LMFDB - Embedded newform 1350.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1350 = 2 \cdot 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1350.a (trivial)

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:	yes
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{11} +5.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +3.00000 q^{22} -3.00000 q^{23} -5.00000 q^{26} +2.00000 q^{28} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +11.0000 q^{37} +4.00000 q^{38} +6.00000 q^{41} -4.00000 q^{43} -3.00000 q^{44} +3.00000 q^{46} +3.00000 q^{47} -3.00000 q^{49} +5.00000 q^{52} -12.0000 q^{53} -2.00000 q^{56} +9.00000 q^{59} +11.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +14.0000 q^{67} +6.00000 q^{68} -15.0000 q^{71} +2.00000 q^{73} -11.0000 q^{74} -4.00000 q^{76} -6.00000 q^{77} -10.0000 q^{79} -6.00000 q^{82} +4.00000 q^{86} +3.00000 q^{88} +6.00000 q^{89} +10.0000 q^{91} -3.00000 q^{92} -3.00000 q^{94} -7.00000 q^{97} +3.00000 q^{98} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	0	0
$26$	−5.00000	−0.980581
$27$	0	0
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	5.00000	0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−11.0000	−1.27872
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	12.0000	1.16554
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	2.00000	0.188982
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−9.00000	−0.828517
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−11.0000	−0.995893
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−14.0000	−1.20942
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	15.0000	1.25877
$143$	−15.0000	−1.25436
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	11.0000	0.904194
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−10.0000	−0.741249
$183$	0	0
$184$	3.00000	0.221163
$185$	0	0
$186$	0	0
$187$	−18.0000	−1.31629
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	7.00000	0.502571
$195$	0	0
$196$	−3.00000	−0.214286
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	−18.0000	−1.26648
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	5.00000	0.346688
$209$	12.0000	0.830057
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−15.0000	−1.02538
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	10.0000	0.677285
$219$	0	0
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	9.00000	0.585850
$237$	0	0
$238$	−12.0000	−0.777844
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	11.0000	0.704203
$245$	0	0
$246$	0	0
$247$	−20.0000	−1.27257
$248$	−2.00000	−0.127000
$249$	0	0
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	22.0000	1.36701
$260$	0	0
$261$	0	0
$262$	−21.0000	−1.29738
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	14.0000	0.855186
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	15.0000	0.886969
$287$	12.0000	0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−11.0000	−0.639362
$297$	0	0
$298$	−18.0000	−1.04271
$299$	−15.0000	−0.867472
$300$	0	0
$301$	−8.00000	−0.461112
$302$	16.0000	0.920697
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−6.00000	−0.341882
$309$	0	0
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	6.00000	0.334367
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	−2.00000	−0.110770
$327$	0	0
$328$	−6.00000	−0.331295
$329$	6.00000	0.330791
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	0	0
$333$	0	0
$334$	−3.00000	−0.164153
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	3.00000	0.159901
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	15.0000	0.792775
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	7.00000	0.367912
$363$	0	0
$364$	10.0000	0.524142
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	−7.00000	−0.355371
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	3.00000	0.151523
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	−33.0000	−1.63575
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	0	0
$412$	14.0000	0.689730
$413$	18.0000	0.885722
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	0	0
$418$	−12.0000	−0.586939
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	22.0000	1.06465
$428$	15.0000	0.725052
$429$	0	0
$430$	0	0
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	0	0
$433$	35.0000	1.68199	0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000	1.68199	0.840996	+	0.541041i	$0.181970\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	12.0000	0.574038
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	0	0
$442$	−30.0000	−1.42695
$443$	−27.0000	−1.28281	−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000	−1.28281	−0.641404	+	0.767203i	$0.721648\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	2.00000	0.0944911
$449$	−42.0000	−1.98210	−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000	−1.98210	−0.991051	+	0.133482i	$0.957384\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	18.0000	0.846649
$453$	0	0
$454$	−21.0000	−0.985579
$455$	0	0
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	7.00000	0.327089
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	28.0000	1.29292
$470$	0	0
$471$	0	0
$472$	−9.00000	−0.414259
$473$	12.0000	0.551761
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	9.00000	0.411650
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	55.0000	2.50778
$482$	19.0000	0.865426
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−11.0000	−0.497947
$489$	0	0
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	0	0
$494$	20.0000	0.899843
$495$	0	0
$496$	2.00000	0.0898027
$497$	−30.0000	−1.34568
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	27.0000	1.20507
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	−9.00000	−0.400099
$507$	0	0
$508$	2.00000	0.0887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	−9.00000	−0.395820
$518$	−22.0000	−0.966625
$519$	0	0
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	21.0000	0.917389
$525$	0	0
$526$	3.00000	0.130806
$527$	12.0000	0.522728
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	−8.00000	−0.346844
$533$	30.0000	1.29944
$534$	0	0
$535$	0	0
$536$	−14.0000	−0.604708
$537$	0	0
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	22.0000	0.934690
$555$	0	0
$556$	14.0000	0.593732
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	0	0
$566$	16.0000	0.672530
$567$	0	0
$568$	15.0000	0.629386
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	−15.0000	−0.627182
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	11.0000	0.452097
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	15.0000	0.613396
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	6.00000	0.241747
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	0	0
$622$	27.0000	1.08260
$623$	12.0000	0.480770
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	2.00000	0.0798087
$629$	66.0000	2.63159
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−12.0000	−0.476581
$635$	0	0
$636$	0	0
$637$	−15.0000	−0.594322
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	50.0000	1.97181	0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000	1.97181	0.985904	+	0.167313i	$0.0535092\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	24.0000	0.944267
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	0	0
$649$	−27.0000	−1.05984
$650$	0	0
$651$	0	0
$652$	2.00000	0.0783260
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	3.00000	0.116073
$669$	0	0
$670$	0	0
$671$	−33.0000	−1.27395
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	12.0000	0.461538
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	6.00000	0.229752
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−60.0000	−2.28582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	36.0000	1.36360
$698$	−26.0000	−0.984115
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−44.0000	−1.65949
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−24.0000	−0.903252
$707$	36.0000	1.35392
$708$	0	0
$709$	−43.0000	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$710$	0	0
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	−15.0000	−0.560576
$717$	0	0
$718$	−3.00000	−0.111959
$719$	−21.0000	−0.783168	−0.391584	−	0.920142i	$-0.628073\pi$
$719$	−21.0000	−0.783168	−0.391584	+	0.920142i	$0.628073\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	3.00000	0.111648
$723$	0	0
$724$	−7.00000	−0.260153
$725$	0	0
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−10.0000	−0.370625
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	3.00000	0.110581
$737$	−42.0000	−1.54709
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	−18.0000	−0.658145
$749$	30.0000	1.09618
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.0000	1.49017	0.745085	−	0.666969i	$-0.232409\pi$
$757$	41.0000	1.49017	0.745085	+	0.666969i	$0.232409\pi$
$758$	34.0000	1.23494
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	0	0
$766$	21.0000	0.758761
$767$	45.0000	1.62486
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	0	0
$776$	7.00000	0.251285
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−24.0000	−0.859889
$780$	0	0
$781$	45.0000	1.61023
$782$	18.0000	0.643679
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	36.0000	1.28001
$792$	0	0
$793$	55.0000	1.95311
$794$	1.00000	0.0354887
$795$	0	0
$796$	2.00000	0.0708881
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	−10.0000	−0.352235
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	0	0
$814$	33.0000	1.15665
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	31.0000	1.08389
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−18.0000	−0.626300
$827$	−51.0000	−1.77344	−0.886722	−	0.462303i	$-0.847023\pi$
$827$	−51.0000	−1.77344	−0.886722	+	0.462303i	$0.847023\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	0	0
$832$	5.00000	0.173344
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	0	0
$838$	24.0000	0.829066
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	19.0000	0.654783
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	−4.00000	−0.137442
$848$	−12.0000	−0.412082
$849$	0	0
$850$	0	0
$851$	−33.0000	−1.13123
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	−22.0000	−0.752825
$855$	0	0
$856$	−15.0000	−0.512689
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	9.00000	0.306541
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	0	0
$866$	−35.0000	−1.18935
$867$	0	0
$868$	4.00000	0.135769
$869$	30.0000	1.01768
$870$	0	0
$871$	70.0000	2.37186
$872$	10.0000	0.338643
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	−1.00000	−0.0337676	−0.0168838	−	0.999857i	$-0.505375\pi$
$877$	−1.00000	−0.0337676	−0.0168838	+	0.999857i	$0.505375\pi$
$878$	−14.0000	−0.472477
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	27.0000	0.907083
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	42.0000	1.40156
$899$	0	0
$900$	0	0
$901$	−72.0000	−2.39867
$902$	18.0000	0.599334
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	21.0000	0.696909
$909$	0	0
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	0	0
$913$	0	0
$914$	−17.0000	−0.562310
$915$	0	0
$916$	−7.00000	−0.231287
$917$	42.0000	1.38696
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	0	0
$922$	−12.0000	−0.395199
$923$	−75.0000	−2.46866
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	6.00000	0.196537
$933$	0	0
$934$	−21.0000	−0.687141
$935$	0	0
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	−28.0000	−0.914232
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	9.00000	0.292925
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	−12.0000	−0.388922
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	0	0
$955$	0	0
$956$	−9.00000	−0.291081
$957$	0	0
$958$	24.0000	0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−55.0000	−1.77327
$963$	0	0
$964$	−19.0000	−0.611949
$965$	0	0
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	0	0
$973$	28.0000	0.897639
$974$	−38.0000	−1.21760
$975$	0	0
$976$	11.0000	0.352101
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	24.0000	0.765871
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−20.0000	−0.636285
$989$	12.0000	0.381578
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	30.0000	0.951542
$995$	0	0
$996$	0	0
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	−20.0000	−0.633089
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.a.i.1.1	yes	1
3.2	odd	2		1350.2.a.t.1.1	yes	1
5.2	odd	4		1350.2.c.d.649.1		2
5.3	odd	4		1350.2.c.d.649.2		2
5.4	even	2		1350.2.a.n.1.1	yes	1
15.2	even	4		1350.2.c.i.649.2		2
15.8	even	4		1350.2.c.i.649.1		2
15.14	odd	2		1350.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.2.a.e.1.1	✓	1	15.14	odd	2
1350.2.a.i.1.1	yes	1	1.1	even	1	trivial
1350.2.a.n.1.1	yes	1	5.4	even	2
1350.2.a.t.1.1	yes	1	3.2	odd	2
1350.2.c.d.649.1		2	5.2	odd	4
1350.2.c.d.649.2		2	5.3	odd	4
1350.2.c.i.649.1		2	15.8	even	4
1350.2.c.i.649.2		2	15.2	even	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1350.2.a.i.1.1

Level	$1350$
Weight	$2$
Character	1350.1
Self dual	yes
Analytic conductor	$10.780$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$