LMFDB - Embedded newform 2240.4.a.cl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$0$
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} - 2x^{3} - 68x^{2} - 168x - 99 $ x^4 - 2x^3 - 68x^2 - 168*x - 99
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.43874$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.92058 q^{3} -5.00000 q^{5} +7.00000 q^{7} +8.05323 q^{9} +O(q^{10})$	$q-5.92058 q^{3} -5.00000 q^{5} +7.00000 q^{7} +8.05323 q^{9} +28.7744 q^{11} +75.4778 q^{13} +29.6029 q^{15} +29.6282 q^{17} +42.8876 q^{19} -41.4440 q^{21} -13.1124 q^{23} +25.0000 q^{25} +112.176 q^{27} +223.019 q^{29} +228.478 q^{31} -170.361 q^{33} -35.0000 q^{35} +54.5411 q^{37} -446.872 q^{39} -344.490 q^{41} +400.681 q^{43} -40.2661 q^{45} -298.871 q^{47} +49.0000 q^{49} -175.416 q^{51} -338.028 q^{53} -143.872 q^{55} -253.919 q^{57} -39.8581 q^{59} +519.466 q^{61} +56.3726 q^{63} -377.389 q^{65} +28.6689 q^{67} +77.6329 q^{69} +483.529 q^{71} +280.535 q^{73} -148.014 q^{75} +201.421 q^{77} +395.739 q^{79} -881.583 q^{81} -646.985 q^{83} -148.141 q^{85} -1320.40 q^{87} -872.092 q^{89} +528.345 q^{91} -1352.72 q^{93} -214.438 q^{95} +768.173 q^{97} +231.727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 9 q^{3} - 20 q^{5} + 28 q^{7} + 55 q^{9}+O(q^{10}) $ 4 * q + 9 * q^3 - 20 * q^5 + 28 * q^7 + 55 * q^9	$ 4 q + 9 q^{3} - 20 q^{5} + 28 q^{7} + 55 q^{9} + 33 q^{11} + 83 q^{13} - 45 q^{15} - 13 q^{17} + 132 q^{19} + 63 q^{21} - 92 q^{23} + 100 q^{25} + 495 q^{27} + 113 q^{29} - 94 q^{31} + 11 q^{33} - 140 q^{35} + 54 q^{37} - 145 q^{39} + 428 q^{41} + 604 q^{43} - 275 q^{45} - 709 q^{47} + 196 q^{49} - 13 q^{51} - 554 q^{53} - 165 q^{55} + 1044 q^{57} - 100 q^{59} + 588 q^{61} + 385 q^{63} - 415 q^{65} + 484 q^{67} + 540 q^{69} + 128 q^{71} + 1508 q^{73} + 225 q^{75} + 231 q^{77} - 587 q^{79} + 2548 q^{81} + 220 q^{83} + 65 q^{85} - 2887 q^{87} - 1392 q^{89} + 581 q^{91} - 2074 q^{93} - 660 q^{95} + 643 q^{97} + 690 q^{99}+O(q^{100}) $ 4 * q + 9 * q^3 - 20 * q^5 + 28 * q^7 + 55 * q^9 + 33 * q^11 + 83 * q^13 - 45 * q^15 - 13 * q^17 + 132 * q^19 + 63 * q^21 - 92 * q^23 + 100 * q^25 + 495 * q^27 + 113 * q^29 - 94 * q^31 + 11 * q^33 - 140 * q^35 + 54 * q^37 - 145 * q^39 + 428 * q^41 + 604 * q^43 - 275 * q^45 - 709 * q^47 + 196 * q^49 - 13 * q^51 - 554 * q^53 - 165 * q^55 + 1044 * q^57 - 100 * q^59 + 588 * q^61 + 385 * q^63 - 415 * q^65 + 484 * q^67 + 540 * q^69 + 128 * q^71 + 1508 * q^73 + 225 * q^75 + 231 * q^77 - 587 * q^79 + 2548 * q^81 + 220 * q^83 + 65 * q^85 - 2887 * q^87 - 1392 * q^89 + 581 * q^91 - 2074 * q^93 - 660 * q^95 + 643 * q^97 + 690 * 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
$3$	−5.92058	−1.13942	−0.569708	−	0.821847i	$-0.692944\pi$
$3$	−5.92058	−1.13942	−0.569708	+	0.821847i	$0.692944\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	8.05323	0.298268
$10$	0	0
$11$	28.7744	0.788709	0.394355	−	0.918958i	$-0.370968\pi$
$11$	28.7744	0.788709	0.394355	+	0.918958i	$0.370968\pi$
$12$	0	0
$13$	75.4778	1.61029	0.805146	−	0.593077i	$-0.202087\pi$
$13$	75.4778	1.61029	0.805146	+	0.593077i	$0.202087\pi$
$14$	0	0
$15$	29.6029	0.509562
$16$	0	0
$17$	29.6282	0.422699	0.211350	−	0.977411i	$-0.432214\pi$
$17$	29.6282	0.422699	0.211350	+	0.977411i	$0.432214\pi$
$18$	0	0
$19$	42.8876	0.517847	0.258923	−	0.965898i	$-0.416632\pi$
$19$	42.8876	0.517847	0.258923	+	0.965898i	$0.416632\pi$
$20$	0	0
$21$	−41.4440	−0.430659
$22$	0	0
$23$	−13.1124	−0.118875	−0.0594375	−	0.998232i	$-0.518931\pi$
$23$	−13.1124	−0.118875	−0.0594375	+	0.998232i	$0.518931\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	112.176	0.799565
$28$	0	0
$29$	223.019	1.42805	0.714026	−	0.700119i	$-0.246870\pi$
$29$	223.019	1.42805	0.714026	+	0.700119i	$0.246870\pi$
$30$	0	0
$31$	228.478	1.32374	0.661868	−	0.749620i	$-0.269764\pi$
$31$	228.478	1.32374	0.661868	+	0.749620i	$0.269764\pi$
$32$	0	0
$33$	−170.361	−0.898667
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	54.5411	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$37$	54.5411	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$38$	0	0
$39$	−446.872	−1.83479
$40$	0	0
$41$	−344.490	−1.31220	−0.656102	−	0.754672i	$-0.727796\pi$
$41$	−344.490	−1.31220	−0.656102	+	0.754672i	$0.727796\pi$
$42$	0	0
$43$	400.681	1.42101	0.710503	−	0.703694i	$-0.248468\pi$
$43$	400.681	1.42101	0.710503	+	0.703694i	$0.248468\pi$
$44$	0	0
$45$	−40.2661	−0.133389
$46$	0	0
$47$	−298.871	−0.927550	−0.463775	−	0.885953i	$-0.653505\pi$
$47$	−298.871	−0.927550	−0.463775	+	0.885953i	$0.653505\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−175.416	−0.481630
$52$	0	0
$53$	−338.028	−0.876070	−0.438035	−	0.898958i	$-0.644325\pi$
$53$	−338.028	−0.876070	−0.438035	+	0.898958i	$0.644325\pi$
$54$	0	0
$55$	−143.872	−0.352721
$56$	0	0
$57$	−253.919	−0.590043
$58$	0	0
$59$	−39.8581	−0.0879504	−0.0439752	−	0.999033i	$-0.514002\pi$
$59$	−39.8581	−0.0879504	−0.0439752	+	0.999033i	$0.514002\pi$
$60$	0	0
$61$	519.466	1.09034	0.545170	−	0.838325i	$-0.316465\pi$
$61$	519.466	1.09034	0.545170	+	0.838325i	$0.316465\pi$
$62$	0	0
$63$	56.3726	0.112735
$64$	0	0
$65$	−377.389	−0.720144
$66$	0	0
$67$	28.6689	0.0522756	0.0261378	−	0.999658i	$-0.491679\pi$
$67$	28.6689	0.0522756	0.0261378	+	0.999658i	$0.491679\pi$
$68$	0	0
$69$	77.6329	0.135448
$70$	0	0
$71$	483.529	0.808229	0.404115	−	0.914708i	$-0.367580\pi$
$71$	483.529	0.808229	0.404115	+	0.914708i	$0.367580\pi$
$72$	0	0
$73$	280.535	0.449783	0.224892	−	0.974384i	$-0.427797\pi$
$73$	280.535	0.449783	0.224892	+	0.974384i	$0.427797\pi$
$74$	0	0
$75$	−148.014	−0.227883
$76$	0	0
$77$	201.421	0.298104
$78$	0	0
$79$	395.739	0.563596	0.281798	−	0.959474i	$-0.409069\pi$
$79$	395.739	0.563596	0.281798	+	0.959474i	$0.409069\pi$
$80$	0	0
$81$	−881.583	−1.20930
$82$	0	0
$83$	−646.985	−0.855613	−0.427806	−	0.903870i	$-0.640713\pi$
$83$	−646.985	−0.855613	−0.427806	+	0.903870i	$0.640713\pi$
$84$	0	0
$85$	−148.141	−0.189037
$86$	0	0
$87$	−1320.40	−1.62714
$88$	0	0
$89$	−872.092	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$89$	−872.092	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$90$	0	0
$91$	528.345	0.608633
$92$	0	0
$93$	−1352.72	−1.50829
$94$	0	0
$95$	−214.438	−0.231588
$96$	0	0
$97$	768.173	0.804085	0.402042	−	0.915621i	$-0.368300\pi$
$97$	768.173	0.804085	0.402042	+	0.915621i	$0.368300\pi$
$98$	0	0
$99$	231.727	0.235246
$100$	0	0
$101$	−1051.64	−1.03606	−0.518029	−	0.855363i	$-0.673334\pi$
$101$	−1051.64	−1.03606	−0.518029	+	0.855363i	$0.673334\pi$
$102$	0	0
$103$	−811.035	−0.775861	−0.387930	−	0.921689i	$-0.626810\pi$
$103$	−811.035	−0.775861	−0.387930	+	0.921689i	$0.626810\pi$
$104$	0	0
$105$	207.220	0.192596
$106$	0	0
$107$	518.420	0.468388	0.234194	−	0.972190i	$-0.424755\pi$
$107$	518.420	0.468388	0.234194	+	0.972190i	$0.424755\pi$
$108$	0	0
$109$	−1443.11	−1.26811	−0.634057	−	0.773286i	$-0.718612\pi$
$109$	−1443.11	−1.26811	−0.634057	+	0.773286i	$0.718612\pi$
$110$	0	0
$111$	−322.915	−0.276124
$112$	0	0
$113$	−264.239	−0.219978	−0.109989	−	0.993933i	$-0.535082\pi$
$113$	−264.239	−0.219978	−0.109989	+	0.993933i	$0.535082\pi$
$114$	0	0
$115$	65.5620	0.0531625
$116$	0	0
$117$	607.840	0.480298
$118$	0	0
$119$	207.397	0.159765
$120$	0	0
$121$	−503.035	−0.377938
$122$	0	0
$123$	2039.58	1.49515
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1365.48	−0.954065	−0.477033	−	0.878886i	$-0.658288\pi$
$127$	−1365.48	−0.954065	−0.477033	+	0.878886i	$0.658288\pi$
$128$	0	0
$129$	−2372.26	−1.61912
$130$	0	0
$131$	1371.99	0.915046	0.457523	−	0.889198i	$-0.348737\pi$
$131$	1371.99	0.915046	0.457523	+	0.889198i	$0.348737\pi$
$132$	0	0
$133$	300.213	0.195728
$134$	0	0
$135$	−560.879	−0.357576
$136$	0	0
$137$	1869.92	1.16612	0.583058	−	0.812430i	$-0.301856\pi$
$137$	1869.92	1.16612	0.583058	+	0.812430i	$0.301856\pi$
$138$	0	0
$139$	3181.93	1.94164	0.970820	−	0.239808i	$-0.0770846\pi$
$139$	3181.93	1.94164	0.970820	+	0.239808i	$0.0770846\pi$
$140$	0	0
$141$	1769.49	1.05686
$142$	0	0
$143$	2171.83	1.27005
$144$	0	0
$145$	−1115.09	−0.638644
$146$	0	0
$147$	−290.108	−0.162774
$148$	0	0
$149$	−1843.88	−1.01380	−0.506902	−	0.862004i	$-0.669209\pi$
$149$	−1843.88	−1.01380	−0.506902	+	0.862004i	$0.669209\pi$
$150$	0	0
$151$	−1135.50	−0.611959	−0.305979	−	0.952038i	$-0.598984\pi$
$151$	−1135.50	−0.611959	−0.305979	+	0.952038i	$0.598984\pi$
$152$	0	0
$153$	238.602	0.126078
$154$	0	0
$155$	−1142.39	−0.591993
$156$	0	0
$157$	−2625.37	−1.33457	−0.667285	−	0.744803i	$-0.732544\pi$
$157$	−2625.37	−1.33457	−0.667285	+	0.744803i	$0.732544\pi$
$158$	0	0
$159$	2001.32	0.998208
$160$	0	0
$161$	−91.7868	−0.0449305
$162$	0	0
$163$	3151.79	1.51452	0.757261	−	0.653112i	$-0.226537\pi$
$163$	3151.79	1.51452	0.757261	+	0.653112i	$0.226537\pi$
$164$	0	0
$165$	851.804	0.401896
$166$	0	0
$167$	3028.13	1.40313	0.701567	−	0.712603i	$-0.252484\pi$
$167$	3028.13	1.40313	0.701567	+	0.712603i	$0.252484\pi$
$168$	0	0
$169$	3499.91	1.59304
$170$	0	0
$171$	345.384	0.154457
$172$	0	0
$173$	519.038	0.228102	0.114051	−	0.993475i	$-0.463617\pi$
$173$	519.038	0.228102	0.114051	+	0.993475i	$0.463617\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	235.983	0.100212
$178$	0	0
$179$	3202.56	1.33726	0.668632	−	0.743593i	$-0.266880\pi$
$179$	3202.56	1.33726	0.668632	+	0.743593i	$0.266880\pi$
$180$	0	0
$181$	−1904.21	−0.781983	−0.390991	−	0.920394i	$-0.627868\pi$
$181$	−1904.21	−0.781983	−0.390991	+	0.920394i	$0.627868\pi$
$182$	0	0
$183$	−3075.54	−1.24235
$184$	0	0
$185$	−272.706	−0.108377
$186$	0	0
$187$	852.532	0.333387
$188$	0	0
$189$	785.231	0.302207
$190$	0	0
$191$	−642.586	−0.243434	−0.121717	−	0.992565i	$-0.538840\pi$
$191$	−642.586	−0.243434	−0.121717	+	0.992565i	$0.538840\pi$
$192$	0	0
$193$	392.541	0.146403	0.0732014	−	0.997317i	$-0.476678\pi$
$193$	392.541	0.146403	0.0732014	+	0.997317i	$0.476678\pi$
$194$	0	0
$195$	2234.36	0.820543
$196$	0	0
$197$	−1459.98	−0.528016	−0.264008	−	0.964520i	$-0.585045\pi$
$197$	−1459.98	−0.528016	−0.264008	+	0.964520i	$0.585045\pi$
$198$	0	0
$199$	3077.11	1.09613	0.548066	−	0.836435i	$-0.315364\pi$
$199$	3077.11	1.09613	0.548066	+	0.836435i	$0.315364\pi$
$200$	0	0
$201$	−169.737	−0.0595636
$202$	0	0
$203$	1561.13	0.539753
$204$	0	0
$205$	1722.45	0.586836
$206$	0	0
$207$	−105.597	−0.0354566
$208$	0	0
$209$	1234.06	0.408431
$210$	0	0
$211$	1676.42	0.546965	0.273483	−	0.961877i	$-0.411824\pi$
$211$	1676.42	0.546965	0.273483	+	0.961877i	$0.411824\pi$
$212$	0	0
$213$	−2862.77	−0.920909
$214$	0	0
$215$	−2003.40	−0.635493
$216$	0	0
$217$	1599.34	0.500325
$218$	0	0
$219$	−1660.93	−0.512490
$220$	0	0
$221$	2236.27	0.680669
$222$	0	0
$223$	−414.004	−0.124322	−0.0621609	−	0.998066i	$-0.519799\pi$
$223$	−414.004	−0.124322	−0.0621609	+	0.998066i	$0.519799\pi$
$224$	0	0
$225$	201.331	0.0596535
$226$	0	0
$227$	−3186.37	−0.931658	−0.465829	−	0.884875i	$-0.654244\pi$
$227$	−3186.37	−0.931658	−0.465829	+	0.884875i	$0.654244\pi$
$228$	0	0
$229$	−5006.36	−1.44467	−0.722335	−	0.691543i	$-0.756931\pi$
$229$	−5006.36	−1.44467	−0.722335	+	0.691543i	$0.756931\pi$
$230$	0	0
$231$	−1192.53	−0.339664
$232$	0	0
$233$	−1804.62	−0.507400	−0.253700	−	0.967283i	$-0.581648\pi$
$233$	−1804.62	−0.507400	−0.253700	+	0.967283i	$0.581648\pi$
$234$	0	0
$235$	1494.36	0.414813
$236$	0	0
$237$	−2343.00	−0.642170
$238$	0	0
$239$	−7008.97	−1.89696	−0.948478	−	0.316842i	$-0.897377\pi$
$239$	−7008.97	−1.89696	−0.948478	+	0.316842i	$0.897377\pi$
$240$	0	0
$241$	−4267.66	−1.14068	−0.570340	−	0.821409i	$-0.693189\pi$
$241$	−4267.66	−1.14068	−0.570340	+	0.821409i	$0.693189\pi$
$242$	0	0
$243$	2190.73	0.578335
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	3237.06	0.833884
$248$	0	0
$249$	3830.53	0.974898
$250$	0	0
$251$	6562.51	1.65029	0.825144	−	0.564923i	$-0.191094\pi$
$251$	6562.51	1.65029	0.825144	+	0.564923i	$0.191094\pi$
$252$	0	0
$253$	−377.301	−0.0937577
$254$	0	0
$255$	877.079	0.215392
$256$	0	0
$257$	6243.95	1.51551	0.757757	−	0.652537i	$-0.226295\pi$
$257$	6243.95	1.51551	0.757757	+	0.652537i	$0.226295\pi$
$258$	0	0
$259$	381.788	0.0915952
$260$	0	0
$261$	1796.02	0.425942
$262$	0	0
$263$	−1798.52	−0.421678	−0.210839	−	0.977521i	$-0.567620\pi$
$263$	−1798.52	−0.421678	−0.210839	+	0.977521i	$0.567620\pi$
$264$	0	0
$265$	1690.14	0.391790
$266$	0	0
$267$	5163.29	1.18348
$268$	0	0
$269$	7637.04	1.73100	0.865499	−	0.500911i	$-0.167002\pi$
$269$	7637.04	1.73100	0.865499	+	0.500911i	$0.167002\pi$
$270$	0	0
$271$	3307.38	0.741362	0.370681	−	0.928760i	$-0.379124\pi$
$271$	3307.38	0.741362	0.370681	+	0.928760i	$0.379124\pi$
$272$	0	0
$273$	−3128.11	−0.693486
$274$	0	0
$275$	719.359	0.157742
$276$	0	0
$277$	7032.60	1.52544	0.762722	−	0.646727i	$-0.223863\pi$
$277$	7032.60	1.52544	0.762722	+	0.646727i	$0.223863\pi$
$278$	0	0
$279$	1839.98	0.394828
$280$	0	0
$281$	−1942.60	−0.412406	−0.206203	−	0.978509i	$-0.566111\pi$
$281$	−1942.60	−0.412406	−0.206203	+	0.978509i	$0.566111\pi$
$282$	0	0
$283$	2342.60	0.492061	0.246030	−	0.969262i	$-0.420874\pi$
$283$	2342.60	0.492061	0.246030	+	0.969262i	$0.420874\pi$
$284$	0	0
$285$	1269.60	0.263875
$286$	0	0
$287$	−2411.43	−0.495967
$288$	0	0
$289$	−4035.17	−0.821325
$290$	0	0
$291$	−4548.03	−0.916186
$292$	0	0
$293$	4502.42	0.897728	0.448864	−	0.893600i	$-0.351829\pi$
$293$	4502.42	0.897728	0.448864	+	0.893600i	$0.351829\pi$
$294$	0	0
$295$	199.290	0.0393326
$296$	0	0
$297$	3227.79	0.630624
$298$	0	0
$299$	−989.695	−0.191423
$300$	0	0
$301$	2804.77	0.537090
$302$	0	0
$303$	6226.30	1.18050
$304$	0	0
$305$	−2597.33	−0.487615
$306$	0	0
$307$	7598.12	1.41253	0.706267	−	0.707946i	$-0.250378\pi$
$307$	7598.12	1.41253	0.706267	+	0.707946i	$0.250378\pi$
$308$	0	0
$309$	4801.79	0.884028
$310$	0	0
$311$	6091.42	1.11065	0.555326	−	0.831633i	$-0.312594\pi$
$311$	6091.42	1.11065	0.555326	+	0.831633i	$0.312594\pi$
$312$	0	0
$313$	1129.40	0.203953	0.101977	−	0.994787i	$-0.467483\pi$
$313$	1129.40	0.203953	0.101977	+	0.994787i	$0.467483\pi$
$314$	0	0
$315$	−281.863	−0.0504164
$316$	0	0
$317$	2079.12	0.368375	0.184187	−	0.982891i	$-0.441035\pi$
$317$	2079.12	0.368375	0.184187	+	0.982891i	$0.441035\pi$
$318$	0	0
$319$	6417.22	1.12632
$320$	0	0
$321$	−3069.34	−0.533689
$322$	0	0
$323$	1270.68	0.218893
$324$	0	0
$325$	1886.95	0.322058
$326$	0	0
$327$	8544.02	1.44491
$328$	0	0
$329$	−2092.10	−0.350581
$330$	0	0
$331$	−10003.3	−1.66112	−0.830560	−	0.556929i	$-0.811980\pi$
$331$	−10003.3	−1.66112	−0.830560	+	0.556929i	$0.811980\pi$
$332$	0	0
$333$	439.232	0.0722816
$334$	0	0
$335$	−143.345	−0.0233784
$336$	0	0
$337$	6747.66	1.09071	0.545354	−	0.838206i	$-0.316395\pi$
$337$	6747.66	1.09071	0.545354	+	0.838206i	$0.316395\pi$
$338$	0	0
$339$	1564.45	0.250646
$340$	0	0
$341$	6574.31	1.04404
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−388.165	−0.0605742
$346$	0	0
$347$	−192.984	−0.0298557	−0.0149278	−	0.999889i	$-0.504752\pi$
$347$	−192.984	−0.0298557	−0.0149278	+	0.999889i	$0.504752\pi$
$348$	0	0
$349$	−6674.43	−1.02371	−0.511854	−	0.859072i	$-0.671041\pi$
$349$	−6674.43	−1.02371	−0.511854	+	0.859072i	$0.671041\pi$
$350$	0	0
$351$	8466.79	1.28753
$352$	0	0
$353$	−4095.84	−0.617563	−0.308781	−	0.951133i	$-0.599921\pi$
$353$	−4095.84	−0.617563	−0.308781	+	0.951133i	$0.599921\pi$
$354$	0	0
$355$	−2417.64	−0.361451
$356$	0	0
$357$	−1227.91	−0.182039
$358$	0	0
$359$	−9030.36	−1.32759	−0.663794	−	0.747915i	$-0.731055\pi$
$359$	−9030.36	−1.32759	−0.663794	+	0.747915i	$0.731055\pi$
$360$	0	0
$361$	−5019.65	−0.731835
$362$	0	0
$363$	2978.26	0.430628
$364$	0	0
$365$	−1402.68	−0.201149
$366$	0	0
$367$	10000.4	1.42239	0.711197	−	0.702993i	$-0.248153\pi$
$367$	10000.4	1.42239	0.711197	+	0.702993i	$0.248153\pi$
$368$	0	0
$369$	−2774.26	−0.391388
$370$	0	0
$371$	−2366.20	−0.331123
$372$	0	0
$373$	−9394.95	−1.30416	−0.652081	−	0.758150i	$-0.726104\pi$
$373$	−9394.95	−1.30416	−0.652081	+	0.758150i	$0.726104\pi$
$374$	0	0
$375$	740.072	0.101912
$376$	0	0
$377$	16833.0	2.29958
$378$	0	0
$379$	−808.071	−0.109519	−0.0547596	−	0.998500i	$-0.517439\pi$
$379$	−808.071	−0.109519	−0.0547596	+	0.998500i	$0.517439\pi$
$380$	0	0
$381$	8084.40	1.08708
$382$	0	0
$383$	−2064.01	−0.275368	−0.137684	−	0.990476i	$-0.543966\pi$
$383$	−2064.01	−0.275368	−0.137684	+	0.990476i	$0.543966\pi$
$384$	0	0
$385$	−1007.10	−0.133316
$386$	0	0
$387$	3226.77	0.423840
$388$	0	0
$389$	−12659.2	−1.64999	−0.824993	−	0.565143i	$-0.808821\pi$
$389$	−12659.2	−1.64999	−0.824993	+	0.565143i	$0.808821\pi$
$390$	0	0
$391$	−388.496	−0.0502483
$392$	0	0
$393$	−8122.95	−1.04262
$394$	0	0
$395$	−1978.69	−0.252048
$396$	0	0
$397$	14246.9	1.80108	0.900542	−	0.434769i	$-0.143170\pi$
$397$	14246.9	1.80108	0.900542	+	0.434769i	$0.143170\pi$
$398$	0	0
$399$	−1777.44	−0.223015
$400$	0	0
$401$	9487.55	1.18151	0.590755	−	0.806851i	$-0.298830\pi$
$401$	9487.55	1.18151	0.590755	+	0.806851i	$0.298830\pi$
$402$	0	0
$403$	17245.0	2.13160
$404$	0	0
$405$	4407.91	0.540817
$406$	0	0
$407$	1569.39	0.191134
$408$	0	0
$409$	7464.32	0.902412	0.451206	−	0.892420i	$-0.350994\pi$
$409$	7464.32	0.902412	0.451206	+	0.892420i	$0.350994\pi$
$410$	0	0
$411$	−11071.0	−1.32869
$412$	0	0
$413$	−279.006	−0.0332421
$414$	0	0
$415$	3234.93	0.382642
$416$	0	0
$417$	−18838.9	−2.21234
$418$	0	0
$419$	−5031.28	−0.586621	−0.293311	−	0.956017i	$-0.594757\pi$
$419$	−5031.28	−0.586621	−0.293311	+	0.956017i	$0.594757\pi$
$420$	0	0
$421$	9333.73	1.08052	0.540259	−	0.841499i	$-0.318326\pi$
$421$	9333.73	1.08052	0.540259	+	0.841499i	$0.318326\pi$
$422$	0	0
$423$	−2406.88	−0.276658
$424$	0	0
$425$	740.704	0.0845398
$426$	0	0
$427$	3636.26	0.412110
$428$	0	0
$429$	−12858.5	−1.44712
$430$	0	0
$431$	−4530.28	−0.506301	−0.253150	−	0.967427i	$-0.581467\pi$
$431$	−4530.28	−0.506301	−0.253150	+	0.967427i	$0.581467\pi$
$432$	0	0
$433$	7115.45	0.789715	0.394857	−	0.918742i	$-0.370794\pi$
$433$	7115.45	0.789715	0.394857	+	0.918742i	$0.370794\pi$
$434$	0	0
$435$	6601.99	0.727681
$436$	0	0
$437$	−562.359	−0.0615590
$438$	0	0
$439$	4177.67	0.454190	0.227095	−	0.973873i	$-0.427077\pi$
$439$	4177.67	0.454190	0.227095	+	0.973873i	$0.427077\pi$
$440$	0	0
$441$	394.608	0.0426097
$442$	0	0
$443$	−9245.42	−0.991565	−0.495782	−	0.868447i	$-0.665119\pi$
$443$	−9245.42	−0.991565	−0.495782	+	0.868447i	$0.665119\pi$
$444$	0	0
$445$	4360.46	0.464507
$446$	0	0
$447$	10916.8	1.15514
$448$	0	0
$449$	476.465	0.0500797	0.0250398	−	0.999686i	$-0.492029\pi$
$449$	476.465	0.0500797	0.0250398	+	0.999686i	$0.492029\pi$
$450$	0	0
$451$	−9912.50	−1.03495
$452$	0	0
$453$	6722.82	0.697275
$454$	0	0
$455$	−2641.72	−0.272189
$456$	0	0
$457$	10383.2	1.06282	0.531408	−	0.847116i	$-0.321663\pi$
$457$	10383.2	1.06282	0.531408	+	0.847116i	$0.321663\pi$
$458$	0	0
$459$	3323.56	0.337975
$460$	0	0
$461$	6973.02	0.704481	0.352241	−	0.935909i	$-0.385420\pi$
$461$	6973.02	0.704481	0.352241	+	0.935909i	$0.385420\pi$
$462$	0	0
$463$	−8171.02	−0.820172	−0.410086	−	0.912047i	$-0.634501\pi$
$463$	−8171.02	−0.820172	−0.410086	+	0.912047i	$0.634501\pi$
$464$	0	0
$465$	6763.60	0.674526
$466$	0	0
$467$	11294.1	1.11912	0.559561	−	0.828789i	$-0.310970\pi$
$467$	11294.1	1.11912	0.559561	+	0.828789i	$0.310970\pi$
$468$	0	0
$469$	200.682	0.0197583
$470$	0	0
$471$	15543.7	1.52063
$472$	0	0
$473$	11529.3	1.12076
$474$	0	0
$475$	1072.19	0.103569
$476$	0	0
$477$	−2722.22	−0.261303
$478$	0	0
$479$	15380.9	1.46716	0.733580	−	0.679603i	$-0.237848\pi$
$479$	15380.9	1.46716	0.733580	+	0.679603i	$0.237848\pi$
$480$	0	0
$481$	4116.65	0.390235
$482$	0	0
$483$	543.431	0.0511945
$484$	0	0
$485$	−3840.87	−0.359598
$486$	0	0
$487$	5382.97	0.500874	0.250437	−	0.968133i	$-0.419426\pi$
$487$	5382.97	0.500874	0.250437	+	0.968133i	$0.419426\pi$
$488$	0	0
$489$	−18660.4	−1.72567
$490$	0	0
$491$	−166.749	−0.0153264	−0.00766322	−	0.999971i	$-0.502439\pi$
$491$	−166.749	−0.0153264	−0.00766322	+	0.999971i	$0.502439\pi$
$492$	0	0
$493$	6607.63	0.603636
$494$	0	0
$495$	−1158.63	−0.105205
$496$	0	0
$497$	3384.70	0.305482
$498$	0	0
$499$	−6370.97	−0.571550	−0.285775	−	0.958297i	$-0.592251\pi$
$499$	−6370.97	−0.571550	−0.285775	+	0.958297i	$0.592251\pi$
$500$	0	0
$501$	−17928.3	−1.59875
$502$	0	0
$503$	−9100.42	−0.806695	−0.403348	−	0.915047i	$-0.632153\pi$
$503$	−9100.42	−0.806695	−0.403348	+	0.915047i	$0.632153\pi$
$504$	0	0
$505$	5258.19	0.463339
$506$	0	0
$507$	−20721.5	−1.81513
$508$	0	0
$509$	−14741.3	−1.28369	−0.641845	−	0.766834i	$-0.721831\pi$
$509$	−14741.3	−1.28369	−0.641845	+	0.766834i	$0.721831\pi$
$510$	0	0
$511$	1963.75	0.170002
$512$	0	0
$513$	4810.95	0.414052
$514$	0	0
$515$	4055.17	0.346975
$516$	0	0
$517$	−8599.83	−0.731567
$518$	0	0
$519$	−3073.00	−0.259903
$520$	0	0
$521$	−5761.20	−0.484458	−0.242229	−	0.970219i	$-0.577879\pi$
$521$	−5761.20	−0.484458	−0.242229	+	0.970219i	$0.577879\pi$
$522$	0	0
$523$	−7324.39	−0.612377	−0.306189	−	0.951971i	$-0.599054\pi$
$523$	−7324.39	−0.612377	−0.306189	+	0.951971i	$0.599054\pi$
$524$	0	0
$525$	−1036.10	−0.0861317
$526$	0	0
$527$	6769.38	0.559542
$528$	0	0
$529$	−11995.1	−0.985869
$530$	0	0
$531$	−320.986	−0.0262328
$532$	0	0
$533$	−26001.4	−2.11303
$534$	0	0
$535$	−2592.10	−0.209469
$536$	0	0
$537$	−18961.0	−1.52370
$538$	0	0
$539$	1409.94	0.112673
$540$	0	0
$541$	−2841.35	−0.225803	−0.112901	−	0.993606i	$-0.536014\pi$
$541$	−2841.35	−0.225803	−0.112901	+	0.993606i	$0.536014\pi$
$542$	0	0
$543$	11274.0	0.891004
$544$	0	0
$545$	7215.53	0.567118
$546$	0	0
$547$	−14148.5	−1.10593	−0.552967	−	0.833203i	$-0.686504\pi$
$547$	−14148.5	−1.10593	−0.552967	+	0.833203i	$0.686504\pi$
$548$	0	0
$549$	4183.38	0.325213
$550$	0	0
$551$	9564.73	0.739512
$552$	0	0
$553$	2770.17	0.213019
$554$	0	0
$555$	1614.57	0.123486
$556$	0	0
$557$	2907.20	0.221152	0.110576	−	0.993868i	$-0.464730\pi$
$557$	2907.20	0.221152	0.110576	+	0.993868i	$0.464730\pi$
$558$	0	0
$559$	30242.5	2.28823
$560$	0	0
$561$	−5047.48	−0.379866
$562$	0	0
$563$	3510.61	0.262797	0.131398	−	0.991330i	$-0.458053\pi$
$563$	3510.61	0.262797	0.131398	+	0.991330i	$0.458053\pi$
$564$	0	0
$565$	1321.19	0.0983771
$566$	0	0
$567$	−6171.08	−0.457074
$568$	0	0
$569$	6760.31	0.498079	0.249040	−	0.968493i	$-0.419885\pi$
$569$	6760.31	0.498079	0.249040	+	0.968493i	$0.419885\pi$
$570$	0	0
$571$	13391.6	0.981469	0.490735	−	0.871309i	$-0.336728\pi$
$571$	13391.6	0.981469	0.490735	+	0.871309i	$0.336728\pi$
$572$	0	0
$573$	3804.48	0.277372
$574$	0	0
$575$	−327.810	−0.0237750
$576$	0	0
$577$	10351.1	0.746834	0.373417	−	0.927664i	$-0.378186\pi$
$577$	10351.1	0.746834	0.373417	+	0.927664i	$0.378186\pi$
$578$	0	0
$579$	−2324.07	−0.166814
$580$	0	0
$581$	−4528.90	−0.323391
$582$	0	0
$583$	−9726.54	−0.690964
$584$	0	0
$585$	−3039.20	−0.214796
$586$	0	0
$587$	3994.52	0.280871	0.140436	−	0.990090i	$-0.455150\pi$
$587$	3994.52	0.280871	0.140436	+	0.990090i	$0.455150\pi$
$588$	0	0
$589$	9798.87	0.685493
$590$	0	0
$591$	8643.92	0.601630
$592$	0	0
$593$	−7925.99	−0.548873	−0.274436	−	0.961605i	$-0.588491\pi$
$593$	−7925.99	−0.548873	−0.274436	+	0.961605i	$0.588491\pi$
$594$	0	0
$595$	−1036.99	−0.0714492
$596$	0	0
$597$	−18218.3	−1.24895
$598$	0	0
$599$	13948.7	0.951468	0.475734	−	0.879589i	$-0.342182\pi$
$599$	13948.7	0.951468	0.475734	+	0.879589i	$0.342182\pi$
$600$	0	0
$601$	5357.60	0.363629	0.181815	−	0.983333i	$-0.441803\pi$
$601$	5357.60	0.363629	0.181815	+	0.983333i	$0.441803\pi$
$602$	0	0
$603$	230.877	0.0155921
$604$	0	0
$605$	2515.18	0.169019
$606$	0	0
$607$	−7672.08	−0.513015	−0.256507	−	0.966542i	$-0.582572\pi$
$607$	−7672.08	−0.513015	−0.256507	+	0.966542i	$0.582572\pi$
$608$	0	0
$609$	−9242.79	−0.615003
$610$	0	0
$611$	−22558.2	−1.49363
$612$	0	0
$613$	−18803.8	−1.23896	−0.619478	−	0.785014i	$-0.712656\pi$
$613$	−18803.8	−1.23896	−0.619478	+	0.785014i	$0.712656\pi$
$614$	0	0
$615$	−10197.9	−0.668650
$616$	0	0
$617$	12134.1	0.791733	0.395866	−	0.918308i	$-0.370444\pi$
$617$	12134.1	0.791733	0.395866	+	0.918308i	$0.370444\pi$
$618$	0	0
$619$	−6663.20	−0.432660	−0.216330	−	0.976320i	$-0.569409\pi$
$619$	−6663.20	−0.432660	−0.216330	+	0.976320i	$0.569409\pi$
$620$	0	0
$621$	−1470.89	−0.0950482
$622$	0	0
$623$	−6104.64	−0.392580
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−7306.37	−0.465372
$628$	0	0
$629$	1615.95	0.102436
$630$	0	0
$631$	−14976.4	−0.944854	−0.472427	−	0.881370i	$-0.656622\pi$
$631$	−14976.4	−0.944854	−0.472427	+	0.881370i	$0.656622\pi$
$632$	0	0
$633$	−9925.38	−0.623221
$634$	0	0
$635$	6827.38	0.426671
$636$	0	0
$637$	3698.41	0.230042
$638$	0	0
$639$	3893.97	0.241069
$640$	0	0
$641$	432.617	0.0266573	0.0133287	−	0.999911i	$-0.495757\pi$
$641$	432.617	0.0266573	0.0133287	+	0.999911i	$0.495757\pi$
$642$	0	0
$643$	22272.3	1.36599	0.682997	−	0.730421i	$-0.260676\pi$
$643$	22272.3	1.36599	0.682997	+	0.730421i	$0.260676\pi$
$644$	0	0
$645$	11861.3	0.724091
$646$	0	0
$647$	19784.7	1.20219	0.601094	−	0.799179i	$-0.294732\pi$
$647$	19784.7	1.20219	0.601094	+	0.799179i	$0.294732\pi$
$648$	0	0
$649$	−1146.89	−0.0693673
$650$	0	0
$651$	−9469.04	−0.570078
$652$	0	0
$653$	10496.2	0.629014	0.314507	−	0.949255i	$-0.398161\pi$
$653$	10496.2	0.629014	0.314507	+	0.949255i	$0.398161\pi$
$654$	0	0
$655$	−6859.93	−0.409221
$656$	0	0
$657$	2259.21	0.134156
$658$	0	0
$659$	14548.7	0.859995	0.429998	−	0.902830i	$-0.358514\pi$
$659$	14548.7	0.859995	0.429998	+	0.902830i	$0.358514\pi$
$660$	0	0
$661$	−18669.1	−1.09855	−0.549276	−	0.835641i	$-0.685097\pi$
$661$	−18669.1	−1.09855	−0.549276	+	0.835641i	$0.685097\pi$
$662$	0	0
$663$	−13240.0	−0.775565
$664$	0	0
$665$	−1501.07	−0.0875321
$666$	0	0
$667$	−2924.31	−0.169760
$668$	0	0
$669$	2451.14	0.141654
$670$	0	0
$671$	14947.3	0.859962
$672$	0	0
$673$	−16896.4	−0.967771	−0.483885	−	0.875131i	$-0.660775\pi$
$673$	−16896.4	−0.967771	−0.483885	+	0.875131i	$0.660775\pi$
$674$	0	0
$675$	2804.40	0.159913
$676$	0	0
$677$	23202.4	1.31719	0.658597	−	0.752496i	$-0.271150\pi$
$677$	23202.4	1.31719	0.658597	+	0.752496i	$0.271150\pi$
$678$	0	0
$679$	5377.21	0.303915
$680$	0	0
$681$	18865.1	1.06155
$682$	0	0
$683$	4004.96	0.224371	0.112185	−	0.993687i	$-0.464215\pi$
$683$	4004.96	0.224371	0.112185	+	0.993687i	$0.464215\pi$
$684$	0	0
$685$	−9349.59	−0.521503
$686$	0	0
$687$	29640.5	1.64608
$688$	0	0
$689$	−25513.6	−1.41073
$690$	0	0
$691$	−33364.5	−1.83682	−0.918412	−	0.395625i	$-0.870528\pi$
$691$	−33364.5	−1.83682	−0.918412	+	0.395625i	$0.870528\pi$
$692$	0	0
$693$	1622.09	0.0889148
$694$	0	0
$695$	−15909.7	−0.868328
$696$	0	0
$697$	−10206.6	−0.554668
$698$	0	0
$699$	10684.4	0.578140
$700$	0	0
$701$	−33371.0	−1.79801	−0.899005	−	0.437939i	$-0.855708\pi$
$701$	−33371.0	−1.79801	−0.899005	+	0.437939i	$0.855708\pi$
$702$	0	0
$703$	2339.14	0.125494
$704$	0	0
$705$	−8847.45	−0.472644
$706$	0	0
$707$	−7361.46	−0.391593
$708$	0	0
$709$	−25272.6	−1.33869	−0.669345	−	0.742952i	$-0.733425\pi$
$709$	−25272.6	−1.33869	−0.669345	+	0.742952i	$0.733425\pi$
$710$	0	0
$711$	3186.97	0.168102
$712$	0	0
$713$	−2995.89	−0.157359
$714$	0	0
$715$	−10859.1	−0.567984
$716$	0	0
$717$	41497.1	2.16142
$718$	0	0
$719$	−12346.2	−0.640385	−0.320192	−	0.947353i	$-0.603748\pi$
$719$	−12346.2	−0.640385	−0.320192	+	0.947353i	$0.603748\pi$
$720$	0	0
$721$	−5677.24	−0.293248
$722$	0	0
$723$	25267.0	1.29971
$724$	0	0
$725$	5575.46	0.285610
$726$	0	0
$727$	28159.7	1.43657	0.718285	−	0.695748i	$-0.244927\pi$
$727$	28159.7	1.43657	0.718285	+	0.695748i	$0.244927\pi$
$728$	0	0
$729$	10832.3	0.550340
$730$	0	0
$731$	11871.4	0.600658
$732$	0	0
$733$	6975.52	0.351496	0.175748	−	0.984435i	$-0.443766\pi$
$733$	6975.52	0.351496	0.175748	+	0.984435i	$0.443766\pi$
$734$	0	0
$735$	1450.54	0.0727946
$736$	0	0
$737$	824.930	0.0412303
$738$	0	0
$739$	17407.3	0.866492	0.433246	−	0.901276i	$-0.357368\pi$
$739$	17407.3	0.866492	0.433246	+	0.901276i	$0.357368\pi$
$740$	0	0
$741$	−19165.3	−0.950141
$742$	0	0
$743$	−28553.4	−1.40986	−0.704928	−	0.709279i	$-0.749021\pi$
$743$	−28553.4	−1.40986	−0.704928	+	0.709279i	$0.749021\pi$
$744$	0	0
$745$	9219.41	0.453386
$746$	0	0
$747$	−5210.32	−0.255202
$748$	0	0
$749$	3628.94	0.177034
$750$	0	0
$751$	−20462.7	−0.994266	−0.497133	−	0.867674i	$-0.665614\pi$
$751$	−20462.7	−0.994266	−0.497133	+	0.867674i	$0.665614\pi$
$752$	0	0
$753$	−38853.9	−1.88036
$754$	0	0
$755$	5677.51	0.273676
$756$	0	0
$757$	8915.94	0.428078	0.214039	−	0.976825i	$-0.431338\pi$
$757$	8915.94	0.428078	0.214039	+	0.976825i	$0.431338\pi$
$758$	0	0
$759$	2233.84	0.106829
$760$	0	0
$761$	−19866.0	−0.946312	−0.473156	−	0.880979i	$-0.656885\pi$
$761$	−19866.0	−0.946312	−0.473156	+	0.880979i	$0.656885\pi$
$762$	0	0
$763$	−10101.7	−0.479302
$764$	0	0
$765$	−1193.01	−0.0563836
$766$	0	0
$767$	−3008.40	−0.141626
$768$	0	0
$769$	−11977.3	−0.561655	−0.280828	−	0.959758i	$-0.590609\pi$
$769$	−11977.3	−0.561655	−0.280828	+	0.959758i	$0.590609\pi$
$770$	0	0
$771$	−36967.8	−1.72680
$772$	0	0
$773$	35108.0	1.63357	0.816784	−	0.576944i	$-0.195755\pi$
$773$	35108.0	1.63357	0.816784	+	0.576944i	$0.195755\pi$
$774$	0	0
$775$	5711.95	0.264747
$776$	0	0
$777$	−2260.40	−0.104365
$778$	0	0
$779$	−14774.4	−0.679521
$780$	0	0
$781$	13913.2	0.637458
$782$	0	0
$783$	25017.3	1.14182
$784$	0	0
$785$	13126.9	0.596838
$786$	0	0
$787$	21809.9	0.987851	0.493926	−	0.869504i	$-0.335562\pi$
$787$	21809.9	0.987851	0.493926	+	0.869504i	$0.335562\pi$
$788$	0	0
$789$	10648.3	0.480467
$790$	0	0
$791$	−1849.67	−0.0831439
$792$	0	0
$793$	39208.2	1.75577
$794$	0	0
$795$	−10006.6	−0.446412
$796$	0	0
$797$	4045.36	0.179792	0.0898958	−	0.995951i	$-0.471347\pi$
$797$	4045.36	0.179792	0.0898958	+	0.995951i	$0.471347\pi$
$798$	0	0
$799$	−8855.01	−0.392075
$800$	0	0
$801$	−7023.16	−0.309802
$802$	0	0
$803$	8072.23	0.354748
$804$	0	0
$805$	458.934	0.0200935
$806$	0	0
$807$	−45215.7	−1.97233
$808$	0	0
$809$	−10664.9	−0.463485	−0.231742	−	0.972777i	$-0.574443\pi$
$809$	−10664.9	−0.463485	−0.231742	+	0.972777i	$0.574443\pi$
$810$	0	0
$811$	−28075.6	−1.21562	−0.607811	−	0.794082i	$-0.707952\pi$
$811$	−28075.6	−1.21562	−0.607811	+	0.794082i	$0.707952\pi$
$812$	0	0
$813$	−19581.6	−0.844719
$814$	0	0
$815$	−15758.9	−0.677315
$816$	0	0
$817$	17184.2	0.735863
$818$	0	0
$819$	4254.88	0.181536
$820$	0	0
$821$	−27468.5	−1.16767	−0.583836	−	0.811872i	$-0.698449\pi$
$821$	−27468.5	−1.16767	−0.583836	+	0.811872i	$0.698449\pi$
$822$	0	0
$823$	−14722.0	−0.623542	−0.311771	−	0.950157i	$-0.600922\pi$
$823$	−14722.0	−0.623542	−0.311771	+	0.950157i	$0.600922\pi$
$824$	0	0
$825$	−4259.02	−0.179733
$826$	0	0
$827$	−14331.7	−0.602615	−0.301308	−	0.953527i	$-0.597423\pi$
$827$	−14331.7	−0.602615	−0.301308	+	0.953527i	$0.597423\pi$
$828$	0	0
$829$	2464.33	0.103245	0.0516223	−	0.998667i	$-0.483561\pi$
$829$	2464.33	0.103245	0.0516223	+	0.998667i	$0.483561\pi$
$830$	0	0
$831$	−41637.0	−1.73811
$832$	0	0
$833$	1451.78	0.0603856
$834$	0	0
$835$	−15140.6	−0.627501
$836$	0	0
$837$	25629.7	1.05841
$838$	0	0
$839$	−41384.0	−1.70290	−0.851450	−	0.524435i	$-0.824277\pi$
$839$	−41384.0	−1.70290	−0.851450	+	0.524435i	$0.824277\pi$
$840$	0	0
$841$	25348.3	1.03933
$842$	0	0
$843$	11501.3	0.469901
$844$	0	0
$845$	−17499.5	−0.712428
$846$	0	0
$847$	−3521.25	−0.142847
$848$	0	0
$849$	−13869.5	−0.560662
$850$	0	0
$851$	−715.165	−0.0288079
$852$	0	0
$853$	11670.5	0.468451	0.234225	−	0.972182i	$-0.424745\pi$
$853$	11670.5	0.468451	0.234225	+	0.972182i	$0.424745\pi$
$854$	0	0
$855$	−1726.92	−0.0690753
$856$	0	0
$857$	−30135.2	−1.20117	−0.600584	−	0.799562i	$-0.705065\pi$
$857$	−30135.2	−1.20117	−0.600584	+	0.799562i	$0.705065\pi$
$858$	0	0
$859$	−34074.0	−1.35342	−0.676711	−	0.736249i	$-0.736595\pi$
$859$	−34074.0	−1.35342	−0.676711	+	0.736249i	$0.736595\pi$
$860$	0	0
$861$	14277.1	0.565112
$862$	0	0
$863$	−41246.6	−1.62694	−0.813472	−	0.581605i	$-0.802425\pi$
$863$	−41246.6	−1.62694	−0.813472	+	0.581605i	$0.802425\pi$
$864$	0	0
$865$	−2595.19	−0.102011
$866$	0	0
$867$	23890.5	0.935831
$868$	0	0
$869$	11387.1	0.444513
$870$	0	0
$871$	2163.87	0.0841790
$872$	0	0
$873$	6186.28	0.239832
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	21517.7	0.828508	0.414254	−	0.910161i	$-0.364042\pi$
$877$	21517.7	0.828508	0.414254	+	0.910161i	$0.364042\pi$
$878$	0	0
$879$	−26656.9	−1.02288
$880$	0	0
$881$	37304.1	1.42657	0.713284	−	0.700875i	$-0.247207\pi$
$881$	37304.1	1.42657	0.713284	+	0.700875i	$0.247207\pi$
$882$	0	0
$883$	50160.7	1.91171	0.955856	−	0.293837i	$-0.0949322\pi$
$883$	50160.7	1.91171	0.955856	+	0.293837i	$0.0949322\pi$
$884$	0	0
$885$	−1179.91	−0.0448162
$886$	0	0
$887$	−30674.5	−1.16116	−0.580580	−	0.814203i	$-0.697174\pi$
$887$	−30674.5	−1.16116	−0.580580	+	0.814203i	$0.697174\pi$
$888$	0	0
$889$	−9558.33	−0.360603
$890$	0	0
$891$	−25367.0	−0.953789
$892$	0	0
$893$	−12817.9	−0.480329
$894$	0	0
$895$	−16012.8	−0.598043
$896$	0	0
$897$	5859.57	0.218111
$898$	0	0
$899$	50954.8	1.89036
$900$	0	0
$901$	−10015.1	−0.370314
$902$	0	0
$903$	−16605.8	−0.611968
$904$	0	0
$905$	9521.06	0.349713
$906$	0	0
$907$	−33054.4	−1.21009	−0.605047	−	0.796190i	$-0.706846\pi$
$907$	−33054.4	−1.21009	−0.605047	+	0.796190i	$0.706846\pi$
$908$	0	0
$909$	−8469.07	−0.309022
$910$	0	0
$911$	15659.0	0.569491	0.284745	−	0.958603i	$-0.408091\pi$
$911$	15659.0	0.569491	0.284745	+	0.958603i	$0.408091\pi$
$912$	0	0
$913$	−18616.6	−0.674829
$914$	0	0
$915$	15377.7	0.555596
$916$	0	0
$917$	9603.91	0.345855
$918$	0	0
$919$	46207.8	1.65860	0.829300	−	0.558803i	$-0.188739\pi$
$919$	46207.8	1.65860	0.829300	+	0.558803i	$0.188739\pi$
$920$	0	0
$921$	−44985.3	−1.60946
$922$	0	0
$923$	36495.7	1.30148
$924$	0	0
$925$	1363.53	0.0484676
$926$	0	0
$927$	−6531.45	−0.231414
$928$	0	0
$929$	−16247.3	−0.573798	−0.286899	−	0.957961i	$-0.592624\pi$
$929$	−16247.3	−0.573798	−0.286899	+	0.957961i	$0.592624\pi$
$930$	0	0
$931$	2101.49	0.0739781
$932$	0	0
$933$	−36064.7	−1.26549
$934$	0	0
$935$	−4262.66	−0.149095
$936$	0	0
$937$	−12913.2	−0.450219	−0.225109	−	0.974334i	$-0.572274\pi$
$937$	−12913.2	−0.450219	−0.225109	+	0.974334i	$0.572274\pi$
$938$	0	0
$939$	−6686.69	−0.232388
$940$	0	0
$941$	−2767.01	−0.0958577	−0.0479288	−	0.998851i	$-0.515262\pi$
$941$	−2767.01	−0.0958577	−0.0479288	+	0.998851i	$0.515262\pi$
$942$	0	0
$943$	4517.10	0.155988
$944$	0	0
$945$	−3926.15	−0.135151
$946$	0	0
$947$	2630.98	0.0902801	0.0451400	−	0.998981i	$-0.485627\pi$
$947$	2630.98	0.0902801	0.0451400	+	0.998981i	$0.485627\pi$
$948$	0	0
$949$	21174.2	0.724282
$950$	0	0
$951$	−12309.6	−0.419732
$952$	0	0
$953$	−4256.89	−0.144695	−0.0723475	−	0.997379i	$-0.523049\pi$
$953$	−4256.89	−0.144695	−0.0723475	+	0.997379i	$0.523049\pi$
$954$	0	0
$955$	3212.93	0.108867
$956$	0	0
$957$	−37993.6	−1.28334
$958$	0	0
$959$	13089.4	0.440751
$960$	0	0
$961$	22411.1	0.752278
$962$	0	0
$963$	4174.95	0.139705
$964$	0	0
$965$	−1962.71	−0.0654733
$966$	0	0
$967$	−711.661	−0.0236665	−0.0118332	−	0.999930i	$-0.503767\pi$
$967$	−711.661	−0.0236665	−0.0118332	+	0.999930i	$0.503767\pi$
$968$	0	0
$969$	−7523.17	−0.249411
$970$	0	0
$971$	−37564.1	−1.24149	−0.620747	−	0.784011i	$-0.713171\pi$
$971$	−37564.1	−1.24149	−0.620747	+	0.784011i	$0.713171\pi$
$972$	0	0
$973$	22273.5	0.733871
$974$	0	0
$975$	−11171.8	−0.366958
$976$	0	0
$977$	29112.6	0.953322	0.476661	−	0.879087i	$-0.341847\pi$
$977$	29112.6	0.953322	0.476661	+	0.879087i	$0.341847\pi$
$978$	0	0
$979$	−25093.9	−0.819208
$980$	0	0
$981$	−11621.7	−0.378238
$982$	0	0
$983$	−59681.0	−1.93645	−0.968225	−	0.250082i	$-0.919542\pi$
$983$	−59681.0	−1.93645	−0.968225	+	0.250082i	$0.919542\pi$
$984$	0	0
$985$	7299.90	0.236136
$986$	0	0
$987$	12386.4	0.399457
$988$	0	0
$989$	−5253.88	−0.168922
$990$	0	0
$991$	46358.4	1.48600	0.742998	−	0.669293i	$-0.233403\pi$
$991$	46358.4	1.48600	0.742998	+	0.669293i	$0.233403\pi$
$992$	0	0
$993$	59225.3	1.89271
$994$	0	0
$995$	−15385.5	−0.490206
$996$	0	0
$997$	9026.51	0.286733	0.143366	−	0.989670i	$-0.454207\pi$
$997$	9026.51	0.286733	0.143366	+	0.989670i	$0.454207\pi$
$998$	0	0
$999$	6118.20	0.193765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.cl.1.1		4
4.3	odd	2		2240.4.a.ca.1.4		4
8.3	odd	2		1120.4.a.p.1.1	yes	4
8.5	even	2		1120.4.a.e.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.e.1.4	✓	4	8.5	even	2
1120.4.a.p.1.1	yes	4	8.3	odd	2
2240.4.a.ca.1.4		4	4.3	odd	2
2240.4.a.cl.1.1		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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q-5.92058 q^{3} -5.00000 q^{5} +7.00000 q^{7} +8.05323 q^{9} +O(q^{10})\)	\(q-5.92058 q^{3} -5.00000 q^{5} +7.00000 q^{7} +8.05323 q^{9} +28.7744 q^{11} +75.4778 q^{13} +29.6029 q^{15} +29.6282 q^{17} +42.8876 q^{19} -41.4440 q^{21} -13.1124 q^{23} +25.0000 q^{25} +112.176 q^{27} +223.019 q^{29} +228.478 q^{31} -170.361 q^{33} -35.0000 q^{35} +54.5411 q^{37} -446.872 q^{39} -344.490 q^{41} +400.681 q^{43} -40.2661 q^{45} -298.871 q^{47} +49.0000 q^{49} -175.416 q^{51} -338.028 q^{53} -143.872 q^{55} -253.919 q^{57} -39.8581 q^{59} +519.466 q^{61} +56.3726 q^{63} -377.389 q^{65} +28.6689 q^{67} +77.6329 q^{69} +483.529 q^{71} +280.535 q^{73} -148.014 q^{75} +201.421 q^{77} +395.739 q^{79} -881.583 q^{81} -646.985 q^{83} -148.141 q^{85} -1320.40 q^{87} -872.092 q^{89} +528.345 q^{91} -1352.72 q^{93} -214.438 q^{95} +768.173 q^{97} +231.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 9 q^{3} - 20 q^{5} + 28 q^{7} + 55 q^{9}+O(q^{10}) \) 4 * q + 9 * q^3 - 20 * q^5 + 28 * q^7 + 55 * q^9	\( 4 q + 9 q^{3} - 20 q^{5} + 28 q^{7} + 55 q^{9} + 33 q^{11} + 83 q^{13} - 45 q^{15} - 13 q^{17} + 132 q^{19} + 63 q^{21} - 92 q^{23} + 100 q^{25} + 495 q^{27} + 113 q^{29} - 94 q^{31} + 11 q^{33} - 140 q^{35} + 54 q^{37} - 145 q^{39} + 428 q^{41} + 604 q^{43} - 275 q^{45} - 709 q^{47} + 196 q^{49} - 13 q^{51} - 554 q^{53} - 165 q^{55} + 1044 q^{57} - 100 q^{59} + 588 q^{61} + 385 q^{63} - 415 q^{65} + 484 q^{67} + 540 q^{69} + 128 q^{71} + 1508 q^{73} + 225 q^{75} + 231 q^{77} - 587 q^{79} + 2548 q^{81} + 220 q^{83} + 65 q^{85} - 2887 q^{87} - 1392 q^{89} + 581 q^{91} - 2074 q^{93} - 660 q^{95} + 643 q^{97} + 690 q^{99}+O(q^{100}) \) 4 * q + 9 * q^3 - 20 * q^5 + 28 * q^7 + 55 * q^9 + 33 * q^11 + 83 * q^13 - 45 * q^15 - 13 * q^17 + 132 * q^19 + 63 * q^21 - 92 * q^23 + 100 * q^25 + 495 * q^27 + 113 * q^29 - 94 * q^31 + 11 * q^33 - 140 * q^35 + 54 * q^37 - 145 * q^39 + 428 * q^41 + 604 * q^43 - 275 * q^45 - 709 * q^47 + 196 * q^49 - 13 * q^51 - 554 * q^53 - 165 * q^55 + 1044 * q^57 - 100 * q^59 + 588 * q^61 + 385 * q^63 - 415 * q^65 + 484 * q^67 + 540 * q^69 + 128 * q^71 + 1508 * q^73 + 225 * q^75 + 231 * q^77 - 587 * q^79 + 2548 * q^81 + 220 * q^83 + 65 * q^85 - 2887 * q^87 - 1392 * q^89 + 581 * q^91 - 2074 * q^93 - 660 * q^95 + 643 * q^97 + 690 * q^99

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