LMFDB - Embedded newform 450.6.a.h.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +22.0000 q^{7} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} +22.0000 q^{7} -64.0000 q^{8} +768.000 q^{11} +46.0000 q^{13} -88.0000 q^{14} +256.000 q^{16} +378.000 q^{17} +1100.00 q^{19} -3072.00 q^{22} -1986.00 q^{23} -184.000 q^{26} +352.000 q^{28} +5610.00 q^{29} -3988.00 q^{31} -1024.00 q^{32} -1512.00 q^{34} +142.000 q^{37} -4400.00 q^{38} -1542.00 q^{41} +5026.00 q^{43} +12288.0 q^{44} +7944.00 q^{46} +24738.0 q^{47} -16323.0 q^{49} +736.000 q^{52} -14166.0 q^{53} -1408.00 q^{56} -22440.0 q^{58} -28380.0 q^{59} +5522.00 q^{61} +15952.0 q^{62} +4096.00 q^{64} +24742.0 q^{67} +6048.00 q^{68} -42372.0 q^{71} +52126.0 q^{73} -568.000 q^{74} +17600.0 q^{76} +16896.0 q^{77} -39640.0 q^{79} +6168.00 q^{82} -59826.0 q^{83} -20104.0 q^{86} -49152.0 q^{88} -57690.0 q^{89} +1012.00 q^{91} -31776.0 q^{92} -98952.0 q^{94} +144382. q^{97} +65292.0 q^{98} +O(q^{100})$

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
$7$	22.0000	0.169698	0.0848492	−	0.996394i	$-0.472959\pi$
$7$	22.0000	0.169698	0.0848492	+	0.996394i	$0.472959\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	768.000	1.91372	0.956862	−	0.290541i	$-0.0938354\pi$
$11$	768.000	1.91372	0.956862	+	0.290541i	$0.0938354\pi$
$12$	0	0
$13$	46.0000	0.0754917	0.0377459	−	0.999287i	$-0.487982\pi$
$13$	46.0000	0.0754917	0.0377459	+	0.999287i	$0.487982\pi$
$14$	−88.0000	−0.119995
$15$	0	0
$16$	256.000	0.250000
$17$	378.000	0.317227	0.158613	−	0.987341i	$-0.449298\pi$
$17$	378.000	0.317227	0.158613	+	0.987341i	$0.449298\pi$
$18$	0	0
$19$	1100.00	0.699051	0.349525	−	0.936927i	$-0.386343\pi$
$19$	1100.00	0.699051	0.349525	+	0.936927i	$0.386343\pi$
$20$	0	0
$21$	0	0
$22$	−3072.00	−1.35321
$23$	−1986.00	−0.782816	−0.391408	−	0.920217i	$-0.628012\pi$
$23$	−1986.00	−0.782816	−0.391408	+	0.920217i	$0.628012\pi$
$24$	0	0
$25$	0	0
$26$	−184.000	−0.0533807
$27$	0	0
$28$	352.000	0.0848492
$29$	5610.00	1.23870	0.619352	−	0.785113i	$-0.287395\pi$
$29$	5610.00	1.23870	0.619352	+	0.785113i	$0.287395\pi$
$30$	0	0
$31$	−3988.00	−0.745334	−0.372667	−	0.927965i	$-0.621557\pi$
$31$	−3988.00	−0.745334	−0.372667	+	0.927965i	$0.621557\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−1512.00	−0.224313
$35$	0	0
$36$	0	0
$37$	142.000	0.0170523	0.00852617	−	0.999964i	$-0.497286\pi$
$37$	142.000	0.0170523	0.00852617	+	0.999964i	$0.497286\pi$
$38$	−4400.00	−0.494303
$39$	0	0
$40$	0	0
$41$	−1542.00	−0.143260	−0.0716300	−	0.997431i	$-0.522820\pi$
$41$	−1542.00	−0.143260	−0.0716300	+	0.997431i	$0.522820\pi$
$42$	0	0
$43$	5026.00	0.414526	0.207263	−	0.978285i	$-0.433544\pi$
$43$	5026.00	0.414526	0.207263	+	0.978285i	$0.433544\pi$
$44$	12288.0	0.956862
$45$	0	0
$46$	7944.00	0.553534
$47$	24738.0	1.63350	0.816752	−	0.576990i	$-0.195773\pi$
$47$	24738.0	1.63350	0.816752	+	0.576990i	$0.195773\pi$
$48$	0	0
$49$	−16323.0	−0.971202
$50$	0	0
$51$	0	0
$52$	736.000	0.0377459
$53$	−14166.0	−0.692720	−0.346360	−	0.938102i	$-0.612582\pi$
$53$	−14166.0	−0.692720	−0.346360	+	0.938102i	$0.612582\pi$
$54$	0	0
$55$	0	0
$56$	−1408.00	−0.0599974
$57$	0	0
$58$	−22440.0	−0.875897
$59$	−28380.0	−1.06141	−0.530704	−	0.847557i	$-0.678072\pi$
$59$	−28380.0	−1.06141	−0.530704	+	0.847557i	$0.678072\pi$
$60$	0	0
$61$	5522.00	0.190008	0.0950040	−	0.995477i	$-0.469714\pi$
$61$	5522.00	0.190008	0.0950040	+	0.995477i	$0.469714\pi$
$62$	15952.0	0.527031
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	24742.0	0.673361	0.336680	−	0.941619i	$-0.390696\pi$
$67$	24742.0	0.673361	0.336680	+	0.941619i	$0.390696\pi$
$68$	6048.00	0.158613
$69$	0	0
$70$	0	0
$71$	−42372.0	−0.997546	−0.498773	−	0.866733i	$-0.666216\pi$
$71$	−42372.0	−0.997546	−0.498773	+	0.866733i	$0.666216\pi$
$72$	0	0
$73$	52126.0	1.14485	0.572423	−	0.819958i	$-0.306003\pi$
$73$	52126.0	1.14485	0.572423	+	0.819958i	$0.306003\pi$
$74$	−568.000	−0.0120578
$75$	0	0
$76$	17600.0	0.349525
$77$	16896.0	0.324756
$78$	0	0
$79$	−39640.0	−0.714605	−0.357302	−	0.933989i	$-0.616303\pi$
$79$	−39640.0	−0.714605	−0.357302	+	0.933989i	$0.616303\pi$
$80$	0	0
$81$	0	0
$82$	6168.00	0.101300
$83$	−59826.0	−0.953223	−0.476612	−	0.879114i	$-0.658135\pi$
$83$	−59826.0	−0.953223	−0.476612	+	0.879114i	$0.658135\pi$
$84$	0	0
$85$	0	0
$86$	−20104.0	−0.293114
$87$	0	0
$88$	−49152.0	−0.676604
$89$	−57690.0	−0.772015	−0.386007	−	0.922496i	$-0.626146\pi$
$89$	−57690.0	−0.772015	−0.386007	+	0.922496i	$0.626146\pi$
$90$	0	0
$91$	1012.00	0.0128108
$92$	−31776.0	−0.391408
$93$	0	0
$94$	−98952.0	−1.15506
$95$	0	0
$96$	0	0
$97$	144382.	1.55806	0.779029	−	0.626988i	$-0.215712\pi$
$97$	144382.	1.55806	0.779029	+	0.626988i	$0.215712\pi$
$98$	65292.0	0.686744
$99$	0	0
$100$	0	0
$101$	141258.	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$101$	141258.	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$102$	0	0
$103$	−139814.	−1.29855	−0.649273	−	0.760555i	$-0.724927\pi$
$103$	−139814.	−1.29855	−0.649273	+	0.760555i	$0.724927\pi$
$104$	−2944.00	−0.0266904
$105$	0	0
$106$	56664.0	0.489827
$107$	86418.0	0.729701	0.364850	−	0.931066i	$-0.381120\pi$
$107$	86418.0	0.729701	0.364850	+	0.931066i	$0.381120\pi$
$108$	0	0
$109$	218450.	1.76111	0.880554	−	0.473947i	$-0.157171\pi$
$109$	218450.	1.76111	0.880554	+	0.473947i	$0.157171\pi$
$110$	0	0
$111$	0	0
$112$	5632.00	0.0424246
$113$	−28806.0	−0.212220	−0.106110	−	0.994354i	$-0.533840\pi$
$113$	−28806.0	−0.212220	−0.106110	+	0.994354i	$0.533840\pi$
$114$	0	0
$115$	0	0
$116$	89760.0	0.619352
$117$	0	0
$118$	113520.	0.750529
$119$	8316.00	0.0538328
$120$	0	0
$121$	428773.	2.66234
$122$	−22088.0	−0.134356
$123$	0	0
$124$	−63808.0	−0.372667
$125$	0	0
$126$	0	0
$127$	216502.	1.19111	0.595556	−	0.803314i	$-0.296932\pi$
$127$	216502.	1.19111	0.595556	+	0.803314i	$0.296932\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	244608.	1.24535	0.622676	−	0.782479i	$-0.286045\pi$
$131$	244608.	1.24535	0.622676	+	0.782479i	$0.286045\pi$
$132$	0	0
$133$	24200.0	0.118628
$134$	−98968.0	−0.476138
$135$	0	0
$136$	−24192.0	−0.112157
$137$	−239502.	−1.09020	−0.545102	−	0.838370i	$-0.683509\pi$
$137$	−239502.	−1.09020	−0.545102	+	0.838370i	$0.683509\pi$
$138$	0	0
$139$	30860.0	0.135475	0.0677375	−	0.997703i	$-0.478422\pi$
$139$	30860.0	0.135475	0.0677375	+	0.997703i	$0.478422\pi$
$140$	0	0
$141$	0	0
$142$	169488.	0.705372
$143$	35328.0	0.144470
$144$	0	0
$145$	0	0
$146$	−208504.	−0.809529
$147$	0	0
$148$	2272.00	0.00852617
$149$	100950.	0.372512	0.186256	−	0.982501i	$-0.440365\pi$
$149$	100950.	0.372512	0.186256	+	0.982501i	$0.440365\pi$
$150$	0	0
$151$	12452.0	0.0444423	0.0222212	−	0.999753i	$-0.492926\pi$
$151$	12452.0	0.0444423	0.0222212	+	0.999753i	$0.492926\pi$
$152$	−70400.0	−0.247152
$153$	0	0
$154$	−67584.0	−0.229637
$155$	0	0
$156$	0	0
$157$	6022.00	0.0194981	0.00974903	−	0.999952i	$-0.496897\pi$
$157$	6022.00	0.0194981	0.00974903	+	0.999952i	$0.496897\pi$
$158$	158560.	0.505302
$159$	0	0
$160$	0	0
$161$	−43692.0	−0.132843
$162$	0	0
$163$	500866.	1.47656	0.738282	−	0.674492i	$-0.235637\pi$
$163$	500866.	1.47656	0.738282	+	0.674492i	$0.235637\pi$
$164$	−24672.0	−0.0716300
$165$	0	0
$166$	239304.	0.674031
$167$	555258.	1.54065	0.770324	−	0.637652i	$-0.220094\pi$
$167$	555258.	1.54065	0.770324	+	0.637652i	$0.220094\pi$
$168$	0	0
$169$	−369177.	−0.994301
$170$	0	0
$171$	0	0
$172$	80416.0	0.207263
$173$	417354.	1.06020	0.530102	−	0.847934i	$-0.322154\pi$
$173$	417354.	1.06020	0.530102	+	0.847934i	$0.322154\pi$
$174$	0	0
$175$	0	0
$176$	196608.	0.478431
$177$	0	0
$178$	230760.	0.545897
$179$	52380.0	0.122189	0.0610946	−	0.998132i	$-0.480541\pi$
$179$	52380.0	0.122189	0.0610946	+	0.998132i	$0.480541\pi$
$180$	0	0
$181$	546662.	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$181$	546662.	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$182$	−4048.00	−0.00905862
$183$	0	0
$184$	127104.	0.276767
$185$	0	0
$186$	0	0
$187$	290304.	0.607084
$188$	395808.	0.816752
$189$	0	0
$190$	0	0
$191$	452028.	0.896565	0.448283	−	0.893892i	$-0.352036\pi$
$191$	452028.	0.896565	0.448283	+	0.893892i	$0.352036\pi$
$192$	0	0
$193$	−485594.	−0.938383	−0.469191	−	0.883097i	$-0.655455\pi$
$193$	−485594.	−0.938383	−0.469191	+	0.883097i	$0.655455\pi$
$194$	−577528.	−1.10171
$195$	0	0
$196$	−261168.	−0.485601
$197$	1.01018e6	1.85452	0.927262	−	0.374414i	$-0.122156\pi$
$197$	1.01018e6	1.85452	0.927262	+	0.374414i	$0.122156\pi$
$198$	0	0
$199$	−807640.	−1.44572	−0.722862	−	0.690993i	$-0.757174\pi$
$199$	−807640.	−1.44572	−0.722862	+	0.690993i	$0.757174\pi$
$200$	0	0
$201$	0	0
$202$	−565032.	−0.974304
$203$	123420.	0.210206
$204$	0	0
$205$	0	0
$206$	559256.	0.918211
$207$	0	0
$208$	11776.0	0.0188729
$209$	844800.	1.33779
$210$	0	0
$211$	149552.	0.231252	0.115626	−	0.993293i	$-0.463113\pi$
$211$	149552.	0.231252	0.115626	+	0.993293i	$0.463113\pi$
$212$	−226656.	−0.346360
$213$	0	0
$214$	−345672.	−0.515976
$215$	0	0
$216$	0	0
$217$	−87736.0	−0.126482
$218$	−873800.	−1.24529
$219$	0	0
$220$	0	0
$221$	17388.0	0.0239480
$222$	0	0
$223$	443506.	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$223$	443506.	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$224$	−22528.0	−0.0299987
$225$	0	0
$226$	115224.	0.150062
$227$	420018.	0.541007	0.270504	−	0.962719i	$-0.412810\pi$
$227$	420018.	0.541007	0.270504	+	0.962719i	$0.412810\pi$
$228$	0	0
$229$	1.05875e6	1.33415	0.667075	−	0.744990i	$-0.267546\pi$
$229$	1.05875e6	1.33415	0.667075	+	0.744990i	$0.267546\pi$
$230$	0	0
$231$	0	0
$232$	−359040.	−0.437948
$233$	−1.27345e6	−1.53671	−0.768353	−	0.640026i	$-0.778923\pi$
$233$	−1.27345e6	−1.53671	−0.768353	+	0.640026i	$0.778923\pi$
$234$	0	0
$235$	0	0
$236$	−454080.	−0.530704
$237$	0	0
$238$	−33264.0	−0.0380655
$239$	370680.	0.419763	0.209882	−	0.977727i	$-0.432692\pi$
$239$	370680.	0.419763	0.209882	+	0.977727i	$0.432692\pi$
$240$	0	0
$241$	−561298.	−0.622517	−0.311258	−	0.950325i	$-0.600750\pi$
$241$	−561298.	−0.622517	−0.311258	+	0.950325i	$0.600750\pi$
$242$	−1.71509e6	−1.88256
$243$	0	0
$244$	88352.0	0.0950040
$245$	0	0
$246$	0	0
$247$	50600.0	0.0527726
$248$	255232.	0.263515
$249$	0	0
$250$	0	0
$251$	−577152.	−0.578237	−0.289119	−	0.957293i	$-0.593362\pi$
$251$	−577152.	−0.578237	−0.289119	+	0.957293i	$0.593362\pi$
$252$	0	0
$253$	−1.52525e6	−1.49809
$254$	−866008.	−0.842243
$255$	0	0
$256$	65536.0	0.0625000
$257$	−651462.	−0.615257	−0.307628	−	0.951507i	$-0.599535\pi$
$257$	−651462.	−0.615257	−0.307628	+	0.951507i	$0.599535\pi$
$258$	0	0
$259$	3124.00	0.00289375
$260$	0	0
$261$	0	0
$262$	−978432.	−0.880597
$263$	917574.	0.817997	0.408999	−	0.912535i	$-0.365878\pi$
$263$	917574.	0.817997	0.408999	+	0.912535i	$0.365878\pi$
$264$	0	0
$265$	0	0
$266$	−96800.0	−0.0838825
$267$	0	0
$268$	395872.	0.336680
$269$	735390.	0.619637	0.309818	−	0.950796i	$-0.399732\pi$
$269$	735390.	0.619637	0.309818	+	0.950796i	$0.399732\pi$
$270$	0	0
$271$	−1.12131e6	−0.927474	−0.463737	−	0.885973i	$-0.653492\pi$
$271$	−1.12131e6	−0.927474	−0.463737	+	0.885973i	$0.653492\pi$
$272$	96768.0	0.0793066
$273$	0	0
$274$	958008.	0.770891
$275$	0	0
$276$	0	0
$277$	1.66034e6	1.30016	0.650082	−	0.759864i	$-0.274735\pi$
$277$	1.66034e6	1.30016	0.650082	+	0.759864i	$0.274735\pi$
$278$	−123440.	−0.0957952
$279$	0	0
$280$	0	0
$281$	−1.45210e6	−1.09706	−0.548531	−	0.836130i	$-0.684813\pi$
$281$	−1.45210e6	−1.09706	−0.548531	+	0.836130i	$0.684813\pi$
$282$	0	0
$283$	−309014.	−0.229357	−0.114679	−	0.993403i	$-0.536584\pi$
$283$	−309014.	−0.229357	−0.114679	+	0.993403i	$0.536584\pi$
$284$	−677952.	−0.498773
$285$	0	0
$286$	−141312.	−0.102156
$287$	−33924.0	−0.0243110
$288$	0	0
$289$	−1.27697e6	−0.899367
$290$	0	0
$291$	0	0
$292$	834016.	0.572423
$293$	−1.59301e6	−1.08405	−0.542024	−	0.840363i	$-0.682342\pi$
$293$	−1.59301e6	−1.08405	−0.542024	+	0.840363i	$0.682342\pi$
$294$	0	0
$295$	0	0
$296$	−9088.00	−0.00602891
$297$	0	0
$298$	−403800.	−0.263406
$299$	−91356.0	−0.0590961
$300$	0	0
$301$	110572.	0.0703443
$302$	−49808.0	−0.0314255
$303$	0	0
$304$	281600.	0.174763
$305$	0	0
$306$	0	0
$307$	−1.24726e6	−0.755284	−0.377642	−	0.925952i	$-0.623265\pi$
$307$	−1.24726e6	−0.755284	−0.377642	+	0.925952i	$0.623265\pi$
$308$	270336.	0.162378
$309$	0	0
$310$	0	0
$311$	665988.	0.390450	0.195225	−	0.980758i	$-0.437456\pi$
$311$	665988.	0.390450	0.195225	+	0.980758i	$0.437456\pi$
$312$	0	0
$313$	591286.	0.341143	0.170572	−	0.985345i	$-0.445439\pi$
$313$	591286.	0.341143	0.170572	+	0.985345i	$0.445439\pi$
$314$	−24088.0	−0.0137872
$315$	0	0
$316$	−634240.	−0.357302
$317$	−516342.	−0.288595	−0.144298	−	0.989534i	$-0.546092\pi$
$317$	−516342.	−0.288595	−0.144298	+	0.989534i	$0.546092\pi$
$318$	0	0
$319$	4.30848e6	2.37054
$320$	0	0
$321$	0	0
$322$	174768.	0.0939339
$323$	415800.	0.221757
$324$	0	0
$325$	0	0
$326$	−2.00346e6	−1.04409
$327$	0	0
$328$	98688.0	0.0506500
$329$	544236.	0.277203
$330$	0	0
$331$	−3.29577e6	−1.65343	−0.826717	−	0.562619i	$-0.809794\pi$
$331$	−3.29577e6	−1.65343	−0.826717	+	0.562619i	$0.809794\pi$
$332$	−957216.	−0.476612
$333$	0	0
$334$	−2.22103e6	−1.08940
$335$	0	0
$336$	0	0
$337$	−1.91098e6	−0.916602	−0.458301	−	0.888797i	$-0.651542\pi$
$337$	−1.91098e6	−0.916602	−0.458301	+	0.888797i	$0.651542\pi$
$338$	1.47671e6	0.703077
$339$	0	0
$340$	0	0
$341$	−3.06278e6	−1.42636
$342$	0	0
$343$	−728860.	−0.334510
$344$	−321664.	−0.146557
$345$	0	0
$346$	−1.66942e6	−0.749677
$347$	2.42006e6	1.07895	0.539476	−	0.842001i	$-0.318622\pi$
$347$	2.42006e6	1.07895	0.539476	+	0.842001i	$0.318622\pi$
$348$	0	0
$349$	2.50727e6	1.10189	0.550944	−	0.834542i	$-0.314268\pi$
$349$	2.50727e6	1.10189	0.550944	+	0.834542i	$0.314268\pi$
$350$	0	0
$351$	0	0
$352$	−786432.	−0.338302
$353$	−413166.	−0.176477	−0.0882384	−	0.996099i	$-0.528124\pi$
$353$	−413166.	−0.176477	−0.0882384	+	0.996099i	$0.528124\pi$
$354$	0	0
$355$	0	0
$356$	−923040.	−0.386007
$357$	0	0
$358$	−209520.	−0.0864008
$359$	−1.73772e6	−0.711613	−0.355806	−	0.934560i	$-0.615794\pi$
$359$	−1.73772e6	−0.711613	−0.355806	+	0.934560i	$0.615794\pi$
$360$	0	0
$361$	−1.26610e6	−0.511328
$362$	−2.18665e6	−0.877016
$363$	0	0
$364$	16192.0	0.00640541
$365$	0	0
$366$	0	0
$367$	−1.16098e6	−0.449944	−0.224972	−	0.974365i	$-0.572229\pi$
$367$	−1.16098e6	−0.449944	−0.224972	+	0.974365i	$0.572229\pi$
$368$	−508416.	−0.195704
$369$	0	0
$370$	0	0
$371$	−311652.	−0.117553
$372$	0	0
$373$	−343754.	−0.127931	−0.0639655	−	0.997952i	$-0.520375\pi$
$373$	−343754.	−0.127931	−0.0639655	+	0.997952i	$0.520375\pi$
$374$	−1.16122e6	−0.429273
$375$	0	0
$376$	−1.58323e6	−0.577531
$377$	258060.	0.0935120
$378$	0	0
$379$	573140.	0.204957	0.102478	−	0.994735i	$-0.467323\pi$
$379$	573140.	0.204957	0.102478	+	0.994735i	$0.467323\pi$
$380$	0	0
$381$	0	0
$382$	−1.80811e6	−0.633967
$383$	−2.88055e6	−1.00341	−0.501704	−	0.865039i	$-0.667293\pi$
$383$	−2.88055e6	−1.00341	−0.501704	+	0.865039i	$0.667293\pi$
$384$	0	0
$385$	0	0
$386$	1.94238e6	0.663537
$387$	0	0
$388$	2.31011e6	0.779029
$389$	3.08559e6	1.03387	0.516933	−	0.856026i	$-0.327074\pi$
$389$	3.08559e6	1.03387	0.516933	+	0.856026i	$0.327074\pi$
$390$	0	0
$391$	−750708.	−0.248330
$392$	1.04467e6	0.343372
$393$	0	0
$394$	−4.04071e6	−1.31135
$395$	0	0
$396$	0	0
$397$	−885458.	−0.281963	−0.140981	−	0.990012i	$-0.545026\pi$
$397$	−885458.	−0.281963	−0.140981	+	0.990012i	$0.545026\pi$
$398$	3.23056e6	1.02228
$399$	0	0
$400$	0	0
$401$	3.75344e6	1.16565	0.582825	−	0.812598i	$-0.301947\pi$
$401$	3.75344e6	1.16565	0.582825	+	0.812598i	$0.301947\pi$
$402$	0	0
$403$	−183448.	−0.0562666
$404$	2.26013e6	0.688937
$405$	0	0
$406$	−493680.	−0.148638
$407$	109056.	0.0326335
$408$	0	0
$409$	−1.94653e6	−0.575377	−0.287689	−	0.957724i	$-0.592887\pi$
$409$	−1.94653e6	−0.575377	−0.287689	+	0.957724i	$0.592887\pi$
$410$	0	0
$411$	0	0
$412$	−2.23702e6	−0.649273
$413$	−624360.	−0.180119
$414$	0	0
$415$	0	0
$416$	−47104.0	−0.0133452
$417$	0	0
$418$	−3.37920e6	−0.945961
$419$	2.99166e6	0.832486	0.416243	−	0.909253i	$-0.363346\pi$
$419$	2.99166e6	0.832486	0.416243	+	0.909253i	$0.363346\pi$
$420$	0	0
$421$	3.96660e6	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$421$	3.96660e6	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$422$	−598208.	−0.163520
$423$	0	0
$424$	906624.	0.244913
$425$	0	0
$426$	0	0
$427$	121484.	0.0322440
$428$	1.38269e6	0.364850
$429$	0	0
$430$	0	0
$431$	5.17115e6	1.34089	0.670446	−	0.741958i	$-0.266103\pi$
$431$	5.17115e6	1.34089	0.670446	+	0.741958i	$0.266103\pi$
$432$	0	0
$433$	4.53485e6	1.16237	0.581183	−	0.813773i	$-0.302590\pi$
$433$	4.53485e6	1.16237	0.581183	+	0.813773i	$0.302590\pi$
$434$	350944.	0.0894362
$435$	0	0
$436$	3.49520e6	0.880554
$437$	−2.18460e6	−0.547228
$438$	0	0
$439$	−1.08220e6	−0.268007	−0.134004	−	0.990981i	$-0.542783\pi$
$439$	−1.08220e6	−0.268007	−0.134004	+	0.990981i	$0.542783\pi$
$440$	0	0
$441$	0	0
$442$	−69552.0	−0.0169338
$443$	−1.08079e6	−0.261656	−0.130828	−	0.991405i	$-0.541764\pi$
$443$	−1.08079e6	−0.261656	−0.130828	+	0.991405i	$0.541764\pi$
$444$	0	0
$445$	0	0
$446$	−1.77402e6	−0.422301
$447$	0	0
$448$	90112.0	0.0212123
$449$	−2.61783e6	−0.612810	−0.306405	−	0.951901i	$-0.599126\pi$
$449$	−2.61783e6	−0.612810	−0.306405	+	0.951901i	$0.599126\pi$
$450$	0	0
$451$	−1.18426e6	−0.274160
$452$	−460896.	−0.106110
$453$	0	0
$454$	−1.68007e6	−0.382550
$455$	0	0
$456$	0	0
$457$	−1.59046e6	−0.356231	−0.178115	−	0.984010i	$-0.557000\pi$
$457$	−1.59046e6	−0.356231	−0.178115	+	0.984010i	$0.557000\pi$
$458$	−4.23500e6	−0.943387
$459$	0	0
$460$	0	0
$461$	−4.25470e6	−0.932431	−0.466216	−	0.884671i	$-0.654383\pi$
$461$	−4.25470e6	−0.932431	−0.466216	+	0.884671i	$0.654383\pi$
$462$	0	0
$463$	−3.26605e6	−0.708061	−0.354031	−	0.935234i	$-0.615189\pi$
$463$	−3.26605e6	−0.708061	−0.354031	+	0.935234i	$0.615189\pi$
$464$	1.43616e6	0.309676
$465$	0	0
$466$	5.09378e6	1.08662
$467$	−601542.	−0.127636	−0.0638181	−	0.997962i	$-0.520328\pi$
$467$	−601542.	−0.127636	−0.0638181	+	0.997962i	$0.520328\pi$
$468$	0	0
$469$	544324.	0.114268
$470$	0	0
$471$	0	0
$472$	1.81632e6	0.375264
$473$	3.85997e6	0.793288
$474$	0	0
$475$	0	0
$476$	133056.	0.0269164
$477$	0	0
$478$	−1.48272e6	−0.296817
$479$	4.57932e6	0.911931	0.455966	−	0.889997i	$-0.349294\pi$
$479$	4.57932e6	0.911931	0.455966	+	0.889997i	$0.349294\pi$
$480$	0	0
$481$	6532.00	0.00128731
$482$	2.24519e6	0.440186
$483$	0	0
$484$	6.86037e6	1.33117
$485$	0	0
$486$	0	0
$487$	−7.05226e6	−1.34743	−0.673714	−	0.738992i	$-0.735302\pi$
$487$	−7.05226e6	−1.34743	−0.673714	+	0.738992i	$0.735302\pi$
$488$	−353408.	−0.0671780
$489$	0	0
$490$	0	0
$491$	2.62349e6	0.491106	0.245553	−	0.969383i	$-0.421030\pi$
$491$	2.62349e6	0.491106	0.245553	+	0.969383i	$0.421030\pi$
$492$	0	0
$493$	2.12058e6	0.392950
$494$	−202400.	−0.0373158
$495$	0	0
$496$	−1.02093e6	−0.186333
$497$	−932184.	−0.169282
$498$	0	0
$499$	−3.61234e6	−0.649437	−0.324719	−	0.945811i	$-0.605270\pi$
$499$	−3.61234e6	−0.649437	−0.324719	+	0.945811i	$0.605270\pi$
$500$	0	0
$501$	0	0
$502$	2.30861e6	0.408875
$503$	9.15629e6	1.61361	0.806807	−	0.590815i	$-0.201194\pi$
$503$	9.15629e6	1.61361	0.806807	+	0.590815i	$0.201194\pi$
$504$	0	0
$505$	0	0
$506$	6.10099e6	1.05931
$507$	0	0
$508$	3.46403e6	0.595556
$509$	−7.26159e6	−1.24233	−0.621165	−	0.783679i	$-0.713340\pi$
$509$	−7.26159e6	−1.24233	−0.621165	+	0.783679i	$0.713340\pi$
$510$	0	0
$511$	1.14677e6	0.194279
$512$	−262144.	−0.0441942
$513$	0	0
$514$	2.60585e6	0.435052
$515$	0	0
$516$	0	0
$517$	1.89988e7	3.12608
$518$	−12496.0	−0.00204619
$519$	0	0
$520$	0	0
$521$	−5.81020e6	−0.937771	−0.468886	−	0.883259i	$-0.655344\pi$
$521$	−5.81020e6	−0.937771	−0.468886	+	0.883259i	$0.655344\pi$
$522$	0	0
$523$	8.17067e6	1.30618	0.653090	−	0.757280i	$-0.273472\pi$
$523$	8.17067e6	1.30618	0.653090	+	0.757280i	$0.273472\pi$
$524$	3.91373e6	0.622676
$525$	0	0
$526$	−3.67030e6	−0.578411
$527$	−1.50746e6	−0.236440
$528$	0	0
$529$	−2.49215e6	−0.387199
$530$	0	0
$531$	0	0
$532$	387200.	0.0593139
$533$	−70932.0	−0.0108149
$534$	0	0
$535$	0	0
$536$	−1.58349e6	−0.238069
$537$	0	0
$538$	−2.94156e6	−0.438149
$539$	−1.25361e7	−1.85861
$540$	0	0
$541$	−817378.	−0.120069	−0.0600343	−	0.998196i	$-0.519121\pi$
$541$	−817378.	−0.120069	−0.0600343	+	0.998196i	$0.519121\pi$
$542$	4.48523e6	0.655823
$543$	0	0
$544$	−387072.	−0.0560783
$545$	0	0
$546$	0	0
$547$	3.50750e6	0.501221	0.250611	−	0.968088i	$-0.419369\pi$
$547$	3.50750e6	0.501221	0.250611	+	0.968088i	$0.419369\pi$
$548$	−3.83203e6	−0.545102
$549$	0	0
$550$	0	0
$551$	6.17100e6	0.865918
$552$	0	0
$553$	−872080.	−0.121267
$554$	−6.64137e6	−0.919355
$555$	0	0
$556$	493760.	0.0677375
$557$	9.61490e6	1.31313	0.656563	−	0.754271i	$-0.272009\pi$
$557$	9.61490e6	1.31313	0.656563	+	0.754271i	$0.272009\pi$
$558$	0	0
$559$	231196.	0.0312933
$560$	0	0
$561$	0	0
$562$	5.80841e6	0.775740
$563$	2.01941e6	0.268506	0.134253	−	0.990947i	$-0.457136\pi$
$563$	2.01941e6	0.268506	0.134253	+	0.990947i	$0.457136\pi$
$564$	0	0
$565$	0	0
$566$	1.23606e6	0.162180
$567$	0	0
$568$	2.71181e6	0.352686
$569$	−1.37859e6	−0.178507	−0.0892533	−	0.996009i	$-0.528448\pi$
$569$	−1.37859e6	−0.178507	−0.0892533	+	0.996009i	$0.528448\pi$
$570$	0	0
$571$	8.54295e6	1.09652	0.548261	−	0.836307i	$-0.315290\pi$
$571$	8.54295e6	1.09652	0.548261	+	0.836307i	$0.315290\pi$
$572$	565248.	0.0722352
$573$	0	0
$574$	135696.	0.0171905
$575$	0	0
$576$	0	0
$577$	2.31458e6	0.289423	0.144711	−	0.989474i	$-0.453775\pi$
$577$	2.31458e6	0.289423	0.144711	+	0.989474i	$0.453775\pi$
$578$	5.10789e6	0.635949
$579$	0	0
$580$	0	0
$581$	−1.31617e6	−0.161760
$582$	0	0
$583$	−1.08795e7	−1.32568
$584$	−3.33606e6	−0.404764
$585$	0	0
$586$	6.37202e6	0.766537
$587$	928338.	0.111202	0.0556008	−	0.998453i	$-0.482293\pi$
$587$	928338.	0.111202	0.0556008	+	0.998453i	$0.482293\pi$
$588$	0	0
$589$	−4.38680e6	−0.521026
$590$	0	0
$591$	0	0
$592$	36352.0	0.00426309
$593$	−909486.	−0.106209	−0.0531043	−	0.998589i	$-0.516912\pi$
$593$	−909486.	−0.106209	−0.0531043	+	0.998589i	$0.516912\pi$
$594$	0	0
$595$	0	0
$596$	1.61520e6	0.186256
$597$	0	0
$598$	365424.	0.0417873
$599$	8.51136e6	0.969241	0.484621	−	0.874724i	$-0.338958\pi$
$599$	8.51136e6	0.969241	0.484621	+	0.874724i	$0.338958\pi$
$600$	0	0
$601$	6.12498e6	0.691701	0.345851	−	0.938290i	$-0.387590\pi$
$601$	6.12498e6	0.691701	0.345851	+	0.938290i	$0.387590\pi$
$602$	−442288.	−0.0497409
$603$	0	0
$604$	199232.	0.0222212
$605$	0	0
$606$	0	0
$607$	4.51646e6	0.497538	0.248769	−	0.968563i	$-0.419974\pi$
$607$	4.51646e6	0.497538	0.248769	+	0.968563i	$0.419974\pi$
$608$	−1.12640e6	−0.123576
$609$	0	0
$610$	0	0
$611$	1.13795e6	0.123316
$612$	0	0
$613$	−9.63979e6	−1.03614	−0.518068	−	0.855340i	$-0.673349\pi$
$613$	−9.63979e6	−1.03614	−0.518068	+	0.855340i	$0.673349\pi$
$614$	4.98903e6	0.534067
$615$	0	0
$616$	−1.08134e6	−0.114819
$617$	−9.92650e6	−1.04974	−0.524872	−	0.851181i	$-0.675887\pi$
$617$	−9.92650e6	−1.04974	−0.524872	+	0.851181i	$0.675887\pi$
$618$	0	0
$619$	7.63322e6	0.800721	0.400360	−	0.916358i	$-0.368885\pi$
$619$	7.63322e6	0.800721	0.400360	+	0.916358i	$0.368885\pi$
$620$	0	0
$621$	0	0
$622$	−2.66395e6	−0.276090
$623$	−1.26918e6	−0.131010
$624$	0	0
$625$	0	0
$626$	−2.36514e6	−0.241225
$627$	0	0
$628$	96352.0	0.00974903
$629$	53676.0	0.00540946
$630$	0	0
$631$	1.80314e7	1.80284	0.901418	−	0.432949i	$-0.142527\pi$
$631$	1.80314e7	1.80284	0.901418	+	0.432949i	$0.142527\pi$
$632$	2.53696e6	0.252651
$633$	0	0
$634$	2.06537e6	0.204068
$635$	0	0
$636$	0	0
$637$	−750858.	−0.0733178
$638$	−1.72339e7	−1.67623
$639$	0	0
$640$	0	0
$641$	−9.30190e6	−0.894184	−0.447092	−	0.894488i	$-0.647540\pi$
$641$	−9.30190e6	−0.894184	−0.447092	+	0.894488i	$0.647540\pi$
$642$	0	0
$643$	1.38332e7	1.31946	0.659730	−	0.751503i	$-0.270671\pi$
$643$	1.38332e7	1.31946	0.659730	+	0.751503i	$0.270671\pi$
$644$	−699072.	−0.0664213
$645$	0	0
$646$	−1.66320e6	−0.156806
$647$	−1.48997e7	−1.39932	−0.699658	−	0.714478i	$-0.746664\pi$
$647$	−1.48997e7	−1.39932	−0.699658	+	0.714478i	$0.746664\pi$
$648$	0	0
$649$	−2.17958e7	−2.03124
$650$	0	0
$651$	0	0
$652$	8.01386e6	0.738282
$653$	−1.93306e7	−1.77403	−0.887016	−	0.461738i	$-0.847226\pi$
$653$	−1.93306e7	−1.77403	−0.887016	+	0.461738i	$0.847226\pi$
$654$	0	0
$655$	0	0
$656$	−394752.	−0.0358150
$657$	0	0
$658$	−2.17694e6	−0.196012
$659$	4.06110e6	0.364276	0.182138	−	0.983273i	$-0.441698\pi$
$659$	4.06110e6	0.364276	0.182138	+	0.983273i	$0.441698\pi$
$660$	0	0
$661$	−1.35152e7	−1.20315	−0.601575	−	0.798816i	$-0.705460\pi$
$661$	−1.35152e7	−1.20315	−0.601575	+	0.798816i	$0.705460\pi$
$662$	1.31831e7	1.16915
$663$	0	0
$664$	3.82886e6	0.337015
$665$	0	0
$666$	0	0
$667$	−1.11415e7	−0.969678
$668$	8.88413e6	0.770324
$669$	0	0
$670$	0	0
$671$	4.24090e6	0.363623
$672$	0	0
$673$	−1.43520e7	−1.22144	−0.610722	−	0.791845i	$-0.709121\pi$
$673$	−1.43520e7	−1.22144	−0.610722	+	0.791845i	$0.709121\pi$
$674$	7.64391e6	0.648136
$675$	0	0
$676$	−5.90683e6	−0.497150
$677$	1.89530e6	0.158930	0.0794650	−	0.996838i	$-0.474679\pi$
$677$	1.89530e6	0.158930	0.0794650	+	0.996838i	$0.474679\pi$
$678$	0	0
$679$	3.17640e6	0.264400
$680$	0	0
$681$	0	0
$682$	1.22511e7	1.00859
$683$	2.91641e6	0.239220	0.119610	−	0.992821i	$-0.461836\pi$
$683$	2.91641e6	0.239220	0.119610	+	0.992821i	$0.461836\pi$
$684$	0	0
$685$	0	0
$686$	2.91544e6	0.236534
$687$	0	0
$688$	1.28666e6	0.103631
$689$	−651636.	−0.0522946
$690$	0	0
$691$	1.44278e7	1.14949	0.574743	−	0.818334i	$-0.305102\pi$
$691$	1.44278e7	1.14949	0.574743	+	0.818334i	$0.305102\pi$
$692$	6.67766e6	0.530102
$693$	0	0
$694$	−9.68023e6	−0.762934
$695$	0	0
$696$	0	0
$697$	−582876.	−0.0454458
$698$	−1.00291e7	−0.779153
$699$	0	0
$700$	0	0
$701$	1.58679e7	1.21962	0.609811	−	0.792547i	$-0.291246\pi$
$701$	1.58679e7	1.21962	0.609811	+	0.792547i	$0.291246\pi$
$702$	0	0
$703$	156200.	0.0119205
$704$	3.14573e6	0.239216
$705$	0	0
$706$	1.65266e6	0.124788
$707$	3.10768e6	0.233823
$708$	0	0
$709$	−301810.	−0.0225485	−0.0112743	−	0.999936i	$-0.503589\pi$
$709$	−301810.	−0.0225485	−0.0112743	+	0.999936i	$0.503589\pi$
$710$	0	0
$711$	0	0
$712$	3.69216e6	0.272948
$713$	7.92017e6	0.583459
$714$	0	0
$715$	0	0
$716$	838080.	0.0610946
$717$	0	0
$718$	6.95088e6	0.503186
$719$	−2.12677e7	−1.53426	−0.767130	−	0.641492i	$-0.778316\pi$
$719$	−2.12677e7	−1.53426	−0.767130	+	0.641492i	$0.778316\pi$
$720$	0	0
$721$	−3.07591e6	−0.220361
$722$	5.06440e6	0.361564
$723$	0	0
$724$	8.74659e6	0.620144
$725$	0	0
$726$	0	0
$727$	−1.55009e7	−1.08773	−0.543863	−	0.839174i	$-0.683039\pi$
$727$	−1.55009e7	−1.08773	−0.543863	+	0.839174i	$0.683039\pi$
$728$	−64768.0	−0.00452931
$729$	0	0
$730$	0	0
$731$	1.89983e6	0.131499
$732$	0	0
$733$	1.21850e7	0.837653	0.418827	−	0.908066i	$-0.362441\pi$
$733$	1.21850e7	0.837653	0.418827	+	0.908066i	$0.362441\pi$
$734$	4.64391e6	0.318159
$735$	0	0
$736$	2.03366e6	0.138384
$737$	1.90019e7	1.28863
$738$	0	0
$739$	−2.90282e7	−1.95528	−0.977641	−	0.210282i	$-0.932562\pi$
$739$	−2.90282e7	−1.95528	−0.977641	+	0.210282i	$0.932562\pi$
$740$	0	0
$741$	0	0
$742$	1.24661e6	0.0831228
$743$	1.61145e7	1.07089	0.535445	−	0.844570i	$-0.320144\pi$
$743$	1.61145e7	1.07089	0.535445	+	0.844570i	$0.320144\pi$
$744$	0	0
$745$	0	0
$746$	1.37502e6	0.0904609
$747$	0	0
$748$	4.64486e6	0.303542
$749$	1.90120e6	0.123829
$750$	0	0
$751$	−2.92431e6	−0.189201	−0.0946005	−	0.995515i	$-0.530157\pi$
$751$	−2.92431e6	−0.189201	−0.0946005	+	0.995515i	$0.530157\pi$
$752$	6.33293e6	0.408376
$753$	0	0
$754$	−1.03224e6	−0.0661230
$755$	0	0
$756$	0	0
$757$	−2.60325e7	−1.65111	−0.825557	−	0.564319i	$-0.809139\pi$
$757$	−2.60325e7	−1.65111	−0.825557	+	0.564319i	$0.809139\pi$
$758$	−2.29256e6	−0.144926
$759$	0	0
$760$	0	0
$761$	−1.63263e7	−1.02194	−0.510970	−	0.859598i	$-0.670714\pi$
$761$	−1.63263e7	−1.02194	−0.510970	+	0.859598i	$0.670714\pi$
$762$	0	0
$763$	4.80590e6	0.298857
$764$	7.23245e6	0.448283
$765$	0	0
$766$	1.15222e7	0.709517
$767$	−1.30548e6	−0.0801275
$768$	0	0
$769$	2.58132e7	1.57408	0.787040	−	0.616902i	$-0.211612\pi$
$769$	2.58132e7	1.57408	0.787040	+	0.616902i	$0.211612\pi$
$770$	0	0
$771$	0	0
$772$	−7.76950e6	−0.469191
$773$	−1.90592e7	−1.14725	−0.573624	−	0.819119i	$-0.694463\pi$
$773$	−1.90592e7	−1.14725	−0.573624	+	0.819119i	$0.694463\pi$
$774$	0	0
$775$	0	0
$776$	−9.24045e6	−0.550857
$777$	0	0
$778$	−1.23424e7	−0.731054
$779$	−1.69620e6	−0.100146
$780$	0	0
$781$	−3.25417e7	−1.90903
$782$	3.00283e6	0.175596
$783$	0	0
$784$	−4.17869e6	−0.242801
$785$	0	0
$786$	0	0
$787$	1.73411e7	0.998021	0.499011	−	0.866596i	$-0.333697\pi$
$787$	1.73411e7	0.998021	0.499011	+	0.866596i	$0.333697\pi$
$788$	1.61628e7	0.927262
$789$	0	0
$790$	0	0
$791$	−633732.	−0.0360134
$792$	0	0
$793$	254012.	0.0143440
$794$	3.54183e6	0.199378
$795$	0	0
$796$	−1.29222e7	−0.722862
$797$	−2.58169e7	−1.43965	−0.719827	−	0.694153i	$-0.755779\pi$
$797$	−2.58169e7	−1.43965	−0.719827	+	0.694153i	$0.755779\pi$
$798$	0	0
$799$	9.35096e6	0.518190
$800$	0	0
$801$	0	0
$802$	−1.50138e7	−0.824239
$803$	4.00328e7	2.19092
$804$	0	0
$805$	0	0
$806$	733792.	0.0397865
$807$	0	0
$808$	−9.04051e6	−0.487152
$809$	−8.88489e6	−0.477288	−0.238644	−	0.971107i	$-0.576703\pi$
$809$	−8.88489e6	−0.477288	−0.238644	+	0.971107i	$0.576703\pi$
$810$	0	0
$811$	−2.46396e7	−1.31547	−0.657735	−	0.753249i	$-0.728485\pi$
$811$	−2.46396e7	−1.31547	−0.657735	+	0.753249i	$0.728485\pi$
$812$	1.97472e6	0.105103
$813$	0	0
$814$	−436224.	−0.0230754
$815$	0	0
$816$	0	0
$817$	5.52860e6	0.289774
$818$	7.78612e6	0.406853
$819$	0	0
$820$	0	0
$821$	−1.13768e7	−0.589062	−0.294531	−	0.955642i	$-0.595163\pi$
$821$	−1.13768e7	−0.589062	−0.294531	+	0.955642i	$0.595163\pi$
$822$	0	0
$823$	1.30783e7	0.673057	0.336529	−	0.941673i	$-0.390747\pi$
$823$	1.30783e7	0.673057	0.336529	+	0.941673i	$0.390747\pi$
$824$	8.94810e6	0.459106
$825$	0	0
$826$	2.49744e6	0.127363
$827$	−3.57188e7	−1.81607	−0.908037	−	0.418891i	$-0.862419\pi$
$827$	−3.57188e7	−1.81607	−0.908037	+	0.418891i	$0.862419\pi$
$828$	0	0
$829$	1.61880e7	0.818103	0.409052	−	0.912511i	$-0.365860\pi$
$829$	1.61880e7	0.818103	0.409052	+	0.912511i	$0.365860\pi$
$830$	0	0
$831$	0	0
$832$	188416.	0.00943647
$833$	−6.17009e6	−0.308091
$834$	0	0
$835$	0	0
$836$	1.35168e7	0.668895
$837$	0	0
$838$	−1.19666e7	−0.588657
$839$	2.55497e7	1.25309	0.626543	−	0.779387i	$-0.284469\pi$
$839$	2.55497e7	1.25309	0.626543	+	0.779387i	$0.284469\pi$
$840$	0	0
$841$	1.09610e7	0.534390
$842$	−1.58664e7	−0.771256
$843$	0	0
$844$	2.39283e6	0.115626
$845$	0	0
$846$	0	0
$847$	9.43301e6	0.451795
$848$	−3.62650e6	−0.173180
$849$	0	0
$850$	0	0
$851$	−282012.	−0.0133488
$852$	0	0
$853$	2.22953e7	1.04916	0.524579	−	0.851362i	$-0.324223\pi$
$853$	2.22953e7	1.04916	0.524579	+	0.851362i	$0.324223\pi$
$854$	−485936.	−0.0228000
$855$	0	0
$856$	−5.53075e6	−0.257988
$857$	1.96872e7	0.915656	0.457828	−	0.889041i	$-0.348628\pi$
$857$	1.96872e7	0.915656	0.457828	+	0.889041i	$0.348628\pi$
$858$	0	0
$859$	6.77582e6	0.313313	0.156657	−	0.987653i	$-0.449928\pi$
$859$	6.77582e6	0.313313	0.156657	+	0.987653i	$0.449928\pi$
$860$	0	0
$861$	0	0
$862$	−2.06846e7	−0.948154
$863$	−2.63804e7	−1.20574	−0.602871	−	0.797839i	$-0.705977\pi$
$863$	−2.63804e7	−1.20574	−0.602871	+	0.797839i	$0.705977\pi$
$864$	0	0
$865$	0	0
$866$	−1.81394e7	−0.821917
$867$	0	0
$868$	−1.40378e6	−0.0632410
$869$	−3.04435e7	−1.36756
$870$	0	0
$871$	1.13813e6	0.0508332
$872$	−1.39808e7	−0.622645
$873$	0	0
$874$	8.73840e6	0.386949
$875$	0	0
$876$	0	0
$877$	−2.95161e7	−1.29587	−0.647934	−	0.761697i	$-0.724367\pi$
$877$	−2.95161e7	−1.29587	−0.647934	+	0.761697i	$0.724367\pi$
$878$	4.32880e6	0.189510
$879$	0	0
$880$	0	0
$881$	1.48565e7	0.644877	0.322438	−	0.946590i	$-0.395498\pi$
$881$	1.48565e7	0.644877	0.322438	+	0.946590i	$0.395498\pi$
$882$	0	0
$883$	1.45340e7	0.627313	0.313656	−	0.949537i	$-0.398446\pi$
$883$	1.45340e7	0.627313	0.313656	+	0.949537i	$0.398446\pi$
$884$	278208.	0.0119740
$885$	0	0
$886$	4.32314e6	0.185019
$887$	−1.72028e7	−0.734160	−0.367080	−	0.930189i	$-0.619642\pi$
$887$	−1.72028e7	−0.734160	−0.367080	+	0.930189i	$0.619642\pi$
$888$	0	0
$889$	4.76304e6	0.202130
$890$	0	0
$891$	0	0
$892$	7.09610e6	0.298612
$893$	2.72118e7	1.14190
$894$	0	0
$895$	0	0
$896$	−360448.	−0.0149994
$897$	0	0
$898$	1.04713e7	0.433322
$899$	−2.23727e7	−0.923249
$900$	0	0
$901$	−5.35475e6	−0.219749
$902$	4.73702e6	0.193860
$903$	0	0
$904$	1.84358e6	0.0750312
$905$	0	0
$906$	0	0
$907$	3.44434e7	1.39023	0.695116	−	0.718897i	$-0.255353\pi$
$907$	3.44434e7	1.39023	0.695116	+	0.718897i	$0.255353\pi$
$908$	6.72029e6	0.270504
$909$	0	0
$910$	0	0
$911$	983748.	0.0392724	0.0196362	−	0.999807i	$-0.493749\pi$
$911$	983748.	0.0392724	0.0196362	+	0.999807i	$0.493749\pi$
$912$	0	0
$913$	−4.59464e7	−1.82421
$914$	6.36183e6	0.251893
$915$	0	0
$916$	1.69400e7	0.667075
$917$	5.38138e6	0.211334
$918$	0	0
$919$	3.08857e7	1.20634	0.603168	−	0.797614i	$-0.293905\pi$
$919$	3.08857e7	1.20634	0.603168	+	0.797614i	$0.293905\pi$
$920$	0	0
$921$	0	0
$922$	1.70188e7	0.659328
$923$	−1.94911e6	−0.0753065
$924$	0	0
$925$	0	0
$926$	1.30642e7	0.500675
$927$	0	0
$928$	−5.74464e6	−0.218974
$929$	3.20874e7	1.21982	0.609909	−	0.792472i	$-0.291206\pi$
$929$	3.20874e7	1.21982	0.609909	+	0.792472i	$0.291206\pi$
$930$	0	0
$931$	−1.79553e7	−0.678920
$932$	−2.03751e7	−0.768353
$933$	0	0
$934$	2.40617e6	0.0902524
$935$	0	0
$936$	0	0
$937$	−1.52520e7	−0.567515	−0.283757	−	0.958896i	$-0.591581\pi$
$937$	−1.52520e7	−0.567515	−0.283757	+	0.958896i	$0.591581\pi$
$938$	−2.17730e6	−0.0807998
$939$	0	0
$940$	0	0
$941$	−3.48166e6	−0.128178	−0.0640889	−	0.997944i	$-0.520414\pi$
$941$	−3.48166e6	−0.128178	−0.0640889	+	0.997944i	$0.520414\pi$
$942$	0	0
$943$	3.06241e6	0.112146
$944$	−7.26528e6	−0.265352
$945$	0	0
$946$	−1.54399e7	−0.560939
$947$	−2.54010e7	−0.920398	−0.460199	−	0.887816i	$-0.652222\pi$
$947$	−2.54010e7	−0.920398	−0.460199	+	0.887816i	$0.652222\pi$
$948$	0	0
$949$	2.39780e6	0.0864265
$950$	0	0
$951$	0	0
$952$	−532224.	−0.0190328
$953$	−4.97352e7	−1.77391	−0.886955	−	0.461856i	$-0.847184\pi$
$953$	−4.97352e7	−1.77391	−0.886955	+	0.461856i	$0.847184\pi$
$954$	0	0
$955$	0	0
$956$	5.93088e6	0.209882
$957$	0	0
$958$	−1.83173e7	−0.644833
$959$	−5.26904e6	−0.185006
$960$	0	0
$961$	−1.27250e7	−0.444477
$962$	−26128.0	−0.000910266 0
$963$	0	0
$964$	−8.98077e6	−0.311258
$965$	0	0
$966$	0	0
$967$	−3.05173e7	−1.04949	−0.524747	−	0.851258i	$-0.675840\pi$
$967$	−3.05173e7	−1.04949	−0.524747	+	0.851258i	$0.675840\pi$
$968$	−2.74415e7	−0.941280
$969$	0	0
$970$	0	0
$971$	−3.19854e7	−1.08869	−0.544344	−	0.838862i	$-0.683221\pi$
$971$	−3.19854e7	−1.08869	−0.544344	+	0.838862i	$0.683221\pi$
$972$	0	0
$973$	678920.	0.0229899
$974$	2.82090e7	0.952776
$975$	0	0
$976$	1.41363e6	0.0475020
$977$	2.90786e6	0.0974623	0.0487312	−	0.998812i	$-0.484482\pi$
$977$	2.90786e6	0.0974623	0.0487312	+	0.998812i	$0.484482\pi$
$978$	0	0
$979$	−4.43059e7	−1.47742
$980$	0	0
$981$	0	0
$982$	−1.04940e7	−0.347264
$983$	3.49621e7	1.15402	0.577010	−	0.816737i	$-0.304219\pi$
$983$	3.49621e7	1.15402	0.577010	+	0.816737i	$0.304219\pi$
$984$	0	0
$985$	0	0
$986$	−8.48232e6	−0.277858
$987$	0	0
$988$	809600.	0.0263863
$989$	−9.98164e6	−0.324497
$990$	0	0
$991$	3.00465e6	0.0971874	0.0485937	−	0.998819i	$-0.484526\pi$
$991$	3.00465e6	0.0971874	0.0485937	+	0.998819i	$0.484526\pi$
$992$	4.08371e6	0.131758
$993$	0	0
$994$	3.72874e6	0.119700
$995$	0	0
$996$	0	0
$997$	−3.20789e7	−1.02207	−0.511035	−	0.859560i	$-0.670738\pi$
$997$	−3.20789e7	−1.02207	−0.511035	+	0.859560i	$0.670738\pi$
$998$	1.44494e7	0.459222
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.h.1.1		1
3.2	odd	2		50.6.a.g.1.1		1
5.2	odd	4		450.6.c.o.199.1		2
5.3	odd	4		450.6.c.o.199.2		2
5.4	even	2		90.6.a.f.1.1		1
12.11	even	2		400.6.a.a.1.1		1
15.2	even	4		50.6.b.d.49.2		2
15.8	even	4		50.6.b.d.49.1		2
15.14	odd	2		10.6.a.a.1.1	✓	1
20.19	odd	2		720.6.a.r.1.1		1
60.23	odd	4		400.6.c.a.49.1		2
60.47	odd	4		400.6.c.a.49.2		2
60.59	even	2		80.6.a.h.1.1		1
105.104	even	2		490.6.a.j.1.1		1
120.29	odd	2		320.6.a.p.1.1		1
120.59	even	2		320.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.a.1.1	✓	1	15.14	odd	2
50.6.a.g.1.1		1	3.2	odd	2
50.6.b.d.49.1		2	15.8	even	4
50.6.b.d.49.2		2	15.2	even	4
80.6.a.h.1.1		1	60.59	even	2
90.6.a.f.1.1		1	5.4	even	2
320.6.a.a.1.1		1	120.59	even	2
320.6.a.p.1.1		1	120.29	odd	2
400.6.a.a.1.1		1	12.11	even	2
400.6.c.a.49.1		2	60.23	odd	4
400.6.c.a.49.2		2	60.47	odd	4
450.6.a.h.1.1		1	1.1	even	1	trivial
450.6.c.o.199.1		2	5.2	odd	4
450.6.c.o.199.2		2	5.3	odd	4
490.6.a.j.1.1		1	105.104	even	2
720.6.a.r.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

Related objects

Downloads

Learn more