LMFDB - Embedded newform 2700.3.g.s.701.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(701,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2700.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.g (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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.488455618816.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 $ x^8 - 4x^7 + 14x^5 + 105x^4 - 238x^3 - 426x^2 + 548x + 3140
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.5
Root			$2.67945 - 2.15831i$ of defining polynomial
Character	$\chi$	$=$	2700.701
Dual form			2700.3.g.s.701.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79549 q^{7} +O(q^{10})$	$q+2.79549 q^{7} -18.1465i q^{11} +23.0266 q^{13} +5.72842i q^{17} -23.1852 q^{19} -0.271584i q^{23} -39.7995i q^{29} +47.3705 q^{31} -34.8712 q^{37} +13.2665i q^{41} +46.7158 q^{43} -40.9137i q^{47} -41.1852 q^{49} -91.3705i q^{53} +78.8398i q^{59} +31.1852 q^{61} -6.91631 q^{67} +81.5870i q^{71} -106.084 q^{73} -50.7284i q^{77} -63.5557 q^{79} +0.284161i q^{83} -28.5210i q^{89} +64.3705 q^{91} -92.9138 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100}) $ 8 * q - 12 * q^19 + 32 * q^31 - 156 * q^49 + 76 * q^61 + 12 * q^79 + 168 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.79549	0.399355	0.199678	−	0.979862i	$-0.436010\pi$
$7$	2.79549	0.399355	0.199678	+	0.979862i	$0.436010\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 18.1465i	− 1.64969i	−0.565363	−	0.824843i	$-0.691264\pi$
$11$	− 18.1465i	− 1.64969i	0.565363	−	0.824843i	$-0.308736\pi$
$12$	0	0
$13$	23.0266	1.77127	0.885637	−	0.464378i	$-0.153722\pi$
$13$	23.0266	1.77127	0.885637	+	0.464378i	$0.153722\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.72842i	0.336966i	0.985705	+	0.168483i	$0.0538868\pi$
$17$	5.72842i	0.336966i	−0.985705	+	0.168483i	$0.946113\pi$
$18$	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
$23$	− 0.271584i	− 0.0118080i	−0.999983	−	0.00590400i	$-0.998121\pi$
$23$	− 0.271584i	− 0.0118080i	0.999983	−	0.00590400i	$-0.00187931\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 39.7995i	− 1.37240i	−0.727415	−	0.686198i	$-0.759278\pi$
$29$	− 39.7995i	− 1.37240i	0.727415	−	0.686198i	$-0.240722\pi$
$30$	0	0
$31$	47.3705	1.52808	0.764040	−	0.645169i	$-0.223213\pi$
$31$	47.3705	1.52808	0.764040	+	0.645169i	$0.223213\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−34.8712	−0.942465	−0.471232	−	0.882009i	$-0.656191\pi$
$37$	−34.8712	−0.942465	−0.471232	+	0.882009i	$0.656191\pi$
$38$	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
$43$	46.7158	1.08641	0.543207	−	0.839599i	$-0.317210\pi$
$43$	46.7158	1.08641	0.543207	+	0.839599i	$0.317210\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 40.9137i	− 0.870504i	−0.900309	−	0.435252i	$-0.856659\pi$
$47$	− 40.9137i	− 0.870504i	0.900309	−	0.435252i	$-0.143341\pi$
$48$	0	0
$49$	−41.1852	−0.840515
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 91.3705i	− 1.72397i	−0.506932	−	0.861986i	$-0.669221\pi$
$53$	− 91.3705i	− 1.72397i	0.506932	−	0.861986i	$-0.330779\pi$
$54$	0	0
$55$	0	0
$56$	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
$65$	0	0
$66$	0	0
$67$	−6.91631	−0.103229	−0.0516143	−	0.998667i	$-0.516437\pi$
$67$	−6.91631	−0.103229	−0.0516143	+	0.998667i	$0.516437\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	81.5870i	1.14911i	0.818465	+	0.574556i	$0.194825\pi$
$71$	81.5870i	1.14911i	−0.818465	+	0.574556i	$0.805175\pi$
$72$	0	0
$73$	−106.084	−1.45320	−0.726601	−	0.687060i	$-0.758901\pi$
$73$	−106.084	−1.45320	−0.726601	+	0.687060i	$0.758901\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 50.7284i	− 0.658811i
$78$	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
$83$	0.284161i	0.00342363i	0.999999	+	0.00171182i	$0.000544888\pi$
$83$	0.284161i	0.00342363i	−0.999999	+	0.00171182i	$0.999455\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 28.5210i	− 0.320461i	−0.987080	−	0.160230i	$-0.948776\pi$
$89$	− 28.5210i	− 0.320461i	0.987080	−	0.160230i	$-0.0512237\pi$
$90$	0	0
$91$	64.3705	0.707368
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−92.9138	−0.957874	−0.478937	−	0.877849i	$-0.658978\pi$
$97$	−92.9138	−0.957874	−0.478937	+	0.877849i	$0.658978\pi$
$98$	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
$103$	−11.1820	−0.108563	−0.0542813	−	0.998526i	$-0.517287\pi$
$103$	−11.1820	−0.108563	−0.0542813	+	0.998526i	$0.517287\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	147.297i	1.37661i	0.725424	+	0.688303i	$0.241644\pi$
$107$	147.297i	1.37661i	−0.725424	+	0.688303i	$0.758356\pi$
$108$	0	0
$109$	121.556	1.11519	0.557595	−	0.830113i	$-0.311724\pi$
$109$	121.556	1.11519	0.557595	+	0.830113i	$0.311724\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 97.6295i	− 0.863978i	−0.901879	−	0.431989i	$-0.857812\pi$
$113$	− 97.6295i	− 0.863978i	0.901879	−	0.431989i	$-0.142188\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	88.6481	0.698017	0.349008	−	0.937120i	$-0.386518\pi$
$127$	88.6481	0.698017	0.349008	+	0.937120i	$0.386518\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 89.9735i	− 0.686820i	−0.939186	−	0.343410i	$-0.888418\pi$
$131$	− 89.9735i	− 0.686820i	0.939186	−	0.343410i	$-0.111582\pi$
$132$	0	0
$133$	−64.8141	−0.487324
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 229.926i	− 1.67829i	−0.543905	−	0.839147i	$-0.683055\pi$
$137$	− 229.926i	− 1.67829i	0.543905	−	0.839147i	$-0.316945\pi$
$138$	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
$143$	− 417.852i	− 2.92205i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.1337i	− 0.0747227i	−0.999302	−	0.0373613i	$-0.988105\pi$
$149$	− 11.1337i	− 0.0747227i	0.999302	−	0.0373613i	$-0.0118953\pi$
$150$	0	0
$151$	−22.1852	−0.146922	−0.0734611	−	0.997298i	$-0.523404\pi$
$151$	−22.1852	−0.146922	−0.0734611	+	0.997298i	$0.523404\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	225.337	1.43527	0.717635	−	0.696419i	$-0.245225\pi$
$157$	225.337	1.43527	0.717635	+	0.696419i	$0.245225\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.759209i	− 0.00471559i
$162$	0	0
$163$	302.948	1.85858	0.929290	−	0.369352i	$-0.120420\pi$
$163$	302.948	1.85858	0.929290	+	0.369352i	$0.120420\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 238.667i	− 1.42915i	−0.699561	−	0.714573i	$-0.746621\pi$
$167$	− 238.667i	− 1.42915i	0.699561	−	0.714573i	$-0.253379\pi$
$168$	0	0
$169$	361.223	2.13741
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 108.802i	− 0.628914i	−0.949272	−	0.314457i	$-0.898178\pi$
$173$	− 108.802i	− 0.628914i	0.949272	−	0.314457i	$-0.101822\pi$
$174$	0	0
$175$	0	0
$176$	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
$185$	0	0
$186$	0	0
$187$	103.951	0.555887
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 154.932i	− 0.811164i	−0.914059	−	0.405582i	$-0.867069\pi$
$191$	− 154.932i	− 0.811164i	0.914059	−	0.405582i	$-0.132931\pi$
$192$	0	0
$193$	−64.0066	−0.331640	−0.165820	−	0.986156i	$-0.553027\pi$
$193$	−64.0066	−0.331640	−0.165820	+	0.986156i	$0.553027\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.56832i	0.0485702i	0.999705	+	0.0242851i	$0.00773094\pi$
$197$	9.56832i	0.0485702i	−0.999705	+	0.0242851i	$0.992269\pi$
$198$	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
$203$	− 111.259i	− 0.548074i
$204$	0	0
$205$	0	0
$206$	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	$-0.536903\pi$
$211$	−48.8148	−0.231350	−0.115675	+	0.993287i	$0.536903\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	132.424	0.610247
$218$	0	0
$219$	0	0
$220$	0	0
$221$	131.906i	0.596859i
$222$	0	0
$223$	−319.721	−1.43373	−0.716864	−	0.697213i	$-0.754423\pi$
$223$	−319.721	−1.43373	−0.716864	+	0.697213i	$0.754423\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.2716i	− 0.0804916i	−0.999190	−	0.0402458i	$-0.987186\pi$
$227$	− 18.2716i	− 0.0804916i	0.999190	−	0.0402458i	$-0.0128141\pi$
$228$	0	0
$229$	−134.074	−0.585475	−0.292737	−	0.956193i	$-0.594566\pi$
$229$	−134.074	−0.585475	−0.292737	+	0.956193i	$0.594566\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 284.173i	− 1.21963i	−0.792546	−	0.609813i	$-0.791245\pi$
$233$	− 284.173i	− 1.21963i	0.792546	−	0.609813i	$-0.208755\pi$
$234$	0	0
$235$	0	0
$236$	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
$245$	0	0
$246$	0	0
$247$	−533.877	−2.16144
$248$	0	0
$249$	0	0
$250$	0	0
$251$	347.387i	1.38401i	0.721893	+	0.692005i	$0.243272\pi$
$251$	347.387i	1.38401i	−0.721893	+	0.692005i	$0.756728\pi$
$252$	0	0
$253$	−4.92831	−0.0194795
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.0377i	0.0818589i	0.999162	+	0.0409294i	$0.0130319\pi$
$257$	21.0377i	0.0818589i	−0.999162	+	0.0409294i	$0.986968\pi$
$258$	0	0
$259$	−97.4820	−0.376378
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 215.457i	− 0.819227i	−0.912259	−	0.409614i	$-0.865663\pi$
$263$	− 215.457i	− 0.819227i	0.912259	−	0.409614i	$-0.134337\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 180.237i	− 0.670024i	−0.942214	−	0.335012i	$-0.891260\pi$
$269$	− 180.237i	− 0.670024i	0.942214	−	0.335012i	$-0.108740\pi$
$270$	0	0
$271$	231.741	0.855133	0.427566	−	0.903984i	$-0.359371\pi$
$271$	231.741	0.855133	0.427566	+	0.903984i	$0.359371\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	105.566	0.381104	0.190552	−	0.981677i	$-0.438972\pi$
$277$	105.566	0.381104	0.190552	+	0.981677i	$0.438972\pi$
$278$	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
$283$	−151.619	−0.535756	−0.267878	−	0.963453i	$-0.586322\pi$
$283$	−151.619	−0.535756	−0.267878	+	0.963453i	$0.586322\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	37.0863i	0.129221i
$288$	0	0
$289$	256.185	0.886454
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 549.038i	− 1.87385i	−0.349532	−	0.936924i	$-0.613659\pi$
$293$	− 549.038i	− 1.87385i	0.349532	−	0.936924i	$-0.386341\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	−146.401	−0.476876	−0.238438	−	0.971158i	$-0.576635\pi$
$307$	−146.401	−0.476876	−0.238438	+	0.971158i	$0.576635\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 487.824i	− 1.56856i	−0.620404	−	0.784282i	$-0.713031\pi$
$311$	− 487.824i	− 1.56856i	0.620404	−	0.784282i	$-0.286969\pi$
$312$	0	0
$313$	109.687	0.350437	0.175218	−	0.984530i	$-0.443937\pi$
$313$	109.687	0.350437	0.175218	+	0.984530i	$0.443937\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 363.568i	− 1.14690i	−0.819239	−	0.573452i	$-0.805604\pi$
$317$	− 363.568i	− 1.14690i	0.819239	−	0.573452i	$-0.194396\pi$
$318$	0	0
$319$	−722.223	−2.26402
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 132.815i	− 0.411191i
$324$	0	0
$325$	0	0
$326$	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	$-0.350172\pi$
$331$	300.223	0.907018	0.453509	+	0.891252i	$0.350172\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	373.353	1.10787	0.553937	−	0.832559i	$-0.313125\pi$
$337$	373.353	1.10787	0.553937	+	0.832559i	$0.313125\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 859.610i	− 2.52085i
$342$	0	0
$343$	−252.112	−0.735020
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.4443i	0.0502716i	0.999684	+	0.0251358i	$0.00800182\pi$
$347$	17.4443i	0.0502716i	−0.999684	+	0.0251358i	$0.991998\pi$
$348$	0	0
$349$	−234.149	−0.670915	−0.335457	−	0.942055i	$-0.608891\pi$
$349$	−234.149	−0.670915	−0.335457	+	0.942055i	$0.608891\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 381.284i	− 1.08013i	−0.841625	−	0.540063i	$-0.818401\pi$
$353$	− 381.284i	− 1.08013i	0.841625	−	0.540063i	$-0.181599\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	−449.205	−1.22399	−0.611995	−	0.790861i	$-0.709633\pi$
$367$	−449.205	−1.22399	−0.611995	+	0.790861i	$0.709633\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 255.425i	− 0.688477i
$372$	0	0
$373$	322.082	0.863492	0.431746	−	0.901995i	$-0.357898\pi$
$373$	322.082	0.863492	0.431746	+	0.901995i	$0.357898\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 916.446i	− 2.43089i
$378$	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
$383$	373.977i	0.976440i	0.872721	+	0.488220i	$0.162354\pi$
$383$	373.977i	0.976440i	−0.872721	+	0.488220i	$0.837646\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 223.108i	− 0.573543i	−0.957999	−	0.286771i	$-0.907418\pi$
$389$	− 223.108i	− 0.573543i	0.957999	−	0.286771i	$-0.0925820\pi$
$390$	0	0
$391$	1.55575	0.00397889
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−610.535	−1.53787	−0.768936	−	0.639325i	$-0.779214\pi$
$397$	−610.535	−1.53787	−0.768936	+	0.639325i	$0.779214\pi$
$398$	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
$403$	1090.78	2.70665
$404$	0	0
$405$	0	0
$406$	0	0
$407$	632.791i	1.55477i
$408$	0	0
$409$	−590.630	−1.44408	−0.722041	−	0.691850i	$-0.756796\pi$
$409$	−590.630	−1.44408	−0.722041	+	0.691850i	$0.756796\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	220.396i	0.533646i
$414$	0	0
$415$	0	0
$416$	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
$425$	0	0
$426$	0	0
$427$	87.1780	0.204164
$428$	0	0
$429$	0	0
$430$	0	0
$431$	436.566i	1.01291i	0.862265	+	0.506457i	$0.169045\pi$
$431$	436.566i	1.01291i	−0.862265	+	0.506457i	$0.830955\pi$
$432$	0	0
$433$	−231.736	−0.535187	−0.267593	−	0.963532i	$-0.586228\pi$
$433$	−231.736	−0.535187	−0.267593	+	0.963532i	$0.586228\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.29674i	0.0144090i
$438$	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
$443$	− 446.520i	− 1.00795i	−0.863720	−	0.503973i	$-0.831871\pi$
$443$	− 446.520i	− 1.00795i	0.863720	−	0.503973i	$-0.168129\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	495.776i	1.10418i	0.833785	+	0.552089i	$0.186169\pi$
$449$	495.776i	1.10418i	−0.833785	+	0.552089i	$0.813831\pi$
$450$	0	0
$451$	240.741	0.533794
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	683.363	1.49532	0.747662	−	0.664080i	$-0.231176\pi$
$457$	683.363	1.49532	0.747662	+	0.664080i	$0.231176\pi$
$458$	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
$463$	316.263	0.683074	0.341537	−	0.939868i	$-0.389053\pi$
$463$	316.263	0.683074	0.341537	+	0.939868i	$0.389053\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	419.741i	0.898803i	0.893330	+	0.449401i	$0.148363\pi$
$467$	419.741i	0.898803i	−0.893330	+	0.449401i	$0.851637\pi$
$468$	0	0
$469$	−19.3345	−0.0412249
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 847.730i	− 1.79224i
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	546.384	1.12194	0.560969	−	0.827837i	$-0.310429\pi$
$487$	546.384	1.12194	0.560969	+	0.827837i	$0.310429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	647.456i	1.31865i	0.751859	+	0.659324i	$0.229157\pi$
$491$	647.456i	1.31865i	−0.751859	+	0.659324i	$0.770843\pi$
$492$	0	0
$493$	227.988	0.462450
$494$	0	0
$495$	0	0
$496$	0	0
$497$	228.075i	0.458904i
$498$	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
$503$	752.520i	1.49606i	0.663663	+	0.748032i	$0.269001\pi$
$503$	752.520i	1.49606i	−0.663663	+	0.748032i	$0.730999\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	291.538i	0.572767i	0.958115	+	0.286383i	$0.0924531\pi$
$509$	291.538i	0.572767i	−0.958115	+	0.286383i	$0.907547\pi$
$510$	0	0
$511$	−296.556	−0.580344
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−742.441	−1.43606
$518$	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
$523$	−330.987	−0.632862	−0.316431	−	0.948616i	$-0.602485\pi$
$523$	−330.987	−0.632862	−0.316431	+	0.948616i	$0.602485\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	271.358i	0.514911i
$528$	0	0
$529$	528.926	0.999861
$530$	0	0
$531$	0	0
$532$	0	0
$533$	305.482i	0.573137i
$534$	0	0
$535$	0	0
$536$	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
$545$	0	0
$546$	0	0
$547$	465.170	0.850402	0.425201	−	0.905099i	$-0.360203\pi$
$547$	465.170	0.850402	0.425201	+	0.905099i	$0.360203\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	922.761i	1.67470i
$552$	0	0
$553$	−177.669	−0.321283
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1059.37i	− 1.90192i	−0.309306	−	0.950962i	$-0.600097\pi$
$557$	− 1059.37i	− 1.90192i	0.309306	−	0.950962i	$-0.399903\pi$
$558$	0	0
$559$	1075.70	1.92434
$560$	0	0
$561$	0	0
$562$	0	0
$563$	386.160i	0.685897i	0.939354	+	0.342949i	$0.111426\pi$
$563$	386.160i	0.685897i	−0.939354	+	0.342949i	$0.888574\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 686.062i	− 1.20573i	−0.797842	−	0.602866i	$-0.794025\pi$
$569$	− 686.062i	− 1.20573i	0.797842	−	0.602866i	$-0.205975\pi$
$570$	0	0
$571$	−821.631	−1.43893	−0.719467	−	0.694527i	$-0.755614\pi$
$571$	−821.631	−1.43893	−0.719467	+	0.694527i	$0.755614\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−183.840	−0.318613	−0.159306	−	0.987229i	$-0.550926\pi$
$577$	−183.840	−0.318613	−0.159306	+	0.987229i	$0.550926\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.794370i	0.00136725i
$582$	0	0
$583$	−1658.06	−2.84401
$584$	0	0
$585$	0	0
$586$	0	0
$587$	320.407i	0.545837i	0.962037	+	0.272919i	$0.0879890\pi$
$587$	320.407i	0.545837i	−0.962037	+	0.272919i	$0.912011\pi$
$588$	0	0
$589$	−1098.30	−1.86468
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1061.34i	1.78977i	0.446292	+	0.894887i	$0.352744\pi$
$593$	1061.34i	1.78977i	−0.446292	+	0.894887i	$0.647256\pi$
$594$	0	0
$595$	0	0
$596$	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
$605$	0	0
$606$	0	0
$607$	668.806	1.10182	0.550911	−	0.834564i	$-0.314280\pi$
$607$	668.806	1.10182	0.550911	+	0.834564i	$0.314280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 942.101i	− 1.54190i
$612$	0	0
$613$	220.119	0.359086	0.179543	−	0.983750i	$-0.442538\pi$
$613$	220.119	0.359086	0.179543	+	0.983750i	$0.442538\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	321.063i	0.520361i	0.965560	+	0.260181i	$0.0837821\pi$
$617$	321.063i	0.520361i	−0.965560	+	0.260181i	$0.916218\pi$
$618$	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
$623$	− 79.7301i	− 0.127978i
$624$	0	0
$625$	0	0
$626$	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	$-0.653046\pi$
$631$	−583.669	−0.924990	−0.462495	+	0.886622i	$0.653046\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−948.355	−1.48878
$638$	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
$643$	−801.664	−1.24676	−0.623378	−	0.781920i	$-0.714240\pi$
$643$	−801.664	−1.24676	−0.623378	+	0.781920i	$0.714240\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	216.221i	0.334191i	0.985941	+	0.167095i	$0.0534387\pi$
$647$	216.221i	0.334191i	−0.985941	+	0.167095i	$0.946561\pi$
$648$	0	0
$649$	1430.67	2.20442
$650$	0	0
$651$	0	0
$652$	0	0
$653$	438.802i	0.671979i	0.941866	+	0.335989i	$0.109071\pi$
$653$	438.802i	0.671979i	−0.941866	+	0.335989i	$0.890929\pi$
$654$	0	0
$655$	0	0
$656$	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
$665$	0	0
$666$	0	0
$667$	−10.8089	−0.0162052
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 565.904i	− 0.843374i
$672$	0	0
$673$	−648.575	−0.963708	−0.481854	−	0.876252i	$-0.660036\pi$
$673$	−648.575	−0.963708	−0.481854	+	0.876252i	$0.660036\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 202.345i	− 0.298885i	−0.988770	−	0.149443i	$-0.952252\pi$
$677$	− 202.345i	− 0.298885i	0.988770	−	0.149443i	$-0.0477479\pi$
$678$	0	0
$679$	−259.739	−0.382532
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 173.829i	− 0.254508i	−0.991870	−	0.127254i	$-0.959384\pi$
$683$	− 173.829i	− 0.254508i	0.991870	−	0.127254i	$-0.0406164\pi$
$684$	0	0
$685$	0	0
$686$	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	$-0.709914\pi$
$691$	−846.743	−1.22539	−0.612694	+	0.790320i	$0.709914\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−75.9960	−0.109033
$698$	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
$703$	808.497	1.15007
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 130.459i	− 0.184524i
$708$	0	0
$709$	−287.557	−0.405582	−0.202791	−	0.979222i	$-0.565001\pi$
$709$	−287.557	−0.405582	−0.202791	+	0.979222i	$0.565001\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.8651i	− 0.0180436i
$714$	0	0
$715$	0	0
$716$	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
$725$	0	0
$726$	0	0
$727$	311.418	0.428361	0.214180	−	0.976794i	$-0.431292\pi$
$727$	311.418	0.428361	0.214180	+	0.976794i	$0.431292\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	267.608i	0.366084i
$732$	0	0
$733$	−321.047	−0.437990	−0.218995	−	0.975726i	$-0.570278\pi$
$733$	−321.047	−0.437990	−0.218995	+	0.975726i	$0.570278\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	125.507i	0.170295i
$738$	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
$743$	1024.96i	1.37949i	0.724051	+	0.689747i	$0.242278\pi$
$743$	1024.96i	1.37949i	−0.724051	+	0.689747i	$0.757722\pi$
$744$	0	0
$745$	0	0
$746$	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	$-0.513386\pi$
$751$	−63.1475	−0.0840846	−0.0420423	+	0.999116i	$0.513386\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1452.21	1.91837	0.959185	−	0.282781i	$-0.0912568\pi$
$757$	1452.21	1.91837	0.959185	+	0.282781i	$0.0912568\pi$
$758$	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
$763$	339.808	0.445357
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1815.41i	2.36690i
$768$	0	0
$769$	−677.148	−0.880556	−0.440278	−	0.897862i	$-0.645120\pi$
$769$	−677.148	−0.880556	−0.440278	+	0.897862i	$0.645120\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	949.926i	1.22888i	0.788963	+	0.614441i	$0.210619\pi$
$773$	949.926i	1.22888i	−0.788963	+	0.614441i	$0.789381\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	1144.33	1.45404	0.727020	−	0.686617i	$-0.240905\pi$
$787$	1144.33	1.45404	0.727020	+	0.686617i	$0.240905\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 272.922i	− 0.345034i
$792$	0	0
$793$	718.089	0.905535
$794$	0	0
$795$	0	0
$796$	0	0
$797$	651.101i	0.816939i	0.912772	+	0.408470i	$0.133937\pi$
$797$	651.101i	0.816939i	−0.912772	+	0.408470i	$0.866063\pi$
$798$	0	0
$799$	234.370	0.293330
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1925.05i	2.39733i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1296.03i	1.60202i	0.598653	+	0.801008i	$0.295703\pi$
$809$	1296.03i	1.60202i	−0.598653	+	0.801008i	$0.704297\pi$
$810$	0	0
$811$	69.9245	0.0862201	0.0431101	−	0.999070i	$-0.486273\pi$
$811$	69.9245	0.0862201	0.0431101	+	0.999070i	$0.486273\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1083.12	−1.32573
$818$	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
$823$	1080.41	1.31277	0.656383	−	0.754428i	$-0.272086\pi$
$823$	1080.41	1.31277	0.656383	+	0.754428i	$0.272086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1105.43i	1.33668i	0.743856	+	0.668339i	$0.232995\pi$
$827$	1105.43i	1.33668i	−0.743856	+	0.668339i	$0.767005\pi$
$828$	0	0
$829$	−899.633	−1.08520	−0.542601	−	0.839990i	$-0.682561\pi$
$829$	−899.633	−1.08520	−0.542601	+	0.839990i	$0.682561\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 235.926i	− 0.283225i
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	−582.291	−0.687475
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.47045i	0.0111286i
$852$	0	0
$853$	1223.43	1.43427	0.717135	−	0.696934i	$-0.245453\pi$
$853$	1223.43	1.43427	0.717135	+	0.696934i	$0.245453\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1670.08i	− 1.94875i	−0.224934	−	0.974374i	$-0.572217\pi$
$857$	− 1670.08i	− 1.94875i	0.224934	−	0.974374i	$-0.427783\pi$
$858$	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
$863$	883.333i	1.02356i	0.859116	+	0.511780i	$0.171014\pi$
$863$	883.333i	1.02356i	−0.859116	+	0.511780i	$0.828986\pi$
$864$	0	0
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	−1507.33	−1.71873	−0.859367	−	0.511359i	$-0.829142\pi$
$877$	−1507.33	−1.71873	−0.859367	+	0.511359i	$0.829142\pi$
$878$	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
$883$	−23.3162	−0.0264057	−0.0132028	−	0.999913i	$-0.504203\pi$
$883$	−23.3162	−0.0264057	−0.0132028	+	0.999913i	$0.504203\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	529.333i	0.596767i	0.954446	+	0.298384i	$0.0964475\pi$
$887$	529.333i	0.596767i	−0.954446	+	0.298384i	$0.903552\pi$
$888$	0	0
$889$	247.815	0.278757
$890$	0	0
$891$	0	0
$892$	0	0
$893$	948.593i	1.06225i
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	−380.932	−0.419992	−0.209996	−	0.977702i	$-0.567345\pi$
$907$	−380.932	−0.419992	−0.209996	+	0.977702i	$0.567345\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 115.133i	− 0.126381i	−0.998001	−	0.0631904i	$-0.979872\pi$
$911$	− 115.133i	− 0.126381i	0.998001	−	0.0631904i	$-0.0201275\pi$
$912$	0	0
$913$	5.15655	0.00564791
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 251.520i	− 0.274285i
$918$	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
$923$	1878.67i	2.03539i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1087.09i	1.17018i	0.810970	+	0.585088i	$0.198940\pi$
$929$	1087.09i	1.17018i	−0.810970	+	0.585088i	$0.801060\pi$
$930$	0	0
$931$	954.890	1.02566
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.9800	0.0437354	0.0218677	−	0.999761i	$-0.493039\pi$
$937$	40.9800	0.0437354	0.0218677	+	0.999761i	$0.493039\pi$
$938$	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
$943$	3.60297	0.00382075
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 364.780i	− 0.385196i	−0.981278	−	0.192598i	$-0.938309\pi$
$947$	− 364.780i	− 0.385196i	0.981278	−	0.192598i	$-0.0616913\pi$
$948$	0	0
$949$	−2442.74	−2.57402
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1529.14i	− 1.60455i	−0.596952	−	0.802277i	$-0.703622\pi$
$953$	− 1529.14i	− 1.60455i	0.596952	−	0.802277i	$-0.296378\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	698.521	0.722359	0.361179	−	0.932496i	$-0.382374\pi$
$967$	698.521	0.722359	0.361179	+	0.932496i	$0.382374\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	554.481i	0.571041i	0.958373	+	0.285521i	$0.0921665\pi$
$971$	554.481i	0.571041i	−0.958373	+	0.285521i	$0.907834\pi$
$972$	0	0
$973$	−161.103	−0.165573
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1570.45i	− 1.60742i	−0.595024	−	0.803708i	$-0.702857\pi$
$977$	− 1570.45i	− 1.60742i	0.595024	−	0.803708i	$-0.297143\pi$
$978$	0	0
$979$	−517.557	−0.528659
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 291.656i	− 0.296700i	−0.988935	−	0.148350i	$-0.952604\pi$
$983$	− 291.656i	− 0.296700i	0.988935	−	0.148350i	$-0.0473963\pi$
$984$	0	0
$985$	0	0
$986$	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	$-0.213117\pi$
$991$	1554.11	1.56823	0.784114	+	0.620616i	$0.213117\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	890.663	0.893344	0.446672	−	0.894698i	$-0.352609\pi$
$997$	890.663	0.893344	0.446672	+	0.894698i	$0.352609\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.g.s.701.5		8
3.2	odd	2	inner	2700.3.g.s.701.6		8
5.2	odd	4		540.3.b.b.269.4	yes	4
5.3	odd	4		540.3.b.a.269.2	yes	4
5.4	even	2	inner	2700.3.g.s.701.3		8
15.2	even	4		540.3.b.a.269.1	✓	4
15.8	even	4		540.3.b.b.269.3	yes	4
15.14	odd	2	inner	2700.3.g.s.701.4		8
20.3	even	4		2160.3.c.h.1889.2		4
20.7	even	4		2160.3.c.l.1889.4		4
45.2	even	12		1620.3.t.d.1349.4		8
45.7	odd	12		1620.3.t.a.1349.1		8
45.13	odd	12		1620.3.t.d.269.4		8
45.22	odd	12		1620.3.t.a.269.3		8
45.23	even	12		1620.3.t.a.269.1		8
45.32	even	12		1620.3.t.d.269.2		8
45.38	even	12		1620.3.t.a.1349.3		8
45.43	odd	12		1620.3.t.d.1349.2		8
60.23	odd	4		2160.3.c.l.1889.3		4
60.47	odd	4		2160.3.c.h.1889.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.a.269.1	✓	4	15.2	even	4
540.3.b.a.269.2	yes	4	5.3	odd	4
540.3.b.b.269.3	yes	4	15.8	even	4
540.3.b.b.269.4	yes	4	5.2	odd	4
1620.3.t.a.269.1		8	45.23	even	12
1620.3.t.a.269.3		8	45.22	odd	12
1620.3.t.a.1349.1		8	45.7	odd	12
1620.3.t.a.1349.3		8	45.38	even	12
1620.3.t.d.269.2		8	45.32	even	12
1620.3.t.d.269.4		8	45.13	odd	12
1620.3.t.d.1349.2		8	45.43	odd	12
1620.3.t.d.1349.4		8	45.2	even	12
2160.3.c.h.1889.1		4	60.47	odd	4
2160.3.c.h.1889.2		4	20.3	even	4
2160.3.c.l.1889.3		4	60.23	odd	4
2160.3.c.l.1889.4		4	20.7	even	4
2700.3.g.s.701.3		8	5.4	even	2	inner
2700.3.g.s.701.4		8	15.14	odd	2	inner
2700.3.g.s.701.5		8	1.1	even	1	trivial
2700.3.g.s.701.6		8	3.2	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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.79549 q^{7} +O(q^{10})\)	\(q+2.79549 q^{7} -18.1465i q^{11} +23.0266 q^{13} +5.72842i q^{17} -23.1852 q^{19} -0.271584i q^{23} -39.7995i q^{29} +47.3705 q^{31} -34.8712 q^{37} +13.2665i q^{41} +46.7158 q^{43} -40.9137i q^{47} -41.1852 q^{49} -91.3705i q^{53} +78.8398i q^{59} +31.1852 q^{61} -6.91631 q^{67} +81.5870i q^{71} -106.084 q^{73} -50.7284i q^{77} -63.5557 q^{79} +0.284161i q^{83} -28.5210i q^{89} +64.3705 q^{91} -92.9138 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100}) \) 8 * q - 12 * q^19 + 32 * q^31 - 156 * q^49 + 76 * q^61 + 12 * q^79 + 168 * q^91

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