LMFDB - Embedded newform 6825.2.a.bi.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			$2.28400$ 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)

$f(q)$	$=$	\(q-2.28400 q^{2} +1.00000 q^{3} +3.21665 q^{4} -2.28400 q^{6} +1.00000 q^{7} -2.77882 q^{8} +1.00000 q^{9} +0.505180 q^{11} +3.21665 q^{12} +1.00000 q^{13} -2.28400 q^{14} -0.0864793 q^{16} -2.34099 q^{17} -2.28400 q^{18} -7.65448 q^{19} +1.00000 q^{21} -1.15383 q^{22} -3.87566 q^{23} -2.77882 q^{24} -2.28400 q^{26} +1.00000 q^{27} +3.21665 q^{28} +4.51977 q^{29} -5.77882 q^{31} +5.75515 q^{32} +0.505180 q^{33} +5.34681 q^{34} +3.21665 q^{36} +6.05828 q^{37} +17.4828 q^{38} +1.00000 q^{39} +2.96598 q^{41} -2.28400 q^{42} +0.538507 q^{43} +1.62499 q^{44} +8.85199 q^{46} +13.1884 q^{47} -0.0864793 q^{48} +1.00000 q^{49} -2.34099 q^{51} +3.21665 q^{52} +1.40380 q^{53} -2.28400 q^{54} -2.77882 q^{56} -7.65448 q^{57} -10.3231 q^{58} +3.66867 q^{59} -12.1930 q^{61} +13.1988 q^{62} +1.00000 q^{63} -12.9718 q^{64} -1.15383 q^{66} +1.39120 q^{67} -7.53013 q^{68} -3.87566 q^{69} +1.26941 q^{71} -2.77882 q^{72} -13.0828 q^{73} -13.8371 q^{74} -24.6217 q^{76} +0.505180 q^{77} -2.28400 q^{78} -10.0495 q^{79} +1.00000 q^{81} -6.77428 q^{82} -12.0725 q^{83} +3.21665 q^{84} -1.22995 q^{86} +4.51977 q^{87} -1.40380 q^{88} -2.95337 q^{89} +1.00000 q^{91} -12.4666 q^{92} -5.77882 q^{93} -30.1224 q^{94} +5.75515 q^{96} +18.2911 q^{97} -2.28400 q^{98} +0.505180 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.28400	−1.61503	−0.807515	−	0.589847i	$-0.799188\pi$
$2$	−2.28400	−1.61503	−0.807515	+	0.589847i	$0.799188\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.21665	1.60832
$5$	0	0
$5$	0	0
$6$	−2.28400	−0.932438
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.77882	−0.982460
$9$	1.00000	0.333333
$10$	0	0
$11$	0.505180	0.152318	0.0761588	−	0.997096i	$-0.475734\pi$
$11$	0.505180	0.152318	0.0761588	+	0.997096i	$0.475734\pi$
$12$	3.21665	0.928566
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.28400	−0.610424
$15$	0	0
$16$	−0.0864793	−0.0216198
$17$	−2.34099	−0.567773	−0.283887	−	0.958858i	$-0.591624\pi$
$17$	−2.34099	−0.567773	−0.283887	+	0.958858i	$0.591624\pi$
$18$	−2.28400	−0.538343
$19$	−7.65448	−1.75606	−0.878029	−	0.478608i	$-0.841141\pi$
$19$	−7.65448	−1.75606	−0.878029	+	0.478608i	$0.841141\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−1.15383	−0.245998
$23$	−3.87566	−0.808130	−0.404065	−	0.914730i	$-0.632403\pi$
$23$	−3.87566	−0.808130	−0.404065	+	0.914730i	$0.632403\pi$
$24$	−2.77882	−0.567224
$25$	0	0
$26$	−2.28400	−0.447929
$27$	1.00000	0.192450
$28$	3.21665	0.607889
$29$	4.51977	0.839301	0.419650	−	0.907686i	$-0.362153\pi$
$29$	4.51977	0.839301	0.419650	+	0.907686i	$0.362153\pi$
$30$	0	0
$31$	−5.77882	−1.03791	−0.518953	−	0.854803i	$-0.673678\pi$
$31$	−5.77882	−1.03791	−0.518953	+	0.854803i	$0.673678\pi$
$32$	5.75515	1.01738
$33$	0.505180	0.0879406
$34$	5.34681	0.916971
$35$	0	0
$36$	3.21665	0.536108
$37$	6.05828	0.995975	0.497987	−	0.867184i	$-0.334073\pi$
$37$	6.05828	0.995975	0.497987	+	0.867184i	$0.334073\pi$
$38$	17.4828	2.83609
$39$	1.00000	0.160128
$40$	0	0
$41$	2.96598	0.463208	0.231604	−	0.972810i	$-0.425603\pi$
$41$	2.96598	0.463208	0.231604	+	0.972810i	$0.425603\pi$
$42$	−2.28400	−0.352429
$43$	0.538507	0.0821216	0.0410608	−	0.999157i	$-0.486926\pi$
$43$	0.538507	0.0821216	0.0410608	+	0.999157i	$0.486926\pi$
$44$	1.62499	0.244976
$45$	0	0
$46$	8.85199	1.30516
$47$	13.1884	1.92373	0.961866	−	0.273520i	$-0.0881880\pi$
$47$	13.1884	1.92373	0.961866	+	0.273520i	$0.0881880\pi$
$48$	−0.0864793	−0.0124822
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.34099	−0.327804
$52$	3.21665	0.446069
$53$	1.40380	0.192827	0.0964137	−	0.995341i	$-0.469263\pi$
$53$	1.40380	0.192827	0.0964137	+	0.995341i	$0.469263\pi$
$54$	−2.28400	−0.310813
$55$	0	0
$56$	−2.77882	−0.371335
$57$	−7.65448	−1.01386
$58$	−10.3231	−1.35550
$59$	3.66867	0.477621	0.238810	−	0.971066i	$-0.423243\pi$
$59$	3.66867	0.477621	0.238810	+	0.971066i	$0.423243\pi$
$60$	0	0
$61$	−12.1930	−1.56115	−0.780576	−	0.625061i	$-0.785074\pi$
$61$	−12.1930	−1.56115	−0.780576	+	0.625061i	$0.785074\pi$
$62$	13.1988	1.67625
$63$	1.00000	0.125988
$64$	−12.9718	−1.62147
$65$	0	0
$66$	−1.15383	−0.142027
$67$	1.39120	0.169962	0.0849810	−	0.996383i	$-0.472917\pi$
$67$	1.39120	0.169962	0.0849810	+	0.996383i	$0.472917\pi$
$68$	−7.53013	−0.913163
$69$	−3.87566	−0.466574
$70$	0	0
$71$	1.26941	0.150651	0.0753254	−	0.997159i	$-0.476000\pi$
$71$	1.26941	0.150651	0.0753254	+	0.997159i	$0.476000\pi$
$72$	−2.77882	−0.327487
$73$	−13.0828	−1.53123	−0.765615	−	0.643299i	$-0.777565\pi$
$73$	−13.0828	−1.53123	−0.765615	+	0.643299i	$0.777565\pi$
$74$	−13.8371	−1.60853
$75$	0	0
$76$	−24.6217	−2.82431
$77$	0.505180	0.0575706
$78$	−2.28400	−0.258612
$79$	−10.0495	−1.13066	−0.565329	−	0.824865i	$-0.691251\pi$
$79$	−10.0495	−1.13066	−0.565329	+	0.824865i	$0.691251\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.77428	−0.748094
$83$	−12.0725	−1.32513	−0.662563	−	0.749006i	$-0.730531\pi$
$83$	−12.0725	−1.32513	−0.662563	+	0.749006i	$0.730531\pi$
$84$	3.21665	0.350965
$85$	0	0
$86$	−1.22995	−0.132629
$87$	4.51977	0.484570
$88$	−1.40380	−0.149646
$89$	−2.95337	−0.313057	−0.156528	−	0.987673i	$-0.550030\pi$
$89$	−2.95337	−0.313057	−0.156528	+	0.987673i	$0.550030\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−12.4666	−1.29973
$93$	−5.77882	−0.599235
$94$	−30.1224	−3.10689
$95$	0	0
$96$	5.75515	0.587383
$97$	18.2911	1.85718	0.928590	−	0.371106i	$-0.121021\pi$
$97$	18.2911	1.85718	0.928590	+	0.371106i	$0.121021\pi$
$98$	−2.28400	−0.230719
$99$	0.505180	0.0507725
$100$	0	0
$101$	4.66484	0.464169	0.232084	−	0.972696i	$-0.425445\pi$
$101$	4.66484	0.464169	0.232084	+	0.972696i	$0.425445\pi$
$102$	5.34681	0.529413
$103$	4.09032	0.403031	0.201515	−	0.979485i	$-0.435413\pi$
$103$	4.09032	0.403031	0.201515	+	0.979485i	$0.435413\pi$
$104$	−2.77882	−0.272485
$105$	0	0
$106$	−3.20629	−0.311422
$107$	−3.50518	−0.338859	−0.169429	−	0.985542i	$-0.554192\pi$
$107$	−3.50518	−0.338859	−0.169429	+	0.985542i	$0.554192\pi$
$108$	3.21665	0.309522
$109$	−13.0981	−1.25457	−0.627287	−	0.778789i	$-0.715835\pi$
$109$	−13.0981	−1.25457	−0.627287	+	0.778789i	$0.715835\pi$
$110$	0	0
$111$	6.05828	0.575026
$112$	−0.0864793	−0.00817153
$113$	−4.27562	−0.402217	−0.201108	−	0.979569i	$-0.564454\pi$
$113$	−4.27562	−0.402217	−0.201108	+	0.979569i	$0.564454\pi$
$114$	17.4828	1.63741
$115$	0	0
$116$	14.5385	1.34987
$117$	1.00000	0.0924500
$118$	−8.37924	−0.771372
$119$	−2.34099	−0.214598
$120$	0	0
$121$	−10.7448	−0.976799
$122$	27.8487	2.52131
$123$	2.96598	0.267433
$124$	−18.5884	−1.66929
$125$	0	0
$126$	−2.28400	−0.203475
$127$	5.34003	0.473851	0.236926	−	0.971528i	$-0.423860\pi$
$127$	5.34003	0.473851	0.236926	+	0.971528i	$0.423860\pi$
$128$	18.1173	1.60135
$129$	0.538507	0.0474129
$130$	0	0
$131$	−19.3294	−1.68881	−0.844407	−	0.535702i	$-0.820047\pi$
$131$	−19.3294	−1.68881	−0.844407	+	0.535702i	$0.820047\pi$
$132$	1.62499	0.141437
$133$	−7.65448	−0.663727
$134$	−3.17749	−0.274494
$135$	0	0
$136$	6.50518	0.557815
$137$	−11.2412	−0.960401	−0.480201	−	0.877159i	$-0.659436\pi$
$137$	−11.2412	−0.960401	−0.480201	+	0.877159i	$0.659436\pi$
$138$	8.85199	0.753532
$139$	−7.42747	−0.629990	−0.314995	−	0.949093i	$-0.602003\pi$
$139$	−7.42747	−0.629990	−0.314995	+	0.949093i	$0.602003\pi$
$140$	0	0
$141$	13.1884	1.11067
$142$	−2.89932	−0.243306
$143$	0.505180	0.0422453
$144$	−0.0864793	−0.00720661
$145$	0	0
$146$	29.8812	2.47298
$147$	1.00000	0.0824786
$148$	19.4873	1.60185
$149$	13.0166	1.06636	0.533180	−	0.846002i	$-0.320997\pi$
$149$	13.0166	1.06636	0.533180	+	0.846002i	$0.320997\pi$
$150$	0	0
$151$	−1.42324	−0.115821	−0.0579107	−	0.998322i	$-0.518444\pi$
$151$	−1.42324	−0.115821	−0.0579107	+	0.998322i	$0.518444\pi$
$152$	21.2704	1.72526
$153$	−2.34099	−0.189258
$154$	−1.15383	−0.0929783
$155$	0	0
$156$	3.21665	0.257538
$157$	7.73731	0.617505	0.308752	−	0.951142i	$-0.400089\pi$
$157$	7.73731	0.617505	0.308752	+	0.951142i	$0.400089\pi$
$158$	22.9531	1.82605
$159$	1.40380	0.111329
$160$	0	0
$161$	−3.87566	−0.305445
$162$	−2.28400	−0.179448
$163$	−17.6474	−1.38225	−0.691125	−	0.722735i	$-0.742885\pi$
$163$	−17.6474	−1.38225	−0.691125	+	0.722735i	$0.742885\pi$
$164$	9.54049	0.744987
$165$	0	0
$166$	27.5735	2.14012
$167$	8.46662	0.655167	0.327583	−	0.944822i	$-0.393766\pi$
$167$	8.46662	0.655167	0.327583	+	0.944822i	$0.393766\pi$
$168$	−2.77882	−0.214390
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.65448	−0.585352
$172$	1.73219	0.132078
$173$	18.2008	1.38378	0.691890	−	0.722003i	$-0.256778\pi$
$173$	18.2008	1.38378	0.691890	+	0.722003i	$0.256778\pi$
$174$	−10.3231	−0.782596
$175$	0	0
$176$	−0.0436877	−0.00329308
$177$	3.66867	0.275754
$178$	6.74549	0.505596
$179$	−14.1739	−1.05940	−0.529702	−	0.848184i	$-0.677696\pi$
$179$	−14.1739	−1.05940	−0.529702	+	0.848184i	$0.677696\pi$
$180$	0	0
$181$	−1.67973	−0.124854	−0.0624268	−	0.998050i	$-0.519884\pi$
$181$	−1.67973	−0.124854	−0.0624268	+	0.998050i	$0.519884\pi$
$182$	−2.28400	−0.169301
$183$	−12.1930	−0.901331
$184$	10.7697	0.793956
$185$	0	0
$186$	13.1988	0.967783
$187$	−1.18262	−0.0864819
$188$	42.4226	3.09398
$189$	1.00000	0.0727393
$190$	0	0
$191$	−2.29730	−0.166227	−0.0831134	−	0.996540i	$-0.526486\pi$
$191$	−2.29730	−0.166227	−0.0831134	+	0.996540i	$0.526486\pi$
$192$	−12.9718	−0.936159
$193$	1.24230	0.0894224	0.0447112	−	0.999000i	$-0.485763\pi$
$193$	1.24230	0.0894224	0.0447112	+	0.999000i	$0.485763\pi$
$194$	−41.7769	−2.99940
$195$	0	0
$196$	3.21665	0.229760
$197$	−19.7519	−1.40727	−0.703633	−	0.710564i	$-0.748440\pi$
$197$	−19.7519	−1.40727	−0.703633	+	0.710564i	$0.748440\pi$
$198$	−1.15383	−0.0819992
$199$	−22.7004	−1.60919	−0.804595	−	0.593824i	$-0.797617\pi$
$199$	−22.7004	−1.60919	−0.804595	+	0.593824i	$0.797617\pi$
$200$	0	0
$201$	1.39120	0.0981276
$202$	−10.6545	−0.749646
$203$	4.51977	0.317226
$204$	−7.53013	−0.527215
$205$	0	0
$206$	−9.34228	−0.650907
$207$	−3.87566	−0.269377
$208$	−0.0864793	−0.00599626
$209$	−3.86689	−0.267478
$210$	0	0
$211$	−0.953370	−0.0656327	−0.0328163	−	0.999461i	$-0.510448\pi$
$211$	−0.953370	−0.0656327	−0.0328163	+	0.999461i	$0.510448\pi$
$212$	4.51554	0.310129
$213$	1.26941	0.0869783
$214$	8.00582	0.547267
$215$	0	0
$216$	−2.77882	−0.189075
$217$	−5.77882	−0.392292
$218$	29.9161	2.02617
$219$	−13.0828	−0.884056
$220$	0	0
$221$	−2.34099	−0.157472
$222$	−13.8371	−0.928685
$223$	11.7353	0.785856	0.392928	−	0.919569i	$-0.371462\pi$
$223$	11.7353	0.785856	0.392928	+	0.919569i	$0.371462\pi$
$224$	5.75515	0.384532
$225$	0	0
$226$	9.76551	0.649592
$227$	2.49482	0.165587	0.0827935	−	0.996567i	$-0.473616\pi$
$227$	2.49482	0.165587	0.0827935	+	0.996567i	$0.473616\pi$
$228$	−24.6217	−1.63061
$229$	−2.86650	−0.189424	−0.0947118	−	0.995505i	$-0.530193\pi$
$229$	−2.86650	−0.189424	−0.0947118	+	0.995505i	$0.530193\pi$
$230$	0	0
$231$	0.505180	0.0332384
$232$	−12.5596	−0.824580
$233$	2.42817	0.159074	0.0795372	−	0.996832i	$-0.474656\pi$
$233$	2.42817	0.159074	0.0795372	+	0.996832i	$0.474656\pi$
$234$	−2.28400	−0.149310
$235$	0	0
$236$	11.8008	0.768168
$237$	−10.0495	−0.652786
$238$	5.34681	0.346582
$239$	−23.8137	−1.54038	−0.770191	−	0.637813i	$-0.779839\pi$
$239$	−23.8137	−1.54038	−0.770191	+	0.637813i	$0.779839\pi$
$240$	0	0
$241$	19.2089	1.23735	0.618676	−	0.785646i	$-0.287669\pi$
$241$	19.2089	1.23735	0.618676	+	0.785646i	$0.287669\pi$
$242$	24.5411	1.57756
$243$	1.00000	0.0641500
$244$	−39.2205	−2.51084
$245$	0	0
$246$	−6.77428	−0.431912
$247$	−7.65448	−0.487043
$248$	16.0583	1.01970
$249$	−12.0725	−0.765062
$250$	0	0
$251$	27.2059	1.71722	0.858611	−	0.512627i	$-0.171328\pi$
$251$	27.2059	1.71722	0.858611	+	0.512627i	$0.171328\pi$
$252$	3.21665	0.202630
$253$	−1.95791	−0.123093
$254$	−12.1966	−0.765284
$255$	0	0
$256$	−15.4362	−0.964761
$257$	−28.2678	−1.76330	−0.881650	−	0.471904i	$-0.843567\pi$
$257$	−28.2678	−1.76330	−0.881650	+	0.471904i	$0.843567\pi$
$258$	−1.22995	−0.0765733
$259$	6.05828	0.376443
$260$	0	0
$261$	4.51977	0.279767
$262$	44.1482	2.72749
$263$	−23.8737	−1.47212	−0.736058	−	0.676919i	$-0.763315\pi$
$263$	−23.8737	−1.47212	−0.736058	+	0.676919i	$0.763315\pi$
$264$	−1.40380	−0.0863982
$265$	0	0
$266$	17.4828	1.07194
$267$	−2.95337	−0.180743
$268$	4.47499	0.273354
$269$	28.2497	1.72241	0.861206	−	0.508256i	$-0.169709\pi$
$269$	28.2497	1.72241	0.861206	+	0.508256i	$0.169709\pi$
$270$	0	0
$271$	−6.81214	−0.413808	−0.206904	−	0.978361i	$-0.566339\pi$
$271$	−6.81214	−0.413808	−0.206904	+	0.978361i	$0.566339\pi$
$272$	0.202447	0.0122752
$273$	1.00000	0.0605228
$274$	25.6749	1.55108
$275$	0	0
$276$	−12.4666	−0.750402
$277$	−17.0246	−1.02291	−0.511453	−	0.859311i	$-0.670893\pi$
$277$	−17.0246	−1.02291	−0.511453	+	0.859311i	$0.670893\pi$
$278$	16.9643	1.01745
$279$	−5.77882	−0.345969
$280$	0	0
$281$	−2.77588	−0.165595	−0.0827974	−	0.996566i	$-0.526385\pi$
$281$	−2.77588	−0.165595	−0.0827974	+	0.996566i	$0.526385\pi$
$282$	−30.1224	−1.79376
$283$	−19.6039	−1.16533	−0.582665	−	0.812712i	$-0.697990\pi$
$283$	−19.6039	−1.16533	−0.582665	+	0.812712i	$0.697990\pi$
$284$	4.08323	0.242295
$285$	0	0
$286$	−1.15383	−0.0682275
$287$	2.96598	0.175076
$288$	5.75515	0.339126
$289$	−11.5198	−0.677634
$290$	0	0
$291$	18.2911	1.07224
$292$	−42.0829	−2.46271
$293$	29.4890	1.72277	0.861384	−	0.507955i	$-0.169598\pi$
$293$	29.4890	1.72277	0.861384	+	0.507955i	$0.169598\pi$
$294$	−2.28400	−0.133205
$295$	0	0
$296$	−16.8349	−0.978506
$297$	0.505180	0.0293135
$298$	−29.7298	−1.72220
$299$	−3.87566	−0.224135
$300$	0	0
$301$	0.538507	0.0310390
$302$	3.25067	0.187055
$303$	4.66484	0.267988
$304$	0.661954	0.0379656
$305$	0	0
$306$	5.34681	0.305657
$307$	12.3679	0.705875	0.352937	−	0.935647i	$-0.385183\pi$
$307$	12.3679	0.705875	0.352937	+	0.935647i	$0.385183\pi$
$308$	1.62499	0.0925922
$309$	4.09032	0.232690
$310$	0	0
$311$	−28.2742	−1.60328	−0.801642	−	0.597804i	$-0.796040\pi$
$311$	−28.2742	−1.60328	−0.801642	+	0.597804i	$0.796040\pi$
$312$	−2.77882	−0.157320
$313$	−28.0722	−1.58673	−0.793367	−	0.608744i	$-0.791674\pi$
$313$	−28.0722	−1.58673	−0.793367	+	0.608744i	$0.791674\pi$
$314$	−17.6720	−0.997289
$315$	0	0
$316$	−32.3257	−1.81846
$317$	20.8054	1.16855	0.584273	−	0.811557i	$-0.301380\pi$
$317$	20.8054	1.16855	0.584273	+	0.811557i	$0.301380\pi$
$318$	−3.20629	−0.179800
$319$	2.28330	0.127840
$320$	0	0
$321$	−3.50518	−0.195640
$322$	8.85199	0.493302
$323$	17.9190	0.997042
$324$	3.21665	0.178703
$325$	0	0
$326$	40.3066	2.23238
$327$	−13.0981	−0.724328
$328$	−8.24190	−0.455083
$329$	13.1884	0.727103
$330$	0	0
$331$	−14.4353	−0.793435	−0.396717	−	0.917941i	$-0.629851\pi$
$331$	−14.4353	−0.793435	−0.396717	+	0.917941i	$0.629851\pi$
$332$	−38.8329	−2.13123
$333$	6.05828	0.331992
$334$	−19.3377	−1.05811
$335$	0	0
$336$	−0.0864793	−0.00471783
$337$	14.5573	0.792985	0.396492	−	0.918038i	$-0.370227\pi$
$337$	14.5573	0.792985	0.396492	+	0.918038i	$0.370227\pi$
$338$	−2.28400	−0.124233
$339$	−4.27562	−0.232220
$340$	0	0
$341$	−2.91935	−0.158091
$342$	17.4828	0.945362
$343$	1.00000	0.0539949
$344$	−1.49641	−0.0806812
$345$	0	0
$346$	−41.5706	−2.23485
$347$	−5.06282	−0.271786	−0.135893	−	0.990724i	$-0.543390\pi$
$347$	−5.06282	−0.271786	−0.135893	+	0.990724i	$0.543390\pi$
$348$	14.5385	0.779346
$349$	11.4142	0.610986	0.305493	−	0.952194i	$-0.401179\pi$
$349$	11.4142	0.610986	0.305493	+	0.952194i	$0.401179\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	2.90739	0.154964
$353$	7.06895	0.376242	0.188121	−	0.982146i	$-0.439760\pi$
$353$	7.06895	0.376242	0.188121	+	0.982146i	$0.439760\pi$
$354$	−8.37924	−0.445352
$355$	0	0
$356$	−9.49995	−0.503496
$357$	−2.34099	−0.123898
$358$	32.3731	1.71097
$359$	18.4472	0.973605	0.486803	−	0.873512i	$-0.338163\pi$
$359$	18.4472	0.973605	0.486803	+	0.873512i	$0.338163\pi$
$360$	0	0
$361$	39.5910	2.08374
$362$	3.83651	0.201642
$363$	−10.7448	−0.563955
$364$	3.21665	0.168598
$365$	0	0
$366$	27.8487	1.45568
$367$	−13.6733	−0.713740	−0.356870	−	0.934154i	$-0.616156\pi$
$367$	−13.6733	−0.713740	−0.356870	+	0.934154i	$0.616156\pi$
$368$	0.335164	0.0174716
$369$	2.96598	0.154403
$370$	0	0
$371$	1.40380	0.0728819
$372$	−18.5884	−0.963764
$373$	−32.0048	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$373$	−32.0048	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$374$	2.70111	0.139671
$375$	0	0
$376$	−36.6483	−1.88999
$377$	4.51977	0.232780
$378$	−2.28400	−0.117476
$379$	−31.5431	−1.62026	−0.810129	−	0.586251i	$-0.800603\pi$
$379$	−31.5431	−1.62026	−0.810129	+	0.586251i	$0.800603\pi$
$380$	0	0
$381$	5.34003	0.273578
$382$	5.24703	0.268461
$383$	−6.62787	−0.338668	−0.169334	−	0.985559i	$-0.554162\pi$
$383$	−6.62787	−0.338668	−0.169334	+	0.985559i	$0.554162\pi$
$384$	18.1173	0.924542
$385$	0	0
$386$	−2.83740	−0.144420
$387$	0.538507	0.0273739
$388$	58.8360	2.98695
$389$	4.69847	0.238222	0.119111	−	0.992881i	$-0.461996\pi$
$389$	4.69847	0.238222	0.119111	+	0.992881i	$0.461996\pi$
$390$	0	0
$391$	9.07287	0.458835
$392$	−2.77882	−0.140351
$393$	−19.3294	−0.975038
$394$	45.1133	2.27278
$395$	0	0
$396$	1.62499	0.0816586
$397$	−5.60944	−0.281530	−0.140765	−	0.990043i	$-0.544956\pi$
$397$	−5.60944	−0.281530	−0.140765	+	0.990043i	$0.544956\pi$
$398$	51.8477	2.59889
$399$	−7.65448	−0.383203
$400$	0	0
$401$	2.22989	0.111355	0.0556777	−	0.998449i	$-0.482268\pi$
$401$	2.22989	0.111355	0.0556777	+	0.998449i	$0.482268\pi$
$402$	−3.17749	−0.158479
$403$	−5.77882	−0.287863
$404$	15.0051	0.746533
$405$	0	0
$406$	−10.3231	−0.512329
$407$	3.06052	0.151705
$408$	6.50518	0.322054
$409$	1.94460	0.0961544	0.0480772	−	0.998844i	$-0.484691\pi$
$409$	1.94460	0.0961544	0.0480772	+	0.998844i	$0.484691\pi$
$410$	0	0
$411$	−11.2412	−0.554488
$412$	13.1571	0.648204
$413$	3.66867	0.180524
$414$	8.85199	0.435052
$415$	0	0
$416$	5.75515	0.282170
$417$	−7.42747	−0.363725
$418$	8.83197	0.431986
$419$	−0.390867	−0.0190951	−0.00954755	−	0.999954i	$-0.503039\pi$
$419$	−0.390867	−0.0190951	−0.00954755	+	0.999954i	$0.503039\pi$
$420$	0	0
$421$	−4.28439	−0.208808	−0.104404	−	0.994535i	$-0.533294\pi$
$421$	−4.28439	−0.208808	−0.104404	+	0.994535i	$0.533294\pi$
$422$	2.17749	0.105999
$423$	13.1884	0.641244
$424$	−3.90092	−0.189445
$425$	0	0
$426$	−2.89932	−0.140473
$427$	−12.1930	−0.590060
$428$	−11.2749	−0.544994
$429$	0.505180	0.0243903
$430$	0	0
$431$	−13.1147	−0.631712	−0.315856	−	0.948807i	$-0.602292\pi$
$431$	−13.1147	−0.631712	−0.315856	+	0.948807i	$0.602292\pi$
$432$	−0.0864793	−0.00416074
$433$	23.7863	1.14310	0.571549	−	0.820568i	$-0.306343\pi$
$433$	23.7863	1.14310	0.571549	+	0.820568i	$0.306343\pi$
$434$	13.1988	0.633563
$435$	0	0
$436$	−42.1320	−2.01776
$437$	29.6661	1.41912
$438$	29.8812	1.42778
$439$	−13.7901	−0.658164	−0.329082	−	0.944301i	$-0.606739\pi$
$439$	−13.7901	−0.658164	−0.329082	+	0.944301i	$0.606739\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	5.34681	0.254322
$443$	−40.4705	−1.92281	−0.961405	−	0.275136i	$-0.911277\pi$
$443$	−40.4705	−1.92281	−0.961405	+	0.275136i	$0.911277\pi$
$444$	19.4873	0.924828
$445$	0	0
$446$	−26.8035	−1.26918
$447$	13.0166	0.615663
$448$	−12.9718	−0.612860
$449$	−12.7597	−0.602167	−0.301083	−	0.953598i	$-0.597348\pi$
$449$	−12.7597	−0.602167	−0.301083	+	0.953598i	$0.597348\pi$
$450$	0	0
$451$	1.49835	0.0705547
$452$	−13.7532	−0.646895
$453$	−1.42324	−0.0668695
$454$	−5.69816	−0.267428
$455$	0	0
$456$	21.2704	0.996077
$457$	−11.9764	−0.560233	−0.280117	−	0.959966i	$-0.590373\pi$
$457$	−11.9764	−0.560233	−0.280117	+	0.959966i	$0.590373\pi$
$458$	6.54708	0.305925
$459$	−2.34099	−0.109268
$460$	0	0
$461$	19.4009	0.903592	0.451796	−	0.892121i	$-0.350783\pi$
$461$	19.4009	0.903592	0.451796	+	0.892121i	$0.350783\pi$
$462$	−1.15383	−0.0536811
$463$	−5.24435	−0.243725	−0.121863	−	0.992547i	$-0.538887\pi$
$463$	−5.24435	−0.243725	−0.121863	+	0.992547i	$0.538887\pi$
$464$	−0.390867	−0.0181455
$465$	0	0
$466$	−5.54593	−0.256910
$467$	−32.4293	−1.50065	−0.750326	−	0.661068i	$-0.770103\pi$
$467$	−32.4293	−1.50065	−0.750326	+	0.661068i	$0.770103\pi$
$468$	3.21665	0.148690
$469$	1.39120	0.0642396
$470$	0	0
$471$	7.73731	0.356517
$472$	−10.1946	−0.469243
$473$	0.272043	0.0125086
$474$	22.9531	1.05427
$475$	0	0
$476$	−7.53013	−0.345143
$477$	1.40380	0.0642758
$478$	54.3905	2.48776
$479$	−12.1288	−0.554178	−0.277089	−	0.960844i	$-0.589370\pi$
$479$	−12.1288	−0.554178	−0.277089	+	0.960844i	$0.589370\pi$
$480$	0	0
$481$	6.05828	0.276234
$482$	−43.8730	−1.99836
$483$	−3.87566	−0.176349
$484$	−34.5622	−1.57101
$485$	0	0
$486$	−2.28400	−0.103604
$487$	18.6722	0.846118	0.423059	−	0.906102i	$-0.360956\pi$
$487$	18.6722	0.846118	0.423059	+	0.906102i	$0.360956\pi$
$488$	33.8821	1.53377
$489$	−17.6474	−0.798043
$490$	0	0
$491$	19.4340	0.877044	0.438522	−	0.898721i	$-0.355502\pi$
$491$	19.4340	0.877044	0.438522	+	0.898721i	$0.355502\pi$
$492$	9.54049	0.430119
$493$	−10.5807	−0.476532
$494$	17.4828	0.786589
$495$	0	0
$496$	0.499748	0.0224394
$497$	1.26941	0.0569407
$498$	27.5735	1.23560
$499$	39.4746	1.76713	0.883564	−	0.468311i	$-0.155137\pi$
$499$	39.4746	1.76713	0.883564	+	0.468311i	$0.155137\pi$
$500$	0	0
$501$	8.46662	0.378261
$502$	−62.1383	−2.77337
$503$	40.0884	1.78745	0.893726	−	0.448614i	$-0.148082\pi$
$503$	40.0884	1.78745	0.893726	+	0.448614i	$0.148082\pi$
$504$	−2.77882	−0.123778
$505$	0	0
$506$	4.47185	0.198798
$507$	1.00000	0.0444116
$508$	17.1770	0.762106
$509$	−35.5604	−1.57619	−0.788094	−	0.615555i	$-0.788932\pi$
$509$	−35.5604	−1.57619	−0.788094	+	0.615555i	$0.788932\pi$
$510$	0	0
$511$	−13.0828	−0.578751
$512$	−0.978322	−0.0432361
$513$	−7.65448	−0.337953
$514$	64.5637	2.84778
$515$	0	0
$516$	1.73219	0.0762553
$517$	6.66254	0.293018
$518$	−13.8371	−0.607967
$519$	18.2008	0.798926
$520$	0	0
$521$	28.9877	1.26997	0.634987	−	0.772523i	$-0.281006\pi$
$521$	28.9877	1.26997	0.634987	+	0.772523i	$0.281006\pi$
$522$	−10.3231	−0.451832
$523$	0.990729	0.0433216	0.0216608	−	0.999765i	$-0.493105\pi$
$523$	0.990729	0.0433216	0.0216608	+	0.999765i	$0.493105\pi$
$524$	−62.1757	−2.71616
$525$	0	0
$526$	54.5275	2.37751
$527$	13.5281	0.589295
$528$	−0.0436877	−0.00190126
$529$	−7.97928	−0.346925
$530$	0	0
$531$	3.66867	0.159207
$532$	−24.6217	−1.06749
$533$	2.96598	0.128471
$534$	6.74549	0.291906
$535$	0	0
$536$	−3.86589	−0.166981
$537$	−14.1739	−0.611647
$538$	−64.5222	−2.78175
$539$	0.505180	0.0217597
$540$	0	0
$541$	−17.2130	−0.740045	−0.370022	−	0.929023i	$-0.620650\pi$
$541$	−17.2130	−0.740045	−0.370022	+	0.929023i	$0.620650\pi$
$542$	15.5589	0.668313
$543$	−1.67973	−0.0720842
$544$	−13.4727	−0.577639
$545$	0	0
$546$	−2.28400	−0.0977461
$547$	0.100286	0.00428793	0.00214396	−	0.999998i	$-0.499318\pi$
$547$	0.100286	0.00428793	0.00214396	+	0.999998i	$0.499318\pi$
$548$	−36.1590	−1.54464
$549$	−12.1930	−0.520384
$550$	0	0
$551$	−34.5965	−1.47386
$552$	10.7697	0.458391
$553$	−10.0495	−0.427349
$554$	38.8841	1.65203
$555$	0	0
$556$	−23.8915	−1.01323
$557$	31.0819	1.31698	0.658492	−	0.752588i	$-0.271195\pi$
$557$	31.0819	1.31698	0.658492	+	0.752588i	$0.271195\pi$
$558$	13.1988	0.558750
$559$	0.538507	0.0227764
$560$	0	0
$561$	−1.18262	−0.0499303
$562$	6.34009	0.267441
$563$	10.6376	0.448321	0.224161	−	0.974552i	$-0.428036\pi$
$563$	10.6376	0.448321	0.224161	+	0.974552i	$0.428036\pi$
$564$	42.4226	1.78631
$565$	0	0
$566$	44.7753	1.88204
$567$	1.00000	0.0419961
$568$	−3.52745	−0.148008
$569$	−5.17206	−0.216824	−0.108412	−	0.994106i	$-0.534577\pi$
$569$	−5.17206	−0.216824	−0.108412	+	0.994106i	$0.534577\pi$
$570$	0	0
$571$	−0.944663	−0.0395329	−0.0197665	−	0.999805i	$-0.506292\pi$
$571$	−0.944663	−0.0395329	−0.0197665	+	0.999805i	$0.506292\pi$
$572$	1.62499	0.0679441
$573$	−2.29730	−0.0959711
$574$	−6.77428	−0.282753
$575$	0	0
$576$	−12.9718	−0.540492
$577$	12.5781	0.523631	0.261816	−	0.965118i	$-0.415679\pi$
$577$	12.5781	0.523631	0.261816	+	0.965118i	$0.415679\pi$
$578$	26.3111	1.09440
$579$	1.24230	0.0516281
$580$	0	0
$581$	−12.0725	−0.500851
$582$	−41.7769	−1.73171
$583$	0.709174	0.0293710
$584$	36.3548	1.50437
$585$	0	0
$586$	−67.3529	−2.78232
$587$	−6.21538	−0.256536	−0.128268	−	0.991740i	$-0.540942\pi$
$587$	−6.21538	−0.256536	−0.128268	+	0.991740i	$0.540942\pi$
$588$	3.21665	0.132652
$589$	44.2338	1.82262
$590$	0	0
$591$	−19.7519	−0.812485
$592$	−0.523916	−0.0215328
$593$	−6.57836	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$593$	−6.57836	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$594$	−1.15383	−0.0473423
$595$	0	0
$596$	41.8697	1.71505
$597$	−22.7004	−0.929066
$598$	8.85199	0.361985
$599$	30.6031	1.25041	0.625205	−	0.780461i	$-0.285015\pi$
$599$	30.6031	1.25041	0.625205	+	0.780461i	$0.285015\pi$
$600$	0	0
$601$	−41.1945	−1.68036	−0.840179	−	0.542310i	$-0.817550\pi$
$601$	−41.1945	−1.68036	−0.840179	+	0.542310i	$0.817550\pi$
$602$	−1.22995	−0.0501290
$603$	1.39120	0.0566540
$604$	−4.57805	−0.186278
$605$	0	0
$606$	−10.6545	−0.432808
$607$	24.2843	0.985669	0.492834	−	0.870123i	$-0.335961\pi$
$607$	24.2843	0.985669	0.492834	+	0.870123i	$0.335961\pi$
$608$	−44.0527	−1.78657
$609$	4.51977	0.183150
$610$	0	0
$611$	13.1884	0.533547
$612$	−7.53013	−0.304388
$613$	−36.8116	−1.48681	−0.743403	−	0.668844i	$-0.766789\pi$
$613$	−36.8116	−1.48681	−0.743403	+	0.668844i	$0.766789\pi$
$614$	−28.2483	−1.14001
$615$	0	0
$616$	−1.40380	−0.0565609
$617$	−23.3177	−0.938734	−0.469367	−	0.883003i	$-0.655518\pi$
$617$	−23.3177	−0.938734	−0.469367	+	0.883003i	$0.655518\pi$
$618$	−9.34228	−0.375801
$619$	19.0336	0.765026	0.382513	−	0.923950i	$-0.375059\pi$
$619$	19.0336	0.765026	0.382513	+	0.923950i	$0.375059\pi$
$620$	0	0
$621$	−3.87566	−0.155525
$622$	64.5783	2.58935
$623$	−2.95337	−0.118324
$624$	−0.0864793	−0.00346194
$625$	0	0
$626$	64.1168	2.56262
$627$	−3.86689	−0.154429
$628$	24.8882	0.993147
$629$	−14.1824	−0.565488
$630$	0	0
$631$	15.5891	0.620593	0.310296	−	0.950640i	$-0.399572\pi$
$631$	15.5891	0.620593	0.310296	+	0.950640i	$0.399572\pi$
$632$	27.9258	1.11083
$633$	−0.953370	−0.0378931
$634$	−47.5194	−1.88724
$635$	0	0
$636$	4.51554	0.179053
$637$	1.00000	0.0396214
$638$	−5.21505	−0.206466
$639$	1.26941	0.0502169
$640$	0	0
$641$	−11.9386	−0.471545	−0.235772	−	0.971808i	$-0.575762\pi$
$641$	−11.9386	−0.471545	−0.235772	+	0.971808i	$0.575762\pi$
$642$	8.00582	0.315965
$643$	23.5663	0.929366	0.464683	−	0.885477i	$-0.346168\pi$
$643$	23.5663	0.929366	0.464683	+	0.885477i	$0.346168\pi$
$644$	−12.4666	−0.491254
$645$	0	0
$646$	−40.9270	−1.61025
$647$	−22.5213	−0.885402	−0.442701	−	0.896669i	$-0.645980\pi$
$647$	−22.5213	−0.885402	−0.442701	+	0.896669i	$0.645980\pi$
$648$	−2.77882	−0.109162
$649$	1.85334	0.0727500
$650$	0	0
$651$	−5.77882	−0.226490
$652$	−56.7654	−2.22310
$653$	−19.2248	−0.752326	−0.376163	−	0.926553i	$-0.622757\pi$
$653$	−19.2248	−0.752326	−0.376163	+	0.926553i	$0.622757\pi$
$654$	29.9161	1.16981
$655$	0	0
$656$	−0.256495	−0.0100145
$657$	−13.0828	−0.510410
$658$	−30.1224	−1.17429
$659$	−15.4009	−0.599932	−0.299966	−	0.953950i	$-0.596975\pi$
$659$	−15.4009	−0.599932	−0.299966	+	0.953950i	$0.596975\pi$
$660$	0	0
$661$	−25.0836	−0.975640	−0.487820	−	0.872944i	$-0.662208\pi$
$661$	−25.0836	−0.975640	−0.487820	+	0.872944i	$0.662208\pi$
$662$	32.9701	1.28142
$663$	−2.34099	−0.0909165
$664$	33.5472	1.30188
$665$	0	0
$666$	−13.8371	−0.536177
$667$	−17.5171	−0.678264
$668$	27.2341	1.05372
$669$	11.7353	0.453714
$670$	0	0
$671$	−6.15966	−0.237791
$672$	5.75515	0.222010
$673$	−25.2799	−0.974467	−0.487233	−	0.873272i	$-0.661994\pi$
$673$	−25.2799	−0.974467	−0.487233	+	0.873272i	$0.661994\pi$
$674$	−33.2488	−1.28069
$675$	0	0
$676$	3.21665	0.123717
$677$	22.5651	0.867248	0.433624	−	0.901094i	$-0.357235\pi$
$677$	22.5651	0.867248	0.433624	+	0.901094i	$0.357235\pi$
$678$	9.76551	0.375042
$679$	18.2911	0.701948
$680$	0	0
$681$	2.49482	0.0956017
$682$	6.66778	0.255322
$683$	−24.5440	−0.939151	−0.469576	−	0.882892i	$-0.655593\pi$
$683$	−24.5440	−0.939151	−0.469576	+	0.882892i	$0.655593\pi$
$684$	−24.6217	−0.941436
$685$	0	0
$686$	−2.28400	−0.0872034
$687$	−2.86650	−0.109364
$688$	−0.0465697	−0.00177545
$689$	1.40380	0.0534807
$690$	0	0
$691$	20.7329	0.788715	0.394358	−	0.918957i	$-0.370967\pi$
$691$	20.7329	0.788715	0.394358	+	0.918957i	$0.370967\pi$
$692$	58.5455	2.22557
$693$	0.505180	0.0191902
$694$	11.5635	0.438943
$695$	0	0
$696$	−12.5596	−0.476071
$697$	−6.94331	−0.262997
$698$	−26.0699	−0.986761
$699$	2.42817	0.0918417
$700$	0	0
$701$	−29.4570	−1.11257	−0.556287	−	0.830990i	$-0.687775\pi$
$701$	−29.4570	−1.11257	−0.556287	+	0.830990i	$0.687775\pi$
$702$	−2.28400	−0.0862039
$703$	−46.3729	−1.74899
$704$	−6.55310	−0.246979
$705$	0	0
$706$	−16.1455	−0.607642
$707$	4.66484	0.175439
$708$	11.8008	0.443502
$709$	40.7608	1.53080	0.765402	−	0.643552i	$-0.222540\pi$
$709$	40.7608	1.53080	0.765402	+	0.643552i	$0.222540\pi$
$710$	0	0
$711$	−10.0495	−0.376886
$712$	8.20688	0.307566
$713$	22.3967	0.838764
$714$	5.34681	0.200099
$715$	0	0
$716$	−45.5923	−1.70386
$717$	−23.8137	−0.889340
$718$	−42.1333	−1.57240
$719$	34.7788	1.29703	0.648516	−	0.761201i	$-0.275390\pi$
$719$	34.7788	1.29703	0.648516	+	0.761201i	$0.275390\pi$
$720$	0	0
$721$	4.09032	0.152331
$722$	−90.4257	−3.36530
$723$	19.2089	0.714385
$724$	−5.40311	−0.200805
$725$	0	0
$726$	24.5411	0.910805
$727$	1.71665	0.0636671	0.0318336	−	0.999493i	$-0.489865\pi$
$727$	1.71665	0.0636671	0.0318336	+	0.999493i	$0.489865\pi$
$728$	−2.77882	−0.102990
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.26064	−0.0466264
$732$	−39.2205	−1.44963
$733$	−25.4622	−0.940468	−0.470234	−	0.882542i	$-0.655831\pi$
$733$	−25.4622	−0.940468	−0.470234	+	0.882542i	$0.655831\pi$
$734$	31.2298	1.15271
$735$	0	0
$736$	−22.3050	−0.822173
$737$	0.702806	0.0258882
$738$	−6.77428	−0.249365
$739$	41.9993	1.54497	0.772485	−	0.635033i	$-0.219013\pi$
$739$	41.9993	1.54497	0.772485	+	0.635033i	$0.219013\pi$
$740$	0	0
$741$	−7.65448	−0.281194
$742$	−3.20629	−0.117706
$743$	−29.3212	−1.07569	−0.537846	−	0.843043i	$-0.680762\pi$
$743$	−29.3212	−1.07569	−0.537846	+	0.843043i	$0.680762\pi$
$744$	16.0583	0.588725
$745$	0	0
$746$	73.0988	2.67634
$747$	−12.0725	−0.441709
$748$	−3.80408	−0.139091
$749$	−3.50518	−0.128077
$750$	0	0
$751$	−7.31220	−0.266826	−0.133413	−	0.991061i	$-0.542594\pi$
$751$	−7.31220	−0.266826	−0.133413	+	0.991061i	$0.542594\pi$
$752$	−1.14053	−0.0415908
$753$	27.2059	0.991439
$754$	−10.3231	−0.375947
$755$	0	0
$756$	3.21665	0.116988
$757$	−45.6236	−1.65822	−0.829109	−	0.559087i	$-0.811152\pi$
$757$	−45.6236	−1.65822	−0.829109	+	0.559087i	$0.811152\pi$
$758$	72.0443	2.61677
$759$	−1.95791	−0.0710675
$760$	0	0
$761$	12.4921	0.452837	0.226419	−	0.974030i	$-0.427298\pi$
$761$	12.4921	0.452837	0.226419	+	0.974030i	$0.427298\pi$
$762$	−12.1966	−0.441837
$763$	−13.0981	−0.474184
$764$	−7.38960	−0.267347
$765$	0	0
$766$	15.1380	0.546960
$767$	3.66867	0.132468
$768$	−15.4362	−0.557005
$769$	−16.7836	−0.605233	−0.302617	−	0.953112i	$-0.597860\pi$
$769$	−16.7836	−0.605233	−0.302617	+	0.953112i	$0.597860\pi$
$770$	0	0
$771$	−28.2678	−1.01804
$772$	3.99603	0.143820
$773$	−7.39761	−0.266074	−0.133037	−	0.991111i	$-0.542473\pi$
$773$	−7.39761	−0.266074	−0.133037	+	0.991111i	$0.542473\pi$
$774$	−1.22995	−0.0442096
$775$	0	0
$776$	−50.8277	−1.82461
$777$	6.05828	0.217340
$778$	−10.7313	−0.384735
$779$	−22.7030	−0.813419
$780$	0	0
$781$	0.641279	0.0229468
$782$	−20.7224	−0.741032
$783$	4.51977	0.161523
$784$	−0.0864793	−0.00308855
$785$	0	0
$786$	44.1482	1.57472
$787$	35.1733	1.25379	0.626896	−	0.779103i	$-0.284325\pi$
$787$	35.1733	1.25379	0.626896	+	0.779103i	$0.284325\pi$
$788$	−63.5349	−2.26334
$789$	−23.8737	−0.849926
$790$	0	0
$791$	−4.27562	−0.152024
$792$	−1.40380	−0.0498820
$793$	−12.1930	−0.432985
$794$	12.8119	0.454679
$795$	0	0
$796$	−73.0192	−2.58810
$797$	−31.6333	−1.12051	−0.560255	−	0.828321i	$-0.689297\pi$
$797$	−31.6333	−1.12051	−0.560255	+	0.828321i	$0.689297\pi$
$798$	17.4828	0.618885
$799$	−30.8740	−1.09224
$800$	0	0
$801$	−2.95337	−0.104352
$802$	−5.09306	−0.179842
$803$	−6.60919	−0.233233
$804$	4.47499	0.157821
$805$	0	0
$806$	13.1988	0.464908
$807$	28.2497	0.994435
$808$	−12.9627	−0.456027
$809$	−28.0660	−0.986746	−0.493373	−	0.869818i	$-0.664236\pi$
$809$	−28.0660	−0.986746	−0.493373	+	0.869818i	$0.664236\pi$
$810$	0	0
$811$	33.5290	1.17736	0.588681	−	0.808366i	$-0.299647\pi$
$811$	33.5290	1.17736	0.588681	+	0.808366i	$0.299647\pi$
$812$	14.5385	0.510202
$813$	−6.81214	−0.238912
$814$	−6.99023	−0.245007
$815$	0	0
$816$	0.202447	0.00708707
$817$	−4.12199	−0.144210
$818$	−4.44147	−0.155292
$819$	1.00000	0.0349428
$820$	0	0
$821$	−45.0951	−1.57383	−0.786916	−	0.617061i	$-0.788323\pi$
$821$	−45.0951	−1.57383	−0.786916	+	0.617061i	$0.788323\pi$
$822$	25.6749	0.895515
$823$	−33.1083	−1.15408	−0.577041	−	0.816715i	$-0.695793\pi$
$823$	−33.1083	−1.15408	−0.577041	+	0.816715i	$0.695793\pi$
$824$	−11.3662	−0.395962
$825$	0	0
$826$	−8.37924	−0.291551
$827$	−42.0818	−1.46333	−0.731663	−	0.681666i	$-0.761256\pi$
$827$	−42.0818	−1.46333	−0.731663	+	0.681666i	$0.761256\pi$
$828$	−12.4666	−0.433245
$829$	9.11124	0.316446	0.158223	−	0.987403i	$-0.449423\pi$
$829$	9.11124	0.316446	0.158223	+	0.987403i	$0.449423\pi$
$830$	0	0
$831$	−17.0246	−0.590575
$832$	−12.9718	−0.449716
$833$	−2.34099	−0.0811105
$834$	16.9643	0.587426
$835$	0	0
$836$	−12.4384	−0.430192
$837$	−5.77882	−0.199745
$838$	0.892739	0.0308392
$839$	−27.9339	−0.964387	−0.482193	−	0.876065i	$-0.660160\pi$
$839$	−27.9339	−0.964387	−0.482193	+	0.876065i	$0.660160\pi$
$840$	0	0
$841$	−8.57166	−0.295575
$842$	9.78554	0.337232
$843$	−2.77588	−0.0956062
$844$	−3.06665	−0.105559
$845$	0	0
$846$	−30.1224	−1.03563
$847$	−10.7448	−0.369195
$848$	−0.121400	−0.00416889
$849$	−19.6039	−0.672804
$850$	0	0
$851$	−23.4798	−0.804878
$852$	4.08323	0.139889
$853$	−46.2159	−1.58240	−0.791201	−	0.611556i	$-0.790544\pi$
$853$	−46.2159	−1.58240	−0.791201	+	0.611556i	$0.790544\pi$
$854$	27.8487	0.952964
$855$	0	0
$856$	9.74026	0.332915
$857$	−8.88347	−0.303453	−0.151727	−	0.988422i	$-0.548483\pi$
$857$	−8.88347	−0.303453	−0.151727	+	0.988422i	$0.548483\pi$
$858$	−1.15383	−0.0393911
$859$	−6.38834	−0.217967	−0.108984	−	0.994044i	$-0.534760\pi$
$859$	−6.38834	−0.217967	−0.108984	+	0.994044i	$0.534760\pi$
$860$	0	0
$861$	2.96598	0.101080
$862$	29.9539	1.02023
$863$	15.3238	0.521630	0.260815	−	0.965389i	$-0.416009\pi$
$863$	15.3238	0.521630	0.260815	+	0.965389i	$0.416009\pi$
$864$	5.75515	0.195794
$865$	0	0
$866$	−54.3279	−1.84614
$867$	−11.5198	−0.391232
$868$	−18.5884	−0.630932
$869$	−5.07682	−0.172219
$870$	0	0
$871$	1.39120	0.0471390
$872$	36.3973	1.23257
$873$	18.2911	0.619060
$874$	−67.7574	−2.29193
$875$	0	0
$876$	−42.0829	−1.42185
$877$	25.5592	0.863074	0.431537	−	0.902095i	$-0.357971\pi$
$877$	25.5592	0.863074	0.431537	+	0.902095i	$0.357971\pi$
$878$	31.4965	1.06296
$879$	29.4890	0.994640
$880$	0	0
$881$	45.6787	1.53895	0.769477	−	0.638675i	$-0.220517\pi$
$881$	45.6787	1.53895	0.769477	+	0.638675i	$0.220517\pi$
$882$	−2.28400	−0.0769062
$883$	−10.2232	−0.344037	−0.172019	−	0.985094i	$-0.555029\pi$
$883$	−10.2232	−0.344037	−0.172019	+	0.985094i	$0.555029\pi$
$884$	−7.53013	−0.253266
$885$	0	0
$886$	92.4345	3.10540
$887$	2.91005	0.0977099	0.0488549	−	0.998806i	$-0.484443\pi$
$887$	2.91005	0.0977099	0.0488549	+	0.998806i	$0.484443\pi$
$888$	−16.8349	−0.564941
$889$	5.34003	0.179099
$890$	0	0
$891$	0.505180	0.0169242
$892$	37.7484	1.26391
$893$	−100.951	−3.37818
$894$	−29.7298	−0.994314
$895$	0	0
$896$	18.1173	0.605255
$897$	−3.87566	−0.129404
$898$	29.1431	0.972518
$899$	−26.1189	−0.871115
$900$	0	0
$901$	−3.28629	−0.109482
$902$	−3.42223	−0.113948
$903$	0.538507	0.0179204
$904$	11.8812	0.395162
$905$	0	0
$906$	3.25067	0.107996
$907$	2.08255	0.0691501	0.0345750	−	0.999402i	$-0.488992\pi$
$907$	2.08255	0.0691501	0.0345750	+	0.999402i	$0.488992\pi$
$908$	8.02495	0.266317
$909$	4.66484	0.154723
$910$	0	0
$911$	4.85463	0.160841	0.0804205	−	0.996761i	$-0.474374\pi$
$911$	4.85463	0.160841	0.0804205	+	0.996761i	$0.474374\pi$
$912$	0.661954	0.0219195
$913$	−6.09878	−0.201840
$914$	27.3541	0.904794
$915$	0	0
$916$	−9.22051	−0.304654
$917$	−19.3294	−0.638312
$918$	5.34681	0.176471
$919$	32.2327	1.06326	0.531630	−	0.846977i	$-0.321580\pi$
$919$	32.2327	1.06326	0.531630	+	0.846977i	$0.321580\pi$
$920$	0	0
$921$	12.3679	0.407537
$922$	−44.3117	−1.45933
$923$	1.26941	0.0417830
$924$	1.62499	0.0534581
$925$	0	0
$926$	11.9781	0.393624
$927$	4.09032	0.134344
$928$	26.0120	0.853885
$929$	5.83223	0.191349	0.0956746	−	0.995413i	$-0.469499\pi$
$929$	5.83223	0.191349	0.0956746	+	0.995413i	$0.469499\pi$
$930$	0	0
$931$	−7.65448	−0.250865
$932$	7.81055	0.255843
$933$	−28.2742	−0.925657
$934$	74.0686	2.42360
$935$	0	0
$936$	−2.77882	−0.0908285
$937$	1.98460	0.0648340	0.0324170	−	0.999474i	$-0.489680\pi$
$937$	1.98460	0.0648340	0.0324170	+	0.999474i	$0.489680\pi$
$938$	−3.17749	−0.103749
$939$	−28.0722	−0.916101
$940$	0	0
$941$	51.6855	1.68490	0.842450	−	0.538774i	$-0.181112\pi$
$941$	51.6855	1.68490	0.842450	+	0.538774i	$0.181112\pi$
$942$	−17.6720	−0.575785
$943$	−11.4951	−0.374332
$944$	−0.317264	−0.0103261
$945$	0	0
$946$	−0.621346	−0.0202017
$947$	−9.11477	−0.296190	−0.148095	−	0.988973i	$-0.547314\pi$
$947$	−9.11477	−0.296190	−0.148095	+	0.988973i	$0.547314\pi$
$948$	−32.3257	−1.04989
$949$	−13.0828	−0.424687
$950$	0	0
$951$	20.8054	0.674660
$952$	6.50518	0.210834
$953$	10.5003	0.340137	0.170069	−	0.985432i	$-0.445601\pi$
$953$	10.5003	0.340137	0.170069	+	0.985432i	$0.445601\pi$
$954$	−3.20629	−0.103807
$955$	0	0
$956$	−76.6004	−2.47743
$957$	2.28330	0.0738086
$958$	27.7021	0.895014
$959$	−11.2412	−0.362997
$960$	0	0
$961$	2.39473	0.0772494
$962$	−13.8371	−0.446126
$963$	−3.50518	−0.112953
$964$	61.7881	1.99006
$965$	0	0
$966$	8.85199	0.284808
$967$	22.8979	0.736348	0.368174	−	0.929757i	$-0.379983\pi$
$967$	22.8979	0.736348	0.368174	+	0.929757i	$0.379983\pi$
$968$	29.8578	0.959667
$969$	17.9190	0.575642
$970$	0	0
$971$	−2.98022	−0.0956399	−0.0478199	−	0.998856i	$-0.515227\pi$
$971$	−2.98022	−0.0956399	−0.0478199	+	0.998856i	$0.515227\pi$
$972$	3.21665	0.103174
$973$	−7.42747	−0.238114
$974$	−42.6473	−1.36651
$975$	0	0
$976$	1.05444	0.0337518
$977$	50.6224	1.61955	0.809777	−	0.586738i	$-0.199588\pi$
$977$	50.6224	1.61955	0.809777	+	0.586738i	$0.199588\pi$
$978$	40.3066	1.28886
$979$	−1.49198	−0.0476840
$980$	0	0
$981$	−13.0981	−0.418191
$982$	−44.3872	−1.41645
$983$	8.51580	0.271612	0.135806	−	0.990735i	$-0.456638\pi$
$983$	8.51580	0.271612	0.135806	+	0.990735i	$0.456638\pi$
$984$	−8.24190	−0.262742
$985$	0	0
$986$	24.1664	0.769614
$987$	13.1884	0.419793
$988$	−24.6217	−0.783322
$989$	−2.08707	−0.0663650
$990$	0	0
$991$	−16.0775	−0.510720	−0.255360	−	0.966846i	$-0.582194\pi$
$991$	−16.0775	−0.510720	−0.255360	+	0.966846i	$0.582194\pi$
$992$	−33.2580	−1.05594
$993$	−14.4353	−0.458090
$994$	−2.89932	−0.0919609
$995$	0	0
$996$	−38.8329	−1.23047
$997$	−32.5794	−1.03180	−0.515901	−	0.856648i	$-0.672543\pi$
$997$	−32.5794	−1.03180	−0.515901	+	0.856648i	$0.672543\pi$
$998$	−90.1600	−2.85396
$999$	6.05828	0.191675

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6825.2.a.bi.1.1	✓	4	1.1	even	1	trivial
6825.2.a.bj.1.4	yes	4	5.4	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:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28400 q^{2} +1.00000 q^{3} +3.21665 q^{4} -2.28400 q^{6} +1.00000 q^{7} -2.77882 q^{8} +1.00000 q^{9} +0.505180 q^{11} +3.21665 q^{12} +1.00000 q^{13} -2.28400 q^{14} -0.0864793 q^{16} -2.34099 q^{17} -2.28400 q^{18} -7.65448 q^{19} +1.00000 q^{21} -1.15383 q^{22} -3.87566 q^{23} -2.77882 q^{24} -2.28400 q^{26} +1.00000 q^{27} +3.21665 q^{28} +4.51977 q^{29} -5.77882 q^{31} +5.75515 q^{32} +0.505180 q^{33} +5.34681 q^{34} +3.21665 q^{36} +6.05828 q^{37} +17.4828 q^{38} +1.00000 q^{39} +2.96598 q^{41} -2.28400 q^{42} +0.538507 q^{43} +1.62499 q^{44} +8.85199 q^{46} +13.1884 q^{47} -0.0864793 q^{48} +1.00000 q^{49} -2.34099 q^{51} +3.21665 q^{52} +1.40380 q^{53} -2.28400 q^{54} -2.77882 q^{56} -7.65448 q^{57} -10.3231 q^{58} +3.66867 q^{59} -12.1930 q^{61} +13.1988 q^{62} +1.00000 q^{63} -12.9718 q^{64} -1.15383 q^{66} +1.39120 q^{67} -7.53013 q^{68} -3.87566 q^{69} +1.26941 q^{71} -2.77882 q^{72} -13.0828 q^{73} -13.8371 q^{74} -24.6217 q^{76} +0.505180 q^{77} -2.28400 q^{78} -10.0495 q^{79} +1.00000 q^{81} -6.77428 q^{82} -12.0725 q^{83} +3.21665 q^{84} -1.22995 q^{86} +4.51977 q^{87} -1.40380 q^{88} -2.95337 q^{89} +1.00000 q^{91} -12.4666 q^{92} -5.77882 q^{93} -30.1224 q^{94} +5.75515 q^{96} +18.2911 q^{97} -2.28400 q^{98} +0.505180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 4 q^{3} + q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{16} - 5 q^{17} - q^{18} - 15 q^{19} + 4 q^{21} - 9 q^{22} - 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 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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Representations

Groups

Database

Properties

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