LMFDB - Embedded newform 2550.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
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	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} -2.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +10.0000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +9.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} +11.0000 q^{41} +4.00000 q^{42} +10.0000 q^{43} -2.00000 q^{44} +5.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{51} -6.00000 q^{52} -11.0000 q^{53} -1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} -15.0000 q^{59} -1.00000 q^{61} +10.0000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -14.0000 q^{67} -1.00000 q^{68} -5.00000 q^{69} +11.0000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +9.00000 q^{74} +4.00000 q^{76} +8.00000 q^{77} +6.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +11.0000 q^{82} -5.00000 q^{83} +4.00000 q^{84} +10.0000 q^{86} -2.00000 q^{88} -6.00000 q^{89} +24.0000 q^{91} +5.00000 q^{92} -10.0000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} +9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	−2.00000	−0.426401
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	4.00000	0.648886
$39$	6.00000	0.960769
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	4.00000	0.617213
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	5.00000	0.737210
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	1.00000	0.140028
$52$	−6.00000	−0.832050
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	10.0000	1.27000
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−1.00000	−0.121268
$69$	−5.00000	−0.601929
$70$	0	0
$71$	11.0000	1.30546	0.652730	−	0.757591i	$-0.273624\pi$
$71$	11.0000	1.30546	0.652730	+	0.757591i	$0.273624\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	9.00000	1.04623
$75$	0	0
$76$	4.00000	0.458831
$77$	8.00000	0.911685
$78$	6.00000	0.679366
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.0000	1.21475
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	5.00000	0.521286
$93$	−10.0000	−1.03695
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	9.00000	0.909137
$99$	−2.00000	−0.201008
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	1.00000	0.0990148
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−11.0000	−1.06841
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	−4.00000	−0.377964
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	−6.00000	−0.554700
$118$	−15.0000	−1.38086
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−1.00000	−0.0905357
$123$	−11.0000	−0.991837
$124$	10.0000	0.898027
$125$	0	0
$126$	−4.00000	−0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	2.00000	0.174078
$133$	−16.0000	−1.38738
$134$	−14.0000	−1.20942
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−5.00000	−0.425628
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	11.0000	0.923099
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	−9.00000	−0.742307
$148$	9.00000	0.739795
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	4.00000	0.324443
$153$	−1.00000	−0.0808452
$154$	8.00000	0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	11.0000	0.872357
$160$	0	0
$161$	−20.0000	−1.57622
$162$	1.00000	0.0785674
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	11.0000	0.858956
$165$	0	0
$166$	−5.00000	−0.388075
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	4.00000	0.308607
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	10.0000	0.762493
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	15.0000	1.12747
$178$	−6.00000	−0.449719
$179$	23.0000	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$179$	23.0000	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	24.0000	1.77900
$183$	1.00000	0.0739221
$184$	5.00000	0.368605
$185$	0	0
$186$	−10.0000	−0.733236
$187$	2.00000	0.146254
$188$	8.00000	0.583460
$189$	4.00000	0.290957
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	−1.00000	−0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−2.00000	−0.142134
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	18.0000	1.26648
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	11.0000	0.766406
$207$	5.00000	0.347524
$208$	−6.00000	−0.416025
$209$	−8.00000	−0.553372
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−11.0000	−0.755483
$213$	−11.0000	−0.753708
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−40.0000	−2.71538
$218$	−14.0000	−0.948200
$219$	−8.00000	−0.540590
$220$	0	0
$221$	6.00000	0.403604
$222$	−9.00000	−0.604040
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−3.00000	−0.199557
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−4.00000	−0.264906
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−15.0000	−0.976417
$237$	8.00000	0.519656
$238$	4.00000	0.259281
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	−11.0000	−0.701334
$247$	−24.0000	−1.52708
$248$	10.0000	0.635001
$249$	5.00000	0.316862
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−4.00000	−0.251976
$253$	−10.0000	−0.628695
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	−10.0000	−0.622573
$259$	−36.0000	−2.23693
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−16.0000	−0.981023
$267$	6.00000	0.367194
$268$	−14.0000	−0.855186
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	−1.00000	−0.0606339
$273$	−24.0000	−1.45255
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−5.00000	−0.300965
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	11.0000	0.659736
$279$	10.0000	0.598684
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	−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	$0.471580\pi$
$284$	11.0000	0.652730
$285$	0	0
$286$	12.0000	0.709575
$287$	−44.0000	−2.59724
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	8.00000	0.468165
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	9.00000	0.523114
$297$	2.00000	0.116052
$298$	−1.00000	−0.0579284
$299$	−30.0000	−1.73494
$300$	0	0
$301$	−40.0000	−2.30556
$302$	−7.00000	−0.402805
$303$	−18.0000	−1.03407
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	8.00000	0.455842
$309$	−11.0000	−0.625768
$310$	0	0
$311$	1.00000	0.0567048	0.0283524	−	0.999598i	$-0.490974\pi$
$311$	1.00000	0.0567048	0.0283524	+	0.999598i	$0.490974\pi$
$312$	6.00000	0.339683
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	11.0000	0.616849
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	−20.0000	−1.11456
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.00000	−0.276924
$327$	14.0000	0.774202
$328$	11.0000	0.607373
$329$	−32.0000	−1.76422
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−5.00000	−0.274411
$333$	9.00000	0.493197
$334$	16.0000	0.875481
$335$	0	0
$336$	4.00000	0.218218
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	23.0000	1.25104
$339$	3.00000	0.162938
$340$	0	0
$341$	−20.0000	−1.08306
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	10.0000	0.539164
$345$	0	0
$346$	4.00000	0.215041
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−2.00000	−0.106600
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	15.0000	0.797241
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	23.0000	1.21559
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−7.00000	−0.367912
$363$	7.00000	0.367405
$364$	24.0000	1.25794
$365$	0	0
$366$	1.00000	0.0522708
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	5.00000	0.260643
$369$	11.0000	0.572637
$370$	0	0
$371$	44.0000	2.28437
$372$	−10.0000	−0.518476
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	4.00000	0.205738
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	−14.0000	−0.716302
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−4.00000	−0.203595
$387$	10.0000	0.508329
$388$	8.00000	0.406138
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	9.00000	0.454569
$393$	−12.0000	−0.605320
$394$	6.00000	0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	10.0000	0.501255
$399$	16.0000	0.801002
$400$	0	0
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	14.0000	0.698257
$403$	−60.0000	−2.98881
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	−18.0000	−0.892227
$408$	1.00000	0.0495074
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	11.0000	0.541931
$413$	60.0000	2.95241
$414$	5.00000	0.245737
$415$	0	0
$416$	−6.00000	−0.294174
$417$	−11.0000	−0.538672
$418$	−8.00000	−0.391293
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	−11.0000	−0.534207
$425$	0	0
$426$	−11.0000	−0.532952
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−1.00000	−0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	−14.0000	−0.670478
$437$	20.0000	0.956730
$438$	−8.00000	−0.382255
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	6.00000	0.285391
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	−9.00000	−0.427121
$445$	0	0
$446$	−7.00000	−0.331460
$447$	1.00000	0.0472984
$448$	−4.00000	−0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−22.0000	−1.03594
$452$	−3.00000	−0.141108
$453$	7.00000	0.328889
$454$	18.0000	0.844782
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	−2.00000	−0.0934539
$459$	1.00000	0.0466760
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	−8.00000	−0.372194
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	0	0
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	37.0000	1.71216	0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000	1.71216	0.856078	+	0.516847i	$0.172894\pi$
$468$	−6.00000	−0.277350
$469$	56.0000	2.58584
$470$	0	0
$471$	18.0000	0.829396
$472$	−15.0000	−0.690431
$473$	−20.0000	−0.919601
$474$	8.00000	0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	−11.0000	−0.503655
$478$	6.00000	0.274434
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−54.0000	−2.46219
$482$	−14.0000	−0.637683
$483$	20.0000	0.910032
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	−1.00000	−0.0452679
$489$	5.00000	0.226108
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	−11.0000	−0.495918
$493$	0	0
$494$	−24.0000	−1.07981
$495$	0	0
$496$	10.0000	0.449013
$497$	−44.0000	−1.97367
$498$	5.00000	0.224055
$499$	−27.0000	−1.20869	−0.604343	−	0.796724i	$-0.706564\pi$
$499$	−27.0000	−1.20869	−0.604343	+	0.796724i	$0.706564\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	28.0000	1.24970
$503$	−5.00000	−0.222939	−0.111469	−	0.993768i	$-0.535556\pi$
$503$	−5.00000	−0.222939	−0.111469	+	0.993768i	$0.535556\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	−10.0000	−0.444554
$507$	−23.0000	−1.02147
$508$	8.00000	0.354943
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	−10.0000	−0.440225
$517$	−16.0000	−0.703679
$518$	−36.0000	−1.58175
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	−10.0000	−0.435607
$528$	2.00000	0.0870388
$529$	2.00000	0.0869565
$530$	0	0
$531$	−15.0000	−0.650945
$532$	−16.0000	−0.693688
$533$	−66.0000	−2.85878
$534$	6.00000	0.259645
$535$	0	0
$536$	−14.0000	−0.604708
$537$	−23.0000	−0.992523
$538$	10.0000	0.431131
$539$	−18.0000	−0.775315
$540$	0	0
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	19.0000	0.816120
$543$	7.00000	0.300399
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−24.0000	−1.02711
$547$	15.0000	0.641354	0.320677	−	0.947189i	$-0.396090\pi$
$547$	15.0000	0.641354	0.320677	+	0.947189i	$0.396090\pi$
$548$	−12.0000	−0.512615
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	0	0
$552$	−5.00000	−0.212814
$553$	32.0000	1.36078
$554$	−3.00000	−0.127458
$555$	0	0
$556$	11.0000	0.466504
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	10.0000	0.423334
$559$	−60.0000	−2.53773
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	14.0000	0.590554
$563$	41.0000	1.72794	0.863972	−	0.503540i	$-0.167969\pi$
$563$	41.0000	1.72794	0.863972	+	0.503540i	$0.167969\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	3.00000	0.126099
$567$	−4.00000	−0.167984
$568$	11.0000	0.461550
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	12.0000	0.501745
$573$	14.0000	0.584858
$574$	−44.0000	−1.83652
$575$	0	0
$576$	1.00000	0.0416667
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	1.00000	0.0415945
$579$	4.00000	0.166234
$580$	0	0
$581$	20.0000	0.829740
$582$	−8.00000	−0.331611
$583$	22.0000	0.911147
$584$	8.00000	0.331042
$585$	0	0
$586$	15.0000	0.619644
$587$	23.0000	0.949312	0.474656	−	0.880172i	$-0.342573\pi$
$587$	23.0000	0.949312	0.474656	+	0.880172i	$0.342573\pi$
$588$	−9.00000	−0.371154
$589$	40.0000	1.64817
$590$	0	0
$591$	−6.00000	−0.246807
$592$	9.00000	0.369898
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−1.00000	−0.0409616
$597$	−10.0000	−0.409273
$598$	−30.0000	−1.22679
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−40.0000	−1.63028
$603$	−14.0000	−0.570124
$604$	−7.00000	−0.284826
$605$	0	0
$606$	−18.0000	−0.731200
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	−1.00000	−0.0404226
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	8.00000	0.322329
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	−11.0000	−0.442485
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	1.00000	0.0400963
$623$	24.0000	0.961540
$624$	6.00000	0.240192
$625$	0	0
$626$	8.00000	0.319744
$627$	8.00000	0.319489
$628$	−18.0000	−0.718278
$629$	−9.00000	−0.358854
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	−8.00000	−0.318223
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	0	0
$636$	11.0000	0.436178
$637$	−54.0000	−2.13956
$638$	0	0
$639$	11.0000	0.435153
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−4.00000	−0.157867
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	−20.0000	−0.788110
$645$	0	0
$646$	−4.00000	−0.157378
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	1.00000	0.0392837
$649$	30.0000	1.17760
$650$	0	0
$651$	40.0000	1.56772
$652$	−5.00000	−0.195815
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	11.0000	0.429478
$657$	8.00000	0.312110
$658$	−32.0000	−1.24749
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	6.00000	0.233197
$663$	−6.00000	−0.233021
$664$	−5.00000	−0.194038
$665$	0	0
$666$	9.00000	0.348743
$667$	0	0
$668$	16.0000	0.619059
$669$	7.00000	0.270636
$670$	0	0
$671$	2.00000	0.0772091
$672$	4.00000	0.154303
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	23.0000	0.884615
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	3.00000	0.115214
$679$	−32.0000	−1.22805
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−20.0000	−0.765840
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	2.00000	0.0763048
$688$	10.0000	0.381246
$689$	66.0000	2.51440
$690$	0	0
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	4.00000	0.152057
$693$	8.00000	0.303895
$694$	8.00000	0.303676
$695$	0	0
$696$	0	0
$697$	−11.0000	−0.416655
$698$	−14.0000	−0.529908
$699$	1.00000	0.0378235
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	6.00000	0.226455
$703$	36.0000	1.35777
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	0	0
$707$	−72.0000	−2.70784
$708$	15.0000	0.563735
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−6.00000	−0.224860
$713$	50.0000	1.87251
$714$	−4.00000	−0.149696
$715$	0	0
$716$	23.0000	0.859550
$717$	−6.00000	−0.224074
$718$	8.00000	0.298557
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	−44.0000	−1.63865
$722$	−3.00000	−0.111648
$723$	14.0000	0.520666
$724$	−7.00000	−0.260153
$725$	0	0
$726$	7.00000	0.259794
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	24.0000	0.889499
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	1.00000	0.0369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	5.00000	0.184302
$737$	28.0000	1.03139
$738$	11.0000	0.404916
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	44.0000	1.61529
$743$	37.0000	1.35740	0.678699	−	0.734416i	$-0.262544\pi$
$743$	37.0000	1.35740	0.678699	+	0.734416i	$0.262544\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	−24.0000	−0.878702
$747$	−5.00000	−0.182940
$748$	2.00000	0.0731272
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	8.00000	0.291730
$753$	−28.0000	−1.02038
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	29.0000	1.05333
$759$	10.0000	0.362977
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	−8.00000	−0.289809
$763$	56.0000	2.02734
$764$	−14.0000	−0.506502
$765$	0	0
$766$	16.0000	0.578103
$767$	90.0000	3.24971
$768$	−1.00000	−0.0360844
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−17.0000	−0.611448	−0.305724	−	0.952120i	$-0.598898\pi$
$773$	−17.0000	−0.611448	−0.305724	+	0.952120i	$0.598898\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	8.00000	0.287183
$777$	36.0000	1.29149
$778$	−33.0000	−1.18311
$779$	44.0000	1.57646
$780$	0	0
$781$	−22.0000	−0.787222
$782$	−5.00000	−0.178800
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	−12.0000	−0.428026
$787$	1.00000	0.0356462	0.0178231	−	0.999841i	$-0.494326\pi$
$787$	1.00000	0.0356462	0.0178231	+	0.999841i	$0.494326\pi$
$788$	6.00000	0.213741
$789$	−8.00000	−0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	−2.00000	−0.0710669
$793$	6.00000	0.213066
$794$	−7.00000	−0.248421
$795$	0	0
$796$	10.0000	0.354441
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	16.0000	0.566394
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−6.00000	−0.212000
$802$	23.0000	0.812158
$803$	−16.0000	−0.564628
$804$	14.0000	0.493742
$805$	0	0
$806$	−60.0000	−2.11341
$807$	−10.0000	−0.352017
$808$	18.0000	0.633238
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	−19.0000	−0.666359
$814$	−18.0000	−0.630900
$815$	0	0
$816$	1.00000	0.0350070
$817$	40.0000	1.39942
$818$	19.0000	0.664319
$819$	24.0000	0.838628
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	12.0000	0.418548
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	60.0000	2.08767
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	5.00000	0.173762
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	3.00000	0.104069
$832$	−6.00000	−0.208013
$833$	−9.00000	−0.311832
$834$	−11.0000	−0.380899
$835$	0	0
$836$	−8.00000	−0.276686
$837$	−10.0000	−0.345651
$838$	−4.00000	−0.138178
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	8.00000	0.275698
$843$	−14.0000	−0.482186
$844$	4.00000	0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	28.0000	0.962091
$848$	−11.0000	−0.377742
$849$	−3.00000	−0.102960
$850$	0	0
$851$	45.0000	1.54258
$852$	−11.0000	−0.376854
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	4.00000	0.136717
$857$	19.0000	0.649028	0.324514	−	0.945881i	$-0.394799\pi$
$857$	19.0000	0.649028	0.324514	+	0.945881i	$0.394799\pi$
$858$	−12.0000	−0.409673
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	44.0000	1.49952
$862$	−20.0000	−0.681203
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	30.0000	1.01944
$867$	−1.00000	−0.0339618
$868$	−40.0000	−1.35769
$869$	16.0000	0.542763
$870$	0	0
$871$	84.0000	2.84623
$872$	−14.0000	−0.474100
$873$	8.00000	0.270759
$874$	20.0000	0.676510
$875$	0	0
$876$	−8.00000	−0.270295
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−28.0000	−0.944954
$879$	−15.0000	−0.505937
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	9.00000	0.303046
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−9.00000	−0.302361
$887$	27.0000	0.906571	0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000	0.906571	0.453286	+	0.891365i	$0.350252\pi$
$888$	−9.00000	−0.302020
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	−7.00000	−0.234377
$893$	32.0000	1.07084
$894$	1.00000	0.0334450
$895$	0	0
$896$	−4.00000	−0.133631
$897$	30.0000	1.00167
$898$	6.00000	0.200223
$899$	0	0
$900$	0	0
$901$	11.0000	0.366463
$902$	−22.0000	−0.732520
$903$	40.0000	1.33112
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	7.00000	0.232559
$907$	13.0000	0.431658	0.215829	−	0.976431i	$-0.430755\pi$
$907$	13.0000	0.431658	0.215829	+	0.976431i	$0.430755\pi$
$908$	18.0000	0.597351
$909$	18.0000	0.597022
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	10.0000	0.330952
$914$	−15.0000	−0.496156
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	−48.0000	−1.58510
$918$	1.00000	0.0330049
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	−15.0000	−0.493999
$923$	−66.0000	−2.17242
$924$	−8.00000	−0.263181
$925$	0	0
$926$	19.0000	0.624379
$927$	11.0000	0.361287
$928$	0	0
$929$	37.0000	1.21393	0.606965	−	0.794728i	$-0.292387\pi$
$929$	37.0000	1.21393	0.606965	+	0.794728i	$0.292387\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−1.00000	−0.0327561
$933$	−1.00000	−0.0327385
$934$	37.0000	1.21068
$935$	0	0
$936$	−6.00000	−0.196116
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	56.0000	1.82846
$939$	−8.00000	−0.261070
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	18.0000	0.586472
$943$	55.0000	1.79105
$944$	−15.0000	−0.488208
$945$	0	0
$946$	−20.0000	−0.650256
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	8.00000	0.259828
$949$	−48.0000	−1.55815
$950$	0	0
$951$	18.0000	0.583690
$952$	4.00000	0.129641
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−11.0000	−0.356138
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	−24.0000	−0.775405
$959$	48.0000	1.55000
$960$	0	0
$961$	69.0000	2.22581
$962$	−54.0000	−1.74103
$963$	4.00000	0.128898
$964$	−14.0000	−0.450910
$965$	0	0
$966$	20.0000	0.643489
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	−7.00000	−0.224989
$969$	4.00000	0.128499
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	−1.00000	−0.0320750
$973$	−44.0000	−1.41058
$974$	6.00000	0.192252
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	5.00000	0.159882
$979$	12.0000	0.383522
$980$	0	0
$981$	−14.0000	−0.446986
$982$	33.0000	1.05307
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−11.0000	−0.350667
$985$	0	0
$986$	0	0
$987$	32.0000	1.01857
$988$	−24.0000	−0.763542
$989$	50.0000	1.58991
$990$	0	0
$991$	−60.0000	−1.90596	−0.952981	−	0.303029i	$-0.902002\pi$
$991$	−60.0000	−1.90596	−0.952981	+	0.303029i	$0.902002\pi$
$992$	10.0000	0.317500
$993$	−6.00000	−0.190404
$994$	−44.0000	−1.39560
$995$	0	0
$996$	5.00000	0.158431
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−27.0000	−0.854670
$999$	−9.00000	−0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.r.1.1	yes	1
3.2	odd	2		7650.2.a.d.1.1		1
5.2	odd	4		2550.2.d.e.2449.2		2
5.3	odd	4		2550.2.d.e.2449.1		2
5.4	even	2		2550.2.a.p.1.1	✓	1
15.14	odd	2		7650.2.a.cm.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.p.1.1	✓	1	5.4	even	2
2550.2.a.r.1.1	yes	1	1.1	even	1	trivial
2550.2.d.e.2449.1		2	5.3	odd	4
2550.2.d.e.2449.2		2	5.2	odd	4
7650.2.a.d.1.1		1	3.2	odd	2
7650.2.a.cm.1.1		1	15.14	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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