LMFDB - Embedded newform 4998.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4998,2,Mod(1,4998)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4998, 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("4998.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4998.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:	$39.9092309302$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 714)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4998.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +5.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +1.00000 q^{20} +2.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} -7.00000 q^{29} -1.00000 q^{30} +9.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -5.00000 q^{39} +1.00000 q^{40} +11.0000 q^{41} -8.00000 q^{43} +1.00000 q^{45} +2.00000 q^{46} +11.0000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -1.00000 q^{51} +5.00000 q^{52} -1.00000 q^{54} -7.00000 q^{58} -11.0000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +9.00000 q^{62} +1.00000 q^{64} +5.00000 q^{65} +2.00000 q^{67} +1.00000 q^{68} -2.00000 q^{69} +10.0000 q^{71} +1.00000 q^{72} -16.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} -5.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +11.0000 q^{82} +3.00000 q^{83} +1.00000 q^{85} -8.00000 q^{86} +7.00000 q^{87} +8.00000 q^{89} +1.00000 q^{90} +2.00000 q^{92} -9.00000 q^{93} +11.0000 q^{94} -1.00000 q^{96} +12.0000 q^{97} +O(q^{100})\)

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
$5$	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
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	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	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	5.00000	0.980581
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	−1.00000	−0.182574
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	1.00000	0.158114
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	2.00000	0.294884
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−1.00000	−0.140028
$52$	5.00000	0.693375
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−7.00000	−0.919145
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	9.00000	1.14300
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	1.00000	0.121268
$69$	−2.00000	−0.240772
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	1.00000	0.117851
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	−5.00000	−0.566139
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	11.0000	1.21475
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−8.00000	−0.862662
$87$	7.00000	0.750479
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	2.00000	0.208514
$93$	−9.00000	−0.933257
$94$	11.0000	1.13456
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	−4.00000	−0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−1.00000	−0.0990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	0	0
$107$	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
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−17.0000	−1.59923	−0.799613	−	0.600516i	$-0.794962\pi$
$113$	−17.0000	−1.59923	−0.799613	+	0.600516i	$0.794962\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	−7.00000	−0.649934
$117$	5.00000	0.462250
$118$	−11.0000	−1.01263
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	−11.0000	−0.991837
$124$	9.00000	0.808224
$125$	−9.00000	−0.804984
$126$	0	0
$127$	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
$129$	8.00000	0.704361
$130$	5.00000	0.438529
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	0	0
$134$	2.00000	0.172774
$135$	−1.00000	−0.0860663
$136$	1.00000	0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−2.00000	−0.170251
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−11.0000	−0.926367
$142$	10.0000	0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	−7.00000	−0.581318
$146$	−16.0000	−1.32417
$147$	0	0
$148$	4.00000	0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	4.00000	0.326599
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	9.00000	0.722897
$156$	−5.00000	−0.400320
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	11.0000	0.858956
$165$	0	0
$166$	3.00000	0.232845
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	1.00000	0.0766965
$171$	0	0
$172$	−8.00000	−0.609994
$173$	17.0000	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$173$	17.0000	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$174$	7.00000	0.530669
$175$	0	0
$176$	0	0
$177$	11.0000	0.826811
$178$	8.00000	0.599625
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	1.00000	0.0745356
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	2.00000	0.147442
$185$	4.00000	0.294086
$186$	−9.00000	−0.659912
$187$	0	0
$188$	11.0000	0.802257
$189$	0	0
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	−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	$0.545985\pi$
$194$	12.0000	0.861550
$195$	−5.00000	−0.358057
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	−4.00000	−0.282843
$201$	−2.00000	−0.141069
$202$	0	0
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	11.0000	0.768273
$206$	8.00000	0.557386
$207$	2.00000	0.139010
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	16.0000	1.08118
$220$	0	0
$221$	5.00000	0.336336
$222$	−4.00000	−0.268462
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−17.0000	−1.13082
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−7.00000	−0.459573
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	5.00000	0.326860
$235$	11.0000	0.717561
$236$	−11.0000	−0.716039
$237$	−12.0000	−0.779484
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	−1.00000	−0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−11.0000	−0.701334
$247$	0	0
$248$	9.00000	0.571501
$249$	−3.00000	−0.190117
$250$	−9.00000	−0.569210
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	5.00000	0.310087
$261$	−7.00000	−0.433289
$262$	16.0000	0.988483
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.00000	−0.489592
$268$	2.00000	0.122169
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	−1.00000	−0.0608581
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	−2.00000	−0.120386
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	4.00000	0.239904
$279$	9.00000	0.538816
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	−11.0000	−0.655040
$283$	−27.0000	−1.60498	−0.802492	−	0.596663i	$-0.796493\pi$
$283$	−27.0000	−1.60498	−0.802492	+	0.596663i	$0.796493\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−7.00000	−0.411054
$291$	−12.0000	−0.703452
$292$	−16.0000	−0.936329
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	4.00000	0.232495
$297$	0	0
$298$	10.0000	0.579284
$299$	10.0000	0.578315
$300$	4.00000	0.230940
$301$	0	0
$302$	10.0000	0.575435
$303$	0	0
$304$	0	0
$305$	−10.0000	−0.572598
$306$	1.00000	0.0571662
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	9.00000	0.511166
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	−5.00000	−0.283069
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	17.0000	0.959366
$315$	0	0
$316$	12.0000	0.675053
$317$	−5.00000	−0.280828	−0.140414	−	0.990093i	$-0.544843\pi$
$317$	−5.00000	−0.280828	−0.140414	+	0.990093i	$0.544843\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−20.0000	−1.10940
$326$	−8.00000	−0.443079
$327$	14.0000	0.774202
$328$	11.0000	0.607373
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	3.00000	0.164646
$333$	4.00000	0.219199
$334$	−20.0000	−1.09435
$335$	2.00000	0.109272
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	12.0000	0.652714
$339$	17.0000	0.923313
$340$	1.00000	0.0542326
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−8.00000	−0.431331
$345$	−2.00000	−0.107676
$346$	17.0000	0.913926
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	7.00000	0.375239
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	11.0000	0.584643
$355$	10.0000	0.530745
$356$	8.00000	0.423999
$357$	0	0
$358$	3.00000	0.158555
$359$	13.0000	0.686114	0.343057	−	0.939315i	$-0.388538\pi$
$359$	13.0000	0.686114	0.343057	+	0.939315i	$0.388538\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−16.0000	−0.837478
$366$	10.0000	0.522708
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	2.00000	0.104257
$369$	11.0000	0.572637
$370$	4.00000	0.207950
$371$	0	0
$372$	−9.00000	−0.466628
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	11.0000	0.567282
$377$	−35.0000	−1.80259
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	21.0000	1.07445
$383$	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
$387$	−8.00000	−0.406663
$388$	12.0000	0.609208
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	−5.00000	−0.253185
$391$	2.00000	0.101144
$392$	0	0
$393$	−16.0000	−0.807093
$394$	−9.00000	−0.453413
$395$	12.0000	0.603786
$396$	0	0
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	−4.00000	−0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−2.00000	−0.0997509
$403$	45.0000	2.24161
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	−1.00000	−0.0495074
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	11.0000	0.543251
$411$	−2.00000	−0.0986527
$412$	8.00000	0.394132
$413$	0	0
$414$	2.00000	0.0982946
$415$	3.00000	0.147264
$416$	5.00000	0.245145
$417$	−4.00000	−0.195881
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	−25.0000	−1.21698
$423$	11.0000	0.534838
$424$	0	0
$425$	−4.00000	−0.194029
$426$	−10.0000	−0.484502
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	7.00000	0.335624
$436$	−14.0000	−0.670478
$437$	0	0
$438$	16.0000	0.764510
$439$	−41.0000	−1.95682	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−41.0000	−1.95682	−0.978412	+	0.206666i	$0.933739\pi$
$440$	0	0
$441$	0	0
$442$	5.00000	0.237826
$443$	5.00000	0.237557	0.118779	−	0.992921i	$-0.462102\pi$
$443$	5.00000	0.237557	0.118779	+	0.992921i	$0.462102\pi$
$444$	−4.00000	−0.189832
$445$	8.00000	0.379236
$446$	6.00000	0.284108
$447$	−10.0000	−0.472984
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	−17.0000	−0.799613
$453$	−10.0000	−0.469841
$454$	14.0000	0.657053
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	7.00000	0.327089
$459$	−1.00000	−0.0466760
$460$	2.00000	0.0932505
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−7.00000	−0.324967
$465$	−9.00000	−0.417365
$466$	1.00000	0.0463241
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	5.00000	0.231125
$469$	0	0
$470$	11.0000	0.507392
$471$	−17.0000	−0.783319
$472$	−11.0000	−0.506316
$473$	0	0
$474$	−12.0000	−0.551178
$475$	0	0
$476$	0	0
$477$	0	0
$478$	11.0000	0.503128
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	−1.00000	−0.0456435
$481$	20.0000	0.911922
$482$	14.0000	0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	12.0000	0.544892
$486$	−1.00000	−0.0453609
$487$	−5.00000	−0.226572	−0.113286	−	0.993562i	$-0.536138\pi$
$487$	−5.00000	−0.226572	−0.113286	+	0.993562i	$0.536138\pi$
$488$	−10.0000	−0.452679
$489$	8.00000	0.361773
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−11.0000	−0.495918
$493$	−7.00000	−0.315264
$494$	0	0
$495$	0	0
$496$	9.00000	0.404112
$497$	0	0
$498$	−3.00000	−0.134433
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	−9.00000	−0.402492
$501$	20.0000	0.893534
$502$	−3.00000	−0.133897
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	8.00000	0.354943
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	−1.00000	−0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	8.00000	0.352522
$516$	8.00000	0.352180
$517$	0	0
$518$	0	0
$519$	−17.0000	−0.746217
$520$	5.00000	0.219265
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−7.00000	−0.306382
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	28.0000	1.22086
$527$	9.00000	0.392046
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−11.0000	−0.477359
$532$	0	0
$533$	55.0000	2.38231
$534$	−8.00000	−0.346194
$535$	4.00000	0.172935
$536$	2.00000	0.0863868
$537$	−3.00000	−0.129460
$538$	−21.0000	−0.905374
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	18.0000	0.773166
$543$	0	0
$544$	1.00000	0.0428746
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	2.00000	0.0854358
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	−2.00000	−0.0851257
$553$	0	0
$554$	26.0000	1.10463
$555$	−4.00000	−0.169791
$556$	4.00000	0.169638
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	9.00000	0.381000
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	8.00000	0.337460
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−11.0000	−0.463184
$565$	−17.0000	−0.715195
$566$	−27.0000	−1.13489
$567$	0	0
$568$	10.0000	0.419591
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	−21.0000	−0.877288
$574$	0	0
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	1.00000	0.0415945
$579$	4.00000	0.166234
$580$	−7.00000	−0.290659
$581$	0	0
$582$	−12.0000	−0.497416
$583$	0	0
$584$	−16.0000	−0.662085
$585$	5.00000	0.206725
$586$	6.00000	0.247858
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	0	0
$590$	−11.0000	−0.452863
$591$	9.00000	0.370211
$592$	4.00000	0.164399
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	−7.00000	−0.286491
$598$	10.0000	0.408930
$599$	−31.0000	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$599$	−31.0000	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$600$	4.00000	0.163299
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	10.0000	0.406894
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	0	0
$610$	−10.0000	−0.404888
$611$	55.0000	2.22506
$612$	1.00000	0.0404226
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	2.00000	0.0807134
$615$	−11.0000	−0.443563
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	−8.00000	−0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	9.00000	0.361449
$621$	−2.00000	−0.0802572
$622$	−26.0000	−1.04251
$623$	0	0
$624$	−5.00000	−0.200160
$625$	11.0000	0.440000
$626$	6.00000	0.239808
$627$	0	0
$628$	17.0000	0.678374
$629$	4.00000	0.159490
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	12.0000	0.477334
$633$	25.0000	0.993661
$634$	−5.00000	−0.198575
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.0000	0.395594
$640$	1.00000	0.0395285
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	−4.00000	−0.157867
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−19.0000	−0.746967	−0.373484	−	0.927637i	$-0.621837\pi$
$647$	−19.0000	−0.746967	−0.373484	+	0.927637i	$0.621837\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−20.0000	−0.784465
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	14.0000	0.547443
$655$	16.0000	0.625172
$656$	11.0000	0.429478
$657$	−16.0000	−0.624219
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	−12.0000	−0.466393
$663$	−5.00000	−0.194184
$664$	3.00000	0.116423
$665$	0	0
$666$	4.00000	0.154997
$667$	−14.0000	−0.542082
$668$	−20.0000	−0.773823
$669$	−6.00000	−0.231973
$670$	2.00000	0.0772667
$671$	0	0
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−28.0000	−1.07852
$675$	4.00000	0.153960
$676$	12.0000	0.461538
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	17.0000	0.652881
$679$	0	0
$680$	1.00000	0.0383482
$681$	−14.0000	−0.536481
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−7.00000	−0.267067
$688$	−8.00000	−0.304997
$689$	0	0
$690$	−2.00000	−0.0761387
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	17.0000	0.646243
$693$	0	0
$694$	−6.00000	−0.227757
$695$	4.00000	0.151729
$696$	7.00000	0.265334
$697$	11.0000	0.416655
$698$	23.0000	0.870563
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	−5.00000	−0.188713
$703$	0	0
$704$	0	0
$705$	−11.0000	−0.414284
$706$	−36.0000	−1.35488
$707$	0	0
$708$	11.0000	0.413405
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	10.0000	0.375293
$711$	12.0000	0.450035
$712$	8.00000	0.299813
$713$	18.0000	0.674105
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	−11.0000	−0.410803
$718$	13.0000	0.485156
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−14.0000	−0.520666
$724$	0	0
$725$	28.0000	1.03989
$726$	11.0000	0.408248
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	−8.00000	−0.295891
$732$	10.0000	0.369611
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	2.00000	0.0737210
$737$	0	0
$738$	11.0000	0.404916
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−9.00000	−0.329956
$745$	10.0000	0.366372
$746$	13.0000	0.475964
$747$	3.00000	0.109764
$748$	0	0
$749$	0	0
$750$	9.00000	0.328634
$751$	−45.0000	−1.64207	−0.821037	−	0.570875i	$-0.806604\pi$
$751$	−45.0000	−1.64207	−0.821037	+	0.570875i	$0.806604\pi$
$752$	11.0000	0.401129
$753$	3.00000	0.109326
$754$	−35.0000	−1.27462
$755$	10.0000	0.363937
$756$	0	0
$757$	45.0000	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$757$	45.0000	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	21.0000	0.759753
$765$	1.00000	0.0361551
$766$	16.0000	0.578103
$767$	−55.0000	−1.98593
$768$	−1.00000	−0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−8.00000	−0.287554
$775$	−36.0000	−1.29316
$776$	12.0000	0.430775
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−5.00000	−0.179029
$781$	0	0
$782$	2.00000	0.0715199
$783$	7.00000	0.250160
$784$	0	0
$785$	17.0000	0.606756
$786$	−16.0000	−0.570701
$787$	−19.0000	−0.677277	−0.338638	−	0.940917i	$-0.609966\pi$
$787$	−19.0000	−0.677277	−0.338638	+	0.940917i	$0.609966\pi$
$788$	−9.00000	−0.320612
$789$	−28.0000	−0.996826
$790$	12.0000	0.426941
$791$	0	0
$792$	0	0
$793$	−50.0000	−1.77555
$794$	−38.0000	−1.34857
$795$	0	0
$796$	7.00000	0.248108
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	11.0000	0.389152
$800$	−4.00000	−0.141421
$801$	8.00000	0.282666
$802$	18.0000	0.635602
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	45.0000	1.58506
$807$	21.0000	0.739235
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	1.00000	0.0351364
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	0	0
$813$	−18.0000	−0.631288
$814$	0	0
$815$	−8.00000	−0.280228
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−2.00000	−0.0699284
$819$	0	0
$820$	11.0000	0.384137
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−2.00000	−0.0697580
$823$	9.00000	0.313720	0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000	0.313720	0.156860	+	0.987621i	$0.449863\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	2.00000	0.0695048
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	3.00000	0.104132
$831$	−26.0000	−0.901930
$832$	5.00000	0.173344
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−20.0000	−0.692129
$836$	0	0
$837$	−9.00000	−0.311086
$838$	14.0000	0.483622
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−21.0000	−0.723708
$843$	−8.00000	−0.275535
$844$	−25.0000	−0.860535
$845$	12.0000	0.412813
$846$	11.0000	0.378188
$847$	0	0
$848$	0	0
$849$	27.0000	0.926638
$850$	−4.00000	−0.137199
$851$	8.00000	0.274236
$852$	−10.0000	−0.342594
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	53.0000	1.81045	0.905223	−	0.424937i	$-0.139704\pi$
$857$	53.0000	1.81045	0.905223	+	0.424937i	$0.139704\pi$
$858$	0	0
$859$	−42.0000	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$859$	−42.0000	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−1.00000	−0.0340207
$865$	17.0000	0.578017
$866$	−11.0000	−0.373795
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	7.00000	0.237322
$871$	10.0000	0.338837
$872$	−14.0000	−0.474100
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	16.0000	0.540590
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−41.0000	−1.38368
$879$	−6.00000	−0.202375
$880$	0	0
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	5.00000	0.168168
$885$	11.0000	0.369761
$886$	5.00000	0.167978
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	8.00000	0.268161
$891$	0	0
$892$	6.00000	0.200895
$893$	0	0
$894$	−10.0000	−0.334450
$895$	3.00000	0.100279
$896$	0	0
$897$	−10.0000	−0.333890
$898$	18.0000	0.600668
$899$	−63.0000	−2.10117
$900$	−4.00000	−0.133333
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−17.0000	−0.565412
$905$	0	0
$906$	−10.0000	−0.332228
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	0	0
$914$	−11.0000	−0.363848
$915$	10.0000	0.330590
$916$	7.00000	0.231287
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	2.00000	0.0659380
$921$	−2.00000	−0.0659022
$922$	−6.00000	−0.197599
$923$	50.0000	1.64577
$924$	0	0
$925$	−16.0000	−0.526077
$926$	8.00000	0.262896
$927$	8.00000	0.262754
$928$	−7.00000	−0.229786
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	−9.00000	−0.295122
$931$	0	0
$932$	1.00000	0.0327561
$933$	26.0000	0.851202
$934$	−5.00000	−0.163605
$935$	0	0
$936$	5.00000	0.163430
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	11.0000	0.358780
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	−17.0000	−0.553890
$943$	22.0000	0.716419
$944$	−11.0000	−0.358020
$945$	0	0
$946$	0	0
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	−12.0000	−0.389742
$949$	−80.0000	−2.59691
$950$	0	0
$951$	5.00000	0.162136
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	21.0000	0.679544
$956$	11.0000	0.355765
$957$	0	0
$958$	22.0000	0.710788
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	50.0000	1.61290
$962$	20.0000	0.644826
$963$	4.00000	0.128898
$964$	14.0000	0.450910
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	12.0000	0.385297
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−5.00000	−0.160210
$975$	20.0000	0.640513
$976$	−10.0000	−0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	8.00000	0.255812
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	36.0000	1.14881
$983$	−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
$985$	−9.00000	−0.286764
$986$	−7.00000	−0.222925
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	3.00000	0.0952981	0.0476491	−	0.998864i	$-0.484827\pi$
$991$	3.00000	0.0952981	0.0476491	+	0.998864i	$0.484827\pi$
$992$	9.00000	0.285750
$993$	12.0000	0.380808
$994$	0	0
$995$	7.00000	0.221915
$996$	−3.00000	−0.0950586
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	−3.00000	−0.0949633
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.bc.1.1		1
7.3	odd	6		714.2.i.a.205.1	✓	2
7.5	odd	6		714.2.i.a.613.1	yes	2
7.6	odd	2		4998.2.a.bm.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.a.205.1	✓	2	7.3	odd	6
714.2.i.a.613.1	yes	2	7.5	odd	6
4998.2.a.bc.1.1		1	1.1	even	1	trivial
4998.2.a.bm.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more