399
$88$		0			0
$89$			103.447i			1.16233i	0.813786	+	0.581164i	$0.197403\pi$
$89$			103.447i			1.16233i	−0.813786	+	0.581164i	$0.802597\pi$
$90$		0			0
$91$			8.25286i			0.0906907i
$92$		0			0
$93$	42.9001			0.461292
$94$		0			0
$95$	45.9629			0.483820
$96$		0			0
$97$			73.2599i			0.755257i	0.925957	+	0.377629i	$0.123260\pi$
$97$			73.2599i			0.755257i	−0.925957	+	0.377629i	$0.876740\pi$
$98$		0			0
$99$		−	493.418i		−	4.98402i
$100$		0			0
$101$	−12.7274			−0.126014			−0.0630069	−	0.998013i	$-0.520069\pi$
$101$	−12.7274			−0.126014			−0.0630069	+	0.998013i	$0.520069\pi$
$102$		0			0
$103$		−	99.7786i		−	0.968725i	−0.874867	−	0.484362i	$-0.839052\pi$
$103$		−	99.7786i		−	0.968725i	0.874867	−	0.484362i	$-0.160948\pi$
$104$		0			0
$105$	−16.0075			−0.152453
$106$		0			0
$107$			185.420i			1.73290i	0.499263	+	0.866450i	$0.333604\pi$
$107$			185.420i			1.73290i	−0.499263	+	0.866450i	$0.666396\pi$
$108$		0			0
$109$		−	47.7532i		−	0.438102i	−0.975713	−	0.219051i	$-0.929704\pi$
$109$		−	47.7532i		−	0.438102i	0.975713	−	0.219051i	$-0.0702962\pi$
$110$		0			0
$111$			303.277i			2.73223i
$112$		0			0
$113$		−	60.3411i		−	0.533992i	−0.963698	−	0.266996i	$-0.913969\pi$
$113$		−	60.3411i		−	0.533992i	0.963698	−	0.266996i	$-0.0860310\pi$
$114$		0			0
$115$	−42.3010	+	29.2511i	−0.367835	+	0.254357i
$116$		0			0
$117$	−174.778			−1.49383
$118$		0			0
$119$	−25.9610			−0.218160
$120$		0			0
$121$	−246.835			−2.03996
$122$		0			0
$123$	−83.6052			−0.679717
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	91.1969			0.718086			0.359043	−	0.933321i	$-0.383103\pi$
$127$	91.1969			0.718086			0.359043	+	0.933321i	$0.383103\pi$
$128$		0			0
$129$		−	220.943i		−	1.71274i
$130$		0			0
$131$	−227.631			−1.73764			−0.868822	−	0.495125i	$-0.835122\pi$
$131$	−227.631			−1.73764			−0.868822	+	0.495125i	$0.835122\pi$
$132$		0			0
$133$	−24.9705			−0.187748
$134$		0			0
$135$		−	220.412i		−	1.63268i
$136$		0			0
$137$			60.9616i			0.444975i	0.974936	+	0.222488i	$0.0714177\pi$
$137$			60.9616i			0.444975i	−0.974936	+	0.222488i	$0.928582\pi$
$138$		0			0
$139$	153.735			1.10601			0.553003	−	0.833180i	$-0.313482\pi$
$139$	153.735			1.10601			0.553003	+	0.833180i	$0.313482\pi$
$140$		0			0
$141$	−345.991			−2.45383
$142$		0			0
$143$			130.294i			0.911148i
$144$		0			0
$145$			33.8039i			0.233130i
$146$		0			0
$147$	−280.058			−1.90516
$148$		0			0
$149$		−	126.021i		−	0.845776i	−0.906182	−	0.422888i	$-0.861016\pi$
$149$		−	126.021i		−	0.845776i	0.906182	−	0.422888i	$-0.138984\pi$
$150$		0			0
$151$	243.700			1.61391			0.806955	−	0.590613i	$-0.201114\pi$
$151$	243.700			1.61391			0.806955	+	0.590613i	$0.201114\pi$
$152$		0			0
$153$		−	549.799i		−	3.59346i
$154$		0			0
$155$		−	16.2784i		−	0.105022i
$156$		0			0
$157$		−	158.005i		−	1.00640i	−0.864170	−	0.503200i	$-0.832156\pi$
$157$		−	158.005i		−	1.00640i	0.864170	−	0.503200i	$-0.167844\pi$
$158$		0			0
$159$		−	145.177i		−	0.913063i
$160$		0			0
$161$	22.9811	−	15.8914i	0.142740	−	0.0987045i
$162$		0			0
$163$	−67.8522			−0.416271			−0.208136	−	0.978100i	$-0.566740\pi$
$163$	−67.8522			−0.416271			−0.208136	+	0.978100i	$0.566740\pi$
$164$		0			0
$165$	−252.723			−1.53166
$166$		0			0
$167$	−163.592			−0.979591			−0.489795	−	0.871837i	$-0.662929\pi$
$167$	−163.592			−0.979591			−0.489795	+	0.871837i	$0.662929\pi$
$168$		0			0
$169$	−122.847			−0.726907
$170$		0			0
$171$		−	528.823i		−	3.09253i
$172$		0			0
$173$	−252.094			−1.45719			−0.728597	−	0.684943i	$-0.759827\pi$
$173$	−252.094			−1.45719			−0.728597	+	0.684943i	$0.759827\pi$
$174$		0			0
$175$			6.07401i			0.0347086i
$176$		0			0
$177$	−406.514			−2.29669
$178$		0			0
$179$	−106.081			−0.592630			−0.296315	−	0.955090i	$-0.595758\pi$
$179$	−106.081			−0.592630			−0.296315	+	0.955090i	$0.595758\pi$
$180$		0			0
$181$		−	63.8813i		−	0.352936i	−0.984306	−	0.176468i	$-0.943533\pi$
$181$		−	63.8813i		−	0.352936i	0.984306	−	0.176468i	$-0.0564671\pi$
$182$		0			0
$183$			583.053i			3.18608i
$184$		0			0
$185$	115.078			0.622042
$186$		0			0
$187$	−409.867			−2.19180
$188$		0			0
$189$			119.744i			0.633569i
$190$		0			0
$191$		−	247.573i		−	1.29619i	−0.761559	−	0.648096i	$-0.775566\pi$
$191$		−	247.573i		−	1.29619i	0.761559	−	0.648096i	$-0.224434\pi$
$192$		0			0
$193$	−108.671			−0.563063			−0.281531	−	0.959552i	$-0.590842\pi$
$193$	−108.671			−0.563063			−0.281531	+	0.959552i	$0.590842\pi$
$194$		0			0
$195$			89.5193i			0.459073i
$196$		0			0
$197$	−127.968			−0.649586			−0.324793	−	0.945785i	$-0.605295\pi$
$197$	−127.968			−0.649586			−0.324793	+	0.945785i	$0.605295\pi$
$198$		0			0
$199$		−	187.445i		−	0.941937i	−0.882150	−	0.470968i	$-0.843905\pi$
$199$		−	187.445i		−	0.941937i	0.882150	−	0.470968i	$-0.156095\pi$
$200$		0			0
$201$		−	307.398i		−	1.52935i
$202$		0			0
$203$		−	18.3649i		−	0.0904673i
$204$		0			0
$205$			31.7238i			0.154750i
$206$		0			0
$207$	336.547	+	486.691i	1.62583	+	2.35117i
$208$		0			0
$209$	−394.229			−1.88626
$210$		0			0
$211$	204.271			0.968111			0.484055	−	0.875037i	$-0.339163\pi$
$211$	204.271			0.968111			0.484055	+	0.875037i	$0.339163\pi$
$212$		0			0
$213$	488.938			2.29549
$214$		0			0
$215$	−83.8363			−0.389936
$216$		0			0
$217$			8.84364i			0.0407541i
$218$		0			0
$219$	401.140			1.83169
$220$		0			0
$221$			145.183i			0.656935i
$222$		0			0
$223$	225.207			1.00990			0.504948	−	0.863150i	$-0.331512\pi$
$223$	225.207			1.00990			0.504948	+	0.863150i	$0.331512\pi$
$224$		0			0
$225$	−128.635			−0.571710
$226$		0			0
$227$		−	148.291i		−	0.653264i	−0.945152	−	0.326632i	$-0.894086\pi$
$227$		−	148.291i		−	0.653264i	0.945152	−	0.326632i	$-0.105914\pi$
$228$		0			0
$229$		−	69.3950i		−	0.303035i	−0.988455	−	0.151518i	$-0.951584\pi$
$229$		−	69.3950i		−	0.303035i	0.988455	−	0.151518i	$-0.0484160\pi$
$230$		0			0
$231$	137.298			0.594366
$232$		0			0
$233$	134.731			0.578244			0.289122	−	0.957292i	$-0.406637\pi$
$233$	134.731			0.578244			0.289122	+	0.957292i	$0.406637\pi$
$234$		0			0
$235$			131.285i			0.558661i
$236$		0			0
$237$			21.8928i			0.0923748i
$238$		0			0
$239$	156.166			0.653413			0.326707	−	0.945126i	$-0.394061\pi$
$239$	156.166			0.653413			0.326707	+	0.945126i	$0.394061\pi$
$240$		0			0
$241$			3.20765i			0.0133097i	0.999978	+	0.00665487i	$0.00211833\pi$
$241$			3.20765i			0.0133097i	−0.999978	+	0.00665487i	$0.997882\pi$
$242$		0			0
$243$	−1171.46			−4.82084
$244$		0			0
$245$			106.267i			0.433745i
$246$		0			0
$247$			139.644i			0.565358i
$248$		0			0
$249$			531.191i			2.13330i
$250$		0			0
$251$		−	172.125i		−	0.685758i	−0.939380	−	0.342879i	$-0.888598\pi$
$251$		−	172.125i		−	0.685758i	0.939380	−	0.342879i	$-0.111402\pi$
$252$		0			0
$253$	362.821	−	250.890i	1.43407	−	0.991661i
$254$		0			0
$255$	−281.601			−1.10432
$256$		0			0
$257$	184.115			0.716402			0.358201	−	0.933644i	$-0.383390\pi$
$257$	184.115			0.716402			0.358201	+	0.933644i	$0.383390\pi$
$258$		0			0
$259$	−62.5191			−0.241386
$260$		0			0
$261$	388.929			1.49015
$262$		0			0
$263$		−	344.445i		−	1.30968i	−0.755768	−	0.654839i	$-0.772736\pi$
$263$		−	344.445i		−	1.30968i	0.755768	−	0.654839i	$-0.227264\pi$
$264$		0			0
$265$	−55.0871			−0.207876
$266$		0			0
$267$		−	609.610i		−	2.28318i
$268$		0			0
$269$	367.014			1.36437			0.682183	−	0.731181i	$-0.261031\pi$
$269$	367.014			1.36437			0.682183	+	0.731181i	$0.261031\pi$
$270$		0			0
$271$	−314.590			−1.16085			−0.580425	−	0.814314i	$-0.697114\pi$
$271$	−314.590			−1.16085			−0.580425	+	0.814314i	$0.697114\pi$
$272$		0			0
$273$		−	48.6337i		−	0.178145i
$274$		0			0
$275$			95.8952i			0.348710i
$276$		0			0
$277$	−26.5629			−0.0958951			−0.0479476	−	0.998850i	$-0.515268\pi$
$277$	−26.5629			−0.0958951			−0.0479476	+	0.998850i	$0.515268\pi$
$278$		0			0
$279$	−187.290			−0.671289
$280$		0			0
$281$			63.1722i			0.224812i	0.993662	+	0.112406i	$0.0358557\pi$
$281$			63.1722i			0.224812i	−0.993662	+	0.112406i	$0.964144\pi$
$282$		0			0
$283$			215.448i			0.761302i	0.924719	+	0.380651i	$0.124300\pi$
$283$			215.448i			0.761302i	−0.924719	+	0.380651i	$0.875700\pi$
$284$		0			0
$285$	−270.857			−0.950376
$286$		0			0
$287$		−	17.2348i		−	0.0600515i
$288$		0			0
$289$	−167.701			−0.580280
$290$		0			0
$291$		−	431.718i		−	1.48357i
$292$		0			0
$293$		−	293.367i		−	1.00125i	−0.865663	−	0.500626i	$-0.833103\pi$
$293$		−	293.367i		−	1.00125i	0.865663	−	0.500626i	$-0.166897\pi$
$294$		0			0
$295$			154.251i			0.522884i
$296$		0			0
$297$			1890.50i			6.36532i
$298$		0			0
$299$	−88.8701	−	128.518i	−0.297224	−	0.429826i
$300$		0			0
$301$	45.5462			0.151316
$302$		0			0
$303$	75.0020			0.247531
$304$		0			0
$305$	221.238			0.725371
$306$		0			0
$307$	50.1619			0.163394			0.0816970	−	0.996657i	$-0.473966\pi$
$307$	50.1619			0.163394			0.0816970	+	0.996657i	$0.473966\pi$
$308$		0			0
$309$			587.991i			1.90288i
$310$		0			0
$311$	−547.613			−1.76081			−0.880407	−	0.474219i	$-0.842731\pi$
$311$	−547.613			−1.76081			−0.880407	+	0.474219i	$0.842731\pi$
$312$		0			0
$313$		−	18.7281i		−	0.0598342i	−0.999552	−	0.0299171i	$-0.990476\pi$
$313$		−	18.7281i		−	0.0598342i	0.999552	−	0.0299171i	$-0.00952433\pi$
$314$		0			0
$315$	69.8842			0.221855
$316$		0			0
$317$	73.9497			0.233280			0.116640	−	0.993174i	$-0.462788\pi$
$317$	73.9497			0.233280			0.116640	+	0.993174i	$0.462788\pi$
$318$		0			0
$319$		−	289.940i		−	0.908904i
$320$		0			0
$321$		−	1092.67i		−	3.40397i
$322$		0			0
$323$	−439.277			−1.35999
$324$		0			0
$325$	33.9679			0.104517
$326$		0			0
$327$			281.407i			0.860573i
$328$		0			0
$329$		−	71.3242i		−	0.216791i
$330$		0			0
$331$	−332.772			−1.00535			−0.502677	−	0.864474i	$-0.667651\pi$
$331$	−332.772			−1.00535			−0.502677	+	0.864474i	$0.667651\pi$
$332$		0			0
$333$		−	1324.02i		−	3.97604i
$334$		0			0
$335$	−116.642			−0.348184
$336$		0			0
$337$		−	316.494i		−	0.939152i	−0.882892	−	0.469576i	$-0.844407\pi$
$337$		−	316.494i		−	0.939152i	0.882892	−	0.469576i	$-0.155593\pi$
$338$		0			0
$339$			355.587i			1.04893i
$340$		0			0
$341$			139.621i			0.409447i
$342$		0			0
$343$		−	117.258i		−	0.341860i
$344$		0			0
$345$	249.278	−	172.375i	0.722545	−	0.499639i
$346$		0			0
$347$	176.028			0.507286			0.253643	−	0.967298i	$-0.418371\pi$
$347$	176.028			0.507286			0.253643	+	0.967298i	$0.418371\pi$
$348$		0			0
$349$	−458.207			−1.31291			−0.656457	−	0.754363i	$-0.727946\pi$
$349$	−458.207			−1.31291			−0.656457	+	0.754363i	$0.727946\pi$
$350$		0			0
$351$	669.651			1.90784
$352$		0			0
$353$	62.0444			0.175763			0.0878816	−	0.996131i	$-0.471990\pi$
$353$	62.0444			0.175763			0.0878816	+	0.996131i	$0.471990\pi$
$354$		0			0
$355$		−	185.526i		−	0.522610i
$356$		0			0
$357$	152.987			0.428535
$358$		0			0
$359$		−	190.457i		−	0.530520i	−0.964177	−	0.265260i	$-0.914542\pi$
$359$		−	190.457i		−	0.530520i	0.964177	−	0.265260i	$-0.0854578\pi$
$360$		0			0
$361$	−61.5173			−0.170408
$362$		0			0
$363$	1454.59			4.00713
$364$		0			0
$365$		−	152.211i		−	0.417018i
$366$		0			0
$367$			491.688i			1.33975i	0.742474	+	0.669875i	$0.233652\pi$
$367$			491.688i			1.33975i	−0.742474	+	0.669875i	$0.766348\pi$
$368$		0			0
$369$	364.996			0.989149
$370$		0			0
$371$	29.9275			0.0806671
$372$		0			0
$373$			726.490i			1.94769i	0.227204	+	0.973847i	$0.427042\pi$
$373$			726.490i			1.94769i	−0.227204	+	0.973847i	$0.572958\pi$
$374$		0			0
$375$			65.8853i			0.175694i
$376$		0			0
$377$	−102.702			−0.272420
$378$		0			0
$379$			165.365i			0.436320i	0.975913	+	0.218160i	$0.0700055\pi$
$379$			165.365i			0.436320i	−0.975913	+	0.218160i	$0.929995\pi$
$380$		0			0
$381$	−537.420			−1.41055
$382$		0			0
$383$		−	654.544i		−	1.70899i	−0.519457	−	0.854497i	$-0.673866\pi$
$383$		−	654.544i		−	1.70899i	0.519457	−	0.854497i	$-0.326134\pi$
$384$		0			0
$385$		−	52.0976i		−	0.135318i
$386$		0			0
$387$			964.573i			2.49244i
$388$		0			0
$389$			660.920i			1.69902i	0.527570	+	0.849512i	$0.323103\pi$
$389$			660.920i			1.69902i	−0.527570	+	0.849512i	$0.676897\pi$
$390$		0			0
$391$	404.279	−	279.559i	1.03396	−	0.714984i
$392$		0			0
$393$	1341.42			3.41329
$394$		0			0
$395$	8.30718			0.0210308
$396$		0			0
$397$	−255.774			−0.644267			−0.322133	−	0.946694i	$-0.604400\pi$
$397$	−255.774			−0.644267			−0.322133	+	0.946694i	$0.604400\pi$
$398$		0			0
$399$	147.150			0.368798
$400$		0			0
$401$		−	607.765i		−	1.51562i	−0.652474	−	0.757811i	$-0.726269\pi$
$401$		−	607.765i		−	1.51562i	0.652474	−	0.757811i	$-0.273731\pi$
$402$		0			0
$403$	49.4566			0.122721
$404$		0			0
$405$			781.132i			1.92872i
$406$		0			0
$407$	−987.037			−2.42515
$408$		0			0
$409$	−49.7513			−0.121641			−0.0608207	−	0.998149i	$-0.519372\pi$
$409$	−49.7513			−0.121641			−0.0608207	+	0.998149i	$0.519372\pi$
$410$		0			0
$411$		−	359.244i		−	0.874073i
$412$		0			0
$413$		−	83.8008i		−	0.202908i
$414$		0			0
$415$	201.559			0.485685
$416$		0			0
$417$	−905.952			−2.17255
$418$		0			0
$419$		−	613.574i		−	1.46438i	−0.681102	−	0.732188i	$-0.738499\pi$
$419$		−	613.574i		−	1.46438i	0.681102	−	0.732188i	$-0.261501\pi$
$420$		0			0
$421$			780.040i			1.85283i	0.376507	+	0.926414i	$0.377125\pi$
$421$			780.040i			1.85283i	−0.376507	+	0.926414i	$0.622875\pi$
$422$		0			0
$423$	1510.49			3.57091
$424$		0			0
$425$			106.853i			0.251418i
$426$		0			0
$427$	−120.193			−0.281483
$428$		0			0
$429$		−	767.818i		−	1.78979i
$430$		0			0
$431$			421.761i			0.978565i	0.872125	+	0.489282i	$0.162741\pi$
$431$			421.761i			0.978565i	−0.872125	+	0.489282i	$0.837259\pi$
$432$		0			0
$433$			198.107i			0.457521i	0.973483	+	0.228761i	$0.0734673\pi$
$433$			198.107i			0.457521i	−0.973483	+	0.228761i	$0.926533\pi$
$434$		0			0
$435$		−	199.205i		−	0.457943i
$436$		0			0
$437$	388.855	−	268.893i	0.889829	−	0.615315i
$438$		0			0
$439$	−382.455			−0.871195			−0.435598	−	0.900141i	$-0.643463\pi$
$439$	−382.455			−0.871195			−0.435598	+	0.900141i	$0.643463\pi$
$440$		0			0
$441$	1222.65			2.77246
$442$		0			0
$443$	−589.236			−1.33010			−0.665052	−	0.746797i	$-0.731591\pi$
$443$	−589.236			−1.33010			−0.665052	+	0.746797i	$0.731591\pi$
$444$		0			0
$445$	−231.315			−0.519809
$446$		0			0
$447$			742.634i			1.66137i
$448$		0			0
$449$	417.233			0.929249			0.464624	−	0.885508i	$-0.346189\pi$
$449$	417.233			0.929249			0.464624	+	0.885508i	$0.346189\pi$
$450$		0			0
$451$		−	272.099i		−	0.603324i
$452$		0			0
$453$	−1436.12			−3.17023
$454$		0			0
$455$	−18.4539			−0.0405581
$456$		0			0
$457$			304.457i			0.666207i	0.942890	+	0.333104i	$0.108096\pi$
$457$			304.457i			0.666207i	−0.942890	+	0.333104i	$0.891904\pi$
$458$		0			0
$459$			2106.52i			4.58937i
$460$		0			0
$461$	−398.642			−0.864734			−0.432367	−	0.901698i	$-0.642322\pi$
$461$	−398.642			−0.864734			−0.432367	+	0.901698i	$0.642322\pi$
$462$		0			0
$463$	254.794			0.550310			0.275155	−	0.961400i	$-0.411271\pi$
$463$	254.794			0.550310			0.275155	+	0.961400i	$0.411271\pi$
$464$		0			0
$465$			95.9276i			0.206296i
$466$		0			0
$467$			170.860i			0.365868i	0.983125	+	0.182934i	$0.0585594\pi$
$467$			170.860i			0.365868i	−0.983125	+	0.182934i	$0.941441\pi$
$468$		0			0
$469$	63.3686			0.135114
$470$		0			0
$471$			931.115i			1.97689i
$472$		0			0
$473$	719.074			1.52024
$474$		0			0
$475$			102.776i			0.216371i
$476$		0			0
$477$			633.801i			1.32872i
$478$		0			0
$479$			254.888i			0.532126i	0.963956	+	0.266063i	$0.0857229\pi$
$479$			254.888i			0.532126i	−0.963956	+	0.266063i	$0.914277\pi$
$480$		0			0
$481$			349.627i			0.726876i
$482$		0			0
$483$	−135.427	+	93.6475i	−0.280387	+	0.193887i
$484$		0			0
$485$	−163.814			−0.337761
$486$		0			0
$487$	−280.911			−0.576819			−0.288410	−	0.957507i	$-0.593126\pi$
$487$	−280.911			−0.576819			−0.288410	+	0.957507i	$0.593126\pi$
$488$		0			0
$489$	399.850			0.817690
$490$		0			0
$491$	365.779			0.744966			0.372483	−	0.928039i	$-0.378506\pi$
$491$	365.779			0.744966			0.372483	+	0.928039i	$0.378506\pi$
$492$		0			0
$493$		−	323.071i		−	0.655317i
$494$		0			0
$495$	1103.32			2.22892
$496$		0			0
$497$			100.792i			0.202801i
$498$		0			0
$499$	−301.893			−0.604996			−0.302498	−	0.953150i	$-0.597821\pi$
$499$	−301.893			−0.604996			−0.302498	+	0.953150i	$0.597821\pi$
$500$		0			0
$501$	964.039			1.92423
$502$		0			0
$503$		−	881.655i		−	1.75279i	−0.481590	−	0.876397i	$-0.659940\pi$
$503$		−	881.655i		−	1.75279i	0.481590	−	0.876397i	$-0.340060\pi$
$504$		0			0
$505$		−	28.4593i		−	0.0563551i
$506$		0			0
$507$	723.934			1.42788
$508$		0			0
$509$	639.641			1.25666			0.628331	−	0.777946i	$-0.283738\pi$
$509$	639.641			1.25666			0.628331	+	0.777946i	$0.283738\pi$
$510$		0			0
$511$			82.6929i			0.161826i
$512$		0			0
$513$			2026.15i			3.94961i
$514$		0			0
$515$	223.112			0.433227
$516$		0			0
$517$		−	1126.05i		−	2.17805i
$518$		0			0
$519$	1485.58			2.86239
$520$		0			0
$521$		−	45.5463i		−	0.0874209i	−0.999044	−	0.0437104i	$-0.986082\pi$
$521$		−	45.5463i		−	0.0874209i	0.999044	−	0.0437104i	$-0.0139179\pi$
$522$		0			0
$523$		−	100.120i		−	0.191434i	−0.995409	−	0.0957172i	$-0.969486\pi$
$523$		−	100.120i		−	0.191434i	0.995409	−	0.0957172i	$-0.0305144\pi$
$524$		0			0
$525$		−	35.7939i		−	0.0681788i
$526$		0			0
$527$			155.576i			0.295210i
$528$		0			0
$529$	−186.749	+	494.940i	−0.353024	+	0.935614i
$530$		0			0
$531$	1774.72			3.34223
$532$		0			0
$533$	−96.3826			−0.180830
$534$		0			0
$535$	−414.613			−0.774977
$536$		0			0
$537$	625.130			1.16411
$538$		0			0
$539$		−	911.469i		−	1.69104i
$540$		0			0
$541$	−89.5038			−0.165441			−0.0827207	−	0.996573i	$-0.526361\pi$
$541$	−89.5038			−0.165441			−0.0827207	+	0.996573i	$0.526361\pi$
$542$		0			0
$543$			376.450i			0.693278i
$544$		0			0
$545$	106.779			0.195925
$546$		0			0
$547$	−536.646			−0.981071			−0.490535	−	0.871421i	$-0.663199\pi$
$547$	−536.646			−0.981071			−0.490535	+	0.871421i	$0.663199\pi$
$548$		0			0
$549$		−	2545.44i		−	4.63650i
$550$		0			0
$551$		−	310.745i		−	0.563966i
$552$		0			0
$553$	−4.51309			−0.00816111
$554$		0			0
$555$	−678.149			−1.22189
$556$		0			0
$557$		−	600.369i		−	1.07786i	−0.842350	−	0.538931i	$-0.818828\pi$
$557$		−	600.369i		−	1.07786i	0.842350	−	0.538931i	$-0.181172\pi$
$558$		0			0
$559$		−	254.710i		−	0.455652i
$560$		0			0
$561$	2415.33			4.30540
$562$		0			0
$563$		−	723.749i		−	1.28552i	−0.766067	−	0.642761i	$-0.777789\pi$
$563$		−	723.749i		−	1.28552i	0.766067	−	0.642761i	$-0.222211\pi$
$564$		0			0
$565$	134.927			0.238808
$566$		0			0
$567$		−	424.371i		−	0.748449i
$568$		0			0
$569$			707.638i			1.24365i	0.783156	+	0.621826i	$0.213609\pi$
$569$			707.638i			1.24365i	−0.783156	+	0.621826i	$0.786391\pi$
$570$		0			0
$571$			1007.22i			1.76396i	0.471287	+	0.881980i	$0.343790\pi$
$571$			1007.22i			1.76396i	−0.471287	+	0.881980i	$0.656210\pi$
$572$		0			0
$573$			1458.93i			2.54613i
$574$		0			0
$575$	−65.4074	−	94.5879i	−0.113752	−	0.164501i
$576$		0			0
$577$	280.046			0.485349			0.242674	−	0.970108i	$-0.421975\pi$
$577$	280.046			0.485349			0.242674	+	0.970108i	$0.421975\pi$
$578$		0			0
$579$	640.394			1.10603
$580$		0			0
$581$	−109.502			−0.188472
$582$		0			0
$583$	472.489			0.810443
$584$		0			0
$585$		−	390.815i		−	0.668060i
$586$		0			0
$587$	−1165.03			−1.98473			−0.992363	−	0.123352i	$-0.960635\pi$
$587$	−1165.03			−1.98473			−0.992363	+	0.123352i	$0.960635\pi$
$588$		0			0
$589$			149.640i			0.254058i
$590$		0			0
$591$	754.112			1.27599
$592$		0			0
$593$	−982.609			−1.65701			−0.828507	−	0.559979i	$-0.810809\pi$
$593$	−982.609			−1.65701			−0.828507	+	0.559979i	$0.810809\pi$
$594$		0			0
$595$		−	58.0506i		−	0.0975640i
$596$		0			0
$597$			1104.61i			1.85026i
$598$		0			0
$599$	22.9515			0.0383163			0.0191581	−	0.999816i	$-0.493901\pi$
$599$	22.9515			0.0383163			0.0191581	+	0.999816i	$0.493901\pi$
$600$		0			0
$601$	475.853			0.791769			0.395885	−	0.918300i	$-0.370438\pi$
$601$	475.853			0.791769			0.395885	+	0.918300i	$0.370438\pi$
$602$		0			0
$603$			1342.01i			2.22556i
$604$		0			0
$605$		−	551.940i		−	0.912298i
$606$		0			0
$607$	−754.060			−1.24227			−0.621137	−	0.783702i	$-0.713329\pi$
$607$	−754.060			−1.24227			−0.621137	+	0.783702i	$0.713329\pi$
$608$		0			0
$609$			108.223i			0.177707i
$610$		0			0
$611$	−398.868			−0.652812
$612$		0			0
$613$			559.933i			0.913431i	0.889613	+	0.456715i	$0.150974\pi$
$613$			559.933i			0.913431i	−0.889613	+	0.456715i	$0.849026\pi$
$614$		0			0
$615$		−	186.947i		−	0.303979i
$616$		0			0
$617$			657.899i			1.06629i	0.846025	+	0.533144i	$0.178989\pi$
$617$			657.899i			1.06629i	−0.846025	+	0.533144i	$0.821011\pi$
$618$		0			0
$619$			648.359i			1.04743i	0.851893	+	0.523715i	$0.175454\pi$
$619$			648.359i			1.04743i	−0.851893	+	0.523715i	$0.824546\pi$
$620$		0			0
$621$	−1289.46	−	1864.73i	−2.07642	−	3.00278i
$622$		0			0
$623$	125.668			0.201714
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	2323.18			3.70523
$628$		0			0
$629$	−1099.82			−1.74853
$630$		0			0
$631$		−	426.772i		−	0.676342i	−0.941085	−	0.338171i	$-0.890192\pi$
$631$		−	426.772i		−	0.676342i	0.941085	−	0.338171i	$-0.109808\pi$
$632$		0			0
$633$	−1203.76			−1.90168
$634$		0			0
$635$			203.923i			0.321138i
$636$		0			0
$637$	−322.860			−0.506844
$638$		0			0
$639$	−2134.56			−3.34047
$640$		0			0
$641$		−	933.107i		−	1.45570i	−0.685734	−	0.727852i	$-0.740519\pi$
$641$		−	933.107i		−	1.45570i	0.685734	−	0.727852i	$-0.259481\pi$
$642$		0			0
$643$			887.579i			1.38037i	0.723632	+	0.690186i	$0.242471\pi$
$643$			887.579i			1.38037i	−0.723632	+	0.690186i	$0.757529\pi$
$644$		0			0
$645$	494.043			0.765959
$646$		0			0
$647$	733.964			1.13441			0.567206	−	0.823576i	$-0.308024\pi$
$647$	733.964			1.13441			0.567206	+	0.823576i	$0.308024\pi$
$648$		0			0
$649$		−	1323.03i		−	2.03856i
$650$		0			0
$651$		−	52.1152i		−	0.0800541i
$652$		0			0
$653$	−1156.08			−1.77041			−0.885207	−	0.465198i	$-0.845983\pi$
$653$	−1156.08			−1.77041			−0.885207	+	0.465198i	$0.845983\pi$
$654$		0			0
$655$		−	508.999i		−	0.777098i
$656$		0			0
$657$	−1751.26			−2.66554
$658$		0			0
$659$		−	21.5959i		−	0.0327707i	−0.999866	−	0.0163853i	$-0.994784\pi$
$659$		−	21.5959i		−	0.0327707i	0.999866	−	0.0163853i	$-0.00521585\pi$
$660$		0			0
$661$		−	529.666i		−	0.801310i	−0.916229	−	0.400655i	$-0.868783\pi$
$661$		−	529.666i		−	0.801310i	0.916229	−	0.400655i	$-0.131217\pi$
$662$		0			0
$663$		−	855.555i		−	1.29043i
$664$		0			0
$665$		−	55.8358i		−	0.0839636i
$666$		0			0
$667$	197.760	+	285.988i	0.296492	+	0.428767i
$668$		0			0
$669$	−1327.13			−1.98376
$670$		0			0
$671$	−1897.59			−2.82800
$672$		0			0
$673$	−551.811			−0.819927			−0.409964	−	0.912102i	$-0.634459\pi$
$673$	−551.811			−0.819927			−0.409964	+	0.912102i	$0.634459\pi$
$674$		0			0
$675$	492.856			0.730156
$676$		0			0
$677$			732.982i			1.08269i	0.840800	+	0.541346i	$0.182085\pi$
$677$			732.982i			1.08269i	−0.840800	+	0.541346i	$0.817915\pi$
$678$		0			0
$679$	88.9964			0.131070
$680$		0			0
$681$			873.872i			1.28322i
$682$		0			0
$683$	−444.506			−0.650814			−0.325407	−	0.945574i	$-0.605501\pi$
$683$	−444.506			−0.650814			−0.325407	+	0.945574i	$0.605501\pi$
$684$		0			0
$685$	−136.314			−0.198999
$686$		0			0
$687$			408.942i			0.595258i
$688$		0			0
$689$		−	167.364i		−	0.242909i
$690$		0			0
$691$	613.573			0.887950			0.443975	−	0.896039i	$-0.353568\pi$
$691$	613.573			0.887950			0.443975	+	0.896039i	$0.353568\pi$
$692$		0			0
$693$	−599.405			−0.864943
$694$		0			0
$695$			343.761i			0.494621i
$696$		0			0
$697$		−	303.191i		−	0.434994i
$698$		0			0
$699$	−793.963			−1.13586
$700$		0			0
$701$			539.961i			0.770272i	0.922860	+	0.385136i	$0.125845\pi$
$701$			539.961i			0.770272i	−0.922860	+	0.385136i	$0.874155\pi$
$702$		0			0
$703$	−1057.86			−1.50478
$704$		0			0
$705$		−	773.659i		−	1.09739i
$706$		0			0
$707$			15.4613i			0.0218689i
$708$		0			0
$709$			2.68407i			0.00378571i	0.999998	+	0.00189286i	$0.000602515\pi$
$709$			2.68407i			0.00378571i	−0.999998	+	0.00189286i	$0.999397\pi$
$710$		0			0
$711$		−	95.5777i		−	0.134427i
$712$		0			0
$713$	−95.2319	−	137.718i	−0.133565	−	0.193153i
$714$		0			0
$715$	−291.347			−0.407478
$716$		0			0
$717$	−920.278			−1.28351
$718$		0			0
$719$	−28.2700			−0.0393185			−0.0196592	−	0.999807i	$-0.506258\pi$
$719$	−28.2700			−0.0393185			−0.0196592	+	0.999807i	$0.506258\pi$
$720$		0			0
$721$	−121.211			−0.168116
$722$		0			0
$723$		−	18.9025i		−	0.0261446i
$724$		0			0
$725$	−75.5879			−0.104259
$726$		0			0
$727$		−	305.976i		−	0.420875i	−0.977607	−	0.210437i	$-0.932511\pi$
$727$		−	305.976i		−	0.420875i	0.977607	−	0.210437i	$-0.0674888\pi$
$728$		0			0
$729$	3759.39			5.15691
$730$		0			0
$731$	801.240			1.09609
$732$		0			0
$733$		−	635.448i		−	0.866914i	−0.901174	−	0.433457i	$-0.857294\pi$
$733$		−	635.448i		−	0.866914i	0.901174	−	0.433457i	$-0.142706\pi$
$734$		0			0
$735$		−	626.230i		−	0.852013i
$736$		0			0
$737$	1000.45			1.35746
$738$		0			0
$739$	1273.79			1.72367			0.861834	−	0.507191i	$-0.169316\pi$
$739$	1273.79			1.72367			0.861834	+	0.507191i	$0.169316\pi$
$740$		0			0
$741$		−	822.913i		−	1.11054i
$742$		0			0
$743$		−	680.588i		−	0.916000i	−0.888952	−	0.458000i	$-0.848566\pi$
$743$		−	680.588i		−	0.916000i	0.888952	−	0.458000i	$-0.151434\pi$
$744$		0			0
$745$	281.791			0.378242
$746$		0			0
$747$		−	2319.03i		−	3.10445i
$748$		0			0
$749$	225.249			0.300733
$750$		0			0
$751$		−	391.564i		−	0.521390i	−0.965421	−	0.260695i	$-0.916048\pi$
$751$		−	391.564i		−	0.521390i	0.965421	−	0.260695i	$-0.0839517\pi$
$752$		0			0
$753$			1014.33i			1.34705i
$754$		0			0
$755$			544.931i			0.721763i
$756$		0			0
$757$		−	1212.06i		−	1.60114i	−0.599238	−	0.800571i	$-0.704530\pi$
$757$		−	1212.06i		−	1.60114i	0.599238	−	0.800571i	$-0.295470\pi$
$758$		0			0
$759$	−2138.09	+	1478.49i	−2.81698	+	1.94794i
$760$		0			0
$761$	−1190.50			−1.56439			−0.782196	−	0.623032i	$-0.785901\pi$
$761$	−1190.50			−1.56439			−0.782196	+	0.623032i	$0.785901\pi$
$762$		0			0
$763$	−58.0107			−0.0760297
$764$		0			0
$765$	1229.39			1.60704
$766$		0			0
$767$	−468.642			−0.611006
$768$		0			0
$769$			228.068i			0.296577i	0.988944	+	0.148289i	$0.0473765\pi$
$769$			228.068i			0.296577i	−0.988944	+	0.148289i	$0.952624\pi$
$770$		0			0
$771$	−1084.98			−1.40724
$772$		0			0
$773$			245.060i			0.317025i	0.987357	+	0.158513i	$0.0506698\pi$
$773$			245.060i			0.317025i	−0.987357	+	0.158513i	$0.949330\pi$
$774$		0			0
$775$	36.3995			0.0469671
$776$		0			0
$777$	368.422			0.474160
$778$		0			0
$779$		−	291.623i		−	0.374356i
$780$		0			0
$781$			1591.28i			2.03750i
$782$		0			0
$783$	−1490.16			−1.90314
$784$		0			0
$785$	353.309			0.450075
$786$		0			0
$787$			749.447i			0.952283i	0.879369	+	0.476141i	$0.157965\pi$
$787$			749.447i			0.952283i	−0.879369	+	0.476141i	$0.842035\pi$
$788$		0			0
$789$			2029.80i			2.57263i
$790$		0			0
$791$	−73.3025			−0.0926707
$792$		0			0
$793$			672.161i			0.847618i
$794$		0			0
$795$	324.626			0.408334
$796$		0			0
$797$		−	223.282i		−	0.280152i	−0.990141	−	0.140076i	$-0.955265\pi$
$797$		−	223.282i		−	0.280152i	0.990141	−	0.140076i	$-0.0447348\pi$
$798$		0			0
$799$		−	1254.72i		−	1.57036i
$800$		0			0
$801$			2661.38i			3.32257i
$802$		0			0
$803$			1305.54i			1.62582i
$804$		0			0
$805$	35.5343	+	51.3874i	0.0441420	+	0.0638352i
$806$		0			0
$807$	−2162.80			−2.68005
$808$		0			0
$809$	−1224.34			−1.51340			−0.756699	−	0.653763i	$-0.773189\pi$
$809$	−1224.34			−1.51340			−0.756699	+	0.653763i	$0.773189\pi$
$810$		0			0
$811$	895.177			1.10379			0.551897	−	0.833912i	$-0.313904\pi$
$811$	895.177			1.10379			0.551897	+	0.833912i	$0.313904\pi$
$812$		0			0
$813$	1853.87			2.28028
$814$		0			0
$815$		−	151.722i		−	0.186162i
$816$		0			0
$817$	770.671			0.943294
$818$		0			0
$819$			212.321i			0.259244i
$820$		0			0
$821$	−55.9736			−0.0681773			−0.0340886	−	0.999419i	$-0.510853\pi$
$821$	−55.9736			−0.0681773			−0.0340886	+	0.999419i	$0.510853\pi$
$822$		0			0
$823$	785.724			0.954707			0.477354	−	0.878711i	$-0.341596\pi$
$823$	785.724			0.954707			0.477354	+	0.878711i	$0.341596\pi$
$824$		0			0
$825$		−	565.106i		−	0.684977i
$826$		0			0
$827$			130.624i			0.157949i	0.996877	+	0.0789745i	$0.0251646\pi$
$827$			130.624i			0.157949i	−0.996877	+	0.0789745i	$0.974835\pi$
$828$		0			0
$829$	166.204			0.200487			0.100243	−	0.994963i	$-0.468038\pi$
$829$	166.204			0.200487			0.100243	+	0.994963i	$0.468038\pi$
$830$		0			0
$831$	156.534			0.188369
$832$		0			0
$833$		−	1015.62i		−	1.21923i
$834$		0			0
$835$		−	365.802i		−	0.438086i
$836$		0			0
$837$	717.588			0.857333
$838$		0			0
$839$			1048.75i			1.25000i	0.780624	+	0.625001i	$0.214902\pi$
$839$			1048.75i			1.25000i	−0.780624	+	0.625001i	$0.785098\pi$
$840$		0			0
$841$	−612.459			−0.728251
$842$		0			0
$843$		−	372.271i		−	0.441602i
$844$		0			0
$845$		−	274.695i		−	0.325083i
$846$		0			0
$847$			299.856i			0.354021i
$848$		0			0
$849$		−	1269.63i		−	1.49544i
$850$		0			0
$851$	973.581	−	673.230i	1.14404	−	0.791105i
$852$		0			0
$853$	1171.23			1.37307			0.686533	−	0.727099i	$-0.259132\pi$
$853$	1171.23			1.37307			0.686533	+	0.727099i	$0.259132\pi$
$854$		0			0
$855$	1182.48			1.38302
$856$		0			0
$857$	−0.578597			−0.000675142			−0.000337571	−	1.00000i	$-0.500107\pi$
$857$	−0.578597			−0.000675142			−0.000337571		1.00000i	$0.500107\pi$
$858$		0			0
$859$	−1095.88			−1.27576			−0.637878	−	0.770137i	$-0.720188\pi$
$859$	−1095.88			−1.27576			−0.637878	+	0.770137i	$0.720188\pi$
$860$		0			0
$861$			101.564i			0.117960i
$862$		0			0
$863$	−494.808			−0.573358			−0.286679	−	0.958027i	$-0.592551\pi$
$863$	−494.808			−0.573358			−0.286679	+	0.958027i	$0.592551\pi$
$864$		0			0
$865$		−	563.700i		−	0.651677i
$866$		0			0
$867$	988.255			1.13986
$868$		0			0
$869$	−71.2517			−0.0819928
$870$		0			0
$871$		−	354.378i		−	0.406864i
$872$		0			0
$873$			1884.75i			2.15894i
$874$		0			0
$875$	−13.5819			−0.0155222
$876$		0			0
$877$	86.7161			0.0988781			0.0494390	−	0.998777i	$-0.484257\pi$
$877$	86.7161			0.0988781			0.0494390	+	0.998777i	$0.484257\pi$
$878$		0			0
$879$			1728.80i			1.96678i
$880$		0			0
$881$			497.562i			0.564770i	0.959301	+	0.282385i	$0.0911255\pi$
$881$			497.562i			0.564770i	−0.959301	+	0.282385i	$0.908874\pi$
$882$		0			0
$883$	70.9475			0.0803483			0.0401741	−	0.999193i	$-0.487209\pi$
$883$	70.9475			0.0803483			0.0401741	+	0.999193i	$0.487209\pi$
$884$		0			0
$885$		−	908.994i		−	1.02711i
$886$		0			0
$887$	−265.641			−0.299482			−0.149741	−	0.988725i	$-0.547844\pi$
$887$	−265.641			−0.299482			−0.149741	+	0.988725i	$0.547844\pi$
$888$		0			0
$889$		−	110.786i		−	0.124619i
$890$		0			0
$891$		−	6699.87i		−	7.51949i
$892$		0			0
$893$		−	1206.85i		−	1.35146i
$894$		0			0
$895$		−	237.204i		−	0.265032i
$896$		0			0
$897$	523.708	+	757.351i	0.583843	+	0.844316i
$898$		0			0
$899$	−110.054			−0.122419
$900$		0			0
$901$	526.478			0.584327
$902$		0			0
$903$	−268.402			−0.297234
$904$		0			0
$905$	142.843			0.157838
$906$		0			0
$907$			845.489i			0.932182i	0.884737	+	0.466091i	$0.154338\pi$
$907$			845.489i			0.932182i	−0.884737	+	0.466091i	$0.845662\pi$
$908$		0			0
$909$	−327.437			−0.360217
$910$		0			0
$911$			1170.60i			1.28496i	0.766303	+	0.642479i	$0.222094\pi$
$911$			1170.60i			1.28496i	−0.766303	+	0.642479i	$0.777906\pi$
$912$		0			0
$913$	−1728.80			−1.89354
$914$		0			0
$915$	−1303.75			−1.42486
$916$		0			0
$917$			276.527i			0.301556i
$918$		0			0
$919$		−	318.340i		−	0.346399i	−0.984887	−	0.173199i	$-0.944590\pi$
$919$		−	318.340i		−	0.346399i	0.984887	−	0.173199i	$-0.0554105\pi$
$920$		0			0
$921$	−295.602			−0.320958
$922$		0			0
$923$	563.663			0.610686
$924$		0			0
$925$			257.322i			0.278186i
$926$		0			0
$927$		−	2567.00i		−	2.76915i
$928$		0			0
$929$	−539.462			−0.580691			−0.290346	−	0.956922i	$-0.593770\pi$
$929$	−539.462			−0.580691			−0.290346	+	0.956922i	$0.593770\pi$
$930$		0			0
$931$		−	976.872i		−	1.04927i
$932$		0			0
$933$	3227.06			3.45880
$934$		0			0
$935$		−	916.490i		−	0.980203i
$936$		0			0
$937$		−	3.93311i		−	0.00419755i	−0.999998	−	0.00209878i	$-0.999332\pi$
$937$		−	3.93311i		−	0.00419755i	0.999998	−	0.00209878i	$-0.000668062\pi$
$938$		0			0
$939$			110.364i			0.117533i
$940$		0			0
$941$		−	805.991i		−	0.856526i	−0.903654	−	0.428263i	$-0.859126\pi$
$941$		−	805.991i		−	0.856526i	0.903654	−	0.428263i	$-0.140874\pi$
$942$		0			0
$943$	185.591	+	268.390i	0.196809	+	0.284613i
$944$		0			0
$945$	−267.757			−0.283340
$946$		0			0
$947$	−184.198			−0.194506			−0.0972532	−	0.995260i	$-0.531006\pi$
$947$	−184.198			−0.194506			−0.0972532	+	0.995260i	$0.531006\pi$
$948$		0			0
$949$	462.446			0.487298
$950$		0			0
$951$	−435.782			−0.458236
$952$		0			0
$953$			15.5674i			0.0163351i	0.999967	+	0.00816757i	$0.00259985\pi$
$953$			15.5674i			0.0163351i	−0.999967	+	0.00816757i	$0.997400\pi$
$954$		0			0
$955$	553.589			0.579675
$956$		0			0
$957$			1708.61i			1.78538i
$958$		0			0
$959$	74.0563			0.0772225
$960$		0			0
$961$	−908.003			−0.944852
$962$		0			0
$963$			4770.30i			4.95358i
$964$		0			0
$965$		−	242.996i		−	0.251809i
$966$		0			0
$967$	1777.81			1.83848			0.919238	−	0.393702i	$-0.128806\pi$
$967$	1777.81			1.83848			0.919238	+	0.393702i	$0.128806\pi$
$968$		0			0
$969$	2588.64			2.67145
$970$		0			0
$971$			111.995i			0.115340i	0.998336	+	0.0576702i	$0.0183672\pi$
$971$			111.995i			0.115340i	−0.998336	+	0.0576702i	$0.981633\pi$
$972$		0			0
$973$		−	186.757i		−	0.191940i
$974$		0			0
$975$	−200.171			−0.205304
$976$		0			0
$977$		−	1090.61i		−	1.11629i	−0.829744	−	0.558144i	$-0.811514\pi$
$977$		−	1090.61i		−	1.11629i	0.829744	−	0.558144i	$-0.188486\pi$
$978$		0			0
$979$	1984.02			2.02658
$980$		0			0
$981$		−	1228.54i		−	1.25234i
$982$		0			0
$983$		−	1464.77i		−	1.49010i	−0.667009	−	0.745049i	$-0.732426\pi$
$983$		−	1464.77i		−	1.49010i	0.667009	−	0.745049i	$-0.267574\pi$
$984$		0			0
$985$		−	286.146i		−	0.290504i
$986$		0			0
$987$			420.310i			0.425846i
$988$		0			0
$989$	−709.271	+	490.460i	−0.717160	+	0.495915i
$990$		0			0
$991$	−57.4257			−0.0579473			−0.0289736	−	0.999580i	$-0.509224\pi$
$991$	−57.4257			−0.0579473			−0.0289736	+	0.999580i	$0.509224\pi$
$992$		0			0
$993$	1961.01			1.97484
$994$		0			0
$995$	419.141			0.421247
$996$		0			0
$997$	859.060			0.861644			0.430822	−	0.902437i	$-0.358224\pi$
$997$	859.060			0.861644			0.430822	+	0.902437i	$0.358224\pi$
$998$		0			0
$999$			5072.90i			5.07798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.c.321.2		16
4.3	odd	2		460.3.f.a.321.16	yes	16
12.11	even	2		4140.3.d.a.2161.5		16
20.3	even	4		2300.3.d.b.1149.22		32
20.7	even	4		2300.3.d.b.1149.11		32
20.19	odd	2		2300.3.f.e.1701.1		16
23.22	odd	2	inner	1840.3.k.c.321.1		16
92.91	even	2		460.3.f.a.321.15	✓	16
276.275	odd	2		4140.3.d.a.2161.12		16
460.183	odd	4		2300.3.d.b.1149.12		32
460.367	odd	4		2300.3.d.b.1149.21		32
460.459	even	2		2300.3.f.e.1701.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.15	✓	16	92.91	even	2
460.3.f.a.321.16	yes	16	4.3	odd	2
1840.3.k.c.321.1		16	23.22	odd	2	inner
1840.3.k.c.321.2		16	1.1	even	1	trivial
2300.3.d.b.1149.11		32	20.7	even	4
2300.3.d.b.1149.12		32	460.183	odd	4
2300.3.d.b.1149.21		32	460.367	odd	4
2300.3.d.b.1149.22		32	20.3	even	4
2300.3.f.e.1701.1		16	20.19	odd	2
2300.3.f.e.1701.2		16	460.459	even	2
4140.3.d.a.2161.5		16	12.11	even	2
4140.3.d.a.2161.12		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)

\(f(q)\)	\(=\)	\(q-5.89296 q^{3} +2.23607i q^{5} -1.21480i q^{7} +25.7269 q^{9} +O(q^{10})\)	\(q-5.89296 q^{3} +2.23607i q^{5} -1.21480i q^{7} +25.7269 q^{9} -19.1790i q^{11} -6.79358 q^{13} -13.1771i q^{15} -21.3706i q^{17} -20.5552i q^{19} +7.15878i q^{21} +(13.0815 + 18.9176i) q^{23} -5.00000 q^{25} -98.5711 q^{27} +15.1176 q^{29} -7.27990 q^{31} +113.021i q^{33} +2.71638 q^{35} -51.4644i q^{37} +40.0343 q^{39} +14.1873 q^{41} +37.4927i q^{43} +57.5272i q^{45} +58.7126 q^{47} +47.5243 q^{49} +125.936i q^{51} +24.6357i q^{53} +42.8856 q^{55} +121.131i q^{57} +68.9831 q^{59} -98.9407i q^{61} -31.2532i q^{63} -15.1909i q^{65} +52.1637i q^{67} +(-77.0886 - 111.480i) q^{69} -82.9700 q^{71} -68.0710 q^{73} +29.4648 q^{75} -23.2987 q^{77} -3.71508i q^{79} +349.333 q^{81} -90.1400i q^{83} +47.7860 q^{85} -89.0872 q^{87} +103.447i q^{89} +8.25286i q^{91} +42.9001 q^{93} +45.9629 q^{95} +73.2599i q^{97} -493.418i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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