LMFDB - Embedded newform 6825.2.a.bg.1.4

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:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.787711$ 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)

$f(q)$	$=$	\(q+2.37951 q^{2} -1.00000 q^{3} +3.66208 q^{4} -2.37951 q^{6} -1.00000 q^{7} +3.95493 q^{8} +1.00000 q^{9} +1.57542 q^{11} -3.66208 q^{12} -1.00000 q^{13} -2.37951 q^{14} +2.08666 q^{16} -4.75902 q^{17} +2.37951 q^{18} -2.23750 q^{19} +1.00000 q^{21} +3.74873 q^{22} -5.84568 q^{23} -3.95493 q^{24} -2.37951 q^{26} -1.00000 q^{27} -3.66208 q^{28} +4.23750 q^{29} +7.28055 q^{31} -2.94464 q^{32} -1.57542 q^{33} -11.3242 q^{34} +3.66208 q^{36} -10.4750 q^{37} -5.32415 q^{38} +1.00000 q^{39} -2.25127 q^{41} +2.37951 q^{42} -0.913344 q^{43} +5.76931 q^{44} -13.9099 q^{46} +2.09695 q^{47} -2.08666 q^{48} +1.00000 q^{49} +4.75902 q^{51} -3.66208 q^{52} -1.08666 q^{53} -2.37951 q^{54} -3.95493 q^{56} +2.23750 q^{57} +10.0832 q^{58} -12.3344 q^{59} -7.51805 q^{61} +17.3242 q^{62} -1.00000 q^{63} -11.1801 q^{64} -3.74873 q^{66} +15.6914 q^{67} -17.4279 q^{68} +5.84568 q^{69} +10.0504 q^{71} +3.95493 q^{72} -15.0797 q^{73} -24.9254 q^{74} -8.19389 q^{76} -1.57542 q^{77} +2.37951 q^{78} -11.7555 q^{79} +1.00000 q^{81} -5.35692 q^{82} -7.42110 q^{83} +3.66208 q^{84} -2.17331 q^{86} -4.23750 q^{87} +6.23069 q^{88} -11.1371 q^{89} +1.00000 q^{91} -21.4073 q^{92} -7.28055 q^{93} +4.98971 q^{94} +2.94464 q^{96} +14.6047 q^{97} +2.37951 q^{98} +1.57542 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * 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 $

$2$	2.37951	1.68257	0.841285	−	0.540593i	$-0.181800\pi$
$2$	2.37951	1.68257	0.841285	+	0.540593i	$0.181800\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.66208	1.83104
$5$	0	0
$5$	0	0
$6$	−2.37951	−0.971432
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.95493	1.39828
$9$	1.00000	0.333333
$10$	0	0
$11$	1.57542	0.475007	0.237504	−	0.971387i	$-0.423671\pi$
$11$	1.57542	0.475007	0.237504	+	0.971387i	$0.423671\pi$
$12$	−3.66208	−1.05715
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.37951	−0.635951
$15$	0	0
$16$	2.08666	0.521664
$17$	−4.75902	−1.15423	−0.577116	−	0.816662i	$-0.695822\pi$
$17$	−4.75902	−1.15423	−0.577116	+	0.816662i	$0.695822\pi$
$18$	2.37951	0.560856
$19$	−2.23750	−0.513317	−0.256659	−	0.966502i	$-0.582622\pi$
$19$	−2.23750	−0.513317	−0.256659	+	0.966502i	$0.582622\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	3.74873	0.799233
$23$	−5.84568	−1.21891	−0.609454	−	0.792821i	$-0.708611\pi$
$23$	−5.84568	−1.21891	−0.609454	+	0.792821i	$0.708611\pi$
$24$	−3.95493	−0.807297
$25$	0	0
$26$	−2.37951	−0.466661
$27$	−1.00000	−0.192450
$28$	−3.66208	−0.692068
$29$	4.23750	0.786884	0.393442	−	0.919349i	$-0.371284\pi$
$29$	4.23750	0.786884	0.393442	+	0.919349i	$0.371284\pi$
$30$	0	0
$31$	7.28055	1.30763	0.653813	−	0.756656i	$-0.273168\pi$
$31$	7.28055	1.30763	0.653813	+	0.756656i	$0.273168\pi$
$32$	−2.94464	−0.520544
$33$	−1.57542	−0.274246
$34$	−11.3242	−1.94208
$35$	0	0
$36$	3.66208	0.610346
$37$	−10.4750	−1.72208	−0.861039	−	0.508538i	$-0.830186\pi$
$37$	−10.4750	−1.72208	−0.861039	+	0.508538i	$0.830186\pi$
$38$	−5.32415	−0.863692
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.25127	−0.351589	−0.175794	−	0.984427i	$-0.556249\pi$
$41$	−2.25127	−0.351589	−0.175794	+	0.984427i	$0.556249\pi$
$42$	2.37951	0.367167
$43$	−0.913344	−0.139284	−0.0696418	−	0.997572i	$-0.522186\pi$
$43$	−0.913344	−0.139284	−0.0696418	+	0.997572i	$0.522186\pi$
$44$	5.76931	0.869757
$45$	0	0
$46$	−13.9099	−2.05090
$47$	2.09695	0.305871	0.152936	−	0.988236i	$-0.451127\pi$
$47$	2.09695	0.305871	0.152936	+	0.988236i	$0.451127\pi$
$48$	−2.08666	−0.301183
$49$	1.00000	0.142857
$50$	0	0
$51$	4.75902	0.666397
$52$	−3.66208	−0.507839
$53$	−1.08666	−0.149264	−0.0746319	−	0.997211i	$-0.523778\pi$
$53$	−1.08666	−0.149264	−0.0746319	+	0.997211i	$0.523778\pi$
$54$	−2.37951	−0.323811
$55$	0	0
$56$	−3.95493	−0.528500
$57$	2.23750	0.296364
$58$	10.0832	1.32399
$59$	−12.3344	−1.60581	−0.802904	−	0.596108i	$-0.796713\pi$
$59$	−12.3344	−1.60581	−0.802904	+	0.596108i	$0.796713\pi$
$60$	0	0
$61$	−7.51805	−0.962587	−0.481294	−	0.876559i	$-0.659833\pi$
$61$	−7.51805	−0.962587	−0.481294	+	0.876559i	$0.659833\pi$
$62$	17.3242	2.20017
$63$	−1.00000	−0.125988
$64$	−11.1801	−1.39752
$65$	0	0
$66$	−3.74873	−0.461437
$67$	15.6914	1.91700	0.958502	−	0.285084i	$-0.0920217\pi$
$67$	15.6914	1.91700	0.958502	+	0.285084i	$0.0920217\pi$
$68$	−17.4279	−2.11345
$69$	5.84568	0.703737
$70$	0	0
$71$	10.0504	1.19277	0.596383	−	0.802700i	$-0.296604\pi$
$71$	10.0504	1.19277	0.596383	+	0.802700i	$0.296604\pi$
$72$	3.95493	0.466093
$73$	−15.0797	−1.76495	−0.882473	−	0.470363i	$-0.844123\pi$
$73$	−15.0797	−1.76495	−0.882473	+	0.470363i	$0.844123\pi$
$74$	−24.9254	−2.89752
$75$	0	0
$76$	−8.19389	−0.939904
$77$	−1.57542	−0.179536
$78$	2.37951	0.269427
$79$	−11.7555	−1.32260	−0.661301	−	0.750121i	$-0.729995\pi$
$79$	−11.7555	−1.32260	−0.661301	+	0.750121i	$0.729995\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.35692	−0.591572
$83$	−7.42110	−0.814572	−0.407286	−	0.913301i	$-0.633525\pi$
$83$	−7.42110	−0.814572	−0.407286	+	0.913301i	$0.633525\pi$
$84$	3.66208	0.399565
$85$	0	0
$86$	−2.17331	−0.234354
$87$	−4.23750	−0.454308
$88$	6.23069	0.664193
$89$	−11.1371	−1.18053	−0.590264	−	0.807210i	$-0.700976\pi$
$89$	−11.1371	−1.18053	−0.590264	+	0.807210i	$0.700976\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−21.4073	−2.23187
$93$	−7.28055	−0.754958
$94$	4.98971	0.514649
$95$	0	0
$96$	2.94464	0.300536
$97$	14.6047	1.48288	0.741442	−	0.671017i	$-0.234142\pi$
$97$	14.6047	1.48288	0.741442	+	0.671017i	$0.234142\pi$
$98$	2.37951	0.240367
$99$	1.57542	0.158336
$100$	0	0
$101$	−11.4349	−1.13781	−0.568906	−	0.822403i	$-0.692633\pi$
$101$	−11.4349	−1.13781	−0.568906	+	0.822403i	$0.692633\pi$
$102$	11.3242	1.12126
$103$	−11.1508	−1.09873	−0.549363	−	0.835584i	$-0.685129\pi$
$103$	−11.1508	−1.09873	−0.549363	+	0.835584i	$0.685129\pi$
$104$	−3.95493	−0.387813
$105$	0	0
$106$	−2.58571	−0.251147
$107$	−2.28403	−0.220805	−0.110403	−	0.993887i	$-0.535214\pi$
$107$	−2.28403	−0.220805	−0.110403	+	0.993887i	$0.535214\pi$
$108$	−3.66208	−0.352384
$109$	14.6689	1.40502	0.702512	−	0.711671i	$-0.252062\pi$
$109$	14.6689	1.40502	0.702512	+	0.711671i	$0.252062\pi$
$110$	0	0
$111$	10.4750	0.994243
$112$	−2.08666	−0.197170
$113$	4.23750	0.398630	0.199315	−	0.979935i	$-0.436128\pi$
$113$	4.23750	0.398630	0.199315	+	0.979935i	$0.436128\pi$
$114$	5.32415	0.498653
$115$	0	0
$116$	15.5180	1.44081
$117$	−1.00000	−0.0924500
$118$	−29.3500	−2.70188
$119$	4.75902	0.436259
$120$	0	0
$121$	−8.51805	−0.774368
$122$	−17.8893	−1.61962
$123$	2.25127	0.202990
$124$	26.6619	2.39431
$125$	0	0
$126$	−2.37951	−0.211984
$127$	−4.84916	−0.430293	−0.215147	−	0.976582i	$-0.569023\pi$
$127$	−4.84916	−0.430293	−0.215147	+	0.976582i	$0.569023\pi$
$128$	−20.7140	−1.83087
$129$	0.913344	0.0804154
$130$	0	0
$131$	−6.36721	−0.556305	−0.278153	−	0.960537i	$-0.589722\pi$
$131$	−6.36721	−0.556305	−0.278153	+	0.960537i	$0.589722\pi$
$132$	−5.76931	−0.502154
$133$	2.23750	0.194016
$134$	37.3378	3.22549
$135$	0	0
$136$	−18.8216	−1.61394
$137$	18.0258	1.54005	0.770024	−	0.638015i	$-0.220244\pi$
$137$	18.0258	1.54005	0.770024	+	0.638015i	$0.220244\pi$
$138$	13.9099	1.18409
$139$	2.67585	0.226962	0.113481	−	0.993540i	$-0.463800\pi$
$139$	2.67585	0.226962	0.113481	+	0.993540i	$0.463800\pi$
$140$	0	0
$141$	−2.09695	−0.176595
$142$	23.9151	2.00691
$143$	−1.57542	−0.131743
$144$	2.08666	0.173888
$145$	0	0
$146$	−35.8823	−2.96964
$147$	−1.00000	−0.0824786
$148$	−38.3603	−3.15319
$149$	−7.46471	−0.611533	−0.305766	−	0.952107i	$-0.598913\pi$
$149$	−7.46471	−0.611533	−0.305766	+	0.952107i	$0.598913\pi$
$150$	0	0
$151$	13.0155	1.05919	0.529594	−	0.848251i	$-0.322344\pi$
$151$	13.0155	1.05919	0.529594	+	0.848251i	$0.322344\pi$
$152$	−8.84916	−0.717761
$153$	−4.75902	−0.384744
$154$	−3.74873	−0.302082
$155$	0	0
$156$	3.66208	0.293201
$157$	17.1233	1.36659	0.683294	−	0.730143i	$-0.260547\pi$
$157$	17.1233	1.36659	0.683294	+	0.730143i	$0.260547\pi$
$158$	−27.9725	−2.22537
$159$	1.08666	0.0861774
$160$	0	0
$161$	5.84568	0.460704
$162$	2.37951	0.186952
$163$	0.849158	0.0665112	0.0332556	−	0.999447i	$-0.489412\pi$
$163$	0.849158	0.0665112	0.0332556	+	0.999447i	$0.489412\pi$
$164$	−8.24431	−0.643773
$165$	0	0
$166$	−17.6586	−1.37057
$167$	−18.3781	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$167$	−18.3781	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$168$	3.95493	0.305130
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.23750	−0.171106
$172$	−3.34474	−0.255034
$173$	12.2565	0.931844	0.465922	−	0.884826i	$-0.345723\pi$
$173$	12.2565	0.931844	0.465922	+	0.884826i	$0.345723\pi$
$174$	−10.0832	−0.764404
$175$	0	0
$176$	3.28736	0.247794
$177$	12.3344	0.927114
$178$	−26.5008	−1.98632
$179$	18.5146	1.38384	0.691922	−	0.721972i	$-0.256764\pi$
$179$	18.5146	1.38384	0.691922	+	0.721972i	$0.256764\pi$
$180$	0	0
$181$	−18.1664	−1.35029	−0.675147	−	0.737683i	$-0.735920\pi$
$181$	−18.1664	−1.35029	−0.675147	+	0.737683i	$0.735920\pi$
$182$	2.37951	0.176381
$183$	7.51805	0.555750
$184$	−23.1193	−1.70438
$185$	0	0
$186$	−17.3242	−1.27027
$187$	−7.49747	−0.548269
$188$	7.67918	0.560062
$189$	1.00000	0.0727393
$190$	0	0
$191$	23.5151	1.70149	0.850747	−	0.525575i	$-0.176150\pi$
$191$	23.5151	1.70149	0.850747	+	0.525575i	$0.176150\pi$
$192$	11.1801	0.806856
$193$	−14.7344	−1.06061	−0.530303	−	0.847808i	$-0.677922\pi$
$193$	−14.7344	−1.06061	−0.530303	+	0.847808i	$0.677922\pi$
$194$	34.7521	2.49505
$195$	0	0
$196$	3.66208	0.261577
$197$	−14.5008	−1.03314	−0.516570	−	0.856245i	$-0.672791\pi$
$197$	−14.5008	−1.03314	−0.516570	+	0.856245i	$0.672791\pi$
$198$	3.74873	0.266411
$199$	−13.5180	−0.958269	−0.479135	−	0.877741i	$-0.659049\pi$
$199$	−13.5180	−0.958269	−0.479135	+	0.877741i	$0.659049\pi$
$200$	0	0
$201$	−15.6914	−1.10678
$202$	−27.2094	−1.91445
$203$	−4.23750	−0.297414
$204$	17.4279	1.22020
$205$	0	0
$206$	−26.5336	−1.84868
$207$	−5.84568	−0.406303
$208$	−2.08666	−0.144684
$209$	−3.52500	−0.243830
$210$	0	0
$211$	4.06419	0.279790	0.139895	−	0.990166i	$-0.455323\pi$
$211$	4.06419	0.279790	0.139895	+	0.990166i	$0.455323\pi$
$212$	−3.97942	−0.273308
$213$	−10.0504	−0.688643
$214$	−5.43487	−0.371520
$215$	0	0
$216$	−3.95493	−0.269099
$217$	−7.28055	−0.494236
$218$	34.9048	2.36405
$219$	15.0797	1.01899
$220$	0	0
$221$	4.75902	0.320127
$222$	24.9254	1.67288
$223$	0.0641862	0.00429822	0.00214911	−	0.999998i	$-0.499316\pi$
$223$	0.0641862	0.00429822	0.00214911	+	0.999998i	$0.499316\pi$
$224$	2.94464	0.196747
$225$	0	0
$226$	10.0832	0.670723
$227$	1.68614	0.111913	0.0559564	−	0.998433i	$-0.482179\pi$
$227$	1.68614	0.111913	0.0559564	+	0.998433i	$0.482179\pi$
$228$	8.19389	0.542654
$229$	0.648310	0.0428415	0.0214208	−	0.999771i	$-0.493181\pi$
$229$	0.648310	0.0428415	0.0214208	+	0.999771i	$0.493181\pi$
$230$	0	0
$231$	1.57542	0.103655
$232$	16.7590	1.10028
$233$	−3.10724	−0.203562	−0.101781	−	0.994807i	$-0.532454\pi$
$233$	−3.10724	−0.203562	−0.101781	+	0.994807i	$0.532454\pi$
$234$	−2.37951	−0.155554
$235$	0	0
$236$	−45.1697	−2.94030
$237$	11.7555	0.763605
$238$	11.3242	0.734036
$239$	−15.0659	−0.974534	−0.487267	−	0.873253i	$-0.662006\pi$
$239$	−15.0659	−0.974534	−0.487267	+	0.873253i	$0.662006\pi$
$240$	0	0
$241$	25.7280	1.65729	0.828643	−	0.559777i	$-0.189113\pi$
$241$	25.7280	1.65729	0.828643	+	0.559777i	$0.189113\pi$
$242$	−20.2688	−1.30293
$243$	−1.00000	−0.0641500
$244$	−27.5317	−1.76253
$245$	0	0
$246$	5.35692	0.341544
$247$	2.23750	0.142369
$248$	28.7941	1.82843
$249$	7.42110	0.470293
$250$	0	0
$251$	−11.6258	−0.733816	−0.366908	−	0.930257i	$-0.619584\pi$
$251$	−11.6258	−0.733816	−0.366908	+	0.930257i	$0.619584\pi$
$252$	−3.66208	−0.230689
$253$	−9.20941	−0.578991
$254$	−11.5386	−0.723998
$255$	0	0
$256$	−26.9289	−1.68305
$257$	12.5651	0.783791	0.391896	−	0.920010i	$-0.371819\pi$
$257$	12.5651	0.783791	0.391896	+	0.920010i	$0.371819\pi$
$258$	2.17331	0.135305
$259$	10.4750	0.650885
$260$	0	0
$261$	4.23750	0.262295
$262$	−15.1508	−0.936022
$263$	−9.37068	−0.577821	−0.288911	−	0.957356i	$-0.593293\pi$
$263$	−9.37068	−0.577821	−0.288911	+	0.957356i	$0.593293\pi$
$264$	−6.23069	−0.383472
$265$	0	0
$266$	5.32415	0.326445
$267$	11.1371	0.681578
$268$	57.4630	3.51011
$269$	−10.3918	−0.633600	−0.316800	−	0.948492i	$-0.602608\pi$
$269$	−10.3918	−0.633600	−0.316800	+	0.948492i	$0.602608\pi$
$270$	0	0
$271$	−5.13026	−0.311641	−0.155821	−	0.987785i	$-0.549802\pi$
$271$	−5.13026	−0.311641	−0.155821	+	0.987785i	$0.549802\pi$
$272$	−9.93045	−0.602122
$273$	−1.00000	−0.0605228
$274$	42.8926	2.59124
$275$	0	0
$276$	21.4073	1.28857
$277$	−8.04361	−0.483293	−0.241647	−	0.970364i	$-0.577688\pi$
$277$	−8.04361	−0.483293	−0.241647	+	0.970364i	$0.577688\pi$
$278$	6.36721	0.381880
$279$	7.28055	0.435875
$280$	0	0
$281$	−27.8525	−1.66154	−0.830770	−	0.556616i	$-0.812100\pi$
$281$	−27.8525	−1.66154	−0.830770	+	0.556616i	$0.812100\pi$
$282$	−4.98971	−0.297133
$283$	−23.8197	−1.41594	−0.707968	−	0.706244i	$-0.750388\pi$
$283$	−23.8197	−1.41594	−0.707968	+	0.706244i	$0.750388\pi$
$284$	36.8054	2.18400
$285$	0	0
$286$	−3.74873	−0.221667
$287$	2.25127	0.132888
$288$	−2.94464	−0.173515
$289$	5.64831	0.332254
$290$	0	0
$291$	−14.6047	−0.856143
$292$	−55.2230	−3.23168
$293$	16.4612	0.961675	0.480838	−	0.876810i	$-0.340333\pi$
$293$	16.4612	0.961675	0.480838	+	0.876810i	$0.340333\pi$
$294$	−2.37951	−0.138776
$295$	0	0
$296$	−41.4279	−2.40795
$297$	−1.57542	−0.0914152
$298$	−17.7624	−1.02895
$299$	5.84568	0.338064
$300$	0	0
$301$	0.913344	0.0526443
$302$	30.9706	1.78216
$303$	11.4349	0.656916
$304$	−4.66889	−0.267779
$305$	0	0
$306$	−11.3242	−0.647359
$307$	−1.95639	−0.111657	−0.0558287	−	0.998440i	$-0.517780\pi$
$307$	−1.95639	−0.111657	−0.0558287	+	0.998440i	$0.517780\pi$
$308$	−5.76931	−0.328737
$309$	11.1508	0.634349
$310$	0	0
$311$	−6.67585	−0.378552	−0.189276	−	0.981924i	$-0.560614\pi$
$311$	−6.67585	−0.378552	−0.189276	+	0.981924i	$0.560614\pi$
$312$	3.95493	0.223904
$313$	6.78364	0.383434	0.191717	−	0.981450i	$-0.438594\pi$
$313$	6.78364	0.383434	0.191717	+	0.981450i	$0.438594\pi$
$314$	40.7451	2.29938
$315$	0	0
$316$	−43.0497	−2.42174
$317$	30.6947	1.72399	0.861993	−	0.506920i	$-0.169216\pi$
$317$	30.6947	1.72399	0.861993	+	0.506920i	$0.169216\pi$
$318$	2.58571	0.145000
$319$	6.67585	0.373776
$320$	0	0
$321$	2.28403	0.127482
$322$	13.9099	0.775167
$323$	10.6483	0.592488
$324$	3.66208	0.203449
$325$	0	0
$326$	2.02058	0.111910
$327$	−14.6689	−0.811191
$328$	−8.90361	−0.491620
$329$	−2.09695	−0.115608
$330$	0	0
$331$	−13.8267	−0.759983	−0.379992	−	0.924990i	$-0.624073\pi$
$331$	−13.8267	−0.759983	−0.379992	+	0.924990i	$0.624073\pi$
$332$	−27.1766	−1.49151
$333$	−10.4750	−0.574026
$334$	−43.7308	−2.39284
$335$	0	0
$336$	2.08666	0.113836
$337$	−23.8633	−1.29992	−0.649959	−	0.759969i	$-0.725214\pi$
$337$	−23.8633	−1.29992	−0.649959	+	0.759969i	$0.725214\pi$
$338$	2.37951	0.129428
$339$	−4.23750	−0.230149
$340$	0	0
$341$	11.4699	0.621132
$342$	−5.32415	−0.287897
$343$	−1.00000	−0.0539949
$344$	−3.61221	−0.194758
$345$	0	0
$346$	29.1645	1.56789
$347$	13.6012	0.730152	0.365076	−	0.930978i	$-0.381043\pi$
$347$	13.6012	0.730152	0.365076	+	0.930978i	$0.381043\pi$
$348$	−15.5180	−0.831855
$349$	−16.4039	−0.878078	−0.439039	−	0.898468i	$-0.644681\pi$
$349$	−16.4039	−0.878078	−0.439039	+	0.898468i	$0.644681\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−4.63905	−0.247262
$353$	4.73322	0.251924	0.125962	−	0.992035i	$-0.459798\pi$
$353$	4.73322	0.251924	0.125962	+	0.992035i	$0.459798\pi$
$354$	29.3500	1.55993
$355$	0	0
$356$	−40.7848	−2.16159
$357$	−4.75902	−0.251874
$358$	44.0556	2.32841
$359$	−17.5479	−0.926142	−0.463071	−	0.886321i	$-0.653253\pi$
$359$	−17.5479	−0.926142	−0.463071	+	0.886321i	$0.653253\pi$
$360$	0	0
$361$	−13.9936	−0.736505
$362$	−43.2271	−2.27196
$363$	8.51805	0.447082
$364$	3.66208	0.191945
$365$	0	0
$366$	17.8893	0.935088
$367$	11.1888	0.584052	0.292026	−	0.956410i	$-0.405671\pi$
$367$	11.1888	0.584052	0.292026	+	0.956410i	$0.405671\pi$
$368$	−12.1979	−0.635861
$369$	−2.25127	−0.117196
$370$	0	0
$371$	1.08666	0.0564164
$372$	−26.6619	−1.38236
$373$	5.43195	0.281256	0.140628	−	0.990063i	$-0.455088\pi$
$373$	5.43195	0.281256	0.140628	+	0.990063i	$0.455088\pi$
$374$	−17.8403	−0.922501
$375$	0	0
$376$	8.29328	0.427693
$377$	−4.23750	−0.218242
$378$	2.37951	0.122389
$379$	10.8422	0.556927	0.278463	−	0.960447i	$-0.410175\pi$
$379$	10.8422	0.556927	0.278463	+	0.960447i	$0.410175\pi$
$380$	0	0
$381$	4.84916	0.248430
$382$	55.9545	2.86288
$383$	−10.5353	−0.538328	−0.269164	−	0.963094i	$-0.586747\pi$
$383$	−10.5353	−0.538328	−0.269164	+	0.963094i	$0.586747\pi$
$384$	20.7140	1.05705
$385$	0	0
$386$	−35.0607	−1.78454
$387$	−0.913344	−0.0464279
$388$	53.4836	2.71522
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	27.8197	1.40690
$392$	3.95493	0.199754
$393$	6.36721	0.321183
$394$	−34.5048	−1.73833
$395$	0	0
$396$	5.76931	0.289919
$397$	35.4400	1.77868	0.889340	−	0.457246i	$-0.151164\pi$
$397$	35.4400	1.77868	0.889340	+	0.457246i	$0.151164\pi$
$398$	−32.1664	−1.61235
$399$	−2.23750	−0.112015
$400$	0	0
$401$	23.7241	1.18473	0.592363	−	0.805671i	$-0.298195\pi$
$401$	23.7241	1.18473	0.592363	+	0.805671i	$0.298195\pi$
$402$	−37.3378	−1.86224
$403$	−7.28055	−0.362670
$404$	−41.8754	−2.08338
$405$	0	0
$406$	−10.0832	−0.500420
$407$	−16.5025	−0.818000
$408$	18.8216	0.931809
$409$	−3.95639	−0.195631	−0.0978156	−	0.995205i	$-0.531186\pi$
$409$	−3.95639	−0.195631	−0.0978156	+	0.995205i	$0.531186\pi$
$410$	0	0
$411$	−18.0258	−0.889147
$412$	−40.8352	−2.01181
$413$	12.3344	0.606938
$414$	−13.9099	−0.683633
$415$	0	0
$416$	2.94464	0.144373
$417$	−2.67585	−0.131037
$418$	−8.38779	−0.410260
$419$	−18.0586	−0.882219	−0.441109	−	0.897453i	$-0.645415\pi$
$419$	−18.0586	−0.882219	−0.441109	+	0.897453i	$0.645415\pi$
$420$	0	0
$421$	24.4956	1.19384	0.596921	−	0.802300i	$-0.296391\pi$
$421$	24.4956	1.19384	0.596921	+	0.802300i	$0.296391\pi$
$422$	9.67078	0.470766
$423$	2.09695	0.101957
$424$	−4.29765	−0.208712
$425$	0	0
$426$	−23.9151	−1.15869
$427$	7.51805	0.363824
$428$	−8.36428	−0.404303
$429$	1.57542	0.0760621
$430$	0	0
$431$	−40.7779	−1.96420	−0.982101	−	0.188357i	$-0.939684\pi$
$431$	−40.7779	−1.96420	−0.982101	+	0.188357i	$0.939684\pi$
$432$	−2.08666	−0.100394
$433$	18.0791	0.868828	0.434414	−	0.900713i	$-0.356955\pi$
$433$	18.0791	0.868828	0.434414	+	0.900713i	$0.356955\pi$
$434$	−17.3242	−0.831586
$435$	0	0
$436$	53.7186	2.57265
$437$	13.0797	0.625687
$438$	35.8823	1.71452
$439$	20.7561	0.990635	0.495317	−	0.868712i	$-0.335052\pi$
$439$	20.7561	0.990635	0.495317	+	0.868712i	$0.335052\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	11.3242	0.538635
$443$	8.24042	0.391514	0.195757	−	0.980652i	$-0.437284\pi$
$443$	8.24042	0.391514	0.195757	+	0.980652i	$0.437284\pi$
$444$	38.3603	1.82050
$445$	0	0
$446$	0.152732	0.00723206
$447$	7.46471	0.353069
$448$	11.1801	0.528211
$449$	36.1061	1.70395	0.851975	−	0.523582i	$-0.175405\pi$
$449$	36.1061	1.70395	0.851975	+	0.523582i	$0.175405\pi$
$450$	0	0
$451$	−3.54669	−0.167007
$452$	15.5180	0.729908
$453$	−13.0155	−0.611522
$454$	4.01218	0.188301
$455$	0	0
$456$	8.84916	0.414400
$457$	6.54052	0.305953	0.152976	−	0.988230i	$-0.451114\pi$
$457$	6.54052	0.305953	0.152976	+	0.988230i	$0.451114\pi$
$458$	1.54266	0.0720838
$459$	4.75902	0.222132
$460$	0	0
$461$	−14.6987	−0.684588	−0.342294	−	0.939593i	$-0.611204\pi$
$461$	−14.6987	−0.684588	−0.342294	+	0.939593i	$0.611204\pi$
$462$	3.74873	0.174407
$463$	−21.7775	−1.01208	−0.506042	−	0.862509i	$-0.668892\pi$
$463$	−21.7775	−1.01208	−0.506042	+	0.862509i	$0.668892\pi$
$464$	8.84220	0.410489
$465$	0	0
$466$	−7.39371	−0.342507
$467$	−35.0086	−1.62000	−0.810001	−	0.586428i	$-0.800534\pi$
$467$	−35.0086	−1.62000	−0.810001	+	0.586428i	$0.800534\pi$
$468$	−3.66208	−0.169280
$469$	−15.6914	−0.724560
$470$	0	0
$471$	−17.1233	−0.789000
$472$	−48.7819	−2.24537
$473$	−1.43890	−0.0661607
$474$	27.9725	1.28482
$475$	0	0
$476$	17.4279	0.798807
$477$	−1.08666	−0.0497546
$478$	−35.8496	−1.63972
$479$	31.7814	1.45213	0.726064	−	0.687628i	$-0.241348\pi$
$479$	31.7814	1.45213	0.726064	+	0.687628i	$0.241348\pi$
$480$	0	0
$481$	10.4750	0.477619
$482$	61.2201	2.78850
$483$	−5.84568	−0.265988
$484$	−31.1938	−1.41790
$485$	0	0
$486$	−2.37951	−0.107937
$487$	28.5817	1.29516	0.647580	−	0.761998i	$-0.275781\pi$
$487$	28.5817	1.29516	0.647580	+	0.761998i	$0.275781\pi$
$488$	−29.7334	−1.34597
$489$	−0.849158	−0.0384002
$490$	0	0
$491$	21.7160	0.980028	0.490014	−	0.871715i	$-0.336992\pi$
$491$	21.7160	0.980028	0.490014	+	0.871715i	$0.336992\pi$
$492$	8.24431	0.371682
$493$	−20.1664	−0.908247
$494$	5.32415	0.239545
$495$	0	0
$496$	15.1920	0.682141
$497$	−10.0504	−0.450823
$498$	17.6586	0.791301
$499$	−42.6895	−1.91104	−0.955522	−	0.294921i	$-0.904707\pi$
$499$	−42.6895	−1.91104	−0.955522	+	0.294921i	$0.904707\pi$
$500$	0	0
$501$	18.3781	0.821071
$502$	−27.6638	−1.23470
$503$	−21.8922	−0.976125	−0.488063	−	0.872809i	$-0.662296\pi$
$503$	−21.8922	−0.976125	−0.488063	+	0.872809i	$0.662296\pi$
$504$	−3.95493	−0.176167
$505$	0	0
$506$	−21.9139	−0.974192
$507$	−1.00000	−0.0444116
$508$	−17.7580	−0.787883
$509$	−5.36546	−0.237820	−0.118910	−	0.992905i	$-0.537940\pi$
$509$	−5.36546	−0.237820	−0.118910	+	0.992905i	$0.537940\pi$
$510$	0	0
$511$	15.0797	0.667087
$512$	−22.6496	−1.00098
$513$	2.23750	0.0987880
$514$	29.8989	1.31878
$515$	0	0
$516$	3.34474	0.147244
$517$	3.30357	0.145291
$518$	24.9254	1.09516
$519$	−12.2565	−0.538000
$520$	0	0
$521$	−11.5221	−0.504791	−0.252396	−	0.967624i	$-0.581218\pi$
$521$	−11.5221	−0.504791	−0.252396	+	0.967624i	$0.581218\pi$
$522$	10.0832	0.441329
$523$	3.62584	0.158547	0.0792734	−	0.996853i	$-0.474740\pi$
$523$	3.62584	0.158547	0.0792734	+	0.996853i	$0.474740\pi$
$524$	−23.3172	−1.01862
$525$	0	0
$526$	−22.2977	−0.972224
$527$	−34.6483	−1.50930
$528$	−3.28736	−0.143064
$529$	11.1720	0.485738
$530$	0	0
$531$	−12.3344	−0.535269
$532$	8.19389	0.355250
$533$	2.25127	0.0975132
$534$	26.5008	1.14680
$535$	0	0
$536$	62.0583	2.68051
$537$	−18.5146	−0.798963
$538$	−24.7275	−1.06608
$539$	1.57542	0.0678582
$540$	0	0
$541$	37.2300	1.60064	0.800321	−	0.599572i	$-0.204662\pi$
$541$	37.2300	1.60064	0.800321	+	0.599572i	$0.204662\pi$
$542$	−12.2075	−0.524358
$543$	18.1664	0.779593
$544$	14.0136	0.600829
$545$	0	0
$546$	−2.37951	−0.101834
$547$	4.34529	0.185791	0.0928956	−	0.995676i	$-0.470388\pi$
$547$	4.34529	0.185791	0.0928956	+	0.995676i	$0.470388\pi$
$548$	66.0119	2.81989
$549$	−7.51805	−0.320862
$550$	0	0
$551$	−9.48140	−0.403921
$552$	23.1193	0.984022
$553$	11.7555	0.499897
$554$	−19.1399	−0.813175
$555$	0	0
$556$	9.79915	0.415577
$557$	−28.0052	−1.18662	−0.593310	−	0.804974i	$-0.702179\pi$
$557$	−28.0052	−1.18662	−0.593310	+	0.804974i	$0.702179\pi$
$558$	17.3242	0.733390
$559$	0.913344	0.0386303
$560$	0	0
$561$	7.49747	0.316543
$562$	−66.2753	−2.79566
$563$	22.2525	0.937829	0.468915	−	0.883243i	$-0.344645\pi$
$563$	22.2525	0.937829	0.468915	+	0.883243i	$0.344645\pi$
$564$	−7.67918	−0.323352
$565$	0	0
$566$	−56.6793	−2.38241
$567$	−1.00000	−0.0419961
$568$	39.7487	1.66782
$569$	3.10724	0.130262	0.0651311	−	0.997877i	$-0.479253\pi$
$569$	3.10724	0.130262	0.0651311	+	0.997877i	$0.479253\pi$
$570$	0	0
$571$	16.5772	0.693733	0.346866	−	0.937915i	$-0.387246\pi$
$571$	16.5772	0.693733	0.346866	+	0.937915i	$0.387246\pi$
$572$	−5.76931	−0.241227
$573$	−23.5151	−0.982358
$574$	5.35692	0.223593
$575$	0	0
$576$	−11.1801	−0.465839
$577$	7.45253	0.310253	0.155126	−	0.987895i	$-0.450422\pi$
$577$	7.45253	0.310253	0.155126	+	0.987895i	$0.450422\pi$
$578$	13.4402	0.559039
$579$	14.7344	0.612341
$580$	0	0
$581$	7.42110	0.307879
$582$	−34.7521	−1.44052
$583$	−1.71194	−0.0709014
$584$	−59.6392	−2.46789
$585$	0	0
$586$	39.1697	1.61809
$587$	−11.9892	−0.494845	−0.247423	−	0.968908i	$-0.579584\pi$
$587$	−11.9892	−0.494845	−0.247423	+	0.968908i	$0.579584\pi$
$588$	−3.66208	−0.151022
$589$	−16.2902	−0.671227
$590$	0	0
$591$	14.5008	0.596483
$592$	−21.8577	−0.898347
$593$	33.4767	1.37473	0.687363	−	0.726314i	$-0.258768\pi$
$593$	33.4767	1.37473	0.687363	+	0.726314i	$0.258768\pi$
$594$	−3.74873	−0.153812
$595$	0	0
$596$	−27.3363	−1.11974
$597$	13.5180	0.553257
$598$	13.9099	0.568817
$599$	−13.5863	−0.555120	−0.277560	−	0.960708i	$-0.589526\pi$
$599$	−13.5863	−0.555120	−0.277560	+	0.960708i	$0.589526\pi$
$600$	0	0
$601$	−6.78364	−0.276710	−0.138355	−	0.990383i	$-0.544182\pi$
$601$	−6.78364	−0.276710	−0.138355	+	0.990383i	$0.544182\pi$
$602$	2.17331	0.0885776
$603$	15.6914	0.639002
$604$	47.6638	1.93941
$605$	0	0
$606$	27.2094	1.10531
$607$	17.4905	0.709918	0.354959	−	0.934882i	$-0.384495\pi$
$607$	17.4905	0.709918	0.354959	+	0.934882i	$0.384495\pi$
$608$	6.58863	0.267204
$609$	4.23750	0.171712
$610$	0	0
$611$	−2.09695	−0.0848334
$612$	−17.4279	−0.704482
$613$	38.2455	1.54472	0.772361	−	0.635184i	$-0.219076\pi$
$613$	38.2455	1.54472	0.772361	+	0.635184i	$0.219076\pi$
$614$	−4.65526	−0.187871
$615$	0	0
$616$	−6.23069	−0.251041
$617$	34.1542	1.37500	0.687498	−	0.726187i	$-0.258709\pi$
$617$	34.1542	1.37500	0.687498	+	0.726187i	$0.258709\pi$
$618$	26.5336	1.06734
$619$	−8.56805	−0.344379	−0.172190	−	0.985064i	$-0.555084\pi$
$619$	−8.56805	−0.344379	−0.172190	+	0.985064i	$0.555084\pi$
$620$	0	0
$621$	5.84568	0.234579
$622$	−15.8853	−0.636941
$623$	11.1371	0.446197
$624$	2.08666	0.0835331
$625$	0	0
$626$	16.1417	0.645154
$627$	3.52500	0.140775
$628$	62.7069	2.50228
$629$	49.8508	1.98768
$630$	0	0
$631$	−17.7119	−0.705101	−0.352551	−	0.935793i	$-0.614686\pi$
$631$	−17.7119	−0.705101	−0.352551	+	0.935793i	$0.614686\pi$
$632$	−46.4924	−1.84937
$633$	−4.06419	−0.161537
$634$	73.0384	2.90073
$635$	0	0
$636$	3.97942	0.157794
$637$	−1.00000	−0.0396214
$638$	15.8853	0.628903
$639$	10.0504	0.397588
$640$	0	0
$641$	16.6117	0.656121	0.328061	−	0.944657i	$-0.393605\pi$
$641$	16.6117	0.656121	0.328061	+	0.944657i	$0.393605\pi$
$642$	5.43487	0.214497
$643$	−12.9500	−0.510698	−0.255349	−	0.966849i	$-0.582190\pi$
$643$	−12.9500	−0.510698	−0.255349	+	0.966849i	$0.582190\pi$
$644$	21.4073	0.843567
$645$	0	0
$646$	25.3378	0.996902
$647$	23.6775	0.930857	0.465428	−	0.885086i	$-0.345900\pi$
$647$	23.6775	0.930857	0.465428	+	0.885086i	$0.345900\pi$
$648$	3.95493	0.155364
$649$	−19.4319	−0.762771
$650$	0	0
$651$	7.28055	0.285347
$652$	3.10968	0.121785
$653$	−22.2811	−0.871927	−0.435963	−	0.899964i	$-0.643592\pi$
$653$	−22.2811	−0.871927	−0.435963	+	0.899964i	$0.643592\pi$
$654$	−34.9048	−1.36489
$655$	0	0
$656$	−4.69762	−0.183411
$657$	−15.0797	−0.588315
$658$	−4.98971	−0.194519
$659$	15.5165	0.604435	0.302218	−	0.953239i	$-0.402273\pi$
$659$	15.5165	0.604435	0.302218	+	0.953239i	$0.402273\pi$
$660$	0	0
$661$	28.1444	1.09469	0.547346	−	0.836906i	$-0.315638\pi$
$661$	28.1444	1.09469	0.547346	+	0.836906i	$0.315638\pi$
$662$	−32.9008	−1.27872
$663$	−4.75902	−0.184825
$664$	−29.3500	−1.13900
$665$	0	0
$666$	−24.9254	−0.965839
$667$	−24.7711	−0.959139
$668$	−67.3018	−2.60399
$669$	−0.0641862	−0.00248158
$670$	0	0
$671$	−11.8441	−0.457236
$672$	−2.94464	−0.113592
$673$	−7.13477	−0.275025	−0.137513	−	0.990500i	$-0.543911\pi$
$673$	−7.13477	−0.275025	−0.137513	+	0.990500i	$0.543911\pi$
$674$	−56.7831	−2.18720
$675$	0	0
$676$	3.66208	0.140849
$677$	10.7660	0.413770	0.206885	−	0.978365i	$-0.433667\pi$
$677$	10.7660	0.413770	0.206885	+	0.978365i	$0.433667\pi$
$678$	−10.0832	−0.387242
$679$	−14.6047	−0.560477
$680$	0	0
$681$	−1.68614	−0.0646129
$682$	27.2928	1.04510
$683$	12.2020	0.466898	0.233449	−	0.972369i	$-0.424999\pi$
$683$	12.2020	0.466898	0.233449	+	0.972369i	$0.424999\pi$
$684$	−8.19389	−0.313301
$685$	0	0
$686$	−2.37951	−0.0908502
$687$	−0.648310	−0.0247346
$688$	−1.90583	−0.0726593
$689$	1.08666	0.0413983
$690$	0	0
$691$	32.0367	1.21873	0.609366	−	0.792889i	$-0.291424\pi$
$691$	32.0367	1.21873	0.609366	+	0.792889i	$0.291424\pi$
$692$	44.8842	1.70624
$693$	−1.57542	−0.0598453
$694$	32.3643	1.22853
$695$	0	0
$696$	−16.7590	−0.635249
$697$	10.7138	0.405815
$698$	−39.0332	−1.47743
$699$	3.10724	0.117526
$700$	0	0
$701$	−14.9513	−0.564704	−0.282352	−	0.959311i	$-0.591115\pi$
$701$	−14.9513	−0.564704	−0.282352	+	0.959311i	$0.591115\pi$
$702$	2.37951	0.0898089
$703$	23.4378	0.883973
$704$	−17.6134	−0.663830
$705$	0	0
$706$	11.2627	0.423879
$707$	11.4349	0.430053
$708$	45.1697	1.69758
$709$	−8.58279	−0.322333	−0.161167	−	0.986927i	$-0.551526\pi$
$709$	−8.58279	−0.322333	−0.161167	+	0.986927i	$0.551526\pi$
$710$	0	0
$711$	−11.7555	−0.440867
$712$	−44.0464	−1.65071
$713$	−42.5598	−1.59388
$714$	−11.3242	−0.423796
$715$	0	0
$716$	67.8018	2.53387
$717$	15.0659	0.562648
$718$	−41.7554	−1.55830
$719$	35.9644	1.34125	0.670623	−	0.741798i	$-0.266027\pi$
$719$	35.9644	1.34125	0.670623	+	0.741798i	$0.266027\pi$
$720$	0	0
$721$	11.1508	0.415279
$722$	−33.2979	−1.23922
$723$	−25.7280	−0.956835
$724$	−66.5266	−2.47244
$725$	0	0
$726$	20.2688	0.752246
$727$	31.5111	1.16868	0.584341	−	0.811508i	$-0.301353\pi$
$727$	31.5111	1.16868	0.584341	+	0.811508i	$0.301353\pi$
$728$	3.95493	0.146580
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.34662	0.160766
$732$	27.5317	1.01760
$733$	26.6047	0.982667	0.491334	−	0.870971i	$-0.336510\pi$
$733$	26.6047	0.982667	0.491334	+	0.870971i	$0.336510\pi$
$734$	26.6240	0.982708
$735$	0	0
$736$	17.2134	0.634496
$737$	24.7205	0.910591
$738$	−5.35692	−0.197191
$739$	−50.3397	−1.85177	−0.925887	−	0.377800i	$-0.876681\pi$
$739$	−50.3397	−1.85177	−0.925887	+	0.377800i	$0.876681\pi$
$740$	0	0
$741$	−2.23750	−0.0821966
$742$	2.58571	0.0949245
$743$	−23.4021	−0.858540	−0.429270	−	0.903176i	$-0.641229\pi$
$743$	−23.4021	−0.858540	−0.429270	+	0.903176i	$0.641229\pi$
$744$	−28.7941	−1.05564
$745$	0	0
$746$	12.9254	0.473232
$747$	−7.42110	−0.271524
$748$	−27.4563	−1.00390
$749$	2.28403	0.0834565
$750$	0	0
$751$	49.0305	1.78915	0.894574	−	0.446920i	$-0.147479\pi$
$751$	49.0305	1.78915	0.894574	+	0.446920i	$0.147479\pi$
$752$	4.37561	0.159562
$753$	11.6258	0.423669
$754$	−10.0832	−0.367208
$755$	0	0
$756$	3.66208	0.133188
$757$	35.3883	1.28621	0.643106	−	0.765778i	$-0.277646\pi$
$757$	35.3883	1.28621	0.643106	+	0.765778i	$0.277646\pi$
$758$	25.7992	0.937067
$759$	9.20941	0.334280
$760$	0	0
$761$	27.7051	1.00431	0.502155	−	0.864778i	$-0.332541\pi$
$761$	27.7051	1.00431	0.502155	+	0.864778i	$0.332541\pi$
$762$	11.5386	0.418000
$763$	−14.6689	−0.531049
$764$	86.1142	3.11550
$765$	0	0
$766$	−25.0689	−0.905775
$767$	12.3344	0.445371
$768$	26.9289	0.971712
$769$	21.1383	0.762265	0.381133	−	0.924520i	$-0.375534\pi$
$769$	21.1383	0.762265	0.381133	+	0.924520i	$0.375534\pi$
$770$	0	0
$771$	−12.5651	−0.452522
$772$	−53.9586	−1.94201
$773$	41.4537	1.49099	0.745493	−	0.666513i	$-0.232214\pi$
$773$	41.4537	1.49099	0.745493	+	0.666513i	$0.232214\pi$
$774$	−2.17331	−0.0781181
$775$	0	0
$776$	57.7606	2.07349
$777$	−10.4750	−0.375788
$778$	14.2771	0.511858
$779$	5.03721	0.180477
$780$	0	0
$781$	15.8336	0.566572
$782$	66.1974	2.36721
$783$	−4.23750	−0.151436
$784$	2.08666	0.0745234
$785$	0	0
$786$	15.1508	0.540413
$787$	28.7711	1.02558	0.512789	−	0.858515i	$-0.328612\pi$
$787$	28.7711	1.02558	0.512789	+	0.858515i	$0.328612\pi$
$788$	−53.1031	−1.89172
$789$	9.37068	0.333605
$790$	0	0
$791$	−4.23750	−0.150668
$792$	6.23069	0.221398
$793$	7.51805	0.266974
$794$	84.3298	2.99275
$795$	0	0
$796$	−49.5041	−1.75463
$797$	−19.5357	−0.691990	−0.345995	−	0.938236i	$-0.612459\pi$
$797$	−19.5357	−0.691990	−0.345995	+	0.938236i	$0.612459\pi$
$798$	−5.32415	−0.188473
$799$	−9.97942	−0.353046
$800$	0	0
$801$	−11.1371	−0.393509
$802$	56.4518	1.99338
$803$	−23.7569	−0.838362
$804$	−57.4630	−2.02656
$805$	0	0
$806$	−17.3242	−0.610217
$807$	10.3918	0.365809
$808$	−45.2241	−1.59098
$809$	−36.1824	−1.27211	−0.636053	−	0.771645i	$-0.719434\pi$
$809$	−36.1824	−1.27211	−0.636053	+	0.771645i	$0.719434\pi$
$810$	0	0
$811$	0.794087	0.0278842	0.0139421	−	0.999903i	$-0.495562\pi$
$811$	0.794087	0.0278842	0.0139421	+	0.999903i	$0.495562\pi$
$812$	−15.5180	−0.544577
$813$	5.13026	0.179926
$814$	−39.2680	−1.37634
$815$	0	0
$816$	9.93045	0.347635
$817$	2.04361	0.0714967
$818$	−9.41429	−0.329163
$819$	1.00000	0.0349428
$820$	0	0
$821$	−15.8525	−0.553256	−0.276628	−	0.960977i	$-0.589217\pi$
$821$	−15.8525	−0.553256	−0.276628	+	0.960977i	$0.589217\pi$
$822$	−42.8926	−1.49605
$823$	20.0412	0.698591	0.349295	−	0.937013i	$-0.386421\pi$
$823$	20.0412	0.698591	0.349295	+	0.937013i	$0.386421\pi$
$824$	−44.1008	−1.53633
$825$	0	0
$826$	29.3500	1.02122
$827$	−34.7848	−1.20959	−0.604794	−	0.796382i	$-0.706744\pi$
$827$	−34.7848	−1.20959	−0.604794	+	0.796382i	$0.706744\pi$
$828$	−21.4073	−0.743956
$829$	−30.6793	−1.06554	−0.532769	−	0.846261i	$-0.678848\pi$
$829$	−30.6793	−1.06554	−0.532769	+	0.846261i	$0.678848\pi$
$830$	0	0
$831$	8.04361	0.279030
$832$	11.1801	0.387601
$833$	−4.75902	−0.164890
$834$	−6.36721	−0.220478
$835$	0	0
$836$	−12.9088	−0.446461
$837$	−7.28055	−0.251653
$838$	−42.9706	−1.48439
$839$	39.8238	1.37487	0.687436	−	0.726245i	$-0.258736\pi$
$839$	39.8238	1.37487	0.687436	+	0.726245i	$0.258736\pi$
$840$	0	0
$841$	−11.0436	−0.380814
$842$	58.2875	2.00872
$843$	27.8525	0.959291
$844$	14.8834	0.512307
$845$	0	0
$846$	4.98971	0.171550
$847$	8.51805	0.292684
$848$	−2.26748	−0.0778655
$849$	23.8197	0.817491
$850$	0	0
$851$	61.2335	2.09906
$852$	−36.8054	−1.26093
$853$	−24.7505	−0.847440	−0.423720	−	0.905793i	$-0.639276\pi$
$853$	−24.7505	−0.847440	−0.423720	+	0.905793i	$0.639276\pi$
$854$	17.8893	0.612159
$855$	0	0
$856$	−9.03317	−0.308748
$857$	−6.47097	−0.221044	−0.110522	−	0.993874i	$-0.535252\pi$
$857$	−6.47097	−0.221044	−0.110522	+	0.993874i	$0.535252\pi$
$858$	3.74873	0.127980
$859$	31.0741	1.06023	0.530117	−	0.847925i	$-0.322148\pi$
$859$	31.0741	1.06023	0.530117	+	0.847925i	$0.322148\pi$
$860$	0	0
$861$	−2.25127	−0.0767230
$862$	−97.0314	−3.30490
$863$	−28.6391	−0.974885	−0.487442	−	0.873155i	$-0.662070\pi$
$863$	−28.6391	−0.974885	−0.487442	+	0.873155i	$0.662070\pi$
$864$	2.94464	0.100179
$865$	0	0
$866$	43.0195	1.46186
$867$	−5.64831	−0.191827
$868$	−26.6619	−0.904965
$869$	−18.5199	−0.628246
$870$	0	0
$871$	−15.6914	−0.531681
$872$	58.0145	1.96462
$873$	14.6047	0.494294
$874$	31.1233	1.05276
$875$	0	0
$876$	55.2230	1.86581
$877$	−0.00110921	−3.74555e−5 0	−1.87278e−5	−	1.00000i	$-0.500006\pi$
$877$	−0.00110921	−3.74555e−5 0	−1.87278e−5		1.00000i	$0.500006\pi$
$878$	49.3894	1.66681
$879$	−16.4612	−0.555223
$880$	0	0
$881$	23.9479	0.806824	0.403412	−	0.915019i	$-0.367824\pi$
$881$	23.9479	0.806824	0.403412	+	0.915019i	$0.367824\pi$
$882$	2.37951	0.0801223
$883$	−51.9965	−1.74982	−0.874911	−	0.484283i	$-0.839081\pi$
$883$	−51.9965	−1.74982	−0.874911	+	0.484283i	$0.839081\pi$
$884$	17.4279	0.586164
$885$	0	0
$886$	19.6082	0.658750
$887$	−10.8697	−0.364970	−0.182485	−	0.983209i	$-0.558414\pi$
$887$	−10.8697	−0.364970	−0.182485	+	0.983209i	$0.558414\pi$
$888$	41.4279	1.39023
$889$	4.84916	0.162636
$890$	0	0
$891$	1.57542	0.0527786
$892$	0.235055	0.00787021
$893$	−4.69192	−0.157009
$894$	17.7624	0.594062
$895$	0	0
$896$	20.7140	0.692005
$897$	−5.84568	−0.195182
$898$	85.9148	2.86701
$899$	30.8513	1.02895
$900$	0	0
$901$	5.17142	0.172285
$902$	−8.43940	−0.281001
$903$	−0.913344	−0.0303942
$904$	16.7590	0.557397
$905$	0	0
$906$	−30.9706	−1.02893
$907$	25.4892	0.846354	0.423177	−	0.906047i	$-0.360915\pi$
$907$	25.4892	0.846354	0.423177	+	0.906047i	$0.360915\pi$
$908$	6.17476	0.204917
$909$	−11.4349	−0.379271
$910$	0	0
$911$	42.2059	1.39834	0.699172	−	0.714953i	$-0.253552\pi$
$911$	42.2059	1.39834	0.699172	+	0.714953i	$0.253552\pi$
$912$	4.66889	0.154602
$913$	−11.6914	−0.386928
$914$	15.5632	0.514786
$915$	0	0
$916$	2.37416	0.0784445
$917$	6.36721	0.210264
$918$	11.3242	0.373753
$919$	−29.4033	−0.969925	−0.484963	−	0.874535i	$-0.661167\pi$
$919$	−29.4033	−0.969925	−0.484963	+	0.874535i	$0.661167\pi$
$920$	0	0
$921$	1.95639	0.0644654
$922$	−34.9758	−1.15187
$923$	−10.0504	−0.330814
$924$	5.76931	0.189797
$925$	0	0
$926$	−51.8197	−1.70290
$927$	−11.1508	−0.366242
$928$	−12.4779	−0.409608
$929$	−13.3759	−0.438849	−0.219425	−	0.975629i	$-0.570418\pi$
$929$	−13.3759	−0.438849	−0.219425	+	0.975629i	$0.570418\pi$
$930$	0	0
$931$	−2.23750	−0.0733311
$932$	−11.3789	−0.372730
$933$	6.67585	0.218557
$934$	−83.3033	−2.72577
$935$	0	0
$936$	−3.95493	−0.129271
$937$	24.9981	0.816653	0.408326	−	0.912836i	$-0.366113\pi$
$937$	24.9981	0.816653	0.408326	+	0.912836i	$0.366113\pi$
$938$	−37.3378	−1.21912
$939$	−6.78364	−0.221376
$940$	0	0
$941$	10.5369	0.343493	0.171746	−	0.985141i	$-0.445059\pi$
$941$	10.5369	0.343493	0.171746	+	0.985141i	$0.445059\pi$
$942$	−40.7451	−1.32755
$943$	13.1602	0.428555
$944$	−25.7377	−0.837692
$945$	0	0
$946$	−3.42388	−0.111320
$947$	52.7779	1.71505	0.857525	−	0.514442i	$-0.172001\pi$
$947$	52.7779	1.71505	0.857525	+	0.514442i	$0.172001\pi$
$948$	43.0497	1.39819
$949$	15.0797	0.489508
$950$	0	0
$951$	−30.6947	−0.995344
$952$	18.8216	0.610012
$953$	−21.8486	−0.707746	−0.353873	−	0.935294i	$-0.615135\pi$
$953$	−21.8486	−0.707746	−0.353873	+	0.935294i	$0.615135\pi$
$954$	−2.58571	−0.0837155
$955$	0	0
$956$	−55.1726	−1.78441
$957$	−6.67585	−0.215799
$958$	75.6241	2.44330
$959$	−18.0258	−0.582084
$960$	0	0
$961$	22.0064	0.709884
$962$	24.9254	0.803627
$963$	−2.28403	−0.0736017
$964$	94.2180	3.03456
$965$	0	0
$966$	−13.9099	−0.447543
$967$	−1.94143	−0.0624323	−0.0312162	−	0.999513i	$-0.509938\pi$
$967$	−1.94143	−0.0624323	−0.0312162	+	0.999513i	$0.509938\pi$
$968$	−33.6883	−1.08278
$969$	−10.6483	−0.342073
$970$	0	0
$971$	−14.9077	−0.478412	−0.239206	−	0.970969i	$-0.576887\pi$
$971$	−14.9077	−0.478412	−0.239206	+	0.970969i	$0.576887\pi$
$972$	−3.66208	−0.117461
$973$	−2.67585	−0.0857837
$974$	68.0105	2.17920
$975$	0	0
$976$	−15.6876	−0.502147
$977$	16.5353	0.529011	0.264505	−	0.964384i	$-0.414791\pi$
$977$	16.5353	0.529011	0.264505	+	0.964384i	$0.414791\pi$
$978$	−2.02058	−0.0646110
$979$	−17.5456	−0.560759
$980$	0	0
$981$	14.6689	0.468342
$982$	51.6734	1.64897
$983$	30.0694	0.959065	0.479533	−	0.877524i	$-0.340806\pi$
$983$	30.0694	0.959065	0.479533	+	0.877524i	$0.340806\pi$
$984$	8.90361	0.283837
$985$	0	0
$986$	−47.9861	−1.52819
$987$	2.09695	0.0667465
$988$	8.19389	0.260682
$989$	5.33912	0.169774
$990$	0	0
$991$	43.5902	1.38469	0.692345	−	0.721567i	$-0.256578\pi$
$991$	43.5902	1.38469	0.692345	+	0.721567i	$0.256578\pi$
$992$	−21.4386	−0.680677
$993$	13.8267	0.438777
$994$	−23.9151	−0.758541
$995$	0	0
$996$	27.1766	0.861125
$997$	−57.8577	−1.83237	−0.916186	−	0.400753i	$-0.868749\pi$
$997$	−57.8577	−1.83237	−0.916186	+	0.400753i	$0.868749\pi$
$998$	−101.580	−3.21546
$999$	10.4750	0.331414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bg.1.4		4
5.4	even	2		273.2.a.e.1.1	✓	4
15.14	odd	2		819.2.a.k.1.4		4
20.19	odd	2		4368.2.a.br.1.2		4
35.34	odd	2		1911.2.a.s.1.1		4
65.64	even	2		3549.2.a.w.1.4		4
105.104	even	2		5733.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.1	✓	4	5.4	even	2
819.2.a.k.1.4		4	15.14	odd	2
1911.2.a.s.1.1		4	35.34	odd	2
3549.2.a.w.1.4		4	65.64	even	2
4368.2.a.br.1.2		4	20.19	odd	2
5733.2.a.bf.1.4		4	105.104	even	2
6825.2.a.bg.1.4		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)

\(f(q)\)	\(=\)	\(q+2.37951 q^{2} -1.00000 q^{3} +3.66208 q^{4} -2.37951 q^{6} -1.00000 q^{7} +3.95493 q^{8} +1.00000 q^{9} +1.57542 q^{11} -3.66208 q^{12} -1.00000 q^{13} -2.37951 q^{14} +2.08666 q^{16} -4.75902 q^{17} +2.37951 q^{18} -2.23750 q^{19} +1.00000 q^{21} +3.74873 q^{22} -5.84568 q^{23} -3.95493 q^{24} -2.37951 q^{26} -1.00000 q^{27} -3.66208 q^{28} +4.23750 q^{29} +7.28055 q^{31} -2.94464 q^{32} -1.57542 q^{33} -11.3242 q^{34} +3.66208 q^{36} -10.4750 q^{37} -5.32415 q^{38} +1.00000 q^{39} -2.25127 q^{41} +2.37951 q^{42} -0.913344 q^{43} +5.76931 q^{44} -13.9099 q^{46} +2.09695 q^{47} -2.08666 q^{48} +1.00000 q^{49} +4.75902 q^{51} -3.66208 q^{52} -1.08666 q^{53} -2.37951 q^{54} -3.95493 q^{56} +2.23750 q^{57} +10.0832 q^{58} -12.3344 q^{59} -7.51805 q^{61} +17.3242 q^{62} -1.00000 q^{63} -11.1801 q^{64} -3.74873 q^{66} +15.6914 q^{67} -17.4279 q^{68} +5.84568 q^{69} +10.0504 q^{71} +3.95493 q^{72} -15.0797 q^{73} -24.9254 q^{74} -8.19389 q^{76} -1.57542 q^{77} +2.37951 q^{78} -11.7555 q^{79} +1.00000 q^{81} -5.35692 q^{82} -7.42110 q^{83} +3.66208 q^{84} -2.17331 q^{86} -4.23750 q^{87} +6.23069 q^{88} -11.1371 q^{89} +1.00000 q^{91} -21.4073 q^{92} -7.28055 q^{93} +4.98971 q^{94} +2.94464 q^{96} +14.6047 q^{97} +2.37951 q^{98} +1.57542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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