LMFDB - Embedded newform 3675.2.a.bc.1.2

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:	$1$
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			1.2
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	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	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	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	0	0
$22$	−3.61803	−0.771367
$23$	4.23607	0.883281	0.441641	−	0.897192i	$-0.354397\pi$
$23$	4.23607	0.883281	0.441641	+	0.897192i	$0.354397\pi$
$24$	−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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−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	$-0.592129\pi$
$37$	−3.47214	−0.570816	−0.285408	+	0.958406i	$0.592129\pi$
$38$	4.00000	0.648886
$39$	−6.47214	−1.03637
$40$	0	0
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	0	0
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	−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	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	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	$-0.638426\pi$
$59$	−6.47214	−0.842600	−0.421300	+	0.906921i	$0.638426\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	−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	$-0.782894\pi$
$67$	−12.7082	−1.55255	−0.776277	+	0.630392i	$0.782894\pi$
$68$	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	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	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	$-0.657201\pi$
$101$	−9.52786	−0.948058	−0.474029	+	0.880509i	$0.657201\pi$
$102$	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	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	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	$-0.335099\pi$
$113$	10.5279	0.990378	0.495189	+	0.868785i	$0.335099\pi$
$114$	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	$-0.823997\pi$
$127$	−19.1803	−1.70198	−0.850990	+	0.525182i	$0.823997\pi$
$128$	13.6180	1.20368
$129$	4.70820	0.414534
$130$	0	0
$131$	−13.5279	−1.18193	−0.590967	−	0.806695i	$-0.701254\pi$
$131$	−13.5279	−1.18193	−0.590967	+	0.806695i	$0.701254\pi$
$132$	−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	$-0.176846\pi$
$137$	19.8885	1.69919	0.849596	+	0.527433i	$0.176846\pi$
$138$	6.85410	0.583460
$139$	11.4164	0.968327	0.484164	−	0.874978i	$-0.339124\pi$
$139$	11.4164	0.968327	0.484164	+	0.874978i	$0.339124\pi$
$140$	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	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−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	$-0.577323\pi$
$181$	−6.47214	−0.481070	−0.240535	+	0.970640i	$0.577323\pi$
$182$	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	$-0.191863\pi$
$193$	22.8885	1.64755	0.823777	+	0.566914i	$0.191863\pi$
$194$	6.47214	0.464672
$195$	0	0
$196$	0	0
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	−3.61803	−0.257122
$199$	21.8885	1.55164	0.775819	−	0.630956i	$-0.217337\pi$
$199$	21.8885	1.55164	0.775819	+	0.630956i	$0.217337\pi$
$200$	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	$-0.905600\pi$
$229$	−28.9443	−1.91269	−0.956346	+	0.292238i	$0.905600\pi$
$230$	0	0
$231$	0	0
$232$	6.70820	0.440415
$233$	4.52786	0.296630	0.148315	−	0.988940i	$-0.452615\pi$
$233$	4.52786	0.296630	0.148315	+	0.988940i	$0.452615\pi$
$234$	−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	$-0.702827\pi$
$241$	−18.4721	−1.18989	−0.594947	+	0.803765i	$0.702827\pi$
$242$	−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	$-0.392783\pi$
$251$	10.4721	0.660995	0.330498	+	0.943807i	$0.392783\pi$
$252$	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	$-0.726497\pi$
$263$	−21.1803	−1.30604	−0.653018	+	0.757343i	$0.726497\pi$
$264$	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	$-0.412093\pi$
$269$	8.94427	0.545342	0.272671	+	0.962107i	$0.412093\pi$
$270$	0	0
$271$	3.41641	0.207532	0.103766	−	0.994602i	$-0.466911\pi$
$271$	3.41641	0.207532	0.103766	+	0.994602i	$0.466911\pi$
$272$	−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	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	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	$-0.695884\pi$
$311$	−20.3607	−1.15455	−0.577274	+	0.816550i	$0.695884\pi$
$312$	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	$-0.571617\pi$
$317$	−7.94427	−0.446195	−0.223097	+	0.974796i	$0.571617\pi$
$318$	−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	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	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	$-0.454634\pi$
$347$	5.29180	0.284078	0.142039	+	0.989861i	$0.454634\pi$
$348$	−1.85410	−0.0993903
$349$	−23.4164	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$349$	−23.4164	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$350$	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	$-0.649409\pi$
$373$	−17.4721	−0.904673	−0.452336	+	0.891847i	$0.649409\pi$
$374$	−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	$-0.558697\pi$
$409$	−7.41641	−0.366718	−0.183359	+	0.983046i	$0.558697\pi$
$410$	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	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	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	$-0.727018\pi$
$439$	−27.4164	−1.30851	−0.654257	+	0.756272i	$0.727018\pi$
$440$	0	0
$441$	0	0
$442$	−25.8885	−1.23139
$443$	15.0557	0.715319	0.357660	−	0.933852i	$-0.383575\pi$
$443$	15.0557	0.715319	0.357660	+	0.933852i	$0.383575\pi$
$444$	−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	$-0.292449\pi$
$457$	25.9443	1.21362	0.606811	+	0.794846i	$0.292449\pi$
$458$	−46.8328	−2.18835
$459$	2.47214	0.115389
$460$	0	0
$461$	−30.4721	−1.41923	−0.709614	−	0.704590i	$-0.751131\pi$
$461$	−30.4721	−1.41923	−0.709614	+	0.704590i	$0.751131\pi$
$462$	0	0
$463$	1.88854	0.0877681	0.0438840	−	0.999037i	$-0.486027\pi$
$463$	1.88854	0.0877681	0.0438840	+	0.999037i	$0.486027\pi$
$464$	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	$-0.254893\pi$
$479$	30.4721	1.39231	0.696154	+	0.717893i	$0.254893\pi$
$480$	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	$-0.608144\pi$
$487$	−14.7082	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$488$	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	$-0.567723\pi$
$509$	−9.52786	−0.422315	−0.211158	+	0.977452i	$0.567723\pi$
$510$	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	$-0.348282\pi$
$521$	20.9443	0.917585	0.458793	+	0.888543i	$0.348282\pi$
$522$	−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	$-0.602616\pi$
$547$	−14.8197	−0.633643	−0.316821	+	0.948485i	$0.602616\pi$
$548$	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	$-0.645922\pi$
$557$	−20.8885	−0.885076	−0.442538	+	0.896750i	$0.645922\pi$
$558$	−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	$-0.771633\pi$
$601$	−36.9443	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$602$	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	$-0.752143\pi$
$613$	−35.2492	−1.42370	−0.711851	+	0.702330i	$0.752143\pi$
$614$	50.8328	2.05145
$615$	0	0
$616$	0	0
$617$	34.4164	1.38555	0.692776	−	0.721153i	$-0.256387\pi$
$617$	34.4164	1.38555	0.692776	+	0.721153i	$0.256387\pi$
$618$	−12.0000	−0.482711
$619$	17.5279	0.704504	0.352252	−	0.935905i	$-0.385416\pi$
$619$	17.5279	0.704504	0.352252	+	0.935905i	$0.385416\pi$
$620$	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	$-0.574729\pi$
$653$	−11.8885	−0.465235	−0.232617	+	0.972568i	$0.574729\pi$
$654$	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	$-0.629591\pi$
$661$	−20.3607	−0.791939	−0.395969	+	0.918264i	$0.629591\pi$
$662$	−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	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	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	$-0.325927\pi$
$683$	27.1803	1.04003	0.520013	+	0.854158i	$0.325927\pi$
$684$	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	$-0.566061\pi$
$691$	−10.8328	−0.412100	−0.206050	+	0.978541i	$0.566061\pi$
$692$	−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	$-0.494395\pi$
$719$	0.944272	0.0352154	0.0176077	+	0.999845i	$0.494395\pi$
$720$	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	$-0.482149\pi$
$743$	3.05573	0.112104	0.0560519	+	0.998428i	$0.482149\pi$
$744$	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	$-0.468294\pi$
$757$	5.47214	0.198888	0.0994441	+	0.995043i	$0.468294\pi$
$758$	14.0902	0.511778
$759$	−9.47214	−0.343817
$760$	0	0
$761$	52.9443	1.91923	0.959614	−	0.281319i	$-0.0907720\pi$
$761$	52.9443	1.91923	0.959614	+	0.281319i	$0.0907720\pi$
$762$	−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	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	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	$-0.738804\pi$
$811$	−38.8328	−1.36360	−0.681802	+	0.731536i	$0.738804\pi$
$812$	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	$-0.401103\pi$
$823$	17.5410	0.611442	0.305721	+	0.952121i	$0.401103\pi$
$824$	16.5836	0.577717
$825$	0	0
$826$	0	0
$827$	14.2361	0.495037	0.247518	−	0.968883i	$-0.420385\pi$
$827$	14.2361	0.495037	0.247518	+	0.968883i	$0.420385\pi$
$828$	2.61803	0.0909830
$829$	2.47214	0.0858608	0.0429304	−	0.999078i	$-0.486331\pi$
$829$	2.47214	0.0858608	0.0429304	+	0.999078i	$0.486331\pi$
$830$	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	$-0.0982453\pi$
$839$	55.1935	1.90549	0.952746	+	0.303770i	$0.0982453\pi$
$840$	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	$-0.354392\pi$
$859$	25.8885	0.883306	0.441653	+	0.897186i	$0.354392\pi$
$860$	0	0
$861$	0	0
$862$	−17.8885	−0.609286
$863$	−2.12461	−0.0723226	−0.0361613	−	0.999346i	$-0.511513\pi$
$863$	−2.12461	−0.0723226	−0.0361613	+	0.999346i	$0.511513\pi$
$864$	−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	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−44.3607	−1.49710
$879$	16.3607	0.551832
$880$	0	0
$881$	−40.9443	−1.37945	−0.689724	−	0.724073i	$-0.742268\pi$
$881$	−40.9443	−1.37945	−0.689724	+	0.724073i	$0.742268\pi$
$882$	0	0
$883$	−43.5410	−1.46527	−0.732636	−	0.680621i	$-0.761710\pi$
$883$	−43.5410	−1.46527	−0.732636	+	0.680621i	$0.761710\pi$
$884$	−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	$-0.224290\pi$
$907$	45.8885	1.52370	0.761852	+	0.647751i	$0.224290\pi$
$908$	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	$-0.476043\pi$
$929$	4.58359	0.150383	0.0751914	+	0.997169i	$0.476043\pi$
$930$	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	$-0.326695\pi$
$941$	31.7771	1.03590	0.517952	+	0.855410i	$0.326695\pi$
$942$	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	$-0.679791\pi$
$947$	−32.9443	−1.07054	−0.535272	+	0.844679i	$0.679791\pi$
$948$	−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	$-0.558906\pi$
$953$	−11.3607	−0.368009	−0.184004	+	0.982925i	$0.558906\pi$
$954$	−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	$-0.251446\pi$
$967$	43.7771	1.40778	0.703888	+	0.710311i	$0.251446\pi$
$968$	13.4164	0.431220
$969$	6.11146	0.196328
$970$	0	0
$971$	−41.3050	−1.32554	−0.662769	−	0.748823i	$-0.730619\pi$
$971$	−41.3050	−1.32554	−0.662769	+	0.748823i	$0.730619\pi$
$972$	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	$-0.461290\pi$
$977$	7.58359	0.242621	0.121310	+	0.992615i	$0.461290\pi$
$978$	−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.bc.1.2	yes	2
5.4	even	2		3675.2.a.u.1.1	✓	2
7.6	odd	2		3675.2.a.ba.1.2	yes	2
35.34	odd	2		3675.2.a.v.1.1	yes	2

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

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

Modular forms

Varieties

Fields

Representations

Groups

Database

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