LMFDB - Embedded newform 2475.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
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	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} +3.00000 q^{8} -1.00000 q^{11} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} +1.00000 q^{22} -1.00000 q^{23} +2.00000 q^{26} -3.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +3.00000 q^{34} -1.00000 q^{37} +1.00000 q^{38} +5.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} +2.00000 q^{52} -10.0000 q^{53} +9.00000 q^{56} +6.00000 q^{58} -11.0000 q^{59} +14.0000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -2.00000 q^{67} +3.00000 q^{68} +5.00000 q^{71} -2.00000 q^{73} +1.00000 q^{74} +1.00000 q^{76} -3.00000 q^{77} +5.00000 q^{79} -5.00000 q^{82} -8.00000 q^{83} +4.00000 q^{86} -3.00000 q^{88} +10.0000 q^{89} -6.00000 q^{91} +1.00000 q^{92} +3.00000 q^{94} +17.0000 q^{97} -2.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	0	0
$56$	9.00000	1.20268
$57$	0	0
$58$	6.00000	0.787839
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	1.00000	0.114708
$77$	−3.00000	−0.341882
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	1.00000	0.104257
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	0	0
$101$	11.0000	1.09454	0.547270	−	0.836956i	$-0.315667\pi$
$101$	11.0000	1.09454	0.547270	+	0.836956i	$0.315667\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	11.0000	1.01263
$119$	−9.00000	−0.825029
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	2.00000	0.172774
$135$	0	0
$136$	−9.00000	−0.771744
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−5.00000	−0.419591
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	1.00000	0.0821995
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	3.00000	0.241747
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	8.00000	0.620920
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−10.0000	−0.749532
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	3.00000	0.219382
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	−19.0000	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$191$	−19.0000	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−17.0000	−1.22053
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−23.0000	−1.63868	−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000	−1.63868	−0.819341	+	0.573306i	$0.805660\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	−11.0000	−0.773957
$203$	−18.0000	−1.26335
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	0	0
$208$	2.00000	0.138675
$209$	1.00000	0.0691714
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	12.0000	0.814613
$218$	12.0000	0.812743
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	18.0000	1.19734
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	11.0000	0.726900	0.363450	−	0.931614i	$-0.381599\pi$
$229$	11.0000	0.726900	0.363450	+	0.931614i	$0.381599\pi$
$230$	0	0
$231$	0	0
$232$	−18.0000	−1.18176
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	0	0
$236$	11.0000	0.716039
$237$	0	0
$238$	9.00000	0.583383
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	12.0000	0.762001
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	−5.00000	−0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	3.00000	0.183942
$267$	0	0
$268$	2.00000	0.122169
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	6.00000	0.362473
$275$	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	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	23.0000	1.36721	0.683604	−	0.729853i	$-0.260412\pi$
$283$	23.0000	1.36721	0.683604	+	0.729853i	$0.260412\pi$
$284$	−5.00000	−0.296695
$285$	0	0
$286$	−2.00000	−0.118262
$287$	15.0000	0.885422
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	−13.0000	−0.759468	−0.379734	−	0.925096i	$-0.623985\pi$
$293$	−13.0000	−0.759468	−0.379734	+	0.925096i	$0.623985\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	7.00000	0.405499
$299$	2.00000	0.115663
$300$	0	0
$301$	−12.0000	−0.691669
$302$	16.0000	0.920697
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−5.00000	−0.281272
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	3.00000	0.167183
$323$	3.00000	0.166924
$324$	0	0
$325$	0	0
$326$	10.0000	0.553849
$327$	0	0
$328$	15.0000	0.828236
$329$	−9.00000	−0.496186
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	10.0000	0.547176
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−9.00000	−0.483843
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	0	0
$352$	5.00000	0.266501
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−1.00000	−0.0528516
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	25.0000	1.31397
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	−9.00000	−0.464140
$377$	12.0000	0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	19.0000	0.972125
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	0	0
$388$	−17.0000	−0.863044
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	6.00000	0.303046
$393$	0	0
$394$	23.0000	1.15872
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	18.0000	0.902258
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−11.0000	−0.547270
$405$	0	0
$406$	18.0000	0.893325
$407$	1.00000	0.0495682
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	0	0
$412$	2.00000	0.0985329
$413$	−33.0000	−1.62382
$414$	0	0
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	−1.00000	−0.0489116
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−30.0000	−1.45693
$425$	0	0
$426$	0	0
$427$	42.0000	2.03252
$428$	18.0000	0.870063
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	12.0000	0.574696
$437$	1.00000	0.0478365
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	31.0000	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	21.0000	0.992157
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	18.0000	0.846649
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−11.0000	−0.513996
$459$	0	0
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	9.00000	0.416917
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	0	0
$471$	0	0
$472$	−33.0000	−1.51895
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	9.00000	0.412514
$477$	0	0
$478$	−20.0000	−0.914779
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−4.00000	−0.182195
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	42.0000	1.90125
$489$	0	0
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	15.0000	0.672842
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	0	0
$502$	12.0000	0.535586
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	0	0
$506$	−1.00000	−0.0444554
$507$	0	0
$508$	−5.00000	−0.221839
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	11.0000	0.486136
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	3.00000	0.131940
$518$	3.00000	0.131812
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	3.00000	0.130066
$533$	−10.0000	−0.433148
$534$	0	0
$535$	0	0
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−24.0000	−1.03471
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	3.00000	0.128861
$543$	0	0
$544$	15.0000	0.643120
$545$	0	0
$546$	0	0
$547$	33.0000	1.41098	0.705489	−	0.708721i	$-0.250727\pi$
$547$	33.0000	1.41098	0.705489	+	0.708721i	$0.250727\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	15.0000	0.637865
$554$	22.0000	0.934690
$555$	0	0
$556$	0	0
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−21.0000	−0.885832
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	0	0
$566$	−23.0000	−0.966762
$567$	0	0
$568$	15.0000	0.629386
$569$	−37.0000	−1.55112	−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000	−1.55112	−0.775560	+	0.631273i	$0.782533\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−15.0000	−0.626088
$575$	0	0
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	10.0000	0.414158
$584$	−6.00000	−0.248282
$585$	0	0
$586$	13.0000	0.537025
$587$	−9.00000	−0.371470	−0.185735	−	0.982600i	$-0.559467\pi$
$587$	−9.00000	−0.371470	−0.185735	+	0.982600i	$0.559467\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	1.00000	0.0410997
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	7.00000	0.286731
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−9.00000	−0.362620
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	30.0000	1.20192
$624$	0	0
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	10.0000	0.399043
$629$	3.00000	0.119618
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	15.0000	0.596668
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−6.00000	−0.237542
$639$	0	0
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	13.0000	0.511083	0.255541	−	0.966798i	$-0.417746\pi$
$647$	13.0000	0.511083	0.255541	+	0.966798i	$0.417746\pi$
$648$	0	0
$649$	11.0000	0.431788
$650$	0	0
$651$	0	0
$652$	10.0000	0.391630
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	0	0
$656$	−5.00000	−0.195217
$657$	0	0
$658$	9.00000	0.350857
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−24.0000	−0.931381
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	10.0000	0.386912
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	51.0000	1.95720
$680$	0	0
$681$	0	0
$682$	4.00000	0.153168
$683$	−47.0000	−1.79841	−0.899203	−	0.437533i	$-0.855852\pi$
$683$	−47.0000	−1.79841	−0.899203	+	0.437533i	$0.855852\pi$
$684$	0	0
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000	0.152499
$689$	20.0000	0.761939
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	−10.0000	−0.379595
$695$	0	0
$696$	0	0
$697$	−15.0000	−0.568166
$698$	−8.00000	−0.302804
$699$	0	0
$700$	0	0
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	−7.00000	−0.263822
$705$	0	0
$706$	6.00000	0.225813
$707$	33.0000	1.24109
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	30.0000	1.12430
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	−1.00000	−0.0373718
$717$	0	0
$718$	−28.0000	−1.04495
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	18.0000	0.669891
$723$	0	0
$724$	25.0000	0.929118
$725$	0	0
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−18.0000	−0.667124
$729$	0	0
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	5.00000	0.184302
$737$	2.00000	0.0736709
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	0	0
$741$	0	0
$742$	30.0000	1.10133
$743$	14.0000	0.513610	0.256805	−	0.966463i	$-0.417330\pi$
$743$	14.0000	0.513610	0.256805	+	0.966463i	$0.417330\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	−3.00000	−0.109691
$749$	−54.0000	−1.97312
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	19.0000	0.687396
$765$	0	0
$766$	0	0
$767$	22.0000	0.794374
$768$	0	0
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	−44.0000	−1.58257	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000	−1.58257	−0.791285	+	0.611448i	$0.790588\pi$
$774$	0	0
$775$	0	0
$776$	51.0000	1.83079
$777$	0	0
$778$	−6.00000	−0.215110
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−5.00000	−0.178914
$782$	−3.00000	−0.107280
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	0	0
$787$	−43.0000	−1.53278	−0.766392	−	0.642373i	$-0.777950\pi$
$787$	−43.0000	−1.53278	−0.766392	+	0.642373i	$0.777950\pi$
$788$	23.0000	0.819341
$789$	0	0
$790$	0	0
$791$	−54.0000	−1.92002
$792$	0	0
$793$	−28.0000	−0.994309
$794$	−6.00000	−0.212932
$795$	0	0
$796$	18.0000	0.637993
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	0	0
$802$	−38.0000	−1.34183
$803$	2.00000	0.0705785
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	33.0000	1.16094
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	18.0000	0.631676
$813$	0	0
$814$	−1.00000	−0.0350500
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	42.0000	1.46403	0.732014	−	0.681290i	$-0.238581\pi$
$823$	42.0000	1.46403	0.732014	+	0.681290i	$0.238581\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	33.0000	1.14822
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	−14.0000	−0.485363
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	−23.0000	−0.794522
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−9.00000	−0.310160
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	3.00000	0.103081
$848$	10.0000	0.343401
$849$	0	0
$850$	0	0
$851$	1.00000	0.0342796
$852$	0	0
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	−42.0000	−1.43721
$855$	0	0
$856$	−54.0000	−1.84568
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	0	0
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	0	0
$861$	0	0
$862$	36.0000	1.22616
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	−12.0000	−0.407307
$869$	−5.00000	−0.169613
$870$	0	0
$871$	4.00000	0.135535
$872$	−36.0000	−1.21911
$873$	0	0
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	0	0
$877$	52.0000	1.75592	0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000	1.75592	0.877958	+	0.478738i	$0.158906\pi$
$878$	35.0000	1.18119
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−31.0000	−1.04147
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	0	0
$891$	0	0
$892$	−22.0000	−0.736614
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	−8.00000	−0.266963
$899$	−24.0000	−0.800445
$900$	0	0
$901$	30.0000	0.999445
$902$	5.00000	0.166482
$903$	0	0
$904$	−54.0000	−1.79601
$905$	0	0
$906$	0	0
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	0	0
$911$	53.0000	1.75597	0.877984	−	0.478690i	$-0.158888\pi$
$911$	53.0000	1.75597	0.877984	+	0.478690i	$0.158888\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	28.0000	0.926158
$915$	0	0
$916$	−11.0000	−0.363450
$917$	−36.0000	−1.18882
$918$	0	0
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	0	0
$921$	0	0
$922$	34.0000	1.11973
$923$	−10.0000	−0.329154
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	0	0
$928$	30.0000	0.984798
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	9.00000	0.294805
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	6.00000	0.195907
$939$	0	0
$940$	0	0
$941$	−5.00000	−0.162995	−0.0814977	−	0.996674i	$-0.525970\pi$
$941$	−5.00000	−0.162995	−0.0814977	+	0.996674i	$0.525970\pi$
$942$	0	0
$943$	−5.00000	−0.162822
$944$	11.0000	0.358020
$945$	0	0
$946$	−4.00000	−0.130051
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	−27.0000	−0.875075
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	0	0
$956$	−20.0000	−0.646846
$957$	0	0
$958$	−18.0000	−0.581554
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−4.00000	−0.128831
$965$	0	0
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	3.00000	0.0964237
$969$	0	0
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	0	0
$973$	0	0
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	0	0
$982$	28.0000	0.893516
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	4.00000	0.127193
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	−15.0000	−0.475771
$995$	0	0
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.d.1.1	yes	1
3.2	odd	2		2475.2.a.k.1.1	yes	1
5.2	odd	4		2475.2.c.c.199.1		2
5.3	odd	4		2475.2.c.c.199.2		2
5.4	even	2		2475.2.a.h.1.1	yes	1
15.2	even	4		2475.2.c.e.199.2		2
15.8	even	4		2475.2.c.e.199.1		2
15.14	odd	2		2475.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.2.a.b.1.1	✓	1	15.14	odd	2
2475.2.a.d.1.1	yes	1	1.1	even	1	trivial
2475.2.a.h.1.1	yes	1	5.4	even	2
2475.2.a.k.1.1	yes	1	3.2	odd	2
2475.2.c.c.199.1		2	5.2	odd	4
2475.2.c.c.199.2		2	5.3	odd	4
2475.2.c.e.199.1		2	15.8	even	4
2475.2.c.e.199.2		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

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