LMFDB - Embedded newform 441.2.c.b.440.6

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:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.6
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.2.c.b.440.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} +1.58579i q^{8} +O(q^{10})$	$q+0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} +1.58579i q^{8} +1.21371i q^{10} +4.82843i q^{11} -2.93015i q^{13} +3.00000 q^{16} -7.07401 q^{17} -5.86030i q^{19} +5.35757 q^{20} -2.00000 q^{22} -2.00000i q^{23} +3.58579 q^{25} +1.21371 q^{26} -0.828427i q^{29} +5.86030i q^{31} +4.41421i q^{32} -2.93015i q^{34} -5.41421 q^{37} +2.42742 q^{38} +4.64659i q^{40} -1.21371 q^{41} +4.48528 q^{43} +8.82843i q^{44} +0.828427 q^{46} +5.86030 q^{47} +1.48528i q^{50} -5.35757i q^{52} -7.07107i q^{53} +14.1480i q^{55} +0.343146 q^{58} +5.86030 q^{59} -1.21371i q^{61} -2.42742 q^{62} +4.17157 q^{64} -8.58579i q^{65} -8.48528 q^{67} -12.9343 q^{68} +0.828427i q^{71} -7.07401i q^{73} -2.24264i q^{74} -10.7151i q^{76} +1.65685 q^{79} +8.79045 q^{80} -0.502734i q^{82} -11.7206 q^{83} -20.7279 q^{85} +1.85786i q^{86} -7.65685 q^{88} -11.2179 q^{89} -3.65685i q^{92} +2.42742i q^{94} -17.1716i q^{95} +7.07401i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 24 q^{16} - 16 q^{22} + 40 q^{25} - 32 q^{37} - 32 q^{43} - 16 q^{46} + 48 q^{58} + 56 q^{64} - 32 q^{79} - 64 q^{85} - 16 q^{88}+O(q^{100}) $ 8 * q - 8 * q^4 + 24 * q^16 - 16 * q^22 + 40 * q^25 - 32 * q^37 - 32 * q^43 - 16 * q^46 + 48 * q^58 + 56 * q^64 - 32 * q^79 - 64 * 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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	0	0
$3$	0	0
$4$	1.82843	0.914214
$5$	2.93015	1.31040	0.655202	−	0.755454i	$-0.272584\pi$
$5$	2.93015	1.31040	0.655202	+	0.755454i	$0.272584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.58579i	0.560660i
$9$	0	0
$10$	1.21371i	0.383808i
$11$	4.82843i	1.45583i	0.685670	+	0.727913i	$0.259509\pi$
$11$	4.82843i	1.45583i	−0.685670	+	0.727913i	$0.740491\pi$
$12$	0	0
$13$	− 2.93015i	− 0.812678i	−0.913722	−	0.406339i	$-0.866805\pi$
$13$	− 2.93015i	− 0.812678i	0.913722	−	0.406339i	$-0.133195\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−7.07401	−1.71570	−0.857850	−	0.513900i	$-0.828200\pi$
$17$	−7.07401	−1.71570	−0.857850	+	0.513900i	$0.828200\pi$
$18$	0	0
$19$	− 5.86030i	− 1.34445i	−0.740349	−	0.672223i	$-0.765340\pi$
$19$	− 5.86030i	− 1.34445i	0.740349	−	0.672223i	$-0.234660\pi$
$20$	5.35757	1.19799
$21$	0	0
$22$	−2.00000	−0.426401
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	0	0
$25$	3.58579	0.717157
$26$	1.21371	0.238028
$27$	0	0
$28$	0	0
$29$	− 0.828427i	− 0.153835i	−0.997037	−	0.0769175i	$-0.975492\pi$
$29$	− 0.828427i	− 0.153835i	0.997037	−	0.0769175i	$-0.0245078\pi$
$30$	0	0
$31$	5.86030i	1.05254i	0.850317	+	0.526271i	$0.176410\pi$
$31$	5.86030i	1.05254i	−0.850317	+	0.526271i	$0.823590\pi$
$32$	4.41421i	0.780330i
$33$	0	0
$34$	− 2.93015i	− 0.502517i
$35$	0	0
$36$	0	0
$37$	−5.41421	−0.890091	−0.445046	−	0.895508i	$-0.646813\pi$
$37$	−5.41421	−0.890091	−0.445046	+	0.895508i	$0.646813\pi$
$38$	2.42742	0.393779
$39$	0	0
$40$	4.64659i	0.734691i
$41$	−1.21371	−0.189549	−0.0947747	−	0.995499i	$-0.530213\pi$
$41$	−1.21371	−0.189549	−0.0947747	+	0.995499i	$0.530213\pi$
$42$	0	0
$43$	4.48528	0.683999	0.341999	−	0.939700i	$-0.388896\pi$
$43$	4.48528	0.683999	0.341999	+	0.939700i	$0.388896\pi$
$44$	8.82843i	1.33094i
$45$	0	0
$46$	0.828427	0.122145
$47$	5.86030	0.854813	0.427406	−	0.904060i	$-0.359427\pi$
$47$	5.86030	0.854813	0.427406	+	0.904060i	$0.359427\pi$
$48$	0	0
$49$	0	0
$50$	1.48528i	0.210051i
$51$	0	0
$52$	− 5.35757i	− 0.742961i
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	14.1480i	1.90772i
$56$	0	0
$57$	0	0
$58$	0.343146	0.0450572
$59$	5.86030	0.762946	0.381473	−	0.924380i	$-0.375417\pi$
$59$	5.86030	0.762946	0.381473	+	0.924380i	$0.375417\pi$
$60$	0	0
$61$	− 1.21371i	− 0.155399i	−0.996977	−	0.0776997i	$-0.975242\pi$
$61$	− 1.21371i	− 0.155399i	0.996977	−	0.0776997i	$-0.0247575\pi$
$62$	−2.42742	−0.308282
$63$	0	0
$64$	4.17157	0.521447
$65$	− 8.58579i	− 1.06494i
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	−12.9343	−1.56852
$69$	0	0
$70$	0	0
$71$	0.828427i	0.0983162i	0.998791	+	0.0491581i	$0.0156538\pi$
$71$	0.828427i	0.0983162i	−0.998791	+	0.0491581i	$0.984346\pi$
$72$	0	0
$73$	− 7.07401i	− 0.827950i	−0.910288	−	0.413975i	$-0.864140\pi$
$73$	− 7.07401i	− 0.827950i	0.910288	−	0.413975i	$-0.135860\pi$
$74$	− 2.24264i	− 0.260702i
$75$	0	0
$76$	− 10.7151i	− 1.22911i
$77$	0	0
$78$	0	0
$79$	1.65685	0.186411	0.0932053	−	0.995647i	$-0.470289\pi$
$79$	1.65685	0.186411	0.0932053	+	0.995647i	$0.470289\pi$
$80$	8.79045	0.982803
$81$	0	0
$82$	− 0.502734i	− 0.0555177i
$83$	−11.7206	−1.28650	−0.643252	−	0.765655i	$-0.722415\pi$
$83$	−11.7206	−1.28650	−0.643252	+	0.765655i	$0.722415\pi$
$84$	0	0
$85$	−20.7279	−2.24826
$86$	1.85786i	0.200339i
$87$	0	0
$88$	−7.65685	−0.816223
$89$	−11.2179	−1.18909	−0.594546	−	0.804062i	$-0.702668\pi$
$89$	−11.2179	−1.18909	−0.594546	+	0.804062i	$0.702668\pi$
$90$	0	0
$91$	0	0
$92$	− 3.65685i	− 0.381253i
$93$	0	0
$94$	2.42742i	0.250369i
$95$	− 17.1716i	− 1.76177i
$96$	0	0
$97$	7.07401i	0.718257i	0.933288	+	0.359128i	$0.116926\pi$
$97$	7.07401i	0.718257i	−0.933288	+	0.359128i	$0.883074\pi$
$98$	0	0
$99$	0	0
$100$	6.55635	0.655635
$101$	−5.35757	−0.533098	−0.266549	−	0.963821i	$-0.585883\pi$
$101$	−5.35757	−0.533098	−0.266549	+	0.963821i	$0.585883\pi$
$102$	0	0
$103$	14.1480i	1.39405i	0.717049	+	0.697023i	$0.245492\pi$
$103$	14.1480i	1.39405i	−0.717049	+	0.697023i	$0.754508\pi$
$104$	4.64659	0.455636
$105$	0	0
$106$	2.92893	0.284483
$107$	− 13.3137i	− 1.28708i	−0.765410	−	0.643542i	$-0.777464\pi$
$107$	− 13.3137i	− 1.28708i	0.765410	−	0.643542i	$-0.222536\pi$
$108$	0	0
$109$	−5.41421	−0.518588	−0.259294	−	0.965798i	$-0.583490\pi$
$109$	−5.41421	−0.518588	−0.259294	+	0.965798i	$0.583490\pi$
$110$	−5.86030	−0.558758
$111$	0	0
$112$	0	0
$113$	− 13.8995i	− 1.30755i	−0.756687	−	0.653777i	$-0.773183\pi$
$113$	− 13.8995i	− 1.30755i	0.756687	−	0.653777i	$-0.226817\pi$
$114$	0	0
$115$	− 5.86030i	− 0.546476i
$116$	− 1.51472i	− 0.140638i
$117$	0	0
$118$	2.42742i	0.223462i
$119$	0	0
$120$	0	0
$121$	−12.3137	−1.11943
$122$	0.502734	0.0455154
$123$	0	0
$124$	10.7151i	0.962248i
$125$	−4.14386	−0.370638
$126$	0	0
$127$	16.4853	1.46283	0.731416	−	0.681931i	$-0.238860\pi$
$127$	16.4853	1.46283	0.731416	+	0.681931i	$0.238860\pi$
$128$	10.5563i	0.933058i
$129$	0	0
$130$	3.55635	0.311912
$131$	8.28772	0.724101	0.362051	−	0.932158i	$-0.382077\pi$
$131$	8.28772	0.724101	0.362051	+	0.932158i	$0.382077\pi$
$132$	0	0
$133$	0	0
$134$	− 3.51472i	− 0.303625i
$135$	0	0
$136$	− 11.2179i	− 0.961924i
$137$	− 4.82843i	− 0.412520i	−0.978497	−	0.206260i	$-0.933871\pi$
$137$	− 4.82843i	− 0.412520i	0.978497	−	0.206260i	$-0.0661293\pi$
$138$	0	0
$139$	− 8.28772i	− 0.702955i	−0.936196	−	0.351478i	$-0.885679\pi$
$139$	− 8.28772i	− 0.702955i	0.936196	−	0.351478i	$-0.114321\pi$
$140$	0	0
$141$	0	0
$142$	−0.343146	−0.0287962
$143$	14.1480	1.18312
$144$	0	0
$145$	− 2.42742i	− 0.201586i
$146$	2.93015	0.242501
$147$	0	0
$148$	−9.89949	−0.813733
$149$	4.24264i	0.347571i	0.984784	+	0.173785i	$0.0555999\pi$
$149$	4.24264i	0.347571i	−0.984784	+	0.173785i	$0.944400\pi$
$150$	0	0
$151$	4.48528	0.365007	0.182504	−	0.983205i	$-0.441580\pi$
$151$	4.48528	0.365007	0.182504	+	0.983205i	$0.441580\pi$
$152$	9.29319	0.753777
$153$	0	0
$154$	0	0
$155$	17.1716i	1.37925i
$156$	0	0
$157$	22.9385i	1.83069i	0.402671	+	0.915345i	$0.368082\pi$
$157$	22.9385i	1.83069i	−0.402671	+	0.915345i	$0.631918\pi$
$158$	0.686292i	0.0545984i
$159$	0	0
$160$	12.9343i	1.02255i
$161$	0	0
$162$	0	0
$163$	9.17157	0.718373	0.359187	−	0.933266i	$-0.383054\pi$
$163$	9.17157	0.718373	0.359187	+	0.933266i	$0.383054\pi$
$164$	−2.21918	−0.173289
$165$	0	0
$166$	− 4.85483i	− 0.376808i
$167$	−14.1480	−1.09481	−0.547403	−	0.836869i	$-0.684384\pi$
$167$	−14.1480	−1.09481	−0.547403	+	0.836869i	$0.684384\pi$
$168$	0	0
$169$	4.41421	0.339555
$170$	− 8.58579i	− 0.658500i
$171$	0	0
$172$	8.20101	0.625321
$173$	−1.21371	−0.0922765	−0.0461383	−	0.998935i	$-0.514691\pi$
$173$	−1.21371	−0.0922765	−0.0461383	+	0.998935i	$0.514691\pi$
$174$	0	0
$175$	0	0
$176$	14.4853i	1.09187i
$177$	0	0
$178$	− 4.64659i	− 0.348277i
$179$	26.4853i	1.97960i	0.142454	+	0.989801i	$0.454501\pi$
$179$	26.4853i	1.97960i	−0.142454	+	0.989801i	$0.545499\pi$
$180$	0	0
$181$	− 9.50143i	− 0.706236i	−0.935579	−	0.353118i	$-0.885121\pi$
$181$	− 9.50143i	− 0.706236i	0.935579	−	0.353118i	$-0.114879\pi$
$182$	0	0
$183$	0	0
$184$	3.17157	0.233811
$185$	−15.8645	−1.16638
$186$	0	0
$187$	− 34.1563i	− 2.49776i
$188$	10.7151	0.781482
$189$	0	0
$190$	7.11270	0.516009
$191$	15.1716i	1.09778i	0.835896	+	0.548888i	$0.184949\pi$
$191$	15.1716i	1.09778i	−0.835896	+	0.548888i	$0.815051\pi$
$192$	0	0
$193$	−6.34315	−0.456590	−0.228295	−	0.973592i	$-0.573315\pi$
$193$	−6.34315	−0.456590	−0.228295	+	0.973592i	$0.573315\pi$
$194$	−2.93015	−0.210373
$195$	0	0
$196$	0	0
$197$	− 7.75736i	− 0.552689i	−0.961059	−	0.276344i	$-0.910877\pi$
$197$	− 7.75736i	− 0.552689i	0.961059	−	0.276344i	$-0.0891231\pi$
$198$	0	0
$199$	− 16.5754i	− 1.17500i	−0.809224	−	0.587501i	$-0.800112\pi$
$199$	− 16.5754i	− 1.17500i	0.809224	−	0.587501i	$-0.199888\pi$
$200$	5.68629i	0.402082i
$201$	0	0
$202$	− 2.21918i	− 0.156141i
$203$	0	0
$204$	0	0
$205$	−3.55635	−0.248386
$206$	−5.86030	−0.408307
$207$	0	0
$208$	− 8.79045i	− 0.609508i
$209$	28.2960	1.95728
$210$	0	0
$211$	−15.3137	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$211$	−15.3137	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$212$	− 12.9289i	− 0.887963i
$213$	0	0
$214$	5.51472	0.376978
$215$	13.1426	0.896315
$216$	0	0
$217$	0	0
$218$	− 2.24264i	− 0.151891i
$219$	0	0
$220$	25.8686i	1.74406i
$221$	20.7279i	1.39431i
$222$	0	0
$223$	− 3.43289i	− 0.229883i	−0.993372	−	0.114942i	$-0.963332\pi$
$223$	− 3.43289i	− 0.229883i	0.993372	−	0.114942i	$-0.0366681\pi$
$224$	0	0
$225$	0	0
$226$	5.75736	0.382974
$227$	−14.1480	−0.939037	−0.469519	−	0.882923i	$-0.655573\pi$
$227$	−14.1480	−0.939037	−0.469519	+	0.882923i	$0.655573\pi$
$228$	0	0
$229$	18.7946i	1.24198i	0.783817	+	0.620992i	$0.213270\pi$
$229$	18.7946i	1.24198i	−0.783817	+	0.620992i	$0.786730\pi$
$230$	2.42742	0.160059
$231$	0	0
$232$	1.31371	0.0862492
$233$	5.17157i	0.338801i	0.985547	+	0.169401i	$0.0541831\pi$
$233$	5.17157i	0.338801i	−0.985547	+	0.169401i	$0.945817\pi$
$234$	0	0
$235$	17.1716	1.12015
$236$	10.7151	0.697496
$237$	0	0
$238$	0	0
$239$	11.6569i	0.754019i	0.926209	+	0.377010i	$0.123048\pi$
$239$	11.6569i	0.754019i	−0.926209	+	0.377010i	$0.876952\pi$
$240$	0	0
$241$	25.3659i	1.63396i	0.576665	+	0.816980i	$0.304354\pi$
$241$	25.3659i	1.63396i	−0.576665	+	0.816980i	$0.695646\pi$
$242$	− 5.10051i	− 0.327873i
$243$	0	0
$244$	− 2.21918i	− 0.142068i
$245$	0	0
$246$	0	0
$247$	−17.1716	−1.09260
$248$	−9.29319	−0.590118
$249$	0	0
$250$	− 1.71644i	− 0.108557i
$251$	10.7151	0.676333	0.338167	−	0.941086i	$-0.390193\pi$
$251$	10.7151	0.676333	0.338167	+	0.941086i	$0.390193\pi$
$252$	0	0
$253$	9.65685	0.607121
$254$	6.82843i	0.428454i
$255$	0	0
$256$	3.97056	0.248160
$257$	10.5069	0.655402	0.327701	−	0.944781i	$-0.393726\pi$
$257$	10.5069	0.655402	0.327701	+	0.944781i	$0.393726\pi$
$258$	0	0
$259$	0	0
$260$	− 15.6985i	− 0.973579i
$261$	0	0
$262$	3.43289i	0.212084i
$263$	4.14214i	0.255415i	0.991812	+	0.127708i	$0.0407619\pi$
$263$	4.14214i	0.255415i	−0.991812	+	0.127708i	$0.959238\pi$
$264$	0	0
$265$	− 20.7193i	− 1.27278i
$266$	0	0
$267$	0	0
$268$	−15.5147	−0.947712
$269$	−2.21918	−0.135306	−0.0676528	−	0.997709i	$-0.521551\pi$
$269$	−2.21918	−0.135306	−0.0676528	+	0.997709i	$0.521551\pi$
$270$	0	0
$271$	5.86030i	0.355988i	0.984032	+	0.177994i	$0.0569608\pi$
$271$	5.86030i	0.355988i	−0.984032	+	0.177994i	$0.943039\pi$
$272$	−21.2220	−1.28677
$273$	0	0
$274$	2.00000	0.120824
$275$	17.3137i	1.04406i
$276$	0	0
$277$	12.9706	0.779326	0.389663	−	0.920958i	$-0.372592\pi$
$277$	12.9706	0.779326	0.389663	+	0.920958i	$0.372592\pi$
$278$	3.43289	0.205891
$279$	0	0
$280$	0	0
$281$	− 13.1716i	− 0.785750i	−0.919592	−	0.392875i	$-0.871480\pi$
$281$	− 13.1716i	− 0.785750i	0.919592	−	0.392875i	$-0.128520\pi$
$282$	0	0
$283$	− 9.29319i	− 0.552423i	−0.961097	−	0.276211i	$-0.910921\pi$
$283$	− 9.29319i	− 0.552423i	0.961097	−	0.276211i	$-0.0890790\pi$
$284$	1.51472i	0.0898820i
$285$	0	0
$286$	5.86030i	0.346527i
$287$	0	0
$288$	0	0
$289$	33.0416	1.94363
$290$	1.00547	0.0590432
$291$	0	0
$292$	− 12.9343i	− 0.756923i
$293$	22.9385	1.34008	0.670040	−	0.742325i	$-0.266277\pi$
$293$	22.9385	1.34008	0.670040	+	0.742325i	$0.266277\pi$
$294$	0	0
$295$	17.1716	0.999768
$296$	− 8.58579i	− 0.499039i
$297$	0	0
$298$	−1.75736	−0.101801
$299$	−5.86030	−0.338910
$300$	0	0
$301$	0	0
$302$	1.85786i	0.106908i
$303$	0	0
$304$	− 17.5809i	− 1.00833i
$305$	− 3.55635i	− 0.203636i
$306$	0	0
$307$	22.4357i	1.28048i	0.768177	+	0.640238i	$0.221164\pi$
$307$	22.4357i	1.28048i	−0.768177	+	0.640238i	$0.778836\pi$
$308$	0	0
$309$	0	0
$310$	−7.11270	−0.403974
$311$	25.8686	1.46688	0.733438	−	0.679757i	$-0.237915\pi$
$311$	25.8686	1.46688	0.733438	+	0.679757i	$0.237915\pi$
$312$	0	0
$313$	5.35757i	0.302828i	0.988470	+	0.151414i	$0.0483826\pi$
$313$	5.35757i	0.302828i	−0.988470	+	0.151414i	$0.951617\pi$
$314$	−9.50143	−0.536197
$315$	0	0
$316$	3.02944	0.170419
$317$	− 32.2426i	− 1.81093i	−0.424424	−	0.905464i	$-0.639523\pi$
$317$	− 32.2426i	− 1.81093i	0.424424	−	0.905464i	$-0.360477\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	12.2233	0.683305
$321$	0	0
$322$	0	0
$323$	41.4558i	2.30666i
$324$	0	0
$325$	− 10.5069i	− 0.582818i
$326$	3.79899i	0.210407i
$327$	0	0
$328$	− 1.92468i	− 0.106273i
$329$	0	0
$330$	0	0
$331$	21.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$331$	21.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$332$	−21.4303	−1.17614
$333$	0	0
$334$	− 5.86030i	− 0.320661i
$335$	−24.8632	−1.35842
$336$	0	0
$337$	−27.0711	−1.47466	−0.737328	−	0.675535i	$-0.763913\pi$
$337$	−27.0711	−1.47466	−0.737328	+	0.675535i	$0.763913\pi$
$338$	1.82843i	0.0994533i
$339$	0	0
$340$	−37.8995	−2.05539
$341$	−28.2960	−1.53232
$342$	0	0
$343$	0	0
$344$	7.11270i	0.383491i
$345$	0	0
$346$	− 0.502734i	− 0.0270272i
$347$	− 14.9706i	− 0.803662i	−0.915714	−	0.401831i	$-0.868374\pi$
$347$	− 14.9706i	− 0.803662i	0.915714	−	0.401831i	$-0.131626\pi$
$348$	0	0
$349$	− 11.2179i	− 0.600479i	−0.953864	−	0.300239i	$-0.902933\pi$
$349$	− 11.2179i	− 0.600479i	0.953864	−	0.300239i	$-0.0970666\pi$
$350$	0	0
$351$	0	0
$352$	−21.3137	−1.13602
$353$	20.5111	1.09169	0.545847	−	0.837885i	$-0.316208\pi$
$353$	20.5111	1.09169	0.545847	+	0.837885i	$0.316208\pi$
$354$	0	0
$355$	2.42742i	0.128834i
$356$	−20.5111	−1.08708
$357$	0	0
$358$	−10.9706	−0.579812
$359$	34.9706i	1.84568i	0.385189	+	0.922838i	$0.374136\pi$
$359$	34.9706i	1.84568i	−0.385189	+	0.922838i	$0.625864\pi$
$360$	0	0
$361$	−15.3431	−0.807534
$362$	3.93562	0.206852
$363$	0	0
$364$	0	0
$365$	− 20.7279i	− 1.08495i
$366$	0	0
$367$	− 20.0083i	− 1.04443i	−0.852815	−	0.522213i	$-0.825107\pi$
$367$	− 20.0083i	− 1.04443i	0.852815	−	0.522213i	$-0.174893\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	− 6.57128i	− 0.341624i
$371$	0	0
$372$	0	0
$373$	22.6274	1.17160	0.585802	−	0.810454i	$-0.300780\pi$
$373$	22.6274	1.17160	0.585802	+	0.810454i	$0.300780\pi$
$374$	14.1480	0.731577
$375$	0	0
$376$	9.29319i	0.479260i
$377$	−2.42742	−0.125018
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	− 31.3970i	− 1.61063i
$381$	0	0
$382$	−6.28427	−0.321531
$383$	−23.4412	−1.19779	−0.598895	−	0.800828i	$-0.704393\pi$
$383$	−23.4412	−1.19779	−0.598895	+	0.800828i	$0.704393\pi$
$384$	0	0
$385$	0	0
$386$	− 2.62742i	− 0.133732i
$387$	0	0
$388$	12.9343i	0.656640i
$389$	− 25.7990i	− 1.30806i	−0.756468	−	0.654030i	$-0.773077\pi$
$389$	− 25.7990i	− 1.30806i	0.756468	−	0.654030i	$-0.226923\pi$
$390$	0	0
$391$	14.1480i	0.715496i
$392$	0	0
$393$	0	0
$394$	3.21320	0.161879
$395$	4.85483	0.244273
$396$	0	0
$397$	17.7891i	0.892812i	0.894831	+	0.446406i	$0.147296\pi$
$397$	17.7891i	0.892812i	−0.894831	+	0.446406i	$0.852704\pi$
$398$	6.86577	0.344150
$399$	0	0
$400$	10.7574	0.537868
$401$	− 5.17157i	− 0.258256i	−0.991628	−	0.129128i	$-0.958782\pi$
$401$	− 5.17157i	− 0.258256i	0.991628	−	0.129128i	$-0.0412178\pi$
$402$	0	0
$403$	17.1716	0.855377
$404$	−9.79592	−0.487365
$405$	0	0
$406$	0	0
$407$	− 26.1421i	− 1.29582i
$408$	0	0
$409$	− 9.50143i	− 0.469815i	−0.972018	−	0.234908i	$-0.924521\pi$
$409$	− 9.50143i	− 0.469815i	0.972018	−	0.234908i	$-0.0754788\pi$
$410$	− 1.47309i	− 0.0727506i
$411$	0	0
$412$	25.8686i	1.27446i
$413$	0	0
$414$	0	0
$415$	−34.3431	−1.68584
$416$	12.9343	0.634157
$417$	0	0
$418$	11.7206i	0.573274i
$419$	19.0029	0.928350	0.464175	−	0.885743i	$-0.346351\pi$
$419$	19.0029	0.928350	0.464175	+	0.885743i	$0.346351\pi$
$420$	0	0
$421$	−0.686292	−0.0334478	−0.0167239	−	0.999860i	$-0.505324\pi$
$421$	−0.686292	−0.0334478	−0.0167239	+	0.999860i	$0.505324\pi$
$422$	− 6.34315i	− 0.308780i
$423$	0	0
$424$	11.2132	0.544561
$425$	−25.3659	−1.23043
$426$	0	0
$427$	0	0
$428$	− 24.3431i	− 1.17667i
$429$	0	0
$430$	5.44382i	0.262524i
$431$	− 35.4558i	− 1.70785i	−0.520398	−	0.853924i	$-0.674216\pi$
$431$	− 35.4558i	− 1.70785i	0.520398	−	0.853924i	$-0.325784\pi$
$432$	0	0
$433$	1.21371i	0.0583271i	0.999575	+	0.0291636i	$0.00928436\pi$
$433$	1.21371i	0.0583271i	−0.999575	+	0.0291636i	$0.990716\pi$
$434$	0	0
$435$	0	0
$436$	−9.89949	−0.474100
$437$	−11.7206	−0.560673
$438$	0	0
$439$	24.8632i	1.18665i	0.804962	+	0.593327i	$0.202186\pi$
$439$	24.8632i	1.18665i	−0.804962	+	0.593327i	$0.797814\pi$
$440$	−22.4357	−1.06958
$441$	0	0
$442$	−8.58579	−0.408384
$443$	19.1716i	0.910869i	0.890269	+	0.455434i	$0.150516\pi$
$443$	19.1716i	0.910869i	−0.890269	+	0.455434i	$0.849484\pi$
$444$	0	0
$445$	−32.8701	−1.55819
$446$	1.42195	0.0673312
$447$	0	0
$448$	0	0
$449$	− 15.7574i	− 0.743636i	−0.928306	−	0.371818i	$-0.878735\pi$
$449$	− 15.7574i	− 0.743636i	0.928306	−	0.371818i	$-0.121265\pi$
$450$	0	0
$451$	− 5.86030i	− 0.275951i
$452$	− 25.4142i	− 1.19538i
$453$	0	0
$454$	− 5.86030i	− 0.275038i
$455$	0	0
$456$	0	0
$457$	−16.6274	−0.777798	−0.388899	−	0.921280i	$-0.627144\pi$
$457$	−16.6274	−0.777798	−0.388899	+	0.921280i	$0.627144\pi$
$458$	−7.78498	−0.363768
$459$	0	0
$460$	− 10.7151i	− 0.499596i
$461$	10.5069	0.489355	0.244677	−	0.969605i	$-0.421318\pi$
$461$	10.5069	0.489355	0.244677	+	0.969605i	$0.421318\pi$
$462$	0	0
$463$	20.4853	0.952032	0.476016	−	0.879437i	$-0.342080\pi$
$463$	20.4853	0.952032	0.476016	+	0.879437i	$0.342080\pi$
$464$	− 2.48528i	− 0.115376i
$465$	0	0
$466$	−2.14214	−0.0992325
$467$	14.1480	0.654692	0.327346	−	0.944904i	$-0.393846\pi$
$467$	14.1480	0.654692	0.327346	+	0.944904i	$0.393846\pi$
$468$	0	0
$469$	0	0
$470$	7.11270i	0.328084i
$471$	0	0
$472$	9.29319i	0.427754i
$473$	21.6569i	0.995783i
$474$	0	0
$475$	− 21.0138i	− 0.964179i
$476$	0	0
$477$	0	0
$478$	−4.82843	−0.220847
$479$	42.4441	1.93932	0.969659	−	0.244460i	$-0.0786106\pi$
$479$	42.4441	1.93932	0.969659	+	0.244460i	$0.0786106\pi$
$480$	0	0
$481$	15.8645i	0.723357i
$482$	−10.5069	−0.478576
$483$	0	0
$484$	−22.5147	−1.02340
$485$	20.7279i	0.941206i
$486$	0	0
$487$	−41.4558	−1.87854	−0.939272	−	0.343174i	$-0.888498\pi$
$487$	−41.4558	−1.87854	−0.939272	+	0.343174i	$0.888498\pi$
$488$	1.92468	0.0871263
$489$	0	0
$490$	0	0
$491$	25.3137i	1.14239i	0.820814	+	0.571196i	$0.193520\pi$
$491$	25.3137i	1.14239i	−0.820814	+	0.571196i	$0.806480\pi$
$492$	0	0
$493$	5.86030i	0.263935i
$494$	− 7.11270i	− 0.320015i
$495$	0	0
$496$	17.5809i	0.789406i
$497$	0	0
$498$	0	0
$499$	26.1421	1.17028	0.585141	−	0.810931i	$-0.301039\pi$
$499$	26.1421	1.17028	0.585141	+	0.810931i	$0.301039\pi$
$500$	−7.57675	−0.338842
$501$	0	0
$502$	4.43835i	0.198093i
$503$	−4.85483	−0.216466	−0.108233	−	0.994126i	$-0.534519\pi$
$503$	−4.85483	−0.216466	−0.108233	+	0.994126i	$0.534519\pi$
$504$	0	0
$505$	−15.6985	−0.698573
$506$	4.00000i	0.177822i
$507$	0	0
$508$	30.1421	1.33734
$509$	−26.0769	−1.15584	−0.577918	−	0.816095i	$-0.696135\pi$
$509$	−26.0769	−1.15584	−0.577918	+	0.816095i	$0.696135\pi$
$510$	0	0
$511$	0	0
$512$	22.7574i	1.00574i
$513$	0	0
$514$	4.35210i	0.191963i
$515$	41.4558i	1.82676i
$516$	0	0
$517$	28.2960i	1.24446i
$518$	0	0
$519$	0	0
$520$	13.6152	0.597067
$521$	−3.93562	−0.172423	−0.0862113	−	0.996277i	$-0.527476\pi$
$521$	−3.93562	−0.172423	−0.0862113	+	0.996277i	$0.527476\pi$
$522$	0	0
$523$	− 31.7289i	− 1.38741i	−0.720260	−	0.693705i	$-0.755977\pi$
$523$	− 31.7289i	− 1.38741i	0.720260	−	0.693705i	$-0.244023\pi$
$524$	15.1535	0.661983
$525$	0	0
$526$	−1.71573	−0.0748093
$527$	− 41.4558i	− 1.80584i
$528$	0	0
$529$	19.0000	0.826087
$530$	8.58221	0.372788
$531$	0	0
$532$	0	0
$533$	3.55635i	0.154043i
$534$	0	0
$535$	− 39.0112i	− 1.68660i
$536$	− 13.4558i	− 0.581204i
$537$	0	0
$538$	− 0.919213i	− 0.0396301i
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	−2.42742	−0.104266
$543$	0	0
$544$	− 31.2262i	− 1.33881i
$545$	−15.8645	−0.679559
$546$	0	0
$547$	18.6274	0.796451	0.398225	−	0.917288i	$-0.369626\pi$
$547$	18.6274	0.796451	0.398225	+	0.917288i	$0.369626\pi$
$548$	− 8.82843i	− 0.377132i
$549$	0	0
$550$	−7.17157	−0.305797
$551$	−4.85483	−0.206823
$552$	0	0
$553$	0	0
$554$	5.37258i	0.228259i
$555$	0	0
$556$	− 15.1535i	− 0.642651i
$557$	13.4142i	0.568378i	0.958768	+	0.284189i	$0.0917244\pi$
$557$	13.4142i	0.568378i	−0.958768	+	0.284189i	$0.908276\pi$
$558$	0	0
$559$	− 13.1426i	− 0.555871i
$560$	0	0
$561$	0	0
$562$	5.45584	0.230141
$563$	−14.1480	−0.596268	−0.298134	−	0.954524i	$-0.596364\pi$
$563$	−14.1480	−0.596268	−0.298134	+	0.954524i	$0.596364\pi$
$564$	0	0
$565$	− 40.7276i	− 1.71342i
$566$	3.84936	0.161801
$567$	0	0
$568$	−1.31371	−0.0551220
$569$	37.1127i	1.55585i	0.628360	+	0.777923i	$0.283726\pi$
$569$	37.1127i	1.55585i	−0.628360	+	0.777923i	$0.716274\pi$
$570$	0	0
$571$	11.3137	0.473464	0.236732	−	0.971575i	$-0.423924\pi$
$571$	11.3137	0.473464	0.236732	+	0.971575i	$0.423924\pi$
$572$	25.8686	1.08162
$573$	0	0
$574$	0	0
$575$	− 7.17157i	− 0.299075i
$576$	0	0
$577$	29.5098i	1.22851i	0.789109	+	0.614254i	$0.210543\pi$
$577$	29.5098i	1.22851i	−0.789109	+	0.614254i	$0.789457\pi$
$578$	13.6863i	0.569275i
$579$	0	0
$580$	− 4.43835i	− 0.184293i
$581$	0	0
$582$	0	0
$583$	34.1421	1.41402
$584$	11.2179	0.464199
$585$	0	0
$586$	9.50143i	0.392500i
$587$	−17.5809	−0.725642	−0.362821	−	0.931859i	$-0.618186\pi$
$587$	−17.5809	−0.725642	−0.362821	+	0.931859i	$0.618186\pi$
$588$	0	0
$589$	34.3431	1.41508
$590$	7.11270i	0.292825i
$591$	0	0
$592$	−16.2426	−0.667568
$593$	23.6494	0.971166	0.485583	−	0.874190i	$-0.338607\pi$
$593$	23.6494	0.971166	0.485583	+	0.874190i	$0.338607\pi$
$594$	0	0
$595$	0	0
$596$	7.75736i	0.317754i
$597$	0	0
$598$	− 2.42742i	− 0.0992645i
$599$	10.4853i	0.428417i	0.976788	+	0.214208i	$0.0687172\pi$
$599$	10.4853i	0.428417i	−0.976788	+	0.214208i	$0.931283\pi$
$600$	0	0
$601$	− 40.5194i	− 1.65282i	−0.563069	−	0.826410i	$-0.690379\pi$
$601$	− 40.5194i	− 1.65282i	0.563069	−	0.826410i	$-0.309621\pi$
$602$	0	0
$603$	0	0
$604$	8.20101	0.333694
$605$	−36.0810	−1.46690
$606$	0	0
$607$	− 3.43289i	− 0.139337i	−0.997570	−	0.0696683i	$-0.977806\pi$
$607$	− 3.43289i	− 0.139337i	0.997570	−	0.0696683i	$-0.0221941\pi$
$608$	25.8686	1.04911
$609$	0	0
$610$	1.47309	0.0596436
$611$	− 17.1716i	− 0.694687i
$612$	0	0
$613$	42.8701	1.73151	0.865753	−	0.500472i	$-0.166840\pi$
$613$	42.8701	1.73151	0.865753	+	0.500472i	$0.166840\pi$
$614$	−9.29319	−0.375043
$615$	0	0
$616$	0	0
$617$	13.5147i	0.544082i	0.962286	+	0.272041i	$0.0876986\pi$
$617$	13.5147i	0.544082i	−0.962286	+	0.272041i	$0.912301\pi$
$618$	0	0
$619$	44.8715i	1.80354i	0.432219	+	0.901769i	$0.357731\pi$
$619$	44.8715i	1.80354i	−0.432219	+	0.901769i	$0.642269\pi$
$620$	31.3970i	1.26093i
$621$	0	0
$622$	10.7151i	0.429638i
$623$	0	0
$624$	0	0
$625$	−30.0711	−1.20284
$626$	−2.21918	−0.0886962
$627$	0	0
$628$	41.9413i	1.67364i
$629$	38.3002	1.52713
$630$	0	0
$631$	−26.8284	−1.06802	−0.534011	−	0.845477i	$-0.679316\pi$
$631$	−26.8284	−1.06802	−0.534011	+	0.845477i	$0.679316\pi$
$632$	2.62742i	0.104513i
$633$	0	0
$634$	13.3553	0.530408
$635$	48.3044	1.91690
$636$	0	0
$637$	0	0
$638$	1.65685i	0.0655955i
$639$	0	0
$640$	30.9317i	1.22268i
$641$	− 24.4853i	− 0.967110i	−0.875314	−	0.483555i	$-0.839345\pi$
$641$	− 24.4853i	− 0.967110i	0.875314	−	0.483555i	$-0.160655\pi$
$642$	0	0
$643$	− 5.86030i	− 0.231108i	−0.993301	−	0.115554i	$-0.963136\pi$
$643$	− 5.86030i	− 0.231108i	0.993301	−	0.115554i	$-0.0368643\pi$
$644$	0	0
$645$	0	0
$646$	−17.1716	−0.675606
$647$	−10.7151	−0.421255	−0.210628	−	0.977566i	$-0.567551\pi$
$647$	−10.7151	−0.421255	−0.210628	+	0.977566i	$0.567551\pi$
$648$	0	0
$649$	28.2960i	1.11072i
$650$	4.35210	0.170703
$651$	0	0
$652$	16.7696	0.656746
$653$	21.5147i	0.841936i	0.907075	+	0.420968i	$0.138310\pi$
$653$	21.5147i	0.841936i	−0.907075	+	0.420968i	$0.861690\pi$
$654$	0	0
$655$	24.2843	0.948865
$656$	−3.64113	−0.142162
$657$	0	0
$658$	0	0
$659$	− 3.45584i	− 0.134621i	−0.997732	−	0.0673103i	$-0.978558\pi$
$659$	− 3.45584i	− 0.134621i	0.997732	−	0.0673103i	$-0.0214417\pi$
$660$	0	0
$661$	− 22.9385i	− 0.892203i	−0.894982	−	0.446102i	$-0.852812\pi$
$661$	− 22.9385i	− 0.892203i	0.894982	−	0.446102i	$-0.147188\pi$
$662$	8.97056i	0.348651i
$663$	0	0
$664$	− 18.5864i	− 0.721291i
$665$	0	0
$666$	0	0
$667$	−1.65685	−0.0641537
$668$	−25.8686	−1.00089
$669$	0	0
$670$	− 10.2987i	− 0.397872i
$671$	5.86030	0.226234
$672$	0	0
$673$	−18.1005	−0.697723	−0.348862	−	0.937174i	$-0.613432\pi$
$673$	−18.1005	−0.697723	−0.348862	+	0.937174i	$0.613432\pi$
$674$	− 11.2132i	− 0.431916i
$675$	0	0
$676$	8.07107	0.310426
$677$	46.0852	1.77120	0.885599	−	0.464451i	$-0.153748\pi$
$677$	46.0852	1.77120	0.885599	+	0.464451i	$0.153748\pi$
$678$	0	0
$679$	0	0
$680$	− 32.8701i	− 1.26051i
$681$	0	0
$682$	− 11.7206i	− 0.448805i
$683$	34.7696i	1.33042i	0.746656	+	0.665210i	$0.231658\pi$
$683$	34.7696i	1.33042i	−0.746656	+	0.665210i	$0.768342\pi$
$684$	0	0
$685$	− 14.1480i	− 0.540568i
$686$	0	0
$687$	0	0
$688$	13.4558	0.512999
$689$	−20.7193	−0.789342
$690$	0	0
$691$	− 8.28772i	− 0.315280i	−0.987497	−	0.157640i	$-0.949611\pi$
$691$	− 8.28772i	− 0.315280i	0.987497	−	0.157640i	$-0.0503885\pi$
$692$	−2.21918	−0.0843605
$693$	0	0
$694$	6.20101	0.235387
$695$	− 24.2843i	− 0.921155i
$696$	0	0
$697$	8.58579	0.325210
$698$	4.64659	0.175876
$699$	0	0
$700$	0	0
$701$	− 32.7696i	− 1.23769i	−0.785514	−	0.618844i	$-0.787601\pi$
$701$	− 32.7696i	− 1.23769i	0.785514	−	0.618844i	$-0.212399\pi$
$702$	0	0
$703$	31.7289i	1.19668i
$704$	20.1421i	0.759135i
$705$	0	0
$706$	8.49596i	0.319750i
$707$	0	0
$708$	0	0
$709$	−7.27208	−0.273109	−0.136554	−	0.990633i	$-0.543603\pi$
$709$	−7.27208	−0.273109	−0.136554	+	0.990633i	$0.543603\pi$
$710$	−1.00547	−0.0377346
$711$	0	0
$712$	− 17.7891i	− 0.666676i
$713$	11.7206	0.438940
$714$	0	0
$715$	41.4558	1.55036
$716$	48.4264i	1.80978i
$717$	0	0
$718$	−14.4853	−0.540586
$719$	−11.7206	−0.437105	−0.218552	−	0.975825i	$-0.570133\pi$
$719$	−11.7206	−0.437105	−0.218552	+	0.975825i	$0.570133\pi$
$720$	0	0
$721$	0	0
$722$	− 6.35534i	− 0.236521i
$723$	0	0
$724$	− 17.3727i	− 0.645650i
$725$	− 2.97056i	− 0.110324i
$726$	0	0
$727$	21.0138i	0.779358i	0.920951	+	0.389679i	$0.127414\pi$
$727$	21.0138i	0.779358i	−0.920951	+	0.389679i	$0.872586\pi$
$728$	0	0
$729$	0	0
$730$	8.58579	0.317774
$731$	−31.7289	−1.17354
$732$	0	0
$733$	− 37.7975i	− 1.39608i	−0.716058	−	0.698041i	$-0.754055\pi$
$733$	− 37.7975i	− 1.39608i	0.716058	−	0.698041i	$-0.245945\pi$
$734$	8.28772	0.305905
$735$	0	0
$736$	8.82843	0.325420
$737$	− 40.9706i	− 1.50917i
$738$	0	0
$739$	−13.1716	−0.484524	−0.242262	−	0.970211i	$-0.577889\pi$
$739$	−13.1716	−0.484524	−0.242262	+	0.970211i	$0.577889\pi$
$740$	−29.0070	−1.06632
$741$	0	0
$742$	0	0
$743$	32.4264i	1.18961i	0.803870	+	0.594805i	$0.202771\pi$
$743$	32.4264i	1.18961i	−0.803870	+	0.594805i	$0.797229\pi$
$744$	0	0
$745$	12.4316i	0.455458i
$746$	9.37258i	0.343155i
$747$	0	0
$748$	− 62.4524i	− 2.28349i
$749$	0	0
$750$	0	0
$751$	43.3137	1.58054	0.790270	−	0.612759i	$-0.209940\pi$
$751$	43.3137	1.58054	0.790270	+	0.612759i	$0.209940\pi$
$752$	17.5809	0.641110
$753$	0	0
$754$	− 1.00547i	− 0.0366170i
$755$	13.1426	0.478306
$756$	0	0
$757$	−46.8701	−1.70352	−0.851761	−	0.523931i	$-0.824465\pi$
$757$	−46.8701	−1.70352	−0.851761	+	0.523931i	$0.824465\pi$
$758$	0	0
$759$	0	0
$760$	27.2304	0.987752
$761$	−17.0782	−0.619083	−0.309542	−	0.950886i	$-0.600176\pi$
$761$	−17.0782	−0.619083	−0.309542	+	0.950886i	$0.600176\pi$
$762$	0	0
$763$	0	0
$764$	27.7401i	1.00360i
$765$	0	0
$766$	− 9.70967i	− 0.350825i
$767$	− 17.1716i	− 0.620030i
$768$	0	0
$769$	48.0961i	1.73439i	0.497968	+	0.867195i	$0.334080\pi$
$769$	48.0961i	1.73439i	−0.497968	+	0.867195i	$0.665920\pi$
$770$	0	0
$771$	0	0
$772$	−11.5980	−0.417420
$773$	24.6549	0.886776	0.443388	−	0.896330i	$-0.353776\pi$
$773$	24.6549	0.886776	0.443388	+	0.896330i	$0.353776\pi$
$774$	0	0
$775$	21.0138i	0.754838i
$776$	−11.2179	−0.402698
$777$	0	0
$778$	10.6863	0.383122
$779$	7.11270i	0.254839i
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−5.86030	−0.209564
$783$	0	0
$784$	0	0
$785$	67.2132i	2.39894i
$786$	0	0
$787$	11.7206i	0.417794i	0.977938	+	0.208897i	$0.0669874\pi$
$787$	11.7206i	0.417794i	−0.977938	+	0.208897i	$0.933013\pi$
$788$	− 14.1838i	− 0.505276i
$789$	0	0
$790$	2.01094i	0.0715460i
$791$	0	0
$792$	0	0
$793$	−3.55635	−0.126290
$794$	−7.36851	−0.261498
$795$	0	0
$796$	− 30.3070i	− 1.07420i
$797$	−51.2345	−1.81482	−0.907410	−	0.420247i	$-0.861944\pi$
$797$	−51.2345	−1.81482	−0.907410	+	0.420247i	$0.861944\pi$
$798$	0	0
$799$	−41.4558	−1.46660
$800$	15.8284i	0.559619i
$801$	0	0
$802$	2.14214	0.0756414
$803$	34.1563	1.20535
$804$	0	0
$805$	0	0
$806$	7.11270i	0.250534i
$807$	0	0
$808$	− 8.49596i	− 0.298887i
$809$	16.0416i	0.563994i	0.959415	+	0.281997i	$0.0909968\pi$
$809$	16.0416i	0.563994i	−0.959415	+	0.281997i	$0.909003\pi$
$810$	0	0
$811$	8.28772i	0.291021i	0.989357	+	0.145511i	$0.0464825\pi$
$811$	8.28772i	0.291021i	−0.989357	+	0.145511i	$0.953518\pi$
$812$	0	0
$813$	0	0
$814$	10.8284	0.379536
$815$	26.8741	0.941359
$816$	0	0
$817$	− 26.2851i	− 0.919599i
$818$	3.93562	0.137606
$819$	0	0
$820$	−6.50253	−0.227078
$821$	− 23.7574i	− 0.829138i	−0.910018	−	0.414569i	$-0.863932\pi$
$821$	− 23.7574i	− 0.829138i	0.910018	−	0.414569i	$-0.136068\pi$
$822$	0	0
$823$	−41.9411	−1.46198	−0.730988	−	0.682390i	$-0.760940\pi$
$823$	−41.9411	−1.46198	−0.730988	+	0.682390i	$0.760940\pi$
$824$	−22.4357	−0.781586
$825$	0	0
$826$	0	0
$827$	− 39.6569i	− 1.37900i	−0.724284	−	0.689502i	$-0.757829\pi$
$827$	− 39.6569i	− 1.37900i	0.724284	−	0.689502i	$-0.242171\pi$
$828$	0	0
$829$	− 36.0810i	− 1.25315i	−0.779363	−	0.626573i	$-0.784457\pi$
$829$	− 36.0810i	− 1.25315i	0.779363	−	0.626573i	$-0.215543\pi$
$830$	− 14.2254i	− 0.493771i
$831$	0	0
$832$	− 12.2233i	− 0.423768i
$833$	0	0
$834$	0	0
$835$	−41.4558	−1.43464
$836$	51.7373	1.78937
$837$	0	0
$838$	7.87124i	0.271907i
$839$	−17.5809	−0.606960	−0.303480	−	0.952838i	$-0.598149\pi$
$839$	−17.5809	−0.606960	−0.303480	+	0.952838i	$0.598149\pi$
$840$	0	0
$841$	28.3137	0.976335
$842$	− 0.284271i	− 0.00979663i
$843$	0	0
$844$	−28.0000	−0.963800
$845$	12.9343	0.444954
$846$	0	0
$847$	0	0
$848$	− 21.2132i	− 0.728464i
$849$	0	0
$850$	− 10.5069i	− 0.360384i
$851$	10.8284i	0.371194i
$852$	0	0
$853$	16.0727i	0.550319i	0.961399	+	0.275159i	$0.0887306\pi$
$853$	16.0727i	0.550319i	−0.961399	+	0.275159i	$0.911269\pi$
$854$	0	0
$855$	0	0
$856$	21.1127	0.721617
$857$	−0.502734	−0.0171731	−0.00858654	−	0.999963i	$-0.502733\pi$
$857$	−0.502734	−0.0171731	−0.00858654	+	0.999963i	$0.502733\pi$
$858$	0	0
$859$	9.29319i	0.317079i	0.987353	+	0.158540i	$0.0506786\pi$
$859$	9.29319i	0.317079i	−0.987353	+	0.158540i	$0.949321\pi$
$860$	24.0302	0.819423
$861$	0	0
$862$	14.6863	0.500217
$863$	10.6863i	0.363766i	0.983320	+	0.181883i	$0.0582191\pi$
$863$	10.6863i	0.363766i	−0.983320	+	0.181883i	$0.941781\pi$
$864$	0	0
$865$	−3.55635	−0.120919
$866$	−0.502734	−0.0170836
$867$	0	0
$868$	0	0
$869$	8.00000i	0.271381i
$870$	0	0
$871$	24.8632i	0.842456i
$872$	− 8.58579i	− 0.290751i
$873$	0	0
$874$	− 4.85483i	− 0.164217i
$875$	0	0
$876$	0	0
$877$	−32.5269	−1.09836	−0.549178	−	0.835705i	$-0.685059\pi$
$877$	−32.5269	−1.09836	−0.549178	+	0.835705i	$0.685059\pi$
$878$	−10.2987	−0.347563
$879$	0	0
$880$	42.4441i	1.43079i
$881$	−23.6494	−0.796770	−0.398385	−	0.917218i	$-0.630429\pi$
$881$	−23.6494	−0.796770	−0.398385	+	0.917218i	$0.630429\pi$
$882$	0	0
$883$	13.4558	0.452825	0.226413	−	0.974031i	$-0.427300\pi$
$883$	13.4558	0.452825	0.226413	+	0.974031i	$0.427300\pi$
$884$	37.8995i	1.27470i
$885$	0	0
$886$	−7.94113	−0.266787
$887$	−50.7318	−1.70341	−0.851703	−	0.524024i	$-0.824430\pi$
$887$	−50.7318	−1.70341	−0.851703	+	0.524024i	$0.824430\pi$
$888$	0	0
$889$	0	0
$890$	− 13.6152i	− 0.456383i
$891$	0	0
$892$	− 6.27678i	− 0.210162i
$893$	− 34.3431i	− 1.14925i
$894$	0	0
$895$	77.6059i	2.59408i
$896$	0	0
$897$	0	0
$898$	6.52691	0.217806
$899$	4.85483	0.161918
$900$	0	0
$901$	50.0208i	1.66643i
$902$	2.42742	0.0808241
$903$	0	0
$904$	22.0416	0.733094
$905$	− 27.8406i	− 0.925454i
$906$	0	0
$907$	28.7696	0.955277	0.477639	−	0.878556i	$-0.341493\pi$
$907$	28.7696	0.955277	0.477639	+	0.878556i	$0.341493\pi$
$908$	−25.8686	−0.858481
$909$	0	0
$910$	0	0
$911$	− 42.4853i	− 1.40760i	−0.710398	−	0.703800i	$-0.751485\pi$
$911$	− 42.4853i	− 1.40760i	0.710398	−	0.703800i	$-0.248515\pi$
$912$	0	0
$913$	− 56.5921i	− 1.87292i
$914$	− 6.88730i	− 0.227812i
$915$	0	0
$916$	34.3646i	1.13544i
$917$	0	0
$918$	0	0
$919$	−41.4558	−1.36750	−0.683751	−	0.729715i	$-0.739653\pi$
$919$	−41.4558	−1.36750	−0.683751	+	0.729715i	$0.739653\pi$
$920$	9.29319	0.306387
$921$	0	0
$922$	4.35210i	0.143329i
$923$	2.42742	0.0798994
$924$	0	0
$925$	−19.4142	−0.638335
$926$	8.48528i	0.278844i
$927$	0	0
$928$	3.65685	0.120042
$929$	−0.502734	−0.0164942	−0.00824709	−	0.999966i	$-0.502625\pi$
$929$	−0.502734	−0.0164942	−0.00824709	+	0.999966i	$0.502625\pi$
$930$	0	0
$931$	0	0
$932$	9.45584i	0.309736i
$933$	0	0
$934$	5.86030i	0.191755i
$935$	− 100.083i	− 3.27307i
$936$	0	0
$937$	− 51.2345i	− 1.67376i	−0.547387	−	0.836879i	$-0.684378\pi$
$937$	− 51.2345i	− 1.67376i	0.547387	−	0.836879i	$-0.315622\pi$
$938$	0	0
$939$	0	0
$940$	31.3970	1.02406
$941$	4.64659	0.151475	0.0757373	−	0.997128i	$-0.475869\pi$
$941$	4.64659	0.151475	0.0757373	+	0.997128i	$0.475869\pi$
$942$	0	0
$943$	2.42742i	0.0790476i
$944$	17.5809	0.572210
$945$	0	0
$946$	−8.97056	−0.291658
$947$	10.9706i	0.356495i	0.983986	+	0.178248i	$0.0570428\pi$
$947$	10.9706i	0.356495i	−0.983986	+	0.178248i	$0.942957\pi$
$948$	0	0
$949$	−20.7279	−0.672857
$950$	8.70420	0.282401
$951$	0	0
$952$	0	0
$953$	− 10.3848i	− 0.336396i	−0.985753	−	0.168198i	$-0.946205\pi$
$953$	− 10.3848i	− 0.336396i	0.985753	−	0.168198i	$-0.0537948\pi$
$954$	0	0
$955$	44.4550i	1.43853i
$956$	21.3137i	0.689335i
$957$	0	0
$958$	17.5809i	0.568013i
$959$	0	0
$960$	0	0
$961$	−3.34315	−0.107843
$962$	−6.57128	−0.211866
$963$	0	0
$964$	46.3797i	1.49379i
$965$	−18.5864	−0.598317
$966$	0	0
$967$	−19.1127	−0.614623	−0.307311	−	0.951609i	$-0.599429\pi$
$967$	−19.1127	−0.614623	−0.307311	+	0.951609i	$0.599429\pi$
$968$	− 19.5269i	− 0.627619i
$969$	0	0
$970$	−8.58579	−0.275673
$971$	−4.85483	−0.155799	−0.0778995	−	0.996961i	$-0.524821\pi$
$971$	−4.85483	−0.155799	−0.0778995	+	0.996961i	$0.524821\pi$
$972$	0	0
$973$	0	0
$974$	− 17.1716i	− 0.550213i
$975$	0	0
$976$	− 3.64113i	− 0.116550i
$977$	− 44.8284i	− 1.43419i	−0.696976	−	0.717094i	$-0.745472\pi$
$977$	− 44.8284i	− 1.43419i	0.696976	−	0.717094i	$-0.254528\pi$
$978$	0	0
$979$	− 54.1647i	− 1.73111i
$980$	0	0
$981$	0	0
$982$	−10.4853	−0.334599
$983$	31.7289	1.01200	0.505998	−	0.862535i	$-0.331124\pi$
$983$	31.7289	1.01200	0.505998	+	0.862535i	$0.331124\pi$
$984$	0	0
$985$	− 22.7302i	− 0.724246i
$986$	−2.42742	−0.0773047
$987$	0	0
$988$	−31.3970	−0.998871
$989$	− 8.97056i	− 0.285247i
$990$	0	0
$991$	−59.3970	−1.88681	−0.943403	−	0.331647i	$-0.892396\pi$
$991$	−59.3970	−1.88681	−0.943403	+	0.331647i	$0.892396\pi$
$992$	−25.8686	−0.821330
$993$	0	0
$994$	0	0
$995$	− 48.5685i	− 1.53973i
$996$	0	0
$997$	39.5139i	1.25142i	0.780056	+	0.625709i	$0.215190\pi$
$997$	39.5139i	1.25142i	−0.780056	+	0.625709i	$0.784810\pi$
$998$	10.8284i	0.342768i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.c.b.440.6	yes	8
3.2	odd	2	inner	441.2.c.b.440.3	✓	8
4.3	odd	2		7056.2.k.g.881.5		8
7.2	even	3		441.2.p.c.80.5		16
7.3	odd	6		441.2.p.c.215.4		16
7.4	even	3		441.2.p.c.215.3		16
7.5	odd	6		441.2.p.c.80.6		16
7.6	odd	2	inner	441.2.c.b.440.5	yes	8
12.11	even	2		7056.2.k.g.881.4		8
21.2	odd	6		441.2.p.c.80.4		16
21.5	even	6		441.2.p.c.80.3		16
21.11	odd	6		441.2.p.c.215.6		16
21.17	even	6		441.2.p.c.215.5		16
21.20	even	2	inner	441.2.c.b.440.4	yes	8
28.27	even	2		7056.2.k.g.881.3		8
84.83	odd	2		7056.2.k.g.881.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.c.b.440.3	✓	8	3.2	odd	2	inner
441.2.c.b.440.4	yes	8	21.20	even	2	inner
441.2.c.b.440.5	yes	8	7.6	odd	2	inner
441.2.c.b.440.6	yes	8	1.1	even	1	trivial
441.2.p.c.80.3		16	21.5	even	6
441.2.p.c.80.4		16	21.2	odd	6
441.2.p.c.80.5		16	7.2	even	3
441.2.p.c.80.6		16	7.5	odd	6
441.2.p.c.215.3		16	7.4	even	3
441.2.p.c.215.4		16	7.3	odd	6
441.2.p.c.215.5		16	21.17	even	6
441.2.p.c.215.6		16	21.11	odd	6
7056.2.k.g.881.3		8	28.27	even	2
7056.2.k.g.881.4		8	12.11	even	2
7056.2.k.g.881.5		8	4.3	odd	2
7056.2.k.g.881.6		8	84.83	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)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} +1.58579i q^{8} +O(q^{10})\)	\(q+0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} +1.58579i q^{8} +1.21371i q^{10} +4.82843i q^{11} -2.93015i q^{13} +3.00000 q^{16} -7.07401 q^{17} -5.86030i q^{19} +5.35757 q^{20} -2.00000 q^{22} -2.00000i q^{23} +3.58579 q^{25} +1.21371 q^{26} -0.828427i q^{29} +5.86030i q^{31} +4.41421i q^{32} -2.93015i q^{34} -5.41421 q^{37} +2.42742 q^{38} +4.64659i q^{40} -1.21371 q^{41} +4.48528 q^{43} +8.82843i q^{44} +0.828427 q^{46} +5.86030 q^{47} +1.48528i q^{50} -5.35757i q^{52} -7.07107i q^{53} +14.1480i q^{55} +0.343146 q^{58} +5.86030 q^{59} -1.21371i q^{61} -2.42742 q^{62} +4.17157 q^{64} -8.58579i q^{65} -8.48528 q^{67} -12.9343 q^{68} +0.828427i q^{71} -7.07401i q^{73} -2.24264i q^{74} -10.7151i q^{76} +1.65685 q^{79} +8.79045 q^{80} -0.502734i q^{82} -11.7206 q^{83} -20.7279 q^{85} +1.85786i q^{86} -7.65685 q^{88} -11.2179 q^{89} -3.65685i q^{92} +2.42742i q^{94} -17.1716i q^{95} +7.07401i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 24 q^{16} - 16 q^{22} + 40 q^{25} - 32 q^{37} - 32 q^{43} - 16 q^{46} + 48 q^{58} + 56 q^{64} - 32 q^{79} - 64 q^{85} - 16 q^{88}+O(q^{100}) \) 8 * q - 8 * q^4 + 24 * q^16 - 16 * q^22 + 40 * q^25 - 32 * q^37 - 32 * q^43 - 16 * q^46 + 48 * q^58 + 56 * q^64 - 32 * q^79 - 64 * q^85 - 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more