LMFDB - Embedded newform 3675.2.a.y.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:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
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)

$f(q)$	$=$	$q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -2.23607 q^{8} +1.00000 q^{9} -2.47214 q^{11} -3.00000 q^{12} -4.47214 q^{13} -1.00000 q^{16} -2.00000 q^{17} -2.23607 q^{18} -6.47214 q^{19} +5.52786 q^{22} -4.00000 q^{23} +2.23607 q^{24} +10.0000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -10.4721 q^{31} +6.70820 q^{32} +2.47214 q^{33} +4.47214 q^{34} +3.00000 q^{36} -10.9443 q^{37} +14.4721 q^{38} +4.47214 q^{39} +2.00000 q^{41} +8.94427 q^{43} -7.41641 q^{44} +8.94427 q^{46} -4.94427 q^{47} +1.00000 q^{48} +2.00000 q^{51} -13.4164 q^{52} +12.4721 q^{53} +2.23607 q^{54} +6.47214 q^{57} +4.47214 q^{58} -8.94427 q^{59} +2.00000 q^{61} +23.4164 q^{62} -13.0000 q^{64} -5.52786 q^{66} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +14.4721 q^{71} -2.23607 q^{72} -3.52786 q^{73} +24.4721 q^{74} -19.4164 q^{76} -10.0000 q^{78} -4.94427 q^{79} +1.00000 q^{81} -4.47214 q^{82} +0.944272 q^{83} -20.0000 q^{86} +2.00000 q^{87} +5.52786 q^{88} +2.00000 q^{89} -12.0000 q^{92} +10.4721 q^{93} +11.0557 q^{94} -6.70820 q^{96} -0.472136 q^{97} -2.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{16} - 4 q^{17} - 4 q^{19} + 20 q^{22} - 8 q^{23} + 20 q^{26} - 2 q^{27} - 4 q^{29} - 12 q^{31} - 4 q^{33} + 6 q^{36} - 4 q^{37} + 20 q^{38} + 4 q^{41} + 12 q^{44} + 8 q^{47} + 2 q^{48} + 4 q^{51} + 16 q^{53} + 4 q^{57} + 4 q^{61} + 20 q^{62} - 26 q^{64} - 20 q^{66} + 8 q^{67} - 12 q^{68} + 8 q^{69} + 20 q^{71} - 16 q^{73} + 40 q^{74} - 12 q^{76} - 20 q^{78} + 8 q^{79} + 2 q^{81} - 16 q^{83} - 40 q^{86} + 4 q^{87} + 20 q^{88} + 4 q^{89} - 24 q^{92} + 12 q^{93} + 40 q^{94} + 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^16 - 4 * q^17 - 4 * q^19 + 20 * q^22 - 8 * q^23 + 20 * q^26 - 2 * q^27 - 4 * q^29 - 12 * q^31 - 4 * q^33 + 6 * q^36 - 4 * q^37 + 20 * q^38 + 4 * q^41 + 12 * q^44 + 8 * q^47 + 2 * q^48 + 4 * q^51 + 16 * q^53 + 4 * q^57 + 4 * q^61 + 20 * q^62 - 26 * q^64 - 20 * q^66 + 8 * q^67 - 12 * q^68 + 8 * q^69 + 20 * q^71 - 16 * q^73 + 40 * q^74 - 12 * q^76 - 20 * q^78 + 8 * q^79 + 2 * q^81 - 16 * q^83 - 40 * q^86 + 4 * q^87 + 20 * q^88 + 4 * q^89 - 24 * q^92 + 12 * q^93 + 40 * q^94 + 8 * q^97 + 4 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	2.23607	0.912871
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.47214	−0.745377	−0.372689	−	0.927957i	$-0.621564\pi$
$11$	−2.47214	−0.745377	−0.372689	+	0.927957i	$0.621564\pi$
$12$	−3.00000	−0.866025
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−2.23607	−0.527046
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	0	0
$22$	5.52786	1.17854
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	10.0000	1.96116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−10.4721	−1.88085	−0.940426	−	0.340000i	$-0.889573\pi$
$31$	−10.4721	−1.88085	−0.940426	+	0.340000i	$0.889573\pi$
$32$	6.70820	1.18585
$33$	2.47214	0.430344
$34$	4.47214	0.766965
$35$	0	0
$36$	3.00000	0.500000
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	14.4721	2.34769
$39$	4.47214	0.716115
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	−7.41641	−1.11807
$45$	0	0
$46$	8.94427	1.31876
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	−13.4164	−1.86052
$53$	12.4721	1.71318	0.856590	−	0.515998i	$-0.172579\pi$
$53$	12.4721	1.71318	0.856590	+	0.515998i	$0.172579\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	0	0
$57$	6.47214	0.857255
$58$	4.47214	0.587220
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	23.4164	2.97389
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	−5.52786	−0.680433
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	4.00000	0.481543
$70$	0	0
$71$	14.4721	1.71753	0.858763	−	0.512373i	$-0.171233\pi$
$71$	14.4721	1.71753	0.858763	+	0.512373i	$0.171233\pi$
$72$	−2.23607	−0.263523
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	24.4721	2.84483
$75$	0	0
$76$	−19.4164	−2.22721
$77$	0	0
$78$	−10.0000	−1.13228
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.47214	−0.493865
$83$	0.944272	0.103647	0.0518237	−	0.998656i	$-0.483497\pi$
$83$	0.944272	0.103647	0.0518237	+	0.998656i	$0.483497\pi$
$84$	0	0
$85$	0	0
$86$	−20.0000	−2.15666
$87$	2.00000	0.214423
$88$	5.52786	0.589272
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	−12.0000	−1.25109
$93$	10.4721	1.08591
$94$	11.0557	1.14031
$95$	0	0
$96$	−6.70820	−0.684653
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	0	0
$99$	−2.47214	−0.248459
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−4.47214	−0.442807
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	10.0000	0.980581
$105$	0	0
$106$	−27.8885	−2.70877
$107$	−4.94427	−0.477981	−0.238990	−	0.971022i	$-0.576816\pi$
$107$	−4.94427	−0.477981	−0.238990	+	0.971022i	$0.576816\pi$
$108$	−3.00000	−0.288675
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	10.9443	1.03878
$112$	0	0
$113$	8.47214	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$113$	8.47214	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$114$	−14.4721	−1.35544
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−4.47214	−0.413449
$118$	20.0000	1.84115
$119$	0	0
$120$	0	0
$121$	−4.88854	−0.444413
$122$	−4.47214	−0.404888
$123$	−2.00000	−0.180334
$124$	−31.4164	−2.82128
$125$	0	0
$126$	0	0
$127$	−12.9443	−1.14862	−0.574309	−	0.818638i	$-0.694729\pi$
$127$	−12.9443	−1.14862	−0.574309	+	0.818638i	$0.694729\pi$
$128$	15.6525	1.38350
$129$	−8.94427	−0.787499
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	7.41641	0.645515
$133$	0	0
$134$	−8.94427	−0.772667
$135$	0	0
$136$	4.47214	0.383482
$137$	−12.4721	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$137$	−12.4721	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$138$	−8.94427	−0.761387
$139$	19.4164	1.64688	0.823439	−	0.567405i	$-0.192052\pi$
$139$	19.4164	1.64688	0.823439	+	0.567405i	$0.192052\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	−32.3607	−2.71565
$143$	11.0557	0.924526
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	7.88854	0.652861
$147$	0	0
$148$	−32.8328	−2.69884
$149$	2.94427	0.241204	0.120602	−	0.992701i	$-0.461517\pi$
$149$	2.94427	0.241204	0.120602	+	0.992701i	$0.461517\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	14.4721	1.17385
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	13.4164	1.07417
$157$	8.47214	0.676150	0.338075	−	0.941119i	$-0.390224\pi$
$157$	8.47214	0.676150	0.338075	+	0.941119i	$0.390224\pi$
$158$	11.0557	0.879547
$159$	−12.4721	−0.989105
$160$	0	0
$161$	0	0
$162$	−2.23607	−0.175682
$163$	−0.944272	−0.0739611	−0.0369805	−	0.999316i	$-0.511774\pi$
$163$	−0.944272	−0.0739611	−0.0369805	+	0.999316i	$0.511774\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−2.11146	−0.163881
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−6.47214	−0.494937
$172$	26.8328	2.04598
$173$	14.9443	1.13619	0.568096	−	0.822962i	$-0.307680\pi$
$173$	14.9443	1.13619	0.568096	+	0.822962i	$0.307680\pi$
$174$	−4.47214	−0.339032
$175$	0	0
$176$	2.47214	0.186344
$177$	8.94427	0.672293
$178$	−4.47214	−0.335201
$179$	−2.47214	−0.184776	−0.0923881	−	0.995723i	$-0.529450\pi$
$179$	−2.47214	−0.184776	−0.0923881	+	0.995723i	$0.529450\pi$
$180$	0	0
$181$	−18.9443	−1.40812	−0.704058	−	0.710142i	$-0.748631\pi$
$181$	−18.9443	−1.40812	−0.704058	+	0.710142i	$0.748631\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	8.94427	0.659380
$185$	0	0
$186$	−23.4164	−1.71697
$187$	4.94427	0.361561
$188$	−14.8328	−1.08179
$189$	0	0
$190$	0	0
$191$	27.4164	1.98378	0.991891	−	0.127093i	$-0.0405646\pi$
$191$	27.4164	1.98378	0.991891	+	0.127093i	$0.0405646\pi$
$192$	13.0000	0.938194
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	1.05573	0.0757969
$195$	0	0
$196$	0	0
$197$	−24.4721	−1.74357	−0.871784	−	0.489891i	$-0.837037\pi$
$197$	−24.4721	−1.74357	−0.871784	+	0.489891i	$0.837037\pi$
$198$	5.52786	0.392848
$199$	−0.583592	−0.0413697	−0.0206849	−	0.999786i	$-0.506585\pi$
$199$	−0.583592	−0.0413697	−0.0206849	+	0.999786i	$0.506585\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−31.3050	−2.20261
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	4.47214	0.310087
$209$	16.0000	1.10674
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	37.4164	2.56977
$213$	−14.4721	−0.991614
$214$	11.0557	0.755754
$215$	0	0
$216$	2.23607	0.152145
$217$	0	0
$218$	4.47214	0.302891
$219$	3.52786	0.238391
$220$	0	0
$221$	8.94427	0.601657
$222$	−24.4721	−1.64246
$223$	4.94427	0.331093	0.165546	−	0.986202i	$-0.447061\pi$
$223$	4.94427	0.331093	0.165546	+	0.986202i	$0.447061\pi$
$224$	0	0
$225$	0	0
$226$	−18.9443	−1.26015
$227$	−16.9443	−1.12463	−0.562315	−	0.826923i	$-0.690089\pi$
$227$	−16.9443	−1.12463	−0.562315	+	0.826923i	$0.690089\pi$
$228$	19.4164	1.28588
$229$	11.8885	0.785617	0.392809	−	0.919620i	$-0.371504\pi$
$229$	11.8885	0.785617	0.392809	+	0.919620i	$0.371504\pi$
$230$	0	0
$231$	0	0
$232$	4.47214	0.293610
$233$	−17.4164	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$233$	−17.4164	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$234$	10.0000	0.653720
$235$	0	0
$236$	−26.8328	−1.74667
$237$	4.94427	0.321165
$238$	0	0
$239$	−1.52786	−0.0988293	−0.0494147	−	0.998778i	$-0.515736\pi$
$239$	−1.52786	−0.0988293	−0.0494147	+	0.998778i	$0.515736\pi$
$240$	0	0
$241$	1.05573	0.0680054	0.0340027	−	0.999422i	$-0.489175\pi$
$241$	1.05573	0.0680054	0.0340027	+	0.999422i	$0.489175\pi$
$242$	10.9311	0.702679
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	4.47214	0.285133
$247$	28.9443	1.84168
$248$	23.4164	1.48694
$249$	−0.944272	−0.0598408
$250$	0	0
$251$	0.944272	0.0596019	0.0298010	−	0.999556i	$-0.490513\pi$
$251$	0.944272	0.0596019	0.0298010	+	0.999556i	$0.490513\pi$
$252$	0	0
$253$	9.88854	0.621687
$254$	28.9443	1.81613
$255$	0	0
$256$	−9.00000	−0.562500
$257$	1.05573	0.0658545	0.0329273	−	0.999458i	$-0.489517\pi$
$257$	1.05573	0.0658545	0.0329273	+	0.999458i	$0.489517\pi$
$258$	20.0000	1.24515
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	8.94427	0.552579
$263$	−24.9443	−1.53813	−0.769065	−	0.639171i	$-0.779278\pi$
$263$	−24.9443	−1.53813	−0.769065	+	0.639171i	$0.779278\pi$
$264$	−5.52786	−0.340217
$265$	0	0
$266$	0	0
$267$	−2.00000	−0.122398
$268$	12.0000	0.733017
$269$	23.8885	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$269$	23.8885	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$270$	0	0
$271$	10.4721	0.636137	0.318068	−	0.948068i	$-0.396966\pi$
$271$	10.4721	0.636137	0.318068	+	0.948068i	$0.396966\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	27.8885	1.68481
$275$	0	0
$276$	12.0000	0.722315
$277$	−1.05573	−0.0634326	−0.0317163	−	0.999497i	$-0.510097\pi$
$277$	−1.05573	−0.0634326	−0.0317163	+	0.999497i	$0.510097\pi$
$278$	−43.4164	−2.60394
$279$	−10.4721	−0.626950
$280$	0	0
$281$	6.94427	0.414261	0.207130	−	0.978313i	$-0.433588\pi$
$281$	6.94427	0.414261	0.207130	+	0.978313i	$0.433588\pi$
$282$	−11.0557	−0.658359
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	43.4164	2.57629
$285$	0	0
$286$	−24.7214	−1.46180
$287$	0	0
$288$	6.70820	0.395285
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0.472136	0.0276771
$292$	−10.5836	−0.619358
$293$	22.9443	1.34042	0.670209	−	0.742172i	$-0.266204\pi$
$293$	22.9443	1.34042	0.670209	+	0.742172i	$0.266204\pi$
$294$	0	0
$295$	0	0
$296$	24.4721	1.42241
$297$	2.47214	0.143448
$298$	−6.58359	−0.381377
$299$	17.8885	1.03452
$300$	0	0
$301$	0	0
$302$	35.7771	2.05874
$303$	−14.0000	−0.804279
$304$	6.47214	0.371202
$305$	0	0
$306$	4.47214	0.255655
$307$	−32.9443	−1.88023	−0.940114	−	0.340859i	$-0.889282\pi$
$307$	−32.9443	−1.88023	−0.940114	+	0.340859i	$0.889282\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.88854	0.560728	0.280364	−	0.959894i	$-0.409545\pi$
$311$	9.88854	0.560728	0.280364	+	0.959894i	$0.409545\pi$
$312$	−10.0000	−0.566139
$313$	9.41641	0.532247	0.266123	−	0.963939i	$-0.414257\pi$
$313$	9.41641	0.532247	0.266123	+	0.963939i	$0.414257\pi$
$314$	−18.9443	−1.06909
$315$	0	0
$316$	−14.8328	−0.834411
$317$	30.3607	1.70523	0.852613	−	0.522543i	$-0.175017\pi$
$317$	30.3607	1.70523	0.852613	+	0.522543i	$0.175017\pi$
$318$	27.8885	1.56391
$319$	4.94427	0.276826
$320$	0	0
$321$	4.94427	0.275962
$322$	0	0
$323$	12.9443	0.720239
$324$	3.00000	0.166667
$325$	0	0
$326$	2.11146	0.116943
$327$	2.00000	0.110600
$328$	−4.47214	−0.246932
$329$	0	0
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	2.83282	0.155471
$333$	−10.9443	−0.599742
$334$	17.8885	0.978818
$335$	0	0
$336$	0	0
$337$	−11.8885	−0.647610	−0.323805	−	0.946124i	$-0.604962\pi$
$337$	−11.8885	−0.647610	−0.323805	+	0.946124i	$0.604962\pi$
$338$	−15.6525	−0.851382
$339$	−8.47214	−0.460143
$340$	0	0
$341$	25.8885	1.40194
$342$	14.4721	0.782563
$343$	0	0
$344$	−20.0000	−1.07833
$345$	0	0
$346$	−33.4164	−1.79648
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	6.00000	0.321634
$349$	−23.8885	−1.27872	−0.639362	−	0.768906i	$-0.720802\pi$
$349$	−23.8885	−1.27872	−0.639362	+	0.768906i	$0.720802\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	−16.5836	−0.883908
$353$	−27.8885	−1.48436	−0.742179	−	0.670202i	$-0.766207\pi$
$353$	−27.8885	−1.48436	−0.742179	+	0.670202i	$0.766207\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	5.52786	0.292157
$359$	9.52786	0.502861	0.251431	−	0.967875i	$-0.419099\pi$
$359$	9.52786	0.502861	0.251431	+	0.967875i	$0.419099\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	42.3607	2.22643
$363$	4.88854	0.256582
$364$	0	0
$365$	0	0
$366$	4.47214	0.233762
$367$	20.9443	1.09328	0.546641	−	0.837367i	$-0.315906\pi$
$367$	20.9443	1.09328	0.546641	+	0.837367i	$0.315906\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	31.4164	1.62886
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−11.0557	−0.571678
$375$	0	0
$376$	11.0557	0.570156
$377$	8.94427	0.460653
$378$	0	0
$379$	−2.11146	−0.108458	−0.0542291	−	0.998529i	$-0.517270\pi$
$379$	−2.11146	−0.108458	−0.0542291	+	0.998529i	$0.517270\pi$
$380$	0	0
$381$	12.9443	0.663155
$382$	−61.3050	−3.13663
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−15.6525	−0.798762
$385$	0	0
$386$	−31.3050	−1.59338
$387$	8.94427	0.454663
$388$	−1.41641	−0.0719072
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	4.00000	0.201773
$394$	54.7214	2.75682
$395$	0	0
$396$	−7.41641	−0.372689
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	1.30495	0.0654113
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.94427	0.446100
$403$	46.8328	2.33291
$404$	42.0000	2.08958
$405$	0	0
$406$	0	0
$407$	27.0557	1.34110
$408$	−4.47214	−0.221404
$409$	23.8885	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$409$	23.8885	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$410$	0	0
$411$	12.4721	0.615205
$412$	0	0
$413$	0	0
$414$	8.94427	0.439587
$415$	0	0
$416$	−30.0000	−1.47087
$417$	−19.4164	−0.950826
$418$	−35.7771	−1.74991
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−2.11146	−0.102784
$423$	−4.94427	−0.240399
$424$	−27.8885	−1.35439
$425$	0	0
$426$	32.3607	1.56788
$427$	0	0
$428$	−14.8328	−0.716971
$429$	−11.0557	−0.533776
$430$	0	0
$431$	−9.52786	−0.458941	−0.229471	−	0.973316i	$-0.573699\pi$
$431$	−9.52786	−0.458941	−0.229471	+	0.973316i	$0.573699\pi$
$432$	1.00000	0.0481125
$433$	7.52786	0.361766	0.180883	−	0.983505i	$-0.442104\pi$
$433$	7.52786	0.361766	0.180883	+	0.983505i	$0.442104\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	25.8885	1.23842
$438$	−7.88854	−0.376929
$439$	−10.4721	−0.499808	−0.249904	−	0.968271i	$-0.580399\pi$
$439$	−10.4721	−0.499808	−0.249904	+	0.968271i	$0.580399\pi$
$440$	0	0
$441$	0	0
$442$	−20.0000	−0.951303
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	32.8328	1.55818
$445$	0	0
$446$	−11.0557	−0.523504
$447$	−2.94427	−0.139259
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−4.94427	−0.232817
$452$	25.4164	1.19549
$453$	16.0000	0.751746
$454$	37.8885	1.77820
$455$	0	0
$456$	−14.4721	−0.677720
$457$	10.9443	0.511951	0.255976	−	0.966683i	$-0.417603\pi$
$457$	10.9443	0.511951	0.255976	+	0.966683i	$0.417603\pi$
$458$	−26.5836	−1.24217
$459$	2.00000	0.0933520
$460$	0	0
$461$	31.8885	1.48520	0.742599	−	0.669737i	$-0.233593\pi$
$461$	31.8885	1.48520	0.742599	+	0.669737i	$0.233593\pi$
$462$	0	0
$463$	−3.05573	−0.142012	−0.0710059	−	0.997476i	$-0.522621\pi$
$463$	−3.05573	−0.142012	−0.0710059	+	0.997476i	$0.522621\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	38.9443	1.80406
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	−13.4164	−0.620174
$469$	0	0
$470$	0	0
$471$	−8.47214	−0.390375
$472$	20.0000	0.920575
$473$	−22.1115	−1.01669
$474$	−11.0557	−0.507806
$475$	0	0
$476$	0	0
$477$	12.4721	0.571060
$478$	3.41641	0.156263
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	48.9443	2.23167
$482$	−2.36068	−0.107526
$483$	0	0
$484$	−14.6656	−0.666620
$485$	0	0
$486$	2.23607	0.101430
$487$	3.05573	0.138468	0.0692341	−	0.997600i	$-0.477944\pi$
$487$	3.05573	0.138468	0.0692341	+	0.997600i	$0.477944\pi$
$488$	−4.47214	−0.202444
$489$	0.944272	0.0427015
$490$	0	0
$491$	41.3050	1.86407	0.932033	−	0.362373i	$-0.118033\pi$
$491$	41.3050	1.86407	0.932033	+	0.362373i	$0.118033\pi$
$492$	−6.00000	−0.270501
$493$	4.00000	0.180151
$494$	−64.7214	−2.91195
$495$	0	0
$496$	10.4721	0.470213
$497$	0	0
$498$	2.11146	0.0946166
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	−2.11146	−0.0942389
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	−22.1115	−0.982974
$507$	−7.00000	−0.310881
$508$	−38.8328	−1.72293
$509$	−11.8885	−0.526950	−0.263475	−	0.964666i	$-0.584869\pi$
$509$	−11.8885	−0.526950	−0.263475	+	0.964666i	$0.584869\pi$
$510$	0	0
$511$	0	0
$512$	−11.1803	−0.494106
$513$	6.47214	0.285752
$514$	−2.36068	−0.104125
$515$	0	0
$516$	−26.8328	−1.18125
$517$	12.2229	0.537563
$518$	0	0
$519$	−14.9443	−0.655981
$520$	0	0
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	4.47214	0.195740
$523$	8.94427	0.391106	0.195553	−	0.980693i	$-0.437350\pi$
$523$	8.94427	0.391106	0.195553	+	0.980693i	$0.437350\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	55.7771	2.43200
$527$	20.9443	0.912347
$528$	−2.47214	−0.107586
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−8.94427	−0.388148
$532$	0	0
$533$	−8.94427	−0.387419
$534$	4.47214	0.193528
$535$	0	0
$536$	−8.94427	−0.386334
$537$	2.47214	0.106681
$538$	−53.4164	−2.30294
$539$	0	0
$540$	0	0
$541$	23.8885	1.02705	0.513524	−	0.858075i	$-0.328340\pi$
$541$	23.8885	1.02705	0.513524	+	0.858075i	$0.328340\pi$
$542$	−23.4164	−1.00582
$543$	18.9443	0.812977
$544$	−13.4164	−0.575224
$545$	0	0
$546$	0	0
$547$	29.8885	1.27794	0.638971	−	0.769231i	$-0.279360\pi$
$547$	29.8885	1.27794	0.638971	+	0.769231i	$0.279360\pi$
$548$	−37.4164	−1.59835
$549$	2.00000	0.0853579
$550$	0	0
$551$	12.9443	0.551445
$552$	−8.94427	−0.380693
$553$	0	0
$554$	2.36068	0.100296
$555$	0	0
$556$	58.2492	2.47032
$557$	−11.5279	−0.488451	−0.244226	−	0.969718i	$-0.578534\pi$
$557$	−11.5279	−0.488451	−0.244226	+	0.969718i	$0.578534\pi$
$558$	23.4164	0.991296
$559$	−40.0000	−1.69182
$560$	0	0
$561$	−4.94427	−0.208747
$562$	−15.5279	−0.655003
$563$	−21.8885	−0.922492	−0.461246	−	0.887272i	$-0.652597\pi$
$563$	−21.8885	−0.922492	−0.461246	+	0.887272i	$0.652597\pi$
$564$	14.8328	0.624574
$565$	0	0
$566$	−26.8328	−1.12787
$567$	0	0
$568$	−32.3607	−1.35782
$569$	−4.11146	−0.172361	−0.0861806	−	0.996280i	$-0.527466\pi$
$569$	−4.11146	−0.172361	−0.0861806	+	0.996280i	$0.527466\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	33.1672	1.38679
$573$	−27.4164	−1.14534
$574$	0	0
$575$	0	0
$576$	−13.0000	−0.541667
$577$	−34.3607	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$577$	−34.3607	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$578$	29.0689	1.20911
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	−1.05573	−0.0437613
$583$	−30.8328	−1.27696
$584$	7.88854	0.326430
$585$	0	0
$586$	−51.3050	−2.11939
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	67.7771	2.79271
$590$	0	0
$591$	24.4721	1.00665
$592$	10.9443	0.449807
$593$	−11.8885	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$593$	−11.8885	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$594$	−5.52786	−0.226811
$595$	0	0
$596$	8.83282	0.361806
$597$	0.583592	0.0238848
$598$	−40.0000	−1.63572
$599$	−32.3607	−1.32222	−0.661111	−	0.750288i	$-0.729915\pi$
$599$	−32.3607	−1.32222	−0.661111	+	0.750288i	$0.729915\pi$
$600$	0	0
$601$	−21.0557	−0.858881	−0.429441	−	0.903095i	$-0.641289\pi$
$601$	−21.0557	−0.858881	−0.429441	+	0.903095i	$0.641289\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−48.0000	−1.95309
$605$	0	0
$606$	31.3050	1.27168
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	−43.4164	−1.76077
$609$	0	0
$610$	0	0
$611$	22.1115	0.894534
$612$	−6.00000	−0.242536
$613$	−10.9443	−0.442035	−0.221017	−	0.975270i	$-0.570938\pi$
$613$	−10.9443	−0.442035	−0.221017	+	0.975270i	$0.570938\pi$
$614$	73.6656	2.97290
$615$	0	0
$616$	0	0
$617$	−7.52786	−0.303060	−0.151530	−	0.988453i	$-0.548420\pi$
$617$	−7.52786	−0.303060	−0.151530	+	0.988453i	$0.548420\pi$
$618$	0	0
$619$	−12.5836	−0.505777	−0.252889	−	0.967495i	$-0.581381\pi$
$619$	−12.5836	−0.505777	−0.252889	+	0.967495i	$0.581381\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−22.1115	−0.886589
$623$	0	0
$624$	−4.47214	−0.179029
$625$	0	0
$626$	−21.0557	−0.841556
$627$	−16.0000	−0.638978
$628$	25.4164	1.01423
$629$	21.8885	0.872753
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	11.0557	0.439773
$633$	−0.944272	−0.0375314
$634$	−67.8885	−2.69620
$635$	0	0
$636$	−37.4164	−1.48366
$637$	0	0
$638$	−11.0557	−0.437700
$639$	14.4721	0.572509
$640$	0	0
$641$	−36.8328	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$641$	−36.8328	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$642$	−11.0557	−0.436335
$643$	−32.9443	−1.29920	−0.649598	−	0.760278i	$-0.725063\pi$
$643$	−32.9443	−1.29920	−0.649598	+	0.760278i	$0.725063\pi$
$644$	0	0
$645$	0	0
$646$	−28.9443	−1.13880
$647$	33.8885	1.33230	0.666148	−	0.745820i	$-0.267942\pi$
$647$	33.8885	1.33230	0.666148	+	0.745820i	$0.267942\pi$
$648$	−2.23607	−0.0878410
$649$	22.1115	0.867951
$650$	0	0
$651$	0	0
$652$	−2.83282	−0.110942
$653$	49.4164	1.93381	0.966907	−	0.255130i	$-0.0821183\pi$
$653$	49.4164	1.93381	0.966907	+	0.255130i	$0.0821183\pi$
$654$	−4.47214	−0.174874
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−3.52786	−0.137635
$658$	0	0
$659$	−41.3050	−1.60901	−0.804506	−	0.593944i	$-0.797570\pi$
$659$	−41.3050	−1.60901	−0.804506	+	0.593944i	$0.797570\pi$
$660$	0	0
$661$	0.111456	0.00433514	0.00216757	−	0.999998i	$-0.499310\pi$
$661$	0.111456	0.00433514	0.00216757	+	0.999998i	$0.499310\pi$
$662$	37.8885	1.47258
$663$	−8.94427	−0.347367
$664$	−2.11146	−0.0819404
$665$	0	0
$666$	24.4721	0.948276
$667$	8.00000	0.309761
$668$	−24.0000	−0.928588
$669$	−4.94427	−0.191157
$670$	0	0
$671$	−4.94427	−0.190872
$672$	0	0
$673$	44.8328	1.72818	0.864089	−	0.503339i	$-0.167895\pi$
$673$	44.8328	1.72818	0.864089	+	0.503339i	$0.167895\pi$
$674$	26.5836	1.02396
$675$	0	0
$676$	21.0000	0.807692
$677$	38.9443	1.49675	0.748375	−	0.663276i	$-0.230834\pi$
$677$	38.9443	1.49675	0.748375	+	0.663276i	$0.230834\pi$
$678$	18.9443	0.727550
$679$	0	0
$680$	0	0
$681$	16.9443	0.649306
$682$	−57.8885	−2.21667
$683$	−33.8885	−1.29671	−0.648355	−	0.761339i	$-0.724543\pi$
$683$	−33.8885	−1.29671	−0.648355	+	0.761339i	$0.724543\pi$
$684$	−19.4164	−0.742405
$685$	0	0
$686$	0	0
$687$	−11.8885	−0.453576
$688$	−8.94427	−0.340997
$689$	−55.7771	−2.12494
$690$	0	0
$691$	0.360680	0.0137209	0.00686045	−	0.999976i	$-0.497816\pi$
$691$	0.360680	0.0137209	0.00686045	+	0.999976i	$0.497816\pi$
$692$	44.8328	1.70429
$693$	0	0
$694$	−17.8885	−0.679040
$695$	0	0
$696$	−4.47214	−0.169516
$697$	−4.00000	−0.151511
$698$	53.4164	2.02184
$699$	17.4164	0.658749
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−10.0000	−0.377426
$703$	70.8328	2.67151
$704$	32.1378	1.21124
$705$	0	0
$706$	62.3607	2.34698
$707$	0	0
$708$	26.8328	1.00844
$709$	−45.7771	−1.71919	−0.859597	−	0.510972i	$-0.829286\pi$
$709$	−45.7771	−1.71919	−0.859597	+	0.510972i	$0.829286\pi$
$710$	0	0
$711$	−4.94427	−0.185425
$712$	−4.47214	−0.167600
$713$	41.8885	1.56874
$714$	0	0
$715$	0	0
$716$	−7.41641	−0.277164
$717$	1.52786	0.0570591
$718$	−21.3050	−0.795094
$719$	−46.8328	−1.74657	−0.873285	−	0.487210i	$-0.838015\pi$
$719$	−46.8328	−1.74657	−0.873285	+	0.487210i	$0.838015\pi$
$720$	0	0
$721$	0	0
$722$	−51.1803	−1.90474
$723$	−1.05573	−0.0392630
$724$	−56.8328	−2.11217
$725$	0	0
$726$	−10.9311	−0.405692
$727$	14.8328	0.550119	0.275059	−	0.961427i	$-0.411302\pi$
$727$	14.8328	0.550119	0.275059	+	0.961427i	$0.411302\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.8885	−0.661632
$732$	−6.00000	−0.221766
$733$	37.4164	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$733$	37.4164	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$734$	−46.8328	−1.72863
$735$	0	0
$736$	−26.8328	−0.989071
$737$	−9.88854	−0.364249
$738$	−4.47214	−0.164622
$739$	29.8885	1.09947	0.549734	−	0.835340i	$-0.314729\pi$
$739$	29.8885	1.09947	0.549734	+	0.835340i	$0.314729\pi$
$740$	0	0
$741$	−28.9443	−1.06329
$742$	0	0
$743$	18.8328	0.690909	0.345455	−	0.938436i	$-0.387725\pi$
$743$	18.8328	0.690909	0.345455	+	0.938436i	$0.387725\pi$
$744$	−23.4164	−0.858487
$745$	0	0
$746$	13.4164	0.491210
$747$	0.944272	0.0345491
$748$	14.8328	0.542341
$749$	0	0
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	4.94427	0.180299
$753$	−0.944272	−0.0344112
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	3.88854	0.141332	0.0706658	−	0.997500i	$-0.477488\pi$
$757$	3.88854	0.141332	0.0706658	+	0.997500i	$0.477488\pi$
$758$	4.72136	0.171488
$759$	−9.88854	−0.358931
$760$	0	0
$761$	−7.88854	−0.285959	−0.142980	−	0.989726i	$-0.545668\pi$
$761$	−7.88854	−0.285959	−0.142980	+	0.989726i	$0.545668\pi$
$762$	−28.9443	−1.04854
$763$	0	0
$764$	82.2492	2.97567
$765$	0	0
$766$	17.8885	0.646339
$767$	40.0000	1.44432
$768$	9.00000	0.324760
$769$	−0.832816	−0.0300321	−0.0150161	−	0.999887i	$-0.504780\pi$
$769$	−0.832816	−0.0300321	−0.0150161	+	0.999887i	$0.504780\pi$
$770$	0	0
$771$	−1.05573	−0.0380211
$772$	42.0000	1.51161
$773$	−25.0557	−0.901192	−0.450596	−	0.892728i	$-0.648788\pi$
$773$	−25.0557	−0.901192	−0.450596	+	0.892728i	$0.648788\pi$
$774$	−20.0000	−0.718885
$775$	0	0
$776$	1.05573	0.0378984
$777$	0	0
$778$	−24.4721	−0.877369
$779$	−12.9443	−0.463777
$780$	0	0
$781$	−35.7771	−1.28020
$782$	−17.8885	−0.639693
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	−8.94427	−0.319032
$787$	48.9443	1.74467	0.872337	−	0.488904i	$-0.162603\pi$
$787$	48.9443	1.74467	0.872337	+	0.488904i	$0.162603\pi$
$788$	−73.4164	−2.61535
$789$	24.9443	0.888040
$790$	0	0
$791$	0	0
$792$	5.52786	0.196424
$793$	−8.94427	−0.317620
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−1.75078	−0.0620546
$797$	−1.05573	−0.0373958	−0.0186979	−	0.999825i	$-0.505952\pi$
$797$	−1.05573	−0.0373958	−0.0186979	+	0.999825i	$0.505952\pi$
$798$	0	0
$799$	9.88854	0.349832
$800$	0	0
$801$	2.00000	0.0706665
$802$	−22.3607	−0.789583
$803$	8.72136	0.307770
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−104.721	−3.68865
$807$	−23.8885	−0.840917
$808$	−31.3050	−1.10130
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	0	0
$811$	−28.5836	−1.00371	−0.501853	−	0.864953i	$-0.667348\pi$
$811$	−28.5836	−1.00371	−0.501853	+	0.864953i	$0.667348\pi$
$812$	0	0
$813$	−10.4721	−0.367274
$814$	−60.4984	−2.12047
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	−57.8885	−2.02526
$818$	−53.4164	−1.86766
$819$	0	0
$820$	0	0
$821$	−37.7771	−1.31843	−0.659215	−	0.751955i	$-0.729111\pi$
$821$	−37.7771	−1.31843	−0.659215	+	0.751955i	$0.729111\pi$
$822$	−27.8885	−0.972725
$823$	−27.0557	−0.943103	−0.471552	−	0.881838i	$-0.656306\pi$
$823$	−27.0557	−0.943103	−0.471552	+	0.881838i	$0.656306\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.94427	0.171929	0.0859646	−	0.996298i	$-0.472603\pi$
$827$	4.94427	0.171929	0.0859646	+	0.996298i	$0.472603\pi$
$828$	−12.0000	−0.417029
$829$	30.9443	1.07474	0.537369	−	0.843347i	$-0.319418\pi$
$829$	30.9443	1.07474	0.537369	+	0.843347i	$0.319418\pi$
$830$	0	0
$831$	1.05573	0.0366228
$832$	58.1378	2.01556
$833$	0	0
$834$	43.4164	1.50339
$835$	0	0
$836$	48.0000	1.66011
$837$	10.4721	0.361970
$838$	13.1672	0.454853
$839$	−1.16718	−0.0402957	−0.0201478	−	0.999797i	$-0.506414\pi$
$839$	−1.16718	−0.0402957	−0.0201478	+	0.999797i	$0.506414\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−49.1935	−1.69532
$843$	−6.94427	−0.239173
$844$	2.83282	0.0975095
$845$	0	0
$846$	11.0557	0.380104
$847$	0	0
$848$	−12.4721	−0.428295
$849$	−12.0000	−0.411839
$850$	0	0
$851$	43.7771	1.50066
$852$	−43.4164	−1.48742
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	0	0
$855$	0	0
$856$	11.0557	0.377877
$857$	−16.8328	−0.574998	−0.287499	−	0.957781i	$-0.592824\pi$
$857$	−16.8328	−0.574998	−0.287499	+	0.957781i	$0.592824\pi$
$858$	24.7214	0.843973
$859$	−41.5279	−1.41691	−0.708456	−	0.705755i	$-0.750608\pi$
$859$	−41.5279	−1.41691	−0.708456	+	0.705755i	$0.750608\pi$
$860$	0	0
$861$	0	0
$862$	21.3050	0.725650
$863$	13.8885	0.472772	0.236386	−	0.971659i	$-0.424037\pi$
$863$	13.8885	0.472772	0.236386	+	0.971659i	$0.424037\pi$
$864$	−6.70820	−0.228218
$865$	0	0
$866$	−16.8328	−0.572002
$867$	13.0000	0.441503
$868$	0	0
$869$	12.2229	0.414634
$870$	0	0
$871$	−17.8885	−0.606130
$872$	4.47214	0.151446
$873$	−0.472136	−0.0159794
$874$	−57.8885	−1.95811
$875$	0	0
$876$	10.5836	0.357586
$877$	3.16718	0.106948	0.0534741	−	0.998569i	$-0.482971\pi$
$877$	3.16718	0.106948	0.0534741	+	0.998569i	$0.482971\pi$
$878$	23.4164	0.790265
$879$	−22.9443	−0.773891
$880$	0	0
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	2.11146	0.0710562	0.0355281	−	0.999369i	$-0.488689\pi$
$883$	2.11146	0.0710562	0.0355281	+	0.999369i	$0.488689\pi$
$884$	26.8328	0.902485
$885$	0	0
$886$	−17.8885	−0.600977
$887$	22.8328	0.766651	0.383325	−	0.923613i	$-0.374779\pi$
$887$	22.8328	0.766651	0.383325	+	0.923613i	$0.374779\pi$
$888$	−24.4721	−0.821231
$889$	0	0
$890$	0	0
$891$	−2.47214	−0.0828197
$892$	14.8328	0.496639
$893$	32.0000	1.07084
$894$	6.58359	0.220188
$895$	0	0
$896$	0	0
$897$	−17.8885	−0.597281
$898$	31.3050	1.04466
$899$	20.9443	0.698531
$900$	0	0
$901$	−24.9443	−0.831014
$902$	11.0557	0.368115
$903$	0	0
$904$	−18.9443	−0.630077
$905$	0	0
$906$	−35.7771	−1.18861
$907$	−18.1115	−0.601381	−0.300691	−	0.953722i	$-0.597217\pi$
$907$	−18.1115	−0.601381	−0.300691	+	0.953722i	$0.597217\pi$
$908$	−50.8328	−1.68695
$909$	14.0000	0.464351
$910$	0	0
$911$	−34.2492	−1.13473	−0.567364	−	0.823467i	$-0.692037\pi$
$911$	−34.2492	−1.13473	−0.567364	+	0.823467i	$0.692037\pi$
$912$	−6.47214	−0.214314
$913$	−2.33437	−0.0772563
$914$	−24.4721	−0.809466
$915$	0	0
$916$	35.6656	1.17843
$917$	0	0
$918$	−4.47214	−0.147602
$919$	−52.9443	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$919$	−52.9443	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$920$	0	0
$921$	32.9443	1.08555
$922$	−71.3050	−2.34830
$923$	−64.7214	−2.13033
$924$	0	0
$925$	0	0
$926$	6.83282	0.224540
$927$	0	0
$928$	−13.4164	−0.440415
$929$	51.8885	1.70241	0.851204	−	0.524835i	$-0.175873\pi$
$929$	51.8885	1.70241	0.851204	+	0.524835i	$0.175873\pi$
$930$	0	0
$931$	0	0
$932$	−52.2492	−1.71148
$933$	−9.88854	−0.323736
$934$	−20.0000	−0.654420
$935$	0	0
$936$	10.0000	0.326860
$937$	−43.5279	−1.42199	−0.710997	−	0.703195i	$-0.751756\pi$
$937$	−43.5279	−1.42199	−0.710997	+	0.703195i	$0.751756\pi$
$938$	0	0
$939$	−9.41641	−0.307293
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	18.9443	0.617238
$943$	−8.00000	−0.260516
$944$	8.94427	0.291111
$945$	0	0
$946$	49.4427	1.60752
$947$	−17.8885	−0.581300	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885	−0.581300	−0.290650	+	0.956830i	$0.593871\pi$
$948$	14.8328	0.481747
$949$	15.7771	0.512146
$950$	0	0
$951$	−30.3607	−0.984512
$952$	0	0
$953$	6.58359	0.213263	0.106632	−	0.994299i	$-0.465993\pi$
$953$	6.58359	0.213263	0.106632	+	0.994299i	$0.465993\pi$
$954$	−27.8885	−0.902925
$955$	0	0
$956$	−4.58359	−0.148244
$957$	−4.94427	−0.159826
$958$	40.0000	1.29234
$959$	0	0
$960$	0	0
$961$	78.6656	2.53760
$962$	−109.443	−3.52857
$963$	−4.94427	−0.159327
$964$	3.16718	0.102008
$965$	0	0
$966$	0	0
$967$	9.88854	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$967$	9.88854	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$968$	10.9311	0.351339
$969$	−12.9443	−0.415830
$970$	0	0
$971$	−23.0557	−0.739894	−0.369947	−	0.929053i	$-0.620624\pi$
$971$	−23.0557	−0.739894	−0.369947	+	0.929053i	$0.620624\pi$
$972$	−3.00000	−0.0962250
$973$	0	0
$974$	−6.83282	−0.218938
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−57.4164	−1.83691	−0.918457	−	0.395521i	$-0.870564\pi$
$977$	−57.4164	−1.83691	−0.918457	+	0.395521i	$0.870564\pi$
$978$	−2.11146	−0.0675169
$979$	−4.94427	−0.158020
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−92.3607	−2.94735
$983$	30.8328	0.983414	0.491707	−	0.870761i	$-0.336373\pi$
$983$	30.8328	0.983414	0.491707	+	0.870761i	$0.336373\pi$
$984$	4.47214	0.142566
$985$	0	0
$986$	−8.94427	−0.284844
$987$	0	0
$988$	86.8328	2.76252
$989$	−35.7771	−1.13765
$990$	0	0
$991$	−12.9443	−0.411188	−0.205594	−	0.978637i	$-0.565913\pi$
$991$	−12.9443	−0.411188	−0.205594	+	0.978637i	$0.565913\pi$
$992$	−70.2492	−2.23042
$993$	16.9443	0.537710
$994$	0	0
$995$	0	0
$996$	−2.83282	−0.0897612
$997$	21.4164	0.678264	0.339132	−	0.940739i	$-0.389867\pi$
$997$	21.4164	0.678264	0.339132	+	0.940739i	$0.389867\pi$
$998$	−48.9443	−1.54930
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.y.1.1		2
5.4	even	2		735.2.a.k.1.2		2
7.6	odd	2		525.2.a.g.1.1		2
15.14	odd	2		2205.2.a.w.1.1		2
21.20	even	2		1575.2.a.r.1.2		2
28.27	even	2		8400.2.a.cx.1.2		2
35.4	even	6		735.2.i.i.226.1		4
35.9	even	6		735.2.i.i.361.1		4
35.13	even	4		525.2.d.c.274.4		4
35.19	odd	6		735.2.i.k.361.1		4
35.24	odd	6		735.2.i.k.226.1		4
35.27	even	4		525.2.d.c.274.1		4
35.34	odd	2		105.2.a.b.1.2	✓	2
105.62	odd	4		1575.2.d.d.1324.3		4
105.83	odd	4		1575.2.d.d.1324.2		4
105.104	even	2		315.2.a.d.1.1		2
140.139	even	2		1680.2.a.v.1.2		2
280.69	odd	2		6720.2.a.cx.1.2		2
280.139	even	2		6720.2.a.cs.1.1		2
420.419	odd	2		5040.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.2	✓	2	35.34	odd	2
315.2.a.d.1.1		2	105.104	even	2
525.2.a.g.1.1		2	7.6	odd	2
525.2.d.c.274.1		4	35.27	even	4
525.2.d.c.274.4		4	35.13	even	4
735.2.a.k.1.2		2	5.4	even	2
735.2.i.i.226.1		4	35.4	even	6
735.2.i.i.361.1		4	35.9	even	6
735.2.i.k.226.1		4	35.24	odd	6
735.2.i.k.361.1		4	35.19	odd	6
1575.2.a.r.1.2		2	21.20	even	2
1575.2.d.d.1324.2		4	105.83	odd	4
1575.2.d.d.1324.3		4	105.62	odd	4
1680.2.a.v.1.2		2	140.139	even	2
2205.2.a.w.1.1		2	15.14	odd	2
3675.2.a.y.1.1		2	1.1	even	1	trivial
5040.2.a.bw.1.1		2	420.419	odd	2
6720.2.a.cs.1.1		2	280.139	even	2
6720.2.a.cx.1.2		2	280.69	odd	2
8400.2.a.cx.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)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -2.23607 q^{8} +1.00000 q^{9} -2.47214 q^{11} -3.00000 q^{12} -4.47214 q^{13} -1.00000 q^{16} -2.00000 q^{17} -2.23607 q^{18} -6.47214 q^{19} +5.52786 q^{22} -4.00000 q^{23} +2.23607 q^{24} +10.0000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -10.4721 q^{31} +6.70820 q^{32} +2.47214 q^{33} +4.47214 q^{34} +3.00000 q^{36} -10.9443 q^{37} +14.4721 q^{38} +4.47214 q^{39} +2.00000 q^{41} +8.94427 q^{43} -7.41641 q^{44} +8.94427 q^{46} -4.94427 q^{47} +1.00000 q^{48} +2.00000 q^{51} -13.4164 q^{52} +12.4721 q^{53} +2.23607 q^{54} +6.47214 q^{57} +4.47214 q^{58} -8.94427 q^{59} +2.00000 q^{61} +23.4164 q^{62} -13.0000 q^{64} -5.52786 q^{66} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +14.4721 q^{71} -2.23607 q^{72} -3.52786 q^{73} +24.4721 q^{74} -19.4164 q^{76} -10.0000 q^{78} -4.94427 q^{79} +1.00000 q^{81} -4.47214 q^{82} +0.944272 q^{83} -20.0000 q^{86} +2.00000 q^{87} +5.52786 q^{88} +2.00000 q^{89} -12.0000 q^{92} +10.4721 q^{93} +11.0557 q^{94} -6.70820 q^{96} -0.472136 q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{16} - 4 q^{17} - 4 q^{19} + 20 q^{22} - 8 q^{23} + 20 q^{26} - 2 q^{27} - 4 q^{29} - 12 q^{31} - 4 q^{33} + 6 q^{36} - 4 q^{37} + 20 q^{38} + 4 q^{41} + 12 q^{44} + 8 q^{47} + 2 q^{48} + 4 q^{51} + 16 q^{53} + 4 q^{57} + 4 q^{61} + 20 q^{62} - 26 q^{64} - 20 q^{66} + 8 q^{67} - 12 q^{68} + 8 q^{69} + 20 q^{71} - 16 q^{73} + 40 q^{74} - 12 q^{76} - 20 q^{78} + 8 q^{79} + 2 q^{81} - 16 q^{83} - 40 q^{86} + 4 q^{87} + 20 q^{88} + 4 q^{89} - 24 q^{92} + 12 q^{93} + 40 q^{94} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^16 - 4 * q^17 - 4 * q^19 + 20 * q^22 - 8 * q^23 + 20 * q^26 - 2 * q^27 - 4 * q^29 - 12 * q^31 - 4 * q^33 + 6 * q^36 - 4 * q^37 + 20 * q^38 + 4 * q^41 + 12 * q^44 + 8 * q^47 + 2 * q^48 + 4 * q^51 + 16 * q^53 + 4 * q^57 + 4 * q^61 + 20 * q^62 - 26 * q^64 - 20 * q^66 + 8 * q^67 - 12 * q^68 + 8 * q^69 + 20 * q^71 - 16 * q^73 + 40 * q^74 - 12 * q^76 - 20 * q^78 + 8 * q^79 + 2 * q^81 - 16 * q^83 - 40 * q^86 + 4 * q^87 + 20 * q^88 + 4 * q^89 - 24 * q^92 + 12 * q^93 + 40 * q^94 + 8 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more