LMFDB - Embedded newform 5225.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-5,-4,17,0,-1,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 $ x^15 - 5x^14 - 11x^13 + 87x^12 - 4x^11 - 545x^10 + 431x^9 + 1480x^8 - 1763x^7 - 1609x^6 + 2516x^5 + 391x^4 - 1081x^3 - 45x^2 + 153x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.61792$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.61792 q^{2} -2.17802 q^{3} +4.85352 q^{4} +5.70188 q^{6} -4.94832 q^{7} -7.47030 q^{8} +1.74376 q^{9} +1.00000 q^{11} -10.5710 q^{12} +4.20557 q^{13} +12.9543 q^{14} +9.84962 q^{16} +6.38851 q^{17} -4.56502 q^{18} +1.00000 q^{19} +10.7775 q^{21} -2.61792 q^{22} -6.66227 q^{23} +16.2704 q^{24} -11.0099 q^{26} +2.73612 q^{27} -24.0168 q^{28} -6.65643 q^{29} -9.53855 q^{31} -10.8449 q^{32} -2.17802 q^{33} -16.7246 q^{34} +8.46336 q^{36} +3.69327 q^{37} -2.61792 q^{38} -9.15980 q^{39} -1.08046 q^{41} -28.2147 q^{42} -9.58018 q^{43} +4.85352 q^{44} +17.4413 q^{46} -2.90169 q^{47} -21.4526 q^{48} +17.4858 q^{49} -13.9143 q^{51} +20.4118 q^{52} +6.75151 q^{53} -7.16294 q^{54} +36.9654 q^{56} -2.17802 q^{57} +17.4260 q^{58} -0.613007 q^{59} +1.33665 q^{61} +24.9712 q^{62} -8.62866 q^{63} +8.69200 q^{64} +5.70188 q^{66} -0.386619 q^{67} +31.0067 q^{68} +14.5105 q^{69} +9.38483 q^{71} -13.0264 q^{72} +1.97456 q^{73} -9.66869 q^{74} +4.85352 q^{76} -4.94832 q^{77} +23.9797 q^{78} -6.61494 q^{79} -11.1906 q^{81} +2.82857 q^{82} +0.100761 q^{83} +52.3089 q^{84} +25.0802 q^{86} +14.4978 q^{87} -7.47030 q^{88} +2.48738 q^{89} -20.8105 q^{91} -32.3355 q^{92} +20.7751 q^{93} +7.59641 q^{94} +23.6205 q^{96} +17.3288 q^{97} -45.7766 q^{98} +1.74376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9} + 15 q^{11} - 11 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 17 q^{17} - 22 q^{18} + 15 q^{19} + 6 q^{21} - 5 q^{22} - 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 21 * q^7 - 9 * q^8 + 15 * q^9 + 15 * q^11 - 11 * q^12 - 13 * q^13 + 9 * q^14 + 21 * q^16 - 17 * q^17 - 22 * q^18 + 15 * q^19 + 6 * q^21 - 5 * q^22 - 26 * q^23 + q^24 + 3 * q^26 - q^27 - 46 * q^28 + 9 * q^29 + 14 * q^31 - 18 * q^32 - 4 * q^33 - 13 * q^34 + 12 * q^36 - 9 * q^37 - 5 * q^38 - 22 * q^39 + 4 * q^41 + 6 * q^42 - 28 * q^43 + 17 * q^44 + 27 * q^46 - 14 * q^47 + 4 * q^48 + 32 * q^49 - 40 * q^51 - 14 * q^52 - 3 * q^53 - 39 * q^54 + 34 * q^56 - 4 * q^57 - 26 * q^58 + q^59 + 2 * q^61 + 3 * q^62 - 45 * q^63 + 5 * q^64 - q^66 - 37 * q^67 - 26 * q^68 - 7 * q^69 - 7 * q^71 - 16 * q^72 - 42 * q^73 - 43 * q^74 + 17 * q^76 - 21 * q^77 + 64 * q^78 - 10 * q^79 + 31 * q^81 - 22 * q^82 - 14 * q^83 - 32 * q^84 + 37 * q^86 - 29 * q^87 - 9 * q^88 + 15 * q^89 - 22 * q^91 - 26 * q^92 + 18 * q^93 - 44 * q^94 + 71 * q^96 - 8 * q^97 + 10 * q^98 + 15 * 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 $

$2$	−2.61792	−1.85115	−0.925575	−	0.378563i	$-0.876418\pi$
$2$	−2.61792	−1.85115	−0.925575	+	0.378563i	$0.876418\pi$
$3$	−2.17802	−1.25748	−0.628739	−	0.777616i	$-0.716429\pi$
$3$	−2.17802	−1.25748	−0.628739	+	0.777616i	$0.716429\pi$
$4$	4.85352	2.42676
$5$	0	0
$5$	0	0
$6$	5.70188	2.32778
$7$	−4.94832	−1.87029	−0.935144	−	0.354268i	$-0.884730\pi$
$7$	−4.94832	−1.87029	−0.935144	+	0.354268i	$0.884730\pi$
$8$	−7.47030	−2.64115
$9$	1.74376	0.581253
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−10.5710	−3.05160
$13$	4.20557	1.16642	0.583208	−	0.812323i	$-0.301797\pi$
$13$	4.20557	1.16642	0.583208	+	0.812323i	$0.301797\pi$
$14$	12.9543	3.46218
$15$	0	0
$16$	9.84962	2.46240
$17$	6.38851	1.54944	0.774720	−	0.632304i	$-0.217891\pi$
$17$	6.38851	1.54944	0.774720	+	0.632304i	$0.217891\pi$
$18$	−4.56502	−1.07599
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	10.7775	2.35185
$22$	−2.61792	−0.558143
$23$	−6.66227	−1.38918	−0.694590	−	0.719406i	$-0.744414\pi$
$23$	−6.66227	−1.38918	−0.694590	+	0.719406i	$0.744414\pi$
$24$	16.2704	3.32119
$25$	0	0
$26$	−11.0099	−2.15921
$27$	2.73612	0.526566
$28$	−24.0168	−4.53874
$29$	−6.65643	−1.23607	−0.618034	−	0.786151i	$-0.712071\pi$
$29$	−6.65643	−1.23607	−0.618034	+	0.786151i	$0.712071\pi$
$30$	0	0
$31$	−9.53855	−1.71317	−0.856587	−	0.516002i	$-0.827420\pi$
$31$	−9.53855	−1.71317	−0.856587	+	0.516002i	$0.827420\pi$
$32$	−10.8449	−1.91713
$33$	−2.17802	−0.379144
$34$	−16.7246	−2.86825
$35$	0	0
$36$	8.46336	1.41056
$37$	3.69327	0.607170	0.303585	−	0.952804i	$-0.401816\pi$
$37$	3.69327	0.607170	0.303585	+	0.952804i	$0.401816\pi$
$38$	−2.61792	−0.424683
$39$	−9.15980	−1.46674
$40$	0	0
$41$	−1.08046	−0.168740	−0.0843701	−	0.996434i	$-0.526888\pi$
$41$	−1.08046	−0.168740	−0.0843701	+	0.996434i	$0.526888\pi$
$42$	−28.2147	−4.35362
$43$	−9.58018	−1.46096	−0.730482	−	0.682932i	$-0.760704\pi$
$43$	−9.58018	−1.46096	−0.730482	+	0.682932i	$0.760704\pi$
$44$	4.85352	0.731696
$45$	0	0
$46$	17.4413	2.57158
$47$	−2.90169	−0.423256	−0.211628	−	0.977350i	$-0.567876\pi$
$47$	−2.90169	−0.423256	−0.211628	+	0.977350i	$0.567876\pi$
$48$	−21.4526	−3.09642
$49$	17.4858	2.49798
$50$	0	0
$51$	−13.9143	−1.94839
$52$	20.4118	2.83061
$53$	6.75151	0.927391	0.463695	−	0.885995i	$-0.346523\pi$
$53$	6.75151	0.927391	0.463695	+	0.885995i	$0.346523\pi$
$54$	−7.16294	−0.974753
$55$	0	0
$56$	36.9654	4.93971
$57$	−2.17802	−0.288485
$58$	17.4260	2.28815
$59$	−0.613007	−0.0798067	−0.0399034	−	0.999204i	$-0.512705\pi$
$59$	−0.613007	−0.0798067	−0.0399034	+	0.999204i	$0.512705\pi$
$60$	0	0
$61$	1.33665	0.171140	0.0855701	−	0.996332i	$-0.472729\pi$
$61$	1.33665	0.171140	0.0855701	+	0.996332i	$0.472729\pi$
$62$	24.9712	3.17134
$63$	−8.62866	−1.08711
$64$	8.69200	1.08650
$65$	0	0
$66$	5.70188	0.701853
$67$	−0.386619	−0.0472330	−0.0236165	−	0.999721i	$-0.507518\pi$
$67$	−0.386619	−0.0472330	−0.0236165	+	0.999721i	$0.507518\pi$
$68$	31.0067	3.76012
$69$	14.5105	1.74686
$70$	0	0
$71$	9.38483	1.11377	0.556887	−	0.830588i	$-0.311996\pi$
$71$	9.38483	1.11377	0.556887	+	0.830588i	$0.311996\pi$
$72$	−13.0264	−1.53517
$73$	1.97456	0.231105	0.115552	−	0.993301i	$-0.463136\pi$
$73$	1.97456	0.231105	0.115552	+	0.993301i	$0.463136\pi$
$74$	−9.66869	−1.12396
$75$	0	0
$76$	4.85352	0.556737
$77$	−4.94832	−0.563913
$78$	23.9797	2.71516
$79$	−6.61494	−0.744238	−0.372119	−	0.928185i	$-0.621369\pi$
$79$	−6.61494	−0.744238	−0.372119	+	0.928185i	$0.621369\pi$
$80$	0	0
$81$	−11.1906	−1.24340
$82$	2.82857	0.312364
$83$	0.100761	0.0110600	0.00553000	−	0.999985i	$-0.498240\pi$
$83$	0.100761	0.0110600	0.00553000	+	0.999985i	$0.498240\pi$
$84$	52.3089	5.70737
$85$	0	0
$86$	25.0802	2.70446
$87$	14.4978	1.55433
$88$	−7.47030	−0.796336
$89$	2.48738	0.263661	0.131831	−	0.991272i	$-0.457914\pi$
$89$	2.48738	0.263661	0.131831	+	0.991272i	$0.457914\pi$
$90$	0	0
$91$	−20.8105	−2.18153
$92$	−32.3355	−3.37121
$93$	20.7751	2.15428
$94$	7.59641	0.783510
$95$	0	0
$96$	23.6205	2.41075
$97$	17.3288	1.75947	0.879737	−	0.475461i	$-0.157719\pi$
$97$	17.3288	1.75947	0.879737	+	0.475461i	$0.157719\pi$
$98$	−45.7766	−4.62413
$99$	1.74376	0.175254
$100$	0	0
$101$	−9.71209	−0.966389	−0.483194	−	0.875513i	$-0.660524\pi$
$101$	−9.71209	−0.966389	−0.483194	+	0.875513i	$0.660524\pi$
$102$	36.4265	3.60676
$103$	−15.6212	−1.53920	−0.769602	−	0.638524i	$-0.779545\pi$
$103$	−15.6212	−1.53920	−0.769602	+	0.638524i	$0.779545\pi$
$104$	−31.4168	−3.08068
$105$	0	0
$106$	−17.6749	−1.71674
$107$	5.49210	0.530941	0.265471	−	0.964119i	$-0.414473\pi$
$107$	5.49210	0.530941	0.265471	+	0.964119i	$0.414473\pi$
$108$	13.2798	1.27785
$109$	13.5177	1.29476	0.647378	−	0.762169i	$-0.275865\pi$
$109$	13.5177	1.29476	0.647378	+	0.762169i	$0.275865\pi$
$110$	0	0
$111$	−8.04400	−0.763503
$112$	−48.7390	−4.60540
$113$	11.1897	1.05264	0.526318	−	0.850288i	$-0.323572\pi$
$113$	11.1897	1.05264	0.526318	+	0.850288i	$0.323572\pi$
$114$	5.70188	0.534030
$115$	0	0
$116$	−32.3071	−2.99964
$117$	7.33349	0.677982
$118$	1.60481	0.147734
$119$	−31.6124	−2.89790
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.49924	−0.316806
$123$	2.35327	0.212187
$124$	−46.2955	−4.15746
$125$	0	0
$126$	22.5892	2.01240
$127$	−6.65255	−0.590319	−0.295159	−	0.955448i	$-0.595373\pi$
$127$	−6.65255	−0.590319	−0.295159	+	0.955448i	$0.595373\pi$
$128$	−1.06509	−0.0941412
$129$	20.8658	1.83713
$130$	0	0
$131$	18.4455	1.61159	0.805797	−	0.592192i	$-0.201737\pi$
$131$	18.4455	1.61159	0.805797	+	0.592192i	$0.201737\pi$
$132$	−10.5710	−0.920092
$133$	−4.94832	−0.429073
$134$	1.01214	0.0874355
$135$	0	0
$136$	−47.7240	−4.09230
$137$	15.3705	1.31319	0.656597	−	0.754242i	$-0.271995\pi$
$137$	15.3705	1.31319	0.656597	+	0.754242i	$0.271995\pi$
$138$	−37.9875	−3.23371
$139$	−2.38661	−0.202430	−0.101215	−	0.994865i	$-0.532273\pi$
$139$	−2.38661	−0.202430	−0.101215	+	0.994865i	$0.532273\pi$
$140$	0	0
$141$	6.31994	0.532235
$142$	−24.5688	−2.06176
$143$	4.20557	0.351687
$144$	17.1753	1.43128
$145$	0	0
$146$	−5.16924	−0.427809
$147$	−38.0844	−3.14115
$148$	17.9254	1.47346
$149$	−10.6114	−0.869317	−0.434659	−	0.900595i	$-0.643131\pi$
$149$	−10.6114	−0.869317	−0.434659	+	0.900595i	$0.643131\pi$
$150$	0	0
$151$	14.8914	1.21185	0.605923	−	0.795523i	$-0.292804\pi$
$151$	14.8914	1.21185	0.605923	+	0.795523i	$0.292804\pi$
$152$	−7.47030	−0.605921
$153$	11.1400	0.900616
$154$	12.9543	1.04389
$155$	0	0
$156$	−44.4573	−3.55943
$157$	−5.84649	−0.466601	−0.233300	−	0.972405i	$-0.574953\pi$
$157$	−5.84649	−0.466601	−0.233300	+	0.972405i	$0.574953\pi$
$158$	17.3174	1.37770
$159$	−14.7049	−1.16617
$160$	0	0
$161$	32.9670	2.59817
$162$	29.2961	2.30172
$163$	3.65131	0.285993	0.142996	−	0.989723i	$-0.454326\pi$
$163$	3.65131	0.285993	0.142996	+	0.989723i	$0.454326\pi$
$164$	−5.24405	−0.409492
$165$	0	0
$166$	−0.263785	−0.0204737
$167$	8.34211	0.645532	0.322766	−	0.946479i	$-0.395387\pi$
$167$	8.34211	0.645532	0.322766	+	0.946479i	$0.395387\pi$
$168$	−80.5112	−6.21158
$169$	4.68682	0.360524
$170$	0	0
$171$	1.74376	0.133348
$172$	−46.4976	−3.54541
$173$	−14.4715	−1.10025	−0.550124	−	0.835083i	$-0.685420\pi$
$173$	−14.4715	−1.10025	−0.550124	+	0.835083i	$0.685420\pi$
$174$	−37.9542	−2.87730
$175$	0	0
$176$	9.84962	0.742443
$177$	1.33514	0.100355
$178$	−6.51176	−0.488077
$179$	23.2230	1.73577	0.867884	−	0.496766i	$-0.165479\pi$
$179$	23.2230	1.73577	0.867884	+	0.496766i	$0.165479\pi$
$180$	0	0
$181$	3.66090	0.272112	0.136056	−	0.990701i	$-0.456557\pi$
$181$	3.66090	0.272112	0.136056	+	0.990701i	$0.456557\pi$
$182$	54.4803	4.03834
$183$	−2.91124	−0.215205
$184$	49.7691	3.66903
$185$	0	0
$186$	−54.3877	−3.98790
$187$	6.38851	0.467174
$188$	−14.0834	−1.02714
$189$	−13.5392	−0.984830
$190$	0	0
$191$	23.3590	1.69020	0.845099	−	0.534610i	$-0.179541\pi$
$191$	23.3590	1.69020	0.845099	+	0.534610i	$0.179541\pi$
$192$	−18.9313	−1.36625
$193$	−25.1709	−1.81184	−0.905922	−	0.423445i	$-0.860821\pi$
$193$	−25.1709	−1.81184	−0.905922	+	0.423445i	$0.860821\pi$
$194$	−45.3655	−3.25705
$195$	0	0
$196$	84.8678	6.06199
$197$	−6.98835	−0.497899	−0.248950	−	0.968516i	$-0.580085\pi$
$197$	−6.98835	−0.497899	−0.248950	+	0.968516i	$0.580085\pi$
$198$	−4.56502	−0.324422
$199$	11.5381	0.817911	0.408955	−	0.912554i	$-0.365893\pi$
$199$	11.5381	0.817911	0.408955	+	0.912554i	$0.365893\pi$
$200$	0	0
$201$	0.842063	0.0593946
$202$	25.4255	1.78893
$203$	32.9381	2.31180
$204$	−67.5332	−4.72827
$205$	0	0
$206$	40.8951	2.84930
$207$	−11.6174	−0.807464
$208$	41.4232	2.87219
$209$	1.00000	0.0691714
$210$	0	0
$211$	13.7660	0.947687	0.473844	−	0.880609i	$-0.342866\pi$
$211$	13.7660	0.947687	0.473844	+	0.880609i	$0.342866\pi$
$212$	32.7686	2.25055
$213$	−20.4403	−1.40055
$214$	−14.3779	−0.982853
$215$	0	0
$216$	−20.4396	−1.39074
$217$	47.1998	3.20413
$218$	−35.3882	−2.39679
$219$	−4.30062	−0.290609
$220$	0	0
$221$	26.8673	1.80729
$222$	21.0586	1.41336
$223$	18.7373	1.25474	0.627371	−	0.778720i	$-0.284131\pi$
$223$	18.7373	1.25474	0.627371	+	0.778720i	$0.284131\pi$
$224$	53.6642	3.58559
$225$	0	0
$226$	−29.2937	−1.94859
$227$	21.7922	1.44640	0.723199	−	0.690640i	$-0.242671\pi$
$227$	21.7922	1.44640	0.723199	+	0.690640i	$0.242671\pi$
$228$	−10.5710	−0.700085
$229$	−15.5530	−1.02777	−0.513886	−	0.857859i	$-0.671795\pi$
$229$	−15.5530	−1.02777	−0.513886	+	0.857859i	$0.671795\pi$
$230$	0	0
$231$	10.7775	0.709109
$232$	49.7255	3.26464
$233$	−27.0173	−1.76996	−0.884982	−	0.465625i	$-0.845830\pi$
$233$	−27.0173	−1.76996	−0.884982	+	0.465625i	$0.845830\pi$
$234$	−19.1985	−1.25505
$235$	0	0
$236$	−2.97524	−0.193672
$237$	14.4074	0.935864
$238$	82.7587	5.36445
$239$	−26.5139	−1.71504	−0.857521	−	0.514449i	$-0.827997\pi$
$239$	−26.5139	−1.71504	−0.857521	+	0.514449i	$0.827997\pi$
$240$	0	0
$241$	−16.1879	−1.04275	−0.521377	−	0.853327i	$-0.674581\pi$
$241$	−16.1879	−1.04275	−0.521377	+	0.853327i	$0.674581\pi$
$242$	−2.61792	−0.168286
$243$	16.1649	1.03698
$244$	6.48745	0.415316
$245$	0	0
$246$	−6.16068	−0.392790
$247$	4.20557	0.267594
$248$	71.2558	4.52475
$249$	−0.219460	−0.0139077
$250$	0	0
$251$	2.48202	0.156664	0.0783319	−	0.996927i	$-0.475041\pi$
$251$	2.48202	0.156664	0.0783319	+	0.996927i	$0.475041\pi$
$252$	−41.8794	−2.63815
$253$	−6.66227	−0.418853
$254$	17.4159	1.09277
$255$	0	0
$256$	−14.5957	−0.912230
$257$	1.75270	0.109330	0.0546651	−	0.998505i	$-0.482591\pi$
$257$	1.75270	0.109330	0.0546651	+	0.998505i	$0.482591\pi$
$258$	−54.6250	−3.40081
$259$	−18.2755	−1.13558
$260$	0	0
$261$	−11.6072	−0.718468
$262$	−48.2890	−2.98330
$263$	15.9328	0.982460	0.491230	−	0.871030i	$-0.336548\pi$
$263$	15.9328	0.982460	0.491230	+	0.871030i	$0.336548\pi$
$264$	16.2704	1.00138
$265$	0	0
$266$	12.9543	0.794280
$267$	−5.41755	−0.331549
$268$	−1.87646	−0.114623
$269$	−5.40061	−0.329281	−0.164641	−	0.986354i	$-0.552646\pi$
$269$	−5.40061	−0.329281	−0.164641	+	0.986354i	$0.552646\pi$
$270$	0	0
$271$	−0.761668	−0.0462680	−0.0231340	−	0.999732i	$-0.507364\pi$
$271$	−0.761668	−0.0462680	−0.0231340	+	0.999732i	$0.507364\pi$
$272$	62.9243	3.81535
$273$	45.3256	2.74323
$274$	−40.2389	−2.43092
$275$	0	0
$276$	70.4272	4.23922
$277$	−26.9579	−1.61974	−0.809872	−	0.586606i	$-0.800464\pi$
$277$	−26.9579	−1.61974	−0.809872	+	0.586606i	$0.800464\pi$
$278$	6.24797	0.374729
$279$	−16.6329	−0.995787
$280$	0	0
$281$	18.4983	1.10352	0.551759	−	0.834004i	$-0.313957\pi$
$281$	18.4983	1.10352	0.551759	+	0.834004i	$0.313957\pi$
$282$	−16.5451	−0.985247
$283$	−18.9028	−1.12365	−0.561827	−	0.827255i	$-0.689901\pi$
$283$	−18.9028	−1.12365	−0.561827	+	0.827255i	$0.689901\pi$
$284$	45.5495	2.70286
$285$	0	0
$286$	−11.0099	−0.651026
$287$	5.34648	0.315593
$288$	−18.9110	−1.11434
$289$	23.8130	1.40077
$290$	0	0
$291$	−37.7424	−2.21250
$292$	9.58356	0.560835
$293$	−13.8515	−0.809214	−0.404607	−	0.914491i	$-0.632592\pi$
$293$	−13.8515	−0.809214	−0.404607	+	0.914491i	$0.632592\pi$
$294$	99.7021	5.81475
$295$	0	0
$296$	−27.5898	−1.60363
$297$	2.73612	0.158766
$298$	27.7797	1.60924
$299$	−28.0186	−1.62036
$300$	0	0
$301$	47.4057	2.73242
$302$	−38.9846	−2.24331
$303$	21.1531	1.21521
$304$	9.84962	0.564914
$305$	0	0
$306$	−29.1637	−1.66718
$307$	−0.233341	−0.0133175	−0.00665874	−	0.999978i	$-0.502120\pi$
$307$	−0.233341	−0.0133175	−0.00665874	+	0.999978i	$0.502120\pi$
$308$	−24.0168	−1.36848
$309$	34.0233	1.93552
$310$	0	0
$311$	7.70278	0.436784	0.218392	−	0.975861i	$-0.429919\pi$
$311$	7.70278	0.436784	0.218392	+	0.975861i	$0.429919\pi$
$312$	68.4264	3.87388
$313$	−14.9273	−0.843742	−0.421871	−	0.906656i	$-0.638626\pi$
$313$	−14.9273	−0.843742	−0.421871	+	0.906656i	$0.638626\pi$
$314$	15.3057	0.863749
$315$	0	0
$316$	−32.1057	−1.80609
$317$	3.39011	0.190408	0.0952038	−	0.995458i	$-0.469650\pi$
$317$	3.39011	0.190408	0.0952038	+	0.995458i	$0.469650\pi$
$318$	38.4963	2.15876
$319$	−6.65643	−0.372689
$320$	0	0
$321$	−11.9619	−0.667648
$322$	−86.3051	−4.80960
$323$	6.38851	0.355466
$324$	−54.3137	−3.01743
$325$	0	0
$326$	−9.55885	−0.529416
$327$	−29.4417	−1.62813
$328$	8.07139	0.445668
$329$	14.3585	0.791610
$330$	0	0
$331$	−8.10646	−0.445571	−0.222786	−	0.974867i	$-0.571515\pi$
$331$	−8.10646	−0.445571	−0.222786	+	0.974867i	$0.571515\pi$
$332$	0.489047	0.0268399
$333$	6.44017	0.352919
$334$	−21.8390	−1.19498
$335$	0	0
$336$	106.154	5.79120
$337$	−13.4558	−0.732984	−0.366492	−	0.930421i	$-0.619441\pi$
$337$	−13.4558	−0.732984	−0.366492	+	0.930421i	$0.619441\pi$
$338$	−12.2697	−0.667385
$339$	−24.3713	−1.32367
$340$	0	0
$341$	−9.53855	−0.516541
$342$	−4.56502	−0.246848
$343$	−51.8872	−2.80165
$344$	71.5667	3.85862
$345$	0	0
$346$	37.8853	2.03673
$347$	15.8117	0.848819	0.424409	−	0.905470i	$-0.360482\pi$
$347$	15.8117	0.848819	0.424409	+	0.905470i	$0.360482\pi$
$348$	70.3654	3.77198
$349$	−7.24809	−0.387981	−0.193991	−	0.981003i	$-0.562143\pi$
$349$	−7.24809	−0.387981	−0.193991	+	0.981003i	$0.562143\pi$
$350$	0	0
$351$	11.5069	0.614194
$352$	−10.8449	−0.578038
$353$	11.6864	0.622005	0.311003	−	0.950409i	$-0.399335\pi$
$353$	11.6864	0.622005	0.311003	+	0.950409i	$0.399335\pi$
$354$	−3.49529	−0.185773
$355$	0	0
$356$	12.0725	0.639843
$357$	68.8522	3.64405
$358$	−60.7960	−3.21317
$359$	27.1065	1.43063	0.715313	−	0.698804i	$-0.246284\pi$
$359$	27.1065	1.43063	0.715313	+	0.698804i	$0.246284\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−9.58395	−0.503721
$363$	−2.17802	−0.114316
$364$	−101.004	−5.29405
$365$	0	0
$366$	7.62140	0.398377
$367$	12.9734	0.677207	0.338604	−	0.940929i	$-0.390045\pi$
$367$	12.9734	0.677207	0.338604	+	0.940929i	$0.390045\pi$
$368$	−65.6208	−3.42072
$369$	−1.88407	−0.0980806
$370$	0	0
$371$	−33.4086	−1.73449
$372$	100.832	5.22792
$373$	25.2889	1.30941	0.654704	−	0.755885i	$-0.272793\pi$
$373$	25.2889	1.30941	0.654704	+	0.755885i	$0.272793\pi$
$374$	−16.7246	−0.864809
$375$	0	0
$376$	21.6765	1.11788
$377$	−27.9941	−1.44177
$378$	35.4445	1.82307
$379$	−25.0016	−1.28425	−0.642123	−	0.766602i	$-0.721946\pi$
$379$	−25.0016	−1.28425	−0.642123	+	0.766602i	$0.721946\pi$
$380$	0	0
$381$	14.4894	0.742313
$382$	−61.1521	−3.12881
$383$	2.66127	0.135985	0.0679923	−	0.997686i	$-0.478341\pi$
$383$	2.66127	0.135985	0.0679923	+	0.997686i	$0.478341\pi$
$384$	2.31978	0.118381
$385$	0	0
$386$	65.8956	3.35400
$387$	−16.7055	−0.849189
$388$	84.1057	4.26982
$389$	6.77412	0.343462	0.171731	−	0.985144i	$-0.445064\pi$
$389$	6.77412	0.343462	0.171731	+	0.985144i	$0.445064\pi$
$390$	0	0
$391$	−42.5620	−2.15245
$392$	−130.624	−6.59753
$393$	−40.1747	−2.02654
$394$	18.2949	0.921686
$395$	0	0
$396$	8.46336	0.425300
$397$	−20.9430	−1.05110	−0.525550	−	0.850763i	$-0.676140\pi$
$397$	−20.9430	−1.05110	−0.525550	+	0.850763i	$0.676140\pi$
$398$	−30.2057	−1.51408
$399$	10.7775	0.539551
$400$	0	0
$401$	15.8352	0.790774	0.395387	−	0.918515i	$-0.370611\pi$
$401$	15.8352	0.790774	0.395387	+	0.918515i	$0.370611\pi$
$402$	−2.20446	−0.109948
$403$	−40.1150	−1.99827
$404$	−47.1378	−2.34519
$405$	0	0
$406$	−86.2295	−4.27950
$407$	3.69327	0.183069
$408$	103.944	5.14598
$409$	−1.16027	−0.0573715	−0.0286857	−	0.999588i	$-0.509132\pi$
$409$	−1.16027	−0.0573715	−0.0286857	+	0.999588i	$0.509132\pi$
$410$	0	0
$411$	−33.4773	−1.65131
$412$	−75.8178	−3.73528
$413$	3.03335	0.149262
$414$	30.4134	1.49474
$415$	0	0
$416$	−45.6092	−2.23617
$417$	5.19809	0.254551
$418$	−2.61792	−0.128047
$419$	0.923724	0.0451269	0.0225634	−	0.999745i	$-0.492817\pi$
$419$	0.923724	0.0451269	0.0225634	+	0.999745i	$0.492817\pi$
$420$	0	0
$421$	13.0507	0.636054	0.318027	−	0.948082i	$-0.396980\pi$
$421$	13.0507	0.636054	0.318027	+	0.948082i	$0.396980\pi$
$422$	−36.0382	−1.75431
$423$	−5.05985	−0.246018
$424$	−50.4357	−2.44938
$425$	0	0
$426$	53.5112	2.59263
$427$	−6.61415	−0.320081
$428$	26.6560	1.28847
$429$	−9.15980	−0.442239
$430$	0	0
$431$	20.3251	0.979025	0.489513	−	0.871996i	$-0.337175\pi$
$431$	20.3251	0.979025	0.489513	+	0.871996i	$0.337175\pi$
$432$	26.9497	1.29662
$433$	20.5451	0.987333	0.493666	−	0.869651i	$-0.335656\pi$
$433$	20.5451	0.987333	0.493666	+	0.869651i	$0.335656\pi$
$434$	−123.565	−5.93133
$435$	0	0
$436$	65.6082	3.14206
$437$	−6.66227	−0.318700
$438$	11.2587	0.537961
$439$	−7.67668	−0.366388	−0.183194	−	0.983077i	$-0.558644\pi$
$439$	−7.67668	−0.366388	−0.183194	+	0.983077i	$0.558644\pi$
$440$	0	0
$441$	30.4911	1.45196
$442$	−70.3365	−3.34557
$443$	−9.04317	−0.429654	−0.214827	−	0.976652i	$-0.568919\pi$
$443$	−9.04317	−0.429654	−0.214827	+	0.976652i	$0.568919\pi$
$444$	−39.0417	−1.85284
$445$	0	0
$446$	−49.0528	−2.32272
$447$	23.1117	1.09315
$448$	−43.0107	−2.03207
$449$	−9.97433	−0.470718	−0.235359	−	0.971909i	$-0.575627\pi$
$449$	−9.97433	−0.470718	−0.235359	+	0.971909i	$0.575627\pi$
$450$	0	0
$451$	−1.08046	−0.0508771
$452$	54.3093	2.55449
$453$	−32.4338	−1.52387
$454$	−57.0502	−2.67750
$455$	0	0
$456$	16.2704	0.761933
$457$	−18.7284	−0.876077	−0.438038	−	0.898956i	$-0.644327\pi$
$457$	−18.7284	−0.876077	−0.438038	+	0.898956i	$0.644327\pi$
$458$	40.7166	1.90256
$459$	17.4797	0.815883
$460$	0	0
$461$	−26.5036	−1.23440	−0.617198	−	0.786808i	$-0.711732\pi$
$461$	−26.5036	−1.23440	−0.617198	+	0.786808i	$0.711732\pi$
$462$	−28.2147	−1.31267
$463$	−18.1944	−0.845564	−0.422782	−	0.906231i	$-0.638946\pi$
$463$	−18.1944	−0.845564	−0.422782	+	0.906231i	$0.638946\pi$
$464$	−65.5633	−3.04370
$465$	0	0
$466$	70.7293	3.27647
$467$	−13.5114	−0.625234	−0.312617	−	0.949879i	$-0.601206\pi$
$467$	−13.5114	−0.625234	−0.312617	+	0.949879i	$0.601206\pi$
$468$	35.5933	1.64530
$469$	1.91311	0.0883394
$470$	0	0
$471$	12.7338	0.586741
$472$	4.57934	0.210781
$473$	−9.58018	−0.440497
$474$	−37.7176	−1.73243
$475$	0	0
$476$	−153.431	−7.03251
$477$	11.7730	0.539048
$478$	69.4114	3.17480
$479$	−15.0739	−0.688743	−0.344371	−	0.938833i	$-0.611908\pi$
$479$	−15.0739	−0.688743	−0.344371	+	0.938833i	$0.611908\pi$
$480$	0	0
$481$	15.5323	0.708212
$482$	42.3786	1.93029
$483$	−71.8027	−3.26714
$484$	4.85352	0.220615
$485$	0	0
$486$	−42.3185	−1.91961
$487$	41.3382	1.87321	0.936607	−	0.350381i	$-0.113948\pi$
$487$	41.3382	1.87321	0.936607	+	0.350381i	$0.113948\pi$
$488$	−9.98515	−0.452007
$489$	−7.95262	−0.359630
$490$	0	0
$491$	−29.4695	−1.32994	−0.664970	−	0.746870i	$-0.731556\pi$
$491$	−29.4695	−1.32994	−0.664970	+	0.746870i	$0.731556\pi$
$492$	11.4216	0.514927
$493$	−42.5246	−1.91521
$494$	−11.0099	−0.495357
$495$	0	0
$496$	−93.9511	−4.21853
$497$	−46.4391	−2.08308
$498$	0.574529	0.0257453
$499$	0.211110	0.00945058	0.00472529	−	0.999989i	$-0.498496\pi$
$499$	0.211110	0.00945058	0.00472529	+	0.999989i	$0.498496\pi$
$500$	0	0
$501$	−18.1693	−0.811743
$502$	−6.49774	−0.290008
$503$	7.94133	0.354086	0.177043	−	0.984203i	$-0.443347\pi$
$503$	7.94133	0.354086	0.177043	+	0.984203i	$0.443347\pi$
$504$	64.4587	2.87122
$505$	0	0
$506$	17.4413	0.775361
$507$	−10.2080	−0.453352
$508$	−32.2883	−1.43256
$509$	7.40748	0.328331	0.164165	−	0.986433i	$-0.447507\pi$
$509$	7.40748	0.328331	0.164165	+	0.986433i	$0.447507\pi$
$510$	0	0
$511$	−9.77074	−0.432232
$512$	40.3405	1.78282
$513$	2.73612	0.120803
$514$	−4.58842	−0.202387
$515$	0	0
$516$	101.273	4.45827
$517$	−2.90169	−0.127616
$518$	47.8438	2.10213
$519$	31.5192	1.38354
$520$	0	0
$521$	14.9680	0.655760	0.327880	−	0.944719i	$-0.393666\pi$
$521$	14.9680	0.655760	0.327880	+	0.944719i	$0.393666\pi$
$522$	30.3868	1.32999
$523$	−6.78937	−0.296878	−0.148439	−	0.988922i	$-0.547425\pi$
$523$	−6.78937	−0.296878	−0.148439	+	0.988922i	$0.547425\pi$
$524$	89.5257	3.91095
$525$	0	0
$526$	−41.7109	−1.81868
$527$	−60.9371	−2.65446
$528$	−21.4526	−0.933606
$529$	21.3859	0.929820
$530$	0	0
$531$	−1.06894	−0.0463879
$532$	−24.0168	−1.04126
$533$	−4.54397	−0.196821
$534$	14.1827	0.613746
$535$	0	0
$536$	2.88816	0.124749
$537$	−50.5801	−2.18269
$538$	14.1384	0.609549
$539$	17.4858	0.753168
$540$	0	0
$541$	−21.8957	−0.941368	−0.470684	−	0.882302i	$-0.655993\pi$
$541$	−21.8957	−0.941368	−0.470684	+	0.882302i	$0.655993\pi$
$542$	1.99399	0.0856491
$543$	−7.97350	−0.342176
$544$	−69.2830	−2.97048
$545$	0	0
$546$	−118.659	−5.07813
$547$	16.7053	0.714266	0.357133	−	0.934054i	$-0.383754\pi$
$547$	16.7053	0.714266	0.357133	+	0.934054i	$0.383754\pi$
$548$	74.6012	3.18681
$549$	2.33079	0.0994757
$550$	0	0
$551$	−6.65643	−0.283573
$552$	−108.398	−4.61373
$553$	32.7328	1.39194
$554$	70.5738	2.99839
$555$	0	0
$556$	−11.5835	−0.491249
$557$	23.8953	1.01248	0.506238	−	0.862394i	$-0.331036\pi$
$557$	23.8953	1.01248	0.506238	+	0.862394i	$0.331036\pi$
$558$	43.5437	1.84335
$559$	−40.2901	−1.70409
$560$	0	0
$561$	−13.9143	−0.587461
$562$	−48.4272	−2.04278
$563$	17.6090	0.742129	0.371065	−	0.928607i	$-0.378993\pi$
$563$	17.6090	0.742129	0.371065	+	0.928607i	$0.378993\pi$
$564$	30.6739	1.29161
$565$	0	0
$566$	49.4860	2.08005
$567$	55.3745	2.32551
$568$	−70.1075	−2.94164
$569$	38.1265	1.59835	0.799174	−	0.601100i	$-0.205271\pi$
$569$	38.1265	1.59835	0.799174	+	0.601100i	$0.205271\pi$
$570$	0	0
$571$	−6.37976	−0.266985	−0.133492	−	0.991050i	$-0.542619\pi$
$571$	−6.37976	−0.266985	−0.133492	+	0.991050i	$0.542619\pi$
$572$	20.4118	0.853461
$573$	−50.8763	−2.12539
$574$	−13.9967	−0.584210
$575$	0	0
$576$	15.1567	0.631531
$577$	19.3120	0.803969	0.401985	−	0.915646i	$-0.368321\pi$
$577$	19.3120	0.803969	0.401985	+	0.915646i	$0.368321\pi$
$578$	−62.3406	−2.59303
$579$	54.8227	2.27835
$580$	0	0
$581$	−0.498599	−0.0206854
$582$	98.8068	4.09567
$583$	6.75151	0.279619
$584$	−14.7505	−0.610381
$585$	0	0
$586$	36.2622	1.49798
$587$	10.2671	0.423769	0.211884	−	0.977295i	$-0.432040\pi$
$587$	10.2671	0.423769	0.211884	+	0.977295i	$0.432040\pi$
$588$	−184.844	−7.62282
$589$	−9.53855	−0.393029
$590$	0	0
$591$	15.2207	0.626097
$592$	36.3773	1.49510
$593$	−28.6849	−1.17795	−0.588975	−	0.808151i	$-0.700468\pi$
$593$	−28.6849	−1.17795	−0.588975	+	0.808151i	$0.700468\pi$
$594$	−7.16294	−0.293899
$595$	0	0
$596$	−51.5025	−2.10962
$597$	−25.1301	−1.02851
$598$	73.3507	2.99953
$599$	−21.3984	−0.874315	−0.437158	−	0.899385i	$-0.644015\pi$
$599$	−21.3984	−0.874315	−0.437158	+	0.899385i	$0.644015\pi$
$600$	0	0
$601$	−36.6616	−1.49546	−0.747729	−	0.664004i	$-0.768856\pi$
$601$	−36.6616	−1.49546	−0.747729	+	0.664004i	$0.768856\pi$
$602$	−124.105	−5.05812
$603$	−0.674170	−0.0274543
$604$	72.2758	2.94086
$605$	0	0
$606$	−55.3772	−2.24954
$607$	−47.9867	−1.94772	−0.973860	−	0.227150i	$-0.927059\pi$
$607$	−47.9867	−1.94772	−0.973860	+	0.227150i	$0.927059\pi$
$608$	−10.8449	−0.439821
$609$	−71.7398	−2.90704
$610$	0	0
$611$	−12.2033	−0.493692
$612$	54.0682	2.18558
$613$	−25.0991	−1.01374	−0.506872	−	0.862021i	$-0.669198\pi$
$613$	−25.0991	−1.01374	−0.506872	+	0.862021i	$0.669198\pi$
$614$	0.610869	0.0246527
$615$	0	0
$616$	36.9654	1.48938
$617$	7.05585	0.284058	0.142029	−	0.989863i	$-0.454637\pi$
$617$	7.05585	0.284058	0.142029	+	0.989863i	$0.454637\pi$
$618$	−89.0703	−3.58293
$619$	15.8586	0.637410	0.318705	−	0.947854i	$-0.396752\pi$
$619$	15.8586	0.637410	0.318705	+	0.947854i	$0.396752\pi$
$620$	0	0
$621$	−18.2288	−0.731495
$622$	−20.1653	−0.808554
$623$	−12.3083	−0.493123
$624$	−90.2205	−3.61171
$625$	0	0
$626$	39.0786	1.56189
$627$	−2.17802	−0.0869816
$628$	−28.3761	−1.13233
$629$	23.5945	0.940773
$630$	0	0
$631$	3.11637	0.124061	0.0620303	−	0.998074i	$-0.480242\pi$
$631$	3.11637	0.124061	0.0620303	+	0.998074i	$0.480242\pi$
$632$	49.4155	1.96564
$633$	−29.9825	−1.19170
$634$	−8.87505	−0.352473
$635$	0	0
$636$	−71.3705	−2.83002
$637$	73.5379	2.91368
$638$	17.4260	0.689903
$639$	16.3649	0.647384
$640$	0	0
$641$	−23.2203	−0.917147	−0.458573	−	0.888657i	$-0.651639\pi$
$641$	−23.2203	−0.917147	−0.458573	+	0.888657i	$0.651639\pi$
$642$	31.3153	1.23592
$643$	−17.3300	−0.683428	−0.341714	−	0.939804i	$-0.611007\pi$
$643$	−17.3300	−0.683428	−0.341714	+	0.939804i	$0.611007\pi$
$644$	160.006	6.30512
$645$	0	0
$646$	−16.7246	−0.658021
$647$	33.8000	1.32881	0.664407	−	0.747371i	$-0.268684\pi$
$647$	33.8000	1.32881	0.664407	+	0.747371i	$0.268684\pi$
$648$	83.5970	3.28400
$649$	−0.613007	−0.0240626
$650$	0	0
$651$	−102.802	−4.02912
$652$	17.7217	0.694036
$653$	−23.8491	−0.933289	−0.466644	−	0.884445i	$-0.654537\pi$
$653$	−23.8491	−0.933289	−0.466644	+	0.884445i	$0.654537\pi$
$654$	77.0761	3.01391
$655$	0	0
$656$	−10.6422	−0.415506
$657$	3.44315	0.134330
$658$	−37.5894	−1.46539
$659$	−34.8028	−1.35572	−0.677862	−	0.735189i	$-0.737093\pi$
$659$	−34.8028	−1.35572	−0.677862	+	0.735189i	$0.737093\pi$
$660$	0	0
$661$	−41.3984	−1.61021	−0.805105	−	0.593132i	$-0.797891\pi$
$661$	−41.3984	−1.61021	−0.805105	+	0.593132i	$0.797891\pi$
$662$	21.2221	0.824820
$663$	−58.5175	−2.27263
$664$	−0.752717	−0.0292111
$665$	0	0
$666$	−16.8599	−0.653306
$667$	44.3469	1.71712
$668$	40.4886	1.56655
$669$	−40.8102	−1.57781
$670$	0	0
$671$	1.33665	0.0516007
$672$	−116.882	−4.50880
$673$	−49.3984	−1.90417	−0.952085	−	0.305835i	$-0.901065\pi$
$673$	−49.3984	−1.90417	−0.952085	+	0.305835i	$0.901065\pi$
$674$	35.2262	1.35686
$675$	0	0
$676$	22.7476	0.874906
$677$	−36.6932	−1.41023	−0.705117	−	0.709091i	$-0.749105\pi$
$677$	−36.6932	−1.41023	−0.705117	+	0.709091i	$0.749105\pi$
$678$	63.8022	2.45031
$679$	−85.7484	−3.29072
$680$	0	0
$681$	−47.4637	−1.81881
$682$	24.9712	0.956196
$683$	26.3703	1.00903	0.504516	−	0.863403i	$-0.331671\pi$
$683$	26.3703	1.00903	0.504516	+	0.863403i	$0.331671\pi$
$684$	8.46336	0.323605
$685$	0	0
$686$	135.837	5.18627
$687$	33.8747	1.29240
$688$	−94.3611	−3.59748
$689$	28.3939	1.08172
$690$	0	0
$691$	13.3427	0.507581	0.253791	−	0.967259i	$-0.418323\pi$
$691$	13.3427	0.507581	0.253791	+	0.967259i	$0.418323\pi$
$692$	−70.2378	−2.67004
$693$	−8.62866	−0.327776
$694$	−41.3939	−1.57129
$695$	0	0
$696$	−108.303	−4.10521
$697$	−6.90255	−0.261453
$698$	18.9749	0.718212
$699$	58.8442	2.22569
$700$	0	0
$701$	−7.00399	−0.264537	−0.132268	−	0.991214i	$-0.542226\pi$
$701$	−7.00399	−0.264537	−0.132268	+	0.991214i	$0.542226\pi$
$702$	−30.1243	−1.13697
$703$	3.69327	0.139294
$704$	8.69200	0.327592
$705$	0	0
$706$	−30.5941	−1.15143
$707$	48.0585	1.80743
$708$	6.48013	0.243538
$709$	−30.5543	−1.14749	−0.573745	−	0.819034i	$-0.694510\pi$
$709$	−30.5543	−1.14749	−0.573745	+	0.819034i	$0.694510\pi$
$710$	0	0
$711$	−11.5348	−0.432591
$712$	−18.5814	−0.696369
$713$	63.5484	2.37991
$714$	−180.250	−6.74568
$715$	0	0
$716$	112.713	4.21229
$717$	57.7478	2.15663
$718$	−70.9627	−2.64831
$719$	−29.1224	−1.08608	−0.543042	−	0.839706i	$-0.682727\pi$
$719$	−29.1224	−1.08608	−0.543042	+	0.839706i	$0.682727\pi$
$720$	0	0
$721$	77.2987	2.87875
$722$	−2.61792	−0.0974290
$723$	35.2575	1.31124
$724$	17.7682	0.660351
$725$	0	0
$726$	5.70188	0.211617
$727$	−20.4576	−0.758731	−0.379366	−	0.925247i	$-0.623858\pi$
$727$	−20.4576	−0.758731	−0.379366	+	0.925247i	$0.623858\pi$
$728$	155.460	5.76175
$729$	−1.63574	−0.0605829
$730$	0	0
$731$	−61.2030	−2.26368
$732$	−14.1298	−0.522251
$733$	43.2636	1.59798	0.798989	−	0.601346i	$-0.205369\pi$
$733$	43.2636	1.59798	0.798989	+	0.601346i	$0.205369\pi$
$734$	−33.9634	−1.25361
$735$	0	0
$736$	72.2520	2.66324
$737$	−0.386619	−0.0142413
$738$	4.93234	0.181562
$739$	−31.1858	−1.14719	−0.573595	−	0.819139i	$-0.694452\pi$
$739$	−31.1858	−1.14719	−0.573595	+	0.819139i	$0.694452\pi$
$740$	0	0
$741$	−9.15980	−0.336494
$742$	87.4611	3.21080
$743$	45.0080	1.65118	0.825592	−	0.564268i	$-0.190842\pi$
$743$	45.0080	1.65118	0.825592	+	0.564268i	$0.190842\pi$
$744$	−155.196	−5.68977
$745$	0	0
$746$	−66.2043	−2.42391
$747$	0.175703	0.00642865
$748$	31.0067	1.13372
$749$	−27.1767	−0.993013
$750$	0	0
$751$	−29.7716	−1.08638	−0.543191	−	0.839609i	$-0.682784\pi$
$751$	−29.7716	−1.08638	−0.543191	+	0.839609i	$0.682784\pi$
$752$	−28.5806	−1.04223
$753$	−5.40588	−0.197001
$754$	73.2863	2.66893
$755$	0	0
$756$	−65.7126	−2.38995
$757$	32.1474	1.16842	0.584209	−	0.811603i	$-0.301405\pi$
$757$	32.1474	1.16842	0.584209	+	0.811603i	$0.301405\pi$
$758$	65.4522	2.37733
$759$	14.5105	0.526699
$760$	0	0
$761$	−2.83297	−0.102695	−0.0513475	−	0.998681i	$-0.516352\pi$
$761$	−2.83297	−0.102695	−0.0513475	+	0.998681i	$0.516352\pi$
$762$	−37.9321	−1.37413
$763$	−66.8896	−2.42157
$764$	113.373	4.10170
$765$	0	0
$766$	−6.96700	−0.251728
$767$	−2.57804	−0.0930878
$768$	31.7896	1.14711
$769$	−0.197238	−0.00711259	−0.00355630	−	0.999994i	$-0.501132\pi$
$769$	−0.197238	−0.00711259	−0.00355630	+	0.999994i	$0.501132\pi$
$770$	0	0
$771$	−3.81740	−0.137480
$772$	−122.168	−4.39691
$773$	−39.5489	−1.42247	−0.711237	−	0.702952i	$-0.751865\pi$
$773$	−39.5489	−1.42247	−0.711237	+	0.702952i	$0.751865\pi$
$774$	43.7337	1.57198
$775$	0	0
$776$	−129.451	−4.64703
$777$	39.8043	1.42797
$778$	−17.7341	−0.635799
$779$	−1.08046	−0.0387116
$780$	0	0
$781$	9.38483	0.335816
$782$	111.424	3.98451
$783$	−18.2128	−0.650871
$784$	172.229	6.15103
$785$	0	0
$786$	105.174	3.75144
$787$	−20.2522	−0.721912	−0.360956	−	0.932583i	$-0.617550\pi$
$787$	−20.2522	−0.721912	−0.360956	+	0.932583i	$0.617550\pi$
$788$	−33.9181	−1.20828
$789$	−34.7019	−1.23542
$790$	0	0
$791$	−55.3700	−1.96873
$792$	−13.0264	−0.462872
$793$	5.62136	0.199620
$794$	54.8272	1.94574
$795$	0	0
$796$	56.0002	1.98487
$797$	−24.5195	−0.868524	−0.434262	−	0.900787i	$-0.642991\pi$
$797$	−24.5195	−0.868524	−0.434262	+	0.900787i	$0.642991\pi$
$798$	−28.2147	−0.998790
$799$	−18.5375	−0.655809
$800$	0	0
$801$	4.33738	0.153254
$802$	−41.4554	−1.46384
$803$	1.97456	0.0696806
$804$	4.08697	0.144136
$805$	0	0
$806$	105.018	3.69910
$807$	11.7626	0.414064
$808$	72.5522	2.55238
$809$	18.8544	0.662885	0.331442	−	0.943475i	$-0.392465\pi$
$809$	18.8544	0.662885	0.331442	+	0.943475i	$0.392465\pi$
$810$	0	0
$811$	30.8152	1.08207	0.541034	−	0.841001i	$-0.318033\pi$
$811$	30.8152	1.08207	0.541034	+	0.841001i	$0.318033\pi$
$812$	159.866	5.61019
$813$	1.65893	0.0581811
$814$	−9.66869	−0.338888
$815$	0	0
$816$	−137.050	−4.79772
$817$	−9.58018	−0.335168
$818$	3.03749	0.106203
$819$	−36.2884	−1.26802
$820$	0	0
$821$	−3.91462	−0.136621	−0.0683106	−	0.997664i	$-0.521761\pi$
$821$	−3.91462	−0.136621	−0.0683106	+	0.997664i	$0.521761\pi$
$822$	87.6410	3.05683
$823$	−30.0384	−1.04707	−0.523537	−	0.852003i	$-0.675388\pi$
$823$	−30.0384	−1.04707	−0.523537	+	0.852003i	$0.675388\pi$
$824$	116.695	4.06526
$825$	0	0
$826$	−7.94108	−0.276306
$827$	19.0520	0.662504	0.331252	−	0.943542i	$-0.392529\pi$
$827$	19.0520	0.662504	0.331252	+	0.943542i	$0.392529\pi$
$828$	−56.3852	−1.95952
$829$	−2.78078	−0.0965806	−0.0482903	−	0.998833i	$-0.515377\pi$
$829$	−2.78078	−0.0965806	−0.0482903	+	0.998833i	$0.515377\pi$
$830$	0	0
$831$	58.7148	2.03679
$832$	36.5548	1.26731
$833$	111.708	3.87047
$834$	−13.6082	−0.471213
$835$	0	0
$836$	4.85352	0.167862
$837$	−26.0986	−0.902099
$838$	−2.41824	−0.0835367
$839$	−12.1680	−0.420087	−0.210044	−	0.977692i	$-0.567361\pi$
$839$	−12.1680	−0.420087	−0.210044	+	0.977692i	$0.567361\pi$
$840$	0	0
$841$	15.3081	0.527864
$842$	−34.1658	−1.17743
$843$	−40.2897	−1.38765
$844$	66.8133	2.29981
$845$	0	0
$846$	13.2463	0.455417
$847$	−4.94832	−0.170026
$848$	66.4997	2.28361
$849$	41.1706	1.41297
$850$	0	0
$851$	−24.6056	−0.843468
$852$	−99.2075	−3.39879
$853$	−30.8061	−1.05478	−0.527390	−	0.849623i	$-0.676829\pi$
$853$	−30.8061	−1.05478	−0.527390	+	0.849623i	$0.676829\pi$
$854$	17.3153	0.592519
$855$	0	0
$856$	−41.0276	−1.40229
$857$	5.90481	0.201705	0.100852	−	0.994901i	$-0.467843\pi$
$857$	5.90481	0.201705	0.100852	+	0.994901i	$0.467843\pi$
$858$	23.9797	0.818652
$859$	28.0485	0.957003	0.478502	−	0.878087i	$-0.341180\pi$
$859$	28.0485	0.957003	0.478502	+	0.878087i	$0.341180\pi$
$860$	0	0
$861$	−11.6447	−0.396851
$862$	−53.2095	−1.81232
$863$	24.0412	0.818374	0.409187	−	0.912451i	$-0.365812\pi$
$863$	24.0412	0.818374	0.409187	+	0.912451i	$0.365812\pi$
$864$	−29.6730	−1.00950
$865$	0	0
$866$	−53.7854	−1.82770
$867$	−51.8652	−1.76143
$868$	229.085	7.77565
$869$	−6.61494	−0.224396
$870$	0	0
$871$	−1.62595	−0.0550933
$872$	−100.981	−3.41964
$873$	30.2172	1.02270
$874$	17.4413	0.589961
$875$	0	0
$876$	−20.8731	−0.705238
$877$	46.1837	1.55951	0.779757	−	0.626083i	$-0.215343\pi$
$877$	46.1837	1.55951	0.779757	+	0.626083i	$0.215343\pi$
$878$	20.0970	0.678240
$879$	30.1688	1.01757
$880$	0	0
$881$	−16.5583	−0.557863	−0.278931	−	0.960311i	$-0.589980\pi$
$881$	−16.5583	−0.557863	−0.278931	+	0.960311i	$0.589980\pi$
$882$	−79.8232	−2.68779
$883$	−39.6210	−1.33335	−0.666676	−	0.745347i	$-0.732284\pi$
$883$	−39.6210	−1.33335	−0.666676	+	0.745347i	$0.732284\pi$
$884$	130.401	4.38586
$885$	0	0
$886$	23.6743	0.795354
$887$	−2.31838	−0.0778436	−0.0389218	−	0.999242i	$-0.512392\pi$
$887$	−2.31838	−0.0778436	−0.0389218	+	0.999242i	$0.512392\pi$
$888$	60.0911	2.01652
$889$	32.9189	1.10407
$890$	0	0
$891$	−11.1906	−0.374899
$892$	90.9419	3.04496
$893$	−2.90169	−0.0971015
$894$	−60.5048	−2.02358
$895$	0	0
$896$	5.27038	0.176071
$897$	61.0251	2.03757
$898$	26.1120	0.871370
$899$	63.4927	2.11760
$900$	0	0
$901$	43.1320	1.43694
$902$	2.82857	0.0941811
$903$	−103.251	−3.43596
$904$	−83.5901	−2.78017
$905$	0	0
$906$	84.9091	2.82092
$907$	−3.05923	−0.101580	−0.0507900	−	0.998709i	$-0.516174\pi$
$907$	−3.05923	−0.101580	−0.0507900	+	0.998709i	$0.516174\pi$
$908$	105.769	3.51006
$909$	−16.9355	−0.561716
$910$	0	0
$911$	−51.4953	−1.70611	−0.853057	−	0.521817i	$-0.825254\pi$
$911$	−51.4953	−1.70611	−0.853057	+	0.521817i	$0.825254\pi$
$912$	−21.4526	−0.710368
$913$	0.100761	0.00333471
$914$	49.0295	1.62175
$915$	0	0
$916$	−75.4868	−2.49416
$917$	−91.2743	−3.01414
$918$	−45.7605	−1.51032
$919$	47.9995	1.58336	0.791678	−	0.610938i	$-0.209208\pi$
$919$	47.9995	1.58336	0.791678	+	0.610938i	$0.209208\pi$
$920$	0	0
$921$	0.508221	0.0167464
$922$	69.3844	2.28505
$923$	39.4686	1.29912
$924$	52.3089	1.72084
$925$	0	0
$926$	47.6315	1.56527
$927$	−27.2396	−0.894666
$928$	72.1886	2.36971
$929$	27.5397	0.903548	0.451774	−	0.892132i	$-0.350791\pi$
$929$	27.5397	0.903548	0.451774	+	0.892132i	$0.350791\pi$
$930$	0	0
$931$	17.4858	0.573075
$932$	−131.129	−4.29528
$933$	−16.7768	−0.549247
$934$	35.3719	1.15740
$935$	0	0
$936$	−54.7834	−1.79065
$937$	−7.50039	−0.245027	−0.122513	−	0.992467i	$-0.539095\pi$
$937$	−7.50039	−0.245027	−0.122513	+	0.992467i	$0.539095\pi$
$938$	−5.00838	−0.163530
$939$	32.5119	1.06099
$940$	0	0
$941$	−42.0396	−1.37045	−0.685226	−	0.728330i	$-0.740297\pi$
$941$	−42.0396	−1.37045	−0.685226	+	0.728330i	$0.740297\pi$
$942$	−33.3360	−1.08615
$943$	7.19835	0.234410
$944$	−6.03789	−0.196516
$945$	0	0
$946$	25.0802	0.815426
$947$	−20.1743	−0.655576	−0.327788	−	0.944751i	$-0.606303\pi$
$947$	−20.1743	−0.655576	−0.327788	+	0.944751i	$0.606303\pi$
$948$	69.9268	2.27112
$949$	8.30414	0.269564
$950$	0	0
$951$	−7.38372	−0.239434
$952$	236.154	7.65378
$953$	1.49342	0.0483766	0.0241883	−	0.999707i	$-0.492300\pi$
$953$	1.49342	0.0483766	0.0241883	+	0.999707i	$0.492300\pi$
$954$	−30.8208	−0.997860
$955$	0	0
$956$	−128.686	−4.16200
$957$	14.4978	0.468648
$958$	39.4622	1.27497
$959$	−76.0583	−2.45605
$960$	0	0
$961$	59.9839	1.93497
$962$	−40.6624	−1.31101
$963$	9.57689	0.308611
$964$	−78.5682	−2.53051
$965$	0	0
$966$	187.974	6.04797
$967$	−37.4587	−1.20459	−0.602295	−	0.798274i	$-0.705747\pi$
$967$	−37.4587	−1.20459	−0.602295	+	0.798274i	$0.705747\pi$
$968$	−7.47030	−0.240104
$969$	−13.9143	−0.446991
$970$	0	0
$971$	−8.24513	−0.264599	−0.132299	−	0.991210i	$-0.542236\pi$
$971$	−8.24513	−0.264599	−0.132299	+	0.991210i	$0.542236\pi$
$972$	78.4568	2.51650
$973$	11.8097	0.378602
$974$	−108.220	−3.46760
$975$	0	0
$976$	13.1655	0.421416
$977$	−29.8558	−0.955171	−0.477586	−	0.878585i	$-0.658488\pi$
$977$	−29.8558	−0.955171	−0.477586	+	0.878585i	$0.658488\pi$
$978$	20.8193	0.665729
$979$	2.48738	0.0794969
$980$	0	0
$981$	23.5715	0.752581
$982$	77.1489	2.46192
$983$	−25.9753	−0.828484	−0.414242	−	0.910167i	$-0.635953\pi$
$983$	−25.9753	−0.828484	−0.414242	+	0.910167i	$0.635953\pi$
$984$	−17.5796	−0.560418
$985$	0	0
$986$	111.326	3.54535
$987$	−31.2731	−0.995432
$988$	20.4118	0.649386
$989$	63.8257	2.02954
$990$	0	0
$991$	−33.3837	−1.06047	−0.530235	−	0.847851i	$-0.677896\pi$
$991$	−33.3837	−1.06047	−0.530235	+	0.847851i	$0.677896\pi$
$992$	103.445	3.28438
$993$	17.6560	0.560297
$994$	121.574	3.85609
$995$	0	0
$996$	−1.06515	−0.0337507
$997$	−11.8321	−0.374725	−0.187363	−	0.982291i	$-0.559994\pi$
$997$	−11.8321	−0.374725	−0.187363	+	0.982291i	$0.559994\pi$
$998$	−0.552670	−0.0174945
$999$	10.1052	0.319715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.r.1.2	✓	15
5.4	even	2		5225.2.a.y.1.14	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.2	✓	15	1.1	even	1	trivial
5225.2.a.y.1.14	yes	15	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61792 q^{2} -2.17802 q^{3} +4.85352 q^{4} +5.70188 q^{6} -4.94832 q^{7} -7.47030 q^{8} +1.74376 q^{9} +1.00000 q^{11} -10.5710 q^{12} +4.20557 q^{13} +12.9543 q^{14} +9.84962 q^{16} +6.38851 q^{17} -4.56502 q^{18} +1.00000 q^{19} +10.7775 q^{21} -2.61792 q^{22} -6.66227 q^{23} +16.2704 q^{24} -11.0099 q^{26} +2.73612 q^{27} -24.0168 q^{28} -6.65643 q^{29} -9.53855 q^{31} -10.8449 q^{32} -2.17802 q^{33} -16.7246 q^{34} +8.46336 q^{36} +3.69327 q^{37} -2.61792 q^{38} -9.15980 q^{39} -1.08046 q^{41} -28.2147 q^{42} -9.58018 q^{43} +4.85352 q^{44} +17.4413 q^{46} -2.90169 q^{47} -21.4526 q^{48} +17.4858 q^{49} -13.9143 q^{51} +20.4118 q^{52} +6.75151 q^{53} -7.16294 q^{54} +36.9654 q^{56} -2.17802 q^{57} +17.4260 q^{58} -0.613007 q^{59} +1.33665 q^{61} +24.9712 q^{62} -8.62866 q^{63} +8.69200 q^{64} +5.70188 q^{66} -0.386619 q^{67} +31.0067 q^{68} +14.5105 q^{69} +9.38483 q^{71} -13.0264 q^{72} +1.97456 q^{73} -9.66869 q^{74} +4.85352 q^{76} -4.94832 q^{77} +23.9797 q^{78} -6.61494 q^{79} -11.1906 q^{81} +2.82857 q^{82} +0.100761 q^{83} +52.3089 q^{84} +25.0802 q^{86} +14.4978 q^{87} -7.47030 q^{88} +2.48738 q^{89} -20.8105 q^{91} -32.3355 q^{92} +20.7751 q^{93} +7.59641 q^{94} +23.6205 q^{96} +17.3288 q^{97} -45.7766 q^{98} +1.74376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9} + 15 q^{11} - 11 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 17 q^{17} - 22 q^{18} + 15 q^{19} + 6 q^{21} - 5 q^{22} - 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 21 * q^7 - 9 * q^8 + 15 * q^9 + 15 * q^11 - 11 * q^12 - 13 * q^13 + 9 * q^14 + 21 * q^16 - 17 * q^17 - 22 * q^18 + 15 * q^19 + 6 * q^21 - 5 * q^22 - 26 * q^23 + q^24 + 3 * q^26 - q^27 - 46 * q^28 + 9 * q^29 + 14 * q^31 - 18 * q^32 - 4 * q^33 - 13 * q^34 + 12 * q^36 - 9 * q^37 - 5 * q^38 - 22 * q^39 + 4 * q^41 + 6 * q^42 - 28 * q^43 + 17 * q^44 + 27 * q^46 - 14 * q^47 + 4 * q^48 + 32 * q^49 - 40 * q^51 - 14 * q^52 - 3 * q^53 - 39 * q^54 + 34 * q^56 - 4 * q^57 - 26 * q^58 + q^59 + 2 * q^61 + 3 * q^62 - 45 * q^63 + 5 * q^64 - q^66 - 37 * q^67 - 26 * q^68 - 7 * q^69 - 7 * q^71 - 16 * q^72 - 42 * q^73 - 43 * q^74 + 17 * q^76 - 21 * q^77 + 64 * q^78 - 10 * q^79 + 31 * q^81 - 22 * q^82 - 14 * q^83 - 32 * q^84 + 37 * q^86 - 29 * q^87 - 9 * q^88 + 15 * q^89 - 22 * q^91 - 26 * q^92 + 18 * q^93 - 44 * q^94 + 71 * q^96 - 8 * q^97 + 10 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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