LMFDB - Embedded newform 1248.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,4,Mod(1,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, 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("1248.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1248.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:	$73.6343836872$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 20x^{3} - 33x^{2} + 17x + 40 $ x^5 - 20x^3 - 33x^2 + 17*x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.05502$ of defining polynomial
Character	$\chi$	$=$	1248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +2.77633 q^{5} -5.23038 q^{7} +9.00000 q^{9} +26.0282 q^{11} -13.0000 q^{13} +8.32898 q^{15} +4.06350 q^{17} -73.9991 q^{19} -15.6911 q^{21} -145.163 q^{23} -117.292 q^{25} +27.0000 q^{27} -259.573 q^{29} +78.5918 q^{31} +78.0845 q^{33} -14.5213 q^{35} +99.8291 q^{37} -39.0000 q^{39} +346.320 q^{41} -137.732 q^{43} +24.9869 q^{45} +56.7561 q^{47} -315.643 q^{49} +12.1905 q^{51} +173.382 q^{53} +72.2627 q^{55} -221.997 q^{57} +675.923 q^{59} -798.265 q^{61} -47.0734 q^{63} -36.0923 q^{65} -292.584 q^{67} -435.489 q^{69} -333.464 q^{71} +243.292 q^{73} -351.876 q^{75} -136.137 q^{77} -670.246 q^{79} +81.0000 q^{81} -786.662 q^{83} +11.2816 q^{85} -778.719 q^{87} -317.044 q^{89} +67.9950 q^{91} +235.775 q^{93} -205.446 q^{95} +886.328 q^{97} +234.253 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15 q^{3} - 10 q^{5} - 14 q^{7} + 45 q^{9} - 22 q^{11} - 65 q^{13} - 30 q^{15} - 34 q^{17} + 90 q^{19} - 42 q^{21} - 96 q^{23} + 107 q^{25} + 135 q^{27} - 54 q^{29} - 378 q^{31} - 66 q^{33} - 84 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 5 * q + 15 * q^3 - 10 * q^5 - 14 * q^7 + 45 * q^9 - 22 * q^11 - 65 * q^13 - 30 * q^15 - 34 * q^17 + 90 * q^19 - 42 * q^21 - 96 * q^23 + 107 * q^25 + 135 * q^27 - 54 * q^29 - 378 * q^31 - 66 * q^33 - 84 * q^35 - 318 * q^37 - 195 * q^39 - 458 * q^41 - 104 * q^43 - 90 * q^45 - 474 * q^47 + 25 * q^49 - 102 * q^51 - 450 * q^53 - 1116 * q^55 + 270 * q^57 - 822 * q^59 - 546 * q^61 - 126 * q^63 + 130 * q^65 + 130 * q^67 - 288 * q^69 - 1646 * q^71 - 166 * q^73 + 321 * q^75 - 900 * q^77 - 196 * q^79 + 405 * q^81 + 118 * q^83 - 3340 * q^85 - 162 * q^87 - 186 * q^89 + 182 * q^91 - 1134 * q^93 - 564 * q^95 - 2582 * q^97 - 198 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	2.77633	0.248322	0.124161	−	0.992262i	$-0.460376\pi$
$5$	2.77633	0.248322	0.124161	+	0.992262i	$0.460376\pi$
$6$	0	0
$7$	−5.23038	−0.282414	−0.141207	−	0.989980i	$-0.545098\pi$
$7$	−5.23038	−0.282414	−0.141207	+	0.989980i	$0.545098\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	26.0282	0.713435	0.356717	−	0.934212i	$-0.383896\pi$
$11$	26.0282	0.713435	0.356717	+	0.934212i	$0.383896\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	8.32898	0.143369
$16$	0	0
$17$	4.06350	0.0579731	0.0289866	−	0.999580i	$-0.490772\pi$
$17$	4.06350	0.0579731	0.0289866	+	0.999580i	$0.490772\pi$
$18$	0	0
$19$	−73.9991	−0.893503	−0.446751	−	0.894658i	$-0.647419\pi$
$19$	−73.9991	−0.893503	−0.446751	+	0.894658i	$0.647419\pi$
$20$	0	0
$21$	−15.6911	−0.163052
$22$	0	0
$23$	−145.163	−1.31603	−0.658013	−	0.753006i	$-0.728603\pi$
$23$	−145.163	−1.31603	−0.658013	+	0.753006i	$0.728603\pi$
$24$	0	0
$25$	−117.292	−0.938336
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−259.573	−1.66212	−0.831060	−	0.556183i	$-0.812265\pi$
$29$	−259.573	−1.66212	−0.831060	+	0.556183i	$0.812265\pi$
$30$	0	0
$31$	78.5918	0.455339	0.227669	−	0.973738i	$-0.426889\pi$
$31$	78.5918	0.455339	0.227669	+	0.973738i	$0.426889\pi$
$32$	0	0
$33$	78.0845	0.411902
$34$	0	0
$35$	−14.5213	−0.0701297
$36$	0	0
$37$	99.8291	0.443562	0.221781	−	0.975096i	$-0.428813\pi$
$37$	99.8291	0.443562	0.221781	+	0.975096i	$0.428813\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	346.320	1.31917	0.659586	−	0.751629i	$-0.270732\pi$
$41$	346.320	1.31917	0.659586	+	0.751629i	$0.270732\pi$
$42$	0	0
$43$	−137.732	−0.488463	−0.244231	−	0.969717i	$-0.578536\pi$
$43$	−137.732	−0.488463	−0.244231	+	0.969717i	$0.578536\pi$
$44$	0	0
$45$	24.9869	0.0827741
$46$	0	0
$47$	56.7561	0.176143	0.0880716	−	0.996114i	$-0.471930\pi$
$47$	56.7561	0.176143	0.0880716	+	0.996114i	$0.471930\pi$
$48$	0	0
$49$	−315.643	−0.920242
$50$	0	0
$51$	12.1905	0.0334708
$52$	0	0
$53$	173.382	0.449355	0.224677	−	0.974433i	$-0.427867\pi$
$53$	173.382	0.449355	0.224677	+	0.974433i	$0.427867\pi$
$54$	0	0
$55$	72.2627	0.177162
$56$	0	0
$57$	−221.997	−0.515864
$58$	0	0
$59$	675.923	1.49149	0.745743	−	0.666234i	$-0.232095\pi$
$59$	675.923	1.49149	0.745743	+	0.666234i	$0.232095\pi$
$60$	0	0
$61$	−798.265	−1.67553	−0.837765	−	0.546030i	$-0.816138\pi$
$61$	−798.265	−1.67553	−0.837765	+	0.546030i	$0.816138\pi$
$62$	0	0
$63$	−47.0734	−0.0941380
$64$	0	0
$65$	−36.0923	−0.0688722
$66$	0	0
$67$	−292.584	−0.533504	−0.266752	−	0.963765i	$-0.585950\pi$
$67$	−292.584	−0.533504	−0.266752	+	0.963765i	$0.585950\pi$
$68$	0	0
$69$	−435.489	−0.759808
$70$	0	0
$71$	−333.464	−0.557393	−0.278697	−	0.960379i	$-0.589902\pi$
$71$	−333.464	−0.557393	−0.278697	+	0.960379i	$0.589902\pi$
$72$	0	0
$73$	243.292	0.390072	0.195036	−	0.980796i	$-0.437518\pi$
$73$	243.292	0.390072	0.195036	+	0.980796i	$0.437518\pi$
$74$	0	0
$75$	−351.876	−0.541749
$76$	0	0
$77$	−136.137	−0.201484
$78$	0	0
$79$	−670.246	−0.954539	−0.477270	−	0.878757i	$-0.658373\pi$
$79$	−670.246	−0.954539	−0.477270	+	0.878757i	$0.658373\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−786.662	−1.04033	−0.520165	−	0.854066i	$-0.674130\pi$
$83$	−786.662	−1.04033	−0.520165	+	0.854066i	$0.674130\pi$
$84$	0	0
$85$	11.2816	0.0143960
$86$	0	0
$87$	−778.719	−0.959625
$88$	0	0
$89$	−317.044	−0.377602	−0.188801	−	0.982015i	$-0.560460\pi$
$89$	−317.044	−0.377602	−0.188801	+	0.982015i	$0.560460\pi$
$90$	0	0
$91$	67.9950	0.0783276
$92$	0	0
$93$	235.775	0.262890
$94$	0	0
$95$	−205.446	−0.221877
$96$	0	0
$97$	886.328	0.927762	0.463881	−	0.885897i	$-0.346456\pi$
$97$	886.328	0.927762	0.463881	+	0.885897i	$0.346456\pi$
$98$	0	0
$99$	234.253	0.237812
$100$	0	0
$101$	207.528	0.204454	0.102227	−	0.994761i	$-0.467403\pi$
$101$	207.528	0.204454	0.102227	+	0.994761i	$0.467403\pi$
$102$	0	0
$103$	1738.86	1.66345	0.831724	−	0.555189i	$-0.187354\pi$
$103$	1738.86	1.66345	0.831724	+	0.555189i	$0.187354\pi$
$104$	0	0
$105$	−43.5638	−0.0404894
$106$	0	0
$107$	−1248.91	−1.12838	−0.564189	−	0.825645i	$-0.690811\pi$
$107$	−1248.91	−1.12838	−0.564189	+	0.825645i	$0.690811\pi$
$108$	0	0
$109$	−1775.18	−1.55992	−0.779961	−	0.625828i	$-0.784761\pi$
$109$	−1775.18	−1.55992	−0.779961	+	0.625828i	$0.784761\pi$
$110$	0	0
$111$	299.487	0.256091
$112$	0	0
$113$	−1385.03	−1.15304	−0.576518	−	0.817085i	$-0.695589\pi$
$113$	−1385.03	−1.15304	−0.576518	+	0.817085i	$0.695589\pi$
$114$	0	0
$115$	−403.020	−0.326799
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−21.2537	−0.0163724
$120$	0	0
$121$	−653.535	−0.491011
$122$	0	0
$123$	1038.96	0.761624
$124$	0	0
$125$	−672.682	−0.481332
$126$	0	0
$127$	−276.727	−0.193351	−0.0966755	−	0.995316i	$-0.530821\pi$
$127$	−276.727	−0.193351	−0.0966755	+	0.995316i	$0.530821\pi$
$128$	0	0
$129$	−413.196	−0.282014
$130$	0	0
$131$	2293.51	1.52966	0.764829	−	0.644233i	$-0.222823\pi$
$131$	2293.51	1.52966	0.764829	+	0.644233i	$0.222823\pi$
$132$	0	0
$133$	387.043	0.252338
$134$	0	0
$135$	74.9608	0.0477896
$136$	0	0
$137$	−2639.75	−1.64620	−0.823098	−	0.567900i	$-0.807756\pi$
$137$	−2639.75	−1.64620	−0.823098	+	0.567900i	$0.807756\pi$
$138$	0	0
$139$	1371.48	0.836888	0.418444	−	0.908243i	$-0.362576\pi$
$139$	1371.48	0.836888	0.418444	+	0.908243i	$0.362576\pi$
$140$	0	0
$141$	170.268	0.101696
$142$	0	0
$143$	−338.366	−0.197871
$144$	0	0
$145$	−720.659	−0.412741
$146$	0	0
$147$	−946.929	−0.531302
$148$	0	0
$149$	−1674.18	−0.920498	−0.460249	−	0.887790i	$-0.652240\pi$
$149$	−1674.18	−0.920498	−0.460249	+	0.887790i	$0.652240\pi$
$150$	0	0
$151$	−2426.56	−1.30775	−0.653876	−	0.756602i	$-0.726858\pi$
$151$	−2426.56	−1.30775	−0.653876	+	0.756602i	$0.726858\pi$
$152$	0	0
$153$	36.5715	0.0193244
$154$	0	0
$155$	218.197	0.113071
$156$	0	0
$157$	−1290.62	−0.656069	−0.328035	−	0.944666i	$-0.606386\pi$
$157$	−1290.62	−0.656069	−0.328035	+	0.944666i	$0.606386\pi$
$158$	0	0
$159$	520.145	0.259435
$160$	0	0
$161$	759.259	0.371664
$162$	0	0
$163$	3787.85	1.82017	0.910083	−	0.414426i	$-0.136018\pi$
$163$	3787.85	1.82017	0.910083	+	0.414426i	$0.136018\pi$
$164$	0	0
$165$	216.788	0.102284
$166$	0	0
$167$	1004.20	0.465314	0.232657	−	0.972559i	$-0.425258\pi$
$167$	1004.20	0.465314	0.232657	+	0.972559i	$0.425258\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−665.992	−0.297834
$172$	0	0
$173$	12.1058	0.00532015	0.00266008	−	0.999996i	$-0.499153\pi$
$173$	12.1058	0.00532015	0.00266008	+	0.999996i	$0.499153\pi$
$174$	0	0
$175$	613.482	0.264999
$176$	0	0
$177$	2027.77	0.861110
$178$	0	0
$179$	−263.726	−0.110122	−0.0550610	−	0.998483i	$-0.517535\pi$
$179$	−263.726	−0.110122	−0.0550610	+	0.998483i	$0.517535\pi$
$180$	0	0
$181$	−1941.94	−0.797478	−0.398739	−	0.917064i	$-0.630552\pi$
$181$	−1941.94	−0.797478	−0.398739	+	0.917064i	$0.630552\pi$
$182$	0	0
$183$	−2394.80	−0.967368
$184$	0	0
$185$	277.158	0.110146
$186$	0	0
$187$	105.765	0.0413601
$188$	0	0
$189$	−141.220	−0.0543506
$190$	0	0
$191$	1925.11	0.729297	0.364649	−	0.931145i	$-0.381189\pi$
$191$	1925.11	0.729297	0.364649	+	0.931145i	$0.381189\pi$
$192$	0	0
$193$	−3317.81	−1.23742	−0.618708	−	0.785621i	$-0.712343\pi$
$193$	−3317.81	−1.23742	−0.618708	+	0.785621i	$0.712343\pi$
$194$	0	0
$195$	−108.277	−0.0397634
$196$	0	0
$197$	−4282.47	−1.54880	−0.774399	−	0.632697i	$-0.781948\pi$
$197$	−4282.47	−1.54880	−0.774399	+	0.632697i	$0.781948\pi$
$198$	0	0
$199$	−1219.10	−0.434271	−0.217135	−	0.976142i	$-0.569671\pi$
$199$	−1219.10	−0.434271	−0.217135	+	0.976142i	$0.569671\pi$
$200$	0	0
$201$	−877.751	−0.308019
$202$	0	0
$203$	1357.67	0.469406
$204$	0	0
$205$	961.496	0.327580
$206$	0	0
$207$	−1306.47	−0.438675
$208$	0	0
$209$	−1926.06	−0.637456
$210$	0	0
$211$	4956.66	1.61721	0.808603	−	0.588355i	$-0.200224\pi$
$211$	4956.66	1.61721	0.808603	+	0.588355i	$0.200224\pi$
$212$	0	0
$213$	−1000.39	−0.321811
$214$	0	0
$215$	−382.389	−0.121296
$216$	0	0
$217$	−411.065	−0.128594
$218$	0	0
$219$	729.877	0.225208
$220$	0	0
$221$	−52.8255	−0.0160789
$222$	0	0
$223$	1918.63	0.576148	0.288074	−	0.957608i	$-0.406985\pi$
$223$	1918.63	0.576148	0.288074	+	0.957608i	$0.406985\pi$
$224$	0	0
$225$	−1055.63	−0.312779
$226$	0	0
$227$	−1268.36	−0.370853	−0.185427	−	0.982658i	$-0.559367\pi$
$227$	−1268.36	−0.370853	−0.185427	+	0.982658i	$0.559367\pi$
$228$	0	0
$229$	−2481.05	−0.715950	−0.357975	−	0.933731i	$-0.616533\pi$
$229$	−2481.05	−0.715950	−0.357975	+	0.933731i	$0.616533\pi$
$230$	0	0
$231$	−408.412	−0.116327
$232$	0	0
$233$	5103.45	1.43493	0.717464	−	0.696595i	$-0.245303\pi$
$233$	5103.45	1.43493	0.717464	+	0.696595i	$0.245303\pi$
$234$	0	0
$235$	157.574	0.0437403
$236$	0	0
$237$	−2010.74	−0.551103
$238$	0	0
$239$	−6552.57	−1.77343	−0.886716	−	0.462314i	$-0.847019\pi$
$239$	−6552.57	−1.77343	−0.886716	+	0.462314i	$0.847019\pi$
$240$	0	0
$241$	2398.64	0.641120	0.320560	−	0.947228i	$-0.396129\pi$
$241$	2398.64	0.641120	0.320560	+	0.947228i	$0.396129\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−876.329	−0.228517
$246$	0	0
$247$	961.988	0.247813
$248$	0	0
$249$	−2359.99	−0.600635
$250$	0	0
$251$	2899.21	0.729069	0.364535	−	0.931190i	$-0.381228\pi$
$251$	2899.21	0.729069	0.364535	+	0.931190i	$0.381228\pi$
$252$	0	0
$253$	−3778.33	−0.938899
$254$	0	0
$255$	33.8448	0.00831155
$256$	0	0
$257$	−5784.67	−1.40404	−0.702019	−	0.712158i	$-0.747718\pi$
$257$	−5784.67	−1.40404	−0.702019	+	0.712158i	$0.747718\pi$
$258$	0	0
$259$	−522.144	−0.125268
$260$	0	0
$261$	−2336.16	−0.554040
$262$	0	0
$263$	−5346.03	−1.25342	−0.626712	−	0.779251i	$-0.715600\pi$
$263$	−5346.03	−1.25342	−0.626712	+	0.779251i	$0.715600\pi$
$264$	0	0
$265$	481.364	0.111585
$266$	0	0
$267$	−951.132	−0.218009
$268$	0	0
$269$	5693.67	1.29052	0.645259	−	0.763964i	$-0.276750\pi$
$269$	5693.67	1.29052	0.645259	+	0.763964i	$0.276750\pi$
$270$	0	0
$271$	−496.345	−0.111258	−0.0556288	−	0.998452i	$-0.517716\pi$
$271$	−496.345	−0.111258	−0.0556288	+	0.998452i	$0.517716\pi$
$272$	0	0
$273$	203.985	0.0452224
$274$	0	0
$275$	−3052.89	−0.669442
$276$	0	0
$277$	2832.42	0.614380	0.307190	−	0.951648i	$-0.400611\pi$
$277$	2832.42	0.614380	0.307190	+	0.951648i	$0.400611\pi$
$278$	0	0
$279$	707.326	0.151780
$280$	0	0
$281$	−5362.12	−1.13835	−0.569176	−	0.822216i	$-0.692738\pi$
$281$	−5362.12	−1.13835	−0.569176	+	0.822216i	$0.692738\pi$
$282$	0	0
$283$	3330.32	0.699530	0.349765	−	0.936838i	$-0.386261\pi$
$283$	3330.32	0.699530	0.349765	+	0.936838i	$0.386261\pi$
$284$	0	0
$285$	−616.337	−0.128100
$286$	0	0
$287$	−1811.38	−0.372553
$288$	0	0
$289$	−4896.49	−0.996639
$290$	0	0
$291$	2658.98	0.535644
$292$	0	0
$293$	−939.941	−0.187413	−0.0937064	−	0.995600i	$-0.529872\pi$
$293$	−939.941	−0.187413	−0.0937064	+	0.995600i	$0.529872\pi$
$294$	0	0
$295$	1876.58	0.370369
$296$	0	0
$297$	702.760	0.137301
$298$	0	0
$299$	1887.12	0.365000
$300$	0	0
$301$	720.390	0.137949
$302$	0	0
$303$	622.585	0.118041
$304$	0	0
$305$	−2216.25	−0.416072
$306$	0	0
$307$	366.403	0.0681165	0.0340582	−	0.999420i	$-0.489157\pi$
$307$	366.403	0.0681165	0.0340582	+	0.999420i	$0.489157\pi$
$308$	0	0
$309$	5216.59	0.960392
$310$	0	0
$311$	−3116.58	−0.568247	−0.284124	−	0.958788i	$-0.591703\pi$
$311$	−3116.58	−0.568247	−0.284124	+	0.958788i	$0.591703\pi$
$312$	0	0
$313$	2360.35	0.426246	0.213123	−	0.977025i	$-0.431636\pi$
$313$	2360.35	0.426246	0.213123	+	0.977025i	$0.431636\pi$
$314$	0	0
$315$	−130.691	−0.0233766
$316$	0	0
$317$	−1279.56	−0.226711	−0.113356	−	0.993554i	$-0.536160\pi$
$317$	−1279.56	−0.226711	−0.113356	+	0.993554i	$0.536160\pi$
$318$	0	0
$319$	−6756.20	−1.18581
$320$	0	0
$321$	−3746.73	−0.651470
$322$	0	0
$323$	−300.695	−0.0517991
$324$	0	0
$325$	1524.80	0.260248
$326$	0	0
$327$	−5325.54	−0.900622
$328$	0	0
$329$	−296.856	−0.0497453
$330$	0	0
$331$	10696.4	1.77621	0.888107	−	0.459638i	$-0.152021\pi$
$331$	10696.4	1.77621	0.888107	+	0.459638i	$0.152021\pi$
$332$	0	0
$333$	898.462	0.147854
$334$	0	0
$335$	−812.308	−0.132481
$336$	0	0
$337$	−1046.80	−0.169208	−0.0846038	−	0.996415i	$-0.526962\pi$
$337$	−1046.80	−0.169208	−0.0846038	+	0.996415i	$0.526962\pi$
$338$	0	0
$339$	−4155.10	−0.665705
$340$	0	0
$341$	2045.60	0.324855
$342$	0	0
$343$	3444.95	0.542303
$344$	0	0
$345$	−1209.06	−0.188677
$346$	0	0
$347$	6330.72	0.979397	0.489698	−	0.871892i	$-0.337107\pi$
$347$	6330.72	0.979397	0.489698	+	0.871892i	$0.337107\pi$
$348$	0	0
$349$	5009.55	0.768353	0.384176	−	0.923260i	$-0.374485\pi$
$349$	5009.55	0.768353	0.384176	+	0.923260i	$0.374485\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	1846.27	0.278376	0.139188	−	0.990266i	$-0.455551\pi$
$353$	1846.27	0.278376	0.139188	+	0.990266i	$0.455551\pi$
$354$	0	0
$355$	−925.806	−0.138413
$356$	0	0
$357$	−63.7610	−0.00945263
$358$	0	0
$359$	−2967.18	−0.436217	−0.218109	−	0.975925i	$-0.569989\pi$
$359$	−2967.18	−0.436217	−0.218109	+	0.975925i	$0.569989\pi$
$360$	0	0
$361$	−1383.14	−0.201653
$362$	0	0
$363$	−1960.61	−0.283485
$364$	0	0
$365$	675.459	0.0968634
$366$	0	0
$367$	−1714.49	−0.243858	−0.121929	−	0.992539i	$-0.538908\pi$
$367$	−1714.49	−0.243858	−0.121929	+	0.992539i	$0.538908\pi$
$368$	0	0
$369$	3116.88	0.439724
$370$	0	0
$371$	−906.852	−0.126904
$372$	0	0
$373$	−12101.2	−1.67983	−0.839917	−	0.542715i	$-0.817396\pi$
$373$	−12101.2	−1.67983	−0.839917	+	0.542715i	$0.817396\pi$
$374$	0	0
$375$	−2018.05	−0.277897
$376$	0	0
$377$	3374.45	0.460989
$378$	0	0
$379$	8525.01	1.15541	0.577705	−	0.816245i	$-0.303948\pi$
$379$	8525.01	1.15541	0.577705	+	0.816245i	$0.303948\pi$
$380$	0	0
$381$	−830.182	−0.111631
$382$	0	0
$383$	−2563.79	−0.342045	−0.171023	−	0.985267i	$-0.554707\pi$
$383$	−2563.79	−0.342045	−0.171023	+	0.985267i	$0.554707\pi$
$384$	0	0
$385$	−377.961	−0.0500330
$386$	0	0
$387$	−1239.59	−0.162821
$388$	0	0
$389$	10963.3	1.42895	0.714475	−	0.699661i	$-0.246666\pi$
$389$	10963.3	1.42895	0.714475	+	0.699661i	$0.246666\pi$
$390$	0	0
$391$	−589.870	−0.0762942
$392$	0	0
$393$	6880.54	0.883149
$394$	0	0
$395$	−1860.82	−0.237033
$396$	0	0
$397$	7970.57	1.00764	0.503818	−	0.863810i	$-0.331928\pi$
$397$	7970.57	1.00764	0.503818	+	0.863810i	$0.331928\pi$
$398$	0	0
$399$	1161.13	0.145687
$400$	0	0
$401$	972.347	0.121089	0.0605445	−	0.998165i	$-0.480716\pi$
$401$	972.347	0.121089	0.0605445	+	0.998165i	$0.480716\pi$
$402$	0	0
$403$	−1021.69	−0.126288
$404$	0	0
$405$	224.883	0.0275914
$406$	0	0
$407$	2598.37	0.316453
$408$	0	0
$409$	13183.5	1.59384	0.796920	−	0.604085i	$-0.206461\pi$
$409$	13183.5	1.59384	0.796920	+	0.604085i	$0.206461\pi$
$410$	0	0
$411$	−7919.24	−0.950431
$412$	0	0
$413$	−3535.33	−0.421216
$414$	0	0
$415$	−2184.03	−0.258337
$416$	0	0
$417$	4114.44	0.483177
$418$	0	0
$419$	−4346.49	−0.506778	−0.253389	−	0.967365i	$-0.581545\pi$
$419$	−4346.49	−0.506778	−0.253389	+	0.967365i	$0.581545\pi$
$420$	0	0
$421$	−2930.12	−0.339205	−0.169603	−	0.985513i	$-0.554248\pi$
$421$	−2930.12	−0.339205	−0.169603	+	0.985513i	$0.554248\pi$
$422$	0	0
$423$	510.805	0.0587144
$424$	0	0
$425$	−476.616	−0.0543983
$426$	0	0
$427$	4175.23	0.473193
$428$	0	0
$429$	−1015.10	−0.114241
$430$	0	0
$431$	−14506.2	−1.62121	−0.810604	−	0.585595i	$-0.800861\pi$
$431$	−14506.2	−1.62121	−0.810604	+	0.585595i	$0.800861\pi$
$432$	0	0
$433$	3672.96	0.407647	0.203823	−	0.979008i	$-0.434663\pi$
$433$	3672.96	0.407647	0.203823	+	0.979008i	$0.434663\pi$
$434$	0	0
$435$	−2161.98	−0.238296
$436$	0	0
$437$	10741.9	1.17587
$438$	0	0
$439$	17003.5	1.84860	0.924299	−	0.381669i	$-0.124650\pi$
$439$	17003.5	1.84860	0.924299	+	0.381669i	$0.124650\pi$
$440$	0	0
$441$	−2840.79	−0.306747
$442$	0	0
$443$	−14173.8	−1.52013	−0.760066	−	0.649846i	$-0.774833\pi$
$443$	−14173.8	−1.52013	−0.760066	+	0.649846i	$0.774833\pi$
$444$	0	0
$445$	−880.218	−0.0937671
$446$	0	0
$447$	−5022.54	−0.531450
$448$	0	0
$449$	−12102.9	−1.27209	−0.636046	−	0.771651i	$-0.719431\pi$
$449$	−12102.9	−1.27209	−0.636046	+	0.771651i	$0.719431\pi$
$450$	0	0
$451$	9014.06	0.941143
$452$	0	0
$453$	−7279.68	−0.755031
$454$	0	0
$455$	188.776	0.0194505
$456$	0	0
$457$	−3341.74	−0.342056	−0.171028	−	0.985266i	$-0.554709\pi$
$457$	−3341.74	−0.342056	−0.171028	+	0.985266i	$0.554709\pi$
$458$	0	0
$459$	109.714	0.0111569
$460$	0	0
$461$	7857.61	0.793851	0.396926	−	0.917851i	$-0.370077\pi$
$461$	7857.61	0.793851	0.396926	+	0.917851i	$0.370077\pi$
$462$	0	0
$463$	7149.73	0.717659	0.358829	−	0.933403i	$-0.383176\pi$
$463$	7149.73	0.717659	0.358829	+	0.933403i	$0.383176\pi$
$464$	0	0
$465$	654.590	0.0652814
$466$	0	0
$467$	11893.0	1.17846	0.589230	−	0.807965i	$-0.299431\pi$
$467$	11893.0	1.17846	0.589230	+	0.807965i	$0.299431\pi$
$468$	0	0
$469$	1530.32	0.150669
$470$	0	0
$471$	−3871.87	−0.378782
$472$	0	0
$473$	−3584.91	−0.348487
$474$	0	0
$475$	8679.50	0.838406
$476$	0	0
$477$	1560.43	0.149785
$478$	0	0
$479$	−12678.9	−1.20942	−0.604710	−	0.796446i	$-0.706711\pi$
$479$	−12678.9	−1.20942	−0.604710	+	0.796446i	$0.706711\pi$
$480$	0	0
$481$	−1297.78	−0.123022
$482$	0	0
$483$	2277.78	0.214581
$484$	0	0
$485$	2460.74	0.230384
$486$	0	0
$487$	−8701.84	−0.809688	−0.404844	−	0.914386i	$-0.632674\pi$
$487$	−8701.84	−0.809688	−0.404844	+	0.914386i	$0.632674\pi$
$488$	0	0
$489$	11363.5	1.05087
$490$	0	0
$491$	5212.84	0.479128	0.239564	−	0.970881i	$-0.422995\pi$
$491$	5212.84	0.479128	0.239564	+	0.970881i	$0.422995\pi$
$492$	0	0
$493$	−1054.77	−0.0963583
$494$	0	0
$495$	650.364	0.0590539
$496$	0	0
$497$	1744.14	0.157416
$498$	0	0
$499$	4138.31	0.371255	0.185627	−	0.982620i	$-0.440568\pi$
$499$	4138.31	0.371255	0.185627	+	0.982620i	$0.440568\pi$
$500$	0	0
$501$	3012.60	0.268649
$502$	0	0
$503$	−17161.5	−1.52126	−0.760630	−	0.649185i	$-0.775110\pi$
$503$	−17161.5	−1.52126	−0.760630	+	0.649185i	$0.775110\pi$
$504$	0	0
$505$	576.166	0.0507704
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	16707.7	1.45492	0.727461	−	0.686149i	$-0.240700\pi$
$509$	16707.7	1.45492	0.727461	+	0.686149i	$0.240700\pi$
$510$	0	0
$511$	−1272.51	−0.110162
$512$	0	0
$513$	−1997.97	−0.171955
$514$	0	0
$515$	4827.65	0.413071
$516$	0	0
$517$	1477.26	0.125667
$518$	0	0
$519$	36.3174	0.00307159
$520$	0	0
$521$	8042.87	0.676324	0.338162	−	0.941088i	$-0.390195\pi$
$521$	8042.87	0.676324	0.338162	+	0.941088i	$0.390195\pi$
$522$	0	0
$523$	3521.49	0.294425	0.147212	−	0.989105i	$-0.452970\pi$
$523$	3521.49	0.294425	0.147212	+	0.989105i	$0.452970\pi$
$524$	0	0
$525$	1840.45	0.152997
$526$	0	0
$527$	319.358	0.0263974
$528$	0	0
$529$	8905.34	0.731926
$530$	0	0
$531$	6083.30	0.497162
$532$	0	0
$533$	−4502.15	−0.365872
$534$	0	0
$535$	−3467.38	−0.280202
$536$	0	0
$537$	−791.179	−0.0635790
$538$	0	0
$539$	−8215.61	−0.656533
$540$	0	0
$541$	20824.9	1.65496	0.827480	−	0.561495i	$-0.189774\pi$
$541$	20824.9	1.65496	0.827480	+	0.561495i	$0.189774\pi$
$542$	0	0
$543$	−5825.83	−0.460424
$544$	0	0
$545$	−4928.49	−0.387363
$546$	0	0
$547$	8219.38	0.642478	0.321239	−	0.946998i	$-0.395901\pi$
$547$	8219.38	0.642478	0.321239	+	0.946998i	$0.395901\pi$
$548$	0	0
$549$	−7184.39	−0.558510
$550$	0	0
$551$	19208.1	1.48511
$552$	0	0
$553$	3505.64	0.269575
$554$	0	0
$555$	831.475	0.0635931
$556$	0	0
$557$	−644.376	−0.0490181	−0.0245091	−	0.999700i	$-0.507802\pi$
$557$	−644.376	−0.0490181	−0.0245091	+	0.999700i	$0.507802\pi$
$558$	0	0
$559$	1790.51	0.135475
$560$	0	0
$561$	317.296	0.0238792
$562$	0	0
$563$	−4735.04	−0.354455	−0.177227	−	0.984170i	$-0.556713\pi$
$563$	−4735.04	−0.354455	−0.177227	+	0.984170i	$0.556713\pi$
$564$	0	0
$565$	−3845.31	−0.286324
$566$	0	0
$567$	−423.661	−0.0313793
$568$	0	0
$569$	3403.32	0.250746	0.125373	−	0.992110i	$-0.459987\pi$
$569$	3403.32	0.250746	0.125373	+	0.992110i	$0.459987\pi$
$570$	0	0
$571$	12107.4	0.887351	0.443676	−	0.896187i	$-0.353674\pi$
$571$	12107.4	0.887351	0.443676	+	0.896187i	$0.353674\pi$
$572$	0	0
$573$	5775.32	0.421060
$574$	0	0
$575$	17026.5	1.23488
$576$	0	0
$577$	3208.73	0.231510	0.115755	−	0.993278i	$-0.463071\pi$
$577$	3208.73	0.231510	0.115755	+	0.993278i	$0.463071\pi$
$578$	0	0
$579$	−9953.43	−0.714422
$580$	0	0
$581$	4114.54	0.293804
$582$	0	0
$583$	4512.80	0.320585
$584$	0	0
$585$	−324.830	−0.0229574
$586$	0	0
$587$	13630.6	0.958427	0.479213	−	0.877698i	$-0.340922\pi$
$587$	13630.6	0.958427	0.479213	+	0.877698i	$0.340922\pi$
$588$	0	0
$589$	−5815.72	−0.406846
$590$	0	0
$591$	−12847.4	−0.894199
$592$	0	0
$593$	3392.81	0.234951	0.117476	−	0.993076i	$-0.462520\pi$
$593$	3392.81	0.234951	0.117476	+	0.993076i	$0.462520\pi$
$594$	0	0
$595$	−59.0071	−0.00406564
$596$	0	0
$597$	−3657.31	−0.250726
$598$	0	0
$599$	−2245.25	−0.153153	−0.0765763	−	0.997064i	$-0.524399\pi$
$599$	−2245.25	−0.153153	−0.0765763	+	0.997064i	$0.524399\pi$
$600$	0	0
$601$	−7307.86	−0.495997	−0.247998	−	0.968760i	$-0.579773\pi$
$601$	−7307.86	−0.495997	−0.247998	+	0.968760i	$0.579773\pi$
$602$	0	0
$603$	−2633.25	−0.177835
$604$	0	0
$605$	−1814.43	−0.121929
$606$	0	0
$607$	8567.55	0.572893	0.286447	−	0.958096i	$-0.407526\pi$
$607$	8567.55	0.572893	0.286447	+	0.958096i	$0.407526\pi$
$608$	0	0
$609$	4073.00	0.271012
$610$	0	0
$611$	−737.830	−0.0488533
$612$	0	0
$613$	20298.2	1.33741	0.668707	−	0.743526i	$-0.266848\pi$
$613$	20298.2	1.33741	0.668707	+	0.743526i	$0.266848\pi$
$614$	0	0
$615$	2884.49	0.189128
$616$	0	0
$617$	−7172.25	−0.467980	−0.233990	−	0.972239i	$-0.575178\pi$
$617$	−7172.25	−0.467980	−0.233990	+	0.972239i	$0.575178\pi$
$618$	0	0
$619$	2861.42	0.185800	0.0928998	−	0.995675i	$-0.470386\pi$
$619$	2861.42	0.185800	0.0928998	+	0.995675i	$0.470386\pi$
$620$	0	0
$621$	−3919.40	−0.253269
$622$	0	0
$623$	1658.26	0.106640
$624$	0	0
$625$	12793.9	0.818811
$626$	0	0
$627$	−5778.18	−0.368035
$628$	0	0
$629$	405.656	0.0257147
$630$	0	0
$631$	30554.9	1.92769	0.963843	−	0.266469i	$-0.0858570\pi$
$631$	30554.9	1.92769	0.963843	+	0.266469i	$0.0858570\pi$
$632$	0	0
$633$	14870.0	0.933694
$634$	0	0
$635$	−768.285	−0.0480133
$636$	0	0
$637$	4103.36	0.255229
$638$	0	0
$639$	−3001.18	−0.185798
$640$	0	0
$641$	18473.6	1.13832	0.569161	−	0.822226i	$-0.307268\pi$
$641$	18473.6	1.13832	0.569161	+	0.822226i	$0.307268\pi$
$642$	0	0
$643$	−22801.7	−1.39846	−0.699232	−	0.714895i	$-0.746474\pi$
$643$	−22801.7	−1.39846	−0.699232	+	0.714895i	$0.746474\pi$
$644$	0	0
$645$	−1147.17	−0.0700304
$646$	0	0
$647$	−11212.0	−0.681280	−0.340640	−	0.940194i	$-0.610644\pi$
$647$	−11212.0	−0.681280	−0.340640	+	0.940194i	$0.610644\pi$
$648$	0	0
$649$	17593.0	1.06408
$650$	0	0
$651$	−1233.20	−0.0742438
$652$	0	0
$653$	1885.87	0.113017	0.0565084	−	0.998402i	$-0.482003\pi$
$653$	1885.87	0.113017	0.0565084	+	0.998402i	$0.482003\pi$
$654$	0	0
$655$	6367.55	0.379848
$656$	0	0
$657$	2189.63	0.130024
$658$	0	0
$659$	525.205	0.0310456	0.0155228	−	0.999880i	$-0.495059\pi$
$659$	525.205	0.0310456	0.0155228	+	0.999880i	$0.495059\pi$
$660$	0	0
$661$	−8036.59	−0.472900	−0.236450	−	0.971644i	$-0.575984\pi$
$661$	−8036.59	−0.472900	−0.236450	+	0.971644i	$0.575984\pi$
$662$	0	0
$663$	−158.476	−0.00928313
$664$	0	0
$665$	1074.56	0.0626611
$666$	0	0
$667$	37680.4	2.18739
$668$	0	0
$669$	5755.89	0.332639
$670$	0	0
$671$	−20777.4	−1.19538
$672$	0	0
$673$	19505.9	1.11723	0.558616	−	0.829426i	$-0.311332\pi$
$673$	19505.9	1.11723	0.558616	+	0.829426i	$0.311332\pi$
$674$	0	0
$675$	−3166.88	−0.180583
$676$	0	0
$677$	−12847.5	−0.729347	−0.364673	−	0.931136i	$-0.618819\pi$
$677$	−12847.5	−0.729347	−0.364673	+	0.931136i	$0.618819\pi$
$678$	0	0
$679$	−4635.83	−0.262013
$680$	0	0
$681$	−3805.07	−0.214112
$682$	0	0
$683$	12286.7	0.688342	0.344171	−	0.938907i	$-0.388160\pi$
$683$	12286.7	0.688342	0.344171	+	0.938907i	$0.388160\pi$
$684$	0	0
$685$	−7328.80	−0.408787
$686$	0	0
$687$	−7443.15	−0.413354
$688$	0	0
$689$	−2253.96	−0.124629
$690$	0	0
$691$	−24058.5	−1.32450	−0.662250	−	0.749283i	$-0.730398\pi$
$691$	−24058.5	−1.32450	−0.662250	+	0.749283i	$0.730398\pi$
$692$	0	0
$693$	−1225.23	−0.0671613
$694$	0	0
$695$	3807.68	0.207818
$696$	0	0
$697$	1407.27	0.0764765
$698$	0	0
$699$	15310.4	0.828456
$700$	0	0
$701$	28829.0	1.55329	0.776644	−	0.629940i	$-0.216920\pi$
$701$	28829.0	1.55329	0.776644	+	0.629940i	$0.216920\pi$
$702$	0	0
$703$	−7387.26	−0.396324
$704$	0	0
$705$	472.721	0.0252535
$706$	0	0
$707$	−1085.45	−0.0577406
$708$	0	0
$709$	4146.60	0.219646	0.109823	−	0.993951i	$-0.464972\pi$
$709$	4146.60	0.219646	0.109823	+	0.993951i	$0.464972\pi$
$710$	0	0
$711$	−6032.22	−0.318180
$712$	0	0
$713$	−11408.6	−0.599238
$714$	0	0
$715$	−939.415	−0.0491358
$716$	0	0
$717$	−19657.7	−1.02389
$718$	0	0
$719$	−9681.67	−0.502177	−0.251089	−	0.967964i	$-0.580789\pi$
$719$	−9681.67	−0.502177	−0.251089	+	0.967964i	$0.580789\pi$
$720$	0	0
$721$	−9094.91	−0.469781
$722$	0	0
$723$	7195.91	0.370151
$724$	0	0
$725$	30445.8	1.55963
$726$	0	0
$727$	10092.3	0.514860	0.257430	−	0.966297i	$-0.417124\pi$
$727$	10092.3	0.514860	0.257430	+	0.966297i	$0.417124\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−559.673	−0.0283177
$732$	0	0
$733$	−7775.50	−0.391807	−0.195904	−	0.980623i	$-0.562764\pi$
$733$	−7775.50	−0.391807	−0.195904	+	0.980623i	$0.562764\pi$
$734$	0	0
$735$	−2628.99	−0.131934
$736$	0	0
$737$	−7615.41	−0.380621
$738$	0	0
$739$	2699.84	0.134392	0.0671958	−	0.997740i	$-0.478595\pi$
$739$	2699.84	0.134392	0.0671958	+	0.997740i	$0.478595\pi$
$740$	0	0
$741$	2885.96	0.143075
$742$	0	0
$743$	−14952.1	−0.738279	−0.369139	−	0.929374i	$-0.620348\pi$
$743$	−14952.1	−0.738279	−0.369139	+	0.929374i	$0.620348\pi$
$744$	0	0
$745$	−4648.07	−0.228580
$746$	0	0
$747$	−7079.96	−0.346777
$748$	0	0
$749$	6532.27	0.318670
$750$	0	0
$751$	21169.6	1.02861	0.514307	−	0.857606i	$-0.328049\pi$
$751$	21169.6	1.02861	0.514307	+	0.857606i	$0.328049\pi$
$752$	0	0
$753$	8697.62	0.420928
$754$	0	0
$755$	−6736.92	−0.324744
$756$	0	0
$757$	−10096.0	−0.484736	−0.242368	−	0.970184i	$-0.577924\pi$
$757$	−10096.0	−0.484736	−0.242368	+	0.970184i	$0.577924\pi$
$758$	0	0
$759$	−11335.0	−0.542074
$760$	0	0
$761$	31395.8	1.49553	0.747764	−	0.663965i	$-0.231127\pi$
$761$	31395.8	1.49553	0.747764	+	0.663965i	$0.231127\pi$
$762$	0	0
$763$	9284.88	0.440544
$764$	0	0
$765$	101.534	0.00479867
$766$	0	0
$767$	−8787.00	−0.413664
$768$	0	0
$769$	−16798.8	−0.787750	−0.393875	−	0.919164i	$-0.628866\pi$
$769$	−16798.8	−0.787750	−0.393875	+	0.919164i	$0.628866\pi$
$770$	0	0
$771$	−17354.0	−0.810622
$772$	0	0
$773$	−18854.0	−0.877271	−0.438635	−	0.898665i	$-0.644538\pi$
$773$	−18854.0	−0.877271	−0.438635	+	0.898665i	$0.644538\pi$
$774$	0	0
$775$	−9218.19	−0.427261
$776$	0	0
$777$	−1566.43	−0.0723237
$778$	0	0
$779$	−25627.3	−1.17868
$780$	0	0
$781$	−8679.46	−0.397664
$782$	0	0
$783$	−7008.47	−0.319875
$784$	0	0
$785$	−3583.19	−0.162917
$786$	0	0
$787$	12420.9	0.562587	0.281294	−	0.959622i	$-0.409237\pi$
$787$	12420.9	0.562587	0.281294	+	0.959622i	$0.409237\pi$
$788$	0	0
$789$	−16038.1	−0.723665
$790$	0	0
$791$	7244.25	0.325633
$792$	0	0
$793$	10377.4	0.464709
$794$	0	0
$795$	1444.09	0.0644235
$796$	0	0
$797$	34417.0	1.52963	0.764814	−	0.644251i	$-0.222831\pi$
$797$	34417.0	1.52963	0.764814	+	0.644251i	$0.222831\pi$
$798$	0	0
$799$	230.628	0.0102116
$800$	0	0
$801$	−2853.40	−0.125867
$802$	0	0
$803$	6332.45	0.278291
$804$	0	0
$805$	2107.95	0.0922925
$806$	0	0
$807$	17081.0	0.745081
$808$	0	0
$809$	−21745.5	−0.945033	−0.472516	−	0.881322i	$-0.656654\pi$
$809$	−21745.5	−0.945033	−0.472516	+	0.881322i	$0.656654\pi$
$810$	0	0
$811$	16487.2	0.713863	0.356932	−	0.934131i	$-0.383823\pi$
$811$	16487.2	0.713863	0.356932	+	0.934131i	$0.383823\pi$
$812$	0	0
$813$	−1489.03	−0.0642346
$814$	0	0
$815$	10516.3	0.451988
$816$	0	0
$817$	10192.0	0.436443
$818$	0	0
$819$	611.955	0.0261092
$820$	0	0
$821$	12274.1	0.521767	0.260883	−	0.965370i	$-0.415986\pi$
$821$	12274.1	0.521767	0.260883	+	0.965370i	$0.415986\pi$
$822$	0	0
$823$	25913.7	1.09756	0.548782	−	0.835966i	$-0.315092\pi$
$823$	25913.7	1.09756	0.548782	+	0.835966i	$0.315092\pi$
$824$	0	0
$825$	−9158.68	−0.386502
$826$	0	0
$827$	−19617.2	−0.824855	−0.412428	−	0.910990i	$-0.635319\pi$
$827$	−19617.2	−0.824855	−0.412428	+	0.910990i	$0.635319\pi$
$828$	0	0
$829$	−40662.5	−1.70358	−0.851790	−	0.523883i	$-0.824483\pi$
$829$	−40662.5	−1.70358	−0.851790	+	0.523883i	$0.824483\pi$
$830$	0	0
$831$	8497.25	0.354713
$832$	0	0
$833$	−1282.62	−0.0533493
$834$	0	0
$835$	2787.99	0.115548
$836$	0	0
$837$	2121.98	0.0876300
$838$	0	0
$839$	16709.7	0.687584	0.343792	−	0.939046i	$-0.388289\pi$
$839$	16709.7	0.687584	0.343792	+	0.939046i	$0.388289\pi$
$840$	0	0
$841$	42989.1	1.76264
$842$	0	0
$843$	−16086.3	−0.657228
$844$	0	0
$845$	469.199	0.0191017
$846$	0	0
$847$	3418.24	0.138668
$848$	0	0
$849$	9990.96	0.403874
$850$	0	0
$851$	−14491.5	−0.583740
$852$	0	0
$853$	−18833.6	−0.755978	−0.377989	−	0.925810i	$-0.623384\pi$
$853$	−18833.6	−0.755978	−0.377989	+	0.925810i	$0.623384\pi$
$854$	0	0
$855$	−1849.01	−0.0739589
$856$	0	0
$857$	−26784.6	−1.06761	−0.533807	−	0.845607i	$-0.679239\pi$
$857$	−26784.6	−1.06761	−0.533807	+	0.845607i	$0.679239\pi$
$858$	0	0
$859$	−10246.9	−0.407007	−0.203504	−	0.979074i	$-0.565233\pi$
$859$	−10246.9	−0.407007	−0.203504	+	0.979074i	$0.565233\pi$
$860$	0	0
$861$	−5434.15	−0.215093
$862$	0	0
$863$	−5000.15	−0.197227	−0.0986136	−	0.995126i	$-0.531441\pi$
$863$	−5000.15	−0.197227	−0.0986136	+	0.995126i	$0.531441\pi$
$864$	0	0
$865$	33.6096	0.00132111
$866$	0	0
$867$	−14689.5	−0.575410
$868$	0	0
$869$	−17445.3	−0.681002
$870$	0	0
$871$	3803.59	0.147967
$872$	0	0
$873$	7976.95	0.309254
$874$	0	0
$875$	3518.38	0.135935
$876$	0	0
$877$	16648.8	0.641039	0.320519	−	0.947242i	$-0.396143\pi$
$877$	16648.8	0.641039	0.320519	+	0.947242i	$0.396143\pi$
$878$	0	0
$879$	−2819.82	−0.108203
$880$	0	0
$881$	−16304.0	−0.623491	−0.311746	−	0.950166i	$-0.600914\pi$
$881$	−16304.0	−0.623491	−0.311746	+	0.950166i	$0.600914\pi$
$882$	0	0
$883$	−17857.7	−0.680589	−0.340295	−	0.940319i	$-0.610527\pi$
$883$	−17857.7	−0.680589	−0.340295	+	0.940319i	$0.610527\pi$
$884$	0	0
$885$	5629.75	0.213833
$886$	0	0
$887$	46797.7	1.77149	0.885745	−	0.464172i	$-0.153648\pi$
$887$	46797.7	1.77149	0.885745	+	0.464172i	$0.153648\pi$
$888$	0	0
$889$	1447.39	0.0546050
$890$	0	0
$891$	2108.28	0.0792705
$892$	0	0
$893$	−4199.90	−0.157384
$894$	0	0
$895$	−732.191	−0.0273458
$896$	0	0
$897$	5661.36	0.210733
$898$	0	0
$899$	−20400.3	−0.756828
$900$	0	0
$901$	704.536	0.0260505
$902$	0	0
$903$	2161.17	0.0796448
$904$	0	0
$905$	−5391.47	−0.198032
$906$	0	0
$907$	−52238.0	−1.91239	−0.956193	−	0.292738i	$-0.905434\pi$
$907$	−52238.0	−1.91239	−0.956193	+	0.292738i	$0.905434\pi$
$908$	0	0
$909$	1867.75	0.0681513
$910$	0	0
$911$	−31047.7	−1.12915	−0.564576	−	0.825381i	$-0.690960\pi$
$911$	−31047.7	−1.12915	−0.564576	+	0.825381i	$0.690960\pi$
$912$	0	0
$913$	−20475.4	−0.742208
$914$	0	0
$915$	−6648.74	−0.240219
$916$	0	0
$917$	−11996.0	−0.431997
$918$	0	0
$919$	13876.1	0.498075	0.249038	−	0.968494i	$-0.419886\pi$
$919$	13876.1	0.498075	0.249038	+	0.968494i	$0.419886\pi$
$920$	0	0
$921$	1099.21	0.0393271
$922$	0	0
$923$	4335.03	0.154593
$924$	0	0
$925$	−11709.2	−0.416211
$926$	0	0
$927$	15649.8	0.554483
$928$	0	0
$929$	−21424.9	−0.756650	−0.378325	−	0.925673i	$-0.623500\pi$
$929$	−21424.9	−0.756650	−0.378325	+	0.925673i	$0.623500\pi$
$930$	0	0
$931$	23357.3	0.822239
$932$	0	0
$933$	−9349.73	−0.328078
$934$	0	0
$935$	293.639	0.0102706
$936$	0	0
$937$	−4812.13	−0.167775	−0.0838876	−	0.996475i	$-0.526734\pi$
$937$	−4812.13	−0.167775	−0.0838876	+	0.996475i	$0.526734\pi$
$938$	0	0
$939$	7081.06	0.246093
$940$	0	0
$941$	−48232.7	−1.67093	−0.835464	−	0.549546i	$-0.814801\pi$
$941$	−48232.7	−1.67093	−0.835464	+	0.549546i	$0.814801\pi$
$942$	0	0
$943$	−50272.8	−1.73606
$944$	0	0
$945$	−392.074	−0.0134965
$946$	0	0
$947$	−33990.6	−1.16636	−0.583181	−	0.812342i	$-0.698192\pi$
$947$	−33990.6	−1.16636	−0.583181	+	0.812342i	$0.698192\pi$
$948$	0	0
$949$	−3162.80	−0.108186
$950$	0	0
$951$	−3838.69	−0.130892
$952$	0	0
$953$	−12836.1	−0.436309	−0.218154	−	0.975914i	$-0.570004\pi$
$953$	−12836.1	−0.436309	−0.218154	+	0.975914i	$0.570004\pi$
$954$	0	0
$955$	5344.72	0.181101
$956$	0	0
$957$	−20268.6	−0.684630
$958$	0	0
$959$	13806.9	0.464909
$960$	0	0
$961$	−23614.3	−0.792667
$962$	0	0
$963$	−11240.2	−0.376126
$964$	0	0
$965$	−9211.32	−0.307278
$966$	0	0
$967$	−5141.56	−0.170984	−0.0854920	−	0.996339i	$-0.527246\pi$
$967$	−5141.56	−0.170984	−0.0854920	+	0.996339i	$0.527246\pi$
$968$	0	0
$969$	−902.085	−0.0299063
$970$	0	0
$971$	−5891.05	−0.194699	−0.0973496	−	0.995250i	$-0.531036\pi$
$971$	−5891.05	−0.194699	−0.0973496	+	0.995250i	$0.531036\pi$
$972$	0	0
$973$	−7173.36	−0.236349
$974$	0	0
$975$	4574.39	0.150254
$976$	0	0
$977$	−10105.7	−0.330922	−0.165461	−	0.986216i	$-0.552911\pi$
$977$	−10105.7	−0.330922	−0.165461	+	0.986216i	$0.552911\pi$
$978$	0	0
$979$	−8252.07	−0.269395
$980$	0	0
$981$	−15976.6	−0.519974
$982$	0	0
$983$	−7957.89	−0.258207	−0.129103	−	0.991631i	$-0.541210\pi$
$983$	−7957.89	−0.258207	−0.129103	+	0.991631i	$0.541210\pi$
$984$	0	0
$985$	−11889.5	−0.384601
$986$	0	0
$987$	−890.568	−0.0287205
$988$	0	0
$989$	19993.6	0.642830
$990$	0	0
$991$	−16996.7	−0.544822	−0.272411	−	0.962181i	$-0.587821\pi$
$991$	−16996.7	−0.544822	−0.272411	+	0.962181i	$0.587821\pi$
$992$	0	0
$993$	32089.2	1.02550
$994$	0	0
$995$	−3384.63	−0.107839
$996$	0	0
$997$	46632.0	1.48130	0.740648	−	0.671894i	$-0.234519\pi$
$997$	46632.0	1.48130	0.740648	+	0.671894i	$0.234519\pi$
$998$	0	0
$999$	2695.39	0.0853636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.4.a.l.1.4	yes	5
4.3	odd	2		1248.4.a.g.1.4	✓	5
8.3	odd	2		2496.4.a.cf.1.2		5
8.5	even	2		2496.4.a.ca.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.4.a.g.1.4	✓	5	4.3	odd	2
1248.4.a.l.1.4	yes	5	1.1	even	1	trivial
2496.4.a.ca.1.2		5	8.5	even	2
2496.4.a.cf.1.2		5	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1248.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +2.77633 q^{5} -5.23038 q^{7} +9.00000 q^{9} +26.0282 q^{11} -13.0000 q^{13} +8.32898 q^{15} +4.06350 q^{17} -73.9991 q^{19} -15.6911 q^{21} -145.163 q^{23} -117.292 q^{25} +27.0000 q^{27} -259.573 q^{29} +78.5918 q^{31} +78.0845 q^{33} -14.5213 q^{35} +99.8291 q^{37} -39.0000 q^{39} +346.320 q^{41} -137.732 q^{43} +24.9869 q^{45} +56.7561 q^{47} -315.643 q^{49} +12.1905 q^{51} +173.382 q^{53} +72.2627 q^{55} -221.997 q^{57} +675.923 q^{59} -798.265 q^{61} -47.0734 q^{63} -36.0923 q^{65} -292.584 q^{67} -435.489 q^{69} -333.464 q^{71} +243.292 q^{73} -351.876 q^{75} -136.137 q^{77} -670.246 q^{79} +81.0000 q^{81} -786.662 q^{83} +11.2816 q^{85} -778.719 q^{87} -317.044 q^{89} +67.9950 q^{91} +235.775 q^{93} -205.446 q^{95} +886.328 q^{97} +234.253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} - 10 q^{5} - 14 q^{7} + 45 q^{9} - 22 q^{11} - 65 q^{13} - 30 q^{15} - 34 q^{17} + 90 q^{19} - 42 q^{21} - 96 q^{23} + 107 q^{25} + 135 q^{27} - 54 q^{29} - 378 q^{31} - 66 q^{33} - 84 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 5 * q + 15 * q^3 - 10 * q^5 - 14 * q^7 + 45 * q^9 - 22 * q^11 - 65 * q^13 - 30 * q^15 - 34 * q^17 + 90 * q^19 - 42 * q^21 - 96 * q^23 + 107 * q^25 + 135 * q^27 - 54 * q^29 - 378 * q^31 - 66 * q^33 - 84 * q^35 - 318 * q^37 - 195 * q^39 - 458 * q^41 - 104 * q^43 - 90 * q^45 - 474 * q^47 + 25 * q^49 - 102 * q^51 - 450 * q^53 - 1116 * q^55 + 270 * q^57 - 822 * q^59 - 546 * q^61 - 126 * q^63 + 130 * q^65 + 130 * q^67 - 288 * q^69 - 1646 * q^71 - 166 * q^73 + 321 * q^75 - 900 * q^77 - 196 * q^79 + 405 * q^81 + 118 * q^83 - 3340 * q^85 - 162 * q^87 - 186 * q^89 + 182 * q^91 - 1134 * q^93 - 564 * q^95 - 2582 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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