LMFDB - Embedded newform 174.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [174,2,Mod(1,174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(174, 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("174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 174 = 2 \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	174.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:	$1.38939699517$
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	$\chi$	$=$	174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} -7.00000 q^{41} +1.00000 q^{42} +9.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +12.0000 q^{67} -3.00000 q^{68} -4.00000 q^{69} -1.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +3.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} -2.00000 q^{77} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -7.00000 q^{82} +1.00000 q^{84} +3.00000 q^{85} +9.00000 q^{86} +1.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -1.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	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	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−1.00000	−0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−2.00000	−0.348155
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	1.00000	0.154303
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	−4.00000	−0.589768
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	1.00000	0.131306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−3.00000	−0.363803
$69$	−4.00000	−0.481543
$70$	−1.00000	−0.119523
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	3.00000	0.348743
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	−2.00000	−0.227921
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	0.109109
$85$	3.00000	0.325396
$86$	9.00000	0.970495
$87$	1.00000	0.107211
$88$	−2.00000	−0.213201
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	−1.00000	−0.103142
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−6.00000	−0.606092
$99$	−2.00000	−0.201008
$100$	−4.00000	−0.400000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−3.00000	−0.297044
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	−2.00000	−0.194257
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	2.00000	0.190693
$111$	3.00000	0.284747
$112$	1.00000	0.0944911
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	−1.00000	−0.0936586
$115$	4.00000	0.373002
$116$	1.00000	0.0928477
$117$	0	0
$118$	−3.00000	−0.276172
$119$	−3.00000	−0.275010
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	−7.00000	−0.631169
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	1.00000	0.0890871
$127$	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
$129$	9.00000	0.792406
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−2.00000	−0.174078
$133$	−1.00000	−0.0867110
$134$	12.0000	1.03664
$135$	−1.00000	−0.0860663
$136$	−3.00000	−0.257248
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−4.00000	−0.340503
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−1.00000	−0.0845154
$141$	−1.00000	−0.0842152
$142$	16.0000	1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	−10.0000	−0.827606
$147$	−6.00000	−0.494872
$148$	3.00000	0.246598
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	−4.00000	−0.326599
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−1.00000	−0.0811107
$153$	−3.00000	−0.242536
$154$	−2.00000	−0.161165
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	10.0000	0.795557
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−7.00000	−0.546608
$165$	2.00000	0.155700
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	1.00000	0.0771517
$169$	−13.0000	−1.00000
$170$	3.00000	0.230089
$171$	−1.00000	−0.0764719
$172$	9.00000	0.686244
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	1.00000	0.0758098
$175$	−4.00000	−0.302372
$176$	−2.00000	−0.150756
$177$	−3.00000	−0.225494
$178$	6.00000	0.449719
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	−1.00000	−0.0745356
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−4.00000	−0.294884
$185$	−3.00000	−0.220564
$186$	4.00000	0.293294
$187$	6.00000	0.438763
$188$	−1.00000	−0.0729325
$189$	1.00000	0.0727393
$190$	1.00000	0.0725476
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	1.00000	0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−5.00000	−0.356235	−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000	−0.356235	−0.178118	+	0.984009i	$0.557001\pi$
$198$	−2.00000	−0.142134
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−4.00000	−0.282843
$201$	12.0000	0.846415
$202$	18.0000	1.26648
$203$	1.00000	0.0701862
$204$	−3.00000	−0.210042
$205$	7.00000	0.488901
$206$	−1.00000	−0.0696733
$207$	−4.00000	−0.278019
$208$	0	0
$209$	2.00000	0.138343
$210$	−1.00000	−0.0690066
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−2.00000	−0.137361
$213$	16.0000	1.09630
$214$	17.0000	1.16210
$215$	−9.00000	−0.613795
$216$	1.00000	0.0680414
$217$	4.00000	0.271538
$218$	−16.0000	−1.08366
$219$	−10.0000	−0.675737
$220$	2.00000	0.134840
$221$	0	0
$222$	3.00000	0.201347
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	−5.00000	−0.332595
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	−1.00000	−0.0662266
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	4.00000	0.263752
$231$	−2.00000	−0.131590
$232$	1.00000	0.0656532
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	−3.00000	−0.195283
$237$	10.0000	0.649570
$238$	−3.00000	−0.194461
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	−1.00000	−0.0645497
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	6.00000	0.383326
$246$	−7.00000	−0.446304
$247$	0	0
$248$	4.00000	0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000	0.0629941
$253$	8.00000	0.502956
$254$	2.00000	0.125491
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	9.00000	0.560316
$259$	3.00000	0.186411
$260$	0	0
$261$	1.00000	0.0618984
$262$	−8.00000	−0.494242
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	−2.00000	−0.123091
$265$	2.00000	0.122859
$266$	−1.00000	−0.0613139
$267$	6.00000	0.367194
$268$	12.0000	0.733017
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−2.00000	−0.120824
$275$	8.00000	0.482418
$276$	−4.00000	−0.240772
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−14.0000	−0.839664
$279$	4.00000	0.239474
$280$	−1.00000	−0.0597614
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−1.00000	−0.0595491
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	16.0000	0.949425
$285$	1.00000	0.0592349
$286$	0	0
$287$	−7.00000	−0.413197
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−1.00000	−0.0587220
$291$	0	0
$292$	−10.0000	−0.585206
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−6.00000	−0.349927
$295$	3.00000	0.174667
$296$	3.00000	0.174371
$297$	−2.00000	−0.116052
$298$	17.0000	0.984784
$299$	0	0
$300$	−4.00000	−0.230940
$301$	9.00000	0.518751
$302$	5.00000	0.287718
$303$	18.0000	1.03407
$304$	−1.00000	−0.0573539
$305$	−6.00000	−0.343559
$306$	−3.00000	−0.171499
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−2.00000	−0.113961
$309$	−1.00000	−0.0568880
$310$	−4.00000	−0.227185
$311$	−17.0000	−0.963982	−0.481991	−	0.876176i	$-0.660086\pi$
$311$	−17.0000	−0.963982	−0.481991	+	0.876176i	$0.660086\pi$
$312$	0	0
$313$	−29.0000	−1.63918	−0.819588	−	0.572953i	$-0.805798\pi$
$313$	−29.0000	−1.63918	−0.819588	+	0.572953i	$0.805798\pi$
$314$	−17.0000	−0.959366
$315$	−1.00000	−0.0563436
$316$	10.0000	0.562544
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	−2.00000	−0.112154
$319$	−2.00000	−0.111979
$320$	−1.00000	−0.0559017
$321$	17.0000	0.948847
$322$	−4.00000	−0.222911
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	−11.0000	−0.609234
$327$	−16.0000	−0.884802
$328$	−7.00000	−0.386510
$329$	−1.00000	−0.0551318
$330$	2.00000	0.110096
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	−14.0000	−0.766046
$335$	−12.0000	−0.655630
$336$	1.00000	0.0545545
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−13.0000	−0.707107
$339$	−5.00000	−0.271563
$340$	3.00000	0.162698
$341$	−8.00000	−0.433224
$342$	−1.00000	−0.0540738
$343$	−13.0000	−0.701934
$344$	9.00000	0.485247
$345$	4.00000	0.215353
$346$	−15.0000	−0.806405
$347$	−37.0000	−1.98626	−0.993132	−	0.116999i	$-0.962673\pi$
$347$	−37.0000	−1.98626	−0.993132	+	0.116999i	$0.962673\pi$
$348$	1.00000	0.0536056
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−3.00000	−0.159448
$355$	−16.0000	−0.849192
$356$	6.00000	0.317999
$357$	−3.00000	−0.158777
$358$	−16.0000	−0.845626
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	−1.00000	−0.0527046
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	10.0000	0.523424
$366$	6.00000	0.313625
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−4.00000	−0.208514
$369$	−7.00000	−0.364405
$370$	−3.00000	−0.155963
$371$	−2.00000	−0.103835
$372$	4.00000	0.207390
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	6.00000	0.310253
$375$	9.00000	0.464758
$376$	−1.00000	−0.0515711
$377$	0	0
$378$	1.00000	0.0514344
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	1.00000	0.0512989
$381$	2.00000	0.102463
$382$	17.0000	0.869796
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	−16.0000	−0.814379
$387$	9.00000	0.457496
$388$	0	0
$389$	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
$391$	12.0000	0.606866
$392$	−6.00000	−0.303046
$393$	−8.00000	−0.403547
$394$	−5.00000	−0.251896
$395$	−10.0000	−0.503155
$396$	−2.00000	−0.100504
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−24.0000	−1.20301
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	18.0000	0.895533
$405$	−1.00000	−0.0496904
$406$	1.00000	0.0496292
$407$	−6.00000	−0.297409
$408$	−3.00000	−0.148522
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	7.00000	0.345705
$411$	−2.00000	−0.0986527
$412$	−1.00000	−0.0492665
$413$	−3.00000	−0.147620
$414$	−4.00000	−0.196589
$415$	0	0
$416$	0	0
$417$	−14.0000	−0.685583
$418$	2.00000	0.0978232
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	−1.00000	−0.0487950
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	23.0000	1.11962
$423$	−1.00000	−0.0486217
$424$	−2.00000	−0.0971286
$425$	12.0000	0.582086
$426$	16.0000	0.775203
$427$	6.00000	0.290360
$428$	17.0000	0.821726
$429$	0	0
$430$	−9.00000	−0.434019
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	4.00000	0.192006
$435$	−1.00000	−0.0479463
$436$	−16.0000	−0.766261
$437$	4.00000	0.191346
$438$	−10.0000	−0.477818
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	2.00000	0.0953463
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	3.00000	0.142374
$445$	−6.00000	−0.284427
$446$	0	0
$447$	17.0000	0.804072
$448$	1.00000	0.0472456
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	−4.00000	−0.188562
$451$	14.0000	0.659234
$452$	−5.00000	−0.235180
$453$	5.00000	0.234920
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	13.0000	0.607450
$459$	−3.00000	−0.140028
$460$	4.00000	0.186501
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	−2.00000	−0.0930484
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	1.00000	0.0464238
$465$	−4.00000	−0.185496
$466$	−4.00000	−0.185296
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	1.00000	0.0461266
$471$	−17.0000	−0.783319
$472$	−3.00000	−0.138086
$473$	−18.0000	−0.827641
$474$	10.0000	0.459315
$475$	4.00000	0.183533
$476$	−3.00000	−0.137505
$477$	−2.00000	−0.0915737
$478$	2.00000	0.0914779
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	25.0000	1.13872
$483$	−4.00000	−0.182006
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−37.0000	−1.67663	−0.838315	−	0.545186i	$-0.816459\pi$
$487$	−37.0000	−1.67663	−0.838315	+	0.545186i	$0.816459\pi$
$488$	6.00000	0.271607
$489$	−11.0000	−0.497437
$490$	6.00000	0.271052
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−7.00000	−0.315584
$493$	−3.00000	−0.135113
$494$	0	0
$495$	2.00000	0.0898933
$496$	4.00000	0.179605
$497$	16.0000	0.717698
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	9.00000	0.402492
$501$	−14.0000	−0.625474
$502$	0	0
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	1.00000	0.0445435
$505$	−18.0000	−0.800989
$506$	8.00000	0.355643
$507$	−13.0000	−0.577350
$508$	2.00000	0.0887357
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	3.00000	0.132842
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−8.00000	−0.352865
$515$	1.00000	0.0440653
$516$	9.00000	0.396203
$517$	2.00000	0.0879599
$518$	3.00000	0.131812
$519$	−15.0000	−0.658427
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	1.00000	0.0437688
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−8.00000	−0.349482
$525$	−4.00000	−0.174574
$526$	−9.00000	−0.392419
$527$	−12.0000	−0.522728
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	−3.00000	−0.130189
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	6.00000	0.259645
$535$	−17.0000	−0.734974
$536$	12.0000	0.518321
$537$	−16.0000	−0.690451
$538$	18.0000	0.776035
$539$	12.0000	0.516877
$540$	−1.00000	−0.0430331
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	−3.00000	−0.128624
$545$	16.0000	0.685365
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−2.00000	−0.0854358
$549$	6.00000	0.256074
$550$	8.00000	0.341121
$551$	−1.00000	−0.0426014
$552$	−4.00000	−0.170251
$553$	10.0000	0.425243
$554$	26.0000	1.10463
$555$	−3.00000	−0.127343
$556$	−14.0000	−0.593732
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	6.00000	0.253320
$562$	30.0000	1.26547
$563$	−10.0000	−0.421450	−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000	−0.421450	−0.210725	+	0.977545i	$0.567582\pi$
$564$	−1.00000	−0.0421076
$565$	5.00000	0.210352
$566$	−10.0000	−0.420331
$567$	1.00000	0.0419961
$568$	16.0000	0.671345
$569$	31.0000	1.29959	0.649794	−	0.760111i	$-0.274855\pi$
$569$	31.0000	1.29959	0.649794	+	0.760111i	$0.274855\pi$
$570$	1.00000	0.0418854
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	17.0000	0.710185
$574$	−7.00000	−0.292174
$575$	16.0000	0.667246
$576$	1.00000	0.0416667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−8.00000	−0.332756
$579$	−16.0000	−0.664937
$580$	−1.00000	−0.0415227
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	−6.00000	−0.247436
$589$	−4.00000	−0.164817
$590$	3.00000	0.123508
$591$	−5.00000	−0.205673
$592$	3.00000	0.123299
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−2.00000	−0.0820610
$595$	3.00000	0.122988
$596$	17.0000	0.696347
$597$	−24.0000	−0.982255
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	−4.00000	−0.163299
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	9.00000	0.366813
$603$	12.0000	0.488678
$604$	5.00000	0.203447
$605$	7.00000	0.284590
$606$	18.0000	0.731200
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−1.00000	−0.0405554
$609$	1.00000	0.0405220
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−3.00000	−0.121268
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	28.0000	1.12999
$615$	7.00000	0.282267
$616$	−2.00000	−0.0805823
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	−1.00000	−0.0402259
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−4.00000	−0.160644
$621$	−4.00000	−0.160514
$622$	−17.0000	−0.681638
$623$	6.00000	0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	−29.0000	−1.15907
$627$	2.00000	0.0798723
$628$	−17.0000	−0.678374
$629$	−9.00000	−0.358854
$630$	−1.00000	−0.0398410
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	10.0000	0.397779
$633$	23.0000	0.914168
$634$	24.0000	0.953162
$635$	−2.00000	−0.0793676
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	16.0000	0.632950
$640$	−1.00000	−0.0395285
$641$	−37.0000	−1.46141	−0.730706	−	0.682692i	$-0.760809\pi$
$641$	−37.0000	−1.46141	−0.730706	+	0.682692i	$0.760809\pi$
$642$	17.0000	0.670936
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−4.00000	−0.157622
$645$	−9.00000	−0.354375
$646$	3.00000	0.118033
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000	0.0392837
$649$	6.00000	0.235521
$650$	0	0
$651$	4.00000	0.156772
$652$	−11.0000	−0.430793
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	−16.0000	−0.625650
$655$	8.00000	0.312586
$656$	−7.00000	−0.273304
$657$	−10.0000	−0.390137
$658$	−1.00000	−0.0389841
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	2.00000	0.0778499
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	17.0000	0.660724
$663$	0	0
$664$	0	0
$665$	1.00000	0.0387783
$666$	3.00000	0.116248
$667$	−4.00000	−0.154881
$668$	−14.0000	−0.541676
$669$	0	0
$670$	−12.0000	−0.463600
$671$	−12.0000	−0.463255
$672$	1.00000	0.0385758
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	2.00000	0.0770371
$675$	−4.00000	−0.153960
$676$	−13.0000	−0.500000
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−5.00000	−0.192024
$679$	0	0
$680$	3.00000	0.115045
$681$	−3.00000	−0.114960
$682$	−8.00000	−0.306336
$683$	−23.0000	−0.880071	−0.440035	−	0.897980i	$-0.645034\pi$
$683$	−23.0000	−0.880071	−0.440035	+	0.897980i	$0.645034\pi$
$684$	−1.00000	−0.0382360
$685$	2.00000	0.0764161
$686$	−13.0000	−0.496342
$687$	13.0000	0.495981
$688$	9.00000	0.343122
$689$	0	0
$690$	4.00000	0.152277
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−15.0000	−0.570214
$693$	−2.00000	−0.0759737
$694$	−37.0000	−1.40450
$695$	14.0000	0.531050
$696$	1.00000	0.0379049
$697$	21.0000	0.795432
$698$	14.0000	0.529908
$699$	−4.00000	−0.151294
$700$	−4.00000	−0.151186
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	−2.00000	−0.0753778
$705$	1.00000	0.0376622
$706$	−10.0000	−0.376355
$707$	18.0000	0.676960
$708$	−3.00000	−0.112747
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	−16.0000	−0.600469
$711$	10.0000	0.375029
$712$	6.00000	0.224860
$713$	−16.0000	−0.599205
$714$	−3.00000	−0.112272
$715$	0	0
$716$	−16.0000	−0.597948
$717$	2.00000	0.0746914
$718$	3.00000	0.111959
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−1.00000	−0.0372678
$721$	−1.00000	−0.0372419
$722$	−18.0000	−0.669891
$723$	25.0000	0.929760
$724$	0	0
$725$	−4.00000	−0.148556
$726$	−7.00000	−0.259794
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−27.0000	−0.998631
$732$	6.00000	0.221766
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−10.0000	−0.369107
$735$	6.00000	0.221313
$736$	−4.00000	−0.147442
$737$	−24.0000	−0.884051
$738$	−7.00000	−0.257674
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	4.00000	0.146647
$745$	−17.0000	−0.622832
$746$	−32.0000	−1.17160
$747$	0	0
$748$	6.00000	0.219382
$749$	17.0000	0.621166
$750$	9.00000	0.328634
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	−1.00000	−0.0364662
$753$	0	0
$754$	0	0
$755$	−5.00000	−0.181969
$756$	1.00000	0.0363696
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	−12.0000	−0.435860
$759$	8.00000	0.290382
$760$	1.00000	0.0362738
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	2.00000	0.0724524
$763$	−16.0000	−0.579239
$764$	17.0000	0.615038
$765$	3.00000	0.108465
$766$	6.00000	0.216789
$767$	0	0
$768$	1.00000	0.0360844
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	2.00000	0.0720750
$771$	−8.00000	−0.288113
$772$	−16.0000	−0.575853
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	9.00000	0.323498
$775$	−16.0000	−0.574737
$776$	0	0
$777$	3.00000	0.107624
$778$	12.0000	0.430221
$779$	7.00000	0.250801
$780$	0	0
$781$	−32.0000	−1.14505
$782$	12.0000	0.429119
$783$	1.00000	0.0357371
$784$	−6.00000	−0.214286
$785$	17.0000	0.606756
$786$	−8.00000	−0.285351
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−5.00000	−0.178118
$789$	−9.00000	−0.320408
$790$	−10.0000	−0.355784
$791$	−5.00000	−0.177780
$792$	−2.00000	−0.0710669
$793$	0	0
$794$	−22.0000	−0.780751
$795$	2.00000	0.0709327
$796$	−24.0000	−0.850657
$797$	−56.0000	−1.98362	−0.991811	−	0.127715i	$-0.959236\pi$
$797$	−56.0000	−1.98362	−0.991811	+	0.127715i	$0.959236\pi$
$798$	−1.00000	−0.0353996
$799$	3.00000	0.106132
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	10.0000	0.353112
$803$	20.0000	0.705785
$804$	12.0000	0.423207
$805$	4.00000	0.140981
$806$	0	0
$807$	18.0000	0.633630
$808$	18.0000	0.633238
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−1.00000	−0.0351364
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	1.00000	0.0350931
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	11.0000	0.385313
$816$	−3.00000	−0.105021
$817$	−9.00000	−0.314870
$818$	4.00000	0.139857
$819$	0	0
$820$	7.00000	0.244451
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−2.00000	−0.0697580
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−1.00000	−0.0348367
$825$	8.00000	0.278524
$826$	−3.00000	−0.104383
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	−4.00000	−0.139010
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	18.0000	0.623663
$834$	−14.0000	−0.484780
$835$	14.0000	0.484490
$836$	2.00000	0.0691714
$837$	4.00000	0.138260
$838$	35.0000	1.20905
$839$	−49.0000	−1.69167	−0.845834	−	0.533446i	$-0.820897\pi$
$839$	−49.0000	−1.69167	−0.845834	+	0.533446i	$0.820897\pi$
$840$	−1.00000	−0.0345033
$841$	1.00000	0.0344828
$842$	30.0000	1.03387
$843$	30.0000	1.03325
$844$	23.0000	0.791693
$845$	13.0000	0.447214
$846$	−1.00000	−0.0343807
$847$	−7.00000	−0.240523
$848$	−2.00000	−0.0686803
$849$	−10.0000	−0.343199
$850$	12.0000	0.411597
$851$	−12.0000	−0.411355
$852$	16.0000	0.548151
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	6.00000	0.205316
$855$	1.00000	0.0341993
$856$	17.0000	0.581048
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	−9.00000	−0.306897
$861$	−7.00000	−0.238559
$862$	12.0000	0.408722
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	1.00000	0.0340207
$865$	15.0000	0.510015
$866$	14.0000	0.475739
$867$	−8.00000	−0.271694
$868$	4.00000	0.135769
$869$	−20.0000	−0.678454
$870$	−1.00000	−0.0339032
$871$	0	0
$872$	−16.0000	−0.541828
$873$	0	0
$874$	4.00000	0.135302
$875$	9.00000	0.304256
$876$	−10.0000	−0.337869
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	−29.0000	−0.978703
$879$	−14.0000	−0.472208
$880$	2.00000	0.0674200
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	−6.00000	−0.202031
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	3.00000	0.100673
$889$	2.00000	0.0670778
$890$	−6.00000	−0.201120
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	1.00000	0.0334637
$894$	17.0000	0.568565
$895$	16.0000	0.534821
$896$	1.00000	0.0334077
$897$	0	0
$898$	9.00000	0.300334
$899$	4.00000	0.133407
$900$	−4.00000	−0.133333
$901$	6.00000	0.199889
$902$	14.0000	0.466149
$903$	9.00000	0.299501
$904$	−5.00000	−0.166298
$905$	0	0
$906$	5.00000	0.166114
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−3.00000	−0.0995585
$909$	18.0000	0.597022
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	3.00000	0.0992312
$915$	−6.00000	−0.198354
$916$	13.0000	0.429532
$917$	−8.00000	−0.264183
$918$	−3.00000	−0.0990148
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	4.00000	0.131876
$921$	28.0000	0.922631
$922$	0	0
$923$	0	0
$924$	−2.00000	−0.0657952
$925$	−12.0000	−0.394558
$926$	16.0000	0.525793
$927$	−1.00000	−0.0328443
$928$	1.00000	0.0328266
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−4.00000	−0.131165
$931$	6.00000	0.196642
$932$	−4.00000	−0.131024
$933$	−17.0000	−0.556555
$934$	−8.00000	−0.261768
$935$	−6.00000	−0.196221
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	12.0000	0.391814
$939$	−29.0000	−0.946379
$940$	1.00000	0.0326164
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−17.0000	−0.553890
$943$	28.0000	0.911805
$944$	−3.00000	−0.0976417
$945$	−1.00000	−0.0325300
$946$	−18.0000	−0.585230
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	10.0000	0.324785
$949$	0	0
$950$	4.00000	0.129777
$951$	24.0000	0.778253
$952$	−3.00000	−0.0972306
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	−2.00000	−0.0647524
$955$	−17.0000	−0.550107
$956$	2.00000	0.0646846
$957$	−2.00000	−0.0646508
$958$	32.0000	1.03387
$959$	−2.00000	−0.0645834
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	0	0
$963$	17.0000	0.547817
$964$	25.0000	0.805196
$965$	16.0000	0.515058
$966$	−4.00000	−0.128698
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−7.00000	−0.224989
$969$	3.00000	0.0963739
$970$	0	0
$971$	−22.0000	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$972$	1.00000	0.0320750
$973$	−14.0000	−0.448819
$974$	−37.0000	−1.18556
$975$	0	0
$976$	6.00000	0.192055
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−11.0000	−0.351741
$979$	−12.0000	−0.383522
$980$	6.00000	0.191663
$981$	−16.0000	−0.510841
$982$	2.00000	0.0638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−7.00000	−0.223152
$985$	5.00000	0.159313
$986$	−3.00000	−0.0955395
$987$	−1.00000	−0.0318304
$988$	0	0
$989$	−36.0000	−1.14473
$990$	2.00000	0.0635642
$991$	19.0000	0.603555	0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000	0.603555	0.301777	+	0.953378i	$0.402420\pi$
$992$	4.00000	0.127000
$993$	17.0000	0.539479
$994$	16.0000	0.507489
$995$	24.0000	0.760851
$996$	0	0
$997$	25.0000	0.791758	0.395879	−	0.918303i	$-0.370440\pi$
$997$	25.0000	0.791758	0.395879	+	0.918303i	$0.370440\pi$
$998$	38.0000	1.20287
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	174.2.a.e.1.1	✓	1
3.2	odd	2		522.2.a.d.1.1		1
4.3	odd	2		1392.2.a.e.1.1		1
5.2	odd	4		4350.2.e.l.349.2		2
5.3	odd	4		4350.2.e.l.349.1		2
5.4	even	2		4350.2.a.e.1.1		1
7.6	odd	2		8526.2.a.s.1.1		1
8.3	odd	2		5568.2.a.bc.1.1		1
8.5	even	2		5568.2.a.l.1.1		1
12.11	even	2		4176.2.a.w.1.1		1
29.28	even	2		5046.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
174.2.a.e.1.1	✓	1	1.1	even	1	trivial
522.2.a.d.1.1		1	3.2	odd	2
1392.2.a.e.1.1		1	4.3	odd	2
4176.2.a.w.1.1		1	12.11	even	2
4350.2.a.e.1.1		1	5.4	even	2
4350.2.e.l.349.1		2	5.3	odd	4
4350.2.e.l.349.2		2	5.2	odd	4
5046.2.a.a.1.1		1	29.28	even	2
5568.2.a.l.1.1		1	8.5	even	2
5568.2.a.bc.1.1		1	8.3	odd	2
8526.2.a.s.1.1		1	7.6	odd	2

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:	\( N \)	\(=\)	\( 174 = 2 \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	174.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more