LMFDB - Embedded newform 450.6.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +79.0000 q^{7} +64.0000 q^{8} +O(q^{10})$

$q+4.00000 q^{2} +16.0000 q^{4} +79.0000 q^{7} +64.0000 q^{8} -150.000 q^{11} -137.000 q^{13} +316.000 q^{14} +256.000 q^{16} -2034.00 q^{17} -1969.00 q^{19} -600.000 q^{22} -1350.00 q^{23} -548.000 q^{26} +1264.00 q^{28} +2946.00 q^{29} +713.000 q^{31} +1024.00 q^{32} -8136.00 q^{34} +3238.00 q^{37} -7876.00 q^{38} -6564.00 q^{41} +19579.0 q^{43} -2400.00 q^{44} -5400.00 q^{46} -21150.0 q^{47} -10566.0 q^{49} -2192.00 q^{52} -25896.0 q^{53} +5056.00 q^{56} +11784.0 q^{58} -25350.0 q^{59} +50615.0 q^{61} +2852.00 q^{62} +4096.00 q^{64} +22519.0 q^{67} -32544.0 q^{68} -33900.0 q^{71} -82442.0 q^{73} +12952.0 q^{74} -31504.0 q^{76} -11850.0 q^{77} -81472.0 q^{79} -26256.0 q^{82} +25782.0 q^{83} +78316.0 q^{86} -9600.00 q^{88} -103728. q^{89} -10823.0 q^{91} -21600.0 q^{92} -84600.0 q^{94} +57343.0 q^{97} -42264.0 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	79.0000	0.609371	0.304686	−	0.952453i	$-0.401449\pi$
$7$	79.0000	0.609371	0.304686	+	0.952453i	$0.401449\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	−150.000	−0.373774	−0.186887	−	0.982381i	$-0.559840\pi$
$11$	−150.000	−0.373774	−0.186887	+	0.982381i	$0.559840\pi$
$12$	0	0
$13$	−137.000	−0.224834	−0.112417	−	0.993661i	$-0.535859\pi$
$13$	−137.000	−0.224834	−0.112417	+	0.993661i	$0.535859\pi$
$14$	316.000	0.430891
$15$	0	0
$16$	256.000	0.250000
$17$	−2034.00	−1.70698	−0.853490	−	0.521109i	$-0.825519\pi$
$17$	−2034.00	−1.70698	−0.853490	+	0.521109i	$0.825519\pi$
$18$	0	0
$19$	−1969.00	−1.25130	−0.625650	−	0.780104i	$-0.715166\pi$
$19$	−1969.00	−1.25130	−0.625650	+	0.780104i	$0.715166\pi$
$20$	0	0
$21$	0	0
$22$	−600.000	−0.264298
$23$	−1350.00	−0.532126	−0.266063	−	0.963956i	$-0.585723\pi$
$23$	−1350.00	−0.532126	−0.266063	+	0.963956i	$0.585723\pi$
$24$	0	0
$25$	0	0
$26$	−548.000	−0.158982
$27$	0	0
$28$	1264.00	0.304686
$29$	2946.00	0.650486	0.325243	−	0.945631i	$-0.394554\pi$
$29$	2946.00	0.650486	0.325243	+	0.945631i	$0.394554\pi$
$30$	0	0
$31$	713.000	0.133256	0.0666278	−	0.997778i	$-0.478776\pi$
$31$	713.000	0.133256	0.0666278	+	0.997778i	$0.478776\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−8136.00	−1.20702
$35$	0	0
$36$	0	0
$37$	3238.00	0.388841	0.194421	−	0.980918i	$-0.437717\pi$
$37$	3238.00	0.388841	0.194421	+	0.980918i	$0.437717\pi$
$38$	−7876.00	−0.884803
$39$	0	0
$40$	0	0
$41$	−6564.00	−0.609830	−0.304915	−	0.952380i	$-0.598628\pi$
$41$	−6564.00	−0.609830	−0.304915	+	0.952380i	$0.598628\pi$
$42$	0	0
$43$	19579.0	1.61480	0.807401	−	0.590003i	$-0.200873\pi$
$43$	19579.0	1.61480	0.807401	+	0.590003i	$0.200873\pi$
$44$	−2400.00	−0.186887
$45$	0	0
$46$	−5400.00	−0.376270
$47$	−21150.0	−1.39658	−0.698290	−	0.715815i	$-0.746055\pi$
$47$	−21150.0	−1.39658	−0.698290	+	0.715815i	$0.746055\pi$
$48$	0	0
$49$	−10566.0	−0.628667
$50$	0	0
$51$	0	0
$52$	−2192.00	−0.112417
$53$	−25896.0	−1.26632	−0.633159	−	0.774021i	$-0.718242\pi$
$53$	−25896.0	−1.26632	−0.633159	+	0.774021i	$0.718242\pi$
$54$	0	0
$55$	0	0
$56$	5056.00	0.215445
$57$	0	0
$58$	11784.0	0.459963
$59$	−25350.0	−0.948086	−0.474043	−	0.880502i	$-0.657206\pi$
$59$	−25350.0	−0.948086	−0.474043	+	0.880502i	$0.657206\pi$
$60$	0	0
$61$	50615.0	1.74163	0.870813	−	0.491615i	$-0.163593\pi$
$61$	50615.0	1.74163	0.870813	+	0.491615i	$0.163593\pi$
$62$	2852.00	0.0942259
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	22519.0	0.612861	0.306431	−	0.951893i	$-0.400865\pi$
$67$	22519.0	0.612861	0.306431	+	0.951893i	$0.400865\pi$
$68$	−32544.0	−0.853490
$69$	0	0
$70$	0	0
$71$	−33900.0	−0.798094	−0.399047	−	0.916931i	$-0.630659\pi$
$71$	−33900.0	−0.798094	−0.399047	+	0.916931i	$0.630659\pi$
$72$	0	0
$73$	−82442.0	−1.81068	−0.905339	−	0.424689i	$-0.860383\pi$
$73$	−82442.0	−1.81068	−0.905339	+	0.424689i	$0.860383\pi$
$74$	12952.0	0.274952
$75$	0	0
$76$	−31504.0	−0.625650
$77$	−11850.0	−0.227767
$78$	0	0
$79$	−81472.0	−1.46873	−0.734363	−	0.678757i	$-0.762519\pi$
$79$	−81472.0	−1.46873	−0.734363	+	0.678757i	$0.762519\pi$
$80$	0	0
$81$	0	0
$82$	−26256.0	−0.431215
$83$	25782.0	0.410791	0.205396	−	0.978679i	$-0.434152\pi$
$83$	25782.0	0.410791	0.205396	+	0.978679i	$0.434152\pi$
$84$	0	0
$85$	0	0
$86$	78316.0	1.14184
$87$	0	0
$88$	−9600.00	−0.132149
$89$	−103728.	−1.38810	−0.694050	−	0.719926i	$-0.744176\pi$
$89$	−103728.	−1.38810	−0.694050	+	0.719926i	$0.744176\pi$
$90$	0	0
$91$	−10823.0	−0.137007
$92$	−21600.0	−0.266063
$93$	0	0
$94$	−84600.0	−0.987531
$95$	0	0
$96$	0	0
$97$	57343.0	0.618801	0.309401	−	0.950932i	$-0.399872\pi$
$97$	57343.0	0.618801	0.309401	+	0.950932i	$0.399872\pi$
$98$	−42264.0	−0.444534
$99$	0	0
$100$	0	0
$101$	91032.0	0.887954	0.443977	−	0.896038i	$-0.353567\pi$
$101$	91032.0	0.887954	0.443977	+	0.896038i	$0.353567\pi$
$102$	0	0
$103$	−191636.	−1.77985	−0.889926	−	0.456104i	$-0.849244\pi$
$103$	−191636.	−1.77985	−0.889926	+	0.456104i	$0.849244\pi$
$104$	−8768.00	−0.0794909
$105$	0	0
$106$	−103584.	−0.895423
$107$	9288.00	0.0784265	0.0392132	−	0.999231i	$-0.487515\pi$
$107$	9288.00	0.0784265	0.0392132	+	0.999231i	$0.487515\pi$
$108$	0	0
$109$	−20635.0	−0.166356	−0.0831780	−	0.996535i	$-0.526507\pi$
$109$	−20635.0	−0.166356	−0.0831780	+	0.996535i	$0.526507\pi$
$110$	0	0
$111$	0	0
$112$	20224.0	0.152343
$113$	176892.	1.30320	0.651602	−	0.758561i	$-0.274097\pi$
$113$	176892.	1.30320	0.651602	+	0.758561i	$0.274097\pi$
$114$	0	0
$115$	0	0
$116$	47136.0	0.325243
$117$	0	0
$118$	−101400.	−0.670398
$119$	−160686.	−1.04019
$120$	0	0
$121$	−138551.	−0.860293
$122$	202460.	1.23151
$123$	0	0
$124$	11408.0	0.0666278
$125$	0	0
$126$	0	0
$127$	256480.	1.41106	0.705528	−	0.708682i	$-0.250710\pi$
$127$	256480.	1.41106	0.705528	+	0.708682i	$0.250710\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	−177600.	−0.904200	−0.452100	−	0.891967i	$-0.649325\pi$
$131$	−177600.	−0.904200	−0.452100	+	0.891967i	$0.649325\pi$
$132$	0	0
$133$	−155551.	−0.762507
$134$	90076.0	0.433358
$135$	0	0
$136$	−130176.	−0.603509
$137$	260886.	1.18754	0.593772	−	0.804634i	$-0.297638\pi$
$137$	260886.	1.18754	0.593772	+	0.804634i	$0.297638\pi$
$138$	0	0
$139$	−217684.	−0.955629	−0.477815	−	0.878461i	$-0.658571\pi$
$139$	−217684.	−0.955629	−0.477815	+	0.878461i	$0.658571\pi$
$140$	0	0
$141$	0	0
$142$	−135600.	−0.564337
$143$	20550.0	0.0840372
$144$	0	0
$145$	0	0
$146$	−329768.	−1.28034
$147$	0	0
$148$	51808.0	0.194421
$149$	−421794.	−1.55645	−0.778224	−	0.627987i	$-0.783879\pi$
$149$	−421794.	−1.55645	−0.778224	+	0.627987i	$0.783879\pi$
$150$	0	0
$151$	−101917.	−0.363751	−0.181876	−	0.983322i	$-0.558217\pi$
$151$	−101917.	−0.363751	−0.181876	+	0.983322i	$0.558217\pi$
$152$	−126016.	−0.442402
$153$	0	0
$154$	−47400.0	−0.161056
$155$	0	0
$156$	0	0
$157$	101773.	0.329521	0.164761	−	0.986334i	$-0.447315\pi$
$157$	101773.	0.329521	0.164761	+	0.986334i	$0.447315\pi$
$158$	−325888.	−1.03855
$159$	0	0
$160$	0	0
$161$	−106650.	−0.324262
$162$	0	0
$163$	202249.	0.596235	0.298117	−	0.954529i	$-0.403641\pi$
$163$	202249.	0.596235	0.298117	+	0.954529i	$0.403641\pi$
$164$	−105024.	−0.304915
$165$	0	0
$166$	103128.	0.290473
$167$	231600.	0.642610	0.321305	−	0.946976i	$-0.395879\pi$
$167$	231600.	0.642610	0.321305	+	0.946976i	$0.395879\pi$
$168$	0	0
$169$	−352524.	−0.949450
$170$	0	0
$171$	0	0
$172$	313264.	0.807401
$173$	−174222.	−0.442576	−0.221288	−	0.975209i	$-0.571026\pi$
$173$	−174222.	−0.442576	−0.221288	+	0.975209i	$0.571026\pi$
$174$	0	0
$175$	0	0
$176$	−38400.0	−0.0934436
$177$	0	0
$178$	−414912.	−0.981535
$179$	642066.	1.49778	0.748888	−	0.662696i	$-0.230588\pi$
$179$	642066.	1.49778	0.748888	+	0.662696i	$0.230588\pi$
$180$	0	0
$181$	−56071.0	−0.127216	−0.0636080	−	0.997975i	$-0.520261\pi$
$181$	−56071.0	−0.127216	−0.0636080	+	0.997975i	$0.520261\pi$
$182$	−43292.0	−0.0968789
$183$	0	0
$184$	−86400.0	−0.188135
$185$	0	0
$186$	0	0
$187$	305100.	0.638026
$188$	−338400.	−0.698290
$189$	0	0
$190$	0	0
$191$	209694.	0.415913	0.207957	−	0.978138i	$-0.433319\pi$
$191$	209694.	0.415913	0.207957	+	0.978138i	$0.433319\pi$
$192$	0	0
$193$	715597.	1.38285	0.691425	−	0.722448i	$-0.256983\pi$
$193$	715597.	1.38285	0.691425	+	0.722448i	$0.256983\pi$
$194$	229372.	0.437558
$195$	0	0
$196$	−169056.	−0.314333
$197$	508854.	0.934174	0.467087	−	0.884211i	$-0.345304\pi$
$197$	508854.	0.934174	0.467087	+	0.884211i	$0.345304\pi$
$198$	0	0
$199$	−986017.	−1.76503	−0.882514	−	0.470286i	$-0.844151\pi$
$199$	−986017.	−1.76503	−0.882514	+	0.470286i	$0.844151\pi$
$200$	0	0
$201$	0	0
$202$	364128.	0.627879
$203$	232734.	0.396387
$204$	0	0
$205$	0	0
$206$	−766544.	−1.25855
$207$	0	0
$208$	−35072.0	−0.0562085
$209$	295350.	0.467704
$210$	0	0
$211$	119495.	0.184775	0.0923876	−	0.995723i	$-0.470550\pi$
$211$	119495.	0.184775	0.0923876	+	0.995723i	$0.470550\pi$
$212$	−414336.	−0.633159
$213$	0	0
$214$	37152.0	0.0554559
$215$	0	0
$216$	0	0
$217$	56327.0	0.0812021
$218$	−82540.0	−0.117631
$219$	0	0
$220$	0	0
$221$	278658.	0.383788
$222$	0	0
$223$	−48545.0	−0.0653706	−0.0326853	−	0.999466i	$-0.510406\pi$
$223$	−48545.0	−0.0653706	−0.0326853	+	0.999466i	$0.510406\pi$
$224$	80896.0	0.107723
$225$	0	0
$226$	707568.	0.921504
$227$	−287652.	−0.370512	−0.185256	−	0.982690i	$-0.559311\pi$
$227$	−287652.	−0.370512	−0.185256	+	0.982690i	$0.559311\pi$
$228$	0	0
$229$	72065.0	0.0908104	0.0454052	−	0.998969i	$-0.485542\pi$
$229$	72065.0	0.0908104	0.0454052	+	0.998969i	$0.485542\pi$
$230$	0	0
$231$	0	0
$232$	188544.	0.229981
$233$	−569148.	−0.686808	−0.343404	−	0.939188i	$-0.611580\pi$
$233$	−569148.	−0.686808	−0.343404	+	0.939188i	$0.611580\pi$
$234$	0	0
$235$	0	0
$236$	−405600.	−0.474043
$237$	0	0
$238$	−642744.	−0.735522
$239$	−696504.	−0.788731	−0.394365	−	0.918954i	$-0.629036\pi$
$239$	−696504.	−0.788731	−0.394365	+	0.918954i	$0.629036\pi$
$240$	0	0
$241$	576137.	0.638974	0.319487	−	0.947591i	$-0.396489\pi$
$241$	576137.	0.638974	0.319487	+	0.947591i	$0.396489\pi$
$242$	−554204.	−0.608319
$243$	0	0
$244$	809840.	0.870813
$245$	0	0
$246$	0	0
$247$	269753.	0.281335
$248$	45632.0	0.0471130
$249$	0	0
$250$	0	0
$251$	1.22492e6	1.22723	0.613613	−	0.789607i	$-0.289715\pi$
$251$	1.22492e6	1.22723	0.613613	+	0.789607i	$0.289715\pi$
$252$	0	0
$253$	202500.	0.198895
$254$	1.02592e6	0.997767
$255$	0	0
$256$	65536.0	0.0625000
$257$	972132.	0.918105	0.459053	−	0.888409i	$-0.348189\pi$
$257$	972132.	0.918105	0.459053	+	0.888409i	$0.348189\pi$
$258$	0	0
$259$	255802.	0.236949
$260$	0	0
$261$	0	0
$262$	−710400.	−0.639366
$263$	1.76190e6	1.57070	0.785348	−	0.619055i	$-0.212484\pi$
$263$	1.76190e6	1.57070	0.785348	+	0.619055i	$0.212484\pi$
$264$	0	0
$265$	0	0
$266$	−622204.	−0.539174
$267$	0	0
$268$	360304.	0.306431
$269$	−1.74383e6	−1.46935	−0.734674	−	0.678421i	$-0.762665\pi$
$269$	−1.74383e6	−1.46935	−0.734674	+	0.678421i	$0.762665\pi$
$270$	0	0
$271$	−1.70426e6	−1.40965	−0.704826	−	0.709381i	$-0.748975\pi$
$271$	−1.70426e6	−1.40965	−0.704826	+	0.709381i	$0.748975\pi$
$272$	−520704.	−0.426745
$273$	0	0
$274$	1.04354e6	0.839720
$275$	0	0
$276$	0	0
$277$	−1.46972e6	−1.15089	−0.575446	−	0.817840i	$-0.695171\pi$
$277$	−1.46972e6	−1.15089	−0.575446	+	0.817840i	$0.695171\pi$
$278$	−870736.	−0.675732
$279$	0	0
$280$	0	0
$281$	1.22530e6	0.925715	0.462858	−	0.886433i	$-0.346824\pi$
$281$	1.22530e6	0.925715	0.462858	+	0.886433i	$0.346824\pi$
$282$	0	0
$283$	637333.	0.473043	0.236521	−	0.971626i	$-0.423993\pi$
$283$	637333.	0.473043	0.236521	+	0.971626i	$0.423993\pi$
$284$	−542400.	−0.399047
$285$	0	0
$286$	82200.0	0.0594233
$287$	−518556.	−0.371613
$288$	0	0
$289$	2.71730e6	1.91378
$290$	0	0
$291$	0	0
$292$	−1.31907e6	−0.905339
$293$	35094.0	0.0238816	0.0119408	−	0.999929i	$-0.496199\pi$
$293$	35094.0	0.0238816	0.0119408	+	0.999929i	$0.496199\pi$
$294$	0	0
$295$	0	0
$296$	207232.	0.137476
$297$	0	0
$298$	−1.68718e6	−1.10058
$299$	184950.	0.119640
$300$	0	0
$301$	1.54674e6	0.984014
$302$	−407668.	−0.257211
$303$	0	0
$304$	−504064.	−0.312825
$305$	0	0
$306$	0	0
$307$	−2.88040e6	−1.74424	−0.872120	−	0.489293i	$-0.837255\pi$
$307$	−2.88040e6	−1.74424	−0.872120	+	0.489293i	$0.837255\pi$
$308$	−189600.	−0.113884
$309$	0	0
$310$	0	0
$311$	2.38305e6	1.39712	0.698558	−	0.715554i	$-0.253826\pi$
$311$	2.38305e6	1.39712	0.698558	+	0.715554i	$0.253826\pi$
$312$	0	0
$313$	1.93081e6	1.11399	0.556993	−	0.830517i	$-0.311955\pi$
$313$	1.93081e6	1.11399	0.556993	+	0.830517i	$0.311955\pi$
$314$	407092.	0.233007
$315$	0	0
$316$	−1.30355e6	−0.734363
$317$	−2.18995e6	−1.22401	−0.612007	−	0.790852i	$-0.709638\pi$
$317$	−2.18995e6	−1.22401	−0.612007	+	0.790852i	$0.709638\pi$
$318$	0	0
$319$	−441900.	−0.243135
$320$	0	0
$321$	0	0
$322$	−426600.	−0.229288
$323$	4.00495e6	2.13595
$324$	0	0
$325$	0	0
$326$	808996.	0.421602
$327$	0	0
$328$	−420096.	−0.215608
$329$	−1.67085e6	−0.851036
$330$	0	0
$331$	1.41429e6	0.709527	0.354764	−	0.934956i	$-0.384561\pi$
$331$	1.41429e6	0.709527	0.354764	+	0.934956i	$0.384561\pi$
$332$	412512.	0.205396
$333$	0	0
$334$	926400.	0.454394
$335$	0	0
$336$	0	0
$337$	−1.38208e6	−0.662916	−0.331458	−	0.943470i	$-0.607541\pi$
$337$	−1.38208e6	−0.662916	−0.331458	+	0.943470i	$0.607541\pi$
$338$	−1.41010e6	−0.671362
$339$	0	0
$340$	0	0
$341$	−106950.	−0.0498075
$342$	0	0
$343$	−2.16247e6	−0.992463
$344$	1.25306e6	0.570919
$345$	0	0
$346$	−696888.	−0.312948
$347$	−283758.	−0.126510	−0.0632549	−	0.997997i	$-0.520148\pi$
$347$	−283758.	−0.126510	−0.0632549	+	0.997997i	$0.520148\pi$
$348$	0	0
$349$	2.13809e6	0.939642	0.469821	−	0.882762i	$-0.344318\pi$
$349$	2.13809e6	0.939642	0.469821	+	0.882762i	$0.344318\pi$
$350$	0	0
$351$	0	0
$352$	−153600.	−0.0660746
$353$	3.16015e6	1.34980	0.674901	−	0.737908i	$-0.264186\pi$
$353$	3.16015e6	1.34980	0.674901	+	0.737908i	$0.264186\pi$
$354$	0	0
$355$	0	0
$356$	−1.65965e6	−0.694050
$357$	0	0
$358$	2.56826e6	1.05909
$359$	3.59564e6	1.47245	0.736225	−	0.676737i	$-0.236607\pi$
$359$	3.59564e6	1.47245	0.736225	+	0.676737i	$0.236607\pi$
$360$	0	0
$361$	1.40086e6	0.565754
$362$	−224284.	−0.0899553
$363$	0	0
$364$	−173168.	−0.0685037
$365$	0	0
$366$	0	0
$367$	4.62043e6	1.79068	0.895339	−	0.445385i	$-0.146933\pi$
$367$	4.62043e6	1.79068	0.895339	+	0.445385i	$0.146933\pi$
$368$	−345600.	−0.133031
$369$	0	0
$370$	0	0
$371$	−2.04578e6	−0.771658
$372$	0	0
$373$	3.51983e6	1.30993	0.654967	−	0.755657i	$-0.272682\pi$
$373$	3.51983e6	1.30993	0.654967	+	0.755657i	$0.272682\pi$
$374$	1.22040e6	0.451152
$375$	0	0
$376$	−1.35360e6	−0.493765
$377$	−403602.	−0.146251
$378$	0	0
$379$	595061.	0.212796	0.106398	−	0.994324i	$-0.466068\pi$
$379$	595061.	0.212796	0.106398	+	0.994324i	$0.466068\pi$
$380$	0	0
$381$	0	0
$382$	838776.	0.294095
$383$	−1.30050e6	−0.453016	−0.226508	−	0.974009i	$-0.572731\pi$
$383$	−1.30050e6	−0.453016	−0.226508	+	0.974009i	$0.572731\pi$
$384$	0	0
$385$	0	0
$386$	2.86239e6	0.977823
$387$	0	0
$388$	917488.	0.309401
$389$	−620724.	−0.207981	−0.103991	−	0.994578i	$-0.533161\pi$
$389$	−620724.	−0.207981	−0.103991	+	0.994578i	$0.533161\pi$
$390$	0	0
$391$	2.74590e6	0.908328
$392$	−676224.	−0.222267
$393$	0	0
$394$	2.03542e6	0.660561
$395$	0	0
$396$	0	0
$397$	−1.18622e6	−0.377737	−0.188869	−	0.982002i	$-0.560482\pi$
$397$	−1.18622e6	−0.377737	−0.188869	+	0.982002i	$0.560482\pi$
$398$	−3.94407e6	−1.24806
$399$	0	0
$400$	0	0
$401$	−3.13334e6	−0.973077	−0.486538	−	0.873659i	$-0.661741\pi$
$401$	−3.13334e6	−0.973077	−0.486538	+	0.873659i	$0.661741\pi$
$402$	0	0
$403$	−97681.0	−0.0299604
$404$	1.45651e6	0.443977
$405$	0	0
$406$	930936.	0.280288
$407$	−485700.	−0.145339
$408$	0	0
$409$	567581.	0.167772	0.0838860	−	0.996475i	$-0.473267\pi$
$409$	567581.	0.167772	0.0838860	+	0.996475i	$0.473267\pi$
$410$	0	0
$411$	0	0
$412$	−3.06618e6	−0.889926
$413$	−2.00265e6	−0.577737
$414$	0	0
$415$	0	0
$416$	−140288.	−0.0397454
$417$	0	0
$418$	1.18140e6	0.330717
$419$	6.09000e6	1.69466	0.847329	−	0.531068i	$-0.178209\pi$
$419$	6.09000e6	1.69466	0.847329	+	0.531068i	$0.178209\pi$
$420$	0	0
$421$	3.07139e6	0.844558	0.422279	−	0.906466i	$-0.361230\pi$
$421$	3.07139e6	0.844558	0.422279	+	0.906466i	$0.361230\pi$
$422$	477980.	0.130656
$423$	0	0
$424$	−1.65734e6	−0.447711
$425$	0	0
$426$	0	0
$427$	3.99858e6	1.06130
$428$	148608.	0.0392132
$429$	0	0
$430$	0	0
$431$	669150.	0.173512	0.0867562	−	0.996230i	$-0.472350\pi$
$431$	669150.	0.173512	0.0867562	+	0.996230i	$0.472350\pi$
$432$	0	0
$433$	3.25439e6	0.834161	0.417080	−	0.908870i	$-0.363053\pi$
$433$	3.25439e6	0.834161	0.417080	+	0.908870i	$0.363053\pi$
$434$	225308.	0.0574186
$435$	0	0
$436$	−330160.	−0.0831780
$437$	2.65815e6	0.665849
$438$	0	0
$439$	1.58759e6	0.393168	0.196584	−	0.980487i	$-0.437015\pi$
$439$	1.58759e6	0.393168	0.196584	+	0.980487i	$0.437015\pi$
$440$	0	0
$441$	0	0
$442$	1.11463e6	0.271379
$443$	−213888.	−0.0517818	−0.0258909	−	0.999665i	$-0.508242\pi$
$443$	−213888.	−0.0517818	−0.0258909	+	0.999665i	$0.508242\pi$
$444$	0	0
$445$	0	0
$446$	−194180.	−0.0462240
$447$	0	0
$448$	323584.	0.0761714
$449$	−3.70724e6	−0.867831	−0.433916	−	0.900953i	$-0.642868\pi$
$449$	−3.70724e6	−0.867831	−0.433916	+	0.900953i	$0.642868\pi$
$450$	0	0
$451$	984600.	0.227939
$452$	2.83027e6	0.651602
$453$	0	0
$454$	−1.15061e6	−0.261992
$455$	0	0
$456$	0	0
$457$	−3.83193e6	−0.858275	−0.429138	−	0.903239i	$-0.641182\pi$
$457$	−3.83193e6	−0.858275	−0.429138	+	0.903239i	$0.641182\pi$
$458$	288260.	0.0642127
$459$	0	0
$460$	0	0
$461$	−5.58672e6	−1.22435	−0.612174	−	0.790723i	$-0.709705\pi$
$461$	−5.58672e6	−1.22435	−0.612174	+	0.790723i	$0.709705\pi$
$462$	0	0
$463$	4.40142e6	0.954203	0.477101	−	0.878848i	$-0.341687\pi$
$463$	4.40142e6	0.954203	0.477101	+	0.878848i	$0.341687\pi$
$464$	754176.	0.162621
$465$	0	0
$466$	−2.27659e6	−0.485647
$467$	3.35225e6	0.711287	0.355643	−	0.934622i	$-0.384262\pi$
$467$	3.35225e6	0.711287	0.355643	+	0.934622i	$0.384262\pi$
$468$	0	0
$469$	1.77900e6	0.373460
$470$	0	0
$471$	0	0
$472$	−1.62240e6	−0.335199
$473$	−2.93685e6	−0.603572
$474$	0	0
$475$	0	0
$476$	−2.57098e6	−0.520093
$477$	0	0
$478$	−2.78602e6	−0.557717
$479$	−909966.	−0.181212	−0.0906059	−	0.995887i	$-0.528880\pi$
$479$	−909966.	−0.181212	−0.0906059	+	0.995887i	$0.528880\pi$
$480$	0	0
$481$	−443606.	−0.0874248
$482$	2.30455e6	0.451823
$483$	0	0
$484$	−2.21682e6	−0.430146
$485$	0	0
$486$	0	0
$487$	4.03579e6	0.771093	0.385546	−	0.922689i	$-0.374013\pi$
$487$	4.03579e6	0.771093	0.385546	+	0.922689i	$0.374013\pi$
$488$	3.23936e6	0.615757
$489$	0	0
$490$	0	0
$491$	−379944.	−0.0711239	−0.0355620	−	0.999367i	$-0.511322\pi$
$491$	−379944.	−0.0711239	−0.0355620	+	0.999367i	$0.511322\pi$
$492$	0	0
$493$	−5.99216e6	−1.11037
$494$	1.07901e6	0.198934
$495$	0	0
$496$	182528.	0.0333139
$497$	−2.67810e6	−0.486335
$498$	0	0
$499$	−7.42433e6	−1.33477	−0.667384	−	0.744714i	$-0.732586\pi$
$499$	−7.42433e6	−1.33477	−0.667384	+	0.744714i	$0.732586\pi$
$500$	0	0
$501$	0	0
$502$	4.89970e6	0.867780
$503$	2.29988e6	0.405309	0.202654	−	0.979250i	$-0.435043\pi$
$503$	2.29988e6	0.405309	0.202654	+	0.979250i	$0.435043\pi$
$504$	0	0
$505$	0	0
$506$	810000.	0.140640
$507$	0	0
$508$	4.10368e6	0.705528
$509$	−6.73721e6	−1.15262	−0.576310	−	0.817231i	$-0.695508\pi$
$509$	−6.73721e6	−1.15262	−0.576310	+	0.817231i	$0.695508\pi$
$510$	0	0
$511$	−6.51292e6	−1.10338
$512$	262144.	0.0441942
$513$	0	0
$514$	3.88853e6	0.649198
$515$	0	0
$516$	0	0
$517$	3.17250e6	0.522006
$518$	1.02321e6	0.167548
$519$	0	0
$520$	0	0
$521$	9.26806e6	1.49587	0.747936	−	0.663770i	$-0.231045\pi$
$521$	9.26806e6	1.49587	0.747936	+	0.663770i	$0.231045\pi$
$522$	0	0
$523$	−1.10620e7	−1.76839	−0.884197	−	0.467113i	$-0.845294\pi$
$523$	−1.10620e7	−1.76839	−0.884197	+	0.467113i	$0.845294\pi$
$524$	−2.84160e6	−0.452100
$525$	0	0
$526$	7.04760e6	1.11065
$527$	−1.45024e6	−0.227465
$528$	0	0
$529$	−4.61384e6	−0.716842
$530$	0	0
$531$	0	0
$532$	−2.48882e6	−0.381253
$533$	899268.	0.137111
$534$	0	0
$535$	0	0
$536$	1.44122e6	0.216679
$537$	0	0
$538$	−6.97534e6	−1.03899
$539$	1.58490e6	0.234979
$540$	0	0
$541$	−1.30754e7	−1.92072	−0.960358	−	0.278771i	$-0.910073\pi$
$541$	−1.30754e7	−1.92072	−0.960358	+	0.278771i	$0.910073\pi$
$542$	−6.81702e6	−0.996774
$543$	0	0
$544$	−2.08282e6	−0.301754
$545$	0	0
$546$	0	0
$547$	1.84605e6	0.263800	0.131900	−	0.991263i	$-0.457892\pi$
$547$	1.84605e6	0.263800	0.131900	+	0.991263i	$0.457892\pi$
$548$	4.17418e6	0.593772
$549$	0	0
$550$	0	0
$551$	−5.80067e6	−0.813953
$552$	0	0
$553$	−6.43629e6	−0.894999
$554$	−5.87887e6	−0.813803
$555$	0	0
$556$	−3.48294e6	−0.477815
$557$	6.48782e6	0.886055	0.443027	−	0.896508i	$-0.353904\pi$
$557$	6.48782e6	0.886055	0.443027	+	0.896508i	$0.353904\pi$
$558$	0	0
$559$	−2.68232e6	−0.363063
$560$	0	0
$561$	0	0
$562$	4.90121e6	0.654579
$563$	3.01538e6	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$563$	3.01538e6	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$564$	0	0
$565$	0	0
$566$	2.54933e6	0.334492
$567$	0	0
$568$	−2.16960e6	−0.282169
$569$	3.49810e6	0.452952	0.226476	−	0.974017i	$-0.427280\pi$
$569$	3.49810e6	0.452952	0.226476	+	0.974017i	$0.427280\pi$
$570$	0	0
$571$	−1.23749e7	−1.58837	−0.794185	−	0.607676i	$-0.792102\pi$
$571$	−1.23749e7	−1.58837	−0.794185	+	0.607676i	$0.792102\pi$
$572$	328800.	0.0420186
$573$	0	0
$574$	−2.07422e6	−0.262770
$575$	0	0
$576$	0	0
$577$	−6.80687e6	−0.851153	−0.425577	−	0.904922i	$-0.639929\pi$
$577$	−6.80687e6	−0.851153	−0.425577	+	0.904922i	$0.639929\pi$
$578$	1.08692e7	1.35325
$579$	0	0
$580$	0	0
$581$	2.03678e6	0.250325
$582$	0	0
$583$	3.88440e6	0.473317
$584$	−5.27629e6	−0.640172
$585$	0	0
$586$	140376.	0.0168869
$587$	7.77113e6	0.930870	0.465435	−	0.885082i	$-0.345898\pi$
$587$	7.77113e6	0.930870	0.465435	+	0.885082i	$0.345898\pi$
$588$	0	0
$589$	−1.40390e6	−0.166743
$590$	0	0
$591$	0	0
$592$	828928.	0.0972104
$593$	−1.51222e7	−1.76595	−0.882976	−	0.469418i	$-0.844464\pi$
$593$	−1.51222e7	−1.76595	−0.882976	+	0.469418i	$0.844464\pi$
$594$	0	0
$595$	0	0
$596$	−6.74870e6	−0.778224
$597$	0	0
$598$	739800.	0.0845982
$599$	1.67012e6	0.190187	0.0950937	−	0.995468i	$-0.469685\pi$
$599$	1.67012e6	0.190187	0.0950937	+	0.995468i	$0.469685\pi$
$600$	0	0
$601$	4.12220e6	0.465525	0.232763	−	0.972534i	$-0.425223\pi$
$601$	4.12220e6	0.465525	0.232763	+	0.972534i	$0.425223\pi$
$602$	6.18696e6	0.695803
$603$	0	0
$604$	−1.63067e6	−0.181876
$605$	0	0
$606$	0	0
$607$	−6.81870e6	−0.751155	−0.375578	−	0.926791i	$-0.622556\pi$
$607$	−6.81870e6	−0.751155	−0.375578	+	0.926791i	$0.622556\pi$
$608$	−2.01626e6	−0.221201
$609$	0	0
$610$	0	0
$611$	2.89755e6	0.313999
$612$	0	0
$613$	−1.59466e7	−1.71402	−0.857010	−	0.515300i	$-0.827681\pi$
$613$	−1.59466e7	−1.71402	−0.857010	+	0.515300i	$0.827681\pi$
$614$	−1.15216e7	−1.23336
$615$	0	0
$616$	−758400.	−0.0805279
$617$	−8.21952e6	−0.869228	−0.434614	−	0.900617i	$-0.643115\pi$
$617$	−8.21952e6	−0.869228	−0.434614	+	0.900617i	$0.643115\pi$
$618$	0	0
$619$	−1.23323e7	−1.29366	−0.646828	−	0.762636i	$-0.723905\pi$
$619$	−1.23323e7	−1.29366	−0.646828	+	0.762636i	$0.723905\pi$
$620$	0	0
$621$	0	0
$622$	9.53220e6	0.987910
$623$	−8.19451e6	−0.845869
$624$	0	0
$625$	0	0
$626$	7.72325e6	0.787706
$627$	0	0
$628$	1.62837e6	0.164761
$629$	−6.58609e6	−0.663745
$630$	0	0
$631$	−3.01912e6	−0.301861	−0.150931	−	0.988544i	$-0.548227\pi$
$631$	−3.01912e6	−0.301861	−0.150931	+	0.988544i	$0.548227\pi$
$632$	−5.21421e6	−0.519273
$633$	0	0
$634$	−8.75981e6	−0.865509
$635$	0	0
$636$	0	0
$637$	1.44754e6	0.141346
$638$	−1.76760e6	−0.171922
$639$	0	0
$640$	0	0
$641$	1.18052e7	1.13482	0.567411	−	0.823435i	$-0.307945\pi$
$641$	1.18052e7	1.13482	0.567411	+	0.823435i	$0.307945\pi$
$642$	0	0
$643$	2.85360e6	0.272186	0.136093	−	0.990696i	$-0.456545\pi$
$643$	2.85360e6	0.272186	0.136093	+	0.990696i	$0.456545\pi$
$644$	−1.70640e6	−0.162131
$645$	0	0
$646$	1.60198e7	1.51034
$647$	−3.40192e6	−0.319494	−0.159747	−	0.987158i	$-0.551068\pi$
$647$	−3.40192e6	−0.319494	−0.159747	+	0.987158i	$0.551068\pi$
$648$	0	0
$649$	3.80250e6	0.354370
$650$	0	0
$651$	0	0
$652$	3.23598e6	0.298117
$653$	−9.39166e6	−0.861905	−0.430953	−	0.902375i	$-0.641822\pi$
$653$	−9.39166e6	−0.861905	−0.430953	+	0.902375i	$0.641822\pi$
$654$	0	0
$655$	0	0
$656$	−1.68038e6	−0.152458
$657$	0	0
$658$	−6.68340e6	−0.601773
$659$	−9.36659e6	−0.840171	−0.420086	−	0.907484i	$-0.638000\pi$
$659$	−9.36659e6	−0.840171	−0.420086	+	0.907484i	$0.638000\pi$
$660$	0	0
$661$	1.57748e7	1.40430	0.702150	−	0.712029i	$-0.252223\pi$
$661$	1.57748e7	1.40430	0.702150	+	0.712029i	$0.252223\pi$
$662$	5.65717e6	0.501712
$663$	0	0
$664$	1.65005e6	0.145237
$665$	0	0
$666$	0	0
$667$	−3.97710e6	−0.346140
$668$	3.70560e6	0.321305
$669$	0	0
$670$	0	0
$671$	−7.59225e6	−0.650975
$672$	0	0
$673$	6.64515e6	0.565545	0.282773	−	0.959187i	$-0.408746\pi$
$673$	6.64515e6	0.565545	0.282773	+	0.959187i	$0.408746\pi$
$674$	−5.52832e6	−0.468753
$675$	0	0
$676$	−5.64038e6	−0.474725
$677$	−1.57780e7	−1.32306	−0.661532	−	0.749917i	$-0.730094\pi$
$677$	−1.57780e7	−1.32306	−0.661532	+	0.749917i	$0.730094\pi$
$678$	0	0
$679$	4.53010e6	0.377080
$680$	0	0
$681$	0	0
$682$	−427800.	−0.0352192
$683$	−1.96654e7	−1.61306	−0.806532	−	0.591190i	$-0.798658\pi$
$683$	−1.96654e7	−1.61306	−0.806532	+	0.591190i	$0.798658\pi$
$684$	0	0
$685$	0	0
$686$	−8.64987e6	−0.701777
$687$	0	0
$688$	5.01222e6	0.403701
$689$	3.54775e6	0.284712
$690$	0	0
$691$	990464.	0.0789121	0.0394560	−	0.999221i	$-0.487437\pi$
$691$	990464.	0.0789121	0.0394560	+	0.999221i	$0.487437\pi$
$692$	−2.78755e6	−0.221288
$693$	0	0
$694$	−1.13503e6	−0.0894560
$695$	0	0
$696$	0	0
$697$	1.33512e7	1.04097
$698$	8.55236e6	0.664427
$699$	0	0
$700$	0	0
$701$	−6.52919e6	−0.501839	−0.250920	−	0.968008i	$-0.580733\pi$
$701$	−6.52919e6	−0.501839	−0.250920	+	0.968008i	$0.580733\pi$
$702$	0	0
$703$	−6.37562e6	−0.486558
$704$	−614400.	−0.0467218
$705$	0	0
$706$	1.26406e7	0.954455
$707$	7.19153e6	0.541094
$708$	0	0
$709$	−2.02106e7	−1.50996	−0.754979	−	0.655749i	$-0.772353\pi$
$709$	−2.02106e7	−1.50996	−0.754979	+	0.655749i	$0.772353\pi$
$710$	0	0
$711$	0	0
$712$	−6.63859e6	−0.490768
$713$	−962550.	−0.0709087
$714$	0	0
$715$	0	0
$716$	1.02731e7	0.748888
$717$	0	0
$718$	1.43826e7	1.04118
$719$	2.27720e7	1.64277	0.821387	−	0.570371i	$-0.193200\pi$
$719$	2.27720e7	1.64277	0.821387	+	0.570371i	$0.193200\pi$
$720$	0	0
$721$	−1.51392e7	−1.08459
$722$	5.60345e6	0.400048
$723$	0	0
$724$	−897136.	−0.0636080
$725$	0	0
$726$	0	0
$727$	7.09186e6	0.497650	0.248825	−	0.968548i	$-0.419956\pi$
$727$	7.09186e6	0.497650	0.248825	+	0.968548i	$0.419956\pi$
$728$	−692672.	−0.0484394
$729$	0	0
$730$	0	0
$731$	−3.98237e7	−2.75644
$732$	0	0
$733$	1.38982e7	0.955432	0.477716	−	0.878514i	$-0.341465\pi$
$733$	1.38982e7	0.955432	0.477716	+	0.878514i	$0.341465\pi$
$734$	1.84817e7	1.26620
$735$	0	0
$736$	−1.38240e6	−0.0940674
$737$	−3.37785e6	−0.229072
$738$	0	0
$739$	1.54857e7	1.04309	0.521543	−	0.853225i	$-0.325357\pi$
$739$	1.54857e7	1.04309	0.521543	+	0.853225i	$0.325357\pi$
$740$	0	0
$741$	0	0
$742$	−8.18314e6	−0.545645
$743$	159276.	0.0105847	0.00529235	−	0.999986i	$-0.498315\pi$
$743$	159276.	0.0105847	0.00529235	+	0.999986i	$0.498315\pi$
$744$	0	0
$745$	0	0
$746$	1.40793e7	0.926263
$747$	0	0
$748$	4.88160e6	0.319013
$749$	733752.	0.0477908
$750$	0	0
$751$	1.15554e7	0.747625	0.373812	−	0.927504i	$-0.378050\pi$
$751$	1.15554e7	0.747625	0.373812	+	0.927504i	$0.378050\pi$
$752$	−5.41440e6	−0.349145
$753$	0	0
$754$	−1.61441e6	−0.103415
$755$	0	0
$756$	0	0
$757$	6.34402e6	0.402369	0.201185	−	0.979553i	$-0.435521\pi$
$757$	6.34402e6	0.402369	0.201185	+	0.979553i	$0.435521\pi$
$758$	2.38024e6	0.150469
$759$	0	0
$760$	0	0
$761$	1.17748e7	0.737040	0.368520	−	0.929620i	$-0.379865\pi$
$761$	1.17748e7	0.737040	0.368520	+	0.929620i	$0.379865\pi$
$762$	0	0
$763$	−1.63016e6	−0.101373
$764$	3.35510e6	0.207957
$765$	0	0
$766$	−5.20200e6	−0.320331
$767$	3.47295e6	0.213162
$768$	0	0
$769$	−2.47869e7	−1.51149	−0.755745	−	0.654865i	$-0.772725\pi$
$769$	−2.47869e7	−1.51149	−0.755745	+	0.654865i	$0.772725\pi$
$770$	0	0
$771$	0	0
$772$	1.14496e7	0.691425
$773$	−1.38017e7	−0.830774	−0.415387	−	0.909645i	$-0.636354\pi$
$773$	−1.38017e7	−0.830774	−0.415387	+	0.909645i	$0.636354\pi$
$774$	0	0
$775$	0	0
$776$	3.66995e6	0.218779
$777$	0	0
$778$	−2.48290e6	−0.147065
$779$	1.29245e7	0.763081
$780$	0	0
$781$	5.08500e6	0.298307
$782$	1.09836e7	0.642285
$783$	0	0
$784$	−2.70490e6	−0.157167
$785$	0	0
$786$	0	0
$787$	1.53795e7	0.885129	0.442565	−	0.896737i	$-0.354069\pi$
$787$	1.53795e7	0.885129	0.442565	+	0.896737i	$0.354069\pi$
$788$	8.14166e6	0.467087
$789$	0	0
$790$	0	0
$791$	1.39745e7	0.794135
$792$	0	0
$793$	−6.93426e6	−0.391577
$794$	−4.74489e6	−0.267101
$795$	0	0
$796$	−1.57763e7	−0.882514
$797$	−2.33978e7	−1.30475	−0.652377	−	0.757894i	$-0.726228\pi$
$797$	−2.33978e7	−1.30475	−0.652377	+	0.757894i	$0.726228\pi$
$798$	0	0
$799$	4.30191e7	2.38393
$800$	0	0
$801$	0	0
$802$	−1.25334e7	−0.688069
$803$	1.23663e7	0.676785
$804$	0	0
$805$	0	0
$806$	−390724.	−0.0211852
$807$	0	0
$808$	5.82605e6	0.313939
$809$	2.80188e7	1.50515	0.752573	−	0.658509i	$-0.228813\pi$
$809$	2.80188e7	1.50515	0.752573	+	0.658509i	$0.228813\pi$
$810$	0	0
$811$	−1.65641e6	−0.0884332	−0.0442166	−	0.999022i	$-0.514079\pi$
$811$	−1.65641e6	−0.0884332	−0.0442166	+	0.999022i	$0.514079\pi$
$812$	3.72374e6	0.198194
$813$	0	0
$814$	−1.94280e6	−0.102770
$815$	0	0
$816$	0	0
$817$	−3.85511e7	−2.02060
$818$	2.27032e6	0.118633
$819$	0	0
$820$	0	0
$821$	−2.97382e7	−1.53978	−0.769888	−	0.638179i	$-0.779688\pi$
$821$	−2.97382e7	−1.53978	−0.769888	+	0.638179i	$0.779688\pi$
$822$	0	0
$823$	1.60108e7	0.823972	0.411986	−	0.911190i	$-0.364835\pi$
$823$	1.60108e7	0.823972	0.411986	+	0.911190i	$0.364835\pi$
$824$	−1.22647e7	−0.629273
$825$	0	0
$826$	−8.01060e6	−0.408522
$827$	2.54748e7	1.29523	0.647614	−	0.761969i	$-0.275767\pi$
$827$	2.54748e7	1.29523	0.647614	+	0.761969i	$0.275767\pi$
$828$	0	0
$829$	1.89971e7	0.960064	0.480032	−	0.877251i	$-0.340625\pi$
$829$	1.89971e7	0.960064	0.480032	+	0.877251i	$0.340625\pi$
$830$	0	0
$831$	0	0
$832$	−561152.	−0.0281043
$833$	2.14912e7	1.07312
$834$	0	0
$835$	0	0
$836$	4.72560e6	0.233852
$837$	0	0
$838$	2.43600e7	1.19830
$839$	−3.98539e7	−1.95463	−0.977317	−	0.211782i	$-0.932073\pi$
$839$	−3.98539e7	−1.95463	−0.977317	+	0.211782i	$0.932073\pi$
$840$	0	0
$841$	−1.18322e7	−0.576868
$842$	1.22856e7	0.597193
$843$	0	0
$844$	1.91192e6	0.0923876
$845$	0	0
$846$	0	0
$847$	−1.09455e7	−0.524238
$848$	−6.62938e6	−0.316580
$849$	0	0
$850$	0	0
$851$	−4.37130e6	−0.206912
$852$	0	0
$853$	−2.19179e7	−1.03140	−0.515700	−	0.856769i	$-0.672468\pi$
$853$	−2.19179e7	−1.03140	−0.515700	+	0.856769i	$0.672468\pi$
$854$	1.59943e7	0.750450
$855$	0	0
$856$	594432.	0.0277280
$857$	−2.72213e7	−1.26607	−0.633033	−	0.774125i	$-0.718190\pi$
$857$	−2.72213e7	−1.26607	−0.633033	+	0.774125i	$0.718190\pi$
$858$	0	0
$859$	3.76214e7	1.73961	0.869805	−	0.493396i	$-0.164245\pi$
$859$	3.76214e7	1.73961	0.869805	+	0.493396i	$0.164245\pi$
$860$	0	0
$861$	0	0
$862$	2.67660e6	0.122692
$863$	2.51995e7	1.15177	0.575885	−	0.817531i	$-0.304658\pi$
$863$	2.51995e7	1.15177	0.575885	+	0.817531i	$0.304658\pi$
$864$	0	0
$865$	0	0
$866$	1.30176e7	0.589841
$867$	0	0
$868$	901232.	0.0406011
$869$	1.22208e7	0.548972
$870$	0	0
$871$	−3.08510e6	−0.137792
$872$	−1.32064e6	−0.0588157
$873$	0	0
$874$	1.06326e7	0.470826
$875$	0	0
$876$	0	0
$877$	−2.77307e6	−0.121748	−0.0608739	−	0.998145i	$-0.519389\pi$
$877$	−2.77307e6	−0.121748	−0.0608739	+	0.998145i	$0.519389\pi$
$878$	6.35037e6	0.278012
$879$	0	0
$880$	0	0
$881$	1.37827e7	0.598268	0.299134	−	0.954211i	$-0.403302\pi$
$881$	1.37827e7	0.598268	0.299134	+	0.954211i	$0.403302\pi$
$882$	0	0
$883$	4.96322e6	0.214221	0.107110	−	0.994247i	$-0.465840\pi$
$883$	4.96322e6	0.214221	0.107110	+	0.994247i	$0.465840\pi$
$884$	4.45853e6	0.191894
$885$	0	0
$886$	−855552.	−0.0366153
$887$	4.12640e7	1.76101	0.880505	−	0.474037i	$-0.157204\pi$
$887$	4.12640e7	1.76101	0.880505	+	0.474037i	$0.157204\pi$
$888$	0	0
$889$	2.02619e7	0.859857
$890$	0	0
$891$	0	0
$892$	−776720.	−0.0326853
$893$	4.16444e7	1.74754
$894$	0	0
$895$	0	0
$896$	1.29434e6	0.0538613
$897$	0	0
$898$	−1.48290e7	−0.613649
$899$	2.10050e6	0.0866808
$900$	0	0
$901$	5.26725e7	2.16158
$902$	3.93840e6	0.161177
$903$	0	0
$904$	1.13211e7	0.460752
$905$	0	0
$906$	0	0
$907$	−1.66308e7	−0.671268	−0.335634	−	0.941992i	$-0.608951\pi$
$907$	−1.66308e7	−0.671268	−0.335634	+	0.941992i	$0.608951\pi$
$908$	−4.60243e6	−0.185256
$909$	0	0
$910$	0	0
$911$	−3.64995e7	−1.45711	−0.728554	−	0.684989i	$-0.759807\pi$
$911$	−3.64995e7	−1.45711	−0.728554	+	0.684989i	$0.759807\pi$
$912$	0	0
$913$	−3.86730e6	−0.153543
$914$	−1.53277e7	−0.606892
$915$	0	0
$916$	1.15304e6	0.0454052
$917$	−1.40304e7	−0.550994
$918$	0	0
$919$	6.83190e6	0.266841	0.133421	−	0.991060i	$-0.457404\pi$
$919$	6.83190e6	0.266841	0.133421	+	0.991060i	$0.457404\pi$
$920$	0	0
$921$	0	0
$922$	−2.23469e7	−0.865744
$923$	4.64430e6	0.179439
$924$	0	0
$925$	0	0
$926$	1.76057e7	0.674723
$927$	0	0
$928$	3.01670e6	0.114991
$929$	−5.05115e6	−0.192022	−0.0960111	−	0.995380i	$-0.530608\pi$
$929$	−5.05115e6	−0.192022	−0.0960111	+	0.995380i	$0.530608\pi$
$930$	0	0
$931$	2.08045e7	0.786651
$932$	−9.10637e6	−0.343404
$933$	0	0
$934$	1.34090e7	0.502956
$935$	0	0
$936$	0	0
$937$	−8.30023e6	−0.308845	−0.154423	−	0.988005i	$-0.549352\pi$
$937$	−8.30023e6	−0.308845	−0.154423	+	0.988005i	$0.549352\pi$
$938$	7.11600e6	0.264076
$939$	0	0
$940$	0	0
$941$	4.87332e7	1.79412	0.897059	−	0.441910i	$-0.145699\pi$
$941$	4.87332e7	1.79412	0.897059	+	0.441910i	$0.145699\pi$
$942$	0	0
$943$	8.86140e6	0.324506
$944$	−6.48960e6	−0.237022
$945$	0	0
$946$	−1.17474e7	−0.426790
$947$	−7.34777e6	−0.266245	−0.133122	−	0.991100i	$-0.542500\pi$
$947$	−7.34777e6	−0.266245	−0.133122	+	0.991100i	$0.542500\pi$
$948$	0	0
$949$	1.12946e7	0.407102
$950$	0	0
$951$	0	0
$952$	−1.02839e7	−0.367761
$953$	−3.51437e7	−1.25347	−0.626737	−	0.779231i	$-0.715610\pi$
$953$	−3.51437e7	−1.25347	−0.626737	+	0.779231i	$0.715610\pi$
$954$	0	0
$955$	0	0
$956$	−1.11441e7	−0.394365
$957$	0	0
$958$	−3.63986e6	−0.128136
$959$	2.06100e7	0.723655
$960$	0	0
$961$	−2.81208e7	−0.982243
$962$	−1.77442e6	−0.0618187
$963$	0	0
$964$	9.21819e6	0.319487
$965$	0	0
$966$	0	0
$967$	−2.31186e6	−0.0795050	−0.0397525	−	0.999210i	$-0.512657\pi$
$967$	−2.31186e6	−0.0795050	−0.0397525	+	0.999210i	$0.512657\pi$
$968$	−8.86726e6	−0.304159
$969$	0	0
$970$	0	0
$971$	3.73588e7	1.27158	0.635791	−	0.771861i	$-0.280674\pi$
$971$	3.73588e7	1.27158	0.635791	+	0.771861i	$0.280674\pi$
$972$	0	0
$973$	−1.71970e7	−0.582333
$974$	1.61432e7	0.545245
$975$	0	0
$976$	1.29574e7	0.435406
$977$	−1.02666e7	−0.344104	−0.172052	−	0.985088i	$-0.555040\pi$
$977$	−1.02666e7	−0.344104	−0.172052	+	0.985088i	$0.555040\pi$
$978$	0	0
$979$	1.55592e7	0.518837
$980$	0	0
$981$	0	0
$982$	−1.51978e6	−0.0502922
$983$	2.98404e7	0.984966	0.492483	−	0.870322i	$-0.336089\pi$
$983$	2.98404e7	0.984966	0.492483	+	0.870322i	$0.336089\pi$
$984$	0	0
$985$	0	0
$986$	−2.39687e7	−0.785148
$987$	0	0
$988$	4.31605e6	0.140668
$989$	−2.64316e7	−0.859278
$990$	0	0
$991$	−2.34222e7	−0.757605	−0.378803	−	0.925477i	$-0.623664\pi$
$991$	−2.34222e7	−0.757605	−0.378803	+	0.925477i	$0.623664\pi$
$992$	730112.	0.0235565
$993$	0	0
$994$	−1.07124e7	−0.343891
$995$	0	0
$996$	0	0
$997$	2.60709e7	0.830651	0.415326	−	0.909673i	$-0.363668\pi$
$997$	2.60709e7	0.830651	0.415326	+	0.909673i	$0.363668\pi$
$998$	−2.96973e7	−0.943824
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.t.1.1		1
3.2	odd	2		150.6.a.g.1.1	✓	1
5.2	odd	4		450.6.c.g.199.2		2
5.3	odd	4		450.6.c.g.199.1		2
5.4	even	2		450.6.a.e.1.1		1
15.2	even	4		150.6.c.c.49.1		2
15.8	even	4		150.6.c.c.49.2		2
15.14	odd	2		150.6.a.i.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.g.1.1	✓	1	3.2	odd	2
150.6.a.i.1.1	yes	1	15.14	odd	2
150.6.c.c.49.1		2	15.2	even	4
150.6.c.c.49.2		2	15.8	even	4
450.6.a.e.1.1		1	5.4	even	2
450.6.a.t.1.1		1	1.1	even	1	trivial
450.6.c.g.199.1		2	5.3	odd	4
450.6.c.g.199.2		2	5.2	odd	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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