LMFDB - Embedded newform 1339.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$1$
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	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -4.00000 q^{22} +3.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} -4.00000 q^{28} -3.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} +4.00000 q^{33} +3.00000 q^{34} +4.00000 q^{35} +2.00000 q^{36} -5.00000 q^{37} -5.00000 q^{38} +1.00000 q^{39} -3.00000 q^{40} -4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +1.00000 q^{52} -12.0000 q^{53} +5.00000 q^{54} -4.00000 q^{55} -12.0000 q^{56} +5.00000 q^{57} -3.00000 q^{58} -5.00000 q^{59} +1.00000 q^{60} -5.00000 q^{61} -8.00000 q^{63} +7.00000 q^{64} -1.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} -3.00000 q^{68} +4.00000 q^{70} -6.00000 q^{71} +6.00000 q^{72} +5.00000 q^{73} -5.00000 q^{74} +4.00000 q^{75} +5.00000 q^{76} -16.0000 q^{77} +1.00000 q^{78} +6.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{83} +4.00000 q^{84} +3.00000 q^{85} -12.0000 q^{86} +3.00000 q^{87} +12.0000 q^{88} -9.00000 q^{89} -2.00000 q^{90} -4.00000 q^{91} +6.00000 q^{94} -5.00000 q^{95} -5.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +8.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	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	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.00000	1.06904
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	−4.00000	−0.800000
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	−4.00000	−0.755929
$29$	−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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	3.00000	0.514496
$35$	4.00000	0.676123
$36$	2.00000	0.333333
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	−5.00000	−0.811107
$39$	1.00000	0.160128
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−4.00000	−0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	4.00000	0.603023
$45$	−2.00000	−0.298142
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	1.00000	0.138675
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	5.00000	0.680414
$55$	−4.00000	−0.539360
$56$	−12.0000	−1.60357
$57$	5.00000	0.662266
$58$	−3.00000	−0.393919
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	1.00000	0.129099
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−8.00000	−1.00791
$64$	7.00000	0.875000
$65$	−1.00000	−0.124035
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	4.00000	0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	6.00000	0.707107
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	−5.00000	−0.581238
$75$	4.00000	0.461880
$76$	5.00000	0.573539
$77$	−16.0000	−1.82337
$78$	1.00000	0.113228
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	4.00000	0.436436
$85$	3.00000	0.325396
$86$	−12.0000	−1.29399
$87$	3.00000	0.321634
$88$	12.0000	1.27920
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	−2.00000	−0.210819
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	6.00000	0.618853
$95$	−5.00000	−0.512989
$96$	−5.00000	−0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	9.00000	0.909137
$99$	8.00000	0.804030
$100$	4.00000	0.400000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−3.00000	−0.297044
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	3.00000	0.294174
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	−5.00000	−0.481125
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−4.00000	−0.381385
$111$	5.00000	0.474579
$112$	−4.00000	−0.377964
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	3.00000	0.278543
$117$	2.00000	0.184900
$118$	−5.00000	−0.460287
$119$	12.0000	1.10004
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	−5.00000	−0.452679
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	−8.00000	−0.712697
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	−3.00000	−0.265165
$129$	12.0000	1.05654
$130$	−1.00000	−0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−4.00000	−0.348155
$133$	−20.0000	−1.73422
$134$	−4.00000	−0.345547
$135$	5.00000	0.430331
$136$	−9.00000	−0.771744
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−4.00000	−0.338062
$141$	−6.00000	−0.505291
$142$	−6.00000	−0.503509
$143$	4.00000	0.334497
$144$	2.00000	0.166667
$145$	−3.00000	−0.249136
$146$	5.00000	0.413803
$147$	−9.00000	−0.742307
$148$	5.00000	0.410997
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	4.00000	0.326599
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	15.0000	1.21666
$153$	−6.00000	−0.485071
$154$	−16.0000	−1.28932
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	6.00000	0.477334
$159$	12.0000	0.951662
$160$	5.00000	0.395285
$161$	0	0
$162$	1.00000	0.0785674
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	−1.00000	−0.0776151
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	12.0000	0.925820
$169$	1.00000	0.0769231
$170$	3.00000	0.230089
$171$	10.0000	0.764719
$172$	12.0000	0.914991
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	3.00000	0.227429
$175$	−16.0000	−1.20949
$176$	4.00000	0.301511
$177$	5.00000	0.375823
$178$	−9.00000	−0.674579
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	2.00000	0.149071
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	−4.00000	−0.296500
$183$	5.00000	0.369611
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−6.00000	−0.437595
$189$	20.0000	1.45479
$190$	−5.00000	−0.362738
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	−7.00000	−0.505181
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	10.0000	0.717958
$195$	1.00000	0.0716115
$196$	−9.00000	−0.642857
$197$	−23.0000	−1.63868	−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000	−1.63868	−0.819341	+	0.573306i	$0.805660\pi$
$198$	8.00000	0.568535
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	12.0000	0.848528
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	−12.0000	−0.842235
$204$	3.00000	0.210042
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	1.00000	0.0693375
$209$	20.0000	1.38343
$210$	−4.00000	−0.276026
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	12.0000	0.824163
$213$	6.00000	0.411113
$214$	−14.0000	−0.957020
$215$	−12.0000	−0.818393
$216$	−15.0000	−1.02062
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−5.00000	−0.337869
$220$	4.00000	0.269680
$221$	−3.00000	−0.201802
$222$	5.00000	0.335578
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	20.0000	1.33631
$225$	8.00000	0.533333
$226$	−10.0000	−0.665190
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−5.00000	−0.331133
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	9.00000	0.590879
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	2.00000	0.130744
$235$	6.00000	0.391397
$236$	5.00000	0.325472
$237$	−6.00000	−0.389742
$238$	12.0000	0.777844
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	1.00000	0.0645497
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	5.00000	0.321412
$243$	−16.0000	−1.02640
$244$	5.00000	0.320092
$245$	9.00000	0.574989
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	1.00000	0.0633724
$250$	−9.00000	−0.569210
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	8.00000	0.503953
$253$	0	0
$254$	−9.00000	−0.564710
$255$	−3.00000	−0.187867
$256$	−17.0000	−1.06250
$257$	−4.00000	−0.249513	−0.124757	−	0.992187i	$-0.539815\pi$
$257$	−4.00000	−0.249513	−0.124757	+	0.992187i	$0.539815\pi$
$258$	12.0000	0.747087
$259$	−20.0000	−1.24274
$260$	1.00000	0.0620174
$261$	6.00000	0.371391
$262$	0	0
$263$	23.0000	1.41824	0.709120	−	0.705087i	$-0.249092\pi$
$263$	23.0000	1.41824	0.709120	+	0.705087i	$0.249092\pi$
$264$	−12.0000	−0.738549
$265$	−12.0000	−0.737154
$266$	−20.0000	−1.22628
$267$	9.00000	0.550791
$268$	4.00000	0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	5.00000	0.304290
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−3.00000	−0.181902
$273$	4.00000	0.242091
$274$	4.00000	0.241649
$275$	16.0000	0.964836
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	−12.0000	−0.717137
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	−6.00000	−0.357295
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	6.00000	0.356034
$285$	5.00000	0.296174
$286$	4.00000	0.236525
$287$	0	0
$288$	−10.0000	−0.589256
$289$	−8.00000	−0.470588
$290$	−3.00000	−0.176166
$291$	−10.0000	−0.586210
$292$	−5.00000	−0.292603
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−9.00000	−0.524891
$295$	−5.00000	−0.291111
$296$	15.0000	0.871857
$297$	−20.0000	−1.16052
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−4.00000	−0.230940
$301$	−48.0000	−2.76667
$302$	16.0000	0.920697
$303$	−10.0000	−0.574485
$304$	5.00000	0.286770
$305$	−5.00000	−0.286299
$306$	−6.00000	−0.342997
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	16.0000	0.911685
$309$	1.00000	0.0568880
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−3.00000	−0.169842
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	10.0000	0.564333
$315$	−8.00000	−0.450749
$316$	−6.00000	−0.337526
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	12.0000	0.672927
$319$	12.0000	0.671871
$320$	7.00000	0.391312
$321$	14.0000	0.781404
$322$	0	0
$323$	−15.0000	−0.834622
$324$	−1.00000	−0.0555556
$325$	4.00000	0.221880
$326$	11.0000	0.609234
$327$	2.00000	0.110600
$328$	0	0
$329$	24.0000	1.32316
$330$	4.00000	0.220193
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	1.00000	0.0548821
$333$	10.0000	0.547997
$334$	9.00000	0.492458
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	1.00000	0.0543928
$339$	10.0000	0.543125
$340$	−3.00000	−0.162698
$341$	0	0
$342$	10.0000	0.540738
$343$	8.00000	0.431959
$344$	36.0000	1.94099
$345$	0	0
$346$	24.0000	1.29025
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−3.00000	−0.160817
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	−16.0000	−0.855236
$351$	−5.00000	−0.266880
$352$	−20.0000	−1.06600
$353$	−33.0000	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$353$	−33.0000	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$354$	5.00000	0.265747
$355$	−6.00000	−0.318447
$356$	9.00000	0.476999
$357$	−12.0000	−0.635107
$358$	8.00000	0.422813
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	6.00000	0.315789
$362$	18.0000	0.946059
$363$	−5.00000	−0.262432
$364$	4.00000	0.209657
$365$	5.00000	0.261712
$366$	5.00000	0.261354
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	0	0
$369$	0	0
$370$	−5.00000	−0.259938
$371$	−48.0000	−2.49204
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−12.0000	−0.620505
$375$	9.00000	0.464758
$376$	−18.0000	−0.928279
$377$	3.00000	0.154508
$378$	20.0000	1.02869
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	5.00000	0.256495
$381$	9.00000	0.461084
$382$	17.0000	0.869796
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	3.00000	0.153093
$385$	−16.0000	−0.815436
$386$	−25.0000	−1.27247
$387$	24.0000	1.21999
$388$	−10.0000	−0.507673
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	1.00000	0.0506370
$391$	0	0
$392$	−27.0000	−1.36371
$393$	0	0
$394$	−23.0000	−1.15872
$395$	6.00000	0.301893
$396$	−8.00000	−0.402015
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	25.0000	1.25314
$399$	20.0000	1.00125
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	−12.0000	−0.595550
$407$	20.0000	0.991363
$408$	9.00000	0.445566
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	1.00000	0.0492665
$413$	−20.0000	−0.984136
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	−5.00000	−0.245145
$417$	−8.00000	−0.391762
$418$	20.0000	0.978232
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	4.00000	0.195180
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	19.0000	0.924906
$423$	−12.0000	−0.583460
$424$	36.0000	1.74831
$425$	−12.0000	−0.582086
$426$	6.00000	0.290701
$427$	−20.0000	−0.967868
$428$	14.0000	0.676716
$429$	−4.00000	−0.193122
$430$	−12.0000	−0.578691
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−5.00000	−0.240563
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	2.00000	0.0957826
$437$	0	0
$438$	−5.00000	−0.238909
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	12.0000	0.572078
$441$	−18.0000	−0.857143
$442$	−3.00000	−0.142695
$443$	−33.0000	−1.56788	−0.783939	−	0.620838i	$-0.786792\pi$
$443$	−33.0000	−1.56788	−0.783939	+	0.620838i	$0.786792\pi$
$444$	−5.00000	−0.237289
$445$	−9.00000	−0.426641
$446$	1.00000	0.0473514
$447$	18.0000	0.851371
$448$	28.0000	1.32288
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	10.0000	0.470360
$453$	−16.0000	−0.751746
$454$	14.0000	0.657053
$455$	−4.00000	−0.187523
$456$	−15.0000	−0.702439
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	20.0000	0.934539
$459$	15.0000	0.700140
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	16.0000	0.744387
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−4.00000	−0.185296
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−2.00000	−0.0924500
$469$	−16.0000	−0.738811
$470$	6.00000	0.276759
$471$	−10.0000	−0.460776
$472$	15.0000	0.690431
$473$	48.0000	2.20704
$474$	−6.00000	−0.275589
$475$	20.0000	0.917663
$476$	−12.0000	−0.550019
$477$	24.0000	1.09888
$478$	−1.00000	−0.0457389
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	−5.00000	−0.228218
$481$	5.00000	0.227980
$482$	11.0000	0.501036
$483$	0	0
$484$	−5.00000	−0.227273
$485$	10.0000	0.454077
$486$	−16.0000	−0.725775
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	15.0000	0.679018
$489$	−11.0000	−0.497437
$490$	9.00000	0.406579
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	5.00000	0.224961
$495$	8.00000	0.359573
$496$	0	0
$497$	−24.0000	−1.07655
$498$	1.00000	0.0448111
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	9.00000	0.402492
$501$	−9.00000	−0.402090
$502$	−21.0000	−0.937276
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	24.0000	1.06904
$505$	10.0000	0.444994
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	9.00000	0.399310
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	−3.00000	−0.132842
$511$	20.0000	0.884748
$512$	−11.0000	−0.486136
$513$	−25.0000	−1.10378
$514$	−4.00000	−0.176432
$515$	−1.00000	−0.0440653
$516$	−12.0000	−0.528271
$517$	−24.0000	−1.05552
$518$	−20.0000	−0.878750
$519$	−24.0000	−1.05348
$520$	3.00000	0.131559
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	6.00000	0.262613
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	16.0000	0.698297
$526$	23.0000	1.00285
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	10.0000	0.433963
$532$	20.0000	0.867110
$533$	0	0
$534$	9.00000	0.389468
$535$	−14.0000	−0.605273
$536$	12.0000	0.518321
$537$	−8.00000	−0.345225
$538$	2.00000	0.0862261
$539$	−36.0000	−1.55063
$540$	−5.00000	−0.215166
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	16.0000	0.687259
$543$	−18.0000	−0.772454
$544$	15.0000	0.643120
$545$	−2.00000	−0.0856706
$546$	4.00000	0.171184
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−4.00000	−0.170872
$549$	10.0000	0.426790
$550$	16.0000	0.682242
$551$	15.0000	0.639021
$552$	0	0
$553$	24.0000	1.02058
$554$	−2.00000	−0.0849719
$555$	5.00000	0.212238
$556$	−8.00000	−0.339276
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	−4.00000	−0.169031
$561$	12.0000	0.506640
$562$	−27.0000	−1.13893
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	6.00000	0.252646
$565$	−10.0000	−0.420703
$566$	−13.0000	−0.546431
$567$	4.00000	0.167984
$568$	18.0000	0.755263
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	5.00000	0.209427
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	−4.00000	−0.167248
$573$	−17.0000	−0.710185
$574$	0	0
$575$	0	0
$576$	−14.0000	−0.583333
$577$	−15.0000	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$577$	−15.0000	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$578$	−8.00000	−0.332756
$579$	25.0000	1.03896
$580$	3.00000	0.124568
$581$	−4.00000	−0.165948
$582$	−10.0000	−0.414513
$583$	48.0000	1.98796
$584$	−15.0000	−0.620704
$585$	2.00000	0.0826898
$586$	−6.00000	−0.247858
$587$	−48.0000	−1.98117	−0.990586	−	0.136892i	$-0.956289\pi$
$587$	−48.0000	−1.98117	−0.990586	+	0.136892i	$0.956289\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	−5.00000	−0.205847
$591$	23.0000	0.946094
$592$	5.00000	0.205499
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	−20.0000	−0.820610
$595$	12.0000	0.491952
$596$	18.0000	0.737309
$597$	−25.0000	−1.02318
$598$	0	0
$599$	45.0000	1.83865	0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000	1.83865	0.919325	+	0.393499i	$0.128735\pi$
$600$	−12.0000	−0.489898
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	−48.0000	−1.95633
$603$	8.00000	0.325785
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	−10.0000	−0.406222
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−25.0000	−1.01388
$609$	12.0000	0.486265
$610$	−5.00000	−0.202444
$611$	−6.00000	−0.242734
$612$	6.00000	0.242536
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	48.0000	1.93398
$617$	−41.0000	−1.65060	−0.825299	−	0.564696i	$-0.808993\pi$
$617$	−41.0000	−1.65060	−0.825299	+	0.564696i	$0.808993\pi$
$618$	1.00000	0.0402259
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	0	0
$622$	−20.0000	−0.801927
$623$	−36.0000	−1.44231
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	13.0000	0.519584
$627$	−20.0000	−0.798723
$628$	−10.0000	−0.399043
$629$	−15.0000	−0.598089
$630$	−8.00000	−0.318728
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	−18.0000	−0.716002
$633$	−19.0000	−0.755182
$634$	10.0000	0.397151
$635$	−9.00000	−0.357154
$636$	−12.0000	−0.475831
$637$	−9.00000	−0.356593
$638$	12.0000	0.475085
$639$	12.0000	0.474713
$640$	−3.00000	−0.118585
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	14.0000	0.552536
$643$	35.0000	1.38027	0.690133	−	0.723683i	$-0.257552\pi$
$643$	35.0000	1.38027	0.690133	+	0.723683i	$0.257552\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	−15.0000	−0.590167
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	−3.00000	−0.117851
$649$	20.0000	0.785069
$650$	4.00000	0.156893
$651$	0	0
$652$	−11.0000	−0.430793
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	24.0000	0.935617
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	−4.00000	−0.155700
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−2.00000	−0.0777322
$663$	3.00000	0.116510
$664$	3.00000	0.116423
$665$	−20.0000	−0.775567
$666$	10.0000	0.387492
$667$	0	0
$668$	−9.00000	−0.348220
$669$	−1.00000	−0.0386622
$670$	−4.00000	−0.154533
$671$	20.0000	0.772091
$672$	−20.0000	−0.771517
$673$	−11.0000	−0.424019	−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000	−0.424019	−0.212009	+	0.977268i	$0.568001\pi$
$674$	−25.0000	−0.962964
$675$	−20.0000	−0.769800
$676$	−1.00000	−0.0384615
$677$	−47.0000	−1.80636	−0.903178	−	0.429265i	$-0.858772\pi$
$677$	−47.0000	−1.80636	−0.903178	+	0.429265i	$0.858772\pi$
$678$	10.0000	0.384048
$679$	40.0000	1.53506
$680$	−9.00000	−0.345134
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	−10.0000	−0.382360
$685$	4.00000	0.152832
$686$	8.00000	0.305441
$687$	−20.0000	−0.763048
$688$	12.0000	0.457496
$689$	12.0000	0.457164
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	−24.0000	−0.912343
$693$	32.0000	1.21558
$694$	28.0000	1.06287
$695$	8.00000	0.303457
$696$	−9.00000	−0.341144
$697$	0	0
$698$	7.00000	0.264954
$699$	4.00000	0.151294
$700$	16.0000	0.604743
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	−5.00000	−0.188713
$703$	25.0000	0.942893
$704$	−28.0000	−1.05529
$705$	−6.00000	−0.225973
$706$	−33.0000	−1.24197
$707$	40.0000	1.50435
$708$	−5.00000	−0.187912
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	−6.00000	−0.225176
$711$	−12.0000	−0.450035
$712$	27.0000	1.01187
$713$	0	0
$714$	−12.0000	−0.449089
$715$	4.00000	0.149592
$716$	−8.00000	−0.298974
$717$	1.00000	0.0373457
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	2.00000	0.0745356
$721$	−4.00000	−0.148968
$722$	6.00000	0.223297
$723$	−11.0000	−0.409094
$724$	−18.0000	−0.668965
$725$	12.0000	0.445669
$726$	−5.00000	−0.185567
$727$	51.0000	1.89149	0.945743	−	0.324917i	$-0.105336\pi$
$727$	51.0000	1.89149	0.945743	+	0.324917i	$0.105336\pi$
$728$	12.0000	0.444750
$729$	13.0000	0.481481
$730$	5.00000	0.185058
$731$	−36.0000	−1.33151
$732$	−5.00000	−0.184805
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−26.0000	−0.959678
$735$	−9.00000	−0.331970
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	5.00000	0.183804
$741$	−5.00000	−0.183680
$742$	−48.0000	−1.76214
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	10.0000	0.366126
$747$	2.00000	0.0731762
$748$	12.0000	0.438763
$749$	−56.0000	−2.04620
$750$	9.00000	0.328634
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−6.00000	−0.218797
$753$	21.0000	0.765283
$754$	3.00000	0.109254
$755$	16.0000	0.582300
$756$	−20.0000	−0.727393
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	−30.0000	−1.08965
$759$	0	0
$760$	15.0000	0.544107
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	9.00000	0.326036
$763$	−8.00000	−0.289619
$764$	−17.0000	−0.615038
$765$	−6.00000	−0.216930
$766$	−18.0000	−0.650366
$767$	5.00000	0.180540
$768$	17.0000	0.613435
$769$	−33.0000	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$769$	−33.0000	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$770$	−16.0000	−0.576600
$771$	4.00000	0.144056
$772$	25.0000	0.899770
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	−30.0000	−1.07694
$777$	20.0000	0.717496
$778$	4.00000	0.143407
$779$	0	0
$780$	−1.00000	−0.0358057
$781$	24.0000	0.858788
$782$	0	0
$783$	−15.0000	−0.536056
$784$	−9.00000	−0.321429
$785$	10.0000	0.356915
$786$	0	0
$787$	15.0000	0.534692	0.267346	−	0.963601i	$-0.413853\pi$
$787$	15.0000	0.534692	0.267346	+	0.963601i	$0.413853\pi$
$788$	23.0000	0.819341
$789$	−23.0000	−0.818822
$790$	6.00000	0.213470
$791$	−40.0000	−1.42224
$792$	−24.0000	−0.852803
$793$	5.00000	0.177555
$794$	5.00000	0.177443
$795$	12.0000	0.425596
$796$	−25.0000	−0.886102
$797$	−35.0000	−1.23976	−0.619882	−	0.784695i	$-0.712819\pi$
$797$	−35.0000	−1.23976	−0.619882	+	0.784695i	$0.712819\pi$
$798$	20.0000	0.707992
$799$	18.0000	0.636794
$800$	−20.0000	−0.707107
$801$	18.0000	0.635999
$802$	−6.00000	−0.211867
$803$	−20.0000	−0.705785
$804$	−4.00000	−0.141069
$805$	0	0
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	−30.0000	−1.05540
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	1.00000	0.0351364
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	12.0000	0.421117
$813$	−16.0000	−0.561144
$814$	20.0000	0.701000
$815$	11.0000	0.385313
$816$	3.00000	0.105021
$817$	60.0000	2.09913
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	28.0000	0.977207	0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000	0.977207	0.488603	+	0.872506i	$0.337507\pi$
$822$	−4.00000	−0.139516
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	3.00000	0.104510
$825$	−16.0000	−0.557048
$826$	−20.0000	−0.695889
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	−1.00000	−0.0347105
$831$	2.00000	0.0693792
$832$	−7.00000	−0.242681
$833$	27.0000	0.935495
$834$	−8.00000	−0.277017
$835$	9.00000	0.311458
$836$	−20.0000	−0.691714
$837$	0	0
$838$	−16.0000	−0.552711
$839$	35.0000	1.20833	0.604167	−	0.796858i	$-0.293506\pi$
$839$	35.0000	1.20833	0.604167	+	0.796858i	$0.293506\pi$
$840$	12.0000	0.414039
$841$	−20.0000	−0.689655
$842$	−2.00000	−0.0689246
$843$	27.0000	0.929929
$844$	−19.0000	−0.654007
$845$	1.00000	0.0344010
$846$	−12.0000	−0.412568
$847$	20.0000	0.687208
$848$	12.0000	0.412082
$849$	13.0000	0.446159
$850$	−12.0000	−0.411597
$851$	0	0
$852$	−6.00000	−0.205557
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−20.0000	−0.684386
$855$	10.0000	0.341993
$856$	42.0000	1.43553
$857$	51.0000	1.74213	0.871063	−	0.491171i	$-0.163431\pi$
$857$	51.0000	1.74213	0.871063	+	0.491171i	$0.163431\pi$
$858$	−4.00000	−0.136558
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	25.0000	0.850517
$865$	24.0000	0.816024
$866$	−4.00000	−0.135926
$867$	8.00000	0.271694
$868$	0	0
$869$	−24.0000	−0.814144
$870$	3.00000	0.101710
$871$	4.00000	0.135535
$872$	6.00000	0.203186
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−36.0000	−1.21702
$876$	5.00000	0.168934
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	−8.00000	−0.269987
$879$	6.00000	0.202375
$880$	4.00000	0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−18.0000	−0.606092
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	3.00000	0.100901
$885$	5.00000	0.168073
$886$	−33.0000	−1.10866
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	−15.0000	−0.503367
$889$	−36.0000	−1.20740
$890$	−9.00000	−0.301681
$891$	−4.00000	−0.134005
$892$	−1.00000	−0.0334825
$893$	−30.0000	−1.00391
$894$	18.0000	0.602010
$895$	8.00000	0.267411
$896$	−12.0000	−0.400892
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	−8.00000	−0.266667
$901$	−36.0000	−1.19933
$902$	0	0
$903$	48.0000	1.59734
$904$	30.0000	0.997785
$905$	18.0000	0.598340
$906$	−16.0000	−0.531564
$907$	−42.0000	−1.39459	−0.697294	−	0.716786i	$-0.745613\pi$
$907$	−42.0000	−1.39459	−0.697294	+	0.716786i	$0.745613\pi$
$908$	−14.0000	−0.464606
$909$	−20.0000	−0.663358
$910$	−4.00000	−0.132599
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	−5.00000	−0.165567
$913$	4.00000	0.132381
$914$	−14.0000	−0.463079
$915$	5.00000	0.165295
$916$	−20.0000	−0.660819
$917$	0	0
$918$	15.0000	0.495074
$919$	−27.0000	−0.890648	−0.445324	−	0.895370i	$-0.646911\pi$
$919$	−27.0000	−0.890648	−0.445324	+	0.895370i	$0.646911\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	14.0000	0.461065
$923$	6.00000	0.197492
$924$	−16.0000	−0.526361
$925$	20.0000	0.657596
$926$	22.0000	0.722965
$927$	2.00000	0.0656886
$928$	−15.0000	−0.492399
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	0	0
$931$	−45.0000	−1.47482
$932$	4.00000	0.131024
$933$	20.0000	0.654771
$934$	−8.00000	−0.261768
$935$	−12.0000	−0.392442
$936$	−6.00000	−0.196116
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−16.0000	−0.522419
$939$	−13.0000	−0.424239
$940$	−6.00000	−0.195698
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−10.0000	−0.325818
$943$	0	0
$944$	5.00000	0.162736
$945$	20.0000	0.650600
$946$	48.0000	1.56061
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	6.00000	0.194871
$949$	−5.00000	−0.162307
$950$	20.0000	0.648886
$951$	−10.0000	−0.324272
$952$	−36.0000	−1.16677
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	24.0000	0.777029
$955$	17.0000	0.550107
$956$	1.00000	0.0323423
$957$	−12.0000	−0.387905
$958$	2.00000	0.0646171
$959$	16.0000	0.516667
$960$	−7.00000	−0.225924
$961$	−31.0000	−1.00000
$962$	5.00000	0.161206
$963$	28.0000	0.902287
$964$	−11.0000	−0.354286
$965$	−25.0000	−0.804778
$966$	0	0
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	−15.0000	−0.482118
$969$	15.0000	0.481869
$970$	10.0000	0.321081
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	16.0000	0.513200
$973$	32.0000	1.02587
$974$	2.00000	0.0640841
$975$	−4.00000	−0.128103
$976$	5.00000	0.160046
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	−11.0000	−0.351741
$979$	36.0000	1.15056
$980$	−9.00000	−0.287494
$981$	4.00000	0.127710
$982$	−20.0000	−0.638226
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	0	0
$985$	−23.0000	−0.732841
$986$	−9.00000	−0.286618
$987$	−24.0000	−0.763928
$988$	−5.00000	−0.159071
$989$	0	0
$990$	8.00000	0.254257
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	2.00000	0.0634681
$994$	−24.0000	−0.761234
$995$	25.0000	0.792553
$996$	−1.00000	−0.0316862
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	4.00000	0.126618
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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