LMFDB - Embedded newform 3450.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
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	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +5.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +5.00000 q^{28} +3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -5.00000 q^{42} +2.00000 q^{43} -1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +18.0000 q^{49} +3.00000 q^{51} +2.00000 q^{52} -12.0000 q^{53} -1.00000 q^{54} -5.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} +6.00000 q^{59} +2.00000 q^{61} -2.00000 q^{62} +5.00000 q^{63} +1.00000 q^{64} +2.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -15.0000 q^{71} -1.00000 q^{72} +11.0000 q^{73} +7.00000 q^{74} +2.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +5.00000 q^{84} -2.00000 q^{86} +3.00000 q^{87} +3.00000 q^{89} +10.0000 q^{91} +1.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -18.0000 q^{98} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	5.00000	0.944911
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−5.00000	−0.771517
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	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	$0.570213\pi$
$48$	1.00000	0.144338
$49$	18.0000	2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	2.00000	0.277350
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−5.00000	−0.668153
$57$	2.00000	0.264906
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	5.00000	0.629941
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	3.00000	0.363803
$69$	1.00000	0.120386
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	−1.00000	−0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	−2.00000	−0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	5.00000	0.545545
$85$	0	0
$86$	−2.00000	−0.215666
$87$	3.00000	0.321634
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	1.00000	0.104257
$93$	2.00000	0.207390
$94$	3.00000	0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−18.0000	−1.81827
$99$	0	0
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	−3.00000	−0.297044
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	5.00000	0.472456
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	3.00000	0.278543
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	15.0000	1.37505
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−2.00000	−0.181071
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	−5.00000	−0.445435
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	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
$133$	10.0000	0.867110
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−1.00000	−0.0851257
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	15.0000	1.25877
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−11.0000	−0.910366
$147$	18.0000	1.48461
$148$	−7.00000	−0.575396
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−2.00000	−0.162221
$153$	3.00000	0.242536
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−0.636446
$159$	−12.0000	−0.951662
$160$	0	0
$161$	5.00000	0.394055
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	9.00000	0.698535
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	−5.00000	−0.385758
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	2.00000	0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	−3.00000	−0.224860
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−10.0000	−0.741249
$183$	2.00000	0.147844
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	−3.00000	−0.218797
$189$	5.00000	0.363696
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.00000	0.0721688
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	3.00000	0.211079
$203$	15.0000	1.05279
$204$	3.00000	0.210042
$205$	0	0
$206$	13.0000	0.905753
$207$	1.00000	0.0695048
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−12.0000	−0.824163
$213$	−15.0000	−1.02778
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	10.0000	0.678844
$218$	−17.0000	−1.15139
$219$	11.0000	0.743311
$220$	0	0
$221$	6.00000	0.403604
$222$	7.00000	0.469809
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	9.00000	0.598671
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	2.00000	0.132453
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	−15.0000	−0.972306
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−2.00000	−0.127000
$249$	−9.00000	−0.570352
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	5.00000	0.314970
$253$	0	0
$254$	16.0000	1.00393
$255$	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	$0.689739\pi$
$258$	−2.00000	−0.124515
$259$	−35.0000	−2.17479
$260$	0	0
$261$	3.00000	0.185695
$262$	12.0000	0.741362
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	0	0
$266$	−10.0000	−0.613139
$267$	3.00000	0.183597
$268$	2.00000	0.122169
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000	0.181902
$273$	10.0000	0.605228
$274$	15.0000	0.906183
$275$	0	0
$276$	1.00000	0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−5.00000	−0.299880
$279$	2.00000	0.119737
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	3.00000	0.178647
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	11.0000	0.643726
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	−12.0000	−0.695141
$299$	2.00000	0.115663
$300$	0	0
$301$	10.0000	0.576390
$302$	4.00000	0.230174
$303$	−3.00000	−0.172345
$304$	2.00000	0.114708
$305$	0	0
$306$	−3.00000	−0.171499
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	−2.00000	−0.113228
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	12.0000	0.672927
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	−5.00000	−0.278639
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	17.0000	0.940102
$328$	0	0
$329$	−15.0000	−0.826977
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−9.00000	−0.493939
$333$	−7.00000	−0.383598
$334$	−15.0000	−0.820763
$335$	0	0
$336$	5.00000	0.272772
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	9.00000	0.489535
$339$	−9.00000	−0.488813
$340$	0	0
$341$	0	0
$342$	−2.00000	−0.108148
$343$	55.0000	2.96972
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	3.00000	0.160817
$349$	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
$351$	2.00000	0.106752
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	3.00000	0.159000
$357$	15.0000	0.793884
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−5.00000	−0.262794
$363$	−11.0000	−0.577350
$364$	10.0000	0.524142
$365$	0	0
$366$	−2.00000	−0.104542
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	0	0
$371$	−60.0000	−3.11504
$372$	2.00000	0.103695
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	6.00000	0.309016
$378$	−5.00000	−0.257172
$379$	−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
$381$	−16.0000	−0.819705
$382$	6.00000	0.306987
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.00000	0.0508987
$387$	2.00000	0.101666
$388$	−10.0000	−0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	−18.0000	−0.909137
$393$	−12.0000	−0.605320
$394$	−3.00000	−0.151138
$395$	0	0
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	19.0000	0.952384
$399$	10.0000	0.500626
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−2.00000	−0.0997509
$403$	4.00000	0.199254
$404$	−3.00000	−0.149256
$405$	0	0
$406$	−15.0000	−0.744438
$407$	0	0
$408$	−3.00000	−0.148522
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	−13.0000	−0.640464
$413$	30.0000	1.47620
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	5.00000	0.244851
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−23.0000	−1.11962
$423$	−3.00000	−0.145865
$424$	12.0000	0.582772
$425$	0	0
$426$	15.0000	0.726752
$427$	10.0000	0.483934
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	17.0000	0.814152
$437$	2.00000	0.0956730
$438$	−11.0000	−0.525600
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	−6.00000	−0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−7.00000	−0.332205
$445$	0	0
$446$	−26.0000	−1.23114
$447$	12.0000	0.567581
$448$	5.00000	0.236228
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	−9.00000	−0.423324
$453$	−4.00000	−0.187936
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−14.0000	−0.654177
$459$	3.00000	0.140028
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	2.00000	0.0924500
$469$	10.0000	0.461757
$470$	0	0
$471$	2.00000	0.0921551
$472$	−6.00000	−0.276172
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	15.0000	0.687524
$477$	−12.0000	−0.549442
$478$	15.0000	0.686084
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	−8.00000	−0.364390
$483$	5.00000	0.227508
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−2.00000	−0.0905357
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	−4.00000	−0.179969
$495$	0	0
$496$	2.00000	0.0898027
$497$	−75.0000	−3.36421
$498$	9.00000	0.403300
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	−3.00000	−0.133897
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−5.00000	−0.222718
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	55.0000	2.43306
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	18.0000	0.793946
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	35.0000	1.53781
$519$	6.00000	0.263371
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	−3.00000	−0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	18.0000	0.784837
$527$	6.00000	0.261364
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.00000	0.260378
$532$	10.0000	0.433555
$533$	0	0
$534$	−3.00000	−0.129823
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	12.0000	0.517838
$538$	−18.0000	−0.776035
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	16.0000	0.687259
$543$	5.00000	0.214571
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−10.0000	−0.427960
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	−15.0000	−0.640768
$549$	2.00000	0.0853579
$550$	0	0
$551$	6.00000	0.255609
$552$	−1.00000	−0.0425628
$553$	40.0000	1.70097
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	5.00000	0.212047
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−2.00000	−0.0846668
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	−27.0000	−1.13893
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	4.00000	0.168133
$567$	5.00000	0.209980
$568$	15.0000	0.629386
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	8.00000	0.332756
$579$	−1.00000	−0.0415586
$580$	0	0
$581$	−45.0000	−1.86691
$582$	10.0000	0.414513
$583$	0	0
$584$	−11.0000	−0.455183
$585$	0	0
$586$	30.0000	1.23929
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	18.0000	0.742307
$589$	4.00000	0.164817
$590$	0	0
$591$	3.00000	0.123404
$592$	−7.00000	−0.287698
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	−19.0000	−0.777618
$598$	−2.00000	−0.0817861
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−43.0000	−1.75401	−0.877003	−	0.480484i	$-0.840461\pi$
$601$	−43.0000	−1.75401	−0.877003	+	0.480484i	$0.840461\pi$
$602$	−10.0000	−0.407570
$603$	2.00000	0.0814463
$604$	−4.00000	−0.162758
$605$	0	0
$606$	3.00000	0.121867
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−2.00000	−0.0811107
$609$	15.0000	0.607831
$610$	0	0
$611$	−6.00000	−0.242734
$612$	3.00000	0.121268
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	−29.0000	−1.17034
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	13.0000	0.522937
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	−27.0000	−1.08260
$623$	15.0000	0.600962
$624$	2.00000	0.0800641
$625$	0	0
$626$	−8.00000	−0.319744
$627$	0	0
$628$	2.00000	0.0798087
$629$	−21.0000	−0.837325
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	−8.00000	−0.318223
$633$	23.0000	0.914168
$634$	9.00000	0.357436
$635$	0	0
$636$	−12.0000	−0.475831
$637$	36.0000	1.42637
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	−12.0000	−0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	−16.0000	−0.626608
$653$	−45.0000	−1.76099	−0.880493	−	0.474059i	$-0.842788\pi$
$653$	−45.0000	−1.76099	−0.880493	+	0.474059i	$0.842788\pi$
$654$	−17.0000	−0.664753
$655$	0	0
$656$	0	0
$657$	11.0000	0.429151
$658$	15.0000	0.584761
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	19.0000	0.738456
$663$	6.00000	0.233021
$664$	9.00000	0.349268
$665$	0	0
$666$	7.00000	0.271244
$667$	3.00000	0.116160
$668$	15.0000	0.580367
$669$	26.0000	1.00522
$670$	0	0
$671$	0	0
$672$	−5.00000	−0.192879
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	9.00000	0.345643
$679$	−50.0000	−1.91882
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	−55.0000	−2.09991
$687$	14.0000	0.534133
$688$	2.00000	0.0762493
$689$	−24.0000	−0.914327
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	−3.00000	−0.113715
$697$	0	0
$698$	−26.0000	−0.984115
$699$	12.0000	0.453882
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	−14.0000	−0.528020
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	−15.0000	−0.564133
$708$	6.00000	0.225494
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	−3.00000	−0.112430
$713$	2.00000	0.0749006
$714$	−15.0000	−0.561361
$715$	0	0
$716$	12.0000	0.448461
$717$	−15.0000	−0.560185
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	−65.0000	−2.42073
$722$	15.0000	0.558242
$723$	8.00000	0.297523
$724$	5.00000	0.185824
$725$	0	0
$726$	11.0000	0.408248
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	−10.0000	−0.370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	2.00000	0.0739221
$733$	−49.0000	−1.80986	−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000	−1.80986	−0.904928	+	0.425564i	$0.860076\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	60.0000	2.20267
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−17.0000	−0.622414
$747$	−9.00000	−0.329293
$748$	0	0
$749$	60.0000	2.19235
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	−3.00000	−0.109399
$753$	3.00000	0.109326
$754$	−6.00000	−0.218507
$755$	0	0
$756$	5.00000	0.181848
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	16.0000	0.579619
$763$	85.0000	3.07721
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−24.0000	−0.867155
$767$	12.0000	0.433295
$768$	1.00000	0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−1.00000	−0.0359908
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	10.0000	0.358979
$777$	−35.0000	−1.25562
$778$	−18.0000	−0.645331
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−3.00000	−0.107280
$783$	3.00000	0.107211
$784$	18.0000	0.642857
$785$	0	0
$786$	12.0000	0.428026
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	3.00000	0.106871
$789$	−18.0000	−0.640817
$790$	0	0
$791$	−45.0000	−1.60002
$792$	0	0
$793$	4.00000	0.142044
$794$	−20.0000	−0.709773
$795$	0	0
$796$	−19.0000	−0.673437
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−10.0000	−0.353996
$799$	−9.00000	−0.318397
$800$	0	0
$801$	3.00000	0.106000
$802$	−6.00000	−0.211867
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	18.0000	0.633630
$808$	3.00000	0.105540
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	15.0000	0.526397
$813$	−16.0000	−0.561144
$814$	0	0
$815$	0	0
$816$	3.00000	0.105021
$817$	4.00000	0.139942
$818$	7.00000	0.244749
$819$	10.0000	0.349428
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	15.0000	0.523185
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	−30.0000	−1.04383
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	1.00000	0.0347524
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	2.00000	0.0693375
$833$	54.0000	1.87099
$834$	−5.00000	−0.173136
$835$	0	0
$836$	0	0
$837$	2.00000	0.0691301
$838$	−15.0000	−0.518166
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.0000	0.344623
$843$	27.0000	0.929929
$844$	23.0000	0.791693
$845$	0	0
$846$	3.00000	0.103142
$847$	−55.0000	−1.88982
$848$	−12.0000	−0.412082
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−7.00000	−0.239957
$852$	−15.0000	−0.513892
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	−8.00000	−0.271694
$868$	10.0000	0.339422
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	−17.0000	−0.575693
$873$	−10.0000	−0.338449
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	11.0000	0.371656
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	28.0000	0.944954
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−18.0000	−0.606092
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	24.0000	0.806296
$887$	−27.0000	−0.906571	−0.453286	−	0.891365i	$-0.649748\pi$
$887$	−27.0000	−0.906571	−0.453286	+	0.891365i	$0.649748\pi$
$888$	7.00000	0.234905
$889$	−80.0000	−2.68311
$890$	0	0
$891$	0	0
$892$	26.0000	0.870544
$893$	−6.00000	−0.200782
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−5.00000	−0.167038
$897$	2.00000	0.0667781
$898$	−30.0000	−1.00111
$899$	6.00000	0.200111
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	10.0000	0.332779
$904$	9.00000	0.299336
$905$	0	0
$906$	4.00000	0.132891
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	3.00000	0.0995585
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	28.0000	0.926158
$915$	0	0
$916$	14.0000	0.462573
$917$	−60.0000	−1.98137
$918$	−3.00000	−0.0990148
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	29.0000	0.955582
$922$	15.0000	0.493999
$923$	−30.0000	−0.987462
$924$	0	0
$925$	0	0
$926$	40.0000	1.31448
$927$	−13.0000	−0.426976
$928$	−3.00000	−0.0984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	12.0000	0.393073
$933$	27.0000	0.883940
$934$	−15.0000	−0.490815
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	−10.0000	−0.326512
$939$	8.00000	0.261070
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	8.00000	0.259828
$949$	22.0000	0.714150
$950$	0	0
$951$	−9.00000	−0.291845
$952$	−15.0000	−0.486153
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−30.0000	−0.969256
$959$	−75.0000	−2.42188
$960$	0	0
$961$	−27.0000	−0.870968
$962$	14.0000	0.451378
$963$	12.0000	0.386695
$964$	8.00000	0.257663
$965$	0	0
$966$	−5.00000	−0.160872
$967$	−58.0000	−1.86515	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000	−1.86515	−0.932577	+	0.360971i	$0.882445\pi$
$968$	11.0000	0.353553
$969$	6.00000	0.192748
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	1.00000	0.0320750
$973$	25.0000	0.801463
$974$	−8.00000	−0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	0	0
$981$	17.0000	0.542768
$982$	6.00000	0.191468
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	0	0
$986$	−9.00000	−0.286618
$987$	−15.0000	−0.477455
$988$	4.00000	0.127257
$989$	2.00000	0.0635963
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−2.00000	−0.0635001
$993$	−19.0000	−0.602947
$994$	75.0000	2.37886
$995$	0	0
$996$	−9.00000	−0.285176
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−5.00000	−0.158272
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.m.1.1	✓	1
5.2	odd	4		3450.2.d.h.2899.1		2
5.3	odd	4		3450.2.d.h.2899.2		2
5.4	even	2		3450.2.a.n.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.m.1.1	✓	1	1.1	even	1	trivial
3450.2.a.n.1.1	yes	1	5.4	even	2
3450.2.d.h.2899.1		2	5.2	odd	4
3450.2.d.h.2899.2		2	5.3	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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