LMFDB - Embedded newform 2160.3.c.l.1889.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-11}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15x^{2} + 4 $ x^4 + 15*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.4
Root			$0.521137i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.l.1889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.36421 + 2.44002i) q^{5} -2.79549i q^{7} +O(q^{10})$	$q+(4.36421 + 2.44002i) q^{5} -2.79549i q^{7} +18.1465i q^{11} -23.0266i q^{13} -5.72842 q^{17} -23.1852 q^{19} +0.271584 q^{23} +(13.0926 + 21.2975i) q^{25} +39.7995i q^{29} -47.3705 q^{31} +(6.82104 - 12.2001i) q^{35} -34.8712i q^{37} +13.2665i q^{41} +46.7158i q^{43} -40.9137 q^{47} +41.1852 q^{49} -91.3705 q^{53} +(-44.2779 + 79.1953i) q^{55} +78.8398i q^{59} +31.1852 q^{61} +(56.1852 - 100.493i) q^{65} +6.91631i q^{67} -81.5870i q^{71} +106.084i q^{73} +50.7284 q^{77} -63.5557 q^{79} -0.284161 q^{83} +(-25.0000 - 13.9774i) q^{85} +28.5210i q^{89} -64.3705 q^{91} +(-101.185 - 56.5724i) q^{95} -92.9138i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5}+O(q^{10}) $ 4 * q + 3 * q^5	$ 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) $ 4 * q + 3 * q^5 + 6 * q^17 - 6 * q^19 + 30 * q^23 + 9 * q^25 - 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 - 192 * q^53 - 47 * q^55 + 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 + 288 * q^83 - 100 * q^85 - 84 * q^91 - 318 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	4.36421	+	2.44002i	0.872842	+	0.488004i
$5$	4.36421	+	2.44002i	0.872842	+	0.488004i
$6$		0			0
$7$		−	2.79549i		−	0.399355i	−0.979862	−	0.199678i	$-0.936010\pi$
$7$		−	2.79549i		−	0.399355i	0.979862	−	0.199678i	$-0.0639895\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			18.1465i			1.64969i	0.565363	+	0.824843i	$0.308736\pi$
$11$			18.1465i			1.64969i	−0.565363	+	0.824843i	$0.691264\pi$
$12$		0			0
$13$		−	23.0266i		−	1.77127i	−0.464378	−	0.885637i	$-0.653722\pi$
$13$		−	23.0266i		−	1.77127i	0.464378	−	0.885637i	$-0.346278\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.72842			−0.336966			−0.168483	−	0.985705i	$-0.553887\pi$
$17$	−5.72842			−0.336966			−0.168483	+	0.985705i	$0.553887\pi$
$18$		0			0
$19$	−23.1852			−1.22028			−0.610138	−	0.792295i	$-0.708886\pi$
$19$	−23.1852			−1.22028			−0.610138	+	0.792295i	$0.708886\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.271584			0.0118080			0.00590400	−	0.999983i	$-0.498121\pi$
$23$	0.271584			0.0118080			0.00590400	+	0.999983i	$0.498121\pi$
$24$		0			0
$25$	13.0926	+	21.2975i	0.523705	+	0.851900i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			39.7995i			1.37240i	0.727415	+	0.686198i	$0.240722\pi$
$29$			39.7995i			1.37240i	−0.727415	+	0.686198i	$0.759278\pi$
$30$		0			0
$31$	−47.3705			−1.52808			−0.764040	−	0.645169i	$-0.776787\pi$
$31$	−47.3705			−1.52808			−0.764040	+	0.645169i	$0.776787\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.82104	−	12.2001i	0.194887	−	0.348574i
$36$		0			0
$37$		−	34.8712i		−	0.942465i	−0.882009	−	0.471232i	$-0.843809\pi$
$37$		−	34.8712i		−	0.942465i	0.882009	−	0.471232i	$-0.156191\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			13.2665i			0.323573i	0.986826	+	0.161787i	$0.0517256\pi$
$41$			13.2665i			0.323573i	−0.986826	+	0.161787i	$0.948274\pi$
$42$		0			0
$43$			46.7158i			1.08641i	0.839599	+	0.543207i	$0.182790\pi$
$43$			46.7158i			1.08641i	−0.839599	+	0.543207i	$0.817210\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−40.9137			−0.870504			−0.435252	−	0.900309i	$-0.643341\pi$
$47$	−40.9137			−0.870504			−0.435252	+	0.900309i	$0.643341\pi$
$48$		0			0
$49$	41.1852			0.840515
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−91.3705			−1.72397			−0.861986	−	0.506932i	$-0.830779\pi$
$53$	−91.3705			−1.72397			−0.861986	+	0.506932i	$0.830779\pi$
$54$		0			0
$55$	−44.2779	+	79.1953i	−0.805052	+	1.43991i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			78.8398i			1.33627i	0.744041	+	0.668134i	$0.232907\pi$
$59$			78.8398i			1.33627i	−0.744041	+	0.668134i	$0.767093\pi$
$60$		0			0
$61$	31.1852			0.511234			0.255617	−	0.966778i	$-0.417721\pi$
$61$	31.1852			0.511234			0.255617	+	0.966778i	$0.417721\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	56.1852	−	100.493i	0.864388	−	1.54604i
$66$		0			0
$67$			6.91631i			0.103229i	0.998667	+	0.0516143i	$0.0164366\pi$
$67$			6.91631i			0.103229i	−0.998667	+	0.0516143i	$0.983563\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	81.5870i		−	1.14911i	−0.818465	−	0.574556i	$-0.805175\pi$
$71$		−	81.5870i		−	1.14911i	0.818465	−	0.574556i	$-0.194825\pi$
$72$		0			0
$73$			106.084i			1.45320i	0.687060	+	0.726601i	$0.258901\pi$
$73$			106.084i			1.45320i	−0.687060	+	0.726601i	$0.741099\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	50.7284			0.658811
$78$		0			0
$79$	−63.5557			−0.804503			−0.402252	−	0.915529i	$-0.631772\pi$
$79$	−63.5557			−0.804503			−0.402252	+	0.915529i	$0.631772\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.284161			−0.00342363			−0.00171182	−	0.999999i	$-0.500545\pi$
$83$	−0.284161			−0.00342363			−0.00171182	+	0.999999i	$0.500545\pi$
$84$		0			0
$85$	−25.0000	−	13.9774i	−0.294118	−	0.164440i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			28.5210i			0.320461i	0.987080	+	0.160230i	$0.0512237\pi$
$89$			28.5210i			0.320461i	−0.987080	+	0.160230i	$0.948776\pi$
$90$		0			0
$91$	−64.3705			−0.707368
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−101.185	−	56.5724i	−1.06511	−	0.595499i
$96$		0			0
$97$		−	92.9138i		−	0.957874i	−0.877849	−	0.478937i	$-0.841022\pi$
$97$		−	92.9138i		−	0.957874i	0.877849	−	0.478937i	$-0.158978\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	46.6675i		−	0.462055i	−0.972947	−	0.231027i	$-0.925791\pi$
$101$		−	46.6675i		−	0.462055i	0.972947	−	0.231027i	$-0.0742087\pi$
$102$		0			0
$103$		−	11.1820i		−	0.108563i	−0.998526	−	0.0542813i	$-0.982713\pi$
$103$		−	11.1820i		−	0.108563i	0.998526	−	0.0542813i	$-0.0172868\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	147.297			1.37661			0.688303	−	0.725424i	$-0.258356\pi$
$107$	147.297			1.37661			0.688303	+	0.725424i	$0.258356\pi$
$108$		0			0
$109$	−121.556			−1.11519			−0.557595	−	0.830113i	$-0.688276\pi$
$109$	−121.556			−1.11519			−0.557595	+	0.830113i	$0.688276\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−97.6295			−0.863978			−0.431989	−	0.901879i	$-0.642188\pi$
$113$	−97.6295			−0.863978			−0.431989	+	0.901879i	$0.642188\pi$
$114$		0			0
$115$	1.18525	+	0.662669i	0.0103065	+	0.00576234i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			16.0137i			0.134569i
$120$		0			0
$121$	−208.297			−1.72146
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.17267	+	124.893i	0.0413814	+	0.999143i
$126$		0			0
$127$		−	88.6481i		−	0.698017i	−0.937120	−	0.349008i	$-0.886518\pi$
$127$		−	88.6481i		−	0.698017i	0.937120	−	0.349008i	$-0.113482\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			89.9735i			0.686820i	0.939186	+	0.343410i	$0.111582\pi$
$131$			89.9735i			0.686820i	−0.939186	+	0.343410i	$0.888418\pi$
$132$		0			0
$133$			64.8141i			0.487324i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	229.926			1.67829			0.839147	−	0.543905i	$-0.183055\pi$
$137$	229.926			1.67829			0.839147	+	0.543905i	$0.183055\pi$
$138$		0			0
$139$	−57.6295			−0.414601			−0.207300	−	0.978277i	$-0.566468\pi$
$139$	−57.6295			−0.414601			−0.207300	+	0.978277i	$0.566468\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	417.852			2.92205
$144$		0			0
$145$	−97.1115	+	173.693i	−0.669734	+	1.19788i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.1337i			0.0747227i	0.999302	+	0.0373613i	$0.0118953\pi$
$149$			11.1337i			0.0747227i	−0.999302	+	0.0373613i	$0.988105\pi$
$150$		0			0
$151$	22.1852			0.146922			0.0734611	−	0.997298i	$-0.476596\pi$
$151$	22.1852			0.146922			0.0734611	+	0.997298i	$0.476596\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−206.735	−	115.585i	−1.33377	−	0.745709i
$156$		0			0
$157$			225.337i			1.43527i	0.696419	+	0.717635i	$0.254775\pi$
$157$			225.337i			1.43527i	−0.696419	+	0.717635i	$0.745225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.759209i		−	0.00471559i
$162$		0			0
$163$			302.948i			1.85858i	0.369352	+	0.929290i	$0.379580\pi$
$163$			302.948i			1.85858i	−0.369352	+	0.929290i	$0.620420\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−238.667			−1.42915			−0.714573	−	0.699561i	$-0.753379\pi$
$167$	−238.667			−1.42915			−0.714573	+	0.699561i	$0.753379\pi$
$168$		0			0
$169$	−361.223			−2.13741
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−108.802			−0.628914			−0.314457	−	0.949272i	$-0.601822\pi$
$173$	−108.802			−0.628914			−0.314457	+	0.949272i	$0.601822\pi$
$174$		0			0
$175$	59.5369	−	36.6003i	0.340211	−	0.209144i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	168.054i		−	0.938849i	−0.882973	−	0.469425i	$-0.844461\pi$
$179$		−	168.054i		−	0.938849i	0.882973	−	0.469425i	$-0.155539\pi$
$180$		0			0
$181$	−154.297			−0.852468			−0.426234	−	0.904613i	$-0.640160\pi$
$181$	−154.297			−0.852468			−0.426234	+	0.904613i	$0.640160\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	85.0863	−	152.185i	0.459926	−	0.822622i
$186$		0			0
$187$		−	103.951i		−	0.555887i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			154.932i			0.811164i	0.914059	+	0.405582i	$0.132931\pi$
$191$			154.932i			0.811164i	−0.914059	+	0.405582i	$0.867069\pi$
$192$		0			0
$193$			64.0066i			0.331640i	0.986156	+	0.165820i	$0.0530271\pi$
$193$			64.0066i			0.331640i	−0.986156	+	0.165820i	$0.946973\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.56832			−0.0485702			−0.0242851	−	0.999705i	$-0.507731\pi$
$197$	−9.56832			−0.0485702			−0.0242851	+	0.999705i	$0.507731\pi$
$198$		0			0
$199$	117.815			0.592034			0.296017	−	0.955183i	$-0.404342\pi$
$199$	117.815			0.592034			0.296017	+	0.955183i	$0.404342\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	111.259			0.548074
$204$		0			0
$205$	−32.3705	+	57.8978i	−0.157905	+	0.282428i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	420.732i		−	2.01307i
$210$		0			0
$211$	48.8148			0.231350			0.115675	−	0.993287i	$-0.463097\pi$
$211$	48.8148			0.231350			0.115675	+	0.993287i	$0.463097\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−113.987	+	203.878i	−0.530174	+	0.948268i
$216$		0			0
$217$			132.424i			0.610247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			131.906i			0.596859i
$222$		0			0
$223$		−	319.721i		−	1.43373i	−0.697213	−	0.716864i	$-0.745577\pi$
$223$		−	319.721i		−	1.43373i	0.697213	−	0.716864i	$-0.254423\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.2716			−0.0804916			−0.0402458	−	0.999190i	$-0.512814\pi$
$227$	−18.2716			−0.0804916			−0.0402458	+	0.999190i	$0.512814\pi$
$228$		0			0
$229$	134.074			0.585475			0.292737	−	0.956193i	$-0.405434\pi$
$229$	134.074			0.585475			0.292737	+	0.956193i	$0.405434\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−284.173			−1.21963			−0.609813	−	0.792546i	$-0.708755\pi$
$233$	−284.173			−1.21963			−0.609813	+	0.792546i	$0.708755\pi$
$234$		0			0
$235$	−178.556	−	99.8301i	−0.759812	−	0.424809i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	206.770i		−	0.865145i	−0.901599	−	0.432572i	$-0.857606\pi$
$239$		−	206.770i		−	0.865145i	0.901599	−	0.432572i	$-0.142394\pi$
$240$		0			0
$241$	−162.667			−0.674968			−0.337484	−	0.941331i	$-0.609576\pi$
$241$	−162.667			−0.674968			−0.337484	+	0.941331i	$0.609576\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	179.741	+	100.493i	0.733637	+	0.410174i
$246$		0			0
$247$			533.877i			2.16144i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	347.387i		−	1.38401i	−0.721893	−	0.692005i	$-0.756728\pi$
$251$		−	347.387i		−	1.38401i	0.721893	−	0.692005i	$-0.243272\pi$
$252$		0			0
$253$			4.92831i			0.0194795i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−21.0377			−0.0818589			−0.0409294	−	0.999162i	$-0.513032\pi$
$257$	−21.0377			−0.0818589			−0.0409294	+	0.999162i	$0.513032\pi$
$258$		0			0
$259$	−97.4820			−0.376378
$260$		0			0
$261$		0			0
$262$		0			0
$263$	215.457			0.819227			0.409614	−	0.912259i	$-0.365663\pi$
$263$	215.457			0.819227			0.409614	+	0.912259i	$0.365663\pi$
$264$		0			0
$265$	−398.760	−	222.946i	−1.50475	−	0.841304i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			180.237i			0.670024i	0.942214	+	0.335012i	$0.108740\pi$
$269$			180.237i			0.670024i	−0.942214	+	0.335012i	$0.891260\pi$
$270$		0			0
$271$	−231.741			−0.855133			−0.427566	−	0.903984i	$-0.640629\pi$
$271$	−231.741			−0.855133			−0.427566	+	0.903984i	$0.640629\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−386.476	+	237.586i	−1.40537	+	0.863948i
$276$		0			0
$277$			105.566i			0.381104i	0.981677	+	0.190552i	$0.0610278\pi$
$277$			105.566i			0.381104i	−0.981677	+	0.190552i	$0.938972\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			504.597i			1.79572i	0.440284	+	0.897859i	$0.354878\pi$
$281$			504.597i			1.79572i	−0.440284	+	0.897859i	$0.645122\pi$
$282$		0			0
$283$		−	151.619i		−	0.535756i	−0.963453	−	0.267878i	$-0.913678\pi$
$283$		−	151.619i		−	0.535756i	0.963453	−	0.267878i	$-0.0863224\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.0863			0.129221
$288$		0			0
$289$	−256.185			−0.886454
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−549.038			−1.87385			−0.936924	−	0.349532i	$-0.886341\pi$
$293$	−549.038			−1.87385			−0.936924	+	0.349532i	$0.886341\pi$
$294$		0			0
$295$	−192.370	+	344.073i	−0.652103	+	1.16635i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	6.25364i		−	0.0209152i
$300$		0			0
$301$	130.593			0.433865
$302$		0			0
$303$		0			0
$304$		0			0
$305$	136.099	+	76.0926i	0.446226	+	0.249484i
$306$		0			0
$307$			146.401i			0.476876i	0.971158	+	0.238438i	$0.0766355\pi$
$307$			146.401i			0.476876i	−0.971158	+	0.238438i	$0.923365\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			487.824i			1.56856i	0.620404	+	0.784282i	$0.286969\pi$
$311$			487.824i			1.56856i	−0.620404	+	0.784282i	$0.713031\pi$
$312$		0			0
$313$		−	109.687i		−	0.350437i	−0.984530	−	0.175218i	$-0.943937\pi$
$313$		−	109.687i		−	0.350437i	0.984530	−	0.175218i	$-0.0560631\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	363.568			1.14690			0.573452	−	0.819239i	$-0.305604\pi$
$317$	363.568			1.14690			0.573452	+	0.819239i	$0.305604\pi$
$318$		0			0
$319$	−722.223			−2.26402
$320$		0			0
$321$		0			0
$322$		0			0
$323$	132.815			0.411191
$324$		0			0
$325$	490.408	−	301.478i	1.50895	−	0.927625i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			114.374i			0.347640i
$330$		0			0
$331$	−300.223			−0.907018			−0.453509	−	0.891252i	$-0.649828\pi$
$331$	−300.223			−0.907018			−0.453509	+	0.891252i	$0.649828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.8759	+	30.1842i	−0.0503759	+	0.0901022i
$336$		0			0
$337$			373.353i			1.10787i	0.832559	+	0.553937i	$0.186875\pi$
$337$			373.353i			1.10787i	−0.832559	+	0.553937i	$0.813125\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	859.610i		−	2.52085i
$342$		0			0
$343$		−	252.112i		−	0.735020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.4443			0.0502716			0.0251358	−	0.999684i	$-0.491998\pi$
$347$	17.4443			0.0502716			0.0251358	+	0.999684i	$0.491998\pi$
$348$		0			0
$349$	234.149			0.670915			0.335457	−	0.942055i	$-0.391109\pi$
$349$	234.149			0.670915			0.335457	+	0.942055i	$0.391109\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−381.284			−1.08013			−0.540063	−	0.841625i	$-0.681599\pi$
$353$	−381.284			−1.08013			−0.540063	+	0.841625i	$0.681599\pi$
$354$		0			0
$355$	199.074	−	356.063i	0.560771	−	1.00299i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			222.024i			0.618451i	0.950989	+	0.309226i	$0.100070\pi$
$359$			222.024i			0.618451i	−0.950989	+	0.309226i	$0.899930\pi$
$360$		0			0
$361$	176.556			0.489074
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−258.846	+	462.971i	−0.709168	+	1.26841i
$366$		0			0
$367$			449.205i			1.22399i	0.790861	+	0.611995i	$0.209633\pi$
$367$			449.205i			1.22399i	−0.790861	+	0.611995i	$0.790367\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			255.425i			0.688477i
$372$		0			0
$373$		−	322.082i		−	0.863492i	−0.901995	−	0.431746i	$-0.857898\pi$
$373$		−	322.082i		−	0.863492i	0.901995	−	0.431746i	$-0.142102\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	916.446			2.43089
$378$		0			0
$379$	216.074			0.570115			0.285058	−	0.958510i	$-0.407987\pi$
$379$	216.074			0.570115			0.285058	+	0.958510i	$0.407987\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−373.977			−0.976440			−0.488220	−	0.872721i	$-0.662354\pi$
$383$	−373.977			−0.976440			−0.488220	+	0.872721i	$0.662354\pi$
$384$		0			0
$385$	221.389	+	123.778i	0.575037	+	0.321502i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			223.108i			0.573543i	0.957999	+	0.286771i	$0.0925820\pi$
$389$			223.108i			0.573543i	−0.957999	+	0.286771i	$0.907418\pi$
$390$		0			0
$391$	−1.55575			−0.00397889
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−277.370	−	155.077i	−0.702204	−	0.392600i
$396$		0			0
$397$		−	610.535i		−	1.53787i	−0.639325	−	0.768936i	$-0.720786\pi$
$397$		−	610.535i		−	1.53787i	0.639325	−	0.768936i	$-0.279214\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			411.441i			1.02604i	0.858377	+	0.513019i	$0.171473\pi$
$401$			411.441i			1.02604i	−0.858377	+	0.513019i	$0.828527\pi$
$402$		0			0
$403$			1090.78i			2.70665i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	632.791			1.55477
$408$		0			0
$409$	590.630			1.44408			0.722041	−	0.691850i	$-0.243204\pi$
$409$	590.630			1.44408			0.722041	+	0.691850i	$0.243204\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	220.396			0.533646
$414$		0			0
$415$	−1.24014	−	0.693359i	−0.00298829	−	0.00167074i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			239.556i			0.571733i	0.958269	+	0.285867i	$0.0922814\pi$
$419$			239.556i			0.571733i	−0.958269	+	0.285867i	$0.907719\pi$
$420$		0			0
$421$	593.408			1.40952			0.704760	−	0.709445i	$-0.251055\pi$
$421$	593.408			1.40952			0.704760	+	0.709445i	$0.251055\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−75.0000	−	122.001i	−0.176471	−	0.287061i
$426$		0			0
$427$		−	87.1780i		−	0.204164i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	436.566i		−	1.01291i	−0.862265	−	0.506457i	$-0.830955\pi$
$431$		−	436.566i		−	1.01291i	0.862265	−	0.506457i	$-0.169045\pi$
$432$		0			0
$433$			231.736i			0.535187i	0.963532	+	0.267593i	$0.0862284\pi$
$433$			231.736i			0.535187i	−0.963532	+	0.267593i	$0.913772\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.29674			−0.0144090
$438$		0			0
$439$	419.223			0.954950			0.477475	−	0.878645i	$-0.341552\pi$
$439$	419.223			0.954950			0.477475	+	0.878645i	$0.341552\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	446.520			1.00795			0.503973	−	0.863720i	$-0.331871\pi$
$443$	446.520			1.00795			0.503973	+	0.863720i	$0.331871\pi$
$444$		0			0
$445$	−69.5918	+	124.472i	−0.156386	+	0.279711i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	495.776i		−	1.10418i	−0.833785	−	0.552089i	$-0.813831\pi$
$449$		−	495.776i		−	1.10418i	0.833785	−	0.552089i	$-0.186169\pi$
$450$		0			0
$451$	−240.741			−0.533794
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−280.926	−	157.065i	−0.617420	−	0.345198i
$456$		0			0
$457$			683.363i			1.49532i	0.664080	+	0.747662i	$0.268824\pi$
$457$			683.363i			1.49532i	−0.664080	+	0.747662i	$0.731176\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	400.128i		−	0.867956i	−0.900923	−	0.433978i	$-0.857110\pi$
$461$		−	400.128i		−	0.867956i	0.900923	−	0.433978i	$-0.142890\pi$
$462$		0			0
$463$			316.263i			0.683074i	0.939868	+	0.341537i	$0.110947\pi$
$463$			316.263i			0.683074i	−0.939868	+	0.341537i	$0.889053\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	419.741			0.898803			0.449401	−	0.893330i	$-0.351637\pi$
$467$	419.741			0.898803			0.449401	+	0.893330i	$0.351637\pi$
$468$		0			0
$469$	19.3345			0.0412249
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−847.730			−1.79224
$474$		0			0
$475$	−303.556	−	493.788i	−0.639065	−	1.03955i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	844.970i		−	1.76403i	−0.471222	−	0.882015i	$-0.656187\pi$
$479$		−	844.970i		−	1.76403i	0.471222	−	0.882015i	$-0.343813\pi$
$480$		0			0
$481$	−802.964			−1.66936
$482$		0			0
$483$		0			0
$484$		0			0
$485$	226.711	−	405.495i	0.467446	−	0.836072i
$486$		0			0
$487$		−	546.384i		−	1.12194i	−0.827837	−	0.560969i	$-0.810429\pi$
$487$		−	546.384i		−	1.12194i	0.827837	−	0.560969i	$-0.189571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	647.456i		−	1.31865i	−0.751859	−	0.659324i	$-0.770843\pi$
$491$		−	647.456i		−	1.31865i	0.751859	−	0.659324i	$-0.229157\pi$
$492$		0			0
$493$		−	227.988i		−	0.462450i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−228.075			−0.458904
$498$		0			0
$499$	−582.815			−1.16797			−0.583983	−	0.811766i	$-0.698506\pi$
$499$	−582.815			−1.16797			−0.583983	+	0.811766i	$0.698506\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−752.520			−1.49606			−0.748032	−	0.663663i	$-0.769001\pi$
$503$	−752.520			−1.49606			−0.748032	+	0.663663i	$0.769001\pi$
$504$		0			0
$505$	113.870	−	203.667i	0.225484	−	0.403301i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	291.538i		−	0.572767i	−0.958115	−	0.286383i	$-0.907547\pi$
$509$		−	291.538i		−	0.572767i	0.958115	−	0.286383i	$-0.0924531\pi$
$510$		0			0
$511$	296.556			0.580344
$512$		0			0
$513$		0			0
$514$		0			0
$515$	27.2842	−	48.8004i	0.0529790	−	0.0947580i
$516$		0			0
$517$		−	742.441i		−	1.43606i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			137.690i			0.264280i	0.991231	+	0.132140i	$0.0421848\pi$
$521$			137.690i			0.264280i	−0.991231	+	0.132140i	$0.957815\pi$
$522$		0			0
$523$		−	330.987i		−	0.632862i	−0.948616	−	0.316431i	$-0.897515\pi$
$523$		−	330.987i		−	0.632862i	0.948616	−	0.316431i	$-0.102485\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	271.358			0.514911
$528$		0			0
$529$	−528.926			−0.999861
$530$		0			0
$531$		0			0
$532$		0			0
$533$	305.482			0.573137
$534$		0			0
$535$	642.834	+	359.407i	1.20156	+	0.671788i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			747.370i			1.38659i
$540$		0			0
$541$	556.593			1.02882			0.514412	−	0.857543i	$-0.328010\pi$
$541$	556.593			1.02882			0.514412	+	0.857543i	$0.328010\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−530.495	−	296.598i	−0.973385	−	0.544217i
$546$		0			0
$547$		−	465.170i		−	0.850402i	−0.905099	−	0.425201i	$-0.860203\pi$
$547$		−	465.170i		−	0.850402i	0.905099	−	0.425201i	$-0.139797\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	922.761i		−	1.67470i
$552$		0			0
$553$			177.669i			0.321283i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1059.37			1.90192			0.950962	−	0.309306i	$-0.100097\pi$
$557$	1059.37			1.90192			0.950962	+	0.309306i	$0.100097\pi$
$558$		0			0
$559$	1075.70			1.92434
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−386.160			−0.685897			−0.342949	−	0.939354i	$-0.611426\pi$
$563$	−386.160			−0.685897			−0.342949	+	0.939354i	$0.611426\pi$
$564$		0			0
$565$	−426.075	−	238.218i	−0.754116	−	0.421624i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			686.062i			1.20573i	0.797842	+	0.602866i	$0.205975\pi$
$569$			686.062i			1.20573i	−0.797842	+	0.602866i	$0.794025\pi$
$570$		0			0
$571$	821.631			1.43893			0.719467	−	0.694527i	$-0.244386\pi$
$571$	821.631			1.43893			0.719467	+	0.694527i	$0.244386\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.55575	+	5.78406i	0.00618390	+	0.0100592i
$576$		0			0
$577$		−	183.840i		−	0.318613i	−0.987229	−	0.159306i	$-0.949074\pi$
$577$		−	183.840i		−	0.318613i	0.987229	−	0.159306i	$-0.0509258\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			0.794370i			0.00136725i
$582$		0			0
$583$		−	1658.06i		−	2.84401i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	320.407			0.545837			0.272919	−	0.962037i	$-0.412011\pi$
$587$	320.407			0.545837			0.272919	+	0.962037i	$0.412011\pi$
$588$		0			0
$589$	1098.30			1.86468
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1061.34			1.78977			0.894887	−	0.446292i	$-0.147256\pi$
$593$	1061.34			1.78977			0.894887	+	0.446292i	$0.147256\pi$
$594$		0			0
$595$	−39.0738	+	69.8872i	−0.0656702	+	0.117457i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			977.056i			1.63115i	0.578655	+	0.815573i	$0.303578\pi$
$599$			977.056i			1.63115i	−0.578655	+	0.815573i	$0.696422\pi$
$600$		0			0
$601$	−261.816			−0.435635			−0.217817	−	0.975990i	$-0.569894\pi$
$601$	−261.816			−0.435635			−0.217817	+	0.975990i	$0.569894\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−909.050	−	508.248i	−1.50256	−	0.840079i
$606$		0			0
$607$		−	668.806i		−	1.10182i	−0.834564	−	0.550911i	$-0.814280\pi$
$607$		−	668.806i		−	1.10182i	0.834564	−	0.550911i	$-0.185720\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			942.101i			1.54190i
$612$		0			0
$613$		−	220.119i		−	0.359086i	−0.983750	−	0.179543i	$-0.942538\pi$
$613$		−	220.119i		−	0.359086i	0.983750	−	0.179543i	$-0.0574618\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−321.063			−0.520361			−0.260181	−	0.965560i	$-0.583782\pi$
$617$	−321.063			−0.520361			−0.260181	+	0.965560i	$0.583782\pi$
$618$		0			0
$619$	891.187			1.43972			0.719860	−	0.694119i	$-0.244206\pi$
$619$	891.187			1.43972			0.719860	+	0.694119i	$0.244206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	79.7301			0.127978
$624$		0			0
$625$	−282.166	+	557.680i	−0.451466	+	0.892288i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			199.757i			0.317578i
$630$		0			0
$631$	583.669			0.924990			0.462495	−	0.886622i	$-0.346954\pi$
$631$	583.669			0.924990			0.462495	+	0.886622i	$0.346954\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	216.303	−	386.879i	0.340635	−	0.609258i
$636$		0			0
$637$		−	948.355i		−	1.48878i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	696.292i		−	1.08626i	−0.839649	−	0.543129i	$-0.817239\pi$
$641$		−	696.292i		−	1.08626i	0.839649	−	0.543129i	$-0.182761\pi$
$642$		0			0
$643$		−	801.664i		−	1.24676i	−0.781920	−	0.623378i	$-0.785760\pi$
$643$		−	801.664i		−	1.24676i	0.781920	−	0.623378i	$-0.214240\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	216.221			0.334191			0.167095	−	0.985941i	$-0.446561\pi$
$647$	216.221			0.334191			0.167095	+	0.985941i	$0.446561\pi$
$648$		0			0
$649$	−1430.67			−2.20442
$650$		0			0
$651$		0			0
$652$		0			0
$653$	438.802			0.671979			0.335989	−	0.941866i	$-0.390929\pi$
$653$	438.802			0.671979			0.335989	+	0.941866i	$0.390929\pi$
$654$		0			0
$655$	−219.537	+	392.663i	−0.335171	+	0.599485i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			366.112i			0.555557i	0.960645	+	0.277779i	$0.0895982\pi$
$659$			366.112i			0.555557i	−0.960645	+	0.277779i	$0.910402\pi$
$660$		0			0
$661$	158.075			0.239146			0.119573	−	0.992825i	$-0.461847\pi$
$661$	158.075			0.239146			0.119573	+	0.992825i	$0.461847\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−158.148	+	282.862i	−0.237816	+	0.425357i
$666$		0			0
$667$			10.8089i			0.0162052i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			565.904i			0.843374i
$672$		0			0
$673$			648.575i			0.963708i	0.876252	+	0.481854i	$0.160036\pi$
$673$			648.575i			0.963708i	−0.876252	+	0.481854i	$0.839964\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	202.345			0.298885			0.149443	−	0.988770i	$-0.452252\pi$
$677$	202.345			0.298885			0.149443	+	0.988770i	$0.452252\pi$
$678$		0			0
$679$	−259.739			−0.382532
$680$		0			0
$681$		0			0
$682$		0			0
$683$	173.829			0.254508			0.127254	−	0.991870i	$-0.459384\pi$
$683$	173.829			0.254508			0.127254	+	0.991870i	$0.459384\pi$
$684$		0			0
$685$	1003.45	+	561.024i	1.46488	+	0.819013i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2103.95i			3.05363i
$690$		0			0
$691$	846.743			1.22539			0.612694	−	0.790320i	$-0.290086\pi$
$691$	846.743			1.22539			0.612694	+	0.790320i	$0.290086\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−251.507	−	140.617i	−0.361881	−	0.202327i
$696$		0			0
$697$		−	75.9960i		−	0.109033i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			72.2614i			0.103083i	0.998671	+	0.0515416i	$0.0164135\pi$
$701$			72.2614i			0.103083i	−0.998671	+	0.0515416i	$0.983587\pi$
$702$		0			0
$703$			808.497i			1.15007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−130.459			−0.184524
$708$		0			0
$709$	287.557			0.405582			0.202791	−	0.979222i	$-0.434999\pi$
$709$	287.557			0.405582			0.202791	+	0.979222i	$0.434999\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.8651			−0.0180436
$714$		0			0
$715$	1823.60	+	1019.57i	2.55048	+	1.42597i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1015.27i		−	1.41205i	−0.708185	−	0.706027i	$-0.750486\pi$
$719$		−	1015.27i		−	1.41205i	0.708185	−	0.706027i	$-0.249514\pi$
$720$		0			0
$721$	−31.2590			−0.0433551
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−847.630	+	521.080i	−1.16914	+	0.718731i
$726$		0			0
$727$		−	311.418i		−	0.428361i	−0.976794	−	0.214180i	$-0.931292\pi$
$727$		−	311.418i		−	0.428361i	0.976794	−	0.214180i	$-0.0687080\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	267.608i		−	0.366084i
$732$		0			0
$733$			321.047i			0.437990i	0.975726	+	0.218995i	$0.0702778\pi$
$733$			321.047i			0.437990i	−0.975726	+	0.218995i	$0.929722\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−125.507			−0.170295
$738$		0			0
$739$	657.185			0.889290			0.444645	−	0.895707i	$-0.353330\pi$
$739$	657.185			0.889290			0.444645	+	0.895707i	$0.353330\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1024.96			−1.37949			−0.689747	−	0.724051i	$-0.742278\pi$
$743$	−1024.96			−1.37949			−0.689747	+	0.724051i	$0.742278\pi$
$744$		0			0
$745$	−27.1664	+	48.5897i	−0.0364649	+	0.0652211i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	411.766i		−	0.549755i
$750$		0			0
$751$	63.1475			0.0840846			0.0420423	−	0.999116i	$-0.486614\pi$
$751$	63.1475			0.0840846			0.0420423	+	0.999116i	$0.486614\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	96.8210	+	54.1324i	0.128240	+	0.0716986i
$756$		0			0
$757$			1452.21i			1.91837i	0.282781	+	0.959185i	$0.408743\pi$
$757$			1452.21i			1.91837i	−0.282781	+	0.959185i	$0.591257\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	487.824i		−	0.641030i	−0.947243	−	0.320515i	$-0.896144\pi$
$761$		−	487.824i		−	0.641030i	0.947243	−	0.320515i	$-0.103856\pi$
$762$		0			0
$763$			339.808i			0.445357i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1815.41			2.36690
$768$		0			0
$769$	677.148			0.880556			0.440278	−	0.897862i	$-0.354880\pi$
$769$	677.148			0.880556			0.440278	+	0.897862i	$0.354880\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	949.926			1.22888			0.614441	−	0.788963i	$-0.289381\pi$
$773$	949.926			1.22888			0.614441	+	0.788963i	$0.289381\pi$
$774$		0			0
$775$	−620.204	−	1008.87i	−0.800263	−	1.30177i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	307.587i		−	0.394849i
$780$		0			0
$781$	1480.52			1.89567
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−549.827	+	983.419i	−0.700417	+	1.25276i
$786$		0			0
$787$		−	1144.33i		−	1.45404i	−0.686617	−	0.727020i	$-0.740905\pi$
$787$		−	1144.33i		−	1.45404i	0.686617	−	0.727020i	$-0.259095\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			272.922i			0.345034i
$792$		0			0
$793$		−	718.089i		−	0.905535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−651.101			−0.816939			−0.408470	−	0.912772i	$-0.633937\pi$
$797$	−651.101			−0.816939			−0.408470	+	0.912772i	$0.633937\pi$
$798$		0			0
$799$	234.370			0.293330
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1925.05			−2.39733
$804$		0			0
$805$	1.85248	−	3.31335i	0.00230122	−	0.00411596i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1296.03i		−	1.60202i	−0.598653	−	0.801008i	$-0.704297\pi$
$809$		−	1296.03i		−	1.60202i	0.598653	−	0.801008i	$-0.295703\pi$
$810$		0			0
$811$	−69.9245			−0.0862201			−0.0431101	−	0.999070i	$-0.513727\pi$
$811$	−69.9245			−0.0862201			−0.0431101	+	0.999070i	$0.513727\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−739.200	+	1322.13i	−0.906993	+	1.62225i
$816$		0			0
$817$		−	1083.12i		−	1.32573i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			935.089i			1.13896i	0.822004	+	0.569481i	$0.192856\pi$
$821$			935.089i			1.13896i	−0.822004	+	0.569481i	$0.807144\pi$
$822$		0			0
$823$			1080.41i			1.31277i	0.754428	+	0.656383i	$0.227914\pi$
$823$			1080.41i			1.31277i	−0.754428	+	0.656383i	$0.772086\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1105.43			1.33668			0.668339	−	0.743856i	$-0.267005\pi$
$827$	1105.43			1.33668			0.668339	+	0.743856i	$0.267005\pi$
$828$		0			0
$829$	899.633			1.08520			0.542601	−	0.839990i	$-0.317439\pi$
$829$	899.633			1.08520			0.542601	+	0.839990i	$0.317439\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−235.926			−0.283225
$834$		0			0
$835$	−1041.59	−	582.352i	−1.24742	−	0.697428i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1140.63i			1.35951i	0.733439	+	0.679755i	$0.237914\pi$
$839$			1140.63i			1.35951i	−0.733439	+	0.679755i	$0.762086\pi$
$840$		0			0
$841$	−743.000			−0.883472
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1576.45	−	881.391i	−1.86562	−	1.04307i
$846$		0			0
$847$			582.291i			0.687475i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	9.47045i		−	0.0111286i
$852$		0			0
$853$		−	1223.43i		−	1.43427i	−0.696934	−	0.717135i	$-0.745453\pi$
$853$		−	1223.43i		−	1.43427i	0.696934	−	0.717135i	$-0.254547\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1670.08			1.94875			0.974374	−	0.224934i	$-0.0722165\pi$
$857$	1670.08			1.94875			0.974374	+	0.224934i	$0.0722165\pi$
$858$		0			0
$859$	−436.221			−0.507825			−0.253912	−	0.967227i	$-0.581717\pi$
$859$	−436.221			−0.507825			−0.253912	+	0.967227i	$0.581717\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−883.333			−1.02356			−0.511780	−	0.859116i	$-0.671014\pi$
$863$	−883.333			−1.02356			−0.511780	+	0.859116i	$0.671014\pi$
$864$		0			0
$865$	−474.835	−	265.479i	−0.548943	−	0.306912i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1153.32i		−	1.32718i
$870$		0			0
$871$	159.259			0.182846
$872$		0			0
$873$		0			0
$874$		0			0
$875$	349.137	−	14.4601i	0.399013	−	0.0165259i
$876$		0			0
$877$		−	1507.33i		−	1.71873i	−0.511359	−	0.859367i	$-0.670858\pi$
$877$		−	1507.33i		−	1.71873i	0.511359	−	0.859367i	$-0.329142\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			626.273i			0.710866i	0.934702	+	0.355433i	$0.115667\pi$
$881$			626.273i			0.710866i	−0.934702	+	0.355433i	$0.884333\pi$
$882$		0			0
$883$		−	23.3162i		−	0.0264057i	−0.999913	−	0.0132028i	$-0.995797\pi$
$883$		−	23.3162i		−	0.0264057i	0.999913	−	0.0132028i	$-0.00420271\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	529.333			0.596767			0.298384	−	0.954446i	$-0.403552\pi$
$887$	529.333			0.596767			0.298384	+	0.954446i	$0.403552\pi$
$888$		0			0
$889$	−247.815			−0.278757
$890$		0			0
$891$		0			0
$892$		0			0
$893$	948.593			1.06225
$894$		0			0
$895$	410.055	−	733.423i	0.458162	−	0.819467i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1885.32i		−	2.09713i
$900$		0			0
$901$	523.408			0.580919
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−673.383	−	376.487i	−0.744070	−	0.416008i
$906$		0			0
$907$			380.932i			0.419992i	0.977702	+	0.209996i	$0.0673450\pi$
$907$			380.932i			0.419992i	−0.977702	+	0.209996i	$0.932655\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			115.133i			0.126381i	0.998001	+	0.0631904i	$0.0201275\pi$
$911$			115.133i			0.126381i	−0.998001	+	0.0631904i	$0.979872\pi$
$912$		0			0
$913$		−	5.15655i		−	0.00564791i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	251.520			0.274285
$918$		0			0
$919$	31.5163			0.0342941			0.0171471	−	0.999853i	$-0.494542\pi$
$919$	31.5163			0.0342941			0.0171471	+	0.999853i	$0.494542\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1878.67			−2.03539
$924$		0			0
$925$	742.669	−	456.555i	0.802885	−	0.493573i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1087.09i		−	1.17018i	−0.810970	−	0.585088i	$-0.801060\pi$
$929$		−	1087.09i		−	1.17018i	0.810970	−	0.585088i	$-0.198940\pi$
$930$		0			0
$931$	−954.890			−1.02566
$932$		0			0
$933$		0			0
$934$		0			0
$935$	253.642	−	453.663i	0.271275	−	0.485201i
$936$		0			0
$937$			40.9800i			0.0437354i	0.999761	+	0.0218677i	$0.00696125\pi$
$937$			40.9800i			0.0437354i	−0.999761	+	0.0218677i	$0.993039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1345.27i			1.42961i	0.699322	+	0.714807i	$0.253485\pi$
$941$			1345.27i			1.42961i	−0.699322	+	0.714807i	$0.746515\pi$
$942$		0			0
$943$			3.60297i			0.00382075i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−364.780			−0.385196			−0.192598	−	0.981278i	$-0.561691\pi$
$947$	−364.780			−0.385196			−0.192598	+	0.981278i	$0.561691\pi$
$948$		0			0
$949$	2442.74			2.57402
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1529.14			−1.60455			−0.802277	−	0.596952i	$-0.796378\pi$
$953$	−1529.14			−1.60455			−0.802277	+	0.596952i	$0.796378\pi$
$954$		0			0
$955$	−378.038	+	676.157i	−0.395851	+	0.708018i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	642.756i		−	0.670236i
$960$		0			0
$961$	1282.96			1.33503
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−156.177	+	279.338i	−0.161842	+	0.289470i
$966$		0			0
$967$		−	698.521i		−	0.722359i	−0.932496	−	0.361179i	$-0.882374\pi$
$967$		−	698.521i		−	0.722359i	0.932496	−	0.361179i	$-0.117626\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	554.481i		−	0.571041i	−0.958373	−	0.285521i	$-0.907834\pi$
$971$		−	554.481i		−	0.571041i	0.958373	−	0.285521i	$-0.0921665\pi$
$972$		0			0
$973$			161.103i			0.165573i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1570.45			1.60742			0.803708	−	0.595024i	$-0.202857\pi$
$977$	1570.45			1.60742			0.803708	+	0.595024i	$0.202857\pi$
$978$		0			0
$979$	−517.557			−0.528659
$980$		0			0
$981$		0			0
$982$		0			0
$983$	291.656			0.296700			0.148350	−	0.988935i	$-0.452604\pi$
$983$	291.656			0.296700			0.148350	+	0.988935i	$0.452604\pi$
$984$		0			0
$985$	−41.7582	−	23.3469i	−0.0423941	−	0.0237024i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			12.6873i			0.0128284i
$990$		0			0
$991$	−1554.11			−1.56823			−0.784114	−	0.620616i	$-0.786883\pi$
$991$	−1554.11			−1.56823			−0.784114	+	0.620616i	$0.786883\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	514.168	+	287.470i	0.516752	+	0.288915i
$996$		0			0
$997$			890.663i			0.893344i	0.894698	+	0.446672i	$0.147391\pi$
$997$			890.663i			0.893344i	−0.894698	+	0.446672i	$0.852609\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.l.1889.4		4
3.2	odd	2		2160.3.c.h.1889.1		4
4.3	odd	2		540.3.b.b.269.4	yes	4
5.4	even	2		2160.3.c.h.1889.2		4
12.11	even	2		540.3.b.a.269.1	✓	4
15.14	odd	2	inner	2160.3.c.l.1889.3		4
20.3	even	4		2700.3.g.s.701.5		8
20.7	even	4		2700.3.g.s.701.3		8
20.19	odd	2		540.3.b.a.269.2	yes	4
36.7	odd	6		1620.3.t.a.1349.1		8
36.11	even	6		1620.3.t.d.1349.4		8
36.23	even	6		1620.3.t.d.269.2		8
36.31	odd	6		1620.3.t.a.269.3		8
60.23	odd	4		2700.3.g.s.701.6		8
60.47	odd	4		2700.3.g.s.701.4		8
60.59	even	2		540.3.b.b.269.3	yes	4
180.59	even	6		1620.3.t.a.269.1		8
180.79	odd	6		1620.3.t.d.1349.2		8
180.119	even	6		1620.3.t.a.1349.3		8
180.139	odd	6		1620.3.t.d.269.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.a.269.1	✓	4	12.11	even	2
540.3.b.a.269.2	yes	4	20.19	odd	2
540.3.b.b.269.3	yes	4	60.59	even	2
540.3.b.b.269.4	yes	4	4.3	odd	2
1620.3.t.a.269.1		8	180.59	even	6
1620.3.t.a.269.3		8	36.31	odd	6
1620.3.t.a.1349.1		8	36.7	odd	6
1620.3.t.a.1349.3		8	180.119	even	6
1620.3.t.d.269.2		8	36.23	even	6
1620.3.t.d.269.4		8	180.139	odd	6
1620.3.t.d.1349.2		8	180.79	odd	6
1620.3.t.d.1349.4		8	36.11	even	6
2160.3.c.h.1889.1		4	3.2	odd	2
2160.3.c.h.1889.2		4	5.4	even	2
2160.3.c.l.1889.3		4	15.14	odd	2	inner
2160.3.c.l.1889.4		4	1.1	even	1	trivial
2700.3.g.s.701.3		8	20.7	even	4
2700.3.g.s.701.4		8	60.47	odd	4
2700.3.g.s.701.5		8	20.3	even	4
2700.3.g.s.701.6		8	60.23	odd	4

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.36421 + 2.44002i) q^{5} -2.79549i q^{7} +O(q^{10})\)	\(q+(4.36421 + 2.44002i) q^{5} -2.79549i q^{7} +18.1465i q^{11} -23.0266i q^{13} -5.72842 q^{17} -23.1852 q^{19} +0.271584 q^{23} +(13.0926 + 21.2975i) q^{25} +39.7995i q^{29} -47.3705 q^{31} +(6.82104 - 12.2001i) q^{35} -34.8712i q^{37} +13.2665i q^{41} +46.7158i q^{43} -40.9137 q^{47} +41.1852 q^{49} -91.3705 q^{53} +(-44.2779 + 79.1953i) q^{55} +78.8398i q^{59} +31.1852 q^{61} +(56.1852 - 100.493i) q^{65} +6.91631i q^{67} -81.5870i q^{71} +106.084i q^{73} +50.7284 q^{77} -63.5557 q^{79} -0.284161 q^{83} +(-25.0000 - 13.9774i) q^{85} +28.5210i q^{89} -64.3705 q^{91} +(-101.185 - 56.5724i) q^{95} -92.9138i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5}+O(q^{10}) \) 4 * q + 3 * q^5	\( 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) \) 4 * q + 3 * q^5 + 6 * q^17 - 6 * q^19 + 30 * q^23 + 9 * q^25 - 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 - 192 * q^53 - 47 * q^55 + 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 + 288 * q^83 - 100 * q^85 - 84 * q^91 - 318 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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