LMFDB - Embedded newform 441.7.d.c.244.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,7,Mod(244,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("441.244");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	441.d (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:	$101.453850876$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 $ x^8 - x^7 + 212x^6 + 473x^5 + 39800x^4 + 36821x^3 + 985651x^2 - 601290x + 21068100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.6
Root			$-2.75320 + 4.76869i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.7.d.c.244.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.50641 q^{2} -43.6923 q^{4} +62.2665i q^{5} -485.305 q^{8} +O(q^{10})$	$q+4.50641 q^{2} -43.6923 q^{4} +62.2665i q^{5} -485.305 q^{8} +280.598i q^{10} -19.4335 q^{11} +1642.65i q^{13} +609.323 q^{16} +214.836i q^{17} +9511.88i q^{19} -2720.57i q^{20} -87.5753 q^{22} +14447.8 q^{23} +11747.9 q^{25} +7402.44i q^{26} -43016.4 q^{29} -9006.23i q^{31} +33805.4 q^{32} +968.141i q^{34} -33129.1 q^{37} +42864.4i q^{38} -30218.3i q^{40} +73712.4i q^{41} +4761.19 q^{43} +849.094 q^{44} +65107.7 q^{46} -73624.3i q^{47} +52940.7 q^{50} -71771.1i q^{52} -238634. q^{53} -1210.06i q^{55} -193850. q^{58} +326562. i q^{59} -404743. i q^{61} -40585.7i q^{62} +113344. q^{64} -102282. q^{65} +219964. q^{67} -9386.70i q^{68} -350228. q^{71} -201617. i q^{73} -149293. q^{74} -415596. i q^{76} +395490. q^{79} +37940.4i q^{80} +332178. i q^{82} -13229.0i q^{83} -13377.1 q^{85} +21455.9 q^{86} +9431.18 q^{88} -230316. i q^{89} -631258. q^{92} -331781. i q^{94} -592271. q^{95} -662517. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8}+O(q^{10}) $ 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8	$ 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95}+O(q^{100}) $ 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8 - 628 * q^11 + 25442 * q^16 + 86106 * q^22 + 7856 * q^23 + 34076 * q^25 + 8300 * q^29 - 372414 * q^32 - 129412 * q^37 + 45740 * q^43 + 185058 * q^44 + 223008 * q^46 - 967216 * q^50 - 1081948 * q^53 - 1079598 * q^58 + 2378626 * q^64 - 828408 * q^65 + 2317804 * q^67 - 1442344 * q^71 - 865880 * q^74 - 1222904 * q^79 - 275112 * q^85 + 1632448 * q^86 + 732882 * q^88 - 678720 * q^92 - 1183584 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.50641	0.563301	0.281650	−	0.959517i	$-0.409118\pi$
$2$	4.50641	0.563301	0.281650	+	0.959517i	$0.409118\pi$
$3$	0	0
$3$	0	0
$4$	−43.6923	−0.682692
$5$	62.2665i	0.498132i	0.968487	+	0.249066i	$0.0801236\pi$
$5$	62.2665i	0.498132i	−0.968487	+	0.249066i	$0.919876\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−485.305	−0.947862
$9$	0	0
$10$	280.598i	0.280598i
$11$	−19.4335	−0.0146007	−0.00730034	−	0.999973i	$-0.502324\pi$
$11$	−19.4335	−0.0146007	−0.00730034	+	0.999973i	$0.502324\pi$
$12$	0	0
$13$	1642.65i	0.747678i	0.927494	+	0.373839i	$0.121959\pi$
$13$	1642.65i	0.747678i	−0.927494	+	0.373839i	$0.878041\pi$
$14$	0	0
$15$	0	0
$16$	609.323	0.148760
$17$	214.836i	0.0437282i	0.999761	+	0.0218641i	$0.00696011\pi$
$17$	214.836i	0.0437282i	−0.999761	+	0.0218641i	$0.993040\pi$
$18$	0	0
$19$	9511.88i	1.38677i	0.720566	+	0.693387i	$0.243882\pi$
$19$	9511.88i	1.38677i	−0.720566	+	0.693387i	$0.756118\pi$
$20$	− 2720.57i	− 0.340071i
$21$	0	0
$22$	−87.5753	−0.00822458
$23$	14447.8	1.18746	0.593729	−	0.804665i	$-0.297655\pi$
$23$	14447.8	1.18746	0.593729	+	0.804665i	$0.297655\pi$
$24$	0	0
$25$	11747.9	0.751864
$26$	7402.44i	0.421168i
$27$	0	0
$28$	0	0
$29$	−43016.4	−1.76376	−0.881882	−	0.471471i	$-0.843723\pi$
$29$	−43016.4	−1.76376	−0.881882	+	0.471471i	$0.843723\pi$
$30$	0	0
$31$	− 9006.23i	− 0.302314i	−0.988510	−	0.151157i	$-0.951700\pi$
$31$	− 9006.23i	− 0.302314i	0.988510	−	0.151157i	$-0.0482999\pi$
$32$	33805.4	1.03166
$33$	0	0
$34$	968.141i	0.0246321i
$35$	0	0
$36$	0	0
$37$	−33129.1	−0.654041	−0.327020	−	0.945017i	$-0.606045\pi$
$37$	−33129.1	−0.654041	−0.327020	+	0.945017i	$0.606045\pi$
$38$	42864.4i	0.781171i
$39$	0	0
$40$	− 30218.3i	− 0.472160i
$41$	73712.4i	1.06952i	0.845004	+	0.534760i	$0.179598\pi$
$41$	73712.4i	1.06952i	−0.845004	+	0.534760i	$0.820402\pi$
$42$	0	0
$43$	4761.19	0.0598839	0.0299420	−	0.999552i	$-0.490468\pi$
$43$	4761.19	0.0598839	0.0299420	+	0.999552i	$0.490468\pi$
$44$	849.094	0.00996777
$45$	0	0
$46$	65107.7	0.668896
$47$	− 73624.3i	− 0.709133i	−0.935031	−	0.354567i	$-0.884628\pi$
$47$	− 73624.3i	− 0.709133i	0.935031	−	0.354567i	$-0.115372\pi$
$48$	0	0
$49$	0	0
$50$	52940.7	0.423526
$51$	0	0
$52$	− 71771.1i	− 0.510434i
$53$	−238634.	−1.60289	−0.801447	−	0.598065i	$-0.795936\pi$
$53$	−238634.	−1.60289	−0.801447	+	0.598065i	$0.795936\pi$
$54$	0	0
$55$	− 1210.06i	− 0.00727307i
$56$	0	0
$57$	0	0
$58$	−193850.	−0.993530
$59$	326562.i	1.59005i	0.606579	+	0.795023i	$0.292541\pi$
$59$	326562.i	1.59005i	−0.606579	+	0.795023i	$0.707459\pi$
$60$	0	0
$61$	− 404743.i	− 1.78316i	−0.452866	−	0.891578i	$-0.649599\pi$
$61$	− 404743.i	− 1.78316i	0.452866	−	0.891578i	$-0.350401\pi$
$62$	− 40585.7i	− 0.170294i
$63$	0	0
$64$	113344.	0.432374
$65$	−102282.	−0.372442
$66$	0	0
$67$	219964.	0.731352	0.365676	−	0.930742i	$-0.380838\pi$
$67$	219964.	0.731352	0.365676	+	0.930742i	$0.380838\pi$
$68$	− 9386.70i	− 0.0298529i
$69$	0	0
$70$	0	0
$71$	−350228.	−0.978535	−0.489267	−	0.872134i	$-0.662736\pi$
$71$	−350228.	−0.978535	−0.489267	+	0.872134i	$0.662736\pi$
$72$	0	0
$73$	− 201617.i	− 0.518274i	−0.965841	−	0.259137i	$-0.916562\pi$
$73$	− 201617.i	− 0.518274i	0.965841	−	0.259137i	$-0.0834381\pi$
$74$	−149293.	−0.368422
$75$	0	0
$76$	− 415596.i	− 0.946739i
$77$	0	0
$78$	0	0
$79$	395490.	0.802148	0.401074	−	0.916046i	$-0.368637\pi$
$79$	395490.	0.802148	0.401074	+	0.916046i	$0.368637\pi$
$80$	37940.4i	0.0741023i
$81$	0	0
$82$	332178.i	0.602462i
$83$	− 13229.0i	− 0.0231362i	−0.999933	−	0.0115681i	$-0.996318\pi$
$83$	− 13229.0i	− 0.0231362i	0.999933	−	0.0115681i	$-0.00368232\pi$
$84$	0	0
$85$	−13377.1	−0.0217824
$86$	21455.9	0.0337327
$87$	0	0
$88$	9431.18	0.0138394
$89$	− 230316.i	− 0.326704i	−0.986568	−	0.163352i	$-0.947769\pi$
$89$	− 230316.i	− 0.326704i	0.986568	−	0.163352i	$-0.0522306\pi$
$90$	0	0
$91$	0	0
$92$	−631258.	−0.810668
$93$	0	0
$94$	− 331781.i	− 0.399455i
$95$	−592271.	−0.690796
$96$	0	0
$97$	− 662517.i	− 0.725909i	−0.931807	−	0.362954i	$-0.881768\pi$
$97$	− 662517.i	− 0.725909i	0.931807	−	0.362954i	$-0.118232\pi$
$98$	0	0
$99$	0	0
$100$	−513292.	−0.513292
$101$	− 1.87761e6i	− 1.82239i	−0.411979	−	0.911193i	$-0.635162\pi$
$101$	− 1.87761e6i	− 1.82239i	0.411979	−	0.911193i	$-0.364838\pi$
$102$	0	0
$103$	− 904000.i	− 0.827288i	−0.910439	−	0.413644i	$-0.864256\pi$
$103$	− 904000.i	− 0.827288i	0.910439	−	0.413644i	$-0.135744\pi$
$104$	− 797186.i	− 0.708696i
$105$	0	0
$106$	−1.07538e6	−0.902912
$107$	−846213.	−0.690761	−0.345381	−	0.938463i	$-0.612250\pi$
$107$	−846213.	−0.690761	−0.345381	+	0.938463i	$0.612250\pi$
$108$	0	0
$109$	−1.69535e6	−1.30912	−0.654560	−	0.756010i	$-0.727146\pi$
$109$	−1.69535e6	−1.30912	−0.654560	+	0.756010i	$0.727146\pi$
$110$	− 5453.01i	− 0.00409693i
$111$	0	0
$112$	0	0
$113$	152753.	0.105866	0.0529328	−	0.998598i	$-0.483143\pi$
$113$	152753.	0.105866	0.0529328	+	0.998598i	$0.483143\pi$
$114$	0	0
$115$	899614.i	0.591511i
$116$	1.87949e6	1.20411
$117$	0	0
$118$	1.47162e6i	0.895675i
$119$	0	0
$120$	0	0
$121$	−1.77118e6	−0.999787
$122$	− 1.82394e6i	− 1.00445i
$123$	0	0
$124$	393503.i	0.206387i
$125$	1.70441e6i	0.872660i
$126$	0	0
$127$	−2.82114e6	−1.37725	−0.688627	−	0.725116i	$-0.741786\pi$
$127$	−2.82114e6	−1.37725	−0.688627	+	0.725116i	$0.741786\pi$
$128$	−1.65277e6	−0.788102
$129$	0	0
$130$	−460924.	−0.209797
$131$	− 4.45866e6i	− 1.98331i	−0.128929	−	0.991654i	$-0.541154\pi$
$131$	− 4.45866e6i	− 1.98331i	0.128929	−	0.991654i	$-0.458846\pi$
$132$	0	0
$133$	0	0
$134$	991245.	0.411971
$135$	0	0
$136$	− 104261.i	− 0.0414483i
$137$	−1.29925e6	−0.505278	−0.252639	−	0.967561i	$-0.581298\pi$
$137$	−1.29925e6	−0.505278	−0.252639	+	0.967561i	$0.581298\pi$
$138$	0	0
$139$	− 119566.i	− 0.0445207i	−0.999752	−	0.0222603i	$-0.992914\pi$
$139$	− 119566.i	− 0.0445207i	0.999752	−	0.0222603i	$-0.00708627\pi$
$140$	0	0
$141$	0	0
$142$	−1.57827e6	−0.551210
$143$	− 31922.4i	− 0.0109166i
$144$	0	0
$145$	− 2.67848e6i	− 0.878587i
$146$	− 908570.i	− 0.291944i
$147$	0	0
$148$	1.44749e6	0.446509
$149$	−6.25829e6	−1.89189	−0.945946	−	0.324323i	$-0.894864\pi$
$149$	−6.25829e6	−1.89189	−0.945946	+	0.324323i	$0.894864\pi$
$150$	0	0
$151$	−1.50794e6	−0.437978	−0.218989	−	0.975727i	$-0.570276\pi$
$151$	−1.50794e6	−0.437978	−0.218989	+	0.975727i	$0.570276\pi$
$152$	− 4.61617e6i	− 1.31447i
$153$	0	0
$154$	0	0
$155$	560786.	0.150592
$156$	0	0
$157$	2.47985e6i	0.640805i	0.947281	+	0.320403i	$0.103818\pi$
$157$	2.47985e6i	0.640805i	−0.947281	+	0.320403i	$0.896182\pi$
$158$	1.78224e6	0.451851
$159$	0	0
$160$	2.10494e6i	0.513902i
$161$	0	0
$162$	0	0
$163$	6.59019e6	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$163$	6.59019e6	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$164$	− 3.22066e6i	− 0.730153i
$165$	0	0
$166$	− 59615.1i	− 0.0130326i
$167$	5.45006e6i	1.17018i	0.810969	+	0.585089i	$0.198941\pi$
$167$	5.45006e6i	1.17018i	−0.810969	+	0.585089i	$0.801059\pi$
$168$	0	0
$169$	2.12851e6	0.440978
$170$	−60282.7	−0.0122700
$171$	0	0
$172$	−208027.	−0.0408823
$173$	2.76144e6i	0.533332i	0.963789	+	0.266666i	$0.0859220\pi$
$173$	2.76144e6i	0.533332i	−0.963789	+	0.266666i	$0.914078\pi$
$174$	0	0
$175$	0	0
$176$	−11841.3	−0.00217200
$177$	0	0
$178$	− 1.03790e6i	− 0.184033i
$179$	1.48773e6	0.259396	0.129698	−	0.991554i	$-0.458599\pi$
$179$	1.48773e6	0.259396	0.129698	+	0.991554i	$0.458599\pi$
$180$	0	0
$181$	6.01679e6i	1.01468i	0.861746	+	0.507340i	$0.169371\pi$
$181$	6.01679e6i	1.01468i	−0.861746	+	0.507340i	$0.830629\pi$
$182$	0	0
$183$	0	0
$184$	−7.01160e6	−1.12555
$185$	− 2.06284e6i	− 0.325799i
$186$	0	0
$187$	− 4175.03i	0 0.000638461i
$188$	3.21682e6i	0.484120i
$189$	0	0
$190$	−2.66902e6	−0.389126
$191$	−6.40344e6	−0.918996	−0.459498	−	0.888179i	$-0.651971\pi$
$191$	−6.40344e6	−0.918996	−0.459498	+	0.888179i	$0.651971\pi$
$192$	0	0
$193$	9.85184e6	1.37039	0.685197	−	0.728358i	$-0.259716\pi$
$193$	9.85184e6	1.37039	0.685197	+	0.728358i	$0.259716\pi$
$194$	− 2.98557e6i	− 0.408905i
$195$	0	0
$196$	0	0
$197$	−2.38883e6	−0.312454	−0.156227	−	0.987721i	$-0.549933\pi$
$197$	−2.38883e6	−0.312454	−0.156227	+	0.987721i	$0.549933\pi$
$198$	0	0
$199$	− 1.57158e7i	− 1.99424i	−0.0758697	−	0.997118i	$-0.524173\pi$
$199$	− 1.57158e7i	− 1.99424i	0.0758697	−	0.997118i	$-0.475827\pi$
$200$	−5.70131e6	−0.712664
$201$	0	0
$202$	− 8.46126e6i	− 1.02655i
$203$	0	0
$204$	0	0
$205$	−4.58981e6	−0.532762
$206$	− 4.07379e6i	− 0.466012i
$207$	0	0
$208$	1.00090e6i	0.111225i
$209$	− 184849.i	− 0.0202478i
$210$	0	0
$211$	1.46740e6	0.156207	0.0781034	−	0.996945i	$-0.475114\pi$
$211$	1.46740e6	0.156207	0.0781034	+	0.996945i	$0.475114\pi$
$212$	1.04265e7	1.09428
$213$	0	0
$214$	−3.81338e6	−0.389107
$215$	296463.i	0.0298301i
$216$	0	0
$217$	0	0
$218$	−7.63993e6	−0.737429
$219$	0	0
$220$	52870.1i	0.00496526i
$221$	−352901.	−0.0326946
$222$	0	0
$223$	3.63341e6i	0.327642i	0.986490	+	0.163821i	$0.0523820\pi$
$223$	3.63341e6i	0.327642i	−0.986490	+	0.163821i	$0.947618\pi$
$224$	0	0
$225$	0	0
$226$	688368.	0.0596342
$227$	− 5.93655e6i	− 0.507524i	−0.967267	−	0.253762i	$-0.918332\pi$
$227$	− 5.93655e6i	− 0.507524i	0.967267	−	0.253762i	$-0.0816680\pi$
$228$	0	0
$229$	− 1.22357e7i	− 1.01888i	−0.860507	−	0.509438i	$-0.829853\pi$
$229$	− 1.22357e7i	− 1.01888i	0.860507	−	0.509438i	$-0.170147\pi$
$230$	4.05403e6i	0.333199i
$231$	0	0
$232$	2.08761e7	1.67180
$233$	−1.21209e7	−0.958226	−0.479113	−	0.877753i	$-0.659042\pi$
$233$	−1.21209e7	−0.958226	−0.479113	+	0.877753i	$0.659042\pi$
$234$	0	0
$235$	4.58433e6	0.353242
$236$	− 1.42683e7i	− 1.08551i
$237$	0	0
$238$	0	0
$239$	−1.19016e7	−0.871790	−0.435895	−	0.899998i	$-0.643568\pi$
$239$	−1.19016e7	−0.871790	−0.435895	+	0.899998i	$0.643568\pi$
$240$	0	0
$241$	− 9.95487e6i	− 0.711188i	−0.934641	−	0.355594i	$-0.884279\pi$
$241$	− 9.95487e6i	− 0.711188i	0.934641	−	0.355594i	$-0.115721\pi$
$242$	−7.98167e6	−0.563181
$243$	0	0
$244$	1.76841e7i	1.21735i
$245$	0	0
$246$	0	0
$247$	−1.56247e7	−1.03686
$248$	4.37077e6i	0.286552i
$249$	0	0
$250$	7.68078e6i	0.491570i
$251$	4.57737e6i	0.289464i	0.989471	+	0.144732i	$0.0462320\pi$
$251$	4.57737e6i	0.289464i	−0.989471	+	0.144732i	$0.953768\pi$
$252$	0	0
$253$	−280772.	−0.0173377
$254$	−1.27132e7	−0.775808
$255$	0	0
$256$	−1.47021e7	−0.876313
$257$	− 4.04640e6i	− 0.238380i	−0.992871	−	0.119190i	$-0.961970\pi$
$257$	− 4.04640e6i	− 0.238380i	0.992871	−	0.119190i	$-0.0380298\pi$
$258$	0	0
$259$	0	0
$260$	4.46893e6	0.254263
$261$	0	0
$262$	− 2.00925e7i	− 1.11720i
$263$	−2.96757e7	−1.63130	−0.815651	−	0.578545i	$-0.803621\pi$
$263$	−2.96757e7	−1.63130	−0.815651	+	0.578545i	$0.803621\pi$
$264$	0	0
$265$	− 1.48589e7i	− 0.798453i
$266$	0	0
$267$	0	0
$268$	−9.61071e6	−0.499288
$269$	2.04711e7i	1.05168i	0.850583	+	0.525841i	$0.176249\pi$
$269$	2.04711e7i	1.05168i	−0.850583	+	0.525841i	$0.823751\pi$
$270$	0	0
$271$	− 2.53211e6i	− 0.127226i	−0.997975	−	0.0636128i	$-0.979738\pi$
$271$	− 2.53211e6i	− 0.127226i	0.997975	−	0.0636128i	$-0.0202623\pi$
$272$	130905.i	0.00650502i
$273$	0	0
$274$	−5.85494e6	−0.284623
$275$	−228303.	−0.0109777
$276$	0	0
$277$	3.05729e7	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$277$	3.05729e7	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$278$	− 538811.i	− 0.0250785i
$279$	0	0
$280$	0	0
$281$	2.27089e7	1.02348	0.511738	−	0.859142i	$-0.329002\pi$
$281$	2.27089e7	1.02348	0.511738	+	0.859142i	$0.329002\pi$
$282$	0	0
$283$	4.09828e7i	1.80818i	0.427339	+	0.904092i	$0.359451\pi$
$283$	4.09828e7i	1.80818i	−0.427339	+	0.904092i	$0.640549\pi$
$284$	1.53023e7	0.668038
$285$	0	0
$286$	− 143855.i	− 0.00614934i
$287$	0	0
$288$	0	0
$289$	2.40914e7	0.998088
$290$	− 1.20703e7i	− 0.494909i
$291$	0	0
$292$	8.80912e6i	0.353821i
$293$	− 2.94333e7i	− 1.17014i	−0.810984	−	0.585069i	$-0.801068\pi$
$293$	− 2.94333e7i	− 1.17014i	0.810984	−	0.585069i	$-0.198932\pi$
$294$	0	0
$295$	−2.03339e7	−0.792053
$296$	1.60777e7	0.619941
$297$	0	0
$298$	−2.82024e7	−1.06571
$299$	2.37327e7i	0.887836i
$300$	0	0
$301$	0	0
$302$	−6.79538e6	−0.246714
$303$	0	0
$304$	5.79580e6i	0.206297i
$305$	2.52019e7	0.888248
$306$	0	0
$307$	4.67295e6i	0.161501i	0.996734	+	0.0807506i	$0.0257317\pi$
$307$	4.67295e6i	0.161501i	−0.996734	+	0.0807506i	$0.974268\pi$
$308$	0	0
$309$	0	0
$310$	2.52713e6	0.0848287
$311$	− 3.20312e7i	− 1.06486i	−0.846474	−	0.532430i	$-0.821279\pi$
$311$	− 3.20312e7i	− 1.06486i	0.846474	−	0.532430i	$-0.178721\pi$
$312$	0	0
$313$	− 2.18013e7i	− 0.710965i	−0.934683	−	0.355483i	$-0.884316\pi$
$313$	− 2.18013e7i	− 0.710965i	0.934683	−	0.355483i	$-0.115684\pi$
$314$	1.11752e7i	0.360966i
$315$	0	0
$316$	−1.72799e7	−0.547620
$317$	−1.05248e7	−0.330397	−0.165199	−	0.986260i	$-0.552827\pi$
$317$	−1.05248e7	−0.330397	−0.165199	+	0.986260i	$0.552827\pi$
$318$	0	0
$319$	835960.	0.0257521
$320$	7.05755e6i	0.215379i
$321$	0	0
$322$	0	0
$323$	−2.04350e6	−0.0606410
$324$	0	0
$325$	1.92976e7i	0.562153i
$326$	2.96981e7	0.857188
$327$	0	0
$328$	− 3.57730e7i	− 1.01376i
$329$	0	0
$330$	0	0
$331$	4.06945e7	1.12215	0.561076	−	0.827764i	$-0.310388\pi$
$331$	4.06945e7	1.12215	0.561076	+	0.827764i	$0.310388\pi$
$332$	578003.i	0.0157949i
$333$	0	0
$334$	2.45602e7i	0.659163i
$335$	1.36964e7i	0.364310i
$336$	0	0
$337$	−4.05258e7	−1.05887	−0.529434	−	0.848351i	$-0.677596\pi$
$337$	−4.05258e7	−1.05887	−0.529434	+	0.848351i	$0.677596\pi$
$338$	9.59195e6	0.248403
$339$	0	0
$340$	584477.	0.0148707
$341$	175023.i	0.00441399i
$342$	0	0
$343$	0	0
$344$	−2.31063e6	−0.0567617
$345$	0	0
$346$	1.24442e7i	0.300426i
$347$	1.65355e7	0.395757	0.197879	−	0.980227i	$-0.436595\pi$
$347$	1.65355e7	0.395757	0.197879	+	0.980227i	$0.436595\pi$
$348$	0	0
$349$	5.07209e6i	0.119319i	0.998219	+	0.0596597i	$0.0190015\pi$
$349$	5.07209e6i	0.119319i	−0.998219	+	0.0596597i	$0.980998\pi$
$350$	0	0
$351$	0	0
$352$	−656957.	−0.0150629
$353$	4.55336e7i	1.03516i	0.855635	+	0.517580i	$0.173167\pi$
$353$	4.55336e7i	1.03516i	−0.855635	+	0.517580i	$0.826833\pi$
$354$	0	0
$355$	− 2.18075e7i	− 0.487439i
$356$	1.00630e7i	0.223038i
$357$	0	0
$358$	6.70430e6	0.146118
$359$	4.00170e7	0.864891	0.432446	−	0.901660i	$-0.357651\pi$
$359$	4.00170e7	0.864891	0.432446	+	0.901660i	$0.357651\pi$
$360$	0	0
$361$	−4.34299e7	−0.923140
$362$	2.71141e7i	0.571570i
$363$	0	0
$364$	0	0
$365$	1.25540e7	0.258169
$366$	0	0
$367$	1.77519e7i	0.359127i	0.983746	+	0.179563i	$0.0574685\pi$
$367$	1.77519e7i	0.359127i	−0.983746	+	0.179563i	$0.942532\pi$
$368$	8.80338e6	0.176647
$369$	0	0
$370$	− 9.29598e6i	− 0.183523i
$371$	0	0
$372$	0	0
$373$	−5.39581e6	−0.103975	−0.0519876	−	0.998648i	$-0.516556\pi$
$373$	−5.39581e6	−0.103975	−0.0519876	+	0.998648i	$0.516556\pi$
$374$	− 18814.4i	0 0.000359646i
$375$	0	0
$376$	3.57303e7i	0.672160i
$377$	− 7.06609e7i	− 1.31873i
$378$	0	0
$379$	3.47845e7	0.638951	0.319475	−	0.947595i	$-0.396493\pi$
$379$	3.47845e7	0.638951	0.319475	+	0.947595i	$0.396493\pi$
$380$	2.58777e7	0.471601
$381$	0	0
$382$	−2.88565e7	−0.517671
$383$	− 5.65804e7i	− 1.00709i	−0.863968	−	0.503546i	$-0.832028\pi$
$383$	− 5.65804e7i	− 1.00709i	0.863968	−	0.503546i	$-0.167972\pi$
$384$	0	0
$385$	0	0
$386$	4.43964e7	0.771944
$387$	0	0
$388$	2.89469e7i	0.495572i
$389$	2.60078e7	0.441830	0.220915	−	0.975293i	$-0.429096\pi$
$389$	2.60078e7	0.441830	0.220915	+	0.975293i	$0.429096\pi$
$390$	0	0
$391$	3.10392e6i	0.0519254i
$392$	0	0
$393$	0	0
$394$	−1.07650e7	−0.176006
$395$	2.46258e7i	0.399576i
$396$	0	0
$397$	1.36299e7i	0.217832i	0.994051	+	0.108916i	$0.0347380\pi$
$397$	1.36299e7i	0.217832i	−0.994051	+	0.108916i	$0.965262\pi$
$398$	− 7.08217e7i	− 1.12335i
$399$	0	0
$400$	7.15825e6	0.111848
$401$	4.01858e7	0.623218	0.311609	−	0.950210i	$-0.399132\pi$
$401$	4.01858e7	0.623218	0.311609	+	0.950210i	$0.399132\pi$
$402$	0	0
$403$	1.47941e7	0.226033
$404$	8.20369e7i	1.24413i
$405$	0	0
$406$	0	0
$407$	643815.	0.00954944
$408$	0	0
$409$	7.56995e6i	0.110643i	0.998469	+	0.0553214i	$0.0176183\pi$
$409$	7.56995e6i	0.110643i	−0.998469	+	0.0553214i	$0.982382\pi$
$410$	−2.06836e7	−0.300106
$411$	0	0
$412$	3.94978e7i	0.564783i
$413$	0	0
$414$	0	0
$415$	823721.	0.0115249
$416$	5.55304e7i	0.771349i
$417$	0	0
$418$	− 833006.i	− 0.0114056i
$419$	3.12413e7i	0.424705i	0.977193	+	0.212353i	$0.0681125\pi$
$419$	3.12413e7i	0.424705i	−0.977193	+	0.212353i	$0.931887\pi$
$420$	0	0
$421$	5.98978e7	0.802721	0.401361	−	0.915920i	$-0.368537\pi$
$421$	5.98978e7	0.802721	0.401361	+	0.915920i	$0.368537\pi$
$422$	6.61269e6	0.0879915
$423$	0	0
$424$	1.15810e8	1.51932
$425$	2.52387e6i	0.0328776i
$426$	0	0
$427$	0	0
$428$	3.69730e7	0.471577
$429$	0	0
$430$	1.33598e6i	0.0168033i
$431$	−9.63020e7	−1.20283	−0.601414	−	0.798938i	$-0.705396\pi$
$431$	−9.63020e7	−1.20283	−0.601414	+	0.798938i	$0.705396\pi$
$432$	0	0
$433$	− 1.11186e8i	− 1.36958i	−0.728741	−	0.684790i	$-0.759894\pi$
$433$	− 1.11186e8i	− 1.36958i	0.728741	−	0.684790i	$-0.240106\pi$
$434$	0	0
$435$	0	0
$436$	7.40737e7	0.893726
$437$	1.37426e8i	1.64674i
$438$	0	0
$439$	9.12810e7i	1.07891i	0.842013	+	0.539457i	$0.181370\pi$
$439$	9.12810e7i	1.07891i	−0.842013	+	0.539457i	$0.818630\pi$
$440$	587247.i	0.00689386i
$441$	0	0
$442$	−1.59031e6	−0.0184169
$443$	1.06327e8	1.22302	0.611509	−	0.791238i	$-0.290563\pi$
$443$	1.06327e8	1.22302	0.611509	+	0.791238i	$0.290563\pi$
$444$	0	0
$445$	1.43410e7	0.162742
$446$	1.63736e7i	0.184561i
$447$	0	0
$448$	0	0
$449$	−7.74221e7	−0.855315	−0.427657	−	0.903941i	$-0.640661\pi$
$449$	−7.74221e7	−0.855315	−0.427657	+	0.903941i	$0.640661\pi$
$450$	0	0
$451$	− 1.43249e6i	− 0.0156157i
$452$	−6.67413e6	−0.0722736
$453$	0	0
$454$	− 2.67525e7i	− 0.285889i
$455$	0	0
$456$	0	0
$457$	1.90094e7	0.199168	0.0995838	−	0.995029i	$-0.468249\pi$
$457$	1.90094e7	0.199168	0.0995838	+	0.995029i	$0.468249\pi$
$458$	− 5.51390e7i	− 0.573934i
$459$	0	0
$460$	− 3.93062e7i	− 0.403820i
$461$	1.05709e8i	1.07897i	0.841996	+	0.539483i	$0.181380\pi$
$461$	1.05709e8i	1.07897i	−0.841996	+	0.539483i	$0.818620\pi$
$462$	0	0
$463$	−2.82212e7	−0.284336	−0.142168	−	0.989843i	$-0.545407\pi$
$463$	−2.82212e7	−0.284336	−0.142168	+	0.989843i	$0.545407\pi$
$464$	−2.62109e7	−0.262378
$465$	0	0
$466$	−5.46218e7	−0.539770
$467$	1.72859e8i	1.69724i	0.529005	+	0.848619i	$0.322565\pi$
$467$	1.72859e8i	1.69724i	−0.529005	+	0.848619i	$0.677435\pi$
$468$	0	0
$469$	0	0
$470$	2.06589e7	0.198982
$471$	0	0
$472$	− 1.58482e8i	− 1.50714i
$473$	−92526.6	−0.000874346 0
$474$	0	0
$475$	1.11744e8i	1.04267i
$476$	0	0
$477$	0	0
$478$	−5.36335e7	−0.491080
$479$	2.79216e7i	0.254059i	0.991899	+	0.127029i	$0.0405442\pi$
$479$	2.79216e7i	0.254059i	−0.991899	+	0.127029i	$0.959456\pi$
$480$	0	0
$481$	− 5.44195e7i	− 0.489012i
$482$	− 4.48607e7i	− 0.400613i
$483$	0	0
$484$	7.73871e7	0.682546
$485$	4.12526e7	0.361598
$486$	0	0
$487$	5.37589e7	0.465440	0.232720	−	0.972544i	$-0.425238\pi$
$487$	5.37589e7	0.465440	0.232720	+	0.972544i	$0.425238\pi$
$488$	1.96424e8i	1.69019i
$489$	0	0
$490$	0	0
$491$	−9.34867e7	−0.789779	−0.394889	−	0.918729i	$-0.629217\pi$
$491$	−9.34867e7	−0.789779	−0.394889	+	0.918729i	$0.629217\pi$
$492$	0	0
$493$	− 9.24150e6i	− 0.0771261i
$494$	−7.04111e7	−0.584064
$495$	0	0
$496$	− 5.48770e6i	− 0.0449723i
$497$	0	0
$498$	0	0
$499$	5.02752e6	0.0404624	0.0202312	−	0.999795i	$-0.493560\pi$
$499$	5.02752e6	0.0404624	0.0202312	+	0.999795i	$0.493560\pi$
$500$	− 7.44697e7i	− 0.595758i
$501$	0	0
$502$	2.06275e7i	0.163055i
$503$	− 1.45016e8i	− 1.13949i	−0.821820	−	0.569747i	$-0.807041\pi$
$503$	− 1.45016e8i	− 1.13949i	0.821820	−	0.569747i	$-0.192959\pi$
$504$	0	0
$505$	1.16912e8	0.907789
$506$	−1.26527e6	−0.00976634
$507$	0	0
$508$	1.23262e8	0.940240
$509$	1.32122e7i	0.100189i	0.998744	+	0.0500946i	$0.0159523\pi$
$509$	1.32122e7i	0.100189i	−0.998744	+	0.0500946i	$0.984048\pi$
$510$	0	0
$511$	0	0
$512$	3.95237e7	0.294474
$513$	0	0
$514$	− 1.82347e7i	− 0.134280i
$515$	5.62889e7	0.412099
$516$	0	0
$517$	1.43078e6i	0.0103538i
$518$	0	0
$519$	0	0
$520$	4.96380e7	0.353024
$521$	7.73845e7i	0.547194i	0.961844	+	0.273597i	$0.0882134\pi$
$521$	7.73845e7i	0.547194i	−0.961844	+	0.273597i	$0.911787\pi$
$522$	0	0
$523$	1.34435e8i	0.939742i	0.882735	+	0.469871i	$0.155700\pi$
$523$	1.34435e8i	0.939742i	−0.882735	+	0.469871i	$0.844300\pi$
$524$	1.94809e8i	1.35399i
$525$	0	0
$526$	−1.33731e8	−0.918913
$527$	1.93487e6	0.0132196
$528$	0	0
$529$	6.07032e7	0.410057
$530$	− 6.69603e7i	− 0.449769i
$531$	0	0
$532$	0	0
$533$	−1.21084e8	−0.799657
$534$	0	0
$535$	− 5.26907e7i	− 0.344090i
$536$	−1.06749e8	−0.693220
$537$	0	0
$538$	9.22511e7i	0.592413i
$539$	0	0
$540$	0	0
$541$	−1.50294e8	−0.949181	−0.474590	−	0.880207i	$-0.657404\pi$
$541$	−1.50294e8	−0.949181	−0.474590	+	0.880207i	$0.657404\pi$
$542$	− 1.14107e7i	− 0.0716663i
$543$	0	0
$544$	7.26263e6i	0.0451125i
$545$	− 1.05563e8i	− 0.652115i
$546$	0	0
$547$	−6.65431e7	−0.406576	−0.203288	−	0.979119i	$-0.565163\pi$
$547$	−6.65431e7	−0.406576	−0.203288	+	0.979119i	$0.565163\pi$
$548$	5.67671e7	0.344949
$549$	0	0
$550$	−1.02882e6	−0.00618377
$551$	− 4.09167e8i	− 2.44594i
$552$	0	0
$553$	0	0
$554$	1.37774e8	0.810284
$555$	0	0
$556$	5.22409e6i	0.0303939i
$557$	2.53685e8	1.46801	0.734007	−	0.679142i	$-0.237648\pi$
$557$	2.53685e8	1.46801	0.734007	+	0.679142i	$0.237648\pi$
$558$	0	0
$559$	7.82096e6i	0.0447739i
$560$	0	0
$561$	0	0
$562$	1.02336e8	0.576525
$563$	− 1.65721e8i	− 0.928652i	−0.885664	−	0.464326i	$-0.846297\pi$
$563$	− 1.65721e8i	− 0.928652i	0.885664	−	0.464326i	$-0.153703\pi$
$564$	0	0
$565$	9.51140e6i	0.0527350i
$566$	1.84685e8i	1.01855i
$567$	0	0
$568$	1.69968e8	0.927516
$569$	−1.63978e8	−0.890122	−0.445061	−	0.895500i	$-0.646818\pi$
$569$	−1.63978e8	−0.890122	−0.445061	+	0.895500i	$0.646818\pi$
$570$	0	0
$571$	−3.40570e8	−1.82935	−0.914677	−	0.404184i	$-0.867555\pi$
$571$	−3.40570e8	−1.82935	−0.914677	+	0.404184i	$0.867555\pi$
$572$	1.39476e6i	0.00745268i
$573$	0	0
$574$	0	0
$575$	1.69731e8	0.892808
$576$	0	0
$577$	2.73797e8i	1.42528i	0.701528	+	0.712642i	$0.252502\pi$
$577$	2.73797e8i	1.42528i	−0.701528	+	0.712642i	$0.747498\pi$
$578$	1.08566e8	0.562224
$579$	0	0
$580$	1.17029e8i	0.599804i
$581$	0	0
$582$	0	0
$583$	4.63750e6	0.0234034
$584$	9.78459e7i	0.491252i
$585$	0	0
$586$	− 1.32639e8i	− 0.659139i
$587$	− 2.96429e8i	− 1.46557i	−0.680460	−	0.732785i	$-0.738220\pi$
$587$	− 2.96429e8i	− 1.46557i	0.680460	−	0.732785i	$-0.261780\pi$
$588$	0	0
$589$	8.56661e7	0.419241
$590$	−9.16328e7	−0.446164
$591$	0	0
$592$	−2.01863e7	−0.0972954
$593$	− 1.29937e8i	− 0.623114i	−0.950227	−	0.311557i	$-0.899149\pi$
$593$	− 1.29937e8i	− 0.623114i	0.950227	−	0.311557i	$-0.100851\pi$
$594$	0	0
$595$	0	0
$596$	2.73439e8	1.29158
$597$	0	0
$598$	1.06949e8i	0.500119i
$599$	7.64852e7	0.355875	0.177937	−	0.984042i	$-0.443058\pi$
$599$	7.64852e7	0.355875	0.177937	+	0.984042i	$0.443058\pi$
$600$	0	0
$601$	− 1.12466e8i	− 0.518080i	−0.965867	−	0.259040i	$-0.916594\pi$
$601$	− 1.12466e8i	− 0.518080i	0.965867	−	0.259040i	$-0.0834061\pi$
$602$	0	0
$603$	0	0
$604$	6.58853e7	0.299004
$605$	− 1.10285e8i	− 0.498026i
$606$	0	0
$607$	4.65527e6i	0.0208151i	0.999946	+	0.0104076i	$0.00331289\pi$
$607$	4.65527e6i	0.0208151i	−0.999946	+	0.0104076i	$0.996687\pi$
$608$	3.21553e8i	1.43068i
$609$	0	0
$610$	1.13570e8	0.500351
$611$	1.20939e8	0.530203
$612$	0	0
$613$	−2.05537e8	−0.892294	−0.446147	−	0.894960i	$-0.647204\pi$
$613$	−2.05537e8	−0.892294	−0.446147	+	0.894960i	$0.647204\pi$
$614$	2.10582e7i	0.0909738i
$615$	0	0
$616$	0	0
$617$	−1.85696e8	−0.790583	−0.395292	−	0.918556i	$-0.629356\pi$
$617$	−1.85696e8	−0.790583	−0.395292	+	0.918556i	$0.629356\pi$
$618$	0	0
$619$	8.77166e7i	0.369837i	0.982754	+	0.184918i	$0.0592020\pi$
$619$	8.77166e7i	0.369837i	−0.982754	+	0.184918i	$0.940798\pi$
$620$	−2.45020e7	−0.102808
$621$	0	0
$622$	− 1.44346e8i	− 0.599837i
$623$	0	0
$624$	0	0
$625$	7.74328e7	0.317165
$626$	− 9.82454e7i	− 0.400488i
$627$	0	0
$628$	− 1.08350e8i	− 0.437473i
$629$	− 7.11735e6i	− 0.0286000i
$630$	0	0
$631$	−4.15732e8	−1.65472	−0.827362	−	0.561670i	$-0.810159\pi$
$631$	−4.15732e8	−1.65472	−0.827362	+	0.561670i	$0.810159\pi$
$632$	−1.91934e8	−0.760326
$633$	0	0
$634$	−4.74291e7	−0.186113
$635$	− 1.75663e8i	− 0.686054i
$636$	0	0
$637$	0	0
$638$	3.76718e6	0.0145062
$639$	0	0
$640$	− 1.02912e8i	− 0.392579i
$641$	−1.79720e8	−0.682374	−0.341187	−	0.939995i	$-0.610829\pi$
$641$	−1.79720e8	−0.682374	−0.341187	+	0.939995i	$0.610829\pi$
$642$	0	0
$643$	8.10561e7i	0.304897i	0.988311	+	0.152448i	$0.0487158\pi$
$643$	8.10561e7i	0.304897i	−0.988311	+	0.152448i	$0.951284\pi$
$644$	0	0
$645$	0	0
$646$	−9.20883e6	−0.0341592
$647$	− 1.04644e8i	− 0.386369i	−0.981162	−	0.193185i	$-0.938118\pi$
$647$	− 1.04644e8i	− 0.386369i	0.981162	−	0.193185i	$-0.0618816\pi$
$648$	0	0
$649$	− 6.34625e6i	− 0.0232158i
$650$	8.69630e7i	0.316661i
$651$	0	0
$652$	−2.87941e8	−1.03887
$653$	−4.72357e8	−1.69641	−0.848206	−	0.529667i	$-0.822317\pi$
$653$	−4.72357e8	−1.69641	−0.848206	+	0.529667i	$0.822317\pi$
$654$	0	0
$655$	2.77625e8	0.987949
$656$	4.49146e7i	0.159102i
$657$	0	0
$658$	0	0
$659$	−1.73758e8	−0.607141	−0.303570	−	0.952809i	$-0.598179\pi$
$659$	−1.73758e8	−0.607141	−0.303570	+	0.952809i	$0.598179\pi$
$660$	0	0
$661$	− 2.21673e7i	− 0.0767553i	−0.999263	−	0.0383776i	$-0.987781\pi$
$661$	− 2.21673e7i	− 0.0767553i	0.999263	−	0.0383776i	$-0.0122190\pi$
$662$	1.83386e8	0.632109
$663$	0	0
$664$	6.42008e6i	0.0219299i
$665$	0	0
$666$	0	0
$667$	−6.21493e8	−2.09440
$668$	− 2.38126e8i	− 0.798872i
$669$	0	0
$670$	6.17214e7i	0.205216i
$671$	7.86557e6i	0.0260353i
$672$	0	0
$673$	−4.00045e8	−1.31239	−0.656195	−	0.754591i	$-0.727835\pi$
$673$	−4.00045e8	−1.31239	−0.656195	+	0.754591i	$0.727835\pi$
$674$	−1.82626e8	−0.596461
$675$	0	0
$676$	−9.29997e7	−0.301052
$677$	− 2.90088e8i	− 0.934898i	−0.884020	−	0.467449i	$-0.845173\pi$
$677$	− 2.90088e8i	− 0.934898i	0.884020	−	0.467449i	$-0.154827\pi$
$678$	0	0
$679$	0	0
$680$	6.49199e6	0.0206467
$681$	0	0
$682$	788723.i	0.00248640i
$683$	−1.15403e8	−0.362205	−0.181102	−	0.983464i	$-0.557967\pi$
$683$	−1.15403e8	−0.362205	−0.181102	+	0.983464i	$0.557967\pi$
$684$	0	0
$685$	− 8.08996e7i	− 0.251695i
$686$	0	0
$687$	0	0
$688$	2.90110e6	0.00890836
$689$	− 3.91992e8i	− 1.19845i
$690$	0	0
$691$	− 4.14900e8i	− 1.25750i	−0.777606	−	0.628752i	$-0.783566\pi$
$691$	− 4.14900e8i	− 1.25750i	0.777606	−	0.628752i	$-0.216434\pi$
$692$	− 1.20654e8i	− 0.364101i
$693$	0	0
$694$	7.45157e7	0.222930
$695$	7.44493e6	0.0221772
$696$	0	0
$697$	−1.58361e7	−0.0467682
$698$	2.28569e7i	0.0672127i
$699$	0	0
$700$	0	0
$701$	−5.13427e8	−1.49047	−0.745237	−	0.666800i	$-0.767664\pi$
$701$	−5.13427e8	−1.49047	−0.745237	+	0.666800i	$0.767664\pi$
$702$	0	0
$703$	− 3.15120e8i	− 0.907006i
$704$	−2.20268e6	−0.00631295
$705$	0	0
$706$	2.05193e8i	0.583107i
$707$	0	0
$708$	0	0
$709$	3.28399e8	0.921432	0.460716	−	0.887547i	$-0.347593\pi$
$709$	3.28399e8	0.921432	0.460716	+	0.887547i	$0.347593\pi$
$710$	− 9.82735e7i	− 0.274575i
$711$	0	0
$712$	1.11774e8i	0.309670i
$713$	− 1.30120e8i	− 0.358985i
$714$	0	0
$715$	1.98770e6	0.00543791
$716$	−6.50022e7	−0.177088
$717$	0	0
$718$	1.80333e8	0.487194
$719$	1.13124e8i	0.304347i	0.988354	+	0.152174i	$0.0486273\pi$
$719$	1.13124e8i	0.304347i	−0.988354	+	0.152174i	$0.951373\pi$
$720$	0	0
$721$	0	0
$722$	−1.95713e8	−0.520006
$723$	0	0
$724$	− 2.62887e8i	− 0.692713i
$725$	−5.05352e8	−1.32611
$726$	0	0
$727$	9.97740e7i	0.259666i	0.991536	+	0.129833i	$0.0414440\pi$
$727$	9.97740e7i	0.259666i	−0.991536	+	0.129833i	$0.958556\pi$
$728$	0	0
$729$	0	0
$730$	5.65734e7	0.145427
$731$	1.02288e6i	0.00261861i
$732$	0	0
$733$	− 3.62533e8i	− 0.920524i	−0.887783	−	0.460262i	$-0.847755\pi$
$733$	− 3.62533e8i	− 0.920524i	0.887783	−	0.460262i	$-0.152245\pi$
$734$	7.99975e7i	0.202296i
$735$	0	0
$736$	4.88414e8	1.22505
$737$	−4.27466e6	−0.0106782
$738$	0	0
$739$	−2.04280e8	−0.506166	−0.253083	−	0.967445i	$-0.581445\pi$
$739$	−2.04280e8	−0.506166	−0.253083	+	0.967445i	$0.581445\pi$
$740$	9.01300e7i	0.222420i
$741$	0	0
$742$	0	0
$743$	−2.63900e8	−0.643389	−0.321695	−	0.946844i	$-0.604252\pi$
$743$	−2.63900e8	−0.643389	−0.321695	+	0.946844i	$0.604252\pi$
$744$	0	0
$745$	− 3.89682e8i	− 0.942413i
$746$	−2.43157e7	−0.0585694
$747$	0	0
$748$	182416.i	0 0.000435872i
$749$	0	0
$750$	0	0
$751$	−7.37876e8	−1.74206	−0.871031	−	0.491228i	$-0.836548\pi$
$751$	−7.37876e8	−1.74206	−0.871031	+	0.491228i	$0.836548\pi$
$752$	− 4.48610e7i	− 0.105491i
$753$	0	0
$754$	− 3.18427e8i	− 0.742840i
$755$	− 9.38940e7i	− 0.218171i
$756$	0	0
$757$	−1.43307e8	−0.330355	−0.165177	−	0.986264i	$-0.552820\pi$
$757$	−1.43307e8	−0.330355	−0.165177	+	0.986264i	$0.552820\pi$
$758$	1.56753e8	0.359922
$759$	0	0
$760$	2.87432e8	0.654779
$761$	− 7.70680e8i	− 1.74872i	−0.485278	−	0.874360i	$-0.661282\pi$
$761$	− 7.70680e8i	− 1.74872i	0.485278	−	0.874360i	$-0.338718\pi$
$762$	0	0
$763$	0	0
$764$	2.79781e8	0.627391
$765$	0	0
$766$	− 2.54974e8i	− 0.567296i
$767$	−5.36427e8	−1.18884
$768$	0	0
$769$	− 6.97335e8i	− 1.53343i	−0.641990	−	0.766713i	$-0.721891\pi$
$769$	− 6.97335e8i	− 1.53343i	0.641990	−	0.766713i	$-0.278109\pi$
$770$	0	0
$771$	0	0
$772$	−4.30449e8	−0.935557
$773$	8.13388e8i	1.76100i	0.474046	+	0.880500i	$0.342793\pi$
$773$	8.13388e8i	1.76100i	−0.474046	+	0.880500i	$0.657207\pi$
$774$	0	0
$775$	− 1.05804e8i	− 0.227299i
$776$	3.21523e8i	0.688061i
$777$	0	0
$778$	1.17202e8	0.248883
$779$	−7.01143e8	−1.48318
$780$	0	0
$781$	6.80616e6	0.0142873
$782$	1.39875e7i	0.0292496i
$783$	0	0
$784$	0	0
$785$	−1.54411e8	−0.319206
$786$	0	0
$787$	3.72885e8i	0.764982i	0.923959	+	0.382491i	$0.124934\pi$
$787$	3.72885e8i	0.764982i	−0.923959	+	0.382491i	$0.875066\pi$
$788$	1.04373e8	0.213310
$789$	0	0
$790$	1.10974e8i	0.225081i
$791$	0	0
$792$	0	0
$793$	6.64850e8	1.33323
$794$	6.14220e7i	0.122705i
$795$	0	0
$796$	6.86658e8i	1.36145i
$797$	4.15047e8i	0.819827i	0.912124	+	0.409913i	$0.134441\pi$
$797$	4.15047e8i	0.819827i	−0.912124	+	0.409913i	$0.865559\pi$
$798$	0	0
$799$	1.58172e7	0.0310091
$800$	3.97142e8	0.775668
$801$	0	0
$802$	1.81094e8	0.351059
$803$	3.91813e6i	0.00756715i
$804$	0	0
$805$	0	0
$806$	6.66681e7	0.127325
$807$	0	0
$808$	9.11213e8i	1.72737i
$809$	2.34767e8	0.443396	0.221698	−	0.975115i	$-0.428840\pi$
$809$	2.34767e8	0.443396	0.221698	+	0.975115i	$0.428840\pi$
$810$	0	0
$811$	− 8.08164e7i	− 0.151508i	−0.997127	−	0.0757542i	$-0.975864\pi$
$811$	− 8.08164e7i	− 0.151508i	0.997127	−	0.0757542i	$-0.0241364\pi$
$812$	0	0
$813$	0	0
$814$	2.90129e6	0.00537921
$815$	4.10348e8i	0.758019i
$816$	0	0
$817$	4.52879e7i	0.0830454i
$818$	3.41133e7i	0.0623252i
$819$	0	0
$820$	2.00539e8	0.363713
$821$	6.45128e7	0.116578	0.0582890	−	0.998300i	$-0.481436\pi$
$821$	6.45128e7	0.116578	0.0582890	+	0.998300i	$0.481436\pi$
$822$	0	0
$823$	−1.56722e8	−0.281145	−0.140572	−	0.990070i	$-0.544894\pi$
$823$	−1.56722e8	−0.281145	−0.140572	+	0.990070i	$0.544894\pi$
$824$	4.38716e8i	0.784155i
$825$	0	0
$826$	0	0
$827$	6.46564e8	1.14313	0.571565	−	0.820557i	$-0.306337\pi$
$827$	6.46564e8	1.14313	0.571565	+	0.820557i	$0.306337\pi$
$828$	0	0
$829$	− 9.37160e8i	− 1.64494i	−0.568809	−	0.822470i	$-0.692596\pi$
$829$	− 9.37160e8i	− 1.64494i	0.568809	−	0.822470i	$-0.307404\pi$
$830$	3.71202e6	0.00649197
$831$	0	0
$832$	1.86185e8i	0.323277i
$833$	0	0
$834$	0	0
$835$	−3.39356e8	−0.582904
$836$	8.07648e6i	0.0138230i
$837$	0	0
$838$	1.40786e8i	0.239237i
$839$	− 7.25284e8i	− 1.22807i	−0.789280	−	0.614034i	$-0.789546\pi$
$839$	− 7.25284e8i	− 1.22807i	0.789280	−	0.614034i	$-0.210454\pi$
$840$	0	0
$841$	1.25559e9	2.11086
$842$	2.69924e8	0.452174
$843$	0	0
$844$	−6.41139e7	−0.106641
$845$	1.32535e8i	0.219665i
$846$	0	0
$847$	0	0
$848$	−1.45405e8	−0.238447
$849$	0	0
$850$	1.13736e7i	0.0185200i
$851$	−4.78643e8	−0.776646
$852$	0	0
$853$	− 1.38063e8i	− 0.222449i	−0.993795	−	0.111224i	$-0.964523\pi$
$853$	− 1.38063e8i	− 0.222449i	0.993795	−	0.111224i	$-0.0354772\pi$
$854$	0	0
$855$	0	0
$856$	4.10671e8	0.654747
$857$	− 7.06952e8i	− 1.12318i	−0.827417	−	0.561588i	$-0.810191\pi$
$857$	− 7.06952e8i	− 1.12318i	0.827417	−	0.561588i	$-0.189809\pi$
$858$	0	0
$859$	− 3.29684e8i	− 0.520138i	−0.965590	−	0.260069i	$-0.916255\pi$
$859$	− 3.29684e8i	− 0.520138i	0.965590	−	0.260069i	$-0.0837453\pi$
$860$	− 1.29531e7i	− 0.0203648i
$861$	0	0
$862$	−4.33976e8	−0.677554
$863$	9.68305e8	1.50654	0.753268	−	0.657713i	$-0.228476\pi$
$863$	9.68305e8	1.50654	0.753268	+	0.657713i	$0.228476\pi$
$864$	0	0
$865$	−1.71945e8	−0.265670
$866$	− 5.01051e8i	− 0.771486i
$867$	0	0
$868$	0	0
$869$	−7.68577e6	−0.0117119
$870$	0	0
$871$	3.61323e8i	0.546816i
$872$	8.22762e8	1.24087
$873$	0	0
$874$	6.19296e8i	0.927608i
$875$	0	0
$876$	0	0
$877$	2.66765e8	0.395485	0.197742	−	0.980254i	$-0.436639\pi$
$877$	2.66765e8	0.395485	0.197742	+	0.980254i	$0.436639\pi$
$878$	4.11349e8i	0.607753i
$879$	0	0
$880$	− 737315.i	− 0.00108194i
$881$	6.08909e8i	0.890481i	0.895411	+	0.445241i	$0.146882\pi$
$881$	6.08909e8i	0.890481i	−0.895411	+	0.445241i	$0.853118\pi$
$882$	0	0
$883$	−6.56488e8	−0.953553	−0.476777	−	0.879025i	$-0.658195\pi$
$883$	−6.56488e8	−0.953553	−0.476777	+	0.879025i	$0.658195\pi$
$884$	1.54190e7	0.0223203
$885$	0	0
$886$	4.79153e8	0.688927
$887$	− 6.97665e7i	− 0.0999714i	−0.998750	−	0.0499857i	$-0.984082\pi$
$887$	− 6.97665e7i	− 0.0999714i	0.998750	−	0.0499857i	$-0.0159176\pi$
$888$	0	0
$889$	0	0
$890$	6.46263e7	0.0916726
$891$	0	0
$892$	− 1.58752e8i	− 0.223679i
$893$	7.00306e8	0.983407
$894$	0	0
$895$	9.26355e7i	0.129214i
$896$	0	0
$897$	0	0
$898$	−3.48896e8	−0.481800
$899$	3.87416e8i	0.533210i
$900$	0	0
$901$	− 5.12673e7i	− 0.0700916i
$902$	− 6.45539e6i	− 0.00879635i
$903$	0	0
$904$	−7.41319e7	−0.100346
$905$	−3.74644e8	−0.505444
$906$	0	0
$907$	8.62722e8	1.15624	0.578122	−	0.815951i	$-0.303786\pi$
$907$	8.62722e8	1.15624	0.578122	+	0.815951i	$0.303786\pi$
$908$	2.59382e8i	0.346483i
$909$	0	0
$910$	0	0
$911$	1.41296e8	0.186885	0.0934425	−	0.995625i	$-0.470213\pi$
$911$	1.41296e8	0.186885	0.0934425	+	0.995625i	$0.470213\pi$
$912$	0	0
$913$	257085.i	0 0.000337804i
$914$	8.56639e7	0.112191
$915$	0	0
$916$	5.34605e8i	0.695579i
$917$	0	0
$918$	0	0
$919$	1.11969e9	1.44262	0.721309	−	0.692613i	$-0.243541\pi$
$919$	1.11969e9	1.44262	0.721309	+	0.692613i	$0.243541\pi$
$920$	− 4.36588e8i	− 0.560671i
$921$	0	0
$922$	4.76367e8i	0.607783i
$923$	− 5.75302e8i	− 0.731629i
$924$	0	0
$925$	−3.89197e8	−0.491750
$926$	−1.27176e8	−0.160167
$927$	0	0
$928$	−1.45419e9	−1.81960
$929$	1.31899e9i	1.64511i	0.568689	+	0.822553i	$0.307451\pi$
$929$	1.31899e9i	1.64511i	−0.568689	+	0.822553i	$0.692549\pi$
$930$	0	0
$931$	0	0
$932$	5.29591e8	0.654173
$933$	0	0
$934$	7.78975e8i	0.956055i
$935$	259964.	0.000318038 0
$936$	0	0
$937$	1.04041e6i	0.00126469i	1.00000		0.000632345i	$0.000201282\pi$
$937$	1.04041e6i	0.00126469i	−1.00000		0.000632345i	$0.999799\pi$
$938$	0	0
$939$	0	0
$940$	−2.00300e8	−0.241155
$941$	− 6.85709e8i	− 0.822946i	−0.911422	−	0.411473i	$-0.865014\pi$
$941$	− 6.85709e8i	− 0.822946i	0.911422	−	0.411473i	$-0.134986\pi$
$942$	0	0
$943$	1.06498e9i	1.27001i
$944$	1.98982e8i	0.236536i
$945$	0	0
$946$	−416963.	−0.000492520 0
$947$	−5.34076e8	−0.628859	−0.314429	−	0.949281i	$-0.601813\pi$
$947$	−5.34076e8	−0.628859	−0.314429	+	0.949281i	$0.601813\pi$
$948$	0	0
$949$	3.31186e8	0.387502
$950$	5.03566e8i	0.587334i
$951$	0	0
$952$	0	0
$953$	−8.66817e8	−1.00150	−0.500748	−	0.865593i	$-0.666942\pi$
$953$	−8.66817e8	−1.00150	−0.500748	+	0.865593i	$0.666942\pi$
$954$	0	0
$955$	− 3.98720e8i	− 0.457781i
$956$	5.20008e8	0.595164
$957$	0	0
$958$	1.25826e8i	0.143112i
$959$	0	0
$960$	0	0
$961$	8.06392e8	0.908606
$962$	− 2.45237e8i	− 0.275461i
$963$	0	0
$964$	4.34951e8i	0.485522i
$965$	6.13440e8i	0.682637i
$966$	0	0
$967$	1.71422e8	0.189577	0.0947886	−	0.995497i	$-0.469782\pi$
$967$	1.71422e8	0.189577	0.0947886	+	0.995497i	$0.469782\pi$
$968$	8.59565e8	0.947660
$969$	0	0
$970$	1.85901e8	0.203689
$971$	1.12846e9i	1.23262i	0.787506	+	0.616308i	$0.211372\pi$
$971$	1.12846e9i	1.23262i	−0.787506	+	0.616308i	$0.788628\pi$
$972$	0	0
$973$	0	0
$974$	2.42259e8	0.262183
$975$	0	0
$976$	− 2.46619e8i	− 0.265263i
$977$	4.19418e8	0.449742	0.224871	−	0.974389i	$-0.427804\pi$
$977$	4.19418e8	0.449742	0.224871	+	0.974389i	$0.427804\pi$
$978$	0	0
$979$	4.47585e6i	0.00477010i
$980$	0	0
$981$	0	0
$982$	−4.21289e8	−0.444883
$983$	6.92053e8i	0.728582i	0.931285	+	0.364291i	$0.118689\pi$
$983$	6.92053e8i	0.728582i	−0.931285	+	0.364291i	$0.881311\pi$
$984$	0	0
$985$	− 1.48744e8i	− 0.155643i
$986$	− 4.16459e7i	− 0.0434452i
$987$	0	0
$988$	6.82678e8	0.707856
$989$	6.87888e7	0.0711097
$990$	0	0
$991$	−8.26871e8	−0.849605	−0.424802	−	0.905286i	$-0.639656\pi$
$991$	−8.26871e8	−0.849605	−0.424802	+	0.905286i	$0.639656\pi$
$992$	− 3.04459e8i	− 0.311885i
$993$	0	0
$994$	0	0
$995$	9.78566e8	0.993393
$996$	0	0
$997$	1.71699e9i	1.73254i	0.499579	+	0.866269i	$0.333488\pi$
$997$	1.71699e9i	1.73254i	−0.499579	+	0.866269i	$0.666512\pi$
$998$	2.26561e7	0.0227925
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.7.d.c.244.6		8
3.2	odd	2		147.7.d.b.97.3		8
7.4	even	3		63.7.m.d.19.2		8
7.5	odd	6		63.7.m.d.10.2		8
7.6	odd	2	inner	441.7.d.c.244.5		8
21.2	odd	6		147.7.f.d.31.3		8
21.5	even	6		21.7.f.a.10.3	✓	8
21.11	odd	6		21.7.f.a.19.3	yes	8
21.17	even	6		147.7.f.d.19.3		8
21.20	even	2		147.7.d.b.97.4		8
84.11	even	6		336.7.bh.d.145.4		8
84.47	odd	6		336.7.bh.d.241.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.f.a.10.3	✓	8	21.5	even	6
21.7.f.a.19.3	yes	8	21.11	odd	6
63.7.m.d.10.2		8	7.5	odd	6
63.7.m.d.19.2		8	7.4	even	3
147.7.d.b.97.3		8	3.2	odd	2
147.7.d.b.97.4		8	21.20	even	2
147.7.f.d.19.3		8	21.17	even	6
147.7.f.d.31.3		8	21.2	odd	6
336.7.bh.d.145.4		8	84.11	even	6
336.7.bh.d.241.4		8	84.47	odd	6
441.7.d.c.244.5		8	7.6	odd	2	inner
441.7.d.c.244.6		8	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.50641 q^{2} -43.6923 q^{4} +62.2665i q^{5} -485.305 q^{8} +O(q^{10})\)	\(q+4.50641 q^{2} -43.6923 q^{4} +62.2665i q^{5} -485.305 q^{8} +280.598i q^{10} -19.4335 q^{11} +1642.65i q^{13} +609.323 q^{16} +214.836i q^{17} +9511.88i q^{19} -2720.57i q^{20} -87.5753 q^{22} +14447.8 q^{23} +11747.9 q^{25} +7402.44i q^{26} -43016.4 q^{29} -9006.23i q^{31} +33805.4 q^{32} +968.141i q^{34} -33129.1 q^{37} +42864.4i q^{38} -30218.3i q^{40} +73712.4i q^{41} +4761.19 q^{43} +849.094 q^{44} +65107.7 q^{46} -73624.3i q^{47} +52940.7 q^{50} -71771.1i q^{52} -238634. q^{53} -1210.06i q^{55} -193850. q^{58} +326562. i q^{59} -404743. i q^{61} -40585.7i q^{62} +113344. q^{64} -102282. q^{65} +219964. q^{67} -9386.70i q^{68} -350228. q^{71} -201617. i q^{73} -149293. q^{74} -415596. i q^{76} +395490. q^{79} +37940.4i q^{80} +332178. i q^{82} -13229.0i q^{83} -13377.1 q^{85} +21455.9 q^{86} +9431.18 q^{88} -230316. i q^{89} -631258. q^{92} -331781. i q^{94} -592271. q^{95} -662517. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8}+O(q^{10}) \) 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8	\( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95}+O(q^{100}) \) 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8 - 628 * q^11 + 25442 * q^16 + 86106 * q^22 + 7856 * q^23 + 34076 * q^25 + 8300 * q^29 - 372414 * q^32 - 129412 * q^37 + 45740 * q^43 + 185058 * q^44 + 223008 * q^46 - 967216 * q^50 - 1081948 * q^53 - 1079598 * q^58 + 2378626 * q^64 - 828408 * q^65 + 2317804 * q^67 - 1442344 * q^71 - 865880 * q^74 - 1222904 * q^79 - 275112 * q^85 + 1632448 * q^86 + 732882 * q^88 - 678720 * q^92 - 1183584 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more