LMFDB - Embedded newform 3675.2.a.v.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:	yes
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)

$f(q)$	$=$	$q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{11} +0.618034 q^{12} -6.47214 q^{13} -4.85410 q^{16} +2.47214 q^{17} -1.61803 q^{18} -2.47214 q^{19} +3.61803 q^{22} -4.23607 q^{23} +2.23607 q^{24} +10.4721 q^{26} +1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} +3.38197 q^{32} -2.23607 q^{33} -4.00000 q^{34} +0.618034 q^{36} +3.47214 q^{37} +4.00000 q^{38} -6.47214 q^{39} +8.94427 q^{41} -4.70820 q^{43} -1.38197 q^{44} +6.85410 q^{46} -10.4721 q^{47} -4.85410 q^{48} +2.47214 q^{51} -4.00000 q^{52} +6.00000 q^{53} -1.61803 q^{54} -2.47214 q^{57} +4.85410 q^{58} +6.47214 q^{59} +12.0000 q^{61} -6.47214 q^{62} +4.23607 q^{64} +3.61803 q^{66} +12.7082 q^{67} +1.52786 q^{68} -4.23607 q^{69} +8.23607 q^{71} +2.23607 q^{72} +14.4721 q^{73} -5.61803 q^{74} -1.52786 q^{76} +10.4721 q^{78} -2.70820 q^{79} +1.00000 q^{81} -14.4721 q^{82} -16.9443 q^{83} +7.61803 q^{86} -3.00000 q^{87} -5.00000 q^{88} +1.52786 q^{89} -2.61803 q^{92} +4.00000 q^{93} +16.9443 q^{94} +3.38197 q^{96} +4.00000 q^{97} -2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9} - q^{12} - 4 q^{13} - 3 q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + 5 q^{22} - 4 q^{23} + 12 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 9 q^{32} - 8 q^{34} - q^{36} - 2 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{43} - 5 q^{44} + 7 q^{46} - 12 q^{47} - 3 q^{48} - 4 q^{51} - 8 q^{52} + 12 q^{53} - q^{54} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 5 q^{66} + 12 q^{67} + 12 q^{68} - 4 q^{69} + 12 q^{71} + 20 q^{73} - 9 q^{74} - 12 q^{76} + 12 q^{78} + 8 q^{79} + 2 q^{81} - 20 q^{82} - 16 q^{83} + 13 q^{86} - 6 q^{87} - 10 q^{88} + 12 q^{89} - 3 q^{92} + 8 q^{93} + 16 q^{94} + 9 q^{96} + 8 q^{97}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9 - q^12 - 4 * q^13 - 3 * q^16 - 4 * q^17 - q^18 + 4 * q^19 + 5 * q^22 - 4 * q^23 + 12 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 9 * q^32 - 8 * q^34 - q^36 - 2 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^43 - 5 * q^44 + 7 * q^46 - 12 * q^47 - 3 * q^48 - 4 * q^51 - 8 * q^52 + 12 * q^53 - q^54 + 4 * q^57 + 3 * q^58 + 4 * q^59 + 24 * q^61 - 4 * q^62 + 4 * q^64 + 5 * q^66 + 12 * q^67 + 12 * q^68 - 4 * q^69 + 12 * q^71 + 20 * q^73 - 9 * q^74 - 12 * q^76 + 12 * q^78 + 8 * q^79 + 2 * q^81 - 20 * q^82 - 16 * q^83 + 13 * q^86 - 6 * q^87 - 10 * q^88 + 12 * q^89 - 3 * q^92 + 8 * q^93 + 16 * q^94 + 9 * q^96 + 8 * q^97

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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0.618034	0.178411
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	0	0
$15$	0	0
$16$	−4.85410	−1.21353
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	−1.61803	−0.381374
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	0	0
$22$	3.61803	0.771367
$23$	−4.23607	−0.883281	−0.441641	−	0.897192i	$-0.645603\pi$
$23$	−4.23607	−0.883281	−0.441641	+	0.897192i	$0.645603\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	10.4721	2.05375
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	3.38197	0.597853
$33$	−2.23607	−0.389249
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0.618034	0.103006
$37$	3.47214	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$37$	3.47214	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$38$	4.00000	0.648886
$39$	−6.47214	−1.03637
$40$	0	0
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	0	0
$43$	−4.70820	−0.717994	−0.358997	−	0.933339i	$-0.616881\pi$
$43$	−4.70820	−0.717994	−0.358997	+	0.933339i	$0.616881\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	6.85410	1.01058
$47$	−10.4721	−1.52752	−0.763759	−	0.645501i	$-0.776648\pi$
$47$	−10.4721	−1.52752	−0.763759	+	0.645501i	$0.776648\pi$
$48$	−4.85410	−0.700629
$49$	0	0
$50$	0	0
$51$	2.47214	0.346168
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	0	0
$57$	−2.47214	−0.327442
$58$	4.85410	0.637375
$59$	6.47214	0.842600	0.421300	−	0.906921i	$-0.361574\pi$
$59$	6.47214	0.842600	0.421300	+	0.906921i	$0.361574\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−6.47214	−0.821962
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	3.61803	0.445349
$67$	12.7082	1.55255	0.776277	−	0.630392i	$-0.217106\pi$
$67$	12.7082	1.55255	0.776277	+	0.630392i	$0.217106\pi$
$68$	1.52786	0.185281
$69$	−4.23607	−0.509963
$70$	0	0
$71$	8.23607	0.977441	0.488721	−	0.872440i	$-0.337464\pi$
$71$	8.23607	0.977441	0.488721	+	0.872440i	$0.337464\pi$
$72$	2.23607	0.263523
$73$	14.4721	1.69384	0.846918	−	0.531724i	$-0.178456\pi$
$73$	14.4721	1.69384	0.846918	+	0.531724i	$0.178456\pi$
$74$	−5.61803	−0.653083
$75$	0	0
$76$	−1.52786	−0.175258
$77$	0	0
$78$	10.4721	1.18574
$79$	−2.70820	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$79$	−2.70820	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.4721	−1.59818
$83$	−16.9443	−1.85988	−0.929938	−	0.367717i	$-0.880140\pi$
$83$	−16.9443	−1.85988	−0.929938	+	0.367717i	$0.880140\pi$
$84$	0	0
$85$	0	0
$86$	7.61803	0.821474
$87$	−3.00000	−0.321634
$88$	−5.00000	−0.533002
$89$	1.52786	0.161953	0.0809766	−	0.996716i	$-0.474196\pi$
$89$	1.52786	0.161953	0.0809766	+	0.996716i	$0.474196\pi$
$90$	0	0
$91$	0	0
$92$	−2.61803	−0.272949
$93$	4.00000	0.414781
$94$	16.9443	1.74767
$95$	0	0
$96$	3.38197	0.345170
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−2.23607	−0.224733
$100$	0	0
$101$	9.52786	0.948058	0.474029	−	0.880509i	$-0.342799\pi$
$101$	9.52786	0.948058	0.474029	+	0.880509i	$0.342799\pi$
$102$	−4.00000	−0.396059
$103$	−7.41641	−0.730760	−0.365380	−	0.930858i	$-0.619061\pi$
$103$	−7.41641	−0.730760	−0.365380	+	0.930858i	$0.619061\pi$
$104$	−14.4721	−1.41911
$105$	0	0
$106$	−9.70820	−0.942944
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0.618034	0.0594703
$109$	16.4164	1.57241	0.786203	−	0.617968i	$-0.212044\pi$
$109$	16.4164	1.57241	0.786203	+	0.617968i	$0.212044\pi$
$110$	0	0
$111$	3.47214	0.329561
$112$	0	0
$113$	−10.5279	−0.990378	−0.495189	−	0.868785i	$-0.664901\pi$
$113$	−10.5279	−0.990378	−0.495189	+	0.868785i	$0.664901\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−1.85410	−0.172149
$117$	−6.47214	−0.598349
$118$	−10.4721	−0.964038
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	−19.4164	−1.75788
$123$	8.94427	0.806478
$124$	2.47214	0.222004
$125$	0	0
$126$	0	0
$127$	19.1803	1.70198	0.850990	−	0.525182i	$-0.176003\pi$
$127$	19.1803	1.70198	0.850990	+	0.525182i	$0.176003\pi$
$128$	−13.6180	−1.20368
$129$	−4.70820	−0.414534
$130$	0	0
$131$	13.5279	1.18193	0.590967	−	0.806695i	$-0.298746\pi$
$131$	13.5279	1.18193	0.590967	+	0.806695i	$0.298746\pi$
$132$	−1.38197	−0.120285
$133$	0	0
$134$	−20.5623	−1.77631
$135$	0	0
$136$	5.52786	0.474010
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	6.85410	0.583460
$139$	−11.4164	−0.968327	−0.484164	−	0.874978i	$-0.660876\pi$
$139$	−11.4164	−0.968327	−0.484164	+	0.874978i	$0.660876\pi$
$140$	0	0
$141$	−10.4721	−0.881913
$142$	−13.3262	−1.11831
$143$	14.4721	1.21022
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−23.4164	−1.93796
$147$	0	0
$148$	2.14590	0.176392
$149$	12.8885	1.05587	0.527935	−	0.849285i	$-0.322966\pi$
$149$	12.8885	1.05587	0.527935	+	0.849285i	$0.322966\pi$
$150$	0	0
$151$	6.70820	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$151$	6.70820	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$152$	−5.52786	−0.448369
$153$	2.47214	0.199860
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	0.944272	0.0753611	0.0376806	−	0.999290i	$-0.488003\pi$
$157$	0.944272	0.0753611	0.0376806	+	0.999290i	$0.488003\pi$
$158$	4.38197	0.348610
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	−1.61803	−0.127125
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	5.52786	0.431654
$165$	0	0
$166$	27.4164	2.12793
$167$	22.4721	1.73895	0.869473	−	0.493980i	$-0.164459\pi$
$167$	22.4721	1.73895	0.869473	+	0.493980i	$0.164459\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	−2.47214	−0.189049
$172$	−2.90983	−0.221872
$173$	−2.47214	−0.187953	−0.0939765	−	0.995574i	$-0.529958\pi$
$173$	−2.47214	−0.187953	−0.0939765	+	0.995574i	$0.529958\pi$
$174$	4.85410	0.367989
$175$	0	0
$176$	10.8541	0.818159
$177$	6.47214	0.486476
$178$	−2.47214	−0.185294
$179$	−0.944272	−0.0705782	−0.0352891	−	0.999377i	$-0.511235\pi$
$179$	−0.944272	−0.0705782	−0.0352891	+	0.999377i	$0.511235\pi$
$180$	0	0
$181$	6.47214	0.481070	0.240535	−	0.970640i	$-0.422677\pi$
$181$	6.47214	0.481070	0.240535	+	0.970640i	$0.422677\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	−9.47214	−0.698295
$185$	0	0
$186$	−6.47214	−0.474560
$187$	−5.52786	−0.404237
$188$	−6.47214	−0.472029
$189$	0	0
$190$	0	0
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	4.23607	0.305712
$193$	−22.8885	−1.64755	−0.823777	−	0.566914i	$-0.808137\pi$
$193$	−22.8885	−1.64755	−0.823777	+	0.566914i	$0.808137\pi$
$194$	−6.47214	−0.464672
$195$	0	0
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	3.61803	0.257122
$199$	−21.8885	−1.55164	−0.775819	−	0.630956i	$-0.782663\pi$
$199$	−21.8885	−1.55164	−0.775819	+	0.630956i	$0.782663\pi$
$200$	0	0
$201$	12.7082	0.896368
$202$	−15.4164	−1.08469
$203$	0	0
$204$	1.52786	0.106972
$205$	0	0
$206$	12.0000	0.836080
$207$	−4.23607	−0.294427
$208$	31.4164	2.17834
$209$	5.52786	0.382370
$210$	0	0
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	3.70820	0.254680
$213$	8.23607	0.564326
$214$	14.4721	0.989295
$215$	0	0
$216$	2.23607	0.152145
$217$	0	0
$218$	−26.5623	−1.79903
$219$	14.4721	0.977936
$220$	0	0
$221$	−16.0000	−1.07628
$222$	−5.61803	−0.377058
$223$	3.41641	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$223$	3.41641	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$224$	0	0
$225$	0	0
$226$	17.0344	1.13311
$227$	7.41641	0.492244	0.246122	−	0.969239i	$-0.420844\pi$
$227$	7.41641	0.492244	0.246122	+	0.969239i	$0.420844\pi$
$228$	−1.52786	−0.101185
$229$	28.9443	1.91269	0.956346	−	0.292238i	$-0.0943999\pi$
$229$	28.9443	1.91269	0.956346	+	0.292238i	$0.0943999\pi$
$230$	0	0
$231$	0	0
$232$	−6.70820	−0.440415
$233$	−4.52786	−0.296630	−0.148315	−	0.988940i	$-0.547385\pi$
$233$	−4.52786	−0.296630	−0.148315	+	0.988940i	$0.547385\pi$
$234$	10.4721	0.684585
$235$	0	0
$236$	4.00000	0.260378
$237$	−2.70820	−0.175917
$238$	0	0
$239$	−20.9443	−1.35477	−0.677386	−	0.735628i	$-0.736887\pi$
$239$	−20.9443	−1.35477	−0.677386	+	0.735628i	$0.736887\pi$
$240$	0	0
$241$	18.4721	1.18989	0.594947	−	0.803765i	$-0.297173\pi$
$241$	18.4721	1.18989	0.594947	+	0.803765i	$0.297173\pi$
$242$	9.70820	0.624067
$243$	1.00000	0.0641500
$244$	7.41641	0.474787
$245$	0	0
$246$	−14.4721	−0.922710
$247$	16.0000	1.01806
$248$	8.94427	0.567962
$249$	−16.9443	−1.07380
$250$	0	0
$251$	−10.4721	−0.660995	−0.330498	−	0.943807i	$-0.607217\pi$
$251$	−10.4721	−0.660995	−0.330498	+	0.943807i	$0.607217\pi$
$252$	0	0
$253$	9.47214	0.595508
$254$	−31.0344	−1.94727
$255$	0	0
$256$	13.5623	0.847644
$257$	8.94427	0.557928	0.278964	−	0.960302i	$-0.410009\pi$
$257$	8.94427	0.557928	0.278964	+	0.960302i	$0.410009\pi$
$258$	7.61803	0.474278
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−21.8885	−1.35228
$263$	21.1803	1.30604	0.653018	−	0.757343i	$-0.273503\pi$
$263$	21.1803	1.30604	0.653018	+	0.757343i	$0.273503\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	0	0
$267$	1.52786	0.0935038
$268$	7.85410	0.479766
$269$	−8.94427	−0.545342	−0.272671	−	0.962107i	$-0.587907\pi$
$269$	−8.94427	−0.545342	−0.272671	+	0.962107i	$0.587907\pi$
$270$	0	0
$271$	−3.41641	−0.207532	−0.103766	−	0.994602i	$-0.533089\pi$
$271$	−3.41641	−0.207532	−0.103766	+	0.994602i	$0.533089\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	32.1803	1.94409
$275$	0	0
$276$	−2.61803	−0.157587
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	18.4721	1.10789
$279$	4.00000	0.239474
$280$	0	0
$281$	−19.3607	−1.15496	−0.577481	−	0.816404i	$-0.695964\pi$
$281$	−19.3607	−1.15496	−0.577481	+	0.816404i	$0.695964\pi$
$282$	16.9443	1.00902
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	5.09017	0.302046
$285$	0	0
$286$	−23.4164	−1.38464
$287$	0	0
$288$	3.38197	0.199284
$289$	−10.8885	−0.640503
$290$	0	0
$291$	4.00000	0.234484
$292$	8.94427	0.523424
$293$	16.3607	0.955801	0.477901	−	0.878414i	$-0.341398\pi$
$293$	16.3607	0.955801	0.477901	+	0.878414i	$0.341398\pi$
$294$	0	0
$295$	0	0
$296$	7.76393	0.451269
$297$	−2.23607	−0.129750
$298$	−20.8541	−1.20805
$299$	27.4164	1.58553
$300$	0	0
$301$	0	0
$302$	−10.8541	−0.624583
$303$	9.52786	0.547361
$304$	12.0000	0.688247
$305$	0	0
$306$	−4.00000	−0.228665
$307$	31.4164	1.79303	0.896515	−	0.443014i	$-0.146091\pi$
$307$	31.4164	1.79303	0.896515	+	0.443014i	$0.146091\pi$
$308$	0	0
$309$	−7.41641	−0.421905
$310$	0	0
$311$	20.3607	1.15455	0.577274	−	0.816550i	$-0.304116\pi$
$311$	20.3607	1.15455	0.577274	+	0.816550i	$0.304116\pi$
$312$	−14.4721	−0.819323
$313$	7.41641	0.419200	0.209600	−	0.977787i	$-0.432784\pi$
$313$	7.41641	0.419200	0.209600	+	0.977787i	$0.432784\pi$
$314$	−1.52786	−0.0862224
$315$	0	0
$316$	−1.67376	−0.0941565
$317$	7.94427	0.446195	0.223097	−	0.974796i	$-0.428383\pi$
$317$	7.94427	0.446195	0.223097	+	0.974796i	$0.428383\pi$
$318$	−9.70820	−0.544409
$319$	6.70820	0.375587
$320$	0	0
$321$	−8.94427	−0.499221
$322$	0	0
$323$	−6.11146	−0.340051
$324$	0.618034	0.0343352
$325$	0	0
$326$	−19.4164	−1.07538
$327$	16.4164	0.907829
$328$	20.0000	1.10432
$329$	0	0
$330$	0	0
$331$	−33.1803	−1.82376	−0.911878	−	0.410461i	$-0.865368\pi$
$331$	−33.1803	−1.82376	−0.911878	+	0.410461i	$0.865368\pi$
$332$	−10.4721	−0.574733
$333$	3.47214	0.190272
$334$	−36.3607	−1.98957
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−46.7426	−2.54246
$339$	−10.5279	−0.571795
$340$	0	0
$341$	−8.94427	−0.484359
$342$	4.00000	0.216295
$343$	0	0
$344$	−10.5279	−0.567624
$345$	0	0
$346$	4.00000	0.215041
$347$	−5.29180	−0.284078	−0.142039	−	0.989861i	$-0.545366\pi$
$347$	−5.29180	−0.284078	−0.142039	+	0.989861i	$0.545366\pi$
$348$	−1.85410	−0.0993903
$349$	23.4164	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$349$	23.4164	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$350$	0	0
$351$	−6.47214	−0.345457
$352$	−7.56231	−0.403072
$353$	−13.8885	−0.739213	−0.369606	−	0.929188i	$-0.620507\pi$
$353$	−13.8885	−0.739213	−0.369606	+	0.929188i	$0.620507\pi$
$354$	−10.4721	−0.556588
$355$	0	0
$356$	0.944272	0.0500463
$357$	0	0
$358$	1.52786	0.0807501
$359$	−11.7639	−0.620877	−0.310438	−	0.950594i	$-0.600476\pi$
$359$	−11.7639	−0.620877	−0.310438	+	0.950594i	$0.600476\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	−10.4721	−0.550403
$363$	−6.00000	−0.314918
$364$	0	0
$365$	0	0
$366$	−19.4164	−1.01491
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	20.5623	1.07188
$369$	8.94427	0.465620
$370$	0	0
$371$	0	0
$372$	2.47214	0.128174
$373$	17.4721	0.904673	0.452336	−	0.891847i	$-0.350591\pi$
$373$	17.4721	0.904673	0.452336	+	0.891847i	$0.350591\pi$
$374$	8.94427	0.462497
$375$	0	0
$376$	−23.4164	−1.20761
$377$	19.4164	0.999996
$378$	0	0
$379$	8.70820	0.447310	0.223655	−	0.974668i	$-0.428201\pi$
$379$	8.70820	0.447310	0.223655	+	0.974668i	$0.428201\pi$
$380$	0	0
$381$	19.1803	0.982639
$382$	−4.94427	−0.252971
$383$	19.4164	0.992132	0.496066	−	0.868285i	$-0.334777\pi$
$383$	19.4164	0.992132	0.496066	+	0.868285i	$0.334777\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	37.0344	1.88500
$387$	−4.70820	−0.239331
$388$	2.47214	0.125504
$389$	12.0557	0.611250	0.305625	−	0.952152i	$-0.401135\pi$
$389$	12.0557	0.611250	0.305625	+	0.952152i	$0.401135\pi$
$390$	0	0
$391$	−10.4721	−0.529599
$392$	0	0
$393$	13.5279	0.682390
$394$	33.9787	1.71182
$395$	0	0
$396$	−1.38197	−0.0694464
$397$	−9.88854	−0.496292	−0.248146	−	0.968723i	$-0.579821\pi$
$397$	−9.88854	−0.496292	−0.248146	+	0.968723i	$0.579821\pi$
$398$	35.4164	1.77526
$399$	0	0
$400$	0	0
$401$	28.4164	1.41905	0.709524	−	0.704681i	$-0.248910\pi$
$401$	28.4164	1.41905	0.709524	+	0.704681i	$0.248910\pi$
$402$	−20.5623	−1.02555
$403$	−25.8885	−1.28960
$404$	5.88854	0.292966
$405$	0	0
$406$	0	0
$407$	−7.76393	−0.384844
$408$	5.52786	0.273670
$409$	7.41641	0.366718	0.183359	−	0.983046i	$-0.441303\pi$
$409$	7.41641	0.366718	0.183359	+	0.983046i	$0.441303\pi$
$410$	0	0
$411$	−19.8885	−0.981030
$412$	−4.58359	−0.225817
$413$	0	0
$414$	6.85410	0.336861
$415$	0	0
$416$	−21.8885	−1.07317
$417$	−11.4164	−0.559064
$418$	−8.94427	−0.437479
$419$	29.8885	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$419$	29.8885	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$420$	0	0
$421$	10.4164	0.507665	0.253832	−	0.967248i	$-0.418309\pi$
$421$	10.4164	0.507665	0.253832	+	0.967248i	$0.418309\pi$
$422$	−22.4721	−1.09393
$423$	−10.4721	−0.509173
$424$	13.4164	0.651558
$425$	0	0
$426$	−13.3262	−0.645658
$427$	0	0
$428$	−5.52786	−0.267199
$429$	14.4721	0.698721
$430$	0	0
$431$	−11.0557	−0.532536	−0.266268	−	0.963899i	$-0.585791\pi$
$431$	−11.0557	−0.532536	−0.266268	+	0.963899i	$0.585791\pi$
$432$	−4.85410	−0.233543
$433$	12.9443	0.622062	0.311031	−	0.950400i	$-0.399326\pi$
$433$	12.9443	0.622062	0.311031	+	0.950400i	$0.399326\pi$
$434$	0	0
$435$	0	0
$436$	10.1459	0.485900
$437$	10.4721	0.500950
$438$	−23.4164	−1.11888
$439$	27.4164	1.30851	0.654257	−	0.756272i	$-0.272982\pi$
$439$	27.4164	1.30851	0.654257	+	0.756272i	$0.272982\pi$
$440$	0	0
$441$	0	0
$442$	25.8885	1.23139
$443$	−15.0557	−0.715319	−0.357660	−	0.933852i	$-0.616425\pi$
$443$	−15.0557	−0.715319	−0.357660	+	0.933852i	$0.616425\pi$
$444$	2.14590	0.101840
$445$	0	0
$446$	−5.52786	−0.261752
$447$	12.8885	0.609607
$448$	0	0
$449$	−6.52786	−0.308069	−0.154034	−	0.988065i	$-0.549227\pi$
$449$	−6.52786	−0.308069	−0.154034	+	0.988065i	$0.549227\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−6.50658	−0.306044
$453$	6.70820	0.315179
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−5.52786	−0.258866
$457$	−25.9443	−1.21362	−0.606811	−	0.794846i	$-0.707551\pi$
$457$	−25.9443	−1.21362	−0.606811	+	0.794846i	$0.707551\pi$
$458$	−46.8328	−2.18835
$459$	2.47214	0.115389
$460$	0	0
$461$	30.4721	1.41923	0.709614	−	0.704590i	$-0.248869\pi$
$461$	30.4721	1.41923	0.709614	+	0.704590i	$0.248869\pi$
$462$	0	0
$463$	−1.88854	−0.0877681	−0.0438840	−	0.999037i	$-0.513973\pi$
$463$	−1.88854	−0.0877681	−0.0438840	+	0.999037i	$0.513973\pi$
$464$	14.5623	0.676038
$465$	0	0
$466$	7.32624	0.339381
$467$	−1.88854	−0.0873914	−0.0436957	−	0.999045i	$-0.513913\pi$
$467$	−1.88854	−0.0873914	−0.0436957	+	0.999045i	$0.513913\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	0.944272	0.0435098
$472$	14.4721	0.666134
$473$	10.5279	0.484072
$474$	4.38197	0.201270
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	33.8885	1.55003
$479$	−30.4721	−1.39231	−0.696154	−	0.717893i	$-0.745107\pi$
$479$	−30.4721	−1.39231	−0.696154	+	0.717893i	$0.745107\pi$
$480$	0	0
$481$	−22.4721	−1.02464
$482$	−29.8885	−1.36139
$483$	0	0
$484$	−3.70820	−0.168555
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	14.7082	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$487$	14.7082	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$488$	26.8328	1.21466
$489$	12.0000	0.542659
$490$	0	0
$491$	4.81966	0.217508	0.108754	−	0.994069i	$-0.465314\pi$
$491$	4.81966	0.217508	0.108754	+	0.994069i	$0.465314\pi$
$492$	5.52786	0.249215
$493$	−7.41641	−0.334018
$494$	−25.8885	−1.16478
$495$	0	0
$496$	−19.4164	−0.871822
$497$	0	0
$498$	27.4164	1.22856
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	22.4721	1.00398
$502$	16.9443	0.756260
$503$	12.9443	0.577157	0.288578	−	0.957456i	$-0.406817\pi$
$503$	12.9443	0.577157	0.288578	+	0.957456i	$0.406817\pi$
$504$	0	0
$505$	0	0
$506$	−15.3262	−0.681334
$507$	28.8885	1.28299
$508$	11.8541	0.525941
$509$	9.52786	0.422315	0.211158	−	0.977452i	$-0.432277\pi$
$509$	9.52786	0.422315	0.211158	+	0.977452i	$0.432277\pi$
$510$	0	0
$511$	0	0
$512$	5.29180	0.233867
$513$	−2.47214	−0.109147
$514$	−14.4721	−0.638339
$515$	0	0
$516$	−2.90983	−0.128098
$517$	23.4164	1.02985
$518$	0	0
$519$	−2.47214	−0.108515
$520$	0	0
$521$	−20.9443	−0.917585	−0.458793	−	0.888543i	$-0.651718\pi$
$521$	−20.9443	−0.917585	−0.458793	+	0.888543i	$0.651718\pi$
$522$	4.85410	0.212458
$523$	1.88854	0.0825803	0.0412901	−	0.999147i	$-0.486853\pi$
$523$	1.88854	0.0825803	0.0412901	+	0.999147i	$0.486853\pi$
$524$	8.36068	0.365238
$525$	0	0
$526$	−34.2705	−1.49427
$527$	9.88854	0.430752
$528$	10.8541	0.472364
$529$	−5.05573	−0.219814
$530$	0	0
$531$	6.47214	0.280867
$532$	0	0
$533$	−57.8885	−2.50743
$534$	−2.47214	−0.106980
$535$	0	0
$536$	28.4164	1.22740
$537$	−0.944272	−0.0407483
$538$	14.4721	0.623938
$539$	0	0
$540$	0	0
$541$	−28.4164	−1.22172	−0.610858	−	0.791740i	$-0.709176\pi$
$541$	−28.4164	−1.22172	−0.610858	+	0.791740i	$0.709176\pi$
$542$	5.52786	0.237442
$543$	6.47214	0.277746
$544$	8.36068	0.358461
$545$	0	0
$546$	0	0
$547$	14.8197	0.633643	0.316821	−	0.948485i	$-0.397384\pi$
$547$	14.8197	0.633643	0.316821	+	0.948485i	$0.397384\pi$
$548$	−12.2918	−0.525080
$549$	12.0000	0.512148
$550$	0	0
$551$	7.41641	0.315950
$552$	−9.47214	−0.403161
$553$	0	0
$554$	−25.7082	−1.09224
$555$	0	0
$556$	−7.05573	−0.299230
$557$	20.8885	0.885076	0.442538	−	0.896750i	$-0.354078\pi$
$557$	20.8885	0.885076	0.442538	+	0.896750i	$0.354078\pi$
$558$	−6.47214	−0.273987
$559$	30.4721	1.28883
$560$	0	0
$561$	−5.52786	−0.233387
$562$	31.3262	1.32142
$563$	4.94427	0.208376	0.104188	−	0.994558i	$-0.466776\pi$
$563$	4.94427	0.208376	0.104188	+	0.994558i	$0.466776\pi$
$564$	−6.47214	−0.272526
$565$	0	0
$566$	0	0
$567$	0	0
$568$	18.4164	0.772735
$569$	−36.3050	−1.52198	−0.760991	−	0.648762i	$-0.775287\pi$
$569$	−36.3050	−1.52198	−0.760991	+	0.648762i	$0.775287\pi$
$570$	0	0
$571$	11.7639	0.492305	0.246153	−	0.969231i	$-0.420834\pi$
$571$	11.7639	0.492305	0.246153	+	0.969231i	$0.420834\pi$
$572$	8.94427	0.373979
$573$	3.05573	0.127655
$574$	0	0
$575$	0	0
$576$	4.23607	0.176503
$577$	−0.583592	−0.0242953	−0.0121476	−	0.999926i	$-0.503867\pi$
$577$	−0.583592	−0.0242953	−0.0121476	+	0.999926i	$0.503867\pi$
$578$	17.6180	0.732814
$579$	−22.8885	−0.951215
$580$	0	0
$581$	0	0
$582$	−6.47214	−0.268279
$583$	−13.4164	−0.555651
$584$	32.3607	1.33909
$585$	0	0
$586$	−26.4721	−1.09355
$587$	14.4721	0.597329	0.298664	−	0.954358i	$-0.403459\pi$
$587$	14.4721	0.597329	0.298664	+	0.954358i	$0.403459\pi$
$588$	0	0
$589$	−9.88854	−0.407450
$590$	0	0
$591$	−21.0000	−0.863825
$592$	−16.8541	−0.692699
$593$	−44.9443	−1.84564	−0.922820	−	0.385231i	$-0.874122\pi$
$593$	−44.9443	−1.84564	−0.922820	+	0.385231i	$0.874122\pi$
$594$	3.61803	0.148450
$595$	0	0
$596$	7.96556	0.326282
$597$	−21.8885	−0.895838
$598$	−44.3607	−1.81404
$599$	12.7082	0.519243	0.259622	−	0.965710i	$-0.416402\pi$
$599$	12.7082	0.519243	0.259622	+	0.965710i	$0.416402\pi$
$600$	0	0
$601$	36.9443	1.50699	0.753494	−	0.657455i	$-0.228367\pi$
$601$	36.9443	1.50699	0.753494	+	0.657455i	$0.228367\pi$
$602$	0	0
$603$	12.7082	0.517518
$604$	4.14590	0.168694
$605$	0	0
$606$	−15.4164	−0.626249
$607$	35.4164	1.43751	0.718754	−	0.695265i	$-0.244713\pi$
$607$	35.4164	1.43751	0.718754	+	0.695265i	$0.244713\pi$
$608$	−8.36068	−0.339070
$609$	0	0
$610$	0	0
$611$	67.7771	2.74197
$612$	1.52786	0.0617602
$613$	35.2492	1.42370	0.711851	−	0.702330i	$-0.247857\pi$
$613$	35.2492	1.42370	0.711851	+	0.702330i	$0.247857\pi$
$614$	−50.8328	−2.05145
$615$	0	0
$616$	0	0
$617$	−34.4164	−1.38555	−0.692776	−	0.721153i	$-0.743613\pi$
$617$	−34.4164	−1.38555	−0.692776	+	0.721153i	$0.743613\pi$
$618$	12.0000	0.482711
$619$	−17.5279	−0.704504	−0.352252	−	0.935905i	$-0.614584\pi$
$619$	−17.5279	−0.704504	−0.352252	+	0.935905i	$0.614584\pi$
$620$	0	0
$621$	−4.23607	−0.169988
$622$	−32.9443	−1.32094
$623$	0	0
$624$	31.4164	1.25766
$625$	0	0
$626$	−12.0000	−0.479616
$627$	5.52786	0.220762
$628$	0.583592	0.0232879
$629$	8.58359	0.342250
$630$	0	0
$631$	0.347524	0.0138347	0.00691736	−	0.999976i	$-0.497798\pi$
$631$	0.347524	0.0138347	0.00691736	+	0.999976i	$0.497798\pi$
$632$	−6.05573	−0.240884
$633$	13.8885	0.552020
$634$	−12.8541	−0.510502
$635$	0	0
$636$	3.70820	0.147040
$637$	0	0
$638$	−10.8541	−0.429718
$639$	8.23607	0.325814
$640$	0	0
$641$	25.3607	1.00169	0.500843	−	0.865538i	$-0.333023\pi$
$641$	25.3607	1.00169	0.500843	+	0.865538i	$0.333023\pi$
$642$	14.4721	0.571170
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	0	0
$645$	0	0
$646$	9.88854	0.389060
$647$	−1.88854	−0.0742463	−0.0371232	−	0.999311i	$-0.511819\pi$
$647$	−1.88854	−0.0742463	−0.0371232	+	0.999311i	$0.511819\pi$
$648$	2.23607	0.0878410
$649$	−14.4721	−0.568081
$650$	0	0
$651$	0	0
$652$	7.41641	0.290449
$653$	11.8885	0.465235	0.232617	−	0.972568i	$-0.425271\pi$
$653$	11.8885	0.465235	0.232617	+	0.972568i	$0.425271\pi$
$654$	−26.5623	−1.03867
$655$	0	0
$656$	−43.4164	−1.69513
$657$	14.4721	0.564612
$658$	0	0
$659$	−15.0557	−0.586488	−0.293244	−	0.956038i	$-0.594735\pi$
$659$	−15.0557	−0.586488	−0.293244	+	0.956038i	$0.594735\pi$
$660$	0	0
$661$	20.3607	0.791939	0.395969	−	0.918264i	$-0.370409\pi$
$661$	20.3607	0.791939	0.395969	+	0.918264i	$0.370409\pi$
$662$	53.6869	2.08660
$663$	−16.0000	−0.621389
$664$	−37.8885	−1.47036
$665$	0	0
$666$	−5.61803	−0.217694
$667$	12.7082	0.492064
$668$	13.8885	0.537364
$669$	3.41641	0.132086
$670$	0	0
$671$	−26.8328	−1.03587
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	3.23607	0.124649
$675$	0	0
$676$	17.8541	0.686696
$677$	49.8885	1.91737	0.958686	−	0.284466i	$-0.0918162\pi$
$677$	49.8885	1.91737	0.958686	+	0.284466i	$0.0918162\pi$
$678$	17.0344	0.654204
$679$	0	0
$680$	0	0
$681$	7.41641	0.284197
$682$	14.4721	0.554167
$683$	−27.1803	−1.04003	−0.520013	−	0.854158i	$-0.674073\pi$
$683$	−27.1803	−1.04003	−0.520013	+	0.854158i	$0.674073\pi$
$684$	−1.52786	−0.0584193
$685$	0	0
$686$	0	0
$687$	28.9443	1.10429
$688$	22.8541	0.871304
$689$	−38.8328	−1.47941
$690$	0	0
$691$	10.8328	0.412100	0.206050	−	0.978541i	$-0.433939\pi$
$691$	10.8328	0.412100	0.206050	+	0.978541i	$0.433939\pi$
$692$	−1.52786	−0.0580807
$693$	0	0
$694$	8.56231	0.325021
$695$	0	0
$696$	−6.70820	−0.254274
$697$	22.1115	0.837531
$698$	−37.8885	−1.43410
$699$	−4.52786	−0.171260
$700$	0	0
$701$	37.7771	1.42682	0.713410	−	0.700746i	$-0.247150\pi$
$701$	37.7771	1.42682	0.713410	+	0.700746i	$0.247150\pi$
$702$	10.4721	0.395245
$703$	−8.58359	−0.323736
$704$	−9.47214	−0.356995
$705$	0	0
$706$	22.4721	0.845750
$707$	0	0
$708$	4.00000	0.150329
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−2.70820	−0.101566
$712$	3.41641	0.128035
$713$	−16.9443	−0.634568
$714$	0	0
$715$	0	0
$716$	−0.583592	−0.0218099
$717$	−20.9443	−0.782178
$718$	19.0344	0.710359
$719$	−0.944272	−0.0352154	−0.0176077	−	0.999845i	$-0.505605\pi$
$719$	−0.944272	−0.0352154	−0.0176077	+	0.999845i	$0.505605\pi$
$720$	0	0
$721$	0	0
$722$	20.8541	0.776109
$723$	18.4721	0.686986
$724$	4.00000	0.148659
$725$	0	0
$726$	9.70820	0.360305
$727$	−17.3050	−0.641805	−0.320903	−	0.947112i	$-0.603986\pi$
$727$	−17.3050	−0.641805	−0.320903	+	0.947112i	$0.603986\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.6393	−0.430496
$732$	7.41641	0.274118
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−6.47214	−0.238891
$735$	0	0
$736$	−14.3262	−0.528072
$737$	−28.4164	−1.04673
$738$	−14.4721	−0.532727
$739$	−20.7082	−0.761764	−0.380882	−	0.924624i	$-0.624380\pi$
$739$	−20.7082	−0.761764	−0.380882	+	0.924624i	$0.624380\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−3.05573	−0.112104	−0.0560519	−	0.998428i	$-0.517851\pi$
$743$	−3.05573	−0.112104	−0.0560519	+	0.998428i	$0.517851\pi$
$744$	8.94427	0.327913
$745$	0	0
$746$	−28.2705	−1.03506
$747$	−16.9443	−0.619958
$748$	−3.41641	−0.124916
$749$	0	0
$750$	0	0
$751$	−19.7771	−0.721676	−0.360838	−	0.932628i	$-0.617509\pi$
$751$	−19.7771	−0.721676	−0.360838	+	0.932628i	$0.617509\pi$
$752$	50.8328	1.85368
$753$	−10.4721	−0.381626
$754$	−31.4164	−1.14412
$755$	0	0
$756$	0	0
$757$	−5.47214	−0.198888	−0.0994441	−	0.995043i	$-0.531706\pi$
$757$	−5.47214	−0.198888	−0.0994441	+	0.995043i	$0.531706\pi$
$758$	−14.0902	−0.511778
$759$	9.47214	0.343817
$760$	0	0
$761$	−52.9443	−1.91923	−0.959614	−	0.281319i	$-0.909228\pi$
$761$	−52.9443	−1.91923	−0.959614	+	0.281319i	$0.909228\pi$
$762$	−31.0344	−1.12426
$763$	0	0
$764$	1.88854	0.0683251
$765$	0	0
$766$	−31.4164	−1.13512
$767$	−41.8885	−1.51251
$768$	13.5623	0.489388
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	8.94427	0.322120
$772$	−14.1459	−0.509122
$773$	39.7771	1.43068	0.715341	−	0.698775i	$-0.246271\pi$
$773$	39.7771	1.43068	0.715341	+	0.698775i	$0.246271\pi$
$774$	7.61803	0.273825
$775$	0	0
$776$	8.94427	0.321081
$777$	0	0
$778$	−19.5066	−0.699345
$779$	−22.1115	−0.792225
$780$	0	0
$781$	−18.4164	−0.658991
$782$	16.9443	0.605926
$783$	−3.00000	−0.107211
$784$	0	0
$785$	0	0
$786$	−21.8885	−0.780739
$787$	33.5279	1.19514	0.597570	−	0.801817i	$-0.296133\pi$
$787$	33.5279	1.19514	0.597570	+	0.801817i	$0.296133\pi$
$788$	−12.9787	−0.462348
$789$	21.1803	0.754040
$790$	0	0
$791$	0	0
$792$	−5.00000	−0.177667
$793$	−77.6656	−2.75799
$794$	16.0000	0.567819
$795$	0	0
$796$	−13.5279	−0.479482
$797$	−34.4721	−1.22107	−0.610533	−	0.791991i	$-0.709045\pi$
$797$	−34.4721	−1.22107	−0.610533	+	0.791991i	$0.709045\pi$
$798$	0	0
$799$	−25.8885	−0.915871
$800$	0	0
$801$	1.52786	0.0539844
$802$	−45.9787	−1.62356
$803$	−32.3607	−1.14198
$804$	7.85410	0.276993
$805$	0	0
$806$	41.8885	1.47546
$807$	−8.94427	−0.314853
$808$	21.3050	0.749506
$809$	8.52786	0.299824	0.149912	−	0.988699i	$-0.452101\pi$
$809$	8.52786	0.299824	0.149912	+	0.988699i	$0.452101\pi$
$810$	0	0
$811$	38.8328	1.36360	0.681802	−	0.731536i	$-0.261196\pi$
$811$	38.8328	1.36360	0.681802	+	0.731536i	$0.261196\pi$
$812$	0	0
$813$	−3.41641	−0.119819
$814$	12.5623	0.440309
$815$	0	0
$816$	−12.0000	−0.420084
$817$	11.6393	0.407208
$818$	−12.0000	−0.419570
$819$	0	0
$820$	0	0
$821$	−45.7771	−1.59763	−0.798816	−	0.601576i	$-0.794540\pi$
$821$	−45.7771	−1.59763	−0.798816	+	0.601576i	$0.794540\pi$
$822$	32.1803	1.12242
$823$	−17.5410	−0.611442	−0.305721	−	0.952121i	$-0.598897\pi$
$823$	−17.5410	−0.611442	−0.305721	+	0.952121i	$0.598897\pi$
$824$	−16.5836	−0.577717
$825$	0	0
$826$	0	0
$827$	−14.2361	−0.495037	−0.247518	−	0.968883i	$-0.579615\pi$
$827$	−14.2361	−0.495037	−0.247518	+	0.968883i	$0.579615\pi$
$828$	−2.61803	−0.0909830
$829$	−2.47214	−0.0858608	−0.0429304	−	0.999078i	$-0.513669\pi$
$829$	−2.47214	−0.0858608	−0.0429304	+	0.999078i	$0.513669\pi$
$830$	0	0
$831$	15.8885	0.551167
$832$	−27.4164	−0.950493
$833$	0	0
$834$	18.4721	0.639638
$835$	0	0
$836$	3.41641	0.118159
$837$	4.00000	0.138260
$838$	−48.3607	−1.67059
$839$	−55.1935	−1.90549	−0.952746	−	0.303770i	$-0.901755\pi$
$839$	−55.1935	−1.90549	−0.952746	+	0.303770i	$0.901755\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−16.8541	−0.580831
$843$	−19.3607	−0.666817
$844$	8.58359	0.295459
$845$	0	0
$846$	16.9443	0.582556
$847$	0	0
$848$	−29.1246	−1.00014
$849$	0	0
$850$	0	0
$851$	−14.7082	−0.504191
$852$	5.09017	0.174386
$853$	−6.83282	−0.233951	−0.116976	−	0.993135i	$-0.537320\pi$
$853$	−6.83282	−0.233951	−0.116976	+	0.993135i	$0.537320\pi$
$854$	0	0
$855$	0	0
$856$	−20.0000	−0.683586
$857$	−37.8885	−1.29425	−0.647124	−	0.762385i	$-0.724028\pi$
$857$	−37.8885	−1.29425	−0.647124	+	0.762385i	$0.724028\pi$
$858$	−23.4164	−0.799423
$859$	−25.8885	−0.883306	−0.441653	−	0.897186i	$-0.645608\pi$
$859$	−25.8885	−0.883306	−0.441653	+	0.897186i	$0.645608\pi$
$860$	0	0
$861$	0	0
$862$	17.8885	0.609286
$863$	2.12461	0.0723226	0.0361613	−	0.999346i	$-0.488487\pi$
$863$	2.12461	0.0723226	0.0361613	+	0.999346i	$0.488487\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	−20.9443	−0.711715
$867$	−10.8885	−0.369794
$868$	0	0
$869$	6.05573	0.205427
$870$	0	0
$871$	−82.2492	−2.78691
$872$	36.7082	1.24310
$873$	4.00000	0.135379
$874$	−16.9443	−0.573149
$875$	0	0
$876$	8.94427	0.302199
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−44.3607	−1.49710
$879$	16.3607	0.551832
$880$	0	0
$881$	40.9443	1.37945	0.689724	−	0.724073i	$-0.257732\pi$
$881$	40.9443	1.37945	0.689724	+	0.724073i	$0.257732\pi$
$882$	0	0
$883$	43.5410	1.46527	0.732636	−	0.680621i	$-0.238290\pi$
$883$	43.5410	1.46527	0.732636	+	0.680621i	$0.238290\pi$
$884$	−9.88854	−0.332588
$885$	0	0
$886$	24.3607	0.818413
$887$	−26.8328	−0.900958	−0.450479	−	0.892787i	$-0.648747\pi$
$887$	−26.8328	−0.900958	−0.450479	+	0.892787i	$0.648747\pi$
$888$	7.76393	0.260540
$889$	0	0
$890$	0	0
$891$	−2.23607	−0.0749111
$892$	2.11146	0.0706968
$893$	25.8885	0.866327
$894$	−20.8541	−0.697466
$895$	0	0
$896$	0	0
$897$	27.4164	0.915407
$898$	10.5623	0.352469
$899$	−12.0000	−0.400222
$900$	0	0
$901$	14.8328	0.494153
$902$	32.3607	1.07749
$903$	0	0
$904$	−23.5410	−0.782963
$905$	0	0
$906$	−10.8541	−0.360603
$907$	−45.8885	−1.52370	−0.761852	−	0.647751i	$-0.775710\pi$
$907$	−45.8885	−1.52370	−0.761852	+	0.647751i	$0.775710\pi$
$908$	4.58359	0.152112
$909$	9.52786	0.316019
$910$	0	0
$911$	−56.0132	−1.85580	−0.927899	−	0.372831i	$-0.878387\pi$
$911$	−56.0132	−1.85580	−0.927899	+	0.372831i	$0.878387\pi$
$912$	12.0000	0.397360
$913$	37.8885	1.25393
$914$	41.9787	1.38853
$915$	0	0
$916$	17.8885	0.591054
$917$	0	0
$918$	−4.00000	−0.132020
$919$	15.1803	0.500753	0.250377	−	0.968149i	$-0.419446\pi$
$919$	15.1803	0.500753	0.250377	+	0.968149i	$0.419446\pi$
$920$	0	0
$921$	31.4164	1.03521
$922$	−49.3050	−1.62377
$923$	−53.3050	−1.75455
$924$	0	0
$925$	0	0
$926$	3.05573	0.100417
$927$	−7.41641	−0.243587
$928$	−10.1459	−0.333055
$929$	−4.58359	−0.150383	−0.0751914	−	0.997169i	$-0.523957\pi$
$929$	−4.58359	−0.150383	−0.0751914	+	0.997169i	$0.523957\pi$
$930$	0	0
$931$	0	0
$932$	−2.79837	−0.0916638
$933$	20.3607	0.666579
$934$	3.05573	0.0999865
$935$	0	0
$936$	−14.4721	−0.473037
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	7.41641	0.242025
$940$	0	0
$941$	−31.7771	−1.03590	−0.517952	−	0.855410i	$-0.673305\pi$
$941$	−31.7771	−1.03590	−0.517952	+	0.855410i	$0.673305\pi$
$942$	−1.52786	−0.0497805
$943$	−37.8885	−1.23382
$944$	−31.4164	−1.02252
$945$	0	0
$946$	−17.0344	−0.553837
$947$	32.9443	1.07054	0.535272	−	0.844679i	$-0.320209\pi$
$947$	32.9443	1.07054	0.535272	+	0.844679i	$0.320209\pi$
$948$	−1.67376	−0.0543613
$949$	−93.6656	−3.04052
$950$	0	0
$951$	7.94427	0.257611
$952$	0	0
$953$	11.3607	0.368009	0.184004	−	0.982925i	$-0.441094\pi$
$953$	11.3607	0.368009	0.184004	+	0.982925i	$0.441094\pi$
$954$	−9.70820	−0.314315
$955$	0	0
$956$	−12.9443	−0.418648
$957$	6.70820	0.216845
$958$	49.3050	1.59297
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	36.3607	1.17232
$963$	−8.94427	−0.288225
$964$	11.4164	0.367698
$965$	0	0
$966$	0	0
$967$	−43.7771	−1.40778	−0.703888	−	0.710311i	$-0.748554\pi$
$967$	−43.7771	−1.40778	−0.703888	+	0.710311i	$0.748554\pi$
$968$	−13.4164	−0.431220
$969$	−6.11146	−0.196328
$970$	0	0
$971$	41.3050	1.32554	0.662769	−	0.748823i	$-0.269381\pi$
$971$	41.3050	1.32554	0.662769	+	0.748823i	$0.269381\pi$
$972$	0.618034	0.0198234
$973$	0	0
$974$	−23.7984	−0.762549
$975$	0	0
$976$	−58.2492	−1.86451
$977$	−7.58359	−0.242621	−0.121310	−	0.992615i	$-0.538710\pi$
$977$	−7.58359	−0.242621	−0.121310	+	0.992615i	$0.538710\pi$
$978$	−19.4164	−0.620868
$979$	−3.41641	−0.109189
$980$	0	0
$981$	16.4164	0.524136
$982$	−7.79837	−0.248856
$983$	−23.0557	−0.735364	−0.367682	−	0.929952i	$-0.619848\pi$
$983$	−23.0557	−0.735364	−0.367682	+	0.929952i	$0.619848\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	9.88854	0.314596
$989$	19.9443	0.634191
$990$	0	0
$991$	−57.5410	−1.82785	−0.913925	−	0.405882i	$-0.866964\pi$
$991$	−57.5410	−1.82785	−0.913925	+	0.405882i	$0.866964\pi$
$992$	13.5279	0.429510
$993$	−33.1803	−1.05295
$994$	0	0
$995$	0	0
$996$	−10.4721	−0.331822
$997$	−7.41641	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$997$	−7.41641	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$998$	19.4164	0.614616
$999$	3.47214	0.109854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.v.1.1	yes	2
5.4	even	2		3675.2.a.ba.1.2	yes	2
7.6	odd	2		3675.2.a.u.1.1	✓	2
35.34	odd	2		3675.2.a.bc.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.2.a.u.1.1	✓	2	7.6	odd	2
3675.2.a.v.1.1	yes	2	1.1	even	1	trivial
3675.2.a.ba.1.2	yes	2	5.4	even	2
3675.2.a.bc.1.2	yes	2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{11} +0.618034 q^{12} -6.47214 q^{13} -4.85410 q^{16} +2.47214 q^{17} -1.61803 q^{18} -2.47214 q^{19} +3.61803 q^{22} -4.23607 q^{23} +2.23607 q^{24} +10.4721 q^{26} +1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} +3.38197 q^{32} -2.23607 q^{33} -4.00000 q^{34} +0.618034 q^{36} +3.47214 q^{37} +4.00000 q^{38} -6.47214 q^{39} +8.94427 q^{41} -4.70820 q^{43} -1.38197 q^{44} +6.85410 q^{46} -10.4721 q^{47} -4.85410 q^{48} +2.47214 q^{51} -4.00000 q^{52} +6.00000 q^{53} -1.61803 q^{54} -2.47214 q^{57} +4.85410 q^{58} +6.47214 q^{59} +12.0000 q^{61} -6.47214 q^{62} +4.23607 q^{64} +3.61803 q^{66} +12.7082 q^{67} +1.52786 q^{68} -4.23607 q^{69} +8.23607 q^{71} +2.23607 q^{72} +14.4721 q^{73} -5.61803 q^{74} -1.52786 q^{76} +10.4721 q^{78} -2.70820 q^{79} +1.00000 q^{81} -14.4721 q^{82} -16.9443 q^{83} +7.61803 q^{86} -3.00000 q^{87} -5.00000 q^{88} +1.52786 q^{89} -2.61803 q^{92} +4.00000 q^{93} +16.9443 q^{94} +3.38197 q^{96} +4.00000 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + 2 q^{9} - q^{12} - 4 q^{13} - 3 q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + 5 q^{22} - 4 q^{23} + 12 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 9 q^{32} - 8 q^{34} - q^{36} - 2 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{43} - 5 q^{44} + 7 q^{46} - 12 q^{47} - 3 q^{48} - 4 q^{51} - 8 q^{52} + 12 q^{53} - q^{54} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 5 q^{66} + 12 q^{67} + 12 q^{68} - 4 q^{69} + 12 q^{71} + 20 q^{73} - 9 q^{74} - 12 q^{76} + 12 q^{78} + 8 q^{79} + 2 q^{81} - 20 q^{82} - 16 q^{83} + 13 q^{86} - 6 q^{87} - 10 q^{88} + 12 q^{89} - 3 q^{92} + 8 q^{93} + 16 q^{94} + 9 q^{96} + 8 q^{97}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + 2 * q^9 - q^12 - 4 * q^13 - 3 * q^16 - 4 * q^17 - q^18 + 4 * q^19 + 5 * q^22 - 4 * q^23 + 12 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 9 * q^32 - 8 * q^34 - q^36 - 2 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^43 - 5 * q^44 + 7 * q^46 - 12 * q^47 - 3 * q^48 - 4 * q^51 - 8 * q^52 + 12 * q^53 - q^54 + 4 * q^57 + 3 * q^58 + 4 * q^59 + 24 * q^61 - 4 * q^62 + 4 * q^64 + 5 * q^66 + 12 * q^67 + 12 * q^68 - 4 * q^69 + 12 * q^71 + 20 * q^73 - 9 * q^74 - 12 * q^76 + 12 * q^78 + 8 * q^79 + 2 * q^81 - 20 * q^82 - 16 * q^83 + 13 * q^86 - 6 * q^87 - 10 * q^88 + 12 * q^89 - 3 * q^92 + 8 * q^93 + 16 * q^94 + 9 * q^96 + 8 * q^97

Introduction

L-functions

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Database

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