LMFDB - Embedded newform 624.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -5.46410 q^{5} -12.3923 q^{7} +9.00000 q^{9} +6.78461 q^{11} -13.0000 q^{13} -16.3923 q^{15} +72.0666 q^{17} +99.2436 q^{19} -37.1769 q^{21} -120.708 q^{23} -95.1436 q^{25} +27.0000 q^{27} -185.138 q^{29} +85.4641 q^{31} +20.3538 q^{33} +67.7128 q^{35} -340.928 q^{37} -39.0000 q^{39} -427.587 q^{41} -64.9179 q^{43} -49.1769 q^{45} +39.2820 q^{47} -189.431 q^{49} +216.200 q^{51} -21.4462 q^{53} -37.0718 q^{55} +297.731 q^{57} -62.4205 q^{59} -423.149 q^{61} -111.531 q^{63} +71.0333 q^{65} -451.643 q^{67} -362.123 q^{69} -335.779 q^{71} -1016.60 q^{73} -285.431 q^{75} -84.0770 q^{77} +398.390 q^{79} +81.0000 q^{81} +865.672 q^{83} -393.779 q^{85} -555.415 q^{87} +641.577 q^{89} +161.100 q^{91} +256.392 q^{93} -542.277 q^{95} +1381.71 q^{97} +61.0615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 4 q^{5} - 4 q^{7} + 18 q^{9} - 28 q^{11} - 26 q^{13} - 12 q^{15} - 36 q^{17} - 44 q^{19} - 12 q^{21} + 8 q^{23} - 218 q^{25} + 54 q^{27} - 204 q^{29} + 164 q^{31} - 84 q^{33} + 80 q^{35}+ \cdots - 252 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 4 * q^5 - 4 * q^7 + 18 * q^9 - 28 * q^11 - 26 * q^13 - 12 * q^15 - 36 * q^17 - 44 * q^19 - 12 * q^21 + 8 * q^23 - 218 * q^25 + 54 * q^27 - 204 * q^29 + 164 * q^31 - 84 * q^33 + 80 * q^35 - 668 * q^37 - 78 * q^39 - 100 * q^41 + 272 * q^43 - 36 * q^45 - 60 * q^47 - 462 * q^49 - 108 * q^51 - 708 * q^53 - 88 * q^55 - 132 * q^57 + 180 * q^59 - 1068 * q^61 - 36 * q^63 + 52 * q^65 + 420 * q^67 + 24 * q^69 - 436 * q^71 - 412 * q^73 - 654 * q^75 - 376 * q^77 - 672 * q^79 + 162 * q^81 + 124 * q^83 - 552 * q^85 - 612 * q^87 + 140 * q^89 + 52 * q^91 + 492 * q^93 - 752 * q^95 - 188 * q^97 - 252 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.46410	−0.488724	−0.244362	−	0.969684i	$-0.578579\pi$
$5$	−5.46410	−0.488724	−0.244362	+	0.969684i	$0.578579\pi$
$6$	0	0
$7$	−12.3923	−0.669122	−0.334561	−	0.942374i	$-0.608588\pi$
$7$	−12.3923	−0.669122	−0.334561	+	0.942374i	$0.608588\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	6.78461	0.185967	0.0929835	−	0.995668i	$-0.470360\pi$
$11$	6.78461	0.185967	0.0929835	+	0.995668i	$0.470360\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−16.3923	−0.282165
$16$	0	0
$17$	72.0666	1.02816	0.514080	−	0.857742i	$-0.328133\pi$
$17$	72.0666	1.02816	0.514080	+	0.857742i	$0.328133\pi$
$18$	0	0
$19$	99.2436	1.19832	0.599159	−	0.800630i	$-0.295502\pi$
$19$	99.2436	1.19832	0.599159	+	0.800630i	$0.295502\pi$
$20$	0	0
$21$	−37.1769	−0.386318
$22$	0	0
$23$	−120.708	−1.09432	−0.547158	−	0.837029i	$-0.684290\pi$
$23$	−120.708	−1.09432	−0.547158	+	0.837029i	$0.684290\pi$
$24$	0	0
$25$	−95.1436	−0.761149
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−185.138	−1.18549	−0.592747	−	0.805388i	$-0.701957\pi$
$29$	−185.138	−1.18549	−0.592747	+	0.805388i	$0.701957\pi$
$30$	0	0
$31$	85.4641	0.495155	0.247578	−	0.968868i	$-0.420366\pi$
$31$	85.4641	0.495155	0.247578	+	0.968868i	$0.420366\pi$
$32$	0	0
$33$	20.3538	0.107368
$34$	0	0
$35$	67.7128	0.327016
$36$	0	0
$37$	−340.928	−1.51482	−0.757409	−	0.652941i	$-0.773535\pi$
$37$	−340.928	−1.51482	−0.757409	+	0.652941i	$0.773535\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−427.587	−1.62873	−0.814364	−	0.580354i	$-0.802914\pi$
$41$	−427.587	−1.62873	−0.814364	+	0.580354i	$0.802914\pi$
$42$	0	0
$43$	−64.9179	−0.230230	−0.115115	−	0.993352i	$-0.536724\pi$
$43$	−64.9179	−0.230230	−0.115115	+	0.993352i	$0.536724\pi$
$44$	0	0
$45$	−49.1769	−0.162908
$46$	0	0
$47$	39.2820	0.121912	0.0609561	−	0.998140i	$-0.480585\pi$
$47$	39.2820	0.121912	0.0609561	+	0.998140i	$0.480585\pi$
$48$	0	0
$49$	−189.431	−0.552276
$50$	0	0
$51$	216.200	0.593609
$52$	0	0
$53$	−21.4462	−0.0555824	−0.0277912	−	0.999614i	$-0.508847\pi$
$53$	−21.4462	−0.0555824	−0.0277912	+	0.999614i	$0.508847\pi$
$54$	0	0
$55$	−37.0718	−0.0908865
$56$	0	0
$57$	297.731	0.691849
$58$	0	0
$59$	−62.4205	−0.137736	−0.0688682	−	0.997626i	$-0.521939\pi$
$59$	−62.4205	−0.137736	−0.0688682	+	0.997626i	$0.521939\pi$
$60$	0	0
$61$	−423.149	−0.888175	−0.444087	−	0.895984i	$-0.646472\pi$
$61$	−423.149	−0.888175	−0.444087	+	0.895984i	$0.646472\pi$
$62$	0	0
$63$	−111.531	−0.223041
$64$	0	0
$65$	71.0333	0.135548
$66$	0	0
$67$	−451.643	−0.823538	−0.411769	−	0.911288i	$-0.635089\pi$
$67$	−451.643	−0.823538	−0.411769	+	0.911288i	$0.635089\pi$
$68$	0	0
$69$	−362.123	−0.631804
$70$	0	0
$71$	−335.779	−0.561263	−0.280632	−	0.959816i	$-0.590544\pi$
$71$	−335.779	−0.561263	−0.280632	+	0.959816i	$0.590544\pi$
$72$	0	0
$73$	−1016.60	−1.62992	−0.814959	−	0.579519i	$-0.803241\pi$
$73$	−1016.60	−1.62992	−0.814959	+	0.579519i	$0.803241\pi$
$74$	0	0
$75$	−285.431	−0.439449
$76$	0	0
$77$	−84.0770	−0.124435
$78$	0	0
$79$	398.390	0.567371	0.283686	−	0.958917i	$-0.408443\pi$
$79$	398.390	0.567371	0.283686	+	0.958917i	$0.408443\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	865.672	1.14482	0.572408	−	0.819969i	$-0.306009\pi$
$83$	865.672	1.14482	0.572408	+	0.819969i	$0.306009\pi$
$84$	0	0
$85$	−393.779	−0.502487
$86$	0	0
$87$	−555.415	−0.684446
$88$	0	0
$89$	641.577	0.764124	0.382062	−	0.924137i	$-0.375214\pi$
$89$	641.577	0.764124	0.382062	+	0.924137i	$0.375214\pi$
$90$	0	0
$91$	161.100	0.185581
$92$	0	0
$93$	256.392	0.285878
$94$	0	0
$95$	−542.277	−0.585647
$96$	0	0
$97$	1381.71	1.44630	0.723150	−	0.690691i	$-0.242693\pi$
$97$	1381.71	1.44630	0.723150	+	0.690691i	$0.242693\pi$
$98$	0	0
$99$	61.0615	0.0619890
$100$	0	0
$101$	932.662	0.918845	0.459422	−	0.888218i	$-0.348056\pi$
$101$	932.662	0.918845	0.459422	+	0.888218i	$0.348056\pi$
$102$	0	0
$103$	−70.3332	−0.0672829	−0.0336414	−	0.999434i	$-0.510710\pi$
$103$	−70.3332	−0.0672829	−0.0336414	+	0.999434i	$0.510710\pi$
$104$	0	0
$105$	203.138	0.188803
$106$	0	0
$107$	283.672	0.256295	0.128148	−	0.991755i	$-0.459097\pi$
$107$	283.672	0.256295	0.128148	+	0.991755i	$0.459097\pi$
$108$	0	0
$109$	−897.559	−0.788721	−0.394360	−	0.918956i	$-0.629034\pi$
$109$	−897.559	−0.788721	−0.394360	+	0.918956i	$0.629034\pi$
$110$	0	0
$111$	−1022.78	−0.874580
$112$	0	0
$113$	−640.656	−0.533344	−0.266672	−	0.963787i	$-0.585924\pi$
$113$	−640.656	−0.533344	−0.266672	+	0.963787i	$0.585924\pi$
$114$	0	0
$115$	659.559	0.534819
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−893.072	−0.687964
$120$	0	0
$121$	−1284.97	−0.965416
$122$	0	0
$123$	−1282.76	−0.940347
$124$	0	0
$125$	1202.89	0.860716
$126$	0	0
$127$	−999.369	−0.698265	−0.349133	−	0.937073i	$-0.613524\pi$
$127$	−999.369	−0.698265	−0.349133	+	0.937073i	$0.613524\pi$
$128$	0	0
$129$	−194.754	−0.132923
$130$	0	0
$131$	−2638.88	−1.76000	−0.880001	−	0.474973i	$-0.842458\pi$
$131$	−2638.88	−1.76000	−0.880001	+	0.474973i	$0.842458\pi$
$132$	0	0
$133$	−1229.86	−0.801820
$134$	0	0
$135$	−147.531	−0.0940550
$136$	0	0
$137$	1850.07	1.15374	0.576870	−	0.816836i	$-0.304274\pi$
$137$	1850.07	1.15374	0.576870	+	0.816836i	$0.304274\pi$
$138$	0	0
$139$	476.687	0.290878	0.145439	−	0.989367i	$-0.453541\pi$
$139$	476.687	0.290878	0.145439	+	0.989367i	$0.453541\pi$
$140$	0	0
$141$	117.846	0.0703860
$142$	0	0
$143$	−88.1999	−0.0515780
$144$	0	0
$145$	1011.62	0.579380
$146$	0	0
$147$	−568.292	−0.318857
$148$	0	0
$149$	−955.889	−0.525567	−0.262784	−	0.964855i	$-0.584641\pi$
$149$	−955.889	−0.525567	−0.262784	+	0.964855i	$0.584641\pi$
$150$	0	0
$151$	−1558.42	−0.839882	−0.419941	−	0.907551i	$-0.637949\pi$
$151$	−1558.42	−0.839882	−0.419941	+	0.907551i	$0.637949\pi$
$152$	0	0
$153$	648.600	0.342720
$154$	0	0
$155$	−466.985	−0.241994
$156$	0	0
$157$	−3292.06	−1.67347	−0.836736	−	0.547607i	$-0.815539\pi$
$157$	−3292.06	−1.67347	−0.836736	+	0.547607i	$0.815539\pi$
$158$	0	0
$159$	−64.3387	−0.0320905
$160$	0	0
$161$	1495.85	0.732231
$162$	0	0
$163$	255.279	0.122669	0.0613344	−	0.998117i	$-0.480464\pi$
$163$	255.279	0.122669	0.0613344	+	0.998117i	$0.480464\pi$
$164$	0	0
$165$	−111.215	−0.0524734
$166$	0	0
$167$	3433.68	1.59106	0.795528	−	0.605917i	$-0.207194\pi$
$167$	3433.68	1.59106	0.795528	+	0.605917i	$0.207194\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	893.192	0.399439
$172$	0	0
$173$	347.543	0.152735	0.0763677	−	0.997080i	$-0.475668\pi$
$173$	347.543	0.152735	0.0763677	+	0.997080i	$0.475668\pi$
$174$	0	0
$175$	1179.05	0.509301
$176$	0	0
$177$	−187.261	−0.0795222
$178$	0	0
$179$	−2267.54	−0.946837	−0.473418	−	0.880838i	$-0.656980\pi$
$179$	−2267.54	−0.946837	−0.473418	+	0.880838i	$0.656980\pi$
$180$	0	0
$181$	670.605	0.275390	0.137695	−	0.990475i	$-0.456031\pi$
$181$	670.605	0.275390	0.137695	+	0.990475i	$0.456031\pi$
$182$	0	0
$183$	−1269.45	−0.512788
$184$	0	0
$185$	1862.87	0.740328
$186$	0	0
$187$	488.944	0.191204
$188$	0	0
$189$	−334.592	−0.128773
$190$	0	0
$191$	−1446.88	−0.548129	−0.274065	−	0.961711i	$-0.588368\pi$
$191$	−1446.88	−0.548129	−0.274065	+	0.961711i	$0.588368\pi$
$192$	0	0
$193$	−1815.79	−0.677222	−0.338611	−	0.940927i	$-0.609957\pi$
$193$	−1815.79	−0.677222	−0.338611	+	0.940927i	$0.609957\pi$
$194$	0	0
$195$	213.100	0.0782585
$196$	0	0
$197$	1933.30	0.699196	0.349598	−	0.936900i	$-0.386318\pi$
$197$	1933.30	0.699196	0.349598	+	0.936900i	$0.386318\pi$
$198$	0	0
$199$	257.595	0.0917611	0.0458805	−	0.998947i	$-0.485391\pi$
$199$	257.595	0.0917611	0.0458805	+	0.998947i	$0.485391\pi$
$200$	0	0
$201$	−1354.93	−0.475470
$202$	0	0
$203$	2294.29	0.793240
$204$	0	0
$205$	2336.38	0.795999
$206$	0	0
$207$	−1086.37	−0.364772
$208$	0	0
$209$	673.329	0.222847
$210$	0	0
$211$	−1990.83	−0.649547	−0.324774	−	0.945792i	$-0.605288\pi$
$211$	−1990.83	−0.649547	−0.324774	+	0.945792i	$0.605288\pi$
$212$	0	0
$213$	−1007.34	−0.324045
$214$	0	0
$215$	354.718	0.112519
$216$	0	0
$217$	−1059.10	−0.331319
$218$	0	0
$219$	−3049.80	−0.941034
$220$	0	0
$221$	−936.866	−0.285160
$222$	0	0
$223$	3481.70	1.04552	0.522762	−	0.852479i	$-0.324902\pi$
$223$	3481.70	1.04552	0.522762	+	0.852479i	$0.324902\pi$
$224$	0	0
$225$	−856.292	−0.253716
$226$	0	0
$227$	3265.64	0.954836	0.477418	−	0.878676i	$-0.341573\pi$
$227$	3265.64	0.954836	0.477418	+	0.878676i	$0.341573\pi$
$228$	0	0
$229$	−771.939	−0.222756	−0.111378	−	0.993778i	$-0.535526\pi$
$229$	−771.939	−0.222756	−0.111378	+	0.993778i	$0.535526\pi$
$230$	0	0
$231$	−252.231	−0.0718423
$232$	0	0
$233$	5.41053	0.00152127	0.000760635	−	1.00000i	$-0.499758\pi$
$233$	5.41053	0.00152127	0.000760635		1.00000i	$0.499758\pi$
$234$	0	0
$235$	−214.641	−0.0595814
$236$	0	0
$237$	1195.17	0.327572
$238$	0	0
$239$	1328.61	0.359584	0.179792	−	0.983705i	$-0.442457\pi$
$239$	1328.61	0.359584	0.179792	+	0.983705i	$0.442457\pi$
$240$	0	0
$241$	3454.13	0.923236	0.461618	−	0.887079i	$-0.347269\pi$
$241$	3454.13	0.923236	0.461618	+	0.887079i	$0.347269\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1035.07	0.269911
$246$	0	0
$247$	−1290.17	−0.332353
$248$	0	0
$249$	2597.01	0.660960
$250$	0	0
$251$	7667.13	1.92807	0.964033	−	0.265781i	$-0.0856298\pi$
$251$	7667.13	1.92807	0.964033	+	0.265781i	$0.0856298\pi$
$252$	0	0
$253$	−818.954	−0.203507
$254$	0	0
$255$	−1181.34	−0.290111
$256$	0	0
$257$	6033.77	1.46450	0.732249	−	0.681037i	$-0.238471\pi$
$257$	6033.77	1.46450	0.732249	+	0.681037i	$0.238471\pi$
$258$	0	0
$259$	4224.89	1.01360
$260$	0	0
$261$	−1666.25	−0.395165
$262$	0	0
$263$	−120.625	−0.0282816	−0.0141408	−	0.999900i	$-0.504501\pi$
$263$	−120.625	−0.0282816	−0.0141408	+	0.999900i	$0.504501\pi$
$264$	0	0
$265$	117.184	0.0271645
$266$	0	0
$267$	1924.73	0.441167
$268$	0	0
$269$	−1207.01	−0.273578	−0.136789	−	0.990600i	$-0.543678\pi$
$269$	−1207.01	−0.273578	−0.136789	+	0.990600i	$0.543678\pi$
$270$	0	0
$271$	4969.39	1.11391	0.556954	−	0.830543i	$-0.311970\pi$
$271$	4969.39	1.11391	0.556954	+	0.830543i	$0.311970\pi$
$272$	0	0
$273$	483.300	0.107145
$274$	0	0
$275$	−645.512	−0.141549
$276$	0	0
$277$	−8498.78	−1.84347	−0.921737	−	0.387817i	$-0.873229\pi$
$277$	−8498.78	−1.84347	−0.921737	+	0.387817i	$0.873229\pi$
$278$	0	0
$279$	769.177	0.165052
$280$	0	0
$281$	−1782.60	−0.378438	−0.189219	−	0.981935i	$-0.560596\pi$
$281$	−1782.60	−0.378438	−0.189219	+	0.981935i	$0.560596\pi$
$282$	0	0
$283$	2119.57	0.445214	0.222607	−	0.974908i	$-0.428543\pi$
$283$	2119.57	0.445214	0.222607	+	0.974908i	$0.428543\pi$
$284$	0	0
$285$	−1626.83	−0.338123
$286$	0	0
$287$	5298.79	1.08982
$288$	0	0
$289$	280.601	0.0571140
$290$	0	0
$291$	4145.12	0.835022
$292$	0	0
$293$	−3426.59	−0.683220	−0.341610	−	0.939842i	$-0.610972\pi$
$293$	−3426.59	−0.683220	−0.341610	+	0.939842i	$0.610972\pi$
$294$	0	0
$295$	341.072	0.0673151
$296$	0	0
$297$	183.184	0.0357894
$298$	0	0
$299$	1569.20	0.303509
$300$	0	0
$301$	804.482	0.154052
$302$	0	0
$303$	2797.98	0.530495
$304$	0	0
$305$	2312.13	0.434072
$306$	0	0
$307$	−6328.81	−1.17656	−0.588280	−	0.808657i	$-0.700195\pi$
$307$	−6328.81	−1.17656	−0.588280	+	0.808657i	$0.700195\pi$
$308$	0	0
$309$	−211.000	−0.0388458
$310$	0	0
$311$	6256.63	1.14077	0.570387	−	0.821376i	$-0.306793\pi$
$311$	6256.63	1.14077	0.570387	+	0.821376i	$0.306793\pi$
$312$	0	0
$313$	5273.42	0.952305	0.476153	−	0.879363i	$-0.342031\pi$
$313$	5273.42	0.952305	0.476153	+	0.879363i	$0.342031\pi$
$314$	0	0
$315$	609.415	0.109005
$316$	0	0
$317$	1697.23	0.300713	0.150357	−	0.988632i	$-0.451958\pi$
$317$	1697.23	0.300713	0.150357	+	0.988632i	$0.451958\pi$
$318$	0	0
$319$	−1256.09	−0.220463
$320$	0	0
$321$	851.015	0.147972
$322$	0	0
$323$	7152.15	1.23206
$324$	0	0
$325$	1236.87	0.211105
$326$	0	0
$327$	−2692.68	−0.455368
$328$	0	0
$329$	−486.795	−0.0815741
$330$	0	0
$331$	−1118.14	−0.185675	−0.0928373	−	0.995681i	$-0.529594\pi$
$331$	−1118.14	−0.185675	−0.0928373	+	0.995681i	$0.529594\pi$
$332$	0	0
$333$	−3068.35	−0.504939
$334$	0	0
$335$	2467.83	0.402483
$336$	0	0
$337$	3523.47	0.569541	0.284771	−	0.958596i	$-0.408083\pi$
$337$	3523.47	0.569541	0.284771	+	0.958596i	$0.408083\pi$
$338$	0	0
$339$	−1921.97	−0.307926
$340$	0	0
$341$	579.841	0.0920825
$342$	0	0
$343$	6598.04	1.03866
$344$	0	0
$345$	1978.68	0.308778
$346$	0	0
$347$	799.968	0.123760	0.0618798	−	0.998084i	$-0.480290\pi$
$347$	799.968	0.123760	0.0618798	+	0.998084i	$0.480290\pi$
$348$	0	0
$349$	−4341.88	−0.665947	−0.332974	−	0.942936i	$-0.608052\pi$
$349$	−4341.88	−0.665947	−0.332974	+	0.942936i	$0.608052\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	9306.07	1.40315	0.701576	−	0.712595i	$-0.252480\pi$
$353$	9306.07	1.40315	0.701576	+	0.712595i	$0.252480\pi$
$354$	0	0
$355$	1834.73	0.274303
$356$	0	0
$357$	−2679.22	−0.397196
$358$	0	0
$359$	1263.82	0.185798	0.0928992	−	0.995676i	$-0.470387\pi$
$359$	1263.82	0.185798	0.0928992	+	0.995676i	$0.470387\pi$
$360$	0	0
$361$	2990.28	0.435965
$362$	0	0
$363$	−3854.91	−0.557383
$364$	0	0
$365$	5554.80	0.796580
$366$	0	0
$367$	−12720.9	−1.80934	−0.904670	−	0.426113i	$-0.859883\pi$
$367$	−12720.9	−1.80934	−0.904670	+	0.426113i	$0.859883\pi$
$368$	0	0
$369$	−3848.28	−0.542910
$370$	0	0
$371$	265.768	0.0371914
$372$	0	0
$373$	3519.00	0.488491	0.244245	−	0.969713i	$-0.421460\pi$
$373$	3519.00	0.488491	0.244245	+	0.969713i	$0.421460\pi$
$374$	0	0
$375$	3608.66	0.496935
$376$	0	0
$377$	2406.80	0.328797
$378$	0	0
$379$	−13881.2	−1.88135	−0.940673	−	0.339314i	$-0.889805\pi$
$379$	−13881.2	−1.88135	−0.940673	+	0.339314i	$0.889805\pi$
$380$	0	0
$381$	−2998.11	−0.403144
$382$	0	0
$383$	9275.31	1.23746	0.618729	−	0.785605i	$-0.287648\pi$
$383$	9275.31	1.23746	0.618729	+	0.785605i	$0.287648\pi$
$384$	0	0
$385$	459.405	0.0608141
$386$	0	0
$387$	−584.261	−0.0767433
$388$	0	0
$389$	1016.09	0.132436	0.0662181	−	0.997805i	$-0.478907\pi$
$389$	1016.09	0.132436	0.0662181	+	0.997805i	$0.478907\pi$
$390$	0	0
$391$	−8699.00	−1.12513
$392$	0	0
$393$	−7916.64	−1.01614
$394$	0	0
$395$	−2176.84	−0.277288
$396$	0	0
$397$	−8300.95	−1.04940	−0.524701	−	0.851287i	$-0.675823\pi$
$397$	−8300.95	−1.04940	−0.524701	+	0.851287i	$0.675823\pi$
$398$	0	0
$399$	−3689.57	−0.462931
$400$	0	0
$401$	9367.49	1.16656	0.583280	−	0.812271i	$-0.301769\pi$
$401$	9367.49	1.16656	0.583280	+	0.812271i	$0.301769\pi$
$402$	0	0
$403$	−1111.03	−0.137331
$404$	0	0
$405$	−442.592	−0.0543027
$406$	0	0
$407$	−2313.06	−0.281706
$408$	0	0
$409$	984.560	0.119030	0.0595151	−	0.998227i	$-0.481045\pi$
$409$	984.560	0.119030	0.0595151	+	0.998227i	$0.481045\pi$
$410$	0	0
$411$	5550.22	0.666113
$412$	0	0
$413$	773.534	0.0921625
$414$	0	0
$415$	−4730.12	−0.559500
$416$	0	0
$417$	1430.06	0.167939
$418$	0	0
$419$	−13998.0	−1.63210	−0.816049	−	0.577983i	$-0.803840\pi$
$419$	−13998.0	−1.63210	−0.816049	+	0.577983i	$0.803840\pi$
$420$	0	0
$421$	12495.1	1.44649	0.723247	−	0.690590i	$-0.242649\pi$
$421$	12495.1	1.44649	0.723247	+	0.690590i	$0.242649\pi$
$422$	0	0
$423$	353.538	0.0406374
$424$	0	0
$425$	−6856.68	−0.782583
$426$	0	0
$427$	5243.79	0.594297
$428$	0	0
$429$	−264.600	−0.0297785
$430$	0	0
$431$	3688.03	0.412172	0.206086	−	0.978534i	$-0.433927\pi$
$431$	3688.03	0.412172	0.206086	+	0.978534i	$0.433927\pi$
$432$	0	0
$433$	7132.01	0.791553	0.395776	−	0.918347i	$-0.370476\pi$
$433$	7132.01	0.791553	0.395776	+	0.918347i	$0.370476\pi$
$434$	0	0
$435$	3034.85	0.334505
$436$	0	0
$437$	−11979.5	−1.31134
$438$	0	0
$439$	−16089.0	−1.74917	−0.874583	−	0.484875i	$-0.838865\pi$
$439$	−16089.0	−1.74917	−0.874583	+	0.484875i	$0.838865\pi$
$440$	0	0
$441$	−1704.88	−0.184092
$442$	0	0
$443$	−8126.71	−0.871584	−0.435792	−	0.900047i	$-0.643532\pi$
$443$	−8126.71	−0.871584	−0.435792	+	0.900047i	$0.643532\pi$
$444$	0	0
$445$	−3505.64	−0.373446
$446$	0	0
$447$	−2867.67	−0.303436
$448$	0	0
$449$	5032.96	0.528998	0.264499	−	0.964386i	$-0.414793\pi$
$449$	5032.96	0.528998	0.264499	+	0.964386i	$0.414793\pi$
$450$	0	0
$451$	−2901.01	−0.302890
$452$	0	0
$453$	−4675.25	−0.484906
$454$	0	0
$455$	−880.267	−0.0906979
$456$	0	0
$457$	13560.8	1.38807	0.694035	−	0.719941i	$-0.255831\pi$
$457$	13560.8	1.38807	0.694035	+	0.719941i	$0.255831\pi$
$458$	0	0
$459$	1945.80	0.197870
$460$	0	0
$461$	−12346.8	−1.24739	−0.623694	−	0.781669i	$-0.714369\pi$
$461$	−12346.8	−1.24739	−0.623694	+	0.781669i	$0.714369\pi$
$462$	0	0
$463$	11593.0	1.16366	0.581829	−	0.813311i	$-0.302337\pi$
$463$	11593.0	1.16366	0.581829	+	0.813311i	$0.302337\pi$
$464$	0	0
$465$	−1400.95	−0.139715
$466$	0	0
$467$	−8356.87	−0.828073	−0.414036	−	0.910260i	$-0.635881\pi$
$467$	−8356.87	−0.828073	−0.414036	+	0.910260i	$0.635881\pi$
$468$	0	0
$469$	5596.90	0.551047
$470$	0	0
$471$	−9876.18	−0.966179
$472$	0	0
$473$	−440.443	−0.0428152
$474$	0	0
$475$	−9442.39	−0.912098
$476$	0	0
$477$	−193.016	−0.0185275
$478$	0	0
$479$	9082.15	0.866334	0.433167	−	0.901314i	$-0.357396\pi$
$479$	9082.15	0.866334	0.433167	+	0.901314i	$0.357396\pi$
$480$	0	0
$481$	4432.07	0.420135
$482$	0	0
$483$	4487.54	0.422754
$484$	0	0
$485$	−7549.79	−0.706842
$486$	0	0
$487$	−12068.8	−1.12297	−0.561487	−	0.827486i	$-0.689770\pi$
$487$	−12068.8	−1.12297	−0.561487	+	0.827486i	$0.689770\pi$
$488$	0	0
$489$	765.838	0.0708229
$490$	0	0
$491$	−8466.25	−0.778159	−0.389080	−	0.921204i	$-0.627207\pi$
$491$	−8466.25	−0.778159	−0.389080	+	0.921204i	$0.627207\pi$
$492$	0	0
$493$	−13342.3	−1.21888
$494$	0	0
$495$	−333.646	−0.0302955
$496$	0	0
$497$	4161.08	0.375553
$498$	0	0
$499$	−2026.69	−0.181818	−0.0909089	−	0.995859i	$-0.528977\pi$
$499$	−2026.69	−0.181818	−0.0909089	+	0.995859i	$0.528977\pi$
$500$	0	0
$501$	10301.0	0.918596
$502$	0	0
$503$	3781.23	0.335182	0.167591	−	0.985857i	$-0.446401\pi$
$503$	3781.23	0.335182	0.167591	+	0.985857i	$0.446401\pi$
$504$	0	0
$505$	−5096.16	−0.449061
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−1661.22	−0.144660	−0.0723302	−	0.997381i	$-0.523044\pi$
$509$	−1661.22	−0.144660	−0.0723302	+	0.997381i	$0.523044\pi$
$510$	0	0
$511$	12598.0	1.09061
$512$	0	0
$513$	2679.58	0.230616
$514$	0	0
$515$	384.308	0.0328828
$516$	0	0
$517$	266.513	0.0226716
$518$	0	0
$519$	1042.63	0.0881818
$520$	0	0
$521$	−3172.82	−0.266802	−0.133401	−	0.991062i	$-0.542590\pi$
$521$	−3172.82	−0.266802	−0.133401	+	0.991062i	$0.542590\pi$
$522$	0	0
$523$	−5582.08	−0.466706	−0.233353	−	0.972392i	$-0.574970\pi$
$523$	−5582.08	−0.466706	−0.233353	+	0.972392i	$0.574970\pi$
$524$	0	0
$525$	3537.15	0.294045
$526$	0	0
$527$	6159.11	0.509099
$528$	0	0
$529$	2403.34	0.197529
$530$	0	0
$531$	−561.784	−0.0459122
$532$	0	0
$533$	5558.63	0.451728
$534$	0	0
$535$	−1550.01	−0.125258
$536$	0	0
$537$	−6802.61	−0.546656
$538$	0	0
$539$	−1285.21	−0.102705
$540$	0	0
$541$	−14569.5	−1.15784	−0.578920	−	0.815385i	$-0.696526\pi$
$541$	−14569.5	−1.15784	−0.578920	+	0.815385i	$0.696526\pi$
$542$	0	0
$543$	2011.81	0.158997
$544$	0	0
$545$	4904.35	0.385467
$546$	0	0
$547$	13175.0	1.02984	0.514920	−	0.857238i	$-0.327822\pi$
$547$	13175.0	1.02984	0.514920	+	0.857238i	$0.327822\pi$
$548$	0	0
$549$	−3808.34	−0.296058
$550$	0	0
$551$	−18373.8	−1.42060
$552$	0	0
$553$	−4936.96	−0.379640
$554$	0	0
$555$	5588.60	0.427429
$556$	0	0
$557$	−22466.6	−1.70905	−0.854524	−	0.519412i	$-0.826151\pi$
$557$	−22466.6	−1.70905	−0.854524	+	0.519412i	$0.826151\pi$
$558$	0	0
$559$	843.933	0.0638543
$560$	0	0
$561$	1466.83	0.110392
$562$	0	0
$563$	15302.0	1.14547	0.572736	−	0.819740i	$-0.305882\pi$
$563$	15302.0	1.14547	0.572736	+	0.819740i	$0.305882\pi$
$564$	0	0
$565$	3500.61	0.260658
$566$	0	0
$567$	−1003.78	−0.0743468
$568$	0	0
$569$	23986.5	1.76725	0.883625	−	0.468196i	$-0.155096\pi$
$569$	23986.5	1.76725	0.883625	+	0.468196i	$0.155096\pi$
$570$	0	0
$571$	−16827.0	−1.23326	−0.616628	−	0.787254i	$-0.711502\pi$
$571$	−16827.0	−1.23326	−0.616628	+	0.787254i	$0.711502\pi$
$572$	0	0
$573$	−4340.64	−0.316463
$574$	0	0
$575$	11484.6	0.832938
$576$	0	0
$577$	−15544.9	−1.12157	−0.560783	−	0.827963i	$-0.689500\pi$
$577$	−15544.9	−1.12157	−0.560783	+	0.827963i	$0.689500\pi$
$578$	0	0
$579$	−5447.38	−0.390994
$580$	0	0
$581$	−10727.7	−0.766022
$582$	0	0
$583$	−145.504	−0.0103365
$584$	0	0
$585$	639.300	0.0451826
$586$	0	0
$587$	−12905.7	−0.907455	−0.453727	−	0.891141i	$-0.649906\pi$
$587$	−12905.7	−0.907455	−0.453727	+	0.891141i	$0.649906\pi$
$588$	0	0
$589$	8481.76	0.593353
$590$	0	0
$591$	5799.89	0.403681
$592$	0	0
$593$	−23367.5	−1.61819	−0.809097	−	0.587675i	$-0.800043\pi$
$593$	−23367.5	−1.61819	−0.809097	+	0.587675i	$0.800043\pi$
$594$	0	0
$595$	4879.84	0.336225
$596$	0	0
$597$	772.786	0.0529783
$598$	0	0
$599$	−18586.6	−1.26783	−0.633913	−	0.773404i	$-0.718552\pi$
$599$	−18586.6	−1.26783	−0.633913	+	0.773404i	$0.718552\pi$
$600$	0	0
$601$	10855.6	0.736790	0.368395	−	0.929669i	$-0.379907\pi$
$601$	10855.6	0.736790	0.368395	+	0.929669i	$0.379907\pi$
$602$	0	0
$603$	−4064.79	−0.274513
$604$	0	0
$605$	7021.20	0.471822
$606$	0	0
$607$	−27366.9	−1.82997	−0.914983	−	0.403493i	$-0.867796\pi$
$607$	−27366.9	−1.82997	−0.914983	+	0.403493i	$0.867796\pi$
$608$	0	0
$609$	6882.88	0.457977
$610$	0	0
$611$	−510.666	−0.0338124
$612$	0	0
$613$	17849.1	1.17605	0.588025	−	0.808843i	$-0.299906\pi$
$613$	17849.1	1.17605	0.588025	+	0.808843i	$0.299906\pi$
$614$	0	0
$615$	7009.14	0.459570
$616$	0	0
$617$	−6364.18	−0.415255	−0.207628	−	0.978208i	$-0.566574\pi$
$617$	−6364.18	−0.415255	−0.207628	+	0.978208i	$0.566574\pi$
$618$	0	0
$619$	10366.6	0.673133	0.336567	−	0.941660i	$-0.390734\pi$
$619$	10366.6	0.673133	0.336567	+	0.941660i	$0.390734\pi$
$620$	0	0
$621$	−3259.11	−0.210601
$622$	0	0
$623$	−7950.61	−0.511292
$624$	0	0
$625$	5320.25	0.340496
$626$	0	0
$627$	2019.99	0.128661
$628$	0	0
$629$	−24569.6	−1.55748
$630$	0	0
$631$	19163.5	1.20902	0.604508	−	0.796599i	$-0.293370\pi$
$631$	19163.5	1.20902	0.604508	+	0.796599i	$0.293370\pi$
$632$	0	0
$633$	−5972.49	−0.375016
$634$	0	0
$635$	5460.66	0.341259
$636$	0	0
$637$	2462.60	0.153174
$638$	0	0
$639$	−3022.02	−0.187088
$640$	0	0
$641$	−19112.1	−1.17766	−0.588832	−	0.808255i	$-0.700412\pi$
$641$	−19112.1	−1.17766	−0.588832	+	0.808255i	$0.700412\pi$
$642$	0	0
$643$	21612.9	1.32555	0.662775	−	0.748818i	$-0.269378\pi$
$643$	21612.9	1.32555	0.662775	+	0.748818i	$0.269378\pi$
$644$	0	0
$645$	1064.15	0.0649628
$646$	0	0
$647$	20487.9	1.24492	0.622460	−	0.782652i	$-0.286133\pi$
$647$	20487.9	1.24492	0.622460	+	0.782652i	$0.286133\pi$
$648$	0	0
$649$	−423.499	−0.0256144
$650$	0	0
$651$	−3177.29	−0.191287
$652$	0	0
$653$	−27263.9	−1.63387	−0.816937	−	0.576727i	$-0.804330\pi$
$653$	−27263.9	−1.63387	−0.816937	+	0.576727i	$0.804330\pi$
$654$	0	0
$655$	14419.1	0.860155
$656$	0	0
$657$	−9149.40	−0.543306
$658$	0	0
$659$	−25701.3	−1.51924	−0.759620	−	0.650367i	$-0.774615\pi$
$659$	−25701.3	−1.51924	−0.759620	+	0.650367i	$0.774615\pi$
$660$	0	0
$661$	−14294.0	−0.841107	−0.420553	−	0.907268i	$-0.638164\pi$
$661$	−14294.0	−0.841107	−0.420553	+	0.907268i	$0.638164\pi$
$662$	0	0
$663$	−2810.60	−0.164637
$664$	0	0
$665$	6720.06	0.391869
$666$	0	0
$667$	22347.6	1.29731
$668$	0	0
$669$	10445.1	0.603633
$670$	0	0
$671$	−2870.90	−0.165171
$672$	0	0
$673$	−15167.5	−0.868742	−0.434371	−	0.900734i	$-0.643029\pi$
$673$	−15167.5	−0.868742	−0.434371	+	0.900734i	$0.643029\pi$
$674$	0	0
$675$	−2568.88	−0.146483
$676$	0	0
$677$	26749.8	1.51858	0.759291	−	0.650751i	$-0.225546\pi$
$677$	26749.8	1.51858	0.759291	+	0.650751i	$0.225546\pi$
$678$	0	0
$679$	−17122.5	−0.967751
$680$	0	0
$681$	9796.91	0.551275
$682$	0	0
$683$	−4152.73	−0.232650	−0.116325	−	0.993211i	$-0.537111\pi$
$683$	−4152.73	−0.232650	−0.116325	+	0.993211i	$0.537111\pi$
$684$	0	0
$685$	−10109.0	−0.563861
$686$	0	0
$687$	−2315.82	−0.128608
$688$	0	0
$689$	278.801	0.0154158
$690$	0	0
$691$	22309.0	1.22818	0.614091	−	0.789235i	$-0.289523\pi$
$691$	22309.0	1.22818	0.614091	+	0.789235i	$0.289523\pi$
$692$	0	0
$693$	−756.693	−0.0414782
$694$	0	0
$695$	−2604.67	−0.142159
$696$	0	0
$697$	−30814.8	−1.67459
$698$	0	0
$699$	16.2316	0.000878305 0
$700$	0	0
$701$	12201.2	0.657392	0.328696	−	0.944436i	$-0.393391\pi$
$701$	12201.2	0.657392	0.328696	+	0.944436i	$0.393391\pi$
$702$	0	0
$703$	−33834.9	−1.81523
$704$	0	0
$705$	−643.923	−0.0343994
$706$	0	0
$707$	−11557.8	−0.614819
$708$	0	0
$709$	348.094	0.0184386	0.00921929	−	0.999958i	$-0.497065\pi$
$709$	348.094	0.0184386	0.00921929	+	0.999958i	$0.497065\pi$
$710$	0	0
$711$	3585.51	0.189124
$712$	0	0
$713$	−10316.2	−0.541856
$714$	0	0
$715$	481.933	0.0252074
$716$	0	0
$717$	3985.83	0.207606
$718$	0	0
$719$	32069.3	1.66340	0.831698	−	0.555228i	$-0.187369\pi$
$719$	32069.3	1.66340	0.831698	+	0.555228i	$0.187369\pi$
$720$	0	0
$721$	871.591	0.0450204
$722$	0	0
$723$	10362.4	0.533031
$724$	0	0
$725$	17614.7	0.902338
$726$	0	0
$727$	33679.4	1.71816	0.859079	−	0.511844i	$-0.171037\pi$
$727$	33679.4	1.71816	0.859079	+	0.511844i	$0.171037\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4678.41	−0.236713
$732$	0	0
$733$	3811.99	0.192086	0.0960429	−	0.995377i	$-0.469381\pi$
$733$	3811.99	0.192086	0.0960429	+	0.995377i	$0.469381\pi$
$734$	0	0
$735$	3105.21	0.155833
$736$	0	0
$737$	−3064.22	−0.153151
$738$	0	0
$739$	30425.3	1.51449	0.757247	−	0.653129i	$-0.226544\pi$
$739$	30425.3	1.51449	0.757247	+	0.653129i	$0.226544\pi$
$740$	0	0
$741$	−3870.50	−0.191884
$742$	0	0
$743$	−3628.76	−0.179174	−0.0895871	−	0.995979i	$-0.528555\pi$
$743$	−3628.76	−0.179174	−0.0895871	+	0.995979i	$0.528555\pi$
$744$	0	0
$745$	5223.08	0.256857
$746$	0	0
$747$	7791.04	0.381606
$748$	0	0
$749$	−3515.34	−0.171493
$750$	0	0
$751$	−30912.3	−1.50200	−0.751002	−	0.660300i	$-0.770429\pi$
$751$	−30912.3	−1.50200	−0.751002	+	0.660300i	$0.770429\pi$
$752$	0	0
$753$	23001.4	1.11317
$754$	0	0
$755$	8515.35	0.410471
$756$	0	0
$757$	−1575.52	−0.0756448	−0.0378224	−	0.999284i	$-0.512042\pi$
$757$	−1575.52	−0.0756448	−0.0378224	+	0.999284i	$0.512042\pi$
$758$	0	0
$759$	−2456.86	−0.117495
$760$	0	0
$761$	−14688.6	−0.699686	−0.349843	−	0.936808i	$-0.613765\pi$
$761$	−14688.6	−0.699686	−0.349843	+	0.936808i	$0.613765\pi$
$762$	0	0
$763$	11122.8	0.527750
$764$	0	0
$765$	−3544.02	−0.167496
$766$	0	0
$767$	811.466	0.0382012
$768$	0	0
$769$	−15519.7	−0.727769	−0.363885	−	0.931444i	$-0.618550\pi$
$769$	−15519.7	−0.727769	−0.363885	+	0.931444i	$0.618550\pi$
$770$	0	0
$771$	18101.3	0.845529
$772$	0	0
$773$	9956.57	0.463277	0.231638	−	0.972802i	$-0.425591\pi$
$773$	9956.57	0.463277	0.231638	+	0.972802i	$0.425591\pi$
$774$	0	0
$775$	−8131.36	−0.376887
$776$	0	0
$777$	12674.7	0.585201
$778$	0	0
$779$	−42435.3	−1.95173
$780$	0	0
$781$	−2278.13	−0.104376
$782$	0	0
$783$	−4998.74	−0.228149
$784$	0	0
$785$	17988.2	0.817866
$786$	0	0
$787$	24811.8	1.12382	0.561910	−	0.827199i	$-0.310067\pi$
$787$	24811.8	1.12382	0.561910	+	0.827199i	$0.310067\pi$
$788$	0	0
$789$	−361.876	−0.0163284
$790$	0	0
$791$	7939.21	0.356872
$792$	0	0
$793$	5500.93	0.246335
$794$	0	0
$795$	351.553	0.0156834
$796$	0	0
$797$	12326.4	0.547836	0.273918	−	0.961753i	$-0.411680\pi$
$797$	12326.4	0.547836	0.273918	+	0.961753i	$0.411680\pi$
$798$	0	0
$799$	2830.92	0.125345
$800$	0	0
$801$	5774.19	0.254708
$802$	0	0
$803$	−6897.23	−0.303111
$804$	0	0
$805$	−8173.46	−0.357859
$806$	0	0
$807$	−3621.02	−0.157950
$808$	0	0
$809$	16164.0	0.702468	0.351234	−	0.936288i	$-0.385762\pi$
$809$	16164.0	0.702468	0.351234	+	0.936288i	$0.385762\pi$
$810$	0	0
$811$	17158.8	0.742944	0.371472	−	0.928444i	$-0.378853\pi$
$811$	17158.8	0.742944	0.371472	+	0.928444i	$0.378853\pi$
$812$	0	0
$813$	14908.2	0.643115
$814$	0	0
$815$	−1394.87	−0.0599512
$816$	0	0
$817$	−6442.68	−0.275889
$818$	0	0
$819$	1449.90	0.0618603
$820$	0	0
$821$	28545.0	1.21343	0.606716	−	0.794919i	$-0.292487\pi$
$821$	28545.0	1.21343	0.606716	+	0.794919i	$0.292487\pi$
$822$	0	0
$823$	6976.67	0.295494	0.147747	−	0.989025i	$-0.452798\pi$
$823$	6976.67	0.295494	0.147747	+	0.989025i	$0.452798\pi$
$824$	0	0
$825$	−1936.54	−0.0817231
$826$	0	0
$827$	−21024.3	−0.884022	−0.442011	−	0.897010i	$-0.645735\pi$
$827$	−21024.3	−0.884022	−0.442011	+	0.897010i	$0.645735\pi$
$828$	0	0
$829$	29235.9	1.22485	0.612427	−	0.790527i	$-0.290193\pi$
$829$	29235.9	1.22485	0.612427	+	0.790527i	$0.290193\pi$
$830$	0	0
$831$	−25496.3	−1.06433
$832$	0	0
$833$	−13651.6	−0.567829
$834$	0	0
$835$	−18762.0	−0.777587
$836$	0	0
$837$	2307.53	0.0952926
$838$	0	0
$839$	−17066.4	−0.702261	−0.351131	−	0.936326i	$-0.614203\pi$
$839$	−17066.4	−0.702261	−0.351131	+	0.936326i	$0.614203\pi$
$840$	0	0
$841$	9887.24	0.405398
$842$	0	0
$843$	−5347.81	−0.218491
$844$	0	0
$845$	−923.433	−0.0375942
$846$	0	0
$847$	15923.7	0.645981
$848$	0	0
$849$	6358.72	0.257044
$850$	0	0
$851$	41152.6	1.65769
$852$	0	0
$853$	−37910.5	−1.52172	−0.760862	−	0.648913i	$-0.775224\pi$
$853$	−37910.5	−1.52172	−0.760862	+	0.648913i	$0.775224\pi$
$854$	0	0
$855$	−4880.49	−0.195216
$856$	0	0
$857$	−25489.3	−1.01598	−0.507991	−	0.861362i	$-0.669612\pi$
$857$	−25489.3	−1.01598	−0.507991	+	0.861362i	$0.669612\pi$
$858$	0	0
$859$	10266.7	0.407796	0.203898	−	0.978992i	$-0.434639\pi$
$859$	10266.7	0.407796	0.203898	+	0.978992i	$0.434639\pi$
$860$	0	0
$861$	15896.4	0.629206
$862$	0	0
$863$	15681.1	0.618529	0.309265	−	0.950976i	$-0.399917\pi$
$863$	15681.1	0.618529	0.309265	+	0.950976i	$0.399917\pi$
$864$	0	0
$865$	−1899.01	−0.0746454
$866$	0	0
$867$	841.803	0.0329748
$868$	0	0
$869$	2702.92	0.105512
$870$	0	0
$871$	5871.36	0.228408
$872$	0	0
$873$	12435.4	0.482100
$874$	0	0
$875$	−14906.5	−0.575924
$876$	0	0
$877$	19487.6	0.750341	0.375171	−	0.926956i	$-0.377584\pi$
$877$	19487.6	0.750341	0.375171	+	0.926956i	$0.377584\pi$
$878$	0	0
$879$	−10279.8	−0.394457
$880$	0	0
$881$	−24792.5	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$881$	−24792.5	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$882$	0	0
$883$	21275.9	0.810861	0.405430	−	0.914126i	$-0.367122\pi$
$883$	21275.9	0.810861	0.405430	+	0.914126i	$0.367122\pi$
$884$	0	0
$885$	1023.22	0.0388644
$886$	0	0
$887$	−39291.8	−1.48736	−0.743682	−	0.668534i	$-0.766922\pi$
$887$	−39291.8	−1.48736	−0.743682	+	0.668534i	$0.766922\pi$
$888$	0	0
$889$	12384.5	0.467224
$890$	0	0
$891$	549.553	0.0206630
$892$	0	0
$893$	3898.49	0.146090
$894$	0	0
$895$	12390.1	0.462742
$896$	0	0
$897$	4707.60	0.175231
$898$	0	0
$899$	−15822.7	−0.587004
$900$	0	0
$901$	−1545.56	−0.0571476
$902$	0	0
$903$	2413.45	0.0889418
$904$	0	0
$905$	−3664.25	−0.134590
$906$	0	0
$907$	−40944.8	−1.49895	−0.749477	−	0.662031i	$-0.769695\pi$
$907$	−40944.8	−1.49895	−0.749477	+	0.662031i	$0.769695\pi$
$908$	0	0
$909$	8393.95	0.306282
$910$	0	0
$911$	23017.5	0.837104	0.418552	−	0.908193i	$-0.362538\pi$
$911$	23017.5	0.837104	0.418552	+	0.908193i	$0.362538\pi$
$912$	0	0
$913$	5873.24	0.212898
$914$	0	0
$915$	6936.38	0.250612
$916$	0	0
$917$	32701.8	1.17765
$918$	0	0
$919$	−29694.7	−1.06587	−0.532936	−	0.846155i	$-0.678911\pi$
$919$	−29694.7	−1.06587	−0.532936	+	0.846155i	$0.678911\pi$
$920$	0	0
$921$	−18986.4	−0.679287
$922$	0	0
$923$	4365.13	0.155666
$924$	0	0
$925$	32437.1	1.15300
$926$	0	0
$927$	−632.999	−0.0224276
$928$	0	0
$929$	24057.2	0.849613	0.424807	−	0.905284i	$-0.360342\pi$
$929$	24057.2	0.849613	0.424807	+	0.905284i	$0.360342\pi$
$930$	0	0
$931$	−18799.8	−0.661802
$932$	0	0
$933$	18769.9	0.658626
$934$	0	0
$935$	−2671.64	−0.0934460
$936$	0	0
$937$	39359.6	1.37228	0.686138	−	0.727472i	$-0.259305\pi$
$937$	39359.6	1.37228	0.686138	+	0.727472i	$0.259305\pi$
$938$	0	0
$939$	15820.3	0.549814
$940$	0	0
$941$	20967.7	0.726384	0.363192	−	0.931714i	$-0.381687\pi$
$941$	20967.7	0.726384	0.363192	+	0.931714i	$0.381687\pi$
$942$	0	0
$943$	51613.0	1.78235
$944$	0	0
$945$	1828.25	0.0629342
$946$	0	0
$947$	−28580.9	−0.980733	−0.490366	−	0.871516i	$-0.663137\pi$
$947$	−28580.9	−0.980733	−0.490366	+	0.871516i	$0.663137\pi$
$948$	0	0
$949$	13215.8	0.452058
$950$	0	0
$951$	5091.70	0.173617
$952$	0	0
$953$	−21949.5	−0.746081	−0.373041	−	0.927815i	$-0.621685\pi$
$953$	−21949.5	−0.746081	−0.373041	+	0.927815i	$0.621685\pi$
$954$	0	0
$955$	7905.91	0.267884
$956$	0	0
$957$	−3768.28	−0.127284
$958$	0	0
$959$	−22926.7	−0.771993
$960$	0	0
$961$	−22486.9	−0.754822
$962$	0	0
$963$	2553.04	0.0854317
$964$	0	0
$965$	9921.69	0.330974
$966$	0	0
$967$	−12955.2	−0.430829	−0.215415	−	0.976523i	$-0.569110\pi$
$967$	−12955.2	−0.430829	−0.215415	+	0.976523i	$0.569110\pi$
$968$	0	0
$969$	21456.4	0.711332
$970$	0	0
$971$	−36777.4	−1.21549	−0.607745	−	0.794132i	$-0.707926\pi$
$971$	−36777.4	−1.21549	−0.607745	+	0.794132i	$0.707926\pi$
$972$	0	0
$973$	−5907.25	−0.194633
$974$	0	0
$975$	3710.60	0.121881
$976$	0	0
$977$	−23701.7	−0.776136	−0.388068	−	0.921631i	$-0.626857\pi$
$977$	−23701.7	−0.776136	−0.388068	+	0.921631i	$0.626857\pi$
$978$	0	0
$979$	4352.85	0.142102
$980$	0	0
$981$	−8078.03	−0.262907
$982$	0	0
$983$	−9338.35	−0.302998	−0.151499	−	0.988457i	$-0.548410\pi$
$983$	−9338.35	−0.302998	−0.151499	+	0.988457i	$0.548410\pi$
$984$	0	0
$985$	−10563.7	−0.341714
$986$	0	0
$987$	−1460.38	−0.0470968
$988$	0	0
$989$	7836.09	0.251944
$990$	0	0
$991$	35647.5	1.14267	0.571333	−	0.820719i	$-0.306427\pi$
$991$	35647.5	1.14267	0.571333	+	0.820719i	$0.306427\pi$
$992$	0	0
$993$	−3354.41	−0.107199
$994$	0	0
$995$	−1407.53	−0.0448458
$996$	0	0
$997$	32855.1	1.04366	0.521832	−	0.853048i	$-0.325249\pi$
$997$	32855.1	1.04366	0.521832	+	0.853048i	$0.325249\pi$
$998$	0	0
$999$	−9205.06	−0.291527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.o.1.1		2
3.2	odd	2		1872.4.a.be.1.2		2
4.3	odd	2		312.4.a.a.1.1	✓	2
8.3	odd	2		2496.4.a.bg.1.2		2
8.5	even	2		2496.4.a.z.1.2		2
12.11	even	2		936.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.a.1.1	✓	2	4.3	odd	2
624.4.a.o.1.1		2	1.1	even	1	trivial
936.4.a.g.1.2		2	12.11	even	2
1872.4.a.be.1.2		2	3.2	odd	2
2496.4.a.z.1.2		2	8.5	even	2
2496.4.a.bg.1.2		2	8.3	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.46410 q^{5} -12.3923 q^{7} +9.00000 q^{9} +6.78461 q^{11} -13.0000 q^{13} -16.3923 q^{15} +72.0666 q^{17} +99.2436 q^{19} -37.1769 q^{21} -120.708 q^{23} -95.1436 q^{25} +27.0000 q^{27} -185.138 q^{29} +85.4641 q^{31} +20.3538 q^{33} +67.7128 q^{35} -340.928 q^{37} -39.0000 q^{39} -427.587 q^{41} -64.9179 q^{43} -49.1769 q^{45} +39.2820 q^{47} -189.431 q^{49} +216.200 q^{51} -21.4462 q^{53} -37.0718 q^{55} +297.731 q^{57} -62.4205 q^{59} -423.149 q^{61} -111.531 q^{63} +71.0333 q^{65} -451.643 q^{67} -362.123 q^{69} -335.779 q^{71} -1016.60 q^{73} -285.431 q^{75} -84.0770 q^{77} +398.390 q^{79} +81.0000 q^{81} +865.672 q^{83} -393.779 q^{85} -555.415 q^{87} +641.577 q^{89} +161.100 q^{91} +256.392 q^{93} -542.277 q^{95} +1381.71 q^{97} +61.0615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 4 q^{5} - 4 q^{7} + 18 q^{9} - 28 q^{11} - 26 q^{13} - 12 q^{15} - 36 q^{17} - 44 q^{19} - 12 q^{21} + 8 q^{23} - 218 q^{25} + 54 q^{27} - 204 q^{29} + 164 q^{31} - 84 q^{33} + 80 q^{35}+ \cdots - 252 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 4 * q^5 - 4 * q^7 + 18 * q^9 - 28 * q^11 - 26 * q^13 - 12 * q^15 - 36 * q^17 - 44 * q^19 - 12 * q^21 + 8 * q^23 - 218 * q^25 + 54 * q^27 - 204 * q^29 + 164 * q^31 - 84 * q^33 + 80 * q^35 - 668 * q^37 - 78 * q^39 - 100 * q^41 + 272 * q^43 - 36 * q^45 - 60 * q^47 - 462 * q^49 - 108 * q^51 - 708 * q^53 - 88 * q^55 - 132 * q^57 + 180 * q^59 - 1068 * q^61 - 36 * q^63 + 52 * q^65 + 420 * q^67 + 24 * q^69 - 436 * q^71 - 412 * q^73 - 654 * q^75 - 376 * q^77 - 672 * q^79 + 162 * q^81 + 124 * q^83 - 552 * q^85 - 612 * q^87 + 140 * q^89 + 52 * q^91 + 492 * q^93 - 752 * q^95 - 188 * q^97 - 252 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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