LMFDB - Embedded newform 825.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -12.0000 q^{6} +21.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -12.0000 q^{6} +21.0000 q^{7} +9.00000 q^{9} +11.0000 q^{11} +24.0000 q^{12} -68.0000 q^{13} -84.0000 q^{14} -64.0000 q^{16} +21.0000 q^{17} -36.0000 q^{18} +125.000 q^{19} +63.0000 q^{21} -44.0000 q^{22} +137.000 q^{23} +272.000 q^{26} +27.0000 q^{27} +168.000 q^{28} -150.000 q^{29} +292.000 q^{31} +256.000 q^{32} +33.0000 q^{33} -84.0000 q^{34} +72.0000 q^{36} -349.000 q^{37} -500.000 q^{38} -204.000 q^{39} +497.000 q^{41} -252.000 q^{42} -208.000 q^{43} +88.0000 q^{44} -548.000 q^{46} -369.000 q^{47} -192.000 q^{48} +98.0000 q^{49} +63.0000 q^{51} -544.000 q^{52} +542.000 q^{53} -108.000 q^{54} +375.000 q^{57} +600.000 q^{58} +235.000 q^{59} +482.000 q^{61} -1168.00 q^{62} +189.000 q^{63} -512.000 q^{64} -132.000 q^{66} -734.000 q^{67} +168.000 q^{68} +411.000 q^{69} +587.000 q^{71} -518.000 q^{73} +1396.00 q^{74} +1000.00 q^{76} +231.000 q^{77} +816.000 q^{78} -1045.00 q^{79} +81.0000 q^{81} -1988.00 q^{82} -608.000 q^{83} +504.000 q^{84} +832.000 q^{86} -450.000 q^{87} -770.000 q^{89} -1428.00 q^{91} +1096.00 q^{92} +876.000 q^{93} +1476.00 q^{94} +768.000 q^{96} +1541.00 q^{97} -392.000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	−12.0000	−0.816497
$7$	21.0000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	21.0000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	24.0000	0.577350
$13$	−68.0000	−1.45075	−0.725377	−	0.688352i	$-0.758335\pi$
$13$	−68.0000	−1.45075	−0.725377	+	0.688352i	$0.758335\pi$
$14$	−84.0000	−1.60357
$15$	0	0
$16$	−64.0000	−1.00000
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	−36.0000	−0.471405
$19$	125.000	1.50931	0.754657	−	0.656119i	$-0.227803\pi$
$19$	125.000	1.50931	0.754657	+	0.656119i	$0.227803\pi$
$20$	0	0
$21$	63.0000	0.654654
$22$	−44.0000	−0.426401
$23$	137.000	1.24202	0.621010	−	0.783802i	$-0.286722\pi$
$23$	137.000	1.24202	0.621010	+	0.783802i	$0.286722\pi$
$24$	0	0
$25$	0	0
$26$	272.000	2.05168
$27$	27.0000	0.192450
$28$	168.000	1.13389
$29$	−150.000	−0.960493	−0.480247	−	0.877134i	$-0.659453\pi$
$29$	−150.000	−0.960493	−0.480247	+	0.877134i	$0.659453\pi$
$30$	0	0
$31$	292.000	1.69177	0.845883	−	0.533368i	$-0.179074\pi$
$31$	292.000	1.69177	0.845883	+	0.533368i	$0.179074\pi$
$32$	256.000	1.41421
$33$	33.0000	0.174078
$34$	−84.0000	−0.423702
$35$	0	0
$36$	72.0000	0.333333
$37$	−349.000	−1.55068	−0.775341	−	0.631543i	$-0.782422\pi$
$37$	−349.000	−1.55068	−0.775341	+	0.631543i	$0.782422\pi$
$38$	−500.000	−2.13449
$39$	−204.000	−0.837593
$40$	0	0
$41$	497.000	1.89313	0.946565	−	0.322512i	$-0.104527\pi$
$41$	497.000	1.89313	0.946565	+	0.322512i	$0.104527\pi$
$42$	−252.000	−0.925820
$43$	−208.000	−0.737668	−0.368834	−	0.929495i	$-0.620243\pi$
$43$	−208.000	−0.737668	−0.368834	+	0.929495i	$0.620243\pi$
$44$	88.0000	0.301511
$45$	0	0
$46$	−548.000	−1.75648
$47$	−369.000	−1.14520	−0.572598	−	0.819837i	$-0.694064\pi$
$47$	−369.000	−1.14520	−0.572598	+	0.819837i	$0.694064\pi$
$48$	−192.000	−0.577350
$49$	98.0000	0.285714
$50$	0	0
$51$	63.0000	0.172976
$52$	−544.000	−1.45075
$53$	542.000	1.40471	0.702353	−	0.711829i	$-0.252133\pi$
$53$	542.000	1.40471	0.702353	+	0.711829i	$0.252133\pi$
$54$	−108.000	−0.272166
$55$	0	0
$56$	0	0
$57$	375.000	0.871403
$58$	600.000	1.35834
$59$	235.000	0.518549	0.259275	−	0.965804i	$-0.416517\pi$
$59$	235.000	0.518549	0.259275	+	0.965804i	$0.416517\pi$
$60$	0	0
$61$	482.000	1.01170	0.505851	−	0.862621i	$-0.331179\pi$
$61$	482.000	1.01170	0.505851	+	0.862621i	$0.331179\pi$
$62$	−1168.00	−2.39252
$63$	189.000	0.377964
$64$	−512.000	−1.00000
$65$	0	0
$66$	−132.000	−0.246183
$67$	−734.000	−1.33839	−0.669197	−	0.743085i	$-0.733362\pi$
$67$	−734.000	−1.33839	−0.669197	+	0.743085i	$0.733362\pi$
$68$	168.000	0.299603
$69$	411.000	0.717081
$70$	0	0
$71$	587.000	0.981184	0.490592	−	0.871389i	$-0.336781\pi$
$71$	587.000	0.981184	0.490592	+	0.871389i	$0.336781\pi$
$72$	0	0
$73$	−518.000	−0.830511	−0.415256	−	0.909705i	$-0.636308\pi$
$73$	−518.000	−0.830511	−0.415256	+	0.909705i	$0.636308\pi$
$74$	1396.00	2.19300
$75$	0	0
$76$	1000.00	1.50931
$77$	231.000	0.341882
$78$	816.000	1.18454
$79$	−1045.00	−1.48825	−0.744125	−	0.668041i	$-0.767133\pi$
$79$	−1045.00	−1.48825	−0.744125	+	0.668041i	$0.767133\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−1988.00	−2.67729
$83$	−608.000	−0.804056	−0.402028	−	0.915627i	$-0.631695\pi$
$83$	−608.000	−0.804056	−0.402028	+	0.915627i	$0.631695\pi$
$84$	504.000	0.654654
$85$	0	0
$86$	832.000	1.04322
$87$	−450.000	−0.554541
$88$	0	0
$89$	−770.000	−0.917077	−0.458538	−	0.888675i	$-0.651627\pi$
$89$	−770.000	−0.917077	−0.458538	+	0.888675i	$0.651627\pi$
$90$	0	0
$91$	−1428.00	−1.64500
$92$	1096.00	1.24202
$93$	876.000	0.976742
$94$	1476.00	1.61955
$95$	0	0
$96$	768.000	0.816497
$97$	1541.00	1.61304	0.806520	−	0.591207i	$-0.201348\pi$
$97$	1541.00	1.61304	0.806520	+	0.591207i	$0.201348\pi$
$98$	−392.000	−0.404061
$99$	99.0000	0.100504
$100$	0	0
$101$	827.000	0.814748	0.407374	−	0.913261i	$-0.366445\pi$
$101$	827.000	0.814748	0.407374	+	0.913261i	$0.366445\pi$
$102$	−252.000	−0.244625
$103$	−248.000	−0.237244	−0.118622	−	0.992939i	$-0.537848\pi$
$103$	−248.000	−0.237244	−0.118622	+	0.992939i	$0.537848\pi$
$104$	0	0
$105$	0	0
$106$	−2168.00	−1.98655
$107$	366.000	0.330678	0.165339	−	0.986237i	$-0.447128\pi$
$107$	366.000	0.330678	0.165339	+	0.986237i	$0.447128\pi$
$108$	216.000	0.192450
$109$	270.000	0.237260	0.118630	−	0.992939i	$-0.462150\pi$
$109$	270.000	0.237260	0.118630	+	0.992939i	$0.462150\pi$
$110$	0	0
$111$	−1047.00	−0.895287
$112$	−1344.00	−1.13389
$113$	1002.00	0.834161	0.417081	−	0.908869i	$-0.363053\pi$
$113$	1002.00	0.834161	0.417081	+	0.908869i	$0.363053\pi$
$114$	−1500.00	−1.23235
$115$	0	0
$116$	−1200.00	−0.960493
$117$	−612.000	−0.483585
$118$	−940.000	−0.733339
$119$	441.000	0.339718
$120$	0	0
$121$	121.000	0.0909091
$122$	−1928.00	−1.43076
$123$	1491.00	1.09300
$124$	2336.00	1.69177
$125$	0	0
$126$	−756.000	−0.534522
$127$	−469.000	−0.327693	−0.163847	−	0.986486i	$-0.552390\pi$
$127$	−469.000	−0.327693	−0.163847	+	0.986486i	$0.552390\pi$
$128$	0	0
$129$	−624.000	−0.425893
$130$	0	0
$131$	−408.000	−0.272115	−0.136058	−	0.990701i	$-0.543443\pi$
$131$	−408.000	−0.272115	−0.136058	+	0.990701i	$0.543443\pi$
$132$	264.000	0.174078
$133$	2625.00	1.71140
$134$	2936.00	1.89277
$135$	0	0
$136$	0	0
$137$	2466.00	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	2466.00	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−1644.00	−1.01411
$139$	1020.00	0.622412	0.311206	−	0.950342i	$-0.399267\pi$
$139$	1020.00	0.622412	0.311206	+	0.950342i	$0.399267\pi$
$140$	0	0
$141$	−1107.00	−0.661179
$142$	−2348.00	−1.38760
$143$	−748.000	−0.437419
$144$	−576.000	−0.333333
$145$	0	0
$146$	2072.00	1.17452
$147$	294.000	0.164957
$148$	−2792.00	−1.55068
$149$	5.00000	0.00274910	0.00137455	−	0.999999i	$-0.499562\pi$
$149$	5.00000	0.00274910	0.00137455	+	0.999999i	$0.499562\pi$
$150$	0	0
$151$	452.000	0.243598	0.121799	−	0.992555i	$-0.461134\pi$
$151$	452.000	0.243598	0.121799	+	0.992555i	$0.461134\pi$
$152$	0	0
$153$	189.000	0.0998676
$154$	−924.000	−0.483494
$155$	0	0
$156$	−1632.00	−0.837593
$157$	1766.00	0.897721	0.448860	−	0.893602i	$-0.351830\pi$
$157$	1766.00	0.897721	0.448860	+	0.893602i	$0.351830\pi$
$158$	4180.00	2.10470
$159$	1626.00	0.811007
$160$	0	0
$161$	2877.00	1.40832
$162$	−324.000	−0.157135
$163$	−2068.00	−0.993732	−0.496866	−	0.867827i	$-0.665516\pi$
$163$	−2068.00	−0.993732	−0.496866	+	0.867827i	$0.665516\pi$
$164$	3976.00	1.89313
$165$	0	0
$166$	2432.00	1.13711
$167$	3386.00	1.56896	0.784481	−	0.620153i	$-0.212930\pi$
$167$	3386.00	1.56896	0.784481	+	0.620153i	$0.212930\pi$
$168$	0	0
$169$	2427.00	1.10469
$170$	0	0
$171$	1125.00	0.503105
$172$	−1664.00	−0.737668
$173$	117.000	0.0514182	0.0257091	−	0.999669i	$-0.491816\pi$
$173$	117.000	0.0514182	0.0257091	+	0.999669i	$0.491816\pi$
$174$	1800.00	0.784239
$175$	0	0
$176$	−704.000	−0.301511
$177$	705.000	0.299384
$178$	3080.00	1.29694
$179$	2995.00	1.25060	0.625298	−	0.780386i	$-0.284977\pi$
$179$	2995.00	1.25060	0.625298	+	0.780386i	$0.284977\pi$
$180$	0	0
$181$	4067.00	1.67015	0.835077	−	0.550134i	$-0.185423\pi$
$181$	4067.00	1.67015	0.835077	+	0.550134i	$0.185423\pi$
$182$	5712.00	2.32638
$183$	1446.00	0.584106
$184$	0	0
$185$	0	0
$186$	−3504.00	−1.38132
$187$	231.000	0.0903337
$188$	−2952.00	−1.14520
$189$	567.000	0.218218
$190$	0	0
$191$	3047.00	1.15431	0.577155	−	0.816635i	$-0.304163\pi$
$191$	3047.00	1.15431	0.577155	+	0.816635i	$0.304163\pi$
$192$	−1536.00	−0.577350
$193$	1232.00	0.459489	0.229744	−	0.973251i	$-0.426211\pi$
$193$	1232.00	0.459489	0.229744	+	0.973251i	$0.426211\pi$
$194$	−6164.00	−2.28118
$195$	0	0
$196$	784.000	0.285714
$197$	−4979.00	−1.80071	−0.900353	−	0.435160i	$-0.856692\pi$
$197$	−4979.00	−1.80071	−0.900353	+	0.435160i	$0.856692\pi$
$198$	−396.000	−0.142134
$199$	600.000	0.213733	0.106867	−	0.994273i	$-0.465918\pi$
$199$	600.000	0.213733	0.106867	+	0.994273i	$0.465918\pi$
$200$	0	0
$201$	−2202.00	−0.772722
$202$	−3308.00	−1.15223
$203$	−3150.00	−1.08910
$204$	504.000	0.172976
$205$	0	0
$206$	992.000	0.335514
$207$	1233.00	0.414007
$208$	4352.00	1.45075
$209$	1375.00	0.455075
$210$	0	0
$211$	−2468.00	−0.805233	−0.402616	−	0.915369i	$-0.631899\pi$
$211$	−2468.00	−0.805233	−0.402616	+	0.915369i	$0.631899\pi$
$212$	4336.00	1.40471
$213$	1761.00	0.566487
$214$	−1464.00	−0.467649
$215$	0	0
$216$	0	0
$217$	6132.00	1.91828
$218$	−1080.00	−0.335536
$219$	−1554.00	−0.479496
$220$	0	0
$221$	−1428.00	−0.434650
$222$	4188.00	1.26613
$223$	5392.00	1.61917	0.809585	−	0.587002i	$-0.199692\pi$
$223$	5392.00	1.61917	0.809585	+	0.587002i	$0.199692\pi$
$224$	5376.00	1.60357
$225$	0	0
$226$	−4008.00	−1.17968
$227$	2366.00	0.691793	0.345896	−	0.938273i	$-0.387575\pi$
$227$	2366.00	0.691793	0.345896	+	0.938273i	$0.387575\pi$
$228$	3000.00	0.871403
$229$	−4645.00	−1.34039	−0.670197	−	0.742183i	$-0.733790\pi$
$229$	−4645.00	−1.34039	−0.670197	+	0.742183i	$0.733790\pi$
$230$	0	0
$231$	693.000	0.197386
$232$	0	0
$233$	−513.000	−0.144239	−0.0721196	−	0.997396i	$-0.522976\pi$
$233$	−513.000	−0.144239	−0.0721196	+	0.997396i	$0.522976\pi$
$234$	2448.00	0.683892
$235$	0	0
$236$	1880.00	0.518549
$237$	−3135.00	−0.859241
$238$	−1764.00	−0.480433
$239$	2690.00	0.728040	0.364020	−	0.931391i	$-0.381404\pi$
$239$	2690.00	0.728040	0.364020	+	0.931391i	$0.381404\pi$
$240$	0	0
$241$	−3728.00	−0.996438	−0.498219	−	0.867051i	$-0.666012\pi$
$241$	−3728.00	−0.996438	−0.498219	+	0.867051i	$0.666012\pi$
$242$	−484.000	−0.128565
$243$	243.000	0.0641500
$244$	3856.00	1.01170
$245$	0	0
$246$	−5964.00	−1.54573
$247$	−8500.00	−2.18964
$248$	0	0
$249$	−1824.00	−0.464222
$250$	0	0
$251$	2352.00	0.591462	0.295731	−	0.955271i	$-0.404437\pi$
$251$	2352.00	0.591462	0.295731	+	0.955271i	$0.404437\pi$
$252$	1512.00	0.377964
$253$	1507.00	0.374483
$254$	1876.00	0.463428
$255$	0	0
$256$	4096.00	1.00000
$257$	3846.00	0.933490	0.466745	−	0.884392i	$-0.345427\pi$
$257$	3846.00	0.933490	0.466745	+	0.884392i	$0.345427\pi$
$258$	2496.00	0.602303
$259$	−7329.00	−1.75831
$260$	0	0
$261$	−1350.00	−0.320164
$262$	1632.00	0.384829
$263$	522.000	0.122387	0.0611937	−	0.998126i	$-0.480509\pi$
$263$	522.000	0.122387	0.0611937	+	0.998126i	$0.480509\pi$
$264$	0	0
$265$	0	0
$266$	−10500.0	−2.42029
$267$	−2310.00	−0.529475
$268$	−5872.00	−1.33839
$269$	4020.00	0.911166	0.455583	−	0.890193i	$-0.349431\pi$
$269$	4020.00	0.911166	0.455583	+	0.890193i	$0.349431\pi$
$270$	0	0
$271$	6687.00	1.49892	0.749458	−	0.662052i	$-0.230314\pi$
$271$	6687.00	1.49892	0.749458	+	0.662052i	$0.230314\pi$
$272$	−1344.00	−0.299603
$273$	−4284.00	−0.949742
$274$	−9864.00	−2.17484
$275$	0	0
$276$	3288.00	0.717081
$277$	3746.00	0.812546	0.406273	−	0.913752i	$-0.366828\pi$
$277$	3746.00	0.812546	0.406273	+	0.913752i	$0.366828\pi$
$278$	−4080.00	−0.880224
$279$	2628.00	0.563922
$280$	0	0
$281$	−5883.00	−1.24893	−0.624467	−	0.781051i	$-0.714684\pi$
$281$	−5883.00	−1.24893	−0.624467	+	0.781051i	$0.714684\pi$
$282$	4428.00	0.935048
$283$	−3943.00	−0.828223	−0.414111	−	0.910226i	$-0.635908\pi$
$283$	−3943.00	−0.828223	−0.414111	+	0.910226i	$0.635908\pi$
$284$	4696.00	0.981184
$285$	0	0
$286$	2992.00	0.618604
$287$	10437.0	2.14661
$288$	2304.00	0.471405
$289$	−4472.00	−0.910238
$290$	0	0
$291$	4623.00	0.931289
$292$	−4144.00	−0.830511
$293$	1487.00	0.296490	0.148245	−	0.988951i	$-0.452638\pi$
$293$	1487.00	0.296490	0.148245	+	0.988951i	$0.452638\pi$
$294$	−1176.00	−0.233285
$295$	0	0
$296$	0	0
$297$	297.000	0.0580259
$298$	−20.0000	−0.00388782
$299$	−9316.00	−1.80187
$300$	0	0
$301$	−4368.00	−0.836436
$302$	−1808.00	−0.344499
$303$	2481.00	0.470395
$304$	−8000.00	−1.50931
$305$	0	0
$306$	−756.000	−0.141234
$307$	−4844.00	−0.900527	−0.450263	−	0.892896i	$-0.648670\pi$
$307$	−4844.00	−0.900527	−0.450263	+	0.892896i	$0.648670\pi$
$308$	1848.00	0.341882
$309$	−744.000	−0.136973
$310$	0	0
$311$	4632.00	0.844555	0.422278	−	0.906467i	$-0.361231\pi$
$311$	4632.00	0.844555	0.422278	+	0.906467i	$0.361231\pi$
$312$	0	0
$313$	8437.00	1.52360	0.761801	−	0.647811i	$-0.224315\pi$
$313$	8437.00	1.52360	0.761801	+	0.647811i	$0.224315\pi$
$314$	−7064.00	−1.26957
$315$	0	0
$316$	−8360.00	−1.48825
$317$	3636.00	0.644221	0.322111	−	0.946702i	$-0.395608\pi$
$317$	3636.00	0.644221	0.322111	+	0.946702i	$0.395608\pi$
$318$	−6504.00	−1.14694
$319$	−1650.00	−0.289600
$320$	0	0
$321$	1098.00	0.190917
$322$	−11508.0	−1.99166
$323$	2625.00	0.452195
$324$	648.000	0.111111
$325$	0	0
$326$	8272.00	1.40535
$327$	810.000	0.136982
$328$	0	0
$329$	−7749.00	−1.29853
$330$	0	0
$331$	−758.000	−0.125871	−0.0629357	−	0.998018i	$-0.520046\pi$
$331$	−758.000	−0.125871	−0.0629357	+	0.998018i	$0.520046\pi$
$332$	−4864.00	−0.804056
$333$	−3141.00	−0.516894
$334$	−13544.0	−2.21885
$335$	0	0
$336$	−4032.00	−0.654654
$337$	−7374.00	−1.19195	−0.595975	−	0.803003i	$-0.703234\pi$
$337$	−7374.00	−1.19195	−0.595975	+	0.803003i	$0.703234\pi$
$338$	−9708.00	−1.56227
$339$	3006.00	0.481603
$340$	0	0
$341$	3212.00	0.510087
$342$	−4500.00	−0.711497
$343$	−5145.00	−0.809924
$344$	0	0
$345$	0	0
$346$	−468.000	−0.0727163
$347$	−6524.00	−1.00930	−0.504649	−	0.863324i	$-0.668378\pi$
$347$	−6524.00	−1.00930	−0.504649	+	0.863324i	$0.668378\pi$
$348$	−3600.00	−0.554541
$349$	3710.00	0.569031	0.284515	−	0.958671i	$-0.408167\pi$
$349$	3710.00	0.569031	0.284515	+	0.958671i	$0.408167\pi$
$350$	0	0
$351$	−1836.00	−0.279198
$352$	2816.00	0.426401
$353$	2832.00	0.427003	0.213502	−	0.976943i	$-0.431513\pi$
$353$	2832.00	0.427003	0.213502	+	0.976943i	$0.431513\pi$
$354$	−2820.00	−0.423394
$355$	0	0
$356$	−6160.00	−0.917077
$357$	1323.00	0.196136
$358$	−11980.0	−1.76861
$359$	−7040.00	−1.03498	−0.517489	−	0.855690i	$-0.673133\pi$
$359$	−7040.00	−1.03498	−0.517489	+	0.855690i	$0.673133\pi$
$360$	0	0
$361$	8766.00	1.27803
$362$	−16268.0	−2.36195
$363$	363.000	0.0524864
$364$	−11424.0	−1.64500
$365$	0	0
$366$	−5784.00	−0.826051
$367$	6206.00	0.882699	0.441350	−	0.897335i	$-0.354500\pi$
$367$	6206.00	0.882699	0.441350	+	0.897335i	$0.354500\pi$
$368$	−8768.00	−1.24202
$369$	4473.00	0.631044
$370$	0	0
$371$	11382.0	1.59279
$372$	7008.00	0.976742
$373$	1962.00	0.272355	0.136178	−	0.990684i	$-0.456518\pi$
$373$	1962.00	0.272355	0.136178	+	0.990684i	$0.456518\pi$
$374$	−924.000	−0.127751
$375$	0	0
$376$	0	0
$377$	10200.0	1.39344
$378$	−2268.00	−0.308607
$379$	−7960.00	−1.07883	−0.539417	−	0.842039i	$-0.681355\pi$
$379$	−7960.00	−1.07883	−0.539417	+	0.842039i	$0.681355\pi$
$380$	0	0
$381$	−1407.00	−0.189194
$382$	−12188.0	−1.63244
$383$	−7188.00	−0.958981	−0.479490	−	0.877547i	$-0.659178\pi$
$383$	−7188.00	−0.958981	−0.479490	+	0.877547i	$0.659178\pi$
$384$	0	0
$385$	0	0
$386$	−4928.00	−0.649815
$387$	−1872.00	−0.245889
$388$	12328.0	1.61304
$389$	7920.00	1.03229	0.516144	−	0.856502i	$-0.327367\pi$
$389$	7920.00	1.03229	0.516144	+	0.856502i	$0.327367\pi$
$390$	0	0
$391$	2877.00	0.372113
$392$	0	0
$393$	−1224.00	−0.157106
$394$	19916.0	2.54658
$395$	0	0
$396$	792.000	0.100504
$397$	−9654.00	−1.22045	−0.610227	−	0.792226i	$-0.708922\pi$
$397$	−9654.00	−1.22045	−0.610227	+	0.792226i	$0.708922\pi$
$398$	−2400.00	−0.302264
$399$	7875.00	0.988078
$400$	0	0
$401$	1952.00	0.243088	0.121544	−	0.992586i	$-0.461215\pi$
$401$	1952.00	0.243088	0.121544	+	0.992586i	$0.461215\pi$
$402$	8808.00	1.09279
$403$	−19856.0	−2.45434
$404$	6616.00	0.814748
$405$	0	0
$406$	12600.0	1.54022
$407$	−3839.00	−0.467548
$408$	0	0
$409$	9690.00	1.17149	0.585745	−	0.810495i	$-0.300802\pi$
$409$	9690.00	1.17149	0.585745	+	0.810495i	$0.300802\pi$
$410$	0	0
$411$	7398.00	0.887875
$412$	−1984.00	−0.237244
$413$	4935.00	0.587979
$414$	−4932.00	−0.585494
$415$	0	0
$416$	−17408.0	−2.05168
$417$	3060.00	0.359350
$418$	−5500.00	−0.643574
$419$	−2935.00	−0.342206	−0.171103	−	0.985253i	$-0.554733\pi$
$419$	−2935.00	−0.342206	−0.171103	+	0.985253i	$0.554733\pi$
$420$	0	0
$421$	12837.0	1.48607	0.743037	−	0.669250i	$-0.233385\pi$
$421$	12837.0	1.48607	0.743037	+	0.669250i	$0.233385\pi$
$422$	9872.00	1.13877
$423$	−3321.00	−0.381732
$424$	0	0
$425$	0	0
$426$	−7044.00	−0.801134
$427$	10122.0	1.14716
$428$	2928.00	0.330678
$429$	−2244.00	−0.252544
$430$	0	0
$431$	−6108.00	−0.682626	−0.341313	−	0.939950i	$-0.610872\pi$
$431$	−6108.00	−0.682626	−0.341313	+	0.939950i	$0.610872\pi$
$432$	−1728.00	−0.192450
$433$	−9278.00	−1.02973	−0.514864	−	0.857272i	$-0.672158\pi$
$433$	−9278.00	−1.02973	−0.514864	+	0.857272i	$0.672158\pi$
$434$	−24528.0	−2.71286
$435$	0	0
$436$	2160.00	0.237260
$437$	17125.0	1.87460
$438$	6216.00	0.678110
$439$	2455.00	0.266904	0.133452	−	0.991055i	$-0.457394\pi$
$439$	2455.00	0.266904	0.133452	+	0.991055i	$0.457394\pi$
$440$	0	0
$441$	882.000	0.0952381
$442$	5712.00	0.614688
$443$	−3503.00	−0.375694	−0.187847	−	0.982198i	$-0.560151\pi$
$443$	−3503.00	−0.375694	−0.187847	+	0.982198i	$0.560151\pi$
$444$	−8376.00	−0.895287
$445$	0	0
$446$	−21568.0	−2.28985
$447$	15.0000	0.00158719
$448$	−10752.0	−1.13389
$449$	−7630.00	−0.801964	−0.400982	−	0.916086i	$-0.631331\pi$
$449$	−7630.00	−0.801964	−0.400982	+	0.916086i	$0.631331\pi$
$450$	0	0
$451$	5467.00	0.570800
$452$	8016.00	0.834161
$453$	1356.00	0.140641
$454$	−9464.00	−0.978343
$455$	0	0
$456$	0	0
$457$	−7414.00	−0.758889	−0.379445	−	0.925214i	$-0.623885\pi$
$457$	−7414.00	−0.758889	−0.379445	+	0.925214i	$0.623885\pi$
$458$	18580.0	1.89560
$459$	567.000	0.0576586
$460$	0	0
$461$	4982.00	0.503329	0.251665	−	0.967814i	$-0.419022\pi$
$461$	4982.00	0.503329	0.251665	+	0.967814i	$0.419022\pi$
$462$	−2772.00	−0.279145
$463$	13422.0	1.34724	0.673621	−	0.739077i	$-0.264738\pi$
$463$	13422.0	1.34724	0.673621	+	0.739077i	$0.264738\pi$
$464$	9600.00	0.960493
$465$	0	0
$466$	2052.00	0.203985
$467$	−15804.0	−1.56600	−0.783000	−	0.622022i	$-0.786311\pi$
$467$	−15804.0	−1.56600	−0.783000	+	0.622022i	$0.786311\pi$
$468$	−4896.00	−0.483585
$469$	−15414.0	−1.51760
$470$	0	0
$471$	5298.00	0.518299
$472$	0	0
$473$	−2288.00	−0.222415
$474$	12540.0	1.21515
$475$	0	0
$476$	3528.00	0.339718
$477$	4878.00	0.468235
$478$	−10760.0	−1.02960
$479$	−9060.00	−0.864221	−0.432111	−	0.901821i	$-0.642231\pi$
$479$	−9060.00	−0.864221	−0.432111	+	0.901821i	$0.642231\pi$
$480$	0	0
$481$	23732.0	2.24966
$482$	14912.0	1.40918
$483$	8631.00	0.813093
$484$	968.000	0.0909091
$485$	0	0
$486$	−972.000	−0.0907218
$487$	−2854.00	−0.265559	−0.132779	−	0.991146i	$-0.542390\pi$
$487$	−2854.00	−0.265559	−0.132779	+	0.991146i	$0.542390\pi$
$488$	0	0
$489$	−6204.00	−0.573731
$490$	0	0
$491$	−2278.00	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$491$	−2278.00	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$492$	11928.0	1.09300
$493$	−3150.00	−0.287766
$494$	34000.0	3.09662
$495$	0	0
$496$	−18688.0	−1.69177
$497$	12327.0	1.11256
$498$	7296.00	0.656509
$499$	13290.0	1.19227	0.596134	−	0.802885i	$-0.296703\pi$
$499$	13290.0	1.19227	0.596134	+	0.802885i	$0.296703\pi$
$500$	0	0
$501$	10158.0	0.905840
$502$	−9408.00	−0.836453
$503$	10762.0	0.953984	0.476992	−	0.878908i	$-0.341727\pi$
$503$	10762.0	0.953984	0.476992	+	0.878908i	$0.341727\pi$
$504$	0	0
$505$	0	0
$506$	−6028.00	−0.529599
$507$	7281.00	0.637792
$508$	−3752.00	−0.327693
$509$	−1570.00	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$509$	−1570.00	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$510$	0	0
$511$	−10878.0	−0.941711
$512$	−16384.0	−1.41421
$513$	3375.00	0.290468
$514$	−15384.0	−1.32015
$515$	0	0
$516$	−4992.00	−0.425893
$517$	−4059.00	−0.345289
$518$	29316.0	2.48662
$519$	351.000	0.0296863
$520$	0	0
$521$	−22638.0	−1.90363	−0.951813	−	0.306680i	$-0.900782\pi$
$521$	−22638.0	−1.90363	−0.951813	+	0.306680i	$0.900782\pi$
$522$	5400.00	0.452781
$523$	−10273.0	−0.858904	−0.429452	−	0.903090i	$-0.641293\pi$
$523$	−10273.0	−0.858904	−0.429452	+	0.903090i	$0.641293\pi$
$524$	−3264.00	−0.272115
$525$	0	0
$526$	−2088.00	−0.173082
$527$	6132.00	0.506858
$528$	−2112.00	−0.174078
$529$	6602.00	0.542615
$530$	0	0
$531$	2115.00	0.172850
$532$	21000.0	1.71140
$533$	−33796.0	−2.74647
$534$	9240.00	0.748790
$535$	0	0
$536$	0	0
$537$	8985.00	0.722032
$538$	−16080.0	−1.28858
$539$	1078.00	0.0861461
$540$	0	0
$541$	−6628.00	−0.526728	−0.263364	−	0.964697i	$-0.584832\pi$
$541$	−6628.00	−0.526728	−0.263364	+	0.964697i	$0.584832\pi$
$542$	−26748.0	−2.11979
$543$	12201.0	0.964263
$544$	5376.00	0.423702
$545$	0	0
$546$	17136.0	1.34314
$547$	1131.00	0.0884060	0.0442030	−	0.999023i	$-0.485925\pi$
$547$	1131.00	0.0884060	0.0442030	+	0.999023i	$0.485925\pi$
$548$	19728.0	1.53784
$549$	4338.00	0.337234
$550$	0	0
$551$	−18750.0	−1.44969
$552$	0	0
$553$	−21945.0	−1.68752
$554$	−14984.0	−1.14911
$555$	0	0
$556$	8160.00	0.622412
$557$	−22954.0	−1.74613	−0.873063	−	0.487607i	$-0.837870\pi$
$557$	−22954.0	−1.74613	−0.873063	+	0.487607i	$0.837870\pi$
$558$	−10512.0	−0.797506
$559$	14144.0	1.07017
$560$	0	0
$561$	693.000	0.0521542
$562$	23532.0	1.76626
$563$	5532.00	0.414114	0.207057	−	0.978329i	$-0.433611\pi$
$563$	5532.00	0.414114	0.207057	+	0.978329i	$0.433611\pi$
$564$	−8856.00	−0.661179
$565$	0	0
$566$	15772.0	1.17128
$567$	1701.00	0.125988
$568$	0	0
$569$	−25225.0	−1.85850	−0.929250	−	0.369450i	$-0.879546\pi$
$569$	−25225.0	−1.85850	−0.929250	+	0.369450i	$0.879546\pi$
$570$	0	0
$571$	−2088.00	−0.153030	−0.0765150	−	0.997068i	$-0.524379\pi$
$571$	−2088.00	−0.153030	−0.0765150	+	0.997068i	$0.524379\pi$
$572$	−5984.00	−0.437419
$573$	9141.00	0.666441
$574$	−41748.0	−3.03576
$575$	0	0
$576$	−4608.00	−0.333333
$577$	7831.00	0.565007	0.282503	−	0.959266i	$-0.408835\pi$
$577$	7831.00	0.565007	0.282503	+	0.959266i	$0.408835\pi$
$578$	17888.0	1.28727
$579$	3696.00	0.265286
$580$	0	0
$581$	−12768.0	−0.911714
$582$	−18492.0	−1.31704
$583$	5962.00	0.423535
$584$	0	0
$585$	0	0
$586$	−5948.00	−0.419300
$587$	−8199.00	−0.576506	−0.288253	−	0.957554i	$-0.593074\pi$
$587$	−8199.00	−0.576506	−0.288253	+	0.957554i	$0.593074\pi$
$588$	2352.00	0.164957
$589$	36500.0	2.55341
$590$	0	0
$591$	−14937.0	−1.03964
$592$	22336.0	1.55068
$593$	9542.00	0.660781	0.330390	−	0.943844i	$-0.392820\pi$
$593$	9542.00	0.660781	0.330390	+	0.943844i	$0.392820\pi$
$594$	−1188.00	−0.0820610
$595$	0	0
$596$	40.0000	0.00274910
$597$	1800.00	0.123399
$598$	37264.0	2.54822
$599$	24705.0	1.68517	0.842587	−	0.538561i	$-0.181032\pi$
$599$	24705.0	1.68517	0.842587	+	0.538561i	$0.181032\pi$
$600$	0	0
$601$	15452.0	1.04875	0.524376	−	0.851487i	$-0.324299\pi$
$601$	15452.0	1.04875	0.524376	+	0.851487i	$0.324299\pi$
$602$	17472.0	1.18290
$603$	−6606.00	−0.446131
$604$	3616.00	0.243598
$605$	0	0
$606$	−9924.00	−0.665239
$607$	6176.00	0.412975	0.206488	−	0.978449i	$-0.433797\pi$
$607$	6176.00	0.412975	0.206488	+	0.978449i	$0.433797\pi$
$608$	32000.0	2.13449
$609$	−9450.00	−0.628790
$610$	0	0
$611$	25092.0	1.66140
$612$	1512.00	0.0998676
$613$	−13198.0	−0.869596	−0.434798	−	0.900528i	$-0.643180\pi$
$613$	−13198.0	−0.869596	−0.434798	+	0.900528i	$0.643180\pi$
$614$	19376.0	1.27354
$615$	0	0
$616$	0	0
$617$	19216.0	1.25382	0.626910	−	0.779092i	$-0.284319\pi$
$617$	19216.0	1.25382	0.626910	+	0.779092i	$0.284319\pi$
$618$	2976.00	0.193709
$619$	27700.0	1.79864	0.899319	−	0.437293i	$-0.144063\pi$
$619$	27700.0	1.79864	0.899319	+	0.437293i	$0.144063\pi$
$620$	0	0
$621$	3699.00	0.239027
$622$	−18528.0	−1.19438
$623$	−16170.0	−1.03987
$624$	13056.0	0.837593
$625$	0	0
$626$	−33748.0	−2.15470
$627$	4125.00	0.262738
$628$	14128.0	0.897721
$629$	−7329.00	−0.464589
$630$	0	0
$631$	−5108.00	−0.322260	−0.161130	−	0.986933i	$-0.551514\pi$
$631$	−5108.00	−0.322260	−0.161130	+	0.986933i	$0.551514\pi$
$632$	0	0
$633$	−7404.00	−0.464901
$634$	−14544.0	−0.911066
$635$	0	0
$636$	13008.0	0.811007
$637$	−6664.00	−0.414501
$638$	6600.00	0.409556
$639$	5283.00	0.327061
$640$	0	0
$641$	322.000	0.0198412	0.00992062	−	0.999951i	$-0.496842\pi$
$641$	322.000	0.0198412	0.00992062	+	0.999951i	$0.496842\pi$
$642$	−4392.00	−0.269998
$643$	7432.00	0.455816	0.227908	−	0.973683i	$-0.426812\pi$
$643$	7432.00	0.455816	0.227908	+	0.973683i	$0.426812\pi$
$644$	23016.0	1.40832
$645$	0	0
$646$	−10500.0	−0.639500
$647$	−1409.00	−0.0856159	−0.0428080	−	0.999083i	$-0.513630\pi$
$647$	−1409.00	−0.0856159	−0.0428080	+	0.999083i	$0.513630\pi$
$648$	0	0
$649$	2585.00	0.156348
$650$	0	0
$651$	18396.0	1.10752
$652$	−16544.0	−0.993732
$653$	−21548.0	−1.29133	−0.645665	−	0.763621i	$-0.723420\pi$
$653$	−21548.0	−1.29133	−0.645665	+	0.763621i	$0.723420\pi$
$654$	−3240.00	−0.193722
$655$	0	0
$656$	−31808.0	−1.89313
$657$	−4662.00	−0.276837
$658$	30996.0	1.83640
$659$	−13380.0	−0.790912	−0.395456	−	0.918485i	$-0.629413\pi$
$659$	−13380.0	−0.790912	−0.395456	+	0.918485i	$0.629413\pi$
$660$	0	0
$661$	10907.0	0.641805	0.320903	−	0.947112i	$-0.396014\pi$
$661$	10907.0	0.641805	0.320903	+	0.947112i	$0.396014\pi$
$662$	3032.00	0.178009
$663$	−4284.00	−0.250945
$664$	0	0
$665$	0	0
$666$	12564.0	0.730999
$667$	−20550.0	−1.19295
$668$	27088.0	1.56896
$669$	16176.0	0.934829
$670$	0	0
$671$	5302.00	0.305039
$672$	16128.0	0.925820
$673$	17522.0	1.00360	0.501800	−	0.864983i	$-0.332671\pi$
$673$	17522.0	1.00360	0.501800	+	0.864983i	$0.332671\pi$
$674$	29496.0	1.68567
$675$	0	0
$676$	19416.0	1.10469
$677$	8306.00	0.471530	0.235765	−	0.971810i	$-0.424241\pi$
$677$	8306.00	0.471530	0.235765	+	0.971810i	$0.424241\pi$
$678$	−12024.0	−0.681090
$679$	32361.0	1.82902
$680$	0	0
$681$	7098.00	0.399407
$682$	−12848.0	−0.721371
$683$	18427.0	1.03234	0.516171	−	0.856486i	$-0.327357\pi$
$683$	18427.0	1.03234	0.516171	+	0.856486i	$0.327357\pi$
$684$	9000.00	0.503105
$685$	0	0
$686$	20580.0	1.14541
$687$	−13935.0	−0.773877
$688$	13312.0	0.737668
$689$	−36856.0	−2.03788
$690$	0	0
$691$	−8278.00	−0.455731	−0.227865	−	0.973693i	$-0.573175\pi$
$691$	−8278.00	−0.455731	−0.227865	+	0.973693i	$0.573175\pi$
$692$	936.000	0.0514182
$693$	2079.00	0.113961
$694$	26096.0	1.42736
$695$	0	0
$696$	0	0
$697$	10437.0	0.567187
$698$	−14840.0	−0.804731
$699$	−1539.00	−0.0832766
$700$	0	0
$701$	−21923.0	−1.18120	−0.590599	−	0.806965i	$-0.701109\pi$
$701$	−21923.0	−1.18120	−0.590599	+	0.806965i	$0.701109\pi$
$702$	7344.00	0.394845
$703$	−43625.0	−2.34047
$704$	−5632.00	−0.301511
$705$	0	0
$706$	−11328.0	−0.603874
$707$	17367.0	0.923838
$708$	5640.00	0.299384
$709$	−11425.0	−0.605183	−0.302592	−	0.953120i	$-0.597852\pi$
$709$	−11425.0	−0.605183	−0.302592	+	0.953120i	$0.597852\pi$
$710$	0	0
$711$	−9405.00	−0.496083
$712$	0	0
$713$	40004.0	2.10121
$714$	−5292.00	−0.277378
$715$	0	0
$716$	23960.0	1.25060
$717$	8070.00	0.420334
$718$	28160.0	1.46368
$719$	7380.00	0.382792	0.191396	−	0.981513i	$-0.438699\pi$
$719$	7380.00	0.382792	0.191396	+	0.981513i	$0.438699\pi$
$720$	0	0
$721$	−5208.00	−0.269010
$722$	−35064.0	−1.80741
$723$	−11184.0	−0.575294
$724$	32536.0	1.67015
$725$	0	0
$726$	−1452.00	−0.0742270
$727$	−10924.0	−0.557288	−0.278644	−	0.960394i	$-0.589885\pi$
$727$	−10924.0	−0.557288	−0.278644	+	0.960394i	$0.589885\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4368.00	−0.221007
$732$	11568.0	0.584106
$733$	−13578.0	−0.684195	−0.342097	−	0.939664i	$-0.611137\pi$
$733$	−13578.0	−0.684195	−0.342097	+	0.939664i	$0.611137\pi$
$734$	−24824.0	−1.24833
$735$	0	0
$736$	35072.0	1.75648
$737$	−8074.00	−0.403541
$738$	−17892.0	−0.892430
$739$	2875.00	0.143110	0.0715552	−	0.997437i	$-0.477204\pi$
$739$	2875.00	0.143110	0.0715552	+	0.997437i	$0.477204\pi$
$740$	0	0
$741$	−25500.0	−1.26419
$742$	−45528.0	−2.25254
$743$	−9568.00	−0.472431	−0.236215	−	0.971701i	$-0.575907\pi$
$743$	−9568.00	−0.472431	−0.236215	+	0.971701i	$0.575907\pi$
$744$	0	0
$745$	0	0
$746$	−7848.00	−0.385168
$747$	−5472.00	−0.268019
$748$	1848.00	0.0903337
$749$	7686.00	0.374954
$750$	0	0
$751$	−35048.0	−1.70296	−0.851478	−	0.524391i	$-0.824293\pi$
$751$	−35048.0	−1.70296	−0.851478	+	0.524391i	$0.824293\pi$
$752$	23616.0	1.14520
$753$	7056.00	0.341481
$754$	−40800.0	−1.97062
$755$	0	0
$756$	4536.00	0.218218
$757$	8226.00	0.394953	0.197476	−	0.980308i	$-0.436725\pi$
$757$	8226.00	0.394953	0.197476	+	0.980308i	$0.436725\pi$
$758$	31840.0	1.52570
$759$	4521.00	0.216208
$760$	0	0
$761$	−6818.00	−0.324773	−0.162387	−	0.986727i	$-0.551919\pi$
$761$	−6818.00	−0.324773	−0.162387	+	0.986727i	$0.551919\pi$
$762$	5628.00	0.267560
$763$	5670.00	0.269027
$764$	24376.0	1.15431
$765$	0	0
$766$	28752.0	1.35620
$767$	−15980.0	−0.752287
$768$	12288.0	0.577350
$769$	29390.0	1.37819	0.689097	−	0.724670i	$-0.258008\pi$
$769$	29390.0	1.37819	0.689097	+	0.724670i	$0.258008\pi$
$770$	0	0
$771$	11538.0	0.538951
$772$	9856.00	0.459489
$773$	−34358.0	−1.59867	−0.799335	−	0.600886i	$-0.794815\pi$
$773$	−34358.0	−1.59867	−0.799335	+	0.600886i	$0.794815\pi$
$774$	7488.00	0.347740
$775$	0	0
$776$	0	0
$777$	−21987.0	−1.01516
$778$	−31680.0	−1.45988
$779$	62125.0	2.85733
$780$	0	0
$781$	6457.00	0.295838
$782$	−11508.0	−0.526247
$783$	−4050.00	−0.184847
$784$	−6272.00	−0.285714
$785$	0	0
$786$	4896.00	0.222181
$787$	14291.0	0.647292	0.323646	−	0.946178i	$-0.395091\pi$
$787$	14291.0	0.647292	0.323646	+	0.946178i	$0.395091\pi$
$788$	−39832.0	−1.80071
$789$	1566.00	0.0706604
$790$	0	0
$791$	21042.0	0.945850
$792$	0	0
$793$	−32776.0	−1.46773
$794$	38616.0	1.72598
$795$	0	0
$796$	4800.00	0.213733
$797$	11576.0	0.514483	0.257242	−	0.966347i	$-0.417186\pi$
$797$	11576.0	0.514483	0.257242	+	0.966347i	$0.417186\pi$
$798$	−31500.0	−1.39735
$799$	−7749.00	−0.343104
$800$	0	0
$801$	−6930.00	−0.305692
$802$	−7808.00	−0.343778
$803$	−5698.00	−0.250409
$804$	−17616.0	−0.772722
$805$	0	0
$806$	79424.0	3.47096
$807$	12060.0	0.526062
$808$	0	0
$809$	−12825.0	−0.557358	−0.278679	−	0.960384i	$-0.589897\pi$
$809$	−12825.0	−0.557358	−0.278679	+	0.960384i	$0.589897\pi$
$810$	0	0
$811$	−36843.0	−1.59523	−0.797616	−	0.603166i	$-0.793906\pi$
$811$	−36843.0	−1.59523	−0.797616	+	0.603166i	$0.793906\pi$
$812$	−25200.0	−1.08910
$813$	20061.0	0.865400
$814$	15356.0	0.661213
$815$	0	0
$816$	−4032.00	−0.172976
$817$	−26000.0	−1.11337
$818$	−38760.0	−1.65674
$819$	−12852.0	−0.548334
$820$	0	0
$821$	43962.0	1.86880	0.934400	−	0.356226i	$-0.115937\pi$
$821$	43962.0	1.86880	0.934400	+	0.356226i	$0.115937\pi$
$822$	−29592.0	−1.25564
$823$	33522.0	1.41981	0.709905	−	0.704298i	$-0.248738\pi$
$823$	33522.0	1.41981	0.709905	+	0.704298i	$0.248738\pi$
$824$	0	0
$825$	0	0
$826$	−19740.0	−0.831528
$827$	−1704.00	−0.0716492	−0.0358246	−	0.999358i	$-0.511406\pi$
$827$	−1704.00	−0.0716492	−0.0358246	+	0.999358i	$0.511406\pi$
$828$	9864.00	0.414007
$829$	1750.00	0.0733173	0.0366586	−	0.999328i	$-0.488329\pi$
$829$	1750.00	0.0733173	0.0366586	+	0.999328i	$0.488329\pi$
$830$	0	0
$831$	11238.0	0.469124
$832$	34816.0	1.45075
$833$	2058.00	0.0856008
$834$	−12240.0	−0.508197
$835$	0	0
$836$	11000.0	0.455075
$837$	7884.00	0.325581
$838$	11740.0	0.483952
$839$	−15260.0	−0.627931	−0.313965	−	0.949434i	$-0.601658\pi$
$839$	−15260.0	−0.627931	−0.313965	+	0.949434i	$0.601658\pi$
$840$	0	0
$841$	−1889.00	−0.0774530
$842$	−51348.0	−2.10163
$843$	−17649.0	−0.721072
$844$	−19744.0	−0.805233
$845$	0	0
$846$	13284.0	0.539850
$847$	2541.00	0.103081
$848$	−34688.0	−1.40471
$849$	−11829.0	−0.478175
$850$	0	0
$851$	−47813.0	−1.92598
$852$	14088.0	0.566487
$853$	−878.000	−0.0352428	−0.0176214	−	0.999845i	$-0.505609\pi$
$853$	−878.000	−0.0352428	−0.0176214	+	0.999845i	$0.505609\pi$
$854$	−40488.0	−1.62233
$855$	0	0
$856$	0	0
$857$	−35019.0	−1.39583	−0.697915	−	0.716181i	$-0.745889\pi$
$857$	−35019.0	−1.39583	−0.697915	+	0.716181i	$0.745889\pi$
$858$	8976.00	0.357151
$859$	−1280.00	−0.0508417	−0.0254209	−	0.999677i	$-0.508093\pi$
$859$	−1280.00	−0.0508417	−0.0254209	+	0.999677i	$0.508093\pi$
$860$	0	0
$861$	31311.0	1.23934
$862$	24432.0	0.965380
$863$	−16888.0	−0.666135	−0.333067	−	0.942903i	$-0.608084\pi$
$863$	−16888.0	−0.666135	−0.333067	+	0.942903i	$0.608084\pi$
$864$	6912.00	0.272166
$865$	0	0
$866$	37112.0	1.45626
$867$	−13416.0	−0.525526
$868$	49056.0	1.91828
$869$	−11495.0	−0.448724
$870$	0	0
$871$	49912.0	1.94168
$872$	0	0
$873$	13869.0	0.537680
$874$	−68500.0	−2.65108
$875$	0	0
$876$	−12432.0	−0.479496
$877$	29836.0	1.14879	0.574396	−	0.818578i	$-0.305237\pi$
$877$	29836.0	1.14879	0.574396	+	0.818578i	$0.305237\pi$
$878$	−9820.00	−0.377459
$879$	4461.00	0.171178
$880$	0	0
$881$	29292.0	1.12017	0.560087	−	0.828434i	$-0.310768\pi$
$881$	29292.0	1.12017	0.560087	+	0.828434i	$0.310768\pi$
$882$	−3528.00	−0.134687
$883$	6532.00	0.248946	0.124473	−	0.992223i	$-0.460276\pi$
$883$	6532.00	0.248946	0.124473	+	0.992223i	$0.460276\pi$
$884$	−11424.0	−0.434650
$885$	0	0
$886$	14012.0	0.531312
$887$	20476.0	0.775103	0.387552	−	0.921848i	$-0.373321\pi$
$887$	20476.0	0.775103	0.387552	+	0.921848i	$0.373321\pi$
$888$	0	0
$889$	−9849.00	−0.371569
$890$	0	0
$891$	891.000	0.0335013
$892$	43136.0	1.61917
$893$	−46125.0	−1.72846
$894$	−60.0000	−0.00224463
$895$	0	0
$896$	0	0
$897$	−27948.0	−1.04031
$898$	30520.0	1.13415
$899$	−43800.0	−1.62493
$900$	0	0
$901$	11382.0	0.420854
$902$	−21868.0	−0.807234
$903$	−13104.0	−0.482917
$904$	0	0
$905$	0	0
$906$	−5424.00	−0.198897
$907$	−51914.0	−1.90052	−0.950262	−	0.311450i	$-0.899185\pi$
$907$	−51914.0	−1.90052	−0.950262	+	0.311450i	$0.899185\pi$
$908$	18928.0	0.691793
$909$	7443.00	0.271583
$910$	0	0
$911$	−41893.0	−1.52358	−0.761788	−	0.647827i	$-0.775678\pi$
$911$	−41893.0	−1.52358	−0.761788	+	0.647827i	$0.775678\pi$
$912$	−24000.0	−0.871403
$913$	−6688.00	−0.242432
$914$	29656.0	1.07323
$915$	0	0
$916$	−37160.0	−1.34039
$917$	−8568.00	−0.308550
$918$	−2268.00	−0.0815416
$919$	495.000	0.0177677	0.00888386	−	0.999961i	$-0.497172\pi$
$919$	495.000	0.0177677	0.00888386	+	0.999961i	$0.497172\pi$
$920$	0	0
$921$	−14532.0	−0.519919
$922$	−19928.0	−0.711815
$923$	−39916.0	−1.42346
$924$	5544.00	0.197386
$925$	0	0
$926$	−53688.0	−1.90529
$927$	−2232.00	−0.0790814
$928$	−38400.0	−1.35834
$929$	−16310.0	−0.576010	−0.288005	−	0.957629i	$-0.592992\pi$
$929$	−16310.0	−0.576010	−0.288005	+	0.957629i	$0.592992\pi$
$930$	0	0
$931$	12250.0	0.431233
$932$	−4104.00	−0.144239
$933$	13896.0	0.487604
$934$	63216.0	2.21466
$935$	0	0
$936$	0	0
$937$	−18744.0	−0.653511	−0.326755	−	0.945109i	$-0.605955\pi$
$937$	−18744.0	−0.653511	−0.326755	+	0.945109i	$0.605955\pi$
$938$	61656.0	2.14620
$939$	25311.0	0.879652
$940$	0	0
$941$	−25553.0	−0.885233	−0.442616	−	0.896711i	$-0.645950\pi$
$941$	−25553.0	−0.885233	−0.442616	+	0.896711i	$0.645950\pi$
$942$	−21192.0	−0.732986
$943$	68089.0	2.35131
$944$	−15040.0	−0.518549
$945$	0	0
$946$	9152.00	0.314542
$947$	−6879.00	−0.236048	−0.118024	−	0.993011i	$-0.537656\pi$
$947$	−6879.00	−0.236048	−0.118024	+	0.993011i	$0.537656\pi$
$948$	−25080.0	−0.859241
$949$	35224.0	1.20487
$950$	0	0
$951$	10908.0	0.371941
$952$	0	0
$953$	13677.0	0.464891	0.232446	−	0.972609i	$-0.425327\pi$
$953$	13677.0	0.464891	0.232446	+	0.972609i	$0.425327\pi$
$954$	−19512.0	−0.662185
$955$	0	0
$956$	21520.0	0.728040
$957$	−4950.00	−0.167200
$958$	36240.0	1.22219
$959$	51786.0	1.74375
$960$	0	0
$961$	55473.0	1.86207
$962$	−94928.0	−3.18150
$963$	3294.00	0.110226
$964$	−29824.0	−0.996438
$965$	0	0
$966$	−34524.0	−1.14989
$967$	−26984.0	−0.897360	−0.448680	−	0.893693i	$-0.648106\pi$
$967$	−26984.0	−0.897360	−0.448680	+	0.893693i	$0.648106\pi$
$968$	0	0
$969$	7875.00	0.261075
$970$	0	0
$971$	41937.0	1.38602	0.693008	−	0.720929i	$-0.256285\pi$
$971$	41937.0	1.38602	0.693008	+	0.720929i	$0.256285\pi$
$972$	1944.00	0.0641500
$973$	21420.0	0.705749
$974$	11416.0	0.375557
$975$	0	0
$976$	−30848.0	−1.01170
$977$	−13504.0	−0.442202	−0.221101	−	0.975251i	$-0.570965\pi$
$977$	−13504.0	−0.442202	−0.221101	+	0.975251i	$0.570965\pi$
$978$	24816.0	0.811379
$979$	−8470.00	−0.276509
$980$	0	0
$981$	2430.00	0.0790866
$982$	9112.00	0.296106
$983$	−33353.0	−1.08219	−0.541096	−	0.840961i	$-0.681991\pi$
$983$	−33353.0	−1.08219	−0.541096	+	0.840961i	$0.681991\pi$
$984$	0	0
$985$	0	0
$986$	12600.0	0.406963
$987$	−23247.0	−0.749706
$988$	−68000.0	−2.18964
$989$	−28496.0	−0.916198
$990$	0	0
$991$	−16978.0	−0.544222	−0.272111	−	0.962266i	$-0.587722\pi$
$991$	−16978.0	−0.544222	−0.272111	+	0.962266i	$0.587722\pi$
$992$	74752.0	2.39252
$993$	−2274.00	−0.0726719
$994$	−49308.0	−1.57340
$995$	0	0
$996$	−14592.0	−0.464222
$997$	−2714.00	−0.0862119	−0.0431059	−	0.999071i	$-0.513725\pi$
$997$	−2714.00	−0.0862119	−0.0431059	+	0.999071i	$0.513725\pi$
$998$	−53160.0	−1.68612
$999$	−9423.00	−0.298429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.b.1.1	✓	1
3.2	odd	2		2475.4.a.j.1.1		1
5.2	odd	4		825.4.c.c.199.1		2
5.3	odd	4		825.4.c.c.199.2		2
5.4	even	2		825.4.a.h.1.1	yes	1
15.14	odd	2		2475.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.b.1.1	✓	1	1.1	even	1	trivial
825.4.a.h.1.1	yes	1	5.4	even	2
825.4.c.c.199.1		2	5.2	odd	4
825.4.c.c.199.2		2	5.3	odd	4
2475.4.a.c.1.1		1	15.14	odd	2
2475.4.a.j.1.1		1	3.2	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more