LMFDB - Embedded newform 1050.4.a.w.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -44.0000 q^{11} +12.0000 q^{12} -54.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -98.0000 q^{17} +18.0000 q^{18} -60.0000 q^{19} +21.0000 q^{21} -88.0000 q^{22} +144.000 q^{23} +24.0000 q^{24} -108.000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -210.000 q^{29} -208.000 q^{31} +32.0000 q^{32} -132.000 q^{33} -196.000 q^{34} +36.0000 q^{36} +226.000 q^{37} -120.000 q^{38} -162.000 q^{39} -502.000 q^{41} +42.0000 q^{42} -484.000 q^{43} -176.000 q^{44} +288.000 q^{46} +232.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -294.000 q^{51} -216.000 q^{52} +530.000 q^{53} +54.0000 q^{54} +56.0000 q^{56} -180.000 q^{57} -420.000 q^{58} -764.000 q^{59} +814.000 q^{61} -416.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -264.000 q^{66} -60.0000 q^{67} -392.000 q^{68} +432.000 q^{69} +848.000 q^{71} +72.0000 q^{72} +958.000 q^{73} +452.000 q^{74} -240.000 q^{76} -308.000 q^{77} -324.000 q^{78} -152.000 q^{79} +81.0000 q^{81} -1004.00 q^{82} -308.000 q^{83} +84.0000 q^{84} -968.000 q^{86} -630.000 q^{87} -352.000 q^{88} -1094.00 q^{89} -378.000 q^{91} +576.000 q^{92} -624.000 q^{93} +464.000 q^{94} +96.0000 q^{96} -554.000 q^{97} +98.0000 q^{98} -396.000 q^{99} +O(q^{100})\)

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
$11$	−44.0000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−44.0000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	12.0000	0.288675
$13$	−54.0000	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−98.0000	−1.39815	−0.699073	−	0.715050i	$-0.746404\pi$
$17$	−98.0000	−1.39815	−0.699073	+	0.715050i	$0.746404\pi$
$18$	18.0000	0.235702
$19$	−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
$21$	21.0000	0.218218
$22$	−88.0000	−0.852803
$23$	144.000	1.30548	0.652741	−	0.757581i	$-0.273619\pi$
$23$	144.000	1.30548	0.652741	+	0.757581i	$0.273619\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−108.000	−0.814636
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−208.000	−1.20509	−0.602547	−	0.798084i	$-0.705847\pi$
$31$	−208.000	−1.20509	−0.602547	+	0.798084i	$0.705847\pi$
$32$	32.0000	0.176777
$33$	−132.000	−0.696311
$34$	−196.000	−0.988639
$35$	0	0
$36$	36.0000	0.166667
$37$	226.000	1.00417	0.502083	−	0.864819i	$-0.332567\pi$
$37$	226.000	1.00417	0.502083	+	0.864819i	$0.332567\pi$
$38$	−120.000	−0.512278
$39$	−162.000	−0.665148
$40$	0	0
$41$	−502.000	−1.91218	−0.956088	−	0.293079i	$-0.905320\pi$
$41$	−502.000	−1.91218	−0.956088	+	0.293079i	$0.905320\pi$
$42$	42.0000	0.154303
$43$	−484.000	−1.71650	−0.858248	−	0.513236i	$-0.828447\pi$
$43$	−484.000	−1.71650	−0.858248	+	0.513236i	$0.828447\pi$
$44$	−176.000	−0.603023
$45$	0	0
$46$	288.000	0.923115
$47$	232.000	0.720014	0.360007	−	0.932950i	$-0.382774\pi$
$47$	232.000	0.720014	0.360007	+	0.932950i	$0.382774\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	−294.000	−0.807220
$52$	−216.000	−0.576035
$53$	530.000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	530.000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	56.0000	0.133631
$57$	−180.000	−0.418273
$58$	−420.000	−0.950840
$59$	−764.000	−1.68584	−0.842918	−	0.538042i	$-0.819164\pi$
$59$	−764.000	−1.68584	−0.842918	+	0.538042i	$0.819164\pi$
$60$	0	0
$61$	814.000	1.70856	0.854279	−	0.519815i	$-0.173999\pi$
$61$	814.000	1.70856	0.854279	+	0.519815i	$0.173999\pi$
$62$	−416.000	−0.852130
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	−264.000	−0.492366
$67$	−60.0000	−0.109405	−0.0547027	−	0.998503i	$-0.517421\pi$
$67$	−60.0000	−0.109405	−0.0547027	+	0.998503i	$0.517421\pi$
$68$	−392.000	−0.699073
$69$	432.000	0.753720
$70$	0	0
$71$	848.000	1.41745	0.708726	−	0.705484i	$-0.249270\pi$
$71$	848.000	1.41745	0.708726	+	0.705484i	$0.249270\pi$
$72$	72.0000	0.117851
$73$	958.000	1.53596	0.767982	−	0.640471i	$-0.221261\pi$
$73$	958.000	1.53596	0.767982	+	0.640471i	$0.221261\pi$
$74$	452.000	0.710053
$75$	0	0
$76$	−240.000	−0.362235
$77$	−308.000	−0.455842
$78$	−324.000	−0.470330
$79$	−152.000	−0.216473	−0.108236	−	0.994125i	$-0.534520\pi$
$79$	−152.000	−0.216473	−0.108236	+	0.994125i	$0.534520\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−1004.00	−1.35211
$83$	−308.000	−0.407318	−0.203659	−	0.979042i	$-0.565283\pi$
$83$	−308.000	−0.407318	−0.203659	+	0.979042i	$0.565283\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	−968.000	−1.21375
$87$	−630.000	−0.776357
$88$	−352.000	−0.426401
$89$	−1094.00	−1.30296	−0.651482	−	0.758664i	$-0.725852\pi$
$89$	−1094.00	−1.30296	−0.651482	+	0.758664i	$0.725852\pi$
$90$	0	0
$91$	−378.000	−0.435441
$92$	576.000	0.652741
$93$	−624.000	−0.695761
$94$	464.000	0.509127
$95$	0	0
$96$	96.0000	0.102062
$97$	−554.000	−0.579899	−0.289949	−	0.957042i	$-0.593638\pi$
$97$	−554.000	−0.579899	−0.289949	+	0.957042i	$0.593638\pi$
$98$	98.0000	0.101015
$99$	−396.000	−0.402015
$100$	0	0
$101$	1134.00	1.11720	0.558600	−	0.829437i	$-0.311339\pi$
$101$	1134.00	1.11720	0.558600	+	0.829437i	$0.311339\pi$
$102$	−588.000	−0.570791
$103$	488.000	0.466836	0.233418	−	0.972377i	$-0.425009\pi$
$103$	488.000	0.466836	0.233418	+	0.972377i	$0.425009\pi$
$104$	−432.000	−0.407318
$105$	0	0
$106$	1060.00	0.971286
$107$	828.000	0.748091	0.374046	−	0.927410i	$-0.377970\pi$
$107$	828.000	0.748091	0.374046	+	0.927410i	$0.377970\pi$
$108$	108.000	0.0962250
$109$	−42.0000	−0.0369071	−0.0184535	−	0.999830i	$-0.505874\pi$
$109$	−42.0000	−0.0369071	−0.0184535	+	0.999830i	$0.505874\pi$
$110$	0	0
$111$	678.000	0.579756
$112$	112.000	0.0944911
$113$	−2138.00	−1.77988	−0.889939	−	0.456080i	$-0.849253\pi$
$113$	−2138.00	−1.77988	−0.889939	+	0.456080i	$0.849253\pi$
$114$	−360.000	−0.295764
$115$	0	0
$116$	−840.000	−0.672345
$117$	−486.000	−0.384023
$118$	−1528.00	−1.19207
$119$	−686.000	−0.528450
$120$	0	0
$121$	605.000	0.454545
$122$	1628.00	1.20813
$123$	−1506.00	−1.10400
$124$	−832.000	−0.602547
$125$	0	0
$126$	126.000	0.0890871
$127$	−1072.00	−0.749013	−0.374506	−	0.927224i	$-0.622188\pi$
$127$	−1072.00	−0.749013	−0.374506	+	0.927224i	$0.622188\pi$
$128$	128.000	0.0883883
$129$	−1452.00	−0.991019
$130$	0	0
$131$	1004.00	0.669617	0.334809	−	0.942286i	$-0.391328\pi$
$131$	1004.00	0.669617	0.334809	+	0.942286i	$0.391328\pi$
$132$	−528.000	−0.348155
$133$	−420.000	−0.273824
$134$	−120.000	−0.0773614
$135$	0	0
$136$	−784.000	−0.494319
$137$	−642.000	−0.400363	−0.200182	−	0.979759i	$-0.564153\pi$
$137$	−642.000	−0.400363	−0.200182	+	0.979759i	$0.564153\pi$
$138$	864.000	0.532961
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	0	0
$141$	696.000	0.415701
$142$	1696.00	1.00229
$143$	2376.00	1.38945
$144$	144.000	0.0833333
$145$	0	0
$146$	1916.00	1.08609
$147$	147.000	0.0824786
$148$	904.000	0.502083
$149$	−2394.00	−1.31627	−0.658135	−	0.752900i	$-0.728654\pi$
$149$	−2394.00	−1.31627	−0.658135	+	0.752900i	$0.728654\pi$
$150$	0	0
$151$	800.000	0.431146	0.215573	−	0.976488i	$-0.430838\pi$
$151$	800.000	0.431146	0.215573	+	0.976488i	$0.430838\pi$
$152$	−480.000	−0.256139
$153$	−882.000	−0.466049
$154$	−616.000	−0.322329
$155$	0	0
$156$	−648.000	−0.332574
$157$	890.000	0.452419	0.226209	−	0.974079i	$-0.427367\pi$
$157$	890.000	0.452419	0.226209	+	0.974079i	$0.427367\pi$
$158$	−304.000	−0.153069
$159$	1590.00	0.793052
$160$	0	0
$161$	1008.00	0.493426
$162$	162.000	0.0785674
$163$	−668.000	−0.320993	−0.160496	−	0.987036i	$-0.551309\pi$
$163$	−668.000	−0.320993	−0.160496	+	0.987036i	$0.551309\pi$
$164$	−2008.00	−0.956088
$165$	0	0
$166$	−616.000	−0.288017
$167$	−816.000	−0.378108	−0.189054	−	0.981967i	$-0.560542\pi$
$167$	−816.000	−0.378108	−0.189054	+	0.981967i	$0.560542\pi$
$168$	168.000	0.0771517
$169$	719.000	0.327264
$170$	0	0
$171$	−540.000	−0.241490
$172$	−1936.00	−0.858248
$173$	−1334.00	−0.586255	−0.293128	−	0.956073i	$-0.594696\pi$
$173$	−1334.00	−0.586255	−0.293128	+	0.956073i	$0.594696\pi$
$174$	−1260.00	−0.548968
$175$	0	0
$176$	−704.000	−0.301511
$177$	−2292.00	−0.973318
$178$	−2188.00	−0.921334
$179$	−1236.00	−0.516106	−0.258053	−	0.966131i	$-0.583081\pi$
$179$	−1236.00	−0.516106	−0.258053	+	0.966131i	$0.583081\pi$
$180$	0	0
$181$	358.000	0.147016	0.0735081	−	0.997295i	$-0.476581\pi$
$181$	358.000	0.147016	0.0735081	+	0.997295i	$0.476581\pi$
$182$	−756.000	−0.307904
$183$	2442.00	0.986436
$184$	1152.00	0.461557
$185$	0	0
$186$	−1248.00	−0.491977
$187$	4312.00	1.68623
$188$	928.000	0.360007
$189$	189.000	0.0727393
$190$	0	0
$191$	−888.000	−0.336405	−0.168203	−	0.985752i	$-0.553796\pi$
$191$	−888.000	−0.336405	−0.168203	+	0.985752i	$0.553796\pi$
$192$	192.000	0.0721688
$193$	4030.00	1.50303	0.751517	−	0.659713i	$-0.229322\pi$
$193$	4030.00	1.50303	0.751517	+	0.659713i	$0.229322\pi$
$194$	−1108.00	−0.410050
$195$	0	0
$196$	196.000	0.0714286
$197$	1714.00	0.619886	0.309943	−	0.950755i	$-0.399690\pi$
$197$	1714.00	0.619886	0.309943	+	0.950755i	$0.399690\pi$
$198$	−792.000	−0.284268
$199$	−4264.00	−1.51893	−0.759465	−	0.650549i	$-0.774539\pi$
$199$	−4264.00	−1.51893	−0.759465	+	0.650549i	$0.774539\pi$
$200$	0	0
$201$	−180.000	−0.0631653
$202$	2268.00	0.789980
$203$	−1470.00	−0.508245
$204$	−1176.00	−0.403610
$205$	0	0
$206$	976.000	0.330103
$207$	1296.00	0.435161
$208$	−864.000	−0.288017
$209$	2640.00	0.873745
$210$	0	0
$211$	3316.00	1.08191	0.540955	−	0.841052i	$-0.318063\pi$
$211$	3316.00	1.08191	0.540955	+	0.841052i	$0.318063\pi$
$212$	2120.00	0.686803
$213$	2544.00	0.818366
$214$	1656.00	0.528981
$215$	0	0
$216$	216.000	0.0680414
$217$	−1456.00	−0.455483
$218$	−84.0000	−0.0260972
$219$	2874.00	0.886790
$220$	0	0
$221$	5292.00	1.61076
$222$	1356.00	0.409949
$223$	−2128.00	−0.639020	−0.319510	−	0.947583i	$-0.603518\pi$
$223$	−2128.00	−0.639020	−0.319510	+	0.947583i	$0.603518\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−4276.00	−1.25856
$227$	−980.000	−0.286541	−0.143271	−	0.989684i	$-0.545762\pi$
$227$	−980.000	−0.286541	−0.143271	+	0.989684i	$0.545762\pi$
$228$	−720.000	−0.209137
$229$	4102.00	1.18370	0.591851	−	0.806047i	$-0.298397\pi$
$229$	4102.00	1.18370	0.591851	+	0.806047i	$0.298397\pi$
$230$	0	0
$231$	−924.000	−0.263181
$232$	−1680.00	−0.475420
$233$	−2466.00	−0.693361	−0.346680	−	0.937983i	$-0.612691\pi$
$233$	−2466.00	−0.693361	−0.346680	+	0.937983i	$0.612691\pi$
$234$	−972.000	−0.271545
$235$	0	0
$236$	−3056.00	−0.842918
$237$	−456.000	−0.124981
$238$	−1372.00	−0.373670
$239$	1192.00	0.322611	0.161306	−	0.986905i	$-0.448430\pi$
$239$	1192.00	0.322611	0.161306	+	0.986905i	$0.448430\pi$
$240$	0	0
$241$	−3854.00	−1.03012	−0.515058	−	0.857155i	$-0.672230\pi$
$241$	−3854.00	−1.03012	−0.515058	+	0.857155i	$0.672230\pi$
$242$	1210.00	0.321412
$243$	243.000	0.0641500
$244$	3256.00	0.854279
$245$	0	0
$246$	−3012.00	−0.780643
$247$	3240.00	0.834641
$248$	−1664.00	−0.426065
$249$	−924.000	−0.235165
$250$	0	0
$251$	4020.00	1.01092	0.505458	−	0.862851i	$-0.331323\pi$
$251$	4020.00	1.01092	0.505458	+	0.862851i	$0.331323\pi$
$252$	252.000	0.0629941
$253$	−6336.00	−1.57447
$254$	−2144.00	−0.529632
$255$	0	0
$256$	256.000	0.0625000
$257$	−1922.00	−0.466502	−0.233251	−	0.972417i	$-0.574936\pi$
$257$	−1922.00	−0.466502	−0.233251	+	0.972417i	$0.574936\pi$
$258$	−2904.00	−0.700756
$259$	1582.00	0.379539
$260$	0	0
$261$	−1890.00	−0.448230
$262$	2008.00	0.473491
$263$	−2480.00	−0.581458	−0.290729	−	0.956805i	$-0.593898\pi$
$263$	−2480.00	−0.581458	−0.290729	+	0.956805i	$0.593898\pi$
$264$	−1056.00	−0.246183
$265$	0	0
$266$	−840.000	−0.193623
$267$	−3282.00	−0.752266
$268$	−240.000	−0.0547027
$269$	−1002.00	−0.227112	−0.113556	−	0.993532i	$-0.536224\pi$
$269$	−1002.00	−0.227112	−0.113556	+	0.993532i	$0.536224\pi$
$270$	0	0
$271$	−2832.00	−0.634804	−0.317402	−	0.948291i	$-0.602810\pi$
$271$	−2832.00	−0.634804	−0.317402	+	0.948291i	$0.602810\pi$
$272$	−1568.00	−0.349537
$273$	−1134.00	−0.251402
$274$	−1284.00	−0.283100
$275$	0	0
$276$	1728.00	0.376860
$277$	−4206.00	−0.912325	−0.456163	−	0.889896i	$-0.650776\pi$
$277$	−4206.00	−0.912325	−0.456163	+	0.889896i	$0.650776\pi$
$278$	−3208.00	−0.692097
$279$	−1872.00	−0.401698
$280$	0	0
$281$	5978.00	1.26910	0.634551	−	0.772881i	$-0.281185\pi$
$281$	5978.00	1.26910	0.634551	+	0.772881i	$0.281185\pi$
$282$	1392.00	0.293945
$283$	−2684.00	−0.563771	−0.281886	−	0.959448i	$-0.590960\pi$
$283$	−2684.00	−0.563771	−0.281886	+	0.959448i	$0.590960\pi$
$284$	3392.00	0.708726
$285$	0	0
$286$	4752.00	0.982488
$287$	−3514.00	−0.722735
$288$	288.000	0.0589256
$289$	4691.00	0.954814
$290$	0	0
$291$	−1662.00	−0.334805
$292$	3832.00	0.767982
$293$	7490.00	1.49341	0.746707	−	0.665153i	$-0.231634\pi$
$293$	7490.00	1.49341	0.746707	+	0.665153i	$0.231634\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	1808.00	0.355027
$297$	−1188.00	−0.232104
$298$	−4788.00	−0.930743
$299$	−7776.00	−1.50401
$300$	0	0
$301$	−3388.00	−0.648774
$302$	1600.00	0.304866
$303$	3402.00	0.645016
$304$	−960.000	−0.181118
$305$	0	0
$306$	−1764.00	−0.329546
$307$	6860.00	1.27531	0.637656	−	0.770321i	$-0.279904\pi$
$307$	6860.00	1.27531	0.637656	+	0.770321i	$0.279904\pi$
$308$	−1232.00	−0.227921
$309$	1464.00	0.269528
$310$	0	0
$311$	−4920.00	−0.897066	−0.448533	−	0.893766i	$-0.648053\pi$
$311$	−4920.00	−0.897066	−0.448533	+	0.893766i	$0.648053\pi$
$312$	−1296.00	−0.235165
$313$	4558.00	0.823110	0.411555	−	0.911385i	$-0.364986\pi$
$313$	4558.00	0.823110	0.411555	+	0.911385i	$0.364986\pi$
$314$	1780.00	0.319908
$315$	0	0
$316$	−608.000	−0.108236
$317$	−5830.00	−1.03295	−0.516475	−	0.856302i	$-0.672756\pi$
$317$	−5830.00	−1.03295	−0.516475	+	0.856302i	$0.672756\pi$
$318$	3180.00	0.560772
$319$	9240.00	1.62176
$320$	0	0
$321$	2484.00	0.431911
$322$	2016.00	0.348905
$323$	5880.00	1.01292
$324$	324.000	0.0555556
$325$	0	0
$326$	−1336.00	−0.226976
$327$	−126.000	−0.0213083
$328$	−4016.00	−0.676056
$329$	1624.00	0.272140
$330$	0	0
$331$	5020.00	0.833608	0.416804	−	0.908996i	$-0.363150\pi$
$331$	5020.00	0.833608	0.416804	+	0.908996i	$0.363150\pi$
$332$	−1232.00	−0.203659
$333$	2034.00	0.334722
$334$	−1632.00	−0.267362
$335$	0	0
$336$	336.000	0.0545545
$337$	3662.00	0.591934	0.295967	−	0.955198i	$-0.404358\pi$
$337$	3662.00	0.591934	0.295967	+	0.955198i	$0.404358\pi$
$338$	1438.00	0.231411
$339$	−6414.00	−1.02761
$340$	0	0
$341$	9152.00	1.45340
$342$	−1080.00	−0.170759
$343$	343.000	0.0539949
$344$	−3872.00	−0.606873
$345$	0	0
$346$	−2668.00	−0.414545
$347$	860.000	0.133047	0.0665234	−	0.997785i	$-0.478809\pi$
$347$	860.000	0.133047	0.0665234	+	0.997785i	$0.478809\pi$
$348$	−2520.00	−0.388179
$349$	−3458.00	−0.530380	−0.265190	−	0.964196i	$-0.585435\pi$
$349$	−3458.00	−0.530380	−0.265190	+	0.964196i	$0.585435\pi$
$350$	0	0
$351$	−1458.00	−0.221716
$352$	−1408.00	−0.213201
$353$	−2994.00	−0.451429	−0.225715	−	0.974193i	$-0.572472\pi$
$353$	−2994.00	−0.451429	−0.225715	+	0.974193i	$0.572472\pi$
$354$	−4584.00	−0.688240
$355$	0	0
$356$	−4376.00	−0.651482
$357$	−2058.00	−0.305101
$358$	−2472.00	−0.364942
$359$	1760.00	0.258744	0.129372	−	0.991596i	$-0.458704\pi$
$359$	1760.00	0.258744	0.129372	+	0.991596i	$0.458704\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	716.000	0.103956
$363$	1815.00	0.262432
$364$	−1512.00	−0.217721
$365$	0	0
$366$	4884.00	0.697516
$367$	608.000	0.0864778	0.0432389	−	0.999065i	$-0.486232\pi$
$367$	608.000	0.0864778	0.0432389	+	0.999065i	$0.486232\pi$
$368$	2304.00	0.326370
$369$	−4518.00	−0.637392
$370$	0	0
$371$	3710.00	0.519174
$372$	−2496.00	−0.347881
$373$	7090.00	0.984199	0.492100	−	0.870539i	$-0.336230\pi$
$373$	7090.00	0.984199	0.492100	+	0.870539i	$0.336230\pi$
$374$	8624.00	1.19234
$375$	0	0
$376$	1856.00	0.254564
$377$	11340.0	1.54918
$378$	378.000	0.0514344
$379$	14268.0	1.93377	0.966884	−	0.255217i	$-0.0821469\pi$
$379$	14268.0	1.93377	0.966884	+	0.255217i	$0.0821469\pi$
$380$	0	0
$381$	−3216.00	−0.432443
$382$	−1776.00	−0.237875
$383$	4104.00	0.547532	0.273766	−	0.961796i	$-0.411731\pi$
$383$	4104.00	0.547532	0.273766	+	0.961796i	$0.411731\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	8060.00	1.06281
$387$	−4356.00	−0.572165
$388$	−2216.00	−0.289949
$389$	15110.0	1.96943	0.984714	−	0.174180i	$-0.0557273\pi$
$389$	15110.0	1.96943	0.984714	+	0.174180i	$0.0557273\pi$
$390$	0	0
$391$	−14112.0	−1.82525
$392$	392.000	0.0505076
$393$	3012.00	0.386604
$394$	3428.00	0.438325
$395$	0	0
$396$	−1584.00	−0.201008
$397$	10682.0	1.35041	0.675207	−	0.737628i	$-0.264054\pi$
$397$	10682.0	1.35041	0.675207	+	0.737628i	$0.264054\pi$
$398$	−8528.00	−1.07405
$399$	−1260.00	−0.158092
$400$	0	0
$401$	−10526.0	−1.31083	−0.655416	−	0.755268i	$-0.727507\pi$
$401$	−10526.0	−1.31083	−0.655416	+	0.755268i	$0.727507\pi$
$402$	−360.000	−0.0446646
$403$	11232.0	1.38835
$404$	4536.00	0.558600
$405$	0	0
$406$	−2940.00	−0.359384
$407$	−9944.00	−1.21107
$408$	−2352.00	−0.285395
$409$	−6.00000	−0.000725381 0	−0.000362691	−	1.00000i	$-0.500115\pi$
$409$	−6.00000	−0.000725381 0	−0.000362691		1.00000i	$0.500115\pi$
$410$	0	0
$411$	−1926.00	−0.231150
$412$	1952.00	0.233418
$413$	−5348.00	−0.637186
$414$	2592.00	0.307705
$415$	0	0
$416$	−1728.00	−0.203659
$417$	−4812.00	−0.565095
$418$	5280.00	0.617831
$419$	−15108.0	−1.76151	−0.880757	−	0.473569i	$-0.842965\pi$
$419$	−15108.0	−1.76151	−0.880757	+	0.473569i	$0.842965\pi$
$420$	0	0
$421$	10094.0	1.16853	0.584265	−	0.811563i	$-0.301383\pi$
$421$	10094.0	1.16853	0.584265	+	0.811563i	$0.301383\pi$
$422$	6632.00	0.765025
$423$	2088.00	0.240005
$424$	4240.00	0.485643
$425$	0	0
$426$	5088.00	0.578672
$427$	5698.00	0.645774
$428$	3312.00	0.374046
$429$	7128.00	0.802198
$430$	0	0
$431$	−12696.0	−1.41890	−0.709449	−	0.704757i	$-0.751056\pi$
$431$	−12696.0	−1.41890	−0.709449	+	0.704757i	$0.751056\pi$
$432$	432.000	0.0481125
$433$	−12122.0	−1.34537	−0.672686	−	0.739928i	$-0.734860\pi$
$433$	−12122.0	−1.34537	−0.672686	+	0.739928i	$0.734860\pi$
$434$	−2912.00	−0.322075
$435$	0	0
$436$	−168.000	−0.0184535
$437$	−8640.00	−0.945783
$438$	5748.00	0.627055
$439$	−8552.00	−0.929760	−0.464880	−	0.885374i	$-0.653903\pi$
$439$	−8552.00	−0.929760	−0.464880	+	0.885374i	$0.653903\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	10584.0	1.13898
$443$	3548.00	0.380520	0.190260	−	0.981734i	$-0.439067\pi$
$443$	3548.00	0.380520	0.190260	+	0.981734i	$0.439067\pi$
$444$	2712.00	0.289878
$445$	0	0
$446$	−4256.00	−0.451855
$447$	−7182.00	−0.759948
$448$	448.000	0.0472456
$449$	−16686.0	−1.75381	−0.876905	−	0.480663i	$-0.840396\pi$
$449$	−16686.0	−1.75381	−0.876905	+	0.480663i	$0.840396\pi$
$450$	0	0
$451$	22088.0	2.30617
$452$	−8552.00	−0.889939
$453$	2400.00	0.248922
$454$	−1960.00	−0.202615
$455$	0	0
$456$	−1440.00	−0.147882
$457$	−778.000	−0.0796352	−0.0398176	−	0.999207i	$-0.512678\pi$
$457$	−778.000	−0.0796352	−0.0398176	+	0.999207i	$0.512678\pi$
$458$	8204.00	0.837004
$459$	−2646.00	−0.269073
$460$	0	0
$461$	838.000	0.0846628	0.0423314	−	0.999104i	$-0.486521\pi$
$461$	838.000	0.0846628	0.0423314	+	0.999104i	$0.486521\pi$
$462$	−1848.00	−0.186097
$463$	7456.00	0.748401	0.374201	−	0.927348i	$-0.377917\pi$
$463$	7456.00	0.748401	0.374201	+	0.927348i	$0.377917\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	−4932.00	−0.490280
$467$	−10724.0	−1.06263	−0.531314	−	0.847175i	$-0.678302\pi$
$467$	−10724.0	−1.06263	−0.531314	+	0.847175i	$0.678302\pi$
$468$	−1944.00	−0.192012
$469$	−420.000	−0.0413514
$470$	0	0
$471$	2670.00	0.261204
$472$	−6112.00	−0.596033
$473$	21296.0	2.07017
$474$	−912.000	−0.0883746
$475$	0	0
$476$	−2744.00	−0.264225
$477$	4770.00	0.457869
$478$	2384.00	0.228121
$479$	768.000	0.0732585	0.0366292	−	0.999329i	$-0.488338\pi$
$479$	768.000	0.0732585	0.0366292	+	0.999329i	$0.488338\pi$
$480$	0	0
$481$	−12204.0	−1.15687
$482$	−7708.00	−0.728402
$483$	3024.00	0.284879
$484$	2420.00	0.227273
$485$	0	0
$486$	486.000	0.0453609
$487$	10744.0	0.999707	0.499853	−	0.866110i	$-0.333387\pi$
$487$	10744.0	0.999707	0.499853	+	0.866110i	$0.333387\pi$
$488$	6512.00	0.604066
$489$	−2004.00	−0.185325
$490$	0	0
$491$	3540.00	0.325373	0.162686	−	0.986678i	$-0.447984\pi$
$491$	3540.00	0.325373	0.162686	+	0.986678i	$0.447984\pi$
$492$	−6024.00	−0.551998
$493$	20580.0	1.88007
$494$	6480.00	0.590180
$495$	0	0
$496$	−3328.00	−0.301273
$497$	5936.00	0.535746
$498$	−1848.00	−0.166287
$499$	15732.0	1.41134	0.705672	−	0.708538i	$-0.250645\pi$
$499$	15732.0	1.41134	0.705672	+	0.708538i	$0.250645\pi$
$500$	0	0
$501$	−2448.00	−0.218301
$502$	8040.00	0.714826
$503$	8496.00	0.753117	0.376559	−	0.926393i	$-0.377107\pi$
$503$	8496.00	0.753117	0.376559	+	0.926393i	$0.377107\pi$
$504$	504.000	0.0445435
$505$	0	0
$506$	−12672.0	−1.11332
$507$	2157.00	0.188946
$508$	−4288.00	−0.374506
$509$	17638.0	1.53593	0.767967	−	0.640489i	$-0.221268\pi$
$509$	17638.0	1.53593	0.767967	+	0.640489i	$0.221268\pi$
$510$	0	0
$511$	6706.00	0.580540
$512$	512.000	0.0441942
$513$	−1620.00	−0.139424
$514$	−3844.00	−0.329867
$515$	0	0
$516$	−5808.00	−0.495510
$517$	−10208.0	−0.868370
$518$	3164.00	0.268375
$519$	−4002.00	−0.338475
$520$	0	0
$521$	1802.00	0.151530	0.0757649	−	0.997126i	$-0.475860\pi$
$521$	1802.00	0.151530	0.0757649	+	0.997126i	$0.475860\pi$
$522$	−3780.00	−0.316947
$523$	−9196.00	−0.768859	−0.384429	−	0.923154i	$-0.625602\pi$
$523$	−9196.00	−0.768859	−0.384429	+	0.923154i	$0.625602\pi$
$524$	4016.00	0.334809
$525$	0	0
$526$	−4960.00	−0.411153
$527$	20384.0	1.68490
$528$	−2112.00	−0.174078
$529$	8569.00	0.704282
$530$	0	0
$531$	−6876.00	−0.561945
$532$	−1680.00	−0.136912
$533$	27108.0	2.20296
$534$	−6564.00	−0.531933
$535$	0	0
$536$	−480.000	−0.0386807
$537$	−3708.00	−0.297974
$538$	−2004.00	−0.160592
$539$	−2156.00	−0.172292
$540$	0	0
$541$	−5130.00	−0.407682	−0.203841	−	0.979004i	$-0.565343\pi$
$541$	−5130.00	−0.407682	−0.203841	+	0.979004i	$0.565343\pi$
$542$	−5664.00	−0.448874
$543$	1074.00	0.0848798
$544$	−3136.00	−0.247160
$545$	0	0
$546$	−2268.00	−0.177768
$547$	−5916.00	−0.462431	−0.231216	−	0.972903i	$-0.574270\pi$
$547$	−5916.00	−0.462431	−0.231216	+	0.972903i	$0.574270\pi$
$548$	−2568.00	−0.200182
$549$	7326.00	0.569519
$550$	0	0
$551$	12600.0	0.974189
$552$	3456.00	0.266480
$553$	−1064.00	−0.0818190
$554$	−8412.00	−0.645111
$555$	0	0
$556$	−6416.00	−0.489387
$557$	−24166.0	−1.83832	−0.919162	−	0.393880i	$-0.871133\pi$
$557$	−24166.0	−1.83832	−0.919162	+	0.393880i	$0.871133\pi$
$558$	−3744.00	−0.284043
$559$	26136.0	1.97752
$560$	0	0
$561$	12936.0	0.973544
$562$	11956.0	0.897390
$563$	−15572.0	−1.16569	−0.582843	−	0.812585i	$-0.698060\pi$
$563$	−15572.0	−1.16569	−0.582843	+	0.812585i	$0.698060\pi$
$564$	2784.00	0.207850
$565$	0	0
$566$	−5368.00	−0.398646
$567$	567.000	0.0419961
$568$	6784.00	0.501145
$569$	−12294.0	−0.905784	−0.452892	−	0.891565i	$-0.649608\pi$
$569$	−12294.0	−0.905784	−0.452892	+	0.891565i	$0.649608\pi$
$570$	0	0
$571$	−4740.00	−0.347395	−0.173698	−	0.984799i	$-0.555572\pi$
$571$	−4740.00	−0.347395	−0.173698	+	0.984799i	$0.555572\pi$
$572$	9504.00	0.694724
$573$	−2664.00	−0.194224
$574$	−7028.00	−0.511051
$575$	0	0
$576$	576.000	0.0416667
$577$	−12442.0	−0.897690	−0.448845	−	0.893610i	$-0.648165\pi$
$577$	−12442.0	−0.897690	−0.448845	+	0.893610i	$0.648165\pi$
$578$	9382.00	0.675155
$579$	12090.0	0.867778
$580$	0	0
$581$	−2156.00	−0.153952
$582$	−3324.00	−0.236743
$583$	−23320.0	−1.65663
$584$	7664.00	0.543046
$585$	0	0
$586$	14980.0	1.05600
$587$	22036.0	1.54944	0.774722	−	0.632303i	$-0.217890\pi$
$587$	22036.0	1.54944	0.774722	+	0.632303i	$0.217890\pi$
$588$	588.000	0.0412393
$589$	12480.0	0.873055
$590$	0	0
$591$	5142.00	0.357891
$592$	3616.00	0.251042
$593$	−9666.00	−0.669368	−0.334684	−	0.942330i	$-0.608630\pi$
$593$	−9666.00	−0.669368	−0.334684	+	0.942330i	$0.608630\pi$
$594$	−2376.00	−0.164122
$595$	0	0
$596$	−9576.00	−0.658135
$597$	−12792.0	−0.876954
$598$	−15552.0	−1.06349
$599$	−4800.00	−0.327417	−0.163708	−	0.986509i	$-0.552346\pi$
$599$	−4800.00	−0.327417	−0.163708	+	0.986509i	$0.552346\pi$
$600$	0	0
$601$	−6150.00	−0.417411	−0.208705	−	0.977979i	$-0.566925\pi$
$601$	−6150.00	−0.417411	−0.208705	+	0.977979i	$0.566925\pi$
$602$	−6776.00	−0.458753
$603$	−540.000	−0.0364685
$604$	3200.00	0.215573
$605$	0	0
$606$	6804.00	0.456095
$607$	9920.00	0.663328	0.331664	−	0.943397i	$-0.392390\pi$
$607$	9920.00	0.663328	0.331664	+	0.943397i	$0.392390\pi$
$608$	−1920.00	−0.128070
$609$	−4410.00	−0.293435
$610$	0	0
$611$	−12528.0	−0.829507
$612$	−3528.00	−0.233024
$613$	17442.0	1.14923	0.574613	−	0.818425i	$-0.305153\pi$
$613$	17442.0	1.14923	0.574613	+	0.818425i	$0.305153\pi$
$614$	13720.0	0.901782
$615$	0	0
$616$	−2464.00	−0.161165
$617$	−27746.0	−1.81039	−0.905196	−	0.424994i	$-0.860276\pi$
$617$	−27746.0	−1.81039	−0.905196	+	0.424994i	$0.860276\pi$
$618$	2928.00	0.190585
$619$	28172.0	1.82929	0.914643	−	0.404262i	$-0.132472\pi$
$619$	28172.0	1.82929	0.914643	+	0.404262i	$0.132472\pi$
$620$	0	0
$621$	3888.00	0.251240
$622$	−9840.00	−0.634322
$623$	−7658.00	−0.492474
$624$	−2592.00	−0.166287
$625$	0	0
$626$	9116.00	0.582027
$627$	7920.00	0.504457
$628$	3560.00	0.226209
$629$	−22148.0	−1.40397
$630$	0	0
$631$	30752.0	1.94012	0.970062	−	0.242859i	$-0.0780852\pi$
$631$	30752.0	1.94012	0.970062	+	0.242859i	$0.0780852\pi$
$632$	−1216.00	−0.0765346
$633$	9948.00	0.624641
$634$	−11660.0	−0.730407
$635$	0	0
$636$	6360.00	0.396526
$637$	−2646.00	−0.164581
$638$	18480.0	1.14676
$639$	7632.00	0.472484
$640$	0	0
$641$	1154.00	0.0711080	0.0355540	−	0.999368i	$-0.488680\pi$
$641$	1154.00	0.0711080	0.0355540	+	0.999368i	$0.488680\pi$
$642$	4968.00	0.305407
$643$	−20308.0	−1.24552	−0.622760	−	0.782413i	$-0.713989\pi$
$643$	−20308.0	−1.24552	−0.622760	+	0.782413i	$0.713989\pi$
$644$	4032.00	0.246713
$645$	0	0
$646$	11760.0	0.716240
$647$	−12256.0	−0.744719	−0.372359	−	0.928089i	$-0.621451\pi$
$647$	−12256.0	−0.744719	−0.372359	+	0.928089i	$0.621451\pi$
$648$	648.000	0.0392837
$649$	33616.0	2.03319
$650$	0	0
$651$	−4368.00	−0.262973
$652$	−2672.00	−0.160496
$653$	−4838.00	−0.289932	−0.144966	−	0.989437i	$-0.546307\pi$
$653$	−4838.00	−0.289932	−0.144966	+	0.989437i	$0.546307\pi$
$654$	−252.000	−0.0150672
$655$	0	0
$656$	−8032.00	−0.478044
$657$	8622.00	0.511988
$658$	3248.00	0.192432
$659$	−5588.00	−0.330315	−0.165157	−	0.986267i	$-0.552813\pi$
$659$	−5588.00	−0.330315	−0.165157	+	0.986267i	$0.552813\pi$
$660$	0	0
$661$	9430.00	0.554893	0.277447	−	0.960741i	$-0.410512\pi$
$661$	9430.00	0.554893	0.277447	+	0.960741i	$0.410512\pi$
$662$	10040.0	0.589450
$663$	15876.0	0.929974
$664$	−2464.00	−0.144009
$665$	0	0
$666$	4068.00	0.236684
$667$	−30240.0	−1.75547
$668$	−3264.00	−0.189054
$669$	−6384.00	−0.368938
$670$	0	0
$671$	−35816.0	−2.06060
$672$	672.000	0.0385758
$673$	16430.0	0.941055	0.470527	−	0.882385i	$-0.344064\pi$
$673$	16430.0	0.941055	0.470527	+	0.882385i	$0.344064\pi$
$674$	7324.00	0.418561
$675$	0	0
$676$	2876.00	0.163632
$677$	−12606.0	−0.715639	−0.357820	−	0.933791i	$-0.616480\pi$
$677$	−12606.0	−0.715639	−0.357820	+	0.933791i	$0.616480\pi$
$678$	−12828.0	−0.726632
$679$	−3878.00	−0.219181
$680$	0	0
$681$	−2940.00	−0.165435
$682$	18304.0	1.02771
$683$	−23444.0	−1.31341	−0.656706	−	0.754147i	$-0.728051\pi$
$683$	−23444.0	−1.31341	−0.656706	+	0.754147i	$0.728051\pi$
$684$	−2160.00	−0.120745
$685$	0	0
$686$	686.000	0.0381802
$687$	12306.0	0.683411
$688$	−7744.00	−0.429124
$689$	−28620.0	−1.58249
$690$	0	0
$691$	14980.0	0.824698	0.412349	−	0.911026i	$-0.364709\pi$
$691$	14980.0	0.824698	0.412349	+	0.911026i	$0.364709\pi$
$692$	−5336.00	−0.293128
$693$	−2772.00	−0.151947
$694$	1720.00	0.0940783
$695$	0	0
$696$	−5040.00	−0.274484
$697$	49196.0	2.67350
$698$	−6916.00	−0.375035
$699$	−7398.00	−0.400312
$700$	0	0
$701$	−25298.0	−1.36304	−0.681521	−	0.731799i	$-0.738681\pi$
$701$	−25298.0	−1.36304	−0.681521	+	0.731799i	$0.738681\pi$
$702$	−2916.00	−0.156777
$703$	−13560.0	−0.727489
$704$	−2816.00	−0.150756
$705$	0	0
$706$	−5988.00	−0.319209
$707$	7938.00	0.422262
$708$	−9168.00	−0.486659
$709$	−25634.0	−1.35784	−0.678918	−	0.734215i	$-0.737551\pi$
$709$	−25634.0	−1.35784	−0.678918	+	0.734215i	$0.737551\pi$
$710$	0	0
$711$	−1368.00	−0.0721575
$712$	−8752.00	−0.460667
$713$	−29952.0	−1.57323
$714$	−4116.00	−0.215739
$715$	0	0
$716$	−4944.00	−0.258053
$717$	3576.00	0.186260
$718$	3520.00	0.182960
$719$	−8448.00	−0.438188	−0.219094	−	0.975704i	$-0.570310\pi$
$719$	−8448.00	−0.438188	−0.219094	+	0.975704i	$0.570310\pi$
$720$	0	0
$721$	3416.00	0.176447
$722$	−6518.00	−0.335976
$723$	−11562.0	−0.594738
$724$	1432.00	0.0735081
$725$	0	0
$726$	3630.00	0.185567
$727$	23112.0	1.17906	0.589530	−	0.807746i	$-0.299313\pi$
$727$	23112.0	1.17906	0.589530	+	0.807746i	$0.299313\pi$
$728$	−3024.00	−0.153952
$729$	729.000	0.0370370
$730$	0	0
$731$	47432.0	2.39991
$732$	9768.00	0.493218
$733$	−10582.0	−0.533227	−0.266613	−	0.963804i	$-0.585905\pi$
$733$	−10582.0	−0.533227	−0.266613	+	0.963804i	$0.585905\pi$
$734$	1216.00	0.0611490
$735$	0	0
$736$	4608.00	0.230779
$737$	2640.00	0.131948
$738$	−9036.00	−0.450704
$739$	27124.0	1.35017	0.675083	−	0.737742i	$-0.264108\pi$
$739$	27124.0	1.35017	0.675083	+	0.737742i	$0.264108\pi$
$740$	0	0
$741$	9720.00	0.481880
$742$	7420.00	0.367112
$743$	−11616.0	−0.573553	−0.286776	−	0.957998i	$-0.592584\pi$
$743$	−11616.0	−0.573553	−0.286776	+	0.957998i	$0.592584\pi$
$744$	−4992.00	−0.245989
$745$	0	0
$746$	14180.0	0.695934
$747$	−2772.00	−0.135773
$748$	17248.0	0.843114
$749$	5796.00	0.282752
$750$	0	0
$751$	1544.00	0.0750218	0.0375109	−	0.999296i	$-0.488057\pi$
$751$	1544.00	0.0750218	0.0375109	+	0.999296i	$0.488057\pi$
$752$	3712.00	0.180004
$753$	12060.0	0.583653
$754$	22680.0	1.09543
$755$	0	0
$756$	756.000	0.0363696
$757$	39506.0	1.89679	0.948395	−	0.317091i	$-0.102706\pi$
$757$	39506.0	1.89679	0.948395	+	0.317091i	$0.102706\pi$
$758$	28536.0	1.36738
$759$	−19008.0	−0.909021
$760$	0	0
$761$	−26022.0	−1.23955	−0.619774	−	0.784780i	$-0.712776\pi$
$761$	−26022.0	−1.23955	−0.619774	+	0.784780i	$0.712776\pi$
$762$	−6432.00	−0.305783
$763$	−294.000	−0.0139496
$764$	−3552.00	−0.168203
$765$	0	0
$766$	8208.00	0.387163
$767$	41256.0	1.94220
$768$	768.000	0.0360844
$769$	−27230.0	−1.27690	−0.638452	−	0.769662i	$-0.720425\pi$
$769$	−27230.0	−1.27690	−0.638452	+	0.769662i	$0.720425\pi$
$770$	0	0
$771$	−5766.00	−0.269335
$772$	16120.0	0.751517
$773$	4866.00	0.226414	0.113207	−	0.993571i	$-0.463888\pi$
$773$	4866.00	0.226414	0.113207	+	0.993571i	$0.463888\pi$
$774$	−8712.00	−0.404582
$775$	0	0
$776$	−4432.00	−0.205025
$777$	4746.00	0.219127
$778$	30220.0	1.39260
$779$	30120.0	1.38532
$780$	0	0
$781$	−37312.0	−1.70951
$782$	−28224.0	−1.29065
$783$	−5670.00	−0.258786
$784$	784.000	0.0357143
$785$	0	0
$786$	6024.00	0.273370
$787$	5852.00	0.265059	0.132529	−	0.991179i	$-0.457690\pi$
$787$	5852.00	0.265059	0.132529	+	0.991179i	$0.457690\pi$
$788$	6856.00	0.309943
$789$	−7440.00	−0.335705
$790$	0	0
$791$	−14966.0	−0.672730
$792$	−3168.00	−0.142134
$793$	−43956.0	−1.96838
$794$	21364.0	0.954887
$795$	0	0
$796$	−17056.0	−0.759465
$797$	−16326.0	−0.725592	−0.362796	−	0.931869i	$-0.618178\pi$
$797$	−16326.0	−0.725592	−0.362796	+	0.931869i	$0.618178\pi$
$798$	−2520.00	−0.111788
$799$	−22736.0	−1.00669
$800$	0	0
$801$	−9846.00	−0.434321
$802$	−21052.0	−0.926898
$803$	−42152.0	−1.85244
$804$	−720.000	−0.0315826
$805$	0	0
$806$	22464.0	0.981713
$807$	−3006.00	−0.131123
$808$	9072.00	0.394990
$809$	7578.00	0.329330	0.164665	−	0.986350i	$-0.447346\pi$
$809$	7578.00	0.329330	0.164665	+	0.986350i	$0.447346\pi$
$810$	0	0
$811$	6860.00	0.297025	0.148512	−	0.988911i	$-0.452551\pi$
$811$	6860.00	0.297025	0.148512	+	0.988911i	$0.452551\pi$
$812$	−5880.00	−0.254123
$813$	−8496.00	−0.366504
$814$	−19888.0	−0.856356
$815$	0	0
$816$	−4704.00	−0.201805
$817$	29040.0	1.24355
$818$	−12.0000	−0.000512922 0
$819$	−3402.00	−0.145147
$820$	0	0
$821$	−25722.0	−1.09343	−0.546714	−	0.837320i	$-0.684121\pi$
$821$	−25722.0	−1.09343	−0.546714	+	0.837320i	$0.684121\pi$
$822$	−3852.00	−0.163448
$823$	21320.0	0.902999	0.451500	−	0.892271i	$-0.350889\pi$
$823$	21320.0	0.902999	0.451500	+	0.892271i	$0.350889\pi$
$824$	3904.00	0.165051
$825$	0	0
$826$	−10696.0	−0.450559
$827$	22156.0	0.931608	0.465804	−	0.884888i	$-0.345765\pi$
$827$	22156.0	0.931608	0.465804	+	0.884888i	$0.345765\pi$
$828$	5184.00	0.217580
$829$	−45394.0	−1.90181	−0.950904	−	0.309486i	$-0.899843\pi$
$829$	−45394.0	−1.90181	−0.950904	+	0.309486i	$0.899843\pi$
$830$	0	0
$831$	−12618.0	−0.526731
$832$	−3456.00	−0.144009
$833$	−4802.00	−0.199735
$834$	−9624.00	−0.399583
$835$	0	0
$836$	10560.0	0.436872
$837$	−5616.00	−0.231920
$838$	−30216.0	−1.24558
$839$	−31048.0	−1.27759	−0.638794	−	0.769378i	$-0.720566\pi$
$839$	−31048.0	−1.27759	−0.638794	+	0.769378i	$0.720566\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	20188.0	0.826276
$843$	17934.0	0.732716
$844$	13264.0	0.540955
$845$	0	0
$846$	4176.00	0.169709
$847$	4235.00	0.171802
$848$	8480.00	0.343401
$849$	−8052.00	−0.325493
$850$	0	0
$851$	32544.0	1.31092
$852$	10176.0	0.409183
$853$	16354.0	0.656448	0.328224	−	0.944600i	$-0.393550\pi$
$853$	16354.0	0.656448	0.328224	+	0.944600i	$0.393550\pi$
$854$	11396.0	0.456631
$855$	0	0
$856$	6624.00	0.264490
$857$	−11274.0	−0.449373	−0.224686	−	0.974431i	$-0.572136\pi$
$857$	−11274.0	−0.449373	−0.224686	+	0.974431i	$0.572136\pi$
$858$	14256.0	0.567240
$859$	−21204.0	−0.842225	−0.421112	−	0.907008i	$-0.638360\pi$
$859$	−21204.0	−0.842225	−0.421112	+	0.907008i	$0.638360\pi$
$860$	0	0
$861$	−10542.0	−0.417271
$862$	−25392.0	−1.00331
$863$	31304.0	1.23476	0.617382	−	0.786664i	$-0.288193\pi$
$863$	31304.0	1.23476	0.617382	+	0.786664i	$0.288193\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−24244.0	−0.951322
$867$	14073.0	0.551262
$868$	−5824.00	−0.227741
$869$	6688.00	0.261076
$870$	0	0
$871$	3240.00	0.126043
$872$	−336.000	−0.0130486
$873$	−4986.00	−0.193300
$874$	−17280.0	−0.668770
$875$	0	0
$876$	11496.0	0.443395
$877$	−23894.0	−0.920003	−0.460002	−	0.887918i	$-0.652151\pi$
$877$	−23894.0	−0.920003	−0.460002	+	0.887918i	$0.652151\pi$
$878$	−17104.0	−0.657440
$879$	22470.0	0.862223
$880$	0	0
$881$	15458.0	0.591139	0.295569	−	0.955321i	$-0.404491\pi$
$881$	15458.0	0.591139	0.295569	+	0.955321i	$0.404491\pi$
$882$	882.000	0.0336718
$883$	24276.0	0.925201	0.462600	−	0.886567i	$-0.346916\pi$
$883$	24276.0	0.925201	0.462600	+	0.886567i	$0.346916\pi$
$884$	21168.0	0.805381
$885$	0	0
$886$	7096.00	0.269069
$887$	−80.0000	−0.00302834	−0.00151417	−	0.999999i	$-0.500482\pi$
$887$	−80.0000	−0.00302834	−0.00151417	+	0.999999i	$0.500482\pi$
$888$	5424.00	0.204975
$889$	−7504.00	−0.283100
$890$	0	0
$891$	−3564.00	−0.134005
$892$	−8512.00	−0.319510
$893$	−13920.0	−0.521629
$894$	−14364.0	−0.537365
$895$	0	0
$896$	896.000	0.0334077
$897$	−23328.0	−0.868338
$898$	−33372.0	−1.24013
$899$	43680.0	1.62048
$900$	0	0
$901$	−51940.0	−1.92050
$902$	44176.0	1.63071
$903$	−10164.0	−0.374570
$904$	−17104.0	−0.629282
$905$	0	0
$906$	4800.00	0.176015
$907$	−21716.0	−0.795003	−0.397502	−	0.917601i	$-0.630123\pi$
$907$	−21716.0	−0.795003	−0.397502	+	0.917601i	$0.630123\pi$
$908$	−3920.00	−0.143271
$909$	10206.0	0.372400
$910$	0	0
$911$	26760.0	0.973214	0.486607	−	0.873621i	$-0.338234\pi$
$911$	26760.0	0.973214	0.486607	+	0.873621i	$0.338234\pi$
$912$	−2880.00	−0.104568
$913$	13552.0	0.491244
$914$	−1556.00	−0.0563106
$915$	0	0
$916$	16408.0	0.591851
$917$	7028.00	0.253092
$918$	−5292.00	−0.190264
$919$	11984.0	0.430159	0.215079	−	0.976597i	$-0.430999\pi$
$919$	11984.0	0.430159	0.215079	+	0.976597i	$0.430999\pi$
$920$	0	0
$921$	20580.0	0.736302
$922$	1676.00	0.0598656
$923$	−45792.0	−1.63300
$924$	−3696.00	−0.131590
$925$	0	0
$926$	14912.0	0.529199
$927$	4392.00	0.155612
$928$	−6720.00	−0.237710
$929$	−28622.0	−1.01083	−0.505413	−	0.862878i	$-0.668660\pi$
$929$	−28622.0	−1.01083	−0.505413	+	0.862878i	$0.668660\pi$
$930$	0	0
$931$	−2940.00	−0.103496
$932$	−9864.00	−0.346680
$933$	−14760.0	−0.517921
$934$	−21448.0	−0.751392
$935$	0	0
$936$	−3888.00	−0.135773
$937$	−17474.0	−0.609232	−0.304616	−	0.952475i	$-0.598528\pi$
$937$	−17474.0	−0.609232	−0.304616	+	0.952475i	$0.598528\pi$
$938$	−840.000	−0.0292398
$939$	13674.0	0.475223
$940$	0	0
$941$	6998.00	0.242432	0.121216	−	0.992626i	$-0.461321\pi$
$941$	6998.00	0.242432	0.121216	+	0.992626i	$0.461321\pi$
$942$	5340.00	0.184699
$943$	−72288.0	−2.49631
$944$	−12224.0	−0.421459
$945$	0	0
$946$	42592.0	1.46383
$947$	−44204.0	−1.51683	−0.758414	−	0.651773i	$-0.774026\pi$
$947$	−44204.0	−1.51683	−0.758414	+	0.651773i	$0.774026\pi$
$948$	−1824.00	−0.0624903
$949$	−51732.0	−1.76954
$950$	0	0
$951$	−17490.0	−0.596374
$952$	−5488.00	−0.186835
$953$	30.0000	0.00101972	0.000509861	−	1.00000i	$-0.499838\pi$
$953$	30.0000	0.00101972	0.000509861		1.00000i	$0.499838\pi$
$954$	9540.00	0.323762
$955$	0	0
$956$	4768.00	0.161306
$957$	27720.0	0.936322
$958$	1536.00	0.0518016
$959$	−4494.00	−0.151323
$960$	0	0
$961$	13473.0	0.452251
$962$	−24408.0	−0.818031
$963$	7452.00	0.249364
$964$	−15416.0	−0.515058
$965$	0	0
$966$	6048.00	0.201440
$967$	−20104.0	−0.668564	−0.334282	−	0.942473i	$-0.608494\pi$
$967$	−20104.0	−0.668564	−0.334282	+	0.942473i	$0.608494\pi$
$968$	4840.00	0.160706
$969$	17640.0	0.584807
$970$	0	0
$971$	−18156.0	−0.600055	−0.300028	−	0.953930i	$-0.596996\pi$
$971$	−18156.0	−0.600055	−0.300028	+	0.953930i	$0.596996\pi$
$972$	972.000	0.0320750
$973$	−11228.0	−0.369942
$974$	21488.0	0.706899
$975$	0	0
$976$	13024.0	0.427139
$977$	−13002.0	−0.425763	−0.212882	−	0.977078i	$-0.568285\pi$
$977$	−13002.0	−0.425763	−0.212882	+	0.977078i	$0.568285\pi$
$978$	−4008.00	−0.131045
$979$	48136.0	1.57143
$980$	0	0
$981$	−378.000	−0.0123024
$982$	7080.00	0.230073
$983$	−11264.0	−0.365479	−0.182739	−	0.983161i	$-0.558496\pi$
$983$	−11264.0	−0.365479	−0.182739	+	0.983161i	$0.558496\pi$
$984$	−12048.0	−0.390321
$985$	0	0
$986$	41160.0	1.32941
$987$	4872.00	0.157120
$988$	12960.0	0.417320
$989$	−69696.0	−2.24085
$990$	0	0
$991$	−10424.0	−0.334137	−0.167068	−	0.985945i	$-0.553430\pi$
$991$	−10424.0	−0.334137	−0.167068	+	0.985945i	$0.553430\pi$
$992$	−6656.00	−0.213032
$993$	15060.0	0.481284
$994$	11872.0	0.378830
$995$	0	0
$996$	−3696.00	−0.117583
$997$	9730.00	0.309079	0.154540	−	0.987987i	$-0.450611\pi$
$997$	9730.00	0.309079	0.154540	+	0.987987i	$0.450611\pi$
$998$	31464.0	0.997971
$999$	6102.00	0.193252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.w.1.1		1
5.2	odd	4		1050.4.g.k.799.2		2
5.3	odd	4		1050.4.g.k.799.1		2
5.4	even	2		210.4.a.b.1.1	✓	1
15.14	odd	2		630.4.a.n.1.1		1
20.19	odd	2		1680.4.a.x.1.1		1
35.34	odd	2		1470.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.b.1.1	✓	1	5.4	even	2
630.4.a.n.1.1		1	15.14	odd	2
1050.4.a.w.1.1		1	1.1	even	1	trivial
1050.4.g.k.799.1		2	5.3	odd	4
1050.4.g.k.799.2		2	5.2	odd	4
1470.4.a.i.1.1		1	35.34	odd	2
1680.4.a.x.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
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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