LMFDB - Embedded newform 784.4.f.g.783.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.6
Root			$0.426664 - 0.739004i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.g.783.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +O(q^{10})$	$q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +4.57968i q^{11} -41.0611i q^{13} +8.64436i q^{15} +119.455i q^{17} +60.1826 q^{19} +8.35305i q^{23} +123.502 q^{25} -29.0976 q^{27} +164.382 q^{29} +287.559 q^{31} +32.3443i q^{33} -148.742 q^{37} -289.998i q^{39} +358.778i q^{41} -360.388i q^{43} +28.0044i q^{45} +225.434 q^{47} +843.660i q^{51} +684.724 q^{53} -5.60537 q^{55} +425.044 q^{57} +85.0522 q^{59} -209.127i q^{61} +50.2574 q^{65} +417.225i q^{67} +58.9941i q^{69} +982.974i q^{71} -341.526i q^{73} +872.242 q^{75} -76.2934i q^{79} -823.265 q^{81} -523.643 q^{83} -146.209 q^{85} +1160.96 q^{87} -972.055i q^{89} +2030.91 q^{93} +73.6614i q^{95} -676.730i q^{97} +104.783i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{3} + 156 q^{9}+O(q^{10}) $ 6 * q - 14 * q^3 + 156 * q^9	$ 6 q - 14 q^{3} + 156 q^{9} + 286 q^{19} - 612 q^{25} - 362 q^{27} - 348 q^{29} + 410 q^{31} + 498 q^{37} + 150 q^{47} + 1290 q^{53} - 918 q^{55} - 6 q^{57} + 642 q^{59} + 1224 q^{65} + 8276 q^{75} - 450 q^{81} - 24 q^{83} + 3786 q^{85} + 7284 q^{87} + 5982 q^{93}+O(q^{100}) $ 6 * q - 14 * q^3 + 156 * q^9 + 286 * q^19 - 612 * q^25 - 362 * q^27 - 348 * q^29 + 410 * q^31 + 498 * q^37 + 150 * q^47 + 1290 * q^53 - 918 * q^55 - 6 * q^57 + 642 * q^59 + 1224 * q^65 + 8276 * q^75 - 450 * q^81 - 24 * q^83 + 3786 * q^85 + 7284 * q^87 + 5982 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	7.06258	1.35919	0.679597	−	0.733586i	$-0.262155\pi$
$3$	7.06258	1.35919	0.679597	+	0.733586i	$0.262155\pi$
$4$	0	0
$5$	1.22397i	0.109475i	0.998501	+	0.0547374i	$0.0174322\pi$
$5$	1.22397i	0.109475i	−0.998501	+	0.0547374i	$0.982568\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	22.8800	0.847408
$10$	0	0
$11$	4.57968i	0.125529i	0.998028	+	0.0627647i	$0.0199918\pi$
$11$	4.57968i	0.125529i	−0.998028	+	0.0627647i	$0.980008\pi$
$12$	0	0
$13$	− 41.0611i	− 0.876024i	−0.898969	−	0.438012i	$-0.855683\pi$
$13$	− 41.0611i	− 0.876024i	0.898969	−	0.438012i	$-0.144317\pi$
$14$	0	0
$15$	8.64436i	0.148798i
$16$	0	0
$17$	119.455i	1.70424i	0.523346	+	0.852120i	$0.324683\pi$
$17$	119.455i	1.70424i	−0.523346	+	0.852120i	$0.675317\pi$
$18$	0	0
$19$	60.1826	0.726675	0.363337	−	0.931658i	$-0.381637\pi$
$19$	60.1826	0.726675	0.363337	+	0.931658i	$0.381637\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.35305i	0.0757275i	0.999283	+	0.0378637i	$0.0120553\pi$
$23$	8.35305i	0.0757275i	−0.999283	+	0.0378637i	$0.987945\pi$
$24$	0	0
$25$	123.502	0.988015
$26$	0	0
$27$	−29.0976	−0.207401
$28$	0	0
$29$	164.382	1.05258	0.526292	−	0.850304i	$-0.323582\pi$
$29$	164.382	1.05258	0.526292	+	0.850304i	$0.323582\pi$
$30$	0	0
$31$	287.559	1.66604	0.833019	−	0.553244i	$-0.186610\pi$
$31$	287.559	1.66604	0.833019	+	0.553244i	$0.186610\pi$
$32$	0	0
$33$	32.3443i	0.170619i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−148.742	−0.660892	−0.330446	−	0.943825i	$-0.607199\pi$
$37$	−148.742	−0.660892	−0.330446	+	0.943825i	$0.607199\pi$
$38$	0	0
$39$	− 289.998i	− 1.19069i
$40$	0	0
$41$	358.778i	1.36663i	0.730125	+	0.683314i	$0.239462\pi$
$41$	358.778i	1.36663i	−0.730125	+	0.683314i	$0.760538\pi$
$42$	0	0
$43$	− 360.388i	− 1.27811i	−0.769161	−	0.639055i	$-0.779326\pi$
$43$	− 360.388i	− 1.27811i	0.769161	−	0.639055i	$-0.220674\pi$
$44$	0	0
$45$	28.0044i	0.0927699i
$46$	0	0
$47$	225.434	0.699637	0.349819	−	0.936817i	$-0.386243\pi$
$47$	225.434	0.699637	0.349819	+	0.936817i	$0.386243\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	843.660i	2.31639i
$52$	0	0
$53$	684.724	1.77460	0.887302	−	0.461189i	$-0.152577\pi$
$53$	684.724	1.77460	0.887302	+	0.461189i	$0.152577\pi$
$54$	0	0
$55$	−5.60537	−0.0137423
$56$	0	0
$57$	425.044	0.987692
$58$	0	0
$59$	85.0522	0.187675	0.0938377	−	0.995588i	$-0.470087\pi$
$59$	85.0522	0.187675	0.0938377	+	0.995588i	$0.470087\pi$
$60$	0	0
$61$	− 209.127i	− 0.438949i	−0.975618	−	0.219475i	$-0.929566\pi$
$61$	− 209.127i	− 0.438949i	0.975618	−	0.219475i	$-0.0704344\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	50.2574	0.0959026
$66$	0	0
$67$	417.225i	0.760778i	0.924827	+	0.380389i	$0.124210\pi$
$67$	417.225i	0.760778i	−0.924827	+	0.380389i	$0.875790\pi$
$68$	0	0
$69$	58.9941i	0.102928i
$70$	0	0
$71$	982.974i	1.64306i	0.570162	+	0.821532i	$0.306880\pi$
$71$	982.974i	1.64306i	−0.570162	+	0.821532i	$0.693120\pi$
$72$	0	0
$73$	− 341.526i	− 0.547570i	−0.961791	−	0.273785i	$-0.911724\pi$
$73$	− 341.526i	− 0.547570i	0.961791	−	0.273785i	$-0.0882757\pi$
$74$	0	0
$75$	872.242	1.34290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 76.2934i	− 0.108654i	−0.998523	−	0.0543271i	$-0.982699\pi$
$79$	− 76.2934i	− 0.108654i	0.998523	−	0.0543271i	$-0.0173014\pi$
$80$	0	0
$81$	−823.265	−1.12931
$82$	0	0
$83$	−523.643	−0.692498	−0.346249	−	0.938143i	$-0.612545\pi$
$83$	−523.643	−0.692498	−0.346249	+	0.938143i	$0.612545\pi$
$84$	0	0
$85$	−146.209	−0.186571
$86$	0	0
$87$	1160.96	1.43067
$88$	0	0
$89$	− 972.055i	− 1.15773i	−0.815425	−	0.578863i	$-0.803497\pi$
$89$	− 972.055i	− 1.15773i	0.815425	−	0.578863i	$-0.196503\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2030.91	2.26447
$94$	0	0
$95$	73.6614i	0.0795526i
$96$	0	0
$97$	− 676.730i	− 0.708367i	−0.935176	−	0.354183i	$-0.884759\pi$
$97$	− 676.730i	− 0.708367i	0.935176	−	0.354183i	$-0.115241\pi$
$98$	0	0
$99$	104.783i	0.106375i
$100$	0	0
$101$	− 838.786i	− 0.826360i	−0.910649	−	0.413180i	$-0.864418\pi$
$101$	− 838.786i	− 0.826360i	0.910649	−	0.413180i	$-0.135582\pi$
$102$	0	0
$103$	1515.20	1.44949	0.724744	−	0.689018i	$-0.241958\pi$
$103$	1515.20	1.44949	0.724744	+	0.689018i	$0.241958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1687.67i	1.52480i	0.647107	+	0.762399i	$0.275979\pi$
$107$	1687.67i	1.52480i	−0.647107	+	0.762399i	$0.724021\pi$
$108$	0	0
$109$	−1576.23	−1.38510	−0.692550	−	0.721370i	$-0.743513\pi$
$109$	−1576.23	−1.38510	−0.692550	+	0.721370i	$0.743513\pi$
$110$	0	0
$111$	−1050.50	−0.898281
$112$	0	0
$113$	−2192.01	−1.82484	−0.912422	−	0.409251i	$-0.865790\pi$
$113$	−2192.01	−1.82484	−0.912422	+	0.409251i	$0.865790\pi$
$114$	0	0
$115$	−10.2239	−0.00829025
$116$	0	0
$117$	− 939.480i	− 0.742350i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1310.03	0.984242
$122$	0	0
$123$	2533.90i	1.85751i
$124$	0	0
$125$	304.158i	0.217638i
$126$	0	0
$127$	1085.60i	0.758515i	0.925291	+	0.379258i	$0.123821\pi$
$127$	1085.60i	0.758515i	−0.925291	+	0.379258i	$0.876179\pi$
$128$	0	0
$129$	− 2545.27i	− 1.73720i
$130$	0	0
$131$	−1533.35	−1.02267	−0.511334	−	0.859382i	$-0.670849\pi$
$131$	−1533.35	−1.02267	−0.511334	+	0.859382i	$0.670849\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 35.6145i	− 0.0227052i
$136$	0	0
$137$	920.175	0.573838	0.286919	−	0.957955i	$-0.407369\pi$
$137$	920.175	0.573838	0.286919	+	0.957955i	$0.407369\pi$
$138$	0	0
$139$	−1424.66	−0.869339	−0.434669	−	0.900590i	$-0.643135\pi$
$139$	−1424.66	−0.869339	−0.434669	+	0.900590i	$0.643135\pi$
$140$	0	0
$141$	1592.15	0.950943
$142$	0	0
$143$	188.047	0.109967
$144$	0	0
$145$	201.198i	0.115232i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1378.25	−0.757791	−0.378896	−	0.925439i	$-0.623696\pi$
$149$	−1378.25	−0.757791	−0.378896	+	0.925439i	$0.623696\pi$
$150$	0	0
$151$	− 1333.78i	− 0.718817i	−0.933180	−	0.359409i	$-0.882978\pi$
$151$	− 1333.78i	− 0.718817i	0.933180	−	0.359409i	$-0.117022\pi$
$152$	0	0
$153$	2733.13i	1.44419i
$154$	0	0
$155$	351.963i	0.182389i
$156$	0	0
$157$	− 1255.89i	− 0.638416i	−0.947685	−	0.319208i	$-0.896583\pi$
$157$	− 1255.89i	− 0.638416i	0.947685	−	0.319208i	$-0.103417\pi$
$158$	0	0
$159$	4835.92	2.41203
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3254.58i	1.56392i	0.623330	+	0.781959i	$0.285779\pi$
$163$	3254.58i	1.56392i	−0.623330	+	0.781959i	$0.714221\pi$
$164$	0	0
$165$	−39.5884	−0.0186785
$166$	0	0
$167$	2736.24	1.26788	0.633941	−	0.773381i	$-0.281436\pi$
$167$	2736.24	1.26788	0.633941	+	0.773381i	$0.281436\pi$
$168$	0	0
$169$	510.983	0.232582
$170$	0	0
$171$	1376.98	0.615791
$172$	0	0
$173$	− 3380.13i	− 1.48547i	−0.669584	−	0.742737i	$-0.733527\pi$
$173$	− 3380.13i	− 1.48547i	0.669584	−	0.742737i	$-0.266473\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	600.688	0.255087
$178$	0	0
$179$	− 1217.98i	− 0.508584i	−0.967128	−	0.254292i	$-0.918158\pi$
$179$	− 1217.98i	− 0.508584i	0.967128	−	0.254292i	$-0.0818423\pi$
$180$	0	0
$181$	− 1757.61i	− 0.721781i	−0.932608	−	0.360890i	$-0.882473\pi$
$181$	− 1757.61i	− 0.721781i	0.932608	−	0.360890i	$-0.117527\pi$
$182$	0	0
$183$	− 1476.97i	− 0.596618i
$184$	0	0
$185$	− 182.055i	− 0.0723511i
$186$	0	0
$187$	−547.065	−0.213932
$188$	0	0
$189$	0	0
$190$	0	0
$191$	956.433i	0.362330i	0.983453	+	0.181165i	$0.0579868\pi$
$191$	956.433i	0.362330i	−0.983453	+	0.181165i	$0.942013\pi$
$192$	0	0
$193$	1333.55	0.497362	0.248681	−	0.968585i	$-0.420003\pi$
$193$	1333.55	0.497362	0.248681	+	0.968585i	$0.420003\pi$
$194$	0	0
$195$	354.947	0.130350
$196$	0	0
$197$	1098.75	0.397373	0.198686	−	0.980063i	$-0.436332\pi$
$197$	1098.75	0.397373	0.198686	+	0.980063i	$0.436332\pi$
$198$	0	0
$199$	−248.579	−0.0885491	−0.0442745	−	0.999019i	$-0.514098\pi$
$199$	−248.579	−0.0885491	−0.0442745	+	0.999019i	$0.514098\pi$
$200$	0	0
$201$	2946.68i	1.03405i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−439.132	−0.149611
$206$	0	0
$207$	191.118i	0.0641721i
$208$	0	0
$209$	275.617i	0.0912191i
$210$	0	0
$211$	2029.24i	0.662078i	0.943617	+	0.331039i	$0.107399\pi$
$211$	2029.24i	0.662078i	−0.943617	+	0.331039i	$0.892601\pi$
$212$	0	0
$213$	6942.33i	2.23324i
$214$	0	0
$215$	441.103	0.139921
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 2412.06i	− 0.744254i
$220$	0	0
$221$	4904.96	1.49295
$222$	0	0
$223$	−3904.24	−1.17241	−0.586204	−	0.810164i	$-0.699378\pi$
$223$	−3904.24	−1.17241	−0.586204	+	0.810164i	$0.699378\pi$
$224$	0	0
$225$	2825.73	0.837253
$226$	0	0
$227$	4255.75	1.24434	0.622168	−	0.782884i	$-0.286252\pi$
$227$	4255.75	1.24434	0.622168	+	0.782884i	$0.286252\pi$
$228$	0	0
$229$	1732.25i	0.499871i	0.968262	+	0.249936i	$0.0804095\pi$
$229$	1732.25i	0.499871i	−0.968262	+	0.249936i	$0.919591\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−874.910	−0.245997	−0.122998	−	0.992407i	$-0.539251\pi$
$233$	−874.910	−0.245997	−0.122998	+	0.992407i	$0.539251\pi$
$234$	0	0
$235$	275.924i	0.0765927i
$236$	0	0
$237$	− 538.828i	− 0.147682i
$238$	0	0
$239$	− 1842.84i	− 0.498760i	−0.968406	−	0.249380i	$-0.919773\pi$
$239$	− 1842.84i	− 0.498760i	0.968406	−	0.249380i	$-0.0802269\pi$
$240$	0	0
$241$	4683.60i	1.25185i	0.779881	+	0.625927i	$0.215279\pi$
$241$	4683.60i	1.25185i	−0.779881	+	0.625927i	$0.784721\pi$
$242$	0	0
$243$	−5028.74	−1.32755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2471.16i	− 0.636585i
$248$	0	0
$249$	−3698.27	−0.941239
$250$	0	0
$251$	−1015.59	−0.255393	−0.127696	−	0.991813i	$-0.540758\pi$
$251$	−1015.59	−0.255393	−0.127696	+	0.991813i	$0.540758\pi$
$252$	0	0
$253$	−38.2543	−0.00950603
$254$	0	0
$255$	−1032.61	−0.253587
$256$	0	0
$257$	3016.56i	0.732171i	0.930581	+	0.366085i	$0.119302\pi$
$257$	3016.56i	0.732171i	−0.930581	+	0.366085i	$0.880698\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3761.06	0.891969
$262$	0	0
$263$	− 2651.05i	− 0.621561i	−0.950482	−	0.310780i	$-0.899410\pi$
$263$	− 2651.05i	− 0.621561i	0.950482	−	0.310780i	$-0.100590\pi$
$264$	0	0
$265$	838.078i	0.194275i
$266$	0	0
$267$	− 6865.21i	− 1.57357i
$268$	0	0
$269$	− 452.582i	− 0.102581i	−0.998684	−	0.0512907i	$-0.983666\pi$
$269$	− 452.582i	− 0.102581i	0.998684	−	0.0512907i	$-0.0163335\pi$
$270$	0	0
$271$	−164.726	−0.0369239	−0.0184620	−	0.999830i	$-0.505877\pi$
$271$	−164.726	−0.0369239	−0.0184620	+	0.999830i	$0.505877\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	565.599i	0.124025i
$276$	0	0
$277$	1939.61	0.420721	0.210360	−	0.977624i	$-0.432536\pi$
$277$	1939.61	0.420721	0.210360	+	0.977624i	$0.432536\pi$
$278$	0	0
$279$	6579.37	1.41181
$280$	0	0
$281$	−7152.79	−1.51850	−0.759252	−	0.650796i	$-0.774435\pi$
$281$	−7152.79	−1.51850	−0.759252	+	0.650796i	$0.774435\pi$
$282$	0	0
$283$	−3295.55	−0.692226	−0.346113	−	0.938193i	$-0.612499\pi$
$283$	−3295.55	−0.692226	−0.346113	+	0.938193i	$0.612499\pi$
$284$	0	0
$285$	520.239i	0.108127i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9356.49	−1.90443
$290$	0	0
$291$	− 4779.46i	− 0.962808i
$292$	0	0
$293$	− 7368.65i	− 1.46922i	−0.678491	−	0.734609i	$-0.737366\pi$
$293$	− 7368.65i	− 1.46922i	0.678491	−	0.734609i	$-0.262634\pi$
$294$	0	0
$295$	104.101i	0.0205457i
$296$	0	0
$297$	− 133.258i	− 0.0260350i
$298$	0	0
$299$	342.986	0.0663391
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 5923.99i	− 1.12318i
$304$	0	0
$305$	255.964	0.0480539
$306$	0	0
$307$	2692.96	0.500636	0.250318	−	0.968164i	$-0.419465\pi$
$307$	2692.96	0.500636	0.250318	+	0.968164i	$0.419465\pi$
$308$	0	0
$309$	10701.2	1.97014
$310$	0	0
$311$	−2971.99	−0.541885	−0.270943	−	0.962596i	$-0.587335\pi$
$311$	−2971.99	−0.541885	−0.270943	+	0.962596i	$0.587335\pi$
$312$	0	0
$313$	− 1140.00i	− 0.205868i	−0.994688	−	0.102934i	$-0.967177\pi$
$313$	− 1140.00i	− 0.205868i	0.994688	−	0.102934i	$-0.0328230\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5645.11	−1.00019	−0.500097	−	0.865970i	$-0.666702\pi$
$317$	−5645.11	−1.00019	−0.500097	+	0.865970i	$0.666702\pi$
$318$	0	0
$319$	752.816i	0.132130i
$320$	0	0
$321$	11919.3i	2.07250i
$322$	0	0
$323$	7189.10i	1.23843i
$324$	0	0
$325$	− 5071.13i	− 0.865525i
$326$	0	0
$327$	−11132.3	−1.88262
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 7359.98i	− 1.22218i	−0.791562	−	0.611089i	$-0.790732\pi$
$331$	− 7359.98i	− 1.22218i	0.791562	−	0.611089i	$-0.209268\pi$
$332$	0	0
$333$	−3403.22	−0.560046
$334$	0	0
$335$	−510.669	−0.0832861
$336$	0	0
$337$	−10163.9	−1.64292	−0.821459	−	0.570268i	$-0.806839\pi$
$337$	−10163.9	−1.64292	−0.821459	+	0.570268i	$0.806839\pi$
$338$	0	0
$339$	−15481.3	−2.48032
$340$	0	0
$341$	1316.93i	0.209137i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−72.2068	−0.0112681
$346$	0	0
$347$	10544.8i	1.63133i	0.578522	+	0.815666i	$0.303630\pi$
$347$	10544.8i	1.63133i	−0.578522	+	0.815666i	$0.696370\pi$
$348$	0	0
$349$	− 6048.02i	− 0.927631i	−0.885932	−	0.463816i	$-0.846480\pi$
$349$	− 6048.02i	− 0.927631i	0.885932	−	0.463816i	$-0.153520\pi$
$350$	0	0
$351$	1194.78i	0.181689i
$352$	0	0
$353$	− 6289.12i	− 0.948261i	−0.880454	−	0.474131i	$-0.842762\pi$
$353$	− 6289.12i	− 0.948261i	0.880454	−	0.474131i	$-0.157238\pi$
$354$	0	0
$355$	−1203.13	−0.179874
$356$	0	0
$357$	0	0
$358$	0	0
$359$	154.662i	0.0227375i	0.999935	+	0.0113687i	$0.00361886\pi$
$359$	154.662i	0.0227375i	−0.999935	+	0.0113687i	$0.996381\pi$
$360$	0	0
$361$	−3237.06	−0.471944
$362$	0	0
$363$	9252.17	1.33778
$364$	0	0
$365$	418.016	0.0599451
$366$	0	0
$367$	−5890.42	−0.837813	−0.418907	−	0.908029i	$-0.637587\pi$
$367$	−5890.42	−0.837813	−0.418907	+	0.908029i	$0.637587\pi$
$368$	0	0
$369$	8208.86i	1.15809i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	350.300	0.0486270	0.0243135	−	0.999704i	$-0.492260\pi$
$373$	350.300	0.0486270	0.0243135	+	0.999704i	$0.492260\pi$
$374$	0	0
$375$	2148.14i	0.295812i
$376$	0	0
$377$	− 6749.71i	− 0.922089i
$378$	0	0
$379$	− 8247.17i	− 1.11775i	−0.829251	−	0.558877i	$-0.811233\pi$
$379$	− 8247.17i	− 1.11775i	0.829251	−	0.558877i	$-0.188767\pi$
$380$	0	0
$381$	7667.14i	1.03097i
$382$	0	0
$383$	−12680.2	−1.69171	−0.845856	−	0.533411i	$-0.820910\pi$
$383$	−12680.2	−1.69171	−0.845856	+	0.533411i	$0.820910\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8245.69i	− 1.08308i
$388$	0	0
$389$	−7727.53	−1.00720	−0.503601	−	0.863937i	$-0.667992\pi$
$389$	−7727.53	−1.00720	−0.503601	+	0.863937i	$0.667992\pi$
$390$	0	0
$391$	−997.814	−0.129058
$392$	0	0
$393$	−10829.4	−1.39000
$394$	0	0
$395$	93.3806	0.0118949
$396$	0	0
$397$	− 10096.2i	− 1.27636i	−0.769889	−	0.638178i	$-0.779688\pi$
$397$	− 10096.2i	− 1.27636i	0.769889	−	0.638178i	$-0.220312\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7678.50	0.956224	0.478112	−	0.878299i	$-0.341321\pi$
$401$	7678.50	0.956224	0.478112	+	0.878299i	$0.341321\pi$
$402$	0	0
$403$	− 11807.5i	− 1.45949i
$404$	0	0
$405$	− 1007.65i	− 0.123631i
$406$	0	0
$407$	− 681.190i	− 0.0829615i
$408$	0	0
$409$	− 3711.55i	− 0.448715i	−0.974507	−	0.224358i	$-0.927972\pi$
$409$	− 3711.55i	− 0.448715i	0.974507	−	0.224358i	$-0.0720284\pi$
$410$	0	0
$411$	6498.81	0.779957
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 640.921i	− 0.0758111i
$416$	0	0
$417$	−10061.8	−1.18160
$418$	0	0
$419$	−8868.09	−1.03397	−0.516986	−	0.855994i	$-0.672946\pi$
$419$	−8868.09	−1.03397	−0.516986	+	0.855994i	$0.672946\pi$
$420$	0	0
$421$	−9571.99	−1.10810	−0.554050	−	0.832483i	$-0.686919\pi$
$421$	−9571.99	−1.10810	−0.554050	+	0.832483i	$0.686919\pi$
$422$	0	0
$423$	5157.94	0.592879
$424$	0	0
$425$	14752.9i	1.68382i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1328.09	0.149466
$430$	0	0
$431$	7712.13i	0.861903i	0.902375	+	0.430952i	$0.141822\pi$
$431$	7712.13i	0.861903i	−0.902375	+	0.430952i	$0.858178\pi$
$432$	0	0
$433$	− 2222.71i	− 0.246690i	−0.992364	−	0.123345i	$-0.960638\pi$
$433$	− 2222.71i	− 0.246690i	0.992364	−	0.123345i	$-0.0393622\pi$
$434$	0	0
$435$	1420.98i	0.156622i
$436$	0	0
$437$	502.708i	0.0550293i
$438$	0	0
$439$	−10612.6	−1.15378	−0.576890	−	0.816822i	$-0.695734\pi$
$439$	−10612.6	−1.15378	−0.576890	+	0.816822i	$0.695734\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 15591.1i	− 1.67213i	−0.548630	−	0.836065i	$-0.684851\pi$
$443$	− 15591.1i	− 1.67213i	0.548630	−	0.836065i	$-0.315149\pi$
$444$	0	0
$445$	1189.76	0.126742
$446$	0	0
$447$	−9734.03	−1.02999
$448$	0	0
$449$	2759.74	0.290067	0.145033	−	0.989427i	$-0.453671\pi$
$449$	2759.74	0.290067	0.145033	+	0.989427i	$0.453671\pi$
$450$	0	0
$451$	−1643.09	−0.171552
$452$	0	0
$453$	− 9419.92i	− 0.977012i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7672.29	−0.785327	−0.392664	−	0.919682i	$-0.628446\pi$
$457$	−7672.29	−0.785327	−0.392664	+	0.919682i	$0.628446\pi$
$458$	0	0
$459$	− 3475.86i	− 0.353462i
$460$	0	0
$461$	− 13864.8i	− 1.40076i	−0.713771	−	0.700379i	$-0.753014\pi$
$461$	− 13864.8i	− 1.40076i	0.713771	−	0.700379i	$-0.246986\pi$
$462$	0	0
$463$	11518.3i	1.15616i	0.815981	+	0.578079i	$0.196197\pi$
$463$	11518.3i	1.15616i	−0.815981	+	0.578079i	$0.803803\pi$
$464$	0	0
$465$	2485.77i	0.247902i
$466$	0	0
$467$	2815.92	0.279027	0.139513	−	0.990220i	$-0.455446\pi$
$467$	2815.92	0.279027	0.139513	+	0.990220i	$0.455446\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 8869.86i	− 0.867731i
$472$	0	0
$473$	1650.46	0.160440
$474$	0	0
$475$	7432.66	0.717966
$476$	0	0
$477$	15666.5	1.50381
$478$	0	0
$479$	−3120.15	−0.297627	−0.148813	−	0.988865i	$-0.547545\pi$
$479$	−3120.15	−0.297627	−0.148813	+	0.988865i	$0.547545\pi$
$480$	0	0
$481$	6107.51i	0.578957i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	828.295	0.0775483
$486$	0	0
$487$	1988.38i	0.185015i	0.995712	+	0.0925074i	$0.0294882\pi$
$487$	1988.38i	0.185015i	−0.995712	+	0.0925074i	$0.970512\pi$
$488$	0	0
$489$	22985.8i	2.12567i
$490$	0	0
$491$	− 3035.51i	− 0.279003i	−0.990222	−	0.139502i	$-0.955450\pi$
$491$	− 3035.51i	− 0.279003i	0.990222	−	0.139502i	$-0.0445501\pi$
$492$	0	0
$493$	19636.2i	1.79386i
$494$	0	0
$495$	−128.251	−0.0116454
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16519.7i	1.48201i	0.671498	+	0.741007i	$0.265651\pi$
$499$	16519.7i	1.48201i	−0.671498	+	0.741007i	$0.734349\pi$
$500$	0	0
$501$	19324.9	1.72330
$502$	0	0
$503$	16446.0	1.45783	0.728917	−	0.684602i	$-0.240024\pi$
$503$	16446.0	1.45783	0.728917	+	0.684602i	$0.240024\pi$
$504$	0	0
$505$	1026.65	0.0904656
$506$	0	0
$507$	3608.86	0.316124
$508$	0	0
$509$	4685.12i	0.407985i	0.978972	+	0.203992i	$0.0653917\pi$
$509$	4685.12i	0.407985i	−0.978972	+	0.203992i	$0.934608\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1751.17	−0.150713
$514$	0	0
$515$	1854.56i	0.158682i
$516$	0	0
$517$	1032.42i	0.0878251i
$518$	0	0
$519$	− 23872.5i	− 2.01905i
$520$	0	0
$521$	− 2199.68i	− 0.184971i	−0.995714	−	0.0924853i	$-0.970519\pi$
$521$	− 2199.68i	− 0.184971i	0.995714	−	0.0924853i	$-0.0294811\pi$
$522$	0	0
$523$	10465.4	0.874988	0.437494	−	0.899221i	$-0.355866\pi$
$523$	10465.4	0.874988	0.437494	+	0.899221i	$0.355866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34350.4i	2.83933i
$528$	0	0
$529$	12097.2	0.994265
$530$	0	0
$531$	1946.00	0.159038
$532$	0	0
$533$	14731.8	1.19720
$534$	0	0
$535$	−2065.65	−0.166927
$536$	0	0
$537$	− 8602.11i	− 0.691264i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2934.94	−0.233240	−0.116620	−	0.993177i	$-0.537206\pi$
$541$	−2934.94	−0.233240	−0.116620	+	0.993177i	$0.537206\pi$
$542$	0	0
$543$	− 12413.3i	− 0.981040i
$544$	0	0
$545$	− 1929.26i	− 0.151634i
$546$	0	0
$547$	9168.13i	0.716638i	0.933599	+	0.358319i	$0.116650\pi$
$547$	9168.13i	0.716638i	−0.933599	+	0.358319i	$0.883350\pi$
$548$	0	0
$549$	− 4784.82i	− 0.371970i
$550$	0	0
$551$	9892.92	0.764887
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 1285.78i	− 0.0983391i
$556$	0	0
$557$	4618.14	0.351305	0.175652	−	0.984452i	$-0.443797\pi$
$557$	4618.14	0.351305	0.175652	+	0.984452i	$0.443797\pi$
$558$	0	0
$559$	−14797.9	−1.11965
$560$	0	0
$561$	−3863.69	−0.290776
$562$	0	0
$563$	23789.6	1.78084	0.890418	−	0.455144i	$-0.150412\pi$
$563$	23789.6	1.78084	0.890418	+	0.455144i	$0.150412\pi$
$564$	0	0
$565$	− 2682.95i	− 0.199775i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14772.6	−1.08840	−0.544200	−	0.838955i	$-0.683167\pi$
$569$	−14772.6	−1.08840	−0.544200	+	0.838955i	$0.683167\pi$
$570$	0	0
$571$	− 24295.7i	− 1.78064i	−0.455339	−	0.890318i	$-0.650482\pi$
$571$	− 24295.7i	− 1.78064i	0.455339	−	0.890318i	$-0.349518\pi$
$572$	0	0
$573$	6754.88i	0.492477i
$574$	0	0
$575$	1031.62i	0.0748199i
$576$	0	0
$577$	− 10024.6i	− 0.723275i	−0.932319	−	0.361638i	$-0.882218\pi$
$577$	− 10024.6i	− 0.723275i	0.932319	−	0.361638i	$-0.117782\pi$
$578$	0	0
$579$	9418.30	0.676012
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3135.81i	0.222765i
$584$	0	0
$585$	1149.89	0.0812686
$586$	0	0
$587$	10134.7	0.712613	0.356306	−	0.934369i	$-0.384036\pi$
$587$	10134.7	0.712613	0.356306	+	0.934369i	$0.384036\pi$
$588$	0	0
$589$	17306.1	1.21067
$590$	0	0
$591$	7759.98	0.540107
$592$	0	0
$593$	− 2197.51i	− 0.152177i	−0.997101	−	0.0760885i	$-0.975757\pi$
$593$	− 2197.51i	− 0.152177i	0.997101	−	0.0760885i	$-0.0242431\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1755.61	−0.120355
$598$	0	0
$599$	24663.6i	1.68235i	0.540765	+	0.841174i	$0.318135\pi$
$599$	24663.6i	1.68235i	−0.540765	+	0.841174i	$0.681865\pi$
$600$	0	0
$601$	21532.0i	1.46142i	0.682691	+	0.730708i	$0.260810\pi$
$601$	21532.0i	1.46142i	−0.682691	+	0.730708i	$0.739190\pi$
$602$	0	0
$603$	9546.12i	0.644690i
$604$	0	0
$605$	1603.43i	0.107750i
$606$	0	0
$607$	987.430	0.0660273	0.0330136	−	0.999455i	$-0.489490\pi$
$607$	987.430	0.0660273	0.0330136	+	0.999455i	$0.489490\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9256.58i	− 0.612899i
$612$	0	0
$613$	7278.51	0.479570	0.239785	−	0.970826i	$-0.422923\pi$
$613$	7278.51	0.479570	0.239785	+	0.970826i	$0.422923\pi$
$614$	0	0
$615$	−3101.41	−0.203351
$616$	0	0
$617$	−2543.97	−0.165991	−0.0829953	−	0.996550i	$-0.526449\pi$
$617$	−2543.97	−0.165991	−0.0829953	+	0.996550i	$0.526449\pi$
$618$	0	0
$619$	7853.15	0.509927	0.254963	−	0.966951i	$-0.417937\pi$
$619$	7853.15	0.509927	0.254963	+	0.966951i	$0.417937\pi$
$620$	0	0
$621$	− 243.054i	− 0.0157060i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15065.5	0.964189
$626$	0	0
$627$	1946.56i	0.123985i
$628$	0	0
$629$	− 17768.0i	− 1.12632i
$630$	0	0
$631$	− 16369.0i	− 1.03271i	−0.856375	−	0.516355i	$-0.827289\pi$
$631$	− 16369.0i	− 1.03271i	0.856375	−	0.516355i	$-0.172711\pi$
$632$	0	0
$633$	14331.7i	0.899893i
$634$	0	0
$635$	−1328.74	−0.0830383
$636$	0	0
$637$	0	0
$638$	0	0
$639$	22490.5i	1.39235i
$640$	0	0
$641$	14497.2	0.893302	0.446651	−	0.894708i	$-0.352617\pi$
$641$	14497.2	0.893302	0.446651	+	0.894708i	$0.352617\pi$
$642$	0	0
$643$	−29771.6	−1.82594	−0.912969	−	0.408029i	$-0.866216\pi$
$643$	−29771.6	−1.82594	−0.912969	+	0.408029i	$0.866216\pi$
$644$	0	0
$645$	3115.32	0.190179
$646$	0	0
$647$	11215.5	0.681496	0.340748	−	0.940155i	$-0.389320\pi$
$647$	11215.5	0.681496	0.340748	+	0.940155i	$0.389320\pi$
$648$	0	0
$649$	389.512i	0.0235588i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20401.0	1.22259	0.611295	−	0.791403i	$-0.290649\pi$
$653$	20401.0	1.22259	0.611295	+	0.791403i	$0.290649\pi$
$654$	0	0
$655$	− 1876.77i	− 0.111956i
$656$	0	0
$657$	− 7814.13i	− 0.464016i
$658$	0	0
$659$	13427.1i	0.793693i	0.917885	+	0.396846i	$0.129895\pi$
$659$	13427.1i	0.793693i	−0.917885	+	0.396846i	$0.870105\pi$
$660$	0	0
$661$	2313.84i	0.136154i	0.997680	+	0.0680771i	$0.0216864\pi$
$661$	2313.84i	0.136154i	−0.997680	+	0.0680771i	$0.978314\pi$
$662$	0	0
$663$	34641.6	2.02922
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1373.09i	0.0797096i
$668$	0	0
$669$	−27574.0	−1.59353
$670$	0	0
$671$	957.732	0.0551011
$672$	0	0
$673$	−16395.5	−0.939081	−0.469540	−	0.882911i	$-0.655580\pi$
$673$	−16395.5	−0.939081	−0.469540	+	0.882911i	$0.655580\pi$
$674$	0	0
$675$	−3593.61	−0.204916
$676$	0	0
$677$	− 24962.8i	− 1.41713i	−0.705645	−	0.708566i	$-0.749342\pi$
$677$	− 24962.8i	− 1.41713i	0.705645	−	0.708566i	$-0.250658\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30056.6	1.69129
$682$	0	0
$683$	20374.5i	1.14145i	0.821142	+	0.570724i	$0.193337\pi$
$683$	20374.5i	1.14145i	−0.821142	+	0.570724i	$0.806663\pi$
$684$	0	0
$685$	1126.26i	0.0628208i
$686$	0	0
$687$	12234.2i	0.679422i
$688$	0	0
$689$	− 28115.5i	− 1.55460i
$690$	0	0
$691$	6393.32	0.351973	0.175986	−	0.984393i	$-0.443689\pi$
$691$	6393.32	0.351973	0.175986	+	0.984393i	$0.443689\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1743.74i	− 0.0951707i
$696$	0	0
$697$	−42857.8	−2.32906
$698$	0	0
$699$	−6179.12	−0.334357
$700$	0	0
$701$	−19401.6	−1.04535	−0.522674	−	0.852533i	$-0.675066\pi$
$701$	−19401.6	−1.04535	−0.522674	+	0.852533i	$0.675066\pi$
$702$	0	0
$703$	−8951.66	−0.480254
$704$	0	0
$705$	1948.73i	0.104104i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14777.7	0.782776	0.391388	−	0.920226i	$-0.371995\pi$
$709$	14777.7	0.782776	0.391388	+	0.920226i	$0.371995\pi$
$710$	0	0
$711$	− 1745.60i	− 0.0920745i
$712$	0	0
$713$	2402.00i	0.126165i
$714$	0	0
$715$	230.163i	0.0120386i
$716$	0	0
$717$	− 13015.2i	− 0.677912i
$718$	0	0
$719$	−6437.04	−0.333882	−0.166941	−	0.985967i	$-0.553389\pi$
$719$	−6437.04	−0.333882	−0.166941	+	0.985967i	$0.553389\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	33078.3i	1.70151i
$724$	0	0
$725$	20301.5	1.03997
$726$	0	0
$727$	−13490.8	−0.688235	−0.344118	−	0.938927i	$-0.611822\pi$
$727$	−13490.8	−0.688235	−0.344118	+	0.938927i	$0.611822\pi$
$728$	0	0
$729$	−13287.7	−0.675086
$730$	0	0
$731$	43050.2	2.17820
$732$	0	0
$733$	− 20806.6i	− 1.04844i	−0.851582	−	0.524221i	$-0.824356\pi$
$733$	− 20806.6i	− 1.04844i	0.851582	−	0.524221i	$-0.175644\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1910.75	−0.0955001
$738$	0	0
$739$	17283.7i	0.860342i	0.902747	+	0.430171i	$0.141547\pi$
$739$	17283.7i	0.860342i	−0.902747	+	0.430171i	$0.858453\pi$
$740$	0	0
$741$	− 17452.8i	− 0.865242i
$742$	0	0
$743$	11763.6i	0.580839i	0.956899	+	0.290420i	$0.0937948\pi$
$743$	11763.6i	0.580839i	−0.956899	+	0.290420i	$0.906205\pi$
$744$	0	0
$745$	− 1686.94i	− 0.0829591i
$746$	0	0
$747$	−11981.0	−0.586828
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 12377.8i	− 0.601427i	−0.953715	−	0.300713i	$-0.902775\pi$
$751$	− 12377.8i	− 0.601427i	0.953715	−	0.300713i	$-0.0972247\pi$
$752$	0	0
$753$	−7172.70	−0.347129
$754$	0	0
$755$	1632.50	0.0786924
$756$	0	0
$757$	18350.8	0.881071	0.440536	−	0.897735i	$-0.354789\pi$
$757$	18350.8	0.881071	0.440536	+	0.897735i	$0.354789\pi$
$758$	0	0
$759$	−270.174	−0.0129205
$760$	0	0
$761$	16558.4i	0.788754i	0.918949	+	0.394377i	$0.129040\pi$
$761$	16558.4i	0.788754i	−0.918949	+	0.394377i	$0.870960\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3345.26	−0.158102
$766$	0	0
$767$	− 3492.34i	− 0.164408i
$768$	0	0
$769$	3663.72i	0.171804i	0.996304	+	0.0859019i	$0.0273772\pi$
$769$	3663.72i	0.171804i	−0.996304	+	0.0859019i	$0.972623\pi$
$770$	0	0
$771$	21304.7i	0.995162i
$772$	0	0
$773$	− 39653.9i	− 1.84509i	−0.385893	−	0.922544i	$-0.626106\pi$
$773$	− 39653.9i	− 1.84509i	0.385893	−	0.922544i	$-0.373894\pi$
$774$	0	0
$775$	35514.1	1.64607
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21592.2i	0.993094i
$780$	0	0
$781$	−4501.70	−0.206253
$782$	0	0
$783$	−4783.12	−0.218308
$784$	0	0
$785$	1537.17	0.0698905
$786$	0	0
$787$	−35468.7	−1.60651	−0.803254	−	0.595637i	$-0.796900\pi$
$787$	−35468.7	−1.60651	−0.803254	+	0.595637i	$0.796900\pi$
$788$	0	0
$789$	− 18723.2i	− 0.844821i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8586.98	−0.384530
$794$	0	0
$795$	5919.00i	0.264057i
$796$	0	0
$797$	32026.0i	1.42336i	0.702503	+	0.711681i	$0.252066\pi$
$797$	32026.0i	1.42336i	−0.702503	+	0.711681i	$0.747934\pi$
$798$	0	0
$799$	26929.2i	1.19235i
$800$	0	0
$801$	− 22240.6i	− 0.981067i
$802$	0	0
$803$	1564.08	0.0687362
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3196.39i	− 0.139428i
$808$	0	0
$809$	3417.20	0.148507	0.0742537	−	0.997239i	$-0.476343\pi$
$809$	3417.20	0.148507	0.0742537	+	0.997239i	$0.476343\pi$
$810$	0	0
$811$	27667.8	1.19796	0.598982	−	0.800763i	$-0.295572\pi$
$811$	27667.8	1.19796	0.598982	+	0.800763i	$0.295572\pi$
$812$	0	0
$813$	−1163.39	−0.0501868
$814$	0	0
$815$	−3983.50	−0.171210
$816$	0	0
$817$	− 21689.1i	− 0.928770i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28684.9	1.21938	0.609690	−	0.792640i	$-0.291294\pi$
$821$	28684.9	1.21938	0.609690	+	0.792640i	$0.291294\pi$
$822$	0	0
$823$	21516.6i	0.911326i	0.890152	+	0.455663i	$0.150598\pi$
$823$	21516.6i	0.911326i	−0.890152	+	0.455663i	$0.849402\pi$
$824$	0	0
$825$	3994.59i	0.168574i
$826$	0	0
$827$	− 28539.8i	− 1.20003i	−0.799988	−	0.600016i	$-0.795161\pi$
$827$	− 28539.8i	− 1.20003i	0.799988	−	0.600016i	$-0.204839\pi$
$828$	0	0
$829$	4362.57i	0.182772i	0.995816	+	0.0913862i	$0.0291298\pi$
$829$	4362.57i	0.182772i	−0.995816	+	0.0913862i	$0.970870\pi$
$830$	0	0
$831$	13698.6	0.571841
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3349.06i	0.138801i
$836$	0	0
$837$	−8367.29	−0.345539
$838$	0	0
$839$	34373.2	1.41442	0.707208	−	0.707006i	$-0.249955\pi$
$839$	34373.2	1.41442	0.707208	+	0.707006i	$0.249955\pi$
$840$	0	0
$841$	2632.42	0.107935
$842$	0	0
$843$	−50517.2	−2.06394
$844$	0	0
$845$	625.426i	0.0254619i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−23275.1	−0.940869
$850$	0	0
$851$	− 1242.45i	− 0.0500477i
$852$	0	0
$853$	− 9468.45i	− 0.380063i	−0.981778	−	0.190031i	$-0.939141\pi$
$853$	− 9468.45i	− 0.380063i	0.981778	−	0.190031i	$-0.0608590\pi$
$854$	0	0
$855$	1685.37i	0.0674136i
$856$	0	0
$857$	11389.0i	0.453957i	0.973900	+	0.226978i	$0.0728847\pi$
$857$	11389.0i	0.453957i	−0.973900	+	0.226978i	$0.927115\pi$
$858$	0	0
$859$	−8455.90	−0.335869	−0.167935	−	0.985798i	$-0.553710\pi$
$859$	−8455.90	−0.335869	−0.167935	+	0.985798i	$0.553710\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7474.49i	0.294826i	0.989075	+	0.147413i	$0.0470946\pi$
$863$	7474.49i	0.294826i	−0.989075	+	0.147413i	$0.952905\pi$
$864$	0	0
$865$	4137.17	0.162622
$866$	0	0
$867$	−66080.9	−2.58850
$868$	0	0
$869$	349.399	0.0136393
$870$	0	0
$871$	17131.7	0.666460
$872$	0	0
$873$	− 15483.6i	− 0.600276i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32056.6	−1.23429	−0.617147	−	0.786848i	$-0.711712\pi$
$877$	−32056.6	−1.23429	−0.617147	+	0.786848i	$0.711712\pi$
$878$	0	0
$879$	− 52041.7i	− 1.99695i
$880$	0	0
$881$	27139.7i	1.03787i	0.854815	+	0.518933i	$0.173671\pi$
$881$	27139.7i	1.03787i	−0.854815	+	0.518933i	$0.826329\pi$
$882$	0	0
$883$	− 43490.8i	− 1.65751i	−0.559611	−	0.828755i	$-0.689049\pi$
$883$	− 43490.8i	− 1.65751i	0.559611	−	0.828755i	$-0.310951\pi$
$884$	0	0
$885$	735.222i	0.0279257i
$886$	0	0
$887$	−33511.2	−1.26854	−0.634271	−	0.773111i	$-0.718700\pi$
$887$	−33511.2	−1.26854	−0.634271	+	0.773111i	$0.718700\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 3770.29i	− 0.141761i
$892$	0	0
$893$	13567.2	0.508409
$894$	0	0
$895$	1490.77	0.0556771
$896$	0	0
$897$	2422.36	0.0901677
$898$	0	0
$899$	47269.6	1.75365
$900$	0	0
$901$	81793.6i	3.02435i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2151.26	0.0790168
$906$	0	0
$907$	− 7530.44i	− 0.275683i	−0.990454	−	0.137841i	$-0.955984\pi$
$907$	− 7530.44i	− 0.275683i	0.990454	−	0.137841i	$-0.0440164\pi$
$908$	0	0
$909$	− 19191.4i	− 0.700264i
$910$	0	0
$911$	19971.6i	0.726332i	0.931724	+	0.363166i	$0.118304\pi$
$911$	19971.6i	0.726332i	−0.931724	+	0.363166i	$0.881696\pi$
$912$	0	0
$913$	− 2398.12i	− 0.0869289i
$914$	0	0
$915$	1807.76	0.0653146
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 46751.5i	− 1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$919$	− 46751.5i	− 1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$920$	0	0
$921$	19019.2	0.680461
$922$	0	0
$923$	40362.0	1.43936
$924$	0	0
$925$	−18369.9	−0.652972
$926$	0	0
$927$	34667.9	1.22831
$928$	0	0
$929$	− 18772.2i	− 0.662968i	−0.943461	−	0.331484i	$-0.892451\pi$
$929$	− 18772.2i	− 0.662968i	0.943461	−	0.331484i	$-0.107549\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20989.9	−0.736527
$934$	0	0
$935$	− 669.589i	− 0.0234202i
$936$	0	0
$937$	− 13717.6i	− 0.478263i	−0.970987	−	0.239132i	$-0.923137\pi$
$937$	− 13717.6i	− 0.478263i	0.970987	−	0.239132i	$-0.0768628\pi$
$938$	0	0
$939$	− 8051.34i	− 0.279814i
$940$	0	0
$941$	18443.8i	0.638949i	0.947595	+	0.319475i	$0.103506\pi$
$941$	18443.8i	0.638949i	−0.947595	+	0.319475i	$0.896494\pi$
$942$	0	0
$943$	−2996.89	−0.103491
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33239.1i	1.14058i	0.821444	+	0.570289i	$0.193169\pi$
$947$	33239.1i	1.14058i	−0.821444	+	0.570289i	$0.806831\pi$
$948$	0	0
$949$	−14023.5	−0.479684
$950$	0	0
$951$	−39869.1	−1.35946
$952$	0	0
$953$	32914.7	1.11880	0.559398	−	0.828899i	$-0.311033\pi$
$953$	32914.7	1.11880	0.559398	+	0.828899i	$0.311033\pi$
$954$	0	0
$955$	−1170.64	−0.0396660
$956$	0	0
$957$	5316.82i	0.179591i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	52899.4	1.77568
$962$	0	0
$963$	38614.0i	1.29213i
$964$	0	0
$965$	1632.22i	0.0544487i
$966$	0	0
$967$	− 29368.7i	− 0.976663i	−0.872658	−	0.488332i	$-0.837606\pi$
$967$	− 29368.7i	− 0.976663i	0.872658	−	0.488332i	$-0.162394\pi$
$968$	0	0
$969$	50773.6i	1.68326i
$970$	0	0
$971$	−31567.2	−1.04329	−0.521647	−	0.853161i	$-0.674682\pi$
$971$	−31567.2	−1.04329	−0.521647	+	0.853161i	$0.674682\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 35815.2i	− 1.17642i
$976$	0	0
$977$	23250.0	0.761344	0.380672	−	0.924710i	$-0.375693\pi$
$977$	23250.0	0.761344	0.380672	+	0.924710i	$0.375693\pi$
$978$	0	0
$979$	4451.70	0.145329
$980$	0	0
$981$	−36064.3	−1.17374
$982$	0	0
$983$	4057.54	0.131654	0.0658268	−	0.997831i	$-0.479032\pi$
$983$	4057.54	0.131654	0.0658268	+	0.997831i	$0.479032\pi$
$984$	0	0
$985$	1344.83i	0.0435023i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3010.34	0.0967880
$990$	0	0
$991$	− 42406.5i	− 1.35932i	−0.733526	−	0.679661i	$-0.762127\pi$
$991$	− 42406.5i	− 1.35932i	0.733526	−	0.679661i	$-0.237873\pi$
$992$	0	0
$993$	− 51980.4i	− 1.66118i
$994$	0	0
$995$	− 304.252i	− 0.00969389i
$996$	0	0
$997$	57689.0i	1.83253i	0.400576	+	0.916263i	$0.368810\pi$
$997$	57689.0i	1.83253i	−0.400576	+	0.916263i	$0.631190\pi$
$998$	0	0
$999$	4328.03	0.137070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.g.783.6		6
4.3	odd	2		784.4.f.h.783.2		6
7.2	even	3		112.4.p.g.31.1	yes	6
7.3	odd	6		112.4.p.f.47.3	yes	6
7.6	odd	2		784.4.f.h.783.1		6
28.3	even	6		112.4.p.g.47.1	yes	6
28.23	odd	6		112.4.p.f.31.3	✓	6
28.27	even	2	inner	784.4.f.g.783.5		6
56.3	even	6		448.4.p.f.383.3		6
56.37	even	6		448.4.p.f.255.3		6
56.45	odd	6		448.4.p.g.383.1		6
56.51	odd	6		448.4.p.g.255.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.3	✓	6	28.23	odd	6
112.4.p.f.47.3	yes	6	7.3	odd	6
112.4.p.g.31.1	yes	6	7.2	even	3
112.4.p.g.47.1	yes	6	28.3	even	6
448.4.p.f.255.3		6	56.37	even	6
448.4.p.f.383.3		6	56.3	even	6
448.4.p.g.255.1		6	56.51	odd	6
448.4.p.g.383.1		6	56.45	odd	6
784.4.f.g.783.5		6	28.27	even	2	inner
784.4.f.g.783.6		6	1.1	even	1	trivial
784.4.f.h.783.1		6	7.6	odd	2
784.4.f.h.783.2		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +O(q^{10})\)	\(q+7.06258 q^{3} +1.22397i q^{5} +22.8800 q^{9} +4.57968i q^{11} -41.0611i q^{13} +8.64436i q^{15} +119.455i q^{17} +60.1826 q^{19} +8.35305i q^{23} +123.502 q^{25} -29.0976 q^{27} +164.382 q^{29} +287.559 q^{31} +32.3443i q^{33} -148.742 q^{37} -289.998i q^{39} +358.778i q^{41} -360.388i q^{43} +28.0044i q^{45} +225.434 q^{47} +843.660i q^{51} +684.724 q^{53} -5.60537 q^{55} +425.044 q^{57} +85.0522 q^{59} -209.127i q^{61} +50.2574 q^{65} +417.225i q^{67} +58.9941i q^{69} +982.974i q^{71} -341.526i q^{73} +872.242 q^{75} -76.2934i q^{79} -823.265 q^{81} -523.643 q^{83} -146.209 q^{85} +1160.96 q^{87} -972.055i q^{89} +2030.91 q^{93} +73.6614i q^{95} -676.730i q^{97} +104.783i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{3} + 156 q^{9}+O(q^{10}) \) 6 * q - 14 * q^3 + 156 * q^9	\( 6 q - 14 q^{3} + 156 q^{9} + 286 q^{19} - 612 q^{25} - 362 q^{27} - 348 q^{29} + 410 q^{31} + 498 q^{37} + 150 q^{47} + 1290 q^{53} - 918 q^{55} - 6 q^{57} + 642 q^{59} + 1224 q^{65} + 8276 q^{75} - 450 q^{81} - 24 q^{83} + 3786 q^{85} + 7284 q^{87} + 5982 q^{93}+O(q^{100}) \) 6 * q - 14 * q^3 + 156 * q^9 + 286 * q^19 - 612 * q^25 - 362 * q^27 - 348 * q^29 + 410 * q^31 + 498 * q^37 + 150 * q^47 + 1290 * q^53 - 918 * q^55 - 6 * q^57 + 642 * q^59 + 1224 * q^65 + 8276 * q^75 - 450 * q^81 - 24 * q^83 + 3786 * q^85 + 7284 * q^87 + 5982 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more