LMFDB - Embedded newform 342.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,6,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	342.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:	$54.8512663760$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} +16.0000 q^{4} +54.0000 q^{5} +104.000 q^{7} +64.0000 q^{8} +216.000 q^{10} +330.000 q^{11} -46.0000 q^{13} +416.000 q^{14} +256.000 q^{16} +618.000 q^{17} +361.000 q^{19} +864.000 q^{20} +1320.00 q^{22} +402.000 q^{23} -209.000 q^{25} -184.000 q^{26} +1664.00 q^{28} +2628.00 q^{29} -2368.00 q^{31} +1024.00 q^{32} +2472.00 q^{34} +5616.00 q^{35} -12130.0 q^{37} +1444.00 q^{38} +3456.00 q^{40} +18864.0 q^{41} -10408.0 q^{43} +5280.00 q^{44} +1608.00 q^{46} +4770.00 q^{47} -5991.00 q^{49} -836.000 q^{50} -736.000 q^{52} +19452.0 q^{53} +17820.0 q^{55} +6656.00 q^{56} +10512.0 q^{58} -30528.0 q^{59} +11138.0 q^{61} -9472.00 q^{62} +4096.00 q^{64} -2484.00 q^{65} +49508.0 q^{67} +9888.00 q^{68} +22464.0 q^{70} -7572.00 q^{71} +2342.00 q^{73} -48520.0 q^{74} +5776.00 q^{76} +34320.0 q^{77} +22424.0 q^{79} +13824.0 q^{80} +75456.0 q^{82} +46734.0 q^{83} +33372.0 q^{85} -41632.0 q^{86} +21120.0 q^{88} +70104.0 q^{89} -4784.00 q^{91} +6432.00 q^{92} +19080.0 q^{94} +19494.0 q^{95} +105710. q^{97} -23964.0 q^{98} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	54.0000	0.965981	0.482991	−	0.875625i	$-0.339550\pi$
$5$	54.0000	0.965981	0.482991	+	0.875625i	$0.339550\pi$
$6$	0	0
$7$	104.000	0.802210	0.401105	−	0.916032i	$-0.368626\pi$
$7$	104.000	0.802210	0.401105	+	0.916032i	$0.368626\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	216.000	0.683052
$11$	330.000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	330.000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	−46.0000	−0.0754917	−0.0377459	−	0.999287i	$-0.512018\pi$
$13$	−46.0000	−0.0754917	−0.0377459	+	0.999287i	$0.512018\pi$
$14$	416.000	0.567248
$15$	0	0
$16$	256.000	0.250000
$17$	618.000	0.518640	0.259320	−	0.965791i	$-0.416502\pi$
$17$	618.000	0.518640	0.259320	+	0.965791i	$0.416502\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	864.000	0.482991
$21$	0	0
$22$	1320.00	0.581456
$23$	402.000	0.158455	0.0792276	−	0.996857i	$-0.474755\pi$
$23$	402.000	0.158455	0.0792276	+	0.996857i	$0.474755\pi$
$24$	0	0
$25$	−209.000	−0.0668800
$26$	−184.000	−0.0533807
$27$	0	0
$28$	1664.00	0.401105
$29$	2628.00	0.580270	0.290135	−	0.956986i	$-0.406300\pi$
$29$	2628.00	0.580270	0.290135	+	0.956986i	$0.406300\pi$
$30$	0	0
$31$	−2368.00	−0.442565	−0.221283	−	0.975210i	$-0.571024\pi$
$31$	−2368.00	−0.442565	−0.221283	+	0.975210i	$0.571024\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	2472.00	0.366734
$35$	5616.00	0.774920
$36$	0	0
$37$	−12130.0	−1.45665	−0.728327	−	0.685230i	$-0.759702\pi$
$37$	−12130.0	−1.45665	−0.728327	+	0.685230i	$0.759702\pi$
$38$	1444.00	0.162221
$39$	0	0
$40$	3456.00	0.341526
$41$	18864.0	1.75257	0.876283	−	0.481798i	$-0.160016\pi$
$41$	18864.0	1.75257	0.876283	+	0.481798i	$0.160016\pi$
$42$	0	0
$43$	−10408.0	−0.858413	−0.429206	−	0.903206i	$-0.641207\pi$
$43$	−10408.0	−0.858413	−0.429206	+	0.903206i	$0.641207\pi$
$44$	5280.00	0.411152
$45$	0	0
$46$	1608.00	0.112045
$47$	4770.00	0.314973	0.157487	−	0.987521i	$-0.449661\pi$
$47$	4770.00	0.314973	0.157487	+	0.987521i	$0.449661\pi$
$48$	0	0
$49$	−5991.00	−0.356459
$50$	−836.000	−0.0472913
$51$	0	0
$52$	−736.000	−0.0377459
$53$	19452.0	0.951206	0.475603	−	0.879660i	$-0.342230\pi$
$53$	19452.0	0.951206	0.475603	+	0.879660i	$0.342230\pi$
$54$	0	0
$55$	17820.0	0.794330
$56$	6656.00	0.283624
$57$	0	0
$58$	10512.0	0.410313
$59$	−30528.0	−1.14174	−0.570871	−	0.821039i	$-0.693395\pi$
$59$	−30528.0	−1.14174	−0.570871	+	0.821039i	$0.693395\pi$
$60$	0	0
$61$	11138.0	0.383250	0.191625	−	0.981468i	$-0.438624\pi$
$61$	11138.0	0.383250	0.191625	+	0.981468i	$0.438624\pi$
$62$	−9472.00	−0.312941
$63$	0	0
$64$	4096.00	0.125000
$65$	−2484.00	−0.0729236
$66$	0	0
$67$	49508.0	1.34737	0.673687	−	0.739016i	$-0.264709\pi$
$67$	49508.0	1.34737	0.673687	+	0.739016i	$0.264709\pi$
$68$	9888.00	0.259320
$69$	0	0
$70$	22464.0	0.547951
$71$	−7572.00	−0.178264	−0.0891322	−	0.996020i	$-0.528409\pi$
$71$	−7572.00	−0.178264	−0.0891322	+	0.996020i	$0.528409\pi$
$72$	0	0
$73$	2342.00	0.0514375	0.0257187	−	0.999669i	$-0.491813\pi$
$73$	2342.00	0.0514375	0.0257187	+	0.999669i	$0.491813\pi$
$74$	−48520.0	−1.03001
$75$	0	0
$76$	5776.00	0.114708
$77$	34320.0	0.659660
$78$	0	0
$79$	22424.0	0.404246	0.202123	−	0.979360i	$-0.435216\pi$
$79$	22424.0	0.404246	0.202123	+	0.979360i	$0.435216\pi$
$80$	13824.0	0.241495
$81$	0	0
$82$	75456.0	1.23925
$83$	46734.0	0.744625	0.372313	−	0.928107i	$-0.378565\pi$
$83$	46734.0	0.744625	0.372313	+	0.928107i	$0.378565\pi$
$84$	0	0
$85$	33372.0	0.500997
$86$	−41632.0	−0.606989
$87$	0	0
$88$	21120.0	0.290728
$89$	70104.0	0.938140	0.469070	−	0.883161i	$-0.344589\pi$
$89$	70104.0	0.938140	0.469070	+	0.883161i	$0.344589\pi$
$90$	0	0
$91$	−4784.00	−0.0605603
$92$	6432.00	0.0792276
$93$	0	0
$94$	19080.0	0.222720
$95$	19494.0	0.221611
$96$	0	0
$97$	105710.	1.14074	0.570370	−	0.821388i	$-0.306800\pi$
$97$	105710.	1.14074	0.570370	+	0.821388i	$0.306800\pi$
$98$	−23964.0	−0.252054
$99$	0	0
$100$	−3344.00	−0.0334400
$101$	−124542.	−1.21482	−0.607411	−	0.794388i	$-0.707792\pi$
$101$	−124542.	−1.21482	−0.607411	+	0.794388i	$0.707792\pi$
$102$	0	0
$103$	6488.00	0.0602584	0.0301292	−	0.999546i	$-0.490408\pi$
$103$	6488.00	0.0602584	0.0301292	+	0.999546i	$0.490408\pi$
$104$	−2944.00	−0.0266904
$105$	0	0
$106$	77808.0	0.672604
$107$	−50040.0	−0.422530	−0.211265	−	0.977429i	$-0.567758\pi$
$107$	−50040.0	−0.422530	−0.211265	+	0.977429i	$0.567758\pi$
$108$	0	0
$109$	88706.0	0.715133	0.357566	−	0.933888i	$-0.383607\pi$
$109$	88706.0	0.715133	0.357566	+	0.933888i	$0.383607\pi$
$110$	71280.0	0.561676
$111$	0	0
$112$	26624.0	0.200553
$113$	156792.	1.15512	0.577561	−	0.816348i	$-0.304005\pi$
$113$	156792.	1.15512	0.577561	+	0.816348i	$0.304005\pi$
$114$	0	0
$115$	21708.0	0.153065
$116$	42048.0	0.290135
$117$	0	0
$118$	−122112.	−0.807334
$119$	64272.0	0.416059
$120$	0	0
$121$	−52151.0	−0.323817
$122$	44552.0	0.270999
$123$	0	0
$124$	−37888.0	−0.221283
$125$	−180036.	−1.03059
$126$	0	0
$127$	−211768.	−1.16507	−0.582534	−	0.812807i	$-0.697939\pi$
$127$	−211768.	−1.16507	−0.582534	+	0.812807i	$0.697939\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−9936.00	−0.0515648
$131$	4410.00	0.0224523	0.0112261	−	0.999937i	$-0.496427\pi$
$131$	4410.00	0.0224523	0.0112261	+	0.999937i	$0.496427\pi$
$132$	0	0
$133$	37544.0	0.184040
$134$	198032.	0.952738
$135$	0	0
$136$	39552.0	0.183367
$137$	−205974.	−0.937586	−0.468793	−	0.883308i	$-0.655311\pi$
$137$	−205974.	−0.937586	−0.468793	+	0.883308i	$0.655311\pi$
$138$	0	0
$139$	315908.	1.38683	0.693416	−	0.720538i	$-0.256105\pi$
$139$	315908.	1.38683	0.693416	+	0.720538i	$0.256105\pi$
$140$	89856.0	0.387460
$141$	0	0
$142$	−30288.0	−0.126052
$143$	−15180.0	−0.0620771
$144$	0	0
$145$	141912.	0.560530
$146$	9368.00	0.0363718
$147$	0	0
$148$	−194080.	−0.728327
$149$	−211158.	−0.779187	−0.389594	−	0.920987i	$-0.627385\pi$
$149$	−211158.	−0.779187	−0.389594	+	0.920987i	$0.627385\pi$
$150$	0	0
$151$	−118120.	−0.421581	−0.210791	−	0.977531i	$-0.567604\pi$
$151$	−118120.	−0.421581	−0.210791	+	0.977531i	$0.567604\pi$
$152$	23104.0	0.0811107
$153$	0	0
$154$	137280.	0.466450
$155$	−127872.	−0.427510
$156$	0	0
$157$	−163990.	−0.530968	−0.265484	−	0.964115i	$-0.585532\pi$
$157$	−163990.	−0.530968	−0.265484	+	0.964115i	$0.585532\pi$
$158$	89696.0	0.285845
$159$	0	0
$160$	55296.0	0.170763
$161$	41808.0	0.127114
$162$	0	0
$163$	26948.0	0.0794433	0.0397217	−	0.999211i	$-0.487353\pi$
$163$	26948.0	0.0794433	0.0397217	+	0.999211i	$0.487353\pi$
$164$	301824.	0.876283
$165$	0	0
$166$	186936.	0.526530
$167$	−5808.00	−0.0161152	−0.00805759	−	0.999968i	$-0.502565\pi$
$167$	−5808.00	−0.0161152	−0.00805759	+	0.999968i	$0.502565\pi$
$168$	0	0
$169$	−369177.	−0.994301
$170$	133488.	0.354258
$171$	0	0
$172$	−166528.	−0.429206
$173$	−37104.0	−0.0942552	−0.0471276	−	0.998889i	$-0.515007\pi$
$173$	−37104.0	−0.0942552	−0.0471276	+	0.998889i	$0.515007\pi$
$174$	0	0
$175$	−21736.0	−0.0536518
$176$	84480.0	0.205576
$177$	0	0
$178$	280416.	0.663365
$179$	−198360.	−0.462723	−0.231362	−	0.972868i	$-0.574318\pi$
$179$	−198360.	−0.462723	−0.231362	+	0.972868i	$0.574318\pi$
$180$	0	0
$181$	−276838.	−0.628101	−0.314050	−	0.949406i	$-0.601686\pi$
$181$	−276838.	−0.628101	−0.314050	+	0.949406i	$0.601686\pi$
$182$	−19136.0	−0.0428226
$183$	0	0
$184$	25728.0	0.0560224
$185$	−655020.	−1.40710
$186$	0	0
$187$	203940.	0.426480
$188$	76320.0	0.157487
$189$	0	0
$190$	77976.0	0.156703
$191$	531222.	1.05364	0.526820	−	0.849977i	$-0.323384\pi$
$191$	531222.	1.05364	0.526820	+	0.849977i	$0.323384\pi$
$192$	0	0
$193$	−245062.	−0.473568	−0.236784	−	0.971562i	$-0.576093\pi$
$193$	−245062.	−0.473568	−0.236784	+	0.971562i	$0.576093\pi$
$194$	422840.	0.806625
$195$	0	0
$196$	−95856.0	−0.178229
$197$	−219906.	−0.403712	−0.201856	−	0.979415i	$-0.564697\pi$
$197$	−219906.	−0.403712	−0.201856	+	0.979415i	$0.564697\pi$
$198$	0	0
$199$	−1.01820e6	−1.82265	−0.911323	−	0.411693i	$-0.864938\pi$
$199$	−1.01820e6	−1.82265	−0.911323	+	0.411693i	$0.864938\pi$
$200$	−13376.0	−0.0236457
$201$	0	0
$202$	−498168.	−0.859008
$203$	273312.	0.465499
$204$	0	0
$205$	1.01866e6	1.69295
$206$	25952.0	0.0426091
$207$	0	0
$208$	−11776.0	−0.0188729
$209$	119130.	0.188649
$210$	0	0
$211$	105020.	0.162392	0.0811962	−	0.996698i	$-0.474126\pi$
$211$	105020.	0.162392	0.0811962	+	0.996698i	$0.474126\pi$
$212$	311232.	0.475603
$213$	0	0
$214$	−200160.	−0.298774
$215$	−562032.	−0.829211
$216$	0	0
$217$	−246272.	−0.355031
$218$	354824.	0.505675
$219$	0	0
$220$	285120.	0.397165
$221$	−28428.0	−0.0391531
$222$	0	0
$223$	988304.	1.33085	0.665424	−	0.746466i	$-0.268251\pi$
$223$	988304.	1.33085	0.665424	+	0.746466i	$0.268251\pi$
$224$	106496.	0.141812
$225$	0	0
$226$	627168.	0.816794
$227$	515940.	0.664561	0.332280	−	0.943181i	$-0.392182\pi$
$227$	515940.	0.664561	0.332280	+	0.943181i	$0.392182\pi$
$228$	0	0
$229$	−277354.	−0.349499	−0.174749	−	0.984613i	$-0.555912\pi$
$229$	−277354.	−0.349499	−0.174749	+	0.984613i	$0.555912\pi$
$230$	86832.0	0.108233
$231$	0	0
$232$	168192.	0.205157
$233$	−1.54151e6	−1.86019	−0.930096	−	0.367317i	$-0.880276\pi$
$233$	−1.54151e6	−1.86019	−0.930096	+	0.367317i	$0.880276\pi$
$234$	0	0
$235$	257580.	0.304258
$236$	−488448.	−0.570871
$237$	0	0
$238$	257088.	0.294198
$239$	314310.	0.355929	0.177965	−	0.984037i	$-0.443049\pi$
$239$	314310.	0.355929	0.177965	+	0.984037i	$0.443049\pi$
$240$	0	0
$241$	24566.0	0.0272453	0.0136227	−	0.999907i	$-0.495664\pi$
$241$	24566.0	0.0272453	0.0136227	+	0.999907i	$0.495664\pi$
$242$	−208604.	−0.228973
$243$	0	0
$244$	178208.	0.191625
$245$	−323514.	−0.344332
$246$	0	0
$247$	−16606.0	−0.0173190
$248$	−151552.	−0.156470
$249$	0	0
$250$	−720144.	−0.728734
$251$	−661290.	−0.662533	−0.331267	−	0.943537i	$-0.607476\pi$
$251$	−661290.	−0.662533	−0.331267	+	0.943537i	$0.607476\pi$
$252$	0	0
$253$	132660.	0.130298
$254$	−847072.	−0.823827
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.05035e6	−0.991974	−0.495987	−	0.868330i	$-0.665194\pi$
$257$	−1.05035e6	−0.991974	−0.495987	+	0.868330i	$0.665194\pi$
$258$	0	0
$259$	−1.26152e6	−1.16854
$260$	−39744.0	−0.0364618
$261$	0	0
$262$	17640.0	0.0158762
$263$	465510.	0.414992	0.207496	−	0.978236i	$-0.433469\pi$
$263$	465510.	0.414992	0.207496	+	0.978236i	$0.433469\pi$
$264$	0	0
$265$	1.05041e6	0.918847
$266$	150176.	0.130136
$267$	0	0
$268$	792128.	0.673687
$269$	317112.	0.267197	0.133599	−	0.991036i	$-0.457347\pi$
$269$	317112.	0.267197	0.133599	+	0.991036i	$0.457347\pi$
$270$	0	0
$271$	−1.48901e6	−1.23162	−0.615808	−	0.787897i	$-0.711170\pi$
$271$	−1.48901e6	−1.23162	−0.615808	+	0.787897i	$0.711170\pi$
$272$	158208.	0.129660
$273$	0	0
$274$	−823896.	−0.662973
$275$	−68970.0	−0.0549957
$276$	0	0
$277$	−1.54028e6	−1.20615	−0.603074	−	0.797685i	$-0.706058\pi$
$277$	−1.54028e6	−1.20615	−0.603074	+	0.797685i	$0.706058\pi$
$278$	1.26363e6	0.980638
$279$	0	0
$280$	359424.	0.273976
$281$	−1.99595e6	−1.50794	−0.753969	−	0.656910i	$-0.771863\pi$
$281$	−1.99595e6	−1.50794	−0.753969	+	0.656910i	$0.771863\pi$
$282$	0	0
$283$	−946936.	−0.702837	−0.351418	−	0.936219i	$-0.614301\pi$
$283$	−946936.	−0.702837	−0.351418	+	0.936219i	$0.614301\pi$
$284$	−121152.	−0.0891322
$285$	0	0
$286$	−60720.0	−0.0438952
$287$	1.96186e6	1.40593
$288$	0	0
$289$	−1.03793e6	−0.731012
$290$	567648.	0.396355
$291$	0	0
$292$	37472.0	0.0257187
$293$	−2.18170e6	−1.48465	−0.742327	−	0.670038i	$-0.766278\pi$
$293$	−2.18170e6	−1.48465	−0.742327	+	0.670038i	$0.766278\pi$
$294$	0	0
$295$	−1.64851e6	−1.10290
$296$	−776320.	−0.515005
$297$	0	0
$298$	−844632.	−0.550969
$299$	−18492.0	−0.0119621
$300$	0	0
$301$	−1.08243e6	−0.688628
$302$	−472480.	−0.298103
$303$	0	0
$304$	92416.0	0.0573539
$305$	601452.	0.370213
$306$	0	0
$307$	−2.50414e6	−1.51640	−0.758198	−	0.652024i	$-0.773920\pi$
$307$	−2.50414e6	−1.51640	−0.758198	+	0.652024i	$0.773920\pi$
$308$	549120.	0.329830
$309$	0	0
$310$	−511488.	−0.302295
$311$	785658.	0.460609	0.230305	−	0.973119i	$-0.426028\pi$
$311$	785658.	0.460609	0.230305	+	0.973119i	$0.426028\pi$
$312$	0	0
$313$	364334.	0.210203	0.105101	−	0.994462i	$-0.466483\pi$
$313$	364334.	0.210203	0.105101	+	0.994462i	$0.466483\pi$
$314$	−655960.	−0.375451
$315$	0	0
$316$	358784.	0.202123
$317$	1.46938e6	0.821268	0.410634	−	0.911800i	$-0.365307\pi$
$317$	1.46938e6	0.821268	0.410634	+	0.911800i	$0.365307\pi$
$318$	0	0
$319$	867240.	0.477158
$320$	221184.	0.120748
$321$	0	0
$322$	167232.	0.0898834
$323$	223098.	0.118984
$324$	0	0
$325$	9614.00	0.00504889
$326$	107792.	0.0561749
$327$	0	0
$328$	1.20730e6	0.619625
$329$	496080.	0.252675
$330$	0	0
$331$	1.67414e6	0.839889	0.419944	−	0.907550i	$-0.362050\pi$
$331$	1.67414e6	0.839889	0.419944	+	0.907550i	$0.362050\pi$
$332$	747744.	0.372313
$333$	0	0
$334$	−23232.0	−0.0113952
$335$	2.67343e6	1.30154
$336$	0	0
$337$	3.02010e6	1.44859	0.724297	−	0.689488i	$-0.242164\pi$
$337$	3.02010e6	1.44859	0.724297	+	0.689488i	$0.242164\pi$
$338$	−1.47671e6	−0.703077
$339$	0	0
$340$	533952.	0.250498
$341$	−781440.	−0.363923
$342$	0	0
$343$	−2.37099e6	−1.08817
$344$	−666112.	−0.303495
$345$	0	0
$346$	−148416.	−0.0666485
$347$	−2.79531e6	−1.24625	−0.623127	−	0.782121i	$-0.714138\pi$
$347$	−2.79531e6	−1.24625	−0.623127	+	0.782121i	$0.714138\pi$
$348$	0	0
$349$	−3.92975e6	−1.72703	−0.863517	−	0.504320i	$-0.831743\pi$
$349$	−3.92975e6	−1.72703	−0.863517	+	0.504320i	$0.831743\pi$
$350$	−86944.0	−0.0379376
$351$	0	0
$352$	337920.	0.145364
$353$	1.56554e6	0.668693	0.334347	−	0.942450i	$-0.391484\pi$
$353$	1.56554e6	0.668693	0.334347	+	0.942450i	$0.391484\pi$
$354$	0	0
$355$	−408888.	−0.172200
$356$	1.12166e6	0.469070
$357$	0	0
$358$	−793440.	−0.327195
$359$	2.19680e6	0.899609	0.449805	−	0.893127i	$-0.351494\pi$
$359$	2.19680e6	0.899609	0.449805	+	0.893127i	$0.351494\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−1.10735e6	−0.444134
$363$	0	0
$364$	−76544.0	−0.0302801
$365$	126468.	0.0496877
$366$	0	0
$367$	−4.24772e6	−1.64623	−0.823115	−	0.567875i	$-0.807766\pi$
$367$	−4.24772e6	−1.64623	−0.823115	+	0.567875i	$0.807766\pi$
$368$	102912.	0.0396138
$369$	0	0
$370$	−2.62008e6	−0.994971
$371$	2.02301e6	0.763067
$372$	0	0
$373$	4.24597e6	1.58018	0.790088	−	0.612994i	$-0.210035\pi$
$373$	4.24597e6	1.58018	0.790088	+	0.612994i	$0.210035\pi$
$374$	815760.	0.301567
$375$	0	0
$376$	305280.	0.111360
$377$	−120888.	−0.0438056
$378$	0	0
$379$	3.06794e6	1.09711	0.548553	−	0.836116i	$-0.315179\pi$
$379$	3.06794e6	1.09711	0.548553	+	0.836116i	$0.315179\pi$
$380$	311904.	0.110806
$381$	0	0
$382$	2.12489e6	0.745037
$383$	−5.23012e6	−1.82186	−0.910929	−	0.412564i	$-0.864633\pi$
$383$	−5.23012e6	−1.82186	−0.910929	+	0.412564i	$0.864633\pi$
$384$	0	0
$385$	1.85328e6	0.637220
$386$	−980248.	−0.334863
$387$	0	0
$388$	1.69136e6	0.570370
$389$	3.36209e6	1.12651	0.563256	−	0.826282i	$-0.309548\pi$
$389$	3.36209e6	1.12651	0.563256	+	0.826282i	$0.309548\pi$
$390$	0	0
$391$	248436.	0.0821812
$392$	−383424.	−0.126027
$393$	0	0
$394$	−879624.	−0.285467
$395$	1.21090e6	0.390494
$396$	0	0
$397$	2.92432e6	0.931211	0.465606	−	0.884992i	$-0.345836\pi$
$397$	2.92432e6	0.931211	0.465606	+	0.884992i	$0.345836\pi$
$398$	−4.07282e6	−1.28880
$399$	0	0
$400$	−53504.0	−0.0167200
$401$	−5.80702e6	−1.80340	−0.901700	−	0.432362i	$-0.857680\pi$
$401$	−5.80702e6	−1.80340	−0.901700	+	0.432362i	$0.857680\pi$
$402$	0	0
$403$	108928.	0.0334100
$404$	−1.99267e6	−0.607411
$405$	0	0
$406$	1.09325e6	0.329157
$407$	−4.00290e6	−1.19781
$408$	0	0
$409$	1.80028e6	0.532147	0.266073	−	0.963953i	$-0.414274\pi$
$409$	1.80028e6	0.532147	0.266073	+	0.963953i	$0.414274\pi$
$410$	4.07462e6	1.19709
$411$	0	0
$412$	103808.	0.0301292
$413$	−3.17491e6	−0.915918
$414$	0	0
$415$	2.52364e6	0.719294
$416$	−47104.0	−0.0133452
$417$	0	0
$418$	476520.	0.133395
$419$	1.38477e6	0.385339	0.192669	−	0.981264i	$-0.438286\pi$
$419$	1.38477e6	0.385339	0.192669	+	0.981264i	$0.438286\pi$
$420$	0	0
$421$	−315598.	−0.0867819	−0.0433909	−	0.999058i	$-0.513816\pi$
$421$	−315598.	−0.0867819	−0.0433909	+	0.999058i	$0.513816\pi$
$422$	420080.	0.114829
$423$	0	0
$424$	1.24493e6	0.336302
$425$	−129162.	−0.0346867
$426$	0	0
$427$	1.15835e6	0.307447
$428$	−800640.	−0.211265
$429$	0	0
$430$	−2.24813e6	−0.586341
$431$	6.52696e6	1.69246	0.846228	−	0.532821i	$-0.178868\pi$
$431$	6.52696e6	1.69246	0.846228	+	0.532821i	$0.178868\pi$
$432$	0	0
$433$	2.63139e6	0.674473	0.337237	−	0.941420i	$-0.390508\pi$
$433$	2.63139e6	0.674473	0.337237	+	0.941420i	$0.390508\pi$
$434$	−985088.	−0.251044
$435$	0	0
$436$	1.41930e6	0.357566
$437$	145122.	0.0363521
$438$	0	0
$439$	3.09445e6	0.766341	0.383170	−	0.923678i	$-0.374832\pi$
$439$	3.09445e6	0.766341	0.383170	+	0.923678i	$0.374832\pi$
$440$	1.14048e6	0.280838
$441$	0	0
$442$	−113712.	−0.0276854
$443$	−985086.	−0.238487	−0.119244	−	0.992865i	$-0.538047\pi$
$443$	−985086.	−0.238487	−0.119244	+	0.992865i	$0.538047\pi$
$444$	0	0
$445$	3.78562e6	0.906226
$446$	3.95322e6	0.941052
$447$	0	0
$448$	425984.	0.100276
$449$	2.91582e6	0.682566	0.341283	−	0.939961i	$-0.389138\pi$
$449$	2.91582e6	0.682566	0.341283	+	0.939961i	$0.389138\pi$
$450$	0	0
$451$	6.22512e6	1.44114
$452$	2.50867e6	0.577561
$453$	0	0
$454$	2.06376e6	0.469915
$455$	−258336.	−0.0585001
$456$	0	0
$457$	−3.54721e6	−0.794505	−0.397252	−	0.917709i	$-0.630036\pi$
$457$	−3.54721e6	−0.794505	−0.397252	+	0.917709i	$0.630036\pi$
$458$	−1.10942e6	−0.247133
$459$	0	0
$460$	347328.	0.0765324
$461$	−4.57057e6	−1.00165	−0.500827	−	0.865547i	$-0.666971\pi$
$461$	−4.57057e6	−1.00165	−0.500827	+	0.865547i	$0.666971\pi$
$462$	0	0
$463$	5.91304e6	1.28191	0.640957	−	0.767577i	$-0.278538\pi$
$463$	5.91304e6	1.28191	0.640957	+	0.767577i	$0.278538\pi$
$464$	672768.	0.145068
$465$	0	0
$466$	−6.16606e6	−1.31535
$467$	3.73808e6	0.793152	0.396576	−	0.918002i	$-0.370198\pi$
$467$	3.73808e6	0.793152	0.396576	+	0.918002i	$0.370198\pi$
$468$	0	0
$469$	5.14883e6	1.08088
$470$	1.03032e6	0.215143
$471$	0	0
$472$	−1.95379e6	−0.403667
$473$	−3.43464e6	−0.705876
$474$	0	0
$475$	−75449.0	−0.0153433
$476$	1.02835e6	0.208029
$477$	0	0
$478$	1.25724e6	0.251680
$479$	3.79193e6	0.755130	0.377565	−	0.925983i	$-0.376761\pi$
$479$	3.79193e6	0.755130	0.377565	+	0.925983i	$0.376761\pi$
$480$	0	0
$481$	557980.	0.109965
$482$	98264.0	0.0192653
$483$	0	0
$484$	−834416.	−0.161908
$485$	5.70834e6	1.10193
$486$	0	0
$487$	2.78559e6	0.532225	0.266112	−	0.963942i	$-0.414261\pi$
$487$	2.78559e6	0.532225	0.266112	+	0.963942i	$0.414261\pi$
$488$	712832.	0.135499
$489$	0	0
$490$	−1.29406e6	−0.243480
$491$	−1.70523e6	−0.319212	−0.159606	−	0.987181i	$-0.551022\pi$
$491$	−1.70523e6	−0.319212	−0.159606	+	0.987181i	$0.551022\pi$
$492$	0	0
$493$	1.62410e6	0.300952
$494$	−66424.0	−0.0122464
$495$	0	0
$496$	−606208.	−0.110641
$497$	−787488.	−0.143006
$498$	0	0
$499$	3.60026e6	0.647266	0.323633	−	0.946183i	$-0.395096\pi$
$499$	3.60026e6	0.647266	0.323633	+	0.946183i	$0.395096\pi$
$500$	−2.88058e6	−0.515293
$501$	0	0
$502$	−2.64516e6	−0.468482
$503$	8.68129e6	1.52990	0.764952	−	0.644087i	$-0.222763\pi$
$503$	8.68129e6	1.52990	0.764952	+	0.644087i	$0.222763\pi$
$504$	0	0
$505$	−6.72527e6	−1.17349
$506$	530640.	0.0921348
$507$	0	0
$508$	−3.38829e6	−0.582534
$509$	1.09767e7	1.87792	0.938962	−	0.344022i	$-0.111789\pi$
$509$	1.09767e7	1.87792	0.938962	+	0.344022i	$0.111789\pi$
$510$	0	0
$511$	243568.	0.0412637
$512$	262144.	0.0441942
$513$	0	0
$514$	−4.20139e6	−0.701432
$515$	350352.	0.0582085
$516$	0	0
$517$	1.57410e6	0.259004
$518$	−5.04608e6	−0.826285
$519$	0	0
$520$	−158976.	−0.0257824
$521$	9.09036e6	1.46719	0.733596	−	0.679586i	$-0.237841\pi$
$521$	9.09036e6	1.46719	0.733596	+	0.679586i	$0.237841\pi$
$522$	0	0
$523$	8.46276e6	1.35287	0.676437	−	0.736500i	$-0.263523\pi$
$523$	8.46276e6	1.35287	0.676437	+	0.736500i	$0.263523\pi$
$524$	70560.0	0.0112261
$525$	0	0
$526$	1.86204e6	0.293444
$527$	−1.46342e6	−0.229532
$528$	0	0
$529$	−6.27474e6	−0.974892
$530$	4.20163e6	0.649723
$531$	0	0
$532$	600704.	0.0920198
$533$	−867744.	−0.132304
$534$	0	0
$535$	−2.70216e6	−0.408156
$536$	3.16851e6	0.476369
$537$	0	0
$538$	1.26845e6	0.188937
$539$	−1.97703e6	−0.293117
$540$	0	0
$541$	−8.03851e6	−1.18082	−0.590408	−	0.807105i	$-0.701033\pi$
$541$	−8.03851e6	−1.18082	−0.590408	+	0.807105i	$0.701033\pi$
$542$	−5.95605e6	−0.870883
$543$	0	0
$544$	632832.	0.0916835
$545$	4.79012e6	0.690805
$546$	0	0
$547$	−1.32779e7	−1.89741	−0.948704	−	0.316167i	$-0.897604\pi$
$547$	−1.32779e7	−1.89741	−0.948704	+	0.316167i	$0.897604\pi$
$548$	−3.29558e6	−0.468793
$549$	0	0
$550$	−275880.	−0.0388878
$551$	948708.	0.133123
$552$	0	0
$553$	2.33210e6	0.324290
$554$	−6.16113e6	−0.852876
$555$	0	0
$556$	5.05453e6	0.693416
$557$	9.11080e6	1.24428	0.622141	−	0.782905i	$-0.286263\pi$
$557$	9.11080e6	1.24428	0.622141	+	0.782905i	$0.286263\pi$
$558$	0	0
$559$	478768.	0.0648031
$560$	1.43770e6	0.193730
$561$	0	0
$562$	−7.98379e6	−1.06627
$563$	5.07313e6	0.674536	0.337268	−	0.941409i	$-0.390497\pi$
$563$	5.07313e6	0.674536	0.337268	+	0.941409i	$0.390497\pi$
$564$	0	0
$565$	8.46677e6	1.11583
$566$	−3.78774e6	−0.496981
$567$	0	0
$568$	−484608.	−0.0630260
$569$	−2.30192e6	−0.298065	−0.149032	−	0.988832i	$-0.547616\pi$
$569$	−2.30192e6	−0.298065	−0.149032	+	0.988832i	$0.547616\pi$
$570$	0	0
$571$	−1.14948e7	−1.47540	−0.737702	−	0.675127i	$-0.764089\pi$
$571$	−1.14948e7	−1.47540	−0.737702	+	0.675127i	$0.764089\pi$
$572$	−242880.	−0.0310386
$573$	0	0
$574$	7.84742e6	0.994140
$575$	−84018.0	−0.0105975
$576$	0	0
$577$	308198.	0.0385381	0.0192690	−	0.999814i	$-0.493866\pi$
$577$	308198.	0.0385381	0.0192690	+	0.999814i	$0.493866\pi$
$578$	−4.15173e6	−0.516904
$579$	0	0
$580$	2.27059e6	0.280265
$581$	4.86034e6	0.597346
$582$	0	0
$583$	6.41916e6	0.782180
$584$	149888.	0.0181859
$585$	0	0
$586$	−8.72678e6	−1.04981
$587$	−5.02053e6	−0.601387	−0.300694	−	0.953721i	$-0.597218\pi$
$587$	−5.02053e6	−0.601387	−0.300694	+	0.953721i	$0.597218\pi$
$588$	0	0
$589$	−854848.	−0.101531
$590$	−6.59405e6	−0.779870
$591$	0	0
$592$	−3.10528e6	−0.364164
$593$	−1.70294e7	−1.98867	−0.994335	−	0.106288i	$-0.966103\pi$
$593$	−1.70294e7	−1.98867	−0.994335	+	0.106288i	$0.966103\pi$
$594$	0	0
$595$	3.47069e6	0.401905
$596$	−3.37853e6	−0.389594
$597$	0	0
$598$	−73968.0	−0.00845845
$599$	−2.86572e6	−0.326337	−0.163169	−	0.986598i	$-0.552171\pi$
$599$	−2.86572e6	−0.326337	−0.163169	+	0.986598i	$0.552171\pi$
$600$	0	0
$601$	−1.03461e7	−1.16840	−0.584200	−	0.811609i	$-0.698592\pi$
$601$	−1.03461e7	−1.16840	−0.584200	+	0.811609i	$0.698592\pi$
$602$	−4.32973e6	−0.486933
$603$	0	0
$604$	−1.88992e6	−0.210791
$605$	−2.81615e6	−0.312801
$606$	0	0
$607$	3.56084e6	0.392266	0.196133	−	0.980577i	$-0.437162\pi$
$607$	3.56084e6	0.392266	0.196133	+	0.980577i	$0.437162\pi$
$608$	369664.	0.0405554
$609$	0	0
$610$	2.40581e6	0.261780
$611$	−219420.	−0.0237779
$612$	0	0
$613$	7.56115e6	0.812712	0.406356	−	0.913715i	$-0.366799\pi$
$613$	7.56115e6	0.812712	0.406356	+	0.913715i	$0.366799\pi$
$614$	−1.00166e7	−1.07225
$615$	0	0
$616$	2.19648e6	0.233225
$617$	−5.03233e6	−0.532177	−0.266088	−	0.963949i	$-0.585731\pi$
$617$	−5.03233e6	−0.532177	−0.266088	+	0.963949i	$0.585731\pi$
$618$	0	0
$619$	−5.87528e6	−0.616313	−0.308157	−	0.951336i	$-0.599712\pi$
$619$	−5.87528e6	−0.616313	−0.308157	+	0.951336i	$0.599712\pi$
$620$	−2.04595e6	−0.213755
$621$	0	0
$622$	3.14263e6	0.325700
$623$	7.29082e6	0.752586
$624$	0	0
$625$	−9.06882e6	−0.928647
$626$	1.45734e6	0.148636
$627$	0	0
$628$	−2.62384e6	−0.265484
$629$	−7.49634e6	−0.755479
$630$	0	0
$631$	−4.33809e6	−0.433735	−0.216868	−	0.976201i	$-0.569584\pi$
$631$	−4.33809e6	−0.433735	−0.216868	+	0.976201i	$0.569584\pi$
$632$	1.43514e6	0.142922
$633$	0	0
$634$	5.87750e6	0.580724
$635$	−1.14355e7	−1.12543
$636$	0	0
$637$	275586.	0.0269097
$638$	3.46896e6	0.337402
$639$	0	0
$640$	884736.	0.0853815
$641$	−1.35603e7	−1.30354	−0.651769	−	0.758417i	$-0.725973\pi$
$641$	−1.35603e7	−1.30354	−0.651769	+	0.758417i	$0.725973\pi$
$642$	0	0
$643$	6.02612e6	0.574792	0.287396	−	0.957812i	$-0.407211\pi$
$643$	6.02612e6	0.574792	0.287396	+	0.957812i	$0.407211\pi$
$644$	668928.	0.0635572
$645$	0	0
$646$	892392.	0.0841345
$647$	8.69474e6	0.816574	0.408287	−	0.912854i	$-0.366126\pi$
$647$	8.69474e6	0.816574	0.408287	+	0.912854i	$0.366126\pi$
$648$	0	0
$649$	−1.00742e7	−0.938859
$650$	38456.0	0.00357010
$651$	0	0
$652$	431168.	0.0397217
$653$	1.09929e6	0.100886	0.0504428	−	0.998727i	$-0.483937\pi$
$653$	1.09929e6	0.100886	0.0504428	+	0.998727i	$0.483937\pi$
$654$	0	0
$655$	238140.	0.0216885
$656$	4.82918e6	0.438141
$657$	0	0
$658$	1.98432e6	0.178668
$659$	3.94808e6	0.354138	0.177069	−	0.984198i	$-0.443338\pi$
$659$	3.94808e6	0.354138	0.177069	+	0.984198i	$0.443338\pi$
$660$	0	0
$661$	−1.96958e7	−1.75335	−0.876676	−	0.481082i	$-0.840244\pi$
$661$	−1.96958e7	−1.75335	−0.876676	+	0.481082i	$0.840244\pi$
$662$	6.69656e6	0.593891
$663$	0	0
$664$	2.99098e6	0.263265
$665$	2.02738e6	0.177779
$666$	0	0
$667$	1.05646e6	0.0919468
$668$	−92928.0	−0.00805759
$669$	0	0
$670$	1.06937e7	0.920327
$671$	3.67554e6	0.315148
$672$	0	0
$673$	−2.06650e7	−1.75873	−0.879363	−	0.476151i	$-0.842031\pi$
$673$	−2.06650e7	−1.75873	−0.879363	+	0.476151i	$0.842031\pi$
$674$	1.20804e7	1.02431
$675$	0	0
$676$	−5.90683e6	−0.497150
$677$	−1.64591e7	−1.38018	−0.690090	−	0.723724i	$-0.742429\pi$
$677$	−1.64591e7	−1.38018	−0.690090	+	0.723724i	$0.742429\pi$
$678$	0	0
$679$	1.09938e7	0.915114
$680$	2.13581e6	0.177129
$681$	0	0
$682$	−3.12576e6	−0.257333
$683$	876672.	0.0719094	0.0359547	−	0.999353i	$-0.488553\pi$
$683$	876672.	0.0719094	0.0359547	+	0.999353i	$0.488553\pi$
$684$	0	0
$685$	−1.11226e7	−0.905690
$686$	−9.48397e6	−0.769449
$687$	0	0
$688$	−2.66445e6	−0.214603
$689$	−894792.	−0.0718082
$690$	0	0
$691$	1.64070e7	1.30717	0.653586	−	0.756852i	$-0.273264\pi$
$691$	1.64070e7	1.30717	0.653586	+	0.756852i	$0.273264\pi$
$692$	−593664.	−0.0471276
$693$	0	0
$694$	−1.11812e7	−0.881234
$695$	1.70590e7	1.33965
$696$	0	0
$697$	1.16580e7	0.908951
$698$	−1.57190e7	−1.22120
$699$	0	0
$700$	−347776.	−0.0268259
$701$	−1.03676e6	−0.0796861	−0.0398430	−	0.999206i	$-0.512686\pi$
$701$	−1.03676e6	−0.0796861	−0.0398430	+	0.999206i	$0.512686\pi$
$702$	0	0
$703$	−4.37893e6	−0.334179
$704$	1.35168e6	0.102788
$705$	0	0
$706$	6.26215e6	0.472837
$707$	−1.29524e7	−0.974542
$708$	0	0
$709$	−2.00637e6	−0.149898	−0.0749491	−	0.997187i	$-0.523879\pi$
$709$	−2.00637e6	−0.149898	−0.0749491	+	0.997187i	$0.523879\pi$
$710$	−1.63555e6	−0.121764
$711$	0	0
$712$	4.48666e6	0.331683
$713$	−951936.	−0.0701268
$714$	0	0
$715$	−819720.	−0.0599654
$716$	−3.17376e6	−0.231362
$717$	0	0
$718$	8.78719e6	0.636120
$719$	8.95036e6	0.645682	0.322841	−	0.946453i	$-0.395362\pi$
$719$	8.95036e6	0.645682	0.322841	+	0.946453i	$0.395362\pi$
$720$	0	0
$721$	674752.	0.0483399
$722$	521284.	0.0372161
$723$	0	0
$724$	−4.42941e6	−0.314050
$725$	−549252.	−0.0388085
$726$	0	0
$727$	−614668.	−0.0431325	−0.0215662	−	0.999767i	$-0.506865\pi$
$727$	−614668.	−0.0431325	−0.0215662	+	0.999767i	$0.506865\pi$
$728$	−306176.	−0.0214113
$729$	0	0
$730$	505872.	0.0351345
$731$	−6.43214e6	−0.445207
$732$	0	0
$733$	−8.47041e6	−0.582297	−0.291149	−	0.956678i	$-0.594037\pi$
$733$	−8.47041e6	−0.582297	−0.291149	+	0.956678i	$0.594037\pi$
$734$	−1.69909e7	−1.16406
$735$	0	0
$736$	411648.	0.0280112
$737$	1.63376e7	1.10795
$738$	0	0
$739$	−9.40491e6	−0.633495	−0.316748	−	0.948510i	$-0.602591\pi$
$739$	−9.40491e6	−0.633495	−0.316748	+	0.948510i	$0.602591\pi$
$740$	−1.04803e7	−0.703550
$741$	0	0
$742$	8.09203e6	0.539570
$743$	−2.73198e7	−1.81554	−0.907769	−	0.419471i	$-0.862216\pi$
$743$	−2.73198e7	−1.81554	−0.907769	+	0.419471i	$0.862216\pi$
$744$	0	0
$745$	−1.14025e7	−0.752680
$746$	1.69839e7	1.11735
$747$	0	0
$748$	3.26304e6	0.213240
$749$	−5.20416e6	−0.338958
$750$	0	0
$751$	−1.07808e7	−0.697509	−0.348755	−	0.937214i	$-0.613395\pi$
$751$	−1.07808e7	−0.697509	−0.348755	+	0.937214i	$0.613395\pi$
$752$	1.22112e6	0.0787433
$753$	0	0
$754$	−483552.	−0.0309752
$755$	−6.37848e6	−0.407239
$756$	0	0
$757$	−543058.	−0.0344434	−0.0172217	−	0.999852i	$-0.505482\pi$
$757$	−543058.	−0.0344434	−0.0172217	+	0.999852i	$0.505482\pi$
$758$	1.22718e7	0.775772
$759$	0	0
$760$	1.24762e6	0.0783514
$761$	−9.07913e6	−0.568307	−0.284153	−	0.958779i	$-0.591712\pi$
$761$	−9.07913e6	−0.568307	−0.284153	+	0.958779i	$0.591712\pi$
$762$	0	0
$763$	9.22542e6	0.573687
$764$	8.49955e6	0.526820
$765$	0	0
$766$	−2.09205e7	−1.28825
$767$	1.40429e6	0.0861922
$768$	0	0
$769$	5.14583e6	0.313790	0.156895	−	0.987615i	$-0.449852\pi$
$769$	5.14583e6	0.313790	0.156895	+	0.987615i	$0.449852\pi$
$770$	7.41312e6	0.450582
$771$	0	0
$772$	−3.92099e6	−0.236784
$773$	−7.50322e6	−0.451647	−0.225823	−	0.974168i	$-0.572507\pi$
$773$	−7.50322e6	−0.451647	−0.225823	+	0.974168i	$0.572507\pi$
$774$	0	0
$775$	494912.	0.0295988
$776$	6.76544e6	0.403313
$777$	0	0
$778$	1.34484e7	0.796564
$779$	6.80990e6	0.402066
$780$	0	0
$781$	−2.49876e6	−0.146588
$782$	993744.	0.0581109
$783$	0	0
$784$	−1.53370e6	−0.0891147
$785$	−8.85546e6	−0.512905
$786$	0	0
$787$	1.48747e7	0.856076	0.428038	−	0.903761i	$-0.359205\pi$
$787$	1.48747e7	0.856076	0.428038	+	0.903761i	$0.359205\pi$
$788$	−3.51850e6	−0.201856
$789$	0	0
$790$	4.84358e6	0.276121
$791$	1.63064e7	0.926651
$792$	0	0
$793$	−512348.	−0.0289322
$794$	1.16973e7	0.658466
$795$	0	0
$796$	−1.62913e7	−0.911323
$797$	−2.77279e7	−1.54622	−0.773109	−	0.634273i	$-0.781299\pi$
$797$	−2.77279e7	−1.54622	−0.773109	+	0.634273i	$0.781299\pi$
$798$	0	0
$799$	2.94786e6	0.163358
$800$	−214016.	−0.0118228
$801$	0	0
$802$	−2.32281e7	−1.27520
$803$	772860.	0.0422972
$804$	0	0
$805$	2.25763e6	0.122790
$806$	435712.	0.0236245
$807$	0	0
$808$	−7.97069e6	−0.429504
$809$	3.46147e7	1.85947	0.929735	−	0.368229i	$-0.120036\pi$
$809$	3.46147e7	1.85947	0.929735	+	0.368229i	$0.120036\pi$
$810$	0	0
$811$	1.42801e7	0.762392	0.381196	−	0.924494i	$-0.375512\pi$
$811$	1.42801e7	0.762392	0.381196	+	0.924494i	$0.375512\pi$
$812$	4.37299e6	0.232749
$813$	0	0
$814$	−1.60116e7	−0.846981
$815$	1.45519e6	0.0767408
$816$	0	0
$817$	−3.75729e6	−0.196933
$818$	7.20111e6	0.376284
$819$	0	0
$820$	1.62985e7	0.846473
$821$	−3.69632e6	−0.191387	−0.0956933	−	0.995411i	$-0.530507\pi$
$821$	−3.69632e6	−0.191387	−0.0956933	+	0.995411i	$0.530507\pi$
$822$	0	0
$823$	2.05703e7	1.05862	0.529310	−	0.848428i	$-0.322451\pi$
$823$	2.05703e7	1.05862	0.529310	+	0.848428i	$0.322451\pi$
$824$	415232.	0.0213046
$825$	0	0
$826$	−1.26996e7	−0.647652
$827$	2.42257e7	1.23172	0.615860	−	0.787856i	$-0.288809\pi$
$827$	2.42257e7	1.23172	0.615860	+	0.787856i	$0.288809\pi$
$828$	0	0
$829$	−1.93936e7	−0.980102	−0.490051	−	0.871694i	$-0.663022\pi$
$829$	−1.93936e7	−0.980102	−0.490051	+	0.871694i	$0.663022\pi$
$830$	1.00945e7	0.508618
$831$	0	0
$832$	−188416.	−0.00943647
$833$	−3.70244e6	−0.184874
$834$	0	0
$835$	−313632.	−0.0155670
$836$	1.90608e6	0.0943247
$837$	0	0
$838$	5.53908e6	0.272476
$839$	3.44560e7	1.68990	0.844949	−	0.534847i	$-0.179631\pi$
$839$	3.44560e7	1.68990	0.844949	+	0.534847i	$0.179631\pi$
$840$	0	0
$841$	−1.36048e7	−0.663286
$842$	−1.26239e6	−0.0613640
$843$	0	0
$844$	1.68032e6	0.0811962
$845$	−1.99356e7	−0.960476
$846$	0	0
$847$	−5.42370e6	−0.259769
$848$	4.97971e6	0.237802
$849$	0	0
$850$	−516648.	−0.0245272
$851$	−4.87626e6	−0.230814
$852$	0	0
$853$	1.43023e7	0.673029	0.336514	−	0.941678i	$-0.390752\pi$
$853$	1.43023e7	0.673029	0.336514	+	0.941678i	$0.390752\pi$
$854$	4.63341e6	0.217398
$855$	0	0
$856$	−3.20256e6	−0.149387
$857$	−1.96706e7	−0.914882	−0.457441	−	0.889240i	$-0.651234\pi$
$857$	−1.96706e7	−0.914882	−0.457441	+	0.889240i	$0.651234\pi$
$858$	0	0
$859$	9.60088e6	0.443944	0.221972	−	0.975053i	$-0.428751\pi$
$859$	9.60088e6	0.443944	0.221972	+	0.975053i	$0.428751\pi$
$860$	−8.99251e6	−0.414605
$861$	0	0
$862$	2.61078e7	1.19675
$863$	−1.46481e7	−0.669508	−0.334754	−	0.942306i	$-0.608653\pi$
$863$	−1.46481e7	−0.669508	−0.334754	+	0.942306i	$0.608653\pi$
$864$	0	0
$865$	−2.00362e6	−0.0910488
$866$	1.05255e7	0.476925
$867$	0	0
$868$	−3.94035e6	−0.177515
$869$	7.39992e6	0.332413
$870$	0	0
$871$	−2.27737e6	−0.101716
$872$	5.67718e6	0.252838
$873$	0	0
$874$	580488.	0.0257048
$875$	−1.87237e7	−0.826747
$876$	0	0
$877$	−1.17035e7	−0.513825	−0.256913	−	0.966435i	$-0.582705\pi$
$877$	−1.17035e7	−0.513825	−0.256913	+	0.966435i	$0.582705\pi$
$878$	1.23778e7	0.541885
$879$	0	0
$880$	4.56192e6	0.198583
$881$	4.52787e7	1.96541	0.982706	−	0.185171i	$-0.0592838\pi$
$881$	4.52787e7	1.96541	0.982706	+	0.185171i	$0.0592838\pi$
$882$	0	0
$883$	2.59473e7	1.11993	0.559965	−	0.828517i	$-0.310815\pi$
$883$	2.59473e7	1.11993	0.559965	+	0.828517i	$0.310815\pi$
$884$	−454848.	−0.0195765
$885$	0	0
$886$	−3.94034e6	−0.168636
$887$	2.07015e7	0.883473	0.441737	−	0.897145i	$-0.354363\pi$
$887$	2.07015e7	0.883473	0.441737	+	0.897145i	$0.354363\pi$
$888$	0	0
$889$	−2.20239e7	−0.934629
$890$	1.51425e7	0.640799
$891$	0	0
$892$	1.58129e7	0.665424
$893$	1.72197e6	0.0722598
$894$	0	0
$895$	−1.07114e7	−0.446982
$896$	1.70394e6	0.0709060
$897$	0	0
$898$	1.16633e7	0.482647
$899$	−6.22310e6	−0.256808
$900$	0	0
$901$	1.20213e7	0.493334
$902$	2.49005e7	1.01904
$903$	0	0
$904$	1.00347e7	0.408397
$905$	−1.49493e7	−0.606734
$906$	0	0
$907$	4.76595e7	1.92367	0.961836	−	0.273628i	$-0.0882236\pi$
$907$	4.76595e7	1.92367	0.961836	+	0.273628i	$0.0882236\pi$
$908$	8.25504e6	0.332280
$909$	0	0
$910$	−1.03334e6	−0.0413658
$911$	387816.	0.0154821	0.00774105	−	0.999970i	$-0.497536\pi$
$911$	387816.	0.0154821	0.00774105	+	0.999970i	$0.497536\pi$
$912$	0	0
$913$	1.54222e7	0.612308
$914$	−1.41888e7	−0.561800
$915$	0	0
$916$	−4.43766e6	−0.174749
$917$	458640.	0.0180114
$918$	0	0
$919$	3.45269e7	1.34855	0.674277	−	0.738479i	$-0.264455\pi$
$919$	3.45269e7	1.34855	0.674277	+	0.738479i	$0.264455\pi$
$920$	1.38931e6	0.0541166
$921$	0	0
$922$	−1.82823e7	−0.708276
$923$	348312.	0.0134575
$924$	0	0
$925$	2.53517e6	0.0974210
$926$	2.36522e7	0.906450
$927$	0	0
$928$	2.69107e6	0.102578
$929$	−3.96164e7	−1.50604	−0.753018	−	0.657999i	$-0.771403\pi$
$929$	−3.96164e7	−1.50604	−0.753018	+	0.657999i	$0.771403\pi$
$930$	0	0
$931$	−2.16275e6	−0.0817772
$932$	−2.46642e7	−0.930096
$933$	0	0
$934$	1.49523e7	0.560843
$935$	1.10128e7	0.411971
$936$	0	0
$937$	−1.42780e7	−0.531272	−0.265636	−	0.964073i	$-0.585582\pi$
$937$	−1.42780e7	−0.531272	−0.265636	+	0.964073i	$0.585582\pi$
$938$	2.05953e7	0.764296
$939$	0	0
$940$	4.12128e6	0.152129
$941$	2.33088e6	0.0858116	0.0429058	−	0.999079i	$-0.486338\pi$
$941$	2.33088e6	0.0858116	0.0429058	+	0.999079i	$0.486338\pi$
$942$	0	0
$943$	7.58333e6	0.277703
$944$	−7.81517e6	−0.285436
$945$	0	0
$946$	−1.37386e7	−0.499130
$947$	2.93987e7	1.06526	0.532628	−	0.846350i	$-0.321205\pi$
$947$	2.93987e7	1.06526	0.532628	+	0.846350i	$0.321205\pi$
$948$	0	0
$949$	−107732.	−0.00388311
$950$	−301796.	−0.0108494
$951$	0	0
$952$	4.11341e6	0.147099
$953$	2.26835e7	0.809055	0.404528	−	0.914526i	$-0.367436\pi$
$953$	2.26835e7	0.809055	0.404528	+	0.914526i	$0.367436\pi$
$954$	0	0
$955$	2.86860e7	1.01780
$956$	5.02896e6	0.177965
$957$	0	0
$958$	1.51677e7	0.533958
$959$	−2.14213e7	−0.752141
$960$	0	0
$961$	−2.30217e7	−0.804136
$962$	2.23192e6	0.0777573
$963$	0	0
$964$	393056.	0.0136227
$965$	−1.32333e7	−0.457458
$966$	0	0
$967$	−2.50637e7	−0.861944	−0.430972	−	0.902365i	$-0.641829\pi$
$967$	−2.50637e7	−0.861944	−0.430972	+	0.902365i	$0.641829\pi$
$968$	−3.33766e6	−0.114486
$969$	0	0
$970$	2.28334e7	0.779185
$971$	−2.93328e7	−0.998402	−0.499201	−	0.866486i	$-0.666373\pi$
$971$	−2.93328e7	−0.998402	−0.499201	+	0.866486i	$0.666373\pi$
$972$	0	0
$973$	3.28544e7	1.11253
$974$	1.11424e7	0.376340
$975$	0	0
$976$	2.85133e6	0.0958126
$977$	−4.53306e7	−1.51934	−0.759670	−	0.650309i	$-0.774639\pi$
$977$	−4.53306e7	−1.51934	−0.759670	+	0.650309i	$0.774639\pi$
$978$	0	0
$979$	2.31343e7	0.771436
$980$	−5.17622e6	−0.172166
$981$	0	0
$982$	−6.82092e6	−0.225717
$983$	2.13167e7	0.703615	0.351808	−	0.936072i	$-0.385567\pi$
$983$	2.13167e7	0.703615	0.351808	+	0.936072i	$0.385567\pi$
$984$	0	0
$985$	−1.18749e7	−0.389978
$986$	6.49642e6	0.212805
$987$	0	0
$988$	−265696.	−0.00865950
$989$	−4.18402e6	−0.136020
$990$	0	0
$991$	−3.14104e7	−1.01599	−0.507996	−	0.861360i	$-0.669613\pi$
$991$	−3.14104e7	−1.01599	−0.507996	+	0.861360i	$0.669613\pi$
$992$	−2.42483e6	−0.0782352
$993$	0	0
$994$	−3.14995e6	−0.101120
$995$	−5.49830e7	−1.76064
$996$	0	0
$997$	9.54091e6	0.303985	0.151992	−	0.988382i	$-0.451431\pi$
$997$	9.54091e6	0.303985	0.151992	+	0.988382i	$0.451431\pi$
$998$	1.44010e7	0.457686
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.6.a.f.1.1		1
3.2	odd	2		114.6.a.a.1.1	✓	1
12.11	even	2		912.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.a.1.1	✓	1	3.2	odd	2
342.6.a.f.1.1		1	1.1	even	1	trivial
912.6.a.c.1.1		1	12.11	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more