LMFDB - Embedded newform 725.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(1,725)] 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("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.337383424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 1 $ x^6 - 13x^4 + 41x^2 - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.77035$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.77035 q^{2} -0.269894 q^{3} +5.67486 q^{4} +0.747703 q^{6} -1.86960 q^{7} -10.1807 q^{8} -2.92716 q^{9} -3.25230 q^{11} -1.53161 q^{12} +3.40121 q^{13} +5.17945 q^{14} +16.8543 q^{16} -2.40939 q^{17} +8.10926 q^{18} -0.674860 q^{19} +0.504595 q^{21} +9.01001 q^{22} +7.41031 q^{23} +2.74770 q^{24} -9.42256 q^{26} +1.59971 q^{27} -10.6097 q^{28} -1.00000 q^{29} +5.25230 q^{31} -26.3311 q^{32} +0.877777 q^{33} +6.67486 q^{34} -16.6112 q^{36} +1.86960 q^{37} +1.86960 q^{38} -0.917968 q^{39} -1.39791 q^{42} -3.46931 q^{43} -18.4563 q^{44} -20.5292 q^{46} +4.00910 q^{47} -4.54888 q^{48} -3.50459 q^{49} +0.650280 q^{51} +19.3014 q^{52} +0.877777 q^{53} -4.43175 q^{54} +19.0338 q^{56} +0.182141 q^{57} +2.77035 q^{58} +10.0000 q^{59} +11.8543 q^{61} -14.5507 q^{62} +5.47261 q^{63} +39.2378 q^{64} -2.43175 q^{66} +7.95010 q^{67} -13.6729 q^{68} -2.00000 q^{69} +2.00000 q^{71} +29.8004 q^{72} -0.607882 q^{73} -5.17945 q^{74} -3.82973 q^{76} +6.08050 q^{77} +2.54310 q^{78} +8.60202 q^{79} +8.34972 q^{81} -2.40939 q^{83} +2.86350 q^{84} +9.61121 q^{86} +0.269894 q^{87} +33.1105 q^{88} -8.50459 q^{89} -6.35891 q^{91} +42.0525 q^{92} -1.41757 q^{93} -11.1066 q^{94} +7.10661 q^{96} -13.1332 q^{97} +9.70897 q^{98} +9.51998 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} + 14 q^{6} + 12 q^{9} - 10 q^{11} - 8 q^{14} + 42 q^{16} + 16 q^{19} - 16 q^{21} + 26 q^{24} - 46 q^{26} - 6 q^{29} + 22 q^{31} + 20 q^{34} - 12 q^{36} - 14 q^{39} - 2 q^{44} - 44 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q + 14 * q^4 + 14 * q^6 + 12 * q^9 - 10 * q^11 - 8 * q^14 + 42 * q^16 + 16 * q^19 - 16 * q^21 + 26 * q^24 - 46 * q^26 - 6 * q^29 + 22 * q^31 + 20 * q^34 - 12 * q^36 - 14 * q^39 - 2 * q^44 - 44 * q^46 - 2 * q^49 + 44 * q^51 + 22 * q^54 + 16 * q^56 + 60 * q^59 + 12 * q^61 + 38 * q^64 + 34 * q^66 - 12 * q^69 + 12 * q^71 + 8 * q^74 - 24 * q^76 + 2 * q^79 + 10 * q^81 - 80 * q^84 - 30 * q^86 - 32 * q^89 + 40 * q^91 + 2 * q^94 - 26 * q^96 + 16 * 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 $

$2$	−2.77035	−1.95894	−0.979468	−	0.201600i	$-0.935386\pi$
$2$	−2.77035	−1.95894	−0.979468	+	0.201600i	$0.935386\pi$
$3$	−0.269894	−0.155824	−0.0779118	−	0.996960i	$-0.524825\pi$
$3$	−0.269894	−0.155824	−0.0779118	+	0.996960i	$0.524825\pi$
$4$	5.67486	2.83743
$5$	0	0
$5$	0	0
$6$	0.747703	0.305248
$7$	−1.86960	−0.706643	−0.353321	−	0.935502i	$-0.614948\pi$
$7$	−1.86960	−0.706643	−0.353321	+	0.935502i	$0.614948\pi$
$8$	−10.1807	−3.59941
$9$	−2.92716	−0.975719
$10$	0	0
$11$	−3.25230	−0.980605	−0.490302	−	0.871552i	$-0.663114\pi$
$11$	−3.25230	−0.980605	−0.490302	+	0.871552i	$0.663114\pi$
$12$	−1.53161	−0.442138
$13$	3.40121	0.943327	0.471663	−	0.881779i	$-0.343654\pi$
$13$	3.40121	0.943327	0.471663	+	0.881779i	$0.343654\pi$
$14$	5.17945	1.38427
$15$	0	0
$16$	16.8543	4.21358
$17$	−2.40939	−0.584363	−0.292181	−	0.956363i	$-0.594381\pi$
$17$	−2.40939	−0.584363	−0.292181	+	0.956363i	$0.594381\pi$
$18$	8.10926	1.91137
$19$	−0.674860	−0.154823	−0.0774117	−	0.996999i	$-0.524666\pi$
$19$	−0.674860	−0.154823	−0.0774117	+	0.996999i	$0.524666\pi$
$20$	0	0
$21$	0.504595	0.110112
$22$	9.01001	1.92094
$23$	7.41031	1.54516	0.772578	−	0.634920i	$-0.218967\pi$
$23$	7.41031	1.54516	0.772578	+	0.634920i	$0.218967\pi$
$24$	2.74770	0.560872
$25$	0	0
$26$	−9.42256	−1.84792
$27$	1.59971	0.307864
$28$	−10.6097	−2.00505
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.25230	0.943340	0.471670	−	0.881775i	$-0.343651\pi$
$31$	5.25230	0.943340	0.471670	+	0.881775i	$0.343651\pi$
$32$	−26.3311	−4.65472
$33$	0.877777	0.152801
$34$	6.67486	1.14473
$35$	0	0
$36$	−16.6112	−2.76853
$37$	1.86960	0.307360	0.153680	−	0.988121i	$-0.450887\pi$
$37$	1.86960	0.307360	0.153680	+	0.988121i	$0.450887\pi$
$38$	1.86960	0.303289
$39$	−0.917968	−0.146993
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−1.39791	−0.215701
$43$	−3.46931	−0.529064	−0.264532	−	0.964377i	$-0.585218\pi$
$43$	−3.46931	−0.529064	−0.264532	+	0.964377i	$0.585218\pi$
$44$	−18.4563	−2.78240
$45$	0	0
$46$	−20.5292	−3.02686
$47$	4.00910	0.584787	0.292393	−	0.956298i	$-0.405548\pi$
$47$	4.00910	0.584787	0.292393	+	0.956298i	$0.405548\pi$
$48$	−4.54888	−0.656575
$49$	−3.50459	−0.500656
$50$	0	0
$51$	0.650280	0.0910575
$52$	19.3014	2.67662
$53$	0.877777	0.120572	0.0602859	−	0.998181i	$-0.480799\pi$
$53$	0.877777	0.120572	0.0602859	+	0.998181i	$0.480799\pi$
$54$	−4.43175	−0.603085
$55$	0	0
$56$	19.0338	2.54349
$57$	0.182141	0.0241251
$58$	2.77035	0.363765
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	11.8543	1.51779	0.758895	−	0.651213i	$-0.225740\pi$
$61$	11.8543	1.51779	0.758895	+	0.651213i	$0.225740\pi$
$62$	−14.5507	−1.84794
$63$	5.47261	0.689485
$64$	39.2378	4.90473
$65$	0	0
$66$	−2.43175	−0.299328
$67$	7.95010	0.971259	0.485629	−	0.874165i	$-0.338590\pi$
$67$	7.95010	0.971259	0.485629	+	0.874165i	$0.338590\pi$
$68$	−13.6729	−1.65809
$69$	−2.00000	−0.240772
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	29.8004	3.51201
$73$	−0.607882	−0.0711472	−0.0355736	−	0.999367i	$-0.511326\pi$
$73$	−0.607882	−0.0711472	−0.0355736	+	0.999367i	$0.511326\pi$
$74$	−5.17945	−0.602099
$75$	0	0
$76$	−3.82973	−0.439301
$77$	6.08050	0.692937
$78$	2.54310	0.287949
$79$	8.60202	0.967803	0.483901	−	0.875123i	$-0.339219\pi$
$79$	8.60202	0.967803	0.483901	+	0.875123i	$0.339219\pi$
$80$	0	0
$81$	8.34972	0.927747
$82$	0	0
$83$	−2.40939	−0.264465	−0.132232	−	0.991219i	$-0.542215\pi$
$83$	−2.40939	−0.264465	−0.132232	+	0.991219i	$0.542215\pi$
$84$	2.86350	0.312434
$85$	0	0
$86$	9.61121	1.03640
$87$	0.269894	0.0289357
$88$	33.1105	3.52960
$89$	−8.50459	−0.901485	−0.450743	−	0.892654i	$-0.648841\pi$
$89$	−8.50459	−0.901485	−0.450743	+	0.892654i	$0.648841\pi$
$90$	0	0
$91$	−6.35891	−0.666595
$92$	42.0525	4.38427
$93$	−1.41757	−0.146995
$94$	−11.1066	−1.14556
$95$	0	0
$96$	7.10661	0.725315
$97$	−13.1332	−1.33347	−0.666735	−	0.745295i	$-0.732309\pi$
$97$	−13.1332	−1.33347	−0.666735	+	0.745295i	$0.732309\pi$
$98$	9.70897	0.980754
$99$	9.51998	0.956794
$100$	0	0
$101$	4.84513	0.482108	0.241054	−	0.970512i	$-0.422507\pi$
$101$	4.84513	0.482108	0.241054	+	0.970512i	$0.422507\pi$
$102$	−1.80151	−0.178376
$103$	14.8887	1.46703	0.733514	−	0.679674i	$-0.237879\pi$
$103$	14.8887	1.46703	0.733514	+	0.679674i	$0.237879\pi$
$104$	−34.6266	−3.39542
$105$	0	0
$106$	−2.43175	−0.236193
$107$	1.86960	0.180741	0.0903705	−	0.995908i	$-0.471195\pi$
$107$	1.86960	0.180741	0.0903705	+	0.995908i	$0.471195\pi$
$108$	9.07811	0.873541
$109$	−8.77228	−0.840232	−0.420116	−	0.907470i	$-0.638011\pi$
$109$	−8.77228	−0.840232	−0.420116	+	0.907470i	$0.638011\pi$
$110$	0	0
$111$	−0.504595	−0.0478940
$112$	−31.5108	−2.97749
$113$	10.0699	0.947299	0.473650	−	0.880713i	$-0.342936\pi$
$113$	10.0699	0.947299	0.473650	+	0.880713i	$0.342936\pi$
$114$	−0.504595	−0.0472596
$115$	0	0
$116$	−5.67486	−0.526898
$117$	−9.95588	−0.920422
$118$	−27.7035	−2.55032
$119$	4.50459	0.412936
$120$	0	0
$121$	−0.422563	−0.0384148
$122$	−32.8406	−2.97325
$123$	0	0
$124$	29.8061	2.67666
$125$	0	0
$126$	−15.1611	−1.35066
$127$	18.6739	1.65704	0.828519	−	0.559961i	$-0.189184\pi$
$127$	18.6739	1.65704	0.828519	+	0.559961i	$0.189184\pi$
$128$	−56.0404	−4.95332
$129$	0.936346	0.0824407
$130$	0	0
$131$	−11.0338	−0.964025	−0.482012	−	0.876164i	$-0.660094\pi$
$131$	−11.0338	−0.964025	−0.482012	+	0.876164i	$0.660094\pi$
$132$	4.98126	0.433563
$133$	1.26172	0.109405
$134$	−22.0246	−1.90263
$135$	0	0
$136$	24.5292	2.10336
$137$	11.8714	1.01425	0.507123	−	0.861874i	$-0.330709\pi$
$137$	11.8714	1.01425	0.507123	+	0.861874i	$0.330709\pi$
$138$	5.54071	0.471656
$139$	14.8451	1.25915	0.629574	−	0.776941i	$-0.283230\pi$
$139$	14.8451	1.25915	0.629574	+	0.776941i	$0.283230\pi$
$140$	0	0
$141$	−1.08203	−0.0911235
$142$	−5.54071	−0.464966
$143$	−11.0618	−0.925030
$144$	−49.3352	−4.11127
$145$	0	0
$146$	1.68405	0.139373
$147$	0.945870	0.0780141
$148$	10.6097	0.872114
$149$	17.2769	1.41538	0.707688	−	0.706525i	$-0.249738\pi$
$149$	17.2769	1.41538	0.707688	+	0.706525i	$0.249738\pi$
$150$	0	0
$151$	0.504595	0.0410633	0.0205317	−	0.999789i	$-0.493464\pi$
$151$	0.504595	0.0410633	0.0205317	+	0.999789i	$0.493464\pi$
$152$	6.87052	0.557273
$153$	7.05266	0.570174
$154$	−16.8451	−1.35742
$155$	0	0
$156$	−5.20934	−0.417081
$157$	−17.9519	−1.43272	−0.716360	−	0.697731i	$-0.754193\pi$
$157$	−17.9519	−1.43272	−0.716360	+	0.697731i	$0.754193\pi$
$158$	−23.8306	−1.89586
$159$	−0.236907	−0.0187879
$160$	0	0
$161$	−13.8543	−1.09187
$162$	−23.1317	−1.81740
$163$	−5.45295	−0.427108	−0.213554	−	0.976931i	$-0.568504\pi$
$163$	−5.45295	−0.427108	−0.213554	+	0.976931i	$0.568504\pi$
$164$	0	0
$165$	0	0
$166$	6.67486	0.518070
$167$	−1.51195	−0.116998	−0.0584992	−	0.998287i	$-0.518632\pi$
$167$	−1.51195	−0.116998	−0.0584992	+	0.998287i	$0.518632\pi$
$168$	−5.13711	−0.396336
$169$	−1.43175	−0.110135
$170$	0	0
$171$	1.97542	0.151064
$172$	−19.6878	−1.50118
$173$	8.74012	0.664499	0.332249	−	0.943192i	$-0.392192\pi$
$173$	8.74012	0.664499	0.332249	+	0.943192i	$0.392192\pi$
$174$	−0.747703	−0.0566832
$175$	0	0
$176$	−54.8152	−4.13185
$177$	−2.69894	−0.202865
$178$	23.5607	1.76595
$179$	−9.49541	−0.709720	−0.354860	−	0.934919i	$-0.615471\pi$
$179$	−9.49541	−0.709720	−0.354860	+	0.934919i	$0.615471\pi$
$180$	0	0
$181$	14.9363	1.11021	0.555105	−	0.831780i	$-0.312678\pi$
$181$	14.9363	1.11021	0.555105	+	0.831780i	$0.312678\pi$
$182$	17.6164	1.30582
$183$	−3.19941	−0.236507
$184$	−75.4418	−5.56165
$185$	0	0
$186$	3.92716	0.287953
$187$	7.83605	0.573029
$188$	22.7511	1.65929
$189$	−2.99081	−0.217549
$190$	0	0
$191$	10.5292	0.761864	0.380932	−	0.924603i	$-0.375603\pi$
$191$	10.5292	0.761864	0.380932	+	0.924603i	$0.375603\pi$
$192$	−10.5901	−0.764272
$193$	−18.1341	−1.30532	−0.652660	−	0.757651i	$-0.726347\pi$
$193$	−18.1341	−1.30532	−0.652660	+	0.757651i	$0.726347\pi$
$194$	36.3835	2.61218
$195$	0	0
$196$	−19.8881	−1.42058
$197$	16.9798	1.20976	0.604879	−	0.796317i	$-0.293221\pi$
$197$	16.9798	1.20976	0.604879	+	0.796317i	$0.293221\pi$
$198$	−26.3737	−1.87430
$199$	7.49541	0.531335	0.265668	−	0.964065i	$-0.414408\pi$
$199$	7.49541	0.531335	0.265668	+	0.964065i	$0.414408\pi$
$200$	0	0
$201$	−2.14569	−0.151345
$202$	−13.4227	−0.944419
$203$	1.86960	0.131220
$204$	3.69025	0.258369
$205$	0	0
$206$	−41.2470	−2.87381
$207$	−21.6911	−1.50764
$208$	57.3251	3.97478
$209$	2.19484	0.151821
$210$	0	0
$211$	18.0974	1.24588	0.622939	−	0.782270i	$-0.285938\pi$
$211$	18.0974	1.24588	0.622939	+	0.782270i	$0.285938\pi$
$212$	4.98126	0.342114
$213$	−0.539789	−0.0369857
$214$	−5.17945	−0.354060
$215$	0	0
$216$	−16.2861	−1.10813
$217$	−9.81970	−0.666604
$218$	24.3023	1.64596
$219$	0.164064	0.0110864
$220$	0	0
$221$	−8.19484	−0.551245
$222$	1.39791	0.0938213
$223$	−6.33073	−0.423937	−0.211969	−	0.977276i	$-0.567987\pi$
$223$	−6.33073	−0.423937	−0.211969	+	0.977276i	$0.567987\pi$
$224$	49.2286	3.28923
$225$	0	0
$226$	−27.8973	−1.85570
$227$	−24.8905	−1.65204	−0.826022	−	0.563638i	$-0.809401\pi$
$227$	−24.8905	−1.65204	−0.826022	+	0.563638i	$0.809401\pi$
$228$	1.03362	0.0684534
$229$	2.14569	0.141791	0.0708955	−	0.997484i	$-0.477414\pi$
$229$	2.14569	0.141791	0.0708955	+	0.997484i	$0.477414\pi$
$230$	0	0
$231$	−1.64109	−0.107976
$232$	10.1807	0.668393
$233$	−5.33891	−0.349763	−0.174882	−	0.984589i	$-0.555954\pi$
$233$	−5.33891	−0.349763	−0.174882	+	0.984589i	$0.555954\pi$
$234$	27.5813	1.80305
$235$	0	0
$236$	56.7486	3.69402
$237$	−2.32164	−0.150806
$238$	−12.4793	−0.808914
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	−6.77228	−0.436241	−0.218121	−	0.975922i	$-0.569993\pi$
$241$	−6.77228	−0.436241	−0.218121	+	0.975922i	$0.569993\pi$
$242$	1.17065	0.0752521
$243$	−7.05266	−0.452428
$244$	67.2716	4.30662
$245$	0	0
$246$	0	0
$247$	−2.29534	−0.146049
$248$	−53.4719	−3.39547
$249$	0.650280	0.0412098
$250$	0	0
$251$	20.2615	1.27889	0.639447	−	0.768835i	$-0.279163\pi$
$251$	20.2615	1.27889	0.639447	+	0.768835i	$0.279163\pi$
$252$	31.0563	1.95636
$253$	−24.1005	−1.51519
$254$	−51.7332	−3.24603
$255$	0	0
$256$	76.7762	4.79851
$257$	19.4376	1.21248	0.606242	−	0.795280i	$-0.292676\pi$
$257$	19.4376	1.21248	0.606242	+	0.795280i	$0.292676\pi$
$258$	−2.59401	−0.161496
$259$	−3.49541	−0.217194
$260$	0	0
$261$	2.92716	0.181186
$262$	30.5674	1.88846
$263$	−8.28808	−0.511065	−0.255533	−	0.966800i	$-0.582251\pi$
$263$	−8.28808	−0.511065	−0.255533	+	0.966800i	$0.582251\pi$
$264$	−8.93635	−0.549994
$265$	0	0
$266$	−3.49541	−0.214317
$267$	2.29534	0.140473
$268$	45.1157	2.75588
$269$	−23.8543	−1.45442	−0.727212	−	0.686413i	$-0.759184\pi$
$269$	−23.8543	−1.45442	−0.727212	+	0.686413i	$0.759184\pi$
$270$	0	0
$271$	−0.893389	−0.0542695	−0.0271347	−	0.999632i	$-0.508638\pi$
$271$	−0.893389	−0.0542695	−0.0271347	+	0.999632i	$0.508638\pi$
$272$	−40.6086	−2.46226
$273$	1.71623	0.103871
$274$	−32.8881	−1.98684
$275$	0	0
$276$	−11.3497	−0.683173
$277$	−3.55706	−0.213723	−0.106862	−	0.994274i	$-0.534080\pi$
$277$	−3.55706	−0.213723	−0.106862	+	0.994274i	$0.534080\pi$
$278$	−41.1262	−2.46659
$279$	−15.3743	−0.920435
$280$	0	0
$281$	7.76309	0.463107	0.231554	−	0.972822i	$-0.425619\pi$
$281$	7.76309	0.463107	0.231554	+	0.972822i	$0.425619\pi$
$282$	2.99761	0.178505
$283$	−30.7889	−1.83021	−0.915105	−	0.403215i	$-0.867893\pi$
$283$	−30.7889	−1.83021	−0.915105	+	0.403215i	$0.867893\pi$
$284$	11.3497	0.673482
$285$	0	0
$286$	30.6450	1.81208
$287$	0	0
$288$	77.0752	4.54170
$289$	−11.1948	−0.658520
$290$	0	0
$291$	3.54456	0.207786
$292$	−3.44965	−0.201875
$293$	−6.33073	−0.369845	−0.184923	−	0.982753i	$-0.559203\pi$
$293$	−6.33073	−0.369845	−0.184923	+	0.982753i	$0.559203\pi$
$294$	−2.62039	−0.152825
$295$	0	0
$296$	−19.0338	−1.10632
$297$	−5.20272	−0.301892
$298$	−47.8631	−2.77263
$299$	25.2040	1.45759
$300$	0	0
$301$	6.48622	0.373859
$302$	−1.39791	−0.0804404
$303$	−1.30767	−0.0751238
$304$	−11.3743	−0.652361
$305$	0	0
$306$	−19.5384	−1.11693
$307$	−12.5671	−0.717241	−0.358620	−	0.933483i	$-0.616753\pi$
$307$	−12.5671	−0.717241	−0.358620	+	0.933483i	$0.616753\pi$
$308$	34.5060	1.96616
$309$	−4.01838	−0.228598
$310$	0	0
$311$	24.3835	1.38266	0.691330	−	0.722539i	$-0.257025\pi$
$311$	24.3835	1.38266	0.691330	+	0.722539i	$0.257025\pi$
$312$	9.34552	0.529086
$313$	34.6618	1.95920	0.979601	−	0.200954i	$-0.0644042\pi$
$313$	34.6618	1.95920	0.979601	+	0.200954i	$0.0644042\pi$
$314$	49.7332	2.80661
$315$	0	0
$316$	48.8152	2.74607
$317$	25.7880	1.44840	0.724199	−	0.689591i	$-0.242210\pi$
$317$	25.7880	1.44840	0.724199	+	0.689591i	$0.242210\pi$
$318$	0.656316	0.0368044
$319$	3.25230	0.182094
$320$	0	0
$321$	−0.504595	−0.0281637
$322$	38.3814	2.13891
$323$	1.62600	0.0904730
$324$	47.3835	2.63242
$325$	0	0
$326$	15.1066	0.836678
$327$	2.36759	0.130928
$328$	0	0
$329$	−7.49541	−0.413235
$330$	0	0
$331$	−16.9609	−0.932257	−0.466128	−	0.884717i	$-0.654352\pi$
$331$	−16.9609	−0.932257	−0.466128	+	0.884717i	$0.654352\pi$
$332$	−13.6729	−0.750400
$333$	−5.47261	−0.299897
$334$	4.18864	0.229192
$335$	0	0
$336$	8.50459	0.463964
$337$	22.5952	1.23084	0.615420	−	0.788200i	$-0.288987\pi$
$337$	22.5952	1.23084	0.615420	+	0.788200i	$0.288987\pi$
$338$	3.96646	0.215747
$339$	−2.71782	−0.147612
$340$	0	0
$341$	−17.0820	−0.925044
$342$	−5.47261	−0.295925
$343$	19.6394	1.06043
$344$	35.3198	1.90432
$345$	0	0
$346$	−24.2132	−1.30171
$347$	−12.0469	−0.646714	−0.323357	−	0.946277i	$-0.604811\pi$
$347$	−12.0469	−0.646714	−0.323357	+	0.946277i	$0.604811\pi$
$348$	1.53161	0.0821030
$349$	−1.56825	−0.0839464	−0.0419732	−	0.999119i	$-0.513364\pi$
$349$	−1.56825	−0.0839464	−0.0419732	+	0.999119i	$0.513364\pi$
$350$	0	0
$351$	5.44094	0.290416
$352$	85.6365	4.56444
$353$	−7.66054	−0.407730	−0.203865	−	0.978999i	$-0.565350\pi$
$353$	−7.66054	−0.407730	−0.203865	+	0.978999i	$0.565350\pi$
$354$	7.47703	0.397400
$355$	0	0
$356$	−48.2624	−2.55790
$357$	−1.21576	−0.0643451
$358$	26.3056	1.39030
$359$	17.5928	0.928514	0.464257	−	0.885701i	$-0.346321\pi$
$359$	17.5928	0.928514	0.464257	+	0.885701i	$0.346321\pi$
$360$	0	0
$361$	−18.5446	−0.976030
$362$	−41.3790	−2.17483
$363$	0.114047	0.00598593
$364$	−36.0859	−1.89142
$365$	0	0
$366$	8.86350	0.463303
$367$	9.88779	0.516138	0.258069	−	0.966126i	$-0.416914\pi$
$367$	9.88779	0.516138	0.258069	+	0.966126i	$0.416914\pi$
$368$	124.896	6.51064
$369$	0	0
$370$	0	0
$371$	−1.64109	−0.0852012
$372$	−8.04448	−0.417087
$373$	−16.2841	−0.843161	−0.421580	−	0.906791i	$-0.638524\pi$
$373$	−16.2841	−0.843161	−0.421580	+	0.906791i	$0.638524\pi$
$374$	−21.7086	−1.12253
$375$	0	0
$376$	−40.8152	−2.10489
$377$	−3.40121	−0.175171
$378$	8.28560	0.426166
$379$	−2.52917	−0.129915	−0.0649575	−	0.997888i	$-0.520691\pi$
$379$	−2.52917	−0.129915	−0.0649575	+	0.997888i	$0.520691\pi$
$380$	0	0
$381$	−5.03997	−0.258205
$382$	−29.1695	−1.49244
$383$	5.60880	0.286596	0.143298	−	0.989680i	$-0.454229\pi$
$383$	5.60880	0.286596	0.143298	+	0.989680i	$0.454229\pi$
$384$	15.1250	0.771844
$385$	0	0
$386$	50.2378	2.55704
$387$	10.1552	0.516218
$388$	−74.5288	−3.78363
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−17.8543	−0.902931
$392$	35.6791	1.80207
$393$	2.97795	0.150218
$394$	−47.0400	−2.36984
$395$	0	0
$396$	54.0246	2.71484
$397$	−27.8660	−1.39856	−0.699278	−	0.714850i	$-0.746495\pi$
$397$	−27.8660	−1.39856	−0.699278	+	0.714850i	$0.746495\pi$
$398$	−20.7649	−1.04085
$399$	−0.340531	−0.0170479
$400$	0	0
$401$	−15.6174	−0.779896	−0.389948	−	0.920837i	$-0.627507\pi$
$401$	−15.6174	−0.779896	−0.389948	+	0.920837i	$0.627507\pi$
$402$	5.94431	0.296475
$403$	17.8642	0.889878
$404$	27.4954	1.36795
$405$	0	0
$406$	−5.17945	−0.257052
$407$	−6.08050	−0.301399
$408$	−6.62028	−0.327753
$409$	−30.5538	−1.51079	−0.755393	−	0.655272i	$-0.772554\pi$
$409$	−30.5538	−1.51079	−0.755393	+	0.655272i	$0.772554\pi$
$410$	0	0
$411$	−3.20403	−0.158043
$412$	84.4913	4.16259
$413$	−18.6960	−0.919970
$414$	60.0921	2.95337
$415$	0	0
$416$	−89.5576	−4.39092
$417$	−4.00661	−0.196205
$418$	−6.08050	−0.297407
$419$	−10.0492	−0.490934	−0.245467	−	0.969405i	$-0.578941\pi$
$419$	−10.0492	−0.490934	−0.245467	+	0.969405i	$0.578941\pi$
$420$	0	0
$421$	−16.1948	−0.789288	−0.394644	−	0.918834i	$-0.629132\pi$
$421$	−16.1948	−0.789288	−0.394644	+	0.918834i	$0.629132\pi$
$422$	−50.1363	−2.44060
$423$	−11.7353	−0.570587
$424$	−8.93635	−0.433987
$425$	0	0
$426$	1.49541	0.0724526
$427$	−22.1628	−1.07253
$428$	10.6097	0.512840
$429$	2.98550	0.144142
$430$	0	0
$431$	11.1857	0.538794	0.269397	−	0.963029i	$-0.413176\pi$
$431$	11.1857	0.538794	0.269397	+	0.963029i	$0.413176\pi$
$432$	26.9619	1.29721
$433$	−14.0306	−0.674267	−0.337134	−	0.941457i	$-0.609457\pi$
$433$	−14.0306	−0.674267	−0.337134	+	0.941457i	$0.609457\pi$
$434$	27.2040	1.30584
$435$	0	0
$436$	−49.7815	−2.38410
$437$	−5.00092	−0.239226
$438$	−0.454515	−0.0217176
$439$	9.65947	0.461021	0.230511	−	0.973070i	$-0.425960\pi$
$439$	9.65947	0.461021	0.230511	+	0.973070i	$0.425960\pi$
$440$	0	0
$441$	10.2585	0.488500
$442$	22.7026	1.07985
$443$	9.75160	0.463313	0.231656	−	0.972798i	$-0.425586\pi$
$443$	9.75160	0.463313	0.231656	+	0.972798i	$0.425586\pi$
$444$	−2.86350	−0.135896
$445$	0	0
$446$	17.5384	0.830466
$447$	−4.66293	−0.220549
$448$	−73.3590	−3.46589
$449$	8.86350	0.418295	0.209147	−	0.977884i	$-0.432931\pi$
$449$	8.86350	0.418295	0.209147	+	0.977884i	$0.432931\pi$
$450$	0	0
$451$	0	0
$452$	57.1454	2.68790
$453$	−0.136187	−0.00639863
$454$	68.9556	3.23625
$455$	0	0
$456$	−1.85431	−0.0868362
$457$	−24.5041	−1.14625	−0.573127	−	0.819466i	$-0.694270\pi$
$457$	−24.5041	−1.14625	−0.573127	+	0.819466i	$0.694270\pi$
$458$	−5.94431	−0.277759
$459$	−3.85431	−0.179904
$460$	0	0
$461$	−37.4173	−1.74270	−0.871348	−	0.490666i	$-0.836753\pi$
$461$	−37.4173	−1.74270	−0.871348	+	0.490666i	$0.836753\pi$
$462$	4.54640	0.211518
$463$	−2.45534	−0.114109	−0.0570547	−	0.998371i	$-0.518171\pi$
$463$	−2.45534	−0.114109	−0.0570547	+	0.998371i	$0.518171\pi$
$464$	−16.8543	−0.782442
$465$	0	0
$466$	14.7907	0.685164
$467$	34.4182	1.59268	0.796342	−	0.604846i	$-0.206765\pi$
$467$	34.4182	1.59268	0.796342	+	0.604846i	$0.206765\pi$
$468$	−56.4982	−2.61163
$469$	−14.8635	−0.686333
$470$	0	0
$471$	4.84513	0.223252
$472$	−101.807	−4.68603
$473$	11.2832	0.518803
$474$	6.43175	0.295420
$475$	0	0
$476$	25.5629	1.17168
$477$	−2.56939	−0.117644
$478$	5.54071	0.253426
$479$	−9.41636	−0.430245	−0.215122	−	0.976587i	$-0.569015\pi$
$479$	−9.41636	−0.430245	−0.215122	+	0.976587i	$0.569015\pi$
$480$	0	0
$481$	6.35891	0.289941
$482$	18.7616	0.854568
$483$	3.73920	0.170140
$484$	−2.39798	−0.108999
$485$	0	0
$486$	19.5384	0.886278
$487$	−33.0909	−1.49949	−0.749745	−	0.661726i	$-0.769824\pi$
$487$	−33.0909	−1.49949	−0.749745	+	0.661726i	$0.769824\pi$
$488$	−120.685	−5.46314
$489$	1.47172	0.0665535
$490$	0	0
$491$	28.9609	1.30699	0.653494	−	0.756932i	$-0.273302\pi$
$491$	28.9609	1.30699	0.653494	+	0.756932i	$0.273302\pi$
$492$	0	0
$493$	2.40939	0.108513
$494$	6.35891	0.286101
$495$	0	0
$496$	88.5239	3.97484
$497$	−3.73920	−0.167726
$498$	−1.80151	−0.0807274
$499$	33.8727	1.51635	0.758175	−	0.652051i	$-0.226091\pi$
$499$	33.8727	1.51635	0.758175	+	0.652051i	$0.226091\pi$
$500$	0	0
$501$	0.408067	0.0182311
$502$	−56.1315	−2.50527
$503$	0.945870	0.0421743	0.0210871	−	0.999778i	$-0.493287\pi$
$503$	0.945870	0.0421743	0.0210871	+	0.999778i	$0.493287\pi$
$504$	−55.7148	−2.48174
$505$	0	0
$506$	66.7670	2.96815
$507$	0.386422	0.0171616
$508$	105.972	4.70173
$509$	32.1404	1.42460	0.712299	−	0.701877i	$-0.247654\pi$
$509$	32.1404	1.42460	0.712299	+	0.701877i	$0.247654\pi$
$510$	0	0
$511$	1.13650	0.0502757
$512$	−100.616	−4.44665
$513$	−1.07958	−0.0476645
$514$	−53.8490	−2.37518
$515$	0	0
$516$	5.31363	0.233920
$517$	−13.0388	−0.573444
$518$	9.68351	0.425469
$519$	−2.35891	−0.103545
$520$	0	0
$521$	−8.43175	−0.369402	−0.184701	−	0.982795i	$-0.559132\pi$
$521$	−8.43175	−0.369402	−0.184701	+	0.982795i	$0.559132\pi$
$522$	−8.10926	−0.354933
$523$	21.8273	0.954442	0.477221	−	0.878783i	$-0.341644\pi$
$523$	21.8273	0.954442	0.477221	+	0.878783i	$0.341644\pi$
$524$	−62.6151	−2.73535
$525$	0	0
$526$	22.9609	1.00114
$527$	−12.6548	−0.551253
$528$	14.7943	0.643840
$529$	31.9127	1.38751
$530$	0	0
$531$	−29.2716	−1.27028
$532$	7.16007	0.310429
$533$	0	0
$534$	−6.35891	−0.275177
$535$	0	0
$536$	−80.9372	−3.49596
$537$	2.56276	0.110591
$538$	66.0849	2.84912
$539$	11.3980	0.490946
$540$	0	0
$541$	−25.5138	−1.09692	−0.548462	−	0.836176i	$-0.684786\pi$
$541$	−25.5138	−1.09692	−0.548462	+	0.836176i	$0.684786\pi$
$542$	2.47500	0.106310
$543$	−4.03123	−0.172997
$544$	63.4418	2.72005
$545$	0	0
$546$	−4.75457	−0.203477
$547$	17.7698	0.759781	0.379891	−	0.925031i	$-0.375962\pi$
$547$	17.7698	0.759781	0.379891	+	0.925031i	$0.375962\pi$
$548$	67.3687	2.87785
$549$	−34.6994	−1.48094
$550$	0	0
$551$	0.674860	0.0287500
$552$	20.3613	0.866635
$553$	−16.0823	−0.683890
$554$	9.85431	0.418670
$555$	0	0
$556$	84.2440	3.57274
$557$	42.1665	1.78665	0.893326	−	0.449409i	$-0.148365\pi$
$557$	42.1665	1.78665	0.893326	+	0.449409i	$0.148365\pi$
$558$	42.5922	1.80307
$559$	−11.7998	−0.499080
$560$	0	0
$561$	−2.11491	−0.0892914
$562$	−21.5065	−0.907198
$563$	5.76465	0.242951	0.121475	−	0.992594i	$-0.461237\pi$
$563$	5.76465	0.242951	0.121475	+	0.992594i	$0.461237\pi$
$564$	−6.14038	−0.258557
$565$	0	0
$566$	85.2961	3.58526
$567$	−15.6106	−0.655585
$568$	−20.3613	−0.854342
$569$	1.49541	0.0626907	0.0313453	−	0.999509i	$-0.490021\pi$
$569$	1.49541	0.0626907	0.0313453	+	0.999509i	$0.490021\pi$
$570$	0	0
$571$	−7.03997	−0.294614	−0.147307	−	0.989091i	$-0.547060\pi$
$571$	−7.03997	−0.294614	−0.147307	+	0.989091i	$0.547060\pi$
$572$	−62.7739	−2.62471
$573$	−2.84176	−0.118716
$574$	0	0
$575$	0	0
$576$	−114.855	−4.78563
$577$	34.7102	1.44501	0.722503	−	0.691368i	$-0.242991\pi$
$577$	34.7102	1.44501	0.722503	+	0.691368i	$0.242991\pi$
$578$	31.0137	1.29000
$579$	4.89428	0.203399
$580$	0	0
$581$	4.50459	0.186882
$582$	−9.81970	−0.407040
$583$	−2.85479	−0.118233
$584$	6.18864	0.256088
$585$	0	0
$586$	17.5384	0.724503
$587$	37.4158	1.54432	0.772158	−	0.635430i	$-0.219177\pi$
$587$	37.4158	1.54432	0.772158	+	0.635430i	$0.219177\pi$
$588$	5.36768	0.221359
$589$	−3.54456	−0.146051
$590$	0	0
$591$	−4.58274	−0.188509
$592$	31.5108	1.29509
$593$	−32.6388	−1.34032	−0.670158	−	0.742218i	$-0.733774\pi$
$593$	−32.6388	−1.34032	−0.670158	+	0.742218i	$0.733774\pi$
$594$	14.4134	0.591388
$595$	0	0
$596$	98.0439	4.01603
$597$	−2.02297	−0.0827945
$598$	−69.8241	−2.85532
$599$	−38.1466	−1.55863	−0.779314	−	0.626634i	$-0.784432\pi$
$599$	−38.1466	−1.55863	−0.779314	+	0.626634i	$0.784432\pi$
$600$	0	0
$601$	−16.7178	−0.681934	−0.340967	−	0.940075i	$-0.610754\pi$
$601$	−16.7178	−0.681934	−0.340967	+	0.940075i	$0.610754\pi$
$602$	−17.9691	−0.732366
$603$	−23.2712	−0.947676
$604$	2.86350	0.116514
$605$	0	0
$606$	3.62271	0.147163
$607$	−0.673496	−0.0273364	−0.0136682	−	0.999907i	$-0.504351\pi$
$607$	−0.673496	−0.0273364	−0.0136682	+	0.999907i	$0.504351\pi$
$608$	17.7698	0.720660
$609$	−0.504595	−0.0204472
$610$	0	0
$611$	13.6358	0.551645
$612$	40.0229	1.61783
$613$	26.4615	1.06877	0.534385	−	0.845241i	$-0.320543\pi$
$613$	26.4615	1.06877	0.534385	+	0.845241i	$0.320543\pi$
$614$	34.8152	1.40503
$615$	0	0
$616$	−61.9035	−2.49416
$617$	27.9538	1.12538	0.562688	−	0.826669i	$-0.309767\pi$
$617$	27.9538	1.12538	0.562688	+	0.826669i	$0.309767\pi$
$618$	11.1323	0.447808
$619$	24.6512	0.990814	0.495407	−	0.868661i	$-0.335019\pi$
$619$	24.6512	0.990814	0.495407	+	0.868661i	$0.335019\pi$
$620$	0	0
$621$	11.8543	0.475697
$622$	−67.5509	−2.70854
$623$	15.9002	0.637028
$624$	−15.4717	−0.619365
$625$	0	0
$626$	−96.0255	−3.83795
$627$	−0.592376	−0.0236572
$628$	−101.875	−4.06524
$629$	−4.50459	−0.179610
$630$	0	0
$631$	11.8052	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$631$	11.8052	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$632$	−87.5742	−3.48352
$633$	−4.88439	−0.194137
$634$	−71.4418	−2.83732
$635$	0	0
$636$	−1.34441	−0.0533095
$637$	−11.9199	−0.472283
$638$	−9.01001	−0.356710
$639$	−5.85431	−0.231593
$640$	0	0
$641$	13.8727	0.547938	0.273969	−	0.961738i	$-0.411663\pi$
$641$	13.8727	0.547938	0.273969	+	0.961738i	$0.411663\pi$
$642$	1.39791	0.0551709
$643$	5.29047	0.208636	0.104318	−	0.994544i	$-0.466734\pi$
$643$	5.29047	0.208636	0.104318	+	0.994544i	$0.466734\pi$
$644$	−78.6213	−3.09811
$645$	0	0
$646$	−4.50459	−0.177231
$647$	40.7448	1.60184	0.800921	−	0.598769i	$-0.204343\pi$
$647$	40.7448	1.60184	0.800921	+	0.598769i	$0.204343\pi$
$648$	−85.0057	−3.33934
$649$	−32.5230	−1.27664
$650$	0	0
$651$	2.65028	0.103873
$652$	−30.9447	−1.21189
$653$	26.8742	1.05167	0.525834	−	0.850587i	$-0.323753\pi$
$653$	26.8742	1.05167	0.525834	+	0.850587i	$0.323753\pi$
$654$	−6.55906	−0.256480
$655$	0	0
$656$	0	0
$657$	1.77937	0.0694197
$658$	20.7649	0.809501
$659$	29.1558	1.13575	0.567874	−	0.823116i	$-0.307766\pi$
$659$	29.1558	1.13575	0.567874	+	0.823116i	$0.307766\pi$
$660$	0	0
$661$	19.9035	0.774155	0.387078	−	0.922047i	$-0.373485\pi$
$661$	19.9035	0.774155	0.387078	+	0.922047i	$0.373485\pi$
$662$	46.9878	1.82623
$663$	2.21174	0.0858969
$664$	24.5292	0.951917
$665$	0	0
$666$	15.1611	0.587480
$667$	−7.41031	−0.286928
$668$	−8.58012	−0.331975
$669$	1.70863	0.0660594
$670$	0	0
$671$	−38.5538	−1.48835
$672$	−13.2865	−0.512539
$673$	−8.71383	−0.335893	−0.167947	−	0.985796i	$-0.553714\pi$
$673$	−8.71383	−0.335893	−0.167947	+	0.985796i	$0.553714\pi$
$674$	−62.5967	−2.41114
$675$	0	0
$676$	−8.12499	−0.312500
$677$	9.70565	0.373018	0.186509	−	0.982453i	$-0.440283\pi$
$677$	9.70565	0.373018	0.186509	+	0.982453i	$0.440283\pi$
$678$	7.52932	0.289162
$679$	24.5538	0.942287
$680$	0	0
$681$	6.71782	0.257427
$682$	47.3233	1.81210
$683$	−9.53014	−0.364661	−0.182330	−	0.983237i	$-0.558364\pi$
$683$	−9.53014	−0.364661	−0.182330	+	0.983237i	$0.558364\pi$
$684$	11.2102	0.428634
$685$	0	0
$686$	−54.4081	−2.07731
$687$	−0.579108	−0.0220944
$688$	−58.4728	−2.22925
$689$	2.98550	0.113739
$690$	0	0
$691$	8.50459	0.323530	0.161765	−	0.986829i	$-0.448281\pi$
$691$	8.50459	0.323530	0.161765	+	0.986829i	$0.448281\pi$
$692$	49.5990	1.88547
$693$	−17.7986	−0.676112
$694$	33.3743	1.26687
$695$	0	0
$696$	−2.74770	−0.104151
$697$	0	0
$698$	4.34460	0.164446
$699$	1.44094	0.0545014
$700$	0	0
$701$	−28.6266	−1.08121	−0.540606	−	0.841276i	$-0.681805\pi$
$701$	−28.6266	−1.08121	−0.540606	+	0.841276i	$0.681805\pi$
$702$	−15.0733	−0.568906
$703$	−1.26172	−0.0475866
$704$	−127.613	−4.80960
$705$	0	0
$706$	21.2224	0.798716
$707$	−9.05845	−0.340678
$708$	−15.3161	−0.575615
$709$	14.9855	0.562792	0.281396	−	0.959592i	$-0.409202\pi$
$709$	14.9855	0.562792	0.281396	+	0.959592i	$0.409202\pi$
$710$	0	0
$711$	−25.1795	−0.944303
$712$	86.5824	3.24481
$713$	38.9211	1.45761
$714$	3.36810	0.126048
$715$	0	0
$716$	−53.8851	−2.01378
$717$	0.539789	0.0201588
$718$	−48.7384	−1.81890
$719$	−36.7486	−1.37049	−0.685246	−	0.728312i	$-0.740305\pi$
$719$	−36.7486	−1.37049	−0.685246	+	0.728312i	$0.740305\pi$
$720$	0	0
$721$	−27.8359	−1.03666
$722$	51.3750	1.91198
$723$	1.82780	0.0679766
$724$	84.7617	3.15014
$725$	0	0
$726$	−0.315951	−0.0117260
$727$	−39.1647	−1.45254	−0.726270	−	0.687410i	$-0.758748\pi$
$727$	−39.1647	−1.45254	−0.726270	+	0.687410i	$0.758748\pi$
$728$	64.7379	2.39935
$729$	−23.1457	−0.857248
$730$	0	0
$731$	8.35891	0.309165
$732$	−18.1562	−0.671073
$733$	5.38071	0.198741	0.0993705	−	0.995051i	$-0.468317\pi$
$733$	5.38071	0.198741	0.0993705	+	0.995051i	$0.468317\pi$
$734$	−27.3927	−1.01108
$735$	0	0
$736$	−195.121	−7.19227
$737$	−25.8561	−0.952421
$738$	0	0
$739$	−9.93336	−0.365404	−0.182702	−	0.983168i	$-0.558484\pi$
$739$	−9.93336	−0.365404	−0.182702	+	0.983168i	$0.558484\pi$
$740$	0	0
$741$	0.619500	0.0227579
$742$	4.54640	0.166904
$743$	43.4963	1.59573	0.797863	−	0.602839i	$-0.205964\pi$
$743$	43.4963	1.59573	0.797863	+	0.602839i	$0.205964\pi$
$744$	14.4318	0.529094
$745$	0	0
$746$	45.1128	1.65170
$747$	7.05266	0.258043
$748$	44.4685	1.62593
$749$	−3.49541	−0.127719
$750$	0	0
$751$	11.3743	0.415054	0.207527	−	0.978229i	$-0.433459\pi$
$751$	11.3743	0.415054	0.207527	+	0.978229i	$0.433459\pi$
$752$	67.5705	2.46404
$753$	−5.46846	−0.199282
$754$	9.42256	0.343149
$755$	0	0
$756$	−16.9724	−0.617281
$757$	−10.0699	−0.365998	−0.182999	−	0.983113i	$-0.558580\pi$
$757$	−10.0699	−0.365998	−0.182999	+	0.983113i	$0.558580\pi$
$758$	7.00671	0.254495
$759$	6.50459	0.236102
$760$	0	0
$761$	−27.0092	−0.979082	−0.489541	−	0.871980i	$-0.662836\pi$
$761$	−27.0092	−0.979082	−0.489541	+	0.871980i	$0.662836\pi$
$762$	13.9625	0.505808
$763$	16.4007	0.593744
$764$	59.7516	2.16174
$765$	0	0
$766$	−15.5384	−0.561424
$767$	34.0121	1.22811
$768$	−20.7215	−0.747721
$769$	25.7086	0.927077	0.463538	−	0.886077i	$-0.346580\pi$
$769$	25.7086	0.927077	0.463538	+	0.886077i	$0.346580\pi$
$770$	0	0
$771$	−5.24610	−0.188934
$772$	−102.908	−3.70375
$773$	32.8185	1.18040	0.590200	−	0.807257i	$-0.299049\pi$
$773$	32.8185	1.18040	0.590200	+	0.807257i	$0.299049\pi$
$774$	−28.1335	−1.01124
$775$	0	0
$776$	133.704	4.79970
$777$	0.943390	0.0338439
$778$	33.2442	1.19186
$779$	0	0
$780$	0	0
$781$	−6.50459	−0.232753
$782$	49.4628	1.76878
$783$	−1.59971	−0.0571688
$784$	−59.0675	−2.10955
$785$	0	0
$786$	−8.24998	−0.294267
$787$	−21.1973	−0.755602	−0.377801	−	0.925887i	$-0.623320\pi$
$787$	−21.1973	−0.755602	−0.377801	+	0.925887i	$0.623320\pi$
$788$	96.3578	3.43261
$789$	2.23691	0.0796360
$790$	0	0
$791$	−18.8267	−0.669402
$792$	−96.9197	−3.44389
$793$	40.3190	1.43177
$794$	77.1987	2.73968
$795$	0	0
$796$	42.5354	1.50763
$797$	21.7305	0.769732	0.384866	−	0.922972i	$-0.374248\pi$
$797$	21.7305	0.769732	0.384866	+	0.922972i	$0.374248\pi$
$798$	0.943390	0.0333956
$799$	−9.65947	−0.341727
$800$	0	0
$801$	24.8943	0.879596
$802$	43.2657	1.52777
$803$	1.97701	0.0697673
$804$	−12.1765	−0.429431
$805$	0	0
$806$	−49.4901	−1.74321
$807$	6.43814	0.226633
$808$	−49.3266	−1.73530
$809$	−7.20403	−0.253280	−0.126640	−	0.991949i	$-0.540419\pi$
$809$	−7.20403	−0.253280	−0.126640	+	0.991949i	$0.540419\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	10.6097	0.372328
$813$	0.241121	0.00845647
$814$	16.8451	0.590421
$815$	0	0
$816$	10.9600	0.383678
$817$	2.34130	0.0819116
$818$	84.6447	2.95953
$819$	18.6135	0.650409
$820$	0	0
$821$	24.1220	0.841864	0.420932	−	0.907092i	$-0.361703\pi$
$821$	24.1220	0.841864	0.420932	+	0.907092i	$0.361703\pi$
$822$	8.87631	0.309597
$823$	−17.2300	−0.600600	−0.300300	−	0.953845i	$-0.597087\pi$
$823$	−17.2300	−0.600600	−0.300300	+	0.953845i	$0.597087\pi$
$824$	−151.577	−5.28043
$825$	0	0
$826$	51.7945	1.80216
$827$	3.46931	0.120640	0.0603198	−	0.998179i	$-0.480788\pi$
$827$	3.46931	0.120640	0.0603198	+	0.998179i	$0.480788\pi$
$828$	−123.094	−4.27782
$829$	7.52619	0.261395	0.130698	−	0.991422i	$-0.458278\pi$
$829$	7.52619	0.261395	0.130698	+	0.991422i	$0.458278\pi$
$830$	0	0
$831$	0.960030	0.0333031
$832$	133.456	4.62676
$833$	8.44393	0.292565
$834$	11.0997	0.384353
$835$	0	0
$836$	12.4554	0.430780
$837$	8.40213	0.290420
$838$	27.8397	0.961707
$839$	−9.61121	−0.331816	−0.165908	−	0.986141i	$-0.553055\pi$
$839$	−9.61121	−0.331816	−0.165908	+	0.986141i	$0.553055\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	44.8654	1.54617
$843$	−2.09521	−0.0721630
$844$	102.700	3.53509
$845$	0	0
$846$	32.5108	1.11774
$847$	0.790023	0.0271455
$848$	14.7943	0.508039
$849$	8.30975	0.285190
$850$	0	0
$851$	13.8543	0.474920
$852$	−3.06322	−0.104944
$853$	20.8330	0.713309	0.356654	−	0.934236i	$-0.383917\pi$
$853$	20.8330	0.713309	0.356654	+	0.934236i	$0.383917\pi$
$854$	61.3989	2.10103
$855$	0	0
$856$	−19.0338	−0.650561
$857$	−33.0424	−1.12871	−0.564354	−	0.825533i	$-0.690875\pi$
$857$	−33.0424	−1.12871	−0.564354	+	0.825533i	$0.690875\pi$
$858$	−8.27090	−0.282364
$859$	49.8061	1.69936	0.849680	−	0.527298i	$-0.176795\pi$
$859$	49.8061	1.69936	0.849680	+	0.527298i	$0.176795\pi$
$860$	0	0
$861$	0	0
$862$	−30.9882	−1.05546
$863$	−9.03631	−0.307599	−0.153800	−	0.988102i	$-0.549151\pi$
$863$	−9.03631	−0.307599	−0.153800	+	0.988102i	$0.549151\pi$
$864$	−42.1220	−1.43302
$865$	0	0
$866$	38.8697	1.32085
$867$	3.02143	0.102613
$868$	−55.7254	−1.89144
$869$	−27.9763	−0.949032
$870$	0	0
$871$	27.0400	0.916214
$872$	89.3076	3.02434
$873$	38.4428	1.30109
$874$	13.8543	0.468629
$875$	0	0
$876$	0.931040	0.0314569
$877$	−24.0809	−0.813153	−0.406576	−	0.913617i	$-0.633278\pi$
$877$	−24.0809	−0.813153	−0.406576	+	0.913617i	$0.633278\pi$
$878$	−26.7601	−0.903111
$879$	1.70863	0.0576306
$880$	0	0
$881$	−0.814344	−0.0274360	−0.0137180	−	0.999906i	$-0.504367\pi$
$881$	−0.814344	−0.0274360	−0.0137180	+	0.999906i	$0.504367\pi$
$882$	−28.4197	−0.956940
$883$	31.8751	1.07268	0.536341	−	0.844001i	$-0.319806\pi$
$883$	31.8751	1.07268	0.536341	+	0.844001i	$0.319806\pi$
$884$	−46.5046	−1.56412
$885$	0	0
$886$	−27.0154	−0.907600
$887$	−45.6358	−1.53230	−0.766150	−	0.642661i	$-0.777830\pi$
$887$	−45.6358	−1.53230	−0.766150	+	0.642661i	$0.777830\pi$
$888$	5.13711	0.172390
$889$	−34.9127	−1.17093
$890$	0	0
$891$	−27.1558	−0.909753
$892$	−35.9260	−1.20289
$893$	−2.70558	−0.0905387
$894$	12.9180	0.432041
$895$	0	0
$896$	104.773	3.50023
$897$	−6.80243	−0.227126
$898$	−24.5550	−0.819412
$899$	−5.25230	−0.175174
$900$	0	0
$901$	−2.11491	−0.0704577
$902$	0	0
$903$	−1.75059	−0.0582561
$904$	−102.519	−3.40972
$905$	0	0
$906$	0.377287	0.0125345
$907$	18.5820	0.617004	0.308502	−	0.951224i	$-0.400172\pi$
$907$	18.5820	0.617004	0.308502	+	0.951224i	$0.400172\pi$
$908$	−141.250	−4.68756
$909$	−14.1824	−0.470402
$910$	0	0
$911$	−45.6296	−1.51178	−0.755888	−	0.654701i	$-0.772794\pi$
$911$	−45.6296	−1.51178	−0.755888	+	0.654701i	$0.772794\pi$
$912$	3.06986	0.101653
$913$	7.83605	0.259335
$914$	67.8851	2.24544
$915$	0	0
$916$	12.1765	0.402322
$917$	20.6287	0.681221
$918$	10.6778	0.352420
$919$	24.6994	0.814759	0.407380	−	0.913259i	$-0.366443\pi$
$919$	24.6994	0.814759	0.407380	+	0.913259i	$0.366443\pi$
$920$	0	0
$921$	3.39178	0.111763
$922$	103.659	3.41383
$923$	6.80243	0.223905
$924$	−9.31296	−0.306374
$925$	0	0
$926$	6.80217	0.223533
$927$	−43.5816	−1.43141
$928$	26.3311	0.864360
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	2.36511	0.0775133
$932$	−30.2975	−0.992429
$933$	−6.58096	−0.215451
$934$	−95.3506	−3.11997
$935$	0	0
$936$	101.357	3.31297
$937$	46.4389	1.51709	0.758546	−	0.651620i	$-0.225910\pi$
$937$	46.4389	1.51709	0.758546	+	0.651620i	$0.225910\pi$
$938$	41.1772	1.34448
$939$	−9.35503	−0.305290
$940$	0	0
$941$	−14.2861	−0.465712	−0.232856	−	0.972511i	$-0.574807\pi$
$941$	−14.2861	−0.465712	−0.232856	+	0.972511i	$0.574807\pi$
$942$	−13.4227	−0.437336
$943$	0	0
$944$	168.543	5.48561
$945$	0	0
$946$	−31.2585	−1.01630
$947$	29.3713	0.954440	0.477220	−	0.878784i	$-0.341644\pi$
$947$	29.3713	0.954440	0.477220	+	0.878784i	$0.341644\pi$
$948$	−13.1750	−0.427903
$949$	−2.06754	−0.0671151
$950$	0	0
$951$	−6.96003	−0.225694
$952$	−45.8598	−1.48632
$953$	−22.5911	−0.731796	−0.365898	−	0.930655i	$-0.619238\pi$
$953$	−22.5911	−0.731796	−0.365898	+	0.930655i	$0.619238\pi$
$954$	7.11812	0.230458
$955$	0	0
$956$	−11.3497	−0.367076
$957$	−0.877777	−0.0283745
$958$	26.0867	0.842821
$959$	−22.1948	−0.716709
$960$	0	0
$961$	−3.41337	−0.110109
$962$	−17.6164	−0.567976
$963$	−5.47261	−0.176353
$964$	−38.4318	−1.23780
$965$	0	0
$966$	−10.3589	−0.333292
$967$	48.4316	1.55746	0.778728	−	0.627362i	$-0.215865\pi$
$967$	48.4316	1.55746	0.778728	+	0.627362i	$0.215865\pi$
$968$	4.30197	0.138270
$969$	−0.438848	−0.0140978
$970$	0	0
$971$	47.0829	1.51096	0.755482	−	0.655170i	$-0.227403\pi$
$971$	47.0829	1.51096	0.755482	+	0.655170i	$0.227403\pi$
$972$	−40.0229	−1.28373
$973$	−27.7545	−0.889767
$974$	91.6734	2.93741
$975$	0	0
$976$	199.796	6.39532
$977$	−51.1987	−1.63799	−0.818995	−	0.573800i	$-0.805468\pi$
$977$	−51.1987	−1.63799	−0.818995	+	0.573800i	$0.805468\pi$
$978$	−4.07719	−0.130374
$979$	27.6595	0.884000
$980$	0	0
$981$	25.6778	0.819831
$982$	−80.2320	−2.56031
$983$	−23.5123	−0.749926	−0.374963	−	0.927040i	$-0.622345\pi$
$983$	−23.5123	−0.749926	−0.374963	+	0.927040i	$0.622345\pi$
$984$	0	0
$985$	0	0
$986$	−6.67486	−0.212571
$987$	2.02297	0.0643918
$988$	−13.0257	−0.414404
$989$	−25.7086	−0.817487
$990$	0	0
$991$	−7.64109	−0.242727	−0.121364	−	0.992608i	$-0.538727\pi$
$991$	−7.64109	−0.242727	−0.121364	+	0.992608i	$0.538727\pi$
$992$	−138.299	−4.39099
$993$	4.57766	0.145268
$994$	10.3589	0.328565
$995$	0	0
$996$	3.69025	0.116930
$997$	3.71043	0.117510	0.0587552	−	0.998272i	$-0.481287\pi$
$997$	3.71043	0.117510	0.0587552	+	0.998272i	$0.481287\pi$
$998$	−93.8393	−2.97043
$999$	2.99081	0.0946251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.l.1.1		6
3.2	odd	2		6525.2.a.bt.1.6		6
5.2	odd	4		145.2.b.c.59.1	✓	6
5.3	odd	4		145.2.b.c.59.6	yes	6
5.4	even	2	inner	725.2.a.l.1.6		6
15.2	even	4		1305.2.c.h.784.6		6
15.8	even	4		1305.2.c.h.784.1		6
15.14	odd	2		6525.2.a.bt.1.1		6
20.3	even	4		2320.2.d.g.929.4		6
20.7	even	4		2320.2.d.g.929.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.1	✓	6	5.2	odd	4
145.2.b.c.59.6	yes	6	5.3	odd	4
725.2.a.l.1.1		6	1.1	even	1	trivial
725.2.a.l.1.6		6	5.4	even	2	inner
1305.2.c.h.784.1		6	15.8	even	4
1305.2.c.h.784.6		6	15.2	even	4
2320.2.d.g.929.3		6	20.7	even	4
2320.2.d.g.929.4		6	20.3	even	4
6525.2.a.bt.1.1		6	15.14	odd	2
6525.2.a.bt.1.6		6	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77035 q^{2} -0.269894 q^{3} +5.67486 q^{4} +0.747703 q^{6} -1.86960 q^{7} -10.1807 q^{8} -2.92716 q^{9} -3.25230 q^{11} -1.53161 q^{12} +3.40121 q^{13} +5.17945 q^{14} +16.8543 q^{16} -2.40939 q^{17} +8.10926 q^{18} -0.674860 q^{19} +0.504595 q^{21} +9.01001 q^{22} +7.41031 q^{23} +2.74770 q^{24} -9.42256 q^{26} +1.59971 q^{27} -10.6097 q^{28} -1.00000 q^{29} +5.25230 q^{31} -26.3311 q^{32} +0.877777 q^{33} +6.67486 q^{34} -16.6112 q^{36} +1.86960 q^{37} +1.86960 q^{38} -0.917968 q^{39} -1.39791 q^{42} -3.46931 q^{43} -18.4563 q^{44} -20.5292 q^{46} +4.00910 q^{47} -4.54888 q^{48} -3.50459 q^{49} +0.650280 q^{51} +19.3014 q^{52} +0.877777 q^{53} -4.43175 q^{54} +19.0338 q^{56} +0.182141 q^{57} +2.77035 q^{58} +10.0000 q^{59} +11.8543 q^{61} -14.5507 q^{62} +5.47261 q^{63} +39.2378 q^{64} -2.43175 q^{66} +7.95010 q^{67} -13.6729 q^{68} -2.00000 q^{69} +2.00000 q^{71} +29.8004 q^{72} -0.607882 q^{73} -5.17945 q^{74} -3.82973 q^{76} +6.08050 q^{77} +2.54310 q^{78} +8.60202 q^{79} +8.34972 q^{81} -2.40939 q^{83} +2.86350 q^{84} +9.61121 q^{86} +0.269894 q^{87} +33.1105 q^{88} -8.50459 q^{89} -6.35891 q^{91} +42.0525 q^{92} -1.41757 q^{93} -11.1066 q^{94} +7.10661 q^{96} -13.1332 q^{97} +9.70897 q^{98} +9.51998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 14 q^{6} + 12 q^{9} - 10 q^{11} - 8 q^{14} + 42 q^{16} + 16 q^{19} - 16 q^{21} + 26 q^{24} - 46 q^{26} - 6 q^{29} + 22 q^{31} + 20 q^{34} - 12 q^{36} - 14 q^{39} - 2 q^{44} - 44 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q + 14 * q^4 + 14 * q^6 + 12 * q^9 - 10 * q^11 - 8 * q^14 + 42 * q^16 + 16 * q^19 - 16 * q^21 + 26 * q^24 - 46 * q^26 - 6 * q^29 + 22 * q^31 + 20 * q^34 - 12 * q^36 - 14 * q^39 - 2 * q^44 - 44 * q^46 - 2 * q^49 + 44 * q^51 + 22 * q^54 + 16 * q^56 + 60 * q^59 + 12 * q^61 + 38 * q^64 + 34 * q^66 - 12 * q^69 + 12 * q^71 + 8 * q^74 - 24 * q^76 + 2 * q^79 + 10 * q^81 - 80 * q^84 - 30 * q^86 - 32 * q^89 + 40 * q^91 + 2 * q^94 - 26 * q^96 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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