LMFDB - Embedded newform 4998.2.a.bi.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -3.00000 q^{20} -1.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -3.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} +6.00000 q^{38} -3.00000 q^{39} -3.00000 q^{40} +11.0000 q^{43} -1.00000 q^{44} -3.00000 q^{45} -2.00000 q^{46} +1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} -3.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -3.00000 q^{60} +14.0000 q^{61} +1.00000 q^{64} +9.00000 q^{65} -1.00000 q^{66} -7.00000 q^{67} -1.00000 q^{68} -2.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} +1.00000 q^{73} +3.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} -3.00000 q^{78} -5.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{83} +3.00000 q^{85} +11.0000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +1.00000 q^{89} -3.00000 q^{90} -2.00000 q^{92} -18.0000 q^{95} +1.00000 q^{96} +13.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−3.00000	−0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	6.00000	0.973329
$39$	−3.00000	−0.480384
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−1.00000	−0.150756
$45$	−3.00000	−0.447214
$46$	−2.00000	−0.294884
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	−1.00000	−0.140028
$52$	−3.00000	−0.416025
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	0	0
$57$	6.00000	0.794719
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−3.00000	−0.387298
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	9.00000	1.11631
$66$	−1.00000	−0.123091
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−1.00000	−0.121268
$69$	−2.00000	−0.240772
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000	0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	3.00000	0.348743
$75$	4.00000	0.461880
$76$	6.00000	0.688247
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	11.0000	1.18616
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	−2.00000	−0.208514
$93$	0	0
$94$	0	0
$95$	−18.0000	−1.84676
$96$	1.00000	0.102062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	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
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	3.00000	0.286039
$111$	3.00000	0.284747
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	6.00000	0.561951
$115$	6.00000	0.559503
$116$	6.00000	0.557086
$117$	−3.00000	−0.277350
$118$	4.00000	0.368230
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−10.0000	−0.909091
$122$	14.0000	1.26750
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000	0.0883883
$129$	11.0000	0.968496
$130$	9.00000	0.789352
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−7.00000	−0.604708
$135$	−3.00000	−0.258199
$136$	−1.00000	−0.0857493
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−2.00000	−0.170251
$139$	17.0000	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	−18.0000	−1.49482
$146$	1.00000	0.0827606
$147$	0	0
$148$	3.00000	0.246598
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	4.00000	0.326599
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	6.00000	0.486664
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	−3.00000	−0.240192
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−5.00000	−0.397779
$159$	−9.00000	−0.713746
$160$	−3.00000	−0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	−3.00000	−0.232845
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	3.00000	0.230089
$171$	6.00000	0.458831
$172$	11.0000	0.838742
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	1.00000	0.0749532
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	−3.00000	−0.223607
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	−2.00000	−0.147442
$185$	−9.00000	−0.661693
$186$	0	0
$187$	1.00000	0.0731272
$188$	0	0
$189$	0	0
$190$	−18.0000	−1.30586
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	1.00000	0.0721688
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	13.0000	0.933346
$195$	9.00000	0.644503
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−1.00000	−0.0710669
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	4.00000	0.282843
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	17.0000	1.18445
$207$	−2.00000	−0.139010
$208$	−3.00000	−0.208013
$209$	−6.00000	−0.415029
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−9.00000	−0.618123
$213$	12.0000	0.822226
$214$	8.00000	0.546869
$215$	−33.0000	−2.25058
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	1.00000	0.0675737
$220$	3.00000	0.202260
$221$	3.00000	0.201802
$222$	3.00000	0.201347
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	13.0000	0.864747
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	6.00000	0.397360
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	6.00000	0.393919
$233$	25.0000	1.63780	0.818902	−	0.573933i	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	4.00000	0.260378
$237$	−5.00000	−0.324785
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	−3.00000	−0.193649
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	−18.0000	−1.14531
$248$	0	0
$249$	−3.00000	−0.190117
$250$	3.00000	0.189737
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−14.0000	−0.878438
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	11.0000	0.684830
$259$	0	0
$260$	9.00000	0.558156
$261$	6.00000	0.371391
$262$	−2.00000	−0.123560
$263$	7.00000	0.431638	0.215819	−	0.976433i	$-0.430758\pi$
$263$	7.00000	0.431638	0.215819	+	0.976433i	$0.430758\pi$
$264$	−1.00000	−0.0615457
$265$	27.0000	1.65860
$266$	0	0
$267$	1.00000	0.0611990
$268$	−7.00000	−0.427593
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−3.00000	−0.182574
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	8.00000	0.483298
$275$	−4.00000	−0.241209
$276$	−2.00000	−0.120386
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	17.0000	1.01959
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	12.0000	0.712069
$285$	−18.0000	−1.06623
$286$	3.00000	0.177394
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−18.0000	−1.05700
$291$	13.0000	0.762073
$292$	1.00000	0.0585206
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	3.00000	0.174371
$297$	−1.00000	−0.0580259
$298$	−13.0000	−0.753070
$299$	6.00000	0.346989
$300$	4.00000	0.230940
$301$	0	0
$302$	−12.0000	−0.690522
$303$	0	0
$304$	6.00000	0.344124
$305$	−42.0000	−2.40491
$306$	−1.00000	−0.0571662
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	17.0000	0.967096
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	−3.00000	−0.169842
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−5.00000	−0.281272
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−9.00000	−0.504695
$319$	−6.00000	−0.335936
$320$	−3.00000	−0.167705
$321$	8.00000	0.446516
$322$	0	0
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	−12.0000	−0.665640
$326$	−10.0000	−0.553849
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	3.00000	0.165145
$331$	21.0000	1.15426	0.577132	−	0.816651i	$-0.304172\pi$
$331$	21.0000	1.15426	0.577132	+	0.816651i	$0.304172\pi$
$332$	−3.00000	−0.164646
$333$	3.00000	0.164399
$334$	7.00000	0.383023
$335$	21.0000	1.14735
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−4.00000	−0.217571
$339$	13.0000	0.706063
$340$	3.00000	0.162698
$341$	0	0
$342$	6.00000	0.324443
$343$	0	0
$344$	11.0000	0.593080
$345$	6.00000	0.323029
$346$	−6.00000	−0.322562
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	6.00000	0.321634
$349$	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
$351$	−3.00000	−0.160128
$352$	−1.00000	−0.0533002
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	4.00000	0.212598
$355$	−36.0000	−1.91068
$356$	1.00000	0.0529999
$357$	0	0
$358$	−22.0000	−1.16274
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	−8.00000	−0.420471
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−3.00000	−0.157027
$366$	14.0000	0.731792
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−2.00000	−0.104257
$369$	0	0
$370$	−9.00000	−0.467888
$371$	0	0
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	1.00000	0.0517088
$375$	3.00000	0.154919
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	−18.0000	−0.923381
$381$	−14.0000	−0.717242
$382$	15.0000	0.767467
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	11.0000	0.559161
$388$	13.0000	0.659975
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	9.00000	0.455733
$391$	2.00000	0.101144
$392$	0	0
$393$	−2.00000	−0.100887
$394$	−12.0000	−0.604551
$395$	15.0000	0.754732
$396$	−1.00000	−0.0502519
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	4.00000	0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−7.00000	−0.349128
$403$	0	0
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−3.00000	−0.148704
$408$	−1.00000	−0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	17.0000	0.837530
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	9.00000	0.441793
$416$	−3.00000	−0.147087
$417$	17.0000	0.832494
$418$	−6.00000	−0.293470
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	−9.00000	−0.437079
$425$	−4.00000	−0.194029
$426$	12.0000	0.581402
$427$	0	0
$428$	8.00000	0.386695
$429$	3.00000	0.144841
$430$	−33.0000	−1.59140
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000	0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	10.0000	0.478913
$437$	−12.0000	−0.574038
$438$	1.00000	0.0477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	3.00000	0.142695
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	3.00000	0.142374
$445$	−3.00000	−0.142214
$446$	16.0000	0.757622
$447$	−13.0000	−0.614879
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	13.0000	0.611469
$453$	−12.0000	−0.563809
$454$	0	0
$455$	0	0
$456$	6.00000	0.280976
$457$	15.0000	0.701670	0.350835	−	0.936437i	$-0.385898\pi$
$457$	15.0000	0.701670	0.350835	+	0.936437i	$0.385898\pi$
$458$	−21.0000	−0.981266
$459$	−1.00000	−0.0466760
$460$	6.00000	0.279751
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	25.0000	1.15810
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−3.00000	−0.138675
$469$	0	0
$470$	0	0
$471$	22.0000	1.01371
$472$	4.00000	0.184115
$473$	−11.0000	−0.505781
$474$	−5.00000	−0.229658
$475$	24.0000	1.10120
$476$	0	0
$477$	−9.00000	−0.412082
$478$	1.00000	0.0457389
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	−3.00000	−0.136931
$481$	−9.00000	−0.410365
$482$	2.00000	0.0910975
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−39.0000	−1.77090
$486$	1.00000	0.0453609
$487$	29.0000	1.31412	0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000	1.31412	0.657058	+	0.753840i	$0.271801\pi$
$488$	14.0000	0.633750
$489$	−10.0000	−0.452216
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	−18.0000	−0.809858
$495$	3.00000	0.134840
$496$	0	0
$497$	0	0
$498$	−3.00000	−0.134433
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	3.00000	0.134164
$501$	7.00000	0.312737
$502$	−21.0000	−0.937276
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	0.0889108
$507$	−4.00000	−0.177646
$508$	−14.0000	−0.621150
$509$	32.0000	1.41838	0.709188	−	0.705020i	$-0.249062\pi$
$509$	32.0000	1.41838	0.709188	+	0.705020i	$0.249062\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	−17.0000	−0.749838
$515$	−51.0000	−2.24733
$516$	11.0000	0.484248
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	9.00000	0.394676
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	6.00000	0.262613
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	7.00000	0.305215
$527$	0	0
$528$	−1.00000	−0.0435194
$529$	−19.0000	−0.826087
$530$	27.0000	1.17281
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	1.00000	0.0432742
$535$	−24.0000	−1.03761
$536$	−7.00000	−0.302354
$537$	−22.0000	−0.949370
$538$	−14.0000	−0.603583
$539$	0	0
$540$	−3.00000	−0.129099
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	−25.0000	−1.07384
$543$	−8.00000	−0.343313
$544$	−1.00000	−0.0428746
$545$	−30.0000	−1.28506
$546$	0	0
$547$	−30.0000	−1.28271	−0.641354	−	0.767245i	$-0.721627\pi$
$547$	−30.0000	−1.28271	−0.641354	+	0.767245i	$0.721627\pi$
$548$	8.00000	0.341743
$549$	14.0000	0.597505
$550$	−4.00000	−0.170561
$551$	36.0000	1.53365
$552$	−2.00000	−0.0851257
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	−9.00000	−0.382029
$556$	17.0000	0.720961
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−33.0000	−1.39575
$560$	0	0
$561$	1.00000	0.0422200
$562$	6.00000	0.253095
$563$	17.0000	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$563$	17.0000	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$564$	0	0
$565$	−39.0000	−1.64074
$566$	5.00000	0.210166
$567$	0	0
$568$	12.0000	0.503509
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	−18.0000	−0.753937
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	3.00000	0.125436
$573$	15.0000	0.626634
$574$	0	0
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	16.0000	0.664937
$580$	−18.0000	−0.747409
$581$	0	0
$582$	13.0000	0.538867
$583$	9.00000	0.372742
$584$	1.00000	0.0413803
$585$	9.00000	0.372104
$586$	4.00000	0.165238
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	0	0
$590$	−12.0000	−0.494032
$591$	−12.0000	−0.493614
$592$	3.00000	0.123299
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−13.0000	−0.532501
$597$	−12.0000	−0.491127
$598$	6.00000	0.245358
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	4.00000	0.163299
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−7.00000	−0.285062
$604$	−12.0000	−0.488273
$605$	30.0000	1.21967
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−42.0000	−1.70053
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	−36.0000	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−36.0000	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	17.0000	0.683840
$619$	−21.0000	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$619$	−21.0000	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	−9.00000	−0.360867
$623$	0	0
$624$	−3.00000	−0.120096
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	−6.00000	−0.239617
$628$	22.0000	0.877896
$629$	−3.00000	−0.119618
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	−5.00000	−0.198889
$633$	22.0000	0.874421
$634$	2.00000	0.0794301
$635$	42.0000	1.66672
$636$	−9.00000	−0.356873
$637$	0	0
$638$	−6.00000	−0.237542
$639$	12.0000	0.474713
$640$	−3.00000	−0.118585
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	8.00000	0.315735
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	−33.0000	−1.29937
$646$	−6.00000	−0.236067
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	1.00000	0.0392837
$649$	−4.00000	−0.157014
$650$	−12.0000	−0.470679
$651$	0	0
$652$	−10.0000	−0.391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	10.0000	0.391031
$655$	6.00000	0.234439
$656$	0	0
$657$	1.00000	0.0390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	3.00000	0.116775
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	21.0000	0.816188
$663$	3.00000	0.116510
$664$	−3.00000	−0.116423
$665$	0	0
$666$	3.00000	0.116248
$667$	−12.0000	−0.464642
$668$	7.00000	0.270838
$669$	16.0000	0.618596
$670$	21.0000	0.811301
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	6.00000	0.231111
$675$	4.00000	0.153960
$676$	−4.00000	−0.153846
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	13.0000	0.499262
$679$	0	0
$680$	3.00000	0.115045
$681$	0	0
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	6.00000	0.229416
$685$	−24.0000	−0.916993
$686$	0	0
$687$	−21.0000	−0.801200
$688$	11.0000	0.419371
$689$	27.0000	1.02862
$690$	6.00000	0.228416
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−3.00000	−0.113878
$695$	−51.0000	−1.93454
$696$	6.00000	0.227429
$697$	0	0
$698$	23.0000	0.870563
$699$	25.0000	0.945587
$700$	0	0
$701$	−51.0000	−1.92624	−0.963122	−	0.269066i	$-0.913285\pi$
$701$	−51.0000	−1.92624	−0.963122	+	0.269066i	$0.913285\pi$
$702$	−3.00000	−0.113228
$703$	18.0000	0.678883
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	21.0000	0.790345
$707$	0	0
$708$	4.00000	0.150329
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	−36.0000	−1.35106
$711$	−5.00000	−0.187515
$712$	1.00000	0.0374766
$713$	0	0
$714$	0	0
$715$	−9.00000	−0.336581
$716$	−22.0000	−0.822179
$717$	1.00000	0.0373457
$718$	−24.0000	−0.895672
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	17.0000	0.632674
$723$	2.00000	0.0743808
$724$	−8.00000	−0.297318
$725$	24.0000	0.891338
$726$	−10.0000	−0.371135
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−3.00000	−0.111035
$731$	−11.0000	−0.406850
$732$	14.0000	0.517455
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	7.00000	0.257848
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−9.00000	−0.330847
$741$	−18.0000	−0.661247
$742$	0	0
$743$	38.0000	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$744$	0	0
$745$	39.0000	1.42885
$746$	−16.0000	−0.585802
$747$	−3.00000	−0.109764
$748$	1.00000	0.0365636
$749$	0	0
$750$	3.00000	0.109545
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	−18.0000	−0.655521
$755$	36.0000	1.31017
$756$	0	0
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	30.0000	1.08965
$759$	2.00000	0.0725954
$760$	−18.0000	−0.652929
$761$	−29.0000	−1.05125	−0.525625	−	0.850717i	$-0.676168\pi$
$761$	−29.0000	−1.05125	−0.525625	+	0.850717i	$0.676168\pi$
$762$	−14.0000	−0.507166
$763$	0	0
$764$	15.0000	0.542681
$765$	3.00000	0.108465
$766$	−20.0000	−0.722629
$767$	−12.0000	−0.433295
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−17.0000	−0.612240
$772$	16.0000	0.575853
$773$	−44.0000	−1.58257	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000	−1.58257	−0.791285	+	0.611448i	$0.790588\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	13.0000	0.466673
$777$	0	0
$778$	−6.00000	−0.215110
$779$	0	0
$780$	9.00000	0.322252
$781$	−12.0000	−0.429394
$782$	2.00000	0.0715199
$783$	6.00000	0.214423
$784$	0	0
$785$	−66.0000	−2.35564
$786$	−2.00000	−0.0713376
$787$	27.0000	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$787$	27.0000	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$788$	−12.0000	−0.427482
$789$	7.00000	0.249207
$790$	15.0000	0.533676
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−42.0000	−1.49146
$794$	28.0000	0.993683
$795$	27.0000	0.957591
$796$	−12.0000	−0.425329
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	1.00000	0.0353333
$802$	30.0000	1.05934
$803$	−1.00000	−0.0352892
$804$	−7.00000	−0.246871
$805$	0	0
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	−3.00000	−0.105409
$811$	13.0000	0.456492	0.228246	−	0.973604i	$-0.426701\pi$
$811$	13.0000	0.456492	0.228246	+	0.973604i	$0.426701\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	−3.00000	−0.105150
$815$	30.0000	1.05085
$816$	−1.00000	−0.0350070
$817$	66.0000	2.30905
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	8.00000	0.279032
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	17.0000	0.592223
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−5.00000	−0.173867	−0.0869335	−	0.996214i	$-0.527707\pi$
$827$	−5.00000	−0.173867	−0.0869335	+	0.996214i	$0.527707\pi$
$828$	−2.00000	−0.0695048
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	9.00000	0.312395
$831$	−2.00000	−0.0693792
$832$	−3.00000	−0.104006
$833$	0	0
$834$	17.0000	0.588662
$835$	−21.0000	−0.726735
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−30.0000	−1.03633
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	6.00000	0.206651
$844$	22.0000	0.757271
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	−9.00000	−0.309061
$849$	5.00000	0.171600
$850$	−4.00000	−0.137199
$851$	−6.00000	−0.205677
$852$	12.0000	0.411113
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	8.00000	0.273434
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	3.00000	0.102418
$859$	42.0000	1.43302	0.716511	−	0.697576i	$-0.245738\pi$
$859$	42.0000	1.43302	0.716511	+	0.697576i	$0.245738\pi$
$860$	−33.0000	−1.12529
$861$	0	0
$862$	−40.0000	−1.36241
$863$	27.0000	0.919091	0.459545	−	0.888154i	$-0.348012\pi$
$863$	27.0000	0.919091	0.459545	+	0.888154i	$0.348012\pi$
$864$	1.00000	0.0340207
$865$	18.0000	0.612018
$866$	−16.0000	−0.543702
$867$	1.00000	0.0339618
$868$	0	0
$869$	5.00000	0.169613
$870$	−18.0000	−0.610257
$871$	21.0000	0.711558
$872$	10.0000	0.338643
$873$	13.0000	0.439983
$874$	−12.0000	−0.405906
$875$	0	0
$876$	1.00000	0.0337869
$877$	−31.0000	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$878$	−28.0000	−0.944954
$879$	4.00000	0.134917
$880$	3.00000	0.101130
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	3.00000	0.100901
$885$	−12.0000	−0.403376
$886$	−28.0000	−0.940678
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	3.00000	0.100673
$889$	0	0
$890$	−3.00000	−0.100560
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	0	0
$894$	−13.0000	−0.434785
$895$	66.0000	2.20614
$896$	0	0
$897$	6.00000	0.200334
$898$	−27.0000	−0.901002
$899$	0	0
$900$	4.00000	0.133333
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	13.0000	0.432374
$905$	24.0000	0.797787
$906$	−12.0000	−0.398673
$907$	−48.0000	−1.59381	−0.796907	−	0.604102i	$-0.793532\pi$
$907$	−48.0000	−1.59381	−0.796907	+	0.604102i	$0.793532\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	6.00000	0.198680
$913$	3.00000	0.0992855
$914$	15.0000	0.496156
$915$	−42.0000	−1.38848
$916$	−21.0000	−0.693860
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	6.00000	0.197814
$921$	8.00000	0.263609
$922$	−18.0000	−0.592798
$923$	−36.0000	−1.18495
$924$	0	0
$925$	12.0000	0.394558
$926$	20.0000	0.657241
$927$	17.0000	0.558353
$928$	6.00000	0.196960
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	25.0000	0.818902
$933$	−9.00000	−0.294647
$934$	24.0000	0.785304
$935$	−3.00000	−0.0981105
$936$	−3.00000	−0.0980581
$937$	−60.0000	−1.96011	−0.980057	−	0.198715i	$-0.936323\pi$
$937$	−60.0000	−1.96011	−0.980057	+	0.198715i	$0.936323\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	−57.0000	−1.85815	−0.929073	−	0.369895i	$-0.879394\pi$
$941$	−57.0000	−1.85815	−0.929073	+	0.369895i	$0.879394\pi$
$942$	22.0000	0.716799
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	−11.0000	−0.357641
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−5.00000	−0.162392
$949$	−3.00000	−0.0973841
$950$	24.0000	0.778663
$951$	2.00000	0.0648544
$952$	0	0
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	−9.00000	−0.291386
$955$	−45.0000	−1.45617
$956$	1.00000	0.0323423
$957$	−6.00000	−0.193952
$958$	−39.0000	−1.26003
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	−31.0000	−1.00000
$962$	−9.00000	−0.290172
$963$	8.00000	0.257796
$964$	2.00000	0.0644157
$965$	−48.0000	−1.54517
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−10.0000	−0.321412
$969$	−6.00000	−0.192748
$970$	−39.0000	−1.25221
$971$	−23.0000	−0.738105	−0.369053	−	0.929409i	$-0.620318\pi$
$971$	−23.0000	−0.738105	−0.369053	+	0.929409i	$0.620318\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	29.0000	0.929220
$975$	−12.0000	−0.384308
$976$	14.0000	0.448129
$977$	60.0000	1.91957	0.959785	−	0.280736i	$-0.0905785\pi$
$977$	60.0000	1.91957	0.959785	+	0.280736i	$0.0905785\pi$
$978$	−10.0000	−0.319765
$979$	−1.00000	−0.0319601
$980$	0	0
$981$	10.0000	0.319275
$982$	30.0000	0.957338
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	−6.00000	−0.191079
$987$	0	0
$988$	−18.0000	−0.572656
$989$	−22.0000	−0.699559
$990$	3.00000	0.0953463
$991$	51.0000	1.62007	0.810034	−	0.586383i	$-0.199448\pi$
$991$	51.0000	1.62007	0.810034	+	0.586383i	$0.199448\pi$
$992$	0	0
$993$	21.0000	0.666415
$994$	0	0
$995$	36.0000	1.14128
$996$	−3.00000	−0.0950586
$997$	48.0000	1.52018	0.760088	−	0.649821i	$-0.225156\pi$
$997$	48.0000	1.52018	0.760088	+	0.649821i	$0.225156\pi$
$998$	−20.0000	−0.633089
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.bi.1.1		1
7.6	odd	2		714.2.a.h.1.1	✓	1
21.20	even	2		2142.2.a.b.1.1		1
28.27	even	2		5712.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.a.h.1.1	✓	1	7.6	odd	2
2142.2.a.b.1.1		1	21.20	even	2
4998.2.a.bi.1.1		1	1.1	even	1	trivial
5712.2.a.ba.1.1		1	28.27	even	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

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