LMFDB - Embedded newform 784.4.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 13 $ x^3 - x^2 - 10*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.36922$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.261560 q^{3} +19.5009 q^{5} -26.9316 q^{9} +O(q^{10})$	$q+0.261560 q^{3} +19.5009 q^{5} -26.9316 q^{9} +56.2875 q^{11} -66.0260 q^{13} +5.10067 q^{15} -18.8372 q^{17} +80.8106 q^{19} +89.3559 q^{23} +255.287 q^{25} -14.1063 q^{27} +104.118 q^{29} +148.437 q^{31} +14.7226 q^{33} +13.8119 q^{37} -17.2697 q^{39} +174.403 q^{41} -205.353 q^{43} -525.191 q^{45} +116.634 q^{47} -4.92706 q^{51} -316.249 q^{53} +1097.66 q^{55} +21.1368 q^{57} +539.585 q^{59} +145.313 q^{61} -1287.57 q^{65} -476.094 q^{67} +23.3719 q^{69} -131.763 q^{71} +349.608 q^{73} +66.7728 q^{75} +806.906 q^{79} +723.463 q^{81} +233.648 q^{83} -367.344 q^{85} +27.2332 q^{87} -535.483 q^{89} +38.8252 q^{93} +1575.88 q^{95} +80.7221 q^{97} -1515.91 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9}+O(q^{10}) $ 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9	$ 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9} + 3 q^{11} - 26 q^{13} - 127 q^{15} - 31 q^{17} + 89 q^{19} + 201 q^{23} + 300 q^{25} + 469 q^{27} + 190 q^{29} + 339 q^{31} - 105 q^{33} - 535 q^{37} + 134 q^{39} + 58 q^{41} - 268 q^{43} - 1410 q^{45} + 205 q^{47} + 965 q^{51} + 757 q^{53} + 1653 q^{55} + 261 q^{57} + 1799 q^{59} + 625 q^{61} - 1750 q^{65} + 495 q^{67} + 973 q^{69} - 640 q^{71} - 443 q^{73} + 1484 q^{75} - 79 q^{79} + 2523 q^{81} + 2372 q^{83} - 977 q^{85} + 910 q^{87} + 821 q^{89} - 1321 q^{93} + 1327 q^{95} - 342 q^{97} - 2310 q^{99}+O(q^{100}) $ 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9 + 3 * q^11 - 26 * q^13 - 127 * q^15 - 31 * q^17 + 89 * q^19 + 201 * q^23 + 300 * q^25 + 469 * q^27 + 190 * q^29 + 339 * q^31 - 105 * q^33 - 535 * q^37 + 134 * q^39 + 58 * q^41 - 268 * q^43 - 1410 * q^45 + 205 * q^47 + 965 * q^51 + 757 * q^53 + 1653 * q^55 + 261 * q^57 + 1799 * q^59 + 625 * q^61 - 1750 * q^65 + 495 * q^67 + 973 * q^69 - 640 * q^71 - 443 * q^73 + 1484 * q^75 - 79 * q^79 + 2523 * q^81 + 2372 * q^83 - 977 * q^85 + 910 * q^87 + 821 * q^89 - 1321 * q^93 + 1327 * q^95 - 342 * q^97 - 2310 * 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
$3$	0.261560	0.0503372	0.0251686	−	0.999683i	$-0.491988\pi$
$3$	0.261560	0.0503372	0.0251686	+	0.999683i	$0.491988\pi$
$4$	0	0
$5$	19.5009	1.74422	0.872109	−	0.489312i	$-0.162752\pi$
$5$	19.5009	1.74422	0.872109	+	0.489312i	$0.162752\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−26.9316	−0.997466
$10$	0	0
$11$	56.2875	1.54285	0.771424	−	0.636322i	$-0.219545\pi$
$11$	56.2875	1.54285	0.771424	+	0.636322i	$0.219545\pi$
$12$	0	0
$13$	−66.0260	−1.40864	−0.704319	−	0.709883i	$-0.748748\pi$
$13$	−66.0260	−1.40864	−0.704319	+	0.709883i	$0.748748\pi$
$14$	0	0
$15$	5.10067	0.0877991
$16$	0	0
$17$	−18.8372	−0.268747	−0.134373	−	0.990931i	$-0.542902\pi$
$17$	−18.8372	−0.268747	−0.134373	+	0.990931i	$0.542902\pi$
$18$	0	0
$19$	80.8106	0.975749	0.487875	−	0.872914i	$-0.337772\pi$
$19$	80.8106	0.975749	0.487875	+	0.872914i	$0.337772\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	89.3559	0.810087	0.405043	−	0.914297i	$-0.367256\pi$
$23$	89.3559	0.810087	0.405043	+	0.914297i	$0.367256\pi$
$24$	0	0
$25$	255.287	2.04230
$26$	0	0
$27$	−14.1063	−0.100547
$28$	0	0
$29$	104.118	0.666700	0.333350	−	0.942803i	$-0.391821\pi$
$29$	104.118	0.666700	0.333350	+	0.942803i	$0.391821\pi$
$30$	0	0
$31$	148.437	0.860002	0.430001	−	0.902828i	$-0.358513\pi$
$31$	148.437	0.860002	0.430001	+	0.902828i	$0.358513\pi$
$32$	0	0
$33$	14.7226	0.0776627
$34$	0	0
$35$	0	0
$36$	0	0
$37$	13.8119	0.0613693	0.0306847	−	0.999529i	$-0.490231\pi$
$37$	13.8119	0.0613693	0.0306847	+	0.999529i	$0.490231\pi$
$38$	0	0
$39$	−17.2697	−0.0709070
$40$	0	0
$41$	174.403	0.664323	0.332161	−	0.943223i	$-0.392222\pi$
$41$	174.403	0.664323	0.332161	+	0.943223i	$0.392222\pi$
$42$	0	0
$43$	−205.353	−0.728279	−0.364139	−	0.931344i	$-0.618637\pi$
$43$	−205.353	−0.728279	−0.364139	+	0.931344i	$0.618637\pi$
$44$	0	0
$45$	−525.191	−1.73980
$46$	0	0
$47$	116.634	0.361976	0.180988	−	0.983485i	$-0.442070\pi$
$47$	116.634	0.361976	0.180988	+	0.983485i	$0.442070\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.92706	−0.0135280
$52$	0	0
$53$	−316.249	−0.819625	−0.409813	−	0.912170i	$-0.634406\pi$
$53$	−316.249	−0.819625	−0.409813	+	0.912170i	$0.634406\pi$
$54$	0	0
$55$	1097.66	2.69106
$56$	0	0
$57$	21.1368	0.0491165
$58$	0	0
$59$	539.585	1.19064	0.595322	−	0.803487i	$-0.297024\pi$
$59$	539.585	1.19064	0.595322	+	0.803487i	$0.297024\pi$
$60$	0	0
$61$	145.313	0.305007	0.152503	−	0.988303i	$-0.451266\pi$
$61$	145.313	0.305007	0.152503	+	0.988303i	$0.451266\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1287.57	−2.45697
$66$	0	0
$67$	−476.094	−0.868122	−0.434061	−	0.900883i	$-0.642920\pi$
$67$	−476.094	−0.868122	−0.434061	+	0.900883i	$0.642920\pi$
$68$	0	0
$69$	23.3719	0.0407775
$70$	0	0
$71$	−131.763	−0.220245	−0.110123	−	0.993918i	$-0.535124\pi$
$71$	−131.763	−0.220245	−0.110123	+	0.993918i	$0.535124\pi$
$72$	0	0
$73$	349.608	0.560528	0.280264	−	0.959923i	$-0.409578\pi$
$73$	349.608	0.560528	0.280264	+	0.959923i	$0.409578\pi$
$74$	0	0
$75$	66.7728	0.102803
$76$	0	0
$77$	0	0
$78$	0	0
$79$	806.906	1.14917	0.574583	−	0.818447i	$-0.305164\pi$
$79$	806.906	1.14917	0.574583	+	0.818447i	$0.305164\pi$
$80$	0	0
$81$	723.463	0.992405
$82$	0	0
$83$	233.648	0.308990	0.154495	−	0.987994i	$-0.450625\pi$
$83$	233.648	0.308990	0.154495	+	0.987994i	$0.450625\pi$
$84$	0	0
$85$	−367.344	−0.468753
$86$	0	0
$87$	27.2332	0.0335599
$88$	0	0
$89$	−535.483	−0.637765	−0.318882	−	0.947794i	$-0.603307\pi$
$89$	−535.483	−0.637765	−0.318882	+	0.947794i	$0.603307\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	38.8252	0.0432901
$94$	0	0
$95$	1575.88	1.70192
$96$	0	0
$97$	80.7221	0.0844958	0.0422479	−	0.999107i	$-0.486548\pi$
$97$	80.7221	0.0844958	0.0422479	+	0.999107i	$0.486548\pi$
$98$	0	0
$99$	−1515.91	−1.53894
$100$	0	0
$101$	−173.472	−0.170902	−0.0854509	−	0.996342i	$-0.527233\pi$
$101$	−173.472	−0.170902	−0.0854509	+	0.996342i	$0.527233\pi$
$102$	0	0
$103$	195.751	0.187262	0.0936308	−	0.995607i	$-0.470153\pi$
$103$	195.751	0.187262	0.0936308	+	0.995607i	$0.470153\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2048.92	−1.85118	−0.925591	−	0.378524i	$-0.876432\pi$
$107$	−2048.92	−1.85118	−0.925591	+	0.378524i	$0.876432\pi$
$108$	0	0
$109$	1557.94	1.36902	0.684512	−	0.729001i	$-0.260015\pi$
$109$	1557.94	1.36902	0.684512	+	0.729001i	$0.260015\pi$
$110$	0	0
$111$	3.61264	0.00308916
$112$	0	0
$113$	−652.982	−0.543605	−0.271802	−	0.962353i	$-0.587620\pi$
$113$	−652.982	−0.543605	−0.271802	+	0.962353i	$0.587620\pi$
$114$	0	0
$115$	1742.53	1.41297
$116$	0	0
$117$	1778.18	1.40507
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1837.28	1.38038
$122$	0	0
$123$	45.6170	0.0334402
$124$	0	0
$125$	2540.72	1.81799
$126$	0	0
$127$	819.069	0.572288	0.286144	−	0.958187i	$-0.407626\pi$
$127$	819.069	0.572288	0.286144	+	0.958187i	$0.407626\pi$
$128$	0	0
$129$	−53.7120	−0.0366595
$130$	0	0
$131$	2428.30	1.61955	0.809777	−	0.586737i	$-0.199588\pi$
$131$	2428.30	1.61955	0.809777	+	0.586737i	$0.199588\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−275.087	−0.175376
$136$	0	0
$137$	100.337	0.0625719	0.0312859	−	0.999510i	$-0.490040\pi$
$137$	100.337	0.0625719	0.0312859	+	0.999510i	$0.490040\pi$
$138$	0	0
$139$	634.299	0.387054	0.193527	−	0.981095i	$-0.438007\pi$
$139$	634.299	0.387054	0.193527	+	0.981095i	$0.438007\pi$
$140$	0	0
$141$	30.5069	0.0182209
$142$	0	0
$143$	−3716.44	−2.17332
$144$	0	0
$145$	2030.41	1.16287
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2105.19	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$149$	−2105.19	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$150$	0	0
$151$	2868.57	1.54596	0.772982	−	0.634428i	$-0.218764\pi$
$151$	2868.57	1.54596	0.772982	+	0.634428i	$0.218764\pi$
$152$	0	0
$153$	507.316	0.268066
$154$	0	0
$155$	2894.66	1.50003
$156$	0	0
$157$	3087.39	1.56943	0.784714	−	0.619858i	$-0.212810\pi$
$157$	3087.39	1.56943	0.784714	+	0.619858i	$0.212810\pi$
$158$	0	0
$159$	−82.7180	−0.0412577
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1011.12	0.485869	0.242935	−	0.970043i	$-0.421890\pi$
$163$	1011.12	0.485869	0.242935	+	0.970043i	$0.421890\pi$
$164$	0	0
$165$	287.104	0.135461
$166$	0	0
$167$	1052.69	0.487784	0.243892	−	0.969802i	$-0.421576\pi$
$167$	1052.69	0.487784	0.243892	+	0.969802i	$0.421576\pi$
$168$	0	0
$169$	2162.43	0.984264
$170$	0	0
$171$	−2176.36	−0.973277
$172$	0	0
$173$	−306.144	−0.134542	−0.0672708	−	0.997735i	$-0.521429\pi$
$173$	−306.144	−0.134542	−0.0672708	+	0.997735i	$0.521429\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	141.134	0.0599337
$178$	0	0
$179$	188.964	0.0789040	0.0394520	−	0.999221i	$-0.487439\pi$
$179$	188.964	0.0789040	0.0394520	+	0.999221i	$0.487439\pi$
$180$	0	0
$181$	−468.520	−0.192402	−0.0962011	−	0.995362i	$-0.530669\pi$
$181$	−468.520	−0.192402	−0.0962011	+	0.995362i	$0.530669\pi$
$182$	0	0
$183$	38.0081	0.0153532
$184$	0	0
$185$	269.345	0.107041
$186$	0	0
$187$	−1060.30	−0.414635
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4200.00	−1.59111	−0.795553	−	0.605884i	$-0.792820\pi$
$191$	−4200.00	−1.59111	−0.795553	+	0.605884i	$0.792820\pi$
$192$	0	0
$193$	−2925.26	−1.09101	−0.545505	−	0.838108i	$-0.683662\pi$
$193$	−2925.26	−1.09101	−0.545505	+	0.838108i	$0.683662\pi$
$194$	0	0
$195$	−336.776	−0.123677
$196$	0	0
$197$	−3402.39	−1.23051	−0.615254	−	0.788329i	$-0.710947\pi$
$197$	−3402.39	−1.23051	−0.615254	+	0.788329i	$0.710947\pi$
$198$	0	0
$199$	98.3992	0.0350519	0.0175260	−	0.999846i	$-0.494421\pi$
$199$	98.3992	0.0350519	0.0175260	+	0.999846i	$0.494421\pi$
$200$	0	0
$201$	−124.527	−0.0436989
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3401.03	1.15872
$206$	0	0
$207$	−2406.50	−0.808034
$208$	0	0
$209$	4548.63	1.50543
$210$	0	0
$211$	−2692.24	−0.878396	−0.439198	−	0.898390i	$-0.644737\pi$
$211$	−2692.24	−0.878396	−0.439198	+	0.898390i	$0.644737\pi$
$212$	0	0
$213$	−34.4640	−0.0110865
$214$	0	0
$215$	−4004.57	−1.27028
$216$	0	0
$217$	0	0
$218$	0	0
$219$	91.4435	0.0282154
$220$	0	0
$221$	1243.75	0.378567
$222$	0	0
$223$	−6075.65	−1.82447	−0.912233	−	0.409672i	$-0.865643\pi$
$223$	−6075.65	−1.82447	−0.912233	+	0.409672i	$0.865643\pi$
$224$	0	0
$225$	−6875.28	−2.03712
$226$	0	0
$227$	2991.15	0.874581	0.437290	−	0.899320i	$-0.355938\pi$
$227$	2991.15	0.874581	0.437290	+	0.899320i	$0.355938\pi$
$228$	0	0
$229$	−6181.17	−1.78368	−0.891841	−	0.452349i	$-0.850586\pi$
$229$	−6181.17	−1.78368	−0.891841	+	0.452349i	$0.850586\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2913.44	0.819166	0.409583	−	0.912273i	$-0.365674\pi$
$233$	2913.44	0.819166	0.409583	+	0.912273i	$0.365674\pi$
$234$	0	0
$235$	2274.48	0.631366
$236$	0	0
$237$	211.054	0.0578458
$238$	0	0
$239$	−3407.25	−0.922163	−0.461081	−	0.887358i	$-0.652538\pi$
$239$	−3407.25	−0.922163	−0.461081	+	0.887358i	$0.652538\pi$
$240$	0	0
$241$	244.124	0.0652508	0.0326254	−	0.999468i	$-0.489613\pi$
$241$	244.124	0.0652508	0.0326254	+	0.999468i	$0.489613\pi$
$242$	0	0
$243$	570.100	0.150502
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5335.60	−1.37448
$248$	0	0
$249$	61.1129	0.0155537
$250$	0	0
$251$	4795.49	1.20593	0.602966	−	0.797767i	$-0.293986\pi$
$251$	4795.49	1.20593	0.602966	+	0.797767i	$0.293986\pi$
$252$	0	0
$253$	5029.62	1.24984
$254$	0	0
$255$	−96.0824	−0.0235957
$256$	0	0
$257$	4782.35	1.16076	0.580380	−	0.814346i	$-0.302904\pi$
$257$	4782.35	1.16076	0.580380	+	0.814346i	$0.302904\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2804.07	−0.665011
$262$	0	0
$263$	−7649.05	−1.79339	−0.896693	−	0.442652i	$-0.854038\pi$
$263$	−7649.05	−1.79339	−0.896693	+	0.442652i	$0.854038\pi$
$264$	0	0
$265$	−6167.15	−1.42960
$266$	0	0
$267$	−140.061	−0.0321033
$268$	0	0
$269$	4200.71	0.952125	0.476062	−	0.879412i	$-0.342064\pi$
$269$	4200.71	0.952125	0.476062	+	0.879412i	$0.342064\pi$
$270$	0	0
$271$	5063.22	1.13494	0.567470	−	0.823394i	$-0.307922\pi$
$271$	5063.22	1.13494	0.567470	+	0.823394i	$0.307922\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14369.5	3.15095
$276$	0	0
$277$	−5346.39	−1.15969	−0.579843	−	0.814728i	$-0.696886\pi$
$277$	−5346.39	−1.15969	−0.579843	+	0.814728i	$0.696886\pi$
$278$	0	0
$279$	−3997.64	−0.857823
$280$	0	0
$281$	6367.90	1.35188	0.675938	−	0.736958i	$-0.263739\pi$
$281$	6367.90	1.35188	0.675938	+	0.736958i	$0.263739\pi$
$282$	0	0
$283$	2124.38	0.446224	0.223112	−	0.974793i	$-0.428378\pi$
$283$	2124.38	0.446224	0.223112	+	0.974793i	$0.428378\pi$
$284$	0	0
$285$	412.188	0.0856699
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4558.16	−0.927775
$290$	0	0
$291$	21.1137	0.00425328
$292$	0	0
$293$	−5921.90	−1.18075	−0.590377	−	0.807128i	$-0.701021\pi$
$293$	−5921.90	−1.18075	−0.590377	+	0.807128i	$0.701021\pi$
$294$	0	0
$295$	10522.4	2.07674
$296$	0	0
$297$	−794.011	−0.155129
$298$	0	0
$299$	−5899.81	−1.14112
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−45.3733	−0.00860273
$304$	0	0
$305$	2833.74	0.531998
$306$	0	0
$307$	6630.60	1.23267	0.616333	−	0.787486i	$-0.288618\pi$
$307$	6630.60	1.23267	0.616333	+	0.787486i	$0.288618\pi$
$308$	0	0
$309$	51.2007	0.00942623
$310$	0	0
$311$	−2181.45	−0.397746	−0.198873	−	0.980025i	$-0.563728\pi$
$311$	−2181.45	−0.397746	−0.198873	+	0.980025i	$0.563728\pi$
$312$	0	0
$313$	−8996.44	−1.62463	−0.812314	−	0.583220i	$-0.801792\pi$
$313$	−8996.44	−1.62463	−0.812314	+	0.583220i	$0.801792\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−314.569	−0.0557348	−0.0278674	−	0.999612i	$-0.508872\pi$
$317$	−314.569	−0.0557348	−0.0278674	+	0.999612i	$0.508872\pi$
$318$	0	0
$319$	5860.57	1.02862
$320$	0	0
$321$	−535.915	−0.0931834
$322$	0	0
$323$	−1522.25	−0.262229
$324$	0	0
$325$	−16855.6	−2.87686
$326$	0	0
$327$	407.495	0.0689129
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7006.07	1.16341	0.581704	−	0.813400i	$-0.302386\pi$
$331$	7006.07	1.16341	0.581704	+	0.813400i	$0.302386\pi$
$332$	0	0
$333$	−371.977	−0.0612138
$334$	0	0
$335$	−9284.29	−1.51419
$336$	0	0
$337$	4697.83	0.759368	0.379684	−	0.925116i	$-0.376033\pi$
$337$	4697.83	0.759368	0.379684	+	0.925116i	$0.376033\pi$
$338$	0	0
$339$	−170.794	−0.0273636
$340$	0	0
$341$	8355.15	1.32685
$342$	0	0
$343$	0	0
$344$	0	0
$345$	455.775	0.0711249
$346$	0	0
$347$	−8094.65	−1.25229	−0.626143	−	0.779708i	$-0.715368\pi$
$347$	−8094.65	−1.25229	−0.626143	+	0.779708i	$0.715368\pi$
$348$	0	0
$349$	−10531.7	−1.61532	−0.807662	−	0.589646i	$-0.799267\pi$
$349$	−10531.7	−1.61532	−0.807662	+	0.589646i	$0.799267\pi$
$350$	0	0
$351$	931.385	0.141634
$352$	0	0
$353$	−11986.5	−1.80730	−0.903648	−	0.428276i	$-0.859121\pi$
$353$	−11986.5	−1.80730	−0.903648	+	0.428276i	$0.859121\pi$
$354$	0	0
$355$	−2569.51	−0.384155
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8073.90	−1.18698	−0.593488	−	0.804843i	$-0.702250\pi$
$359$	−8073.90	−1.18698	−0.593488	+	0.804843i	$0.702250\pi$
$360$	0	0
$361$	−328.641	−0.0479138
$362$	0	0
$363$	480.560	0.0694845
$364$	0	0
$365$	6817.69	0.977683
$366$	0	0
$367$	−2904.85	−0.413166	−0.206583	−	0.978429i	$-0.566234\pi$
$367$	−2904.85	−0.413166	−0.206583	+	0.978429i	$0.566234\pi$
$368$	0	0
$369$	−4696.96	−0.662640
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5116.71	0.710277	0.355139	−	0.934814i	$-0.384434\pi$
$373$	5116.71	0.710277	0.355139	+	0.934814i	$0.384434\pi$
$374$	0	0
$375$	664.550	0.0915126
$376$	0	0
$377$	−6874.52	−0.939140
$378$	0	0
$379$	−7841.91	−1.06283	−0.531414	−	0.847112i	$-0.678339\pi$
$379$	−7841.91	−1.06283	−0.531414	+	0.847112i	$0.678339\pi$
$380$	0	0
$381$	214.236	0.0288074
$382$	0	0
$383$	4959.17	0.661623	0.330812	−	0.943697i	$-0.392678\pi$
$383$	4959.17	0.661623	0.330812	+	0.943697i	$0.392678\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5530.47	0.726433
$388$	0	0
$389$	2188.94	0.285306	0.142653	−	0.989773i	$-0.454437\pi$
$389$	2188.94	0.285306	0.142653	+	0.989773i	$0.454437\pi$
$390$	0	0
$391$	−1683.22	−0.217708
$392$	0	0
$393$	635.146	0.0815239
$394$	0	0
$395$	15735.4	2.00439
$396$	0	0
$397$	−6953.85	−0.879103	−0.439551	−	0.898217i	$-0.644863\pi$
$397$	−6953.85	−0.879103	−0.439551	+	0.898217i	$0.644863\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3768.06	−0.469246	−0.234623	−	0.972086i	$-0.575386\pi$
$401$	−3768.06	−0.469246	−0.234623	+	0.972086i	$0.575386\pi$
$402$	0	0
$403$	−9800.69	−1.21143
$404$	0	0
$405$	14108.2	1.73097
$406$	0	0
$407$	777.438	0.0946835
$408$	0	0
$409$	16138.5	1.95110	0.975548	−	0.219787i	$-0.0705362\pi$
$409$	16138.5	1.95110	0.975548	+	0.219787i	$0.0705362\pi$
$410$	0	0
$411$	26.2441	0.00314969
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4556.36	0.538946
$416$	0	0
$417$	165.907	0.0194832
$418$	0	0
$419$	−9445.77	−1.10133	−0.550664	−	0.834727i	$-0.685625\pi$
$419$	−9445.77	−1.10133	−0.550664	+	0.834727i	$0.685625\pi$
$420$	0	0
$421$	−11555.5	−1.33772	−0.668858	−	0.743390i	$-0.733217\pi$
$421$	−11555.5	−1.33772	−0.668858	+	0.743390i	$0.733217\pi$
$422$	0	0
$423$	−3141.15	−0.361059
$424$	0	0
$425$	−4808.89	−0.548860
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−972.071	−0.109399
$430$	0	0
$431$	−9497.23	−1.06141	−0.530703	−	0.847558i	$-0.678072\pi$
$431$	−9497.23	−1.06141	−0.530703	+	0.847558i	$0.678072\pi$
$432$	0	0
$433$	−10826.5	−1.20160	−0.600798	−	0.799401i	$-0.705150\pi$
$433$	−10826.5	−1.20160	−0.600798	+	0.799401i	$0.705150\pi$
$434$	0	0
$435$	531.073	0.0585357
$436$	0	0
$437$	7220.91	0.790442
$438$	0	0
$439$	−12635.7	−1.37373	−0.686867	−	0.726783i	$-0.741015\pi$
$439$	−12635.7	−1.37373	−0.686867	+	0.726783i	$0.741015\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4875.92	0.522939	0.261470	−	0.965212i	$-0.415793\pi$
$443$	4875.92	0.522939	0.261470	+	0.965212i	$0.415793\pi$
$444$	0	0
$445$	−10442.4	−1.11240
$446$	0	0
$447$	−550.632	−0.0582640
$448$	0	0
$449$	−5294.85	−0.556524	−0.278262	−	0.960505i	$-0.589758\pi$
$449$	−5294.85	−0.556524	−0.278262	+	0.960505i	$0.589758\pi$
$450$	0	0
$451$	9816.74	1.02495
$452$	0	0
$453$	750.302	0.0778196
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1899.61	0.194442	0.0972211	−	0.995263i	$-0.469005\pi$
$457$	1899.61	0.194442	0.0972211	+	0.995263i	$0.469005\pi$
$458$	0	0
$459$	265.724	0.0270217
$460$	0	0
$461$	−4665.78	−0.471382	−0.235691	−	0.971828i	$-0.575735\pi$
$461$	−4665.78	−0.471382	−0.235691	+	0.971828i	$0.575735\pi$
$462$	0	0
$463$	−1960.56	−0.196792	−0.0983962	−	0.995147i	$-0.531371\pi$
$463$	−1960.56	−0.196792	−0.0983962	+	0.995147i	$0.531371\pi$
$464$	0	0
$465$	757.127	0.0755074
$466$	0	0
$467$	10206.4	1.01134	0.505671	−	0.862726i	$-0.331245\pi$
$467$	10206.4	1.01134	0.505671	+	0.862726i	$0.331245\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	807.537	0.0790007
$472$	0	0
$473$	−11558.8	−1.12362
$474$	0	0
$475$	20629.9	1.99277
$476$	0	0
$477$	8517.08	0.817548
$478$	0	0
$479$	−10961.2	−1.04558	−0.522789	−	0.852462i	$-0.675108\pi$
$479$	−10961.2	−1.04558	−0.522789	+	0.852462i	$0.675108\pi$
$480$	0	0
$481$	−911.945	−0.0864472
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1574.16	0.147379
$486$	0	0
$487$	17409.7	1.61994	0.809970	−	0.586471i	$-0.199483\pi$
$487$	17409.7	1.61994	0.809970	+	0.586471i	$0.199483\pi$
$488$	0	0
$489$	264.467	0.0244573
$490$	0	0
$491$	5282.62	0.485543	0.242771	−	0.970084i	$-0.421944\pi$
$491$	5282.62	0.485543	0.242771	+	0.970084i	$0.421944\pi$
$492$	0	0
$493$	−1961.30	−0.179174
$494$	0	0
$495$	−29561.7	−2.68424
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18977.1	−1.70246	−0.851232	−	0.524789i	$-0.824144\pi$
$499$	−18977.1	−1.70246	−0.851232	+	0.524789i	$0.824144\pi$
$500$	0	0
$501$	275.343	0.0245537
$502$	0	0
$503$	−8525.73	−0.755753	−0.377876	−	0.925856i	$-0.623346\pi$
$503$	−8525.73	−0.755753	−0.377876	+	0.925856i	$0.623346\pi$
$504$	0	0
$505$	−3382.86	−0.298090
$506$	0	0
$507$	565.604	0.0495451
$508$	0	0
$509$	10235.7	0.891333	0.445666	−	0.895199i	$-0.352967\pi$
$509$	10235.7	0.891333	0.445666	+	0.895199i	$0.352967\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1139.94	−0.0981086
$514$	0	0
$515$	3817.34	0.326625
$516$	0	0
$517$	6565.07	0.558474
$518$	0	0
$519$	−80.0750	−0.00677245
$520$	0	0
$521$	1592.37	0.133902	0.0669511	−	0.997756i	$-0.478673\pi$
$521$	1592.37	0.133902	0.0669511	+	0.997756i	$0.478673\pi$
$522$	0	0
$523$	14008.0	1.17118	0.585589	−	0.810608i	$-0.300863\pi$
$523$	14008.0	1.17118	0.585589	+	0.810608i	$0.300863\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2796.14	−0.231123
$528$	0	0
$529$	−4182.52	−0.343759
$530$	0	0
$531$	−14531.9	−1.18763
$532$	0	0
$533$	−11515.2	−0.935791
$534$	0	0
$535$	−39955.9	−3.22887
$536$	0	0
$537$	49.4254	0.00397181
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8928.02	0.709511	0.354755	−	0.934959i	$-0.384564\pi$
$541$	8928.02	0.709511	0.354755	+	0.934959i	$0.384564\pi$
$542$	0	0
$543$	−122.546	−0.00968500
$544$	0	0
$545$	30381.3	2.38788
$546$	0	0
$547$	−9736.48	−0.761063	−0.380532	−	0.924768i	$-0.624259\pi$
$547$	−9736.48	−0.761063	−0.380532	+	0.924768i	$0.624259\pi$
$548$	0	0
$549$	−3913.51	−0.304234
$550$	0	0
$551$	8413.88	0.650532
$552$	0	0
$553$	0	0
$554$	0	0
$555$	70.4500	0.00538817
$556$	0	0
$557$	4432.76	0.337203	0.168602	−	0.985684i	$-0.446075\pi$
$557$	4432.76	0.337203	0.168602	+	0.985684i	$0.446075\pi$
$558$	0	0
$559$	13558.6	1.02588
$560$	0	0
$561$	−277.332	−0.0208716
$562$	0	0
$563$	−1402.96	−0.105022	−0.0525112	−	0.998620i	$-0.516723\pi$
$563$	−1402.96	−0.105022	−0.0525112	+	0.998620i	$0.516723\pi$
$564$	0	0
$565$	−12733.8	−0.948165
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5798.54	0.427218	0.213609	−	0.976919i	$-0.431478\pi$
$569$	5798.54	0.427218	0.213609	+	0.976919i	$0.431478\pi$
$570$	0	0
$571$	−13761.8	−1.00861	−0.504303	−	0.863527i	$-0.668250\pi$
$571$	−13761.8	−1.00861	−0.504303	+	0.863527i	$0.668250\pi$
$572$	0	0
$573$	−1098.55	−0.0800919
$574$	0	0
$575$	22811.4	1.65444
$576$	0	0
$577$	−7420.50	−0.535389	−0.267695	−	0.963504i	$-0.586262\pi$
$577$	−7420.50	−0.535389	−0.267695	+	0.963504i	$0.586262\pi$
$578$	0	0
$579$	−765.131	−0.0549184
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−17800.9	−1.26456
$584$	0	0
$585$	34676.3	2.45075
$586$	0	0
$587$	−19685.1	−1.38414	−0.692072	−	0.721828i	$-0.743302\pi$
$587$	−19685.1	−1.38414	−0.692072	+	0.721828i	$0.743302\pi$
$588$	0	0
$589$	11995.3	0.839146
$590$	0	0
$591$	−889.928	−0.0619404
$592$	0	0
$593$	24079.3	1.66749	0.833743	−	0.552153i	$-0.186193\pi$
$593$	24079.3	1.66749	0.833743	+	0.552153i	$0.186193\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.7373	0.00176442
$598$	0	0
$599$	17966.4	1.22552	0.612759	−	0.790270i	$-0.290060\pi$
$599$	17966.4	1.22552	0.612759	+	0.790270i	$0.290060\pi$
$600$	0	0
$601$	12893.6	0.875111	0.437555	−	0.899192i	$-0.355844\pi$
$601$	12893.6	0.875111	0.437555	+	0.899192i	$0.355844\pi$
$602$	0	0
$603$	12822.0	0.865922
$604$	0	0
$605$	35828.8	2.40768
$606$	0	0
$607$	25200.5	1.68510	0.842550	−	0.538619i	$-0.181054\pi$
$607$	25200.5	1.68510	0.842550	+	0.538619i	$0.181054\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7700.90	−0.509894
$612$	0	0
$613$	12814.9	0.844357	0.422178	−	0.906513i	$-0.361266\pi$
$613$	12814.9	0.844357	0.422178	+	0.906513i	$0.361266\pi$
$614$	0	0
$615$	889.574	0.0583270
$616$	0	0
$617$	7597.02	0.495696	0.247848	−	0.968799i	$-0.420277\pi$
$617$	7597.02	0.495696	0.247848	+	0.968799i	$0.420277\pi$
$618$	0	0
$619$	22412.8	1.45533	0.727664	−	0.685934i	$-0.240606\pi$
$619$	22412.8	1.45533	0.727664	+	0.685934i	$0.240606\pi$
$620$	0	0
$621$	−1260.49	−0.0814518
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17635.5	1.12867
$626$	0	0
$627$	1189.74	0.0757793
$628$	0	0
$629$	−260.178	−0.0164928
$630$	0	0
$631$	−8993.43	−0.567389	−0.283695	−	0.958915i	$-0.591560\pi$
$631$	−8993.43	−0.567389	−0.283695	+	0.958915i	$0.591560\pi$
$632$	0	0
$633$	−704.183	−0.0442160
$634$	0	0
$635$	15972.6	0.998195
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3548.59	0.219687
$640$	0	0
$641$	−18152.7	−1.11855	−0.559273	−	0.828983i	$-0.688920\pi$
$641$	−18152.7	−1.11855	−0.559273	+	0.828983i	$0.688920\pi$
$642$	0	0
$643$	25767.3	1.58034	0.790172	−	0.612885i	$-0.209991\pi$
$643$	25767.3	1.58034	0.790172	+	0.612885i	$0.209991\pi$
$644$	0	0
$645$	−1047.44	−0.0639422
$646$	0	0
$647$	6962.52	0.423068	0.211534	−	0.977371i	$-0.432154\pi$
$647$	6962.52	0.423068	0.211534	+	0.977371i	$0.432154\pi$
$648$	0	0
$649$	30371.9	1.83698
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12157.8	−0.728595	−0.364297	−	0.931283i	$-0.618691\pi$
$653$	−12157.8	−0.728595	−0.364297	+	0.931283i	$0.618691\pi$
$654$	0	0
$655$	47354.2	2.82486
$656$	0	0
$657$	−9415.50	−0.559108
$658$	0	0
$659$	3853.23	0.227770	0.113885	−	0.993494i	$-0.463670\pi$
$659$	3853.23	0.227770	0.113885	+	0.993494i	$0.463670\pi$
$660$	0	0
$661$	−4579.02	−0.269445	−0.134723	−	0.990883i	$-0.543014\pi$
$661$	−4579.02	−0.269445	−0.134723	+	0.990883i	$0.543014\pi$
$662$	0	0
$663$	325.314	0.0190560
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9303.60	0.540085
$668$	0	0
$669$	−1589.15	−0.0918386
$670$	0	0
$671$	8179.31	0.470579
$672$	0	0
$673$	−21847.0	−1.25132	−0.625661	−	0.780095i	$-0.715171\pi$
$673$	−21847.0	−1.25132	−0.625661	+	0.780095i	$0.715171\pi$
$674$	0	0
$675$	−3601.16	−0.205347
$676$	0	0
$677$	4823.04	0.273803	0.136901	−	0.990585i	$-0.456286\pi$
$677$	4823.04	0.273803	0.136901	+	0.990585i	$0.456286\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	782.366	0.0440240
$682$	0	0
$683$	16222.7	0.908850	0.454425	−	0.890785i	$-0.349845\pi$
$683$	16222.7	0.908850	0.454425	+	0.890785i	$0.349845\pi$
$684$	0	0
$685$	1956.66	0.109139
$686$	0	0
$687$	−1616.75	−0.0897856
$688$	0	0
$689$	20880.6	1.15456
$690$	0	0
$691$	−25356.2	−1.39594	−0.697971	−	0.716126i	$-0.745914\pi$
$691$	−25356.2	−1.39594	−0.697971	+	0.716126i	$0.745914\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12369.4	0.675107
$696$	0	0
$697$	−3285.28	−0.178535
$698$	0	0
$699$	762.039	0.0412345
$700$	0	0
$701$	−11538.7	−0.621701	−0.310850	−	0.950459i	$-0.600614\pi$
$701$	−11538.7	−0.621701	−0.310850	+	0.950459i	$0.600614\pi$
$702$	0	0
$703$	1116.15	0.0598810
$704$	0	0
$705$	594.914	0.0317812
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33771.6	−1.78888	−0.894442	−	0.447185i	$-0.852427\pi$
$709$	−33771.6	−1.78888	−0.894442	+	0.447185i	$0.852427\pi$
$710$	0	0
$711$	−21731.3	−1.14625
$712$	0	0
$713$	13263.7	0.696676
$714$	0	0
$715$	−72474.0	−3.79074
$716$	0	0
$717$	−891.201	−0.0464191
$718$	0	0
$719$	−14047.9	−0.728647	−0.364324	−	0.931272i	$-0.618700\pi$
$719$	−14047.9	−0.728647	−0.364324	+	0.931272i	$0.618700\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	63.8532	0.00328454
$724$	0	0
$725$	26580.1	1.36160
$726$	0	0
$727$	−10726.8	−0.547226	−0.273613	−	0.961840i	$-0.588219\pi$
$727$	−10726.8	−0.547226	−0.273613	+	0.961840i	$0.588219\pi$
$728$	0	0
$729$	−19384.4	−0.984829
$730$	0	0
$731$	3868.27	0.195723
$732$	0	0
$733$	−10039.3	−0.505880	−0.252940	−	0.967482i	$-0.581397\pi$
$733$	−10039.3	−0.505880	−0.252940	+	0.967482i	$0.581397\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−26798.2	−1.33938
$738$	0	0
$739$	−6984.31	−0.347661	−0.173831	−	0.984776i	$-0.555615\pi$
$739$	−6984.31	−0.347661	−0.173831	+	0.984776i	$0.555615\pi$
$740$	0	0
$741$	−1395.58	−0.0691874
$742$	0	0
$743$	20657.2	1.01997	0.509986	−	0.860183i	$-0.329651\pi$
$743$	20657.2	1.01997	0.509986	+	0.860183i	$0.329651\pi$
$744$	0	0
$745$	−41053.1	−2.01889
$746$	0	0
$747$	−6292.51	−0.308207
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29782.2	−1.44709	−0.723547	−	0.690275i	$-0.757489\pi$
$751$	−29782.2	−1.44709	−0.723547	+	0.690275i	$0.757489\pi$
$752$	0	0
$753$	1254.31	0.0607032
$754$	0	0
$755$	55939.7	2.69650
$756$	0	0
$757$	−17157.0	−0.823754	−0.411877	−	0.911240i	$-0.635127\pi$
$757$	−17157.0	−0.823754	−0.411877	+	0.911240i	$0.635127\pi$
$758$	0	0
$759$	1315.55	0.0629135
$760$	0	0
$761$	−38481.4	−1.83305	−0.916523	−	0.399981i	$-0.869017\pi$
$761$	−38481.4	−1.83305	−0.916523	+	0.399981i	$0.869017\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9893.14	0.467565
$766$	0	0
$767$	−35626.6	−1.67719
$768$	0	0
$769$	−17934.4	−0.841002	−0.420501	−	0.907292i	$-0.638146\pi$
$769$	−17934.4	−0.841002	−0.420501	+	0.907292i	$0.638146\pi$
$770$	0	0
$771$	1250.87	0.0584294
$772$	0	0
$773$	−11443.2	−0.532451	−0.266226	−	0.963911i	$-0.585777\pi$
$773$	−11443.2	−0.532451	−0.266226	+	0.963911i	$0.585777\pi$
$774$	0	0
$775$	37894.0	1.75638
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14093.7	0.648213
$780$	0	0
$781$	−7416.62	−0.339805
$782$	0	0
$783$	−1468.73	−0.0670347
$784$	0	0
$785$	60207.0	2.73743
$786$	0	0
$787$	11578.5	0.524431	0.262216	−	0.965009i	$-0.415547\pi$
$787$	11578.5	0.524431	0.262216	+	0.965009i	$0.415547\pi$
$788$	0	0
$789$	−2000.69	−0.0902741
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9594.43	−0.429645
$794$	0	0
$795$	−1613.08	−0.0719623
$796$	0	0
$797$	36578.7	1.62570	0.812850	−	0.582473i	$-0.197915\pi$
$797$	36578.7	1.62570	0.812850	+	0.582473i	$0.197915\pi$
$798$	0	0
$799$	−2197.07	−0.0972800
$800$	0	0
$801$	14421.4	0.636149
$802$	0	0
$803$	19678.6	0.864809
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1098.74	0.0479273
$808$	0	0
$809$	5876.16	0.255370	0.127685	−	0.991815i	$-0.459245\pi$
$809$	5876.16	0.255370	0.127685	+	0.991815i	$0.459245\pi$
$810$	0	0
$811$	3272.77	0.141705	0.0708523	−	0.997487i	$-0.477428\pi$
$811$	3272.77	0.141705	0.0708523	+	0.997487i	$0.477428\pi$
$812$	0	0
$813$	1324.33	0.0571297
$814$	0	0
$815$	19717.7	0.847462
$816$	0	0
$817$	−16594.7	−0.710617
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11801.6	0.501679	0.250840	−	0.968029i	$-0.419293\pi$
$821$	11801.6	0.501679	0.250840	+	0.968029i	$0.419293\pi$
$822$	0	0
$823$	−9636.61	−0.408154	−0.204077	−	0.978955i	$-0.565419\pi$
$823$	−9636.61	−0.408154	−0.204077	+	0.978955i	$0.565419\pi$
$824$	0	0
$825$	3758.48	0.158610
$826$	0	0
$827$	28192.3	1.18542	0.592710	−	0.805416i	$-0.298058\pi$
$827$	28192.3	1.18542	0.592710	+	0.805416i	$0.298058\pi$
$828$	0	0
$829$	−24633.1	−1.03202	−0.516010	−	0.856583i	$-0.672583\pi$
$829$	−24633.1	−1.03202	−0.516010	+	0.856583i	$0.672583\pi$
$830$	0	0
$831$	−1398.40	−0.0583754
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20528.5	0.850801
$836$	0	0
$837$	−2093.90	−0.0864706
$838$	0	0
$839$	3682.43	0.151527	0.0757637	−	0.997126i	$-0.475861\pi$
$839$	3682.43	0.151527	0.0757637	+	0.997126i	$0.475861\pi$
$840$	0	0
$841$	−13548.4	−0.555511
$842$	0	0
$843$	1665.59	0.0680497
$844$	0	0
$845$	42169.4	1.71677
$846$	0	0
$847$	0	0
$848$	0	0
$849$	555.654	0.0224617
$850$	0	0
$851$	1234.18	0.0497145
$852$	0	0
$853$	−790.015	−0.0317112	−0.0158556	−	0.999874i	$-0.505047\pi$
$853$	−790.015	−0.0317112	−0.0158556	+	0.999874i	$0.505047\pi$
$854$	0	0
$855$	−42441.1	−1.69761
$856$	0	0
$857$	−15450.3	−0.615835	−0.307918	−	0.951413i	$-0.599632\pi$
$857$	−15450.3	−0.615835	−0.307918	+	0.951413i	$0.599632\pi$
$858$	0	0
$859$	−19601.6	−0.778579	−0.389289	−	0.921116i	$-0.627279\pi$
$859$	−19601.6	−0.778579	−0.389289	+	0.921116i	$0.627279\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38572.6	−1.52147	−0.760734	−	0.649064i	$-0.775161\pi$
$863$	−38572.6	−1.52147	−0.760734	+	0.649064i	$0.775161\pi$
$864$	0	0
$865$	−5970.10	−0.234670
$866$	0	0
$867$	−1192.23	−0.0467016
$868$	0	0
$869$	45418.7	1.77299
$870$	0	0
$871$	31434.6	1.22287
$872$	0	0
$873$	−2173.97	−0.0842817
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6319.27	−0.243314	−0.121657	−	0.992572i	$-0.538821\pi$
$877$	−6319.27	−0.243314	−0.121657	+	0.992572i	$0.538821\pi$
$878$	0	0
$879$	−1548.93	−0.0594359
$880$	0	0
$881$	−15224.6	−0.582213	−0.291106	−	0.956691i	$-0.594023\pi$
$881$	−15224.6	−0.582213	−0.291106	+	0.956691i	$0.594023\pi$
$882$	0	0
$883$	33244.2	1.26700	0.633498	−	0.773744i	$-0.281618\pi$
$883$	33244.2	1.26700	0.633498	+	0.773744i	$0.281618\pi$
$884$	0	0
$885$	2752.24	0.104537
$886$	0	0
$887$	−15940.1	−0.603401	−0.301700	−	0.953403i	$-0.597554\pi$
$887$	−15940.1	−0.603401	−0.301700	+	0.953403i	$0.597554\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	40721.9	1.53113
$892$	0	0
$893$	9425.31	0.353198
$894$	0	0
$895$	3684.97	0.137626
$896$	0	0
$897$	−1543.15	−0.0574408
$898$	0	0
$899$	15455.0	0.573364
$900$	0	0
$901$	5957.25	0.220272
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9136.58	−0.335591
$906$	0	0
$907$	22038.3	0.806804	0.403402	−	0.915023i	$-0.367828\pi$
$907$	22038.3	0.806804	0.403402	+	0.915023i	$0.367828\pi$
$908$	0	0
$909$	4671.87	0.170469
$910$	0	0
$911$	−2241.91	−0.0815342	−0.0407671	−	0.999169i	$-0.512980\pi$
$911$	−2241.91	−0.0815342	−0.0407671	+	0.999169i	$0.512980\pi$
$912$	0	0
$913$	13151.5	0.476725
$914$	0	0
$915$	741.193	0.0267793
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35585.4	1.27732	0.638658	−	0.769491i	$-0.279490\pi$
$919$	35585.4	1.27732	0.638658	+	0.769491i	$0.279490\pi$
$920$	0	0
$921$	1734.30	0.0620490
$922$	0	0
$923$	8699.79	0.310246
$924$	0	0
$925$	3526.00	0.125334
$926$	0	0
$927$	−5271.89	−0.186787
$928$	0	0
$929$	45914.2	1.62153	0.810763	−	0.585375i	$-0.199053\pi$
$929$	45914.2	1.62153	0.810763	+	0.585375i	$0.199053\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−570.581	−0.0200214
$934$	0	0
$935$	−20676.9	−0.723214
$936$	0	0
$937$	−42963.4	−1.49792	−0.748962	−	0.662613i	$-0.769447\pi$
$937$	−42963.4	−1.49792	−0.748962	+	0.662613i	$0.769447\pi$
$938$	0	0
$939$	−2353.11	−0.0817793
$940$	0	0
$941$	−21420.7	−0.742077	−0.371039	−	0.928617i	$-0.620998\pi$
$941$	−21420.7	−0.742077	−0.371039	+	0.928617i	$0.620998\pi$
$942$	0	0
$943$	15584.0	0.538159
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6523.67	0.223855	0.111928	−	0.993716i	$-0.464298\pi$
$947$	6523.67	0.223855	0.111928	+	0.993716i	$0.464298\pi$
$948$	0	0
$949$	−23083.2	−0.789581
$950$	0	0
$951$	−82.2786	−0.00280554
$952$	0	0
$953$	−32537.1	−1.10596	−0.552979	−	0.833195i	$-0.686509\pi$
$953$	−32537.1	−1.10596	−0.552979	+	0.833195i	$0.686509\pi$
$954$	0	0
$955$	−81904.0	−2.77524
$956$	0	0
$957$	1532.89	0.0517777
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7757.48	−0.260397
$962$	0	0
$963$	55180.7	1.84649
$964$	0	0
$965$	−57045.4	−1.90296
$966$	0	0
$967$	−9013.22	−0.299737	−0.149868	−	0.988706i	$-0.547885\pi$
$967$	−9013.22	−0.299737	−0.149868	+	0.988706i	$0.547885\pi$
$968$	0	0
$969$	−398.159	−0.0131999
$970$	0	0
$971$	47164.6	1.55879	0.779395	−	0.626533i	$-0.215527\pi$
$971$	47164.6	1.55879	0.779395	+	0.626533i	$0.215527\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4408.74	−0.144813
$976$	0	0
$977$	29115.2	0.953407	0.476703	−	0.879064i	$-0.341832\pi$
$977$	29115.2	0.953407	0.476703	+	0.879064i	$0.341832\pi$
$978$	0	0
$979$	−30141.0	−0.983974
$980$	0	0
$981$	−41957.8	−1.36556
$982$	0	0
$983$	38771.1	1.25799	0.628996	−	0.777409i	$-0.283466\pi$
$983$	38771.1	1.25799	0.628996	+	0.777409i	$0.283466\pi$
$984$	0	0
$985$	−66349.8	−2.14627
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18349.5	−0.589969
$990$	0	0
$991$	18648.2	0.597760	0.298880	−	0.954291i	$-0.403387\pi$
$991$	18648.2	0.597760	0.298880	+	0.954291i	$0.403387\pi$
$992$	0	0
$993$	1832.51	0.0585628
$994$	0	0
$995$	1918.88	0.0611382
$996$	0	0
$997$	16939.6	0.538097	0.269049	−	0.963127i	$-0.413291\pi$
$997$	16939.6	0.538097	0.269049	+	0.963127i	$0.413291\pi$
$998$	0	0
$999$	−194.836	−0.00617050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.be.1.2		3
4.3	odd	2		392.4.a.i.1.2		3
7.2	even	3		112.4.i.e.81.2		6
7.4	even	3		112.4.i.e.65.2		6
7.6	odd	2		784.4.a.bb.1.2		3
28.3	even	6		392.4.i.m.177.2		6
28.11	odd	6		56.4.i.b.9.2	✓	6
28.19	even	6		392.4.i.m.361.2		6
28.23	odd	6		56.4.i.b.25.2	yes	6
28.27	even	2		392.4.a.l.1.2		3
56.11	odd	6		448.4.i.j.65.2		6
56.37	even	6		448.4.i.m.193.2		6
56.51	odd	6		448.4.i.j.193.2		6
56.53	even	6		448.4.i.m.65.2		6
84.11	even	6		504.4.s.h.289.3		6
84.23	even	6		504.4.s.h.361.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.i.b.9.2	✓	6	28.11	odd	6
56.4.i.b.25.2	yes	6	28.23	odd	6
112.4.i.e.65.2		6	7.4	even	3
112.4.i.e.81.2		6	7.2	even	3
392.4.a.i.1.2		3	4.3	odd	2
392.4.a.l.1.2		3	28.27	even	2
392.4.i.m.177.2		6	28.3	even	6
392.4.i.m.361.2		6	28.19	even	6
448.4.i.j.65.2		6	56.11	odd	6
448.4.i.j.193.2		6	56.51	odd	6
448.4.i.m.65.2		6	56.53	even	6
448.4.i.m.193.2		6	56.37	even	6
504.4.s.h.289.3		6	84.11	even	6
504.4.s.h.361.3		6	84.23	even	6
784.4.a.bb.1.2		3	7.6	odd	2
784.4.a.be.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+0.261560 q^{3} +19.5009 q^{5} -26.9316 q^{9} +O(q^{10})\)	\(q+0.261560 q^{3} +19.5009 q^{5} -26.9316 q^{9} +56.2875 q^{11} -66.0260 q^{13} +5.10067 q^{15} -18.8372 q^{17} +80.8106 q^{19} +89.3559 q^{23} +255.287 q^{25} -14.1063 q^{27} +104.118 q^{29} +148.437 q^{31} +14.7226 q^{33} +13.8119 q^{37} -17.2697 q^{39} +174.403 q^{41} -205.353 q^{43} -525.191 q^{45} +116.634 q^{47} -4.92706 q^{51} -316.249 q^{53} +1097.66 q^{55} +21.1368 q^{57} +539.585 q^{59} +145.313 q^{61} -1287.57 q^{65} -476.094 q^{67} +23.3719 q^{69} -131.763 q^{71} +349.608 q^{73} +66.7728 q^{75} +806.906 q^{79} +723.463 q^{81} +233.648 q^{83} -367.344 q^{85} +27.2332 q^{87} -535.483 q^{89} +38.8252 q^{93} +1575.88 q^{95} +80.7221 q^{97} -1515.91 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9}+O(q^{10}) \) 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9	\( 3 q + 7 q^{3} - 3 q^{5} + 18 q^{9} + 3 q^{11} - 26 q^{13} - 127 q^{15} - 31 q^{17} + 89 q^{19} + 201 q^{23} + 300 q^{25} + 469 q^{27} + 190 q^{29} + 339 q^{31} - 105 q^{33} - 535 q^{37} + 134 q^{39} + 58 q^{41} - 268 q^{43} - 1410 q^{45} + 205 q^{47} + 965 q^{51} + 757 q^{53} + 1653 q^{55} + 261 q^{57} + 1799 q^{59} + 625 q^{61} - 1750 q^{65} + 495 q^{67} + 973 q^{69} - 640 q^{71} - 443 q^{73} + 1484 q^{75} - 79 q^{79} + 2523 q^{81} + 2372 q^{83} - 977 q^{85} + 910 q^{87} + 821 q^{89} - 1321 q^{93} + 1327 q^{95} - 342 q^{97} - 2310 q^{99}+O(q^{100}) \) 3 * q + 7 * q^3 - 3 * q^5 + 18 * q^9 + 3 * q^11 - 26 * q^13 - 127 * q^15 - 31 * q^17 + 89 * q^19 + 201 * q^23 + 300 * q^25 + 469 * q^27 + 190 * q^29 + 339 * q^31 - 105 * q^33 - 535 * q^37 + 134 * q^39 + 58 * q^41 - 268 * q^43 - 1410 * q^45 + 205 * q^47 + 965 * q^51 + 757 * q^53 + 1653 * q^55 + 261 * q^57 + 1799 * q^59 + 625 * q^61 - 1750 * q^65 + 495 * q^67 + 973 * q^69 - 640 * q^71 - 443 * q^73 + 1484 * q^75 - 79 * q^79 + 2523 * q^81 + 2372 * q^83 - 977 * q^85 + 910 * q^87 + 821 * q^89 - 1321 * q^93 + 1327 * q^95 - 342 * q^97 - 2310 * q^99

Introduction

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