LMFDB - Embedded newform 704.4.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.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:	$41.5373446440$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.11109.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x - 6 $ x^3 - x^2 - 15*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.56976$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82655 q^{3} -17.4525 q^{5} +4.62596 q^{7} -19.0106 q^{9} +11.0000 q^{11} -85.1298 q^{13} -49.3304 q^{15} -6.08908 q^{17} +52.0891 q^{19} +13.0755 q^{21} +183.964 q^{23} +179.590 q^{25} -130.051 q^{27} -140.498 q^{29} +250.685 q^{31} +31.0920 q^{33} -80.7345 q^{35} +203.522 q^{37} -240.624 q^{39} -22.3275 q^{41} -117.409 q^{43} +331.783 q^{45} +275.744 q^{47} -321.601 q^{49} -17.2111 q^{51} -26.6548 q^{53} -191.978 q^{55} +147.232 q^{57} +515.922 q^{59} +693.071 q^{61} -87.9423 q^{63} +1485.73 q^{65} +341.545 q^{67} +519.983 q^{69} -831.425 q^{71} +251.591 q^{73} +507.620 q^{75} +50.8855 q^{77} +917.036 q^{79} +145.690 q^{81} -456.749 q^{83} +106.270 q^{85} -397.124 q^{87} +91.4974 q^{89} -393.807 q^{91} +708.574 q^{93} -909.085 q^{95} +146.569 q^{97} -209.117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 8 q^{5} - 24 q^{7} + 89 q^{9} + 33 q^{11} - 66 q^{13} + 10 q^{15} + 210 q^{17} - 72 q^{19} - 200 q^{21} + 50 q^{23} - q^{25} - 286 q^{27} + 50 q^{29} + 298 q^{31} + 22 q^{33} - 304 q^{35}+ \cdots + 979 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 8 * q^5 - 24 * q^7 + 89 * q^9 + 33 * q^11 - 66 * q^13 + 10 * q^15 + 210 * q^17 - 72 * q^19 - 200 * q^21 + 50 * q^23 - q^25 - 286 * q^27 + 50 * q^29 + 298 * q^31 + 22 * q^33 - 304 * q^35 + 4 * q^37 + 876 * q^39 + 254 * q^41 - 112 * q^43 + 790 * q^45 - 40 * q^47 - 285 * q^49 + 48 * q^51 + 550 * q^53 - 88 * q^55 + 44 * q^57 + 1630 * q^59 + 906 * q^61 - 1448 * q^63 + 2016 * q^65 - 482 * q^67 + 1398 * q^69 - 242 * q^71 + 998 * q^73 + 1180 * q^75 - 264 * q^77 - 324 * q^79 + 1715 * q^81 + 1360 * q^83 + 1188 * q^85 - 2880 * q^87 + 1200 * q^89 - 2232 * q^91 + 4998 * q^93 - 1556 * q^95 + 180 * q^97 + 979 * 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
$3$	2.82655	0.543970	0.271985	−	0.962302i	$-0.412320\pi$
$3$	2.82655	0.543970	0.271985	+	0.962302i	$0.412320\pi$
$4$	0	0
$5$	−17.4525	−1.56100	−0.780500	−	0.625156i	$-0.785035\pi$
$5$	−17.4525	−1.56100	−0.780500	+	0.625156i	$0.785035\pi$
$6$	0	0
$7$	4.62596	0.249778	0.124889	−	0.992171i	$-0.460142\pi$
$7$	4.62596	0.249778	0.124889	+	0.992171i	$0.460142\pi$
$8$	0	0
$9$	−19.0106	−0.704097
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−85.1298	−1.81621	−0.908106	−	0.418741i	$-0.862472\pi$
$13$	−85.1298	−1.81621	−0.908106	+	0.418741i	$0.862472\pi$
$14$	0	0
$15$	−49.3304	−0.849137
$16$	0	0
$17$	−6.08908	−0.0868717	−0.0434358	−	0.999056i	$-0.513830\pi$
$17$	−6.08908	−0.0868717	−0.0434358	+	0.999056i	$0.513830\pi$
$18$	0	0
$19$	52.0891	0.628950	0.314475	−	0.949266i	$-0.398172\pi$
$19$	52.0891	0.628950	0.314475	+	0.949266i	$0.398172\pi$
$20$	0	0
$21$	13.0755	0.135872
$22$	0	0
$23$	183.964	1.66779	0.833894	−	0.551924i	$-0.186106\pi$
$23$	183.964	1.66779	0.833894	+	0.551924i	$0.186106\pi$
$24$	0	0
$25$	179.590	1.43672
$26$	0	0
$27$	−130.051	−0.926977
$28$	0	0
$29$	−140.498	−0.899649	−0.449824	−	0.893117i	$-0.648513\pi$
$29$	−140.498	−0.899649	−0.449824	+	0.893117i	$0.648513\pi$
$30$	0	0
$31$	250.685	1.45240	0.726199	−	0.687484i	$-0.241285\pi$
$31$	250.685	1.45240	0.726199	+	0.687484i	$0.241285\pi$
$32$	0	0
$33$	31.0920	0.164013
$34$	0	0
$35$	−80.7345	−0.389904
$36$	0	0
$37$	203.522	0.904293	0.452146	−	0.891944i	$-0.350658\pi$
$37$	203.522	0.904293	0.452146	+	0.891944i	$0.350658\pi$
$38$	0	0
$39$	−240.624	−0.987964
$40$	0	0
$41$	−22.3275	−0.0850479	−0.0425239	−	0.999095i	$-0.513540\pi$
$41$	−22.3275	−0.0850479	−0.0425239	+	0.999095i	$0.513540\pi$
$42$	0	0
$43$	−117.409	−0.416388	−0.208194	−	0.978088i	$-0.566759\pi$
$43$	−117.409	−0.416388	−0.208194	+	0.978088i	$0.566759\pi$
$44$	0	0
$45$	331.783	1.09910
$46$	0	0
$47$	275.744	0.855774	0.427887	−	0.903832i	$-0.359258\pi$
$47$	275.744	0.855774	0.427887	+	0.903832i	$0.359258\pi$
$48$	0	0
$49$	−321.601	−0.937611
$50$	0	0
$51$	−17.2111	−0.0472556
$52$	0	0
$53$	−26.6548	−0.0690815	−0.0345408	−	0.999403i	$-0.510997\pi$
$53$	−26.6548	−0.0690815	−0.0345408	+	0.999403i	$0.510997\pi$
$54$	0	0
$55$	−191.978	−0.470659
$56$	0	0
$57$	147.232	0.342130
$58$	0	0
$59$	515.922	1.13843	0.569214	−	0.822189i	$-0.307248\pi$
$59$	515.922	1.13843	0.569214	+	0.822189i	$0.307248\pi$
$60$	0	0
$61$	693.071	1.45473	0.727366	−	0.686249i	$-0.240744\pi$
$61$	693.071	1.45473	0.727366	+	0.686249i	$0.240744\pi$
$62$	0	0
$63$	−87.9423	−0.175868
$64$	0	0
$65$	1485.73	2.83511
$66$	0	0
$67$	341.545	0.622782	0.311391	−	0.950282i	$-0.399205\pi$
$67$	341.545	0.622782	0.311391	+	0.950282i	$0.399205\pi$
$68$	0	0
$69$	519.983	0.907227
$70$	0	0
$71$	−831.425	−1.38975	−0.694873	−	0.719133i	$-0.744539\pi$
$71$	−831.425	−1.38975	−0.694873	+	0.719133i	$0.744539\pi$
$72$	0	0
$73$	251.591	0.403377	0.201688	−	0.979450i	$-0.435357\pi$
$73$	251.591	0.403377	0.201688	+	0.979450i	$0.435357\pi$
$74$	0	0
$75$	507.620	0.781532
$76$	0	0
$77$	50.8855	0.0753109
$78$	0	0
$79$	917.036	1.30601	0.653004	−	0.757355i	$-0.273509\pi$
$79$	917.036	1.30601	0.653004	+	0.757355i	$0.273509\pi$
$80$	0	0
$81$	145.690	0.199849
$82$	0	0
$83$	−456.749	−0.604033	−0.302017	−	0.953303i	$-0.597660\pi$
$83$	−456.749	−0.604033	−0.302017	+	0.953303i	$0.597660\pi$
$84$	0	0
$85$	106.270	0.135607
$86$	0	0
$87$	−397.124	−0.489382
$88$	0	0
$89$	91.4974	0.108974	0.0544871	−	0.998514i	$-0.482648\pi$
$89$	91.4974	0.108974	0.0544871	+	0.998514i	$0.482648\pi$
$90$	0	0
$91$	−393.807	−0.453650
$92$	0	0
$93$	708.574	0.790061
$94$	0	0
$95$	−909.085	−0.981791
$96$	0	0
$97$	146.569	0.153421	0.0767104	−	0.997053i	$-0.475558\pi$
$97$	146.569	0.153421	0.0767104	+	0.997053i	$0.475558\pi$
$98$	0	0
$99$	−209.117	−0.212293
$100$	0	0
$101$	1163.16	1.14593	0.572963	−	0.819581i	$-0.305794\pi$
$101$	1163.16	1.14593	0.572963	+	0.819581i	$0.305794\pi$
$102$	0	0
$103$	−1703.51	−1.62963	−0.814815	−	0.579721i	$-0.803161\pi$
$103$	−1703.51	−1.62963	−0.814815	+	0.579721i	$0.803161\pi$
$104$	0	0
$105$	−228.200	−0.212096
$106$	0	0
$107$	1997.18	1.80444	0.902218	−	0.431281i	$-0.141938\pi$
$107$	1997.18	1.80444	0.902218	+	0.431281i	$0.141938\pi$
$108$	0	0
$109$	−1350.72	−1.18694	−0.593468	−	0.804858i	$-0.702242\pi$
$109$	−1350.72	−1.18694	−0.593468	+	0.804858i	$0.702242\pi$
$110$	0	0
$111$	575.265	0.491908
$112$	0	0
$113$	−451.353	−0.375750	−0.187875	−	0.982193i	$-0.560160\pi$
$113$	−451.353	−0.375750	−0.187875	+	0.982193i	$0.560160\pi$
$114$	0	0
$115$	−3210.63	−2.60342
$116$	0	0
$117$	1618.37	1.27879
$118$	0	0
$119$	−28.1678	−0.0216986
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−63.1097	−0.0462635
$124$	0	0
$125$	−952.732	−0.681719
$126$	0	0
$127$	2737.44	1.91267	0.956333	−	0.292279i	$-0.0944135\pi$
$127$	2737.44	1.91267	0.956333	+	0.292279i	$0.0944135\pi$
$128$	0	0
$129$	−331.862	−0.226502
$130$	0	0
$131$	978.943	0.652906	0.326453	−	0.945213i	$-0.394147\pi$
$131$	978.943	0.652906	0.326453	+	0.945213i	$0.394147\pi$
$132$	0	0
$133$	240.962	0.157098
$134$	0	0
$135$	2269.72	1.44701
$136$	0	0
$137$	−1104.14	−0.688565	−0.344283	−	0.938866i	$-0.611878\pi$
$137$	−1104.14	−0.688565	−0.344283	+	0.938866i	$0.611878\pi$
$138$	0	0
$139$	−129.542	−0.0790474	−0.0395237	−	0.999219i	$-0.512584\pi$
$139$	−129.542	−0.0790474	−0.0395237	+	0.999219i	$0.512584\pi$
$140$	0	0
$141$	779.404	0.465515
$142$	0	0
$143$	−936.428	−0.547608
$144$	0	0
$145$	2452.04	1.40435
$146$	0	0
$147$	−909.020	−0.510032
$148$	0	0
$149$	−1891.15	−1.03979	−0.519897	−	0.854229i	$-0.674030\pi$
$149$	−1891.15	−1.03979	−0.519897	+	0.854229i	$0.674030\pi$
$150$	0	0
$151$	−7.13044	−0.00384283	−0.00192141	−	0.999998i	$-0.500612\pi$
$151$	−7.13044	−0.00384283	−0.00192141	+	0.999998i	$0.500612\pi$
$152$	0	0
$153$	115.757	0.0611661
$154$	0	0
$155$	−4375.08	−2.26719
$156$	0	0
$157$	16.2914	0.00828151	0.00414075	−	0.999991i	$-0.498682\pi$
$157$	16.2914	0.00828151	0.00414075	+	0.999991i	$0.498682\pi$
$158$	0	0
$159$	−75.3412	−0.0375783
$160$	0	0
$161$	851.009	0.416577
$162$	0	0
$163$	2947.53	1.41637	0.708186	−	0.706026i	$-0.249514\pi$
$163$	2947.53	1.41637	0.708186	+	0.706026i	$0.249514\pi$
$164$	0	0
$165$	−542.634	−0.256024
$166$	0	0
$167$	−3567.73	−1.65317	−0.826585	−	0.562813i	$-0.809719\pi$
$167$	−3567.73	−1.65317	−0.826585	+	0.562813i	$0.809719\pi$
$168$	0	0
$169$	5050.08	2.29863
$170$	0	0
$171$	−990.246	−0.442842
$172$	0	0
$173$	647.739	0.284663	0.142331	−	0.989819i	$-0.454540\pi$
$173$	647.739	0.284663	0.142331	+	0.989819i	$0.454540\pi$
$174$	0	0
$175$	830.775	0.358861
$176$	0	0
$177$	1458.28	0.619270
$178$	0	0
$179$	3061.71	1.27845	0.639225	−	0.769019i	$-0.279255\pi$
$179$	3061.71	1.27845	0.639225	+	0.769019i	$0.279255\pi$
$180$	0	0
$181$	4403.36	1.80828	0.904141	−	0.427233i	$-0.140512\pi$
$181$	4403.36	1.80828	0.904141	+	0.427233i	$0.140512\pi$
$182$	0	0
$183$	1959.00	0.791330
$184$	0	0
$185$	−3551.97	−1.41160
$186$	0	0
$187$	−66.9799	−0.0261928
$188$	0	0
$189$	−601.612	−0.231539
$190$	0	0
$191$	−904.321	−0.342588	−0.171294	−	0.985220i	$-0.554795\pi$
$191$	−904.321	−0.342588	−0.171294	+	0.985220i	$0.554795\pi$
$192$	0	0
$193$	1344.72	0.501530	0.250765	−	0.968048i	$-0.419318\pi$
$193$	1344.72	0.501530	0.250765	+	0.968048i	$0.419318\pi$
$194$	0	0
$195$	4199.48	1.54221
$196$	0	0
$197$	−5213.66	−1.88557	−0.942787	−	0.333395i	$-0.891806\pi$
$197$	−5213.66	−1.88557	−0.942787	+	0.333395i	$0.891806\pi$
$198$	0	0
$199$	3145.18	1.12038	0.560192	−	0.828363i	$-0.310728\pi$
$199$	3145.18	1.12038	0.560192	+	0.828363i	$0.310728\pi$
$200$	0	0
$201$	965.394	0.338774
$202$	0	0
$203$	−649.937	−0.224713
$204$	0	0
$205$	389.670	0.132760
$206$	0	0
$207$	−3497.27	−1.17429
$208$	0	0
$209$	572.980	0.189636
$210$	0	0
$211$	−3878.22	−1.26535	−0.632673	−	0.774419i	$-0.718042\pi$
$211$	−3878.22	−1.26535	−0.632673	+	0.774419i	$0.718042\pi$
$212$	0	0
$213$	−2350.06	−0.755980
$214$	0	0
$215$	2049.08	0.649981
$216$	0	0
$217$	1159.66	0.362777
$218$	0	0
$219$	711.135	0.219425
$220$	0	0
$221$	518.362	0.157777
$222$	0	0
$223$	367.736	0.110428	0.0552140	−	0.998475i	$-0.482416\pi$
$223$	367.736	0.110428	0.0552140	+	0.998475i	$0.482416\pi$
$224$	0	0
$225$	−3414.12	−1.01159
$226$	0	0
$227$	6061.59	1.77234	0.886171	−	0.463358i	$-0.153356\pi$
$227$	6061.59	1.77234	0.886171	+	0.463358i	$0.153356\pi$
$228$	0	0
$229$	4244.17	1.22473	0.612365	−	0.790576i	$-0.290218\pi$
$229$	4244.17	1.22473	0.612365	+	0.790576i	$0.290218\pi$
$230$	0	0
$231$	143.830	0.0409669
$232$	0	0
$233$	−5000.36	−1.40594	−0.702971	−	0.711218i	$-0.748144\pi$
$233$	−5000.36	−1.40594	−0.702971	+	0.711218i	$0.748144\pi$
$234$	0	0
$235$	−4812.42	−1.33586
$236$	0	0
$237$	2592.05	0.710429
$238$	0	0
$239$	733.915	0.198632	0.0993160	−	0.995056i	$-0.468335\pi$
$239$	733.915	0.198632	0.0993160	+	0.995056i	$0.468335\pi$
$240$	0	0
$241$	424.106	0.113357	0.0566785	−	0.998392i	$-0.481949\pi$
$241$	424.106	0.113357	0.0566785	+	0.998392i	$0.481949\pi$
$242$	0	0
$243$	3923.19	1.03569
$244$	0	0
$245$	5612.74	1.46361
$246$	0	0
$247$	−4434.33	−1.14231
$248$	0	0
$249$	−1291.02	−0.328576
$250$	0	0
$251$	−950.127	−0.238930	−0.119465	−	0.992838i	$-0.538118\pi$
$251$	−950.127	−0.238930	−0.119465	+	0.992838i	$0.538118\pi$
$252$	0	0
$253$	2023.60	0.502857
$254$	0	0
$255$	300.377	0.0737659
$256$	0	0
$257$	−4822.67	−1.17054	−0.585272	−	0.810837i	$-0.699012\pi$
$257$	−4822.67	−1.17054	−0.585272	+	0.810837i	$0.699012\pi$
$258$	0	0
$259$	941.485	0.225873
$260$	0	0
$261$	2670.95	0.633440
$262$	0	0
$263$	1628.73	0.381871	0.190935	−	0.981603i	$-0.438848\pi$
$263$	1628.73	0.381871	0.190935	+	0.981603i	$0.438848\pi$
$264$	0	0
$265$	465.193	0.107836
$266$	0	0
$267$	258.622	0.0592787
$268$	0	0
$269$	−8432.58	−1.91131	−0.955657	−	0.294483i	$-0.904852\pi$
$269$	−8432.58	−1.91131	−0.955657	+	0.294483i	$0.904852\pi$
$270$	0	0
$271$	5763.53	1.29192	0.645958	−	0.763373i	$-0.276458\pi$
$271$	5763.53	1.29192	0.645958	+	0.763373i	$0.276458\pi$
$272$	0	0
$273$	−1113.11	−0.246772
$274$	0	0
$275$	1975.49	0.433187
$276$	0	0
$277$	−5869.73	−1.27320	−0.636602	−	0.771192i	$-0.719661\pi$
$277$	−5869.73	−1.27320	−0.636602	+	0.771192i	$0.719661\pi$
$278$	0	0
$279$	−4765.68	−1.02263
$280$	0	0
$281$	−4339.50	−0.921256	−0.460628	−	0.887593i	$-0.652376\pi$
$281$	−4339.50	−0.921256	−0.460628	+	0.887593i	$0.652376\pi$
$282$	0	0
$283$	1177.65	0.247364	0.123682	−	0.992322i	$-0.460530\pi$
$283$	1177.65	0.247364	0.123682	+	0.992322i	$0.460530\pi$
$284$	0	0
$285$	−2569.57	−0.534065
$286$	0	0
$287$	−103.286	−0.0212431
$288$	0	0
$289$	−4875.92	−0.992453
$290$	0	0
$291$	414.284	0.0834562
$292$	0	0
$293$	−6012.35	−1.19879	−0.599394	−	0.800454i	$-0.704592\pi$
$293$	−6012.35	−1.19879	−0.599394	+	0.800454i	$0.704592\pi$
$294$	0	0
$295$	−9004.12	−1.77709
$296$	0	0
$297$	−1430.56	−0.279494
$298$	0	0
$299$	−15660.8	−3.02906
$300$	0	0
$301$	−543.128	−0.104005
$302$	0	0
$303$	3287.72	0.623349
$304$	0	0
$305$	−12095.8	−2.27084
$306$	0	0
$307$	−203.484	−0.0378287	−0.0189144	−	0.999821i	$-0.506021\pi$
$307$	−203.484	−0.0378287	−0.0189144	+	0.999821i	$0.506021\pi$
$308$	0	0
$309$	−4815.06	−0.886469
$310$	0	0
$311$	4801.67	0.875492	0.437746	−	0.899099i	$-0.355777\pi$
$311$	4801.67	0.875492	0.437746	+	0.899099i	$0.355777\pi$
$312$	0	0
$313$	731.096	0.132026	0.0660128	−	0.997819i	$-0.478972\pi$
$313$	731.096	0.132026	0.0660128	+	0.997819i	$0.478972\pi$
$314$	0	0
$315$	1534.81	0.274530
$316$	0	0
$317$	1344.21	0.238165	0.119083	−	0.992884i	$-0.462005\pi$
$317$	1344.21	0.238165	0.119083	+	0.992884i	$0.462005\pi$
$318$	0	0
$319$	−1545.48	−0.271254
$320$	0	0
$321$	5645.13	0.981558
$322$	0	0
$323$	−317.175	−0.0546380
$324$	0	0
$325$	−15288.5	−2.60939
$326$	0	0
$327$	−3817.89	−0.645657
$328$	0	0
$329$	1275.58	0.213754
$330$	0	0
$331$	−1953.29	−0.324359	−0.162179	−	0.986761i	$-0.551852\pi$
$331$	−1953.29	−0.324359	−0.162179	+	0.986761i	$0.551852\pi$
$332$	0	0
$333$	−3869.08	−0.636710
$334$	0	0
$335$	−5960.82	−0.972162
$336$	0	0
$337$	6114.96	0.988436	0.494218	−	0.869338i	$-0.335454\pi$
$337$	6114.96	0.988436	0.494218	+	0.869338i	$0.335454\pi$
$338$	0	0
$339$	−1275.77	−0.204397
$340$	0	0
$341$	2757.53	0.437915
$342$	0	0
$343$	−3074.41	−0.483973
$344$	0	0
$345$	−9075.01	−1.41618
$346$	0	0
$347$	5262.87	0.814195	0.407098	−	0.913385i	$-0.366541\pi$
$347$	5262.87	0.814195	0.407098	+	0.913385i	$0.366541\pi$
$348$	0	0
$349$	−6426.98	−0.985755	−0.492877	−	0.870099i	$-0.664055\pi$
$349$	−6426.98	−0.985755	−0.492877	+	0.870099i	$0.664055\pi$
$350$	0	0
$351$	11071.2	1.68359
$352$	0	0
$353$	4196.42	0.632728	0.316364	−	0.948638i	$-0.397538\pi$
$353$	4196.42	0.632728	0.316364	+	0.948638i	$0.397538\pi$
$354$	0	0
$355$	14510.4	2.16939
$356$	0	0
$357$	−79.6177	−0.0118034
$358$	0	0
$359$	−4010.77	−0.589639	−0.294820	−	0.955553i	$-0.595260\pi$
$359$	−4010.77	−0.589639	−0.294820	+	0.955553i	$0.595260\pi$
$360$	0	0
$361$	−4145.73	−0.604422
$362$	0	0
$363$	342.013	0.0494518
$364$	0	0
$365$	−4390.90	−0.629671
$366$	0	0
$367$	3543.59	0.504016	0.252008	−	0.967725i	$-0.418909\pi$
$367$	3543.59	0.504016	0.252008	+	0.967725i	$0.418909\pi$
$368$	0	0
$369$	424.459	0.0598820
$370$	0	0
$371$	−123.304	−0.0172551
$372$	0	0
$373$	3804.50	0.528122	0.264061	−	0.964506i	$-0.414938\pi$
$373$	3804.50	0.528122	0.264061	+	0.964506i	$0.414938\pi$
$374$	0	0
$375$	−2692.94	−0.370835
$376$	0	0
$377$	11960.6	1.63395
$378$	0	0
$379$	328.702	0.0445496	0.0222748	−	0.999752i	$-0.492909\pi$
$379$	328.702	0.0445496	0.0222748	+	0.999752i	$0.492909\pi$
$380$	0	0
$381$	7737.51	1.04043
$382$	0	0
$383$	6321.67	0.843400	0.421700	−	0.906735i	$-0.361434\pi$
$383$	6321.67	0.843400	0.421700	+	0.906735i	$0.361434\pi$
$384$	0	0
$385$	−888.080	−0.117560
$386$	0	0
$387$	2232.01	0.293177
$388$	0	0
$389$	9498.91	1.23808	0.619041	−	0.785359i	$-0.287521\pi$
$389$	9498.91	1.23808	0.619041	+	0.785359i	$0.287521\pi$
$390$	0	0
$391$	−1120.17	−0.144884
$392$	0	0
$393$	2767.03	0.355161
$394$	0	0
$395$	−16004.6	−2.03868
$396$	0	0
$397$	5007.59	0.633057	0.316529	−	0.948583i	$-0.397483\pi$
$397$	5007.59	0.633057	0.316529	+	0.948583i	$0.397483\pi$
$398$	0	0
$399$	681.090	0.0854566
$400$	0	0
$401$	−5614.60	−0.699202	−0.349601	−	0.936899i	$-0.613683\pi$
$401$	−5614.60	−0.699202	−0.349601	+	0.936899i	$0.613683\pi$
$402$	0	0
$403$	−21340.8	−2.63786
$404$	0	0
$405$	−2542.66	−0.311965
$406$	0	0
$407$	2238.74	0.272655
$408$	0	0
$409$	10346.6	1.25087	0.625437	−	0.780274i	$-0.284921\pi$
$409$	10346.6	1.25087	0.625437	+	0.780274i	$0.284921\pi$
$410$	0	0
$411$	−3120.92	−0.374559
$412$	0	0
$413$	2386.63	0.284354
$414$	0	0
$415$	7971.42	0.942895
$416$	0	0
$417$	−366.156	−0.0429994
$418$	0	0
$419$	−2420.02	−0.282162	−0.141081	−	0.989998i	$-0.545058\pi$
$419$	−2420.02	−0.282162	−0.141081	+	0.989998i	$0.545058\pi$
$420$	0	0
$421$	−4934.72	−0.571268	−0.285634	−	0.958339i	$-0.592204\pi$
$421$	−4934.72	−0.571268	−0.285634	+	0.958339i	$0.592204\pi$
$422$	0	0
$423$	−5242.06	−0.602548
$424$	0	0
$425$	−1093.54	−0.124810
$426$	0	0
$427$	3206.12	0.363360
$428$	0	0
$429$	−2646.86	−0.297882
$430$	0	0
$431$	−2393.56	−0.267503	−0.133751	−	0.991015i	$-0.542702\pi$
$431$	−2393.56	−0.267503	−0.133751	+	0.991015i	$0.542702\pi$
$432$	0	0
$433$	10706.5	1.18827	0.594135	−	0.804365i	$-0.297494\pi$
$433$	10706.5	1.18827	0.594135	+	0.804365i	$0.297494\pi$
$434$	0	0
$435$	6930.82	0.763925
$436$	0	0
$437$	9582.52	1.04896
$438$	0	0
$439$	11485.6	1.24869	0.624346	−	0.781148i	$-0.285366\pi$
$439$	11485.6	1.24869	0.624346	+	0.781148i	$0.285366\pi$
$440$	0	0
$441$	6113.82	0.660169
$442$	0	0
$443$	2111.07	0.226410	0.113205	−	0.993572i	$-0.463888\pi$
$443$	2111.07	0.226410	0.113205	+	0.993572i	$0.463888\pi$
$444$	0	0
$445$	−1596.86	−0.170109
$446$	0	0
$447$	−5345.44	−0.565617
$448$	0	0
$449$	−8476.58	−0.890946	−0.445473	−	0.895295i	$-0.646964\pi$
$449$	−8476.58	−0.890946	−0.445473	+	0.895295i	$0.646964\pi$
$450$	0	0
$451$	−245.602	−0.0256429
$452$	0	0
$453$	−20.1545	−0.00209038
$454$	0	0
$455$	6872.91	0.708148
$456$	0	0
$457$	−3199.98	−0.327546	−0.163773	−	0.986498i	$-0.552366\pi$
$457$	−3199.98	−0.327546	−0.163773	+	0.986498i	$0.552366\pi$
$458$	0	0
$459$	791.893	0.0805281
$460$	0	0
$461$	6745.98	0.681544	0.340772	−	0.940146i	$-0.389312\pi$
$461$	6745.98	0.681544	0.340772	+	0.940146i	$0.389312\pi$
$462$	0	0
$463$	5410.30	0.543063	0.271531	−	0.962430i	$-0.412470\pi$
$463$	5410.30	0.543063	0.271531	+	0.962430i	$0.412470\pi$
$464$	0	0
$465$	−12366.4	−1.23328
$466$	0	0
$467$	14305.6	1.41752	0.708761	−	0.705448i	$-0.249254\pi$
$467$	14305.6	1.41752	0.708761	+	0.705448i	$0.249254\pi$
$468$	0	0
$469$	1579.97	0.155557
$470$	0	0
$471$	46.0485	0.00450489
$472$	0	0
$473$	−1291.50	−0.125546
$474$	0	0
$475$	9354.68	0.903625
$476$	0	0
$477$	506.724	0.0486401
$478$	0	0
$479$	−1718.44	−0.163919	−0.0819596	−	0.996636i	$-0.526118\pi$
$479$	−1718.44	−0.163919	−0.0819596	+	0.996636i	$0.526118\pi$
$480$	0	0
$481$	−17325.8	−1.64239
$482$	0	0
$483$	2405.42	0.226605
$484$	0	0
$485$	−2557.99	−0.239490
$486$	0	0
$487$	−13434.9	−1.25009	−0.625046	−	0.780588i	$-0.714920\pi$
$487$	−13434.9	−1.25009	−0.625046	+	0.780588i	$0.714920\pi$
$488$	0	0
$489$	8331.35	0.770463
$490$	0	0
$491$	−6739.73	−0.619470	−0.309735	−	0.950823i	$-0.600240\pi$
$491$	−6739.73	−0.619470	−0.309735	+	0.950823i	$0.600240\pi$
$492$	0	0
$493$	855.503	0.0781540
$494$	0	0
$495$	3649.61	0.331390
$496$	0	0
$497$	−3846.13	−0.347128
$498$	0	0
$499$	−5609.08	−0.503200	−0.251600	−	0.967831i	$-0.580957\pi$
$499$	−5609.08	−0.503200	−0.251600	+	0.967831i	$0.580957\pi$
$500$	0	0
$501$	−10084.4	−0.899274
$502$	0	0
$503$	−15395.3	−1.36470	−0.682350	−	0.731026i	$-0.739042\pi$
$503$	−15395.3	−1.36470	−0.682350	+	0.731026i	$0.739042\pi$
$504$	0	0
$505$	−20300.0	−1.78879
$506$	0	0
$507$	14274.3	1.25038
$508$	0	0
$509$	−7701.34	−0.670640	−0.335320	−	0.942104i	$-0.608844\pi$
$509$	−7701.34	−0.670640	−0.335320	+	0.942104i	$0.608844\pi$
$510$	0	0
$511$	1163.85	0.100755
$512$	0	0
$513$	−6774.25	−0.583022
$514$	0	0
$515$	29730.5	2.54385
$516$	0	0
$517$	3033.18	0.258026
$518$	0	0
$519$	1830.87	0.154848
$520$	0	0
$521$	−1370.94	−0.115282	−0.0576412	−	0.998337i	$-0.518358\pi$
$521$	−1370.94	−0.115282	−0.0576412	+	0.998337i	$0.518358\pi$
$522$	0	0
$523$	−5394.92	−0.451058	−0.225529	−	0.974236i	$-0.572411\pi$
$523$	−5394.92	−0.451058	−0.225529	+	0.974236i	$0.572411\pi$
$524$	0	0
$525$	2348.23	0.195210
$526$	0	0
$527$	−1526.44	−0.126172
$528$	0	0
$529$	21675.8	1.78152
$530$	0	0
$531$	−9807.99	−0.801564
$532$	0	0
$533$	1900.73	0.154465
$534$	0	0
$535$	−34855.8	−2.81672
$536$	0	0
$537$	8654.07	0.695439
$538$	0	0
$539$	−3537.61	−0.282700
$540$	0	0
$541$	16039.6	1.27467	0.637335	−	0.770587i	$-0.280037\pi$
$541$	16039.6	1.27467	0.637335	+	0.770587i	$0.280037\pi$
$542$	0	0
$543$	12446.3	0.983651
$544$	0	0
$545$	23573.5	1.85281
$546$	0	0
$547$	−7648.69	−0.597869	−0.298935	−	0.954274i	$-0.596631\pi$
$547$	−7648.69	−0.597869	−0.298935	+	0.954274i	$0.596631\pi$
$548$	0	0
$549$	−13175.7	−1.02427
$550$	0	0
$551$	−7318.41	−0.565834
$552$	0	0
$553$	4242.17	0.326212
$554$	0	0
$555$	−10039.8	−0.767868
$556$	0	0
$557$	18116.4	1.37812	0.689062	−	0.724702i	$-0.258023\pi$
$557$	18116.4	1.37812	0.689062	+	0.724702i	$0.258023\pi$
$558$	0	0
$559$	9994.99	0.756249
$560$	0	0
$561$	−189.322	−0.0142481
$562$	0	0
$563$	4828.86	0.361478	0.180739	−	0.983531i	$-0.442151\pi$
$563$	4828.86	0.361478	0.180739	+	0.983531i	$0.442151\pi$
$564$	0	0
$565$	7877.24	0.586545
$566$	0	0
$567$	673.957	0.0499180
$568$	0	0
$569$	18277.3	1.34661	0.673307	−	0.739363i	$-0.264873\pi$
$569$	18277.3	1.34661	0.673307	+	0.739363i	$0.264873\pi$
$570$	0	0
$571$	14940.9	1.09502	0.547511	−	0.836798i	$-0.315575\pi$
$571$	14940.9	1.09502	0.547511	+	0.836798i	$0.315575\pi$
$572$	0	0
$573$	−2556.11	−0.186358
$574$	0	0
$575$	33038.1	2.39615
$576$	0	0
$577$	10861.2	0.783635	0.391818	−	0.920043i	$-0.371846\pi$
$577$	10861.2	0.783635	0.391818	+	0.920043i	$0.371846\pi$
$578$	0	0
$579$	3800.93	0.272817
$580$	0	0
$581$	−2112.90	−0.150874
$582$	0	0
$583$	−293.203	−0.0208289
$584$	0	0
$585$	−28244.6	−1.99619
$586$	0	0
$587$	13347.4	0.938515	0.469257	−	0.883062i	$-0.344522\pi$
$587$	13347.4	0.938515	0.469257	+	0.883062i	$0.344522\pi$
$588$	0	0
$589$	13057.9	0.913486
$590$	0	0
$591$	−14736.7	−1.02570
$592$	0	0
$593$	6260.84	0.433562	0.216781	−	0.976220i	$-0.430444\pi$
$593$	6260.84	0.433562	0.216781	+	0.976220i	$0.430444\pi$
$594$	0	0
$595$	491.599	0.0338716
$596$	0	0
$597$	8890.02	0.609455
$598$	0	0
$599$	1235.84	0.0842991	0.0421496	−	0.999111i	$-0.486579\pi$
$599$	1235.84	0.0842991	0.0421496	+	0.999111i	$0.486579\pi$
$600$	0	0
$601$	11558.8	0.784515	0.392257	−	0.919855i	$-0.371694\pi$
$601$	11558.8	0.784515	0.392257	+	0.919855i	$0.371694\pi$
$602$	0	0
$603$	−6492.98	−0.438499
$604$	0	0
$605$	−2111.75	−0.141909
$606$	0	0
$607$	1054.63	0.0705205	0.0352603	−	0.999378i	$-0.488774\pi$
$607$	1054.63	0.0705205	0.0352603	+	0.999378i	$0.488774\pi$
$608$	0	0
$609$	−1837.08	−0.122237
$610$	0	0
$611$	−23474.0	−1.55427
$612$	0	0
$613$	−1203.04	−0.0792664	−0.0396332	−	0.999214i	$-0.512619\pi$
$613$	−1203.04	−0.0792664	−0.0396332	+	0.999214i	$0.512619\pi$
$614$	0	0
$615$	1101.42	0.0722173
$616$	0	0
$617$	−959.742	−0.0626220	−0.0313110	−	0.999510i	$-0.509968\pi$
$617$	−959.742	−0.0626220	−0.0313110	+	0.999510i	$0.509968\pi$
$618$	0	0
$619$	14011.0	0.909774	0.454887	−	0.890549i	$-0.349680\pi$
$619$	14011.0	0.909774	0.454887	+	0.890549i	$0.349680\pi$
$620$	0	0
$621$	−23924.8	−1.54600
$622$	0	0
$623$	423.263	0.0272194
$624$	0	0
$625$	−5821.19	−0.372556
$626$	0	0
$627$	1619.56	0.103156
$628$	0	0
$629$	−1239.26	−0.0785574
$630$	0	0
$631$	−5897.24	−0.372053	−0.186026	−	0.982545i	$-0.559561\pi$
$631$	−5897.24	−0.372053	−0.186026	+	0.982545i	$0.559561\pi$
$632$	0	0
$633$	−10962.0	−0.688310
$634$	0	0
$635$	−47775.2	−2.98567
$636$	0	0
$637$	27377.8	1.70290
$638$	0	0
$639$	15805.9	0.978516
$640$	0	0
$641$	23075.3	1.42187	0.710937	−	0.703256i	$-0.248271\pi$
$641$	23075.3	1.42187	0.710937	+	0.703256i	$0.248271\pi$
$642$	0	0
$643$	8295.98	0.508805	0.254402	−	0.967098i	$-0.418121\pi$
$643$	8295.98	0.508805	0.254402	+	0.967098i	$0.418121\pi$
$644$	0	0
$645$	5791.82	0.353570
$646$	0	0
$647$	17869.7	1.08583	0.542913	−	0.839789i	$-0.317321\pi$
$647$	17869.7	1.08583	0.542913	+	0.839789i	$0.317321\pi$
$648$	0	0
$649$	5675.14	0.343249
$650$	0	0
$651$	3277.83	0.197340
$652$	0	0
$653$	−12150.0	−0.728125	−0.364063	−	0.931374i	$-0.618611\pi$
$653$	−12150.0	−0.728125	−0.364063	+	0.931374i	$0.618611\pi$
$654$	0	0
$655$	−17085.0	−1.01919
$656$	0	0
$657$	−4782.90	−0.284016
$658$	0	0
$659$	16104.7	0.951975	0.475987	−	0.879452i	$-0.342091\pi$
$659$	16104.7	0.951975	0.475987	+	0.879452i	$0.342091\pi$
$660$	0	0
$661$	28742.7	1.69132	0.845660	−	0.533721i	$-0.179207\pi$
$661$	28742.7	1.69132	0.845660	+	0.533721i	$0.179207\pi$
$662$	0	0
$663$	1465.18	0.0858261
$664$	0	0
$665$	−4205.39	−0.245230
$666$	0	0
$667$	−25846.6	−1.50042
$668$	0	0
$669$	1039.42	0.0600694
$670$	0	0
$671$	7623.78	0.438618
$672$	0	0
$673$	18091.3	1.03621	0.518106	−	0.855317i	$-0.326637\pi$
$673$	18091.3	1.03621	0.518106	+	0.855317i	$0.326637\pi$
$674$	0	0
$675$	−23355.9	−1.33181
$676$	0	0
$677$	1915.04	0.108716	0.0543581	−	0.998522i	$-0.482689\pi$
$677$	1915.04	0.108716	0.0543581	+	0.998522i	$0.482689\pi$
$678$	0	0
$679$	678.021	0.0383211
$680$	0	0
$681$	17133.4	0.964100
$682$	0	0
$683$	4576.76	0.256405	0.128203	−	0.991748i	$-0.459079\pi$
$683$	4576.76	0.256405	0.128203	+	0.991748i	$0.459079\pi$
$684$	0	0
$685$	19270.1	1.07485
$686$	0	0
$687$	11996.4	0.666216
$688$	0	0
$689$	2269.12	0.125467
$690$	0	0
$691$	−19493.4	−1.07317	−0.536587	−	0.843845i	$-0.680287\pi$
$691$	−19493.4	−1.07317	−0.536587	+	0.843845i	$0.680287\pi$
$692$	0	0
$693$	−967.365	−0.0530262
$694$	0	0
$695$	2260.83	0.123393
$696$	0	0
$697$	135.954	0.00738825
$698$	0	0
$699$	−14133.8	−0.764790
$700$	0	0
$701$	−16641.1	−0.896613	−0.448306	−	0.893880i	$-0.647973\pi$
$701$	−16641.1	−0.896613	−0.448306	+	0.893880i	$0.647973\pi$
$702$	0	0
$703$	10601.3	0.568755
$704$	0	0
$705$	−13602.6	−0.726669
$706$	0	0
$707$	5380.72	0.286227
$708$	0	0
$709$	34690.1	1.83754	0.918768	−	0.394799i	$-0.129186\pi$
$709$	34690.1	1.83754	0.918768	+	0.394799i	$0.129186\pi$
$710$	0	0
$711$	−17433.4	−0.919556
$712$	0	0
$713$	46117.0	2.42229
$714$	0	0
$715$	16343.0	0.854817
$716$	0	0
$717$	2074.45	0.108050
$718$	0	0
$719$	−31316.5	−1.62435	−0.812175	−	0.583414i	$-0.801716\pi$
$719$	−31316.5	−1.62435	−0.812175	+	0.583414i	$0.801716\pi$
$720$	0	0
$721$	−7880.36	−0.407046
$722$	0	0
$723$	1198.76	0.0616628
$724$	0	0
$725$	−25232.0	−1.29254
$726$	0	0
$727$	6829.67	0.348416	0.174208	−	0.984709i	$-0.444264\pi$
$727$	6829.67	0.348416	0.174208	+	0.984709i	$0.444264\pi$
$728$	0	0
$729$	7155.44	0.363534
$730$	0	0
$731$	714.912	0.0361723
$732$	0	0
$733$	−30580.1	−1.54093	−0.770465	−	0.637482i	$-0.779976\pi$
$733$	−30580.1	−1.54093	−0.770465	+	0.637482i	$0.779976\pi$
$734$	0	0
$735$	15864.7	0.796160
$736$	0	0
$737$	3757.00	0.187776
$738$	0	0
$739$	24945.0	1.24170	0.620850	−	0.783929i	$-0.286788\pi$
$739$	24945.0	1.24170	0.620850	+	0.783929i	$0.286788\pi$
$740$	0	0
$741$	−12533.9	−0.621380
$742$	0	0
$743$	−20683.3	−1.02126	−0.510630	−	0.859801i	$-0.670588\pi$
$743$	−20683.3	−1.02126	−0.510630	+	0.859801i	$0.670588\pi$
$744$	0	0
$745$	33005.4	1.62312
$746$	0	0
$747$	8683.08	0.425298
$748$	0	0
$749$	9238.86	0.450709
$750$	0	0
$751$	−12716.9	−0.617902	−0.308951	−	0.951078i	$-0.599978\pi$
$751$	−12716.9	−0.617902	−0.308951	+	0.951078i	$0.599978\pi$
$752$	0	0
$753$	−2685.58	−0.129971
$754$	0	0
$755$	124.444	0.00599865
$756$	0	0
$757$	37126.5	1.78254	0.891271	−	0.453472i	$-0.149815\pi$
$757$	37126.5	1.78254	0.891271	+	0.453472i	$0.149815\pi$
$758$	0	0
$759$	5719.82	0.273539
$760$	0	0
$761$	4198.08	0.199974	0.0999870	−	0.994989i	$-0.468120\pi$
$761$	4198.08	0.199974	0.0999870	+	0.994989i	$0.468120\pi$
$762$	0	0
$763$	−6248.39	−0.296471
$764$	0	0
$765$	−2020.25	−0.0954802
$766$	0	0
$767$	−43920.3	−2.06763
$768$	0	0
$769$	16085.3	0.754294	0.377147	−	0.926153i	$-0.376905\pi$
$769$	16085.3	0.754294	0.377147	+	0.926153i	$0.376905\pi$
$770$	0	0
$771$	−13631.5	−0.636741
$772$	0	0
$773$	−3557.93	−0.165550	−0.0827748	−	0.996568i	$-0.526378\pi$
$773$	−3557.93	−0.165550	−0.0827748	+	0.996568i	$0.526378\pi$
$774$	0	0
$775$	45020.5	2.08669
$776$	0	0
$777$	2661.15	0.122868
$778$	0	0
$779$	−1163.02	−0.0534909
$780$	0	0
$781$	−9145.67	−0.419024
$782$	0	0
$783$	18271.9	0.833954
$784$	0	0
$785$	−284.326	−0.0129274
$786$	0	0
$787$	34848.3	1.57841	0.789204	−	0.614132i	$-0.210494\pi$
$787$	34848.3	1.57841	0.789204	+	0.614132i	$0.210494\pi$
$788$	0	0
$789$	4603.70	0.207726
$790$	0	0
$791$	−2087.94	−0.0938541
$792$	0	0
$793$	−59001.0	−2.64210
$794$	0	0
$795$	1314.89	0.0586596
$796$	0	0
$797$	19212.8	0.853892	0.426946	−	0.904277i	$-0.359589\pi$
$797$	19212.8	0.853892	0.426946	+	0.904277i	$0.359589\pi$
$798$	0	0
$799$	−1679.03	−0.0743426
$800$	0	0
$801$	−1739.42	−0.0767284
$802$	0	0
$803$	2767.50	0.121623
$804$	0	0
$805$	−14852.2	−0.650277
$806$	0	0
$807$	−23835.1	−1.03970
$808$	0	0
$809$	−35866.9	−1.55873	−0.779365	−	0.626570i	$-0.784458\pi$
$809$	−35866.9	−1.55873	−0.779365	+	0.626570i	$0.784458\pi$
$810$	0	0
$811$	−12295.8	−0.532386	−0.266193	−	0.963920i	$-0.585766\pi$
$811$	−12295.8	−0.532386	−0.266193	+	0.963920i	$0.585766\pi$
$812$	0	0
$813$	16290.9	0.702764
$814$	0	0
$815$	−51441.8	−2.21096
$816$	0	0
$817$	−6115.72	−0.261887
$818$	0	0
$819$	7486.51	0.319414
$820$	0	0
$821$	−8301.98	−0.352913	−0.176456	−	0.984308i	$-0.556463\pi$
$821$	−8301.98	−0.352913	−0.176456	+	0.984308i	$0.556463\pi$
$822$	0	0
$823$	19260.2	0.815757	0.407879	−	0.913036i	$-0.366269\pi$
$823$	19260.2	0.815757	0.407879	+	0.913036i	$0.366269\pi$
$824$	0	0
$825$	5583.82	0.235641
$826$	0	0
$827$	−15002.2	−0.630806	−0.315403	−	0.948958i	$-0.602140\pi$
$827$	−15002.2	−0.630806	−0.315403	+	0.948958i	$0.602140\pi$
$828$	0	0
$829$	15589.7	0.653141	0.326570	−	0.945173i	$-0.394107\pi$
$829$	15589.7	0.653141	0.326570	+	0.945173i	$0.394107\pi$
$830$	0	0
$831$	−16591.1	−0.692585
$832$	0	0
$833$	1958.25	0.0814518
$834$	0	0
$835$	62265.8	2.58060
$836$	0	0
$837$	−32601.9	−1.34634
$838$	0	0
$839$	42573.2	1.75183	0.875917	−	0.482461i	$-0.160257\pi$
$839$	42573.2	1.75183	0.875917	+	0.482461i	$0.160257\pi$
$840$	0	0
$841$	−4649.33	−0.190632
$842$	0	0
$843$	−12265.8	−0.501135
$844$	0	0
$845$	−88136.5	−3.58815
$846$	0	0
$847$	559.741	0.0227071
$848$	0	0
$849$	3328.69	0.134559
$850$	0	0
$851$	37440.7	1.50817
$852$	0	0
$853$	23827.7	0.956441	0.478221	−	0.878240i	$-0.341282\pi$
$853$	23827.7	0.956441	0.478221	+	0.878240i	$0.341282\pi$
$854$	0	0
$855$	17282.3	0.691276
$856$	0	0
$857$	−42582.9	−1.69732	−0.848660	−	0.528938i	$-0.822590\pi$
$857$	−42582.9	−1.69732	−0.848660	+	0.528938i	$0.822590\pi$
$858$	0	0
$859$	−37604.0	−1.49364	−0.746818	−	0.665029i	$-0.768419\pi$
$859$	−37604.0	−1.49364	−0.746818	+	0.665029i	$0.768419\pi$
$860$	0	0
$861$	−291.943	−0.0115556
$862$	0	0
$863$	36934.0	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$863$	36934.0	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$864$	0	0
$865$	−11304.7	−0.444359
$866$	0	0
$867$	−13782.0	−0.539865
$868$	0	0
$869$	10087.4	0.393776
$870$	0	0
$871$	−29075.7	−1.13110
$872$	0	0
$873$	−2786.36	−0.108023
$874$	0	0
$875$	−4407.29	−0.170279
$876$	0	0
$877$	−18212.3	−0.701238	−0.350619	−	0.936518i	$-0.614029\pi$
$877$	−18212.3	−0.701238	−0.350619	+	0.936518i	$0.614029\pi$
$878$	0	0
$879$	−16994.2	−0.652105
$880$	0	0
$881$	5878.65	0.224809	0.112405	−	0.993663i	$-0.464145\pi$
$881$	5878.65	0.224809	0.112405	+	0.993663i	$0.464145\pi$
$882$	0	0
$883$	6399.06	0.243879	0.121940	−	0.992538i	$-0.461089\pi$
$883$	6399.06	0.243879	0.121940	+	0.992538i	$0.461089\pi$
$884$	0	0
$885$	−25450.6	−0.966681
$886$	0	0
$887$	33294.9	1.26035	0.630177	−	0.776452i	$-0.282982\pi$
$887$	33294.9	1.26035	0.630177	+	0.776452i	$0.282982\pi$
$888$	0	0
$889$	12663.3	0.477742
$890$	0	0
$891$	1602.59	0.0602569
$892$	0	0
$893$	14363.2	0.538239
$894$	0	0
$895$	−53434.5	−1.99566
$896$	0	0
$897$	−44266.1	−1.64772
$898$	0	0
$899$	−35220.7	−1.30665
$900$	0	0
$901$	162.303	0.00600123
$902$	0	0
$903$	−1535.18	−0.0565754
$904$	0	0
$905$	−76849.7	−2.82273
$906$	0	0
$907$	20701.9	0.757878	0.378939	−	0.925422i	$-0.376289\pi$
$907$	20701.9	0.757878	0.378939	+	0.925422i	$0.376289\pi$
$908$	0	0
$909$	−22112.4	−0.806843
$910$	0	0
$911$	2496.01	0.0907756	0.0453878	−	0.998969i	$-0.485548\pi$
$911$	2496.01	0.0907756	0.0453878	+	0.998969i	$0.485548\pi$
$912$	0	0
$913$	−5024.24	−0.182123
$914$	0	0
$915$	−34189.5	−1.23527
$916$	0	0
$917$	4528.55	0.163082
$918$	0	0
$919$	45604.5	1.63695	0.818473	−	0.574546i	$-0.194821\pi$
$919$	45604.5	1.63695	0.818473	+	0.574546i	$0.194821\pi$
$920$	0	0
$921$	−575.156	−0.0205777
$922$	0	0
$923$	70779.0	2.52407
$924$	0	0
$925$	36550.5	1.29922
$926$	0	0
$927$	32384.8	1.14742
$928$	0	0
$929$	−21550.0	−0.761067	−0.380534	−	0.924767i	$-0.624260\pi$
$929$	−21550.0	−0.761067	−0.380534	+	0.924767i	$0.624260\pi$
$930$	0	0
$931$	−16751.9	−0.589711
$932$	0	0
$933$	13572.2	0.476241
$934$	0	0
$935$	1168.97	0.0408869
$936$	0	0
$937$	−33457.9	−1.16651	−0.583257	−	0.812288i	$-0.698222\pi$
$937$	−33457.9	−1.16651	−0.583257	+	0.812288i	$0.698222\pi$
$938$	0	0
$939$	2066.48	0.0718179
$940$	0	0
$941$	45107.8	1.56267	0.781334	−	0.624113i	$-0.214539\pi$
$941$	45107.8	1.56267	0.781334	+	0.624113i	$0.214539\pi$
$942$	0	0
$943$	−4107.45	−0.141842
$944$	0	0
$945$	10499.6	0.361432
$946$	0	0
$947$	30270.6	1.03871	0.519357	−	0.854557i	$-0.326172\pi$
$947$	30270.6	1.03871	0.519357	+	0.854557i	$0.326172\pi$
$948$	0	0
$949$	−21417.9	−0.732618
$950$	0	0
$951$	3799.48	0.129555
$952$	0	0
$953$	−4145.41	−0.140905	−0.0704527	−	0.997515i	$-0.522444\pi$
$953$	−4145.41	−0.140905	−0.0704527	+	0.997515i	$0.522444\pi$
$954$	0	0
$955$	15782.7	0.534780
$956$	0	0
$957$	−4368.37	−0.147554
$958$	0	0
$959$	−5107.72	−0.171989
$960$	0	0
$961$	33052.0	1.10946
$962$	0	0
$963$	−37967.6	−1.27050
$964$	0	0
$965$	−23468.8	−0.782888
$966$	0	0
$967$	−1369.06	−0.0455285	−0.0227643	−	0.999741i	$-0.507247\pi$
$967$	−1369.06	−0.0455285	−0.0227643	+	0.999741i	$0.507247\pi$
$968$	0	0
$969$	−896.510	−0.0297214
$970$	0	0
$971$	29221.5	0.965770	0.482885	−	0.875684i	$-0.339589\pi$
$971$	29221.5	0.965770	0.482885	+	0.875684i	$0.339589\pi$
$972$	0	0
$973$	−599.255	−0.0197443
$974$	0	0
$975$	−43213.6	−1.41943
$976$	0	0
$977$	−33896.8	−1.10999	−0.554993	−	0.831855i	$-0.687279\pi$
$977$	−33896.8	−1.10999	−0.554993	+	0.831855i	$0.687279\pi$
$978$	0	0
$979$	1006.47	0.0328570
$980$	0	0
$981$	25678.1	0.835718
$982$	0	0
$983$	11859.9	0.384815	0.192408	−	0.981315i	$-0.438370\pi$
$983$	11859.9	0.384815	0.192408	+	0.981315i	$0.438370\pi$
$984$	0	0
$985$	90991.5	2.94338
$986$	0	0
$987$	3605.49	0.116276
$988$	0	0
$989$	−21599.0	−0.694447
$990$	0	0
$991$	2037.86	0.0653227	0.0326614	−	0.999466i	$-0.489602\pi$
$991$	2037.86	0.0653227	0.0326614	+	0.999466i	$0.489602\pi$
$992$	0	0
$993$	−5521.08	−0.176441
$994$	0	0
$995$	−54891.4	−1.74892
$996$	0	0
$997$	35940.6	1.14168	0.570838	−	0.821063i	$-0.306618\pi$
$997$	35940.6	1.14168	0.570838	+	0.821063i	$0.306618\pi$
$998$	0	0
$999$	−26468.3	−0.838259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.a.u.1.2		3
4.3	odd	2		704.4.a.t.1.2		3
8.3	odd	2		88.4.a.d.1.2	✓	3
8.5	even	2		176.4.a.j.1.2		3
24.5	odd	2		1584.4.a.bl.1.1		3
24.11	even	2		792.4.a.l.1.1		3
40.19	odd	2		2200.4.a.m.1.2		3
88.21	odd	2		1936.4.a.bh.1.2		3
88.43	even	2		968.4.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.a.d.1.2	✓	3	8.3	odd	2
176.4.a.j.1.2		3	8.5	even	2
704.4.a.t.1.2		3	4.3	odd	2
704.4.a.u.1.2		3	1.1	even	1	trivial
792.4.a.l.1.1		3	24.11	even	2
968.4.a.i.1.2		3	88.43	even	2
1584.4.a.bl.1.1		3	24.5	odd	2
1936.4.a.bh.1.2		3	88.21	odd	2
2200.4.a.m.1.2		3	40.19	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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82655 q^{3} -17.4525 q^{5} +4.62596 q^{7} -19.0106 q^{9} +11.0000 q^{11} -85.1298 q^{13} -49.3304 q^{15} -6.08908 q^{17} +52.0891 q^{19} +13.0755 q^{21} +183.964 q^{23} +179.590 q^{25} -130.051 q^{27} -140.498 q^{29} +250.685 q^{31} +31.0920 q^{33} -80.7345 q^{35} +203.522 q^{37} -240.624 q^{39} -22.3275 q^{41} -117.409 q^{43} +331.783 q^{45} +275.744 q^{47} -321.601 q^{49} -17.2111 q^{51} -26.6548 q^{53} -191.978 q^{55} +147.232 q^{57} +515.922 q^{59} +693.071 q^{61} -87.9423 q^{63} +1485.73 q^{65} +341.545 q^{67} +519.983 q^{69} -831.425 q^{71} +251.591 q^{73} +507.620 q^{75} +50.8855 q^{77} +917.036 q^{79} +145.690 q^{81} -456.749 q^{83} +106.270 q^{85} -397.124 q^{87} +91.4974 q^{89} -393.807 q^{91} +708.574 q^{93} -909.085 q^{95} +146.569 q^{97} -209.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 8 q^{5} - 24 q^{7} + 89 q^{9} + 33 q^{11} - 66 q^{13} + 10 q^{15} + 210 q^{17} - 72 q^{19} - 200 q^{21} + 50 q^{23} - q^{25} - 286 q^{27} + 50 q^{29} + 298 q^{31} + 22 q^{33} - 304 q^{35}+ \cdots + 979 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 8 * q^5 - 24 * q^7 + 89 * q^9 + 33 * q^11 - 66 * q^13 + 10 * q^15 + 210 * q^17 - 72 * q^19 - 200 * q^21 + 50 * q^23 - q^25 - 286 * q^27 + 50 * q^29 + 298 * q^31 + 22 * q^33 - 304 * q^35 + 4 * q^37 + 876 * q^39 + 254 * q^41 - 112 * q^43 + 790 * q^45 - 40 * q^47 - 285 * q^49 + 48 * q^51 + 550 * q^53 - 88 * q^55 + 44 * q^57 + 1630 * q^59 + 906 * q^61 - 1448 * q^63 + 2016 * q^65 - 482 * q^67 + 1398 * q^69 - 242 * q^71 + 998 * q^73 + 1180 * q^75 - 264 * q^77 - 324 * q^79 + 1715 * q^81 + 1360 * q^83 + 1188 * q^85 - 2880 * q^87 + 1200 * q^89 - 2232 * q^91 + 4998 * q^93 - 1556 * q^95 + 180 * q^97 + 979 * q^99

Introduction

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