LMFDB - Embedded newform 825.4.a.d.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
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)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} -36.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} -36.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +11.0000 q^{11} +21.0000 q^{12} -2.00000 q^{13} +36.0000 q^{14} +41.0000 q^{16} -66.0000 q^{17} -9.00000 q^{18} +140.000 q^{19} +108.000 q^{21} -11.0000 q^{22} +68.0000 q^{23} -45.0000 q^{24} +2.00000 q^{26} -27.0000 q^{27} +252.000 q^{28} +150.000 q^{29} -128.000 q^{31} -161.000 q^{32} -33.0000 q^{33} +66.0000 q^{34} -63.0000 q^{36} +314.000 q^{37} -140.000 q^{38} +6.00000 q^{39} -118.000 q^{41} -108.000 q^{42} -172.000 q^{43} -77.0000 q^{44} -68.0000 q^{46} +324.000 q^{47} -123.000 q^{48} +953.000 q^{49} +198.000 q^{51} +14.0000 q^{52} -82.0000 q^{53} +27.0000 q^{54} -540.000 q^{56} -420.000 q^{57} -150.000 q^{58} -740.000 q^{59} +122.000 q^{61} +128.000 q^{62} -324.000 q^{63} -167.000 q^{64} +33.0000 q^{66} +124.000 q^{67} +462.000 q^{68} -204.000 q^{69} -988.000 q^{71} +135.000 q^{72} -2.00000 q^{73} -314.000 q^{74} -980.000 q^{76} -396.000 q^{77} -6.00000 q^{78} +1100.00 q^{79} +81.0000 q^{81} +118.000 q^{82} +868.000 q^{83} -756.000 q^{84} +172.000 q^{86} -450.000 q^{87} +165.000 q^{88} -470.000 q^{89} +72.0000 q^{91} -476.000 q^{92} +384.000 q^{93} -324.000 q^{94} +483.000 q^{96} -1186.00 q^{97} -953.000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	3.00000	0.204124
$7$	−36.0000	−1.94382	−0.971909	−	0.235358i	$-0.924374\pi$
$7$	−36.0000	−1.94382	−0.971909	+	0.235358i	$0.924374\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	21.0000	0.505181
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	36.0000	0.687243
$15$	0	0
$16$	41.0000	0.640625
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	−9.00000	−0.117851
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	108.000	1.12226
$22$	−11.0000	−0.106600
$23$	68.0000	0.616477	0.308239	−	0.951309i	$-0.400260\pi$
$23$	68.0000	0.616477	0.308239	+	0.951309i	$0.400260\pi$
$24$	−45.0000	−0.382733
$25$	0	0
$26$	2.00000	0.0150859
$27$	−27.0000	−0.192450
$28$	252.000	1.70084
$29$	150.000	0.960493	0.480247	−	0.877134i	$-0.340547\pi$
$29$	150.000	0.960493	0.480247	+	0.877134i	$0.340547\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	−161.000	−0.889408
$33$	−33.0000	−0.174078
$34$	66.0000	0.332909
$35$	0	0
$36$	−63.0000	−0.291667
$37$	314.000	1.39517	0.697585	−	0.716502i	$-0.254258\pi$
$37$	314.000	1.39517	0.697585	+	0.716502i	$0.254258\pi$
$38$	−140.000	−0.597658
$39$	6.00000	0.0246351
$40$	0	0
$41$	−118.000	−0.449476	−0.224738	−	0.974419i	$-0.572153\pi$
$41$	−118.000	−0.449476	−0.224738	+	0.974419i	$0.572153\pi$
$42$	−108.000	−0.396780
$43$	−172.000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−77.0000	−0.263822
$45$	0	0
$46$	−68.0000	−0.217958
$47$	324.000	1.00554	0.502769	−	0.864421i	$-0.332315\pi$
$47$	324.000	1.00554	0.502769	+	0.864421i	$0.332315\pi$
$48$	−123.000	−0.369865
$49$	953.000	2.77843
$50$	0	0
$51$	198.000	0.543638
$52$	14.0000	0.0373356
$53$	−82.0000	−0.212520	−0.106260	−	0.994338i	$-0.533888\pi$
$53$	−82.0000	−0.212520	−0.106260	+	0.994338i	$0.533888\pi$
$54$	27.0000	0.0680414
$55$	0	0
$56$	−540.000	−1.28858
$57$	−420.000	−0.975971
$58$	−150.000	−0.339586
$59$	−740.000	−1.63288	−0.816439	−	0.577432i	$-0.804055\pi$
$59$	−740.000	−1.63288	−0.816439	+	0.577432i	$0.804055\pi$
$60$	0	0
$61$	122.000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	122.000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	128.000	0.262194
$63$	−324.000	−0.647939
$64$	−167.000	−0.326172
$65$	0	0
$66$	33.0000	0.0615457
$67$	124.000	0.226105	0.113052	−	0.993589i	$-0.463937\pi$
$67$	124.000	0.226105	0.113052	+	0.993589i	$0.463937\pi$
$68$	462.000	0.823908
$69$	−204.000	−0.355923
$70$	0	0
$71$	−988.000	−1.65147	−0.825733	−	0.564062i	$-0.809238\pi$
$71$	−988.000	−1.65147	−0.825733	+	0.564062i	$0.809238\pi$
$72$	135.000	0.220971
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	−314.000	−0.493267
$75$	0	0
$76$	−980.000	−1.47913
$77$	−396.000	−0.586083
$78$	−6.00000	−0.00870982
$79$	1100.00	1.56658	0.783289	−	0.621658i	$-0.213540\pi$
$79$	1100.00	1.56658	0.783289	+	0.621658i	$0.213540\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	118.000	0.158914
$83$	868.000	1.14790	0.573948	−	0.818892i	$-0.305411\pi$
$83$	868.000	1.14790	0.573948	+	0.818892i	$0.305411\pi$
$84$	−756.000	−0.981981
$85$	0	0
$86$	172.000	0.215666
$87$	−450.000	−0.554541
$88$	165.000	0.199876
$89$	−470.000	−0.559774	−0.279887	−	0.960033i	$-0.590297\pi$
$89$	−470.000	−0.559774	−0.279887	+	0.960033i	$0.590297\pi$
$90$	0	0
$91$	72.0000	0.0829412
$92$	−476.000	−0.539418
$93$	384.000	0.428161
$94$	−324.000	−0.355511
$95$	0	0
$96$	483.000	0.513500
$97$	−1186.00	−1.24144	−0.620722	−	0.784031i	$-0.713160\pi$
$97$	−1186.00	−1.24144	−0.620722	+	0.784031i	$0.713160\pi$
$98$	−953.000	−0.982322
$99$	99.0000	0.100504
$100$	0	0
$101$	1502.00	1.47975	0.739874	−	0.672745i	$-0.234885\pi$
$101$	1502.00	1.47975	0.739874	+	0.672745i	$0.234885\pi$
$102$	−198.000	−0.192205
$103$	−32.0000	−0.0306122	−0.0153061	−	0.999883i	$-0.504872\pi$
$103$	−32.0000	−0.0306122	−0.0153061	+	0.999883i	$0.504872\pi$
$104$	−30.0000	−0.0282860
$105$	0	0
$106$	82.0000	0.0751372
$107$	−1116.00	−1.00830	−0.504149	−	0.863617i	$-0.668194\pi$
$107$	−1116.00	−1.00830	−0.504149	+	0.863617i	$0.668194\pi$
$108$	189.000	0.168394
$109$	−2190.00	−1.92444	−0.962220	−	0.272273i	$-0.912225\pi$
$109$	−2190.00	−1.92444	−0.962220	+	0.272273i	$0.912225\pi$
$110$	0	0
$111$	−942.000	−0.805502
$112$	−1476.00	−1.24526
$113$	1398.00	1.16383	0.581915	−	0.813250i	$-0.302304\pi$
$113$	1398.00	1.16383	0.581915	+	0.813250i	$0.302304\pi$
$114$	420.000	0.345058
$115$	0	0
$116$	−1050.00	−0.840431
$117$	−18.0000	−0.0142231
$118$	740.000	0.577310
$119$	2376.00	1.83032
$120$	0	0
$121$	121.000	0.0909091
$122$	−122.000	−0.0905357
$123$	354.000	0.259505
$124$	896.000	0.648897
$125$	0	0
$126$	324.000	0.229081
$127$	44.0000	0.0307431	0.0153715	−	0.999882i	$-0.495107\pi$
$127$	44.0000	0.0307431	0.0153715	+	0.999882i	$0.495107\pi$
$128$	1455.00	1.00473
$129$	516.000	0.352180
$130$	0	0
$131$	−1308.00	−0.872370	−0.436185	−	0.899857i	$-0.643671\pi$
$131$	−1308.00	−0.872370	−0.436185	+	0.899857i	$0.643671\pi$
$132$	231.000	0.152318
$133$	−5040.00	−3.28589
$134$	−124.000	−0.0799401
$135$	0	0
$136$	−990.000	−0.624204
$137$	−1626.00	−1.01400	−0.507002	−	0.861945i	$-0.669246\pi$
$137$	−1626.00	−1.01400	−0.507002	+	0.861945i	$0.669246\pi$
$138$	204.000	0.125838
$139$	180.000	0.109837	0.0549187	−	0.998491i	$-0.482510\pi$
$139$	180.000	0.109837	0.0549187	+	0.998491i	$0.482510\pi$
$140$	0	0
$141$	−972.000	−0.580547
$142$	988.000	0.583881
$143$	−22.0000	−0.0128653
$144$	369.000	0.213542
$145$	0	0
$146$	2.00000	0.00113371
$147$	−2859.00	−1.60412
$148$	−2198.00	−1.22077
$149$	1430.00	0.786243	0.393121	−	0.919487i	$-0.371395\pi$
$149$	1430.00	0.786243	0.393121	+	0.919487i	$0.371395\pi$
$150$	0	0
$151$	−1948.00	−1.04984	−0.524921	−	0.851151i	$-0.675905\pi$
$151$	−1948.00	−1.04984	−0.524921	+	0.851151i	$0.675905\pi$
$152$	2100.00	1.12061
$153$	−594.000	−0.313870
$154$	396.000	0.207212
$155$	0	0
$156$	−42.0000	−0.0215557
$157$	−646.000	−0.328385	−0.164192	−	0.986428i	$-0.552502\pi$
$157$	−646.000	−0.328385	−0.164192	+	0.986428i	$0.552502\pi$
$158$	−1100.00	−0.553869
$159$	246.000	0.122699
$160$	0	0
$161$	−2448.00	−1.19832
$162$	−81.0000	−0.0392837
$163$	−3052.00	−1.46657	−0.733286	−	0.679921i	$-0.762014\pi$
$163$	−3052.00	−1.46657	−0.733286	+	0.679921i	$0.762014\pi$
$164$	826.000	0.393291
$165$	0	0
$166$	−868.000	−0.405843
$167$	−1216.00	−0.563455	−0.281727	−	0.959495i	$-0.590907\pi$
$167$	−1216.00	−0.563455	−0.281727	+	0.959495i	$0.590907\pi$
$168$	1620.00	0.743963
$169$	−2193.00	−0.998179
$170$	0	0
$171$	1260.00	0.563477
$172$	1204.00	0.533745
$173$	3858.00	1.69548	0.847741	−	0.530411i	$-0.177962\pi$
$173$	3858.00	1.69548	0.847741	+	0.530411i	$0.177962\pi$
$174$	450.000	0.196060
$175$	0	0
$176$	451.000	0.193156
$177$	2220.00	0.942742
$178$	470.000	0.197910
$179$	−380.000	−0.158673	−0.0793367	−	0.996848i	$-0.525280\pi$
$179$	−380.000	−0.158673	−0.0793367	+	0.996848i	$0.525280\pi$
$180$	0	0
$181$	−538.000	−0.220935	−0.110467	−	0.993880i	$-0.535235\pi$
$181$	−538.000	−0.220935	−0.110467	+	0.993880i	$0.535235\pi$
$182$	−72.0000	−0.0293241
$183$	−366.000	−0.147844
$184$	1020.00	0.408671
$185$	0	0
$186$	−384.000	−0.151378
$187$	−726.000	−0.283906
$188$	−2268.00	−0.879845
$189$	972.000	0.374088
$190$	0	0
$191$	1412.00	0.534915	0.267457	−	0.963570i	$-0.413817\pi$
$191$	1412.00	0.534915	0.267457	+	0.963570i	$0.413817\pi$
$192$	501.000	0.188315
$193$	638.000	0.237949	0.118975	−	0.992897i	$-0.462039\pi$
$193$	638.000	0.237949	0.118975	+	0.992897i	$0.462039\pi$
$194$	1186.00	0.438917
$195$	0	0
$196$	−6671.00	−2.43112
$197$	−3686.00	−1.33308	−0.666540	−	0.745470i	$-0.732225\pi$
$197$	−3686.00	−1.33308	−0.666540	+	0.745470i	$0.732225\pi$
$198$	−99.0000	−0.0355335
$199$	−240.000	−0.0854932	−0.0427466	−	0.999086i	$-0.513611\pi$
$199$	−240.000	−0.0854932	−0.0427466	+	0.999086i	$0.513611\pi$
$200$	0	0
$201$	−372.000	−0.130542
$202$	−1502.00	−0.523170
$203$	−5400.00	−1.86702
$204$	−1386.00	−0.475683
$205$	0	0
$206$	32.0000	0.0108230
$207$	612.000	0.205492
$208$	−82.0000	−0.0273350
$209$	1540.00	0.509684
$210$	0	0
$211$	5092.00	1.66136	0.830682	−	0.556747i	$-0.187951\pi$
$211$	5092.00	1.66136	0.830682	+	0.556747i	$0.187951\pi$
$212$	574.000	0.185955
$213$	2964.00	0.953474
$214$	1116.00	0.356487
$215$	0	0
$216$	−405.000	−0.127578
$217$	4608.00	1.44153
$218$	2190.00	0.680392
$219$	6.00000	0.00185134
$220$	0	0
$221$	132.000	0.0401777
$222$	942.000	0.284788
$223$	3808.00	1.14351	0.571755	−	0.820425i	$-0.306263\pi$
$223$	3808.00	1.14351	0.571755	+	0.820425i	$0.306263\pi$
$224$	5796.00	1.72885
$225$	0	0
$226$	−1398.00	−0.411476
$227$	44.0000	0.0128651	0.00643256	−	0.999979i	$-0.497952\pi$
$227$	44.0000	0.0128651	0.00643256	+	0.999979i	$0.497952\pi$
$228$	2940.00	0.853975
$229$	−2650.00	−0.764703	−0.382351	−	0.924017i	$-0.624886\pi$
$229$	−2650.00	−0.764703	−0.382351	+	0.924017i	$0.624886\pi$
$230$	0	0
$231$	1188.00	0.338375
$232$	2250.00	0.636723
$233$	5718.00	1.60772	0.803860	−	0.594819i	$-0.202776\pi$
$233$	5718.00	1.60772	0.803860	+	0.594819i	$0.202776\pi$
$234$	18.0000	0.00502862
$235$	0	0
$236$	5180.00	1.42877
$237$	−3300.00	−0.904464
$238$	−2376.00	−0.647114
$239$	−6520.00	−1.76462	−0.882309	−	0.470670i	$-0.844012\pi$
$239$	−6520.00	−1.76462	−0.882309	+	0.470670i	$0.844012\pi$
$240$	0	0
$241$	−2438.00	−0.651641	−0.325820	−	0.945432i	$-0.605640\pi$
$241$	−2438.00	−0.651641	−0.325820	+	0.945432i	$0.605640\pi$
$242$	−121.000	−0.0321412
$243$	−243.000	−0.0641500
$244$	−854.000	−0.224065
$245$	0	0
$246$	−354.000	−0.0917488
$247$	−280.000	−0.0721294
$248$	−1920.00	−0.491613
$249$	−2604.00	−0.662738
$250$	0	0
$251$	−5268.00	−1.32475	−0.662377	−	0.749171i	$-0.730452\pi$
$251$	−5268.00	−1.32475	−0.662377	+	0.749171i	$0.730452\pi$
$252$	2268.00	0.566947
$253$	748.000	0.185875
$254$	−44.0000	−0.0108693
$255$	0	0
$256$	−119.000	−0.0290527
$257$	5574.00	1.35290	0.676452	−	0.736486i	$-0.263516\pi$
$257$	5574.00	1.35290	0.676452	+	0.736486i	$0.263516\pi$
$258$	−516.000	−0.124515
$259$	−11304.0	−2.71196
$260$	0	0
$261$	1350.00	0.320164
$262$	1308.00	0.308429
$263$	−2472.00	−0.579582	−0.289791	−	0.957090i	$-0.593586\pi$
$263$	−2472.00	−0.579582	−0.289791	+	0.957090i	$0.593586\pi$
$264$	−495.000	−0.115398
$265$	0	0
$266$	5040.00	1.16174
$267$	1410.00	0.323186
$268$	−868.000	−0.197842
$269$	4410.00	0.999563	0.499781	−	0.866152i	$-0.333414\pi$
$269$	4410.00	0.999563	0.499781	+	0.866152i	$0.333414\pi$
$270$	0	0
$271$	−1308.00	−0.293193	−0.146597	−	0.989196i	$-0.546832\pi$
$271$	−1308.00	−0.293193	−0.146597	+	0.989196i	$0.546832\pi$
$272$	−2706.00	−0.603218
$273$	−216.000	−0.0478861
$274$	1626.00	0.358505
$275$	0	0
$276$	1428.00	0.311433
$277$	−226.000	−0.0490217	−0.0245109	−	0.999700i	$-0.507803\pi$
$277$	−226.000	−0.0490217	−0.0245109	+	0.999700i	$0.507803\pi$
$278$	−180.000	−0.0388334
$279$	−1152.00	−0.247199
$280$	0	0
$281$	522.000	0.110818	0.0554091	−	0.998464i	$-0.482354\pi$
$281$	522.000	0.110818	0.0554091	+	0.998464i	$0.482354\pi$
$282$	972.000	0.205254
$283$	1508.00	0.316754	0.158377	−	0.987379i	$-0.449374\pi$
$283$	1508.00	0.316754	0.158377	+	0.987379i	$0.449374\pi$
$284$	6916.00	1.44503
$285$	0	0
$286$	22.0000	0.00454856
$287$	4248.00	0.873699
$288$	−1449.00	−0.296469
$289$	−557.000	−0.113373
$290$	0	0
$291$	3558.00	0.716748
$292$	14.0000	0.00280578
$293$	4658.00	0.928748	0.464374	−	0.885639i	$-0.346279\pi$
$293$	4658.00	0.928748	0.464374	+	0.885639i	$0.346279\pi$
$294$	2859.00	0.567144
$295$	0	0
$296$	4710.00	0.924876
$297$	−297.000	−0.0580259
$298$	−1430.00	−0.277979
$299$	−136.000	−0.0263046
$300$	0	0
$301$	6192.00	1.18572
$302$	1948.00	0.371175
$303$	−4506.00	−0.854333
$304$	5740.00	1.08293
$305$	0	0
$306$	594.000	0.110970
$307$	−6476.00	−1.20392	−0.601962	−	0.798525i	$-0.705614\pi$
$307$	−6476.00	−1.20392	−0.601962	+	0.798525i	$0.705614\pi$
$308$	2772.00	0.512823
$309$	96.0000	0.0176739
$310$	0	0
$311$	5652.00	1.03053	0.515266	−	0.857030i	$-0.327693\pi$
$311$	5652.00	1.03053	0.515266	+	0.857030i	$0.327693\pi$
$312$	90.0000	0.0163309
$313$	5638.00	1.01814	0.509071	−	0.860724i	$-0.329989\pi$
$313$	5638.00	1.01814	0.509071	+	0.860724i	$0.329989\pi$
$314$	646.000	0.116102
$315$	0	0
$316$	−7700.00	−1.37076
$317$	−1506.00	−0.266831	−0.133415	−	0.991060i	$-0.542594\pi$
$317$	−1506.00	−0.266831	−0.133415	+	0.991060i	$0.542594\pi$
$318$	−246.000	−0.0433805
$319$	1650.00	0.289600
$320$	0	0
$321$	3348.00	0.582141
$322$	2448.00	0.423670
$323$	−9240.00	−1.59173
$324$	−567.000	−0.0972222
$325$	0	0
$326$	3052.00	0.518511
$327$	6570.00	1.11108
$328$	−1770.00	−0.297963
$329$	−11664.0	−1.95458
$330$	0	0
$331$	−7268.00	−1.20690	−0.603452	−	0.797399i	$-0.706209\pi$
$331$	−7268.00	−1.20690	−0.603452	+	0.797399i	$0.706209\pi$
$332$	−6076.00	−1.00441
$333$	2826.00	0.465057
$334$	1216.00	0.199211
$335$	0	0
$336$	4428.00	0.718950
$337$	7254.00	1.17255	0.586277	−	0.810111i	$-0.300593\pi$
$337$	7254.00	1.17255	0.586277	+	0.810111i	$0.300593\pi$
$338$	2193.00	0.352910
$339$	−4194.00	−0.671937
$340$	0	0
$341$	−1408.00	−0.223600
$342$	−1260.00	−0.199219
$343$	−21960.0	−3.45693
$344$	−2580.00	−0.404373
$345$	0	0
$346$	−3858.00	−0.599443
$347$	−2276.00	−0.352110	−0.176055	−	0.984380i	$-0.556334\pi$
$347$	−2276.00	−0.352110	−0.176055	+	0.984380i	$0.556334\pi$
$348$	3150.00	0.485223
$349$	4610.00	0.707071	0.353535	−	0.935421i	$-0.384979\pi$
$349$	4610.00	0.707071	0.353535	+	0.935421i	$0.384979\pi$
$350$	0	0
$351$	54.0000	0.00821170
$352$	−1771.00	−0.268167
$353$	−4602.00	−0.693880	−0.346940	−	0.937887i	$-0.612779\pi$
$353$	−4602.00	−0.693880	−0.346940	+	0.937887i	$0.612779\pi$
$354$	−2220.00	−0.333310
$355$	0	0
$356$	3290.00	0.489802
$357$	−7128.00	−1.05673
$358$	380.000	0.0560995
$359$	−4160.00	−0.611578	−0.305789	−	0.952099i	$-0.598920\pi$
$359$	−4160.00	−0.611578	−0.305789	+	0.952099i	$0.598920\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	538.000	0.0781123
$363$	−363.000	−0.0524864
$364$	−504.000	−0.0725736
$365$	0	0
$366$	366.000	0.0522708
$367$	5024.00	0.714579	0.357290	−	0.933994i	$-0.383701\pi$
$367$	5024.00	0.714579	0.357290	+	0.933994i	$0.383701\pi$
$368$	2788.00	0.394931
$369$	−1062.00	−0.149825
$370$	0	0
$371$	2952.00	0.413100
$372$	−2688.00	−0.374641
$373$	−10842.0	−1.50503	−0.752517	−	0.658573i	$-0.771160\pi$
$373$	−10842.0	−1.50503	−0.752517	+	0.658573i	$0.771160\pi$
$374$	726.000	0.100376
$375$	0	0
$376$	4860.00	0.666583
$377$	−300.000	−0.0409835
$378$	−972.000	−0.132260
$379$	−1540.00	−0.208719	−0.104359	−	0.994540i	$-0.533279\pi$
$379$	−1540.00	−0.208719	−0.104359	+	0.994540i	$0.533279\pi$
$380$	0	0
$381$	−132.000	−0.0177495
$382$	−1412.00	−0.189121
$383$	9468.00	1.26317	0.631583	−	0.775309i	$-0.282406\pi$
$383$	9468.00	1.26317	0.631583	+	0.775309i	$0.282406\pi$
$384$	−4365.00	−0.580079
$385$	0	0
$386$	−638.000	−0.0841278
$387$	−1548.00	−0.203331
$388$	8302.00	1.08626
$389$	−10590.0	−1.38029	−0.690147	−	0.723669i	$-0.742454\pi$
$389$	−10590.0	−1.38029	−0.690147	+	0.723669i	$0.742454\pi$
$390$	0	0
$391$	−4488.00	−0.580481
$392$	14295.0	1.84185
$393$	3924.00	0.503663
$394$	3686.00	0.471315
$395$	0	0
$396$	−693.000	−0.0879408
$397$	7434.00	0.939803	0.469901	−	0.882719i	$-0.344289\pi$
$397$	7434.00	0.939803	0.469901	+	0.882719i	$0.344289\pi$
$398$	240.000	0.0302264
$399$	15120.0	1.89711
$400$	0	0
$401$	11402.0	1.41992	0.709961	−	0.704241i	$-0.248713\pi$
$401$	11402.0	1.41992	0.709961	+	0.704241i	$0.248713\pi$
$402$	372.000	0.0461534
$403$	256.000	0.0316433
$404$	−10514.0	−1.29478
$405$	0	0
$406$	5400.00	0.660092
$407$	3454.00	0.420660
$408$	2970.00	0.360385
$409$	−510.000	−0.0616574	−0.0308287	−	0.999525i	$-0.509815\pi$
$409$	−510.000	−0.0616574	−0.0308287	+	0.999525i	$0.509815\pi$
$410$	0	0
$411$	4878.00	0.585436
$412$	224.000	0.0267857
$413$	26640.0	3.17402
$414$	−612.000	−0.0726526
$415$	0	0
$416$	322.000	0.0379504
$417$	−540.000	−0.0634147
$418$	−1540.00	−0.180201
$419$	2420.00	0.282159	0.141080	−	0.989998i	$-0.454943\pi$
$419$	2420.00	0.282159	0.141080	+	0.989998i	$0.454943\pi$
$420$	0	0
$421$	−8178.00	−0.946725	−0.473363	−	0.880868i	$-0.656960\pi$
$421$	−8178.00	−0.946725	−0.473363	+	0.880868i	$0.656960\pi$
$422$	−5092.00	−0.587381
$423$	2916.00	0.335179
$424$	−1230.00	−0.140882
$425$	0	0
$426$	−2964.00	−0.337104
$427$	−4392.00	−0.497761
$428$	7812.00	0.882260
$429$	66.0000	0.00742776
$430$	0	0
$431$	−6768.00	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$431$	−6768.00	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$432$	−1107.00	−0.123288
$433$	478.000	0.0530513	0.0265257	−	0.999648i	$-0.491556\pi$
$433$	478.000	0.0530513	0.0265257	+	0.999648i	$0.491556\pi$
$434$	−4608.00	−0.509657
$435$	0	0
$436$	15330.0	1.68388
$437$	9520.00	1.04211
$438$	−6.00000	−0.000654546 0
$439$	−3260.00	−0.354422	−0.177211	−	0.984173i	$-0.556708\pi$
$439$	−3260.00	−0.354422	−0.177211	+	0.984173i	$0.556708\pi$
$440$	0	0
$441$	8577.00	0.926142
$442$	−132.000	−0.0142050
$443$	−6812.00	−0.730582	−0.365291	−	0.930893i	$-0.619031\pi$
$443$	−6812.00	−0.730582	−0.365291	+	0.930893i	$0.619031\pi$
$444$	6594.00	0.704814
$445$	0	0
$446$	−3808.00	−0.404292
$447$	−4290.00	−0.453937
$448$	6012.00	0.634019
$449$	15290.0	1.60708	0.803541	−	0.595250i	$-0.202947\pi$
$449$	15290.0	1.60708	0.803541	+	0.595250i	$0.202947\pi$
$450$	0	0
$451$	−1298.00	−0.135522
$452$	−9786.00	−1.01835
$453$	5844.00	0.606126
$454$	−44.0000	−0.00454851
$455$	0	0
$456$	−6300.00	−0.646984
$457$	12854.0	1.31572	0.657861	−	0.753140i	$-0.271462\pi$
$457$	12854.0	1.31572	0.657861	+	0.753140i	$0.271462\pi$
$458$	2650.00	0.270363
$459$	1782.00	0.181213
$460$	0	0
$461$	6782.00	0.685183	0.342591	−	0.939485i	$-0.388695\pi$
$461$	6782.00	0.685183	0.342591	+	0.939485i	$0.388695\pi$
$462$	−1188.00	−0.119634
$463$	6408.00	0.643207	0.321604	−	0.946874i	$-0.395778\pi$
$463$	6408.00	0.643207	0.321604	+	0.946874i	$0.395778\pi$
$464$	6150.00	0.615316
$465$	0	0
$466$	−5718.00	−0.568415
$467$	−10476.0	−1.03805	−0.519027	−	0.854758i	$-0.673706\pi$
$467$	−10476.0	−1.03805	−0.519027	+	0.854758i	$0.673706\pi$
$468$	126.000	0.0124452
$469$	−4464.00	−0.439506
$470$	0	0
$471$	1938.00	0.189593
$472$	−11100.0	−1.08246
$473$	−1892.00	−0.183920
$474$	3300.00	0.319776
$475$	0	0
$476$	−16632.0	−1.60153
$477$	−738.000	−0.0708400
$478$	6520.00	0.623887
$479$	1920.00	0.183146	0.0915731	−	0.995798i	$-0.470810\pi$
$479$	1920.00	0.183146	0.0915731	+	0.995798i	$0.470810\pi$
$480$	0	0
$481$	−628.000	−0.0595308
$482$	2438.00	0.230390
$483$	7344.00	0.691850
$484$	−847.000	−0.0795455
$485$	0	0
$486$	243.000	0.0226805
$487$	−8416.00	−0.783091	−0.391546	−	0.920159i	$-0.628059\pi$
$487$	−8416.00	−0.783091	−0.391546	+	0.920159i	$0.628059\pi$
$488$	1830.00	0.169755
$489$	9156.00	0.846725
$490$	0	0
$491$	17732.0	1.62980	0.814902	−	0.579598i	$-0.196791\pi$
$491$	17732.0	1.62980	0.814902	+	0.579598i	$0.196791\pi$
$492$	−2478.00	−0.227067
$493$	−9900.00	−0.904409
$494$	280.000	0.0255016
$495$	0	0
$496$	−5248.00	−0.475085
$497$	35568.0	3.21015
$498$	2604.00	0.234313
$499$	−5580.00	−0.500591	−0.250296	−	0.968169i	$-0.580528\pi$
$499$	−5580.00	−0.500591	−0.250296	+	0.968169i	$0.580528\pi$
$500$	0	0
$501$	3648.00	0.325311
$502$	5268.00	0.468371
$503$	−12512.0	−1.10911	−0.554555	−	0.832147i	$-0.687112\pi$
$503$	−12512.0	−1.10911	−0.554555	+	0.832147i	$0.687112\pi$
$504$	−4860.00	−0.429527
$505$	0	0
$506$	−748.000	−0.0657167
$507$	6579.00	0.576299
$508$	−308.000	−0.0269002
$509$	−22390.0	−1.94974	−0.974872	−	0.222767i	$-0.928491\pi$
$509$	−22390.0	−1.94974	−0.974872	+	0.222767i	$0.928491\pi$
$510$	0	0
$511$	72.0000	0.00623306
$512$	−11521.0	−0.994455
$513$	−3780.00	−0.325324
$514$	−5574.00	−0.478324
$515$	0	0
$516$	−3612.00	−0.308158
$517$	3564.00	0.303181
$518$	11304.0	0.958821
$519$	−11574.0	−0.978887
$520$	0	0
$521$	882.000	0.0741672	0.0370836	−	0.999312i	$-0.488193\pi$
$521$	882.000	0.0741672	0.0370836	+	0.999312i	$0.488193\pi$
$522$	−1350.00	−0.113195
$523$	−18172.0	−1.51932	−0.759662	−	0.650319i	$-0.774635\pi$
$523$	−18172.0	−1.51932	−0.759662	+	0.650319i	$0.774635\pi$
$524$	9156.00	0.763324
$525$	0	0
$526$	2472.00	0.204913
$527$	8448.00	0.698293
$528$	−1353.00	−0.111518
$529$	−7543.00	−0.619956
$530$	0	0
$531$	−6660.00	−0.544293
$532$	35280.0	2.87515
$533$	236.000	0.0191788
$534$	−1410.00	−0.114263
$535$	0	0
$536$	1860.00	0.149888
$537$	1140.00	0.0916101
$538$	−4410.00	−0.353399
$539$	10483.0	0.837727
$540$	0	0
$541$	−17998.0	−1.43030	−0.715152	−	0.698969i	$-0.753643\pi$
$541$	−17998.0	−1.43030	−0.715152	+	0.698969i	$0.753643\pi$
$542$	1308.00	0.103659
$543$	1614.00	0.127557
$544$	10626.0	0.837474
$545$	0	0
$546$	216.000	0.0169303
$547$	4164.00	0.325484	0.162742	−	0.986669i	$-0.447966\pi$
$547$	4164.00	0.325484	0.162742	+	0.986669i	$0.447966\pi$
$548$	11382.0	0.887254
$549$	1098.00	0.0853579
$550$	0	0
$551$	21000.0	1.62365
$552$	−3060.00	−0.235946
$553$	−39600.0	−3.04514
$554$	226.000	0.0173318
$555$	0	0
$556$	−1260.00	−0.0961077
$557$	−13366.0	−1.01676	−0.508380	−	0.861133i	$-0.669756\pi$
$557$	−13366.0	−1.01676	−0.508380	+	0.861133i	$0.669756\pi$
$558$	1152.00	0.0873979
$559$	344.000	0.0260280
$560$	0	0
$561$	2178.00	0.163913
$562$	−522.000	−0.0391801
$563$	−24612.0	−1.84240	−0.921201	−	0.389087i	$-0.872790\pi$
$563$	−24612.0	−1.84240	−0.921201	+	0.389087i	$0.872790\pi$
$564$	6804.00	0.507979
$565$	0	0
$566$	−1508.00	−0.111989
$567$	−2916.00	−0.215980
$568$	−14820.0	−1.09478
$569$	−70.0000	−0.00515739	−0.00257869	−	0.999997i	$-0.500821\pi$
$569$	−70.0000	−0.00515739	−0.00257869	+	0.999997i	$0.500821\pi$
$570$	0	0
$571$	−12348.0	−0.904987	−0.452494	−	0.891768i	$-0.649465\pi$
$571$	−12348.0	−0.904987	−0.452494	+	0.891768i	$0.649465\pi$
$572$	154.000	0.0112571
$573$	−4236.00	−0.308833
$574$	−4248.00	−0.308899
$575$	0	0
$576$	−1503.00	−0.108724
$577$	−23426.0	−1.69019	−0.845093	−	0.534620i	$-0.820455\pi$
$577$	−23426.0	−1.69019	−0.845093	+	0.534620i	$0.820455\pi$
$578$	557.000	0.0400833
$579$	−1914.00	−0.137380
$580$	0	0
$581$	−31248.0	−2.23130
$582$	−3558.00	−0.253409
$583$	−902.000	−0.0640772
$584$	−30.0000	−0.00212570
$585$	0	0
$586$	−4658.00	−0.328362
$587$	−21036.0	−1.47913	−0.739564	−	0.673086i	$-0.764968\pi$
$587$	−21036.0	−1.47913	−0.739564	+	0.673086i	$0.764968\pi$
$588$	20013.0	1.40361
$589$	−17920.0	−1.25362
$590$	0	0
$591$	11058.0	0.769654
$592$	12874.0	0.893781
$593$	2798.00	0.193761	0.0968803	−	0.995296i	$-0.469114\pi$
$593$	2798.00	0.193761	0.0968803	+	0.995296i	$0.469114\pi$
$594$	297.000	0.0205152
$595$	0	0
$596$	−10010.0	−0.687962
$597$	720.000	0.0493595
$598$	136.000	0.00930009
$599$	−11100.0	−0.757151	−0.378576	−	0.925570i	$-0.623586\pi$
$599$	−11100.0	−0.757151	−0.378576	+	0.925570i	$0.623586\pi$
$600$	0	0
$601$	15242.0	1.03450	0.517250	−	0.855835i	$-0.326956\pi$
$601$	15242.0	1.03450	0.517250	+	0.855835i	$0.326956\pi$
$602$	−6192.00	−0.419214
$603$	1116.00	0.0753682
$604$	13636.0	0.918611
$605$	0	0
$606$	4506.00	0.302052
$607$	−13876.0	−0.927857	−0.463929	−	0.885873i	$-0.653561\pi$
$607$	−13876.0	−0.927857	−0.463929	+	0.885873i	$0.653561\pi$
$608$	−22540.0	−1.50348
$609$	16200.0	1.07793
$610$	0	0
$611$	−648.000	−0.0429055
$612$	4158.00	0.274636
$613$	−8722.00	−0.574679	−0.287340	−	0.957829i	$-0.592771\pi$
$613$	−8722.00	−0.574679	−0.287340	+	0.957829i	$0.592771\pi$
$614$	6476.00	0.425652
$615$	0	0
$616$	−5940.00	−0.388522
$617$	8014.00	0.522904	0.261452	−	0.965217i	$-0.415799\pi$
$617$	8014.00	0.522904	0.261452	+	0.965217i	$0.415799\pi$
$618$	−96.0000	−0.00624868
$619$	26020.0	1.68955	0.844776	−	0.535121i	$-0.179734\pi$
$619$	26020.0	1.68955	0.844776	+	0.535121i	$0.179734\pi$
$620$	0	0
$621$	−1836.00	−0.118641
$622$	−5652.00	−0.364348
$623$	16920.0	1.08810
$624$	246.000	0.0157819
$625$	0	0
$626$	−5638.00	−0.359968
$627$	−4620.00	−0.294266
$628$	4522.00	0.287337
$629$	−20724.0	−1.31370
$630$	0	0
$631$	−23528.0	−1.48437	−0.742183	−	0.670197i	$-0.766209\pi$
$631$	−23528.0	−1.48437	−0.742183	+	0.670197i	$0.766209\pi$
$632$	16500.0	1.03850
$633$	−15276.0	−0.959189
$634$	1506.00	0.0943390
$635$	0	0
$636$	−1722.00	−0.107361
$637$	−1906.00	−0.118553
$638$	−1650.00	−0.102389
$639$	−8892.00	−0.550488
$640$	0	0
$641$	−18998.0	−1.17063	−0.585317	−	0.810805i	$-0.699030\pi$
$641$	−18998.0	−1.17063	−0.585317	+	0.810805i	$0.699030\pi$
$642$	−3348.00	−0.205818
$643$	4348.00	0.266669	0.133335	−	0.991071i	$-0.457431\pi$
$643$	4348.00	0.266669	0.133335	+	0.991071i	$0.457431\pi$
$644$	17136.0	1.04853
$645$	0	0
$646$	9240.00	0.562760
$647$	−31916.0	−1.93933	−0.969666	−	0.244435i	$-0.921398\pi$
$647$	−31916.0	−1.93933	−0.969666	+	0.244435i	$0.921398\pi$
$648$	1215.00	0.0736570
$649$	−8140.00	−0.492331
$650$	0	0
$651$	−13824.0	−0.832266
$652$	21364.0	1.28325
$653$	−25682.0	−1.53907	−0.769536	−	0.638603i	$-0.779513\pi$
$653$	−25682.0	−1.53907	−0.769536	+	0.638603i	$0.779513\pi$
$654$	−6570.00	−0.392825
$655$	0	0
$656$	−4838.00	−0.287945
$657$	−18.0000	−0.00106887
$658$	11664.0	0.691049
$659$	14940.0	0.883126	0.441563	−	0.897230i	$-0.354424\pi$
$659$	14940.0	0.883126	0.441563	+	0.897230i	$0.354424\pi$
$660$	0	0
$661$	4502.00	0.264913	0.132457	−	0.991189i	$-0.457714\pi$
$661$	4502.00	0.264913	0.132457	+	0.991189i	$0.457714\pi$
$662$	7268.00	0.426705
$663$	−396.000	−0.0231966
$664$	13020.0	0.760955
$665$	0	0
$666$	−2826.00	−0.164422
$667$	10200.0	0.592122
$668$	8512.00	0.493023
$669$	−11424.0	−0.660205
$670$	0	0
$671$	1342.00	0.0772091
$672$	−17388.0	−0.998150
$673$	−5362.00	−0.307117	−0.153559	−	0.988140i	$-0.549073\pi$
$673$	−5362.00	−0.307117	−0.153559	+	0.988140i	$0.549073\pi$
$674$	−7254.00	−0.414560
$675$	0	0
$676$	15351.0	0.873407
$677$	−14566.0	−0.826908	−0.413454	−	0.910525i	$-0.635678\pi$
$677$	−14566.0	−0.826908	−0.413454	+	0.910525i	$0.635678\pi$
$678$	4194.00	0.237566
$679$	42696.0	2.41314
$680$	0	0
$681$	−132.000	−0.00742768
$682$	1408.00	0.0790544
$683$	−11852.0	−0.663989	−0.331994	−	0.943281i	$-0.607721\pi$
$683$	−11852.0	−0.663989	−0.331994	+	0.943281i	$0.607721\pi$
$684$	−8820.00	−0.493043
$685$	0	0
$686$	21960.0	1.22221
$687$	7950.00	0.441501
$688$	−7052.00	−0.390778
$689$	164.000	0.00906807
$690$	0	0
$691$	−1828.00	−0.100637	−0.0503187	−	0.998733i	$-0.516024\pi$
$691$	−1828.00	−0.100637	−0.0503187	+	0.998733i	$0.516024\pi$
$692$	−27006.0	−1.48355
$693$	−3564.00	−0.195361
$694$	2276.00	0.124490
$695$	0	0
$696$	−6750.00	−0.367612
$697$	7788.00	0.423230
$698$	−4610.00	−0.249987
$699$	−17154.0	−0.928217
$700$	0	0
$701$	23182.0	1.24903	0.624516	−	0.781012i	$-0.285296\pi$
$701$	23182.0	1.24903	0.624516	+	0.781012i	$0.285296\pi$
$702$	−54.0000	−0.00290327
$703$	43960.0	2.35844
$704$	−1837.00	−0.0983445
$705$	0	0
$706$	4602.00	0.245324
$707$	−54072.0	−2.87636
$708$	−15540.0	−0.824900
$709$	−33850.0	−1.79304	−0.896519	−	0.443006i	$-0.853912\pi$
$709$	−33850.0	−1.79304	−0.896519	+	0.443006i	$0.853912\pi$
$710$	0	0
$711$	9900.00	0.522193
$712$	−7050.00	−0.371081
$713$	−8704.00	−0.457177
$714$	7128.00	0.373612
$715$	0	0
$716$	2660.00	0.138839
$717$	19560.0	1.01880
$718$	4160.00	0.216225
$719$	−7140.00	−0.370344	−0.185172	−	0.982706i	$-0.559284\pi$
$719$	−7140.00	−0.370344	−0.185172	+	0.982706i	$0.559284\pi$
$720$	0	0
$721$	1152.00	0.0595045
$722$	−12741.0	−0.656746
$723$	7314.00	0.376225
$724$	3766.00	0.193318
$725$	0	0
$726$	363.000	0.0185567
$727$	−8896.00	−0.453830	−0.226915	−	0.973915i	$-0.572864\pi$
$727$	−8896.00	−0.453830	−0.226915	+	0.973915i	$0.572864\pi$
$728$	1080.00	0.0549828
$729$	729.000	0.0370370
$730$	0	0
$731$	11352.0	0.574376
$732$	2562.00	0.129364
$733$	13038.0	0.656984	0.328492	−	0.944507i	$-0.393460\pi$
$733$	13038.0	0.656984	0.328492	+	0.944507i	$0.393460\pi$
$734$	−5024.00	−0.252642
$735$	0	0
$736$	−10948.0	−0.548300
$737$	1364.00	0.0681731
$738$	1062.00	0.0529712
$739$	11620.0	0.578415	0.289207	−	0.957266i	$-0.406608\pi$
$739$	11620.0	0.578415	0.289207	+	0.957266i	$0.406608\pi$
$740$	0	0
$741$	840.000	0.0416440
$742$	−2952.00	−0.146053
$743$	1328.00	0.0655715	0.0327857	−	0.999462i	$-0.489562\pi$
$743$	1328.00	0.0655715	0.0327857	+	0.999462i	$0.489562\pi$
$744$	5760.00	0.283833
$745$	0	0
$746$	10842.0	0.532110
$747$	7812.00	0.382632
$748$	5082.00	0.248418
$749$	40176.0	1.95995
$750$	0	0
$751$	−37808.0	−1.83706	−0.918531	−	0.395349i	$-0.870624\pi$
$751$	−37808.0	−1.83706	−0.918531	+	0.395349i	$0.870624\pi$
$752$	13284.0	0.644172
$753$	15804.0	0.764847
$754$	300.000	0.0144899
$755$	0	0
$756$	−6804.00	−0.327327
$757$	−34326.0	−1.64808	−0.824042	−	0.566529i	$-0.808286\pi$
$757$	−34326.0	−1.64808	−0.824042	+	0.566529i	$0.808286\pi$
$758$	1540.00	0.0737933
$759$	−2244.00	−0.107315
$760$	0	0
$761$	4282.00	0.203972	0.101986	−	0.994786i	$-0.467480\pi$
$761$	4282.00	0.203972	0.101986	+	0.994786i	$0.467480\pi$
$762$	132.000	0.00627540
$763$	78840.0	3.74076
$764$	−9884.00	−0.468050
$765$	0	0
$766$	−9468.00	−0.446596
$767$	1480.00	0.0696737
$768$	357.000	0.0167736
$769$	24410.0	1.14466	0.572332	−	0.820022i	$-0.306039\pi$
$769$	24410.0	1.14466	0.572332	+	0.820022i	$0.306039\pi$
$770$	0	0
$771$	−16722.0	−0.781100
$772$	−4466.00	−0.208206
$773$	−8162.00	−0.379776	−0.189888	−	0.981806i	$-0.560812\pi$
$773$	−8162.00	−0.379776	−0.189888	+	0.981806i	$0.560812\pi$
$774$	1548.00	0.0718885
$775$	0	0
$776$	−17790.0	−0.822969
$777$	33912.0	1.56575
$778$	10590.0	0.488008
$779$	−16520.0	−0.759808
$780$	0	0
$781$	−10868.0	−0.497935
$782$	4488.00	0.205231
$783$	−4050.00	−0.184847
$784$	39073.0	1.77993
$785$	0	0
$786$	−3924.00	−0.178072
$787$	−26596.0	−1.20463	−0.602316	−	0.798258i	$-0.705755\pi$
$787$	−26596.0	−1.20463	−0.602316	+	0.798258i	$0.705755\pi$
$788$	25802.0	1.16644
$789$	7416.00	0.334622
$790$	0	0
$791$	−50328.0	−2.26227
$792$	1485.00	0.0666252
$793$	−244.000	−0.0109265
$794$	−7434.00	−0.332271
$795$	0	0
$796$	1680.00	0.0748066
$797$	24614.0	1.09394	0.546972	−	0.837151i	$-0.315781\pi$
$797$	24614.0	1.09394	0.546972	+	0.837151i	$0.315781\pi$
$798$	−15120.0	−0.670730
$799$	−21384.0	−0.946823
$800$	0	0
$801$	−4230.00	−0.186591
$802$	−11402.0	−0.502018
$803$	−22.0000	−0.000966828 0
$804$	2604.00	0.114224
$805$	0	0
$806$	−256.000	−0.0111876
$807$	−13230.0	−0.577098
$808$	22530.0	0.980944
$809$	39930.0	1.73531	0.867654	−	0.497169i	$-0.165627\pi$
$809$	39930.0	1.73531	0.867654	+	0.497169i	$0.165627\pi$
$810$	0	0
$811$	2412.00	0.104435	0.0522175	−	0.998636i	$-0.483371\pi$
$811$	2412.00	0.104435	0.0522175	+	0.998636i	$0.483371\pi$
$812$	37800.0	1.63365
$813$	3924.00	0.169275
$814$	−3454.00	−0.148726
$815$	0	0
$816$	8118.00	0.348268
$817$	−24080.0	−1.03115
$818$	510.000	0.0217992
$819$	648.000	0.0276471
$820$	0	0
$821$	−6018.00	−0.255822	−0.127911	−	0.991786i	$-0.540827\pi$
$821$	−6018.00	−0.255822	−0.127911	+	0.991786i	$0.540827\pi$
$822$	−4878.00	−0.206983
$823$	−34632.0	−1.46682	−0.733412	−	0.679785i	$-0.762073\pi$
$823$	−34632.0	−1.46682	−0.733412	+	0.679785i	$0.762073\pi$
$824$	−480.000	−0.0202932
$825$	0	0
$826$	−26640.0	−1.12218
$827$	40044.0	1.68376	0.841878	−	0.539668i	$-0.181450\pi$
$827$	40044.0	1.68376	0.841878	+	0.539668i	$0.181450\pi$
$828$	−4284.00	−0.179806
$829$	−44090.0	−1.84718	−0.923588	−	0.383386i	$-0.874758\pi$
$829$	−44090.0	−1.84718	−0.923588	+	0.383386i	$0.874758\pi$
$830$	0	0
$831$	678.000	0.0283027
$832$	334.000	0.0139175
$833$	−62898.0	−2.61619
$834$	540.000	0.0224205
$835$	0	0
$836$	−10780.0	−0.445974
$837$	3456.00	0.142720
$838$	−2420.00	−0.0997584
$839$	−23780.0	−0.978518	−0.489259	−	0.872138i	$-0.662733\pi$
$839$	−23780.0	−0.978518	−0.489259	+	0.872138i	$0.662733\pi$
$840$	0	0
$841$	−1889.00	−0.0774530
$842$	8178.00	0.334718
$843$	−1566.00	−0.0639809
$844$	−35644.0	−1.45369
$845$	0	0
$846$	−2916.00	−0.118504
$847$	−4356.00	−0.176711
$848$	−3362.00	−0.136146
$849$	−4524.00	−0.182878
$850$	0	0
$851$	21352.0	0.860091
$852$	−20748.0	−0.834290
$853$	4078.00	0.163691	0.0818453	−	0.996645i	$-0.473919\pi$
$853$	4078.00	0.163691	0.0818453	+	0.996645i	$0.473919\pi$
$854$	4392.00	0.175985
$855$	0	0
$856$	−16740.0	−0.668413
$857$	−30666.0	−1.22232	−0.611161	−	0.791506i	$-0.709297\pi$
$857$	−30666.0	−1.22232	−0.611161	+	0.791506i	$0.709297\pi$
$858$	−66.0000	−0.00262611
$859$	4780.00	0.189862	0.0949310	−	0.995484i	$-0.469737\pi$
$859$	4780.00	0.189862	0.0949310	+	0.995484i	$0.469737\pi$
$860$	0	0
$861$	−12744.0	−0.504430
$862$	6768.00	0.267423
$863$	9428.00	0.371880	0.185940	−	0.982561i	$-0.440467\pi$
$863$	9428.00	0.371880	0.185940	+	0.982561i	$0.440467\pi$
$864$	4347.00	0.171167
$865$	0	0
$866$	−478.000	−0.0187565
$867$	1671.00	0.0654558
$868$	−32256.0	−1.26134
$869$	12100.0	0.472341
$870$	0	0
$871$	−248.000	−0.00964771
$872$	−32850.0	−1.27574
$873$	−10674.0	−0.413815
$874$	−9520.00	−0.368443
$875$	0	0
$876$	−42.0000	−0.00161992
$877$	−15266.0	−0.587795	−0.293897	−	0.955837i	$-0.594952\pi$
$877$	−15266.0	−0.587795	−0.293897	+	0.955837i	$0.594952\pi$
$878$	3260.00	0.125307
$879$	−13974.0	−0.536213
$880$	0	0
$881$	−5118.00	−0.195721	−0.0978603	−	0.995200i	$-0.531200\pi$
$881$	−5118.00	−0.195721	−0.0978603	+	0.995200i	$0.531200\pi$
$882$	−8577.00	−0.327441
$883$	44188.0	1.68408	0.842041	−	0.539413i	$-0.181354\pi$
$883$	44188.0	1.68408	0.842041	+	0.539413i	$0.181354\pi$
$884$	−924.000	−0.0351555
$885$	0	0
$886$	6812.00	0.258300
$887$	16864.0	0.638374	0.319187	−	0.947692i	$-0.396590\pi$
$887$	16864.0	0.638374	0.319187	+	0.947692i	$0.396590\pi$
$888$	−14130.0	−0.533977
$889$	−1584.00	−0.0597589
$890$	0	0
$891$	891.000	0.0335013
$892$	−26656.0	−1.00057
$893$	45360.0	1.69979
$894$	4290.00	0.160491
$895$	0	0
$896$	−52380.0	−1.95301
$897$	408.000	0.0151870
$898$	−15290.0	−0.568189
$899$	−19200.0	−0.712298
$900$	0	0
$901$	5412.00	0.200111
$902$	1298.00	0.0479143
$903$	−18576.0	−0.684574
$904$	20970.0	0.771518
$905$	0	0
$906$	−5844.00	−0.214298
$907$	37924.0	1.38836	0.694182	−	0.719800i	$-0.255766\pi$
$907$	37924.0	1.38836	0.694182	+	0.719800i	$0.255766\pi$
$908$	−308.000	−0.0112570
$909$	13518.0	0.493249
$910$	0	0
$911$	−36628.0	−1.33210	−0.666048	−	0.745909i	$-0.732015\pi$
$911$	−36628.0	−1.33210	−0.666048	+	0.745909i	$0.732015\pi$
$912$	−17220.0	−0.625232
$913$	9548.00	0.346104
$914$	−12854.0	−0.465178
$915$	0	0
$916$	18550.0	0.669115
$917$	47088.0	1.69573
$918$	−1782.00	−0.0640684
$919$	−21300.0	−0.764551	−0.382275	−	0.924048i	$-0.624859\pi$
$919$	−21300.0	−0.764551	−0.382275	+	0.924048i	$0.624859\pi$
$920$	0	0
$921$	19428.0	0.695086
$922$	−6782.00	−0.242249
$923$	1976.00	0.0704668
$924$	−8316.00	−0.296078
$925$	0	0
$926$	−6408.00	−0.227408
$927$	−288.000	−0.0102041
$928$	−24150.0	−0.854270
$929$	31450.0	1.11070	0.555350	−	0.831616i	$-0.312584\pi$
$929$	31450.0	1.11070	0.555350	+	0.831616i	$0.312584\pi$
$930$	0	0
$931$	133420.	4.69674
$932$	−40026.0	−1.40675
$933$	−16956.0	−0.594978
$934$	10476.0	0.367008
$935$	0	0
$936$	−270.000	−0.00942866
$937$	6174.00	0.215257	0.107628	−	0.994191i	$-0.465674\pi$
$937$	6174.00	0.215257	0.107628	+	0.994191i	$0.465674\pi$
$938$	4464.00	0.155389
$939$	−16914.0	−0.587825
$940$	0	0
$941$	20422.0	0.707479	0.353740	−	0.935344i	$-0.384910\pi$
$941$	20422.0	0.707479	0.353740	+	0.935344i	$0.384910\pi$
$942$	−1938.00	−0.0670313
$943$	−8024.00	−0.277092
$944$	−30340.0	−1.04606
$945$	0	0
$946$	1892.00	0.0650256
$947$	−12156.0	−0.417125	−0.208562	−	0.978009i	$-0.566878\pi$
$947$	−12156.0	−0.417125	−0.208562	+	0.978009i	$0.566878\pi$
$948$	23100.0	0.791406
$949$	4.00000	0.000136823 0
$950$	0	0
$951$	4518.00	0.154055
$952$	35640.0	1.21334
$953$	−14442.0	−0.490894	−0.245447	−	0.969410i	$-0.578935\pi$
$953$	−14442.0	−0.490894	−0.245447	+	0.969410i	$0.578935\pi$
$954$	738.000	0.0250457
$955$	0	0
$956$	45640.0	1.54404
$957$	−4950.00	−0.167200
$958$	−1920.00	−0.0647520
$959$	58536.0	1.97104
$960$	0	0
$961$	−13407.0	−0.450035
$962$	628.000	0.0210473
$963$	−10044.0	−0.336099
$964$	17066.0	0.570186
$965$	0	0
$966$	−7344.00	−0.244606
$967$	−33356.0	−1.10926	−0.554631	−	0.832096i	$-0.687141\pi$
$967$	−33356.0	−1.10926	−0.554631	+	0.832096i	$0.687141\pi$
$968$	1815.00	0.0602648
$969$	27720.0	0.918983
$970$	0	0
$971$	3852.00	0.127309	0.0636543	−	0.997972i	$-0.479725\pi$
$971$	3852.00	0.127309	0.0636543	+	0.997972i	$0.479725\pi$
$972$	1701.00	0.0561313
$973$	−6480.00	−0.213504
$974$	8416.00	0.276865
$975$	0	0
$976$	5002.00	0.164047
$977$	37454.0	1.22647	0.613234	−	0.789901i	$-0.289868\pi$
$977$	37454.0	1.22647	0.613234	+	0.789901i	$0.289868\pi$
$978$	−9156.00	−0.299363
$979$	−5170.00	−0.168778
$980$	0	0
$981$	−19710.0	−0.641480
$982$	−17732.0	−0.576223
$983$	16228.0	0.526544	0.263272	−	0.964722i	$-0.415198\pi$
$983$	16228.0	0.526544	0.263272	+	0.964722i	$0.415198\pi$
$984$	5310.00	0.172029
$985$	0	0
$986$	9900.00	0.319757
$987$	34992.0	1.12848
$988$	1960.00	0.0631133
$989$	−11696.0	−0.376048
$990$	0	0
$991$	−23728.0	−0.760590	−0.380295	−	0.924865i	$-0.624178\pi$
$991$	−23728.0	−0.760590	−0.380295	+	0.924865i	$0.624178\pi$
$992$	20608.0	0.659581
$993$	21804.0	0.696807
$994$	−35568.0	−1.13496
$995$	0	0
$996$	18228.0	0.579896
$997$	−41306.0	−1.31211	−0.656055	−	0.754713i	$-0.727776\pi$
$997$	−41306.0	−1.31211	−0.656055	+	0.754713i	$0.727776\pi$
$998$	5580.00	0.176986
$999$	−8478.00	−0.268501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.d.1.1		1
3.2	odd	2		2475.4.a.g.1.1		1
5.2	odd	4		825.4.c.e.199.1		2
5.3	odd	4		825.4.c.e.199.2		2
5.4	even	2		165.4.a.b.1.1	✓	1
15.14	odd	2		495.4.a.b.1.1		1
55.54	odd	2		1815.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.b.1.1	✓	1	5.4	even	2
495.4.a.b.1.1		1	15.14	odd	2
825.4.a.d.1.1		1	1.1	even	1	trivial
825.4.c.e.199.1		2	5.2	odd	4
825.4.c.e.199.2		2	5.3	odd	4
1815.4.a.c.1.1		1	55.54	odd	2
2475.4.a.g.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

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