LMFDB - Embedded newform 1305.4.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 $ x^7 - x^6 - 35x^5 + 18x^4 + 329x^3 - 167x^2 - 767*x + 638
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.12367$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.12367 q^{2} +18.2520 q^{4} +5.00000 q^{5} -21.7657 q^{7} +52.5281 q^{8} +25.6184 q^{10} -57.1430 q^{11} -41.2271 q^{13} -111.520 q^{14} +123.120 q^{16} -73.0149 q^{17} -0.658616 q^{19} +91.2602 q^{20} -292.782 q^{22} +96.1672 q^{23} +25.0000 q^{25} -211.234 q^{26} -397.267 q^{28} +29.0000 q^{29} -2.01051 q^{31} +210.604 q^{32} -374.104 q^{34} -108.828 q^{35} -315.774 q^{37} -3.37453 q^{38} +262.640 q^{40} -219.873 q^{41} +81.2960 q^{43} -1042.98 q^{44} +492.730 q^{46} -440.820 q^{47} +130.744 q^{49} +128.092 q^{50} -752.479 q^{52} -65.8584 q^{53} -285.715 q^{55} -1143.31 q^{56} +148.587 q^{58} +551.520 q^{59} -149.536 q^{61} -10.3012 q^{62} +94.1041 q^{64} -206.136 q^{65} -888.233 q^{67} -1332.67 q^{68} -557.601 q^{70} +570.702 q^{71} +664.264 q^{73} -1617.92 q^{74} -12.0211 q^{76} +1243.75 q^{77} +221.956 q^{79} +615.602 q^{80} -1126.56 q^{82} +740.432 q^{83} -365.074 q^{85} +416.534 q^{86} -3001.61 q^{88} +895.415 q^{89} +897.336 q^{91} +1755.25 q^{92} -2258.62 q^{94} -3.29308 q^{95} -705.666 q^{97} +669.888 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

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 $

$2$	5.12367	1.81149	0.905746	−	0.423821i	$-0.139311\pi$
$2$	5.12367	1.81149	0.905746	+	0.423821i	$0.139311\pi$
$3$	0	0
$3$	0	0
$4$	18.2520	2.28150
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−21.7657	−1.17523	−0.587617	−	0.809139i	$-0.699934\pi$
$7$	−21.7657	−1.17523	−0.587617	+	0.809139i	$0.699934\pi$
$8$	52.5281	2.32143
$9$	0	0
$10$	25.6184	0.810124
$11$	−57.1430	−1.56630	−0.783148	−	0.621835i	$-0.786387\pi$
$11$	−57.1430	−1.56630	−0.783148	+	0.621835i	$0.786387\pi$
$12$	0	0
$13$	−41.2271	−0.879565	−0.439783	−	0.898104i	$-0.644945\pi$
$13$	−41.2271	−0.879565	−0.439783	+	0.898104i	$0.644945\pi$
$14$	−111.520	−2.12893
$15$	0	0
$16$	123.120	1.92376
$17$	−73.0149	−1.04169	−0.520844	−	0.853652i	$-0.674383\pi$
$17$	−73.0149	−1.04169	−0.520844	+	0.853652i	$0.674383\pi$
$18$	0	0
$19$	−0.658616	−0.00795246	−0.00397623	−	0.999992i	$-0.501266\pi$
$19$	−0.658616	−0.00795246	−0.00397623	+	0.999992i	$0.501266\pi$
$20$	91.2602	1.02032
$21$	0	0
$22$	−292.782	−2.83733
$23$	96.1672	0.871837	0.435919	−	0.899986i	$-0.356424\pi$
$23$	96.1672	0.871837	0.435919	+	0.899986i	$0.356424\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−211.234	−1.59333
$27$	0	0
$28$	−397.267	−2.68130
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−2.01051	−0.0116483	−0.00582415	−	0.999983i	$-0.501854\pi$
$31$	−2.01051	−0.0116483	−0.00582415	+	0.999983i	$0.501854\pi$
$32$	210.604	1.16343
$33$	0	0
$34$	−374.104	−1.88701
$35$	−108.828	−0.525581
$36$	0	0
$37$	−315.774	−1.40305	−0.701526	−	0.712644i	$-0.747498\pi$
$37$	−315.774	−1.40305	−0.701526	+	0.712644i	$0.747498\pi$
$38$	−3.37453	−0.0144058
$39$	0	0
$40$	262.640	1.03818
$41$	−219.873	−0.837522	−0.418761	−	0.908097i	$-0.637535\pi$
$41$	−219.873	−0.837522	−0.418761	+	0.908097i	$0.637535\pi$
$42$	0	0
$43$	81.2960	0.288315	0.144157	−	0.989555i	$-0.453953\pi$
$43$	81.2960	0.288315	0.144157	+	0.989555i	$0.453953\pi$
$44$	−1042.98	−3.57351
$45$	0	0
$46$	492.730	1.57933
$47$	−440.820	−1.36809	−0.684044	−	0.729441i	$-0.739780\pi$
$47$	−440.820	−1.36809	−0.684044	+	0.729441i	$0.739780\pi$
$48$	0	0
$49$	130.744	0.381177
$50$	128.092	0.362298
$51$	0	0
$52$	−752.479	−2.00673
$53$	−65.8584	−0.170686	−0.0853429	−	0.996352i	$-0.527199\pi$
$53$	−65.8584	−0.170686	−0.0853429	+	0.996352i	$0.527199\pi$
$54$	0	0
$55$	−285.715	−0.700469
$56$	−1143.31	−2.72823
$57$	0	0
$58$	148.587	0.336386
$59$	551.520	1.21698	0.608489	−	0.793562i	$-0.291776\pi$
$59$	551.520	1.21698	0.608489	+	0.793562i	$0.291776\pi$
$60$	0	0
$61$	−149.536	−0.313871	−0.156936	−	0.987609i	$-0.550161\pi$
$61$	−149.536	−0.313871	−0.156936	+	0.987609i	$0.550161\pi$
$62$	−10.3012	−0.0211008
$63$	0	0
$64$	94.1041	0.183797
$65$	−206.136	−0.393354
$66$	0	0
$67$	−888.233	−1.61963	−0.809813	−	0.586689i	$-0.800431\pi$
$67$	−888.233	−1.61963	−0.809813	+	0.586689i	$0.800431\pi$
$68$	−1332.67	−2.37662
$69$	0	0
$70$	−557.601	−0.952086
$71$	570.702	0.953942	0.476971	−	0.878919i	$-0.341735\pi$
$71$	570.702	0.953942	0.476971	+	0.878919i	$0.341735\pi$
$72$	0	0
$73$	664.264	1.06502	0.532509	−	0.846425i	$-0.321249\pi$
$73$	664.264	1.06502	0.532509	+	0.846425i	$0.321249\pi$
$74$	−1617.92	−2.54162
$75$	0	0
$76$	−12.0211	−0.0181436
$77$	1243.75	1.84077
$78$	0	0
$79$	221.956	0.316102	0.158051	−	0.987431i	$-0.449479\pi$
$79$	221.956	0.316102	0.158051	+	0.987431i	$0.449479\pi$
$80$	615.602	0.860330
$81$	0	0
$82$	−1126.56	−1.51716
$83$	740.432	0.979192	0.489596	−	0.871949i	$-0.337144\pi$
$83$	740.432	0.979192	0.489596	+	0.871949i	$0.337144\pi$
$84$	0	0
$85$	−365.074	−0.465857
$86$	416.534	0.522280
$87$	0	0
$88$	−3001.61	−3.63605
$89$	895.415	1.06645	0.533224	−	0.845974i	$-0.320980\pi$
$89$	895.415	1.06645	0.533224	+	0.845974i	$0.320980\pi$
$90$	0	0
$91$	897.336	1.03370
$92$	1755.25	1.98910
$93$	0	0
$94$	−2258.62	−2.47828
$95$	−3.29308	−0.00355645
$96$	0	0
$97$	−705.666	−0.738655	−0.369327	−	0.929299i	$-0.620412\pi$
$97$	−705.666	−0.738655	−0.369327	+	0.929299i	$0.620412\pi$
$98$	669.888	0.690499
$99$	0	0
$100$	456.301	0.456301
$101$	1290.41	1.27129	0.635647	−	0.771980i	$-0.280734\pi$
$101$	1290.41	1.27129	0.635647	+	0.771980i	$0.280734\pi$
$102$	0	0
$103$	897.912	0.858970	0.429485	−	0.903074i	$-0.358695\pi$
$103$	897.912	0.858970	0.429485	+	0.903074i	$0.358695\pi$
$104$	−2165.58	−2.04185
$105$	0	0
$106$	−337.437	−0.309196
$107$	2080.08	1.87934	0.939669	−	0.342086i	$-0.111133\pi$
$107$	2080.08	1.87934	0.939669	+	0.342086i	$0.111133\pi$
$108$	0	0
$109$	−1386.38	−1.21827	−0.609133	−	0.793068i	$-0.708482\pi$
$109$	−1386.38	−1.21827	−0.609133	+	0.793068i	$0.708482\pi$
$110$	−1463.91	−1.26889
$111$	0	0
$112$	−2679.80	−2.26087
$113$	−477.103	−0.397187	−0.198593	−	0.980082i	$-0.563637\pi$
$113$	−477.103	−0.397187	−0.198593	+	0.980082i	$0.563637\pi$
$114$	0	0
$115$	480.836	0.389897
$116$	529.309	0.423665
$117$	0	0
$118$	2825.81	2.20455
$119$	1589.22	1.22423
$120$	0	0
$121$	1934.32	1.45328
$122$	−766.174	−0.568575
$123$	0	0
$124$	−36.6958	−0.0265757
$125$	125.000	0.0894427
$126$	0	0
$127$	−1357.50	−0.948495	−0.474247	−	0.880392i	$-0.657280\pi$
$127$	−1357.50	−0.948495	−0.474247	+	0.880392i	$0.657280\pi$
$128$	−1202.67	−0.830488
$129$	0	0
$130$	−1056.17	−0.712557
$131$	−2994.84	−1.99741	−0.998704	−	0.0508962i	$-0.983792\pi$
$131$	−2994.84	−1.99741	−0.998704	+	0.0508962i	$0.983792\pi$
$132$	0	0
$133$	14.3352	0.00934601
$134$	−4551.01	−2.93394
$135$	0	0
$136$	−3835.33	−2.41821
$137$	1067.77	0.665878	0.332939	−	0.942948i	$-0.391960\pi$
$137$	1067.77	0.665878	0.332939	+	0.942948i	$0.391960\pi$
$138$	0	0
$139$	−843.314	−0.514597	−0.257298	−	0.966332i	$-0.582832\pi$
$139$	−843.314	−0.514597	−0.257298	+	0.966332i	$0.582832\pi$
$140$	−1986.34	−1.19912
$141$	0	0
$142$	2924.09	1.72806
$143$	2355.84	1.37766
$144$	0	0
$145$	145.000	0.0830455
$146$	3403.47	1.92927
$147$	0	0
$148$	−5763.52	−3.20107
$149$	−1273.71	−0.700313	−0.350157	−	0.936691i	$-0.613872\pi$
$149$	−1273.71	−0.700313	−0.350157	+	0.936691i	$0.613872\pi$
$150$	0	0
$151$	−3030.58	−1.63328	−0.816639	−	0.577149i	$-0.804165\pi$
$151$	−3030.58	−1.63328	−0.816639	+	0.577149i	$0.804165\pi$
$152$	−34.5958	−0.0184611
$153$	0	0
$154$	6372.59	3.33453
$155$	−10.0525	−0.00520928
$156$	0	0
$157$	−1513.57	−0.769399	−0.384700	−	0.923042i	$-0.625695\pi$
$157$	−1513.57	−0.769399	−0.384700	+	0.923042i	$0.625695\pi$
$158$	1137.23	0.572616
$159$	0	0
$160$	1053.02	0.520304
$161$	−2093.14	−1.02461
$162$	0	0
$163$	−3134.88	−1.50640	−0.753200	−	0.657792i	$-0.771491\pi$
$163$	−3134.88	−1.50640	−0.753200	+	0.657792i	$0.771491\pi$
$164$	−4013.13	−1.91081
$165$	0	0
$166$	3793.73	1.77380
$167$	−703.416	−0.325940	−0.162970	−	0.986631i	$-0.552107\pi$
$167$	−703.416	−0.325940	−0.162970	+	0.986631i	$0.552107\pi$
$168$	0	0
$169$	−497.323	−0.226365
$170$	−1870.52	−0.843897
$171$	0	0
$172$	1483.82	0.657791
$173$	1699.75	0.746993	0.373497	−	0.927632i	$-0.378159\pi$
$173$	1699.75	0.746993	0.373497	+	0.927632i	$0.378159\pi$
$174$	0	0
$175$	−544.141	−0.235047
$176$	−7035.47	−3.01317
$177$	0	0
$178$	4587.81	1.93186
$179$	1616.87	0.675143	0.337572	−	0.941300i	$-0.390394\pi$
$179$	1616.87	0.675143	0.337572	+	0.941300i	$0.390394\pi$
$180$	0	0
$181$	790.412	0.324590	0.162295	−	0.986742i	$-0.448110\pi$
$181$	790.412	0.324590	0.162295	+	0.986742i	$0.448110\pi$
$182$	4597.65	1.87253
$183$	0	0
$184$	5051.48	2.02391
$185$	−1578.87	−0.627464
$186$	0	0
$187$	4172.29	1.63159
$188$	−8045.85	−3.12130
$189$	0	0
$190$	−16.8727	−0.00644248
$191$	2966.34	1.12375	0.561877	−	0.827221i	$-0.310079\pi$
$191$	2966.34	1.12375	0.561877	+	0.827221i	$0.310079\pi$
$192$	0	0
$193$	−1871.55	−0.698018	−0.349009	−	0.937119i	$-0.613482\pi$
$193$	−1871.55	−0.698018	−0.349009	+	0.937119i	$0.613482\pi$
$194$	−3615.60	−1.33807
$195$	0	0
$196$	2386.34	0.869656
$197$	1765.75	0.638603	0.319301	−	0.947653i	$-0.396552\pi$
$197$	1765.75	0.638603	0.319301	+	0.947653i	$0.396552\pi$
$198$	0	0
$199$	−153.070	−0.0545268	−0.0272634	−	0.999628i	$-0.508679\pi$
$199$	−153.070	−0.0545268	−0.0272634	+	0.999628i	$0.508679\pi$
$200$	1313.20	0.464287
$201$	0	0
$202$	6611.64	2.30294
$203$	−631.204	−0.218236
$204$	0	0
$205$	−1099.37	−0.374551
$206$	4600.61	1.55602
$207$	0	0
$208$	−5075.90	−1.69207
$209$	37.6353	0.0124559
$210$	0	0
$211$	5632.85	1.83783	0.918913	−	0.394460i	$-0.129068\pi$
$211$	5632.85	1.83783	0.918913	+	0.394460i	$0.129068\pi$
$212$	−1202.05	−0.389420
$213$	0	0
$214$	10657.7	3.40440
$215$	406.480	0.128938
$216$	0	0
$217$	43.7600	0.0136895
$218$	−7103.35	−2.20688
$219$	0	0
$220$	−5214.88	−1.59812
$221$	3010.19	0.916233
$222$	0	0
$223$	−2962.89	−0.889730	−0.444865	−	0.895598i	$-0.646748\pi$
$223$	−2962.89	−0.889730	−0.444865	+	0.895598i	$0.646748\pi$
$224$	−4583.94	−1.36731
$225$	0	0
$226$	−2444.52	−0.719501
$227$	1006.81	0.294381	0.147190	−	0.989108i	$-0.452977\pi$
$227$	1006.81	0.294381	0.147190	+	0.989108i	$0.452977\pi$
$228$	0	0
$229$	−1297.30	−0.374359	−0.187179	−	0.982326i	$-0.559935\pi$
$229$	−1297.30	−0.374359	−0.187179	+	0.982326i	$0.559935\pi$
$230$	2463.65	0.706296
$231$	0	0
$232$	1523.31	0.431080
$233$	1649.87	0.463891	0.231946	−	0.972729i	$-0.425491\pi$
$233$	1649.87	0.463891	0.231946	+	0.972729i	$0.425491\pi$
$234$	0	0
$235$	−2204.10	−0.611828
$236$	10066.4	2.77654
$237$	0	0
$238$	8142.63	2.21768
$239$	−5704.40	−1.54388	−0.771939	−	0.635696i	$-0.780713\pi$
$239$	−5704.40	−1.54388	−0.771939	+	0.635696i	$0.780713\pi$
$240$	0	0
$241$	7233.99	1.93354	0.966768	−	0.255657i	$-0.0822916\pi$
$241$	7233.99	1.93354	0.966768	+	0.255657i	$0.0822916\pi$
$242$	9910.83	2.63261
$243$	0	0
$244$	−2729.34	−0.716098
$245$	653.718	0.170467
$246$	0	0
$247$	27.1528	0.00699471
$248$	−105.608	−0.0270408
$249$	0	0
$250$	640.459	0.162025
$251$	−5887.94	−1.48065	−0.740326	−	0.672248i	$-0.765329\pi$
$251$	−5887.94	−1.48065	−0.740326	+	0.672248i	$0.765329\pi$
$252$	0	0
$253$	−5495.28	−1.36556
$254$	−6955.40	−1.71819
$255$	0	0
$256$	−6914.95	−1.68822
$257$	−5751.33	−1.39595	−0.697973	−	0.716124i	$-0.745914\pi$
$257$	−5751.33	−1.39595	−0.697973	+	0.716124i	$0.745914\pi$
$258$	0	0
$259$	6873.03	1.64892
$260$	−3762.39	−0.897438
$261$	0	0
$262$	−15344.6	−3.61829
$263$	−5196.53	−1.21837	−0.609186	−	0.793027i	$-0.708504\pi$
$263$	−5196.53	−1.21837	−0.609186	+	0.793027i	$0.708504\pi$
$264$	0	0
$265$	−329.292	−0.0763330
$266$	73.4489	0.0169302
$267$	0	0
$268$	−16212.1	−3.69518
$269$	7201.66	1.63232	0.816158	−	0.577829i	$-0.196100\pi$
$269$	7201.66	1.63232	0.816158	+	0.577829i	$0.196100\pi$
$270$	0	0
$271$	−1510.97	−0.338690	−0.169345	−	0.985557i	$-0.554165\pi$
$271$	−1510.97	−0.338690	−0.169345	+	0.985557i	$0.554165\pi$
$272$	−8989.62	−2.00396
$273$	0	0
$274$	5470.88	1.20623
$275$	−1428.57	−0.313259
$276$	0	0
$277$	6906.59	1.49811	0.749056	−	0.662507i	$-0.230507\pi$
$277$	6906.59	1.49811	0.749056	+	0.662507i	$0.230507\pi$
$278$	−4320.87	−0.932188
$279$	0	0
$280$	−5716.54	−1.22010
$281$	5553.49	1.17898	0.589490	−	0.807776i	$-0.299329\pi$
$281$	5553.49	1.17898	0.589490	+	0.807776i	$0.299329\pi$
$282$	0	0
$283$	−906.811	−0.190475	−0.0952373	−	0.995455i	$-0.530361\pi$
$283$	−906.811	−0.190475	−0.0952373	+	0.995455i	$0.530361\pi$
$284$	10416.5	2.17642
$285$	0	0
$286$	12070.6	2.49562
$287$	4785.68	0.984285
$288$	0	0
$289$	418.172	0.0851154
$290$	742.933	0.150436
$291$	0	0
$292$	12124.2	2.42984
$293$	−478.454	−0.0953979	−0.0476989	−	0.998862i	$-0.515189\pi$
$293$	−478.454	−0.0953979	−0.0476989	+	0.998862i	$0.515189\pi$
$294$	0	0
$295$	2757.60	0.544249
$296$	−16587.0	−3.25709
$297$	0	0
$298$	−6526.09	−1.26861
$299$	−3964.70	−0.766838
$300$	0	0
$301$	−1769.46	−0.338837
$302$	−15527.7	−2.95867
$303$	0	0
$304$	−81.0890	−0.0152986
$305$	−747.680	−0.140367
$306$	0	0
$307$	−8071.13	−1.50047	−0.750235	−	0.661172i	$-0.770059\pi$
$307$	−8071.13	−1.50047	−0.750235	+	0.661172i	$0.770059\pi$
$308$	22701.0	4.19971
$309$	0	0
$310$	−51.5059	−0.00943657
$311$	2191.99	0.399667	0.199834	−	0.979830i	$-0.435960\pi$
$311$	2191.99	0.399667	0.199834	+	0.979830i	$0.435960\pi$
$312$	0	0
$313$	10935.0	1.97471	0.987353	−	0.158535i	$-0.0506771\pi$
$313$	10935.0	1.97471	0.987353	+	0.158535i	$0.0506771\pi$
$314$	−7755.02	−1.39376
$315$	0	0
$316$	4051.15	0.721187
$317$	−8640.32	−1.53088	−0.765440	−	0.643508i	$-0.777478\pi$
$317$	−8640.32	−1.53088	−0.765440	+	0.643508i	$0.777478\pi$
$318$	0	0
$319$	−1657.15	−0.290854
$320$	470.520	0.0821965
$321$	0	0
$322$	−10724.6	−1.85608
$323$	48.0887	0.00828399
$324$	0	0
$325$	−1030.68	−0.175913
$326$	−16062.1	−2.72883
$327$	0	0
$328$	−11549.5	−1.94425
$329$	9594.72	1.60782
$330$	0	0
$331$	−10031.0	−1.66571	−0.832857	−	0.553488i	$-0.813297\pi$
$331$	−10031.0	−1.66571	−0.832857	+	0.553488i	$0.813297\pi$
$332$	13514.4	2.23403
$333$	0	0
$334$	−3604.07	−0.590437
$335$	−4441.16	−0.724318
$336$	0	0
$337$	2538.16	0.410274	0.205137	−	0.978733i	$-0.434236\pi$
$337$	2538.16	0.410274	0.205137	+	0.978733i	$0.434236\pi$
$338$	−2548.12	−0.410058
$339$	0	0
$340$	−6663.35	−1.06286
$341$	114.886	0.0182447
$342$	0	0
$343$	4619.90	0.727263
$344$	4270.32	0.669304
$345$	0	0
$346$	8708.98	1.35317
$347$	−8139.07	−1.25916	−0.629579	−	0.776936i	$-0.716773\pi$
$347$	−8139.07	−1.25916	−0.629579	+	0.776936i	$0.716773\pi$
$348$	0	0
$349$	3054.95	0.468561	0.234280	−	0.972169i	$-0.424727\pi$
$349$	3054.95	0.468561	0.234280	+	0.972169i	$0.424727\pi$
$350$	−2788.00	−0.425786
$351$	0	0
$352$	−12034.6	−1.82228
$353$	4993.98	0.752982	0.376491	−	0.926420i	$-0.377131\pi$
$353$	4993.98	0.752982	0.376491	+	0.926420i	$0.377131\pi$
$354$	0	0
$355$	2853.51	0.426616
$356$	16343.1	2.43310
$357$	0	0
$358$	8284.32	1.22302
$359$	3701.99	0.544245	0.272122	−	0.962263i	$-0.412274\pi$
$359$	3701.99	0.544245	0.272122	+	0.962263i	$0.412274\pi$
$360$	0	0
$361$	−6858.57	−0.999937
$362$	4049.81	0.587993
$363$	0	0
$364$	16378.2	2.35838
$365$	3321.32	0.476290
$366$	0	0
$367$	−1936.81	−0.275478	−0.137739	−	0.990469i	$-0.543984\pi$
$367$	−1936.81	−0.275478	−0.137739	+	0.990469i	$0.543984\pi$
$368$	11840.1	1.67720
$369$	0	0
$370$	−8089.61	−1.13665
$371$	1433.45	0.200596
$372$	0	0
$373$	12540.1	1.74075	0.870376	−	0.492387i	$-0.163876\pi$
$373$	12540.1	1.74075	0.870376	+	0.492387i	$0.163876\pi$
$374$	21377.4	2.95562
$375$	0	0
$376$	−23155.4	−3.17593
$377$	−1195.59	−0.163331
$378$	0	0
$379$	7644.03	1.03601	0.518005	−	0.855378i	$-0.326675\pi$
$379$	7644.03	1.03601	0.518005	+	0.855378i	$0.326675\pi$
$380$	−60.1054	−0.00811405
$381$	0	0
$382$	15198.6	2.03567
$383$	−13834.6	−1.84574	−0.922868	−	0.385115i	$-0.874162\pi$
$383$	−13834.6	−1.84574	−0.922868	+	0.385115i	$0.874162\pi$
$384$	0	0
$385$	6218.77	0.823215
$386$	−9589.24	−1.26445
$387$	0	0
$388$	−12879.8	−1.68524
$389$	14818.0	1.93137	0.965684	−	0.259719i	$-0.0836298\pi$
$389$	14818.0	1.93137	0.965684	+	0.259719i	$0.0836298\pi$
$390$	0	0
$391$	−7021.64	−0.908183
$392$	6867.71	0.884877
$393$	0	0
$394$	9047.14	1.15682
$395$	1109.78	0.141365
$396$	0	0
$397$	−2846.02	−0.359792	−0.179896	−	0.983686i	$-0.557576\pi$
$397$	−2846.02	−0.359792	−0.179896	+	0.983686i	$0.557576\pi$
$398$	−784.281	−0.0987750
$399$	0	0
$400$	3078.01	0.384751
$401$	−7701.88	−0.959136	−0.479568	−	0.877505i	$-0.659207\pi$
$401$	−7701.88	−0.959136	−0.479568	+	0.877505i	$0.659207\pi$
$402$	0	0
$403$	82.8874	0.0102454
$404$	23552.6	2.90046
$405$	0	0
$406$	−3234.08	−0.395332
$407$	18044.3	2.19759
$408$	0	0
$409$	−10969.5	−1.32618	−0.663088	−	0.748541i	$-0.730755\pi$
$409$	−10969.5	−1.32618	−0.663088	+	0.748541i	$0.730755\pi$
$410$	−5632.79	−0.678497
$411$	0	0
$412$	16388.7	1.95974
$413$	−12004.2	−1.43024
$414$	0	0
$415$	3702.16	0.437908
$416$	−8682.61	−1.02332
$417$	0	0
$418$	192.831	0.0225638
$419$	10760.9	1.25467	0.627335	−	0.778750i	$-0.284146\pi$
$419$	10760.9	1.25467	0.627335	+	0.778750i	$0.284146\pi$
$420$	0	0
$421$	−3562.20	−0.412378	−0.206189	−	0.978512i	$-0.566106\pi$
$421$	−3562.20	−0.412378	−0.206189	+	0.978512i	$0.566106\pi$
$422$	28860.9	3.32921
$423$	0	0
$424$	−3459.41	−0.396236
$425$	−1825.37	−0.208338
$426$	0	0
$427$	3254.75	0.368872
$428$	37965.7	4.28772
$429$	0	0
$430$	2082.67	0.233571
$431$	1086.78	0.121458	0.0607290	−	0.998154i	$-0.480657\pi$
$431$	1086.78	0.121458	0.0607290	+	0.998154i	$0.480657\pi$
$432$	0	0
$433$	4185.80	0.464565	0.232282	−	0.972648i	$-0.425381\pi$
$433$	4185.80	0.464565	0.232282	+	0.972648i	$0.425381\pi$
$434$	224.212	0.0247984
$435$	0	0
$436$	−25304.2	−2.77948
$437$	−63.3372	−0.00693325
$438$	0	0
$439$	3960.69	0.430600	0.215300	−	0.976548i	$-0.430927\pi$
$439$	3960.69	0.430600	0.215300	+	0.976548i	$0.430927\pi$
$440$	−15008.1	−1.62609
$441$	0	0
$442$	15423.3	1.65975
$443$	−1695.58	−0.181850	−0.0909250	−	0.995858i	$-0.528982\pi$
$443$	−1695.58	−0.181850	−0.0909250	+	0.995858i	$0.528982\pi$
$444$	0	0
$445$	4477.07	0.476930
$446$	−15180.9	−1.61174
$447$	0	0
$448$	−2048.24	−0.216005
$449$	−655.044	−0.0688495	−0.0344248	−	0.999407i	$-0.510960\pi$
$449$	−655.044	−0.0688495	−0.0344248	+	0.999407i	$0.510960\pi$
$450$	0	0
$451$	12564.2	1.31181
$452$	−8708.11	−0.906183
$453$	0	0
$454$	5158.58	0.533269
$455$	4486.68	0.462283
$456$	0	0
$457$	10691.4	1.09437	0.547183	−	0.837013i	$-0.315700\pi$
$457$	10691.4	1.09437	0.547183	+	0.837013i	$0.315700\pi$
$458$	−6646.96	−0.678148
$459$	0	0
$460$	8776.24	0.889553
$461$	−15258.6	−1.54157	−0.770783	−	0.637098i	$-0.780135\pi$
$461$	−15258.6	−1.54157	−0.770783	+	0.637098i	$0.780135\pi$
$462$	0	0
$463$	−9359.99	−0.939515	−0.469758	−	0.882795i	$-0.655659\pi$
$463$	−9359.99	−0.939515	−0.469758	+	0.882795i	$0.655659\pi$
$464$	3570.49	0.357233
$465$	0	0
$466$	8453.40	0.840335
$467$	−11840.9	−1.17330	−0.586651	−	0.809840i	$-0.699554\pi$
$467$	−11840.9	−1.17330	−0.586651	+	0.809840i	$0.699554\pi$
$468$	0	0
$469$	19333.0	1.90344
$470$	−11293.1	−1.10832
$471$	0	0
$472$	28970.3	2.82514
$473$	−4645.50	−0.451586
$474$	0	0
$475$	−16.4654	−0.00159049
$476$	29006.4	2.79308
$477$	0	0
$478$	−29227.5	−2.79672
$479$	−581.072	−0.0554277	−0.0277139	−	0.999616i	$-0.508823\pi$
$479$	−581.072	−0.0554277	−0.0277139	+	0.999616i	$0.508823\pi$
$480$	0	0
$481$	13018.5	1.23408
$482$	37064.6	3.50258
$483$	0	0
$484$	35305.3	3.31567
$485$	−3528.33	−0.330336
$486$	0	0
$487$	−5996.94	−0.558003	−0.279001	−	0.960291i	$-0.590003\pi$
$487$	−5996.94	−0.558003	−0.279001	+	0.960291i	$0.590003\pi$
$488$	−7854.84	−0.728631
$489$	0	0
$490$	3349.44	0.308800
$491$	−18570.0	−1.70683	−0.853414	−	0.521234i	$-0.825472\pi$
$491$	−18570.0	−1.70683	−0.853414	+	0.521234i	$0.825472\pi$
$492$	0	0
$493$	−2117.43	−0.193437
$494$	139.122	0.0126709
$495$	0	0
$496$	−247.534	−0.0224085
$497$	−12421.7	−1.12111
$498$	0	0
$499$	3462.43	0.310621	0.155310	−	0.987866i	$-0.450362\pi$
$499$	3462.43	0.310621	0.155310	+	0.987866i	$0.450362\pi$
$500$	2281.50	0.204064
$501$	0	0
$502$	−30167.9	−2.68219
$503$	2437.40	0.216060	0.108030	−	0.994148i	$-0.465546\pi$
$503$	2437.40	0.216060	0.108030	+	0.994148i	$0.465546\pi$
$504$	0	0
$505$	6452.05	0.568540
$506$	−28156.0	−2.47369
$507$	0	0
$508$	−24777.2	−2.16399
$509$	17387.0	1.51407	0.757037	−	0.653372i	$-0.226646\pi$
$509$	17387.0	1.51407	0.757037	+	0.653372i	$0.226646\pi$
$510$	0	0
$511$	−14458.1	−1.25165
$512$	−25808.5	−2.22771
$513$	0	0
$514$	−29467.9	−2.52874
$515$	4489.56	0.384143
$516$	0	0
$517$	25189.7	2.14283
$518$	35215.1	2.98700
$519$	0	0
$520$	−10827.9	−0.913145
$521$	13912.4	1.16989	0.584945	−	0.811073i	$-0.301116\pi$
$521$	13912.4	1.16989	0.584945	+	0.811073i	$0.301116\pi$
$522$	0	0
$523$	4178.69	0.349371	0.174686	−	0.984624i	$-0.444109\pi$
$523$	4178.69	0.349371	0.174686	+	0.984624i	$0.444109\pi$
$524$	−54661.9	−4.55709
$525$	0	0
$526$	−26625.3	−2.20707
$527$	146.797	0.0121339
$528$	0	0
$529$	−2918.86	−0.239900
$530$	−1687.18	−0.138277
$531$	0	0
$532$	261.647	0.0213230
$533$	9064.73	0.736655
$534$	0	0
$535$	10400.4	0.840465
$536$	−46657.1	−3.75985
$537$	0	0
$538$	36899.0	2.95693
$539$	−7471.08	−0.597036
$540$	0	0
$541$	15727.3	1.24985	0.624927	−	0.780683i	$-0.285129\pi$
$541$	15727.3	1.24985	0.624927	+	0.780683i	$0.285129\pi$
$542$	−7741.72	−0.613534
$543$	0	0
$544$	−15377.2	−1.21194
$545$	−6931.89	−0.544825
$546$	0	0
$547$	−17384.0	−1.35884	−0.679420	−	0.733750i	$-0.737769\pi$
$547$	−17384.0	−1.35884	−0.679420	+	0.733750i	$0.737769\pi$
$548$	19488.9	1.51920
$549$	0	0
$550$	−7319.55	−0.567467
$551$	−19.0999	−0.00147674
$552$	0	0
$553$	−4831.02	−0.371494
$554$	35387.1	2.71382
$555$	0	0
$556$	−15392.2	−1.17405
$557$	18853.0	1.43416	0.717080	−	0.696991i	$-0.245478\pi$
$557$	18853.0	1.43416	0.717080	+	0.696991i	$0.245478\pi$
$558$	0	0
$559$	−3351.60	−0.253592
$560$	−13399.0	−1.01109
$561$	0	0
$562$	28454.3	2.13571
$563$	−6897.37	−0.516323	−0.258161	−	0.966102i	$-0.583117\pi$
$563$	−6897.37	−0.516323	−0.258161	+	0.966102i	$0.583117\pi$
$564$	0	0
$565$	−2385.52	−0.177627
$566$	−4646.20	−0.345043
$567$	0	0
$568$	29977.9	2.21451
$569$	−4008.52	−0.295336	−0.147668	−	0.989037i	$-0.547177\pi$
$569$	−4008.52	−0.295336	−0.147668	+	0.989037i	$0.547177\pi$
$570$	0	0
$571$	−6259.59	−0.458766	−0.229383	−	0.973336i	$-0.573671\pi$
$571$	−6259.59	−0.458766	−0.229383	+	0.973336i	$0.573671\pi$
$572$	42998.9	3.14314
$573$	0	0
$574$	24520.3	1.78302
$575$	2404.18	0.174367
$576$	0	0
$577$	−12282.9	−0.886208	−0.443104	−	0.896470i	$-0.646123\pi$
$577$	−12282.9	−0.886208	−0.443104	+	0.896470i	$0.646123\pi$
$578$	2142.58	0.154186
$579$	0	0
$580$	2646.54	0.189469
$581$	−16116.0	−1.15078
$582$	0	0
$583$	3763.34	0.267344
$584$	34892.5	2.47237
$585$	0	0
$586$	−2451.44	−0.172813
$587$	3526.91	0.247992	0.123996	−	0.992283i	$-0.460429\pi$
$587$	3526.91	0.247992	0.123996	+	0.992283i	$0.460429\pi$
$588$	0	0
$589$	1.32415	9.26327e−5 0
$590$	14129.0	0.985903
$591$	0	0
$592$	−38878.2	−2.69913
$593$	14783.4	1.02375	0.511875	−	0.859060i	$-0.328951\pi$
$593$	14783.4	1.02375	0.511875	+	0.859060i	$0.328951\pi$
$594$	0	0
$595$	7946.08	0.547492
$596$	−23247.9	−1.59777
$597$	0	0
$598$	−20313.8	−1.38912
$599$	−23249.3	−1.58588	−0.792940	−	0.609300i	$-0.791451\pi$
$599$	−23249.3	−1.58588	−0.792940	+	0.609300i	$0.791451\pi$
$600$	0	0
$601$	−15134.4	−1.02720	−0.513599	−	0.858031i	$-0.671688\pi$
$601$	−15134.4	−1.02720	−0.513599	+	0.858031i	$0.671688\pi$
$602$	−9066.14	−0.613801
$603$	0	0
$604$	−55314.2	−3.72633
$605$	9671.60	0.649928
$606$	0	0
$607$	−26346.6	−1.76174	−0.880870	−	0.473358i	$-0.843042\pi$
$607$	−26346.6	−1.76174	−0.880870	+	0.473358i	$0.843042\pi$
$608$	−138.707	−0.00925217
$609$	0	0
$610$	−3830.87	−0.254274
$611$	18173.7	1.20332
$612$	0	0
$613$	3259.35	0.214754	0.107377	−	0.994218i	$-0.465755\pi$
$613$	3259.35	0.214754	0.107377	+	0.994218i	$0.465755\pi$
$614$	−41353.9	−2.71809
$615$	0	0
$616$	65332.0	4.27322
$617$	6482.74	0.422990	0.211495	−	0.977379i	$-0.432167\pi$
$617$	6482.74	0.422990	0.211495	+	0.977379i	$0.432167\pi$
$618$	0	0
$619$	−3720.63	−0.241591	−0.120796	−	0.992677i	$-0.538545\pi$
$619$	−3720.63	−0.241591	−0.120796	+	0.992677i	$0.538545\pi$
$620$	−183.479	−0.0118850
$621$	0	0
$622$	11231.1	0.723994
$623$	−19489.3	−1.25333
$624$	0	0
$625$	625.000	0.0400000
$626$	56027.4	3.57717
$627$	0	0
$628$	−27625.6	−1.75539
$629$	23056.2	1.46154
$630$	0	0
$631$	−13113.6	−0.827328	−0.413664	−	0.910430i	$-0.635751\pi$
$631$	−13113.6	−0.827328	−0.413664	+	0.910430i	$0.635751\pi$
$632$	11658.9	0.733809
$633$	0	0
$634$	−44270.2	−2.77318
$635$	−6787.51	−0.424180
$636$	0	0
$637$	−5390.19	−0.335270
$638$	−8490.68	−0.526880
$639$	0	0
$640$	−6013.37	−0.371405
$641$	−25815.2	−1.59070	−0.795352	−	0.606148i	$-0.792714\pi$
$641$	−25815.2	−1.59070	−0.795352	+	0.606148i	$0.792714\pi$
$642$	0	0
$643$	9425.88	0.578103	0.289052	−	0.957313i	$-0.406660\pi$
$643$	9425.88	0.578103	0.289052	+	0.957313i	$0.406660\pi$
$644$	−38204.1	−2.33766
$645$	0	0
$646$	246.391	0.0150064
$647$	−2734.65	−0.166167	−0.0830836	−	0.996543i	$-0.526477\pi$
$647$	−2734.65	−0.166167	−0.0830836	+	0.996543i	$0.526477\pi$
$648$	0	0
$649$	−31515.5	−1.90615
$650$	−5280.86	−0.318665
$651$	0	0
$652$	−57218.0	−3.43686
$653$	−602.998	−0.0361365	−0.0180683	−	0.999837i	$-0.505752\pi$
$653$	−602.998	−0.0361365	−0.0180683	+	0.999837i	$0.505752\pi$
$654$	0	0
$655$	−14974.2	−0.893268
$656$	−27070.9	−1.61119
$657$	0	0
$658$	49160.2	2.91256
$659$	23932.8	1.41470	0.707352	−	0.706862i	$-0.249890\pi$
$659$	23932.8	1.41470	0.707352	+	0.706862i	$0.249890\pi$
$660$	0	0
$661$	−12965.0	−0.762905	−0.381452	−	0.924388i	$-0.624576\pi$
$661$	−12965.0	−0.762905	−0.381452	+	0.924388i	$0.624576\pi$
$662$	−51395.4	−3.01743
$663$	0	0
$664$	38893.4	2.27313
$665$	71.6760	0.00417966
$666$	0	0
$667$	2788.85	0.161896
$668$	−12838.8	−0.743633
$669$	0	0
$670$	−22755.1	−1.31210
$671$	8544.94	0.491615
$672$	0	0
$673$	−4523.17	−0.259072	−0.129536	−	0.991575i	$-0.541349\pi$
$673$	−4523.17	−0.259072	−0.129536	+	0.991575i	$0.541349\pi$
$674$	13004.7	0.743208
$675$	0	0
$676$	−9077.16	−0.516452
$677$	17685.7	1.00401	0.502007	−	0.864863i	$-0.332595\pi$
$677$	17685.7	1.00401	0.502007	+	0.864863i	$0.332595\pi$
$678$	0	0
$679$	15359.3	0.868093
$680$	−19176.6	−1.08146
$681$	0	0
$682$	588.640	0.0330501
$683$	−31140.8	−1.74461	−0.872307	−	0.488958i	$-0.837377\pi$
$683$	−31140.8	−1.74461	−0.872307	+	0.488958i	$0.837377\pi$
$684$	0	0
$685$	5338.83	0.297790
$686$	23670.8	1.31743
$687$	0	0
$688$	10009.2	0.554647
$689$	2715.15	0.150129
$690$	0	0
$691$	−18166.3	−1.00012	−0.500058	−	0.865992i	$-0.666688\pi$
$691$	−18166.3	−1.00012	−0.500058	+	0.865992i	$0.666688\pi$
$692$	31024.0	1.70427
$693$	0	0
$694$	−41701.9	−2.28096
$695$	−4216.57	−0.230135
$696$	0	0
$697$	16054.0	0.872437
$698$	15652.6	0.848794
$699$	0	0
$700$	−9931.68	−0.536261
$701$	11583.6	0.624120	0.312060	−	0.950062i	$-0.398981\pi$
$701$	11583.6	0.624120	0.312060	+	0.950062i	$0.398981\pi$
$702$	0	0
$703$	207.974	0.0111577
$704$	−5377.39	−0.287881
$705$	0	0
$706$	25587.5	1.36402
$707$	−28086.6	−1.49407
$708$	0	0
$709$	−20232.9	−1.07174	−0.535869	−	0.844301i	$-0.680016\pi$
$709$	−20232.9	−1.07174	−0.535869	+	0.844301i	$0.680016\pi$
$710$	14620.5	0.772811
$711$	0	0
$712$	47034.4	2.47569
$713$	−193.345	−0.0101554
$714$	0	0
$715$	11779.2	0.616108
$716$	29511.2	1.54034
$717$	0	0
$718$	18967.8	0.985895
$719$	−16489.7	−0.855304	−0.427652	−	0.903943i	$-0.640659\pi$
$719$	−16489.7	−0.855304	−0.427652	+	0.903943i	$0.640659\pi$
$720$	0	0
$721$	−19543.6	−1.00949
$722$	−35141.1	−1.81138
$723$	0	0
$724$	14426.6	0.740554
$725$	725.000	0.0371391
$726$	0	0
$727$	27239.4	1.38962	0.694810	−	0.719194i	$-0.255489\pi$
$727$	27239.4	1.38962	0.694810	+	0.719194i	$0.255489\pi$
$728$	47135.3	2.39966
$729$	0	0
$730$	17017.4	0.862796
$731$	−5935.82	−0.300334
$732$	0	0
$733$	−8145.16	−0.410434	−0.205217	−	0.978716i	$-0.565790\pi$
$733$	−8145.16	−0.410434	−0.205217	+	0.978716i	$0.565790\pi$
$734$	−9923.57	−0.499027
$735$	0	0
$736$	20253.2	1.01433
$737$	50756.3	2.53681
$738$	0	0
$739$	23926.8	1.19102	0.595508	−	0.803349i	$-0.296951\pi$
$739$	23926.8	1.19102	0.595508	+	0.803349i	$0.296951\pi$
$740$	−28817.6	−1.43156
$741$	0	0
$742$	7344.53	0.363378
$743$	2938.74	0.145103	0.0725517	−	0.997365i	$-0.476886\pi$
$743$	2938.74	0.145103	0.0725517	+	0.997365i	$0.476886\pi$
$744$	0	0
$745$	−6368.57	−0.313190
$746$	64251.3	3.15336
$747$	0	0
$748$	76152.7	3.72249
$749$	−45274.3	−2.20866
$750$	0	0
$751$	7625.31	0.370508	0.185254	−	0.982691i	$-0.440689\pi$
$751$	7625.31	0.370508	0.185254	+	0.982691i	$0.440689\pi$
$752$	−54273.9	−2.63187
$753$	0	0
$754$	−6125.80	−0.295873
$755$	−15152.9	−0.730424
$756$	0	0
$757$	−1641.02	−0.0787898	−0.0393949	−	0.999224i	$-0.512543\pi$
$757$	−1641.02	−0.0787898	−0.0393949	+	0.999224i	$0.512543\pi$
$758$	39165.5	1.87672
$759$	0	0
$760$	−172.979	−0.00825606
$761$	18376.8	0.875374	0.437687	−	0.899127i	$-0.355798\pi$
$761$	18376.8	0.875374	0.437687	+	0.899127i	$0.355798\pi$
$762$	0	0
$763$	30175.4	1.43175
$764$	54141.8	2.56385
$765$	0	0
$766$	−70884.2	−3.34354
$767$	−22737.6	−1.07041
$768$	0	0
$769$	−16315.5	−0.765085	−0.382543	−	0.923938i	$-0.624951\pi$
$769$	−16315.5	−0.765085	−0.382543	+	0.923938i	$0.624951\pi$
$770$	31863.0	1.49125
$771$	0	0
$772$	−34159.7	−1.59253
$773$	1255.81	0.0584325	0.0292163	−	0.999573i	$-0.490699\pi$
$773$	1255.81	0.0584325	0.0292163	+	0.999573i	$0.490699\pi$
$774$	0	0
$775$	−50.2626	−0.00232966
$776$	−37067.3	−1.71474
$777$	0	0
$778$	75922.6	3.49866
$779$	144.812	0.00666036
$780$	0	0
$781$	−32611.6	−1.49415
$782$	−35976.6	−1.64517
$783$	0	0
$784$	16097.2	0.733291
$785$	−7567.83	−0.344086
$786$	0	0
$787$	−18667.8	−0.845534	−0.422767	−	0.906238i	$-0.638941\pi$
$787$	−18667.8	−0.845534	−0.422767	+	0.906238i	$0.638941\pi$
$788$	32228.6	1.45697
$789$	0	0
$790$	5686.16	0.256082
$791$	10384.5	0.466788
$792$	0	0
$793$	6164.94	0.276070
$794$	−14582.1	−0.651761
$795$	0	0
$796$	−2793.84	−0.124403
$797$	7280.08	0.323556	0.161778	−	0.986827i	$-0.448277\pi$
$797$	7280.08	0.323556	0.161778	+	0.986827i	$0.448277\pi$
$798$	0	0
$799$	32186.4	1.42512
$800$	5265.11	0.232687
$801$	0	0
$802$	−39461.9	−1.73747
$803$	−37958.1	−1.66813
$804$	0	0
$805$	−10465.7	−0.458221
$806$	424.688	0.0185595
$807$	0	0
$808$	67782.8	2.95122
$809$	−44097.9	−1.91644	−0.958221	−	0.286030i	$-0.907664\pi$
$809$	−44097.9	−1.91644	−0.958221	+	0.286030i	$0.907664\pi$
$810$	0	0
$811$	−17249.1	−0.746854	−0.373427	−	0.927660i	$-0.621817\pi$
$811$	−17249.1	−0.746854	−0.373427	+	0.927660i	$0.621817\pi$
$812$	−11520.8	−0.497905
$813$	0	0
$814$	92452.9	3.98093
$815$	−15674.4	−0.673682
$816$	0	0
$817$	−53.5428	−0.00229281
$818$	−56204.1	−2.40236
$819$	0	0
$820$	−20065.6	−0.854540
$821$	−11201.7	−0.476177	−0.238088	−	0.971244i	$-0.576521\pi$
$821$	−11201.7	−0.476177	−0.238088	+	0.971244i	$0.576521\pi$
$822$	0	0
$823$	12114.8	0.513118	0.256559	−	0.966529i	$-0.417411\pi$
$823$	12114.8	0.513118	0.256559	+	0.966529i	$0.417411\pi$
$824$	47165.6	1.99404
$825$	0	0
$826$	−61505.5	−2.59086
$827$	−37074.2	−1.55888	−0.779442	−	0.626474i	$-0.784498\pi$
$827$	−37074.2	−1.55888	−0.779442	+	0.626474i	$0.784498\pi$
$828$	0	0
$829$	611.209	0.0256069	0.0128035	−	0.999918i	$-0.495924\pi$
$829$	611.209	0.0256069	0.0128035	+	0.999918i	$0.495924\pi$
$830$	18968.7	0.793267
$831$	0	0
$832$	−3879.64	−0.161662
$833$	−9546.23	−0.397068
$834$	0	0
$835$	−3517.08	−0.145765
$836$	686.920	0.0284182
$837$	0	0
$838$	55135.6	2.27282
$839$	43306.8	1.78202	0.891010	−	0.453983i	$-0.149997\pi$
$839$	43306.8	1.78202	0.891010	+	0.453983i	$0.149997\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−18251.6	−0.747020
$843$	0	0
$844$	102811.	4.19301
$845$	−2486.62	−0.101233
$846$	0	0
$847$	−42101.7	−1.70795
$848$	−8108.51	−0.328358
$849$	0	0
$850$	−9352.61	−0.377402
$851$	−30367.1	−1.22323
$852$	0	0
$853$	−3447.15	−0.138368	−0.0691842	−	0.997604i	$-0.522040\pi$
$853$	−3447.15	−0.138368	−0.0691842	+	0.997604i	$0.522040\pi$
$854$	16676.3	0.668209
$855$	0	0
$856$	109263.	4.36276
$857$	42416.7	1.69069	0.845347	−	0.534217i	$-0.179393\pi$
$857$	42416.7	1.69069	0.845347	+	0.534217i	$0.179393\pi$
$858$	0	0
$859$	29061.5	1.15433	0.577163	−	0.816629i	$-0.304160\pi$
$859$	29061.5	1.15433	0.577163	+	0.816629i	$0.304160\pi$
$860$	7419.09	0.294173
$861$	0	0
$862$	5568.31	0.220020
$863$	−20744.0	−0.818232	−0.409116	−	0.912482i	$-0.634163\pi$
$863$	−20744.0	−0.818232	−0.409116	+	0.912482i	$0.634163\pi$
$864$	0	0
$865$	8498.77	0.334066
$866$	21446.7	0.841556
$867$	0	0
$868$	798.708	0.0312326
$869$	−12683.2	−0.495109
$870$	0	0
$871$	36619.3	1.42457
$872$	−72823.8	−2.82812
$873$	0	0
$874$	−324.519	−0.0125595
$875$	−2720.71	−0.105116
$876$	0	0
$877$	−26329.7	−1.01379	−0.506893	−	0.862009i	$-0.669206\pi$
$877$	−26329.7	−1.01379	−0.506893	+	0.862009i	$0.669206\pi$
$878$	20293.3	0.780028
$879$	0	0
$880$	−35177.3	−1.34753
$881$	7199.40	0.275317	0.137658	−	0.990480i	$-0.456042\pi$
$881$	7199.40	0.275317	0.137658	+	0.990480i	$0.456042\pi$
$882$	0	0
$883$	64.9911	0.00247693	0.00123846	−	0.999999i	$-0.499606\pi$
$883$	64.9911	0.00247693	0.00123846	+	0.999999i	$0.499606\pi$
$884$	54942.2	2.09039
$885$	0	0
$886$	−8687.61	−0.329420
$887$	−12243.2	−0.463456	−0.231728	−	0.972781i	$-0.574438\pi$
$887$	−12243.2	−0.463456	−0.231728	+	0.972781i	$0.574438\pi$
$888$	0	0
$889$	29546.9	1.11470
$890$	22939.1	0.863954
$891$	0	0
$892$	−54078.8	−2.02992
$893$	290.331	0.0108797
$894$	0	0
$895$	8084.36	0.301933
$896$	26177.0	0.976018
$897$	0	0
$898$	−3356.23	−0.124720
$899$	−58.3047	−0.00216304
$900$	0	0
$901$	4808.64	0.177801
$902$	64374.9	2.37633
$903$	0	0
$904$	−25061.3	−0.922043
$905$	3952.06	0.145161
$906$	0	0
$907$	30619.9	1.12097	0.560483	−	0.828166i	$-0.310616\pi$
$907$	30619.9	1.12097	0.560483	+	0.828166i	$0.310616\pi$
$908$	18376.4	0.671631
$909$	0	0
$910$	22988.3	0.837422
$911$	−3062.05	−0.111362	−0.0556808	−	0.998449i	$-0.517733\pi$
$911$	−3062.05	−0.111362	−0.0556808	+	0.998449i	$0.517733\pi$
$912$	0	0
$913$	−42310.5	−1.53370
$914$	54779.5	1.98243
$915$	0	0
$916$	−23678.4	−0.854101
$917$	65184.7	2.34742
$918$	0	0
$919$	−1527.71	−0.0548361	−0.0274180	−	0.999624i	$-0.508729\pi$
$919$	−1527.71	−0.0548361	−0.0274180	+	0.999624i	$0.508729\pi$
$920$	25257.4	0.905121
$921$	0	0
$922$	−78179.9	−2.79253
$923$	−23528.4	−0.839054
$924$	0	0
$925$	−7894.35	−0.280610
$926$	−47957.5	−1.70192
$927$	0	0
$928$	6107.52	0.216044
$929$	968.476	0.0342031	0.0171015	−	0.999854i	$-0.494556\pi$
$929$	968.476	0.0342031	0.0171015	+	0.999854i	$0.494556\pi$
$930$	0	0
$931$	−86.1098	−0.00303129
$932$	30113.5	1.05837
$933$	0	0
$934$	−60669.0	−2.12543
$935$	20861.4	0.729671
$936$	0	0
$937$	−47051.9	−1.64047	−0.820234	−	0.572029i	$-0.806157\pi$
$937$	−47051.9	−1.64047	−0.820234	+	0.572029i	$0.806157\pi$
$938$	99055.8	3.44807
$939$	0	0
$940$	−40229.3	−1.39589
$941$	31285.3	1.08382	0.541908	−	0.840438i	$-0.317702\pi$
$941$	31285.3	1.08382	0.541908	+	0.840438i	$0.317702\pi$
$942$	0	0
$943$	−21144.6	−0.730183
$944$	67903.3	2.34117
$945$	0	0
$946$	−23802.0	−0.818045
$947$	1155.25	0.0396416	0.0198208	−	0.999804i	$-0.493690\pi$
$947$	1155.25	0.0396416	0.0198208	+	0.999804i	$0.493690\pi$
$948$	0	0
$949$	−27385.7	−0.936752
$950$	−84.3633	−0.00288116
$951$	0	0
$952$	83478.5	2.84197
$953$	−8073.69	−0.274431	−0.137215	−	0.990541i	$-0.543815\pi$
$953$	−8073.69	−0.274431	−0.137215	+	0.990541i	$0.543815\pi$
$954$	0	0
$955$	14831.7	0.502558
$956$	−104117.	−3.52236
$957$	0	0
$958$	−2977.22	−0.100407
$959$	−23240.6	−0.782563
$960$	0	0
$961$	−29787.0	−0.999864
$962$	66702.3	2.23552
$963$	0	0
$964$	132035.	4.41137
$965$	−9357.77	−0.312163
$966$	0	0
$967$	30587.8	1.01721	0.508603	−	0.861001i	$-0.330162\pi$
$967$	30587.8	1.01721	0.508603	+	0.861001i	$0.330162\pi$
$968$	101606.	3.37370
$969$	0	0
$970$	−18078.0	−0.598402
$971$	37585.6	1.24220	0.621101	−	0.783730i	$-0.286686\pi$
$971$	37585.6	1.24220	0.621101	+	0.783730i	$0.286686\pi$
$972$	0	0
$973$	18355.3	0.604772
$974$	−30726.4	−1.01082
$975$	0	0
$976$	−18410.9	−0.603811
$977$	−605.357	−0.0198230	−0.00991151	−	0.999951i	$-0.503155\pi$
$977$	−605.357	−0.0198230	−0.00991151	+	0.999951i	$0.503155\pi$
$978$	0	0
$979$	−51166.7	−1.67037
$980$	11931.7	0.388922
$981$	0	0
$982$	−95146.6	−3.09190
$983$	−24370.9	−0.790755	−0.395378	−	0.918519i	$-0.629386\pi$
$983$	−24370.9	−0.790755	−0.395378	+	0.918519i	$0.629386\pi$
$984$	0	0
$985$	8828.77	0.285592
$986$	−10849.0	−0.350409
$987$	0	0
$988$	495.594	0.0159585
$989$	7818.02	0.251363
$990$	0	0
$991$	53287.8	1.70812	0.854058	−	0.520178i	$-0.174134\pi$
$991$	53287.8	1.70812	0.854058	+	0.520178i	$0.174134\pi$
$992$	−423.421	−0.0135520
$993$	0	0
$994$	−63644.7	−2.03087
$995$	−765.350	−0.0243851
$996$	0	0
$997$	880.741	0.0279773	0.0139886	−	0.999902i	$-0.495547\pi$
$997$	880.741	0.0279773	0.0139886	+	0.999902i	$0.495547\pi$
$998$	17740.4	0.562687
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.m.1.7		7
3.2	odd	2		435.4.a.j.1.1	✓	7
15.14	odd	2		2175.4.a.m.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.j.1.1	✓	7	3.2	odd	2
1305.4.a.m.1.7		7	1.1	even	1	trivial
2175.4.a.m.1.7		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+5.12367 q^{2} +18.2520 q^{4} +5.00000 q^{5} -21.7657 q^{7} +52.5281 q^{8} +25.6184 q^{10} -57.1430 q^{11} -41.2271 q^{13} -111.520 q^{14} +123.120 q^{16} -73.0149 q^{17} -0.658616 q^{19} +91.2602 q^{20} -292.782 q^{22} +96.1672 q^{23} +25.0000 q^{25} -211.234 q^{26} -397.267 q^{28} +29.0000 q^{29} -2.01051 q^{31} +210.604 q^{32} -374.104 q^{34} -108.828 q^{35} -315.774 q^{37} -3.37453 q^{38} +262.640 q^{40} -219.873 q^{41} +81.2960 q^{43} -1042.98 q^{44} +492.730 q^{46} -440.820 q^{47} +130.744 q^{49} +128.092 q^{50} -752.479 q^{52} -65.8584 q^{53} -285.715 q^{55} -1143.31 q^{56} +148.587 q^{58} +551.520 q^{59} -149.536 q^{61} -10.3012 q^{62} +94.1041 q^{64} -206.136 q^{65} -888.233 q^{67} -1332.67 q^{68} -557.601 q^{70} +570.702 q^{71} +664.264 q^{73} -1617.92 q^{74} -12.0211 q^{76} +1243.75 q^{77} +221.956 q^{79} +615.602 q^{80} -1126.56 q^{82} +740.432 q^{83} -365.074 q^{85} +416.534 q^{86} -3001.61 q^{88} +895.415 q^{89} +897.336 q^{91} +1755.25 q^{92} -2258.62 q^{94} -3.29308 q^{95} -705.666 q^{97} +669.888 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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