LMFDB - Embedded newform 2600.2.k.f.2001.3

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14926 q^{3} -2.12171i q^{7} +1.61930 q^{9} -0.796315i q^{11} +(0.479016 + 3.57359i) q^{13} -4.56010 q^{17} -0.220019i q^{19} +4.56010i q^{21} -4.73499 q^{23} +2.96747 q^{27} +8.17671 q^{29} +1.17805i q^{31} +1.71149i q^{33} -0.121710i q^{37} +(-1.02953 - 7.68056i) q^{39} +4.87481i q^{41} -3.16559 q^{43} +2.12171i q^{47} +2.49835 q^{49} +9.80082 q^{51} -11.0043 q^{53} +0.472877i q^{57} +3.72871i q^{59} -3.36438 q^{61} -3.43569i q^{63} -7.43569i q^{67} +10.1767 q^{69} -15.2081i q^{71} +5.43569i q^{73} -1.68955 q^{77} +9.05575 q^{79} -11.2358 q^{81} -14.1563i q^{83} -17.5738 q^{87} +15.6846i q^{89} +(7.58212 - 1.01633i) q^{91} -2.53193i q^{93} -2.09143i q^{97} -1.28948i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.868675\pi$
$7$		−	2.12171i		−	0.801931i	0.916093	−	0.400966i	$-0.131325\pi$
$8$		0			0
$9$	1.61930			0.539768
$10$		0			0
$11$		−	0.796315i		−	0.240098i	−0.992768	−	0.120049i	$-0.961695\pi$
$11$		−	0.796315i		−	0.240098i	0.992768	−	0.120049i	$-0.0383052\pi$
$12$		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	$-0.991966\pi$
$19$		−	0.220019i		−	0.0504758i	0.999681	−	0.0252379i	$-0.00803432\pi$
$20$		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	$0.0337378\pi$
$31$			1.17805i			0.211584i	−0.994388	+	0.105792i	$0.966262\pi$
$32$		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	$-0.996815\pi$
$37$		−	0.121710i		−	0.0200090i	0.999950	−	0.0100045i	$-0.00318459\pi$
$38$		0			0
$39$	−1.02953	−	7.68056i	−0.164857	−	1.22987i
$40$		0			0
$41$			4.87481i			0.761317i	0.924716	+	0.380659i	$0.124303\pi$
$41$			4.87481i			0.761317i	−0.924716	+	0.380659i	$0.875697\pi$
$42$		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	$0.0494545\pi$
$47$			2.12171i			0.309483i	−0.987955	+	0.154742i	$0.950545\pi$
$48$		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	$0.0780392\pi$
$59$			3.72871i			0.485437i	−0.970097	+	0.242719i	$0.921961\pi$
$60$		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	$-0.849923\pi$
$67$		−	7.43569i		−	0.908415i	0.890896	−	0.454207i	$-0.150077\pi$
$68$		0			0
$69$	10.1767			1.22513
$70$		0			0
$71$		−	15.2081i		−	1.80487i	−0.430823	−	0.902437i	$-0.641777\pi$
$71$		−	15.2081i		−	1.80487i	0.430823	−	0.902437i	$-0.358223\pi$
$72$		0			0
$73$			5.43569i			0.636200i	0.948057	+	0.318100i	$0.103045\pi$
$73$			5.43569i			0.636200i	−0.948057	+	0.318100i	$0.896955\pi$
$74$		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	$-0.716776\pi$
$83$		−	14.1563i		−	1.55386i	0.629587	−	0.776930i	$-0.283224\pi$
$84$		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	$0.312391\pi$
$89$			15.6846i			1.66256i	−0.555854	+	0.831280i	$0.687609\pi$
$90$		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	$-0.966139\pi$
$97$		−	2.09143i		−	0.212352i	0.994347	−	0.106176i	$-0.0338608\pi$
$98$		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	$0.123956\pi$
$109$			7.92736i			0.759303i	−0.925130	+	0.379652i	$0.876044\pi$
$110$		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	$0.215902\pi$
$137$			14.6883i			1.25490i	−0.778656	+	0.627452i	$0.784098\pi$
$138$		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	$-0.760768\pi$
$149$		−	16.6689i		−	1.36557i	0.730618	−	0.682786i	$-0.239232\pi$
$150$		0			0
$151$			0.999600i			0.0813463i	0.999173	+	0.0406732i	$0.0129502\pi$
$151$			0.999600i			0.0813463i	−0.999173	+	0.0406732i	$0.987050\pi$
$152$		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	$0.0364330\pi$
$163$			2.91622i			0.228416i	−0.993457	+	0.114208i	$0.963567\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			19.9225i			1.54165i	0.637046	+	0.770826i	$0.280156\pi$
$167$			19.9225i			1.54165i	−0.637046	+	0.770826i	$0.719844\pi$
$168$		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	$0.224604\pi$
$193$			18.0186i			1.29700i	−0.761213	+	0.648502i	$0.775396\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.75883i			0.552794i	0.961044	+	0.276397i	$0.0891404\pi$
$197$			7.75883i			0.552794i	−0.961044	+	0.276397i	$0.910860\pi$
$198$		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	$0.0942769\pi$
$223$			8.71704i			0.583736i	−0.956459	+	0.291868i	$0.905723\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	26.6758i		−	1.77054i	−0.465082	−	0.885268i	$-0.653975\pi$
$227$		−	26.6758i		−	1.77054i	0.465082	−	0.885268i	$-0.346025\pi$
$228$		0			0
$229$		−	28.6735i		−	1.89480i	−0.320058	−	0.947398i	$-0.603702\pi$
$229$		−	28.6735i		−	1.89480i	0.320058	−	0.947398i	$-0.396298\pi$
$230$		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	$0.235888\pi$
$239$			20.8728i			1.35015i	−0.737751	+	0.675073i	$0.764112\pi$
$240$		0			0
$241$			0.466816i			0.0300703i	0.999887	+	0.0150351i	$0.00478601\pi$
$241$			0.466816i			0.0300703i	−0.999887	+	0.0150351i	$0.995214\pi$
$242$		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	$0.224954\pi$
$271$			21.3789i			1.29868i	−0.760500	+	0.649338i	$0.775046\pi$
$272$		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	$-0.739047\pi$
$281$		−	24.5081i		−	1.46203i	0.682361	−	0.731015i	$-0.260953\pi$
$282$		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	$-0.987273\pi$
$293$		−	1.36844i		−	0.0799450i	0.999201	−	0.0399725i	$-0.0127270\pi$
$294$		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	$0.0819808\pi$
$307$			8.92584i			0.509425i	−0.967017	+	0.254712i	$0.918019\pi$
$308$		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	$0.00407464\pi$
$317$			0.455813i			0.0256010i	−0.999918	+	0.0128005i	$0.995925\pi$
$318$		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	$0.0322088\pi$
$331$			3.67559i			0.202029i	−0.994885	+	0.101014i	$0.967791\pi$
$332$		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	$-0.842585\pi$
$349$		−	17.7333i		−	0.949244i	0.880190	−	0.474622i	$-0.157415\pi$
$350$		0			0
$351$	1.42147	+	10.6045i	0.0758723	+	0.566027i
$352$		0			0
$353$			1.80442i			0.0960395i	0.998846	+	0.0480198i	$0.0152910\pi$
$353$			1.80442i			0.0960395i	−0.998846	+	0.0480198i	$0.984709\pi$
$354$		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	$-0.753774\pi$
$359$		−	26.4760i		−	1.39735i	0.715440	−	0.698674i	$-0.246226\pi$
$360$		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	$0.314695\pi$
$379$			32.5223i			1.67056i	−0.549825	+	0.835280i	$0.685305\pi$
$380$		0			0
$381$	−25.9531			−1.32962
$382$		0			0
$383$		−	24.7170i		−	1.26298i	−0.775383	−	0.631491i	$-0.782443\pi$
$383$		−	24.7170i		−	1.26298i	0.775383	−	0.631491i	$-0.217557\pi$
$384$		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	$0.0413300\pi$
$397$			5.15964i			0.258955i	−0.991582	+	0.129477i	$0.958670\pi$
$398$		0			0
$399$	1.00331			0.0502282
$400$		0			0
$401$		−	11.7358i		−	0.586059i	−0.956103	−	0.293030i	$-0.905337\pi$
$401$		−	11.7358i		−	0.586059i	0.956103	−	0.293030i	$-0.0946634\pi$
$402$		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	$-0.822171\pi$
$409$		−	21.4394i		−	1.06011i	0.847963	−	0.530056i	$-0.177829\pi$
$410$		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	$-0.912712\pi$
$421$		−	11.1126i		−	0.541596i	0.962636	−	0.270798i	$-0.0872875\pi$
$422$		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	$0.114109\pi$
$431$			14.5679i			0.701712i	−0.936429	+	0.350856i	$0.885891\pi$
$432$		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	$0.0478614\pi$
$449$			6.34820i			0.299590i	−0.988717	+	0.149795i	$0.952139\pi$
$450$		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	$-0.888986\pi$
$457$		−	14.6109i		−	0.683469i	0.939796	−	0.341735i	$-0.111014\pi$
$458$		0			0
$459$	−13.5320			−0.631618
$460$		0			0
$461$		−	2.52938i		−	0.117805i	−0.998264	−	0.0589024i	$-0.981240\pi$
$461$		−	2.52938i		−	0.117805i	0.998264	−	0.0589024i	$-0.0187601\pi$
$462$		0			0
$463$			11.4357i			0.531462i	0.964047	+	0.265731i	$0.0856133\pi$
$463$			11.4357i			0.531462i	−0.964047	+	0.265731i	$0.914387\pi$
$464$		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	$0.262704\pi$
$479$			32.1619i			1.46951i	−0.678330	+	0.734757i	$0.737296\pi$
$480$		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	$0.116137\pi$
$487$			15.7484i			0.713626i	−0.934176	+	0.356813i	$0.883863\pi$
$488$		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	$0.371502\pi$
$499$			41.0855i			1.83924i	−0.392812	+	0.919619i	$0.628498\pi$
$500$		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	$0.0494352\pi$
$509$			6.97956i			0.309364i	−0.987964	+	0.154682i	$0.950565\pi$
$510$		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	$0.0132249\pi$
$541$			1.93217i			0.0830706i	−0.999137	+	0.0415353i	$0.986775\pi$
$542$		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	$0.293596\pi$
$557$			37.6211i			1.59406i	−0.603943	+	0.797028i	$0.706404\pi$
$558$		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	$0.123666\pi$
$577$			18.1986i			0.757617i	−0.925475	+	0.378809i	$0.876334\pi$
$578$		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	$-0.783858\pi$
$587$		−	30.4324i		−	1.25608i	0.778181	−	0.628040i	$-0.216142\pi$
$588$		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	$0.122694\pi$
$593$			18.3114i			0.751960i	−0.926628	+	0.375980i	$0.877306\pi$
$594$		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	$-0.586240\pi$
$613$		−	47.7114i		−	1.92704i	0.267628	−	0.963522i	$-0.413760\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	43.5087i		−	1.75159i	−0.482679	−	0.875797i	$-0.660336\pi$
$617$		−	43.5087i		−	1.75159i	0.482679	−	0.875797i	$-0.339664\pi$
$618$		0			0
$619$		−	27.8570i		−	1.11967i	−0.828605	−	0.559834i	$-0.810865\pi$
$619$		−	27.8570i		−	1.11967i	0.828605	−	0.559834i	$-0.189135\pi$
$620$		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	$-0.823160\pi$
$631$		−	26.4973i		−	1.05484i	0.849605	−	0.527420i	$-0.176840\pi$
$632$		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	$-0.968829\pi$
$643$		−	4.95848i		−	0.195543i	0.995209	−	0.0977716i	$-0.0311715\pi$
$644$		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	$0.274542\pi$
$661$			39.0519i			1.51894i	−0.650541	+	0.759472i	$0.725458\pi$
$662$		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	$0.0959697\pi$
$683$			15.5212i			0.593901i	−0.954893	+	0.296951i	$0.904030\pi$
$684$		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	$0.264768\pi$
$691$			38.8594i			1.47828i	−0.673551	+	0.739140i	$0.735232\pi$
$692$		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	$0.192709\pi$
$709$			30.3070i			1.13820i	−0.822267	+	0.569102i	$0.807291\pi$
$710$		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	$0.207738\pi$
$733$			32.8827i			1.21455i	−0.794491	+	0.607276i	$0.792262\pi$
$734$		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	$0.0877867\pi$
$739$			14.8051i			0.544615i	−0.962210	+	0.272307i	$0.912213\pi$
$740$		0			0
$741$	−1.68987	+	0.226516i	−0.0620789	+	0.00832126i
$742$		0			0
$743$			23.5884i			0.865376i	0.901544	+	0.432688i	$0.142435\pi$
$743$			23.5884i			0.865376i	−0.901544	+	0.432688i	$0.857565\pi$
$744$		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	$0.121261\pi$
$761$			20.5134i			0.743609i	−0.928311	+	0.371804i	$0.878739\pi$
$762$		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	$0.245375\pi$
$769$			38.6434i			1.39352i	−0.717306	+	0.696758i	$0.754625\pi$
$770$		0			0
$771$	−38.3895			−1.38257
$772$		0			0
$773$			37.2827i			1.34097i	0.741924	+	0.670484i	$0.233913\pi$
$773$			37.2827i			1.34097i	−0.741924	+	0.670484i	$0.766087\pi$
$774$		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	$-0.665198\pi$
$787$		−	48.7190i		−	1.73665i	0.495999	−	0.868323i	$-0.334802\pi$
$788$		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	$-0.729847\pi$
$811$		−	42.7416i		−	1.50086i	0.660950	−	0.750430i	$-0.270153\pi$
$812$		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	$0.0176889\pi$
$821$			3.18294i			0.111085i	−0.998456	+	0.0555426i	$0.982311\pi$
$822$		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	$-0.987189\pi$
$827$		−	2.31427i		−	0.0804751i	0.999190	−	0.0402375i	$-0.0128115\pi$
$828$		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	$0.0612633\pi$
$839$			11.0809i			0.382557i	−0.981536	+	0.191278i	$0.938737\pi$
$840$		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	$0.0499300\pi$
$853$			9.12502i			0.312435i	−0.987723	+	0.156217i	$0.950070\pi$
$854$		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	$0.322215\pi$
$863$			49.8252i			1.69607i	−0.529940	+	0.848035i	$0.677785\pi$
$864$		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	$-0.739155\pi$
$877$		−	43.2831i		−	1.46157i	0.682609	−	0.730784i	$-0.260845\pi$
$878$		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	$-0.935512\pi$
$929$		−	12.2657i		−	0.402424i	0.979548	−	0.201212i	$-0.0644880\pi$
$930$		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	$0.0391703\pi$
$941$			7.53068i			0.245493i	−0.992438	+	0.122747i	$0.960830\pi$
$942$		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	$-0.815548\pi$
$947$		−	33.7019i		−	1.09517i	0.836752	−	0.547583i	$-0.184452\pi$
$948$		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	$0.322958\pi$
$967$			52.8189i			1.69854i	−0.527957	+	0.849271i	$0.677042\pi$
$968$		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	$0.000796011\pi$
$977$			0.156331i			0.00500148i	−0.999997	+	0.00250074i	$0.999204\pi$
$978$		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	$0.122622\pi$
$983$			23.5630i			0.751542i	−0.926712	+	0.375771i	$0.877378\pi$
$984$		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

Twists

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

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

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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.k (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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