LMFDB - Embedded newform 174.2.a.b.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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -5.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} +5.00000 q^{21} -6.00000 q^{22} -1.00000 q^{24} +4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +5.00000 q^{28} -1.00000 q^{29} +3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} -15.0000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} +3.00000 q^{40} -9.00000 q^{41} -5.00000 q^{42} -7.00000 q^{43} +6.00000 q^{44} -3.00000 q^{45} -3.00000 q^{47} +1.00000 q^{48} +18.0000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -18.0000 q^{55} -5.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} -3.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +5.00000 q^{63} +1.00000 q^{64} +12.0000 q^{65} -6.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} +15.0000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +1.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} +30.0000 q^{77} +4.00000 q^{78} +14.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +5.00000 q^{84} -9.00000 q^{85} +7.00000 q^{86} -1.00000 q^{87} -6.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -20.0000 q^{91} -4.00000 q^{93} +3.00000 q^{94} +3.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -18.0000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−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
$10$	3.00000	0.948683
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−5.00000	−1.33631
$15$	−3.00000	−0.774597
$16$	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
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−3.00000	−0.670820
$21$	5.00000	1.09109
$22$	−6.00000	−1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	5.00000	0.944911
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.00000	0.547723
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	−3.00000	−0.514496
$35$	−15.0000	−2.53546
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	1.00000	0.162221
$39$	−4.00000	−0.640513
$40$	3.00000	0.474342
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−5.00000	−0.771517
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	6.00000	0.904534
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−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
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	−18.0000	−2.42712
$56$	−5.00000	−0.668153
$57$	−1.00000	−0.132453
$58$	1.00000	0.131306
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−3.00000	−0.387298
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	5.00000	0.629941
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	−6.00000	−0.738549
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	15.0000	1.79284
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−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
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	30.0000	3.41882
$78$	4.00000	0.452911
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	5.00000	0.545545
$85$	−9.00000	−0.976187
$86$	7.00000	0.754829
$87$	−1.00000	−0.107211
$88$	−6.00000	−0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	3.00000	0.316228
$91$	−20.0000	−2.09657
$92$	0	0
$93$	−4.00000	−0.414781
$94$	3.00000	0.309426
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−18.0000	−1.81827
$99$	6.00000	0.603023
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−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	$0.721258\pi$
$104$	4.00000	0.392232
$105$	−15.0000	−1.46385
$106$	6.00000	0.582772
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	1.00000	0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	18.0000	1.71623
$111$	−1.00000	−0.0949158
$112$	5.00000	0.472456
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−4.00000	−0.369800
$118$	−3.00000	−0.276172
$119$	15.0000	1.37505
$120$	3.00000	0.273861
$121$	25.0000	2.27273
$122$	10.0000	0.905357
$123$	−9.00000	−0.811503
$124$	−4.00000	−0.359211
$125$	3.00000	0.268328
$126$	−5.00000	−0.445435
$127$	−22.0000	−1.95218	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000	−1.95218	−0.976092	+	0.217357i	$0.930256\pi$
$128$	−1.00000	−0.0883883
$129$	−7.00000	−0.616316
$130$	−12.0000	−1.05247
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000	0.522233
$133$	−5.00000	−0.433555
$134$	4.00000	0.345547
$135$	−3.00000	−0.258199
$136$	−3.00000	−0.257248
$137$	−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
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−15.0000	−1.26773
$141$	−3.00000	−0.252646
$142$	−12.0000	−1.00702
$143$	−24.0000	−2.00698
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	−2.00000	−0.165521
$147$	18.0000	1.48461
$148$	−1.00000	−0.0821995
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	−4.00000	−0.326599
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	1.00000	0.0811107
$153$	3.00000	0.242536
$154$	−30.0000	−2.41747
$155$	12.0000	0.963863
$156$	−4.00000	−0.320256
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−14.0000	−1.11378
$159$	−6.00000	−0.475831
$160$	3.00000	0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	−9.00000	−0.702782
$165$	−18.0000	−1.40130
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	−5.00000	−0.385758
$169$	3.00000	0.230769
$170$	9.00000	0.690268
$171$	−1.00000	−0.0764719
$172$	−7.00000	−0.533745
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	1.00000	0.0758098
$175$	20.0000	1.51186
$176$	6.00000	0.452267
$177$	3.00000	0.225494
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−3.00000	−0.223607
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	20.0000	1.48250
$183$	−10.0000	−0.739221
$184$	0	0
$185$	3.00000	0.220564
$186$	4.00000	0.293294
$187$	18.0000	1.31629
$188$	−3.00000	−0.218797
$189$	5.00000	0.363696
$190$	−3.00000	−0.217643
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−8.00000	−0.574367
$195$	12.0000	0.859338
$196$	18.0000	1.28571
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	−6.00000	−0.426401
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−4.00000	−0.282843
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	−5.00000	−0.350931
$204$	3.00000	0.210042
$205$	27.0000	1.88576
$206$	13.0000	0.905753
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−6.00000	−0.415029
$210$	15.0000	1.03510
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	−6.00000	−0.412082
$213$	12.0000	0.822226
$214$	9.00000	0.615227
$215$	21.0000	1.43219
$216$	−1.00000	−0.0680414
$217$	−20.0000	−1.35769
$218$	−8.00000	−0.541828
$219$	2.00000	0.135147
$220$	−18.0000	−1.21356
$221$	−12.0000	−0.807207
$222$	1.00000	0.0671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−5.00000	−0.334077
$225$	4.00000	0.266667
$226$	3.00000	0.199557
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−1.00000	−0.0662266
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	30.0000	1.97386
$232$	1.00000	0.0656532
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	4.00000	0.261488
$235$	9.00000	0.587095
$236$	3.00000	0.195283
$237$	14.0000	0.909398
$238$	−15.0000	−0.972306
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−3.00000	−0.193649
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	−25.0000	−1.60706
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	−54.0000	−3.44993
$246$	9.00000	0.573819
$247$	4.00000	0.254514
$248$	4.00000	0.254000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	5.00000	0.314970
$253$	0	0
$254$	22.0000	1.38040
$255$	−9.00000	−0.563602
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	7.00000	0.435801
$259$	−5.00000	−0.310685
$260$	12.0000	0.744208
$261$	−1.00000	−0.0618984
$262$	−12.0000	−0.741362
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	−6.00000	−0.369274
$265$	18.0000	1.10573
$266$	5.00000	0.306570
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	3.00000	0.182574
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	3.00000	0.181902
$273$	−20.0000	−1.21046
$274$	6.00000	0.362473
$275$	24.0000	1.44725
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−14.0000	−0.839664
$279$	−4.00000	−0.239474
$280$	15.0000	0.896421
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	3.00000	0.178647
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	12.0000	0.712069
$285$	3.00000	0.177705
$286$	24.0000	1.41915
$287$	−45.0000	−2.65627
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	−3.00000	−0.176166
$291$	8.00000	0.468968
$292$	2.00000	0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−18.0000	−1.04978
$295$	−9.00000	−0.524000
$296$	1.00000	0.0581238
$297$	6.00000	0.348155
$298$	−3.00000	−0.173785
$299$	0	0
$300$	4.00000	0.230940
$301$	−35.0000	−2.01737
$302$	−17.0000	−0.978240
$303$	−6.00000	−0.344691
$304$	−1.00000	−0.0573539
$305$	30.0000	1.71780
$306$	−3.00000	−0.171499
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	30.0000	1.70941
$309$	−13.0000	−0.739544
$310$	−12.0000	−0.681554
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	4.00000	0.226455
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	13.0000	0.733632
$315$	−15.0000	−0.845154
$316$	14.0000	0.787562
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	6.00000	0.336463
$319$	−6.00000	−0.335936
$320$	−3.00000	−0.167705
$321$	−9.00000	−0.502331
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	−16.0000	−0.887520
$326$	−5.00000	−0.276924
$327$	8.00000	0.442401
$328$	9.00000	0.496942
$329$	−15.0000	−0.826977
$330$	18.0000	0.990867
$331$	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
$333$	−1.00000	−0.0547997
$334$	−6.00000	−0.328305
$335$	12.0000	0.655630
$336$	5.00000	0.272772
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−3.00000	−0.163178
$339$	−3.00000	−0.162938
$340$	−9.00000	−0.488094
$341$	−24.0000	−1.29967
$342$	1.00000	0.0540738
$343$	55.0000	2.96972
$344$	7.00000	0.377415
$345$	0	0
$346$	21.0000	1.12897
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	−1.00000	−0.0536056
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−20.0000	−1.06904
$351$	−4.00000	−0.213504
$352$	−6.00000	−0.319801
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−3.00000	−0.159448
$355$	−36.0000	−1.91068
$356$	−6.00000	−0.317999
$357$	15.0000	0.793884
$358$	0	0
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	3.00000	0.158114
$361$	−18.0000	−0.947368
$362$	−8.00000	−0.420471
$363$	25.0000	1.31216
$364$	−20.0000	−1.04828
$365$	−6.00000	−0.314054
$366$	10.0000	0.522708
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	−3.00000	−0.155963
$371$	−30.0000	−1.55752
$372$	−4.00000	−0.207390
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−18.0000	−0.930758
$375$	3.00000	0.154919
$376$	3.00000	0.154713
$377$	4.00000	0.206010
$378$	−5.00000	−0.257172
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	3.00000	0.153897
$381$	−22.0000	−1.12709
$382$	21.0000	1.07445
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	−1.00000	−0.0510310
$385$	−90.0000	−4.58682
$386$	4.00000	0.203595
$387$	−7.00000	−0.355830
$388$	8.00000	0.406138
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	−12.0000	−0.607644
$391$	0	0
$392$	−18.0000	−0.909137
$393$	12.0000	0.605320
$394$	−9.00000	−0.453413
$395$	−42.0000	−2.11325
$396$	6.00000	0.301511
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	16.0000	0.802008
$399$	−5.00000	−0.250313
$400$	4.00000	0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000	0.199502
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	−3.00000	−0.149071
$406$	5.00000	0.248146
$407$	−6.00000	−0.297409
$408$	−3.00000	−0.148522
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	−27.0000	−1.33343
$411$	−6.00000	−0.295958
$412$	−13.0000	−0.640464
$413$	15.0000	0.738102
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	14.0000	0.685583
$418$	6.00000	0.293470
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	−15.0000	−0.731925
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	25.0000	1.21698
$423$	−3.00000	−0.145865
$424$	6.00000	0.291386
$425$	12.0000	0.582086
$426$	−12.0000	−0.581402
$427$	−50.0000	−2.41967
$428$	−9.00000	−0.435031
$429$	−24.0000	−1.15873
$430$	−21.0000	−1.01271
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	1.00000	0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	20.0000	0.960031
$435$	3.00000	0.143839
$436$	8.00000	0.383131
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	18.0000	0.858116
$441$	18.0000	0.857143
$442$	12.0000	0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−1.00000	−0.0474579
$445$	18.0000	0.853282
$446$	16.0000	0.757622
$447$	3.00000	0.141895
$448$	5.00000	0.236228
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	−4.00000	−0.188562
$451$	−54.0000	−2.54276
$452$	−3.00000	−0.141108
$453$	17.0000	0.798730
$454$	−3.00000	−0.140797
$455$	60.0000	2.81284
$456$	1.00000	0.0468293
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	7.00000	0.327089
$459$	3.00000	0.140028
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−30.0000	−1.39573
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−1.00000	−0.0464238
$465$	12.0000	0.556487
$466$	−24.0000	−1.11178
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−4.00000	−0.184900
$469$	−20.0000	−0.923514
$470$	−9.00000	−0.415139
$471$	−13.0000	−0.599008
$472$	−3.00000	−0.138086
$473$	−42.0000	−1.93116
$474$	−14.0000	−0.643041
$475$	−4.00000	−0.183533
$476$	15.0000	0.687524
$477$	−6.00000	−0.274721
$478$	−6.00000	−0.274434
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	3.00000	0.136931
$481$	4.00000	0.182384
$482$	−17.0000	−0.774329
$483$	0	0
$484$	25.0000	1.13636
$485$	−24.0000	−1.08978
$486$	−1.00000	−0.0453609
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	10.0000	0.452679
$489$	5.00000	0.226108
$490$	54.0000	2.43947
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−9.00000	−0.405751
$493$	−3.00000	−0.135113
$494$	−4.00000	−0.179969
$495$	−18.0000	−0.809040
$496$	−4.00000	−0.179605
$497$	60.0000	2.69137
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	3.00000	0.134164
$501$	6.00000	0.268060
$502$	0	0
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	−5.00000	−0.222718
$505$	18.0000	0.800989
$506$	0	0
$507$	3.00000	0.133235
$508$	−22.0000	−0.976092
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	9.00000	0.398527
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	39.0000	1.71855
$516$	−7.00000	−0.308158
$517$	−18.0000	−0.791639
$518$	5.00000	0.219687
$519$	−21.0000	−0.921798
$520$	−12.0000	−0.526235
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	1.00000	0.0437688
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	12.0000	0.524222
$525$	20.0000	0.872872
$526$	3.00000	0.130806
$527$	−12.0000	−0.522728
$528$	6.00000	0.261116
$529$	−23.0000	−1.00000
$530$	−18.0000	−0.781870
$531$	3.00000	0.130189
$532$	−5.00000	−0.216777
$533$	36.0000	1.55933
$534$	6.00000	0.259645
$535$	27.0000	1.16731
$536$	4.00000	0.172774
$537$	0	0
$538$	−6.00000	−0.258678
$539$	108.000	4.65189
$540$	−3.00000	−0.129099
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	−20.0000	−0.859074
$543$	8.00000	0.343313
$544$	−3.00000	−0.128624
$545$	−24.0000	−1.02805
$546$	20.0000	0.855921
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	−24.0000	−1.02336
$551$	1.00000	0.0426014
$552$	0	0
$553$	70.0000	2.97670
$554$	−14.0000	−0.594803
$555$	3.00000	0.127343
$556$	14.0000	0.593732
$557$	45.0000	1.90671	0.953356	−	0.301849i	$-0.0976040\pi$
$557$	45.0000	1.90671	0.953356	+	0.301849i	$0.0976040\pi$
$558$	4.00000	0.169334
$559$	28.0000	1.18427
$560$	−15.0000	−0.633866
$561$	18.0000	0.759961
$562$	6.00000	0.253095
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	−3.00000	−0.126323
$565$	9.00000	0.378633
$566$	−14.0000	−0.588464
$567$	5.00000	0.209980
$568$	−12.0000	−0.503509
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	−3.00000	−0.125656
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−24.0000	−1.00349
$573$	−21.0000	−0.877288
$574$	45.0000	1.87826
$575$	0	0
$576$	1.00000	0.0416667
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	8.00000	0.332756
$579$	−4.00000	−0.166234
$580$	3.00000	0.124568
$581$	0	0
$582$	−8.00000	−0.331611
$583$	−36.0000	−1.49097
$584$	−2.00000	−0.0827606
$585$	12.0000	0.496139
$586$	6.00000	0.247858
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	18.0000	0.742307
$589$	4.00000	0.164817
$590$	9.00000	0.370524
$591$	9.00000	0.370211
$592$	−1.00000	−0.0410997
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	−6.00000	−0.246183
$595$	−45.0000	−1.84482
$596$	3.00000	0.122885
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	−4.00000	−0.163299
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	35.0000	1.42649
$603$	−4.00000	−0.162893
$604$	17.0000	0.691720
$605$	−75.0000	−3.04918
$606$	6.00000	0.243733
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	1.00000	0.0405554
$609$	−5.00000	−0.202610
$610$	−30.0000	−1.21466
$611$	12.0000	0.485468
$612$	3.00000	0.121268
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−20.0000	−0.807134
$615$	27.0000	1.08875
$616$	−30.0000	−1.20873
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	13.0000	0.522937
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	−21.0000	−0.842023
$623$	−30.0000	−1.20192
$624$	−4.00000	−0.160128
$625$	−29.0000	−1.16000
$626$	−11.0000	−0.439648
$627$	−6.00000	−0.239617
$628$	−13.0000	−0.518756
$629$	−3.00000	−0.119618
$630$	15.0000	0.597614
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	−14.0000	−0.556890
$633$	−25.0000	−0.993661
$634$	0	0
$635$	66.0000	2.61913
$636$	−6.00000	−0.237915
$637$	−72.0000	−2.85274
$638$	6.00000	0.237542
$639$	12.0000	0.474713
$640$	3.00000	0.118585
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	9.00000	0.355202
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	21.0000	0.826874
$646$	3.00000	0.118033
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	16.0000	0.627572
$651$	−20.0000	−0.783862
$652$	5.00000	0.195815
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−8.00000	−0.312825
$655$	−36.0000	−1.40664
$656$	−9.00000	−0.351391
$657$	2.00000	0.0780274
$658$	15.0000	0.584761
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−18.0000	−0.700649
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−17.0000	−0.660724
$663$	−12.0000	−0.466041
$664$	0	0
$665$	15.0000	0.581675
$666$	1.00000	0.0387492
$667$	0	0
$668$	6.00000	0.232147
$669$	−16.0000	−0.618596
$670$	−12.0000	−0.463600
$671$	−60.0000	−2.31627
$672$	−5.00000	−0.192879
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	10.0000	0.385186
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	3.00000	0.115214
$679$	40.0000	1.53506
$680$	9.00000	0.345134
$681$	3.00000	0.114960
$682$	24.0000	0.919007
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−1.00000	−0.0382360
$685$	18.0000	0.687745
$686$	−55.0000	−2.09991
$687$	−7.00000	−0.267067
$688$	−7.00000	−0.266872
$689$	24.0000	0.914327
$690$	0	0
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	−21.0000	−0.798300
$693$	30.0000	1.13961
$694$	3.00000	0.113878
$695$	−42.0000	−1.59315
$696$	1.00000	0.0379049
$697$	−27.0000	−1.02270
$698$	−2.00000	−0.0757011
$699$	24.0000	0.907763
$700$	20.0000	0.755929
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	4.00000	0.150970
$703$	1.00000	0.0377157
$704$	6.00000	0.226134
$705$	9.00000	0.338960
$706$	30.0000	1.12906
$707$	−30.0000	−1.12827
$708$	3.00000	0.112747
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	36.0000	1.35106
$711$	14.0000	0.525041
$712$	6.00000	0.224860
$713$	0	0
$714$	−15.0000	−0.561361
$715$	72.0000	2.69265
$716$	0	0
$717$	6.00000	0.224074
$718$	−9.00000	−0.335877
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	−3.00000	−0.111803
$721$	−65.0000	−2.42073
$722$	18.0000	0.669891
$723$	17.0000	0.632237
$724$	8.00000	0.297318
$725$	−4.00000	−0.148556
$726$	−25.0000	−0.927837
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	20.0000	0.741249
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	−21.0000	−0.776713
$732$	−10.0000	−0.369611
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	−26.0000	−0.959678
$735$	−54.0000	−1.99182
$736$	0	0
$737$	−24.0000	−0.884051
$738$	9.00000	0.331295
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	3.00000	0.110282
$741$	4.00000	0.146944
$742$	30.0000	1.10133
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	4.00000	0.146647
$745$	−9.00000	−0.329734
$746$	−20.0000	−0.732252
$747$	0	0
$748$	18.0000	0.658145
$749$	−45.0000	−1.64426
$750$	−3.00000	−0.109545
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	−4.00000	−0.145671
$755$	−51.0000	−1.85608
$756$	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	$0.540602\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−3.00000	−0.108821
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	22.0000	0.796976
$763$	40.0000	1.44810
$764$	−21.0000	−0.759753
$765$	−9.00000	−0.325396
$766$	6.00000	0.216789
$767$	−12.0000	−0.433295
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	90.0000	3.24337
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	7.00000	0.251610
$775$	−16.0000	−0.574737
$776$	−8.00000	−0.287183
$777$	−5.00000	−0.179374
$778$	12.0000	0.430221
$779$	9.00000	0.322458
$780$	12.0000	0.429669
$781$	72.0000	2.57636
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	18.0000	0.642857
$785$	39.0000	1.39197
$786$	−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	$0.306812\pi$
$788$	9.00000	0.320612
$789$	−3.00000	−0.106803
$790$	42.0000	1.49429
$791$	−15.0000	−0.533339
$792$	−6.00000	−0.213201
$793$	40.0000	1.42044
$794$	−26.0000	−0.922705
$795$	18.0000	0.638394
$796$	−16.0000	−0.567105
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	5.00000	0.176998
$799$	−9.00000	−0.318397
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	−30.0000	−1.05934
$803$	12.0000	0.423471
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	6.00000	0.211210
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	3.00000	0.105409
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−5.00000	−0.175466
$813$	20.0000	0.701431
$814$	6.00000	0.210300
$815$	−15.0000	−0.525427
$816$	3.00000	0.105021
$817$	7.00000	0.244899
$818$	−8.00000	−0.279713
$819$	−20.0000	−0.698857
$820$	27.0000	0.942881
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000	0.209274
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	13.0000	0.452876
$825$	24.0000	0.835573
$826$	−15.0000	−0.521917
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−49.0000	−1.70184	−0.850920	−	0.525295i	$-0.823955\pi$
$829$	−49.0000	−1.70184	−0.850920	+	0.525295i	$0.823955\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	−4.00000	−0.138675
$833$	54.0000	1.87099
$834$	−14.0000	−0.484780
$835$	−18.0000	−0.622916
$836$	−6.00000	−0.207514
$837$	−4.00000	−0.138260
$838$	−21.0000	−0.725433
$839$	−3.00000	−0.103572	−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000	−0.103572	−0.0517858	+	0.998658i	$0.516491\pi$
$840$	15.0000	0.517549
$841$	1.00000	0.0344828
$842$	−14.0000	−0.482472
$843$	−6.00000	−0.206651
$844$	−25.0000	−0.860535
$845$	−9.00000	−0.309609
$846$	3.00000	0.103142
$847$	125.000	4.29505
$848$	−6.00000	−0.206041
$849$	14.0000	0.480479
$850$	−12.0000	−0.411597
$851$	0	0
$852$	12.0000	0.411113
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	50.0000	1.71096
$855$	3.00000	0.102598
$856$	9.00000	0.307614
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	24.0000	0.819346
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	21.0000	0.716094
$861$	−45.0000	−1.53360
$862$	−36.0000	−1.22616
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	63.0000	2.14206
$866$	−26.0000	−0.883516
$867$	−8.00000	−0.271694
$868$	−20.0000	−0.678844
$869$	84.0000	2.84950
$870$	−3.00000	−0.101710
$871$	16.0000	0.542139
$872$	−8.00000	−0.270914
$873$	8.00000	0.270759
$874$	0	0
$875$	15.0000	0.507093
$876$	2.00000	0.0675737
$877$	−16.0000	−0.540282	−0.270141	−	0.962821i	$-0.587070\pi$
$877$	−16.0000	−0.540282	−0.270141	+	0.962821i	$0.587070\pi$
$878$	1.00000	0.0337484
$879$	−6.00000	−0.202375
$880$	−18.0000	−0.606780
$881$	−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
$883$	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
$885$	−9.00000	−0.302532
$886$	−12.0000	−0.403148
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	1.00000	0.0335578
$889$	−110.000	−3.68928
$890$	−18.0000	−0.603361
$891$	6.00000	0.201008
$892$	−16.0000	−0.535720
$893$	3.00000	0.100391
$894$	−3.00000	−0.100335
$895$	0	0
$896$	−5.00000	−0.167038
$897$	0	0
$898$	−15.0000	−0.500556
$899$	4.00000	0.133407
$900$	4.00000	0.133333
$901$	−18.0000	−0.599667
$902$	54.0000	1.79800
$903$	−35.0000	−1.16473
$904$	3.00000	0.0997785
$905$	−24.0000	−0.797787
$906$	−17.0000	−0.564787
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	3.00000	0.0995585
$909$	−6.00000	−0.199007
$910$	−60.0000	−1.98898
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	−11.0000	−0.363848
$915$	30.0000	0.991769
$916$	−7.00000	−0.231287
$917$	60.0000	1.98137
$918$	−3.00000	−0.0990148
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−12.0000	−0.395199
$923$	−48.0000	−1.57994
$924$	30.0000	0.986928
$925$	−4.00000	−0.131519
$926$	16.0000	0.525793
$927$	−13.0000	−0.426976
$928$	1.00000	0.0328266
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−12.0000	−0.393496
$931$	−18.0000	−0.589926
$932$	24.0000	0.786146
$933$	21.0000	0.687509
$934$	12.0000	0.392652
$935$	−54.0000	−1.76599
$936$	4.00000	0.130744
$937$	−49.0000	−1.60076	−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000	−1.60076	−0.800380	+	0.599493i	$0.795369\pi$
$938$	20.0000	0.653023
$939$	11.0000	0.358971
$940$	9.00000	0.293548
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	13.0000	0.423563
$943$	0	0
$944$	3.00000	0.0976417
$945$	−15.0000	−0.487950
$946$	42.0000	1.36554
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	14.0000	0.454699
$949$	−8.00000	−0.259691
$950$	4.00000	0.129777
$951$	0	0
$952$	−15.0000	−0.486153
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	6.00000	0.194257
$955$	63.0000	2.03863
$956$	6.00000	0.194054
$957$	−6.00000	−0.193952
$958$	−24.0000	−0.775405
$959$	−30.0000	−0.968751
$960$	−3.00000	−0.0968246
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	−9.00000	−0.290021
$964$	17.0000	0.547533
$965$	12.0000	0.386294
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−25.0000	−0.803530
$969$	−3.00000	−0.0963739
$970$	24.0000	0.770594
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	1.00000	0.0320750
$973$	70.0000	2.24410
$974$	−23.0000	−0.736968
$975$	−16.0000	−0.512410
$976$	−10.0000	−0.320092
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−5.00000	−0.159882
$979$	−36.0000	−1.15056
$980$	−54.0000	−1.72497
$981$	8.00000	0.255420
$982$	30.0000	0.957338
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	9.00000	0.286910
$985$	−27.0000	−0.860292
$986$	3.00000	0.0955395
$987$	−15.0000	−0.477455
$988$	4.00000	0.127257
$989$	0	0
$990$	18.0000	0.572078
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	4.00000	0.127000
$993$	17.0000	0.539479
$994$	−60.0000	−1.90308
$995$	48.0000	1.52170
$996$	0	0
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	22.0000	0.696398
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	174.2.a.b.1.1	✓	1
3.2	odd	2		522.2.a.m.1.1		1
4.3	odd	2		1392.2.a.b.1.1		1
5.2	odd	4		4350.2.e.j.349.1		2
5.3	odd	4		4350.2.e.j.349.2		2
5.4	even	2		4350.2.a.o.1.1		1
7.6	odd	2		8526.2.a.f.1.1		1
8.3	odd	2		5568.2.a.bh.1.1		1
8.5	even	2		5568.2.a.q.1.1		1
12.11	even	2		4176.2.a.bd.1.1		1
29.28	even	2		5046.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
174.2.a.b.1.1	✓	1	1.1	even	1	trivial
522.2.a.m.1.1		1	3.2	odd	2
1392.2.a.b.1.1		1	4.3	odd	2
4176.2.a.bd.1.1		1	12.11	even	2
4350.2.a.o.1.1		1	5.4	even	2
4350.2.e.j.349.1		2	5.2	odd	4
4350.2.e.j.349.2		2	5.3	odd	4
5046.2.a.i.1.1		1	29.28	even	2
5568.2.a.q.1.1		1	8.5	even	2
5568.2.a.bh.1.1		1	8.3	odd	2
8526.2.a.f.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

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