LMFDB - Embedded newform 1425.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.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:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} -4.13941 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} -4.13941 q^{7} +0.665102 q^{8} +1.00000 q^{9} -4.80451 q^{11} +1.97196 q^{12} -4.00000 q^{13} +0.693141 q^{14} +3.83255 q^{16} -5.94392 q^{17} -0.167449 q^{18} -1.00000 q^{19} +4.13941 q^{21} +0.804512 q^{22} +7.13941 q^{23} -0.665102 q^{24} +0.669797 q^{26} -1.00000 q^{27} +8.16275 q^{28} -5.00000 q^{29} +8.80451 q^{31} -1.97196 q^{32} +4.80451 q^{33} +0.995305 q^{34} -1.97196 q^{36} -3.66510 q^{37} +0.167449 q^{38} +4.00000 q^{39} +0.195488 q^{41} -0.693141 q^{42} -4.00000 q^{43} +9.47431 q^{44} -1.19549 q^{46} -11.6090 q^{47} -3.83255 q^{48} +10.1347 q^{49} +5.94392 q^{51} +7.88784 q^{52} +2.33020 q^{53} +0.167449 q^{54} -2.75313 q^{56} +1.00000 q^{57} +0.837246 q^{58} +7.46961 q^{59} -6.60902 q^{61} -1.47431 q^{62} -4.13941 q^{63} -7.33490 q^{64} -0.804512 q^{66} -1.19549 q^{67} +11.7212 q^{68} -7.13941 q^{69} +9.46961 q^{71} +0.665102 q^{72} -0.330203 q^{73} +0.613718 q^{74} +1.97196 q^{76} +19.8878 q^{77} -0.669797 q^{78} +13.4182 q^{79} +1.00000 q^{81} -0.0327344 q^{82} +2.80451 q^{83} -8.16275 q^{84} +0.669797 q^{86} +5.00000 q^{87} -3.19549 q^{88} +7.27882 q^{89} +16.5576 q^{91} -14.0786 q^{92} -8.80451 q^{93} +1.94392 q^{94} +1.97196 q^{96} +11.2741 q^{97} -1.69705 q^{98} -4.80451 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} - 12 q^{13} + 15 q^{14} + 12 q^{16} + 6 q^{17} - 3 q^{19} - 9 q^{22} + 9 q^{23} - 3 q^{24} - 3 q^{27} + 27 q^{28} - 15 q^{29} + 15 q^{31} + 6 q^{32} + 3 q^{33} + 6 q^{34} + 6 q^{36} - 12 q^{37} + 12 q^{39} + 12 q^{41} - 15 q^{42} - 12 q^{43} + 15 q^{44} - 15 q^{46} - 12 q^{47} - 12 q^{48} + 21 q^{49} - 6 q^{51} - 24 q^{52} + 9 q^{53} + 30 q^{56} + 3 q^{57} + 12 q^{59} + 3 q^{61} + 9 q^{62} - 21 q^{64} + 9 q^{66} - 15 q^{67} + 60 q^{68} - 9 q^{69} + 18 q^{71} + 3 q^{72} - 3 q^{73} - 24 q^{74} - 6 q^{76} + 12 q^{77} + 3 q^{79} + 3 q^{81} - 9 q^{82} - 3 q^{83} - 27 q^{84} + 15 q^{87} - 21 q^{88} - 3 q^{89} - 9 q^{92} - 15 q^{93} - 18 q^{94} - 6 q^{96} + 12 q^{97} + 57 q^{98} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 6 * q^12 - 12 * q^13 + 15 * q^14 + 12 * q^16 + 6 * q^17 - 3 * q^19 - 9 * q^22 + 9 * q^23 - 3 * q^24 - 3 * q^27 + 27 * q^28 - 15 * q^29 + 15 * q^31 + 6 * q^32 + 3 * q^33 + 6 * q^34 + 6 * q^36 - 12 * q^37 + 12 * q^39 + 12 * q^41 - 15 * q^42 - 12 * q^43 + 15 * q^44 - 15 * q^46 - 12 * q^47 - 12 * q^48 + 21 * q^49 - 6 * q^51 - 24 * q^52 + 9 * q^53 + 30 * q^56 + 3 * q^57 + 12 * q^59 + 3 * q^61 + 9 * q^62 - 21 * q^64 + 9 * q^66 - 15 * q^67 + 60 * q^68 - 9 * q^69 + 18 * q^71 + 3 * q^72 - 3 * q^73 - 24 * q^74 - 6 * q^76 + 12 * q^77 + 3 * q^79 + 3 * q^81 - 9 * q^82 - 3 * q^83 - 27 * q^84 + 15 * q^87 - 21 * q^88 - 3 * q^89 - 9 * q^92 - 15 * q^93 - 18 * q^94 - 6 * q^96 + 12 * q^97 + 57 * q^98 - 3 * 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 $

$2$	−0.167449	−0.118404	−0.0592022	−	0.998246i	$-0.518856\pi$
$2$	−0.167449	−0.118404	−0.0592022	+	0.998246i	$0.518856\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.97196	−0.985980
$5$	0	0
$5$	0	0
$6$	0.167449	0.0683608
$7$	−4.13941	−1.56455	−0.782275	−	0.622933i	$-0.785941\pi$
$7$	−4.13941	−1.56455	−0.782275	+	0.622933i	$0.785941\pi$
$8$	0.665102	0.235149
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.80451	−1.44861	−0.724307	−	0.689477i	$-0.757840\pi$
$11$	−4.80451	−1.44861	−0.724307	+	0.689477i	$0.757840\pi$
$12$	1.97196	0.569256
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0.693141	0.185250
$15$	0	0
$16$	3.83255	0.958138
$17$	−5.94392	−1.44161	−0.720806	−	0.693136i	$-0.756228\pi$
$17$	−5.94392	−1.44161	−0.720806	+	0.693136i	$0.756228\pi$
$18$	−0.167449	−0.0394682
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.13941	0.903293
$22$	0.804512	0.171522
$23$	7.13941	1.48867	0.744335	−	0.667806i	$-0.232767\pi$
$23$	7.13941	1.48867	0.744335	+	0.667806i	$0.232767\pi$
$24$	−0.665102	−0.135763
$25$	0	0
$26$	0.669797	0.131358
$27$	−1.00000	−0.192450
$28$	8.16275	1.54262
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	8.80451	1.58134	0.790668	−	0.612245i	$-0.209733\pi$
$31$	8.80451	1.58134	0.790668	+	0.612245i	$0.209733\pi$
$32$	−1.97196	−0.348597
$33$	4.80451	0.836358
$34$	0.995305	0.170693
$35$	0	0
$36$	−1.97196	−0.328660
$37$	−3.66510	−0.602539	−0.301269	−	0.953539i	$-0.597410\pi$
$37$	−3.66510	−0.602539	−0.301269	+	0.953539i	$0.597410\pi$
$38$	0.167449	0.0271638
$39$	4.00000	0.640513
$40$	0	0
$41$	0.195488	0.0305302	0.0152651	−	0.999883i	$-0.495141\pi$
$41$	0.195488	0.0305302	0.0152651	+	0.999883i	$0.495141\pi$
$42$	−0.693141	−0.106954
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	9.47431	1.42831
$45$	0	0
$46$	−1.19549	−0.176265
$47$	−11.6090	−1.69335	−0.846675	−	0.532110i	$-0.821399\pi$
$47$	−11.6090	−1.69335	−0.846675	+	0.532110i	$0.821399\pi$
$48$	−3.83255	−0.553181
$49$	10.1347	1.44782
$50$	0	0
$51$	5.94392	0.832315
$52$	7.88784	1.09385
$53$	2.33020	0.320078	0.160039	−	0.987111i	$-0.448838\pi$
$53$	2.33020	0.320078	0.160039	+	0.987111i	$0.448838\pi$
$54$	0.167449	0.0227869
$55$	0	0
$56$	−2.75313	−0.367902
$57$	1.00000	0.132453
$58$	0.837246	0.109936
$59$	7.46961	0.972461	0.486230	−	0.873831i	$-0.338372\pi$
$59$	7.46961	0.972461	0.486230	+	0.873831i	$0.338372\pi$
$60$	0	0
$61$	−6.60902	−0.846199	−0.423099	−	0.906083i	$-0.639058\pi$
$61$	−6.60902	−0.846199	−0.423099	+	0.906083i	$0.639058\pi$
$62$	−1.47431	−0.187237
$63$	−4.13941	−0.521517
$64$	−7.33490	−0.916862
$65$	0	0
$66$	−0.804512	−0.0990285
$67$	−1.19549	−0.146052	−0.0730261	−	0.997330i	$-0.523266\pi$
$67$	−1.19549	−0.146052	−0.0730261	+	0.997330i	$0.523266\pi$
$68$	11.7212	1.42140
$69$	−7.13941	−0.859484
$70$	0	0
$71$	9.46961	1.12384	0.561918	−	0.827193i	$-0.310064\pi$
$71$	9.46961	1.12384	0.561918	+	0.827193i	$0.310064\pi$
$72$	0.665102	0.0783830
$73$	−0.330203	−0.0386474	−0.0193237	−	0.999813i	$-0.506151\pi$
$73$	−0.330203	−0.0386474	−0.0193237	+	0.999813i	$0.506151\pi$
$74$	0.613718	0.0713433
$75$	0	0
$76$	1.97196	0.226199
$77$	19.8878	2.26643
$78$	−0.669797	−0.0758395
$79$	13.4182	1.50967	0.754834	−	0.655915i	$-0.227717\pi$
$79$	13.4182	1.50967	0.754834	+	0.655915i	$0.227717\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.0327344	−0.00361491
$83$	2.80451	0.307835	0.153918	−	0.988084i	$-0.450811\pi$
$83$	2.80451	0.307835	0.153918	+	0.988084i	$0.450811\pi$
$84$	−8.16275	−0.890629
$85$	0	0
$86$	0.669797	0.0722260
$87$	5.00000	0.536056
$88$	−3.19549	−0.340640
$89$	7.27882	0.771553	0.385777	−	0.922592i	$-0.373934\pi$
$89$	7.27882	0.771553	0.385777	+	0.922592i	$0.373934\pi$
$90$	0	0
$91$	16.5576	1.73571
$92$	−14.0786	−1.46780
$93$	−8.80451	−0.912985
$94$	1.94392	0.200500
$95$	0	0
$96$	1.97196	0.201262
$97$	11.2741	1.14471	0.572357	−	0.820005i	$-0.306029\pi$
$97$	11.2741	1.14471	0.572357	+	0.820005i	$0.306029\pi$
$98$	−1.69705	−0.171428
$99$	−4.80451	−0.482872
$100$	0	0
$101$	9.94392	0.989457	0.494729	−	0.869048i	$-0.335267\pi$
$101$	9.94392	0.989457	0.494729	+	0.869048i	$0.335267\pi$
$102$	−0.995305	−0.0985499
$103$	10.1441	0.999528	0.499764	−	0.866162i	$-0.333420\pi$
$103$	10.1441	0.999528	0.499764	+	0.866162i	$0.333420\pi$
$104$	−2.66041	−0.260874
$105$	0	0
$106$	−0.390191	−0.0378987
$107$	6.80921	0.658271	0.329135	−	0.944283i	$-0.393243\pi$
$107$	6.80921	0.658271	0.329135	+	0.944283i	$0.393243\pi$
$108$	1.97196	0.189752
$109$	−20.2227	−1.93699	−0.968494	−	0.249038i	$-0.919886\pi$
$109$	−20.2227	−1.93699	−0.968494	+	0.249038i	$0.919886\pi$
$110$	0	0
$111$	3.66510	0.347876
$112$	−15.8645	−1.49905
$113$	−18.0226	−1.69542	−0.847710	−	0.530460i	$-0.822019\pi$
$113$	−18.0226	−1.69542	−0.847710	+	0.530460i	$0.822019\pi$
$114$	−0.167449	−0.0156831
$115$	0	0
$116$	9.85980	0.915460
$117$	−4.00000	−0.369800
$118$	−1.25078	−0.115144
$119$	24.6043	2.25548
$120$	0	0
$121$	12.0833	1.09848
$122$	1.10668	0.100194
$123$	−0.195488	−0.0176266
$124$	−17.3622	−1.55917
$125$	0	0
$126$	0.693141	0.0617499
$127$	−14.0786	−1.24928	−0.624638	−	0.780914i	$-0.714754\pi$
$127$	−14.0786	−1.24928	−0.624638	+	0.780914i	$0.714754\pi$
$128$	5.17214	0.457157
$129$	4.00000	0.352180
$130$	0	0
$131$	−3.47431	−0.303552	−0.151776	−	0.988415i	$-0.548499\pi$
$131$	−3.47431	−0.303552	−0.151776	+	0.988415i	$0.548499\pi$
$132$	−9.47431	−0.824633
$133$	4.13941	0.358932
$134$	0.200184	0.0172932
$135$	0	0
$136$	−3.95331	−0.338994
$137$	18.6137	1.59028	0.795139	−	0.606428i	$-0.207398\pi$
$137$	18.6137	1.59028	0.795139	+	0.606428i	$0.207398\pi$
$138$	1.19549	0.101767
$139$	−6.13941	−0.520738	−0.260369	−	0.965509i	$-0.583844\pi$
$139$	−6.13941	−0.520738	−0.260369	+	0.965509i	$0.583844\pi$
$140$	0	0
$141$	11.6090	0.977656
$142$	−1.58568	−0.133067
$143$	19.2180	1.60709
$144$	3.83255	0.319379
$145$	0	0
$146$	0.0552923	0.00457602
$147$	−10.1347	−0.835897
$148$	7.22744	0.594092
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	2.72588	0.221829	0.110914	−	0.993830i	$-0.464622\pi$
$151$	2.72588	0.221829	0.110914	+	0.993830i	$0.464622\pi$
$152$	−0.665102	−0.0539469
$153$	−5.94392	−0.480538
$154$	−3.33020	−0.268355
$155$	0	0
$156$	−7.88784	−0.631533
$157$	−15.6924	−1.25239	−0.626193	−	0.779668i	$-0.715388\pi$
$157$	−15.6924	−1.25239	−0.626193	+	0.779668i	$0.715388\pi$
$158$	−2.24687	−0.178752
$159$	−2.33020	−0.184797
$160$	0	0
$161$	−29.5529	−2.32910
$162$	−0.167449	−0.0131561
$163$	−16.4088	−1.28524	−0.642620	−	0.766185i	$-0.722152\pi$
$163$	−16.4088	−1.28524	−0.642620	+	0.766185i	$0.722152\pi$
$164$	−0.385496	−0.0301021
$165$	0	0
$166$	−0.469613	−0.0364491
$167$	18.1394	1.40367	0.701835	−	0.712340i	$-0.252364\pi$
$167$	18.1394	1.40367	0.701835	+	0.712340i	$0.252364\pi$
$168$	2.75313	0.212408
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	7.88784	0.601442
$173$	8.21805	0.624806	0.312403	−	0.949950i	$-0.398866\pi$
$173$	8.21805	0.624806	0.312403	+	0.949950i	$0.398866\pi$
$174$	−0.837246	−0.0634715
$175$	0	0
$176$	−18.4135	−1.38797
$177$	−7.46961	−0.561451
$178$	−1.21883	−0.0913554
$179$	1.20018	0.0897059	0.0448530	−	0.998994i	$-0.485718\pi$
$179$	1.20018	0.0897059	0.0448530	+	0.998994i	$0.485718\pi$
$180$	0	0
$181$	−26.5482	−1.97332	−0.986658	−	0.162807i	$-0.947945\pi$
$181$	−26.5482	−1.97332	−0.986658	+	0.162807i	$0.947945\pi$
$182$	−2.77256	−0.205516
$183$	6.60902	0.488553
$184$	4.74843	0.350059
$185$	0	0
$186$	1.47431	0.108102
$187$	28.5576	2.08834
$188$	22.8925	1.66961
$189$	4.13941	0.301098
$190$	0	0
$191$	−22.9712	−1.66214	−0.831068	−	0.556171i	$-0.812270\pi$
$191$	−22.9712	−1.66214	−0.831068	+	0.556171i	$0.812270\pi$
$192$	7.33490	0.529351
$193$	−4.72588	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$193$	−4.72588	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$194$	−1.88784	−0.135539
$195$	0	0
$196$	−19.9853	−1.42752
$197$	3.67449	0.261797	0.130898	−	0.991396i	$-0.458214\pi$
$197$	3.67449	0.261797	0.130898	+	0.991396i	$0.458214\pi$
$198$	0.804512	0.0571741
$199$	5.19079	0.367966	0.183983	−	0.982929i	$-0.441101\pi$
$199$	5.19079	0.367966	0.183983	+	0.982929i	$0.441101\pi$
$200$	0	0
$201$	1.19549	0.0843233
$202$	−1.66510	−0.117156
$203$	20.6970	1.45265
$204$	−11.7212	−0.820647
$205$	0	0
$206$	−1.69862	−0.118349
$207$	7.13941	0.496223
$208$	−15.3302	−1.06296
$209$	4.80451	0.332335
$210$	0	0
$211$	15.4182	1.06143	0.530717	−	0.847549i	$-0.321923\pi$
$211$	15.4182	1.06143	0.530717	+	0.847549i	$0.321923\pi$
$212$	−4.59507	−0.315591
$213$	−9.46961	−0.648847
$214$	−1.14020	−0.0779422
$215$	0	0
$216$	−0.665102	−0.0452544
$217$	−36.4455	−2.47408
$218$	3.38628	0.229348
$219$	0.330203	0.0223131
$220$	0	0
$221$	23.7757	1.59933
$222$	−0.613718	−0.0411901
$223$	−5.13941	−0.344160	−0.172080	−	0.985083i	$-0.555049\pi$
$223$	−5.13941	−0.344160	−0.172080	+	0.985083i	$0.555049\pi$
$224$	8.16275	0.545397
$225$	0	0
$226$	3.01786	0.200745
$227$	25.3575	1.68303	0.841517	−	0.540231i	$-0.181663\pi$
$227$	25.3575	1.68303	0.841517	+	0.540231i	$0.181663\pi$
$228$	−1.97196	−0.130596
$229$	−27.6316	−1.82595	−0.912973	−	0.408020i	$-0.866219\pi$
$229$	−27.6316	−1.82595	−0.912973	+	0.408020i	$0.866219\pi$
$230$	0	0
$231$	−19.8878	−1.30852
$232$	−3.32551	−0.218330
$233$	−16.2788	−1.06646	−0.533230	−	0.845970i	$-0.679022\pi$
$233$	−16.2788	−1.06646	−0.533230	+	0.845970i	$0.679022\pi$
$234$	0.669797	0.0437860
$235$	0	0
$236$	−14.7298	−0.958827
$237$	−13.4182	−0.871608
$238$	−4.11997	−0.267058
$239$	−25.1620	−1.62759	−0.813796	−	0.581150i	$-0.802603\pi$
$239$	−25.1620	−1.62759	−0.813796	+	0.581150i	$0.802603\pi$
$240$	0	0
$241$	−13.6651	−0.880247	−0.440123	−	0.897937i	$-0.645065\pi$
$241$	−13.6651	−0.880247	−0.440123	+	0.897937i	$0.645065\pi$
$242$	−2.02334	−0.130065
$243$	−1.00000	−0.0641500
$244$	13.0327	0.834335
$245$	0	0
$246$	0.0327344	0.00208707
$247$	4.00000	0.254514
$248$	5.85589	0.371850
$249$	−2.80451	−0.177729
$250$	0	0
$251$	14.0561	0.887212	0.443606	−	0.896222i	$-0.353699\pi$
$251$	14.0561	0.887212	0.443606	+	0.896222i	$0.353699\pi$
$252$	8.16275	0.514205
$253$	−34.3014	−2.15651
$254$	2.35746	0.147920
$255$	0	0
$256$	13.8037	0.862733
$257$	16.4969	1.02905	0.514523	−	0.857477i	$-0.327969\pi$
$257$	16.4969	1.02905	0.514523	+	0.857477i	$0.327969\pi$
$258$	−0.669797	−0.0416997
$259$	15.1714	0.942702
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0.581770	0.0359419
$263$	7.13941	0.440235	0.220117	−	0.975473i	$-0.429356\pi$
$263$	7.13941	0.440235	0.220117	+	0.975473i	$0.429356\pi$
$264$	3.19549	0.196669
$265$	0	0
$266$	−0.693141	−0.0424992
$267$	−7.27882	−0.445457
$268$	2.35746	0.144005
$269$	−5.88784	−0.358988	−0.179494	−	0.983759i	$-0.557446\pi$
$269$	−5.88784	−0.358988	−0.179494	+	0.983759i	$0.557446\pi$
$270$	0	0
$271$	9.46961	0.575238	0.287619	−	0.957745i	$-0.407136\pi$
$271$	9.46961	0.575238	0.287619	+	0.957745i	$0.407136\pi$
$272$	−22.7804	−1.38126
$273$	−16.5576	−1.00211
$274$	−3.11685	−0.188296
$275$	0	0
$276$	14.0786	0.847434
$277$	16.4969	0.991201	0.495600	−	0.868551i	$-0.334948\pi$
$277$	16.4969	0.991201	0.495600	+	0.868551i	$0.334948\pi$
$278$	1.02804	0.0616577
$279$	8.80451	0.527112
$280$	0	0
$281$	−9.86529	−0.588514	−0.294257	−	0.955726i	$-0.595072\pi$
$281$	−9.86529	−0.588514	−0.294257	+	0.955726i	$0.595072\pi$
$282$	−1.94392	−0.115759
$283$	−1.33959	−0.0796306	−0.0398153	−	0.999207i	$-0.512677\pi$
$283$	−1.33959	−0.0796306	−0.0398153	+	0.999207i	$0.512677\pi$
$284$	−18.6737	−1.10808
$285$	0	0
$286$	−3.21805	−0.190287
$287$	−0.809207	−0.0477660
$288$	−1.97196	−0.116199
$289$	18.3302	1.07825
$290$	0	0
$291$	−11.2741	−0.660901
$292$	0.651148	0.0381055
$293$	−8.52569	−0.498076	−0.249038	−	0.968494i	$-0.580114\pi$
$293$	−8.52569	−0.498076	−0.249038	+	0.968494i	$0.580114\pi$
$294$	1.69705	0.0989740
$295$	0	0
$296$	−2.43767	−0.141686
$297$	4.80451	0.278786
$298$	0	0
$299$	−28.5576	−1.65153
$300$	0	0
$301$	16.5576	0.954366
$302$	−0.456446	−0.0262655
$303$	−9.94392	−0.571263
$304$	−3.83255	−0.219812
$305$	0	0
$306$	0.995305	0.0568978
$307$	2.41353	0.137748	0.0688739	−	0.997625i	$-0.478059\pi$
$307$	2.41353	0.137748	0.0688739	+	0.997625i	$0.478059\pi$
$308$	−39.2180	−2.23466
$309$	−10.1441	−0.577078
$310$	0	0
$311$	9.33959	0.529600	0.264800	−	0.964303i	$-0.414694\pi$
$311$	9.33959	0.529600	0.264800	+	0.964303i	$0.414694\pi$
$312$	2.66041	0.150616
$313$	7.74374	0.437702	0.218851	−	0.975758i	$-0.429769\pi$
$313$	7.74374	0.437702	0.218851	+	0.975758i	$0.429769\pi$
$314$	2.62767	0.148288
$315$	0	0
$316$	−26.4602	−1.48850
$317$	1.11216	0.0624650	0.0312325	−	0.999512i	$-0.490057\pi$
$317$	1.11216	0.0624650	0.0312325	+	0.999512i	$0.490057\pi$
$318$	0.390191	0.0218808
$319$	24.0226	1.34501
$320$	0	0
$321$	−6.80921	−0.380053
$322$	4.94862	0.275776
$323$	5.94392	0.330729
$324$	−1.97196	−0.109553
$325$	0	0
$326$	2.74765	0.152178
$327$	20.2227	1.11832
$328$	0.130020	0.00717914
$329$	48.0545	2.64933
$330$	0	0
$331$	1.46492	0.0805192	0.0402596	−	0.999189i	$-0.487182\pi$
$331$	1.46492	0.0805192	0.0402596	+	0.999189i	$0.487182\pi$
$332$	−5.53039	−0.303519
$333$	−3.66510	−0.200846
$334$	−3.03743	−0.166201
$335$	0	0
$336$	15.8645	0.865479
$337$	−0.278820	−0.0151883	−0.00759414	−	0.999971i	$-0.502417\pi$
$337$	−0.278820	−0.0151883	−0.00759414	+	0.999971i	$0.502417\pi$
$338$	−0.502348	−0.0273241
$339$	18.0226	0.978851
$340$	0	0
$341$	−42.3014	−2.29075
$342$	0.167449	0.00905462
$343$	−12.9759	−0.700631
$344$	−2.66041	−0.143440
$345$	0	0
$346$	−1.37611	−0.0739799
$347$	−13.6651	−0.733581	−0.366791	−	0.930304i	$-0.619543\pi$
$347$	−13.6651	−0.733581	−0.366791	+	0.930304i	$0.619543\pi$
$348$	−9.85980	−0.528541
$349$	−5.94862	−0.318422	−0.159211	−	0.987245i	$-0.550895\pi$
$349$	−5.94862	−0.318422	−0.159211	+	0.987245i	$0.550895\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	9.47431	0.504982
$353$	0.112157	0.00596951	0.00298476	−	0.999996i	$-0.499050\pi$
$353$	0.112157	0.00596951	0.00298476	+	0.999996i	$0.499050\pi$
$354$	1.25078	0.0664782
$355$	0	0
$356$	−14.3535	−0.760736
$357$	−24.6043	−1.30220
$358$	−0.200970	−0.0106216
$359$	−29.2741	−1.54503	−0.772515	−	0.634997i	$-0.781001\pi$
$359$	−29.2741	−1.54503	−0.772515	+	0.634997i	$0.781001\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.44548	0.233649
$363$	−12.0833	−0.634210
$364$	−32.6510	−1.71138
$365$	0	0
$366$	−1.10668	−0.0578469
$367$	16.9392	0.884220	0.442110	−	0.896961i	$-0.354230\pi$
$367$	16.9392	0.884220	0.442110	+	0.896961i	$0.354230\pi$
$368$	27.3622	1.42635
$369$	0.195488	0.0101767
$370$	0	0
$371$	−9.64567	−0.500778
$372$	17.3622	0.900186
$373$	17.5529	0.908857	0.454429	−	0.890783i	$-0.349844\pi$
$373$	17.5529	0.908857	0.454429	+	0.890783i	$0.349844\pi$
$374$	−4.78195	−0.247269
$375$	0	0
$376$	−7.72118	−0.398189
$377$	20.0000	1.03005
$378$	−0.693141	−0.0356513
$379$	−0.604328	−0.0310422	−0.0155211	−	0.999880i	$-0.504941\pi$
$379$	−0.604328	−0.0310422	−0.0155211	+	0.999880i	$0.504941\pi$
$380$	0	0
$381$	14.0786	0.721270
$382$	3.84650	0.196804
$383$	19.4790	0.995331	0.497665	−	0.867369i	$-0.334191\pi$
$383$	19.4790	0.995331	0.497665	+	0.867369i	$0.334191\pi$
$384$	−5.17214	−0.263940
$385$	0	0
$386$	0.791344	0.0402783
$387$	−4.00000	−0.203331
$388$	−22.2321	−1.12867
$389$	24.9392	1.26447	0.632234	−	0.774777i	$-0.282138\pi$
$389$	24.9392	1.26447	0.632234	+	0.774777i	$0.282138\pi$
$390$	0	0
$391$	−42.4361	−2.14609
$392$	6.74062	0.340452
$393$	3.47431	0.175256
$394$	−0.615291	−0.0309979
$395$	0	0
$396$	9.47431	0.476102
$397$	12.4135	0.623017	0.311509	−	0.950243i	$-0.399166\pi$
$397$	12.4135	0.623017	0.311509	+	0.950243i	$0.399166\pi$
$398$	−0.869194	−0.0435688
$399$	−4.13941	−0.207230
$400$	0	0
$401$	3.74374	0.186953	0.0934767	−	0.995621i	$-0.470202\pi$
$401$	3.74374	0.186953	0.0934767	+	0.995621i	$0.470202\pi$
$402$	−0.200184	−0.00998425
$403$	−35.2180	−1.75434
$404$	−19.6090	−0.975585
$405$	0	0
$406$	−3.46570	−0.172000
$407$	17.6090	0.872847
$408$	3.95331	0.195718
$409$	13.9533	0.689947	0.344973	−	0.938612i	$-0.387888\pi$
$409$	13.9533	0.689947	0.344973	+	0.938612i	$0.387888\pi$
$410$	0	0
$411$	−18.6137	−0.918147
$412$	−20.0038	−0.985515
$413$	−30.9198	−1.52146
$414$	−1.19549	−0.0587551
$415$	0	0
$416$	7.88784	0.386733
$417$	6.13941	0.300648
$418$	−0.804512	−0.0393499
$419$	6.26943	0.306282	0.153141	−	0.988204i	$-0.451061\pi$
$419$	6.26943	0.306282	0.153141	+	0.988204i	$0.451061\pi$
$420$	0	0
$421$	−32.4361	−1.58084	−0.790419	−	0.612566i	$-0.790137\pi$
$421$	−32.4361	−1.58084	−0.790419	+	0.612566i	$0.790137\pi$
$422$	−2.58177	−0.125679
$423$	−11.6090	−0.564450
$424$	1.54982	0.0752660
$425$	0	0
$426$	1.58568	0.0768264
$427$	27.3575	1.32392
$428$	−13.4275	−0.649042
$429$	−19.2180	−0.927856
$430$	0	0
$431$	21.7484	1.04759	0.523793	−	0.851846i	$-0.324517\pi$
$431$	21.7484	1.04759	0.523793	+	0.851846i	$0.324517\pi$
$432$	−3.83255	−0.184394
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	6.10277	0.292942
$435$	0	0
$436$	39.8785	1.90983
$437$	−7.13941	−0.341524
$438$	−0.0552923	−0.00264197
$439$	−9.68766	−0.462367	−0.231183	−	0.972910i	$-0.574260\pi$
$439$	−9.68766	−0.462367	−0.231183	+	0.972910i	$0.574260\pi$
$440$	0	0
$441$	10.1347	0.482605
$442$	−3.98122	−0.189367
$443$	32.6830	1.55281	0.776407	−	0.630232i	$-0.217040\pi$
$443$	32.6830	1.55281	0.776407	+	0.630232i	$0.217040\pi$
$444$	−7.22744	−0.342999
$445$	0	0
$446$	0.860590	0.0407501
$447$	0	0
$448$	30.3622	1.43448
$449$	8.60902	0.406285	0.203142	−	0.979149i	$-0.434885\pi$
$449$	8.60902	0.406285	0.203142	+	0.979149i	$0.434885\pi$
$450$	0	0
$451$	−0.939226	−0.0442264
$452$	35.5398	1.67165
$453$	−2.72588	−0.128073
$454$	−4.24609	−0.199279
$455$	0	0
$456$	0.665102	0.0311462
$457$	−1.25626	−0.0587655	−0.0293827	−	0.999568i	$-0.509354\pi$
$457$	−1.25626	−0.0587655	−0.0293827	+	0.999568i	$0.509354\pi$
$458$	4.62689	0.216200
$459$	5.94392	0.277438
$460$	0	0
$461$	18.6137	0.866927	0.433464	−	0.901171i	$-0.357291\pi$
$461$	18.6137	0.866927	0.433464	+	0.901171i	$0.357291\pi$
$462$	3.33020	0.154935
$463$	24.6698	1.14650	0.573251	−	0.819380i	$-0.305682\pi$
$463$	24.6698	1.14650	0.573251	+	0.819380i	$0.305682\pi$
$464$	−19.1628	−0.889609
$465$	0	0
$466$	2.72588	0.126274
$467$	−5.02725	−0.232634	−0.116317	−	0.993212i	$-0.537109\pi$
$467$	−5.02725	−0.232634	−0.116317	+	0.993212i	$0.537109\pi$
$468$	7.88784	0.364616
$469$	4.94862	0.228506
$470$	0	0
$471$	15.6924	0.723066
$472$	4.96805	0.228673
$473$	19.2180	0.883647
$474$	2.24687	0.103202
$475$	0	0
$476$	−48.5188	−2.22385
$477$	2.33020	0.106693
$478$	4.21335	0.192714
$479$	41.9665	1.91750	0.958749	−	0.284255i	$-0.0917463\pi$
$479$	41.9665	1.91750	0.958749	+	0.284255i	$0.0917463\pi$
$480$	0	0
$481$	14.6604	0.668457
$482$	2.28821	0.104225
$483$	29.5529	1.34471
$484$	−23.8279	−1.08308
$485$	0	0
$486$	0.167449	0.00759565
$487$	−28.4455	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$487$	−28.4455	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$488$	−4.39567	−0.198983
$489$	16.4088	0.742033
$490$	0	0
$491$	18.6137	0.840025	0.420013	−	0.907518i	$-0.362026\pi$
$491$	18.6137	0.840025	0.420013	+	0.907518i	$0.362026\pi$
$492$	0.385496	0.0173795
$493$	29.7196	1.33850
$494$	−0.669797	−0.0301356
$495$	0	0
$496$	33.7437	1.51514
$497$	−39.1986	−1.75830
$498$	0.469613	0.0210439
$499$	−18.2967	−0.819072	−0.409536	−	0.912294i	$-0.634309\pi$
$499$	−18.2967	−0.819072	−0.409536	+	0.912294i	$0.634309\pi$
$500$	0	0
$501$	−18.1394	−0.810409
$502$	−2.35368	−0.105050
$503$	−28.5576	−1.27332	−0.636661	−	0.771144i	$-0.719685\pi$
$503$	−28.5576	−1.27332	−0.636661	+	0.771144i	$0.719685\pi$
$504$	−2.75313	−0.122634
$505$	0	0
$506$	5.74374	0.255340
$507$	−3.00000	−0.133235
$508$	27.7625	1.23176
$509$	16.3302	0.723824	0.361912	−	0.932212i	$-0.382124\pi$
$509$	16.3302	0.723824	0.361912	+	0.932212i	$0.382124\pi$
$510$	0	0
$511$	1.36685	0.0604657
$512$	−12.6557	−0.559309
$513$	1.00000	0.0441511
$514$	−2.76239	−0.121844
$515$	0	0
$516$	−7.88784	−0.347243
$517$	55.7757	2.45301
$518$	−2.54043	−0.111620
$519$	−8.21805	−0.360732
$520$	0	0
$521$	−22.6184	−0.990931	−0.495465	−	0.868628i	$-0.665002\pi$
$521$	−22.6184	−0.990931	−0.495465	+	0.868628i	$0.665002\pi$
$522$	0.837246	0.0366453
$523$	−8.21335	−0.359145	−0.179572	−	0.983745i	$-0.557471\pi$
$523$	−8.21335	−0.359145	−0.179572	+	0.983745i	$0.557471\pi$
$524$	6.85120	0.299296
$525$	0	0
$526$	−1.19549	−0.0521258
$527$	−52.3333	−2.27968
$528$	18.4135	0.801346
$529$	27.9712	1.21614
$530$	0	0
$531$	7.46961	0.324154
$532$	−8.16275	−0.353900
$533$	−0.781954	−0.0338702
$534$	1.21883	0.0527440
$535$	0	0
$536$	−0.795121	−0.0343440
$537$	−1.20018	−0.0517917
$538$	0.985915	0.0425058
$539$	−48.6924	−2.09733
$540$	0	0
$541$	6.40414	0.275336	0.137668	−	0.990478i	$-0.456039\pi$
$541$	6.40414	0.275336	0.137668	+	0.990478i	$0.456039\pi$
$542$	−1.58568	−0.0681107
$543$	26.5482	1.13929
$544$	11.7212	0.502541
$545$	0	0
$546$	2.77256	0.118655
$547$	−17.8092	−0.761467	−0.380733	−	0.924685i	$-0.624328\pi$
$547$	−17.8092	−0.761467	−0.380733	+	0.924685i	$0.624328\pi$
$548$	−36.7055	−1.56798
$549$	−6.60902	−0.282066
$550$	0	0
$551$	5.00000	0.213007
$552$	−4.74843	−0.202107
$553$	−55.5436	−2.36195
$554$	−2.76239	−0.117363
$555$	0	0
$556$	12.1067	0.513437
$557$	39.2180	1.66172	0.830861	−	0.556480i	$-0.187848\pi$
$557$	39.2180	1.66172	0.830861	+	0.556480i	$0.187848\pi$
$558$	−1.47431	−0.0624124
$559$	16.0000	0.676728
$560$	0	0
$561$	−28.5576	−1.20570
$562$	1.65193	0.0696826
$563$	−21.9151	−0.923611	−0.461806	−	0.886981i	$-0.652798\pi$
$563$	−21.9151	−0.923611	−0.461806	+	0.886981i	$0.652798\pi$
$564$	−22.8925	−0.963950
$565$	0	0
$566$	0.224314	0.00942861
$567$	−4.13941	−0.173839
$568$	6.29826	0.264269
$569$	−35.4135	−1.48461	−0.742306	−	0.670061i	$-0.766268\pi$
$569$	−35.4135	−1.48461	−0.742306	+	0.670061i	$0.766268\pi$
$570$	0	0
$571$	6.80921	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$571$	6.80921	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$572$	−37.8972	−1.58456
$573$	22.9712	0.959635
$574$	0.135501	0.00565570
$575$	0	0
$576$	−7.33490	−0.305621
$577$	−12.1441	−0.505566	−0.252783	−	0.967523i	$-0.581346\pi$
$577$	−12.1441	−0.505566	−0.252783	+	0.967523i	$0.581346\pi$
$578$	−3.06938	−0.127669
$579$	4.72588	0.196401
$580$	0	0
$581$	−11.6090	−0.481623
$582$	1.88784	0.0782536
$583$	−11.1955	−0.463670
$584$	−0.219619	−0.00908789
$585$	0	0
$586$	1.42762	0.0589744
$587$	7.13002	0.294287	0.147144	−	0.989115i	$-0.452992\pi$
$587$	7.13002	0.294287	0.147144	+	0.989115i	$0.452992\pi$
$588$	19.9853	0.824178
$589$	−8.80451	−0.362784
$590$	0	0
$591$	−3.67449	−0.151148
$592$	−14.0467	−0.577315
$593$	2.00939	0.0825158	0.0412579	−	0.999149i	$-0.486863\pi$
$593$	2.00939	0.0825158	0.0412579	+	0.999149i	$0.486863\pi$
$594$	−0.804512	−0.0330095
$595$	0	0
$596$	0	0
$597$	−5.19079	−0.212445
$598$	4.78195	0.195549
$599$	29.1153	1.18962	0.594809	−	0.803867i	$-0.297228\pi$
$599$	29.1153	1.18962	0.594809	+	0.803867i	$0.297228\pi$
$600$	0	0
$601$	−10.2788	−0.419282	−0.209641	−	0.977778i	$-0.567229\pi$
$601$	−10.2788	−0.419282	−0.209641	+	0.977778i	$0.567229\pi$
$602$	−2.77256	−0.113001
$603$	−1.19549	−0.0486841
$604$	−5.37532	−0.218719
$605$	0	0
$606$	1.66510	0.0676401
$607$	−15.5304	−0.630359	−0.315179	−	0.949032i	$-0.602065\pi$
$607$	−15.5304	−0.630359	−0.315179	+	0.949032i	$0.602065\pi$
$608$	1.97196	0.0799736
$609$	−20.6970	−0.838687
$610$	0	0
$611$	46.4361	1.87860
$612$	11.7212	0.473801
$613$	−4.47431	−0.180716	−0.0903578	−	0.995909i	$-0.528801\pi$
$613$	−4.47431	−0.180716	−0.0903578	+	0.995909i	$0.528801\pi$
$614$	−0.404144	−0.0163099
$615$	0	0
$616$	13.2274	0.532949
$617$	10.6698	0.429550	0.214775	−	0.976664i	$-0.431098\pi$
$617$	10.6698	0.429550	0.214775	+	0.976664i	$0.431098\pi$
$618$	1.69862	0.0683286
$619$	−11.0786	−0.445288	−0.222644	−	0.974900i	$-0.571469\pi$
$619$	−11.0786	−0.445288	−0.222644	+	0.974900i	$0.571469\pi$
$620$	0	0
$621$	−7.13941	−0.286495
$622$	−1.56391	−0.0627070
$623$	−30.1300	−1.20713
$624$	15.3302	0.613699
$625$	0	0
$626$	−1.29668	−0.0518259
$627$	−4.80451	−0.191874
$628$	30.9447	1.23483
$629$	21.7851	0.868628
$630$	0	0
$631$	−1.22744	−0.0488635	−0.0244317	−	0.999702i	$-0.507778\pi$
$631$	−1.22744	−0.0488635	−0.0244317	+	0.999702i	$0.507778\pi$
$632$	8.92449	0.354997
$633$	−15.4182	−0.612820
$634$	−0.186230	−0.00739613
$635$	0	0
$636$	4.59507	0.182206
$637$	−40.5389	−1.60621
$638$	−4.02256	−0.159255
$639$	9.46961	0.374612
$640$	0	0
$641$	13.7757	0.544107	0.272053	−	0.962282i	$-0.412297\pi$
$641$	13.7757	0.544107	0.272053	+	0.962282i	$0.412297\pi$
$642$	1.14020	0.0450000
$643$	17.0786	0.673516	0.336758	−	0.941591i	$-0.390670\pi$
$643$	17.0786	0.673516	0.336758	+	0.941591i	$0.390670\pi$
$644$	58.2772	2.29645
$645$	0	0
$646$	−0.995305	−0.0391597
$647$	−26.7017	−1.04975	−0.524877	−	0.851178i	$-0.675889\pi$
$647$	−26.7017	−1.04975	−0.524877	+	0.851178i	$0.675889\pi$
$648$	0.665102	0.0261277
$649$	−35.8878	−1.40872
$650$	0	0
$651$	36.4455	1.42841
$652$	32.3576	1.26722
$653$	36.4455	1.42622	0.713111	−	0.701051i	$-0.247286\pi$
$653$	36.4455	1.42622	0.713111	+	0.701051i	$0.247286\pi$
$654$	−3.38628	−0.132414
$655$	0	0
$656$	0.749219	0.0292521
$657$	−0.330203	−0.0128825
$658$	−8.04669	−0.313693
$659$	−46.4549	−1.80962	−0.904812	−	0.425810i	$-0.859989\pi$
$659$	−46.4549	−1.80962	−0.904812	+	0.425810i	$0.859989\pi$
$660$	0	0
$661$	34.5016	1.34196	0.670978	−	0.741478i	$-0.265875\pi$
$661$	34.5016	1.34196	0.670978	+	0.741478i	$0.265875\pi$
$662$	−0.245299	−0.00953383
$663$	−23.7757	−0.923371
$664$	1.86529	0.0723871
$665$	0	0
$666$	0.613718	0.0237811
$667$	−35.6970	−1.38220
$668$	−35.7702	−1.38399
$669$	5.13941	0.198701
$670$	0	0
$671$	31.7531	1.22582
$672$	−8.16275	−0.314885
$673$	20.5576	0.792439	0.396219	−	0.918156i	$-0.370322\pi$
$673$	20.5576	0.792439	0.396219	+	0.918156i	$0.370322\pi$
$674$	0.0466882	0.00179836
$675$	0	0
$676$	−5.91588	−0.227534
$677$	20.2274	0.777404	0.388702	−	0.921364i	$-0.372924\pi$
$677$	20.2274	0.777404	0.388702	+	0.921364i	$0.372924\pi$
$678$	−3.01786	−0.115900
$679$	−46.6682	−1.79096
$680$	0	0
$681$	−25.3575	−0.971700
$682$	7.08333	0.271235
$683$	−26.9759	−1.03220	−0.516101	−	0.856527i	$-0.672617\pi$
$683$	−26.9759	−1.03220	−0.516101	+	0.856527i	$0.672617\pi$
$684$	1.97196	0.0753998
$685$	0	0
$686$	2.17280	0.0829578
$687$	27.6316	1.05421
$688$	−15.3302	−0.584459
$689$	−9.32081	−0.355095
$690$	0	0
$691$	−8.82707	−0.335798	−0.167899	−	0.985804i	$-0.553698\pi$
$691$	−8.82707	−0.335798	−0.167899	+	0.985804i	$0.553698\pi$
$692$	−16.2057	−0.616047
$693$	19.8878	0.755477
$694$	2.28821	0.0868593
$695$	0	0
$696$	3.32551	0.126053
$697$	−1.16197	−0.0440127
$698$	0.996091	0.0377026
$699$	16.2788	0.615722
$700$	0	0
$701$	36.2134	1.36776	0.683880	−	0.729595i	$-0.260291\pi$
$701$	36.2134	1.36776	0.683880	+	0.729595i	$0.260291\pi$
$702$	−0.669797	−0.0252798
$703$	3.66510	0.138232
$704$	35.2406	1.32818
$705$	0	0
$706$	−0.0187806	−0.000706817 0
$707$	−41.1620	−1.54806
$708$	14.7298	0.553579
$709$	27.1573	1.01991	0.509956	−	0.860200i	$-0.329662\pi$
$709$	27.1573	1.01991	0.509956	+	0.860200i	$0.329662\pi$
$710$	0	0
$711$	13.4182	0.503223
$712$	4.84115	0.181430
$713$	62.8590	2.35409
$714$	4.11997	0.154186
$715$	0	0
$716$	−2.36671	−0.0884483
$717$	25.1620	0.939691
$718$	4.90193	0.182938
$719$	41.3622	1.54255	0.771274	−	0.636503i	$-0.219620\pi$
$719$	41.3622	1.54255	0.771274	+	0.636503i	$0.219620\pi$
$720$	0	0
$721$	−41.9906	−1.56381
$722$	−0.167449	−0.00623181
$723$	13.6651	0.508211
$724$	52.3521	1.94565
$725$	0	0
$726$	2.02334	0.0750933
$727$	47.2547	1.75258	0.876290	−	0.481785i	$-0.160011\pi$
$727$	47.2547	1.75258	0.876290	+	0.481785i	$0.160011\pi$
$728$	11.0125	0.408151
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.7757	0.879376
$732$	−13.0327	−0.481704
$733$	17.9580	0.663294	0.331647	−	0.943404i	$-0.392396\pi$
$733$	17.9580	0.663294	0.331647	+	0.943404i	$0.392396\pi$
$734$	−2.83646	−0.104696
$735$	0	0
$736$	−14.0786	−0.518945
$737$	5.74374	0.211573
$738$	−0.0327344	−0.00120497
$739$	31.0786	1.14325	0.571623	−	0.820516i	$-0.306314\pi$
$739$	31.0786	1.14325	0.571623	+	0.820516i	$0.306314\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	1.61516	0.0592944
$743$	−19.6363	−0.720385	−0.360193	−	0.932878i	$-0.617289\pi$
$743$	−19.6363	−0.720385	−0.360193	+	0.932878i	$0.617289\pi$
$744$	−5.85589	−0.214688
$745$	0	0
$746$	−2.93923	−0.107613
$747$	2.80451	0.102612
$748$	−56.3145	−2.05906
$749$	−28.1861	−1.02990
$750$	0	0
$751$	25.4875	0.930051	0.465026	−	0.885297i	$-0.346045\pi$
$751$	25.4875	0.930051	0.465026	+	0.885297i	$0.346045\pi$
$752$	−44.4922	−1.62246
$753$	−14.0561	−0.512232
$754$	−3.34898	−0.121963
$755$	0	0
$756$	−8.16275	−0.296876
$757$	−29.3910	−1.06823	−0.534117	−	0.845411i	$-0.679356\pi$
$757$	−29.3910	−1.06823	−0.534117	+	0.845411i	$0.679356\pi$
$758$	0.101194	0.00367554
$759$	34.3014	1.24506
$760$	0	0
$761$	−17.1526	−0.621780	−0.310890	−	0.950446i	$-0.600627\pi$
$761$	−17.1526	−0.621780	−0.310890	+	0.950446i	$0.600627\pi$
$762$	−2.35746	−0.0854016
$763$	83.7102	3.03051
$764$	45.2983	1.63883
$765$	0	0
$766$	−3.26174	−0.117852
$767$	−29.8785	−1.07885
$768$	−13.8037	−0.498099
$769$	−9.11216	−0.328593	−0.164296	−	0.986411i	$-0.552535\pi$
$769$	−9.11216	−0.328593	−0.164296	+	0.986411i	$0.552535\pi$
$770$	0	0
$771$	−16.4969	−0.594120
$772$	9.31924	0.335407
$773$	26.0927	0.938490	0.469245	−	0.883068i	$-0.344526\pi$
$773$	26.0927	0.938490	0.469245	+	0.883068i	$0.344526\pi$
$774$	0.669797	0.0240753
$775$	0	0
$776$	7.49844	0.269178
$777$	−15.1714	−0.544269
$778$	−4.17605	−0.149719
$779$	−0.195488	−0.00700410
$780$	0	0
$781$	−45.4969	−1.62801
$782$	7.10589	0.254106
$783$	5.00000	0.178685
$784$	38.8418	1.38721
$785$	0	0
$786$	−0.581770	−0.0207511
$787$	−1.92136	−0.0684892	−0.0342446	−	0.999413i	$-0.510903\pi$
$787$	−1.92136	−0.0684892	−0.0342446	+	0.999413i	$0.510903\pi$
$788$	−7.24595	−0.258126
$789$	−7.13941	−0.254170
$790$	0	0
$791$	74.6028	2.65257
$792$	−3.19549	−0.113547
$793$	26.4361	0.938773
$794$	−2.07864	−0.0737680
$795$	0	0
$796$	−10.2360	−0.362807
$797$	40.6410	1.43958	0.719789	−	0.694193i	$-0.244239\pi$
$797$	40.6410	1.43958	0.719789	+	0.694193i	$0.244239\pi$
$798$	0.693141	0.0245369
$799$	69.0031	2.44115
$800$	0	0
$801$	7.27882	0.257184
$802$	−0.626886	−0.0221361
$803$	1.58647	0.0559851
$804$	−2.35746	−0.0831411
$805$	0	0
$806$	5.89723	0.207721
$807$	5.88784	0.207262
$808$	6.61372	0.232670
$809$	12.2788	0.431700	0.215850	−	0.976426i	$-0.430748\pi$
$809$	12.2788	0.431700	0.215850	+	0.976426i	$0.430748\pi$
$810$	0	0
$811$	50.5802	1.77611	0.888055	−	0.459736i	$-0.152056\pi$
$811$	50.5802	1.77611	0.888055	+	0.459736i	$0.152056\pi$
$812$	−40.8138	−1.43228
$813$	−9.46961	−0.332114
$814$	−2.94862	−0.103349
$815$	0	0
$816$	22.7804	0.797473
$817$	4.00000	0.139942
$818$	−2.33647	−0.0816928
$819$	16.5576	0.578571
$820$	0	0
$821$	9.82237	0.342803	0.171402	−	0.985201i	$-0.445170\pi$
$821$	9.82237	0.342803	0.171402	+	0.985201i	$0.445170\pi$
$822$	3.11685	0.108713
$823$	−10.1394	−0.353438	−0.176719	−	0.984261i	$-0.556548\pi$
$823$	−10.1394	−0.353438	−0.176719	+	0.984261i	$0.556548\pi$
$824$	6.74686	0.235038
$825$	0	0
$826$	5.17749	0.180148
$827$	9.33959	0.324769	0.162385	−	0.986728i	$-0.448081\pi$
$827$	9.33959	0.324769	0.162385	+	0.986728i	$0.448081\pi$
$828$	−14.0786	−0.489266
$829$	20.5670	0.714322	0.357161	−	0.934043i	$-0.383745\pi$
$829$	20.5670	0.714322	0.357161	+	0.934043i	$0.383745\pi$
$830$	0	0
$831$	−16.4969	−0.572270
$832$	29.3396	1.01717
$833$	−60.2399	−2.08719
$834$	−1.02804	−0.0355981
$835$	0	0
$836$	−9.47431	−0.327676
$837$	−8.80451	−0.304328
$838$	−1.04981	−0.0362651
$839$	−43.8029	−1.51225	−0.756123	−	0.654430i	$-0.772909\pi$
$839$	−43.8029	−1.51225	−0.756123	+	0.654430i	$0.772909\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	5.43140	0.187178
$843$	9.86529	0.339778
$844$	−30.4041	−1.04655
$845$	0	0
$846$	1.94392	0.0668334
$847$	−50.0179	−1.71863
$848$	8.93062	0.306679
$849$	1.33959	0.0459747
$850$	0	0
$851$	−26.1667	−0.896982
$852$	18.6737	0.639751
$853$	−32.6410	−1.11761	−0.558803	−	0.829301i	$-0.688739\pi$
$853$	−32.6410	−1.11761	−0.558803	+	0.829301i	$0.688739\pi$
$854$	−4.58098	−0.156758
$855$	0	0
$856$	4.52881	0.154792
$857$	−17.1441	−0.585631	−0.292816	−	0.956169i	$-0.594592\pi$
$857$	−17.1441	−0.585631	−0.292816	+	0.956169i	$0.594592\pi$
$858$	3.21805	0.109862
$859$	21.5241	0.734393	0.367197	−	0.930143i	$-0.380318\pi$
$859$	21.5241	0.734393	0.367197	+	0.930143i	$0.380318\pi$
$860$	0	0
$861$	0.809207	0.0275777
$862$	−3.64176	−0.124039
$863$	38.8543	1.32262	0.661308	−	0.750114i	$-0.270002\pi$
$863$	38.8543	1.32262	0.661308	+	0.750114i	$0.270002\pi$
$864$	1.97196	0.0670875
$865$	0	0
$866$	−1.00470	−0.0341409
$867$	−18.3302	−0.622526
$868$	71.8691	2.43939
$869$	−64.4680	−2.18693
$870$	0	0
$871$	4.78195	0.162030
$872$	−13.4502	−0.455481
$873$	11.2741	0.381571
$874$	1.19549	0.0404380
$875$	0	0
$876$	−0.651148	−0.0220002
$877$	24.1573	0.815733	0.407867	−	0.913042i	$-0.366273\pi$
$877$	24.1573	0.815733	0.407867	+	0.913042i	$0.366273\pi$
$878$	1.62219	0.0547463
$879$	8.52569	0.287564
$880$	0	0
$881$	3.33020	0.112197	0.0560987	−	0.998425i	$-0.482134\pi$
$881$	3.33020	0.112197	0.0560987	+	0.998425i	$0.482134\pi$
$882$	−1.69705	−0.0571426
$883$	48.8543	1.64408	0.822039	−	0.569431i	$-0.192836\pi$
$883$	48.8543	1.64408	0.822039	+	0.569431i	$0.192836\pi$
$884$	−46.8847	−1.57690
$885$	0	0
$886$	−5.47274	−0.183860
$887$	30.9937	1.04067	0.520334	−	0.853963i	$-0.325808\pi$
$887$	30.9937	1.04067	0.520334	+	0.853963i	$0.325808\pi$
$888$	2.43767	0.0818027
$889$	58.2772	1.95456
$890$	0	0
$891$	−4.80451	−0.160957
$892$	10.1347	0.339335
$893$	11.6090	0.388481
$894$	0	0
$895$	0	0
$896$	−21.4096	−0.715245
$897$	28.5576	0.953512
$898$	−1.44157	−0.0481059
$899$	−44.0226	−1.46823
$900$	0	0
$901$	−13.8505	−0.461429
$902$	0.157273	0.00523661
$903$	−16.5576	−0.551004
$904$	−11.9868	−0.398676
$905$	0	0
$906$	0.456446	0.0151644
$907$	−46.3894	−1.54033	−0.770167	−	0.637842i	$-0.779827\pi$
$907$	−46.3894	−1.54033	−0.770167	+	0.637842i	$0.779827\pi$
$908$	−50.0039	−1.65944
$909$	9.94392	0.329819
$910$	0	0
$911$	46.3061	1.53419	0.767094	−	0.641534i	$-0.221702\pi$
$911$	46.3061	1.53419	0.767094	+	0.641534i	$0.221702\pi$
$912$	3.83255	0.126908
$913$	−13.4743	−0.445935
$914$	0.210360	0.00695809
$915$	0	0
$916$	54.4884	1.80035
$917$	14.3816	0.474922
$918$	−0.995305	−0.0328500
$919$	−27.0880	−0.893552	−0.446776	−	0.894646i	$-0.647428\pi$
$919$	−27.0880	−0.893552	−0.446776	+	0.894646i	$0.647428\pi$
$920$	0	0
$921$	−2.41353	−0.0795287
$922$	−3.11685	−0.102648
$923$	−37.8785	−1.24678
$924$	39.2180	1.29018
$925$	0	0
$926$	−4.13094	−0.135751
$927$	10.1441	0.333176
$928$	9.85980	0.323664
$929$	−18.8925	−0.619844	−0.309922	−	0.950762i	$-0.600303\pi$
$929$	−18.8925	−0.619844	−0.309922	+	0.950762i	$0.600303\pi$
$930$	0	0
$931$	−10.1347	−0.332152
$932$	32.1012	1.05151
$933$	−9.33959	−0.305765
$934$	0.841809	0.0275448
$935$	0	0
$936$	−2.66041	−0.0869581
$937$	31.5257	1.02990	0.514950	−	0.857220i	$-0.327811\pi$
$937$	31.5257	1.02990	0.514950	+	0.857220i	$0.327811\pi$
$938$	−0.828642	−0.0270561
$939$	−7.74374	−0.252707
$940$	0	0
$941$	23.3622	0.761584	0.380792	−	0.924661i	$-0.375651\pi$
$941$	23.3622	0.761584	0.380792	+	0.924661i	$0.375651\pi$
$942$	−2.62767	−0.0856142
$943$	1.39567	0.0454493
$944$	28.6277	0.931751
$945$	0	0
$946$	−3.21805	−0.104628
$947$	19.2180	0.624503	0.312251	−	0.950000i	$-0.398917\pi$
$947$	19.2180	0.624503	0.312251	+	0.950000i	$0.398917\pi$
$948$	26.4602	0.859388
$949$	1.32081	0.0428754
$950$	0	0
$951$	−1.11216	−0.0360642
$952$	16.3644	0.530373
$953$	−7.16666	−0.232151	−0.116075	−	0.993240i	$-0.537031\pi$
$953$	−7.16666	−0.232151	−0.116075	+	0.993240i	$0.537031\pi$
$954$	−0.390191	−0.0126329
$955$	0	0
$956$	49.6184	1.60477
$957$	−24.0226	−0.776539
$958$	−7.02725	−0.227040
$959$	−77.0498	−2.48807
$960$	0	0
$961$	46.5194	1.50063
$962$	−2.45487	−0.0791483
$963$	6.80921	0.219424
$964$	26.9470	0.867906
$965$	0	0
$966$	−4.94862	−0.159219
$967$	7.75782	0.249475	0.124737	−	0.992190i	$-0.460191\pi$
$967$	7.75782	0.249475	0.124737	+	0.992190i	$0.460191\pi$
$968$	8.03664	0.258307
$969$	−5.94392	−0.190946
$970$	0	0
$971$	−42.1845	−1.35377	−0.676883	−	0.736091i	$-0.736670\pi$
$971$	−42.1845	−1.35377	−0.676883	+	0.736091i	$0.736670\pi$
$972$	1.97196	0.0632507
$973$	25.4135	0.814721
$974$	4.76317	0.152622
$975$	0	0
$976$	−25.3294	−0.810775
$977$	−20.6604	−0.660985	−0.330492	−	0.943809i	$-0.607215\pi$
$977$	−20.6604	−0.660985	−0.330492	+	0.943809i	$0.607215\pi$
$978$	−2.74765	−0.0878601
$979$	−34.9712	−1.11768
$980$	0	0
$981$	−20.2227	−0.645662
$982$	−3.11685	−0.0994627
$983$	−3.61841	−0.115409	−0.0577047	−	0.998334i	$-0.518378\pi$
$983$	−3.61841	−0.115409	−0.0577047	+	0.998334i	$0.518378\pi$
$984$	−0.130020	−0.00414488
$985$	0	0
$986$	−4.97652	−0.158485
$987$	−48.0545	−1.52959
$988$	−7.88784	−0.250946
$989$	−28.5576	−0.908080
$990$	0	0
$991$	1.46492	0.0465347	0.0232673	−	0.999729i	$-0.492593\pi$
$991$	1.46492	0.0465347	0.0232673	+	0.999729i	$0.492593\pi$
$992$	−17.3622	−0.551249
$993$	−1.46492	−0.0464878
$994$	6.56378	0.208190
$995$	0	0
$996$	5.53039	0.175237
$997$	−19.3622	−0.613205	−0.306603	−	0.951838i	$-0.599192\pi$
$997$	−19.3622	−0.613205	−0.306603	+	0.951838i	$0.599192\pi$
$998$	3.06376	0.0969818
$999$	3.66510	0.115959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.a.t.1.2	✓	3
3.2	odd	2		4275.2.a.bf.1.2		3
5.2	odd	4		1425.2.c.o.799.3		6
5.3	odd	4		1425.2.c.o.799.4		6
5.4	even	2		1425.2.a.w.1.2	yes	3
15.14	odd	2		4275.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.t.1.2	✓	3	1.1	even	1	trivial
1425.2.a.w.1.2	yes	3	5.4	even	2
1425.2.c.o.799.3		6	5.2	odd	4
1425.2.c.o.799.4		6	5.3	odd	4
4275.2.a.bf.1.2		3	3.2	odd	2
4275.2.a.bg.1.2		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} -4.13941 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} -4.13941 q^{7} +0.665102 q^{8} +1.00000 q^{9} -4.80451 q^{11} +1.97196 q^{12} -4.00000 q^{13} +0.693141 q^{14} +3.83255 q^{16} -5.94392 q^{17} -0.167449 q^{18} -1.00000 q^{19} +4.13941 q^{21} +0.804512 q^{22} +7.13941 q^{23} -0.665102 q^{24} +0.669797 q^{26} -1.00000 q^{27} +8.16275 q^{28} -5.00000 q^{29} +8.80451 q^{31} -1.97196 q^{32} +4.80451 q^{33} +0.995305 q^{34} -1.97196 q^{36} -3.66510 q^{37} +0.167449 q^{38} +4.00000 q^{39} +0.195488 q^{41} -0.693141 q^{42} -4.00000 q^{43} +9.47431 q^{44} -1.19549 q^{46} -11.6090 q^{47} -3.83255 q^{48} +10.1347 q^{49} +5.94392 q^{51} +7.88784 q^{52} +2.33020 q^{53} +0.167449 q^{54} -2.75313 q^{56} +1.00000 q^{57} +0.837246 q^{58} +7.46961 q^{59} -6.60902 q^{61} -1.47431 q^{62} -4.13941 q^{63} -7.33490 q^{64} -0.804512 q^{66} -1.19549 q^{67} +11.7212 q^{68} -7.13941 q^{69} +9.46961 q^{71} +0.665102 q^{72} -0.330203 q^{73} +0.613718 q^{74} +1.97196 q^{76} +19.8878 q^{77} -0.669797 q^{78} +13.4182 q^{79} +1.00000 q^{81} -0.0327344 q^{82} +2.80451 q^{83} -8.16275 q^{84} +0.669797 q^{86} +5.00000 q^{87} -3.19549 q^{88} +7.27882 q^{89} +16.5576 q^{91} -14.0786 q^{92} -8.80451 q^{93} +1.94392 q^{94} +1.97196 q^{96} +11.2741 q^{97} -1.69705 q^{98} -4.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} - 12 q^{13} + 15 q^{14} + 12 q^{16} + 6 q^{17} - 3 q^{19} - 9 q^{22} + 9 q^{23} - 3 q^{24} - 3 q^{27} + 27 q^{28} - 15 q^{29} + 15 q^{31} + 6 q^{32} + 3 q^{33} + 6 q^{34} + 6 q^{36} - 12 q^{37} + 12 q^{39} + 12 q^{41} - 15 q^{42} - 12 q^{43} + 15 q^{44} - 15 q^{46} - 12 q^{47} - 12 q^{48} + 21 q^{49} - 6 q^{51} - 24 q^{52} + 9 q^{53} + 30 q^{56} + 3 q^{57} + 12 q^{59} + 3 q^{61} + 9 q^{62} - 21 q^{64} + 9 q^{66} - 15 q^{67} + 60 q^{68} - 9 q^{69} + 18 q^{71} + 3 q^{72} - 3 q^{73} - 24 q^{74} - 6 q^{76} + 12 q^{77} + 3 q^{79} + 3 q^{81} - 9 q^{82} - 3 q^{83} - 27 q^{84} + 15 q^{87} - 21 q^{88} - 3 q^{89} - 9 q^{92} - 15 q^{93} - 18 q^{94} - 6 q^{96} + 12 q^{97} + 57 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 6 * q^12 - 12 * q^13 + 15 * q^14 + 12 * q^16 + 6 * q^17 - 3 * q^19 - 9 * q^22 + 9 * q^23 - 3 * q^24 - 3 * q^27 + 27 * q^28 - 15 * q^29 + 15 * q^31 + 6 * q^32 + 3 * q^33 + 6 * q^34 + 6 * q^36 - 12 * q^37 + 12 * q^39 + 12 * q^41 - 15 * q^42 - 12 * q^43 + 15 * q^44 - 15 * q^46 - 12 * q^47 - 12 * q^48 + 21 * q^49 - 6 * q^51 - 24 * q^52 + 9 * q^53 + 30 * q^56 + 3 * q^57 + 12 * q^59 + 3 * q^61 + 9 * q^62 - 21 * q^64 + 9 * q^66 - 15 * q^67 + 60 * q^68 - 9 * q^69 + 18 * q^71 + 3 * q^72 - 3 * q^73 - 24 * q^74 - 6 * q^76 + 12 * q^77 + 3 * q^79 + 3 * q^81 - 9 * q^82 - 3 * q^83 - 27 * q^84 + 15 * q^87 - 21 * q^88 - 3 * q^89 - 9 * q^92 - 15 * q^93 - 18 * q^94 - 6 * q^96 + 12 * q^97 + 57 * q^98 - 3 * q^99

Introduction

L-functions

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