LMFDB - Embedded newform 825.4.a.f.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:	no (minimal twist has level 33)
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)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +11.0000 q^{11} -21.0000 q^{12} +32.0000 q^{13} +26.0000 q^{14} +41.0000 q^{16} -74.0000 q^{17} +9.00000 q^{18} -60.0000 q^{19} +78.0000 q^{21} +11.0000 q^{22} +182.000 q^{23} -45.0000 q^{24} +32.0000 q^{26} +27.0000 q^{27} -182.000 q^{28} -90.0000 q^{29} -8.00000 q^{31} +161.000 q^{32} +33.0000 q^{33} -74.0000 q^{34} -63.0000 q^{36} +66.0000 q^{37} -60.0000 q^{38} +96.0000 q^{39} +422.000 q^{41} +78.0000 q^{42} -408.000 q^{43} -77.0000 q^{44} +182.000 q^{46} +506.000 q^{47} +123.000 q^{48} +333.000 q^{49} -222.000 q^{51} -224.000 q^{52} -348.000 q^{53} +27.0000 q^{54} -390.000 q^{56} -180.000 q^{57} -90.0000 q^{58} -200.000 q^{59} +132.000 q^{61} -8.00000 q^{62} +234.000 q^{63} -167.000 q^{64} +33.0000 q^{66} +1036.00 q^{67} +518.000 q^{68} +546.000 q^{69} +762.000 q^{71} -135.000 q^{72} +542.000 q^{73} +66.0000 q^{74} +420.000 q^{76} +286.000 q^{77} +96.0000 q^{78} -550.000 q^{79} +81.0000 q^{81} +422.000 q^{82} +132.000 q^{83} -546.000 q^{84} -408.000 q^{86} -270.000 q^{87} -165.000 q^{88} +570.000 q^{89} +832.000 q^{91} -1274.00 q^{92} -24.0000 q^{93} +506.000 q^{94} +483.000 q^{96} -14.0000 q^{97} +333.000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	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
$7$	26.0000	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$7$	26.0000	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$8$	−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
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	26.0000	0.496342
$15$	0	0
$16$	41.0000	0.640625
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	9.00000	0.117851
$19$	−60.0000	−0.724471	−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000	−0.724471	−0.362235	+	0.932087i	$0.617986\pi$
$20$	0	0
$21$	78.0000	0.810524
$22$	11.0000	0.106600
$23$	182.000	1.64998	0.824992	−	0.565145i	$-0.191180\pi$
$23$	182.000	1.64998	0.824992	+	0.565145i	$0.191180\pi$
$24$	−45.0000	−0.382733
$25$	0	0
$26$	32.0000	0.241374
$27$	27.0000	0.192450
$28$	−182.000	−1.22838
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	161.000	0.889408
$33$	33.0000	0.174078
$34$	−74.0000	−0.373262
$35$	0	0
$36$	−63.0000	−0.291667
$37$	66.0000	0.293252	0.146626	−	0.989192i	$-0.453159\pi$
$37$	66.0000	0.293252	0.146626	+	0.989192i	$0.453159\pi$
$38$	−60.0000	−0.256139
$39$	96.0000	0.394162
$40$	0	0
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	78.0000	0.286563
$43$	−408.000	−1.44696	−0.723482	−	0.690344i	$-0.757459\pi$
$43$	−408.000	−1.44696	−0.723482	+	0.690344i	$0.757459\pi$
$44$	−77.0000	−0.263822
$45$	0	0
$46$	182.000	0.583357
$47$	506.000	1.57038	0.785188	−	0.619257i	$-0.212566\pi$
$47$	506.000	1.57038	0.785188	+	0.619257i	$0.212566\pi$
$48$	123.000	0.369865
$49$	333.000	0.970845
$50$	0	0
$51$	−222.000	−0.609534
$52$	−224.000	−0.597369
$53$	−348.000	−0.901915	−0.450957	−	0.892546i	$-0.648917\pi$
$53$	−348.000	−0.901915	−0.450957	+	0.892546i	$0.648917\pi$
$54$	27.0000	0.0680414
$55$	0	0
$56$	−390.000	−0.930642
$57$	−180.000	−0.418273
$58$	−90.0000	−0.203751
$59$	−200.000	−0.441318	−0.220659	−	0.975351i	$-0.570821\pi$
$59$	−200.000	−0.441318	−0.220659	+	0.975351i	$0.570821\pi$
$60$	0	0
$61$	132.000	0.277063	0.138532	−	0.990358i	$-0.455762\pi$
$61$	132.000	0.277063	0.138532	+	0.990358i	$0.455762\pi$
$62$	−8.00000	−0.0163871
$63$	234.000	0.467956
$64$	−167.000	−0.326172
$65$	0	0
$66$	33.0000	0.0615457
$67$	1036.00	1.88907	0.944534	−	0.328414i	$-0.106514\pi$
$67$	1036.00	1.88907	0.944534	+	0.328414i	$0.106514\pi$
$68$	518.000	0.923775
$69$	546.000	0.952618
$70$	0	0
$71$	762.000	1.27370	0.636850	−	0.770987i	$-0.280237\pi$
$71$	762.000	1.27370	0.636850	+	0.770987i	$0.280237\pi$
$72$	−135.000	−0.220971
$73$	542.000	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$73$	542.000	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$74$	66.0000	0.103680
$75$	0	0
$76$	420.000	0.633912
$77$	286.000	0.423282
$78$	96.0000	0.139357
$79$	−550.000	−0.783289	−0.391645	−	0.920117i	$-0.628094\pi$
$79$	−550.000	−0.783289	−0.391645	+	0.920117i	$0.628094\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	422.000	0.568318
$83$	132.000	0.174565	0.0872824	−	0.996184i	$-0.472182\pi$
$83$	132.000	0.174565	0.0872824	+	0.996184i	$0.472182\pi$
$84$	−546.000	−0.709208
$85$	0	0
$86$	−408.000	−0.511579
$87$	−270.000	−0.332725
$88$	−165.000	−0.199876
$89$	570.000	0.678875	0.339438	−	0.940629i	$-0.389763\pi$
$89$	570.000	0.678875	0.339438	+	0.940629i	$0.389763\pi$
$90$	0	0
$91$	832.000	0.958432
$92$	−1274.00	−1.44374
$93$	−24.0000	−0.0267600
$94$	506.000	0.555212
$95$	0	0
$96$	483.000	0.513500
$97$	−14.0000	−0.0146545	−0.00732724	−	0.999973i	$-0.502332\pi$
$97$	−14.0000	−0.0146545	−0.00732724	+	0.999973i	$0.502332\pi$
$98$	333.000	0.343246
$99$	99.0000	0.100504
$100$	0	0
$101$	1702.00	1.67679	0.838393	−	0.545067i	$-0.183496\pi$
$101$	1702.00	1.67679	0.838393	+	0.545067i	$0.183496\pi$
$102$	−222.000	−0.215503
$103$	1132.00	1.08291	0.541453	−	0.840731i	$-0.317874\pi$
$103$	1132.00	1.08291	0.541453	+	0.840731i	$0.317874\pi$
$104$	−480.000	−0.452576
$105$	0	0
$106$	−348.000	−0.318875
$107$	−564.000	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$108$	−189.000	−0.168394
$109$	−320.000	−0.281197	−0.140598	−	0.990067i	$-0.544903\pi$
$109$	−320.000	−0.281197	−0.140598	+	0.990067i	$0.544903\pi$
$110$	0	0
$111$	198.000	0.169309
$112$	1066.00	0.899353
$113$	2142.00	1.78321	0.891604	−	0.452817i	$-0.149581\pi$
$113$	2142.00	1.78321	0.891604	+	0.452817i	$0.149581\pi$
$114$	−180.000	−0.147882
$115$	0	0
$116$	630.000	0.504259
$117$	288.000	0.227569
$118$	−200.000	−0.156030
$119$	−1924.00	−1.48212
$120$	0	0
$121$	121.000	0.0909091
$122$	132.000	0.0979567
$123$	1266.00	0.928060
$124$	56.0000	0.0405560
$125$	0	0
$126$	234.000	0.165447
$127$	1606.00	1.12212	0.561061	−	0.827775i	$-0.310393\pi$
$127$	1606.00	1.12212	0.561061	+	0.827775i	$0.310393\pi$
$128$	−1455.00	−1.00473
$129$	−1224.00	−0.835405
$130$	0	0
$131$	−1908.00	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$131$	−1908.00	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$132$	−231.000	−0.152318
$133$	−1560.00	−1.01706
$134$	1036.00	0.667886
$135$	0	0
$136$	1110.00	0.699866
$137$	2186.00	1.36323	0.681615	−	0.731711i	$-0.261278\pi$
$137$	2186.00	1.36323	0.681615	+	0.731711i	$0.261278\pi$
$138$	546.000	0.336801
$139$	2740.00	1.67197	0.835985	−	0.548753i	$-0.184897\pi$
$139$	2740.00	1.67197	0.835985	+	0.548753i	$0.184897\pi$
$140$	0	0
$141$	1518.00	0.906657
$142$	762.000	0.450321
$143$	352.000	0.205844
$144$	369.000	0.213542
$145$	0	0
$146$	542.000	0.307235
$147$	999.000	0.560518
$148$	−462.000	−0.256596
$149$	−1310.00	−0.720264	−0.360132	−	0.932901i	$-0.617268\pi$
$149$	−1310.00	−0.720264	−0.360132	+	0.932901i	$0.617268\pi$
$150$	0	0
$151$	−1198.00	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$151$	−1198.00	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$152$	900.000	0.480261
$153$	−666.000	−0.351914
$154$	286.000	0.149653
$155$	0	0
$156$	−672.000	−0.344891
$157$	−2114.00	−1.07462	−0.537311	−	0.843384i	$-0.680560\pi$
$157$	−2114.00	−1.07462	−0.537311	+	0.843384i	$0.680560\pi$
$158$	−550.000	−0.276934
$159$	−1044.00	−0.520721
$160$	0	0
$161$	4732.00	2.31636
$162$	81.0000	0.0392837
$163$	−3868.00	−1.85868	−0.929341	−	0.369223i	$-0.879624\pi$
$163$	−3868.00	−1.85868	−0.929341	+	0.369223i	$0.879624\pi$
$164$	−2954.00	−1.40652
$165$	0	0
$166$	132.000	0.0617180
$167$	−2004.00	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−2004.00	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1170.00	−0.537306
$169$	−1173.00	−0.533910
$170$	0	0
$171$	−540.000	−0.241490
$172$	2856.00	1.26609
$173$	−678.000	−0.297962	−0.148981	−	0.988840i	$-0.547599\pi$
$173$	−678.000	−0.297962	−0.148981	+	0.988840i	$0.547599\pi$
$174$	−270.000	−0.117636
$175$	0	0
$176$	451.000	0.193156
$177$	−600.000	−0.254795
$178$	570.000	0.240019
$179$	−1680.00	−0.701503	−0.350752	−	0.936469i	$-0.614074\pi$
$179$	−1680.00	−0.701503	−0.350752	+	0.936469i	$0.614074\pi$
$180$	0	0
$181$	−4358.00	−1.78966	−0.894828	−	0.446412i	$-0.852702\pi$
$181$	−4358.00	−1.78966	−0.894828	+	0.446412i	$0.852702\pi$
$182$	832.000	0.338857
$183$	396.000	0.159963
$184$	−2730.00	−1.09379
$185$	0	0
$186$	−24.0000	−0.00946110
$187$	−814.000	−0.318319
$188$	−3542.00	−1.37408
$189$	702.000	0.270175
$190$	0	0
$191$	−1778.00	−0.673568	−0.336784	−	0.941582i	$-0.609339\pi$
$191$	−1778.00	−0.673568	−0.336784	+	0.941582i	$0.609339\pi$
$192$	−501.000	−0.188315
$193$	3962.00	1.47767	0.738837	−	0.673884i	$-0.235375\pi$
$193$	3962.00	1.47767	0.738837	+	0.673884i	$0.235375\pi$
$194$	−14.0000	−0.00518114
$195$	0	0
$196$	−2331.00	−0.849490
$197$	−374.000	−0.135261	−0.0676304	−	0.997710i	$-0.521544\pi$
$197$	−374.000	−0.135261	−0.0676304	+	0.997710i	$0.521544\pi$
$198$	99.0000	0.0355335
$199$	2100.00	0.748066	0.374033	−	0.927415i	$-0.377975\pi$
$199$	2100.00	0.748066	0.374033	+	0.927415i	$0.377975\pi$
$200$	0	0
$201$	3108.00	1.09065
$202$	1702.00	0.592833
$203$	−2340.00	−0.809043
$204$	1554.00	0.533342
$205$	0	0
$206$	1132.00	0.382865
$207$	1638.00	0.549995
$208$	1312.00	0.437360
$209$	−660.000	−0.218436
$210$	0	0
$211$	2232.00	0.728233	0.364117	−	0.931353i	$-0.381371\pi$
$211$	2232.00	0.728233	0.364117	+	0.931353i	$0.381371\pi$
$212$	2436.00	0.789175
$213$	2286.00	0.735372
$214$	−564.000	−0.180160
$215$	0	0
$216$	−405.000	−0.127578
$217$	−208.000	−0.0650689
$218$	−320.000	−0.0994180
$219$	1626.00	0.501712
$220$	0	0
$221$	−2368.00	−0.720764
$222$	198.000	0.0598599
$223$	−2128.00	−0.639020	−0.319510	−	0.947583i	$-0.603518\pi$
$223$	−2128.00	−0.639020	−0.319510	+	0.947583i	$0.603518\pi$
$224$	4186.00	1.24861
$225$	0	0
$226$	2142.00	0.630459
$227$	−2964.00	−0.866641	−0.433321	−	0.901240i	$-0.642658\pi$
$227$	−2964.00	−0.866641	−0.433321	+	0.901240i	$0.642658\pi$
$228$	1260.00	0.365989
$229$	−2550.00	−0.735846	−0.367923	−	0.929856i	$-0.619931\pi$
$229$	−2550.00	−0.735846	−0.367923	+	0.929856i	$0.619931\pi$
$230$	0	0
$231$	858.000	0.244382
$232$	1350.00	0.382034
$233$	3042.00	0.855314	0.427657	−	0.903941i	$-0.359339\pi$
$233$	3042.00	0.855314	0.427657	+	0.903941i	$0.359339\pi$
$234$	288.000	0.0804579
$235$	0	0
$236$	1400.00	0.386154
$237$	−1650.00	−0.452232
$238$	−1924.00	−0.524010
$239$	2700.00	0.730747	0.365373	−	0.930861i	$-0.380941\pi$
$239$	2700.00	0.730747	0.365373	+	0.930861i	$0.380941\pi$
$240$	0	0
$241$	−578.000	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$241$	−578.000	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$242$	121.000	0.0321412
$243$	243.000	0.0641500
$244$	−924.000	−0.242430
$245$	0	0
$246$	1266.00	0.328119
$247$	−1920.00	−0.494602
$248$	120.000	0.0307258
$249$	396.000	0.100785
$250$	0	0
$251$	3752.00	0.943522	0.471761	−	0.881726i	$-0.343618\pi$
$251$	3752.00	0.943522	0.471761	+	0.881726i	$0.343618\pi$
$252$	−1638.00	−0.409462
$253$	2002.00	0.497489
$254$	1606.00	0.396730
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−674.000	−0.163591	−0.0817957	−	0.996649i	$-0.526065\pi$
$257$	−674.000	−0.163591	−0.0817957	+	0.996649i	$0.526065\pi$
$258$	−1224.00	−0.295360
$259$	1716.00	0.411687
$260$	0	0
$261$	−810.000	−0.192099
$262$	−1908.00	−0.449911
$263$	4352.00	1.02036	0.510182	−	0.860066i	$-0.329578\pi$
$263$	4352.00	1.02036	0.510182	+	0.860066i	$0.329578\pi$
$264$	−495.000	−0.115398
$265$	0	0
$266$	−1560.00	−0.359585
$267$	1710.00	0.391949
$268$	−7252.00	−1.65293
$269$	500.000	0.113329	0.0566646	−	0.998393i	$-0.481953\pi$
$269$	500.000	0.113329	0.0566646	+	0.998393i	$0.481953\pi$
$270$	0	0
$271$	−6538.00	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$271$	−6538.00	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$272$	−3034.00	−0.676336
$273$	2496.00	0.553351
$274$	2186.00	0.481975
$275$	0	0
$276$	−3822.00	−0.833541
$277$	−124.000	−0.0268969	−0.0134484	−	0.999910i	$-0.504281\pi$
$277$	−124.000	−0.0268969	−0.0134484	+	0.999910i	$0.504281\pi$
$278$	2740.00	0.591131
$279$	−72.0000	−0.0154499
$280$	0	0
$281$	3642.00	0.773180	0.386590	−	0.922252i	$-0.373653\pi$
$281$	3642.00	0.773180	0.386590	+	0.922252i	$0.373653\pi$
$282$	1518.00	0.320552
$283$	−4648.00	−0.976307	−0.488154	−	0.872758i	$-0.662329\pi$
$283$	−4648.00	−0.976307	−0.488154	+	0.872758i	$0.662329\pi$
$284$	−5334.00	−1.11449
$285$	0	0
$286$	352.000	0.0727769
$287$	10972.0	2.25664
$288$	1449.00	0.296469
$289$	563.000	0.114594
$290$	0	0
$291$	−42.0000	−0.00846077
$292$	−3794.00	−0.760367
$293$	3102.00	0.618501	0.309250	−	0.950981i	$-0.399922\pi$
$293$	3102.00	0.618501	0.309250	+	0.950981i	$0.399922\pi$
$294$	999.000	0.198173
$295$	0	0
$296$	−990.000	−0.194401
$297$	297.000	0.0580259
$298$	−1310.00	−0.254652
$299$	5824.00	1.12646
$300$	0	0
$301$	−10608.0	−2.03135
$302$	−1198.00	−0.228269
$303$	5106.00	0.968093
$304$	−2460.00	−0.464114
$305$	0	0
$306$	−666.000	−0.124421
$307$	−1244.00	−0.231267	−0.115633	−	0.993292i	$-0.536890\pi$
$307$	−1244.00	−0.231267	−0.115633	+	0.993292i	$0.536890\pi$
$308$	−2002.00	−0.370372
$309$	3396.00	0.625216
$310$	0	0
$311$	2082.00	0.379612	0.189806	−	0.981822i	$-0.439214\pi$
$311$	2082.00	0.379612	0.189806	+	0.981822i	$0.439214\pi$
$312$	−1440.00	−0.261295
$313$	−2378.00	−0.429433	−0.214716	−	0.976676i	$-0.568883\pi$
$313$	−2378.00	−0.429433	−0.214716	+	0.976676i	$0.568883\pi$
$314$	−2114.00	−0.379936
$315$	0	0
$316$	3850.00	0.685378
$317$	496.000	0.0878806	0.0439403	−	0.999034i	$-0.486009\pi$
$317$	496.000	0.0878806	0.0439403	+	0.999034i	$0.486009\pi$
$318$	−1044.00	−0.184103
$319$	−990.000	−0.173760
$320$	0	0
$321$	−1692.00	−0.294200
$322$	4732.00	0.818957
$323$	4440.00	0.764855
$324$	−567.000	−0.0972222
$325$	0	0
$326$	−3868.00	−0.657143
$327$	−960.000	−0.162349
$328$	−6330.00	−1.06560
$329$	13156.0	2.20460
$330$	0	0
$331$	−2708.00	−0.449683	−0.224842	−	0.974395i	$-0.572186\pi$
$331$	−2708.00	−0.449683	−0.224842	+	0.974395i	$0.572186\pi$
$332$	−924.000	−0.152744
$333$	594.000	0.0977507
$334$	−2004.00	−0.328305
$335$	0	0
$336$	3198.00	0.519242
$337$	−4034.00	−0.652065	−0.326033	−	0.945359i	$-0.605712\pi$
$337$	−4034.00	−0.652065	−0.326033	+	0.945359i	$0.605712\pi$
$338$	−1173.00	−0.188766
$339$	6426.00	1.02954
$340$	0	0
$341$	−88.0000	−0.0139750
$342$	−540.000	−0.0853797
$343$	−260.000	−0.0409291
$344$	6120.00	0.959210
$345$	0	0
$346$	−678.000	−0.105345
$347$	−11084.0	−1.71476	−0.857378	−	0.514687i	$-0.827908\pi$
$347$	−11084.0	−1.71476	−0.857378	+	0.514687i	$0.827908\pi$
$348$	1890.00	0.291134
$349$	−3120.00	−0.478538	−0.239269	−	0.970953i	$-0.576908\pi$
$349$	−3120.00	−0.478538	−0.239269	+	0.970953i	$0.576908\pi$
$350$	0	0
$351$	864.000	0.131387
$352$	1771.00	0.268167
$353$	5622.00	0.847674	0.423837	−	0.905739i	$-0.360683\pi$
$353$	5622.00	0.847674	0.423837	+	0.905739i	$0.360683\pi$
$354$	−600.000	−0.0900837
$355$	0	0
$356$	−3990.00	−0.594016
$357$	−5772.00	−0.855705
$358$	−1680.00	−0.248019
$359$	−8500.00	−1.24962	−0.624809	−	0.780778i	$-0.714823\pi$
$359$	−8500.00	−1.24962	−0.624809	+	0.780778i	$0.714823\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	−4358.00	−0.632739
$363$	363.000	0.0524864
$364$	−5824.00	−0.838628
$365$	0	0
$366$	396.000	0.0565553
$367$	−7144.00	−1.01611	−0.508057	−	0.861324i	$-0.669636\pi$
$367$	−7144.00	−1.01611	−0.508057	+	0.861324i	$0.669636\pi$
$368$	7462.00	1.05702
$369$	3798.00	0.535816
$370$	0	0
$371$	−9048.00	−1.26617
$372$	168.000	0.0234150
$373$	632.000	0.0877312	0.0438656	−	0.999037i	$-0.486033\pi$
$373$	632.000	0.0877312	0.0438656	+	0.999037i	$0.486033\pi$
$374$	−814.000	−0.112543
$375$	0	0
$376$	−7590.00	−1.04102
$377$	−2880.00	−0.393442
$378$	702.000	0.0955211
$379$	−4220.00	−0.571944	−0.285972	−	0.958238i	$-0.592316\pi$
$379$	−4220.00	−0.571944	−0.285972	+	0.958238i	$0.592316\pi$
$380$	0	0
$381$	4818.00	0.647857
$382$	−1778.00	−0.238142
$383$	−8458.00	−1.12842	−0.564208	−	0.825632i	$-0.690819\pi$
$383$	−8458.00	−1.12842	−0.564208	+	0.825632i	$0.690819\pi$
$384$	−4365.00	−0.580079
$385$	0	0
$386$	3962.00	0.522437
$387$	−3672.00	−0.482321
$388$	98.0000	0.0128227
$389$	1740.00	0.226790	0.113395	−	0.993550i	$-0.463827\pi$
$389$	1740.00	0.226790	0.113395	+	0.993550i	$0.463827\pi$
$390$	0	0
$391$	−13468.0	−1.74196
$392$	−4995.00	−0.643586
$393$	−5724.00	−0.734701
$394$	−374.000	−0.0478219
$395$	0	0
$396$	−693.000	−0.0879408
$397$	5126.00	0.648027	0.324013	−	0.946053i	$-0.394968\pi$
$397$	5126.00	0.648027	0.324013	+	0.946053i	$0.394968\pi$
$398$	2100.00	0.264481
$399$	−4680.00	−0.587201
$400$	0	0
$401$	−3098.00	−0.385802	−0.192901	−	0.981218i	$-0.561790\pi$
$401$	−3098.00	−0.385802	−0.192901	+	0.981218i	$0.561790\pi$
$402$	3108.00	0.385604
$403$	−256.000	−0.0316433
$404$	−11914.0	−1.46719
$405$	0	0
$406$	−2340.00	−0.286040
$407$	726.000	0.0884189
$408$	3330.00	0.404068
$409$	6390.00	0.772531	0.386265	−	0.922388i	$-0.373765\pi$
$409$	6390.00	0.772531	0.386265	+	0.922388i	$0.373765\pi$
$410$	0	0
$411$	6558.00	0.787062
$412$	−7924.00	−0.947542
$413$	−5200.00	−0.619553
$414$	1638.00	0.194452
$415$	0	0
$416$	5152.00	0.607206
$417$	8220.00	0.965312
$418$	−660.000	−0.0772288
$419$	9760.00	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$419$	9760.00	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$420$	0	0
$421$	−5138.00	−0.594800	−0.297400	−	0.954753i	$-0.596119\pi$
$421$	−5138.00	−0.594800	−0.297400	+	0.954753i	$0.596119\pi$
$422$	2232.00	0.257469
$423$	4554.00	0.523459
$424$	5220.00	0.597891
$425$	0	0
$426$	2286.00	0.259993
$427$	3432.00	0.388960
$428$	3948.00	0.445873
$429$	1056.00	0.118844
$430$	0	0
$431$	−7008.00	−0.783210	−0.391605	−	0.920133i	$-0.628080\pi$
$431$	−7008.00	−0.783210	−0.391605	+	0.920133i	$0.628080\pi$
$432$	1107.00	0.123288
$433$	−5578.00	−0.619080	−0.309540	−	0.950886i	$-0.600175\pi$
$433$	−5578.00	−0.619080	−0.309540	+	0.950886i	$0.600175\pi$
$434$	−208.000	−0.0230053
$435$	0	0
$436$	2240.00	0.246047
$437$	−10920.0	−1.19536
$438$	1626.00	0.177382
$439$	−10430.0	−1.13393	−0.566967	−	0.823741i	$-0.691883\pi$
$439$	−10430.0	−1.13393	−0.566967	+	0.823741i	$0.691883\pi$
$440$	0	0
$441$	2997.00	0.323615
$442$	−2368.00	−0.254829
$443$	4432.00	0.475329	0.237664	−	0.971347i	$-0.423618\pi$
$443$	4432.00	0.475329	0.237664	+	0.971347i	$0.423618\pi$
$444$	−1386.00	−0.148146
$445$	0	0
$446$	−2128.00	−0.225928
$447$	−3930.00	−0.415845
$448$	−4342.00	−0.457902
$449$	−6290.00	−0.661121	−0.330561	−	0.943785i	$-0.607238\pi$
$449$	−6290.00	−0.661121	−0.330561	+	0.943785i	$0.607238\pi$
$450$	0	0
$451$	4642.00	0.484664
$452$	−14994.0	−1.56031
$453$	−3594.00	−0.372761
$454$	−2964.00	−0.306404
$455$	0	0
$456$	2700.00	0.277279
$457$	−3054.00	−0.312604	−0.156302	−	0.987709i	$-0.549957\pi$
$457$	−3054.00	−0.312604	−0.156302	+	0.987709i	$0.549957\pi$
$458$	−2550.00	−0.260161
$459$	−1998.00	−0.203178
$460$	0	0
$461$	12882.0	1.30146	0.650732	−	0.759308i	$-0.274462\pi$
$461$	12882.0	1.30146	0.650732	+	0.759308i	$0.274462\pi$
$462$	858.000	0.0864021
$463$	−6148.00	−0.617110	−0.308555	−	0.951207i	$-0.599845\pi$
$463$	−6148.00	−0.617110	−0.308555	+	0.951207i	$0.599845\pi$
$464$	−3690.00	−0.369190
$465$	0	0
$466$	3042.00	0.302399
$467$	−5124.00	−0.507731	−0.253866	−	0.967240i	$-0.581702\pi$
$467$	−5124.00	−0.507731	−0.253866	+	0.967240i	$0.581702\pi$
$468$	−2016.00	−0.199123
$469$	26936.0	2.65200
$470$	0	0
$471$	−6342.00	−0.620433
$472$	3000.00	0.292555
$473$	−4488.00	−0.436276
$474$	−1650.00	−0.159888
$475$	0	0
$476$	13468.0	1.29686
$477$	−3132.00	−0.300638
$478$	2700.00	0.258358
$479$	−16520.0	−1.57582	−0.787910	−	0.615790i	$-0.788837\pi$
$479$	−16520.0	−1.57582	−0.787910	+	0.615790i	$0.788837\pi$
$480$	0	0
$481$	2112.00	0.200206
$482$	−578.000	−0.0546207
$483$	14196.0	1.33735
$484$	−847.000	−0.0795455
$485$	0	0
$486$	243.000	0.0226805
$487$	−524.000	−0.0487571	−0.0243785	−	0.999703i	$-0.507761\pi$
$487$	−524.000	−0.0487571	−0.0243785	+	0.999703i	$0.507761\pi$
$488$	−1980.00	−0.183669
$489$	−11604.0	−1.07311
$490$	0	0
$491$	−15028.0	−1.38127	−0.690636	−	0.723203i	$-0.742669\pi$
$491$	−15028.0	−1.38127	−0.690636	+	0.723203i	$0.742669\pi$
$492$	−8862.00	−0.812052
$493$	6660.00	0.608421
$494$	−1920.00	−0.174868
$495$	0	0
$496$	−328.000	−0.0296928
$497$	19812.0	1.78811
$498$	396.000	0.0356329
$499$	9020.00	0.809200	0.404600	−	0.914494i	$-0.367411\pi$
$499$	9020.00	0.809200	0.404600	+	0.914494i	$0.367411\pi$
$500$	0	0
$501$	−6012.00	−0.536120
$502$	3752.00	0.333586
$503$	14812.0	1.31299	0.656495	−	0.754330i	$-0.272038\pi$
$503$	14812.0	1.31299	0.656495	+	0.754330i	$0.272038\pi$
$504$	−3510.00	−0.310214
$505$	0	0
$506$	2002.00	0.175889
$507$	−3519.00	−0.308253
$508$	−11242.0	−0.981856
$509$	12660.0	1.10245	0.551223	−	0.834358i	$-0.314161\pi$
$509$	12660.0	1.10245	0.551223	+	0.834358i	$0.314161\pi$
$510$	0	0
$511$	14092.0	1.21995
$512$	11521.0	0.994455
$513$	−1620.00	−0.139424
$514$	−674.000	−0.0578383
$515$	0	0
$516$	8568.00	0.730979
$517$	5566.00	0.473486
$518$	1716.00	0.145553
$519$	−2034.00	−0.172028
$520$	0	0
$521$	−3738.00	−0.314328	−0.157164	−	0.987573i	$-0.550235\pi$
$521$	−3738.00	−0.314328	−0.157164	+	0.987573i	$0.550235\pi$
$522$	−810.000	−0.0679171
$523$	6352.00	0.531078	0.265539	−	0.964100i	$-0.414450\pi$
$523$	6352.00	0.531078	0.265539	+	0.964100i	$0.414450\pi$
$524$	13356.0	1.11347
$525$	0	0
$526$	4352.00	0.360753
$527$	592.000	0.0489334
$528$	1353.00	0.111518
$529$	20957.0	1.72245
$530$	0	0
$531$	−1800.00	−0.147106
$532$	10920.0	0.889929
$533$	13504.0	1.09742
$534$	1710.00	0.138575
$535$	0	0
$536$	−15540.0	−1.25229
$537$	−5040.00	−0.405013
$538$	500.000	0.0400679
$539$	3663.00	0.292721
$540$	0	0
$541$	−24728.0	−1.96514	−0.982569	−	0.185898i	$-0.940481\pi$
$541$	−24728.0	−1.96514	−0.982569	+	0.185898i	$0.940481\pi$
$542$	−6538.00	−0.518139
$543$	−13074.0	−1.03326
$544$	−11914.0	−0.938986
$545$	0	0
$546$	2496.00	0.195639
$547$	22756.0	1.77875	0.889375	−	0.457178i	$-0.151140\pi$
$547$	22756.0	1.77875	0.889375	+	0.457178i	$0.151140\pi$
$548$	−15302.0	−1.19283
$549$	1188.00	0.0923545
$550$	0	0
$551$	5400.00	0.417509
$552$	−8190.00	−0.631503
$553$	−14300.0	−1.09963
$554$	−124.000	−0.00950949
$555$	0	0
$556$	−19180.0	−1.46297
$557$	9526.00	0.724649	0.362325	−	0.932052i	$-0.381983\pi$
$557$	9526.00	0.724649	0.362325	+	0.932052i	$0.381983\pi$
$558$	−72.0000	−0.00546237
$559$	−13056.0	−0.987853
$560$	0	0
$561$	−2442.00	−0.183781
$562$	3642.00	0.273360
$563$	−12068.0	−0.903385	−0.451692	−	0.892174i	$-0.649180\pi$
$563$	−12068.0	−0.903385	−0.451692	+	0.892174i	$0.649180\pi$
$564$	−10626.0	−0.793325
$565$	0	0
$566$	−4648.00	−0.345177
$567$	2106.00	0.155985
$568$	−11430.0	−0.844352
$569$	15090.0	1.11179	0.555893	−	0.831254i	$-0.312377\pi$
$569$	15090.0	1.11179	0.555893	+	0.831254i	$0.312377\pi$
$570$	0	0
$571$	4412.00	0.323356	0.161678	−	0.986844i	$-0.448309\pi$
$571$	4412.00	0.323356	0.161678	+	0.986844i	$0.448309\pi$
$572$	−2464.00	−0.180114
$573$	−5334.00	−0.388885
$574$	10972.0	0.797844
$575$	0	0
$576$	−1503.00	−0.108724
$577$	3906.00	0.281818	0.140909	−	0.990023i	$-0.454998\pi$
$577$	3906.00	0.281818	0.140909	+	0.990023i	$0.454998\pi$
$578$	563.000	0.0405151
$579$	11886.0	0.853135
$580$	0	0
$581$	3432.00	0.245066
$582$	−42.0000	−0.00299133
$583$	−3828.00	−0.271937
$584$	−8130.00	−0.576065
$585$	0	0
$586$	3102.00	0.218673
$587$	12016.0	0.844895	0.422448	−	0.906387i	$-0.361171\pi$
$587$	12016.0	0.844895	0.422448	+	0.906387i	$0.361171\pi$
$588$	−6993.00	−0.490453
$589$	480.000	0.0335790
$590$	0	0
$591$	−1122.00	−0.0780929
$592$	2706.00	0.187865
$593$	11342.0	0.785430	0.392715	−	0.919660i	$-0.371536\pi$
$593$	11342.0	0.785430	0.392715	+	0.919660i	$0.371536\pi$
$594$	297.000	0.0205152
$595$	0	0
$596$	9170.00	0.630231
$597$	6300.00	0.431896
$598$	5824.00	0.398263
$599$	20690.0	1.41130	0.705651	−	0.708559i	$-0.250654\pi$
$599$	20690.0	1.41130	0.705651	+	0.708559i	$0.250654\pi$
$600$	0	0
$601$	−598.000	−0.0405872	−0.0202936	−	0.999794i	$-0.506460\pi$
$601$	−598.000	−0.0405872	−0.0202936	+	0.999794i	$0.506460\pi$
$602$	−10608.0	−0.718189
$603$	9324.00	0.629689
$604$	8386.00	0.564936
$605$	0	0
$606$	5106.00	0.342272
$607$	166.000	0.0111001	0.00555003	−	0.999985i	$-0.498233\pi$
$607$	166.000	0.0111001	0.00555003	+	0.999985i	$0.498233\pi$
$608$	−9660.00	−0.644350
$609$	−7020.00	−0.467101
$610$	0	0
$611$	16192.0	1.07211
$612$	4662.00	0.307925
$613$	−20108.0	−1.32488	−0.662442	−	0.749113i	$-0.730480\pi$
$613$	−20108.0	−1.32488	−0.662442	+	0.749113i	$0.730480\pi$
$614$	−1244.00	−0.0817651
$615$	0	0
$616$	−4290.00	−0.280599
$617$	2286.00	0.149159	0.0745793	−	0.997215i	$-0.476239\pi$
$617$	2286.00	0.149159	0.0745793	+	0.997215i	$0.476239\pi$
$618$	3396.00	0.221047
$619$	−25660.0	−1.66618	−0.833088	−	0.553141i	$-0.813429\pi$
$619$	−25660.0	−1.66618	−0.833088	+	0.553141i	$0.813429\pi$
$620$	0	0
$621$	4914.00	0.317539
$622$	2082.00	0.134213
$623$	14820.0	0.953051
$624$	3936.00	0.252510
$625$	0	0
$626$	−2378.00	−0.151827
$627$	−1980.00	−0.126114
$628$	14798.0	0.940294
$629$	−4884.00	−0.309599
$630$	0	0
$631$	−11408.0	−0.719723	−0.359862	−	0.933006i	$-0.617176\pi$
$631$	−11408.0	−0.719723	−0.359862	+	0.933006i	$0.617176\pi$
$632$	8250.00	0.519252
$633$	6696.00	0.420446
$634$	496.000	0.0310705
$635$	0	0
$636$	7308.00	0.455631
$637$	10656.0	0.662804
$638$	−990.000	−0.0614333
$639$	6858.00	0.424567
$640$	0	0
$641$	−3378.00	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$641$	−3378.00	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$642$	−1692.00	−0.104015
$643$	11212.0	0.687649	0.343824	−	0.939034i	$-0.388278\pi$
$643$	11212.0	0.687649	0.343824	+	0.939034i	$0.388278\pi$
$644$	−33124.0	−2.02681
$645$	0	0
$646$	4440.00	0.270417
$647$	86.0000	0.00522567	0.00261284	−	0.999997i	$-0.499168\pi$
$647$	86.0000	0.00522567	0.00261284	+	0.999997i	$0.499168\pi$
$648$	−1215.00	−0.0736570
$649$	−2200.00	−0.133062
$650$	0	0
$651$	−624.000	−0.0375676
$652$	27076.0	1.62635
$653$	4432.00	0.265601	0.132801	−	0.991143i	$-0.457603\pi$
$653$	4432.00	0.265601	0.132801	+	0.991143i	$0.457603\pi$
$654$	−960.000	−0.0573990
$655$	0	0
$656$	17302.0	1.02977
$657$	4878.00	0.289663
$658$	13156.0	0.779444
$659$	4580.00	0.270731	0.135365	−	0.990796i	$-0.456779\pi$
$659$	4580.00	0.270731	0.135365	+	0.990796i	$0.456779\pi$
$660$	0	0
$661$	4282.00	0.251967	0.125984	−	0.992032i	$-0.459791\pi$
$661$	4282.00	0.251967	0.125984	+	0.992032i	$0.459791\pi$
$662$	−2708.00	−0.158987
$663$	−7104.00	−0.416133
$664$	−1980.00	−0.115721
$665$	0	0
$666$	594.000	0.0345601
$667$	−16380.0	−0.950879
$668$	14028.0	0.812514
$669$	−6384.00	−0.368938
$670$	0	0
$671$	1452.00	0.0835378
$672$	12558.0	0.720886
$673$	−8438.00	−0.483300	−0.241650	−	0.970363i	$-0.577689\pi$
$673$	−8438.00	−0.483300	−0.241650	+	0.970363i	$0.577689\pi$
$674$	−4034.00	−0.230540
$675$	0	0
$676$	8211.00	0.467171
$677$	−34494.0	−1.95822	−0.979108	−	0.203341i	$-0.934820\pi$
$677$	−34494.0	−1.95822	−0.979108	+	0.203341i	$0.934820\pi$
$678$	6426.00	0.363996
$679$	−364.000	−0.0205730
$680$	0	0
$681$	−8892.00	−0.500356
$682$	−88.0000	−0.00494090
$683$	13712.0	0.768192	0.384096	−	0.923293i	$-0.374513\pi$
$683$	13712.0	0.768192	0.384096	+	0.923293i	$0.374513\pi$
$684$	3780.00	0.211304
$685$	0	0
$686$	−260.000	−0.0144706
$687$	−7650.00	−0.424841
$688$	−16728.0	−0.926961
$689$	−11136.0	−0.615744
$690$	0	0
$691$	11372.0	0.626066	0.313033	−	0.949742i	$-0.398655\pi$
$691$	11372.0	0.626066	0.313033	+	0.949742i	$0.398655\pi$
$692$	4746.00	0.260717
$693$	2574.00	0.141094
$694$	−11084.0	−0.606258
$695$	0	0
$696$	4050.00	0.220567
$697$	−31228.0	−1.69705
$698$	−3120.00	−0.169189
$699$	9126.00	0.493815
$700$	0	0
$701$	−6398.00	−0.344721	−0.172360	−	0.985034i	$-0.555139\pi$
$701$	−6398.00	−0.344721	−0.172360	+	0.985034i	$0.555139\pi$
$702$	864.000	0.0464524
$703$	−3960.00	−0.212453
$704$	−1837.00	−0.0983445
$705$	0	0
$706$	5622.00	0.299698
$707$	44252.0	2.35399
$708$	4200.00	0.222946
$709$	−5830.00	−0.308816	−0.154408	−	0.988007i	$-0.549347\pi$
$709$	−5830.00	−0.308816	−0.154408	+	0.988007i	$0.549347\pi$
$710$	0	0
$711$	−4950.00	−0.261096
$712$	−8550.00	−0.450035
$713$	−1456.00	−0.0764763
$714$	−5772.00	−0.302537
$715$	0	0
$716$	11760.0	0.613815
$717$	8100.00	0.421897
$718$	−8500.00	−0.441807
$719$	34530.0	1.79103	0.895516	−	0.445030i	$-0.146807\pi$
$719$	34530.0	1.79103	0.895516	+	0.445030i	$0.146807\pi$
$720$	0	0
$721$	29432.0	1.52026
$722$	−3259.00	−0.167988
$723$	−1734.00	−0.0891952
$724$	30506.0	1.56595
$725$	0	0
$726$	363.000	0.0185567
$727$	17316.0	0.883377	0.441688	−	0.897169i	$-0.354380\pi$
$727$	17316.0	0.883377	0.441688	+	0.897169i	$0.354380\pi$
$728$	−12480.0	−0.635357
$729$	729.000	0.0370370
$730$	0	0
$731$	30192.0	1.52762
$732$	−2772.00	−0.139967
$733$	27072.0	1.36416	0.682079	−	0.731279i	$-0.261076\pi$
$733$	27072.0	1.36416	0.682079	+	0.731279i	$0.261076\pi$
$734$	−7144.00	−0.359250
$735$	0	0
$736$	29302.0	1.46751
$737$	11396.0	0.569575
$738$	3798.00	0.189439
$739$	−17320.0	−0.862147	−0.431073	−	0.902317i	$-0.641865\pi$
$739$	−17320.0	−0.862147	−0.431073	+	0.902317i	$0.641865\pi$
$740$	0	0
$741$	−5760.00	−0.285559
$742$	−9048.00	−0.447658
$743$	−14588.0	−0.720299	−0.360149	−	0.932895i	$-0.617274\pi$
$743$	−14588.0	−0.720299	−0.360149	+	0.932895i	$0.617274\pi$
$744$	360.000	0.0177396
$745$	0	0
$746$	632.000	0.0310176
$747$	1188.00	0.0581883
$748$	5698.00	0.278529
$749$	−14664.0	−0.715368
$750$	0	0
$751$	26152.0	1.27071	0.635353	−	0.772222i	$-0.280855\pi$
$751$	26152.0	1.27071	0.635353	+	0.772222i	$0.280855\pi$
$752$	20746.0	1.00602
$753$	11256.0	0.544743
$754$	−2880.00	−0.139103
$755$	0	0
$756$	−4914.00	−0.236403
$757$	1066.00	0.0511815	0.0255908	−	0.999673i	$-0.491853\pi$
$757$	1066.00	0.0511815	0.0255908	+	0.999673i	$0.491853\pi$
$758$	−4220.00	−0.202213
$759$	6006.00	0.287225
$760$	0	0
$761$	−37518.0	−1.78716	−0.893578	−	0.448907i	$-0.851813\pi$
$761$	−37518.0	−1.78716	−0.893578	+	0.448907i	$0.851813\pi$
$762$	4818.00	0.229052
$763$	−8320.00	−0.394763
$764$	12446.0	0.589372
$765$	0	0
$766$	−8458.00	−0.398956
$767$	−6400.00	−0.301292
$768$	−357.000	−0.0167736
$769$	−17290.0	−0.810785	−0.405392	−	0.914143i	$-0.632865\pi$
$769$	−17290.0	−0.810785	−0.405392	+	0.914143i	$0.632865\pi$
$770$	0	0
$771$	−2022.00	−0.0944495
$772$	−27734.0	−1.29296
$773$	17172.0	0.799009	0.399504	−	0.916731i	$-0.369182\pi$
$773$	17172.0	0.799009	0.399504	+	0.916731i	$0.369182\pi$
$774$	−3672.00	−0.170526
$775$	0	0
$776$	210.000	0.00971464
$777$	5148.00	0.237688
$778$	1740.00	0.0801825
$779$	−25320.0	−1.16455
$780$	0	0
$781$	8382.00	0.384035
$782$	−13468.0	−0.615876
$783$	−2430.00	−0.110908
$784$	13653.0	0.621948
$785$	0	0
$786$	−5724.00	−0.259756
$787$	9536.00	0.431921	0.215960	−	0.976402i	$-0.430712\pi$
$787$	9536.00	0.431921	0.215960	+	0.976402i	$0.430712\pi$
$788$	2618.00	0.118353
$789$	13056.0	0.589108
$790$	0	0
$791$	55692.0	2.50339
$792$	−1485.00	−0.0666252
$793$	4224.00	0.189153
$794$	5126.00	0.229112
$795$	0	0
$796$	−14700.0	−0.654557
$797$	20516.0	0.911812	0.455906	−	0.890028i	$-0.349315\pi$
$797$	20516.0	0.911812	0.455906	+	0.890028i	$0.349315\pi$
$798$	−4680.00	−0.207607
$799$	−37444.0	−1.65791
$800$	0	0
$801$	5130.00	0.226292
$802$	−3098.00	−0.136402
$803$	5962.00	0.262010
$804$	−21756.0	−0.954322
$805$	0	0
$806$	−256.000	−0.0111876
$807$	1500.00	0.0654306
$808$	−25530.0	−1.11156
$809$	22470.0	0.976518	0.488259	−	0.872699i	$-0.337632\pi$
$809$	22470.0	0.976518	0.488259	+	0.872699i	$0.337632\pi$
$810$	0	0
$811$	−3368.00	−0.145828	−0.0729140	−	0.997338i	$-0.523230\pi$
$811$	−3368.00	−0.145828	−0.0729140	+	0.997338i	$0.523230\pi$
$812$	16380.0	0.707913
$813$	−19614.0	−0.846117
$814$	726.000	0.0312608
$815$	0	0
$816$	−9102.00	−0.390483
$817$	24480.0	1.04828
$818$	6390.00	0.273131
$819$	7488.00	0.319477
$820$	0	0
$821$	−10738.0	−0.456466	−0.228233	−	0.973607i	$-0.573295\pi$
$821$	−10738.0	−0.456466	−0.228233	+	0.973607i	$0.573295\pi$
$822$	6558.00	0.278268
$823$	15912.0	0.673946	0.336973	−	0.941514i	$-0.390597\pi$
$823$	15912.0	0.673946	0.336973	+	0.941514i	$0.390597\pi$
$824$	−16980.0	−0.717872
$825$	0	0
$826$	−5200.00	−0.219045
$827$	−22924.0	−0.963900	−0.481950	−	0.876199i	$-0.660071\pi$
$827$	−22924.0	−0.963900	−0.481950	+	0.876199i	$0.660071\pi$
$828$	−11466.0	−0.481245
$829$	−41690.0	−1.74663	−0.873313	−	0.487159i	$-0.838033\pi$
$829$	−41690.0	−1.74663	−0.873313	+	0.487159i	$0.838033\pi$
$830$	0	0
$831$	−372.000	−0.0155289
$832$	−5344.00	−0.222680
$833$	−24642.0	−1.02496
$834$	8220.00	0.341289
$835$	0	0
$836$	4620.00	0.191132
$837$	−216.000	−0.00892001
$838$	9760.00	0.402331
$839$	−16450.0	−0.676898	−0.338449	−	0.940985i	$-0.609902\pi$
$839$	−16450.0	−0.676898	−0.338449	+	0.940985i	$0.609902\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	−5138.00	−0.210294
$843$	10926.0	0.446396
$844$	−15624.0	−0.637204
$845$	0	0
$846$	4554.00	0.185071
$847$	3146.00	0.127624
$848$	−14268.0	−0.577789
$849$	−13944.0	−0.563671
$850$	0	0
$851$	12012.0	0.483861
$852$	−16002.0	−0.643450
$853$	30892.0	1.24000	0.620001	−	0.784601i	$-0.287132\pi$
$853$	30892.0	1.24000	0.620001	+	0.784601i	$0.287132\pi$
$854$	3432.00	0.137518
$855$	0	0
$856$	8460.00	0.337800
$857$	38906.0	1.55076	0.775381	−	0.631493i	$-0.217558\pi$
$857$	38906.0	1.55076	0.775381	+	0.631493i	$0.217558\pi$
$858$	1056.00	0.0420178
$859$	−1020.00	−0.0405145	−0.0202572	−	0.999795i	$-0.506449\pi$
$859$	−1020.00	−0.0405145	−0.0202572	+	0.999795i	$0.506449\pi$
$860$	0	0
$861$	32916.0	1.30287
$862$	−7008.00	−0.276907
$863$	−15078.0	−0.594741	−0.297370	−	0.954762i	$-0.596110\pi$
$863$	−15078.0	−0.594741	−0.297370	+	0.954762i	$0.596110\pi$
$864$	4347.00	0.171167
$865$	0	0
$866$	−5578.00	−0.218878
$867$	1689.00	0.0661608
$868$	1456.00	0.0569353
$869$	−6050.00	−0.236171
$870$	0	0
$871$	33152.0	1.28968
$872$	4800.00	0.186409
$873$	−126.000	−0.00488483
$874$	−10920.0	−0.422625
$875$	0	0
$876$	−11382.0	−0.438998
$877$	−22704.0	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$877$	−22704.0	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$878$	−10430.0	−0.400906
$879$	9306.00	0.357092
$880$	0	0
$881$	−19358.0	−0.740281	−0.370141	−	0.928976i	$-0.620690\pi$
$881$	−19358.0	−0.740281	−0.370141	+	0.928976i	$0.620690\pi$
$882$	2997.00	0.114415
$883$	11252.0	0.428833	0.214417	−	0.976742i	$-0.431215\pi$
$883$	11252.0	0.428833	0.214417	+	0.976742i	$0.431215\pi$
$884$	16576.0	0.630669
$885$	0	0
$886$	4432.00	0.168054
$887$	−43684.0	−1.65362	−0.826812	−	0.562478i	$-0.809848\pi$
$887$	−43684.0	−1.65362	−0.826812	+	0.562478i	$0.809848\pi$
$888$	−2970.00	−0.112237
$889$	41756.0	1.57531
$890$	0	0
$891$	891.000	0.0335013
$892$	14896.0	0.559142
$893$	−30360.0	−1.13769
$894$	−3930.00	−0.147023
$895$	0	0
$896$	−37830.0	−1.41050
$897$	17472.0	0.650360
$898$	−6290.00	−0.233742
$899$	720.000	0.0267112
$900$	0	0
$901$	25752.0	0.952190
$902$	4642.00	0.171354
$903$	−31824.0	−1.17280
$904$	−32130.0	−1.18211
$905$	0	0
$906$	−3594.00	−0.131791
$907$	−45804.0	−1.67684	−0.838422	−	0.545022i	$-0.816521\pi$
$907$	−45804.0	−1.67684	−0.838422	+	0.545022i	$0.816521\pi$
$908$	20748.0	0.758311
$909$	15318.0	0.558928
$910$	0	0
$911$	−15318.0	−0.557089	−0.278544	−	0.960423i	$-0.589852\pi$
$911$	−15318.0	−0.557089	−0.278544	+	0.960423i	$0.589852\pi$
$912$	−7380.00	−0.267956
$913$	1452.00	0.0526333
$914$	−3054.00	−0.110522
$915$	0	0
$916$	17850.0	0.643865
$917$	−49608.0	−1.78648
$918$	−1998.00	−0.0718342
$919$	11350.0	0.407401	0.203701	−	0.979033i	$-0.434703\pi$
$919$	11350.0	0.407401	0.203701	+	0.979033i	$0.434703\pi$
$920$	0	0
$921$	−3732.00	−0.133522
$922$	12882.0	0.460137
$923$	24384.0	0.869566
$924$	−6006.00	−0.213834
$925$	0	0
$926$	−6148.00	−0.218181
$927$	10188.0	0.360969
$928$	−14490.0	−0.512562
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	0	0
$931$	−19980.0	−0.703349
$932$	−21294.0	−0.748399
$933$	6246.00	0.219169
$934$	−5124.00	−0.179510
$935$	0	0
$936$	−4320.00	−0.150859
$937$	10006.0	0.348860	0.174430	−	0.984670i	$-0.444192\pi$
$937$	10006.0	0.348860	0.174430	+	0.984670i	$0.444192\pi$
$938$	26936.0	0.937624
$939$	−7134.00	−0.247933
$940$	0	0
$941$	2622.00	0.0908340	0.0454170	−	0.998968i	$-0.485538\pi$
$941$	2622.00	0.0908340	0.0454170	+	0.998968i	$0.485538\pi$
$942$	−6342.00	−0.219356
$943$	76804.0	2.65226
$944$	−8200.00	−0.282720
$945$	0	0
$946$	−4488.00	−0.154247
$947$	39876.0	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$947$	39876.0	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$948$	11550.0	0.395703
$949$	17344.0	0.593267
$950$	0	0
$951$	1488.00	0.0507379
$952$	28860.0	0.982519
$953$	−38918.0	−1.32285	−0.661426	−	0.750011i	$-0.730048\pi$
$953$	−38918.0	−1.32285	−0.661426	+	0.750011i	$0.730048\pi$
$954$	−3132.00	−0.106292
$955$	0	0
$956$	−18900.0	−0.639403
$957$	−2970.00	−0.100320
$958$	−16520.0	−0.557137
$959$	56836.0	1.91380
$960$	0	0
$961$	−29727.0	−0.997852
$962$	2112.00	0.0707834
$963$	−5076.00	−0.169857
$964$	4046.00	0.135179
$965$	0	0
$966$	14196.0	0.472825
$967$	−1114.00	−0.0370464	−0.0185232	−	0.999828i	$-0.505896\pi$
$967$	−1114.00	−0.0370464	−0.0185232	+	0.999828i	$0.505896\pi$
$968$	−1815.00	−0.0602648
$969$	13320.0	0.441589
$970$	0	0
$971$	−1688.00	−0.0557884	−0.0278942	−	0.999611i	$-0.508880\pi$
$971$	−1688.00	−0.0557884	−0.0278942	+	0.999611i	$0.508880\pi$
$972$	−1701.00	−0.0561313
$973$	71240.0	2.34722
$974$	−524.000	−0.0172382
$975$	0	0
$976$	5412.00	0.177494
$977$	41826.0	1.36963	0.684817	−	0.728715i	$-0.259882\pi$
$977$	41826.0	1.36963	0.684817	+	0.728715i	$0.259882\pi$
$978$	−11604.0	−0.379402
$979$	6270.00	0.204689
$980$	0	0
$981$	−2880.00	−0.0937322
$982$	−15028.0	−0.488353
$983$	−978.000	−0.0317328	−0.0158664	−	0.999874i	$-0.505051\pi$
$983$	−978.000	−0.0317328	−0.0158664	+	0.999874i	$0.505051\pi$
$984$	−18990.0	−0.615223
$985$	0	0
$986$	6660.00	0.215109
$987$	39468.0	1.27283
$988$	13440.0	0.432777
$989$	−74256.0	−2.38747
$990$	0	0
$991$	47272.0	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$991$	47272.0	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$992$	−1288.00	−0.0412238
$993$	−8124.00	−0.259625
$994$	19812.0	0.632192
$995$	0	0
$996$	−2772.00	−0.0881869
$997$	−51104.0	−1.62335	−0.811675	−	0.584109i	$-0.801444\pi$
$997$	−51104.0	−1.62335	−0.811675	+	0.584109i	$0.801444\pi$
$998$	9020.00	0.286095
$999$	1782.00	0.0564364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.f.1.1		1
3.2	odd	2		2475.4.a.e.1.1		1
5.2	odd	4		825.4.c.f.199.2		2
5.3	odd	4		825.4.c.f.199.1		2
5.4	even	2		33.4.a.b.1.1	✓	1
15.14	odd	2		99.4.a.a.1.1		1
20.19	odd	2		528.4.a.h.1.1		1
35.34	odd	2		1617.4.a.d.1.1		1
40.19	odd	2		2112.4.a.h.1.1		1
40.29	even	2		2112.4.a.u.1.1		1
55.54	odd	2		363.4.a.d.1.1		1
60.59	even	2		1584.4.a.l.1.1		1
165.164	even	2		1089.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.b.1.1	✓	1	5.4	even	2
99.4.a.a.1.1		1	15.14	odd	2
363.4.a.d.1.1		1	55.54	odd	2
528.4.a.h.1.1		1	20.19	odd	2
825.4.a.f.1.1		1	1.1	even	1	trivial
825.4.c.f.199.1		2	5.3	odd	4
825.4.c.f.199.2		2	5.2	odd	4
1089.4.a.e.1.1		1	165.164	even	2
1584.4.a.l.1.1		1	60.59	even	2
1617.4.a.d.1.1		1	35.34	odd	2
2112.4.a.h.1.1		1	40.19	odd	2
2112.4.a.u.1.1		1	40.29	even	2
2475.4.a.e.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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