LMFDB - Embedded newform 928.4.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,4,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, 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("928.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	928.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:	$54.7537724853$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.95108$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95108 q^{3} +18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} +O(q^{10})$	$q+1.95108 q^{3} +18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} -11.1881 q^{11} -0.532875 q^{13} +35.2419 q^{15} +56.2274 q^{17} -26.5240 q^{19} -42.9588 q^{21} -66.9071 q^{23} +201.265 q^{25} -97.9310 q^{27} -29.0000 q^{29} -85.2968 q^{31} -21.8288 q^{33} -397.706 q^{35} -180.274 q^{37} -1.03968 q^{39} +46.4366 q^{41} -212.454 q^{43} -418.936 q^{45} -108.280 q^{47} +141.792 q^{49} +109.704 q^{51} -444.412 q^{53} -202.088 q^{55} -51.7503 q^{57} -43.3659 q^{59} +33.1252 q^{61} +510.670 q^{63} -9.62522 q^{65} -988.140 q^{67} -130.541 q^{69} +154.560 q^{71} -1030.98 q^{73} +392.683 q^{75} +246.339 q^{77} +631.568 q^{79} +435.148 q^{81} +145.157 q^{83} +1015.62 q^{85} -56.5812 q^{87} +169.477 q^{89} +11.7328 q^{91} -166.421 q^{93} -479.097 q^{95} -29.4573 q^{97} +259.489 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 4 q^{3} + 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q - 4 * q^3 + 10 * q^5 - 12 * q^7 + 49 * q^9	$ 9 q - 4 q^{3} + 10 q^{5} - 12 q^{7} + 49 q^{9} - 64 q^{11} + 70 q^{13} - 170 q^{15} - 66 q^{17} - 42 q^{19} + 76 q^{21} - 40 q^{23} + 111 q^{25} - 322 q^{27} - 261 q^{29} + 64 q^{31} - 52 q^{33} - 496 q^{35} - 54 q^{37} - 590 q^{39} - 378 q^{41} + 32 q^{43} + 1046 q^{45} - 1164 q^{47} - 351 q^{49} - 376 q^{51} + 278 q^{53} - 614 q^{55} + 28 q^{57} - 640 q^{59} + 1054 q^{61} - 1660 q^{63} - 708 q^{65} - 1184 q^{67} + 188 q^{69} - 1988 q^{71} - 750 q^{73} - 3126 q^{75} + 1260 q^{77} - 2916 q^{79} + 293 q^{81} - 2832 q^{83} + 56 q^{85} + 116 q^{87} - 370 q^{89} - 3016 q^{91} - 1696 q^{93} - 4412 q^{95} - 2234 q^{97} - 4118 q^{99}+O(q^{100}) $ 9 * q - 4 * q^3 + 10 * q^5 - 12 * q^7 + 49 * q^9 - 64 * q^11 + 70 * q^13 - 170 * q^15 - 66 * q^17 - 42 * q^19 + 76 * q^21 - 40 * q^23 + 111 * q^25 - 322 * q^27 - 261 * q^29 + 64 * q^31 - 52 * q^33 - 496 * q^35 - 54 * q^37 - 590 * q^39 - 378 * q^41 + 32 * q^43 + 1046 * q^45 - 1164 * q^47 - 351 * q^49 - 376 * q^51 + 278 * q^53 - 614 * q^55 + 28 * q^57 - 640 * q^59 + 1054 * q^61 - 1660 * q^63 - 708 * q^65 - 1184 * q^67 + 188 * q^69 - 1988 * q^71 - 750 * q^73 - 3126 * q^75 + 1260 * q^77 - 2916 * q^79 + 293 * q^81 - 2832 * q^83 + 56 * q^85 + 116 * q^87 - 370 * q^89 - 3016 * q^91 - 1696 * q^93 - 4412 * q^95 - 2234 * q^97 - 4118 * 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
$3$	1.95108	0.375485	0.187743	−	0.982218i	$-0.439883\pi$
$3$	1.95108	0.375485	0.187743	+	0.982218i	$0.439883\pi$
$4$	0	0
$5$	18.0628	1.61559	0.807793	−	0.589466i	$-0.200662\pi$
$5$	18.0628	1.61559	0.807793	+	0.589466i	$0.200662\pi$
$6$	0	0
$7$	−22.0180	−1.18886	−0.594430	−	0.804148i	$-0.702622\pi$
$7$	−22.0180	−1.18886	−0.594430	+	0.804148i	$0.702622\pi$
$8$	0	0
$9$	−23.1933	−0.859011
$10$	0	0
$11$	−11.1881	−0.306667	−0.153334	−	0.988174i	$-0.549001\pi$
$11$	−11.1881	−0.306667	−0.153334	+	0.988174i	$0.549001\pi$
$12$	0	0
$13$	−0.532875	−0.0113687	−0.00568435	−	0.999984i	$-0.501809\pi$
$13$	−0.532875	−0.0113687	−0.00568435	+	0.999984i	$0.501809\pi$
$14$	0	0
$15$	35.2419	0.606628
$16$	0	0
$17$	56.2274	0.802185	0.401092	−	0.916038i	$-0.368631\pi$
$17$	56.2274	0.802185	0.401092	+	0.916038i	$0.368631\pi$
$18$	0	0
$19$	−26.5240	−0.320264	−0.160132	−	0.987096i	$-0.551192\pi$
$19$	−26.5240	−0.320264	−0.160132	+	0.987096i	$0.551192\pi$
$20$	0	0
$21$	−42.9588	−0.446399
$22$	0	0
$23$	−66.9071	−0.606570	−0.303285	−	0.952900i	$-0.598083\pi$
$23$	−66.9071	−0.606570	−0.303285	+	0.952900i	$0.598083\pi$
$24$	0	0
$25$	201.265	1.61012
$26$	0	0
$27$	−97.9310	−0.698031
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−85.2968	−0.494186	−0.247093	−	0.968992i	$-0.579475\pi$
$31$	−85.2968	−0.494186	−0.247093	+	0.968992i	$0.579475\pi$
$32$	0	0
$33$	−21.8288	−0.115149
$34$	0	0
$35$	−397.706	−1.92070
$36$	0	0
$37$	−180.274	−0.800995	−0.400498	−	0.916298i	$-0.631163\pi$
$37$	−180.274	−0.800995	−0.400498	+	0.916298i	$0.631163\pi$
$38$	0	0
$39$	−1.03968	−0.00426877
$40$	0	0
$41$	46.4366	0.176882	0.0884412	−	0.996081i	$-0.471811\pi$
$41$	46.4366	0.176882	0.0884412	+	0.996081i	$0.471811\pi$
$42$	0	0
$43$	−212.454	−0.753464	−0.376732	−	0.926322i	$-0.622952\pi$
$43$	−212.454	−0.753464	−0.376732	+	0.926322i	$0.622952\pi$
$44$	0	0
$45$	−418.936	−1.38781
$46$	0	0
$47$	−108.280	−0.336048	−0.168024	−	0.985783i	$-0.553739\pi$
$47$	−108.280	−0.336048	−0.168024	+	0.985783i	$0.553739\pi$
$48$	0	0
$49$	141.792	0.413386
$50$	0	0
$51$	109.704	0.301208
$52$	0	0
$53$	−444.412	−1.15179	−0.575893	−	0.817525i	$-0.695345\pi$
$53$	−444.412	−1.15179	−0.575893	+	0.817525i	$0.695345\pi$
$54$	0	0
$55$	−202.088	−0.495447
$56$	0	0
$57$	−51.7503	−0.120254
$58$	0	0
$59$	−43.3659	−0.0956908	−0.0478454	−	0.998855i	$-0.515235\pi$
$59$	−43.3659	−0.0956908	−0.0478454	+	0.998855i	$0.515235\pi$
$60$	0	0
$61$	33.1252	0.0695286	0.0347643	−	0.999396i	$-0.488932\pi$
$61$	33.1252	0.0695286	0.0347643	+	0.999396i	$0.488932\pi$
$62$	0	0
$63$	510.670	1.02124
$64$	0	0
$65$	−9.62522	−0.0183671
$66$	0	0
$67$	−988.140	−1.80180	−0.900899	−	0.434028i	$-0.857092\pi$
$67$	−988.140	−1.80180	−0.900899	+	0.434028i	$0.857092\pi$
$68$	0	0
$69$	−130.541	−0.227758
$70$	0	0
$71$	154.560	0.258350	0.129175	−	0.991622i	$-0.458767\pi$
$71$	154.560	0.258350	0.129175	+	0.991622i	$0.458767\pi$
$72$	0	0
$73$	−1030.98	−1.65297	−0.826484	−	0.562961i	$-0.809662\pi$
$73$	−1030.98	−1.65297	−0.826484	+	0.562961i	$0.809662\pi$
$74$	0	0
$75$	392.683	0.604575
$76$	0	0
$77$	246.339	0.364584
$78$	0	0
$79$	631.568	0.899455	0.449728	−	0.893166i	$-0.351521\pi$
$79$	631.568	0.899455	0.449728	+	0.893166i	$0.351521\pi$
$80$	0	0
$81$	435.148	0.596911
$82$	0	0
$83$	145.157	0.191965	0.0959824	−	0.995383i	$-0.469401\pi$
$83$	145.157	0.191965	0.0959824	+	0.995383i	$0.469401\pi$
$84$	0	0
$85$	1015.62	1.29600
$86$	0	0
$87$	−56.5812	−0.0697258
$88$	0	0
$89$	169.477	0.201849	0.100924	−	0.994894i	$-0.467820\pi$
$89$	169.477	0.201849	0.100924	+	0.994894i	$0.467820\pi$
$90$	0	0
$91$	11.7328	0.0135158
$92$	0	0
$93$	−166.421	−0.185559
$94$	0	0
$95$	−479.097	−0.517414
$96$	0	0
$97$	−29.4573	−0.0308344	−0.0154172	−	0.999881i	$-0.504908\pi$
$97$	−29.4573	−0.0308344	−0.0154172	+	0.999881i	$0.504908\pi$
$98$	0	0
$99$	259.489	0.263430
$100$	0	0
$101$	887.027	0.873886	0.436943	−	0.899489i	$-0.356061\pi$
$101$	887.027	0.873886	0.436943	+	0.899489i	$0.356061\pi$
$102$	0	0
$103$	−778.724	−0.744951	−0.372475	−	0.928042i	$-0.621491\pi$
$103$	−778.724	−0.744951	−0.372475	+	0.928042i	$0.621491\pi$
$104$	0	0
$105$	−775.956	−0.721196
$106$	0	0
$107$	1837.16	1.65986	0.829930	−	0.557868i	$-0.188380\pi$
$107$	1837.16	1.65986	0.829930	+	0.557868i	$0.188380\pi$
$108$	0	0
$109$	232.609	0.204403	0.102201	−	0.994764i	$-0.467411\pi$
$109$	232.609	0.204403	0.102201	+	0.994764i	$0.467411\pi$
$110$	0	0
$111$	−351.728	−0.300762
$112$	0	0
$113$	1873.74	1.55988	0.779939	−	0.625855i	$-0.215250\pi$
$113$	1873.74	1.55988	0.779939	+	0.625855i	$0.215250\pi$
$114$	0	0
$115$	−1208.53	−0.979966
$116$	0	0
$117$	12.3591	0.00976583
$118$	0	0
$119$	−1238.01	−0.953685
$120$	0	0
$121$	−1205.83	−0.905955
$122$	0	0
$123$	90.6014	0.0664167
$124$	0	0
$125$	1377.56	0.985699
$126$	0	0
$127$	−1672.06	−1.16828	−0.584141	−	0.811652i	$-0.698568\pi$
$127$	−1672.06	−1.16828	−0.584141	+	0.811652i	$0.698568\pi$
$128$	0	0
$129$	−414.515	−0.282915
$130$	0	0
$131$	−2611.20	−1.74154	−0.870769	−	0.491693i	$-0.836378\pi$
$131$	−2611.20	−1.74154	−0.870769	+	0.491693i	$0.836378\pi$
$132$	0	0
$133$	584.004	0.380749
$134$	0	0
$135$	−1768.91	−1.12773
$136$	0	0
$137$	−1506.03	−0.939186	−0.469593	−	0.882883i	$-0.655599\pi$
$137$	−1506.03	−0.939186	−0.469593	+	0.882883i	$0.655599\pi$
$138$	0	0
$139$	−2818.87	−1.72009	−0.860047	−	0.510214i	$-0.829566\pi$
$139$	−2818.87	−1.72009	−0.860047	+	0.510214i	$0.829566\pi$
$140$	0	0
$141$	−211.263	−0.126181
$142$	0	0
$143$	5.96186	0.00348640
$144$	0	0
$145$	−523.821	−0.300007
$146$	0	0
$147$	276.646	0.155220
$148$	0	0
$149$	1672.65	0.919658	0.459829	−	0.888007i	$-0.347911\pi$
$149$	1672.65	0.919658	0.459829	+	0.888007i	$0.347911\pi$
$150$	0	0
$151$	−1065.84	−0.574417	−0.287209	−	0.957868i	$-0.592727\pi$
$151$	−1065.84	−0.574417	−0.287209	+	0.957868i	$0.592727\pi$
$152$	0	0
$153$	−1304.10	−0.689085
$154$	0	0
$155$	−1540.70	−0.798400
$156$	0	0
$157$	2554.74	1.29866	0.649332	−	0.760505i	$-0.275048\pi$
$157$	2554.74	1.29866	0.649332	+	0.760505i	$0.275048\pi$
$158$	0	0
$159$	−867.081	−0.432478
$160$	0	0
$161$	1473.16	0.721126
$162$	0	0
$163$	−1345.04	−0.646327	−0.323164	−	0.946343i	$-0.604746\pi$
$163$	−1345.04	−0.646327	−0.323164	+	0.946343i	$0.604746\pi$
$164$	0	0
$165$	−394.290	−0.186033
$166$	0	0
$167$	801.061	0.371185	0.185593	−	0.982627i	$-0.440579\pi$
$167$	801.061	0.371185	0.185593	+	0.982627i	$0.440579\pi$
$168$	0	0
$169$	−2196.72	−0.999871
$170$	0	0
$171$	615.178	0.275110
$172$	0	0
$173$	1999.30	0.878635	0.439318	−	0.898332i	$-0.355220\pi$
$173$	1999.30	0.878635	0.439318	+	0.898332i	$0.355220\pi$
$174$	0	0
$175$	−4431.45	−1.91420
$176$	0	0
$177$	−84.6102	−0.0359305
$178$	0	0
$179$	−1183.93	−0.494362	−0.247181	−	0.968969i	$-0.579504\pi$
$179$	−1183.93	−0.494362	−0.247181	+	0.968969i	$0.579504\pi$
$180$	0	0
$181$	−1164.04	−0.478025	−0.239013	−	0.971016i	$-0.576824\pi$
$181$	−1164.04	−0.478025	−0.239013	+	0.971016i	$0.576824\pi$
$182$	0	0
$183$	64.6297	0.0261069
$184$	0	0
$185$	−3256.25	−1.29408
$186$	0	0
$187$	−629.077	−0.246004
$188$	0	0
$189$	2156.24	0.829860
$190$	0	0
$191$	1613.60	0.611288	0.305644	−	0.952146i	$-0.401128\pi$
$191$	1613.60	0.611288	0.305644	+	0.952146i	$0.401128\pi$
$192$	0	0
$193$	−1048.74	−0.391139	−0.195570	−	0.980690i	$-0.562656\pi$
$193$	−1048.74	−0.391139	−0.195570	+	0.980690i	$0.562656\pi$
$194$	0	0
$195$	−18.7795	−0.00689657
$196$	0	0
$197$	−1487.86	−0.538099	−0.269050	−	0.963126i	$-0.586710\pi$
$197$	−1487.86	−0.538099	−0.269050	+	0.963126i	$0.586710\pi$
$198$	0	0
$199$	297.754	0.106067	0.0530333	−	0.998593i	$-0.483111\pi$
$199$	297.754	0.106067	0.0530333	+	0.998593i	$0.483111\pi$
$200$	0	0
$201$	−1927.94	−0.676549
$202$	0	0
$203$	638.521	0.220766
$204$	0	0
$205$	838.775	0.285769
$206$	0	0
$207$	1551.80	0.521050
$208$	0	0
$209$	296.753	0.0982145
$210$	0	0
$211$	−3947.96	−1.28810	−0.644049	−	0.764984i	$-0.722747\pi$
$211$	−3947.96	−1.28810	−0.644049	+	0.764984i	$0.722747\pi$
$212$	0	0
$213$	301.558	0.0970065
$214$	0	0
$215$	−3837.52	−1.21729
$216$	0	0
$217$	1878.06	0.587517
$218$	0	0
$219$	−2011.51	−0.620664
$220$	0	0
$221$	−29.9622	−0.00911979
$222$	0	0
$223$	3790.90	1.13837	0.569187	−	0.822208i	$-0.307258\pi$
$223$	3790.90	1.13837	0.569187	+	0.822208i	$0.307258\pi$
$224$	0	0
$225$	−4668.00	−1.38311
$226$	0	0
$227$	1952.76	0.570966	0.285483	−	0.958384i	$-0.407846\pi$
$227$	1952.76	0.570966	0.285483	+	0.958384i	$0.407846\pi$
$228$	0	0
$229$	−878.288	−0.253445	−0.126723	−	0.991938i	$-0.540446\pi$
$229$	−878.288	−0.253445	−0.126723	+	0.991938i	$0.540446\pi$
$230$	0	0
$231$	480.627	0.136896
$232$	0	0
$233$	−2556.16	−0.718711	−0.359355	−	0.933201i	$-0.617003\pi$
$233$	−2556.16	−0.718711	−0.359355	+	0.933201i	$0.617003\pi$
$234$	0	0
$235$	−1955.84	−0.542914
$236$	0	0
$237$	1232.24	0.337732
$238$	0	0
$239$	13.0160	0.00352274	0.00176137	−	0.999998i	$-0.499439\pi$
$239$	13.0160	0.00352274	0.00176137	+	0.999998i	$0.499439\pi$
$240$	0	0
$241$	1987.42	0.531207	0.265603	−	0.964082i	$-0.414429\pi$
$241$	1987.42	0.531207	0.265603	+	0.964082i	$0.414429\pi$
$242$	0	0
$243$	3493.14	0.922162
$244$	0	0
$245$	2561.15	0.667861
$246$	0	0
$247$	14.1340	0.00364098
$248$	0	0
$249$	283.213	0.0720799
$250$	0	0
$251$	5553.49	1.39655	0.698273	−	0.715831i	$-0.253952\pi$
$251$	5553.49	1.39655	0.698273	+	0.715831i	$0.253952\pi$
$252$	0	0
$253$	748.564	0.186015
$254$	0	0
$255$	1981.56	0.486628
$256$	0	0
$257$	−1089.82	−0.264518	−0.132259	−	0.991215i	$-0.542223\pi$
$257$	−1089.82	−0.264518	−0.132259	+	0.991215i	$0.542223\pi$
$258$	0	0
$259$	3969.27	0.952271
$260$	0	0
$261$	672.606	0.159514
$262$	0	0
$263$	6808.83	1.59639	0.798195	−	0.602399i	$-0.205788\pi$
$263$	6808.83	1.59639	0.798195	+	0.602399i	$0.205788\pi$
$264$	0	0
$265$	−8027.32	−1.86081
$266$	0	0
$267$	330.663	0.0757912
$268$	0	0
$269$	4978.43	1.12840	0.564201	−	0.825637i	$-0.309184\pi$
$269$	4978.43	1.12840	0.564201	+	0.825637i	$0.309184\pi$
$270$	0	0
$271$	−231.445	−0.0518793	−0.0259397	−	0.999664i	$-0.508258\pi$
$271$	−231.445	−0.0518793	−0.0259397	+	0.999664i	$0.508258\pi$
$272$	0	0
$273$	22.8917	0.00507497
$274$	0	0
$275$	−2251.77	−0.493771
$276$	0	0
$277$	2395.60	0.519630	0.259815	−	0.965658i	$-0.416338\pi$
$277$	2395.60	0.519630	0.259815	+	0.965658i	$0.416338\pi$
$278$	0	0
$279$	1978.31	0.424511
$280$	0	0
$281$	7597.72	1.61296	0.806480	−	0.591261i	$-0.201370\pi$
$281$	7597.72	1.61296	0.806480	+	0.591261i	$0.201370\pi$
$282$	0	0
$283$	5203.18	1.09292	0.546461	−	0.837484i	$-0.315974\pi$
$283$	5203.18	1.09292	0.546461	+	0.837484i	$0.315974\pi$
$284$	0	0
$285$	−934.756	−0.194281
$286$	0	0
$287$	−1022.44	−0.210288
$288$	0	0
$289$	−1751.48	−0.356500
$290$	0	0
$291$	−57.4735	−0.0115779
$292$	0	0
$293$	−6628.25	−1.32159	−0.660796	−	0.750566i	$-0.729781\pi$
$293$	−6628.25	−1.32159	−0.660796	+	0.750566i	$0.729781\pi$
$294$	0	0
$295$	−783.309	−0.154597
$296$	0	0
$297$	1095.66	0.214063
$298$	0	0
$299$	35.6532	0.00689590
$300$	0	0
$301$	4677.81	0.895763
$302$	0	0
$303$	1730.66	0.328131
$304$	0	0
$305$	598.333	0.112329
$306$	0	0
$307$	−8142.39	−1.51371	−0.756857	−	0.653580i	$-0.773266\pi$
$307$	−8142.39	−1.51371	−0.756857	+	0.653580i	$0.773266\pi$
$308$	0	0
$309$	−1519.35	−0.279718
$310$	0	0
$311$	1020.13	0.186000	0.0930000	−	0.995666i	$-0.470354\pi$
$311$	1020.13	0.186000	0.0930000	+	0.995666i	$0.470354\pi$
$312$	0	0
$313$	−6853.77	−1.23769	−0.618846	−	0.785512i	$-0.712400\pi$
$313$	−6853.77	−1.23769	−0.618846	+	0.785512i	$0.712400\pi$
$314$	0	0
$315$	9224.12	1.64991
$316$	0	0
$317$	7542.08	1.33630	0.668148	−	0.744029i	$-0.267087\pi$
$317$	7542.08	1.33630	0.668148	+	0.744029i	$0.267087\pi$
$318$	0	0
$319$	324.455	0.0569467
$320$	0	0
$321$	3584.44	0.623252
$322$	0	0
$323$	−1491.37	−0.256911
$324$	0	0
$325$	−107.249	−0.0183049
$326$	0	0
$327$	453.838	0.0767501
$328$	0	0
$329$	2384.11	0.399514
$330$	0	0
$331$	849.318	0.141035	0.0705177	−	0.997511i	$-0.477535\pi$
$331$	849.318	0.141035	0.0705177	+	0.997511i	$0.477535\pi$
$332$	0	0
$333$	4181.14	0.688064
$334$	0	0
$335$	−17848.6	−2.91096
$336$	0	0
$337$	8174.38	1.32133	0.660663	−	0.750682i	$-0.270275\pi$
$337$	8174.38	1.32133	0.660663	+	0.750682i	$0.270275\pi$
$338$	0	0
$339$	3655.81	0.585711
$340$	0	0
$341$	954.309	0.151551
$342$	0	0
$343$	4430.20	0.697401
$344$	0	0
$345$	−2357.94	−0.367962
$346$	0	0
$347$	−2726.57	−0.421816	−0.210908	−	0.977506i	$-0.567642\pi$
$347$	−2726.57	−0.421816	−0.210908	+	0.977506i	$0.567642\pi$
$348$	0	0
$349$	−7409.06	−1.13638	−0.568191	−	0.822896i	$-0.692357\pi$
$349$	−7409.06	−1.13638	−0.568191	+	0.822896i	$0.692357\pi$
$350$	0	0
$351$	52.1850	0.00793570
$352$	0	0
$353$	−5535.82	−0.834679	−0.417340	−	0.908751i	$-0.637037\pi$
$353$	−5535.82	−0.834679	−0.417340	+	0.908751i	$0.637037\pi$
$354$	0	0
$355$	2791.78	0.417387
$356$	0	0
$357$	−2415.46	−0.358094
$358$	0	0
$359$	−3735.97	−0.549239	−0.274619	−	0.961553i	$-0.588552\pi$
$359$	−3735.97	−0.549239	−0.274619	+	0.961553i	$0.588552\pi$
$360$	0	0
$361$	−6155.48	−0.897431
$362$	0	0
$363$	−2352.66	−0.340173
$364$	0	0
$365$	−18622.3	−2.67051
$366$	0	0
$367$	11094.8	1.57805	0.789027	−	0.614359i	$-0.210585\pi$
$367$	11094.8	1.57805	0.789027	+	0.614359i	$0.210585\pi$
$368$	0	0
$369$	−1077.02	−0.151944
$370$	0	0
$371$	9785.05	1.36931
$372$	0	0
$373$	4753.81	0.659901	0.329951	−	0.943998i	$-0.392968\pi$
$373$	4753.81	0.659901	0.329951	+	0.943998i	$0.392968\pi$
$374$	0	0
$375$	2687.72	0.370115
$376$	0	0
$377$	15.4534	0.00211111
$378$	0	0
$379$	−6164.88	−0.835537	−0.417769	−	0.908553i	$-0.637188\pi$
$379$	−6164.88	−0.835537	−0.417769	+	0.908553i	$0.637188\pi$
$380$	0	0
$381$	−3262.33	−0.438672
$382$	0	0
$383$	2734.38	0.364805	0.182402	−	0.983224i	$-0.441613\pi$
$383$	2734.38	0.364805	0.182402	+	0.983224i	$0.441613\pi$
$384$	0	0
$385$	4449.58	0.589017
$386$	0	0
$387$	4927.51	0.647234
$388$	0	0
$389$	−2349.93	−0.306289	−0.153144	−	0.988204i	$-0.548940\pi$
$389$	−2349.93	−0.306289	−0.153144	+	0.988204i	$0.548940\pi$
$390$	0	0
$391$	−3762.01	−0.486581
$392$	0	0
$393$	−5094.65	−0.653921
$394$	0	0
$395$	11407.9	1.45315
$396$	0	0
$397$	5065.74	0.640409	0.320204	−	0.947348i	$-0.396248\pi$
$397$	5065.74	0.640409	0.320204	+	0.947348i	$0.396248\pi$
$398$	0	0
$399$	1139.44	0.142965
$400$	0	0
$401$	5031.88	0.626634	0.313317	−	0.949649i	$-0.398560\pi$
$401$	5031.88	0.626634	0.313317	+	0.949649i	$0.398560\pi$
$402$	0	0
$403$	45.4526	0.00561825
$404$	0	0
$405$	7859.99	0.964361
$406$	0	0
$407$	2016.92	0.245639
$408$	0	0
$409$	−8107.90	−0.980220	−0.490110	−	0.871661i	$-0.663043\pi$
$409$	−8107.90	−0.980220	−0.490110	+	0.871661i	$0.663043\pi$
$410$	0	0
$411$	−2938.37	−0.352650
$412$	0	0
$413$	954.829	0.113763
$414$	0	0
$415$	2621.95	0.310136
$416$	0	0
$417$	−5499.83	−0.645870
$418$	0	0
$419$	12818.4	1.49456	0.747279	−	0.664510i	$-0.231360\pi$
$419$	12818.4	1.49456	0.747279	+	0.664510i	$0.231360\pi$
$420$	0	0
$421$	6405.87	0.741575	0.370788	−	0.928718i	$-0.379088\pi$
$421$	6405.87	0.741575	0.370788	+	0.928718i	$0.379088\pi$
$422$	0	0
$423$	2511.37	0.288669
$424$	0	0
$425$	11316.6	1.29161
$426$	0	0
$427$	−729.349	−0.0826597
$428$	0	0
$429$	11.6321	0.00130909
$430$	0	0
$431$	1873.09	0.209336	0.104668	−	0.994507i	$-0.466622\pi$
$431$	1873.09	0.209336	0.104668	+	0.994507i	$0.466622\pi$
$432$	0	0
$433$	492.001	0.0546052	0.0273026	−	0.999627i	$-0.491308\pi$
$433$	492.001	0.0546052	0.0273026	+	0.999627i	$0.491308\pi$
$434$	0	0
$435$	−1022.02	−0.112648
$436$	0	0
$437$	1774.64	0.194262
$438$	0	0
$439$	−9655.65	−1.04975	−0.524874	−	0.851180i	$-0.675887\pi$
$439$	−9655.65	−1.04975	−0.524874	+	0.851180i	$0.675887\pi$
$440$	0	0
$441$	−3288.61	−0.355104
$442$	0	0
$443$	−3021.05	−0.324005	−0.162003	−	0.986790i	$-0.551795\pi$
$443$	−3021.05	−0.324005	−0.162003	+	0.986790i	$0.551795\pi$
$444$	0	0
$445$	3061.23	0.326104
$446$	0	0
$447$	3263.47	0.345318
$448$	0	0
$449$	1335.44	0.140364	0.0701820	−	0.997534i	$-0.477642\pi$
$449$	1335.44	0.140364	0.0701820	+	0.997534i	$0.477642\pi$
$450$	0	0
$451$	−519.537	−0.0542440
$452$	0	0
$453$	−2079.54	−0.215685
$454$	0	0
$455$	211.928	0.0218359
$456$	0	0
$457$	−8984.92	−0.919687	−0.459843	−	0.888000i	$-0.652094\pi$
$457$	−8984.92	−0.919687	−0.459843	+	0.888000i	$0.652094\pi$
$458$	0	0
$459$	−5506.40	−0.559950
$460$	0	0
$461$	−8468.56	−0.855575	−0.427788	−	0.903879i	$-0.640707\pi$
$461$	−8468.56	−0.855575	−0.427788	+	0.903879i	$0.640707\pi$
$462$	0	0
$463$	−13946.0	−1.39983	−0.699917	−	0.714224i	$-0.746780\pi$
$463$	−13946.0	−1.39983	−0.699917	+	0.714224i	$0.746780\pi$
$464$	0	0
$465$	−3006.02	−0.299787
$466$	0	0
$467$	1242.51	0.123119	0.0615594	−	0.998103i	$-0.480393\pi$
$467$	1242.51	0.123119	0.0615594	+	0.998103i	$0.480393\pi$
$468$	0	0
$469$	21756.9	2.14209
$470$	0	0
$471$	4984.49	0.487629
$472$	0	0
$473$	2376.96	0.231063
$474$	0	0
$475$	−5338.34	−0.515663
$476$	0	0
$477$	10307.4	0.989396
$478$	0	0
$479$	−1054.91	−0.100626	−0.0503131	−	0.998733i	$-0.516022\pi$
$479$	−1054.91	−0.100626	−0.0503131	+	0.998733i	$0.516022\pi$
$480$	0	0
$481$	96.0634	0.00910627
$482$	0	0
$483$	2874.25	0.270772
$484$	0	0
$485$	−532.082	−0.0498156
$486$	0	0
$487$	8140.23	0.757432	0.378716	−	0.925513i	$-0.376366\pi$
$487$	8140.23	0.757432	0.378716	+	0.925513i	$0.376366\pi$
$488$	0	0
$489$	−2624.27	−0.242686
$490$	0	0
$491$	19696.4	1.81035	0.905177	−	0.425034i	$-0.139738\pi$
$491$	19696.4	1.81035	0.905177	+	0.425034i	$0.139738\pi$
$492$	0	0
$493$	−1630.59	−0.148962
$494$	0	0
$495$	4687.10	0.425595
$496$	0	0
$497$	−3403.09	−0.307142
$498$	0	0
$499$	3554.81	0.318908	0.159454	−	0.987205i	$-0.449027\pi$
$499$	3554.81	0.318908	0.159454	+	0.987205i	$0.449027\pi$
$500$	0	0
$501$	1562.93	0.139375
$502$	0	0
$503$	21333.8	1.89111	0.945555	−	0.325463i	$-0.105520\pi$
$503$	21333.8	1.89111	0.945555	+	0.325463i	$0.105520\pi$
$504$	0	0
$505$	16022.2	1.41184
$506$	0	0
$507$	−4285.96	−0.375436
$508$	0	0
$509$	11816.7	1.02901	0.514506	−	0.857487i	$-0.327975\pi$
$509$	11816.7	1.02901	0.514506	+	0.857487i	$0.327975\pi$
$510$	0	0
$511$	22700.0	1.96515
$512$	0	0
$513$	2597.52	0.223554
$514$	0	0
$515$	−14065.9	−1.20353
$516$	0	0
$517$	1211.45	0.103055
$518$	0	0
$519$	3900.79	0.329914
$520$	0	0
$521$	12945.2	1.08856	0.544278	−	0.838905i	$-0.316804\pi$
$521$	12945.2	1.08856	0.544278	+	0.838905i	$0.316804\pi$
$522$	0	0
$523$	−2842.92	−0.237690	−0.118845	−	0.992913i	$-0.537919\pi$
$523$	−2842.92	−0.237690	−0.118845	+	0.992913i	$0.537919\pi$
$524$	0	0
$525$	−8646.09	−0.718755
$526$	0	0
$527$	−4796.01	−0.396428
$528$	0	0
$529$	−7690.44	−0.632073
$530$	0	0
$531$	1005.80	0.0821994
$532$	0	0
$533$	−24.7449	−0.00201092
$534$	0	0
$535$	33184.3	2.68165
$536$	0	0
$537$	−2309.93	−0.185625
$538$	0	0
$539$	−1586.38	−0.126772
$540$	0	0
$541$	564.312	0.0448459	0.0224230	−	0.999749i	$-0.492862\pi$
$541$	564.312	0.0448459	0.0224230	+	0.999749i	$0.492862\pi$
$542$	0	0
$543$	−2271.14	−0.179491
$544$	0	0
$545$	4201.57	0.330230
$546$	0	0
$547$	21063.9	1.64648	0.823241	−	0.567692i	$-0.192164\pi$
$547$	21063.9	1.64648	0.823241	+	0.567692i	$0.192164\pi$
$548$	0	0
$549$	−768.282	−0.0597258
$550$	0	0
$551$	769.195	0.0594715
$552$	0	0
$553$	−13905.9	−1.06933
$554$	0	0
$555$	−6353.20	−0.485907
$556$	0	0
$557$	−23042.5	−1.75286	−0.876430	−	0.481530i	$-0.840081\pi$
$557$	−23042.5	−1.75286	−0.876430	+	0.481530i	$0.840081\pi$
$558$	0	0
$559$	113.212	0.00856590
$560$	0	0
$561$	−1227.38	−0.0923707
$562$	0	0
$563$	−9753.28	−0.730109	−0.365055	−	0.930986i	$-0.618950\pi$
$563$	−9753.28	−0.730109	−0.365055	+	0.930986i	$0.618950\pi$
$564$	0	0
$565$	33844.9	2.52012
$566$	0	0
$567$	−9581.08	−0.709643
$568$	0	0
$569$	−11380.6	−0.838485	−0.419242	−	0.907874i	$-0.637704\pi$
$569$	−11380.6	−0.838485	−0.419242	+	0.907874i	$0.637704\pi$
$570$	0	0
$571$	12977.3	0.951109	0.475555	−	0.879686i	$-0.342247\pi$
$571$	12977.3	0.951109	0.475555	+	0.879686i	$0.342247\pi$
$572$	0	0
$573$	3148.26	0.229529
$574$	0	0
$575$	−13466.1	−0.976649
$576$	0	0
$577$	12229.5	0.882359	0.441180	−	0.897419i	$-0.354560\pi$
$577$	12229.5	0.882359	0.441180	+	0.897419i	$0.354560\pi$
$578$	0	0
$579$	−2046.17	−0.146867
$580$	0	0
$581$	−3196.07	−0.228219
$582$	0	0
$583$	4972.12	0.353215
$584$	0	0
$585$	223.241	0.0157775
$586$	0	0
$587$	−13884.3	−0.976263	−0.488132	−	0.872770i	$-0.662321\pi$
$587$	−13884.3	−0.976263	−0.488132	+	0.872770i	$0.662321\pi$
$588$	0	0
$589$	2262.41	0.158270
$590$	0	0
$591$	−2902.93	−0.202048
$592$	0	0
$593$	−22356.1	−1.54815	−0.774077	−	0.633092i	$-0.781786\pi$
$593$	−22356.1	−1.54815	−0.774077	+	0.633092i	$0.781786\pi$
$594$	0	0
$595$	−22362.0	−1.54076
$596$	0	0
$597$	580.942	0.0398264
$598$	0	0
$599$	3966.27	0.270546	0.135273	−	0.990808i	$-0.456809\pi$
$599$	3966.27	0.270546	0.135273	+	0.990808i	$0.456809\pi$
$600$	0	0
$601$	−17343.0	−1.17710	−0.588550	−	0.808461i	$-0.700301\pi$
$601$	−17343.0	−1.17710	−0.588550	+	0.808461i	$0.700301\pi$
$602$	0	0
$603$	22918.2	1.54777
$604$	0	0
$605$	−21780.6	−1.46365
$606$	0	0
$607$	17468.7	1.16810	0.584048	−	0.811719i	$-0.301468\pi$
$607$	17468.7	1.16810	0.584048	+	0.811719i	$0.301468\pi$
$608$	0	0
$609$	1245.80	0.0828942
$610$	0	0
$611$	57.6997	0.00382042
$612$	0	0
$613$	−5605.90	−0.369364	−0.184682	−	0.982798i	$-0.559125\pi$
$613$	−5605.90	−0.369364	−0.184682	+	0.982798i	$0.559125\pi$
$614$	0	0
$615$	1636.52	0.107302
$616$	0	0
$617$	12413.8	0.809987	0.404993	−	0.914320i	$-0.367274\pi$
$617$	12413.8	0.809987	0.404993	+	0.914320i	$0.367274\pi$
$618$	0	0
$619$	24588.8	1.59662	0.798310	−	0.602247i	$-0.205728\pi$
$619$	24588.8	1.59662	0.798310	+	0.602247i	$0.205728\pi$
$620$	0	0
$621$	6552.28	0.423404
$622$	0	0
$623$	−3731.54	−0.239970
$624$	0	0
$625$	−275.570	−0.0176365
$626$	0	0
$627$	578.988	0.0368781
$628$	0	0
$629$	−10136.3	−0.642546
$630$	0	0
$631$	15144.9	0.955481	0.477740	−	0.878501i	$-0.341456\pi$
$631$	15144.9	0.955481	0.477740	+	0.878501i	$0.341456\pi$
$632$	0	0
$633$	−7702.78	−0.483662
$634$	0	0
$635$	−30202.2	−1.88746
$636$	0	0
$637$	−75.5572	−0.00469966
$638$	0	0
$639$	−3584.75	−0.221925
$640$	0	0
$641$	−1087.26	−0.0669957	−0.0334979	−	0.999439i	$-0.510665\pi$
$641$	−1087.26	−0.0669957	−0.0334979	+	0.999439i	$0.510665\pi$
$642$	0	0
$643$	14014.7	0.859541	0.429771	−	0.902938i	$-0.358594\pi$
$643$	14014.7	0.859541	0.429771	+	0.902938i	$0.358594\pi$
$644$	0	0
$645$	−7487.30	−0.457073
$646$	0	0
$647$	−9390.59	−0.570606	−0.285303	−	0.958437i	$-0.592094\pi$
$647$	−9390.59	−0.570606	−0.285303	+	0.958437i	$0.592094\pi$
$648$	0	0
$649$	485.182	0.0293452
$650$	0	0
$651$	3664.25	0.220604
$652$	0	0
$653$	−8357.09	−0.500824	−0.250412	−	0.968139i	$-0.580566\pi$
$653$	−8357.09	−0.500824	−0.250412	+	0.968139i	$0.580566\pi$
$654$	0	0
$655$	−47165.5	−2.81360
$656$	0	0
$657$	23911.7	1.41992
$658$	0	0
$659$	−13965.0	−0.825489	−0.412745	−	0.910847i	$-0.635430\pi$
$659$	−13965.0	−0.825489	−0.412745	+	0.910847i	$0.635430\pi$
$660$	0	0
$661$	10501.3	0.617934	0.308967	−	0.951073i	$-0.400017\pi$
$661$	10501.3	0.617934	0.308967	+	0.951073i	$0.400017\pi$
$662$	0	0
$663$	−58.4585	−0.00342434
$664$	0	0
$665$	10548.8	0.615133
$666$	0	0
$667$	1940.31	0.112637
$668$	0	0
$669$	7396.34	0.427443
$670$	0	0
$671$	−370.608	−0.0213221
$672$	0	0
$673$	−30084.5	−1.72314	−0.861569	−	0.507641i	$-0.830518\pi$
$673$	−30084.5	−1.72314	−0.861569	+	0.507641i	$0.830518\pi$
$674$	0	0
$675$	−19710.1	−1.12391
$676$	0	0
$677$	14825.2	0.841621	0.420810	−	0.907149i	$-0.361746\pi$
$677$	14825.2	0.841621	0.420810	+	0.907149i	$0.361746\pi$
$678$	0	0
$679$	648.591	0.0366578
$680$	0	0
$681$	3809.99	0.214389
$682$	0	0
$683$	−30850.4	−1.72834	−0.864171	−	0.503199i	$-0.832156\pi$
$683$	−30850.4	−1.72834	−0.864171	+	0.503199i	$0.832156\pi$
$684$	0	0
$685$	−27203.1	−1.51734
$686$	0	0
$687$	−1713.61	−0.0951648
$688$	0	0
$689$	236.816	0.0130943
$690$	0	0
$691$	27669.0	1.52327	0.761635	−	0.648006i	$-0.224397\pi$
$691$	27669.0	1.52327	0.761635	+	0.648006i	$0.224397\pi$
$692$	0	0
$693$	−5713.42	−0.313182
$694$	0	0
$695$	−50916.6	−2.77896
$696$	0	0
$697$	2611.01	0.141892
$698$	0	0
$699$	−4987.26	−0.269865
$700$	0	0
$701$	−18155.9	−0.978229	−0.489115	−	0.872220i	$-0.662680\pi$
$701$	−18155.9	−0.978229	−0.489115	+	0.872220i	$0.662680\pi$
$702$	0	0
$703$	4781.58	0.256530
$704$	0	0
$705$	−3815.99	−0.203856
$706$	0	0
$707$	−19530.5	−1.03893
$708$	0	0
$709$	−12971.8	−0.687118	−0.343559	−	0.939131i	$-0.611633\pi$
$709$	−12971.8	−0.687118	−0.343559	+	0.939131i	$0.611633\pi$
$710$	0	0
$711$	−14648.1	−0.772642
$712$	0	0
$713$	5706.97	0.299758
$714$	0	0
$715$	107.688	0.00563259
$716$	0	0
$717$	25.3952	0.00132274
$718$	0	0
$719$	11163.6	0.579044	0.289522	−	0.957171i	$-0.406504\pi$
$719$	11163.6	0.579044	0.289522	+	0.957171i	$0.406504\pi$
$720$	0	0
$721$	17145.9	0.885642
$722$	0	0
$723$	3877.60	0.199460
$724$	0	0
$725$	−5836.68	−0.298992
$726$	0	0
$727$	23149.5	1.18097	0.590487	−	0.807047i	$-0.298936\pi$
$727$	23149.5	1.18097	0.590487	+	0.807047i	$0.298936\pi$
$728$	0	0
$729$	−4933.60	−0.250653
$730$	0	0
$731$	−11945.7	−0.604418
$732$	0	0
$733$	−30271.8	−1.52539	−0.762697	−	0.646756i	$-0.776125\pi$
$733$	−30271.8	−1.52539	−0.762697	+	0.646756i	$0.776125\pi$
$734$	0	0
$735$	4997.01	0.250772
$736$	0	0
$737$	11055.4	0.552553
$738$	0	0
$739$	−31154.1	−1.55077	−0.775387	−	0.631487i	$-0.782445\pi$
$739$	−31154.1	−1.55077	−0.775387	+	0.631487i	$0.782445\pi$
$740$	0	0
$741$	27.5765	0.00136713
$742$	0	0
$743$	−32817.8	−1.62041	−0.810207	−	0.586144i	$-0.800645\pi$
$743$	−32817.8	−1.62041	−0.810207	+	0.586144i	$0.800645\pi$
$744$	0	0
$745$	30212.8	1.48579
$746$	0	0
$747$	−3366.68	−0.164900
$748$	0	0
$749$	−40450.6	−1.97334
$750$	0	0
$751$	−26390.0	−1.28227	−0.641135	−	0.767428i	$-0.721536\pi$
$751$	−26390.0	−1.28227	−0.641135	+	0.767428i	$0.721536\pi$
$752$	0	0
$753$	10835.3	0.524382
$754$	0	0
$755$	−19252.1	−0.928021
$756$	0	0
$757$	2340.44	0.112371	0.0561853	−	0.998420i	$-0.482106\pi$
$757$	2340.44	0.112371	0.0561853	+	0.998420i	$0.482106\pi$
$758$	0	0
$759$	1460.51	0.0698458
$760$	0	0
$761$	34510.4	1.64389	0.821946	−	0.569565i	$-0.192888\pi$
$761$	34510.4	1.64389	0.821946	+	0.569565i	$0.192888\pi$
$762$	0	0
$763$	−5121.58	−0.243006
$764$	0	0
$765$	−23555.7	−1.11328
$766$	0	0
$767$	23.1086	0.00108788
$768$	0	0
$769$	24619.2	1.15447	0.577237	−	0.816577i	$-0.304131\pi$
$769$	24619.2	1.15447	0.577237	+	0.816577i	$0.304131\pi$
$770$	0	0
$771$	−2126.32	−0.0993225
$772$	0	0
$773$	4506.63	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$773$	4506.63	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$774$	0	0
$775$	−17167.2	−0.795698
$776$	0	0
$777$	7744.34	0.357563
$778$	0	0
$779$	−1231.68	−0.0566491
$780$	0	0
$781$	−1729.23	−0.0792274
$782$	0	0
$783$	2840.00	0.129621
$784$	0	0
$785$	46145.7	2.09810
$786$	0	0
$787$	12917.4	0.585076	0.292538	−	0.956254i	$-0.405500\pi$
$787$	12917.4	0.585076	0.292538	+	0.956254i	$0.405500\pi$
$788$	0	0
$789$	13284.6	0.599421
$790$	0	0
$791$	−41255.9	−1.85448
$792$	0	0
$793$	−17.6516	−0.000790449 0
$794$	0	0
$795$	−15661.9	−0.698706
$796$	0	0
$797$	−14608.1	−0.649243	−0.324621	−	0.945844i	$-0.605237\pi$
$797$	−14608.1	−0.649243	−0.324621	+	0.945844i	$0.605237\pi$
$798$	0	0
$799$	−6088.29	−0.269572
$800$	0	0
$801$	−3930.73	−0.173390
$802$	0	0
$803$	11534.7	0.506911
$804$	0	0
$805$	26609.4	1.16504
$806$	0	0
$807$	9713.31	0.423698
$808$	0	0
$809$	16506.3	0.717342	0.358671	−	0.933464i	$-0.383230\pi$
$809$	16506.3	0.717342	0.358671	+	0.933464i	$0.383230\pi$
$810$	0	0
$811$	43809.7	1.89688	0.948438	−	0.316964i	$-0.102663\pi$
$811$	43809.7	1.89688	0.948438	+	0.316964i	$0.102663\pi$
$812$	0	0
$813$	−451.568	−0.0194799
$814$	0	0
$815$	−24295.1	−1.04420
$816$	0	0
$817$	5635.13	0.241308
$818$	0	0
$819$	−272.123	−0.0116102
$820$	0	0
$821$	17285.6	0.734802	0.367401	−	0.930063i	$-0.380248\pi$
$821$	17285.6	0.734802	0.367401	+	0.930063i	$0.380248\pi$
$822$	0	0
$823$	−39612.1	−1.67775	−0.838876	−	0.544323i	$-0.816787\pi$
$823$	−39612.1	−1.67775	−0.838876	+	0.544323i	$0.816787\pi$
$824$	0	0
$825$	−4393.38	−0.185403
$826$	0	0
$827$	−23037.5	−0.968674	−0.484337	−	0.874882i	$-0.660939\pi$
$827$	−23037.5	−0.968674	−0.484337	+	0.874882i	$0.660939\pi$
$828$	0	0
$829$	−32090.7	−1.34446	−0.672228	−	0.740344i	$-0.734663\pi$
$829$	−32090.7	−1.34446	−0.672228	+	0.740344i	$0.734663\pi$
$830$	0	0
$831$	4674.00	0.195113
$832$	0	0
$833$	7972.57	0.331612
$834$	0	0
$835$	14469.4	0.599682
$836$	0	0
$837$	8353.20	0.344957
$838$	0	0
$839$	−19295.2	−0.793973	−0.396987	−	0.917824i	$-0.629944\pi$
$839$	−19295.2	−0.793973	−0.396987	+	0.917824i	$0.629944\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	14823.7	0.605643
$844$	0	0
$845$	−39678.8	−1.61538
$846$	0	0
$847$	26549.9	1.07705
$848$	0	0
$849$	10151.8	0.410376
$850$	0	0
$851$	12061.6	0.485860
$852$	0	0
$853$	19213.6	0.771234	0.385617	−	0.922659i	$-0.373989\pi$
$853$	19213.6	0.771234	0.385617	+	0.922659i	$0.373989\pi$
$854$	0	0
$855$	11111.8	0.444464
$856$	0	0
$857$	−38335.0	−1.52800	−0.764001	−	0.645214i	$-0.776768\pi$
$857$	−38335.0	−1.52800	−0.764001	+	0.645214i	$0.776768\pi$
$858$	0	0
$859$	34344.3	1.36416	0.682080	−	0.731278i	$-0.261076\pi$
$859$	34344.3	1.36416	0.682080	+	0.731278i	$0.261076\pi$
$860$	0	0
$861$	−1994.86	−0.0789601
$862$	0	0
$863$	39162.2	1.54472	0.772362	−	0.635183i	$-0.219075\pi$
$863$	39162.2	1.54472	0.772362	+	0.635183i	$0.219075\pi$
$864$	0	0
$865$	36112.9	1.41951
$866$	0	0
$867$	−3417.28	−0.133860
$868$	0	0
$869$	−7066.05	−0.275833
$870$	0	0
$871$	526.555	0.0204841
$872$	0	0
$873$	683.212	0.0264871
$874$	0	0
$875$	−30331.0	−1.17186
$876$	0	0
$877$	−15235.0	−0.586601	−0.293301	−	0.956020i	$-0.594754\pi$
$877$	−15235.0	−0.586601	−0.293301	+	0.956020i	$0.594754\pi$
$878$	0	0
$879$	−12932.2	−0.496238
$880$	0	0
$881$	29455.1	1.12641	0.563206	−	0.826317i	$-0.309568\pi$
$881$	29455.1	1.12641	0.563206	+	0.826317i	$0.309568\pi$
$882$	0	0
$883$	37608.0	1.43331	0.716653	−	0.697430i	$-0.245673\pi$
$883$	37608.0	1.43331	0.716653	+	0.697430i	$0.245673\pi$
$884$	0	0
$885$	−1528.30	−0.0580487
$886$	0	0
$887$	−20517.3	−0.776666	−0.388333	−	0.921519i	$-0.626949\pi$
$887$	−20517.3	−0.776666	−0.388333	+	0.921519i	$0.626949\pi$
$888$	0	0
$889$	36815.5	1.38892
$890$	0	0
$891$	−4868.48	−0.183053
$892$	0	0
$893$	2872.01	0.107624
$894$	0	0
$895$	−21385.0	−0.798684
$896$	0	0
$897$	69.5621	0.00258931
$898$	0	0
$899$	2473.61	0.0917680
$900$	0	0
$901$	−24988.1	−0.923944
$902$	0	0
$903$	9126.78	0.336346
$904$	0	0
$905$	−21025.9	−0.772291
$906$	0	0
$907$	−49117.1	−1.79813	−0.899067	−	0.437811i	$-0.855754\pi$
$907$	−49117.1	−1.79813	−0.899067	+	0.437811i	$0.855754\pi$
$908$	0	0
$909$	−20573.1	−0.750678
$910$	0	0
$911$	23792.4	0.865289	0.432645	−	0.901564i	$-0.357580\pi$
$911$	23792.4	0.865289	0.432645	+	0.901564i	$0.357580\pi$
$912$	0	0
$913$	−1624.03	−0.0588693
$914$	0	0
$915$	1167.39	0.0421780
$916$	0	0
$917$	57493.3	2.07044
$918$	0	0
$919$	43361.1	1.55642	0.778211	−	0.628003i	$-0.216127\pi$
$919$	43361.1	1.55642	0.778211	+	0.628003i	$0.216127\pi$
$920$	0	0
$921$	−15886.4	−0.568377
$922$	0	0
$923$	−82.3610	−0.00293710
$924$	0	0
$925$	−36282.8	−1.28970
$926$	0	0
$927$	18061.2	0.639921
$928$	0	0
$929$	−24782.7	−0.875235	−0.437617	−	0.899161i	$-0.644178\pi$
$929$	−24782.7	−0.875235	−0.437617	+	0.899161i	$0.644178\pi$
$930$	0	0
$931$	−3760.88	−0.132393
$932$	0	0
$933$	1990.34	0.0698402
$934$	0	0
$935$	−11362.9	−0.397440
$936$	0	0
$937$	−31441.0	−1.09619	−0.548097	−	0.836415i	$-0.684647\pi$
$937$	−31441.0	−1.09619	−0.548097	+	0.836415i	$0.684647\pi$
$938$	0	0
$939$	−13372.2	−0.464735
$940$	0	0
$941$	−2094.24	−0.0725509	−0.0362754	−	0.999342i	$-0.511549\pi$
$941$	−2094.24	−0.0725509	−0.0362754	+	0.999342i	$0.511549\pi$
$942$	0	0
$943$	−3106.94	−0.107292
$944$	0	0
$945$	38947.8	1.34071
$946$	0	0
$947$	−29337.6	−1.00670	−0.503350	−	0.864083i	$-0.667899\pi$
$947$	−29337.6	−1.00670	−0.503350	+	0.864083i	$0.667899\pi$
$948$	0	0
$949$	549.381	0.0187921
$950$	0	0
$951$	14715.2	0.501759
$952$	0	0
$953$	−13702.6	−0.465763	−0.232881	−	0.972505i	$-0.574815\pi$
$953$	−13702.6	−0.465763	−0.232881	+	0.972505i	$0.574815\pi$
$954$	0	0
$955$	29146.1	0.987588
$956$	0	0
$957$	633.037	0.0213826
$958$	0	0
$959$	33159.7	1.11656
$960$	0	0
$961$	−22515.5	−0.755780
$962$	0	0
$963$	−42609.8	−1.42584
$964$	0	0
$965$	−18943.2	−0.631919
$966$	0	0
$967$	15477.4	0.514706	0.257353	−	0.966318i	$-0.417150\pi$
$967$	15477.4	0.514706	0.257353	+	0.966318i	$0.417150\pi$
$968$	0	0
$969$	−2909.78	−0.0964662
$970$	0	0
$971$	−943.820	−0.0311932	−0.0155966	−	0.999878i	$-0.504965\pi$
$971$	−943.820	−0.0311932	−0.0155966	+	0.999878i	$0.504965\pi$
$972$	0	0
$973$	62065.8	2.04495
$974$	0	0
$975$	−209.251	−0.00687323
$976$	0	0
$977$	41392.6	1.35544	0.677720	−	0.735320i	$-0.262968\pi$
$977$	41392.6	1.35544	0.677720	+	0.735320i	$0.262968\pi$
$978$	0	0
$979$	−1896.13	−0.0619004
$980$	0	0
$981$	−5394.96	−0.175584
$982$	0	0
$983$	9580.92	0.310869	0.155434	−	0.987846i	$-0.450322\pi$
$983$	9580.92	0.310869	0.155434	+	0.987846i	$0.450322\pi$
$984$	0	0
$985$	−26874.9	−0.869346
$986$	0	0
$987$	4651.57	0.150011
$988$	0	0
$989$	14214.7	0.457029
$990$	0	0
$991$	−25374.3	−0.813361	−0.406680	−	0.913570i	$-0.633314\pi$
$991$	−25374.3	−0.813361	−0.406680	+	0.913570i	$0.633314\pi$
$992$	0	0
$993$	1657.08	0.0529567
$994$	0	0
$995$	5378.28	0.171360
$996$	0	0
$997$	−51564.5	−1.63798	−0.818989	−	0.573810i	$-0.805465\pi$
$997$	−51564.5	−1.63798	−0.818989	+	0.573810i	$0.805465\pi$
$998$	0	0
$999$	17654.4	0.559119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.4.a.d.1.6	✓	9
4.3	odd	2		928.4.a.e.1.4	yes	9
8.3	odd	2		1856.4.a.bf.1.6		9
8.5	even	2		1856.4.a.bg.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.6	✓	9	1.1	even	1	trivial
928.4.a.e.1.4	yes	9	4.3	odd	2
1856.4.a.bf.1.6		9	8.3	odd	2
1856.4.a.bg.1.4		9	8.5	even	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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95108 q^{3} +18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} +O(q^{10})\)	\(q+1.95108 q^{3} +18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} -11.1881 q^{11} -0.532875 q^{13} +35.2419 q^{15} +56.2274 q^{17} -26.5240 q^{19} -42.9588 q^{21} -66.9071 q^{23} +201.265 q^{25} -97.9310 q^{27} -29.0000 q^{29} -85.2968 q^{31} -21.8288 q^{33} -397.706 q^{35} -180.274 q^{37} -1.03968 q^{39} +46.4366 q^{41} -212.454 q^{43} -418.936 q^{45} -108.280 q^{47} +141.792 q^{49} +109.704 q^{51} -444.412 q^{53} -202.088 q^{55} -51.7503 q^{57} -43.3659 q^{59} +33.1252 q^{61} +510.670 q^{63} -9.62522 q^{65} -988.140 q^{67} -130.541 q^{69} +154.560 q^{71} -1030.98 q^{73} +392.683 q^{75} +246.339 q^{77} +631.568 q^{79} +435.148 q^{81} +145.157 q^{83} +1015.62 q^{85} -56.5812 q^{87} +169.477 q^{89} +11.7328 q^{91} -166.421 q^{93} -479.097 q^{95} -29.4573 q^{97} +259.489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 4 q^{3} + 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q - 4 * q^3 + 10 * q^5 - 12 * q^7 + 49 * q^9	\( 9 q - 4 q^{3} + 10 q^{5} - 12 q^{7} + 49 q^{9} - 64 q^{11} + 70 q^{13} - 170 q^{15} - 66 q^{17} - 42 q^{19} + 76 q^{21} - 40 q^{23} + 111 q^{25} - 322 q^{27} - 261 q^{29} + 64 q^{31} - 52 q^{33} - 496 q^{35} - 54 q^{37} - 590 q^{39} - 378 q^{41} + 32 q^{43} + 1046 q^{45} - 1164 q^{47} - 351 q^{49} - 376 q^{51} + 278 q^{53} - 614 q^{55} + 28 q^{57} - 640 q^{59} + 1054 q^{61} - 1660 q^{63} - 708 q^{65} - 1184 q^{67} + 188 q^{69} - 1988 q^{71} - 750 q^{73} - 3126 q^{75} + 1260 q^{77} - 2916 q^{79} + 293 q^{81} - 2832 q^{83} + 56 q^{85} + 116 q^{87} - 370 q^{89} - 3016 q^{91} - 1696 q^{93} - 4412 q^{95} - 2234 q^{97} - 4118 q^{99}+O(q^{100}) \) 9 * q - 4 * q^3 + 10 * q^5 - 12 * q^7 + 49 * q^9 - 64 * q^11 + 70 * q^13 - 170 * q^15 - 66 * q^17 - 42 * q^19 + 76 * q^21 - 40 * q^23 + 111 * q^25 - 322 * q^27 - 261 * q^29 + 64 * q^31 - 52 * q^33 - 496 * q^35 - 54 * q^37 - 590 * q^39 - 378 * q^41 + 32 * q^43 + 1046 * q^45 - 1164 * q^47 - 351 * q^49 - 376 * q^51 + 278 * q^53 - 614 * q^55 + 28 * q^57 - 640 * q^59 + 1054 * q^61 - 1660 * q^63 - 708 * q^65 - 1184 * q^67 + 188 * q^69 - 1988 * q^71 - 750 * q^73 - 3126 * q^75 + 1260 * q^77 - 2916 * q^79 + 293 * q^81 - 2832 * q^83 + 56 * q^85 + 116 * q^87 - 370 * q^89 - 3016 * q^91 - 1696 * q^93 - 4412 * q^95 - 2234 * q^97 - 4118 * q^99

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