LMFDB - Embedded newform 2325.2.a.bd.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] 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("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,3,11,15,0,3,8,9,11,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 $ x^11 - 3x^10 - 14x^9 + 44x^8 + 61x^7 - 211x^6 - 83x^5 + 369x^4 + 10x^3 - 168x^2 - 31x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.60996$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.60996 q^{2} +1.00000 q^{3} +4.81187 q^{4} +2.60996 q^{6} +1.35174 q^{7} +7.33885 q^{8} +1.00000 q^{9} +2.89708 q^{11} +4.81187 q^{12} -4.64772 q^{13} +3.52797 q^{14} +9.53034 q^{16} +6.57693 q^{17} +2.60996 q^{18} -4.94592 q^{19} +1.35174 q^{21} +7.56126 q^{22} -3.70309 q^{23} +7.33885 q^{24} -12.1303 q^{26} +1.00000 q^{27} +6.50437 q^{28} -2.99394 q^{29} +1.00000 q^{31} +10.1961 q^{32} +2.89708 q^{33} +17.1655 q^{34} +4.81187 q^{36} -9.62684 q^{37} -12.9086 q^{38} -4.64772 q^{39} -7.00839 q^{41} +3.52797 q^{42} -2.99944 q^{43} +13.9404 q^{44} -9.66491 q^{46} +7.98377 q^{47} +9.53034 q^{48} -5.17281 q^{49} +6.57693 q^{51} -22.3642 q^{52} +13.0752 q^{53} +2.60996 q^{54} +9.92018 q^{56} -4.94592 q^{57} -7.81406 q^{58} +4.50584 q^{59} +7.87056 q^{61} +2.60996 q^{62} +1.35174 q^{63} +7.55057 q^{64} +7.56126 q^{66} -4.37424 q^{67} +31.6473 q^{68} -3.70309 q^{69} -11.0706 q^{71} +7.33885 q^{72} +4.81757 q^{73} -25.1256 q^{74} -23.7991 q^{76} +3.91609 q^{77} -12.1303 q^{78} +5.87005 q^{79} +1.00000 q^{81} -18.2916 q^{82} -0.458163 q^{83} +6.50437 q^{84} -7.82841 q^{86} -2.99394 q^{87} +21.2613 q^{88} +8.57129 q^{89} -6.28249 q^{91} -17.8188 q^{92} +1.00000 q^{93} +20.8373 q^{94} +10.1961 q^{96} -0.397729 q^{97} -13.5008 q^{98} +2.89708 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{2} + 11 q^{3} + 15 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 11 q^{9} + 15 q^{12} + 14 q^{13} - 14 q^{14} + 27 q^{16} + 12 q^{17} + 3 q^{18} + 12 q^{19} + 8 q^{21} + 10 q^{22} + 12 q^{23} + 9 q^{24}+ \cdots + 17 q^{98}+O(q^{100}) $ 11 * q + 3 * q^2 + 11 * q^3 + 15 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 11 * q^9 + 15 * q^12 + 14 * q^13 - 14 * q^14 + 27 * q^16 + 12 * q^17 + 3 * q^18 + 12 * q^19 + 8 * q^21 + 10 * q^22 + 12 * q^23 + 9 * q^24 - 6 * q^26 + 11 * q^27 + 22 * q^28 - 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 + 15 * q^36 + 16 * q^37 + 26 * q^38 + 14 * q^39 - 8 * q^41 - 14 * q^42 + 28 * q^43 - 26 * q^44 + 8 * q^46 + 14 * q^47 + 27 * q^48 + 23 * q^49 + 12 * q^51 + 18 * q^52 + 26 * q^53 + 3 * q^54 - 42 * q^56 + 12 * q^57 + 6 * q^58 - 6 * q^59 + 14 * q^61 + 3 * q^62 + 8 * q^63 + 29 * q^64 + 10 * q^66 + 6 * q^67 + 54 * q^68 + 12 * q^69 - 18 * q^71 + 9 * q^72 + 26 * q^73 - 40 * q^74 + 10 * q^76 + 2 * q^77 - 6 * q^78 + 8 * q^79 + 11 * q^81 - 30 * q^82 + 48 * q^83 + 22 * q^84 - 12 * q^86 - 8 * q^87 + 40 * q^88 - 14 * q^89 - 4 * q^91 - 30 * q^92 + 11 * q^93 - 4 * q^94 + 21 * q^96 + 16 * q^97 + 17 * q^98

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 $

$2$	2.60996	1.84552	0.922759	−	0.385378i	$-0.125929\pi$
$2$	2.60996	1.84552	0.922759	+	0.385378i	$0.125929\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.81187	2.40593
$5$	0	0
$5$	0	0
$6$	2.60996	1.06551
$7$	1.35174	0.510908	0.255454	−	0.966821i	$-0.417775\pi$
$7$	1.35174	0.510908	0.255454	+	0.966821i	$0.417775\pi$
$8$	7.33885	2.59468
$9$	1.00000	0.333333
$10$	0	0
$11$	2.89708	0.873503	0.436752	−	0.899582i	$-0.356129\pi$
$11$	2.89708	0.873503	0.436752	+	0.899582i	$0.356129\pi$
$12$	4.81187	1.38907
$13$	−4.64772	−1.28905	−0.644523	−	0.764585i	$-0.722944\pi$
$13$	−4.64772	−1.28905	−0.644523	+	0.764585i	$0.722944\pi$
$14$	3.52797	0.942889
$15$	0	0
$16$	9.53034	2.38258
$17$	6.57693	1.59514	0.797570	−	0.603226i	$-0.206118\pi$
$17$	6.57693	1.59514	0.797570	+	0.603226i	$0.206118\pi$
$18$	2.60996	0.615172
$19$	−4.94592	−1.13467	−0.567335	−	0.823487i	$-0.692026\pi$
$19$	−4.94592	−1.13467	−0.567335	+	0.823487i	$0.692026\pi$
$20$	0	0
$21$	1.35174	0.294973
$22$	7.56126	1.61207
$23$	−3.70309	−0.772149	−0.386074	−	0.922468i	$-0.626169\pi$
$23$	−3.70309	−0.772149	−0.386074	+	0.922468i	$0.626169\pi$
$24$	7.33885	1.49804
$25$	0	0
$26$	−12.1303	−2.37896
$27$	1.00000	0.192450
$28$	6.50437	1.22921
$29$	−2.99394	−0.555961	−0.277981	−	0.960587i	$-0.589665\pi$
$29$	−2.99394	−0.555961	−0.277981	+	0.960587i	$0.589665\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	10.1961	1.80242
$33$	2.89708	0.504317
$34$	17.1655	2.94386
$35$	0	0
$36$	4.81187	0.801978
$37$	−9.62684	−1.58264	−0.791321	−	0.611400i	$-0.790607\pi$
$37$	−9.62684	−1.58264	−0.791321	+	0.611400i	$0.790607\pi$
$38$	−12.9086	−2.09405
$39$	−4.64772	−0.744231
$40$	0	0
$41$	−7.00839	−1.09453	−0.547264	−	0.836960i	$-0.684330\pi$
$41$	−7.00839	−1.09453	−0.547264	+	0.836960i	$0.684330\pi$
$42$	3.52797	0.544377
$43$	−2.99944	−0.457411	−0.228705	−	0.973496i	$-0.573449\pi$
$43$	−2.99944	−0.457411	−0.228705	+	0.973496i	$0.573449\pi$
$44$	13.9404	2.10159
$45$	0	0
$46$	−9.66491	−1.42501
$47$	7.98377	1.16455	0.582276	−	0.812991i	$-0.302162\pi$
$47$	7.98377	1.16455	0.582276	+	0.812991i	$0.302162\pi$
$48$	9.53034	1.37559
$49$	−5.17281	−0.738973
$50$	0	0
$51$	6.57693	0.920955
$52$	−22.3642	−3.10136
$53$	13.0752	1.79601	0.898006	−	0.439984i	$-0.145016\pi$
$53$	13.0752	1.79601	0.898006	+	0.439984i	$0.145016\pi$
$54$	2.60996	0.355170
$55$	0	0
$56$	9.92018	1.32564
$57$	−4.94592	−0.655103
$58$	−7.81406	−1.02604
$59$	4.50584	0.586610	0.293305	−	0.956019i	$-0.405245\pi$
$59$	4.50584	0.586610	0.293305	+	0.956019i	$0.405245\pi$
$60$	0	0
$61$	7.87056	1.00772	0.503861	−	0.863785i	$-0.331912\pi$
$61$	7.87056	1.00772	0.503861	+	0.863785i	$0.331912\pi$
$62$	2.60996	0.331465
$63$	1.35174	0.170303
$64$	7.55057	0.943821
$65$	0	0
$66$	7.56126	0.930726
$67$	−4.37424	−0.534398	−0.267199	−	0.963641i	$-0.586098\pi$
$67$	−4.37424	−0.534398	−0.267199	+	0.963641i	$0.586098\pi$
$68$	31.6473	3.83780
$69$	−3.70309	−0.445800
$70$	0	0
$71$	−11.0706	−1.31384	−0.656918	−	0.753962i	$-0.728140\pi$
$71$	−11.0706	−1.31384	−0.656918	+	0.753962i	$0.728140\pi$
$72$	7.33885	0.864892
$73$	4.81757	0.563854	0.281927	−	0.959436i	$-0.409026\pi$
$73$	4.81757	0.563854	0.281927	+	0.959436i	$0.409026\pi$
$74$	−25.1256	−2.92079
$75$	0	0
$76$	−23.7991	−2.72994
$77$	3.91609	0.446280
$78$	−12.1303	−1.37349
$79$	5.87005	0.660433	0.330216	−	0.943905i	$-0.392878\pi$
$79$	5.87005	0.660433	0.330216	+	0.943905i	$0.392878\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−18.2916	−2.01997
$83$	−0.458163	−0.0502900	−0.0251450	−	0.999684i	$-0.508005\pi$
$83$	−0.458163	−0.0502900	−0.0251450	+	0.999684i	$0.508005\pi$
$84$	6.50437	0.709685
$85$	0	0
$86$	−7.82841	−0.844159
$87$	−2.99394	−0.320985
$88$	21.2613	2.26646
$89$	8.57129	0.908555	0.454277	−	0.890860i	$-0.349898\pi$
$89$	8.57129	0.908555	0.454277	+	0.890860i	$0.349898\pi$
$90$	0	0
$91$	−6.28249	−0.658584
$92$	−17.8188	−1.85774
$93$	1.00000	0.103695
$94$	20.8373	2.14920
$95$	0	0
$96$	10.1961	1.04063
$97$	−0.397729	−0.0403832	−0.0201916	−	0.999796i	$-0.506428\pi$
$97$	−0.397729	−0.0403832	−0.0201916	+	0.999796i	$0.506428\pi$
$98$	−13.5008	−1.36379
$99$	2.89708	0.291168
$100$	0	0
$101$	11.8759	1.18170	0.590848	−	0.806783i	$-0.298793\pi$
$101$	11.8759	1.18170	0.590848	+	0.806783i	$0.298793\pi$
$102$	17.1655	1.69964
$103$	1.83276	0.180587	0.0902937	−	0.995915i	$-0.471219\pi$
$103$	1.83276	0.180587	0.0902937	+	0.995915i	$0.471219\pi$
$104$	−34.1089	−3.34466
$105$	0	0
$106$	34.1256	3.31457
$107$	−5.21399	−0.504055	−0.252028	−	0.967720i	$-0.581097\pi$
$107$	−5.21399	−0.504055	−0.252028	+	0.967720i	$0.581097\pi$
$108$	4.81187	0.463022
$109$	−2.46632	−0.236231	−0.118115	−	0.993000i	$-0.537685\pi$
$109$	−2.46632	−0.236231	−0.118115	+	0.993000i	$0.537685\pi$
$110$	0	0
$111$	−9.62684	−0.913739
$112$	12.8825	1.21728
$113$	−19.3409	−1.81944	−0.909722	−	0.415219i	$-0.863705\pi$
$113$	−19.3409	−1.81944	−0.909722	+	0.415219i	$0.863705\pi$
$114$	−12.9086	−1.20900
$115$	0	0
$116$	−14.4065	−1.33761
$117$	−4.64772	−0.429682
$118$	11.7600	1.08260
$119$	8.89027	0.814970
$120$	0	0
$121$	−2.60691	−0.236992
$122$	20.5418	1.85977
$123$	−7.00839	−0.631926
$124$	4.81187	0.432118
$125$	0	0
$126$	3.52797	0.314296
$127$	14.3830	1.27629	0.638144	−	0.769917i	$-0.279702\pi$
$127$	14.3830	1.27629	0.638144	+	0.769917i	$0.279702\pi$
$128$	−0.685450	−0.0605858
$129$	−2.99944	−0.264086
$130$	0	0
$131$	−12.9938	−1.13527	−0.567635	−	0.823280i	$-0.692141\pi$
$131$	−12.9938	−1.13527	−0.567635	+	0.823280i	$0.692141\pi$
$132$	13.9404	1.21335
$133$	−6.68557	−0.579712
$134$	−11.4166	−0.986241
$135$	0	0
$136$	48.2671	4.13887
$137$	−14.3381	−1.22498	−0.612491	−	0.790477i	$-0.709833\pi$
$137$	−14.3381	−1.22498	−0.612491	+	0.790477i	$0.709833\pi$
$138$	−9.66491	−0.822732
$139$	−2.98385	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$139$	−2.98385	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$140$	0	0
$141$	7.98377	0.672355
$142$	−28.8937	−2.42471
$143$	−13.4648	−1.12599
$144$	9.53034	0.794195
$145$	0	0
$146$	12.5736	1.04060
$147$	−5.17281	−0.426646
$148$	−46.3231	−3.80773
$149$	−7.49650	−0.614137	−0.307069	−	0.951687i	$-0.599348\pi$
$149$	−7.49650	−0.614137	−0.307069	+	0.951687i	$0.599348\pi$
$150$	0	0
$151$	−3.97530	−0.323505	−0.161753	−	0.986831i	$-0.551715\pi$
$151$	−3.97530	−0.323505	−0.161753	+	0.986831i	$0.551715\pi$
$152$	−36.2973	−2.94410
$153$	6.57693	0.531713
$154$	10.2208	0.823617
$155$	0	0
$156$	−22.3642	−1.79057
$157$	−16.8595	−1.34554	−0.672768	−	0.739854i	$-0.734895\pi$
$157$	−16.8595	−1.34554	−0.672768	+	0.739854i	$0.734895\pi$
$158$	15.3206	1.21884
$159$	13.0752	1.03693
$160$	0	0
$161$	−5.00560	−0.394497
$162$	2.60996	0.205057
$163$	17.8435	1.39761	0.698805	−	0.715312i	$-0.253715\pi$
$163$	17.8435	1.39761	0.698805	+	0.715312i	$0.253715\pi$
$164$	−33.7235	−2.63336
$165$	0	0
$166$	−1.19579	−0.0928110
$167$	1.16540	0.0901810	0.0450905	−	0.998983i	$-0.485642\pi$
$167$	1.16540	0.0901810	0.0450905	+	0.998983i	$0.485642\pi$
$168$	9.92018	0.765359
$169$	8.60132	0.661640
$170$	0	0
$171$	−4.94592	−0.378224
$172$	−14.4329	−1.10050
$173$	22.3609	1.70007	0.850035	−	0.526726i	$-0.176581\pi$
$173$	22.3609	1.70007	0.850035	+	0.526726i	$0.176581\pi$
$174$	−7.81406	−0.592382
$175$	0	0
$176$	27.6102	2.08119
$177$	4.50584	0.338680
$178$	22.3707	1.67675
$179$	15.0243	1.12297	0.561483	−	0.827488i	$-0.310231\pi$
$179$	15.0243	1.12297	0.561483	+	0.827488i	$0.310231\pi$
$180$	0	0
$181$	9.66473	0.718374	0.359187	−	0.933266i	$-0.383054\pi$
$181$	9.66473	0.718374	0.359187	+	0.933266i	$0.383054\pi$
$182$	−16.3970	−1.21543
$183$	7.87056	0.581809
$184$	−27.1765	−2.00347
$185$	0	0
$186$	2.60996	0.191371
$187$	19.0539	1.39336
$188$	38.4168	2.80184
$189$	1.35174	0.0983243
$190$	0	0
$191$	−25.1736	−1.82150	−0.910750	−	0.412959i	$-0.864495\pi$
$191$	−25.1736	−1.82150	−0.910750	+	0.412959i	$0.864495\pi$
$192$	7.55057	0.544916
$193$	20.3877	1.46754	0.733771	−	0.679397i	$-0.237758\pi$
$193$	20.3877	1.46754	0.733771	+	0.679397i	$0.237758\pi$
$194$	−1.03805	−0.0745280
$195$	0	0
$196$	−24.8909	−1.77792
$197$	−11.0317	−0.785974	−0.392987	−	0.919544i	$-0.628558\pi$
$197$	−11.0317	−0.785974	−0.392987	+	0.919544i	$0.628558\pi$
$198$	7.56126	0.537355
$199$	−27.7798	−1.96926	−0.984628	−	0.174665i	$-0.944116\pi$
$199$	−27.7798	−1.96926	−0.984628	+	0.174665i	$0.944116\pi$
$200$	0	0
$201$	−4.37424	−0.308535
$202$	30.9956	2.18084
$203$	−4.04702	−0.284045
$204$	31.6473	2.21576
$205$	0	0
$206$	4.78343	0.333277
$207$	−3.70309	−0.257383
$208$	−44.2943	−3.07126
$209$	−14.3287	−0.991139
$210$	0	0
$211$	−6.73018	−0.463324	−0.231662	−	0.972796i	$-0.574416\pi$
$211$	−6.73018	−0.463324	−0.231662	+	0.972796i	$0.574416\pi$
$212$	62.9159	4.32108
$213$	−11.0706	−0.758543
$214$	−13.6083	−0.930243
$215$	0	0
$216$	7.33885	0.499345
$217$	1.35174	0.0917618
$218$	−6.43698	−0.435968
$219$	4.81757	0.325541
$220$	0	0
$221$	−30.5677	−2.05621
$222$	−25.1256	−1.68632
$223$	20.2081	1.35323	0.676617	−	0.736335i	$-0.263445\pi$
$223$	20.2081	1.35323	0.676617	+	0.736335i	$0.263445\pi$
$224$	13.7824	0.920873
$225$	0	0
$226$	−50.4790	−3.35781
$227$	11.4787	0.761871	0.380936	−	0.924602i	$-0.375602\pi$
$227$	11.4787	0.761871	0.380936	+	0.924602i	$0.375602\pi$
$228$	−23.7991	−1.57613
$229$	8.45308	0.558595	0.279298	−	0.960205i	$-0.409898\pi$
$229$	8.45308	0.558595	0.279298	+	0.960205i	$0.409898\pi$
$230$	0	0
$231$	3.91609	0.257660
$232$	−21.9721	−1.44254
$233$	6.39116	0.418699	0.209349	−	0.977841i	$-0.432865\pi$
$233$	6.39116	0.418699	0.209349	+	0.977841i	$0.432865\pi$
$234$	−12.1303	−0.792986
$235$	0	0
$236$	21.6815	1.41135
$237$	5.87005	0.381301
$238$	23.2032	1.50404
$239$	−18.1098	−1.17143	−0.585714	−	0.810518i	$-0.699186\pi$
$239$	−18.1098	−1.17143	−0.585714	+	0.810518i	$0.699186\pi$
$240$	0	0
$241$	6.62015	0.426442	0.213221	−	0.977004i	$-0.431605\pi$
$241$	6.62015	0.426442	0.213221	+	0.977004i	$0.431605\pi$
$242$	−6.80392	−0.437373
$243$	1.00000	0.0641500
$244$	37.8721	2.42451
$245$	0	0
$246$	−18.2916	−1.16623
$247$	22.9872	1.46264
$248$	7.33885	0.466017
$249$	−0.458163	−0.0290349
$250$	0	0
$251$	20.4490	1.29073	0.645363	−	0.763876i	$-0.276706\pi$
$251$	20.4490	1.29073	0.645363	+	0.763876i	$0.276706\pi$
$252$	6.50437	0.409737
$253$	−10.7282	−0.674474
$254$	37.5391	2.35541
$255$	0	0
$256$	−16.8901	−1.05563
$257$	16.8684	1.05222	0.526110	−	0.850417i	$-0.323650\pi$
$257$	16.8684	1.05222	0.526110	+	0.850417i	$0.323650\pi$
$258$	−7.82841	−0.487375
$259$	−13.0129	−0.808585
$260$	0	0
$261$	−2.99394	−0.185320
$262$	−33.9131	−2.09516
$263$	−19.6858	−1.21388	−0.606940	−	0.794748i	$-0.707603\pi$
$263$	−19.6858	−1.21388	−0.606940	+	0.794748i	$0.707603\pi$
$264$	21.2613	1.30854
$265$	0	0
$266$	−17.4490	−1.06987
$267$	8.57129	0.524554
$268$	−21.0483	−1.28573
$269$	−23.0747	−1.40689	−0.703443	−	0.710751i	$-0.748355\pi$
$269$	−23.0747	−1.40689	−0.703443	+	0.710751i	$0.748355\pi$
$270$	0	0
$271$	−23.0339	−1.39921	−0.699604	−	0.714530i	$-0.746640\pi$
$271$	−23.0339	−1.39921	−0.699604	+	0.714530i	$0.746640\pi$
$272$	62.6804	3.80055
$273$	−6.28249	−0.380234
$274$	−37.4217	−2.26073
$275$	0	0
$276$	−17.8188	−1.07257
$277$	−1.34884	−0.0810437	−0.0405219	−	0.999179i	$-0.512902\pi$
$277$	−1.34884	−0.0810437	−0.0405219	+	0.999179i	$0.512902\pi$
$278$	−7.78772	−0.467077
$279$	1.00000	0.0598684
$280$	0	0
$281$	−18.8851	−1.12659	−0.563296	−	0.826256i	$-0.690467\pi$
$281$	−18.8851	−1.12659	−0.563296	+	0.826256i	$0.690467\pi$
$282$	20.8373	1.24084
$283$	8.54033	0.507670	0.253835	−	0.967248i	$-0.418308\pi$
$283$	8.54033	0.507670	0.253835	+	0.967248i	$0.418308\pi$
$284$	−53.2701	−3.16100
$285$	0	0
$286$	−35.1426	−2.07803
$287$	−9.47349	−0.559203
$288$	10.1961	0.600808
$289$	26.2560	1.54447
$290$	0	0
$291$	−0.397729	−0.0233153
$292$	23.1815	1.35659
$293$	−1.42571	−0.0832907	−0.0416453	−	0.999132i	$-0.513260\pi$
$293$	−1.42571	−0.0832907	−0.0416453	+	0.999132i	$0.513260\pi$
$294$	−13.5008	−0.787383
$295$	0	0
$296$	−70.6499	−4.10644
$297$	2.89708	0.168106
$298$	−19.5655	−1.13340
$299$	17.2110	0.995335
$300$	0	0
$301$	−4.05445	−0.233695
$302$	−10.3754	−0.597035
$303$	11.8759	0.682253
$304$	−47.1362	−2.70345
$305$	0	0
$306$	17.1655	0.981286
$307$	18.1183	1.03406	0.517032	−	0.855966i	$-0.327037\pi$
$307$	18.1183	1.03406	0.517032	+	0.855966i	$0.327037\pi$
$308$	18.8437	1.07372
$309$	1.83276	0.104262
$310$	0	0
$311$	−13.5982	−0.771081	−0.385540	−	0.922691i	$-0.625985\pi$
$311$	−13.5982	−0.771081	−0.385540	+	0.922691i	$0.625985\pi$
$312$	−34.1089	−1.93104
$313$	17.1840	0.971298	0.485649	−	0.874154i	$-0.338583\pi$
$313$	17.1840	0.971298	0.485649	+	0.874154i	$0.338583\pi$
$314$	−44.0026	−2.48321
$315$	0	0
$316$	28.2459	1.58896
$317$	18.9517	1.06443	0.532217	−	0.846608i	$-0.321359\pi$
$317$	18.9517	1.06443	0.532217	+	0.846608i	$0.321359\pi$
$318$	34.1256	1.91367
$319$	−8.67370	−0.485634
$320$	0	0
$321$	−5.21399	−0.291016
$322$	−13.0644	−0.728051
$323$	−32.5289	−1.80996
$324$	4.81187	0.267326
$325$	0	0
$326$	46.5707	2.57931
$327$	−2.46632	−0.136388
$328$	−51.4335	−2.83994
$329$	10.7919	0.594979
$330$	0	0
$331$	22.5785	1.24103	0.620513	−	0.784196i	$-0.286924\pi$
$331$	22.5785	1.24103	0.620513	+	0.784196i	$0.286924\pi$
$332$	−2.20462	−0.120994
$333$	−9.62684	−0.527548
$334$	3.04163	0.166431
$335$	0	0
$336$	12.8825	0.702797
$337$	−20.6429	−1.12449	−0.562245	−	0.826971i	$-0.690062\pi$
$337$	−20.6429	−1.12449	−0.562245	+	0.826971i	$0.690062\pi$
$338$	22.4491	1.22107
$339$	−19.3409	−1.05046
$340$	0	0
$341$	2.89708	0.156886
$342$	−12.9086	−0.698018
$343$	−16.4544	−0.888455
$344$	−22.0125	−1.18683
$345$	0	0
$346$	58.3611	3.13751
$347$	17.8082	0.955992	0.477996	−	0.878362i	$-0.341363\pi$
$347$	17.8082	0.955992	0.477996	+	0.878362i	$0.341363\pi$
$348$	−14.4065	−0.772267
$349$	5.54713	0.296931	0.148466	−	0.988918i	$-0.452567\pi$
$349$	5.54713	0.296931	0.148466	+	0.988918i	$0.452567\pi$
$350$	0	0
$351$	−4.64772	−0.248077
$352$	29.5388	1.57442
$353$	0.894986	0.0476353	0.0238176	−	0.999716i	$-0.492418\pi$
$353$	0.894986	0.0476353	0.0238176	+	0.999716i	$0.492418\pi$
$354$	11.7600	0.625039
$355$	0	0
$356$	41.2439	2.18592
$357$	8.89027	0.470523
$358$	39.2127	2.07245
$359$	−22.6591	−1.19590	−0.597950	−	0.801534i	$-0.704018\pi$
$359$	−22.6591	−1.19590	−0.597950	+	0.801534i	$0.704018\pi$
$360$	0	0
$361$	5.46208	0.287478
$362$	25.2245	1.32577
$363$	−2.60691	−0.136827
$364$	−30.2305	−1.58451
$365$	0	0
$366$	20.5418	1.07374
$367$	4.07528	0.212728	0.106364	−	0.994327i	$-0.466079\pi$
$367$	4.07528	0.212728	0.106364	+	0.994327i	$0.466079\pi$
$368$	−35.2917	−1.83971
$369$	−7.00839	−0.364842
$370$	0	0
$371$	17.6742	0.917596
$372$	4.81187	0.249484
$373$	−2.55182	−0.132128	−0.0660642	−	0.997815i	$-0.521044\pi$
$373$	−2.55182	−0.132128	−0.0660642	+	0.997815i	$0.521044\pi$
$374$	49.7299	2.57147
$375$	0	0
$376$	58.5917	3.02163
$377$	13.9150	0.716660
$378$	3.52797	0.181459
$379$	9.03832	0.464267	0.232134	−	0.972684i	$-0.425429\pi$
$379$	9.03832	0.464267	0.232134	+	0.972684i	$0.425429\pi$
$380$	0	0
$381$	14.3830	0.736865
$382$	−65.7020	−3.36161
$383$	19.3181	0.987110	0.493555	−	0.869715i	$-0.335697\pi$
$383$	19.3181	0.987110	0.493555	+	0.869715i	$0.335697\pi$
$384$	−0.685450	−0.0349792
$385$	0	0
$386$	53.2111	2.70837
$387$	−2.99944	−0.152470
$388$	−1.91382	−0.0971594
$389$	16.5174	0.837468	0.418734	−	0.908109i	$-0.362474\pi$
$389$	16.5174	0.837468	0.418734	+	0.908109i	$0.362474\pi$
$390$	0	0
$391$	−24.3550	−1.23169
$392$	−37.9625	−1.91740
$393$	−12.9938	−0.655448
$394$	−28.7922	−1.45053
$395$	0	0
$396$	13.9404	0.700530
$397$	22.1351	1.11093	0.555464	−	0.831541i	$-0.312541\pi$
$397$	22.1351	1.11093	0.555464	+	0.831541i	$0.312541\pi$
$398$	−72.5040	−3.63430
$399$	−6.68557	−0.334697
$400$	0	0
$401$	24.6373	1.23033	0.615163	−	0.788400i	$-0.289090\pi$
$401$	24.6373	1.23033	0.615163	+	0.788400i	$0.289090\pi$
$402$	−11.4166	−0.569407
$403$	−4.64772	−0.231520
$404$	57.1453	2.84308
$405$	0	0
$406$	−10.5625	−0.524210
$407$	−27.8898	−1.38244
$408$	48.2671	2.38958
$409$	−22.7649	−1.12565	−0.562825	−	0.826576i	$-0.690286\pi$
$409$	−22.7649	−1.12565	−0.562825	+	0.826576i	$0.690286\pi$
$410$	0	0
$411$	−14.3381	−0.707244
$412$	8.81901	0.434481
$413$	6.09070	0.299704
$414$	−9.66491	−0.475005
$415$	0	0
$416$	−47.3884	−2.32341
$417$	−2.98385	−0.146120
$418$	−37.3973	−1.82916
$419$	29.0185	1.41765	0.708823	−	0.705387i	$-0.249227\pi$
$419$	29.0185	1.41765	0.708823	+	0.705387i	$0.249227\pi$
$420$	0	0
$421$	−24.2411	−1.18144	−0.590720	−	0.806876i	$-0.701156\pi$
$421$	−24.2411	−1.18144	−0.590720	+	0.806876i	$0.701156\pi$
$422$	−17.5655	−0.855073
$423$	7.98377	0.388184
$424$	95.9566	4.66007
$425$	0	0
$426$	−28.8937	−1.39990
$427$	10.6389	0.514853
$428$	−25.0890	−1.21272
$429$	−13.4648	−0.650088
$430$	0	0
$431$	−23.7355	−1.14330	−0.571649	−	0.820498i	$-0.693696\pi$
$431$	−23.7355	−1.14330	−0.571649	+	0.820498i	$0.693696\pi$
$432$	9.53034	0.458528
$433$	7.26850	0.349302	0.174651	−	0.984630i	$-0.444120\pi$
$433$	7.26850	0.349302	0.174651	+	0.984630i	$0.444120\pi$
$434$	3.52797	0.169348
$435$	0	0
$436$	−11.8676	−0.568355
$437$	18.3152	0.876135
$438$	12.5736	0.600792
$439$	−24.3863	−1.16390	−0.581948	−	0.813226i	$-0.697709\pi$
$439$	−24.3863	−1.16390	−0.581948	+	0.813226i	$0.697709\pi$
$440$	0	0
$441$	−5.17281	−0.246324
$442$	−79.7805	−3.79477
$443$	12.6288	0.600012	0.300006	−	0.953937i	$-0.403011\pi$
$443$	12.6288	0.600012	0.300006	+	0.953937i	$0.403011\pi$
$444$	−46.3231	−2.19840
$445$	0	0
$446$	52.7422	2.49742
$447$	−7.49650	−0.354572
$448$	10.2064	0.482206
$449$	35.7239	1.68592	0.842958	−	0.537979i	$-0.180812\pi$
$449$	35.7239	1.68592	0.842958	+	0.537979i	$0.180812\pi$
$450$	0	0
$451$	−20.3039	−0.956073
$452$	−93.0661	−4.37746
$453$	−3.97530	−0.186776
$454$	29.9590	1.40605
$455$	0	0
$456$	−36.2973	−1.69978
$457$	20.1012	0.940296	0.470148	−	0.882588i	$-0.344201\pi$
$457$	20.1012	0.940296	0.470148	+	0.882588i	$0.344201\pi$
$458$	22.0622	1.03090
$459$	6.57693	0.306985
$460$	0	0
$461$	17.4892	0.814553	0.407276	−	0.913305i	$-0.366479\pi$
$461$	17.4892	0.814553	0.407276	+	0.913305i	$0.366479\pi$
$462$	10.2208	0.475515
$463$	−20.5115	−0.953250	−0.476625	−	0.879107i	$-0.658140\pi$
$463$	−20.5115	−0.953250	−0.476625	+	0.879107i	$0.658140\pi$
$464$	−28.5333	−1.32462
$465$	0	0
$466$	16.6806	0.772716
$467$	−3.72615	−0.172426	−0.0862129	−	0.996277i	$-0.527477\pi$
$467$	−3.72615	−0.172426	−0.0862129	+	0.996277i	$0.527477\pi$
$468$	−22.3642	−1.03379
$469$	−5.91281	−0.273028
$470$	0	0
$471$	−16.8595	−0.776845
$472$	33.0677	1.52206
$473$	−8.68963	−0.399550
$474$	15.3206	0.703697
$475$	0	0
$476$	42.7788	1.96076
$477$	13.0752	0.598670
$478$	−47.2658	−2.16189
$479$	−10.5607	−0.482529	−0.241265	−	0.970459i	$-0.577562\pi$
$479$	−10.5607	−0.482529	−0.241265	+	0.970459i	$0.577562\pi$
$480$	0	0
$481$	44.7429	2.04010
$482$	17.2783	0.787005
$483$	−5.00560	−0.227763
$484$	−12.5441	−0.570187
$485$	0	0
$486$	2.60996	0.118390
$487$	32.7564	1.48434	0.742168	−	0.670214i	$-0.233798\pi$
$487$	32.7564	1.48434	0.742168	+	0.670214i	$0.233798\pi$
$488$	57.7609	2.61471
$489$	17.8435	0.806911
$490$	0	0
$491$	8.34967	0.376815	0.188408	−	0.982091i	$-0.439667\pi$
$491$	8.34967	0.376815	0.188408	+	0.982091i	$0.439667\pi$
$492$	−33.7235	−1.52037
$493$	−19.6910	−0.886836
$494$	59.9957	2.69933
$495$	0	0
$496$	9.53034	0.427925
$497$	−14.9645	−0.671249
$498$	−1.19579	−0.0535845
$499$	−9.06078	−0.405616	−0.202808	−	0.979218i	$-0.565007\pi$
$499$	−9.06078	−0.405616	−0.202808	+	0.979218i	$0.565007\pi$
$500$	0	0
$501$	1.16540	0.0520660
$502$	53.3709	2.38206
$503$	−13.5436	−0.603879	−0.301939	−	0.953327i	$-0.597634\pi$
$503$	−13.5436	−0.603879	−0.301939	+	0.953327i	$0.597634\pi$
$504$	9.92018	0.441880
$505$	0	0
$506$	−28.0001	−1.24475
$507$	8.60132	0.381998
$508$	69.2093	3.07067
$509$	−5.99518	−0.265732	−0.132866	−	0.991134i	$-0.542418\pi$
$509$	−5.99518	−0.265732	−0.132866	+	0.991134i	$0.542418\pi$
$510$	0	0
$511$	6.51208	0.288077
$512$	−42.7116	−1.88760
$513$	−4.94592	−0.218368
$514$	44.0257	1.94189
$515$	0	0
$516$	−14.4329	−0.635374
$517$	23.1296	1.01724
$518$	−33.9632	−1.49226
$519$	22.3609	0.981536
$520$	0	0
$521$	0.831312	0.0364204	0.0182102	−	0.999834i	$-0.494203\pi$
$521$	0.831312	0.0364204	0.0182102	+	0.999834i	$0.494203\pi$
$522$	−7.81406	−0.342012
$523$	34.3566	1.50231	0.751154	−	0.660127i	$-0.229498\pi$
$523$	34.3566	1.50231	0.751154	+	0.660127i	$0.229498\pi$
$524$	−62.5242	−2.73138
$525$	0	0
$526$	−51.3791	−2.24024
$527$	6.57693	0.286496
$528$	27.6102	1.20158
$529$	−9.28709	−0.403786
$530$	0	0
$531$	4.50584	0.195537
$532$	−32.1701	−1.39475
$533$	32.5731	1.41090
$534$	22.3707	0.968074
$535$	0	0
$536$	−32.1019	−1.38659
$537$	15.0243	0.648345
$538$	−60.2238	−2.59643
$539$	−14.9861	−0.645495
$540$	0	0
$541$	30.0236	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$541$	30.0236	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$542$	−60.1174	−2.58226
$543$	9.66473	0.414753
$544$	67.0587	2.87512
$545$	0	0
$546$	−16.3970	−0.701728
$547$	32.3911	1.38494	0.692472	−	0.721445i	$-0.256522\pi$
$547$	32.3911	1.38494	0.692472	+	0.721445i	$0.256522\pi$
$548$	−68.9928	−2.94723
$549$	7.87056	0.335908
$550$	0	0
$551$	14.8078	0.630833
$552$	−27.1765	−1.15671
$553$	7.93476	0.337420
$554$	−3.52040	−0.149568
$555$	0	0
$556$	−14.3579	−0.608911
$557$	−4.01126	−0.169963	−0.0849814	−	0.996383i	$-0.527083\pi$
$557$	−4.01126	−0.169963	−0.0849814	+	0.996383i	$0.527083\pi$
$558$	2.60996	0.110488
$559$	13.9406	0.589623
$560$	0	0
$561$	19.0539	0.804457
$562$	−49.2893	−2.07914
$563$	21.6277	0.911498	0.455749	−	0.890108i	$-0.349371\pi$
$563$	21.6277	0.911498	0.455749	+	0.890108i	$0.349371\pi$
$564$	38.4168	1.61764
$565$	0	0
$566$	22.2899	0.936914
$567$	1.35174	0.0567675
$568$	−81.2453	−3.40898
$569$	7.83046	0.328270	0.164135	−	0.986438i	$-0.447517\pi$
$569$	7.83046	0.328270	0.164135	+	0.986438i	$0.447517\pi$
$570$	0	0
$571$	16.8246	0.704089	0.352045	−	0.935983i	$-0.385487\pi$
$571$	16.8246	0.704089	0.352045	+	0.935983i	$0.385487\pi$
$572$	−64.7910	−2.70905
$573$	−25.1736	−1.05164
$574$	−24.7254	−1.03202
$575$	0	0
$576$	7.55057	0.314607
$577$	−11.5651	−0.481463	−0.240732	−	0.970592i	$-0.577387\pi$
$577$	−11.5651	−0.481463	−0.240732	+	0.970592i	$0.577387\pi$
$578$	68.5270	2.85035
$579$	20.3877	0.847286
$580$	0	0
$581$	−0.619316	−0.0256935
$582$	−1.03805	−0.0430288
$583$	37.8798	1.56882
$584$	35.3554	1.46302
$585$	0	0
$586$	−3.72103	−0.153714
$587$	4.42457	0.182622	0.0913108	−	0.995822i	$-0.470894\pi$
$587$	4.42457	0.182622	0.0913108	+	0.995822i	$0.470894\pi$
$588$	−24.8909	−1.02648
$589$	−4.94592	−0.203793
$590$	0	0
$591$	−11.0317	−0.453782
$592$	−91.7470	−3.77078
$593$	−22.0225	−0.904356	−0.452178	−	0.891928i	$-0.649353\pi$
$593$	−22.0225	−0.904356	−0.452178	+	0.891928i	$0.649353\pi$
$594$	7.56126	0.310242
$595$	0	0
$596$	−36.0722	−1.47757
$597$	−27.7798	−1.13695
$598$	44.9198	1.83691
$599$	−36.0976	−1.47491	−0.737453	−	0.675398i	$-0.763972\pi$
$599$	−36.0976	−1.47491	−0.737453	+	0.675398i	$0.763972\pi$
$600$	0	0
$601$	−38.4272	−1.56748	−0.783738	−	0.621091i	$-0.786690\pi$
$601$	−38.4272	−1.56748	−0.783738	+	0.621091i	$0.786690\pi$
$602$	−10.5819	−0.431288
$603$	−4.37424	−0.178133
$604$	−19.1286	−0.778332
$605$	0	0
$606$	30.9956	1.25911
$607$	−4.75673	−0.193070	−0.0965349	−	0.995330i	$-0.530776\pi$
$607$	−4.75673	−0.193070	−0.0965349	+	0.995330i	$0.530776\pi$
$608$	−50.4288	−2.04516
$609$	−4.04702	−0.163994
$610$	0	0
$611$	−37.1063	−1.50116
$612$	31.6473	1.27927
$613$	35.7017	1.44198	0.720990	−	0.692946i	$-0.243687\pi$
$613$	35.7017	1.44198	0.720990	+	0.692946i	$0.243687\pi$
$614$	47.2879	1.90838
$615$	0	0
$616$	28.7396	1.15795
$617$	−14.8874	−0.599345	−0.299673	−	0.954042i	$-0.596877\pi$
$617$	−14.8874	−0.599345	−0.299673	+	0.954042i	$0.596877\pi$
$618$	4.78343	0.192418
$619$	−2.66707	−0.107199	−0.0535993	−	0.998563i	$-0.517069\pi$
$619$	−2.66707	−0.107199	−0.0535993	+	0.998563i	$0.517069\pi$
$620$	0	0
$621$	−3.70309	−0.148600
$622$	−35.4906	−1.42304
$623$	11.5861	0.464188
$624$	−44.2943	−1.77319
$625$	0	0
$626$	44.8495	1.79255
$627$	−14.3287	−0.572234
$628$	−81.1257	−3.23727
$629$	−63.3151	−2.52454
$630$	0	0
$631$	33.4741	1.33258	0.666292	−	0.745691i	$-0.267880\pi$
$631$	33.4741	1.33258	0.666292	+	0.745691i	$0.267880\pi$
$632$	43.0794	1.71361
$633$	−6.73018	−0.267500
$634$	49.4631	1.96443
$635$	0	0
$636$	62.9159	2.49478
$637$	24.0418	0.952570
$638$	−22.6380	−0.896246
$639$	−11.0706	−0.437945
$640$	0	0
$641$	−14.9709	−0.591316	−0.295658	−	0.955294i	$-0.595539\pi$
$641$	−14.9709	−0.591316	−0.295658	+	0.955294i	$0.595539\pi$
$642$	−13.6083	−0.537076
$643$	−12.3887	−0.488563	−0.244281	−	0.969704i	$-0.578552\pi$
$643$	−12.3887	−0.488563	−0.244281	+	0.969704i	$0.578552\pi$
$644$	−24.0863	−0.949133
$645$	0	0
$646$	−84.8991	−3.34031
$647$	−22.7051	−0.892630	−0.446315	−	0.894876i	$-0.647264\pi$
$647$	−22.7051	−0.892630	−0.446315	+	0.894876i	$0.647264\pi$
$648$	7.33885	0.288297
$649$	13.0538	0.512406
$650$	0	0
$651$	1.35174	0.0529787
$652$	85.8605	3.36256
$653$	−39.7149	−1.55416	−0.777082	−	0.629399i	$-0.783301\pi$
$653$	−39.7149	−1.55416	−0.777082	+	0.629399i	$0.783301\pi$
$654$	−6.43698	−0.251706
$655$	0	0
$656$	−66.7923	−2.60780
$657$	4.81757	0.187951
$658$	28.1665	1.09804
$659$	46.4837	1.81075	0.905374	−	0.424615i	$-0.139591\pi$
$659$	46.4837	1.81075	0.905374	+	0.424615i	$0.139591\pi$
$660$	0	0
$661$	−16.8797	−0.656545	−0.328272	−	0.944583i	$-0.606466\pi$
$661$	−16.8797	−0.656545	−0.328272	+	0.944583i	$0.606466\pi$
$662$	58.9289	2.29034
$663$	−30.5677	−1.18715
$664$	−3.36239	−0.130486
$665$	0	0
$666$	−25.1256	−0.973598
$667$	11.0869	0.429285
$668$	5.60773	0.216970
$669$	20.2081	0.781290
$670$	0	0
$671$	22.8017	0.880249
$672$	13.7824	0.531666
$673$	−16.3739	−0.631167	−0.315583	−	0.948898i	$-0.602200\pi$
$673$	−16.3739	−0.631167	−0.315583	+	0.948898i	$0.602200\pi$
$674$	−53.8770	−2.07527
$675$	0	0
$676$	41.3884	1.59186
$677$	−7.91943	−0.304368	−0.152184	−	0.988352i	$-0.548631\pi$
$677$	−7.91943	−0.304368	−0.152184	+	0.988352i	$0.548631\pi$
$678$	−50.4790	−1.93863
$679$	−0.537624	−0.0206321
$680$	0	0
$681$	11.4787	0.439867
$682$	7.56126	0.289535
$683$	0.937540	0.0358740	0.0179370	−	0.999839i	$-0.494290\pi$
$683$	0.937540	0.0358740	0.0179370	+	0.999839i	$0.494290\pi$
$684$	−23.7991	−0.909981
$685$	0	0
$686$	−42.9453	−1.63966
$687$	8.45308	0.322505
$688$	−28.5857	−1.08982
$689$	−60.7697	−2.31514
$690$	0	0
$691$	−23.9525	−0.911196	−0.455598	−	0.890186i	$-0.650575\pi$
$691$	−23.9525	−0.911196	−0.455598	+	0.890186i	$0.650575\pi$
$692$	107.598	4.09026
$693$	3.91609	0.148760
$694$	46.4785	1.76430
$695$	0	0
$696$	−21.9721	−0.832851
$697$	−46.0937	−1.74592
$698$	14.4778	0.547992
$699$	6.39116	0.241736
$700$	0	0
$701$	33.9432	1.28202	0.641009	−	0.767534i	$-0.278516\pi$
$701$	33.9432	1.28202	0.641009	+	0.767534i	$0.278516\pi$
$702$	−12.1303	−0.457830
$703$	47.6135	1.79578
$704$	21.8746	0.824431
$705$	0	0
$706$	2.33587	0.0879117
$707$	16.0531	0.603738
$708$	21.6815	0.814841
$709$	34.5150	1.29624	0.648118	−	0.761540i	$-0.275556\pi$
$709$	34.5150	1.29624	0.648118	+	0.761540i	$0.275556\pi$
$710$	0	0
$711$	5.87005	0.220144
$712$	62.9034	2.35740
$713$	−3.70309	−0.138682
$714$	23.2032	0.868358
$715$	0	0
$716$	72.2948	2.70178
$717$	−18.1098	−0.676324
$718$	−59.1391	−2.20705
$719$	14.8414	0.553489	0.276745	−	0.960943i	$-0.410744\pi$
$719$	14.8414	0.553489	0.276745	+	0.960943i	$0.410744\pi$
$720$	0	0
$721$	2.47741	0.0922636
$722$	14.2558	0.530545
$723$	6.62015	0.246206
$724$	46.5054	1.72836
$725$	0	0
$726$	−6.80392	−0.252517
$727$	−27.3231	−1.01336	−0.506679	−	0.862135i	$-0.669127\pi$
$727$	−27.3231	−1.01336	−0.506679	+	0.862135i	$0.669127\pi$
$728$	−46.1062	−1.70881
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.7271	−0.729634
$732$	37.8721	1.39979
$733$	41.6761	1.53934	0.769670	−	0.638441i	$-0.220421\pi$
$733$	41.6761	1.53934	0.769670	+	0.638441i	$0.220421\pi$
$734$	10.6363	0.392593
$735$	0	0
$736$	−37.7569	−1.39174
$737$	−12.6725	−0.466799
$738$	−18.2916	−0.673323
$739$	3.72185	0.136910	0.0684552	−	0.997654i	$-0.478193\pi$
$739$	3.72185	0.136910	0.0684552	+	0.997654i	$0.478193\pi$
$740$	0	0
$741$	22.9872	0.844457
$742$	46.1287	1.69344
$743$	21.7070	0.796351	0.398175	−	0.917309i	$-0.369644\pi$
$743$	21.7070	0.796351	0.398175	+	0.917309i	$0.369644\pi$
$744$	7.33885	0.269055
$745$	0	0
$746$	−6.66014	−0.243845
$747$	−0.458163	−0.0167633
$748$	91.6849	3.35233
$749$	−7.04793	−0.257526
$750$	0	0
$751$	24.8253	0.905887	0.452943	−	0.891539i	$-0.350374\pi$
$751$	24.8253	0.905887	0.452943	+	0.891539i	$0.350374\pi$
$752$	76.0880	2.77464
$753$	20.4490	0.745201
$754$	36.3176	1.32261
$755$	0	0
$756$	6.50437	0.236562
$757$	−20.0128	−0.727377	−0.363688	−	0.931521i	$-0.618483\pi$
$757$	−20.0128	−0.727377	−0.363688	+	0.931521i	$0.618483\pi$
$758$	23.5896	0.856814
$759$	−10.7282	−0.389408
$760$	0	0
$761$	−1.88654	−0.0683871	−0.0341935	−	0.999415i	$-0.510886\pi$
$761$	−1.88654	−0.0683871	−0.0341935	+	0.999415i	$0.510886\pi$
$762$	37.5391	1.35990
$763$	−3.33381	−0.120692
$764$	−121.132	−4.38241
$765$	0	0
$766$	50.4194	1.82173
$767$	−20.9419	−0.756168
$768$	−16.8901	−0.609470
$769$	53.3296	1.92311	0.961557	−	0.274604i	$-0.0885467\pi$
$769$	53.3296	1.92311	0.961557	+	0.274604i	$0.0885467\pi$
$770$	0	0
$771$	16.8684	0.607499
$772$	98.1031	3.53081
$773$	20.2423	0.728065	0.364033	−	0.931386i	$-0.381400\pi$
$773$	20.2423	0.728065	0.364033	+	0.931386i	$0.381400\pi$
$774$	−7.82841	−0.281386
$775$	0	0
$776$	−2.91887	−0.104781
$777$	−13.0129	−0.466837
$778$	43.1098	1.54556
$779$	34.6629	1.24193
$780$	0	0
$781$	−32.0724	−1.14764
$782$	−63.5655	−2.27310
$783$	−2.99394	−0.106995
$784$	−49.2986	−1.76067
$785$	0	0
$786$	−33.9131	−1.20964
$787$	49.6194	1.76874	0.884370	−	0.466787i	$-0.154589\pi$
$787$	49.6194	1.76874	0.884370	+	0.466787i	$0.154589\pi$
$788$	−53.0829	−1.89100
$789$	−19.6858	−0.700834
$790$	0	0
$791$	−26.1438	−0.929568
$792$	21.2613	0.755486
$793$	−36.5802	−1.29900
$794$	57.7716	2.05024
$795$	0	0
$796$	−133.673	−4.73790
$797$	−37.6997	−1.33539	−0.667695	−	0.744435i	$-0.732719\pi$
$797$	−37.6997	−1.33539	−0.667695	+	0.744435i	$0.732719\pi$
$798$	−17.4490	−0.617689
$799$	52.5087	1.85762
$800$	0	0
$801$	8.57129	0.302852
$802$	64.3022	2.27059
$803$	13.9569	0.492528
$804$	−21.0483	−0.742315
$805$	0	0
$806$	−12.1303	−0.427273
$807$	−23.0747	−0.812266
$808$	87.1555	3.06612
$809$	−5.44384	−0.191395	−0.0956977	−	0.995410i	$-0.530508\pi$
$809$	−5.44384	−0.191395	−0.0956977	+	0.995410i	$0.530508\pi$
$810$	0	0
$811$	29.4222	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$811$	29.4222	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$812$	−19.4737	−0.683394
$813$	−23.0339	−0.807834
$814$	−72.7910	−2.55132
$815$	0	0
$816$	62.6804	2.19425
$817$	14.8350	0.519010
$818$	−59.4153	−2.07741
$819$	−6.28249	−0.219528
$820$	0	0
$821$	3.90102	0.136146	0.0680732	−	0.997680i	$-0.478315\pi$
$821$	3.90102	0.136146	0.0680732	+	0.997680i	$0.478315\pi$
$822$	−37.4217	−1.30523
$823$	−45.1123	−1.57252	−0.786258	−	0.617898i	$-0.787984\pi$
$823$	−45.1123	−1.57252	−0.786258	+	0.617898i	$0.787984\pi$
$824$	13.4504	0.468566
$825$	0	0
$826$	15.8965	0.553108
$827$	−9.38902	−0.326488	−0.163244	−	0.986586i	$-0.552196\pi$
$827$	−9.38902	−0.326488	−0.163244	+	0.986586i	$0.552196\pi$
$828$	−17.8188	−0.619246
$829$	−19.6667	−0.683054	−0.341527	−	0.939872i	$-0.610944\pi$
$829$	−19.6667	−0.683054	−0.341527	+	0.939872i	$0.610944\pi$
$830$	0	0
$831$	−1.34884	−0.0467906
$832$	−35.0930	−1.21663
$833$	−34.0212	−1.17877
$834$	−7.78772	−0.269667
$835$	0	0
$836$	−68.9479	−2.38461
$837$	1.00000	0.0345651
$838$	75.7370	2.61629
$839$	12.8568	0.443865	0.221933	−	0.975062i	$-0.428763\pi$
$839$	12.8568	0.443865	0.221933	+	0.975062i	$0.428763\pi$
$840$	0	0
$841$	−20.0363	−0.690907
$842$	−63.2683	−2.18037
$843$	−18.8851	−0.650438
$844$	−32.3847	−1.11473
$845$	0	0
$846$	20.8373	0.716400
$847$	−3.52385	−0.121081
$848$	124.611	4.27915
$849$	8.54033	0.293103
$850$	0	0
$851$	35.6491	1.22204
$852$	−53.2701	−1.82500
$853$	10.3662	0.354933	0.177467	−	0.984127i	$-0.443210\pi$
$853$	10.3662	0.354933	0.177467	+	0.984127i	$0.443210\pi$
$854$	27.7671	0.950171
$855$	0	0
$856$	−38.2647	−1.30786
$857$	−10.2071	−0.348667	−0.174333	−	0.984687i	$-0.555777\pi$
$857$	−10.2071	−0.348667	−0.174333	+	0.984687i	$0.555777\pi$
$858$	−35.1426	−1.19975
$859$	−3.34065	−0.113981	−0.0569907	−	0.998375i	$-0.518151\pi$
$859$	−3.34065	−0.113981	−0.0569907	+	0.998375i	$0.518151\pi$
$860$	0	0
$861$	−9.47349	−0.322856
$862$	−61.9485	−2.10998
$863$	29.7744	1.01353	0.506765	−	0.862084i	$-0.330841\pi$
$863$	29.7744	1.01353	0.506765	+	0.862084i	$0.330841\pi$
$864$	10.1961	0.346877
$865$	0	0
$866$	18.9705	0.644643
$867$	26.2560	0.891701
$868$	6.50437	0.220773
$869$	17.0060	0.576890
$870$	0	0
$871$	20.3302	0.688864
$872$	−18.0999	−0.612941
$873$	−0.397729	−0.0134611
$874$	47.8018	1.61692
$875$	0	0
$876$	23.1815	0.783230
$877$	−11.2457	−0.379741	−0.189870	−	0.981809i	$-0.560807\pi$
$877$	−11.2457	−0.379741	−0.189870	+	0.981809i	$0.560807\pi$
$878$	−63.6473	−2.14799
$879$	−1.42571	−0.0480879
$880$	0	0
$881$	37.8238	1.27432	0.637158	−	0.770733i	$-0.280110\pi$
$881$	37.8238	1.27432	0.637158	+	0.770733i	$0.280110\pi$
$882$	−13.5008	−0.454596
$883$	4.37102	0.147097	0.0735483	−	0.997292i	$-0.476568\pi$
$883$	4.37102	0.147097	0.0735483	+	0.997292i	$0.476568\pi$
$884$	−147.088	−4.94710
$885$	0	0
$886$	32.9606	1.10733
$887$	−17.0810	−0.573525	−0.286763	−	0.958002i	$-0.592579\pi$
$887$	−17.0810	−0.573525	−0.286763	+	0.958002i	$0.592579\pi$
$888$	−70.6499	−2.37086
$889$	19.4421	0.652066
$890$	0	0
$891$	2.89708	0.0970559
$892$	97.2386	3.25579
$893$	−39.4870	−1.32138
$894$	−19.5655	−0.654369
$895$	0	0
$896$	−0.926547	−0.0309538
$897$	17.2110	0.574657
$898$	93.2378	3.11139
$899$	−2.99394	−0.0998536
$900$	0	0
$901$	85.9944	2.86489
$902$	−52.9923	−1.76445
$903$	−4.05445	−0.134924
$904$	−141.940	−4.72086
$905$	0	0
$906$	−10.3754	−0.344698
$907$	4.72624	0.156932	0.0784661	−	0.996917i	$-0.474998\pi$
$907$	4.72624	0.156932	0.0784661	+	0.996917i	$0.474998\pi$
$908$	55.2342	1.83301
$909$	11.8759	0.393899
$910$	0	0
$911$	31.3184	1.03762	0.518812	−	0.854888i	$-0.326374\pi$
$911$	31.3184	1.03762	0.518812	+	0.854888i	$0.326374\pi$
$912$	−47.1362	−1.56084
$913$	−1.32734	−0.0439285
$914$	52.4633	1.73533
$915$	0	0
$916$	40.6751	1.34394
$917$	−17.5641	−0.580018
$918$	17.1655	0.566546
$919$	6.50359	0.214534	0.107267	−	0.994230i	$-0.465790\pi$
$919$	6.50359	0.214534	0.107267	+	0.994230i	$0.465790\pi$
$920$	0	0
$921$	18.1183	0.597017
$922$	45.6460	1.50327
$923$	51.4529	1.69359
$924$	18.8437	0.619912
$925$	0	0
$926$	−53.5341	−1.75924
$927$	1.83276	0.0601958
$928$	−30.5264	−1.00208
$929$	−21.6418	−0.710044	−0.355022	−	0.934858i	$-0.615527\pi$
$929$	−21.6418	−0.710044	−0.355022	+	0.934858i	$0.615527\pi$
$930$	0	0
$931$	25.5843	0.838491
$932$	30.7534	1.00736
$933$	−13.5982	−0.445184
$934$	−9.72509	−0.318215
$935$	0	0
$936$	−34.1089	−1.11489
$937$	11.6829	0.381663	0.190831	−	0.981623i	$-0.438882\pi$
$937$	11.6829	0.381663	0.190831	+	0.981623i	$0.438882\pi$
$938$	−15.4322	−0.503878
$939$	17.1840	0.560779
$940$	0	0
$941$	−24.2911	−0.791866	−0.395933	−	0.918279i	$-0.629579\pi$
$941$	−24.2911	−0.791866	−0.395933	+	0.918279i	$0.629579\pi$
$942$	−44.0026	−1.43368
$943$	25.9527	0.845138
$944$	42.9421	1.39765
$945$	0	0
$946$	−22.6795	−0.737376
$947$	55.8068	1.81348	0.906739	−	0.421692i	$-0.138564\pi$
$947$	55.8068	1.81348	0.906739	+	0.421692i	$0.138564\pi$
$948$	28.2459	0.917385
$949$	−22.3907	−0.726833
$950$	0	0
$951$	18.9517	0.614551
$952$	65.2443	2.11458
$953$	−27.5385	−0.892059	−0.446029	−	0.895018i	$-0.647162\pi$
$953$	−27.5385	−0.892059	−0.446029	+	0.895018i	$0.647162\pi$
$954$	34.1256	1.10486
$955$	0	0
$956$	−87.1421	−2.81838
$957$	−8.67370	−0.280381
$958$	−27.5629	−0.890516
$959$	−19.3813	−0.625853
$960$	0	0
$961$	1.00000	0.0322581
$962$	116.777	3.76504
$963$	−5.21399	−0.168018
$964$	31.8553	1.02599
$965$	0	0
$966$	−13.0644	−0.420340
$967$	−40.2348	−1.29386	−0.646931	−	0.762548i	$-0.723948\pi$
$967$	−40.2348	−1.29386	−0.646931	+	0.762548i	$0.723948\pi$
$968$	−19.1317	−0.614917
$969$	−32.5289	−1.04498
$970$	0	0
$971$	−44.6382	−1.43251	−0.716254	−	0.697840i	$-0.754145\pi$
$971$	−44.6382	−1.43251	−0.716254	+	0.697840i	$0.754145\pi$
$972$	4.81187	0.154341
$973$	−4.03338	−0.129304
$974$	85.4928	2.73937
$975$	0	0
$976$	75.0091	2.40098
$977$	59.0405	1.88887	0.944436	−	0.328696i	$-0.106609\pi$
$977$	59.0405	1.88887	0.944436	+	0.328696i	$0.106609\pi$
$978$	46.5707	1.48917
$979$	24.8317	0.793625
$980$	0	0
$981$	−2.46632	−0.0787435
$982$	21.7923	0.695419
$983$	16.1265	0.514354	0.257177	−	0.966364i	$-0.417208\pi$
$983$	16.1265	0.514354	0.257177	+	0.966364i	$0.417208\pi$
$984$	−51.4335	−1.63964
$985$	0	0
$986$	−51.3925	−1.63667
$987$	10.7919	0.343511
$988$	110.612	3.51902
$989$	11.1072	0.353189
$990$	0	0
$991$	62.0192	1.97011	0.985053	−	0.172253i	$-0.0551048\pi$
$991$	62.0192	1.97011	0.985053	+	0.172253i	$0.0551048\pi$
$992$	10.1961	0.323725
$993$	22.5785	0.716507
$994$	−39.0566	−1.23880
$995$	0	0
$996$	−2.20462	−0.0698561
$997$	0.590602	0.0187046	0.00935228	−	0.999956i	$-0.497023\pi$
$997$	0.590602	0.0187046	0.00935228	+	0.999956i	$0.497023\pi$
$998$	−23.6482	−0.748572
$999$	−9.62684	−0.304580

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.bd.1.10		11
3.2	odd	2		6975.2.a.ci.1.2		11
5.2	odd	4		465.2.c.b.94.20	yes	22
5.3	odd	4		465.2.c.b.94.3	✓	22
5.4	even	2		2325.2.a.bc.1.2		11
15.2	even	4		1395.2.c.h.559.3		22
15.8	even	4		1395.2.c.h.559.20		22
15.14	odd	2		6975.2.a.cj.1.10		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.b.94.3	✓	22	5.3	odd	4
465.2.c.b.94.20	yes	22	5.2	odd	4
1395.2.c.h.559.3		22	15.2	even	4
1395.2.c.h.559.20		22	15.8	even	4
2325.2.a.bc.1.2		11	5.4	even	2
2325.2.a.bd.1.10		11	1.1	even	1	trivial
6975.2.a.ci.1.2		11	3.2	odd	2
6975.2.a.cj.1.10		11	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:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60996 q^{2} +1.00000 q^{3} +4.81187 q^{4} +2.60996 q^{6} +1.35174 q^{7} +7.33885 q^{8} +1.00000 q^{9} +2.89708 q^{11} +4.81187 q^{12} -4.64772 q^{13} +3.52797 q^{14} +9.53034 q^{16} +6.57693 q^{17} +2.60996 q^{18} -4.94592 q^{19} +1.35174 q^{21} +7.56126 q^{22} -3.70309 q^{23} +7.33885 q^{24} -12.1303 q^{26} +1.00000 q^{27} +6.50437 q^{28} -2.99394 q^{29} +1.00000 q^{31} +10.1961 q^{32} +2.89708 q^{33} +17.1655 q^{34} +4.81187 q^{36} -9.62684 q^{37} -12.9086 q^{38} -4.64772 q^{39} -7.00839 q^{41} +3.52797 q^{42} -2.99944 q^{43} +13.9404 q^{44} -9.66491 q^{46} +7.98377 q^{47} +9.53034 q^{48} -5.17281 q^{49} +6.57693 q^{51} -22.3642 q^{52} +13.0752 q^{53} +2.60996 q^{54} +9.92018 q^{56} -4.94592 q^{57} -7.81406 q^{58} +4.50584 q^{59} +7.87056 q^{61} +2.60996 q^{62} +1.35174 q^{63} +7.55057 q^{64} +7.56126 q^{66} -4.37424 q^{67} +31.6473 q^{68} -3.70309 q^{69} -11.0706 q^{71} +7.33885 q^{72} +4.81757 q^{73} -25.1256 q^{74} -23.7991 q^{76} +3.91609 q^{77} -12.1303 q^{78} +5.87005 q^{79} +1.00000 q^{81} -18.2916 q^{82} -0.458163 q^{83} +6.50437 q^{84} -7.82841 q^{86} -2.99394 q^{87} +21.2613 q^{88} +8.57129 q^{89} -6.28249 q^{91} -17.8188 q^{92} +1.00000 q^{93} +20.8373 q^{94} +10.1961 q^{96} -0.397729 q^{97} -13.5008 q^{98} +2.89708 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{2} + 11 q^{3} + 15 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 11 q^{9} + 15 q^{12} + 14 q^{13} - 14 q^{14} + 27 q^{16} + 12 q^{17} + 3 q^{18} + 12 q^{19} + 8 q^{21} + 10 q^{22} + 12 q^{23} + 9 q^{24}+ \cdots + 17 q^{98}+O(q^{100}) \) 11 * q + 3 * q^2 + 11 * q^3 + 15 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 11 * q^9 + 15 * q^12 + 14 * q^13 - 14 * q^14 + 27 * q^16 + 12 * q^17 + 3 * q^18 + 12 * q^19 + 8 * q^21 + 10 * q^22 + 12 * q^23 + 9 * q^24 - 6 * q^26 + 11 * q^27 + 22 * q^28 - 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 + 15 * q^36 + 16 * q^37 + 26 * q^38 + 14 * q^39 - 8 * q^41 - 14 * q^42 + 28 * q^43 - 26 * q^44 + 8 * q^46 + 14 * q^47 + 27 * q^48 + 23 * q^49 + 12 * q^51 + 18 * q^52 + 26 * q^53 + 3 * q^54 - 42 * q^56 + 12 * q^57 + 6 * q^58 - 6 * q^59 + 14 * q^61 + 3 * q^62 + 8 * q^63 + 29 * q^64 + 10 * q^66 + 6 * q^67 + 54 * q^68 + 12 * q^69 - 18 * q^71 + 9 * q^72 + 26 * q^73 - 40 * q^74 + 10 * q^76 + 2 * q^77 - 6 * q^78 + 8 * q^79 + 11 * q^81 - 30 * q^82 + 48 * q^83 + 22 * q^84 - 12 * q^86 - 8 * q^87 + 40 * q^88 - 14 * q^89 - 4 * q^91 - 30 * q^92 + 11 * q^93 - 4 * q^94 + 21 * q^96 + 16 * q^97 + 17 * q^98

Introduction

L-functions

Modular forms

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Database

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