LMFDB - Embedded newform 4140.3.d.a.2161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.d (of order $2$, degree $1$, 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:	$112.806829445$
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:	$\Z[a_1, \ldots, a_{13}]$
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			2161.5
Root			$5.89296 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.a.2161.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +1.21480i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +1.21480i q^{7} -19.1790i q^{11} -6.79358 q^{13} +21.3706i q^{17} +20.5552i q^{19} +(13.0815 + 18.9176i) q^{23} -5.00000 q^{25} -15.1176 q^{29} +7.27990 q^{31} +2.71638 q^{35} -51.4644i q^{37} -14.1873 q^{41} -37.4927i q^{43} +58.7126 q^{47} +47.5243 q^{49} -24.6357i q^{53} -42.8856 q^{55} +68.9831 q^{59} -98.9407i q^{61} +15.1909i q^{65} -52.1637i q^{67} -82.9700 q^{71} -68.0710 q^{73} +23.2987 q^{77} +3.71508i q^{79} -90.1400i q^{83} +47.7860 q^{85} -103.447i q^{89} -8.25286i q^{91} +45.9629 q^{95} +73.2599i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) $ 16 * q - 12 * q^13 + 14 * q^23 - 80 * q^25 - 90 * q^29 + 10 * q^31 - 30 * q^35 - 186 * q^41 + 320 * q^47 + 2 * q^49 - 120 * q^55 + 90 * q^59 + 238 * q^71 - 280 * q^73 - 324 * q^77 - 30 * q^85 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			1.21480i			0.173543i	0.996228	+	0.0867716i	$0.0276550\pi$
$7$			1.21480i			0.173543i	−0.996228	+	0.0867716i	$0.972345\pi$
$8$		0			0
$9$		0			0
$10$		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
$15$		0			0
$16$		0			0
$17$			21.3706i			1.25709i	0.777772	+	0.628546i	$0.216350\pi$
$17$			21.3706i			1.25709i	−0.777772	+	0.628546i	$0.783650\pi$
$18$		0			0
$19$			20.5552i			1.08185i	0.841070	+	0.540927i	$0.181926\pi$
$19$			20.5552i			1.08185i	−0.841070	+	0.540927i	$0.818074\pi$
$20$		0			0
$21$		0			0
$22$		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
$27$		0			0
$28$		0			0
$29$	−15.1176			−0.521296			−0.260648	−	0.965434i	$-0.583936\pi$
$29$	−15.1176			−0.521296			−0.260648	+	0.965434i	$0.583936\pi$
$30$		0			0
$31$	7.27990			0.234836			0.117418	−	0.993083i	$-0.462538\pi$
$31$	7.27990			0.234836			0.117418	+	0.993083i	$0.462538\pi$
$32$		0			0
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	−14.1873			−0.346032			−0.173016	−	0.984919i	$-0.555351\pi$
$41$	−14.1873			−0.346032			−0.173016	+	0.984919i	$0.555351\pi$
$42$		0			0
$43$		−	37.4927i		−	0.871924i	−0.899965	−	0.435962i	$-0.856408\pi$
$43$		−	37.4927i		−	0.871924i	0.899965	−	0.435962i	$-0.143592\pi$
$44$		0			0
$45$		0			0
$46$		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
$51$		0			0
$52$		0			0
$53$		−	24.6357i		−	0.464824i	−0.972617	−	0.232412i	$-0.925338\pi$
$53$		−	24.6357i		−	0.464824i	0.972617	−	0.232412i	$-0.0746618\pi$
$54$		0			0
$55$	−42.8856			−0.779739
$56$		0			0
$57$		0			0
$58$		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
$63$		0			0
$64$		0			0
$65$			15.1909i			0.233706i
$66$		0			0
$67$		−	52.1637i		−	0.778563i	−0.921119	−	0.389281i	$-0.872723\pi$
$67$		−	52.1637i		−	0.778563i	0.921119	−	0.389281i	$-0.127277\pi$
$68$		0			0
$69$		0			0
$70$		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
$75$		0			0
$76$		0			0
$77$	23.2987			0.302581
$78$		0			0
$79$			3.71508i			0.0470264i	0.999724	+	0.0235132i	$0.00748517\pi$
$79$			3.71508i			0.0470264i	−0.999724	+	0.0235132i	$0.992515\pi$
$80$		0			0
$81$		0			0
$82$		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
$87$		0			0
$88$		0			0
$89$		−	103.447i		−	1.16233i	−0.813786	−	0.581164i	$-0.802597\pi$
$89$		−	103.447i		−	1.16233i	0.813786	−	0.581164i	$-0.197403\pi$
$90$		0			0
$91$		−	8.25286i		−	0.0906907i
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$	12.7274			0.126014			0.0630069	−	0.998013i	$-0.479931\pi$
$101$	12.7274			0.126014			0.0630069	+	0.998013i	$0.479931\pi$
$102$		0			0
$103$			99.7786i			0.968725i	0.874867	+	0.484362i	$0.160948\pi$
$103$			99.7786i			0.968725i	−0.874867	+	0.484362i	$0.839052\pi$
$104$		0			0
$105$		0			0
$106$		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
$111$		0			0
$112$		0			0
$113$			60.3411i			0.533992i	0.963698	+	0.266996i	$0.0860310\pi$
$113$			60.3411i			0.533992i	−0.963698	+	0.266996i	$0.913969\pi$
$114$		0			0
$115$	42.3010	−	29.2511i	0.367835	−	0.254357i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−25.9610			−0.218160
$120$		0			0
$121$	−246.835			−2.03996
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−91.1969			−0.718086			−0.359043	−	0.933321i	$-0.616897\pi$
$127$	−91.1969			−0.718086			−0.359043	+	0.933321i	$0.616897\pi$
$128$		0			0
$129$		0			0
$130$		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
$135$		0			0
$136$		0			0
$137$		−	60.9616i		−	0.444975i	−0.974936	−	0.222488i	$-0.928582\pi$
$137$		−	60.9616i		−	0.444975i	0.974936	−	0.222488i	$-0.0714177\pi$
$138$		0			0
$139$	−153.735			−1.10601			−0.553003	−	0.833180i	$-0.686518\pi$
$139$	−153.735			−1.10601			−0.553003	+	0.833180i	$0.686518\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			130.294i			0.911148i
$144$		0			0
$145$			33.8039i			0.233130i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			126.021i			0.845776i	0.906182	+	0.422888i	$0.138984\pi$
$149$			126.021i			0.845776i	−0.906182	+	0.422888i	$0.861016\pi$
$150$		0			0
$151$	−243.700			−1.61391			−0.806955	−	0.590613i	$-0.798886\pi$
$151$	−243.700			−1.61391			−0.806955	+	0.590613i	$0.798886\pi$
$152$		0			0
$153$		0			0
$154$		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
$159$		0			0
$160$		0			0
$161$	−22.9811	+	15.8914i	−0.142740	+	0.0987045i
$162$		0			0
$163$	67.8522			0.416271			0.208136	−	0.978100i	$-0.433260\pi$
$163$	67.8522			0.416271			0.208136	+	0.978100i	$0.433260\pi$
$164$		0			0
$165$		0			0
$166$		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
$171$		0			0
$172$		0			0
$173$	252.094			1.45719			0.728597	−	0.684943i	$-0.240173\pi$
$173$	252.094			1.45719			0.728597	+	0.684943i	$0.240173\pi$
$174$		0			0
$175$		−	6.07401i		−	0.0347086i
$176$		0			0
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	−115.078			−0.622042
$186$		0			0
$187$	409.867			2.19180
$188$		0			0
$189$		0			0
$190$		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
$195$		0			0
$196$		0			0
$197$	127.968			0.649586			0.324793	−	0.945785i	$-0.394705\pi$
$197$	127.968			0.649586			0.324793	+	0.945785i	$0.394705\pi$
$198$		0			0
$199$			187.445i			0.941937i	0.882150	+	0.470968i	$0.156095\pi$
$199$			187.445i			0.941937i	−0.882150	+	0.470968i	$0.843905\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	18.3649i		−	0.0904673i
$204$		0			0
$205$			31.7238i			0.154750i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	394.229			1.88626
$210$		0			0
$211$	−204.271			−0.968111			−0.484055	−	0.875037i	$-0.660837\pi$
$211$	−204.271			−0.968111			−0.484055	+	0.875037i	$0.660837\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−83.8363			−0.389936
$216$		0			0
$217$			8.84364i			0.0407541i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	145.183i		−	0.656935i
$222$		0			0
$223$	−225.207			−1.00990			−0.504948	−	0.863150i	$-0.668488\pi$
$223$	−225.207			−1.00990			−0.504948	+	0.863150i	$0.668488\pi$
$224$		0			0
$225$		0			0
$226$		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
$231$		0			0
$232$		0			0
$233$	−134.731			−0.578244			−0.289122	−	0.957292i	$-0.593363\pi$
$233$	−134.731			−0.578244			−0.289122	+	0.957292i	$0.593363\pi$
$234$		0			0
$235$		−	131.285i		−	0.558661i
$236$		0			0
$237$		0			0
$238$		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
$243$		0			0
$244$		0			0
$245$		−	106.267i		−	0.433745i
$246$		0			0
$247$		−	139.644i		−	0.565358i
$248$		0			0
$249$		0			0
$250$		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
$255$		0			0
$256$		0			0
$257$	−184.115			−0.716402			−0.358201	−	0.933644i	$-0.616610\pi$
$257$	−184.115			−0.716402			−0.358201	+	0.933644i	$0.616610\pi$
$258$		0			0
$259$	62.5191			0.241386
$260$		0			0
$261$		0			0
$262$		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
$267$		0			0
$268$		0			0
$269$	−367.014			−1.36437			−0.682183	−	0.731181i	$-0.738969\pi$
$269$	−367.014			−1.36437			−0.682183	+	0.731181i	$0.738969\pi$
$270$		0			0
$271$	314.590			1.16085			0.580425	−	0.814314i	$-0.302886\pi$
$271$	314.590			1.16085			0.580425	+	0.814314i	$0.302886\pi$
$272$		0			0
$273$		0			0
$274$		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
$279$		0			0
$280$		0			0
$281$		−	63.1722i		−	0.224812i	−0.993662	−	0.112406i	$-0.964144\pi$
$281$		−	63.1722i		−	0.224812i	0.993662	−	0.112406i	$-0.0358557\pi$
$282$		0			0
$283$		−	215.448i		−	0.761302i	−0.924719	−	0.380651i	$-0.875700\pi$
$283$		−	215.448i		−	0.761302i	0.924719	−	0.380651i	$-0.124300\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	17.2348i		−	0.0600515i
$288$		0			0
$289$	−167.701			−0.580280
$290$		0			0
$291$		0			0
$292$		0			0
$293$			293.367i			1.00125i	0.865663	+	0.500626i	$0.166897\pi$
$293$			293.367i			1.00125i	−0.865663	+	0.500626i	$0.833103\pi$
$294$		0			0
$295$		−	154.251i		−	0.522884i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−88.8701	−	128.518i	−0.297224	−	0.429826i
$300$		0			0
$301$	45.5462			0.151316
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−221.238			−0.725371
$306$		0			0
$307$	−50.1619			−0.163394			−0.0816970	−	0.996657i	$-0.526034\pi$
$307$	−50.1619			−0.163394			−0.0816970	+	0.996657i	$0.526034\pi$
$308$		0			0
$309$		0			0
$310$		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
$315$		0			0
$316$		0			0
$317$	−73.9497			−0.233280			−0.116640	−	0.993174i	$-0.537212\pi$
$317$	−73.9497			−0.233280			−0.116640	+	0.993174i	$0.537212\pi$
$318$		0			0
$319$			289.940i			0.908904i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−439.277			−1.35999
$324$		0			0
$325$	33.9679			0.104517
$326$		0			0
$327$		0			0
$328$		0			0
$329$			71.3242i			0.216791i
$330$		0			0
$331$	332.772			1.00535			0.502677	−	0.864474i	$-0.332349\pi$
$331$	332.772			1.00535			0.502677	+	0.864474i	$0.332349\pi$
$332$		0			0
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$		−	139.621i		−	0.409447i
$342$		0			0
$343$			117.258i			0.341860i
$344$		0			0
$345$		0			0
$346$		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
$351$		0			0
$352$		0			0
$353$	−62.0444			−0.175763			−0.0878816	−	0.996131i	$-0.528010\pi$
$353$	−62.0444			−0.175763			−0.0878816	+	0.996131i	$0.528010\pi$
$354$		0			0
$355$			185.526i			0.522610i
$356$		0			0
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$			152.211i			0.417018i
$366$		0			0
$367$		−	491.688i		−	1.33975i	−0.742474	−	0.669875i	$-0.766348\pi$
$367$		−	491.688i		−	1.33975i	0.742474	−	0.669875i	$-0.233652\pi$
$368$		0			0
$369$		0			0
$370$		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
$375$		0			0
$376$		0			0
$377$	102.702			0.272420
$378$		0			0
$379$		−	165.365i		−	0.436320i	−0.975913	−	0.218160i	$-0.929995\pi$
$379$		−	165.365i		−	0.436320i	0.975913	−	0.218160i	$-0.0700055\pi$
$380$		0			0
$381$		0			0
$382$		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
$387$		0			0
$388$		0			0
$389$		−	660.920i		−	1.69902i	−0.527570	−	0.849512i	$-0.676897\pi$
$389$		−	660.920i		−	1.69902i	0.527570	−	0.849512i	$-0.323103\pi$
$390$		0			0
$391$	−404.279	+	279.559i	−1.03396	+	0.714984i
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$			607.765i			1.51562i	0.652474	+	0.757811i	$0.273731\pi$
$401$			607.765i			1.51562i	−0.652474	+	0.757811i	$0.726269\pi$
$402$		0			0
$403$	−49.4566			−0.122721
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			83.8008i			0.202908i
$414$		0			0
$415$	−201.559			−0.485685
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$		−	106.853i		−	0.251418i
$426$		0			0
$427$	120.193			0.281483
$428$		0			0
$429$		0			0
$430$		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
$435$		0			0
$436$		0			0
$437$	−388.855	+	268.893i	−0.889829	+	0.615315i
$438$		0			0
$439$	382.455			0.871195			0.435598	−	0.900141i	$-0.356537\pi$
$439$	382.455			0.871195			0.435598	+	0.900141i	$0.356537\pi$
$440$		0			0
$441$		0			0
$442$		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
$447$		0			0
$448$		0			0
$449$	−417.233			−0.929249			−0.464624	−	0.885508i	$-0.653811\pi$
$449$	−417.233			−0.929249			−0.464624	+	0.885508i	$0.653811\pi$
$450$		0			0
$451$			272.099i			0.603324i
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	398.642			0.864734			0.432367	−	0.901698i	$-0.357678\pi$
$461$	398.642			0.864734			0.432367	+	0.901698i	$0.357678\pi$
$462$		0			0
$463$	−254.794			−0.550310			−0.275155	−	0.961400i	$-0.588729\pi$
$463$	−254.794			−0.550310			−0.275155	+	0.961400i	$0.588729\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$	−719.074			−1.52024
$474$		0			0
$475$		−	102.776i		−	0.216371i
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		0			0
$485$	163.814			0.337761
$486$		0			0
$487$	280.911			0.576819			0.288410	−	0.957507i	$-0.406874\pi$
$487$	280.911			0.576819			0.288410	+	0.957507i	$0.406874\pi$
$488$		0			0
$489$		0			0
$490$		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
$495$		0			0
$496$		0			0
$497$		−	100.792i		−	0.202801i
$498$		0			0
$499$	301.893			0.604996			0.302498	−	0.953150i	$-0.402179\pi$
$499$	301.893			0.604996			0.302498	+	0.953150i	$0.402179\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$	−639.641			−1.25666			−0.628331	−	0.777946i	$-0.716262\pi$
$509$	−639.641			−1.25666			−0.628331	+	0.777946i	$0.716262\pi$
$510$		0			0
$511$		−	82.6929i		−	0.161826i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	223.112			0.433227
$516$		0			0
$517$		−	1126.05i		−	2.17805i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			45.5463i			0.0874209i	0.999044	+	0.0437104i	$0.0139179\pi$
$521$			45.5463i			0.0874209i	−0.999044	+	0.0437104i	$0.986082\pi$
$522$		0			0
$523$			100.120i			0.191434i	0.995409	+	0.0957172i	$0.0305144\pi$
$523$			100.120i			0.191434i	−0.995409	+	0.0957172i	$0.969486\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			155.576i			0.295210i
$528$		0			0
$529$	−186.749	+	494.940i	−0.353024	+	0.935614i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	96.3826			0.180830
$534$		0			0
$535$	414.613			0.774977
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	−106.779			−0.195925
$546$		0			0
$547$	536.646			0.981071			0.490535	−	0.871421i	$-0.336801\pi$
$547$	536.646			0.981071			0.490535	+	0.871421i	$0.336801\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	310.745i		−	0.563966i
$552$		0			0
$553$	−4.51309			−0.00816111
$554$		0			0
$555$		0			0
$556$		0			0
$557$			600.369i			1.07786i	0.842350	+	0.538931i	$0.181172\pi$
$557$			600.369i			1.07786i	−0.842350	+	0.538931i	$0.818828\pi$
$558$		0			0
$559$			254.710i			0.455652i
$560$		0			0
$561$		0			0
$562$		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
$567$		0			0
$568$		0			0
$569$		−	707.638i		−	1.24365i	−0.783156	−	0.621826i	$-0.786391\pi$
$569$		−	707.638i		−	1.24365i	0.783156	−	0.621826i	$-0.213609\pi$
$570$		0			0
$571$		−	1007.22i		−	1.76396i	−0.471287	−	0.881980i	$-0.656210\pi$
$571$		−	1007.22i		−	1.76396i	0.471287	−	0.881980i	$-0.343790\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	109.502			0.188472
$582$		0			0
$583$	−472.489			−0.810443
$584$		0			0
$585$		0			0
$586$		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
$591$		0			0
$592$		0			0
$593$	982.609			1.65701			0.828507	−	0.559979i	$-0.189191\pi$
$593$	982.609			1.65701			0.828507	+	0.559979i	$0.189191\pi$
$594$		0			0
$595$			58.0506i			0.0975640i
$596$		0			0
$597$		0			0
$598$		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
$603$		0			0
$604$		0			0
$605$			551.940i			0.912298i
$606$		0			0
$607$	754.060			1.24227			0.621137	−	0.783702i	$-0.286671\pi$
$607$	754.060			1.24227			0.621137	+	0.783702i	$0.286671\pi$
$608$		0			0
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$		−	657.899i		−	1.06629i	−0.846025	−	0.533144i	$-0.821011\pi$
$617$		−	657.899i		−	1.06629i	0.846025	−	0.533144i	$-0.178989\pi$
$618$		0			0
$619$		−	648.359i		−	1.04743i	−0.851893	−	0.523715i	$-0.824546\pi$
$619$		−	648.359i		−	1.04743i	0.851893	−	0.523715i	$-0.175454\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	125.668			0.201714
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1099.82			1.74853
$630$		0			0
$631$			426.772i			0.676342i	0.941085	+	0.338171i	$0.109808\pi$
$631$			426.772i			0.676342i	−0.941085	+	0.338171i	$0.890192\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			203.923i			0.321138i
$636$		0			0
$637$	−322.860			−0.506844
$638$		0			0
$639$		0			0
$640$		0			0
$641$			933.107i			1.45570i	0.685734	+	0.727852i	$0.259481\pi$
$641$			933.107i			1.45570i	−0.685734	+	0.727852i	$0.740519\pi$
$642$		0			0
$643$		−	887.579i		−	1.38037i	−0.723632	−	0.690186i	$-0.757529\pi$
$643$		−	887.579i		−	1.38037i	0.723632	−	0.690186i	$-0.242471\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		0			0
$653$	1156.08			1.77041			0.885207	−	0.465198i	$-0.154017\pi$
$653$	1156.08			1.77041			0.885207	+	0.465198i	$0.154017\pi$
$654$		0			0
$655$			508.999i			0.777098i
$656$		0			0
$657$		0			0
$658$		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
$663$		0			0
$664$		0			0
$665$			55.8358i			0.0839636i
$666$		0			0
$667$	−197.760	−	285.988i	−0.296492	−	0.428767i
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$		−	732.982i		−	1.08269i	−0.840800	−	0.541346i	$-0.817915\pi$
$677$		−	732.982i		−	1.08269i	0.840800	−	0.541346i	$-0.182085\pi$
$678$		0			0
$679$	−88.9964			−0.131070
$680$		0			0
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$			167.364i			0.242909i
$690$		0			0
$691$	−613.573			−0.887950			−0.443975	−	0.896039i	$-0.646432\pi$
$691$	−613.573			−0.887950			−0.443975	+	0.896039i	$0.646432\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			343.761i			0.494621i
$696$		0			0
$697$		−	303.191i		−	0.434994i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	539.961i		−	0.770272i	−0.922860	−	0.385136i	$-0.874155\pi$
$701$		−	539.961i		−	0.770272i	0.922860	−	0.385136i	$-0.125845\pi$
$702$		0			0
$703$	1057.86			1.50478
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$	95.2319	+	137.718i	0.133565	+	0.193153i
$714$		0			0
$715$	291.347			0.407478
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$	75.5879			0.104259
$726$		0			0
$727$			305.976i			0.420875i	0.977607	+	0.210437i	$0.0674888\pi$
$727$			305.976i			0.420875i	−0.977607	+	0.210437i	$0.932511\pi$
$728$		0			0
$729$		0			0
$730$		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
$735$		0			0
$736$		0			0
$737$	−1000.45			−1.35746
$738$		0			0
$739$	−1273.79			−1.72367			−0.861834	−	0.507191i	$-0.830684\pi$
$739$	−1273.79			−1.72367			−0.861834	+	0.507191i	$0.830684\pi$
$740$		0			0
$741$		0			0
$742$		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
$747$		0			0
$748$		0			0
$749$	−225.249			−0.300733
$750$		0			0
$751$			391.564i			0.521390i	0.965421	+	0.260695i	$0.0839517\pi$
$751$			391.564i			0.521390i	−0.965421	+	0.260695i	$0.916048\pi$
$752$		0			0
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	1190.50			1.56439			0.782196	−	0.623032i	$-0.214099\pi$
$761$	1190.50			1.56439			0.782196	+	0.623032i	$0.214099\pi$
$762$		0			0
$763$	58.0107			0.0760297
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$		−	245.060i		−	0.317025i	−0.987357	−	0.158513i	$-0.949330\pi$
$773$		−	245.060i		−	0.317025i	0.987357	−	0.158513i	$-0.0506698\pi$
$774$		0			0
$775$	−36.3995			−0.0469671
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	291.623i		−	0.374356i
$780$		0			0
$781$			1591.28i			2.03750i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−353.309			−0.450075
$786$		0			0
$787$		−	749.447i		−	0.952283i	−0.879369	−	0.476141i	$-0.842035\pi$
$787$		−	749.447i		−	0.952283i	0.879369	−	0.476141i	$-0.157965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−73.3025			−0.0926707
$792$		0			0
$793$			672.161i			0.847618i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			223.282i			0.280152i	0.990141	+	0.140076i	$0.0447348\pi$
$797$			223.282i			0.280152i	−0.990141	+	0.140076i	$0.955265\pi$
$798$		0			0
$799$			1254.72i			1.57036i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1305.54i			1.62582i
$804$		0			0
$805$	35.5343	+	51.3874i	0.0441420	+	0.0638352i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1224.34			1.51340			0.756699	−	0.653763i	$-0.226811\pi$
$809$	1224.34			1.51340			0.756699	+	0.653763i	$0.226811\pi$
$810$		0			0
$811$	−895.177			−1.10379			−0.551897	−	0.833912i	$-0.686096\pi$
$811$	−895.177			−1.10379			−0.551897	+	0.833912i	$0.686096\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	151.722i		−	0.186162i
$816$		0			0
$817$	770.671			0.943294
$818$		0			0
$819$		0			0
$820$		0			0
$821$	55.9736			0.0681773			0.0340886	−	0.999419i	$-0.489147\pi$
$821$	55.9736			0.0681773			0.0340886	+	0.999419i	$0.489147\pi$
$822$		0			0
$823$	−785.724			−0.954707			−0.477354	−	0.878711i	$-0.658404\pi$
$823$	−785.724			−0.954707			−0.477354	+	0.878711i	$0.658404\pi$
$824$		0			0
$825$		0			0
$826$		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
$831$		0			0
$832$		0			0
$833$			1015.62i			1.21923i
$834$		0			0
$835$			365.802i			0.438086i
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$			274.695i			0.325083i
$846$		0			0
$847$		−	299.856i		−	0.354021i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$	0.578597			0.000675142			0.000337571	−	1.00000i	$-0.499893\pi$
$857$	0.578597			0.000675142			0.000337571		1.00000i	$0.499893\pi$
$858$		0			0
$859$	1095.88			1.27576			0.637878	−	0.770137i	$-0.279812\pi$
$859$	1095.88			1.27576			0.637878	+	0.770137i	$0.279812\pi$
$860$		0			0
$861$		0			0
$862$		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
$867$		0			0
$868$		0			0
$869$	71.2517			0.0819928
$870$		0			0
$871$			354.378i			0.406864i
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$		−	497.562i		−	0.564770i	−0.959301	−	0.282385i	$-0.908874\pi$
$881$		−	497.562i		−	0.564770i	0.959301	−	0.282385i	$-0.0911255\pi$
$882$		0			0
$883$	−70.9475			−0.0803483			−0.0401741	−	0.999193i	$-0.512791\pi$
$883$	−70.9475			−0.0803483			−0.0401741	+	0.999193i	$0.512791\pi$
$884$		0			0
$885$		0			0
$886$		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
$891$		0			0
$892$		0			0
$893$			1206.85i			1.35146i
$894$		0			0
$895$			237.204i			0.265032i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−110.054			−0.122419
$900$		0			0
$901$	526.478			0.584327
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−142.843			−0.157838
$906$		0			0
$907$		−	845.489i		−	0.932182i	−0.884737	−	0.466091i	$-0.845662\pi$
$907$		−	845.489i		−	0.932182i	0.884737	−	0.466091i	$-0.154338\pi$
$908$		0			0
$909$		0			0
$910$		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
$915$		0			0
$916$		0			0
$917$		−	276.527i		−	0.301556i
$918$		0			0
$919$			318.340i			0.346399i	0.984887	+	0.173199i	$0.0554105\pi$
$919$			318.340i			0.346399i	−0.984887	+	0.173199i	$0.944590\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	563.663			0.610686
$924$		0			0
$925$			257.322i			0.278186i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	539.462			0.580691			0.290346	−	0.956922i	$-0.406230\pi$
$929$	539.462			0.580691			0.290346	+	0.956922i	$0.406230\pi$
$930$		0			0
$931$			976.872i			1.04927i
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$			805.991i			0.856526i	0.903654	+	0.428263i	$0.140874\pi$
$941$			805.991i			0.856526i	−0.903654	+	0.428263i	$0.859126\pi$
$942$		0			0
$943$	−185.591	−	268.390i	−0.196809	−	0.284613i
$944$		0			0
$945$		0			0
$946$		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
$951$		0			0
$952$		0			0
$953$		−	15.5674i		−	0.0163351i	−0.999967	−	0.00816757i	$-0.997400\pi$
$953$		−	15.5674i		−	0.0163351i	0.999967	−	0.00816757i	$-0.00259985\pi$
$954$		0			0
$955$	−553.589			−0.579675
$956$		0			0
$957$		0			0
$958$		0			0
$959$	74.0563			0.0772225
$960$		0			0
$961$	−908.003			−0.944852
$962$		0			0
$963$		0			0
$964$		0			0
$965$			242.996i			0.251809i
$966$		0			0
$967$	−1777.81			−1.83848			−0.919238	−	0.393702i	$-0.871194\pi$
$967$	−1777.81			−1.83848			−0.919238	+	0.393702i	$0.871194\pi$
$968$		0			0
$969$		0			0
$970$		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
$975$		0			0
$976$		0			0
$977$			1090.61i			1.11629i	0.829744	+	0.558144i	$0.188486\pi$
$977$			1090.61i			1.11629i	−0.829744	+	0.558144i	$0.811514\pi$
$978$		0			0
$979$	−1984.02			−2.02658
$980$		0			0
$981$		0			0
$982$		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
$987$		0			0
$988$		0			0
$989$	709.271	−	490.460i	0.717160	−	0.495915i
$990$		0			0
$991$	57.4257			0.0579473			0.0289736	−	0.999580i	$-0.490776\pi$
$991$	57.4257			0.0579473			0.0289736	+	0.999580i	$0.490776\pi$
$992$		0			0
$993$		0			0
$994$		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
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.a.2161.5		16
3.2	odd	2		460.3.f.a.321.16	yes	16
12.11	even	2		1840.3.k.c.321.2		16
15.2	even	4		2300.3.d.b.1149.11		32
15.8	even	4		2300.3.d.b.1149.22		32
15.14	odd	2		2300.3.f.e.1701.1		16
23.22	odd	2	inner	4140.3.d.a.2161.12		16
69.68	even	2		460.3.f.a.321.15	✓	16
276.275	odd	2		1840.3.k.c.321.1		16
345.68	odd	4		2300.3.d.b.1149.12		32
345.137	odd	4		2300.3.d.b.1149.21		32
345.344	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	69.68	even	2
460.3.f.a.321.16	yes	16	3.2	odd	2
1840.3.k.c.321.1		16	276.275	odd	2
1840.3.k.c.321.2		16	12.11	even	2
2300.3.d.b.1149.11		32	15.2	even	4
2300.3.d.b.1149.12		32	345.68	odd	4
2300.3.d.b.1149.21		32	345.137	odd	4
2300.3.d.b.1149.22		32	15.8	even	4
2300.3.f.e.1701.1		16	15.14	odd	2
2300.3.f.e.1701.2		16	345.344	even	2
4140.3.d.a.2161.5		16	1.1	even	1	trivial
4140.3.d.a.2161.12		16	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +1.21480i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +1.21480i q^{7} -19.1790i q^{11} -6.79358 q^{13} +21.3706i q^{17} +20.5552i q^{19} +(13.0815 + 18.9176i) q^{23} -5.00000 q^{25} -15.1176 q^{29} +7.27990 q^{31} +2.71638 q^{35} -51.4644i q^{37} -14.1873 q^{41} -37.4927i q^{43} +58.7126 q^{47} +47.5243 q^{49} -24.6357i q^{53} -42.8856 q^{55} +68.9831 q^{59} -98.9407i q^{61} +15.1909i q^{65} -52.1637i q^{67} -82.9700 q^{71} -68.0710 q^{73} +23.2987 q^{77} +3.71508i q^{79} -90.1400i q^{83} +47.7860 q^{85} -103.447i q^{89} -8.25286i q^{91} +45.9629 q^{95} +73.2599i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) \) 16 * q - 12 * q^13 + 14 * q^23 - 80 * q^25 - 90 * q^29 + 10 * q^31 - 30 * q^35 - 186 * q^41 + 320 * q^47 + 2 * q^49 - 120 * q^55 + 90 * q^59 + 238 * q^71 - 280 * q^73 - 324 * q^77 - 30 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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