LMFDB - Embedded newform 2600.2.k.f.2001.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(2001,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2600.2001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2600 = 2^{3} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2600.k (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:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} + 60x^{16} + 1134x^{12} + 6924x^{8} + 3545x^{4} + 64$ x^20 + 60x^16 + 1134x^12 + 6924x^8 + 3545x^4 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{30}$
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.4
Root			$1.47826 + 1.47826i$ of defining polynomial
Character	$\chi$	$=$	2600.2001
Dual form			2600.2.k.f.2001.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14926 q^{3} +2.12171i q^{7} +1.61930 q^{9} +0.796315i q^{11} +(0.479016 - 3.57359i) q^{13} -4.56010 q^{17} +0.220019i q^{19} -4.56010i q^{21} -4.73499 q^{23} +2.96747 q^{27} +8.17671 q^{29} -1.17805i q^{31} -1.71149i q^{33} +0.121710i q^{37} +(-1.02953 + 7.68056i) q^{39} -4.87481i q^{41} -3.16559 q^{43} -2.12171i q^{47} +2.49835 q^{49} +9.80082 q^{51} -11.0043 q^{53} -0.472877i q^{57} -3.72871i q^{59} -3.36438 q^{61} +3.43569i q^{63} +7.43569i q^{67} +10.1767 q^{69} +15.2081i q^{71} -5.43569i q^{73} -1.68955 q^{77} +9.05575 q^{79} -11.2358 q^{81} +14.1563i q^{83} -17.5738 q^{87} -15.6846i q^{89} +(7.58212 + 1.01633i) q^{91} +2.53193i q^{93} +2.09143i q^{97} +1.28948i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 16 q^{9} - 16 q^{29} + 24 q^{39} - 24 q^{49} - 60 q^{51} - 32 q^{61} + 24 q^{69} - 56 q^{79} + 44 q^{81} + 4 q^{91}+O(q^{100})$ 20 * q + 16 * q^9 - 16 * q^29 + 24 * q^39 - 24 * q^49 - 60 * q^51 - 32 * q^61 + 24 * q^69 - 56 * q^79 + 44 * q^81 + 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$ .

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$	−2.14926			−1.24087			−0.620437	−	0.784256i	$-0.713045\pi$
$3$	−2.14926			−1.24087			−0.620437	+	0.784256i	$0.713045\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			2.12171i			0.801931i	0.916093	+	0.400966i	$0.131325\pi$
$7$			2.12171i			0.801931i	−0.916093	+	0.400966i	$0.868675\pi$
$8$		0			0
$9$	1.61930			0.539768
$10$		0			0
$11$			0.796315i			0.240098i	0.992768	+	0.120049i	$0.0383052\pi$
$11$			0.796315i			0.240098i	−0.992768	+	0.120049i	$0.961695\pi$
$12$		0			0
$13$	0.479016	−	3.57359i	0.132855	−	0.991135i
$13$	0.479016	−	3.57359i	0.132855	−	0.991135i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.56010			−1.10599			−0.552993	−	0.833186i	$-0.686514\pi$
$17$	−4.56010			−1.10599			−0.552993	+	0.833186i	$0.686514\pi$
$18$		0			0
$19$			0.220019i			0.0504758i	0.999681	+	0.0252379i	$0.00803432\pi$
$19$			0.220019i			0.0504758i	−0.999681	+	0.0252379i	$0.991966\pi$
$20$		0			0
$21$		−	4.56010i		−	0.995095i
$22$		0			0
$23$	−4.73499			−0.987314			−0.493657	−	0.869657i	$-0.664340\pi$
$23$	−4.73499			−0.987314			−0.493657	+	0.869657i	$0.664340\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.96747			0.571090
$28$		0			0
$29$	8.17671			1.51838			0.759188	−	0.650871i	$-0.225596\pi$
$29$	8.17671			1.51838			0.759188	+	0.650871i	$0.225596\pi$
$30$		0			0
$31$		−	1.17805i		−	0.211584i	−0.994388	−	0.105792i	$-0.966262\pi$
$31$		−	1.17805i		−	0.211584i	0.994388	−	0.105792i	$-0.0337378\pi$
$32$		0			0
$33$		−	1.71149i		−	0.297931i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.121710i			0.0200090i	0.999950	+	0.0100045i	$0.00318459\pi$
$37$			0.121710i			0.0200090i	−0.999950	+	0.0100045i	$0.996815\pi$
$38$		0			0
$39$	−1.02953	+	7.68056i	−0.164857	+	1.22987i
$40$		0			0
$41$		−	4.87481i		−	0.761317i	−0.924716	−	0.380659i	$-0.875697\pi$
$41$		−	4.87481i		−	0.761317i	0.924716	−	0.380659i	$-0.124303\pi$
$42$		0			0
$43$	−3.16559			−0.482748			−0.241374	−	0.970432i	$-0.577598\pi$
$43$	−3.16559			−0.482748			−0.241374	+	0.970432i	$0.577598\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.12171i		−	0.309483i	−0.987955	−	0.154742i	$-0.950545\pi$
$47$		−	2.12171i		−	0.309483i	0.987955	−	0.154742i	$-0.0494545\pi$
$48$		0			0
$49$	2.49835			0.356907
$50$		0			0
$51$	9.80082			1.37239
$52$		0			0
$53$	−11.0043			−1.51156			−0.755779	−	0.654827i	$-0.772742\pi$
$53$	−11.0043			−1.51156			−0.755779	+	0.654827i	$0.772742\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	0.472877i		−	0.0626341i
$58$		0			0
$59$		−	3.72871i		−	0.485437i	−0.970097	−	0.242719i	$-0.921961\pi$
$59$		−	3.72871i		−	0.485437i	0.970097	−	0.242719i	$-0.0780392\pi$
$60$		0			0
$61$	−3.36438			−0.430764			−0.215382	−	0.976530i	$-0.569100\pi$
$61$	−3.36438			−0.430764			−0.215382	+	0.976530i	$0.569100\pi$
$62$		0			0
$63$			3.43569i			0.432857i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			7.43569i			0.908415i	0.890896	+	0.454207i	$0.150077\pi$
$67$			7.43569i			0.908415i	−0.890896	+	0.454207i	$0.849923\pi$
$68$		0			0
$69$	10.1767			1.22513
$70$		0			0
$71$			15.2081i			1.80487i	0.430823	+	0.902437i	$0.358223\pi$
$71$			15.2081i			1.80487i	−0.430823	+	0.902437i	$0.641777\pi$
$72$		0			0
$73$		−	5.43569i		−	0.636200i	−0.948057	−	0.318100i	$-0.896955\pi$
$73$		−	5.43569i		−	0.636200i	0.948057	−	0.318100i	$-0.103045\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.68955			−0.192542
$78$		0			0
$79$	9.05575			1.01885			0.509426	−	0.860515i	$-0.329858\pi$
$79$	9.05575			1.01885			0.509426	+	0.860515i	$0.329858\pi$
$80$		0			0
$81$	−11.2358			−1.24842
$82$		0			0
$83$			14.1563i			1.55386i	0.629587	+	0.776930i	$0.283224\pi$
$83$			14.1563i			1.55386i	−0.629587	+	0.776930i	$0.716776\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.5738			−1.88411
$88$		0			0
$89$		−	15.6846i		−	1.66256i	−0.555854	−	0.831280i	$-0.687609\pi$
$89$		−	15.6846i		−	1.66256i	0.555854	−	0.831280i	$-0.312391\pi$
$90$		0			0
$91$	7.58212	+	1.01633i	0.794822	+	0.106541i
$92$		0			0
$93$			2.53193i			0.262549i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			2.09143i			0.212352i	0.994347	+	0.106176i	$0.0338608\pi$
$97$			2.09143i			0.212352i	−0.994347	+	0.106176i	$0.966139\pi$
$98$		0			0
$99$			1.28948i			0.129597i
$100$		0			0
$101$	6.29285			0.626162			0.313081	−	0.949726i	$-0.398639\pi$
$101$	6.29285			0.626162			0.313081	+	0.949726i	$0.398639\pi$
$102$		0			0
$103$	11.9659			1.17904			0.589518	−	0.807755i	$-0.299318\pi$
$103$	11.9659			1.17904			0.589518	+	0.807755i	$0.299318\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.5166			1.40337			0.701685	−	0.712488i	$-0.252432\pi$
$107$	14.5166			1.40337			0.701685	+	0.712488i	$0.252432\pi$
$108$		0			0
$109$		−	7.92736i		−	0.759303i	−0.925130	−	0.379652i	$-0.876044\pi$
$109$		−	7.92736i		−	0.759303i	0.925130	−	0.379652i	$-0.123956\pi$
$110$		0			0
$111$		−	0.261586i		−	0.0248287i
$112$		0			0
$113$	14.0651			1.32314			0.661569	−	0.749884i	$-0.269891\pi$
$113$	14.0651			1.32314			0.661569	+	0.749884i	$0.269891\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.775673	−	5.78673i	0.0717110	−	0.534983i
$118$		0			0
$119$		−	9.67521i		−	0.886925i
$120$		0			0
$121$	10.3659			0.942353
$122$		0			0
$123$			10.4772i			0.944699i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.0754			1.07152			0.535758	−	0.844371i	$-0.320026\pi$
$127$	12.0754			1.07152			0.535758	+	0.844371i	$0.320026\pi$
$128$		0			0
$129$	6.80367			0.599029
$130$		0			0
$131$	7.22229			0.631014			0.315507	−	0.948923i	$-0.397825\pi$
$131$	7.22229			0.631014			0.315507	+	0.948923i	$0.397825\pi$
$132$		0			0
$133$	−0.466816			−0.0404781
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.6883i		−	1.25490i	−0.778656	−	0.627452i	$-0.784098\pi$
$137$		−	14.6883i		−	1.25490i	0.778656	−	0.627452i	$-0.215902\pi$
$138$		0			0
$139$	17.8503			1.51404			0.757020	−	0.653392i	$-0.226655\pi$
$139$	17.8503			1.51404			0.757020	+	0.653392i	$0.226655\pi$
$140$		0			0
$141$			4.56010i			0.384030i
$142$		0			0
$143$	2.84570	+	0.381448i	0.237970	+	0.0318983i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.36959			−0.442876
$148$		0			0
$149$			16.6689i			1.36557i	0.730618	+	0.682786i	$0.239232\pi$
$149$			16.6689i			1.36557i	−0.730618	+	0.682786i	$0.760768\pi$
$150$		0			0
$151$		−	0.999600i		−	0.0813463i	−0.999173	−	0.0406732i	$-0.987050\pi$
$151$		−	0.999600i		−	0.0813463i	0.999173	−	0.0406732i	$-0.0129502\pi$
$152$		0			0
$153$	−7.38419			−0.596976
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.7778			0.860162			0.430081	−	0.902790i	$-0.358485\pi$
$157$	10.7778			0.860162			0.430081	+	0.902790i	$0.358485\pi$
$158$		0			0
$159$	23.6511			1.87565
$160$		0			0
$161$		−	10.0463i		−	0.791757i
$162$		0			0
$163$		−	2.91622i		−	0.228416i	−0.993457	−	0.114208i	$-0.963567\pi$
$163$		−	2.91622i		−	0.228416i	0.993457	−	0.114208i	$-0.0364330\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.9225i		−	1.54165i	−0.637046	−	0.770826i	$-0.719844\pi$
$167$		−	19.9225i		−	1.54165i	0.637046	−	0.770826i	$-0.280156\pi$
$168$		0			0
$169$	−12.5411	−	3.42362i	−0.964699	−	0.263355i
$170$		0			0
$171$			0.356277i			0.0272452i
$172$		0			0
$173$	17.6853			1.34459			0.672293	−	0.740285i	$-0.265309\pi$
$173$	17.6853			1.34459			0.672293	+	0.740285i	$0.265309\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			8.01396i			0.602366i
$178$		0			0
$179$	−9.28135			−0.693721			−0.346860	−	0.937917i	$-0.612752\pi$
$179$	−9.28135			−0.693721			−0.346860	+	0.937917i	$0.612752\pi$
$180$		0			0
$181$	21.9022			1.62797			0.813987	−	0.580883i	$-0.197293\pi$
$181$	21.9022			1.62797			0.813987	+	0.580883i	$0.197293\pi$
$182$		0			0
$183$	7.23091			0.534524
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	3.63128i		−	0.265545i
$188$		0			0
$189$			6.29611i			0.457975i
$190$		0			0
$191$	−23.0822			−1.67017			−0.835084	−	0.550123i	$-0.814581\pi$
$191$	−23.0822			−1.67017			−0.835084	+	0.550123i	$0.814581\pi$
$192$		0			0
$193$		−	18.0186i		−	1.29700i	−0.761213	−	0.648502i	$-0.775396\pi$
$193$		−	18.0186i		−	1.29700i	0.761213	−	0.648502i	$-0.224604\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.75883i		−	0.552794i	−0.961044	−	0.276397i	$-0.910860\pi$
$197$		−	7.75883i		−	0.552794i	0.961044	−	0.276397i	$-0.0891404\pi$
$198$		0			0
$199$	10.9641			0.777221			0.388611	−	0.921402i	$-0.372955\pi$
$199$	10.9641			0.777221			0.388611	+	0.921402i	$0.372955\pi$
$200$		0			0
$201$		−	15.9812i		−	1.12723i
$202$		0			0
$203$			17.3486i			1.21763i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.66739			−0.532920
$208$		0			0
$209$	−0.175204			−0.0121191
$210$		0			0
$211$	−9.28135			−0.638954			−0.319477	−	0.947594i	$-0.603507\pi$
$211$	−9.28135			−0.638954			−0.319477	+	0.947594i	$0.603507\pi$
$212$		0			0
$213$		−	32.6862i		−	2.23962i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.49948			0.169676
$218$		0			0
$219$			11.6827i			0.789444i
$220$		0			0
$221$	−2.18436	+	16.2959i	−0.146936	+	1.09618i
$222$		0			0
$223$		−	8.71704i		−	0.583736i	−0.956459	−	0.291868i	$-0.905723\pi$
$223$		−	8.71704i		−	0.583736i	0.956459	−	0.291868i	$-0.0942769\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			26.6758i			1.77054i	0.465082	+	0.885268i	$0.346025\pi$
$227$			26.6758i			1.77054i	−0.465082	+	0.885268i	$0.653975\pi$
$228$		0			0
$229$			28.6735i			1.89480i	0.320058	+	0.947398i	$0.396298\pi$
$229$			28.6735i			1.89480i	−0.320058	+	0.947398i	$0.603702\pi$
$230$		0			0
$231$	3.63128			0.238920
$232$		0			0
$233$	−1.07818			−0.0706341			−0.0353171	−	0.999376i	$-0.511244\pi$
$233$	−1.07818			−0.0706341			−0.0353171	+	0.999376i	$0.511244\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.4631			−1.26427
$238$		0			0
$239$		−	20.8728i		−	1.35015i	−0.737751	−	0.675073i	$-0.764112\pi$
$239$		−	20.8728i		−	1.35015i	0.737751	−	0.675073i	$-0.235888\pi$
$240$		0			0
$241$		−	0.466816i		−	0.0300703i	−0.999887	−	0.0150351i	$-0.995214\pi$
$241$		−	0.466816i		−	0.0300703i	0.999887	−	0.0150351i	$-0.00478601\pi$
$242$		0			0
$243$	15.2461			0.978040
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.786257	+	0.105393i	0.0500283	+	0.00670597i
$248$		0			0
$249$		−	30.4256i		−	1.92814i
$250$		0			0
$251$	2.42778			0.153240			0.0766201	−	0.997060i	$-0.475587\pi$
$251$	2.42778			0.153240			0.0766201	+	0.997060i	$0.475587\pi$
$252$		0			0
$253$		−	3.77054i		−	0.237052i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.8618			1.11419			0.557094	−	0.830450i	$-0.311916\pi$
$257$	17.8618			1.11419			0.557094	+	0.830450i	$0.311916\pi$
$258$		0			0
$259$	−0.258234			−0.0160459
$260$		0			0
$261$	13.2406			0.819571
$262$		0			0
$263$	25.7344			1.58685			0.793425	−	0.608668i	$-0.208296\pi$
$263$	25.7344			1.58685			0.793425	+	0.608668i	$0.208296\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			33.7101i			2.06303i
$268$		0			0
$269$	6.41097			0.390884			0.195442	−	0.980715i	$-0.437386\pi$
$269$	6.41097			0.390884			0.195442	+	0.980715i	$0.437386\pi$
$270$		0			0
$271$		−	21.3789i		−	1.29868i	−0.760500	−	0.649338i	$-0.775046\pi$
$271$		−	21.3789i		−	1.29868i	0.760500	−	0.649338i	$-0.224954\pi$
$272$		0			0
$273$	−16.2959	−	2.18436i	−0.986274	−	0.132204i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.71292			0.403340			0.201670	−	0.979454i	$-0.435363\pi$
$277$	6.71292			0.403340			0.201670	+	0.979454i	$0.435363\pi$
$278$		0			0
$279$		−	1.90762i		−	0.114206i
$280$		0			0
$281$			24.5081i			1.46203i	0.682361	+	0.731015i	$0.260953\pi$
$281$			24.5081i			1.46203i	−0.682361	+	0.731015i	$0.739047\pi$
$282$		0			0
$283$	−3.55044			−0.211052			−0.105526	−	0.994417i	$-0.533653\pi$
$283$	−3.55044			−0.211052			−0.105526	+	0.994417i	$0.533653\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.3429			0.610524
$288$		0			0
$289$	3.79451			0.223206
$290$		0			0
$291$		−	4.49502i		−	0.263503i
$292$		0			0
$293$			1.36844i			0.0799450i	0.999201	+	0.0399725i	$0.0127270\pi$
$293$			1.36844i			0.0799450i	−0.999201	+	0.0399725i	$0.987273\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			2.36304i			0.137118i
$298$		0			0
$299$	−2.26814	+	16.9209i	−0.131170	+	0.978562i
$300$		0			0
$301$		−	6.71647i		−	0.387131i
$302$		0			0
$303$	−13.5250			−0.776989
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	8.92584i		−	0.509425i	−0.967017	−	0.254712i	$-0.918019\pi$
$307$		−	8.92584i		−	0.509425i	0.967017	−	0.254712i	$-0.0819808\pi$
$308$		0			0
$309$	−25.7178			−1.46303
$310$		0			0
$311$	−13.1052			−0.743127			−0.371563	−	0.928408i	$-0.621178\pi$
$311$	−13.1052			−0.743127			−0.371563	+	0.928408i	$0.621178\pi$
$312$		0			0
$313$	−12.5277			−0.708108			−0.354054	−	0.935225i	$-0.615197\pi$
$313$	−12.5277			−0.708108			−0.354054	+	0.935225i	$0.615197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	0.455813i		−	0.0256010i	−0.999918	−	0.0128005i	$-0.995925\pi$
$317$		−	0.455813i		−	0.0256010i	0.999918	−	0.0128005i	$-0.00407464\pi$
$318$		0			0
$319$			6.51124i			0.364559i
$320$		0			0
$321$	−31.1998			−1.74140
$322$		0			0
$323$		−	1.00331i		−	0.0558255i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			17.0379i			0.942200i
$328$		0			0
$329$	4.50165			0.248184
$330$		0			0
$331$		−	3.67559i		−	0.202029i	−0.994885	−	0.101014i	$-0.967791\pi$
$331$		−	3.67559i		−	0.202029i	0.994885	−	0.101014i	$-0.0322088\pi$
$332$		0			0
$333$			0.197086i			0.0108002i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	16.2959			0.887696			0.443848	−	0.896102i	$-0.353613\pi$
$337$	16.2959			0.887696			0.443848	+	0.896102i	$0.353613\pi$
$338$		0			0
$339$	−30.2296			−1.64185
$340$		0			0
$341$	0.938100			0.0508010
$342$		0			0
$343$			20.1527i			1.08815i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.809921			0.0434788			0.0217394	−	0.999764i	$-0.493080\pi$
$347$	0.809921			0.0434788			0.0217394	+	0.999764i	$0.493080\pi$
$348$		0			0
$349$			17.7333i			0.949244i	0.880190	+	0.474622i	$0.157415\pi$
$349$			17.7333i			0.949244i	−0.880190	+	0.474622i	$0.842585\pi$
$350$		0			0
$351$	1.42147	−	10.6045i	0.0758723	−	0.566027i
$352$		0			0
$353$		−	1.80442i		−	0.0960395i	−0.998846	−	0.0480198i	$-0.984709\pi$
$353$		−	1.80442i		−	0.0960395i	0.998846	−	0.0480198i	$-0.0152910\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			20.7945i			1.10056i
$358$		0			0
$359$			26.4760i			1.39735i	0.715440	+	0.698674i	$0.246226\pi$
$359$			26.4760i			1.39735i	−0.715440	+	0.698674i	$0.753774\pi$
$360$		0			0
$361$	18.9516			0.997452
$362$		0			0
$363$	−22.2789			−1.16934
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−30.5490			−1.59464			−0.797321	−	0.603555i	$-0.793750\pi$
$367$	−30.5490			−1.59464			−0.797321	+	0.603555i	$0.793750\pi$
$368$		0			0
$369$		−	7.89380i		−	0.410935i
$370$		0			0
$371$		−	23.3480i		−	1.21217i
$372$		0			0
$373$	−1.72151			−0.0891362			−0.0445681	−	0.999006i	$-0.514191\pi$
$373$	−1.72151			−0.0891362			−0.0445681	+	0.999006i	$0.514191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.91678	−	29.2202i	0.201724	−	1.50492i
$378$		0			0
$379$		−	32.5223i		−	1.67056i	−0.549825	−	0.835280i	$-0.685305\pi$
$379$		−	32.5223i		−	1.67056i	0.549825	−	0.835280i	$-0.314695\pi$
$380$		0			0
$381$	−25.9531			−1.32962
$382$		0			0
$383$			24.7170i			1.26298i	0.775383	+	0.631491i	$0.217557\pi$
$383$			24.7170i			1.26298i	−0.775383	+	0.631491i	$0.782443\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.12605			−0.260572
$388$		0			0
$389$	14.1181			0.715817			0.357908	−	0.933757i	$-0.383490\pi$
$389$	14.1181			0.715817			0.357908	+	0.933757i	$0.383490\pi$
$390$		0			0
$391$	21.5920			1.09196
$392$		0			0
$393$	−15.5226			−0.783009
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	5.15964i		−	0.258955i	−0.991582	−	0.129477i	$-0.958670\pi$
$397$		−	5.15964i		−	0.258955i	0.991582	−	0.129477i	$-0.0413300\pi$
$398$		0			0
$399$	1.00331			0.0502282
$400$		0			0
$401$			11.7358i			0.586059i	0.956103	+	0.293030i	$0.0946634\pi$
$401$			11.7358i			0.586059i	−0.956103	+	0.293030i	$0.905337\pi$
$402$		0			0
$403$	−4.20987	−	0.564306i	−0.209709	−	0.0281101i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.0969196			−0.00480413
$408$		0			0
$409$			21.4394i			1.06011i	0.847963	+	0.530056i	$0.177829\pi$
$409$			21.4394i			1.06011i	−0.847963	+	0.530056i	$0.822171\pi$
$410$		0			0
$411$			31.5689i			1.55718i
$412$		0			0
$413$	7.91125			0.389287
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−38.3648			−1.87873
$418$		0			0
$419$	−27.9618			−1.36602			−0.683011	−	0.730408i	$-0.739330\pi$
$419$	−27.9618			−1.36602			−0.683011	+	0.730408i	$0.739330\pi$
$420$		0			0
$421$			11.1126i			0.541596i	0.962636	+	0.270798i	$0.0872875\pi$
$421$			11.1126i			0.541596i	−0.962636	+	0.270798i	$0.912712\pi$
$422$		0			0
$423$		−	3.43569i		−	0.167049i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	7.13824i		−	0.345443i
$428$		0			0
$429$	−6.11615	−	0.819829i	−0.295290	−	0.0395817i
$430$		0			0
$431$		−	14.5679i		−	0.701712i	−0.936429	−	0.350856i	$-0.885891\pi$
$431$		−	14.5679i		−	0.701712i	0.936429	−	0.350856i	$-0.114109\pi$
$432$		0			0
$433$	15.7268			0.755779			0.377890	−	0.925851i	$-0.376650\pi$
$433$	15.7268			0.755779			0.377890	+	0.925851i	$0.376650\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.04179i		−	0.0498354i
$438$		0			0
$439$	−7.16424			−0.341931			−0.170965	−	0.985277i	$-0.554689\pi$
$439$	−7.16424			−0.341931			−0.170965	+	0.985277i	$0.554689\pi$
$440$		0			0
$441$	4.04558			0.192647
$442$		0			0
$443$	−19.6963			−0.935801			−0.467900	−	0.883781i	$-0.654989\pi$
$443$	−19.6963			−0.935801			−0.467900	+	0.883781i	$0.654989\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	35.8258i		−	1.69450i
$448$		0			0
$449$		−	6.34820i		−	0.299590i	−0.988717	−	0.149795i	$-0.952139\pi$
$449$		−	6.34820i		−	0.299590i	0.988717	−	0.149795i	$-0.0478614\pi$
$450$		0			0
$451$	3.88188			0.182791
$452$		0			0
$453$			2.14840i			0.100941i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			14.6109i			0.683469i	0.939796	+	0.341735i	$0.111014\pi$
$457$			14.6109i			0.683469i	−0.939796	+	0.341735i	$0.888986\pi$
$458$		0			0
$459$	−13.5320			−0.631618
$460$		0			0
$461$			2.52938i			0.117805i	0.998264	+	0.0589024i	$0.0187601\pi$
$461$			2.52938i			0.117805i	−0.998264	+	0.0589024i	$0.981240\pi$
$462$		0			0
$463$		−	11.4357i		−	0.531462i	−0.964047	−	0.265731i	$-0.914387\pi$
$463$		−	11.4357i		−	0.531462i	0.964047	−	0.265731i	$-0.0856133\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.8931			0.689172			0.344586	−	0.938755i	$-0.388019\pi$
$467$	14.8931			0.689172			0.344586	+	0.938755i	$0.388019\pi$
$468$		0			0
$469$	−15.7764			−0.728486
$470$		0			0
$471$	−23.1642			−1.06735
$472$		0			0
$473$		−	2.52081i		−	0.115907i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−17.8193			−0.815891
$478$		0			0
$479$		−	32.1619i		−	1.46951i	−0.678330	−	0.734757i	$-0.737296\pi$
$479$		−	32.1619i		−	1.46951i	0.678330	−	0.734757i	$-0.262704\pi$
$480$		0			0
$481$	0.434942	+	0.0583011i	0.0198316	+	0.00265830i
$482$		0			0
$483$			21.5920i			0.982471i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	15.7484i		−	0.713626i	−0.934176	−	0.356813i	$-0.883863\pi$
$487$		−	15.7484i		−	0.713626i	0.934176	−	0.356813i	$-0.116137\pi$
$488$		0			0
$489$			6.26770i			0.283435i
$490$		0			0
$491$	−1.19587			−0.0539688			−0.0269844	−	0.999636i	$-0.508590\pi$
$491$	−1.19587			−0.0539688			−0.0269844	+	0.999636i	$0.508590\pi$
$492$		0			0
$493$	−37.2866			−1.67930
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−32.2672			−1.44738
$498$		0			0
$499$		−	41.0855i		−	1.83924i	−0.392812	−	0.919619i	$-0.628498\pi$
$499$		−	41.0855i		−	1.83924i	0.392812	−	0.919619i	$-0.371502\pi$
$500$		0			0
$501$			42.8186i			1.91300i
$502$		0			0
$503$	−34.9278			−1.55736			−0.778678	−	0.627424i	$-0.784109\pi$
$503$	−34.9278			−1.55736			−0.778678	+	0.627424i	$0.784109\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	26.9540	+	7.35823i	1.19707	+	0.326790i
$508$		0			0
$509$		−	6.97956i		−	0.309364i	−0.987964	−	0.154682i	$-0.950565\pi$
$509$		−	6.97956i		−	0.309364i	0.987964	−	0.154682i	$-0.0494352\pi$
$510$		0			0
$511$	11.5330			0.510188
$512$		0			0
$513$			0.652899i			0.0288262i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.68955			0.0743063
$518$		0			0
$519$	−38.0102			−1.66846
$520$		0			0
$521$	−7.80036			−0.341740			−0.170870	−	0.985294i	$-0.554658\pi$
$521$	−7.80036			−0.341740			−0.170870	+	0.985294i	$0.554658\pi$
$522$		0			0
$523$	40.2344			1.75933			0.879664	−	0.475596i	$-0.157768\pi$
$523$	40.2344			1.75933			0.879664	+	0.475596i	$0.157768\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.37203i			0.234009i
$528$		0			0
$529$	−0.579871			−0.0252118
$530$		0			0
$531$		−	6.03792i		−	0.262024i
$532$		0			0
$533$	−17.4206	−	2.33511i	−0.754569	−	0.101145i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	19.9480			0.860820
$538$		0			0
$539$			1.98947i			0.0856926i
$540$		0			0
$541$		−	1.93217i		−	0.0830706i	−0.999137	−	0.0415353i	$-0.986775\pi$
$541$		−	1.93217i		−	0.0830706i	0.999137	−	0.0415353i	$-0.0132249\pi$
$542$		0			0
$543$	−47.0734			−2.02011
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.299309			0.0127975			0.00639876	−	0.999980i	$-0.497963\pi$
$547$	0.299309			0.0127975			0.00639876	+	0.999980i	$0.497963\pi$
$548$		0			0
$549$	−5.44795			−0.232513
$550$		0			0
$551$			1.79903i			0.0766412i
$552$		0			0
$553$			19.2137i			0.817049i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	37.6211i		−	1.59406i	−0.603943	−	0.797028i	$-0.706404\pi$
$557$		−	37.6211i		−	1.59406i	0.603943	−	0.797028i	$-0.293596\pi$
$558$		0			0
$559$	−1.51637	+	11.3125i	−0.0641356	+	0.478469i
$560$		0			0
$561$			7.80454i			0.329508i
$562$		0			0
$563$	−7.42476			−0.312917			−0.156458	−	0.987685i	$-0.550008\pi$
$563$	−7.42476			−0.312917			−0.156458	+	0.987685i	$0.550008\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	23.8390i		−	1.00115i
$568$		0			0
$569$	−42.0875			−1.76440			−0.882201	−	0.470874i	$-0.843939\pi$
$569$	−42.0875			−1.76440			−0.882201	+	0.470874i	$0.843939\pi$
$570$		0			0
$571$	−20.2286			−0.846541			−0.423270	−	0.906003i	$-0.639118\pi$
$571$	−20.2286			−0.846541			−0.423270	+	0.906003i	$0.639118\pi$
$572$		0			0
$573$	49.6095			2.07247
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	18.1986i		−	0.757617i	−0.925475	−	0.378809i	$-0.876334\pi$
$577$		−	18.1986i		−	0.757617i	0.925475	−	0.378809i	$-0.123666\pi$
$578$		0			0
$579$			38.7265i			1.60942i
$580$		0			0
$581$	−30.0356			−1.24609
$582$		0			0
$583$		−	8.76290i		−	0.362922i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			30.4324i			1.25608i	0.778181	+	0.628040i	$0.216142\pi$
$587$			30.4324i			1.25608i	−0.778181	+	0.628040i	$0.783858\pi$
$588$		0			0
$589$	0.259193			0.0106799
$590$		0			0
$591$			16.6757i			0.685947i
$592$		0			0
$593$		−	18.3114i		−	0.751960i	−0.926628	−	0.375980i	$-0.877306\pi$
$593$		−	18.3114i		−	0.751960i	0.926628	−	0.375980i	$-0.122694\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−23.5646			−0.964434
$598$		0			0
$599$	−32.0692			−1.31031			−0.655157	−	0.755493i	$-0.727397\pi$
$599$	−32.0692			−1.31031			−0.655157	+	0.755493i	$0.727397\pi$
$600$		0			0
$601$	21.4255			0.873964			0.436982	−	0.899470i	$-0.356047\pi$
$601$	21.4255			0.873964			0.436982	+	0.899470i	$0.356047\pi$
$602$		0			0
$603$			12.0407i			0.490333i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	25.7076			1.04344			0.521720	−	0.853117i	$-0.325291\pi$
$607$	25.7076			1.04344			0.521720	+	0.853117i	$0.325291\pi$
$608$		0			0
$609$		−	37.2866i		−	1.51093i
$610$		0			0
$611$	−7.58212	−	1.01633i	−0.306740	−	0.0411165i
$612$		0			0
$613$			47.7114i			1.92704i	0.267628	+	0.963522i	$0.413760\pi$
$613$			47.7114i			1.92704i	−0.267628	+	0.963522i	$0.586240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			43.5087i			1.75159i	0.482679	+	0.875797i	$0.339664\pi$
$617$			43.5087i			1.75159i	−0.482679	+	0.875797i	$0.660336\pi$
$618$		0			0
$619$			27.8570i			1.11967i	0.828605	+	0.559834i	$0.189135\pi$
$619$			27.8570i			1.11967i	−0.828605	+	0.559834i	$0.810865\pi$
$620$		0			0
$621$	−14.0509			−0.563845
$622$		0			0
$623$	33.2781			1.33326
$624$		0			0
$625$		0			0
$626$		0			0
$627$	0.376559			0.0150383
$628$		0			0
$629$		−	0.555010i		−	0.0221297i
$630$		0			0
$631$			26.4973i			1.05484i	0.849605	+	0.527420i	$0.176840\pi$
$631$			26.4973i			1.05484i	−0.849605	+	0.527420i	$0.823160\pi$
$632$		0			0
$633$	19.9480			0.792862
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.19675	−	8.92806i	0.0474169	−	0.353743i
$638$		0			0
$639$			24.6266i			0.974213i
$640$		0			0
$641$	−8.96406			−0.354059			−0.177029	−	0.984206i	$-0.556649\pi$
$641$	−8.96406			−0.354059			−0.177029	+	0.984206i	$0.556649\pi$
$642$		0			0
$643$			4.95848i			0.195543i	0.995209	+	0.0977716i	$0.0311715\pi$
$643$			4.95848i			0.195543i	−0.995209	+	0.0977716i	$0.968829\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.3038			−0.640967			−0.320483	−	0.947254i	$-0.603845\pi$
$647$	−16.3038			−0.640967			−0.320483	+	0.947254i	$0.603845\pi$
$648$		0			0
$649$	2.96923			0.116553
$650$		0			0
$651$	−5.37203			−0.210547
$652$		0			0
$653$	−16.8293			−0.658583			−0.329291	−	0.944228i	$-0.606810\pi$
$653$	−16.8293			−0.658583			−0.329291	+	0.944228i	$0.606810\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	8.80204i		−	0.343400i
$658$		0			0
$659$	33.0231			1.28640			0.643199	−	0.765699i	$-0.277607\pi$
$659$	33.0231			1.28640			0.643199	+	0.765699i	$0.277607\pi$
$660$		0			0
$661$		−	39.0519i		−	1.51894i	−0.650541	−	0.759472i	$-0.725458\pi$
$661$		−	39.0519i		−	1.51894i	0.650541	−	0.759472i	$-0.274542\pi$
$662$		0			0
$663$	4.69475	−	35.0241i	0.182329	−	1.36022i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−38.7166			−1.49911
$668$		0			0
$669$			18.7352i			0.724343i
$670$		0			0
$671$		−	2.67910i		−	0.103426i
$672$		0			0
$673$	41.8467			1.61307			0.806536	−	0.591185i	$-0.201340\pi$
$673$	41.8467			1.61307			0.806536	+	0.591185i	$0.201340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	42.6066			1.63751			0.818753	−	0.574146i	$-0.194666\pi$
$677$	42.6066			1.63751			0.818753	+	0.574146i	$0.194666\pi$
$678$		0			0
$679$	−4.43741			−0.170292
$680$		0			0
$681$		−	57.3332i		−	2.19701i
$682$		0			0
$683$		−	15.5212i		−	0.593901i	−0.954893	−	0.296951i	$-0.904030\pi$
$683$		−	15.5212i		−	0.593901i	0.954893	−	0.296951i	$-0.0959697\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	61.6266i		−	2.35120i
$688$		0			0
$689$	−5.27124	+	39.3249i	−0.200818	+	1.49816i
$690$		0			0
$691$		−	38.8594i		−	1.47828i	−0.673551	−	0.739140i	$-0.735232\pi$
$691$		−	38.8594i		−	1.47828i	0.673551	−	0.739140i	$-0.264768\pi$
$692$		0			0
$693$	−2.73589			−0.103928
$694$		0			0
$695$		0			0
$696$		0			0
$697$			22.2296i			0.842007i
$698$		0			0
$699$	2.31729			0.0876480
$700$		0			0
$701$	−30.3813			−1.14749			−0.573744	−	0.819035i	$-0.694509\pi$
$701$	−30.3813			−1.14749			−0.573744	+	0.819035i	$0.694509\pi$
$702$		0			0
$703$	−0.0267785			−0.00100997
$704$		0			0
$705$		0			0
$706$		0			0
$707$			13.3516i			0.502139i
$708$		0			0
$709$		−	30.3070i		−	1.13820i	−0.822267	−	0.569102i	$-0.807291\pi$
$709$		−	30.3070i		−	1.13820i	0.822267	−	0.569102i	$-0.192709\pi$
$710$		0			0
$711$	14.6640			0.549944
$712$		0			0
$713$			5.57806i			0.208900i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			44.8609i			1.67536i
$718$		0			0
$719$	−42.1802			−1.57306			−0.786528	−	0.617555i	$-0.788123\pi$
$719$	−42.1802			−1.57306			−0.786528	+	0.617555i	$0.788123\pi$
$720$		0			0
$721$			25.3882i			0.945505i
$722$		0			0
$723$			1.00331i			0.0373134i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	33.6752			1.24894			0.624472	−	0.781047i	$-0.285314\pi$
$727$	33.6752			1.24894			0.624472	+	0.781047i	$0.285314\pi$
$728$		0			0
$729$	0.939438			0.0347940
$730$		0			0
$731$	14.4354			0.533913
$732$		0			0
$733$		−	32.8827i		−	1.21455i	−0.794491	−	0.607276i	$-0.792262\pi$
$733$		−	32.8827i		−	1.21455i	0.794491	−	0.607276i	$-0.207738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.92115			−0.218109
$738$		0			0
$739$		−	14.8051i		−	0.544615i	−0.962210	−	0.272307i	$-0.912213\pi$
$739$		−	14.8051i		−	0.544615i	0.962210	−	0.272307i	$-0.0877867\pi$
$740$		0			0
$741$	−1.68987	−	0.226516i	−0.0620789	−	0.00832126i
$742$		0			0
$743$		−	23.5884i		−	0.865376i	−0.901544	−	0.432688i	$-0.857565\pi$
$743$		−	23.5884i		−	0.865376i	0.901544	−	0.432688i	$-0.142435\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			22.9234i			0.838724i
$748$		0			0
$749$			30.7999i			1.12541i
$750$		0			0
$751$	40.5393			1.47930			0.739650	−	0.672992i	$-0.234991\pi$
$751$	40.5393			1.47930			0.739650	+	0.672992i	$0.234991\pi$
$752$		0			0
$753$	−5.21793			−0.190152
$754$		0			0
$755$		0			0
$756$		0			0
$757$	43.8231			1.59278			0.796389	−	0.604784i	$-0.206741\pi$
$757$	43.8231			1.59278			0.796389	+	0.604784i	$0.206741\pi$
$758$		0			0
$759$			8.10387i			0.294152i
$760$		0			0
$761$		−	20.5134i		−	0.743609i	−0.928311	−	0.371804i	$-0.878739\pi$
$761$		−	20.5134i		−	0.743609i	0.928311	−	0.371804i	$-0.121261\pi$
$762$		0			0
$763$	16.8196			0.608909
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.3249	−	1.78611i	−0.481134	−	0.0644929i
$768$		0			0
$769$		−	38.6434i		−	1.39352i	−0.717306	−	0.696758i	$-0.754625\pi$
$769$		−	38.6434i		−	1.39352i	0.717306	−	0.696758i	$-0.245375\pi$
$770$		0			0
$771$	−38.3895			−1.38257
$772$		0			0
$773$		−	37.2827i		−	1.34097i	−0.741924	−	0.670484i	$-0.766087\pi$
$773$		−	37.2827i		−	1.34097i	0.741924	−	0.670484i	$-0.233913\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.555010			0.0199109
$778$		0			0
$779$	1.07255			0.0384281
$780$		0			0
$781$	−12.1105			−0.433346
$782$		0			0
$783$	24.2641			0.867129
$784$		0			0
$785$		0			0
$786$		0			0
$787$			48.7190i			1.73665i	0.495999	+	0.868323i	$0.334802\pi$
$787$			48.7190i			1.73665i	−0.495999	+	0.868323i	$0.665198\pi$
$788$		0			0
$789$	−55.3098			−1.96908
$790$		0			0
$791$			29.8422i			1.06107i
$792$		0			0
$793$	−1.61159	+	12.0229i	−0.0572293	+	0.426946i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.328613			−0.0116401			−0.00582003	−	0.999983i	$-0.501853\pi$
$797$	−0.328613			−0.0116401			−0.00582003	+	0.999983i	$0.501853\pi$
$798$		0			0
$799$			9.67521i			0.342284i
$800$		0			0
$801$		−	25.3981i		−	0.897397i
$802$		0			0
$803$	4.32852			0.152750
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−13.7788			−0.485037
$808$		0			0
$809$	−24.2689			−0.853249			−0.426624	−	0.904429i	$-0.640297\pi$
$809$	−24.2689			−0.853249			−0.426624	+	0.904429i	$0.640297\pi$
$810$		0			0
$811$			42.7416i			1.50086i	0.660950	+	0.750430i	$0.270153\pi$
$811$			42.7416i			1.50086i	−0.660950	+	0.750430i	$0.729847\pi$
$812$		0			0
$813$			45.9488i			1.61149i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	0.696489i		−	0.0243671i
$818$		0			0
$819$	12.2778	+	1.64575i	0.429020	+	0.0575073i
$820$		0			0
$821$		−	3.18294i		−	0.111085i	−0.998456	−	0.0555426i	$-0.982311\pi$
$821$		−	3.18294i		−	0.111085i	0.998456	−	0.0555426i	$-0.0176889\pi$
$822$		0			0
$823$	−13.8891			−0.484143			−0.242072	−	0.970258i	$-0.577827\pi$
$823$	−13.8891			−0.484143			−0.242072	+	0.970258i	$0.577827\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.31427i			0.0804751i	0.999190	+	0.0402375i	$0.0128115\pi$
$827$			2.31427i			0.0804751i	−0.999190	+	0.0402375i	$0.987189\pi$
$828$		0			0
$829$	36.5581			1.26971			0.634857	−	0.772630i	$-0.281059\pi$
$829$	36.5581			1.26971			0.634857	+	0.772630i	$0.281059\pi$
$830$		0			0
$831$	−14.4278			−0.500494
$832$		0			0
$833$	−11.3927			−0.394734
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	3.49583i		−	0.120834i
$838$		0			0
$839$		−	11.0809i		−	0.382557i	−0.981536	−	0.191278i	$-0.938737\pi$
$839$		−	11.0809i		−	0.382557i	0.981536	−	0.191278i	$-0.0612633\pi$
$840$		0			0
$841$	37.8586			1.30547
$842$		0			0
$843$		−	52.6742i		−	1.81420i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			21.9934i			0.755702i
$848$		0			0
$849$	7.63081			0.261889
$850$		0			0
$851$		−	0.576296i		−	0.0197552i
$852$		0			0
$853$		−	9.12502i		−	0.312435i	−0.987723	−	0.156217i	$-0.950070\pi$
$853$		−	9.12502i		−	0.312435i	0.987723	−	0.156217i	$-0.0499300\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.81456			−0.164462			−0.0822312	−	0.996613i	$-0.526205\pi$
$857$	−4.81456			−0.164462			−0.0822312	+	0.996613i	$0.526205\pi$
$858$		0			0
$859$	42.1481			1.43807			0.719037	−	0.694971i	$-0.244583\pi$
$859$	42.1481			1.43807			0.719037	+	0.694971i	$0.244583\pi$
$860$		0			0
$861$	−22.2296			−0.757583
$862$		0			0
$863$		−	49.8252i		−	1.69607i	−0.529940	−	0.848035i	$-0.677785\pi$
$863$		−	49.8252i		−	1.69607i	0.529940	−	0.848035i	$-0.322215\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−8.15537			−0.276971
$868$		0			0
$869$			7.21123i			0.244624i
$870$		0			0
$871$	26.5721	+	3.56182i	0.900362	+	0.120688i
$872$		0			0
$873$			3.38666i			0.114621i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			43.2831i			1.46157i	0.682609	+	0.730784i	$0.260845\pi$
$877$			43.2831i			1.46157i	−0.682609	+	0.730784i	$0.739155\pi$
$878$		0			0
$879$		−	2.94112i		−	0.0992017i
$880$		0			0
$881$	−20.2113			−0.680937			−0.340468	−	0.940256i	$-0.610586\pi$
$881$	−20.2113			−0.680937			−0.340468	+	0.940256i	$0.610586\pi$
$882$		0			0
$883$	20.3203			0.683833			0.341916	−	0.939730i	$-0.388924\pi$
$883$	20.3203			0.683833			0.341916	+	0.939730i	$0.388924\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	37.6498			1.26416			0.632079	−	0.774904i	$-0.282202\pi$
$887$	37.6498			1.26416			0.632079	+	0.774904i	$0.282202\pi$
$888$		0			0
$889$			25.6205i			0.859283i
$890$		0			0
$891$		−	8.94721i		−	0.299743i
$892$		0			0
$893$	0.466816			0.0156214
$894$		0			0
$895$		0			0
$896$		0			0
$897$	4.87481	−	36.3674i	0.162765	−	1.21427i
$898$		0			0
$899$		−	9.63258i		−	0.321265i
$900$		0			0
$901$	50.1807			1.67176
$902$		0			0
$903$			14.4354i			0.480380i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−20.9901			−0.696965			−0.348482	−	0.937315i	$-0.613303\pi$
$907$	−20.9901			−0.696965			−0.348482	+	0.937315i	$0.613303\pi$
$908$		0			0
$909$	10.1900			0.337982
$910$		0			0
$911$	−20.5887			−0.682135			−0.341067	−	0.940039i	$-0.610788\pi$
$911$	−20.5887			−0.682135			−0.341067	+	0.940039i	$0.610788\pi$
$912$		0			0
$913$	−11.2729			−0.373078
$914$		0			0
$915$		0			0
$916$		0			0
$917$			15.3236i			0.506030i
$918$		0			0
$919$	−3.06293			−0.101037			−0.0505183	−	0.998723i	$-0.516087\pi$
$919$	−3.06293			−0.101037			−0.0505183	+	0.998723i	$0.516087\pi$
$920$		0			0
$921$			19.1839i			0.632132i
$922$		0			0
$923$	54.3476	+	7.28494i	1.78887	+	0.239787i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	19.3764			0.636406
$928$		0			0
$929$			12.2657i			0.402424i	0.979548	+	0.201212i	$0.0644880\pi$
$929$			12.2657i			0.402424i	−0.979548	+	0.201212i	$0.935512\pi$
$930$		0			0
$931$			0.549683i			0.0180151i
$932$		0			0
$933$	28.1664			0.922127
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.98472			0.162844			0.0814219	−	0.996680i	$-0.474054\pi$
$937$	4.98472			0.162844			0.0814219	+	0.996680i	$0.474054\pi$
$938$		0			0
$939$	26.9253			0.878673
$940$		0			0
$941$		−	7.53068i		−	0.245493i	−0.992438	−	0.122747i	$-0.960830\pi$
$941$		−	7.53068i		−	0.245493i	0.992438	−	0.122747i	$-0.0391703\pi$
$942$		0			0
$943$			23.0822i			0.751659i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			33.7019i			1.09517i	0.836752	+	0.547583i	$0.184452\pi$
$947$			33.7019i			1.09517i	−0.836752	+	0.547583i	$0.815548\pi$
$948$		0			0
$949$	−19.4249	−	2.60379i	−0.630560	−	0.0845225i
$950$		0			0
$951$			0.979660i			0.0317677i
$952$		0			0
$953$	31.0905			1.00712			0.503560	−	0.863960i	$-0.332023\pi$
$953$	31.0905			1.00712			0.503560	+	0.863960i	$0.332023\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	13.9943i		−	0.452372i
$958$		0			0
$959$	31.1642			1.00635
$960$		0			0
$961$	29.6122			0.955232
$962$		0			0
$963$	23.5067			0.757494
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	52.8189i		−	1.69854i	−0.527957	−	0.849271i	$-0.677042\pi$
$967$		−	52.8189i		−	1.69854i	0.527957	−	0.849271i	$-0.322958\pi$
$968$		0			0
$969$			2.15637i			0.0692724i
$970$		0			0
$971$	22.8041			0.731819			0.365910	−	0.930650i	$-0.380758\pi$
$971$	22.8041			0.731819			0.365910	+	0.930650i	$0.380758\pi$
$972$		0			0
$973$			37.8731i			1.21416i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	0.156331i		−	0.00500148i	−0.999997	−	0.00250074i	$-0.999204\pi$
$977$		−	0.156331i		−	0.00500148i	0.999997	−	0.00250074i	$-0.000796011\pi$
$978$		0			0
$979$	12.4898			0.399177
$980$		0			0
$981$		−	12.8368i		−	0.409848i
$982$		0			0
$983$		−	23.5630i		−	0.751542i	−0.926712	−	0.375771i	$-0.877378\pi$
$983$		−	23.5630i		−	0.751542i	0.926712	−	0.375771i	$-0.122622\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−9.67521			−0.307965
$988$		0			0
$989$	14.9890			0.476624
$990$		0			0
$991$	14.0102			0.445048			0.222524	−	0.974927i	$-0.428570\pi$
$991$	14.0102			0.445048			0.222524	+	0.974927i	$0.428570\pi$
$992$		0			0
$993$			7.89979i			0.250692i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.625270			0.0198025			0.00990124	−	0.999951i	$-0.496848\pi$
$997$	0.625270			0.0198025			0.00990124	+	0.999951i	$0.496848\pi$
$998$		0			0
$999$			0.361171i			0.0114269i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.k.f.2001.4		20
5.2	odd	4		520.2.f.a.129.9	yes	10
5.3	odd	4		520.2.f.b.129.2	yes	10
5.4	even	2	inner	2600.2.k.f.2001.17		20
13.12	even	2	inner	2600.2.k.f.2001.3		20
20.3	even	4		1040.2.f.g.129.9		10
20.7	even	4		1040.2.f.f.129.2		10
65.12	odd	4		520.2.f.b.129.9	yes	10
65.38	odd	4		520.2.f.a.129.2	✓	10
65.64	even	2	inner	2600.2.k.f.2001.18		20
260.103	even	4		1040.2.f.f.129.9		10
260.207	even	4		1040.2.f.g.129.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.2	✓	10	65.38	odd	4
520.2.f.a.129.9	yes	10	5.2	odd	4
520.2.f.b.129.2	yes	10	5.3	odd	4
520.2.f.b.129.9	yes	10	65.12	odd	4
1040.2.f.f.129.2		10	20.7	even	4
1040.2.f.f.129.9		10	260.103	even	4
1040.2.f.g.129.2		10	260.207	even	4
1040.2.f.g.129.9		10	20.3	even	4
2600.2.k.f.2001.3		20	13.12	even	2	inner
2600.2.k.f.2001.4		20	1.1	even	1	trivial
2600.2.k.f.2001.17		20	5.4	even	2	inner
2600.2.k.f.2001.18		20	65.64	even	2	inner

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2600.2.k.f.2001.4

Level	$2600$
Weight	$2$
Character	2600.2001
Analytic conductor	$20.761$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$