LMFDB - Embedded newform 170.2.a.d.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)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +3.00000 q^{12} -3.00000 q^{13} -2.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{18} +3.00000 q^{19} -1.00000 q^{20} +6.00000 q^{21} +4.00000 q^{22} -6.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +3.00000 q^{26} +9.00000 q^{27} +2.00000 q^{28} +9.00000 q^{29} +3.00000 q^{30} -3.00000 q^{31} -1.00000 q^{32} -12.0000 q^{33} -1.00000 q^{34} -2.00000 q^{35} +6.00000 q^{36} -8.00000 q^{37} -3.00000 q^{38} -9.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -6.00000 q^{42} +6.00000 q^{43} -4.00000 q^{44} -6.00000 q^{45} +6.00000 q^{46} -13.0000 q^{47} +3.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +3.00000 q^{51} -3.00000 q^{52} -9.00000 q^{53} -9.00000 q^{54} +4.00000 q^{55} -2.00000 q^{56} +9.00000 q^{57} -9.00000 q^{58} +15.0000 q^{59} -3.00000 q^{60} +7.00000 q^{61} +3.00000 q^{62} +12.0000 q^{63} +1.00000 q^{64} +3.00000 q^{65} +12.0000 q^{66} -2.00000 q^{67} +1.00000 q^{68} -18.0000 q^{69} +2.00000 q^{70} +9.00000 q^{71} -6.00000 q^{72} -3.00000 q^{73} +8.00000 q^{74} +3.00000 q^{75} +3.00000 q^{76} -8.00000 q^{77} +9.00000 q^{78} -1.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +6.00000 q^{84} -1.00000 q^{85} -6.00000 q^{86} +27.0000 q^{87} +4.00000 q^{88} -9.00000 q^{89} +6.00000 q^{90} -6.00000 q^{91} -6.00000 q^{92} -9.00000 q^{93} +13.0000 q^{94} -3.00000 q^{95} -3.00000 q^{96} +7.00000 q^{97} +3.00000 q^{98} -24.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.00000	−1.22474
$7$	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
$9$	6.00000	2.00000
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	3.00000	0.866025
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−2.00000	−0.534522
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−6.00000	−1.41421
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−1.00000	−0.223607
$21$	6.00000	1.30931
$22$	4.00000	0.852803
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	3.00000	0.588348
$27$	9.00000	1.73205
$28$	2.00000	0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	3.00000	0.547723
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	−12.0000	−2.08893
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	6.00000	1.00000
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−3.00000	−0.486664
$39$	−9.00000	−1.44115
$40$	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	$0.655213\pi$
$42$	−6.00000	−0.925820
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−4.00000	−0.603023
$45$	−6.00000	−0.894427
$46$	6.00000	0.884652
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	3.00000	0.433013
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	3.00000	0.420084
$52$	−3.00000	−0.416025
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−9.00000	−1.22474
$55$	4.00000	0.539360
$56$	−2.00000	−0.267261
$57$	9.00000	1.19208
$58$	−9.00000	−1.18176
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	−3.00000	−0.387298
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	3.00000	0.381000
$63$	12.0000	1.51186
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	12.0000	1.47710
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	1.00000	0.121268
$69$	−18.0000	−2.16695
$70$	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	$0.320669\pi$
$72$	−6.00000	−0.707107
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	8.00000	0.929981
$75$	3.00000	0.346410
$76$	3.00000	0.344124
$77$	−8.00000	−0.911685
$78$	9.00000	1.01905
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	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	$0.271155\pi$
$84$	6.00000	0.654654
$85$	−1.00000	−0.108465
$86$	−6.00000	−0.646997
$87$	27.0000	2.89470
$88$	4.00000	0.426401
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	6.00000	0.632456
$91$	−6.00000	−0.628971
$92$	−6.00000	−0.625543
$93$	−9.00000	−0.933257
$94$	13.0000	1.34085
$95$	−3.00000	−0.307794
$96$	−3.00000	−0.306186
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	3.00000	0.303046
$99$	−24.0000	−2.41209
$100$	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	$0.596490\pi$
$102$	−3.00000	−0.297044
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	3.00000	0.294174
$105$	−6.00000	−0.585540
$106$	9.00000	0.874157
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	9.00000	0.866025
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−4.00000	−0.381385
$111$	−24.0000	−2.27798
$112$	2.00000	0.188982
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−9.00000	−0.842927
$115$	6.00000	0.559503
$116$	9.00000	0.835629
$117$	−18.0000	−1.66410
$118$	−15.0000	−1.38086
$119$	2.00000	0.183340
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	−7.00000	−0.633750
$123$	−18.0000	−1.62301
$124$	−3.00000	−0.269408
$125$	−1.00000	−0.0894427
$126$	−12.0000	−1.06904
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	−1.00000	−0.0883883
$129$	18.0000	1.58481
$130$	−3.00000	−0.263117
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−12.0000	−1.04447
$133$	6.00000	0.520266
$134$	2.00000	0.172774
$135$	−9.00000	−0.774597
$136$	−1.00000	−0.0857493
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	18.0000	1.53226
$139$	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
$141$	−39.0000	−3.28439
$142$	−9.00000	−0.755263
$143$	12.0000	1.00349
$144$	6.00000	0.500000
$145$	−9.00000	−0.747409
$146$	3.00000	0.248282
$147$	−9.00000	−0.742307
$148$	−8.00000	−0.657596
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	−3.00000	−0.244949
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−3.00000	−0.243332
$153$	6.00000	0.485071
$154$	8.00000	0.644658
$155$	3.00000	0.240966
$156$	−9.00000	−0.720577
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−27.0000	−2.14124
$160$	1.00000	0.0790569
$161$	−12.0000	−0.945732
$162$	−9.00000	−0.707107
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−6.00000	−0.468521
$165$	12.0000	0.934199
$166$	−12.0000	−0.931381
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	−6.00000	−0.462910
$169$	−4.00000	−0.307692
$170$	1.00000	0.0766965
$171$	18.0000	1.37649
$172$	6.00000	0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−27.0000	−2.04686
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	45.0000	3.38241
$178$	9.00000	0.674579
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−6.00000	−0.447214
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	6.00000	0.444750
$183$	21.0000	1.55236
$184$	6.00000	0.442326
$185$	8.00000	0.588172
$186$	9.00000	0.659912
$187$	−4.00000	−0.292509
$188$	−13.0000	−0.948122
$189$	18.0000	1.30931
$190$	3.00000	0.217643
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	3.00000	0.216506
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−7.00000	−0.502571
$195$	9.00000	0.644503
$196$	−3.00000	−0.214286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	24.0000	1.70561
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	−1.00000	−0.0707107
$201$	−6.00000	−0.423207
$202$	6.00000	0.422159
$203$	18.0000	1.26335
$204$	3.00000	0.210042
$205$	6.00000	0.419058
$206$	−16.0000	−1.11477
$207$	−36.0000	−2.50217
$208$	−3.00000	−0.208013
$209$	−12.0000	−0.830057
$210$	6.00000	0.414039
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−9.00000	−0.618123
$213$	27.0000	1.85001
$214$	−12.0000	−0.820303
$215$	−6.00000	−0.409197
$216$	−9.00000	−0.612372
$217$	−6.00000	−0.407307
$218$	−7.00000	−0.474100
$219$	−9.00000	−0.608164
$220$	4.00000	0.269680
$221$	−3.00000	−0.201802
$222$	24.0000	1.61077
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−2.00000	−0.133631
$225$	6.00000	0.400000
$226$	1.00000	0.0665190
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	9.00000	0.596040
$229$	−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
$232$	−9.00000	−0.590879
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	18.0000	1.17670
$235$	13.0000	0.848026
$236$	15.0000	0.976417
$237$	0	0
$238$	−2.00000	−0.129641
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	−3.00000	−0.193649
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	7.00000	0.448129
$245$	3.00000	0.191663
$246$	18.0000	1.14764
$247$	−9.00000	−0.572656
$248$	3.00000	0.190500
$249$	36.0000	2.28141
$250$	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	$0.376365\pi$
$252$	12.0000	0.755929
$253$	24.0000	1.50887
$254$	11.0000	0.690201
$255$	−3.00000	−0.187867
$256$	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	$0.689739\pi$
$258$	−18.0000	−1.12063
$259$	−16.0000	−0.994192
$260$	3.00000	0.186052
$261$	54.0000	3.34252
$262$	2.00000	0.123560
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	12.0000	0.738549
$265$	9.00000	0.552866
$266$	−6.00000	−0.367884
$267$	−27.0000	−1.65237
$268$	−2.00000	−0.122169
$269$	19.0000	1.15845	0.579225	−	0.815168i	$-0.303355\pi$
$269$	19.0000	1.15845	0.579225	+	0.815168i	$0.303355\pi$
$270$	9.00000	0.547723
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	1.00000	0.0606339
$273$	−18.0000	−1.08941
$274$	−22.0000	−1.32907
$275$	−4.00000	−0.241209
$276$	−18.0000	−1.08347
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−8.00000	−0.479808
$279$	−18.0000	−1.07763
$280$	2.00000	0.119523
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	39.0000	2.32242
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	9.00000	0.534052
$285$	−9.00000	−0.533114
$286$	−12.0000	−0.709575
$287$	−12.0000	−0.708338
$288$	−6.00000	−0.353553
$289$	1.00000	0.0588235
$290$	9.00000	0.528498
$291$	21.0000	1.23104
$292$	−3.00000	−0.175562
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	9.00000	0.524891
$295$	−15.0000	−0.873334
$296$	8.00000	0.464991
$297$	−36.0000	−2.08893
$298$	−16.0000	−0.926855
$299$	18.0000	1.04097
$300$	3.00000	0.173205
$301$	12.0000	0.691669
$302$	−6.00000	−0.345261
$303$	−18.0000	−1.03407
$304$	3.00000	0.172062
$305$	−7.00000	−0.400819
$306$	−6.00000	−0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−8.00000	−0.455842
$309$	48.0000	2.73062
$310$	−3.00000	−0.170389
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	9.00000	0.509525
$313$	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
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−16.0000	−0.898650	−0.449325	−	0.893368i	$-0.648335\pi$
$317$	−16.0000	−0.898650	−0.449325	+	0.893368i	$0.648335\pi$
$318$	27.0000	1.51408
$319$	−36.0000	−2.01561
$320$	−1.00000	−0.0559017
$321$	36.0000	2.00932
$322$	12.0000	0.668734
$323$	3.00000	0.166924
$324$	9.00000	0.500000
$325$	−3.00000	−0.166410
$326$	8.00000	0.443079
$327$	21.0000	1.16130
$328$	6.00000	0.331295
$329$	−26.0000	−1.43343
$330$	−12.0000	−0.660578
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	12.0000	0.658586
$333$	−48.0000	−2.63038
$334$	−2.00000	−0.109435
$335$	2.00000	0.109272
$336$	6.00000	0.327327
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	4.00000	0.217571
$339$	−3.00000	−0.162938
$340$	−1.00000	−0.0542326
$341$	12.0000	0.649836
$342$	−18.0000	−0.973329
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	18.0000	0.969087
$346$	6.00000	0.322562
$347$	−21.0000	−1.12734	−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000	−1.12734	−0.563670	+	0.826000i	$0.690611\pi$
$348$	27.0000	1.44735
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	−2.00000	−0.106904
$351$	−27.0000	−1.44115
$352$	4.00000	0.213201
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−45.0000	−2.39172
$355$	−9.00000	−0.477670
$356$	−9.00000	−0.476999
$357$	6.00000	0.317554
$358$	−20.0000	−1.05703
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	6.00000	0.316228
$361$	−10.0000	−0.526316
$362$	2.00000	0.105118
$363$	15.0000	0.787296
$364$	−6.00000	−0.314485
$365$	3.00000	0.157027
$366$	−21.0000	−1.09769
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−6.00000	−0.312772
$369$	−36.0000	−1.87409
$370$	−8.00000	−0.415900
$371$	−18.0000	−0.934513
$372$	−9.00000	−0.466628
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	4.00000	0.206835
$375$	−3.00000	−0.154919
$376$	13.0000	0.670424
$377$	−27.0000	−1.39057
$378$	−18.0000	−0.925820
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	−3.00000	−0.153897
$381$	−33.0000	−1.69064
$382$	−4.00000	−0.204658
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	−3.00000	−0.153093
$385$	8.00000	0.407718
$386$	18.0000	0.916176
$387$	36.0000	1.82998
$388$	7.00000	0.355371
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	−9.00000	−0.455733
$391$	−6.00000	−0.303433
$392$	3.00000	0.151523
$393$	−6.00000	−0.302660
$394$	12.0000	0.604551
$395$	0	0
$396$	−24.0000	−1.20605
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	13.0000	0.651631
$399$	18.0000	0.901127
$400$	1.00000	0.0500000
$401$	−40.0000	−1.99750	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−40.0000	−1.99750	−0.998752	+	0.0499376i	$0.984098\pi$
$402$	6.00000	0.299253
$403$	9.00000	0.448322
$404$	−6.00000	−0.298511
$405$	−9.00000	−0.447214
$406$	−18.0000	−0.893325
$407$	32.0000	1.58618
$408$	−3.00000	−0.148522
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	−6.00000	−0.296319
$411$	66.0000	3.25554
$412$	16.0000	0.788263
$413$	30.0000	1.47620
$414$	36.0000	1.76930
$415$	−12.0000	−0.589057
$416$	3.00000	0.147087
$417$	24.0000	1.17529
$418$	12.0000	0.586939
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	−6.00000	−0.292770
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	10.0000	0.486792
$423$	−78.0000	−3.79249
$424$	9.00000	0.437079
$425$	1.00000	0.0485071
$426$	−27.0000	−1.30815
$427$	14.0000	0.677507
$428$	12.0000	0.580042
$429$	36.0000	1.73810
$430$	6.00000	0.289346
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	9.00000	0.433013
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	6.00000	0.288009
$435$	−27.0000	−1.29455
$436$	7.00000	0.335239
$437$	−18.0000	−0.861057
$438$	9.00000	0.430037
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−4.00000	−0.190693
$441$	−18.0000	−0.857143
$442$	3.00000	0.142695
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−24.0000	−1.13899
$445$	9.00000	0.426641
$446$	7.00000	0.331460
$447$	48.0000	2.27032
$448$	2.00000	0.0944911
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−6.00000	−0.282843
$451$	24.0000	1.13012
$452$	−1.00000	−0.0470360
$453$	18.0000	0.845714
$454$	7.00000	0.328526
$455$	6.00000	0.281284
$456$	−9.00000	−0.421464
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	22.0000	1.02799
$459$	9.00000	0.420084
$460$	6.00000	0.279751
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	24.0000	1.11658
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	9.00000	0.417815
$465$	9.00000	0.417365
$466$	17.0000	0.787510
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−18.0000	−0.832050
$469$	−4.00000	−0.184703
$470$	−13.0000	−0.599645
$471$	30.0000	1.38233
$472$	−15.0000	−0.690431
$473$	−24.0000	−1.10352
$474$	0	0
$475$	3.00000	0.137649
$476$	2.00000	0.0916698
$477$	−54.0000	−2.47249
$478$	−14.0000	−0.640345
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	3.00000	0.136931
$481$	24.0000	1.09431
$482$	14.0000	0.637683
$483$	−36.0000	−1.63806
$484$	5.00000	0.227273
$485$	−7.00000	−0.317854
$486$	0	0
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	−7.00000	−0.316875
$489$	−24.0000	−1.08532
$490$	−3.00000	−0.135526
$491$	−11.0000	−0.496423	−0.248212	−	0.968706i	$-0.579843\pi$
$491$	−11.0000	−0.496423	−0.248212	+	0.968706i	$0.579843\pi$
$492$	−18.0000	−0.811503
$493$	9.00000	0.405340
$494$	9.00000	0.404929
$495$	24.0000	1.07872
$496$	−3.00000	−0.134704
$497$	18.0000	0.807410
$498$	−36.0000	−1.61320
$499$	−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
$501$	6.00000	0.268060
$502$	−12.0000	−0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−12.0000	−0.534522
$505$	6.00000	0.266996
$506$	−24.0000	−1.06693
$507$	−12.0000	−0.532939
$508$	−11.0000	−0.488046
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	3.00000	0.132842
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	27.0000	1.19208
$514$	18.0000	0.793946
$515$	−16.0000	−0.705044
$516$	18.0000	0.792406
$517$	52.0000	2.28696
$518$	16.0000	0.703000
$519$	−18.0000	−0.790112
$520$	−3.00000	−0.131559
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	−54.0000	−2.36352
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−2.00000	−0.0873704
$525$	6.00000	0.261861
$526$	−15.0000	−0.654031
$527$	−3.00000	−0.130682
$528$	−12.0000	−0.522233
$529$	13.0000	0.565217
$530$	−9.00000	−0.390935
$531$	90.0000	3.90567
$532$	6.00000	0.260133
$533$	18.0000	0.779667
$534$	27.0000	1.16840
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	60.0000	2.58919
$538$	−19.0000	−0.819148
$539$	12.0000	0.516877
$540$	−9.00000	−0.387298
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−4.00000	−0.171815
$543$	−6.00000	−0.257485
$544$	−1.00000	−0.0428746
$545$	−7.00000	−0.299847
$546$	18.0000	0.770329
$547$	−5.00000	−0.213785	−0.106892	−	0.994271i	$-0.534090\pi$
$547$	−5.00000	−0.213785	−0.106892	+	0.994271i	$0.534090\pi$
$548$	22.0000	0.939793
$549$	42.0000	1.79252
$550$	4.00000	0.170561
$551$	27.0000	1.15024
$552$	18.0000	0.766131
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	24.0000	1.01874
$556$	8.00000	0.339276
$557$	5.00000	0.211857	0.105928	−	0.994374i	$-0.466219\pi$
$557$	5.00000	0.211857	0.105928	+	0.994374i	$0.466219\pi$
$558$	18.0000	0.762001
$559$	−18.0000	−0.761319
$560$	−2.00000	−0.0845154
$561$	−12.0000	−0.506640
$562$	−19.0000	−0.801467
$563$	−38.0000	−1.60151	−0.800755	−	0.598993i	$-0.795568\pi$
$563$	−38.0000	−1.60151	−0.800755	+	0.598993i	$0.795568\pi$
$564$	−39.0000	−1.64220
$565$	1.00000	0.0420703
$566$	−1.00000	−0.0420331
$567$	18.0000	0.755929
$568$	−9.00000	−0.377632
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	9.00000	0.376969
$571$	−42.0000	−1.75765	−0.878823	−	0.477149i	$-0.841670\pi$
$571$	−42.0000	−1.75765	−0.878823	+	0.477149i	$0.841670\pi$
$572$	12.0000	0.501745
$573$	12.0000	0.501307
$574$	12.0000	0.500870
$575$	−6.00000	−0.250217
$576$	6.00000	0.250000
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	−1.00000	−0.0415945
$579$	−54.0000	−2.24416
$580$	−9.00000	−0.373705
$581$	24.0000	0.995688
$582$	−21.0000	−0.870478
$583$	36.0000	1.49097
$584$	3.00000	0.124141
$585$	18.0000	0.744208
$586$	−7.00000	−0.289167
$587$	22.0000	0.908037	0.454019	−	0.890992i	$-0.349990\pi$
$587$	22.0000	0.908037	0.454019	+	0.890992i	$0.349990\pi$
$588$	−9.00000	−0.371154
$589$	−9.00000	−0.370839
$590$	15.0000	0.617540
$591$	−36.0000	−1.48084
$592$	−8.00000	−0.328798
$593$	10.0000	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$594$	36.0000	1.47710
$595$	−2.00000	−0.0819920
$596$	16.0000	0.655386
$597$	−39.0000	−1.59616
$598$	−18.0000	−0.736075
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−3.00000	−0.122474
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	−12.0000	−0.489083
$603$	−12.0000	−0.488678
$604$	6.00000	0.244137
$605$	−5.00000	−0.203279
$606$	18.0000	0.731200
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−3.00000	−0.121666
$609$	54.0000	2.18819
$610$	7.00000	0.283422
$611$	39.0000	1.57777
$612$	6.00000	0.242536
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	−4.00000	−0.161427
$615$	18.0000	0.725830
$616$	8.00000	0.322329
$617$	−41.0000	−1.65060	−0.825299	−	0.564696i	$-0.808993\pi$
$617$	−41.0000	−1.65060	−0.825299	+	0.564696i	$0.808993\pi$
$618$	−48.0000	−1.93084
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	3.00000	0.120483
$621$	−54.0000	−2.16695
$622$	0	0
$623$	−18.0000	−0.721155
$624$	−9.00000	−0.360288
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	−36.0000	−1.43770
$628$	10.0000	0.399043
$629$	−8.00000	−0.318981
$630$	12.0000	0.478091
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	0	0
$633$	−30.0000	−1.19239
$634$	16.0000	0.635441
$635$	11.0000	0.436522
$636$	−27.0000	−1.07062
$637$	9.00000	0.356593
$638$	36.0000	1.42525
$639$	54.0000	2.13621
$640$	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
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−12.0000	−0.472866
$645$	−18.0000	−0.708749
$646$	−3.00000	−0.118033
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	−9.00000	−0.353553
$649$	−60.0000	−2.35521
$650$	3.00000	0.117670
$651$	−18.0000	−0.705476
$652$	−8.00000	−0.313304
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−21.0000	−0.821165
$655$	2.00000	0.0781465
$656$	−6.00000	−0.234261
$657$	−18.0000	−0.702247
$658$	26.0000	1.01359
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	12.0000	0.467099
$661$	44.0000	1.71140	0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000	1.71140	0.855701	+	0.517471i	$0.173126\pi$
$662$	−13.0000	−0.505259
$663$	−9.00000	−0.349531
$664$	−12.0000	−0.465690
$665$	−6.00000	−0.232670
$666$	48.0000	1.85996
$667$	−54.0000	−2.09089
$668$	2.00000	0.0773823
$669$	−21.0000	−0.811907
$670$	−2.00000	−0.0772667
$671$	−28.0000	−1.08093
$672$	−6.00000	−0.231455
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	−13.0000	−0.500741
$675$	9.00000	0.346410
$676$	−4.00000	−0.153846
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	3.00000	0.115214
$679$	14.0000	0.537271
$680$	1.00000	0.0383482
$681$	−21.0000	−0.804722
$682$	−12.0000	−0.459504
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	18.0000	0.688247
$685$	−22.0000	−0.840577
$686$	20.0000	0.763604
$687$	−66.0000	−2.51806
$688$	6.00000	0.228748
$689$	27.0000	1.02862
$690$	−18.0000	−0.685248
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	−48.0000	−1.82337
$694$	21.0000	0.797149
$695$	−8.00000	−0.303457
$696$	−27.0000	−1.02343
$697$	−6.00000	−0.227266
$698$	0	0
$699$	−51.0000	−1.92900
$700$	2.00000	0.0755929
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	27.0000	1.01905
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	39.0000	1.46882
$706$	−18.0000	−0.677439
$707$	−12.0000	−0.451306
$708$	45.0000	1.69120
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	9.00000	0.337764
$711$	0	0
$712$	9.00000	0.337289
$713$	18.0000	0.674105
$714$	−6.00000	−0.224544
$715$	−12.0000	−0.448775
$716$	20.0000	0.747435
$717$	42.0000	1.56852
$718$	6.00000	0.223918
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	−6.00000	−0.223607
$721$	32.0000	1.19174
$722$	10.0000	0.372161
$723$	−42.0000	−1.56200
$724$	−2.00000	−0.0743294
$725$	9.00000	0.334252
$726$	−15.0000	−0.556702
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	6.00000	0.222375
$729$	−27.0000	−1.00000
$730$	−3.00000	−0.111035
$731$	6.00000	0.221918
$732$	21.0000	0.776182
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−22.0000	−0.812035
$735$	9.00000	0.331970
$736$	6.00000	0.221163
$737$	8.00000	0.294684
$738$	36.0000	1.32518
$739$	49.0000	1.80249	0.901247	−	0.433306i	$-0.142653\pi$
$739$	49.0000	1.80249	0.901247	+	0.433306i	$0.142653\pi$
$740$	8.00000	0.294086
$741$	−27.0000	−0.991870
$742$	18.0000	0.660801
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	9.00000	0.329956
$745$	−16.0000	−0.586195
$746$	2.00000	0.0732252
$747$	72.0000	2.63434
$748$	−4.00000	−0.146254
$749$	24.0000	0.876941
$750$	3.00000	0.109545
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	−13.0000	−0.474061
$753$	36.0000	1.31191
$754$	27.0000	0.983282
$755$	−6.00000	−0.218362
$756$	18.0000	0.654654
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	32.0000	1.16229
$759$	72.0000	2.61343
$760$	3.00000	0.108821
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	33.0000	1.19546
$763$	14.0000	0.506834
$764$	4.00000	0.144715
$765$	−6.00000	−0.216930
$766$	−1.00000	−0.0361315
$767$	−45.0000	−1.62486
$768$	3.00000	0.108253
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	−8.00000	−0.288300
$771$	−54.0000	−1.94476
$772$	−18.0000	−0.647834
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−36.0000	−1.29399
$775$	−3.00000	−0.107763
$776$	−7.00000	−0.251285
$777$	−48.0000	−1.72199
$778$	−24.0000	−0.860442
$779$	−18.0000	−0.644917
$780$	9.00000	0.322252
$781$	−36.0000	−1.28818
$782$	6.00000	0.214560
$783$	81.0000	2.89470
$784$	−3.00000	−0.107143
$785$	−10.0000	−0.356915
$786$	6.00000	0.214013
$787$	3.00000	0.106938	0.0534692	−	0.998569i	$-0.482972\pi$
$787$	3.00000	0.106938	0.0534692	+	0.998569i	$0.482972\pi$
$788$	−12.0000	−0.427482
$789$	45.0000	1.60204
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	24.0000	0.852803
$793$	−21.0000	−0.745732
$794$	8.00000	0.283909
$795$	27.0000	0.957591
$796$	−13.0000	−0.460773
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−18.0000	−0.637193
$799$	−13.0000	−0.459907
$800$	−1.00000	−0.0353553
$801$	−54.0000	−1.90800
$802$	40.0000	1.41245
$803$	12.0000	0.423471
$804$	−6.00000	−0.211604
$805$	12.0000	0.422944
$806$	−9.00000	−0.317011
$807$	57.0000	2.00650
$808$	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	$0.533636\pi$
$810$	9.00000	0.316228
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	18.0000	0.631676
$813$	12.0000	0.420858
$814$	−32.0000	−1.12160
$815$	8.00000	0.280228
$816$	3.00000	0.105021
$817$	18.0000	0.629740
$818$	19.0000	0.664319
$819$	−36.0000	−1.25794
$820$	6.00000	0.209529
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	−66.0000	−2.30201
$823$	42.0000	1.46403	0.732014	−	0.681290i	$-0.238581\pi$
$823$	42.0000	1.46403	0.732014	+	0.681290i	$0.238581\pi$
$824$	−16.0000	−0.557386
$825$	−12.0000	−0.417786
$826$	−30.0000	−1.04383
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	−36.0000	−1.25109
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	12.0000	0.416526
$831$	6.00000	0.208138
$832$	−3.00000	−0.104006
$833$	−3.00000	−0.103944
$834$	−24.0000	−0.831052
$835$	−2.00000	−0.0692129
$836$	−12.0000	−0.415029
$837$	−27.0000	−0.933257
$838$	14.0000	0.483622
$839$	27.0000	0.932144	0.466072	−	0.884747i	$-0.345669\pi$
$839$	27.0000	0.932144	0.466072	+	0.884747i	$0.345669\pi$
$840$	6.00000	0.207020
$841$	52.0000	1.79310
$842$	4.00000	0.137849
$843$	57.0000	1.96318
$844$	−10.0000	−0.344214
$845$	4.00000	0.137604
$846$	78.0000	2.68170
$847$	10.0000	0.343604
$848$	−9.00000	−0.309061
$849$	3.00000	0.102960
$850$	−1.00000	−0.0342997
$851$	48.0000	1.64542
$852$	27.0000	0.925005
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	−14.0000	−0.479070
$855$	−18.0000	−0.615587
$856$	−12.0000	−0.410152
$857$	55.0000	1.87876	0.939382	−	0.342872i	$-0.111400\pi$
$857$	55.0000	1.87876	0.939382	+	0.342872i	$0.111400\pi$
$858$	−36.0000	−1.22902
$859$	−53.0000	−1.80834	−0.904168	−	0.427176i	$-0.859508\pi$
$859$	−53.0000	−1.80834	−0.904168	+	0.427176i	$0.859508\pi$
$860$	−6.00000	−0.204598
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−9.00000	−0.306186
$865$	6.00000	0.204006
$866$	26.0000	0.883516
$867$	3.00000	0.101885
$868$	−6.00000	−0.203653
$869$	0	0
$870$	27.0000	0.915386
$871$	6.00000	0.203302
$872$	−7.00000	−0.237050
$873$	42.0000	1.42148
$874$	18.0000	0.608859
$875$	−2.00000	−0.0676123
$876$	−9.00000	−0.304082
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	−32.0000	−1.07995
$879$	21.0000	0.708312
$880$	4.00000	0.134840
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	18.0000	0.606092
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	−3.00000	−0.100901
$885$	−45.0000	−1.51266
$886$	−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	$0.749103\pi$
$888$	24.0000	0.805387
$889$	−22.0000	−0.737856
$890$	−9.00000	−0.301681
$891$	−36.0000	−1.20605
$892$	−7.00000	−0.234377
$893$	−39.0000	−1.30509
$894$	−48.0000	−1.60536
$895$	−20.0000	−0.668526
$896$	−2.00000	−0.0668153
$897$	54.0000	1.80301
$898$	12.0000	0.400445
$899$	−27.0000	−0.900500
$900$	6.00000	0.200000
$901$	−9.00000	−0.299833
$902$	−24.0000	−0.799113
$903$	36.0000	1.19800
$904$	1.00000	0.0332595
$905$	2.00000	0.0664822
$906$	−18.0000	−0.598010
$907$	−11.0000	−0.365249	−0.182625	−	0.983183i	$-0.558459\pi$
$907$	−11.0000	−0.365249	−0.182625	+	0.983183i	$0.558459\pi$
$908$	−7.00000	−0.232303
$909$	−36.0000	−1.19404
$910$	−6.00000	−0.198898
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	9.00000	0.298020
$913$	−48.0000	−1.58857
$914$	30.0000	0.992312
$915$	−21.0000	−0.694239
$916$	−22.0000	−0.726900
$917$	−4.00000	−0.132092
$918$	−9.00000	−0.297044
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	−6.00000	−0.197814
$921$	12.0000	0.395413
$922$	14.0000	0.461065
$923$	−27.0000	−0.888716
$924$	−24.0000	−0.789542
$925$	−8.00000	−0.263038
$926$	−11.0000	−0.361482
$927$	96.0000	3.15305
$928$	−9.00000	−0.295439
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−9.00000	−0.295122
$931$	−9.00000	−0.294963
$932$	−17.0000	−0.556854
$933$	0	0
$934$	12.0000	0.392652
$935$	4.00000	0.130814
$936$	18.0000	0.588348
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	4.00000	0.130605
$939$	6.00000	0.195803
$940$	13.0000	0.424013
$941$	5.00000	0.162995	0.0814977	−	0.996674i	$-0.474030\pi$
$941$	5.00000	0.162995	0.0814977	+	0.996674i	$0.474030\pi$
$942$	−30.0000	−0.977453
$943$	36.0000	1.17232
$944$	15.0000	0.488208
$945$	−18.0000	−0.585540
$946$	24.0000	0.780307
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	−3.00000	−0.0973329
$951$	−48.0000	−1.55651
$952$	−2.00000	−0.0648204
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	54.0000	1.74831
$955$	−4.00000	−0.129437
$956$	14.0000	0.452792
$957$	−108.000	−3.49114
$958$	−27.0000	−0.872330
$959$	44.0000	1.42083
$960$	−3.00000	−0.0968246
$961$	−22.0000	−0.709677
$962$	−24.0000	−0.773791
$963$	72.0000	2.32017
$964$	−14.0000	−0.450910
$965$	18.0000	0.579441
$966$	36.0000	1.15828
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−5.00000	−0.160706
$969$	9.00000	0.289122
$970$	7.00000	0.224756
$971$	55.0000	1.76503	0.882517	−	0.470281i	$-0.155847\pi$
$971$	55.0000	1.76503	0.882517	+	0.470281i	$0.155847\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	−6.00000	−0.192252
$975$	−9.00000	−0.288231
$976$	7.00000	0.224065
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	24.0000	0.767435
$979$	36.0000	1.15056
$980$	3.00000	0.0958315
$981$	42.0000	1.34096
$982$	11.0000	0.351024
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	18.0000	0.573819
$985$	12.0000	0.382352
$986$	−9.00000	−0.286618
$987$	−78.0000	−2.48277
$988$	−9.00000	−0.286328
$989$	−36.0000	−1.14473
$990$	−24.0000	−0.762770
$991$	−43.0000	−1.36594	−0.682970	−	0.730446i	$-0.739312\pi$
$991$	−43.0000	−1.36594	−0.682970	+	0.730446i	$0.739312\pi$
$992$	3.00000	0.0952501
$993$	39.0000	1.23763
$994$	−18.0000	−0.570925
$995$	13.0000	0.412128
$996$	36.0000	1.14070
$997$	34.0000	1.07679	0.538395	−	0.842692i	$-0.319031\pi$
$997$	34.0000	1.07679	0.538395	+	0.842692i	$0.319031\pi$
$998$	10.0000	0.316544
$999$	−72.0000	−2.27798

Twists

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

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