LMFDB - Embedded newform 21.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,4,Mod(1,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("21.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	21.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:	$1.23904011012$
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	$\chi$	$=$	21.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -4.00000 q^{5} -12.0000 q^{6} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -4.00000 q^{5} -12.0000 q^{6} -7.00000 q^{7} +9.00000 q^{9} -16.0000 q^{10} +62.0000 q^{11} -24.0000 q^{12} -62.0000 q^{13} -28.0000 q^{14} +12.0000 q^{15} -64.0000 q^{16} +84.0000 q^{17} +36.0000 q^{18} +100.000 q^{19} -32.0000 q^{20} +21.0000 q^{21} +248.000 q^{22} -42.0000 q^{23} -109.000 q^{25} -248.000 q^{26} -27.0000 q^{27} -56.0000 q^{28} -10.0000 q^{29} +48.0000 q^{30} -48.0000 q^{31} -256.000 q^{32} -186.000 q^{33} +336.000 q^{34} +28.0000 q^{35} +72.0000 q^{36} -246.000 q^{37} +400.000 q^{38} +186.000 q^{39} -248.000 q^{41} +84.0000 q^{42} +68.0000 q^{43} +496.000 q^{44} -36.0000 q^{45} -168.000 q^{46} +324.000 q^{47} +192.000 q^{48} +49.0000 q^{49} -436.000 q^{50} -252.000 q^{51} -496.000 q^{52} +258.000 q^{53} -108.000 q^{54} -248.000 q^{55} -300.000 q^{57} -40.0000 q^{58} +120.000 q^{59} +96.0000 q^{60} +622.000 q^{61} -192.000 q^{62} -63.0000 q^{63} -512.000 q^{64} +248.000 q^{65} -744.000 q^{66} +904.000 q^{67} +672.000 q^{68} +126.000 q^{69} +112.000 q^{70} -678.000 q^{71} -642.000 q^{73} -984.000 q^{74} +327.000 q^{75} +800.000 q^{76} -434.000 q^{77} +744.000 q^{78} +740.000 q^{79} +256.000 q^{80} +81.0000 q^{81} -992.000 q^{82} +468.000 q^{83} +168.000 q^{84} -336.000 q^{85} +272.000 q^{86} +30.0000 q^{87} +200.000 q^{89} -144.000 q^{90} +434.000 q^{91} -336.000 q^{92} +144.000 q^{93} +1296.00 q^{94} -400.000 q^{95} +768.000 q^{96} -1266.00 q^{97} +196.000 q^{98} +558.000 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	4.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	8.00000	1.00000
$5$	−4.00000	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$5$	−4.00000	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$6$	−12.0000	−0.816497
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	−16.0000	−0.505964
$11$	62.0000	1.69943	0.849714	−	0.527244i	$-0.176775\pi$
$11$	62.0000	1.69943	0.849714	+	0.527244i	$0.176775\pi$
$12$	−24.0000	−0.577350
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	−28.0000	−0.534522
$15$	12.0000	0.206559
$16$	−64.0000	−1.00000
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	36.0000	0.471405
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	−32.0000	−0.357771
$21$	21.0000	0.218218
$22$	248.000	2.40335
$23$	−42.0000	−0.380765	−0.190383	−	0.981710i	$-0.560973\pi$
$23$	−42.0000	−0.380765	−0.190383	+	0.981710i	$0.560973\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	−248.000	−1.87065
$27$	−27.0000	−0.192450
$28$	−56.0000	−0.377964
$29$	−10.0000	−0.0640329	−0.0320164	−	0.999487i	$-0.510193\pi$
$29$	−10.0000	−0.0640329	−0.0320164	+	0.999487i	$0.510193\pi$
$30$	48.0000	0.292119
$31$	−48.0000	−0.278099	−0.139049	−	0.990285i	$-0.544405\pi$
$31$	−48.0000	−0.278099	−0.139049	+	0.990285i	$0.544405\pi$
$32$	−256.000	−1.41421
$33$	−186.000	−0.981165
$34$	336.000	1.69481
$35$	28.0000	0.135225
$36$	72.0000	0.333333
$37$	−246.000	−1.09303	−0.546516	−	0.837449i	$-0.684046\pi$
$37$	−246.000	−1.09303	−0.546516	+	0.837449i	$0.684046\pi$
$38$	400.000	1.70759
$39$	186.000	0.763688
$40$	0	0
$41$	−248.000	−0.944661	−0.472330	−	0.881422i	$-0.656587\pi$
$41$	−248.000	−0.944661	−0.472330	+	0.881422i	$0.656587\pi$
$42$	84.0000	0.308607
$43$	68.0000	0.241161	0.120580	−	0.992704i	$-0.461524\pi$
$43$	68.0000	0.241161	0.120580	+	0.992704i	$0.461524\pi$
$44$	496.000	1.69943
$45$	−36.0000	−0.119257
$46$	−168.000	−0.538484
$47$	324.000	1.00554	0.502769	−	0.864421i	$-0.332315\pi$
$47$	324.000	1.00554	0.502769	+	0.864421i	$0.332315\pi$
$48$	192.000	0.577350
$49$	49.0000	0.142857
$50$	−436.000	−1.23319
$51$	−252.000	−0.691903
$52$	−496.000	−1.32275
$53$	258.000	0.668661	0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000	0.668661	0.334330	+	0.942456i	$0.391490\pi$
$54$	−108.000	−0.272166
$55$	−248.000	−0.608006
$56$	0	0
$57$	−300.000	−0.697122
$58$	−40.0000	−0.0905562
$59$	120.000	0.264791	0.132396	−	0.991197i	$-0.457733\pi$
$59$	120.000	0.264791	0.132396	+	0.991197i	$0.457733\pi$
$60$	96.0000	0.206559
$61$	622.000	1.30556	0.652778	−	0.757549i	$-0.273603\pi$
$61$	622.000	1.30556	0.652778	+	0.757549i	$0.273603\pi$
$62$	−192.000	−0.393291
$63$	−63.0000	−0.125988
$64$	−512.000	−1.00000
$65$	248.000	0.473240
$66$	−744.000	−1.38758
$67$	904.000	1.64838	0.824188	−	0.566316i	$-0.191632\pi$
$67$	904.000	1.64838	0.824188	+	0.566316i	$0.191632\pi$
$68$	672.000	1.19841
$69$	126.000	0.219835
$70$	112.000	0.191237
$71$	−678.000	−1.13329	−0.566646	−	0.823961i	$-0.691759\pi$
$71$	−678.000	−1.13329	−0.566646	+	0.823961i	$0.691759\pi$
$72$	0	0
$73$	−642.000	−1.02932	−0.514660	−	0.857394i	$-0.672082\pi$
$73$	−642.000	−1.02932	−0.514660	+	0.857394i	$0.672082\pi$
$74$	−984.000	−1.54578
$75$	327.000	0.503449
$76$	800.000	1.20745
$77$	−434.000	−0.642323
$78$	744.000	1.08002
$79$	740.000	1.05388	0.526940	−	0.849903i	$-0.323339\pi$
$79$	740.000	1.05388	0.526940	+	0.849903i	$0.323339\pi$
$80$	256.000	0.357771
$81$	81.0000	0.111111
$82$	−992.000	−1.33595
$83$	468.000	0.618912	0.309456	−	0.950914i	$-0.399853\pi$
$83$	468.000	0.618912	0.309456	+	0.950914i	$0.399853\pi$
$84$	168.000	0.218218
$85$	−336.000	−0.428757
$86$	272.000	0.341052
$87$	30.0000	0.0369694
$88$	0	0
$89$	200.000	0.238202	0.119101	−	0.992882i	$-0.461999\pi$
$89$	200.000	0.238202	0.119101	+	0.992882i	$0.461999\pi$
$90$	−144.000	−0.168655
$91$	434.000	0.499951
$92$	−336.000	−0.380765
$93$	144.000	0.160560
$94$	1296.00	1.42204
$95$	−400.000	−0.431991
$96$	768.000	0.816497
$97$	−1266.00	−1.32518	−0.662592	−	0.748981i	$-0.730544\pi$
$97$	−1266.00	−1.32518	−0.662592	+	0.748981i	$0.730544\pi$
$98$	196.000	0.202031
$99$	558.000	0.566476
$100$	−872.000	−0.872000
$101$	232.000	0.228563	0.114281	−	0.993448i	$-0.463543\pi$
$101$	232.000	0.228563	0.114281	+	0.993448i	$0.463543\pi$
$102$	−1008.00	−0.978499
$103$	−1792.00	−1.71428	−0.857141	−	0.515082i	$-0.827761\pi$
$103$	−1792.00	−1.71428	−0.857141	+	0.515082i	$0.827761\pi$
$104$	0	0
$105$	−84.0000	−0.0780720
$106$	1032.00	0.945629
$107$	−1906.00	−1.72206	−0.861028	−	0.508558i	$-0.830179\pi$
$107$	−1906.00	−1.72206	−0.861028	+	0.508558i	$0.830179\pi$
$108$	−216.000	−0.192450
$109$	−90.0000	−0.0790866	−0.0395433	−	0.999218i	$-0.512590\pi$
$109$	−90.0000	−0.0790866	−0.0395433	+	0.999218i	$0.512590\pi$
$110$	−992.000	−0.859850
$111$	738.000	0.631062
$112$	448.000	0.377964
$113$	458.000	0.381283	0.190642	−	0.981660i	$-0.438943\pi$
$113$	458.000	0.381283	0.190642	+	0.981660i	$0.438943\pi$
$114$	−1200.00	−0.985880
$115$	168.000	0.136227
$116$	−80.0000	−0.0640329
$117$	−558.000	−0.440916
$118$	480.000	0.374471
$119$	−588.000	−0.452957
$120$	0	0
$121$	2513.00	1.88805
$122$	2488.00	1.84634
$123$	744.000	0.545400
$124$	−384.000	−0.278099
$125$	936.000	0.669747
$126$	−252.000	−0.178174
$127$	804.000	0.561760	0.280880	−	0.959743i	$-0.409374\pi$
$127$	804.000	0.561760	0.280880	+	0.959743i	$0.409374\pi$
$128$	0	0
$129$	−204.000	−0.139234
$130$	992.000	0.669263
$131$	812.000	0.541563	0.270782	−	0.962641i	$-0.412718\pi$
$131$	812.000	0.541563	0.270782	+	0.962641i	$0.412718\pi$
$132$	−1488.00	−0.981165
$133$	−700.000	−0.456374
$134$	3616.00	2.33116
$135$	108.000	0.0688530
$136$	0	0
$137$	414.000	0.258178	0.129089	−	0.991633i	$-0.458795\pi$
$137$	414.000	0.258178	0.129089	+	0.991633i	$0.458795\pi$
$138$	504.000	0.310894
$139$	−1620.00	−0.988537	−0.494268	−	0.869309i	$-0.664564\pi$
$139$	−1620.00	−0.988537	−0.494268	+	0.869309i	$0.664564\pi$
$140$	224.000	0.135225
$141$	−972.000	−0.580547
$142$	−2712.00	−1.60272
$143$	−3844.00	−2.24791
$144$	−576.000	−0.333333
$145$	40.0000	0.0229091
$146$	−2568.00	−1.45568
$147$	−147.000	−0.0824786
$148$	−1968.00	−1.09303
$149$	2370.00	1.30307	0.651537	−	0.758617i	$-0.274125\pi$
$149$	2370.00	1.30307	0.651537	+	0.758617i	$0.274125\pi$
$150$	1308.00	0.711985
$151$	−568.000	−0.306114	−0.153057	−	0.988217i	$-0.548912\pi$
$151$	−568.000	−0.306114	−0.153057	+	0.988217i	$0.548912\pi$
$152$	0	0
$153$	756.000	0.399470
$154$	−1736.00	−0.908382
$155$	192.000	0.0994956
$156$	1488.00	0.763688
$157$	−266.000	−0.135217	−0.0676086	−	0.997712i	$-0.521537\pi$
$157$	−266.000	−0.135217	−0.0676086	+	0.997712i	$0.521537\pi$
$158$	2960.00	1.49041
$159$	−774.000	−0.386052
$160$	1024.00	0.505964
$161$	294.000	0.143916
$162$	324.000	0.157135
$163$	−272.000	−0.130704	−0.0653518	−	0.997862i	$-0.520817\pi$
$163$	−272.000	−0.130704	−0.0653518	+	0.997862i	$0.520817\pi$
$164$	−1984.00	−0.944661
$165$	744.000	0.351032
$166$	1872.00	0.875273
$167$	−1876.00	−0.869277	−0.434638	−	0.900605i	$-0.643124\pi$
$167$	−1876.00	−0.869277	−0.434638	+	0.900605i	$0.643124\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	−1344.00	−0.606353
$171$	900.000	0.402484
$172$	544.000	0.241161
$173$	−152.000	−0.0667997	−0.0333998	−	0.999442i	$-0.510633\pi$
$173$	−152.000	−0.0667997	−0.0333998	+	0.999442i	$0.510633\pi$
$174$	120.000	0.0522826
$175$	763.000	0.329585
$176$	−3968.00	−1.69943
$177$	−360.000	−0.152877
$178$	800.000	0.336868
$179$	610.000	0.254713	0.127356	−	0.991857i	$-0.459351\pi$
$179$	610.000	0.254713	0.127356	+	0.991857i	$0.459351\pi$
$180$	−288.000	−0.119257
$181$	1042.00	0.427907	0.213954	−	0.976844i	$-0.431366\pi$
$181$	1042.00	0.427907	0.213954	+	0.976844i	$0.431366\pi$
$182$	1736.00	0.707038
$183$	−1866.00	−0.753763
$184$	0	0
$185$	984.000	0.391055
$186$	576.000	0.227067
$187$	5208.00	2.03661
$188$	2592.00	1.00554
$189$	189.000	0.0727393
$190$	−1600.00	−0.610927
$191$	−2038.00	−0.772065	−0.386033	−	0.922485i	$-0.626155\pi$
$191$	−2038.00	−0.772065	−0.386033	+	0.922485i	$0.626155\pi$
$192$	1536.00	0.577350
$193$	−2602.00	−0.970446	−0.485223	−	0.874390i	$-0.661262\pi$
$193$	−2602.00	−0.970446	−0.485223	+	0.874390i	$0.661262\pi$
$194$	−5064.00	−1.87409
$195$	−744.000	−0.273225
$196$	392.000	0.142857
$197$	2354.00	0.851348	0.425674	−	0.904877i	$-0.360037\pi$
$197$	2354.00	0.851348	0.425674	+	0.904877i	$0.360037\pi$
$198$	2232.00	0.801118
$199$	1680.00	0.598452	0.299226	−	0.954182i	$-0.403271\pi$
$199$	1680.00	0.598452	0.299226	+	0.954182i	$0.403271\pi$
$200$	0	0
$201$	−2712.00	−0.951690
$202$	928.000	0.323237
$203$	70.0000	0.0242022
$204$	−2016.00	−0.691903
$205$	992.000	0.337972
$206$	−7168.00	−2.42436
$207$	−378.000	−0.126922
$208$	3968.00	1.32275
$209$	6200.00	2.05198
$210$	−336.000	−0.110410
$211$	−668.000	−0.217948	−0.108974	−	0.994045i	$-0.534757\pi$
$211$	−668.000	−0.217948	−0.108974	+	0.994045i	$0.534757\pi$
$212$	2064.00	0.668661
$213$	2034.00	0.654307
$214$	−7624.00	−2.43535
$215$	−272.000	−0.0862802
$216$	0	0
$217$	336.000	0.105111
$218$	−360.000	−0.111845
$219$	1926.00	0.594279
$220$	−1984.00	−0.608006
$221$	−5208.00	−1.58519
$222$	2952.00	0.892456
$223$	−1832.00	−0.550134	−0.275067	−	0.961425i	$-0.588700\pi$
$223$	−1832.00	−0.550134	−0.275067	+	0.961425i	$0.588700\pi$
$224$	1792.00	0.534522
$225$	−981.000	−0.290667
$226$	1832.00	0.539216
$227$	4944.00	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$227$	4944.00	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$228$	−2400.00	−0.697122
$229$	−5470.00	−1.57846	−0.789231	−	0.614096i	$-0.789521\pi$
$229$	−5470.00	−1.57846	−0.789231	+	0.614096i	$0.789521\pi$
$230$	672.000	0.192654
$231$	1302.00	0.370846
$232$	0	0
$233$	−2802.00	−0.787833	−0.393917	−	0.919146i	$-0.628880\pi$
$233$	−2802.00	−0.787833	−0.393917	+	0.919146i	$0.628880\pi$
$234$	−2232.00	−0.623549
$235$	−1296.00	−0.359752
$236$	960.000	0.264791
$237$	−2220.00	−0.608458
$238$	−2352.00	−0.640578
$239$	−1170.00	−0.316657	−0.158328	−	0.987386i	$-0.550610\pi$
$239$	−1170.00	−0.316657	−0.158328	+	0.987386i	$0.550610\pi$
$240$	−768.000	−0.206559
$241$	−2338.00	−0.624912	−0.312456	−	0.949932i	$-0.601152\pi$
$241$	−2338.00	−0.624912	−0.312456	+	0.949932i	$0.601152\pi$
$242$	10052.0	2.67011
$243$	−243.000	−0.0641500
$244$	4976.00	1.30556
$245$	−196.000	−0.0511101
$246$	2976.00	0.771312
$247$	−6200.00	−1.59715
$248$	0	0
$249$	−1404.00	−0.357329
$250$	3744.00	0.947165
$251$	2792.00	0.702109	0.351055	−	0.936355i	$-0.385823\pi$
$251$	2792.00	0.702109	0.351055	+	0.936355i	$0.385823\pi$
$252$	−504.000	−0.125988
$253$	−2604.00	−0.647083
$254$	3216.00	0.794448
$255$	1008.00	0.247543
$256$	4096.00	1.00000
$257$	7024.00	1.70484	0.852422	−	0.522854i	$-0.175133\pi$
$257$	7024.00	1.70484	0.852422	+	0.522854i	$0.175133\pi$
$258$	−816.000	−0.196907
$259$	1722.00	0.413127
$260$	1984.00	0.473240
$261$	−90.0000	−0.0213443
$262$	3248.00	0.765886
$263$	2438.00	0.571610	0.285805	−	0.958288i	$-0.407739\pi$
$263$	2438.00	0.571610	0.285805	+	0.958288i	$0.407739\pi$
$264$	0	0
$265$	−1032.00	−0.239227
$266$	−2800.00	−0.645410
$267$	−600.000	−0.137526
$268$	7232.00	1.64838
$269$	−6780.00	−1.53674	−0.768372	−	0.640004i	$-0.778933\pi$
$269$	−6780.00	−1.53674	−0.768372	+	0.640004i	$0.778933\pi$
$270$	432.000	0.0973729
$271$	−1928.00	−0.432168	−0.216084	−	0.976375i	$-0.569329\pi$
$271$	−1928.00	−0.432168	−0.216084	+	0.976375i	$0.569329\pi$
$272$	−5376.00	−1.19841
$273$	−1302.00	−0.288647
$274$	1656.00	0.365119
$275$	−6758.00	−1.48190
$276$	1008.00	0.219835
$277$	5554.00	1.20472	0.602360	−	0.798224i	$-0.294227\pi$
$277$	5554.00	1.20472	0.602360	+	0.798224i	$0.294227\pi$
$278$	−6480.00	−1.39800
$279$	−432.000	−0.0926995
$280$	0	0
$281$	1942.00	0.412278	0.206139	−	0.978523i	$-0.433910\pi$
$281$	1942.00	0.412278	0.206139	+	0.978523i	$0.433910\pi$
$282$	−3888.00	−0.821018
$283$	4828.00	1.01412	0.507058	−	0.861912i	$-0.330733\pi$
$283$	4828.00	1.01412	0.507058	+	0.861912i	$0.330733\pi$
$284$	−5424.00	−1.13329
$285$	1200.00	0.249410
$286$	−15376.0	−3.17903
$287$	1736.00	0.357048
$288$	−2304.00	−0.471405
$289$	2143.00	0.436190
$290$	160.000	0.0323984
$291$	3798.00	0.765095
$292$	−5136.00	−1.02932
$293$	−6152.00	−1.22663	−0.613317	−	0.789837i	$-0.710165\pi$
$293$	−6152.00	−1.22663	−0.613317	+	0.789837i	$0.710165\pi$
$294$	−588.000	−0.116642
$295$	−480.000	−0.0947345
$296$	0	0
$297$	−1674.00	−0.327055
$298$	9480.00	1.84282
$299$	2604.00	0.503656
$300$	2616.00	0.503449
$301$	−476.000	−0.0911501
$302$	−2272.00	−0.432910
$303$	−696.000	−0.131961
$304$	−6400.00	−1.20745
$305$	−2488.00	−0.467090
$306$	3024.00	0.564937
$307$	5884.00	1.09387	0.546934	−	0.837176i	$-0.315795\pi$
$307$	5884.00	1.09387	0.546934	+	0.837176i	$0.315795\pi$
$308$	−3472.00	−0.642323
$309$	5376.00	0.989741
$310$	768.000	0.140708
$311$	9132.00	1.66504	0.832521	−	0.553993i	$-0.186897\pi$
$311$	9132.00	1.66504	0.832521	+	0.553993i	$0.186897\pi$
$312$	0	0
$313$	−9382.00	−1.69426	−0.847128	−	0.531389i	$-0.821670\pi$
$313$	−9382.00	−1.69426	−0.847128	+	0.531389i	$0.821670\pi$
$314$	−1064.00	−0.191226
$315$	252.000	0.0450749
$316$	5920.00	1.05388
$317$	3114.00	0.551734	0.275867	−	0.961196i	$-0.411035\pi$
$317$	3114.00	0.551734	0.275867	+	0.961196i	$0.411035\pi$
$318$	−3096.00	−0.545959
$319$	−620.000	−0.108819
$320$	2048.00	0.357771
$321$	5718.00	0.994229
$322$	1176.00	0.203528
$323$	8400.00	1.44702
$324$	648.000	0.111111
$325$	6758.00	1.15344
$326$	−1088.00	−0.184843
$327$	270.000	0.0456607
$328$	0	0
$329$	−2268.00	−0.380057
$330$	2976.00	0.496435
$331$	1532.00	0.254400	0.127200	−	0.991877i	$-0.459401\pi$
$331$	1532.00	0.254400	0.127200	+	0.991877i	$0.459401\pi$
$332$	3744.00	0.618912
$333$	−2214.00	−0.364344
$334$	−7504.00	−1.22934
$335$	−3616.00	−0.589741
$336$	−1344.00	−0.218218
$337$	−4166.00	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$337$	−4166.00	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$338$	6588.00	1.06018
$339$	−1374.00	−0.220134
$340$	−2688.00	−0.428757
$341$	−2976.00	−0.472608
$342$	3600.00	0.569198
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−504.000	−0.0786506
$346$	−608.000	−0.0944690
$347$	−11366.0	−1.75838	−0.879191	−	0.476469i	$-0.841917\pi$
$347$	−11366.0	−1.75838	−0.879191	+	0.476469i	$0.841917\pi$
$348$	240.000	0.0369694
$349$	9310.00	1.42795	0.713973	−	0.700174i	$-0.246894\pi$
$349$	9310.00	1.42795	0.713973	+	0.700174i	$0.246894\pi$
$350$	3052.00	0.466104
$351$	1674.00	0.254563
$352$	−15872.0	−2.40335
$353$	−8572.00	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$353$	−8572.00	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$354$	−1440.00	−0.216201
$355$	2712.00	0.405459
$356$	1600.00	0.238202
$357$	1764.00	0.261515
$358$	2440.00	0.360218
$359$	−4790.00	−0.704196	−0.352098	−	0.935963i	$-0.614532\pi$
$359$	−4790.00	−0.704196	−0.352098	+	0.935963i	$0.614532\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	4168.00	0.605153
$363$	−7539.00	−1.09007
$364$	3472.00	0.499951
$365$	2568.00	0.368261
$366$	−7464.00	−1.06598
$367$	5424.00	0.771473	0.385736	−	0.922609i	$-0.373947\pi$
$367$	5424.00	0.771473	0.385736	+	0.922609i	$0.373947\pi$
$368$	2688.00	0.380765
$369$	−2232.00	−0.314887
$370$	3936.00	0.553035
$371$	−1806.00	−0.252730
$372$	1152.00	0.160560
$373$	1838.00	0.255142	0.127571	−	0.991829i	$-0.459282\pi$
$373$	1838.00	0.255142	0.127571	+	0.991829i	$0.459282\pi$
$374$	20832.0	2.88021
$375$	−2808.00	−0.386679
$376$	0	0
$377$	620.000	0.0846993
$378$	756.000	0.102869
$379$	−4260.00	−0.577365	−0.288683	−	0.957425i	$-0.593217\pi$
$379$	−4260.00	−0.577365	−0.288683	+	0.957425i	$0.593217\pi$
$380$	−3200.00	−0.431991
$381$	−2412.00	−0.324332
$382$	−8152.00	−1.09187
$383$	9048.00	1.20713	0.603566	−	0.797313i	$-0.293746\pi$
$383$	9048.00	1.20713	0.603566	+	0.797313i	$0.293746\pi$
$384$	0	0
$385$	1736.00	0.229805
$386$	−10408.0	−1.37242
$387$	612.000	0.0803868
$388$	−10128.0	−1.32518
$389$	−11490.0	−1.49760	−0.748800	−	0.662796i	$-0.769369\pi$
$389$	−11490.0	−1.49760	−0.748800	+	0.662796i	$0.769369\pi$
$390$	−2976.00	−0.386399
$391$	−3528.00	−0.456314
$392$	0	0
$393$	−2436.00	−0.312672
$394$	9416.00	1.20399
$395$	−2960.00	−0.377048
$396$	4464.00	0.566476
$397$	−1866.00	−0.235899	−0.117949	−	0.993020i	$-0.537632\pi$
$397$	−1866.00	−0.235899	−0.117949	+	0.993020i	$0.537632\pi$
$398$	6720.00	0.846340
$399$	2100.00	0.263487
$400$	6976.00	0.872000
$401$	13662.0	1.70137	0.850683	−	0.525679i	$-0.176189\pi$
$401$	13662.0	1.70137	0.850683	+	0.525679i	$0.176189\pi$
$402$	−10848.0	−1.34589
$403$	2976.00	0.367854
$404$	1856.00	0.228563
$405$	−324.000	−0.0397523
$406$	280.000	0.0342270
$407$	−15252.0	−1.85753
$408$	0	0
$409$	−13210.0	−1.59705	−0.798524	−	0.601963i	$-0.794385\pi$
$409$	−13210.0	−1.59705	−0.798524	+	0.601963i	$0.794385\pi$
$410$	3968.00	0.477965
$411$	−1242.00	−0.149059
$412$	−14336.0	−1.71428
$413$	−840.000	−0.100082
$414$	−1512.00	−0.179495
$415$	−1872.00	−0.221429
$416$	15872.0	1.87065
$417$	4860.00	0.570732
$418$	24800.0	2.90193
$419$	6960.00	0.811499	0.405750	−	0.913984i	$-0.367010\pi$
$419$	6960.00	0.811499	0.405750	+	0.913984i	$0.367010\pi$
$420$	−672.000	−0.0780720
$421$	8162.00	0.944873	0.472437	−	0.881365i	$-0.343375\pi$
$421$	8162.00	0.944873	0.472437	+	0.881365i	$0.343375\pi$
$422$	−2672.00	−0.308225
$423$	2916.00	0.335179
$424$	0	0
$425$	−9156.00	−1.04501
$426$	8136.00	0.925330
$427$	−4354.00	−0.493454
$428$	−15248.0	−1.72206
$429$	11532.0	1.29783
$430$	−1088.00	−0.122019
$431$	16602.0	1.85543	0.927715	−	0.373290i	$-0.121770\pi$
$431$	16602.0	1.85543	0.927715	+	0.373290i	$0.121770\pi$
$432$	1728.00	0.192450
$433$	7738.00	0.858810	0.429405	−	0.903112i	$-0.358723\pi$
$433$	7738.00	0.858810	0.429405	+	0.903112i	$0.358723\pi$
$434$	1344.00	0.148650
$435$	−120.000	−0.0132266
$436$	−720.000	−0.0790866
$437$	−4200.00	−0.459756
$438$	7704.00	0.840437
$439$	−840.000	−0.0913235	−0.0456617	−	0.998957i	$-0.514540\pi$
$439$	−840.000	−0.0913235	−0.0456617	+	0.998957i	$0.514540\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−20832.0	−2.24180
$443$	6618.00	0.709776	0.354888	−	0.934909i	$-0.384519\pi$
$443$	6618.00	0.709776	0.354888	+	0.934909i	$0.384519\pi$
$444$	5904.00	0.631062
$445$	−800.000	−0.0852217
$446$	−7328.00	−0.778006
$447$	−7110.00	−0.752330
$448$	3584.00	0.377964
$449$	3090.00	0.324780	0.162390	−	0.986727i	$-0.448080\pi$
$449$	3090.00	0.324780	0.162390	+	0.986727i	$0.448080\pi$
$450$	−3924.00	−0.411065
$451$	−15376.0	−1.60538
$452$	3664.00	0.381283
$453$	1704.00	0.176735
$454$	19776.0	2.04435
$455$	−1736.00	−0.178868
$456$	0	0
$457$	5914.00	0.605351	0.302675	−	0.953094i	$-0.402120\pi$
$457$	5914.00	0.605351	0.302675	+	0.953094i	$0.402120\pi$
$458$	−21880.0	−2.23228
$459$	−2268.00	−0.230634
$460$	1344.00	0.136227
$461$	−15968.0	−1.61324	−0.806620	−	0.591070i	$-0.798706\pi$
$461$	−15968.0	−1.61324	−0.806620	+	0.591070i	$0.798706\pi$
$462$	5208.00	0.524455
$463$	−1172.00	−0.117640	−0.0588202	−	0.998269i	$-0.518734\pi$
$463$	−1172.00	−0.117640	−0.0588202	+	0.998269i	$0.518734\pi$
$464$	640.000	0.0640329
$465$	−576.000	−0.0574438
$466$	−11208.0	−1.11416
$467$	5304.00	0.525567	0.262784	−	0.964855i	$-0.415359\pi$
$467$	5304.00	0.525567	0.262784	+	0.964855i	$0.415359\pi$
$468$	−4464.00	−0.440916
$469$	−6328.00	−0.623027
$470$	−5184.00	−0.508766
$471$	798.000	0.0780677
$472$	0	0
$473$	4216.00	0.409835
$474$	−8880.00	−0.860489
$475$	−10900.0	−1.05290
$476$	−4704.00	−0.452957
$477$	2322.00	0.222887
$478$	−4680.00	−0.447821
$479$	5740.00	0.547531	0.273765	−	0.961796i	$-0.411731\pi$
$479$	5740.00	0.547531	0.273765	+	0.961796i	$0.411731\pi$
$480$	−3072.00	−0.292119
$481$	15252.0	1.44580
$482$	−9352.00	−0.883759
$483$	−882.000	−0.0830898
$484$	20104.0	1.88805
$485$	5064.00	0.474112
$486$	−972.000	−0.0907218
$487$	8944.00	0.832220	0.416110	−	0.909314i	$-0.363393\pi$
$487$	8944.00	0.832220	0.416110	+	0.909314i	$0.363393\pi$
$488$	0	0
$489$	816.000	0.0754617
$490$	−784.000	−0.0722806
$491$	−5558.00	−0.510853	−0.255427	−	0.966828i	$-0.582216\pi$
$491$	−5558.00	−0.510853	−0.255427	+	0.966828i	$0.582216\pi$
$492$	5952.00	0.545400
$493$	−840.000	−0.0767377
$494$	−24800.0	−2.25871
$495$	−2232.00	−0.202669
$496$	3072.00	0.278099
$497$	4746.00	0.428344
$498$	−5616.00	−0.505339
$499$	−19820.0	−1.77809	−0.889043	−	0.457823i	$-0.848629\pi$
$499$	−19820.0	−1.77809	−0.889043	+	0.457823i	$0.848629\pi$
$500$	7488.00	0.669747
$501$	5628.00	0.501877
$502$	11168.0	0.992933
$503$	1848.00	0.163814	0.0819068	−	0.996640i	$-0.473899\pi$
$503$	1848.00	0.163814	0.0819068	+	0.996640i	$0.473899\pi$
$504$	0	0
$505$	−928.000	−0.0817732
$506$	−10416.0	−0.915114
$507$	−4941.00	−0.432816
$508$	6432.00	0.561760
$509$	340.000	0.0296075	0.0148038	−	0.999890i	$-0.495288\pi$
$509$	340.000	0.0296075	0.0148038	+	0.999890i	$0.495288\pi$
$510$	4032.00	0.350078
$511$	4494.00	0.389047
$512$	16384.0	1.41421
$513$	−2700.00	−0.232374
$514$	28096.0	2.41101
$515$	7168.00	0.613320
$516$	−1632.00	−0.139234
$517$	20088.0	1.70884
$518$	6888.00	0.584250
$519$	456.000	0.0385668
$520$	0	0
$521$	10212.0	0.858725	0.429363	−	0.903132i	$-0.358738\pi$
$521$	10212.0	0.858725	0.429363	+	0.903132i	$0.358738\pi$
$522$	−360.000	−0.0301854
$523$	−9332.00	−0.780229	−0.390115	−	0.920766i	$-0.627565\pi$
$523$	−9332.00	−0.780229	−0.390115	+	0.920766i	$0.627565\pi$
$524$	6496.00	0.541563
$525$	−2289.00	−0.190286
$526$	9752.00	0.808379
$527$	−4032.00	−0.333276
$528$	11904.0	0.981165
$529$	−10403.0	−0.855018
$530$	−4128.00	−0.338319
$531$	1080.00	0.0882637
$532$	−5600.00	−0.456374
$533$	15376.0	1.24955
$534$	−2400.00	−0.194491
$535$	7624.00	0.616101
$536$	0	0
$537$	−1830.00	−0.147058
$538$	−27120.0	−2.17328
$539$	3038.00	0.242775
$540$	864.000	0.0688530
$541$	−8998.00	−0.715073	−0.357536	−	0.933899i	$-0.616383\pi$
$541$	−8998.00	−0.715073	−0.357536	+	0.933899i	$0.616383\pi$
$542$	−7712.00	−0.611179
$543$	−3126.00	−0.247052
$544$	−21504.0	−1.69481
$545$	360.000	0.0282949
$546$	−5208.00	−0.408208
$547$	−3416.00	−0.267016	−0.133508	−	0.991048i	$-0.542624\pi$
$547$	−3416.00	−0.267016	−0.133508	+	0.991048i	$0.542624\pi$
$548$	3312.00	0.258178
$549$	5598.00	0.435185
$550$	−27032.0	−2.09572
$551$	−1000.00	−0.0773166
$552$	0	0
$553$	−5180.00	−0.398329
$554$	22216.0	1.70373
$555$	−2952.00	−0.225776
$556$	−12960.0	−0.988537
$557$	−526.000	−0.0400132	−0.0200066	−	0.999800i	$-0.506369\pi$
$557$	−526.000	−0.0400132	−0.0200066	+	0.999800i	$0.506369\pi$
$558$	−1728.00	−0.131097
$559$	−4216.00	−0.318994
$560$	−1792.00	−0.135225
$561$	−15624.0	−1.17584
$562$	7768.00	0.583049
$563$	−6712.00	−0.502446	−0.251223	−	0.967929i	$-0.580833\pi$
$563$	−6712.00	−0.502446	−0.251223	+	0.967929i	$0.580833\pi$
$564$	−7776.00	−0.580547
$565$	−1832.00	−0.136412
$566$	19312.0	1.43418
$567$	−567.000	−0.0419961
$568$	0	0
$569$	4190.00	0.308706	0.154353	−	0.988016i	$-0.450671\pi$
$569$	4190.00	0.308706	0.154353	+	0.988016i	$0.450671\pi$
$570$	4800.00	0.352719
$571$	3032.00	0.222216	0.111108	−	0.993808i	$-0.464560\pi$
$571$	3032.00	0.222216	0.111108	+	0.993808i	$0.464560\pi$
$572$	−30752.0	−2.24791
$573$	6114.00	0.445752
$574$	6944.00	0.504942
$575$	4578.00	0.332027
$576$	−4608.00	−0.333333
$577$	5434.00	0.392063	0.196032	−	0.980598i	$-0.437195\pi$
$577$	5434.00	0.392063	0.196032	+	0.980598i	$0.437195\pi$
$578$	8572.00	0.616865
$579$	7806.00	0.560287
$580$	320.000	0.0229091
$581$	−3276.00	−0.233927
$582$	15192.0	1.08201
$583$	15996.0	1.13634
$584$	0	0
$585$	2232.00	0.157747
$586$	−24608.0	−1.73472
$587$	464.000	0.0326258	0.0163129	−	0.999867i	$-0.494807\pi$
$587$	464.000	0.0326258	0.0163129	+	0.999867i	$0.494807\pi$
$588$	−1176.00	−0.0824786
$589$	−4800.00	−0.335790
$590$	−1920.00	−0.133975
$591$	−7062.00	−0.491526
$592$	15744.0	1.09303
$593$	11748.0	0.813546	0.406773	−	0.913529i	$-0.366654\pi$
$593$	11748.0	0.813546	0.406773	+	0.913529i	$0.366654\pi$
$594$	−6696.00	−0.462526
$595$	2352.00	0.162055
$596$	18960.0	1.30307
$597$	−5040.00	−0.345517
$598$	10416.0	0.712277
$599$	7650.00	0.521821	0.260910	−	0.965363i	$-0.415977\pi$
$599$	7650.00	0.521821	0.260910	+	0.965363i	$0.415977\pi$
$600$	0	0
$601$	−22878.0	−1.55277	−0.776384	−	0.630261i	$-0.782948\pi$
$601$	−22878.0	−1.55277	−0.776384	+	0.630261i	$0.782948\pi$
$602$	−1904.00	−0.128906
$603$	8136.00	0.549459
$604$	−4544.00	−0.306114
$605$	−10052.0	−0.675491
$606$	−2784.00	−0.186621
$607$	704.000	0.0470749	0.0235375	−	0.999723i	$-0.492507\pi$
$607$	704.000	0.0470749	0.0235375	+	0.999723i	$0.492507\pi$
$608$	−25600.0	−1.70759
$609$	−210.000	−0.0139731
$610$	−9952.00	−0.660565
$611$	−20088.0	−1.33007
$612$	6048.00	0.399470
$613$	24958.0	1.64444	0.822222	−	0.569167i	$-0.192734\pi$
$613$	24958.0	1.64444	0.822222	+	0.569167i	$0.192734\pi$
$614$	23536.0	1.54696
$615$	−2976.00	−0.195128
$616$	0	0
$617$	−8826.00	−0.575886	−0.287943	−	0.957648i	$-0.592971\pi$
$617$	−8826.00	−0.575886	−0.287943	+	0.957648i	$0.592971\pi$
$618$	21504.0	1.39971
$619$	21220.0	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$620$	1536.00	0.0994956
$621$	1134.00	0.0732783
$622$	36528.0	2.35473
$623$	−1400.00	−0.0900318
$624$	−11904.0	−0.763688
$625$	9881.00	0.632384
$626$	−37528.0	−2.39604
$627$	−18600.0	−1.18471
$628$	−2128.00	−0.135217
$629$	−20664.0	−1.30990
$630$	1008.00	0.0637455
$631$	−3268.00	−0.206176	−0.103088	−	0.994672i	$-0.532872\pi$
$631$	−3268.00	−0.206176	−0.103088	+	0.994672i	$0.532872\pi$
$632$	0	0
$633$	2004.00	0.125832
$634$	12456.0	0.780270
$635$	−3216.00	−0.200981
$636$	−6192.00	−0.386052
$637$	−3038.00	−0.188964
$638$	−2480.00	−0.153894
$639$	−6102.00	−0.377764
$640$	0	0
$641$	13062.0	0.804864	0.402432	−	0.915450i	$-0.368165\pi$
$641$	13062.0	0.804864	0.402432	+	0.915450i	$0.368165\pi$
$642$	22872.0	1.40605
$643$	−28012.0	−1.71802	−0.859009	−	0.511961i	$-0.828919\pi$
$643$	−28012.0	−1.71802	−0.859009	+	0.511961i	$0.828919\pi$
$644$	2352.00	0.143916
$645$	816.000	0.0498139
$646$	33600.0	2.04640
$647$	3844.00	0.233575	0.116788	−	0.993157i	$-0.462740\pi$
$647$	3844.00	0.233575	0.116788	+	0.993157i	$0.462740\pi$
$648$	0	0
$649$	7440.00	0.449993
$650$	27032.0	1.63120
$651$	−1008.00	−0.0606861
$652$	−2176.00	−0.130704
$653$	−28482.0	−1.70687	−0.853436	−	0.521198i	$-0.825485\pi$
$653$	−28482.0	−1.70687	−0.853436	+	0.521198i	$0.825485\pi$
$654$	1080.00	0.0645739
$655$	−3248.00	−0.193756
$656$	15872.0	0.944661
$657$	−5778.00	−0.343107
$658$	−9072.00	−0.537482
$659$	−9330.00	−0.551510	−0.275755	−	0.961228i	$-0.588928\pi$
$659$	−9330.00	−0.551510	−0.275755	+	0.961228i	$0.588928\pi$
$660$	5952.00	0.351032
$661$	8782.00	0.516763	0.258381	−	0.966043i	$-0.416811\pi$
$661$	8782.00	0.516763	0.258381	+	0.966043i	$0.416811\pi$
$662$	6128.00	0.359776
$663$	15624.0	0.915212
$664$	0	0
$665$	2800.00	0.163277
$666$	−8856.00	−0.515260
$667$	420.000	0.0243815
$668$	−15008.0	−0.869277
$669$	5496.00	0.317620
$670$	−14464.0	−0.834020
$671$	38564.0	2.21870
$672$	−5376.00	−0.308607
$673$	−10562.0	−0.604956	−0.302478	−	0.953156i	$-0.597814\pi$
$673$	−10562.0	−0.604956	−0.302478	+	0.953156i	$0.597814\pi$
$674$	−16664.0	−0.952334
$675$	2943.00	0.167816
$676$	13176.0	0.749659
$677$	−26016.0	−1.47692	−0.738461	−	0.674296i	$-0.764447\pi$
$677$	−26016.0	−1.47692	−0.738461	+	0.674296i	$0.764447\pi$
$678$	−5496.00	−0.311317
$679$	8862.00	0.500872
$680$	0	0
$681$	−14832.0	−0.834601
$682$	−11904.0	−0.668369
$683$	8898.00	0.498496	0.249248	−	0.968440i	$-0.419817\pi$
$683$	8898.00	0.498496	0.249248	+	0.968440i	$0.419817\pi$
$684$	7200.00	0.402484
$685$	−1656.00	−0.0923686
$686$	−1372.00	−0.0763604
$687$	16410.0	0.911325
$688$	−4352.00	−0.241161
$689$	−15996.0	−0.884469
$690$	−2016.00	−0.111229
$691$	30572.0	1.68309	0.841544	−	0.540189i	$-0.181647\pi$
$691$	30572.0	1.68309	0.841544	+	0.540189i	$0.181647\pi$
$692$	−1216.00	−0.0667997
$693$	−3906.00	−0.214108
$694$	−45464.0	−2.48673
$695$	6480.00	0.353670
$696$	0	0
$697$	−20832.0	−1.13209
$698$	37240.0	2.01942
$699$	8406.00	0.454856
$700$	6104.00	0.329585
$701$	−30618.0	−1.64968	−0.824840	−	0.565366i	$-0.808735\pi$
$701$	−30618.0	−1.64968	−0.824840	+	0.565366i	$0.808735\pi$
$702$	6696.00	0.360006
$703$	−24600.0	−1.31978
$704$	−31744.0	−1.69943
$705$	3888.00	0.207703
$706$	−34288.0	−1.82783
$707$	−1624.00	−0.0863887
$708$	−2880.00	−0.152877
$709$	−8130.00	−0.430647	−0.215323	−	0.976543i	$-0.569081\pi$
$709$	−8130.00	−0.430647	−0.215323	+	0.976543i	$0.569081\pi$
$710$	10848.0	0.573406
$711$	6660.00	0.351293
$712$	0	0
$713$	2016.00	0.105890
$714$	7056.00	0.369838
$715$	15376.0	0.804237
$716$	4880.00	0.254713
$717$	3510.00	0.182822
$718$	−19160.0	−0.995884
$719$	−27840.0	−1.44403	−0.722014	−	0.691878i	$-0.756784\pi$
$719$	−27840.0	−1.44403	−0.722014	+	0.691878i	$0.756784\pi$
$720$	2304.00	0.119257
$721$	12544.0	0.647938
$722$	12564.0	0.647623
$723$	7014.00	0.360793
$724$	8336.00	0.427907
$725$	1090.00	0.0558367
$726$	−30156.0	−1.54159
$727$	14624.0	0.746044	0.373022	−	0.927822i	$-0.378322\pi$
$727$	14624.0	0.746044	0.373022	+	0.927822i	$0.378322\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	10272.0	0.520800
$731$	5712.00	0.289010
$732$	−14928.0	−0.753763
$733$	−20862.0	−1.05124	−0.525618	−	0.850721i	$-0.676166\pi$
$733$	−20862.0	−1.05124	−0.525618	+	0.850721i	$0.676166\pi$
$734$	21696.0	1.09103
$735$	588.000	0.0295084
$736$	10752.0	0.538484
$737$	56048.0	2.80130
$738$	−8928.00	−0.445317
$739$	−13920.0	−0.692903	−0.346452	−	0.938068i	$-0.612614\pi$
$739$	−13920.0	−0.692903	−0.346452	+	0.938068i	$0.612614\pi$
$740$	7872.00	0.391055
$741$	18600.0	0.922116
$742$	−7224.00	−0.357414
$743$	25578.0	1.26294	0.631471	−	0.775400i	$-0.282452\pi$
$743$	25578.0	1.26294	0.631471	+	0.775400i	$0.282452\pi$
$744$	0	0
$745$	−9480.00	−0.466202
$746$	7352.00	0.360826
$747$	4212.00	0.206304
$748$	41664.0	2.03661
$749$	13342.0	0.650876
$750$	−11232.0	−0.546846
$751$	33472.0	1.62638	0.813189	−	0.581999i	$-0.197729\pi$
$751$	33472.0	1.62638	0.813189	+	0.581999i	$0.197729\pi$
$752$	−20736.0	−1.00554
$753$	−8376.00	−0.405363
$754$	2480.00	0.119783
$755$	2272.00	0.109519
$756$	1512.00	0.0727393
$757$	25934.0	1.24516	0.622581	−	0.782556i	$-0.286084\pi$
$757$	25934.0	1.24516	0.622581	+	0.782556i	$0.286084\pi$
$758$	−17040.0	−0.816518
$759$	7812.00	0.373594
$760$	0	0
$761$	26952.0	1.28385	0.641925	−	0.766768i	$-0.278136\pi$
$761$	26952.0	1.28385	0.641925	+	0.766768i	$0.278136\pi$
$762$	−9648.00	−0.458675
$763$	630.000	0.0298919
$764$	−16304.0	−0.772065
$765$	−3024.00	−0.142919
$766$	36192.0	1.70714
$767$	−7440.00	−0.350251
$768$	−12288.0	−0.577350
$769$	23450.0	1.09965	0.549824	−	0.835281i	$-0.314695\pi$
$769$	23450.0	1.09965	0.549824	+	0.835281i	$0.314695\pi$
$770$	6944.00	0.324993
$771$	−21072.0	−0.984293
$772$	−20816.0	−0.970446
$773$	39568.0	1.84109	0.920545	−	0.390637i	$-0.127745\pi$
$773$	39568.0	1.84109	0.920545	+	0.390637i	$0.127745\pi$
$774$	2448.00	0.113684
$775$	5232.00	0.242502
$776$	0	0
$777$	−5166.00	−0.238519
$778$	−45960.0	−2.11793
$779$	−24800.0	−1.14063
$780$	−5952.00	−0.273225
$781$	−42036.0	−1.92595
$782$	−14112.0	−0.645325
$783$	270.000	0.0123231
$784$	−3136.00	−0.142857
$785$	1064.00	0.0483768
$786$	−9744.00	−0.442184
$787$	−12356.0	−0.559649	−0.279825	−	0.960051i	$-0.590276\pi$
$787$	−12356.0	−0.559649	−0.279825	+	0.960051i	$0.590276\pi$
$788$	18832.0	0.851348
$789$	−7314.00	−0.330019
$790$	−11840.0	−0.533226
$791$	−3206.00	−0.144112
$792$	0	0
$793$	−38564.0	−1.72692
$794$	−7464.00	−0.333611
$795$	3096.00	0.138118
$796$	13440.0	0.598452
$797$	−21736.0	−0.966033	−0.483017	−	0.875611i	$-0.660459\pi$
$797$	−21736.0	−0.966033	−0.483017	+	0.875611i	$0.660459\pi$
$798$	8400.00	0.372628
$799$	27216.0	1.20505
$800$	27904.0	1.23319
$801$	1800.00	0.0794006
$802$	54648.0	2.40609
$803$	−39804.0	−1.74926
$804$	−21696.0	−0.951690
$805$	−1176.00	−0.0514889
$806$	11904.0	0.520224
$807$	20340.0	0.887239
$808$	0	0
$809$	−38310.0	−1.66490	−0.832452	−	0.554097i	$-0.813064\pi$
$809$	−38310.0	−1.66490	−0.832452	+	0.554097i	$0.813064\pi$
$810$	−1296.00	−0.0562183
$811$	2132.00	0.0923115	0.0461558	−	0.998934i	$-0.485303\pi$
$811$	2132.00	0.0923115	0.0461558	+	0.998934i	$0.485303\pi$
$812$	560.000	0.0242022
$813$	5784.00	0.249513
$814$	−61008.0	−2.62694
$815$	1088.00	0.0467619
$816$	16128.0	0.691903
$817$	6800.00	0.291190
$818$	−52840.0	−2.25857
$819$	3906.00	0.166650
$820$	7936.00	0.337972
$821$	5002.00	0.212632	0.106316	−	0.994332i	$-0.466094\pi$
$821$	5002.00	0.212632	0.106316	+	0.994332i	$0.466094\pi$
$822$	−4968.00	−0.210802
$823$	−3612.00	−0.152985	−0.0764923	−	0.997070i	$-0.524372\pi$
$823$	−3612.00	−0.152985	−0.0764923	+	0.997070i	$0.524372\pi$
$824$	0	0
$825$	20274.0	0.855576
$826$	−3360.00	−0.141537
$827$	−27666.0	−1.16329	−0.581645	−	0.813443i	$-0.697591\pi$
$827$	−27666.0	−1.16329	−0.581645	+	0.813443i	$0.697591\pi$
$828$	−3024.00	−0.126922
$829$	12890.0	0.540034	0.270017	−	0.962856i	$-0.412971\pi$
$829$	12890.0	0.540034	0.270017	+	0.962856i	$0.412971\pi$
$830$	−7488.00	−0.313147
$831$	−16662.0	−0.695546
$832$	31744.0	1.32275
$833$	4116.00	0.171202
$834$	19440.0	0.807137
$835$	7504.00	0.311002
$836$	49600.0	2.05198
$837$	1296.00	0.0535201
$838$	27840.0	1.14763
$839$	−9340.00	−0.384330	−0.192165	−	0.981363i	$-0.561551\pi$
$839$	−9340.00	−0.384330	−0.192165	+	0.981363i	$0.561551\pi$
$840$	0	0
$841$	−24289.0	−0.995900
$842$	32648.0	1.33625
$843$	−5826.00	−0.238029
$844$	−5344.00	−0.217948
$845$	−6588.00	−0.268206
$846$	11664.0	0.474015
$847$	−17591.0	−0.713617
$848$	−16512.0	−0.668661
$849$	−14484.0	−0.585500
$850$	−36624.0	−1.47787
$851$	10332.0	0.416188
$852$	16272.0	0.654307
$853$	−33082.0	−1.32791	−0.663954	−	0.747773i	$-0.731123\pi$
$853$	−33082.0	−1.32791	−0.663954	+	0.747773i	$0.731123\pi$
$854$	−17416.0	−0.697849
$855$	−3600.00	−0.143997
$856$	0	0
$857$	7544.00	0.300698	0.150349	−	0.988633i	$-0.451960\pi$
$857$	7544.00	0.300698	0.150349	+	0.988633i	$0.451960\pi$
$858$	46128.0	1.83541
$859$	8180.00	0.324910	0.162455	−	0.986716i	$-0.448059\pi$
$859$	8180.00	0.324910	0.162455	+	0.986716i	$0.448059\pi$
$860$	−2176.00	−0.0862802
$861$	−5208.00	−0.206142
$862$	66408.0	2.62397
$863$	10518.0	0.414875	0.207437	−	0.978248i	$-0.433488\pi$
$863$	10518.0	0.414875	0.207437	+	0.978248i	$0.433488\pi$
$864$	6912.00	0.272166
$865$	608.000	0.0238990
$866$	30952.0	1.21454
$867$	−6429.00	−0.251834
$868$	2688.00	0.105111
$869$	45880.0	1.79099
$870$	−480.000	−0.0187052
$871$	−56048.0	−2.18038
$872$	0	0
$873$	−11394.0	−0.441728
$874$	−16800.0	−0.650193
$875$	−6552.00	−0.253141
$876$	15408.0	0.594279
$877$	14134.0	0.544209	0.272104	−	0.962268i	$-0.412280\pi$
$877$	14134.0	0.544209	0.272104	+	0.962268i	$0.412280\pi$
$878$	−3360.00	−0.129151
$879$	18456.0	0.708197
$880$	15872.0	0.608006
$881$	6492.00	0.248265	0.124132	−	0.992266i	$-0.460385\pi$
$881$	6492.00	0.248265	0.124132	+	0.992266i	$0.460385\pi$
$882$	1764.00	0.0673435
$883$	38228.0	1.45694	0.728468	−	0.685080i	$-0.240233\pi$
$883$	38228.0	1.45694	0.728468	+	0.685080i	$0.240233\pi$
$884$	−41664.0	−1.58519
$885$	1440.00	0.0546950
$886$	26472.0	1.00377
$887$	−43076.0	−1.63061	−0.815305	−	0.579032i	$-0.803431\pi$
$887$	−43076.0	−1.63061	−0.815305	+	0.579032i	$0.803431\pi$
$888$	0	0
$889$	−5628.00	−0.212325
$890$	−3200.00	−0.120522
$891$	5022.00	0.188825
$892$	−14656.0	−0.550134
$893$	32400.0	1.21414
$894$	−28440.0	−1.06396
$895$	−2440.00	−0.0911287
$896$	0	0
$897$	−7812.00	−0.290786
$898$	12360.0	0.459308
$899$	480.000	0.0178074
$900$	−7848.00	−0.290667
$901$	21672.0	0.801331
$902$	−61504.0	−2.27035
$903$	1428.00	0.0526255
$904$	0	0
$905$	−4168.00	−0.153093
$906$	6816.00	0.249941
$907$	−32236.0	−1.18013	−0.590065	−	0.807355i	$-0.700898\pi$
$907$	−32236.0	−1.18013	−0.590065	+	0.807355i	$0.700898\pi$
$908$	39552.0	1.44557
$909$	2088.00	0.0761877
$910$	−6944.00	−0.252958
$911$	−46518.0	−1.69178	−0.845889	−	0.533359i	$-0.820930\pi$
$911$	−46518.0	−1.69178	−0.845889	+	0.533359i	$0.820930\pi$
$912$	19200.0	0.697122
$913$	29016.0	1.05180
$914$	23656.0	0.856095
$915$	7464.00	0.269675
$916$	−43760.0	−1.57846
$917$	−5684.00	−0.204692
$918$	−9072.00	−0.326166
$919$	17840.0	0.640356	0.320178	−	0.947357i	$-0.396257\pi$
$919$	17840.0	0.640356	0.320178	+	0.947357i	$0.396257\pi$
$920$	0	0
$921$	−17652.0	−0.631545
$922$	−63872.0	−2.28147
$923$	42036.0	1.49906
$924$	10416.0	0.370846
$925$	26814.0	0.953123
$926$	−4688.00	−0.166369
$927$	−16128.0	−0.571427
$928$	2560.00	0.0905562
$929$	7000.00	0.247215	0.123607	−	0.992331i	$-0.460554\pi$
$929$	7000.00	0.247215	0.123607	+	0.992331i	$0.460554\pi$
$930$	−2304.00	−0.0812378
$931$	4900.00	0.172493
$932$	−22416.0	−0.787833
$933$	−27396.0	−0.961313
$934$	21216.0	0.743264
$935$	−20832.0	−0.728641
$936$	0	0
$937$	36114.0	1.25912	0.629559	−	0.776953i	$-0.283236\pi$
$937$	36114.0	1.25912	0.629559	+	0.776953i	$0.283236\pi$
$938$	−25312.0	−0.881094
$939$	28146.0	0.978179
$940$	−10368.0	−0.359752
$941$	−4748.00	−0.164485	−0.0822425	−	0.996612i	$-0.526208\pi$
$941$	−4748.00	−0.164485	−0.0822425	+	0.996612i	$0.526208\pi$
$942$	3192.00	0.110404
$943$	10416.0	0.359694
$944$	−7680.00	−0.264791
$945$	−756.000	−0.0260240
$946$	16864.0	0.579594
$947$	42694.0	1.46501	0.732507	−	0.680759i	$-0.238350\pi$
$947$	42694.0	1.46501	0.732507	+	0.680759i	$0.238350\pi$
$948$	−17760.0	−0.608458
$949$	39804.0	1.36153
$950$	−43600.0	−1.48902
$951$	−9342.00	−0.318544
$952$	0	0
$953$	−16742.0	−0.569073	−0.284537	−	0.958665i	$-0.591840\pi$
$953$	−16742.0	−0.569073	−0.284537	+	0.958665i	$0.591840\pi$
$954$	9288.00	0.315210
$955$	8152.00	0.276223
$956$	−9360.00	−0.316657
$957$	1860.00	0.0628268
$958$	22960.0	0.774326
$959$	−2898.00	−0.0975822
$960$	−6144.00	−0.206559
$961$	−27487.0	−0.922661
$962$	61008.0	2.04467
$963$	−17154.0	−0.574019
$964$	−18704.0	−0.624912
$965$	10408.0	0.347197
$966$	−3528.00	−0.117507
$967$	−9956.00	−0.331089	−0.165545	−	0.986202i	$-0.552938\pi$
$967$	−9956.00	−0.331089	−0.165545	+	0.986202i	$0.552938\pi$
$968$	0	0
$969$	−25200.0	−0.835439
$970$	20256.0	0.670496
$971$	−26388.0	−0.872123	−0.436061	−	0.899917i	$-0.643627\pi$
$971$	−26388.0	−0.872123	−0.436061	+	0.899917i	$0.643627\pi$
$972$	−1944.00	−0.0641500
$973$	11340.0	0.373632
$974$	35776.0	1.17694
$975$	−20274.0	−0.665936
$976$	−39808.0	−1.30556
$977$	−786.000	−0.0257383	−0.0128692	−	0.999917i	$-0.504096\pi$
$977$	−786.000	−0.0257383	−0.0128692	+	0.999917i	$0.504096\pi$
$978$	3264.00	0.106719
$979$	12400.0	0.404807
$980$	−1568.00	−0.0511101
$981$	−810.000	−0.0263622
$982$	−22232.0	−0.722456
$983$	51888.0	1.68359	0.841796	−	0.539796i	$-0.181499\pi$
$983$	51888.0	1.68359	0.841796	+	0.539796i	$0.181499\pi$
$984$	0	0
$985$	−9416.00	−0.304588
$986$	−3360.00	−0.108524
$987$	6804.00	0.219426
$988$	−49600.0	−1.59715
$989$	−2856.00	−0.0918256
$990$	−8928.00	−0.286617
$991$	−51928.0	−1.66453	−0.832264	−	0.554379i	$-0.812956\pi$
$991$	−51928.0	−1.66453	−0.832264	+	0.554379i	$0.812956\pi$
$992$	12288.0	0.393291
$993$	−4596.00	−0.146878
$994$	18984.0	0.605771
$995$	−6720.00	−0.214109
$996$	−11232.0	−0.357329
$997$	−386.000	−0.0122615	−0.00613076	−	0.999981i	$-0.501951\pi$
$997$	−386.000	−0.0122615	−0.00613076	+	0.999981i	$0.501951\pi$
$998$	−79280.0	−2.51459
$999$	6642.00	0.210354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.4.a.b.1.1	✓	1
3.2	odd	2		63.4.a.a.1.1		1
4.3	odd	2		336.4.a.h.1.1		1
5.2	odd	4		525.4.d.b.274.2		2
5.3	odd	4		525.4.d.b.274.1		2
5.4	even	2		525.4.a.b.1.1		1
7.2	even	3		147.4.e.c.67.1		2
7.3	odd	6		147.4.e.b.79.1		2
7.4	even	3		147.4.e.c.79.1		2
7.5	odd	6		147.4.e.b.67.1		2
7.6	odd	2		147.4.a.g.1.1		1
8.3	odd	2		1344.4.a.i.1.1		1
8.5	even	2		1344.4.a.w.1.1		1
12.11	even	2		1008.4.a.m.1.1		1
15.14	odd	2		1575.4.a.k.1.1		1
21.2	odd	6		441.4.e.m.361.1		2
21.5	even	6		441.4.e.n.361.1		2
21.11	odd	6		441.4.e.m.226.1		2
21.17	even	6		441.4.e.n.226.1		2
21.20	even	2		441.4.a.b.1.1		1
28.27	even	2		2352.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.b.1.1	✓	1	1.1	even	1	trivial
63.4.a.a.1.1		1	3.2	odd	2
147.4.a.g.1.1		1	7.6	odd	2
147.4.e.b.67.1		2	7.5	odd	6
147.4.e.b.79.1		2	7.3	odd	6
147.4.e.c.67.1		2	7.2	even	3
147.4.e.c.79.1		2	7.4	even	3
336.4.a.h.1.1		1	4.3	odd	2
441.4.a.b.1.1		1	21.20	even	2
441.4.e.m.226.1		2	21.11	odd	6
441.4.e.m.361.1		2	21.2	odd	6
441.4.e.n.226.1		2	21.17	even	6
441.4.e.n.361.1		2	21.5	even	6
525.4.a.b.1.1		1	5.4	even	2
525.4.d.b.274.1		2	5.3	odd	4
525.4.d.b.274.2		2	5.2	odd	4
1008.4.a.m.1.1		1	12.11	even	2
1344.4.a.i.1.1		1	8.3	odd	2
1344.4.a.w.1.1		1	8.5	even	2
1575.4.a.k.1.1		1	15.14	odd	2
2352.4.a.l.1.1		1	28.27	even	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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