LMFDB - Embedded newform 4600.2.e.w.4049.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 $ x^10 + 18x^8 + 117x^6 + 333x^4 + 396x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.6
Root			$0.794805i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.w.4049.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.794805i q^{3} +2.47193i q^{7} +2.36829 q^{9} +O(q^{10})$	$q+0.794805i q^{3} +2.47193i q^{7} +2.36829 q^{9} -2.29993 q^{11} +3.84022i q^{13} -7.74682i q^{17} +2.29993 q^{19} -1.96471 q^{21} +1.00000i q^{23} +4.26674i q^{27} -5.28380 q^{29} -6.40148 q^{31} -1.82800i q^{33} +8.56457i q^{37} -3.05223 q^{39} +4.27699 q^{41} +1.88954i q^{43} +12.3432i q^{47} +0.889540 q^{49} +6.15721 q^{51} +7.57482i q^{53} +1.82800i q^{57} -6.07180 q^{59} -0.635155 q^{61} +5.85425i q^{63} +11.1333i q^{67} -0.794805 q^{69} +8.58163 q^{71} -16.5849i q^{73} -5.68528i q^{77} -0.335225 q^{79} +3.71363 q^{81} -15.1937i q^{83} -4.19959i q^{87} -5.55735 q^{89} -9.49277 q^{91} -5.08792i q^{93} +6.42786i q^{97} -5.44689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{9}+O(q^{10}) $ 10 * q - 6 * q^9	$ 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) $ 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.794805i	0.458881i	0.973323	+	0.229440i	$0.0736896\pi$
$3$	0.794805i	0.458881i	−0.973323	+	0.229440i	$0.926310\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.47193i	0.934303i	0.884177	+	0.467152i	$0.154720\pi$
$7$	2.47193i	0.934303i	−0.884177	+	0.467152i	$0.845280\pi$
$8$	0	0
$9$	2.36829	0.789428
$10$	0	0
$11$	−2.29993	−0.693455	−0.346728	−	0.937966i	$-0.612707\pi$
$11$	−2.29993	−0.693455	−0.346728	+	0.937966i	$0.612707\pi$
$12$	0	0
$13$	3.84022i	1.06509i	0.846403	+	0.532543i	$0.178763\pi$
$13$	3.84022i	1.06509i	−0.846403	+	0.532543i	$0.821237\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.74682i	− 1.87888i	−0.342713	−	0.939440i	$-0.611346\pi$
$17$	− 7.74682i	− 1.87888i	0.342713	−	0.939440i	$-0.388654\pi$
$18$	0	0
$19$	2.29993	0.527640	0.263820	−	0.964572i	$-0.415017\pi$
$19$	2.29993	0.527640	0.263820	+	0.964572i	$0.415017\pi$
$20$	0	0
$21$	−1.96471	−0.428734
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.26674i	0.821134i
$28$	0	0
$29$	−5.28380	−0.981177	−0.490589	−	0.871391i	$-0.663218\pi$
$29$	−5.28380	−0.981177	−0.490589	+	0.871391i	$0.663218\pi$
$30$	0	0
$31$	−6.40148	−1.14974	−0.574870	−	0.818245i	$-0.694947\pi$
$31$	−6.40148	−1.14974	−0.574870	+	0.818245i	$0.694947\pi$
$32$	0	0
$33$	− 1.82800i	− 0.318213i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56457i	1.40801i	0.710197	+	0.704003i	$0.248606\pi$
$37$	8.56457i	1.40801i	−0.710197	+	0.704003i	$0.751394\pi$
$38$	0	0
$39$	−3.05223	−0.488747
$40$	0	0
$41$	4.27699	0.667954	0.333977	−	0.942581i	$-0.391609\pi$
$41$	4.27699	0.667954	0.333977	+	0.942581i	$0.391609\pi$
$42$	0	0
$43$	1.88954i	0.288152i	0.989567	+	0.144076i	$0.0460210\pi$
$43$	1.88954i	0.288152i	−0.989567	+	0.144076i	$0.953979\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.3432i	1.80045i	0.435428	+	0.900223i	$0.356597\pi$
$47$	12.3432i	1.80045i	−0.435428	+	0.900223i	$0.643403\pi$
$48$	0	0
$49$	0.889540	0.127077
$50$	0	0
$51$	6.15721	0.862182
$52$	0	0
$53$	7.57482i	1.04048i	0.854020	+	0.520241i	$0.174158\pi$
$53$	7.57482i	1.04048i	−0.854020	+	0.520241i	$0.825842\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.82800i	0.242124i
$58$	0	0
$59$	−6.07180	−0.790480	−0.395240	−	0.918578i	$-0.629339\pi$
$59$	−6.07180	−0.790480	−0.395240	+	0.918578i	$0.629339\pi$
$60$	0	0
$61$	−0.635155	−0.0813233	−0.0406616	−	0.999173i	$-0.512947\pi$
$61$	−0.635155	−0.0813233	−0.0406616	+	0.999173i	$0.512947\pi$
$62$	0	0
$63$	5.85425i	0.737566i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.1333i	1.36015i	0.733141	+	0.680077i	$0.238054\pi$
$67$	11.1333i	1.36015i	−0.733141	+	0.680077i	$0.761946\pi$
$68$	0	0
$69$	−0.794805	−0.0956833
$70$	0	0
$71$	8.58163	1.01845	0.509226	−	0.860633i	$-0.329932\pi$
$71$	8.58163	1.01845	0.509226	+	0.860633i	$0.329932\pi$
$72$	0	0
$73$	− 16.5849i	− 1.94112i	−0.240859	−	0.970560i	$-0.577429\pi$
$73$	− 16.5849i	− 1.94112i	0.240859	−	0.970560i	$-0.422571\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.68528i	− 0.647897i
$78$	0	0
$79$	−0.335225	−0.0377157	−0.0188579	−	0.999822i	$-0.506003\pi$
$79$	−0.335225	−0.0377157	−0.0188579	+	0.999822i	$0.506003\pi$
$80$	0	0
$81$	3.71363	0.412626
$82$	0	0
$83$	− 15.1937i	− 1.66773i	−0.551971	−	0.833863i	$-0.686124\pi$
$83$	− 15.1937i	− 1.66773i	0.551971	−	0.833863i	$-0.313876\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.19959i	− 0.450243i
$88$	0	0
$89$	−5.55735	−0.589078	−0.294539	−	0.955639i	$-0.595166\pi$
$89$	−5.55735	−0.589078	−0.294539	+	0.955639i	$0.595166\pi$
$90$	0	0
$91$	−9.49277	−0.995113
$92$	0	0
$93$	− 5.08792i	− 0.527593i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.42786i	0.652650i	0.945258	+	0.326325i	$0.105810\pi$
$97$	6.42786i	0.652650i	−0.945258	+	0.326325i	$0.894190\pi$
$98$	0	0
$99$	−5.44689	−0.547433
$100$	0	0
$101$	−14.6041	−1.45316	−0.726581	−	0.687081i	$-0.758892\pi$
$101$	−14.6041	−1.45316	−0.726581	+	0.687081i	$0.758892\pi$
$102$	0	0
$103$	13.3980i	1.32014i	0.751203	+	0.660071i	$0.229474\pi$
$103$	13.3980i	1.32014i	−0.751203	+	0.660071i	$0.770526\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.63549i	0.158109i	0.996870	+	0.0790544i	$0.0251901\pi$
$107$	1.63549i	0.158109i	−0.996870	+	0.0790544i	$0.974810\pi$
$108$	0	0
$109$	−1.50182	−0.143848	−0.0719239	−	0.997410i	$-0.522914\pi$
$109$	−1.50182	−0.143848	−0.0719239	+	0.997410i	$0.522914\pi$
$110$	0	0
$111$	−6.80716	−0.646107
$112$	0	0
$113$	− 17.6451i	− 1.65992i	−0.557826	−	0.829958i	$-0.688364\pi$
$113$	− 17.6451i	− 1.65992i	0.557826	−	0.829958i	$-0.311636\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.09474i	0.840809i
$118$	0	0
$119$	19.1496	1.75544
$120$	0	0
$121$	−5.71032	−0.519120
$122$	0	0
$123$	3.39937i	0.306511i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.97362i	0.352602i	0.984336	+	0.176301i	$0.0564132\pi$
$127$	3.97362i	0.352602i	−0.984336	+	0.176301i	$0.943587\pi$
$128$	0	0
$129$	−1.50182	−0.132228
$130$	0	0
$131$	−14.3032	−1.24968	−0.624840	−	0.780753i	$-0.714836\pi$
$131$	−14.3032	−1.24968	−0.624840	+	0.780753i	$0.714836\pi$
$132$	0	0
$133$	5.68528i	0.492976i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.25585i	0.363602i	0.983335	+	0.181801i	$0.0581927\pi$
$137$	4.25585i	0.363602i	−0.983335	+	0.181801i	$0.941807\pi$
$138$	0	0
$139$	9.86950	0.837120	0.418560	−	0.908189i	$-0.362535\pi$
$139$	9.86950	0.837120	0.418560	+	0.908189i	$0.362535\pi$
$140$	0	0
$141$	−9.81047	−0.826191
$142$	0	0
$143$	− 8.83224i	− 0.738589i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.707011i	0.0583133i
$148$	0	0
$149$	−9.55735	−0.782969	−0.391484	−	0.920185i	$-0.628038\pi$
$149$	−9.55735	−0.782969	−0.391484	+	0.920185i	$0.628038\pi$
$150$	0	0
$151$	−24.4557	−1.99018	−0.995090	−	0.0989730i	$-0.968444\pi$
$151$	−24.4557	−1.99018	−0.995090	+	0.0989730i	$0.968444\pi$
$152$	0	0
$153$	− 18.3467i	− 1.48324i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.1922i	0.973047i	0.873667	+	0.486524i	$0.161735\pi$
$157$	12.1922i	0.973047i	−0.873667	+	0.486524i	$0.838265\pi$
$158$	0	0
$159$	−6.02050	−0.477457
$160$	0	0
$161$	−2.47193	−0.194816
$162$	0	0
$163$	18.5762i	1.45500i	0.686109	+	0.727498i	$0.259317\pi$
$163$	18.5762i	1.45500i	−0.686109	+	0.727498i	$0.740683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.8639i	− 1.30497i	−0.757803	−	0.652484i	$-0.773727\pi$
$167$	− 16.8639i	− 1.30497i	0.757803	−	0.652484i	$-0.226273\pi$
$168$	0	0
$169$	−1.74729	−0.134407
$170$	0	0
$171$	5.44689	0.416534
$172$	0	0
$173$	− 24.4626i	− 1.85985i	−0.367745	−	0.929927i	$-0.619870\pi$
$173$	− 24.4626i	− 1.85985i	0.367745	−	0.929927i	$-0.380130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.82589i	− 0.362736i
$178$	0	0
$179$	−2.26173	−0.169050	−0.0845249	−	0.996421i	$-0.526937\pi$
$179$	−2.26173	−0.169050	−0.0845249	+	0.996421i	$0.526937\pi$
$180$	0	0
$181$	14.7187	1.09404	0.547018	−	0.837121i	$-0.315763\pi$
$181$	14.7187	1.09404	0.547018	+	0.837121i	$0.315763\pi$
$182$	0	0
$183$	− 0.504824i	− 0.0373177i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.8171i	1.30292i
$188$	0	0
$189$	−10.5471	−0.767189
$190$	0	0
$191$	−23.7055	−1.71527	−0.857636	−	0.514258i	$-0.828067\pi$
$191$	−23.7055	−1.71527	−0.857636	+	0.514258i	$0.828067\pi$
$192$	0	0
$193$	20.4146i	1.46947i	0.678353	+	0.734736i	$0.262694\pi$
$193$	20.4146i	1.46947i	−0.678353	+	0.734736i	$0.737306\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.7607i	1.26540i	0.774398	+	0.632699i	$0.218053\pi$
$197$	17.7607i	1.26540i	−0.774398	+	0.632699i	$0.781947\pi$
$198$	0	0
$199$	−11.1379	−0.789546	−0.394773	−	0.918779i	$-0.629177\pi$
$199$	−11.1379	−0.789546	−0.394773	+	0.918779i	$0.629177\pi$
$200$	0	0
$201$	−8.84883	−0.624149
$202$	0	0
$203$	− 13.0612i	− 0.916717i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.36829i	0.164607i
$208$	0	0
$209$	−5.28968	−0.365895
$210$	0	0
$211$	4.17347	0.287314	0.143657	−	0.989628i	$-0.454114\pi$
$211$	4.17347	0.287314	0.143657	+	0.989628i	$0.454114\pi$
$212$	0	0
$213$	6.82072i	0.467348i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 15.8240i	− 1.07421i
$218$	0	0
$219$	13.1818	0.890743
$220$	0	0
$221$	29.7495	2.00117
$222$	0	0
$223$	− 11.4218i	− 0.764863i	−0.923984	−	0.382432i	$-0.875087\pi$
$223$	− 11.4218i	− 0.764863i	0.923984	−	0.382432i	$-0.124913\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.40014i	0.358420i	0.983811	+	0.179210i	$0.0573541\pi$
$227$	5.40014i	0.358420i	−0.983811	+	0.179210i	$0.942646\pi$
$228$	0	0
$229$	18.4333	1.21811	0.609054	−	0.793129i	$-0.291549\pi$
$229$	18.4333	1.21811	0.609054	+	0.793129i	$0.291549\pi$
$230$	0	0
$231$	4.51869	0.297308
$232$	0	0
$233$	7.82122i	0.512385i	0.966626	+	0.256193i	$0.0824681\pi$
$233$	7.82122i	0.512385i	−0.966626	+	0.256193i	$0.917532\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.266438i	− 0.0173070i
$238$	0	0
$239$	2.26133	0.146273	0.0731365	−	0.997322i	$-0.476699\pi$
$239$	2.26133	0.146273	0.0731365	+	0.997322i	$0.476699\pi$
$240$	0	0
$241$	13.4906	0.869006	0.434503	−	0.900670i	$-0.356924\pi$
$241$	13.4906	0.869006	0.434503	+	0.900670i	$0.356924\pi$
$242$	0	0
$243$	15.7518i	1.01048i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.83224i	0.561982i
$248$	0	0
$249$	12.0760	0.765288
$250$	0	0
$251$	15.5642	0.982406	0.491203	−	0.871045i	$-0.336557\pi$
$251$	15.5642	0.982406	0.491203	+	0.871045i	$0.336557\pi$
$252$	0	0
$253$	− 2.29993i	− 0.144595i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.2354i	1.57414i	0.616865	+	0.787069i	$0.288402\pi$
$257$	25.2354i	1.57414i	−0.616865	+	0.787069i	$0.711598\pi$
$258$	0	0
$259$	−21.1710	−1.31550
$260$	0	0
$261$	−12.5135	−0.774569
$262$	0	0
$263$	12.6243i	0.778448i	0.921143	+	0.389224i	$0.127257\pi$
$263$	12.6243i	0.778448i	−0.921143	+	0.389224i	$0.872743\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.41701i	− 0.270317i
$268$	0	0
$269$	−25.0644	−1.52820	−0.764101	−	0.645096i	$-0.776817\pi$
$269$	−25.0644	−1.52820	−0.764101	+	0.645096i	$0.776817\pi$
$270$	0	0
$271$	−17.6777	−1.07384	−0.536922	−	0.843632i	$-0.680413\pi$
$271$	−17.6777	−1.07384	−0.536922	+	0.843632i	$0.680413\pi$
$272$	0	0
$273$	− 7.54490i	− 0.456638i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.22724i	− 0.434243i	−0.976145	−	0.217121i	$-0.930333\pi$
$277$	− 7.22724i	− 0.434243i	0.976145	−	0.217121i	$-0.0696667\pi$
$278$	0	0
$279$	−15.1605	−0.907637
$280$	0	0
$281$	−5.61315	−0.334852	−0.167426	−	0.985885i	$-0.553546\pi$
$281$	−5.61315	−0.334852	−0.167426	+	0.985885i	$0.553546\pi$
$282$	0	0
$283$	0.873541i	0.0519266i	0.999663	+	0.0259633i	$0.00826531\pi$
$283$	0.873541i	0.0519266i	−0.999663	+	0.0259633i	$0.991735\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.5724i	0.624071i
$288$	0	0
$289$	−43.0132	−2.53019
$290$	0	0
$291$	−5.10889	−0.299489
$292$	0	0
$293$	− 10.2737i	− 0.600195i	−0.953909	−	0.300097i	$-0.902981\pi$
$293$	− 10.2737i	− 0.600195i	0.953909	−	0.300097i	$-0.0970192\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.81320i	− 0.569420i
$298$	0	0
$299$	−3.84022	−0.222086
$300$	0	0
$301$	−4.67082	−0.269222
$302$	0	0
$303$	− 11.6074i	− 0.666828i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.3994i	1.50670i	0.657622	+	0.753348i	$0.271562\pi$
$307$	26.3994i	1.50670i	−0.657622	+	0.753348i	$0.728438\pi$
$308$	0	0
$309$	−10.6488	−0.605788
$310$	0	0
$311$	31.6429	1.79430	0.897151	−	0.441725i	$-0.145633\pi$
$311$	31.6429	1.79430	0.897151	+	0.441725i	$0.145633\pi$
$312$	0	0
$313$	24.1327i	1.36406i	0.731323	+	0.682032i	$0.238903\pi$
$313$	24.1327i	1.36406i	−0.731323	+	0.682032i	$0.761097\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.973154i	− 0.0546578i	−0.999626	−	0.0273289i	$-0.991300\pi$
$317$	− 0.973154i	− 0.0546578i	0.999626	−	0.0273289i	$-0.00870014\pi$
$318$	0	0
$319$	12.1524	0.680402
$320$	0	0
$321$	−1.29990	−0.0725531
$322$	0	0
$323$	− 17.8171i	− 0.991373i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.19365i	− 0.0660090i
$328$	0	0
$329$	−30.5117	−1.68216
$330$	0	0
$331$	1.10352	0.0606549	0.0303274	−	0.999540i	$-0.490345\pi$
$331$	1.10352	0.0606549	0.0303274	+	0.999540i	$0.490345\pi$
$332$	0	0
$333$	20.2833i	1.11152i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.33526i	− 0.508524i	−0.967135	−	0.254262i	$-0.918167\pi$
$337$	− 9.33526i	− 0.508524i	0.967135	−	0.254262i	$-0.0818326\pi$
$338$	0	0
$339$	14.0244	0.761703
$340$	0	0
$341$	14.7229	0.797292
$342$	0	0
$343$	19.5024i	1.05303i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.4052i	1.04173i	0.853641	+	0.520863i	$0.174390\pi$
$347$	19.4052i	1.04173i	−0.853641	+	0.520863i	$0.825610\pi$
$348$	0	0
$349$	−26.0825	−1.39616	−0.698081	−	0.716019i	$-0.745962\pi$
$349$	−26.0825	−1.39616	−0.698081	+	0.716019i	$0.745962\pi$
$350$	0	0
$351$	−16.3852	−0.874578
$352$	0	0
$353$	− 24.5504i	− 1.30669i	−0.757062	−	0.653343i	$-0.773366\pi$
$353$	− 24.5504i	− 1.30669i	0.757062	−	0.653343i	$-0.226634\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.2202i	0.805540i
$358$	0	0
$359$	10.2809	0.542607	0.271303	−	0.962494i	$-0.412545\pi$
$359$	10.2809	0.542607	0.271303	+	0.962494i	$0.412545\pi$
$360$	0	0
$361$	−13.7103	−0.721596
$362$	0	0
$363$	− 4.53859i	− 0.238214i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.2290i	− 0.690549i	−0.938502	−	0.345274i	$-0.887786\pi$
$367$	− 13.2290i	− 0.690549i	0.938502	−	0.345274i	$-0.112214\pi$
$368$	0	0
$369$	10.1291	0.527302
$370$	0	0
$371$	−18.7245	−0.972125
$372$	0	0
$373$	− 34.2573i	− 1.77378i	−0.461983	−	0.886889i	$-0.652862\pi$
$373$	− 34.2573i	− 1.77378i	0.461983	−	0.886889i	$-0.347138\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 20.2910i	− 1.04504i
$378$	0	0
$379$	10.2939	0.528763	0.264382	−	0.964418i	$-0.414832\pi$
$379$	10.2939	0.528763	0.264382	+	0.964418i	$0.414832\pi$
$380$	0	0
$381$	−3.15825	−0.161802
$382$	0	0
$383$	26.2028i	1.33890i	0.742857	+	0.669450i	$0.233470\pi$
$383$	26.2028i	1.33890i	−0.742857	+	0.669450i	$0.766530\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.47497i	0.227476i
$388$	0	0
$389$	−5.59078	−0.283464	−0.141732	−	0.989905i	$-0.545267\pi$
$389$	−5.59078	−0.283464	−0.141732	+	0.989905i	$0.545267\pi$
$390$	0	0
$391$	7.74682	0.391774
$392$	0	0
$393$	− 11.3683i	− 0.573454i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.98084i	− 0.149604i	−0.997198	−	0.0748019i	$-0.976168\pi$
$397$	− 2.98084i	− 0.149604i	0.997198	−	0.0748019i	$-0.0238324\pi$
$398$	0	0
$399$	−4.51869	−0.226217
$400$	0	0
$401$	−19.8229	−0.989906	−0.494953	−	0.868920i	$-0.664815\pi$
$401$	−19.8229	−0.989906	−0.494953	+	0.868920i	$0.664815\pi$
$402$	0	0
$403$	− 24.5831i	− 1.22457i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 19.6979i	− 0.976389i
$408$	0	0
$409$	−12.8521	−0.635498	−0.317749	−	0.948175i	$-0.602927\pi$
$409$	−12.8521	−0.635498	−0.317749	+	0.948175i	$0.602927\pi$
$410$	0	0
$411$	−3.38257	−0.166850
$412$	0	0
$413$	− 15.0091i	− 0.738549i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.84433i	0.384139i
$418$	0	0
$419$	5.42665	0.265109	0.132555	−	0.991176i	$-0.457682\pi$
$419$	5.42665	0.265109	0.132555	+	0.991176i	$0.457682\pi$
$420$	0	0
$421$	19.2456	0.937973	0.468987	−	0.883205i	$-0.344619\pi$
$421$	19.2456	0.937973	0.468987	+	0.883205i	$0.344619\pi$
$422$	0	0
$423$	29.2323i	1.42132i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.57006i	− 0.0759806i
$428$	0	0
$429$	7.01991	0.338924
$430$	0	0
$431$	8.53348	0.411043	0.205522	−	0.978653i	$-0.434111\pi$
$431$	8.53348	0.411043	0.205522	+	0.978653i	$0.434111\pi$
$432$	0	0
$433$	− 22.7745i	− 1.09447i	−0.836978	−	0.547237i	$-0.815680\pi$
$433$	− 22.7745i	− 1.09447i	0.836978	−	0.547237i	$-0.184320\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.29993i	0.110021i
$438$	0	0
$439$	−4.67135	−0.222952	−0.111476	−	0.993767i	$-0.535558\pi$
$439$	−4.67135	−0.222952	−0.111476	+	0.993767i	$0.535558\pi$
$440$	0	0
$441$	2.10668	0.100318
$442$	0	0
$443$	− 3.00378i	− 0.142714i	−0.997451	−	0.0713568i	$-0.977267\pi$
$443$	− 3.00378i	− 0.142714i	0.997451	−	0.0713568i	$-0.0227329\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7.59623i	− 0.359289i
$448$	0	0
$449$	−5.56914	−0.262824	−0.131412	−	0.991328i	$-0.541951\pi$
$449$	−5.56914	−0.262824	−0.131412	+	0.991328i	$0.541951\pi$
$450$	0	0
$451$	−9.83678	−0.463196
$452$	0	0
$453$	− 19.4375i	− 0.913256i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31.0180i	− 1.45096i	−0.688242	−	0.725481i	$-0.741617\pi$
$457$	− 31.0180i	− 1.45096i	0.688242	−	0.725481i	$-0.258383\pi$
$458$	0	0
$459$	33.0537	1.54281
$460$	0	0
$461$	−9.92290	−0.462155	−0.231078	−	0.972935i	$-0.574225\pi$
$461$	−9.92290	−0.462155	−0.231078	+	0.972935i	$0.574225\pi$
$462$	0	0
$463$	− 10.0111i	− 0.465256i	−0.972566	−	0.232628i	$-0.925267\pi$
$463$	− 10.0111i	− 0.465256i	0.972566	−	0.232628i	$-0.0747325\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.9345i	1.70912i	0.519350	+	0.854562i	$0.326174\pi$
$467$	36.9345i	1.70912i	−0.519350	+	0.854562i	$0.673826\pi$
$468$	0	0
$469$	−27.5209	−1.27080
$470$	0	0
$471$	−9.69046	−0.446513
$472$	0	0
$473$	− 4.34581i	− 0.199821i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	17.9393i	0.821385i
$478$	0	0
$479$	16.7734	0.766396	0.383198	−	0.923666i	$-0.374823\pi$
$479$	16.7734	0.766396	0.383198	+	0.923666i	$0.374823\pi$
$480$	0	0
$481$	−32.8898	−1.49965
$482$	0	0
$483$	− 1.96471i	− 0.0893972i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.47391i	0.248046i	0.992279	+	0.124023i	$0.0395797\pi$
$487$	5.47391i	0.248046i	−0.992279	+	0.124023i	$0.960420\pi$
$488$	0	0
$489$	−14.7644	−0.667670
$490$	0	0
$491$	35.4536	1.60000	0.800000	−	0.600001i	$-0.204833\pi$
$491$	35.4536	1.60000	0.800000	+	0.600001i	$0.204833\pi$
$492$	0	0
$493$	40.9327i	1.84351i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.2132i	0.951543i
$498$	0	0
$499$	33.4110	1.49568	0.747840	−	0.663879i	$-0.231091\pi$
$499$	33.4110	1.49568	0.747840	+	0.663879i	$0.231091\pi$
$500$	0	0
$501$	13.4035	0.598825
$502$	0	0
$503$	1.97733i	0.0881650i	0.999028	+	0.0440825i	$0.0140364\pi$
$503$	1.97733i	0.0881650i	−0.999028	+	0.0440825i	$0.985964\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.38875i	− 0.0616766i
$508$	0	0
$509$	23.0984	1.02382	0.511909	−	0.859040i	$-0.328939\pi$
$509$	23.0984	1.02382	0.511909	+	0.859040i	$0.328939\pi$
$510$	0	0
$511$	40.9969	1.81360
$512$	0	0
$513$	9.81320i	0.433264i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 28.3886i	− 1.24853i
$518$	0	0
$519$	19.4430	0.853451
$520$	0	0
$521$	21.2867	0.932588	0.466294	−	0.884630i	$-0.345589\pi$
$521$	21.2867	0.932588	0.466294	+	0.884630i	$0.345589\pi$
$522$	0	0
$523$	− 28.1943i	− 1.23285i	−0.787414	−	0.616425i	$-0.788580\pi$
$523$	− 28.1943i	− 1.23285i	0.787414	−	0.616425i	$-0.211420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	49.5911i	2.16022i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−14.3797	−0.624028
$532$	0	0
$533$	16.4246i	0.711428i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.79763i	− 0.0775737i
$538$	0	0
$539$	−2.04588	−0.0881223
$540$	0	0
$541$	14.8271	0.637468	0.318734	−	0.947844i	$-0.396742\pi$
$541$	14.8271	0.637468	0.318734	+	0.947844i	$0.396742\pi$
$542$	0	0
$543$	11.6985i	0.502032i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.09950i	− 0.389067i	−0.980896	−	0.194533i	$-0.937681\pi$
$547$	− 9.09950i	− 0.389067i	0.980896	−	0.194533i	$-0.0623192\pi$
$548$	0	0
$549$	−1.50423	−0.0641989
$550$	0	0
$551$	−12.1524	−0.517709
$552$	0	0
$553$	− 0.828654i	− 0.0352379i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.9824i	− 1.39751i	−0.715362	−	0.698754i	$-0.753738\pi$
$557$	− 32.9824i	− 1.39751i	0.715362	−	0.698754i	$-0.246262\pi$
$558$	0	0
$559$	−7.25625	−0.306907
$560$	0	0
$561$	−14.1612	−0.597885
$562$	0	0
$563$	21.9384i	0.924595i	0.886725	+	0.462297i	$0.152975\pi$
$563$	21.9384i	0.924595i	−0.886725	+	0.462297i	$0.847025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.17985i	0.385517i
$568$	0	0
$569$	28.2664	1.18499	0.592495	−	0.805574i	$-0.298143\pi$
$569$	28.2664	1.18499	0.592495	+	0.805574i	$0.298143\pi$
$570$	0	0
$571$	−4.13671	−0.173116	−0.0865580	−	0.996247i	$-0.527587\pi$
$571$	−4.13671	−0.173116	−0.0865580	+	0.996247i	$0.527587\pi$
$572$	0	0
$573$	− 18.8413i	− 0.787105i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.8146i	0.575111i	0.957764	+	0.287556i	$0.0928426\pi$
$577$	13.8146i	0.575111i	−0.957764	+	0.287556i	$0.907157\pi$
$578$	0	0
$579$	−16.2256	−0.674313
$580$	0	0
$581$	37.5579	1.55816
$582$	0	0
$583$	− 17.4216i	− 0.721527i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17.2506i	− 0.712010i	−0.934484	−	0.356005i	$-0.884139\pi$
$587$	− 17.2506i	− 0.712010i	0.934484	−	0.356005i	$-0.115861\pi$
$588$	0	0
$589$	−14.7229	−0.606649
$590$	0	0
$591$	−14.1163	−0.580667
$592$	0	0
$593$	− 27.0042i	− 1.10893i	−0.832207	−	0.554466i	$-0.812923\pi$
$593$	− 27.0042i	− 1.10893i	0.832207	−	0.554466i	$-0.187077\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.85247i	− 0.362307i
$598$	0	0
$599$	11.5259	0.470934	0.235467	−	0.971882i	$-0.424338\pi$
$599$	11.5259	0.470934	0.235467	+	0.971882i	$0.424338\pi$
$600$	0	0
$601$	−10.8541	−0.442750	−0.221375	−	0.975189i	$-0.571054\pi$
$601$	−10.8541	−0.442750	−0.221375	+	0.975189i	$0.571054\pi$
$602$	0	0
$603$	26.3669i	1.07374i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5.16930i	− 0.209816i	−0.994482	−	0.104908i	$-0.966545\pi$
$607$	− 5.16930i	− 0.209816i	0.994482	−	0.104908i	$-0.0334547\pi$
$608$	0	0
$609$	10.3811	0.420664
$610$	0	0
$611$	−47.4008	−1.91763
$612$	0	0
$613$	27.3190i	1.10340i	0.834042	+	0.551701i	$0.186021\pi$
$613$	27.3190i	1.10340i	−0.834042	+	0.551701i	$0.813979\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.08325i	0.204644i	0.994751	+	0.102322i	$0.0326272\pi$
$617$	5.08325i	0.204644i	−0.994751	+	0.102322i	$0.967373\pi$
$618$	0	0
$619$	0.207299	0.00833203	0.00416602	−	0.999991i	$-0.498674\pi$
$619$	0.207299	0.00833203	0.00416602	+	0.999991i	$0.498674\pi$
$620$	0	0
$621$	−4.26674	−0.171218
$622$	0	0
$623$	− 13.7374i	− 0.550378i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.20426i	− 0.167902i
$628$	0	0
$629$	66.3482	2.64547
$630$	0	0
$631$	40.7410	1.62187	0.810937	−	0.585133i	$-0.198958\pi$
$631$	40.7410	1.62187	0.810937	+	0.585133i	$0.198958\pi$
$632$	0	0
$633$	3.31710i	0.131843i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.41603i	0.135348i
$638$	0	0
$639$	20.3237	0.803995
$640$	0	0
$641$	−29.7426	−1.17476	−0.587381	−	0.809310i	$-0.699841\pi$
$641$	−29.7426	−1.17476	−0.587381	+	0.809310i	$0.699841\pi$
$642$	0	0
$643$	34.3329i	1.35396i	0.736003	+	0.676978i	$0.236711\pi$
$643$	34.3329i	1.35396i	−0.736003	+	0.676978i	$0.763289\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.0002i	− 0.589717i	−0.955541	−	0.294859i	$-0.904727\pi$
$647$	− 15.0002i	− 0.589717i	0.955541	−	0.294859i	$-0.0952726\pi$
$648$	0	0
$649$	13.9647	0.548163
$650$	0	0
$651$	12.5770	0.492932
$652$	0	0
$653$	14.2038i	0.555837i	0.960605	+	0.277918i	$0.0896444\pi$
$653$	14.2038i	0.555837i	−0.960605	+	0.277918i	$0.910356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 39.2779i	− 1.53238i
$658$	0	0
$659$	−47.9718	−1.86872	−0.934359	−	0.356334i	$-0.884027\pi$
$659$	−47.9718	−1.86872	−0.934359	+	0.356334i	$0.884027\pi$
$660$	0	0
$661$	43.6518	1.69786	0.848928	−	0.528508i	$-0.177248\pi$
$661$	43.6518	1.69786	0.848928	+	0.528508i	$0.177248\pi$
$662$	0	0
$663$	23.6450i	0.918297i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.28380i	− 0.204590i
$668$	0	0
$669$	9.07814	0.350981
$670$	0	0
$671$	1.46081	0.0563940
$672$	0	0
$673$	11.9330i	0.459983i	0.973193	+	0.229991i	$0.0738698\pi$
$673$	11.9330i	0.459983i	−0.973193	+	0.229991i	$0.926130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.7598i	− 0.720999i	−0.932759	−	0.360500i	$-0.882606\pi$
$677$	− 18.7598i	− 0.720999i	0.932759	−	0.360500i	$-0.117394\pi$
$678$	0	0
$679$	−15.8892	−0.609773
$680$	0	0
$681$	−4.29206	−0.164472
$682$	0	0
$683$	− 10.4542i	− 0.400020i	−0.979794	−	0.200010i	$-0.935902\pi$
$683$	− 10.4542i	− 0.400020i	0.979794	−	0.200010i	$-0.0640975\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.6509i	0.558966i
$688$	0	0
$689$	−29.0890	−1.10820
$690$	0	0
$691$	−20.8518	−0.793240	−0.396620	−	0.917983i	$-0.629817\pi$
$691$	−20.8518	−0.793240	−0.396620	+	0.917983i	$0.629817\pi$
$692$	0	0
$693$	− 13.4644i	− 0.511469i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 33.1331i	− 1.25500i
$698$	0	0
$699$	−6.21634	−0.235124
$700$	0	0
$701$	22.1231	0.835578	0.417789	−	0.908544i	$-0.362805\pi$
$701$	22.1231	0.835578	0.417789	+	0.908544i	$0.362805\pi$
$702$	0	0
$703$	19.6979i	0.742921i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 36.1004i	− 1.35769i
$708$	0	0
$709$	−14.1669	−0.532049	−0.266024	−	0.963966i	$-0.585710\pi$
$709$	−14.1669	−0.532049	−0.266024	+	0.963966i	$0.585710\pi$
$710$	0	0
$711$	−0.793908	−0.0297739
$712$	0	0
$713$	− 6.40148i	− 0.239737i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.79731i	0.0671219i
$718$	0	0
$719$	6.86389	0.255980	0.127990	−	0.991775i	$-0.459147\pi$
$719$	6.86389	0.255980	0.127990	+	0.991775i	$0.459147\pi$
$720$	0	0
$721$	−33.1189	−1.23341
$722$	0	0
$723$	10.7224i	0.398770i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.72298i	− 0.0639016i	−0.999489	−	0.0319508i	$-0.989828\pi$
$727$	− 1.72298i	− 0.0639016i	0.999489	−	0.0319508i	$-0.0101720\pi$
$728$	0	0
$729$	−1.37874	−0.0510645
$730$	0	0
$731$	14.6379	0.541403
$732$	0	0
$733$	− 13.2855i	− 0.490712i	−0.969433	−	0.245356i	$-0.921095\pi$
$733$	− 13.2855i	− 0.490712i	0.969433	−	0.245356i	$-0.0789049\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 25.6059i	− 0.943206i
$738$	0	0
$739$	28.8043	1.05958	0.529791	−	0.848128i	$-0.322270\pi$
$739$	28.8043	1.05958	0.529791	+	0.848128i	$0.322270\pi$
$740$	0	0
$741$	−7.01991	−0.257883
$742$	0	0
$743$	− 15.7966i	− 0.579521i	−0.957099	−	0.289761i	$-0.906424\pi$
$743$	− 15.7966i	− 0.579521i	0.957099	−	0.289761i	$-0.0935757\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 35.9830i	− 1.31655i
$748$	0	0
$749$	−4.04282	−0.147722
$750$	0	0
$751$	−12.7384	−0.464829	−0.232415	−	0.972617i	$-0.574663\pi$
$751$	−12.7384	−0.464829	−0.232415	+	0.972617i	$0.574663\pi$
$752$	0	0
$753$	12.3705i	0.450807i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.07420i	− 0.111734i	−0.998438	−	0.0558668i	$-0.982208\pi$
$757$	− 3.07420i	− 0.111734i	0.998438	−	0.0558668i	$-0.0177922\pi$
$758$	0	0
$759$	1.82800	0.0663520
$760$	0	0
$761$	30.6818	1.11222	0.556108	−	0.831110i	$-0.312294\pi$
$761$	30.6818	1.11222	0.556108	+	0.831110i	$0.312294\pi$
$762$	0	0
$763$	− 3.71239i	− 0.134398i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 23.3170i	− 0.841929i
$768$	0	0
$769$	34.5153	1.24465	0.622327	−	0.782758i	$-0.286188\pi$
$769$	34.5153	1.24465	0.622327	+	0.782758i	$0.286188\pi$
$770$	0	0
$771$	−20.0572	−0.722342
$772$	0	0
$773$	36.6662i	1.31879i	0.751796	+	0.659396i	$0.229188\pi$
$773$	36.6662i	1.31879i	−0.751796	+	0.659396i	$0.770812\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16.8269i	− 0.603660i
$778$	0	0
$779$	9.83678	0.352439
$780$	0	0
$781$	−19.7371	−0.706251
$782$	0	0
$783$	− 22.5446i	− 0.805678i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.6991i	0.630906i	0.948941	+	0.315453i	$0.102156\pi$
$787$	17.6991i	0.630906i	−0.948941	+	0.315453i	$0.897844\pi$
$788$	0	0
$789$	−10.0339	−0.357215
$790$	0	0
$791$	43.6176	1.55086
$792$	0	0
$793$	− 2.43913i	− 0.0866162i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.6535i	0.944117i	0.881567	+	0.472058i	$0.156489\pi$
$797$	26.6535i	0.944117i	−0.881567	+	0.472058i	$0.843511\pi$
$798$	0	0
$799$	95.6209	3.38282
$800$	0	0
$801$	−13.1614	−0.465035
$802$	0	0
$803$	38.1442i	1.34608i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 19.9213i	− 0.701263i
$808$	0	0
$809$	24.1080	0.847591	0.423795	−	0.905758i	$-0.360698\pi$
$809$	24.1080	0.847591	0.423795	+	0.905758i	$0.360698\pi$
$810$	0	0
$811$	−11.3363	−0.398071	−0.199035	−	0.979992i	$-0.563781\pi$
$811$	−11.3363	−0.398071	−0.199035	+	0.979992i	$0.563781\pi$
$812$	0	0
$813$	− 14.0503i	− 0.492766i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.34581i	0.152041i
$818$	0	0
$819$	−22.4816	−0.785570
$820$	0	0
$821$	−3.95739	−0.138114	−0.0690569	−	0.997613i	$-0.521999\pi$
$821$	−3.95739	−0.138114	−0.0690569	+	0.997613i	$0.521999\pi$
$822$	0	0
$823$	− 46.5023i	− 1.62097i	−0.585761	−	0.810484i	$-0.699204\pi$
$823$	− 46.5023i	− 1.62097i	0.585761	−	0.810484i	$-0.300796\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.7619i	0.965375i	0.875793	+	0.482687i	$0.160339\pi$
$827$	27.7619i	0.965375i	−0.875793	+	0.482687i	$0.839661\pi$
$828$	0	0
$829$	−37.8234	−1.31366	−0.656831	−	0.754038i	$-0.728104\pi$
$829$	−37.8234	−1.31366	−0.656831	+	0.754038i	$0.728104\pi$
$830$	0	0
$831$	5.74424	0.199266
$832$	0	0
$833$	− 6.89111i	− 0.238763i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 27.3134i	− 0.944090i
$838$	0	0
$839$	−20.0661	−0.692758	−0.346379	−	0.938095i	$-0.612589\pi$
$839$	−20.0661	−0.692758	−0.346379	+	0.938095i	$0.612589\pi$
$840$	0	0
$841$	−1.08145	−0.0372913
$842$	0	0
$843$	− 4.46136i	− 0.153657i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.1155i	− 0.485016i
$848$	0	0
$849$	−0.694295	−0.0238281
$850$	0	0
$851$	−8.56457	−0.293590
$852$	0	0
$853$	8.35933i	0.286218i	0.989707	+	0.143109i	$0.0457100\pi$
$853$	8.35933i	0.286218i	−0.989707	+	0.143109i	$0.954290\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 17.0931i	− 0.583889i	−0.956435	−	0.291945i	$-0.905698\pi$
$857$	− 17.0931i	− 0.583889i	0.956435	−	0.291945i	$-0.0943023\pi$
$858$	0	0
$859$	−23.0254	−0.785617	−0.392809	−	0.919620i	$-0.628497\pi$
$859$	−23.0254	−0.785617	−0.392809	+	0.919620i	$0.628497\pi$
$860$	0	0
$861$	−8.40303	−0.286374
$862$	0	0
$863$	− 5.49197i	− 0.186949i	−0.995622	−	0.0934745i	$-0.970203\pi$
$863$	− 5.49197i	− 0.186949i	0.995622	−	0.0934745i	$-0.0297974\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 34.1871i	− 1.16106i
$868$	0	0
$869$	0.770994	0.0261542
$870$	0	0
$871$	−42.7545	−1.44868
$872$	0	0
$873$	15.2230i	0.515220i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.98307i	0.269569i	0.990875	+	0.134785i	$0.0430342\pi$
$877$	7.98307i	0.269569i	−0.990875	+	0.134785i	$0.956966\pi$
$878$	0	0
$879$	8.16557	0.275418
$880$	0	0
$881$	44.5925	1.50236	0.751180	−	0.660098i	$-0.229485\pi$
$881$	44.5925	1.50236	0.751180	+	0.660098i	$0.229485\pi$
$882$	0	0
$883$	− 45.9134i	− 1.54511i	−0.634949	−	0.772554i	$-0.718979\pi$
$883$	− 45.9134i	− 1.54511i	0.634949	−	0.772554i	$-0.281021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.6109i	1.80008i	0.435810	+	0.900039i	$0.356462\pi$
$887$	53.6109i	1.80008i	−0.435810	+	0.900039i	$0.643538\pi$
$888$	0	0
$889$	−9.82253	−0.329437
$890$	0	0
$891$	−8.54109	−0.286137
$892$	0	0
$893$	28.3886i	0.949988i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.05223i	− 0.101911i
$898$	0	0
$899$	33.8241	1.12810
$900$	0	0
$901$	58.6808	1.95494
$902$	0	0
$903$	− 3.71239i	− 0.123541i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.1277i	1.49844i	0.662322	+	0.749220i	$0.269571\pi$
$907$	45.1277i	1.49844i	−0.662322	+	0.749220i	$0.730429\pi$
$908$	0	0
$909$	−34.5867	−1.14717
$910$	0	0
$911$	6.06248	0.200859	0.100429	−	0.994944i	$-0.467978\pi$
$911$	6.06248	0.200859	0.100429	+	0.994944i	$0.467978\pi$
$912$	0	0
$913$	34.9445i	1.15649i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 35.3567i	− 1.16758i
$918$	0	0
$919$	−38.8692	−1.28218	−0.641089	−	0.767467i	$-0.721517\pi$
$919$	−38.8692	−1.28218	−0.641089	+	0.767467i	$0.721517\pi$
$920$	0	0
$921$	−20.9824	−0.691394
$922$	0	0
$923$	32.9553i	1.08474i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31.7302i	1.04216i
$928$	0	0
$929$	43.4632	1.42598	0.712991	−	0.701173i	$-0.247340\pi$
$929$	43.4632	1.42598	0.712991	+	0.701173i	$0.247340\pi$
$930$	0	0
$931$	2.04588	0.0670510
$932$	0	0
$933$	25.1499i	0.823371i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.7812i	0.646223i	0.946361	+	0.323112i	$0.104729\pi$
$937$	19.7812i	0.646223i	−0.946361	+	0.323112i	$0.895271\pi$
$938$	0	0
$939$	−19.1808	−0.625942
$940$	0	0
$941$	31.1805	1.01645	0.508227	−	0.861223i	$-0.330301\pi$
$941$	31.1805	1.01645	0.508227	+	0.861223i	$0.330301\pi$
$942$	0	0
$943$	4.27699i	0.139278i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.65195i	0.118673i	0.998238	+	0.0593363i	$0.0188984\pi$
$947$	3.65195i	0.118673i	−0.998238	+	0.0593363i	$0.981102\pi$
$948$	0	0
$949$	63.6898	2.06746
$950$	0	0
$951$	0.773468	0.0250814
$952$	0	0
$953$	− 4.81987i	− 0.156131i	−0.996948	−	0.0780655i	$-0.975126\pi$
$953$	− 4.81987i	− 0.156131i	0.996948	−	0.0780655i	$-0.0248743\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.65877i	0.312224i
$958$	0	0
$959$	−10.5202	−0.339715
$960$	0	0
$961$	9.97890	0.321900
$962$	0	0
$963$	3.87331i	0.124816i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 28.9718i	− 0.931670i	−0.884872	−	0.465835i	$-0.845754\pi$
$967$	− 28.9718i	− 0.931670i	0.884872	−	0.465835i	$-0.154246\pi$
$968$	0	0
$969$	14.1612	0.454922
$970$	0	0
$971$	32.8376	1.05381	0.526905	−	0.849924i	$-0.323352\pi$
$971$	32.8376	1.05381	0.526905	+	0.849924i	$0.323352\pi$
$972$	0	0
$973$	24.3968i	0.782125i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.0091i	1.85587i	0.372737	+	0.927937i	$0.378419\pi$
$977$	58.0091i	1.85587i	−0.372737	+	0.927937i	$0.621581\pi$
$978$	0	0
$979$	12.7815	0.408499
$980$	0	0
$981$	−3.55673	−0.113558
$982$	0	0
$983$	− 4.41757i	− 0.140899i	−0.997515	−	0.0704493i	$-0.977557\pi$
$983$	− 4.41757i	− 0.140899i	0.997515	−	0.0704493i	$-0.0224433\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.2508i	− 0.771913i
$988$	0	0
$989$	−1.88954	−0.0600839
$990$	0	0
$991$	36.1617	1.14871	0.574357	−	0.818605i	$-0.305252\pi$
$991$	36.1617	1.14871	0.574357	+	0.818605i	$0.305252\pi$
$992$	0	0
$993$	0.877082i	0.0278334i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.6357i	0.368506i	0.982879	+	0.184253i	$0.0589866\pi$
$997$	11.6357i	0.368506i	−0.982879	+	0.184253i	$0.941013\pi$
$998$	0	0
$999$	−36.5428	−1.15616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.w.4049.6		10
5.2	odd	4		4600.2.a.bd.1.3	✓	5
5.3	odd	4		4600.2.a.bf.1.3	yes	5
5.4	even	2	inner	4600.2.e.w.4049.5		10
20.3	even	4		9200.2.a.ct.1.3		5
20.7	even	4		9200.2.a.cv.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.3	✓	5	5.2	odd	4
4600.2.a.bf.1.3	yes	5	5.3	odd	4
4600.2.e.w.4049.5		10	5.4	even	2	inner
4600.2.e.w.4049.6		10	1.1	even	1	trivial
9200.2.a.ct.1.3		5	20.3	even	4
9200.2.a.cv.1.3		5	20.7	even	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.794805i q^{3} +2.47193i q^{7} +2.36829 q^{9} +O(q^{10})\)	\(q+0.794805i q^{3} +2.47193i q^{7} +2.36829 q^{9} -2.29993 q^{11} +3.84022i q^{13} -7.74682i q^{17} +2.29993 q^{19} -1.96471 q^{21} +1.00000i q^{23} +4.26674i q^{27} -5.28380 q^{29} -6.40148 q^{31} -1.82800i q^{33} +8.56457i q^{37} -3.05223 q^{39} +4.27699 q^{41} +1.88954i q^{43} +12.3432i q^{47} +0.889540 q^{49} +6.15721 q^{51} +7.57482i q^{53} +1.82800i q^{57} -6.07180 q^{59} -0.635155 q^{61} +5.85425i q^{63} +11.1333i q^{67} -0.794805 q^{69} +8.58163 q^{71} -16.5849i q^{73} -5.68528i q^{77} -0.335225 q^{79} +3.71363 q^{81} -15.1937i q^{83} -4.19959i q^{87} -5.55735 q^{89} -9.49277 q^{91} -5.08792i q^{93} +6.42786i q^{97} -5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{9}+O(q^{10}) \) 10 * q - 6 * q^9	\( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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