LMFDB - Embedded newform 4600.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

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:	yes
Analytic conductor:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.521397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 18x + 12 $ x^5 - 9x^3 - 3x^2 + 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.794805$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.794805 q^{3} -2.47193 q^{7} -2.36829 q^{9} -2.29993 q^{11} +3.84022 q^{13} +7.74682 q^{17} -2.29993 q^{19} -1.96471 q^{21} +1.00000 q^{23} -4.26674 q^{27} +5.28380 q^{29} -6.40148 q^{31} -1.82800 q^{33} -8.56457 q^{37} +3.05223 q^{39} +4.27699 q^{41} +1.88954 q^{43} -12.3432 q^{47} -0.889540 q^{49} +6.15721 q^{51} +7.57482 q^{53} -1.82800 q^{57} +6.07180 q^{59} -0.635155 q^{61} +5.85425 q^{63} -11.1333 q^{67} +0.794805 q^{69} +8.58163 q^{71} -16.5849 q^{73} +5.68528 q^{77} +0.335225 q^{79} +3.71363 q^{81} -15.1937 q^{83} +4.19959 q^{87} +5.55735 q^{89} -9.49277 q^{91} -5.08792 q^{93} -6.42786 q^{97} +5.44689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} - 18 q^{31} - 6 q^{33} - 10 q^{37} + 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} - 6 q^{51} - 10 q^{53} - 6 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) $ 5 * q - 4 * q^7 + 3 * q^9 - 4 * q^13 - 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 - 18 * q^31 - 6 * q^33 - 10 * q^37 + 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 - 6 * q^51 - 10 * q^53 - 6 * q^57 - q^59 + 10 * q^61 - 8 * q^67 + 8 * q^71 - 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 - 46 * q^91 + 3 * q^93 - 6 * q^97 - 6 * q^99

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
$3$	0.794805	0.458881	0.229440	−	0.973323i	$-0.426310\pi$
$3$	0.794805	0.458881	0.229440	+	0.973323i	$0.426310\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.47193	−0.934303	−0.467152	−	0.884177i	$-0.654720\pi$
$7$	−2.47193	−0.934303	−0.467152	+	0.884177i	$0.654720\pi$
$8$	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
$13$	3.84022	1.06509	0.532543	−	0.846403i	$-0.321237\pi$
$13$	3.84022	1.06509	0.532543	+	0.846403i	$0.321237\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.74682	1.87888	0.939440	−	0.342713i	$-0.111346\pi$
$17$	7.74682	1.87888	0.939440	+	0.342713i	$0.111346\pi$
$18$	0	0
$19$	−2.29993	−0.527640	−0.263820	−	0.964572i	$-0.584983\pi$
$19$	−2.29993	−0.527640	−0.263820	+	0.964572i	$0.584983\pi$
$20$	0	0
$21$	−1.96471	−0.428734
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.26674	−0.821134
$28$	0	0
$29$	5.28380	0.981177	0.490589	−	0.871391i	$-0.336782\pi$
$29$	5.28380	0.981177	0.490589	+	0.871391i	$0.336782\pi$
$30$	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
$33$	−1.82800	−0.318213
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.56457	−1.40801	−0.704003	−	0.710197i	$-0.748606\pi$
$37$	−8.56457	−1.40801	−0.704003	+	0.710197i	$0.748606\pi$
$38$	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
$43$	1.88954	0.288152	0.144076	−	0.989567i	$-0.453979\pi$
$43$	1.88954	0.288152	0.144076	+	0.989567i	$0.453979\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.3432	−1.80045	−0.900223	−	0.435428i	$-0.856597\pi$
$47$	−12.3432	−1.80045	−0.900223	+	0.435428i	$0.856597\pi$
$48$	0	0
$49$	−0.889540	−0.127077
$50$	0	0
$51$	6.15721	0.862182
$52$	0	0
$53$	7.57482	1.04048	0.520241	−	0.854020i	$-0.325842\pi$
$53$	7.57482	1.04048	0.520241	+	0.854020i	$0.325842\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.82800	−0.242124
$58$	0	0
$59$	6.07180	0.790480	0.395240	−	0.918578i	$-0.370661\pi$
$59$	6.07180	0.790480	0.395240	+	0.918578i	$0.370661\pi$
$60$	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
$63$	5.85425	0.737566
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.1333	−1.36015	−0.680077	−	0.733141i	$-0.738054\pi$
$67$	−11.1333	−1.36015	−0.680077	+	0.733141i	$0.738054\pi$
$68$	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
$73$	−16.5849	−1.94112	−0.970560	−	0.240859i	$-0.922571\pi$
$73$	−16.5849	−1.94112	−0.970560	+	0.240859i	$0.922571\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.68528	0.647897
$78$	0	0
$79$	0.335225	0.0377157	0.0188579	−	0.999822i	$-0.493997\pi$
$79$	0.335225	0.0377157	0.0188579	+	0.999822i	$0.493997\pi$
$80$	0	0
$81$	3.71363	0.412626
$82$	0	0
$83$	−15.1937	−1.66773	−0.833863	−	0.551971i	$-0.813876\pi$
$83$	−15.1937	−1.66773	−0.833863	+	0.551971i	$0.813876\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.19959	0.450243
$88$	0	0
$89$	5.55735	0.589078	0.294539	−	0.955639i	$-0.404834\pi$
$89$	5.55735	0.589078	0.294539	+	0.955639i	$0.404834\pi$
$90$	0	0
$91$	−9.49277	−0.995113
$92$	0	0
$93$	−5.08792	−0.527593
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.42786	−0.652650	−0.326325	−	0.945258i	$-0.605810\pi$
$97$	−6.42786	−0.652650	−0.326325	+	0.945258i	$0.605810\pi$
$98$	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
$103$	13.3980	1.32014	0.660071	−	0.751203i	$-0.270526\pi$
$103$	13.3980	1.32014	0.660071	+	0.751203i	$0.270526\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.63549	−0.158109	−0.0790544	−	0.996870i	$-0.525190\pi$
$107$	−1.63549	−0.158109	−0.0790544	+	0.996870i	$0.525190\pi$
$108$	0	0
$109$	1.50182	0.143848	0.0719239	−	0.997410i	$-0.477086\pi$
$109$	1.50182	0.143848	0.0719239	+	0.997410i	$0.477086\pi$
$110$	0	0
$111$	−6.80716	−0.646107
$112$	0	0
$113$	−17.6451	−1.65992	−0.829958	−	0.557826i	$-0.811636\pi$
$113$	−17.6451	−1.65992	−0.829958	+	0.557826i	$0.811636\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.09474	−0.840809
$118$	0	0
$119$	−19.1496	−1.75544
$120$	0	0
$121$	−5.71032	−0.519120
$122$	0	0
$123$	3.39937	0.306511
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.97362	−0.352602	−0.176301	−	0.984336i	$-0.556413\pi$
$127$	−3.97362	−0.352602	−0.176301	+	0.984336i	$0.556413\pi$
$128$	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
$133$	5.68528	0.492976
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.25585	−0.363602	−0.181801	−	0.983335i	$-0.558193\pi$
$137$	−4.25585	−0.363602	−0.181801	+	0.983335i	$0.558193\pi$
$138$	0	0
$139$	−9.86950	−0.837120	−0.418560	−	0.908189i	$-0.637465\pi$
$139$	−9.86950	−0.837120	−0.418560	+	0.908189i	$0.637465\pi$
$140$	0	0
$141$	−9.81047	−0.826191
$142$	0	0
$143$	−8.83224	−0.738589
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.707011	−0.0583133
$148$	0	0
$149$	9.55735	0.782969	0.391484	−	0.920185i	$-0.371962\pi$
$149$	9.55735	0.782969	0.391484	+	0.920185i	$0.371962\pi$
$150$	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
$153$	−18.3467	−1.48324
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.1922	−0.973047	−0.486524	−	0.873667i	$-0.661735\pi$
$157$	−12.1922	−0.973047	−0.486524	+	0.873667i	$0.661735\pi$
$158$	0	0
$159$	6.02050	0.477457
$160$	0	0
$161$	−2.47193	−0.194816
$162$	0	0
$163$	18.5762	1.45500	0.727498	−	0.686109i	$-0.240683\pi$
$163$	18.5762	1.45500	0.727498	+	0.686109i	$0.240683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.8639	1.30497	0.652484	−	0.757803i	$-0.273727\pi$
$167$	16.8639	1.30497	0.652484	+	0.757803i	$0.273727\pi$
$168$	0	0
$169$	1.74729	0.134407
$170$	0	0
$171$	5.44689	0.416534
$172$	0	0
$173$	−24.4626	−1.85985	−0.929927	−	0.367745i	$-0.880130\pi$
$173$	−24.4626	−1.85985	−0.929927	+	0.367745i	$0.880130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.82589	0.362736
$178$	0	0
$179$	2.26173	0.169050	0.0845249	−	0.996421i	$-0.473063\pi$
$179$	2.26173	0.169050	0.0845249	+	0.996421i	$0.473063\pi$
$180$	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
$183$	−0.504824	−0.0373177
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.8171	−1.30292
$188$	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
$193$	20.4146	1.46947	0.734736	−	0.678353i	$-0.237306\pi$
$193$	20.4146	1.46947	0.734736	+	0.678353i	$0.237306\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.7607	−1.26540	−0.632699	−	0.774398i	$-0.718053\pi$
$197$	−17.7607	−1.26540	−0.632699	+	0.774398i	$0.718053\pi$
$198$	0	0
$199$	11.1379	0.789546	0.394773	−	0.918779i	$-0.370823\pi$
$199$	11.1379	0.789546	0.394773	+	0.918779i	$0.370823\pi$
$200$	0	0
$201$	−8.84883	−0.624149
$202$	0	0
$203$	−13.0612	−0.916717
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.36829	−0.164607
$208$	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
$213$	6.82072	0.467348
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.8240	1.07421
$218$	0	0
$219$	−13.1818	−0.890743
$220$	0	0
$221$	29.7495	2.00117
$222$	0	0
$223$	−11.4218	−0.764863	−0.382432	−	0.923984i	$-0.624913\pi$
$223$	−11.4218	−0.764863	−0.382432	+	0.923984i	$0.624913\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.40014	−0.358420	−0.179210	−	0.983811i	$-0.557354\pi$
$227$	−5.40014	−0.358420	−0.179210	+	0.983811i	$0.557354\pi$
$228$	0	0
$229$	−18.4333	−1.21811	−0.609054	−	0.793129i	$-0.708451\pi$
$229$	−18.4333	−1.21811	−0.609054	+	0.793129i	$0.708451\pi$
$230$	0	0
$231$	4.51869	0.297308
$232$	0	0
$233$	7.82122	0.512385	0.256193	−	0.966626i	$-0.417532\pi$
$233$	7.82122	0.512385	0.256193	+	0.966626i	$0.417532\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.266438	0.0173070
$238$	0	0
$239$	−2.26133	−0.146273	−0.0731365	−	0.997322i	$-0.523301\pi$
$239$	−2.26133	−0.146273	−0.0731365	+	0.997322i	$0.523301\pi$
$240$	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
$243$	15.7518	1.01048
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.83224	−0.561982
$248$	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
$253$	−2.29993	−0.144595
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.2354	−1.57414	−0.787069	−	0.616865i	$-0.788402\pi$
$257$	−25.2354	−1.57414	−0.787069	+	0.616865i	$0.788402\pi$
$258$	0	0
$259$	21.1710	1.31550
$260$	0	0
$261$	−12.5135	−0.774569
$262$	0	0
$263$	12.6243	0.778448	0.389224	−	0.921143i	$-0.372743\pi$
$263$	12.6243	0.778448	0.389224	+	0.921143i	$0.372743\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.41701	0.270317
$268$	0	0
$269$	25.0644	1.52820	0.764101	−	0.645096i	$-0.223183\pi$
$269$	25.0644	1.52820	0.764101	+	0.645096i	$0.223183\pi$
$270$	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
$273$	−7.54490	−0.456638
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.22724	0.434243	0.217121	−	0.976145i	$-0.430333\pi$
$277$	7.22724	0.434243	0.217121	+	0.976145i	$0.430333\pi$
$278$	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
$283$	0.873541	0.0519266	0.0259633	−	0.999663i	$-0.491735\pi$
$283$	0.873541	0.0519266	0.0259633	+	0.999663i	$0.491735\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.5724	−0.624071
$288$	0	0
$289$	43.0132	2.53019
$290$	0	0
$291$	−5.10889	−0.299489
$292$	0	0
$293$	−10.2737	−0.600195	−0.300097	−	0.953909i	$-0.597019\pi$
$293$	−10.2737	−0.600195	−0.300097	+	0.953909i	$0.597019\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.81320	0.569420
$298$	0	0
$299$	3.84022	0.222086
$300$	0	0
$301$	−4.67082	−0.269222
$302$	0	0
$303$	−11.6074	−0.666828
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.3994	−1.50670	−0.753348	−	0.657622i	$-0.771562\pi$
$307$	−26.3994	−1.50670	−0.753348	+	0.657622i	$0.771562\pi$
$308$	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
$313$	24.1327	1.36406	0.682032	−	0.731323i	$-0.261097\pi$
$313$	24.1327	1.36406	0.682032	+	0.731323i	$0.261097\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.973154	0.0546578	0.0273289	−	0.999626i	$-0.491300\pi$
$317$	0.973154	0.0546578	0.0273289	+	0.999626i	$0.491300\pi$
$318$	0	0
$319$	−12.1524	−0.680402
$320$	0	0
$321$	−1.29990	−0.0725531
$322$	0	0
$323$	−17.8171	−0.991373
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.19365	0.0660090
$328$	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
$333$	20.2833	1.11152
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.33526	0.508524	0.254262	−	0.967135i	$-0.418167\pi$
$337$	9.33526	0.508524	0.254262	+	0.967135i	$0.418167\pi$
$338$	0	0
$339$	−14.0244	−0.761703
$340$	0	0
$341$	14.7229	0.797292
$342$	0	0
$343$	19.5024	1.05303
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.4052	−1.04173	−0.520863	−	0.853641i	$-0.674390\pi$
$347$	−19.4052	−1.04173	−0.520863	+	0.853641i	$0.674390\pi$
$348$	0	0
$349$	26.0825	1.39616	0.698081	−	0.716019i	$-0.254038\pi$
$349$	26.0825	1.39616	0.698081	+	0.716019i	$0.254038\pi$
$350$	0	0
$351$	−16.3852	−0.874578
$352$	0	0
$353$	−24.5504	−1.30669	−0.653343	−	0.757062i	$-0.726634\pi$
$353$	−24.5504	−1.30669	−0.653343	+	0.757062i	$0.726634\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−15.2202	−0.805540
$358$	0	0
$359$	−10.2809	−0.542607	−0.271303	−	0.962494i	$-0.587455\pi$
$359$	−10.2809	−0.542607	−0.271303	+	0.962494i	$0.587455\pi$
$360$	0	0
$361$	−13.7103	−0.721596
$362$	0	0
$363$	−4.53859	−0.238214
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.2290	0.690549	0.345274	−	0.938502i	$-0.387786\pi$
$367$	13.2290	0.690549	0.345274	+	0.938502i	$0.387786\pi$
$368$	0	0
$369$	−10.1291	−0.527302
$370$	0	0
$371$	−18.7245	−0.972125
$372$	0	0
$373$	−34.2573	−1.77378	−0.886889	−	0.461983i	$-0.847138\pi$
$373$	−34.2573	−1.77378	−0.886889	+	0.461983i	$0.847138\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.2910	1.04504
$378$	0	0
$379$	−10.2939	−0.528763	−0.264382	−	0.964418i	$-0.585168\pi$
$379$	−10.2939	−0.528763	−0.264382	+	0.964418i	$0.585168\pi$
$380$	0	0
$381$	−3.15825	−0.161802
$382$	0	0
$383$	26.2028	1.33890	0.669450	−	0.742857i	$-0.266530\pi$
$383$	26.2028	1.33890	0.669450	+	0.742857i	$0.266530\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.47497	−0.227476
$388$	0	0
$389$	5.59078	0.283464	0.141732	−	0.989905i	$-0.454733\pi$
$389$	5.59078	0.283464	0.141732	+	0.989905i	$0.454733\pi$
$390$	0	0
$391$	7.74682	0.391774
$392$	0	0
$393$	−11.3683	−0.573454
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.98084	0.149604	0.0748019	−	0.997198i	$-0.476168\pi$
$397$	2.98084	0.149604	0.0748019	+	0.997198i	$0.476168\pi$
$398$	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
$403$	−24.5831	−1.22457
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.6979	0.976389
$408$	0	0
$409$	12.8521	0.635498	0.317749	−	0.948175i	$-0.397073\pi$
$409$	12.8521	0.635498	0.317749	+	0.948175i	$0.397073\pi$
$410$	0	0
$411$	−3.38257	−0.166850
$412$	0	0
$413$	−15.0091	−0.738549
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.84433	−0.384139
$418$	0	0
$419$	−5.42665	−0.265109	−0.132555	−	0.991176i	$-0.542318\pi$
$419$	−5.42665	−0.265109	−0.132555	+	0.991176i	$0.542318\pi$
$420$	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
$423$	29.2323	1.42132
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.57006	0.0759806
$428$	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
$433$	−22.7745	−1.09447	−0.547237	−	0.836978i	$-0.684320\pi$
$433$	−22.7745	−1.09447	−0.547237	+	0.836978i	$0.684320\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.29993	−0.110021
$438$	0	0
$439$	4.67135	0.222952	0.111476	−	0.993767i	$-0.464442\pi$
$439$	4.67135	0.222952	0.111476	+	0.993767i	$0.464442\pi$
$440$	0	0
$441$	2.10668	0.100318
$442$	0	0
$443$	−3.00378	−0.142714	−0.0713568	−	0.997451i	$-0.522733\pi$
$443$	−3.00378	−0.142714	−0.0713568	+	0.997451i	$0.522733\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.59623	0.359289
$448$	0	0
$449$	5.56914	0.262824	0.131412	−	0.991328i	$-0.458049\pi$
$449$	5.56914	0.262824	0.131412	+	0.991328i	$0.458049\pi$
$450$	0	0
$451$	−9.83678	−0.463196
$452$	0	0
$453$	−19.4375	−0.913256
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.0180	1.45096	0.725481	−	0.688242i	$-0.241617\pi$
$457$	31.0180	1.45096	0.725481	+	0.688242i	$0.241617\pi$
$458$	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
$463$	−10.0111	−0.465256	−0.232628	−	0.972566i	$-0.574733\pi$
$463$	−10.0111	−0.465256	−0.232628	+	0.972566i	$0.574733\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.9345	−1.70912	−0.854562	−	0.519350i	$-0.826174\pi$
$467$	−36.9345	−1.70912	−0.854562	+	0.519350i	$0.826174\pi$
$468$	0	0
$469$	27.5209	1.27080
$470$	0	0
$471$	−9.69046	−0.446513
$472$	0	0
$473$	−4.34581	−0.199821
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−17.9393	−0.821385
$478$	0	0
$479$	−16.7734	−0.766396	−0.383198	−	0.923666i	$-0.625177\pi$
$479$	−16.7734	−0.766396	−0.383198	+	0.923666i	$0.625177\pi$
$480$	0	0
$481$	−32.8898	−1.49965
$482$	0	0
$483$	−1.96471	−0.0893972
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.47391	−0.248046	−0.124023	−	0.992279i	$-0.539580\pi$
$487$	−5.47391	−0.248046	−0.124023	+	0.992279i	$0.539580\pi$
$488$	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
$493$	40.9327	1.84351
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21.2132	−0.951543
$498$	0	0
$499$	−33.4110	−1.49568	−0.747840	−	0.663879i	$-0.768909\pi$
$499$	−33.4110	−1.49568	−0.747840	+	0.663879i	$0.768909\pi$
$500$	0	0
$501$	13.4035	0.598825
$502$	0	0
$503$	1.97733	0.0881650	0.0440825	−	0.999028i	$-0.485964\pi$
$503$	1.97733	0.0881650	0.0440825	+	0.999028i	$0.485964\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.38875	0.0616766
$508$	0	0
$509$	−23.0984	−1.02382	−0.511909	−	0.859040i	$-0.671061\pi$
$509$	−23.0984	−1.02382	−0.511909	+	0.859040i	$0.671061\pi$
$510$	0	0
$511$	40.9969	1.81360
$512$	0	0
$513$	9.81320	0.433264
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28.3886	1.24853
$518$	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
$523$	−28.1943	−1.23285	−0.616425	−	0.787414i	$-0.711420\pi$
$523$	−28.1943	−1.23285	−0.616425	+	0.787414i	$0.711420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−49.5911	−2.16022
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.3797	−0.624028
$532$	0	0
$533$	16.4246	0.711428
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.79763	0.0775737
$538$	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
$543$	11.6985	0.502032
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.09950	0.389067	0.194533	−	0.980896i	$-0.437681\pi$
$547$	9.09950	0.389067	0.194533	+	0.980896i	$0.437681\pi$
$548$	0	0
$549$	1.50423	0.0641989
$550$	0	0
$551$	−12.1524	−0.517709
$552$	0	0
$553$	−0.828654	−0.0352379
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.9824	1.39751	0.698754	−	0.715362i	$-0.253738\pi$
$557$	32.9824	1.39751	0.698754	+	0.715362i	$0.253738\pi$
$558$	0	0
$559$	7.25625	0.306907
$560$	0	0
$561$	−14.1612	−0.597885
$562$	0	0
$563$	21.9384	0.924595	0.462297	−	0.886725i	$-0.347025\pi$
$563$	21.9384	0.924595	0.462297	+	0.886725i	$0.347025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−9.17985	−0.385517
$568$	0	0
$569$	−28.2664	−1.18499	−0.592495	−	0.805574i	$-0.701857\pi$
$569$	−28.2664	−1.18499	−0.592495	+	0.805574i	$0.701857\pi$
$570$	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
$573$	−18.8413	−0.787105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13.8146	−0.575111	−0.287556	−	0.957764i	$-0.592843\pi$
$577$	−13.8146	−0.575111	−0.287556	+	0.957764i	$0.592843\pi$
$578$	0	0
$579$	16.2256	0.674313
$580$	0	0
$581$	37.5579	1.55816
$582$	0	0
$583$	−17.4216	−0.721527
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.2506	0.712010	0.356005	−	0.934484i	$-0.384139\pi$
$587$	17.2506	0.712010	0.356005	+	0.934484i	$0.384139\pi$
$588$	0	0
$589$	14.7229	0.606649
$590$	0	0
$591$	−14.1163	−0.580667
$592$	0	0
$593$	−27.0042	−1.10893	−0.554466	−	0.832207i	$-0.687077\pi$
$593$	−27.0042	−1.10893	−0.554466	+	0.832207i	$0.687077\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.85247	0.362307
$598$	0	0
$599$	−11.5259	−0.470934	−0.235467	−	0.971882i	$-0.575662\pi$
$599$	−11.5259	−0.470934	−0.235467	+	0.971882i	$0.575662\pi$
$600$	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
$603$	26.3669	1.07374
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.16930	0.209816	0.104908	−	0.994482i	$-0.466545\pi$
$607$	5.16930	0.209816	0.104908	+	0.994482i	$0.466545\pi$
$608$	0	0
$609$	−10.3811	−0.420664
$610$	0	0
$611$	−47.4008	−1.91763
$612$	0	0
$613$	27.3190	1.10340	0.551701	−	0.834042i	$-0.313979\pi$
$613$	27.3190	1.10340	0.551701	+	0.834042i	$0.313979\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.08325	−0.204644	−0.102322	−	0.994751i	$-0.532627\pi$
$617$	−5.08325	−0.204644	−0.102322	+	0.994751i	$0.532627\pi$
$618$	0	0
$619$	−0.207299	−0.00833203	−0.00416602	−	0.999991i	$-0.501326\pi$
$619$	−0.207299	−0.00833203	−0.00416602	+	0.999991i	$0.501326\pi$
$620$	0	0
$621$	−4.26674	−0.171218
$622$	0	0
$623$	−13.7374	−0.550378
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.20426	0.167902
$628$	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
$633$	3.31710	0.131843
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.41603	−0.135348
$638$	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
$643$	34.3329	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$643$	34.3329	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.0002	0.589717	0.294859	−	0.955541i	$-0.404727\pi$
$647$	15.0002	0.589717	0.294859	+	0.955541i	$0.404727\pi$
$648$	0	0
$649$	−13.9647	−0.548163
$650$	0	0
$651$	12.5770	0.492932
$652$	0	0
$653$	14.2038	0.555837	0.277918	−	0.960605i	$-0.410356\pi$
$653$	14.2038	0.555837	0.277918	+	0.960605i	$0.410356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	39.2779	1.53238
$658$	0	0
$659$	47.9718	1.86872	0.934359	−	0.356334i	$-0.115973\pi$
$659$	47.9718	1.86872	0.934359	+	0.356334i	$0.115973\pi$
$660$	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
$663$	23.6450	0.918297
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.28380	0.204590
$668$	0	0
$669$	−9.07814	−0.350981
$670$	0	0
$671$	1.46081	0.0563940
$672$	0	0
$673$	11.9330	0.459983	0.229991	−	0.973193i	$-0.426130\pi$
$673$	11.9330	0.459983	0.229991	+	0.973193i	$0.426130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.7598	0.720999	0.360500	−	0.932759i	$-0.382606\pi$
$677$	18.7598	0.720999	0.360500	+	0.932759i	$0.382606\pi$
$678$	0	0
$679$	15.8892	0.609773
$680$	0	0
$681$	−4.29206	−0.164472
$682$	0	0
$683$	−10.4542	−0.400020	−0.200010	−	0.979794i	$-0.564098\pi$
$683$	−10.4542	−0.400020	−0.200010	+	0.979794i	$0.564098\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.6509	−0.558966
$688$	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
$693$	−13.4644	−0.511469
$694$	0	0
$695$	0	0
$696$	0	0
$697$	33.1331	1.25500
$698$	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
$703$	19.6979	0.742921
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.1004	1.35769
$708$	0	0
$709$	14.1669	0.532049	0.266024	−	0.963966i	$-0.414290\pi$
$709$	14.1669	0.532049	0.266024	+	0.963966i	$0.414290\pi$
$710$	0	0
$711$	−0.793908	−0.0297739
$712$	0	0
$713$	−6.40148	−0.239737
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.79731	−0.0671219
$718$	0	0
$719$	−6.86389	−0.255980	−0.127990	−	0.991775i	$-0.540853\pi$
$719$	−6.86389	−0.255980	−0.127990	+	0.991775i	$0.540853\pi$
$720$	0	0
$721$	−33.1189	−1.23341
$722$	0	0
$723$	10.7224	0.398770
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.72298	0.0639016	0.0319508	−	0.999489i	$-0.489828\pi$
$727$	1.72298	0.0639016	0.0319508	+	0.999489i	$0.489828\pi$
$728$	0	0
$729$	1.37874	0.0510645
$730$	0	0
$731$	14.6379	0.541403
$732$	0	0
$733$	−13.2855	−0.490712	−0.245356	−	0.969433i	$-0.578905\pi$
$733$	−13.2855	−0.490712	−0.245356	+	0.969433i	$0.578905\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.6059	0.943206
$738$	0	0
$739$	−28.8043	−1.05958	−0.529791	−	0.848128i	$-0.677730\pi$
$739$	−28.8043	−1.05958	−0.529791	+	0.848128i	$0.677730\pi$
$740$	0	0
$741$	−7.01991	−0.257883
$742$	0	0
$743$	−15.7966	−0.579521	−0.289761	−	0.957099i	$-0.593576\pi$
$743$	−15.7966	−0.579521	−0.289761	+	0.957099i	$0.593576\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	35.9830	1.31655
$748$	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
$753$	12.3705	0.450807
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.07420	0.111734	0.0558668	−	0.998438i	$-0.482208\pi$
$757$	3.07420	0.111734	0.0558668	+	0.998438i	$0.482208\pi$
$758$	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
$763$	−3.71239	−0.134398
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.3170	0.841929
$768$	0	0
$769$	−34.5153	−1.24465	−0.622327	−	0.782758i	$-0.713812\pi$
$769$	−34.5153	−1.24465	−0.622327	+	0.782758i	$0.713812\pi$
$770$	0	0
$771$	−20.0572	−0.722342
$772$	0	0
$773$	36.6662	1.31879	0.659396	−	0.751796i	$-0.270812\pi$
$773$	36.6662	1.31879	0.659396	+	0.751796i	$0.270812\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.8269	0.603660
$778$	0	0
$779$	−9.83678	−0.352439
$780$	0	0
$781$	−19.7371	−0.706251
$782$	0	0
$783$	−22.5446	−0.805678
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−17.6991	−0.630906	−0.315453	−	0.948941i	$-0.602156\pi$
$787$	−17.6991	−0.630906	−0.315453	+	0.948941i	$0.602156\pi$
$788$	0	0
$789$	10.0339	0.357215
$790$	0	0
$791$	43.6176	1.55086
$792$	0	0
$793$	−2.43913	−0.0866162
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.6535	−0.944117	−0.472058	−	0.881567i	$-0.656489\pi$
$797$	−26.6535	−0.944117	−0.472058	+	0.881567i	$0.656489\pi$
$798$	0	0
$799$	−95.6209	−3.38282
$800$	0	0
$801$	−13.1614	−0.465035
$802$	0	0
$803$	38.1442	1.34608
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.9213	0.701263
$808$	0	0
$809$	−24.1080	−0.847591	−0.423795	−	0.905758i	$-0.639302\pi$
$809$	−24.1080	−0.847591	−0.423795	+	0.905758i	$0.639302\pi$
$810$	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
$813$	−14.0503	−0.492766
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.34581	−0.152041
$818$	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
$823$	−46.5023	−1.62097	−0.810484	−	0.585761i	$-0.800796\pi$
$823$	−46.5023	−1.62097	−0.810484	+	0.585761i	$0.800796\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.7619	−0.965375	−0.482687	−	0.875793i	$-0.660339\pi$
$827$	−27.7619	−0.965375	−0.482687	+	0.875793i	$0.660339\pi$
$828$	0	0
$829$	37.8234	1.31366	0.656831	−	0.754038i	$-0.271896\pi$
$829$	37.8234	1.31366	0.656831	+	0.754038i	$0.271896\pi$
$830$	0	0
$831$	5.74424	0.199266
$832$	0	0
$833$	−6.89111	−0.238763
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.3134	0.944090
$838$	0	0
$839$	20.0661	0.692758	0.346379	−	0.938095i	$-0.387411\pi$
$839$	20.0661	0.692758	0.346379	+	0.938095i	$0.387411\pi$
$840$	0	0
$841$	−1.08145	−0.0372913
$842$	0	0
$843$	−4.46136	−0.153657
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.1155	0.485016
$848$	0	0
$849$	0.694295	0.0238281
$850$	0	0
$851$	−8.56457	−0.293590
$852$	0	0
$853$	8.35933	0.286218	0.143109	−	0.989707i	$-0.454290\pi$
$853$	8.35933	0.286218	0.143109	+	0.989707i	$0.454290\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.0931	0.583889	0.291945	−	0.956435i	$-0.405698\pi$
$857$	17.0931	0.583889	0.291945	+	0.956435i	$0.405698\pi$
$858$	0	0
$859$	23.0254	0.785617	0.392809	−	0.919620i	$-0.371503\pi$
$859$	23.0254	0.785617	0.392809	+	0.919620i	$0.371503\pi$
$860$	0	0
$861$	−8.40303	−0.286374
$862$	0	0
$863$	−5.49197	−0.186949	−0.0934745	−	0.995622i	$-0.529797\pi$
$863$	−5.49197	−0.186949	−0.0934745	+	0.995622i	$0.529797\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	34.1871	1.16106
$868$	0	0
$869$	−0.770994	−0.0261542
$870$	0	0
$871$	−42.7545	−1.44868
$872$	0	0
$873$	15.2230	0.515220
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.98307	−0.269569	−0.134785	−	0.990875i	$-0.543034\pi$
$877$	−7.98307	−0.269569	−0.134785	+	0.990875i	$0.543034\pi$
$878$	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
$883$	−45.9134	−1.54511	−0.772554	−	0.634949i	$-0.781021\pi$
$883$	−45.9134	−1.54511	−0.772554	+	0.634949i	$0.781021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.6109	−1.80008	−0.900039	−	0.435810i	$-0.856462\pi$
$887$	−53.6109	−1.80008	−0.900039	+	0.435810i	$0.856462\pi$
$888$	0	0
$889$	9.82253	0.329437
$890$	0	0
$891$	−8.54109	−0.286137
$892$	0	0
$893$	28.3886	0.949988
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.05223	0.101911
$898$	0	0
$899$	−33.8241	−1.12810
$900$	0	0
$901$	58.6808	1.95494
$902$	0	0
$903$	−3.71239	−0.123541
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45.1277	−1.49844	−0.749220	−	0.662322i	$-0.769571\pi$
$907$	−45.1277	−1.49844	−0.749220	+	0.662322i	$0.769571\pi$
$908$	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
$913$	34.9445	1.15649
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35.3567	1.16758
$918$	0	0
$919$	38.8692	1.28218	0.641089	−	0.767467i	$-0.278483\pi$
$919$	38.8692	1.28218	0.641089	+	0.767467i	$0.278483\pi$
$920$	0	0
$921$	−20.9824	−0.691394
$922$	0	0
$923$	32.9553	1.08474
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−31.7302	−1.04216
$928$	0	0
$929$	−43.4632	−1.42598	−0.712991	−	0.701173i	$-0.752660\pi$
$929$	−43.4632	−1.42598	−0.712991	+	0.701173i	$0.752660\pi$
$930$	0	0
$931$	2.04588	0.0670510
$932$	0	0
$933$	25.1499	0.823371
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.7812	−0.646223	−0.323112	−	0.946361i	$-0.604729\pi$
$937$	−19.7812	−0.646223	−0.323112	+	0.946361i	$0.604729\pi$
$938$	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
$943$	4.27699	0.139278
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.65195	−0.118673	−0.0593363	−	0.998238i	$-0.518898\pi$
$947$	−3.65195	−0.118673	−0.0593363	+	0.998238i	$0.518898\pi$
$948$	0	0
$949$	−63.6898	−2.06746
$950$	0	0
$951$	0.773468	0.0250814
$952$	0	0
$953$	−4.81987	−0.156131	−0.0780655	−	0.996948i	$-0.524874\pi$
$953$	−4.81987	−0.156131	−0.0780655	+	0.996948i	$0.524874\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.65877	−0.312224
$958$	0	0
$959$	10.5202	0.339715
$960$	0	0
$961$	9.97890	0.321900
$962$	0	0
$963$	3.87331	0.124816
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.9718	0.931670	0.465835	−	0.884872i	$-0.345754\pi$
$967$	28.9718	0.931670	0.465835	+	0.884872i	$0.345754\pi$
$968$	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
$973$	24.3968	0.782125
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.0091	−1.85587	−0.927937	−	0.372737i	$-0.878419\pi$
$977$	−58.0091	−1.85587	−0.927937	+	0.372737i	$0.878419\pi$
$978$	0	0
$979$	−12.7815	−0.408499
$980$	0	0
$981$	−3.55673	−0.113558
$982$	0	0
$983$	−4.41757	−0.140899	−0.0704493	−	0.997515i	$-0.522443\pi$
$983$	−4.41757	−0.140899	−0.0704493	+	0.997515i	$0.522443\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.2508	0.771913
$988$	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
$993$	0.877082	0.0278334
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.6357	−0.368506	−0.184253	−	0.982879i	$-0.558987\pi$
$997$	−11.6357	−0.368506	−0.184253	+	0.982879i	$0.558987\pi$
$998$	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.a.bd.1.3	✓	5
4.3	odd	2		9200.2.a.cv.1.3		5
5.2	odd	4		4600.2.e.w.4049.5		10
5.3	odd	4		4600.2.e.w.4049.6		10
5.4	even	2		4600.2.a.bf.1.3	yes	5
20.19	odd	2		9200.2.a.ct.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.3	✓	5	1.1	even	1	trivial
4600.2.a.bf.1.3	yes	5	5.4	even	2
4600.2.e.w.4049.5		10	5.2	odd	4
4600.2.e.w.4049.6		10	5.3	odd	4
9200.2.a.ct.1.3		5	20.19	odd	2
9200.2.a.cv.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+0.794805 q^{3} -2.47193 q^{7} -2.36829 q^{9} -2.29993 q^{11} +3.84022 q^{13} +7.74682 q^{17} -2.29993 q^{19} -1.96471 q^{21} +1.00000 q^{23} -4.26674 q^{27} +5.28380 q^{29} -6.40148 q^{31} -1.82800 q^{33} -8.56457 q^{37} +3.05223 q^{39} +4.27699 q^{41} +1.88954 q^{43} -12.3432 q^{47} -0.889540 q^{49} +6.15721 q^{51} +7.57482 q^{53} -1.82800 q^{57} +6.07180 q^{59} -0.635155 q^{61} +5.85425 q^{63} -11.1333 q^{67} +0.794805 q^{69} +8.58163 q^{71} -16.5849 q^{73} +5.68528 q^{77} +0.335225 q^{79} +3.71363 q^{81} -15.1937 q^{83} +4.19959 q^{87} +5.55735 q^{89} -9.49277 q^{91} -5.08792 q^{93} -6.42786 q^{97} +5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} - 18 q^{31} - 6 q^{33} - 10 q^{37} + 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} - 6 q^{51} - 10 q^{53} - 6 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) 5 * q - 4 * q^7 + 3 * q^9 - 4 * q^13 - 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 - 18 * q^31 - 6 * q^33 - 10 * q^37 + 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 - 6 * q^51 - 10 * q^53 - 6 * q^57 - q^59 + 10 * q^61 - 8 * q^67 + 8 * q^71 - 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 - 46 * q^91 + 3 * q^93 - 6 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more