LMFDB - Embedded newform 2700.3.b.i.1349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(1349,2700)]

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, 1]))

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

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

chi := DirichletCharacter("2700.1349");

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.b (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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1349.2
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2700.1349
Dual form			2700.3.b.i.1349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{7} +O(q^{10})$	$q-4.00000i q^{7} +6.70820i q^{11} +7.00000i q^{13} -20.1246 q^{17} -8.00000 q^{19} +6.70820 q^{23} -46.9574i q^{29} +29.0000 q^{31} +2.00000i q^{37} +13.4164i q^{41} +7.00000i q^{43} -33.5410 q^{47} +33.0000 q^{49} +93.9149 q^{53} +40.2492i q^{59} +62.0000 q^{61} -58.0000i q^{67} +53.6656i q^{71} +52.0000i q^{73} +26.8328 q^{77} +49.0000 q^{79} -13.4164 q^{83} -107.331i q^{89} +28.0000 q^{91} -34.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 32 q^{19} + 116 q^{31} + 132 q^{49} + 248 q^{61} + 196 q^{79} + 112 q^{91}+O(q^{100}) $ 4 * q - 32 * q^19 + 116 * q^31 + 132 * q^49 + 248 * q^61 + 196 * q^79 + 112 * 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
$7$	− 4.00000i	− 0.571429i	−0.958315	−	0.285714i	$-0.907769\pi$
$7$	− 4.00000i	− 0.571429i	0.958315	−	0.285714i	$-0.0922308\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.70820i	0.609837i	0.952378	+	0.304918i	$0.0986292\pi$
$11$	6.70820i	0.609837i	−0.952378	+	0.304918i	$0.901371\pi$
$12$	0	0
$13$	7.00000i	0.538462i	0.963076	+	0.269231i	$0.0867694\pi$
$13$	7.00000i	0.538462i	−0.963076	+	0.269231i	$0.913231\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−20.1246	−1.18380	−0.591900	−	0.806011i	$-0.701622\pi$
$17$	−20.1246	−1.18380	−0.591900	+	0.806011i	$0.701622\pi$
$18$	0	0
$19$	−8.00000	−0.421053	−0.210526	−	0.977588i	$-0.567518\pi$
$19$	−8.00000	−0.421053	−0.210526	+	0.977588i	$0.567518\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.70820	0.291661	0.145831	−	0.989310i	$-0.453415\pi$
$23$	6.70820	0.291661	0.145831	+	0.989310i	$0.453415\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 46.9574i	− 1.61922i	−0.586967	−	0.809611i	$-0.699678\pi$
$29$	− 46.9574i	− 1.61922i	0.586967	−	0.809611i	$-0.300322\pi$
$30$	0	0
$31$	29.0000	0.935484	0.467742	−	0.883865i	$-0.345068\pi$
$31$	29.0000	0.935484	0.467742	+	0.883865i	$0.345068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.0540541i	0.999635	+	0.0270270i	$0.00860402\pi$
$37$	2.00000i	0.0540541i	−0.999635	+	0.0270270i	$0.991396\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	13.4164i	0.327229i	0.986524	+	0.163615i	$0.0523154\pi$
$41$	13.4164i	0.327229i	−0.986524	+	0.163615i	$0.947685\pi$
$42$	0	0
$43$	7.00000i	0.162791i	0.996682	+	0.0813953i	$0.0259376\pi$
$43$	7.00000i	0.162791i	−0.996682	+	0.0813953i	$0.974062\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−33.5410	−0.713639	−0.356819	−	0.934173i	$-0.616139\pi$
$47$	−33.5410	−0.713639	−0.356819	+	0.934173i	$0.616139\pi$
$48$	0	0
$49$	33.0000	0.673469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	93.9149	1.77198	0.885989	−	0.463706i	$-0.153481\pi$
$53$	93.9149	1.77198	0.885989	+	0.463706i	$0.153481\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	40.2492i	0.682190i	0.940029	+	0.341095i	$0.110798\pi$
$59$	40.2492i	0.682190i	−0.940029	+	0.341095i	$0.889202\pi$
$60$	0	0
$61$	62.0000	1.01639	0.508197	−	0.861241i	$-0.330312\pi$
$61$	62.0000	1.01639	0.508197	+	0.861241i	$0.330312\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 58.0000i	− 0.865672i	−0.901473	−	0.432836i	$-0.857513\pi$
$67$	− 58.0000i	− 0.865672i	0.901473	−	0.432836i	$-0.142487\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	53.6656i	0.755854i	0.925835	+	0.377927i	$0.123363\pi$
$71$	53.6656i	0.755854i	−0.925835	+	0.377927i	$0.876637\pi$
$72$	0	0
$73$	52.0000i	0.712329i	0.934423	+	0.356164i	$0.115916\pi$
$73$	52.0000i	0.712329i	−0.934423	+	0.356164i	$0.884084\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	26.8328	0.348478
$78$	0	0
$79$	49.0000	0.620253	0.310127	−	0.950695i	$-0.399629\pi$
$79$	49.0000	0.620253	0.310127	+	0.950695i	$0.399629\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.4164	−0.161643	−0.0808217	−	0.996729i	$-0.525754\pi$
$83$	−13.4164	−0.161643	−0.0808217	+	0.996729i	$0.525754\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 107.331i	− 1.20597i	−0.797753	−	0.602985i	$-0.793978\pi$
$89$	− 107.331i	− 1.20597i	0.797753	−	0.602985i	$-0.206022\pi$
$90$	0	0
$91$	28.0000	0.307692
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 34.0000i	− 0.350515i	−0.984523	−	0.175258i	$-0.943924\pi$
$97$	− 34.0000i	− 0.350515i	0.984523	−	0.175258i	$-0.0560759\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	100.623i	0.996268i	0.867100	+	0.498134i	$0.165981\pi$
$101$	100.623i	0.996268i	−0.867100	+	0.498134i	$0.834019\pi$
$102$	0	0
$103$	− 98.0000i	− 0.951456i	−0.879592	−	0.475728i	$-0.842185\pi$
$103$	− 98.0000i	− 0.951456i	0.879592	−	0.475728i	$-0.157815\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−26.0000	−0.238532	−0.119266	−	0.992862i	$-0.538054\pi$
$109$	−26.0000	−0.238532	−0.119266	+	0.992862i	$0.538054\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−181.122	−1.60285	−0.801423	−	0.598099i	$-0.795923\pi$
$113$	−181.122	−1.60285	−0.801423	+	0.598099i	$0.795923\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	80.4984i	0.676458i
$120$	0	0
$121$	76.0000	0.628099
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 214.000i	− 1.68504i	−0.538666	−	0.842520i	$-0.681071\pi$
$127$	− 214.000i	− 1.68504i	0.538666	−	0.842520i	$-0.318929\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 248.204i	− 1.89468i	−0.320224	−	0.947342i	$-0.603758\pi$
$131$	− 248.204i	− 1.89468i	0.320224	−	0.947342i	$-0.396242\pi$
$132$	0	0
$133$	32.0000i	0.240602i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−40.2492	−0.293790	−0.146895	−	0.989152i	$-0.546928\pi$
$137$	−40.2492	−0.293790	−0.146895	+	0.989152i	$0.546928\pi$
$138$	0	0
$139$	64.0000	0.460432	0.230216	−	0.973140i	$-0.426057\pi$
$139$	64.0000	0.460432	0.230216	+	0.973140i	$0.426057\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−46.9574	−0.328374
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	154.289i	1.03549i	0.855534	+	0.517747i	$0.173229\pi$
$149$	154.289i	1.03549i	−0.855534	+	0.517747i	$0.826771\pi$
$150$	0	0
$151$	287.000	1.90066	0.950331	−	0.311241i	$-0.100745\pi$
$151$	287.000	1.90066	0.950331	+	0.311241i	$0.100745\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 193.000i	− 1.22930i	−0.788800	−	0.614650i	$-0.789297\pi$
$157$	− 193.000i	− 1.22930i	0.788800	−	0.614650i	$-0.210703\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 26.8328i	− 0.166663i
$162$	0	0
$163$	− 83.0000i	− 0.509202i	−0.967046	−	0.254601i	$-0.918056\pi$
$163$	− 83.0000i	− 0.509202i	0.967046	−	0.254601i	$-0.0819442\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	308.577	1.84777	0.923884	−	0.382672i	$-0.124996\pi$
$167$	308.577	1.84777	0.923884	+	0.382672i	$0.124996\pi$
$168$	0	0
$169$	120.000	0.710059
$170$	0	0
$171$	0	0
$172$	0	0
$173$	214.663	1.24082	0.620412	−	0.784276i	$-0.286965\pi$
$173$	214.663	1.24082	0.620412	+	0.784276i	$0.286965\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	93.9149i	0.524664i	0.964978	+	0.262332i	$0.0844916\pi$
$179$	93.9149i	0.524664i	−0.964978	+	0.262332i	$0.915508\pi$
$180$	0	0
$181$	164.000	0.906077	0.453039	−	0.891491i	$-0.350340\pi$
$181$	164.000	0.906077	0.453039	+	0.891491i	$0.350340\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 135.000i	− 0.721925i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 214.663i	− 1.12389i	−0.827175	−	0.561944i	$-0.810054\pi$
$191$	− 214.663i	− 1.12389i	0.827175	−	0.561944i	$-0.189946\pi$
$192$	0	0
$193$	16.0000i	0.0829016i	0.999141	+	0.0414508i	$0.0131980\pi$
$193$	16.0000i	0.0829016i	−0.999141	+	0.0414508i	$0.986802\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−187.830	−0.953450	−0.476725	−	0.879052i	$-0.658176\pi$
$197$	−187.830	−0.953450	−0.476725	+	0.879052i	$0.658176\pi$
$198$	0	0
$199$	139.000	0.698492	0.349246	−	0.937031i	$-0.386438\pi$
$199$	139.000	0.698492	0.349246	+	0.937031i	$0.386438\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−187.830	−0.925270
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 53.6656i	− 0.256773i
$210$	0	0
$211$	74.0000	0.350711	0.175355	−	0.984505i	$-0.443893\pi$
$211$	74.0000	0.350711	0.175355	+	0.984505i	$0.443893\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 116.000i	− 0.534562i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 140.872i	− 0.637431i
$222$	0	0
$223$	352.000i	1.57848i	0.614088	+	0.789238i	$0.289524\pi$
$223$	352.000i	1.57848i	−0.614088	+	0.789238i	$0.710476\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	214.663	0.945650	0.472825	−	0.881156i	$-0.343234\pi$
$227$	214.663	0.945650	0.472825	+	0.881156i	$0.343234\pi$
$228$	0	0
$229$	34.0000	0.148472	0.0742358	−	0.997241i	$-0.476348\pi$
$229$	34.0000	0.148472	0.0742358	+	0.997241i	$0.476348\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−93.9149	−0.403068	−0.201534	−	0.979482i	$-0.564593\pi$
$233$	−93.9149	−0.403068	−0.201534	+	0.979482i	$0.564593\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.8328i	0.112271i	0.998423	+	0.0561356i	$0.0178779\pi$
$239$	26.8328i	0.112271i	−0.998423	+	0.0561356i	$0.982122\pi$
$240$	0	0
$241$	419.000	1.73859	0.869295	−	0.494294i	$-0.164573\pi$
$241$	419.000	1.73859	0.869295	+	0.494294i	$0.164573\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 56.0000i	− 0.226721i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	167.705i	0.668148i	0.942547	+	0.334074i	$0.108424\pi$
$251$	167.705i	0.668148i	−0.942547	+	0.334074i	$0.891576\pi$
$252$	0	0
$253$	45.0000i	0.177866i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.1246	−0.0783059	−0.0391529	−	0.999233i	$-0.512466\pi$
$257$	−20.1246	−0.0783059	−0.0391529	+	0.999233i	$0.512466\pi$
$258$	0	0
$259$	8.00000	0.0308880
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−254.912	−0.969246	−0.484623	−	0.874723i	$-0.661043\pi$
$263$	−254.912	−0.969246	−0.484623	+	0.874723i	$0.661043\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 127.456i	− 0.473814i	−0.971532	−	0.236907i	$-0.923866\pi$
$269$	− 127.456i	− 0.473814i	0.971532	−	0.236907i	$-0.0761336\pi$
$270$	0	0
$271$	434.000	1.60148	0.800738	−	0.599015i	$-0.204441\pi$
$271$	434.000	1.60148	0.800738	+	0.599015i	$0.204441\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 118.000i	− 0.425993i	−0.977053	−	0.212996i	$-0.931678\pi$
$277$	− 118.000i	− 0.425993i	0.977053	−	0.212996i	$-0.0683223\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	281.745i	1.00265i	0.865259	+	0.501325i	$0.167154\pi$
$281$	281.745i	1.00265i	−0.865259	+	0.501325i	$0.832846\pi$
$282$	0	0
$283$	− 218.000i	− 0.770318i	−0.922850	−	0.385159i	$-0.874147\pi$
$283$	− 218.000i	− 0.770318i	0.922850	−	0.385159i	$-0.125853\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	53.6656	0.186988
$288$	0	0
$289$	116.000	0.401384
$290$	0	0
$291$	0	0
$292$	0	0
$293$	295.161	1.00738	0.503688	−	0.863886i	$-0.331976\pi$
$293$	295.161	1.00738	0.503688	+	0.863886i	$0.331976\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	46.9574i	0.157048i
$300$	0	0
$301$	28.0000	0.0930233
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 223.000i	− 0.726384i	−0.931714	−	0.363192i	$-0.881687\pi$
$307$	− 223.000i	− 0.726384i	0.931714	−	0.363192i	$-0.118313\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	442.741i	1.42361i	0.702379	+	0.711803i	$0.252121\pi$
$311$	442.741i	1.42361i	−0.702379	+	0.711803i	$0.747879\pi$
$312$	0	0
$313$	562.000i	1.79553i	0.440478	+	0.897764i	$0.354809\pi$
$313$	562.000i	1.79553i	−0.440478	+	0.897764i	$0.645191\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−335.410	−1.05808	−0.529038	−	0.848598i	$-0.677447\pi$
$317$	−335.410	−1.05808	−0.529038	+	0.848598i	$0.677447\pi$
$318$	0	0
$319$	315.000	0.987461
$320$	0	0
$321$	0	0
$322$	0	0
$323$	160.997	0.498442
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	134.164i	0.407794i
$330$	0	0
$331$	644.000	1.94562	0.972810	−	0.231606i	$-0.0743980\pi$
$331$	644.000	1.94562	0.972810	+	0.231606i	$0.0743980\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 478.000i	− 1.41840i	−0.705009	−	0.709199i	$-0.749057\pi$
$337$	− 478.000i	− 1.41840i	0.705009	−	0.709199i	$-0.250943\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	194.538i	0.570492i
$342$	0	0
$343$	− 328.000i	− 0.956268i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−496.407	−1.43057	−0.715284	−	0.698834i	$-0.753703\pi$
$347$	−496.407	−1.43057	−0.715284	+	0.698834i	$0.753703\pi$
$348$	0	0
$349$	−356.000	−1.02006	−0.510029	−	0.860157i	$-0.670365\pi$
$349$	−356.000	−1.02006	−0.510029	+	0.860157i	$0.670365\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	140.872	0.399072	0.199536	−	0.979891i	$-0.436057\pi$
$353$	140.872	0.399072	0.199536	+	0.979891i	$0.436057\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 201.246i	− 0.560574i	−0.959916	−	0.280287i	$-0.909570\pi$
$359$	− 201.246i	− 0.560574i	0.959916	−	0.280287i	$-0.0904297\pi$
$360$	0	0
$361$	−297.000	−0.822715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	92.0000i	0.250681i	0.992114	+	0.125341i	$0.0400024\pi$
$367$	92.0000i	0.250681i	−0.992114	+	0.125341i	$0.959998\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 375.659i	− 1.01256i
$372$	0	0
$373$	− 239.000i	− 0.640751i	−0.947291	−	0.320375i	$-0.896191\pi$
$373$	− 239.000i	− 0.640751i	0.947291	−	0.320375i	$-0.103809\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	328.702	0.871889
$378$	0	0
$379$	484.000	1.27704	0.638522	−	0.769603i	$-0.279546\pi$
$379$	484.000	1.27704	0.638522	+	0.769603i	$0.279546\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	449.450	1.17350	0.586749	−	0.809769i	$-0.300408\pi$
$383$	449.450	1.17350	0.586749	+	0.809769i	$0.300408\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 623.863i	− 1.60376i	−0.597485	−	0.801880i	$-0.703833\pi$
$389$	− 623.863i	− 1.60376i	0.597485	−	0.801880i	$-0.296167\pi$
$390$	0	0
$391$	−135.000	−0.345269
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 193.000i	− 0.486146i	−0.970008	−	0.243073i	$-0.921845\pi$
$397$	− 193.000i	− 0.486146i	0.970008	−	0.243073i	$-0.0781555\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.4164i	0.0334574i	0.999860	+	0.0167287i	$0.00532516\pi$
$401$	13.4164i	0.0334574i	−0.999860	+	0.0167287i	$0.994675\pi$
$402$	0	0
$403$	203.000i	0.503722i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.4164	−0.0329641
$408$	0	0
$409$	−431.000	−1.05379	−0.526895	−	0.849930i	$-0.676644\pi$
$409$	−431.000	−1.05379	−0.526895	+	0.849930i	$0.676644\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	160.997	0.389823
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 677.529i	− 1.61701i	−0.588487	−	0.808507i	$-0.700276\pi$
$419$	− 677.529i	− 1.61701i	0.588487	−	0.808507i	$-0.299724\pi$
$420$	0	0
$421$	194.000	0.460808	0.230404	−	0.973095i	$-0.425995\pi$
$421$	194.000	0.460808	0.230404	+	0.973095i	$0.425995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 248.000i	− 0.580796i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 442.741i	− 1.02724i	−0.858017	−	0.513621i	$-0.828304\pi$
$431$	− 442.741i	− 1.02724i	0.858017	−	0.513621i	$-0.171696\pi$
$432$	0	0
$433$	− 278.000i	− 0.642032i	−0.947074	−	0.321016i	$-0.895976\pi$
$433$	− 278.000i	− 0.642032i	0.947074	−	0.321016i	$-0.104024\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−53.6656	−0.122805
$438$	0	0
$439$	274.000	0.624146	0.312073	−	0.950058i	$-0.398977\pi$
$439$	274.000	0.624146	0.312073	+	0.950058i	$0.398977\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−93.9149	−0.211997	−0.105999	−	0.994366i	$-0.533804\pi$
$443$	−93.9149	−0.211997	−0.105999	+	0.994366i	$0.533804\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	643.988i	1.43427i	0.696934	+	0.717135i	$0.254547\pi$
$449$	643.988i	1.43427i	−0.696934	+	0.717135i	$0.745453\pi$
$450$	0	0
$451$	−90.0000	−0.199557
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	266.000i	0.582057i	0.956714	+	0.291028i	$0.0939974\pi$
$457$	266.000i	0.582057i	−0.956714	+	0.291028i	$0.906003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	308.577i	0.669365i	0.942331	+	0.334683i	$0.108629\pi$
$461$	308.577i	0.669365i	−0.942331	+	0.334683i	$0.891371\pi$
$462$	0	0
$463$	− 458.000i	− 0.989201i	−0.869120	−	0.494600i	$-0.835314\pi$
$463$	− 458.000i	− 0.989201i	0.869120	−	0.494600i	$-0.164686\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−496.407	−1.06297	−0.531485	−	0.847068i	$-0.678366\pi$
$467$	−496.407	−1.06297	−0.531485	+	0.847068i	$0.678366\pi$
$468$	0	0
$469$	−232.000	−0.494670
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−46.9574	−0.0992757
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	603.738i	1.26041i	0.776427	+	0.630207i	$0.217030\pi$
$479$	603.738i	1.26041i	−0.776427	+	0.630207i	$0.782970\pi$
$480$	0	0
$481$	−14.0000	−0.0291060
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000i	0.0657084i	0.999460	+	0.0328542i	$0.0104597\pi$
$487$	32.0000i	0.0657084i	−0.999460	+	0.0328542i	$0.989540\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	469.574i	0.956363i	0.878261	+	0.478182i	$0.158704\pi$
$491$	469.574i	0.956363i	−0.878261	+	0.478182i	$0.841296\pi$
$492$	0	0
$493$	945.000i	1.91684i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	214.663	0.431917
$498$	0	0
$499$	514.000	1.03006	0.515030	−	0.857172i	$-0.327781\pi$
$499$	514.000	1.03006	0.515030	+	0.857172i	$0.327781\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−288.453	−0.573465	−0.286732	−	0.958011i	$-0.592569\pi$
$503$	−288.453	−0.573465	−0.286732	+	0.958011i	$0.592569\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 476.282i	− 0.935722i	−0.883802	−	0.467861i	$-0.845025\pi$
$509$	− 476.282i	− 0.935722i	0.883802	−	0.467861i	$-0.154975\pi$
$510$	0	0
$511$	208.000	0.407045
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 225.000i	− 0.435203i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 576.906i	− 1.10730i	−0.832748	−	0.553652i	$-0.813234\pi$
$521$	− 576.906i	− 1.10730i	0.832748	−	0.553652i	$-0.186766\pi$
$522$	0	0
$523$	− 689.000i	− 1.31740i	−0.752406	−	0.658700i	$-0.771107\pi$
$523$	− 689.000i	− 1.31740i	0.752406	−	0.658700i	$-0.228893\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−583.614	−1.10743
$528$	0	0
$529$	−484.000	−0.914934
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−93.9149	−0.176200
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	221.371i	0.410706i
$540$	0	0
$541$	−628.000	−1.16081	−0.580407	−	0.814327i	$-0.697106\pi$
$541$	−628.000	−1.16081	−0.580407	+	0.814327i	$0.697106\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	617.000i	1.12797i	0.825785	+	0.563985i	$0.190733\pi$
$547$	617.000i	1.12797i	−0.825785	+	0.563985i	$0.809267\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	375.659i	0.681778i
$552$	0	0
$553$	− 196.000i	− 0.354430i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	711.070	1.27661	0.638303	−	0.769785i	$-0.279637\pi$
$557$	711.070	1.27661	0.638303	+	0.769785i	$0.279637\pi$
$558$	0	0
$559$	−49.0000	−0.0876565
$560$	0	0
$561$	0	0
$562$	0	0
$563$	187.830	0.333623	0.166811	−	0.985989i	$-0.446653\pi$
$563$	187.830	0.333623	0.166811	+	0.985989i	$0.446653\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	670.820i	1.17895i	0.807788	+	0.589473i	$0.200665\pi$
$569$	670.820i	1.17895i	−0.807788	+	0.589473i	$0.799335\pi$
$570$	0	0
$571$	−1018.00	−1.78284	−0.891419	−	0.453181i	$-0.850289\pi$
$571$	−1018.00	−1.78284	−0.891419	+	0.453181i	$0.850289\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	356.000i	0.616984i	0.951227	+	0.308492i	$0.0998244\pi$
$577$	356.000i	0.616984i	−0.951227	+	0.308492i	$0.900176\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	53.6656i	0.0923677i
$582$	0	0
$583$	630.000i	1.08062i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	818.401	1.39421	0.697105	−	0.716969i	$-0.254471\pi$
$587$	818.401	1.39421	0.697105	+	0.716969i	$0.254471\pi$
$588$	0	0
$589$	−232.000	−0.393888
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1039.77	−1.75341	−0.876705	−	0.481029i	$-0.840263\pi$
$593$	−1039.77	−1.75341	−0.876705	+	0.481029i	$0.840263\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 160.997i	− 0.268776i	−0.990929	−	0.134388i	$-0.957093\pi$
$599$	− 160.997i	− 0.268776i	0.990929	−	0.134388i	$-0.0429069\pi$
$600$	0	0
$601$	569.000	0.946755	0.473378	−	0.880860i	$-0.343035\pi$
$601$	569.000	0.946755	0.473378	+	0.880860i	$0.343035\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	602.000i	0.991763i	0.868390	+	0.495881i	$0.165155\pi$
$607$	602.000i	0.991763i	−0.868390	+	0.495881i	$0.834845\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 234.787i	− 0.384267i
$612$	0	0
$613$	277.000i	0.451876i	0.974142	+	0.225938i	$0.0725447\pi$
$613$	277.000i	0.451876i	−0.974142	+	0.225938i	$0.927455\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	315.286	0.510998	0.255499	−	0.966809i	$-0.417760\pi$
$617$	315.286	0.510998	0.255499	+	0.966809i	$0.417760\pi$
$618$	0	0
$619$	622.000	1.00485	0.502423	−	0.864622i	$-0.332442\pi$
$619$	622.000	1.00485	0.502423	+	0.864622i	$0.332442\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−429.325	−0.689125
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 40.2492i	− 0.0639892i
$630$	0	0
$631$	−106.000	−0.167987	−0.0839937	−	0.996466i	$-0.526768\pi$
$631$	−106.000	−0.167987	−0.0839937	+	0.996466i	$0.526768\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	231.000i	0.362637i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1207.48i	− 1.88374i	−0.335979	−	0.941870i	$-0.609067\pi$
$641$	− 1207.48i	− 1.88374i	0.335979	−	0.941870i	$-0.390933\pi$
$642$	0	0
$643$	− 1079.00i	− 1.67807i	−0.544076	−	0.839036i	$-0.683120\pi$
$643$	− 1079.00i	− 1.67807i	0.544076	−	0.839036i	$-0.316880\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	737.902	1.14050	0.570249	−	0.821472i	$-0.306847\pi$
$647$	737.902	1.14050	0.570249	+	0.821472i	$0.306847\pi$
$648$	0	0
$649$	−270.000	−0.416025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−737.902	−1.13002	−0.565010	−	0.825084i	$-0.691128\pi$
$653$	−737.902	−1.13002	−0.565010	+	0.825084i	$0.691128\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	308.577i	0.468251i	0.972206	+	0.234125i	$0.0752227\pi$
$659$	308.577i	0.468251i	−0.972206	+	0.234125i	$0.924777\pi$
$660$	0	0
$661$	−1126.00	−1.70348	−0.851740	−	0.523965i	$-0.824452\pi$
$661$	−1126.00	−1.70348	−0.851740	+	0.523965i	$0.824452\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 315.000i	− 0.472264i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	415.909i	0.619834i
$672$	0	0
$673$	922.000i	1.36999i	0.728550	+	0.684993i	$0.240195\pi$
$673$	922.000i	1.36999i	−0.728550	+	0.684993i	$0.759805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−617.155	−0.911602	−0.455801	−	0.890082i	$-0.650647\pi$
$677$	−617.155	−0.911602	−0.455801	+	0.890082i	$0.650647\pi$
$678$	0	0
$679$	−136.000	−0.200295
$680$	0	0
$681$	0	0
$682$	0	0
$683$	845.234	1.23753	0.618766	−	0.785576i	$-0.287633\pi$
$683$	845.234	1.23753	0.618766	+	0.785576i	$0.287633\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	657.404i	0.954142i
$690$	0	0
$691$	752.000	1.08828	0.544139	−	0.838995i	$-0.316856\pi$
$691$	752.000	1.08828	0.544139	+	0.838995i	$0.316856\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 270.000i	− 0.387374i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 181.122i	− 0.258376i	−0.991620	−	0.129188i	$-0.958763\pi$
$701$	− 181.122i	− 0.258376i	0.991620	−	0.129188i	$-0.0412371\pi$
$702$	0	0
$703$	− 16.0000i	− 0.0227596i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	402.492	0.569296
$708$	0	0
$709$	−1298.00	−1.83075	−0.915374	−	0.402605i	$-0.868105\pi$
$709$	−1298.00	−1.83075	−0.915374	+	0.402605i	$0.868105\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	194.538	0.272844
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 67.0820i	− 0.0932991i	−0.998911	−	0.0466495i	$-0.985146\pi$
$719$	− 67.0820i	− 0.0932991i	0.998911	−	0.0466495i	$-0.0148544\pi$
$720$	0	0
$721$	−392.000	−0.543689
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1202.00i	1.65337i	0.562665	+	0.826685i	$0.309776\pi$
$727$	1202.00i	1.65337i	−0.562665	+	0.826685i	$0.690224\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 140.872i	− 0.192712i
$732$	0	0
$733$	− 314.000i	− 0.428377i	−0.976792	−	0.214188i	$-0.931289\pi$
$733$	− 314.000i	− 0.428377i	0.976792	−	0.214188i	$-0.0687106\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	389.076	0.527918
$738$	0	0
$739$	−488.000	−0.660352	−0.330176	−	0.943919i	$-0.607108\pi$
$739$	−488.000	−0.660352	−0.330176	+	0.943919i	$0.607108\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1321.52	1.77862	0.889311	−	0.457303i	$-0.151184\pi$
$743$	1321.52	1.77862	0.889311	+	0.457303i	$0.151184\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−631.000	−0.840213	−0.420107	−	0.907475i	$-0.638007\pi$
$751$	−631.000	−0.840213	−0.420107	+	0.907475i	$0.638007\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	941.000i	1.24306i	0.783388	+	0.621532i	$0.213490\pi$
$757$	941.000i	1.24306i	−0.783388	+	0.621532i	$0.786510\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 979.398i	− 1.28699i	−0.765451	−	0.643494i	$-0.777484\pi$
$761$	− 979.398i	− 1.28699i	0.765451	−	0.643494i	$-0.222516\pi$
$762$	0	0
$763$	104.000i	0.136304i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−281.745	−0.367333
$768$	0	0
$769$	−281.000	−0.365410	−0.182705	−	0.983168i	$-0.558485\pi$
$769$	−281.000	−0.365410	−0.182705	+	0.983168i	$0.558485\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−214.663	−0.277701	−0.138850	−	0.990313i	$-0.544341\pi$
$773$	−214.663	−0.277701	−0.138850	+	0.990313i	$0.544341\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 107.331i	− 0.137781i
$780$	0	0
$781$	−360.000	−0.460948
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	641.000i	0.814485i	0.913320	+	0.407243i	$0.133510\pi$
$787$	641.000i	0.814485i	−0.913320	+	0.407243i	$0.866490\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	724.486i	0.915912i
$792$	0	0
$793$	434.000i	0.547289i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−402.492	−0.505009	−0.252505	−	0.967596i	$-0.581254\pi$
$797$	−402.492	−0.505009	−0.252505	+	0.967596i	$0.581254\pi$
$798$	0	0
$799$	675.000	0.844806
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−348.827	−0.434404
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	657.404i	0.812613i	0.913737	+	0.406307i	$0.133183\pi$
$809$	657.404i	0.812613i	−0.913737	+	0.406307i	$0.866817\pi$
$810$	0	0
$811$	−1336.00	−1.64735	−0.823674	−	0.567063i	$-0.808080\pi$
$811$	−1336.00	−1.64735	−0.823674	+	0.567063i	$0.808080\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 56.0000i	− 0.0685435i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	872.067i	1.06220i	0.847309	+	0.531100i	$0.178221\pi$
$821$	872.067i	1.06220i	−0.847309	+	0.531100i	$0.821779\pi$
$822$	0	0
$823$	− 848.000i	− 1.03038i	−0.857077	−	0.515188i	$-0.827722\pi$
$823$	− 848.000i	− 1.03038i	0.857077	−	0.515188i	$-0.172278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1287.98	−1.55741	−0.778703	−	0.627392i	$-0.784122\pi$
$827$	−1287.98	−1.55741	−0.778703	+	0.627392i	$0.784122\pi$
$828$	0	0
$829$	−416.000	−0.501809	−0.250905	−	0.968012i	$-0.580728\pi$
$829$	−416.000	−0.501809	−0.250905	+	0.968012i	$0.580728\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−664.112	−0.797254
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1261.14i	− 1.50315i	−0.659648	−	0.751575i	$-0.729295\pi$
$839$	− 1261.14i	− 1.50315i	0.659648	−	0.751575i	$-0.270705\pi$
$840$	0	0
$841$	−1364.00	−1.62188
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 304.000i	− 0.358914i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.4164i	0.0157655i
$852$	0	0
$853$	997.000i	1.16882i	0.811460	+	0.584408i	$0.198673\pi$
$853$	997.000i	1.16882i	−0.811460	+	0.584408i	$0.801327\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	496.407	0.579238	0.289619	−	0.957142i	$-0.406471\pi$
$857$	496.407	0.579238	0.289619	+	0.957142i	$0.406471\pi$
$858$	0	0
$859$	334.000	0.388824	0.194412	−	0.980920i	$-0.437720\pi$
$859$	334.000	0.388824	0.194412	+	0.980920i	$0.437720\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−597.030	−0.691808	−0.345904	−	0.938270i	$-0.612428\pi$
$863$	−597.030	−0.691808	−0.345904	+	0.938270i	$0.612428\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	328.702i	0.378253i
$870$	0	0
$871$	406.000	0.466131
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1331.00i	1.51767i	0.651281	+	0.758837i	$0.274232\pi$
$877$	1331.00i	1.51767i	−0.651281	+	0.758837i	$0.725768\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.2492i	0.0456858i	0.999739	+	0.0228429i	$0.00727176\pi$
$881$	40.2492i	0.0456858i	−0.999739	+	0.0228429i	$0.992728\pi$
$882$	0	0
$883$	− 1214.00i	− 1.37486i	−0.726251	−	0.687429i	$-0.758739\pi$
$883$	− 1214.00i	− 1.37486i	0.726251	−	0.687429i	$-0.241261\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−878.775	−0.990727	−0.495363	−	0.868686i	$-0.664965\pi$
$887$	−878.775	−0.990727	−0.495363	+	0.868686i	$0.664965\pi$
$888$	0	0
$889$	−856.000	−0.962880
$890$	0	0
$891$	0	0
$892$	0	0
$893$	268.328	0.300479
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1361.77i	− 1.51476i
$900$	0	0
$901$	−1890.00	−2.09767
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1247.00i	1.37486i	0.726250	+	0.687431i	$0.241262\pi$
$907$	1247.00i	1.37486i	−0.726250	+	0.687431i	$0.758738\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 254.912i	− 0.279815i	−0.990165	−	0.139908i	$-0.955319\pi$
$911$	− 254.912i	− 0.279815i	0.990165	−	0.139908i	$-0.0446806\pi$
$912$	0	0
$913$	− 90.0000i	− 0.0985761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−992.814	−1.08268
$918$	0	0
$919$	259.000	0.281828	0.140914	−	0.990022i	$-0.454996\pi$
$919$	259.000	0.281828	0.140914	+	0.990022i	$0.454996\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−375.659	−0.406998
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1489.22i	− 1.60304i	−0.597970	−	0.801518i	$-0.704026\pi$
$929$	− 1489.22i	− 1.60304i	0.597970	−	0.801518i	$-0.295974\pi$
$930$	0	0
$931$	−264.000	−0.283566
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	236.000i	0.251868i	0.992039	+	0.125934i	$0.0401927\pi$
$937$	236.000i	0.251868i	−0.992039	+	0.125934i	$0.959807\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	597.030i	0.634463i	0.948348	+	0.317232i	$0.102753\pi$
$941$	597.030i	0.634463i	−0.948348	+	0.317232i	$0.897247\pi$
$942$	0	0
$943$	90.0000i	0.0954401i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	93.9149	0.0991709	0.0495855	−	0.998770i	$-0.484210\pi$
$947$	93.9149	0.0991709	0.0495855	+	0.998770i	$0.484210\pi$
$948$	0	0
$949$	−364.000	−0.383562
$950$	0	0
$951$	0	0
$952$	0	0
$953$	986.106	1.03474	0.517369	−	0.855762i	$-0.326911\pi$
$953$	986.106	1.03474	0.517369	+	0.855762i	$0.326911\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	160.997i	0.167880i
$960$	0	0
$961$	−120.000	−0.124870
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 568.000i	− 0.587384i	−0.955900	−	0.293692i	$-0.905116\pi$
$967$	− 568.000i	− 0.587384i	0.955900	−	0.293692i	$-0.0948839\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1428.85i	1.47152i	0.677242	+	0.735761i	$0.263175\pi$
$971$	1428.85i	1.47152i	−0.677242	+	0.735761i	$0.736825\pi$
$972$	0	0
$973$	− 256.000i	− 0.263104i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1831.34	1.87445	0.937226	−	0.348722i	$-0.113384\pi$
$977$	1831.34	1.87445	0.937226	+	0.348722i	$0.113384\pi$
$978$	0	0
$979$	720.000	0.735444
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1402.01	1.42626	0.713131	−	0.701031i	$-0.247277\pi$
$983$	1402.01	1.42626	0.713131	+	0.701031i	$0.247277\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46.9574i	0.0474797i
$990$	0	0
$991$	77.0000	0.0776993	0.0388496	−	0.999245i	$-0.487631\pi$
$991$	77.0000	0.0776993	0.0388496	+	0.999245i	$0.487631\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	941.000i	0.943831i	0.881644	+	0.471916i	$0.156437\pi$
$997$	941.000i	0.943831i	−0.881644	+	0.471916i	$0.843563\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.b.i.1349.2		4
3.2	odd	2	inner	2700.3.b.i.1349.1		4
5.2	odd	4		2700.3.g.k.701.2		2
5.3	odd	4		540.3.g.b.161.1	✓	2
5.4	even	2	inner	2700.3.b.i.1349.4		4
15.2	even	4		2700.3.g.k.701.1		2
15.8	even	4		540.3.g.b.161.2	yes	2
15.14	odd	2	inner	2700.3.b.i.1349.3		4
20.3	even	4		2160.3.l.c.161.1		2
45.13	odd	12		1620.3.o.c.1241.1		4
45.23	even	12		1620.3.o.c.1241.2		4
45.38	even	12		1620.3.o.c.701.1		4
45.43	odd	12		1620.3.o.c.701.2		4
60.23	odd	4		2160.3.l.c.161.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.g.b.161.1	✓	2	5.3	odd	4
540.3.g.b.161.2	yes	2	15.8	even	4
1620.3.o.c.701.1		4	45.38	even	12
1620.3.o.c.701.2		4	45.43	odd	12
1620.3.o.c.1241.1		4	45.13	odd	12
1620.3.o.c.1241.2		4	45.23	even	12
2160.3.l.c.161.1		2	20.3	even	4
2160.3.l.c.161.2		2	60.23	odd	4
2700.3.b.i.1349.1		4	3.2	odd	2	inner
2700.3.b.i.1349.2		4	1.1	even	1	trivial
2700.3.b.i.1349.3		4	15.14	odd	2	inner
2700.3.b.i.1349.4		4	5.4	even	2	inner
2700.3.g.k.701.1		2	15.2	even	4
2700.3.g.k.701.2		2	5.2	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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{7} +O(q^{10})\)	\(q-4.00000i q^{7} +6.70820i q^{11} +7.00000i q^{13} -20.1246 q^{17} -8.00000 q^{19} +6.70820 q^{23} -46.9574i q^{29} +29.0000 q^{31} +2.00000i q^{37} +13.4164i q^{41} +7.00000i q^{43} -33.5410 q^{47} +33.0000 q^{49} +93.9149 q^{53} +40.2492i q^{59} +62.0000 q^{61} -58.0000i q^{67} +53.6656i q^{71} +52.0000i q^{73} +26.8328 q^{77} +49.0000 q^{79} -13.4164 q^{83} -107.331i q^{89} +28.0000 q^{91} -34.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 32 q^{19} + 116 q^{31} + 132 q^{49} + 248 q^{61} + 196 q^{79} + 112 q^{91}+O(q^{100}) \) 4 * q - 32 * q^19 + 116 * q^31 + 132 * q^49 + 248 * q^61 + 196 * q^79 + 112 * q^91

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