LMFDB - Embedded newform 4140.3.d.a.2161.13

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			2161.13
Root			$-1.83366 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.a.2161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -0.167846i q^{7} +O(q^{10})$	$q+2.23607i q^{5} -0.167846i q^{7} -11.8263i q^{11} +9.42841 q^{13} +0.812526i q^{17} -12.4039i q^{19} +(-11.0031 - 20.1973i) q^{23} -5.00000 q^{25} -30.8080 q^{29} +31.2013 q^{31} +0.375316 q^{35} -0.0361243i q^{37} +11.4571 q^{41} +59.8367i q^{43} -35.5771 q^{47} +48.9718 q^{49} -94.9170i q^{53} +26.4445 q^{55} -82.3693 q^{59} -32.9697i q^{61} +21.0826i q^{65} +3.42158i q^{67} -105.890 q^{71} -118.034 q^{73} -1.98501 q^{77} +111.883i q^{79} +42.9999i q^{83} -1.81686 q^{85} +120.503i q^{89} -1.58252i q^{91} +27.7361 q^{95} -6.47565i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	0.167846i		−	0.0239781i	−0.999928	−	0.0119890i	$-0.996184\pi$
$7$		−	0.167846i		−	0.0239781i	0.999928	−	0.0119890i	$-0.00381632\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	11.8263i		−	1.07512i	−0.843225	−	0.537561i	$-0.819346\pi$
$11$		−	11.8263i		−	1.07512i	0.843225	−	0.537561i	$-0.180654\pi$
$12$		0			0
$13$	9.42841			0.725262			0.362631	−	0.931933i	$-0.381879\pi$
$13$	9.42841			0.725262			0.362631	+	0.931933i	$0.381879\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.812526i			0.0477956i	0.999714	+	0.0238978i	$0.00760763\pi$
$17$			0.812526i			0.0477956i	−0.999714	+	0.0238978i	$0.992392\pi$
$18$		0			0
$19$		−	12.4039i		−	0.652839i	−0.945225	−	0.326419i	$-0.894158\pi$
$19$		−	12.4039i		−	0.652839i	0.945225	−	0.326419i	$-0.105842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11.0031	−	20.1973i	−0.478395	−	0.878145i
$23$	−11.0031	−	20.1973i	−0.478395	−	0.878145i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−30.8080			−1.06234			−0.531172	−	0.847264i	$-0.678248\pi$
$29$	−30.8080			−1.06234			−0.531172	+	0.847264i	$0.678248\pi$
$30$		0			0
$31$	31.2013			1.00649			0.503247	−	0.864143i	$-0.332139\pi$
$31$	31.2013			1.00649			0.503247	+	0.864143i	$0.332139\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.375316			0.0107233
$36$		0			0
$37$		−	0.0361243i		−	0.000976333i	−1.00000		0.000488166i	$-0.999845\pi$
$37$		−	0.0361243i		−	0.000976333i	1.00000		0.000488166i	$-0.000155388\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.4571			0.279441			0.139720	−	0.990191i	$-0.455380\pi$
$41$	11.4571			0.279441			0.139720	+	0.990191i	$0.455380\pi$
$42$		0			0
$43$			59.8367i			1.39155i	0.718259	+	0.695776i	$0.244939\pi$
$43$			59.8367i			1.39155i	−0.718259	+	0.695776i	$0.755061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−35.5771			−0.756959			−0.378480	−	0.925610i	$-0.623553\pi$
$47$	−35.5771			−0.756959			−0.378480	+	0.925610i	$0.623553\pi$
$48$		0			0
$49$	48.9718			0.999425
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	94.9170i		−	1.79089i	−0.445176	−	0.895443i	$-0.646859\pi$
$53$		−	94.9170i		−	1.79089i	0.445176	−	0.895443i	$-0.353141\pi$
$54$		0			0
$55$	26.4445			0.480809
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−82.3693			−1.39609			−0.698045	−	0.716054i	$-0.745946\pi$
$59$	−82.3693			−1.39609			−0.698045	+	0.716054i	$0.745946\pi$
$60$		0			0
$61$		−	32.9697i		−	0.540487i	−0.962792	−	0.270244i	$-0.912896\pi$
$61$		−	32.9697i		−	0.540487i	0.962792	−	0.270244i	$-0.0871043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			21.0826i			0.324347i
$66$		0			0
$67$			3.42158i			0.0510684i	0.999674	+	0.0255342i	$0.00812867\pi$
$67$			3.42158i			0.0510684i	−0.999674	+	0.0255342i	$0.991871\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−105.890			−1.49140			−0.745702	−	0.666280i	$-0.767886\pi$
$71$	−105.890			−1.49140			−0.745702	+	0.666280i	$0.767886\pi$
$72$		0			0
$73$	−118.034			−1.61691			−0.808453	−	0.588561i	$-0.799695\pi$
$73$	−118.034			−1.61691			−0.808453	+	0.588561i	$0.799695\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.98501			−0.0257793
$78$		0			0
$79$			111.883i			1.41624i	0.706094	+	0.708118i	$0.250456\pi$
$79$			111.883i			1.41624i	−0.706094	+	0.708118i	$0.749544\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			42.9999i			0.518072i	0.965868	+	0.259036i	$0.0834048\pi$
$83$			42.9999i			0.518072i	−0.965868	+	0.259036i	$0.916595\pi$
$84$		0			0
$85$	−1.81686			−0.0213749
$86$		0			0
$87$		0			0
$88$		0			0
$89$			120.503i			1.35396i	0.736000	+	0.676981i	$0.236712\pi$
$89$			120.503i			1.35396i	−0.736000	+	0.676981i	$0.763288\pi$
$90$		0			0
$91$		−	1.58252i		−	0.0173904i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	27.7361			0.291958
$96$		0			0
$97$		−	6.47565i		−	0.0667593i	−0.999443	−	0.0333796i	$-0.989373\pi$
$97$		−	6.47565i		−	0.0667593i	0.999443	−	0.0333796i	$-0.0106270\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	163.394			1.61776			0.808882	−	0.587971i	$-0.200073\pi$
$101$	163.394			1.61776			0.808882	+	0.587971i	$0.200073\pi$
$102$		0			0
$103$			90.4540i			0.878194i	0.898440	+	0.439097i	$0.144702\pi$
$103$			90.4540i			0.878194i	−0.898440	+	0.439097i	$0.855298\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			48.3482i			0.451852i	0.974144	+	0.225926i	$0.0725408\pi$
$107$			48.3482i			0.451852i	−0.974144	+	0.225926i	$0.927459\pi$
$108$		0			0
$109$		−	143.411i		−	1.31570i	−0.753151	−	0.657848i	$-0.771467\pi$
$109$		−	143.411i		−	1.31570i	0.753151	−	0.657848i	$-0.228533\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	79.7469i		−	0.705725i	−0.935675	−	0.352862i	$-0.885208\pi$
$113$		−	79.7469i		−	0.705725i	0.935675	−	0.352862i	$-0.114792\pi$
$114$		0			0
$115$	45.1626	−	24.6037i	0.392718	−	0.213945i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.136379			0.00114605
$120$		0			0
$121$	−18.8625			−0.155888
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−59.5178			−0.468644			−0.234322	−	0.972159i	$-0.575287\pi$
$127$	−59.5178			−0.468644			−0.234322	+	0.972159i	$0.575287\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	20.3080			0.155023			0.0775113	−	0.996991i	$-0.475303\pi$
$131$	20.3080			0.155023			0.0775113	+	0.996991i	$0.475303\pi$
$132$		0			0
$133$	−2.08196			−0.0156538
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	82.7905i		−	0.604310i	−0.953259	−	0.302155i	$-0.902294\pi$
$137$		−	82.7905i		−	0.604310i	0.953259	−	0.302155i	$-0.0977061\pi$
$138$		0			0
$139$	−56.5216			−0.406630			−0.203315	−	0.979113i	$-0.565172\pi$
$139$	−56.5216			−0.406630			−0.203315	+	0.979113i	$0.565172\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	111.504i		−	0.779745i
$144$		0			0
$145$		−	68.8887i		−	0.475094i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			50.9828i			0.342167i	0.985257	+	0.171083i	$0.0547267\pi$
$149$			50.9828i			0.342167i	−0.985257	+	0.171083i	$0.945273\pi$
$150$		0			0
$151$	−71.6646			−0.474600			−0.237300	−	0.971436i	$-0.576262\pi$
$151$	−71.6646			−0.474600			−0.237300	+	0.971436i	$0.576262\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			69.7683i			0.450118i
$156$		0			0
$157$		−	93.3559i		−	0.594624i	−0.954780	−	0.297312i	$-0.903910\pi$
$157$		−	93.3559i		−	0.594624i	0.954780	−	0.297312i	$-0.0960901\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.39005	+	1.84683i	−0.0210562	+	0.0114710i
$162$		0			0
$163$	−214.778			−1.31765			−0.658827	−	0.752295i	$-0.728947\pi$
$163$	−214.778			−1.31765			−0.658827	+	0.752295i	$0.728947\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−291.228			−1.74388			−0.871941	−	0.489611i	$-0.837139\pi$
$167$	−291.228			−1.74388			−0.871941	+	0.489611i	$0.837139\pi$
$168$		0			0
$169$	−80.1052			−0.473995
$170$		0			0
$171$		0			0
$172$		0			0
$173$	93.2029			0.538745			0.269373	−	0.963036i	$-0.413184\pi$
$173$	93.2029			0.538745			0.269373	+	0.963036i	$0.413184\pi$
$174$		0			0
$175$			0.839232i			0.00479561i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−105.100			−0.587148			−0.293574	−	0.955936i	$-0.594845\pi$
$179$	−105.100			−0.587148			−0.293574	+	0.955936i	$0.594845\pi$
$180$		0			0
$181$			318.705i			1.76080i	0.474228	+	0.880402i	$0.342727\pi$
$181$			318.705i			1.76080i	−0.474228	+	0.880402i	$0.657273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0807764			0.000436629
$186$		0			0
$187$	9.60921			0.0513861
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	87.5629i		−	0.458445i	−0.973374	−	0.229222i	$-0.926382\pi$
$191$		−	87.5629i		−	0.458445i	0.973374	−	0.229222i	$-0.0736183\pi$
$192$		0			0
$193$	55.5462			0.287804			0.143902	−	0.989592i	$-0.454035\pi$
$193$	55.5462			0.287804			0.143902	+	0.989592i	$0.454035\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	78.4332			0.398138			0.199069	−	0.979985i	$-0.436208\pi$
$197$	78.4332			0.398138			0.199069	+	0.979985i	$0.436208\pi$
$198$		0			0
$199$		−	205.377i		−	1.03205i	−0.856575	−	0.516023i	$-0.827412\pi$
$199$		−	205.377i		−	1.03205i	0.856575	−	0.516023i	$-0.172588\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			5.17100i			0.0254729i
$204$		0			0
$205$			25.6188i			0.124970i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−146.693			−0.701882
$210$		0			0
$211$	−126.367			−0.598896			−0.299448	−	0.954113i	$-0.596802\pi$
$211$	−126.367			−0.598896			−0.299448	+	0.954113i	$0.596802\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−133.799			−0.622321
$216$		0			0
$217$		−	5.23703i		−	0.0241338i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			7.66082i			0.0346643i
$222$		0			0
$223$	163.336			0.732448			0.366224	−	0.930527i	$-0.380650\pi$
$223$	163.336			0.732448			0.366224	+	0.930527i	$0.380650\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	423.813i		−	1.86702i	−0.358555	−	0.933509i	$-0.616730\pi$
$227$		−	423.813i		−	1.86702i	0.358555	−	0.933509i	$-0.383270\pi$
$228$		0			0
$229$		−	376.488i		−	1.64405i	−0.569449	−	0.822027i	$-0.692843\pi$
$229$		−	376.488i		−	1.64405i	0.569449	−	0.822027i	$-0.307157\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−277.322			−1.19022			−0.595112	−	0.803643i	$-0.702892\pi$
$233$	−277.322			−1.19022			−0.595112	+	0.803643i	$0.702892\pi$
$234$		0			0
$235$		−	79.5528i		−	0.338523i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−215.835			−0.903074			−0.451537	−	0.892252i	$-0.649124\pi$
$239$	−215.835			−0.903074			−0.451537	+	0.892252i	$0.649124\pi$
$240$		0			0
$241$		−	105.315i		−	0.436991i	−0.975838	−	0.218495i	$-0.929885\pi$
$241$		−	105.315i		−	0.436991i	0.975838	−	0.218495i	$-0.0701148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			109.504i			0.446956i
$246$		0			0
$247$		−	116.949i		−	0.473479i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			69.8236i			0.278182i	0.990280	+	0.139091i	$0.0444180\pi$
$251$			69.8236i			0.278182i	−0.990280	+	0.139091i	$0.955582\pi$
$252$		0			0
$253$	−238.861	+	130.126i	−0.944113	+	0.514333i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−44.2881			−0.172327			−0.0861637	−	0.996281i	$-0.527461\pi$
$257$	−44.2881			−0.172327			−0.0861637	+	0.996281i	$0.527461\pi$
$258$		0			0
$259$	−0.00606333			−2.34106e−5
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	401.139i		−	1.52525i	−0.646844	−	0.762623i	$-0.723911\pi$
$263$		−	401.139i		−	1.52525i	0.646844	−	0.762623i	$-0.276089\pi$
$264$		0			0
$265$	212.241			0.800909
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0096			−0.0372102			−0.0186051	−	0.999827i	$-0.505923\pi$
$269$	−10.0096			−0.0372102			−0.0186051	+	0.999827i	$0.505923\pi$
$270$		0			0
$271$	101.695			0.375258			0.187629	−	0.982240i	$-0.439920\pi$
$271$	101.695			0.375258			0.187629	+	0.982240i	$0.439920\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			59.1317i			0.215024i
$276$		0			0
$277$	−181.805			−0.656335			−0.328167	−	0.944620i	$-0.606431\pi$
$277$	−181.805			−0.656335			−0.328167	+	0.944620i	$0.606431\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	366.625i		−	1.30472i	−0.757911	−	0.652358i	$-0.773780\pi$
$281$		−	366.625i		−	1.30472i	0.757911	−	0.652358i	$-0.226220\pi$
$282$		0			0
$283$			28.8256i			0.101857i	0.998702	+	0.0509286i	$0.0162181\pi$
$283$			28.8256i			0.101857i	−0.998702	+	0.0509286i	$0.983782\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.92303i		−	0.00670044i
$288$		0			0
$289$	288.340			0.997716
$290$		0			0
$291$		0			0
$292$		0			0
$293$			322.471i			1.10058i	0.834973	+	0.550291i	$0.185483\pi$
$293$			322.471i			1.10058i	−0.834973	+	0.550291i	$0.814517\pi$
$294$		0			0
$295$		−	184.183i		−	0.624350i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−103.742	−	190.429i	−0.346962	−	0.636885i
$300$		0			0
$301$	10.0434			0.0333667
$302$		0			0
$303$		0			0
$304$		0			0
$305$	73.7226			0.241713
$306$		0			0
$307$	129.613			0.422192			0.211096	−	0.977465i	$-0.432297\pi$
$307$	129.613			0.422192			0.211096	+	0.977465i	$0.432297\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−69.5283			−0.223564			−0.111782	−	0.993733i	$-0.535656\pi$
$311$	−69.5283			−0.223564			−0.111782	+	0.993733i	$0.535656\pi$
$312$		0			0
$313$		−	106.050i		−	0.338818i	−0.985546	−	0.169409i	$-0.945814\pi$
$313$		−	106.050i		−	0.338818i	0.985546	−	0.169409i	$-0.0541859\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−355.098			−1.12018			−0.560091	−	0.828431i	$-0.689234\pi$
$317$	−355.098			−1.12018			−0.560091	+	0.828431i	$0.689234\pi$
$318$		0			0
$319$			364.346i			1.14215i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.0785			0.0312028
$324$		0			0
$325$	−47.1420			−0.145052
$326$		0			0
$327$		0			0
$328$		0			0
$329$			5.97149i			0.0181504i
$330$		0			0
$331$	−67.1078			−0.202743			−0.101371	−	0.994849i	$-0.532323\pi$
$331$	−67.1078			−0.202743			−0.101371	+	0.994849i	$0.532323\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.65089			−0.0228385
$336$		0			0
$337$		−	221.211i		−	0.656413i	−0.944606	−	0.328207i	$-0.893556\pi$
$337$		−	221.211i		−	0.656413i	0.944606	−	0.328207i	$-0.106444\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	368.998i		−	1.08210i
$342$		0			0
$343$		−	16.4442i		−	0.0479423i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.4672			−0.0330466			−0.0165233	−	0.999863i	$-0.505260\pi$
$347$	−11.4672			−0.0330466			−0.0165233	+	0.999863i	$0.505260\pi$
$348$		0			0
$349$	149.056			0.427095			0.213547	−	0.976933i	$-0.431498\pi$
$349$	149.056			0.427095			0.213547	+	0.976933i	$0.431498\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−408.203			−1.15638			−0.578191	−	0.815901i	$-0.696241\pi$
$353$	−408.203			−1.15638			−0.578191	+	0.815901i	$0.696241\pi$
$354$		0			0
$355$		−	236.777i		−	0.666976i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			496.320i			1.38251i	0.722612	+	0.691253i	$0.242941\pi$
$359$			496.320i			1.38251i	−0.722612	+	0.691253i	$0.757059\pi$
$360$		0			0
$361$	207.142			0.573801
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	263.932i		−	0.723102i
$366$		0			0
$367$			367.612i			1.00167i	0.865543	+	0.500834i	$0.166973\pi$
$367$			367.612i			1.00167i	−0.865543	+	0.500834i	$0.833027\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.9315			−0.0429420
$372$		0			0
$373$			476.357i			1.27710i	0.769582	+	0.638548i	$0.220465\pi$
$373$			476.357i			1.27710i	−0.769582	+	0.638548i	$0.779535\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−290.470			−0.770477
$378$		0			0
$379$		−	132.999i		−	0.350921i	−0.984486	−	0.175461i	$-0.943859\pi$
$379$		−	132.999i		−	0.350921i	0.984486	−	0.175461i	$-0.0561415\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	293.636i		−	0.766674i	−0.923609	−	0.383337i	$-0.874775\pi$
$383$		−	293.636i		−	0.766674i	0.923609	−	0.383337i	$-0.125225\pi$
$384$		0			0
$385$		−	4.43862i		−	0.0115289i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			83.9353i			0.215772i	0.994163	+	0.107886i	$0.0344082\pi$
$389$			83.9353i			0.215772i	−0.994163	+	0.107886i	$0.965592\pi$
$390$		0			0
$391$	16.4108	−	8.94029i	0.0419715	−	0.0228652i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−250.177			−0.633360
$396$		0			0
$397$	−535.742			−1.34948			−0.674738	−	0.738057i	$-0.735743\pi$
$397$	−535.742			−1.34948			−0.674738	+	0.738057i	$0.735743\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	606.395i		−	1.51221i	−0.654452	−	0.756103i	$-0.727101\pi$
$401$		−	606.395i		−	1.51221i	0.654452	−	0.756103i	$-0.272899\pi$
$402$		0			0
$403$	294.179			0.729972
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.427219			−0.00104968
$408$		0			0
$409$	−251.269			−0.614349			−0.307175	−	0.951653i	$-0.599384\pi$
$409$	−251.269			−0.614349			−0.307175	+	0.951653i	$0.599384\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			13.8254i			0.0334755i
$414$		0			0
$415$	−96.1508			−0.231689
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	531.560i		−	1.26864i	−0.773071	−	0.634319i	$-0.781280\pi$
$419$		−	531.560i		−	1.26864i	0.773071	−	0.634319i	$-0.218720\pi$
$420$		0			0
$421$		−	682.107i		−	1.62021i	−0.586287	−	0.810103i	$-0.699411\pi$
$421$		−	682.107i		−	1.62021i	0.586287	−	0.810103i	$-0.300589\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.06263i		−	0.00955913i
$426$		0			0
$427$	−5.53385			−0.0129598
$428$		0			0
$429$		0			0
$430$		0			0
$431$			531.119i			1.23229i	0.787631	+	0.616147i	$0.211307\pi$
$431$			531.119i			1.23229i	−0.787631	+	0.616147i	$0.788693\pi$
$432$		0			0
$433$			108.426i			0.250407i	0.992131	+	0.125204i	$0.0399584\pi$
$433$			108.426i			0.250407i	−0.992131	+	0.125204i	$0.960042\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−250.526	+	136.482i	−0.573287	+	0.312315i
$438$		0			0
$439$	−345.284			−0.786524			−0.393262	−	0.919427i	$-0.628653\pi$
$439$	−345.284			−0.786524			−0.393262	+	0.919427i	$0.628653\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	237.450			0.536005			0.268003	−	0.963418i	$-0.413636\pi$
$443$	237.450			0.536005			0.268003	+	0.963418i	$0.413636\pi$
$444$		0			0
$445$	−269.452			−0.605511
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−225.599			−0.502447			−0.251223	−	0.967929i	$-0.580833\pi$
$449$	−225.599			−0.502447			−0.251223	+	0.967929i	$0.580833\pi$
$450$		0			0
$451$		−	135.495i		−	0.300433i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.53863			0.00777721
$456$		0			0
$457$			143.797i			0.314653i	0.987547	+	0.157327i	$0.0502876\pi$
$457$			143.797i			0.314653i	−0.987547	+	0.157327i	$0.949712\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−264.233			−0.573174			−0.286587	−	0.958054i	$-0.592521\pi$
$461$	−264.233			−0.573174			−0.286587	+	0.958054i	$0.592521\pi$
$462$		0			0
$463$	−344.466			−0.743987			−0.371993	−	0.928235i	$-0.621326\pi$
$463$	−344.466			−0.743987			−0.371993	+	0.928235i	$0.621326\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			875.578i			1.87490i	0.348119	+	0.937450i	$0.386820\pi$
$467$			875.578i			1.87490i	−0.348119	+	0.937450i	$0.613180\pi$
$468$		0			0
$469$	0.574300			0.00122452
$470$		0			0
$471$		0			0
$472$		0			0
$473$	707.650			1.49609
$474$		0			0
$475$			62.0197i			0.130568i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	28.3088i		−	0.0590998i	−0.999563	−	0.0295499i	$-0.990593\pi$
$479$		−	28.3088i		−	0.0590998i	0.999563	−	0.0295499i	$-0.00940739\pi$
$480$		0			0
$481$		−	0.340595i		−	0.000708097i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.4800			0.0298557
$486$		0			0
$487$	−286.586			−0.588472			−0.294236	−	0.955733i	$-0.595065\pi$
$487$	−286.586			−0.588472			−0.294236	+	0.955733i	$0.595065\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−326.729			−0.665435			−0.332718	−	0.943026i	$-0.607966\pi$
$491$	−326.729			−0.665435			−0.332718	+	0.943026i	$0.607966\pi$
$492$		0			0
$493$		−	25.0323i		−	0.0507754i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			17.7732i			0.0357610i
$498$		0			0
$499$	−725.905			−1.45472			−0.727360	−	0.686256i	$-0.759253\pi$
$499$	−725.905			−1.45472			−0.727360	+	0.686256i	$0.759253\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	83.0457i		−	0.165101i	−0.996587	−	0.0825503i	$-0.973693\pi$
$503$		−	83.0457i		−	0.165101i	0.996587	−	0.0825503i	$-0.0263065\pi$
$504$		0			0
$505$			365.361i			0.723486i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	262.561			0.515836			0.257918	−	0.966167i	$-0.416964\pi$
$509$	262.561			0.515836			0.257918	+	0.966167i	$0.416964\pi$
$510$		0			0
$511$			19.8116i			0.0387703i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−202.261			−0.392740
$516$		0			0
$517$			420.747i			0.813824i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			273.816i			0.525559i	0.964856	+	0.262780i	$0.0846392\pi$
$521$			273.816i			0.525559i	−0.964856	+	0.262780i	$0.915361\pi$
$522$		0			0
$523$		−	605.284i		−	1.15733i	−0.815565	−	0.578666i	$-0.803574\pi$
$523$		−	605.284i		−	1.15733i	0.815565	−	0.578666i	$-0.196426\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			25.3519i			0.0481060i
$528$		0			0
$529$	−286.864	+	444.466i	−0.542276	+	0.840200i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	108.022			0.202668
$534$		0			0
$535$	−108.110			−0.202074
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	579.158i		−	1.07450i
$540$		0			0
$541$	−994.614			−1.83847			−0.919236	−	0.393706i	$-0.871193\pi$
$541$	−994.614			−1.83847			−0.919236	+	0.393706i	$0.871193\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	320.676			0.588397
$546$		0			0
$547$	−495.815			−0.906426			−0.453213	−	0.891402i	$-0.649722\pi$
$547$	−495.815			−0.906426			−0.453213	+	0.891402i	$0.649722\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			382.140i			0.693539i
$552$		0			0
$553$	18.7791			0.0339586
$554$		0			0
$555$		0			0
$556$		0			0
$557$			217.121i			0.389804i	0.980823	+	0.194902i	$0.0624389\pi$
$557$			217.121i			0.389804i	−0.980823	+	0.194902i	$0.937561\pi$
$558$		0			0
$559$			564.165i			1.00924i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	399.971i		−	0.710428i	−0.934785	−	0.355214i	$-0.884408\pi$
$563$		−	399.971i		−	0.710428i	0.934785	−	0.355214i	$-0.115592\pi$
$564$		0			0
$565$	178.319			0.315610
$566$		0			0
$567$		0			0
$568$		0			0
$569$			581.833i			1.02255i	0.859416	+	0.511277i	$0.170827\pi$
$569$			581.833i			1.02255i	−0.859416	+	0.511277i	$0.829173\pi$
$570$		0			0
$571$		−	944.467i		−	1.65406i	−0.562160	−	0.827029i	$-0.690029\pi$
$571$		−	944.467i		−	1.65406i	0.562160	−	0.827029i	$-0.309971\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	55.0154	+	100.987i	0.0956790	+	0.175629i
$576$		0			0
$577$	994.281			1.72319			0.861596	−	0.507595i	$-0.169465\pi$
$577$	994.281			1.72319			0.861596	+	0.507595i	$0.169465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.21738			0.0124223
$582$		0			0
$583$	−1122.52			−1.92542
$584$		0			0
$585$		0			0
$586$		0			0
$587$	936.823			1.59595			0.797975	−	0.602690i	$-0.205905\pi$
$587$	936.823			1.59595			0.797975	+	0.602690i	$0.205905\pi$
$588$		0			0
$589$		−	387.019i		−	0.657078i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−644.340			−1.08658			−0.543288	−	0.839546i	$-0.682821\pi$
$593$	−644.340			−1.08658			−0.543288	+	0.839546i	$0.682821\pi$
$594$		0			0
$595$			0.304954i			0.000512527i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	190.360			0.317796			0.158898	−	0.987295i	$-0.449206\pi$
$599$	190.360			0.317796			0.158898	+	0.987295i	$0.449206\pi$
$600$		0			0
$601$	869.009			1.44594			0.722969	−	0.690880i	$-0.242777\pi$
$601$	869.009			1.44594			0.722969	+	0.690880i	$0.242777\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	42.1777i		−	0.0697153i
$606$		0			0
$607$	−720.505			−1.18699			−0.593496	−	0.804837i	$-0.702253\pi$
$607$	−720.505			−1.18699			−0.593496	+	0.804837i	$0.702253\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−335.435			−0.548994
$612$		0			0
$613$		−	978.166i		−	1.59570i	−0.602854	−	0.797852i	$-0.705970\pi$
$613$		−	978.166i		−	1.59570i	0.602854	−	0.797852i	$-0.294030\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			401.759i			0.651149i	0.945516	+	0.325574i	$0.105558\pi$
$617$			401.759i			0.651149i	−0.945516	+	0.325574i	$0.894442\pi$
$618$		0			0
$619$		−	423.880i		−	0.684782i	−0.939557	−	0.342391i	$-0.888763\pi$
$619$		−	423.880i		−	0.684782i	0.939557	−	0.342391i	$-0.111237\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20.2259			0.0324654
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.0293519			4.66644e−5
$630$		0			0
$631$		−	852.860i		−	1.35160i	−0.737084	−	0.675801i	$-0.763798\pi$
$631$		−	852.860i		−	1.35160i	0.737084	−	0.675801i	$-0.236202\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	133.086i		−	0.209584i
$636$		0			0
$637$	461.726			0.724845
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	756.377i		−	1.17999i	−0.807405	−	0.589997i	$-0.799129\pi$
$641$		−	756.377i		−	1.17999i	0.807405	−	0.589997i	$-0.200871\pi$
$642$		0			0
$643$		−	209.381i		−	0.325632i	−0.986656	−	0.162816i	$-0.947942\pi$
$643$		−	209.381i		−	0.325632i	0.986656	−	0.162816i	$-0.0520577\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−84.3179			−0.130321			−0.0651607	−	0.997875i	$-0.520756\pi$
$647$	−84.3179			−0.130321			−0.0651607	+	0.997875i	$0.520756\pi$
$648$		0			0
$649$			974.128i			1.50097i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−323.963			−0.496116			−0.248058	−	0.968745i	$-0.579792\pi$
$653$	−323.963			−0.496116			−0.248058	+	0.968745i	$0.579792\pi$
$654$		0			0
$655$			45.4100i			0.0693282i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	879.605i		−	1.33476i	−0.744718	−	0.667379i	$-0.767416\pi$
$659$		−	879.605i		−	1.33476i	0.744718	−	0.667379i	$-0.232584\pi$
$660$		0			0
$661$		−	464.571i		−	0.702831i	−0.936220	−	0.351416i	$-0.885700\pi$
$661$		−	464.571i		−	0.702831i	0.936220	−	0.351416i	$-0.114300\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	4.65540i		−	0.00700059i
$666$		0			0
$667$	338.983	+	622.238i	0.508220	+	0.932891i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−389.911			−0.581090
$672$		0			0
$673$	1009.25			1.49964			0.749818	−	0.661644i	$-0.230141\pi$
$673$	1009.25			1.49964			0.749818	+	0.661644i	$0.230141\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			796.672i			1.17677i	0.808581	+	0.588384i	$0.200236\pi$
$677$			796.672i			1.17677i	−0.808581	+	0.588384i	$0.799764\pi$
$678$		0			0
$679$	−1.08691			−0.00160076
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−824.076			−1.20655			−0.603276	−	0.797532i	$-0.706138\pi$
$683$	−824.076			−1.20655			−0.603276	+	0.797532i	$0.706138\pi$
$684$		0			0
$685$	185.125			0.270256
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	894.916i		−	1.29886i
$690$		0			0
$691$	245.973			0.355967			0.177984	−	0.984033i	$-0.443043\pi$
$691$	245.973			0.355967			0.177984	+	0.984033i	$0.443043\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	126.386i		−	0.181850i
$696$		0			0
$697$			9.30916i			0.0133560i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	707.702i		−	1.00956i	−0.863248	−	0.504781i	$-0.831573\pi$
$701$		−	707.702i		−	1.00956i	0.863248	−	0.504781i	$-0.168427\pi$
$702$		0			0
$703$	−0.448084			−0.000637388
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	27.4251i		−	0.0387908i
$708$		0			0
$709$			662.227i			0.934030i	0.884249	+	0.467015i	$0.154671\pi$
$709$			662.227i			0.934030i	−0.884249	+	0.467015i	$0.845329\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−343.311	−	630.183i	−0.481502	−	0.883847i
$714$		0			0
$715$	249.330			0.348713
$716$		0			0
$717$		0			0
$718$		0			0
$719$	756.461			1.05210			0.526051	−	0.850453i	$-0.323672\pi$
$719$	756.461			1.05210			0.526051	+	0.850453i	$0.323672\pi$
$720$		0			0
$721$	15.1824			0.0210574
$722$		0			0
$723$		0			0
$724$		0			0
$725$	154.040			0.212469
$726$		0			0
$727$		−	505.968i		−	0.695966i	−0.937501	−	0.347983i	$-0.886867\pi$
$727$		−	505.968i		−	0.695966i	0.937501	−	0.347983i	$-0.113133\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.6189			−0.0665101
$732$		0			0
$733$			690.764i			0.942379i	0.882032	+	0.471190i	$0.156175\pi$
$733$			690.764i			0.942379i	−0.882032	+	0.471190i	$0.843825\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	40.4648			0.0549048
$738$		0			0
$739$	−732.402			−0.991071			−0.495536	−	0.868588i	$-0.665028\pi$
$739$	−732.402			−0.991071			−0.495536	+	0.868588i	$0.665028\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1196.62i			1.61052i	0.592922	+	0.805260i	$0.297974\pi$
$743$			1196.62i			1.61052i	−0.592922	+	0.805260i	$0.702026\pi$
$744$		0			0
$745$	−114.001			−0.153022
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.11507			0.0108345
$750$		0			0
$751$		−	639.498i		−	0.851529i	−0.904834	−	0.425764i	$-0.860005\pi$
$751$		−	639.498i		−	0.851529i	0.904834	−	0.425764i	$-0.139995\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	160.247i		−	0.212248i
$756$		0			0
$757$		−	615.162i		−	0.812632i	−0.913733	−	0.406316i	$-0.866813\pi$
$757$		−	615.162i		−	0.812632i	0.913733	−	0.406316i	$-0.133187\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−605.335			−0.795447			−0.397724	−	0.917505i	$-0.630200\pi$
$761$	−605.335			−0.795447			−0.397724	+	0.917505i	$0.630200\pi$
$762$		0			0
$763$	−24.0710			−0.0315478
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−776.611			−1.01253
$768$		0			0
$769$			919.319i			1.19547i	0.801692	+	0.597737i	$0.203933\pi$
$769$			919.319i			1.19547i	−0.801692	+	0.597737i	$0.796067\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			512.921i			0.663545i	0.943359	+	0.331773i	$0.107647\pi$
$773$			512.921i			0.663545i	−0.943359	+	0.331773i	$0.892353\pi$
$774$		0			0
$775$	−156.007			−0.201299
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	142.113i		−	0.182430i
$780$		0			0
$781$			1252.29i			1.60344i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	208.750			0.265924
$786$		0			0
$787$			1123.52i			1.42760i	0.700352	+	0.713798i	$0.253026\pi$
$787$			1123.52i			1.42760i	−0.700352	+	0.713798i	$0.746974\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13.3852			−0.0169219
$792$		0			0
$793$		−	310.852i		−	0.391995i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			699.854i			0.878110i	0.898460	+	0.439055i	$0.144687\pi$
$797$			699.854i			0.878110i	−0.898460	+	0.439055i	$0.855313\pi$
$798$		0			0
$799$		−	28.9073i		−	0.0361793i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1395.91i			1.73837i
$804$		0			0
$805$	−4.12963	−	7.58038i	−0.00512998	−	0.00941662i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−371.766			−0.459538			−0.229769	−	0.973245i	$-0.573797\pi$
$809$	−371.766			−0.459538			−0.229769	+	0.973245i	$0.573797\pi$
$810$		0			0
$811$	577.533			0.712124			0.356062	−	0.934462i	$-0.384119\pi$
$811$	577.533			0.712124			0.356062	+	0.934462i	$0.384119\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	480.257i		−	0.589273i
$816$		0			0
$817$	742.211			0.908459
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1041.07			1.26805			0.634025	−	0.773313i	$-0.281402\pi$
$821$	1041.07			1.26805			0.634025	+	0.773313i	$0.281402\pi$
$822$		0			0
$823$	741.598			0.901091			0.450546	−	0.892753i	$-0.351229\pi$
$823$	741.598			0.901091			0.450546	+	0.892753i	$0.351229\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	896.999i		−	1.08464i	−0.840172	−	0.542321i	$-0.817546\pi$
$827$		−	896.999i		−	1.08464i	0.840172	−	0.542321i	$-0.182454\pi$
$828$		0			0
$829$	1519.97			1.83350			0.916750	−	0.399461i	$-0.130803\pi$
$829$	1519.97			1.83350			0.916750	+	0.399461i	$0.130803\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			39.7909i			0.0477681i
$834$		0			0
$835$		−	651.206i		−	0.779888i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1367.51i		−	1.62992i	−0.579515	−	0.814961i	$-0.696758\pi$
$839$		−	1367.51i		−	1.62992i	0.579515	−	0.814961i	$-0.303242\pi$
$840$		0			0
$841$	108.130			0.128574
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	179.121i		−	0.211977i
$846$		0			0
$847$			3.16600i			0.00373789i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.729615	+	0.397479i	−0.000857361	+	0.000467073i
$852$		0			0
$853$	−394.255			−0.462198			−0.231099	−	0.972930i	$-0.574232\pi$
$853$	−394.255			−0.462198			−0.231099	+	0.972930i	$0.574232\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−583.972			−0.681415			−0.340707	−	0.940169i	$-0.610667\pi$
$857$	−583.972			−0.681415			−0.340707	+	0.940169i	$0.610667\pi$
$858$		0			0
$859$	−232.794			−0.271006			−0.135503	−	0.990777i	$-0.543265\pi$
$859$	−232.794			−0.271006			−0.135503	+	0.990777i	$0.543265\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	504.254			0.584304			0.292152	−	0.956372i	$-0.405629\pi$
$863$	504.254			0.584304			0.292152	+	0.956372i	$0.405629\pi$
$864$		0			0
$865$			208.408i			0.240934i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1323.16			1.52263
$870$		0			0
$871$			32.2601i			0.0370380i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.87658			−0.00214466
$876$		0			0
$877$	−332.149			−0.378733			−0.189366	−	0.981907i	$-0.560643\pi$
$877$	−332.149			−0.378733			−0.189366	+	0.981907i	$0.560643\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			27.0454i			0.0306986i	0.999882	+	0.0153493i	$0.00488602\pi$
$881$			27.0454i			0.0306986i	−0.999882	+	0.0153493i	$0.995114\pi$
$882$		0			0
$883$	736.220			0.833771			0.416886	−	0.908959i	$-0.363122\pi$
$883$	736.220			0.833771			0.416886	+	0.908959i	$0.363122\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	882.965			0.995451			0.497726	−	0.867335i	$-0.334169\pi$
$887$	882.965			0.995451			0.497726	+	0.867335i	$0.334169\pi$
$888$		0			0
$889$			9.98984i			0.0112372i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			441.296i			0.494172i
$894$		0			0
$895$		−	235.010i		−	0.262581i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−961.249			−1.06924
$900$		0			0
$901$	77.1225			0.0855965
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−712.647			−0.787455
$906$		0			0
$907$			935.513i			1.03144i	0.856758	+	0.515718i	$0.172475\pi$
$907$			935.513i			1.03144i	−0.856758	+	0.515718i	$0.827525\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			807.073i			0.885920i	0.896541	+	0.442960i	$0.146072\pi$
$911$			807.073i			0.885920i	−0.896541	+	0.442960i	$0.853928\pi$
$912$		0			0
$913$	508.532			0.556990
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	3.40862i		−	0.00371714i
$918$		0			0
$919$			1301.33i			1.41603i	0.706196	+	0.708017i	$0.250410\pi$
$919$			1301.33i			1.41603i	−0.706196	+	0.708017i	$0.749590\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−998.371			−1.08166
$924$		0			0
$925$			0.180622i			0.000195267i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	726.093			0.781586			0.390793	−	0.920479i	$-0.372201\pi$
$929$	726.093			0.781586			0.390793	+	0.920479i	$0.372201\pi$
$930$		0			0
$931$		−	607.444i		−	0.652464i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			21.4868i			0.0229806i
$936$		0			0
$937$		−	935.259i		−	0.998141i	−0.866561	−	0.499071i	$-0.833675\pi$
$937$		−	935.259i		−	0.998141i	0.866561	−	0.499071i	$-0.166325\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	119.809i		−	0.127321i	−0.997972	−	0.0636606i	$-0.979722\pi$
$941$		−	119.809i		−	0.127321i	0.997972	−	0.0636606i	$-0.0202775\pi$
$942$		0			0
$943$	−126.063	−	231.402i	−0.133683	−	0.245389i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−672.752			−0.710404			−0.355202	−	0.934790i	$-0.615588\pi$
$947$	−672.752			−0.710404			−0.355202	+	0.934790i	$0.615588\pi$
$948$		0			0
$949$	−1112.87			−1.17268
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1142.06i			1.19839i	0.800604	+	0.599194i	$0.204512\pi$
$953$			1142.06i			1.19839i	−0.800604	+	0.599194i	$0.795488\pi$
$954$		0			0
$955$	195.797			0.205023
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.8961			−0.0144902
$960$		0			0
$961$	12.5218			0.0130300
$962$		0			0
$963$		0			0
$964$		0			0
$965$			124.205i			0.128710i
$966$		0			0
$967$	−987.693			−1.02140			−0.510700	−	0.859759i	$-0.670614\pi$
$967$	−987.693			−1.02140			−0.510700	+	0.859759i	$0.670614\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1444.10i		−	1.48723i	−0.668606	−	0.743617i	$-0.733109\pi$
$971$		−	1444.10i		−	1.48723i	0.668606	−	0.743617i	$-0.266891\pi$
$972$		0			0
$973$			9.48694i			0.00975019i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	344.734i		−	0.352850i	−0.984314	−	0.176425i	$-0.943547\pi$
$977$		−	344.734i		−	0.352850i	0.984314	−	0.176425i	$-0.0564533\pi$
$978$		0			0
$979$	1425.11			1.45568
$980$		0			0
$981$		0			0
$982$		0			0
$983$			147.854i			0.150411i	0.997168	+	0.0752053i	$0.0239612\pi$
$983$			147.854i			0.150411i	−0.997168	+	0.0752053i	$0.976039\pi$
$984$		0			0
$985$			175.382i			0.178053i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1208.54	−	658.389i	1.22198	−	0.665711i
$990$		0			0
$991$	−289.664			−0.292295			−0.146147	−	0.989263i	$-0.546687\pi$
$991$	−289.664			−0.292295			−0.146147	+	0.989263i	$0.546687\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	459.237			0.461545
$996$		0			0
$997$	−94.0845			−0.0943676			−0.0471838	−	0.998886i	$-0.515025\pi$
$997$	−94.0845			−0.0943676			−0.0471838	+	0.998886i	$0.515025\pi$
$998$		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.2161.13		16
3.2	odd	2		460.3.f.a.321.5	✓	16
12.11	even	2		1840.3.k.c.321.11		16
15.2	even	4		2300.3.d.b.1149.18		32
15.8	even	4		2300.3.d.b.1149.15		32
15.14	odd	2		2300.3.f.e.1701.12		16
23.22	odd	2	inner	4140.3.d.a.2161.4		16
69.68	even	2		460.3.f.a.321.6	yes	16
276.275	odd	2		1840.3.k.c.321.12		16
345.68	odd	4		2300.3.d.b.1149.17		32
345.137	odd	4		2300.3.d.b.1149.16		32
345.344	even	2		2300.3.f.e.1701.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.5	✓	16	3.2	odd	2
460.3.f.a.321.6	yes	16	69.68	even	2
1840.3.k.c.321.11		16	12.11	even	2
1840.3.k.c.321.12		16	276.275	odd	2
2300.3.d.b.1149.15		32	15.8	even	4
2300.3.d.b.1149.16		32	345.137	odd	4
2300.3.d.b.1149.17		32	345.68	odd	4
2300.3.d.b.1149.18		32	15.2	even	4
2300.3.f.e.1701.11		16	345.344	even	2
2300.3.f.e.1701.12		16	15.14	odd	2
4140.3.d.a.2161.4		16	23.22	odd	2	inner
4140.3.d.a.2161.13		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -0.167846i q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} -0.167846i q^{7} -11.8263i q^{11} +9.42841 q^{13} +0.812526i q^{17} -12.4039i q^{19} +(-11.0031 - 20.1973i) q^{23} -5.00000 q^{25} -30.8080 q^{29} +31.2013 q^{31} +0.375316 q^{35} -0.0361243i q^{37} +11.4571 q^{41} +59.8367i q^{43} -35.5771 q^{47} +48.9718 q^{49} -94.9170i q^{53} +26.4445 q^{55} -82.3693 q^{59} -32.9697i q^{61} +21.0826i q^{65} +3.42158i q^{67} -105.890 q^{71} -118.034 q^{73} -1.98501 q^{77} +111.883i q^{79} +42.9999i q^{83} -1.81686 q^{85} +120.503i q^{89} -1.58252i q^{91} +27.7361 q^{95} -6.47565i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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