LMFDB - Embedded newform 4600.2.a.bi.1.7

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.98707$ 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)

$f(q)$	$=$	$q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} +O(q^{10})$	$q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} -6.14721 q^{11} -6.37400 q^{13} +3.36098 q^{17} +1.08276 q^{19} -2.92900 q^{21} -1.00000 q^{23} +8.72993 q^{27} -0.271042 q^{29} -8.77792 q^{31} -18.3621 q^{33} -8.84665 q^{37} -19.0396 q^{39} -4.85308 q^{41} +1.87756 q^{43} +0.196089 q^{47} -6.03850 q^{49} +10.0395 q^{51} -1.93157 q^{53} +3.23429 q^{57} +13.0036 q^{59} +6.00189 q^{61} -5.80744 q^{63} +2.26847 q^{67} -2.98707 q^{69} +10.2677 q^{71} +1.38188 q^{73} +6.02771 q^{77} -4.67459 q^{79} +8.30917 q^{81} -15.7171 q^{83} -0.809622 q^{87} -11.1548 q^{89} +6.25008 q^{91} -26.2202 q^{93} -10.2868 q^{97} -36.4073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} - 7 q^{11} - 7 q^{13} - 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} - 10 q^{31} - 19 q^{33} - 19 q^{37} - 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} - 7 q^{51} - 15 q^{53} - 8 q^{57} - 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} - 14 q^{71} - 10 q^{73} - 6 q^{77} - 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} - 7 q^{91} - 26 q^{93} + 7 q^{97} - 61 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 - 7 * q^11 - 7 * q^13 - 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 - 10 * q^31 - 19 * q^33 - 19 * q^37 - 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 - 7 * q^51 - 15 * q^53 - 8 * q^57 - 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 - 14 * q^71 - 10 * q^73 - 6 * q^77 - 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 - 7 * q^91 - 26 * q^93 + 7 * q^97 - 61 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.98707	1.72458	0.862292	−	0.506411i	$-0.169028\pi$
$3$	2.98707	1.72458	0.862292	+	0.506411i	$0.169028\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.980560	−0.370617	−0.185308	−	0.982680i	$-0.559328\pi$
$7$	−0.980560	−0.370617	−0.185308	+	0.982680i	$0.559328\pi$
$8$	0	0
$9$	5.92257	1.97419
$10$	0	0
$11$	−6.14721	−1.85345	−0.926727	−	0.375734i	$-0.877391\pi$
$11$	−6.14721	−1.85345	−0.926727	+	0.375734i	$0.877391\pi$
$12$	0	0
$13$	−6.37400	−1.76783	−0.883914	−	0.467649i	$-0.845101\pi$
$13$	−6.37400	−1.76783	−0.883914	+	0.467649i	$0.845101\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.36098	0.815157	0.407579	−	0.913170i	$-0.366373\pi$
$17$	3.36098	0.815157	0.407579	+	0.913170i	$0.366373\pi$
$18$	0	0
$19$	1.08276	0.248403	0.124202	−	0.992257i	$-0.460363\pi$
$19$	1.08276	0.248403	0.124202	+	0.992257i	$0.460363\pi$
$20$	0	0
$21$	−2.92900	−0.639160
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	8.72993	1.68008
$28$	0	0
$29$	−0.271042	−0.0503313	−0.0251657	−	0.999683i	$-0.508011\pi$
$29$	−0.271042	−0.0503313	−0.0251657	+	0.999683i	$0.508011\pi$
$30$	0	0
$31$	−8.77792	−1.57656	−0.788280	−	0.615316i	$-0.789028\pi$
$31$	−8.77792	−1.57656	−0.788280	+	0.615316i	$0.789028\pi$
$32$	0	0
$33$	−18.3621	−3.19644
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.84665	−1.45438	−0.727190	−	0.686436i	$-0.759174\pi$
$37$	−8.84665	−1.45438	−0.727190	+	0.686436i	$0.759174\pi$
$38$	0	0
$39$	−19.0396	−3.04877
$40$	0	0
$41$	−4.85308	−0.757923	−0.378962	−	0.925412i	$-0.623719\pi$
$41$	−4.85308	−0.757923	−0.378962	+	0.925412i	$0.623719\pi$
$42$	0	0
$43$	1.87756	0.286326	0.143163	−	0.989699i	$-0.454273\pi$
$43$	1.87756	0.286326	0.143163	+	0.989699i	$0.454273\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.196089	0.0286026	0.0143013	−	0.999898i	$-0.495448\pi$
$47$	0.196089	0.0286026	0.0143013	+	0.999898i	$0.495448\pi$
$48$	0	0
$49$	−6.03850	−0.862643
$50$	0	0
$51$	10.0395	1.40581
$52$	0	0
$53$	−1.93157	−0.265321	−0.132661	−	0.991162i	$-0.542352\pi$
$53$	−1.93157	−0.265321	−0.132661	+	0.991162i	$0.542352\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.23429	0.428392
$58$	0	0
$59$	13.0036	1.69293	0.846465	−	0.532444i	$-0.178726\pi$
$59$	13.0036	1.69293	0.846465	+	0.532444i	$0.178726\pi$
$60$	0	0
$61$	6.00189	0.768464	0.384232	−	0.923237i	$-0.374466\pi$
$61$	6.00189	0.768464	0.384232	+	0.923237i	$0.374466\pi$
$62$	0	0
$63$	−5.80744	−0.731668
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.26847	0.277138	0.138569	−	0.990353i	$-0.455750\pi$
$67$	2.26847	0.277138	0.138569	+	0.990353i	$0.455750\pi$
$68$	0	0
$69$	−2.98707	−0.359601
$70$	0	0
$71$	10.2677	1.21855	0.609277	−	0.792958i	$-0.291460\pi$
$71$	10.2677	1.21855	0.609277	+	0.792958i	$0.291460\pi$
$72$	0	0
$73$	1.38188	0.161737	0.0808685	−	0.996725i	$-0.474231\pi$
$73$	1.38188	0.161737	0.0808685	+	0.996725i	$0.474231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.02771	0.686921
$78$	0	0
$79$	−4.67459	−0.525932	−0.262966	−	0.964805i	$-0.584701\pi$
$79$	−4.67459	−0.525932	−0.262966	+	0.964805i	$0.584701\pi$
$80$	0	0
$81$	8.30917	0.923241
$82$	0	0
$83$	−15.7171	−1.72517	−0.862587	−	0.505909i	$-0.831157\pi$
$83$	−15.7171	−1.72517	−0.862587	+	0.505909i	$0.831157\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.809622	−0.0868006
$88$	0	0
$89$	−11.1548	−1.18241	−0.591206	−	0.806521i	$-0.701348\pi$
$89$	−11.1548	−1.18241	−0.591206	+	0.806521i	$0.701348\pi$
$90$	0	0
$91$	6.25008	0.655187
$92$	0	0
$93$	−26.2202	−2.71891
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.2868	−1.04447	−0.522234	−	0.852802i	$-0.674901\pi$
$97$	−10.2868	−1.04447	−0.522234	+	0.852802i	$0.674901\pi$
$98$	0	0
$99$	−36.4073	−3.65908
$100$	0	0
$101$	−15.2625	−1.51868	−0.759339	−	0.650695i	$-0.774478\pi$
$101$	−15.2625	−1.51868	−0.759339	+	0.650695i	$0.774478\pi$
$102$	0	0
$103$	−7.60795	−0.749634	−0.374817	−	0.927099i	$-0.622294\pi$
$103$	−7.60795	−0.749634	−0.374817	+	0.927099i	$0.622294\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3081	1.09320	0.546599	−	0.837395i	$-0.315922\pi$
$107$	11.3081	1.09320	0.546599	+	0.837395i	$0.315922\pi$
$108$	0	0
$109$	3.45336	0.330772	0.165386	−	0.986229i	$-0.447113\pi$
$109$	3.45336	0.330772	0.165386	+	0.986229i	$0.447113\pi$
$110$	0	0
$111$	−26.4256	−2.50820
$112$	0	0
$113$	14.6889	1.38181	0.690907	−	0.722944i	$-0.257212\pi$
$113$	14.6889	1.38181	0.690907	+	0.722944i	$0.257212\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−37.7505	−3.49003
$118$	0	0
$119$	−3.29564	−0.302111
$120$	0	0
$121$	26.7882	2.43530
$122$	0	0
$123$	−14.4965	−1.30710
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.19312	−0.105873	−0.0529363	−	0.998598i	$-0.516858\pi$
$127$	−1.19312	−0.105873	−0.0529363	+	0.998598i	$0.516858\pi$
$128$	0	0
$129$	5.60841	0.493793
$130$	0	0
$131$	−11.7326	−1.02509	−0.512543	−	0.858662i	$-0.671296\pi$
$131$	−11.7326	−1.02509	−0.512543	+	0.858662i	$0.671296\pi$
$132$	0	0
$133$	−1.06171	−0.0920623
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.4665	−0.894210	−0.447105	−	0.894481i	$-0.647545\pi$
$137$	−10.4665	−0.894210	−0.447105	+	0.894481i	$0.647545\pi$
$138$	0	0
$139$	4.11349	0.348902	0.174451	−	0.984666i	$-0.444185\pi$
$139$	4.11349	0.348902	0.174451	+	0.984666i	$0.444185\pi$
$140$	0	0
$141$	0.585732	0.0493275
$142$	0	0
$143$	39.1823	3.27659
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.0374	−1.48770
$148$	0	0
$149$	1.37793	0.112885	0.0564424	−	0.998406i	$-0.482024\pi$
$149$	1.37793	0.112885	0.0564424	+	0.998406i	$0.482024\pi$
$150$	0	0
$151$	−3.49384	−0.284325	−0.142162	−	0.989843i	$-0.545406\pi$
$151$	−3.49384	−0.284325	−0.142162	+	0.989843i	$0.545406\pi$
$152$	0	0
$153$	19.9057	1.60928
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.32373	−0.265262	−0.132631	−	0.991165i	$-0.542343\pi$
$157$	−3.32373	−0.265262	−0.132631	+	0.991165i	$0.542343\pi$
$158$	0	0
$159$	−5.76973	−0.457569
$160$	0	0
$161$	0.980560	0.0772789
$162$	0	0
$163$	−8.52075	−0.667397	−0.333698	−	0.942680i	$-0.608297\pi$
$163$	−8.52075	−0.667397	−0.333698	+	0.942680i	$0.608297\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.6190	1.20864	0.604319	−	0.796742i	$-0.293445\pi$
$167$	15.6190	1.20864	0.604319	+	0.796742i	$0.293445\pi$
$168$	0	0
$169$	27.6278	2.12522
$170$	0	0
$171$	6.41275	0.490395
$172$	0	0
$173$	−8.42727	−0.640713	−0.320357	−	0.947297i	$-0.603803\pi$
$173$	−8.42727	−0.640713	−0.320357	+	0.947297i	$0.603803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	38.8428	2.91960
$178$	0	0
$179$	10.8047	0.807580	0.403790	−	0.914852i	$-0.367693\pi$
$179$	10.8047	0.807580	0.403790	+	0.914852i	$0.367693\pi$
$180$	0	0
$181$	−15.4282	−1.14677	−0.573385	−	0.819286i	$-0.694370\pi$
$181$	−15.4282	−1.14677	−0.573385	+	0.819286i	$0.694370\pi$
$182$	0	0
$183$	17.9281	1.32528
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.6607	−1.51086
$188$	0	0
$189$	−8.56022	−0.622664
$190$	0	0
$191$	−22.2634	−1.61092	−0.805462	−	0.592648i	$-0.798083\pi$
$191$	−22.2634	−1.61092	−0.805462	+	0.592648i	$0.798083\pi$
$192$	0	0
$193$	16.1399	1.16178	0.580889	−	0.813983i	$-0.302705\pi$
$193$	16.1399	1.16178	0.580889	+	0.813983i	$0.302705\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.9936	0.997004	0.498502	−	0.866889i	$-0.333884\pi$
$197$	13.9936	0.997004	0.498502	+	0.866889i	$0.333884\pi$
$198$	0	0
$199$	−6.91926	−0.490493	−0.245246	−	0.969461i	$-0.578869\pi$
$199$	−6.91926	−0.490493	−0.245246	+	0.969461i	$0.578869\pi$
$200$	0	0
$201$	6.77608	0.477948
$202$	0	0
$203$	0.265773	0.0186536
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.92257	−0.411647
$208$	0	0
$209$	−6.65598	−0.460404
$210$	0	0
$211$	12.9170	0.889246	0.444623	−	0.895718i	$-0.353338\pi$
$211$	12.9170	0.889246	0.444623	+	0.895718i	$0.353338\pi$
$212$	0	0
$213$	30.6704	2.10150
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.60728	0.584300
$218$	0	0
$219$	4.12777	0.278929
$220$	0	0
$221$	−21.4229	−1.44106
$222$	0	0
$223$	26.6579	1.78514	0.892572	−	0.450905i	$-0.148899\pi$
$223$	26.6579	1.78514	0.892572	+	0.450905i	$0.148899\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.44407	−0.626825	−0.313412	−	0.949617i	$-0.601472\pi$
$227$	−9.44407	−0.626825	−0.313412	+	0.949617i	$0.601472\pi$
$228$	0	0
$229$	14.1994	0.938322	0.469161	−	0.883113i	$-0.344556\pi$
$229$	14.1994	0.938322	0.469161	+	0.883113i	$0.344556\pi$
$230$	0	0
$231$	18.0052	1.18465
$232$	0	0
$233$	−9.41134	−0.616557	−0.308279	−	0.951296i	$-0.599753\pi$
$233$	−9.41134	−0.616557	−0.308279	+	0.951296i	$0.599753\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.9633	−0.907014
$238$	0	0
$239$	−15.4781	−1.00119	−0.500597	−	0.865680i	$-0.666886\pi$
$239$	−15.4781	−1.00119	−0.500597	+	0.865680i	$0.666886\pi$
$240$	0	0
$241$	−3.51052	−0.226133	−0.113066	−	0.993587i	$-0.536067\pi$
$241$	−3.51052	−0.226133	−0.113066	+	0.993587i	$0.536067\pi$
$242$	0	0
$243$	−1.36974	−0.0878689
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.90153	−0.439134
$248$	0	0
$249$	−46.9480	−2.97521
$250$	0	0
$251$	−23.6649	−1.49372	−0.746859	−	0.664982i	$-0.768439\pi$
$251$	−23.6649	−1.49372	−0.746859	+	0.664982i	$0.768439\pi$
$252$	0	0
$253$	6.14721	0.386472
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.8326	1.92328	0.961641	−	0.274311i	$-0.0884497\pi$
$257$	30.8326	1.92328	0.961641	+	0.274311i	$0.0884497\pi$
$258$	0	0
$259$	8.67467	0.539018
$260$	0	0
$261$	−1.60527	−0.0993636
$262$	0	0
$263$	20.5880	1.26951	0.634755	−	0.772714i	$-0.281101\pi$
$263$	20.5880	1.26951	0.634755	+	0.772714i	$0.281101\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−33.3203	−2.03917
$268$	0	0
$269$	−17.0277	−1.03820	−0.519099	−	0.854714i	$-0.673732\pi$
$269$	−17.0277	−1.03820	−0.519099	+	0.854714i	$0.673732\pi$
$270$	0	0
$271$	13.6709	0.830450	0.415225	−	0.909719i	$-0.363703\pi$
$271$	13.6709	0.830450	0.415225	+	0.909719i	$0.363703\pi$
$272$	0	0
$273$	18.6694	1.12993
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.2380	0.975647	0.487824	−	0.872942i	$-0.337791\pi$
$277$	16.2380	0.975647	0.487824	+	0.872942i	$0.337791\pi$
$278$	0	0
$279$	−51.9879	−3.11243
$280$	0	0
$281$	−0.362407	−0.0216194	−0.0108097	−	0.999942i	$-0.503441\pi$
$281$	−0.362407	−0.0216194	−0.0108097	+	0.999942i	$0.503441\pi$
$282$	0	0
$283$	6.80039	0.404241	0.202120	−	0.979361i	$-0.435217\pi$
$283$	6.80039	0.404241	0.202120	+	0.979361i	$0.435217\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.75873	0.280899
$288$	0	0
$289$	−5.70382	−0.335519
$290$	0	0
$291$	−30.7274	−1.80127
$292$	0	0
$293$	−26.2172	−1.53162	−0.765812	−	0.643065i	$-0.777663\pi$
$293$	−26.2172	−1.53162	−0.765812	+	0.643065i	$0.777663\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−53.6647	−3.11394
$298$	0	0
$299$	6.37400	0.368618
$300$	0	0
$301$	−1.84106	−0.106117
$302$	0	0
$303$	−45.5902	−2.61909
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.7146	1.12517	0.562587	−	0.826738i	$-0.309806\pi$
$307$	19.7146	1.12517	0.562587	+	0.826738i	$0.309806\pi$
$308$	0	0
$309$	−22.7255	−1.29281
$310$	0	0
$311$	15.1003	0.856258	0.428129	−	0.903718i	$-0.359173\pi$
$311$	15.1003	0.856258	0.428129	+	0.903718i	$0.359173\pi$
$312$	0	0
$313$	12.9813	0.733747	0.366874	−	0.930271i	$-0.380428\pi$
$313$	12.9813	0.733747	0.366874	+	0.930271i	$0.380428\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.3503	0.974489	0.487244	−	0.873266i	$-0.338002\pi$
$317$	17.3503	0.974489	0.487244	+	0.873266i	$0.338002\pi$
$318$	0	0
$319$	1.66616	0.0932868
$320$	0	0
$321$	33.7781	1.88531
$322$	0	0
$323$	3.63915	0.202488
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.3154	0.570444
$328$	0	0
$329$	−0.192277	−0.0106006
$330$	0	0
$331$	0.984756	0.0541271	0.0270635	−	0.999634i	$-0.491384\pi$
$331$	0.984756	0.0541271	0.0270635	+	0.999634i	$0.491384\pi$
$332$	0	0
$333$	−52.3950	−2.87123
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.5575	1.28326	0.641629	−	0.767015i	$-0.278259\pi$
$337$	23.5575	1.28326	0.641629	+	0.767015i	$0.278259\pi$
$338$	0	0
$339$	43.8767	2.38305
$340$	0	0
$341$	53.9598	2.92208
$342$	0	0
$343$	12.7850	0.690327
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.0916	1.40067	0.700337	−	0.713813i	$-0.253033\pi$
$347$	26.0916	1.40067	0.700337	+	0.713813i	$0.253033\pi$
$348$	0	0
$349$	−17.8609	−0.956070	−0.478035	−	0.878341i	$-0.658651\pi$
$349$	−17.8609	−0.956070	−0.478035	+	0.878341i	$0.658651\pi$
$350$	0	0
$351$	−55.6445	−2.97009
$352$	0	0
$353$	−13.1080	−0.697667	−0.348833	−	0.937185i	$-0.613422\pi$
$353$	−13.1080	−0.697667	−0.348833	+	0.937185i	$0.613422\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.84430	−0.521016
$358$	0	0
$359$	22.1119	1.16702	0.583510	−	0.812106i	$-0.301679\pi$
$359$	22.1119	1.16702	0.583510	+	0.812106i	$0.301679\pi$
$360$	0	0
$361$	−17.8276	−0.938296
$362$	0	0
$363$	80.0183	4.19987
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.98495	0.416811	0.208406	−	0.978043i	$-0.433173\pi$
$367$	7.98495	0.416811	0.208406	+	0.978043i	$0.433173\pi$
$368$	0	0
$369$	−28.7427	−1.49629
$370$	0	0
$371$	1.89402	0.0983326
$372$	0	0
$373$	23.8712	1.23600	0.618002	−	0.786177i	$-0.287942\pi$
$373$	23.8712	1.23600	0.618002	+	0.786177i	$0.287942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.72762	0.0889771
$378$	0	0
$379$	−34.5647	−1.77547	−0.887735	−	0.460355i	$-0.847722\pi$
$379$	−34.5647	−1.77547	−0.887735	+	0.460355i	$0.847722\pi$
$380$	0	0
$381$	−3.56394	−0.182586
$382$	0	0
$383$	12.3139	0.629211	0.314606	−	0.949223i	$-0.398128\pi$
$383$	12.3139	0.629211	0.314606	+	0.949223i	$0.398128\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.1200	0.565262
$388$	0	0
$389$	−27.0447	−1.37122	−0.685611	−	0.727968i	$-0.740465\pi$
$389$	−27.0447	−1.37122	−0.685611	+	0.727968i	$0.740465\pi$
$390$	0	0
$391$	−3.36098	−0.169972
$392$	0	0
$393$	−35.0462	−1.76785
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9.07631	−0.455527	−0.227763	−	0.973717i	$-0.573141\pi$
$397$	−9.07631	−0.455527	−0.227763	+	0.973717i	$0.573141\pi$
$398$	0	0
$399$	−3.17141	−0.158769
$400$	0	0
$401$	6.73762	0.336461	0.168230	−	0.985748i	$-0.446195\pi$
$401$	6.73762	0.336461	0.168230	+	0.985748i	$0.446195\pi$
$402$	0	0
$403$	55.9504	2.78709
$404$	0	0
$405$	0	0
$406$	0	0
$407$	54.3823	2.69563
$408$	0	0
$409$	25.7205	1.27180	0.635898	−	0.771773i	$-0.280630\pi$
$409$	25.7205	1.27180	0.635898	+	0.771773i	$0.280630\pi$
$410$	0	0
$411$	−31.2640	−1.54214
$412$	0	0
$413$	−12.7509	−0.627428
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.2873	0.601711
$418$	0	0
$419$	−16.2666	−0.794676	−0.397338	−	0.917672i	$-0.630066\pi$
$419$	−16.2666	−0.794676	−0.397338	+	0.917672i	$0.630066\pi$
$420$	0	0
$421$	−18.9106	−0.921645	−0.460822	−	0.887492i	$-0.652445\pi$
$421$	−18.9106	−0.921645	−0.460822	+	0.887492i	$0.652445\pi$
$422$	0	0
$423$	1.16135	0.0564669
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.88522	−0.284806
$428$	0	0
$429$	117.040	5.65076
$430$	0	0
$431$	−11.4086	−0.549535	−0.274767	−	0.961511i	$-0.588601\pi$
$431$	−11.4086	−0.549535	−0.274767	+	0.961511i	$0.588601\pi$
$432$	0	0
$433$	−23.2756	−1.11856	−0.559278	−	0.828980i	$-0.688922\pi$
$433$	−23.2756	−1.11856	−0.559278	+	0.828980i	$0.688922\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.08276	−0.0517956
$438$	0	0
$439$	−13.0581	−0.623231	−0.311615	−	0.950208i	$-0.600870\pi$
$439$	−13.0581	−0.623231	−0.311615	+	0.950208i	$0.600870\pi$
$440$	0	0
$441$	−35.7635	−1.70302
$442$	0	0
$443$	−0.113844	−0.00540891	−0.00270446	−	0.999996i	$-0.500861\pi$
$443$	−0.113844	−0.00540891	−0.00270446	+	0.999996i	$0.500861\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.11598	0.194679
$448$	0	0
$449$	−11.0371	−0.520874	−0.260437	−	0.965491i	$-0.583867\pi$
$449$	−11.0371	−0.520874	−0.260437	+	0.965491i	$0.583867\pi$
$450$	0	0
$451$	29.8329	1.40478
$452$	0	0
$453$	−10.4363	−0.490342
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.1638	−1.41100	−0.705501	−	0.708709i	$-0.749278\pi$
$457$	−30.1638	−1.41100	−0.705501	+	0.708709i	$0.749278\pi$
$458$	0	0
$459$	29.3411	1.36953
$460$	0	0
$461$	−37.3908	−1.74146	−0.870732	−	0.491759i	$-0.836354\pi$
$461$	−37.3908	−1.74146	−0.870732	+	0.491759i	$0.836354\pi$
$462$	0	0
$463$	−16.7365	−0.777810	−0.388905	−	0.921278i	$-0.627147\pi$
$463$	−16.7365	−0.777810	−0.388905	+	0.921278i	$0.627147\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.2260	−0.843399	−0.421699	−	0.906736i	$-0.638566\pi$
$467$	−18.2260	−0.843399	−0.421699	+	0.906736i	$0.638566\pi$
$468$	0	0
$469$	−2.22437	−0.102712
$470$	0	0
$471$	−9.92820	−0.457467
$472$	0	0
$473$	−11.5418	−0.530692
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.4399	−0.523795
$478$	0	0
$479$	3.31948	0.151671	0.0758355	−	0.997120i	$-0.475838\pi$
$479$	3.31948	0.151671	0.0758355	+	0.997120i	$0.475838\pi$
$480$	0	0
$481$	56.3885	2.57110
$482$	0	0
$483$	2.92900	0.133274
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.7412	−0.894558	−0.447279	−	0.894395i	$-0.647607\pi$
$487$	−19.7412	−0.894558	−0.447279	+	0.894395i	$0.647607\pi$
$488$	0	0
$489$	−25.4521	−1.15098
$490$	0	0
$491$	18.1353	0.818432	0.409216	−	0.912437i	$-0.365802\pi$
$491$	18.1353	0.818432	0.409216	+	0.912437i	$0.365802\pi$
$492$	0	0
$493$	−0.910968	−0.0410279
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.0681	−0.451616
$498$	0	0
$499$	14.9797	0.670582	0.335291	−	0.942115i	$-0.391165\pi$
$499$	14.9797	0.670582	0.335291	+	0.942115i	$0.391165\pi$
$500$	0	0
$501$	46.6552	2.08440
$502$	0	0
$503$	7.90238	0.352350	0.176175	−	0.984359i	$-0.443628\pi$
$503$	7.90238	0.352350	0.176175	+	0.984359i	$0.443628\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	82.5262	3.66512
$508$	0	0
$509$	−21.1882	−0.939152	−0.469576	−	0.882892i	$-0.655593\pi$
$509$	−21.1882	−0.939152	−0.469576	+	0.882892i	$0.655593\pi$
$510$	0	0
$511$	−1.35502	−0.0599425
$512$	0	0
$513$	9.45245	0.417336
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.20540	−0.0530136
$518$	0	0
$519$	−25.1728	−1.10496
$520$	0	0
$521$	−11.1210	−0.487219	−0.243610	−	0.969873i	$-0.578332\pi$
$521$	−11.1210	−0.487219	−0.243610	+	0.969873i	$0.578332\pi$
$522$	0	0
$523$	16.2670	0.711304	0.355652	−	0.934618i	$-0.384259\pi$
$523$	16.2670	0.711304	0.355652	+	0.934618i	$0.384259\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.5024	−1.28515
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	77.0151	3.34217
$532$	0	0
$533$	30.9335	1.33988
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.2743	1.39274
$538$	0	0
$539$	37.1200	1.59887
$540$	0	0
$541$	−13.8797	−0.596733	−0.298367	−	0.954451i	$-0.596442\pi$
$541$	−13.8797	−0.596733	−0.298367	+	0.954451i	$0.596442\pi$
$542$	0	0
$543$	−46.0851	−1.97770
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.91647	0.252970	0.126485	−	0.991969i	$-0.459630\pi$
$547$	5.91647	0.252970	0.126485	+	0.991969i	$0.459630\pi$
$548$	0	0
$549$	35.5467	1.51709
$550$	0	0
$551$	−0.293475	−0.0125025
$552$	0	0
$553$	4.58371	0.194919
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.2857	−1.19850	−0.599252	−	0.800560i	$-0.704535\pi$
$557$	−28.2857	−1.19850	−0.599252	+	0.800560i	$0.704535\pi$
$558$	0	0
$559$	−11.9676	−0.506175
$560$	0	0
$561$	−61.7148	−2.60560
$562$	0	0
$563$	−31.7025	−1.33610	−0.668050	−	0.744117i	$-0.732871\pi$
$563$	−31.7025	−1.33610	−0.668050	+	0.744117i	$0.732871\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8.14763	−0.342169
$568$	0	0
$569$	−24.0702	−1.00907	−0.504537	−	0.863390i	$-0.668337\pi$
$569$	−24.0702	−1.00907	−0.504537	+	0.863390i	$0.668337\pi$
$570$	0	0
$571$	14.3935	0.602349	0.301175	−	0.953569i	$-0.402621\pi$
$571$	14.3935	0.602349	0.301175	+	0.953569i	$0.402621\pi$
$572$	0	0
$573$	−66.5023	−2.77817
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.77998	−0.240624	−0.120312	−	0.992736i	$-0.538389\pi$
$577$	−5.77998	−0.240624	−0.120312	+	0.992736i	$0.538389\pi$
$578$	0	0
$579$	48.2111	2.00358
$580$	0	0
$581$	15.4115	0.639378
$582$	0	0
$583$	11.8738	0.491761
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.7897	1.14700	0.573502	−	0.819204i	$-0.305584\pi$
$587$	27.7897	1.14700	0.573502	+	0.819204i	$0.305584\pi$
$588$	0	0
$589$	−9.50441	−0.391623
$590$	0	0
$591$	41.7999	1.71942
$592$	0	0
$593$	−22.1872	−0.911118	−0.455559	−	0.890205i	$-0.650561\pi$
$593$	−22.1872	−0.911118	−0.455559	+	0.890205i	$0.650561\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.6683	−0.845897
$598$	0	0
$599$	15.1380	0.618521	0.309260	−	0.950977i	$-0.399919\pi$
$599$	15.1380	0.618521	0.309260	+	0.950977i	$0.399919\pi$
$600$	0	0
$601$	20.5353	0.837652	0.418826	−	0.908067i	$-0.362442\pi$
$601$	20.5353	0.837652	0.418826	+	0.908067i	$0.362442\pi$
$602$	0	0
$603$	13.4352	0.547123
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.9906	1.17669	0.588345	−	0.808610i	$-0.299780\pi$
$607$	28.9906	1.17669	0.588345	+	0.808610i	$0.299780\pi$
$608$	0	0
$609$	0.793883	0.0321698
$610$	0	0
$611$	−1.24987	−0.0505644
$612$	0	0
$613$	−9.03621	−0.364969	−0.182485	−	0.983209i	$-0.558414\pi$
$613$	−9.03621	−0.364969	−0.182485	+	0.983209i	$0.558414\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.42741	−0.178241	−0.0891203	−	0.996021i	$-0.528406\pi$
$617$	−4.42741	−0.178241	−0.0891203	+	0.996021i	$0.528406\pi$
$618$	0	0
$619$	14.8403	0.596483	0.298241	−	0.954490i	$-0.403600\pi$
$619$	14.8403	0.596483	0.298241	+	0.954490i	$0.403600\pi$
$620$	0	0
$621$	−8.72993	−0.350320
$622$	0	0
$623$	10.9380	0.438222
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−19.8819	−0.794005
$628$	0	0
$629$	−29.7334	−1.18555
$630$	0	0
$631$	41.8369	1.66550	0.832751	−	0.553648i	$-0.186765\pi$
$631$	41.8369	1.66550	0.832751	+	0.553648i	$0.186765\pi$
$632$	0	0
$633$	38.5841	1.53358
$634$	0	0
$635$	0	0
$636$	0	0
$637$	38.4894	1.52501
$638$	0	0
$639$	60.8113	2.40566
$640$	0	0
$641$	5.86478	0.231645	0.115822	−	0.993270i	$-0.463050\pi$
$641$	5.86478	0.231645	0.115822	+	0.993270i	$0.463050\pi$
$642$	0	0
$643$	−0.460745	−0.0181700	−0.00908500	−	0.999959i	$-0.502892\pi$
$643$	−0.460745	−0.0181700	−0.00908500	+	0.999959i	$0.502892\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.5690	0.612079	0.306040	−	0.952019i	$-0.400996\pi$
$647$	15.5690	0.612079	0.306040	+	0.952019i	$0.400996\pi$
$648$	0	0
$649$	−79.9362	−3.13777
$650$	0	0
$651$	25.7105	1.00767
$652$	0	0
$653$	29.4538	1.15262	0.576308	−	0.817233i	$-0.304493\pi$
$653$	29.4538	1.15262	0.576308	+	0.817233i	$0.304493\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.18430	0.319300
$658$	0	0
$659$	22.5604	0.878829	0.439415	−	0.898284i	$-0.355186\pi$
$659$	22.5604	0.878829	0.439415	+	0.898284i	$0.355186\pi$
$660$	0	0
$661$	−39.7530	−1.54621	−0.773106	−	0.634277i	$-0.781298\pi$
$661$	−39.7530	−1.54621	−0.773106	+	0.634277i	$0.781298\pi$
$662$	0	0
$663$	−63.9916	−2.48523
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.271042	0.0104948
$668$	0	0
$669$	79.6289	3.07863
$670$	0	0
$671$	−36.8949	−1.42431
$672$	0	0
$673$	−11.5047	−0.443475	−0.221737	−	0.975106i	$-0.571173\pi$
$673$	−11.5047	−0.443475	−0.221737	+	0.975106i	$0.571173\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.7703	−1.22103	−0.610515	−	0.792005i	$-0.709038\pi$
$677$	−31.7703	−1.22103	−0.610515	+	0.792005i	$0.709038\pi$
$678$	0	0
$679$	10.0868	0.387098
$680$	0	0
$681$	−28.2101	−1.08101
$682$	0	0
$683$	−7.22683	−0.276527	−0.138264	−	0.990395i	$-0.544152\pi$
$683$	−7.22683	−0.276527	−0.138264	+	0.990395i	$0.544152\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.4145	1.61822
$688$	0	0
$689$	12.3118	0.469043
$690$	0	0
$691$	3.29788	0.125457	0.0627286	−	0.998031i	$-0.480020\pi$
$691$	3.29788	0.125457	0.0627286	+	0.998031i	$0.480020\pi$
$692$	0	0
$693$	35.6996	1.35611
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.3111	−0.617827
$698$	0	0
$699$	−28.1123	−1.06331
$700$	0	0
$701$	18.7936	0.709824	0.354912	−	0.934900i	$-0.384511\pi$
$701$	18.7936	0.709824	0.354912	+	0.934900i	$0.384511\pi$
$702$	0	0
$703$	−9.57884	−0.361273
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.9658	0.562848
$708$	0	0
$709$	14.1039	0.529685	0.264842	−	0.964292i	$-0.414680\pi$
$709$	14.1039	0.529685	0.264842	+	0.964292i	$0.414680\pi$
$710$	0	0
$711$	−27.6856	−1.03829
$712$	0	0
$713$	8.77792	0.328736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−46.2341	−1.72664
$718$	0	0
$719$	−23.5631	−0.878754	−0.439377	−	0.898303i	$-0.644801\pi$
$719$	−23.5631	−0.878754	−0.439377	+	0.898303i	$0.644801\pi$
$720$	0	0
$721$	7.46005	0.277827
$722$	0	0
$723$	−10.4862	−0.389985
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.57021	0.132412	0.0662059	−	0.997806i	$-0.478911\pi$
$727$	3.57021	0.132412	0.0662059	+	0.997806i	$0.478911\pi$
$728$	0	0
$729$	−29.0190	−1.07478
$730$	0	0
$731$	6.31045	0.233401
$732$	0	0
$733$	−10.3158	−0.381022	−0.190511	−	0.981685i	$-0.561015\pi$
$733$	−10.3158	−0.381022	−0.190511	+	0.981685i	$0.561015\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.9448	−0.513663
$738$	0	0
$739$	−36.2056	−1.33185	−0.665923	−	0.746021i	$-0.731962\pi$
$739$	−36.2056	−1.33185	−0.665923	+	0.746021i	$0.731962\pi$
$740$	0	0
$741$	−20.6153	−0.757324
$742$	0	0
$743$	−3.24466	−0.119035	−0.0595174	−	0.998227i	$-0.518956\pi$
$743$	−3.24466	−0.119035	−0.0595174	+	0.998227i	$0.518956\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−93.0856	−3.40582
$748$	0	0
$749$	−11.0883	−0.405157
$750$	0	0
$751$	26.9796	0.984499	0.492250	−	0.870454i	$-0.336175\pi$
$751$	26.9796	0.984499	0.492250	+	0.870454i	$0.336175\pi$
$752$	0	0
$753$	−70.6888	−2.57604
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.7249	0.426148	0.213074	−	0.977036i	$-0.431652\pi$
$757$	11.7249	0.426148	0.213074	+	0.977036i	$0.431652\pi$
$758$	0	0
$759$	18.3621	0.666504
$760$	0	0
$761$	−24.4303	−0.885599	−0.442800	−	0.896621i	$-0.646015\pi$
$761$	−24.4303	−0.885599	−0.442800	+	0.896621i	$0.646015\pi$
$762$	0	0
$763$	−3.38623	−0.122590
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−82.8852	−2.99281
$768$	0	0
$769$	0.804588	0.0290142	0.0145071	−	0.999895i	$-0.495382\pi$
$769$	0.804588	0.0290142	0.0145071	+	0.999895i	$0.495382\pi$
$770$	0	0
$771$	92.0989	3.31686
$772$	0	0
$773$	−6.02477	−0.216696	−0.108348	−	0.994113i	$-0.534556\pi$
$773$	−6.02477	−0.216696	−0.108348	+	0.994113i	$0.534556\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	25.9118	0.929582
$778$	0	0
$779$	−5.25474	−0.188270
$780$	0	0
$781$	−63.1179	−2.25853
$782$	0	0
$783$	−2.36618	−0.0845604
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.9326	0.460996	0.230498	−	0.973073i	$-0.425964\pi$
$787$	12.9326	0.460996	0.230498	+	0.973073i	$0.425964\pi$
$788$	0	0
$789$	61.4977	2.18938
$790$	0	0
$791$	−14.4033	−0.512123
$792$	0	0
$793$	−38.2561	−1.35851
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.3660	−1.28815	−0.644075	−	0.764962i	$-0.722758\pi$
$797$	−36.3660	−1.28815	−0.644075	+	0.764962i	$0.722758\pi$
$798$	0	0
$799$	0.659052	0.0233156
$800$	0	0
$801$	−66.0654	−2.33431
$802$	0	0
$803$	−8.49472	−0.299772
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−50.8629	−1.79046
$808$	0	0
$809$	26.5840	0.934642	0.467321	−	0.884088i	$-0.345219\pi$
$809$	26.5840	0.934642	0.467321	+	0.884088i	$0.345219\pi$
$810$	0	0
$811$	−44.7712	−1.57213	−0.786064	−	0.618145i	$-0.787884\pi$
$811$	−44.7712	−1.57213	−0.786064	+	0.618145i	$0.787884\pi$
$812$	0	0
$813$	40.8360	1.43218
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.03296	0.0711242
$818$	0	0
$819$	37.0166	1.29346
$820$	0	0
$821$	22.3949	0.781587	0.390794	−	0.920478i	$-0.372201\pi$
$821$	22.3949	0.781587	0.390794	+	0.920478i	$0.372201\pi$
$822$	0	0
$823$	−2.18913	−0.0763081	−0.0381541	−	0.999272i	$-0.512148\pi$
$823$	−2.18913	−0.0763081	−0.0381541	+	0.999272i	$0.512148\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.4367	−1.37135	−0.685674	−	0.727909i	$-0.740492\pi$
$827$	−39.4367	−1.37135	−0.685674	+	0.727909i	$0.740492\pi$
$828$	0	0
$829$	−11.0629	−0.384229	−0.192115	−	0.981373i	$-0.561535\pi$
$829$	−11.0629	−0.384229	−0.192115	+	0.981373i	$0.561535\pi$
$830$	0	0
$831$	48.5040	1.68259
$832$	0	0
$833$	−20.2953	−0.703190
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−76.6306	−2.64874
$838$	0	0
$839$	−37.8510	−1.30676	−0.653381	−	0.757030i	$-0.726650\pi$
$839$	−37.8510	−1.30676	−0.653381	+	0.757030i	$0.726650\pi$
$840$	0	0
$841$	−28.9265	−0.997467
$842$	0	0
$843$	−1.08254	−0.0372845
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.2675	−0.902561
$848$	0	0
$849$	20.3132	0.697148
$850$	0	0
$851$	8.84665	0.303259
$852$	0	0
$853$	−28.9770	−0.992155	−0.496078	−	0.868278i	$-0.665227\pi$
$853$	−28.9770	−0.992155	−0.496078	+	0.868278i	$0.665227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	56.0462	1.91450	0.957250	−	0.289261i	$-0.0934095\pi$
$857$	56.0462	1.91450	0.957250	+	0.289261i	$0.0934095\pi$
$858$	0	0
$859$	43.8866	1.49739	0.748695	−	0.662914i	$-0.230681\pi$
$859$	43.8866	1.49739	0.748695	+	0.662914i	$0.230681\pi$
$860$	0	0
$861$	14.2147	0.484434
$862$	0	0
$863$	−22.1955	−0.755542	−0.377771	−	0.925899i	$-0.623309\pi$
$863$	−22.1955	−0.755542	−0.377771	+	0.925899i	$0.623309\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−17.0377	−0.578630
$868$	0	0
$869$	28.7357	0.974791
$870$	0	0
$871$	−14.4592	−0.489932
$872$	0	0
$873$	−60.9245	−2.06198
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.44979	0.184026	0.0920131	−	0.995758i	$-0.470670\pi$
$877$	5.44979	0.184026	0.0920131	+	0.995758i	$0.470670\pi$
$878$	0	0
$879$	−78.3125	−2.64141
$880$	0	0
$881$	−11.7854	−0.397060	−0.198530	−	0.980095i	$-0.563617\pi$
$881$	−11.7854	−0.397060	−0.198530	+	0.980095i	$0.563617\pi$
$882$	0	0
$883$	−23.7945	−0.800748	−0.400374	−	0.916352i	$-0.631120\pi$
$883$	−23.7945	−0.800748	−0.400374	+	0.916352i	$0.631120\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.4300	1.49181	0.745907	−	0.666050i	$-0.232016\pi$
$887$	44.4300	1.49181	0.745907	+	0.666050i	$0.232016\pi$
$888$	0	0
$889$	1.16993	0.0392381
$890$	0	0
$891$	−51.0782	−1.71119
$892$	0	0
$893$	0.212318	0.00710496
$894$	0	0
$895$	0	0
$896$	0	0
$897$	19.0396	0.635712
$898$	0	0
$899$	2.37919	0.0793504
$900$	0	0
$901$	−6.49196	−0.216279
$902$	0	0
$903$	−5.49938	−0.183008
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.7542	1.35322	0.676611	−	0.736341i	$-0.263448\pi$
$907$	40.7542	1.35322	0.676611	+	0.736341i	$0.263448\pi$
$908$	0	0
$909$	−90.3934	−2.99816
$910$	0	0
$911$	−2.71520	−0.0899587	−0.0449794	−	0.998988i	$-0.514322\pi$
$911$	−2.71520	−0.0899587	−0.0449794	+	0.998988i	$0.514322\pi$
$912$	0	0
$913$	96.6163	3.19753
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.5046	0.379914
$918$	0	0
$919$	10.7227	0.353708	0.176854	−	0.984237i	$-0.443408\pi$
$919$	10.7227	0.353708	0.176854	+	0.984237i	$0.443408\pi$
$920$	0	0
$921$	58.8890	1.94046
$922$	0	0
$923$	−65.4464	−2.15419
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−45.0587	−1.47992
$928$	0	0
$929$	38.8670	1.27518	0.637592	−	0.770374i	$-0.279930\pi$
$929$	38.8670	1.27518	0.637592	+	0.770374i	$0.279930\pi$
$930$	0	0
$931$	−6.53827	−0.214283
$932$	0	0
$933$	45.1055	1.47669
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.6117	1.26139	0.630694	−	0.776032i	$-0.282770\pi$
$937$	38.6117	1.26139	0.630694	+	0.776032i	$0.282770\pi$
$938$	0	0
$939$	38.7761	1.26541
$940$	0	0
$941$	26.3928	0.860381	0.430191	−	0.902738i	$-0.358446\pi$
$941$	26.3928	0.860381	0.430191	+	0.902738i	$0.358446\pi$
$942$	0	0
$943$	4.85308	0.158038
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.3777	−0.727179	−0.363589	−	0.931559i	$-0.618449\pi$
$947$	−22.3777	−0.727179	−0.363589	+	0.931559i	$0.618449\pi$
$948$	0	0
$949$	−8.80811	−0.285923
$950$	0	0
$951$	51.8265	1.68059
$952$	0	0
$953$	−17.4362	−0.564815	−0.282407	−	0.959295i	$-0.591133\pi$
$953$	−17.4362	−0.564815	−0.282407	+	0.959295i	$0.591133\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.97692	0.160881
$958$	0	0
$959$	10.2630	0.331409
$960$	0	0
$961$	46.0519	1.48554
$962$	0	0
$963$	66.9732	2.15818
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−45.1587	−1.45221	−0.726103	−	0.687586i	$-0.758670\pi$
$967$	−45.1587	−1.45221	−0.726103	+	0.687586i	$0.758670\pi$
$968$	0	0
$969$	10.8704	0.349207
$970$	0	0
$971$	−34.3123	−1.10113	−0.550567	−	0.834791i	$-0.685588\pi$
$971$	−34.3123	−1.10113	−0.550567	+	0.834791i	$0.685588\pi$
$972$	0	0
$973$	−4.03353	−0.129309
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.51320	−0.0484115	−0.0242058	−	0.999707i	$-0.507706\pi$
$977$	−1.51320	−0.0484115	−0.0242058	+	0.999707i	$0.507706\pi$
$978$	0	0
$979$	68.5713	2.19155
$980$	0	0
$981$	20.4528	0.653007
$982$	0	0
$983$	−15.1715	−0.483895	−0.241947	−	0.970289i	$-0.577786\pi$
$983$	−15.1715	−0.483895	−0.241947	+	0.970289i	$0.577786\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.574345	−0.0182816
$988$	0	0
$989$	−1.87756	−0.0597031
$990$	0	0
$991$	−39.4470	−1.25307	−0.626537	−	0.779392i	$-0.715528\pi$
$991$	−39.4470	−1.25307	−0.626537	+	0.779392i	$0.715528\pi$
$992$	0	0
$993$	2.94153	0.0933467
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.33319	0.0738928	0.0369464	−	0.999317i	$-0.488237\pi$
$997$	2.33319	0.0738928	0.0369464	+	0.999317i	$0.488237\pi$
$998$	0	0
$999$	−77.2307	−2.44347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bi.1.7		7
4.3	odd	2		9200.2.a.cz.1.1		7
5.2	odd	4		920.2.e.b.369.1	✓	14
5.3	odd	4		920.2.e.b.369.14	yes	14
5.4	even	2		4600.2.a.bh.1.1		7
20.3	even	4		1840.2.e.g.369.1		14
20.7	even	4		1840.2.e.g.369.14		14
20.19	odd	2		9200.2.a.dc.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.1	✓	14	5.2	odd	4
920.2.e.b.369.14	yes	14	5.3	odd	4
1840.2.e.g.369.1		14	20.3	even	4
1840.2.e.g.369.14		14	20.7	even	4
4600.2.a.bh.1.1		7	5.4	even	2
4600.2.a.bi.1.7		7	1.1	even	1	trivial
9200.2.a.cz.1.1		7	4.3	odd	2
9200.2.a.dc.1.7		7	20.19	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)

\(f(q)\)	\(=\)	\(q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} +O(q^{10})\)	\(q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} -6.14721 q^{11} -6.37400 q^{13} +3.36098 q^{17} +1.08276 q^{19} -2.92900 q^{21} -1.00000 q^{23} +8.72993 q^{27} -0.271042 q^{29} -8.77792 q^{31} -18.3621 q^{33} -8.84665 q^{37} -19.0396 q^{39} -4.85308 q^{41} +1.87756 q^{43} +0.196089 q^{47} -6.03850 q^{49} +10.0395 q^{51} -1.93157 q^{53} +3.23429 q^{57} +13.0036 q^{59} +6.00189 q^{61} -5.80744 q^{63} +2.26847 q^{67} -2.98707 q^{69} +10.2677 q^{71} +1.38188 q^{73} +6.02771 q^{77} -4.67459 q^{79} +8.30917 q^{81} -15.7171 q^{83} -0.809622 q^{87} -11.1548 q^{89} +6.25008 q^{91} -26.2202 q^{93} -10.2868 q^{97} -36.4073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} - 7 q^{11} - 7 q^{13} - 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} - 10 q^{31} - 19 q^{33} - 19 q^{37} - 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} - 7 q^{51} - 15 q^{53} - 8 q^{57} - 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} - 14 q^{71} - 10 q^{73} - 6 q^{77} - 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} - 7 q^{91} - 26 q^{93} + 7 q^{97} - 61 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 - 7 * q^11 - 7 * q^13 - 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 - 10 * q^31 - 19 * q^33 - 19 * q^37 - 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 - 7 * q^51 - 15 * q^53 - 8 * q^57 - 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 - 14 * q^71 - 10 * q^73 - 6 * q^77 - 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 - 7 * q^91 - 26 * q^93 + 7 * q^97 - 61 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more