LMFDB - Embedded newform 1560.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 112x^{2} - 28x + 1648 $ x^4 - x^3 - 112x^2 - 28x + 1648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.4982$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -14.3589 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} -14.3589 q^{7} +9.00000 q^{9} -17.0410 q^{11} -13.0000 q^{13} +15.0000 q^{15} -15.6554 q^{17} +16.7926 q^{19} +43.0766 q^{21} +127.794 q^{23} +25.0000 q^{25} -27.0000 q^{27} +293.400 q^{29} +280.164 q^{31} +51.1229 q^{33} +71.7944 q^{35} -257.108 q^{37} +39.0000 q^{39} -39.0837 q^{41} -59.7647 q^{43} -45.0000 q^{45} -488.078 q^{47} -136.823 q^{49} +46.9661 q^{51} -235.012 q^{53} +85.2048 q^{55} -50.3778 q^{57} +469.840 q^{59} +24.7022 q^{61} -129.230 q^{63} +65.0000 q^{65} +649.946 q^{67} -383.383 q^{69} -286.928 q^{71} +377.207 q^{73} -75.0000 q^{75} +244.689 q^{77} +67.2993 q^{79} +81.0000 q^{81} +1505.85 q^{83} +78.2769 q^{85} -880.199 q^{87} -990.860 q^{89} +186.665 q^{91} -840.491 q^{93} -83.9629 q^{95} +1053.81 q^{97} -153.369 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−14.3589	−0.775307	−0.387653	−	0.921805i	$-0.626714\pi$
$7$	−14.3589	−0.775307	−0.387653	+	0.921805i	$0.626714\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−17.0410	−0.467095	−0.233547	−	0.972345i	$-0.575033\pi$
$11$	−17.0410	−0.467095	−0.233547	+	0.972345i	$0.575033\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−15.6554	−0.223352	−0.111676	−	0.993745i	$-0.535622\pi$
$17$	−15.6554	−0.223352	−0.111676	+	0.993745i	$0.535622\pi$
$18$	0	0
$19$	16.7926	0.202762	0.101381	−	0.994848i	$-0.467674\pi$
$19$	16.7926	0.202762	0.101381	+	0.994848i	$0.467674\pi$
$20$	0	0
$21$	43.0766	0.447623
$22$	0	0
$23$	127.794	1.15856	0.579282	−	0.815127i	$-0.303333\pi$
$23$	127.794	1.15856	0.579282	+	0.815127i	$0.303333\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	293.400	1.87872	0.939361	−	0.342929i	$-0.111419\pi$
$29$	293.400	1.87872	0.939361	+	0.342929i	$0.111419\pi$
$30$	0	0
$31$	280.164	1.62319	0.811595	−	0.584220i	$-0.198600\pi$
$31$	280.164	1.62319	0.811595	+	0.584220i	$0.198600\pi$
$32$	0	0
$33$	51.1229	0.269677
$34$	0	0
$35$	71.7944	0.346728
$36$	0	0
$37$	−257.108	−1.14239	−0.571194	−	0.820815i	$-0.693520\pi$
$37$	−257.108	−1.14239	−0.571194	+	0.820815i	$0.693520\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−39.0837	−0.148874	−0.0744372	−	0.997226i	$-0.523716\pi$
$41$	−39.0837	−0.148874	−0.0744372	+	0.997226i	$0.523716\pi$
$42$	0	0
$43$	−59.7647	−0.211954	−0.105977	−	0.994369i	$-0.533797\pi$
$43$	−59.7647	−0.211954	−0.105977	+	0.994369i	$0.533797\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−488.078	−1.51475	−0.757377	−	0.652977i	$-0.773520\pi$
$47$	−488.078	−1.51475	−0.757377	+	0.652977i	$0.773520\pi$
$48$	0	0
$49$	−136.823	−0.398900
$50$	0	0
$51$	46.9661	0.128952
$52$	0	0
$53$	−235.012	−0.609083	−0.304542	−	0.952499i	$-0.598503\pi$
$53$	−235.012	−0.609083	−0.304542	+	0.952499i	$0.598503\pi$
$54$	0	0
$55$	85.2048	0.208891
$56$	0	0
$57$	−50.3778	−0.117065
$58$	0	0
$59$	469.840	1.03674	0.518372	−	0.855155i	$-0.326538\pi$
$59$	469.840	1.03674	0.518372	+	0.855155i	$0.326538\pi$
$60$	0	0
$61$	24.7022	0.0518491	0.0259245	−	0.999664i	$-0.491747\pi$
$61$	24.7022	0.0518491	0.0259245	+	0.999664i	$0.491747\pi$
$62$	0	0
$63$	−129.230	−0.258436
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	649.946	1.18513	0.592564	−	0.805524i	$-0.298116\pi$
$67$	649.946	1.18513	0.592564	+	0.805524i	$0.298116\pi$
$68$	0	0
$69$	−383.383	−0.668897
$70$	0	0
$71$	−286.928	−0.479607	−0.239803	−	0.970822i	$-0.577083\pi$
$71$	−286.928	−0.479607	−0.239803	+	0.970822i	$0.577083\pi$
$72$	0	0
$73$	377.207	0.604777	0.302389	−	0.953185i	$-0.402216\pi$
$73$	377.207	0.604777	0.302389	+	0.953185i	$0.402216\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	244.689	0.362142
$78$	0	0
$79$	67.2993	0.0958450	0.0479225	−	0.998851i	$-0.484740\pi$
$79$	67.2993	0.0958450	0.0479225	+	0.998851i	$0.484740\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1505.85	1.99142	0.995711	−	0.0925181i	$-0.0294916\pi$
$83$	1505.85	1.99142	0.995711	+	0.0925181i	$0.0294916\pi$
$84$	0	0
$85$	78.2769	0.0998861
$86$	0	0
$87$	−880.199	−1.08468
$88$	0	0
$89$	−990.860	−1.18012	−0.590061	−	0.807358i	$-0.700896\pi$
$89$	−990.860	−1.18012	−0.590061	+	0.807358i	$0.700896\pi$
$90$	0	0
$91$	186.665	0.215031
$92$	0	0
$93$	−840.491	−0.937149
$94$	0	0
$95$	−83.9629	−0.0906780
$96$	0	0
$97$	1053.81	1.10308	0.551540	−	0.834149i	$-0.314041\pi$
$97$	1053.81	1.10308	0.551540	+	0.834149i	$0.314041\pi$
$98$	0	0
$99$	−153.369	−0.155698
$100$	0	0
$101$	−570.997	−0.562538	−0.281269	−	0.959629i	$-0.590755\pi$
$101$	−570.997	−0.562538	−0.281269	+	0.959629i	$0.590755\pi$
$102$	0	0
$103$	−1017.13	−0.973016	−0.486508	−	0.873676i	$-0.661729\pi$
$103$	−1017.13	−0.973016	−0.486508	+	0.873676i	$0.661729\pi$
$104$	0	0
$105$	−215.383	−0.200183
$106$	0	0
$107$	−545.878	−0.493196	−0.246598	−	0.969118i	$-0.579313\pi$
$107$	−545.878	−0.493196	−0.246598	+	0.969118i	$0.579313\pi$
$108$	0	0
$109$	−490.698	−0.431195	−0.215598	−	0.976482i	$-0.569170\pi$
$109$	−490.698	−0.431195	−0.215598	+	0.976482i	$0.569170\pi$
$110$	0	0
$111$	771.325	0.659558
$112$	0	0
$113$	−669.460	−0.557323	−0.278662	−	0.960389i	$-0.589891\pi$
$113$	−669.460	−0.557323	−0.278662	+	0.960389i	$0.589891\pi$
$114$	0	0
$115$	−638.972	−0.518125
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	224.794	0.173166
$120$	0	0
$121$	−1040.61	−0.781822
$122$	0	0
$123$	117.251	0.0859527
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−198.097	−0.138411	−0.0692057	−	0.997602i	$-0.522046\pi$
$127$	−198.097	−0.138411	−0.0692057	+	0.997602i	$0.522046\pi$
$128$	0	0
$129$	179.294	0.122372
$130$	0	0
$131$	240.952	0.160703	0.0803515	−	0.996767i	$-0.474396\pi$
$131$	240.952	0.160703	0.0803515	+	0.996767i	$0.474396\pi$
$132$	0	0
$133$	−241.123	−0.157203
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1146.39	−0.714907	−0.357454	−	0.933931i	$-0.616355\pi$
$137$	−1146.39	−0.714907	−0.357454	+	0.933931i	$0.616355\pi$
$138$	0	0
$139$	−1757.01	−1.07214	−0.536070	−	0.844174i	$-0.680092\pi$
$139$	−1757.01	−1.07214	−0.536070	+	0.844174i	$0.680092\pi$
$140$	0	0
$141$	1464.23	0.874544
$142$	0	0
$143$	221.532	0.129549
$144$	0	0
$145$	−1467.00	−0.840190
$146$	0	0
$147$	410.468	0.230305
$148$	0	0
$149$	1243.89	0.683913	0.341957	−	0.939716i	$-0.388910\pi$
$149$	1243.89	0.683913	0.341957	+	0.939716i	$0.388910\pi$
$150$	0	0
$151$	1597.34	0.860858	0.430429	−	0.902624i	$-0.358362\pi$
$151$	1597.34	0.860858	0.430429	+	0.902624i	$0.358362\pi$
$152$	0	0
$153$	−140.898	−0.0744507
$154$	0	0
$155$	−1400.82	−0.725913
$156$	0	0
$157$	95.7994	0.0486982	0.0243491	−	0.999704i	$-0.492249\pi$
$157$	95.7994	0.0486982	0.0243491	+	0.999704i	$0.492249\pi$
$158$	0	0
$159$	705.037	0.351654
$160$	0	0
$161$	−1834.98	−0.898242
$162$	0	0
$163$	−3000.46	−1.44181	−0.720903	−	0.693036i	$-0.756273\pi$
$163$	−3000.46	−1.44181	−0.720903	+	0.693036i	$0.756273\pi$
$164$	0	0
$165$	−255.614	−0.120603
$166$	0	0
$167$	−2889.70	−1.33899	−0.669497	−	0.742815i	$-0.733490\pi$
$167$	−2889.70	−1.33899	−0.669497	+	0.742815i	$0.733490\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	151.133	0.0675874
$172$	0	0
$173$	−2042.91	−0.897803	−0.448901	−	0.893581i	$-0.648185\pi$
$173$	−2042.91	−0.897803	−0.448901	+	0.893581i	$0.648185\pi$
$174$	0	0
$175$	−358.972	−0.155061
$176$	0	0
$177$	−1409.52	−0.598564
$178$	0	0
$179$	1299.03	0.542425	0.271212	−	0.962520i	$-0.412575\pi$
$179$	1299.03	0.542425	0.271212	+	0.962520i	$0.412575\pi$
$180$	0	0
$181$	−1104.52	−0.453581	−0.226791	−	0.973944i	$-0.572823\pi$
$181$	−1104.52	−0.453581	−0.226791	+	0.973944i	$0.572823\pi$
$182$	0	0
$183$	−74.1066	−0.0299351
$184$	0	0
$185$	1285.54	0.510892
$186$	0	0
$187$	266.783	0.104327
$188$	0	0
$189$	387.690	0.149208
$190$	0	0
$191$	851.877	0.322721	0.161360	−	0.986896i	$-0.448412\pi$
$191$	851.877	0.322721	0.161360	+	0.986896i	$0.448412\pi$
$192$	0	0
$193$	3104.68	1.15792	0.578962	−	0.815354i	$-0.303458\pi$
$193$	3104.68	1.15792	0.578962	+	0.815354i	$0.303458\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	−3413.82	−1.23464	−0.617322	−	0.786711i	$-0.711782\pi$
$197$	−3413.82	−1.23464	−0.617322	+	0.786711i	$0.711782\pi$
$198$	0	0
$199$	−1441.56	−0.513516	−0.256758	−	0.966476i	$-0.582654\pi$
$199$	−1441.56	−0.513516	−0.256758	+	0.966476i	$0.582654\pi$
$200$	0	0
$201$	−1949.84	−0.684233
$202$	0	0
$203$	−4212.89	−1.45659
$204$	0	0
$205$	195.419	0.0665787
$206$	0	0
$207$	1150.15	0.386188
$208$	0	0
$209$	−286.162	−0.0947092
$210$	0	0
$211$	−1755.46	−0.572754	−0.286377	−	0.958117i	$-0.592451\pi$
$211$	−1755.46	−0.572754	−0.286377	+	0.958117i	$0.592451\pi$
$212$	0	0
$213$	860.783	0.276901
$214$	0	0
$215$	298.823	0.0947887
$216$	0	0
$217$	−4022.84	−1.25847
$218$	0	0
$219$	−1131.62	−0.349168
$220$	0	0
$221$	203.520	0.0619467
$222$	0	0
$223$	2896.63	0.869832	0.434916	−	0.900471i	$-0.356778\pi$
$223$	2896.63	0.869832	0.434916	+	0.900471i	$0.356778\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	1880.86	0.549944	0.274972	−	0.961452i	$-0.411331\pi$
$227$	1880.86	0.549944	0.274972	+	0.961452i	$0.411331\pi$
$228$	0	0
$229$	2355.91	0.679839	0.339920	−	0.940455i	$-0.389600\pi$
$229$	2355.91	0.679839	0.339920	+	0.940455i	$0.389600\pi$
$230$	0	0
$231$	−734.067	−0.209083
$232$	0	0
$233$	−4473.38	−1.25777	−0.628885	−	0.777498i	$-0.716489\pi$
$233$	−4473.38	−1.25777	−0.628885	+	0.777498i	$0.716489\pi$
$234$	0	0
$235$	2440.39	0.677419
$236$	0	0
$237$	−201.898	−0.0553362
$238$	0	0
$239$	−398.688	−0.107904	−0.0539518	−	0.998544i	$-0.517182\pi$
$239$	−398.688	−0.107904	−0.0539518	+	0.998544i	$0.517182\pi$
$240$	0	0
$241$	−4525.85	−1.20969	−0.604846	−	0.796343i	$-0.706765\pi$
$241$	−4525.85	−1.20969	−0.604846	+	0.796343i	$0.706765\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	684.113	0.178393
$246$	0	0
$247$	−218.304	−0.0562361
$248$	0	0
$249$	−4517.54	−1.14975
$250$	0	0
$251$	1744.09	0.438589	0.219294	−	0.975659i	$-0.429624\pi$
$251$	1744.09	0.438589	0.219294	+	0.975659i	$0.429624\pi$
$252$	0	0
$253$	−2177.74	−0.541159
$254$	0	0
$255$	−234.831	−0.0576693
$256$	0	0
$257$	−5271.05	−1.27937	−0.639687	−	0.768635i	$-0.720936\pi$
$257$	−5271.05	−1.27937	−0.639687	+	0.768635i	$0.720936\pi$
$258$	0	0
$259$	3691.79	0.885701
$260$	0	0
$261$	2640.60	0.626241
$262$	0	0
$263$	−1483.22	−0.347753	−0.173876	−	0.984767i	$-0.555629\pi$
$263$	−1483.22	−0.347753	−0.173876	+	0.984767i	$0.555629\pi$
$264$	0	0
$265$	1175.06	0.272390
$266$	0	0
$267$	2972.58	0.681344
$268$	0	0
$269$	−1296.91	−0.293956	−0.146978	−	0.989140i	$-0.546955\pi$
$269$	−1296.91	−0.293956	−0.146978	+	0.989140i	$0.546955\pi$
$270$	0	0
$271$	8385.50	1.87964	0.939821	−	0.341667i	$-0.110991\pi$
$271$	8385.50	1.87964	0.939821	+	0.341667i	$0.110991\pi$
$272$	0	0
$273$	−559.996	−0.124148
$274$	0	0
$275$	−426.024	−0.0934190
$276$	0	0
$277$	813.294	0.176412	0.0882059	−	0.996102i	$-0.471887\pi$
$277$	813.294	0.176412	0.0882059	+	0.996102i	$0.471887\pi$
$278$	0	0
$279$	2521.47	0.541063
$280$	0	0
$281$	−1729.77	−0.367222	−0.183611	−	0.982999i	$-0.558779\pi$
$281$	−1729.77	−0.367222	−0.183611	+	0.982999i	$0.558779\pi$
$282$	0	0
$283$	−5332.95	−1.12018	−0.560090	−	0.828432i	$-0.689233\pi$
$283$	−5332.95	−1.12018	−0.560090	+	0.828432i	$0.689233\pi$
$284$	0	0
$285$	251.889	0.0523530
$286$	0	0
$287$	561.198	0.115423
$288$	0	0
$289$	−4667.91	−0.950114
$290$	0	0
$291$	−3161.44	−0.636863
$292$	0	0
$293$	−7668.21	−1.52895	−0.764474	−	0.644655i	$-0.777001\pi$
$293$	−7668.21	−1.52895	−0.764474	+	0.644655i	$0.777001\pi$
$294$	0	0
$295$	−2349.20	−0.463646
$296$	0	0
$297$	460.106	0.0898924
$298$	0	0
$299$	−1661.33	−0.321328
$300$	0	0
$301$	858.153	0.164329
$302$	0	0
$303$	1712.99	0.324782
$304$	0	0
$305$	−123.511	−0.0231876
$306$	0	0
$307$	−1089.37	−0.202521	−0.101260	−	0.994860i	$-0.532288\pi$
$307$	−1089.37	−0.202521	−0.101260	+	0.994860i	$0.532288\pi$
$308$	0	0
$309$	3051.38	0.561771
$310$	0	0
$311$	−6502.31	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$311$	−6502.31	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$312$	0	0
$313$	−4052.09	−0.731750	−0.365875	−	0.930664i	$-0.619230\pi$
$313$	−4052.09	−0.731750	−0.365875	+	0.930664i	$0.619230\pi$
$314$	0	0
$315$	646.150	0.115576
$316$	0	0
$317$	−8602.85	−1.52424	−0.762120	−	0.647436i	$-0.775841\pi$
$317$	−8602.85	−1.52424	−0.762120	+	0.647436i	$0.775841\pi$
$318$	0	0
$319$	−4999.81	−0.877541
$320$	0	0
$321$	1637.63	0.284747
$322$	0	0
$323$	−262.894	−0.0452874
$324$	0	0
$325$	−325.000	−0.0554700
$326$	0	0
$327$	1472.09	0.248951
$328$	0	0
$329$	7008.25	1.17440
$330$	0	0
$331$	−7234.18	−1.20129	−0.600645	−	0.799516i	$-0.705089\pi$
$331$	−7234.18	−1.20129	−0.600645	+	0.799516i	$0.705089\pi$
$332$	0	0
$333$	−2313.98	−0.380796
$334$	0	0
$335$	−3249.73	−0.530005
$336$	0	0
$337$	2526.24	0.408347	0.204173	−	0.978935i	$-0.434549\pi$
$337$	2526.24	0.408347	0.204173	+	0.978935i	$0.434549\pi$
$338$	0	0
$339$	2008.38	0.321771
$340$	0	0
$341$	−4774.26	−0.758184
$342$	0	0
$343$	6889.71	1.08458
$344$	0	0
$345$	1916.91	0.299140
$346$	0	0
$347$	8846.22	1.36856	0.684280	−	0.729220i	$-0.260117\pi$
$347$	8846.22	1.36856	0.684280	+	0.729220i	$0.260117\pi$
$348$	0	0
$349$	9235.17	1.41647	0.708234	−	0.705978i	$-0.249492\pi$
$349$	9235.17	1.41647	0.708234	+	0.705978i	$0.249492\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−5365.53	−0.809004	−0.404502	−	0.914537i	$-0.632555\pi$
$353$	−5365.53	−0.809004	−0.404502	+	0.914537i	$0.632555\pi$
$354$	0	0
$355$	1434.64	0.214487
$356$	0	0
$357$	−674.381	−0.0999777
$358$	0	0
$359$	−4098.58	−0.602548	−0.301274	−	0.953538i	$-0.597412\pi$
$359$	−4098.58	−0.602548	−0.301274	+	0.953538i	$0.597412\pi$
$360$	0	0
$361$	−6577.01	−0.958887
$362$	0	0
$363$	3121.82	0.451385
$364$	0	0
$365$	−1886.03	−0.270465
$366$	0	0
$367$	−6690.85	−0.951660	−0.475830	−	0.879537i	$-0.657852\pi$
$367$	−6690.85	−0.951660	−0.475830	+	0.879537i	$0.657852\pi$
$368$	0	0
$369$	−351.754	−0.0496248
$370$	0	0
$371$	3374.51	0.472226
$372$	0	0
$373$	661.594	0.0918393	0.0459196	−	0.998945i	$-0.485378\pi$
$373$	661.594	0.0918393	0.0459196	+	0.998945i	$0.485378\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−3814.20	−0.521064
$378$	0	0
$379$	13194.0	1.78821	0.894106	−	0.447856i	$-0.147812\pi$
$379$	13194.0	1.78821	0.894106	+	0.447856i	$0.147812\pi$
$380$	0	0
$381$	594.291	0.0799119
$382$	0	0
$383$	10800.3	1.44092	0.720458	−	0.693498i	$-0.243932\pi$
$383$	10800.3	1.44092	0.720458	+	0.693498i	$0.243932\pi$
$384$	0	0
$385$	−1223.45	−0.161955
$386$	0	0
$387$	−537.882	−0.0706514
$388$	0	0
$389$	−12633.2	−1.64661	−0.823303	−	0.567603i	$-0.807871\pi$
$389$	−12633.2	−1.64661	−0.823303	+	0.567603i	$0.807871\pi$
$390$	0	0
$391$	−2000.67	−0.258768
$392$	0	0
$393$	−722.857	−0.0927819
$394$	0	0
$395$	−336.496	−0.0428632
$396$	0	0
$397$	8669.51	1.09600	0.547998	−	0.836480i	$-0.315390\pi$
$397$	8669.51	1.09600	0.547998	+	0.836480i	$0.315390\pi$
$398$	0	0
$399$	723.368	0.0907611
$400$	0	0
$401$	−3828.05	−0.476717	−0.238359	−	0.971177i	$-0.576609\pi$
$401$	−3828.05	−0.476717	−0.238359	+	0.971177i	$0.576609\pi$
$402$	0	0
$403$	−3642.13	−0.450192
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	4381.37	0.533604
$408$	0	0
$409$	−2202.00	−0.266215	−0.133107	−	0.991102i	$-0.542496\pi$
$409$	−2202.00	−0.266215	−0.133107	+	0.991102i	$0.542496\pi$
$410$	0	0
$411$	3439.16	0.412752
$412$	0	0
$413$	−6746.37	−0.803794
$414$	0	0
$415$	−7529.23	−0.890591
$416$	0	0
$417$	5271.02	0.619000
$418$	0	0
$419$	−2186.17	−0.254896	−0.127448	−	0.991845i	$-0.540679\pi$
$419$	−2186.17	−0.254896	−0.127448	+	0.991845i	$0.540679\pi$
$420$	0	0
$421$	2708.04	0.313496	0.156748	−	0.987639i	$-0.449899\pi$
$421$	2708.04	0.313496	0.156748	+	0.987639i	$0.449899\pi$
$422$	0	0
$423$	−4392.70	−0.504918
$424$	0	0
$425$	−391.384	−0.0446704
$426$	0	0
$427$	−354.696	−0.0401989
$428$	0	0
$429$	−664.597	−0.0747950
$430$	0	0
$431$	2657.22	0.296969	0.148485	−	0.988915i	$-0.452560\pi$
$431$	2657.22	0.296969	0.148485	+	0.988915i	$0.452560\pi$
$432$	0	0
$433$	−5243.87	−0.581996	−0.290998	−	0.956724i	$-0.593987\pi$
$433$	−5243.87	−0.581996	−0.290998	+	0.956724i	$0.593987\pi$
$434$	0	0
$435$	4401.00	0.485084
$436$	0	0
$437$	2146.00	0.234913
$438$	0	0
$439$	14392.1	1.56468	0.782341	−	0.622850i	$-0.214025\pi$
$439$	14392.1	1.56468	0.782341	+	0.622850i	$0.214025\pi$
$440$	0	0
$441$	−1231.40	−0.132967
$442$	0	0
$443$	−7795.64	−0.836077	−0.418039	−	0.908429i	$-0.637282\pi$
$443$	−7795.64	−0.836077	−0.418039	+	0.908429i	$0.637282\pi$
$444$	0	0
$445$	4954.30	0.527767
$446$	0	0
$447$	−3731.66	−0.394857
$448$	0	0
$449$	−11443.2	−1.20275	−0.601377	−	0.798965i	$-0.705381\pi$
$449$	−11443.2	−1.20275	−0.601377	+	0.798965i	$0.705381\pi$
$450$	0	0
$451$	666.024	0.0695385
$452$	0	0
$453$	−4792.02	−0.497017
$454$	0	0
$455$	−933.327	−0.0961649
$456$	0	0
$457$	6876.61	0.703882	0.351941	−	0.936022i	$-0.385522\pi$
$457$	6876.61	0.703882	0.351941	+	0.936022i	$0.385522\pi$
$458$	0	0
$459$	422.695	0.0429841
$460$	0	0
$461$	6753.92	0.682345	0.341173	−	0.940001i	$-0.389176\pi$
$461$	6753.92	0.682345	0.341173	+	0.940001i	$0.389176\pi$
$462$	0	0
$463$	−17866.8	−1.79339	−0.896695	−	0.442648i	$-0.854039\pi$
$463$	−17866.8	−1.79339	−0.896695	+	0.442648i	$0.854039\pi$
$464$	0	0
$465$	4202.46	0.419106
$466$	0	0
$467$	9541.71	0.945476	0.472738	−	0.881203i	$-0.343266\pi$
$467$	9541.71	0.945476	0.472738	+	0.881203i	$0.343266\pi$
$468$	0	0
$469$	−9332.49	−0.918837
$470$	0	0
$471$	−287.398	−0.0281159
$472$	0	0
$473$	1018.45	0.0990026
$474$	0	0
$475$	419.815	0.0405525
$476$	0	0
$477$	−2115.11	−0.203028
$478$	0	0
$479$	12554.5	1.19756	0.598778	−	0.800915i	$-0.295653\pi$
$479$	12554.5	1.19756	0.598778	+	0.800915i	$0.295653\pi$
$480$	0	0
$481$	3342.41	0.316842
$482$	0	0
$483$	5504.95	0.518600
$484$	0	0
$485$	−5269.07	−0.493312
$486$	0	0
$487$	14273.8	1.32815	0.664075	−	0.747666i	$-0.268825\pi$
$487$	14273.8	1.32815	0.664075	+	0.747666i	$0.268825\pi$
$488$	0	0
$489$	9001.39	0.832427
$490$	0	0
$491$	−1401.23	−0.128792	−0.0643958	−	0.997924i	$-0.520512\pi$
$491$	−1401.23	−0.128792	−0.0643958	+	0.997924i	$0.520512\pi$
$492$	0	0
$493$	−4593.28	−0.419617
$494$	0	0
$495$	766.843	0.0696304
$496$	0	0
$497$	4119.96	0.371842
$498$	0	0
$499$	−2560.46	−0.229703	−0.114852	−	0.993383i	$-0.536639\pi$
$499$	−2560.46	−0.229703	−0.114852	+	0.993383i	$0.536639\pi$
$500$	0	0
$501$	8669.11	0.773068
$502$	0	0
$503$	−8487.36	−0.752351	−0.376176	−	0.926548i	$-0.622761\pi$
$503$	−8487.36	−0.752351	−0.376176	+	0.926548i	$0.622761\pi$
$504$	0	0
$505$	2854.99	0.251575
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	20934.3	1.82298	0.911491	−	0.411321i	$-0.134933\pi$
$509$	20934.3	1.82298	0.911491	+	0.411321i	$0.134933\pi$
$510$	0	0
$511$	−5416.27	−0.468888
$512$	0	0
$513$	−453.400	−0.0390216
$514$	0	0
$515$	5085.64	0.435146
$516$	0	0
$517$	8317.32	0.707534
$518$	0	0
$519$	6128.74	0.518347
$520$	0	0
$521$	22149.8	1.86258	0.931288	−	0.364283i	$-0.118686\pi$
$521$	22149.8	1.86258	0.931288	+	0.364283i	$0.118686\pi$
$522$	0	0
$523$	17584.2	1.47018	0.735089	−	0.677971i	$-0.237140\pi$
$523$	17584.2	1.47018	0.735089	+	0.677971i	$0.237140\pi$
$524$	0	0
$525$	1076.92	0.0895247
$526$	0	0
$527$	−4386.07	−0.362543
$528$	0	0
$529$	4164.39	0.342269
$530$	0	0
$531$	4228.56	0.345581
$532$	0	0
$533$	508.088	0.0412903
$534$	0	0
$535$	2729.39	0.220564
$536$	0	0
$537$	−3897.09	−0.313169
$538$	0	0
$539$	2331.59	0.186324
$540$	0	0
$541$	14870.4	1.18175	0.590877	−	0.806762i	$-0.298782\pi$
$541$	14870.4	1.18175	0.590877	+	0.806762i	$0.298782\pi$
$542$	0	0
$543$	3313.56	0.261875
$544$	0	0
$545$	2453.49	0.192836
$546$	0	0
$547$	11586.5	0.905673	0.452837	−	0.891594i	$-0.350412\pi$
$547$	11586.5	0.905673	0.452837	+	0.891594i	$0.350412\pi$
$548$	0	0
$549$	222.320	0.0172830
$550$	0	0
$551$	4926.94	0.380934
$552$	0	0
$553$	−966.342	−0.0743093
$554$	0	0
$555$	−3856.63	−0.294963
$556$	0	0
$557$	−6081.49	−0.462623	−0.231311	−	0.972880i	$-0.574302\pi$
$557$	−6081.49	−0.462623	−0.231311	+	0.972880i	$0.574302\pi$
$558$	0	0
$559$	776.941	0.0587855
$560$	0	0
$561$	−800.348	−0.0602330
$562$	0	0
$563$	−21261.9	−1.59162	−0.795809	−	0.605548i	$-0.792954\pi$
$563$	−21261.9	−1.59162	−0.795809	+	0.605548i	$0.792954\pi$
$564$	0	0
$565$	3347.30	0.249243
$566$	0	0
$567$	−1163.07	−0.0861452
$568$	0	0
$569$	56.6126	0.00417104	0.00208552	−	0.999998i	$-0.499336\pi$
$569$	56.6126	0.00417104	0.00208552	+	0.999998i	$0.499336\pi$
$570$	0	0
$571$	4646.44	0.340538	0.170269	−	0.985398i	$-0.445536\pi$
$571$	4646.44	0.340538	0.170269	+	0.985398i	$0.445536\pi$
$572$	0	0
$573$	−2555.63	−0.186323
$574$	0	0
$575$	3194.86	0.231713
$576$	0	0
$577$	3205.13	0.231250	0.115625	−	0.993293i	$-0.463113\pi$
$577$	3205.13	0.231250	0.115625	+	0.993293i	$0.463113\pi$
$578$	0	0
$579$	−9314.03	−0.668528
$580$	0	0
$581$	−21622.3	−1.54396
$582$	0	0
$583$	4004.83	0.284500
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	−23097.5	−1.62408	−0.812041	−	0.583600i	$-0.801644\pi$
$587$	−23097.5	−1.62408	−0.812041	+	0.583600i	$0.801644\pi$
$588$	0	0
$589$	4704.67	0.329122
$590$	0	0
$591$	10241.5	0.712822
$592$	0	0
$593$	−18320.2	−1.26867	−0.634333	−	0.773060i	$-0.718725\pi$
$593$	−18320.2	−1.26867	−0.634333	+	0.773060i	$0.718725\pi$
$594$	0	0
$595$	−1123.97	−0.0774424
$596$	0	0
$597$	4324.69	0.296479
$598$	0	0
$599$	−7924.66	−0.540556	−0.270278	−	0.962782i	$-0.587116\pi$
$599$	−7924.66	−0.540556	−0.270278	+	0.962782i	$0.587116\pi$
$600$	0	0
$601$	−19153.9	−1.30001	−0.650003	−	0.759932i	$-0.725232\pi$
$601$	−19153.9	−1.30001	−0.650003	+	0.759932i	$0.725232\pi$
$602$	0	0
$603$	5849.51	0.395042
$604$	0	0
$605$	5203.03	0.349642
$606$	0	0
$607$	11249.9	0.752254	0.376127	−	0.926568i	$-0.377256\pi$
$607$	11249.9	0.752254	0.376127	+	0.926568i	$0.377256\pi$
$608$	0	0
$609$	12638.7	0.840960
$610$	0	0
$611$	6345.01	0.420117
$612$	0	0
$613$	7498.86	0.494088	0.247044	−	0.969004i	$-0.420541\pi$
$613$	7498.86	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$614$	0	0
$615$	−586.256	−0.0384392
$616$	0	0
$617$	−26649.9	−1.73887	−0.869436	−	0.494045i	$-0.835518\pi$
$617$	−26649.9	−1.73887	−0.869436	+	0.494045i	$0.835518\pi$
$618$	0	0
$619$	18002.8	1.16897	0.584486	−	0.811404i	$-0.301296\pi$
$619$	18002.8	1.16897	0.584486	+	0.811404i	$0.301296\pi$
$620$	0	0
$621$	−3450.45	−0.222966
$622$	0	0
$623$	14227.6	0.914957
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	858.485	0.0546804
$628$	0	0
$629$	4025.13	0.255155
$630$	0	0
$631$	−5138.51	−0.324185	−0.162093	−	0.986776i	$-0.551824\pi$
$631$	−5138.51	−0.324185	−0.162093	+	0.986776i	$0.551824\pi$
$632$	0	0
$633$	5266.39	0.330679
$634$	0	0
$635$	990.484	0.0618995
$636$	0	0
$637$	1778.69	0.110635
$638$	0	0
$639$	−2582.35	−0.159869
$640$	0	0
$641$	1644.82	0.101352	0.0506759	−	0.998715i	$-0.483862\pi$
$641$	1644.82	0.101352	0.0506759	+	0.998715i	$0.483862\pi$
$642$	0	0
$643$	−19549.2	−1.19898	−0.599491	−	0.800381i	$-0.704630\pi$
$643$	−19549.2	−1.19898	−0.599491	+	0.800381i	$0.704630\pi$
$644$	0	0
$645$	−896.470	−0.0547263
$646$	0	0
$647$	8554.97	0.519831	0.259915	−	0.965631i	$-0.416305\pi$
$647$	8554.97	0.519831	0.259915	+	0.965631i	$0.416305\pi$
$648$	0	0
$649$	−8006.52	−0.484258
$650$	0	0
$651$	12068.5	0.726578
$652$	0	0
$653$	2885.08	0.172897	0.0864486	−	0.996256i	$-0.472448\pi$
$653$	2885.08	0.172897	0.0864486	+	0.996256i	$0.472448\pi$
$654$	0	0
$655$	−1204.76	−0.0718686
$656$	0	0
$657$	3394.86	0.201592
$658$	0	0
$659$	−7179.95	−0.424417	−0.212209	−	0.977224i	$-0.568066\pi$
$659$	−7179.95	−0.424417	−0.212209	+	0.977224i	$0.568066\pi$
$660$	0	0
$661$	−16312.9	−0.959908	−0.479954	−	0.877294i	$-0.659347\pi$
$661$	−16312.9	−0.959908	−0.479954	+	0.877294i	$0.659347\pi$
$662$	0	0
$663$	−610.560	−0.0357650
$664$	0	0
$665$	1205.61	0.0703033
$666$	0	0
$667$	37494.8	2.17662
$668$	0	0
$669$	−8689.88	−0.502198
$670$	0	0
$671$	−420.949	−0.0242184
$672$	0	0
$673$	−23422.3	−1.34155	−0.670775	−	0.741661i	$-0.734038\pi$
$673$	−23422.3	−1.34155	−0.670775	+	0.741661i	$0.734038\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	6619.14	0.375767	0.187883	−	0.982191i	$-0.439837\pi$
$677$	6619.14	0.375767	0.187883	+	0.982191i	$0.439837\pi$
$678$	0	0
$679$	−15131.6	−0.855224
$680$	0	0
$681$	−5642.59	−0.317510
$682$	0	0
$683$	4435.72	0.248504	0.124252	−	0.992251i	$-0.460347\pi$
$683$	4435.72	0.248504	0.124252	+	0.992251i	$0.460347\pi$
$684$	0	0
$685$	5731.93	0.319716
$686$	0	0
$687$	−7067.74	−0.392505
$688$	0	0
$689$	3055.16	0.168929
$690$	0	0
$691$	579.781	0.0319188	0.0159594	−	0.999873i	$-0.494920\pi$
$691$	579.781	0.0319188	0.0159594	+	0.999873i	$0.494920\pi$
$692$	0	0
$693$	2202.20	0.120714
$694$	0	0
$695$	8785.03	0.479475
$696$	0	0
$697$	611.870	0.0332514
$698$	0	0
$699$	13420.1	0.726174
$700$	0	0
$701$	970.166	0.0522720	0.0261360	−	0.999658i	$-0.491680\pi$
$701$	970.166	0.0522720	0.0261360	+	0.999658i	$0.491680\pi$
$702$	0	0
$703$	−4317.52	−0.231633
$704$	0	0
$705$	−7321.17	−0.391108
$706$	0	0
$707$	8198.88	0.436140
$708$	0	0
$709$	26032.6	1.37895	0.689475	−	0.724310i	$-0.257841\pi$
$709$	26032.6	1.37895	0.689475	+	0.724310i	$0.257841\pi$
$710$	0	0
$711$	605.693	0.0319483
$712$	0	0
$713$	35803.3	1.88057
$714$	0	0
$715$	−1107.66	−0.0579360
$716$	0	0
$717$	1196.06	0.0622982
$718$	0	0
$719$	−1469.70	−0.0762318	−0.0381159	−	0.999273i	$-0.512136\pi$
$719$	−1469.70	−0.0762318	−0.0381159	+	0.999273i	$0.512136\pi$
$720$	0	0
$721$	14604.8	0.754385
$722$	0	0
$723$	13577.5	0.698416
$724$	0	0
$725$	7334.99	0.375745
$726$	0	0
$727$	−2043.46	−0.104247	−0.0521236	−	0.998641i	$-0.516599\pi$
$727$	−2043.46	−0.104247	−0.0521236	+	0.998641i	$0.516599\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	935.638	0.0473404
$732$	0	0
$733$	4595.58	0.231571	0.115785	−	0.993274i	$-0.463061\pi$
$733$	4595.58	0.231571	0.115785	+	0.993274i	$0.463061\pi$
$734$	0	0
$735$	−2052.34	−0.102995
$736$	0	0
$737$	−11075.7	−0.553567
$738$	0	0
$739$	16877.2	0.840106	0.420053	−	0.907499i	$-0.362011\pi$
$739$	16877.2	0.840106	0.420053	+	0.907499i	$0.362011\pi$
$740$	0	0
$741$	654.911	0.0324679
$742$	0	0
$743$	−32603.6	−1.60984	−0.804920	−	0.593383i	$-0.797792\pi$
$743$	−32603.6	−1.60984	−0.804920	+	0.593383i	$0.797792\pi$
$744$	0	0
$745$	−6219.43	−0.305855
$746$	0	0
$747$	13552.6	0.663807
$748$	0	0
$749$	7838.19	0.382378
$750$	0	0
$751$	1967.92	0.0956196	0.0478098	−	0.998856i	$-0.484776\pi$
$751$	1967.92	0.0956196	0.0478098	+	0.998856i	$0.484776\pi$
$752$	0	0
$753$	−5232.26	−0.253219
$754$	0	0
$755$	−7986.70	−0.384988
$756$	0	0
$757$	631.853	0.0303370	0.0151685	−	0.999885i	$-0.495172\pi$
$757$	631.853	0.0303370	0.0151685	+	0.999885i	$0.495172\pi$
$758$	0	0
$759$	6533.21	0.312438
$760$	0	0
$761$	−26806.7	−1.27693	−0.638463	−	0.769653i	$-0.720429\pi$
$761$	−26806.7	−1.27693	−0.638463	+	0.769653i	$0.720429\pi$
$762$	0	0
$763$	7045.87	0.334309
$764$	0	0
$765$	704.492	0.0332954
$766$	0	0
$767$	−6107.91	−0.287541
$768$	0	0
$769$	−7598.66	−0.356326	−0.178163	−	0.984001i	$-0.557015\pi$
$769$	−7598.66	−0.356326	−0.178163	+	0.984001i	$0.557015\pi$
$770$	0	0
$771$	15813.2	0.738647
$772$	0	0
$773$	35095.5	1.63299	0.816493	−	0.577356i	$-0.195915\pi$
$773$	35095.5	1.63299	0.816493	+	0.577356i	$0.195915\pi$
$774$	0	0
$775$	7004.09	0.324638
$776$	0	0
$777$	−11075.4	−0.511360
$778$	0	0
$779$	−656.317	−0.0301861
$780$	0	0
$781$	4889.52	0.224022
$782$	0	0
$783$	−7921.79	−0.361560
$784$	0	0
$785$	−478.997	−0.0217785
$786$	0	0
$787$	−3200.94	−0.144983	−0.0724913	−	0.997369i	$-0.523095\pi$
$787$	−3200.94	−0.144983	−0.0724913	+	0.997369i	$0.523095\pi$
$788$	0	0
$789$	4449.65	0.200775
$790$	0	0
$791$	9612.70	0.432096
$792$	0	0
$793$	−321.129	−0.0143804
$794$	0	0
$795$	−3525.18	−0.157265
$796$	0	0
$797$	−12558.9	−0.558169	−0.279084	−	0.960267i	$-0.590031\pi$
$797$	−12558.9	−0.558169	−0.279084	+	0.960267i	$0.590031\pi$
$798$	0	0
$799$	7641.04	0.338324
$800$	0	0
$801$	−8917.74	−0.393374
$802$	0	0
$803$	−6427.97	−0.282488
$804$	0	0
$805$	9174.92	0.401706
$806$	0	0
$807$	3890.74	0.169716
$808$	0	0
$809$	−6110.17	−0.265541	−0.132770	−	0.991147i	$-0.542387\pi$
$809$	−6110.17	−0.265541	−0.132770	+	0.991147i	$0.542387\pi$
$810$	0	0
$811$	−29878.8	−1.29370	−0.646848	−	0.762619i	$-0.723913\pi$
$811$	−29878.8	−1.29370	−0.646848	+	0.762619i	$0.723913\pi$
$812$	0	0
$813$	−25156.5	−1.08521
$814$	0	0
$815$	15002.3	0.644795
$816$	0	0
$817$	−1003.60	−0.0429763
$818$	0	0
$819$	1679.99	0.0716771
$820$	0	0
$821$	−36617.6	−1.55659	−0.778296	−	0.627898i	$-0.783916\pi$
$821$	−36617.6	−1.55659	−0.778296	+	0.627898i	$0.783916\pi$
$822$	0	0
$823$	14007.9	0.593296	0.296648	−	0.954987i	$-0.404131\pi$
$823$	14007.9	0.593296	0.296648	+	0.954987i	$0.404131\pi$
$824$	0	0
$825$	1278.07	0.0539355
$826$	0	0
$827$	10184.3	0.428224	0.214112	−	0.976809i	$-0.431314\pi$
$827$	10184.3	0.428224	0.214112	+	0.976809i	$0.431314\pi$
$828$	0	0
$829$	29169.8	1.22209	0.611044	−	0.791597i	$-0.290750\pi$
$829$	29169.8	1.22209	0.611044	+	0.791597i	$0.290750\pi$
$830$	0	0
$831$	−2439.88	−0.101851
$832$	0	0
$833$	2142.01	0.0890951
$834$	0	0
$835$	14448.5	0.598816
$836$	0	0
$837$	−7564.42	−0.312383
$838$	0	0
$839$	12067.6	0.496568	0.248284	−	0.968687i	$-0.420133\pi$
$839$	12067.6	0.496568	0.248284	+	0.968687i	$0.420133\pi$
$840$	0	0
$841$	61694.4	2.52960
$842$	0	0
$843$	5189.30	0.212016
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	14941.9	0.606152
$848$	0	0
$849$	15998.8	0.646736
$850$	0	0
$851$	−32857.0	−1.32353
$852$	0	0
$853$	36116.7	1.44972	0.724860	−	0.688896i	$-0.241904\pi$
$853$	36116.7	1.44972	0.724860	+	0.688896i	$0.241904\pi$
$854$	0	0
$855$	−755.666	−0.0302260
$856$	0	0
$857$	−47865.4	−1.90788	−0.953938	−	0.300004i	$-0.903012\pi$
$857$	−47865.4	−1.90788	−0.953938	+	0.300004i	$0.903012\pi$
$858$	0	0
$859$	41748.0	1.65823	0.829117	−	0.559075i	$-0.188843\pi$
$859$	41748.0	1.65823	0.829117	+	0.559075i	$0.188843\pi$
$860$	0	0
$861$	−1683.60	−0.0666397
$862$	0	0
$863$	−23774.7	−0.937774	−0.468887	−	0.883258i	$-0.655345\pi$
$863$	−23774.7	−0.937774	−0.468887	+	0.883258i	$0.655345\pi$
$864$	0	0
$865$	10214.6	0.401510
$866$	0	0
$867$	14003.7	0.548548
$868$	0	0
$869$	−1146.84	−0.0447687
$870$	0	0
$871$	−8449.30	−0.328695
$872$	0	0
$873$	9484.33	0.367693
$874$	0	0
$875$	1794.86	0.0693455
$876$	0	0
$877$	−35670.5	−1.37344	−0.686720	−	0.726922i	$-0.740950\pi$
$877$	−35670.5	−1.37344	−0.686720	+	0.726922i	$0.740950\pi$
$878$	0	0
$879$	23004.6	0.882738
$880$	0	0
$881$	15650.2	0.598488	0.299244	−	0.954177i	$-0.403265\pi$
$881$	15650.2	0.598488	0.299244	+	0.954177i	$0.403265\pi$
$882$	0	0
$883$	−30872.3	−1.17660	−0.588298	−	0.808644i	$-0.700202\pi$
$883$	−30872.3	−1.17660	−0.588298	+	0.808644i	$0.700202\pi$
$884$	0	0
$885$	7047.59	0.267686
$886$	0	0
$887$	−17242.8	−0.652712	−0.326356	−	0.945247i	$-0.605821\pi$
$887$	−17242.8	−0.652712	−0.326356	+	0.945247i	$0.605821\pi$
$888$	0	0
$889$	2844.45	0.107311
$890$	0	0
$891$	−1380.32	−0.0518994
$892$	0	0
$893$	−8196.09	−0.307135
$894$	0	0
$895$	−6495.15	−0.242580
$896$	0	0
$897$	4983.98	0.185519
$898$	0	0
$899$	82200.0	3.04952
$900$	0	0
$901$	3679.20	0.136040
$902$	0	0
$903$	−2574.46	−0.0948756
$904$	0	0
$905$	5522.59	0.202848
$906$	0	0
$907$	−13479.8	−0.493484	−0.246742	−	0.969081i	$-0.579360\pi$
$907$	−13479.8	−0.493484	−0.246742	+	0.969081i	$0.579360\pi$
$908$	0	0
$909$	−5138.98	−0.187513
$910$	0	0
$911$	−39291.5	−1.42896	−0.714481	−	0.699655i	$-0.753337\pi$
$911$	−39291.5	−1.42896	−0.714481	+	0.699655i	$0.753337\pi$
$912$	0	0
$913$	−25661.1	−0.930183
$914$	0	0
$915$	370.533	0.0133874
$916$	0	0
$917$	−3459.80	−0.124594
$918$	0	0
$919$	−2166.17	−0.0777535	−0.0388768	−	0.999244i	$-0.512378\pi$
$919$	−2166.17	−0.0777535	−0.0388768	+	0.999244i	$0.512378\pi$
$920$	0	0
$921$	3268.12	0.116925
$922$	0	0
$923$	3730.06	0.133019
$924$	0	0
$925$	−6427.71	−0.228478
$926$	0	0
$927$	−9154.15	−0.324339
$928$	0	0
$929$	−36317.0	−1.28258	−0.641292	−	0.767297i	$-0.721601\pi$
$929$	−36317.0	−1.28258	−0.641292	+	0.767297i	$0.721601\pi$
$930$	0	0
$931$	−2297.61	−0.0808818
$932$	0	0
$933$	19506.9	0.684489
$934$	0	0
$935$	−1333.91	−0.0466563
$936$	0	0
$937$	37120.7	1.29422	0.647108	−	0.762398i	$-0.275978\pi$
$937$	37120.7	1.29422	0.647108	+	0.762398i	$0.275978\pi$
$938$	0	0
$939$	12156.3	0.422476
$940$	0	0
$941$	−6793.24	−0.235338	−0.117669	−	0.993053i	$-0.537542\pi$
$941$	−6793.24	−0.235338	−0.117669	+	0.993053i	$0.537542\pi$
$942$	0	0
$943$	−4994.68	−0.172480
$944$	0	0
$945$	−1938.45	−0.0667278
$946$	0	0
$947$	56033.0	1.92273	0.961367	−	0.275271i	$-0.0887677\pi$
$947$	56033.0	1.92273	0.961367	+	0.275271i	$0.0887677\pi$
$948$	0	0
$949$	−4903.69	−0.167735
$950$	0	0
$951$	25808.5	0.880020
$952$	0	0
$953$	−50571.3	−1.71896	−0.859478	−	0.511172i	$-0.829211\pi$
$953$	−50571.3	−1.71896	−0.859478	+	0.511172i	$0.829211\pi$
$954$	0	0
$955$	−4259.39	−0.144325
$956$	0	0
$957$	14999.4	0.506649
$958$	0	0
$959$	16460.8	0.554272
$960$	0	0
$961$	48700.7	1.63475
$962$	0	0
$963$	−4912.90	−0.164399
$964$	0	0
$965$	−15523.4	−0.517840
$966$	0	0
$967$	16678.9	0.554662	0.277331	−	0.960774i	$-0.410550\pi$
$967$	16678.9	0.554662	0.277331	+	0.960774i	$0.410550\pi$
$968$	0	0
$969$	788.683	0.0261467
$970$	0	0
$971$	−2913.63	−0.0962953	−0.0481477	−	0.998840i	$-0.515332\pi$
$971$	−2913.63	−0.0962953	−0.0481477	+	0.998840i	$0.515332\pi$
$972$	0	0
$973$	25228.6	0.831237
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	−46971.1	−1.53812	−0.769058	−	0.639180i	$-0.779274\pi$
$977$	−46971.1	−1.53812	−0.769058	+	0.639180i	$0.779274\pi$
$978$	0	0
$979$	16885.2	0.551229
$980$	0	0
$981$	−4416.28	−0.143732
$982$	0	0
$983$	−54144.9	−1.75682	−0.878410	−	0.477908i	$-0.841395\pi$
$983$	−54144.9	−1.75682	−0.878410	+	0.477908i	$0.841395\pi$
$984$	0	0
$985$	17069.1	0.552149
$986$	0	0
$987$	−21024.8	−0.678040
$988$	0	0
$989$	−7637.58	−0.245562
$990$	0	0
$991$	−24633.3	−0.789608	−0.394804	−	0.918765i	$-0.629187\pi$
$991$	−24633.3	−0.789608	−0.394804	+	0.918765i	$0.629187\pi$
$992$	0	0
$993$	21702.6	0.693565
$994$	0	0
$995$	7207.82	0.229651
$996$	0	0
$997$	−21015.6	−0.667574	−0.333787	−	0.942648i	$-0.608327\pi$
$997$	−21015.6	−0.667574	−0.333787	+	0.942648i	$0.608327\pi$
$998$	0	0
$999$	6941.93	0.219853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.l.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.l.1.2	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -14.3589 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} -14.3589 q^{7} +9.00000 q^{9} -17.0410 q^{11} -13.0000 q^{13} +15.0000 q^{15} -15.6554 q^{17} +16.7926 q^{19} +43.0766 q^{21} +127.794 q^{23} +25.0000 q^{25} -27.0000 q^{27} +293.400 q^{29} +280.164 q^{31} +51.1229 q^{33} +71.7944 q^{35} -257.108 q^{37} +39.0000 q^{39} -39.0837 q^{41} -59.7647 q^{43} -45.0000 q^{45} -488.078 q^{47} -136.823 q^{49} +46.9661 q^{51} -235.012 q^{53} +85.2048 q^{55} -50.3778 q^{57} +469.840 q^{59} +24.7022 q^{61} -129.230 q^{63} +65.0000 q^{65} +649.946 q^{67} -383.383 q^{69} -286.928 q^{71} +377.207 q^{73} -75.0000 q^{75} +244.689 q^{77} +67.2993 q^{79} +81.0000 q^{81} +1505.85 q^{83} +78.2769 q^{85} -880.199 q^{87} -990.860 q^{89} +186.665 q^{91} -840.491 q^{93} -83.9629 q^{95} +1053.81 q^{97} -153.369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

Introduction

L-functions

Modular forms

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Database

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