LMFDB - Embedded newform 4225.2.a.bt.1.7

Error: table mf_hecke_newspace_traces does not exist

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 = [9,3,-7,17,0,2,7] 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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 $ x^9 - 17x^7 - 9x^6 + 59x^5 + 32x^4 - 44x^3 - 23x^2 + x + 1
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			1.7
Root			$1.76052$ 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)

$f(q)$	$=$	\(q+2.20556 q^{2} +0.0130567 q^{3} +2.86449 q^{4} +0.0287972 q^{6} +4.60897 q^{7} +1.90669 q^{8} -2.99983 q^{9} -2.93232 q^{11} +0.0374007 q^{12} +10.1654 q^{14} -1.52367 q^{16} +3.35206 q^{17} -6.61630 q^{18} +2.46021 q^{19} +0.0601778 q^{21} -6.46741 q^{22} +1.58329 q^{23} +0.0248950 q^{24} -0.0783378 q^{27} +13.2024 q^{28} +8.26322 q^{29} +9.77959 q^{31} -7.17392 q^{32} -0.0382863 q^{33} +7.39317 q^{34} -8.59299 q^{36} +4.12917 q^{37} +5.42613 q^{38} -7.26508 q^{41} +0.132726 q^{42} +0.705329 q^{43} -8.39961 q^{44} +3.49204 q^{46} +8.57322 q^{47} -0.0198940 q^{48} +14.2426 q^{49} +0.0437667 q^{51} +12.4639 q^{53} -0.172779 q^{54} +8.78787 q^{56} +0.0321221 q^{57} +18.2250 q^{58} +5.33445 q^{59} +1.92092 q^{61} +21.5695 q^{62} -13.8261 q^{63} -12.7752 q^{64} -0.0844427 q^{66} -7.29793 q^{67} +9.60195 q^{68} +0.0206725 q^{69} +6.68083 q^{71} -5.71974 q^{72} -12.4541 q^{73} +9.10713 q^{74} +7.04724 q^{76} -13.5150 q^{77} +0.984840 q^{79} +8.99847 q^{81} -16.0236 q^{82} -7.84405 q^{83} +0.172379 q^{84} +1.55565 q^{86} +0.107890 q^{87} -5.59102 q^{88} +0.412834 q^{89} +4.53532 q^{92} +0.127689 q^{93} +18.9087 q^{94} -0.0936675 q^{96} -3.38416 q^{97} +31.4129 q^{98} +8.79646 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{2} - 7 q^{3} + 17 q^{4} + 2 q^{6} + 7 q^{7} + 12 q^{8} + 16 q^{9} + 9 q^{11} - 12 q^{12} - 2 q^{14} + 37 q^{16} + q^{17} - 10 q^{18} + 4 q^{19} + q^{21} - 12 q^{22} - 14 q^{23} + 35 q^{24}+ \cdots + q^{99}+O(q^{100}) $ 9 * q + 3 * q^2 - 7 * q^3 + 17 * q^4 + 2 * q^6 + 7 * q^7 + 12 * q^8 + 16 * q^9 + 9 * q^11 - 12 * q^12 - 2 * q^14 + 37 * q^16 + q^17 - 10 * q^18 + 4 * q^19 + q^21 - 12 * q^22 - 14 * q^23 + 35 * q^24 - 22 * q^27 + 18 * q^28 + 12 * q^29 + 7 * q^31 + 22 * q^32 - 8 * q^33 + 30 * q^34 + 3 * q^36 - 5 * q^37 + 47 * q^38 + 10 * q^41 + 11 * q^42 - 39 * q^43 + 25 * q^44 - 6 * q^46 + 36 * q^47 + 3 * q^48 + 16 * q^49 + 43 * q^51 + 8 * q^53 + 2 * q^54 - 29 * q^56 - 32 * q^57 + 21 * q^58 + 21 * q^59 - 3 * q^61 + 10 * q^62 + 35 * q^63 + 34 * q^64 - 49 * q^66 + q^67 + 20 * q^68 - 13 * q^69 + q^71 - 3 * q^72 - 15 * q^74 + 5 * q^76 + 4 * q^77 + 39 * q^79 + 29 * q^81 + 4 * q^82 + 7 * q^83 - 12 * q^84 + 24 * q^86 - 16 * q^87 - 42 * q^88 + 19 * q^89 + 27 * q^92 + 31 * q^93 + 16 * q^94 + 7 * q^96 - 34 * q^97 + 48 * q^98 + 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 $

$2$	2.20556	1.55957	0.779783	−	0.626050i	$-0.215329\pi$
$2$	2.20556	1.55957	0.779783	+	0.626050i	$0.215329\pi$
$3$	0.0130567	0.00753827	0.00376913	−	0.999993i	$-0.498800\pi$
$3$	0.0130567	0.00753827	0.00376913	+	0.999993i	$0.498800\pi$
$4$	2.86449	1.43225
$5$	0	0
$5$	0	0
$6$	0.0287972	0.0117564
$7$	4.60897	1.74203	0.871013	−	0.491259i	$-0.163463\pi$
$7$	4.60897	1.74203	0.871013	+	0.491259i	$0.163463\pi$
$8$	1.90669	0.674116
$9$	−2.99983	−0.999943
$10$	0	0
$11$	−2.93232	−0.884128	−0.442064	−	0.896984i	$-0.645754\pi$
$11$	−2.93232	−0.884128	−0.442064	+	0.896984i	$0.645754\pi$
$12$	0.0374007	0.0107967
$13$	0	0
$13$	0	0
$14$	10.1654	2.71681
$15$	0	0
$16$	−1.52367	−0.380917
$17$	3.35206	0.812994	0.406497	−	0.913652i	$-0.366750\pi$
$17$	3.35206	0.812994	0.406497	+	0.913652i	$0.366750\pi$
$18$	−6.61630	−1.55948
$19$	2.46021	0.564410	0.282205	−	0.959354i	$-0.408934\pi$
$19$	2.46021	0.564410	0.282205	+	0.959354i	$0.408934\pi$
$20$	0	0
$21$	0.0601778	0.0131319
$22$	−6.46741	−1.37886
$23$	1.58329	0.330139	0.165069	−	0.986282i	$-0.447215\pi$
$23$	1.58329	0.330139	0.165069	+	0.986282i	$0.447215\pi$
$24$	0.0248950	0.00508167
$25$	0	0
$26$	0	0
$27$	−0.0783378	−0.0150761
$28$	13.2024	2.49501
$29$	8.26322	1.53444	0.767221	−	0.641383i	$-0.221639\pi$
$29$	8.26322	1.53444	0.767221	+	0.641383i	$0.221639\pi$
$30$	0	0
$31$	9.77959	1.75647	0.878233	−	0.478233i	$-0.158723\pi$
$31$	9.77959	1.75647	0.878233	+	0.478233i	$0.158723\pi$
$32$	−7.17392	−1.26818
$33$	−0.0382863	−0.00666479
$34$	7.39317	1.26792
$35$	0	0
$36$	−8.59299	−1.43216
$37$	4.12917	0.678831	0.339416	−	0.940637i	$-0.389771\pi$
$37$	4.12917	0.678831	0.339416	+	0.940637i	$0.389771\pi$
$38$	5.42613	0.880235
$39$	0	0
$40$	0	0
$41$	−7.26508	−1.13462	−0.567308	−	0.823506i	$-0.692015\pi$
$41$	−7.26508	−1.13462	−0.567308	+	0.823506i	$0.692015\pi$
$42$	0.132726	0.0204800
$43$	0.705329	0.107562	0.0537809	−	0.998553i	$-0.482873\pi$
$43$	0.705329	0.107562	0.0537809	+	0.998553i	$0.482873\pi$
$44$	−8.39961	−1.26629
$45$	0	0
$46$	3.49204	0.514873
$47$	8.57322	1.25053	0.625266	−	0.780411i	$-0.284990\pi$
$47$	8.57322	1.25053	0.625266	+	0.780411i	$0.284990\pi$
$48$	−0.0198940	−0.00287146
$49$	14.2426	2.03466
$50$	0	0
$51$	0.0437667	0.00612857
$52$	0	0
$53$	12.4639	1.71204	0.856022	−	0.516940i	$-0.172929\pi$
$53$	12.4639	1.71204	0.856022	+	0.516940i	$0.172929\pi$
$54$	−0.172779	−0.0235122
$55$	0	0
$56$	8.78787	1.17433
$57$	0.0321221	0.00425467
$58$	18.2250	2.39306
$59$	5.33445	0.694487	0.347243	−	0.937775i	$-0.387118\pi$
$59$	5.33445	0.694487	0.347243	+	0.937775i	$0.387118\pi$
$60$	0	0
$61$	1.92092	0.245949	0.122974	−	0.992410i	$-0.460757\pi$
$61$	1.92092	0.245949	0.122974	+	0.992410i	$0.460757\pi$
$62$	21.5695	2.73932
$63$	−13.8261	−1.74193
$64$	−12.7752	−1.59690
$65$	0	0
$66$	−0.0844427	−0.0103942
$67$	−7.29793	−0.891584	−0.445792	−	0.895137i	$-0.647078\pi$
$67$	−7.29793	−0.891584	−0.445792	+	0.895137i	$0.647078\pi$
$68$	9.60195	1.16441
$69$	0.0206725	0.00248867
$70$	0	0
$71$	6.68083	0.792868	0.396434	−	0.918063i	$-0.370248\pi$
$71$	6.68083	0.792868	0.396434	+	0.918063i	$0.370248\pi$
$72$	−5.71974	−0.674078
$73$	−12.4541	−1.45765	−0.728824	−	0.684702i	$-0.759933\pi$
$73$	−12.4541	−1.45765	−0.728824	+	0.684702i	$0.759933\pi$
$74$	9.10713	1.05868
$75$	0	0
$76$	7.04724	0.808374
$77$	−13.5150	−1.54017
$78$	0	0
$79$	0.984840	0.110803	0.0554016	−	0.998464i	$-0.482356\pi$
$79$	0.984840	0.110803	0.0554016	+	0.998464i	$0.482356\pi$
$80$	0	0
$81$	8.99847	0.999830
$82$	−16.0236	−1.76951
$83$	−7.84405	−0.860997	−0.430498	−	0.902591i	$-0.641662\pi$
$83$	−7.84405	−0.860997	−0.430498	+	0.902591i	$0.641662\pi$
$84$	0.172379	0.0188081
$85$	0	0
$86$	1.55565	0.167750
$87$	0.107890	0.0115670
$88$	−5.59102	−0.596005
$89$	0.412834	0.0437604	0.0218802	−	0.999761i	$-0.493035\pi$
$89$	0.412834	0.0437604	0.0218802	+	0.999761i	$0.493035\pi$
$90$	0	0
$91$	0	0
$92$	4.53532	0.472840
$93$	0.127689	0.0132407
$94$	18.9087	1.95029
$95$	0	0
$96$	−0.0936675	−0.00955989
$97$	−3.38416	−0.343609	−0.171805	−	0.985131i	$-0.554960\pi$
$97$	−3.38416	−0.343609	−0.171805	+	0.985131i	$0.554960\pi$
$98$	31.4129	3.17318
$99$	8.79646	0.884077
$100$	0	0
$101$	−10.5432	−1.04909	−0.524545	−	0.851383i	$-0.675765\pi$
$101$	−10.5432	−1.04909	−0.524545	+	0.851383i	$0.675765\pi$
$102$	0.0965301	0.00955790
$103$	−14.1152	−1.39081	−0.695404	−	0.718619i	$-0.744774\pi$
$103$	−14.1152	−1.39081	−0.695404	+	0.718619i	$0.744774\pi$
$104$	0	0
$105$	0	0
$106$	27.4898	2.67004
$107$	3.97165	0.383954	0.191977	−	0.981399i	$-0.438510\pi$
$107$	3.97165	0.383954	0.191977	+	0.981399i	$0.438510\pi$
$108$	−0.224398	−0.0215927
$109$	−10.1737	−0.974460	−0.487230	−	0.873274i	$-0.661993\pi$
$109$	−10.1737	−0.974460	−0.487230	+	0.873274i	$0.661993\pi$
$110$	0	0
$111$	0.0539132	0.00511721
$112$	−7.02254	−0.663568
$113$	2.59984	0.244572	0.122286	−	0.992495i	$-0.460977\pi$
$113$	2.59984	0.244572	0.122286	+	0.992495i	$0.460977\pi$
$114$	0.0708472	0.00663545
$115$	0	0
$116$	23.6699	2.19770
$117$	0	0
$118$	11.7655	1.08310
$119$	15.4495	1.41626
$120$	0	0
$121$	−2.40150	−0.218318
$122$	4.23670	0.383573
$123$	−0.0948578	−0.00855304
$124$	28.0135	2.51569
$125$	0	0
$126$	−30.4943	−2.71665
$127$	−19.9719	−1.77222	−0.886109	−	0.463476i	$-0.846602\pi$
$127$	−19.9719	−1.77222	−0.886109	+	0.463476i	$0.846602\pi$
$128$	−13.8286	−1.22228
$129$	0.00920925	0.000810829 0
$130$	0	0
$131$	−11.4400	−0.999514	−0.499757	−	0.866166i	$-0.666577\pi$
$131$	−11.4400	−0.999514	−0.499757	+	0.866166i	$0.666577\pi$
$132$	−0.109671	−0.00954562
$133$	11.3390	0.983217
$134$	−16.0960	−1.39048
$135$	0	0
$136$	6.39133	0.548052
$137$	3.12520	0.267004	0.133502	−	0.991049i	$-0.457378\pi$
$137$	3.12520	0.267004	0.133502	+	0.991049i	$0.457378\pi$
$138$	0.0455944	0.00388125
$139$	6.09526	0.516993	0.258497	−	0.966012i	$-0.416773\pi$
$139$	6.09526	0.516993	0.258497	+	0.966012i	$0.416773\pi$
$140$	0	0
$141$	0.111938	0.00942685
$142$	14.7350	1.23653
$143$	0	0
$144$	4.57075	0.380896
$145$	0	0
$146$	−27.4684	−2.27330
$147$	0.185961	0.0153378
$148$	11.8280	0.972253
$149$	14.6196	1.19769	0.598844	−	0.800866i	$-0.295627\pi$
$149$	14.6196	1.19769	0.598844	+	0.800866i	$0.295627\pi$
$150$	0	0
$151$	−12.2510	−0.996973	−0.498487	−	0.866897i	$-0.666111\pi$
$151$	−12.2510	−0.996973	−0.498487	+	0.866897i	$0.666111\pi$
$152$	4.69085	0.380478
$153$	−10.0556	−0.812948
$154$	−29.8081	−2.40200
$155$	0	0
$156$	0	0
$157$	−7.13991	−0.569827	−0.284914	−	0.958553i	$-0.591965\pi$
$157$	−7.13991	−0.569827	−0.284914	+	0.958553i	$0.591965\pi$
$158$	2.17212	0.172805
$159$	0.162736	0.0129058
$160$	0	0
$161$	7.29733	0.575110
$162$	19.8467	1.55930
$163$	2.39002	0.187201	0.0936004	−	0.995610i	$-0.470162\pi$
$163$	2.39002	0.187201	0.0936004	+	0.995610i	$0.470162\pi$
$164$	−20.8108	−1.62505
$165$	0	0
$166$	−17.3005	−1.34278
$167$	−7.43841	−0.575601	−0.287801	−	0.957690i	$-0.592924\pi$
$167$	−7.43841	−0.575601	−0.287801	+	0.957690i	$0.592924\pi$
$168$	0.114740	0.00885240
$169$	0	0
$170$	0	0
$171$	−7.38020	−0.564378
$172$	2.02041	0.154055
$173$	−2.54119	−0.193203	−0.0966015	−	0.995323i	$-0.530797\pi$
$173$	−2.54119	−0.193203	−0.0966015	+	0.995323i	$0.530797\pi$
$174$	0.237958	0.0180396
$175$	0	0
$176$	4.46788	0.336779
$177$	0.0696502	0.00523523
$178$	0.910531	0.0682472
$179$	−10.6913	−0.799105	−0.399553	−	0.916710i	$-0.630835\pi$
$179$	−10.6913	−0.799105	−0.399553	+	0.916710i	$0.630835\pi$
$180$	0	0
$181$	−11.7908	−0.876402	−0.438201	−	0.898877i	$-0.644384\pi$
$181$	−11.7908	−0.876402	−0.438201	+	0.898877i	$0.644384\pi$
$182$	0	0
$183$	0.0250808	0.00185403
$184$	3.01884	0.222552
$185$	0	0
$186$	0.281625	0.0206498
$187$	−9.82931	−0.718790
$188$	24.5579	1.79107
$189$	−0.361056	−0.0262630
$190$	0	0
$191$	13.4746	0.974990	0.487495	−	0.873126i	$-0.337911\pi$
$191$	13.4746	0.974990	0.487495	+	0.873126i	$0.337911\pi$
$192$	−0.166801	−0.0120378
$193$	6.98737	0.502962	0.251481	−	0.967862i	$-0.419082\pi$
$193$	6.98737	0.502962	0.251481	+	0.967862i	$0.419082\pi$
$194$	−7.46397	−0.535882
$195$	0	0
$196$	40.7978	2.91413
$197$	1.43795	0.102450	0.0512248	−	0.998687i	$-0.483687\pi$
$197$	1.43795	0.102450	0.0512248	+	0.998687i	$0.483687\pi$
$198$	19.4011	1.37878
$199$	−2.42852	−0.172153	−0.0860765	−	0.996289i	$-0.527433\pi$
$199$	−2.42852	−0.172153	−0.0860765	+	0.996289i	$0.527433\pi$
$200$	0	0
$201$	−0.0952866	−0.00672100
$202$	−23.2537	−1.63613
$203$	38.0849	2.67304
$204$	0.125369	0.00877761
$205$	0	0
$206$	−31.1318	−2.16906
$207$	−4.74960	−0.330120
$208$	0	0
$209$	−7.21411	−0.499011
$210$	0	0
$211$	18.2640	1.25735	0.628674	−	0.777669i	$-0.283598\pi$
$211$	18.2640	1.25735	0.628674	+	0.777669i	$0.283598\pi$
$212$	35.7026	2.45207
$213$	0.0872293	0.00597685
$214$	8.75970	0.598801
$215$	0	0
$216$	−0.149366	−0.0101631
$217$	45.0738	3.05981
$218$	−22.4386	−1.51973
$219$	−0.162610	−0.0109881
$220$	0	0
$221$	0	0
$222$	0.118909	0.00798063
$223$	15.5841	1.04359	0.521793	−	0.853072i	$-0.325263\pi$
$223$	15.5841	1.04359	0.521793	+	0.853072i	$0.325263\pi$
$224$	−33.0644	−2.20921
$225$	0	0
$226$	5.73409	0.381426
$227$	−4.03380	−0.267733	−0.133866	−	0.990999i	$-0.542739\pi$
$227$	−4.03380	−0.267733	−0.133866	+	0.990999i	$0.542739\pi$
$228$	0.0920135	0.00609374
$229$	−22.5147	−1.48781	−0.743906	−	0.668284i	$-0.767029\pi$
$229$	−22.5147	−1.48781	−0.743906	+	0.668284i	$0.767029\pi$
$230$	0	0
$231$	−0.176460	−0.0116102
$232$	15.7554	1.03439
$233$	7.69759	0.504286	0.252143	−	0.967690i	$-0.418865\pi$
$233$	7.69759	0.504286	0.252143	+	0.967690i	$0.418865\pi$
$234$	0	0
$235$	0	0
$236$	15.2805	0.994676
$237$	0.0128587	0.000835264 0
$238$	34.0749	2.20875
$239$	−18.7203	−1.21092	−0.605459	−	0.795877i	$-0.707010\pi$
$239$	−18.7203	−1.21092	−0.605459	+	0.795877i	$0.707010\pi$
$240$	0	0
$241$	22.6183	1.45697	0.728486	−	0.685061i	$-0.240225\pi$
$241$	22.6183	1.45697	0.728486	+	0.685061i	$0.240225\pi$
$242$	−5.29665	−0.340482
$243$	0.352503	0.0226131
$244$	5.50246	0.352259
$245$	0	0
$246$	−0.209214	−0.0133390
$247$	0	0
$248$	18.6466	1.18406
$249$	−0.102417	−0.00649042
$250$	0	0
$251$	1.78412	0.112612	0.0563062	−	0.998414i	$-0.482068\pi$
$251$	1.78412	0.112612	0.0563062	+	0.998414i	$0.482068\pi$
$252$	−39.6048	−2.49487
$253$	−4.64271	−0.291885
$254$	−44.0492	−2.76389
$255$	0	0
$256$	−4.94936	−0.309335
$257$	4.59847	0.286845	0.143422	−	0.989662i	$-0.454189\pi$
$257$	4.59847	0.286845	0.143422	+	0.989662i	$0.454189\pi$
$258$	0.0203115	0.00126454
$259$	19.0312	1.18254
$260$	0	0
$261$	−24.7883	−1.53435
$262$	−25.2315	−1.55881
$263$	−18.8127	−1.16004	−0.580022	−	0.814601i	$-0.696956\pi$
$263$	−18.8127	−1.16004	−0.580022	+	0.814601i	$0.696956\pi$
$264$	−0.0730001	−0.00449284
$265$	0	0
$266$	25.0089	1.53339
$267$	0.00539024	0.000329877 0
$268$	−20.9049	−1.27697
$269$	1.85412	0.113048	0.0565238	−	0.998401i	$-0.481998\pi$
$269$	1.85412	0.113048	0.0565238	+	0.998401i	$0.481998\pi$
$270$	0	0
$271$	−0.149698	−0.00909352	−0.00454676	−	0.999990i	$-0.501447\pi$
$271$	−0.149698	−0.00909352	−0.00454676	+	0.999990i	$0.501447\pi$
$272$	−5.10743	−0.309683
$273$	0	0
$274$	6.89281	0.416410
$275$	0	0
$276$	0.0592161	0.00356439
$277$	−22.0184	−1.32296	−0.661478	−	0.749965i	$-0.730071\pi$
$277$	−22.0184	−1.32296	−0.661478	+	0.749965i	$0.730071\pi$
$278$	13.4435	0.806285
$279$	−29.3371	−1.75637
$280$	0	0
$281$	0.438029	0.0261306	0.0130653	−	0.999915i	$-0.495841\pi$
$281$	0.438029	0.0261306	0.0130653	+	0.999915i	$0.495841\pi$
$282$	0.246885	0.0147018
$283$	−13.8290	−0.822049	−0.411025	−	0.911624i	$-0.634829\pi$
$283$	−13.8290	−0.822049	−0.411025	+	0.911624i	$0.634829\pi$
$284$	19.1372	1.13558
$285$	0	0
$286$	0	0
$287$	−33.4845	−1.97653
$288$	21.5205	1.26811
$289$	−5.76370	−0.339041
$290$	0	0
$291$	−0.0441858	−0.00259022
$292$	−35.6748	−2.08771
$293$	14.5656	0.850929	0.425464	−	0.904975i	$-0.360111\pi$
$293$	14.5656	0.850929	0.425464	+	0.904975i	$0.360111\pi$
$294$	0.410148	0.0239203
$295$	0	0
$296$	7.87304	0.457611
$297$	0.229711	0.0133292
$298$	32.2445	1.86787
$299$	0	0
$300$	0	0
$301$	3.25084	0.187375
$302$	−27.0203	−1.55485
$303$	−0.137659	−0.00790833
$304$	−3.74854	−0.214994
$305$	0	0
$306$	−22.1782	−1.26785
$307$	−9.05062	−0.516546	−0.258273	−	0.966072i	$-0.583153\pi$
$307$	−9.05062	−0.516546	−0.258273	+	0.966072i	$0.583153\pi$
$308$	−38.7135	−2.20591
$309$	−0.184297	−0.0104843
$310$	0	0
$311$	7.28812	0.413272	0.206636	−	0.978418i	$-0.433748\pi$
$311$	7.28812	0.413272	0.206636	+	0.978418i	$0.433748\pi$
$312$	0	0
$313$	−12.9725	−0.733250	−0.366625	−	0.930369i	$-0.619487\pi$
$313$	−12.9725	−0.733250	−0.366625	+	0.930369i	$0.619487\pi$
$314$	−15.7475	−0.888683
$315$	0	0
$316$	2.82107	0.158697
$317$	15.5048	0.870838	0.435419	−	0.900228i	$-0.356600\pi$
$317$	15.5048	0.870838	0.435419	+	0.900228i	$0.356600\pi$
$318$	0.358925	0.0201275
$319$	−24.2304	−1.35664
$320$	0	0
$321$	0.0518565	0.00289434
$322$	16.0947	0.896922
$323$	8.24676	0.458862
$324$	25.7760	1.43200
$325$	0	0
$326$	5.27133	0.291952
$327$	−0.132834	−0.00734574
$328$	−13.8523	−0.764863
$329$	39.5137	2.17846
$330$	0	0
$331$	−26.0853	−1.43378	−0.716890	−	0.697186i	$-0.754435\pi$
$331$	−26.0853	−1.43378	−0.716890	+	0.697186i	$0.754435\pi$
$332$	−22.4692	−1.23316
$333$	−12.3868	−0.678793
$334$	−16.4058	−0.897688
$335$	0	0
$336$	−0.0916910	−0.00500215
$337$	14.4785	0.788697	0.394348	−	0.918961i	$-0.370970\pi$
$337$	14.4785	0.788697	0.394348	+	0.918961i	$0.370970\pi$
$338$	0	0
$339$	0.0339452	0.00184365
$340$	0	0
$341$	−28.6769	−1.55294
$342$	−16.2775	−0.880185
$343$	33.3809	1.80240
$344$	1.34484	0.0725091
$345$	0	0
$346$	−5.60474	−0.301313
$347$	−6.81412	−0.365801	−0.182901	−	0.983131i	$-0.558549\pi$
$347$	−6.81412	−0.365801	−0.182901	+	0.983131i	$0.558549\pi$
$348$	0.309050	0.0165668
$349$	33.7357	1.80583	0.902915	−	0.429818i	$-0.141422\pi$
$349$	33.7357	1.80583	0.902915	+	0.429818i	$0.141422\pi$
$350$	0	0
$351$	0	0
$352$	21.0362	1.12123
$353$	−18.7215	−0.996447	−0.498223	−	0.867049i	$-0.666014\pi$
$353$	−18.7215	−0.996447	−0.498223	+	0.867049i	$0.666014\pi$
$354$	0.153618	0.00816468
$355$	0	0
$356$	1.18256	0.0626756
$357$	0.201719	0.0106761
$358$	−23.5803	−1.24626
$359$	27.9262	1.47389	0.736945	−	0.675952i	$-0.236268\pi$
$359$	27.9262	1.47389	0.736945	+	0.675952i	$0.236268\pi$
$360$	0	0
$361$	−12.9474	−0.681441
$362$	−26.0053	−1.36681
$363$	−0.0313556	−0.00164574
$364$	0	0
$365$	0	0
$366$	0.0553172	0.00289148
$367$	−2.50988	−0.131015	−0.0655074	−	0.997852i	$-0.520867\pi$
$367$	−2.50988	−0.131015	−0.0655074	+	0.997852i	$0.520867\pi$
$368$	−2.41241	−0.125755
$369$	21.7940	1.13455
$370$	0	0
$371$	57.4456	2.98243
$372$	0.365763	0.0189640
$373$	−22.1779	−1.14833	−0.574165	−	0.818739i	$-0.694673\pi$
$373$	−22.1779	−1.14833	−0.574165	+	0.818739i	$0.694673\pi$
$374$	−21.6791	−1.12100
$375$	0	0
$376$	16.3465	0.843005
$377$	0	0
$378$	−0.796331	−0.0409589
$379$	1.42116	0.0730001	0.0365001	−	0.999334i	$-0.488379\pi$
$379$	1.42116	0.0730001	0.0365001	+	0.999334i	$0.488379\pi$
$380$	0	0
$381$	−0.260766	−0.0133595
$382$	29.7191	1.52056
$383$	21.0733	1.07679	0.538397	−	0.842691i	$-0.319030\pi$
$383$	21.0733	1.07679	0.538397	+	0.842691i	$0.319030\pi$
$384$	−0.180555	−0.00921390
$385$	0	0
$386$	15.4111	0.784402
$387$	−2.11587	−0.107556
$388$	−9.69390	−0.492133
$389$	13.4713	0.683021	0.341511	−	0.939878i	$-0.389061\pi$
$389$	13.4713	0.683021	0.341511	+	0.939878i	$0.389061\pi$
$390$	0	0
$391$	5.30728	0.268401
$392$	27.1562	1.37160
$393$	−0.149368	−0.00753461
$394$	3.17148	0.159777
$395$	0	0
$396$	25.1974	1.26622
$397$	27.1859	1.36442	0.682210	−	0.731156i	$-0.261019\pi$
$397$	27.1859	1.36442	0.682210	+	0.731156i	$0.261019\pi$
$398$	−5.35624	−0.268484
$399$	0.148050	0.00741176
$400$	0	0
$401$	3.27520	0.163556	0.0817779	−	0.996651i	$-0.473940\pi$
$401$	3.27520	0.163556	0.0817779	+	0.996651i	$0.473940\pi$
$402$	−0.210160	−0.0104818
$403$	0	0
$404$	−30.2010	−1.50256
$405$	0	0
$406$	83.9986	4.16878
$407$	−12.1080	−0.600173
$408$	0.0834495	0.00413137
$409$	19.1677	0.947782	0.473891	−	0.880583i	$-0.342849\pi$
$409$	19.1677	0.947782	0.473891	+	0.880583i	$0.342849\pi$
$410$	0	0
$411$	0.0408047	0.00201275
$412$	−40.4327	−1.99198
$413$	24.5863	1.20981
$414$	−10.4755	−0.514844
$415$	0	0
$416$	0	0
$417$	0.0795837	0.00389723
$418$	−15.9112	−0.778240
$419$	0.392463	0.0191731	0.00958653	−	0.999954i	$-0.496948\pi$
$419$	0.392463	0.0191731	0.00958653	+	0.999954i	$0.496948\pi$
$420$	0	0
$421$	−35.1816	−1.71465	−0.857323	−	0.514779i	$-0.827874\pi$
$421$	−35.1816	−1.71465	−0.857323	+	0.514779i	$0.827874\pi$
$422$	40.2824	1.96092
$423$	−25.7182	−1.25046
$424$	23.7647	1.15412
$425$	0	0
$426$	0.192389	0.00932130
$427$	8.85346	0.428449
$428$	11.3768	0.549916
$429$	0	0
$430$	0	0
$431$	−2.27454	−0.109561	−0.0547803	−	0.998498i	$-0.517446\pi$
$431$	−2.27454	−0.109561	−0.0547803	+	0.998498i	$0.517446\pi$
$432$	0.119361	0.00574275
$433$	−11.8827	−0.571048	−0.285524	−	0.958372i	$-0.592168\pi$
$433$	−11.8827	−0.571048	−0.285524	+	0.958372i	$0.592168\pi$
$434$	99.4130	4.77197
$435$	0	0
$436$	−29.1424	−1.39567
$437$	3.89522	0.186334
$438$	−0.358645	−0.0171367
$439$	−35.7273	−1.70517	−0.852585	−	0.522588i	$-0.824967\pi$
$439$	−35.7273	−1.70517	−0.852585	+	0.522588i	$0.824967\pi$
$440$	0	0
$441$	−42.7254	−2.03454
$442$	0	0
$443$	−4.98904	−0.237036	−0.118518	−	0.992952i	$-0.537814\pi$
$443$	−4.98904	−0.237036	−0.118518	+	0.992952i	$0.537814\pi$
$444$	0.154434	0.00732911
$445$	0	0
$446$	34.3716	1.62754
$447$	0.190884	0.00902849
$448$	−58.8804	−2.78184
$449$	23.4886	1.10850	0.554248	−	0.832352i	$-0.313006\pi$
$449$	23.4886	1.10850	0.554248	+	0.832352i	$0.313006\pi$
$450$	0	0
$451$	21.3035	1.00314
$452$	7.44721	0.350287
$453$	−0.159957	−0.00751545
$454$	−8.89679	−0.417547
$455$	0	0
$456$	0.0612468	0.00286815
$457$	40.2510	1.88286	0.941430	−	0.337208i	$-0.109482\pi$
$457$	40.2510	1.88286	0.941430	+	0.337208i	$0.109482\pi$
$458$	−49.6575	−2.32034
$459$	−0.262593	−0.0122568
$460$	0	0
$461$	31.2140	1.45378	0.726890	−	0.686754i	$-0.240965\pi$
$461$	31.2140	1.45378	0.726890	+	0.686754i	$0.240965\pi$
$462$	−0.389194	−0.0181069
$463$	−17.7491	−0.824872	−0.412436	−	0.910987i	$-0.635322\pi$
$463$	−17.7491	−0.824872	−0.412436	+	0.910987i	$0.635322\pi$
$464$	−12.5904	−0.584495
$465$	0	0
$466$	16.9775	0.786467
$467$	−26.6645	−1.23389	−0.616944	−	0.787007i	$-0.711630\pi$
$467$	−26.6645	−1.23389	−0.616944	+	0.787007i	$0.711630\pi$
$468$	0	0
$469$	−33.6359	−1.55316
$470$	0	0
$471$	−0.0932234	−0.00429551
$472$	10.1711	0.468165
$473$	−2.06825	−0.0950983
$474$	0.0283607	0.00130265
$475$	0	0
$476$	44.2551	2.02843
$477$	−37.3895	−1.71195
$478$	−41.2888	−1.88851
$479$	20.4630	0.934980	0.467490	−	0.883998i	$-0.345158\pi$
$479$	20.4630	0.934980	0.467490	+	0.883998i	$0.345158\pi$
$480$	0	0
$481$	0	0
$482$	49.8860	2.27224
$483$	0.0952788	0.00433534
$484$	−6.87908	−0.312686
$485$	0	0
$486$	0.777467	0.0352666
$487$	−8.11789	−0.367857	−0.183928	−	0.982940i	$-0.558881\pi$
$487$	−8.11789	−0.367857	−0.183928	+	0.982940i	$0.558881\pi$
$488$	3.66260	0.165798
$489$	0.0312057	0.00141117
$490$	0	0
$491$	3.26996	0.147571	0.0737857	−	0.997274i	$-0.476492\pi$
$491$	3.26996	0.147571	0.0737857	+	0.997274i	$0.476492\pi$
$492$	−0.271719	−0.0122501
$493$	27.6988	1.24749
$494$	0	0
$495$	0	0
$496$	−14.9008	−0.669068
$497$	30.7917	1.38120
$498$	−0.225887	−0.0101222
$499$	8.89340	0.398123	0.199062	−	0.979987i	$-0.436211\pi$
$499$	8.89340	0.398123	0.199062	+	0.979987i	$0.436211\pi$
$500$	0	0
$501$	−0.0971208	−0.00433904
$502$	3.93498	0.175627
$503$	23.7974	1.06107	0.530537	−	0.847662i	$-0.321990\pi$
$503$	23.7974	1.06107	0.530537	+	0.847662i	$0.321990\pi$
$504$	−26.3621	−1.17426
$505$	0	0
$506$	−10.2398	−0.455213
$507$	0	0
$508$	−57.2093	−2.53825
$509$	−1.26770	−0.0561898	−0.0280949	−	0.999605i	$-0.508944\pi$
$509$	−1.26770	−0.0561898	−0.0280949	+	0.999605i	$0.508944\pi$
$510$	0	0
$511$	−57.4008	−2.53926
$512$	16.7410	0.739855
$513$	−0.192727	−0.00850911
$514$	10.1422	0.447354
$515$	0	0
$516$	0.0263798	0.00116131
$517$	−25.1394	−1.10563
$518$	41.9745	1.84425
$519$	−0.0331794	−0.00145642
$520$	0	0
$521$	14.3916	0.630506	0.315253	−	0.949008i	$-0.397911\pi$
$521$	14.3916	0.630506	0.315253	+	0.949008i	$0.397911\pi$
$522$	−54.6720	−2.39293
$523$	−19.0185	−0.831622	−0.415811	−	0.909451i	$-0.636502\pi$
$523$	−19.0185	−0.831622	−0.415811	+	0.909451i	$0.636502\pi$
$524$	−32.7697	−1.43155
$525$	0	0
$526$	−41.4926	−1.80916
$527$	32.7818	1.42800
$528$	0.0583357	0.00253873
$529$	−20.4932	−0.891008
$530$	0	0
$531$	−16.0024	−0.694447
$532$	32.4805	1.40821
$533$	0	0
$534$	0.0118885	0.000514465 0
$535$	0	0
$536$	−13.9149	−0.601031
$537$	−0.139593	−0.00602387
$538$	4.08937	0.176305
$539$	−41.7638	−1.79890
$540$	0	0
$541$	−13.8969	−0.597475	−0.298738	−	0.954335i	$-0.596566\pi$
$541$	−13.8969	−0.597475	−0.298738	+	0.954335i	$0.596566\pi$
$542$	−0.330168	−0.0141819
$543$	−0.153948	−0.00660656
$544$	−24.0474	−1.03102
$545$	0	0
$546$	0	0
$547$	−37.0840	−1.58560	−0.792798	−	0.609484i	$-0.791377\pi$
$547$	−37.0840	−1.58560	−0.792798	+	0.609484i	$0.791377\pi$
$548$	8.95211	0.382415
$549$	−5.76243	−0.245935
$550$	0	0
$551$	20.3292	0.866054
$552$	0.0394160	0.00167766
$553$	4.53910	0.193022
$554$	−48.5628	−2.06324
$555$	0	0
$556$	17.4598	0.740461
$557$	31.8065	1.34768	0.673842	−	0.738875i	$-0.264643\pi$
$557$	31.8065	1.34768	0.673842	+	0.738875i	$0.264643\pi$
$558$	−64.7047	−2.73917
$559$	0	0
$560$	0	0
$561$	−0.128338	−0.00541843
$562$	0.966100	0.0407525
$563$	−26.3595	−1.11092	−0.555461	−	0.831543i	$-0.687458\pi$
$563$	−26.3595	−1.11092	−0.555461	+	0.831543i	$0.687458\pi$
$564$	0.320645	0.0135016
$565$	0	0
$566$	−30.5007	−1.28204
$567$	41.4736	1.74173
$568$	12.7383	0.534485
$569$	17.7846	0.745569	0.372785	−	0.927918i	$-0.378403\pi$
$569$	17.7846	0.745569	0.372785	+	0.927918i	$0.378403\pi$
$570$	0	0
$571$	19.6399	0.821903	0.410952	−	0.911657i	$-0.365196\pi$
$571$	19.6399	0.821903	0.410952	+	0.911657i	$0.365196\pi$
$572$	0	0
$573$	0.175934	0.00734974
$574$	−73.8522	−3.08253
$575$	0	0
$576$	38.3233	1.59681
$577$	18.1454	0.755403	0.377701	−	0.925927i	$-0.376715\pi$
$577$	18.1454	0.755403	0.377701	+	0.925927i	$0.376715\pi$
$578$	−12.7122	−0.528757
$579$	0.0912317	0.00379146
$580$	0	0
$581$	−36.1530	−1.49988
$582$	−0.0974545	−0.00403962
$583$	−36.5480	−1.51366
$584$	−23.7462	−0.982624
$585$	0	0
$586$	32.1252	1.32708
$587$	−0.423843	−0.0174939	−0.00874694	−	0.999962i	$-0.502784\pi$
$587$	−0.423843	−0.0174939	−0.00874694	+	0.999962i	$0.502784\pi$
$588$	0.532683	0.0219675
$589$	24.0598	0.991367
$590$	0	0
$591$	0.0187748	0.000772293 0
$592$	−6.29149	−0.258578
$593$	−47.5064	−1.95085	−0.975426	−	0.220327i	$-0.929288\pi$
$593$	−47.5064	−1.95085	−0.975426	+	0.220327i	$0.929288\pi$
$594$	0.506642	0.0207878
$595$	0	0
$596$	41.8778	1.71538
$597$	−0.0317083	−0.00129774
$598$	0	0
$599$	25.7871	1.05363	0.526816	−	0.849979i	$-0.323386\pi$
$599$	25.7871	1.05363	0.526816	+	0.849979i	$0.323386\pi$
$600$	0	0
$601$	−12.6776	−0.517130	−0.258565	−	0.965994i	$-0.583250\pi$
$601$	−12.6776	−0.517130	−0.258565	+	0.965994i	$0.583250\pi$
$602$	7.16992	0.292224
$603$	21.8925	0.891533
$604$	−35.0929	−1.42791
$605$	0	0
$606$	−0.303616	−0.0123336
$607$	4.40448	0.178772	0.0893862	−	0.995997i	$-0.471509\pi$
$607$	4.40448	0.178772	0.0893862	+	0.995997i	$0.471509\pi$
$608$	−17.6493	−0.715775
$609$	0.497262	0.0201501
$610$	0	0
$611$	0	0
$612$	−28.8042	−1.16434
$613$	−33.3599	−1.34739	−0.673696	−	0.739008i	$-0.735294\pi$
$613$	−33.3599	−1.34739	−0.673696	+	0.739008i	$0.735294\pi$
$614$	−19.9617	−0.805588
$615$	0	0
$616$	−25.7688	−1.03826
$617$	−32.5535	−1.31055	−0.655277	−	0.755389i	$-0.727448\pi$
$617$	−32.5535	−1.31055	−0.655277	+	0.755389i	$0.727448\pi$
$618$	−0.406477	−0.0163509
$619$	35.5813	1.43013	0.715067	−	0.699056i	$-0.246396\pi$
$619$	35.5813	1.43013	0.715067	+	0.699056i	$0.246396\pi$
$620$	0	0
$621$	−0.124031	−0.00497721
$622$	16.0744	0.644524
$623$	1.90274	0.0762317
$624$	0	0
$625$	0	0
$626$	−28.6116	−1.14355
$627$	−0.0941922	−0.00376168
$628$	−20.4522	−0.816133
$629$	13.8412	0.551886
$630$	0	0
$631$	−25.4117	−1.01162	−0.505812	−	0.862644i	$-0.668807\pi$
$631$	−25.4117	−1.01162	−0.505812	+	0.862644i	$0.668807\pi$
$632$	1.87778	0.0746942
$633$	0.238467	0.00947822
$634$	34.1968	1.35813
$635$	0	0
$636$	0.466157	0.0184843
$637$	0	0
$638$	−53.4416	−2.11577
$639$	−20.0413	−0.792823
$640$	0	0
$641$	−10.1479	−0.400817	−0.200408	−	0.979712i	$-0.564227\pi$
$641$	−10.1479	−0.400817	−0.200408	+	0.979712i	$0.564227\pi$
$642$	0.114372	0.00451392
$643$	1.85439	0.0731298	0.0365649	−	0.999331i	$-0.488358\pi$
$643$	1.85439	0.0731298	0.0365649	+	0.999331i	$0.488358\pi$
$644$	20.9032	0.823700
$645$	0	0
$646$	18.1887	0.715626
$647$	29.6034	1.16383	0.581914	−	0.813250i	$-0.302304\pi$
$647$	29.6034	1.16383	0.581914	+	0.813250i	$0.302304\pi$
$648$	17.1573	0.674001
$649$	−15.6423	−0.614015
$650$	0	0
$651$	0.588514	0.0230657
$652$	6.84619	0.268118
$653$	−6.78882	−0.265667	−0.132834	−	0.991138i	$-0.542408\pi$
$653$	−6.78882	−0.265667	−0.132834	+	0.991138i	$0.542408\pi$
$654$	−0.292973	−0.0114562
$655$	0	0
$656$	11.0696	0.432195
$657$	37.3603	1.45756
$658$	87.1498	3.39745
$659$	7.94959	0.309672	0.154836	−	0.987940i	$-0.450515\pi$
$659$	7.94959	0.309672	0.154836	+	0.987940i	$0.450515\pi$
$660$	0	0
$661$	−40.3832	−1.57072	−0.785361	−	0.619038i	$-0.787523\pi$
$661$	−40.3832	−1.57072	−0.785361	+	0.619038i	$0.787523\pi$
$662$	−57.5328	−2.23607
$663$	0	0
$664$	−14.9562	−0.580412
$665$	0	0
$666$	−27.3198	−1.05862
$667$	13.0831	0.506579
$668$	−21.3073	−0.824402
$669$	0.203476	0.00786684
$670$	0	0
$671$	−5.63275	−0.217450
$672$	−0.431710	−0.0166536
$673$	−9.63920	−0.371564	−0.185782	−	0.982591i	$-0.559482\pi$
$673$	−9.63920	−0.371564	−0.185782	+	0.982591i	$0.559482\pi$
$674$	31.9333	1.23002
$675$	0	0
$676$	0	0
$677$	−51.1565	−1.96610	−0.983051	−	0.183330i	$-0.941312\pi$
$677$	−51.1565	−1.96610	−0.983051	+	0.183330i	$0.941312\pi$
$678$	0.0748681	0.00287529
$679$	−15.5975	−0.598577
$680$	0	0
$681$	−0.0526680	−0.00201824
$682$	−63.2485	−2.42191
$683$	−8.34502	−0.319313	−0.159657	−	0.987173i	$-0.551039\pi$
$683$	−8.34502	−0.319313	−0.159657	+	0.987173i	$0.551039\pi$
$684$	−21.1405	−0.808328
$685$	0	0
$686$	73.6236	2.81096
$687$	−0.293967	−0.0112155
$688$	−1.07469	−0.0409721
$689$	0	0
$690$	0	0
$691$	36.1243	1.37423	0.687117	−	0.726547i	$-0.258876\pi$
$691$	36.1243	1.37423	0.687117	+	0.726547i	$0.258876\pi$
$692$	−7.27921	−0.276714
$693$	40.5426	1.54009
$694$	−15.0289	−0.570491
$695$	0	0
$696$	0.205713	0.00779753
$697$	−24.3530	−0.922435
$698$	74.4061	2.81631
$699$	0.100505	0.00380144
$700$	0	0
$701$	−45.9823	−1.73673	−0.868364	−	0.495928i	$-0.834828\pi$
$701$	−45.9823	−1.73673	−0.868364	+	0.495928i	$0.834828\pi$
$702$	0	0
$703$	10.1586	0.383139
$704$	37.4609	1.41186
$705$	0	0
$706$	−41.2915	−1.55402
$707$	−48.5934	−1.82754
$708$	0.199512	0.00749813
$709$	−38.0949	−1.43069	−0.715343	−	0.698774i	$-0.753729\pi$
$709$	−38.0949	−1.43069	−0.715343	+	0.698774i	$0.753729\pi$
$710$	0	0
$711$	−2.95435	−0.110797
$712$	0.787147	0.0294996
$713$	15.4839	0.579877
$714$	0.444904	0.0166501
$715$	0	0
$716$	−30.6251	−1.14452
$717$	−0.244425	−0.00912822
$718$	61.5930	2.29863
$719$	−27.8672	−1.03927	−0.519635	−	0.854388i	$-0.673932\pi$
$719$	−27.8672	−1.03927	−0.519635	+	0.854388i	$0.673932\pi$
$720$	0	0
$721$	−65.0563	−2.42282
$722$	−28.5562	−1.06275
$723$	0.295319	0.0109830
$724$	−33.7746	−1.25522
$725$	0	0
$726$	−0.0691566	−0.00256664
$727$	13.7750	0.510885	0.255443	−	0.966824i	$-0.417779\pi$
$727$	13.7750	0.510885	0.255443	+	0.966824i	$0.417779\pi$
$728$	0	0
$729$	−26.9908	−0.999659
$730$	0	0
$731$	2.36431	0.0874470
$732$	0.0718438	0.00265542
$733$	−6.58392	−0.243183	−0.121591	−	0.992580i	$-0.538800\pi$
$733$	−6.58392	−0.243183	−0.121591	+	0.992580i	$0.538800\pi$
$734$	−5.53570	−0.204326
$735$	0	0
$736$	−11.3584	−0.418676
$737$	21.3999	0.788274
$738$	48.0680	1.76941
$739$	−5.47501	−0.201401	−0.100701	−	0.994917i	$-0.532108\pi$
$739$	−5.47501	−0.201401	−0.100701	+	0.994917i	$0.532108\pi$
$740$	0	0
$741$	0	0
$742$	126.700	4.65129
$743$	−26.6714	−0.978479	−0.489239	−	0.872150i	$-0.662726\pi$
$743$	−26.6714	−0.978479	−0.489239	+	0.872150i	$0.662726\pi$
$744$	0.243463	0.00892578
$745$	0	0
$746$	−48.9148	−1.79090
$747$	23.5308	0.860948
$748$	−28.1560	−1.02948
$749$	18.3052	0.668857
$750$	0	0
$751$	21.8067	0.795740	0.397870	−	0.917442i	$-0.369750\pi$
$751$	21.8067	0.795740	0.397870	+	0.917442i	$0.369750\pi$
$752$	−13.0627	−0.476349
$753$	0.0232946	0.000848903 0
$754$	0	0
$755$	0	0
$756$	−1.03424	−0.0376151
$757$	8.33422	0.302912	0.151456	−	0.988464i	$-0.451604\pi$
$757$	8.33422	0.302912	0.151456	+	0.988464i	$0.451604\pi$
$758$	3.13446	0.113849
$759$	−0.0606183	−0.00220031
$760$	0	0
$761$	7.33096	0.265747	0.132874	−	0.991133i	$-0.457580\pi$
$761$	7.33096	0.265747	0.132874	+	0.991133i	$0.457580\pi$
$762$	−0.575135	−0.0208350
$763$	−46.8901	−1.69753
$764$	38.5980	1.39643
$765$	0	0
$766$	46.4783	1.67933
$767$	0	0
$768$	−0.0646221	−0.00233185
$769$	22.3346	0.805407	0.402703	−	0.915331i	$-0.368071\pi$
$769$	22.3346	0.805407	0.402703	+	0.915331i	$0.368071\pi$
$770$	0	0
$771$	0.0600407	0.00216231
$772$	20.0153	0.720365
$773$	−27.1591	−0.976844	−0.488422	−	0.872608i	$-0.662427\pi$
$773$	−27.1591	−0.976844	−0.488422	+	0.872608i	$0.662427\pi$
$774$	−4.66667	−0.167740
$775$	0	0
$776$	−6.45254	−0.231633
$777$	0.248484	0.00891432
$778$	29.7117	1.06522
$779$	−17.8736	−0.640388
$780$	0	0
$781$	−19.5903	−0.700997
$782$	11.7055	0.418589
$783$	−0.647322	−0.0231334
$784$	−21.7010	−0.775036
$785$	0	0
$786$	−0.329440	−0.0117507
$787$	11.5806	0.412805	0.206403	−	0.978467i	$-0.433824\pi$
$787$	11.5806	0.412805	0.206403	+	0.978467i	$0.433824\pi$
$788$	4.11899	0.146733
$789$	−0.245632	−0.00874472
$790$	0	0
$791$	11.9826	0.426051
$792$	16.7721	0.595971
$793$	0	0
$794$	59.9601	2.12790
$795$	0	0
$796$	−6.95647	−0.246566
$797$	47.7004	1.68963	0.844817	−	0.535055i	$-0.179709\pi$
$797$	47.7004	1.68963	0.844817	+	0.535055i	$0.179709\pi$
$798$	0.326532	0.0115591
$799$	28.7379	1.01668
$800$	0	0
$801$	−1.23843	−0.0437579
$802$	7.22365	0.255076
$803$	36.5195	1.28875
$804$	−0.272948	−0.00962612
$805$	0	0
$806$	0	0
$807$	0.0242086	0.000852184 0
$808$	−20.1027	−0.707209
$809$	36.3910	1.27944	0.639720	−	0.768608i	$-0.279050\pi$
$809$	36.3910	1.27944	0.639720	+	0.768608i	$0.279050\pi$
$810$	0	0
$811$	36.8088	1.29253	0.646266	−	0.763112i	$-0.276330\pi$
$811$	36.8088	1.29253	0.646266	+	0.763112i	$0.276330\pi$
$812$	109.094	3.82845
$813$	−0.00195456	−6.85494e−5 0
$814$	−26.7050	−0.936010
$815$	0	0
$816$	−0.0666860	−0.00233448
$817$	1.73526	0.0607089
$818$	42.2755	1.47813
$819$	0	0
$820$	0	0
$821$	−33.6668	−1.17498	−0.587491	−	0.809231i	$-0.699884\pi$
$821$	−33.6668	−1.17498	−0.587491	+	0.809231i	$0.699884\pi$
$822$	0.0899971	0.00313901
$823$	−31.8809	−1.11130	−0.555649	−	0.831417i	$-0.687530\pi$
$823$	−31.8809	−1.11130	−0.555649	+	0.831417i	$0.687530\pi$
$824$	−26.9132	−0.937566
$825$	0	0
$826$	54.2266	1.88678
$827$	−18.4625	−0.642003	−0.321002	−	0.947079i	$-0.604019\pi$
$827$	−18.4625	−0.642003	−0.321002	+	0.947079i	$0.604019\pi$
$828$	−13.6052	−0.472813
$829$	−34.9663	−1.21443	−0.607214	−	0.794538i	$-0.707713\pi$
$829$	−34.9663	−1.21443	−0.607214	+	0.794538i	$0.707713\pi$
$830$	0	0
$831$	−0.287487	−0.00997280
$832$	0	0
$833$	47.7420	1.65416
$834$	0.175527	0.00607799
$835$	0	0
$836$	−20.6648	−0.714706
$837$	−0.766111	−0.0264807
$838$	0.865600	0.0299017
$839$	28.7234	0.991641	0.495820	−	0.868425i	$-0.334867\pi$
$839$	28.7234	0.991641	0.495820	+	0.868425i	$0.334867\pi$
$840$	0	0
$841$	39.2808	1.35451
$842$	−77.5951	−2.67410
$843$	0.00571920	0.000196980 0
$844$	52.3172	1.80083
$845$	0	0
$846$	−56.7230	−1.95018
$847$	−11.0684	−0.380316
$848$	−18.9908	−0.652147
$849$	−0.180561	−0.00619683
$850$	0	0
$851$	6.53767	0.224108
$852$	0.249868	0.00856032
$853$	−13.7453	−0.470629	−0.235315	−	0.971919i	$-0.575612\pi$
$853$	−13.7453	−0.470629	−0.235315	+	0.971919i	$0.575612\pi$
$854$	19.5268	0.668195
$855$	0	0
$856$	7.57269	0.258829
$857$	12.6495	0.432099	0.216049	−	0.976382i	$-0.430683\pi$
$857$	12.6495	0.432099	0.216049	+	0.976382i	$0.430683\pi$
$858$	0	0
$859$	−11.0929	−0.378485	−0.189243	−	0.981930i	$-0.560603\pi$
$859$	−11.0929	−0.378485	−0.189243	+	0.981930i	$0.560603\pi$
$860$	0	0
$861$	−0.437196	−0.0148996
$862$	−5.01663	−0.170867
$863$	−33.0295	−1.12434	−0.562169	−	0.827022i	$-0.690033\pi$
$863$	−33.0295	−1.12434	−0.562169	+	0.827022i	$0.690033\pi$
$864$	0.561989	0.0191192
$865$	0	0
$866$	−26.2081	−0.890587
$867$	−0.0752546	−0.00255578
$868$	129.114	4.38240
$869$	−2.88787	−0.0979641
$870$	0	0
$871$	0	0
$872$	−19.3980	−0.656899
$873$	10.1519	0.343590
$874$	8.59114	0.290600
$875$	0	0
$876$	−0.465794	−0.0157377
$877$	15.4397	0.521362	0.260681	−	0.965425i	$-0.416053\pi$
$877$	15.4397	0.521362	0.260681	+	0.965425i	$0.416053\pi$
$878$	−78.7987	−2.65933
$879$	0.190178	0.00641453
$880$	0	0
$881$	−16.6260	−0.560143	−0.280072	−	0.959979i	$-0.590358\pi$
$881$	−16.6260	−0.560143	−0.280072	+	0.959979i	$0.590358\pi$
$882$	−94.2333	−3.17300
$883$	−57.2845	−1.92778	−0.963888	−	0.266309i	$-0.914196\pi$
$883$	−57.2845	−1.92778	−0.963888	+	0.266309i	$0.914196\pi$
$884$	0	0
$885$	0	0
$886$	−11.0036	−0.369674
$887$	−18.4657	−0.620018	−0.310009	−	0.950734i	$-0.600332\pi$
$887$	−18.4657	−0.620018	−0.310009	+	0.950734i	$0.600332\pi$
$888$	0.102796	0.00344960
$889$	−92.0498	−3.08725
$890$	0	0
$891$	−26.3864	−0.883977
$892$	44.6405	1.49467
$893$	21.0919	0.705813
$894$	0.421005	0.0140805
$895$	0	0
$896$	−63.7354	−2.12925
$897$	0	0
$898$	51.8055	1.72877
$899$	80.8109	2.69519
$900$	0	0
$901$	41.7796	1.39188
$902$	46.9862	1.56447
$903$	0.0424451	0.00141249
$904$	4.95708	0.164870
$905$	0	0
$906$	−0.352795	−0.0117208
$907$	−44.9489	−1.49250	−0.746251	−	0.665664i	$-0.768148\pi$
$907$	−44.9489	−1.49250	−0.746251	+	0.665664i	$0.768148\pi$
$908$	−11.5548	−0.383459
$909$	31.6279	1.04903
$910$	0	0
$911$	−44.3728	−1.47014	−0.735069	−	0.677992i	$-0.762850\pi$
$911$	−44.3728	−1.47014	−0.735069	+	0.677992i	$0.762850\pi$
$912$	−0.0489434	−0.00162068
$913$	23.0013	0.761231
$914$	88.7759	2.93645
$915$	0	0
$916$	−64.4931	−2.13091
$917$	−52.7264	−1.74118
$918$	−0.579164	−0.0191153
$919$	25.1680	0.830217	0.415108	−	0.909772i	$-0.363744\pi$
$919$	25.1680	0.830217	0.415108	+	0.909772i	$0.363744\pi$
$920$	0	0
$921$	−0.118171	−0.00389386
$922$	68.8443	2.26726
$923$	0	0
$924$	−0.505469	−0.0166287
$925$	0	0
$926$	−39.1468	−1.28644
$927$	42.3430	1.39073
$928$	−59.2797	−1.94595
$929$	37.5601	1.23231	0.616153	−	0.787626i	$-0.288690\pi$
$929$	37.5601	1.23231	0.616153	+	0.787626i	$0.288690\pi$
$930$	0	0
$931$	35.0397	1.14838
$932$	22.0497	0.722262
$933$	0.0951586	0.00311535
$934$	−58.8102	−1.92433
$935$	0	0
$936$	0	0
$937$	17.1212	0.559326	0.279663	−	0.960098i	$-0.409777\pi$
$937$	17.1212	0.559326	0.279663	+	0.960098i	$0.409777\pi$
$938$	−74.1860	−2.42226
$939$	−0.169378	−0.00552743
$940$	0	0
$941$	−0.208568	−0.00679913	−0.00339956	−	0.999994i	$-0.501082\pi$
$941$	−0.208568	−0.00679913	−0.00339956	+	0.999994i	$0.501082\pi$
$942$	−0.205610	−0.00669913
$943$	−11.5027	−0.374580
$944$	−8.12794	−0.264542
$945$	0	0
$946$	−4.56165	−0.148312
$947$	−0.414059	−0.0134551	−0.00672755	−	0.999977i	$-0.502141\pi$
$947$	−0.414059	−0.0134551	−0.00672755	+	0.999977i	$0.502141\pi$
$948$	0.0368337	0.00119630
$949$	0	0
$950$	0	0
$951$	0.202441	0.00656461
$952$	29.4575	0.954722
$953$	33.8593	1.09681	0.548404	−	0.836213i	$-0.315235\pi$
$953$	33.8593	1.09681	0.548404	+	0.836213i	$0.315235\pi$
$954$	−82.4647	−2.66989
$955$	0	0
$956$	−53.6243	−1.73433
$957$	−0.316368	−0.0102267
$958$	45.1324	1.45816
$959$	14.4039	0.465128
$960$	0	0
$961$	64.6403	2.08517
$962$	0	0
$963$	−11.9143	−0.383932
$964$	64.7899	2.08674
$965$	0	0
$966$	0.210143	0.00676124
$967$	−13.2125	−0.424885	−0.212442	−	0.977174i	$-0.568142\pi$
$967$	−13.2125	−0.424885	−0.212442	+	0.977174i	$0.568142\pi$
$968$	−4.57892	−0.147172
$969$	0.107675	0.00345902
$970$	0	0
$971$	−17.6019	−0.564872	−0.282436	−	0.959286i	$-0.591142\pi$
$971$	−17.6019	−0.564872	−0.282436	+	0.959286i	$0.591142\pi$
$972$	1.00974	0.0323875
$973$	28.0929	0.900616
$974$	−17.9045	−0.573697
$975$	0	0
$976$	−2.92685	−0.0936861
$977$	11.7812	0.376913	0.188457	−	0.982082i	$-0.439652\pi$
$977$	11.7812	0.376913	0.188457	+	0.982082i	$0.439652\pi$
$978$	0.0688260	0.00220081
$979$	−1.21056	−0.0386897
$980$	0	0
$981$	30.5192	0.974404
$982$	7.21210	0.230147
$983$	45.4985	1.45118	0.725589	−	0.688128i	$-0.241567\pi$
$983$	45.4985	1.45118	0.725589	+	0.688128i	$0.241567\pi$
$984$	−0.180864	−0.00576574
$985$	0	0
$986$	61.0914	1.94555
$987$	0.515917	0.0164218
$988$	0	0
$989$	1.11674	0.0355103
$990$	0	0
$991$	33.2685	1.05681	0.528405	−	0.848993i	$-0.322790\pi$
$991$	33.2685	1.05681	0.528405	+	0.848993i	$0.322790\pi$
$992$	−70.1580	−2.22752
$993$	−0.340588	−0.0108082
$994$	67.9130	2.15407
$995$	0	0
$996$	−0.293373	−0.00929589
$997$	39.7870	1.26007	0.630033	−	0.776568i	$-0.283041\pi$
$997$	39.7870	1.26007	0.630033	+	0.776568i	$0.283041\pi$
$998$	19.6149	0.620899
$999$	−0.323470	−0.0102341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bt.1.7		9
5.4	even	2		845.2.a.n.1.3	✓	9
13.12	even	2		4225.2.a.bs.1.3		9
15.14	odd	2		7605.2.a.cs.1.7		9
65.4	even	6		845.2.e.o.146.3		18
65.9	even	6		845.2.e.p.146.7		18
65.19	odd	12		845.2.m.j.361.14		36
65.24	odd	12		845.2.m.j.316.14		36
65.29	even	6		845.2.e.p.191.7		18
65.34	odd	4		845.2.c.h.506.5		18
65.44	odd	4		845.2.c.h.506.14		18
65.49	even	6		845.2.e.o.191.3		18
65.54	odd	12		845.2.m.j.316.5		36
65.59	odd	12		845.2.m.j.361.5		36
65.64	even	2		845.2.a.o.1.7	yes	9
195.194	odd	2		7605.2.a.cp.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.n.1.3	✓	9	5.4	even	2
845.2.a.o.1.7	yes	9	65.64	even	2
845.2.c.h.506.5		18	65.34	odd	4
845.2.c.h.506.14		18	65.44	odd	4
845.2.e.o.146.3		18	65.4	even	6
845.2.e.o.191.3		18	65.49	even	6
845.2.e.p.146.7		18	65.9	even	6
845.2.e.p.191.7		18	65.29	even	6
845.2.m.j.316.5		36	65.54	odd	12
845.2.m.j.316.14		36	65.24	odd	12
845.2.m.j.361.5		36	65.59	odd	12
845.2.m.j.361.14		36	65.19	odd	12
4225.2.a.bs.1.3		9	13.12	even	2
4225.2.a.bt.1.7		9	1.1	even	1	trivial
7605.2.a.cp.1.3		9	195.194	odd	2
7605.2.a.cs.1.7		9	15.14	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20556 q^{2} +0.0130567 q^{3} +2.86449 q^{4} +0.0287972 q^{6} +4.60897 q^{7} +1.90669 q^{8} -2.99983 q^{9} -2.93232 q^{11} +0.0374007 q^{12} +10.1654 q^{14} -1.52367 q^{16} +3.35206 q^{17} -6.61630 q^{18} +2.46021 q^{19} +0.0601778 q^{21} -6.46741 q^{22} +1.58329 q^{23} +0.0248950 q^{24} -0.0783378 q^{27} +13.2024 q^{28} +8.26322 q^{29} +9.77959 q^{31} -7.17392 q^{32} -0.0382863 q^{33} +7.39317 q^{34} -8.59299 q^{36} +4.12917 q^{37} +5.42613 q^{38} -7.26508 q^{41} +0.132726 q^{42} +0.705329 q^{43} -8.39961 q^{44} +3.49204 q^{46} +8.57322 q^{47} -0.0198940 q^{48} +14.2426 q^{49} +0.0437667 q^{51} +12.4639 q^{53} -0.172779 q^{54} +8.78787 q^{56} +0.0321221 q^{57} +18.2250 q^{58} +5.33445 q^{59} +1.92092 q^{61} +21.5695 q^{62} -13.8261 q^{63} -12.7752 q^{64} -0.0844427 q^{66} -7.29793 q^{67} +9.60195 q^{68} +0.0206725 q^{69} +6.68083 q^{71} -5.71974 q^{72} -12.4541 q^{73} +9.10713 q^{74} +7.04724 q^{76} -13.5150 q^{77} +0.984840 q^{79} +8.99847 q^{81} -16.0236 q^{82} -7.84405 q^{83} +0.172379 q^{84} +1.55565 q^{86} +0.107890 q^{87} -5.59102 q^{88} +0.412834 q^{89} +4.53532 q^{92} +0.127689 q^{93} +18.9087 q^{94} -0.0936675 q^{96} -3.38416 q^{97} +31.4129 q^{98} +8.79646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{2} - 7 q^{3} + 17 q^{4} + 2 q^{6} + 7 q^{7} + 12 q^{8} + 16 q^{9} + 9 q^{11} - 12 q^{12} - 2 q^{14} + 37 q^{16} + q^{17} - 10 q^{18} + 4 q^{19} + q^{21} - 12 q^{22} - 14 q^{23} + 35 q^{24}+ \cdots + q^{99}+O(q^{100}) \) 9 * q + 3 * q^2 - 7 * q^3 + 17 * q^4 + 2 * q^6 + 7 * q^7 + 12 * q^8 + 16 * q^9 + 9 * q^11 - 12 * q^12 - 2 * q^14 + 37 * q^16 + q^17 - 10 * q^18 + 4 * q^19 + q^21 - 12 * q^22 - 14 * q^23 + 35 * q^24 - 22 * q^27 + 18 * q^28 + 12 * q^29 + 7 * q^31 + 22 * q^32 - 8 * q^33 + 30 * q^34 + 3 * q^36 - 5 * q^37 + 47 * q^38 + 10 * q^41 + 11 * q^42 - 39 * q^43 + 25 * q^44 - 6 * q^46 + 36 * q^47 + 3 * q^48 + 16 * q^49 + 43 * q^51 + 8 * q^53 + 2 * q^54 - 29 * q^56 - 32 * q^57 + 21 * q^58 + 21 * q^59 - 3 * q^61 + 10 * q^62 + 35 * q^63 + 34 * q^64 - 49 * q^66 + q^67 + 20 * q^68 - 13 * q^69 + q^71 - 3 * q^72 - 15 * q^74 + 5 * q^76 + 4 * q^77 + 39 * q^79 + 29 * q^81 + 4 * q^82 + 7 * q^83 - 12 * q^84 + 24 * q^86 - 16 * q^87 - 42 * q^88 + 19 * q^89 + 27 * q^92 + 31 * q^93 + 16 * q^94 + 7 * q^96 - 34 * q^97 + 48 * q^98 + q^99

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