LMFDB - Embedded newform 6975.2.a.bo.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,0,8,0,0,-4,0,0,0,-6,0,-2,-4,0,12,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
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			1.3
Root			$-0.704624$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79888 q^{2} +1.23597 q^{4} -4.20813 q^{7} -1.37440 q^{8} +3.00701 q^{11} -7.14544 q^{13} -7.56992 q^{14} -4.94432 q^{16} -5.17328 q^{17} +4.00000 q^{19} +5.40925 q^{22} +8.29910 q^{23} -12.8538 q^{26} -5.20112 q^{28} +3.97216 q^{29} -1.00000 q^{31} -6.14544 q^{32} -9.30611 q^{34} +3.80589 q^{37} +7.19552 q^{38} +2.18851 q^{41} +9.35357 q^{43} +3.71657 q^{44} +14.9291 q^{46} -1.95254 q^{47} +10.7083 q^{49} -8.83155 q^{52} -6.77104 q^{53} +5.78365 q^{56} +7.14544 q^{58} -3.90246 q^{59} -14.3606 q^{61} -1.79888 q^{62} -1.16627 q^{64} +4.39664 q^{67} -6.39402 q^{68} +4.50723 q^{71} +7.38963 q^{73} +6.84634 q^{74} +4.94388 q^{76} -12.6539 q^{77} +14.5573 q^{79} +3.93687 q^{82} -5.24298 q^{83} +16.8259 q^{86} -4.13283 q^{88} +4.49321 q^{89} +30.0689 q^{91} +10.2574 q^{92} -3.51239 q^{94} -1.00701 q^{97} +19.2630 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{11} - 2 q^{13} - 4 q^{14} + 12 q^{16} - 10 q^{17} + 16 q^{19} + 14 q^{22} + 6 q^{23} + 10 q^{26} - 26 q^{28} - 4 q^{31} + 2 q^{32} + 8 q^{34} - 8 q^{37} + 8 q^{38}+ \cdots + 8 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 - 4 * q^7 - 6 * q^11 - 2 * q^13 - 4 * q^14 + 12 * q^16 - 10 * q^17 + 16 * q^19 + 14 * q^22 + 6 * q^23 + 10 * q^26 - 26 * q^28 - 4 * q^31 + 2 * q^32 + 8 * q^34 - 8 * q^37 + 8 * q^38 + 6 * q^41 - 2 * q^43 + 6 * q^44 + 20 * q^46 - 2 * q^47 + 8 * q^49 + 16 * q^52 - 6 * q^53 + 10 * q^56 + 2 * q^58 - 4 * q^59 - 2 * q^62 - 12 * q^64 + 2 * q^67 - 10 * q^68 - 22 * q^71 + 32 * q^73 + 28 * q^74 + 32 * q^76 - 28 * q^77 + 24 * q^79 + 46 * q^82 - 6 * q^83 - 10 * q^86 + 18 * q^88 + 14 * q^89 + 28 * q^91 - 54 * q^92 - 44 * q^94 + 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	$-0.280588\pi$
$2$	1.79888	1.27200	0.636000	+	0.771689i	$0.280588\pi$
$3$	0	0
$3$	0	0
$4$	1.23597	0.617985
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.20813	−1.59052	−0.795262	−	0.606266i	$-0.792667\pi$
$7$	−4.20813	−1.59052	−0.795262	+	0.606266i	$0.792667\pi$
$8$	−1.37440	−0.485923
$9$	0	0
$10$	0	0
$11$	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	$-0.957011\pi$
$13$	−7.14544	−1.98179	−0.990894	+	0.134644i	$0.957011\pi$
$14$	−7.56992	−2.02315
$15$	0	0
$16$	−4.94432	−1.23608
$17$	−5.17328	−1.25470	−0.627352	−	0.778736i	$-0.715861\pi$
$17$	−5.17328	−1.25470	−0.627352	+	0.778736i	$0.715861\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	5.40925	1.15326
$23$	8.29910	1.73048	0.865241	−	0.501356i	$-0.167165\pi$
$23$	8.29910	1.73048	0.865241	+	0.501356i	$0.167165\pi$
$24$	0	0
$25$	0	0
$26$	−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
$35$	0	0
$36$	0	0
$37$	3.80589	0.625684	0.312842	−	0.949805i	$-0.398719\pi$
$37$	3.80589	0.625684	0.312842	+	0.949805i	$0.398719\pi$
$38$	7.19552	1.16727
$39$	0	0
$40$	0	0
$41$	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	$-0.247244\pi$
$43$	9.35357	1.42641	0.713203	+	0.700958i	$0.247244\pi$
$44$	3.71657	0.560294
$45$	0	0
$46$	14.9291	2.20117
$47$	−1.95254	−0.284807	−0.142404	−	0.989809i	$-0.545483\pi$
$47$	−1.95254	−0.284807	−0.142404	+	0.989809i	$0.545483\pi$
$48$	0	0
$49$	10.7083	1.52976
$50$	0	0
$51$	0	0
$52$	−8.83155	−1.22471
$53$	−6.77104	−0.930074	−0.465037	−	0.885291i	$-0.653959\pi$
$53$	−6.77104	−0.930074	−0.465037	+	0.885291i	$0.653959\pi$
$54$	0	0
$55$	0	0
$56$	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
$65$	0	0
$66$	0	0
$67$	4.39664	0.537135	0.268568	−	0.963261i	$-0.413450\pi$
$67$	4.39664	0.537135	0.268568	+	0.963261i	$0.413450\pi$
$68$	−6.39402	−0.775388
$69$	0	0
$70$	0	0
$71$	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	$-0.357651\pi$
$73$	7.38963	0.864891	0.432445	+	0.901660i	$0.357651\pi$
$74$	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	$0.194577\pi$
$80$	0	0
$81$	0	0
$82$	3.93687	0.434755
$83$	−5.24298	−0.575492	−0.287746	−	0.957707i	$-0.592906\pi$
$83$	−5.24298	−0.575492	−0.287746	+	0.957707i	$0.592906\pi$
$84$	0	0
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1.00701	−0.102246	−0.0511231	−	0.998692i	$-0.516280\pi$
$97$	−1.00701	−0.102246	−0.0511231	+	0.998692i	$0.516280\pi$
$98$	19.2630	1.94586
$99$	0	0
$100$	0	0
$101$	−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	$-0.253027\pi$
$103$	14.2156	1.40070	0.700351	+	0.713798i	$0.253027\pi$
$104$	9.82068	0.962997
$105$	0	0
$106$	−12.1803	−1.18305
$107$	13.9918	1.35264	0.676318	−	0.736610i	$-0.263575\pi$
$107$	13.9918	1.35264	0.676318	+	0.736610i	$0.263575\pi$
$108$	0	0
$109$	6.95955	0.666604	0.333302	−	0.942820i	$-0.391837\pi$
$109$	6.95955	0.666604	0.333302	+	0.942820i	$0.391837\pi$
$110$	0	0
$111$	0	0
$112$	20.8063	1.96601
$113$	7.26522	0.683454	0.341727	−	0.939799i	$-0.388988\pi$
$113$	7.26522	0.683454	0.341727	+	0.939799i	$0.388988\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	5.48640	0.486839	0.243419	−	0.969921i	$-0.421731\pi$
$127$	5.48640	0.486839	0.243419	+	0.969921i	$0.421731\pi$
$128$	10.1929	0.900933
$129$	0	0
$130$	0	0
$131$	−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	$-0.484040\pi$
$137$	1.17328	0.100240	0.0501200	+	0.998743i	$0.484040\pi$
$138$	0	0
$139$	13.3601	1.13319	0.566596	−	0.823996i	$-0.308260\pi$
$139$	13.3601	1.13319	0.566596	+	0.823996i	$0.308260\pi$
$140$	0	0
$141$	0	0
$142$	8.10796	0.680405
$143$	−21.4864	−1.79678
$144$	0	0
$145$	0	0
$146$	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
$155$	0	0
$156$	0	0
$157$	−2.89807	−0.231292	−0.115646	−	0.993291i	$-0.536894\pi$
$157$	−2.89807	−0.231292	−0.115646	+	0.993291i	$0.536894\pi$
$158$	26.1869	2.08331
$159$	0	0
$160$	0	0
$161$	−34.9237	−2.75237
$162$	0	0
$163$	−14.3550	−1.12437	−0.562184	−	0.827012i	$-0.690039\pi$
$163$	−14.3550	−1.12437	−0.562184	+	0.827012i	$0.690039\pi$
$164$	2.70493	0.211220
$165$	0	0
$166$	−9.43149	−0.732026
$167$	5.93030	0.458900	0.229450	−	0.973320i	$-0.426307\pi$
$167$	5.93030	0.458900	0.229450	+	0.973320i	$0.426307\pi$
$168$	0	0
$169$	38.0573	2.92748
$170$	0	0
$171$	0	0
$172$	11.5607	0.881497
$173$	9.00044	0.684291	0.342145	−	0.939647i	$-0.388846\pi$
$173$	9.00044	0.684291	0.342145	+	0.939647i	$0.388846\pi$
$174$	0	0
$175$	0	0
$176$	−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
$185$	0	0
$186$	0	0
$187$	−15.5561	−1.13757
$188$	−2.41328	−0.176007
$189$	0	0
$190$	0	0
$191$	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	$-0.434039\pi$
$193$	5.71657	0.411488	0.205744	+	0.978606i	$0.434039\pi$
$194$	−1.81149	−0.130057
$195$	0	0
$196$	13.2352	0.945371
$197$	−23.5643	−1.67889	−0.839444	−	0.543446i	$-0.817119\pi$
$197$	−23.5643	−1.67889	−0.839444	+	0.543446i	$0.817119\pi$
$198$	0	0
$199$	−9.74758	−0.690988	−0.345494	−	0.938421i	$-0.612289\pi$
$199$	−9.74758	−0.690988	−0.345494	+	0.938421i	$0.612289\pi$
$200$	0	0
$201$	0	0
$202$	−0.962526	−0.0677231
$203$	−16.7154	−1.17319
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	0	0
$217$	4.20813	0.285666
$218$	12.5194	0.847921
$219$	0	0
$220$	0	0
$221$	36.9653	2.48656
$222$	0	0
$223$	5.88119	0.393833	0.196917	−	0.980420i	$-0.436907\pi$
$223$	5.88119	0.393833	0.196917	+	0.980420i	$0.436907\pi$
$224$	25.8608	1.72790
$225$	0	0
$226$	13.0693	0.869354
$227$	−23.2013	−1.53993	−0.769963	−	0.638089i	$-0.779725\pi$
$227$	−23.2013	−1.53993	−0.769963	+	0.638089i	$0.779725\pi$
$228$	0	0
$229$	4.30732	0.284636	0.142318	−	0.989821i	$-0.454544\pi$
$229$	4.30732	0.284636	0.142318	+	0.989821i	$0.454544\pi$
$230$	0	0
$231$	0	0
$232$	−5.45933	−0.358423
$233$	12.9357	0.847443	0.423721	−	0.905793i	$-0.360724\pi$
$233$	12.9357	0.847443	0.423721	+	0.905793i	$0.360724\pi$
$234$	0	0
$235$	0	0
$236$	−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
$245$	0	0
$246$	0	0
$247$	−28.5818	−1.81861
$248$	1.37440	0.0872744
$249$	0	0
$250$	0	0
$251$	−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	$-0.609467\pi$
$257$	−10.8103	−0.674326	−0.337163	+	0.941446i	$0.609467\pi$
$258$	0	0
$259$	−16.0157	−0.995165
$260$	0	0
$261$	0	0
$262$	−20.8129	−1.28583
$263$	6.52850	0.402565	0.201282	−	0.979533i	$-0.435489\pi$
$263$	6.52850	0.402565	0.201282	+	0.979533i	$0.435489\pi$
$264$	0	0
$265$	0	0
$266$	−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
$275$	0	0
$276$	0	0
$277$	−11.0014	−0.661011	−0.330505	−	0.943804i	$-0.607219\pi$
$277$	−11.0014	−0.661011	−0.330505	+	0.943804i	$0.607219\pi$
$278$	24.0333	1.44142
$279$	0	0
$280$	0	0
$281$	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	$-0.162371\pi$
$283$	29.3620	1.74539	0.872694	+	0.488267i	$0.162371\pi$
$284$	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
$290$	0	0
$291$	0	0
$292$	9.13336	0.534490
$293$	−12.9357	−0.755709	−0.377855	−	0.925865i	$-0.623338\pi$
$293$	−12.9357	−0.755709	−0.377855	+	0.925865i	$0.623338\pi$
$294$	0	0
$295$	0	0
$296$	−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
$305$	0	0
$306$	0	0
$307$	18.0827	1.03204	0.516018	−	0.856577i	$-0.327414\pi$
$307$	18.0827	1.03204	0.516018	+	0.856577i	$0.327414\pi$
$308$	−15.6398	−0.891161
$309$	0	0
$310$	0	0
$311$	−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	$-0.425671\pi$
$313$	8.18754	0.462787	0.231394	+	0.972860i	$0.425671\pi$
$314$	−5.21329	−0.294203
$315$	0	0
$316$	17.9924	1.01215
$317$	−19.3804	−1.08851	−0.544257	−	0.838919i	$-0.683188\pi$
$317$	−19.3804	−1.08851	−0.544257	+	0.838919i	$0.683188\pi$
$318$	0	0
$319$	11.9443	0.668753
$320$	0	0
$321$	0	0
$322$	−62.8235	−3.50102
$323$	−20.6931	−1.15140
$324$	0	0
$325$	0	0
$326$	−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
$335$	0	0
$336$	0	0
$337$	2.16889	0.118147	0.0590736	−	0.998254i	$-0.481185\pi$
$337$	2.16889	0.118147	0.0590736	+	0.998254i	$0.481185\pi$
$338$	68.4605	3.72376
$339$	0	0
$340$	0	0
$341$	−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	$-0.241617\pi$
$347$	27.0285	1.45096	0.725482	+	0.688241i	$0.241617\pi$
$348$	0	0
$349$	−14.7088	−0.787343	−0.393672	−	0.919251i	$-0.628795\pi$
$349$	−14.7088	−0.787343	−0.393672	+	0.919251i	$0.628795\pi$
$350$	0	0
$351$	0	0
$352$	−18.4794	−0.984955
$353$	23.0315	1.22584	0.612920	−	0.790145i	$-0.289995\pi$
$353$	23.0315	1.22584	0.612920	+	0.790145i	$0.289995\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	1.60896	0.0839870	0.0419935	−	0.999118i	$-0.486629\pi$
$367$	1.60896	0.0839870	0.0419935	+	0.999118i	$0.486629\pi$
$368$	−41.0334	−2.13901
$369$	0	0
$370$	0	0
$371$	28.4934	1.47930
$372$	0	0
$373$	−0.0696997	−0.00360891	−0.00180446	−	0.999998i	$-0.500574\pi$
$373$	−0.0696997	−0.00360891	−0.00180446	+	0.999998i	$0.500574\pi$
$374$	−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	$0.223026\pi$
$380$	0	0
$381$	0	0
$382$	23.3361	1.19398
$383$	26.6737	1.36296	0.681481	−	0.731836i	$-0.261336\pi$
$383$	26.6737	1.36296	0.681481	+	0.731836i	$0.261336\pi$
$384$	0	0
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	6.73760	0.338150	0.169075	−	0.985603i	$-0.445922\pi$
$397$	6.73760	0.338150	0.169075	+	0.985603i	$0.445922\pi$
$398$	−17.5347	−0.878937
$399$	0	0
$400$	0	0
$401$	−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	$0.632671\pi$
$410$	0	0
$411$	0	0
$412$	17.5700	0.865613
$413$	16.4221	0.808076
$414$	0	0
$415$	0	0
$416$	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
$425$	0	0
$426$	0	0
$427$	60.4312	2.92447
$428$	17.2934	0.835909
$429$	0	0
$430$	0	0
$431$	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	$-0.734632\pi$
$433$	−27.9733	−1.34431	−0.672156	+	0.740409i	$0.734632\pi$
$434$	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	$0.269623\pi$
$440$	0	0
$441$	0	0
$442$	66.4962	3.16290
$443$	−2.57508	−0.122346	−0.0611729	−	0.998127i	$-0.519484\pi$
$443$	−2.57508	−0.122346	−0.0611729	+	0.998127i	$0.519484\pi$
$444$	0	0
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	5.63797	0.263733	0.131866	−	0.991267i	$-0.457903\pi$
$457$	5.63797	0.263733	0.131866	+	0.991267i	$0.457903\pi$
$458$	7.74836	0.362057
$459$	0	0
$460$	0	0
$461$	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	$-0.473180\pi$
$463$	3.62166	0.168313	0.0841563	+	0.996453i	$0.473180\pi$
$464$	−19.6396	−0.911746
$465$	0	0
$466$	23.2697	1.07795
$467$	23.0507	1.06666	0.533330	−	0.845907i	$-0.320940\pi$
$467$	23.0507	1.06666	0.533330	+	0.845907i	$0.320940\pi$
$468$	0	0
$469$	−18.5016	−0.854326
$470$	0	0
$471$	0	0
$472$	5.36354	0.246877
$473$	28.1263	1.29325
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	−16.4607	−0.745907	−0.372954	−	0.927850i	$-0.621655\pi$
$487$	−16.4607	−0.745907	−0.372954	+	0.927850i	$0.621655\pi$
$488$	19.7372	0.893459
$489$	0	0
$490$	0	0
$491$	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	$0.450521\pi$
$500$	0	0
$501$	0	0
$502$	−44.2996	−1.97719
$503$	6.87997	0.306763	0.153381	−	0.988167i	$-0.450984\pi$
$503$	6.87997	0.306763	0.153381	+	0.988167i	$0.450984\pi$
$504$	0	0
$505$	0	0
$506$	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
$515$	0	0
$516$	0	0
$517$	−5.87131	−0.258220
$518$	−28.8103	−1.26585
$519$	0	0
$520$	0	0
$521$	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	$-0.272671\pi$
$523$	29.9583	1.30999	0.654993	+	0.755635i	$0.272671\pi$
$524$	−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
$530$	0	0
$531$	0	0
$532$	−20.8045	−0.901989
$533$	−15.6379	−0.677352
$534$	0	0
$535$	0	0
$536$	−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
$545$	0	0
$546$	0	0
$547$	6.88260	0.294279	0.147139	−	0.989116i	$-0.452993\pi$
$547$	6.88260	0.294279	0.147139	+	0.989116i	$0.452993\pi$
$548$	1.45014	0.0619468
$549$	0	0
$550$	0	0
$551$	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	$-0.631729\pi$
$557$	−18.9810	−0.804252	−0.402126	+	0.915584i	$0.631729\pi$
$558$	0	0
$559$	−66.8353	−2.82683
$560$	0	0
$561$	0	0
$562$	2.22118	0.0936947
$563$	−17.6428	−0.743555	−0.371777	−	0.928322i	$-0.621252\pi$
$563$	−17.6428	−0.743555	−0.371777	+	0.928322i	$0.621252\pi$
$564$	0	0
$565$	0	0
$566$	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
$575$	0	0
$576$	0	0
$577$	−31.1585	−1.29714	−0.648572	−	0.761153i	$-0.724634\pi$
$577$	−31.1585	−1.29714	−0.648572	+	0.761153i	$0.724634\pi$
$578$	17.5621	0.730489
$579$	0	0
$580$	0	0
$581$	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	$-0.348568\pi$
$587$	22.1927	0.915989	0.457994	+	0.888955i	$0.348568\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	−18.8175	−0.773396
$593$	−15.3601	−0.630765	−0.315383	−	0.948965i	$-0.602133\pi$
$593$	−15.3601	−0.630765	−0.315383	+	0.948965i	$0.602133\pi$
$594$	0	0
$595$	0	0
$596$	−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
$605$	0	0
$606$	0	0
$607$	−5.24560	−0.212912	−0.106456	−	0.994317i	$-0.533950\pi$
$607$	−5.24560	−0.212912	−0.106456	+	0.994317i	$0.533950\pi$
$608$	−24.5818	−0.996922
$609$	0	0
$610$	0	0
$611$	13.9518	0.564428
$612$	0	0
$613$	12.8434	0.518739	0.259369	−	0.965778i	$-0.416485\pi$
$613$	12.8434	0.518739	0.259369	+	0.965778i	$0.416485\pi$
$614$	32.5287	1.31275
$615$	0	0
$616$	17.3915	0.700723
$617$	32.5677	1.31113	0.655564	−	0.755140i	$-0.272431\pi$
$617$	32.5677	1.31113	0.655564	+	0.755140i	$0.272431\pi$
$618$	0	0
$619$	−35.8991	−1.44291	−0.721453	−	0.692464i	$-0.756525\pi$
$619$	−35.8991	−1.44291	−0.721453	+	0.692464i	$0.756525\pi$
$620$	0	0
$621$	0	0
$622$	−48.8083	−1.95703
$623$	−18.9080	−0.757533
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	−76.5158	−3.03167
$638$	21.4864	0.850655
$639$	0	0
$640$	0	0
$641$	−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	$-0.719049\pi$
$643$	−32.2100	−1.27024	−0.635119	+	0.772414i	$0.719049\pi$
$644$	−43.1646	−1.70092
$645$	0	0
$646$	−37.2244	−1.46458
$647$	−30.5140	−1.19963	−0.599814	−	0.800139i	$-0.704759\pi$
$647$	−30.5140	−1.19963	−0.599814	+	0.800139i	$0.704759\pi$
$648$	0	0
$649$	−11.7347	−0.460628
$650$	0	0
$651$	0	0
$652$	−17.7423	−0.694843
$653$	−20.4786	−0.801390	−0.400695	−	0.916211i	$-0.631231\pi$
$653$	−20.4786	−0.801390	−0.400695	+	0.916211i	$0.631231\pi$
$654$	0	0
$655$	0	0
$656$	−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
$665$	0	0
$666$	0	0
$667$	32.9653	1.27642
$668$	7.32967	0.283594
$669$	0	0
$670$	0	0
$671$	−43.1824	−1.66704
$672$	0	0
$673$	−20.7857	−0.801231	−0.400616	−	0.916246i	$-0.631204\pi$
$673$	−20.7857	−0.801231	−0.400616	+	0.916246i	$0.631204\pi$
$674$	3.90158	0.150283
$675$	0	0
$676$	47.0377	1.80914
$677$	30.6432	1.17771	0.588857	−	0.808237i	$-0.299578\pi$
$677$	30.6432	1.17771	0.588857	+	0.808237i	$0.299578\pi$
$678$	0	0
$679$	4.23762	0.162625
$680$	0	0
$681$	0	0
$682$	−5.40925	−0.207131
$683$	−6.24585	−0.238991	−0.119495	−	0.992835i	$-0.538128\pi$
$683$	−6.24585	−0.238991	−0.119495	+	0.992835i	$0.538128\pi$
$684$	0	0
$685$	0	0
$686$	−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
$695$	0	0
$696$	0	0
$697$	−11.3218	−0.428843
$698$	−26.4593	−1.00150
$699$	0	0
$700$	0	0
$701$	−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	$0.358955\pi$
$710$	0	0
$711$	0	0
$712$	−6.17546	−0.231435
$713$	−8.29910	−0.310804
$714$	0	0
$715$	0	0
$716$	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
$725$	0	0
$726$	0	0
$727$	13.6566	0.506496	0.253248	−	0.967401i	$-0.418501\pi$
$727$	13.6566	0.506496	0.253248	+	0.967401i	$0.418501\pi$
$728$	−41.3267	−1.53167
$729$	0	0
$730$	0	0
$731$	−48.3886	−1.78972
$732$	0	0
$733$	7.14861	0.264040	0.132020	−	0.991247i	$-0.457854\pi$
$733$	7.14861	0.264040	0.132020	+	0.991247i	$0.457854\pi$
$734$	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	$0.501299\pi$
$740$	0	0
$741$	0	0
$742$	51.2562	1.88168
$743$	38.5285	1.41347	0.706737	−	0.707477i	$-0.250167\pi$
$743$	38.5285	1.41347	0.706737	+	0.707477i	$0.250167\pi$
$744$	0	0
$745$	0	0
$746$	−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
$755$	0	0
$756$	0	0
$757$	−40.9532	−1.48847	−0.744234	−	0.667919i	$-0.767185\pi$
$757$	−40.9532	−1.48847	−0.744234	+	0.667919i	$0.767185\pi$
$758$	53.5405	1.94468
$759$	0	0
$760$	0	0
$761$	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	$0.116967\pi$
$770$	0	0
$771$	0	0
$772$	7.06551	0.254293
$773$	−34.8383	−1.25305	−0.626523	−	0.779403i	$-0.715523\pi$
$773$	−34.8383	−1.25305	−0.626523	+	0.779403i	$0.715523\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	−42.1782	−1.50349	−0.751745	−	0.659454i	$-0.770788\pi$
$787$	−42.1782	−1.50349	−0.751745	+	0.659454i	$0.770788\pi$
$788$	−29.1248	−1.03753
$789$	0	0
$790$	0	0
$791$	−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	$-0.551055\pi$
$797$	−9.01743	−0.319414	−0.159707	+	0.987164i	$0.551055\pi$
$798$	0	0
$799$	10.1010	0.357349
$800$	0	0
$801$	0	0
$802$	−39.0911	−1.38035
$803$	22.2207	0.784151
$804$	0	0
$805$	0	0
$806$	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
$815$	0	0
$816$	0	0
$817$	37.4143	1.30896
$818$	−29.4559	−1.02990
$819$	0	0
$820$	0	0
$821$	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	$-0.441598\pi$
$823$	10.4682	0.364898	0.182449	+	0.983215i	$0.441598\pi$
$824$	−19.5379	−0.680634
$825$	0	0
$826$	29.5413	1.02787
$827$	0.407155	0.0141582	0.00707909	−	0.999975i	$-0.497747\pi$
$827$	0.407155	0.0141582	0.00707909	+	0.999975i	$0.497747\pi$
$828$	0	0
$829$	3.66464	0.127278	0.0636391	−	0.997973i	$-0.479729\pi$
$829$	3.66464	0.127278	0.0636391	+	0.997973i	$0.479729\pi$
$830$	0	0
$831$	0	0
$832$	8.33351	0.288912
$833$	−55.3973	−1.91940
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	8.23908	0.283098
$848$	33.4782	1.14965
$849$	0	0
$850$	0	0
$851$	31.5855	1.08274
$852$	0	0
$853$	20.0074	0.685042	0.342521	−	0.939510i	$-0.388719\pi$
$853$	20.0074	0.685042	0.342521	+	0.939510i	$0.388719\pi$
$854$	108.708	3.71992
$855$	0	0
$856$	−19.2303	−0.657278
$857$	9.62823	0.328894	0.164447	−	0.986386i	$-0.447416\pi$
$857$	9.62823	0.328894	0.164447	+	0.986386i	$0.447416\pi$
$858$	0	0
$859$	0.804479	0.0274485	0.0137242	−	0.999906i	$-0.495631\pi$
$859$	0.804479	0.0274485	0.0137242	+	0.999906i	$0.495631\pi$
$860$	0	0
$861$	0	0
$862$	22.5696	0.768723
$863$	17.8746	0.608459	0.304230	−	0.952599i	$-0.401601\pi$
$863$	17.8746	0.608459	0.304230	+	0.952599i	$0.401601\pi$
$864$	0	0
$865$	0	0
$866$	−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
$875$	0	0
$876$	0	0
$877$	24.0041	0.810562	0.405281	−	0.914192i	$-0.367174\pi$
$877$	24.0041	0.810562	0.405281	+	0.914192i	$0.367174\pi$
$878$	49.9176	1.68464
$879$	0	0
$880$	0	0
$881$	−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	$-0.446588\pi$
$883$	9.92567	0.334025	0.167013	+	0.985955i	$0.446588\pi$
$884$	45.6880	1.53666
$885$	0	0
$886$	−4.63226	−0.155624
$887$	24.4352	0.820453	0.410227	−	0.911984i	$-0.365450\pi$
$887$	24.4352	0.820453	0.410227	+	0.911984i	$0.365450\pi$
$888$	0	0
$889$	−23.0875	−0.774329
$890$	0	0
$891$	0	0
$892$	7.26897	0.243383
$893$	−7.81017	−0.261357
$894$	0	0
$895$	0	0
$896$	−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
$905$	0	0
$906$	0	0
$907$	−8.66887	−0.287845	−0.143923	−	0.989589i	$-0.545972\pi$
$907$	−8.66887	−0.287845	−0.143923	+	0.989589i	$0.545972\pi$
$908$	−28.6761	−0.951651
$909$	0	0
$910$	0	0
$911$	−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	$0.785960\pi$
$920$	0	0
$921$	0	0
$922$	66.7348	2.19779
$923$	−32.2061	−1.06008
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	25.5888	0.835948	0.417974	−	0.908459i	$-0.362740\pi$
$937$	25.5888	0.835948	0.417974	+	0.908459i	$0.362740\pi$
$938$	−33.2822	−1.08670
$939$	0	0
$940$	0	0
$941$	−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	$-0.777630\pi$
$947$	−47.1292	−1.53149	−0.765746	+	0.643143i	$0.777630\pi$
$948$	0	0
$949$	−52.8022	−1.71403
$950$	0	0
$951$	0	0
$952$	−29.9204	−0.969726
$953$	39.0091	1.26363	0.631814	−	0.775120i	$-0.282311\pi$
$953$	39.0091	1.26363	0.631814	+	0.775120i	$0.282311\pi$
$954$	0	0
$955$	0	0
$956$	−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
$965$	0	0
$966$	0	0
$967$	61.6585	1.98280	0.991401	−	0.130857i	$-0.0417730\pi$
$967$	61.6585	1.98280	0.991401	+	0.130857i	$0.0417730\pi$
$968$	2.69093	0.0864898
$969$	0	0
$970$	0	0
$971$	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	$-0.273958\pi$
$977$	40.7549	1.30387	0.651933	+	0.758277i	$0.273958\pi$
$978$	0	0
$979$	13.5111	0.431817
$980$	0	0
$981$	0	0
$982$	62.1927	1.98465
$983$	31.8349	1.01538	0.507688	−	0.861541i	$-0.330500\pi$
$983$	31.8349	1.01538	0.507688	+	0.861541i	$0.330500\pi$
$984$	0	0
$985$	0	0
$986$	−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
$995$	0	0
$996$	0	0
$997$	−26.6001	−0.842435	−0.421217	−	0.906960i	$-0.638397\pi$
$997$	−26.6001	−0.842435	−0.421217	+	0.906960i	$0.638397\pi$
$998$	12.4424	0.393856
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.h.1.3	✓	4	15.14	odd	2
1395.2.a.k.1.2		4	5.4	even	2
2325.2.a.v.1.2		4	3.2	odd	2
2325.2.c.p.1024.3		8	15.2	even	4
2325.2.c.p.1024.6		8	15.8	even	4
6975.2.a.bo.1.3		4	1.1	even	1	trivial
7440.2.a.bz.1.1		4	60.59	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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

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

Introduction

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