LMFDB - Embedded newform 6825.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.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:	$54.4978993795$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 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			$-1.75080$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.75080 q^{2} -1.00000 q^{3} +1.06530 q^{4} +1.75080 q^{6} -1.00000 q^{7} +1.63647 q^{8} +1.00000 q^{9} -2.38727 q^{11} -1.06530 q^{12} -1.00000 q^{13} +1.75080 q^{14} -4.99574 q^{16} +2.20764 q^{17} -1.75080 q^{18} -4.49413 q^{19} +1.00000 q^{21} +4.17963 q^{22} +1.85767 q^{23} -1.63647 q^{24} +1.75080 q^{26} -1.00000 q^{27} -1.06530 q^{28} +5.12634 q^{29} -4.63647 q^{31} +5.47360 q^{32} +2.38727 q^{33} -3.86513 q^{34} +1.06530 q^{36} +2.41256 q^{37} +7.86833 q^{38} +1.00000 q^{39} -1.33553 q^{41} -1.75080 q^{42} +8.53890 q^{43} -2.54316 q^{44} -3.25240 q^{46} +2.34299 q^{47} +4.99574 q^{48} +1.00000 q^{49} -2.20764 q^{51} -1.06530 q^{52} +3.90669 q^{53} +1.75080 q^{54} -1.63647 q^{56} +4.49413 q^{57} -8.97520 q^{58} -12.4693 q^{59} +0.0447645 q^{61} +8.11753 q^{62} -1.00000 q^{63} +0.408295 q^{64} -4.17963 q^{66} +9.30414 q^{67} +2.35180 q^{68} -1.85767 q^{69} -6.26441 q^{71} +1.63647 q^{72} -14.1506 q^{73} -4.22391 q^{74} -4.78761 q^{76} +2.38727 q^{77} -1.75080 q^{78} +10.9989 q^{79} +1.00000 q^{81} +2.33824 q^{82} -9.37603 q^{83} +1.06530 q^{84} -14.9499 q^{86} -5.12634 q^{87} -3.90669 q^{88} +6.73298 q^{89} +1.00000 q^{91} +1.97897 q^{92} +4.63647 q^{93} -4.10211 q^{94} -5.47360 q^{96} +7.19457 q^{97} -1.75080 q^{98} -2.38727 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{13} - q^{14} - q^{16} + 5 q^{17} + q^{18} - 15 q^{19} + 4 q^{21} + 9 q^{22} + 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q + q^2 - 4 * q^3 + q^4 - q^6 - 4 * q^7 + 3 * q^8 + 4 * q^9 + 2 * q^11 - q^12 - 4 * q^13 - q^14 - q^16 + 5 * q^17 + q^18 - 15 * q^19 + 4 * q^21 + 9 * q^22 + 8 * q^23 - 3 * q^24 - q^26 - 4 * q^27 - q^28 - 5 * q^29 - 15 * q^31 + 2 * q^32 - 2 * q^33 - 3 * q^34 + q^36 + 12 * q^37 - 13 * q^38 + 4 * q^39 - 5 * q^41 + q^42 + 11 * q^43 - 6 * q^44 + 11 * q^46 - 4 * q^47 + q^48 + 4 * q^49 - 5 * q^51 - q^52 + 15 * q^53 - q^54 - 3 * q^56 + 15 * q^57 - 11 * q^59 - 20 * q^61 - 4 * q^62 - 4 * q^63 - 15 * q^64 - 9 * q^66 + 2 * q^67 + 7 * q^68 - 8 * q^69 + 20 * q^71 + 3 * q^72 + 6 * q^73 - 11 * q^74 - 26 * q^76 - 2 * q^77 + q^78 - 7 * q^79 + 4 * q^81 - q^82 + 2 * q^83 + q^84 - 10 * q^86 + 5 * q^87 - 15 * q^88 - 8 * q^89 + 4 * q^91 + 13 * q^92 + 15 * q^93 - 28 * q^94 - 2 * q^96 - 5 * q^97 + q^98 + 2 * q^99

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 $

$2$	−1.75080	−1.23800	−0.619001	−	0.785390i	$-0.712462\pi$
$2$	−1.75080	−1.23800	−0.619001	+	0.785390i	$0.712462\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.06530	0.532651
$5$	0	0
$5$	0	0
$6$	1.75080	0.714761
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.63647	0.578579
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.38727	−0.719789	−0.359894	−	0.932993i	$-0.617187\pi$
$11$	−2.38727	−0.719789	−0.359894	+	0.932993i	$0.617187\pi$
$12$	−1.06530	−0.307526
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.75080	0.467921
$15$	0	0
$16$	−4.99574	−1.24893
$17$	2.20764	0.535431	0.267715	−	0.963498i	$-0.413731\pi$
$17$	2.20764	0.535431	0.267715	+	0.963498i	$0.413731\pi$
$18$	−1.75080	−0.412668
$19$	−4.49413	−1.03103	−0.515513	−	0.856882i	$-0.672398\pi$
$19$	−4.49413	−1.03103	−0.515513	+	0.856882i	$0.672398\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	4.17963	0.891101
$23$	1.85767	0.387350	0.193675	−	0.981066i	$-0.437959\pi$
$23$	1.85767	0.387350	0.193675	+	0.981066i	$0.437959\pi$
$24$	−1.63647	−0.334043
$25$	0	0
$26$	1.75080	0.343360
$27$	−1.00000	−0.192450
$28$	−1.06530	−0.201323
$29$	5.12634	0.951937	0.475969	−	0.879462i	$-0.342098\pi$
$29$	5.12634	0.951937	0.475969	+	0.879462i	$0.342098\pi$
$30$	0	0
$31$	−4.63647	−0.832734	−0.416367	−	0.909197i	$-0.636697\pi$
$31$	−4.63647	−0.832734	−0.416367	+	0.909197i	$0.636697\pi$
$32$	5.47360	0.967604
$33$	2.38727	0.415570
$34$	−3.86513	−0.662865
$35$	0	0
$36$	1.06530	0.177550
$37$	2.41256	0.396622	0.198311	−	0.980139i	$-0.436454\pi$
$37$	2.41256	0.396622	0.198311	+	0.980139i	$0.436454\pi$
$38$	7.86833	1.27641
$39$	1.00000	0.160128
$40$	0	0
$41$	−1.33553	−0.208574	−0.104287	−	0.994547i	$-0.533256\pi$
$41$	−1.33553	−0.208574	−0.104287	+	0.994547i	$0.533256\pi$
$42$	−1.75080	−0.270154
$43$	8.53890	1.30217	0.651085	−	0.759005i	$-0.274314\pi$
$43$	8.53890	1.30217	0.651085	+	0.759005i	$0.274314\pi$
$44$	−2.54316	−0.383396
$45$	0	0
$46$	−3.25240	−0.479540
$47$	2.34299	0.341761	0.170880	−	0.985292i	$-0.445339\pi$
$47$	2.34299	0.341761	0.170880	+	0.985292i	$0.445339\pi$
$48$	4.99574	0.721072
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.20764	−0.309131
$52$	−1.06530	−0.147731
$53$	3.90669	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$53$	3.90669	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$54$	1.75080	0.238254
$55$	0	0
$56$	−1.63647	−0.218682
$57$	4.49413	0.595263
$58$	−8.97520	−1.17850
$59$	−12.4693	−1.62337	−0.811684	−	0.584096i	$-0.801449\pi$
$59$	−12.4693	−1.62337	−0.811684	+	0.584096i	$0.801449\pi$
$60$	0	0
$61$	0.0447645	0.00573151	0.00286575	−	0.999996i	$-0.499088\pi$
$61$	0.0447645	0.00573151	0.00286575	+	0.999996i	$0.499088\pi$
$62$	8.11753	1.03093
$63$	−1.00000	−0.125988
$64$	0.408295	0.0510369
$65$	0	0
$66$	−4.17963	−0.514477
$67$	9.30414	1.13668	0.568341	−	0.822793i	$-0.307585\pi$
$67$	9.30414	1.13668	0.568341	+	0.822793i	$0.307585\pi$
$68$	2.35180	0.285198
$69$	−1.85767	−0.223637
$70$	0	0
$71$	−6.26441	−0.743449	−0.371724	−	0.928343i	$-0.621233\pi$
$71$	−6.26441	−0.743449	−0.371724	+	0.928343i	$0.621233\pi$
$72$	1.63647	0.192860
$73$	−14.1506	−1.65620	−0.828099	−	0.560581i	$-0.810578\pi$
$73$	−14.1506	−1.65620	−0.828099	+	0.560581i	$0.810578\pi$
$74$	−4.22391	−0.491020
$75$	0	0
$76$	−4.78761	−0.549177
$77$	2.38727	0.272055
$78$	−1.75080	−0.198239
$79$	10.9989	1.23748	0.618739	−	0.785597i	$-0.287644\pi$
$79$	10.9989	1.23748	0.618739	+	0.785597i	$0.287644\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.33824	0.258216
$83$	−9.37603	−1.02915	−0.514576	−	0.857445i	$-0.672051\pi$
$83$	−9.37603	−1.02915	−0.514576	+	0.857445i	$0.672051\pi$
$84$	1.06530	0.116234
$85$	0	0
$86$	−14.9499	−1.61209
$87$	−5.12634	−0.549601
$88$	−3.90669	−0.416455
$89$	6.73298	0.713694	0.356847	−	0.934163i	$-0.383852\pi$
$89$	6.73298	0.713694	0.356847	+	0.934163i	$0.383852\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	1.97897	0.206322
$93$	4.63647	0.480779
$94$	−4.10211	−0.423101
$95$	0	0
$96$	−5.47360	−0.558647
$97$	7.19457	0.730497	0.365249	−	0.930910i	$-0.380984\pi$
$97$	7.19457	0.730497	0.365249	+	0.930910i	$0.380984\pi$
$98$	−1.75080	−0.176858
$99$	−2.38727	−0.239930
$100$	0	0
$101$	−4.28040	−0.425916	−0.212958	−	0.977061i	$-0.568310\pi$
$101$	−4.28040	−0.425916	−0.212958	+	0.977061i	$0.568310\pi$
$102$	3.86513	0.382705
$103$	−1.80681	−0.178030	−0.0890150	−	0.996030i	$-0.528372\pi$
$103$	−1.80681	−0.178030	−0.0890150	+	0.996030i	$0.528372\pi$
$104$	−1.63647	−0.160469
$105$	0	0
$106$	−6.83984	−0.664344
$107$	0.612730	0.0592349	0.0296174	−	0.999561i	$-0.490571\pi$
$107$	0.612730	0.0592349	0.0296174	+	0.999561i	$0.490571\pi$
$108$	−1.06530	−0.102509
$109$	0.149801	0.0143483	0.00717415	−	0.999974i	$-0.497716\pi$
$109$	0.149801	0.0143483	0.00717415	+	0.999974i	$0.497716\pi$
$110$	0	0
$111$	−2.41256	−0.228990
$112$	4.99574	0.472053
$113$	4.13990	0.389449	0.194724	−	0.980858i	$-0.437619\pi$
$113$	4.13990	0.389449	0.194724	+	0.980858i	$0.437619\pi$
$114$	−7.86833	−0.736937
$115$	0	0
$116$	5.46110	0.507050
$117$	−1.00000	−0.0924500
$118$	21.8313	2.00974
$119$	−2.20764	−0.202374
$120$	0	0
$121$	−5.30094	−0.481904
$122$	−0.0783737	−0.00709562
$123$	1.33553	0.120420
$124$	−4.93924	−0.443557
$125$	0	0
$126$	1.75080	0.155974
$127$	18.3353	1.62700	0.813500	−	0.581565i	$-0.197560\pi$
$127$	18.3353	1.62700	0.813500	+	0.581565i	$0.197560\pi$
$128$	−11.6620	−1.03079
$129$	−8.53890	−0.751808
$130$	0	0
$131$	−7.42911	−0.649084	−0.324542	−	0.945871i	$-0.605210\pi$
$131$	−7.42911	−0.649084	−0.324542	+	0.945871i	$0.605210\pi$
$132$	2.54316	0.221354
$133$	4.49413	0.389691
$134$	−16.2897	−1.40722
$135$	0	0
$136$	3.61273	0.309789
$137$	−9.67271	−0.826395	−0.413197	−	0.910641i	$-0.635588\pi$
$137$	−9.67271	−0.826395	−0.413197	+	0.910641i	$0.635588\pi$
$138$	3.25240	0.276863
$139$	−12.2034	−1.03508	−0.517538	−	0.855660i	$-0.673151\pi$
$139$	−12.2034	−1.03508	−0.517538	+	0.855660i	$0.673151\pi$
$140$	0	0
$141$	−2.34299	−0.197316
$142$	10.9677	0.920392
$143$	2.38727	0.199634
$144$	−4.99574	−0.416311
$145$	0	0
$146$	24.7748	2.05038
$147$	−1.00000	−0.0824786
$148$	2.57010	0.211261
$149$	14.6298	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$149$	14.6298	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$150$	0	0
$151$	3.08478	0.251036	0.125518	−	0.992091i	$-0.459941\pi$
$151$	3.08478	0.251036	0.125518	+	0.992091i	$0.459941\pi$
$152$	−7.35451	−0.596530
$153$	2.20764	0.178477
$154$	−4.17963	−0.336804
$155$	0	0
$156$	1.06530	0.0852924
$157$	22.6564	1.80818	0.904090	−	0.427342i	$-0.140550\pi$
$157$	22.6564	1.80818	0.904090	+	0.427342i	$0.140550\pi$
$158$	−19.2569	−1.53200
$159$	−3.90669	−0.309821
$160$	0	0
$161$	−1.85767	−0.146405
$162$	−1.75080	−0.137556
$163$	6.01384	0.471040	0.235520	−	0.971869i	$-0.424321\pi$
$163$	6.01384	0.471040	0.235520	+	0.971869i	$0.424321\pi$
$164$	−1.42274	−0.111097
$165$	0	0
$166$	16.4156	1.27409
$167$	2.02103	0.156392	0.0781958	−	0.996938i	$-0.475084\pi$
$167$	2.02103	0.156392	0.0781958	+	0.996938i	$0.475084\pi$
$168$	1.63647	0.126256
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.49413	−0.343675
$172$	9.09651	0.693602
$173$	5.00137	0.380247	0.190124	−	0.981760i	$-0.439111\pi$
$173$	5.00137	0.380247	0.190124	+	0.981760i	$0.439111\pi$
$174$	8.97520	0.680408
$175$	0	0
$176$	11.9262	0.898969
$177$	12.4693	0.937252
$178$	−11.7881	−0.883555
$179$	4.85660	0.363000	0.181500	−	0.983391i	$-0.441905\pi$
$179$	4.85660	0.363000	0.181500	+	0.983391i	$0.441905\pi$
$180$	0	0
$181$	9.75672	0.725211	0.362606	−	0.931943i	$-0.381887\pi$
$181$	9.75672	0.725211	0.362606	+	0.931943i	$0.381887\pi$
$182$	−1.75080	−0.129778
$183$	−0.0447645	−0.00330909
$184$	3.04001	0.224113
$185$	0	0
$186$	−8.11753	−0.595206
$187$	−5.27022	−0.385397
$188$	2.49600	0.182039
$189$	1.00000	0.0727393
$190$	0	0
$191$	−14.1338	−1.02269	−0.511343	−	0.859377i	$-0.670852\pi$
$191$	−14.1338	−1.02269	−0.511343	+	0.859377i	$0.670852\pi$
$192$	−0.408295	−0.0294662
$193$	−7.29153	−0.524856	−0.262428	−	0.964952i	$-0.584523\pi$
$193$	−7.29153	−0.524856	−0.262428	+	0.964952i	$0.584523\pi$
$194$	−12.5962	−0.904358
$195$	0	0
$196$	1.06530	0.0760930
$197$	−7.14282	−0.508905	−0.254453	−	0.967085i	$-0.581895\pi$
$197$	−7.14282	−0.508905	−0.254453	+	0.967085i	$0.581895\pi$
$198$	4.17963	0.297034
$199$	−18.7400	−1.32844	−0.664220	−	0.747537i	$-0.731236\pi$
$199$	−18.7400	−1.32844	−0.664220	+	0.747537i	$0.731236\pi$
$200$	0	0
$201$	−9.30414	−0.656264
$202$	7.49413	0.527285
$203$	−5.12634	−0.359799
$204$	−2.35180	−0.164659
$205$	0	0
$206$	3.16336	0.220402
$207$	1.85767	0.129117
$208$	4.99574	0.346392
$209$	10.7287	0.742121
$210$	0	0
$211$	8.73298	0.601203	0.300601	−	0.953750i	$-0.402813\pi$
$211$	8.73298	0.601203	0.300601	+	0.953750i	$0.402813\pi$
$212$	4.16181	0.285834
$213$	6.26441	0.429230
$214$	−1.07277	−0.0733329
$215$	0	0
$216$	−1.63647	−0.111348
$217$	4.63647	0.314744
$218$	−0.262271	−0.0177632
$219$	14.1506	0.956207
$220$	0	0
$221$	−2.20764	−0.148502
$222$	4.22391	0.283490
$223$	16.7726	1.12318	0.561588	−	0.827417i	$-0.310191\pi$
$223$	16.7726	1.12318	0.561588	+	0.827417i	$0.310191\pi$
$224$	−5.47360	−0.365720
$225$	0	0
$226$	−7.24814	−0.482139
$227$	−5.38727	−0.357566	−0.178783	−	0.983889i	$-0.557216\pi$
$227$	−5.38727	−0.357566	−0.178783	+	0.983889i	$0.557216\pi$
$228$	4.78761	0.317067
$229$	28.2058	1.86389	0.931946	−	0.362598i	$-0.118110\pi$
$229$	28.2058	1.86389	0.931946	+	0.362598i	$0.118110\pi$
$230$	0	0
$231$	−2.38727	−0.157071
$232$	8.38910	0.550771
$233$	−17.6905	−1.15895	−0.579473	−	0.814992i	$-0.696741\pi$
$233$	−17.6905	−1.15895	−0.579473	+	0.814992i	$0.696741\pi$
$234$	1.75080	0.114453
$235$	0	0
$236$	−13.2836	−0.864689
$237$	−10.9989	−0.714458
$238$	3.86513	0.250539
$239$	1.87608	0.121353	0.0606767	−	0.998157i	$-0.480674\pi$
$239$	1.87608	0.121353	0.0606767	+	0.998157i	$0.480674\pi$
$240$	0	0
$241$	−1.90215	−0.122528	−0.0612642	−	0.998122i	$-0.519513\pi$
$241$	−1.90215	−0.122528	−0.0612642	+	0.998122i	$0.519513\pi$
$242$	9.28089	0.596598
$243$	−1.00000	−0.0641500
$244$	0.0476877	0.00305289
$245$	0	0
$246$	−2.33824	−0.149081
$247$	4.49413	0.285955
$248$	−7.58744	−0.481803
$249$	9.37603	0.594182
$250$	0	0
$251$	−15.5613	−0.982220	−0.491110	−	0.871097i	$-0.663409\pi$
$251$	−15.5613	−0.982220	−0.491110	+	0.871097i	$0.663409\pi$
$252$	−1.06530	−0.0671077
$253$	−4.43475	−0.278810
$254$	−32.1015	−2.01423
$255$	0	0
$256$	19.6013	1.22508
$257$	9.17245	0.572161	0.286081	−	0.958206i	$-0.407647\pi$
$257$	9.17245	0.572161	0.286081	+	0.958206i	$0.407647\pi$
$258$	14.9499	0.930741
$259$	−2.41256	−0.149909
$260$	0	0
$261$	5.12634	0.317312
$262$	13.0069	0.803568
$263$	−6.18256	−0.381233	−0.190616	−	0.981665i	$-0.561049\pi$
$263$	−6.18256	−0.381233	−0.190616	+	0.981665i	$0.561049\pi$
$264$	3.90669	0.240440
$265$	0	0
$266$	−7.86833	−0.482438
$267$	−6.73298	−0.412051
$268$	9.91172	0.605455
$269$	25.9036	1.57937	0.789685	−	0.613513i	$-0.210244\pi$
$269$	25.9036	1.57937	0.789685	+	0.613513i	$0.210244\pi$
$270$	0	0
$271$	0.515160	0.0312937	0.0156469	−	0.999878i	$-0.495019\pi$
$271$	0.515160	0.0312937	0.0156469	+	0.999878i	$0.495019\pi$
$272$	−11.0288	−0.668717
$273$	−1.00000	−0.0605228
$274$	16.9350	1.02308
$275$	0	0
$276$	−1.97897	−0.119120
$277$	−1.73801	−0.104427	−0.0522134	−	0.998636i	$-0.516628\pi$
$277$	−1.73801	−0.104427	−0.0522134	+	0.998636i	$0.516628\pi$
$278$	21.3657	1.28143
$279$	−4.63647	−0.277578
$280$	0	0
$281$	20.0227	1.19445	0.597226	−	0.802073i	$-0.296269\pi$
$281$	20.0227	1.19445	0.597226	+	0.802073i	$0.296269\pi$
$282$	4.10211	0.244277
$283$	−19.3952	−1.15293	−0.576463	−	0.817123i	$-0.695568\pi$
$283$	−19.3952	−1.15293	−0.576463	+	0.817123i	$0.695568\pi$
$284$	−6.67349	−0.395999
$285$	0	0
$286$	−4.17963	−0.247147
$287$	1.33553	0.0788337
$288$	5.47360	0.322535
$289$	−12.1263	−0.713314
$290$	0	0
$291$	−7.19457	−0.421753
$292$	−15.0746	−0.882176
$293$	−11.5360	−0.673939	−0.336969	−	0.941516i	$-0.609402\pi$
$293$	−11.5360	−0.673939	−0.336969	+	0.941516i	$0.609402\pi$
$294$	1.75080	0.102109
$295$	0	0
$296$	3.94808	0.229477
$297$	2.38727	0.138523
$298$	−25.6138	−1.48377
$299$	−1.85767	−0.107432
$300$	0	0
$301$	−8.53890	−0.492174
$302$	−5.40083	−0.310783
$303$	4.28040	0.245903
$304$	22.4515	1.28768
$305$	0	0
$306$	−3.86513	−0.220955
$307$	−8.06287	−0.460172	−0.230086	−	0.973170i	$-0.573901\pi$
$307$	−8.06287	−0.460172	−0.230086	+	0.973170i	$0.573901\pi$
$308$	2.54316	0.144910
$309$	1.80681	0.102786
$310$	0	0
$311$	−7.16559	−0.406323	−0.203162	−	0.979145i	$-0.565122\pi$
$311$	−7.16559	−0.406323	−0.203162	+	0.979145i	$0.565122\pi$
$312$	1.63647	0.0926468
$313$	12.6139	0.712979	0.356490	−	0.934299i	$-0.383973\pi$
$313$	12.6139	0.712979	0.356490	+	0.934299i	$0.383973\pi$
$314$	−39.6669	−2.23853
$315$	0	0
$316$	11.7172	0.659144
$317$	0.985378	0.0553444	0.0276722	−	0.999617i	$-0.491191\pi$
$317$	0.985378	0.0553444	0.0276722	+	0.999617i	$0.491191\pi$
$318$	6.83984	0.383559
$319$	−12.2380	−0.685194
$320$	0	0
$321$	−0.612730	−0.0341993
$322$	3.25240	0.181249
$323$	−9.92142	−0.552042
$324$	1.06530	0.0591835
$325$	0	0
$326$	−10.5290	−0.583149
$327$	−0.149801	−0.00828400
$328$	−2.18555	−0.120677
$329$	−2.34299	−0.129173
$330$	0	0
$331$	−8.24676	−0.453283	−0.226642	−	0.973978i	$-0.572775\pi$
$331$	−8.24676	−0.453283	−0.226642	+	0.973978i	$0.572775\pi$
$332$	−9.98830	−0.548179
$333$	2.41256	0.132207
$334$	−3.53841	−0.193613
$335$	0	0
$336$	−4.99574	−0.272540
$337$	34.1282	1.85908	0.929541	−	0.368719i	$-0.120204\pi$
$337$	34.1282	1.85908	0.929541	+	0.368719i	$0.120204\pi$
$338$	−1.75080	−0.0952310
$339$	−4.13990	−0.224848
$340$	0	0
$341$	11.0685	0.599393
$342$	7.86833	0.425471
$343$	−1.00000	−0.0539949
$344$	13.9736	0.753409
$345$	0	0
$346$	−8.75641	−0.470747
$347$	−0.114331	−0.00613761	−0.00306880	−	0.999995i	$-0.500977\pi$
$347$	−0.114331	−0.00613761	−0.00306880	+	0.999995i	$0.500977\pi$
$348$	−5.46110	−0.292746
$349$	0.318766	0.0170632	0.00853158	−	0.999964i	$-0.497284\pi$
$349$	0.318766	0.0170632	0.00853158	+	0.999964i	$0.497284\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−13.0670	−0.696471
$353$	13.1770	0.701342	0.350671	−	0.936499i	$-0.385954\pi$
$353$	13.1770	0.701342	0.350671	+	0.936499i	$0.385954\pi$
$354$	−21.8313	−1.16032
$355$	0	0
$356$	7.17265	0.380150
$357$	2.20764	0.116841
$358$	−8.50294	−0.449394
$359$	7.15706	0.377735	0.188868	−	0.982003i	$-0.439518\pi$
$359$	7.15706	0.377735	0.188868	+	0.982003i	$0.439518\pi$
$360$	0	0
$361$	1.19725	0.0630130
$362$	−17.0821	−0.897813
$363$	5.30094	0.278227
$364$	1.06530	0.0558370
$365$	0	0
$366$	0.0783737	0.00409666
$367$	23.2959	1.21604	0.608018	−	0.793923i	$-0.291965\pi$
$367$	23.2959	1.21604	0.608018	+	0.793923i	$0.291965\pi$
$368$	−9.28040	−0.483775
$369$	−1.33553	−0.0695248
$370$	0	0
$371$	−3.90669	−0.202825
$372$	4.93924	0.256088
$373$	−33.9611	−1.75844	−0.879219	−	0.476419i	$-0.841935\pi$
$373$	−33.9611	−1.75844	−0.879219	+	0.476419i	$0.841935\pi$
$374$	9.22711	0.477123
$375$	0	0
$376$	3.83424	0.197736
$377$	−5.12634	−0.264020
$378$	−1.75080	−0.0900515
$379$	4.16473	0.213928	0.106964	−	0.994263i	$-0.465887\pi$
$379$	4.16473	0.213928	0.106964	+	0.994263i	$0.465887\pi$
$380$	0	0
$381$	−18.3353	−0.939348
$382$	24.7455	1.26609
$383$	−28.2751	−1.44479	−0.722394	−	0.691481i	$-0.756958\pi$
$383$	−28.2751	−1.44479	−0.722394	+	0.691481i	$0.756958\pi$
$384$	11.6620	0.595126
$385$	0	0
$386$	12.7660	0.649773
$387$	8.53890	0.434057
$388$	7.66439	0.389100
$389$	−16.4220	−0.832626	−0.416313	−	0.909221i	$-0.636678\pi$
$389$	−16.4220	−0.832626	−0.416313	+	0.909221i	$0.636678\pi$
$390$	0	0
$391$	4.10105	0.207399
$392$	1.63647	0.0826542
$393$	7.42911	0.374749
$394$	12.5057	0.630026
$395$	0	0
$396$	−2.54316	−0.127799
$397$	−25.5998	−1.28482	−0.642408	−	0.766363i	$-0.722064\pi$
$397$	−25.5998	−1.28482	−0.642408	+	0.766363i	$0.722064\pi$
$398$	32.8099	1.64461
$399$	−4.49413	−0.224988
$400$	0	0
$401$	−36.4412	−1.81978	−0.909892	−	0.414845i	$-0.863836\pi$
$401$	−36.4412	−1.81978	−0.909892	+	0.414845i	$0.863836\pi$
$402$	16.2897	0.812456
$403$	4.63647	0.230959
$404$	−4.55992	−0.226865
$405$	0	0
$406$	8.97520	0.445432
$407$	−5.75943	−0.285484
$408$	−3.61273	−0.178857
$409$	−20.3194	−1.00473	−0.502364	−	0.864656i	$-0.667536\pi$
$409$	−20.3194	−1.00473	−0.502364	+	0.864656i	$0.667536\pi$
$410$	0	0
$411$	9.67271	0.477119
$412$	−1.92480	−0.0948279
$413$	12.4693	0.613576
$414$	−3.25240	−0.159847
$415$	0	0
$416$	−5.47360	−0.268365
$417$	12.2034	0.597602
$418$	−18.7838	−0.918747
$419$	−25.6098	−1.25112	−0.625561	−	0.780175i	$-0.715130\pi$
$419$	−25.6098	−1.25112	−0.625561	+	0.780175i	$0.715130\pi$
$420$	0	0
$421$	−16.7263	−0.815189	−0.407594	−	0.913163i	$-0.633632\pi$
$421$	−16.7263	−0.815189	−0.407594	+	0.913163i	$0.633632\pi$
$422$	−15.2897	−0.744291
$423$	2.34299	0.113920
$424$	6.39319	0.310481
$425$	0	0
$426$	−10.9677	−0.531388
$427$	−0.0447645	−0.00216631
$428$	0.652743	0.0315515
$429$	−2.38727	−0.115258
$430$	0	0
$431$	−31.4040	−1.51268	−0.756339	−	0.654179i	$-0.773014\pi$
$431$	−31.4040	−1.51268	−0.756339	+	0.654179i	$0.773014\pi$
$432$	4.99574	0.240357
$433$	−17.6698	−0.849156	−0.424578	−	0.905391i	$-0.639577\pi$
$433$	−17.6698	−0.849156	−0.424578	+	0.905391i	$0.639577\pi$
$434$	−8.11753	−0.389654
$435$	0	0
$436$	0.159583	0.00764264
$437$	−8.34860	−0.399368
$438$	−24.7748	−1.18379
$439$	21.9861	1.04934	0.524671	−	0.851305i	$-0.324188\pi$
$439$	21.9861	1.04934	0.524671	+	0.851305i	$0.324188\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	3.86513	0.183846
$443$	−7.13402	−0.338947	−0.169474	−	0.985535i	$-0.554207\pi$
$443$	−7.13402	−0.338947	−0.169474	+	0.985535i	$0.554207\pi$
$444$	−2.57010	−0.121972
$445$	0	0
$446$	−29.3655	−1.39049
$447$	−14.6298	−0.691964
$448$	−0.408295	−0.0192901
$449$	−4.82463	−0.227688	−0.113844	−	0.993499i	$-0.536316\pi$
$449$	−4.82463	−0.227688	−0.113844	+	0.993499i	$0.536316\pi$
$450$	0	0
$451$	3.18826	0.150129
$452$	4.41024	0.207440
$453$	−3.08478	−0.144935
$454$	9.43203	0.442667
$455$	0	0
$456$	7.35451	0.344407
$457$	24.3569	1.13937	0.569685	−	0.821863i	$-0.307065\pi$
$457$	24.3569	1.13937	0.569685	+	0.821863i	$0.307065\pi$
$458$	−49.3827	−2.30750
$459$	−2.20764	−0.103044
$460$	0	0
$461$	14.9012	0.694016	0.347008	−	0.937862i	$-0.387198\pi$
$461$	14.9012	0.694016	0.347008	+	0.937862i	$0.387198\pi$
$462$	4.17963	0.194454
$463$	−33.3488	−1.54985	−0.774925	−	0.632053i	$-0.782212\pi$
$463$	−33.3488	−1.54985	−0.774925	+	0.632053i	$0.782212\pi$
$464$	−25.6098	−1.18891
$465$	0	0
$466$	30.9726	1.43478
$467$	1.97422	0.0913561	0.0456781	−	0.998956i	$-0.485455\pi$
$467$	1.97422	0.0913561	0.0456781	+	0.998956i	$0.485455\pi$
$468$	−1.06530	−0.0492436
$469$	−9.30414	−0.429625
$470$	0	0
$471$	−22.6564	−1.04395
$472$	−20.4057	−0.939248
$473$	−20.3847	−0.937288
$474$	19.2569	0.884501
$475$	0	0
$476$	−2.35180	−0.107795
$477$	3.90669	0.178875
$478$	−3.28464	−0.150236
$479$	15.9047	0.726706	0.363353	−	0.931651i	$-0.381632\pi$
$479$	15.9047	0.726706	0.363353	+	0.931651i	$0.381632\pi$
$480$	0	0
$481$	−2.41256	−0.110003
$482$	3.33029	0.151690
$483$	1.85767	0.0845267
$484$	−5.64711	−0.256687
$485$	0	0
$486$	1.75080	0.0794179
$487$	−1.33166	−0.0603433	−0.0301716	−	0.999545i	$-0.509605\pi$
$487$	−1.33166	−0.0603433	−0.0301716	+	0.999545i	$0.509605\pi$
$488$	0.0732557	0.00331613
$489$	−6.01384	−0.271955
$490$	0	0
$491$	25.6178	1.15611	0.578057	−	0.815997i	$-0.303811\pi$
$491$	25.6178	1.15611	0.578057	+	0.815997i	$0.303811\pi$
$492$	1.42274	0.0641421
$493$	11.3171	0.509696
$494$	−7.86833	−0.354013
$495$	0	0
$496$	23.1626	1.04003
$497$	6.26441	0.280997
$498$	−16.4156	−0.735598
$499$	−21.3129	−0.954095	−0.477047	−	0.878878i	$-0.658293\pi$
$499$	−21.3129	−0.954095	−0.477047	+	0.878878i	$0.658293\pi$
$500$	0	0
$501$	−2.02103	−0.0902927
$502$	27.2447	1.21599
$503$	−9.76659	−0.435471	−0.217735	−	0.976008i	$-0.569867\pi$
$503$	−9.76659	−0.435471	−0.217735	+	0.976008i	$0.569867\pi$
$504$	−1.63647	−0.0728941
$505$	0	0
$506$	7.76436	0.345168
$507$	−1.00000	−0.0444116
$508$	19.5327	0.866623
$509$	30.8043	1.36538	0.682688	−	0.730710i	$-0.260811\pi$
$509$	30.8043	1.36538	0.682688	+	0.730710i	$0.260811\pi$
$510$	0	0
$511$	14.1506	0.625984
$512$	−10.9939	−0.485867
$513$	4.49413	0.198421
$514$	−16.0591	−0.708337
$515$	0	0
$516$	−9.09651	−0.400451
$517$	−5.59336	−0.245996
$518$	4.22391	0.185588
$519$	−5.00137	−0.219536
$520$	0	0
$521$	6.73432	0.295036	0.147518	−	0.989059i	$-0.452872\pi$
$521$	6.73432	0.295036	0.147518	+	0.989059i	$0.452872\pi$
$522$	−8.97520	−0.392834
$523$	−33.7388	−1.47529	−0.737647	−	0.675187i	$-0.764063\pi$
$523$	−33.7388	−1.47529	−0.737647	+	0.675187i	$0.764063\pi$
$524$	−7.91425	−0.345736
$525$	0	0
$526$	10.8244	0.471967
$527$	−10.2356	−0.445872
$528$	−11.9262	−0.519020
$529$	−19.5491	−0.849960
$530$	0	0
$531$	−12.4693	−0.541123
$532$	4.78761	0.207569
$533$	1.33553	0.0578481
$534$	11.7881	0.510121
$535$	0	0
$536$	15.2259	0.657661
$537$	−4.85660	−0.209578
$538$	−45.3520	−1.95526
$539$	−2.38727	−0.102827
$540$	0	0
$541$	17.0810	0.734370	0.367185	−	0.930148i	$-0.380322\pi$
$541$	17.0810	0.734370	0.367185	+	0.930148i	$0.380322\pi$
$542$	−0.901942	−0.0387417
$543$	−9.75672	−0.418701
$544$	12.0837	0.518085
$545$	0	0
$546$	1.75080	0.0749273
$547$	24.4448	1.04518	0.522592	−	0.852583i	$-0.324965\pi$
$547$	24.4448	1.04518	0.522592	+	0.852583i	$0.324965\pi$
$548$	−10.3044	−0.440180
$549$	0.0447645	0.00191050
$550$	0	0
$551$	−23.0385	−0.981471
$552$	−3.04001	−0.129392
$553$	−10.9989	−0.467722
$554$	3.04290	0.129281
$555$	0	0
$556$	−13.0003	−0.551335
$557$	−32.6975	−1.38544	−0.692719	−	0.721208i	$-0.743587\pi$
$557$	−32.6975	−1.38544	−0.692719	+	0.721208i	$0.743587\pi$
$558$	8.11753	0.343643
$559$	−8.53890	−0.361157
$560$	0	0
$561$	5.27022	0.222509
$562$	−35.0557	−1.47874
$563$	−11.8543	−0.499599	−0.249799	−	0.968298i	$-0.580365\pi$
$563$	−11.8543	−0.499599	−0.249799	+	0.968298i	$0.580365\pi$
$564$	−2.49600	−0.105100
$565$	0	0
$566$	33.9572	1.42733
$567$	−1.00000	−0.0419961
$568$	−10.2515	−0.430144
$569$	−43.8395	−1.83785	−0.918925	−	0.394433i	$-0.870941\pi$
$569$	−43.8395	−1.83785	−0.918925	+	0.394433i	$0.870941\pi$
$570$	0	0
$571$	−31.0717	−1.30031	−0.650155	−	0.759801i	$-0.725296\pi$
$571$	−31.0717	−1.30031	−0.650155	+	0.759801i	$0.725296\pi$
$572$	2.54316	0.106335
$573$	14.1338	0.590448
$574$	−2.33824	−0.0975963
$575$	0	0
$576$	0.408295	0.0170123
$577$	−4.71378	−0.196237	−0.0981186	−	0.995175i	$-0.531282\pi$
$577$	−4.71378	−0.196237	−0.0981186	+	0.995175i	$0.531282\pi$
$578$	21.2308	0.883085
$579$	7.29153	0.303026
$580$	0	0
$581$	9.37603	0.388983
$582$	12.5962	0.522131
$583$	−9.32633	−0.386257
$584$	−23.1570	−0.958242
$585$	0	0
$586$	20.1972	0.834338
$587$	−13.4878	−0.556700	−0.278350	−	0.960480i	$-0.589787\pi$
$587$	−13.4878	−0.556700	−0.278350	+	0.960480i	$0.589787\pi$
$588$	−1.06530	−0.0439323
$589$	20.8369	0.858570
$590$	0	0
$591$	7.14282	0.293816
$592$	−12.0525	−0.495355
$593$	−7.27614	−0.298795	−0.149398	−	0.988777i	$-0.547733\pi$
$593$	−7.27614	−0.298795	−0.149398	+	0.988777i	$0.547733\pi$
$594$	−4.17963	−0.171492
$595$	0	0
$596$	15.5851	0.638392
$597$	18.7400	0.766975
$598$	3.25240	0.133001
$599$	−41.3494	−1.68949	−0.844745	−	0.535169i	$-0.820248\pi$
$599$	−41.3494	−1.68949	−0.844745	+	0.535169i	$0.820248\pi$
$600$	0	0
$601$	22.7510	0.928032	0.464016	−	0.885827i	$-0.346408\pi$
$601$	22.7510	0.928032	0.464016	+	0.885827i	$0.346408\pi$
$602$	14.9499	0.609313
$603$	9.30414	0.378894
$604$	3.28622	0.133714
$605$	0	0
$606$	−7.49413	−0.304428
$607$	−3.38097	−0.137229	−0.0686146	−	0.997643i	$-0.521858\pi$
$607$	−3.38097	−0.137229	−0.0686146	+	0.997643i	$0.521858\pi$
$608$	−24.5991	−0.997625
$609$	5.12634	0.207730
$610$	0	0
$611$	−2.34299	−0.0947873
$612$	2.35180	0.0950659
$613$	22.4189	0.905492	0.452746	−	0.891640i	$-0.350444\pi$
$613$	22.4189	0.905492	0.452746	+	0.891640i	$0.350444\pi$
$614$	14.1165	0.569694
$615$	0	0
$616$	3.90669	0.157405
$617$	−22.8164	−0.918554	−0.459277	−	0.888293i	$-0.651891\pi$
$617$	−22.8164	−0.918554	−0.459277	+	0.888293i	$0.651891\pi$
$618$	−3.16336	−0.127249
$619$	6.85845	0.275664	0.137832	−	0.990456i	$-0.455987\pi$
$619$	6.85845	0.275664	0.137832	+	0.990456i	$0.455987\pi$
$620$	0	0
$621$	−1.85767	−0.0745455
$622$	12.5455	0.503029
$623$	−6.73298	−0.269751
$624$	−4.99574	−0.199989
$625$	0	0
$626$	−22.0844	−0.882671
$627$	−10.7287	−0.428464
$628$	24.1359	0.963129
$629$	5.32605	0.212364
$630$	0	0
$631$	12.4264	0.494687	0.247344	−	0.968928i	$-0.420442\pi$
$631$	12.4264	0.494687	0.247344	+	0.968928i	$0.420442\pi$
$632$	17.9994	0.715979
$633$	−8.73298	−0.347105
$634$	−1.72520	−0.0685165
$635$	0	0
$636$	−4.16181	−0.165026
$637$	−1.00000	−0.0396214
$638$	21.4262	0.848272
$639$	−6.26441	−0.247816
$640$	0	0
$641$	−27.2103	−1.07474	−0.537372	−	0.843345i	$-0.680583\pi$
$641$	−27.2103	−1.07474	−0.537372	+	0.843345i	$0.680583\pi$
$642$	1.07277	0.0423388
$643$	18.5317	0.730821	0.365410	−	0.930847i	$-0.380929\pi$
$643$	18.5317	0.730821	0.365410	+	0.930847i	$0.380929\pi$
$644$	−1.97897	−0.0779825
$645$	0	0
$646$	17.3704	0.683430
$647$	−28.5799	−1.12359	−0.561796	−	0.827276i	$-0.689889\pi$
$647$	−28.5799	−1.12359	−0.561796	+	0.827276i	$0.689889\pi$
$648$	1.63647	0.0642866
$649$	29.7677	1.16848
$650$	0	0
$651$	−4.63647	−0.181718
$652$	6.40655	0.250900
$653$	22.5858	0.883850	0.441925	−	0.897052i	$-0.354296\pi$
$653$	22.5858	0.883850	0.441925	+	0.897052i	$0.354296\pi$
$654$	0.262271	0.0102556
$655$	0	0
$656$	6.67194	0.260495
$657$	−14.1506	−0.552066
$658$	4.10211	0.159917
$659$	11.5658	0.450541	0.225271	−	0.974296i	$-0.427673\pi$
$659$	11.5658	0.450541	0.225271	+	0.974296i	$0.427673\pi$
$660$	0	0
$661$	−30.8036	−1.19812	−0.599060	−	0.800704i	$-0.704459\pi$
$661$	−30.8036	−1.19812	−0.599060	+	0.800704i	$0.704459\pi$
$662$	14.4384	0.561166
$663$	2.20764	0.0857375
$664$	−15.3436	−0.595446
$665$	0	0
$666$	−4.22391	−0.163673
$667$	9.52302	0.368733
$668$	2.15300	0.0833022
$669$	−16.7726	−0.648466
$670$	0	0
$671$	−0.106865	−0.00412547
$672$	5.47360	0.211149
$673$	34.4280	1.32710	0.663552	−	0.748130i	$-0.269048\pi$
$673$	34.4280	1.32710	0.663552	+	0.748130i	$0.269048\pi$
$674$	−59.7517	−2.30155
$675$	0	0
$676$	1.06530	0.0409732
$677$	−45.2303	−1.73834	−0.869171	−	0.494512i	$-0.835347\pi$
$677$	−45.2303	−1.73834	−0.869171	+	0.494512i	$0.835347\pi$
$678$	7.24814	0.278363
$679$	−7.19457	−0.276102
$680$	0	0
$681$	5.38727	0.206441
$682$	−19.3787	−0.742050
$683$	12.3783	0.473641	0.236820	−	0.971553i	$-0.423895\pi$
$683$	12.3783	0.473641	0.236820	+	0.971553i	$0.423895\pi$
$684$	−4.78761	−0.183059
$685$	0	0
$686$	1.75080	0.0668459
$687$	−28.2058	−1.07612
$688$	−42.6581	−1.62632
$689$	−3.90669	−0.148833
$690$	0	0
$691$	9.51234	0.361866	0.180933	−	0.983495i	$-0.442088\pi$
$691$	9.51234	0.361866	0.180933	+	0.983495i	$0.442088\pi$
$692$	5.32797	0.202539
$693$	2.38727	0.0906849
$694$	0.200171	0.00759838
$695$	0	0
$696$	−8.38910	−0.317988
$697$	−2.94836	−0.111677
$698$	−0.558096	−0.0211242
$699$	17.6905	0.669117
$700$	0	0
$701$	−5.31661	−0.200806	−0.100403	−	0.994947i	$-0.532013\pi$
$701$	−5.31661	−0.200806	−0.100403	+	0.994947i	$0.532013\pi$
$702$	−1.75080	−0.0660797
$703$	−10.8424	−0.408928
$704$	−0.974710	−0.0367358
$705$	0	0
$706$	−23.0703	−0.868263
$707$	4.28040	0.160981
$708$	13.2836	0.499229
$709$	29.8648	1.12160	0.560798	−	0.827953i	$-0.310494\pi$
$709$	29.8648	1.12160	0.560798	+	0.827953i	$0.310494\pi$
$710$	0	0
$711$	10.9989	0.412492
$712$	11.0183	0.412929
$713$	−8.61301	−0.322560
$714$	−3.86513	−0.144649
$715$	0	0
$716$	5.17375	0.193352
$717$	−1.87608	−0.0700634
$718$	−12.5306	−0.467637
$719$	3.71241	0.138449	0.0692247	−	0.997601i	$-0.477947\pi$
$719$	3.71241	0.138449	0.0692247	+	0.997601i	$0.477947\pi$
$720$	0	0
$721$	1.80681	0.0672890
$722$	−2.09614	−0.0780103
$723$	1.90215	0.0707418
$724$	10.3939	0.386284
$725$	0	0
$726$	−9.28089	−0.344446
$727$	−35.2837	−1.30860	−0.654300	−	0.756235i	$-0.727037\pi$
$727$	−35.2837	−1.30860	−0.654300	+	0.756235i	$0.727037\pi$
$728$	1.63647	0.0606516
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.8508	0.697222
$732$	−0.0476877	−0.00176259
$733$	15.1020	0.557805	0.278902	−	0.960319i	$-0.410029\pi$
$733$	15.1020	0.557805	0.278902	+	0.960319i	$0.410029\pi$
$734$	−40.7865	−1.50546
$735$	0	0
$736$	10.1681	0.374802
$737$	−22.2115	−0.818171
$738$	2.33824	0.0860718
$739$	1.58878	0.0584443	0.0292221	−	0.999573i	$-0.490697\pi$
$739$	1.58878	0.0584443	0.0292221	+	0.999573i	$0.490697\pi$
$740$	0	0
$741$	−4.49413	−0.165096
$742$	6.83984	0.251098
$743$	34.3756	1.26112	0.630560	−	0.776141i	$-0.282825\pi$
$743$	34.3756	1.26112	0.630560	+	0.776141i	$0.282825\pi$
$744$	7.58744	0.278169
$745$	0	0
$746$	59.4590	2.17695
$747$	−9.37603	−0.343051
$748$	−5.61438	−0.205282
$749$	−0.612730	−0.0223887
$750$	0	0
$751$	−16.6575	−0.607841	−0.303920	−	0.952697i	$-0.598296\pi$
$751$	−16.6575	−0.607841	−0.303920	+	0.952697i	$0.598296\pi$
$752$	−11.7050	−0.426836
$753$	15.5613	0.567085
$754$	8.97520	0.326857
$755$	0	0
$756$	1.06530	0.0387447
$757$	−9.44154	−0.343159	−0.171579	−	0.985170i	$-0.554887\pi$
$757$	−9.44154	−0.343159	−0.171579	+	0.985170i	$0.554887\pi$
$758$	−7.29162	−0.264843
$759$	4.43475	0.160971
$760$	0	0
$761$	−45.6071	−1.65326	−0.826628	−	0.562749i	$-0.809744\pi$
$761$	−45.6071	−1.65326	−0.826628	+	0.562749i	$0.809744\pi$
$762$	32.1015	1.16292
$763$	−0.149801	−0.00542315
$764$	−15.0568	−0.544735
$765$	0	0
$766$	49.5040	1.78865
$767$	12.4693	0.450242
$768$	−19.6013	−0.707301
$769$	−2.06647	−0.0745187	−0.0372593	−	0.999306i	$-0.511863\pi$
$769$	−2.06647	−0.0745187	−0.0372593	+	0.999306i	$0.511863\pi$
$770$	0	0
$771$	−9.17245	−0.330338
$772$	−7.76768	−0.279565
$773$	−35.2351	−1.26732	−0.633659	−	0.773613i	$-0.718448\pi$
$773$	−35.2351	−1.26732	−0.633659	+	0.773613i	$0.718448\pi$
$774$	−14.9499	−0.537363
$775$	0	0
$776$	11.7737	0.422651
$777$	2.41256	0.0865501
$778$	28.7516	1.03079
$779$	6.00204	0.215045
$780$	0	0
$781$	14.9548	0.535126
$782$	−7.18012	−0.256761
$783$	−5.12634	−0.183200
$784$	−4.99574	−0.178419
$785$	0	0
$786$	−13.0069	−0.463940
$787$	−9.07749	−0.323578	−0.161789	−	0.986825i	$-0.551726\pi$
$787$	−9.07749	−0.323578	−0.161789	+	0.986825i	$0.551726\pi$
$788$	−7.60926	−0.271069
$789$	6.18256	0.220105
$790$	0	0
$791$	−4.13990	−0.147198
$792$	−3.90669	−0.138818
$793$	−0.0447645	−0.00158963
$794$	44.8201	1.59060
$795$	0	0
$796$	−19.9637	−0.707595
$797$	−28.8249	−1.02103	−0.510516	−	0.859868i	$-0.670545\pi$
$797$	−28.8249	−1.02103	−0.510516	+	0.859868i	$0.670545\pi$
$798$	7.86833	0.278536
$799$	5.17248	0.182989
$800$	0	0
$801$	6.73298	0.237898
$802$	63.8012	2.25290
$803$	33.7812	1.19211
$804$	−9.91172	−0.349560
$805$	0	0
$806$	−8.11753	−0.285928
$807$	−25.9036	−0.911850
$808$	−7.00475	−0.246426
$809$	−5.20958	−0.183159	−0.0915796	−	0.995798i	$-0.529192\pi$
$809$	−5.20958	−0.183159	−0.0915796	+	0.995798i	$0.529192\pi$
$810$	0	0
$811$	−51.0035	−1.79097	−0.895487	−	0.445087i	$-0.853173\pi$
$811$	−51.0035	−1.79097	−0.895487	+	0.445087i	$0.853173\pi$
$812$	−5.46110	−0.191647
$813$	−0.515160	−0.0180674
$814$	10.0836	0.353430
$815$	0	0
$816$	11.0288	0.386084
$817$	−38.3750	−1.34257
$818$	35.5751	1.24386
$819$	1.00000	0.0349428
$820$	0	0
$821$	−29.2368	−1.02037	−0.510186	−	0.860064i	$-0.670423\pi$
$821$	−29.2368	−1.02037	−0.510186	+	0.860064i	$0.670423\pi$
$822$	−16.9350	−0.590675
$823$	−41.7366	−1.45485	−0.727424	−	0.686188i	$-0.759283\pi$
$823$	−41.7366	−1.45485	−0.727424	+	0.686188i	$0.759283\pi$
$824$	−2.95679	−0.103005
$825$	0	0
$826$	−21.8313	−0.759609
$827$	−42.0389	−1.46183	−0.730917	−	0.682466i	$-0.760907\pi$
$827$	−42.0389	−1.46183	−0.730917	+	0.682466i	$0.760907\pi$
$828$	1.97897	0.0687741
$829$	−44.0775	−1.53087	−0.765437	−	0.643510i	$-0.777477\pi$
$829$	−44.0775	−1.53087	−0.765437	+	0.643510i	$0.777477\pi$
$830$	0	0
$831$	1.73801	0.0602908
$832$	−0.408295	−0.0141551
$833$	2.20764	0.0764901
$834$	−21.3657	−0.739833
$835$	0	0
$836$	11.4293	0.395291
$837$	4.63647	0.160260
$838$	44.8377	1.54889
$839$	−17.4451	−0.602272	−0.301136	−	0.953581i	$-0.597366\pi$
$839$	−17.4451	−0.602272	−0.301136	+	0.953581i	$0.597366\pi$
$840$	0	0
$841$	−2.72064	−0.0938152
$842$	29.2844	1.00921
$843$	−20.0227	−0.689618
$844$	9.30326	0.320231
$845$	0	0
$846$	−4.10211	−0.141034
$847$	5.30094	0.182143
$848$	−19.5168	−0.670210
$849$	19.3952	0.665642
$850$	0	0
$851$	4.48173	0.153632
$852$	6.67349	0.228630
$853$	−18.8129	−0.644142	−0.322071	−	0.946716i	$-0.604379\pi$
$853$	−18.8129	−0.644142	−0.322071	+	0.946716i	$0.604379\pi$
$854$	0.0783737	0.00268189
$855$	0	0
$856$	1.00271	0.0342721
$857$	−4.09894	−0.140017	−0.0700086	−	0.997546i	$-0.522303\pi$
$857$	−4.09894	−0.140017	−0.0700086	+	0.997546i	$0.522303\pi$
$858$	4.17963	0.142690
$859$	3.49629	0.119292	0.0596460	−	0.998220i	$-0.481003\pi$
$859$	3.49629	0.119292	0.0596460	+	0.998220i	$0.481003\pi$
$860$	0	0
$861$	−1.33553	−0.0455146
$862$	54.9822	1.87270
$863$	−6.51196	−0.221670	−0.110835	−	0.993839i	$-0.535352\pi$
$863$	−6.51196	−0.221670	−0.110835	+	0.993839i	$0.535352\pi$
$864$	−5.47360	−0.186216
$865$	0	0
$866$	30.9363	1.05126
$867$	12.1263	0.411832
$868$	4.93924	0.167649
$869$	−26.2574	−0.890722
$870$	0	0
$871$	−9.30414	−0.315259
$872$	0.245144	0.00830163
$873$	7.19457	0.243499
$874$	14.6167	0.494418
$875$	0	0
$876$	15.0746	0.509325
$877$	−4.91202	−0.165867	−0.0829336	−	0.996555i	$-0.526429\pi$
$877$	−4.91202	−0.165867	−0.0829336	+	0.996555i	$0.526429\pi$
$878$	−38.4934	−1.29909
$879$	11.5360	0.389099
$880$	0	0
$881$	−22.6450	−0.762930	−0.381465	−	0.924383i	$-0.624580\pi$
$881$	−22.6450	−0.762930	−0.381465	+	0.924383i	$0.624580\pi$
$882$	−1.75080	−0.0589525
$883$	9.47969	0.319017	0.159508	−	0.987197i	$-0.449009\pi$
$883$	9.47969	0.319017	0.159508	+	0.987197i	$0.449009\pi$
$884$	−2.35180	−0.0790996
$885$	0	0
$886$	12.4902	0.419618
$887$	−51.5943	−1.73237	−0.866184	−	0.499725i	$-0.833434\pi$
$887$	−51.5943	−1.73237	−0.866184	+	0.499725i	$0.833434\pi$
$888$	−3.94808	−0.132489
$889$	−18.3353	−0.614948
$890$	0	0
$891$	−2.38727	−0.0799765
$892$	17.8679	0.598261
$893$	−10.5297	−0.352364
$894$	25.6138	0.856654
$895$	0	0
$896$	11.6620	0.389601
$897$	1.85767	0.0620256
$898$	8.44697	0.281879
$899$	−23.7681	−0.792711
$900$	0	0
$901$	8.62456	0.287326
$902$	−5.58201	−0.185861
$903$	8.53890	0.284157
$904$	6.77482	0.225327
$905$	0	0
$906$	5.40083	0.179431
$907$	8.28233	0.275010	0.137505	−	0.990501i	$-0.456092\pi$
$907$	8.28233	0.275010	0.137505	+	0.990501i	$0.456092\pi$
$908$	−5.73907	−0.190458
$909$	−4.28040	−0.141972
$910$	0	0
$911$	20.3967	0.675772	0.337886	−	0.941187i	$-0.390288\pi$
$911$	20.3967	0.675772	0.337886	+	0.941187i	$0.390288\pi$
$912$	−22.4515	−0.743444
$913$	22.3831	0.740773
$914$	−42.6441	−1.41054
$915$	0	0
$916$	30.0477	0.992804
$917$	7.42911	0.245331
$918$	3.86513	0.127568
$919$	−18.1273	−0.597965	−0.298982	−	0.954259i	$-0.596647\pi$
$919$	−18.1273	−0.597965	−0.298982	+	0.954259i	$0.596647\pi$
$920$	0	0
$921$	8.06287	0.265680
$922$	−26.0890	−0.859194
$923$	6.26441	0.206196
$924$	−2.54316	−0.0836639
$925$	0	0
$926$	58.3871	1.91872
$927$	−1.80681	−0.0593434
$928$	28.0595	0.921099
$929$	28.8397	0.946199	0.473099	−	0.881009i	$-0.343135\pi$
$929$	28.8397	0.946199	0.473099	+	0.881009i	$0.343135\pi$
$930$	0	0
$931$	−4.49413	−0.147289
$932$	−18.8458	−0.617313
$933$	7.16559	0.234591
$934$	−3.45647	−0.113099
$935$	0	0
$936$	−1.63647	−0.0534897
$937$	−49.8015	−1.62694	−0.813471	−	0.581605i	$-0.802425\pi$
$937$	−49.8015	−1.62694	−0.813471	+	0.581605i	$0.802425\pi$
$938$	16.2897	0.531878
$939$	−12.6139	−0.411639
$940$	0	0
$941$	20.2922	0.661507	0.330753	−	0.943717i	$-0.392697\pi$
$941$	20.2922	0.661507	0.330753	+	0.943717i	$0.392697\pi$
$942$	39.6669	1.29242
$943$	−2.48096	−0.0807912
$944$	62.2935	2.02748
$945$	0	0
$946$	35.6895	1.16036
$947$	49.8710	1.62059	0.810295	−	0.586022i	$-0.199307\pi$
$947$	49.8710	1.62059	0.810295	+	0.586022i	$0.199307\pi$
$948$	−11.7172	−0.380557
$949$	14.1506	0.459347
$950$	0	0
$951$	−0.985378	−0.0319531
$952$	−3.61273	−0.117089
$953$	42.0866	1.36332	0.681660	−	0.731669i	$-0.261258\pi$
$953$	42.0866	1.36332	0.681660	+	0.731669i	$0.261258\pi$
$954$	−6.83984	−0.221448
$955$	0	0
$956$	1.99859	0.0646390
$957$	12.2380	0.395597
$958$	−27.8460	−0.899665
$959$	9.67271	0.312348
$960$	0	0
$961$	−9.50315	−0.306553
$962$	4.22391	0.136184
$963$	0.612730	0.0197450
$964$	−2.02637	−0.0652649
$965$	0	0
$966$	−3.25240	−0.104644
$967$	−9.99342	−0.321367	−0.160683	−	0.987006i	$-0.551370\pi$
$967$	−9.99342	−0.321367	−0.160683	+	0.987006i	$0.551370\pi$
$968$	−8.67483	−0.278820
$969$	9.92142	0.318722
$970$	0	0
$971$	33.3447	1.07008	0.535042	−	0.844826i	$-0.320296\pi$
$971$	33.3447	1.07008	0.535042	+	0.844826i	$0.320296\pi$
$972$	−1.06530	−0.0341696
$973$	12.2034	0.391222
$974$	2.33147	0.0747052
$975$	0	0
$976$	−0.223632	−0.00715827
$977$	−60.3205	−1.92982	−0.964912	−	0.262573i	$-0.915429\pi$
$977$	−60.3205	−1.92982	−0.964912	+	0.262573i	$0.915429\pi$
$978$	10.5290	0.336681
$979$	−16.0734	−0.513709
$980$	0	0
$981$	0.149801	0.00478277
$982$	−44.8516	−1.43127
$983$	−12.8940	−0.411255	−0.205628	−	0.978630i	$-0.565924\pi$
$983$	−12.8940	−0.411255	−0.205628	+	0.978630i	$0.565924\pi$
$984$	2.18555	0.0696728
$985$	0	0
$986$	−19.8140	−0.631006
$987$	2.34299	0.0745783
$988$	4.78761	0.152314
$989$	15.8624	0.504396
$990$	0	0
$991$	−17.8205	−0.566088	−0.283044	−	0.959107i	$-0.591344\pi$
$991$	−17.8205	−0.566088	−0.283044	+	0.959107i	$0.591344\pi$
$992$	−25.3782	−0.805758
$993$	8.24676	0.261703
$994$	−10.9677	−0.347875
$995$	0	0
$996$	9.98830	0.316491
$997$	45.6881	1.44696	0.723478	−	0.690347i	$-0.242542\pi$
$997$	45.6881	1.44696	0.723478	+	0.690347i	$0.242542\pi$
$998$	37.3146	1.18117
$999$	−2.41256	−0.0763300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bj.1.1	yes	4
5.4	even	2		6825.2.a.bi.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6825.2.a.bi.1.4	✓	4	5.4	even	2
6825.2.a.bj.1.1	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75080 q^{2} -1.00000 q^{3} +1.06530 q^{4} +1.75080 q^{6} -1.00000 q^{7} +1.63647 q^{8} +1.00000 q^{9} -2.38727 q^{11} -1.06530 q^{12} -1.00000 q^{13} +1.75080 q^{14} -4.99574 q^{16} +2.20764 q^{17} -1.75080 q^{18} -4.49413 q^{19} +1.00000 q^{21} +4.17963 q^{22} +1.85767 q^{23} -1.63647 q^{24} +1.75080 q^{26} -1.00000 q^{27} -1.06530 q^{28} +5.12634 q^{29} -4.63647 q^{31} +5.47360 q^{32} +2.38727 q^{33} -3.86513 q^{34} +1.06530 q^{36} +2.41256 q^{37} +7.86833 q^{38} +1.00000 q^{39} -1.33553 q^{41} -1.75080 q^{42} +8.53890 q^{43} -2.54316 q^{44} -3.25240 q^{46} +2.34299 q^{47} +4.99574 q^{48} +1.00000 q^{49} -2.20764 q^{51} -1.06530 q^{52} +3.90669 q^{53} +1.75080 q^{54} -1.63647 q^{56} +4.49413 q^{57} -8.97520 q^{58} -12.4693 q^{59} +0.0447645 q^{61} +8.11753 q^{62} -1.00000 q^{63} +0.408295 q^{64} -4.17963 q^{66} +9.30414 q^{67} +2.35180 q^{68} -1.85767 q^{69} -6.26441 q^{71} +1.63647 q^{72} -14.1506 q^{73} -4.22391 q^{74} -4.78761 q^{76} +2.38727 q^{77} -1.75080 q^{78} +10.9989 q^{79} +1.00000 q^{81} +2.33824 q^{82} -9.37603 q^{83} +1.06530 q^{84} -14.9499 q^{86} -5.12634 q^{87} -3.90669 q^{88} +6.73298 q^{89} +1.00000 q^{91} +1.97897 q^{92} +4.63647 q^{93} -4.10211 q^{94} -5.47360 q^{96} +7.19457 q^{97} -1.75080 q^{98} -2.38727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{13} - q^{14} - q^{16} + 5 q^{17} + q^{18} - 15 q^{19} + 4 q^{21} + 9 q^{22} + 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q + q^2 - 4 * q^3 + q^4 - q^6 - 4 * q^7 + 3 * q^8 + 4 * q^9 + 2 * q^11 - q^12 - 4 * q^13 - q^14 - q^16 + 5 * q^17 + q^18 - 15 * q^19 + 4 * q^21 + 9 * q^22 + 8 * q^23 - 3 * q^24 - q^26 - 4 * q^27 - q^28 - 5 * q^29 - 15 * q^31 + 2 * q^32 - 2 * q^33 - 3 * q^34 + q^36 + 12 * q^37 - 13 * q^38 + 4 * q^39 - 5 * q^41 + q^42 + 11 * q^43 - 6 * q^44 + 11 * q^46 - 4 * q^47 + q^48 + 4 * q^49 - 5 * q^51 - q^52 + 15 * q^53 - q^54 - 3 * q^56 + 15 * q^57 - 11 * q^59 - 20 * q^61 - 4 * q^62 - 4 * q^63 - 15 * q^64 - 9 * q^66 + 2 * q^67 + 7 * q^68 - 8 * q^69 + 20 * q^71 + 3 * q^72 + 6 * q^73 - 11 * q^74 - 26 * q^76 - 2 * q^77 + q^78 - 7 * q^79 + 4 * q^81 - q^82 + 2 * q^83 + q^84 - 10 * q^86 + 5 * q^87 - 15 * q^88 - 8 * q^89 + 4 * q^91 + 13 * q^92 + 15 * q^93 - 28 * q^94 - 2 * q^96 - 5 * q^97 + q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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