LMFDB - Embedded newform 768.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -10.4222 q^{5} -6.42221 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -10.4222 q^{5} -6.42221 q^{7} +9.00000 q^{9} -61.6888 q^{11} -64.8444 q^{13} +31.2666 q^{15} -75.6888 q^{17} +10.3112 q^{19} +19.2666 q^{21} +156.844 q^{23} -16.3776 q^{25} -27.0000 q^{27} +53.7998 q^{29} -227.489 q^{31} +185.066 q^{33} +66.9335 q^{35} -10.3112 q^{37} +194.533 q^{39} +70.4441 q^{41} -298.311 q^{43} -93.7998 q^{45} +89.9109 q^{47} -301.755 q^{49} +227.066 q^{51} -388.333 q^{53} +642.934 q^{55} -30.9335 q^{57} -324.000 q^{59} -324.000 q^{61} -57.7998 q^{63} +675.822 q^{65} +920.266 q^{67} -470.533 q^{69} +995.156 q^{71} -362.266 q^{73} +49.1329 q^{75} +396.178 q^{77} +1098.91 q^{79} +81.0000 q^{81} +791.822 q^{83} +788.844 q^{85} -161.400 q^{87} +150.622 q^{89} +416.444 q^{91} +682.466 q^{93} -107.465 q^{95} -1879.15 q^{97} -555.199 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9} - 8 q^{11} - 72 q^{13} - 24 q^{15} - 36 q^{17} + 136 q^{19} - 48 q^{21} + 256 q^{23} + 198 q^{25} - 54 q^{27} - 152 q^{29} - 80 q^{31} + 24 q^{33} + 480 q^{35} - 136 q^{37} + 216 q^{39} - 436 q^{41} - 712 q^{43} + 72 q^{45} - 224 q^{47} - 142 q^{49} + 108 q^{51} - 344 q^{53} + 1632 q^{55} - 408 q^{57} - 648 q^{59} - 648 q^{61} + 144 q^{63} + 544 q^{65} + 456 q^{67} - 768 q^{69} + 2048 q^{71} + 660 q^{73} - 594 q^{75} + 1600 q^{77} + 496 q^{79} + 162 q^{81} + 776 q^{83} + 1520 q^{85} + 456 q^{87} + 532 q^{89} + 256 q^{91} + 240 q^{93} + 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9 - 8 * q^11 - 72 * q^13 - 24 * q^15 - 36 * q^17 + 136 * q^19 - 48 * q^21 + 256 * q^23 + 198 * q^25 - 54 * q^27 - 152 * q^29 - 80 * q^31 + 24 * q^33 + 480 * q^35 - 136 * q^37 + 216 * q^39 - 436 * q^41 - 712 * q^43 + 72 * q^45 - 224 * q^47 - 142 * q^49 + 108 * q^51 - 344 * q^53 + 1632 * q^55 - 408 * q^57 - 648 * q^59 - 648 * q^61 + 144 * q^63 + 544 * q^65 + 456 * q^67 - 768 * q^69 + 2048 * q^71 + 660 * q^73 - 594 * q^75 + 1600 * q^77 + 496 * q^79 + 162 * q^81 + 776 * q^83 + 1520 * q^85 + 456 * q^87 + 532 * q^89 + 256 * q^91 + 240 * q^93 + 2208 * q^95 - 1220 * q^97 - 72 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−10.4222	−0.932190	−0.466095	−	0.884735i	$-0.654340\pi$
$5$	−10.4222	−0.932190	−0.466095	+	0.884735i	$0.654340\pi$
$6$	0	0
$7$	−6.42221	−0.346766	−0.173383	−	0.984854i	$-0.555470\pi$
$7$	−6.42221	−0.346766	−0.173383	+	0.984854i	$0.555470\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−61.6888	−1.69090	−0.845449	−	0.534056i	$-0.820667\pi$
$11$	−61.6888	−1.69090	−0.845449	+	0.534056i	$0.820667\pi$
$12$	0	0
$13$	−64.8444	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$13$	−64.8444	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$14$	0	0
$15$	31.2666	0.538200
$16$	0	0
$17$	−75.6888	−1.07984	−0.539919	−	0.841717i	$-0.681545\pi$
$17$	−75.6888	−1.07984	−0.539919	+	0.841717i	$0.681545\pi$
$18$	0	0
$19$	10.3112	0.124502	0.0622512	−	0.998061i	$-0.480172\pi$
$19$	10.3112	0.124502	0.0622512	+	0.998061i	$0.480172\pi$
$20$	0	0
$21$	19.2666	0.200206
$22$	0	0
$23$	156.844	1.42193	0.710963	−	0.703229i	$-0.248259\pi$
$23$	156.844	1.42193	0.710963	+	0.703229i	$0.248259\pi$
$24$	0	0
$25$	−16.3776	−0.131021
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	53.7998	0.344496	0.172248	−	0.985054i	$-0.444897\pi$
$29$	53.7998	0.344496	0.172248	+	0.985054i	$0.444897\pi$
$30$	0	0
$31$	−227.489	−1.31801	−0.659003	−	0.752141i	$-0.729021\pi$
$31$	−227.489	−1.31801	−0.659003	+	0.752141i	$0.729021\pi$
$32$	0	0
$33$	185.066	0.976240
$34$	0	0
$35$	66.9335	0.323252
$36$	0	0
$37$	−10.3112	−0.0458148	−0.0229074	−	0.999738i	$-0.507292\pi$
$37$	−10.3112	−0.0458148	−0.0229074	+	0.999738i	$0.507292\pi$
$38$	0	0
$39$	194.533	0.798724
$40$	0	0
$41$	70.4441	0.268330	0.134165	−	0.990959i	$-0.457165\pi$
$41$	70.4441	0.268330	0.134165	+	0.990959i	$0.457165\pi$
$42$	0	0
$43$	−298.311	−1.05795	−0.528977	−	0.848636i	$-0.677424\pi$
$43$	−298.311	−1.05795	−0.528977	+	0.848636i	$0.677424\pi$
$44$	0	0
$45$	−93.7998	−0.310730
$46$	0	0
$47$	89.9109	0.279039	0.139520	−	0.990219i	$-0.455444\pi$
$47$	89.9109	0.279039	0.139520	+	0.990219i	$0.455444\pi$
$48$	0	0
$49$	−301.755	−0.879753
$50$	0	0
$51$	227.066	0.623444
$52$	0	0
$53$	−388.333	−1.00645	−0.503223	−	0.864157i	$-0.667853\pi$
$53$	−388.333	−1.00645	−0.503223	+	0.864157i	$0.667853\pi$
$54$	0	0
$55$	642.934	1.57624
$56$	0	0
$57$	−30.9335	−0.0718815
$58$	0	0
$59$	−324.000	−0.714936	−0.357468	−	0.933925i	$-0.616360\pi$
$59$	−324.000	−0.714936	−0.357468	+	0.933925i	$0.616360\pi$
$60$	0	0
$61$	−324.000	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$61$	−324.000	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$62$	0	0
$63$	−57.7998	−0.115589
$64$	0	0
$65$	675.822	1.28962
$66$	0	0
$67$	920.266	1.67804	0.839018	−	0.544104i	$-0.183130\pi$
$67$	920.266	1.67804	0.839018	+	0.544104i	$0.183130\pi$
$68$	0	0
$69$	−470.533	−0.820950
$70$	0	0
$71$	995.156	1.66343	0.831713	−	0.555206i	$-0.187361\pi$
$71$	995.156	1.66343	0.831713	+	0.555206i	$0.187361\pi$
$72$	0	0
$73$	−362.266	−0.580822	−0.290411	−	0.956902i	$-0.593792\pi$
$73$	−362.266	−0.580822	−0.290411	+	0.956902i	$0.593792\pi$
$74$	0	0
$75$	49.1329	0.0756451
$76$	0	0
$77$	396.178	0.586347
$78$	0	0
$79$	1098.91	1.56503	0.782513	−	0.622634i	$-0.213938\pi$
$79$	1098.91	1.56503	0.782513	+	0.622634i	$0.213938\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	791.822	1.04715	0.523577	−	0.851979i	$-0.324597\pi$
$83$	791.822	1.04715	0.523577	+	0.851979i	$0.324597\pi$
$84$	0	0
$85$	788.844	1.00661
$86$	0	0
$87$	−161.400	−0.198895
$88$	0	0
$89$	150.622	0.179393	0.0896963	−	0.995969i	$-0.471410\pi$
$89$	150.622	0.179393	0.0896963	+	0.995969i	$0.471410\pi$
$90$	0	0
$91$	416.444	0.479728
$92$	0	0
$93$	682.466	0.760951
$94$	0	0
$95$	−107.465	−0.116060
$96$	0	0
$97$	−1879.15	−1.96700	−0.983501	−	0.180903i	$-0.942098\pi$
$97$	−1879.15	−1.96700	−0.983501	+	0.180903i	$0.942098\pi$
$98$	0	0
$99$	−555.199	−0.563633
$100$	0	0
$101$	1722.51	1.69699	0.848496	−	0.529202i	$-0.177509\pi$
$101$	1722.51	1.69699	0.848496	+	0.529202i	$0.177509\pi$
$102$	0	0
$103$	1908.82	1.82604	0.913018	−	0.407919i	$-0.133745\pi$
$103$	1908.82	1.82604	0.913018	+	0.407919i	$0.133745\pi$
$104$	0	0
$105$	−200.801	−0.186630
$106$	0	0
$107$	−128.622	−0.116209	−0.0581046	−	0.998310i	$-0.518506\pi$
$107$	−128.622	−0.116209	−0.0581046	+	0.998310i	$0.518506\pi$
$108$	0	0
$109$	−758.267	−0.666320	−0.333160	−	0.942870i	$-0.608115\pi$
$109$	−758.267	−0.666320	−0.333160	+	0.942870i	$0.608115\pi$
$110$	0	0
$111$	30.9335	0.0264512
$112$	0	0
$113$	921.643	0.767265	0.383632	−	0.923486i	$-0.374673\pi$
$113$	921.643	0.767265	0.383632	+	0.923486i	$0.374673\pi$
$114$	0	0
$115$	−1634.66	−1.32551
$116$	0	0
$117$	−583.600	−0.461144
$118$	0	0
$119$	486.089	0.374451
$120$	0	0
$121$	2474.51	1.85914
$122$	0	0
$123$	−211.332	−0.154920
$124$	0	0
$125$	1473.47	1.05433
$126$	0	0
$127$	1647.22	1.15092	0.575462	−	0.817828i	$-0.304822\pi$
$127$	1647.22	1.15092	0.575462	+	0.817828i	$0.304822\pi$
$128$	0	0
$129$	894.934	0.610810
$130$	0	0
$131$	−2000.27	−1.33408	−0.667038	−	0.745023i	$-0.732438\pi$
$131$	−2000.27	−1.33408	−0.667038	+	0.745023i	$0.732438\pi$
$132$	0	0
$133$	−66.2205	−0.0431733
$134$	0	0
$135$	281.400	0.179400
$136$	0	0
$137$	−2574.71	−1.60564	−0.802819	−	0.596223i	$-0.796667\pi$
$137$	−2574.71	−1.60564	−0.802819	+	0.596223i	$0.796667\pi$
$138$	0	0
$139$	−1331.29	−0.812362	−0.406181	−	0.913793i	$-0.633140\pi$
$139$	−1331.29	−0.812362	−0.406181	+	0.913793i	$0.633140\pi$
$140$	0	0
$141$	−269.733	−0.161103
$142$	0	0
$143$	4000.18	2.33924
$144$	0	0
$145$	−560.713	−0.321136
$146$	0	0
$147$	905.266	0.507926
$148$	0	0
$149$	−552.377	−0.303708	−0.151854	−	0.988403i	$-0.548524\pi$
$149$	−552.377	−0.303708	−0.151854	+	0.988403i	$0.548524\pi$
$150$	0	0
$151$	−1307.13	−0.704456	−0.352228	−	0.935914i	$-0.614576\pi$
$151$	−1307.13	−0.704456	−0.352228	+	0.935914i	$0.614576\pi$
$152$	0	0
$153$	−681.199	−0.359946
$154$	0	0
$155$	2370.93	1.22863
$156$	0	0
$157$	−3527.11	−1.79295	−0.896477	−	0.443089i	$-0.853882\pi$
$157$	−3527.11	−1.79295	−0.896477	+	0.443089i	$0.853882\pi$
$158$	0	0
$159$	1165.00	0.581072
$160$	0	0
$161$	−1007.29	−0.493077
$162$	0	0
$163$	−926.045	−0.444991	−0.222495	−	0.974934i	$-0.571420\pi$
$163$	−926.045	−0.444991	−0.222495	+	0.974934i	$0.571420\pi$
$164$	0	0
$165$	−1928.80	−0.910042
$166$	0	0
$167$	−1393.33	−0.645624	−0.322812	−	0.946463i	$-0.604628\pi$
$167$	−1393.33	−0.645624	−0.322812	+	0.946463i	$0.604628\pi$
$168$	0	0
$169$	2007.80	0.913881
$170$	0	0
$171$	92.8006	0.0415008
$172$	0	0
$173$	3311.71	1.45540	0.727701	−	0.685894i	$-0.240589\pi$
$173$	3311.71	1.45540	0.727701	+	0.685894i	$0.240589\pi$
$174$	0	0
$175$	105.181	0.0454337
$176$	0	0
$177$	972.000	0.412768
$178$	0	0
$179$	323.287	0.134992	0.0674961	−	0.997720i	$-0.478499\pi$
$179$	323.287	0.134992	0.0674961	+	0.997720i	$0.478499\pi$
$180$	0	0
$181$	3066.80	1.25941	0.629705	−	0.776834i	$-0.283176\pi$
$181$	3066.80	1.25941	0.629705	+	0.776834i	$0.283176\pi$
$182$	0	0
$183$	972.000	0.392636
$184$	0	0
$185$	107.465	0.0427081
$186$	0	0
$187$	4669.15	1.82589
$188$	0	0
$189$	173.400	0.0667352
$190$	0	0
$191$	1856.44	0.703286	0.351643	−	0.936134i	$-0.385623\pi$
$191$	1856.44	0.703286	0.351643	+	0.936134i	$0.385623\pi$
$192$	0	0
$193$	3008.84	1.12218	0.561091	−	0.827754i	$-0.310382\pi$
$193$	3008.84	1.12218	0.561091	+	0.827754i	$0.310382\pi$
$194$	0	0
$195$	−2027.47	−0.744563
$196$	0	0
$197$	−909.309	−0.328861	−0.164430	−	0.986389i	$-0.552579\pi$
$197$	−909.309	−0.328861	−0.164430	+	0.986389i	$0.552579\pi$
$198$	0	0
$199$	−2780.24	−0.990383	−0.495191	−	0.868784i	$-0.664902\pi$
$199$	−2780.24	−0.990383	−0.495191	+	0.868784i	$0.664902\pi$
$200$	0	0
$201$	−2760.80	−0.968814
$202$	0	0
$203$	−345.514	−0.119460
$204$	0	0
$205$	−734.183	−0.250134
$206$	0	0
$207$	1411.60	0.473976
$208$	0	0
$209$	−636.085	−0.210521
$210$	0	0
$211$	385.511	0.125780	0.0628901	−	0.998020i	$-0.479968\pi$
$211$	385.511	0.125780	0.0628901	+	0.998020i	$0.479968\pi$
$212$	0	0
$213$	−2985.47	−0.960379
$214$	0	0
$215$	3109.06	0.986215
$216$	0	0
$217$	1460.98	0.457040
$218$	0	0
$219$	1086.80	0.335338
$220$	0	0
$221$	4908.00	1.49388
$222$	0	0
$223$	601.975	0.180768	0.0903839	−	0.995907i	$-0.471191\pi$
$223$	601.975	0.180768	0.0903839	+	0.995907i	$0.471191\pi$
$224$	0	0
$225$	−147.399	−0.0436737
$226$	0	0
$227$	−185.779	−0.0543199	−0.0271600	−	0.999631i	$-0.508646\pi$
$227$	−185.779	−0.0543199	−0.0271600	+	0.999631i	$0.508646\pi$
$228$	0	0
$229$	−1237.11	−0.356989	−0.178495	−	0.983941i	$-0.557123\pi$
$229$	−1237.11	−0.356989	−0.178495	+	0.983941i	$0.557123\pi$
$230$	0	0
$231$	−1188.53	−0.338527
$232$	0	0
$233$	−217.378	−0.0611197	−0.0305598	−	0.999533i	$-0.509729\pi$
$233$	−217.378	−0.0611197	−0.0305598	+	0.999533i	$0.509729\pi$
$234$	0	0
$235$	−937.070	−0.260118
$236$	0	0
$237$	−3296.73	−0.903568
$238$	0	0
$239$	−6055.99	−1.63904	−0.819518	−	0.573053i	$-0.805759\pi$
$239$	−6055.99	−1.63904	−0.819518	+	0.573053i	$0.805759\pi$
$240$	0	0
$241$	−6815.69	−1.82173	−0.910865	−	0.412705i	$-0.864584\pi$
$241$	−6815.69	−1.82173	−0.910865	+	0.412705i	$0.864584\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3144.96	0.820097
$246$	0	0
$247$	−668.622	−0.172241
$248$	0	0
$249$	−2375.47	−0.604574
$250$	0	0
$251$	2448.53	0.615738	0.307869	−	0.951429i	$-0.400384\pi$
$251$	2448.53	0.615738	0.307869	+	0.951429i	$0.400384\pi$
$252$	0	0
$253$	−9675.55	−2.40433
$254$	0	0
$255$	−2366.53	−0.581169
$256$	0	0
$257$	2949.55	0.715907	0.357953	−	0.933739i	$-0.383475\pi$
$257$	2949.55	0.715907	0.357953	+	0.933739i	$0.383475\pi$
$258$	0	0
$259$	66.2205	0.0158870
$260$	0	0
$261$	484.199	0.114832
$262$	0	0
$263$	−4831.64	−1.13282	−0.566410	−	0.824124i	$-0.691668\pi$
$263$	−4831.64	−1.13282	−0.566410	+	0.824124i	$0.691668\pi$
$264$	0	0
$265$	4047.29	0.938199
$266$	0	0
$267$	−451.867	−0.103572
$268$	0	0
$269$	−1093.84	−0.247928	−0.123964	−	0.992287i	$-0.539561\pi$
$269$	−1093.84	−0.247928	−0.123964	+	0.992287i	$0.539561\pi$
$270$	0	0
$271$	−3079.80	−0.690349	−0.345175	−	0.938539i	$-0.612180\pi$
$271$	−3079.80	−0.690349	−0.345175	+	0.938539i	$0.612180\pi$
$272$	0	0
$273$	−1249.33	−0.276971
$274$	0	0
$275$	1010.32	0.221543
$276$	0	0
$277$	−5600.75	−1.21486	−0.607430	−	0.794373i	$-0.707800\pi$
$277$	−5600.75	−1.21486	−0.607430	+	0.794373i	$0.707800\pi$
$278$	0	0
$279$	−2047.40	−0.439335
$280$	0	0
$281$	203.426	0.0431864	0.0215932	−	0.999767i	$-0.493126\pi$
$281$	203.426	0.0431864	0.0215932	+	0.999767i	$0.493126\pi$
$282$	0	0
$283$	−2214.84	−0.465225	−0.232612	−	0.972570i	$-0.574727\pi$
$283$	−2214.84	−0.465225	−0.232612	+	0.972570i	$0.574727\pi$
$284$	0	0
$285$	322.396	0.0670073
$286$	0	0
$287$	−452.406	−0.0930478
$288$	0	0
$289$	815.798	0.166049
$290$	0	0
$291$	5637.46	1.13565
$292$	0	0
$293$	5812.37	1.15892	0.579458	−	0.815002i	$-0.303264\pi$
$293$	5812.37	1.15892	0.579458	+	0.815002i	$0.303264\pi$
$294$	0	0
$295$	3376.79	0.666456
$296$	0	0
$297$	1665.60	0.325413
$298$	0	0
$299$	−10170.5	−1.96714
$300$	0	0
$301$	1915.82	0.366863
$302$	0	0
$303$	−5167.53	−0.979758
$304$	0	0
$305$	3376.79	0.633950
$306$	0	0
$307$	7337.33	1.36405	0.682025	−	0.731329i	$-0.261099\pi$
$307$	7337.33	1.36405	0.682025	+	0.731329i	$0.261099\pi$
$308$	0	0
$309$	−5726.46	−1.05426
$310$	0	0
$311$	−6575.91	−1.19899	−0.599494	−	0.800379i	$-0.704632\pi$
$311$	−6575.91	−1.19899	−0.599494	+	0.800379i	$0.704632\pi$
$312$	0	0
$313$	−1556.67	−0.281113	−0.140556	−	0.990073i	$-0.544889\pi$
$313$	−1556.67	−0.281113	−0.140556	+	0.990073i	$0.544889\pi$
$314$	0	0
$315$	602.402	0.107751
$316$	0	0
$317$	9457.27	1.67562	0.837812	−	0.545959i	$-0.183834\pi$
$317$	9457.27	1.67562	0.837812	+	0.545959i	$0.183834\pi$
$318$	0	0
$319$	−3318.85	−0.582507
$320$	0	0
$321$	385.867	0.0670935
$322$	0	0
$323$	−780.441	−0.134442
$324$	0	0
$325$	1062.00	0.181259
$326$	0	0
$327$	2274.80	0.384700
$328$	0	0
$329$	−577.426	−0.0967615
$330$	0	0
$331$	−10907.6	−1.81129	−0.905647	−	0.424032i	$-0.860614\pi$
$331$	−10907.6	−1.81129	−0.905647	+	0.424032i	$0.860614\pi$
$332$	0	0
$333$	−92.8006	−0.0152716
$334$	0	0
$335$	−9591.20	−1.56425
$336$	0	0
$337$	4256.22	0.687986	0.343993	−	0.938972i	$-0.388220\pi$
$337$	4256.22	0.687986	0.343993	+	0.938972i	$0.388220\pi$
$338$	0	0
$339$	−2764.93	−0.442981
$340$	0	0
$341$	14033.5	2.22861
$342$	0	0
$343$	4140.75	0.651835
$344$	0	0
$345$	4903.99	0.765282
$346$	0	0
$347$	4217.07	0.652403	0.326202	−	0.945300i	$-0.394231\pi$
$347$	4217.07	0.652403	0.326202	+	0.945300i	$0.394231\pi$
$348$	0	0
$349$	2986.57	0.458073	0.229037	−	0.973418i	$-0.426442\pi$
$349$	2986.57	0.458073	0.229037	+	0.973418i	$0.426442\pi$
$350$	0	0
$351$	1750.80	0.266241
$352$	0	0
$353$	−8231.42	−1.24112	−0.620558	−	0.784160i	$-0.713094\pi$
$353$	−8231.42	−1.24112	−0.620558	+	0.784160i	$0.713094\pi$
$354$	0	0
$355$	−10371.7	−1.55063
$356$	0	0
$357$	−1458.27	−0.216190
$358$	0	0
$359$	5857.19	0.861088	0.430544	−	0.902569i	$-0.358322\pi$
$359$	5857.19	0.861088	0.430544	+	0.902569i	$0.358322\pi$
$360$	0	0
$361$	−6752.68	−0.984499
$362$	0	0
$363$	−7423.53	−1.07337
$364$	0	0
$365$	3775.61	0.541437
$366$	0	0
$367$	−10715.7	−1.52412	−0.762061	−	0.647505i	$-0.775812\pi$
$367$	−10715.7	−1.52412	−0.762061	+	0.647505i	$0.775812\pi$
$368$	0	0
$369$	633.997	0.0894433
$370$	0	0
$371$	2493.95	0.349002
$372$	0	0
$373$	−1817.52	−0.252299	−0.126149	−	0.992011i	$-0.540262\pi$
$373$	−1817.52	−0.252299	−0.126149	+	0.992011i	$0.540262\pi$
$374$	0	0
$375$	−4420.40	−0.608716
$376$	0	0
$377$	−3488.62	−0.476586
$378$	0	0
$379$	9789.68	1.32681	0.663407	−	0.748259i	$-0.269110\pi$
$379$	9789.68	1.32681	0.663407	+	0.748259i	$0.269110\pi$
$380$	0	0
$381$	−4941.67	−0.664486
$382$	0	0
$383$	−11502.3	−1.53457	−0.767285	−	0.641306i	$-0.778393\pi$
$383$	−11502.3	−1.53457	−0.767285	+	0.641306i	$0.778393\pi$
$384$	0	0
$385$	−4129.05	−0.546587
$386$	0	0
$387$	−2684.80	−0.352651
$388$	0	0
$389$	−6.06261	−0.000790197 0	−0.000395098	−	1.00000i	$-0.500126\pi$
$389$	−6.06261	−0.000790197 0	−0.000395098		1.00000i	$0.500126\pi$
$390$	0	0
$391$	−11871.4	−1.53545
$392$	0	0
$393$	6000.80	0.770230
$394$	0	0
$395$	−11453.1	−1.45890
$396$	0	0
$397$	5982.31	0.756281	0.378141	−	0.925748i	$-0.376564\pi$
$397$	5982.31	0.756281	0.378141	+	0.925748i	$0.376564\pi$
$398$	0	0
$399$	198.662	0.0249261
$400$	0	0
$401$	10443.6	1.30057	0.650284	−	0.759691i	$-0.274650\pi$
$401$	10443.6	1.30057	0.650284	+	0.759691i	$0.274650\pi$
$402$	0	0
$403$	14751.4	1.82337
$404$	0	0
$405$	−844.199	−0.103577
$406$	0	0
$407$	636.085	0.0774682
$408$	0	0
$409$	−8141.01	−0.984223	−0.492112	−	0.870532i	$-0.663775\pi$
$409$	−8141.01	−0.984223	−0.492112	+	0.870532i	$0.663775\pi$
$410$	0	0
$411$	7724.13	0.927015
$412$	0	0
$413$	2080.79	0.247916
$414$	0	0
$415$	−8252.53	−0.976146
$416$	0	0
$417$	3993.86	0.469017
$418$	0	0
$419$	8716.96	1.01635	0.508176	−	0.861253i	$-0.330320\pi$
$419$	8716.96	1.01635	0.508176	+	0.861253i	$0.330320\pi$
$420$	0	0
$421$	−13437.5	−1.55560	−0.777798	−	0.628514i	$-0.783663\pi$
$421$	−13437.5	−1.55560	−0.777798	+	0.628514i	$0.783663\pi$
$422$	0	0
$423$	809.198	0.0930131
$424$	0	0
$425$	1239.60	0.141482
$426$	0	0
$427$	2080.79	0.235824
$428$	0	0
$429$	−12000.5	−1.35056
$430$	0	0
$431$	7343.86	0.820745	0.410373	−	0.911918i	$-0.365399\pi$
$431$	7343.86	0.820745	0.410373	+	0.911918i	$0.365399\pi$
$432$	0	0
$433$	4490.80	0.498416	0.249208	−	0.968450i	$-0.419830\pi$
$433$	4490.80	0.498416	0.249208	+	0.968450i	$0.419830\pi$
$434$	0	0
$435$	1682.14	0.185408
$436$	0	0
$437$	1617.25	0.177033
$438$	0	0
$439$	−9437.93	−1.02608	−0.513038	−	0.858366i	$-0.671480\pi$
$439$	−9437.93	−1.02608	−0.513038	+	0.858366i	$0.671480\pi$
$440$	0	0
$441$	−2715.80	−0.293251
$442$	0	0
$443$	12668.3	1.35866	0.679331	−	0.733832i	$-0.262270\pi$
$443$	12668.3	1.35866	0.679331	+	0.733832i	$0.262270\pi$
$444$	0	0
$445$	−1569.82	−0.167228
$446$	0	0
$447$	1657.13	0.175346
$448$	0	0
$449$	−13052.0	−1.37186	−0.685929	−	0.727669i	$-0.740604\pi$
$449$	−13052.0	−1.37186	−0.685929	+	0.727669i	$0.740604\pi$
$450$	0	0
$451$	−4345.61	−0.453718
$452$	0	0
$453$	3921.40	0.406718
$454$	0	0
$455$	−4340.27	−0.447197
$456$	0	0
$457$	−1313.64	−0.134463	−0.0672316	−	0.997737i	$-0.521417\pi$
$457$	−1313.64	−0.134463	−0.0672316	+	0.997737i	$0.521417\pi$
$458$	0	0
$459$	2043.60	0.207815
$460$	0	0
$461$	627.883	0.0634347	0.0317174	−	0.999497i	$-0.489902\pi$
$461$	627.883	0.0634347	0.0317174	+	0.999497i	$0.489902\pi$
$462$	0	0
$463$	7315.03	0.734251	0.367126	−	0.930171i	$-0.380342\pi$
$463$	7315.03	0.734251	0.367126	+	0.930171i	$0.380342\pi$
$464$	0	0
$465$	−7112.80	−0.709351
$466$	0	0
$467$	759.997	0.0753072	0.0376536	−	0.999291i	$-0.488012\pi$
$467$	759.997	0.0753072	0.0376536	+	0.999291i	$0.488012\pi$
$468$	0	0
$469$	−5910.14	−0.581886
$470$	0	0
$471$	10581.3	1.03516
$472$	0	0
$473$	18402.5	1.78889
$474$	0	0
$475$	−168.873	−0.0163125
$476$	0	0
$477$	−3495.00	−0.335482
$478$	0	0
$479$	7403.51	0.706211	0.353106	−	0.935583i	$-0.385126\pi$
$479$	7403.51	0.706211	0.353106	+	0.935583i	$0.385126\pi$
$480$	0	0
$481$	668.622	0.0633816
$482$	0	0
$483$	3021.86	0.284678
$484$	0	0
$485$	19584.9	1.83362
$486$	0	0
$487$	−3488.11	−0.324561	−0.162281	−	0.986745i	$-0.551885\pi$
$487$	−3488.11	−0.324561	−0.162281	+	0.986745i	$0.551885\pi$
$488$	0	0
$489$	2778.14	0.256915
$490$	0	0
$491$	6575.73	0.604396	0.302198	−	0.953245i	$-0.402280\pi$
$491$	6575.73	0.604396	0.302198	+	0.953245i	$0.402280\pi$
$492$	0	0
$493$	−4072.05	−0.372000
$494$	0	0
$495$	5786.40	0.525413
$496$	0	0
$497$	−6391.09	−0.576820
$498$	0	0
$499$	−5187.82	−0.465408	−0.232704	−	0.972548i	$-0.574757\pi$
$499$	−5187.82	−0.465408	−0.232704	+	0.972548i	$0.574757\pi$
$500$	0	0
$501$	4180.00	0.372751
$502$	0	0
$503$	−7248.38	−0.642524	−0.321262	−	0.946990i	$-0.604107\pi$
$503$	−7248.38	−0.642524	−0.321262	+	0.946990i	$0.604107\pi$
$504$	0	0
$505$	−17952.4	−1.58192
$506$	0	0
$507$	−6023.39	−0.527630
$508$	0	0
$509$	6613.44	0.575905	0.287952	−	0.957645i	$-0.407026\pi$
$509$	6613.44	0.575905	0.287952	+	0.957645i	$0.407026\pi$
$510$	0	0
$511$	2326.55	0.201410
$512$	0	0
$513$	−278.402	−0.0239605
$514$	0	0
$515$	−19894.1	−1.70221
$516$	0	0
$517$	−5546.50	−0.471827
$518$	0	0
$519$	−9935.13	−0.840277
$520$	0	0
$521$	20761.3	1.74581	0.872906	−	0.487888i	$-0.162233\pi$
$521$	20761.3	1.74581	0.872906	+	0.487888i	$0.162233\pi$
$522$	0	0
$523$	−13494.2	−1.12822	−0.564112	−	0.825698i	$-0.690781\pi$
$523$	−13494.2	−1.12822	−0.564112	+	0.825698i	$0.690781\pi$
$524$	0	0
$525$	−315.542	−0.0262312
$526$	0	0
$527$	17218.3	1.42323
$528$	0	0
$529$	12433.2	1.02188
$530$	0	0
$531$	−2916.00	−0.238312
$532$	0	0
$533$	−4567.91	−0.371216
$534$	0	0
$535$	1340.53	0.108329
$536$	0	0
$537$	−969.861	−0.0779378
$538$	0	0
$539$	18614.9	1.48757
$540$	0	0
$541$	2708.58	0.215251	0.107626	−	0.994191i	$-0.465675\pi$
$541$	2708.58	0.215251	0.107626	+	0.994191i	$0.465675\pi$
$542$	0	0
$543$	−9200.39	−0.727121
$544$	0	0
$545$	7902.82	0.621137
$546$	0	0
$547$	−15783.5	−1.23373	−0.616866	−	0.787068i	$-0.711598\pi$
$547$	−15783.5	−1.23373	−0.616866	+	0.787068i	$0.711598\pi$
$548$	0	0
$549$	−2916.00	−0.226688
$550$	0	0
$551$	554.740	0.0428906
$552$	0	0
$553$	−7057.43	−0.542699
$554$	0	0
$555$	−322.396	−0.0246575
$556$	0	0
$557$	1892.77	0.143984	0.0719922	−	0.997405i	$-0.477064\pi$
$557$	1892.77	0.143984	0.0719922	+	0.997405i	$0.477064\pi$
$558$	0	0
$559$	19343.8	1.46361
$560$	0	0
$561$	−14007.5	−1.05418
$562$	0	0
$563$	−3876.26	−0.290169	−0.145084	−	0.989419i	$-0.546345\pi$
$563$	−3876.26	−0.290169	−0.145084	+	0.989419i	$0.546345\pi$
$564$	0	0
$565$	−9605.56	−0.715237
$566$	0	0
$567$	−520.199	−0.0385296
$568$	0	0
$569$	−14900.9	−1.09785	−0.548927	−	0.835870i	$-0.684963\pi$
$569$	−14900.9	−1.09785	−0.548927	+	0.835870i	$0.684963\pi$
$570$	0	0
$571$	24926.0	1.82683	0.913414	−	0.407032i	$-0.133436\pi$
$571$	24926.0	1.82683	0.913414	+	0.407032i	$0.133436\pi$
$572$	0	0
$573$	−5569.33	−0.406042
$574$	0	0
$575$	−2568.74	−0.186302
$576$	0	0
$577$	21022.4	1.51677	0.758384	−	0.651808i	$-0.225989\pi$
$577$	21022.4	1.51677	0.758384	+	0.651808i	$0.225989\pi$
$578$	0	0
$579$	−9026.52	−0.647892
$580$	0	0
$581$	−5085.24	−0.363118
$582$	0	0
$583$	23955.8	1.70180
$584$	0	0
$585$	6082.40	0.429874
$586$	0	0
$587$	−13847.0	−0.973641	−0.486821	−	0.873502i	$-0.661843\pi$
$587$	−13847.0	−0.973641	−0.486821	+	0.873502i	$0.661843\pi$
$588$	0	0
$589$	−2345.68	−0.164095
$590$	0	0
$591$	2727.93	0.189868
$592$	0	0
$593$	−4113.01	−0.284825	−0.142413	−	0.989807i	$-0.545486\pi$
$593$	−4113.01	−0.284825	−0.142413	+	0.989807i	$0.545486\pi$
$594$	0	0
$595$	−5066.12	−0.349060
$596$	0	0
$597$	8340.73	0.571798
$598$	0	0
$599$	4863.60	0.331755	0.165878	−	0.986146i	$-0.446954\pi$
$599$	4863.60	0.331755	0.165878	+	0.986146i	$0.446954\pi$
$600$	0	0
$601$	6827.76	0.463411	0.231706	−	0.972786i	$-0.425569\pi$
$601$	6827.76	0.463411	0.231706	+	0.972786i	$0.425569\pi$
$602$	0	0
$603$	8282.39	0.559345
$604$	0	0
$605$	−25789.9	−1.73307
$606$	0	0
$607$	18178.3	1.21554	0.607770	−	0.794113i	$-0.292064\pi$
$607$	18178.3	1.21554	0.607770	+	0.794113i	$0.292064\pi$
$608$	0	0
$609$	1036.54	0.0689700
$610$	0	0
$611$	−5830.22	−0.386032
$612$	0	0
$613$	−15687.2	−1.03360	−0.516802	−	0.856105i	$-0.672878\pi$
$613$	−15687.2	−1.03360	−0.516802	+	0.856105i	$0.672878\pi$
$614$	0	0
$615$	2202.55	0.144415
$616$	0	0
$617$	−16420.2	−1.07140	−0.535700	−	0.844409i	$-0.679952\pi$
$617$	−16420.2	−1.07140	−0.535700	+	0.844409i	$0.679952\pi$
$618$	0	0
$619$	5517.78	0.358285	0.179142	−	0.983823i	$-0.442668\pi$
$619$	5517.78	0.358285	0.179142	+	0.983823i	$0.442668\pi$
$620$	0	0
$621$	−4234.80	−0.273650
$622$	0	0
$623$	−967.328	−0.0622073
$624$	0	0
$625$	−13309.6	−0.851812
$626$	0	0
$627$	1908.25	0.121544
$628$	0	0
$629$	780.441	0.0494725
$630$	0	0
$631$	1559.17	0.0983670	0.0491835	−	0.998790i	$-0.484338\pi$
$631$	1559.17	0.0983670	0.0491835	+	0.998790i	$0.484338\pi$
$632$	0	0
$633$	−1156.53	−0.0726193
$634$	0	0
$635$	−17167.7	−1.07288
$636$	0	0
$637$	19567.1	1.21708
$638$	0	0
$639$	8956.40	0.554475
$640$	0	0
$641$	−15188.6	−0.935900	−0.467950	−	0.883755i	$-0.655007\pi$
$641$	−15188.6	−0.935900	−0.467950	+	0.883755i	$0.655007\pi$
$642$	0	0
$643$	16666.6	1.02219	0.511095	−	0.859524i	$-0.329240\pi$
$643$	16666.6	1.02219	0.511095	+	0.859524i	$0.329240\pi$
$644$	0	0
$645$	−9327.18	−0.569391
$646$	0	0
$647$	3038.27	0.184616	0.0923081	−	0.995730i	$-0.470576\pi$
$647$	3038.27	0.184616	0.0923081	+	0.995730i	$0.470576\pi$
$648$	0	0
$649$	19987.2	1.20888
$650$	0	0
$651$	−4382.94	−0.263872
$652$	0	0
$653$	9078.99	0.544086	0.272043	−	0.962285i	$-0.412301\pi$
$653$	9078.99	0.544086	0.272043	+	0.962285i	$0.412301\pi$
$654$	0	0
$655$	20847.2	1.24361
$656$	0	0
$657$	−3260.39	−0.193607
$658$	0	0
$659$	16892.0	0.998511	0.499255	−	0.866455i	$-0.333607\pi$
$659$	16892.0	0.998511	0.499255	+	0.866455i	$0.333607\pi$
$660$	0	0
$661$	−5064.16	−0.297992	−0.148996	−	0.988838i	$-0.547604\pi$
$661$	−5064.16	−0.297992	−0.148996	+	0.988838i	$0.547604\pi$
$662$	0	0
$663$	−14724.0	−0.862492
$664$	0	0
$665$	690.164	0.0402457
$666$	0	0
$667$	8438.21	0.489848
$668$	0	0
$669$	−1805.93	−0.104366
$670$	0	0
$671$	19987.2	1.14992
$672$	0	0
$673$	28625.9	1.63959	0.819797	−	0.572654i	$-0.194086\pi$
$673$	28625.9	1.63959	0.819797	+	0.572654i	$0.194086\pi$
$674$	0	0
$675$	442.196	0.0252150
$676$	0	0
$677$	16061.2	0.911791	0.455895	−	0.890033i	$-0.349319\pi$
$677$	16061.2	0.911791	0.455895	+	0.890033i	$0.349319\pi$
$678$	0	0
$679$	12068.3	0.682090
$680$	0	0
$681$	557.338	0.0313616
$682$	0	0
$683$	−7868.09	−0.440796	−0.220398	−	0.975410i	$-0.570736\pi$
$683$	−7868.09	−0.440796	−0.220398	+	0.975410i	$0.570736\pi$
$684$	0	0
$685$	26834.2	1.49676
$686$	0	0
$687$	3711.33	0.206108
$688$	0	0
$689$	25181.2	1.39235
$690$	0	0
$691$	−16886.2	−0.929641	−0.464820	−	0.885405i	$-0.653881\pi$
$691$	−16886.2	−0.929641	−0.464820	+	0.885405i	$0.653881\pi$
$692$	0	0
$693$	3565.60	0.195449
$694$	0	0
$695$	13874.9	0.757276
$696$	0	0
$697$	−5331.83	−0.289753
$698$	0	0
$699$	652.133	0.0352875
$700$	0	0
$701$	−18293.6	−0.985647	−0.492824	−	0.870129i	$-0.664035\pi$
$701$	−18293.6	−0.985647	−0.492824	+	0.870129i	$0.664035\pi$
$702$	0	0
$703$	−106.320	−0.00570406
$704$	0	0
$705$	2811.21	0.150179
$706$	0	0
$707$	−11062.3	−0.588460
$708$	0	0
$709$	21555.6	1.14180	0.570900	−	0.821019i	$-0.306594\pi$
$709$	21555.6	1.14180	0.570900	+	0.821019i	$0.306594\pi$
$710$	0	0
$711$	9890.19	0.521675
$712$	0	0
$713$	−35680.3	−1.87411
$714$	0	0
$715$	−41690.6	−2.18062
$716$	0	0
$717$	18168.0	0.946298
$718$	0	0
$719$	−9059.31	−0.469896	−0.234948	−	0.972008i	$-0.575492\pi$
$719$	−9059.31	−0.469896	−0.234948	+	0.972008i	$0.575492\pi$
$720$	0	0
$721$	−12258.8	−0.633208
$722$	0	0
$723$	20447.1	1.05178
$724$	0	0
$725$	−881.115	−0.0451362
$726$	0	0
$727$	7074.83	0.360923	0.180462	−	0.983582i	$-0.442241\pi$
$727$	7074.83	0.360923	0.180462	+	0.983582i	$0.442241\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	22578.8	1.14242
$732$	0	0
$733$	17095.9	0.861459	0.430730	−	0.902481i	$-0.358256\pi$
$733$	17095.9	0.861459	0.430730	+	0.902481i	$0.358256\pi$
$734$	0	0
$735$	−9434.87	−0.473483
$736$	0	0
$737$	−56770.1	−2.83739
$738$	0	0
$739$	28157.4	1.40161	0.700803	−	0.713355i	$-0.252825\pi$
$739$	28157.4	1.40161	0.700803	+	0.713355i	$0.252825\pi$
$740$	0	0
$741$	2005.87	0.0994431
$742$	0	0
$743$	34033.2	1.68043	0.840214	−	0.542255i	$-0.182429\pi$
$743$	34033.2	1.68043	0.840214	+	0.542255i	$0.182429\pi$
$744$	0	0
$745$	5756.99	0.283114
$746$	0	0
$747$	7126.40	0.349051
$748$	0	0
$749$	826.039	0.0402975
$750$	0	0
$751$	−34356.8	−1.66937	−0.834685	−	0.550728i	$-0.814350\pi$
$751$	−34356.8	−1.66937	−0.834685	+	0.550728i	$0.814350\pi$
$752$	0	0
$753$	−7345.60	−0.355496
$754$	0	0
$755$	13623.2	0.656687
$756$	0	0
$757$	29464.6	1.41467	0.707337	−	0.706876i	$-0.249896\pi$
$757$	29464.6	1.41467	0.707337	+	0.706876i	$0.249896\pi$
$758$	0	0
$759$	29026.6	1.38814
$760$	0	0
$761$	27284.5	1.29969	0.649843	−	0.760069i	$-0.274835\pi$
$761$	27284.5	1.29969	0.649843	+	0.760069i	$0.274835\pi$
$762$	0	0
$763$	4869.75	0.231057
$764$	0	0
$765$	7099.60	0.335538
$766$	0	0
$767$	21009.6	0.989064
$768$	0	0
$769$	2320.13	0.108798	0.0543992	−	0.998519i	$-0.482676\pi$
$769$	2320.13	0.108798	0.0543992	+	0.998519i	$0.482676\pi$
$770$	0	0
$771$	−8848.66	−0.413329
$772$	0	0
$773$	−4217.83	−0.196254	−0.0981272	−	0.995174i	$-0.531285\pi$
$773$	−4217.83	−0.196254	−0.0981272	+	0.995174i	$0.531285\pi$
$774$	0	0
$775$	3725.73	0.172687
$776$	0	0
$777$	−198.662	−0.00917238
$778$	0	0
$779$	726.362	0.0334077
$780$	0	0
$781$	−61390.0	−2.81268
$782$	0	0
$783$	−1452.60	−0.0662983
$784$	0	0
$785$	36760.3	1.67138
$786$	0	0
$787$	11436.3	0.517991	0.258995	−	0.965879i	$-0.416609\pi$
$787$	11436.3	0.517991	0.258995	+	0.965879i	$0.416609\pi$
$788$	0	0
$789$	14494.9	0.654034
$790$	0	0
$791$	−5918.98	−0.266062
$792$	0	0
$793$	21009.6	0.940823
$794$	0	0
$795$	−12141.9	−0.541670
$796$	0	0
$797$	14102.3	0.626761	0.313381	−	0.949628i	$-0.398538\pi$
$797$	14102.3	0.626761	0.313381	+	0.949628i	$0.398538\pi$
$798$	0	0
$799$	−6805.25	−0.301317
$800$	0	0
$801$	1355.60	0.0597975
$802$	0	0
$803$	22347.8	0.982111
$804$	0	0
$805$	10498.2	0.459641
$806$	0	0
$807$	3281.52	0.143141
$808$	0	0
$809$	18791.3	0.816647	0.408323	−	0.912837i	$-0.366114\pi$
$809$	18791.3	0.816647	0.408323	+	0.912837i	$0.366114\pi$
$810$	0	0
$811$	14452.6	0.625771	0.312886	−	0.949791i	$-0.398704\pi$
$811$	14452.6	0.625771	0.312886	+	0.949791i	$0.398704\pi$
$812$	0	0
$813$	9239.40	0.398573
$814$	0	0
$815$	9651.43	0.414816
$816$	0	0
$817$	−3075.94	−0.131718
$818$	0	0
$819$	3748.00	0.159909
$820$	0	0
$821$	−27786.1	−1.18117	−0.590585	−	0.806975i	$-0.701103\pi$
$821$	−27786.1	−1.18117	−0.590585	+	0.806975i	$0.701103\pi$
$822$	0	0
$823$	39205.8	1.66055	0.830273	−	0.557357i	$-0.188185\pi$
$823$	39205.8	1.66055	0.830273	+	0.557357i	$0.188185\pi$
$824$	0	0
$825$	−3030.95	−0.127908
$826$	0	0
$827$	−5929.15	−0.249307	−0.124653	−	0.992200i	$-0.539782\pi$
$827$	−5929.15	−0.249307	−0.124653	+	0.992200i	$0.539782\pi$
$828$	0	0
$829$	1269.44	0.0531840	0.0265920	−	0.999646i	$-0.491535\pi$
$829$	1269.44	0.0531840	0.0265920	+	0.999646i	$0.491535\pi$
$830$	0	0
$831$	16802.3	0.701400
$832$	0	0
$833$	22839.5	0.949990
$834$	0	0
$835$	14521.6	0.601845
$836$	0	0
$837$	6142.19	0.253650
$838$	0	0
$839$	17884.1	0.735910	0.367955	−	0.929844i	$-0.380058\pi$
$839$	17884.1	0.735910	0.367955	+	0.929844i	$0.380058\pi$
$840$	0	0
$841$	−21494.6	−0.881323
$842$	0	0
$843$	−610.278	−0.0249337
$844$	0	0
$845$	−20925.7	−0.851911
$846$	0	0
$847$	−15891.8	−0.644686
$848$	0	0
$849$	6644.52	0.268598
$850$	0	0
$851$	−1617.25	−0.0651453
$852$	0	0
$853$	−16083.4	−0.645585	−0.322793	−	0.946470i	$-0.604622\pi$
$853$	−16083.4	−0.645585	−0.322793	+	0.946470i	$0.604622\pi$
$854$	0	0
$855$	−967.187	−0.0386867
$856$	0	0
$857$	17203.6	0.685722	0.342861	−	0.939386i	$-0.388604\pi$
$857$	17203.6	0.685722	0.342861	+	0.939386i	$0.388604\pi$
$858$	0	0
$859$	−24427.5	−0.970264	−0.485132	−	0.874441i	$-0.661228\pi$
$859$	−24427.5	−0.970264	−0.485132	+	0.874441i	$0.661228\pi$
$860$	0	0
$861$	1357.22	0.0537212
$862$	0	0
$863$	−46584.1	−1.83747	−0.918737	−	0.394870i	$-0.870790\pi$
$863$	−46584.1	−1.83747	−0.918737	+	0.394870i	$0.870790\pi$
$864$	0	0
$865$	−34515.3	−1.35671
$866$	0	0
$867$	−2447.39	−0.0958683
$868$	0	0
$869$	−67790.5	−2.64630
$870$	0	0
$871$	−59674.1	−2.32145
$872$	0	0
$873$	−16912.4	−0.655667
$874$	0	0
$875$	−9462.91	−0.365605
$876$	0	0
$877$	7402.05	0.285005	0.142503	−	0.989794i	$-0.454485\pi$
$877$	7402.05	0.285005	0.142503	+	0.989794i	$0.454485\pi$
$878$	0	0
$879$	−17437.1	−0.669101
$880$	0	0
$881$	12044.7	0.460608	0.230304	−	0.973119i	$-0.426028\pi$
$881$	12044.7	0.460608	0.230304	+	0.973119i	$0.426028\pi$
$882$	0	0
$883$	−7150.02	−0.272500	−0.136250	−	0.990674i	$-0.543505\pi$
$883$	−7150.02	−0.272500	−0.136250	+	0.990674i	$0.543505\pi$
$884$	0	0
$885$	−10130.4	−0.384779
$886$	0	0
$887$	21897.6	0.828916	0.414458	−	0.910068i	$-0.363971\pi$
$887$	21897.6	0.828916	0.414458	+	0.910068i	$0.363971\pi$
$888$	0	0
$889$	−10578.8	−0.399102
$890$	0	0
$891$	−4996.79	−0.187878
$892$	0	0
$893$	927.087	0.0347411
$894$	0	0
$895$	−3369.36	−0.125838
$896$	0	0
$897$	30511.4	1.13573
$898$	0	0
$899$	−12238.9	−0.454047
$900$	0	0
$901$	29392.5	1.08680
$902$	0	0
$903$	−5747.45	−0.211808
$904$	0	0
$905$	−31962.8	−1.17401
$906$	0	0
$907$	800.885	0.0293197	0.0146598	−	0.999893i	$-0.495333\pi$
$907$	800.885	0.0293197	0.0146598	+	0.999893i	$0.495333\pi$
$908$	0	0
$909$	15502.6	0.565664
$910$	0	0
$911$	−26742.9	−0.972593	−0.486296	−	0.873794i	$-0.661652\pi$
$911$	−26742.9	−0.972593	−0.486296	+	0.873794i	$0.661652\pi$
$912$	0	0
$913$	−48846.5	−1.77063
$914$	0	0
$915$	−10130.4	−0.366011
$916$	0	0
$917$	12846.1	0.462613
$918$	0	0
$919$	−3744.72	−0.134414	−0.0672072	−	0.997739i	$-0.521409\pi$
$919$	−3744.72	−0.134414	−0.0672072	+	0.997739i	$0.521409\pi$
$920$	0	0
$921$	−22012.0	−0.787535
$922$	0	0
$923$	−64530.3	−2.30124
$924$	0	0
$925$	168.873	0.00600271
$926$	0	0
$927$	17179.4	0.608679
$928$	0	0
$929$	−25667.3	−0.906477	−0.453238	−	0.891389i	$-0.649731\pi$
$929$	−25667.3	−0.906477	−0.453238	+	0.891389i	$0.649731\pi$
$930$	0	0
$931$	−3111.45	−0.109531
$932$	0	0
$933$	19727.7	0.692236
$934$	0	0
$935$	−48662.9	−1.70208
$936$	0	0
$937$	17978.7	0.626829	0.313414	−	0.949616i	$-0.398527\pi$
$937$	17978.7	0.626829	0.313414	+	0.949616i	$0.398527\pi$
$938$	0	0
$939$	4670.01	0.162300
$940$	0	0
$941$	−8120.01	−0.281302	−0.140651	−	0.990059i	$-0.544919\pi$
$941$	−8120.01	−0.281302	−0.140651	+	0.990059i	$0.544919\pi$
$942$	0	0
$943$	11048.8	0.381545
$944$	0	0
$945$	−1807.21	−0.0622099
$946$	0	0
$947$	−7930.95	−0.272145	−0.136072	−	0.990699i	$-0.543448\pi$
$947$	−7930.95	−0.272145	−0.136072	+	0.990699i	$0.543448\pi$
$948$	0	0
$949$	23490.9	0.803527
$950$	0	0
$951$	−28371.8	−0.967422
$952$	0	0
$953$	29833.3	1.01406	0.507028	−	0.861930i	$-0.330744\pi$
$953$	29833.3	1.01406	0.507028	+	0.861930i	$0.330744\pi$
$954$	0	0
$955$	−19348.2	−0.655596
$956$	0	0
$957$	9956.55	0.336311
$958$	0	0
$959$	16535.3	0.556781
$960$	0	0
$961$	21960.1	0.737139
$962$	0	0
$963$	−1157.60	−0.0387364
$964$	0	0
$965$	−31358.7	−1.04609
$966$	0	0
$967$	7650.56	0.254421	0.127211	−	0.991876i	$-0.459398\pi$
$967$	7650.56	0.254421	0.127211	+	0.991876i	$0.459398\pi$
$968$	0	0
$969$	2341.32	0.0776204
$970$	0	0
$971$	5634.51	0.186220	0.0931102	−	0.995656i	$-0.470319\pi$
$971$	5634.51	0.186220	0.0931102	+	0.995656i	$0.470319\pi$
$972$	0	0
$973$	8549.80	0.281700
$974$	0	0
$975$	−3186.00	−0.104650
$976$	0	0
$977$	43983.5	1.44028	0.720142	−	0.693827i	$-0.244077\pi$
$977$	43983.5	1.44028	0.720142	+	0.693827i	$0.244077\pi$
$978$	0	0
$979$	−9291.72	−0.303335
$980$	0	0
$981$	−6824.41	−0.222107
$982$	0	0
$983$	−7759.30	−0.251763	−0.125882	−	0.992045i	$-0.540176\pi$
$983$	−7759.30	−0.251763	−0.125882	+	0.992045i	$0.540176\pi$
$984$	0	0
$985$	9477.00	0.306561
$986$	0	0
$987$	1732.28	0.0558653
$988$	0	0
$989$	−46788.4	−1.50433
$990$	0	0
$991$	10114.8	0.324226	0.162113	−	0.986772i	$-0.448169\pi$
$991$	10114.8	0.324226	0.162113	+	0.986772i	$0.448169\pi$
$992$	0	0
$993$	32722.9	1.04575
$994$	0	0
$995$	28976.3	0.923225
$996$	0	0
$997$	43842.2	1.39267	0.696337	−	0.717715i	$-0.254812\pi$
$997$	43842.2	1.39267	0.696337	+	0.717715i	$0.254812\pi$
$998$	0	0
$999$	278.402	0.00881706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.j.1.1		2
3.2	odd	2		2304.4.a.t.1.2		2
4.3	odd	2		768.4.a.p.1.1		2
8.3	odd	2		768.4.a.e.1.2		2
8.5	even	2		768.4.a.k.1.2		2
12.11	even	2		2304.4.a.s.1.2		2
16.3	odd	4		384.4.d.e.193.4	yes	4
16.5	even	4		384.4.d.c.193.3	yes	4
16.11	odd	4		384.4.d.e.193.1	yes	4
16.13	even	4		384.4.d.c.193.2	✓	4
24.5	odd	2		2304.4.a.bq.1.1		2
24.11	even	2		2304.4.a.bp.1.1		2
48.5	odd	4		1152.4.d.i.577.3		4
48.11	even	4		1152.4.d.o.577.3		4
48.29	odd	4		1152.4.d.i.577.2		4
48.35	even	4		1152.4.d.o.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.c.193.2	✓	4	16.13	even	4
384.4.d.c.193.3	yes	4	16.5	even	4
384.4.d.e.193.1	yes	4	16.11	odd	4
384.4.d.e.193.4	yes	4	16.3	odd	4
768.4.a.e.1.2		2	8.3	odd	2
768.4.a.j.1.1		2	1.1	even	1	trivial
768.4.a.k.1.2		2	8.5	even	2
768.4.a.p.1.1		2	4.3	odd	2
1152.4.d.i.577.2		4	48.29	odd	4
1152.4.d.i.577.3		4	48.5	odd	4
1152.4.d.o.577.2		4	48.35	even	4
1152.4.d.o.577.3		4	48.11	even	4
2304.4.a.s.1.2		2	12.11	even	2
2304.4.a.t.1.2		2	3.2	odd	2
2304.4.a.bp.1.1		2	24.11	even	2
2304.4.a.bq.1.1		2	24.5	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.4222 q^{5} -6.42221 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -10.4222 q^{5} -6.42221 q^{7} +9.00000 q^{9} -61.6888 q^{11} -64.8444 q^{13} +31.2666 q^{15} -75.6888 q^{17} +10.3112 q^{19} +19.2666 q^{21} +156.844 q^{23} -16.3776 q^{25} -27.0000 q^{27} +53.7998 q^{29} -227.489 q^{31} +185.066 q^{33} +66.9335 q^{35} -10.3112 q^{37} +194.533 q^{39} +70.4441 q^{41} -298.311 q^{43} -93.7998 q^{45} +89.9109 q^{47} -301.755 q^{49} +227.066 q^{51} -388.333 q^{53} +642.934 q^{55} -30.9335 q^{57} -324.000 q^{59} -324.000 q^{61} -57.7998 q^{63} +675.822 q^{65} +920.266 q^{67} -470.533 q^{69} +995.156 q^{71} -362.266 q^{73} +49.1329 q^{75} +396.178 q^{77} +1098.91 q^{79} +81.0000 q^{81} +791.822 q^{83} +788.844 q^{85} -161.400 q^{87} +150.622 q^{89} +416.444 q^{91} +682.466 q^{93} -107.465 q^{95} -1879.15 q^{97} -555.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9} - 8 q^{11} - 72 q^{13} - 24 q^{15} - 36 q^{17} + 136 q^{19} - 48 q^{21} + 256 q^{23} + 198 q^{25} - 54 q^{27} - 152 q^{29} - 80 q^{31} + 24 q^{33} + 480 q^{35} - 136 q^{37} + 216 q^{39} - 436 q^{41} - 712 q^{43} + 72 q^{45} - 224 q^{47} - 142 q^{49} + 108 q^{51} - 344 q^{53} + 1632 q^{55} - 408 q^{57} - 648 q^{59} - 648 q^{61} + 144 q^{63} + 544 q^{65} + 456 q^{67} - 768 q^{69} + 2048 q^{71} + 660 q^{73} - 594 q^{75} + 1600 q^{77} + 496 q^{79} + 162 q^{81} + 776 q^{83} + 1520 q^{85} + 456 q^{87} + 532 q^{89} + 256 q^{91} + 240 q^{93} + 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9 - 8 * q^11 - 72 * q^13 - 24 * q^15 - 36 * q^17 + 136 * q^19 - 48 * q^21 + 256 * q^23 + 198 * q^25 - 54 * q^27 - 152 * q^29 - 80 * q^31 + 24 * q^33 + 480 * q^35 - 136 * q^37 + 216 * q^39 - 436 * q^41 - 712 * q^43 + 72 * q^45 - 224 * q^47 - 142 * q^49 + 108 * q^51 - 344 * q^53 + 1632 * q^55 - 408 * q^57 - 648 * q^59 - 648 * q^61 + 144 * q^63 + 544 * q^65 + 456 * q^67 - 768 * q^69 + 2048 * q^71 + 660 * q^73 - 594 * q^75 + 1600 * q^77 + 496 * q^79 + 162 * q^81 + 776 * q^83 + 1520 * q^85 + 456 * q^87 + 532 * q^89 + 256 * q^91 + 240 * q^93 + 2208 * q^95 - 1220 * q^97 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more