LMFDB - Embedded newform 6825.2.a.bm.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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1365)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.17557$ 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)

$f(q)$	$=$	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.35114 q^{11} +1.61803 q^{12} -1.00000 q^{13} +0.618034 q^{14} +1.85410 q^{16} -2.63522 q^{17} -0.618034 q^{18} +3.69572 q^{19} +1.00000 q^{21} +2.07112 q^{22} +6.13818 q^{23} -2.23607 q^{24} +0.618034 q^{26} -1.00000 q^{27} +1.61803 q^{28} -1.10851 q^{29} -9.67551 q^{31} -5.61803 q^{32} +3.35114 q^{33} +1.62866 q^{34} -1.61803 q^{36} +0.206396 q^{37} -2.28408 q^{38} +1.00000 q^{39} +0.726543 q^{41} -0.618034 q^{42} +0.360723 q^{43} +5.42226 q^{44} -3.79360 q^{46} -2.31375 q^{47} -1.85410 q^{48} +1.00000 q^{49} +2.63522 q^{51} +1.61803 q^{52} -0.874305 q^{53} +0.618034 q^{54} -2.23607 q^{56} -3.69572 q^{57} +0.685096 q^{58} +2.29064 q^{59} -2.03739 q^{61} +5.97980 q^{62} -1.00000 q^{63} -0.236068 q^{64} -2.07112 q^{66} -0.691660 q^{67} +4.26388 q^{68} -6.13818 q^{69} -3.71998 q^{71} +2.23607 q^{72} +2.60085 q^{73} -0.127559 q^{74} -5.97980 q^{76} +3.35114 q^{77} -0.618034 q^{78} -11.5373 q^{79} +1.00000 q^{81} -0.449028 q^{82} +5.80236 q^{83} -1.61803 q^{84} -0.222939 q^{86} +1.10851 q^{87} -7.49338 q^{88} +2.48642 q^{89} +1.00000 q^{91} -9.93179 q^{92} +9.67551 q^{93} +1.42998 q^{94} +5.61803 q^{96} -0.864724 q^{97} -0.618034 q^{98} -3.35114 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} - 2 q^{4} - 2 q^{6} - 4 q^{7} + 4 q^{9} - 4 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} - 6 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{21} - 2 q^{22} + 8 q^{23} - 2 q^{26} - 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 - 2 * q^4 - 2 * q^6 - 4 * q^7 + 4 * q^9 - 4 * q^11 + 2 * q^12 - 4 * q^13 - 2 * q^14 - 6 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^19 + 4 * q^21 - 2 * q^22 + 8 * q^23 - 2 * q^26 - 4 * q^27 + 2 * q^28 - 6 * q^29 - 2 * q^31 - 18 * q^32 + 4 * q^33 - 4 * q^34 - 2 * q^36 + 10 * q^37 - 6 * q^38 + 4 * q^39 + 2 * q^42 + 18 * q^43 + 2 * q^44 - 6 * q^46 + 12 * q^47 + 6 * q^48 + 4 * q^49 - 2 * q^51 + 2 * q^52 - 10 * q^53 - 2 * q^54 + 2 * q^57 - 8 * q^58 + 4 * q^59 - 20 * q^61 + 4 * q^62 - 4 * q^63 + 8 * q^64 + 2 * q^66 + 10 * q^67 - 6 * q^68 - 8 * q^69 - 14 * q^71 + 14 * q^73 + 10 * q^74 - 4 * q^76 + 4 * q^77 + 2 * q^78 - 26 * q^79 + 4 * q^81 + 12 * q^83 - 2 * q^84 + 4 * q^86 + 6 * q^87 + 20 * q^89 + 4 * q^91 - 14 * q^92 + 2 * q^93 + 16 * q^94 + 18 * q^96 + 16 * q^97 + 2 * q^98 - 4 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.35114	−1.01041	−0.505204	−	0.863000i	$-0.668583\pi$
$11$	−3.35114	−1.01041	−0.505204	+	0.863000i	$0.668583\pi$
$12$	1.61803	0.467086
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−2.63522	−0.639135	−0.319567	−	0.947564i	$-0.603538\pi$
$17$	−2.63522	−0.639135	−0.319567	+	0.947564i	$0.603538\pi$
$18$	−0.618034	−0.145672
$19$	3.69572	0.847856	0.423928	−	0.905696i	$-0.360651\pi$
$19$	3.69572	0.847856	0.423928	+	0.905696i	$0.360651\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	2.07112	0.441564
$23$	6.13818	1.27990	0.639950	−	0.768417i	$-0.278955\pi$
$23$	6.13818	1.27990	0.639950	+	0.768417i	$0.278955\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	0.618034	0.121206
$27$	−1.00000	−0.192450
$28$	1.61803	0.305780
$29$	−1.10851	−0.205845	−0.102922	−	0.994689i	$-0.532819\pi$
$29$	−1.10851	−0.205845	−0.102922	+	0.994689i	$0.532819\pi$
$30$	0	0
$31$	−9.67551	−1.73777	−0.868887	−	0.495011i	$-0.835164\pi$
$31$	−9.67551	−1.73777	−0.868887	+	0.495011i	$0.835164\pi$
$32$	−5.61803	−0.993137
$33$	3.35114	0.583359
$34$	1.62866	0.279312
$35$	0	0
$36$	−1.61803	−0.269672
$37$	0.206396	0.0339312	0.0169656	−	0.999856i	$-0.494599\pi$
$37$	0.206396	0.0339312	0.0169656	+	0.999856i	$0.494599\pi$
$38$	−2.28408	−0.370527
$39$	1.00000	0.160128
$40$	0	0
$41$	0.726543	0.113467	0.0567334	−	0.998389i	$-0.481931\pi$
$41$	0.726543	0.113467	0.0567334	+	0.998389i	$0.481931\pi$
$42$	−0.618034	−0.0953647
$43$	0.360723	0.0550097	0.0275049	−	0.999622i	$-0.491244\pi$
$43$	0.360723	0.0550097	0.0275049	+	0.999622i	$0.491244\pi$
$44$	5.42226	0.817436
$45$	0	0
$46$	−3.79360	−0.559336
$47$	−2.31375	−0.337495	−0.168748	−	0.985659i	$-0.553972\pi$
$47$	−2.31375	−0.337495	−0.168748	+	0.985659i	$0.553972\pi$
$48$	−1.85410	−0.267617
$49$	1.00000	0.142857
$50$	0	0
$51$	2.63522	0.369005
$52$	1.61803	0.224381
$53$	−0.874305	−0.120095	−0.0600475	−	0.998196i	$-0.519125\pi$
$53$	−0.874305	−0.120095	−0.0600475	+	0.998196i	$0.519125\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	−2.23607	−0.298807
$57$	−3.69572	−0.489510
$58$	0.685096	0.0899575
$59$	2.29064	0.298216	0.149108	−	0.988821i	$-0.452360\pi$
$59$	2.29064	0.298216	0.149108	+	0.988821i	$0.452360\pi$
$60$	0	0
$61$	−2.03739	−0.260861	−0.130430	−	0.991457i	$-0.541636\pi$
$61$	−2.03739	−0.260861	−0.130430	+	0.991457i	$0.541636\pi$
$62$	5.97980	0.759435
$63$	−1.00000	−0.125988
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−2.07112	−0.254937
$67$	−0.691660	−0.0844998	−0.0422499	−	0.999107i	$-0.513453\pi$
$67$	−0.691660	−0.0844998	−0.0422499	+	0.999107i	$0.513453\pi$
$68$	4.26388	0.517071
$69$	−6.13818	−0.738950
$70$	0	0
$71$	−3.71998	−0.441480	−0.220740	−	0.975333i	$-0.570847\pi$
$71$	−3.71998	−0.441480	−0.220740	+	0.975333i	$0.570847\pi$
$72$	2.23607	0.263523
$73$	2.60085	0.304406	0.152203	−	0.988349i	$-0.451363\pi$
$73$	2.60085	0.304406	0.152203	+	0.988349i	$0.451363\pi$
$74$	−0.127559	−0.0148285
$75$	0	0
$76$	−5.97980	−0.685930
$77$	3.35114	0.381898
$78$	−0.618034	−0.0699786
$79$	−11.5373	−1.29805	−0.649026	−	0.760766i	$-0.724823\pi$
$79$	−11.5373	−1.29805	−0.649026	+	0.760766i	$0.724823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.449028	−0.0495868
$83$	5.80236	0.636892	0.318446	−	0.947941i	$-0.396839\pi$
$83$	5.80236	0.636892	0.318446	+	0.947941i	$0.396839\pi$
$84$	−1.61803	−0.176542
$85$	0	0
$86$	−0.222939	−0.0240401
$87$	1.10851	0.118845
$88$	−7.49338	−0.798797
$89$	2.48642	0.263560	0.131780	−	0.991279i	$-0.457931\pi$
$89$	2.48642	0.263560	0.131780	+	0.991279i	$0.457931\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−9.93179	−1.03546
$93$	9.67551	1.00330
$94$	1.42998	0.147491
$95$	0	0
$96$	5.61803	0.573388
$97$	−0.864724	−0.0877994	−0.0438997	−	0.999036i	$-0.513978\pi$
$97$	−0.864724	−0.0877994	−0.0438997	+	0.999036i	$0.513978\pi$
$98$	−0.618034	−0.0624309
$99$	−3.35114	−0.336802
$100$	0	0
$101$	−10.3434	−1.02921	−0.514605	−	0.857428i	$-0.672061\pi$
$101$	−10.3434	−1.02921	−0.514605	+	0.857428i	$0.672061\pi$
$102$	−1.62866	−0.161261
$103$	15.3552	1.51299	0.756496	−	0.653998i	$-0.226910\pi$
$103$	15.3552	1.51299	0.756496	+	0.653998i	$0.226910\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	0.540350	0.0524835
$107$	−16.6487	−1.60950	−0.804748	−	0.593617i	$-0.797699\pi$
$107$	−16.6487	−1.60950	−0.804748	+	0.593617i	$0.797699\pi$
$108$	1.61803	0.155695
$109$	−18.9615	−1.81618	−0.908089	−	0.418777i	$-0.862459\pi$
$109$	−18.9615	−1.81618	−0.908089	+	0.418777i	$0.862459\pi$
$110$	0	0
$111$	−0.206396	−0.0195902
$112$	−1.85410	−0.175196
$113$	−6.85108	−0.644496	−0.322248	−	0.946655i	$-0.604438\pi$
$113$	−6.85108	−0.644496	−0.322248	+	0.946655i	$0.604438\pi$
$114$	2.28408	0.213924
$115$	0	0
$116$	1.79360	0.166532
$117$	−1.00000	−0.0924500
$118$	−1.41570	−0.130325
$119$	2.63522	0.241570
$120$	0	0
$121$	0.230146	0.0209224
$122$	1.25918	0.114000
$123$	−0.726543	−0.0655101
$124$	15.6553	1.40589
$125$	0	0
$126$	0.618034	0.0550588
$127$	0.211095	0.0187317	0.00936583	−	0.999956i	$-0.497019\pi$
$127$	0.211095	0.0187317	0.00936583	+	0.999956i	$0.497019\pi$
$128$	11.3820	1.00603
$129$	−0.360723	−0.0317599
$130$	0	0
$131$	6.50587	0.568420	0.284210	−	0.958762i	$-0.408269\pi$
$131$	6.50587	0.568420	0.284210	+	0.958762i	$0.408269\pi$
$132$	−5.42226	−0.471947
$133$	−3.69572	−0.320459
$134$	0.427470	0.0369278
$135$	0	0
$136$	−5.89253	−0.505280
$137$	−13.4934	−1.15282	−0.576409	−	0.817162i	$-0.695546\pi$
$137$	−13.4934	−1.15282	−0.576409	+	0.817162i	$0.695546\pi$
$138$	3.79360	0.322933
$139$	−5.10266	−0.432802	−0.216401	−	0.976305i	$-0.569432\pi$
$139$	−5.10266	−0.432802	−0.216401	+	0.976305i	$0.569432\pi$
$140$	0	0
$141$	2.31375	0.194853
$142$	2.29907	0.192934
$143$	3.35114	0.280236
$144$	1.85410	0.154508
$145$	0	0
$146$	−1.60741	−0.133030
$147$	−1.00000	−0.0824786
$148$	−0.333955	−0.0274509
$149$	15.2976	1.25323	0.626614	−	0.779330i	$-0.284440\pi$
$149$	15.2976	1.25323	0.626614	+	0.779330i	$0.284440\pi$
$150$	0	0
$151$	−2.18515	−0.177825	−0.0889126	−	0.996039i	$-0.528339\pi$
$151$	−2.18515	−0.177825	−0.0889126	+	0.996039i	$0.528339\pi$
$152$	8.26388	0.670289
$153$	−2.63522	−0.213045
$154$	−2.07112	−0.166896
$155$	0	0
$156$	−1.61803	−0.129546
$157$	21.6267	1.72600	0.862998	−	0.505207i	$-0.168584\pi$
$157$	21.6267	1.72600	0.862998	+	0.505207i	$0.168584\pi$
$158$	7.13046	0.567269
$159$	0.874305	0.0693369
$160$	0	0
$161$	−6.13818	−0.483756
$162$	−0.618034	−0.0485573
$163$	20.1440	1.57780	0.788900	−	0.614522i	$-0.210651\pi$
$163$	20.1440	1.57780	0.788900	+	0.614522i	$0.210651\pi$
$164$	−1.17557	−0.0917966
$165$	0	0
$166$	−3.58606	−0.278332
$167$	11.1532	0.863059	0.431529	−	0.902099i	$-0.357974\pi$
$167$	11.1532	0.863059	0.431529	+	0.902099i	$0.357974\pi$
$168$	2.23607	0.172516
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.69572	0.282619
$172$	−0.583662	−0.0445038
$173$	−15.7763	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$173$	−15.7763	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$174$	−0.685096	−0.0519370
$175$	0	0
$176$	−6.21336	−0.468349
$177$	−2.29064	−0.172175
$178$	−1.53669	−0.115180
$179$	−21.9229	−1.63860	−0.819298	−	0.573368i	$-0.805636\pi$
$179$	−21.9229	−1.63860	−0.819298	+	0.573368i	$0.805636\pi$
$180$	0	0
$181$	−10.9929	−0.817098	−0.408549	−	0.912736i	$-0.633965\pi$
$181$	−10.9929	−0.817098	−0.408549	+	0.912736i	$0.633965\pi$
$182$	−0.618034	−0.0458117
$183$	2.03739	0.150608
$184$	13.7254	1.01185
$185$	0	0
$186$	−5.97980	−0.438460
$187$	8.83099	0.645786
$188$	3.74373	0.273039
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.6382	0.842113	0.421057	−	0.907034i	$-0.361659\pi$
$191$	11.6382	0.842113	0.421057	+	0.907034i	$0.361659\pi$
$192$	0.236068	0.0170367
$193$	−9.78101	−0.704052	−0.352026	−	0.935990i	$-0.614507\pi$
$193$	−9.78101	−0.704052	−0.352026	+	0.935990i	$0.614507\pi$
$194$	0.534429	0.0383697
$195$	0	0
$196$	−1.61803	−0.115574
$197$	15.3606	1.09440	0.547199	−	0.837002i	$-0.315694\pi$
$197$	15.3606	1.09440	0.547199	+	0.837002i	$0.315694\pi$
$198$	2.07112	0.147188
$199$	−17.7788	−1.26031	−0.630153	−	0.776471i	$-0.717008\pi$
$199$	−17.7788	−1.26031	−0.630153	+	0.776471i	$0.717008\pi$
$200$	0	0
$201$	0.691660	0.0487860
$202$	6.39259	0.449781
$203$	1.10851	0.0778020
$204$	−4.26388	−0.298531
$205$	0	0
$206$	−9.49003	−0.661202
$207$	6.13818	0.426633
$208$	−1.85410	−0.128559
$209$	−12.3849	−0.856679
$210$	0	0
$211$	19.3414	1.33152	0.665760	−	0.746166i	$-0.268107\pi$
$211$	19.3414	1.33152	0.665760	+	0.746166i	$0.268107\pi$
$212$	1.41466	0.0971590
$213$	3.71998	0.254889
$214$	10.2895	0.703375
$215$	0	0
$216$	−2.23607	−0.152145
$217$	9.67551	0.656817
$218$	11.7188	0.793699
$219$	−2.60085	−0.175749
$220$	0	0
$221$	2.63522	0.177264
$222$	0.127559	0.00856123
$223$	19.6755	1.31757	0.658785	−	0.752331i	$-0.271071\pi$
$223$	19.6755	1.31757	0.658785	+	0.752331i	$0.271071\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	4.23420	0.281655
$227$	13.4981	0.895904	0.447952	−	0.894058i	$-0.352154\pi$
$227$	13.4981	0.895904	0.447952	+	0.894058i	$0.352154\pi$
$228$	5.97980	0.396022
$229$	−13.9138	−0.919448	−0.459724	−	0.888062i	$-0.652052\pi$
$229$	−13.9138	−0.919448	−0.459724	+	0.888062i	$0.652052\pi$
$230$	0	0
$231$	−3.35114	−0.220489
$232$	−2.47870	−0.162735
$233$	21.7103	1.42229	0.711144	−	0.703046i	$-0.248177\pi$
$233$	21.7103	1.42229	0.711144	+	0.703046i	$0.248177\pi$
$234$	0.618034	0.0404021
$235$	0	0
$236$	−3.70634	−0.241262
$237$	11.5373	0.749430
$238$	−1.62866	−0.105570
$239$	−2.40090	−0.155302	−0.0776508	−	0.996981i	$-0.524742\pi$
$239$	−2.40090	−0.155302	−0.0776508	+	0.996981i	$0.524742\pi$
$240$	0	0
$241$	15.5953	1.00458	0.502292	−	0.864698i	$-0.332490\pi$
$241$	15.5953	1.00458	0.502292	+	0.864698i	$0.332490\pi$
$242$	−0.142238	−0.00914341
$243$	−1.00000	−0.0641500
$244$	3.29657	0.211041
$245$	0	0
$246$	0.449028	0.0286290
$247$	−3.69572	−0.235153
$248$	−21.6351	−1.37383
$249$	−5.80236	−0.367710
$250$	0	0
$251$	−28.5238	−1.80040	−0.900202	−	0.435473i	$-0.856581\pi$
$251$	−28.5238	−1.80040	−0.900202	+	0.435473i	$0.856581\pi$
$252$	1.61803	0.101927
$253$	−20.5699	−1.29322
$254$	−0.130464	−0.00818604
$255$	0	0
$256$	−6.56231	−0.410144
$257$	8.56108	0.534026	0.267013	−	0.963693i	$-0.413963\pi$
$257$	8.56108	0.534026	0.267013	+	0.963693i	$0.413963\pi$
$258$	0.222939	0.0138796
$259$	−0.206396	−0.0128248
$260$	0	0
$261$	−1.10851	−0.0686150
$262$	−4.02085	−0.248409
$263$	13.9681	0.861312	0.430656	−	0.902516i	$-0.358282\pi$
$263$	13.9681	0.861312	0.430656	+	0.902516i	$0.358282\pi$
$264$	7.49338	0.461186
$265$	0	0
$266$	2.28408	0.140046
$267$	−2.48642	−0.152166
$268$	1.11913	0.0683618
$269$	7.68760	0.468721	0.234361	−	0.972150i	$-0.424700\pi$
$269$	7.68760	0.468721	0.234361	+	0.972150i	$0.424700\pi$
$270$	0	0
$271$	−24.0948	−1.46365	−0.731826	−	0.681491i	$-0.761332\pi$
$271$	−24.0948	−1.46365	−0.731826	+	0.681491i	$0.761332\pi$
$272$	−4.88597	−0.296255
$273$	−1.00000	−0.0605228
$274$	8.33937	0.503800
$275$	0	0
$276$	9.93179	0.597823
$277$	11.9524	0.718149	0.359075	−	0.933309i	$-0.383092\pi$
$277$	11.9524	0.718149	0.359075	+	0.933309i	$0.383092\pi$
$278$	3.15362	0.189141
$279$	−9.67551	−0.579258
$280$	0	0
$281$	0.437581	0.0261039	0.0130520	−	0.999915i	$-0.495845\pi$
$281$	0.437581	0.0261039	0.0130520	+	0.999915i	$0.495845\pi$
$282$	−1.42998	−0.0851539
$283$	9.50733	0.565152	0.282576	−	0.959245i	$-0.408811\pi$
$283$	9.50733	0.565152	0.282576	+	0.959245i	$0.408811\pi$
$284$	6.01905	0.357165
$285$	0	0
$286$	−2.07112	−0.122468
$287$	−0.726543	−0.0428864
$288$	−5.61803	−0.331046
$289$	−10.0556	−0.591507
$290$	0	0
$291$	0.864724	0.0506910
$292$	−4.20826	−0.246270
$293$	8.55805	0.499966	0.249983	−	0.968250i	$-0.419575\pi$
$293$	8.55805	0.499966	0.249983	+	0.968250i	$0.419575\pi$
$294$	0.618034	0.0360445
$295$	0	0
$296$	0.461514	0.0268250
$297$	3.35114	0.194453
$298$	−9.45444	−0.547681
$299$	−6.13818	−0.354980
$300$	0	0
$301$	−0.360723	−0.0207917
$302$	1.35050	0.0777125
$303$	10.3434	0.594214
$304$	6.85224	0.393003
$305$	0	0
$306$	1.62866	0.0931040
$307$	−23.3100	−1.33037	−0.665185	−	0.746678i	$-0.731647\pi$
$307$	−23.3100	−1.33037	−0.665185	+	0.746678i	$0.731647\pi$
$308$	−5.42226	−0.308962
$309$	−15.3552	−0.873527
$310$	0	0
$311$	22.1564	1.25637	0.628187	−	0.778062i	$-0.283797\pi$
$311$	22.1564	1.25637	0.628187	+	0.778062i	$0.283797\pi$
$312$	2.23607	0.126592
$313$	−33.9218	−1.91737	−0.958686	−	0.284467i	$-0.908183\pi$
$313$	−33.9218	−1.91737	−0.958686	+	0.284467i	$0.908183\pi$
$314$	−13.3660	−0.754288
$315$	0	0
$316$	18.6678	1.05015
$317$	1.74861	0.0982118	0.0491059	−	0.998794i	$-0.484363\pi$
$317$	1.74861	0.0982118	0.0491059	+	0.998794i	$0.484363\pi$
$318$	−0.540350	−0.0303013
$319$	3.71477	0.207987
$320$	0	0
$321$	16.6487	0.929242
$322$	3.79360	0.211409
$323$	−9.73903	−0.541894
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	−12.4497	−0.689523
$327$	18.9615	1.04857
$328$	1.62460	0.0897034
$329$	2.31375	0.127561
$330$	0	0
$331$	31.0973	1.70926	0.854632	−	0.519234i	$-0.173783\pi$
$331$	31.0973	1.70926	0.854632	+	0.519234i	$0.173783\pi$
$332$	−9.38842	−0.515256
$333$	0.206396	0.0113104
$334$	−6.89304	−0.377170
$335$	0	0
$336$	1.85410	0.101150
$337$	−22.0366	−1.20041	−0.600206	−	0.799845i	$-0.704915\pi$
$337$	−22.0366	−1.20041	−0.600206	+	0.799845i	$0.704915\pi$
$338$	−0.618034	−0.0336166
$339$	6.85108	0.372100
$340$	0	0
$341$	32.4240	1.75586
$342$	−2.28408	−0.123509
$343$	−1.00000	−0.0539949
$344$	0.806601	0.0434890
$345$	0	0
$346$	9.75029	0.524179
$347$	28.9108	1.55201	0.776005	−	0.630726i	$-0.217243\pi$
$347$	28.9108	1.55201	0.776005	+	0.630726i	$0.217243\pi$
$348$	−1.79360	−0.0961473
$349$	−0.741404	−0.0396864	−0.0198432	−	0.999803i	$-0.506317\pi$
$349$	−0.741404	−0.0396864	−0.0198432	+	0.999803i	$0.506317\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	18.8268	1.00347
$353$	3.65259	0.194408	0.0972038	−	0.995264i	$-0.469010\pi$
$353$	3.65259	0.194408	0.0972038	+	0.995264i	$0.469010\pi$
$354$	1.41570	0.0752434
$355$	0	0
$356$	−4.02311	−0.213224
$357$	−2.63522	−0.139471
$358$	13.5491	0.716093
$359$	−18.4661	−0.974604	−0.487302	−	0.873234i	$-0.662019\pi$
$359$	−18.4661	−0.974604	−0.487302	+	0.873234i	$0.662019\pi$
$360$	0	0
$361$	−5.34167	−0.281141
$362$	6.79400	0.357085
$363$	−0.230146	−0.0120795
$364$	−1.61803	−0.0848080
$365$	0	0
$366$	−1.25918	−0.0658182
$367$	27.8760	1.45511	0.727557	−	0.686047i	$-0.240656\pi$
$367$	27.8760	1.45511	0.727557	+	0.686047i	$0.240656\pi$
$368$	11.3808	0.593266
$369$	0.726543	0.0378223
$370$	0	0
$371$	0.874305	0.0453917
$372$	−15.6553	−0.811690
$373$	−9.46018	−0.489830	−0.244915	−	0.969545i	$-0.578760\pi$
$373$	−9.46018	−0.489830	−0.244915	+	0.969545i	$0.578760\pi$
$374$	−5.45785	−0.282219
$375$	0	0
$376$	−5.17371	−0.266813
$377$	1.10851	0.0570911
$378$	−0.618034	−0.0317882
$379$	4.05812	0.208452	0.104226	−	0.994554i	$-0.466764\pi$
$379$	4.05812	0.208452	0.104226	+	0.994554i	$0.466764\pi$
$380$	0	0
$381$	−0.211095	−0.0108147
$382$	−7.19283	−0.368017
$383$	−6.46725	−0.330461	−0.165231	−	0.986255i	$-0.552837\pi$
$383$	−6.46725	−0.330461	−0.165231	+	0.986255i	$0.552837\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	6.04499	0.307682
$387$	0.360723	0.0183366
$388$	1.39915	0.0710312
$389$	−12.7738	−0.647657	−0.323829	−	0.946116i	$-0.604970\pi$
$389$	−12.7738	−0.647657	−0.323829	+	0.946116i	$0.604970\pi$
$390$	0	0
$391$	−16.1755	−0.818028
$392$	2.23607	0.112938
$393$	−6.50587	−0.328178
$394$	−9.49338	−0.478270
$395$	0	0
$396$	5.42226	0.272479
$397$	19.9100	0.999256	0.499628	−	0.866240i	$-0.333470\pi$
$397$	19.9100	0.999256	0.499628	+	0.866240i	$0.333470\pi$
$398$	10.9879	0.550774
$399$	3.69572	0.185017
$400$	0	0
$401$	7.50870	0.374967	0.187483	−	0.982268i	$-0.439967\pi$
$401$	7.50870	0.374967	0.187483	+	0.982268i	$0.439967\pi$
$402$	−0.427470	−0.0213202
$403$	9.67551	0.481972
$404$	16.7360	0.832648
$405$	0	0
$406$	−0.685096	−0.0340007
$407$	−0.691660	−0.0342843
$408$	5.89253	0.291724
$409$	19.4922	0.963829	0.481914	−	0.876218i	$-0.339942\pi$
$409$	19.4922	0.963829	0.481914	+	0.876218i	$0.339942\pi$
$410$	0	0
$411$	13.4934	0.665579
$412$	−24.8452	−1.22404
$413$	−2.29064	−0.112715
$414$	−3.79360	−0.186445
$415$	0	0
$416$	5.61803	0.275447
$417$	5.10266	0.249878
$418$	7.65427	0.374383
$419$	1.47265	0.0719435	0.0359717	−	0.999353i	$-0.488547\pi$
$419$	1.47265	0.0719435	0.0359717	+	0.999353i	$0.488547\pi$
$420$	0	0
$421$	30.6014	1.49142	0.745712	−	0.666269i	$-0.232110\pi$
$421$	30.6014	1.49142	0.745712	+	0.666269i	$0.232110\pi$
$422$	−11.9537	−0.581896
$423$	−2.31375	−0.112498
$424$	−1.95501	−0.0949435
$425$	0	0
$426$	−2.29907	−0.111390
$427$	2.03739	0.0985962
$428$	26.9382	1.30211
$429$	−3.35114	−0.161795
$430$	0	0
$431$	34.8057	1.67653	0.838265	−	0.545264i	$-0.183570\pi$
$431$	34.8057	1.67653	0.838265	+	0.545264i	$0.183570\pi$
$432$	−1.85410	−0.0892055
$433$	39.9493	1.91984	0.959919	−	0.280276i	$-0.0904261\pi$
$433$	39.9493	1.91984	0.959919	+	0.280276i	$0.0904261\pi$
$434$	−5.97980	−0.287039
$435$	0	0
$436$	30.6803	1.46932
$437$	22.6850	1.08517
$438$	1.60741	0.0768051
$439$	29.4245	1.40436	0.702178	−	0.712002i	$-0.252211\pi$
$439$	29.4245	1.40436	0.702178	+	0.712002i	$0.252211\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−1.62866	−0.0774672
$443$	−3.02060	−0.143513	−0.0717565	−	0.997422i	$-0.522860\pi$
$443$	−3.02060	−0.143513	−0.0717565	+	0.997422i	$0.522860\pi$
$444$	0.333955	0.0158488
$445$	0	0
$446$	−12.1601	−0.575799
$447$	−15.2976	−0.723552
$448$	0.236068	0.0111532
$449$	25.7734	1.21632	0.608160	−	0.793814i	$-0.291908\pi$
$449$	25.7734	1.21632	0.608160	+	0.793814i	$0.291908\pi$
$450$	0	0
$451$	−2.43475	−0.114648
$452$	11.0853	0.521408
$453$	2.18515	0.102667
$454$	−8.34231	−0.391524
$455$	0	0
$456$	−8.26388	−0.386991
$457$	0.0785087	0.00367248	0.00183624	−	0.999998i	$-0.499416\pi$
$457$	0.0785087	0.00367248	0.00183624	+	0.999998i	$0.499416\pi$
$458$	8.59919	0.401814
$459$	2.63522	0.123002
$460$	0	0
$461$	22.0395	1.02648	0.513240	−	0.858245i	$-0.328445\pi$
$461$	22.0395	1.02648	0.513240	+	0.858245i	$0.328445\pi$
$462$	2.07112	0.0963572
$463$	12.5200	0.581852	0.290926	−	0.956746i	$-0.406037\pi$
$463$	12.5200	0.581852	0.290926	+	0.956746i	$0.406037\pi$
$464$	−2.05529	−0.0954143
$465$	0	0
$466$	−13.4177	−0.621563
$467$	−30.7347	−1.42223	−0.711117	−	0.703073i	$-0.751810\pi$
$467$	−30.7347	−1.42223	−0.711117	+	0.703073i	$0.751810\pi$
$468$	1.61803	0.0747936
$469$	0.691660	0.0319379
$470$	0	0
$471$	−21.6267	−0.996505
$472$	5.12203	0.235761
$473$	−1.20883	−0.0555822
$474$	−7.13046	−0.327513
$475$	0	0
$476$	−4.26388	−0.195434
$477$	−0.874305	−0.0400317
$478$	1.48384	0.0678693
$479$	32.4830	1.48418	0.742092	−	0.670298i	$-0.233834\pi$
$479$	32.4830	1.48418	0.742092	+	0.670298i	$0.233834\pi$
$480$	0	0
$481$	−0.206396	−0.00941083
$482$	−9.63844	−0.439019
$483$	6.13818	0.279297
$484$	−0.372384	−0.0169266
$485$	0	0
$486$	0.618034	0.0280346
$487$	−1.01783	−0.0461222	−0.0230611	−	0.999734i	$-0.507341\pi$
$487$	−1.01783	−0.0461222	−0.0230611	+	0.999734i	$0.507341\pi$
$488$	−4.55574	−0.206229
$489$	−20.1440	−0.910943
$490$	0	0
$491$	9.36330	0.422560	0.211280	−	0.977426i	$-0.432237\pi$
$491$	9.36330	0.422560	0.211280	+	0.977426i	$0.432237\pi$
$492$	1.17557	0.0529988
$493$	2.92116	0.131563
$494$	2.28408	0.102766
$495$	0	0
$496$	−17.9394	−0.805502
$497$	3.71998	0.166864
$498$	3.58606	0.160695
$499$	−16.5095	−0.739069	−0.369534	−	0.929217i	$-0.620483\pi$
$499$	−16.5095	−0.739069	−0.369534	+	0.929217i	$0.620483\pi$
$500$	0	0
$501$	−11.1532	−0.498287
$502$	17.6287	0.786805
$503$	36.9170	1.64605	0.823024	−	0.568007i	$-0.192285\pi$
$503$	36.9170	1.64605	0.823024	+	0.568007i	$0.192285\pi$
$504$	−2.23607	−0.0996024
$505$	0	0
$506$	12.7129	0.565157
$507$	−1.00000	−0.0444116
$508$	−0.341559	−0.0151542
$509$	20.9424	0.928256	0.464128	−	0.885768i	$-0.346368\pi$
$509$	20.9424	0.928256	0.464128	+	0.885768i	$0.346368\pi$
$510$	0	0
$511$	−2.60085	−0.115055
$512$	−18.7082	−0.826794
$513$	−3.69572	−0.163170
$514$	−5.29104	−0.233378
$515$	0	0
$516$	0.583662	0.0256943
$517$	7.75371	0.341008
$518$	0.127559	0.00560464
$519$	15.7763	0.692503
$520$	0	0
$521$	22.7354	0.996055	0.498027	−	0.867161i	$-0.334058\pi$
$521$	22.7354	0.996055	0.498027	+	0.867161i	$0.334058\pi$
$522$	0.685096	0.0299858
$523$	−12.1732	−0.532295	−0.266147	−	0.963932i	$-0.585751\pi$
$523$	−12.1732	−0.532295	−0.266147	+	0.963932i	$0.585751\pi$
$524$	−10.5267	−0.459862
$525$	0	0
$526$	−8.63278	−0.376407
$527$	25.4971	1.11067
$528$	6.21336	0.270402
$529$	14.6773	0.638142
$530$	0	0
$531$	2.29064	0.0994055
$532$	5.97980	0.259257
$533$	−0.726543	−0.0314701
$534$	1.53669	0.0664991
$535$	0	0
$536$	−1.54660	−0.0668029
$537$	21.9229	0.946044
$538$	−4.75120	−0.204839
$539$	−3.35114	−0.144344
$540$	0	0
$541$	3.44016	0.147904	0.0739520	−	0.997262i	$-0.476439\pi$
$541$	3.44016	0.147904	0.0739520	+	0.997262i	$0.476439\pi$
$542$	14.8914	0.639639
$543$	10.9929	0.471752
$544$	14.8048	0.634749
$545$	0	0
$546$	0.618034	0.0264494
$547$	12.0570	0.515519	0.257759	−	0.966209i	$-0.417016\pi$
$547$	12.0570	0.515519	0.257759	+	0.966209i	$0.417016\pi$
$548$	21.8327	0.932649
$549$	−2.03739	−0.0869537
$550$	0	0
$551$	−4.09673	−0.174527
$552$	−13.7254	−0.584191
$553$	11.5373	0.490617
$554$	−7.38698	−0.313843
$555$	0	0
$556$	8.25627	0.350144
$557$	23.9499	1.01479	0.507395	−	0.861714i	$-0.330609\pi$
$557$	23.9499	1.01479	0.507395	+	0.861714i	$0.330609\pi$
$558$	5.97980	0.253145
$559$	−0.360723	−0.0152570
$560$	0	0
$561$	−8.83099	−0.372845
$562$	−0.270440	−0.0114078
$563$	29.7854	1.25531	0.627653	−	0.778493i	$-0.284015\pi$
$563$	29.7854	1.25531	0.627653	+	0.778493i	$0.284015\pi$
$564$	−3.74373	−0.157639
$565$	0	0
$566$	−5.87586	−0.246981
$567$	−1.00000	−0.0419961
$568$	−8.31812	−0.349021
$569$	26.0033	1.09012	0.545058	−	0.838399i	$-0.316508\pi$
$569$	26.0033	1.09012	0.545058	+	0.838399i	$0.316508\pi$
$570$	0	0
$571$	−28.8335	−1.20664	−0.603322	−	0.797498i	$-0.706156\pi$
$571$	−28.8335	−1.20664	−0.603322	+	0.797498i	$0.706156\pi$
$572$	−5.42226	−0.226716
$573$	−11.6382	−0.486194
$574$	0.449028	0.0187421
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−35.8525	−1.49256	−0.746280	−	0.665632i	$-0.768162\pi$
$577$	−35.8525	−1.49256	−0.746280	+	0.665632i	$0.768162\pi$
$578$	6.21471	0.258498
$579$	9.78101	0.406485
$580$	0	0
$581$	−5.80236	−0.240723
$582$	−0.534429	−0.0221528
$583$	2.92992	0.121345
$584$	5.81567	0.240654
$585$	0	0
$586$	−5.28916	−0.218493
$587$	40.2255	1.66029	0.830143	−	0.557551i	$-0.188259\pi$
$587$	40.2255	1.66029	0.830143	+	0.557551i	$0.188259\pi$
$588$	1.61803	0.0667266
$589$	−35.7580	−1.47338
$590$	0	0
$591$	−15.3606	−0.631851
$592$	0.382678	0.0157280
$593$	4.57569	0.187901	0.0939506	−	0.995577i	$-0.470050\pi$
$593$	4.57569	0.187901	0.0939506	+	0.995577i	$0.470050\pi$
$594$	−2.07112	−0.0849790
$595$	0	0
$596$	−24.7520	−1.01388
$597$	17.7788	0.727638
$598$	3.79360	0.155132
$599$	45.7310	1.86852	0.934258	−	0.356597i	$-0.116063\pi$
$599$	45.7310	1.86852	0.934258	+	0.356597i	$0.116063\pi$
$600$	0	0
$601$	−3.01519	−0.122992	−0.0614961	−	0.998107i	$-0.519587\pi$
$601$	−3.01519	−0.122992	−0.0614961	+	0.998107i	$0.519587\pi$
$602$	0.222939	0.00908632
$603$	−0.691660	−0.0281666
$604$	3.53565	0.143864
$605$	0	0
$606$	−6.39259	−0.259681
$607$	−8.89145	−0.360893	−0.180446	−	0.983585i	$-0.557754\pi$
$607$	−8.89145	−0.360893	−0.180446	+	0.983585i	$0.557754\pi$
$608$	−20.7627	−0.842037
$609$	−1.10851	−0.0449190
$610$	0	0
$611$	2.31375	0.0936044
$612$	4.26388	0.172357
$613$	−5.00295	−0.202067	−0.101034	−	0.994883i	$-0.532215\pi$
$613$	−5.00295	−0.202067	−0.101034	+	0.994883i	$0.532215\pi$
$614$	14.4064	0.581393
$615$	0	0
$616$	7.49338	0.301917
$617$	7.38558	0.297332	0.148666	−	0.988887i	$-0.452502\pi$
$617$	7.38558	0.297332	0.148666	+	0.988887i	$0.452502\pi$
$618$	9.49003	0.381745
$619$	−26.6538	−1.07131	−0.535653	−	0.844438i	$-0.679934\pi$
$619$	−26.6538	−1.07131	−0.535653	+	0.844438i	$0.679934\pi$
$620$	0	0
$621$	−6.13818	−0.246317
$622$	−13.6934	−0.549056
$623$	−2.48642	−0.0996162
$624$	1.85410	0.0742235
$625$	0	0
$626$	20.9648	0.837922
$627$	12.3849	0.494604
$628$	−34.9927	−1.39636
$629$	−0.543898	−0.0216866
$630$	0	0
$631$	−28.4686	−1.13332	−0.566659	−	0.823953i	$-0.691764\pi$
$631$	−28.4686	−1.13332	−0.566659	+	0.823953i	$0.691764\pi$
$632$	−25.7983	−1.02620
$633$	−19.3414	−0.768753
$634$	−1.08070	−0.0429201
$635$	0	0
$636$	−1.41466	−0.0560948
$637$	−1.00000	−0.0396214
$638$	−2.29585	−0.0908937
$639$	−3.71998	−0.147160
$640$	0	0
$641$	−36.9595	−1.45981	−0.729906	−	0.683548i	$-0.760436\pi$
$641$	−36.9595	−1.45981	−0.729906	+	0.683548i	$0.760436\pi$
$642$	−10.2895	−0.406094
$643$	38.6067	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$643$	38.6067	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$644$	9.93179	0.391367
$645$	0	0
$646$	6.01905	0.236816
$647$	7.11213	0.279607	0.139803	−	0.990179i	$-0.455353\pi$
$647$	7.11213	0.279607	0.139803	+	0.990179i	$0.455353\pi$
$648$	2.23607	0.0878410
$649$	−7.67627	−0.301320
$650$	0	0
$651$	−9.67551	−0.379213
$652$	−32.5937	−1.27647
$653$	28.7474	1.12497	0.562487	−	0.826806i	$-0.309845\pi$
$653$	28.7474	1.12497	0.562487	+	0.826806i	$0.309845\pi$
$654$	−11.7188	−0.458242
$655$	0	0
$656$	1.34708	0.0525948
$657$	2.60085	0.101469
$658$	−1.42998	−0.0557463
$659$	47.7956	1.86185	0.930925	−	0.365210i	$-0.119003\pi$
$659$	47.7956	1.86185	0.930925	+	0.365210i	$0.119003\pi$
$660$	0	0
$661$	−21.8950	−0.851617	−0.425808	−	0.904813i	$-0.640010\pi$
$661$	−21.8950	−0.851617	−0.425808	+	0.904813i	$0.640010\pi$
$662$	−19.2192	−0.746976
$663$	−2.63522	−0.102343
$664$	12.9745	0.503507
$665$	0	0
$666$	−0.127559	−0.00494283
$667$	−6.80423	−0.263461
$668$	−18.0462	−0.698229
$669$	−19.6755	−0.760699
$670$	0	0
$671$	6.82758	0.263576
$672$	−5.61803	−0.216720
$673$	−16.5433	−0.637696	−0.318848	−	0.947806i	$-0.603296\pi$
$673$	−16.5433	−0.637696	−0.318848	+	0.947806i	$0.603296\pi$
$674$	13.6194	0.524599
$675$	0	0
$676$	−1.61803	−0.0622321
$677$	−6.40173	−0.246038	−0.123019	−	0.992404i	$-0.539258\pi$
$677$	−6.40173	−0.246038	−0.123019	+	0.992404i	$0.539258\pi$
$678$	−4.23420	−0.162614
$679$	0.864724	0.0331850
$680$	0	0
$681$	−13.4981	−0.517250
$682$	−20.0391	−0.767338
$683$	−34.5834	−1.32330	−0.661649	−	0.749814i	$-0.730143\pi$
$683$	−34.5834	−1.32330	−0.661649	+	0.749814i	$0.730143\pi$
$684$	−5.97980	−0.228643
$685$	0	0
$686$	0.618034	0.0235966
$687$	13.9138	0.530844
$688$	0.668817	0.0254984
$689$	0.874305	0.0333084
$690$	0	0
$691$	34.0179	1.29410	0.647051	−	0.762447i	$-0.276002\pi$
$691$	34.0179	1.29410	0.647051	+	0.762447i	$0.276002\pi$
$692$	25.5266	0.970376
$693$	3.35114	0.127299
$694$	−17.8678	−0.678254
$695$	0	0
$696$	2.47870	0.0939549
$697$	−1.91460	−0.0725206
$698$	0.458213	0.0173436
$699$	−21.7103	−0.821158
$700$	0	0
$701$	−28.4911	−1.07609	−0.538047	−	0.842915i	$-0.680838\pi$
$701$	−28.4911	−1.07609	−0.538047	+	0.842915i	$0.680838\pi$
$702$	−0.618034	−0.0233262
$703$	0.762779	0.0287688
$704$	0.791097	0.0298156
$705$	0	0
$706$	−2.25742	−0.0849593
$707$	10.3434	0.389004
$708$	3.70634	0.139293
$709$	−19.2329	−0.722306	−0.361153	−	0.932507i	$-0.617617\pi$
$709$	−19.2329	−0.722306	−0.361153	+	0.932507i	$0.617617\pi$
$710$	0	0
$711$	−11.5373	−0.432684
$712$	5.55980	0.208362
$713$	−59.3901	−2.22418
$714$	1.62866	0.0609509
$715$	0	0
$716$	35.4720	1.32565
$717$	2.40090	0.0896634
$718$	11.4127	0.425917
$719$	18.3317	0.683658	0.341829	−	0.939762i	$-0.388954\pi$
$719$	18.3317	0.683658	0.341829	+	0.939762i	$0.388954\pi$
$720$	0	0
$721$	−15.3552	−0.571857
$722$	3.30133	0.122863
$723$	−15.5953	−0.579996
$724$	17.7869	0.661046
$725$	0	0
$726$	0.142238	0.00527895
$727$	−42.6667	−1.58242	−0.791209	−	0.611545i	$-0.790548\pi$
$727$	−42.6667	−1.58242	−0.791209	+	0.611545i	$0.790548\pi$
$728$	2.23607	0.0828742
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.950584	−0.0351586
$732$	−3.29657	−0.121845
$733$	6.05562	0.223669	0.111835	−	0.993727i	$-0.464327\pi$
$733$	6.05562	0.223669	0.111835	+	0.993727i	$0.464327\pi$
$734$	−17.2283	−0.635908
$735$	0	0
$736$	−34.4845	−1.27112
$737$	2.31785	0.0853792
$738$	−0.449028	−0.0165289
$739$	25.7066	0.945631	0.472815	−	0.881161i	$-0.343238\pi$
$739$	25.7066	0.945631	0.472815	+	0.881161i	$0.343238\pi$
$740$	0	0
$741$	3.69572	0.135766
$742$	−0.540350	−0.0198369
$743$	−13.8400	−0.507741	−0.253871	−	0.967238i	$-0.581704\pi$
$743$	−13.8400	−0.507741	−0.253871	+	0.967238i	$0.581704\pi$
$744$	21.6351	0.793182
$745$	0	0
$746$	5.84671	0.214063
$747$	5.80236	0.212297
$748$	−14.2888	−0.522452
$749$	16.6487	0.608332
$750$	0	0
$751$	35.2777	1.28730	0.643651	−	0.765319i	$-0.277419\pi$
$751$	35.2777	1.28730	0.643651	+	0.765319i	$0.277419\pi$
$752$	−4.28993	−0.156438
$753$	28.5238	1.03946
$754$	−0.685096	−0.0249497
$755$	0	0
$756$	−1.61803	−0.0588473
$757$	−27.4699	−0.998411	−0.499205	−	0.866484i	$-0.666375\pi$
$757$	−27.4699	−0.998411	−0.499205	+	0.866484i	$0.666375\pi$
$758$	−2.50806	−0.0910968
$759$	20.5699	0.746640
$760$	0	0
$761$	−8.60763	−0.312026	−0.156013	−	0.987755i	$-0.549864\pi$
$761$	−8.60763	−0.312026	−0.156013	+	0.987755i	$0.549864\pi$
$762$	0.130464	0.00472621
$763$	18.9615	0.686451
$764$	−18.8311	−0.681284
$765$	0	0
$766$	3.99698	0.144417
$767$	−2.29064	−0.0827103
$768$	6.56231	0.236797
$769$	4.50149	0.162328	0.0811640	−	0.996701i	$-0.474136\pi$
$769$	4.50149	0.162328	0.0811640	+	0.996701i	$0.474136\pi$
$770$	0	0
$771$	−8.56108	−0.308320
$772$	15.8260	0.569590
$773$	25.7292	0.925416	0.462708	−	0.886511i	$-0.346878\pi$
$773$	25.7292	0.925416	0.462708	+	0.886511i	$0.346878\pi$
$774$	−0.222939	−0.00801338
$775$	0	0
$776$	−1.93358	−0.0694115
$777$	0.206396	0.00740440
$778$	7.89464	0.283037
$779$	2.68510	0.0962035
$780$	0	0
$781$	12.4662	0.446075
$782$	9.99698	0.357491
$783$	1.10851	0.0396149
$784$	1.85410	0.0662179
$785$	0	0
$786$	4.02085	0.143419
$787$	8.36864	0.298310	0.149155	−	0.988814i	$-0.452345\pi$
$787$	8.36864	0.298310	0.149155	+	0.988814i	$0.452345\pi$
$788$	−24.8540	−0.885387
$789$	−13.9681	−0.497279
$790$	0	0
$791$	6.85108	0.243597
$792$	−7.49338	−0.266266
$793$	2.03739	0.0723498
$794$	−12.3051	−0.436691
$795$	0	0
$796$	28.7667	1.01961
$797$	21.6426	0.766619	0.383310	−	0.923620i	$-0.374784\pi$
$797$	21.6426	0.766619	0.383310	+	0.923620i	$0.374784\pi$
$798$	−2.28408	−0.0808555
$799$	6.09724	0.215705
$800$	0	0
$801$	2.48642	0.0878532
$802$	−4.64063	−0.163866
$803$	−8.71581	−0.307574
$804$	−1.11913	−0.0394687
$805$	0	0
$806$	−5.97980	−0.210629
$807$	−7.68760	−0.270616
$808$	−23.1286	−0.813661
$809$	24.2264	0.851753	0.425877	−	0.904781i	$-0.359966\pi$
$809$	24.2264	0.851753	0.425877	+	0.904781i	$0.359966\pi$
$810$	0	0
$811$	−13.1236	−0.460831	−0.230416	−	0.973092i	$-0.574009\pi$
$811$	−13.1236	−0.460831	−0.230416	+	0.973092i	$0.574009\pi$
$812$	−1.79360	−0.0629432
$813$	24.0948	0.845040
$814$	0.427470	0.0149828
$815$	0	0
$816$	4.88597	0.171043
$817$	1.33313	0.0466403
$818$	−12.0469	−0.421209
$819$	1.00000	0.0349428
$820$	0	0
$821$	2.93114	0.102298	0.0511488	−	0.998691i	$-0.483712\pi$
$821$	2.93114	0.102298	0.0511488	+	0.998691i	$0.483712\pi$
$822$	−8.33937	−0.290869
$823$	19.5519	0.681536	0.340768	−	0.940147i	$-0.389313\pi$
$823$	19.5519	0.681536	0.340768	+	0.940147i	$0.389313\pi$
$824$	34.3353	1.19613
$825$	0	0
$826$	1.41570	0.0492583
$827$	−28.9405	−1.00636	−0.503180	−	0.864182i	$-0.667837\pi$
$827$	−28.9405	−1.00636	−0.503180	+	0.864182i	$0.667837\pi$
$828$	−9.93179	−0.345153
$829$	7.12403	0.247428	0.123714	−	0.992318i	$-0.460519\pi$
$829$	7.12403	0.247428	0.123714	+	0.992318i	$0.460519\pi$
$830$	0	0
$831$	−11.9524	−0.414624
$832$	0.236068	0.00818418
$833$	−2.63522	−0.0913050
$834$	−3.15362	−0.109201
$835$	0	0
$836$	20.0391	0.693068
$837$	9.67551	0.334435
$838$	−0.910145	−0.0314404
$839$	49.6349	1.71359	0.856794	−	0.515659i	$-0.172453\pi$
$839$	49.6349	1.71359	0.856794	+	0.515659i	$0.172453\pi$
$840$	0	0
$841$	−27.7712	−0.957628
$842$	−18.9127	−0.651776
$843$	−0.437581	−0.0150711
$844$	−31.2951	−1.07722
$845$	0	0
$846$	1.42998	0.0491636
$847$	−0.230146	−0.00790791
$848$	−1.62105	−0.0556671
$849$	−9.50733	−0.326291
$850$	0	0
$851$	1.26689	0.0434285
$852$	−6.01905	−0.206209
$853$	−14.2794	−0.488919	−0.244460	−	0.969660i	$-0.578611\pi$
$853$	−14.2794	−0.488919	−0.244460	+	0.969660i	$0.578611\pi$
$854$	−1.25918	−0.0430881
$855$	0	0
$856$	−37.2277	−1.27242
$857$	−13.4333	−0.458872	−0.229436	−	0.973324i	$-0.573688\pi$
$857$	−13.4333	−0.458872	−0.229436	+	0.973324i	$0.573688\pi$
$858$	2.07112	0.0707068
$859$	−40.3190	−1.37567	−0.687833	−	0.725869i	$-0.741438\pi$
$859$	−40.3190	−1.37567	−0.687833	+	0.725869i	$0.741438\pi$
$860$	0	0
$861$	0.726543	0.0247605
$862$	−21.5111	−0.732670
$863$	42.9000	1.46033	0.730166	−	0.683270i	$-0.239443\pi$
$863$	42.9000	1.46033	0.730166	+	0.683270i	$0.239443\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	−24.6900	−0.839000
$867$	10.0556	0.341507
$868$	−15.6553	−0.531376
$869$	38.6632	1.31156
$870$	0	0
$871$	0.691660	0.0234360
$872$	−42.3991	−1.43582
$873$	−0.864724	−0.0292665
$874$	−14.0201	−0.474237
$875$	0	0
$876$	4.20826	0.142184
$877$	−50.8231	−1.71617	−0.858086	−	0.513506i	$-0.828347\pi$
$877$	−50.8231	−1.71617	−0.858086	+	0.513506i	$0.828347\pi$
$878$	−18.1854	−0.613726
$879$	−8.55805	−0.288656
$880$	0	0
$881$	44.4756	1.49842	0.749211	−	0.662331i	$-0.230433\pi$
$881$	44.4756	1.49842	0.749211	+	0.662331i	$0.230433\pi$
$882$	−0.618034	−0.0208103
$883$	41.0957	1.38298	0.691490	−	0.722386i	$-0.256955\pi$
$883$	41.0957	1.38298	0.691490	+	0.722386i	$0.256955\pi$
$884$	−4.26388	−0.143410
$885$	0	0
$886$	1.86683	0.0627175
$887$	−4.72082	−0.158510	−0.0792549	−	0.996854i	$-0.525254\pi$
$887$	−4.72082	−0.158510	−0.0792549	+	0.996854i	$0.525254\pi$
$888$	−0.461514	−0.0154874
$889$	−0.211095	−0.00707990
$890$	0	0
$891$	−3.35114	−0.112267
$892$	−31.8357	−1.06594
$893$	−8.55097	−0.286147
$894$	9.45444	0.316204
$895$	0	0
$896$	−11.3820	−0.380245
$897$	6.13818	0.204948
$898$	−15.9288	−0.531551
$899$	10.7254	0.357712
$900$	0	0
$901$	2.30399	0.0767569
$902$	1.50476	0.0501029
$903$	0.360723	0.0120041
$904$	−15.3195	−0.509519
$905$	0	0
$906$	−1.35050	−0.0448673
$907$	−43.5266	−1.44528	−0.722639	−	0.691226i	$-0.757071\pi$
$907$	−43.5266	−1.44528	−0.722639	+	0.691226i	$0.757071\pi$
$908$	−21.8405	−0.724801
$909$	−10.3434	−0.343070
$910$	0	0
$911$	−43.9500	−1.45613	−0.728065	−	0.685508i	$-0.759580\pi$
$911$	−43.9500	−1.45613	−0.728065	+	0.685508i	$0.759580\pi$
$912$	−6.85224	−0.226900
$913$	−19.4445	−0.643520
$914$	−0.0485210	−0.00160493
$915$	0	0
$916$	22.5130	0.743849
$917$	−6.50587	−0.214843
$918$	−1.62866	−0.0537536
$919$	28.6274	0.944330	0.472165	−	0.881510i	$-0.343473\pi$
$919$	28.6274	0.944330	0.472165	+	0.881510i	$0.343473\pi$
$920$	0	0
$921$	23.3100	0.768090
$922$	−13.6211	−0.448588
$923$	3.71998	0.122445
$924$	5.42226	0.178379
$925$	0	0
$926$	−7.73776	−0.254279
$927$	15.3552	0.504331
$928$	6.22764	0.204432
$929$	−20.0266	−0.657052	−0.328526	−	0.944495i	$-0.606552\pi$
$929$	−20.0266	−0.657052	−0.328526	+	0.944495i	$0.606552\pi$
$930$	0	0
$931$	3.69572	0.121122
$932$	−35.1280	−1.15066
$933$	−22.1564	−0.725368
$934$	18.9951	0.621539
$935$	0	0
$936$	−2.23607	−0.0730882
$937$	14.1187	0.461237	0.230619	−	0.973044i	$-0.425925\pi$
$937$	14.1187	0.461237	0.230619	+	0.973044i	$0.425925\pi$
$938$	−0.427470	−0.0139574
$939$	33.9218	1.10699
$940$	0	0
$941$	55.0872	1.79579	0.897896	−	0.440208i	$-0.145095\pi$
$941$	55.0872	1.79579	0.897896	+	0.440208i	$0.145095\pi$
$942$	13.3660	0.435489
$943$	4.45965	0.145226
$944$	4.24709	0.138231
$945$	0	0
$946$	0.747100	0.0242903
$947$	29.0137	0.942820	0.471410	−	0.881914i	$-0.343745\pi$
$947$	29.0137	0.942820	0.471410	+	0.881914i	$0.343745\pi$
$948$	−18.6678	−0.606302
$949$	−2.60085	−0.0844271
$950$	0	0
$951$	−1.74861	−0.0567026
$952$	5.89253	0.190978
$953$	28.8354	0.934072	0.467036	−	0.884238i	$-0.345322\pi$
$953$	28.8354	0.934072	0.467036	+	0.884238i	$0.345322\pi$
$954$	0.540350	0.0174945
$955$	0	0
$956$	3.88474	0.125642
$957$	−3.71477	−0.120081
$958$	−20.0756	−0.648612
$959$	13.4934	0.435724
$960$	0	0
$961$	62.6156	2.01986
$962$	0.127559	0.00411268
$963$	−16.6487	−0.536498
$964$	−25.2338	−0.812725
$965$	0	0
$966$	−3.79360	−0.122057
$967$	−7.99397	−0.257069	−0.128534	−	0.991705i	$-0.541027\pi$
$967$	−7.99397	−0.257069	−0.128534	+	0.991705i	$0.541027\pi$
$968$	0.514622	0.0165406
$969$	9.73903	0.312863
$970$	0	0
$971$	15.7672	0.505992	0.252996	−	0.967467i	$-0.418584\pi$
$971$	15.7672	0.505992	0.252996	+	0.967467i	$0.418584\pi$
$972$	1.61803	0.0518985
$973$	5.10266	0.163584
$974$	0.629053	0.0201561
$975$	0	0
$976$	−3.77753	−0.120916
$977$	8.57921	0.274473	0.137237	−	0.990538i	$-0.456178\pi$
$977$	8.57921	0.274473	0.137237	+	0.990538i	$0.456178\pi$
$978$	12.4497	0.398097
$979$	−8.33234	−0.266303
$980$	0	0
$981$	−18.9615	−0.605393
$982$	−5.78684	−0.184665
$983$	−20.3519	−0.649123	−0.324562	−	0.945865i	$-0.605217\pi$
$983$	−20.3519	−0.649123	−0.324562	+	0.945865i	$0.605217\pi$
$984$	−1.62460	−0.0517903
$985$	0	0
$986$	−1.80538	−0.0574950
$987$	−2.31375	−0.0736475
$988$	5.97980	0.190243
$989$	2.21418	0.0704069
$990$	0	0
$991$	−11.2052	−0.355946	−0.177973	−	0.984035i	$-0.556954\pi$
$991$	−11.2052	−0.355946	−0.177973	+	0.984035i	$0.556954\pi$
$992$	54.3574	1.72585
$993$	−31.0973	−0.986844
$994$	−2.29907	−0.0729222
$995$	0	0
$996$	9.38842	0.297483
$997$	−26.6899	−0.845277	−0.422638	−	0.906298i	$-0.638896\pi$
$997$	−26.6899	−0.845277	−0.422638	+	0.906298i	$0.638896\pi$
$998$	10.2035	0.322985
$999$	−0.206396	−0.00653006

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bm.1.1		4
5.2	odd	4		1365.2.f.c.274.3	✓	8
5.3	odd	4		1365.2.f.c.274.5	yes	8
5.4	even	2		6825.2.a.be.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.c.274.3	✓	8	5.2	odd	4
1365.2.f.c.274.5	yes	8	5.3	odd	4
6825.2.a.be.1.3		4	5.4	even	2
6825.2.a.bm.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.35114 q^{11} +1.61803 q^{12} -1.00000 q^{13} +0.618034 q^{14} +1.85410 q^{16} -2.63522 q^{17} -0.618034 q^{18} +3.69572 q^{19} +1.00000 q^{21} +2.07112 q^{22} +6.13818 q^{23} -2.23607 q^{24} +0.618034 q^{26} -1.00000 q^{27} +1.61803 q^{28} -1.10851 q^{29} -9.67551 q^{31} -5.61803 q^{32} +3.35114 q^{33} +1.62866 q^{34} -1.61803 q^{36} +0.206396 q^{37} -2.28408 q^{38} +1.00000 q^{39} +0.726543 q^{41} -0.618034 q^{42} +0.360723 q^{43} +5.42226 q^{44} -3.79360 q^{46} -2.31375 q^{47} -1.85410 q^{48} +1.00000 q^{49} +2.63522 q^{51} +1.61803 q^{52} -0.874305 q^{53} +0.618034 q^{54} -2.23607 q^{56} -3.69572 q^{57} +0.685096 q^{58} +2.29064 q^{59} -2.03739 q^{61} +5.97980 q^{62} -1.00000 q^{63} -0.236068 q^{64} -2.07112 q^{66} -0.691660 q^{67} +4.26388 q^{68} -6.13818 q^{69} -3.71998 q^{71} +2.23607 q^{72} +2.60085 q^{73} -0.127559 q^{74} -5.97980 q^{76} +3.35114 q^{77} -0.618034 q^{78} -11.5373 q^{79} +1.00000 q^{81} -0.449028 q^{82} +5.80236 q^{83} -1.61803 q^{84} -0.222939 q^{86} +1.10851 q^{87} -7.49338 q^{88} +2.48642 q^{89} +1.00000 q^{91} -9.93179 q^{92} +9.67551 q^{93} +1.42998 q^{94} +5.61803 q^{96} -0.864724 q^{97} -0.618034 q^{98} -3.35114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} - 2 q^{4} - 2 q^{6} - 4 q^{7} + 4 q^{9} - 4 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} - 6 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{21} - 2 q^{22} + 8 q^{23} - 2 q^{26} - 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 - 2 * q^4 - 2 * q^6 - 4 * q^7 + 4 * q^9 - 4 * q^11 + 2 * q^12 - 4 * q^13 - 2 * q^14 - 6 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^19 + 4 * q^21 - 2 * q^22 + 8 * q^23 - 2 * q^26 - 4 * q^27 + 2 * q^28 - 6 * q^29 - 2 * q^31 - 18 * q^32 + 4 * q^33 - 4 * q^34 - 2 * q^36 + 10 * q^37 - 6 * q^38 + 4 * q^39 + 2 * q^42 + 18 * q^43 + 2 * q^44 - 6 * q^46 + 12 * q^47 + 6 * q^48 + 4 * q^49 - 2 * q^51 + 2 * q^52 - 10 * q^53 - 2 * q^54 + 2 * q^57 - 8 * q^58 + 4 * q^59 - 20 * q^61 + 4 * q^62 - 4 * q^63 + 8 * q^64 + 2 * q^66 + 10 * q^67 - 6 * q^68 - 8 * q^69 - 14 * q^71 + 14 * q^73 + 10 * q^74 - 4 * q^76 + 4 * q^77 + 2 * q^78 - 26 * q^79 + 4 * q^81 + 12 * q^83 - 2 * q^84 + 4 * q^86 + 6 * q^87 + 20 * q^89 + 4 * q^91 - 14 * q^92 + 2 * q^93 + 16 * q^94 + 18 * q^96 + 16 * q^97 + 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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