LMFDB - Embedded newform 370.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [370,2,Mod(1,370)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("370.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.95446487479$
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	$\chi$	$=$	370.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -6.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +1.00000 q^{72} +2.00000 q^{73} +1.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} -12.0000 q^{87} -6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +20.0000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -2.00000 q^{96} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	2.00000	0.534522
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	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	$0.426318\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	4.00000	0.769800
$28$	2.00000	0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−12.0000	−1.68034
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	2.00000	0.267261
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−2.00000	−0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−10.0000	−1.27000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	1.00000	0.116248
$75$	−2.00000	−0.230940
$76$	2.00000	0.229416
$77$	0	0
$78$	−4.00000	−0.452911
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−4.00000	−0.436436
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	−12.0000	−1.28654
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	4.00000	0.419314
$92$	0	0
$93$	20.0000	2.07390
$94$	−6.00000	−0.618853
$95$	2.00000	0.205196
$96$	−2.00000	−0.204124
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−12.0000	−1.18818
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.00000	0.196116
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	2.00000	0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	12.0000	1.10004
$120$	−2.00000	−0.182574
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	12.0000	1.08200
$124$	−10.0000	−0.898027
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	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
$129$	8.00000	0.704361
$130$	2.00000	0.175412
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	2.00000	0.172774
$135$	4.00000	0.344265
$136$	6.00000	0.514496
$137$	−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
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	2.00000	0.169031
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	2.00000	0.165521
$147$	6.00000	0.494872
$148$	1.00000	0.0821995
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−2.00000	−0.163299
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	2.00000	0.162221
$153$	6.00000	0.485071
$154$	0	0
$155$	−10.0000	−0.803219
$156$	−4.00000	−0.320256
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−10.0000	−0.795557
$159$	−12.0000	−0.951662
$160$	1.00000	0.0790569
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	2.00000	0.152944
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−12.0000	−0.909718
$175$	2.00000	0.151186
$176$	0	0
$177$	12.0000	0.901975
$178$	−6.00000	−0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	1.00000	0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000	0.296500
$183$	20.0000	1.47844
$184$	0	0
$185$	1.00000	0.0735215
$186$	20.0000	1.46647
$187$	0	0
$188$	−6.00000	−0.437595
$189$	8.00000	0.581914
$190$	2.00000	0.145095
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−2.00000	−0.144338
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	−4.00000	−0.286446
$196$	−3.00000	−0.214286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	18.0000	1.26648
$203$	12.0000	0.842235
$204$	−12.0000	−0.840168
$205$	−6.00000	−0.419058
$206$	−4.00000	−0.278693
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	−4.00000	−0.276026
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	6.00000	0.410152
$215$	−4.00000	−0.272798
$216$	4.00000	0.272166
$217$	−20.0000	−1.35769
$218$	14.0000	0.948200
$219$	−4.00000	−0.270295
$220$	0	0
$221$	12.0000	0.807207
$222$	−2.00000	−0.134231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−4.00000	−0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	2.00000	0.130744
$235$	−6.00000	−0.391397
$236$	−6.00000	−0.390567
$237$	20.0000	1.29914
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−2.00000	−0.129099
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−11.0000	−0.707107
$243$	10.0000	0.641500
$244$	−10.0000	−0.640184
$245$	−3.00000	−0.191663
$246$	12.0000	0.765092
$247$	4.00000	0.254514
$248$	−10.0000	−0.635001
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	2.00000	0.125491
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	8.00000	0.498058
$259$	2.00000	0.124274
$260$	2.00000	0.124035
$261$	6.00000	0.371391
$262$	−6.00000	−0.370681
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	4.00000	0.245256
$267$	12.0000	0.734388
$268$	2.00000	0.122169
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	4.00000	0.243432
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	6.00000	0.363803
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−4.00000	−0.239904
$279$	−10.0000	−0.598684
$280$	2.00000	0.119523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	12.0000	0.714590
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	−4.00000	−0.234484
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	6.00000	0.349927
$295$	−6.00000	−0.349334
$296$	1.00000	0.0581238
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	−2.00000	−0.115470
$301$	−8.00000	−0.461112
$302$	20.0000	1.15087
$303$	−36.0000	−2.06815
$304$	2.00000	0.114708
$305$	−10.0000	−0.572598
$306$	6.00000	0.342997
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−10.0000	−0.567962
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−10.0000	−0.564333
$315$	2.00000	0.112687
$316$	−10.0000	−0.562544
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	12.0000	0.667698
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	−16.0000	−0.886158
$327$	−28.0000	−1.54840
$328$	−6.00000	−0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−6.00000	−0.329293
$333$	1.00000	0.0547997
$334$	12.0000	0.656611
$335$	2.00000	0.109272
$336$	−4.00000	−0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	−12.0000	−0.651751
$340$	6.00000	0.325396
$341$	0	0
$342$	2.00000	0.108148
$343$	−20.0000	−1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−12.0000	−0.643268
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	2.00000	0.106904
$351$	8.00000	0.427008
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−24.0000	−1.27021
$358$	6.00000	0.317110
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	−22.0000	−1.15629
$363$	22.0000	1.15470
$364$	4.00000	0.209657
$365$	2.00000	0.104685
$366$	20.0000	1.04542
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	1.00000	0.0519875
$371$	12.0000	0.623009
$372$	20.0000	1.03695
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	−6.00000	−0.309426
$377$	12.0000	0.618031
$378$	8.00000	0.411476
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	2.00000	0.102598
$381$	−4.00000	−0.204926
$382$	−6.00000	−0.306987
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	−3.00000	−0.151523
$393$	12.0000	0.605320
$394$	−18.0000	−0.906827
$395$	−10.0000	−0.503155
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−22.0000	−1.10276
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	−20.0000	−0.996271
$404$	18.0000	0.895533
$405$	−11.0000	−0.546594
$406$	12.0000	0.595550
$407$	0	0
$408$	−12.0000	−0.594089
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	−6.00000	−0.296319
$411$	12.0000	0.591916
$412$	−4.00000	−0.197066
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−6.00000	−0.294528
$416$	2.00000	0.0980581
$417$	8.00000	0.391762
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	−4.00000	−0.195180
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000	0.973585
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	−20.0000	−0.967868
$428$	6.00000	0.290021
$429$	0	0
$430$	−4.00000	−0.192897
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	4.00000	0.192450
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−20.0000	−0.960031
$435$	−12.0000	−0.575356
$436$	14.0000	0.670478
$437$	0	0
$438$	−4.00000	−0.191127
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−2.00000	−0.0949158
$445$	−6.00000	−0.284427
$446$	14.0000	0.662919
$447$	−36.0000	−1.70274
$448$	2.00000	0.0944911
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	6.00000	0.282216
$453$	−40.0000	−1.87936
$454$	24.0000	1.12638
$455$	4.00000	0.187523
$456$	−4.00000	−0.187317
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−10.0000	−0.467269
$459$	24.0000	1.12022
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	6.00000	0.278543
$465$	20.0000	0.927478
$466$	18.0000	0.833834
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	−6.00000	−0.276759
$471$	20.0000	0.921551
$472$	−6.00000	−0.276172
$473$	0	0
$474$	20.0000	0.918630
$475$	2.00000	0.0917663
$476$	12.0000	0.550019
$477$	6.00000	0.274721
$478$	6.00000	0.274434
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−2.00000	−0.0912871
$481$	2.00000	0.0911922
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−11.0000	−0.500000
$485$	2.00000	0.0908153
$486$	10.0000	0.453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	32.0000	1.44709
$490$	−3.00000	−0.135526
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	12.0000	0.541002
$493$	36.0000	1.62136
$494$	4.00000	0.179969
$495$	0	0
$496$	−10.0000	−0.449013
$497$	0	0
$498$	12.0000	0.537733
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	1.00000	0.0447214
$501$	−24.0000	−1.07224
$502$	18.0000	0.803379
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	2.00000	0.0890871
$505$	18.0000	0.800989
$506$	0	0
$507$	18.0000	0.799408
$508$	2.00000	0.0887357
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−12.0000	−0.531369
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	8.00000	0.353209
$514$	6.00000	0.264649
$515$	−4.00000	−0.176261
$516$	8.00000	0.352180
$517$	0	0
$518$	2.00000	0.0878750
$519$	36.0000	1.58022
$520$	2.00000	0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	6.00000	0.262613
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−6.00000	−0.262111
$525$	−4.00000	−0.174574
$526$	−6.00000	−0.261612
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	−6.00000	−0.260378
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	12.0000	0.519291
$535$	6.00000	0.259403
$536$	2.00000	0.0863868
$537$	−12.0000	−0.517838
$538$	−6.00000	−0.258678
$539$	0	0
$540$	4.00000	0.172133
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−16.0000	−0.687259
$543$	44.0000	1.88822
$544$	6.00000	0.257248
$545$	14.0000	0.599694
$546$	−8.00000	−0.342368
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−20.0000	−0.850487
$554$	26.0000	1.10463
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−10.0000	−0.423334
$559$	−8.00000	−0.338364
$560$	2.00000	0.0845154
$561$	0	0
$562$	18.0000	0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	12.0000	0.505291
$565$	6.00000	0.252422
$566$	−16.0000	−0.672530
$567$	−22.0000	−0.923913
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−4.00000	−0.167542
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	19.0000	0.790296
$579$	−4.00000	−0.166234
$580$	6.00000	0.249136
$581$	−12.0000	−0.497844
$582$	−4.00000	−0.165805
$583$	0	0
$584$	2.00000	0.0827606
$585$	2.00000	0.0826898
$586$	6.00000	0.247858
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	6.00000	0.247436
$589$	−20.0000	−0.824086
$590$	−6.00000	−0.247016
$591$	36.0000	1.48084
$592$	1.00000	0.0410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	18.0000	0.737309
$597$	44.0000	1.80080
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	−2.00000	−0.0816497
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	−8.00000	−0.326056
$603$	2.00000	0.0814463
$604$	20.0000	0.813788
$605$	−11.0000	−0.447214
$606$	−36.0000	−1.46240
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	−24.0000	−0.972529
$610$	−10.0000	−0.404888
$611$	−12.0000	−0.485468
$612$	6.00000	0.242536
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	26.0000	1.04927
$615$	12.0000	0.483887
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	8.00000	0.321807
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−10.0000	−0.401610
$621$	0	0
$622$	−18.0000	−0.721734
$623$	−12.0000	−0.480770
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−10.0000	−0.399043
$629$	6.00000	0.239236
$630$	2.00000	0.0796819
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−10.0000	−0.397779
$633$	−40.0000	−1.58986
$634$	−18.0000	−0.714871
$635$	2.00000	0.0793676
$636$	−12.0000	−0.475831
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−12.0000	−0.473602
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	12.0000	0.472134
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	2.00000	0.0784465
$651$	40.0000	1.56772
$652$	−16.0000	−0.626608
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−28.0000	−1.09489
$655$	−6.00000	−0.234439
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	−12.0000	−0.467809
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−10.0000	−0.388661
$663$	−24.0000	−0.932083
$664$	−6.00000	−0.232845
$665$	4.00000	0.155113
$666$	1.00000	0.0387492
$667$	0	0
$668$	12.0000	0.464294
$669$	−28.0000	−1.08254
$670$	2.00000	0.0772667
$671$	0	0
$672$	−4.00000	−0.154303
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	2.00000	0.0770371
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−12.0000	−0.460857
$679$	4.00000	0.153506
$680$	6.00000	0.230089
$681$	−48.0000	−1.83936
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	2.00000	0.0764719
$685$	−6.00000	−0.229248
$686$	−20.0000	−0.763604
$687$	20.0000	0.763048
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−4.00000	−0.151729
$696$	−12.0000	−0.454859
$697$	−36.0000	−1.36360
$698$	26.0000	0.984115
$699$	−36.0000	−1.36165
$700$	2.00000	0.0755929
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	8.00000	0.301941
$703$	2.00000	0.0754314
$704$	0	0
$705$	12.0000	0.451946
$706$	−30.0000	−1.12906
$707$	36.0000	1.35392
$708$	12.0000	0.450988
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	0	0
$714$	−24.0000	−0.898177
$715$	0	0
$716$	6.00000	0.224231
$717$	−12.0000	−0.448148
$718$	36.0000	1.34351
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	1.00000	0.0372678
$721$	−8.00000	−0.297936
$722$	−15.0000	−0.558242
$723$	44.0000	1.63638
$724$	−22.0000	−0.817624
$725$	6.00000	0.222834
$726$	22.0000	0.816497
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	4.00000	0.148250
$729$	13.0000	0.481481
$730$	2.00000	0.0740233
$731$	−24.0000	−0.887672
$732$	20.0000	0.739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	2.00000	0.0738213
$735$	6.00000	0.221313
$736$	0	0
$737$	0	0
$738$	−6.00000	−0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	1.00000	0.0367607
$741$	−8.00000	−0.293887
$742$	12.0000	0.440534
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	20.0000	0.733236
$745$	18.0000	0.659469
$746$	−34.0000	−1.24483
$747$	−6.00000	−0.219529
$748$	0	0
$749$	12.0000	0.438470
$750$	−2.00000	−0.0730297
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−6.00000	−0.218797
$753$	−36.0000	−1.31191
$754$	12.0000	0.437014
$755$	20.0000	0.727875
$756$	8.00000	0.290957
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	2.00000	0.0725476
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−4.00000	−0.144905
$763$	28.0000	1.01367
$764$	−6.00000	−0.217072
$765$	6.00000	0.216930
$766$	0	0
$767$	−12.0000	−0.433295
$768$	−2.00000	−0.0721688
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	2.00000	0.0719816
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000	−0.143777
$775$	−10.0000	−0.359211
$776$	2.00000	0.0717958
$777$	−4.00000	−0.143499
$778$	6.00000	0.215110
$779$	−12.0000	−0.429945
$780$	−4.00000	−0.143223
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	−3.00000	−0.107143
$785$	−10.0000	−0.356915
$786$	12.0000	0.428026
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	−18.0000	−0.641223
$789$	12.0000	0.427211
$790$	−10.0000	−0.355784
$791$	12.0000	0.426671
$792$	0	0
$793$	−20.0000	−0.710221
$794$	14.0000	0.496841
$795$	−12.0000	−0.425596
$796$	−22.0000	−0.779769
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−8.00000	−0.283197
$799$	−36.0000	−1.27359
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	−6.00000	−0.211867
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−20.0000	−0.704470
$807$	12.0000	0.422420
$808$	18.0000	0.633238
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−11.0000	−0.386501
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	12.0000	0.421117
$813$	32.0000	1.12229
$814$	0	0
$815$	−16.0000	−0.560456
$816$	−12.0000	−0.420084
$817$	−8.00000	−0.279885
$818$	2.00000	0.0699284
$819$	4.00000	0.139771
$820$	−6.00000	−0.209529
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	12.0000	0.418548
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−6.00000	−0.208263
$831$	−52.0000	−1.80386
$832$	2.00000	0.0693375
$833$	−18.0000	−0.623663
$834$	8.00000	0.277017
$835$	12.0000	0.415277
$836$	0	0
$837$	−40.0000	−1.38260
$838$	24.0000	0.829066
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−4.00000	−0.138013
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	−36.0000	−1.23991
$844$	20.0000	0.688428
$845$	−9.00000	−0.309609
$846$	−6.00000	−0.206284
$847$	−22.0000	−0.755929
$848$	6.00000	0.206041
$849$	32.0000	1.09824
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−20.0000	−0.684386
$855$	2.00000	0.0683986
$856$	6.00000	0.205076
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	−4.00000	−0.136399
$861$	24.0000	0.817918
$862$	30.0000	1.02180
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	4.00000	0.136083
$865$	−18.0000	−0.612018
$866$	2.00000	0.0679628
$867$	−38.0000	−1.29055
$868$	−20.0000	−0.678844
$869$	0	0
$870$	−12.0000	−0.406838
$871$	4.00000	0.135535
$872$	14.0000	0.474100
$873$	2.00000	0.0676897
$874$	0	0
$875$	2.00000	0.0676123
$876$	−4.00000	−0.135147
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−22.0000	−0.742464
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−3.00000	−0.101015
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	12.0000	0.403604
$885$	12.0000	0.403376
$886$	6.00000	0.201574
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	−2.00000	−0.0671156
$889$	4.00000	0.134156
$890$	−6.00000	−0.201120
$891$	0	0
$892$	14.0000	0.468755
$893$	−12.0000	−0.401565
$894$	−36.0000	−1.20402
$895$	6.00000	0.200558
$896$	2.00000	0.0668153
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−60.0000	−2.00111
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	0	0
$903$	16.0000	0.532447
$904$	6.00000	0.199557
$905$	−22.0000	−0.731305
$906$	−40.0000	−1.32891
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	24.0000	0.796468
$909$	18.0000	0.597022
$910$	4.00000	0.132599
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	26.0000	0.860004
$915$	20.0000	0.661180
$916$	−10.0000	−0.330409
$917$	−12.0000	−0.396275
$918$	24.0000	0.792118
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	0	0
$921$	−52.0000	−1.71346
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	1.00000	0.0328798
$926$	−40.0000	−1.31448
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	20.0000	0.655826
$931$	−6.00000	−0.196642
$932$	18.0000	0.589610
$933$	36.0000	1.17859
$934$	0	0
$935$	0	0
$936$	2.00000	0.0653720
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	4.00000	0.130605
$939$	44.0000	1.43589
$940$	−6.00000	−0.195698
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	20.0000	0.651635
$943$	0	0
$944$	−6.00000	−0.195283
$945$	8.00000	0.260240
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	20.0000	0.649570
$949$	4.00000	0.129845
$950$	2.00000	0.0648886
$951$	36.0000	1.16738
$952$	12.0000	0.388922
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	6.00000	0.194257
$955$	−6.00000	−0.194155
$956$	6.00000	0.194054
$957$	0	0
$958$	−18.0000	−0.581554
$959$	−12.0000	−0.387500
$960$	−2.00000	−0.0645497
$961$	69.0000	2.22581
$962$	2.00000	0.0644826
$963$	6.00000	0.193347
$964$	−22.0000	−0.708572
$965$	2.00000	0.0643823
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−11.0000	−0.353553
$969$	−24.0000	−0.770991
$970$	2.00000	0.0642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	10.0000	0.320750
$973$	−8.00000	−0.256468
$974$	20.0000	0.640841
$975$	−4.00000	−0.128103
$976$	−10.0000	−0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	32.0000	1.02325
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	14.0000	0.446986
$982$	36.0000	1.14881
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	12.0000	0.382546
$985$	−18.0000	−0.573528
$986$	36.0000	1.14647
$987$	24.0000	0.763928
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	−10.0000	−0.317500
$993$	20.0000	0.634681
$994$	0	0
$995$	−22.0000	−0.697447
$996$	12.0000	0.380235
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	14.0000	0.443162
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.a.d.1.1	✓	1
3.2	odd	2		3330.2.a.d.1.1		1
4.3	odd	2		2960.2.a.m.1.1		1
5.2	odd	4		1850.2.b.b.149.2		2
5.3	odd	4		1850.2.b.b.149.1		2
5.4	even	2		1850.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.d.1.1	✓	1	1.1	even	1	trivial
1850.2.a.f.1.1		1	5.4	even	2
1850.2.b.b.149.1		2	5.3	odd	4
1850.2.b.b.149.2		2	5.2	odd	4
2960.2.a.m.1.1		1	4.3	odd	2
3330.2.a.d.1.1		1	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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