LMFDB - Embedded newform 242.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,6,Mod(1,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("242.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	242.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:	$38.8128843947$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -21.0000 q^{3} +16.0000 q^{4} +81.0000 q^{5} -84.0000 q^{6} -98.0000 q^{7} +64.0000 q^{8} +198.000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -21.0000 q^{3} +16.0000 q^{4} +81.0000 q^{5} -84.0000 q^{6} -98.0000 q^{7} +64.0000 q^{8} +198.000 q^{9} +324.000 q^{10} -336.000 q^{12} -824.000 q^{13} -392.000 q^{14} -1701.00 q^{15} +256.000 q^{16} -978.000 q^{17} +792.000 q^{18} +2140.00 q^{19} +1296.00 q^{20} +2058.00 q^{21} +3699.00 q^{23} -1344.00 q^{24} +3436.00 q^{25} -3296.00 q^{26} +945.000 q^{27} -1568.00 q^{28} -3480.00 q^{29} -6804.00 q^{30} -7813.00 q^{31} +1024.00 q^{32} -3912.00 q^{34} -7938.00 q^{35} +3168.00 q^{36} -13597.0 q^{37} +8560.00 q^{38} +17304.0 q^{39} +5184.00 q^{40} -6492.00 q^{41} +8232.00 q^{42} -14234.0 q^{43} +16038.0 q^{45} +14796.0 q^{46} -20352.0 q^{47} -5376.00 q^{48} -7203.00 q^{49} +13744.0 q^{50} +20538.0 q^{51} -13184.0 q^{52} -366.000 q^{53} +3780.00 q^{54} -6272.00 q^{56} -44940.0 q^{57} -13920.0 q^{58} +9825.00 q^{59} -27216.0 q^{60} -26132.0 q^{61} -31252.0 q^{62} -19404.0 q^{63} +4096.00 q^{64} -66744.0 q^{65} +17093.0 q^{67} -15648.0 q^{68} -77679.0 q^{69} -31752.0 q^{70} -23583.0 q^{71} +12672.0 q^{72} +35176.0 q^{73} -54388.0 q^{74} -72156.0 q^{75} +34240.0 q^{76} +69216.0 q^{78} +42490.0 q^{79} +20736.0 q^{80} -67959.0 q^{81} -25968.0 q^{82} -22674.0 q^{83} +32928.0 q^{84} -79218.0 q^{85} -56936.0 q^{86} +73080.0 q^{87} -17145.0 q^{89} +64152.0 q^{90} +80752.0 q^{91} +59184.0 q^{92} +164073. q^{93} -81408.0 q^{94} +173340. q^{95} -21504.0 q^{96} -30727.0 q^{97} -28812.0 q^{98} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−21.0000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−21.0000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	16.0000	0.500000
$5$	81.0000	1.44897	0.724486	−	0.689289i	$-0.242077\pi$
$5$	81.0000	1.44897	0.724486	+	0.689289i	$0.242077\pi$
$6$	−84.0000	−0.952579
$7$	−98.0000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−98.0000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	64.0000	0.353553
$9$	198.000	0.814815
$10$	324.000	1.02458
$11$	0	0
$11$	0	0
$12$	−336.000	−0.673575
$13$	−824.000	−1.35229	−0.676143	−	0.736770i	$-0.736350\pi$
$13$	−824.000	−1.35229	−0.676143	+	0.736770i	$0.736350\pi$
$14$	−392.000	−0.534522
$15$	−1701.00	−1.95198
$16$	256.000	0.250000
$17$	−978.000	−0.820761	−0.410380	−	0.911914i	$-0.634604\pi$
$17$	−978.000	−0.820761	−0.410380	+	0.911914i	$0.634604\pi$
$18$	792.000	0.576161
$19$	2140.00	1.35997	0.679986	−	0.733225i	$-0.261986\pi$
$19$	2140.00	1.35997	0.679986	+	0.733225i	$0.261986\pi$
$20$	1296.00	0.724486
$21$	2058.00	1.01835
$22$	0	0
$23$	3699.00	1.45802	0.729012	−	0.684501i	$-0.239980\pi$
$23$	3699.00	1.45802	0.729012	+	0.684501i	$0.239980\pi$
$24$	−1344.00	−0.476290
$25$	3436.00	1.09952
$26$	−3296.00	−0.956211
$27$	945.000	0.249472
$28$	−1568.00	−0.377964
$29$	−3480.00	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−3480.00	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	−6804.00	−1.38026
$31$	−7813.00	−1.46020	−0.730102	−	0.683338i	$-0.760528\pi$
$31$	−7813.00	−1.46020	−0.730102	+	0.683338i	$0.760528\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−3912.00	−0.580365
$35$	−7938.00	−1.09532
$36$	3168.00	0.407407
$37$	−13597.0	−1.63282	−0.816411	−	0.577471i	$-0.804039\pi$
$37$	−13597.0	−1.63282	−0.816411	+	0.577471i	$0.804039\pi$
$38$	8560.00	0.961645
$39$	17304.0	1.82173
$40$	5184.00	0.512289
$41$	−6492.00	−0.603141	−0.301571	−	0.953444i	$-0.597511\pi$
$41$	−6492.00	−0.603141	−0.301571	+	0.953444i	$0.597511\pi$
$42$	8232.00	0.720082
$43$	−14234.0	−1.17397	−0.586983	−	0.809599i	$-0.699685\pi$
$43$	−14234.0	−1.17397	−0.586983	+	0.809599i	$0.699685\pi$
$44$	0	0
$45$	16038.0	1.18064
$46$	14796.0	1.03098
$47$	−20352.0	−1.34389	−0.671943	−	0.740603i	$-0.734540\pi$
$47$	−20352.0	−1.34389	−0.671943	+	0.740603i	$0.734540\pi$
$48$	−5376.00	−0.336788
$49$	−7203.00	−0.428571
$50$	13744.0	0.777478
$51$	20538.0	1.10569
$52$	−13184.0	−0.676143
$53$	−366.000	−0.0178975	−0.00894873	−	0.999960i	$-0.502849\pi$
$53$	−366.000	−0.0178975	−0.00894873	+	0.999960i	$0.502849\pi$
$54$	3780.00	0.176404
$55$	0	0
$56$	−6272.00	−0.267261
$57$	−44940.0	−1.83209
$58$	−13920.0	−0.543337
$59$	9825.00	0.367454	0.183727	−	0.982977i	$-0.441184\pi$
$59$	9825.00	0.367454	0.183727	+	0.982977i	$0.441184\pi$
$60$	−27216.0	−0.975992
$61$	−26132.0	−0.899183	−0.449591	−	0.893234i	$-0.648430\pi$
$61$	−26132.0	−0.899183	−0.449591	+	0.893234i	$0.648430\pi$
$62$	−31252.0	−1.03252
$63$	−19404.0	−0.615942
$64$	4096.00	0.125000
$65$	−66744.0	−1.95943
$66$	0	0
$67$	17093.0	0.465191	0.232595	−	0.972574i	$-0.425278\pi$
$67$	17093.0	0.465191	0.232595	+	0.972574i	$0.425278\pi$
$68$	−15648.0	−0.410380
$69$	−77679.0	−1.96418
$70$	−31752.0	−0.774508
$71$	−23583.0	−0.555205	−0.277602	−	0.960696i	$-0.589540\pi$
$71$	−23583.0	−0.555205	−0.277602	+	0.960696i	$0.589540\pi$
$72$	12672.0	0.288081
$73$	35176.0	0.772573	0.386286	−	0.922379i	$-0.373758\pi$
$73$	35176.0	0.772573	0.386286	+	0.922379i	$0.373758\pi$
$74$	−54388.0	−1.15458
$75$	−72156.0	−1.48122
$76$	34240.0	0.679986
$77$	0	0
$78$	69216.0	1.28816
$79$	42490.0	0.765983	0.382991	−	0.923752i	$-0.374894\pi$
$79$	42490.0	0.765983	0.382991	+	0.923752i	$0.374894\pi$
$80$	20736.0	0.362243
$81$	−67959.0	−1.15089
$82$	−25968.0	−0.426485
$83$	−22674.0	−0.361271	−0.180635	−	0.983550i	$-0.557815\pi$
$83$	−22674.0	−0.361271	−0.180635	+	0.983550i	$0.557815\pi$
$84$	32928.0	0.509175
$85$	−79218.0	−1.18926
$86$	−56936.0	−0.830120
$87$	73080.0	1.03514
$88$	0	0
$89$	−17145.0	−0.229436	−0.114718	−	0.993398i	$-0.536597\pi$
$89$	−17145.0	−0.229436	−0.114718	+	0.993398i	$0.536597\pi$
$90$	64152.0	0.834841
$91$	80752.0	1.02223
$92$	59184.0	0.729012
$93$	164073.	1.96711
$94$	−81408.0	−0.950271
$95$	173340.	1.97056
$96$	−21504.0	−0.238145
$97$	−30727.0	−0.331582	−0.165791	−	0.986161i	$-0.553018\pi$
$97$	−30727.0	−0.331582	−0.165791	+	0.986161i	$0.553018\pi$
$98$	−28812.0	−0.303046
$99$	0	0
$100$	54976.0	0.549760
$101$	−138102.	−1.34709	−0.673545	−	0.739146i	$-0.735229\pi$
$101$	−138102.	−1.34709	−0.673545	+	0.739146i	$0.735229\pi$
$102$	82152.0	0.781840
$103$	11864.0	0.110189	0.0550945	−	0.998481i	$-0.482454\pi$
$103$	11864.0	0.110189	0.0550945	+	0.998481i	$0.482454\pi$
$104$	−52736.0	−0.478106
$105$	166698.	1.47556
$106$	−1464.00	−0.0126554
$107$	−16998.0	−0.143529	−0.0717643	−	0.997422i	$-0.522863\pi$
$107$	−16998.0	−0.143529	−0.0717643	+	0.997422i	$0.522863\pi$
$108$	15120.0	0.124736
$109$	221830.	1.78836	0.894178	−	0.447711i	$-0.147761\pi$
$109$	221830.	1.78836	0.894178	+	0.447711i	$0.147761\pi$
$110$	0	0
$111$	285537.	2.19966
$112$	−25088.0	−0.188982
$113$	−196671.	−1.44892	−0.724460	−	0.689317i	$-0.757911\pi$
$113$	−196671.	−1.44892	−0.724460	+	0.689317i	$0.757911\pi$
$114$	−179760.	−1.29548
$115$	299619.	2.11264
$116$	−55680.0	−0.384197
$117$	−163152.	−1.10186
$118$	39300.0	0.259829
$119$	95844.0	0.620437
$120$	−108864.	−0.690130
$121$	0	0
$122$	−104528.	−0.635818
$123$	136332.	0.812522
$124$	−125008.	−0.730102
$125$	25191.0	0.144202
$126$	−77616.0	−0.435537
$127$	−120548.	−0.663209	−0.331605	−	0.943418i	$-0.607590\pi$
$127$	−120548.	−0.663209	−0.331605	+	0.943418i	$0.607590\pi$
$128$	16384.0	0.0883883
$129$	298914.	1.58151
$130$	−266976.	−1.38552
$131$	−68442.0	−0.348453	−0.174227	−	0.984706i	$-0.555743\pi$
$131$	−68442.0	−0.348453	−0.174227	+	0.984706i	$0.555743\pi$
$132$	0	0
$133$	−209720.	−1.02804
$134$	68372.0	0.328940
$135$	76545.0	0.361478
$136$	−62592.0	−0.290183
$137$	−373647.	−1.70083	−0.850413	−	0.526115i	$-0.823648\pi$
$137$	−373647.	−1.70083	−0.850413	+	0.526115i	$0.823648\pi$
$138$	−310716.	−1.38888
$139$	60610.0	0.266077	0.133038	−	0.991111i	$-0.457527\pi$
$139$	60610.0	0.266077	0.133038	+	0.991111i	$0.457527\pi$
$140$	−127008.	−0.547660
$141$	427392.	1.81042
$142$	−94332.0	−0.392589
$143$	0	0
$144$	50688.0	0.203704
$145$	−281880.	−1.11338
$146$	140704.	0.546291
$147$	151263.	0.577350
$148$	−217552.	−0.816411
$149$	438030.	1.61636	0.808180	−	0.588935i	$-0.200453\pi$
$149$	438030.	1.61636	0.808180	+	0.588935i	$0.200453\pi$
$150$	−288624.	−1.04738
$151$	239398.	0.854433	0.427217	−	0.904149i	$-0.359494\pi$
$151$	239398.	0.854433	0.427217	+	0.904149i	$0.359494\pi$
$152$	136960.	0.480822
$153$	−193644.	−0.668768
$154$	0	0
$155$	−632853.	−2.11580
$156$	276864.	0.910867
$157$	62153.0	0.201239	0.100620	−	0.994925i	$-0.467917\pi$
$157$	62153.0	0.201239	0.100620	+	0.994925i	$0.467917\pi$
$158$	169960.	0.541632
$159$	7686.00	0.0241106
$160$	82944.0	0.256144
$161$	−362502.	−1.10216
$162$	−271836.	−0.813803
$163$	298724.	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$163$	298724.	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$164$	−103872.	−0.301571
$165$	0	0
$166$	−90696.0	−0.255457
$167$	−82728.0	−0.229542	−0.114771	−	0.993392i	$-0.536613\pi$
$167$	−82728.0	−0.229542	−0.114771	+	0.993392i	$0.536613\pi$
$168$	131712.	0.360041
$169$	307683.	0.828680
$170$	−316872.	−0.840933
$171$	423720.	1.10812
$172$	−227744.	−0.586983
$173$	−135834.	−0.345059	−0.172529	−	0.985004i	$-0.555194\pi$
$173$	−135834.	−0.345059	−0.172529	+	0.985004i	$0.555194\pi$
$174$	292320.	0.731957
$175$	−336728.	−0.831159
$176$	0	0
$177$	−206325.	−0.495015
$178$	−68580.0	−0.162236
$179$	112725.	0.262959	0.131479	−	0.991319i	$-0.458027\pi$
$179$	112725.	0.262959	0.131479	+	0.991319i	$0.458027\pi$
$180$	256608.	0.590322
$181$	593807.	1.34725	0.673626	−	0.739072i	$-0.264736\pi$
$181$	593807.	1.34725	0.673626	+	0.739072i	$0.264736\pi$
$182$	323008.	0.722828
$183$	548772.	1.21133
$184$	236736.	0.515489
$185$	−1.10136e6	−2.36591
$186$	656292.	1.39096
$187$	0	0
$188$	−325632.	−0.671943
$189$	−92610.0	−0.188583
$190$	693360.	1.39340
$191$	652557.	1.29430	0.647150	−	0.762363i	$-0.275961\pi$
$191$	652557.	1.29430	0.647150	+	0.762363i	$0.275961\pi$
$192$	−86016.0	−0.168394
$193$	−402164.	−0.777159	−0.388580	−	0.921415i	$-0.627034\pi$
$193$	−402164.	−0.777159	−0.388580	+	0.921415i	$0.627034\pi$
$194$	−122908.	−0.234464
$195$	1.40162e6	2.63964
$196$	−115248.	−0.214286
$197$	268482.	0.492890	0.246445	−	0.969157i	$-0.420738\pi$
$197$	268482.	0.492890	0.246445	+	0.969157i	$0.420738\pi$
$198$	0	0
$199$	−581200.	−1.04038	−0.520191	−	0.854050i	$-0.674139\pi$
$199$	−581200.	−1.04038	−0.520191	+	0.854050i	$0.674139\pi$
$200$	219904.	0.388739
$201$	−358953.	−0.626682
$202$	−552408.	−0.952536
$203$	341040.	0.580852
$204$	328608.	0.552844
$205$	−525852.	−0.873934
$206$	47456.0	0.0779154
$207$	732402.	1.18802
$208$	−210944.	−0.338072
$209$	0	0
$210$	666792.	1.04338
$211$	183988.	0.284501	0.142250	−	0.989831i	$-0.454566\pi$
$211$	183988.	0.284501	0.142250	+	0.989831i	$0.454566\pi$
$212$	−5856.00	−0.00894873
$213$	495243.	0.747944
$214$	−67992.0	−0.101490
$215$	−1.15295e6	−1.70105
$216$	60480.0	0.0882018
$217$	765674.	1.10381
$218$	887320.	1.26456
$219$	−738696.	−1.04077
$220$	0	0
$221$	805872.	1.10990
$222$	1.14215e6	1.55539
$223$	631679.	0.850617	0.425309	−	0.905048i	$-0.360166\pi$
$223$	631679.	0.850617	0.425309	+	0.905048i	$0.360166\pi$
$224$	−100352.	−0.133631
$225$	680328.	0.895905
$226$	−786684.	−1.02454
$227$	−1398.00	−0.00180070	−0.000900352	−	1.00000i	$-0.500287\pi$
$227$	−1398.00	−0.00180070	−0.000900352		1.00000i	$0.500287\pi$
$228$	−719040.	−0.916043
$229$	206135.	0.259754	0.129877	−	0.991530i	$-0.458542\pi$
$229$	206135.	0.259754	0.129877	+	0.991530i	$0.458542\pi$
$230$	1.19848e6	1.49386
$231$	0	0
$232$	−222720.	−0.271668
$233$	−640704.	−0.773157	−0.386578	−	0.922257i	$-0.626343\pi$
$233$	−640704.	−0.773157	−0.386578	+	0.922257i	$0.626343\pi$
$234$	−652608.	−0.779135
$235$	−1.64851e6	−1.94725
$236$	157200.	0.183727
$237$	−892290.	−1.03189
$238$	383376.	0.438715
$239$	−128250.	−0.145232	−0.0726161	−	0.997360i	$-0.523135\pi$
$239$	−128250.	−0.145232	−0.0726161	+	0.997360i	$0.523135\pi$
$240$	−435456.	−0.487996
$241$	1.69749e6	1.88263	0.941313	−	0.337535i	$-0.109593\pi$
$241$	1.69749e6	1.88263	0.941313	+	0.337535i	$0.109593\pi$
$242$	0	0
$243$	1.19750e6	1.30095
$244$	−418112.	−0.449591
$245$	−583443.	−0.620988
$246$	545328.	0.574540
$247$	−1.76336e6	−1.83907
$248$	−500032.	−0.516260
$249$	476154.	0.486686
$250$	100764.	0.101966
$251$	−325323.	−0.325935	−0.162967	−	0.986631i	$-0.552107\pi$
$251$	−325323.	−0.325935	−0.162967	+	0.986631i	$0.552107\pi$
$252$	−310464.	−0.307971
$253$	0	0
$254$	−482192.	−0.468960
$255$	1.66358e6	1.60211
$256$	65536.0	0.0625000
$257$	1.64948e6	1.55781	0.778904	−	0.627144i	$-0.215776\pi$
$257$	1.64948e6	1.55781	0.778904	+	0.627144i	$0.215776\pi$
$258$	1.19566e6	1.11830
$259$	1.33251e6	1.23430
$260$	−1.06790e6	−0.979713
$261$	−689040.	−0.626099
$262$	−273768.	−0.246394
$263$	1.37653e6	1.22714	0.613571	−	0.789639i	$-0.289732\pi$
$263$	1.37653e6	1.22714	0.613571	+	0.789639i	$0.289732\pi$
$264$	0	0
$265$	−29646.0	−0.0259329
$266$	−838880.	−0.726935
$267$	360045.	0.309085
$268$	273488.	0.232595
$269$	75450.0	0.0635739	0.0317869	−	0.999495i	$-0.489880\pi$
$269$	75450.0	0.0635739	0.0317869	+	0.999495i	$0.489880\pi$
$270$	306180.	0.255604
$271$	360568.	0.298239	0.149119	−	0.988819i	$-0.452356\pi$
$271$	360568.	0.298239	0.149119	+	0.988819i	$0.452356\pi$
$272$	−250368.	−0.205190
$273$	−1.69579e6	−1.37710
$274$	−1.49459e6	−1.20267
$275$	0	0
$276$	−1.24286e6	−0.982089
$277$	418522.	0.327732	0.163866	−	0.986483i	$-0.447604\pi$
$277$	418522.	0.327732	0.163866	+	0.986483i	$0.447604\pi$
$278$	242440.	0.188145
$279$	−1.54697e6	−1.18980
$280$	−508032.	−0.387254
$281$	794298.	0.600092	0.300046	−	0.953925i	$-0.402998\pi$
$281$	794298.	0.600092	0.300046	+	0.953925i	$0.402998\pi$
$282$	1.70957e6	1.28016
$283$	−1.80796e6	−1.34191	−0.670955	−	0.741498i	$-0.734116\pi$
$283$	−1.80796e6	−1.34191	−0.670955	+	0.741498i	$0.734116\pi$
$284$	−377328.	−0.277602
$285$	−3.64014e6	−2.65464
$286$	0	0
$287$	636216.	0.455932
$288$	202752.	0.144040
$289$	−463373.	−0.326352
$290$	−1.12752e6	−0.787280
$291$	645267.	0.446691
$292$	562816.	0.386286
$293$	875436.	0.595738	0.297869	−	0.954607i	$-0.403724\pi$
$293$	875436.	0.595738	0.297869	+	0.954607i	$0.403724\pi$
$294$	605052.	0.408248
$295$	795825.	0.532430
$296$	−870208.	−0.577290
$297$	0	0
$298$	1.75212e6	1.14294
$299$	−3.04798e6	−1.97167
$300$	−1.15450e6	−0.740610
$301$	1.39493e6	0.887436
$302$	957592.	0.604176
$303$	2.90014e6	1.81473
$304$	547840.	0.339993
$305$	−2.11669e6	−1.30289
$306$	−774576.	−0.472890
$307$	−137468.	−0.0832445	−0.0416223	−	0.999133i	$-0.513253\pi$
$307$	−137468.	−0.0832445	−0.0416223	+	0.999133i	$0.513253\pi$
$308$	0	0
$309$	−249144.	−0.148441
$310$	−2.53141e6	−1.49609
$311$	1.22629e6	0.718940	0.359470	−	0.933157i	$-0.382957\pi$
$311$	1.22629e6	0.718940	0.359470	+	0.933157i	$0.382957\pi$
$312$	1.10746e6	0.644080
$313$	−3.13692e6	−1.80985	−0.904925	−	0.425570i	$-0.860073\pi$
$313$	−3.13692e6	−1.80985	−0.904925	+	0.425570i	$0.860073\pi$
$314$	248612.	0.142298
$315$	−1.57172e6	−0.892483
$316$	679840.	0.382991
$317$	−1.69119e6	−0.945243	−0.472622	−	0.881266i	$-0.656692\pi$
$317$	−1.69119e6	−0.945243	−0.472622	+	0.881266i	$0.656692\pi$
$318$	30744.0	0.0170488
$319$	0	0
$320$	331776.	0.181122
$321$	356958.	0.193355
$322$	−1.45001e6	−0.779347
$323$	−2.09292e6	−1.11621
$324$	−1.08734e6	−0.575446
$325$	−2.83126e6	−1.48687
$326$	1.19490e6	0.622710
$327$	−4.65843e6	−2.40919
$328$	−415488.	−0.213243
$329$	1.99450e6	1.01588
$330$	0	0
$331$	−2.13901e6	−1.07311	−0.536554	−	0.843866i	$-0.680274\pi$
$331$	−2.13901e6	−1.07311	−0.536554	+	0.843866i	$0.680274\pi$
$332$	−362784.	−0.180635
$333$	−2.69221e6	−1.33045
$334$	−330912.	−0.162310
$335$	1.38453e6	0.674049
$336$	526848.	0.254588
$337$	−553598.	−0.265534	−0.132767	−	0.991147i	$-0.542386\pi$
$337$	−553598.	−0.265534	−0.132767	+	0.991147i	$0.542386\pi$
$338$	1.23073e6	0.585965
$339$	4.13009e6	1.95191
$340$	−1.26749e6	−0.594630
$341$	0	0
$342$	1.69488e6	0.783563
$343$	2.35298e6	1.07990
$344$	−910976.	−0.415060
$345$	−6.29200e6	−2.84604
$346$	−543336.	−0.243993
$347$	−167208.	−0.0745475	−0.0372738	−	0.999305i	$-0.511867\pi$
$347$	−167208.	−0.0745475	−0.0372738	+	0.999305i	$0.511867\pi$
$348$	1.16928e6	0.517572
$349$	−469490.	−0.206330	−0.103165	−	0.994664i	$-0.532897\pi$
$349$	−469490.	−0.206330	−0.103165	+	0.994664i	$0.532897\pi$
$350$	−1.34691e6	−0.587718
$351$	−778680.	−0.337358
$352$	0	0
$353$	2.82154e6	1.20517	0.602586	−	0.798054i	$-0.294137\pi$
$353$	2.82154e6	1.20517	0.602586	+	0.798054i	$0.294137\pi$
$354$	−825300.	−0.350029
$355$	−1.91022e6	−0.804476
$356$	−274320.	−0.114718
$357$	−2.01272e6	−0.835822
$358$	450900.	0.185940
$359$	−1.95696e6	−0.801394	−0.400697	−	0.916211i	$-0.631232\pi$
$359$	−1.95696e6	−0.801394	−0.400697	+	0.916211i	$0.631232\pi$
$360$	1.02643e6	0.417421
$361$	2.10350e6	0.849522
$362$	2.37523e6	0.952651
$363$	0	0
$364$	1.29203e6	0.511116
$365$	2.84926e6	1.11944
$366$	2.19509e6	0.856543
$367$	−2.28592e6	−0.885922	−0.442961	−	0.896541i	$-0.646072\pi$
$367$	−2.28592e6	−0.885922	−0.442961	+	0.896541i	$0.646072\pi$
$368$	946944.	0.364506
$369$	−1.28542e6	−0.491448
$370$	−4.40543e6	−1.67295
$371$	35868.0	0.0135292
$372$	2.62517e6	0.983557
$373$	−583274.	−0.217070	−0.108535	−	0.994093i	$-0.534616\pi$
$373$	−583274.	−0.217070	−0.108535	+	0.994093i	$0.534616\pi$
$374$	0	0
$375$	−529011.	−0.194261
$376$	−1.30253e6	−0.475135
$377$	2.86752e6	1.03909
$378$	−370440.	−0.133349
$379$	2.83629e6	1.01427	0.507135	−	0.861867i	$-0.330705\pi$
$379$	2.83629e6	1.01427	0.507135	+	0.861867i	$0.330705\pi$
$380$	2.77344e6	0.985280
$381$	2.53151e6	0.893443
$382$	2.61023e6	0.915208
$383$	−3.52202e6	−1.22686	−0.613430	−	0.789749i	$-0.710211\pi$
$383$	−3.52202e6	−1.22686	−0.613430	+	0.789749i	$0.710211\pi$
$384$	−344064.	−0.119072
$385$	0	0
$386$	−1.60866e6	−0.549534
$387$	−2.81833e6	−0.956566
$388$	−491632.	−0.165791
$389$	−1.81358e6	−0.607661	−0.303831	−	0.952726i	$-0.598266\pi$
$389$	−1.81358e6	−0.607661	−0.303831	+	0.952726i	$0.598266\pi$
$390$	5.60650e6	1.86651
$391$	−3.61762e6	−1.19669
$392$	−460992.	−0.151523
$393$	1.43728e6	0.469419
$394$	1.07393e6	0.348526
$395$	3.44169e6	1.10989
$396$	0	0
$397$	3.42076e6	1.08930	0.544648	−	0.838665i	$-0.316663\pi$
$397$	3.42076e6	1.08930	0.544648	+	0.838665i	$0.316663\pi$
$398$	−2.32480e6	−0.735661
$399$	4.40412e6	1.38493
$400$	879616.	0.274880
$401$	398442.	0.123738	0.0618692	−	0.998084i	$-0.480294\pi$
$401$	398442.	0.123738	0.0618692	+	0.998084i	$0.480294\pi$
$402$	−1.43581e6	−0.443131
$403$	6.43791e6	1.97461
$404$	−2.20963e6	−0.673545
$405$	−5.50468e6	−1.66761
$406$	1.36416e6	0.410724
$407$	0	0
$408$	1.31443e6	0.390920
$409$	−1.40957e6	−0.416657	−0.208328	−	0.978059i	$-0.566802\pi$
$409$	−1.40957e6	−0.416657	−0.208328	+	0.978059i	$0.566802\pi$
$410$	−2.10341e6	−0.617965
$411$	7.84659e6	2.29127
$412$	189824.	0.0550945
$413$	−962850.	−0.277769
$414$	2.92961e6	0.840057
$415$	−1.83659e6	−0.523471
$416$	−843776.	−0.239053
$417$	−1.27281e6	−0.358446
$418$	0	0
$419$	−1.86618e6	−0.519300	−0.259650	−	0.965703i	$-0.583607\pi$
$419$	−1.86618e6	−0.519300	−0.259650	+	0.965703i	$0.583607\pi$
$420$	2.66717e6	0.737780
$421$	2.07774e6	0.571329	0.285665	−	0.958330i	$-0.407786\pi$
$421$	2.07774e6	0.571329	0.285665	+	0.958330i	$0.407786\pi$
$422$	735952.	0.201172
$423$	−4.02970e6	−1.09502
$424$	−23424.0	−0.00632771
$425$	−3.36041e6	−0.902443
$426$	1.98097e6	0.528877
$427$	2.56094e6	0.679718
$428$	−271968.	−0.0717643
$429$	0	0
$430$	−4.61182e6	−1.20282
$431$	6.28436e6	1.62955	0.814775	−	0.579777i	$-0.196860\pi$
$431$	6.28436e6	1.62955	0.814775	+	0.579777i	$0.196860\pi$
$432$	241920.	0.0623681
$433$	3.26559e6	0.837031	0.418516	−	0.908210i	$-0.362550\pi$
$433$	3.26559e6	0.837031	0.418516	+	0.908210i	$0.362550\pi$
$434$	3.06270e6	0.780512
$435$	5.91948e6	1.49989
$436$	3.54928e6	0.894178
$437$	7.91586e6	1.98287
$438$	−2.95478e6	−0.735937
$439$	2.64568e6	0.655203	0.327602	−	0.944816i	$-0.393760\pi$
$439$	2.64568e6	0.655203	0.327602	+	0.944816i	$0.393760\pi$
$440$	0	0
$441$	−1.42619e6	−0.349206
$442$	3.22349e6	0.784821
$443$	−5.63531e6	−1.36430	−0.682148	−	0.731214i	$-0.738954\pi$
$443$	−5.63531e6	−1.36430	−0.682148	+	0.731214i	$0.738954\pi$
$444$	4.56859e6	1.09983
$445$	−1.38874e6	−0.332447
$446$	2.52672e6	0.601477
$447$	−9.19863e6	−2.17748
$448$	−401408.	−0.0944911
$449$	−370005.	−0.0866147	−0.0433074	−	0.999062i	$-0.513789\pi$
$449$	−370005.	−0.0866147	−0.0433074	+	0.999062i	$0.513789\pi$
$450$	2.72131e6	0.633501
$451$	0	0
$452$	−3.14674e6	−0.724460
$453$	−5.02736e6	−1.15105
$454$	−5592.00	−0.00127329
$455$	6.54091e6	1.48119
$456$	−2.87616e6	−0.647740
$457$	−3.31891e6	−0.743369	−0.371685	−	0.928359i	$-0.621220\pi$
$457$	−3.31891e6	−0.743369	−0.371685	+	0.928359i	$0.621220\pi$
$458$	824540.	0.183674
$459$	−924210.	−0.204757
$460$	4.79390e6	1.05632
$461$	−8.77021e6	−1.92202	−0.961010	−	0.276515i	$-0.910821\pi$
$461$	−8.77021e6	−1.92202	−0.961010	+	0.276515i	$0.910821\pi$
$462$	0	0
$463$	224249.	0.0486159	0.0243079	−	0.999705i	$-0.492262\pi$
$463$	224249.	0.0486159	0.0243079	+	0.999705i	$0.492262\pi$
$464$	−890880.	−0.192099
$465$	1.32899e7	2.85029
$466$	−2.56282e6	−0.546704
$467$	−2.55573e6	−0.542278	−0.271139	−	0.962540i	$-0.587400\pi$
$467$	−2.55573e6	−0.542278	−0.271139	+	0.962540i	$0.587400\pi$
$468$	−2.61043e6	−0.550932
$469$	−1.67511e6	−0.351651
$470$	−6.59405e6	−1.37692
$471$	−1.30521e6	−0.271100
$472$	628800.	0.129914
$473$	0	0
$474$	−3.56916e6	−0.729659
$475$	7.35304e6	1.49532
$476$	1.53350e6	0.310218
$477$	−72468.0	−0.0145831
$478$	−513000.	−0.102695
$479$	−5.36664e6	−1.06872	−0.534360	−	0.845257i	$-0.679447\pi$
$479$	−5.36664e6	−1.06872	−0.534360	+	0.845257i	$0.679447\pi$
$480$	−1.74182e6	−0.345065
$481$	1.12039e7	2.20804
$482$	6.78995e6	1.33122
$483$	7.61254e6	1.48478
$484$	0	0
$485$	−2.48889e6	−0.480453
$486$	4.79002e6	0.919912
$487$	−8.17528e6	−1.56200	−0.780998	−	0.624533i	$-0.785289\pi$
$487$	−8.17528e6	−1.56200	−0.780998	+	0.624533i	$0.785289\pi$
$488$	−1.67245e6	−0.317909
$489$	−6.27320e6	−1.18636
$490$	−2.33377e6	−0.439105
$491$	−3.78007e6	−0.707614	−0.353807	−	0.935318i	$-0.615113\pi$
$491$	−3.78007e6	−0.707614	−0.353807	+	0.935318i	$0.615113\pi$
$492$	2.18131e6	0.406261
$493$	3.40344e6	0.630668
$494$	−7.05344e6	−1.30042
$495$	0	0
$496$	−2.00013e6	−0.365051
$497$	2.31113e6	0.419695
$498$	1.90462e6	0.344139
$499$	3.98186e6	0.715871	0.357935	−	0.933746i	$-0.383481\pi$
$499$	3.98186e6	0.715871	0.357935	+	0.933746i	$0.383481\pi$
$500$	403056.	0.0721008
$501$	1.73729e6	0.309227
$502$	−1.30129e6	−0.230471
$503$	−1.04569e7	−1.84281	−0.921406	−	0.388601i	$-0.872958\pi$
$503$	−1.04569e7	−1.84281	−0.921406	+	0.388601i	$0.872958\pi$
$504$	−1.24186e6	−0.217768
$505$	−1.11863e7	−1.95190
$506$	0	0
$507$	−6.46134e6	−1.11636
$508$	−1.92877e6	−0.331605
$509$	−6.05507e6	−1.03592	−0.517958	−	0.855406i	$-0.673308\pi$
$509$	−6.05507e6	−1.03592	−0.517958	+	0.855406i	$0.673308\pi$
$510$	6.65431e6	1.13286
$511$	−3.44725e6	−0.584010
$512$	262144.	0.0441942
$513$	2.02230e6	0.339275
$514$	6.59791e6	1.10154
$515$	960984.	0.159661
$516$	4.78262e6	0.790755
$517$	0	0
$518$	5.33002e6	0.872780
$519$	2.85251e6	0.464846
$520$	−4.27162e6	−0.692762
$521$	1.05053e7	1.69556	0.847780	−	0.530348i	$-0.177939\pi$
$521$	1.05053e7	1.69556	0.847780	+	0.530348i	$0.177939\pi$
$522$	−2.75616e6	−0.442719
$523$	−6.49492e6	−1.03829	−0.519146	−	0.854685i	$-0.673750\pi$
$523$	−6.49492e6	−1.03829	−0.519146	+	0.854685i	$0.673750\pi$
$524$	−1.09507e6	−0.174227
$525$	7.07129e6	1.11970
$526$	5.50610e6	0.867721
$527$	7.64111e6	1.19848
$528$	0	0
$529$	7.24626e6	1.12583
$530$	−118584.	−0.0183373
$531$	1.94535e6	0.299407
$532$	−3.35552e6	−0.514021
$533$	5.34941e6	0.815620
$534$	1.44018e6	0.218556
$535$	−1.37684e6	−0.207969
$536$	1.09395e6	0.164470
$537$	−2.36722e6	−0.354245
$538$	301800.	0.0449535
$539$	0	0
$540$	1.22472e6	0.180739
$541$	−5.66349e6	−0.831938	−0.415969	−	0.909379i	$-0.636558\pi$
$541$	−5.66349e6	−0.831938	−0.415969	+	0.909379i	$0.636558\pi$
$542$	1.44227e6	0.210887
$543$	−1.24699e7	−1.81495
$544$	−1.00147e6	−0.145091
$545$	1.79682e7	2.59128
$546$	−6.78317e6	−0.973758
$547$	1.33609e7	1.90927	0.954636	−	0.297775i	$-0.0962446\pi$
$547$	1.33609e7	1.90927	0.954636	+	0.297775i	$0.0962446\pi$
$548$	−5.97835e6	−0.850413
$549$	−5.17414e6	−0.732668
$550$	0	0
$551$	−7.44720e6	−1.04499
$552$	−4.97146e6	−0.694442
$553$	−4.16402e6	−0.579029
$554$	1.67409e6	0.231742
$555$	2.31285e7	3.18724
$556$	969760.	0.133038
$557$	1.00947e7	1.37866	0.689330	−	0.724447i	$-0.257905\pi$
$557$	1.00947e7	1.37866	0.689330	+	0.724447i	$0.257905\pi$
$558$	−6.18790e6	−0.841313
$559$	1.17288e7	1.58754
$560$	−2.03213e6	−0.273830
$561$	0	0
$562$	3.17719e6	0.424329
$563$	−5.58692e6	−0.742851	−0.371426	−	0.928463i	$-0.621131\pi$
$563$	−5.58692e6	−0.742851	−0.371426	+	0.928463i	$0.621131\pi$
$564$	6.83827e6	0.905208
$565$	−1.59304e7	−2.09944
$566$	−7.23186e6	−0.948874
$567$	6.65998e6	0.869992
$568$	−1.50931e6	−0.196295
$569$	−2.20884e6	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$569$	−2.20884e6	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$570$	−1.45606e7	−1.87712
$571$	1.05324e7	1.35188	0.675940	−	0.736957i	$-0.263738\pi$
$571$	1.05324e7	1.35188	0.675940	+	0.736957i	$0.263738\pi$
$572$	0	0
$573$	−1.37037e7	−1.74362
$574$	2.54486e6	0.322392
$575$	1.27098e7	1.60313
$576$	811008.	0.101852
$577$	1.84319e6	0.230479	0.115239	−	0.993338i	$-0.463236\pi$
$577$	1.84319e6	0.230479	0.115239	+	0.993338i	$0.463236\pi$
$578$	−1.85349e6	−0.230766
$579$	8.44544e6	1.04695
$580$	−4.51008e6	−0.556691
$581$	2.22205e6	0.273095
$582$	2.58107e6	0.315858
$583$	0	0
$584$	2.25126e6	0.273146
$585$	−1.32153e7	−1.59657
$586$	3.50174e6	0.421250
$587$	1.16959e7	1.40100	0.700501	−	0.713652i	$-0.252960\pi$
$587$	1.16959e7	1.40100	0.700501	+	0.713652i	$0.252960\pi$
$588$	2.42021e6	0.288675
$589$	−1.67198e7	−1.98584
$590$	3.18330e6	0.376485
$591$	−5.63812e6	−0.663996
$592$	−3.48083e6	−0.408205
$593$	−5.81252e6	−0.678778	−0.339389	−	0.940646i	$-0.610220\pi$
$593$	−5.81252e6	−0.678778	−0.339389	+	0.940646i	$0.610220\pi$
$594$	0	0
$595$	7.76336e6	0.898996
$596$	7.00848e6	0.808180
$597$	1.22052e7	1.40155
$598$	−1.21919e7	−1.39418
$599$	7.85604e6	0.894616	0.447308	−	0.894380i	$-0.352383\pi$
$599$	7.85604e6	0.894616	0.447308	+	0.894380i	$0.352383\pi$
$600$	−4.61798e6	−0.523690
$601$	−1.09429e7	−1.23579	−0.617895	−	0.786261i	$-0.712014\pi$
$601$	−1.09429e7	−1.23579	−0.617895	+	0.786261i	$0.712014\pi$
$602$	5.57973e6	0.627512
$603$	3.38441e6	0.379045
$604$	3.83037e6	0.427217
$605$	0	0
$606$	1.16006e7	1.28321
$607$	−185018.	−0.0203818	−0.0101909	−	0.999948i	$-0.503244\pi$
$607$	−185018.	−0.0203818	−0.0101909	+	0.999948i	$0.503244\pi$
$608$	2.19136e6	0.240411
$609$	−7.16184e6	−0.782495
$610$	−8.46677e6	−0.921283
$611$	1.67700e7	1.81732
$612$	−3.09830e6	−0.334384
$613$	−1.77449e7	−1.90731	−0.953655	−	0.300901i	$-0.902713\pi$
$613$	−1.77449e7	−1.90731	−0.953655	+	0.300901i	$0.902713\pi$
$614$	−549872.	−0.0588628
$615$	1.10429e7	1.17732
$616$	0	0
$617$	1.53912e7	1.62765	0.813823	−	0.581113i	$-0.197382\pi$
$617$	1.53912e7	1.62765	0.813823	+	0.581113i	$0.197382\pi$
$618$	−996576.	−0.104964
$619$	−1.75502e7	−1.84101	−0.920504	−	0.390734i	$-0.872221\pi$
$619$	−1.75502e7	−1.84101	−0.920504	+	0.390734i	$0.872221\pi$
$620$	−1.01256e7	−1.05790
$621$	3.49556e6	0.363737
$622$	4.90517e6	0.508368
$623$	1.68021e6	0.173438
$624$	4.42982e6	0.455434
$625$	−8.69703e6	−0.890576
$626$	−1.25477e7	−1.27976
$627$	0	0
$628$	994448.	0.100620
$629$	1.32979e7	1.34016
$630$	−6.28690e6	−0.631081
$631$	−5.64613e6	−0.564518	−0.282259	−	0.959338i	$-0.591084\pi$
$631$	−5.64613e6	−0.564518	−0.282259	+	0.959338i	$0.591084\pi$
$632$	2.71936e6	0.270816
$633$	−3.86375e6	−0.383265
$634$	−6.76475e6	−0.668388
$635$	−9.76439e6	−0.960972
$636$	122976.	0.0120553
$637$	5.93527e6	0.579552
$638$	0	0
$639$	−4.66943e6	−0.452389
$640$	1.32710e6	0.128072
$641$	−4.46307e6	−0.429031	−0.214516	−	0.976721i	$-0.568817\pi$
$641$	−4.46307e6	−0.429031	−0.214516	+	0.976721i	$0.568817\pi$
$642$	1.42783e6	0.136722
$643$	−3.24099e6	−0.309137	−0.154568	−	0.987982i	$-0.549399\pi$
$643$	−3.24099e6	−0.309137	−0.154568	+	0.987982i	$0.549399\pi$
$644$	−5.80003e6	−0.551081
$645$	2.42120e7	2.29156
$646$	−8.37168e6	−0.789280
$647$	1.01885e7	0.956858	0.478429	−	0.878126i	$-0.341206\pi$
$647$	1.01885e7	0.956858	0.478429	+	0.878126i	$0.341206\pi$
$648$	−4.34938e6	−0.406902
$649$	0	0
$650$	−1.13251e7	−1.05137
$651$	−1.60792e7	−1.48700
$652$	4.77958e6	0.440323
$653$	−4.66760e6	−0.428362	−0.214181	−	0.976794i	$-0.568708\pi$
$653$	−4.66760e6	−0.428362	−0.214181	+	0.976794i	$0.568708\pi$
$654$	−1.86337e7	−1.70355
$655$	−5.54380e6	−0.504899
$656$	−1.66195e6	−0.150785
$657$	6.96485e6	0.629504
$658$	7.97798e6	0.718337
$659$	−1.29177e6	−0.115870	−0.0579351	−	0.998320i	$-0.518452\pi$
$659$	−1.29177e6	−0.115870	−0.0579351	+	0.998320i	$0.518452\pi$
$660$	0	0
$661$	−1.15658e7	−1.02960	−0.514802	−	0.857309i	$-0.672135\pi$
$661$	−1.15658e7	−1.02960	−0.514802	+	0.857309i	$0.672135\pi$
$662$	−8.55605e6	−0.758802
$663$	−1.69233e7	−1.49521
$664$	−1.45114e6	−0.127729
$665$	−1.69873e7	−1.48960
$666$	−1.07688e7	−0.940768
$667$	−1.28725e7	−1.12034
$668$	−1.32365e6	−0.114771
$669$	−1.32653e7	−1.14591
$670$	5.53813e6	0.476624
$671$	0	0
$672$	2.10739e6	0.180021
$673$	−1.47706e7	−1.25707	−0.628535	−	0.777781i	$-0.716345\pi$
$673$	−1.47706e7	−1.25707	−0.628535	+	0.777781i	$0.716345\pi$
$674$	−2.21439e6	−0.187761
$675$	3.24702e6	0.274300
$676$	4.92293e6	0.414340
$677$	3.09022e6	0.259130	0.129565	−	0.991571i	$-0.458642\pi$
$677$	3.09022e6	0.259130	0.129565	+	0.991571i	$0.458642\pi$
$678$	1.65204e7	1.38021
$679$	3.01125e6	0.250652
$680$	−5.06995e6	−0.420467
$681$	29358.0	0.00242582
$682$	0	0
$683$	1.47394e7	1.20900	0.604502	−	0.796604i	$-0.293372\pi$
$683$	1.47394e7	1.20900	0.604502	+	0.796604i	$0.293372\pi$
$684$	6.77952e6	0.554062
$685$	−3.02654e7	−2.46445
$686$	9.41192e6	0.763604
$687$	−4.32884e6	−0.349928
$688$	−3.64390e6	−0.293492
$689$	301584.	0.0242025
$690$	−2.51680e7	−2.01245
$691$	2.36276e6	0.188245	0.0941226	−	0.995561i	$-0.469995\pi$
$691$	2.36276e6	0.188245	0.0941226	+	0.995561i	$0.469995\pi$
$692$	−2.17334e6	−0.172529
$693$	0	0
$694$	−668832.	−0.0527131
$695$	4.90941e6	0.385538
$696$	4.67712e6	0.365978
$697$	6.34918e6	0.495034
$698$	−1.87796e6	−0.145897
$699$	1.34548e7	1.04156
$700$	−5.38765e6	−0.415579
$701$	1.78888e7	1.37495	0.687473	−	0.726210i	$-0.258720\pi$
$701$	1.78888e7	1.37495	0.687473	+	0.726210i	$0.258720\pi$
$702$	−3.11472e6	−0.238548
$703$	−2.90976e7	−2.22059
$704$	0	0
$705$	3.46188e7	2.62324
$706$	1.12862e7	0.852186
$707$	1.35340e7	1.01830
$708$	−3.30120e6	−0.247508
$709$	1.22735e7	0.916962	0.458481	−	0.888704i	$-0.348394\pi$
$709$	1.22735e7	0.916962	0.458481	+	0.888704i	$0.348394\pi$
$710$	−7.64089e6	−0.568851
$711$	8.41302e6	0.624134
$712$	−1.09728e6	−0.0811180
$713$	−2.89003e7	−2.12901
$714$	−8.05090e6	−0.591015
$715$	0	0
$716$	1.80360e6	0.131479
$717$	2.69325e6	0.195650
$718$	−7.82784e6	−0.566671
$719$	−7.35232e6	−0.530399	−0.265199	−	0.964194i	$-0.585438\pi$
$719$	−7.35232e6	−0.530399	−0.265199	+	0.964194i	$0.585438\pi$
$720$	4.10573e6	0.295161
$721$	−1.16267e6	−0.0832950
$722$	8.41400e6	0.600703
$723$	−3.56472e7	−2.53618
$724$	9.50091e6	0.673626
$725$	−1.19573e7	−0.844865
$726$	0	0
$727$	−3.16762e6	−0.222278	−0.111139	−	0.993805i	$-0.535450\pi$
$727$	−3.16762e6	−0.222278	−0.111139	+	0.993805i	$0.535450\pi$
$728$	5.16813e6	0.361414
$729$	−8.63355e6	−0.601687
$730$	1.13970e7	0.791561
$731$	1.39209e7	0.963546
$732$	8.78035e6	0.605667
$733$	−857924.	−0.0589778	−0.0294889	−	0.999565i	$-0.509388\pi$
$733$	−857924.	−0.0589778	−0.0294889	+	0.999565i	$0.509388\pi$
$734$	−9.14367e6	−0.626441
$735$	1.22523e7	0.836564
$736$	3.78778e6	0.257745
$737$	0	0
$738$	−5.14166e6	−0.347506
$739$	1.93551e7	1.30372	0.651860	−	0.758339i	$-0.273989\pi$
$739$	1.93551e7	1.30372	0.651860	+	0.758339i	$0.273989\pi$
$740$	−1.76217e7	−1.18296
$741$	3.70306e7	2.47751
$742$	143472.	0.00956660
$743$	2.80305e7	1.86277	0.931383	−	0.364040i	$-0.118603\pi$
$743$	2.80305e7	1.86277	0.931383	+	0.364040i	$0.118603\pi$
$744$	1.05007e7	0.695480
$745$	3.54804e7	2.34206
$746$	−2.33310e6	−0.153492
$747$	−4.48945e6	−0.294369
$748$	0	0
$749$	1.66580e6	0.108497
$750$	−2.11604e6	−0.137364
$751$	2.57014e7	1.66287	0.831434	−	0.555624i	$-0.187521\pi$
$751$	2.57014e7	1.66287	0.831434	+	0.555624i	$0.187521\pi$
$752$	−5.21011e6	−0.335972
$753$	6.83178e6	0.439083
$754$	1.14701e7	0.734747
$755$	1.93912e7	1.23805
$756$	−1.48176e6	−0.0942917
$757$	−1.17223e7	−0.743489	−0.371745	−	0.928335i	$-0.621240\pi$
$757$	−1.17223e7	−0.743489	−0.371745	+	0.928335i	$0.621240\pi$
$758$	1.13452e7	0.717197
$759$	0	0
$760$	1.10938e7	0.696698
$761$	−5.77783e6	−0.361662	−0.180831	−	0.983514i	$-0.557879\pi$
$761$	−5.77783e6	−0.361662	−0.180831	+	0.983514i	$0.557879\pi$
$762$	1.01260e7	0.631760
$763$	−2.17393e7	−1.35187
$764$	1.04409e7	0.647150
$765$	−1.56852e7	−0.969026
$766$	−1.40881e7	−0.867521
$767$	−8.09580e6	−0.496903
$768$	−1.37626e6	−0.0841969
$769$	2.04989e7	1.25001	0.625007	−	0.780619i	$-0.285096\pi$
$769$	2.04989e7	1.25001	0.625007	+	0.780619i	$0.285096\pi$
$770$	0	0
$771$	−3.46390e7	−2.09860
$772$	−6.43462e6	−0.388580
$773$	1.50298e7	0.904697	0.452348	−	0.891841i	$-0.350586\pi$
$773$	1.50298e7	0.904697	0.452348	+	0.891841i	$0.350586\pi$
$774$	−1.12733e7	−0.676394
$775$	−2.68455e7	−1.60552
$776$	−1.96653e6	−0.117232
$777$	−2.79826e7	−1.66278
$778$	−7.25430e6	−0.429681
$779$	−1.38929e7	−0.820255
$780$	2.24260e7	1.31982
$781$	0	0
$782$	−1.44705e7	−0.846187
$783$	−3.28860e6	−0.191693
$784$	−1.84397e6	−0.107143
$785$	5.03439e6	0.291590
$786$	5.74913e6	0.331929
$787$	−2.05325e7	−1.18169	−0.590847	−	0.806784i	$-0.701206\pi$
$787$	−2.05325e7	−1.18169	−0.590847	+	0.806784i	$0.701206\pi$
$788$	4.29571e6	0.246445
$789$	−2.89070e7	−1.65315
$790$	1.37668e7	0.784809
$791$	1.92738e7	1.09528
$792$	0	0
$793$	2.15328e7	1.21595
$794$	1.36830e7	0.770249
$795$	622566.	0.0349355
$796$	−9.29920e6	−0.520191
$797$	−1.76247e6	−0.0982823	−0.0491411	−	0.998792i	$-0.515648\pi$
$797$	−1.76247e6	−0.0982823	−0.0491411	+	0.998792i	$0.515648\pi$
$798$	1.76165e7	0.979291
$799$	1.99043e7	1.10301
$800$	3.51846e6	0.194370
$801$	−3.39471e6	−0.186948
$802$	1.59377e6	0.0874962
$803$	0	0
$804$	−5.74325e6	−0.313341
$805$	−2.93627e7	−1.59700
$806$	2.57516e7	1.39626
$807$	−1.58445e6	−0.0856436
$808$	−8.83853e6	−0.476268
$809$	−2.00289e7	−1.07593	−0.537967	−	0.842966i	$-0.680808\pi$
$809$	−2.00289e7	−1.07593	−0.537967	+	0.842966i	$0.680808\pi$
$810$	−2.20187e7	−1.17918
$811$	2.55409e7	1.36359	0.681796	−	0.731542i	$-0.261199\pi$
$811$	2.55409e7	1.36359	0.681796	+	0.731542i	$0.261199\pi$
$812$	5.45664e6	0.290426
$813$	−7.57193e6	−0.401772
$814$	0	0
$815$	2.41966e7	1.27603
$816$	5.25773e6	0.276422
$817$	−3.04608e7	−1.59656
$818$	−5.63828e6	−0.294621
$819$	1.59889e7	0.832930
$820$	−8.41363e6	−0.436967
$821$	6.73166e6	0.348549	0.174275	−	0.984697i	$-0.444242\pi$
$821$	6.73166e6	0.348549	0.174275	+	0.984697i	$0.444242\pi$
$822$	3.13863e7	1.62017
$823$	1.44880e7	0.745606	0.372803	−	0.927911i	$-0.378397\pi$
$823$	1.44880e7	0.745606	0.372803	+	0.927911i	$0.378397\pi$
$824$	759296.	0.0389577
$825$	0	0
$826$	−3.85140e6	−0.196412
$827$	4.15879e6	0.211448	0.105724	−	0.994396i	$-0.466284\pi$
$827$	4.15879e6	0.211448	0.105724	+	0.994396i	$0.466284\pi$
$828$	1.17184e7	0.594010
$829$	1.02525e7	0.518136	0.259068	−	0.965859i	$-0.416585\pi$
$829$	1.02525e7	0.518136	0.259068	+	0.965859i	$0.416585\pi$
$830$	−7.34638e6	−0.370150
$831$	−8.78896e6	−0.441504
$832$	−3.37510e6	−0.169036
$833$	7.04453e6	0.351755
$834$	−5.09124e6	−0.253459
$835$	−6.70097e6	−0.332599
$836$	0	0
$837$	−7.38328e6	−0.364281
$838$	−7.46472e6	−0.367201
$839$	3.07705e7	1.50914	0.754571	−	0.656218i	$-0.227845\pi$
$839$	3.07705e7	1.50914	0.754571	+	0.656218i	$0.227845\pi$
$840$	1.06687e7	0.521690
$841$	−8.40075e6	−0.409570
$842$	8.31097e6	0.403991
$843$	−1.66803e7	−0.808414
$844$	2.94381e6	0.142250
$845$	2.49223e7	1.20073
$846$	−1.61188e7	−0.774295
$847$	0	0
$848$	−93696.0	−0.00447437
$849$	3.79672e7	1.80776
$850$	−1.34416e7	−0.638123
$851$	−5.02953e7	−2.38069
$852$	7.92389e6	0.373972
$853$	−4.19232e7	−1.97280	−0.986398	−	0.164377i	$-0.947439\pi$
$853$	−4.19232e7	−1.97280	−0.986398	+	0.164377i	$0.947439\pi$
$854$	1.02437e7	0.480634
$855$	3.43213e7	1.60564
$856$	−1.08787e6	−0.0507450
$857$	1.87686e7	0.872929	0.436464	−	0.899722i	$-0.356231\pi$
$857$	1.87686e7	0.872929	0.436464	+	0.899722i	$0.356231\pi$
$858$	0	0
$859$	−3.95520e7	−1.82888	−0.914441	−	0.404720i	$-0.867369\pi$
$859$	−3.95520e7	−1.82888	−0.914441	+	0.404720i	$0.867369\pi$
$860$	−1.84473e7	−0.850523
$861$	−1.33605e7	−0.614209
$862$	2.51374e7	1.15227
$863$	1.50569e7	0.688191	0.344095	−	0.938935i	$-0.388186\pi$
$863$	1.50569e7	0.688191	0.344095	+	0.938935i	$0.388186\pi$
$864$	967680.	0.0441009
$865$	−1.10026e7	−0.499981
$866$	1.30624e7	0.591871
$867$	9.73083e6	0.439645
$868$	1.22508e7	0.551905
$869$	0	0
$870$	2.36779e7	1.06058
$871$	−1.40846e7	−0.629072
$872$	1.41971e7	0.632279
$873$	−6.08395e6	−0.270178
$874$	3.16634e7	1.40210
$875$	−2.46872e6	−0.109006
$876$	−1.18191e7	−0.520386
$877$	−1.58591e7	−0.696272	−0.348136	−	0.937444i	$-0.613185\pi$
$877$	−1.58591e7	−0.696272	−0.348136	+	0.937444i	$0.613185\pi$
$878$	1.05827e7	0.463299
$879$	−1.83842e7	−0.802549
$880$	0	0
$881$	−2.29142e7	−0.994640	−0.497320	−	0.867567i	$-0.665682\pi$
$881$	−2.29142e7	−0.994640	−0.497320	+	0.867567i	$0.665682\pi$
$882$	−5.70478e6	−0.246926
$883$	1.06073e7	0.457830	0.228915	−	0.973446i	$-0.426482\pi$
$883$	1.06073e7	0.457830	0.228915	+	0.973446i	$0.426482\pi$
$884$	1.28940e7	0.554952
$885$	−1.67123e7	−0.717263
$886$	−2.25412e7	−0.964703
$887$	−2.28527e7	−0.975278	−0.487639	−	0.873045i	$-0.662142\pi$
$887$	−2.28527e7	−0.975278	−0.487639	+	0.873045i	$0.662142\pi$
$888$	1.82744e7	0.777696
$889$	1.18137e7	0.501339
$890$	−5.55498e6	−0.235076
$891$	0	0
$892$	1.01069e7	0.425309
$893$	−4.35533e7	−1.82765
$894$	−3.67945e7	−1.53971
$895$	9.13072e6	0.381020
$896$	−1.60563e6	−0.0668153
$897$	6.40075e7	2.65613
$898$	−1.48002e6	−0.0612459
$899$	2.71892e7	1.12201
$900$	1.08852e7	0.447953
$901$	357948.	0.0146895
$902$	0	0
$903$	−2.92936e7	−1.19551
$904$	−1.25869e7	−0.512270
$905$	4.80984e7	1.95213
$906$	−2.01094e7	−0.813915
$907$	−3.21947e7	−1.29947	−0.649736	−	0.760160i	$-0.725120\pi$
$907$	−3.21947e7	−1.29947	−0.649736	+	0.760160i	$0.725120\pi$
$908$	−22368.0	−0.000900352 0
$909$	−2.73442e7	−1.09763
$910$	2.61636e7	1.04736
$911$	2.02254e7	0.807424	0.403712	−	0.914886i	$-0.367720\pi$
$911$	2.02254e7	0.807424	0.403712	+	0.914886i	$0.367720\pi$
$912$	−1.15046e7	−0.458022
$913$	0	0
$914$	−1.32756e7	−0.525642
$915$	4.44505e7	1.75519
$916$	3.29816e6	0.129877
$917$	6.70732e6	0.263406
$918$	−3.69684e6	−0.144785
$919$	2.30019e7	0.898411	0.449206	−	0.893428i	$-0.351707\pi$
$919$	2.30019e7	0.898411	0.449206	+	0.893428i	$0.351707\pi$
$920$	1.91756e7	0.746930
$921$	2.88683e6	0.112143
$922$	−3.50808e7	−1.35907
$923$	1.94324e7	0.750796
$924$	0	0
$925$	−4.67193e7	−1.79532
$926$	896996.	0.0343766
$927$	2.34907e6	0.0897836
$928$	−3.56352e6	−0.135834
$929$	−1.64913e7	−0.626924	−0.313462	−	0.949601i	$-0.601489\pi$
$929$	−1.64913e7	−0.626924	−0.313462	+	0.949601i	$0.601489\pi$
$930$	5.31597e7	2.01546
$931$	−1.54144e7	−0.582845
$932$	−1.02513e7	−0.386578
$933$	−2.57521e7	−0.968521
$934$	−1.02229e7	−0.383449
$935$	0	0
$936$	−1.04417e7	−0.389568
$937$	4.99402e7	1.85824	0.929119	−	0.369780i	$-0.120567\pi$
$937$	4.99402e7	1.85824	0.929119	+	0.369780i	$0.120567\pi$
$938$	−6.70046e6	−0.248655
$939$	6.58753e7	2.43814
$940$	−2.63762e7	−0.973627
$941$	−4.21000e6	−0.154992	−0.0774958	−	0.996993i	$-0.524692\pi$
$941$	−4.21000e6	−0.154992	−0.0774958	+	0.996993i	$0.524692\pi$
$942$	−5.22085e6	−0.191696
$943$	−2.40139e7	−0.879394
$944$	2.51520e6	0.0918634
$945$	−7.50141e6	−0.273252
$946$	0	0
$947$	2.76066e7	1.00032	0.500159	−	0.865933i	$-0.333275\pi$
$947$	2.76066e7	1.00032	0.500159	+	0.865933i	$0.333275\pi$
$948$	−1.42766e7	−0.515947
$949$	−2.89850e7	−1.04474
$950$	2.94122e7	1.05735
$951$	3.55149e7	1.27338
$952$	6.13402e6	0.219358
$953$	−1.69297e7	−0.603832	−0.301916	−	0.953335i	$-0.597626\pi$
$953$	−1.69297e7	−0.603832	−0.301916	+	0.953335i	$0.597626\pi$
$954$	−289872.	−0.0103118
$955$	5.28571e7	1.87540
$956$	−2.05200e6	−0.0726161
$957$	0	0
$958$	−2.14666e7	−0.755699
$959$	3.66174e7	1.28570
$960$	−6.96730e6	−0.243998
$961$	3.24138e7	1.13220
$962$	4.48157e7	1.56132
$963$	−3.36560e6	−0.116949
$964$	2.71598e7	0.941313
$965$	−3.25753e7	−1.12608
$966$	3.04502e7	1.04990
$967$	−1.08793e6	−0.0374140	−0.0187070	−	0.999825i	$-0.505955\pi$
$967$	−1.08793e6	−0.0374140	−0.0187070	+	0.999825i	$0.505955\pi$
$968$	0	0
$969$	4.39513e7	1.50370
$970$	−9.95555e6	−0.339732
$971$	7.15335e6	0.243479	0.121739	−	0.992562i	$-0.461153\pi$
$971$	7.15335e6	0.243479	0.121739	+	0.992562i	$0.461153\pi$
$972$	1.91601e7	0.650476
$973$	−5.93978e6	−0.201135
$974$	−3.27011e7	−1.10450
$975$	5.94565e7	2.00303
$976$	−6.68979e6	−0.224796
$977$	−7.88535e6	−0.264292	−0.132146	−	0.991230i	$-0.542187\pi$
$977$	−7.88535e6	−0.264292	−0.132146	+	0.991230i	$0.542187\pi$
$978$	−2.50928e7	−0.838885
$979$	0	0
$980$	−9.33509e6	−0.310494
$981$	4.39223e7	1.45718
$982$	−1.51203e7	−0.500359
$983$	7.58842e6	0.250477	0.125238	−	0.992127i	$-0.460030\pi$
$983$	7.58842e6	0.250477	0.125238	+	0.992127i	$0.460030\pi$
$984$	8.72525e6	0.287270
$985$	2.17470e7	0.714183
$986$	1.36138e7	0.445950
$987$	−4.18844e7	−1.36855
$988$	−2.82138e7	−0.919536
$989$	−5.26516e7	−1.71167
$990$	0	0
$991$	8.80935e6	0.284944	0.142472	−	0.989799i	$-0.454495\pi$
$991$	8.80935e6	0.284944	0.142472	+	0.989799i	$0.454495\pi$
$992$	−8.00051e6	−0.258130
$993$	4.49193e7	1.44564
$994$	9.24454e6	0.296769
$995$	−4.70772e7	−1.50748
$996$	7.61846e6	0.243343
$997$	−3.53833e7	−1.12735	−0.563677	−	0.825995i	$-0.690614\pi$
$997$	−3.53833e7	−1.12735	−0.563677	+	0.825995i	$0.690614\pi$
$998$	1.59274e7	0.506197
$999$	−1.28492e7	−0.407344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.6.a.c.1.1		1
11.10	odd	2		22.6.a.a.1.1	✓	1
33.32	even	2		198.6.a.d.1.1		1
44.43	even	2		176.6.a.d.1.1		1
55.32	even	4		550.6.b.g.199.1		2
55.43	even	4		550.6.b.g.199.2		2
55.54	odd	2		550.6.a.g.1.1		1
77.76	even	2		1078.6.a.b.1.1		1
88.21	odd	2		704.6.a.i.1.1		1
88.43	even	2		704.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.a.1.1	✓	1	11.10	odd	2
176.6.a.d.1.1		1	44.43	even	2
198.6.a.d.1.1		1	33.32	even	2
242.6.a.c.1.1		1	1.1	even	1	trivial
550.6.a.g.1.1		1	55.54	odd	2
550.6.b.g.199.1		2	55.32	even	4
550.6.b.g.199.2		2	55.43	even	4
704.6.a.b.1.1		1	88.43	even	2
704.6.a.i.1.1		1	88.21	odd	2
1078.6.a.b.1.1		1	77.76	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:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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