LMFDB - Embedded newform 1395.2.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1395,2,Mod(1,1395)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1395, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1395.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1395 = 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1395.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-2,0,8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.1391310820$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.704624$ of defining polynomial
Character	$\chi$	$=$	1395.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.79888 q^{2} +1.23597 q^{4} -1.00000 q^{5} +4.20813 q^{7} +1.37440 q^{8} +1.79888 q^{10} +3.00701 q^{11} +7.14544 q^{13} -7.56992 q^{14} -4.94432 q^{16} +5.17328 q^{17} +4.00000 q^{19} -1.23597 q^{20} -5.40925 q^{22} -8.29910 q^{23} +1.00000 q^{25} -12.8538 q^{26} +5.20112 q^{28} +3.97216 q^{29} -1.00000 q^{31} +6.14544 q^{32} -9.30611 q^{34} -4.20813 q^{35} -3.80589 q^{37} -7.19552 q^{38} -1.37440 q^{40} +2.18851 q^{41} -9.35357 q^{43} +3.71657 q^{44} +14.9291 q^{46} +1.95254 q^{47} +10.7083 q^{49} -1.79888 q^{50} +8.83155 q^{52} +6.77104 q^{53} -3.00701 q^{55} +5.78365 q^{56} -7.14544 q^{58} -3.90246 q^{59} -14.3606 q^{61} +1.79888 q^{62} -1.16627 q^{64} -7.14544 q^{65} -4.39664 q^{67} +6.39402 q^{68} +7.56992 q^{70} +4.50723 q^{71} -7.38963 q^{73} +6.84634 q^{74} +4.94388 q^{76} +12.6539 q^{77} +14.5573 q^{79} +4.94432 q^{80} -3.93687 q^{82} +5.24298 q^{83} -5.17328 q^{85} +16.8259 q^{86} +4.13283 q^{88} +4.49321 q^{89} +30.0689 q^{91} -10.2574 q^{92} -3.51239 q^{94} -4.00000 q^{95} +1.00701 q^{97} -19.2630 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 2 q^{10} - 6 q^{11} + 2 q^{13} - 4 q^{14} + 12 q^{16} + 10 q^{17} + 16 q^{19} - 8 q^{20} - 14 q^{22} - 6 q^{23} + 4 q^{25} + 10 q^{26} + 26 q^{28} - 4 q^{31}+ \cdots - 8 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 2 * q^10 - 6 * q^11 + 2 * q^13 - 4 * q^14 + 12 * q^16 + 10 * q^17 + 16 * q^19 - 8 * q^20 - 14 * q^22 - 6 * q^23 + 4 * q^25 + 10 * q^26 + 26 * q^28 - 4 * q^31 - 2 * q^32 + 8 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 + 6 * q^41 + 2 * q^43 + 6 * q^44 + 20 * q^46 + 2 * q^47 + 8 * q^49 - 2 * q^50 - 16 * q^52 + 6 * q^53 + 6 * q^55 + 10 * q^56 - 2 * q^58 - 4 * q^59 + 2 * q^62 - 12 * q^64 - 2 * q^65 - 2 * q^67 + 10 * q^68 + 4 * q^70 - 22 * q^71 - 32 * q^73 + 28 * q^74 + 32 * q^76 + 28 * q^77 + 24 * q^79 - 12 * q^80 - 46 * q^82 + 6 * q^83 - 10 * q^85 - 10 * q^86 - 18 * q^88 + 14 * q^89 + 28 * q^91 + 54 * q^92 - 44 * q^94 - 16 * q^95 - 14 * q^97 - 8 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.79888	−1.27200	−0.636000	−	0.771689i	$-0.719412\pi$
$2$	−1.79888	−1.27200	−0.636000	+	0.771689i	$0.719412\pi$
$3$	0	0
$3$	0	0
$4$	1.23597	0.617985
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.20813	1.59052	0.795262	−	0.606266i	$-0.207333\pi$
$7$	4.20813	1.59052	0.795262	+	0.606266i	$0.207333\pi$
$8$	1.37440	0.485923
$9$	0	0
$10$	1.79888	0.568856
$11$	3.00701	0.906647	0.453324	−	0.891346i	$-0.350238\pi$
$11$	3.00701	0.906647	0.453324	+	0.891346i	$0.350238\pi$
$12$	0	0
$13$	7.14544	1.98179	0.990894	−	0.134644i	$-0.0429892\pi$
$13$	7.14544	1.98179	0.990894	+	0.134644i	$0.0429892\pi$
$14$	−7.56992	−2.02315
$15$	0	0
$16$	−4.94432	−1.23608
$17$	5.17328	1.25470	0.627352	−	0.778736i	$-0.284139\pi$
$17$	5.17328	1.25470	0.627352	+	0.778736i	$0.284139\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.23597	−0.276371
$21$	0	0
$22$	−5.40925	−1.15326
$23$	−8.29910	−1.73048	−0.865241	−	0.501356i	$-0.832835\pi$
$23$	−8.29910	−1.73048	−0.865241	+	0.501356i	$0.832835\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−12.8538	−2.52083
$27$	0	0
$28$	5.20112	0.982919
$29$	3.97216	0.737611	0.368806	−	0.929507i	$-0.379767\pi$
$29$	3.97216	0.737611	0.368806	+	0.929507i	$0.379767\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	6.14544	1.08637
$33$	0	0
$34$	−9.30611	−1.59598
$35$	−4.20813	−0.711304
$36$	0	0
$37$	−3.80589	−0.625684	−0.312842	−	0.949805i	$-0.601281\pi$
$37$	−3.80589	−0.625684	−0.312842	+	0.949805i	$0.601281\pi$
$38$	−7.19552	−1.16727
$39$	0	0
$40$	−1.37440	−0.217312
$41$	2.18851	0.341788	0.170894	−	0.985289i	$-0.445334\pi$
$41$	2.18851	0.341788	0.170894	+	0.985289i	$0.445334\pi$
$42$	0	0
$43$	−9.35357	−1.42641	−0.713203	−	0.700958i	$-0.752756\pi$
$43$	−9.35357	−1.42641	−0.713203	+	0.700958i	$0.752756\pi$
$44$	3.71657	0.560294
$45$	0	0
$46$	14.9291	2.20117
$47$	1.95254	0.284807	0.142404	−	0.989809i	$-0.454517\pi$
$47$	1.95254	0.284807	0.142404	+	0.989809i	$0.454517\pi$
$48$	0	0
$49$	10.7083	1.52976
$50$	−1.79888	−0.254400
$51$	0	0
$52$	8.83155	1.22471
$53$	6.77104	0.930074	0.465037	−	0.885291i	$-0.346041\pi$
$53$	6.77104	0.930074	0.465037	+	0.885291i	$0.346041\pi$
$54$	0	0
$55$	−3.00701	−0.405465
$56$	5.78365	0.772872
$57$	0	0
$58$	−7.14544	−0.938242
$59$	−3.90246	−0.508057	−0.254028	−	0.967197i	$-0.581756\pi$
$59$	−3.90246	−0.508057	−0.254028	+	0.967197i	$0.581756\pi$
$60$	0	0
$61$	−14.3606	−1.83868	−0.919342	−	0.393460i	$-0.871278\pi$
$61$	−14.3606	−1.83868	−0.919342	+	0.393460i	$0.871278\pi$
$62$	1.79888	0.228458
$63$	0	0
$64$	−1.16627	−0.145784
$65$	−7.14544	−0.886283
$66$	0	0
$67$	−4.39664	−0.537135	−0.268568	−	0.963261i	$-0.586550\pi$
$67$	−4.39664	−0.537135	−0.268568	+	0.963261i	$0.586550\pi$
$68$	6.39402	0.775388
$69$	0	0
$70$	7.56992	0.904778
$71$	4.50723	0.534910	0.267455	−	0.963570i	$-0.413817\pi$
$71$	4.50723	0.534910	0.267455	+	0.963570i	$0.413817\pi$
$72$	0	0
$73$	−7.38963	−0.864891	−0.432445	−	0.901660i	$-0.642349\pi$
$73$	−7.38963	−0.864891	−0.432445	+	0.901660i	$0.642349\pi$
$74$	6.84634	0.795871
$75$	0	0
$76$	4.94388	0.567102
$77$	12.6539	1.44204
$78$	0	0
$79$	14.5573	1.63783	0.818913	−	0.573918i	$-0.194577\pi$
$79$	14.5573	1.63783	0.818913	+	0.573918i	$0.194577\pi$
$80$	4.94432	0.552792
$81$	0	0
$82$	−3.93687	−0.434755
$83$	5.24298	0.575492	0.287746	−	0.957707i	$-0.407094\pi$
$83$	5.24298	0.575492	0.287746	+	0.957707i	$0.407094\pi$
$84$	0	0
$85$	−5.17328	−0.561121
$86$	16.8259	1.81439
$87$	0	0
$88$	4.13283	0.440561
$89$	4.49321	0.476279	0.238140	−	0.971231i	$-0.423462\pi$
$89$	4.49321	0.476279	0.238140	+	0.971231i	$0.423462\pi$
$90$	0	0
$91$	30.0689	3.15208
$92$	−10.2574	−1.06941
$93$	0	0
$94$	−3.51239	−0.362275
$95$	−4.00000	−0.410391
$96$	0	0
$97$	1.00701	0.102246	0.0511231	−	0.998692i	$-0.483720\pi$
$97$	1.00701	0.102246	0.0511231	+	0.998692i	$0.483720\pi$
$98$	−19.2630	−1.94586
$99$	0	0
$100$	1.23597	0.123597
$101$	−0.535070	−0.0532414	−0.0266207	−	0.999646i	$-0.508475\pi$
$101$	−0.535070	−0.0532414	−0.0266207	+	0.999646i	$0.508475\pi$
$102$	0	0
$103$	−14.2156	−1.40070	−0.700351	−	0.713798i	$-0.746973\pi$
$103$	−14.2156	−1.40070	−0.700351	+	0.713798i	$0.746973\pi$
$104$	9.82068	0.962997
$105$	0	0
$106$	−12.1803	−1.18305
$107$	−13.9918	−1.35264	−0.676318	−	0.736610i	$-0.736425\pi$
$107$	−13.9918	−1.35264	−0.676318	+	0.736610i	$0.736425\pi$
$108$	0	0
$109$	6.95955	0.666604	0.333302	−	0.942820i	$-0.391837\pi$
$109$	6.95955	0.666604	0.333302	+	0.942820i	$0.391837\pi$
$110$	5.40925	0.515752
$111$	0	0
$112$	−20.8063	−1.96601
$113$	−7.26522	−0.683454	−0.341727	−	0.939799i	$-0.611012\pi$
$113$	−7.26522	−0.683454	−0.341727	+	0.939799i	$0.611012\pi$
$114$	0	0
$115$	8.29910	0.773895
$116$	4.90947	0.455833
$117$	0	0
$118$	7.02006	0.646249
$119$	21.7698	1.99564
$120$	0	0
$121$	−1.95790	−0.177991
$122$	25.8330	2.33881
$123$	0	0
$124$	−1.23597	−0.110993
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.48640	−0.486839	−0.243419	−	0.969921i	$-0.578269\pi$
$127$	−5.48640	−0.486839	−0.243419	+	0.969921i	$0.578269\pi$
$128$	−10.1929	−0.900933
$129$	0	0
$130$	12.8538	1.12735
$131$	−11.5699	−1.01087	−0.505434	−	0.862865i	$-0.668668\pi$
$131$	−11.5699	−1.01087	−0.505434	+	0.862865i	$0.668668\pi$
$132$	0	0
$133$	16.8325	1.45956
$134$	7.90903	0.683236
$135$	0	0
$136$	7.11015	0.609690
$137$	−1.17328	−0.100240	−0.0501200	−	0.998743i	$-0.515960\pi$
$137$	−1.17328	−0.100240	−0.0501200	+	0.998743i	$0.515960\pi$
$138$	0	0
$139$	13.3601	1.13319	0.566596	−	0.823996i	$-0.308260\pi$
$139$	13.3601	1.13319	0.566596	+	0.823996i	$0.308260\pi$
$140$	−5.20112	−0.439575
$141$	0	0
$142$	−8.10796	−0.680405
$143$	21.4864	1.79678
$144$	0	0
$145$	−3.97216	−0.329870
$146$	13.2931	1.10014
$147$	0	0
$148$	−4.70396	−0.386663
$149$	−0.867610	−0.0710773	−0.0355387	−	0.999368i	$-0.511315\pi$
$149$	−0.867610	−0.0710773	−0.0355387	+	0.999368i	$0.511315\pi$
$150$	0	0
$151$	16.9596	1.38015	0.690074	−	0.723738i	$-0.257578\pi$
$151$	16.9596	1.38015	0.690074	+	0.723738i	$0.257578\pi$
$152$	5.49760	0.445914
$153$	0	0
$154$	−22.7628	−1.83428
$155$	1.00000	0.0803219
$156$	0	0
$157$	2.89807	0.231292	0.115646	−	0.993291i	$-0.463106\pi$
$157$	2.89807	0.231292	0.115646	+	0.993291i	$0.463106\pi$
$158$	−26.1869	−2.08331
$159$	0	0
$160$	−6.14544	−0.485840
$161$	−34.9237	−2.75237
$162$	0	0
$163$	14.3550	1.12437	0.562184	−	0.827012i	$-0.309961\pi$
$163$	14.3550	1.12437	0.562184	+	0.827012i	$0.309961\pi$
$164$	2.70493	0.211220
$165$	0	0
$166$	−9.43149	−0.732026
$167$	−5.93030	−0.458900	−0.229450	−	0.973320i	$-0.573693\pi$
$167$	−5.93030	−0.458900	−0.229450	+	0.973320i	$0.573693\pi$
$168$	0	0
$169$	38.0573	2.92748
$170$	9.30611	0.713746
$171$	0	0
$172$	−11.5607	−0.881497
$173$	−9.00044	−0.684291	−0.342145	−	0.939647i	$-0.611154\pi$
$173$	−9.00044	−0.684291	−0.342145	+	0.939647i	$0.611154\pi$
$174$	0	0
$175$	4.20813	0.318105
$176$	−14.8676	−1.12069
$177$	0	0
$178$	−8.08275	−0.605828
$179$	7.52806	0.562674	0.281337	−	0.959609i	$-0.409222\pi$
$179$	7.52806	0.562674	0.281337	+	0.959609i	$0.409222\pi$
$180$	0	0
$181$	−24.6122	−1.82941	−0.914706	−	0.404120i	$-0.867578\pi$
$181$	−24.6122	−1.82941	−0.914706	+	0.404120i	$0.867578\pi$
$182$	−54.0904	−4.00945
$183$	0	0
$184$	−11.4063	−0.840882
$185$	3.80589	0.279815
$186$	0	0
$187$	15.5561	1.13757
$188$	2.41328	0.176007
$189$	0	0
$190$	7.19552	0.522018
$191$	12.9726	0.938664	0.469332	−	0.883022i	$-0.344495\pi$
$191$	12.9726	0.938664	0.469332	+	0.883022i	$0.344495\pi$
$192$	0	0
$193$	−5.71657	−0.411488	−0.205744	−	0.978606i	$-0.565961\pi$
$193$	−5.71657	−0.411488	−0.205744	+	0.978606i	$0.565961\pi$
$194$	−1.81149	−0.130057
$195$	0	0
$196$	13.2352	0.945371
$197$	23.5643	1.67889	0.839444	−	0.543446i	$-0.182881\pi$
$197$	23.5643	1.67889	0.839444	+	0.543446i	$0.182881\pi$
$198$	0	0
$199$	−9.74758	−0.690988	−0.345494	−	0.938421i	$-0.612289\pi$
$199$	−9.74758	−0.690988	−0.345494	+	0.938421i	$0.612289\pi$
$200$	1.37440	0.0971847
$201$	0	0
$202$	0.962526	0.0677231
$203$	16.7154	1.17319
$204$	0	0
$205$	−2.18851	−0.152852
$206$	25.5721	1.78169
$207$	0	0
$208$	−35.3293	−2.44965
$209$	12.0280	0.831997
$210$	0	0
$211$	−4.08372	−0.281135	−0.140567	−	0.990071i	$-0.544893\pi$
$211$	−4.08372	−0.281135	−0.140567	+	0.990071i	$0.544893\pi$
$212$	8.36880	0.574772
$213$	0	0
$214$	25.1695	1.72055
$215$	9.35357	0.637908
$216$	0	0
$217$	−4.20813	−0.285666
$218$	−12.5194	−0.847921
$219$	0	0
$220$	−3.71657	−0.250571
$221$	36.9653	2.48656
$222$	0	0
$223$	−5.88119	−0.393833	−0.196917	−	0.980420i	$-0.563093\pi$
$223$	−5.88119	−0.393833	−0.196917	+	0.980420i	$0.563093\pi$
$224$	25.8608	1.72790
$225$	0	0
$226$	13.0693	0.869354
$227$	23.2013	1.53993	0.769963	−	0.638089i	$-0.220275\pi$
$227$	23.2013	1.53993	0.769963	+	0.638089i	$0.220275\pi$
$228$	0	0
$229$	4.30732	0.284636	0.142318	−	0.989821i	$-0.454544\pi$
$229$	4.30732	0.284636	0.142318	+	0.989821i	$0.454544\pi$
$230$	−14.9291	−0.984395
$231$	0	0
$232$	5.45933	0.358423
$233$	−12.9357	−0.847443	−0.423721	−	0.905793i	$-0.639276\pi$
$233$	−12.9357	−0.847443	−0.423721	+	0.905793i	$0.639276\pi$
$234$	0	0
$235$	−1.95254	−0.127370
$236$	−4.82332	−0.313972
$237$	0	0
$238$	−39.1613	−2.53845
$239$	−24.5883	−1.59049	−0.795243	−	0.606291i	$-0.792657\pi$
$239$	−24.5883	−1.59049	−0.795243	+	0.606291i	$0.792657\pi$
$240$	0	0
$241$	6.27686	0.404328	0.202164	−	0.979352i	$-0.435203\pi$
$241$	6.27686	0.404328	0.202164	+	0.979352i	$0.435203\pi$
$242$	3.52202	0.226404
$243$	0	0
$244$	−17.7492	−1.13628
$245$	−10.7083	−0.684131
$246$	0	0
$247$	28.5818	1.81861
$248$	−1.37440	−0.0872744
$249$	0	0
$250$	1.79888	0.113771
$251$	−24.6262	−1.55439	−0.777197	−	0.629257i	$-0.783359\pi$
$251$	−24.6262	−1.55439	−0.777197	+	0.629257i	$0.783359\pi$
$252$	0	0
$253$	−24.9555	−1.56894
$254$	9.86937	0.619259
$255$	0	0
$256$	20.6683	1.29177
$257$	10.8103	0.674326	0.337163	−	0.941446i	$-0.390533\pi$
$257$	10.8103	0.674326	0.337163	+	0.941446i	$0.390533\pi$
$258$	0	0
$259$	−16.0157	−0.995165
$260$	−8.83155	−0.547709
$261$	0	0
$262$	20.8129	1.28583
$263$	−6.52850	−0.402565	−0.201282	−	0.979533i	$-0.564511\pi$
$263$	−6.52850	−0.402565	−0.201282	+	0.979533i	$0.564511\pi$
$264$	0	0
$265$	−6.77104	−0.415942
$266$	−30.2797	−1.85657
$267$	0	0
$268$	−5.43411	−0.331941
$269$	−2.41888	−0.147482	−0.0737409	−	0.997277i	$-0.523494\pi$
$269$	−2.41888	−0.147482	−0.0737409	+	0.997277i	$0.523494\pi$
$270$	0	0
$271$	19.1234	1.16166	0.580832	−	0.814024i	$-0.302727\pi$
$271$	19.1234	1.16166	0.580832	+	0.814024i	$0.302727\pi$
$272$	−25.5783	−1.55091
$273$	0	0
$274$	2.11059	0.127505
$275$	3.00701	0.181329
$276$	0	0
$277$	11.0014	0.661011	0.330505	−	0.943804i	$-0.392781\pi$
$277$	11.0014	0.661011	0.330505	+	0.943804i	$0.392781\pi$
$278$	−24.0333	−1.44142
$279$	0	0
$280$	−5.78365	−0.345639
$281$	1.23476	0.0736593	0.0368297	−	0.999322i	$-0.488274\pi$
$281$	1.23476	0.0736593	0.0368297	+	0.999322i	$0.488274\pi$
$282$	0	0
$283$	−29.3620	−1.74539	−0.872694	−	0.488267i	$-0.837629\pi$
$283$	−29.3620	−1.74539	−0.872694	+	0.488267i	$0.837629\pi$
$284$	5.57080	0.330566
$285$	0	0
$286$	−38.6515	−2.28551
$287$	9.20954	0.543622
$288$	0	0
$289$	9.76282	0.574283
$290$	7.14544	0.419595
$291$	0	0
$292$	−9.13336	−0.534490
$293$	12.9357	0.755709	0.377855	−	0.925865i	$-0.376662\pi$
$293$	12.9357	0.755709	0.377855	+	0.925865i	$0.376662\pi$
$294$	0	0
$295$	3.90246	0.227210
$296$	−5.23081	−0.304035
$297$	0	0
$298$	1.56073	0.0904104
$299$	−59.3007	−3.42945
$300$	0	0
$301$	−39.3610	−2.26873
$302$	−30.5082	−1.75555
$303$	0	0
$304$	−19.7773	−1.13430
$305$	14.3606	0.822284
$306$	0	0
$307$	−18.0827	−1.03204	−0.516018	−	0.856577i	$-0.672586\pi$
$307$	−18.0827	−1.03204	−0.516018	+	0.856577i	$0.672586\pi$
$308$	15.6398	0.891161
$309$	0	0
$310$	−1.79888	−0.102170
$311$	−27.1326	−1.53855	−0.769274	−	0.638919i	$-0.779382\pi$
$311$	−27.1326	−1.53855	−0.769274	+	0.638919i	$0.779382\pi$
$312$	0	0
$313$	−8.18754	−0.462787	−0.231394	−	0.972860i	$-0.574329\pi$
$313$	−8.18754	−0.462787	−0.231394	+	0.972860i	$0.574329\pi$
$314$	−5.21329	−0.294203
$315$	0	0
$316$	17.9924	1.01215
$317$	19.3804	1.08851	0.544257	−	0.838919i	$-0.316812\pi$
$317$	19.3804	1.08851	0.544257	+	0.838919i	$0.316812\pi$
$318$	0	0
$319$	11.9443	0.668753
$320$	1.16627	0.0651965
$321$	0	0
$322$	62.8235	3.50102
$323$	20.6931	1.15140
$324$	0	0
$325$	7.14544	0.396358
$326$	−25.8229	−1.43020
$327$	0	0
$328$	3.00789	0.166083
$329$	8.21655	0.452993
$330$	0	0
$331$	15.8618	0.871842	0.435921	−	0.899985i	$-0.356423\pi$
$331$	15.8618	0.871842	0.435921	+	0.899985i	$0.356423\pi$
$332$	6.48016	0.355645
$333$	0	0
$334$	10.6679	0.583721
$335$	4.39664	0.240214
$336$	0	0
$337$	−2.16889	−0.118147	−0.0590736	−	0.998254i	$-0.518815\pi$
$337$	−2.16889	−0.118147	−0.0590736	+	0.998254i	$0.518815\pi$
$338$	−68.4605	−3.72376
$339$	0	0
$340$	−6.39402	−0.346764
$341$	−3.00701	−0.162839
$342$	0	0
$343$	15.6052	0.842602
$344$	−12.8555	−0.693124
$345$	0	0
$346$	16.1907	0.870418
$347$	−27.0285	−1.45096	−0.725482	−	0.688241i	$-0.758383\pi$
$347$	−27.0285	−1.45096	−0.725482	+	0.688241i	$0.758383\pi$
$348$	0	0
$349$	−14.7088	−0.787343	−0.393672	−	0.919251i	$-0.628795\pi$
$349$	−14.7088	−0.787343	−0.393672	+	0.919251i	$0.628795\pi$
$350$	−7.56992	−0.404629
$351$	0	0
$352$	18.4794	0.984955
$353$	−23.0315	−1.22584	−0.612920	−	0.790145i	$-0.710005\pi$
$353$	−23.0315	−1.22584	−0.612920	+	0.790145i	$0.710005\pi$
$354$	0	0
$355$	−4.50723	−0.239219
$356$	5.55347	0.294333
$357$	0	0
$358$	−13.5421	−0.715721
$359$	−10.2065	−0.538677	−0.269339	−	0.963046i	$-0.586805\pi$
$359$	−10.2065	−0.538677	−0.269339	+	0.963046i	$0.586805\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	44.2744	2.32701
$363$	0	0
$364$	37.1643	1.94794
$365$	7.38963	0.386791
$366$	0	0
$367$	−1.60896	−0.0839870	−0.0419935	−	0.999118i	$-0.513371\pi$
$367$	−1.60896	−0.0839870	−0.0419935	+	0.999118i	$0.513371\pi$
$368$	41.0334	2.13901
$369$	0	0
$370$	−6.84634	−0.355924
$371$	28.4934	1.47930
$372$	0	0
$373$	0.0696997	0.00360891	0.00180446	−	0.999998i	$-0.499426\pi$
$373$	0.0696997	0.00360891	0.00180446	+	0.999998i	$0.499426\pi$
$374$	−27.9836	−1.44700
$375$	0	0
$376$	2.68357	0.138395
$377$	28.3828	1.46179
$378$	0	0
$379$	29.7633	1.52884	0.764418	−	0.644721i	$-0.223026\pi$
$379$	29.7633	1.52884	0.764418	+	0.644721i	$0.223026\pi$
$380$	−4.94388	−0.253616
$381$	0	0
$382$	−23.3361	−1.19398
$383$	−26.6737	−1.36296	−0.681481	−	0.731836i	$-0.738664\pi$
$383$	−26.6737	−1.36296	−0.681481	+	0.731836i	$0.738664\pi$
$384$	0	0
$385$	−12.6539	−0.644902
$386$	10.2834	0.523413
$387$	0	0
$388$	1.24463	0.0631866
$389$	−20.7981	−1.05451	−0.527253	−	0.849708i	$-0.676778\pi$
$389$	−20.7981	−1.05451	−0.527253	+	0.849708i	$0.676778\pi$
$390$	0	0
$391$	−42.9336	−2.17124
$392$	14.7175	0.743348
$393$	0	0
$394$	−42.3894	−2.13555
$395$	−14.5573	−0.732458
$396$	0	0
$397$	−6.73760	−0.338150	−0.169075	−	0.985603i	$-0.554078\pi$
$397$	−6.73760	−0.338150	−0.169075	+	0.985603i	$0.554078\pi$
$398$	17.5347	0.878937
$399$	0	0
$400$	−4.94432	−0.247216
$401$	−21.7308	−1.08518	−0.542592	−	0.839997i	$-0.682557\pi$
$401$	−21.7308	−1.08518	−0.542592	+	0.839997i	$0.682557\pi$
$402$	0	0
$403$	−7.14544	−0.355940
$404$	−0.661330	−0.0329024
$405$	0	0
$406$	−30.0689	−1.49230
$407$	−11.4443	−0.567275
$408$	0	0
$409$	−16.3746	−0.809672	−0.404836	−	0.914389i	$-0.632671\pi$
$409$	−16.3746	−0.809672	−0.404836	+	0.914389i	$0.632671\pi$
$410$	3.93687	0.194428
$411$	0	0
$412$	−17.5700	−0.865613
$413$	−16.4221	−0.808076
$414$	0	0
$415$	−5.24298	−0.257368
$416$	43.9118	2.15296
$417$	0	0
$418$	−21.6370	−1.05830
$419$	32.9235	1.60842	0.804209	−	0.594347i	$-0.202589\pi$
$419$	32.9235	1.60842	0.804209	+	0.594347i	$0.202589\pi$
$420$	0	0
$421$	−13.2636	−0.646427	−0.323213	−	0.946326i	$-0.604763\pi$
$421$	−13.2636	−0.646427	−0.323213	+	0.946326i	$0.604763\pi$
$422$	7.34612	0.357603
$423$	0	0
$424$	9.30611	0.451945
$425$	5.17328	0.250941
$426$	0	0
$427$	−60.4312	−2.92447
$428$	−17.2934	−0.835909
$429$	0	0
$430$	−16.8259	−0.811419
$431$	12.5465	0.604342	0.302171	−	0.953254i	$-0.402289\pi$
$431$	12.5465	0.604342	0.302171	+	0.953254i	$0.402289\pi$
$432$	0	0
$433$	27.9733	1.34431	0.672156	−	0.740409i	$-0.265368\pi$
$433$	27.9733	1.34431	0.672156	+	0.740409i	$0.265368\pi$
$434$	7.56992	0.363368
$435$	0	0
$436$	8.60179	0.411951
$437$	−33.1964	−1.58800
$438$	0	0
$439$	27.7492	1.32440	0.662199	−	0.749328i	$-0.269623\pi$
$439$	27.7492	1.32440	0.662199	+	0.749328i	$0.269623\pi$
$440$	−4.13283	−0.197025
$441$	0	0
$442$	−66.4962	−3.16290
$443$	2.57508	0.122346	0.0611729	−	0.998127i	$-0.480516\pi$
$443$	2.57508	0.122346	0.0611729	+	0.998127i	$0.480516\pi$
$444$	0	0
$445$	−4.49321	−0.212999
$446$	10.5796	0.500956
$447$	0	0
$448$	−4.90781	−0.231872
$449$	11.6330	0.548998	0.274499	−	0.961587i	$-0.411488\pi$
$449$	11.6330	0.548998	0.274499	+	0.961587i	$0.411488\pi$
$450$	0	0
$451$	6.58087	0.309881
$452$	−8.97959	−0.422364
$453$	0	0
$454$	−41.7364	−1.95879
$455$	−30.0689	−1.40965
$456$	0	0
$457$	−5.63797	−0.263733	−0.131866	−	0.991267i	$-0.542097\pi$
$457$	−5.63797	−0.263733	−0.131866	+	0.991267i	$0.542097\pi$
$458$	−7.74836	−0.362057
$459$	0	0
$460$	10.2574	0.478255
$461$	37.0980	1.72783	0.863913	−	0.503642i	$-0.168007\pi$
$461$	37.0980	1.72783	0.863913	+	0.503642i	$0.168007\pi$
$462$	0	0
$463$	−3.62166	−0.168313	−0.0841563	−	0.996453i	$-0.526820\pi$
$463$	−3.62166	−0.168313	−0.0841563	+	0.996453i	$0.526820\pi$
$464$	−19.6396	−0.911746
$465$	0	0
$466$	23.2697	1.07795
$467$	−23.0507	−1.06666	−0.533330	−	0.845907i	$-0.679060\pi$
$467$	−23.0507	−1.06666	−0.533330	+	0.845907i	$0.679060\pi$
$468$	0	0
$469$	−18.5016	−0.854326
$470$	3.51239	0.162014
$471$	0	0
$472$	−5.36354	−0.246877
$473$	−28.1263	−1.29325
$474$	0	0
$475$	4.00000	0.183533
$476$	26.9068	1.23327
$477$	0	0
$478$	44.2314	2.02310
$479$	18.4768	0.844225	0.422112	−	0.906543i	$-0.361289\pi$
$479$	18.4768	0.844225	0.422112	+	0.906543i	$0.361289\pi$
$480$	0	0
$481$	−27.1947	−1.23997
$482$	−11.2913	−0.514305
$483$	0	0
$484$	−2.41990	−0.109996
$485$	−1.00701	−0.0457259
$486$	0	0
$487$	16.4607	0.745907	0.372954	−	0.927850i	$-0.378345\pi$
$487$	16.4607	0.745907	0.372954	+	0.927850i	$0.378345\pi$
$488$	−19.7372	−0.893459
$489$	0	0
$490$	19.2630	0.870215
$491$	34.5730	1.56026	0.780128	−	0.625619i	$-0.215154\pi$
$491$	34.5730	1.56026	0.780128	+	0.625619i	$0.215154\pi$
$492$	0	0
$493$	20.5491	0.925484
$494$	−51.4151	−2.31328
$495$	0	0
$496$	4.94432	0.222006
$497$	18.9670	0.850786
$498$	0	0
$499$	6.91672	0.309635	0.154817	−	0.987943i	$-0.450521\pi$
$499$	6.91672	0.309635	0.154817	+	0.987943i	$0.450521\pi$
$500$	−1.23597	−0.0552742
$501$	0	0
$502$	44.2996	1.97719
$503$	−6.87997	−0.306763	−0.153381	−	0.988167i	$-0.549016\pi$
$503$	−6.87997	−0.306763	−0.153381	+	0.988167i	$0.549016\pi$
$504$	0	0
$505$	0.535070	0.0238103
$506$	44.8919	1.99569
$507$	0	0
$508$	−6.78102	−0.300859
$509$	20.7701	0.920617	0.460309	−	0.887759i	$-0.347739\pi$
$509$	20.7701	0.920617	0.460309	+	0.887759i	$0.347739\pi$
$510$	0	0
$511$	−31.0965	−1.37563
$512$	−16.7941	−0.742200
$513$	0	0
$514$	−19.4464	−0.857743
$515$	14.2156	0.626413
$516$	0	0
$517$	5.87131	0.258220
$518$	28.8103	1.26585
$519$	0	0
$520$	−9.82068	−0.430665
$521$	15.6357	0.685011	0.342506	−	0.939516i	$-0.388724\pi$
$521$	15.6357	0.685011	0.342506	+	0.939516i	$0.388724\pi$
$522$	0	0
$523$	−29.9583	−1.30999	−0.654993	−	0.755635i	$-0.727329\pi$
$523$	−29.9583	−1.30999	−0.654993	+	0.755635i	$0.727329\pi$
$524$	−14.3001	−0.624701
$525$	0	0
$526$	11.7440	0.512062
$527$	−5.17328	−0.225352
$528$	0	0
$529$	45.8751	1.99457
$530$	12.1803	0.529078
$531$	0	0
$532$	20.8045	0.901989
$533$	15.6379	0.677352
$534$	0	0
$535$	13.9918	0.604917
$536$	−6.04274	−0.261007
$537$	0	0
$538$	4.35128	0.187597
$539$	32.2001	1.38696
$540$	0	0
$541$	36.1919	1.55601	0.778005	−	0.628258i	$-0.216232\pi$
$541$	36.1919	1.55601	0.778005	+	0.628258i	$0.216232\pi$
$542$	−34.4007	−1.47764
$543$	0	0
$544$	31.7921	1.36307
$545$	−6.95955	−0.298114
$546$	0	0
$547$	−6.88260	−0.294279	−0.147139	−	0.989116i	$-0.547007\pi$
$547$	−6.88260	−0.294279	−0.147139	+	0.989116i	$0.547007\pi$
$548$	−1.45014	−0.0619468
$549$	0	0
$550$	−5.40925	−0.230651
$551$	15.8886	0.676879
$552$	0	0
$553$	61.2590	2.60500
$554$	−19.7902	−0.840806
$555$	0	0
$556$	16.5127	0.700296
$557$	18.9810	0.804252	0.402126	−	0.915584i	$-0.368271\pi$
$557$	18.9810	0.804252	0.402126	+	0.915584i	$0.368271\pi$
$558$	0	0
$559$	−66.8353	−2.82683
$560$	20.8063	0.879228
$561$	0	0
$562$	−2.22118	−0.0936947
$563$	17.6428	0.743555	0.371777	−	0.928322i	$-0.378748\pi$
$563$	17.6428	0.743555	0.371777	+	0.928322i	$0.378748\pi$
$564$	0	0
$565$	7.26522	0.305650
$566$	52.8187	2.22014
$567$	0	0
$568$	6.19473	0.259925
$569$	20.7045	0.867978	0.433989	−	0.900918i	$-0.357106\pi$
$569$	20.7045	0.867978	0.433989	+	0.900918i	$0.357106\pi$
$570$	0	0
$571$	−20.0392	−0.838616	−0.419308	−	0.907844i	$-0.637727\pi$
$571$	−20.0392	−0.838616	−0.419308	+	0.907844i	$0.637727\pi$
$572$	26.5565	1.11038
$573$	0	0
$574$	−16.5669	−0.691487
$575$	−8.29910	−0.346096
$576$	0	0
$577$	31.1585	1.29714	0.648572	−	0.761153i	$-0.275366\pi$
$577$	31.1585	1.29714	0.648572	+	0.761153i	$0.275366\pi$
$578$	−17.5621	−0.730489
$579$	0	0
$580$	−4.90947	−0.203855
$581$	22.0631	0.915333
$582$	0	0
$583$	20.3606	0.843249
$584$	−10.1563	−0.420271
$585$	0	0
$586$	−23.2697	−0.961262
$587$	−22.1927	−0.915989	−0.457994	−	0.888955i	$-0.651432\pi$
$587$	−22.1927	−0.915989	−0.457994	+	0.888955i	$0.651432\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−7.02006	−0.289011
$591$	0	0
$592$	18.8175	0.773396
$593$	15.3601	0.630765	0.315383	−	0.948965i	$-0.397867\pi$
$593$	15.3601	0.630765	0.315383	+	0.948965i	$0.397867\pi$
$594$	0	0
$595$	−21.7698	−0.892476
$596$	−1.07234	−0.0439247
$597$	0	0
$598$	106.675	4.36226
$599$	−14.9487	−0.610787	−0.305394	−	0.952226i	$-0.598788\pi$
$599$	−14.9487	−0.610787	−0.305394	+	0.952226i	$0.598788\pi$
$600$	0	0
$601$	13.6118	0.555236	0.277618	−	0.960692i	$-0.410455\pi$
$601$	13.6118	0.555236	0.277618	+	0.960692i	$0.410455\pi$
$602$	70.8057	2.88583
$603$	0	0
$604$	20.9615	0.852911
$605$	1.95790	0.0795998
$606$	0	0
$607$	5.24560	0.212912	0.106456	−	0.994317i	$-0.466050\pi$
$607$	5.24560	0.212912	0.106456	+	0.994317i	$0.466050\pi$
$608$	24.5818	0.996922
$609$	0	0
$610$	−25.8330	−1.04595
$611$	13.9518	0.564428
$612$	0	0
$613$	−12.8434	−0.518739	−0.259369	−	0.965778i	$-0.583515\pi$
$613$	−12.8434	−0.518739	−0.259369	+	0.965778i	$0.583515\pi$
$614$	32.5287	1.31275
$615$	0	0
$616$	17.3915	0.700723
$617$	−32.5677	−1.31113	−0.655564	−	0.755140i	$-0.727569\pi$
$617$	−32.5677	−1.31113	−0.655564	+	0.755140i	$0.727569\pi$
$618$	0	0
$619$	−35.8991	−1.44291	−0.721453	−	0.692464i	$-0.756525\pi$
$619$	−35.8991	−1.44291	−0.721453	+	0.692464i	$0.756525\pi$
$620$	1.23597	0.0496377
$621$	0	0
$622$	48.8083	1.95703
$623$	18.9080	0.757533
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.7284	0.588665
$627$	0	0
$628$	3.58193	0.142935
$629$	−19.6889	−0.785049
$630$	0	0
$631$	−1.69792	−0.0675933	−0.0337967	−	0.999429i	$-0.510760\pi$
$631$	−1.69792	−0.0675933	−0.0337967	+	0.999429i	$0.510760\pi$
$632$	20.0076	0.795858
$633$	0	0
$634$	−34.8631	−1.38459
$635$	5.48640	0.217721
$636$	0	0
$637$	76.5158	3.03167
$638$	−21.4864	−0.850655
$639$	0	0
$640$	10.1929	0.402910
$641$	−31.7659	−1.25468	−0.627338	−	0.778747i	$-0.715856\pi$
$641$	−31.7659	−1.25468	−0.627338	+	0.778747i	$0.715856\pi$
$642$	0	0
$643$	32.2100	1.27024	0.635119	−	0.772414i	$-0.280951\pi$
$643$	32.2100	1.27024	0.635119	+	0.772414i	$0.280951\pi$
$644$	−43.1646	−1.70092
$645$	0	0
$646$	−37.2244	−1.46458
$647$	30.5140	1.19963	0.599814	−	0.800139i	$-0.295241\pi$
$647$	30.5140	1.19963	0.599814	+	0.800139i	$0.295241\pi$
$648$	0	0
$649$	−11.7347	−0.460628
$650$	−12.8538	−0.504167
$651$	0	0
$652$	17.7423	0.694843
$653$	20.4786	0.801390	0.400695	−	0.916211i	$-0.368769\pi$
$653$	20.4786	0.801390	0.400695	+	0.916211i	$0.368769\pi$
$654$	0	0
$655$	11.5699	0.452074
$656$	−10.8207	−0.422477
$657$	0	0
$658$	−14.7806	−0.576207
$659$	−22.1442	−0.862616	−0.431308	−	0.902205i	$-0.641948\pi$
$659$	−22.1442	−0.862616	−0.431308	+	0.902205i	$0.641948\pi$
$660$	0	0
$661$	28.2744	1.09975	0.549874	−	0.835248i	$-0.314676\pi$
$661$	28.2744	1.09975	0.549874	+	0.835248i	$0.314676\pi$
$662$	−28.5334	−1.10898
$663$	0	0
$664$	7.20594	0.279645
$665$	−16.8325	−0.652737
$666$	0	0
$667$	−32.9653	−1.27642
$668$	−7.32967	−0.283594
$669$	0	0
$670$	−7.90903	−0.305552
$671$	−43.1824	−1.66704
$672$	0	0
$673$	20.7857	0.801231	0.400616	−	0.916246i	$-0.368796\pi$
$673$	20.7857	0.801231	0.400616	+	0.916246i	$0.368796\pi$
$674$	3.90158	0.150283
$675$	0	0
$676$	47.0377	1.80914
$677$	−30.6432	−1.17771	−0.588857	−	0.808237i	$-0.700422\pi$
$677$	−30.6432	−1.17771	−0.588857	+	0.808237i	$0.700422\pi$
$678$	0	0
$679$	4.23762	0.162625
$680$	−7.11015	−0.272662
$681$	0	0
$682$	5.40925	0.207131
$683$	6.24585	0.238991	0.119495	−	0.992835i	$-0.461872\pi$
$683$	6.24585	0.238991	0.119495	+	0.992835i	$0.461872\pi$
$684$	0	0
$685$	1.17328	0.0448287
$686$	−28.0719	−1.07179
$687$	0	0
$688$	46.2470	1.76315
$689$	48.3820	1.84321
$690$	0	0
$691$	−7.39060	−0.281152	−0.140576	−	0.990070i	$-0.544895\pi$
$691$	−7.39060	−0.281152	−0.140576	+	0.990070i	$0.544895\pi$
$692$	−11.1243	−0.422881
$693$	0	0
$694$	48.6210	1.84563
$695$	−13.3601	−0.506779
$696$	0	0
$697$	11.3218	0.428843
$698$	26.4593	1.00150
$699$	0	0
$700$	5.20112	0.196584
$701$	−4.36802	−0.164978	−0.0824890	−	0.996592i	$-0.526287\pi$
$701$	−4.36802	−0.164978	−0.0824890	+	0.996592i	$0.526287\pi$
$702$	0	0
$703$	−15.2236	−0.574167
$704$	−3.50698	−0.132174
$705$	0	0
$706$	41.4308	1.55927
$707$	−2.25164	−0.0846817
$708$	0	0
$709$	22.8325	0.857493	0.428747	−	0.903425i	$-0.358955\pi$
$709$	22.8325	0.857493	0.428747	+	0.903425i	$0.358955\pi$
$710$	8.10796	0.304286
$711$	0	0
$712$	6.17546	0.231435
$713$	8.29910	0.310804
$714$	0	0
$715$	−21.4864	−0.803546
$716$	9.30445	0.347724
$717$	0	0
$718$	18.3602	0.685198
$719$	−21.8404	−0.814510	−0.407255	−	0.913315i	$-0.633514\pi$
$719$	−21.8404	−0.814510	−0.407255	+	0.913315i	$0.633514\pi$
$720$	0	0
$721$	−59.8210	−2.22785
$722$	5.39664	0.200842
$723$	0	0
$724$	−30.4200	−1.13055
$725$	3.97216	0.147522
$726$	0	0
$727$	−13.6566	−0.506496	−0.253248	−	0.967401i	$-0.581499\pi$
$727$	−13.6566	−0.506496	−0.253248	+	0.967401i	$0.581499\pi$
$728$	41.3267	1.53167
$729$	0	0
$730$	−13.2931	−0.491998
$731$	−48.3886	−1.78972
$732$	0	0
$733$	−7.14861	−0.264040	−0.132020	−	0.991247i	$-0.542146\pi$
$733$	−7.14861	−0.264040	−0.132020	+	0.991247i	$0.542146\pi$
$734$	2.89432	0.106831
$735$	0	0
$736$	−51.0016	−1.87994
$737$	−13.2207	−0.486992
$738$	0	0
$739$	−0.221952	−0.00816463	−0.00408231	−	0.999992i	$-0.501299\pi$
$739$	−0.221952	−0.00816463	−0.00408231	+	0.999992i	$0.501299\pi$
$740$	4.70396	0.172921
$741$	0	0
$742$	−51.2562	−1.88168
$743$	−38.5285	−1.41347	−0.706737	−	0.707477i	$-0.749833\pi$
$743$	−38.5285	−1.41347	−0.706737	+	0.707477i	$0.749833\pi$
$744$	0	0
$745$	0.867610	0.0317868
$746$	−0.125381	−0.00459054
$747$	0	0
$748$	19.2269	0.703004
$749$	−58.8792	−2.15140
$750$	0	0
$751$	−15.5508	−0.567459	−0.283729	−	0.958904i	$-0.591572\pi$
$751$	−15.5508	−0.567459	−0.283729	+	0.958904i	$0.591572\pi$
$752$	−9.65399	−0.352045
$753$	0	0
$754$	−51.0573	−1.85940
$755$	−16.9596	−0.617221
$756$	0	0
$757$	40.9532	1.48847	0.744234	−	0.667919i	$-0.232815\pi$
$757$	40.9532	1.48847	0.744234	+	0.667919i	$0.232815\pi$
$758$	−53.5405	−1.94468
$759$	0	0
$760$	−5.49760	−0.199419
$761$	32.3308	1.17199	0.585995	−	0.810315i	$-0.300704\pi$
$761$	32.3308	1.17199	0.585995	+	0.810315i	$0.300704\pi$
$762$	0	0
$763$	29.2867	1.06025
$764$	16.0337	0.580080
$765$	0	0
$766$	47.9828	1.73369
$767$	−27.8848	−1.00686
$768$	0	0
$769$	51.7592	1.86648	0.933242	−	0.359249i	$-0.116967\pi$
$769$	51.7592	1.86648	0.933242	+	0.359249i	$0.116967\pi$
$770$	22.7628	0.820315
$771$	0	0
$772$	−7.06551	−0.254293
$773$	34.8383	1.25305	0.626523	−	0.779403i	$-0.284477\pi$
$773$	34.8383	1.25305	0.626523	+	0.779403i	$0.284477\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	1.38403	0.0496839
$777$	0	0
$778$	37.4133	1.34133
$779$	8.75405	0.313646
$780$	0	0
$781$	13.5533	0.484974
$782$	77.2323	2.76182
$783$	0	0
$784$	−52.9455	−1.89091
$785$	−2.89807	−0.103437
$786$	0	0
$787$	42.1782	1.50349	0.751745	−	0.659454i	$-0.229212\pi$
$787$	42.1782	1.50349	0.751745	+	0.659454i	$0.229212\pi$
$788$	29.1248	1.03753
$789$	0	0
$790$	26.1869	0.931687
$791$	−30.5730	−1.08705
$792$	0	0
$793$	−102.613	−3.64388
$794$	12.1201	0.430127
$795$	0	0
$796$	−12.0477	−0.427020
$797$	9.01743	0.319414	0.159707	−	0.987164i	$-0.448945\pi$
$797$	9.01743	0.319414	0.159707	+	0.987164i	$0.448945\pi$
$798$	0	0
$799$	10.1010	0.357349
$800$	6.14544	0.217274
$801$	0	0
$802$	39.0911	1.38035
$803$	−22.2207	−0.784151
$804$	0	0
$805$	34.9237	1.23090
$806$	12.8538	0.452755
$807$	0	0
$808$	−0.735399	−0.0258712
$809$	−8.91472	−0.313425	−0.156712	−	0.987644i	$-0.550090\pi$
$809$	−8.91472	−0.313425	−0.156712	+	0.987644i	$0.550090\pi$
$810$	0	0
$811$	10.1515	0.356467	0.178233	−	0.983988i	$-0.442962\pi$
$811$	10.1515	0.356467	0.178233	+	0.983988i	$0.442962\pi$
$812$	20.6597	0.725012
$813$	0	0
$814$	20.5870	0.721574
$815$	−14.3550	−0.502833
$816$	0	0
$817$	−37.4143	−1.30896
$818$	29.4559	1.02990
$819$	0	0
$820$	−2.70493	−0.0944604
$821$	16.2720	0.567898	0.283949	−	0.958839i	$-0.408355\pi$
$821$	16.2720	0.567898	0.283949	+	0.958839i	$0.408355\pi$
$822$	0	0
$823$	−10.4682	−0.364898	−0.182449	−	0.983215i	$-0.558402\pi$
$823$	−10.4682	−0.364898	−0.182449	+	0.983215i	$0.558402\pi$
$824$	−19.5379	−0.680634
$825$	0	0
$826$	29.5413	1.02787
$827$	−0.407155	−0.0141582	−0.00707909	−	0.999975i	$-0.502253\pi$
$827$	−0.407155	−0.0141582	−0.00707909	+	0.999975i	$0.502253\pi$
$828$	0	0
$829$	3.66464	0.127278	0.0636391	−	0.997973i	$-0.479729\pi$
$829$	3.66464	0.127278	0.0636391	+	0.997973i	$0.479729\pi$
$830$	9.43149	0.327372
$831$	0	0
$832$	−8.33351	−0.288912
$833$	55.3973	1.91940
$834$	0	0
$835$	5.93030	0.205227
$836$	14.8663	0.514161
$837$	0	0
$838$	−59.2254	−2.04591
$839$	−20.0811	−0.693276	−0.346638	−	0.937999i	$-0.612677\pi$
$839$	−20.0811	−0.693276	−0.346638	+	0.937999i	$0.612677\pi$
$840$	0	0
$841$	−13.2220	−0.455929
$842$	23.8596	0.822255
$843$	0	0
$844$	−5.04735	−0.173737
$845$	−38.0573	−1.30921
$846$	0	0
$847$	−8.23908	−0.283098
$848$	−33.4782	−1.14965
$849$	0	0
$850$	−9.30611	−0.319197
$851$	31.5855	1.08274
$852$	0	0
$853$	−20.0074	−0.685042	−0.342521	−	0.939510i	$-0.611281\pi$
$853$	−20.0074	−0.685042	−0.342521	+	0.939510i	$0.611281\pi$
$854$	108.708	3.71992
$855$	0	0
$856$	−19.2303	−0.657278
$857$	−9.62823	−0.328894	−0.164447	−	0.986386i	$-0.552584\pi$
$857$	−9.62823	−0.328894	−0.164447	+	0.986386i	$0.552584\pi$
$858$	0	0
$859$	0.804479	0.0274485	0.0137242	−	0.999906i	$-0.495631\pi$
$859$	0.804479	0.0274485	0.0137242	+	0.999906i	$0.495631\pi$
$860$	11.5607	0.394217
$861$	0	0
$862$	−22.5696	−0.768723
$863$	−17.8746	−0.608459	−0.304230	−	0.952599i	$-0.598399\pi$
$863$	−17.8746	−0.608459	−0.304230	+	0.952599i	$0.598399\pi$
$864$	0	0
$865$	9.00044	0.306024
$866$	−50.3207	−1.70997
$867$	0	0
$868$	−5.20112	−0.176538
$869$	43.7740	1.48493
$870$	0	0
$871$	−31.4159	−1.06449
$872$	9.56520	0.323919
$873$	0	0
$874$	59.7163	2.01994
$875$	−4.20813	−0.142261
$876$	0	0
$877$	−24.0041	−0.810562	−0.405281	−	0.914192i	$-0.632826\pi$
$877$	−24.0041	−0.810562	−0.405281	+	0.914192i	$0.632826\pi$
$878$	−49.9176	−1.68464
$879$	0	0
$880$	14.8676	0.501187
$881$	−10.6727	−0.359573	−0.179787	−	0.983706i	$-0.557541\pi$
$881$	−10.6727	−0.359573	−0.179787	+	0.983706i	$0.557541\pi$
$882$	0	0
$883$	−9.92567	−0.334025	−0.167013	−	0.985955i	$-0.553412\pi$
$883$	−9.92567	−0.334025	−0.167013	+	0.985955i	$0.553412\pi$
$884$	45.6880	1.53666
$885$	0	0
$886$	−4.63226	−0.155624
$887$	−24.4352	−0.820453	−0.410227	−	0.911984i	$-0.634550\pi$
$887$	−24.4352	−0.820453	−0.410227	+	0.911984i	$0.634550\pi$
$888$	0	0
$889$	−23.0875	−0.774329
$890$	8.08275	0.270934
$891$	0	0
$892$	−7.26897	−0.243383
$893$	7.81017	0.261357
$894$	0	0
$895$	−7.52806	−0.251635
$896$	−42.8930	−1.43296
$897$	0	0
$898$	−20.9265	−0.698325
$899$	−3.97216	−0.132479
$900$	0	0
$901$	35.0285	1.16697
$902$	−11.8382	−0.394169
$903$	0	0
$904$	−9.98531	−0.332106
$905$	24.6122	0.818138
$906$	0	0
$907$	8.66887	0.287845	0.143923	−	0.989589i	$-0.454028\pi$
$907$	8.66887	0.287845	0.143923	+	0.989589i	$0.454028\pi$
$908$	28.6761	0.951651
$909$	0	0
$910$	54.0904	1.79308
$911$	−42.8676	−1.42027	−0.710134	−	0.704067i	$-0.751365\pi$
$911$	−42.8676	−1.42027	−0.710134	+	0.704067i	$0.751365\pi$
$912$	0	0
$913$	15.7657	0.521768
$914$	10.1420	0.335468
$915$	0	0
$916$	5.32372	0.175901
$917$	−48.6877	−1.60781
$918$	0	0
$919$	−47.4316	−1.56462	−0.782312	−	0.622887i	$-0.785960\pi$
$919$	−47.4316	−1.56462	−0.782312	+	0.622887i	$0.785960\pi$
$920$	11.4063	0.376054
$921$	0	0
$922$	−66.7348	−2.19779
$923$	32.2061	1.06008
$924$	0	0
$925$	−3.80589	−0.125137
$926$	6.51492	0.214094
$927$	0	0
$928$	24.4107	0.801319
$929$	−50.0568	−1.64231	−0.821154	−	0.570706i	$-0.806670\pi$
$929$	−50.0568	−1.64231	−0.821154	+	0.570706i	$0.806670\pi$
$930$	0	0
$931$	42.8334	1.40381
$932$	−15.9881	−0.523707
$933$	0	0
$934$	41.4655	1.35679
$935$	−15.5561	−0.508739
$936$	0	0
$937$	−25.5888	−0.835948	−0.417974	−	0.908459i	$-0.637260\pi$
$937$	−25.5888	−0.835948	−0.417974	+	0.908459i	$0.637260\pi$
$938$	33.2822	1.08670
$939$	0	0
$940$	−2.41328	−0.0787126
$941$	−26.3029	−0.857450	−0.428725	−	0.903435i	$-0.641037\pi$
$941$	−26.3029	−0.857450	−0.428725	+	0.903435i	$0.641037\pi$
$942$	0	0
$943$	−18.1627	−0.591458
$944$	19.2950	0.627999
$945$	0	0
$946$	50.5958	1.64501
$947$	47.1292	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$947$	47.1292	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$948$	0	0
$949$	−52.8022	−1.71403
$950$	−7.19552	−0.233454
$951$	0	0
$952$	29.9204	0.969726
$953$	−39.0091	−1.26363	−0.631814	−	0.775120i	$-0.717689\pi$
$953$	−39.0091	−1.26363	−0.631814	+	0.775120i	$0.717689\pi$
$954$	0	0
$955$	−12.9726	−0.419783
$956$	−30.3904	−0.982896
$957$	0	0
$958$	−33.2375	−1.07385
$959$	−4.93731	−0.159434
$960$	0	0
$961$	1.00000	0.0322581
$962$	48.9201	1.57725
$963$	0	0
$964$	7.75801	0.249869
$965$	5.71657	0.184023
$966$	0	0
$967$	−61.6585	−1.98280	−0.991401	−	0.130857i	$-0.958227\pi$
$967$	−61.6585	−1.98280	−0.991401	+	0.130857i	$0.958227\pi$
$968$	−2.69093	−0.0864898
$969$	0	0
$970$	1.81149	0.0581634
$971$	39.4375	1.26561	0.632804	−	0.774312i	$-0.281904\pi$
$971$	39.4375	1.26561	0.632804	+	0.774312i	$0.281904\pi$
$972$	0	0
$973$	56.2212	1.80237
$974$	−29.6109	−0.948794
$975$	0	0
$976$	71.0033	2.27276
$977$	−40.7549	−1.30387	−0.651933	−	0.758277i	$-0.726042\pi$
$977$	−40.7549	−1.30387	−0.651933	+	0.758277i	$0.726042\pi$
$978$	0	0
$979$	13.5111	0.431817
$980$	−13.2352	−0.422783
$981$	0	0
$982$	−62.1927	−1.98465
$983$	−31.8349	−1.01538	−0.507688	−	0.861541i	$-0.669500\pi$
$983$	−31.8349	−1.01538	−0.507688	+	0.861541i	$0.669500\pi$
$984$	0	0
$985$	−23.5643	−0.750822
$986$	−36.9653	−1.17722
$987$	0	0
$988$	35.3262	1.12388
$989$	77.6262	2.46837
$990$	0	0
$991$	−14.9055	−0.473490	−0.236745	−	0.971572i	$-0.576081\pi$
$991$	−14.9055	−0.473490	−0.236745	+	0.971572i	$0.576081\pi$
$992$	−6.14544	−0.195118
$993$	0	0
$994$	−34.1194	−1.08220
$995$	9.74758	0.309019
$996$	0	0
$997$	26.6001	0.842435	0.421217	−	0.906960i	$-0.361603\pi$
$997$	26.6001	0.842435	0.421217	+	0.906960i	$0.361603\pi$
$998$	−12.4424	−0.393856
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1395.2.a.k.1.2		4
3.2	odd	2		465.2.a.h.1.3	✓	4
5.4	even	2		6975.2.a.bo.1.3		4
12.11	even	2		7440.2.a.bz.1.1		4
15.2	even	4		2325.2.c.p.1024.6		8
15.8	even	4		2325.2.c.p.1024.3		8
15.14	odd	2		2325.2.a.v.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.h.1.3	✓	4	3.2	odd	2
1395.2.a.k.1.2		4	1.1	even	1	trivial
2325.2.a.v.1.2		4	15.14	odd	2
2325.2.c.p.1024.3		8	15.8	even	4
2325.2.c.p.1024.6		8	15.2	even	4
6975.2.a.bo.1.3		4	5.4	even	2
7440.2.a.bz.1.1		4	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1395.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79888 q^{2} +1.23597 q^{4} -1.00000 q^{5} +4.20813 q^{7} +1.37440 q^{8} +1.79888 q^{10} +3.00701 q^{11} +7.14544 q^{13} -7.56992 q^{14} -4.94432 q^{16} +5.17328 q^{17} +4.00000 q^{19} -1.23597 q^{20} -5.40925 q^{22} -8.29910 q^{23} +1.00000 q^{25} -12.8538 q^{26} +5.20112 q^{28} +3.97216 q^{29} -1.00000 q^{31} +6.14544 q^{32} -9.30611 q^{34} -4.20813 q^{35} -3.80589 q^{37} -7.19552 q^{38} -1.37440 q^{40} +2.18851 q^{41} -9.35357 q^{43} +3.71657 q^{44} +14.9291 q^{46} +1.95254 q^{47} +10.7083 q^{49} -1.79888 q^{50} +8.83155 q^{52} +6.77104 q^{53} -3.00701 q^{55} +5.78365 q^{56} -7.14544 q^{58} -3.90246 q^{59} -14.3606 q^{61} +1.79888 q^{62} -1.16627 q^{64} -7.14544 q^{65} -4.39664 q^{67} +6.39402 q^{68} +7.56992 q^{70} +4.50723 q^{71} -7.38963 q^{73} +6.84634 q^{74} +4.94388 q^{76} +12.6539 q^{77} +14.5573 q^{79} +4.94432 q^{80} -3.93687 q^{82} +5.24298 q^{83} -5.17328 q^{85} +16.8259 q^{86} +4.13283 q^{88} +4.49321 q^{89} +30.0689 q^{91} -10.2574 q^{92} -3.51239 q^{94} -4.00000 q^{95} +1.00701 q^{97} -19.2630 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 2 q^{10} - 6 q^{11} + 2 q^{13} - 4 q^{14} + 12 q^{16} + 10 q^{17} + 16 q^{19} - 8 q^{20} - 14 q^{22} - 6 q^{23} + 4 q^{25} + 10 q^{26} + 26 q^{28} - 4 q^{31}+ \cdots - 8 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 2 * q^10 - 6 * q^11 + 2 * q^13 - 4 * q^14 + 12 * q^16 + 10 * q^17 + 16 * q^19 - 8 * q^20 - 14 * q^22 - 6 * q^23 + 4 * q^25 + 10 * q^26 + 26 * q^28 - 4 * q^31 - 2 * q^32 + 8 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 + 6 * q^41 + 2 * q^43 + 6 * q^44 + 20 * q^46 + 2 * q^47 + 8 * q^49 - 2 * q^50 - 16 * q^52 + 6 * q^53 + 6 * q^55 + 10 * q^56 - 2 * q^58 - 4 * q^59 + 2 * q^62 - 12 * q^64 - 2 * q^65 - 2 * q^67 + 10 * q^68 + 4 * q^70 - 22 * q^71 - 32 * q^73 + 28 * q^74 + 32 * q^76 + 28 * q^77 + 24 * q^79 - 12 * q^80 - 46 * q^82 + 6 * q^83 - 10 * q^85 - 10 * q^86 - 18 * q^88 + 14 * q^89 + 28 * q^91 + 54 * q^92 - 44 * q^94 - 16 * q^95 - 14 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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