LMFDB - Embedded newform 425.2.b.d.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("425.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.b (of order $2$, degree $1$, not 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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.d.324.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205i q^{2} -2.73205i q^{3} -1.00000 q^{4} -4.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -4.46410 q^{9} +4.73205 q^{11} +2.73205i q^{12} +4.00000i q^{13} -4.73205 q^{14} -5.00000 q^{16} -1.00000i q^{17} +7.73205i q^{18} +1.46410 q^{19} -7.46410 q^{21} -8.19615i q^{22} +8.19615i q^{23} -4.73205 q^{24} +6.92820 q^{26} +4.00000i q^{27} +2.73205i q^{28} +3.46410 q^{29} +3.26795 q^{31} +5.19615i q^{32} -12.9282i q^{33} -1.73205 q^{34} +4.46410 q^{36} -0.535898i q^{37} -2.53590i q^{38} +10.9282 q^{39} -3.46410 q^{41} +12.9282i q^{42} +0.535898i q^{43} -4.73205 q^{44} +14.1962 q^{46} +12.9282i q^{47} +13.6603i q^{48} -0.464102 q^{49} -2.73205 q^{51} -4.00000i q^{52} -6.00000i q^{53} +6.92820 q^{54} -4.73205 q^{56} -4.00000i q^{57} -6.00000i q^{58} -2.53590 q^{59} -4.92820 q^{61} -5.66025i q^{62} +12.1962i q^{63} -1.00000 q^{64} -22.3923 q^{66} -10.0000i q^{67} +1.00000i q^{68} +22.3923 q^{69} +11.6603 q^{71} +7.73205i q^{72} -6.39230i q^{73} -0.928203 q^{74} -1.46410 q^{76} -12.9282i q^{77} -18.9282i q^{78} -14.5885 q^{79} -2.46410 q^{81} +6.00000i q^{82} -8.53590i q^{83} +7.46410 q^{84} +0.928203 q^{86} -9.46410i q^{87} -8.19615i q^{88} -4.39230 q^{89} +10.9282 q^{91} -8.19615i q^{92} -8.92820i q^{93} +22.3923 q^{94} +14.1962 q^{96} -4.92820i q^{97} +0.803848i q^{98} -21.1244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 12 q^{6} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 8 q^{19} - 16 q^{21} - 12 q^{24} + 20 q^{31} + 4 q^{36} + 16 q^{39} - 12 q^{44} + 36 q^{46} + 12 q^{49} - 4 q^{51} - 12 q^{56} - 24 q^{59}+ \cdots - 36 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 12 * q^6 - 4 * q^9 + 12 * q^11 - 12 * q^14 - 20 * q^16 - 8 * q^19 - 16 * q^21 - 12 * q^24 + 20 * q^31 + 4 * q^36 + 16 * q^39 - 12 * q^44 + 36 * q^46 + 12 * q^49 - 4 * q^51 - 12 * q^56 - 24 * q^59 + 8 * q^61 - 4 * q^64 - 48 * q^66 + 48 * q^69 + 12 * q^71 + 24 * q^74 + 8 * q^76 + 4 * q^79 + 4 * q^81 + 16 * q^84 - 24 * q^86 + 24 * q^89 + 16 * q^91 + 48 * q^94 + 36 * q^96 - 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.73205i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 1.73205i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	− 2.73205i	− 1.57735i	−0.614810	−	0.788675i	$-0.710767\pi$
$3$	− 2.73205i	− 1.57735i	0.614810	−	0.788675i	$-0.289233\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−4.73205	−1.93185
$7$	− 2.73205i	− 1.03262i	−0.856402	−	0.516309i	$-0.827306\pi$
$7$	− 2.73205i	− 1.03262i	0.856402	−	0.516309i	$-0.172694\pi$
$8$	− 1.73205i	− 0.612372i
$9$	−4.46410	−1.48803
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	2.73205i	0.788675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−4.73205	−1.26469
$15$	0	0
$16$	−5.00000	−1.25000
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	7.73205i	1.82246i
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	−7.46410	−1.62880
$22$	− 8.19615i	− 1.74743i
$23$	8.19615i	1.70902i	0.519438	+	0.854508i	$0.326141\pi$
$23$	8.19615i	1.70902i	−0.519438	+	0.854508i	$0.673859\pi$
$24$	−4.73205	−0.965926
$25$	0	0
$26$	6.92820	1.35873
$27$	4.00000i	0.769800i
$28$	2.73205i	0.516309i
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	5.19615i	0.918559i
$33$	− 12.9282i	− 2.25051i
$34$	−1.73205	−0.297044
$35$	0	0
$36$	4.46410	0.744017
$37$	− 0.535898i	− 0.0881012i	−0.999029	−	0.0440506i	$-0.985974\pi$
$37$	− 0.535898i	− 0.0881012i	0.999029	−	0.0440506i	$-0.0140263\pi$
$38$	− 2.53590i	− 0.411377i
$39$	10.9282	1.74991
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	12.9282i	1.99487i
$43$	0.535898i	0.0817237i	0.999165	+	0.0408619i	$0.0130104\pi$
$43$	0.535898i	0.0817237i	−0.999165	+	0.0408619i	$0.986990\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	14.1962	2.09311
$47$	12.9282i	1.88577i	0.333115	+	0.942886i	$0.391900\pi$
$47$	12.9282i	1.88577i	−0.333115	+	0.942886i	$0.608100\pi$
$48$	13.6603i	1.97169i
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	−2.73205	−0.382564
$52$	− 4.00000i	− 0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	6.92820	0.942809
$55$	0	0
$56$	−4.73205	−0.632347
$57$	− 4.00000i	− 0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	− 5.66025i	− 0.718853i
$63$	12.1962i	1.53657i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−22.3923	−2.75630
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	1.00000i	0.121268i
$69$	22.3923	2.69572
$70$	0	0
$71$	11.6603	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$71$	11.6603	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$72$	7.73205i	0.911231i
$73$	− 6.39230i	− 0.748163i	−0.927396	−	0.374081i	$-0.877958\pi$
$73$	− 6.39230i	− 0.748163i	0.927396	−	0.374081i	$-0.122042\pi$
$74$	−0.928203	−0.107901
$75$	0	0
$76$	−1.46410	−0.167944
$77$	− 12.9282i	− 1.47331i
$78$	− 18.9282i	− 2.14320i
$79$	−14.5885	−1.64133	−0.820665	−	0.571410i	$-0.806397\pi$
$79$	−14.5885	−1.64133	−0.820665	+	0.571410i	$0.806397\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	6.00000i	0.662589i
$83$	− 8.53590i	− 0.936937i	−0.883480	−	0.468468i	$-0.844806\pi$
$83$	− 8.53590i	− 0.936937i	0.883480	−	0.468468i	$-0.155194\pi$
$84$	7.46410	0.814400
$85$	0	0
$86$	0.928203	0.100091
$87$	− 9.46410i	− 1.01466i
$88$	− 8.19615i	− 0.873713i
$89$	−4.39230	−0.465583	−0.232792	−	0.972527i	$-0.574786\pi$
$89$	−4.39230	−0.465583	−0.232792	+	0.972527i	$0.574786\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	− 8.19615i	− 0.854508i
$93$	− 8.92820i	− 0.925812i
$94$	22.3923	2.30959
$95$	0	0
$96$	14.1962	1.44889
$97$	− 4.92820i	− 0.500383i	−0.968196	−	0.250192i	$-0.919506\pi$
$97$	− 4.92820i	− 0.500383i	0.968196	−	0.250192i	$-0.0804936\pi$
$98$	0.803848i	0.0812009i
$99$	−21.1244	−2.12308
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	4.73205i	0.468543i
$103$	− 8.92820i	− 0.879722i	−0.898066	−	0.439861i	$-0.855028\pi$
$103$	− 8.92820i	− 0.879722i	0.898066	−	0.439861i	$-0.144972\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	−10.3923	−1.00939
$107$	17.6603i	1.70728i	0.520862	+	0.853641i	$0.325610\pi$
$107$	17.6603i	1.70728i	−0.520862	+	0.853641i	$0.674390\pi$
$108$	− 4.00000i	− 0.384900i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−1.46410	−0.138966
$112$	13.6603i	1.29077i
$113$	17.3205i	1.62938i	0.579899	+	0.814688i	$0.303092\pi$
$113$	17.3205i	1.62938i	−0.579899	+	0.814688i	$0.696908\pi$
$114$	−6.92820	−0.648886
$115$	0	0
$116$	−3.46410	−0.321634
$117$	− 17.8564i	− 1.65083i
$118$	4.39230i	0.404344i
$119$	−2.73205	−0.250447
$120$	0	0
$121$	11.3923	1.03566
$122$	8.53590i	0.772804i
$123$	9.46410i	0.853349i
$124$	−3.26795	−0.293471
$125$	0	0
$126$	21.1244	1.88191
$127$	− 14.3923i	− 1.27711i	−0.769576	−	0.638555i	$-0.779532\pi$
$127$	− 14.3923i	− 1.27711i	0.769576	−	0.638555i	$-0.220468\pi$
$128$	12.1244i	1.07165i
$129$	1.46410	0.128907
$130$	0	0
$131$	−2.19615	−0.191879	−0.0959394	−	0.995387i	$-0.530585\pi$
$131$	−2.19615	−0.191879	−0.0959394	+	0.995387i	$0.530585\pi$
$132$	12.9282i	1.12526i
$133$	− 4.00000i	− 0.346844i
$134$	−17.3205	−1.49626
$135$	0	0
$136$	−1.73205	−0.148522
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	− 38.7846i	− 3.30157i
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	0	0
$141$	35.3205	2.97452
$142$	− 20.1962i	− 1.69482i
$143$	18.9282i	1.58286i
$144$	22.3205	1.86004
$145$	0	0
$146$	−11.0718	−0.916308
$147$	1.26795i	0.104579i
$148$	0.535898i	0.0440506i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	− 2.53590i	− 0.205689i
$153$	4.46410i	0.360901i
$154$	−22.3923	−1.80442
$155$	0	0
$156$	−10.9282	−0.874957
$157$	8.92820i	0.712548i	0.934381	+	0.356274i	$0.115953\pi$
$157$	8.92820i	0.712548i	−0.934381	+	0.356274i	$0.884047\pi$
$158$	25.2679i	2.01021i
$159$	−16.3923	−1.29999
$160$	0	0
$161$	22.3923	1.76476
$162$	4.26795i	0.335322i
$163$	0.196152i	0.0153638i	0.999970	+	0.00768192i	$0.00244526\pi$
$163$	0.196152i	0.0153638i	−0.999970	+	0.00768192i	$0.997555\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	12.5885i	0.974124i	0.873367	+	0.487062i	$0.161931\pi$
$167$	12.5885i	0.974124i	−0.873367	+	0.487062i	$0.838069\pi$
$168$	12.9282i	0.997433i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−6.53590	−0.499813
$172$	− 0.535898i	− 0.0408619i
$173$	− 3.46410i	− 0.263371i	−0.991292	−	0.131685i	$-0.957961\pi$
$173$	− 3.46410i	− 0.263371i	0.991292	−	0.131685i	$-0.0420389\pi$
$174$	−16.3923	−1.24270
$175$	0	0
$176$	−23.6603	−1.78346
$177$	6.92820i	0.520756i
$178$	7.60770i	0.570221i
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	− 18.9282i	− 1.40305i
$183$	13.4641i	0.995295i
$184$	14.1962	1.04655
$185$	0	0
$186$	−15.4641	−1.13388
$187$	− 4.73205i	− 0.346042i
$188$	− 12.9282i	− 0.942886i
$189$	10.9282	0.794910
$190$	0	0
$191$	1.85641	0.134325	0.0671624	−	0.997742i	$-0.478605\pi$
$191$	1.85641	0.134325	0.0671624	+	0.997742i	$0.478605\pi$
$192$	2.73205i	0.197169i
$193$	− 16.5359i	− 1.19028i	−0.803622	−	0.595140i	$-0.797097\pi$
$193$	− 16.5359i	− 1.19028i	0.803622	−	0.595140i	$-0.202903\pi$
$194$	−8.53590	−0.612842
$195$	0	0
$196$	0.464102	0.0331501
$197$	17.3205i	1.23404i	0.786949	+	0.617018i	$0.211659\pi$
$197$	17.3205i	1.23404i	−0.786949	+	0.617018i	$0.788341\pi$
$198$	36.5885i	2.60023i
$199$	−10.1962	−0.722786	−0.361393	−	0.932414i	$-0.617699\pi$
$199$	−10.1962	−0.722786	−0.361393	+	0.932414i	$0.617699\pi$
$200$	0	0
$201$	−27.3205	−1.92704
$202$	16.3923i	1.15336i
$203$	− 9.46410i	− 0.664250i
$204$	2.73205	0.191282
$205$	0	0
$206$	−15.4641	−1.07744
$207$	− 36.5885i	− 2.54307i
$208$	− 20.0000i	− 1.38675i
$209$	6.92820	0.479234
$210$	0	0
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	6.00000i	0.412082i
$213$	− 31.8564i	− 2.18277i
$214$	30.5885	2.09098
$215$	0	0
$216$	6.92820	0.471405
$217$	− 8.92820i	− 0.606086i
$218$	− 17.3205i	− 1.17309i
$219$	−17.4641	−1.18011
$220$	0	0
$221$	4.00000	0.269069
$222$	2.53590i	0.170198i
$223$	26.3923i	1.76736i	0.468092	+	0.883680i	$0.344942\pi$
$223$	26.3923i	1.76736i	−0.468092	+	0.883680i	$0.655058\pi$
$224$	14.1962	0.948520
$225$	0	0
$226$	30.0000	1.99557
$227$	22.7321i	1.50878i	0.656427	+	0.754390i	$0.272067\pi$
$227$	22.7321i	1.50878i	−0.656427	+	0.754390i	$0.727933\pi$
$228$	4.00000i	0.264906i
$229$	8.39230	0.554579	0.277290	−	0.960786i	$-0.410564\pi$
$229$	8.39230	0.554579	0.277290	+	0.960786i	$0.410564\pi$
$230$	0	0
$231$	−35.3205	−2.32392
$232$	− 6.00000i	− 0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−30.9282	−2.02184
$235$	0	0
$236$	2.53590	0.165073
$237$	39.8564i	2.58895i
$238$	4.73205i	0.306733i
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	− 19.7321i	− 1.26842i
$243$	18.7321i	1.20166i
$244$	4.92820	0.315496
$245$	0	0
$246$	16.3923	1.04514
$247$	5.85641i	0.372634i
$248$	− 5.66025i	− 0.359426i
$249$	−23.3205	−1.47788
$250$	0	0
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	− 12.1962i	− 0.768285i
$253$	38.7846i	2.43837i
$254$	−24.9282	−1.56413
$255$	0	0
$256$	19.0000	1.18750
$257$	− 6.92820i	− 0.432169i	−0.976375	−	0.216085i	$-0.930671\pi$
$257$	− 6.92820i	− 0.432169i	0.976375	−	0.216085i	$-0.0693287\pi$
$258$	− 2.53590i	− 0.157878i
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	−15.4641	−0.957204
$262$	3.80385i	0.235002i
$263$	1.60770i	0.0991347i	0.998771	+	0.0495674i	$0.0157843\pi$
$263$	1.60770i	0.0991347i	−0.998771	+	0.0495674i	$0.984216\pi$
$264$	−22.3923	−1.37815
$265$	0	0
$266$	−6.92820	−0.424795
$267$	12.0000i	0.734388i
$268$	10.0000i	0.610847i
$269$	−0.928203	−0.0565935	−0.0282968	−	0.999600i	$-0.509008\pi$
$269$	−0.928203	−0.0565935	−0.0282968	+	0.999600i	$0.509008\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	5.00000i	0.303170i
$273$	− 29.8564i	− 1.80699i
$274$	0	0
$275$	0	0
$276$	−22.3923	−1.34786
$277$	20.9282i	1.25745i	0.777626	+	0.628727i	$0.216424\pi$
$277$	20.9282i	1.25745i	−0.777626	+	0.628727i	$0.783576\pi$
$278$	− 6.33975i	− 0.380233i
$279$	−14.5885	−0.873388
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	− 61.1769i	− 3.64303i
$283$	5.26795i	0.313147i	0.987666	+	0.156574i	$0.0500448\pi$
$283$	5.26795i	0.313147i	−0.987666	+	0.156574i	$0.949955\pi$
$284$	−11.6603	−0.691909
$285$	0	0
$286$	32.7846	1.93859
$287$	9.46410i	0.558648i
$288$	− 23.1962i	− 1.36685i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−13.4641	−0.789280
$292$	6.39230i	0.374081i
$293$	0.928203i	0.0542262i	0.999632	+	0.0271131i	$0.00863143\pi$
$293$	0.928203i	0.0542262i	−0.999632	+	0.0271131i	$0.991369\pi$
$294$	2.19615	0.128082
$295$	0	0
$296$	−0.928203	−0.0539507
$297$	18.9282i	1.09833i
$298$	− 10.3923i	− 0.602010i
$299$	−32.7846	−1.89598
$300$	0	0
$301$	1.46410	0.0843894
$302$	2.53590i	0.145925i
$303$	25.8564i	1.48541i
$304$	−7.32051	−0.419860
$305$	0	0
$306$	7.73205	0.442012
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	12.9282i	0.736653i
$309$	−24.3923	−1.38763
$310$	0	0
$311$	−16.0526	−0.910257	−0.455129	−	0.890426i	$-0.650407\pi$
$311$	−16.0526	−0.910257	−0.455129	+	0.890426i	$0.650407\pi$
$312$	− 18.9282i	− 1.07160i
$313$	26.3923i	1.49178i	0.666068	+	0.745891i	$0.267976\pi$
$313$	26.3923i	1.49178i	−0.666068	+	0.745891i	$0.732024\pi$
$314$	15.4641	0.872690
$315$	0	0
$316$	14.5885	0.820665
$317$	− 24.9282i	− 1.40011i	−0.714090	−	0.700054i	$-0.753159\pi$
$317$	− 24.9282i	− 1.40011i	0.714090	−	0.700054i	$-0.246841\pi$
$318$	28.3923i	1.59216i
$319$	16.3923	0.917793
$320$	0	0
$321$	48.2487	2.69298
$322$	− 38.7846i	− 2.16138i
$323$	− 1.46410i	− 0.0814648i
$324$	2.46410	0.136895
$325$	0	0
$326$	0.339746	0.0188168
$327$	− 27.3205i	− 1.51083i
$328$	6.00000i	0.331295i
$329$	35.3205	1.94728
$330$	0	0
$331$	−6.53590	−0.359245	−0.179623	−	0.983736i	$-0.557488\pi$
$331$	−6.53590	−0.359245	−0.179623	+	0.983736i	$0.557488\pi$
$332$	8.53590i	0.468468i
$333$	2.39230i	0.131097i
$334$	21.8038	1.19305
$335$	0	0
$336$	37.3205	2.03600
$337$	− 6.78461i	− 0.369581i	−0.982778	−	0.184791i	$-0.940839\pi$
$337$	− 6.78461i	− 0.369581i	0.982778	−	0.184791i	$-0.0591607\pi$
$338$	5.19615i	0.282633i
$339$	47.3205	2.57010
$340$	0	0
$341$	15.4641	0.837428
$342$	11.3205i	0.612143i
$343$	− 17.8564i	− 0.964155i
$344$	0.928203	0.0500454
$345$	0	0
$346$	−6.00000	−0.322562
$347$	3.80385i	0.204201i	0.994774	+	0.102101i	$0.0325564\pi$
$347$	3.80385i	0.204201i	−0.994774	+	0.102101i	$0.967444\pi$
$348$	9.46410i	0.507329i
$349$	−10.7846	−0.577287	−0.288643	−	0.957437i	$-0.593204\pi$
$349$	−10.7846	−0.577287	−0.288643	+	0.957437i	$0.593204\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	24.5885i	1.31057i
$353$	− 26.7846i	− 1.42560i	−0.701367	−	0.712800i	$-0.747427\pi$
$353$	− 26.7846i	− 1.42560i	0.701367	−	0.712800i	$-0.252573\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	4.39230	0.232792
$357$	7.46410i	0.395042i
$358$	− 19.6077i	− 1.03630i
$359$	21.4641	1.13283	0.566416	−	0.824119i	$-0.308330\pi$
$359$	21.4641	1.13283	0.566416	+	0.824119i	$0.308330\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	4.14359i	0.217782i
$363$	− 31.1244i	− 1.63361i
$364$	−10.9282	−0.572793
$365$	0	0
$366$	23.3205	1.21898
$367$	− 7.80385i	− 0.407358i	−0.979038	−	0.203679i	$-0.934710\pi$
$367$	− 7.80385i	− 0.407358i	0.979038	−	0.203679i	$-0.0652898\pi$
$368$	− 40.9808i	− 2.13627i
$369$	15.4641	0.805029
$370$	0	0
$371$	−16.3923	−0.851046
$372$	8.92820i	0.462906i
$373$	− 20.0000i	− 1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$	− 20.0000i	− 1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	−8.19615	−0.423813
$375$	0	0
$376$	22.3923	1.15479
$377$	13.8564i	0.713641i
$378$	− 18.9282i	− 0.973562i
$379$	−17.8038	−0.914522	−0.457261	−	0.889332i	$-0.651170\pi$
$379$	−17.8038	−0.914522	−0.457261	+	0.889332i	$0.651170\pi$
$380$	0	0
$381$	−39.3205	−2.01445
$382$	− 3.21539i	− 0.164514i
$383$	− 15.4641i	− 0.790179i	−0.918643	−	0.395089i	$-0.870714\pi$
$383$	− 15.4641i	− 0.790179i	0.918643	−	0.395089i	$-0.129286\pi$
$384$	33.1244	1.69037
$385$	0	0
$386$	−28.6410	−1.45779
$387$	− 2.39230i	− 0.121608i
$388$	4.92820i	0.250192i
$389$	4.39230	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$389$	4.39230	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	0.803848i	0.0406004i
$393$	6.00000i	0.302660i
$394$	30.0000	1.51138
$395$	0	0
$396$	21.1244	1.06154
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	17.6603i	0.885229i
$399$	−10.9282	−0.547094
$400$	0	0
$401$	12.9282	0.645604	0.322802	−	0.946467i	$-0.395375\pi$
$401$	12.9282	0.645604	0.322802	+	0.946467i	$0.395375\pi$
$402$	47.3205i	2.36013i
$403$	13.0718i	0.651153i
$404$	9.46410	0.470857
$405$	0	0
$406$	−16.3923	−0.813536
$407$	− 2.53590i	− 0.125700i
$408$	4.73205i	0.234271i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	8.92820i	0.439861i
$413$	6.92820i	0.340915i
$414$	−63.3731	−3.11462
$415$	0	0
$416$	−20.7846	−1.01905
$417$	− 10.0000i	− 0.489702i
$418$	− 12.0000i	− 0.586939i
$419$	−38.1962	−1.86600	−0.933002	−	0.359871i	$-0.882821\pi$
$419$	−38.1962	−1.86600	−0.933002	+	0.359871i	$0.882821\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	− 17.6603i	− 0.859688i
$423$	− 57.7128i	− 2.80609i
$424$	−10.3923	−0.504695
$425$	0	0
$426$	−55.1769	−2.67333
$427$	13.4641i	0.651574i
$428$	− 17.6603i	− 0.853641i
$429$	51.7128	2.49672
$430$	0	0
$431$	9.80385	0.472235	0.236117	−	0.971725i	$-0.424125\pi$
$431$	9.80385	0.472235	0.236117	+	0.971725i	$0.424125\pi$
$432$	− 20.0000i	− 0.962250i
$433$	17.8564i	0.858124i	0.903275	+	0.429062i	$0.141156\pi$
$433$	17.8564i	0.858124i	−0.903275	+	0.429062i	$0.858844\pi$
$434$	−15.4641	−0.742301
$435$	0	0
$436$	−10.0000	−0.478913
$437$	12.0000i	0.574038i
$438$	30.2487i	1.44534i
$439$	20.0526	0.957056	0.478528	−	0.878072i	$-0.341170\pi$
$439$	20.0526	0.957056	0.478528	+	0.878072i	$0.341170\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	− 6.92820i	− 0.329541i
$443$	12.9282i	0.614237i	0.951671	+	0.307119i	$0.0993649\pi$
$443$	12.9282i	0.614237i	−0.951671	+	0.307119i	$0.900635\pi$
$444$	1.46410	0.0694832
$445$	0	0
$446$	45.7128	2.16456
$447$	− 16.3923i	− 0.775329i
$448$	2.73205i	0.129077i
$449$	34.3923	1.62307	0.811537	−	0.584302i	$-0.198631\pi$
$449$	34.3923	1.62307	0.811537	+	0.584302i	$0.198631\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	− 17.3205i	− 0.814688i
$453$	4.00000i	0.187936i
$454$	39.3731	1.84787
$455$	0	0
$456$	−6.92820	−0.324443
$457$	− 36.7846i	− 1.72071i	−0.509694	−	0.860356i	$-0.670241\pi$
$457$	− 36.7846i	− 1.72071i	0.509694	−	0.860356i	$-0.329759\pi$
$458$	− 14.5359i	− 0.679218i
$459$	4.00000	0.186704
$460$	0	0
$461$	−24.9282	−1.16102	−0.580511	−	0.814252i	$-0.697147\pi$
$461$	−24.9282	−1.16102	−0.580511	+	0.814252i	$0.697147\pi$
$462$	61.1769i	2.84621i
$463$	23.8564i	1.10870i	0.832283	+	0.554351i	$0.187033\pi$
$463$	23.8564i	1.10870i	−0.832283	+	0.554351i	$0.812967\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	−10.3923	−0.481414
$467$	1.60770i	0.0743953i	0.999308	+	0.0371976i	$0.0118431\pi$
$467$	1.60770i	0.0743953i	−0.999308	+	0.0371976i	$0.988157\pi$
$468$	17.8564i	0.825413i
$469$	−27.3205	−1.26154
$470$	0	0
$471$	24.3923	1.12394
$472$	4.39230i	0.202172i
$473$	2.53590i	0.116601i
$474$	69.0333	3.17081
$475$	0	0
$476$	2.73205	0.125223
$477$	26.7846i	1.22638i
$478$	− 36.0000i	− 1.64660i
$479$	11.6603	0.532771	0.266385	−	0.963867i	$-0.414171\pi$
$479$	11.6603	0.532771	0.266385	+	0.963867i	$0.414171\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	9.71281i	0.442407i
$483$	− 61.1769i	− 2.78365i
$484$	−11.3923	−0.517832
$485$	0	0
$486$	32.4449	1.47173
$487$	24.9808i	1.13199i	0.824410	+	0.565993i	$0.191507\pi$
$487$	24.9808i	1.13199i	−0.824410	+	0.565993i	$0.808493\pi$
$488$	8.53590i	0.386402i
$489$	0.535898	0.0242342
$490$	0	0
$491$	−19.6077	−0.884883	−0.442441	−	0.896797i	$-0.645888\pi$
$491$	−19.6077	−0.884883	−0.442441	+	0.896797i	$0.645888\pi$
$492$	− 9.46410i	− 0.426675i
$493$	− 3.46410i	− 0.156015i
$494$	10.1436	0.456382
$495$	0	0
$496$	−16.3397	−0.733676
$497$	− 31.8564i	− 1.42896i
$498$	40.3923i	1.81002i
$499$	15.6603	0.701049	0.350525	−	0.936554i	$-0.386003\pi$
$499$	15.6603	0.701049	0.350525	+	0.936554i	$0.386003\pi$
$500$	0	0
$501$	34.3923	1.53653
$502$	− 12.0000i	− 0.535586i
$503$	− 15.1244i	− 0.674362i	−0.941440	−	0.337181i	$-0.890527\pi$
$503$	− 15.1244i	− 0.674362i	0.941440	−	0.337181i	$-0.109473\pi$
$504$	21.1244	0.940954
$505$	0	0
$506$	67.1769	2.98638
$507$	8.19615i	0.364004i
$508$	14.3923i	0.638555i
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−17.4641	−0.772566
$512$	− 8.66025i	− 0.382733i
$513$	5.85641i	0.258567i
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−1.46410	−0.0644535
$517$	61.1769i	2.69056i
$518$	2.53590i	0.111421i
$519$	−9.46410	−0.415428
$520$	0	0
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	26.7846i	1.17233i
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	2.19615	0.0959394
$525$	0	0
$526$	2.78461	0.121415
$527$	− 3.26795i	− 0.142354i
$528$	64.6410i	2.81314i
$529$	−44.1769	−1.92074
$530$	0	0
$531$	11.3205	0.491268
$532$	4.00000i	0.173422i
$533$	− 13.8564i	− 0.600188i
$534$	20.7846	0.899438
$535$	0	0
$536$	−17.3205	−0.748132
$537$	− 30.9282i	− 1.33465i
$538$	1.60770i	0.0693127i
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	39.1769	1.68435	0.842174	−	0.539207i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539207i	$0.181276\pi$
$542$	− 5.07180i	− 0.217852i
$543$	6.53590i	0.280482i
$544$	5.19615	0.222783
$545$	0	0
$546$	−51.7128	−2.21310
$547$	− 39.9090i	− 1.70638i	−0.521597	−	0.853192i	$-0.674663\pi$
$547$	− 39.9090i	− 1.70638i	0.521597	−	0.853192i	$-0.325337\pi$
$548$	0	0
$549$	22.0000	0.938937
$550$	0	0
$551$	5.07180	0.216066
$552$	− 38.7846i	− 1.65078i
$553$	39.8564i	1.69487i
$554$	36.2487	1.54006
$555$	0	0
$556$	−3.66025	−0.155229
$557$	− 6.92820i	− 0.293557i	−0.989169	−	0.146779i	$-0.953109\pi$
$557$	− 6.92820i	− 0.293557i	0.989169	−	0.146779i	$-0.0468905\pi$
$558$	25.2679i	1.06968i
$559$	−2.14359	−0.0906643
$560$	0	0
$561$	−12.9282	−0.545829
$562$	22.3923i	0.944562i
$563$	− 27.4641i	− 1.15747i	−0.815514	−	0.578737i	$-0.803546\pi$
$563$	− 27.4641i	− 1.15747i	0.815514	−	0.578737i	$-0.196454\pi$
$564$	−35.3205	−1.48726
$565$	0	0
$566$	9.12436	0.383525
$567$	6.73205i	0.282720i
$568$	− 20.1962i	− 0.847412i
$569$	−40.6410	−1.70376	−0.851880	−	0.523737i	$-0.824537\pi$
$569$	−40.6410	−1.70376	−0.851880	+	0.523737i	$0.824537\pi$
$570$	0	0
$571$	−36.4449	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$571$	−36.4449	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$572$	− 18.9282i	− 0.791428i
$573$	− 5.07180i	− 0.211877i
$574$	16.3923	0.684202
$575$	0	0
$576$	4.46410	0.186004
$577$	− 38.6410i	− 1.60865i	−0.594192	−	0.804323i	$-0.702528\pi$
$577$	− 38.6410i	− 1.60865i	0.594192	−	0.804323i	$-0.297472\pi$
$578$	1.73205i	0.0720438i
$579$	−45.1769	−1.87749
$580$	0	0
$581$	−23.3205	−0.967498
$582$	23.3205i	0.966666i
$583$	− 28.3923i	− 1.17589i
$584$	−11.0718	−0.458154
$585$	0	0
$586$	1.60770	0.0664133
$587$	46.3923i	1.91482i	0.288740	+	0.957408i	$0.406764\pi$
$587$	46.3923i	1.91482i	−0.288740	+	0.957408i	$0.593236\pi$
$588$	− 1.26795i	− 0.0522893i
$589$	4.78461	0.197146
$590$	0	0
$591$	47.3205	1.94651
$592$	2.67949i	0.110126i
$593$	− 19.8564i	− 0.815405i	−0.913115	−	0.407702i	$-0.866330\pi$
$593$	− 19.8564i	− 0.815405i	0.913115	−	0.407702i	$-0.133670\pi$
$594$	32.7846	1.34517
$595$	0	0
$596$	−6.00000	−0.245770
$597$	27.8564i	1.14009i
$598$	56.7846i	2.32210i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	− 2.53590i	− 0.103356i
$603$	44.6410i	1.81792i
$604$	1.46410	0.0595734
$605$	0	0
$606$	44.7846	1.81925
$607$	− 21.6603i	− 0.879163i	−0.898203	−	0.439581i	$-0.855127\pi$
$607$	− 21.6603i	− 0.879163i	0.898203	−	0.439581i	$-0.144873\pi$
$608$	7.60770i	0.308533i
$609$	−25.8564	−1.04775
$610$	0	0
$611$	−51.7128	−2.09208
$612$	− 4.46410i	− 0.180451i
$613$	− 15.8564i	− 0.640434i	−0.947344	−	0.320217i	$-0.896244\pi$
$613$	− 15.8564i	− 0.640434i	0.947344	−	0.320217i	$-0.103756\pi$
$614$	−17.3205	−0.698999
$615$	0	0
$616$	−22.3923	−0.902212
$617$	− 27.4641i	− 1.10566i	−0.833293	−	0.552832i	$-0.813547\pi$
$617$	− 27.4641i	− 1.10566i	0.833293	−	0.552832i	$-0.186453\pi$
$618$	42.2487i	1.69949i
$619$	−38.5885	−1.55100	−0.775501	−	0.631347i	$-0.782502\pi$
$619$	−38.5885	−1.55100	−0.775501	+	0.631347i	$0.782502\pi$
$620$	0	0
$621$	−32.7846	−1.31560
$622$	27.8038i	1.11483i
$623$	12.0000i	0.480770i
$624$	−54.6410	−2.18739
$625$	0	0
$626$	45.7128	1.82705
$627$	− 18.9282i	− 0.755920i
$628$	− 8.92820i	− 0.356274i
$629$	−0.535898	−0.0213677
$630$	0	0
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	25.2679i	1.00511i
$633$	− 27.8564i	− 1.10719i
$634$	−43.1769	−1.71477
$635$	0	0
$636$	16.3923	0.649997
$637$	− 1.85641i	− 0.0735535i
$638$	− 28.3923i	− 1.12406i
$639$	−52.0526	−2.05917
$640$	0	0
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	− 83.5692i	− 3.29821i
$643$	24.1962i	0.954203i	0.878848	+	0.477102i	$0.158313\pi$
$643$	24.1962i	0.954203i	−0.878848	+	0.477102i	$0.841687\pi$
$644$	−22.3923	−0.882380
$645$	0	0
$646$	−2.53590	−0.0997736
$647$	− 38.7846i	− 1.52478i	−0.647118	−	0.762390i	$-0.724026\pi$
$647$	− 38.7846i	− 1.52478i	0.647118	−	0.762390i	$-0.275974\pi$
$648$	4.26795i	0.167661i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−24.3923	−0.956010
$652$	− 0.196152i	− 0.00768192i
$653$	25.6077i	1.00211i	0.865416	+	0.501053i	$0.167054\pi$
$653$	25.6077i	1.00211i	−0.865416	+	0.501053i	$0.832946\pi$
$654$	−47.3205	−1.85038
$655$	0	0
$656$	17.3205	0.676252
$657$	28.5359i	1.11329i
$658$	− 61.1769i	− 2.38492i
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	0	0
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	11.3205i	0.439984i
$663$	− 10.9282i	− 0.424416i
$664$	−14.7846	−0.573754
$665$	0	0
$666$	4.14359	0.160561
$667$	28.3923i	1.09935i
$668$	− 12.5885i	− 0.487062i
$669$	72.1051	2.78774
$670$	0	0
$671$	−23.3205	−0.900278
$672$	− 38.7846i	− 1.49615i
$673$	− 23.4641i	− 0.904475i	−0.891898	−	0.452237i	$-0.850626\pi$
$673$	− 23.4641i	− 0.904475i	0.891898	−	0.452237i	$-0.149374\pi$
$674$	−11.7513	−0.452643
$675$	0	0
$676$	3.00000	0.115385
$677$	2.78461i	0.107021i	0.998567	+	0.0535106i	$0.0170411\pi$
$677$	2.78461i	0.107021i	−0.998567	+	0.0535106i	$0.982959\pi$
$678$	− 81.9615i	− 3.14771i
$679$	−13.4641	−0.516705
$680$	0	0
$681$	62.1051	2.37987
$682$	− 26.7846i	− 1.02564i
$683$	30.8372i	1.17995i	0.807421	+	0.589976i	$0.200863\pi$
$683$	30.8372i	1.17995i	−0.807421	+	0.589976i	$0.799137\pi$
$684$	6.53590	0.249906
$685$	0	0
$686$	−30.9282	−1.18084
$687$	− 22.9282i	− 0.874766i
$688$	− 2.67949i	− 0.102155i
$689$	24.0000	0.914327
$690$	0	0
$691$	−38.9808	−1.48290	−0.741449	−	0.671009i	$-0.765861\pi$
$691$	−38.9808	−1.48290	−0.741449	+	0.671009i	$0.765861\pi$
$692$	3.46410i	0.131685i
$693$	57.7128i	2.19233i
$694$	6.58846	0.250094
$695$	0	0
$696$	−16.3923	−0.621349
$697$	3.46410i	0.131212i
$698$	18.6795i	0.707029i
$699$	−16.3923	−0.620014
$700$	0	0
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	27.7128i	1.04595i
$703$	− 0.784610i	− 0.0295921i
$704$	−4.73205	−0.178346
$705$	0	0
$706$	−46.3923	−1.74600
$707$	25.8564i	0.972430i
$708$	− 6.92820i	− 0.260378i
$709$	−4.53590	−0.170349	−0.0851746	−	0.996366i	$-0.527145\pi$
$709$	−4.53590	−0.170349	−0.0851746	+	0.996366i	$0.527145\pi$
$710$	0	0
$711$	65.1244	2.44235
$712$	7.60770i	0.285110i
$713$	26.7846i	1.00309i
$714$	12.9282	0.483826
$715$	0	0
$716$	−11.3205	−0.423067
$717$	− 56.7846i	− 2.12066i
$718$	− 37.1769i	− 1.38743i
$719$	5.41154	0.201816	0.100908	−	0.994896i	$-0.467825\pi$
$719$	5.41154	0.201816	0.100908	+	0.994896i	$0.467825\pi$
$720$	0	0
$721$	−24.3923	−0.908417
$722$	29.1962i	1.08657i
$723$	15.3205i	0.569776i
$724$	2.39230	0.0889093
$725$	0	0
$726$	−53.9090	−2.00075
$727$	0.143594i	0.00532559i	0.999996	+	0.00266279i	$0.000847595\pi$
$727$	0.143594i	0.00532559i	−0.999996	+	0.00266279i	$0.999152\pi$
$728$	− 18.9282i	− 0.701526i
$729$	43.7846	1.62165
$730$	0	0
$731$	0.535898	0.0198209
$732$	− 13.4641i	− 0.497648i
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	−13.5167	−0.498909
$735$	0	0
$736$	−42.5885	−1.56983
$737$	− 47.3205i	− 1.74307i
$738$	− 26.7846i	− 0.985955i
$739$	11.6077	0.426996	0.213498	−	0.976944i	$-0.431514\pi$
$739$	11.6077	0.426996	0.213498	+	0.976944i	$0.431514\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	28.3923i	1.04231i
$743$	22.7321i	0.833958i	0.908916	+	0.416979i	$0.136911\pi$
$743$	22.7321i	0.833958i	−0.908916	+	0.416979i	$0.863089\pi$
$744$	−15.4641	−0.566941
$745$	0	0
$746$	−34.6410	−1.26830
$747$	38.1051i	1.39419i
$748$	4.73205i	0.173021i
$749$	48.2487	1.76297
$750$	0	0
$751$	43.6603	1.59319	0.796593	−	0.604516i	$-0.206634\pi$
$751$	43.6603	1.59319	0.796593	+	0.604516i	$0.206634\pi$
$752$	− 64.6410i	− 2.35722i
$753$	− 18.9282i	− 0.689782i
$754$	24.0000	0.874028
$755$	0	0
$756$	−10.9282	−0.397455
$757$	30.6410i	1.11367i	0.830624	+	0.556833i	$0.187984\pi$
$757$	30.6410i	1.11367i	−0.830624	+	0.556833i	$0.812016\pi$
$758$	30.8372i	1.12006i
$759$	105.962	3.84616
$760$	0	0
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	68.1051i	2.46719i
$763$	− 27.3205i	− 0.989069i
$764$	−1.85641	−0.0671624
$765$	0	0
$766$	−26.7846	−0.967767
$767$	− 10.1436i	− 0.366264i
$768$	− 51.9090i	− 1.87310i
$769$	−36.3923	−1.31234	−0.656170	−	0.754613i	$-0.727825\pi$
$769$	−36.3923	−1.31234	−0.656170	+	0.754613i	$0.727825\pi$
$770$	0	0
$771$	−18.9282	−0.681683
$772$	16.5359i	0.595140i
$773$	46.6410i	1.67756i	0.544470	+	0.838780i	$0.316731\pi$
$773$	46.6410i	1.67756i	−0.544470	+	0.838780i	$0.683269\pi$
$774$	−4.14359	−0.148938
$775$	0	0
$776$	−8.53590	−0.306421
$777$	4.00000i	0.143499i
$778$	− 7.60770i	− 0.272749i
$779$	−5.07180	−0.181716
$780$	0	0
$781$	55.1769	1.97439
$782$	− 14.1962i	− 0.507653i
$783$	13.8564i	0.495188i
$784$	2.32051	0.0828753
$785$	0	0
$786$	10.3923	0.370681
$787$	43.9090i	1.56519i	0.622534	+	0.782593i	$0.286103\pi$
$787$	43.9090i	1.56519i	−0.622534	+	0.782593i	$0.713897\pi$
$788$	− 17.3205i	− 0.617018i
$789$	4.39230	0.156370
$790$	0	0
$791$	47.3205	1.68252
$792$	36.5885i	1.30011i
$793$	− 19.7128i	− 0.700023i
$794$	24.2487	0.860555
$795$	0	0
$796$	10.1962	0.361393
$797$	− 24.9282i	− 0.883002i	−0.897261	−	0.441501i	$-0.854446\pi$
$797$	− 24.9282i	− 0.883002i	0.897261	−	0.441501i	$-0.145554\pi$
$798$	18.9282i	0.670051i
$799$	12.9282	0.457367
$800$	0	0
$801$	19.6077	0.692804
$802$	− 22.3923i	− 0.790700i
$803$	− 30.2487i	− 1.06745i
$804$	27.3205	0.963520
$805$	0	0
$806$	22.6410	0.797496
$807$	2.53590i	0.0892679i
$808$	16.3923i	0.576679i
$809$	55.8564	1.96381	0.981903	−	0.189383i	$-0.0606487\pi$
$809$	55.8564	1.96381	0.981903	+	0.189383i	$0.0606487\pi$
$810$	0	0
$811$	44.8372	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$811$	44.8372	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$812$	9.46410i	0.332125i
$813$	− 8.00000i	− 0.280572i
$814$	−4.39230	−0.153950
$815$	0	0
$816$	13.6603	0.478205
$817$	0.784610i	0.0274500i
$818$	45.0333i	1.57455i
$819$	−48.7846	−1.70467
$820$	0	0
$821$	−24.9282	−0.870000	−0.435000	−	0.900430i	$-0.643252\pi$
$821$	−24.9282	−0.870000	−0.435000	+	0.900430i	$0.643252\pi$
$822$	0	0
$823$	8.98076i	0.313050i	0.987674	+	0.156525i	$0.0500291\pi$
$823$	8.98076i	0.313050i	−0.987674	+	0.156525i	$0.949971\pi$
$824$	−15.4641	−0.538718
$825$	0	0
$826$	12.0000	0.417533
$827$	− 15.8038i	− 0.549554i	−0.961508	−	0.274777i	$-0.911396\pi$
$827$	− 15.8038i	− 0.549554i	0.961508	−	0.274777i	$-0.0886040\pi$
$828$	36.5885i	1.27154i
$829$	−17.7128	−0.615191	−0.307596	−	0.951517i	$-0.599524\pi$
$829$	−17.7128	−0.615191	−0.307596	+	0.951517i	$0.599524\pi$
$830$	0	0
$831$	57.1769	1.98345
$832$	− 4.00000i	− 0.138675i
$833$	0.464102i	0.0160802i
$834$	−17.3205	−0.599760
$835$	0	0
$836$	−6.92820	−0.239617
$837$	13.0718i	0.451827i
$838$	66.1577i	2.28538i
$839$	−19.2679	−0.665203	−0.332602	−	0.943067i	$-0.607926\pi$
$839$	−19.2679	−0.665203	−0.332602	+	0.943067i	$0.607926\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	− 9.46410i	− 0.326154i
$843$	35.3205i	1.21650i
$844$	−10.1962	−0.350966
$845$	0	0
$846$	−99.9615	−3.43675
$847$	− 31.1244i	− 1.06945i
$848$	30.0000i	1.03020i
$849$	14.3923	0.493943
$850$	0	0
$851$	4.39230	0.150566
$852$	31.8564i	1.09138i
$853$	23.1769i	0.793562i	0.917913	+	0.396781i	$0.129873\pi$
$853$	23.1769i	0.793562i	−0.917913	+	0.396781i	$0.870127\pi$
$854$	23.3205	0.798011
$855$	0	0
$856$	30.5885	1.04549
$857$	31.1769i	1.06498i	0.846435	+	0.532492i	$0.178744\pi$
$857$	31.1769i	1.06498i	−0.846435	+	0.532492i	$0.821256\pi$
$858$	− 89.5692i	− 3.05784i
$859$	25.4641	0.868824	0.434412	−	0.900714i	$-0.356956\pi$
$859$	25.4641	0.868824	0.434412	+	0.900714i	$0.356956\pi$
$860$	0	0
$861$	25.8564	0.881184
$862$	− 16.9808i	− 0.578367i
$863$	− 23.0718i	− 0.785373i	−0.919672	−	0.392687i	$-0.871546\pi$
$863$	− 23.0718i	− 0.785373i	0.919672	−	0.392687i	$-0.128454\pi$
$864$	−20.7846	−0.707107
$865$	0	0
$866$	30.9282	1.05098
$867$	2.73205i	0.0927853i
$868$	8.92820i	0.303043i
$869$	−69.0333	−2.34180
$870$	0	0
$871$	40.0000	1.35535
$872$	− 17.3205i	− 0.586546i
$873$	22.0000i	0.744587i
$874$	20.7846	0.703050
$875$	0	0
$876$	17.4641	0.590057
$877$	− 1.21539i	− 0.0410408i	−0.999789	−	0.0205204i	$-0.993468\pi$
$877$	− 1.21539i	− 0.0410408i	0.999789	−	0.0205204i	$-0.00653231\pi$
$878$	− 34.7321i	− 1.17215i
$879$	2.53590	0.0855337
$880$	0	0
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	− 3.58846i	− 0.120830i
$883$	10.0000i	0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$	10.0000i	0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	22.3923	0.752284
$887$	3.12436i	0.104906i	0.998623	+	0.0524528i	$0.0167039\pi$
$887$	3.12436i	0.104906i	−0.998623	+	0.0524528i	$0.983296\pi$
$888$	2.53590i	0.0850992i
$889$	−39.3205	−1.31877
$890$	0	0
$891$	−11.6603	−0.390633
$892$	− 26.3923i	− 0.883680i
$893$	18.9282i	0.633408i
$894$	−28.3923	−0.949581
$895$	0	0
$896$	33.1244	1.10661
$897$	89.5692i	2.99063i
$898$	− 59.5692i	− 1.98785i
$899$	11.3205	0.377560
$900$	0	0
$901$	−6.00000	−0.199889
$902$	28.3923i	0.945360i
$903$	− 4.00000i	− 0.133112i
$904$	30.0000	0.997785
$905$	0	0
$906$	6.92820	0.230174
$907$	30.7321i	1.02044i	0.860044	+	0.510220i	$0.170436\pi$
$907$	30.7321i	1.02044i	−0.860044	+	0.510220i	$0.829564\pi$
$908$	− 22.7321i	− 0.754390i
$909$	42.2487	1.40130
$910$	0	0
$911$	36.3397	1.20399	0.601995	−	0.798500i	$-0.294373\pi$
$911$	36.3397	1.20399	0.601995	+	0.798500i	$0.294373\pi$
$912$	20.0000i	0.662266i
$913$	− 40.3923i	− 1.33679i
$914$	−63.7128	−2.10743
$915$	0	0
$916$	−8.39230	−0.277290
$917$	6.00000i	0.198137i
$918$	− 6.92820i	− 0.228665i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−27.3205	−0.900241
$922$	43.1769i	1.42196i
$923$	46.6410i	1.53521i
$924$	35.3205	1.16196
$925$	0	0
$926$	41.3205	1.35788
$927$	39.8564i	1.30906i
$928$	18.0000i	0.590879i
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−0.679492	−0.0222694
$932$	6.00000i	0.196537i
$933$	43.8564i	1.43579i
$934$	2.78461	0.0911152
$935$	0	0
$936$	−30.9282	−1.01092
$937$	− 32.6410i	− 1.06634i	−0.846010	−	0.533168i	$-0.821001\pi$
$937$	− 32.6410i	− 1.06634i	0.846010	−	0.533168i	$-0.178999\pi$
$938$	47.3205i	1.54507i
$939$	72.1051	2.35306
$940$	0	0
$941$	48.2487	1.57286	0.786432	−	0.617677i	$-0.211926\pi$
$941$	48.2487	1.57286	0.786432	+	0.617677i	$0.211926\pi$
$942$	− 42.2487i	− 1.37654i
$943$	− 28.3923i	− 0.924581i
$944$	12.6795	0.412682
$945$	0	0
$946$	4.39230	0.142806
$947$	8.19615i	0.266339i	0.991093	+	0.133170i	$0.0425155\pi$
$947$	8.19615i	0.266339i	−0.991093	+	0.133170i	$0.957485\pi$
$948$	− 39.8564i	− 1.29448i
$949$	25.5692	0.830012
$950$	0	0
$951$	−68.1051	−2.20846
$952$	4.73205i	0.153367i
$953$	8.78461i	0.284561i	0.989826	+	0.142281i	$0.0454436\pi$
$953$	8.78461i	0.284561i	−0.989826	+	0.142281i	$0.954556\pi$
$954$	46.3923	1.50201
$955$	0	0
$956$	−20.7846	−0.672222
$957$	− 44.7846i	− 1.44768i
$958$	− 20.1962i	− 0.652508i
$959$	0	0
$960$	0	0
$961$	−20.3205	−0.655500
$962$	− 3.71281i	− 0.119706i
$963$	− 78.8372i	− 2.54049i
$964$	5.60770	0.180612
$965$	0	0
$966$	−105.962	−3.40926
$967$	39.1769i	1.25984i	0.776658	+	0.629922i	$0.216913\pi$
$967$	39.1769i	1.25984i	−0.776658	+	0.629922i	$0.783087\pi$
$968$	− 19.7321i	− 0.634212i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	54.2487	1.74092	0.870462	−	0.492236i	$-0.163820\pi$
$971$	54.2487	1.74092	0.870462	+	0.492236i	$0.163820\pi$
$972$	− 18.7321i	− 0.600831i
$973$	− 10.0000i	− 0.320585i
$974$	43.2679	1.38639
$975$	0	0
$976$	24.6410	0.788740
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	− 0.928203i	− 0.0296807i
$979$	−20.7846	−0.664279
$980$	0	0
$981$	−44.6410	−1.42528
$982$	33.9615i	1.08376i
$983$	− 58.7321i	− 1.87326i	−0.350318	−	0.936631i	$-0.613927\pi$
$983$	− 58.7321i	− 1.87326i	0.350318	−	0.936631i	$-0.386073\pi$
$984$	16.3923	0.522568
$985$	0	0
$986$	−6.00000	−0.191079
$987$	− 96.4974i	− 3.07155i
$988$	− 5.85641i	− 0.186317i
$989$	−4.39230	−0.139667
$990$	0	0
$991$	−2.98076	−0.0946870	−0.0473435	−	0.998879i	$-0.515076\pi$
$991$	−2.98076	−0.0946870	−0.0473435	+	0.998879i	$0.515076\pi$
$992$	16.9808i	0.539140i
$993$	17.8564i	0.566656i
$994$	−55.1769	−1.75011
$995$	0	0
$996$	23.3205	0.738939
$997$	− 2.39230i	− 0.0757651i	−0.999282	−	0.0378825i	$-0.987939\pi$
$997$	− 2.39230i	− 0.0757651i	0.999282	−	0.0378825i	$-0.0120613\pi$
$998$	− 27.1244i	− 0.858606i
$999$	2.14359	0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.d.324.1		4
5.2	odd	4		425.2.a.e.1.2		2
5.3	odd	4		85.2.a.c.1.1	✓	2
5.4	even	2	inner	425.2.b.d.324.4		4
15.2	even	4		3825.2.a.v.1.1		2
15.8	even	4		765.2.a.g.1.2		2
20.3	even	4		1360.2.a.k.1.1		2
20.7	even	4		6800.2.a.bg.1.2		2
35.13	even	4		4165.2.a.t.1.1		2
40.3	even	4		5440.2.a.bl.1.2		2
40.13	odd	4		5440.2.a.bb.1.1		2
85.13	odd	4		1445.2.d.e.866.4		4
85.33	odd	4		1445.2.a.g.1.1		2
85.38	odd	4		1445.2.d.e.866.3		4
85.67	odd	4		7225.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	5.3	odd	4
425.2.a.e.1.2		2	5.2	odd	4
425.2.b.d.324.1		4	1.1	even	1	trivial
425.2.b.d.324.4		4	5.4	even	2	inner
765.2.a.g.1.2		2	15.8	even	4
1360.2.a.k.1.1		2	20.3	even	4
1445.2.a.g.1.1		2	85.33	odd	4
1445.2.d.e.866.3		4	85.38	odd	4
1445.2.d.e.866.4		4	85.13	odd	4
3825.2.a.v.1.1		2	15.2	even	4
4165.2.a.t.1.1		2	35.13	even	4
5440.2.a.bb.1.1		2	40.13	odd	4
5440.2.a.bl.1.2		2	40.3	even	4
6800.2.a.bg.1.2		2	20.7	even	4
7225.2.a.l.1.2		2	85.67	odd	4

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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -2.73205i q^{3} -1.00000 q^{4} -4.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -4.46410 q^{9} +4.73205 q^{11} +2.73205i q^{12} +4.00000i q^{13} -4.73205 q^{14} -5.00000 q^{16} -1.00000i q^{17} +7.73205i q^{18} +1.46410 q^{19} -7.46410 q^{21} -8.19615i q^{22} +8.19615i q^{23} -4.73205 q^{24} +6.92820 q^{26} +4.00000i q^{27} +2.73205i q^{28} +3.46410 q^{29} +3.26795 q^{31} +5.19615i q^{32} -12.9282i q^{33} -1.73205 q^{34} +4.46410 q^{36} -0.535898i q^{37} -2.53590i q^{38} +10.9282 q^{39} -3.46410 q^{41} +12.9282i q^{42} +0.535898i q^{43} -4.73205 q^{44} +14.1962 q^{46} +12.9282i q^{47} +13.6603i q^{48} -0.464102 q^{49} -2.73205 q^{51} -4.00000i q^{52} -6.00000i q^{53} +6.92820 q^{54} -4.73205 q^{56} -4.00000i q^{57} -6.00000i q^{58} -2.53590 q^{59} -4.92820 q^{61} -5.66025i q^{62} +12.1962i q^{63} -1.00000 q^{64} -22.3923 q^{66} -10.0000i q^{67} +1.00000i q^{68} +22.3923 q^{69} +11.6603 q^{71} +7.73205i q^{72} -6.39230i q^{73} -0.928203 q^{74} -1.46410 q^{76} -12.9282i q^{77} -18.9282i q^{78} -14.5885 q^{79} -2.46410 q^{81} +6.00000i q^{82} -8.53590i q^{83} +7.46410 q^{84} +0.928203 q^{86} -9.46410i q^{87} -8.19615i q^{88} -4.39230 q^{89} +10.9282 q^{91} -8.19615i q^{92} -8.92820i q^{93} +22.3923 q^{94} +14.1962 q^{96} -4.92820i q^{97} +0.803848i q^{98} -21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 12 q^{6} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 8 q^{19} - 16 q^{21} - 12 q^{24} + 20 q^{31} + 4 q^{36} + 16 q^{39} - 12 q^{44} + 36 q^{46} + 12 q^{49} - 4 q^{51} - 12 q^{56} - 24 q^{59}+ \cdots - 36 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 12 * q^6 - 4 * q^9 + 12 * q^11 - 12 * q^14 - 20 * q^16 - 8 * q^19 - 16 * q^21 - 12 * q^24 + 20 * q^31 + 4 * q^36 + 16 * q^39 - 12 * q^44 + 36 * q^46 + 12 * q^49 - 4 * q^51 - 12 * q^56 - 24 * q^59 + 8 * q^61 - 4 * q^64 - 48 * q^66 + 48 * q^69 + 12 * q^71 + 24 * q^74 + 8 * q^76 + 4 * q^79 + 4 * q^81 + 16 * q^84 - 24 * q^86 + 24 * q^89 + 16 * q^91 + 48 * q^94 + 36 * q^96 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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