LMFDB - Embedded newform 4225.2.a.bw.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
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} - 5 x^{11} - 5 x^{10} + 48 x^{9} + 2 x^{8} - 171 x^{7} + 6 x^{6} + 260 x^{5} + 27 x^{4} + \cdots + 1 $ x^12 - 5x^11 - 5x^10 + 48x^9 + 2x^8 - 171x^7 + 6x^6 + 260x^5 + 27x^4 - 138x^3 - 36x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.59533$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.595328 q^{2} -1.48507 q^{3} -1.64558 q^{4} -0.884102 q^{6} -0.0123504 q^{7} -2.17032 q^{8} -0.794576 q^{9} +O(q^{10})$	\(q+0.595328 q^{2} -1.48507 q^{3} -1.64558 q^{4} -0.884102 q^{6} -0.0123504 q^{7} -2.17032 q^{8} -0.794576 q^{9} -4.04974 q^{11} +2.44380 q^{12} -0.00735255 q^{14} +1.99912 q^{16} +3.89327 q^{17} -0.473033 q^{18} +5.83063 q^{19} +0.0183412 q^{21} -2.41092 q^{22} -1.97957 q^{23} +3.22307 q^{24} +5.63520 q^{27} +0.0203236 q^{28} +7.07286 q^{29} -0.0765196 q^{31} +5.53077 q^{32} +6.01413 q^{33} +2.31777 q^{34} +1.30754 q^{36} -5.27523 q^{37} +3.47114 q^{38} -1.54799 q^{41} +0.0109190 q^{42} +9.86444 q^{43} +6.66418 q^{44} -1.17849 q^{46} +7.83718 q^{47} -2.96882 q^{48} -6.99985 q^{49} -5.78177 q^{51} -7.84271 q^{53} +3.35479 q^{54} +0.0268043 q^{56} -8.65888 q^{57} +4.21067 q^{58} +2.79065 q^{59} -12.2172 q^{61} -0.0455543 q^{62} +0.00981334 q^{63} -0.705609 q^{64} +3.58038 q^{66} -16.0395 q^{67} -6.40671 q^{68} +2.93979 q^{69} +8.77623 q^{71} +1.72448 q^{72} -8.12747 q^{73} -3.14049 q^{74} -9.59479 q^{76} +0.0500159 q^{77} +0.418943 q^{79} -5.98492 q^{81} -0.921564 q^{82} -11.4824 q^{83} -0.0301820 q^{84} +5.87258 q^{86} -10.5037 q^{87} +8.78922 q^{88} +10.1140 q^{89} +3.25754 q^{92} +0.113637 q^{93} +4.66570 q^{94} -8.21356 q^{96} +10.3990 q^{97} -4.16721 q^{98} +3.21782 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 7 q^{2} - q^{3} + 13 q^{4} + 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 12 * q - 7 * q^2 - q^3 + 13 * q^4 + 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9	$ 12 q - 7 q^{2} - q^{3} + 13 q^{4} + 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9} - 3 q^{11} - 8 q^{12} + 2 q^{14} + 11 q^{16} + 2 q^{17} - 14 q^{18} + 5 q^{21} + 12 q^{22} - 3 q^{23} + 4 q^{24} + 5 q^{27} - 29 q^{28} + 9 q^{29} + 4 q^{31} - 48 q^{32} - 30 q^{33} + 3 q^{34} + 44 q^{36} - 17 q^{37} + 15 q^{38} - 9 q^{41} - 80 q^{42} + q^{43} - 10 q^{44} - 3 q^{46} - 61 q^{47} + 35 q^{48} + 8 q^{49} - 26 q^{51} + 23 q^{53} + 48 q^{54} + 51 q^{56} - 10 q^{57} + 17 q^{58} + q^{59} + 13 q^{61} + 29 q^{62} - 46 q^{63} + 50 q^{64} + 29 q^{66} - 43 q^{67} - 26 q^{68} - 31 q^{69} - 19 q^{71} - 48 q^{72} - 21 q^{73} + 7 q^{74} - 46 q^{76} - 42 q^{77} - 7 q^{79} - 16 q^{81} + 15 q^{82} - 75 q^{83} + 83 q^{84} - 17 q^{86} + 5 q^{87} + 38 q^{88} + 19 q^{89} - 35 q^{92} - 11 q^{93} + 26 q^{94} - 90 q^{96} - 17 q^{97} - 11 q^{98} + 6 q^{99}+O(q^{100}) $ 12 * q - 7 * q^2 - q^3 + 13 * q^4 + 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9 - 3 * q^11 - 8 * q^12 + 2 * q^14 + 11 * q^16 + 2 * q^17 - 14 * q^18 + 5 * q^21 + 12 * q^22 - 3 * q^23 + 4 * q^24 + 5 * q^27 - 29 * q^28 + 9 * q^29 + 4 * q^31 - 48 * q^32 - 30 * q^33 + 3 * q^34 + 44 * q^36 - 17 * q^37 + 15 * q^38 - 9 * q^41 - 80 * q^42 + q^43 - 10 * q^44 - 3 * q^46 - 61 * q^47 + 35 * q^48 + 8 * q^49 - 26 * q^51 + 23 * q^53 + 48 * q^54 + 51 * q^56 - 10 * q^57 + 17 * q^58 + q^59 + 13 * q^61 + 29 * q^62 - 46 * q^63 + 50 * q^64 + 29 * q^66 - 43 * q^67 - 26 * q^68 - 31 * q^69 - 19 * q^71 - 48 * q^72 - 21 * q^73 + 7 * q^74 - 46 * q^76 - 42 * q^77 - 7 * q^79 - 16 * q^81 + 15 * q^82 - 75 * q^83 + 83 * q^84 - 17 * q^86 + 5 * q^87 + 38 * q^88 + 19 * q^89 - 35 * q^92 - 11 * q^93 + 26 * q^94 - 90 * q^96 - 17 * q^97 - 11 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.595328	0.420961	0.210480	−	0.977598i	$-0.432497\pi$
$2$	0.595328	0.420961	0.210480	+	0.977598i	$0.432497\pi$
$3$	−1.48507	−0.857404	−0.428702	−	0.903446i	$-0.641029\pi$
$3$	−1.48507	−0.857404	−0.428702	+	0.903446i	$0.641029\pi$
$4$	−1.64558	−0.822792
$5$	0	0
$5$	0	0
$6$	−0.884102	−0.360933
$7$	−0.0123504	−0.00466802	−0.00233401	−	0.999997i	$-0.500743\pi$
$7$	−0.0123504	−0.00466802	−0.00233401	+	0.999997i	$0.500743\pi$
$8$	−2.17032	−0.767324
$9$	−0.794576	−0.264859
$10$	0	0
$11$	−4.04974	−1.22104	−0.610521	−	0.792000i	$-0.709040\pi$
$11$	−4.04974	−1.22104	−0.610521	+	0.792000i	$0.709040\pi$
$12$	2.44380	0.705465
$13$	0	0
$13$	0	0
$14$	−0.00735255	−0.00196505
$15$	0	0
$16$	1.99912	0.499779
$17$	3.89327	0.944257	0.472129	−	0.881530i	$-0.343486\pi$
$17$	3.89327	0.944257	0.472129	+	0.881530i	$0.343486\pi$
$18$	−0.473033	−0.111495
$19$	5.83063	1.33764	0.668819	−	0.743425i	$-0.266800\pi$
$19$	5.83063	1.33764	0.668819	+	0.743425i	$0.266800\pi$
$20$	0	0
$21$	0.0183412	0.00400238
$22$	−2.41092	−0.514010
$23$	−1.97957	−0.412768	−0.206384	−	0.978471i	$-0.566170\pi$
$23$	−1.97957	−0.412768	−0.206384	+	0.978471i	$0.566170\pi$
$24$	3.22307	0.657906
$25$	0	0
$26$	0	0
$27$	5.63520	1.08449
$28$	0.0203236	0.00384081
$29$	7.07286	1.31340	0.656698	−	0.754153i	$-0.271952\pi$
$29$	7.07286	1.31340	0.656698	+	0.754153i	$0.271952\pi$
$30$	0	0
$31$	−0.0765196	−0.0137433	−0.00687166	−	0.999976i	$-0.502187\pi$
$31$	−0.0765196	−0.0137433	−0.00687166	+	0.999976i	$0.502187\pi$
$32$	5.53077	0.977711
$33$	6.01413	1.04693
$34$	2.31777	0.397495
$35$	0	0
$36$	1.30754	0.217924
$37$	−5.27523	−0.867243	−0.433621	−	0.901095i	$-0.642764\pi$
$37$	−5.27523	−0.867243	−0.433621	+	0.901095i	$0.642764\pi$
$38$	3.47114	0.563093
$39$	0	0
$40$	0	0
$41$	−1.54799	−0.241756	−0.120878	−	0.992667i	$-0.538571\pi$
$41$	−1.54799	−0.241756	−0.120878	+	0.992667i	$0.538571\pi$
$42$	0.0109190	0.00168484
$43$	9.86444	1.50431	0.752156	−	0.658985i	$-0.229014\pi$
$43$	9.86444	1.50431	0.752156	+	0.658985i	$0.229014\pi$
$44$	6.66418	1.00466
$45$	0	0
$46$	−1.17849	−0.173759
$47$	7.83718	1.14317	0.571585	−	0.820543i	$-0.306329\pi$
$47$	7.83718	1.14317	0.571585	+	0.820543i	$0.306329\pi$
$48$	−2.96882	−0.428512
$49$	−6.99985	−0.999978
$50$	0	0
$51$	−5.78177	−0.809610
$52$	0	0
$53$	−7.84271	−1.07728	−0.538639	−	0.842536i	$-0.681061\pi$
$53$	−7.84271	−1.07728	−0.538639	+	0.842536i	$0.681061\pi$
$54$	3.35479	0.456530
$55$	0	0
$56$	0.0268043	0.00358188
$57$	−8.65888	−1.14690
$58$	4.21067	0.552888
$59$	2.79065	0.363312	0.181656	−	0.983362i	$-0.441854\pi$
$59$	2.79065	0.363312	0.181656	+	0.983362i	$0.441854\pi$
$60$	0	0
$61$	−12.2172	−1.56425	−0.782127	−	0.623119i	$-0.785865\pi$
$61$	−12.2172	−1.56425	−0.782127	+	0.623119i	$0.785865\pi$
$62$	−0.0455543	−0.00578540
$63$	0.00981334	0.00123636
$64$	−0.705609	−0.0882011
$65$	0	0
$66$	3.58038	0.440714
$67$	−16.0395	−1.95954	−0.979768	−	0.200136i	$-0.935862\pi$
$67$	−16.0395	−1.95954	−0.979768	+	0.200136i	$0.935862\pi$
$68$	−6.40671	−0.776927
$69$	2.93979	0.353909
$70$	0	0
$71$	8.77623	1.04155	0.520774	−	0.853695i	$-0.325644\pi$
$71$	8.77623	1.04155	0.520774	+	0.853695i	$0.325644\pi$
$72$	1.72448	0.203232
$73$	−8.12747	−0.951249	−0.475624	−	0.879649i	$-0.657778\pi$
$73$	−8.12747	−0.951249	−0.475624	+	0.879649i	$0.657778\pi$
$74$	−3.14049	−0.365075
$75$	0	0
$76$	−9.59479	−1.10060
$77$	0.0500159	0.00569984
$78$	0	0
$79$	0.418943	0.0471347	0.0235674	−	0.999722i	$-0.492498\pi$
$79$	0.418943	0.0471347	0.0235674	+	0.999722i	$0.492498\pi$
$80$	0	0
$81$	−5.98492	−0.664991
$82$	−0.921564	−0.101770
$83$	−11.4824	−1.26035	−0.630176	−	0.776452i	$-0.717017\pi$
$83$	−11.4824	−1.26035	−0.630176	+	0.776452i	$0.717017\pi$
$84$	−0.0301820	−0.00329312
$85$	0	0
$86$	5.87258	0.633256
$87$	−10.5037	−1.12611
$88$	8.78922	0.936934
$89$	10.1140	1.07209	0.536044	−	0.844190i	$-0.319918\pi$
$89$	10.1140	1.07209	0.536044	+	0.844190i	$0.319918\pi$
$90$	0	0
$91$	0	0
$92$	3.25754	0.339623
$93$	0.113637	0.0117836
$94$	4.66570	0.481230
$95$	0	0
$96$	−8.21356	−0.838293
$97$	10.3990	1.05586	0.527928	−	0.849289i	$-0.322969\pi$
$97$	10.3990	1.05586	0.527928	+	0.849289i	$0.322969\pi$
$98$	−4.16721	−0.420951
$99$	3.21782	0.323403
$100$	0	0
$101$	6.74480	0.671132	0.335566	−	0.942017i	$-0.391072\pi$
$101$	6.74480	0.671132	0.335566	+	0.942017i	$0.391072\pi$
$102$	−3.44205	−0.340814
$103$	−1.85695	−0.182970	−0.0914852	−	0.995806i	$-0.529161\pi$
$103$	−1.85695	−0.182970	−0.0914852	+	0.995806i	$0.529161\pi$
$104$	0	0
$105$	0	0
$106$	−4.66899	−0.453492
$107$	−18.5879	−1.79696	−0.898479	−	0.439017i	$-0.855327\pi$
$107$	−18.5879	−1.79696	−0.898479	+	0.439017i	$0.855327\pi$
$108$	−9.27320	−0.892314
$109$	19.0791	1.82745	0.913723	−	0.406338i	$-0.133195\pi$
$109$	19.0791	1.82745	0.913723	+	0.406338i	$0.133195\pi$
$110$	0	0
$111$	7.83407	0.743577
$112$	−0.0246899	−0.00233298
$113$	−10.2196	−0.961377	−0.480689	−	0.876891i	$-0.659613\pi$
$113$	−10.2196	−0.961377	−0.480689	+	0.876891i	$0.659613\pi$
$114$	−5.15487	−0.482798
$115$	0	0
$116$	−11.6390	−1.08065
$117$	0	0
$118$	1.66135	0.152940
$119$	−0.0480835	−0.00440781
$120$	0	0
$121$	5.40036	0.490942
$122$	−7.27325	−0.658489
$123$	2.29887	0.207282
$124$	0.125919	0.0113079
$125$	0	0
$126$	0.00584216	0.000520461 0
$127$	7.95303	0.705717	0.352858	−	0.935677i	$-0.385210\pi$
$127$	7.95303	0.705717	0.352858	+	0.935677i	$0.385210\pi$
$128$	−11.4816	−1.01484
$129$	−14.6494	−1.28980
$130$	0	0
$131$	−0.859760	−0.0751176	−0.0375588	−	0.999294i	$-0.511958\pi$
$131$	−0.859760	−0.0751176	−0.0375588	+	0.999294i	$0.511958\pi$
$132$	−9.89676	−0.861402
$133$	−0.0720107	−0.00624412
$134$	−9.54876	−0.824888
$135$	0	0
$136$	−8.44964	−0.724551
$137$	−8.84890	−0.756013	−0.378006	−	0.925803i	$-0.623390\pi$
$137$	−8.84890	−0.756013	−0.378006	+	0.925803i	$0.623390\pi$
$138$	1.75014	0.148982
$139$	11.4798	0.973704	0.486852	−	0.873485i	$-0.338145\pi$
$139$	11.4798	0.973704	0.486852	+	0.873485i	$0.338145\pi$
$140$	0	0
$141$	−11.6387	−0.980159
$142$	5.22474	0.438450
$143$	0	0
$144$	−1.58845	−0.132371
$145$	0	0
$146$	−4.83851	−0.400438
$147$	10.3952	0.857385
$148$	8.68084	0.713560
$149$	−12.1258	−0.993380	−0.496690	−	0.867928i	$-0.665451\pi$
$149$	−12.1258	−0.993380	−0.496690	+	0.867928i	$0.665451\pi$
$150$	0	0
$151$	−13.1143	−1.06722	−0.533611	−	0.845730i	$-0.679165\pi$
$151$	−13.1143	−1.06722	−0.533611	+	0.845730i	$0.679165\pi$
$152$	−12.6543	−1.02640
$153$	−3.09350	−0.250095
$154$	0.0297759	0.00239941
$155$	0	0
$156$	0	0
$157$	20.1702	1.60976	0.804879	−	0.593439i	$-0.202230\pi$
$157$	20.1702	1.60976	0.804879	+	0.593439i	$0.202230\pi$
$158$	0.249408	0.0198419
$159$	11.6469	0.923663
$160$	0	0
$161$	0.0244485	0.00192681
$162$	−3.56299	−0.279935
$163$	−7.69960	−0.603079	−0.301539	−	0.953454i	$-0.597500\pi$
$163$	−7.69960	−0.603079	−0.301539	+	0.953454i	$0.597500\pi$
$164$	2.54735	0.198915
$165$	0	0
$166$	−6.83577	−0.530559
$167$	−21.7723	−1.68479	−0.842396	−	0.538858i	$-0.818856\pi$
$167$	−21.7723	−1.68479	−0.842396	+	0.538858i	$0.818856\pi$
$168$	−0.0398062	−0.00307112
$169$	0	0
$170$	0	0
$171$	−4.63288	−0.354285
$172$	−16.2328	−1.23774
$173$	4.74360	0.360649	0.180325	−	0.983607i	$-0.442285\pi$
$173$	4.74360	0.360649	0.180325	+	0.983607i	$0.442285\pi$
$174$	−6.25313	−0.474049
$175$	0	0
$176$	−8.09589	−0.610251
$177$	−4.14430	−0.311505
$178$	6.02118	0.451306
$179$	−3.25210	−0.243073	−0.121537	−	0.992587i	$-0.538782\pi$
$179$	−3.25210	−0.243073	−0.121537	+	0.992587i	$0.538782\pi$
$180$	0	0
$181$	24.0790	1.78978	0.894890	−	0.446287i	$-0.147254\pi$
$181$	24.0790	1.78978	0.894890	+	0.446287i	$0.147254\pi$
$182$	0	0
$183$	18.1434	1.34120
$184$	4.29629	0.316727
$185$	0	0
$186$	0.0676511	0.00496042
$187$	−15.7667	−1.15298
$188$	−12.8967	−0.940592
$189$	−0.0695970	−0.00506244
$190$	0	0
$191$	−20.7685	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$191$	−20.7685	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$192$	1.04788	0.0756240
$193$	−16.5276	−1.18968	−0.594841	−	0.803844i	$-0.702785\pi$
$193$	−16.5276	−1.18968	−0.594841	+	0.803844i	$0.702785\pi$
$194$	6.19080	0.444474
$195$	0	0
$196$	11.5188	0.822774
$197$	8.65096	0.616355	0.308178	−	0.951329i	$-0.400281\pi$
$197$	8.65096	0.616355	0.308178	+	0.951329i	$0.400281\pi$
$198$	1.91566	0.136140
$199$	−3.80351	−0.269624	−0.134812	−	0.990871i	$-0.543043\pi$
$199$	−3.80351	−0.269624	−0.134812	+	0.990871i	$0.543043\pi$
$200$	0	0
$201$	23.8197	1.68011
$202$	4.01537	0.282520
$203$	−0.0873527	−0.00613096
$204$	9.51439	0.666140
$205$	0	0
$206$	−1.10549	−0.0770234
$207$	1.57292	0.109325
$208$	0	0
$209$	−23.6125	−1.63331
$210$	0	0
$211$	−13.7968	−0.949813	−0.474907	−	0.880036i	$-0.657518\pi$
$211$	−13.7968	−0.949813	−0.474907	+	0.880036i	$0.657518\pi$
$212$	12.9058	0.886376
$213$	−13.0333	−0.893027
$214$	−11.0659	−0.756448
$215$	0	0
$216$	−12.2302	−0.832159
$217$	0.000945048 0	6.41541e−5 0
$218$	11.3583	0.769283
$219$	12.0698	0.815604
$220$	0	0
$221$	0	0
$222$	4.66384	0.313017
$223$	−12.5218	−0.838524	−0.419262	−	0.907865i	$-0.637711\pi$
$223$	−12.5218	−0.838524	−0.419262	+	0.907865i	$0.637711\pi$
$224$	−0.0683073	−0.00456397
$225$	0	0
$226$	−6.08401	−0.404702
$227$	−8.50898	−0.564761	−0.282381	−	0.959302i	$-0.591124\pi$
$227$	−8.50898	−0.564761	−0.282381	+	0.959302i	$0.591124\pi$
$228$	14.2489	0.943657
$229$	−16.9636	−1.12098	−0.560492	−	0.828160i	$-0.689388\pi$
$229$	−16.9636	−1.12098	−0.560492	+	0.828160i	$0.689388\pi$
$230$	0	0
$231$	−0.0742770	−0.00488707
$232$	−15.3504	−1.00780
$233$	−11.5317	−0.755466	−0.377733	−	0.925915i	$-0.623296\pi$
$233$	−11.5317	−0.755466	−0.377733	+	0.925915i	$0.623296\pi$
$234$	0	0
$235$	0	0
$236$	−4.59225	−0.298930
$237$	−0.622158	−0.0404135
$238$	−0.0286255	−0.00185551
$239$	−11.5758	−0.748777	−0.374389	−	0.927272i	$-0.622147\pi$
$239$	−11.5758	−0.748777	−0.374389	+	0.927272i	$0.622147\pi$
$240$	0	0
$241$	9.39380	0.605108	0.302554	−	0.953132i	$-0.402161\pi$
$241$	9.39380	0.605108	0.302554	+	0.953132i	$0.402161\pi$
$242$	3.21499	0.206667
$243$	−8.01759	−0.514328
$244$	20.1045	1.28706
$245$	0	0
$246$	1.36858	0.0872577
$247$	0	0
$248$	0.166072	0.0105456
$249$	17.0521	1.08063
$250$	0	0
$251$	24.1062	1.52157	0.760786	−	0.649003i	$-0.224814\pi$
$251$	24.1062	1.52157	0.760786	+	0.649003i	$0.224814\pi$
$252$	−0.0161487	−0.00101727
$253$	8.01672	0.504007
$254$	4.73466	0.297079
$255$	0	0
$256$	−5.42411	−0.339007
$257$	21.9512	1.36928	0.684640	−	0.728881i	$-0.259959\pi$
$257$	21.9512	1.36928	0.684640	+	0.728881i	$0.259959\pi$
$258$	−8.72117	−0.542957
$259$	0.0651513	0.00404830
$260$	0	0
$261$	−5.61992	−0.347864
$262$	−0.511839	−0.0316215
$263$	−7.96389	−0.491075	−0.245537	−	0.969387i	$-0.578964\pi$
$263$	−7.96389	−0.491075	−0.245537	+	0.969387i	$0.578964\pi$
$264$	−13.0526	−0.803331
$265$	0	0
$266$	−0.0428700	−0.00262853
$267$	−15.0200	−0.919212
$268$	26.3943	1.61229
$269$	27.3653	1.66849	0.834246	−	0.551393i	$-0.185904\pi$
$269$	27.3653	1.66849	0.834246	+	0.551393i	$0.185904\pi$
$270$	0	0
$271$	−2.22907	−0.135406	−0.0677030	−	0.997706i	$-0.521567\pi$
$271$	−2.22907	−0.135406	−0.0677030	+	0.997706i	$0.521567\pi$
$272$	7.78310	0.471920
$273$	0	0
$274$	−5.26800	−0.318252
$275$	0	0
$276$	−4.83767	−0.291194
$277$	−10.8468	−0.651721	−0.325861	−	0.945418i	$-0.605654\pi$
$277$	−10.8468	−0.651721	−0.325861	+	0.945418i	$0.605654\pi$
$278$	6.83425	0.409891
$279$	0.0608006	0.00364004
$280$	0	0
$281$	14.4469	0.861828	0.430914	−	0.902393i	$-0.358191\pi$
$281$	14.4469	0.861828	0.430914	+	0.902393i	$0.358191\pi$
$282$	−6.92887	−0.412608
$283$	−18.3946	−1.09345	−0.546723	−	0.837314i	$-0.684125\pi$
$283$	−18.3946	−1.09345	−0.546723	+	0.837314i	$0.684125\pi$
$284$	−14.4420	−0.856977
$285$	0	0
$286$	0	0
$287$	0.0191183	0.00112852
$288$	−4.39461	−0.258955
$289$	−1.84244	−0.108379
$290$	0	0
$291$	−15.4432	−0.905295
$292$	13.3744	0.782680
$293$	−17.6321	−1.03008	−0.515039	−	0.857167i	$-0.672223\pi$
$293$	−17.6321	−1.03008	−0.515039	+	0.857167i	$0.672223\pi$
$294$	6.18858	0.360925
$295$	0	0
$296$	11.4489	0.665456
$297$	−22.8211	−1.32421
$298$	−7.21880	−0.418174
$299$	0	0
$300$	0	0
$301$	−0.121830	−0.00702216
$302$	−7.80729	−0.449259
$303$	−10.0165	−0.575432
$304$	11.6561	0.668524
$305$	0	0
$306$	−1.84165	−0.105280
$307$	−15.9134	−0.908228	−0.454114	−	0.890944i	$-0.650044\pi$
$307$	−15.9134	−0.908228	−0.454114	+	0.890944i	$0.650044\pi$
$308$	−0.0823054	−0.00468978
$309$	2.75769	0.156880
$310$	0	0
$311$	13.6969	0.776682	0.388341	−	0.921516i	$-0.373048\pi$
$311$	13.6969	0.776682	0.388341	+	0.921516i	$0.373048\pi$
$312$	0	0
$313$	21.8663	1.23596	0.617978	−	0.786195i	$-0.287952\pi$
$313$	21.8663	1.23596	0.617978	+	0.786195i	$0.287952\pi$
$314$	12.0079	0.677645
$315$	0	0
$316$	−0.689405	−0.0387821
$317$	−29.6388	−1.66468	−0.832340	−	0.554265i	$-0.813000\pi$
$317$	−29.6388	−1.66468	−0.832340	+	0.554265i	$0.813000\pi$
$318$	6.93376	0.388826
$319$	−28.6432	−1.60371
$320$	0	0
$321$	27.6042	1.54072
$322$	0.0145549	0.000811111 0
$323$	22.7002	1.26307
$324$	9.84869	0.547150
$325$	0	0
$326$	−4.58379	−0.253872
$327$	−28.3337	−1.56686
$328$	3.35964	0.185505
$329$	−0.0967925	−0.00533634
$330$	0	0
$331$	−11.2700	−0.619454	−0.309727	−	0.950826i	$-0.600238\pi$
$331$	−11.2700	−0.619454	−0.309727	+	0.950826i	$0.600238\pi$
$332$	18.8952	1.03701
$333$	4.19157	0.229697
$334$	−12.9617	−0.709232
$335$	0	0
$336$	0.0366662	0.00200030
$337$	−13.1449	−0.716046	−0.358023	−	0.933713i	$-0.616549\pi$
$337$	−13.1449	−0.716046	−0.358023	+	0.933713i	$0.616549\pi$
$338$	0	0
$339$	15.1768	0.824289
$340$	0	0
$341$	0.309884	0.0167812
$342$	−2.75808	−0.149140
$343$	0.172904	0.00933593
$344$	−21.4090	−1.15429
$345$	0	0
$346$	2.82400	0.151819
$347$	−4.13615	−0.222040	−0.111020	−	0.993818i	$-0.535412\pi$
$347$	−4.13615	−0.222040	−0.111020	+	0.993818i	$0.535412\pi$
$348$	17.2847	0.926556
$349$	25.1469	1.34608	0.673042	−	0.739605i	$-0.264987\pi$
$349$	25.1469	1.34608	0.673042	+	0.739605i	$0.264987\pi$
$350$	0	0
$351$	0	0
$352$	−22.3982	−1.19383
$353$	3.29476	0.175363	0.0876813	−	0.996149i	$-0.472054\pi$
$353$	3.29476	0.175363	0.0876813	+	0.996149i	$0.472054\pi$
$354$	−2.46722	−0.131131
$355$	0	0
$356$	−16.6435	−0.882105
$357$	0.0714072	0.00377927
$358$	−1.93607	−0.102324
$359$	−34.2940	−1.80997	−0.904983	−	0.425448i	$-0.860116\pi$
$359$	−34.2940	−1.80997	−0.904983	+	0.425448i	$0.860116\pi$
$360$	0	0
$361$	14.9963	0.789277
$362$	14.3349	0.753427
$363$	−8.01990	−0.420935
$364$	0	0
$365$	0	0
$366$	10.8013	0.564591
$367$	−10.1960	−0.532226	−0.266113	−	0.963942i	$-0.585739\pi$
$367$	−10.1960	−0.532226	−0.266113	+	0.963942i	$0.585739\pi$
$368$	−3.95738	−0.206293
$369$	1.23000	0.0640311
$370$	0	0
$371$	0.0968607	0.00502876
$372$	−0.186999	−0.00969543
$373$	−7.84815	−0.406362	−0.203181	−	0.979141i	$-0.565128\pi$
$373$	−7.84815	−0.406362	−0.203181	+	0.979141i	$0.565128\pi$
$374$	−9.38637	−0.485358
$375$	0	0
$376$	−17.0092	−0.877182
$377$	0	0
$378$	−0.0414331	−0.00213109
$379$	9.39241	0.482456	0.241228	−	0.970469i	$-0.422450\pi$
$379$	9.39241	0.482456	0.241228	+	0.970469i	$0.422450\pi$
$380$	0	0
$381$	−11.8108	−0.605084
$382$	−12.3641	−0.632600
$383$	−4.03099	−0.205974	−0.102987	−	0.994683i	$-0.532840\pi$
$383$	−4.03099	−0.205974	−0.102987	+	0.994683i	$0.532840\pi$
$384$	17.0510	0.870128
$385$	0	0
$386$	−9.83934	−0.500809
$387$	−7.83804	−0.398430
$388$	−17.1124	−0.868750
$389$	10.4119	0.527902	0.263951	−	0.964536i	$-0.414974\pi$
$389$	10.4119	0.527902	0.263951	+	0.964536i	$0.414974\pi$
$390$	0	0
$391$	−7.70699	−0.389759
$392$	15.1919	0.767307
$393$	1.27680	0.0644061
$394$	5.15016	0.259461
$395$	0	0
$396$	−5.29520	−0.266094
$397$	10.5336	0.528666	0.264333	−	0.964431i	$-0.414848\pi$
$397$	10.5336	0.528666	0.264333	+	0.964431i	$0.414848\pi$
$398$	−2.26434	−0.113501
$399$	0.106941	0.00535373
$400$	0	0
$401$	10.8299	0.540821	0.270410	−	0.962745i	$-0.412841\pi$
$401$	10.8299	0.540821	0.270410	+	0.962745i	$0.412841\pi$
$402$	14.1806	0.707262
$403$	0	0
$404$	−11.0991	−0.552203
$405$	0	0
$406$	−0.0520035	−0.00258089
$407$	21.3633	1.05894
$408$	12.5483	0.621233
$409$	−0.321084	−0.0158766	−0.00793828	−	0.999968i	$-0.502527\pi$
$409$	−0.321084	−0.0158766	−0.00793828	+	0.999968i	$0.502527\pi$
$410$	0	0
$411$	13.1412	0.648208
$412$	3.05576	0.150547
$413$	−0.0344657	−0.00169594
$414$	0.936401	0.0460216
$415$	0	0
$416$	0	0
$417$	−17.0483	−0.834857
$418$	−14.0572	−0.687560
$419$	−15.5574	−0.760027	−0.380013	−	0.924981i	$-0.624081\pi$
$419$	−15.5574	−0.760027	−0.380013	+	0.924981i	$0.624081\pi$
$420$	0	0
$421$	4.54317	0.221421	0.110710	−	0.993853i	$-0.464687\pi$
$421$	4.54317	0.221421	0.110710	+	0.993853i	$0.464687\pi$
$422$	−8.21364	−0.399834
$423$	−6.22724	−0.302779
$424$	17.0212	0.826621
$425$	0	0
$426$	−7.75909	−0.375929
$427$	0.150888	0.00730196
$428$	30.5879	1.47852
$429$	0	0
$430$	0	0
$431$	−31.9712	−1.54000	−0.770000	−	0.638043i	$-0.779744\pi$
$431$	−31.9712	−1.54000	−0.770000	+	0.638043i	$0.779744\pi$
$432$	11.2654	0.542008
$433$	17.6683	0.849086	0.424543	−	0.905408i	$-0.360435\pi$
$433$	17.6683	0.849086	0.424543	+	0.905408i	$0.360435\pi$
$434$	0.000562614 0	2.70063e−5 0
$435$	0	0
$436$	−31.3962	−1.50361
$437$	−11.5421	−0.552135
$438$	7.18552	0.343337
$439$	−3.34473	−0.159635	−0.0798177	−	0.996809i	$-0.525434\pi$
$439$	−3.34473	−0.159635	−0.0798177	+	0.996809i	$0.525434\pi$
$440$	0	0
$441$	5.56191	0.264853
$442$	0	0
$443$	−23.8464	−1.13298	−0.566488	−	0.824070i	$-0.691698\pi$
$443$	−23.8464	−1.13298	−0.566488	+	0.824070i	$0.691698\pi$
$444$	−12.8916	−0.611810
$445$	0	0
$446$	−7.45460	−0.352985
$447$	18.0076	0.851728
$448$	0.00871456	0.000411724 0
$449$	−6.44179	−0.304007	−0.152004	−	0.988380i	$-0.548573\pi$
$449$	−6.44179	−0.304007	−0.152004	+	0.988380i	$0.548573\pi$
$450$	0	0
$451$	6.26896	0.295194
$452$	16.8172	0.791014
$453$	19.4755	0.915041
$454$	−5.06564	−0.237742
$455$	0	0
$456$	18.7925	0.880041
$457$	7.96307	0.372497	0.186248	−	0.982503i	$-0.440367\pi$
$457$	7.96307	0.372497	0.186248	+	0.982503i	$0.440367\pi$
$458$	−10.0989	−0.471890
$459$	21.9394	1.02404
$460$	0	0
$461$	−3.82712	−0.178247	−0.0891233	−	0.996021i	$-0.528407\pi$
$461$	−3.82712	−0.178247	−0.0891233	+	0.996021i	$0.528407\pi$
$462$	−0.0442192	−0.00205726
$463$	−1.96955	−0.0915329	−0.0457665	−	0.998952i	$-0.514573\pi$
$463$	−1.96955	−0.0915329	−0.0457665	+	0.998952i	$0.514573\pi$
$464$	14.1395	0.656408
$465$	0	0
$466$	−6.86514	−0.318021
$467$	−25.0811	−1.16061	−0.580307	−	0.814398i	$-0.697067\pi$
$467$	−25.0811	−1.16061	−0.580307	+	0.814398i	$0.697067\pi$
$468$	0	0
$469$	0.198094	0.00914715
$470$	0	0
$471$	−29.9541	−1.38021
$472$	−6.05660	−0.278778
$473$	−39.9484	−1.83683
$474$	−0.370388	−0.0170125
$475$	0	0
$476$	0.0791255	0.00362671
$477$	6.23162	0.285326
$478$	−6.89141	−0.315206
$479$	−15.3209	−0.700030	−0.350015	−	0.936744i	$-0.613824\pi$
$479$	−15.3209	−0.700030	−0.350015	+	0.936744i	$0.613824\pi$
$480$	0	0
$481$	0	0
$482$	5.59239	0.254727
$483$	−0.0363076	−0.00165205
$484$	−8.88675	−0.403943
$485$	0	0
$486$	−4.77310	−0.216512
$487$	−1.70416	−0.0772228	−0.0386114	−	0.999254i	$-0.512293\pi$
$487$	−1.70416	−0.0772228	−0.0386114	+	0.999254i	$0.512293\pi$
$488$	26.5153	1.20029
$489$	11.4344	0.517082
$490$	0	0
$491$	−1.52274	−0.0687201	−0.0343600	−	0.999410i	$-0.510939\pi$
$491$	−1.52274	−0.0687201	−0.0343600	+	0.999410i	$0.510939\pi$
$492$	−3.78299	−0.170550
$493$	27.5366	1.24018
$494$	0	0
$495$	0	0
$496$	−0.152972	−0.00686862
$497$	−0.108390	−0.00486196
$498$	10.1516	0.454903
$499$	−17.1423	−0.767395	−0.383698	−	0.923459i	$-0.625349\pi$
$499$	−17.1423	−0.767395	−0.383698	+	0.923459i	$0.625349\pi$
$500$	0	0
$501$	32.3334	1.44455
$502$	14.3511	0.640522
$503$	14.9270	0.665562	0.332781	−	0.943004i	$-0.392013\pi$
$503$	14.9270	0.665562	0.332781	+	0.943004i	$0.392013\pi$
$504$	−0.0212981	−0.000948692 0
$505$	0	0
$506$	4.77258	0.212167
$507$	0	0
$508$	−13.0874	−0.580658
$509$	−41.3147	−1.83124	−0.915621	−	0.402043i	$-0.868300\pi$
$509$	−41.3147	−1.83124	−0.915621	+	0.402043i	$0.868300\pi$
$510$	0	0
$511$	0.100378	0.00444044
$512$	19.7341	0.872132
$513$	32.8568	1.45066
$514$	13.0682	0.576413
$515$	0	0
$516$	24.1067	1.06124
$517$	−31.7385	−1.39586
$518$	0.0387864	0.00170418
$519$	−7.04457	−0.309222
$520$	0	0
$521$	24.1169	1.05658	0.528290	−	0.849064i	$-0.322833\pi$
$521$	24.1169	1.05658	0.528290	+	0.849064i	$0.322833\pi$
$522$	−3.34570	−0.146437
$523$	44.3064	1.93738	0.968692	−	0.248266i	$-0.0798608\pi$
$523$	44.3064	1.93738	0.968692	+	0.248266i	$0.0798608\pi$
$524$	1.41481	0.0618061
$525$	0	0
$526$	−4.74113	−0.206723
$527$	−0.297911	−0.0129772
$528$	12.0229	0.523231
$529$	−19.0813	−0.829622
$530$	0	0
$531$	−2.21738	−0.0962262
$532$	0.118500	0.00513761
$533$	0	0
$534$	−8.94185	−0.386952
$535$	0	0
$536$	34.8108	1.50360
$537$	4.82958	0.208412
$538$	16.2913	0.702369
$539$	28.3475	1.22101
$540$	0	0
$541$	−9.21416	−0.396148	−0.198074	−	0.980187i	$-0.563469\pi$
$541$	−9.21416	−0.396148	−0.198074	+	0.980187i	$0.563469\pi$
$542$	−1.32703	−0.0570006
$543$	−35.7590	−1.53456
$544$	21.5328	0.923211
$545$	0	0
$546$	0	0
$547$	13.9959	0.598420	0.299210	−	0.954187i	$-0.403277\pi$
$547$	13.9959	0.598420	0.299210	+	0.954187i	$0.403277\pi$
$548$	14.5616	0.622041
$549$	9.70750	0.414306
$550$	0	0
$551$	41.2392	1.75685
$552$	−6.38028	−0.271563
$553$	−0.00517411	−0.000220026 0
$554$	−6.45741	−0.274349
$555$	0	0
$556$	−18.8910	−0.801156
$557$	26.4909	1.12246	0.561228	−	0.827661i	$-0.310329\pi$
$557$	26.4909	1.12246	0.561228	+	0.827661i	$0.310329\pi$
$558$	0.0361963	0.00153231
$559$	0	0
$560$	0	0
$561$	23.4146	0.988567
$562$	8.60063	0.362796
$563$	18.2417	0.768795	0.384397	−	0.923168i	$-0.374409\pi$
$563$	18.2417	0.768795	0.384397	+	0.923168i	$0.374409\pi$
$564$	19.1525	0.806467
$565$	0	0
$566$	−10.9508	−0.460298
$567$	0.0739163	0.00310419
$568$	−19.0472	−0.799204
$569$	−9.82506	−0.411888	−0.205944	−	0.978564i	$-0.566026\pi$
$569$	−9.82506	−0.411888	−0.205944	+	0.978564i	$0.566026\pi$
$570$	0	0
$571$	24.1082	1.00890	0.504448	−	0.863442i	$-0.331696\pi$
$571$	24.1082	1.00890	0.504448	+	0.863442i	$0.331696\pi$
$572$	0	0
$573$	30.8426	1.28847
$574$	0.0113817	0.000475063 0
$575$	0	0
$576$	0.560660	0.0233608
$577$	40.7449	1.69623	0.848116	−	0.529810i	$-0.177737\pi$
$577$	40.7449	1.69623	0.848116	+	0.529810i	$0.177737\pi$
$578$	−1.09685	−0.0456231
$579$	24.5446	1.02004
$580$	0	0
$581$	0.141812	0.00588335
$582$	−9.19376	−0.381093
$583$	31.7609	1.31540
$584$	17.6392	0.729916
$585$	0	0
$586$	−10.4969	−0.433623
$587$	21.6671	0.894297	0.447148	−	0.894460i	$-0.352440\pi$
$587$	21.6671	0.894297	0.447148	+	0.894460i	$0.352440\pi$
$588$	−17.1062	−0.705450
$589$	−0.446157	−0.0183836
$590$	0	0
$591$	−12.8473	−0.528465
$592$	−10.5458	−0.433430
$593$	−43.7853	−1.79805	−0.899023	−	0.437901i	$-0.855722\pi$
$593$	−43.7853	−1.79805	−0.899023	+	0.437901i	$0.855722\pi$
$594$	−13.5860	−0.557441
$595$	0	0
$596$	19.9539	0.817345
$597$	5.64847	0.231177
$598$	0	0
$599$	11.1180	0.454271	0.227135	−	0.973863i	$-0.427064\pi$
$599$	11.1180	0.454271	0.227135	+	0.973863i	$0.427064\pi$
$600$	0	0
$601$	6.72352	0.274258	0.137129	−	0.990553i	$-0.456213\pi$
$601$	6.72352	0.274258	0.137129	+	0.990553i	$0.456213\pi$
$602$	−0.0725288	−0.00295605
$603$	12.7446	0.519000
$604$	21.5806	0.878103
$605$	0	0
$606$	−5.96309	−0.242234
$607$	−19.6086	−0.795890	−0.397945	−	0.917409i	$-0.630276\pi$
$607$	−19.6086	−0.795890	−0.397945	+	0.917409i	$0.630276\pi$
$608$	32.2479	1.30782
$609$	0.129725	0.00525671
$610$	0	0
$611$	0	0
$612$	5.09061	0.205776
$613$	4.70328	0.189964	0.0949818	−	0.995479i	$-0.469721\pi$
$613$	4.70328	0.189964	0.0949818	+	0.995479i	$0.469721\pi$
$614$	−9.47372	−0.382328
$615$	0	0
$616$	−0.108550	−0.00437362
$617$	12.3878	0.498713	0.249356	−	0.968412i	$-0.419781\pi$
$617$	12.3878	0.498713	0.249356	+	0.968412i	$0.419781\pi$
$618$	1.64173	0.0660401
$619$	28.4531	1.14363	0.571814	−	0.820383i	$-0.306240\pi$
$619$	28.4531	1.14363	0.571814	+	0.820383i	$0.306240\pi$
$620$	0	0
$621$	−11.1553	−0.447645
$622$	8.15417	0.326952
$623$	−0.124913	−0.00500452
$624$	0	0
$625$	0	0
$626$	13.0176	0.520289
$627$	35.0662	1.40041
$628$	−33.1918	−1.32450
$629$	−20.5379	−0.818900
$630$	0	0
$631$	−27.5829	−1.09806	−0.549030	−	0.835803i	$-0.685003\pi$
$631$	−27.5829	−1.09806	−0.549030	+	0.835803i	$0.685003\pi$
$632$	−0.909239	−0.0361676
$633$	20.4892	0.814373
$634$	−17.6448	−0.700765
$635$	0	0
$636$	−19.1660	−0.759983
$637$	0	0
$638$	−17.0521	−0.675099
$639$	−6.97338	−0.275863
$640$	0	0
$641$	−23.0335	−0.909767	−0.454884	−	0.890551i	$-0.650319\pi$
$641$	−23.0335	−0.909767	−0.454884	+	0.890551i	$0.650319\pi$
$642$	16.4336	0.648582
$643$	−18.1925	−0.717443	−0.358721	−	0.933445i	$-0.616787\pi$
$643$	−18.1925	−0.717443	−0.358721	+	0.933445i	$0.616787\pi$
$644$	−0.0402320	−0.00158536
$645$	0	0
$646$	13.5141	0.531705
$647$	−4.57203	−0.179745	−0.0898725	−	0.995953i	$-0.528646\pi$
$647$	−4.57203	−0.179745	−0.0898725	+	0.995953i	$0.528646\pi$
$648$	12.9892	0.510264
$649$	−11.3014	−0.443618
$650$	0	0
$651$	−0.00140346	−5.50059e−5 0
$652$	12.6703	0.496209
$653$	−38.2079	−1.49519	−0.747596	−	0.664154i	$-0.768792\pi$
$653$	−38.2079	−1.49519	−0.747596	+	0.664154i	$0.768792\pi$
$654$	−16.8679	−0.659586
$655$	0	0
$656$	−3.09462	−0.120825
$657$	6.45789	0.251946
$658$	−0.0576233	−0.00224639
$659$	−9.35604	−0.364459	−0.182230	−	0.983256i	$-0.558331\pi$
$659$	−9.35604	−0.364459	−0.182230	+	0.983256i	$0.558331\pi$
$660$	0	0
$661$	19.4712	0.757342	0.378671	−	0.925531i	$-0.376381\pi$
$661$	19.4712	0.757342	0.378671	+	0.925531i	$0.376381\pi$
$662$	−6.70934	−0.260766
$663$	0	0
$664$	24.9204	0.967099
$665$	0	0
$666$	2.49536	0.0966933
$667$	−14.0012	−0.542129
$668$	35.8282	1.38623
$669$	18.5958	0.718953
$670$	0	0
$671$	49.4765	1.91002
$672$	0.101441	0.00391317
$673$	−20.4116	−0.786809	−0.393405	−	0.919365i	$-0.628703\pi$
$673$	−20.4116	−0.786809	−0.393405	+	0.919365i	$0.628703\pi$
$674$	−7.82551	−0.301427
$675$	0	0
$676$	0	0
$677$	−5.65928	−0.217504	−0.108752	−	0.994069i	$-0.534685\pi$
$677$	−5.65928	−0.217504	−0.108752	+	0.994069i	$0.534685\pi$
$678$	9.03516	0.346993
$679$	−0.128432	−0.00492875
$680$	0	0
$681$	12.6364	0.484228
$682$	0.184483	0.00706421
$683$	−29.4953	−1.12861	−0.564304	−	0.825567i	$-0.690855\pi$
$683$	−29.4953	−1.12861	−0.564304	+	0.825567i	$0.690855\pi$
$684$	7.62379	0.291503
$685$	0	0
$686$	0.102935	0.00393006
$687$	25.1920	0.961136
$688$	19.7202	0.751824
$689$	0	0
$690$	0	0
$691$	−15.8114	−0.601493	−0.300746	−	0.953704i	$-0.597236\pi$
$691$	−15.8114	−0.601493	−0.300746	+	0.953704i	$0.597236\pi$
$692$	−7.80600	−0.296740
$693$	−0.0397414	−0.00150965
$694$	−2.46237	−0.0934701
$695$	0	0
$696$	22.7963	0.864092
$697$	−6.02676	−0.228280
$698$	14.9707	0.566648
$699$	17.1253	0.647739
$700$	0	0
$701$	−32.7085	−1.23538	−0.617692	−	0.786420i	$-0.711932\pi$
$701$	−32.7085	−1.23538	−0.617692	+	0.786420i	$0.711932\pi$
$702$	0	0
$703$	−30.7579	−1.16006
$704$	2.85753	0.107697
$705$	0	0
$706$	1.96147	0.0738208
$707$	−0.0833010	−0.00313286
$708$	6.81980	0.256304
$709$	−41.3978	−1.55473	−0.777363	−	0.629052i	$-0.783443\pi$
$709$	−41.3978	−1.55473	−0.777363	+	0.629052i	$0.783443\pi$
$710$	0	0
$711$	−0.332882	−0.0124840
$712$	−21.9507	−0.822638
$713$	0.151476	0.00567281
$714$	0.0425107	0.00159092
$715$	0	0
$716$	5.35160	0.199999
$717$	17.1909	0.642005
$718$	−20.4162	−0.761924
$719$	34.1320	1.27291	0.636454	−	0.771314i	$-0.280400\pi$
$719$	34.1320	1.27291	0.636454	+	0.771314i	$0.280400\pi$
$720$	0	0
$721$	0.0229341	0.000854109 0
$722$	8.92769	0.332254
$723$	−13.9504	−0.518822
$724$	−39.6241	−1.47262
$725$	0	0
$726$	−4.77447	−0.177197
$727$	8.52629	0.316223	0.158111	−	0.987421i	$-0.449460\pi$
$727$	8.52629	0.316223	0.158111	+	0.987421i	$0.449460\pi$
$728$	0	0
$729$	29.8614	1.10598
$730$	0	0
$731$	38.4049	1.42046
$732$	−29.8565	−1.10353
$733$	−41.7181	−1.54089	−0.770446	−	0.637506i	$-0.779966\pi$
$733$	−41.7181	−1.54089	−0.770446	+	0.637506i	$0.779966\pi$
$734$	−6.06996	−0.224046
$735$	0	0
$736$	−10.9485	−0.403568
$737$	64.9557	2.39267
$738$	0.732252	0.0269546
$739$	−39.8251	−1.46499	−0.732495	−	0.680772i	$-0.761644\pi$
$739$	−39.8251	−1.46499	−0.732495	+	0.680772i	$0.761644\pi$
$740$	0	0
$741$	0	0
$742$	0.0576639	0.00211691
$743$	29.7624	1.09188	0.545938	−	0.837825i	$-0.316173\pi$
$743$	29.7624	1.09188	0.545938	+	0.837825i	$0.316173\pi$
$744$	−0.246628	−0.00904182
$745$	0	0
$746$	−4.67223	−0.171062
$747$	9.12360	0.333815
$748$	25.9455	0.948660
$749$	0.229568	0.00838823
$750$	0	0
$751$	42.7074	1.55841	0.779207	−	0.626767i	$-0.215622\pi$
$751$	42.7074	1.55841	0.779207	+	0.626767i	$0.215622\pi$
$752$	15.6674	0.571333
$753$	−35.7994	−1.30460
$754$	0	0
$755$	0	0
$756$	0.114528	0.00416534
$757$	45.8109	1.66503	0.832513	−	0.554005i	$-0.186901\pi$
$757$	45.8109	1.66503	0.832513	+	0.554005i	$0.186901\pi$
$758$	5.59157	0.203095
$759$	−11.9054	−0.432138
$760$	0	0
$761$	−28.2519	−1.02413	−0.512064	−	0.858947i	$-0.671119\pi$
$761$	−28.2519	−1.02413	−0.512064	+	0.858947i	$0.671119\pi$
$762$	−7.03129	−0.254717
$763$	−0.235635	−0.00853055
$764$	34.1763	1.23645
$765$	0	0
$766$	−2.39976	−0.0867069
$767$	0	0
$768$	8.05516	0.290666
$769$	−0.292307	−0.0105409	−0.00527043	−	0.999986i	$-0.501678\pi$
$769$	−0.292307	−0.0105409	−0.00527043	+	0.999986i	$0.501678\pi$
$770$	0	0
$771$	−32.5991	−1.17403
$772$	27.1975	0.978860
$773$	−52.8525	−1.90097	−0.950485	−	0.310769i	$-0.899413\pi$
$773$	−52.8525	−1.90097	−0.950485	+	0.310769i	$0.899413\pi$
$774$	−4.66621	−0.167723
$775$	0	0
$776$	−22.5691	−0.810183
$777$	−0.0967540	−0.00347103
$778$	6.19847	0.222226
$779$	−9.02577	−0.323382
$780$	0	0
$781$	−35.5414	−1.27177
$782$	−4.58819	−0.164073
$783$	39.8570	1.42437
$784$	−13.9935	−0.499768
$785$	0	0
$786$	0.760116	0.0271124
$787$	−27.8785	−0.993761	−0.496881	−	0.867819i	$-0.665521\pi$
$787$	−27.8785	−0.993761	−0.496881	+	0.867819i	$0.665521\pi$
$788$	−14.2359	−0.507132
$789$	11.8269	0.421049
$790$	0	0
$791$	0.126216	0.00448773
$792$	−6.98370	−0.248155
$793$	0	0
$794$	6.27095	0.222548
$795$	0	0
$796$	6.25900	0.221844
$797$	−34.8156	−1.23323	−0.616616	−	0.787264i	$-0.711497\pi$
$797$	−34.8156	−1.23323	−0.616616	+	0.787264i	$0.711497\pi$
$798$	0.0636648	0.00225371
$799$	30.5123	1.07945
$800$	0	0
$801$	−8.03638	−0.283951
$802$	6.44736	0.227664
$803$	32.9141	1.16151
$804$	−39.1974	−1.38238
$805$	0	0
$806$	0	0
$807$	−40.6393	−1.43057
$808$	−14.6384	−0.514976
$809$	48.2068	1.69486	0.847430	−	0.530907i	$-0.178149\pi$
$809$	48.2068	1.69486	0.847430	+	0.530907i	$0.178149\pi$
$810$	0	0
$811$	28.5024	1.00086	0.500428	−	0.865778i	$-0.333176\pi$
$811$	28.5024	1.00086	0.500428	+	0.865778i	$0.333176\pi$
$812$	0.143746	0.00504450
$813$	3.31031	0.116098
$814$	12.7182	0.445772
$815$	0	0
$816$	−11.5584	−0.404626
$817$	57.5159	2.01223
$818$	−0.191150	−0.00668341
$819$	0	0
$820$	0	0
$821$	43.7623	1.52732	0.763658	−	0.645621i	$-0.223401\pi$
$821$	43.7623	1.52732	0.763658	+	0.645621i	$0.223401\pi$
$822$	7.82334	0.272870
$823$	0.102737	0.00358119	0.00179059	−	0.999998i	$-0.499430\pi$
$823$	0.102737	0.00358119	0.00179059	+	0.999998i	$0.499430\pi$
$824$	4.03017	0.140398
$825$	0	0
$826$	−0.0205184	−0.000713926 0
$827$	15.7763	0.548597	0.274299	−	0.961645i	$-0.411554\pi$
$827$	15.7763	0.548597	0.274299	+	0.961645i	$0.411554\pi$
$828$	−2.58837	−0.0899519
$829$	−27.2851	−0.947650	−0.473825	−	0.880619i	$-0.657127\pi$
$829$	−27.2851	−0.947650	−0.473825	+	0.880619i	$0.657127\pi$
$830$	0	0
$831$	16.1082	0.558788
$832$	0	0
$833$	−27.2523	−0.944236
$834$	−10.1493	−0.351442
$835$	0	0
$836$	38.8564	1.34388
$837$	−0.431203	−0.0149046
$838$	−9.26173	−0.319941
$839$	18.2102	0.628686	0.314343	−	0.949310i	$-0.398216\pi$
$839$	18.2102	0.628686	0.314343	+	0.949310i	$0.398216\pi$
$840$	0	0
$841$	21.0253	0.725011
$842$	2.70468	0.0932094
$843$	−21.4546	−0.738935
$844$	22.7038	0.781499
$845$	0	0
$846$	−3.70725	−0.127458
$847$	−0.0666967	−0.00229172
$848$	−15.6785	−0.538401
$849$	27.3172	0.937525
$850$	0	0
$851$	10.4427	0.357970
$852$	21.4474	0.734775
$853$	55.2019	1.89008	0.945038	−	0.326960i	$-0.106024\pi$
$853$	55.2019	1.89008	0.945038	+	0.326960i	$0.106024\pi$
$854$	0.0898277	0.00307384
$855$	0	0
$856$	40.3416	1.37885
$857$	−22.2905	−0.761428	−0.380714	−	0.924693i	$-0.624322\pi$
$857$	−22.2905	−0.761428	−0.380714	+	0.924693i	$0.624322\pi$
$858$	0	0
$859$	37.0440	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$859$	37.0440	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$860$	0	0
$861$	−0.0283920	−0.000967598 0
$862$	−19.0334	−0.648280
$863$	−5.88863	−0.200451	−0.100226	−	0.994965i	$-0.531956\pi$
$863$	−5.88863	−0.200451	−0.100226	+	0.994965i	$0.531956\pi$
$864$	31.1670	1.06032
$865$	0	0
$866$	10.5185	0.357432
$867$	2.73614	0.0929243
$868$	−0.00155516	−5.27855e−5 0
$869$	−1.69661	−0.0575534
$870$	0	0
$871$	0	0
$872$	−41.4077	−1.40224
$873$	−8.26277	−0.279652
$874$	−6.87135	−0.232427
$875$	0	0
$876$	−19.8619	−0.671073
$877$	−2.85035	−0.0962494	−0.0481247	−	0.998841i	$-0.515324\pi$
$877$	−2.85035	−0.0962494	−0.0481247	+	0.998841i	$0.515324\pi$
$878$	−1.99121	−0.0672002
$879$	26.1849	0.883194
$880$	0	0
$881$	−39.5446	−1.33229	−0.666145	−	0.745822i	$-0.732057\pi$
$881$	−39.5446	−1.33229	−0.666145	+	0.745822i	$0.732057\pi$
$882$	3.31116	0.111493
$883$	−46.2706	−1.55713	−0.778564	−	0.627565i	$-0.784052\pi$
$883$	−46.2706	−1.55713	−0.778564	+	0.627565i	$0.784052\pi$
$884$	0	0
$885$	0	0
$886$	−14.1964	−0.476938
$887$	52.2173	1.75328	0.876642	−	0.481143i	$-0.159778\pi$
$887$	52.2173	1.75328	0.876642	+	0.481143i	$0.159778\pi$
$888$	−17.0024	−0.570564
$889$	−0.0982232	−0.00329430
$890$	0	0
$891$	24.2374	0.811982
$892$	20.6057	0.689931
$893$	45.6957	1.52915
$894$	10.7204	0.358544
$895$	0	0
$896$	0.141803	0.00473729
$897$	0	0
$898$	−3.83498	−0.127975
$899$	−0.541212	−0.0180504
$900$	0	0
$901$	−30.5338	−1.01723
$902$	3.73209	0.124265
$903$	0.180926	0.00602083
$904$	22.1798	0.737688
$905$	0	0
$906$	11.5943	0.385196
$907$	−0.798878	−0.0265263	−0.0132632	−	0.999912i	$-0.504222\pi$
$907$	−0.798878	−0.0265263	−0.0132632	+	0.999912i	$0.504222\pi$
$908$	14.0022	0.464681
$909$	−5.35925	−0.177755
$910$	0	0
$911$	−24.2428	−0.803201	−0.401600	−	0.915815i	$-0.631546\pi$
$911$	−24.2428	−0.803201	−0.401600	+	0.915815i	$0.631546\pi$
$912$	−17.3101	−0.573195
$913$	46.5005	1.53894
$914$	4.74064	0.156806
$915$	0	0
$916$	27.9150	0.922337
$917$	0.0106184	0.000350650 0
$918$	13.0611	0.431081
$919$	−46.0224	−1.51814	−0.759070	−	0.651009i	$-0.774346\pi$
$919$	−46.0224	−1.51814	−0.759070	+	0.651009i	$0.774346\pi$
$920$	0	0
$921$	23.6325	0.778718
$922$	−2.27839	−0.0750348
$923$	0	0
$924$	0.122229	0.00402104
$925$	0	0
$926$	−1.17253	−0.0385318
$927$	1.47549	0.0484613
$928$	39.1183	1.28412
$929$	−46.8169	−1.53601	−0.768006	−	0.640442i	$-0.778751\pi$
$929$	−46.8169	−1.53601	−0.768006	+	0.640442i	$0.778751\pi$
$930$	0	0
$931$	−40.8135	−1.33761
$932$	18.9764	0.621591
$933$	−20.3409	−0.665930
$934$	−14.9315	−0.488573
$935$	0	0
$936$	0	0
$937$	32.7928	1.07129	0.535646	−	0.844442i	$-0.320068\pi$
$937$	32.7928	1.07129	0.535646	+	0.844442i	$0.320068\pi$
$938$	0.117931	0.00385059
$939$	−32.4729	−1.05971
$940$	0	0
$941$	49.3829	1.60984	0.804918	−	0.593386i	$-0.202209\pi$
$941$	49.3829	1.60984	0.804918	+	0.593386i	$0.202209\pi$
$942$	−17.8325	−0.581015
$943$	3.06436	0.0997892
$944$	5.57883	0.181576
$945$	0	0
$946$	−23.7824	−0.773232
$947$	1.12912	0.0366916	0.0183458	−	0.999832i	$-0.494160\pi$
$947$	1.12912	0.0366916	0.0183458	+	0.999832i	$0.494160\pi$
$948$	1.02381	0.0332519
$949$	0	0
$950$	0	0
$951$	44.0156	1.42730
$952$	0.104357	0.00338222
$953$	−3.76653	−0.122010	−0.0610049	−	0.998137i	$-0.519431\pi$
$953$	−3.76653	−0.122010	−0.0610049	+	0.998137i	$0.519431\pi$
$954$	3.70986	0.120111
$955$	0	0
$956$	19.0490	0.616088
$957$	42.5371	1.37503
$958$	−9.12097	−0.294685
$959$	0.109288	0.00352908
$960$	0	0
$961$	−30.9941	−0.999811
$962$	0	0
$963$	14.7695	0.475940
$964$	−15.4583	−0.497878
$965$	0	0
$966$	−0.0216150	−0.000695450 0
$967$	5.50306	0.176966	0.0884832	−	0.996078i	$-0.471798\pi$
$967$	5.50306	0.176966	0.0884832	+	0.996078i	$0.471798\pi$
$968$	−11.7205	−0.376711
$969$	−33.7114	−1.08296
$970$	0	0
$971$	−8.60436	−0.276127	−0.138064	−	0.990423i	$-0.544088\pi$
$971$	−8.60436	−0.276127	−0.138064	+	0.990423i	$0.544088\pi$
$972$	13.1936	0.423185
$973$	−0.141780	−0.00454527
$974$	−1.01453	−0.0325078
$975$	0	0
$976$	−24.4236	−0.781781
$977$	48.6703	1.55710	0.778551	−	0.627582i	$-0.215955\pi$
$977$	48.6703	1.55710	0.778551	+	0.627582i	$0.215955\pi$
$978$	6.80723	0.217671
$979$	−40.9592	−1.30906
$980$	0	0
$981$	−15.1598	−0.484015
$982$	−0.906527	−0.0289284
$983$	−58.6285	−1.86996	−0.934980	−	0.354701i	$-0.884583\pi$
$983$	−58.6285	−1.86996	−0.934980	+	0.354701i	$0.884583\pi$
$984$	−4.98929	−0.159053
$985$	0	0
$986$	16.3933	0.522069
$987$	0.143743	0.00457540
$988$	0	0
$989$	−19.5273	−0.620933
$990$	0	0
$991$	24.7825	0.787241	0.393620	−	0.919273i	$-0.371222\pi$
$991$	24.7825	0.787241	0.393620	+	0.919273i	$0.371222\pi$
$992$	−0.423212	−0.0134370
$993$	16.7367	0.531122
$994$	−0.0645277	−0.00204669
$995$	0	0
$996$	−28.0606	−0.889135
$997$	44.9485	1.42353	0.711766	−	0.702416i	$-0.247895\pi$
$997$	44.9485	1.42353	0.711766	+	0.702416i	$0.247895\pi$
$998$	−10.2053	−0.323043
$999$	−29.7270	−0.940520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bw.1.9	✓	12
5.4	even	2		4225.2.a.bz.1.4	yes	12
13.12	even	2		4225.2.a.by.1.4	yes	12
65.64	even	2		4225.2.a.bx.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4225.2.a.bw.1.9	✓	12	1.1	even	1	trivial
4225.2.a.bx.1.9	yes	12	65.64	even	2
4225.2.a.by.1.4	yes	12	13.12	even	2
4225.2.a.bz.1.4	yes	12	5.4	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.595328 q^{2} -1.48507 q^{3} -1.64558 q^{4} -0.884102 q^{6} -0.0123504 q^{7} -2.17032 q^{8} -0.794576 q^{9} +O(q^{10})\)	\(q+0.595328 q^{2} -1.48507 q^{3} -1.64558 q^{4} -0.884102 q^{6} -0.0123504 q^{7} -2.17032 q^{8} -0.794576 q^{9} -4.04974 q^{11} +2.44380 q^{12} -0.00735255 q^{14} +1.99912 q^{16} +3.89327 q^{17} -0.473033 q^{18} +5.83063 q^{19} +0.0183412 q^{21} -2.41092 q^{22} -1.97957 q^{23} +3.22307 q^{24} +5.63520 q^{27} +0.0203236 q^{28} +7.07286 q^{29} -0.0765196 q^{31} +5.53077 q^{32} +6.01413 q^{33} +2.31777 q^{34} +1.30754 q^{36} -5.27523 q^{37} +3.47114 q^{38} -1.54799 q^{41} +0.0109190 q^{42} +9.86444 q^{43} +6.66418 q^{44} -1.17849 q^{46} +7.83718 q^{47} -2.96882 q^{48} -6.99985 q^{49} -5.78177 q^{51} -7.84271 q^{53} +3.35479 q^{54} +0.0268043 q^{56} -8.65888 q^{57} +4.21067 q^{58} +2.79065 q^{59} -12.2172 q^{61} -0.0455543 q^{62} +0.00981334 q^{63} -0.705609 q^{64} +3.58038 q^{66} -16.0395 q^{67} -6.40671 q^{68} +2.93979 q^{69} +8.77623 q^{71} +1.72448 q^{72} -8.12747 q^{73} -3.14049 q^{74} -9.59479 q^{76} +0.0500159 q^{77} +0.418943 q^{79} -5.98492 q^{81} -0.921564 q^{82} -11.4824 q^{83} -0.0301820 q^{84} +5.87258 q^{86} -10.5037 q^{87} +8.78922 q^{88} +10.1140 q^{89} +3.25754 q^{92} +0.113637 q^{93} +4.66570 q^{94} -8.21356 q^{96} +10.3990 q^{97} -4.16721 q^{98} +3.21782 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 7 q^{2} - q^{3} + 13 q^{4} + 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 12 * q - 7 * q^2 - q^3 + 13 * q^4 + 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9	\( 12 q - 7 q^{2} - q^{3} + 13 q^{4} + 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9} - 3 q^{11} - 8 q^{12} + 2 q^{14} + 11 q^{16} + 2 q^{17} - 14 q^{18} + 5 q^{21} + 12 q^{22} - 3 q^{23} + 4 q^{24} + 5 q^{27} - 29 q^{28} + 9 q^{29} + 4 q^{31} - 48 q^{32} - 30 q^{33} + 3 q^{34} + 44 q^{36} - 17 q^{37} + 15 q^{38} - 9 q^{41} - 80 q^{42} + q^{43} - 10 q^{44} - 3 q^{46} - 61 q^{47} + 35 q^{48} + 8 q^{49} - 26 q^{51} + 23 q^{53} + 48 q^{54} + 51 q^{56} - 10 q^{57} + 17 q^{58} + q^{59} + 13 q^{61} + 29 q^{62} - 46 q^{63} + 50 q^{64} + 29 q^{66} - 43 q^{67} - 26 q^{68} - 31 q^{69} - 19 q^{71} - 48 q^{72} - 21 q^{73} + 7 q^{74} - 46 q^{76} - 42 q^{77} - 7 q^{79} - 16 q^{81} + 15 q^{82} - 75 q^{83} + 83 q^{84} - 17 q^{86} + 5 q^{87} + 38 q^{88} + 19 q^{89} - 35 q^{92} - 11 q^{93} + 26 q^{94} - 90 q^{96} - 17 q^{97} - 11 q^{98} + 6 q^{99}+O(q^{100}) \) 12 * q - 7 * q^2 - q^3 + 13 * q^4 + 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9 - 3 * q^11 - 8 * q^12 + 2 * q^14 + 11 * q^16 + 2 * q^17 - 14 * q^18 + 5 * q^21 + 12 * q^22 - 3 * q^23 + 4 * q^24 + 5 * q^27 - 29 * q^28 + 9 * q^29 + 4 * q^31 - 48 * q^32 - 30 * q^33 + 3 * q^34 + 44 * q^36 - 17 * q^37 + 15 * q^38 - 9 * q^41 - 80 * q^42 + q^43 - 10 * q^44 - 3 * q^46 - 61 * q^47 + 35 * q^48 + 8 * q^49 - 26 * q^51 + 23 * q^53 + 48 * q^54 + 51 * q^56 - 10 * q^57 + 17 * q^58 + q^59 + 13 * q^61 + 29 * q^62 - 46 * q^63 + 50 * q^64 + 29 * q^66 - 43 * q^67 - 26 * q^68 - 31 * q^69 - 19 * q^71 - 48 * q^72 - 21 * q^73 + 7 * q^74 - 46 * q^76 - 42 * q^77 - 7 * q^79 - 16 * q^81 + 15 * q^82 - 75 * q^83 + 83 * q^84 - 17 * q^86 + 5 * q^87 + 38 * q^88 + 19 * q^89 - 35 * q^92 - 11 * q^93 + 26 * q^94 - 90 * q^96 - 17 * q^97 - 11 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

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Database

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