LMFDB - Embedded newform 441.2.c.a.440.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(440,441)]

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([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("441.440");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.c (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:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.2.c.a.440.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} -3.46410i q^{10} -1.41421i q^{11} +5.19615i q^{13} -4.00000 q^{16} +4.89898 q^{17} -1.73205i q^{19} -2.00000 q^{22} -5.65685i q^{23} +1.00000 q^{25} +7.34847 q^{26} +2.82843i q^{29} +1.73205i q^{31} -6.92820i q^{34} -1.00000 q^{37} -2.44949 q^{38} -6.92820i q^{40} -7.34847 q^{41} -1.00000 q^{43} -8.00000 q^{46} -12.2474 q^{47} -1.41421i q^{50} +2.82843i q^{53} -3.46410i q^{55} +4.00000 q^{58} +4.89898 q^{59} +3.46410i q^{61} +2.44949 q^{62} -8.00000 q^{64} +12.7279i q^{65} +11.0000 q^{67} +7.07107i q^{71} +1.73205i q^{73} +1.41421i q^{74} +5.00000 q^{79} -9.79796 q^{80} +10.3923i q^{82} +7.34847 q^{83} +12.0000 q^{85} +1.41421i q^{86} -4.00000 q^{88} -4.89898 q^{89} +17.3205i q^{94} -4.24264i q^{95} +10.3923i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{37} - 4 q^{43} - 32 q^{46} + 16 q^{58} - 32 q^{64} + 44 q^{67} + 20 q^{79} + 48 q^{85} - 16 q^{88}+O(q^{100}) $ 4 * q - 16 * q^16 - 8 * q^22 + 4 * q^25 - 4 * q^37 - 4 * q^43 - 32 * q^46 + 16 * q^58 - 32 * q^64 + 44 * q^67 + 20 * q^79 + 48 * q^85 - 16 * q^88

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 $

$2$	− 1.41421i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	− 1.41421i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	− 3.46410i	− 1.09545i
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	5.19615i	1.44115i	0.693375	+	0.720577i	$0.256123\pi$
$13$	5.19615i	1.44115i	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	− 1.73205i	− 0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$19$	− 1.73205i	− 0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	− 5.65685i	− 1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$	− 5.65685i	− 1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	7.34847	1.44115
$27$	0	0
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	1.73205i	0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	1.73205i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	0	0
$33$	0	0
$34$	− 6.92820i	− 1.18818i
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−2.44949	−0.397360
$39$	0	0
$40$	− 6.92820i	− 1.09545i
$41$	−7.34847	−1.14764	−0.573819	−	0.818982i	$-0.694539\pi$
$41$	−7.34847	−1.14764	−0.573819	+	0.818982i	$0.694539\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−12.2474	−1.78647	−0.893237	−	0.449586i	$-0.851571\pi$
$47$	−12.2474	−1.78647	−0.893237	+	0.449586i	$0.851571\pi$
$48$	0	0
$49$	0	0
$50$	− 1.41421i	− 0.200000i
$51$	0	0
$52$	0	0
$53$	2.82843i	0.388514i	0.980951	+	0.194257i	$0.0622296\pi$
$53$	2.82843i	0.388514i	−0.980951	+	0.194257i	$0.937770\pi$
$54$	0	0
$55$	− 3.46410i	− 0.467099i
$56$	0	0
$57$	0	0
$58$	4.00000	0.525226
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	0	0
$61$	3.46410i	0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$	3.46410i	0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$	2.44949	0.311086
$63$	0	0
$64$	−8.00000	−1.00000
$65$	12.7279i	1.57870i
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.07107i	0.839181i	0.907713	+	0.419591i	$0.137826\pi$
$71$	7.07107i	0.839181i	−0.907713	+	0.419591i	$0.862174\pi$
$72$	0	0
$73$	1.73205i	0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	1.73205i	0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	1.41421i	0.164399i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−9.79796	−1.09545
$81$	0	0
$82$	10.3923i	1.14764i
$83$	7.34847	0.806599	0.403300	−	0.915068i	$-0.367863\pi$
$83$	7.34847	0.806599	0.403300	+	0.915068i	$0.367863\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	1.41421i	0.152499i
$87$	0	0
$88$	−4.00000	−0.426401
$89$	−4.89898	−0.519291	−0.259645	−	0.965704i	$-0.583606\pi$
$89$	−4.89898	−0.519291	−0.259645	+	0.965704i	$0.583606\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	17.3205i	1.78647i
$95$	− 4.24264i	− 0.435286i
$96$	0	0
$97$	10.3923i	1.05518i	0.849500	+	0.527589i	$0.176904\pi$
$97$	10.3923i	1.05518i	−0.849500	+	0.527589i	$0.823096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.1464	−1.70613	−0.853067	−	0.521802i	$-0.825260\pi$
$101$	−17.1464	−1.70613	−0.853067	+	0.521802i	$0.825260\pi$
$102$	0	0
$103$	8.66025i	0.853320i	0.904412	+	0.426660i	$0.140310\pi$
$103$	8.66025i	0.853320i	−0.904412	+	0.426660i	$0.859690\pi$
$104$	14.6969	1.44115
$105$	0	0
$106$	4.00000	0.388514
$107$	2.82843i	0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$	2.82843i	0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−4.89898	−0.467099
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	− 13.8564i	− 1.29212i
$116$	0	0
$117$	0	0
$118$	− 6.92820i	− 0.637793i
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	4.89898	0.443533
$123$	0	0
$124$	0	0
$125$	−9.79796	−0.876356
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	18.0000	1.57870
$131$	2.44949	0.214013	0.107006	−	0.994258i	$-0.465873\pi$
$131$	2.44949	0.214013	0.107006	+	0.994258i	$0.465873\pi$
$132$	0	0
$133$	0	0
$134$	− 15.5563i	− 1.34386i
$135$	0	0
$136$	− 13.8564i	− 1.18818i
$137$	11.3137i	0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$	11.3137i	0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$	0	0
$139$	5.19615i	0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	5.19615i	0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$	0	0
$141$	0	0
$142$	10.0000	0.839181
$143$	7.34847	0.614510
$144$	0	0
$145$	6.92820i	0.575356i
$146$	2.44949	0.202721
$147$	0	0
$148$	0	0
$149$	− 5.65685i	− 0.463428i	−0.972784	−	0.231714i	$-0.925567\pi$
$149$	− 5.65685i	− 0.463428i	0.972784	−	0.231714i	$-0.0744333\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−4.89898	−0.397360
$153$	0	0
$154$	0	0
$155$	4.24264i	0.340777i
$156$	0	0
$157$	17.3205i	1.38233i	0.722698	+	0.691164i	$0.242902\pi$
$157$	17.3205i	1.38233i	−0.722698	+	0.691164i	$0.757098\pi$
$158$	− 7.07107i	− 0.562544i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	− 10.3923i	− 0.806599i
$167$	−7.34847	−0.568642	−0.284321	−	0.958729i	$-0.591768\pi$
$167$	−7.34847	−0.568642	−0.284321	+	0.958729i	$0.591768\pi$
$168$	0	0
$169$	−14.0000	−1.07692
$170$	− 16.9706i	− 1.30158i
$171$	0	0
$172$	0	0
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	0	0
$175$	0	0
$176$	5.65685i	0.426401i
$177$	0	0
$178$	6.92820i	0.519291i
$179$	− 9.89949i	− 0.739923i	−0.929047	−	0.369961i	$-0.879371\pi$
$179$	− 9.89949i	− 0.739923i	0.929047	−	0.369961i	$-0.120629\pi$
$180$	0	0
$181$	− 15.5885i	− 1.15868i	−0.815086	−	0.579340i	$-0.803310\pi$
$181$	− 15.5885i	− 1.15868i	0.815086	−	0.579340i	$-0.196690\pi$
$182$	0	0
$183$	0	0
$184$	−16.0000	−1.17954
$185$	−2.44949	−0.180090
$186$	0	0
$187$	− 6.92820i	− 0.506640i
$188$	0	0
$189$	0	0
$190$	−6.00000	−0.435286
$191$	− 1.41421i	− 0.102329i	−0.998690	−	0.0511645i	$-0.983707\pi$
$191$	− 1.41421i	− 0.102329i	0.998690	−	0.0511645i	$-0.0162933\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	14.6969	1.05518
$195$	0	0
$196$	0	0
$197$	19.7990i	1.41062i	0.708899	+	0.705310i	$0.249192\pi$
$197$	19.7990i	1.41062i	−0.708899	+	0.705310i	$0.750808\pi$
$198$	0	0
$199$	− 13.8564i	− 0.982255i	−0.871088	−	0.491127i	$-0.836585\pi$
$199$	− 13.8564i	− 0.982255i	0.871088	−	0.491127i	$-0.163415\pi$
$200$	− 2.82843i	− 0.200000i
$201$	0	0
$202$	24.2487i	1.70613i
$203$	0	0
$204$	0	0
$205$	−18.0000	−1.25717
$206$	12.2474	0.853320
$207$	0	0
$208$	− 20.7846i	− 1.44115i
$209$	−2.44949	−0.169435
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	0	0
$213$	0	0
$214$	4.00000	0.273434
$215$	−2.44949	−0.167054
$216$	0	0
$217$	0	0
$218$	1.41421i	0.0957826i
$219$	0	0
$220$	0	0
$221$	25.4558i	1.71235i
$222$	0	0
$223$	− 20.7846i	− 1.39184i	−0.718119	−	0.695920i	$-0.754997\pi$
$223$	− 20.7846i	− 1.39184i	0.718119	−	0.695920i	$-0.245003\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	26.9444	1.78836	0.894181	−	0.447706i	$-0.147759\pi$
$227$	26.9444	1.78836	0.894181	+	0.447706i	$0.147759\pi$
$228$	0	0
$229$	− 22.5167i	− 1.48794i	−0.668211	−	0.743971i	$-0.732940\pi$
$229$	− 22.5167i	− 1.48794i	0.668211	−	0.743971i	$-0.267060\pi$
$230$	−19.5959	−1.29212
$231$	0	0
$232$	8.00000	0.525226
$233$	− 9.89949i	− 0.648537i	−0.945965	−	0.324269i	$-0.894882\pi$
$233$	− 9.89949i	− 0.648537i	0.945965	−	0.324269i	$-0.105118\pi$
$234$	0	0
$235$	−30.0000	−1.95698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 26.8701i	− 1.73808i	−0.494742	−	0.869040i	$-0.664738\pi$
$239$	− 26.8701i	− 1.73808i	0.494742	−	0.869040i	$-0.335262\pi$
$240$	0	0
$241$	− 13.8564i	− 0.892570i	−0.894891	−	0.446285i	$-0.852747\pi$
$241$	− 13.8564i	− 0.892570i	0.894891	−	0.446285i	$-0.147253\pi$
$242$	− 12.7279i	− 0.818182i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.00000	0.572656
$248$	4.89898	0.311086
$249$	0	0
$250$	13.8564i	0.876356i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	− 15.5563i	− 0.976092i
$255$	0	0
$256$	0	0
$257$	2.44949	0.152795	0.0763975	−	0.997077i	$-0.475658\pi$
$257$	2.44949	0.152795	0.0763975	+	0.997077i	$0.475658\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 3.46410i	− 0.214013i
$263$	− 14.1421i	− 0.872041i	−0.899937	−	0.436021i	$-0.856387\pi$
$263$	− 14.1421i	− 0.872041i	0.899937	−	0.436021i	$-0.143613\pi$
$264$	0	0
$265$	6.92820i	0.425596i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.1464	−1.04544	−0.522718	−	0.852506i	$-0.675082\pi$
$269$	−17.1464	−1.04544	−0.522718	+	0.852506i	$0.675082\pi$
$270$	0	0
$271$	13.8564i	0.841717i	0.907126	+	0.420858i	$0.138271\pi$
$271$	13.8564i	0.841717i	−0.907126	+	0.420858i	$0.861729\pi$
$272$	−19.5959	−1.18818
$273$	0	0
$274$	16.0000	0.966595
$275$	− 1.41421i	− 0.0852803i
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	7.34847	0.440732
$279$	0	0
$280$	0	0
$281$	− 22.6274i	− 1.34984i	−0.737892	−	0.674919i	$-0.764178\pi$
$281$	− 22.6274i	− 1.34984i	0.737892	−	0.674919i	$-0.235822\pi$
$282$	0	0
$283$	1.73205i	0.102960i	0.998674	+	0.0514799i	$0.0163938\pi$
$283$	1.73205i	0.102960i	−0.998674	+	0.0514799i	$0.983606\pi$
$284$	0	0
$285$	0	0
$286$	− 10.3923i	− 0.614510i
$287$	0	0
$288$	0	0
$289$	7.00000	0.411765
$290$	9.79796	0.575356
$291$	0	0
$292$	0	0
$293$	14.6969	0.858604	0.429302	−	0.903161i	$-0.358760\pi$
$293$	14.6969	0.858604	0.429302	+	0.903161i	$0.358760\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	2.82843i	0.164399i
$297$	0	0
$298$	−8.00000	−0.463428
$299$	29.3939	1.69989
$300$	0	0
$301$	0	0
$302$	31.1127i	1.79033i
$303$	0	0
$304$	6.92820i	0.397360i
$305$	8.48528i	0.485866i
$306$	0	0
$307$	15.5885i	0.889680i	0.895610	+	0.444840i	$0.146740\pi$
$307$	15.5885i	0.889680i	−0.895610	+	0.444840i	$0.853260\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	−17.1464	−0.972285	−0.486142	−	0.873880i	$-0.661596\pi$
$311$	−17.1464	−0.972285	−0.486142	+	0.873880i	$0.661596\pi$
$312$	0	0
$313$	− 12.1244i	− 0.685309i	−0.939461	−	0.342655i	$-0.888674\pi$
$313$	− 12.1244i	− 0.685309i	0.939461	−	0.342655i	$-0.111326\pi$
$314$	24.4949	1.38233
$315$	0	0
$316$	0	0
$317$	− 14.1421i	− 0.794301i	−0.917753	−	0.397151i	$-0.869999\pi$
$317$	− 14.1421i	− 0.794301i	0.917753	−	0.397151i	$-0.130001\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	−19.5959	−1.09545
$321$	0	0
$322$	0	0
$323$	− 8.48528i	− 0.472134i
$324$	0	0
$325$	5.19615i	0.288231i
$326$	14.1421i	0.783260i
$327$	0	0
$328$	20.7846i	1.14764i
$329$	0	0
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	0	0
$333$	0	0
$334$	10.3923i	0.568642i
$335$	26.9444	1.47213
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	19.7990i	1.07692i
$339$	0	0
$340$	0	0
$341$	2.44949	0.132647
$342$	0	0
$343$	0	0
$344$	2.82843i	0.152499i
$345$	0	0
$346$	− 13.8564i	− 0.744925i
$347$	− 31.1127i	− 1.67022i	−0.550085	−	0.835109i	$-0.685405\pi$
$347$	− 31.1127i	− 1.67022i	0.550085	−	0.835109i	$-0.314595\pi$
$348$	0	0
$349$	10.3923i	0.556287i	0.960539	+	0.278144i	$0.0897191\pi$
$349$	10.3923i	0.556287i	−0.960539	+	0.278144i	$0.910281\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.1464	−0.912612	−0.456306	−	0.889823i	$-0.650828\pi$
$353$	−17.1464	−0.912612	−0.456306	+	0.889823i	$0.650828\pi$
$354$	0	0
$355$	17.3205i	0.919277i
$356$	0	0
$357$	0	0
$358$	−14.0000	−0.739923
$359$	28.2843i	1.49279i	0.665505	+	0.746393i	$0.268216\pi$
$359$	28.2843i	1.49279i	−0.665505	+	0.746393i	$0.731784\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	−22.0454	−1.15868
$363$	0	0
$364$	0	0
$365$	4.24264i	0.222070i
$366$	0	0
$367$	1.73205i	0.0904123i	0.998978	+	0.0452062i	$0.0143945\pi$
$367$	1.73205i	0.0904123i	−0.998978	+	0.0452062i	$0.985606\pi$
$368$	22.6274i	1.17954i
$369$	0	0
$370$	3.46410i	0.180090i
$371$	0	0
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	−9.79796	−0.506640
$375$	0	0
$376$	34.6410i	1.78647i
$377$	−14.6969	−0.756931
$378$	0	0
$379$	−7.00000	−0.359566	−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000	−0.359566	−0.179783	+	0.983706i	$0.557540\pi$
$380$	0	0
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−19.5959	−1.00130	−0.500652	−	0.865648i	$-0.666906\pi$
$383$	−19.5959	−1.00130	−0.500652	+	0.865648i	$0.666906\pi$
$384$	0	0
$385$	0	0
$386$	− 15.5563i	− 0.791797i
$387$	0	0
$388$	0	0
$389$	− 26.8701i	− 1.36237i	−0.732113	−	0.681183i	$-0.761466\pi$
$389$	− 26.8701i	− 1.36237i	0.732113	−	0.681183i	$-0.238534\pi$
$390$	0	0
$391$	− 27.7128i	− 1.40150i
$392$	0	0
$393$	0	0
$394$	28.0000	1.41062
$395$	12.2474	0.616236
$396$	0	0
$397$	− 1.73205i	− 0.0869291i	−0.999055	−	0.0434646i	$-0.986160\pi$
$397$	− 1.73205i	− 0.0869291i	0.999055	−	0.0434646i	$-0.0138396\pi$
$398$	−19.5959	−0.982255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	19.7990i	0.988714i	0.869259	+	0.494357i	$0.164597\pi$
$401$	19.7990i	0.988714i	−0.869259	+	0.494357i	$0.835403\pi$
$402$	0	0
$403$	−9.00000	−0.448322
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.41421i	0.0701000i
$408$	0	0
$409$	32.9090i	1.62724i	0.581394	+	0.813622i	$0.302507\pi$
$409$	32.9090i	1.62724i	−0.581394	+	0.813622i	$0.697493\pi$
$410$	25.4558i	1.25717i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	0	0
$418$	3.46410i	0.169435i
$419$	36.7423	1.79498	0.897491	−	0.441034i	$-0.145388\pi$
$419$	36.7423	1.79498	0.897491	+	0.441034i	$0.145388\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	31.1127i	1.51454i
$423$	0	0
$424$	8.00000	0.388514
$425$	4.89898	0.237635
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	3.46410i	0.167054i
$431$	15.5563i	0.749323i	0.927162	+	0.374661i	$0.122241\pi$
$431$	15.5563i	0.749323i	−0.927162	+	0.374661i	$0.877759\pi$
$432$	0	0
$433$	− 15.5885i	− 0.749133i	−0.927200	−	0.374567i	$-0.877791\pi$
$433$	− 15.5885i	− 0.749133i	0.927200	−	0.374567i	$-0.122209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.79796	−0.468700
$438$	0	0
$439$	− 27.7128i	− 1.32266i	−0.750095	−	0.661330i	$-0.769992\pi$
$439$	− 27.7128i	− 1.32266i	0.750095	−	0.661330i	$-0.230008\pi$
$440$	−9.79796	−0.467099
$441$	0	0
$442$	36.0000	1.71235
$443$	− 39.5980i	− 1.88136i	−0.339300	−	0.940678i	$-0.610190\pi$
$443$	− 39.5980i	− 1.88136i	0.339300	−	0.940678i	$-0.389810\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	−29.3939	−1.39184
$447$	0	0
$448$	0	0
$449$	7.07107i	0.333704i	0.985982	+	0.166852i	$0.0533603\pi$
$449$	7.07107i	0.333704i	−0.985982	+	0.166852i	$0.946640\pi$
$450$	0	0
$451$	10.3923i	0.489355i
$452$	0	0
$453$	0	0
$454$	− 38.1051i	− 1.78836i
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	−31.8434	−1.48794
$459$	0	0
$460$	0	0
$461$	−14.6969	−0.684505	−0.342252	−	0.939608i	$-0.611190\pi$
$461$	−14.6969	−0.684505	−0.342252	+	0.939608i	$0.611190\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	− 11.3137i	− 0.525226i
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−26.9444	−1.24684	−0.623419	−	0.781888i	$-0.714257\pi$
$467$	−26.9444	−1.24684	−0.623419	+	0.781888i	$0.714257\pi$
$468$	0	0
$469$	0	0
$470$	42.4264i	1.95698i
$471$	0	0
$472$	− 13.8564i	− 0.637793i
$473$	1.41421i	0.0650256i
$474$	0	0
$475$	− 1.73205i	− 0.0794719i
$476$	0	0
$477$	0	0
$478$	−38.0000	−1.73808
$479$	4.89898	0.223840	0.111920	−	0.993717i	$-0.464300\pi$
$479$	4.89898	0.223840	0.111920	+	0.993717i	$0.464300\pi$
$480$	0	0
$481$	− 5.19615i	− 0.236924i
$482$	−19.5959	−0.892570
$483$	0	0
$484$	0	0
$485$	25.4558i	1.15589i
$486$	0	0
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	9.79796	0.443533
$489$	0	0
$490$	0	0
$491$	11.3137i	0.510581i	0.966864	+	0.255290i	$0.0821710\pi$
$491$	11.3137i	0.510581i	−0.966864	+	0.255290i	$0.917829\pi$
$492$	0	0
$493$	13.8564i	0.624061i
$494$	− 12.7279i	− 0.572656i
$495$	0	0
$496$	− 6.92820i	− 0.311086i
$497$	0	0
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.0454	0.982956	0.491478	−	0.870890i	$-0.336457\pi$
$503$	22.0454	0.982956	0.491478	+	0.870890i	$0.336457\pi$
$504$	0	0
$505$	−42.0000	−1.86898
$506$	11.3137i	0.502956i
$507$	0	0
$508$	0	0
$509$	2.44949	0.108572	0.0542859	−	0.998525i	$-0.482712\pi$
$509$	2.44949	0.108572	0.0542859	+	0.998525i	$0.482712\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	− 3.46410i	− 0.152795i
$515$	21.2132i	0.934765i
$516$	0	0
$517$	17.3205i	0.761755i
$518$	0	0
$519$	0	0
$520$	36.0000	1.57870
$521$	4.89898	0.214628	0.107314	−	0.994225i	$-0.465775\pi$
$521$	4.89898	0.214628	0.107314	+	0.994225i	$0.465775\pi$
$522$	0	0
$523$	− 1.73205i	− 0.0757373i	−0.999283	−	0.0378686i	$-0.987943\pi$
$523$	− 1.73205i	− 0.0757373i	0.999283	−	0.0378686i	$-0.0120568\pi$
$524$	0	0
$525$	0	0
$526$	−20.0000	−0.872041
$527$	8.48528i	0.369625i
$528$	0	0
$529$	−9.00000	−0.391304
$530$	9.79796	0.425596
$531$	0	0
$532$	0	0
$533$	− 38.1838i	− 1.65392i
$534$	0	0
$535$	6.92820i	0.299532i
$536$	− 31.1127i	− 1.34386i
$537$	0	0
$538$	24.2487i	1.04544i
$539$	0	0
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	19.5959	0.841717
$543$	0	0
$544$	0	0
$545$	−2.44949	−0.104925
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	0	0
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	4.89898	0.208704
$552$	0	0
$553$	0	0
$554$	− 32.5269i	− 1.38194i
$555$	0	0
$556$	0	0
$557$	15.5563i	0.659144i	0.944131	+	0.329572i	$0.106904\pi$
$557$	15.5563i	0.659144i	−0.944131	+	0.329572i	$0.893096\pi$
$558$	0	0
$559$	− 5.19615i	− 0.219774i
$560$	0	0
$561$	0	0
$562$	−32.0000	−1.34984
$563$	26.9444	1.13557	0.567785	−	0.823177i	$-0.307800\pi$
$563$	26.9444	1.13557	0.567785	+	0.823177i	$0.307800\pi$
$564$	0	0
$565$	− 3.46410i	− 0.145736i
$566$	2.44949	0.102960
$567$	0	0
$568$	20.0000	0.839181
$569$	− 1.41421i	− 0.0592869i	−0.999561	−	0.0296435i	$-0.990563\pi$
$569$	− 1.41421i	− 0.0592869i	0.999561	−	0.0296435i	$-0.00943719\pi$
$570$	0	0
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 5.65685i	− 0.235907i
$576$	0	0
$577$	1.73205i	0.0721062i	0.999350	+	0.0360531i	$0.0114785\pi$
$577$	1.73205i	0.0721062i	−0.999350	+	0.0360531i	$0.988521\pi$
$578$	− 9.89949i	− 0.411765i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	4.89898	0.202721
$585$	0	0
$586$	− 20.7846i	− 0.858604i
$587$	−14.6969	−0.606608	−0.303304	−	0.952894i	$-0.598090\pi$
$587$	−14.6969	−0.606608	−0.303304	+	0.952894i	$0.598090\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	− 16.9706i	− 0.698667i
$591$	0	0
$592$	4.00000	0.164399
$593$	17.1464	0.704119	0.352060	−	0.935978i	$-0.385481\pi$
$593$	17.1464	0.704119	0.352060	+	0.935978i	$0.385481\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 41.5692i	− 1.69989i
$599$	− 5.65685i	− 0.231133i	−0.993300	−	0.115566i	$-0.963132\pi$
$599$	− 5.65685i	− 0.231133i	0.993300	−	0.115566i	$-0.0368683\pi$
$600$	0	0
$601$	25.9808i	1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$	25.9808i	1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.0454	0.896273
$606$	0	0
$607$	39.8372i	1.61694i	0.588537	+	0.808470i	$0.299704\pi$
$607$	39.8372i	1.61694i	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	12.0000	0.485866
$611$	− 63.6396i	− 2.57458i
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	22.0454	0.889680
$615$	0	0
$616$	0	0
$617$	24.0416i	0.967880i	0.875101	+	0.483940i	$0.160795\pi$
$617$	24.0416i	0.967880i	−0.875101	+	0.483940i	$0.839205\pi$
$618$	0	0
$619$	− 29.4449i	− 1.18349i	−0.806126	−	0.591744i	$-0.798439\pi$
$619$	− 29.4449i	− 1.18349i	0.806126	−	0.591744i	$-0.201561\pi$
$620$	0	0
$621$	0	0
$622$	24.2487i	0.972285i
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−17.1464	−0.685309
$627$	0	0
$628$	0	0
$629$	−4.89898	−0.195335
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	− 14.1421i	− 0.562544i
$633$	0	0
$634$	−20.0000	−0.794301
$635$	26.9444	1.06926
$636$	0	0
$637$	0	0
$638$	− 5.65685i	− 0.223957i
$639$	0	0
$640$	27.7128i	1.09545i
$641$	− 14.1421i	− 0.558581i	−0.960207	−	0.279290i	$-0.909901\pi$
$641$	− 14.1421i	− 0.558581i	0.960207	−	0.279290i	$-0.0900992\pi$
$642$	0	0
$643$	− 25.9808i	− 1.02458i	−0.858812	−	0.512291i	$-0.828797\pi$
$643$	− 25.9808i	− 1.02458i	0.858812	−	0.512291i	$-0.171203\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−17.1464	−0.674096	−0.337048	−	0.941488i	$-0.609428\pi$
$647$	−17.1464	−0.674096	−0.337048	+	0.941488i	$0.609428\pi$
$648$	0	0
$649$	− 6.92820i	− 0.271956i
$650$	7.34847	0.288231
$651$	0	0
$652$	0	0
$653$	7.07107i	0.276712i	0.990383	+	0.138356i	$0.0441819\pi$
$653$	7.07107i	0.276712i	−0.990383	+	0.138356i	$0.955818\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	29.3939	1.14764
$657$	0	0
$658$	0	0
$659$	− 22.6274i	− 0.881439i	−0.897645	−	0.440720i	$-0.854723\pi$
$659$	− 22.6274i	− 0.881439i	0.897645	−	0.440720i	$-0.145277\pi$
$660$	0	0
$661$	− 29.4449i	− 1.14527i	−0.819810	−	0.572636i	$-0.805921\pi$
$661$	− 29.4449i	− 1.14527i	0.819810	−	0.572636i	$-0.194079\pi$
$662$	43.8406i	1.70391i
$663$	0	0
$664$	− 20.7846i	− 0.806599i
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	0	0
$670$	− 38.1051i	− 1.47213i
$671$	4.89898	0.189123
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	− 32.5269i	− 1.25289i
$675$	0	0
$676$	0	0
$677$	9.79796	0.376566	0.188283	−	0.982115i	$-0.439708\pi$
$677$	9.79796	0.376566	0.188283	+	0.982115i	$0.439708\pi$
$678$	0	0
$679$	0	0
$680$	− 33.9411i	− 1.30158i
$681$	0	0
$682$	− 3.46410i	− 0.132647i
$683$	− 48.0833i	− 1.83985i	−0.392089	−	0.919927i	$-0.628247\pi$
$683$	− 48.0833i	− 1.83985i	0.392089	−	0.919927i	$-0.371753\pi$
$684$	0	0
$685$	27.7128i	1.05885i
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−14.6969	−0.559909
$690$	0	0
$691$	− 43.3013i	− 1.64726i	−0.567129	−	0.823629i	$-0.691946\pi$
$691$	− 43.3013i	− 1.64726i	0.567129	−	0.823629i	$-0.308054\pi$
$692$	0	0
$693$	0	0
$694$	−44.0000	−1.67022
$695$	12.7279i	0.482798i
$696$	0	0
$697$	−36.0000	−1.36360
$698$	14.6969	0.556287
$699$	0	0
$700$	0	0
$701$	− 5.65685i	− 0.213656i	−0.994277	−	0.106828i	$-0.965931\pi$
$701$	− 5.65685i	− 0.213656i	0.994277	−	0.106828i	$-0.0340695\pi$
$702$	0	0
$703$	1.73205i	0.0653255i
$704$	11.3137i	0.426401i
$705$	0	0
$706$	24.2487i	0.912612i
$707$	0	0
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	24.4949	0.919277
$711$	0	0
$712$	13.8564i	0.519291i
$713$	9.79796	0.366936
$714$	0	0
$715$	18.0000	0.673162
$716$	0	0
$717$	0	0
$718$	40.0000	1.49279
$719$	−26.9444	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$719$	−26.9444	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$720$	0	0
$721$	0	0
$722$	− 22.6274i	− 0.842105i
$723$	0	0
$724$	0	0
$725$	2.82843i	0.105045i
$726$	0	0
$727$	25.9808i	0.963573i	0.876289	+	0.481787i	$0.160012\pi$
$727$	25.9808i	0.963573i	−0.876289	+	0.481787i	$0.839988\pi$
$728$	0	0
$729$	0	0
$730$	6.00000	0.222070
$731$	−4.89898	−0.181195
$732$	0	0
$733$	39.8372i	1.47142i	0.677297	+	0.735710i	$0.263151\pi$
$733$	39.8372i	1.47142i	−0.677297	+	0.735710i	$0.736849\pi$
$734$	2.44949	0.0904123
$735$	0	0
$736$	0	0
$737$	− 15.5563i	− 0.573025i
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0416i	0.882002i	0.897507	+	0.441001i	$0.145376\pi$
$743$	24.0416i	0.882002i	−0.897507	+	0.441001i	$0.854624\pi$
$744$	0	0
$745$	− 13.8564i	− 0.507659i
$746$	− 41.0122i	− 1.50156i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	29.0000	1.05823	0.529113	−	0.848552i	$-0.322525\pi$
$751$	29.0000	1.05823	0.529113	+	0.848552i	$0.322525\pi$
$752$	48.9898	1.78647
$753$	0	0
$754$	20.7846i	0.756931i
$755$	−53.8888	−1.96121
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	9.89949i	0.359566i
$759$	0	0
$760$	−12.0000	−0.435286
$761$	24.4949	0.887939	0.443970	−	0.896042i	$-0.353570\pi$
$761$	24.4949	0.887939	0.443970	+	0.896042i	$0.353570\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	27.7128i	1.00130i
$767$	25.4558i	0.919157i
$768$	0	0
$769$	− 25.9808i	− 0.936890i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	− 25.9808i	− 0.936890i	0.883493	−	0.468445i	$-0.155186\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.9444	0.969122	0.484561	−	0.874757i	$-0.338979\pi$
$773$	26.9444	0.969122	0.484561	+	0.874757i	$0.338979\pi$
$774$	0	0
$775$	1.73205i	0.0622171i
$776$	29.3939	1.05518
$777$	0	0
$778$	−38.0000	−1.36237
$779$	12.7279i	0.456025i
$780$	0	0
$781$	10.0000	0.357828
$782$	−39.1918	−1.40150
$783$	0	0
$784$	0	0
$785$	42.4264i	1.51426i
$786$	0	0
$787$	− 45.0333i	− 1.60526i	−0.596474	−	0.802632i	$-0.703432\pi$
$787$	− 45.0333i	− 1.60526i	0.596474	−	0.802632i	$-0.296568\pi$
$788$	0	0
$789$	0	0
$790$	− 17.3205i	− 0.616236i
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	−2.44949	−0.0869291
$795$	0	0
$796$	0	0
$797$	−29.3939	−1.04118	−0.520592	−	0.853805i	$-0.674289\pi$
$797$	−29.3939	−1.04118	−0.520592	+	0.853805i	$0.674289\pi$
$798$	0	0
$799$	−60.0000	−2.12265
$800$	0	0
$801$	0	0
$802$	28.0000	0.988714
$803$	2.44949	0.0864406
$804$	0	0
$805$	0	0
$806$	12.7279i	0.448322i
$807$	0	0
$808$	48.4974i	1.70613i
$809$	41.0122i	1.44191i	0.692981	+	0.720956i	$0.256297\pi$
$809$	41.0122i	1.44191i	−0.692981	+	0.720956i	$0.743703\pi$
$810$	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
$814$	2.00000	0.0701000
$815$	−24.4949	−0.858019
$816$	0	0
$817$	1.73205i	0.0605968i
$818$	46.5403	1.62724
$819$	0	0
$820$	0	0
$821$	24.0416i	0.839059i	0.907742	+	0.419529i	$0.137805\pi$
$821$	24.0416i	0.839059i	−0.907742	+	0.419529i	$0.862195\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	24.4949	0.853320
$825$	0	0
$826$	0	0
$827$	7.07107i	0.245885i	0.992414	+	0.122943i	$0.0392331\pi$
$827$	7.07107i	0.245885i	−0.992414	+	0.122943i	$0.960767\pi$
$828$	0	0
$829$	1.73205i	0.0601566i	0.999548	+	0.0300783i	$0.00957567\pi$
$829$	1.73205i	0.0601566i	−0.999548	+	0.0300783i	$0.990424\pi$
$830$	− 25.4558i	− 0.883585i
$831$	0	0
$832$	− 41.5692i	− 1.44115i
$833$	0	0
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	0	0
$838$	− 51.9615i	− 1.79498i
$839$	−14.6969	−0.507395	−0.253697	−	0.967284i	$-0.581647\pi$
$839$	−14.6969	−0.507395	−0.253697	+	0.967284i	$0.581647\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	1.41421i	0.0487370i
$843$	0	0
$844$	0	0
$845$	−34.2929	−1.17971
$846$	0	0
$847$	0	0
$848$	− 11.3137i	− 0.388514i
$849$	0	0
$850$	− 6.92820i	− 0.237635i
$851$	5.65685i	0.193914i
$852$	0	0
$853$	− 36.3731i	− 1.24539i	−0.782465	−	0.622695i	$-0.786038\pi$
$853$	− 36.3731i	− 1.24539i	0.782465	−	0.622695i	$-0.213962\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	−39.1918	−1.33877	−0.669384	−	0.742917i	$-0.733442\pi$
$857$	−39.1918	−1.33877	−0.669384	+	0.742917i	$0.733442\pi$
$858$	0	0
$859$	− 38.1051i	− 1.30013i	−0.759879	−	0.650065i	$-0.774742\pi$
$859$	− 38.1051i	− 1.30013i	0.759879	−	0.650065i	$-0.225258\pi$
$860$	0	0
$861$	0	0
$862$	22.0000	0.749323
$863$	41.0122i	1.39607i	0.716063	+	0.698036i	$0.245942\pi$
$863$	41.0122i	1.39607i	−0.716063	+	0.698036i	$0.754058\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	−22.0454	−0.749133
$867$	0	0
$868$	0	0
$869$	− 7.07107i	− 0.239870i
$870$	0	0
$871$	57.1577i	1.93671i
$872$	2.82843i	0.0957826i
$873$	0	0
$874$	13.8564i	0.468700i
$875$	0	0
$876$	0	0
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	−39.1918	−1.32266
$879$	0	0
$880$	13.8564i	0.467099i
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	0	0
$886$	−56.0000	−1.88136
$887$	−41.6413	−1.39818	−0.699089	−	0.715034i	$-0.746411\pi$
$887$	−41.6413	−1.39818	−0.699089	+	0.715034i	$0.746411\pi$
$888$	0	0
$889$	0	0
$890$	16.9706i	0.568855i
$891$	0	0
$892$	0	0
$893$	21.2132i	0.709873i
$894$	0	0
$895$	− 24.2487i	− 0.810545i
$896$	0	0
$897$	0	0
$898$	10.0000	0.333704
$899$	−4.89898	−0.163390
$900$	0	0
$901$	13.8564i	0.461624i
$902$	14.6969	0.489355
$903$	0	0
$904$	−4.00000	−0.133038
$905$	− 38.1838i	− 1.26927i
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53.7401i	1.78049i	0.455483	+	0.890245i	$0.349467\pi$
$911$	53.7401i	1.78049i	−0.455483	+	0.890245i	$0.650533\pi$
$912$	0	0
$913$	− 10.3923i	− 0.343935i
$914$	− 7.07107i	− 0.233890i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	−39.1918	−1.29212
$921$	0	0
$922$	20.7846i	0.684505i
$923$	−36.7423	−1.20939
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	18.3848i	0.604161i
$927$	0	0
$928$	0	0
$929$	−56.3383	−1.84840	−0.924199	−	0.381911i	$-0.875266\pi$
$929$	−56.3383	−1.84840	−0.924199	+	0.381911i	$0.875266\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	38.1051i	1.24684i
$935$	− 16.9706i	− 0.554997i
$936$	0	0
$937$	− 46.7654i	− 1.52776i	−0.645359	−	0.763879i	$-0.723292\pi$
$937$	− 46.7654i	− 1.52776i	0.645359	−	0.763879i	$-0.276708\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48.9898	1.59702	0.798511	−	0.601980i	$-0.205621\pi$
$941$	48.9898	1.59702	0.798511	+	0.601980i	$0.205621\pi$
$942$	0	0
$943$	41.5692i	1.35368i
$944$	−19.5959	−0.637793
$945$	0	0
$946$	2.00000	0.0650256
$947$	49.4975i	1.60845i	0.594324	+	0.804226i	$0.297420\pi$
$947$	49.4975i	1.60845i	−0.594324	+	0.804226i	$0.702580\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	−2.44949	−0.0794719
$951$	0	0
$952$	0	0
$953$	− 5.65685i	− 0.183243i	−0.995794	−	0.0916217i	$-0.970795\pi$
$953$	− 5.65685i	− 0.183243i	0.995794	−	0.0916217i	$-0.0292051\pi$
$954$	0	0
$955$	− 3.46410i	− 0.112096i
$956$	0	0
$957$	0	0
$958$	− 6.92820i	− 0.223840i
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	−7.34847	−0.236924
$963$	0	0
$964$	0	0
$965$	26.9444	0.867371
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	− 25.4558i	− 0.818182i
$969$	0	0
$970$	36.0000	1.15589
$971$	39.1918	1.25773	0.628863	−	0.777516i	$-0.283521\pi$
$971$	39.1918	1.25773	0.628863	+	0.777516i	$0.283521\pi$
$972$	0	0
$973$	0	0
$974$	− 24.0416i	− 0.770344i
$975$	0	0
$976$	− 13.8564i	− 0.443533i
$977$	7.07107i	0.226224i	0.993582	+	0.113112i	$0.0360818\pi$
$977$	7.07107i	0.226224i	−0.993582	+	0.113112i	$0.963918\pi$
$978$	0	0
$979$	6.92820i	0.221426i
$980$	0	0
$981$	0	0
$982$	16.0000	0.510581
$983$	4.89898	0.156253	0.0781266	−	0.996943i	$-0.475106\pi$
$983$	4.89898	0.156253	0.0781266	+	0.996943i	$0.475106\pi$
$984$	0	0
$985$	48.4974i	1.54526i
$986$	19.5959	0.624061
$987$	0	0
$988$	0	0
$989$	5.65685i	0.179878i
$990$	0	0
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 33.9411i	− 1.07601i
$996$	0	0
$997$	− 29.4449i	− 0.932528i	−0.884646	−	0.466264i	$-0.845600\pi$
$997$	− 29.4449i	− 0.932528i	0.884646	−	0.466264i	$-0.154400\pi$
$998$	35.3553i	1.11915i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.c.a.440.2		4
3.2	odd	2	inner	441.2.c.a.440.3		4
4.3	odd	2		7056.2.k.b.881.4		4
7.2	even	3		441.2.p.a.80.1		4
7.3	odd	6		441.2.p.a.215.2		4
7.4	even	3		63.2.p.a.26.2	yes	4
7.5	odd	6		63.2.p.a.17.1	✓	4
7.6	odd	2	inner	441.2.c.a.440.1		4
12.11	even	2		7056.2.k.b.881.1		4
21.2	odd	6		441.2.p.a.80.2		4
21.5	even	6		63.2.p.a.17.2	yes	4
21.11	odd	6		63.2.p.a.26.1	yes	4
21.17	even	6		441.2.p.a.215.1		4
21.20	even	2	inner	441.2.c.a.440.4		4
28.11	odd	6		1008.2.bt.b.593.1		4
28.19	even	6		1008.2.bt.b.17.2		4
28.27	even	2		7056.2.k.b.881.2		4
35.4	even	6		1575.2.bk.c.26.1		4
35.12	even	12		1575.2.bc.a.899.1		8
35.18	odd	12		1575.2.bc.a.1349.3		8
35.19	odd	6		1575.2.bk.c.1151.2		4
35.32	odd	12		1575.2.bc.a.1349.2		8
35.33	even	12		1575.2.bc.a.899.4		8
63.4	even	3		567.2.s.d.26.1		4
63.5	even	6		567.2.i.d.269.2		4
63.11	odd	6		567.2.i.d.215.2		4
63.25	even	3		567.2.i.d.215.1		4
63.32	odd	6		567.2.s.d.26.2		4
63.40	odd	6		567.2.i.d.269.1		4
63.47	even	6		567.2.s.d.458.1		4
63.61	odd	6		567.2.s.d.458.2		4
84.11	even	6		1008.2.bt.b.593.2		4
84.47	odd	6		1008.2.bt.b.17.1		4
84.83	odd	2		7056.2.k.b.881.3		4
105.32	even	12		1575.2.bc.a.1349.4		8
105.47	odd	12		1575.2.bc.a.899.3		8
105.53	even	12		1575.2.bc.a.1349.1		8
105.68	odd	12		1575.2.bc.a.899.2		8
105.74	odd	6		1575.2.bk.c.26.2		4
105.89	even	6		1575.2.bk.c.1151.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.p.a.17.1	✓	4	7.5	odd	6
63.2.p.a.17.2	yes	4	21.5	even	6
63.2.p.a.26.1	yes	4	21.11	odd	6
63.2.p.a.26.2	yes	4	7.4	even	3
441.2.c.a.440.1		4	7.6	odd	2	inner
441.2.c.a.440.2		4	1.1	even	1	trivial
441.2.c.a.440.3		4	3.2	odd	2	inner
441.2.c.a.440.4		4	21.20	even	2	inner
441.2.p.a.80.1		4	7.2	even	3
441.2.p.a.80.2		4	21.2	odd	6
441.2.p.a.215.1		4	21.17	even	6
441.2.p.a.215.2		4	7.3	odd	6
567.2.i.d.215.1		4	63.25	even	3
567.2.i.d.215.2		4	63.11	odd	6
567.2.i.d.269.1		4	63.40	odd	6
567.2.i.d.269.2		4	63.5	even	6
567.2.s.d.26.1		4	63.4	even	3
567.2.s.d.26.2		4	63.32	odd	6
567.2.s.d.458.1		4	63.47	even	6
567.2.s.d.458.2		4	63.61	odd	6
1008.2.bt.b.17.1		4	84.47	odd	6
1008.2.bt.b.17.2		4	28.19	even	6
1008.2.bt.b.593.1		4	28.11	odd	6
1008.2.bt.b.593.2		4	84.11	even	6
1575.2.bc.a.899.1		8	35.12	even	12
1575.2.bc.a.899.2		8	105.68	odd	12
1575.2.bc.a.899.3		8	105.47	odd	12
1575.2.bc.a.899.4		8	35.33	even	12
1575.2.bc.a.1349.1		8	105.53	even	12
1575.2.bc.a.1349.2		8	35.32	odd	12
1575.2.bc.a.1349.3		8	35.18	odd	12
1575.2.bc.a.1349.4		8	105.32	even	12
1575.2.bk.c.26.1		4	35.4	even	6
1575.2.bk.c.26.2		4	105.74	odd	6
1575.2.bk.c.1151.1		4	105.89	even	6
1575.2.bk.c.1151.2		4	35.19	odd	6
7056.2.k.b.881.1		4	12.11	even	2
7056.2.k.b.881.2		4	28.27	even	2
7056.2.k.b.881.3		4	84.83	odd	2
7056.2.k.b.881.4		4	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} -3.46410i q^{10} -1.41421i q^{11} +5.19615i q^{13} -4.00000 q^{16} +4.89898 q^{17} -1.73205i q^{19} -2.00000 q^{22} -5.65685i q^{23} +1.00000 q^{25} +7.34847 q^{26} +2.82843i q^{29} +1.73205i q^{31} -6.92820i q^{34} -1.00000 q^{37} -2.44949 q^{38} -6.92820i q^{40} -7.34847 q^{41} -1.00000 q^{43} -8.00000 q^{46} -12.2474 q^{47} -1.41421i q^{50} +2.82843i q^{53} -3.46410i q^{55} +4.00000 q^{58} +4.89898 q^{59} +3.46410i q^{61} +2.44949 q^{62} -8.00000 q^{64} +12.7279i q^{65} +11.0000 q^{67} +7.07107i q^{71} +1.73205i q^{73} +1.41421i q^{74} +5.00000 q^{79} -9.79796 q^{80} +10.3923i q^{82} +7.34847 q^{83} +12.0000 q^{85} +1.41421i q^{86} -4.00000 q^{88} -4.89898 q^{89} +17.3205i q^{94} -4.24264i q^{95} +10.3923i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{37} - 4 q^{43} - 32 q^{46} + 16 q^{58} - 32 q^{64} + 44 q^{67} + 20 q^{79} + 48 q^{85} - 16 q^{88}+O(q^{100}) \) 4 * q - 16 * q^16 - 8 * q^22 + 4 * q^25 - 4 * q^37 - 4 * q^43 - 32 * q^46 + 16 * q^58 - 32 * q^64 + 44 * q^67 + 20 * q^79 + 48 * q^85 - 16 * q^88

Introduction

L-functions

Modular forms

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Database

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