LMFDB - Embedded newform 1050.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,4,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.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:	$61.9520055060$
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	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +42.0000 q^{11} -12.0000 q^{12} +47.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -3.00000 q^{17} -18.0000 q^{18} +56.0000 q^{19} -21.0000 q^{21} -84.0000 q^{22} +9.00000 q^{23} +24.0000 q^{24} -94.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} +189.000 q^{29} +263.000 q^{31} -32.0000 q^{32} -126.000 q^{33} +6.00000 q^{34} +36.0000 q^{36} -58.0000 q^{37} -112.000 q^{38} -141.000 q^{39} -273.000 q^{41} +42.0000 q^{42} -307.000 q^{43} +168.000 q^{44} -18.0000 q^{46} -156.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} +9.00000 q^{51} +188.000 q^{52} +207.000 q^{53} +54.0000 q^{54} -56.0000 q^{56} -168.000 q^{57} -378.000 q^{58} -507.000 q^{59} +635.000 q^{61} -526.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +252.000 q^{66} -556.000 q^{67} -12.0000 q^{68} -27.0000 q^{69} +684.000 q^{71} -72.0000 q^{72} +482.000 q^{73} +116.000 q^{74} +224.000 q^{76} +294.000 q^{77} +282.000 q^{78} +182.000 q^{79} +81.0000 q^{81} +546.000 q^{82} -291.000 q^{83} -84.0000 q^{84} +614.000 q^{86} -567.000 q^{87} -336.000 q^{88} -810.000 q^{89} +329.000 q^{91} +36.0000 q^{92} -789.000 q^{93} +312.000 q^{94} +96.0000 q^{96} -910.000 q^{97} -98.0000 q^{98} +378.000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	−12.0000	−0.288675
$13$	47.0000	1.00273	0.501364	−	0.865237i	$-0.332832\pi$
$13$	47.0000	1.00273	0.501364	+	0.865237i	$0.332832\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−3.00000	−0.0428004	−0.0214002	−	0.999771i	$-0.506812\pi$
$17$	−3.00000	−0.0428004	−0.0214002	+	0.999771i	$0.506812\pi$
$18$	−18.0000	−0.235702
$19$	56.0000	0.676173	0.338086	−	0.941115i	$-0.390220\pi$
$19$	56.0000	0.676173	0.338086	+	0.941115i	$0.390220\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	−84.0000	−0.814039
$23$	9.00000	0.0815926	0.0407963	−	0.999167i	$-0.487011\pi$
$23$	9.00000	0.0815926	0.0407963	+	0.999167i	$0.487011\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−94.0000	−0.709035
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	189.000	1.21022	0.605111	−	0.796141i	$-0.293129\pi$
$29$	189.000	1.21022	0.605111	+	0.796141i	$0.293129\pi$
$30$	0	0
$31$	263.000	1.52375	0.761874	−	0.647725i	$-0.224279\pi$
$31$	263.000	1.52375	0.761874	+	0.647725i	$0.224279\pi$
$32$	−32.0000	−0.176777
$33$	−126.000	−0.664660
$34$	6.00000	0.0302645
$35$	0	0
$36$	36.0000	0.166667
$37$	−58.0000	−0.257707	−0.128853	−	0.991664i	$-0.541130\pi$
$37$	−58.0000	−0.257707	−0.128853	+	0.991664i	$0.541130\pi$
$38$	−112.000	−0.478126
$39$	−141.000	−0.578925
$40$	0	0
$41$	−273.000	−1.03989	−0.519944	−	0.854200i	$-0.674047\pi$
$41$	−273.000	−1.03989	−0.519944	+	0.854200i	$0.674047\pi$
$42$	42.0000	0.154303
$43$	−307.000	−1.08877	−0.544384	−	0.838836i	$-0.683237\pi$
$43$	−307.000	−1.08877	−0.544384	+	0.838836i	$0.683237\pi$
$44$	168.000	0.575613
$45$	0	0
$46$	−18.0000	−0.0576947
$47$	−156.000	−0.484148	−0.242074	−	0.970258i	$-0.577828\pi$
$47$	−156.000	−0.484148	−0.242074	+	0.970258i	$0.577828\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	9.00000	0.0247108
$52$	188.000	0.501364
$53$	207.000	0.536484	0.268242	−	0.963352i	$-0.413557\pi$
$53$	207.000	0.536484	0.268242	+	0.963352i	$0.413557\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−56.0000	−0.133631
$57$	−168.000	−0.390388
$58$	−378.000	−0.855756
$59$	−507.000	−1.11874	−0.559371	−	0.828917i	$-0.688957\pi$
$59$	−507.000	−1.11874	−0.559371	+	0.828917i	$0.688957\pi$
$60$	0	0
$61$	635.000	1.33284	0.666421	−	0.745575i	$-0.267825\pi$
$61$	635.000	1.33284	0.666421	+	0.745575i	$0.267825\pi$
$62$	−526.000	−1.07745
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	252.000	0.469986
$67$	−556.000	−1.01382	−0.506912	−	0.861998i	$-0.669213\pi$
$67$	−556.000	−1.01382	−0.506912	+	0.861998i	$0.669213\pi$
$68$	−12.0000	−0.0214002
$69$	−27.0000	−0.0471075
$70$	0	0
$71$	684.000	1.14332	0.571661	−	0.820490i	$-0.306299\pi$
$71$	684.000	1.14332	0.571661	+	0.820490i	$0.306299\pi$
$72$	−72.0000	−0.117851
$73$	482.000	0.772792	0.386396	−	0.922333i	$-0.373720\pi$
$73$	482.000	0.772792	0.386396	+	0.922333i	$0.373720\pi$
$74$	116.000	0.182226
$75$	0	0
$76$	224.000	0.338086
$77$	294.000	0.435122
$78$	282.000	0.409362
$79$	182.000	0.259197	0.129599	−	0.991567i	$-0.458631\pi$
$79$	182.000	0.259197	0.129599	+	0.991567i	$0.458631\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	546.000	0.735312
$83$	−291.000	−0.384836	−0.192418	−	0.981313i	$-0.561633\pi$
$83$	−291.000	−0.384836	−0.192418	+	0.981313i	$0.561633\pi$
$84$	−84.0000	−0.109109
$85$	0	0
$86$	614.000	0.769876
$87$	−567.000	−0.698722
$88$	−336.000	−0.407020
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	329.000	0.378995
$92$	36.0000	0.0407963
$93$	−789.000	−0.879736
$94$	312.000	0.342344
$95$	0	0
$96$	96.0000	0.102062
$97$	−910.000	−0.952541	−0.476271	−	0.879299i	$-0.658012\pi$
$97$	−910.000	−0.952541	−0.476271	+	0.879299i	$0.658012\pi$
$98$	−98.0000	−0.101015
$99$	378.000	0.383742
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−18.0000	−0.0174732
$103$	611.000	0.584501	0.292251	−	0.956342i	$-0.405596\pi$
$103$	611.000	0.584501	0.292251	+	0.956342i	$0.405596\pi$
$104$	−376.000	−0.354518
$105$	0	0
$106$	−414.000	−0.379351
$107$	−2094.00	−1.89191	−0.945956	−	0.324294i	$-0.894873\pi$
$107$	−2094.00	−1.89191	−0.945956	+	0.324294i	$0.894873\pi$
$108$	−108.000	−0.0962250
$109$	−196.000	−0.172233	−0.0861165	−	0.996285i	$-0.527446\pi$
$109$	−196.000	−0.172233	−0.0861165	+	0.996285i	$0.527446\pi$
$110$	0	0
$111$	174.000	0.148787
$112$	112.000	0.0944911
$113$	162.000	0.134864	0.0674322	−	0.997724i	$-0.478519\pi$
$113$	162.000	0.134864	0.0674322	+	0.997724i	$0.478519\pi$
$114$	336.000	0.276046
$115$	0	0
$116$	756.000	0.605111
$117$	423.000	0.334242
$118$	1014.00	0.791070
$119$	−21.0000	−0.0161770
$120$	0	0
$121$	433.000	0.325319
$122$	−1270.00	−0.942462
$123$	819.000	0.600380
$124$	1052.00	0.761874
$125$	0	0
$126$	−126.000	−0.0890871
$127$	1922.00	1.34291	0.671456	−	0.741044i	$-0.265669\pi$
$127$	1922.00	1.34291	0.671456	+	0.741044i	$0.265669\pi$
$128$	−128.000	−0.0883883
$129$	921.000	0.628601
$130$	0	0
$131$	2448.00	1.63269	0.816346	−	0.577563i	$-0.195996\pi$
$131$	2448.00	1.63269	0.816346	+	0.577563i	$0.195996\pi$
$132$	−504.000	−0.332330
$133$	392.000	0.255569
$134$	1112.00	0.716882
$135$	0	0
$136$	24.0000	0.0151322
$137$	−408.000	−0.254436	−0.127218	−	0.991875i	$-0.540605\pi$
$137$	−408.000	−0.254436	−0.127218	+	0.991875i	$0.540605\pi$
$138$	54.0000	0.0333100
$139$	−976.000	−0.595563	−0.297781	−	0.954634i	$-0.596247\pi$
$139$	−976.000	−0.595563	−0.297781	+	0.954634i	$0.596247\pi$
$140$	0	0
$141$	468.000	0.279523
$142$	−1368.00	−0.808451
$143$	1974.00	1.15436
$144$	144.000	0.0833333
$145$	0	0
$146$	−964.000	−0.546447
$147$	−147.000	−0.0824786
$148$	−232.000	−0.128853
$149$	−1215.00	−0.668031	−0.334016	−	0.942568i	$-0.608404\pi$
$149$	−1215.00	−0.668031	−0.334016	+	0.942568i	$0.608404\pi$
$150$	0	0
$151$	1382.00	0.744805	0.372403	−	0.928071i	$-0.378534\pi$
$151$	1382.00	0.744805	0.372403	+	0.928071i	$0.378534\pi$
$152$	−448.000	−0.239063
$153$	−27.0000	−0.0142668
$154$	−588.000	−0.307678
$155$	0	0
$156$	−564.000	−0.289462
$157$	2894.00	1.47112	0.735562	−	0.677458i	$-0.236918\pi$
$157$	2894.00	1.47112	0.735562	+	0.677458i	$0.236918\pi$
$158$	−364.000	−0.183280
$159$	−621.000	−0.309739
$160$	0	0
$161$	63.0000	0.0308391
$162$	−162.000	−0.0785674
$163$	3407.00	1.63716	0.818579	−	0.574394i	$-0.194762\pi$
$163$	3407.00	1.63716	0.818579	+	0.574394i	$0.194762\pi$
$164$	−1092.00	−0.519944
$165$	0	0
$166$	582.000	0.272120
$167$	414.000	0.191834	0.0959170	−	0.995389i	$-0.469422\pi$
$167$	414.000	0.191834	0.0959170	+	0.995389i	$0.469422\pi$
$168$	168.000	0.0771517
$169$	12.0000	0.00546199
$170$	0	0
$171$	504.000	0.225391
$172$	−1228.00	−0.544384
$173$	2136.00	0.938711	0.469356	−	0.883009i	$-0.344486\pi$
$173$	2136.00	0.938711	0.469356	+	0.883009i	$0.344486\pi$
$174$	1134.00	0.494071
$175$	0	0
$176$	672.000	0.287806
$177$	1521.00	0.645906
$178$	1620.00	0.682158
$179$	−2430.00	−1.01467	−0.507337	−	0.861748i	$-0.669370\pi$
$179$	−2430.00	−1.01467	−0.507337	+	0.861748i	$0.669370\pi$
$180$	0	0
$181$	2762.00	1.13424	0.567121	−	0.823634i	$-0.308057\pi$
$181$	2762.00	1.13424	0.567121	+	0.823634i	$0.308057\pi$
$182$	−658.000	−0.267990
$183$	−1905.00	−0.769517
$184$	−72.0000	−0.0288473
$185$	0	0
$186$	1578.00	0.622068
$187$	−126.000	−0.0492729
$188$	−624.000	−0.242074
$189$	−189.000	−0.0727393
$190$	0	0
$191$	4437.00	1.68089	0.840445	−	0.541897i	$-0.182294\pi$
$191$	4437.00	1.68089	0.840445	+	0.541897i	$0.182294\pi$
$192$	−192.000	−0.0721688
$193$	3194.00	1.19124	0.595620	−	0.803267i	$-0.296907\pi$
$193$	3194.00	1.19124	0.595620	+	0.803267i	$0.296907\pi$
$194$	1820.00	0.673548
$195$	0	0
$196$	196.000	0.0714286
$197$	−3729.00	−1.34863	−0.674315	−	0.738444i	$-0.735561\pi$
$197$	−3729.00	−1.34863	−0.674315	+	0.738444i	$0.735561\pi$
$198$	−756.000	−0.271346
$199$	3824.00	1.36219	0.681096	−	0.732194i	$-0.261504\pi$
$199$	3824.00	1.36219	0.681096	+	0.732194i	$0.261504\pi$
$200$	0	0
$201$	1668.00	0.585332
$202$	0	0
$203$	1323.00	0.457421
$204$	36.0000	0.0123554
$205$	0	0
$206$	−1222.00	−0.413305
$207$	81.0000	0.0271975
$208$	752.000	0.250682
$209$	2352.00	0.778427
$210$	0	0
$211$	1205.00	0.393155	0.196577	−	0.980488i	$-0.437017\pi$
$211$	1205.00	0.393155	0.196577	+	0.980488i	$0.437017\pi$
$212$	828.000	0.268242
$213$	−2052.00	−0.660097
$214$	4188.00	1.33778
$215$	0	0
$216$	216.000	0.0680414
$217$	1841.00	0.575923
$218$	392.000	0.121787
$219$	−1446.00	−0.446172
$220$	0	0
$221$	−141.000	−0.0429171
$222$	−348.000	−0.105208
$223$	479.000	0.143840	0.0719198	−	0.997410i	$-0.477087\pi$
$223$	479.000	0.143840	0.0719198	+	0.997410i	$0.477087\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−324.000	−0.0953635
$227$	951.000	0.278062	0.139031	−	0.990288i	$-0.455601\pi$
$227$	951.000	0.278062	0.139031	+	0.990288i	$0.455601\pi$
$228$	−672.000	−0.195194
$229$	4826.00	1.39262	0.696312	−	0.717739i	$-0.254823\pi$
$229$	4826.00	1.39262	0.696312	+	0.717739i	$0.254823\pi$
$230$	0	0
$231$	−882.000	−0.251218
$232$	−1512.00	−0.427878
$233$	3240.00	0.910985	0.455492	−	0.890240i	$-0.349463\pi$
$233$	3240.00	0.910985	0.455492	+	0.890240i	$0.349463\pi$
$234$	−846.000	−0.236345
$235$	0	0
$236$	−2028.00	−0.559371
$237$	−546.000	−0.149648
$238$	42.0000	0.0114389
$239$	−96.0000	−0.0259821	−0.0129911	−	0.999916i	$-0.504135\pi$
$239$	−96.0000	−0.0259821	−0.0129911	+	0.999916i	$0.504135\pi$
$240$	0	0
$241$	−3046.00	−0.814150	−0.407075	−	0.913395i	$-0.633451\pi$
$241$	−3046.00	−0.814150	−0.407075	+	0.913395i	$0.633451\pi$
$242$	−866.000	−0.230035
$243$	−243.000	−0.0641500
$244$	2540.00	0.666421
$245$	0	0
$246$	−1638.00	−0.424533
$247$	2632.00	0.678017
$248$	−2104.00	−0.538726
$249$	873.000	0.222185
$250$	0	0
$251$	−1095.00	−0.275362	−0.137681	−	0.990477i	$-0.543965\pi$
$251$	−1095.00	−0.275362	−0.137681	+	0.990477i	$0.543965\pi$
$252$	252.000	0.0629941
$253$	378.000	0.0939314
$254$	−3844.00	−0.949583
$255$	0	0
$256$	256.000	0.0625000
$257$	3585.00	0.870141	0.435070	−	0.900396i	$-0.356724\pi$
$257$	3585.00	0.870141	0.435070	+	0.900396i	$0.356724\pi$
$258$	−1842.00	−0.444488
$259$	−406.000	−0.0974039
$260$	0	0
$261$	1701.00	0.403407
$262$	−4896.00	−1.15449
$263$	3717.00	0.871483	0.435742	−	0.900072i	$-0.356486\pi$
$263$	3717.00	0.871483	0.435742	+	0.900072i	$0.356486\pi$
$264$	1008.00	0.234993
$265$	0	0
$266$	−784.000	−0.180715
$267$	2430.00	0.556980
$268$	−2224.00	−0.506912
$269$	282.000	0.0639176	0.0319588	−	0.999489i	$-0.489825\pi$
$269$	282.000	0.0639176	0.0319588	+	0.999489i	$0.489825\pi$
$270$	0	0
$271$	−4300.00	−0.963861	−0.481931	−	0.876209i	$-0.660064\pi$
$271$	−4300.00	−0.963861	−0.481931	+	0.876209i	$0.660064\pi$
$272$	−48.0000	−0.0107001
$273$	−987.000	−0.218813
$274$	816.000	0.179914
$275$	0	0
$276$	−108.000	−0.0235538
$277$	−5326.00	−1.15526	−0.577632	−	0.816297i	$-0.696023\pi$
$277$	−5326.00	−1.15526	−0.577632	+	0.816297i	$0.696023\pi$
$278$	1952.00	0.421127
$279$	2367.00	0.507916
$280$	0	0
$281$	−480.000	−0.101902	−0.0509509	−	0.998701i	$-0.516225\pi$
$281$	−480.000	−0.101902	−0.0509509	+	0.998701i	$0.516225\pi$
$282$	−936.000	−0.197652
$283$	3800.00	0.798186	0.399093	−	0.916911i	$-0.369325\pi$
$283$	3800.00	0.798186	0.399093	+	0.916911i	$0.369325\pi$
$284$	2736.00	0.571661
$285$	0	0
$286$	−3948.00	−0.816259
$287$	−1911.00	−0.393041
$288$	−288.000	−0.0589256
$289$	−4904.00	−0.998168
$290$	0	0
$291$	2730.00	0.549950
$292$	1928.00	0.386396
$293$	−6240.00	−1.24418	−0.622090	−	0.782946i	$-0.713716\pi$
$293$	−6240.00	−1.24418	−0.622090	+	0.782946i	$0.713716\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	464.000	0.0911130
$297$	−1134.00	−0.221553
$298$	2430.00	0.472370
$299$	423.000	0.0818151
$300$	0	0
$301$	−2149.00	−0.411516
$302$	−2764.00	−0.526657
$303$	0	0
$304$	896.000	0.169043
$305$	0	0
$306$	54.0000	0.0100882
$307$	10094.0	1.87653	0.938265	−	0.345916i	$-0.112432\pi$
$307$	10094.0	1.87653	0.938265	+	0.345916i	$0.112432\pi$
$308$	1176.00	0.217561
$309$	−1833.00	−0.337462
$310$	0	0
$311$	−3666.00	−0.668424	−0.334212	−	0.942498i	$-0.608470\pi$
$311$	−3666.00	−0.668424	−0.334212	+	0.942498i	$0.608470\pi$
$312$	1128.00	0.204681
$313$	−1414.00	−0.255348	−0.127674	−	0.991816i	$-0.540751\pi$
$313$	−1414.00	−0.255348	−0.127674	+	0.991816i	$0.540751\pi$
$314$	−5788.00	−1.04024
$315$	0	0
$316$	728.000	0.129599
$317$	699.000	0.123848	0.0619239	−	0.998081i	$-0.480276\pi$
$317$	699.000	0.123848	0.0619239	+	0.998081i	$0.480276\pi$
$318$	1242.00	0.219019
$319$	7938.00	1.39324
$320$	0	0
$321$	6282.00	1.09230
$322$	−126.000	−0.0218065
$323$	−168.000	−0.0289405
$324$	324.000	0.0555556
$325$	0	0
$326$	−6814.00	−1.15765
$327$	588.000	0.0994388
$328$	2184.00	0.367656
$329$	−1092.00	−0.182991
$330$	0	0
$331$	−4273.00	−0.709563	−0.354781	−	0.934949i	$-0.615445\pi$
$331$	−4273.00	−0.709563	−0.354781	+	0.934949i	$0.615445\pi$
$332$	−1164.00	−0.192418
$333$	−522.000	−0.0859022
$334$	−828.000	−0.135647
$335$	0	0
$336$	−336.000	−0.0545545
$337$	3851.00	0.622485	0.311242	−	0.950331i	$-0.399255\pi$
$337$	3851.00	0.622485	0.311242	+	0.950331i	$0.399255\pi$
$338$	−24.0000	−0.00386221
$339$	−486.000	−0.0778640
$340$	0	0
$341$	11046.0	1.75418
$342$	−1008.00	−0.159375
$343$	343.000	0.0539949
$344$	2456.00	0.384938
$345$	0	0
$346$	−4272.00	−0.663769
$347$	−5682.00	−0.879037	−0.439518	−	0.898234i	$-0.644851\pi$
$347$	−5682.00	−0.879037	−0.439518	+	0.898234i	$0.644851\pi$
$348$	−2268.00	−0.349361
$349$	−2599.00	−0.398628	−0.199314	−	0.979936i	$-0.563871\pi$
$349$	−2599.00	−0.398628	−0.199314	+	0.979936i	$0.563871\pi$
$350$	0	0
$351$	−1269.00	−0.192975
$352$	−1344.00	−0.203510
$353$	−5010.00	−0.755398	−0.377699	−	0.925928i	$-0.623285\pi$
$353$	−5010.00	−0.755398	−0.377699	+	0.925928i	$0.623285\pi$
$354$	−3042.00	−0.456725
$355$	0	0
$356$	−3240.00	−0.482359
$357$	63.0000	0.00933981
$358$	4860.00	0.717483
$359$	−8151.00	−1.19831	−0.599155	−	0.800633i	$-0.704497\pi$
$359$	−8151.00	−1.19831	−0.599155	+	0.800633i	$0.704497\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	−5524.00	−0.802030
$363$	−1299.00	−0.187823
$364$	1316.00	0.189498
$365$	0	0
$366$	3810.00	0.544131
$367$	−871.000	−0.123885	−0.0619425	−	0.998080i	$-0.519730\pi$
$367$	−871.000	−0.123885	−0.0619425	+	0.998080i	$0.519730\pi$
$368$	144.000	0.0203981
$369$	−2457.00	−0.346630
$370$	0	0
$371$	1449.00	0.202772
$372$	−3156.00	−0.439868
$373$	−628.000	−0.0871759	−0.0435879	−	0.999050i	$-0.513879\pi$
$373$	−628.000	−0.0871759	−0.0435879	+	0.999050i	$0.513879\pi$
$374$	252.000	0.0348412
$375$	0	0
$376$	1248.00	0.171172
$377$	8883.00	1.21352
$378$	378.000	0.0514344
$379$	11225.0	1.52134	0.760672	−	0.649136i	$-0.224869\pi$
$379$	11225.0	1.52134	0.760672	+	0.649136i	$0.224869\pi$
$380$	0	0
$381$	−5766.00	−0.775331
$382$	−8874.00	−1.18857
$383$	2760.00	0.368223	0.184112	−	0.982905i	$-0.441059\pi$
$383$	2760.00	0.368223	0.184112	+	0.982905i	$0.441059\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−6388.00	−0.842333
$387$	−2763.00	−0.362923
$388$	−3640.00	−0.476271
$389$	−8322.00	−1.08468	−0.542342	−	0.840158i	$-0.682462\pi$
$389$	−8322.00	−1.08468	−0.542342	+	0.840158i	$0.682462\pi$
$390$	0	0
$391$	−27.0000	−0.00349220
$392$	−392.000	−0.0505076
$393$	−7344.00	−0.942636
$394$	7458.00	0.953626
$395$	0	0
$396$	1512.00	0.191871
$397$	8309.00	1.05042	0.525210	−	0.850973i	$-0.323987\pi$
$397$	8309.00	1.05042	0.525210	+	0.850973i	$0.323987\pi$
$398$	−7648.00	−0.963215
$399$	−1176.00	−0.147553
$400$	0	0
$401$	8076.00	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$401$	8076.00	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$402$	−3336.00	−0.413892
$403$	12361.0	1.52790
$404$	0	0
$405$	0	0
$406$	−2646.00	−0.323445
$407$	−2436.00	−0.296678
$408$	−72.0000	−0.00873660
$409$	−8926.00	−1.07913	−0.539563	−	0.841945i	$-0.681410\pi$
$409$	−8926.00	−1.07913	−0.539563	+	0.841945i	$0.681410\pi$
$410$	0	0
$411$	1224.00	0.146899
$412$	2444.00	0.292251
$413$	−3549.00	−0.422845
$414$	−162.000	−0.0192316
$415$	0	0
$416$	−1504.00	−0.177259
$417$	2928.00	0.343848
$418$	−4704.00	−0.550431
$419$	−14385.0	−1.67722	−0.838608	−	0.544736i	$-0.816630\pi$
$419$	−14385.0	−1.67722	−0.838608	+	0.544736i	$0.816630\pi$
$420$	0	0
$421$	−11566.0	−1.33894	−0.669468	−	0.742841i	$-0.733478\pi$
$421$	−11566.0	−1.33894	−0.669468	+	0.742841i	$0.733478\pi$
$422$	−2410.00	−0.278002
$423$	−1404.00	−0.161383
$424$	−1656.00	−0.189676
$425$	0	0
$426$	4104.00	0.466759
$427$	4445.00	0.503767
$428$	−8376.00	−0.945956
$429$	−5922.00	−0.666473
$430$	0	0
$431$	13557.0	1.51512	0.757561	−	0.652764i	$-0.226391\pi$
$431$	13557.0	1.51512	0.757561	+	0.652764i	$0.226391\pi$
$432$	−432.000	−0.0481125
$433$	3158.00	0.350494	0.175247	−	0.984525i	$-0.443928\pi$
$433$	3158.00	0.350494	0.175247	+	0.984525i	$0.443928\pi$
$434$	−3682.00	−0.407239
$435$	0	0
$436$	−784.000	−0.0861165
$437$	504.000	0.0551707
$438$	2892.00	0.315491
$439$	11483.0	1.24841	0.624207	−	0.781259i	$-0.285422\pi$
$439$	11483.0	1.24841	0.624207	+	0.781259i	$0.285422\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	282.000	0.0303470
$443$	−8484.00	−0.909903	−0.454951	−	0.890516i	$-0.650343\pi$
$443$	−8484.00	−0.909903	−0.454951	+	0.890516i	$0.650343\pi$
$444$	696.000	0.0743935
$445$	0	0
$446$	−958.000	−0.101710
$447$	3645.00	0.385688
$448$	448.000	0.0472456
$449$	−12804.0	−1.34579	−0.672893	−	0.739740i	$-0.734949\pi$
$449$	−12804.0	−1.34579	−0.672893	+	0.739740i	$0.734949\pi$
$450$	0	0
$451$	−11466.0	−1.19715
$452$	648.000	0.0674322
$453$	−4146.00	−0.430013
$454$	−1902.00	−0.196620
$455$	0	0
$456$	1344.00	0.138023
$457$	10439.0	1.06852	0.534262	−	0.845319i	$-0.320589\pi$
$457$	10439.0	1.06852	0.534262	+	0.845319i	$0.320589\pi$
$458$	−9652.00	−0.984734
$459$	81.0000	0.00823694
$460$	0	0
$461$	−8688.00	−0.877745	−0.438873	−	0.898549i	$-0.644622\pi$
$461$	−8688.00	−0.877745	−0.438873	+	0.898549i	$0.644622\pi$
$462$	1764.00	0.177638
$463$	−13408.0	−1.34584	−0.672919	−	0.739717i	$-0.734960\pi$
$463$	−13408.0	−1.34584	−0.672919	+	0.739717i	$0.734960\pi$
$464$	3024.00	0.302555
$465$	0	0
$466$	−6480.00	−0.644164
$467$	11673.0	1.15666	0.578332	−	0.815802i	$-0.303704\pi$
$467$	11673.0	1.15666	0.578332	+	0.815802i	$0.303704\pi$
$468$	1692.00	0.167121
$469$	−3892.00	−0.383189
$470$	0	0
$471$	−8682.00	−0.849353
$472$	4056.00	0.395535
$473$	−12894.0	−1.25342
$474$	1092.00	0.105817
$475$	0	0
$476$	−84.0000	−0.00808852
$477$	1863.00	0.178828
$478$	192.000	0.0183721
$479$	−7068.00	−0.674207	−0.337104	−	0.941468i	$-0.609447\pi$
$479$	−7068.00	−0.674207	−0.337104	+	0.941468i	$0.609447\pi$
$480$	0	0
$481$	−2726.00	−0.258409
$482$	6092.00	0.575691
$483$	−189.000	−0.0178050
$484$	1732.00	0.162660
$485$	0	0
$486$	486.000	0.0453609
$487$	9626.00	0.895679	0.447840	−	0.894114i	$-0.352194\pi$
$487$	9626.00	0.895679	0.447840	+	0.894114i	$0.352194\pi$
$488$	−5080.00	−0.471231
$489$	−10221.0	−0.945214
$490$	0	0
$491$	−10590.0	−0.973361	−0.486680	−	0.873580i	$-0.661792\pi$
$491$	−10590.0	−0.973361	−0.486680	+	0.873580i	$0.661792\pi$
$492$	3276.00	0.300190
$493$	−567.000	−0.0517980
$494$	−5264.00	−0.479430
$495$	0	0
$496$	4208.00	0.380937
$497$	4788.00	0.432135
$498$	−1746.00	−0.157109
$499$	4007.00	0.359475	0.179737	−	0.983715i	$-0.442475\pi$
$499$	4007.00	0.359475	0.179737	+	0.983715i	$0.442475\pi$
$500$	0	0
$501$	−1242.00	−0.110755
$502$	2190.00	0.194710
$503$	−2814.00	−0.249443	−0.124722	−	0.992192i	$-0.539804\pi$
$503$	−2814.00	−0.249443	−0.124722	+	0.992192i	$0.539804\pi$
$504$	−504.000	−0.0445435
$505$	0	0
$506$	−756.000	−0.0664196
$507$	−36.0000	−0.00315348
$508$	7688.00	0.671456
$509$	13860.0	1.20694	0.603471	−	0.797385i	$-0.293784\pi$
$509$	13860.0	1.20694	0.603471	+	0.797385i	$0.293784\pi$
$510$	0	0
$511$	3374.00	0.292088
$512$	−512.000	−0.0441942
$513$	−1512.00	−0.130129
$514$	−7170.00	−0.615282
$515$	0	0
$516$	3684.00	0.314300
$517$	−6552.00	−0.557363
$518$	812.000	0.0688750
$519$	−6408.00	−0.541965
$520$	0	0
$521$	−1491.00	−0.125378	−0.0626890	−	0.998033i	$-0.519968\pi$
$521$	−1491.00	−0.125378	−0.0626890	+	0.998033i	$0.519968\pi$
$522$	−3402.00	−0.285252
$523$	12104.0	1.01199	0.505995	−	0.862536i	$-0.331125\pi$
$523$	12104.0	1.01199	0.505995	+	0.862536i	$0.331125\pi$
$524$	9792.00	0.816346
$525$	0	0
$526$	−7434.00	−0.616232
$527$	−789.000	−0.0652170
$528$	−2016.00	−0.166165
$529$	−12086.0	−0.993343
$530$	0	0
$531$	−4563.00	−0.372914
$532$	1568.00	0.127785
$533$	−12831.0	−1.04272
$534$	−4860.00	−0.393844
$535$	0	0
$536$	4448.00	0.358441
$537$	7290.00	0.585823
$538$	−564.000	−0.0451966
$539$	2058.00	0.164461
$540$	0	0
$541$	6032.00	0.479364	0.239682	−	0.970851i	$-0.422957\pi$
$541$	6032.00	0.479364	0.239682	+	0.970851i	$0.422957\pi$
$542$	8600.00	0.681553
$543$	−8286.00	−0.654855
$544$	96.0000	0.00756611
$545$	0	0
$546$	1974.00	0.154724
$547$	−16765.0	−1.31046	−0.655228	−	0.755431i	$-0.727428\pi$
$547$	−16765.0	−1.31046	−0.655228	+	0.755431i	$0.727428\pi$
$548$	−1632.00	−0.127218
$549$	5715.00	0.444281
$550$	0	0
$551$	10584.0	0.818319
$552$	216.000	0.0166550
$553$	1274.00	0.0979674
$554$	10652.0	0.816896
$555$	0	0
$556$	−3904.00	−0.297781
$557$	−15126.0	−1.15064	−0.575322	−	0.817927i	$-0.695123\pi$
$557$	−15126.0	−1.15064	−0.575322	+	0.817927i	$0.695123\pi$
$558$	−4734.00	−0.359151
$559$	−14429.0	−1.09174
$560$	0	0
$561$	378.000	0.0284477
$562$	960.000	0.0720554
$563$	11817.0	0.884595	0.442298	−	0.896868i	$-0.354163\pi$
$563$	11817.0	0.884595	0.442298	+	0.896868i	$0.354163\pi$
$564$	1872.00	0.139761
$565$	0	0
$566$	−7600.00	−0.564403
$567$	567.000	0.0419961
$568$	−5472.00	−0.404225
$569$	2826.00	0.208211	0.104106	−	0.994566i	$-0.466802\pi$
$569$	2826.00	0.208211	0.104106	+	0.994566i	$0.466802\pi$
$570$	0	0
$571$	7769.00	0.569391	0.284696	−	0.958618i	$-0.408107\pi$
$571$	7769.00	0.569391	0.284696	+	0.958618i	$0.408107\pi$
$572$	7896.00	0.577182
$573$	−13311.0	−0.970462
$574$	3822.00	0.277922
$575$	0	0
$576$	576.000	0.0416667
$577$	14462.0	1.04343	0.521717	−	0.853119i	$-0.325292\pi$
$577$	14462.0	1.04343	0.521717	+	0.853119i	$0.325292\pi$
$578$	9808.00	0.705811
$579$	−9582.00	−0.687762
$580$	0	0
$581$	−2037.00	−0.145454
$582$	−5460.00	−0.388873
$583$	8694.00	0.617614
$584$	−3856.00	−0.273223
$585$	0	0
$586$	12480.0	0.879768
$587$	16749.0	1.17769	0.588846	−	0.808245i	$-0.299582\pi$
$587$	16749.0	1.17769	0.588846	+	0.808245i	$0.299582\pi$
$588$	−588.000	−0.0412393
$589$	14728.0	1.03032
$590$	0	0
$591$	11187.0	0.778632
$592$	−928.000	−0.0644266
$593$	1794.00	0.124234	0.0621170	−	0.998069i	$-0.480215\pi$
$593$	1794.00	0.124234	0.0621170	+	0.998069i	$0.480215\pi$
$594$	2268.00	0.156662
$595$	0	0
$596$	−4860.00	−0.334016
$597$	−11472.0	−0.786462
$598$	−846.000	−0.0578520
$599$	18885.0	1.28818	0.644090	−	0.764949i	$-0.277236\pi$
$599$	18885.0	1.28818	0.644090	+	0.764949i	$0.277236\pi$
$600$	0	0
$601$	−21112.0	−1.43291	−0.716453	−	0.697635i	$-0.754236\pi$
$601$	−21112.0	−1.43291	−0.716453	+	0.697635i	$0.754236\pi$
$602$	4298.00	0.290986
$603$	−5004.00	−0.337941
$604$	5528.00	0.372403
$605$	0	0
$606$	0	0
$607$	10484.0	0.701042	0.350521	−	0.936555i	$-0.386005\pi$
$607$	10484.0	0.701042	0.350521	+	0.936555i	$0.386005\pi$
$608$	−1792.00	−0.119532
$609$	−3969.00	−0.264092
$610$	0	0
$611$	−7332.00	−0.485468
$612$	−108.000	−0.00713340
$613$	3386.00	0.223098	0.111549	−	0.993759i	$-0.464419\pi$
$613$	3386.00	0.223098	0.111549	+	0.993759i	$0.464419\pi$
$614$	−20188.0	−1.32691
$615$	0	0
$616$	−2352.00	−0.153839
$617$	−21078.0	−1.37531	−0.687657	−	0.726036i	$-0.741361\pi$
$617$	−21078.0	−1.37531	−0.687657	+	0.726036i	$0.741361\pi$
$618$	3666.00	0.238622
$619$	7262.00	0.471542	0.235771	−	0.971809i	$-0.424238\pi$
$619$	7262.00	0.471542	0.235771	+	0.971809i	$0.424238\pi$
$620$	0	0
$621$	−243.000	−0.0157025
$622$	7332.00	0.472647
$623$	−5670.00	−0.364629
$624$	−2256.00	−0.144731
$625$	0	0
$626$	2828.00	0.180558
$627$	−7056.00	−0.449425
$628$	11576.0	0.735562
$629$	174.000	0.0110299
$630$	0	0
$631$	−27634.0	−1.74341	−0.871705	−	0.490030i	$-0.836986\pi$
$631$	−27634.0	−1.74341	−0.871705	+	0.490030i	$0.836986\pi$
$632$	−1456.00	−0.0916401
$633$	−3615.00	−0.226988
$634$	−1398.00	−0.0875736
$635$	0	0
$636$	−2484.00	−0.154870
$637$	2303.00	0.143247
$638$	−15876.0	−0.985167
$639$	6156.00	0.381107
$640$	0	0
$641$	12714.0	0.783421	0.391710	−	0.920089i	$-0.371883\pi$
$641$	12714.0	0.783421	0.391710	+	0.920089i	$0.371883\pi$
$642$	−12564.0	−0.772370
$643$	26534.0	1.62737	0.813685	−	0.581306i	$-0.197458\pi$
$643$	26534.0	1.62737	0.813685	+	0.581306i	$0.197458\pi$
$644$	252.000	0.0154196
$645$	0	0
$646$	336.000	0.0204640
$647$	−6222.00	−0.378071	−0.189036	−	0.981970i	$-0.560536\pi$
$647$	−6222.00	−0.378071	−0.189036	+	0.981970i	$0.560536\pi$
$648$	−648.000	−0.0392837
$649$	−21294.0	−1.28792
$650$	0	0
$651$	−5523.00	−0.332509
$652$	13628.0	0.818579
$653$	16518.0	0.989892	0.494946	−	0.868924i	$-0.335188\pi$
$653$	16518.0	0.989892	0.494946	+	0.868924i	$0.335188\pi$
$654$	−1176.00	−0.0703138
$655$	0	0
$656$	−4368.00	−0.259972
$657$	4338.00	0.257597
$658$	2184.00	0.129394
$659$	30366.0	1.79498	0.897490	−	0.441035i	$-0.145389\pi$
$659$	30366.0	1.79498	0.897490	+	0.441035i	$0.145389\pi$
$660$	0	0
$661$	−32242.0	−1.89723	−0.948615	−	0.316434i	$-0.897514\pi$
$661$	−32242.0	−1.89723	−0.948615	+	0.316434i	$0.897514\pi$
$662$	8546.00	0.501737
$663$	423.000	0.0247782
$664$	2328.00	0.136060
$665$	0	0
$666$	1044.00	0.0607420
$667$	1701.00	0.0987451
$668$	1656.00	0.0959170
$669$	−1437.00	−0.0830458
$670$	0	0
$671$	26670.0	1.53440
$672$	672.000	0.0385758
$673$	2099.00	0.120224	0.0601118	−	0.998192i	$-0.480854\pi$
$673$	2099.00	0.120224	0.0601118	+	0.998192i	$0.480854\pi$
$674$	−7702.00	−0.440163
$675$	0	0
$676$	48.0000	0.00273100
$677$	15120.0	0.858359	0.429179	−	0.903219i	$-0.358803\pi$
$677$	15120.0	0.858359	0.429179	+	0.903219i	$0.358803\pi$
$678$	972.000	0.0550582
$679$	−6370.00	−0.360027
$680$	0	0
$681$	−2853.00	−0.160539
$682$	−22092.0	−1.24039
$683$	−14982.0	−0.839342	−0.419671	−	0.907676i	$-0.637854\pi$
$683$	−14982.0	−0.839342	−0.419671	+	0.907676i	$0.637854\pi$
$684$	2016.00	0.112695
$685$	0	0
$686$	−686.000	−0.0381802
$687$	−14478.0	−0.804032
$688$	−4912.00	−0.272192
$689$	9729.00	0.537947
$690$	0	0
$691$	29078.0	1.60084	0.800419	−	0.599441i	$-0.204610\pi$
$691$	29078.0	1.60084	0.800419	+	0.599441i	$0.204610\pi$
$692$	8544.00	0.469356
$693$	2646.00	0.145041
$694$	11364.0	0.621573
$695$	0	0
$696$	4536.00	0.247035
$697$	819.000	0.0445077
$698$	5198.00	0.281873
$699$	−9720.00	−0.525957
$700$	0	0
$701$	621.000	0.0334591	0.0167296	−	0.999860i	$-0.494675\pi$
$701$	621.000	0.0334591	0.0167296	+	0.999860i	$0.494675\pi$
$702$	2538.00	0.136454
$703$	−3248.00	−0.174254
$704$	2688.00	0.143903
$705$	0	0
$706$	10020.0	0.534147
$707$	0	0
$708$	6084.00	0.322953
$709$	6776.00	0.358925	0.179463	−	0.983765i	$-0.442564\pi$
$709$	6776.00	0.358925	0.179463	+	0.983765i	$0.442564\pi$
$710$	0	0
$711$	1638.00	0.0863992
$712$	6480.00	0.341079
$713$	2367.00	0.124327
$714$	−126.000	−0.00660425
$715$	0	0
$716$	−9720.00	−0.507337
$717$	288.000	0.0150008
$718$	16302.0	0.847333
$719$	11394.0	0.590994	0.295497	−	0.955344i	$-0.404515\pi$
$719$	11394.0	0.590994	0.295497	+	0.955344i	$0.404515\pi$
$720$	0	0
$721$	4277.00	0.220921
$722$	7446.00	0.383811
$723$	9138.00	0.470050
$724$	11048.0	0.567121
$725$	0	0
$726$	2598.00	0.132811
$727$	18947.0	0.966582	0.483291	−	0.875460i	$-0.339441\pi$
$727$	18947.0	0.966582	0.483291	+	0.875460i	$0.339441\pi$
$728$	−2632.00	−0.133995
$729$	729.000	0.0370370
$730$	0	0
$731$	921.000	0.0465997
$732$	−7620.00	−0.384759
$733$	−11587.0	−0.583868	−0.291934	−	0.956438i	$-0.594299\pi$
$733$	−11587.0	−0.583868	−0.291934	+	0.956438i	$0.594299\pi$
$734$	1742.00	0.0876000
$735$	0	0
$736$	−288.000	−0.0144237
$737$	−23352.0	−1.16714
$738$	4914.00	0.245104
$739$	1769.00	0.0880565	0.0440282	−	0.999030i	$-0.485981\pi$
$739$	1769.00	0.0880565	0.0440282	+	0.999030i	$0.485981\pi$
$740$	0	0
$741$	−7896.00	−0.391453
$742$	−2898.00	−0.143381
$743$	26259.0	1.29657	0.648283	−	0.761399i	$-0.275487\pi$
$743$	26259.0	1.29657	0.648283	+	0.761399i	$0.275487\pi$
$744$	6312.00	0.311034
$745$	0	0
$746$	1256.00	0.0616427
$747$	−2619.00	−0.128279
$748$	−504.000	−0.0246365
$749$	−14658.0	−0.715076
$750$	0	0
$751$	−31060.0	−1.50918	−0.754591	−	0.656196i	$-0.772165\pi$
$751$	−31060.0	−1.50918	−0.754591	+	0.656196i	$0.772165\pi$
$752$	−2496.00	−0.121037
$753$	3285.00	0.158980
$754$	−17766.0	−0.858090
$755$	0	0
$756$	−756.000	−0.0363696
$757$	−29686.0	−1.42531	−0.712653	−	0.701517i	$-0.752506\pi$
$757$	−29686.0	−1.42531	−0.712653	+	0.701517i	$0.752506\pi$
$758$	−22450.0	−1.07575
$759$	−1134.00	−0.0542313
$760$	0	0
$761$	−11346.0	−0.540463	−0.270231	−	0.962795i	$-0.587100\pi$
$761$	−11346.0	−0.540463	−0.270231	+	0.962795i	$0.587100\pi$
$762$	11532.0	0.548242
$763$	−1372.00	−0.0650979
$764$	17748.0	0.840445
$765$	0	0
$766$	−5520.00	−0.260373
$767$	−23829.0	−1.12179
$768$	−768.000	−0.0360844
$769$	−11560.0	−0.542086	−0.271043	−	0.962567i	$-0.587369\pi$
$769$	−11560.0	−0.542086	−0.271043	+	0.962567i	$0.587369\pi$
$770$	0	0
$771$	−10755.0	−0.502376
$772$	12776.0	0.595620
$773$	−34860.0	−1.62203	−0.811014	−	0.585027i	$-0.801084\pi$
$773$	−34860.0	−1.62203	−0.811014	+	0.585027i	$0.801084\pi$
$774$	5526.00	0.256625
$775$	0	0
$776$	7280.00	0.336774
$777$	1218.00	0.0562362
$778$	16644.0	0.766988
$779$	−15288.0	−0.703144
$780$	0	0
$781$	28728.0	1.31622
$782$	54.0000	0.00246936
$783$	−5103.00	−0.232907
$784$	784.000	0.0357143
$785$	0	0
$786$	14688.0	0.666544
$787$	−13306.0	−0.602678	−0.301339	−	0.953517i	$-0.597434\pi$
$787$	−13306.0	−0.602678	−0.301339	+	0.953517i	$0.597434\pi$
$788$	−14916.0	−0.674315
$789$	−11151.0	−0.503151
$790$	0	0
$791$	1134.00	0.0509740
$792$	−3024.00	−0.135673
$793$	29845.0	1.33648
$794$	−16618.0	−0.742759
$795$	0	0
$796$	15296.0	0.681096
$797$	−6414.00	−0.285063	−0.142532	−	0.989790i	$-0.545524\pi$
$797$	−6414.00	−0.285063	−0.142532	+	0.989790i	$0.545524\pi$
$798$	2352.00	0.104336
$799$	468.000	0.0207217
$800$	0	0
$801$	−7290.00	−0.321572
$802$	−16152.0	−0.711156
$803$	20244.0	0.889658
$804$	6672.00	0.292666
$805$	0	0
$806$	−24722.0	−1.08039
$807$	−846.000	−0.0369029
$808$	0	0
$809$	36420.0	1.58277	0.791384	−	0.611320i	$-0.209361\pi$
$809$	36420.0	1.58277	0.791384	+	0.611320i	$0.209361\pi$
$810$	0	0
$811$	−34954.0	−1.51344	−0.756721	−	0.653738i	$-0.773200\pi$
$811$	−34954.0	−1.51344	−0.756721	+	0.653738i	$0.773200\pi$
$812$	5292.00	0.228710
$813$	12900.0	0.556486
$814$	4872.00	0.209783
$815$	0	0
$816$	144.000	0.00617771
$817$	−17192.0	−0.736196
$818$	17852.0	0.763057
$819$	2961.00	0.126332
$820$	0	0
$821$	42042.0	1.78718	0.893591	−	0.448883i	$-0.148178\pi$
$821$	42042.0	1.78718	0.893591	+	0.448883i	$0.148178\pi$
$822$	−2448.00	−0.103873
$823$	24254.0	1.02727	0.513634	−	0.858009i	$-0.328299\pi$
$823$	24254.0	1.02727	0.513634	+	0.858009i	$0.328299\pi$
$824$	−4888.00	−0.206652
$825$	0	0
$826$	7098.00	0.298996
$827$	−31650.0	−1.33081	−0.665404	−	0.746483i	$-0.731741\pi$
$827$	−31650.0	−1.33081	−0.665404	+	0.746483i	$0.731741\pi$
$828$	324.000	0.0135988
$829$	29477.0	1.23496	0.617478	−	0.786588i	$-0.288154\pi$
$829$	29477.0	1.23496	0.617478	+	0.786588i	$0.288154\pi$
$830$	0	0
$831$	15978.0	0.666992
$832$	3008.00	0.125341
$833$	−147.000	−0.00611434
$834$	−5856.00	−0.243138
$835$	0	0
$836$	9408.00	0.389213
$837$	−7101.00	−0.293245
$838$	28770.0	1.18597
$839$	29340.0	1.20731	0.603653	−	0.797247i	$-0.293711\pi$
$839$	29340.0	1.20731	0.603653	+	0.797247i	$0.293711\pi$
$840$	0	0
$841$	11332.0	0.464636
$842$	23132.0	0.946771
$843$	1440.00	0.0588330
$844$	4820.00	0.196577
$845$	0	0
$846$	2808.00	0.114115
$847$	3031.00	0.122959
$848$	3312.00	0.134121
$849$	−11400.0	−0.460833
$850$	0	0
$851$	−522.000	−0.0210269
$852$	−8208.00	−0.330049
$853$	−6379.00	−0.256053	−0.128026	−	0.991771i	$-0.540864\pi$
$853$	−6379.00	−0.256053	−0.128026	+	0.991771i	$0.540864\pi$
$854$	−8890.00	−0.356217
$855$	0	0
$856$	16752.0	0.668892
$857$	−42138.0	−1.67959	−0.839794	−	0.542905i	$-0.817324\pi$
$857$	−42138.0	−1.67959	−0.839794	+	0.542905i	$0.817324\pi$
$858$	11844.0	0.471267
$859$	13880.0	0.551315	0.275657	−	0.961256i	$-0.411104\pi$
$859$	13880.0	0.551315	0.275657	+	0.961256i	$0.411104\pi$
$860$	0	0
$861$	5733.00	0.226922
$862$	−27114.0	−1.07135
$863$	29544.0	1.16534	0.582671	−	0.812708i	$-0.302008\pi$
$863$	29544.0	1.16534	0.582671	+	0.812708i	$0.302008\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−6316.00	−0.247837
$867$	14712.0	0.576293
$868$	7364.00	0.287961
$869$	7644.00	0.298395
$870$	0	0
$871$	−26132.0	−1.01659
$872$	1568.00	0.0608936
$873$	−8190.00	−0.317514
$874$	−1008.00	−0.0390116
$875$	0	0
$876$	−5784.00	−0.223086
$877$	23096.0	0.889278	0.444639	−	0.895710i	$-0.353332\pi$
$877$	23096.0	0.889278	0.444639	+	0.895710i	$0.353332\pi$
$878$	−22966.0	−0.882762
$879$	18720.0	0.718328
$880$	0	0
$881$	37095.0	1.41857	0.709286	−	0.704921i	$-0.249017\pi$
$881$	37095.0	1.41857	0.709286	+	0.704921i	$0.249017\pi$
$882$	−882.000	−0.0336718
$883$	−19177.0	−0.730869	−0.365435	−	0.930837i	$-0.619080\pi$
$883$	−19177.0	−0.730869	−0.365435	+	0.930837i	$0.619080\pi$
$884$	−564.000	−0.0214586
$885$	0	0
$886$	16968.0	0.643399
$887$	−5850.00	−0.221447	−0.110724	−	0.993851i	$-0.535317\pi$
$887$	−5850.00	−0.221447	−0.110724	+	0.993851i	$0.535317\pi$
$888$	−1392.00	−0.0526041
$889$	13454.0	0.507573
$890$	0	0
$891$	3402.00	0.127914
$892$	1916.00	0.0719198
$893$	−8736.00	−0.327367
$894$	−7290.00	−0.272723
$895$	0	0
$896$	−896.000	−0.0334077
$897$	−1269.00	−0.0472360
$898$	25608.0	0.951615
$899$	49707.0	1.84407
$900$	0	0
$901$	−621.000	−0.0229617
$902$	22932.0	0.846510
$903$	6447.00	0.237589
$904$	−1296.00	−0.0476818
$905$	0	0
$906$	8292.00	0.304065
$907$	1127.00	0.0412585	0.0206292	−	0.999787i	$-0.493433\pi$
$907$	1127.00	0.0412585	0.0206292	+	0.999787i	$0.493433\pi$
$908$	3804.00	0.139031
$909$	0	0
$910$	0	0
$911$	−41949.0	−1.52561	−0.762806	−	0.646628i	$-0.776179\pi$
$911$	−41949.0	−1.52561	−0.762806	+	0.646628i	$0.776179\pi$
$912$	−2688.00	−0.0975971
$913$	−12222.0	−0.443033
$914$	−20878.0	−0.755561
$915$	0	0
$916$	19304.0	0.696312
$917$	17136.0	0.617100
$918$	−162.000	−0.00582440
$919$	−7600.00	−0.272797	−0.136399	−	0.990654i	$-0.543553\pi$
$919$	−7600.00	−0.272797	−0.136399	+	0.990654i	$0.543553\pi$
$920$	0	0
$921$	−30282.0	−1.08342
$922$	17376.0	0.620660
$923$	32148.0	1.14644
$924$	−3528.00	−0.125609
$925$	0	0
$926$	26816.0	0.951651
$927$	5499.00	0.194834
$928$	−6048.00	−0.213939
$929$	−33525.0	−1.18398	−0.591991	−	0.805944i	$-0.701658\pi$
$929$	−33525.0	−1.18398	−0.591991	+	0.805944i	$0.701658\pi$
$930$	0	0
$931$	2744.00	0.0965961
$932$	12960.0	0.455492
$933$	10998.0	0.385915
$934$	−23346.0	−0.817885
$935$	0	0
$936$	−3384.00	−0.118173
$937$	−24622.0	−0.858448	−0.429224	−	0.903198i	$-0.641213\pi$
$937$	−24622.0	−0.858448	−0.429224	+	0.903198i	$0.641213\pi$
$938$	7784.00	0.270956
$939$	4242.00	0.147425
$940$	0	0
$941$	29040.0	1.00603	0.503016	−	0.864277i	$-0.332224\pi$
$941$	29040.0	1.00603	0.503016	+	0.864277i	$0.332224\pi$
$942$	17364.0	0.600584
$943$	−2457.00	−0.0848472
$944$	−8112.00	−0.279685
$945$	0	0
$946$	25788.0	0.886300
$947$	55524.0	1.90527	0.952633	−	0.304121i	$-0.0983628\pi$
$947$	55524.0	1.90527	0.952633	+	0.304121i	$0.0983628\pi$
$948$	−2184.00	−0.0748239
$949$	22654.0	0.774900
$950$	0	0
$951$	−2097.00	−0.0715036
$952$	168.000	0.00571944
$953$	−46800.0	−1.59077	−0.795383	−	0.606107i	$-0.792730\pi$
$953$	−46800.0	−1.59077	−0.795383	+	0.606107i	$0.792730\pi$
$954$	−3726.00	−0.126450
$955$	0	0
$956$	−384.000	−0.0129911
$957$	−23814.0	−0.804386
$958$	14136.0	0.476736
$959$	−2856.00	−0.0961679
$960$	0	0
$961$	39378.0	1.32181
$962$	5452.00	0.182723
$963$	−18846.0	−0.630637
$964$	−12184.0	−0.407075
$965$	0	0
$966$	378.000	0.0125900
$967$	−53992.0	−1.79552	−0.897759	−	0.440487i	$-0.854806\pi$
$967$	−53992.0	−1.79552	−0.897759	+	0.440487i	$0.854806\pi$
$968$	−3464.00	−0.115018
$969$	504.000	0.0167088
$970$	0	0
$971$	20916.0	0.691273	0.345637	−	0.938368i	$-0.387663\pi$
$971$	20916.0	0.691273	0.345637	+	0.938368i	$0.387663\pi$
$972$	−972.000	−0.0320750
$973$	−6832.00	−0.225102
$974$	−19252.0	−0.633341
$975$	0	0
$976$	10160.0	0.333211
$977$	−27594.0	−0.903593	−0.451796	−	0.892121i	$-0.649217\pi$
$977$	−27594.0	−0.903593	−0.451796	+	0.892121i	$0.649217\pi$
$978$	20442.0	0.668367
$979$	−34020.0	−1.11061
$980$	0	0
$981$	−1764.00	−0.0574110
$982$	21180.0	0.688270
$983$	19368.0	0.628427	0.314213	−	0.949352i	$-0.398259\pi$
$983$	19368.0	0.628427	0.314213	+	0.949352i	$0.398259\pi$
$984$	−6552.00	−0.212266
$985$	0	0
$986$	1134.00	0.0366267
$987$	3276.00	0.105650
$988$	10528.0	0.339008
$989$	−2763.00	−0.0888355
$990$	0	0
$991$	−2476.00	−0.0793670	−0.0396835	−	0.999212i	$-0.512635\pi$
$991$	−2476.00	−0.0793670	−0.0396835	+	0.999212i	$0.512635\pi$
$992$	−8416.00	−0.269363
$993$	12819.0	0.409666
$994$	−9576.00	−0.305566
$995$	0	0
$996$	3492.00	0.111093
$997$	10154.0	0.322548	0.161274	−	0.986910i	$-0.448440\pi$
$997$	10154.0	0.322548	0.161274	+	0.986910i	$0.448440\pi$
$998$	−8014.00	−0.254187
$999$	1566.00	0.0495956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.e.1.1	✓	1
5.2	odd	4		1050.4.g.q.799.1		2
5.3	odd	4		1050.4.g.q.799.2		2
5.4	even	2		1050.4.a.v.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.4.a.e.1.1	✓	1	1.1	even	1	trivial
1050.4.a.v.1.1	yes	1	5.4	even	2
1050.4.g.q.799.1		2	5.2	odd	4
1050.4.g.q.799.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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