LMFDB - Embedded newform 170.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [170,2,Mod(1,170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(170, 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("170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 170 = 2 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	170.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:	$1.35745683436$
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	$\chi$	$=$	170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} -5.00000 q^{27} +2.00000 q^{28} -3.00000 q^{29} -1.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{42} -10.0000 q^{43} +2.00000 q^{45} -6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -3.00000 q^{53} -5.00000 q^{54} +2.00000 q^{56} -1.00000 q^{57} -3.00000 q^{58} +3.00000 q^{59} -1.00000 q^{60} +11.0000 q^{61} +5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} -2.00000 q^{70} +9.00000 q^{71} -2.00000 q^{72} +11.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +1.00000 q^{85} -10.0000 q^{86} -3.00000 q^{87} +15.0000 q^{89} +2.00000 q^{90} -2.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} -3.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -7.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	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
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.00000	−0.471405
$19$	−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
$21$	2.00000	0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−5.00000	−0.962250
$28$	2.00000	0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−1.00000	−0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	2.00000	0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	−6.00000	−0.884652
$47$	−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
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	2.00000	0.267261
$57$	−1.00000	−0.132453
$58$	−3.00000	−0.393919
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−1.00000	−0.129099
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	5.00000	0.635001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−1.00000	−0.121268
$69$	−6.00000	−0.722315
$70$	−2.00000	−0.239046
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	−2.00000	−0.235702
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	8.00000	0.929981
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	1.00000	0.108465
$86$	−10.0000	−1.07833
$87$	−3.00000	−0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	2.00000	0.210819
$91$	−2.00000	−0.209657
$92$	−6.00000	−0.625543
$93$	5.00000	0.518476
$94$	−3.00000	−0.309426
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−1.00000	−0.0980581
$105$	−2.00000	−0.195180
$106$	−3.00000	−0.291386
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−5.00000	−0.481125
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	2.00000	0.188982
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−1.00000	−0.0936586
$115$	6.00000	0.559503
$116$	−3.00000	−0.278543
$117$	2.00000	0.184900
$118$	3.00000	0.276172
$119$	−2.00000	−0.183340
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	11.0000	0.995893
$123$	6.00000	0.541002
$124$	5.00000	0.449013
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	1.00000	0.0877058
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	2.00000	0.172774
$135$	5.00000	0.430331
$136$	−1.00000	−0.0857493
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−6.00000	−0.510754
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.00000	−0.169031
$141$	−3.00000	−0.252646
$142$	9.00000	0.755263
$143$	0	0
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	11.0000	0.910366
$147$	−3.00000	−0.247436
$148$	8.00000	0.657596
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	1.00000	0.0816497
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−1.00000	−0.0811107
$153$	2.00000	0.161690
$154$	0	0
$155$	−5.00000	−0.401610
$156$	−1.00000	−0.0800641
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	−3.00000	−0.237915
$160$	−1.00000	−0.0790569
$161$	−12.0000	−0.945732
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	2.00000	0.154303
$169$	−12.0000	−0.923077
$170$	1.00000	0.0766965
$171$	2.00000	0.152944
$172$	−10.0000	−0.762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−3.00000	−0.227429
$175$	2.00000	0.151186
$176$	0	0
$177$	3.00000	0.225494
$178$	15.0000	1.12430
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−2.00000	−0.148250
$183$	11.0000	0.813143
$184$	−6.00000	−0.442326
$185$	−8.00000	−0.588172
$186$	5.00000	0.366618
$187$	0	0
$188$	−3.00000	−0.218797
$189$	−10.0000	−0.727393
$190$	1.00000	0.0725476
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−7.00000	−0.502571
$195$	1.00000	0.0716115
$196$	−3.00000	−0.214286
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	1.00000	0.0707107
$201$	2.00000	0.141069
$202$	6.00000	0.422159
$203$	−6.00000	−0.421117
$204$	−1.00000	−0.0700140
$205$	−6.00000	−0.419058
$206$	8.00000	0.557386
$207$	12.0000	0.834058
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	−2.00000	−0.138013
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−3.00000	−0.206041
$213$	9.00000	0.616670
$214$	−12.0000	−0.820303
$215$	10.0000	0.681994
$216$	−5.00000	−0.340207
$217$	10.0000	0.678844
$218$	11.0000	0.745014
$219$	11.0000	0.743311
$220$	0	0
$221$	1.00000	0.0672673
$222$	8.00000	0.536925
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	2.00000	0.133631
$225$	−2.00000	−0.133333
$226$	9.00000	0.598671
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−1.00000	−0.0662266
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−3.00000	−0.196960
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	2.00000	0.130744
$235$	3.00000	0.195698
$236$	3.00000	0.195283
$237$	8.00000	0.519656
$238$	−2.00000	−0.129641
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−1.00000	−0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−11.0000	−0.707107
$243$	16.0000	1.02640
$244$	11.0000	0.704203
$245$	3.00000	0.191663
$246$	6.00000	0.382546
$247$	1.00000	0.0636285
$248$	5.00000	0.317500
$249$	−12.0000	−0.760469
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−4.00000	−0.251976
$253$	0	0
$254$	11.0000	0.690201
$255$	1.00000	0.0626224
$256$	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	$0.440080\pi$
$258$	−10.0000	−0.622573
$259$	16.0000	0.994192
$260$	1.00000	0.0620174
$261$	6.00000	0.371391
$262$	−18.0000	−1.11204
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	−2.00000	−0.122628
$267$	15.0000	0.917985
$268$	2.00000	0.122169
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	5.00000	0.304290
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	−1.00000	−0.0606339
$273$	−2.00000	−0.121046
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	8.00000	0.479808
$279$	−10.0000	−0.598684
$280$	−2.00000	−0.119523
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	−3.00000	−0.178647
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	9.00000	0.534052
$285$	1.00000	0.0592349
$286$	0	0
$287$	12.0000	0.708338
$288$	−2.00000	−0.117851
$289$	1.00000	0.0588235
$290$	3.00000	0.176166
$291$	−7.00000	−0.410347
$292$	11.0000	0.643726
$293$	−3.00000	−0.175262	−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000	−0.175262	−0.0876309	+	0.996153i	$0.527930\pi$
$294$	−3.00000	−0.174964
$295$	−3.00000	−0.174667
$296$	8.00000	0.464991
$297$	0	0
$298$	−12.0000	−0.695141
$299$	6.00000	0.346989
$300$	1.00000	0.0577350
$301$	−20.0000	−1.15278
$302$	−22.0000	−1.26596
$303$	6.00000	0.344691
$304$	−1.00000	−0.0573539
$305$	−11.0000	−0.629858
$306$	2.00000	0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−5.00000	−0.283981
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−1.00000	−0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	14.0000	0.790066
$315$	4.00000	0.225374
$316$	8.00000	0.450035
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	−12.0000	−0.668734
$323$	1.00000	0.0556415
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−16.0000	−0.886158
$327$	11.0000	0.608301
$328$	6.00000	0.331295
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	−12.0000	−0.658586
$333$	−16.0000	−0.876795
$334$	−18.0000	−0.984916
$335$	−2.00000	−0.109272
$336$	2.00000	0.109109
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−12.0000	−0.652714
$339$	9.00000	0.488813
$340$	1.00000	0.0542326
$341$	0	0
$342$	2.00000	0.108148
$343$	−20.0000	−1.07990
$344$	−10.0000	−0.539164
$345$	6.00000	0.323029
$346$	18.0000	0.967686
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	−3.00000	−0.160817
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	2.00000	0.106904
$351$	5.00000	0.266880
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	3.00000	0.159448
$355$	−9.00000	−0.477670
$356$	15.0000	0.794998
$357$	−2.00000	−0.105851
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	2.00000	0.105409
$361$	−18.0000	−0.947368
$362$	−10.0000	−0.525588
$363$	−11.0000	−0.577350
$364$	−2.00000	−0.104828
$365$	−11.0000	−0.575766
$366$	11.0000	0.574979
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−6.00000	−0.312772
$369$	−12.0000	−0.624695
$370$	−8.00000	−0.415900
$371$	−6.00000	−0.311504
$372$	5.00000	0.259238
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−3.00000	−0.154713
$377$	3.00000	0.154508
$378$	−10.0000	−0.514344
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	1.00000	0.0512989
$381$	11.0000	0.563547
$382$	12.0000	0.613973
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	20.0000	1.01666
$388$	−7.00000	−0.355371
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	1.00000	0.0506370
$391$	6.00000	0.303433
$392$	−3.00000	−0.151523
$393$	−18.0000	−0.907980
$394$	−24.0000	−1.20910
$395$	−8.00000	−0.402524
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	11.0000	0.551380
$399$	−2.00000	−0.100125
$400$	1.00000	0.0500000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	2.00000	0.0997509
$403$	−5.00000	−0.249068
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	−6.00000	−0.297775
$407$	0	0
$408$	−1.00000	−0.0495074
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	−6.00000	−0.296319
$411$	−18.0000	−0.887875
$412$	8.00000	0.394132
$413$	6.00000	0.295241
$414$	12.0000	0.589768
$415$	12.0000	0.589057
$416$	−1.00000	−0.0490290
$417$	8.00000	0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	−2.00000	−0.0975900
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	2.00000	0.0973585
$423$	6.00000	0.291730
$424$	−3.00000	−0.145693
$425$	−1.00000	−0.0485071
$426$	9.00000	0.436051
$427$	22.0000	1.06465
$428$	−12.0000	−0.580042
$429$	0	0
$430$	10.0000	0.482243
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−5.00000	−0.240563
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	10.0000	0.480015
$435$	3.00000	0.143839
$436$	11.0000	0.526804
$437$	6.00000	0.287019
$438$	11.0000	0.525600
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	1.00000	0.0475651
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	8.00000	0.379663
$445$	−15.0000	−0.711068
$446$	−1.00000	−0.0473514
$447$	−12.0000	−0.567581
$448$	2.00000	0.0944911
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	9.00000	0.423324
$453$	−22.0000	−1.03365
$454$	3.00000	0.140797
$455$	2.00000	0.0937614
$456$	−1.00000	−0.0468293
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−22.0000	−1.02799
$459$	5.00000	0.233380
$460$	6.00000	0.279751
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	−3.00000	−0.139272
$465$	−5.00000	−0.231869
$466$	9.00000	0.416917
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	3.00000	0.138380
$471$	14.0000	0.645086
$472$	3.00000	0.138086
$473$	0	0
$474$	8.00000	0.367452
$475$	−1.00000	−0.0458831
$476$	−2.00000	−0.0916698
$477$	6.00000	0.274721
$478$	6.00000	0.274434
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	−1.00000	−0.0456435
$481$	−8.00000	−0.364769
$482$	−10.0000	−0.455488
$483$	−12.0000	−0.546019
$484$	−11.0000	−0.500000
$485$	7.00000	0.317854
$486$	16.0000	0.725775
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	11.0000	0.497947
$489$	−16.0000	−0.723545
$490$	3.00000	0.135526
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	6.00000	0.270501
$493$	3.00000	0.135113
$494$	1.00000	0.0449921
$495$	0	0
$496$	5.00000	0.224507
$497$	18.0000	0.807410
$498$	−12.0000	−0.537733
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−1.00000	−0.0447214
$501$	−18.0000	−0.804181
$502$	−12.0000	−0.535586
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−4.00000	−0.178174
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−12.0000	−0.532939
$508$	11.0000	0.488046
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	1.00000	0.0442807
$511$	22.0000	0.973223
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	6.00000	0.264649
$515$	−8.00000	−0.352522
$516$	−10.0000	−0.440225
$517$	0	0
$518$	16.0000	0.703000
$519$	18.0000	0.790112
$520$	1.00000	0.0438529
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	6.00000	0.262613
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−18.0000	−0.786334
$525$	2.00000	0.0872872
$526$	9.00000	0.392419
$527$	−5.00000	−0.217803
$528$	0	0
$529$	13.0000	0.565217
$530$	3.00000	0.130312
$531$	−6.00000	−0.260378
$532$	−2.00000	−0.0867110
$533$	−6.00000	−0.259889
$534$	15.0000	0.649113
$535$	12.0000	0.518805
$536$	2.00000	0.0863868
$537$	−12.0000	−0.517838
$538$	15.0000	0.646696
$539$	0	0
$540$	5.00000	0.215166
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	32.0000	1.37452
$543$	−10.0000	−0.429141
$544$	−1.00000	−0.0428746
$545$	−11.0000	−0.471188
$546$	−2.00000	−0.0855921
$547$	−31.0000	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$547$	−31.0000	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$548$	−18.0000	−0.768922
$549$	−22.0000	−0.938937
$550$	0	0
$551$	3.00000	0.127804
$552$	−6.00000	−0.255377
$553$	16.0000	0.680389
$554$	−10.0000	−0.424859
$555$	−8.00000	−0.339581
$556$	8.00000	0.339276
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	−10.0000	−0.423334
$559$	10.0000	0.422955
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	27.0000	1.13893
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	−3.00000	−0.126323
$565$	−9.00000	−0.378633
$566$	−13.0000	−0.546431
$567$	2.00000	0.0839921
$568$	9.00000	0.377632
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	1.00000	0.0418854
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	12.0000	0.500870
$575$	−6.00000	−0.250217
$576$	−2.00000	−0.0833333
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	1.00000	0.0415945
$579$	2.00000	0.0831172
$580$	3.00000	0.124568
$581$	−24.0000	−0.995688
$582$	−7.00000	−0.290159
$583$	0	0
$584$	11.0000	0.455183
$585$	−2.00000	−0.0826898
$586$	−3.00000	−0.123929
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	−3.00000	−0.123718
$589$	−5.00000	−0.206021
$590$	−3.00000	−0.123508
$591$	−24.0000	−0.987228
$592$	8.00000	0.328798
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	−12.0000	−0.491539
$597$	11.0000	0.450200
$598$	6.00000	0.245358
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	1.00000	0.0408248
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	−20.0000	−0.815139
$603$	−4.00000	−0.162893
$604$	−22.0000	−0.895167
$605$	11.0000	0.447214
$606$	6.00000	0.243733
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	−1.00000	−0.0405554
$609$	−6.00000	−0.243132
$610$	−11.0000	−0.445377
$611$	3.00000	0.121367
$612$	2.00000	0.0808452
$613$	−1.00000	−0.0403896	−0.0201948	−	0.999796i	$-0.506429\pi$
$613$	−1.00000	−0.0403896	−0.0201948	+	0.999796i	$0.506429\pi$
$614$	20.0000	0.807134
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	8.00000	0.321807
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	−5.00000	−0.200805
$621$	30.0000	1.20386
$622$	0	0
$623$	30.0000	1.20192
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	14.0000	0.558661
$629$	−8.00000	−0.318981
$630$	4.00000	0.159364
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	8.00000	0.318223
$633$	2.00000	0.0794929
$634$	0	0
$635$	−11.0000	−0.436522
$636$	−3.00000	−0.118958
$637$	3.00000	0.118864
$638$	0	0
$639$	−18.0000	−0.712069
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−12.0000	−0.472866
$645$	10.0000	0.393750
$646$	1.00000	0.0393445
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	10.0000	0.391931
$652$	−16.0000	−0.626608
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	11.0000	0.430134
$655$	18.0000	0.703318
$656$	6.00000	0.234261
$657$	−22.0000	−0.858302
$658$	−6.00000	−0.233904
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	−31.0000	−1.20485
$663$	1.00000	0.0388368
$664$	−12.0000	−0.465690
$665$	2.00000	0.0775567
$666$	−16.0000	−0.619987
$667$	18.0000	0.696963
$668$	−18.0000	−0.696441
$669$	−1.00000	−0.0386622
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	2.00000	0.0771517
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−13.0000	−0.500741
$675$	−5.00000	−0.192450
$676$	−12.0000	−0.461538
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	9.00000	0.345643
$679$	−14.0000	−0.537271
$680$	1.00000	0.0383482
$681$	3.00000	0.114960
$682$	0	0
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	2.00000	0.0764719
$685$	18.0000	0.687745
$686$	−20.0000	−0.763604
$687$	−22.0000	−0.839352
$688$	−10.0000	−0.381246
$689$	3.00000	0.114291
$690$	6.00000	0.228416
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−15.0000	−0.569392
$695$	−8.00000	−0.303457
$696$	−3.00000	−0.113715
$697$	−6.00000	−0.227266
$698$	−28.0000	−1.05982
$699$	9.00000	0.340411
$700$	2.00000	0.0755929
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	5.00000	0.188713
$703$	−8.00000	−0.301726
$704$	0	0
$705$	3.00000	0.112987
$706$	18.0000	0.677439
$707$	12.0000	0.451306
$708$	3.00000	0.112747
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	−9.00000	−0.337764
$711$	−16.0000	−0.600047
$712$	15.0000	0.562149
$713$	−30.0000	−1.12351
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	6.00000	0.224074
$718$	30.0000	1.11959
$719$	−39.0000	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$719$	−39.0000	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$720$	2.00000	0.0745356
$721$	16.0000	0.595871
$722$	−18.0000	−0.669891
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	−3.00000	−0.111417
$726$	−11.0000	−0.408248
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	−11.0000	−0.407128
$731$	10.0000	0.369863
$732$	11.0000	0.406572
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−22.0000	−0.812035
$735$	3.00000	0.110657
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−12.0000	−0.441726
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	−8.00000	−0.294086
$741$	1.00000	0.0367359
$742$	−6.00000	−0.220267
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	5.00000	0.183309
$745$	12.0000	0.439646
$746$	−22.0000	−0.805477
$747$	24.0000	0.878114
$748$	0	0
$749$	−24.0000	−0.876941
$750$	−1.00000	−0.0365148
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	−3.00000	−0.109399
$753$	−12.0000	−0.437304
$754$	3.00000	0.109254
$755$	22.0000	0.800662
$756$	−10.0000	−0.363696
$757$	−13.0000	−0.472493	−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000	−0.472493	−0.236247	+	0.971693i	$0.575917\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	1.00000	0.0362738
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	11.0000	0.398488
$763$	22.0000	0.796453
$764$	12.0000	0.434145
$765$	−2.00000	−0.0723102
$766$	−9.00000	−0.325183
$767$	−3.00000	−0.108324
$768$	1.00000	0.0360844
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	2.00000	0.0719816
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	20.0000	0.718885
$775$	5.00000	0.179605
$776$	−7.00000	−0.251285
$777$	16.0000	0.573997
$778$	12.0000	0.430221
$779$	−6.00000	−0.214972
$780$	1.00000	0.0358057
$781$	0	0
$782$	6.00000	0.214560
$783$	15.0000	0.536056
$784$	−3.00000	−0.107143
$785$	−14.0000	−0.499681
$786$	−18.0000	−0.642039
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−24.0000	−0.854965
$789$	9.00000	0.320408
$790$	−8.00000	−0.284627
$791$	18.0000	0.640006
$792$	0	0
$793$	−11.0000	−0.390621
$794$	20.0000	0.709773
$795$	3.00000	0.106399
$796$	11.0000	0.389885
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	−2.00000	−0.0707992
$799$	3.00000	0.106132
$800$	1.00000	0.0353553
$801$	−30.0000	−1.06000
$802$	36.0000	1.27120
$803$	0	0
$804$	2.00000	0.0705346
$805$	12.0000	0.422944
$806$	−5.00000	−0.176117
$807$	15.0000	0.528025
$808$	6.00000	0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	−1.00000	−0.0351364
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−6.00000	−0.210559
$813$	32.0000	1.12229
$814$	0	0
$815$	16.0000	0.560456
$816$	−1.00000	−0.0350070
$817$	10.0000	0.349856
$818$	29.0000	1.01396
$819$	4.00000	0.139771
$820$	−6.00000	−0.209529
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	−18.0000	−0.627822
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	6.00000	0.208767
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	12.0000	0.417029
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	12.0000	0.416526
$831$	−10.0000	−0.346896
$832$	−1.00000	−0.0346688
$833$	3.00000	0.103944
$834$	8.00000	0.277017
$835$	18.0000	0.622916
$836$	0	0
$837$	−25.0000	−0.864126
$838$	6.00000	0.207267
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	−2.00000	−0.0690066
$841$	−20.0000	−0.689655
$842$	−4.00000	−0.137849
$843$	27.0000	0.929929
$844$	2.00000	0.0688428
$845$	12.0000	0.412813
$846$	6.00000	0.206284
$847$	−22.0000	−0.755929
$848$	−3.00000	−0.103020
$849$	−13.0000	−0.446159
$850$	−1.00000	−0.0342997
$851$	−48.0000	−1.64542
$852$	9.00000	0.308335
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	22.0000	0.752825
$855$	−2.00000	−0.0683986
$856$	−12.0000	−0.410152
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	10.0000	0.340997
$861$	12.0000	0.408959
$862$	−24.0000	−0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−5.00000	−0.170103
$865$	−18.0000	−0.612018
$866$	−34.0000	−1.15537
$867$	1.00000	0.0339618
$868$	10.0000	0.339422
$869$	0	0
$870$	3.00000	0.101710
$871$	−2.00000	−0.0677674
$872$	11.0000	0.372507
$873$	14.0000	0.473828
$874$	6.00000	0.202953
$875$	−2.00000	−0.0676123
$876$	11.0000	0.371656
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	8.00000	0.269987
$879$	−3.00000	−0.101187
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	6.00000	0.202031
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	1.00000	0.0336336
$885$	−3.00000	−0.100844
$886$	12.0000	0.403148
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	8.00000	0.268462
$889$	22.0000	0.737856
$890$	−15.0000	−0.502801
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	3.00000	0.100391
$894$	−12.0000	−0.401340
$895$	12.0000	0.401116
$896$	2.00000	0.0668153
$897$	6.00000	0.200334
$898$	0	0
$899$	−15.0000	−0.500278
$900$	−2.00000	−0.0666667
$901$	3.00000	0.0999445
$902$	0	0
$903$	−20.0000	−0.665558
$904$	9.00000	0.299336
$905$	10.0000	0.332411
$906$	−22.0000	−0.730901
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	3.00000	0.0995585
$909$	−12.0000	−0.398015
$910$	2.00000	0.0662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	26.0000	0.860004
$915$	−11.0000	−0.363649
$916$	−22.0000	−0.726900
$917$	−36.0000	−1.18882
$918$	5.00000	0.165025
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	6.00000	0.197814
$921$	20.0000	0.659022
$922$	30.0000	0.987997
$923$	−9.00000	−0.296239
$924$	0	0
$925$	8.00000	0.263038
$926$	−19.0000	−0.624379
$927$	−16.0000	−0.525509
$928$	−3.00000	−0.0984798
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	−5.00000	−0.163956
$931$	3.00000	0.0983210
$932$	9.00000	0.294805
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	2.00000	0.0653720
$937$	44.0000	1.43742	0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000	1.43742	0.718709	+	0.695311i	$0.244734\pi$
$938$	4.00000	0.130605
$939$	14.0000	0.456873
$940$	3.00000	0.0978492
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	14.0000	0.456145
$943$	−36.0000	−1.17232
$944$	3.00000	0.0976417
$945$	10.0000	0.325300
$946$	0	0
$947$	51.0000	1.65728	0.828639	−	0.559784i	$-0.189116\pi$
$947$	51.0000	1.65728	0.828639	+	0.559784i	$0.189116\pi$
$948$	8.00000	0.259828
$949$	−11.0000	−0.357075
$950$	−1.00000	−0.0324443
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	6.00000	0.194257
$955$	−12.0000	−0.388311
$956$	6.00000	0.194054
$957$	0	0
$958$	27.0000	0.872330
$959$	−36.0000	−1.16250
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	−8.00000	−0.257930
$963$	24.0000	0.773389
$964$	−10.0000	−0.322078
$965$	−2.00000	−0.0643823
$966$	−12.0000	−0.386094
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−11.0000	−0.353553
$969$	1.00000	0.0321246
$970$	7.00000	0.224756
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	16.0000	0.513200
$973$	16.0000	0.512936
$974$	−22.0000	−0.704925
$975$	−1.00000	−0.0320256
$976$	11.0000	0.352101
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−16.0000	−0.511624
$979$	0	0
$980$	3.00000	0.0958315
$981$	−22.0000	−0.702406
$982$	−39.0000	−1.24454
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	6.00000	0.191273
$985$	24.0000	0.764704
$986$	3.00000	0.0955395
$987$	−6.00000	−0.190982
$988$	1.00000	0.0318142
$989$	60.0000	1.90789
$990$	0	0
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	5.00000	0.158750
$993$	−31.0000	−0.983755
$994$	18.0000	0.570925
$995$	−11.0000	−0.348723
$996$	−12.0000	−0.380235
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−22.0000	−0.696398
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.2.a.e.1.1	✓	1
3.2	odd	2		1530.2.a.g.1.1		1
4.3	odd	2		1360.2.a.d.1.1		1
5.2	odd	4		850.2.c.e.749.2		2
5.3	odd	4		850.2.c.e.749.1		2
5.4	even	2		850.2.a.b.1.1		1
7.6	odd	2		8330.2.a.q.1.1		1
8.3	odd	2		5440.2.a.r.1.1		1
8.5	even	2		5440.2.a.k.1.1		1
15.14	odd	2		7650.2.a.bo.1.1		1
17.4	even	4		2890.2.b.b.2311.1		2
17.13	even	4		2890.2.b.b.2311.2		2
17.16	even	2		2890.2.a.n.1.1		1
20.19	odd	2		6800.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.e.1.1	✓	1	1.1	even	1	trivial
850.2.a.b.1.1		1	5.4	even	2
850.2.c.e.749.1		2	5.3	odd	4
850.2.c.e.749.2		2	5.2	odd	4
1360.2.a.d.1.1		1	4.3	odd	2
1530.2.a.g.1.1		1	3.2	odd	2
2890.2.a.n.1.1		1	17.16	even	2
2890.2.b.b.2311.1		2	17.4	even	4
2890.2.b.b.2311.2		2	17.13	even	4
5440.2.a.k.1.1		1	8.5	even	2
5440.2.a.r.1.1		1	8.3	odd	2
6800.2.a.t.1.1		1	20.19	odd	2
7650.2.a.bo.1.1		1	15.14	odd	2
8330.2.a.q.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:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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