LMFDB - Embedded newform 882.2.d.a.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(881,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("882.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.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:	$7.04280545828$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	882.881
Dual form			882.2.d.a.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.18154 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -4.18154 q^{5} +1.00000i q^{8} +4.18154i q^{10} -3.00000i q^{11} +2.44949i q^{13} +1.00000 q^{16} +1.01461 q^{17} +1.01461i q^{19} +4.18154 q^{20} -3.00000 q^{22} +4.24264i q^{23} +12.4853 q^{25} +2.44949 q^{26} +1.24264i q^{29} +5.61642i q^{31} -1.00000i q^{32} -1.01461i q^{34} +8.24264 q^{37} +1.01461 q^{38} -4.18154i q^{40} +2.02922 q^{41} +8.24264 q^{43} +3.00000i q^{44} +4.24264 q^{46} +1.01461 q^{47} -12.4853i q^{50} -2.44949i q^{52} +1.24264i q^{53} +12.5446i q^{55} +1.24264 q^{58} +11.5300 q^{59} -5.91359i q^{61} +5.61642 q^{62} -1.00000 q^{64} -10.2426i q^{65} -10.0000 q^{67} -1.01461 q^{68} +10.2426i q^{71} +8.36308i q^{73} -8.24264i q^{74} -1.01461i q^{76} -11.2426 q^{79} -4.18154 q^{80} -2.02922i q^{82} -3.16693 q^{83} -4.24264 q^{85} -8.24264i q^{86} +3.00000 q^{88} -10.3923 q^{89} -4.24264i q^{92} -1.01461i q^{94} -4.24264i q^{95} +3.76127i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} - 24 q^{22} + 32 q^{25} + 32 q^{37} + 32 q^{43} - 24 q^{58} - 8 q^{64} - 80 q^{67} - 56 q^{79} + 24 q^{88}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 - 24 * q^22 + 32 * q^25 + 32 * q^37 + 32 * q^43 - 24 * q^58 - 8 * q^64 - 80 * q^67 - 56 * q^79 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−4.18154	−1.87004	−0.935021	−	0.354593i	$-0.884620\pi$
$5$	−4.18154	−1.87004	−0.935021	+	0.354593i	$0.884620\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	4.18154i	1.32232i
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	2.44949i	0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$	2.44949i	0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.01461	0.246080	0.123040	−	0.992402i	$-0.460736\pi$
$17$	1.01461	0.246080	0.123040	+	0.992402i	$0.460736\pi$
$18$	0	0
$19$	1.01461i	0.232768i	0.993204	+	0.116384i	$0.0371303\pi$
$19$	1.01461i	0.232768i	−0.993204	+	0.116384i	$0.962870\pi$
$20$	4.18154	0.935021
$21$	0	0
$22$	−3.00000	−0.639602
$23$	4.24264i	0.884652i	0.896854	+	0.442326i	$0.145847\pi$
$23$	4.24264i	0.884652i	−0.896854	+	0.442326i	$0.854153\pi$
$24$	0	0
$25$	12.4853	2.49706
$26$	2.44949	0.480384
$27$	0	0
$28$	0	0
$29$	1.24264i	0.230753i	0.993322	+	0.115376i	$0.0368074\pi$
$29$	1.24264i	0.230753i	−0.993322	+	0.115376i	$0.963193\pi$
$30$	0	0
$31$	5.61642i	1.00874i	0.863488	+	0.504369i	$0.168275\pi$
$31$	5.61642i	1.00874i	−0.863488	+	0.504369i	$0.831725\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 1.01461i	− 0.174005i
$35$	0	0
$36$	0	0
$37$	8.24264	1.35508	0.677541	−	0.735485i	$-0.263046\pi$
$37$	8.24264	1.35508	0.677541	+	0.735485i	$0.263046\pi$
$38$	1.01461	0.164592
$39$	0	0
$40$	− 4.18154i	− 0.661160i
$41$	2.02922	0.316912	0.158456	−	0.987366i	$-0.449348\pi$
$41$	2.02922	0.316912	0.158456	+	0.987366i	$0.449348\pi$
$42$	0	0
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	3.00000i	0.452267i
$45$	0	0
$46$	4.24264	0.625543
$47$	1.01461	0.147996	0.0739982	−	0.997258i	$-0.476424\pi$
$47$	1.01461	0.147996	0.0739982	+	0.997258i	$0.476424\pi$
$48$	0	0
$49$	0	0
$50$	− 12.4853i	− 1.76569i
$51$	0	0
$52$	− 2.44949i	− 0.339683i
$53$	1.24264i	0.170690i	0.996351	+	0.0853449i	$0.0271992\pi$
$53$	1.24264i	0.170690i	−0.996351	+	0.0853449i	$0.972801\pi$
$54$	0	0
$55$	12.5446i	1.69152i
$56$	0	0
$57$	0	0
$58$	1.24264	0.163167
$59$	11.5300	1.50108	0.750540	−	0.660825i	$-0.229794\pi$
$59$	11.5300	1.50108	0.750540	+	0.660825i	$0.229794\pi$
$60$	0	0
$61$	− 5.91359i	− 0.757158i	−0.925569	−	0.378579i	$-0.876413\pi$
$61$	− 5.91359i	− 0.757158i	0.925569	−	0.378579i	$-0.123587\pi$
$62$	5.61642	0.713286
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 10.2426i	− 1.27044i
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−1.01461	−0.123040
$69$	0	0
$70$	0	0
$71$	10.2426i	1.21558i	0.794099	+	0.607789i	$0.207943\pi$
$71$	10.2426i	1.21558i	−0.794099	+	0.607789i	$0.792057\pi$
$72$	0	0
$73$	8.36308i	0.978825i	0.872053	+	0.489412i	$0.162789\pi$
$73$	8.36308i	0.978825i	−0.872053	+	0.489412i	$0.837211\pi$
$74$	− 8.24264i	− 0.958188i
$75$	0	0
$76$	− 1.01461i	− 0.116384i
$77$	0	0
$78$	0	0
$79$	−11.2426	−1.26490	−0.632448	−	0.774603i	$-0.717950\pi$
$79$	−11.2426	−1.26490	−0.632448	+	0.774603i	$0.717950\pi$
$80$	−4.18154	−0.467510
$81$	0	0
$82$	− 2.02922i	− 0.224090i
$83$	−3.16693	−0.347616	−0.173808	−	0.984780i	$-0.555607\pi$
$83$	−3.16693	−0.347616	−0.173808	+	0.984780i	$0.555607\pi$
$84$	0	0
$85$	−4.24264	−0.460179
$86$	− 8.24264i	− 0.888827i
$87$	0	0
$88$	3.00000	0.319801
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	0	0
$92$	− 4.24264i	− 0.442326i
$93$	0	0
$94$	− 1.01461i	− 0.104649i
$95$	− 4.24264i	− 0.435286i
$96$	0	0
$97$	3.76127i	0.381900i	0.981600	+	0.190950i	$0.0611568\pi$
$97$	3.76127i	0.381900i	−0.981600	+	0.190950i	$0.938843\pi$
$98$	0	0
$99$	0	0
$100$	−12.4853	−1.24853
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	15.2913i	1.50670i	0.657623	+	0.753348i	$0.271562\pi$
$103$	15.2913i	1.50670i	−0.657623	+	0.753348i	$0.728438\pi$
$104$	−2.44949	−0.240192
$105$	0	0
$106$	1.24264	0.120696
$107$	− 5.48528i	− 0.530282i	−0.964210	−	0.265141i	$-0.914581\pi$
$107$	− 5.48528i	− 0.530282i	0.964210	−	0.265141i	$-0.0854185\pi$
$108$	0	0
$109$	1.51472	0.145084	0.0725419	−	0.997365i	$-0.476889\pi$
$109$	1.51472	0.145084	0.0725419	+	0.997365i	$0.476889\pi$
$110$	12.5446	1.19608
$111$	0	0
$112$	0	0
$113$	8.48528i	0.798228i	0.916901	+	0.399114i	$0.130682\pi$
$113$	8.48528i	0.798228i	−0.916901	+	0.399114i	$0.869318\pi$
$114$	0	0
$115$	− 17.7408i	− 1.65434i
$116$	− 1.24264i	− 0.115376i
$117$	0	0
$118$	− 11.5300i	− 1.06142i
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	−5.91359	−0.535391
$123$	0	0
$124$	− 5.61642i	− 0.504369i
$125$	−31.3000	−2.79956
$126$	0	0
$127$	−5.24264	−0.465209	−0.232605	−	0.972571i	$-0.574725\pi$
$127$	−5.24264	−0.465209	−0.232605	+	0.972571i	$0.574725\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−10.2426	−0.898339
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	0	0
$134$	10.0000i	0.863868i
$135$	0	0
$136$	1.01461i	0.0870023i
$137$	14.4853i	1.23756i	0.785564	+	0.618781i	$0.212373\pi$
$137$	14.4853i	1.23756i	−0.785564	+	0.618781i	$0.787627\pi$
$138$	0	0
$139$	− 20.1903i	− 1.71252i	−0.516549	−	0.856258i	$-0.672783\pi$
$139$	− 20.1903i	− 1.71252i	0.516549	−	0.856258i	$-0.327217\pi$
$140$	0	0
$141$	0	0
$142$	10.2426	0.859543
$143$	7.34847	0.614510
$144$	0	0
$145$	− 5.19615i	− 0.431517i
$146$	8.36308	0.692134
$147$	0	0
$148$	−8.24264	−0.677541
$149$	20.4853i	1.67822i	0.543962	+	0.839110i	$0.316924\pi$
$149$	20.4853i	1.67822i	−0.543962	+	0.839110i	$0.683076\pi$
$150$	0	0
$151$	−3.24264	−0.263882	−0.131941	−	0.991258i	$-0.542121\pi$
$151$	−3.24264	−0.263882	−0.131941	+	0.991258i	$0.542121\pi$
$152$	−1.01461	−0.0822959
$153$	0	0
$154$	0	0
$155$	− 23.4853i	− 1.88638i
$156$	0	0
$157$	− 14.6969i	− 1.17294i	−0.809970	−	0.586472i	$-0.800517\pi$
$157$	− 14.6969i	− 1.17294i	0.809970	−	0.586472i	$-0.199483\pi$
$158$	11.2426i	0.894416i
$159$	0	0
$160$	4.18154i	0.330580i
$161$	0	0
$162$	0	0
$163$	6.24264	0.488961	0.244481	−	0.969654i	$-0.421383\pi$
$163$	6.24264	0.488961	0.244481	+	0.969654i	$0.421383\pi$
$164$	−2.02922	−0.158456
$165$	0	0
$166$	3.16693i	0.245801i
$167$	23.0600	1.78444	0.892219	−	0.451603i	$-0.149148\pi$
$167$	23.0600	1.78444	0.892219	+	0.451603i	$0.149148\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	4.24264i	0.325396i
$171$	0	0
$172$	−8.24264	−0.628495
$173$	20.7846	1.58022	0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	0	0
$175$	0	0
$176$	− 3.00000i	− 0.226134i
$177$	0	0
$178$	10.3923i	0.778936i
$179$	9.51472i	0.711163i	0.934645	+	0.355582i	$0.115717\pi$
$179$	9.51472i	0.711163i	−0.934645	+	0.355582i	$0.884283\pi$
$180$	0	0
$181$	2.02922i	0.150831i	0.997152	+	0.0754155i	$0.0240283\pi$
$181$	2.02922i	0.150831i	−0.997152	+	0.0754155i	$0.975972\pi$
$182$	0	0
$183$	0	0
$184$	−4.24264	−0.312772
$185$	−34.4669	−2.53406
$186$	0	0
$187$	− 3.04384i	− 0.222587i
$188$	−1.01461	−0.0739982
$189$	0	0
$190$	−4.24264	−0.307794
$191$	8.48528i	0.613973i	0.951714	+	0.306987i	$0.0993207\pi$
$191$	8.48528i	0.613973i	−0.951714	+	0.306987i	$0.900679\pi$
$192$	0	0
$193$	−7.48528	−0.538802	−0.269401	−	0.963028i	$-0.586826\pi$
$193$	−7.48528	−0.538802	−0.269401	+	0.963028i	$0.586826\pi$
$194$	3.76127	0.270044
$195$	0	0
$196$	0	0
$197$	9.51472i	0.677896i	0.940805	+	0.338948i	$0.110071\pi$
$197$	9.51472i	0.677896i	−0.940805	+	0.338948i	$0.889929\pi$
$198$	0	0
$199$	− 16.1318i	− 1.14355i	−0.820409	−	0.571777i	$-0.806254\pi$
$199$	− 16.1318i	− 1.14355i	0.820409	−	0.571777i	$-0.193746\pi$
$200$	12.4853i	0.882843i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.48528	−0.592638
$206$	15.2913	1.06539
$207$	0	0
$208$	2.44949i	0.169842i
$209$	3.04384	0.210547
$210$	0	0
$211$	8.24264	0.567447	0.283723	−	0.958906i	$-0.408430\pi$
$211$	8.24264	0.567447	0.283723	+	0.958906i	$0.408430\pi$
$212$	− 1.24264i	− 0.0853449i
$213$	0	0
$214$	−5.48528	−0.374966
$215$	−34.4669	−2.35063
$216$	0	0
$217$	0	0
$218$	− 1.51472i	− 0.102590i
$219$	0	0
$220$	− 12.5446i	− 0.845758i
$221$	2.48528i	0.167178i
$222$	0	0
$223$	− 12.5446i	− 0.840050i	−0.907513	−	0.420025i	$-0.862021\pi$
$223$	− 12.5446i	− 0.840050i	0.907513	−	0.420025i	$-0.137979\pi$
$224$	0	0
$225$	0	0
$226$	8.48528	0.564433
$227$	−15.5885	−1.03464	−0.517321	−	0.855791i	$-0.673071\pi$
$227$	−15.5885	−1.03464	−0.517321	+	0.855791i	$0.673071\pi$
$228$	0	0
$229$	13.8564i	0.915657i	0.889041	+	0.457829i	$0.151373\pi$
$229$	13.8564i	0.915657i	−0.889041	+	0.457829i	$0.848627\pi$
$230$	−17.7408	−1.16979
$231$	0	0
$232$	−1.24264	−0.0815834
$233$	6.72792i	0.440761i	0.975414	+	0.220380i	$0.0707299\pi$
$233$	6.72792i	0.440761i	−0.975414	+	0.220380i	$0.929270\pi$
$234$	0	0
$235$	−4.24264	−0.276759
$236$	−11.5300	−0.750540
$237$	0	0
$238$	0	0
$239$	12.7279i	0.823301i	0.911342	+	0.411650i	$0.135048\pi$
$239$	12.7279i	0.823301i	−0.911342	+	0.411650i	$0.864952\pi$
$240$	0	0
$241$	17.0233i	1.09657i	0.836292	+	0.548285i	$0.184719\pi$
$241$	17.0233i	1.09657i	−0.836292	+	0.548285i	$0.815281\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	5.91359i	0.378579i
$245$	0	0
$246$	0	0
$247$	−2.48528	−0.158135
$248$	−5.61642	−0.356643
$249$	0	0
$250$	31.3000i	1.97959i
$251$	−17.6177	−1.11202	−0.556009	−	0.831176i	$-0.687668\pi$
$251$	−17.6177	−1.11202	−0.556009	+	0.831176i	$0.687668\pi$
$252$	0	0
$253$	12.7279	0.800198
$254$	5.24264i	0.328953i
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.0892	1.56502	0.782512	−	0.622636i	$-0.213938\pi$
$257$	25.0892	1.56502	0.782512	+	0.622636i	$0.213938\pi$
$258$	0	0
$259$	0	0
$260$	10.2426i	0.635222i
$261$	0	0
$262$	− 5.19615i	− 0.321019i
$263$	− 27.2132i	− 1.67804i	−0.544102	−	0.839019i	$-0.683130\pi$
$263$	− 27.2132i	− 1.67804i	0.544102	−	0.839019i	$-0.316870\pi$
$264$	0	0
$265$	− 5.19615i	− 0.319197i
$266$	0	0
$267$	0	0
$268$	10.0000	0.610847
$269$	−10.5154	−0.641135	−0.320568	−	0.947226i	$-0.603874\pi$
$269$	−10.5154	−0.641135	−0.320568	+	0.947226i	$0.603874\pi$
$270$	0	0
$271$	11.1097i	0.674869i	0.941349	+	0.337434i	$0.109559\pi$
$271$	11.1097i	0.674869i	−0.941349	+	0.337434i	$0.890441\pi$
$272$	1.01461	0.0615199
$273$	0	0
$274$	14.4853	0.875088
$275$	− 37.4558i	− 2.25867i
$276$	0	0
$277$	−20.9706	−1.26000	−0.630000	−	0.776596i	$-0.716945\pi$
$277$	−20.9706	−1.26000	−0.630000	+	0.776596i	$0.716945\pi$
$278$	−20.1903	−1.21093
$279$	0	0
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	− 6.50794i	− 0.386857i	−0.981114	−	0.193428i	$-0.938039\pi$
$283$	− 6.50794i	− 0.386857i	0.981114	−	0.193428i	$-0.0619607\pi$
$284$	− 10.2426i	− 0.607789i
$285$	0	0
$286$	− 7.34847i	− 0.434524i
$287$	0	0
$288$	0	0
$289$	−15.9706	−0.939445
$290$	−5.19615	−0.305129
$291$	0	0
$292$	− 8.36308i	− 0.489412i
$293$	4.18154	0.244288	0.122144	−	0.992512i	$-0.461023\pi$
$293$	4.18154	0.244288	0.122144	+	0.992512i	$0.461023\pi$
$294$	0	0
$295$	−48.2132	−2.80708
$296$	8.24264i	0.479094i
$297$	0	0
$298$	20.4853	1.18668
$299$	−10.3923	−0.601003
$300$	0	0
$301$	0	0
$302$	3.24264i	0.186593i
$303$	0	0
$304$	1.01461i	0.0581920i
$305$	24.7279i	1.41592i
$306$	0	0
$307$	− 24.6690i	− 1.40793i	−0.710233	−	0.703966i	$-0.751411\pi$
$307$	− 24.6690i	− 1.40793i	0.710233	−	0.703966i	$-0.248589\pi$
$308$	0	0
$309$	0	0
$310$	−23.4853	−1.33387
$311$	18.7554	1.06352	0.531760	−	0.846895i	$-0.321531\pi$
$311$	18.7554	1.06352	0.531760	+	0.846895i	$0.321531\pi$
$312$	0	0
$313$	1.13770i	0.0643069i	0.999483	+	0.0321534i	$0.0102365\pi$
$313$	1.13770i	0.0643069i	−0.999483	+	0.0321534i	$0.989763\pi$
$314$	−14.6969	−0.829396
$315$	0	0
$316$	11.2426	0.632448
$317$	7.24264i	0.406787i	0.979097	+	0.203394i	$0.0651971\pi$
$317$	7.24264i	0.406787i	−0.979097	+	0.203394i	$0.934803\pi$
$318$	0	0
$319$	3.72792	0.208724
$320$	4.18154	0.233755
$321$	0	0
$322$	0	0
$323$	1.02944i	0.0572794i
$324$	0	0
$325$	30.5826i	1.69642i
$326$	− 6.24264i	− 0.345748i
$327$	0	0
$328$	2.02922i	0.112045i
$329$	0	0
$330$	0	0
$331$	17.4558	0.959460	0.479730	−	0.877416i	$-0.340735\pi$
$331$	17.4558	0.959460	0.479730	+	0.877416i	$0.340735\pi$
$332$	3.16693	0.173808
$333$	0	0
$334$	− 23.0600i	− 1.26179i
$335$	41.8154	2.28462
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	− 7.00000i	− 0.380750i
$339$	0	0
$340$	4.24264	0.230089
$341$	16.8493	0.912438
$342$	0	0
$343$	0	0
$344$	8.24264i	0.444413i
$345$	0	0
$346$	− 20.7846i	− 1.11739i
$347$	− 14.4853i	− 0.777611i	−0.921320	−	0.388805i	$-0.872888\pi$
$347$	− 14.4853i	− 0.777611i	0.921320	−	0.388805i	$-0.127112\pi$
$348$	0	0
$349$	36.9164i	1.97609i	0.154163	+	0.988045i	$0.450732\pi$
$349$	36.9164i	1.97609i	−0.154163	+	0.988045i	$0.549268\pi$
$350$	0	0
$351$	0	0
$352$	−3.00000	−0.159901
$353$	18.7554	0.998248	0.499124	−	0.866530i	$-0.333655\pi$
$353$	18.7554	0.998248	0.499124	+	0.866530i	$0.333655\pi$
$354$	0	0
$355$	− 42.8300i	− 2.27318i
$356$	10.3923	0.550791
$357$	0	0
$358$	9.51472	0.502869
$359$	18.0000i	0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$	18.0000i	0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$	0	0
$361$	17.9706	0.945819
$362$	2.02922	0.106654
$363$	0	0
$364$	0	0
$365$	− 34.9706i	− 1.83044i
$366$	0	0
$367$	18.8785i	0.985449i	0.870185	+	0.492724i	$0.163999\pi$
$367$	18.8785i	0.985449i	−0.870185	+	0.492724i	$0.836001\pi$
$368$	4.24264i	0.221163i
$369$	0	0
$370$	34.4669i	1.79185i
$371$	0	0
$372$	0	0
$373$	21.4558	1.11094	0.555471	−	0.831536i	$-0.312538\pi$
$373$	21.4558	1.11094	0.555471	+	0.831536i	$0.312538\pi$
$374$	−3.04384	−0.157393
$375$	0	0
$376$	1.01461i	0.0523246i
$377$	−3.04384	−0.156766
$378$	0	0
$379$	−4.48528	−0.230393	−0.115197	−	0.993343i	$-0.536750\pi$
$379$	−4.48528	−0.230393	−0.115197	+	0.993343i	$0.536750\pi$
$380$	4.24264i	0.217643i
$381$	0	0
$382$	8.48528	0.434145
$383$	12.4215	0.634710	0.317355	−	0.948307i	$-0.397205\pi$
$383$	12.4215	0.634710	0.317355	+	0.948307i	$0.397205\pi$
$384$	0	0
$385$	0	0
$386$	7.48528i	0.380991i
$387$	0	0
$388$	− 3.76127i	− 0.190950i
$389$	19.4558i	0.986450i	0.869902	+	0.493225i	$0.164182\pi$
$389$	19.4558i	0.986450i	−0.869902	+	0.493225i	$0.835818\pi$
$390$	0	0
$391$	4.30463i	0.217695i
$392$	0	0
$393$	0	0
$394$	9.51472	0.479345
$395$	47.0116	2.36541
$396$	0	0
$397$	− 13.8564i	− 0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$	− 13.8564i	− 0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$	−16.1318	−0.808615
$399$	0	0
$400$	12.4853	0.624264
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−13.7574	−0.685303
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 24.7279i	− 1.22572i
$408$	0	0
$409$	− 4.60181i	− 0.227545i	−0.993507	−	0.113772i	$-0.963707\pi$
$409$	− 4.60181i	− 0.227545i	0.993507	−	0.113772i	$-0.0362934\pi$
$410$	8.48528i	0.419058i
$411$	0	0
$412$	− 15.2913i	− 0.753348i
$413$	0	0
$414$	0	0
$415$	13.2426	0.650056
$416$	2.44949	0.120096
$417$	0	0
$418$	− 3.04384i	− 0.148879i
$419$	4.05845	0.198268	0.0991341	−	0.995074i	$-0.468393\pi$
$419$	4.05845	0.198268	0.0991341	+	0.995074i	$0.468393\pi$
$420$	0	0
$421$	−5.75736	−0.280597	−0.140298	−	0.990109i	$-0.544806\pi$
$421$	−5.75736	−0.280597	−0.140298	+	0.990109i	$0.544806\pi$
$422$	− 8.24264i	− 0.401245i
$423$	0	0
$424$	−1.24264	−0.0603480
$425$	12.6677	0.614474
$426$	0	0
$427$	0	0
$428$	5.48528i	0.265141i
$429$	0	0
$430$	34.4669i	1.66214i
$431$	− 20.4853i	− 0.986741i	−0.869819	−	0.493371i	$-0.835765\pi$
$431$	− 20.4853i	− 0.986741i	0.869819	−	0.493371i	$-0.164235\pi$
$432$	0	0
$433$	− 3.46410i	− 0.166474i	−0.996530	−	0.0832370i	$-0.973474\pi$
$433$	− 3.46410i	− 0.166474i	0.996530	−	0.0832370i	$-0.0265259\pi$
$434$	0	0
$435$	0	0
$436$	−1.51472	−0.0725419
$437$	−4.30463	−0.205919
$438$	0	0
$439$	− 27.2416i	− 1.30017i	−0.759862	−	0.650084i	$-0.774734\pi$
$439$	− 27.2416i	− 1.30017i	0.759862	−	0.650084i	$-0.225266\pi$
$440$	−12.5446	−0.598041
$441$	0	0
$442$	2.48528	0.118213
$443$	− 34.4558i	− 1.63705i	−0.574473	−	0.818523i	$-0.694793\pi$
$443$	− 34.4558i	− 1.63705i	0.574473	−	0.818523i	$-0.305207\pi$
$444$	0	0
$445$	43.4558	2.06000
$446$	−12.5446	−0.594005
$447$	0	0
$448$	0	0
$449$	10.2426i	0.483380i	0.970354	+	0.241690i	$0.0777017\pi$
$449$	10.2426i	0.483380i	−0.970354	+	0.241690i	$0.922298\pi$
$450$	0	0
$451$	− 6.08767i	− 0.286657i
$452$	− 8.48528i	− 0.399114i
$453$	0	0
$454$	15.5885i	0.731603i
$455$	0	0
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	13.8564	0.647467
$459$	0	0
$460$	17.7408i	0.827168i
$461$	22.8138	1.06255	0.531273	−	0.847201i	$-0.321714\pi$
$461$	22.8138	1.06255	0.531273	+	0.847201i	$0.321714\pi$
$462$	0	0
$463$	21.4558	0.997138	0.498569	−	0.866850i	$-0.333859\pi$
$463$	21.4558	0.997138	0.498569	+	0.866850i	$0.333859\pi$
$464$	1.24264i	0.0576881i
$465$	0	0
$466$	6.72792	0.311665
$467$	19.0016	0.879288	0.439644	−	0.898172i	$-0.355105\pi$
$467$	19.0016	0.879288	0.439644	+	0.898172i	$0.355105\pi$
$468$	0	0
$469$	0	0
$470$	4.24264i	0.195698i
$471$	0	0
$472$	11.5300i	0.530712i
$473$	− 24.7279i	− 1.13699i
$474$	0	0
$475$	12.6677i	0.581235i
$476$	0	0
$477$	0	0
$478$	12.7279	0.582162
$479$	−36.4962	−1.66755	−0.833776	−	0.552103i	$-0.813826\pi$
$479$	−36.4962	−1.66755	−0.833776	+	0.552103i	$0.813826\pi$
$480$	0	0
$481$	20.1903i	0.920597i
$482$	17.0233	0.775392
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	− 15.7279i	− 0.714168i
$486$	0	0
$487$	28.2132	1.27846	0.639231	−	0.769015i	$-0.279253\pi$
$487$	28.2132	1.27846	0.639231	+	0.769015i	$0.279253\pi$
$488$	5.91359	0.267696
$489$	0	0
$490$	0	0
$491$	− 19.9706i	− 0.901259i	−0.892711	−	0.450629i	$-0.851200\pi$
$491$	− 19.9706i	− 0.901259i	0.892711	−	0.450629i	$-0.148800\pi$
$492$	0	0
$493$	1.26080i	0.0567835i
$494$	2.48528i	0.111818i
$495$	0	0
$496$	5.61642i	0.252185i
$497$	0	0
$498$	0	0
$499$	−35.9411	−1.60895	−0.804473	−	0.593989i	$-0.797552\pi$
$499$	−35.9411	−1.60895	−0.804473	+	0.593989i	$0.797552\pi$
$500$	31.3000	1.39978
$501$	0	0
$502$	17.6177i	0.786316i
$503$	3.29002	0.146695	0.0733474	−	0.997306i	$-0.476632\pi$
$503$	3.29002	0.146695	0.0733474	+	0.997306i	$0.476632\pi$
$504$	0	0
$505$	0	0
$506$	− 12.7279i	− 0.565825i
$507$	0	0
$508$	5.24264	0.232605
$509$	−41.6923	−1.84798	−0.923990	−	0.382418i	$-0.875092\pi$
$509$	−41.6923	−1.84798	−0.923990	+	0.382418i	$0.875092\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 25.0892i	− 1.10664i
$515$	− 63.9411i	− 2.81758i
$516$	0	0
$517$	− 3.04384i	− 0.133868i
$518$	0	0
$519$	0	0
$520$	10.2426	0.449170
$521$	−20.0162	−0.876925	−0.438462	−	0.898749i	$-0.644477\pi$
$521$	−20.0162	−0.876925	−0.438462	+	0.898749i	$0.644477\pi$
$522$	0	0
$523$	27.5387i	1.20418i	0.798426	+	0.602092i	$0.205666\pi$
$523$	27.5387i	1.20418i	−0.798426	+	0.602092i	$0.794334\pi$
$524$	−5.19615	−0.226995
$525$	0	0
$526$	−27.2132	−1.18655
$527$	5.69848i	0.248230i
$528$	0	0
$529$	5.00000	0.217391
$530$	−5.19615	−0.225706
$531$	0	0
$532$	0	0
$533$	4.97056i	0.215299i
$534$	0	0
$535$	22.9369i	0.991650i
$536$	− 10.0000i	− 0.431934i
$537$	0	0
$538$	10.5154i	0.453351i
$539$	0	0
$540$	0	0
$541$	−10.7279	−0.461229	−0.230615	−	0.973045i	$-0.574074\pi$
$541$	−10.7279	−0.461229	−0.230615	+	0.973045i	$0.574074\pi$
$542$	11.1097	0.477204
$543$	0	0
$544$	− 1.01461i	− 0.0435011i
$545$	−6.33386	−0.271313
$546$	0	0
$547$	19.6985	0.842246	0.421123	−	0.907003i	$-0.361636\pi$
$547$	19.6985	0.842246	0.421123	+	0.907003i	$0.361636\pi$
$548$	− 14.4853i	− 0.618781i
$549$	0	0
$550$	−37.4558	−1.59712
$551$	−1.26080	−0.0537118
$552$	0	0
$553$	0	0
$554$	20.9706i	0.890954i
$555$	0	0
$556$	20.1903i	0.856258i
$557$	− 15.7279i	− 0.666413i	−0.942854	−	0.333207i	$-0.891869\pi$
$557$	− 15.7279i	− 0.666413i	0.942854	−	0.333207i	$-0.108131\pi$
$558$	0	0
$559$	20.1903i	0.853957i
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	24.1977	1.01981	0.509906	−	0.860230i	$-0.329680\pi$
$563$	24.1977	1.01981	0.509906	+	0.860230i	$0.329680\pi$
$564$	0	0
$565$	− 35.4815i	− 1.49272i
$566$	−6.50794	−0.273549
$567$	0	0
$568$	−10.2426	−0.429772
$569$	1.75736i	0.0736723i	0.999321	+	0.0368362i	$0.0117280\pi$
$569$	1.75736i	0.0736723i	−0.999321	+	0.0368362i	$0.988272\pi$
$570$	0	0
$571$	−16.7279	−0.700042	−0.350021	−	0.936742i	$-0.613826\pi$
$571$	−16.7279	−0.700042	−0.350021	+	0.936742i	$0.613826\pi$
$572$	−7.34847	−0.307255
$573$	0	0
$574$	0	0
$575$	52.9706i	2.20903i
$576$	0	0
$577$	20.4874i	0.852903i	0.904510	+	0.426452i	$0.140237\pi$
$577$	20.4874i	0.852903i	−0.904510	+	0.426452i	$0.859763\pi$
$578$	15.9706i	0.664288i
$579$	0	0
$580$	5.19615i	0.215758i
$581$	0	0
$582$	0	0
$583$	3.72792	0.154395
$584$	−8.36308	−0.346067
$585$	0	0
$586$	− 4.18154i	− 0.172738i
$587$	5.19615	0.214468	0.107234	−	0.994234i	$-0.465801\pi$
$587$	5.19615	0.214468	0.107234	+	0.994234i	$0.465801\pi$
$588$	0	0
$589$	−5.69848	−0.234802
$590$	48.2132i	1.98491i
$591$	0	0
$592$	8.24264	0.338770
$593$	−30.4085	−1.24873	−0.624363	−	0.781134i	$-0.714642\pi$
$593$	−30.4085	−1.24873	−0.624363	+	0.781134i	$0.714642\pi$
$594$	0	0
$595$	0	0
$596$	− 20.4853i	− 0.839110i
$597$	0	0
$598$	10.3923i	0.424973i
$599$	43.4558i	1.77556i	0.460270	+	0.887779i	$0.347752\pi$
$599$	43.4558i	1.77556i	−0.460270	+	0.887779i	$0.652248\pi$
$600$	0	0
$601$	− 6.03668i	− 0.246241i	−0.992392	−	0.123121i	$-0.960710\pi$
$601$	− 6.03668i	− 0.246241i	0.992392	−	0.123121i	$-0.0392902\pi$
$602$	0	0
$603$	0	0
$604$	3.24264	0.131941
$605$	−8.36308	−0.340008
$606$	0	0
$607$	24.9662i	1.01334i	0.862139	+	0.506672i	$0.169125\pi$
$607$	24.9662i	1.01334i	−0.862139	+	0.506672i	$0.830875\pi$
$608$	1.01461	0.0411479
$609$	0	0
$610$	24.7279	1.00120
$611$	2.48528i	0.100544i
$612$	0	0
$613$	5.21320	0.210559	0.105280	−	0.994443i	$-0.466426\pi$
$613$	5.21320	0.210559	0.105280	+	0.994443i	$0.466426\pi$
$614$	−24.6690	−0.995559
$615$	0	0
$616$	0	0
$617$	− 41.6985i	− 1.67872i	−0.543578	−	0.839359i	$-0.682931\pi$
$617$	− 41.6985i	− 1.67872i	0.543578	−	0.839359i	$-0.317069\pi$
$618$	0	0
$619$	− 47.7290i	− 1.91839i	−0.282745	−	0.959195i	$-0.591245\pi$
$619$	− 47.7290i	− 1.91839i	0.282745	−	0.959195i	$-0.408755\pi$
$620$	23.4853i	0.943192i
$621$	0	0
$622$	− 18.7554i	− 0.752022i
$623$	0	0
$624$	0	0
$625$	68.4558	2.73823
$626$	1.13770	0.0454718
$627$	0	0
$628$	14.6969i	0.586472i
$629$	8.36308	0.333458
$630$	0	0
$631$	33.2426	1.32337	0.661684	−	0.749783i	$-0.269842\pi$
$631$	33.2426	1.32337	0.661684	+	0.749783i	$0.269842\pi$
$632$	− 11.2426i	− 0.447208i
$633$	0	0
$634$	7.24264	0.287642
$635$	21.9223	0.869961
$636$	0	0
$637$	0	0
$638$	− 3.72792i	− 0.147590i
$639$	0	0
$640$	− 4.18154i	− 0.165290i
$641$	41.6985i	1.64699i	0.567323	+	0.823496i	$0.307979\pi$
$641$	41.6985i	1.64699i	−0.567323	+	0.823496i	$0.692021\pi$
$642$	0	0
$643$	2.62357i	0.103463i	0.998661	+	0.0517317i	$0.0164741\pi$
$643$	2.62357i	0.103463i	−0.998661	+	0.0517317i	$0.983526\pi$
$644$	0	0
$645$	0	0
$646$	1.02944	0.0405027
$647$	−11.6531	−0.458131	−0.229065	−	0.973411i	$-0.573567\pi$
$647$	−11.6531	−0.458131	−0.229065	+	0.973411i	$0.573567\pi$
$648$	0	0
$649$	− 34.5900i	− 1.35778i
$650$	30.5826	1.19955
$651$	0	0
$652$	−6.24264	−0.244481
$653$	− 10.7574i	− 0.420968i	−0.977597	−	0.210484i	$-0.932496\pi$
$653$	− 10.7574i	− 0.420968i	0.977597	−	0.210484i	$-0.0675040\pi$
$654$	0	0
$655$	−21.7279	−0.848980
$656$	2.02922	0.0792279
$657$	0	0
$658$	0	0
$659$	− 6.00000i	− 0.233727i	−0.993148	−	0.116863i	$-0.962716\pi$
$659$	− 6.00000i	− 0.233727i	0.993148	−	0.116863i	$-0.0372840\pi$
$660$	0	0
$661$	40.5546i	1.57739i	0.614784	+	0.788696i	$0.289243\pi$
$661$	40.5546i	1.57739i	−0.614784	+	0.788696i	$0.710757\pi$
$662$	− 17.4558i	− 0.678441i
$663$	0	0
$664$	− 3.16693i	− 0.122901i
$665$	0	0
$666$	0	0
$667$	−5.27208	−0.204136
$668$	−23.0600	−0.892219
$669$	0	0
$670$	− 41.8154i	− 1.61547i
$671$	−17.7408	−0.684875
$672$	0	0
$673$	−15.9706	−0.615620	−0.307810	−	0.951448i	$-0.599596\pi$
$673$	−15.9706	−0.615620	−0.307810	+	0.951448i	$0.599596\pi$
$674$	5.00000i	0.192593i
$675$	0	0
$676$	−7.00000	−0.269231
$677$	12.5446	0.482129	0.241064	−	0.970509i	$-0.422503\pi$
$677$	12.5446	0.482129	0.241064	+	0.970509i	$0.422503\pi$
$678$	0	0
$679$	0	0
$680$	− 4.24264i	− 0.162698i
$681$	0	0
$682$	− 16.8493i	− 0.645191i
$683$	25.9706i	0.993736i	0.867826	+	0.496868i	$0.165517\pi$
$683$	25.9706i	0.993736i	−0.867826	+	0.496868i	$0.834483\pi$
$684$	0	0
$685$	− 60.5708i	− 2.31429i
$686$	0	0
$687$	0	0
$688$	8.24264	0.314248
$689$	−3.04384	−0.115961
$690$	0	0
$691$	0.840532i	0.0319753i	0.999872	+	0.0159877i	$0.00508925\pi$
$691$	0.840532i	0.0319753i	−0.999872	+	0.0159877i	$0.994911\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	−14.4853	−0.549854
$695$	84.4264i	3.20248i
$696$	0	0
$697$	2.05887	0.0779854
$698$	36.9164	1.39731
$699$	0	0
$700$	0	0
$701$	38.6985i	1.46162i	0.682580	+	0.730811i	$0.260858\pi$
$701$	38.6985i	1.46162i	−0.682580	+	0.730811i	$0.739142\pi$
$702$	0	0
$703$	8.36308i	0.315420i
$704$	3.00000i	0.113067i
$705$	0	0
$706$	− 18.7554i	− 0.705868i
$707$	0	0
$708$	0	0
$709$	−6.97056	−0.261785	−0.130892	−	0.991397i	$-0.541784\pi$
$709$	−6.97056	−0.261785	−0.130892	+	0.991397i	$0.541784\pi$
$710$	−42.8300	−1.60738
$711$	0	0
$712$	− 10.3923i	− 0.389468i
$713$	−23.8284	−0.892382
$714$	0	0
$715$	−30.7279	−1.14916
$716$	− 9.51472i	− 0.355582i
$717$	0	0
$718$	18.0000	0.671754
$719$	−23.0600	−0.859994	−0.429997	−	0.902830i	$-0.641485\pi$
$719$	−23.0600	−0.859994	−0.429997	+	0.902830i	$0.641485\pi$
$720$	0	0
$721$	0	0
$722$	− 17.9706i	− 0.668795i
$723$	0	0
$724$	− 2.02922i	− 0.0754155i
$725$	15.5147i	0.576202i
$726$	0	0
$727$	− 26.4010i	− 0.979160i	−0.871958	−	0.489580i	$-0.837150\pi$
$727$	− 26.4010i	− 0.979160i	0.871958	−	0.489580i	$-0.162850\pi$
$728$	0	0
$729$	0	0
$730$	−34.9706	−1.29432
$731$	8.36308	0.309320
$732$	0	0
$733$	39.3659i	1.45401i	0.686630	+	0.727007i	$0.259089\pi$
$733$	39.3659i	1.45401i	−0.686630	+	0.727007i	$0.740911\pi$
$734$	18.8785	0.696817
$735$	0	0
$736$	4.24264	0.156386
$737$	30.0000i	1.10506i
$738$	0	0
$739$	−35.4558	−1.30426	−0.652132	−	0.758105i	$-0.726125\pi$
$739$	−35.4558	−1.30426	−0.652132	+	0.758105i	$0.726125\pi$
$740$	34.4669	1.26703
$741$	0	0
$742$	0	0
$743$	− 21.5147i	− 0.789298i	−0.918832	−	0.394649i	$-0.870866\pi$
$743$	− 21.5147i	− 0.789298i	0.918832	−	0.394649i	$-0.129134\pi$
$744$	0	0
$745$	− 85.6600i	− 3.13834i
$746$	− 21.4558i	− 0.785554i
$747$	0	0
$748$	3.04384i	0.111294i
$749$	0	0
$750$	0	0
$751$	26.7574	0.976390	0.488195	−	0.872735i	$-0.337655\pi$
$751$	26.7574	0.976390	0.488195	+	0.872735i	$0.337655\pi$
$752$	1.01461	0.0369991
$753$	0	0
$754$	3.04384i	0.110850i
$755$	13.5592	0.493471
$756$	0	0
$757$	−42.2426	−1.53533	−0.767667	−	0.640848i	$-0.778583\pi$
$757$	−42.2426	−1.53533	−0.767667	+	0.640848i	$0.778583\pi$
$758$	4.48528i	0.162913i
$759$	0	0
$760$	4.24264	0.153897
$761$	5.07306	0.183898	0.0919491	−	0.995764i	$-0.470690\pi$
$761$	5.07306	0.183898	0.0919491	+	0.995764i	$0.470690\pi$
$762$	0	0
$763$	0	0
$764$	− 8.48528i	− 0.306987i
$765$	0	0
$766$	− 12.4215i	− 0.448808i
$767$	28.2426i	1.01978i
$768$	0	0
$769$	49.0408i	1.76846i	0.467056	+	0.884228i	$0.345315\pi$
$769$	49.0408i	1.76846i	−0.467056	+	0.884228i	$0.654685\pi$
$770$	0	0
$771$	0	0
$772$	7.48528	0.269401
$773$	23.0600	0.829411	0.414706	−	0.909956i	$-0.363885\pi$
$773$	23.0600	0.829411	0.414706	+	0.909956i	$0.363885\pi$
$774$	0	0
$775$	70.1226i	2.51888i
$776$	−3.76127	−0.135022
$777$	0	0
$778$	19.4558	0.697526
$779$	2.05887i	0.0737668i
$780$	0	0
$781$	30.7279	1.09953
$782$	4.30463	0.153933
$783$	0	0
$784$	0	0
$785$	61.4558i	2.19345i
$786$	0	0
$787$	37.0905i	1.32213i	0.750327	+	0.661067i	$0.229896\pi$
$787$	37.0905i	1.32213i	−0.750327	+	0.661067i	$0.770104\pi$
$788$	− 9.51472i	− 0.338948i
$789$	0	0
$790$	− 47.0116i	− 1.67260i
$791$	0	0
$792$	0	0
$793$	14.4853	0.514387
$794$	−13.8564	−0.491745
$795$	0	0
$796$	16.1318i	0.571777i
$797$	−37.6339	−1.33306	−0.666530	−	0.745478i	$-0.732221\pi$
$797$	−37.6339	−1.33306	−0.666530	+	0.745478i	$0.732221\pi$
$798$	0	0
$799$	1.02944	0.0364189
$800$	− 12.4853i	− 0.441421i
$801$	0	0
$802$	0	0
$803$	25.0892	0.885380
$804$	0	0
$805$	0	0
$806$	13.7574i	0.484582i
$807$	0	0
$808$	0	0
$809$	− 40.9706i	− 1.44045i	−0.693741	−	0.720224i	$-0.744039\pi$
$809$	− 40.9706i	− 1.44045i	0.693741	−	0.720224i	$-0.255961\pi$
$810$	0	0
$811$	31.1769i	1.09477i	0.836881	+	0.547385i	$0.184377\pi$
$811$	31.1769i	1.09477i	−0.836881	+	0.547385i	$0.815623\pi$
$812$	0	0
$813$	0	0
$814$	−24.7279	−0.866713
$815$	−26.1039	−0.914378
$816$	0	0
$817$	8.36308i	0.292587i
$818$	−4.60181	−0.160898
$819$	0	0
$820$	8.48528	0.296319
$821$	− 49.2426i	− 1.71858i	−0.511489	−	0.859290i	$-0.670906\pi$
$821$	− 49.2426i	− 1.71858i	0.511489	−	0.859290i	$-0.329094\pi$
$822$	0	0
$823$	−37.9411	−1.32254	−0.661272	−	0.750146i	$-0.729983\pi$
$823$	−37.9411	−1.32254	−0.661272	+	0.750146i	$0.729983\pi$
$824$	−15.2913	−0.532697
$825$	0	0
$826$	0	0
$827$	− 4.02944i	− 0.140117i	−0.997543	−	0.0700586i	$-0.977681\pi$
$827$	− 4.02944i	− 0.140117i	0.997543	−	0.0700586i	$-0.0223186\pi$
$828$	0	0
$829$	− 40.8008i	− 1.41707i	−0.705676	−	0.708535i	$-0.749356\pi$
$829$	− 40.8008i	− 1.41707i	0.705676	−	0.708535i	$-0.250644\pi$
$830$	− 13.2426i	− 0.459659i
$831$	0	0
$832$	− 2.44949i	− 0.0849208i
$833$	0	0
$834$	0	0
$835$	−96.4264	−3.33697
$836$	−3.04384	−0.105273
$837$	0	0
$838$	− 4.05845i	− 0.140197i
$839$	−24.0746	−0.831149	−0.415574	−	0.909559i	$-0.636419\pi$
$839$	−24.0746	−0.831149	−0.415574	+	0.909559i	$0.636419\pi$
$840$	0	0
$841$	27.4558	0.946753
$842$	5.75736i	0.198412i
$843$	0	0
$844$	−8.24264	−0.283723
$845$	−29.2708	−1.00695
$846$	0	0
$847$	0	0
$848$	1.24264i	0.0426725i
$849$	0	0
$850$	− 12.6677i	− 0.434499i
$851$	34.9706i	1.19878i
$852$	0	0
$853$	− 2.27541i	− 0.0779085i	−0.999241	−	0.0389543i	$-0.987597\pi$
$853$	− 2.27541i	− 0.0779085i	0.999241	−	0.0389543i	$-0.0124027\pi$
$854$	0	0
$855$	0	0
$856$	5.48528	0.187483
$857$	20.0162	0.683740	0.341870	−	0.939747i	$-0.388940\pi$
$857$	20.0162	0.683740	0.341870	+	0.939747i	$0.388940\pi$
$858$	0	0
$859$	− 4.47871i	− 0.152812i	−0.997077	−	0.0764059i	$-0.975656\pi$
$859$	− 4.47871i	− 0.152812i	0.997077	−	0.0764059i	$-0.0243445\pi$
$860$	34.4669	1.17531
$861$	0	0
$862$	−20.4853	−0.697731
$863$	− 50.4853i	− 1.71854i	−0.511523	−	0.859269i	$-0.670919\pi$
$863$	− 50.4853i	− 1.71854i	0.511523	−	0.859269i	$-0.329081\pi$
$864$	0	0
$865$	−86.9117	−2.95509
$866$	−3.46410	−0.117715
$867$	0	0
$868$	0	0
$869$	33.7279i	1.14414i
$870$	0	0
$871$	− 24.4949i	− 0.829978i
$872$	1.51472i	0.0512948i
$873$	0	0
$874$	4.30463i	0.145606i
$875$	0	0
$876$	0	0
$877$	−12.4853	−0.421598	−0.210799	−	0.977529i	$-0.567607\pi$
$877$	−12.4853	−0.421598	−0.210799	+	0.977529i	$0.567607\pi$
$878$	−27.2416	−0.919358
$879$	0	0
$880$	12.5446i	0.422879i
$881$	−39.7862	−1.34043	−0.670215	−	0.742167i	$-0.733798\pi$
$881$	−39.7862	−1.34043	−0.670215	+	0.742167i	$0.733798\pi$
$882$	0	0
$883$	−9.45584	−0.318214	−0.159107	−	0.987261i	$-0.550862\pi$
$883$	−9.45584	−0.318214	−0.159107	+	0.987261i	$0.550862\pi$
$884$	− 2.48528i	− 0.0835891i
$885$	0	0
$886$	−34.4558	−1.15757
$887$	−44.8592	−1.50623	−0.753113	−	0.657891i	$-0.771449\pi$
$887$	−44.8592	−1.50623	−0.753113	+	0.657891i	$0.771449\pi$
$888$	0	0
$889$	0	0
$890$	− 43.4558i	− 1.45664i
$891$	0	0
$892$	12.5446i	0.420025i
$893$	1.02944i	0.0344488i
$894$	0	0
$895$	− 39.7862i	− 1.32991i
$896$	0	0
$897$	0	0
$898$	10.2426	0.341801
$899$	−6.97919	−0.232769
$900$	0	0
$901$	1.26080i	0.0420033i
$902$	−6.08767	−0.202697
$903$	0	0
$904$	−8.48528	−0.282216
$905$	− 8.48528i	− 0.282060i
$906$	0	0
$907$	27.6985	0.919713	0.459857	−	0.887993i	$-0.347901\pi$
$907$	27.6985	0.919713	0.459857	+	0.887993i	$0.347901\pi$
$908$	15.5885	0.517321
$909$	0	0
$910$	0	0
$911$	− 18.7279i	− 0.620484i	−0.950658	−	0.310242i	$-0.899590\pi$
$911$	− 18.7279i	− 0.620484i	0.950658	−	0.310242i	$-0.100410\pi$
$912$	0	0
$913$	9.50079i	0.314430i
$914$	23.0000i	0.760772i
$915$	0	0
$916$	− 13.8564i	− 0.457829i
$917$	0	0
$918$	0	0
$919$	19.5147	0.643731	0.321866	−	0.946785i	$-0.395690\pi$
$919$	19.5147	0.643731	0.321866	+	0.946785i	$0.395690\pi$
$920$	17.7408	0.584896
$921$	0	0
$922$	− 22.8138i	− 0.751334i
$923$	−25.0892	−0.825823
$924$	0	0
$925$	102.912	3.38372
$926$	− 21.4558i	− 0.705083i
$927$	0	0
$928$	1.24264	0.0407917
$929$	−3.29002	−0.107942	−0.0539711	−	0.998543i	$-0.517188\pi$
$929$	−3.29002	−0.107942	−0.0539711	+	0.998543i	$0.517188\pi$
$930$	0	0
$931$	0	0
$932$	− 6.72792i	− 0.220380i
$933$	0	0
$934$	− 19.0016i	− 0.621750i
$935$	12.7279i	0.416248i
$936$	0	0
$937$	4.00746i	0.130918i	0.997855	+	0.0654590i	$0.0208512\pi$
$937$	4.00746i	0.130918i	−0.997855	+	0.0654590i	$0.979149\pi$
$938$	0	0
$939$	0	0
$940$	4.24264	0.138380
$941$	−52.0846	−1.69791	−0.848955	−	0.528465i	$-0.822768\pi$
$941$	−52.0846	−1.69791	−0.848955	+	0.528465i	$0.822768\pi$
$942$	0	0
$943$	8.60927i	0.280356i
$944$	11.5300	0.375270
$945$	0	0
$946$	−24.7279	−0.803974
$947$	− 22.9706i	− 0.746443i	−0.927742	−	0.373221i	$-0.878253\pi$
$947$	− 22.9706i	− 0.746443i	0.927742	−	0.373221i	$-0.121747\pi$
$948$	0	0
$949$	−20.4853	−0.664980
$950$	12.6677	0.410995
$951$	0	0
$952$	0	0
$953$	− 41.6985i	− 1.35075i	−0.737476	−	0.675373i	$-0.763983\pi$
$953$	− 41.6985i	− 1.35075i	0.737476	−	0.675373i	$-0.236017\pi$
$954$	0	0
$955$	− 35.4815i	− 1.14816i
$956$	− 12.7279i	− 0.411650i
$957$	0	0
$958$	36.4962i	1.17914i
$959$	0	0
$960$	0	0
$961$	−0.544156	−0.0175534
$962$	20.1903	0.650960
$963$	0	0
$964$	− 17.0233i	− 0.548285i
$965$	31.3000	1.00758
$966$	0	0
$967$	22.2721	0.716222	0.358111	−	0.933679i	$-0.383421\pi$
$967$	22.2721	0.716222	0.358111	+	0.933679i	$0.383421\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	−15.7279	−0.504993
$971$	51.3162	1.64681	0.823407	−	0.567451i	$-0.192070\pi$
$971$	51.3162	1.64681	0.823407	+	0.567451i	$0.192070\pi$
$972$	0	0
$973$	0	0
$974$	− 28.2132i	− 0.904009i
$975$	0	0
$976$	− 5.91359i	− 0.189289i
$977$	− 31.7574i	− 1.01601i	−0.861355	−	0.508004i	$-0.830384\pi$
$977$	− 31.7574i	− 1.01601i	0.861355	−	0.508004i	$-0.169616\pi$
$978$	0	0
$979$	31.1769i	0.996419i
$980$	0	0
$981$	0	0
$982$	−19.9706	−0.637286
$983$	8.11689	0.258889	0.129444	−	0.991587i	$-0.458681\pi$
$983$	8.11689	0.258889	0.129444	+	0.991587i	$0.458681\pi$
$984$	0	0
$985$	− 39.7862i	− 1.26769i
$986$	1.26080	0.0401520
$987$	0	0
$988$	2.48528	0.0790673
$989$	34.9706i	1.11200i
$990$	0	0
$991$	9.78680	0.310888	0.155444	−	0.987845i	$-0.450319\pi$
$991$	9.78680	0.310888	0.155444	+	0.987845i	$0.450319\pi$
$992$	5.61642	0.178321
$993$	0	0
$994$	0	0
$995$	67.4558i	2.13849i
$996$	0	0
$997$	− 9.55177i	− 0.302508i	−0.988495	−	0.151254i	$-0.951669\pi$
$997$	− 9.55177i	− 0.302508i	0.988495	−	0.151254i	$-0.0483311\pi$
$998$	35.9411i	1.13770i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.d.a.881.1		8
3.2	odd	2	inner	882.2.d.a.881.8		8
4.3	odd	2		7056.2.k.f.881.2		8
7.2	even	3		882.2.k.a.521.2		8
7.3	odd	6		882.2.k.a.215.3		8
7.4	even	3		126.2.k.a.89.4	yes	8
7.5	odd	6		126.2.k.a.17.1	✓	8
7.6	odd	2	inner	882.2.d.a.881.4		8
12.11	even	2		7056.2.k.f.881.7		8
21.2	odd	6		882.2.k.a.521.3		8
21.5	even	6		126.2.k.a.17.4	yes	8
21.11	odd	6		126.2.k.a.89.1	yes	8
21.17	even	6		882.2.k.a.215.2		8
21.20	even	2	inner	882.2.d.a.881.5		8
28.11	odd	6		1008.2.bt.c.593.4		8
28.19	even	6		1008.2.bt.c.17.1		8
28.27	even	2		7056.2.k.f.881.8		8
35.4	even	6		3150.2.bf.a.1601.2		8
35.12	even	12		3150.2.bp.b.899.4		8
35.18	odd	12		3150.2.bp.e.1349.4		8
35.19	odd	6		3150.2.bf.a.1151.4		8
35.32	odd	12		3150.2.bp.b.1349.1		8
35.33	even	12		3150.2.bp.e.899.1		8
63.4	even	3		1134.2.t.e.593.1		8
63.5	even	6		1134.2.l.f.269.4		8
63.11	odd	6		1134.2.l.f.215.3		8
63.25	even	3		1134.2.l.f.215.2		8
63.32	odd	6		1134.2.t.e.593.4		8
63.40	odd	6		1134.2.l.f.269.1		8
63.47	even	6		1134.2.t.e.1025.1		8
63.61	odd	6		1134.2.t.e.1025.4		8
84.11	even	6		1008.2.bt.c.593.1		8
84.47	odd	6		1008.2.bt.c.17.4		8
84.83	odd	2		7056.2.k.f.881.1		8
105.32	even	12		3150.2.bp.e.1349.1		8
105.47	odd	12		3150.2.bp.e.899.4		8
105.53	even	12		3150.2.bp.b.1349.4		8
105.68	odd	12		3150.2.bp.b.899.1		8
105.74	odd	6		3150.2.bf.a.1601.4		8
105.89	even	6		3150.2.bf.a.1151.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.k.a.17.1	✓	8	7.5	odd	6
126.2.k.a.17.4	yes	8	21.5	even	6
126.2.k.a.89.1	yes	8	21.11	odd	6
126.2.k.a.89.4	yes	8	7.4	even	3
882.2.d.a.881.1		8	1.1	even	1	trivial
882.2.d.a.881.4		8	7.6	odd	2	inner
882.2.d.a.881.5		8	21.20	even	2	inner
882.2.d.a.881.8		8	3.2	odd	2	inner
882.2.k.a.215.2		8	21.17	even	6
882.2.k.a.215.3		8	7.3	odd	6
882.2.k.a.521.2		8	7.2	even	3
882.2.k.a.521.3		8	21.2	odd	6
1008.2.bt.c.17.1		8	28.19	even	6
1008.2.bt.c.17.4		8	84.47	odd	6
1008.2.bt.c.593.1		8	84.11	even	6
1008.2.bt.c.593.4		8	28.11	odd	6
1134.2.l.f.215.2		8	63.25	even	3
1134.2.l.f.215.3		8	63.11	odd	6
1134.2.l.f.269.1		8	63.40	odd	6
1134.2.l.f.269.4		8	63.5	even	6
1134.2.t.e.593.1		8	63.4	even	3
1134.2.t.e.593.4		8	63.32	odd	6
1134.2.t.e.1025.1		8	63.47	even	6
1134.2.t.e.1025.4		8	63.61	odd	6
3150.2.bf.a.1151.2		8	105.89	even	6
3150.2.bf.a.1151.4		8	35.19	odd	6
3150.2.bf.a.1601.2		8	35.4	even	6
3150.2.bf.a.1601.4		8	105.74	odd	6
3150.2.bp.b.899.1		8	105.68	odd	12
3150.2.bp.b.899.4		8	35.12	even	12
3150.2.bp.b.1349.1		8	35.32	odd	12
3150.2.bp.b.1349.4		8	105.53	even	12
3150.2.bp.e.899.1		8	35.33	even	12
3150.2.bp.e.899.4		8	105.47	odd	12
3150.2.bp.e.1349.1		8	105.32	even	12
3150.2.bp.e.1349.4		8	35.18	odd	12
7056.2.k.f.881.1		8	84.83	odd	2
7056.2.k.f.881.2		8	4.3	odd	2
7056.2.k.f.881.7		8	12.11	even	2
7056.2.k.f.881.8		8	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.18154 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.18154 q^{5} +1.00000i q^{8} +4.18154i q^{10} -3.00000i q^{11} +2.44949i q^{13} +1.00000 q^{16} +1.01461 q^{17} +1.01461i q^{19} +4.18154 q^{20} -3.00000 q^{22} +4.24264i q^{23} +12.4853 q^{25} +2.44949 q^{26} +1.24264i q^{29} +5.61642i q^{31} -1.00000i q^{32} -1.01461i q^{34} +8.24264 q^{37} +1.01461 q^{38} -4.18154i q^{40} +2.02922 q^{41} +8.24264 q^{43} +3.00000i q^{44} +4.24264 q^{46} +1.01461 q^{47} -12.4853i q^{50} -2.44949i q^{52} +1.24264i q^{53} +12.5446i q^{55} +1.24264 q^{58} +11.5300 q^{59} -5.91359i q^{61} +5.61642 q^{62} -1.00000 q^{64} -10.2426i q^{65} -10.0000 q^{67} -1.01461 q^{68} +10.2426i q^{71} +8.36308i q^{73} -8.24264i q^{74} -1.01461i q^{76} -11.2426 q^{79} -4.18154 q^{80} -2.02922i q^{82} -3.16693 q^{83} -4.24264 q^{85} -8.24264i q^{86} +3.00000 q^{88} -10.3923 q^{89} -4.24264i q^{92} -1.01461i q^{94} -4.24264i q^{95} +3.76127i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} - 24 q^{22} + 32 q^{25} + 32 q^{37} + 32 q^{43} - 24 q^{58} - 8 q^{64} - 80 q^{67} - 56 q^{79} + 24 q^{88}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 - 24 * q^22 + 32 * q^25 + 32 * q^37 + 32 * q^43 - 24 * q^58 - 8 * q^64 - 80 * q^67 - 56 * q^79 + 24 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more