LMFDB - Embedded newform 1560.4.a.m.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:	$0$
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} - x^{3} - 76x^{2} + 50x + 1156 $ x^4 - x^3 - 76x^2 + 50x + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.35294$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -5.00000 q^{5} -9.18441 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -5.00000 q^{5} -9.18441 q^{7} +9.00000 q^{9} +8.91444 q^{11} +13.0000 q^{13} -15.0000 q^{15} +8.31320 q^{17} -71.9653 q^{19} -27.5532 q^{21} +14.3132 q^{23} +25.0000 q^{25} +27.0000 q^{27} -123.634 q^{29} +253.599 q^{31} +26.7433 q^{33} +45.9220 q^{35} -57.9472 q^{37} +39.0000 q^{39} +130.181 q^{41} -73.9904 q^{43} -45.0000 q^{45} -48.1240 q^{47} -258.647 q^{49} +24.9396 q^{51} +327.200 q^{53} -44.5722 q^{55} -215.896 q^{57} +338.153 q^{59} +558.554 q^{61} -82.6597 q^{63} -65.0000 q^{65} +592.316 q^{67} +42.9396 q^{69} -704.645 q^{71} -948.099 q^{73} +75.0000 q^{75} -81.8738 q^{77} -534.397 q^{79} +81.0000 q^{81} +127.041 q^{83} -41.5660 q^{85} -370.902 q^{87} +1462.06 q^{89} -119.397 q^{91} +760.798 q^{93} +359.826 q^{95} +1510.57 q^{97} +80.2300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9 - 7 * q^11 + 52 * q^13 - 60 * q^15 - 73 * q^17 + 68 * q^19 + 45 * q^21 - 49 * q^23 + 100 * q^25 + 108 * q^27 - 66 * q^29 + 230 * q^31 - 21 * q^33 - 75 * q^35 + 303 * q^37 + 156 * q^39 + 155 * q^41 + 336 * q^43 - 180 * q^45 + 178 * q^47 + 377 * q^49 - 219 * q^51 + 349 * q^53 + 35 * q^55 + 204 * q^57 - 360 * q^59 + 223 * q^61 + 135 * q^63 - 260 * q^65 + 588 * q^67 - 147 * q^69 - 83 * q^71 + 754 * q^73 + 300 * q^75 + 1401 * q^77 + 869 * q^79 + 324 * q^81 - 1044 * q^83 + 365 * q^85 - 198 * q^87 + 1017 * q^89 + 195 * q^91 + 690 * q^93 - 340 * q^95 + 1367 * q^97 - 63 * 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
$7$	−9.18441	−0.495911	−0.247956	−	0.968771i	$-0.579759\pi$
$7$	−9.18441	−0.495911	−0.247956	+	0.968771i	$0.579759\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	8.91444	0.244346	0.122173	−	0.992509i	$-0.461014\pi$
$11$	8.91444	0.244346	0.122173	+	0.992509i	$0.461014\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	8.31320	0.118603	0.0593014	−	0.998240i	$-0.481113\pi$
$17$	8.31320	0.118603	0.0593014	+	0.998240i	$0.481113\pi$
$18$	0	0
$19$	−71.9653	−0.868945	−0.434473	−	0.900685i	$-0.643065\pi$
$19$	−71.9653	−0.868945	−0.434473	+	0.900685i	$0.643065\pi$
$20$	0	0
$21$	−27.5532	−0.286315
$22$	0	0
$23$	14.3132	0.129761	0.0648806	−	0.997893i	$-0.479333\pi$
$23$	14.3132	0.129761	0.0648806	+	0.997893i	$0.479333\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−123.634	−0.791664	−0.395832	−	0.918323i	$-0.629544\pi$
$29$	−123.634	−0.791664	−0.395832	+	0.918323i	$0.629544\pi$
$30$	0	0
$31$	253.599	1.46928	0.734641	−	0.678456i	$-0.237350\pi$
$31$	253.599	1.46928	0.734641	+	0.678456i	$0.237350\pi$
$32$	0	0
$33$	26.7433	0.141073
$34$	0	0
$35$	45.9220	0.221778
$36$	0	0
$37$	−57.9472	−0.257472	−0.128736	−	0.991679i	$-0.541092\pi$
$37$	−57.9472	−0.257472	−0.128736	+	0.991679i	$0.541092\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	130.181	0.495875	0.247937	−	0.968776i	$-0.420247\pi$
$41$	130.181	0.495875	0.247937	+	0.968776i	$0.420247\pi$
$42$	0	0
$43$	−73.9904	−0.262405	−0.131203	−	0.991356i	$-0.541884\pi$
$43$	−73.9904	−0.262405	−0.131203	+	0.991356i	$0.541884\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−48.1240	−0.149353	−0.0746767	−	0.997208i	$-0.523792\pi$
$47$	−48.1240	−0.149353	−0.0746767	+	0.997208i	$0.523792\pi$
$48$	0	0
$49$	−258.647	−0.754072
$50$	0	0
$51$	24.9396	0.0684753
$52$	0	0
$53$	327.200	0.848007	0.424004	−	0.905661i	$-0.360624\pi$
$53$	327.200	0.848007	0.424004	+	0.905661i	$0.360624\pi$
$54$	0	0
$55$	−44.5722	−0.109275
$56$	0	0
$57$	−215.896	−0.501686
$58$	0	0
$59$	338.153	0.746166	0.373083	−	0.927798i	$-0.378301\pi$
$59$	338.153	0.746166	0.373083	+	0.927798i	$0.378301\pi$
$60$	0	0
$61$	558.554	1.17239	0.586193	−	0.810171i	$-0.300626\pi$
$61$	558.554	1.17239	0.586193	+	0.810171i	$0.300626\pi$
$62$	0	0
$63$	−82.6597	−0.165304
$64$	0	0
$65$	−65.0000	−0.124035
$66$	0	0
$67$	592.316	1.08004	0.540022	−	0.841651i	$-0.318416\pi$
$67$	592.316	1.08004	0.540022	+	0.841651i	$0.318416\pi$
$68$	0	0
$69$	42.9396	0.0749177
$70$	0	0
$71$	−704.645	−1.17783	−0.588915	−	0.808195i	$-0.700445\pi$
$71$	−704.645	−1.17783	−0.588915	+	0.808195i	$0.700445\pi$
$72$	0	0
$73$	−948.099	−1.52009	−0.760045	−	0.649870i	$-0.774823\pi$
$73$	−948.099	−1.52009	−0.760045	+	0.649870i	$0.774823\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−81.8738	−0.121174
$78$	0	0
$79$	−534.397	−0.761068	−0.380534	−	0.924767i	$-0.624260\pi$
$79$	−534.397	−0.761068	−0.380534	+	0.924767i	$0.624260\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	127.041	0.168007	0.0840035	−	0.996465i	$-0.473229\pi$
$83$	127.041	0.168007	0.0840035	+	0.996465i	$0.473229\pi$
$84$	0	0
$85$	−41.5660	−0.0530408
$86$	0	0
$87$	−370.902	−0.457067
$88$	0	0
$89$	1462.06	1.74133	0.870666	−	0.491875i	$-0.163688\pi$
$89$	1462.06	1.74133	0.870666	+	0.491875i	$0.163688\pi$
$90$	0	0
$91$	−119.397	−0.137541
$92$	0	0
$93$	760.798	0.848291
$94$	0	0
$95$	359.826	0.388604
$96$	0	0
$97$	1510.57	1.58118	0.790591	−	0.612344i	$-0.209773\pi$
$97$	1510.57	1.58118	0.790591	+	0.612344i	$0.209773\pi$
$98$	0	0
$99$	80.2300	0.0814486
$100$	0	0
$101$	1714.84	1.68943	0.844716	−	0.535215i	$-0.179770\pi$
$101$	1714.84	1.68943	0.844716	+	0.535215i	$0.179770\pi$
$102$	0	0
$103$	1650.98	1.57938	0.789689	−	0.613508i	$-0.210242\pi$
$103$	1650.98	1.57938	0.789689	+	0.613508i	$0.210242\pi$
$104$	0	0
$105$	137.766	0.128044
$106$	0	0
$107$	−66.9290	−0.0604698	−0.0302349	−	0.999543i	$-0.509626\pi$
$107$	−66.9290	−0.0604698	−0.0302349	+	0.999543i	$0.509626\pi$
$108$	0	0
$109$	823.450	0.723599	0.361799	−	0.932256i	$-0.382163\pi$
$109$	823.450	0.723599	0.361799	+	0.932256i	$0.382163\pi$
$110$	0	0
$111$	−173.842	−0.148651
$112$	0	0
$113$	−1595.98	−1.32865	−0.664323	−	0.747446i	$-0.731280\pi$
$113$	−1595.98	−1.32865	−0.664323	+	0.747446i	$0.731280\pi$
$114$	0	0
$115$	−71.5660	−0.0580310
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−76.3518	−0.0588164
$120$	0	0
$121$	−1251.53	−0.940295
$122$	0	0
$123$	390.543	0.286293
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1193.20	0.833698	0.416849	−	0.908976i	$-0.363134\pi$
$127$	1193.20	0.833698	0.416849	+	0.908976i	$0.363134\pi$
$128$	0	0
$129$	−221.971	−0.151500
$130$	0	0
$131$	725.095	0.483602	0.241801	−	0.970326i	$-0.422262\pi$
$131$	725.095	0.483602	0.241801	+	0.970326i	$0.422262\pi$
$132$	0	0
$133$	660.958	0.430920
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	2585.11	1.61212	0.806061	−	0.591833i	$-0.201595\pi$
$137$	2585.11	1.61212	0.806061	+	0.591833i	$0.201595\pi$
$138$	0	0
$139$	1824.73	1.11347	0.556733	−	0.830692i	$-0.312055\pi$
$139$	1824.73	1.11347	0.556733	+	0.830692i	$0.312055\pi$
$140$	0	0
$141$	−144.372	−0.0862292
$142$	0	0
$143$	115.888	0.0677694
$144$	0	0
$145$	618.170	0.354043
$146$	0	0
$147$	−775.940	−0.435364
$148$	0	0
$149$	−2557.90	−1.40638	−0.703192	−	0.711000i	$-0.748243\pi$
$149$	−2557.90	−1.40638	−0.703192	+	0.711000i	$0.748243\pi$
$150$	0	0
$151$	3042.70	1.63981	0.819906	−	0.572498i	$-0.194026\pi$
$151$	3042.70	1.63981	0.819906	+	0.572498i	$0.194026\pi$
$152$	0	0
$153$	74.8188	0.0395342
$154$	0	0
$155$	−1268.00	−0.657083
$156$	0	0
$157$	2467.81	1.25448	0.627238	−	0.778828i	$-0.284185\pi$
$157$	2467.81	1.25448	0.627238	+	0.778828i	$0.284185\pi$
$158$	0	0
$159$	981.600	0.489597
$160$	0	0
$161$	−131.458	−0.0643501
$162$	0	0
$163$	−1281.18	−0.615641	−0.307820	−	0.951444i	$-0.599600\pi$
$163$	−1281.18	−0.615641	−0.307820	+	0.951444i	$0.599600\pi$
$164$	0	0
$165$	−133.717	−0.0630898
$166$	0	0
$167$	418.839	0.194076	0.0970381	−	0.995281i	$-0.469063\pi$
$167$	418.839	0.194076	0.0970381	+	0.995281i	$0.469063\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−647.687	−0.289648
$172$	0	0
$173$	484.852	0.213079	0.106539	−	0.994308i	$-0.466023\pi$
$173$	484.852	0.213079	0.106539	+	0.994308i	$0.466023\pi$
$174$	0	0
$175$	−229.610	−0.0991823
$176$	0	0
$177$	1014.46	0.430799
$178$	0	0
$179$	−2313.88	−0.966187	−0.483094	−	0.875569i	$-0.660487\pi$
$179$	−2313.88	−0.966187	−0.483094	+	0.875569i	$0.660487\pi$
$180$	0	0
$181$	−2221.97	−0.912475	−0.456237	−	0.889858i	$-0.650803\pi$
$181$	−2221.97	−0.912475	−0.456237	+	0.889858i	$0.650803\pi$
$182$	0	0
$183$	1675.66	0.676878
$184$	0	0
$185$	289.736	0.115145
$186$	0	0
$187$	74.1075	0.0289801
$188$	0	0
$189$	−247.979	−0.0954382
$190$	0	0
$191$	1657.40	0.627879	0.313940	−	0.949443i	$-0.398351\pi$
$191$	1657.40	0.627879	0.313940	+	0.949443i	$0.398351\pi$
$192$	0	0
$193$	−234.064	−0.0872969	−0.0436484	−	0.999047i	$-0.513898\pi$
$193$	−234.064	−0.0872969	−0.0436484	+	0.999047i	$0.513898\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	1203.72	0.435337	0.217668	−	0.976023i	$-0.430155\pi$
$197$	1203.72	0.435337	0.217668	+	0.976023i	$0.430155\pi$
$198$	0	0
$199$	558.776	0.199048	0.0995241	−	0.995035i	$-0.468268\pi$
$199$	558.776	0.199048	0.0995241	+	0.995035i	$0.468268\pi$
$200$	0	0
$201$	1776.95	0.623563
$202$	0	0
$203$	1135.50	0.392595
$204$	0	0
$205$	−650.905	−0.221762
$206$	0	0
$207$	128.819	0.0432537
$208$	0	0
$209$	−641.530	−0.212323
$210$	0	0
$211$	−1664.68	−0.543136	−0.271568	−	0.962419i	$-0.587542\pi$
$211$	−1664.68	−0.543136	−0.271568	+	0.962419i	$0.587542\pi$
$212$	0	0
$213$	−2113.94	−0.680021
$214$	0	0
$215$	369.952	0.117351
$216$	0	0
$217$	−2329.16	−0.728634
$218$	0	0
$219$	−2844.30	−0.877625
$220$	0	0
$221$	108.072	0.0328945
$222$	0	0
$223$	26.9328	0.00808768	0.00404384	−	0.999992i	$-0.498713\pi$
$223$	26.9328	0.00808768	0.00404384	+	0.999992i	$0.498713\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3814.74	1.11539	0.557694	−	0.830047i	$-0.311686\pi$
$227$	3814.74	1.11539	0.557694	+	0.830047i	$0.311686\pi$
$228$	0	0
$229$	2303.07	0.664591	0.332296	−	0.943175i	$-0.392177\pi$
$229$	2303.07	0.664591	0.332296	+	0.943175i	$0.392177\pi$
$230$	0	0
$231$	−245.622	−0.0699598
$232$	0	0
$233$	6520.13	1.83325	0.916626	−	0.399746i	$-0.130901\pi$
$233$	6520.13	1.83325	0.916626	+	0.399746i	$0.130901\pi$
$234$	0	0
$235$	240.620	0.0667928
$236$	0	0
$237$	−1603.19	−0.439403
$238$	0	0
$239$	2107.59	0.570414	0.285207	−	0.958466i	$-0.407938\pi$
$239$	2107.59	0.570414	0.285207	+	0.958466i	$0.407938\pi$
$240$	0	0
$241$	3312.85	0.885474	0.442737	−	0.896652i	$-0.354008\pi$
$241$	3312.85	0.885474	0.442737	+	0.896652i	$0.354008\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1293.23	0.337231
$246$	0	0
$247$	−935.548	−0.241002
$248$	0	0
$249$	381.124	0.0969989
$250$	0	0
$251$	−2975.51	−0.748257	−0.374129	−	0.927377i	$-0.622058\pi$
$251$	−2975.51	−0.748257	−0.374129	+	0.927377i	$0.622058\pi$
$252$	0	0
$253$	127.594	0.0317066
$254$	0	0
$255$	−124.698	−0.0306231
$256$	0	0
$257$	−7089.87	−1.72083	−0.860417	−	0.509591i	$-0.829797\pi$
$257$	−7089.87	−1.72083	−0.860417	+	0.509591i	$0.829797\pi$
$258$	0	0
$259$	532.210	0.127683
$260$	0	0
$261$	−1112.71	−0.263888
$262$	0	0
$263$	−6393.07	−1.49891	−0.749455	−	0.662055i	$-0.769684\pi$
$263$	−6393.07	−1.49891	−0.749455	+	0.662055i	$0.769684\pi$
$264$	0	0
$265$	−1636.00	−0.379240
$266$	0	0
$267$	4386.19	1.00536
$268$	0	0
$269$	−2674.07	−0.606100	−0.303050	−	0.952975i	$-0.598005\pi$
$269$	−2674.07	−0.606100	−0.303050	+	0.952975i	$0.598005\pi$
$270$	0	0
$271$	7897.49	1.77025	0.885126	−	0.465352i	$-0.154072\pi$
$271$	7897.49	1.77025	0.885126	+	0.465352i	$0.154072\pi$
$272$	0	0
$273$	−358.192	−0.0794094
$274$	0	0
$275$	222.861	0.0488692
$276$	0	0
$277$	7744.54	1.67987	0.839936	−	0.542686i	$-0.182593\pi$
$277$	7744.54	1.67987	0.839936	+	0.542686i	$0.182593\pi$
$278$	0	0
$279$	2282.39	0.489761
$280$	0	0
$281$	−3133.13	−0.665149	−0.332574	−	0.943077i	$-0.607917\pi$
$281$	−3133.13	−0.665149	−0.332574	+	0.943077i	$0.607917\pi$
$282$	0	0
$283$	3845.98	0.807843	0.403922	−	0.914794i	$-0.367647\pi$
$283$	3845.98	0.807843	0.403922	+	0.914794i	$0.367647\pi$
$284$	0	0
$285$	1079.48	0.224361
$286$	0	0
$287$	−1195.64	−0.245910
$288$	0	0
$289$	−4843.89	−0.985933
$290$	0	0
$291$	4531.70	0.912896
$292$	0	0
$293$	−1516.72	−0.302415	−0.151207	−	0.988502i	$-0.548316\pi$
$293$	−1516.72	−0.302415	−0.151207	+	0.988502i	$0.548316\pi$
$294$	0	0
$295$	−1690.77	−0.333696
$296$	0	0
$297$	240.690	0.0470244
$298$	0	0
$299$	186.072	0.0359893
$300$	0	0
$301$	679.558	0.130130
$302$	0	0
$303$	5144.51	0.975394
$304$	0	0
$305$	−2792.77	−0.524307
$306$	0	0
$307$	1955.60	0.363558	0.181779	−	0.983339i	$-0.441815\pi$
$307$	1955.60	0.363558	0.181779	+	0.983339i	$0.441815\pi$
$308$	0	0
$309$	4952.94	0.911854
$310$	0	0
$311$	6174.63	1.12582	0.562912	−	0.826517i	$-0.309681\pi$
$311$	6174.63	1.12582	0.562912	+	0.826517i	$0.309681\pi$
$312$	0	0
$313$	−8307.50	−1.50022	−0.750108	−	0.661315i	$-0.769999\pi$
$313$	−8307.50	−1.50022	−0.750108	+	0.661315i	$0.769999\pi$
$314$	0	0
$315$	413.298	0.0739261
$316$	0	0
$317$	−296.580	−0.0525476	−0.0262738	−	0.999655i	$-0.508364\pi$
$317$	−296.580	−0.0525476	−0.0262738	+	0.999655i	$0.508364\pi$
$318$	0	0
$319$	−1102.13	−0.193440
$320$	0	0
$321$	−200.787	−0.0349123
$322$	0	0
$323$	−598.261	−0.103059
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	2470.35	0.417770
$328$	0	0
$329$	441.990	0.0740660
$330$	0	0
$331$	6844.48	1.13658	0.568288	−	0.822830i	$-0.307606\pi$
$331$	6844.48	1.13658	0.568288	+	0.822830i	$0.307606\pi$
$332$	0	0
$333$	−521.525	−0.0858239
$334$	0	0
$335$	−2961.58	−0.483010
$336$	0	0
$337$	−3008.63	−0.486321	−0.243161	−	0.969986i	$-0.578184\pi$
$337$	−3008.63	−0.486321	−0.243161	+	0.969986i	$0.578184\pi$
$338$	0	0
$339$	−4787.93	−0.767094
$340$	0	0
$341$	2260.70	0.359013
$342$	0	0
$343$	5525.77	0.869864
$344$	0	0
$345$	−214.698	−0.0335042
$346$	0	0
$347$	−5809.16	−0.898709	−0.449354	−	0.893354i	$-0.648346\pi$
$347$	−5809.16	−0.898709	−0.449354	+	0.893354i	$0.648346\pi$
$348$	0	0
$349$	1808.97	0.277455	0.138727	−	0.990331i	$-0.455699\pi$
$349$	1808.97	0.277455	0.138727	+	0.990331i	$0.455699\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−6901.75	−1.04063	−0.520316	−	0.853974i	$-0.674186\pi$
$353$	−6901.75	−1.04063	−0.520316	+	0.853974i	$0.674186\pi$
$354$	0	0
$355$	3523.23	0.526742
$356$	0	0
$357$	−229.055	−0.0339577
$358$	0	0
$359$	7781.88	1.14404	0.572022	−	0.820238i	$-0.306159\pi$
$359$	7781.88	1.14404	0.572022	+	0.820238i	$0.306159\pi$
$360$	0	0
$361$	−1680.00	−0.244934
$362$	0	0
$363$	−3754.60	−0.542880
$364$	0	0
$365$	4740.50	0.679805
$366$	0	0
$367$	12566.2	1.78732	0.893662	−	0.448740i	$-0.148127\pi$
$367$	12566.2	1.78732	0.893662	+	0.448740i	$0.148127\pi$
$368$	0	0
$369$	1171.63	0.165292
$370$	0	0
$371$	−3005.14	−0.420536
$372$	0	0
$373$	815.595	0.113217	0.0566085	−	0.998396i	$-0.481971\pi$
$373$	815.595	0.113217	0.0566085	+	0.998396i	$0.481971\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−1607.24	−0.219568
$378$	0	0
$379$	4831.28	0.654793	0.327396	−	0.944887i	$-0.393829\pi$
$379$	4831.28	0.654793	0.327396	+	0.944887i	$0.393829\pi$
$380$	0	0
$381$	3579.61	0.481336
$382$	0	0
$383$	2813.08	0.375304	0.187652	−	0.982236i	$-0.439912\pi$
$383$	2813.08	0.375304	0.187652	+	0.982236i	$0.439912\pi$
$384$	0	0
$385$	409.369	0.0541906
$386$	0	0
$387$	−665.914	−0.0874685
$388$	0	0
$389$	−5797.55	−0.755649	−0.377825	−	0.925877i	$-0.623328\pi$
$389$	−5797.55	−0.755649	−0.377825	+	0.925877i	$0.623328\pi$
$390$	0	0
$391$	118.988	0.0153900
$392$	0	0
$393$	2175.28	0.279208
$394$	0	0
$395$	2671.99	0.340360
$396$	0	0
$397$	−8464.73	−1.07011	−0.535053	−	0.844818i	$-0.679709\pi$
$397$	−8464.73	−1.07011	−0.535053	+	0.844818i	$0.679709\pi$
$398$	0	0
$399$	1982.87	0.248792
$400$	0	0
$401$	545.269	0.0679039	0.0339519	−	0.999423i	$-0.489191\pi$
$401$	545.269	0.0679039	0.0339519	+	0.999423i	$0.489191\pi$
$402$	0	0
$403$	3296.79	0.407506
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−516.567	−0.0629122
$408$	0	0
$409$	11275.4	1.36317	0.681583	−	0.731741i	$-0.261292\pi$
$409$	11275.4	1.36317	0.681583	+	0.731741i	$0.261292\pi$
$410$	0	0
$411$	7755.32	0.930759
$412$	0	0
$413$	−3105.74	−0.370032
$414$	0	0
$415$	−635.206	−0.0751350
$416$	0	0
$417$	5474.19	0.642859
$418$	0	0
$419$	−6472.08	−0.754610	−0.377305	−	0.926089i	$-0.623149\pi$
$419$	−6472.08	−0.754610	−0.377305	+	0.926089i	$0.623149\pi$
$420$	0	0
$421$	−8250.77	−0.955150	−0.477575	−	0.878591i	$-0.658484\pi$
$421$	−8250.77	−0.955150	−0.477575	+	0.878591i	$0.658484\pi$
$422$	0	0
$423$	−433.116	−0.0497844
$424$	0	0
$425$	207.830	0.0237205
$426$	0	0
$427$	−5129.99	−0.581400
$428$	0	0
$429$	347.663	0.0391267
$430$	0	0
$431$	5626.53	0.628818	0.314409	−	0.949288i	$-0.398194\pi$
$431$	5626.53	0.628818	0.314409	+	0.949288i	$0.398194\pi$
$432$	0	0
$433$	257.523	0.0285815	0.0142907	−	0.999898i	$-0.495451\pi$
$433$	257.523	0.0285815	0.0142907	+	0.999898i	$0.495451\pi$
$434$	0	0
$435$	1854.51	0.204407
$436$	0	0
$437$	−1030.05	−0.112755
$438$	0	0
$439$	5056.74	0.549761	0.274880	−	0.961478i	$-0.411362\pi$
$439$	5056.74	0.549761	0.274880	+	0.961478i	$0.411362\pi$
$440$	0	0
$441$	−2327.82	−0.251357
$442$	0	0
$443$	11950.0	1.28163	0.640815	−	0.767695i	$-0.278597\pi$
$443$	11950.0	1.28163	0.640815	+	0.767695i	$0.278597\pi$
$444$	0	0
$445$	−7310.32	−0.778747
$446$	0	0
$447$	−7673.70	−0.811976
$448$	0	0
$449$	−13778.2	−1.44818	−0.724091	−	0.689705i	$-0.757740\pi$
$449$	−13778.2	−1.44818	−0.724091	+	0.689705i	$0.757740\pi$
$450$	0	0
$451$	1160.49	0.121165
$452$	0	0
$453$	9128.11	0.946746
$454$	0	0
$455$	596.986	0.0615102
$456$	0	0
$457$	−2969.71	−0.303977	−0.151988	−	0.988382i	$-0.548568\pi$
$457$	−2969.71	−0.303977	−0.151988	+	0.988382i	$0.548568\pi$
$458$	0	0
$459$	224.456	0.0228251
$460$	0	0
$461$	2889.72	0.291947	0.145974	−	0.989288i	$-0.453369\pi$
$461$	2889.72	0.291947	0.145974	+	0.989288i	$0.453369\pi$
$462$	0	0
$463$	8059.68	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$463$	8059.68	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$464$	0	0
$465$	−3803.99	−0.379367
$466$	0	0
$467$	15088.4	1.49509	0.747546	−	0.664210i	$-0.231232\pi$
$467$	15088.4	1.49509	0.747546	+	0.664210i	$0.231232\pi$
$468$	0	0
$469$	−5440.07	−0.535606
$470$	0	0
$471$	7403.43	0.724272
$472$	0	0
$473$	−659.583	−0.0641177
$474$	0	0
$475$	−1799.13	−0.173789
$476$	0	0
$477$	2944.80	0.282669
$478$	0	0
$479$	−13600.0	−1.29729	−0.648644	−	0.761092i	$-0.724664\pi$
$479$	−13600.0	−1.29729	−0.648644	+	0.761092i	$0.724664\pi$
$480$	0	0
$481$	−753.313	−0.0714098
$482$	0	0
$483$	−394.375	−0.0371525
$484$	0	0
$485$	−7552.83	−0.707126
$486$	0	0
$487$	−5556.51	−0.517021	−0.258511	−	0.966008i	$-0.583232\pi$
$487$	−5556.51	−0.517021	−0.258511	+	0.966008i	$0.583232\pi$
$488$	0	0
$489$	−3843.53	−0.355440
$490$	0	0
$491$	2639.02	0.242561	0.121280	−	0.992618i	$-0.461300\pi$
$491$	2639.02	0.242561	0.121280	+	0.992618i	$0.461300\pi$
$492$	0	0
$493$	−1027.79	−0.0938935
$494$	0	0
$495$	−401.150	−0.0364249
$496$	0	0
$497$	6471.75	0.584100
$498$	0	0
$499$	−121.756	−0.0109229	−0.00546146	−	0.999985i	$-0.501738\pi$
$499$	−121.756	−0.0109229	−0.00546146	+	0.999985i	$0.501738\pi$
$500$	0	0
$501$	1256.52	0.112050
$502$	0	0
$503$	−9992.42	−0.885765	−0.442883	−	0.896580i	$-0.646044\pi$
$503$	−9992.42	−0.885765	−0.442883	+	0.896580i	$0.646044\pi$
$504$	0	0
$505$	−8574.18	−0.755537
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	2286.99	0.199153	0.0995765	−	0.995030i	$-0.468251\pi$
$509$	2286.99	0.199153	0.0995765	+	0.995030i	$0.468251\pi$
$510$	0	0
$511$	8707.73	0.753830
$512$	0	0
$513$	−1943.06	−0.167229
$514$	0	0
$515$	−8254.90	−0.706319
$516$	0	0
$517$	−428.999	−0.0364939
$518$	0	0
$519$	1454.56	0.123021
$520$	0	0
$521$	−8723.51	−0.733558	−0.366779	−	0.930308i	$-0.619540\pi$
$521$	−8723.51	−0.733558	−0.366779	+	0.930308i	$0.619540\pi$
$522$	0	0
$523$	−14023.2	−1.17245	−0.586224	−	0.810149i	$-0.699386\pi$
$523$	−14023.2	−1.17245	−0.586224	+	0.810149i	$0.699386\pi$
$524$	0	0
$525$	−688.830	−0.0572629
$526$	0	0
$527$	2108.22	0.174261
$528$	0	0
$529$	−11962.1	−0.983162
$530$	0	0
$531$	3043.38	0.248722
$532$	0	0
$533$	1692.35	0.137531
$534$	0	0
$535$	334.645	0.0270429
$536$	0	0
$537$	−6941.64	−0.557828
$538$	0	0
$539$	−2305.69	−0.184254
$540$	0	0
$541$	1119.47	0.0889646	0.0444823	−	0.999010i	$-0.485836\pi$
$541$	1119.47	0.0889646	0.0444823	+	0.999010i	$0.485836\pi$
$542$	0	0
$543$	−6665.92	−0.526818
$544$	0	0
$545$	−4117.25	−0.323603
$546$	0	0
$547$	10036.6	0.784525	0.392263	−	0.919853i	$-0.371692\pi$
$547$	10036.6	0.784525	0.392263	+	0.919853i	$0.371692\pi$
$548$	0	0
$549$	5026.99	0.390795
$550$	0	0
$551$	8897.35	0.687913
$552$	0	0
$553$	4908.12	0.377422
$554$	0	0
$555$	869.208	0.0664789
$556$	0	0
$557$	−3348.98	−0.254759	−0.127380	−	0.991854i	$-0.540657\pi$
$557$	−3348.98	−0.254759	−0.127380	+	0.991854i	$0.540657\pi$
$558$	0	0
$559$	−961.875	−0.0727782
$560$	0	0
$561$	222.323	0.0167317
$562$	0	0
$563$	11087.4	0.829980	0.414990	−	0.909826i	$-0.363785\pi$
$563$	11087.4	0.829980	0.414990	+	0.909826i	$0.363785\pi$
$564$	0	0
$565$	7979.89	0.594188
$566$	0	0
$567$	−743.937	−0.0551013
$568$	0	0
$569$	−3761.70	−0.277151	−0.138575	−	0.990352i	$-0.544252\pi$
$569$	−3761.70	−0.277151	−0.138575	+	0.990352i	$0.544252\pi$
$570$	0	0
$571$	−2635.68	−0.193170	−0.0965849	−	0.995325i	$-0.530792\pi$
$571$	−2635.68	−0.193170	−0.0965849	+	0.995325i	$0.530792\pi$
$572$	0	0
$573$	4972.19	0.362506
$574$	0	0
$575$	357.830	0.0259522
$576$	0	0
$577$	9175.07	0.661981	0.330991	−	0.943634i	$-0.392617\pi$
$577$	9175.07	0.661981	0.330991	+	0.943634i	$0.392617\pi$
$578$	0	0
$579$	−702.192	−0.0504009
$580$	0	0
$581$	−1166.80	−0.0833166
$582$	0	0
$583$	2916.80	0.207207
$584$	0	0
$585$	−585.000	−0.0413449
$586$	0	0
$587$	−25053.5	−1.76162	−0.880809	−	0.473473i	$-0.843000\pi$
$587$	−25053.5	−1.76162	−0.880809	+	0.473473i	$0.843000\pi$
$588$	0	0
$589$	−18250.3	−1.27673
$590$	0	0
$591$	3611.15	0.251342
$592$	0	0
$593$	−9428.23	−0.652902	−0.326451	−	0.945214i	$-0.605853\pi$
$593$	−9428.23	−0.652902	−0.326451	+	0.945214i	$0.605853\pi$
$594$	0	0
$595$	381.759	0.0263035
$596$	0	0
$597$	1676.33	0.114920
$598$	0	0
$599$	−8329.53	−0.568172	−0.284086	−	0.958799i	$-0.591690\pi$
$599$	−8329.53	−0.568172	−0.284086	+	0.958799i	$0.591690\pi$
$600$	0	0
$601$	−801.516	−0.0544002	−0.0272001	−	0.999630i	$-0.508659\pi$
$601$	−801.516	−0.0544002	−0.0272001	+	0.999630i	$0.508659\pi$
$602$	0	0
$603$	5330.84	0.360014
$604$	0	0
$605$	6257.66	0.420513
$606$	0	0
$607$	22190.7	1.48384	0.741920	−	0.670488i	$-0.233915\pi$
$607$	22190.7	1.48384	0.741920	+	0.670488i	$0.233915\pi$
$608$	0	0
$609$	3406.51	0.226665
$610$	0	0
$611$	−625.612	−0.0414232
$612$	0	0
$613$	26527.6	1.74786	0.873931	−	0.486051i	$-0.161563\pi$
$613$	26527.6	1.74786	0.873931	+	0.486051i	$0.161563\pi$
$614$	0	0
$615$	−1952.72	−0.128034
$616$	0	0
$617$	−11685.3	−0.762451	−0.381225	−	0.924482i	$-0.624498\pi$
$617$	−11685.3	−0.762451	−0.381225	+	0.924482i	$0.624498\pi$
$618$	0	0
$619$	−16693.0	−1.08392	−0.541962	−	0.840403i	$-0.682318\pi$
$619$	−16693.0	−1.08392	−0.541962	+	0.840403i	$0.682318\pi$
$620$	0	0
$621$	386.456	0.0249726
$622$	0	0
$623$	−13428.2	−0.863546
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−1924.59	−0.122585
$628$	0	0
$629$	−481.726	−0.0305369
$630$	0	0
$631$	6102.99	0.385034	0.192517	−	0.981294i	$-0.438335\pi$
$631$	6102.99	0.385034	0.192517	+	0.981294i	$0.438335\pi$
$632$	0	0
$633$	−4994.05	−0.313580
$634$	0	0
$635$	−5966.01	−0.372841
$636$	0	0
$637$	−3362.41	−0.209142
$638$	0	0
$639$	−6341.81	−0.392610
$640$	0	0
$641$	−8775.25	−0.540720	−0.270360	−	0.962759i	$-0.587143\pi$
$641$	−8775.25	−0.540720	−0.270360	+	0.962759i	$0.587143\pi$
$642$	0	0
$643$	12103.5	0.742324	0.371162	−	0.928568i	$-0.378959\pi$
$643$	12103.5	0.742324	0.371162	+	0.928568i	$0.378959\pi$
$644$	0	0
$645$	1109.86	0.0677528
$646$	0	0
$647$	−9408.21	−0.571677	−0.285838	−	0.958278i	$-0.592272\pi$
$647$	−9408.21	−0.571677	−0.285838	+	0.958278i	$0.592272\pi$
$648$	0	0
$649$	3014.45	0.182323
$650$	0	0
$651$	−6987.48	−0.420677
$652$	0	0
$653$	−20291.0	−1.21600	−0.608001	−	0.793936i	$-0.708029\pi$
$653$	−20291.0	−1.21600	−0.608001	+	0.793936i	$0.708029\pi$
$654$	0	0
$655$	−3625.47	−0.216273
$656$	0	0
$657$	−8532.89	−0.506697
$658$	0	0
$659$	−10628.7	−0.628278	−0.314139	−	0.949377i	$-0.601716\pi$
$659$	−10628.7	−0.628278	−0.314139	+	0.949377i	$0.601716\pi$
$660$	0	0
$661$	−22526.3	−1.32552	−0.662761	−	0.748831i	$-0.730616\pi$
$661$	−22526.3	−1.32552	−0.662761	+	0.748831i	$0.730616\pi$
$662$	0	0
$663$	324.215	0.0189916
$664$	0	0
$665$	−3304.79	−0.192713
$666$	0	0
$667$	−1769.60	−0.102727
$668$	0	0
$669$	80.7983	0.00466942
$670$	0	0
$671$	4979.20	0.286468
$672$	0	0
$673$	−5917.49	−0.338934	−0.169467	−	0.985536i	$-0.554205\pi$
$673$	−5917.49	−0.338934	−0.169467	+	0.985536i	$0.554205\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−30940.0	−1.75645	−0.878227	−	0.478244i	$-0.841273\pi$
$677$	−30940.0	−1.75645	−0.878227	+	0.478244i	$0.841273\pi$
$678$	0	0
$679$	−13873.6	−0.784126
$680$	0	0
$681$	11444.2	0.643969
$682$	0	0
$683$	−12357.7	−0.692319	−0.346160	−	0.938176i	$-0.612514\pi$
$683$	−12357.7	−0.692319	−0.346160	+	0.938176i	$0.612514\pi$
$684$	0	0
$685$	−12925.5	−0.720963
$686$	0	0
$687$	6909.22	0.383702
$688$	0	0
$689$	4253.60	0.235195
$690$	0	0
$691$	1692.62	0.0931842	0.0465921	−	0.998914i	$-0.485164\pi$
$691$	1692.62	0.0931842	0.0465921	+	0.998914i	$0.485164\pi$
$692$	0	0
$693$	−736.865	−0.0403913
$694$	0	0
$695$	−9123.65	−0.497957
$696$	0	0
$697$	1082.22	0.0588121
$698$	0	0
$699$	19560.4	1.05843
$700$	0	0
$701$	708.542	0.0381758	0.0190879	−	0.999818i	$-0.493924\pi$
$701$	708.542	0.0381758	0.0190879	+	0.999818i	$0.493924\pi$
$702$	0	0
$703$	4170.18	0.223729
$704$	0	0
$705$	721.860	0.0385629
$706$	0	0
$707$	−15749.7	−0.837808
$708$	0	0
$709$	−19055.3	−1.00936	−0.504679	−	0.863307i	$-0.668389\pi$
$709$	−19055.3	−1.00936	−0.504679	+	0.863307i	$0.668389\pi$
$710$	0	0
$711$	−4809.58	−0.253689
$712$	0	0
$713$	3629.82	0.190656
$714$	0	0
$715$	−579.439	−0.0303074
$716$	0	0
$717$	6322.78	0.329329
$718$	0	0
$719$	−9329.52	−0.483911	−0.241956	−	0.970287i	$-0.577789\pi$
$719$	−9329.52	−0.483911	−0.241956	+	0.970287i	$0.577789\pi$
$720$	0	0
$721$	−15163.3	−0.783231
$722$	0	0
$723$	9938.54	0.511229
$724$	0	0
$725$	−3090.85	−0.158333
$726$	0	0
$727$	−15229.1	−0.776915	−0.388457	−	0.921467i	$-0.626992\pi$
$727$	−15229.1	−0.776915	−0.388457	+	0.921467i	$0.626992\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−615.097	−0.0311220
$732$	0	0
$733$	7791.16	0.392596	0.196298	−	0.980544i	$-0.437108\pi$
$733$	7791.16	0.392596	0.196298	+	0.980544i	$0.437108\pi$
$734$	0	0
$735$	3879.70	0.194701
$736$	0	0
$737$	5280.16	0.263904
$738$	0	0
$739$	24285.6	1.20888	0.604438	−	0.796652i	$-0.293398\pi$
$739$	24285.6	1.20888	0.604438	+	0.796652i	$0.293398\pi$
$740$	0	0
$741$	−2806.65	−0.139143
$742$	0	0
$743$	15755.7	0.777957	0.388978	−	0.921247i	$-0.372828\pi$
$743$	15755.7	0.777957	0.388978	+	0.921247i	$0.372828\pi$
$744$	0	0
$745$	12789.5	0.628954
$746$	0	0
$747$	1143.37	0.0560023
$748$	0	0
$749$	614.703	0.0299877
$750$	0	0
$751$	−12979.4	−0.630661	−0.315330	−	0.948982i	$-0.602115\pi$
$751$	−12979.4	−0.630661	−0.315330	+	0.948982i	$0.602115\pi$
$752$	0	0
$753$	−8926.53	−0.432007
$754$	0	0
$755$	−15213.5	−0.733346
$756$	0	0
$757$	32088.5	1.54066	0.770328	−	0.637648i	$-0.220092\pi$
$757$	32088.5	1.54066	0.770328	+	0.637648i	$0.220092\pi$
$758$	0	0
$759$	382.782	0.0183058
$760$	0	0
$761$	−38923.9	−1.85413	−0.927064	−	0.374903i	$-0.877676\pi$
$761$	−38923.9	−1.85413	−0.927064	+	0.374903i	$0.877676\pi$
$762$	0	0
$763$	−7562.90	−0.358841
$764$	0	0
$765$	−374.094	−0.0176803
$766$	0	0
$767$	4395.99	0.206949
$768$	0	0
$769$	−28176.7	−1.32130	−0.660648	−	0.750696i	$-0.729719\pi$
$769$	−28176.7	−1.32130	−0.660648	+	0.750696i	$0.729719\pi$
$770$	0	0
$771$	−21269.6	−0.993524
$772$	0	0
$773$	13797.1	0.641975	0.320988	−	0.947083i	$-0.395985\pi$
$773$	13797.1	0.641975	0.320988	+	0.947083i	$0.395985\pi$
$774$	0	0
$775$	6339.98	0.293857
$776$	0	0
$777$	1596.63	0.0737179
$778$	0	0
$779$	−9368.51	−0.430888
$780$	0	0
$781$	−6281.52	−0.287798
$782$	0	0
$783$	−3338.12	−0.152356
$784$	0	0
$785$	−12339.1	−0.561019
$786$	0	0
$787$	1674.04	0.0758235	0.0379117	−	0.999281i	$-0.487929\pi$
$787$	1674.04	0.0758235	0.0379117	+	0.999281i	$0.487929\pi$
$788$	0	0
$789$	−19179.2	−0.865396
$790$	0	0
$791$	14658.1	0.658890
$792$	0	0
$793$	7261.21	0.325161
$794$	0	0
$795$	−4908.00	−0.218954
$796$	0	0
$797$	21295.8	0.946468	0.473234	−	0.880937i	$-0.343086\pi$
$797$	21295.8	0.946468	0.473234	+	0.880937i	$0.343086\pi$
$798$	0	0
$799$	−400.064	−0.0177137
$800$	0	0
$801$	13158.6	0.580444
$802$	0	0
$803$	−8451.77	−0.371428
$804$	0	0
$805$	657.291	0.0287782
$806$	0	0
$807$	−8022.21	−0.349932
$808$	0	0
$809$	−30876.3	−1.34185	−0.670924	−	0.741526i	$-0.734102\pi$
$809$	−30876.3	−1.34185	−0.670924	+	0.741526i	$0.734102\pi$
$810$	0	0
$811$	28403.1	1.22980	0.614901	−	0.788604i	$-0.289196\pi$
$811$	28403.1	1.22980	0.614901	+	0.788604i	$0.289196\pi$
$812$	0	0
$813$	23692.5	1.02206
$814$	0	0
$815$	6405.88	0.275323
$816$	0	0
$817$	5324.74	0.228016
$818$	0	0
$819$	−1074.58	−0.0458470
$820$	0	0
$821$	−25881.0	−1.10019	−0.550093	−	0.835104i	$-0.685408\pi$
$821$	−25881.0	−1.10019	−0.550093	+	0.835104i	$0.685408\pi$
$822$	0	0
$823$	−25433.5	−1.07723	−0.538613	−	0.842553i	$-0.681052\pi$
$823$	−25433.5	−1.07723	−0.538613	+	0.842553i	$0.681052\pi$
$824$	0	0
$825$	668.583	0.0282146
$826$	0	0
$827$	9052.43	0.380633	0.190317	−	0.981723i	$-0.439049\pi$
$827$	9052.43	0.380633	0.190317	+	0.981723i	$0.439049\pi$
$828$	0	0
$829$	−23896.4	−1.00115	−0.500577	−	0.865692i	$-0.666879\pi$
$829$	−23896.4	−1.00115	−0.500577	+	0.865692i	$0.666879\pi$
$830$	0	0
$831$	23233.6	0.969874
$832$	0	0
$833$	−2150.18	−0.0894350
$834$	0	0
$835$	−2094.19	−0.0867935
$836$	0	0
$837$	6847.18	0.282764
$838$	0	0
$839$	−19316.8	−0.794862	−0.397431	−	0.917632i	$-0.630098\pi$
$839$	−19316.8	−0.794862	−0.397431	+	0.917632i	$0.630098\pi$
$840$	0	0
$841$	−9103.64	−0.373268
$842$	0	0
$843$	−9399.39	−0.384024
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	11494.6	0.466303
$848$	0	0
$849$	11537.9	0.466409
$850$	0	0
$851$	−829.409	−0.0334099
$852$	0	0
$853$	16154.4	0.648437	0.324219	−	0.945982i	$-0.394899\pi$
$853$	16154.4	0.648437	0.324219	+	0.945982i	$0.394899\pi$
$854$	0	0
$855$	3238.44	0.129535
$856$	0	0
$857$	−17127.3	−0.682682	−0.341341	−	0.939940i	$-0.610881\pi$
$857$	−17127.3	−0.682682	−0.341341	+	0.939940i	$0.610881\pi$
$858$	0	0
$859$	−34893.5	−1.38597	−0.692987	−	0.720950i	$-0.743706\pi$
$859$	−34893.5	−1.38597	−0.692987	+	0.720950i	$0.743706\pi$
$860$	0	0
$861$	−3586.91	−0.141976
$862$	0	0
$863$	−23437.4	−0.924472	−0.462236	−	0.886757i	$-0.652953\pi$
$863$	−23437.4	−0.924472	−0.462236	+	0.886757i	$0.652953\pi$
$864$	0	0
$865$	−2424.26	−0.0952917
$866$	0	0
$867$	−14531.7	−0.569229
$868$	0	0
$869$	−4763.85	−0.185964
$870$	0	0
$871$	7700.11	0.299550
$872$	0	0
$873$	13595.1	0.527061
$874$	0	0
$875$	1148.05	0.0443557
$876$	0	0
$877$	718.222	0.0276541	0.0138270	−	0.999904i	$-0.495599\pi$
$877$	718.222	0.0276541	0.0138270	+	0.999904i	$0.495599\pi$
$878$	0	0
$879$	−4550.15	−0.174599
$880$	0	0
$881$	−5341.66	−0.204273	−0.102137	−	0.994770i	$-0.532568\pi$
$881$	−5341.66	−0.204273	−0.102137	+	0.994770i	$0.532568\pi$
$882$	0	0
$883$	28905.7	1.10165	0.550823	−	0.834622i	$-0.314314\pi$
$883$	28905.7	1.10165	0.550823	+	0.834622i	$0.314314\pi$
$884$	0	0
$885$	−5072.30	−0.192659
$886$	0	0
$887$	−37616.8	−1.42395	−0.711977	−	0.702203i	$-0.752200\pi$
$887$	−37616.8	−1.42395	−0.711977	+	0.702203i	$0.752200\pi$
$888$	0	0
$889$	−10958.9	−0.413440
$890$	0	0
$891$	722.070	0.0271495
$892$	0	0
$893$	3463.26	0.129780
$894$	0	0
$895$	11569.4	0.432092
$896$	0	0
$897$	558.215	0.0207784
$898$	0	0
$899$	−31353.5	−1.16318
$900$	0	0
$901$	2720.08	0.100576
$902$	0	0
$903$	2038.67	0.0751305
$904$	0	0
$905$	11109.9	0.408071
$906$	0	0
$907$	21653.0	0.792696	0.396348	−	0.918100i	$-0.370277\pi$
$907$	21653.0	0.792696	0.396348	+	0.918100i	$0.370277\pi$
$908$	0	0
$909$	15433.5	0.563144
$910$	0	0
$911$	−9265.67	−0.336976	−0.168488	−	0.985704i	$-0.553888\pi$
$911$	−9265.67	−0.336976	−0.168488	+	0.985704i	$0.553888\pi$
$912$	0	0
$913$	1132.50	0.0410518
$914$	0	0
$915$	−8378.32	−0.302709
$916$	0	0
$917$	−6659.56	−0.239824
$918$	0	0
$919$	20343.6	0.730221	0.365111	−	0.930964i	$-0.381031\pi$
$919$	20343.6	0.730221	0.365111	+	0.930964i	$0.381031\pi$
$920$	0	0
$921$	5866.81	0.209900
$922$	0	0
$923$	−9160.39	−0.326671
$924$	0	0
$925$	−1448.68	−0.0514944
$926$	0	0
$927$	14858.8	0.526459
$928$	0	0
$929$	1904.21	0.0672500	0.0336250	−	0.999435i	$-0.489295\pi$
$929$	1904.21	0.0672500	0.0336250	+	0.999435i	$0.489295\pi$
$930$	0	0
$931$	18613.6	0.655247
$932$	0	0
$933$	18523.9	0.649994
$934$	0	0
$935$	−370.538	−0.0129603
$936$	0	0
$937$	−31790.0	−1.10836	−0.554180	−	0.832397i	$-0.686968\pi$
$937$	−31790.0	−1.10836	−0.554180	+	0.832397i	$0.686968\pi$
$938$	0	0
$939$	−24922.5	−0.866150
$940$	0	0
$941$	−10049.0	−0.348128	−0.174064	−	0.984734i	$-0.555690\pi$
$941$	−10049.0	−0.348128	−0.174064	+	0.984734i	$0.555690\pi$
$942$	0	0
$943$	1863.31	0.0643453
$944$	0	0
$945$	1239.89	0.0426812
$946$	0	0
$947$	−30543.4	−1.04808	−0.524038	−	0.851695i	$-0.675575\pi$
$947$	−30543.4	−1.04808	−0.524038	+	0.851695i	$0.675575\pi$
$948$	0	0
$949$	−12325.3	−0.421597
$950$	0	0
$951$	−889.739	−0.0303384
$952$	0	0
$953$	28020.9	0.952452	0.476226	−	0.879323i	$-0.342004\pi$
$953$	28020.9	0.952452	0.476226	+	0.879323i	$0.342004\pi$
$954$	0	0
$955$	−8286.98	−0.280796
$956$	0	0
$957$	−3306.38	−0.111683
$958$	0	0
$959$	−23742.7	−0.799469
$960$	0	0
$961$	34521.6	1.15879
$962$	0	0
$963$	−602.361	−0.0201566
$964$	0	0
$965$	1170.32	0.0390404
$966$	0	0
$967$	42979.6	1.42930	0.714648	−	0.699484i	$-0.246587\pi$
$967$	42979.6	1.42930	0.714648	+	0.699484i	$0.246587\pi$
$968$	0	0
$969$	−1794.78	−0.0595013
$970$	0	0
$971$	47718.1	1.57708	0.788541	−	0.614982i	$-0.210837\pi$
$971$	47718.1	1.57708	0.788541	+	0.614982i	$0.210837\pi$
$972$	0	0
$973$	−16759.1	−0.552180
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	27127.5	0.888317	0.444159	−	0.895948i	$-0.353503\pi$
$977$	27127.5	0.888317	0.444159	+	0.895948i	$0.353503\pi$
$978$	0	0
$979$	13033.5	0.425487
$980$	0	0
$981$	7411.05	0.241200
$982$	0	0
$983$	12265.7	0.397982	0.198991	−	0.980001i	$-0.436233\pi$
$983$	12265.7	0.397982	0.198991	+	0.980001i	$0.436233\pi$
$984$	0	0
$985$	−6018.59	−0.194689
$986$	0	0
$987$	1325.97	0.0427620
$988$	0	0
$989$	−1059.04	−0.0340500
$990$	0	0
$991$	16915.5	0.542218	0.271109	−	0.962549i	$-0.412610\pi$
$991$	16915.5	0.542218	0.271109	+	0.962549i	$0.412610\pi$
$992$	0	0
$993$	20533.4	0.656202
$994$	0	0
$995$	−2793.88	−0.0890170
$996$	0	0
$997$	−38248.8	−1.21500	−0.607498	−	0.794321i	$-0.707827\pi$
$997$	−38248.8	−1.21500	−0.607498	+	0.794321i	$0.707827\pi$
$998$	0	0
$999$	−1564.57	−0.0495505

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.m.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.00000 q^{5} -9.18441 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -5.00000 q^{5} -9.18441 q^{7} +9.00000 q^{9} +8.91444 q^{11} +13.0000 q^{13} -15.0000 q^{15} +8.31320 q^{17} -71.9653 q^{19} -27.5532 q^{21} +14.3132 q^{23} +25.0000 q^{25} +27.0000 q^{27} -123.634 q^{29} +253.599 q^{31} +26.7433 q^{33} +45.9220 q^{35} -57.9472 q^{37} +39.0000 q^{39} +130.181 q^{41} -73.9904 q^{43} -45.0000 q^{45} -48.1240 q^{47} -258.647 q^{49} +24.9396 q^{51} +327.200 q^{53} -44.5722 q^{55} -215.896 q^{57} +338.153 q^{59} +558.554 q^{61} -82.6597 q^{63} -65.0000 q^{65} +592.316 q^{67} +42.9396 q^{69} -704.645 q^{71} -948.099 q^{73} +75.0000 q^{75} -81.8738 q^{77} -534.397 q^{79} +81.0000 q^{81} +127.041 q^{83} -41.5660 q^{85} -370.902 q^{87} +1462.06 q^{89} -119.397 q^{91} +760.798 q^{93} +359.826 q^{95} +1510.57 q^{97} +80.2300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9 - 7 * q^11 + 52 * q^13 - 60 * q^15 - 73 * q^17 + 68 * q^19 + 45 * q^21 - 49 * q^23 + 100 * q^25 + 108 * q^27 - 66 * q^29 + 230 * q^31 - 21 * q^33 - 75 * q^35 + 303 * q^37 + 156 * q^39 + 155 * q^41 + 336 * q^43 - 180 * q^45 + 178 * q^47 + 377 * q^49 - 219 * q^51 + 349 * q^53 + 35 * q^55 + 204 * q^57 - 360 * q^59 + 223 * q^61 + 135 * q^63 - 260 * q^65 + 588 * q^67 - 147 * q^69 - 83 * q^71 + 754 * q^73 + 300 * q^75 + 1401 * q^77 + 869 * q^79 + 324 * q^81 - 1044 * q^83 + 365 * q^85 - 198 * q^87 + 1017 * q^89 + 195 * q^91 + 690 * q^93 - 340 * q^95 + 1367 * q^97 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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