LMFDB - Embedded newform 1849.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} +3.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -5.00000 q^{13} +3.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} -7.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} +5.00000 q^{26} +5.00000 q^{27} +3.00000 q^{28} -3.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -5.00000 q^{32} -3.00000 q^{34} -3.00000 q^{35} +2.00000 q^{36} +9.00000 q^{37} +1.00000 q^{38} +5.00000 q^{39} +3.00000 q^{40} -10.0000 q^{41} -3.00000 q^{42} -2.00000 q^{45} +7.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} -3.00000 q^{51} +5.00000 q^{52} -5.00000 q^{53} -5.00000 q^{54} -9.00000 q^{56} +1.00000 q^{57} +3.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +13.0000 q^{61} -5.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -5.00000 q^{65} -3.00000 q^{67} -3.00000 q^{68} +7.00000 q^{69} +3.00000 q^{70} +1.00000 q^{71} -6.00000 q^{72} -11.0000 q^{73} -9.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -5.00000 q^{78} -5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +9.00000 q^{83} -3.00000 q^{84} +3.00000 q^{85} +3.00000 q^{87} +1.00000 q^{89} +2.00000 q^{90} +15.0000 q^{91} +7.00000 q^{92} -5.00000 q^{93} +8.00000 q^{94} -1.00000 q^{95} +5.00000 q^{96} -2.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	3.00000	1.06066
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.00000	0.801784
$15$	−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
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	−3.00000	−0.612372
$25$	−4.00000	−0.800000
$26$	5.00000	0.980581
$27$	5.00000	0.962250
$28$	3.00000	0.566947
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	1.00000	0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−3.00000	−0.507093
$36$	2.00000	0.333333
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	1.00000	0.162221
$39$	5.00000	0.800641
$40$	3.00000	0.474342
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−3.00000	−0.462910
$43$	0	0
$43$	0	0
$44$	0	0
$45$	−2.00000	−0.298142
$46$	7.00000	1.03209
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	−3.00000	−0.420084
$52$	5.00000	0.693375
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−9.00000	−1.20268
$57$	1.00000	0.132453
$58$	3.00000	0.393919
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	−5.00000	−0.635001
$63$	6.00000	0.755929
$64$	7.00000	0.875000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−3.00000	−0.363803
$69$	7.00000	0.842701
$70$	3.00000	0.358569
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	−6.00000	−0.707107
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−9.00000	−1.04623
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	0	0
$78$	−5.00000	−0.566139
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−3.00000	−0.327327
$85$	3.00000	0.325396
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	2.00000	0.210819
$91$	15.0000	1.57243
$92$	7.00000	0.729800
$93$	−5.00000	−0.518476
$94$	8.00000	0.825137
$95$	−1.00000	−0.102598
$96$	5.00000	0.510310
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	4.00000	0.400000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	3.00000	0.297044
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	−15.0000	−1.47087
$105$	3.00000	0.292770
$106$	5.00000	0.485643
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−5.00000	−0.481125
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	3.00000	0.283473
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−1.00000	−0.0936586
$115$	−7.00000	−0.652753
$116$	3.00000	0.278543
$117$	10.0000	0.924500
$118$	−12.0000	−1.10469
$119$	−9.00000	−0.825029
$120$	−3.00000	−0.273861
$121$	−11.0000	−1.00000
$122$	−13.0000	−1.17696
$123$	10.0000	0.901670
$124$	−5.00000	−0.449013
$125$	−9.00000	−0.804984
$126$	−6.00000	−0.534522
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	5.00000	0.438529
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	3.00000	0.259161
$135$	5.00000	0.430331
$136$	9.00000	0.771744
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−7.00000	−0.595880
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	3.00000	0.253546
$141$	8.00000	0.673722
$142$	−1.00000	−0.0839181
$143$	0	0
$144$	2.00000	0.166667
$145$	−3.00000	−0.249136
$146$	11.0000	0.910366
$147$	−2.00000	−0.164957
$148$	−9.00000	−0.739795
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	−4.00000	−0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−3.00000	−0.243332
$153$	−6.00000	−0.485071
$154$	0	0
$155$	5.00000	0.401610
$156$	−5.00000	−0.400320
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	5.00000	0.397779
$159$	5.00000	0.396526
$160$	−5.00000	−0.395285
$161$	21.0000	1.65503
$162$	−1.00000	−0.0785674
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−9.00000	−0.698535
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	9.00000	0.694365
$169$	12.0000	0.923077
$170$	−3.00000	−0.230089
$171$	2.00000	0.152944
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−3.00000	−0.227429
$175$	12.0000	0.907115
$176$	0	0
$177$	−12.0000	−0.901975
$178$	−1.00000	−0.0749532
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	2.00000	0.149071
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−15.0000	−1.11187
$183$	−13.0000	−0.960988
$184$	−21.0000	−1.54814
$185$	9.00000	0.661693
$186$	5.00000	0.366618
$187$	0	0
$188$	8.00000	0.583460
$189$	−15.0000	−1.09109
$190$	1.00000	0.0725476
$191$	19.0000	1.37479	0.687396	−	0.726283i	$-0.258754\pi$
$191$	19.0000	1.37479	0.687396	+	0.726283i	$0.258754\pi$
$192$	−7.00000	−0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	2.00000	0.143592
$195$	5.00000	0.358057
$196$	−2.00000	−0.142857
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−12.0000	−0.848528
$201$	3.00000	0.211604
$202$	9.00000	0.633238
$203$	9.00000	0.631676
$204$	3.00000	0.210042
$205$	−10.0000	−0.698430
$206$	−7.00000	−0.487713
$207$	14.0000	0.973067
$208$	5.00000	0.346688
$209$	0	0
$210$	−3.00000	−0.207020
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	5.00000	0.343401
$213$	−1.00000	−0.0685189
$214$	−12.0000	−0.820303
$215$	0	0
$216$	15.0000	1.02062
$217$	−15.0000	−1.01827
$218$	−7.00000	−0.474100
$219$	11.0000	0.743311
$220$	0	0
$221$	−15.0000	−1.00901
$222$	9.00000	0.604040
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	15.0000	1.00223
$225$	8.00000	0.533333
$226$	−2.00000	−0.133038
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	−1.00000	−0.0662266
$229$	−9.00000	−0.594737	−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000	−0.594737	−0.297368	+	0.954763i	$0.596109\pi$
$230$	7.00000	0.461566
$231$	0	0
$232$	−9.00000	−0.590879
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	−10.0000	−0.653720
$235$	−8.00000	−0.521862
$236$	−12.0000	−0.781133
$237$	5.00000	0.324785
$238$	9.00000	0.583383
$239$	25.0000	1.61712	0.808558	−	0.588417i	$-0.200249\pi$
$239$	25.0000	1.61712	0.808558	+	0.588417i	$0.200249\pi$
$240$	1.00000	0.0645497
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	11.0000	0.707107
$243$	−16.0000	−1.02640
$244$	−13.0000	−0.832240
$245$	2.00000	0.127775
$246$	−10.0000	−0.637577
$247$	5.00000	0.318142
$248$	15.0000	0.952501
$249$	−9.00000	−0.570352
$250$	9.00000	0.569210
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	−6.00000	−0.377964
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−3.00000	−0.187867
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−27.0000	−1.67770
$260$	5.00000	0.310087
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	−5.00000	−0.307148
$266$	−3.00000	−0.183942
$267$	−1.00000	−0.0611990
$268$	3.00000	0.183254
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−5.00000	−0.304290
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	−3.00000	−0.181902
$273$	−15.0000	−0.907841
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−7.00000	−0.421350
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	−13.0000	−0.779688
$279$	−10.0000	−0.598684
$280$	−9.00000	−0.537853
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	−8.00000	−0.476393
$283$	3.00000	0.178331	0.0891657	−	0.996017i	$-0.471580\pi$
$283$	3.00000	0.178331	0.0891657	+	0.996017i	$0.471580\pi$
$284$	−1.00000	−0.0593391
$285$	1.00000	0.0592349
$286$	0	0
$287$	30.0000	1.77084
$288$	10.0000	0.589256
$289$	−8.00000	−0.470588
$290$	3.00000	0.176166
$291$	2.00000	0.117242
$292$	11.0000	0.643726
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	2.00000	0.116642
$295$	12.0000	0.698667
$296$	27.0000	1.56934
$297$	0	0
$298$	−21.0000	−1.21650
$299$	35.0000	2.02410
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−8.00000	−0.460348
$303$	9.00000	0.517036
$304$	1.00000	0.0573539
$305$	13.0000	0.744378
$306$	6.00000	0.342997
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	0	0
$309$	−7.00000	−0.398216
$310$	−5.00000	−0.283981
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	15.0000	0.849208
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	−1.00000	−0.0564333
$315$	6.00000	0.338062
$316$	5.00000	0.281272
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−5.00000	−0.280386
$319$	0	0
$320$	7.00000	0.391312
$321$	−12.0000	−0.669775
$322$	−21.0000	−1.17028
$323$	−3.00000	−0.166924
$324$	−1.00000	−0.0555556
$325$	20.0000	1.10940
$326$	−1.00000	−0.0553849
$327$	−7.00000	−0.387101
$328$	−30.0000	−1.65647
$329$	24.0000	1.32316
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−9.00000	−0.493939
$333$	−18.0000	−0.986394
$334$	3.00000	0.164153
$335$	−3.00000	−0.163908
$336$	−3.00000	−0.163663
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	−12.0000	−0.652714
$339$	−2.00000	−0.108625
$340$	−3.00000	−0.162698
$341$	0	0
$342$	−2.00000	−0.108148
$343$	15.0000	0.809924
$344$	0	0
$345$	7.00000	0.376867
$346$	6.00000	0.322562
$347$	−37.0000	−1.98626	−0.993132	−	0.116999i	$-0.962673\pi$
$347$	−37.0000	−1.98626	−0.993132	+	0.116999i	$0.962673\pi$
$348$	−3.00000	−0.160817
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	−12.0000	−0.641427
$351$	−25.0000	−1.33440
$352$	0	0
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	12.0000	0.637793
$355$	1.00000	0.0530745
$356$	−1.00000	−0.0529999
$357$	9.00000	0.476331
$358$	−1.00000	−0.0528516
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	−6.00000	−0.316228
$361$	−18.0000	−0.947368
$362$	−7.00000	−0.367912
$363$	11.0000	0.577350
$364$	−15.0000	−0.786214
$365$	−11.0000	−0.575766
$366$	13.0000	0.679521
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	7.00000	0.364900
$369$	20.0000	1.04116
$370$	−9.00000	−0.467888
$371$	15.0000	0.778761
$372$	5.00000	0.259238
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	−24.0000	−1.23771
$377$	15.0000	0.772539
$378$	15.0000	0.771517
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	1.00000	0.0512989
$381$	−16.0000	−0.819705
$382$	−19.0000	−0.972125
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−6.00000	−0.305392
$387$	0	0
$388$	2.00000	0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−5.00000	−0.253185
$391$	−21.0000	−1.06202
$392$	6.00000	0.303046
$393$	−4.00000	−0.201773
$394$	−11.0000	−0.554172
$395$	−5.00000	−0.251577
$396$	0	0
$397$	−21.0000	−1.05396	−0.526980	−	0.849878i	$-0.676676\pi$
$397$	−21.0000	−1.05396	−0.526980	+	0.849878i	$0.676676\pi$
$398$	8.00000	0.401004
$399$	−3.00000	−0.150188
$400$	4.00000	0.200000
$401$	−37.0000	−1.84769	−0.923846	−	0.382765i	$-0.874972\pi$
$401$	−37.0000	−1.84769	−0.923846	+	0.382765i	$0.874972\pi$
$402$	−3.00000	−0.149626
$403$	−25.0000	−1.24534
$404$	9.00000	0.447767
$405$	1.00000	0.0496904
$406$	−9.00000	−0.446663
$407$	0	0
$408$	−9.00000	−0.445566
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	10.0000	0.493865
$411$	−18.0000	−0.887875
$412$	−7.00000	−0.344865
$413$	−36.0000	−1.77144
$414$	−14.0000	−0.688062
$415$	9.00000	0.441793
$416$	25.0000	1.22573
$417$	−13.0000	−0.636613
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−3.00000	−0.146385
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	8.00000	0.389434
$423$	16.0000	0.777947
$424$	−15.0000	−0.728464
$425$	−12.0000	−0.582086
$426$	1.00000	0.0484502
$427$	−39.0000	−1.88734
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−5.00000	−0.240563
$433$	−27.0000	−1.29754	−0.648769	−	0.760986i	$-0.724716\pi$
$433$	−27.0000	−1.29754	−0.648769	+	0.760986i	$0.724716\pi$
$434$	15.0000	0.720023
$435$	3.00000	0.143839
$436$	−7.00000	−0.335239
$437$	7.00000	0.334855
$438$	−11.0000	−0.525600
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	15.0000	0.713477
$443$	−37.0000	−1.75792	−0.878962	−	0.476893i	$-0.841763\pi$
$443$	−37.0000	−1.75792	−0.878962	+	0.476893i	$0.841763\pi$
$444$	9.00000	0.427121
$445$	1.00000	0.0474045
$446$	20.0000	0.947027
$447$	−21.0000	−0.993266
$448$	−21.0000	−0.992157
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	−8.00000	−0.375873
$454$	−7.00000	−0.328526
$455$	15.0000	0.703211
$456$	3.00000	0.140488
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	9.00000	0.420542
$459$	15.0000	0.700140
$460$	7.00000	0.326377
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	3.00000	0.139272
$465$	−5.00000	−0.231869
$466$	−9.00000	−0.416917
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	−10.0000	−0.462250
$469$	9.00000	0.415581
$470$	8.00000	0.369012
$471$	−1.00000	−0.0460776
$472$	36.0000	1.65703
$473$	0	0
$474$	−5.00000	−0.229658
$475$	4.00000	0.183533
$476$	9.00000	0.412514
$477$	10.0000	0.457869
$478$	−25.0000	−1.14347
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	5.00000	0.228218
$481$	−45.0000	−2.05182
$482$	15.0000	0.683231
$483$	−21.0000	−0.955533
$484$	11.0000	0.500000
$485$	−2.00000	−0.0908153
$486$	16.0000	0.725775
$487$	−9.00000	−0.407829	−0.203914	−	0.978989i	$-0.565366\pi$
$487$	−9.00000	−0.407829	−0.203914	+	0.978989i	$0.565366\pi$
$488$	39.0000	1.76545
$489$	−1.00000	−0.0452216
$490$	−2.00000	−0.0903508
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	−10.0000	−0.450835
$493$	−9.00000	−0.405340
$494$	−5.00000	−0.224961
$495$	0	0
$496$	−5.00000	−0.224507
$497$	−3.00000	−0.134568
$498$	9.00000	0.403300
$499$	17.0000	0.761025	0.380512	−	0.924776i	$-0.375748\pi$
$499$	17.0000	0.761025	0.380512	+	0.924776i	$0.375748\pi$
$500$	9.00000	0.402492
$501$	3.00000	0.134030
$502$	−7.00000	−0.312425
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	18.0000	0.801784
$505$	−9.00000	−0.400495
$506$	0	0
$507$	−12.0000	−0.532939
$508$	−16.0000	−0.709885
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	3.00000	0.132842
$511$	33.0000	1.45983
$512$	11.0000	0.486136
$513$	−5.00000	−0.220755
$514$	6.00000	0.264649
$515$	7.00000	0.308457
$516$	0	0
$517$	0	0
$518$	27.0000	1.18631
$519$	6.00000	0.263371
$520$	−15.0000	−0.657794
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	−6.00000	−0.262613
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	−4.00000	−0.174741
$525$	−12.0000	−0.523723
$526$	21.0000	0.915644
$527$	15.0000	0.653410
$528$	0	0
$529$	26.0000	1.13043
$530$	5.00000	0.217186
$531$	−24.0000	−1.04151
$532$	−3.00000	−0.130066
$533$	50.0000	2.16574
$534$	1.00000	0.0432742
$535$	12.0000	0.518805
$536$	−9.00000	−0.388741
$537$	−1.00000	−0.0431532
$538$	−14.0000	−0.603583
$539$	0	0
$540$	−5.00000	−0.215166
$541$	31.0000	1.33279	0.666397	−	0.745597i	$-0.267836\pi$
$541$	31.0000	1.33279	0.666397	+	0.745597i	$0.267836\pi$
$542$	−23.0000	−0.987935
$543$	−7.00000	−0.300399
$544$	−15.0000	−0.643120
$545$	7.00000	0.299847
$546$	15.0000	0.641941
$547$	1.00000	0.0427569	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000	0.0427569	0.0213785	+	0.999771i	$0.493195\pi$
$548$	−18.0000	−0.768922
$549$	−26.0000	−1.10965
$550$	0	0
$551$	3.00000	0.127804
$552$	21.0000	0.893819
$553$	15.0000	0.637865
$554$	7.00000	0.297402
$555$	−9.00000	−0.382029
$556$	−13.0000	−0.551323
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	10.0000	0.423334
$559$	0	0
$560$	3.00000	0.126773
$561$	0	0
$562$	17.0000	0.717102
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	2.00000	0.0841406
$566$	−3.00000	−0.126099
$567$	−3.00000	−0.125988
$568$	3.00000	0.125877
$569$	−17.0000	−0.712677	−0.356339	−	0.934357i	$-0.615975\pi$
$569$	−17.0000	−0.712677	−0.356339	+	0.934357i	$0.615975\pi$
$570$	−1.00000	−0.0418854
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	0	0
$573$	−19.0000	−0.793736
$574$	−30.0000	−1.25218
$575$	28.0000	1.16768
$576$	−14.0000	−0.583333
$577$	5.00000	0.208153	0.104076	−	0.994569i	$-0.466811\pi$
$577$	5.00000	0.208153	0.104076	+	0.994569i	$0.466811\pi$
$578$	8.00000	0.332756
$579$	−6.00000	−0.249351
$580$	3.00000	0.124568
$581$	−27.0000	−1.12015
$582$	−2.00000	−0.0829027
$583$	0	0
$584$	−33.0000	−1.36555
$585$	10.0000	0.413449
$586$	14.0000	0.578335
$587$	13.0000	0.536567	0.268284	−	0.963340i	$-0.413544\pi$
$587$	13.0000	0.536567	0.268284	+	0.963340i	$0.413544\pi$
$588$	2.00000	0.0824786
$589$	−5.00000	−0.206021
$590$	−12.0000	−0.494032
$591$	−11.0000	−0.452480
$592$	−9.00000	−0.369898
$593$	1.00000	0.0410651	0.0205325	−	0.999789i	$-0.493464\pi$
$593$	1.00000	0.0410651	0.0205325	+	0.999789i	$0.493464\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	−21.0000	−0.860194
$597$	8.00000	0.327418
$598$	−35.0000	−1.43126
$599$	−31.0000	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$599$	−31.0000	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$600$	12.0000	0.489898
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	−8.00000	−0.325515
$605$	−11.0000	−0.447214
$606$	−9.00000	−0.365600
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	5.00000	0.202777
$609$	−9.00000	−0.364698
$610$	−13.0000	−0.526355
$611$	40.0000	1.61823
$612$	6.00000	0.242536
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	−5.00000	−0.201784
$615$	10.0000	0.403239
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	7.00000	0.281581
$619$	−21.0000	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$619$	−21.0000	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$620$	−5.00000	−0.200805
$621$	−35.0000	−1.40450
$622$	3.00000	0.120289
$623$	−3.00000	−0.120192
$624$	−5.00000	−0.200160
$625$	11.0000	0.440000
$626$	−17.0000	−0.679457
$627$	0	0
$628$	−1.00000	−0.0399043
$629$	27.0000	1.07656
$630$	−6.00000	−0.239046
$631$	9.00000	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$632$	−15.0000	−0.596668
$633$	8.00000	0.317971
$634$	18.0000	0.714871
$635$	16.0000	0.634941
$636$	−5.00000	−0.198263
$637$	−10.0000	−0.396214
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	3.00000	0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−21.0000	−0.827516
$645$	0	0
$646$	3.00000	0.118033
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	−20.0000	−0.784465
$651$	15.0000	0.587896
$652$	−1.00000	−0.0391630
$653$	50.0000	1.95665	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	50.0000	1.95665	0.978326	+	0.207072i	$0.0663936\pi$
$654$	7.00000	0.273722
$655$	4.00000	0.156293
$656$	10.0000	0.390434
$657$	22.0000	0.858302
$658$	−24.0000	−0.935617
$659$	23.0000	0.895953	0.447976	−	0.894045i	$-0.352145\pi$
$659$	23.0000	0.895953	0.447976	+	0.894045i	$0.352145\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	19.0000	0.738456
$663$	15.0000	0.582552
$664$	27.0000	1.04780
$665$	3.00000	0.116335
$666$	18.0000	0.697486
$667$	21.0000	0.813123
$668$	3.00000	0.116073
$669$	20.0000	0.773245
$670$	3.00000	0.115900
$671$	0	0
$672$	−15.0000	−0.578638
$673$	−47.0000	−1.81172	−0.905858	−	0.423581i	$-0.860773\pi$
$673$	−47.0000	−1.81172	−0.905858	+	0.423581i	$0.860773\pi$
$674$	−3.00000	−0.115556
$675$	−20.0000	−0.769800
$676$	−12.0000	−0.461538
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	2.00000	0.0768095
$679$	6.00000	0.230259
$680$	9.00000	0.345134
$681$	−7.00000	−0.268241
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−2.00000	−0.0764719
$685$	18.0000	0.687745
$686$	−15.0000	−0.572703
$687$	9.00000	0.343371
$688$	0	0
$689$	25.0000	0.952424
$690$	−7.00000	−0.266485
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	37.0000	1.40450
$695$	13.0000	0.493118
$696$	9.00000	0.341144
$697$	−30.0000	−1.13633
$698$	−1.00000	−0.0378506
$699$	−9.00000	−0.340411
$700$	−12.0000	−0.453557
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	25.0000	0.943564
$703$	−9.00000	−0.339441
$704$	0	0
$705$	8.00000	0.301297
$706$	25.0000	0.940887
$707$	27.0000	1.01544
$708$	12.0000	0.450988
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−1.00000	−0.0375293
$711$	10.0000	0.375029
$712$	3.00000	0.112430
$713$	−35.0000	−1.31076
$714$	−9.00000	−0.336817
$715$	0	0
$716$	−1.00000	−0.0373718
$717$	−25.0000	−0.933642
$718$	−19.0000	−0.709074
$719$	−31.0000	−1.15610	−0.578052	−	0.816000i	$-0.696187\pi$
$719$	−31.0000	−1.15610	−0.578052	+	0.816000i	$0.696187\pi$
$720$	2.00000	0.0745356
$721$	−21.0000	−0.782081
$722$	18.0000	0.669891
$723$	15.0000	0.557856
$724$	−7.00000	−0.260153
$725$	12.0000	0.445669
$726$	−11.0000	−0.408248
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	45.0000	1.66781
$729$	13.0000	0.481481
$730$	11.0000	0.407128
$731$	0	0
$732$	13.0000	0.480494
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−13.0000	−0.479839
$735$	−2.00000	−0.0737711
$736$	35.0000	1.29012
$737$	0	0
$738$	−20.0000	−0.736210
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−9.00000	−0.330847
$741$	−5.00000	−0.183680
$742$	−15.0000	−0.550667
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	−15.0000	−0.549927
$745$	21.0000	0.769380
$746$	11.0000	0.402739
$747$	−18.0000	−0.658586
$748$	0	0
$749$	−36.0000	−1.31541
$750$	−9.00000	−0.328634
$751$	27.0000	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$752$	8.00000	0.291730
$753$	−7.00000	−0.255094
$754$	−15.0000	−0.546268
$755$	8.00000	0.291150
$756$	15.0000	0.545545
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	−3.00000	−0.108821
$761$	37.0000	1.34125	0.670624	−	0.741797i	$-0.266026\pi$
$761$	37.0000	1.34125	0.670624	+	0.741797i	$0.266026\pi$
$762$	16.0000	0.579619
$763$	−21.0000	−0.760251
$764$	−19.0000	−0.687396
$765$	−6.00000	−0.216930
$766$	8.00000	0.289052
$767$	−60.0000	−2.16647
$768$	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	$0.702850\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−6.00000	−0.215945
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	−6.00000	−0.215387
$777$	27.0000	0.968620
$778$	−6.00000	−0.215110
$779$	10.0000	0.358287
$780$	−5.00000	−0.179029
$781$	0	0
$782$	21.0000	0.750958
$783$	−15.0000	−0.536056
$784$	−2.00000	−0.0714286
$785$	1.00000	0.0356915
$786$	4.00000	0.142675
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	−11.0000	−0.391859
$789$	21.0000	0.747620
$790$	5.00000	0.177892
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−65.0000	−2.30822
$794$	21.0000	0.745262
$795$	5.00000	0.177332
$796$	8.00000	0.283552
$797$	−21.0000	−0.743858	−0.371929	−	0.928261i	$-0.621304\pi$
$797$	−21.0000	−0.743858	−0.371929	+	0.928261i	$0.621304\pi$
$798$	3.00000	0.106199
$799$	−24.0000	−0.849059
$800$	20.0000	0.707107
$801$	−2.00000	−0.0706665
$802$	37.0000	1.30652
$803$	0	0
$804$	−3.00000	−0.105802
$805$	21.0000	0.740153
$806$	25.0000	0.880587
$807$	−14.0000	−0.492823
$808$	−27.0000	−0.949857
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	−1.00000	−0.0351364
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	−9.00000	−0.315838
$813$	−23.0000	−0.806645
$814$	0	0
$815$	1.00000	0.0350285
$816$	3.00000	0.105021
$817$	0	0
$818$	−18.0000	−0.629355
$819$	−30.0000	−1.04828
$820$	10.0000	0.349215
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	18.0000	0.627822
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	21.0000	0.731570
$825$	0	0
$826$	36.0000	1.25260
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	−14.0000	−0.486534
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	−9.00000	−0.312395
$831$	7.00000	0.242827
$832$	−35.0000	−1.21341
$833$	6.00000	0.207888
$834$	13.0000	0.450153
$835$	−3.00000	−0.103819
$836$	0	0
$837$	25.0000	0.864126
$838$	−28.0000	−0.967244
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	9.00000	0.310530
$841$	−20.0000	−0.689655
$842$	−37.0000	−1.27510
$843$	17.0000	0.585511
$844$	8.00000	0.275371
$845$	12.0000	0.412813
$846$	−16.0000	−0.550091
$847$	33.0000	1.13389
$848$	5.00000	0.171701
$849$	−3.00000	−0.102960
$850$	12.0000	0.411597
$851$	−63.0000	−2.15961
$852$	1.00000	0.0342594
$853$	−5.00000	−0.171197	−0.0855984	−	0.996330i	$-0.527280\pi$
$853$	−5.00000	−0.171197	−0.0855984	+	0.996330i	$0.527280\pi$
$854$	39.0000	1.33455
$855$	2.00000	0.0683986
$856$	36.0000	1.23045
$857$	35.0000	1.19558	0.597789	−	0.801654i	$-0.296046\pi$
$857$	35.0000	1.19558	0.597789	+	0.801654i	$0.296046\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−30.0000	−1.02240
$862$	0	0
$863$	27.0000	0.919091	0.459545	−	0.888154i	$-0.348012\pi$
$863$	27.0000	0.919091	0.459545	+	0.888154i	$0.348012\pi$
$864$	−25.0000	−0.850517
$865$	−6.00000	−0.204006
$866$	27.0000	0.917497
$867$	8.00000	0.271694
$868$	15.0000	0.509133
$869$	0	0
$870$	−3.00000	−0.101710
$871$	15.0000	0.508256
$872$	21.0000	0.711150
$873$	4.00000	0.135379
$874$	−7.00000	−0.236779
$875$	27.0000	0.912767
$876$	−11.0000	−0.371656
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	13.0000	0.438729
$879$	14.0000	0.472208
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	4.00000	0.134687
$883$	13.0000	0.437485	0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	15.0000	0.504505
$885$	−12.0000	−0.403376
$886$	37.0000	1.24304
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	−27.0000	−0.906061
$889$	−48.0000	−1.60987
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	20.0000	0.669650
$893$	8.00000	0.267710
$894$	21.0000	0.702345
$895$	1.00000	0.0334263
$896$	−9.00000	−0.300669
$897$	−35.0000	−1.16862
$898$	−21.0000	−0.700779
$899$	−15.0000	−0.500278
$900$	−8.00000	−0.266667
$901$	−15.0000	−0.499722
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	7.00000	0.232688
$906$	8.00000	0.265782
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−7.00000	−0.232303
$909$	18.0000	0.597022
$910$	−15.0000	−0.497245
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	6.00000	0.198462
$915$	−13.0000	−0.429767
$916$	9.00000	0.297368
$917$	−12.0000	−0.396275
$918$	−15.0000	−0.495074
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−21.0000	−0.692349
$921$	−5.00000	−0.164756
$922$	−3.00000	−0.0987997
$923$	−5.00000	−0.164577
$924$	0	0
$925$	−36.0000	−1.18367
$926$	−23.0000	−0.755827
$927$	−14.0000	−0.459820
$928$	15.0000	0.492399
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	5.00000	0.163956
$931$	−2.00000	−0.0655474
$932$	−9.00000	−0.294805
$933$	3.00000	0.0982156
$934$	21.0000	0.687141
$935$	0	0
$936$	30.0000	0.980581
$937$	−35.0000	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$937$	−35.0000	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$938$	−9.00000	−0.293860
$939$	−17.0000	−0.554774
$940$	8.00000	0.260931
$941$	−9.00000	−0.293392	−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000	−0.293392	−0.146696	+	0.989182i	$0.546864\pi$
$942$	1.00000	0.0325818
$943$	70.0000	2.27951
$944$	−12.0000	−0.390567
$945$	−15.0000	−0.487950
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−5.00000	−0.162392
$949$	55.0000	1.78538
$950$	−4.00000	−0.129777
$951$	18.0000	0.583690
$952$	−27.0000	−0.875075
$953$	41.0000	1.32812	0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000	1.32812	0.664060	+	0.747679i	$0.268832\pi$
$954$	−10.0000	−0.323762
$955$	19.0000	0.614826
$956$	−25.0000	−0.808558
$957$	0	0
$958$	15.0000	0.484628
$959$	−54.0000	−1.74375
$960$	−7.00000	−0.225924
$961$	−6.00000	−0.193548
$962$	45.0000	1.45086
$963$	−24.0000	−0.773389
$964$	15.0000	0.483117
$965$	6.00000	0.193147
$966$	21.0000	0.675664
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−33.0000	−1.06066
$969$	3.00000	0.0963739
$970$	2.00000	0.0642161
$971$	35.0000	1.12320	0.561602	−	0.827408i	$-0.310185\pi$
$971$	35.0000	1.12320	0.561602	+	0.827408i	$0.310185\pi$
$972$	16.0000	0.513200
$973$	−39.0000	−1.25028
$974$	9.00000	0.288379
$975$	−20.0000	−0.640513
$976$	−13.0000	−0.416120
$977$	−41.0000	−1.31171	−0.655853	−	0.754889i	$-0.727691\pi$
$977$	−41.0000	−1.31171	−0.655853	+	0.754889i	$0.727691\pi$
$978$	1.00000	0.0319765
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	−14.0000	−0.446986
$982$	−9.00000	−0.287202
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	30.0000	0.956365
$985$	11.0000	0.350489
$986$	9.00000	0.286618
$987$	−24.0000	−0.763928
$988$	−5.00000	−0.159071
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−25.0000	−0.793751
$993$	19.0000	0.602947
$994$	3.00000	0.0951542
$995$	−8.00000	−0.253617
$996$	9.00000	0.285176
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−17.0000	−0.538126
$999$	45.0000	1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.a.1.1		1
43.7	odd	6		43.2.c.a.6.1	✓	2
43.37	odd	6		43.2.c.a.36.1	yes	2
43.42	odd	2		1849.2.a.c.1.1		1
129.50	even	6		387.2.h.a.307.1		2
129.80	even	6		387.2.h.a.208.1		2
172.7	even	6		688.2.i.d.49.1		2
172.123	even	6		688.2.i.d.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.c.a.6.1	✓	2	43.7	odd	6
43.2.c.a.36.1	yes	2	43.37	odd	6
387.2.h.a.208.1		2	129.80	even	6
387.2.h.a.307.1		2	129.50	even	6
688.2.i.d.49.1		2	172.7	even	6
688.2.i.d.337.1		2	172.123	even	6
1849.2.a.a.1.1		1	1.1	even	1	trivial
1849.2.a.c.1.1		1	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more