LMFDB - Embedded newform 7225.2.a.bq.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.360254$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.360254 q^{2} +0.0542373 q^{3} -1.87022 q^{4} -0.0195392 q^{6} +0.298718 q^{7} +1.39426 q^{8} -2.99706 q^{9} -2.76730 q^{11} -0.101435 q^{12} +1.97956 q^{13} -0.107615 q^{14} +3.23814 q^{16} +1.07970 q^{18} +2.81896 q^{19} +0.0162017 q^{21} +0.996933 q^{22} -6.73975 q^{23} +0.0756210 q^{24} -0.713144 q^{26} -0.325264 q^{27} -0.558668 q^{28} +4.72608 q^{29} +3.02620 q^{31} -3.95508 q^{32} -0.150091 q^{33} +5.60515 q^{36} -9.42129 q^{37} -1.01554 q^{38} +0.107366 q^{39} -3.10972 q^{41} -0.00583672 q^{42} +8.18506 q^{43} +5.17546 q^{44} +2.42803 q^{46} -1.08341 q^{47} +0.175628 q^{48} -6.91077 q^{49} -3.70220 q^{52} -2.68500 q^{53} +0.117178 q^{54} +0.416492 q^{56} +0.152893 q^{57} -1.70259 q^{58} -9.15435 q^{59} +11.1939 q^{61} -1.09020 q^{62} -0.895276 q^{63} -5.05145 q^{64} +0.0540709 q^{66} -12.5585 q^{67} -0.365546 q^{69} +5.88778 q^{71} -4.17869 q^{72} -0.216666 q^{73} +3.39406 q^{74} -5.27208 q^{76} -0.826644 q^{77} -0.0386790 q^{78} -10.6556 q^{79} +8.97353 q^{81} +1.12029 q^{82} +15.5747 q^{83} -0.0303006 q^{84} -2.94870 q^{86} +0.256329 q^{87} -3.85835 q^{88} +1.55264 q^{89} +0.591330 q^{91} +12.6048 q^{92} +0.164133 q^{93} +0.390304 q^{94} -0.214513 q^{96} -8.97035 q^{97} +2.48963 q^{98} +8.29377 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * 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 $

$2$	−0.360254	−0.254738	−0.127369	−	0.991855i	$-0.540653\pi$
$2$	−0.360254	−0.254738	−0.127369	+	0.991855i	$0.540653\pi$
$3$	0.0542373	0.0313139	0.0156569	−	0.999877i	$-0.495016\pi$
$3$	0.0542373	0.0313139	0.0156569	+	0.999877i	$0.495016\pi$
$4$	−1.87022	−0.935108
$5$	0	0
$5$	0	0
$6$	−0.0195392	−0.00797685
$7$	0.298718	0.112905	0.0564524	−	0.998405i	$-0.482021\pi$
$7$	0.298718	0.112905	0.0564524	+	0.998405i	$0.482021\pi$
$8$	1.39426	0.492946
$9$	−2.99706	−0.999019
$10$	0	0
$11$	−2.76730	−0.834373	−0.417187	−	0.908821i	$-0.636984\pi$
$11$	−2.76730	−0.834373	−0.417187	+	0.908821i	$0.636984\pi$
$12$	−0.101435	−0.0292819
$13$	1.97956	0.549030	0.274515	−	0.961583i	$-0.411483\pi$
$13$	1.97956	0.549030	0.274515	+	0.961583i	$0.411483\pi$
$14$	−0.107615	−0.0287612
$15$	0	0
$16$	3.23814	0.809536
$17$	0	0
$17$	0	0
$18$	1.07970	0.254489
$19$	2.81896	0.646715	0.323357	−	0.946277i	$-0.395188\pi$
$19$	2.81896	0.646715	0.323357	+	0.946277i	$0.395188\pi$
$20$	0	0
$21$	0.0162017	0.00353549
$22$	0.996933	0.212547
$23$	−6.73975	−1.40534	−0.702668	−	0.711518i	$-0.748008\pi$
$23$	−6.73975	−1.40534	−0.702668	+	0.711518i	$0.748008\pi$
$24$	0.0756210	0.0154361
$25$	0	0
$26$	−0.713144	−0.139859
$27$	−0.325264	−0.0625971
$28$	−0.558668	−0.105578
$29$	4.72608	0.877610	0.438805	−	0.898582i	$-0.355402\pi$
$29$	4.72608	0.877610	0.438805	+	0.898582i	$0.355402\pi$
$30$	0	0
$31$	3.02620	0.543521	0.271761	−	0.962365i	$-0.412394\pi$
$31$	3.02620	0.543521	0.271761	+	0.962365i	$0.412394\pi$
$32$	−3.95508	−0.699166
$33$	−0.150091	−0.0261275
$34$	0	0
$35$	0	0
$36$	5.60515	0.934191
$37$	−9.42129	−1.54885	−0.774425	−	0.632665i	$-0.781961\pi$
$37$	−9.42129	−1.54885	−0.774425	+	0.632665i	$0.781961\pi$
$38$	−1.01554	−0.164743
$39$	0.107366	0.0171923
$40$	0	0
$41$	−3.10972	−0.485657	−0.242828	−	0.970069i	$-0.578075\pi$
$41$	−3.10972	−0.485657	−0.242828	+	0.970069i	$0.578075\pi$
$42$	−0.00583672	−0.000900625 0
$43$	8.18506	1.24821	0.624105	−	0.781341i	$-0.285464\pi$
$43$	8.18506	1.24821	0.624105	+	0.781341i	$0.285464\pi$
$44$	5.17546	0.780229
$45$	0	0
$46$	2.42803	0.357993
$47$	−1.08341	−0.158032	−0.0790159	−	0.996873i	$-0.525178\pi$
$47$	−1.08341	−0.158032	−0.0790159	+	0.996873i	$0.525178\pi$
$48$	0.175628	0.0253497
$49$	−6.91077	−0.987252
$50$	0	0
$51$	0	0
$52$	−3.70220	−0.513403
$53$	−2.68500	−0.368813	−0.184406	−	0.982850i	$-0.559036\pi$
$53$	−2.68500	−0.368813	−0.184406	+	0.982850i	$0.559036\pi$
$54$	0.117178	0.0159459
$55$	0	0
$56$	0.416492	0.0556560
$57$	0.152893	0.0202512
$58$	−1.70259	−0.223561
$59$	−9.15435	−1.19179	−0.595897	−	0.803061i	$-0.703203\pi$
$59$	−9.15435	−1.19179	−0.595897	+	0.803061i	$0.703203\pi$
$60$	0	0
$61$	11.1939	1.43323	0.716616	−	0.697468i	$-0.245690\pi$
$61$	11.1939	1.43323	0.716616	+	0.697468i	$0.245690\pi$
$62$	−1.09020	−0.138456
$63$	−0.895276	−0.112794
$64$	−5.05145	−0.631432
$65$	0	0
$66$	0.0540709	0.00665567
$67$	−12.5585	−1.53427	−0.767133	−	0.641488i	$-0.778317\pi$
$67$	−12.5585	−1.53427	−0.767133	+	0.641488i	$0.778317\pi$
$68$	0	0
$69$	−0.365546	−0.0440066
$70$	0	0
$71$	5.88778	0.698750	0.349375	−	0.936983i	$-0.386394\pi$
$71$	5.88778	0.698750	0.349375	+	0.936983i	$0.386394\pi$
$72$	−4.17869	−0.492463
$73$	−0.216666	−0.0253588	−0.0126794	−	0.999920i	$-0.504036\pi$
$73$	−0.216666	−0.0253588	−0.0126794	+	0.999920i	$0.504036\pi$
$74$	3.39406	0.394552
$75$	0	0
$76$	−5.27208	−0.604749
$77$	−0.826644	−0.0942048
$78$	−0.0386790	−0.00437953
$79$	−10.6556	−1.19885	−0.599423	−	0.800432i	$-0.704603\pi$
$79$	−10.6556	−1.19885	−0.599423	+	0.800432i	$0.704603\pi$
$80$	0	0
$81$	8.97353	0.997059
$82$	1.12029	0.123715
$83$	15.5747	1.70954	0.854770	−	0.519007i	$-0.173698\pi$
$83$	15.5747	1.70954	0.854770	+	0.519007i	$0.173698\pi$
$84$	−0.0303006	−0.00330607
$85$	0	0
$86$	−2.94870	−0.317967
$87$	0.256329	0.0274814
$88$	−3.85835	−0.411301
$89$	1.55264	0.164579	0.0822897	−	0.996608i	$-0.473777\pi$
$89$	1.55264	0.164579	0.0822897	+	0.996608i	$0.473777\pi$
$90$	0	0
$91$	0.591330	0.0619882
$92$	12.6048	1.31414
$93$	0.164133	0.0170198
$94$	0.390304	0.0402568
$95$	0	0
$96$	−0.214513	−0.0218936
$97$	−8.97035	−0.910801	−0.455401	−	0.890287i	$-0.650504\pi$
$97$	−8.97035	−0.910801	−0.455401	+	0.890287i	$0.650504\pi$
$98$	2.48963	0.251491
$99$	8.29377	0.833555
$100$	0	0
$101$	6.92132	0.688697	0.344349	−	0.938842i	$-0.388100\pi$
$101$	6.92132	0.688697	0.344349	+	0.938842i	$0.388100\pi$
$102$	0	0
$103$	−12.9642	−1.27740	−0.638699	−	0.769457i	$-0.720527\pi$
$103$	−12.9642	−1.27740	−0.638699	+	0.769457i	$0.720527\pi$
$104$	2.76002	0.270643
$105$	0	0
$106$	0.967283	0.0939508
$107$	−3.62363	−0.350309	−0.175155	−	0.984541i	$-0.556043\pi$
$107$	−3.62363	−0.350309	−0.175155	+	0.984541i	$0.556043\pi$
$108$	0.608314	0.0585351
$109$	5.15037	0.493316	0.246658	−	0.969103i	$-0.420668\pi$
$109$	5.15037	0.493316	0.246658	+	0.969103i	$0.420668\pi$
$110$	0	0
$111$	−0.510985	−0.0485005
$112$	0.967293	0.0914006
$113$	−11.0065	−1.03541	−0.517704	−	0.855560i	$-0.673213\pi$
$113$	−11.0065	−1.03541	−0.517704	+	0.855560i	$0.673213\pi$
$114$	−0.0550804	−0.00515875
$115$	0	0
$116$	−8.83879	−0.820661
$117$	−5.93285	−0.548492
$118$	3.29789	0.303596
$119$	0	0
$120$	0	0
$121$	−3.34204	−0.303822
$122$	−4.03265	−0.365099
$123$	−0.168663	−0.0152078
$124$	−5.65965	−0.508251
$125$	0	0
$126$	0.322527	0.0287330
$127$	−7.33231	−0.650637	−0.325319	−	0.945604i	$-0.605472\pi$
$127$	−7.33231	−0.650637	−0.325319	+	0.945604i	$0.605472\pi$
$128$	9.72997	0.860016
$129$	0.443935	0.0390863
$130$	0	0
$131$	17.3220	1.51343	0.756716	−	0.653744i	$-0.226802\pi$
$131$	17.3220	1.51343	0.756716	+	0.653744i	$0.226802\pi$
$132$	0.280703	0.0244320
$133$	0.842076	0.0730173
$134$	4.52426	0.390836
$135$	0	0
$136$	0	0
$137$	13.9745	1.19392	0.596959	−	0.802272i	$-0.296375\pi$
$137$	13.9745	1.19392	0.596959	+	0.802272i	$0.296375\pi$
$138$	0.131690	0.0112102
$139$	3.62061	0.307096	0.153548	−	0.988141i	$-0.450930\pi$
$139$	3.62061	0.307096	0.153548	+	0.988141i	$0.450930\pi$
$140$	0	0
$141$	−0.0587613	−0.00494859
$142$	−2.12110	−0.177999
$143$	−5.47803	−0.458096
$144$	−9.70491	−0.808742
$145$	0	0
$146$	0.0780548	0.00645986
$147$	−0.374821	−0.0309147
$148$	17.6199	1.44834
$149$	12.8580	1.05336	0.526682	−	0.850062i	$-0.323436\pi$
$149$	12.8580	1.05336	0.526682	+	0.850062i	$0.323436\pi$
$150$	0	0
$151$	19.1642	1.55956	0.779779	−	0.626055i	$-0.215331\pi$
$151$	19.1642	1.55956	0.779779	+	0.626055i	$0.215331\pi$
$152$	3.93038	0.318796
$153$	0	0
$154$	0.297802	0.0239976
$155$	0	0
$156$	−0.200797	−0.0160766
$157$	2.70222	0.215661	0.107830	−	0.994169i	$-0.465610\pi$
$157$	2.70222	0.215661	0.107830	+	0.994169i	$0.465610\pi$
$158$	3.83872	0.305392
$159$	−0.145627	−0.0115490
$160$	0	0
$161$	−2.01329	−0.158669
$162$	−3.23276	−0.253989
$163$	17.1057	1.33983	0.669913	−	0.742440i	$-0.266332\pi$
$163$	17.1057	1.33983	0.669913	+	0.742440i	$0.266332\pi$
$164$	5.81585	0.454142
$165$	0	0
$166$	−5.61084	−0.435486
$167$	−8.85257	−0.685032	−0.342516	−	0.939512i	$-0.611279\pi$
$167$	−8.85257	−0.685032	−0.342516	+	0.939512i	$0.611279\pi$
$168$	0.0225894	0.00174281
$169$	−9.08135	−0.698566
$170$	0	0
$171$	−8.44860	−0.646081
$172$	−15.3078	−1.16721
$173$	19.1849	1.45860	0.729301	−	0.684193i	$-0.239845\pi$
$173$	19.1849	1.45860	0.729301	+	0.684193i	$0.239845\pi$
$174$	−0.0923438	−0.00700057
$175$	0	0
$176$	−8.96092	−0.675455
$177$	−0.496507	−0.0373197
$178$	−0.559345	−0.0419247
$179$	4.33533	0.324038	0.162019	−	0.986788i	$-0.448199\pi$
$179$	4.33533	0.324038	0.162019	+	0.986788i	$0.448199\pi$
$180$	0	0
$181$	−23.7783	−1.76743	−0.883715	−	0.468025i	$-0.844966\pi$
$181$	−23.7783	−1.76743	−0.883715	+	0.468025i	$0.844966\pi$
$182$	−0.213029	−0.0157908
$183$	0.607127	0.0448801
$184$	−9.39699	−0.692755
$185$	0	0
$186$	−0.0591296	−0.00433559
$187$	0	0
$188$	2.02621	0.147777
$189$	−0.0971623	−0.00706752
$190$	0	0
$191$	−7.27055	−0.526079	−0.263039	−	0.964785i	$-0.584725\pi$
$191$	−7.27055	−0.526079	−0.263039	+	0.964785i	$0.584725\pi$
$192$	−0.273977	−0.0197726
$193$	−22.1992	−1.59794	−0.798968	−	0.601373i	$-0.794620\pi$
$193$	−22.1992	−1.59794	−0.798968	+	0.601373i	$0.794620\pi$
$194$	3.23161	0.232016
$195$	0	0
$196$	12.9246	0.923188
$197$	−4.48140	−0.319286	−0.159643	−	0.987175i	$-0.551034\pi$
$197$	−4.48140	−0.319286	−0.159643	+	0.987175i	$0.551034\pi$
$198$	−2.98787	−0.212338
$199$	−0.399388	−0.0283119	−0.0141559	−	0.999900i	$-0.504506\pi$
$199$	−0.399388	−0.0283119	−0.0141559	+	0.999900i	$0.504506\pi$
$200$	0	0
$201$	−0.681139	−0.0480438
$202$	−2.49344	−0.175438
$203$	1.41176	0.0990865
$204$	0	0
$205$	0	0
$206$	4.67040	0.325402
$207$	20.1994	1.40396
$208$	6.41009	0.444460
$209$	−7.80093	−0.539602
$210$	0	0
$211$	1.95016	0.134255	0.0671273	−	0.997744i	$-0.478617\pi$
$211$	1.95016	0.134255	0.0671273	+	0.997744i	$0.478617\pi$
$212$	5.02153	0.344880
$213$	0.319337	0.0218806
$214$	1.30543	0.0892372
$215$	0	0
$216$	−0.453504	−0.0308570
$217$	0.903981	0.0613662
$218$	−1.85544	−0.125666
$219$	−0.0117514	−0.000794083 0
$220$	0	0
$221$	0	0
$222$	0.184085	0.0123550
$223$	23.7306	1.58912	0.794558	−	0.607188i	$-0.207703\pi$
$223$	23.7306	1.58912	0.794558	+	0.607188i	$0.207703\pi$
$224$	−1.18145	−0.0789393
$225$	0	0
$226$	3.96516	0.263758
$227$	20.5598	1.36460	0.682301	−	0.731071i	$-0.260979\pi$
$227$	20.5598	1.36460	0.682301	+	0.731071i	$0.260979\pi$
$228$	−0.285943	−0.0189370
$229$	26.5965	1.75755	0.878773	−	0.477239i	$-0.158362\pi$
$229$	26.5965	1.75755	0.878773	+	0.477239i	$0.158362\pi$
$230$	0	0
$231$	−0.0448349	−0.00294992
$232$	6.58939	0.432615
$233$	23.4992	1.53948	0.769741	−	0.638356i	$-0.220385\pi$
$233$	23.4992	1.53948	0.769741	+	0.638356i	$0.220385\pi$
$234$	2.13734	0.139722
$235$	0	0
$236$	17.1206	1.11446
$237$	−0.577929	−0.0375405
$238$	0	0
$239$	5.25737	0.340071	0.170036	−	0.985438i	$-0.445612\pi$
$239$	5.25737	0.340071	0.170036	+	0.985438i	$0.445612\pi$
$240$	0	0
$241$	−5.43912	−0.350364	−0.175182	−	0.984536i	$-0.556051\pi$
$241$	−5.43912	−0.350364	−0.175182	+	0.984536i	$0.556051\pi$
$242$	1.20398	0.0773950
$243$	1.46249	0.0938189
$244$	−20.9350	−1.34023
$245$	0	0
$246$	0.0607615	0.00387401
$247$	5.58030	0.355066
$248$	4.21932	0.267927
$249$	0.844727	0.0535324
$250$	0	0
$251$	26.3864	1.66550	0.832748	−	0.553652i	$-0.186766\pi$
$251$	26.3864	1.66550	0.832748	+	0.553652i	$0.186766\pi$
$252$	1.67436	0.105475
$253$	18.6509	1.17257
$254$	2.64150	0.165742
$255$	0	0
$256$	6.59764	0.412352
$257$	18.7509	1.16965	0.584823	−	0.811161i	$-0.301164\pi$
$257$	18.7509	1.16965	0.584823	+	0.811161i	$0.301164\pi$
$258$	−0.159930	−0.00995678
$259$	−2.81431	−0.174873
$260$	0	0
$261$	−14.1643	−0.876750
$262$	−6.24034	−0.385529
$263$	−16.0011	−0.986668	−0.493334	−	0.869840i	$-0.664222\pi$
$263$	−16.0011	−0.986668	−0.493334	+	0.869840i	$0.664222\pi$
$264$	−0.209266	−0.0128794
$265$	0	0
$266$	−0.303362	−0.0186003
$267$	0.0842109	0.00515362
$268$	23.4871	1.43470
$269$	1.19638	0.0729447	0.0364724	−	0.999335i	$-0.488388\pi$
$269$	1.19638	0.0729447	0.0364724	+	0.999335i	$0.488388\pi$
$270$	0	0
$271$	0.888590	0.0539780	0.0269890	−	0.999636i	$-0.491408\pi$
$271$	0.888590	0.0539780	0.0269890	+	0.999636i	$0.491408\pi$
$272$	0	0
$273$	0.0320721	0.00194109
$274$	−5.03436	−0.304137
$275$	0	0
$276$	0.683650	0.0411509
$277$	−6.17873	−0.371244	−0.185622	−	0.982621i	$-0.559430\pi$
$277$	−6.17873	−0.371244	−0.185622	+	0.982621i	$0.559430\pi$
$278$	−1.30434	−0.0782292
$279$	−9.06970	−0.542989
$280$	0	0
$281$	27.8954	1.66410	0.832051	−	0.554700i	$-0.187167\pi$
$281$	27.8954	1.66410	0.832051	+	0.554700i	$0.187167\pi$
$282$	0.0211690	0.00126060
$283$	−29.7120	−1.76619	−0.883097	−	0.469190i	$-0.844546\pi$
$283$	−29.7120	−1.76619	−0.883097	+	0.469190i	$0.844546\pi$
$284$	−11.0114	−0.653407
$285$	0	0
$286$	1.97349	0.116695
$287$	−0.928930	−0.0548330
$288$	11.8536	0.698481
$289$	0	0
$290$	0	0
$291$	−0.486527	−0.0285207
$292$	0.405212	0.0237132
$293$	−22.4626	−1.31228	−0.656140	−	0.754639i	$-0.727812\pi$
$293$	−22.4626	−1.31228	−0.656140	+	0.754639i	$0.727812\pi$
$294$	0.135031	0.00787517
$295$	0	0
$296$	−13.1358	−0.763500
$297$	0.900104	0.0522293
$298$	−4.63214	−0.268332
$299$	−13.3417	−0.771572
$300$	0	0
$301$	2.44503	0.140929
$302$	−6.90398	−0.397279
$303$	0.375394	0.0215658
$304$	9.12821	0.523539
$305$	0	0
$306$	0	0
$307$	−5.53854	−0.316101	−0.158050	−	0.987431i	$-0.550521\pi$
$307$	−5.53854	−0.316101	−0.158050	+	0.987431i	$0.550521\pi$
$308$	1.54600	0.0880917
$309$	−0.703141	−0.0400003
$310$	0	0
$311$	22.5689	1.27977	0.639883	−	0.768473i	$-0.278983\pi$
$311$	22.5689	1.27977	0.639883	+	0.768473i	$0.278983\pi$
$312$	0.149696	0.00847487
$313$	−12.1857	−0.688775	−0.344387	−	0.938828i	$-0.611913\pi$
$313$	−12.1857	−0.688775	−0.344387	+	0.938828i	$0.611913\pi$
$314$	−0.973486	−0.0549370
$315$	0	0
$316$	19.9282	1.12105
$317$	21.9772	1.23436	0.617181	−	0.786821i	$-0.288275\pi$
$317$	21.9772	1.23436	0.617181	+	0.786821i	$0.288275\pi$
$318$	0.0524628	0.00294197
$319$	−13.0785	−0.732254
$320$	0	0
$321$	−0.196536	−0.0109695
$322$	0.725296	0.0404192
$323$	0	0
$324$	−16.7825	−0.932358
$325$	0	0
$326$	−6.16242	−0.341305
$327$	0.279342	0.0154476
$328$	−4.33577	−0.239403
$329$	−0.323635	−0.0178426
$330$	0	0
$331$	14.4620	0.794905	0.397452	−	0.917623i	$-0.369894\pi$
$331$	14.4620	0.794905	0.397452	+	0.917623i	$0.369894\pi$
$332$	−29.1280	−1.59861
$333$	28.2362	1.54733
$334$	3.18918	0.174504
$335$	0	0
$336$	0.0524633	0.00286211
$337$	24.2460	1.32076	0.660381	−	0.750931i	$-0.270395\pi$
$337$	24.2460	1.32076	0.660381	+	0.750931i	$0.270395\pi$
$338$	3.27160	0.177951
$339$	−0.596965	−0.0324227
$340$	0	0
$341$	−8.37441	−0.453500
$342$	3.04365	0.164582
$343$	−4.15540	−0.224370
$344$	11.4121	0.615300
$345$	0	0
$346$	−6.91145	−0.371562
$347$	−21.4962	−1.15398	−0.576989	−	0.816752i	$-0.695772\pi$
$347$	−21.4962	−1.15398	−0.576989	+	0.816752i	$0.695772\pi$
$348$	−0.479392	−0.0256981
$349$	24.1362	1.29198	0.645991	−	0.763345i	$-0.276444\pi$
$349$	24.1362	1.29198	0.645991	+	0.763345i	$0.276444\pi$
$350$	0	0
$351$	−0.643879	−0.0343677
$352$	10.9449	0.583366
$353$	7.24444	0.385583	0.192791	−	0.981240i	$-0.438246\pi$
$353$	7.24444	0.385583	0.192791	+	0.981240i	$0.438246\pi$
$354$	0.178869	0.00950677
$355$	0	0
$356$	−2.90377	−0.153900
$357$	0	0
$358$	−1.56182	−0.0825449
$359$	−17.2646	−0.911189	−0.455595	−	0.890187i	$-0.650573\pi$
$359$	−17.2646	−0.911189	−0.455595	+	0.890187i	$0.650573\pi$
$360$	0	0
$361$	−11.0534	−0.581760
$362$	8.56626	0.450232
$363$	−0.181263	−0.00951384
$364$	−1.10591	−0.0579657
$365$	0	0
$366$	−0.218720	−0.0114327
$367$	−37.4432	−1.95452	−0.977261	−	0.212042i	$-0.931989\pi$
$367$	−37.4432	−1.95452	−0.977261	+	0.212042i	$0.931989\pi$
$368$	−21.8243	−1.13767
$369$	9.32001	0.485180
$370$	0	0
$371$	−0.802058	−0.0416408
$372$	−0.306964	−0.0159153
$373$	29.8887	1.54758	0.773789	−	0.633443i	$-0.218359\pi$
$373$	29.8887	1.54758	0.773789	+	0.633443i	$0.218359\pi$
$374$	0	0
$375$	0	0
$376$	−1.51056	−0.0779012
$377$	9.35554	0.481835
$378$	0.0350031	0.00180037
$379$	30.0204	1.54205	0.771023	−	0.636807i	$-0.219745\pi$
$379$	30.0204	1.54205	0.771023	+	0.636807i	$0.219745\pi$
$380$	0	0
$381$	−0.397684	−0.0203740
$382$	2.61925	0.134012
$383$	19.7234	1.00782	0.503908	−	0.863757i	$-0.331895\pi$
$383$	19.7234	1.00782	0.503908	+	0.863757i	$0.331895\pi$
$384$	0.527727	0.0269305
$385$	0	0
$386$	7.99738	0.407056
$387$	−24.5311	−1.24699
$388$	16.7765	0.851698
$389$	3.71513	0.188364	0.0941822	−	0.995555i	$-0.469976\pi$
$389$	3.71513	0.188364	0.0941822	+	0.995555i	$0.469976\pi$
$390$	0	0
$391$	0	0
$392$	−9.63543	−0.486663
$393$	0.939499	0.0473915
$394$	1.61444	0.0813345
$395$	0	0
$396$	−15.5111	−0.779464
$397$	−5.17180	−0.259565	−0.129783	−	0.991542i	$-0.541428\pi$
$397$	−5.17180	−0.259565	−0.129783	+	0.991542i	$0.541428\pi$
$398$	0.143881	0.00721212
$399$	0.0456719	0.00228646
$400$	0	0
$401$	0.394746	0.0197127	0.00985635	−	0.999951i	$-0.496863\pi$
$401$	0.394746	0.0197127	0.00985635	+	0.999951i	$0.496863\pi$
$402$	0.245383	0.0122386
$403$	5.99054	0.298410
$404$	−12.9444	−0.644007
$405$	0	0
$406$	−0.508595	−0.0252411
$407$	26.0716	1.29232
$408$	0	0
$409$	0.521080	0.0257657	0.0128829	−	0.999917i	$-0.495899\pi$
$409$	0.521080	0.0257657	0.0128829	+	0.999917i	$0.495899\pi$
$410$	0	0
$411$	0.757936	0.0373862
$412$	24.2458	1.19451
$413$	−2.73457	−0.134559
$414$	−7.27694	−0.357642
$415$	0	0
$416$	−7.82931	−0.383864
$417$	0.196372	0.00961638
$418$	2.81032	0.137457
$419$	4.57224	0.223368	0.111684	−	0.993744i	$-0.464376\pi$
$419$	4.57224	0.223368	0.111684	+	0.993744i	$0.464376\pi$
$420$	0	0
$421$	7.55233	0.368078	0.184039	−	0.982919i	$-0.441083\pi$
$421$	7.55233	0.368078	0.184039	+	0.982919i	$0.441083\pi$
$422$	−0.702554	−0.0341998
$423$	3.24705	0.157877
$424$	−3.74359	−0.181805
$425$	0	0
$426$	−0.115043	−0.00557383
$427$	3.34382	0.161819
$428$	6.77697	0.327577
$429$	−0.297114	−0.0143448
$430$	0	0
$431$	−20.8886	−1.00617	−0.503083	−	0.864238i	$-0.667801\pi$
$431$	−20.8886	−1.00617	−0.503083	+	0.864238i	$0.667801\pi$
$432$	−1.05325	−0.0506746
$433$	−32.8510	−1.57872	−0.789359	−	0.613932i	$-0.789587\pi$
$433$	−32.8510	−1.57872	−0.789359	+	0.613932i	$0.789587\pi$
$434$	−0.325663	−0.0156323
$435$	0	0
$436$	−9.63230	−0.461304
$437$	−18.9991	−0.908852
$438$	0.00423348	0.000202284 0
$439$	24.0164	1.14624	0.573121	−	0.819471i	$-0.305733\pi$
$439$	24.0164	1.14624	0.573121	+	0.819471i	$0.305733\pi$
$440$	0	0
$441$	20.7120	0.986284
$442$	0	0
$443$	−8.36893	−0.397620	−0.198810	−	0.980038i	$-0.563708\pi$
$443$	−8.36893	−0.397620	−0.198810	+	0.980038i	$0.563708\pi$
$444$	0.955653	0.0453533
$445$	0	0
$446$	−8.54904	−0.404809
$447$	0.697380	0.0329850
$448$	−1.50896	−0.0712917
$449$	−24.5964	−1.16078	−0.580389	−	0.814339i	$-0.697099\pi$
$449$	−24.5964	−1.16078	−0.580389	+	0.814339i	$0.697099\pi$
$450$	0	0
$451$	8.60554	0.405219
$452$	20.5846	0.968219
$453$	1.03941	0.0488359
$454$	−7.40676	−0.347617
$455$	0	0
$456$	0.213173	0.00998274
$457$	−7.61053	−0.356005	−0.178003	−	0.984030i	$-0.556964\pi$
$457$	−7.61053	−0.356005	−0.178003	+	0.984030i	$0.556964\pi$
$458$	−9.58151	−0.447715
$459$	0	0
$460$	0	0
$461$	−7.22066	−0.336299	−0.168150	−	0.985761i	$-0.553779\pi$
$461$	−7.22066	−0.336299	−0.168150	+	0.985761i	$0.553779\pi$
$462$	0.0161520	0.000751458 0
$463$	23.0362	1.07058	0.535292	−	0.844667i	$-0.320202\pi$
$463$	23.0362	1.07058	0.535292	+	0.844667i	$0.320202\pi$
$464$	15.3037	0.710457
$465$	0	0
$466$	−8.46569	−0.392165
$467$	29.9045	1.38382	0.691908	−	0.721985i	$-0.256770\pi$
$467$	29.9045	1.38382	0.691908	+	0.721985i	$0.256770\pi$
$468$	11.0957	0.512900
$469$	−3.75145	−0.173226
$470$	0	0
$471$	0.146561	0.00675317
$472$	−12.7636	−0.587491
$473$	−22.6505	−1.04147
$474$	0.208202	0.00956302
$475$	0	0
$476$	0	0
$477$	8.04710	0.368451
$478$	−1.89399	−0.0866292
$479$	−9.65729	−0.441253	−0.220627	−	0.975358i	$-0.570810\pi$
$479$	−9.65729	−0.441253	−0.220627	+	0.975358i	$0.570810\pi$
$480$	0	0
$481$	−18.6500	−0.850366
$482$	1.95947	0.0892512
$483$	−0.109195	−0.00496855
$484$	6.25033	0.284106
$485$	0	0
$486$	−0.526869	−0.0238993
$487$	4.83011	0.218873	0.109436	−	0.993994i	$-0.465095\pi$
$487$	4.83011	0.218873	0.109436	+	0.993994i	$0.465095\pi$
$488$	15.6072	0.706507
$489$	0.927769	0.0419552
$490$	0	0
$491$	21.4747	0.969142	0.484571	−	0.874752i	$-0.338976\pi$
$491$	21.4747	0.969142	0.484571	+	0.874752i	$0.338976\pi$
$492$	0.315436	0.0142209
$493$	0	0
$494$	−2.01033	−0.0904490
$495$	0	0
$496$	9.79927	0.440000
$497$	1.75879	0.0788923
$498$	−0.304317	−0.0136368
$499$	18.6381	0.834357	0.417178	−	0.908825i	$-0.363019\pi$
$499$	18.6381	0.834357	0.417178	+	0.908825i	$0.363019\pi$
$500$	0	0
$501$	−0.480139	−0.0214510
$502$	−9.50583	−0.424266
$503$	18.5788	0.828389	0.414195	−	0.910188i	$-0.364063\pi$
$503$	18.5788	0.828389	0.414195	+	0.910188i	$0.364063\pi$
$504$	−1.24825	−0.0556015
$505$	0	0
$506$	−6.71908	−0.298700
$507$	−0.492548	−0.0218748
$508$	13.7130	0.608416
$509$	28.4343	1.26033	0.630163	−	0.776463i	$-0.282988\pi$
$509$	28.4343	1.26033	0.630163	+	0.776463i	$0.282988\pi$
$510$	0	0
$511$	−0.0647220	−0.00286313
$512$	−21.8368	−0.965058
$513$	−0.916908	−0.0404825
$514$	−6.75508	−0.297954
$515$	0	0
$516$	−0.830255	−0.0365499
$517$	2.99813	0.131857
$518$	1.01387	0.0445468
$519$	1.04054	0.0456745
$520$	0	0
$521$	−18.9733	−0.831236	−0.415618	−	0.909539i	$-0.636435\pi$
$521$	−18.9733	−0.831236	−0.415618	+	0.909539i	$0.636435\pi$
$522$	5.10276	0.223342
$523$	24.4504	1.06914	0.534571	−	0.845124i	$-0.320473\pi$
$523$	24.4504	1.06914	0.534571	+	0.845124i	$0.320473\pi$
$524$	−32.3959	−1.41522
$525$	0	0
$526$	5.76445	0.251342
$527$	0	0
$528$	−0.486016	−0.0211511
$529$	22.4243	0.974969
$530$	0	0
$531$	27.4361	1.19063
$532$	−1.57486	−0.0682791
$533$	−6.15587	−0.266640
$534$	−0.0303374	−0.00131283
$535$	0	0
$536$	−17.5099	−0.756311
$537$	0.235136	0.0101469
$538$	−0.431002	−0.0185818
$539$	19.1242	0.823737
$540$	0	0
$541$	−10.8105	−0.464780	−0.232390	−	0.972623i	$-0.574655\pi$
$541$	−10.8105	−0.464780	−0.232390	+	0.972623i	$0.574655\pi$
$542$	−0.320119	−0.0137503
$543$	−1.28967	−0.0553451
$544$	0	0
$545$	0	0
$546$	−0.0115541	−0.000494471 0
$547$	−7.27247	−0.310948	−0.155474	−	0.987840i	$-0.549691\pi$
$547$	−7.27247	−0.310948	−0.155474	+	0.987840i	$0.549691\pi$
$548$	−26.1353	−1.11644
$549$	−33.5488	−1.43183
$550$	0	0
$551$	13.3226	0.567564
$552$	−0.509667	−0.0216929
$553$	−3.18301	−0.135356
$554$	2.22592	0.0945701
$555$	0	0
$556$	−6.77133	−0.287168
$557$	18.3930	0.779335	0.389667	−	0.920956i	$-0.372590\pi$
$557$	18.3930	0.779335	0.389667	+	0.920956i	$0.372590\pi$
$558$	3.26740	0.138320
$559$	16.2028	0.685305
$560$	0	0
$561$	0	0
$562$	−10.0495	−0.423911
$563$	0.449134	0.0189288	0.00946438	−	0.999955i	$-0.496987\pi$
$563$	0.449134	0.0189288	0.00946438	+	0.999955i	$0.496987\pi$
$564$	0.109896	0.00462747
$565$	0	0
$566$	10.7039	0.449918
$567$	2.68056	0.112573
$568$	8.20911	0.344447
$569$	14.3259	0.600573	0.300287	−	0.953849i	$-0.402918\pi$
$569$	14.3259	0.600573	0.300287	+	0.953849i	$0.402918\pi$
$570$	0	0
$571$	27.9925	1.17145	0.585724	−	0.810510i	$-0.300810\pi$
$571$	27.9925	1.17145	0.585724	+	0.810510i	$0.300810\pi$
$572$	10.2451	0.428370
$573$	−0.394335	−0.0164736
$574$	0.334651	0.0139681
$575$	0	0
$576$	15.1395	0.630812
$577$	−6.76924	−0.281807	−0.140904	−	0.990023i	$-0.545001\pi$
$577$	−6.76924	−0.281807	−0.140904	+	0.990023i	$0.545001\pi$
$578$	0	0
$579$	−1.20403	−0.0500376
$580$	0	0
$581$	4.65243	0.193015
$582$	0.175274	0.00726533
$583$	7.43020	0.307728
$584$	−0.302089	−0.0125005
$585$	0	0
$586$	8.09226	0.334288
$587$	24.7081	1.01981	0.509907	−	0.860230i	$-0.329680\pi$
$587$	24.7081	1.01981	0.509907	+	0.860230i	$0.329680\pi$
$588$	0.700997	0.0289086
$589$	8.53075	0.351503
$590$	0	0
$591$	−0.243059	−0.00999810
$592$	−30.5075	−1.25385
$593$	22.4505	0.921932	0.460966	−	0.887418i	$-0.347503\pi$
$593$	22.4505	0.921932	0.460966	+	0.887418i	$0.347503\pi$
$594$	−0.324266	−0.0133048
$595$	0	0
$596$	−24.0472	−0.985010
$597$	−0.0216617	−0.000886555 0
$598$	4.80642	0.196549
$599$	−19.1639	−0.783018	−0.391509	−	0.920174i	$-0.628047\pi$
$599$	−19.1639	−0.783018	−0.391509	+	0.920174i	$0.628047\pi$
$600$	0	0
$601$	23.2032	0.946477	0.473239	−	0.880934i	$-0.343085\pi$
$601$	23.2032	0.946477	0.473239	+	0.880934i	$0.343085\pi$
$602$	−0.880832	−0.0359000
$603$	37.6386	1.53276
$604$	−35.8412	−1.45836
$605$	0	0
$606$	−0.135237	−0.00549364
$607$	−1.86436	−0.0756720	−0.0378360	−	0.999284i	$-0.512046\pi$
$607$	−1.86436	−0.0756720	−0.0378360	+	0.999284i	$0.512046\pi$
$608$	−11.1492	−0.452161
$609$	0.0765703	0.00310278
$610$	0	0
$611$	−2.14468	−0.0867643
$612$	0	0
$613$	0.383092	0.0154729	0.00773647	−	0.999970i	$-0.497537\pi$
$613$	0.383092	0.0154729	0.00773647	+	0.999970i	$0.497537\pi$
$614$	1.99528	0.0805230
$615$	0	0
$616$	−1.15256	−0.0464379
$617$	11.6676	0.469719	0.234860	−	0.972029i	$-0.424537\pi$
$617$	11.6676	0.469719	0.234860	+	0.972029i	$0.424537\pi$
$618$	0.253310	0.0101896
$619$	13.4856	0.542030	0.271015	−	0.962575i	$-0.412641\pi$
$619$	13.4856	0.542030	0.271015	+	0.962575i	$0.412641\pi$
$620$	0	0
$621$	2.19220	0.0879699
$622$	−8.13055	−0.326005
$623$	0.463802	0.0185818
$624$	0.347666	0.0139178
$625$	0	0
$626$	4.38994	0.175457
$627$	−0.423101	−0.0168970
$628$	−5.05373	−0.201666
$629$	0	0
$630$	0	0
$631$	46.9395	1.86863	0.934316	−	0.356445i	$-0.116011\pi$
$631$	46.9395	1.86863	0.934316	+	0.356445i	$0.116011\pi$
$632$	−14.8567	−0.590967
$633$	0.105771	0.00420403
$634$	−7.91738	−0.314439
$635$	0	0
$636$	0.272354	0.0107995
$637$	−13.6803	−0.542032
$638$	4.71158	0.186533
$639$	−17.6460	−0.698065
$640$	0	0
$641$	39.5785	1.56326	0.781628	−	0.623745i	$-0.214390\pi$
$641$	39.5785	1.56326	0.781628	+	0.623745i	$0.214390\pi$
$642$	0.0708028	0.00279436
$643$	−17.7172	−0.698700	−0.349350	−	0.936992i	$-0.613598\pi$
$643$	−17.7172	−0.698700	−0.349350	+	0.936992i	$0.613598\pi$
$644$	3.76528	0.148373
$645$	0	0
$646$	0	0
$647$	14.8304	0.583045	0.291522	−	0.956564i	$-0.405838\pi$
$647$	14.8304	0.583045	0.291522	+	0.956564i	$0.405838\pi$
$648$	12.5115	0.491497
$649$	25.3328	0.994401
$650$	0	0
$651$	0.0490294	0.00192162
$652$	−31.9915	−1.25288
$653$	10.1986	0.399102	0.199551	−	0.979887i	$-0.436052\pi$
$653$	10.1986	0.399102	0.199551	+	0.979887i	$0.436052\pi$
$654$	−0.100634	−0.00393511
$655$	0	0
$656$	−10.0697	−0.393157
$657$	0.649360	0.0253340
$658$	0.116591	0.00454518
$659$	45.4453	1.77030	0.885148	−	0.465309i	$-0.154057\pi$
$659$	45.4453	1.77030	0.885148	+	0.465309i	$0.154057\pi$
$660$	0	0
$661$	−7.95091	−0.309254	−0.154627	−	0.987973i	$-0.549418\pi$
$661$	−7.95091	−0.309254	−0.154627	+	0.987973i	$0.549418\pi$
$662$	−5.21001	−0.202493
$663$	0	0
$664$	21.7152	0.842712
$665$	0	0
$666$	−10.1722	−0.394165
$667$	−31.8526	−1.23334
$668$	16.5562	0.640580
$669$	1.28708	0.0497614
$670$	0	0
$671$	−30.9769	−1.19585
$672$	−0.0640789	−0.00247190
$673$	−21.9876	−0.847558	−0.423779	−	0.905766i	$-0.639297\pi$
$673$	−21.9876	−0.847558	−0.423779	+	0.905766i	$0.639297\pi$
$674$	−8.73472	−0.336449
$675$	0	0
$676$	16.9841	0.653234
$677$	18.2370	0.700904	0.350452	−	0.936581i	$-0.386028\pi$
$677$	18.2370	0.700904	0.350452	+	0.936581i	$0.386028\pi$
$678$	0.215059	0.00825930
$679$	−2.67961	−0.102834
$680$	0	0
$681$	1.11511	0.0427310
$682$	3.01692	0.115524
$683$	−18.2031	−0.696523	−0.348262	−	0.937397i	$-0.613228\pi$
$683$	−18.2031	−0.696523	−0.348262	+	0.937397i	$0.613228\pi$
$684$	15.8007	0.604156
$685$	0	0
$686$	1.49700	0.0571558
$687$	1.44252	0.0550356
$688$	26.5044	1.01047
$689$	−5.31511	−0.202490
$690$	0	0
$691$	15.5626	0.592031	0.296015	−	0.955183i	$-0.404342\pi$
$691$	15.5626	0.592031	0.296015	+	0.955183i	$0.404342\pi$
$692$	−35.8800	−1.36395
$693$	2.47750	0.0941124
$694$	7.74411	0.293962
$695$	0	0
$696$	0.357391	0.0135469
$697$	0	0
$698$	−8.69519	−0.329118
$699$	1.27453	0.0482072
$700$	0	0
$701$	8.07561	0.305011	0.152506	−	0.988303i	$-0.451266\pi$
$701$	8.07561	0.305011	0.152506	+	0.988303i	$0.451266\pi$
$702$	0.231960	0.00875477
$703$	−26.5583	−1.00166
$704$	13.9789	0.526849
$705$	0	0
$706$	−2.60984	−0.0982227
$707$	2.06753	0.0777573
$708$	0.928575	0.0348980
$709$	−5.29018	−0.198677	−0.0993385	−	0.995054i	$-0.531673\pi$
$709$	−5.29018	−0.198677	−0.0993385	+	0.995054i	$0.531673\pi$
$710$	0	0
$711$	31.9354	1.19767
$712$	2.16479	0.0811289
$713$	−20.3958	−0.763830
$714$	0	0
$715$	0	0
$716$	−8.10801	−0.303010
$717$	0.285146	0.0106490
$718$	6.21964	0.232115
$719$	8.95929	0.334125	0.167063	−	0.985946i	$-0.446572\pi$
$719$	8.95929	0.334125	0.167063	+	0.985946i	$0.446572\pi$
$720$	0	0
$721$	−3.87263	−0.144224
$722$	3.98205	0.148197
$723$	−0.295003	−0.0109713
$724$	44.4707	1.65274
$725$	0	0
$726$	0.0653008	0.00242354
$727$	29.9868	1.11215	0.556074	−	0.831133i	$-0.312307\pi$
$727$	29.9868	1.11215	0.556074	+	0.831133i	$0.312307\pi$
$728$	0.824469	0.0305569
$729$	−26.8413	−0.994121
$730$	0	0
$731$	0	0
$732$	−1.13546	−0.0419678
$733$	−26.9685	−0.996104	−0.498052	−	0.867147i	$-0.665951\pi$
$733$	−26.9685	−0.996104	−0.498052	+	0.867147i	$0.665951\pi$
$734$	13.4891	0.497892
$735$	0	0
$736$	26.6563	0.982564
$737$	34.7532	1.28015
$738$	−3.35758	−0.123594
$739$	−34.8463	−1.28184	−0.640922	−	0.767606i	$-0.721448\pi$
$739$	−34.8463	−1.28184	−0.640922	+	0.767606i	$0.721448\pi$
$740$	0	0
$741$	0.302660	0.0111185
$742$	0.288945	0.0106075
$743$	17.0929	0.627077	0.313539	−	0.949575i	$-0.398486\pi$
$743$	17.0929	0.627077	0.313539	+	0.949575i	$0.398486\pi$
$744$	0.228844	0.00838984
$745$	0	0
$746$	−10.7675	−0.394228
$747$	−46.6781	−1.70786
$748$	0	0
$749$	−1.08244	−0.0395516
$750$	0	0
$751$	32.8813	1.19985	0.599927	−	0.800054i	$-0.295196\pi$
$751$	32.8813	1.19985	0.599927	+	0.800054i	$0.295196\pi$
$752$	−3.50824	−0.127932
$753$	1.43113	0.0521532
$754$	−3.37037	−0.122742
$755$	0	0
$756$	0.181715	0.00660889
$757$	17.4978	0.635968	0.317984	−	0.948096i	$-0.396994\pi$
$757$	17.4978	0.635968	0.317984	+	0.948096i	$0.396994\pi$
$758$	−10.8150	−0.392818
$759$	1.01158	0.0367179
$760$	0	0
$761$	−10.8439	−0.393092	−0.196546	−	0.980495i	$-0.562972\pi$
$761$	−10.8439	−0.393092	−0.196546	+	0.980495i	$0.562972\pi$
$762$	0.143268	0.00519004
$763$	1.53851	0.0556978
$764$	13.5975	0.491941
$765$	0	0
$766$	−7.10543	−0.256730
$767$	−18.1216	−0.654331
$768$	0.357838	0.0129124
$769$	14.8775	0.536497	0.268248	−	0.963350i	$-0.413555\pi$
$769$	14.8775	0.536497	0.268248	+	0.963350i	$0.413555\pi$
$770$	0	0
$771$	1.01699	0.0366262
$772$	41.5174	1.49424
$773$	−7.71506	−0.277491	−0.138746	−	0.990328i	$-0.544307\pi$
$773$	−7.71506	−0.277491	−0.138746	+	0.990328i	$0.544307\pi$
$774$	8.83744	0.317655
$775$	0	0
$776$	−12.5070	−0.448976
$777$	−0.152641	−0.00547595
$778$	−1.33839	−0.0479836
$779$	−8.76619	−0.314081
$780$	0	0
$781$	−16.2933	−0.583019
$782$	0	0
$783$	−1.53722	−0.0549358
$784$	−22.3781	−0.799216
$785$	0	0
$786$	−0.338459	−0.0120724
$787$	−38.2522	−1.36354	−0.681772	−	0.731565i	$-0.738790\pi$
$787$	−38.2522	−1.36354	−0.681772	+	0.731565i	$0.738790\pi$
$788$	8.38119	0.298567
$789$	−0.867854	−0.0308964
$790$	0	0
$791$	−3.28786	−0.116903
$792$	11.5637	0.410898
$793$	22.1590	0.786888
$794$	1.86317	0.0661213
$795$	0	0
$796$	0.746943	0.0264747
$797$	26.6162	0.942796	0.471398	−	0.881921i	$-0.343750\pi$
$797$	26.6162	0.942796	0.471398	+	0.881921i	$0.343750\pi$
$798$	−0.0164535	−0.000582448 0
$799$	0	0
$800$	0	0
$801$	−4.65335	−0.164418
$802$	−0.142209	−0.00502158
$803$	0.599580	0.0211587
$804$	1.27388	0.0449262
$805$	0	0
$806$	−2.15812	−0.0760164
$807$	0.0648885	0.00228418
$808$	9.65014	0.339491
$809$	16.4368	0.577887	0.288943	−	0.957346i	$-0.406696\pi$
$809$	16.4368	0.577887	0.288943	+	0.957346i	$0.406696\pi$
$810$	0	0
$811$	−3.64147	−0.127869	−0.0639347	−	0.997954i	$-0.520365\pi$
$811$	−3.64147	−0.127869	−0.0639347	+	0.997954i	$0.520365\pi$
$812$	−2.64031	−0.0926566
$813$	0.0481947	0.00169026
$814$	−9.39240	−0.329203
$815$	0	0
$816$	0	0
$817$	23.0734	0.807236
$818$	−0.187721	−0.00656352
$819$	−1.77225	−0.0619274
$820$	0	0
$821$	−35.5272	−1.23991	−0.619953	−	0.784639i	$-0.712848\pi$
$821$	−35.5272	−1.23991	−0.619953	+	0.784639i	$0.712848\pi$
$822$	−0.273050	−0.00952371
$823$	5.01622	0.174854	0.0874271	−	0.996171i	$-0.472136\pi$
$823$	5.01622	0.174854	0.0874271	+	0.996171i	$0.472136\pi$
$824$	−18.0755	−0.629689
$825$	0	0
$826$	0.985141	0.0342774
$827$	−39.0885	−1.35924	−0.679621	−	0.733564i	$-0.737856\pi$
$827$	−39.0885	−1.35924	−0.679621	+	0.733564i	$0.737856\pi$
$828$	−37.7773	−1.31285
$829$	−16.3998	−0.569587	−0.284794	−	0.958589i	$-0.591925\pi$
$829$	−16.3998	−0.569587	−0.284794	+	0.958589i	$0.591925\pi$
$830$	0	0
$831$	−0.335118	−0.0116251
$832$	−9.99964	−0.346675
$833$	0	0
$834$	−0.0707439	−0.00244966
$835$	0	0
$836$	14.5894	0.504586
$837$	−0.984314	−0.0340229
$838$	−1.64717	−0.0569005
$839$	29.9127	1.03270	0.516350	−	0.856378i	$-0.327290\pi$
$839$	29.9127	1.03270	0.516350	+	0.856378i	$0.327290\pi$
$840$	0	0
$841$	−6.66421	−0.229800
$842$	−2.72076	−0.0937636
$843$	1.51297	0.0521095
$844$	−3.64722	−0.125543
$845$	0	0
$846$	−1.16976	−0.0402173
$847$	−0.998327	−0.0343029
$848$	−8.69441	−0.298567
$849$	−1.61150	−0.0553064
$850$	0	0
$851$	63.4972	2.17666
$852$	−0.597229	−0.0204607
$853$	−35.9121	−1.22961	−0.614803	−	0.788680i	$-0.710765\pi$
$853$	−35.9121	−1.22961	−0.614803	+	0.788680i	$0.710765\pi$
$854$	−1.20463	−0.0412215
$855$	0	0
$856$	−5.05229	−0.172684
$857$	21.0716	0.719791	0.359896	−	0.932993i	$-0.382812\pi$
$857$	21.0716	0.719791	0.359896	+	0.932993i	$0.382812\pi$
$858$	0.107036	0.00365417
$859$	15.8539	0.540928	0.270464	−	0.962730i	$-0.412823\pi$
$859$	15.8539	0.540928	0.270464	+	0.962730i	$0.412823\pi$
$860$	0	0
$861$	−0.0503826	−0.00171704
$862$	7.52520	0.256309
$863$	−23.8125	−0.810587	−0.405294	−	0.914187i	$-0.632831\pi$
$863$	−23.8125	−0.810587	−0.405294	+	0.914187i	$0.632831\pi$
$864$	1.28645	0.0437658
$865$	0	0
$866$	11.8347	0.402160
$867$	0	0
$868$	−1.69064	−0.0573841
$869$	29.4872	1.00028
$870$	0	0
$871$	−24.8603	−0.842359
$872$	7.18097	0.243178
$873$	26.8847	0.909908
$874$	6.84452	0.231519
$875$	0	0
$876$	0.0219776	0.000742554 0
$877$	−17.9197	−0.605107	−0.302553	−	0.953132i	$-0.597839\pi$
$877$	−17.9197	−0.605107	−0.302553	+	0.953132i	$0.597839\pi$
$878$	−8.65203	−0.291992
$879$	−1.21831	−0.0410926
$880$	0	0
$881$	−33.7879	−1.13834	−0.569171	−	0.822219i	$-0.692736\pi$
$881$	−33.7879	−1.13834	−0.569171	+	0.822219i	$0.692736\pi$
$882$	−7.46158	−0.251245
$883$	26.0211	0.875680	0.437840	−	0.899053i	$-0.355744\pi$
$883$	26.0211	0.875680	0.437840	+	0.899053i	$0.355744\pi$
$884$	0	0
$885$	0	0
$886$	3.01495	0.101289
$887$	−29.6531	−0.995652	−0.497826	−	0.867277i	$-0.665868\pi$
$887$	−29.6531	−0.995652	−0.497826	+	0.867277i	$0.665868\pi$
$888$	−0.712447	−0.0239082
$889$	−2.19029	−0.0734601
$890$	0	0
$891$	−24.8325	−0.831919
$892$	−44.3813	−1.48600
$893$	−3.05410	−0.102202
$894$	−0.251234	−0.00840254
$895$	0	0
$896$	2.90652	0.0971000
$897$	−0.723619	−0.0241609
$898$	8.86098	0.295695
$899$	14.3020	0.477000
$900$	0	0
$901$	0	0
$902$	−3.10018	−0.103225
$903$	0.132612	0.00441303
$904$	−15.3460	−0.510401
$905$	0	0
$906$	−0.374453	−0.0124404
$907$	−6.01268	−0.199648	−0.0998238	−	0.995005i	$-0.531828\pi$
$907$	−6.01268	−0.199648	−0.0998238	+	0.995005i	$0.531828\pi$
$908$	−38.4513	−1.27605
$909$	−20.7436	−0.688022
$910$	0	0
$911$	−36.3470	−1.20423	−0.602115	−	0.798409i	$-0.705675\pi$
$911$	−36.3470	−1.20423	−0.602115	+	0.798409i	$0.705675\pi$
$912$	0.495089	0.0163940
$913$	−43.0998	−1.42639
$914$	2.74173	0.0906883
$915$	0	0
$916$	−49.7412	−1.64350
$917$	5.17440	0.170874
$918$	0	0
$919$	31.4827	1.03852	0.519259	−	0.854617i	$-0.326208\pi$
$919$	31.4827	1.03852	0.519259	+	0.854617i	$0.326208\pi$
$920$	0	0
$921$	−0.300395	−0.00989835
$922$	2.60127	0.0856684
$923$	11.6552	0.383635
$924$	0.0838510	0.00275849
$925$	0	0
$926$	−8.29890	−0.272719
$927$	38.8544	1.27615
$928$	−18.6920	−0.613595
$929$	−43.2005	−1.41736	−0.708682	−	0.705528i	$-0.750710\pi$
$929$	−43.2005	−1.41736	−0.708682	+	0.705528i	$0.750710\pi$
$930$	0	0
$931$	−19.4812	−0.638471
$932$	−43.9486	−1.43958
$933$	1.22408	0.0400744
$934$	−10.7732	−0.352511
$935$	0	0
$936$	−8.27195	−0.270377
$937$	11.5364	0.376877	0.188439	−	0.982085i	$-0.439657\pi$
$937$	11.5364	0.376877	0.188439	+	0.982085i	$0.439657\pi$
$938$	1.35148	0.0441273
$939$	−0.660917	−0.0215682
$940$	0	0
$941$	27.2725	0.889057	0.444528	−	0.895765i	$-0.353371\pi$
$941$	27.2725	0.889057	0.444528	+	0.895765i	$0.353371\pi$
$942$	−0.0527992	−0.00172029
$943$	20.9588	0.682511
$944$	−29.6431	−0.964800
$945$	0	0
$946$	8.15996	0.265303
$947$	22.7925	0.740657	0.370328	−	0.928901i	$-0.379245\pi$
$947$	22.7925	0.740657	0.370328	+	0.928901i	$0.379245\pi$
$948$	1.08085	0.0351045
$949$	−0.428902	−0.0139228
$950$	0	0
$951$	1.19198	0.0386527
$952$	0	0
$953$	28.9548	0.937938	0.468969	−	0.883215i	$-0.344626\pi$
$953$	28.9548	0.937938	0.468969	+	0.883215i	$0.344626\pi$
$954$	−2.89900	−0.0938587
$955$	0	0
$956$	−9.83243	−0.318003
$957$	−0.709341	−0.0229297
$958$	3.47908	0.112404
$959$	4.17443	0.134799
$960$	0	0
$961$	−21.8421	−0.704584
$962$	6.71874	0.216621
$963$	10.8602	0.349966
$964$	10.1723	0.327629
$965$	0	0
$966$	0.0393381	0.00126568
$967$	−50.6105	−1.62752	−0.813762	−	0.581198i	$-0.802584\pi$
$967$	−50.6105	−1.62752	−0.813762	+	0.581198i	$0.802584\pi$
$968$	−4.65968	−0.149768
$969$	0	0
$970$	0	0
$971$	−32.8412	−1.05392	−0.526962	−	0.849889i	$-0.676669\pi$
$971$	−32.8412	−1.05392	−0.526962	+	0.849889i	$0.676669\pi$
$972$	−2.73518	−0.0877308
$973$	1.08154	0.0346727
$974$	−1.74007	−0.0557553
$975$	0	0
$976$	36.2475	1.16025
$977$	21.6326	0.692089	0.346044	−	0.938218i	$-0.387525\pi$
$977$	21.6326	0.692089	0.346044	+	0.938218i	$0.387525\pi$
$978$	−0.334233	−0.0106876
$979$	−4.29662	−0.137321
$980$	0	0
$981$	−15.4360	−0.492832
$982$	−7.73637	−0.246878
$983$	−5.11049	−0.162999	−0.0814996	−	0.996673i	$-0.525971\pi$
$983$	−5.11049	−0.162999	−0.0814996	+	0.996673i	$0.525971\pi$
$984$	−0.235160	−0.00749663
$985$	0	0
$986$	0	0
$987$	−0.0175531	−0.000558720 0
$988$	−10.4364	−0.332025
$989$	−55.1653	−1.75415
$990$	0	0
$991$	−2.51317	−0.0798336	−0.0399168	−	0.999203i	$-0.512709\pi$
$991$	−2.51317	−0.0798336	−0.0399168	+	0.999203i	$0.512709\pi$
$992$	−11.9689	−0.380012
$993$	0.784381	0.0248916
$994$	−0.633611	−0.0200969
$995$	0	0
$996$	−1.57982	−0.0500586
$997$	−36.0472	−1.14163	−0.570813	−	0.821080i	$-0.693372\pi$
$997$	−36.0472	−1.14163	−0.570813	+	0.821080i	$0.693372\pi$
$998$	−6.71447	−0.212543
$999$	3.06441	0.0969535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bq.1.5		12
5.4	even	2		1445.2.a.q.1.8		12
17.5	odd	16		425.2.m.b.76.4		24
17.7	odd	16		425.2.m.b.151.4		24
17.16	even	2		7225.2.a.bs.1.5		12
85.4	even	4		1445.2.d.j.866.10		24
85.7	even	16		425.2.n.f.49.3		24
85.22	even	16		425.2.n.c.399.4		24
85.24	odd	16		85.2.l.a.66.3	✓	24
85.39	odd	16		85.2.l.a.76.3	yes	24
85.58	even	16		425.2.n.c.49.4		24
85.64	even	4		1445.2.d.j.866.9		24
85.73	even	16		425.2.n.f.399.3		24
85.84	even	2		1445.2.a.p.1.8		12
255.194	even	16		765.2.be.b.406.4		24
255.209	even	16		765.2.be.b.586.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.66.3	✓	24	85.24	odd	16
85.2.l.a.76.3	yes	24	85.39	odd	16
425.2.m.b.76.4		24	17.5	odd	16
425.2.m.b.151.4		24	17.7	odd	16
425.2.n.c.49.4		24	85.58	even	16
425.2.n.c.399.4		24	85.22	even	16
425.2.n.f.49.3		24	85.7	even	16
425.2.n.f.399.3		24	85.73	even	16
765.2.be.b.406.4		24	255.194	even	16
765.2.be.b.586.4		24	255.209	even	16
1445.2.a.p.1.8		12	85.84	even	2
1445.2.a.q.1.8		12	5.4	even	2
1445.2.d.j.866.9		24	85.64	even	4
1445.2.d.j.866.10		24	85.4	even	4
7225.2.a.bq.1.5		12	1.1	even	1	trivial
7225.2.a.bs.1.5		12	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.360254 q^{2} +0.0542373 q^{3} -1.87022 q^{4} -0.0195392 q^{6} +0.298718 q^{7} +1.39426 q^{8} -2.99706 q^{9} -2.76730 q^{11} -0.101435 q^{12} +1.97956 q^{13} -0.107615 q^{14} +3.23814 q^{16} +1.07970 q^{18} +2.81896 q^{19} +0.0162017 q^{21} +0.996933 q^{22} -6.73975 q^{23} +0.0756210 q^{24} -0.713144 q^{26} -0.325264 q^{27} -0.558668 q^{28} +4.72608 q^{29} +3.02620 q^{31} -3.95508 q^{32} -0.150091 q^{33} +5.60515 q^{36} -9.42129 q^{37} -1.01554 q^{38} +0.107366 q^{39} -3.10972 q^{41} -0.00583672 q^{42} +8.18506 q^{43} +5.17546 q^{44} +2.42803 q^{46} -1.08341 q^{47} +0.175628 q^{48} -6.91077 q^{49} -3.70220 q^{52} -2.68500 q^{53} +0.117178 q^{54} +0.416492 q^{56} +0.152893 q^{57} -1.70259 q^{58} -9.15435 q^{59} +11.1939 q^{61} -1.09020 q^{62} -0.895276 q^{63} -5.05145 q^{64} +0.0540709 q^{66} -12.5585 q^{67} -0.365546 q^{69} +5.88778 q^{71} -4.17869 q^{72} -0.216666 q^{73} +3.39406 q^{74} -5.27208 q^{76} -0.826644 q^{77} -0.0386790 q^{78} -10.6556 q^{79} +8.97353 q^{81} +1.12029 q^{82} +15.5747 q^{83} -0.0303006 q^{84} -2.94870 q^{86} +0.256329 q^{87} -3.85835 q^{88} +1.55264 q^{89} +0.591330 q^{91} +12.6048 q^{92} +0.164133 q^{93} +0.390304 q^{94} -0.214513 q^{96} -8.97035 q^{97} +2.48963 q^{98} +8.29377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more