LMFDB - Embedded newform 3675.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} -2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} -2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} -5.47214 q^{11} +4.85410 q^{12} +0.763932 q^{13} +9.85410 q^{16} -7.70820 q^{17} -2.61803 q^{18} +3.23607 q^{19} +14.3262 q^{22} -5.00000 q^{23} -7.47214 q^{24} -2.00000 q^{26} +1.00000 q^{27} +4.70820 q^{29} -4.47214 q^{31} -10.8541 q^{32} -5.47214 q^{33} +20.1803 q^{34} +4.85410 q^{36} -5.47214 q^{37} -8.47214 q^{38} +0.763932 q^{39} +8.00000 q^{41} -8.23607 q^{43} -26.5623 q^{44} +13.0902 q^{46} +7.23607 q^{47} +9.85410 q^{48} -7.70820 q^{51} +3.70820 q^{52} -0.472136 q^{53} -2.61803 q^{54} +3.23607 q^{57} -12.3262 q^{58} +0.763932 q^{59} +15.4164 q^{61} +11.7082 q^{62} +8.70820 q^{64} +14.3262 q^{66} +2.70820 q^{67} -37.4164 q^{68} -5.00000 q^{69} -0.527864 q^{71} -7.47214 q^{72} +1.23607 q^{73} +14.3262 q^{74} +15.7082 q^{76} -2.00000 q^{78} +1.76393 q^{79} +1.00000 q^{81} -20.9443 q^{82} +12.4721 q^{83} +21.5623 q^{86} +4.70820 q^{87} +40.8885 q^{88} +5.70820 q^{89} -24.2705 q^{92} -4.47214 q^{93} -18.9443 q^{94} -10.8541 q^{96} +12.4721 q^{97} -5.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} + 3 q^{12} + 6 q^{13} + 13 q^{16} - 2 q^{17} - 3 q^{18} + 2 q^{19} + 13 q^{22} - 10 q^{23} - 6 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{29} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 2 q^{37} - 8 q^{38} + 6 q^{39} + 16 q^{41} - 12 q^{43} - 33 q^{44} + 15 q^{46} + 10 q^{47} + 13 q^{48} - 2 q^{51} - 6 q^{52} + 8 q^{53} - 3 q^{54} + 2 q^{57} - 9 q^{58} + 6 q^{59} + 4 q^{61} + 10 q^{62} + 4 q^{64} + 13 q^{66} - 8 q^{67} - 48 q^{68} - 10 q^{69} - 10 q^{71} - 6 q^{72} - 2 q^{73} + 13 q^{74} + 18 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 24 q^{82} + 16 q^{83} + 23 q^{86} - 4 q^{87} + 46 q^{88} - 2 q^{89} - 15 q^{92} - 20 q^{94} - 15 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 + 3 * q^12 + 6 * q^13 + 13 * q^16 - 2 * q^17 - 3 * q^18 + 2 * q^19 + 13 * q^22 - 10 * q^23 - 6 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^29 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 2 * q^37 - 8 * q^38 + 6 * q^39 + 16 * q^41 - 12 * q^43 - 33 * q^44 + 15 * q^46 + 10 * q^47 + 13 * q^48 - 2 * q^51 - 6 * q^52 + 8 * q^53 - 3 * q^54 + 2 * q^57 - 9 * q^58 + 6 * q^59 + 4 * q^61 + 10 * q^62 + 4 * q^64 + 13 * q^66 - 8 * q^67 - 48 * q^68 - 10 * q^69 - 10 * q^71 - 6 * q^72 - 2 * q^73 + 13 * q^74 + 18 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 24 * q^82 + 16 * q^83 + 23 * q^86 - 4 * q^87 + 46 * q^88 - 2 * q^89 - 15 * q^92 - 20 * q^94 - 15 * q^96 + 16 * q^97 - 2 * 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 $

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	−2.61803	−1.06881
$7$	0	0
$7$	0	0
$8$	−7.47214	−2.64180
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	4.85410	1.40126
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	0	0
$15$	0	0
$16$	9.85410	2.46353
$17$	−7.70820	−1.86951	−0.934757	−	0.355288i	$-0.884383\pi$
$17$	−7.70820	−1.86951	−0.934757	+	0.355288i	$0.884383\pi$
$18$	−2.61803	−0.617077
$19$	3.23607	0.742405	0.371202	−	0.928552i	$-0.378946\pi$
$19$	3.23607	0.742405	0.371202	+	0.928552i	$0.378946\pi$
$20$	0	0
$21$	0	0
$22$	14.3262	3.05436
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−7.47214	−1.52524
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	4.70820	0.874292	0.437146	−	0.899391i	$-0.355989\pi$
$29$	4.70820	0.874292	0.437146	+	0.899391i	$0.355989\pi$
$30$	0	0
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	−10.8541	−1.91875
$33$	−5.47214	−0.952577
$34$	20.1803	3.46090
$35$	0	0
$36$	4.85410	0.809017
$37$	−5.47214	−0.899614	−0.449807	−	0.893126i	$-0.648507\pi$
$37$	−5.47214	−0.899614	−0.449807	+	0.893126i	$0.648507\pi$
$38$	−8.47214	−1.37436
$39$	0.763932	0.122327
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−8.23607	−1.25599	−0.627994	−	0.778218i	$-0.716124\pi$
$43$	−8.23607	−1.25599	−0.627994	+	0.778218i	$0.716124\pi$
$44$	−26.5623	−4.00442
$45$	0	0
$46$	13.0902	1.93004
$47$	7.23607	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$47$	7.23607	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$48$	9.85410	1.42232
$49$	0	0
$50$	0	0
$51$	−7.70820	−1.07936
$52$	3.70820	0.514235
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	−2.61803	−0.356269
$55$	0	0
$56$	0	0
$57$	3.23607	0.428628
$58$	−12.3262	−1.61851
$59$	0.763932	0.0994555	0.0497277	−	0.998763i	$-0.484165\pi$
$59$	0.763932	0.0994555	0.0497277	+	0.998763i	$0.484165\pi$
$60$	0	0
$61$	15.4164	1.97387	0.986934	−	0.161123i	$-0.0515115\pi$
$61$	15.4164	1.97387	0.986934	+	0.161123i	$0.0515115\pi$
$62$	11.7082	1.48694
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	14.3262	1.76344
$67$	2.70820	0.330860	0.165430	−	0.986222i	$-0.447099\pi$
$67$	2.70820	0.330860	0.165430	+	0.986222i	$0.447099\pi$
$68$	−37.4164	−4.53741
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−0.527864	−0.0626459	−0.0313230	−	0.999509i	$-0.509972\pi$
$71$	−0.527864	−0.0626459	−0.0313230	+	0.999509i	$0.509972\pi$
$72$	−7.47214	−0.880600
$73$	1.23607	0.144671	0.0723354	−	0.997380i	$-0.476955\pi$
$73$	1.23607	0.144671	0.0723354	+	0.997380i	$0.476955\pi$
$74$	14.3262	1.66539
$75$	0	0
$76$	15.7082	1.80185
$77$	0	0
$78$	−2.00000	−0.226455
$79$	1.76393	0.198458	0.0992289	−	0.995065i	$-0.468362\pi$
$79$	1.76393	0.198458	0.0992289	+	0.995065i	$0.468362\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−20.9443	−2.31291
$83$	12.4721	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$83$	12.4721	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$84$	0	0
$85$	0	0
$86$	21.5623	2.32512
$87$	4.70820	0.504772
$88$	40.8885	4.35873
$89$	5.70820	0.605068	0.302534	−	0.953139i	$-0.402167\pi$
$89$	5.70820	0.605068	0.302534	+	0.953139i	$0.402167\pi$
$90$	0	0
$91$	0	0
$92$	−24.2705	−2.53038
$93$	−4.47214	−0.463739
$94$	−18.9443	−1.95395
$95$	0	0
$96$	−10.8541	−1.10779
$97$	12.4721	1.26635	0.633177	−	0.774007i	$-0.281751\pi$
$97$	12.4721	1.26635	0.633177	+	0.774007i	$0.281751\pi$
$98$	0	0
$99$	−5.47214	−0.549970
$100$	0	0
$101$	4.29180	0.427050	0.213525	−	0.976938i	$-0.431506\pi$
$101$	4.29180	0.427050	0.213525	+	0.976938i	$0.431506\pi$
$102$	20.1803	1.99815
$103$	9.70820	0.956578	0.478289	−	0.878203i	$-0.341257\pi$
$103$	9.70820	0.956578	0.478289	+	0.878203i	$0.341257\pi$
$104$	−5.70820	−0.559735
$105$	0	0
$106$	1.23607	0.120058
$107$	4.94427	0.477981	0.238990	−	0.971022i	$-0.423184\pi$
$107$	4.94427	0.477981	0.238990	+	0.971022i	$0.423184\pi$
$108$	4.85410	0.467086
$109$	−6.41641	−0.614580	−0.307290	−	0.951616i	$-0.599422\pi$
$109$	−6.41641	−0.614580	−0.307290	+	0.951616i	$0.599422\pi$
$110$	0	0
$111$	−5.47214	−0.519392
$112$	0	0
$113$	16.2361	1.52736	0.763680	−	0.645594i	$-0.223390\pi$
$113$	16.2361	1.52736	0.763680	+	0.645594i	$0.223390\pi$
$114$	−8.47214	−0.793488
$115$	0	0
$116$	22.8541	2.12195
$117$	0.763932	0.0706255
$118$	−2.00000	−0.184115
$119$	0	0
$120$	0	0
$121$	18.9443	1.72221
$122$	−40.3607	−3.65408
$123$	8.00000	0.721336
$124$	−21.7082	−1.94945
$125$	0	0
$126$	0	0
$127$	12.2361	1.08578	0.542888	−	0.839805i	$-0.317331\pi$
$127$	12.2361	1.08578	0.542888	+	0.839805i	$0.317331\pi$
$128$	−1.09017	−0.0963583
$129$	−8.23607	−0.725145
$130$	0	0
$131$	4.29180	0.374976	0.187488	−	0.982267i	$-0.439965\pi$
$131$	4.29180	0.374976	0.187488	+	0.982267i	$0.439965\pi$
$132$	−26.5623	−2.31195
$133$	0	0
$134$	−7.09017	−0.612497
$135$	0	0
$136$	57.5967	4.93888
$137$	0.472136	0.0403373	0.0201686	−	0.999797i	$-0.493580\pi$
$137$	0.472136	0.0403373	0.0201686	+	0.999797i	$0.493580\pi$
$138$	13.0902	1.11431
$139$	−3.70820	−0.314526	−0.157263	−	0.987557i	$-0.550267\pi$
$139$	−3.70820	−0.314526	−0.157263	+	0.987557i	$0.550267\pi$
$140$	0	0
$141$	7.23607	0.609387
$142$	1.38197	0.115972
$143$	−4.18034	−0.349578
$144$	9.85410	0.821175
$145$	0	0
$146$	−3.23607	−0.267819
$147$	0	0
$148$	−26.5623	−2.18341
$149$	8.23607	0.674725	0.337362	−	0.941375i	$-0.390465\pi$
$149$	8.23607	0.674725	0.337362	+	0.941375i	$0.390465\pi$
$150$	0	0
$151$	1.29180	0.105125	0.0525624	−	0.998618i	$-0.483261\pi$
$151$	1.29180	0.105125	0.0525624	+	0.998618i	$0.483261\pi$
$152$	−24.1803	−1.96128
$153$	−7.70820	−0.623171
$154$	0	0
$155$	0	0
$156$	3.70820	0.296894
$157$	16.9443	1.35230	0.676150	−	0.736764i	$-0.263647\pi$
$157$	16.9443	1.35230	0.676150	+	0.736764i	$0.263647\pi$
$158$	−4.61803	−0.367391
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	0	0
$162$	−2.61803	−0.205692
$163$	−18.4721	−1.44685	−0.723425	−	0.690403i	$-0.757433\pi$
$163$	−18.4721	−1.44685	−0.723425	+	0.690403i	$0.757433\pi$
$164$	38.8328	3.03233
$165$	0	0
$166$	−32.6525	−2.53432
$167$	−6.29180	−0.486874	−0.243437	−	0.969917i	$-0.578275\pi$
$167$	−6.29180	−0.486874	−0.243437	+	0.969917i	$0.578275\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	3.23607	0.247468
$172$	−39.9787	−3.04835
$173$	−3.81966	−0.290403	−0.145202	−	0.989402i	$-0.546383\pi$
$173$	−3.81966	−0.290403	−0.145202	+	0.989402i	$0.546383\pi$
$174$	−12.3262	−0.934450
$175$	0	0
$176$	−53.9230	−4.06460
$177$	0.763932	0.0574206
$178$	−14.9443	−1.12012
$179$	4.94427	0.369552	0.184776	−	0.982781i	$-0.440844\pi$
$179$	4.94427	0.369552	0.184776	+	0.982781i	$0.440844\pi$
$180$	0	0
$181$	−4.65248	−0.345816	−0.172908	−	0.984938i	$-0.555316\pi$
$181$	−4.65248	−0.345816	−0.172908	+	0.984938i	$0.555316\pi$
$182$	0	0
$183$	15.4164	1.13961
$184$	37.3607	2.75427
$185$	0	0
$186$	11.7082	0.858487
$187$	42.1803	3.08453
$188$	35.1246	2.56173
$189$	0	0
$190$	0	0
$191$	−11.4164	−0.826062	−0.413031	−	0.910717i	$-0.635530\pi$
$191$	−11.4164	−0.826062	−0.413031	+	0.910717i	$0.635530\pi$
$192$	8.70820	0.628460
$193$	−0.416408	−0.0299737	−0.0149868	−	0.999888i	$-0.504771\pi$
$193$	−0.416408	−0.0299737	−0.0149868	+	0.999888i	$0.504771\pi$
$194$	−32.6525	−2.34431
$195$	0	0
$196$	0	0
$197$	5.76393	0.410663	0.205332	−	0.978692i	$-0.434173\pi$
$197$	5.76393	0.410663	0.205332	+	0.978692i	$0.434173\pi$
$198$	14.3262	1.01812
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	2.70820	0.191022
$202$	−11.2361	−0.790567
$203$	0	0
$204$	−37.4164	−2.61967
$205$	0	0
$206$	−25.4164	−1.77085
$207$	−5.00000	−0.347524
$208$	7.52786	0.521963
$209$	−17.7082	−1.22490
$210$	0	0
$211$	12.9443	0.891120	0.445560	−	0.895252i	$-0.353005\pi$
$211$	12.9443	0.891120	0.445560	+	0.895252i	$0.353005\pi$
$212$	−2.29180	−0.157401
$213$	−0.527864	−0.0361686
$214$	−12.9443	−0.884852
$215$	0	0
$216$	−7.47214	−0.508414
$217$	0	0
$218$	16.7984	1.13773
$219$	1.23607	0.0835257
$220$	0	0
$221$	−5.88854	−0.396106
$222$	14.3262	0.961514
$223$	−1.23607	−0.0827732	−0.0413866	−	0.999143i	$-0.513178\pi$
$223$	−1.23607	−0.0827732	−0.0413866	+	0.999143i	$0.513178\pi$
$224$	0	0
$225$	0	0
$226$	−42.5066	−2.82750
$227$	−4.76393	−0.316193	−0.158097	−	0.987424i	$-0.550536\pi$
$227$	−4.76393	−0.316193	−0.158097	+	0.987424i	$0.550536\pi$
$228$	15.7082	1.04030
$229$	−22.3607	−1.47764	−0.738818	−	0.673905i	$-0.764616\pi$
$229$	−22.3607	−1.47764	−0.738818	+	0.673905i	$0.764616\pi$
$230$	0	0
$231$	0	0
$232$	−35.1803	−2.30970
$233$	−7.29180	−0.477701	−0.238851	−	0.971056i	$-0.576771\pi$
$233$	−7.29180	−0.477701	−0.238851	+	0.971056i	$0.576771\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	3.70820	0.241384
$237$	1.76393	0.114580
$238$	0	0
$239$	−28.3607	−1.83450	−0.917250	−	0.398312i	$-0.869596\pi$
$239$	−28.3607	−1.83450	−0.917250	+	0.398312i	$0.869596\pi$
$240$	0	0
$241$	−19.2361	−1.23910	−0.619552	−	0.784956i	$-0.712686\pi$
$241$	−19.2361	−1.23910	−0.619552	+	0.784956i	$0.712686\pi$
$242$	−49.5967	−3.18820
$243$	1.00000	0.0641500
$244$	74.8328	4.79068
$245$	0	0
$246$	−20.9443	−1.33536
$247$	2.47214	0.157298
$248$	33.4164	2.12194
$249$	12.4721	0.790390
$250$	0	0
$251$	−2.76393	−0.174458	−0.0872289	−	0.996188i	$-0.527801\pi$
$251$	−2.76393	−0.174458	−0.0872289	+	0.996188i	$0.527801\pi$
$252$	0	0
$253$	27.3607	1.72015
$254$	−32.0344	−2.01002
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−8.47214	−0.528477	−0.264239	−	0.964457i	$-0.585121\pi$
$257$	−8.47214	−0.528477	−0.264239	+	0.964457i	$0.585121\pi$
$258$	21.5623	1.34241
$259$	0	0
$260$	0	0
$261$	4.70820	0.291431
$262$	−11.2361	−0.694167
$263$	−2.05573	−0.126762	−0.0633808	−	0.997989i	$-0.520188\pi$
$263$	−2.05573	−0.126762	−0.0633808	+	0.997989i	$0.520188\pi$
$264$	40.8885	2.51652
$265$	0	0
$266$	0	0
$267$	5.70820	0.349336
$268$	13.1459	0.803014
$269$	28.4721	1.73598	0.867988	−	0.496584i	$-0.165413\pi$
$269$	28.4721	1.73598	0.867988	+	0.496584i	$0.165413\pi$
$270$	0	0
$271$	−23.2361	−1.41149	−0.705745	−	0.708466i	$-0.749388\pi$
$271$	−23.2361	−1.41149	−0.705745	+	0.708466i	$0.749388\pi$
$272$	−75.9574	−4.60560
$273$	0	0
$274$	−1.23607	−0.0746736
$275$	0	0
$276$	−24.2705	−1.46091
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	9.70820	0.582259
$279$	−4.47214	−0.267740
$280$	0	0
$281$	13.6525	0.814438	0.407219	−	0.913330i	$-0.366499\pi$
$281$	13.6525	0.814438	0.407219	+	0.913330i	$0.366499\pi$
$282$	−18.9443	−1.12811
$283$	−13.4164	−0.797523	−0.398761	−	0.917055i	$-0.630560\pi$
$283$	−13.4164	−0.797523	−0.398761	+	0.917055i	$0.630560\pi$
$284$	−2.56231	−0.152045
$285$	0	0
$286$	10.9443	0.647148
$287$	0	0
$288$	−10.8541	−0.639584
$289$	42.4164	2.49508
$290$	0	0
$291$	12.4721	0.731130
$292$	6.00000	0.351123
$293$	−26.6525	−1.55705	−0.778527	−	0.627611i	$-0.784033\pi$
$293$	−26.6525	−1.55705	−0.778527	+	0.627611i	$0.784033\pi$
$294$	0	0
$295$	0	0
$296$	40.8885	2.37660
$297$	−5.47214	−0.317526
$298$	−21.5623	−1.24907
$299$	−3.81966	−0.220897
$300$	0	0
$301$	0	0
$302$	−3.38197	−0.194610
$303$	4.29180	0.246557
$304$	31.8885	1.82893
$305$	0	0
$306$	20.1803	1.15363
$307$	24.6525	1.40699	0.703496	−	0.710700i	$-0.251621\pi$
$307$	24.6525	1.40699	0.703496	+	0.710700i	$0.251621\pi$
$308$	0	0
$309$	9.70820	0.552280
$310$	0	0
$311$	−13.2361	−0.750549	−0.375274	−	0.926914i	$-0.622451\pi$
$311$	−13.2361	−0.750549	−0.375274	+	0.926914i	$0.622451\pi$
$312$	−5.70820	−0.323163
$313$	−5.70820	−0.322647	−0.161323	−	0.986902i	$-0.551576\pi$
$313$	−5.70820	−0.322647	−0.161323	+	0.986902i	$0.551576\pi$
$314$	−44.3607	−2.50342
$315$	0	0
$316$	8.56231	0.481667
$317$	−17.6525	−0.991462	−0.495731	−	0.868476i	$-0.665100\pi$
$317$	−17.6525	−0.991462	−0.495731	+	0.868476i	$0.665100\pi$
$318$	1.23607	0.0693153
$319$	−25.7639	−1.44250
$320$	0	0
$321$	4.94427	0.275962
$322$	0	0
$323$	−24.9443	−1.38794
$324$	4.85410	0.269672
$325$	0	0
$326$	48.3607	2.67845
$327$	−6.41641	−0.354828
$328$	−59.7771	−3.30064
$329$	0	0
$330$	0	0
$331$	13.7639	0.756534	0.378267	−	0.925697i	$-0.376520\pi$
$331$	13.7639	0.756534	0.378267	+	0.925697i	$0.376520\pi$
$332$	60.5410	3.32262
$333$	−5.47214	−0.299871
$334$	16.4721	0.901315
$335$	0	0
$336$	0	0
$337$	28.4721	1.55098	0.775488	−	0.631362i	$-0.217504\pi$
$337$	28.4721	1.55098	0.775488	+	0.631362i	$0.217504\pi$
$338$	32.5066	1.76812
$339$	16.2361	0.881822
$340$	0	0
$341$	24.4721	1.32524
$342$	−8.47214	−0.458121
$343$	0	0
$344$	61.5410	3.31807
$345$	0	0
$346$	10.0000	0.537603
$347$	13.0000	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$347$	13.0000	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$348$	22.8541	1.22511
$349$	−0.763932	−0.0408923	−0.0204462	−	0.999791i	$-0.506509\pi$
$349$	−0.763932	−0.0408923	−0.0204462	+	0.999791i	$0.506509\pi$
$350$	0	0
$351$	0.763932	0.0407757
$352$	59.3951	3.16577
$353$	1.05573	0.0561907	0.0280954	−	0.999605i	$-0.491056\pi$
$353$	1.05573	0.0561907	0.0280954	+	0.999605i	$0.491056\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	27.7082	1.46853
$357$	0	0
$358$	−12.9443	−0.684126
$359$	25.9443	1.36929	0.684643	−	0.728878i	$-0.259958\pi$
$359$	25.9443	1.36929	0.684643	+	0.728878i	$0.259958\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	12.1803	0.640184
$363$	18.9443	0.994316
$364$	0	0
$365$	0	0
$366$	−40.3607	−2.10969
$367$	−8.94427	−0.466887	−0.233444	−	0.972370i	$-0.574999\pi$
$367$	−8.94427	−0.466887	−0.233444	+	0.972370i	$0.574999\pi$
$368$	−49.2705	−2.56840
$369$	8.00000	0.416463
$370$	0	0
$371$	0	0
$372$	−21.7082	−1.12552
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−110.430	−5.71018
$375$	0	0
$376$	−54.0689	−2.78839
$377$	3.59675	0.185242
$378$	0	0
$379$	16.5967	0.852518	0.426259	−	0.904601i	$-0.359831\pi$
$379$	16.5967	0.852518	0.426259	+	0.904601i	$0.359831\pi$
$380$	0	0
$381$	12.2361	0.626873
$382$	29.8885	1.52923
$383$	25.1246	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$383$	25.1246	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$384$	−1.09017	−0.0556325
$385$	0	0
$386$	1.09017	0.0554882
$387$	−8.23607	−0.418663
$388$	60.5410	3.07350
$389$	−1.76393	−0.0894349	−0.0447175	−	0.999000i	$-0.514239\pi$
$389$	−1.76393	−0.0894349	−0.0447175	+	0.999000i	$0.514239\pi$
$390$	0	0
$391$	38.5410	1.94910
$392$	0	0
$393$	4.29180	0.216492
$394$	−15.0902	−0.760232
$395$	0	0
$396$	−26.5623	−1.33481
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−41.8885	−2.09968
$399$	0	0
$400$	0	0
$401$	−26.7082	−1.33374	−0.666872	−	0.745172i	$-0.732367\pi$
$401$	−26.7082	−1.33374	−0.666872	+	0.745172i	$0.732367\pi$
$402$	−7.09017	−0.353626
$403$	−3.41641	−0.170183
$404$	20.8328	1.03647
$405$	0	0
$406$	0	0
$407$	29.9443	1.48428
$408$	57.5967	2.85146
$409$	8.18034	0.404492	0.202246	−	0.979335i	$-0.435176\pi$
$409$	8.18034	0.404492	0.202246	+	0.979335i	$0.435176\pi$
$410$	0	0
$411$	0.472136	0.0232887
$412$	47.1246	2.32166
$413$	0	0
$414$	13.0902	0.643347
$415$	0	0
$416$	−8.29180	−0.406539
$417$	−3.70820	−0.181592
$418$	46.3607	2.26757
$419$	−9.05573	−0.442401	−0.221201	−	0.975228i	$-0.570998\pi$
$419$	−9.05573	−0.442401	−0.221201	+	0.975228i	$0.570998\pi$
$420$	0	0
$421$	−0.416408	−0.0202945	−0.0101472	−	0.999949i	$-0.503230\pi$
$421$	−0.416408	−0.0202945	−0.0101472	+	0.999949i	$0.503230\pi$
$422$	−33.8885	−1.64967
$423$	7.23607	0.351830
$424$	3.52786	0.171328
$425$	0	0
$426$	1.38197	0.0669565
$427$	0	0
$428$	24.0000	1.16008
$429$	−4.18034	−0.201829
$430$	0	0
$431$	33.8885	1.63235	0.816177	−	0.577802i	$-0.196089\pi$
$431$	33.8885	1.63235	0.816177	+	0.577802i	$0.196089\pi$
$432$	9.85410	0.474106
$433$	37.3050	1.79276	0.896381	−	0.443285i	$-0.146187\pi$
$433$	37.3050	1.79276	0.896381	+	0.443285i	$0.146187\pi$
$434$	0	0
$435$	0	0
$436$	−31.1459	−1.49162
$437$	−16.1803	−0.774011
$438$	−3.23607	−0.154625
$439$	11.2361	0.536268	0.268134	−	0.963382i	$-0.413593\pi$
$439$	11.2361	0.536268	0.268134	+	0.963382i	$0.413593\pi$
$440$	0	0
$441$	0	0
$442$	15.4164	0.733284
$443$	−12.5836	−0.597865	−0.298932	−	0.954274i	$-0.596630\pi$
$443$	−12.5836	−0.597865	−0.298932	+	0.954274i	$0.596630\pi$
$444$	−26.5623	−1.26059
$445$	0	0
$446$	3.23607	0.153232
$447$	8.23607	0.389553
$448$	0	0
$449$	3.29180	0.155349	0.0776747	−	0.996979i	$-0.475250\pi$
$449$	3.29180	0.155349	0.0776747	+	0.996979i	$0.475250\pi$
$450$	0	0
$451$	−43.7771	−2.06138
$452$	78.8115	3.70698
$453$	1.29180	0.0606939
$454$	12.4721	0.585346
$455$	0	0
$456$	−24.1803	−1.13235
$457$	29.4721	1.37865	0.689324	−	0.724453i	$-0.257908\pi$
$457$	29.4721	1.37865	0.689324	+	0.724453i	$0.257908\pi$
$458$	58.5410	2.73544
$459$	−7.70820	−0.359788
$460$	0	0
$461$	−10.6525	−0.496135	−0.248068	−	0.968743i	$-0.579796\pi$
$461$	−10.6525	−0.496135	−0.248068	+	0.968743i	$0.579796\pi$
$462$	0	0
$463$	−28.3607	−1.31803	−0.659016	−	0.752129i	$-0.729027\pi$
$463$	−28.3607	−1.31803	−0.659016	+	0.752129i	$0.729027\pi$
$464$	46.3951	2.15384
$465$	0	0
$466$	19.0902	0.884335
$467$	23.8885	1.10543	0.552715	−	0.833370i	$-0.313592\pi$
$467$	23.8885	1.10543	0.552715	+	0.833370i	$0.313592\pi$
$468$	3.70820	0.171412
$469$	0	0
$470$	0	0
$471$	16.9443	0.780751
$472$	−5.70820	−0.262741
$473$	45.0689	2.07227
$474$	−4.61803	−0.212113
$475$	0	0
$476$	0	0
$477$	−0.472136	−0.0216176
$478$	74.2492	3.39608
$479$	23.2361	1.06168	0.530842	−	0.847471i	$-0.321876\pi$
$479$	23.2361	1.06168	0.530842	+	0.847471i	$0.321876\pi$
$480$	0	0
$481$	−4.18034	−0.190607
$482$	50.3607	2.29387
$483$	0	0
$484$	91.9574	4.17988
$485$	0	0
$486$	−2.61803	−0.118756
$487$	6.81966	0.309028	0.154514	−	0.987991i	$-0.450619\pi$
$487$	6.81966	0.309028	0.154514	+	0.987991i	$0.450619\pi$
$488$	−115.193	−5.21456
$489$	−18.4721	−0.835339
$490$	0	0
$491$	−0.527864	−0.0238222	−0.0119111	−	0.999929i	$-0.503792\pi$
$491$	−0.527864	−0.0238222	−0.0119111	+	0.999929i	$0.503792\pi$
$492$	38.8328	1.75072
$493$	−36.2918	−1.63450
$494$	−6.47214	−0.291195
$495$	0	0
$496$	−44.0689	−1.97875
$497$	0	0
$498$	−32.6525	−1.46319
$499$	−31.7771	−1.42254	−0.711269	−	0.702920i	$-0.751879\pi$
$499$	−31.7771	−1.42254	−0.711269	+	0.702920i	$0.751879\pi$
$500$	0	0
$501$	−6.29180	−0.281097
$502$	7.23607	0.322962
$503$	17.4164	0.776559	0.388280	−	0.921542i	$-0.373069\pi$
$503$	17.4164	0.776559	0.388280	+	0.921542i	$0.373069\pi$
$504$	0	0
$505$	0	0
$506$	−71.6312	−3.18439
$507$	−12.4164	−0.551432
$508$	59.3951	2.63523
$509$	30.6525	1.35865	0.679324	−	0.733839i	$-0.262273\pi$
$509$	30.6525	1.35865	0.679324	+	0.733839i	$0.262273\pi$
$510$	0	0
$511$	0	0
$512$	40.3050	1.78124
$513$	3.23607	0.142876
$514$	22.1803	0.978333
$515$	0	0
$516$	−39.9787	−1.75996
$517$	−39.5967	−1.74146
$518$	0	0
$519$	−3.81966	−0.167664
$520$	0	0
$521$	37.7771	1.65504	0.827522	−	0.561433i	$-0.189750\pi$
$521$	37.7771	1.65504	0.827522	+	0.561433i	$0.189750\pi$
$522$	−12.3262	−0.539505
$523$	−35.7771	−1.56442	−0.782211	−	0.623013i	$-0.785908\pi$
$523$	−35.7771	−1.56442	−0.782211	+	0.623013i	$0.785908\pi$
$524$	20.8328	0.910086
$525$	0	0
$526$	5.38197	0.234665
$527$	34.4721	1.50163
$528$	−53.9230	−2.34670
$529$	2.00000	0.0869565
$530$	0	0
$531$	0.763932	0.0331518
$532$	0	0
$533$	6.11146	0.264717
$534$	−14.9443	−0.646702
$535$	0	0
$536$	−20.2361	−0.874065
$537$	4.94427	0.213361
$538$	−74.5410	−3.21369
$539$	0	0
$540$	0	0
$541$	−17.9443	−0.771485	−0.385742	−	0.922607i	$-0.626055\pi$
$541$	−17.9443	−0.771485	−0.385742	+	0.922607i	$0.626055\pi$
$542$	60.8328	2.61299
$543$	−4.65248	−0.199657
$544$	83.6656	3.58713
$545$	0	0
$546$	0	0
$547$	35.0689	1.49944	0.749719	−	0.661757i	$-0.230189\pi$
$547$	35.0689	1.49944	0.749719	+	0.661757i	$0.230189\pi$
$548$	2.29180	0.0979007
$549$	15.4164	0.657956
$550$	0	0
$551$	15.2361	0.649078
$552$	37.3607	1.59018
$553$	0	0
$554$	−41.5967	−1.76728
$555$	0	0
$556$	−18.0000	−0.763370
$557$	7.65248	0.324246	0.162123	−	0.986771i	$-0.448166\pi$
$557$	7.65248	0.324246	0.162123	+	0.986771i	$0.448166\pi$
$558$	11.7082	0.495648
$559$	−6.29180	−0.266115
$560$	0	0
$561$	42.1803	1.78086
$562$	−35.7426	−1.50771
$563$	35.3050	1.48793	0.743963	−	0.668221i	$-0.232944\pi$
$563$	35.3050	1.48793	0.743963	+	0.668221i	$0.232944\pi$
$564$	35.1246	1.47901
$565$	0	0
$566$	35.1246	1.47640
$567$	0	0
$568$	3.94427	0.165498
$569$	−12.2361	−0.512963	−0.256481	−	0.966549i	$-0.582563\pi$
$569$	−12.2361	−0.512963	−0.256481	+	0.966549i	$0.582563\pi$
$570$	0	0
$571$	−40.7082	−1.70359	−0.851793	−	0.523879i	$-0.824484\pi$
$571$	−40.7082	−1.70359	−0.851793	+	0.523879i	$0.824484\pi$
$572$	−20.2918	−0.848443
$573$	−11.4164	−0.476927
$574$	0	0
$575$	0	0
$576$	8.70820	0.362842
$577$	11.2361	0.467764	0.233882	−	0.972265i	$-0.424857\pi$
$577$	11.2361	0.467764	0.233882	+	0.972265i	$0.424857\pi$
$578$	−111.048	−4.61897
$579$	−0.416408	−0.0173053
$580$	0	0
$581$	0	0
$582$	−32.6525	−1.35349
$583$	2.58359	0.107001
$584$	−9.23607	−0.382191
$585$	0	0
$586$	69.7771	2.88246
$587$	−7.12461	−0.294064	−0.147032	−	0.989132i	$-0.546972\pi$
$587$	−7.12461	−0.294064	−0.147032	+	0.989132i	$0.546972\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	0	0
$591$	5.76393	0.237096
$592$	−53.9230	−2.21622
$593$	25.3050	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$593$	25.3050	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$594$	14.3262	0.587813
$595$	0	0
$596$	39.9787	1.63759
$597$	16.0000	0.654836
$598$	10.0000	0.408930
$599$	−45.9443	−1.87723	−0.938616	−	0.344964i	$-0.887891\pi$
$599$	−45.9443	−1.87723	−0.938616	+	0.344964i	$0.887891\pi$
$600$	0	0
$601$	28.3607	1.15686	0.578428	−	0.815733i	$-0.303666\pi$
$601$	28.3607	1.15686	0.578428	+	0.815733i	$0.303666\pi$
$602$	0	0
$603$	2.70820	0.110287
$604$	6.27051	0.255143
$605$	0	0
$606$	−11.2361	−0.456434
$607$	2.29180	0.0930211	0.0465106	−	0.998918i	$-0.485190\pi$
$607$	2.29180	0.0930211	0.0465106	+	0.998918i	$0.485190\pi$
$608$	−35.1246	−1.42449
$609$	0	0
$610$	0	0
$611$	5.52786	0.223633
$612$	−37.4164	−1.51247
$613$	−10.0557	−0.406147	−0.203074	−	0.979163i	$-0.565093\pi$
$613$	−10.0557	−0.406147	−0.203074	+	0.979163i	$0.565093\pi$
$614$	−64.5410	−2.60466
$615$	0	0
$616$	0	0
$617$	21.6525	0.871696	0.435848	−	0.900020i	$-0.356449\pi$
$617$	21.6525	0.871696	0.435848	+	0.900020i	$0.356449\pi$
$618$	−25.4164	−1.02240
$619$	−25.1246	−1.00984	−0.504922	−	0.863165i	$-0.668479\pi$
$619$	−25.1246	−1.00984	−0.504922	+	0.863165i	$0.668479\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	34.6525	1.38944
$623$	0	0
$624$	7.52786	0.301356
$625$	0	0
$626$	14.9443	0.597293
$627$	−17.7082	−0.707198
$628$	82.2492	3.28210
$629$	42.1803	1.68184
$630$	0	0
$631$	−40.1246	−1.59734	−0.798668	−	0.601772i	$-0.794462\pi$
$631$	−40.1246	−1.59734	−0.798668	+	0.601772i	$0.794462\pi$
$632$	−13.1803	−0.524286
$633$	12.9443	0.514489
$634$	46.2148	1.83542
$635$	0	0
$636$	−2.29180	−0.0908756
$637$	0	0
$638$	67.4508	2.67040
$639$	−0.527864	−0.0208820
$640$	0	0
$641$	−15.2918	−0.603990	−0.301995	−	0.953310i	$-0.597653\pi$
$641$	−15.2918	−0.603990	−0.301995	+	0.953310i	$0.597653\pi$
$642$	−12.9443	−0.510870
$643$	−44.7214	−1.76364	−0.881819	−	0.471588i	$-0.843681\pi$
$643$	−44.7214	−1.76364	−0.881819	+	0.471588i	$0.843681\pi$
$644$	0	0
$645$	0	0
$646$	65.3050	2.56939
$647$	−1.05573	−0.0415050	−0.0207525	−	0.999785i	$-0.506606\pi$
$647$	−1.05573	−0.0415050	−0.0207525	+	0.999785i	$0.506606\pi$
$648$	−7.47214	−0.293533
$649$	−4.18034	−0.164093
$650$	0	0
$651$	0	0
$652$	−89.6656	−3.51158
$653$	−25.4164	−0.994621	−0.497310	−	0.867573i	$-0.665679\pi$
$653$	−25.4164	−0.994621	−0.497310	+	0.867573i	$0.665679\pi$
$654$	16.7984	0.656868
$655$	0	0
$656$	78.8328	3.07790
$657$	1.23607	0.0482236
$658$	0	0
$659$	40.9443	1.59496	0.797481	−	0.603344i	$-0.206165\pi$
$659$	40.9443	1.59496	0.797481	+	0.603344i	$0.206165\pi$
$660$	0	0
$661$	−8.18034	−0.318178	−0.159089	−	0.987264i	$-0.550856\pi$
$661$	−8.18034	−0.318178	−0.159089	+	0.987264i	$0.550856\pi$
$662$	−36.0344	−1.40052
$663$	−5.88854	−0.228692
$664$	−93.1935	−3.61661
$665$	0	0
$666$	14.3262	0.555130
$667$	−23.5410	−0.911512
$668$	−30.5410	−1.18167
$669$	−1.23607	−0.0477891
$670$	0	0
$671$	−84.3607	−3.25671
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−74.5410	−2.87121
$675$	0	0
$676$	−60.2705	−2.31810
$677$	43.3050	1.66434	0.832172	−	0.554517i	$-0.187097\pi$
$677$	43.3050	1.66434	0.832172	+	0.554517i	$0.187097\pi$
$678$	−42.5066	−1.63246
$679$	0	0
$680$	0	0
$681$	−4.76393	−0.182554
$682$	−64.0689	−2.45332
$683$	−5.11146	−0.195584	−0.0977922	−	0.995207i	$-0.531178\pi$
$683$	−5.11146	−0.195584	−0.0977922	+	0.995207i	$0.531178\pi$
$684$	15.7082	0.600618
$685$	0	0
$686$	0	0
$687$	−22.3607	−0.853113
$688$	−81.1591	−3.09416
$689$	−0.360680	−0.0137408
$690$	0	0
$691$	12.3607	0.470222	0.235111	−	0.971968i	$-0.424455\pi$
$691$	12.3607	0.470222	0.235111	+	0.971968i	$0.424455\pi$
$692$	−18.5410	−0.704824
$693$	0	0
$694$	−34.0344	−1.29193
$695$	0	0
$696$	−35.1803	−1.33351
$697$	−61.6656	−2.33575
$698$	2.00000	0.0757011
$699$	−7.29180	−0.275801
$700$	0	0
$701$	33.4164	1.26212	0.631060	−	0.775734i	$-0.282620\pi$
$701$	33.4164	1.26212	0.631060	+	0.775734i	$0.282620\pi$
$702$	−2.00000	−0.0754851
$703$	−17.7082	−0.667878
$704$	−47.6525	−1.79597
$705$	0	0
$706$	−2.76393	−0.104022
$707$	0	0
$708$	3.70820	0.139363
$709$	43.8885	1.64827	0.824134	−	0.566394i	$-0.191662\pi$
$709$	43.8885	1.64827	0.824134	+	0.566394i	$0.191662\pi$
$710$	0	0
$711$	1.76393	0.0661526
$712$	−42.6525	−1.59847
$713$	22.3607	0.837414
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	−28.3607	−1.05915
$718$	−67.9230	−2.53486
$719$	−42.2492	−1.57563	−0.787815	−	0.615912i	$-0.788788\pi$
$719$	−42.2492	−1.57563	−0.787815	+	0.615912i	$0.788788\pi$
$720$	0	0
$721$	0	0
$722$	22.3262	0.830897
$723$	−19.2361	−0.715397
$724$	−22.5836	−0.839313
$725$	0	0
$726$	−49.5967	−1.84071
$727$	−18.5410	−0.687648	−0.343824	−	0.939034i	$-0.611722\pi$
$727$	−18.5410	−0.687648	−0.343824	+	0.939034i	$0.611722\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	63.4853	2.34809
$732$	74.8328	2.76590
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	23.4164	0.864315
$735$	0	0
$736$	54.2705	2.00044
$737$	−14.8197	−0.545889
$738$	−20.9443	−0.770969
$739$	3.76393	0.138458	0.0692292	−	0.997601i	$-0.477946\pi$
$739$	3.76393	0.138458	0.0692292	+	0.997601i	$0.477946\pi$
$740$	0	0
$741$	2.47214	0.0908162
$742$	0	0
$743$	30.2492	1.10974	0.554868	−	0.831938i	$-0.312769\pi$
$743$	30.2492	1.10974	0.554868	+	0.831938i	$0.312769\pi$
$744$	33.4164	1.22510
$745$	0	0
$746$	−39.2705	−1.43780
$747$	12.4721	0.456332
$748$	204.748	7.48632
$749$	0	0
$750$	0	0
$751$	40.3607	1.47278	0.736391	−	0.676556i	$-0.236528\pi$
$751$	40.3607	1.47278	0.736391	+	0.676556i	$0.236528\pi$
$752$	71.3050	2.60022
$753$	−2.76393	−0.100723
$754$	−9.41641	−0.342925
$755$	0	0
$756$	0	0
$757$	13.9443	0.506813	0.253407	−	0.967360i	$-0.418449\pi$
$757$	13.9443	0.506813	0.253407	+	0.967360i	$0.418449\pi$
$758$	−43.4508	−1.57821
$759$	27.3607	0.993130
$760$	0	0
$761$	−9.30495	−0.337304	−0.168652	−	0.985676i	$-0.553941\pi$
$761$	−9.30495	−0.337304	−0.168652	+	0.985676i	$0.553941\pi$
$762$	−32.0344	−1.16049
$763$	0	0
$764$	−55.4164	−2.00490
$765$	0	0
$766$	−65.7771	−2.37662
$767$	0.583592	0.0210723
$768$	−14.5623	−0.525472
$769$	−35.4164	−1.27715	−0.638574	−	0.769560i	$-0.720475\pi$
$769$	−35.4164	−1.27715	−0.638574	+	0.769560i	$0.720475\pi$
$770$	0	0
$771$	−8.47214	−0.305117
$772$	−2.02129	−0.0727477
$773$	−34.9443	−1.25686	−0.628429	−	0.777867i	$-0.716302\pi$
$773$	−34.9443	−1.25686	−0.628429	+	0.777867i	$0.716302\pi$
$774$	21.5623	0.775041
$775$	0	0
$776$	−93.1935	−3.34545
$777$	0	0
$778$	4.61803	0.165565
$779$	25.8885	0.927553
$780$	0	0
$781$	2.88854	0.103360
$782$	−100.902	−3.60824
$783$	4.70820	0.168257
$784$	0	0
$785$	0	0
$786$	−11.2361	−0.400777
$787$	38.7639	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$787$	38.7639	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$788$	27.9787	0.996700
$789$	−2.05573	−0.0731859
$790$	0	0
$791$	0	0
$792$	40.8885	1.45291
$793$	11.7771	0.418217
$794$	47.1246	1.67239
$795$	0	0
$796$	77.6656	2.75279
$797$	−40.0689	−1.41931	−0.709656	−	0.704548i	$-0.751150\pi$
$797$	−40.0689	−1.41931	−0.709656	+	0.704548i	$0.751150\pi$
$798$	0	0
$799$	−55.7771	−1.97325
$800$	0	0
$801$	5.70820	0.201689
$802$	69.9230	2.46907
$803$	−6.76393	−0.238694
$804$	13.1459	0.463620
$805$	0	0
$806$	8.94427	0.315049
$807$	28.4721	1.00227
$808$	−32.0689	−1.12818
$809$	−6.81966	−0.239766	−0.119883	−	0.992788i	$-0.538252\pi$
$809$	−6.81966	−0.239766	−0.119883	+	0.992788i	$0.538252\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	−23.2361	−0.814924
$814$	−78.3951	−2.74775
$815$	0	0
$816$	−75.9574	−2.65904
$817$	−26.6525	−0.932452
$818$	−21.4164	−0.748807
$819$	0	0
$820$	0	0
$821$	−10.9443	−0.381958	−0.190979	−	0.981594i	$-0.561166\pi$
$821$	−10.9443	−0.381958	−0.190979	+	0.981594i	$0.561166\pi$
$822$	−1.23607	−0.0431128
$823$	44.0132	1.53420	0.767101	−	0.641526i	$-0.221698\pi$
$823$	44.0132	1.53420	0.767101	+	0.641526i	$0.221698\pi$
$824$	−72.5410	−2.52709
$825$	0	0
$826$	0	0
$827$	−31.9443	−1.11081	−0.555406	−	0.831580i	$-0.687437\pi$
$827$	−31.9443	−1.11081	−0.555406	+	0.831580i	$0.687437\pi$
$828$	−24.2705	−0.843459
$829$	−26.7639	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$829$	−26.7639	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$830$	0	0
$831$	15.8885	0.551167
$832$	6.65248	0.230633
$833$	0	0
$834$	9.70820	0.336168
$835$	0	0
$836$	−85.9574	−2.97290
$837$	−4.47214	−0.154580
$838$	23.7082	0.818986
$839$	−27.1246	−0.936446	−0.468223	−	0.883610i	$-0.655106\pi$
$839$	−27.1246	−0.936446	−0.468223	+	0.883610i	$0.655106\pi$
$840$	0	0
$841$	−6.83282	−0.235614
$842$	1.09017	0.0375697
$843$	13.6525	0.470216
$844$	62.8328	2.16279
$845$	0	0
$846$	−18.9443	−0.651317
$847$	0	0
$848$	−4.65248	−0.159767
$849$	−13.4164	−0.460450
$850$	0	0
$851$	27.3607	0.937912
$852$	−2.56231	−0.0877832
$853$	52.3607	1.79280	0.896398	−	0.443251i	$-0.146175\pi$
$853$	52.3607	1.79280	0.896398	+	0.443251i	$0.146175\pi$
$854$	0	0
$855$	0	0
$856$	−36.9443	−1.26273
$857$	39.4164	1.34644	0.673219	−	0.739443i	$-0.264911\pi$
$857$	39.4164	1.34644	0.673219	+	0.739443i	$0.264911\pi$
$858$	10.9443	0.373631
$859$	0.472136	0.0161091	0.00805454	−	0.999968i	$-0.497436\pi$
$859$	0.472136	0.0161091	0.00805454	+	0.999968i	$0.497436\pi$
$860$	0	0
$861$	0	0
$862$	−88.7214	−3.02186
$863$	−2.05573	−0.0699778	−0.0349889	−	0.999388i	$-0.511140\pi$
$863$	−2.05573	−0.0699778	−0.0349889	+	0.999388i	$0.511140\pi$
$864$	−10.8541	−0.369264
$865$	0	0
$866$	−97.6656	−3.31881
$867$	42.4164	1.44054
$868$	0	0
$869$	−9.65248	−0.327438
$870$	0	0
$871$	2.06888	0.0701015
$872$	47.9443	1.62360
$873$	12.4721	0.422118
$874$	42.3607	1.43287
$875$	0	0
$876$	6.00000	0.202721
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−29.4164	−0.992756
$879$	−26.6525	−0.898966
$880$	0	0
$881$	38.3607	1.29240	0.646202	−	0.763166i	$-0.276356\pi$
$881$	38.3607	1.29240	0.646202	+	0.763166i	$0.276356\pi$
$882$	0	0
$883$	−34.2361	−1.15214	−0.576068	−	0.817402i	$-0.695414\pi$
$883$	−34.2361	−1.15214	−0.576068	+	0.817402i	$0.695414\pi$
$884$	−28.5836	−0.961370
$885$	0	0
$886$	32.9443	1.10678
$887$	−15.4164	−0.517632	−0.258816	−	0.965927i	$-0.583332\pi$
$887$	−15.4164	−0.517632	−0.258816	+	0.965927i	$0.583332\pi$
$888$	40.8885	1.37213
$889$	0	0
$890$	0	0
$891$	−5.47214	−0.183323
$892$	−6.00000	−0.200895
$893$	23.4164	0.783600
$894$	−21.5623	−0.721151
$895$	0	0
$896$	0	0
$897$	−3.81966	−0.127535
$898$	−8.61803	−0.287588
$899$	−21.0557	−0.702248
$900$	0	0
$901$	3.63932	0.121243
$902$	114.610	3.81609
$903$	0	0
$904$	−121.318	−4.03498
$905$	0	0
$906$	−3.38197	−0.112358
$907$	16.3607	0.543247	0.271624	−	0.962404i	$-0.412439\pi$
$907$	16.3607	0.543247	0.271624	+	0.962404i	$0.412439\pi$
$908$	−23.1246	−0.767417
$909$	4.29180	0.142350
$910$	0	0
$911$	7.58359	0.251256	0.125628	−	0.992077i	$-0.459905\pi$
$911$	7.58359	0.251256	0.125628	+	0.992077i	$0.459905\pi$
$912$	31.8885	1.05594
$913$	−68.2492	−2.25872
$914$	−77.1591	−2.55219
$915$	0	0
$916$	−108.541	−3.58630
$917$	0	0
$918$	20.1803	0.666050
$919$	36.0132	1.18796	0.593982	−	0.804478i	$-0.297555\pi$
$919$	36.0132	1.18796	0.593982	+	0.804478i	$0.297555\pi$
$920$	0	0
$921$	24.6525	0.812327
$922$	27.8885	0.918460
$923$	−0.403252	−0.0132732
$924$	0	0
$925$	0	0
$926$	74.2492	2.43998
$927$	9.70820	0.318859
$928$	−51.1033	−1.67755
$929$	30.1803	0.990185	0.495092	−	0.868840i	$-0.335134\pi$
$929$	30.1803	0.990185	0.495092	+	0.868840i	$0.335134\pi$
$930$	0	0
$931$	0	0
$932$	−35.3951	−1.15941
$933$	−13.2361	−0.433329
$934$	−62.5410	−2.04640
$935$	0	0
$936$	−5.70820	−0.186578
$937$	4.36068	0.142457	0.0712286	−	0.997460i	$-0.477308\pi$
$937$	4.36068	0.142457	0.0712286	+	0.997460i	$0.477308\pi$
$938$	0	0
$939$	−5.70820	−0.186280
$940$	0	0
$941$	−44.8328	−1.46151	−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328	−1.46151	−0.730754	+	0.682641i	$0.760831\pi$
$942$	−44.3607	−1.44535
$943$	−40.0000	−1.30258
$944$	7.52786	0.245011
$945$	0	0
$946$	−117.992	−3.83625
$947$	−32.9443	−1.07054	−0.535272	−	0.844679i	$-0.679791\pi$
$947$	−32.9443	−1.07054	−0.535272	+	0.844679i	$0.679791\pi$
$948$	8.56231	0.278091
$949$	0.944272	0.0306524
$950$	0	0
$951$	−17.6525	−0.572421
$952$	0	0
$953$	−16.1246	−0.522327	−0.261164	−	0.965295i	$-0.584106\pi$
$953$	−16.1246	−0.522327	−0.261164	+	0.965295i	$0.584106\pi$
$954$	1.23607	0.0400192
$955$	0	0
$956$	−137.666	−4.45242
$957$	−25.7639	−0.832830
$958$	−60.8328	−1.96542
$959$	0	0
$960$	0	0
$961$	−11.0000	−0.354839
$962$	10.9443	0.352857
$963$	4.94427	0.159327
$964$	−93.3738	−3.00737
$965$	0	0
$966$	0	0
$967$	2.11146	0.0678999	0.0339499	−	0.999424i	$-0.489191\pi$
$967$	2.11146	0.0678999	0.0339499	+	0.999424i	$0.489191\pi$
$968$	−141.554	−4.54972
$969$	−24.9443	−0.801325
$970$	0	0
$971$	−11.5967	−0.372157	−0.186079	−	0.982535i	$-0.559578\pi$
$971$	−11.5967	−0.372157	−0.186079	+	0.982535i	$0.559578\pi$
$972$	4.85410	0.155695
$973$	0	0
$974$	−17.8541	−0.572082
$975$	0	0
$976$	151.915	4.86268
$977$	−3.29180	−0.105314	−0.0526569	−	0.998613i	$-0.516769\pi$
$977$	−3.29180	−0.105314	−0.0526569	+	0.998613i	$0.516769\pi$
$978$	48.3607	1.54640
$979$	−31.2361	−0.998309
$980$	0	0
$981$	−6.41641	−0.204860
$982$	1.38197	0.0441003
$983$	−2.58359	−0.0824038	−0.0412019	−	0.999151i	$-0.513119\pi$
$983$	−2.58359	−0.0824038	−0.0412019	+	0.999151i	$0.513119\pi$
$984$	−59.7771	−1.90562
$985$	0	0
$986$	95.0132	3.02584
$987$	0	0
$988$	12.0000	0.381771
$989$	41.1803	1.30946
$990$	0	0
$991$	8.59675	0.273085	0.136542	−	0.990634i	$-0.456401\pi$
$991$	8.59675	0.273085	0.136542	+	0.990634i	$0.456401\pi$
$992$	48.5410	1.54118
$993$	13.7639	0.436785
$994$	0	0
$995$	0	0
$996$	60.5410	1.91832
$997$	52.5410	1.66399	0.831995	−	0.554783i	$-0.187199\pi$
$997$	52.5410	1.66399	0.831995	+	0.554783i	$0.187199\pi$
$998$	83.1935	2.63344
$999$	−5.47214	−0.173131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.r.1.1		2
5.4	even	2		3675.2.a.bh.1.2		2
7.6	odd	2		525.2.a.e.1.1	✓	2
21.20	even	2		1575.2.a.v.1.2		2
28.27	even	2		8400.2.a.da.1.2		2
35.13	even	4		525.2.d.e.274.4		4
35.27	even	4		525.2.d.e.274.1		4
35.34	odd	2		525.2.a.i.1.2	yes	2
105.62	odd	4		1575.2.d.f.1324.4		4
105.83	odd	4		1575.2.d.f.1324.1		4
105.104	even	2		1575.2.a.l.1.1		2
140.139	even	2		8400.2.a.cy.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.e.1.1	✓	2	7.6	odd	2
525.2.a.i.1.2	yes	2	35.34	odd	2
525.2.d.e.274.1		4	35.27	even	4
525.2.d.e.274.4		4	35.13	even	4
1575.2.a.l.1.1		2	105.104	even	2
1575.2.a.v.1.2		2	21.20	even	2
1575.2.d.f.1324.1		4	105.83	odd	4
1575.2.d.f.1324.4		4	105.62	odd	4
3675.2.a.r.1.1		2	1.1	even	1	trivial
3675.2.a.bh.1.2		2	5.4	even	2
8400.2.a.cy.1.2		2	140.139	even	2
8400.2.a.da.1.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} -2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} -2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} -5.47214 q^{11} +4.85410 q^{12} +0.763932 q^{13} +9.85410 q^{16} -7.70820 q^{17} -2.61803 q^{18} +3.23607 q^{19} +14.3262 q^{22} -5.00000 q^{23} -7.47214 q^{24} -2.00000 q^{26} +1.00000 q^{27} +4.70820 q^{29} -4.47214 q^{31} -10.8541 q^{32} -5.47214 q^{33} +20.1803 q^{34} +4.85410 q^{36} -5.47214 q^{37} -8.47214 q^{38} +0.763932 q^{39} +8.00000 q^{41} -8.23607 q^{43} -26.5623 q^{44} +13.0902 q^{46} +7.23607 q^{47} +9.85410 q^{48} -7.70820 q^{51} +3.70820 q^{52} -0.472136 q^{53} -2.61803 q^{54} +3.23607 q^{57} -12.3262 q^{58} +0.763932 q^{59} +15.4164 q^{61} +11.7082 q^{62} +8.70820 q^{64} +14.3262 q^{66} +2.70820 q^{67} -37.4164 q^{68} -5.00000 q^{69} -0.527864 q^{71} -7.47214 q^{72} +1.23607 q^{73} +14.3262 q^{74} +15.7082 q^{76} -2.00000 q^{78} +1.76393 q^{79} +1.00000 q^{81} -20.9443 q^{82} +12.4721 q^{83} +21.5623 q^{86} +4.70820 q^{87} +40.8885 q^{88} +5.70820 q^{89} -24.2705 q^{92} -4.47214 q^{93} -18.9443 q^{94} -10.8541 q^{96} +12.4721 q^{97} -5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} + 3 q^{12} + 6 q^{13} + 13 q^{16} - 2 q^{17} - 3 q^{18} + 2 q^{19} + 13 q^{22} - 10 q^{23} - 6 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{29} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 2 q^{37} - 8 q^{38} + 6 q^{39} + 16 q^{41} - 12 q^{43} - 33 q^{44} + 15 q^{46} + 10 q^{47} + 13 q^{48} - 2 q^{51} - 6 q^{52} + 8 q^{53} - 3 q^{54} + 2 q^{57} - 9 q^{58} + 6 q^{59} + 4 q^{61} + 10 q^{62} + 4 q^{64} + 13 q^{66} - 8 q^{67} - 48 q^{68} - 10 q^{69} - 10 q^{71} - 6 q^{72} - 2 q^{73} + 13 q^{74} + 18 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 24 q^{82} + 16 q^{83} + 23 q^{86} - 4 q^{87} + 46 q^{88} - 2 q^{89} - 15 q^{92} - 20 q^{94} - 15 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 + 3 * q^12 + 6 * q^13 + 13 * q^16 - 2 * q^17 - 3 * q^18 + 2 * q^19 + 13 * q^22 - 10 * q^23 - 6 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^29 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 2 * q^37 - 8 * q^38 + 6 * q^39 + 16 * q^41 - 12 * q^43 - 33 * q^44 + 15 * q^46 + 10 * q^47 + 13 * q^48 - 2 * q^51 - 6 * q^52 + 8 * q^53 - 3 * q^54 + 2 * q^57 - 9 * q^58 + 6 * q^59 + 4 * q^61 + 10 * q^62 + 4 * q^64 + 13 * q^66 - 8 * q^67 - 48 * q^68 - 10 * q^69 - 10 * q^71 - 6 * q^72 - 2 * q^73 + 13 * q^74 + 18 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 24 * q^82 + 16 * q^83 + 23 * q^86 - 4 * q^87 + 46 * q^88 - 2 * q^89 - 15 * q^92 - 20 * q^94 - 15 * q^96 + 16 * q^97 - 2 * q^99

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