LMFDB - Embedded newform 425.6.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+6.00000 q^{2} -10.0000 q^{3} +4.00000 q^{4} -60.0000 q^{6} +196.000 q^{7} -168.000 q^{8} -143.000 q^{9} +450.000 q^{11} -40.0000 q^{12} +142.000 q^{13} +1176.00 q^{14} -1136.00 q^{16} +289.000 q^{17} -858.000 q^{18} -244.000 q^{19} -1960.00 q^{21} +2700.00 q^{22} -2904.00 q^{23} +1680.00 q^{24} +852.000 q^{26} +3860.00 q^{27} +784.000 q^{28} -6984.00 q^{29} -436.000 q^{31} -1440.00 q^{32} -4500.00 q^{33} +1734.00 q^{34} -572.000 q^{36} +8572.00 q^{37} -1464.00 q^{38} -1420.00 q^{39} +16374.0 q^{41} -11760.0 q^{42} +19216.0 q^{43} +1800.00 q^{44} -17424.0 q^{46} +19920.0 q^{47} +11360.0 q^{48} +21609.0 q^{49} -2890.00 q^{51} +568.000 q^{52} -1146.00 q^{53} +23160.0 q^{54} -32928.0 q^{56} +2440.00 q^{57} -41904.0 q^{58} +22008.0 q^{59} +35780.0 q^{61} -2616.00 q^{62} -28028.0 q^{63} +27712.0 q^{64} -27000.0 q^{66} -23264.0 q^{67} +1156.00 q^{68} +29040.0 q^{69} -31704.0 q^{71} +24024.0 q^{72} +13966.0 q^{73} +51432.0 q^{74} -976.000 q^{76} +88200.0 q^{77} -8520.00 q^{78} -51088.0 q^{79} -3851.00 q^{81} +98244.0 q^{82} +64344.0 q^{83} -7840.00 q^{84} +115296. q^{86} +69840.0 q^{87} -75600.0 q^{88} +70650.0 q^{89} +27832.0 q^{91} -11616.0 q^{92} +4360.00 q^{93} +119520. q^{94} +14400.0 q^{96} -62702.0 q^{97} +129654. q^{98} -64350.0 q^{99} +O(q^{100})\)

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 $

$2$	6.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	6.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	−10.0000	−0.641500	−0.320750	−	0.947164i	$-0.603935\pi$
$3$	−10.0000	−0.641500	−0.320750	+	0.947164i	$0.603935\pi$
$4$	4.00000	0.125000
$5$	0	0
$5$	0	0
$6$	−60.0000	−0.680414
$7$	196.000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	196.000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−168.000	−0.928078
$9$	−143.000	−0.588477
$10$	0	0
$11$	450.000	1.12132	0.560662	−	0.828045i	$-0.310547\pi$
$11$	450.000	1.12132	0.560662	+	0.828045i	$0.310547\pi$
$12$	−40.0000	−0.0801875
$13$	142.000	0.233040	0.116520	−	0.993188i	$-0.462826\pi$
$13$	142.000	0.233040	0.116520	+	0.993188i	$0.462826\pi$
$14$	1176.00	1.60357
$15$	0	0
$16$	−1136.00	−1.10938
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	−858.000	−0.624175
$19$	−244.000	−0.155062	−0.0775311	−	0.996990i	$-0.524704\pi$
$19$	−244.000	−0.155062	−0.0775311	+	0.996990i	$0.524704\pi$
$20$	0	0
$21$	−1960.00	−0.969857
$22$	2700.00	1.18934
$23$	−2904.00	−1.14466	−0.572331	−	0.820023i	$-0.693961\pi$
$23$	−2904.00	−1.14466	−0.572331	+	0.820023i	$0.693961\pi$
$24$	1680.00	0.595362
$25$	0	0
$26$	852.000	0.247176
$27$	3860.00	1.01901
$28$	784.000	0.188982
$29$	−6984.00	−1.54209	−0.771044	−	0.636782i	$-0.780265\pi$
$29$	−6984.00	−1.54209	−0.771044	+	0.636782i	$0.780265\pi$
$30$	0	0
$31$	−436.000	−0.0814859	−0.0407429	−	0.999170i	$-0.512972\pi$
$31$	−436.000	−0.0814859	−0.0407429	+	0.999170i	$0.512972\pi$
$32$	−1440.00	−0.248592
$33$	−4500.00	−0.719329
$34$	1734.00	0.257248
$35$	0	0
$36$	−572.000	−0.0735597
$37$	8572.00	1.02939	0.514693	−	0.857375i	$-0.327906\pi$
$37$	8572.00	1.02939	0.514693	+	0.857375i	$0.327906\pi$
$38$	−1464.00	−0.164468
$39$	−1420.00	−0.149495
$40$	0	0
$41$	16374.0	1.52123	0.760615	−	0.649203i	$-0.224897\pi$
$41$	16374.0	1.52123	0.760615	+	0.649203i	$0.224897\pi$
$42$	−11760.0	−1.02869
$43$	19216.0	1.58486	0.792432	−	0.609961i	$-0.208815\pi$
$43$	19216.0	1.58486	0.792432	+	0.609961i	$0.208815\pi$
$44$	1800.00	0.140165
$45$	0	0
$46$	−17424.0	−1.21410
$47$	19920.0	1.31536	0.657680	−	0.753297i	$-0.271538\pi$
$47$	19920.0	1.31536	0.657680	+	0.753297i	$0.271538\pi$
$48$	11360.0	0.711664
$49$	21609.0	1.28571
$50$	0	0
$51$	−2890.00	−0.155587
$52$	568.000	0.0291300
$53$	−1146.00	−0.0560396	−0.0280198	−	0.999607i	$-0.508920\pi$
$53$	−1146.00	−0.0560396	−0.0280198	+	0.999607i	$0.508920\pi$
$54$	23160.0	1.08082
$55$	0	0
$56$	−32928.0	−1.40312
$57$	2440.00	0.0994724
$58$	−41904.0	−1.63563
$59$	22008.0	0.823096	0.411548	−	0.911388i	$-0.364988\pi$
$59$	22008.0	0.823096	0.411548	+	0.911388i	$0.364988\pi$
$60$	0	0
$61$	35780.0	1.23116	0.615582	−	0.788073i	$-0.288921\pi$
$61$	35780.0	1.23116	0.615582	+	0.788073i	$0.288921\pi$
$62$	−2616.00	−0.0864288
$63$	−28028.0	−0.889694
$64$	27712.0	0.845703
$65$	0	0
$66$	−27000.0	−0.762964
$67$	−23264.0	−0.633137	−0.316568	−	0.948570i	$-0.602531\pi$
$67$	−23264.0	−0.633137	−0.316568	+	0.948570i	$0.602531\pi$
$68$	1156.00	0.0303170
$69$	29040.0	0.734301
$70$	0	0
$71$	−31704.0	−0.746394	−0.373197	−	0.927752i	$-0.621738\pi$
$71$	−31704.0	−0.746394	−0.373197	+	0.927752i	$0.621738\pi$
$72$	24024.0	0.546153
$73$	13966.0	0.306736	0.153368	−	0.988169i	$-0.450988\pi$
$73$	13966.0	0.306736	0.153368	+	0.988169i	$0.450988\pi$
$74$	51432.0	1.09183
$75$	0	0
$76$	−976.000	−0.0193828
$77$	88200.0	1.69528
$78$	−8520.00	−0.158563
$79$	−51088.0	−0.920982	−0.460491	−	0.887664i	$-0.652327\pi$
$79$	−51088.0	−0.920982	−0.460491	+	0.887664i	$0.652327\pi$
$80$	0	0
$81$	−3851.00	−0.0652170
$82$	98244.0	1.61351
$83$	64344.0	1.02521	0.512605	−	0.858625i	$-0.328681\pi$
$83$	64344.0	1.02521	0.512605	+	0.858625i	$0.328681\pi$
$84$	−7840.00	−0.121232
$85$	0	0
$86$	115296.	1.68100
$87$	69840.0	0.989250
$88$	−75600.0	−1.04067
$89$	70650.0	0.945447	0.472723	−	0.881211i	$-0.343271\pi$
$89$	70650.0	0.945447	0.472723	+	0.881211i	$0.343271\pi$
$90$	0	0
$91$	27832.0	0.352323
$92$	−11616.0	−0.143083
$93$	4360.00	0.0522732
$94$	119520.	1.39515
$95$	0	0
$96$	14400.0	0.159472
$97$	−62702.0	−0.676631	−0.338316	−	0.941033i	$-0.609857\pi$
$97$	−62702.0	−0.676631	−0.338316	+	0.941033i	$0.609857\pi$
$98$	129654.	1.36371
$99$	−64350.0	−0.659873
$100$	0	0
$101$	−85470.0	−0.833701	−0.416850	−	0.908975i	$-0.636866\pi$
$101$	−85470.0	−0.833701	−0.416850	+	0.908975i	$0.636866\pi$
$102$	−17340.0	−0.165025
$103$	181792.	1.68842	0.844212	−	0.536009i	$-0.180069\pi$
$103$	181792.	1.68842	0.844212	+	0.536009i	$0.180069\pi$
$104$	−23856.0	−0.216279
$105$	0	0
$106$	−6876.00	−0.0594390
$107$	−82638.0	−0.697783	−0.348891	−	0.937163i	$-0.613442\pi$
$107$	−82638.0	−0.697783	−0.348891	+	0.937163i	$0.613442\pi$
$108$	15440.0	0.127376
$109$	245576.	1.97979	0.989896	−	0.141794i	$-0.0452869\pi$
$109$	245576.	1.97979	0.989896	+	0.141794i	$0.0452869\pi$
$110$	0	0
$111$	−85720.0	−0.660351
$112$	−222656.	−1.67722
$113$	187014.	1.37777	0.688887	−	0.724869i	$-0.258100\pi$
$113$	187014.	1.37777	0.688887	+	0.724869i	$0.258100\pi$
$114$	14640.0	0.105506
$115$	0	0
$116$	−27936.0	−0.192761
$117$	−20306.0	−0.137139
$118$	132048.	0.873025
$119$	56644.0	0.366679
$120$	0	0
$121$	41449.0	0.257366
$122$	214680.	1.30585
$123$	−163740.	−0.975870
$124$	−1744.00	−0.0101857
$125$	0	0
$126$	−168168.	−0.943663
$127$	−102848.	−0.565831	−0.282915	−	0.959145i	$-0.591302\pi$
$127$	−102848.	−0.565831	−0.282915	+	0.959145i	$0.591302\pi$
$128$	212352.	1.14560
$129$	−192160.	−1.01669
$130$	0	0
$131$	128286.	0.653132	0.326566	−	0.945174i	$-0.394108\pi$
$131$	128286.	0.653132	0.326566	+	0.945174i	$0.394108\pi$
$132$	−18000.0	−0.0899161
$133$	−47824.0	−0.234432
$134$	−139584.	−0.671543
$135$	0	0
$136$	−48552.0	−0.225092
$137$	−23334.0	−0.106215	−0.0531077	−	0.998589i	$-0.516913\pi$
$137$	−23334.0	−0.106215	−0.0531077	+	0.998589i	$0.516913\pi$
$138$	174240.	0.778843
$139$	−17338.0	−0.0761136	−0.0380568	−	0.999276i	$-0.512117\pi$
$139$	−17338.0	−0.0761136	−0.0380568	+	0.999276i	$0.512117\pi$
$140$	0	0
$141$	−199200.	−0.843804
$142$	−190224.	−0.791670
$143$	63900.0	0.261313
$144$	162448.	0.652842
$145$	0	0
$146$	83796.0	0.325343
$147$	−216090.	−0.824786
$148$	34288.0	0.128673
$149$	51990.0	0.191847	0.0959233	−	0.995389i	$-0.469420\pi$
$149$	51990.0	0.191847	0.0959233	+	0.995389i	$0.469420\pi$
$150$	0	0
$151$	131504.	0.469350	0.234675	−	0.972074i	$-0.424597\pi$
$151$	131504.	0.469350	0.234675	+	0.972074i	$0.424597\pi$
$152$	40992.0	0.143910
$153$	−41327.0	−0.142727
$154$	529200.	1.79812
$155$	0	0
$156$	−5680.00	−0.0186869
$157$	598882.	1.93906	0.969532	−	0.244965i	$-0.0787764\pi$
$157$	598882.	1.93906	0.969532	+	0.244965i	$0.0787764\pi$
$158$	−306528.	−0.976849
$159$	11460.0	0.0359494
$160$	0	0
$161$	−569184.	−1.73057
$162$	−23106.0	−0.0691731
$163$	321946.	0.949104	0.474552	−	0.880227i	$-0.342610\pi$
$163$	321946.	0.949104	0.474552	+	0.880227i	$0.342610\pi$
$164$	65496.0	0.190154
$165$	0	0
$166$	386064.	1.08740
$167$	−339516.	−0.942039	−0.471020	−	0.882123i	$-0.656114\pi$
$167$	−339516.	−0.942039	−0.471020	+	0.882123i	$0.656114\pi$
$168$	329280.	0.900103
$169$	−351129.	−0.945692
$170$	0	0
$171$	34892.0	0.0912506
$172$	76864.0	0.198108
$173$	−252432.	−0.641253	−0.320626	−	0.947206i	$-0.603893\pi$
$173$	−252432.	−0.641253	−0.320626	+	0.947206i	$0.603893\pi$
$174$	419040.	1.04926
$175$	0	0
$176$	−511200.	−1.24397
$177$	−220080.	−0.528016
$178$	423900.	1.00280
$179$	−555108.	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$179$	−555108.	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$180$	0	0
$181$	1508.00	0.00342141	0.00171070	−	0.999999i	$-0.499455\pi$
$181$	1508.00	0.00342141	0.00171070	+	0.999999i	$0.499455\pi$
$182$	166992.	0.373695
$183$	−357800.	−0.789792
$184$	487872.	1.06233
$185$	0	0
$186$	26160.0	0.0554441
$187$	130050.	0.271961
$188$	79680.0	0.164420
$189$	756560.	1.54060
$190$	0	0
$191$	−392856.	−0.779202	−0.389601	−	0.920984i	$-0.627387\pi$
$191$	−392856.	−0.779202	−0.389601	+	0.920984i	$0.627387\pi$
$192$	−277120.	−0.542519
$193$	390538.	0.754692	0.377346	−	0.926072i	$-0.376837\pi$
$193$	390538.	0.754692	0.377346	+	0.926072i	$0.376837\pi$
$194$	−376212.	−0.717676
$195$	0	0
$196$	86436.0	0.160714
$197$	−205212.	−0.376736	−0.188368	−	0.982099i	$-0.560320\pi$
$197$	−205212.	−0.376736	−0.188368	+	0.982099i	$0.560320\pi$
$198$	−386100.	−0.699901
$199$	−405160.	−0.725260	−0.362630	−	0.931933i	$-0.618121\pi$
$199$	−405160.	−0.725260	−0.362630	+	0.931933i	$0.618121\pi$
$200$	0	0
$201$	232640.	0.406157
$202$	−512820.	−0.884273
$203$	−1.36886e6	−2.33142
$204$	−11560.0	−0.0194483
$205$	0	0
$206$	1.09075e6	1.79084
$207$	415272.	0.673607
$208$	−161312.	−0.258528
$209$	−109800.	−0.173875
$210$	0	0
$211$	−269662.	−0.416978	−0.208489	−	0.978025i	$-0.566855\pi$
$211$	−269662.	−0.416978	−0.208489	+	0.978025i	$0.566855\pi$
$212$	−4584.00	−0.00700495
$213$	317040.	0.478812
$214$	−495828.	−0.740111
$215$	0	0
$216$	−648480.	−0.945719
$217$	−85456.0	−0.123195
$218$	1.47346e6	2.09989
$219$	−139660.	−0.196771
$220$	0	0
$221$	41038.0	0.0565204
$222$	−514320.	−0.700408
$223$	455656.	0.613585	0.306793	−	0.951776i	$-0.400744\pi$
$223$	455656.	0.613585	0.306793	+	0.951776i	$0.400744\pi$
$224$	−282240.	−0.375836
$225$	0	0
$226$	1.12208e6	1.46135
$227$	−792366.	−1.02061	−0.510307	−	0.859993i	$-0.670468\pi$
$227$	−792366.	−1.02061	−0.510307	+	0.859993i	$0.670468\pi$
$228$	9760.00	0.0124341
$229$	−179794.	−0.226562	−0.113281	−	0.993563i	$-0.536136\pi$
$229$	−179794.	−0.226562	−0.113281	+	0.993563i	$0.536136\pi$
$230$	0	0
$231$	−882000.	−1.08752
$232$	1.17331e6	1.43118
$233$	−245154.	−0.295835	−0.147917	−	0.989000i	$-0.547257\pi$
$233$	−245154.	−0.295835	−0.147917	+	0.989000i	$0.547257\pi$
$234$	−121836.	−0.145457
$235$	0	0
$236$	88032.0	0.102887
$237$	510880.	0.590810
$238$	339864.	0.388922
$239$	−334536.	−0.378833	−0.189417	−	0.981897i	$-0.560660\pi$
$239$	−334536.	−0.378833	−0.189417	+	0.981897i	$0.560660\pi$
$240$	0	0
$241$	−882922.	−0.979219	−0.489609	−	0.871942i	$-0.662861\pi$
$241$	−882922.	−0.979219	−0.489609	+	0.871942i	$0.662861\pi$
$242$	248694.	0.272978
$243$	−899470.	−0.977172
$244$	143120.	0.153895
$245$	0	0
$246$	−982440.	−1.03507
$247$	−34648.0	−0.0361356
$248$	73248.0	0.0756252
$249$	−643440.	−0.657673
$250$	0	0
$251$	−146928.	−0.147204	−0.0736021	−	0.997288i	$-0.523449\pi$
$251$	−146928.	−0.147204	−0.0736021	+	0.997288i	$0.523449\pi$
$252$	−112112.	−0.111212
$253$	−1.30680e6	−1.28354
$254$	−617088.	−0.600154
$255$	0	0
$256$	387328.	0.369385
$257$	737082.	0.696118	0.348059	−	0.937473i	$-0.386841\pi$
$257$	737082.	0.696118	0.348059	+	0.937473i	$0.386841\pi$
$258$	−1.15296e6	−1.07836
$259$	1.68011e6	1.55628
$260$	0	0
$261$	998712.	0.907484
$262$	769716.	0.692751
$263$	1.08214e6	0.964700	0.482350	−	0.875978i	$-0.339783\pi$
$263$	1.08214e6	0.964700	0.482350	+	0.875978i	$0.339783\pi$
$264$	756000.	0.667593
$265$	0	0
$266$	−286944.	−0.248653
$267$	−706500.	−0.606504
$268$	−93056.0	−0.0791421
$269$	1.51939e6	1.28023	0.640117	−	0.768278i	$-0.278886\pi$
$269$	1.51939e6	1.28023	0.640117	+	0.768278i	$0.278886\pi$
$270$	0	0
$271$	395624.	0.327235	0.163617	−	0.986524i	$-0.447684\pi$
$271$	395624.	0.327235	0.163617	+	0.986524i	$0.447684\pi$
$272$	−328304.	−0.269063
$273$	−278320.	−0.226015
$274$	−140004.	−0.112659
$275$	0	0
$276$	116160.	0.0917876
$277$	1.15178e6	0.901921	0.450961	−	0.892544i	$-0.351081\pi$
$277$	1.15178e6	0.901921	0.450961	+	0.892544i	$0.351081\pi$
$278$	−104028.	−0.0807306
$279$	62348.0	0.0479526
$280$	0	0
$281$	1.37261e6	1.03701	0.518505	−	0.855075i	$-0.326489\pi$
$281$	1.37261e6	1.03701	0.518505	+	0.855075i	$0.326489\pi$
$282$	−1.19520e6	−0.894989
$283$	−229754.	−0.170529	−0.0852643	−	0.996358i	$-0.527173\pi$
$283$	−229754.	−0.170529	−0.0852643	+	0.996358i	$0.527173\pi$
$284$	−126816.	−0.0932993
$285$	0	0
$286$	383400.	0.277164
$287$	3.20930e6	2.29988
$288$	205920.	0.146291
$289$	83521.0	0.0588235
$290$	0	0
$291$	627020.	0.434059
$292$	55864.0	0.0383420
$293$	−761526.	−0.518222	−0.259111	−	0.965848i	$-0.583430\pi$
$293$	−761526.	−0.518222	−0.259111	+	0.965848i	$0.583430\pi$
$294$	−1.29654e6	−0.874818
$295$	0	0
$296$	−1.44010e6	−0.955349
$297$	1.73700e6	1.14264
$298$	311940.	0.203484
$299$	−412368.	−0.266752
$300$	0	0
$301$	3.76634e6	2.39609
$302$	789024.	0.497821
$303$	854700.	0.534819
$304$	277184.	0.172022
$305$	0	0
$306$	−247962.	−0.151385
$307$	−599348.	−0.362939	−0.181469	−	0.983397i	$-0.558085\pi$
$307$	−599348.	−0.362939	−0.181469	+	0.983397i	$0.558085\pi$
$308$	352800.	0.211910
$309$	−1.81792e6	−1.08312
$310$	0	0
$311$	847548.	0.496894	0.248447	−	0.968646i	$-0.420080\pi$
$311$	847548.	0.496894	0.248447	+	0.968646i	$0.420080\pi$
$312$	238560.	0.138743
$313$	−2.00008e6	−1.15395	−0.576974	−	0.816763i	$-0.695767\pi$
$313$	−2.00008e6	−1.15395	−0.576974	+	0.816763i	$0.695767\pi$
$314$	3.59329e6	2.05669
$315$	0	0
$316$	−204352.	−0.115123
$317$	−194460.	−0.108688	−0.0543441	−	0.998522i	$-0.517307\pi$
$317$	−194460.	−0.108688	−0.0543441	+	0.998522i	$0.517307\pi$
$318$	68760.0	0.0381301
$319$	−3.14280e6	−1.72918
$320$	0	0
$321$	826380.	0.447628
$322$	−3.41510e6	−1.83554
$323$	−70516.0	−0.0376081
$324$	−15404.0	−0.00815213
$325$	0	0
$326$	1.93168e6	1.00668
$327$	−2.45576e6	−1.27004
$328$	−2.75083e6	−1.41182
$329$	3.90432e6	1.98864
$330$	0	0
$331$	−1.82342e6	−0.914778	−0.457389	−	0.889267i	$-0.651215\pi$
$331$	−1.82342e6	−0.914778	−0.457389	+	0.889267i	$0.651215\pi$
$332$	257376.	0.128151
$333$	−1.22580e6	−0.605770
$334$	−2.03710e6	−0.999184
$335$	0	0
$336$	2.22656e6	1.07594
$337$	2.59269e6	1.24359	0.621794	−	0.783181i	$-0.286404\pi$
$337$	2.59269e6	1.24359	0.621794	+	0.783181i	$0.286404\pi$
$338$	−2.10677e6	−1.00306
$339$	−1.87014e6	−0.883842
$340$	0	0
$341$	−196200.	−0.0913720
$342$	209352.	0.0967858
$343$	941192.	0.431959
$344$	−3.22829e6	−1.47088
$345$	0	0
$346$	−1.51459e6	−0.680151
$347$	−2.67418e6	−1.19225	−0.596125	−	0.802892i	$-0.703294\pi$
$347$	−2.67418e6	−1.19225	−0.596125	+	0.802892i	$0.703294\pi$
$348$	279360.	0.123656
$349$	504122.	0.221550	0.110775	−	0.993846i	$-0.464667\pi$
$349$	504122.	0.221550	0.110775	+	0.993846i	$0.464667\pi$
$350$	0	0
$351$	548120.	0.237470
$352$	−648000.	−0.278752
$353$	−1.00791e6	−0.430512	−0.215256	−	0.976558i	$-0.569059\pi$
$353$	−1.00791e6	−0.430512	−0.215256	+	0.976558i	$0.569059\pi$
$354$	−1.32048e6	−0.560046
$355$	0	0
$356$	282600.	0.118181
$357$	−566440.	−0.235225
$358$	−3.33065e6	−1.37348
$359$	−4.74516e6	−1.94319	−0.971594	−	0.236655i	$-0.923949\pi$
$359$	−4.74516e6	−1.94319	−0.971594	+	0.236655i	$0.923949\pi$
$360$	0	0
$361$	−2.41656e6	−0.975956
$362$	9048.00	0.00362895
$363$	−414490.	−0.165100
$364$	111328.	0.0440404
$365$	0	0
$366$	−2.14680e6	−0.837701
$367$	−155048.	−0.0600898	−0.0300449	−	0.999549i	$-0.509565\pi$
$367$	−155048.	−0.0600898	−0.0300449	+	0.999549i	$0.509565\pi$
$368$	3.29894e6	1.26986
$369$	−2.34148e6	−0.895210
$370$	0	0
$371$	−224616.	−0.0847239
$372$	17440.0	0.00653415
$373$	−3.89811e6	−1.45071	−0.725357	−	0.688373i	$-0.758325\pi$
$373$	−3.89811e6	−1.45071	−0.725357	+	0.688373i	$0.758325\pi$
$374$	780300.	0.288458
$375$	0	0
$376$	−3.34656e6	−1.22076
$377$	−991728.	−0.359368
$378$	4.53936e6	1.63405
$379$	−2.76828e6	−0.989946	−0.494973	−	0.868908i	$-0.664822\pi$
$379$	−2.76828e6	−0.989946	−0.494973	+	0.868908i	$0.664822\pi$
$380$	0	0
$381$	1.02848e6	0.362981
$382$	−2.35714e6	−0.826468
$383$	3.51362e6	1.22393	0.611967	−	0.790883i	$-0.290378\pi$
$383$	3.51362e6	1.22393	0.611967	+	0.790883i	$0.290378\pi$
$384$	−2.12352e6	−0.734900
$385$	0	0
$386$	2.34323e6	0.800472
$387$	−2.74789e6	−0.932656
$388$	−250808.	−0.0845789
$389$	834366.	0.279565	0.139782	−	0.990182i	$-0.455360\pi$
$389$	834366.	0.279565	0.139782	+	0.990182i	$0.455360\pi$
$390$	0	0
$391$	−839256.	−0.277621
$392$	−3.63031e6	−1.19324
$393$	−1.28286e6	−0.418984
$394$	−1.23127e6	−0.399589
$395$	0	0
$396$	−257400.	−0.0824842
$397$	5.08080e6	1.61792	0.808958	−	0.587866i	$-0.200032\pi$
$397$	5.08080e6	1.61792	0.808958	+	0.587866i	$0.200032\pi$
$398$	−2.43096e6	−0.769255
$399$	478240.	0.150388
$400$	0	0
$401$	649566.	0.201726	0.100863	−	0.994900i	$-0.467840\pi$
$401$	649566.	0.201726	0.100863	+	0.994900i	$0.467840\pi$
$402$	1.39584e6	0.430795
$403$	−61912.0	−0.0189894
$404$	−341880.	−0.104213
$405$	0	0
$406$	−8.21318e6	−2.47284
$407$	3.85740e6	1.15427
$408$	485520.	0.144397
$409$	3.33465e6	0.985693	0.492846	−	0.870116i	$-0.335956\pi$
$409$	3.33465e6	0.985693	0.492846	+	0.870116i	$0.335956\pi$
$410$	0	0
$411$	233340.	0.0681373
$412$	727168.	0.211053
$413$	4.31357e6	1.24440
$414$	2.49163e6	0.714468
$415$	0	0
$416$	−204480.	−0.0579319
$417$	173380.	0.0488269
$418$	−658800.	−0.184422
$419$	1.72532e6	0.480103	0.240051	−	0.970760i	$-0.422836\pi$
$419$	1.72532e6	0.480103	0.240051	+	0.970760i	$0.422836\pi$
$420$	0	0
$421$	924722.	0.254276	0.127138	−	0.991885i	$-0.459421\pi$
$421$	924722.	0.254276	0.127138	+	0.991885i	$0.459421\pi$
$422$	−1.61797e6	−0.442272
$423$	−2.84856e6	−0.774060
$424$	192528.	0.0520091
$425$	0	0
$426$	1.90224e6	0.507857
$427$	7.01288e6	1.86134
$428$	−330552.	−0.0872229
$429$	−639000.	−0.167632
$430$	0	0
$431$	4.83139e6	1.25279	0.626396	−	0.779505i	$-0.284529\pi$
$431$	4.83139e6	1.25279	0.626396	+	0.779505i	$0.284529\pi$
$432$	−4.38496e6	−1.13046
$433$	−5.39952e6	−1.38400	−0.691999	−	0.721898i	$-0.743270\pi$
$433$	−5.39952e6	−1.38400	−0.691999	+	0.721898i	$0.743270\pi$
$434$	−512736.	−0.130668
$435$	0	0
$436$	982304.	0.247474
$437$	708576.	0.177494
$438$	−837960.	−0.208707
$439$	3.78320e6	0.936910	0.468455	−	0.883487i	$-0.344811\pi$
$439$	3.78320e6	0.936910	0.468455	+	0.883487i	$0.344811\pi$
$440$	0	0
$441$	−3.09009e6	−0.756614
$442$	246228.	0.0599490
$443$	4.20235e6	1.01738	0.508690	−	0.860950i	$-0.330130\pi$
$443$	4.20235e6	1.01738	0.508690	+	0.860950i	$0.330130\pi$
$444$	−342880.	−0.0825439
$445$	0	0
$446$	2.73394e6	0.650805
$447$	−519900.	−0.123070
$448$	5.43155e6	1.27858
$449$	−3.72223e6	−0.871338	−0.435669	−	0.900107i	$-0.643488\pi$
$449$	−3.72223e6	−0.871338	−0.435669	+	0.900107i	$0.643488\pi$
$450$	0	0
$451$	7.36830e6	1.70579
$452$	748056.	0.172222
$453$	−1.31504e6	−0.301088
$454$	−4.75420e6	−1.08252
$455$	0	0
$456$	−409920.	−0.0923181
$457$	−6.53743e6	−1.46426	−0.732128	−	0.681167i	$-0.761473\pi$
$457$	−6.53743e6	−1.46426	−0.732128	+	0.681167i	$0.761473\pi$
$458$	−1.07876e6	−0.240305
$459$	1.11554e6	0.247146
$460$	0	0
$461$	−2.83978e6	−0.622347	−0.311174	−	0.950353i	$-0.600722\pi$
$461$	−2.83978e6	−0.622347	−0.311174	+	0.950353i	$0.600722\pi$
$462$	−5.29200e6	−1.15349
$463$	4.22901e6	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$463$	4.22901e6	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$464$	7.93382e6	1.71075
$465$	0	0
$466$	−1.47092e6	−0.313780
$467$	4.02215e6	0.853426	0.426713	−	0.904387i	$-0.359671\pi$
$467$	4.02215e6	0.853426	0.426713	+	0.904387i	$0.359671\pi$
$468$	−81224.0	−0.0171423
$469$	−4.55974e6	−0.957212
$470$	0	0
$471$	−5.98882e6	−1.24391
$472$	−3.69734e6	−0.763897
$473$	8.64720e6	1.77714
$474$	3.06528e6	0.626649
$475$	0	0
$476$	226576.	0.0458349
$477$	163878.	0.0329780
$478$	−2.00722e6	−0.401813
$479$	−3.68606e6	−0.734047	−0.367024	−	0.930212i	$-0.619623\pi$
$479$	−3.68606e6	−0.734047	−0.367024	+	0.930212i	$0.619623\pi$
$480$	0	0
$481$	1.21722e6	0.239888
$482$	−5.29753e6	−1.03862
$483$	5.69184e6	1.11016
$484$	165796.	0.0321707
$485$	0	0
$486$	−5.39682e6	−1.03645
$487$	2.22808e6	0.425705	0.212852	−	0.977084i	$-0.431725\pi$
$487$	2.22808e6	0.425705	0.212852	+	0.977084i	$0.431725\pi$
$488$	−6.01104e6	−1.14262
$489$	−3.21946e6	−0.608851
$490$	0	0
$491$	2.01546e6	0.377286	0.188643	−	0.982046i	$-0.439591\pi$
$491$	2.01546e6	0.377286	0.188643	+	0.982046i	$0.439591\pi$
$492$	−654960.	−0.121984
$493$	−2.01838e6	−0.374011
$494$	−207888.	−0.0383276
$495$	0	0
$496$	495296.	0.0903984
$497$	−6.21398e6	−1.12844
$498$	−3.86064e6	−0.697567
$499$	1.58273e6	0.284548	0.142274	−	0.989827i	$-0.454559\pi$
$499$	1.58273e6	0.284548	0.142274	+	0.989827i	$0.454559\pi$
$500$	0	0
$501$	3.39516e6	0.604319
$502$	−881568.	−0.156134
$503$	−1.64357e6	−0.289646	−0.144823	−	0.989458i	$-0.546261\pi$
$503$	−1.64357e6	−0.289646	−0.144823	+	0.989458i	$0.546261\pi$
$504$	4.70870e6	0.825705
$505$	0	0
$506$	−7.84080e6	−1.36139
$507$	3.51129e6	0.606662
$508$	−411392.	−0.0707288
$509$	5.24456e6	0.897252	0.448626	−	0.893720i	$-0.351914\pi$
$509$	5.24456e6	0.897252	0.448626	+	0.893720i	$0.351914\pi$
$510$	0	0
$511$	2.73734e6	0.463741
$512$	−4.47130e6	−0.753804
$513$	−941840.	−0.158010
$514$	4.42249e6	0.738345
$515$	0	0
$516$	−768640.	−0.127086
$517$	8.96400e6	1.47494
$518$	1.00807e7	1.65069
$519$	2.52432e6	0.411364
$520$	0	0
$521$	4.17805e6	0.674340	0.337170	−	0.941444i	$-0.390530\pi$
$521$	4.17805e6	0.674340	0.337170	+	0.941444i	$0.390530\pi$
$522$	5.99227e6	0.962532
$523$	4.38518e6	0.701024	0.350512	−	0.936558i	$-0.386008\pi$
$523$	4.38518e6	0.701024	0.350512	+	0.936558i	$0.386008\pi$
$524$	513144.	0.0816415
$525$	0	0
$526$	6.49282e6	1.02322
$527$	−126004.	−0.0197632
$528$	5.11200e6	0.798006
$529$	1.99687e6	0.310250
$530$	0	0
$531$	−3.14714e6	−0.484373
$532$	−191296.	−0.0293040
$533$	2.32511e6	0.354507
$534$	−4.23900e6	−0.643295
$535$	0	0
$536$	3.90835e6	0.587600
$537$	5.55108e6	0.830695
$538$	9.11635e6	1.35789
$539$	9.72405e6	1.44170
$540$	0	0
$541$	−4.33056e6	−0.636138	−0.318069	−	0.948068i	$-0.603034\pi$
$541$	−4.33056e6	−0.636138	−0.318069	+	0.948068i	$0.603034\pi$
$542$	2.37374e6	0.347085
$543$	−15080.0	−0.00219483
$544$	−416160.	−0.0602925
$545$	0	0
$546$	−1.66992e6	−0.239725
$547$	2.70020e6	0.385858	0.192929	−	0.981213i	$-0.438201\pi$
$547$	2.70020e6	0.385858	0.192929	+	0.981213i	$0.438201\pi$
$548$	−93336.0	−0.0132769
$549$	−5.11654e6	−0.724512
$550$	0	0
$551$	1.70410e6	0.239120
$552$	−4.87872e6	−0.681488
$553$	−1.00132e7	−1.39239
$554$	6.91066e6	0.956632
$555$	0	0
$556$	−69352.0	−0.00951419
$557$	−9.26997e6	−1.26602	−0.633010	−	0.774144i	$-0.718181\pi$
$557$	−9.26997e6	−1.26602	−0.633010	+	0.774144i	$0.718181\pi$
$558$	374088.	0.0508614
$559$	2.72867e6	0.369336
$560$	0	0
$561$	−1.30050e6	−0.174463
$562$	8.23568e6	1.09991
$563$	−8.11546e6	−1.07905	−0.539525	−	0.841969i	$-0.681396\pi$
$563$	−8.11546e6	−1.07905	−0.539525	+	0.841969i	$0.681396\pi$
$564$	−796800.	−0.105475
$565$	0	0
$566$	−1.37852e6	−0.180873
$567$	−754796.	−0.0985989
$568$	5.32627e6	0.692712
$569$	−1.14498e7	−1.48257	−0.741286	−	0.671190i	$-0.765784\pi$
$569$	−1.14498e7	−1.48257	−0.741286	+	0.671190i	$0.765784\pi$
$570$	0	0
$571$	−429442.	−0.0551206	−0.0275603	−	0.999620i	$-0.508774\pi$
$571$	−429442.	−0.0551206	−0.0275603	+	0.999620i	$0.508774\pi$
$572$	255600.	0.0326641
$573$	3.92856e6	0.499858
$574$	1.92558e7	2.43940
$575$	0	0
$576$	−3.96282e6	−0.497677
$577$	4.38411e6	0.548204	0.274102	−	0.961701i	$-0.411619\pi$
$577$	4.38411e6	0.548204	0.274102	+	0.961701i	$0.411619\pi$
$578$	501126.	0.0623918
$579$	−3.90538e6	−0.484135
$580$	0	0
$581$	1.26114e7	1.54997
$582$	3.76212e6	0.460389
$583$	−515700.	−0.0628385
$584$	−2.34629e6	−0.284675
$585$	0	0
$586$	−4.56916e6	−0.549657
$587$	−1.89188e6	−0.226621	−0.113310	−	0.993560i	$-0.536145\pi$
$587$	−1.89188e6	−0.226621	−0.113310	+	0.993560i	$0.536145\pi$
$588$	−864360.	−0.103098
$589$	106384.	0.0126354
$590$	0	0
$591$	2.05212e6	0.241676
$592$	−9.73779e6	−1.14197
$593$	1.32588e7	1.54834	0.774171	−	0.632977i	$-0.218167\pi$
$593$	1.32588e7	1.54834	0.774171	+	0.632977i	$0.218167\pi$
$594$	1.04220e7	1.21195
$595$	0	0
$596$	207960.	0.0239808
$597$	4.05160e6	0.465255
$598$	−2.47421e6	−0.282933
$599$	9.69070e6	1.10354	0.551770	−	0.833996i	$-0.313953\pi$
$599$	9.69070e6	1.10354	0.551770	+	0.833996i	$0.313953\pi$
$600$	0	0
$601$	−3.95875e6	−0.447066	−0.223533	−	0.974696i	$-0.571759\pi$
$601$	−3.95875e6	−0.447066	−0.223533	+	0.974696i	$0.571759\pi$
$602$	2.25980e7	2.54144
$603$	3.32675e6	0.372587
$604$	526016.	0.0586687
$605$	0	0
$606$	5.12820e6	0.567262
$607$	3.92350e6	0.432217	0.216109	−	0.976369i	$-0.430663\pi$
$607$	3.92350e6	0.432217	0.216109	+	0.976369i	$0.430663\pi$
$608$	351360.	0.0385472
$609$	1.36886e7	1.49561
$610$	0	0
$611$	2.82864e6	0.306531
$612$	−165308.	−0.0178408
$613$	−1.50137e7	−1.61375	−0.806875	−	0.590722i	$-0.798843\pi$
$613$	−1.50137e7	−1.61375	−0.806875	+	0.590722i	$0.798843\pi$
$614$	−3.59609e6	−0.384954
$615$	0	0
$616$	−1.48176e7	−1.57335
$617$	−5.90027e6	−0.623964	−0.311982	−	0.950088i	$-0.600993\pi$
$617$	−5.90027e6	−0.623964	−0.311982	+	0.950088i	$0.600993\pi$
$618$	−1.09075e7	−1.14883
$619$	1.52636e7	1.60114	0.800571	−	0.599238i	$-0.204530\pi$
$619$	1.52636e7	1.60114	0.800571	+	0.599238i	$0.204530\pi$
$620$	0	0
$621$	−1.12094e7	−1.16642
$622$	5.08529e6	0.527035
$623$	1.38474e7	1.42938
$624$	1.61312e6	0.165846
$625$	0	0
$626$	−1.20005e7	−1.22395
$627$	1.09800e6	0.111541
$628$	2.39553e6	0.242383
$629$	2.47731e6	0.249663
$630$	0	0
$631$	1.42765e7	1.42740	0.713702	−	0.700449i	$-0.247017\pi$
$631$	1.42765e7	1.42740	0.713702	+	0.700449i	$0.247017\pi$
$632$	8.58278e6	0.854743
$633$	2.69662e6	0.267492
$634$	−1.16676e6	−0.115281
$635$	0	0
$636$	45840.0	0.00449368
$637$	3.06848e6	0.299623
$638$	−1.88568e7	−1.83407
$639$	4.53367e6	0.439236
$640$	0	0
$641$	−3.51957e6	−0.338333	−0.169167	−	0.985587i	$-0.554108\pi$
$641$	−3.51957e6	−0.338333	−0.169167	+	0.985587i	$0.554108\pi$
$642$	4.95828e6	0.474781
$643$	117478.	0.0112054	0.00560272	−	0.999984i	$-0.498217\pi$
$643$	117478.	0.0112054	0.00560272	+	0.999984i	$0.498217\pi$
$644$	−2.27674e6	−0.216321
$645$	0	0
$646$	−423096.	−0.0398894
$647$	−3.45269e6	−0.324262	−0.162131	−	0.986769i	$-0.551837\pi$
$647$	−3.45269e6	−0.324262	−0.162131	+	0.986769i	$0.551837\pi$
$648$	646968.	0.0605265
$649$	9.90360e6	0.922957
$650$	0	0
$651$	854560.	0.0790297
$652$	1.28778e6	0.118638
$653$	1.40017e7	1.28499	0.642494	−	0.766291i	$-0.277900\pi$
$653$	1.40017e7	1.28499	0.642494	+	0.766291i	$0.277900\pi$
$654$	−1.47346e7	−1.34708
$655$	0	0
$656$	−1.86009e7	−1.68762
$657$	−1.99714e6	−0.180507
$658$	2.34259e7	2.10927
$659$	4.84952e6	0.434996	0.217498	−	0.976061i	$-0.430210\pi$
$659$	4.84952e6	0.434996	0.217498	+	0.976061i	$0.430210\pi$
$660$	0	0
$661$	1.01240e7	0.901258	0.450629	−	0.892711i	$-0.351200\pi$
$661$	1.01240e7	0.901258	0.450629	+	0.892711i	$0.351200\pi$
$662$	−1.09405e7	−0.970269
$663$	−410380.	−0.0362579
$664$	−1.08098e7	−0.951474
$665$	0	0
$666$	−7.35478e6	−0.642516
$667$	2.02815e7	1.76517
$668$	−1.35806e6	−0.117755
$669$	−4.55656e6	−0.393615
$670$	0	0
$671$	1.61010e7	1.38053
$672$	2.82240e6	0.241099
$673$	1.40204e7	1.19322	0.596612	−	0.802530i	$-0.296513\pi$
$673$	1.40204e7	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$674$	1.55562e7	1.31902
$675$	0	0
$676$	−1.40452e6	−0.118212
$677$	−7.67190e6	−0.643326	−0.321663	−	0.946854i	$-0.604242\pi$
$677$	−7.67190e6	−0.643326	−0.321663	+	0.946854i	$0.604242\pi$
$678$	−1.12208e7	−0.937457
$679$	−1.22896e7	−1.02297
$680$	0	0
$681$	7.92366e6	0.654724
$682$	−1.17720e6	−0.0969146
$683$	5.36631e6	0.440174	0.220087	−	0.975480i	$-0.429366\pi$
$683$	5.36631e6	0.440174	0.220087	+	0.975480i	$0.429366\pi$
$684$	139568.	0.0114063
$685$	0	0
$686$	5.64715e6	0.458162
$687$	1.79794e6	0.145339
$688$	−2.18294e7	−1.75821
$689$	−162732.	−0.0130595
$690$	0	0
$691$	−1.31376e7	−1.04670	−0.523348	−	0.852119i	$-0.675317\pi$
$691$	−1.31376e7	−1.04670	−0.523348	+	0.852119i	$0.675317\pi$
$692$	−1.00973e6	−0.0801566
$693$	−1.26126e7	−0.997635
$694$	−1.60451e7	−1.26457
$695$	0	0
$696$	−1.17331e7	−0.918101
$697$	4.73209e6	0.368953
$698$	3.02473e6	0.234989
$699$	2.45154e6	0.189778
$700$	0	0
$701$	−1.78160e6	−0.136935	−0.0684675	−	0.997653i	$-0.521811\pi$
$701$	−1.78160e6	−0.136935	−0.0684675	+	0.997653i	$0.521811\pi$
$702$	3.28872e6	0.251874
$703$	−2.09157e6	−0.159619
$704$	1.24704e7	0.948307
$705$	0	0
$706$	−6.04746e6	−0.456627
$707$	−1.67521e7	−1.26044
$708$	−880320.	−0.0660020
$709$	−1.16819e7	−0.872769	−0.436385	−	0.899760i	$-0.643741\pi$
$709$	−1.16819e7	−0.872769	−0.436385	+	0.899760i	$0.643741\pi$
$710$	0	0
$711$	7.30558e6	0.541977
$712$	−1.18692e7	−0.877448
$713$	1.26614e6	0.0932737
$714$	−3.39864e6	−0.249494
$715$	0	0
$716$	−2.22043e6	−0.161866
$717$	3.34536e6	0.243022
$718$	−2.84710e7	−2.06106
$719$	−1.62177e7	−1.16995	−0.584976	−	0.811051i	$-0.698896\pi$
$719$	−1.62177e7	−1.16995	−0.584976	+	0.811051i	$0.698896\pi$
$720$	0	0
$721$	3.56312e7	2.55266
$722$	−1.44994e7	−1.03516
$723$	8.82922e6	0.628169
$724$	6032.00	0.000427676 0
$725$	0	0
$726$	−2.48694e6	−0.175115
$727$	−1.27353e7	−0.893663	−0.446832	−	0.894618i	$-0.647448\pi$
$727$	−1.27353e7	−0.893663	−0.446832	+	0.894618i	$0.647448\pi$
$728$	−4.67578e6	−0.326983
$729$	9.93049e6	0.692073
$730$	0	0
$731$	5.55342e6	0.384386
$732$	−1.43120e6	−0.0987240
$733$	−5.93825e6	−0.408224	−0.204112	−	0.978948i	$-0.565431\pi$
$733$	−5.93825e6	−0.408224	−0.204112	+	0.978948i	$0.565431\pi$
$734$	−930288.	−0.0637349
$735$	0	0
$736$	4.18176e6	0.284554
$737$	−1.04688e7	−0.709951
$738$	−1.40489e7	−0.949514
$739$	−2.46285e7	−1.65892	−0.829462	−	0.558562i	$-0.811353\pi$
$739$	−2.46285e7	−1.65892	−0.829462	+	0.558562i	$0.811353\pi$
$740$	0	0
$741$	346480.	0.0231810
$742$	−1.34770e6	−0.0898633
$743$	440628.	0.0292820	0.0146410	−	0.999893i	$-0.495339\pi$
$743$	440628.	0.0292820	0.0146410	+	0.999893i	$0.495339\pi$
$744$	−732480.	−0.0485136
$745$	0	0
$746$	−2.33886e7	−1.53871
$747$	−9.20119e6	−0.603313
$748$	520200.	0.0339951
$749$	−1.61970e7	−1.05495
$750$	0	0
$751$	−5.22736e6	−0.338207	−0.169103	−	0.985598i	$-0.554087\pi$
$751$	−5.22736e6	−0.338207	−0.169103	+	0.985598i	$0.554087\pi$
$752$	−2.26291e7	−1.45923
$753$	1.46928e6	0.0944316
$754$	−5.95037e6	−0.381167
$755$	0	0
$756$	3.02624e6	0.192575
$757$	1.60789e7	1.01980	0.509901	−	0.860233i	$-0.329682\pi$
$757$	1.60789e7	1.01980	0.509901	+	0.860233i	$0.329682\pi$
$758$	−1.66097e7	−1.05000
$759$	1.30680e7	0.823388
$760$	0	0
$761$	−1.98848e7	−1.24468	−0.622341	−	0.782746i	$-0.713818\pi$
$761$	−1.98848e7	−1.24468	−0.622341	+	0.782746i	$0.713818\pi$
$762$	6.17088e6	0.384999
$763$	4.81329e7	2.99316
$764$	−1.57142e6	−0.0974002
$765$	0	0
$766$	2.10817e7	1.29818
$767$	3.12514e6	0.191814
$768$	−3.87328e6	−0.236960
$769$	−2.73915e7	−1.67032	−0.835160	−	0.550007i	$-0.814625\pi$
$769$	−2.73915e7	−1.67032	−0.835160	+	0.550007i	$0.814625\pi$
$770$	0	0
$771$	−7.37082e6	−0.446560
$772$	1.56215e6	0.0943366
$773$	−2.30626e7	−1.38822	−0.694112	−	0.719867i	$-0.744203\pi$
$773$	−2.30626e7	−1.38822	−0.694112	+	0.719867i	$0.744203\pi$
$774$	−1.64873e7	−0.989231
$775$	0	0
$776$	1.05339e7	0.627966
$777$	−1.68011e7	−0.998357
$778$	5.00620e6	0.296523
$779$	−3.99526e6	−0.235885
$780$	0	0
$781$	−1.42668e7	−0.836949
$782$	−5.03554e6	−0.294462
$783$	−2.69582e7	−1.57140
$784$	−2.45478e7	−1.42634
$785$	0	0
$786$	−7.69716e6	−0.444400
$787$	642682.	0.0369879	0.0184939	−	0.999829i	$-0.494113\pi$
$787$	642682.	0.0369879	0.0184939	+	0.999829i	$0.494113\pi$
$788$	−820848.	−0.0470920
$789$	−1.08214e7	−0.618856
$790$	0	0
$791$	3.66547e7	2.08300
$792$	1.08108e7	0.612414
$793$	5.08076e6	0.286910
$794$	3.04848e7	1.71606
$795$	0	0
$796$	−1.62064e6	−0.0906575
$797$	−9.10785e6	−0.507890	−0.253945	−	0.967219i	$-0.581728\pi$
$797$	−9.10785e6	−0.507890	−0.253945	+	0.967219i	$0.581728\pi$
$798$	2.86944e6	0.159511
$799$	5.75688e6	0.319022
$800$	0	0
$801$	−1.01030e7	−0.556374
$802$	3.89740e6	0.213963
$803$	6.28470e6	0.343950
$804$	930560.	0.0507697
$805$	0	0
$806$	−371472.	−0.0201413
$807$	−1.51939e7	−0.821270
$808$	1.43590e7	0.773739
$809$	−1.37190e7	−0.736972	−0.368486	−	0.929633i	$-0.620124\pi$
$809$	−1.37190e7	−0.736972	−0.368486	+	0.929633i	$0.620124\pi$
$810$	0	0
$811$	2.74959e7	1.46796	0.733982	−	0.679168i	$-0.237659\pi$
$811$	2.74959e7	1.46796	0.733982	+	0.679168i	$0.237659\pi$
$812$	−5.47546e6	−0.291427
$813$	−3.95624e6	−0.209921
$814$	2.31444e7	1.22429
$815$	0	0
$816$	3.28304e6	0.172604
$817$	−4.68870e6	−0.245752
$818$	2.00079e7	1.04548
$819$	−3.97998e6	−0.207334
$820$	0	0
$821$	2.85504e7	1.47827	0.739136	−	0.673556i	$-0.235234\pi$
$821$	2.85504e7	1.47827	0.739136	+	0.673556i	$0.235234\pi$
$822$	1.40004e6	0.0722705
$823$	58840.0	0.00302812	0.00151406	−	0.999999i	$-0.499518\pi$
$823$	58840.0	0.00302812	0.00151406	+	0.999999i	$0.499518\pi$
$824$	−3.05411e7	−1.56699
$825$	0	0
$826$	2.58814e7	1.31989
$827$	3.14197e7	1.59749	0.798745	−	0.601670i	$-0.205498\pi$
$827$	3.14197e7	1.59749	0.798745	+	0.601670i	$0.205498\pi$
$828$	1.66109e6	0.0842009
$829$	9.55809e6	0.483042	0.241521	−	0.970396i	$-0.422354\pi$
$829$	9.55809e6	0.483042	0.241521	+	0.970396i	$0.422354\pi$
$830$	0	0
$831$	−1.15178e7	−0.578583
$832$	3.93510e6	0.197082
$833$	6.24500e6	0.311832
$834$	1.04028e6	0.0517887
$835$	0	0
$836$	−439200.	−0.0217343
$837$	−1.68296e6	−0.0830348
$838$	1.03519e7	0.509226
$839$	−82980.0	−0.00406976	−0.00203488	−	0.999998i	$-0.500648\pi$
$839$	−82980.0	−0.00406976	−0.00203488	+	0.999998i	$0.500648\pi$
$840$	0	0
$841$	2.82651e7	1.37804
$842$	5.54833e6	0.269701
$843$	−1.37261e7	−0.665242
$844$	−1.07865e6	−0.0521223
$845$	0	0
$846$	−1.70914e7	−0.821014
$847$	8.12400e6	0.389100
$848$	1.30186e6	0.0621689
$849$	2.29754e6	0.109394
$850$	0	0
$851$	−2.48931e7	−1.17830
$852$	1.26816e6	0.0598515
$853$	−2.82115e7	−1.32756	−0.663779	−	0.747929i	$-0.731049\pi$
$853$	−2.82115e7	−1.32756	−0.663779	+	0.747929i	$0.731049\pi$
$854$	4.20773e7	1.97425
$855$	0	0
$856$	1.38832e7	0.647597
$857$	−3.64611e6	−0.169581	−0.0847906	−	0.996399i	$-0.527022\pi$
$857$	−3.64611e6	−0.169581	−0.0847906	+	0.996399i	$0.527022\pi$
$858$	−3.83400e6	−0.177801
$859$	−1.98241e7	−0.916664	−0.458332	−	0.888781i	$-0.651553\pi$
$859$	−1.98241e7	−0.916664	−0.458332	+	0.888781i	$0.651553\pi$
$860$	0	0
$861$	−3.20930e7	−1.47538
$862$	2.89884e7	1.32879
$863$	3.96712e7	1.81321	0.906605	−	0.421980i	$-0.138665\pi$
$863$	3.96712e7	1.81321	0.906605	+	0.421980i	$0.138665\pi$
$864$	−5.55840e6	−0.253318
$865$	0	0
$866$	−3.23971e7	−1.46795
$867$	−835210.	−0.0377353
$868$	−341824.	−0.0153994
$869$	−2.29896e7	−1.03272
$870$	0	0
$871$	−3.30349e6	−0.147546
$872$	−4.12568e7	−1.83740
$873$	8.96639e6	0.398182
$874$	4.25146e6	0.188260
$875$	0	0
$876$	−558640.	−0.0245964
$877$	−3.83641e7	−1.68432	−0.842162	−	0.539225i	$-0.818717\pi$
$877$	−3.83641e7	−1.68432	−0.842162	+	0.539225i	$0.818717\pi$
$878$	2.26992e7	0.993743
$879$	7.61526e6	0.332439
$880$	0	0
$881$	−2.64738e7	−1.14915	−0.574575	−	0.818452i	$-0.694833\pi$
$881$	−2.64738e7	−1.14915	−0.574575	+	0.818452i	$0.694833\pi$
$882$	−1.85405e7	−0.802510
$883$	−2.76564e7	−1.19370	−0.596848	−	0.802354i	$-0.703581\pi$
$883$	−2.76564e7	−1.19370	−0.596848	+	0.802354i	$0.703581\pi$
$884$	164152.	0.00706505
$885$	0	0
$886$	2.52141e7	1.07909
$887$	−5.91883e6	−0.252596	−0.126298	−	0.991992i	$-0.540310\pi$
$887$	−5.91883e6	−0.252596	−0.126298	+	0.991992i	$0.540310\pi$
$888$	1.44010e7	0.612857
$889$	−2.01582e7	−0.855456
$890$	0	0
$891$	−1.73295e6	−0.0731294
$892$	1.82262e6	0.0766982
$893$	−4.86048e6	−0.203963
$894$	−3.11940e6	−0.130535
$895$	0	0
$896$	4.16210e7	1.73198
$897$	4.12368e6	0.171121
$898$	−2.23334e7	−0.924194
$899$	3.04502e6	0.125658
$900$	0	0
$901$	−331194.	−0.0135916
$902$	4.42098e7	1.80927
$903$	−3.76634e7	−1.53709
$904$	−3.14184e7	−1.27868
$905$	0	0
$906$	−7.89024e6	−0.319352
$907$	4.04616e7	1.63315	0.816573	−	0.577243i	$-0.195871\pi$
$907$	4.04616e7	1.63315	0.816573	+	0.577243i	$0.195871\pi$
$908$	−3.16946e6	−0.127577
$909$	1.22222e7	0.490614
$910$	0	0
$911$	−1.02375e7	−0.408695	−0.204347	−	0.978898i	$-0.565507\pi$
$911$	−1.02375e7	−0.408695	−0.204347	+	0.978898i	$0.565507\pi$
$912$	−2.77184e6	−0.110352
$913$	2.89548e7	1.14959
$914$	−3.92246e7	−1.55308
$915$	0	0
$916$	−719176.	−0.0283202
$917$	2.51441e7	0.987443
$918$	6.69324e6	0.262138
$919$	−1.95081e7	−0.761948	−0.380974	−	0.924586i	$-0.624411\pi$
$919$	−1.95081e7	−0.761948	−0.380974	+	0.924586i	$0.624411\pi$
$920$	0	0
$921$	5.99348e6	0.232825
$922$	−1.70387e7	−0.660099
$923$	−4.50197e6	−0.173939
$924$	−3.52800e6	−0.135940
$925$	0	0
$926$	2.53740e7	0.972439
$927$	−2.59963e7	−0.993600
$928$	1.00570e7	0.383351
$929$	−6.77925e6	−0.257717	−0.128858	−	0.991663i	$-0.541131\pi$
$929$	−6.77925e6	−0.257717	−0.128858	+	0.991663i	$0.541131\pi$
$930$	0	0
$931$	−5.27260e6	−0.199366
$932$	−980616.	−0.0369793
$933$	−8.47548e6	−0.318757
$934$	2.41329e7	0.905195
$935$	0	0
$936$	3.41141e6	0.127275
$937$	−4.20612e7	−1.56507	−0.782533	−	0.622609i	$-0.786073\pi$
$937$	−4.20612e7	−1.56507	−0.782533	+	0.622609i	$0.786073\pi$
$938$	−2.73585e7	−1.01528
$939$	2.00008e7	0.740258
$940$	0	0
$941$	−4.41336e7	−1.62478	−0.812391	−	0.583113i	$-0.801834\pi$
$941$	−4.41336e7	−1.62478	−0.812391	+	0.583113i	$0.801834\pi$
$942$	−3.59329e7	−1.31937
$943$	−4.75501e7	−1.74129
$944$	−2.50011e7	−0.913122
$945$	0	0
$946$	5.18832e7	1.88495
$947$	−1.21508e7	−0.440282	−0.220141	−	0.975468i	$-0.570652\pi$
$947$	−1.21508e7	−0.440282	−0.220141	+	0.975468i	$0.570652\pi$
$948$	2.04352e6	0.0738513
$949$	1.98317e6	0.0714817
$950$	0	0
$951$	1.94460e6	0.0697235
$952$	−9.51619e6	−0.340307
$953$	−2.39119e7	−0.852869	−0.426435	−	0.904518i	$-0.640231\pi$
$953$	−2.39119e7	−0.852869	−0.426435	+	0.904518i	$0.640231\pi$
$954$	983268.	0.0349785
$955$	0	0
$956$	−1.33814e6	−0.0473542
$957$	3.14280e7	1.10927
$958$	−2.21164e7	−0.778575
$959$	−4.57346e6	−0.160583
$960$	0	0
$961$	−2.84391e7	−0.993360
$962$	7.30334e6	0.254439
$963$	1.18172e7	0.410629
$964$	−3.53169e6	−0.122402
$965$	0	0
$966$	3.41510e7	1.17750
$967$	4.71653e7	1.62202	0.811011	−	0.585031i	$-0.198918\pi$
$967$	4.71653e7	1.62202	0.811011	+	0.585031i	$0.198918\pi$
$968$	−6.96343e6	−0.238855
$969$	705160.	0.0241256
$970$	0	0
$971$	3.76796e7	1.28250	0.641251	−	0.767331i	$-0.278416\pi$
$971$	3.76796e7	1.28250	0.641251	+	0.767331i	$0.278416\pi$
$972$	−3.59788e6	−0.122146
$973$	−3.39825e6	−0.115073
$974$	1.33685e7	0.451528
$975$	0	0
$976$	−4.06461e7	−1.36582
$977$	4.65997e7	1.56188	0.780938	−	0.624608i	$-0.214741\pi$
$977$	4.65997e7	1.56188	0.780938	+	0.624608i	$0.214741\pi$
$978$	−1.93168e7	−0.645784
$979$	3.17925e7	1.06015
$980$	0	0
$981$	−3.51174e7	−1.16506
$982$	1.20928e7	0.400172
$983$	−3.55297e7	−1.17276	−0.586378	−	0.810038i	$-0.699447\pi$
$983$	−3.55297e7	−1.17276	−0.586378	+	0.810038i	$0.699447\pi$
$984$	2.75083e7	0.905683
$985$	0	0
$986$	−1.21103e7	−0.396699
$987$	−3.90432e7	−1.27571
$988$	−138592.	−0.00451696
$989$	−5.58033e7	−1.81413
$990$	0	0
$991$	3.01289e7	0.974538	0.487269	−	0.873252i	$-0.337993\pi$
$991$	3.01289e7	0.974538	0.487269	+	0.873252i	$0.337993\pi$
$992$	627840.	0.0202568
$993$	1.82342e7	0.586830
$994$	−3.72839e7	−1.19689
$995$	0	0
$996$	−2.57376e6	−0.0822091
$997$	2.76127e7	0.879774	0.439887	−	0.898053i	$-0.355018\pi$
$997$	2.76127e7	0.879774	0.439887	+	0.898053i	$0.355018\pi$
$998$	9.49638e6	0.301809
$999$	3.30879e7	1.04895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.b.1.1		1
5.4	even	2		17.6.a.a.1.1	✓	1
15.14	odd	2		153.6.a.b.1.1		1
20.19	odd	2		272.6.a.a.1.1		1
35.34	odd	2		833.6.a.a.1.1		1
40.19	odd	2		1088.6.a.g.1.1		1
40.29	even	2		1088.6.a.d.1.1		1
85.84	even	2		289.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.a.a.1.1	✓	1	5.4	even	2
153.6.a.b.1.1		1	15.14	odd	2
272.6.a.a.1.1		1	20.19	odd	2
289.6.a.a.1.1		1	85.84	even	2
425.6.a.b.1.1		1	1.1	even	1	trivial
833.6.a.a.1.1		1	35.34	odd	2
1088.6.a.d.1.1		1	40.29	even	2
1088.6.a.g.1.1		1	40.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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