LMFDB - Embedded newform 170.2.a.a.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -12.0000 q^{33} -1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} -6.00000 q^{55} -2.00000 q^{56} -16.0000 q^{57} +6.00000 q^{58} +2.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +12.0000 q^{66} +8.00000 q^{67} +1.00000 q^{68} +12.0000 q^{69} +2.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} +8.00000 q^{76} +12.0000 q^{77} +4.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -1.00000 q^{85} +4.00000 q^{86} +12.0000 q^{87} -6.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} -8.00000 q^{95} +2.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$3$	−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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−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
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	−6.00000	−1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	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	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−12.0000	−2.08893
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−8.00000	−1.29777
$39$	−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	$0.598656\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	6.00000	0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	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
$55$	−6.00000	−0.809040
$56$	−2.00000	−0.267261
$57$	−16.0000	−2.11925
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	12.0000	1.47710
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	1.00000	0.121268
$69$	12.0000	1.44463
$70$	2.00000	0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−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	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	8.00000	0.917663
$77$	12.0000	1.36753
$78$	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
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000	−0.436436
$85$	−1.00000	−0.108465
$86$	4.00000	0.431331
$87$	12.0000	1.28654
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	−12.0000	−1.23771
$95$	−8.00000	−0.820783
$96$	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
$99$	6.00000	0.603023
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000	0.198030
$103$	−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
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	4.00000	0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	6.00000	0.572078
$111$	−4.00000	−0.379663
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	16.0000	1.49854
$115$	6.00000	0.559503
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	0	0
$119$	2.00000	0.183340
$120$	−2.00000	−0.182574
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	12.0000	1.08200
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	2.00000	0.175412
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−12.0000	−1.04447
$133$	16.0000	1.38738
$134$	−8.00000	−0.691095
$135$	−4.00000	−0.344265
$136$	−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	$0.779207\pi$
$138$	−12.0000	−1.02151
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	−2.00000	−0.169031
$141$	−24.0000	−2.02116
$142$	6.00000	0.503509
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	−2.00000	−0.165521
$147$	6.00000	0.494872
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	2.00000	0.163299
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−8.00000	−0.648886
$153$	1.00000	0.0808452
$154$	−12.0000	−0.966988
$155$	−2.00000	−0.160644
$156$	−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
$161$	−12.0000	−0.945732
$162$	11.0000	0.864242
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−6.00000	−0.468521
$165$	12.0000	0.934199
$166$	−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	$0.745234\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	1.00000	0.0766965
$171$	8.00000	0.611775
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−12.0000	−0.909718
$175$	2.00000	0.151186
$176$	6.00000	0.452267
$177$	0	0
$178$	−6.00000	−0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−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
$183$	−4.00000	−0.295689
$184$	6.00000	0.442326
$185$	−2.00000	−0.147043
$186$	4.00000	0.293294
$187$	6.00000	0.438763
$188$	12.0000	0.875190
$189$	8.00000	0.581914
$190$	8.00000	0.580381
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−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	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−6.00000	−0.426401
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−1.00000	−0.0707107
$201$	−16.0000	−1.12855
$202$	6.00000	0.422159
$203$	−12.0000	−0.842235
$204$	−2.00000	−0.140028
$205$	6.00000	0.419058
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	48.0000	3.32023
$210$	−4.00000	−0.276026
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	12.0000	0.822226
$214$	18.0000	1.23045
$215$	4.00000	0.272798
$216$	−4.00000	−0.272166
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	−4.00000	−0.270295
$220$	−6.00000	−0.404520
$221$	2.00000	0.134535
$222$	4.00000	0.268462
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−16.0000	−1.05963
$229$	−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
$231$	−24.0000	−1.57908
$232$	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
$235$	−12.0000	−0.782794
$236$	0	0
$237$	20.0000	1.29914
$238$	−2.00000	−0.129641
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−25.0000	−1.60706
$243$	10.0000	0.641500
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	−12.0000	−0.765092
$247$	16.0000	1.01806
$248$	−2.00000	−0.127000
$249$	−24.0000	−1.52094
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	−36.0000	−2.26330
$254$	16.0000	1.00393
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−8.00000	−0.498058
$259$	4.00000	0.248548
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	−18.0000	−1.11204
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	12.0000	0.738549
$265$	−6.00000	−0.368577
$266$	−16.0000	−0.981023
$267$	−12.0000	−0.734388
$268$	8.00000	0.488678
$269$	−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	$0.661532\pi$
$272$	1.00000	0.0606339
$273$	−8.00000	−0.484182
$274$	18.0000	1.08742
$275$	6.00000	0.361814
$276$	12.0000	0.722315
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	22.0000	1.31947
$279$	2.00000	0.119737
$280$	2.00000	0.119523
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000	1.42918
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	−6.00000	−0.356034
$285$	16.0000	0.947758
$286$	−12.0000	−0.709575
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	−4.00000	−0.234484
$292$	2.00000	0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−2.00000	−0.116248
$297$	24.0000	1.39262
$298$	−6.00000	−0.347571
$299$	−12.0000	−0.693978
$300$	−2.00000	−0.115470
$301$	−8.00000	−0.461112
$302$	4.00000	0.230174
$303$	12.0000	0.689382
$304$	8.00000	0.458831
$305$	−2.00000	−0.114520
$306$	−1.00000	−0.0571662
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	12.0000	0.683763
$309$	8.00000	0.455104
$310$	2.00000	0.113592
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	4.00000	0.226455
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	10.0000	0.564333
$315$	−2.00000	−0.112687
$316$	−10.0000	−0.562544
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	12.0000	0.672927
$319$	−36.0000	−2.01561
$320$	−1.00000	−0.0559017
$321$	36.0000	2.00932
$322$	12.0000	0.668734
$323$	8.00000	0.445132
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	−2.00000	−0.110770
$327$	−4.00000	−0.221201
$328$	6.00000	0.331295
$329$	24.0000	1.32316
$330$	−12.0000	−0.660578
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	18.0000	0.984916
$335$	−8.00000	−0.437087
$336$	−4.00000	−0.218218
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	9.00000	0.489535
$339$	12.0000	0.651751
$340$	−1.00000	−0.0542326
$341$	12.0000	0.649836
$342$	−8.00000	−0.432590
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	−12.0000	−0.646058
$346$	6.00000	0.322562
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	12.0000	0.643268
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−2.00000	−0.106904
$351$	8.00000	0.427008
$352$	−6.00000	−0.319801
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	6.00000	0.317999
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	1.00000	0.0527046
$361$	45.0000	2.36842
$362$	22.0000	1.15629
$363$	−50.0000	−2.62432
$364$	4.00000	0.209657
$365$	−2.00000	−0.104685
$366$	4.00000	0.209083
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−6.00000	−0.312772
$369$	−6.00000	−0.312348
$370$	2.00000	0.103975
$371$	12.0000	0.623009
$372$	−4.00000	−0.207390
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−6.00000	−0.310253
$375$	2.00000	0.103280
$376$	−12.0000	−0.618853
$377$	−12.0000	−0.618031
$378$	−8.00000	−0.411476
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	−8.00000	−0.410391
$381$	32.0000	1.63941
$382$	−24.0000	−1.22795
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	2.00000	0.102062
$385$	−12.0000	−0.611577
$386$	−2.00000	−0.101797
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−4.00000	−0.202548
$391$	−6.00000	−0.303433
$392$	3.00000	0.151523
$393$	−36.0000	−1.81596
$394$	−18.0000	−0.906827
$395$	10.0000	0.503155
$396$	6.00000	0.301511
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−2.00000	−0.100251
$399$	−32.0000	−1.60200
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	16.0000	0.798007
$403$	4.00000	0.199254
$404$	−6.00000	−0.298511
$405$	11.0000	0.546594
$406$	12.0000	0.595550
$407$	12.0000	0.594818
$408$	2.00000	0.0990148
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	36.0000	1.77575
$412$	−4.00000	−0.197066
$413$	0	0
$414$	6.00000	0.294884
$415$	−12.0000	−0.589057
$416$	−2.00000	−0.0980581
$417$	44.0000	2.15469
$418$	−48.0000	−2.34776
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	4.00000	0.195180
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	10.0000	0.486792
$423$	12.0000	0.583460
$424$	−6.00000	−0.291386
$425$	1.00000	0.0485071
$426$	−12.0000	−0.581402
$427$	4.00000	0.193574
$428$	−18.0000	−0.870063
$429$	−24.0000	−1.15873
$430$	−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
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−4.00000	−0.192006
$435$	−12.0000	−0.575356
$436$	2.00000	0.0957826
$437$	−48.0000	−2.29615
$438$	4.00000	0.191127
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	6.00000	0.286039
$441$	−3.00000	−0.142857
$442$	−2.00000	−0.0951303
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−4.00000	−0.189832
$445$	−6.00000	−0.284427
$446$	−8.00000	−0.378811
$447$	−12.0000	−0.567581
$448$	2.00000	0.0944911
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−1.00000	−0.0471405
$451$	−36.0000	−1.69517
$452$	−6.00000	−0.282216
$453$	8.00000	0.375873
$454$	−18.0000	−0.844782
$455$	−4.00000	−0.187523
$456$	16.0000	0.749269
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	22.0000	1.02799
$459$	4.00000	0.186704
$460$	6.00000	0.279751
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	24.0000	1.11658
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−6.00000	−0.278543
$465$	4.00000	0.185496
$466$	−18.0000	−0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	16.0000	0.738811
$470$	12.0000	0.553519
$471$	20.0000	0.921551
$472$	0	0
$473$	−24.0000	−1.10352
$474$	−20.0000	−0.918630
$475$	8.00000	0.367065
$476$	2.00000	0.0916698
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	42.0000	1.91903	0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000	1.91903	0.959514	+	0.281659i	$0.0908848\pi$
$480$	−2.00000	−0.0912871
$481$	4.00000	0.182384
$482$	−26.0000	−1.18427
$483$	24.0000	1.09204
$484$	25.0000	1.13636
$485$	−2.00000	−0.0908153
$486$	−10.0000	−0.453609
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	−2.00000	−0.0905357
$489$	−4.00000	−0.180886
$490$	−3.00000	−0.135526
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	12.0000	0.541002
$493$	−6.00000	−0.270226
$494$	−16.0000	−0.719874
$495$	−6.00000	−0.269680
$496$	2.00000	0.0898027
$497$	−12.0000	−0.538274
$498$	24.0000	1.07547
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	−1.00000	−0.0447214
$501$	36.0000	1.60836
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−2.00000	−0.0890871
$505$	6.00000	0.266996
$506$	36.0000	1.60040
$507$	18.0000	0.799408
$508$	−16.0000	−0.709885
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−2.00000	−0.0885615
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	32.0000	1.41283
$514$	18.0000	0.793946
$515$	4.00000	0.176261
$516$	8.00000	0.352180
$517$	72.0000	3.16656
$518$	−4.00000	−0.175750
$519$	12.0000	0.526742
$520$	2.00000	0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.00000	0.262613
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	18.0000	0.786334
$525$	−4.00000	−0.174574
$526$	0	0
$527$	2.00000	0.0871214
$528$	−12.0000	−0.522233
$529$	13.0000	0.565217
$530$	6.00000	0.260623
$531$	0	0
$532$	16.0000	0.693688
$533$	−12.0000	−0.519778
$534$	12.0000	0.519291
$535$	18.0000	0.778208
$536$	−8.00000	−0.345547
$537$	0	0
$538$	6.00000	0.258678
$539$	−18.0000	−0.775315
$540$	−4.00000	−0.172133
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	16.0000	0.687259
$543$	44.0000	1.88822
$544$	−1.00000	−0.0428746
$545$	−2.00000	−0.0856706
$546$	8.00000	0.342368
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	−18.0000	−0.768922
$549$	2.00000	0.0853579
$550$	−6.00000	−0.255841
$551$	−48.0000	−2.04487
$552$	−12.0000	−0.510754
$553$	−20.0000	−0.850487
$554$	−2.00000	−0.0849719
$555$	4.00000	0.169791
$556$	−22.0000	−0.933008
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−2.00000	−0.0846668
$559$	−8.00000	−0.338364
$560$	−2.00000	−0.0845154
$561$	−12.0000	−0.506640
$562$	6.00000	0.253095
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−24.0000	−1.01058
$565$	6.00000	0.252422
$566$	−26.0000	−1.09286
$567$	−22.0000	−0.923913
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−16.0000	−0.670166
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	12.0000	0.501745
$573$	−48.0000	−2.00523
$574$	12.0000	0.500870
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−1.00000	−0.0415945
$579$	−4.00000	−0.166234
$580$	6.00000	0.249136
$581$	24.0000	0.995688
$582$	4.00000	0.165805
$583$	36.0000	1.49097
$584$	−2.00000	−0.0827606
$585$	−2.00000	−0.0826898
$586$	18.0000	0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	6.00000	0.247436
$589$	16.0000	0.659269
$590$	0	0
$591$	−36.0000	−1.48084
$592$	2.00000	0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−24.0000	−0.984732
$595$	−2.00000	−0.0819920
$596$	6.00000	0.245770
$597$	−4.00000	−0.163709
$598$	12.0000	0.490716
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	2.00000	0.0816497
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	8.00000	0.326056
$603$	8.00000	0.325785
$604$	−4.00000	−0.162758
$605$	−25.0000	−1.01639
$606$	−12.0000	−0.487467
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−8.00000	−0.324443
$609$	24.0000	0.972529
$610$	2.00000	0.0809776
$611$	24.0000	0.970936
$612$	1.00000	0.0404226
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	16.0000	0.645707
$615$	−12.0000	−0.483887
$616$	−12.0000	−0.483494
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−8.00000	−0.321807
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−2.00000	−0.0803219
$621$	−24.0000	−0.963087
$622$	30.0000	1.20289
$623$	12.0000	0.480770
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	−96.0000	−3.83387
$628$	−10.0000	−0.399043
$629$	2.00000	0.0797452
$630$	2.00000	0.0796819
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	10.0000	0.397779
$633$	20.0000	0.794929
$634$	6.00000	0.238290
$635$	16.0000	0.634941
$636$	−12.0000	−0.475831
$637$	−6.00000	−0.237729
$638$	36.0000	1.42525
$639$	−6.00000	−0.237356
$640$	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	$0.701845\pi$
$642$	−36.0000	−1.42081
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−12.0000	−0.472866
$645$	−8.00000	−0.315000
$646$	−8.00000	−0.314756
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−8.00000	−0.313545
$652$	2.00000	0.0783260
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	4.00000	0.156412
$655$	−18.0000	−0.703318
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	−24.0000	−0.935617
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	12.0000	0.467099
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−8.00000	−0.310929
$663$	−4.00000	−0.155347
$664$	−12.0000	−0.465690
$665$	−16.0000	−0.620453
$666$	−2.00000	−0.0774984
$667$	36.0000	1.39393
$668$	−18.0000	−0.696441
$669$	−16.0000	−0.618596
$670$	8.00000	0.309067
$671$	12.0000	0.463255
$672$	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	$0.332924\pi$
$674$	22.0000	0.847408
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−12.0000	−0.460857
$679$	4.00000	0.153506
$680$	1.00000	0.0383482
$681$	−36.0000	−1.37952
$682$	−12.0000	−0.459504
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	8.00000	0.305888
$685$	18.0000	0.687745
$686$	20.0000	0.763604
$687$	44.0000	1.67870
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	12.0000	0.456832
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−6.00000	−0.228086
$693$	12.0000	0.455842
$694$	6.00000	0.227757
$695$	22.0000	0.834508
$696$	−12.0000	−0.454859
$697$	−6.00000	−0.227266
$698$	10.0000	0.378506
$699$	−36.0000	−1.36165
$700$	2.00000	0.0755929
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−8.00000	−0.301941
$703$	16.0000	0.603451
$704$	6.00000	0.226134
$705$	24.0000	0.903892
$706$	−18.0000	−0.677439
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−6.00000	−0.225176
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	−12.0000	−0.449404
$714$	4.00000	0.149696
$715$	−12.0000	−0.448775
$716$	0	0
$717$	−48.0000	−1.79259
$718$	36.0000	1.34351
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	−1.00000	−0.0372678
$721$	−8.00000	−0.297936
$722$	−45.0000	−1.67473
$723$	−52.0000	−1.93390
$724$	−22.0000	−0.817624
$725$	−6.00000	−0.222834
$726$	50.0000	1.85567
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	2.00000	0.0740233
$731$	−4.00000	−0.147945
$732$	−4.00000	−0.147844
$733$	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
$736$	6.00000	0.221163
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−2.00000	−0.0735215
$741$	−32.0000	−1.17555
$742$	−12.0000	−0.440534
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	4.00000	0.146647
$745$	−6.00000	−0.219823
$746$	22.0000	0.805477
$747$	12.0000	0.439057
$748$	6.00000	0.219382
$749$	−36.0000	−1.31541
$750$	−2.00000	−0.0730297
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	12.0000	0.437595
$753$	−24.0000	−0.874609
$754$	12.0000	0.437014
$755$	4.00000	0.145575
$756$	8.00000	0.290957
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	22.0000	0.799076
$759$	72.0000	2.61343
$760$	8.00000	0.290191
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−32.0000	−1.15924
$763$	4.00000	0.144810
$764$	24.0000	0.868290
$765$	−1.00000	−0.0361551
$766$	24.0000	0.867155
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	12.0000	0.432450
$771$	36.0000	1.29651
$772$	2.00000	0.0719816
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000	0.143777
$775$	2.00000	0.0718421
$776$	−2.00000	−0.0717958
$777$	−8.00000	−0.286998
$778$	6.00000	0.215110
$779$	−48.0000	−1.71978
$780$	4.00000	0.143223
$781$	−36.0000	−1.28818
$782$	6.00000	0.214560
$783$	−24.0000	−0.857690
$784$	−3.00000	−0.107143
$785$	10.0000	0.356915
$786$	36.0000	1.28408
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−10.0000	−0.355784
$791$	−12.0000	−0.426671
$792$	−6.00000	−0.213201
$793$	4.00000	0.142044
$794$	−2.00000	−0.0709773
$795$	12.0000	0.425596
$796$	2.00000	0.0708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	32.0000	1.13279
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	30.0000	1.05934
$803$	12.0000	0.423471
$804$	−16.0000	−0.564276
$805$	12.0000	0.422944
$806$	−4.00000	−0.140894
$807$	12.0000	0.422420
$808$	6.00000	0.211079
$809$	−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
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	−12.0000	−0.421117
$813$	32.0000	1.12229
$814$	−12.0000	−0.420600
$815$	−2.00000	−0.0700569
$816$	−2.00000	−0.0700140
$817$	−32.0000	−1.11954
$818$	−26.0000	−0.909069
$819$	4.00000	0.139771
$820$	6.00000	0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−36.0000	−1.25564
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	4.00000	0.139347
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	−6.00000	−0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	12.0000	0.416526
$831$	−4.00000	−0.138758
$832$	2.00000	0.0693375
$833$	−3.00000	−0.103944
$834$	−44.0000	−1.52360
$835$	18.0000	0.622916
$836$	48.0000	1.66011
$837$	8.00000	0.276520
$838$	−6.00000	−0.207267
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	−4.00000	−0.138013
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	12.0000	0.413302
$844$	−10.0000	−0.344214
$845$	9.00000	0.309609
$846$	−12.0000	−0.412568
$847$	50.0000	1.71802
$848$	6.00000	0.206041
$849$	−52.0000	−1.78464
$850$	−1.00000	−0.0342997
$851$	−12.0000	−0.411355
$852$	12.0000	0.411113
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	−4.00000	−0.136877
$855$	−8.00000	−0.273594
$856$	18.0000	0.615227
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	24.0000	0.819346
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	4.00000	0.136399
$861$	24.0000	0.817918
$862$	−30.0000	−1.02180
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−4.00000	−0.136083
$865$	6.00000	0.204006
$866$	−14.0000	−0.475739
$867$	−2.00000	−0.0679236
$868$	4.00000	0.135769
$869$	−60.0000	−2.03536
$870$	12.0000	0.406838
$871$	16.0000	0.542139
$872$	−2.00000	−0.0677285
$873$	2.00000	0.0676897
$874$	48.0000	1.62362
$875$	−2.00000	−0.0676123
$876$	−4.00000	−0.135147
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−2.00000	−0.0674967
$879$	36.0000	1.21425
$880$	−6.00000	−0.202260
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	3.00000	0.101015
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	4.00000	0.134231
$889$	−32.0000	−1.07325
$890$	6.00000	0.201120
$891$	−66.0000	−2.21108
$892$	8.00000	0.267860
$893$	96.0000	3.21252
$894$	12.0000	0.401340
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	24.0000	0.801337
$898$	−18.0000	−0.600668
$899$	−12.0000	−0.400222
$900$	1.00000	0.0333333
$901$	6.00000	0.199889
$902$	36.0000	1.19867
$903$	16.0000	0.532447
$904$	6.00000	0.199557
$905$	22.0000	0.731305
$906$	−8.00000	−0.265782
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	18.0000	0.597351
$909$	−6.00000	−0.199007
$910$	4.00000	0.132599
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	−16.0000	−0.529813
$913$	72.0000	2.38285
$914$	10.0000	0.330771
$915$	4.00000	0.132236
$916$	−22.0000	−0.726900
$917$	36.0000	1.18882
$918$	−4.00000	−0.132020
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−6.00000	−0.197814
$921$	32.0000	1.05444
$922$	−6.00000	−0.197599
$923$	−12.0000	−0.394985
$924$	−24.0000	−0.789542
$925$	2.00000	0.0657596
$926$	4.00000	0.131448
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−4.00000	−0.131165
$931$	−24.0000	−0.786568
$932$	18.0000	0.589610
$933$	60.0000	1.96431
$934$	12.0000	0.392652
$935$	−6.00000	−0.196221
$936$	−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	$0.360383\pi$
$938$	−16.0000	−0.522419
$939$	−4.00000	−0.130535
$940$	−12.0000	−0.391397
$941$	−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
$943$	36.0000	1.17232
$944$	0	0
$945$	−8.00000	−0.260240
$946$	24.0000	0.780307
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	20.0000	0.649570
$949$	4.00000	0.129845
$950$	−8.00000	−0.259554
$951$	12.0000	0.389127
$952$	−2.00000	−0.0648204
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−6.00000	−0.194257
$955$	−24.0000	−0.776622
$956$	24.0000	0.776215
$957$	72.0000	2.32743
$958$	−42.0000	−1.35696
$959$	−36.0000	−1.16250
$960$	2.00000	0.0645497
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	−18.0000	−0.580042
$964$	26.0000	0.837404
$965$	−2.00000	−0.0643823
$966$	−24.0000	−0.772187
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−25.0000	−0.803530
$969$	−16.0000	−0.513994
$970$	2.00000	0.0642161
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	10.0000	0.320750
$973$	−44.0000	−1.41058
$974$	−26.0000	−0.833094
$975$	−4.00000	−0.128103
$976$	2.00000	0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	4.00000	0.127906
$979$	36.0000	1.15056
$980$	3.00000	0.0958315
$981$	2.00000	0.0638551
$982$	−24.0000	−0.765871
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	−12.0000	−0.382546
$985$	−18.0000	−0.573528
$986$	6.00000	0.191079
$987$	−48.0000	−1.52786
$988$	16.0000	0.509028
$989$	24.0000	0.763156
$990$	6.00000	0.190693
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−2.00000	−0.0635001
$993$	−16.0000	−0.507745
$994$	12.0000	0.380617
$995$	−2.00000	−0.0634043
$996$	−24.0000	−0.760469
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	10.0000	0.316544
$999$	8.00000	0.253109

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.a.1.1	✓	1	1.1	even	1	trivial
850.2.a.j.1.1		1	5.4	even	2
850.2.c.g.749.1		2	5.2	odd	4
850.2.c.g.749.2		2	5.3	odd	4
1360.2.a.i.1.1		1	4.3	odd	2
1530.2.a.n.1.1		1	3.2	odd	2
2890.2.a.i.1.1		1	17.16	even	2
2890.2.b.g.2311.1		2	17.13	even	4
2890.2.b.g.2311.2		2	17.4	even	4
5440.2.a.f.1.1		1	8.3	odd	2
5440.2.a.w.1.1		1	8.5	even	2
6800.2.a.e.1.1		1	20.19	odd	2
7650.2.a.g.1.1		1	15.14	odd	2
8330.2.a.m.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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