LMFDB - Embedded newform 4225.2.a.cb.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] 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("4225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,16,0,16,0] 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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.395078$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.395078 q^{2} +2.73651 q^{3} -1.84391 q^{4} -1.08113 q^{6} -0.629405 q^{7} +1.51865 q^{8} +4.48846 q^{9} +5.49747 q^{11} -5.04588 q^{12} +0.248664 q^{14} +3.08784 q^{16} -4.25364 q^{17} -1.77329 q^{18} +3.61686 q^{19} -1.72237 q^{21} -2.17193 q^{22} +7.05869 q^{23} +4.15578 q^{24} +4.07319 q^{27} +1.16057 q^{28} +2.09557 q^{29} +2.97582 q^{31} -4.25723 q^{32} +15.0439 q^{33} +1.68052 q^{34} -8.27634 q^{36} -0.817814 q^{37} -1.42894 q^{38} -4.89390 q^{41} +0.680471 q^{42} -9.91411 q^{43} -10.1369 q^{44} -2.78873 q^{46} +2.29494 q^{47} +8.44990 q^{48} -6.60385 q^{49} -11.6401 q^{51} -5.84288 q^{53} -1.60923 q^{54} -0.955844 q^{56} +9.89756 q^{57} -0.827912 q^{58} +5.19393 q^{59} +5.92410 q^{61} -1.17568 q^{62} -2.82506 q^{63} -4.49375 q^{64} -5.94350 q^{66} +13.7031 q^{67} +7.84335 q^{68} +19.3161 q^{69} +0.210677 q^{71} +6.81639 q^{72} +12.4282 q^{73} +0.323100 q^{74} -6.66918 q^{76} -3.46014 q^{77} -7.19084 q^{79} -2.31909 q^{81} +1.93347 q^{82} +6.61897 q^{83} +3.17590 q^{84} +3.91685 q^{86} +5.73453 q^{87} +8.34872 q^{88} +3.66109 q^{89} -13.0156 q^{92} +8.14336 q^{93} -0.906679 q^{94} -11.6499 q^{96} +4.54596 q^{97} +2.60904 q^{98} +24.6752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.395078	−0.279362	−0.139681	−	0.990197i	$-0.544608\pi$
$2$	−0.395078	−0.279362	−0.139681	+	0.990197i	$0.544608\pi$
$3$	2.73651	1.57992	0.789961	−	0.613157i	$-0.210101\pi$
$3$	2.73651	1.57992	0.789961	+	0.613157i	$0.210101\pi$
$4$	−1.84391	−0.921957
$5$	0	0
$5$	0	0
$6$	−1.08113	−0.441371
$7$	−0.629405	−0.237893	−0.118946	−	0.992901i	$-0.537952\pi$
$7$	−0.629405	−0.237893	−0.118946	+	0.992901i	$0.537952\pi$
$8$	1.51865	0.536922
$9$	4.48846	1.49615
$10$	0	0
$11$	5.49747	1.65755	0.828775	−	0.559582i	$-0.189038\pi$
$11$	5.49747	1.65755	0.828775	+	0.559582i	$0.189038\pi$
$12$	−5.04588	−1.45662
$13$	0	0
$13$	0	0
$14$	0.248664	0.0664583
$15$	0	0
$16$	3.08784	0.771961
$17$	−4.25364	−1.03166	−0.515830	−	0.856691i	$-0.672516\pi$
$17$	−4.25364	−1.03166	−0.515830	+	0.856691i	$0.672516\pi$
$18$	−1.77329	−0.417969
$19$	3.61686	0.829765	0.414883	−	0.909875i	$-0.363823\pi$
$19$	3.61686	0.829765	0.414883	+	0.909875i	$0.363823\pi$
$20$	0	0
$21$	−1.72237	−0.375852
$22$	−2.17193	−0.463057
$23$	7.05869	1.47184	0.735919	−	0.677069i	$-0.236750\pi$
$23$	7.05869	1.47184	0.735919	+	0.677069i	$0.236750\pi$
$24$	4.15578	0.848296
$25$	0	0
$26$	0	0
$27$	4.07319	0.783885
$28$	1.16057	0.219327
$29$	2.09557	0.389137	0.194568	−	0.980889i	$-0.437669\pi$
$29$	2.09557	0.389137	0.194568	+	0.980889i	$0.437669\pi$
$30$	0	0
$31$	2.97582	0.534474	0.267237	−	0.963631i	$-0.413889\pi$
$31$	2.97582	0.534474	0.267237	+	0.963631i	$0.413889\pi$
$32$	−4.25723	−0.752579
$33$	15.0439	2.61880
$34$	1.68052	0.288207
$35$	0	0
$36$	−8.27634	−1.37939
$37$	−0.817814	−0.134448	−0.0672239	−	0.997738i	$-0.521414\pi$
$37$	−0.817814	−0.134448	−0.0672239	+	0.997738i	$0.521414\pi$
$38$	−1.42894	−0.231805
$39$	0	0
$40$	0	0
$41$	−4.89390	−0.764298	−0.382149	−	0.924101i	$-0.624816\pi$
$41$	−4.89390	−0.764298	−0.382149	+	0.924101i	$0.624816\pi$
$42$	0.680471	0.104999
$43$	−9.91411	−1.51189	−0.755944	−	0.654636i	$-0.772822\pi$
$43$	−9.91411	−1.51189	−0.755944	+	0.654636i	$0.772822\pi$
$44$	−10.1369	−1.52819
$45$	0	0
$46$	−2.78873	−0.411176
$47$	2.29494	0.334751	0.167375	−	0.985893i	$-0.446471\pi$
$47$	2.29494	0.334751	0.167375	+	0.985893i	$0.446471\pi$
$48$	8.44990	1.21964
$49$	−6.60385	−0.943407
$50$	0	0
$51$	−11.6401	−1.62994
$52$	0	0
$53$	−5.84288	−0.802581	−0.401291	−	0.915951i	$-0.631438\pi$
$53$	−5.84288	−0.802581	−0.401291	+	0.915951i	$0.631438\pi$
$54$	−1.60923	−0.218988
$55$	0	0
$56$	−0.955844	−0.127730
$57$	9.89756	1.31096
$58$	−0.827912	−0.108710
$59$	5.19393	0.676192	0.338096	−	0.941112i	$-0.390217\pi$
$59$	5.19393	0.676192	0.338096	+	0.941112i	$0.390217\pi$
$60$	0	0
$61$	5.92410	0.758503	0.379252	−	0.925294i	$-0.376181\pi$
$61$	5.92410	0.758503	0.379252	+	0.925294i	$0.376181\pi$
$62$	−1.17568	−0.149312
$63$	−2.82506	−0.355924
$64$	−4.49375	−0.561718
$65$	0	0
$66$	−5.94350	−0.731595
$67$	13.7031	1.67410	0.837052	−	0.547124i	$-0.184277\pi$
$67$	13.7031	1.67410	0.837052	+	0.547124i	$0.184277\pi$
$68$	7.84335	0.951146
$69$	19.3161	2.32539
$70$	0	0
$71$	0.210677	0.0250028	0.0125014	−	0.999922i	$-0.496021\pi$
$71$	0.210677	0.0250028	0.0125014	+	0.999922i	$0.496021\pi$
$72$	6.81639	0.803319
$73$	12.4282	1.45461	0.727306	−	0.686314i	$-0.240772\pi$
$73$	12.4282	1.45461	0.727306	+	0.686314i	$0.240772\pi$
$74$	0.323100	0.0375596
$75$	0	0
$76$	−6.66918	−0.765007
$77$	−3.46014	−0.394319
$78$	0	0
$79$	−7.19084	−0.809032	−0.404516	−	0.914531i	$-0.632560\pi$
$79$	−7.19084	−0.809032	−0.404516	+	0.914531i	$0.632560\pi$
$80$	0	0
$81$	−2.31909	−0.257677
$82$	1.93347	0.213516
$83$	6.61897	0.726526	0.363263	−	0.931687i	$-0.381663\pi$
$83$	6.61897	0.726526	0.363263	+	0.931687i	$0.381663\pi$
$84$	3.17590	0.346519
$85$	0	0
$86$	3.91685	0.422365
$87$	5.73453	0.614806
$88$	8.34872	0.889976
$89$	3.66109	0.388074	0.194037	−	0.980994i	$-0.437842\pi$
$89$	3.66109	0.388074	0.194037	+	0.980994i	$0.437842\pi$
$90$	0	0
$91$	0	0
$92$	−13.0156	−1.35697
$93$	8.14336	0.844427
$94$	−0.906679	−0.0935168
$95$	0	0
$96$	−11.6499	−1.18902
$97$	4.54596	0.461573	0.230786	−	0.973004i	$-0.425870\pi$
$97$	4.54596	0.461573	0.230786	+	0.973004i	$0.425870\pi$
$98$	2.60904	0.263552
$99$	24.6752	2.47995
$100$	0	0
$101$	10.5652	1.05128	0.525639	−	0.850708i	$-0.323826\pi$
$101$	10.5652	1.05128	0.525639	+	0.850708i	$0.323826\pi$
$102$	4.59876	0.455345
$103$	−8.11584	−0.799677	−0.399839	−	0.916586i	$-0.630934\pi$
$103$	−8.11584	−0.799677	−0.399839	+	0.916586i	$0.630934\pi$
$104$	0	0
$105$	0	0
$106$	2.30839	0.224211
$107$	−4.75614	−0.459793	−0.229897	−	0.973215i	$-0.573839\pi$
$107$	−4.75614	−0.459793	−0.229897	+	0.973215i	$0.573839\pi$
$108$	−7.51060	−0.722708
$109$	2.23043	0.213636	0.106818	−	0.994279i	$-0.465934\pi$
$109$	2.23043	0.213636	0.106818	+	0.994279i	$0.465934\pi$
$110$	0	0
$111$	−2.23795	−0.212417
$112$	−1.94350	−0.183644
$113$	−6.66337	−0.626837	−0.313419	−	0.949615i	$-0.601474\pi$
$113$	−6.66337	−0.626837	−0.313419	+	0.949615i	$0.601474\pi$
$114$	−3.91031	−0.366234
$115$	0	0
$116$	−3.86404	−0.358767
$117$	0	0
$118$	−2.05201	−0.188903
$119$	2.67726	0.245424
$120$	0	0
$121$	19.2222	1.74747
$122$	−2.34048	−0.211897
$123$	−13.3922	−1.20753
$124$	−5.48716	−0.492761
$125$	0	0
$126$	1.11612	0.0994319
$127$	10.6090	0.941398	0.470699	−	0.882294i	$-0.344002\pi$
$127$	10.6090	0.941398	0.470699	+	0.882294i	$0.344002\pi$
$128$	10.2898	0.909502
$129$	−27.1300	−2.38867
$130$	0	0
$131$	0.888454	0.0776246	0.0388123	−	0.999247i	$-0.487643\pi$
$131$	0.888454	0.0776246	0.0388123	+	0.999247i	$0.487643\pi$
$132$	−27.7396	−2.41442
$133$	−2.27647	−0.197395
$134$	−5.41380	−0.467682
$135$	0	0
$136$	−6.45978	−0.553921
$137$	−2.66255	−0.227477	−0.113738	−	0.993511i	$-0.536283\pi$
$137$	−2.66255	−0.227477	−0.113738	+	0.993511i	$0.536283\pi$
$138$	−7.63138	−0.649627
$139$	13.9094	1.17978	0.589892	−	0.807482i	$-0.299170\pi$
$139$	13.9094	1.17978	0.589892	+	0.807482i	$0.299170\pi$
$140$	0	0
$141$	6.28011	0.528880
$142$	−0.0832340	−0.00698484
$143$	0	0
$144$	13.8597	1.15497
$145$	0	0
$146$	−4.91011	−0.406364
$147$	−18.0715	−1.49051
$148$	1.50798	0.123955
$149$	11.4154	0.935186	0.467593	−	0.883944i	$-0.345121\pi$
$149$	11.4154	0.935186	0.467593	+	0.883944i	$0.345121\pi$
$150$	0	0
$151$	13.1691	1.07169	0.535843	−	0.844317i	$-0.319994\pi$
$151$	13.1691	1.07169	0.535843	+	0.844317i	$0.319994\pi$
$152$	5.49273	0.445519
$153$	−19.0923	−1.54352
$154$	1.36702	0.110158
$155$	0	0
$156$	0	0
$157$	7.24976	0.578593	0.289297	−	0.957239i	$-0.406579\pi$
$157$	7.24976	0.578593	0.289297	+	0.957239i	$0.406579\pi$
$158$	2.84094	0.226013
$159$	−15.9891	−1.26802
$160$	0	0
$161$	−4.44277	−0.350140
$162$	0.916221	0.0719851
$163$	−14.3271	−1.12218	−0.561091	−	0.827754i	$-0.689618\pi$
$163$	−14.3271	−1.12218	−0.561091	+	0.827754i	$0.689618\pi$
$164$	9.02392	0.704650
$165$	0	0
$166$	−2.61501	−0.202964
$167$	−18.1705	−1.40608	−0.703038	−	0.711152i	$-0.748174\pi$
$167$	−18.1705	−1.40608	−0.703038	+	0.711152i	$0.748174\pi$
$168$	−2.61567	−0.201803
$169$	0	0
$170$	0	0
$171$	16.2342	1.24146
$172$	18.2808	1.39390
$173$	13.3976	1.01860	0.509300	−	0.860589i	$-0.329905\pi$
$173$	13.3976	1.01860	0.509300	+	0.860589i	$0.329905\pi$
$174$	−2.26559	−0.171754
$175$	0	0
$176$	16.9753	1.27956
$177$	14.2132	1.06833
$178$	−1.44642	−0.108413
$179$	−4.87440	−0.364330	−0.182165	−	0.983268i	$-0.558310\pi$
$179$	−4.87440	−0.364330	−0.182165	+	0.983268i	$0.558310\pi$
$180$	0	0
$181$	21.6211	1.60708	0.803541	−	0.595249i	$-0.202947\pi$
$181$	21.6211	1.60708	0.803541	+	0.595249i	$0.202947\pi$
$182$	0	0
$183$	16.2113	1.19838
$184$	10.7196	0.790263
$185$	0	0
$186$	−3.21726	−0.235901
$187$	−23.3843	−1.71003
$188$	−4.23166	−0.308626
$189$	−2.56369	−0.186481
$190$	0	0
$191$	3.40566	0.246425	0.123212	−	0.992380i	$-0.460680\pi$
$191$	3.40566	0.246425	0.123212	+	0.992380i	$0.460680\pi$
$192$	−12.2972	−0.887471
$193$	−12.8482	−0.924832	−0.462416	−	0.886663i	$-0.653017\pi$
$193$	−12.8482	−0.924832	−0.462416	+	0.886663i	$0.653017\pi$
$194$	−1.79601	−0.128946
$195$	0	0
$196$	12.1769	0.869780
$197$	6.29884	0.448774	0.224387	−	0.974500i	$-0.427962\pi$
$197$	6.29884	0.448774	0.224387	+	0.974500i	$0.427962\pi$
$198$	−9.74863	−0.692805
$199$	5.91708	0.419451	0.209725	−	0.977760i	$-0.432743\pi$
$199$	5.91708	0.419451	0.209725	+	0.977760i	$0.432743\pi$
$200$	0	0
$201$	37.4987	2.64495
$202$	−4.17408	−0.293687
$203$	−1.31896	−0.0925728
$204$	21.4634	1.50274
$205$	0	0
$206$	3.20639	0.223400
$207$	31.6827	2.20210
$208$	0	0
$209$	19.8836	1.37538
$210$	0	0
$211$	10.4670	0.720578	0.360289	−	0.932841i	$-0.382678\pi$
$211$	10.4670	0.720578	0.360289	+	0.932841i	$0.382678\pi$
$212$	10.7738	0.739945
$213$	0.576520	0.0395025
$214$	1.87905	0.128449
$215$	0	0
$216$	6.18573	0.420886
$217$	−1.87300	−0.127147
$218$	−0.881193	−0.0596819
$219$	34.0099	2.29817
$220$	0	0
$221$	0	0
$222$	0.884166	0.0593413
$223$	−11.0759	−0.741696	−0.370848	−	0.928693i	$-0.620933\pi$
$223$	−11.0759	−0.741696	−0.370848	+	0.928693i	$0.620933\pi$
$224$	2.67952	0.179033
$225$	0	0
$226$	2.63255	0.175115
$227$	23.2890	1.54574	0.772872	−	0.634562i	$-0.218819\pi$
$227$	23.2890	1.54574	0.772872	+	0.634562i	$0.218819\pi$
$228$	−18.2502	−1.20865
$229$	7.19056	0.475166	0.237583	−	0.971367i	$-0.423645\pi$
$229$	7.19056	0.475166	0.237583	+	0.971367i	$0.423645\pi$
$230$	0	0
$231$	−9.46869	−0.622994
$232$	3.18242	0.208936
$233$	3.46735	0.227154	0.113577	−	0.993529i	$-0.463769\pi$
$233$	3.46735	0.227154	0.113577	+	0.993529i	$0.463769\pi$
$234$	0	0
$235$	0	0
$236$	−9.57716	−0.623420
$237$	−19.6778	−1.27821
$238$	−1.05773	−0.0685624
$239$	−18.4358	−1.19251	−0.596255	−	0.802795i	$-0.703345\pi$
$239$	−18.4358	−1.19251	−0.596255	+	0.802795i	$0.703345\pi$
$240$	0	0
$241$	−8.22700	−0.529948	−0.264974	−	0.964256i	$-0.585363\pi$
$241$	−8.22700	−0.529948	−0.264974	+	0.964256i	$0.585363\pi$
$242$	−7.59427	−0.488178
$243$	−18.5658	−1.19099
$244$	−10.9235	−0.699307
$245$	0	0
$246$	5.29096	0.337339
$247$	0	0
$248$	4.51922	0.286971
$249$	18.1129	1.14786
$250$	0	0
$251$	−29.0068	−1.83089	−0.915446	−	0.402442i	$-0.868162\pi$
$251$	−29.0068	−1.83089	−0.915446	+	0.402442i	$0.868162\pi$
$252$	5.20917	0.328147
$253$	38.8049	2.43965
$254$	−4.19139	−0.262991
$255$	0	0
$256$	4.92220	0.307638
$257$	11.3098	0.705488	0.352744	−	0.935720i	$-0.385249\pi$
$257$	11.3098	0.705488	0.352744	+	0.935720i	$0.385249\pi$
$258$	10.7185	0.667303
$259$	0.514736	0.0319841
$260$	0	0
$261$	9.40587	0.582209
$262$	−0.351009	−0.0216854
$263$	−14.9504	−0.921882	−0.460941	−	0.887431i	$-0.652488\pi$
$263$	−14.9504	−0.921882	−0.460941	+	0.887431i	$0.652488\pi$
$264$	22.8463	1.40609
$265$	0	0
$266$	0.899384	0.0551448
$267$	10.0186	0.613127
$268$	−25.2674	−1.54345
$269$	−19.1369	−1.16680	−0.583398	−	0.812187i	$-0.698277\pi$
$269$	−19.1369	−1.16680	−0.583398	+	0.812187i	$0.698277\pi$
$270$	0	0
$271$	7.95487	0.483224	0.241612	−	0.970373i	$-0.422324\pi$
$271$	7.95487	0.483224	0.241612	+	0.970373i	$0.422324\pi$
$272$	−13.1346	−0.796401
$273$	0	0
$274$	1.05191	0.0635484
$275$	0	0
$276$	−35.6173	−2.14391
$277$	32.0550	1.92600	0.963000	−	0.269502i	$-0.0868590\pi$
$277$	32.0550	1.92600	0.963000	+	0.269502i	$0.0868590\pi$
$278$	−5.49532	−0.329587
$279$	13.3569	0.799655
$280$	0	0
$281$	15.1240	0.902221	0.451110	−	0.892468i	$-0.351028\pi$
$281$	15.1240	0.902221	0.451110	+	0.892468i	$0.351028\pi$
$282$	−2.48113	−0.147749
$283$	−12.0333	−0.715306	−0.357653	−	0.933855i	$-0.616423\pi$
$283$	−12.0333	−0.715306	−0.357653	+	0.933855i	$0.616423\pi$
$284$	−0.388471	−0.0230515
$285$	0	0
$286$	0	0
$287$	3.08024	0.181821
$288$	−19.1084	−1.12597
$289$	1.09348	0.0643224
$290$	0	0
$291$	12.4401	0.729249
$292$	−22.9165	−1.34109
$293$	−23.1522	−1.35257	−0.676283	−	0.736642i	$-0.736410\pi$
$293$	−23.1522	−1.35257	−0.676283	+	0.736642i	$0.736410\pi$
$294$	7.13964	0.416392
$295$	0	0
$296$	−1.24197	−0.0721880
$297$	22.3922	1.29933
$298$	−4.50997	−0.261256
$299$	0	0
$300$	0	0
$301$	6.23999	0.359667
$302$	−5.20283	−0.299389
$303$	28.9117	1.66094
$304$	11.1683	0.640546
$305$	0	0
$306$	7.54296	0.431202
$307$	−24.6921	−1.40925	−0.704627	−	0.709578i	$-0.748886\pi$
$307$	−24.6921	−1.40925	−0.704627	+	0.709578i	$0.748886\pi$
$308$	6.38019	0.363545
$309$	−22.2090	−1.26343
$310$	0	0
$311$	−18.8313	−1.06782	−0.533912	−	0.845540i	$-0.679279\pi$
$311$	−18.8313	−1.06782	−0.533912	+	0.845540i	$0.679279\pi$
$312$	0	0
$313$	−6.10639	−0.345154	−0.172577	−	0.984996i	$-0.555209\pi$
$313$	−6.10639	−0.345154	−0.172577	+	0.984996i	$0.555209\pi$
$314$	−2.86422	−0.161637
$315$	0	0
$316$	13.2593	0.745893
$317$	8.97504	0.504088	0.252044	−	0.967716i	$-0.418897\pi$
$317$	8.97504	0.504088	0.252044	+	0.967716i	$0.418897\pi$
$318$	6.31693	0.354236
$319$	11.5203	0.645014
$320$	0	0
$321$	−13.0152	−0.726437
$322$	1.75524	0.0978159
$323$	−15.3848	−0.856035
$324$	4.27620	0.237567
$325$	0	0
$326$	5.66031	0.313495
$327$	6.10358	0.337529
$328$	−7.43210	−0.410369
$329$	−1.44445	−0.0796348
$330$	0	0
$331$	7.89045	0.433698	0.216849	−	0.976205i	$-0.430422\pi$
$331$	7.89045	0.433698	0.216849	+	0.976205i	$0.430422\pi$
$332$	−12.2048	−0.669826
$333$	−3.67073	−0.201155
$334$	7.17877	0.392805
$335$	0	0
$336$	−5.31841	−0.290143
$337$	0.643374	0.0350468	0.0175234	−	0.999846i	$-0.494422\pi$
$337$	0.643374	0.0350468	0.0175234	+	0.999846i	$0.494422\pi$
$338$	0	0
$339$	−18.2344	−0.990354
$340$	0	0
$341$	16.3595	0.885917
$342$	−6.41376	−0.346816
$343$	8.56233	0.462323
$344$	−15.0560	−0.811767
$345$	0	0
$346$	−5.29309	−0.284558
$347$	−14.2788	−0.766527	−0.383264	−	0.923639i	$-0.625200\pi$
$347$	−14.2788	−0.766527	−0.383264	+	0.923639i	$0.625200\pi$
$348$	−10.5740	−0.566824
$349$	−22.5207	−1.20551	−0.602753	−	0.797927i	$-0.705930\pi$
$349$	−22.5207	−1.20551	−0.602753	+	0.797927i	$0.705930\pi$
$350$	0	0
$351$	0	0
$352$	−23.4040	−1.24744
$353$	1.49751	0.0797047	0.0398523	−	0.999206i	$-0.487311\pi$
$353$	1.49751	0.0797047	0.0398523	+	0.999206i	$0.487311\pi$
$354$	−5.61533	−0.298452
$355$	0	0
$356$	−6.75073	−0.357788
$357$	7.32635	0.387752
$358$	1.92577	0.101780
$359$	31.3295	1.65351	0.826753	−	0.562566i	$-0.190186\pi$
$359$	31.3295	1.65351	0.826753	+	0.562566i	$0.190186\pi$
$360$	0	0
$361$	−5.91831	−0.311490
$362$	−8.54202	−0.448959
$363$	52.6017	2.76087
$364$	0	0
$365$	0	0
$366$	−6.40474	−0.334781
$367$	−8.02558	−0.418932	−0.209466	−	0.977816i	$-0.567173\pi$
$367$	−8.02558	−0.418932	−0.209466	+	0.977816i	$0.567173\pi$
$368$	21.7961	1.13620
$369$	−21.9661	−1.14351
$370$	0	0
$371$	3.67754	0.190928
$372$	−15.0156	−0.778525
$373$	26.3733	1.36556	0.682778	−	0.730626i	$-0.260772\pi$
$373$	26.3733	1.36556	0.682778	+	0.730626i	$0.260772\pi$
$374$	9.23862	0.477718
$375$	0	0
$376$	3.48520	0.179735
$377$	0	0
$378$	1.01286	0.0520957
$379$	−28.5043	−1.46417	−0.732085	−	0.681214i	$-0.761453\pi$
$379$	−28.5043	−1.46417	−0.732085	+	0.681214i	$0.761453\pi$
$380$	0	0
$381$	29.0316	1.48734
$382$	−1.34550	−0.0688418
$383$	−14.8547	−0.759038	−0.379519	−	0.925184i	$-0.623911\pi$
$383$	−14.8547	−0.759038	−0.379519	+	0.925184i	$0.623911\pi$
$384$	28.1582	1.43694
$385$	0	0
$386$	5.07603	0.258363
$387$	−44.4991	−2.26202
$388$	−8.38236	−0.425550
$389$	−21.1231	−1.07098	−0.535491	−	0.844541i	$-0.679873\pi$
$389$	−21.1231	−1.07098	−0.535491	+	0.844541i	$0.679873\pi$
$390$	0	0
$391$	−30.0251	−1.51844
$392$	−10.0289	−0.506536
$393$	2.43126	0.122641
$394$	−2.48854	−0.125371
$395$	0	0
$396$	−45.4989	−2.28641
$397$	−23.0726	−1.15798	−0.578991	−	0.815334i	$-0.696553\pi$
$397$	−23.0726	−1.15798	−0.578991	+	0.815334i	$0.696553\pi$
$398$	−2.33771	−0.117179
$399$	−6.22958	−0.311869
$400$	0	0
$401$	−15.4998	−0.774021	−0.387011	−	0.922075i	$-0.626492\pi$
$401$	−15.4998	−0.774021	−0.387011	+	0.922075i	$0.626492\pi$
$402$	−14.8149	−0.738900
$403$	0	0
$404$	−19.4813	−0.969232
$405$	0	0
$406$	0.521092	0.0258614
$407$	−4.49591	−0.222854
$408$	−17.6772	−0.875153
$409$	35.7519	1.76782	0.883908	−	0.467661i	$-0.154903\pi$
$409$	35.7519	1.76782	0.883908	+	0.467661i	$0.154903\pi$
$410$	0	0
$411$	−7.28607	−0.359395
$412$	14.9649	0.737268
$413$	−3.26909	−0.160861
$414$	−12.5171	−0.615183
$415$	0	0
$416$	0	0
$417$	38.0633	1.86397
$418$	−7.85558	−0.384229
$419$	28.9110	1.41239	0.706196	−	0.708016i	$-0.250410\pi$
$419$	28.9110	1.41239	0.706196	+	0.708016i	$0.250410\pi$
$420$	0	0
$421$	−20.1416	−0.981640	−0.490820	−	0.871261i	$-0.663303\pi$
$421$	−20.1416	−0.981640	−0.490820	+	0.871261i	$0.663303\pi$
$422$	−4.13528	−0.201302
$423$	10.3007	0.500839
$424$	−8.87327	−0.430924
$425$	0	0
$426$	−0.227770	−0.0110355
$427$	−3.72866	−0.180442
$428$	8.76990	0.423909
$429$	0	0
$430$	0	0
$431$	28.3861	1.36731	0.683656	−	0.729805i	$-0.260389\pi$
$431$	28.3861	1.36731	0.683656	+	0.729805i	$0.260389\pi$
$432$	12.5774	0.605129
$433$	−4.76509	−0.228996	−0.114498	−	0.993423i	$-0.536526\pi$
$433$	−4.76509	−0.228996	−0.114498	+	0.993423i	$0.536526\pi$
$434$	0.739981	0.0355202
$435$	0	0
$436$	−4.11271	−0.196963
$437$	25.5303	1.22128
$438$	−13.4365	−0.642023
$439$	16.0627	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$439$	16.0627	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$440$	0	0
$441$	−29.6411	−1.41148
$442$	0	0
$443$	−5.76501	−0.273904	−0.136952	−	0.990578i	$-0.543731\pi$
$443$	−5.76501	−0.273904	−0.136952	+	0.990578i	$0.543731\pi$
$444$	4.12659	0.195839
$445$	0	0
$446$	4.37584	0.207202
$447$	31.2383	1.47752
$448$	2.82839	0.133629
$449$	−16.4687	−0.777208	−0.388604	−	0.921405i	$-0.627043\pi$
$449$	−16.4687	−0.777208	−0.388604	+	0.921405i	$0.627043\pi$
$450$	0	0
$451$	−26.9041	−1.26686
$452$	12.2867	0.577917
$453$	36.0373	1.69318
$454$	−9.20097	−0.431823
$455$	0	0
$456$	15.0309	0.703886
$457$	22.3227	1.04421	0.522106	−	0.852880i	$-0.325146\pi$
$457$	22.3227	1.04421	0.522106	+	0.852880i	$0.325146\pi$
$458$	−2.84083	−0.132743
$459$	−17.3259	−0.808703
$460$	0	0
$461$	−11.6864	−0.544289	−0.272145	−	0.962256i	$-0.587733\pi$
$461$	−11.6864	−0.544289	−0.272145	+	0.962256i	$0.587733\pi$
$462$	3.74087	0.174041
$463$	−13.7716	−0.640020	−0.320010	−	0.947414i	$-0.603686\pi$
$463$	−13.7716	−0.640020	−0.320010	+	0.947414i	$0.603686\pi$
$464$	6.47078	0.300398
$465$	0	0
$466$	−1.36987	−0.0634582
$467$	−9.80830	−0.453874	−0.226937	−	0.973909i	$-0.572871\pi$
$467$	−9.80830	−0.453874	−0.226937	+	0.973909i	$0.572871\pi$
$468$	0	0
$469$	−8.62482	−0.398257
$470$	0	0
$471$	19.8390	0.914133
$472$	7.88774	0.363063
$473$	−54.5026	−2.50603
$474$	7.77425	0.357083
$475$	0	0
$476$	−4.93664	−0.226271
$477$	−26.2255	−1.20079
$478$	7.28357	0.333143
$479$	4.30152	0.196541	0.0982707	−	0.995160i	$-0.468669\pi$
$479$	4.30152	0.196541	0.0982707	+	0.995160i	$0.468669\pi$
$480$	0	0
$481$	0	0
$482$	3.25031	0.148048
$483$	−12.1577	−0.553193
$484$	−35.4441	−1.61109
$485$	0	0
$486$	7.33493	0.332719
$487$	−32.0389	−1.45182	−0.725910	−	0.687790i	$-0.758581\pi$
$487$	−32.0389	−1.45182	−0.725910	+	0.687790i	$0.758581\pi$
$488$	8.99661	0.407257
$489$	−39.2061	−1.77296
$490$	0	0
$491$	16.3710	0.738812	0.369406	−	0.929268i	$-0.379561\pi$
$491$	16.3710	0.738812	0.369406	+	0.929268i	$0.379561\pi$
$492$	24.6940	1.11329
$493$	−8.91379	−0.401457
$494$	0	0
$495$	0	0
$496$	9.18887	0.412593
$497$	−0.132601	−0.00594799
$498$	−7.15599	−0.320668
$499$	13.4370	0.601523	0.300761	−	0.953699i	$-0.402759\pi$
$499$	13.4370	0.601523	0.300761	+	0.953699i	$0.402759\pi$
$500$	0	0
$501$	−49.7237	−2.22149
$502$	11.4599	0.511482
$503$	−12.8188	−0.571561	−0.285781	−	0.958295i	$-0.592253\pi$
$503$	−12.8188	−0.571561	−0.285781	+	0.958295i	$0.592253\pi$
$504$	−4.29027	−0.191104
$505$	0	0
$506$	−15.3310	−0.681545
$507$	0	0
$508$	−19.5621	−0.867928
$509$	4.55682	0.201978	0.100989	−	0.994888i	$-0.467799\pi$
$509$	4.55682	0.201978	0.100989	+	0.994888i	$0.467799\pi$
$510$	0	0
$511$	−7.82238	−0.346042
$512$	−22.5243	−0.995445
$513$	14.7322	0.650441
$514$	−4.46827	−0.197087
$515$	0	0
$516$	50.0254	2.20225
$517$	12.6164	0.554867
$518$	−0.203361	−0.00893517
$519$	36.6626	1.60931
$520$	0	0
$521$	−24.5910	−1.07735	−0.538675	−	0.842513i	$-0.681075\pi$
$521$	−24.5910	−1.07735	−0.538675	+	0.842513i	$0.681075\pi$
$522$	−3.71605	−0.162647
$523$	−0.369116	−0.0161403	−0.00807015	−	0.999967i	$-0.502569\pi$
$523$	−0.369116	−0.0161403	−0.00807015	+	0.999967i	$0.502569\pi$
$524$	−1.63823	−0.0715665
$525$	0	0
$526$	5.90658	0.257539
$527$	−12.6581	−0.551395
$528$	46.4531	2.02161
$529$	26.8251	1.16631
$530$	0	0
$531$	23.3128	1.01169
$532$	4.19762	0.181990
$533$	0	0
$534$	−3.95812	−0.171285
$535$	0	0
$536$	20.8102	0.898864
$537$	−13.3388	−0.575612
$538$	7.56056	0.325959
$539$	−36.3045	−1.56374
$540$	0	0
$541$	−34.1173	−1.46682	−0.733408	−	0.679789i	$-0.762071\pi$
$541$	−34.1173	−1.46682	−0.733408	+	0.679789i	$0.762071\pi$
$542$	−3.14280	−0.134995
$543$	59.1662	2.53907
$544$	18.1087	0.776406
$545$	0	0
$546$	0	0
$547$	−13.8903	−0.593908	−0.296954	−	0.954892i	$-0.595971\pi$
$547$	−13.8903	−0.593908	−0.296954	+	0.954892i	$0.595971\pi$
$548$	4.90950	0.209724
$549$	26.5901	1.13484
$550$	0	0
$551$	7.57937	0.322892
$552$	29.3344	1.24855
$553$	4.52595	0.192463
$554$	−12.6642	−0.538052
$555$	0	0
$556$	−25.6478	−1.08771
$557$	−30.5369	−1.29389	−0.646945	−	0.762537i	$-0.723954\pi$
$557$	−30.5369	−1.29389	−0.646945	+	0.762537i	$0.723954\pi$
$558$	−5.27701	−0.223394
$559$	0	0
$560$	0	0
$561$	−63.9912	−2.70171
$562$	−5.97515	−0.252047
$563$	−35.3205	−1.48858	−0.744292	−	0.667855i	$-0.767213\pi$
$563$	−35.3205	−1.48858	−0.744292	+	0.667855i	$0.767213\pi$
$564$	−11.5800	−0.487605
$565$	0	0
$566$	4.75410	0.199830
$567$	1.45965	0.0612994
$568$	0.319944	0.0134246
$569$	34.7921	1.45856	0.729280	−	0.684215i	$-0.239855\pi$
$569$	34.7921	1.45856	0.729280	+	0.684215i	$0.239855\pi$
$570$	0	0
$571$	0.332839	0.0139289	0.00696445	−	0.999976i	$-0.497783\pi$
$571$	0.332839	0.0139289	0.00696445	+	0.999976i	$0.497783\pi$
$572$	0	0
$573$	9.31960	0.389332
$574$	−1.21694	−0.0507940
$575$	0	0
$576$	−20.1700	−0.840417
$577$	−24.1460	−1.00521	−0.502605	−	0.864516i	$-0.667625\pi$
$577$	−24.1460	−1.00521	−0.502605	+	0.864516i	$0.667625\pi$
$578$	−0.432010	−0.0179693
$579$	−35.1591	−1.46116
$580$	0	0
$581$	−4.16601	−0.172835
$582$	−4.91479	−0.203725
$583$	−32.1211	−1.33032
$584$	18.8740	0.781013
$585$	0	0
$586$	9.14693	0.377856
$587$	12.9251	0.533476	0.266738	−	0.963769i	$-0.414054\pi$
$587$	12.9251	0.533476	0.266738	+	0.963769i	$0.414054\pi$
$588$	33.3222	1.37419
$589$	10.7631	0.443488
$590$	0	0
$591$	17.2368	0.709028
$592$	−2.52528	−0.103788
$593$	23.5928	0.968840	0.484420	−	0.874836i	$-0.339031\pi$
$593$	23.5928	0.968840	0.484420	+	0.874836i	$0.339031\pi$
$594$	−8.84668	−0.362984
$595$	0	0
$596$	−21.0490	−0.862201
$597$	16.1921	0.662699
$598$	0	0
$599$	46.4577	1.89821	0.949105	−	0.314961i	$-0.101991\pi$
$599$	46.4577	1.89821	0.949105	+	0.314961i	$0.101991\pi$
$600$	0	0
$601$	−37.0915	−1.51300	−0.756498	−	0.653996i	$-0.773091\pi$
$601$	−37.0915	−1.51300	−0.756498	+	0.653996i	$0.773091\pi$
$602$	−2.46528	−0.100478
$603$	61.5060	2.50472
$604$	−24.2827	−0.988049
$605$	0	0
$606$	−11.4224	−0.464003
$607$	−20.2943	−0.823722	−0.411861	−	0.911247i	$-0.635121\pi$
$607$	−20.2943	−0.823722	−0.411861	+	0.911247i	$0.635121\pi$
$608$	−15.3978	−0.624464
$609$	−3.60934	−0.146258
$610$	0	0
$611$	0	0
$612$	35.2046	1.42306
$613$	41.3558	1.67035	0.835173	−	0.549987i	$-0.185367\pi$
$613$	41.3558	1.67035	0.835173	+	0.549987i	$0.185367\pi$
$614$	9.75532	0.393693
$615$	0	0
$616$	−5.25472	−0.211719
$617$	−38.5909	−1.55361	−0.776806	−	0.629739i	$-0.783162\pi$
$617$	−38.5909	−1.55361	−0.776806	+	0.629739i	$0.783162\pi$
$618$	8.77430	0.352954
$619$	−13.0016	−0.522579	−0.261289	−	0.965261i	$-0.584148\pi$
$619$	−13.0016	−0.522579	−0.261289	+	0.965261i	$0.584148\pi$
$620$	0	0
$621$	28.7514	1.15375
$622$	7.43983	0.298310
$623$	−2.30431	−0.0923201
$624$	0	0
$625$	0	0
$626$	2.41250	0.0964229
$627$	54.4116	2.17299
$628$	−13.3679	−0.533438
$629$	3.47869	0.138704
$630$	0	0
$631$	21.2897	0.847531	0.423765	−	0.905772i	$-0.360708\pi$
$631$	21.2897	0.847531	0.423765	+	0.905772i	$0.360708\pi$
$632$	−10.9203	−0.434388
$633$	28.6430	1.13846
$634$	−3.54584	−0.140823
$635$	0	0
$636$	29.4825	1.16906
$637$	0	0
$638$	−4.55142	−0.180193
$639$	0.945617	0.0374080
$640$	0	0
$641$	−24.9061	−0.983732	−0.491866	−	0.870671i	$-0.663685\pi$
$641$	−24.9061	−0.983732	−0.491866	+	0.870671i	$0.663685\pi$
$642$	5.14202	0.202939
$643$	−32.6396	−1.28718	−0.643590	−	0.765370i	$-0.722556\pi$
$643$	−32.6396	−1.28718	−0.643590	+	0.765370i	$0.722556\pi$
$644$	8.19209	0.322814
$645$	0	0
$646$	6.07821	0.239144
$647$	20.8586	0.820037	0.410019	−	0.912077i	$-0.365522\pi$
$647$	20.8586	0.820037	0.410019	+	0.912077i	$0.365522\pi$
$648$	−3.52188	−0.138352
$649$	28.5535	1.12082
$650$	0	0
$651$	−5.12547	−0.200883
$652$	26.4179	1.03460
$653$	30.6745	1.20038	0.600192	−	0.799856i	$-0.295091\pi$
$653$	30.6745	1.20038	0.600192	+	0.799856i	$0.295091\pi$
$654$	−2.41139	−0.0942928
$655$	0	0
$656$	−15.1116	−0.590008
$657$	55.7835	2.17632
$658$	0.570669	0.0222470
$659$	−13.5866	−0.529259	−0.264630	−	0.964350i	$-0.585250\pi$
$659$	−13.5866	−0.529259	−0.264630	+	0.964350i	$0.585250\pi$
$660$	0	0
$661$	49.3974	1.92134	0.960668	−	0.277701i	$-0.0895724\pi$
$661$	49.3974	1.92134	0.960668	+	0.277701i	$0.0895724\pi$
$662$	−3.11734	−0.121159
$663$	0	0
$664$	10.0519	0.390088
$665$	0	0
$666$	1.45022	0.0561950
$667$	14.7919	0.572746
$668$	33.5048	1.29634
$669$	−30.3092	−1.17182
$670$	0	0
$671$	32.5676	1.25726
$672$	7.33253	0.282859
$673$	−34.5275	−1.33094	−0.665469	−	0.746426i	$-0.731768\pi$
$673$	−34.5275	−1.33094	−0.665469	+	0.746426i	$0.731768\pi$
$674$	−0.254183	−0.00979077
$675$	0	0
$676$	0	0
$677$	−37.0871	−1.42537	−0.712686	−	0.701483i	$-0.752522\pi$
$677$	−37.0871	−1.42537	−0.712686	+	0.701483i	$0.752522\pi$
$678$	7.20400	0.276668
$679$	−2.86125	−0.109805
$680$	0	0
$681$	63.7304	2.44216
$682$	−6.46328	−0.247492
$683$	23.2145	0.888278	0.444139	−	0.895958i	$-0.353510\pi$
$683$	23.2145	0.888278	0.444139	+	0.895958i	$0.353510\pi$
$684$	−29.9344	−1.14457
$685$	0	0
$686$	−3.38279	−0.129156
$687$	19.6770	0.750725
$688$	−30.6132	−1.16712
$689$	0	0
$690$	0	0
$691$	−45.0405	−1.71342	−0.856710	−	0.515798i	$-0.827496\pi$
$691$	−45.0405	−1.71342	−0.856710	+	0.515798i	$0.827496\pi$
$692$	−24.7040	−0.939104
$693$	−15.5307	−0.589962
$694$	5.64125	0.214139
$695$	0	0
$696$	8.70872	0.330103
$697$	20.8169	0.788496
$698$	8.89745	0.336773
$699$	9.48842	0.358885
$700$	0	0
$701$	6.01190	0.227066	0.113533	−	0.993534i	$-0.463783\pi$
$701$	6.01190	0.227066	0.113533	+	0.993534i	$0.463783\pi$
$702$	0	0
$703$	−2.95792	−0.111560
$704$	−24.7042	−0.931076
$705$	0	0
$706$	−0.591635	−0.0222665
$707$	−6.64979	−0.250091
$708$	−26.2080	−0.984955
$709$	1.93887	0.0728159	0.0364080	−	0.999337i	$-0.488408\pi$
$709$	1.93887	0.0728159	0.0364080	+	0.999337i	$0.488408\pi$
$710$	0	0
$711$	−32.2758	−1.21044
$712$	5.55989	0.208366
$713$	21.0054	0.786659
$714$	−2.89448	−0.108323
$715$	0	0
$716$	8.98797	0.335896
$717$	−50.4496	−1.88407
$718$	−12.3776	−0.461927
$719$	−32.3056	−1.20479	−0.602397	−	0.798197i	$-0.705788\pi$
$719$	−32.3056	−1.20479	−0.602397	+	0.798197i	$0.705788\pi$
$720$	0	0
$721$	5.10815	0.190237
$722$	2.33819	0.0870186
$723$	−22.5132	−0.837276
$724$	−39.8674	−1.48166
$725$	0	0
$726$	−20.7818	−0.771284
$727$	−13.3705	−0.495883	−0.247941	−	0.968775i	$-0.579754\pi$
$727$	−13.3705	−0.495883	−0.247941	+	0.968775i	$0.579754\pi$
$728$	0	0
$729$	−43.8480	−1.62400
$730$	0	0
$731$	42.1711	1.55975
$732$	−29.8923	−1.10485
$733$	−29.5619	−1.09189	−0.545946	−	0.837820i	$-0.683830\pi$
$733$	−29.5619	−1.09189	−0.545946	+	0.837820i	$0.683830\pi$
$734$	3.17073	0.117034
$735$	0	0
$736$	−30.0505	−1.10767
$737$	75.3326	2.77491
$738$	8.67832	0.319453
$739$	8.79626	0.323576	0.161788	−	0.986826i	$-0.448274\pi$
$739$	8.79626	0.323576	0.161788	+	0.986826i	$0.448274\pi$
$740$	0	0
$741$	0	0
$742$	−1.45291	−0.0533382
$743$	19.0280	0.698069	0.349035	−	0.937110i	$-0.386510\pi$
$743$	19.0280	0.698069	0.349035	+	0.937110i	$0.386510\pi$
$744$	12.3669	0.453392
$745$	0	0
$746$	−10.4195	−0.381485
$747$	29.7090	1.08700
$748$	43.1186	1.57657
$749$	2.99354	0.109381
$750$	0	0
$751$	12.7174	0.464066	0.232033	−	0.972708i	$-0.425462\pi$
$751$	12.7174	0.464066	0.232033	+	0.972708i	$0.425462\pi$
$752$	7.08640	0.258415
$753$	−79.3772	−2.89267
$754$	0	0
$755$	0	0
$756$	4.72721	0.171927
$757$	−2.74724	−0.0998501	−0.0499251	−	0.998753i	$-0.515898\pi$
$757$	−2.74724	−0.0998501	−0.0499251	+	0.998753i	$0.515898\pi$
$758$	11.2614	0.409034
$759$	106.190	3.85445
$760$	0	0
$761$	29.8209	1.08101	0.540503	−	0.841342i	$-0.318234\pi$
$761$	29.8209	1.08101	0.540503	+	0.841342i	$0.318234\pi$
$762$	−11.4698	−0.415506
$763$	−1.40384	−0.0508225
$764$	−6.27973	−0.227193
$765$	0	0
$766$	5.86876	0.212047
$767$	0	0
$768$	13.4696	0.486043
$769$	−6.64765	−0.239720	−0.119860	−	0.992791i	$-0.538245\pi$
$769$	−6.64765	−0.239720	−0.119860	+	0.992791i	$0.538245\pi$
$770$	0	0
$771$	30.9494	1.11462
$772$	23.6909	0.852655
$773$	−29.8954	−1.07526	−0.537631	−	0.843180i	$-0.680681\pi$
$773$	−29.8954	−1.07526	−0.537631	+	0.843180i	$0.680681\pi$
$774$	17.5806	0.631923
$775$	0	0
$776$	6.90371	0.247829
$777$	1.40858	0.0505325
$778$	8.34526	0.299192
$779$	−17.7006	−0.634188
$780$	0	0
$781$	1.15819	0.0414434
$782$	11.8623	0.424194
$783$	8.53563	0.305039
$784$	−20.3916	−0.728273
$785$	0	0
$786$	−0.960538	−0.0342613
$787$	42.7877	1.52522	0.762608	−	0.646861i	$-0.223919\pi$
$787$	42.7877	1.52522	0.762608	+	0.646861i	$0.223919\pi$
$788$	−11.6145	−0.413750
$789$	−40.9119	−1.45650
$790$	0	0
$791$	4.19396	0.149120
$792$	37.4729	1.33154
$793$	0	0
$794$	9.11549	0.323497
$795$	0	0
$796$	−10.9106	−0.386715
$797$	30.7061	1.08766	0.543832	−	0.839194i	$-0.316973\pi$
$797$	30.7061	1.08766	0.543832	+	0.839194i	$0.316973\pi$
$798$	2.46117	0.0871245
$799$	−9.76184	−0.345349
$800$	0	0
$801$	16.4327	0.580619
$802$	6.12362	0.216233
$803$	68.3237	2.41109
$804$	−69.1443	−2.43853
$805$	0	0
$806$	0	0
$807$	−52.3681	−1.84345
$808$	16.0448	0.564454
$809$	−26.6310	−0.936296	−0.468148	−	0.883650i	$-0.655079\pi$
$809$	−26.6310	−0.936296	−0.468148	+	0.883650i	$0.655079\pi$
$810$	0	0
$811$	−24.8696	−0.873290	−0.436645	−	0.899634i	$-0.643833\pi$
$811$	−24.8696	−0.873290	−0.436645	+	0.899634i	$0.643833\pi$
$812$	2.43205	0.0853481
$813$	21.7686	0.763457
$814$	1.77624	0.0622570
$815$	0	0
$816$	−35.9429	−1.25825
$817$	−35.8580	−1.25451
$818$	−14.1248	−0.493861
$819$	0	0
$820$	0	0
$821$	44.0239	1.53644	0.768222	−	0.640184i	$-0.221142\pi$
$821$	44.0239	1.53644	0.768222	+	0.640184i	$0.221142\pi$
$822$	2.87857	0.100402
$823$	−27.4490	−0.956812	−0.478406	−	0.878139i	$-0.658785\pi$
$823$	−27.4490	−0.956812	−0.478406	+	0.878139i	$0.658785\pi$
$824$	−12.3251	−0.429365
$825$	0	0
$826$	1.29154	0.0449386
$827$	−15.8828	−0.552299	−0.276150	−	0.961115i	$-0.589059\pi$
$827$	−15.8828	−0.552299	−0.276150	+	0.961115i	$0.589059\pi$
$828$	−58.4201	−2.03024
$829$	23.2845	0.808704	0.404352	−	0.914603i	$-0.367497\pi$
$829$	23.2845	0.808704	0.404352	+	0.914603i	$0.367497\pi$
$830$	0	0
$831$	87.7188	3.04293
$832$	0	0
$833$	28.0904	0.973275
$834$	−15.0380	−0.520722
$835$	0	0
$836$	−36.6636	−1.26804
$837$	12.1211	0.418966
$838$	−11.4221	−0.394569
$839$	20.4521	0.706084	0.353042	−	0.935607i	$-0.385147\pi$
$839$	20.4521	0.706084	0.353042	+	0.935607i	$0.385147\pi$
$840$	0	0
$841$	−24.6086	−0.848573
$842$	7.95749	0.274233
$843$	41.3868	1.42544
$844$	−19.3002	−0.664342
$845$	0	0
$846$	−4.06960	−0.139916
$847$	−12.0986	−0.415711
$848$	−18.0419	−0.619561
$849$	−32.9292	−1.13013
$850$	0	0
$851$	−5.77269	−0.197885
$852$	−1.06305	−0.0364196
$853$	38.9677	1.33423	0.667114	−	0.744956i	$-0.267530\pi$
$853$	38.9677	1.33423	0.667114	+	0.744956i	$0.267530\pi$
$854$	1.47311	0.0504088
$855$	0	0
$856$	−7.22289	−0.246873
$857$	−20.6596	−0.705719	−0.352860	−	0.935676i	$-0.614791\pi$
$857$	−20.6596	−0.705719	−0.352860	+	0.935676i	$0.614791\pi$
$858$	0	0
$859$	−7.07452	−0.241380	−0.120690	−	0.992690i	$-0.538511\pi$
$859$	−7.07452	−0.241380	−0.120690	+	0.992690i	$0.538511\pi$
$860$	0	0
$861$	8.42911	0.287263
$862$	−11.2147	−0.381975
$863$	28.3108	0.963710	0.481855	−	0.876251i	$-0.339963\pi$
$863$	28.3108	0.963710	0.481855	+	0.876251i	$0.339963\pi$
$864$	−17.3405	−0.589936
$865$	0	0
$866$	1.88258	0.0639728
$867$	2.99232	0.101624
$868$	3.45365	0.117224
$869$	−39.5314	−1.34101
$870$	0	0
$871$	0	0
$872$	3.38723	0.114706
$873$	20.4044	0.690584
$874$	−10.0865	−0.341180
$875$	0	0
$876$	−62.7112	−2.11882
$877$	47.3468	1.59879	0.799395	−	0.600806i	$-0.205154\pi$
$877$	47.3468	1.59879	0.799395	+	0.600806i	$0.205154\pi$
$878$	−6.34601	−0.214168
$879$	−63.3562	−2.13695
$880$	0	0
$881$	34.9584	1.17778	0.588890	−	0.808213i	$-0.299565\pi$
$881$	34.9584	1.17778	0.588890	+	0.808213i	$0.299565\pi$
$882$	11.7106	0.394315
$883$	18.1140	0.609583	0.304792	−	0.952419i	$-0.401413\pi$
$883$	18.1140	0.609583	0.304792	+	0.952419i	$0.401413\pi$
$884$	0	0
$885$	0	0
$886$	2.27763	0.0765184
$887$	6.12777	0.205751	0.102875	−	0.994694i	$-0.467196\pi$
$887$	6.12777	0.205751	0.102875	+	0.994694i	$0.467196\pi$
$888$	−3.39866	−0.114051
$889$	−6.67737	−0.223952
$890$	0	0
$891$	−12.7491	−0.427112
$892$	20.4230	0.683812
$893$	8.30047	0.277765
$894$	−12.3416	−0.412764
$895$	0	0
$896$	−6.47648	−0.216364
$897$	0	0
$898$	6.50644	0.217123
$899$	6.23603	0.207983
$900$	0	0
$901$	24.8535	0.827991
$902$	10.6292	0.353914
$903$	17.0758	0.568246
$904$	−10.1193	−0.336563
$905$	0	0
$906$	−14.2376	−0.473011
$907$	11.3755	0.377717	0.188858	−	0.982004i	$-0.439521\pi$
$907$	11.3755	0.377717	0.188858	+	0.982004i	$0.439521\pi$
$908$	−42.9429	−1.42511
$909$	47.4215	1.57287
$910$	0	0
$911$	−1.21214	−0.0401600	−0.0200800	−	0.999798i	$-0.506392\pi$
$911$	−1.21214	−0.0401600	−0.0200800	+	0.999798i	$0.506392\pi$
$912$	30.5621	1.01201
$913$	36.3876	1.20425
$914$	−8.81922	−0.291714
$915$	0	0
$916$	−13.2588	−0.438082
$917$	−0.559198	−0.0184663
$918$	6.84508	0.225921
$919$	−44.7911	−1.47752	−0.738761	−	0.673968i	$-0.764589\pi$
$919$	−44.7911	−1.47752	−0.738761	+	0.673968i	$0.764589\pi$
$920$	0	0
$921$	−67.5701	−2.22651
$922$	4.61704	0.152054
$923$	0	0
$924$	17.4594	0.574373
$925$	0	0
$926$	5.44085	0.178798
$927$	−36.4276	−1.19644
$928$	−8.92131	−0.292856
$929$	1.59021	0.0521731	0.0260865	−	0.999660i	$-0.491695\pi$
$929$	1.59021	0.0521731	0.0260865	+	0.999660i	$0.491695\pi$
$930$	0	0
$931$	−23.8852	−0.782806
$932$	−6.39349	−0.209426
$933$	−51.5320	−1.68708
$934$	3.87505	0.126795
$935$	0	0
$936$	0	0
$937$	13.6224	0.445025	0.222513	−	0.974930i	$-0.428574\pi$
$937$	13.6224	0.445025	0.222513	+	0.974930i	$0.428574\pi$
$938$	3.40748	0.111258
$939$	−16.7102	−0.545316
$940$	0	0
$941$	46.6484	1.52069	0.760346	−	0.649518i	$-0.225029\pi$
$941$	46.6484	1.52069	0.760346	+	0.649518i	$0.225029\pi$
$942$	−7.83795	−0.255374
$943$	−34.5445	−1.12492
$944$	16.0380	0.521994
$945$	0	0
$946$	21.5328	0.700091
$947$	−0.787397	−0.0255870	−0.0127935	−	0.999918i	$-0.504072\pi$
$947$	−0.787397	−0.0255870	−0.0127935	+	0.999918i	$0.504072\pi$
$948$	36.2841	1.17845
$949$	0	0
$950$	0	0
$951$	24.5602	0.796420
$952$	4.06582	0.131774
$953$	13.7138	0.444232	0.222116	−	0.975020i	$-0.428704\pi$
$953$	13.7138	0.444232	0.222116	+	0.975020i	$0.428704\pi$
$954$	10.3611	0.335454
$955$	0	0
$956$	33.9939	1.09944
$957$	31.5254	1.01907
$958$	−1.69944	−0.0549063
$959$	1.67582	0.0541150
$960$	0	0
$961$	−22.1445	−0.714338
$962$	0	0
$963$	−21.3477	−0.687922
$964$	15.1699	0.488589
$965$	0	0
$966$	4.80323	0.154541
$967$	−29.1127	−0.936202	−0.468101	−	0.883675i	$-0.655062\pi$
$967$	−29.1127	−0.936202	−0.468101	+	0.883675i	$0.655062\pi$
$968$	29.1917	0.938258
$969$	−42.1007	−1.35247
$970$	0	0
$971$	3.89669	0.125051	0.0625253	−	0.998043i	$-0.480085\pi$
$971$	3.89669	0.125051	0.0625253	+	0.998043i	$0.480085\pi$
$972$	34.2337	1.09804
$973$	−8.75468	−0.280662
$974$	12.6579	0.405584
$975$	0	0
$976$	18.2927	0.585535
$977$	−42.5418	−1.36103	−0.680516	−	0.732733i	$-0.738244\pi$
$977$	−42.5418	−1.36103	−0.680516	+	0.732733i	$0.738244\pi$
$978$	15.4895	0.495299
$979$	20.1267	0.643253
$980$	0	0
$981$	10.0112	0.319633
$982$	−6.46782	−0.206396
$983$	−21.7356	−0.693257	−0.346629	−	0.938002i	$-0.612674\pi$
$983$	−21.7356	−0.693257	−0.346629	+	0.938002i	$0.612674\pi$
$984$	−20.3380	−0.648351
$985$	0	0
$986$	3.52164	0.112152
$987$	−3.95273	−0.125817
$988$	0	0
$989$	−69.9806	−2.22525
$990$	0	0
$991$	−46.0574	−1.46306	−0.731530	−	0.681809i	$-0.761194\pi$
$991$	−46.0574	−1.46306	−0.731530	+	0.681809i	$0.761194\pi$
$992$	−12.6688	−0.402234
$993$	21.5923	0.685210
$994$	0.0523879	0.00166164
$995$	0	0
$996$	−33.3985	−1.05827
$997$	−40.8120	−1.29253	−0.646265	−	0.763113i	$-0.723670\pi$
$997$	−40.8120	−1.29253	−0.646265	+	0.763113i	$0.723670\pi$
$998$	−5.30867	−0.168043
$999$	−3.33111	−0.105392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.cb.1.8		18
5.2	odd	4		845.2.b.h.339.8	yes	18
5.3	odd	4		845.2.b.h.339.11	yes	18
5.4	even	2	inner	4225.2.a.cb.1.11		18
13.12	even	2		4225.2.a.ca.1.11		18
65.2	even	12		845.2.l.g.654.22		72
65.3	odd	12		845.2.n.h.529.8		36
65.7	even	12		845.2.l.g.699.15		72
65.8	even	4		845.2.d.e.844.16		36
65.12	odd	4		845.2.b.g.339.11	yes	18
65.17	odd	12		845.2.n.i.484.11		36
65.18	even	4		845.2.d.e.844.22		36
65.22	odd	12		845.2.n.h.484.8		36
65.23	odd	12		845.2.n.i.529.11		36
65.28	even	12		845.2.l.g.654.15		72
65.32	even	12		845.2.l.g.699.21		72
65.33	even	12		845.2.l.g.699.22		72
65.37	even	12		845.2.l.g.654.16		72
65.38	odd	4		845.2.b.g.339.8	✓	18
65.42	odd	12		845.2.n.h.529.11		36
65.43	odd	12		845.2.n.i.484.8		36
65.47	even	4		845.2.d.e.844.21		36
65.48	odd	12		845.2.n.h.484.11		36
65.57	even	4		845.2.d.e.844.15		36
65.58	even	12		845.2.l.g.699.16		72
65.62	odd	12		845.2.n.i.529.8		36
65.63	even	12		845.2.l.g.654.21		72
65.64	even	2		4225.2.a.ca.1.8		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.8	✓	18	65.38	odd	4
845.2.b.g.339.11	yes	18	65.12	odd	4
845.2.b.h.339.8	yes	18	5.2	odd	4
845.2.b.h.339.11	yes	18	5.3	odd	4
845.2.d.e.844.15		36	65.57	even	4
845.2.d.e.844.16		36	65.8	even	4
845.2.d.e.844.21		36	65.47	even	4
845.2.d.e.844.22		36	65.18	even	4
845.2.l.g.654.15		72	65.28	even	12
845.2.l.g.654.16		72	65.37	even	12
845.2.l.g.654.21		72	65.63	even	12
845.2.l.g.654.22		72	65.2	even	12
845.2.l.g.699.15		72	65.7	even	12
845.2.l.g.699.16		72	65.58	even	12
845.2.l.g.699.21		72	65.32	even	12
845.2.l.g.699.22		72	65.33	even	12
845.2.n.h.484.8		36	65.22	odd	12
845.2.n.h.484.11		36	65.48	odd	12
845.2.n.h.529.8		36	65.3	odd	12
845.2.n.h.529.11		36	65.42	odd	12
845.2.n.i.484.8		36	65.43	odd	12
845.2.n.i.484.11		36	65.17	odd	12
845.2.n.i.529.8		36	65.62	odd	12
845.2.n.i.529.11		36	65.23	odd	12
4225.2.a.ca.1.8		18	65.64	even	2
4225.2.a.ca.1.11		18	13.12	even	2
4225.2.a.cb.1.8		18	1.1	even	1	trivial
4225.2.a.cb.1.11		18	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.395078 q^{2} +2.73651 q^{3} -1.84391 q^{4} -1.08113 q^{6} -0.629405 q^{7} +1.51865 q^{8} +4.48846 q^{9} +5.49747 q^{11} -5.04588 q^{12} +0.248664 q^{14} +3.08784 q^{16} -4.25364 q^{17} -1.77329 q^{18} +3.61686 q^{19} -1.72237 q^{21} -2.17193 q^{22} +7.05869 q^{23} +4.15578 q^{24} +4.07319 q^{27} +1.16057 q^{28} +2.09557 q^{29} +2.97582 q^{31} -4.25723 q^{32} +15.0439 q^{33} +1.68052 q^{34} -8.27634 q^{36} -0.817814 q^{37} -1.42894 q^{38} -4.89390 q^{41} +0.680471 q^{42} -9.91411 q^{43} -10.1369 q^{44} -2.78873 q^{46} +2.29494 q^{47} +8.44990 q^{48} -6.60385 q^{49} -11.6401 q^{51} -5.84288 q^{53} -1.60923 q^{54} -0.955844 q^{56} +9.89756 q^{57} -0.827912 q^{58} +5.19393 q^{59} +5.92410 q^{61} -1.17568 q^{62} -2.82506 q^{63} -4.49375 q^{64} -5.94350 q^{66} +13.7031 q^{67} +7.84335 q^{68} +19.3161 q^{69} +0.210677 q^{71} +6.81639 q^{72} +12.4282 q^{73} +0.323100 q^{74} -6.66918 q^{76} -3.46014 q^{77} -7.19084 q^{79} -2.31909 q^{81} +1.93347 q^{82} +6.61897 q^{83} +3.17590 q^{84} +3.91685 q^{86} +5.73453 q^{87} +8.34872 q^{88} +3.66109 q^{89} -13.0156 q^{92} +8.14336 q^{93} -0.906679 q^{94} -11.6499 q^{96} +4.54596 q^{97} +2.60904 q^{98} +24.6752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

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