LMFDB - Embedded newform 2550.2.a.o.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:	no (minimal twist has level 510)
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} +2.00000 q^{22} -1.00000 q^{24} +4.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -1.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -4.00000 q^{39} -4.00000 q^{41} -4.00000 q^{42} +10.0000 q^{43} -2.00000 q^{44} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{57} -6.00000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +14.0000 q^{67} +1.00000 q^{68} -14.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000 q^{77} +4.00000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} +4.00000 q^{84} -10.0000 q^{86} +2.00000 q^{88} -6.00000 q^{89} -16.0000 q^{91} -12.0000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−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
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−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
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	1.00000	0.140028
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	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	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	−8.00000	−0.911685
$78$	4.00000	0.452911
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	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
$91$	−16.0000	−1.67726
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−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
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	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
$111$	6.00000	0.569495
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	−4.00000	−0.356348
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	10.0000	0.880451
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	−2.00000	−0.174078
$133$	16.0000	1.38738
$134$	−14.0000	−1.20942
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	14.0000	1.17485
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	9.00000	0.742307
$148$	6.00000	0.493197
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	8.00000	0.644658
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	−12.0000	−0.954669
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	−4.00000	−0.308607
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	10.0000	0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	16.0000	1.18600
$183$	−6.00000	−0.443533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.00000	−0.146254
$188$	12.0000	0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	1.00000	0.0721688
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	2.00000	0.142134
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	−18.0000	−1.26648
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−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	$0.456034\pi$
$212$	6.00000	0.412082
$213$	−14.0000	−0.959264
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	2.00000	0.135147
$220$	0	0
$221$	−4.00000	−0.269069
$222$	−6.00000	−0.402694
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	4.00000	0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	0	0
$237$	12.0000	0.779484
$238$	−4.00000	−0.259281
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	4.00000	0.255031
$247$	−16.0000	−1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−10.0000	−0.622573
$259$	24.0000	1.49129
$260$	0	0
$261$	0	0
$262$	18.0000	1.11204
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−16.0000	−0.981023
$267$	−6.00000	−0.367194
$268$	14.0000	0.855186
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	1.00000	0.0606339
$273$	−16.0000	−0.968364
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−12.0000	−0.714590
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−16.0000	−0.944450
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	2.00000	0.117041
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−6.00000	−0.348743
$297$	−2.00000	−0.116052
$298$	−14.0000	−0.810998
$299$	0	0
$300$	0	0
$301$	40.0000	2.30556
$302$	−8.00000	−0.460348
$303$	18.0000	1.03407
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	−8.00000	−0.455842
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	4.00000	0.226455
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	12.0000	0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−14.0000	−0.774202
$328$	4.00000	0.220863
$329$	48.0000	2.64633
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	−24.0000	−1.31322
$335$	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	$0.412193\pi$
$338$	−3.00000	−0.163178
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	2.00000	0.106600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	4.00000	0.211702
$358$	12.0000	0.634220
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−18.0000	−0.946059
$363$	−7.00000	−0.367405
$364$	−16.0000	−0.838628
$365$	0	0
$366$	6.00000	0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	−4.00000	−0.205738
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−16.0000	−0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	26.0000	1.32337
$387$	10.0000	0.508329
$388$	2.00000	0.101535
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−18.0000	−0.907980
$394$	6.00000	0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	20.0000	1.00251
$399$	16.0000	0.801002
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	−14.0000	−0.698257
$403$	0	0
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	−1.00000	−0.0495074
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−6.00000	−0.295599
$413$	0	0
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	16.0000	0.783523
$418$	8.00000	0.391293
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	−4.00000	−0.194717
$423$	12.0000	0.583460
$424$	−6.00000	−0.291386
$425$	0	0
$426$	14.0000	0.678302
$427$	−24.0000	−1.16144
$428$	−4.00000	−0.193347
$429$	8.00000	0.386244
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	1.00000	0.0481125
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	14.0000	0.662177
$448$	4.00000	0.188982
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	−2.00000	−0.0940721
$453$	8.00000	0.375873
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	22.0000	1.02799
$459$	1.00000	0.0466760
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	8.00000	0.372194
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−4.00000	−0.184900
$469$	56.0000	2.58584
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−20.0000	−0.919601
$474$	−12.0000	−0.551178
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	4.00000	0.182956
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	−26.0000	−1.18427
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	6.00000	0.271607
$489$	20.0000	0.904431
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−4.00000	−0.180334
$493$	0	0
$494$	16.0000	0.719874
$495$	0	0
$496$	0	0
$497$	−56.0000	−2.51194
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	12.0000	0.535586
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	2.00000	0.0887357
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−30.0000	−1.32324
$515$	0	0
$516$	10.0000	0.440225
$517$	−24.0000	−1.05552
$518$	−24.0000	−1.05450
$519$	6.00000	0.263371
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	28.0000	1.22086
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	16.0000	0.693688
$533$	16.0000	0.693037
$534$	6.00000	0.259645
$535$	0	0
$536$	−14.0000	−0.604708
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	−24.0000	−1.03089
$543$	18.0000	0.772454
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	16.0000	0.684737
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−18.0000	−0.768922
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	48.0000	2.04117
$554$	22.0000	0.934690
$555$	0	0
$556$	16.0000	0.678551
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	6.00000	0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	8.00000	0.336265
$567$	4.00000	0.167984
$568$	14.0000	0.587427
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	8.00000	0.334497
$573$	16.0000	0.668410
$574$	16.0000	0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−1.00000	−0.0415945
$579$	−26.0000	−1.08052
$580$	0	0
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	−12.0000	−0.496989
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−10.0000	−0.413096
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	6.00000	0.246598
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	14.0000	0.573462
$597$	−20.0000	−0.818546
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	−40.0000	−1.63028
$603$	14.0000	0.570124
$604$	8.00000	0.325515
$605$	0	0
$606$	−18.0000	−0.731200
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	1.00000	0.0404226
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	8.00000	0.322329
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	6.00000	0.241355
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	34.0000	1.36328
$623$	−24.0000	−0.961540
$624$	−4.00000	−0.160128
$625$	0	0
$626$	18.0000	0.719425
$627$	−8.00000	−0.319489
$628$	−12.0000	−0.478852
$629$	6.00000	0.239236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−12.0000	−0.477334
$633$	4.00000	0.158986
$634$	2.00000	0.0794301
$635$	0	0
$636$	6.00000	0.237915
$637$	−36.0000	−1.42637
$638$	0	0
$639$	−14.0000	−0.553831
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	4.00000	0.157867
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000	0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−4.00000	−0.156174
$657$	2.00000	0.0780274
$658$	−48.0000	−1.87123
$659$	−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
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	4.00000	0.155464
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	0	0
$668$	24.0000	0.928588
$669$	2.00000	0.0773245
$670$	0	0
$671$	12.0000	0.463255
$672$	−4.00000	−0.154303
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	3.00000	0.115385
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	2.00000	0.0768095
$679$	8.00000	0.307012
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	−22.0000	−0.839352
$688$	10.0000	0.381246
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	6.00000	0.228086
$693$	−8.00000	−0.303895
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	34.0000	1.28692
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	4.00000	0.150970
$703$	24.0000	0.905177
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−10.0000	−0.376355
$707$	72.0000	2.70784
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	6.00000	0.224860
$713$	0	0
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−4.00000	−0.149383
$718$	12.0000	0.447836
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	3.00000	0.111648
$723$	26.0000	0.966950
$724$	18.0000	0.668965
$725$	0	0
$726$	7.00000	0.259794
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	16.0000	0.592999
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.0000	0.369863
$732$	−6.00000	−0.221766
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	0	0
$737$	−28.0000	−1.03139
$738$	4.00000	0.147242
$739$	−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
$741$	−16.0000	−0.587775
$742$	−24.0000	−0.881068
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	0	0
$746$	−24.0000	−0.878702
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000	0.437595
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−2.00000	−0.0724524
$763$	−56.0000	−2.02734
$764$	16.0000	0.578860
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	−26.0000	−0.935760
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	24.0000	0.860995
$778$	−22.0000	−0.788738
$779$	−16.0000	−0.573259
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	18.0000	0.642039
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−6.00000	−0.213741
$789$	−28.0000	−0.996826
$790$	0	0
$791$	−8.00000	−0.284447
$792$	2.00000	0.0710669
$793$	24.0000	0.852265
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−20.0000	−0.708881
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	−16.0000	−0.566394
$799$	12.0000	0.424529
$800$	0	0
$801$	−6.00000	−0.212000
$802$	12.0000	0.423735
$803$	−4.00000	−0.141157
$804$	14.0000	0.493742
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	12.0000	0.420600
$815$	0	0
$816$	1.00000	0.0350070
$817$	40.0000	1.39942
$818$	6.00000	0.209785
$819$	−16.0000	−0.559085
$820$	0	0
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	18.0000	0.627822
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−4.00000	−0.138675
$833$	9.00000	0.311832
$834$	−16.0000	−0.554035
$835$	0	0
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−6.00000	−0.207267
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−18.0000	−0.620321
$843$	−6.00000	−0.206651
$844$	4.00000	0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	−28.0000	−0.962091
$848$	6.00000	0.206041
$849$	−8.00000	−0.274559
$850$	0	0
$851$	0	0
$852$	−14.0000	−0.479632
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	4.00000	0.136717
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	−8.00000	−0.273115
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	30.0000	1.02180
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−56.0000	−1.89749
$872$	14.0000	0.474100
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−32.0000	−1.07995
$879$	10.0000	0.337292
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	−9.00000	−0.303046
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−6.00000	−0.201347
$889$	8.00000	0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	2.00000	0.0669650
$893$	48.0000	1.60626
$894$	−14.0000	−0.468230
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	4.00000	0.133482
$899$	0	0
$900$	0	0
$901$	6.00000	0.199889
$902$	−8.00000	−0.266371
$903$	40.0000	1.33112
$904$	2.00000	0.0665190
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	12.0000	0.398234
$909$	18.0000	0.597022
$910$	0	0
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	20.0000	0.661541
$915$	0	0
$916$	−22.0000	−0.726900
$917$	−72.0000	−2.37765
$918$	−1.00000	−0.0330049
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	30.0000	0.987997
$923$	56.0000	1.84326
$924$	−8.00000	−0.263181
$925$	0	0
$926$	34.0000	1.11731
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	6.00000	0.196537
$933$	−34.0000	−1.11311
$934$	12.0000	0.392652
$935$	0	0
$936$	4.00000	0.130744
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	−56.0000	−1.82846
$939$	−18.0000	−0.587408
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	12.0000	0.390981
$943$	0	0
$944$	0	0
$945$	0	0
$946$	20.0000	0.650256
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	12.0000	0.389742
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	−4.00000	−0.129641
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−4.00000	−0.129369
$957$	0	0
$958$	14.0000	0.452319
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	24.0000	0.773791
$963$	−4.00000	−0.128898
$964$	26.0000	0.837404
$965$	0	0
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	7.00000	0.224989
$969$	4.00000	0.128499
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	64.0000	2.05175
$974$	16.0000	0.512673
$975$	0	0
$976$	−6.00000	−0.192055
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−20.0000	−0.639529
$979$	12.0000	0.383522
$980$	0	0
$981$	−14.0000	−0.446986
$982$	−8.00000	−0.255290
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	4.00000	0.127515
$985$	0	0
$986$	0	0
$987$	48.0000	1.52786
$988$	−16.0000	−0.509028
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	56.0000	1.77621
$995$	0	0
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	−8.00000	−0.253236
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.o.1.1		1
3.2	odd	2		7650.2.a.ck.1.1		1
5.2	odd	4		510.2.d.a.409.1	✓	2
5.3	odd	4		510.2.d.a.409.2	yes	2
5.4	even	2		2550.2.a.s.1.1		1
15.2	even	4		1530.2.d.c.919.2		2
15.8	even	4		1530.2.d.c.919.1		2
15.14	odd	2		7650.2.a.e.1.1		1
20.3	even	4		4080.2.m.d.2449.1		2
20.7	even	4		4080.2.m.d.2449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.a.409.1	✓	2	5.2	odd	4
510.2.d.a.409.2	yes	2	5.3	odd	4
1530.2.d.c.919.1		2	15.8	even	4
1530.2.d.c.919.2		2	15.2	even	4
2550.2.a.o.1.1		1	1.1	even	1	trivial
2550.2.a.s.1.1		1	5.4	even	2
4080.2.m.d.2449.1		2	20.3	even	4
4080.2.m.d.2449.2		2	20.7	even	4
7650.2.a.e.1.1		1	15.14	odd	2
7650.2.a.ck.1.1		1	3.2	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

Related objects

Downloads

Learn more