LMFDB - Embedded newform 6825.2.a.bi.1.3

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			1.3
Root			$-0.318459$ 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)

$f(q)$	$=$	\(q+0.318459 q^{2} +1.00000 q^{3} -1.89858 q^{4} +0.318459 q^{6} +1.00000 q^{7} -1.24154 q^{8} +1.00000 q^{9} -0.559998 q^{11} -1.89858 q^{12} +1.00000 q^{13} +0.318459 q^{14} +3.40179 q^{16} -4.38166 q^{17} +0.318459 q^{18} +1.03871 q^{19} +1.00000 q^{21} -0.178336 q^{22} +3.28025 q^{23} -1.24154 q^{24} +0.318459 q^{26} +1.00000 q^{27} -1.89858 q^{28} -9.19896 q^{29} -4.24154 q^{31} +3.56641 q^{32} -0.559998 q^{33} -1.39538 q^{34} -1.89858 q^{36} -4.73397 q^{37} +0.330785 q^{38} +1.00000 q^{39} +4.44486 q^{41} +0.318459 q^{42} +3.46499 q^{43} +1.06320 q^{44} +1.04462 q^{46} +0.769240 q^{47} +3.40179 q^{48} +1.00000 q^{49} -4.38166 q^{51} -1.89858 q^{52} -0.695259 q^{53} +0.318459 q^{54} -1.24154 q^{56} +1.03871 q^{57} -2.92949 q^{58} +4.96820 q^{59} -6.42628 q^{61} -1.35076 q^{62} +1.00000 q^{63} -5.66782 q^{64} -0.178336 q^{66} -8.96024 q^{67} +8.31895 q^{68} +3.28025 q^{69} +11.3205 q^{71} -1.24154 q^{72} -10.1529 q^{73} -1.50757 q^{74} -1.97207 q^{76} -0.559998 q^{77} +0.318459 q^{78} -3.12795 q^{79} +1.00000 q^{81} +1.41551 q^{82} -11.2729 q^{83} -1.89858 q^{84} +1.10346 q^{86} -9.19896 q^{87} +0.695259 q^{88} +3.82012 q^{89} +1.00000 q^{91} -6.22782 q^{92} -4.24154 q^{93} +0.244971 q^{94} +3.56641 q^{96} -8.52959 q^{97} +0.318459 q^{98} -0.559998 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.318459	0.225184	0.112592	−	0.993641i	$-0.464085\pi$
$2$	0.318459	0.225184	0.112592	+	0.993641i	$0.464085\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.89858	−0.949292
$5$	0	0
$5$	0	0
$6$	0.318459	0.130010
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.24154	−0.438950
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.559998	−0.168846	−0.0844228	−	0.996430i	$-0.526905\pi$
$11$	−0.559998	−0.168846	−0.0844228	+	0.996430i	$0.526905\pi$
$12$	−1.89858	−0.548074
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.318459	0.0851117
$15$	0	0
$16$	3.40179	0.850447
$17$	−4.38166	−1.06271	−0.531354	−	0.847150i	$-0.678317\pi$
$17$	−4.38166	−1.06271	−0.531354	+	0.847150i	$0.678317\pi$
$18$	0.318459	0.0750615
$19$	1.03871	0.238296	0.119148	−	0.992877i	$-0.461984\pi$
$19$	1.03871	0.238296	0.119148	+	0.992877i	$0.461984\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−0.178336	−0.0380214
$23$	3.28025	0.683978	0.341989	−	0.939704i	$-0.388899\pi$
$23$	3.28025	0.683978	0.341989	+	0.939704i	$0.388899\pi$
$24$	−1.24154	−0.253428
$25$	0	0
$26$	0.318459	0.0624549
$27$	1.00000	0.192450
$28$	−1.89858	−0.358799
$29$	−9.19896	−1.70820	−0.854102	−	0.520106i	$-0.825892\pi$
$29$	−9.19896	−1.70820	−0.854102	+	0.520106i	$0.825892\pi$
$30$	0	0
$31$	−4.24154	−0.761803	−0.380901	−	0.924616i	$-0.624386\pi$
$31$	−4.24154	−0.761803	−0.380901	+	0.924616i	$0.624386\pi$
$32$	3.56641	0.630458
$33$	−0.559998	−0.0974831
$34$	−1.39538	−0.239306
$35$	0	0
$36$	−1.89858	−0.316431
$37$	−4.73397	−0.778259	−0.389130	−	0.921183i	$-0.627224\pi$
$37$	−4.73397	−0.778259	−0.389130	+	0.921183i	$0.627224\pi$
$38$	0.330785	0.0536605
$39$	1.00000	0.160128
$40$	0	0
$41$	4.44486	0.694171	0.347086	−	0.937833i	$-0.387171\pi$
$41$	4.44486	0.694171	0.347086	+	0.937833i	$0.387171\pi$
$42$	0.318459	0.0491393
$43$	3.46499	0.528406	0.264203	−	0.964467i	$-0.414891\pi$
$43$	3.46499	0.528406	0.264203	+	0.964467i	$0.414891\pi$
$44$	1.06320	0.160284
$45$	0	0
$46$	1.04462	0.154021
$47$	0.769240	0.112205	0.0561026	−	0.998425i	$-0.482133\pi$
$47$	0.769240	0.112205	0.0561026	+	0.998425i	$0.482133\pi$
$48$	3.40179	0.491006
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.38166	−0.613555
$52$	−1.89858	−0.263286
$53$	−0.695259	−0.0955011	−0.0477506	−	0.998859i	$-0.515205\pi$
$53$	−0.695259	−0.0955011	−0.0477506	+	0.998859i	$0.515205\pi$
$54$	0.318459	0.0433368
$55$	0	0
$56$	−1.24154	−0.165908
$57$	1.03871	0.137580
$58$	−2.92949	−0.384661
$59$	4.96820	0.646804	0.323402	−	0.946262i	$-0.395173\pi$
$59$	4.96820	0.646804	0.323402	+	0.946262i	$0.395173\pi$
$60$	0	0
$61$	−6.42628	−0.822801	−0.411401	−	0.911455i	$-0.634960\pi$
$61$	−6.42628	−0.822801	−0.411401	+	0.911455i	$0.634960\pi$
$62$	−1.35076	−0.171546
$63$	1.00000	0.125988
$64$	−5.66782	−0.708478
$65$	0	0
$66$	−0.178336	−0.0219517
$67$	−8.96024	−1.09467	−0.547334	−	0.836914i	$-0.684357\pi$
$67$	−8.96024	−1.09467	−0.547334	+	0.836914i	$0.684357\pi$
$68$	8.31895	1.00882
$69$	3.28025	0.394895
$70$	0	0
$71$	11.3205	1.34350	0.671748	−	0.740780i	$-0.265544\pi$
$71$	11.3205	1.34350	0.671748	+	0.740780i	$0.265544\pi$
$72$	−1.24154	−0.146317
$73$	−10.1529	−1.18831	−0.594156	−	0.804350i	$-0.702514\pi$
$73$	−10.1529	−1.18831	−0.594156	+	0.804350i	$0.702514\pi$
$74$	−1.50757	−0.175252
$75$	0	0
$76$	−1.97207	−0.226212
$77$	−0.559998	−0.0638177
$78$	0.318459	0.0360584
$79$	−3.12795	−0.351922	−0.175961	−	0.984397i	$-0.556303\pi$
$79$	−3.12795	−0.351922	−0.175961	+	0.984397i	$0.556303\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.41551	0.156317
$83$	−11.2729	−1.23737	−0.618683	−	0.785641i	$-0.712333\pi$
$83$	−11.2729	−1.23737	−0.618683	+	0.785641i	$0.712333\pi$
$84$	−1.89858	−0.207152
$85$	0	0
$86$	1.10346	0.118989
$87$	−9.19896	−0.986232
$88$	0.695259	0.0741148
$89$	3.82012	0.404931	0.202466	−	0.979289i	$-0.435105\pi$
$89$	3.82012	0.404931	0.202466	+	0.979289i	$0.435105\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−6.22782	−0.649295
$93$	−4.24154	−0.439827
$94$	0.244971	0.0252668
$95$	0	0
$96$	3.56641	0.363995
$97$	−8.52959	−0.866048	−0.433024	−	0.901382i	$-0.642553\pi$
$97$	−8.52959	−0.866048	−0.433024	+	0.901382i	$0.642553\pi$
$98$	0.318459	0.0321692
$99$	−0.559998	−0.0562819
$100$	0	0
$101$	−6.15870	−0.612814	−0.306407	−	0.951901i	$-0.599127\pi$
$101$	−6.15870	−0.612814	−0.306407	+	0.951901i	$0.599127\pi$
$102$	−1.39538	−0.138163
$103$	12.7251	1.25384	0.626921	−	0.779083i	$-0.284315\pi$
$103$	12.7251	1.25384	0.626921	+	0.779083i	$0.284315\pi$
$104$	−1.24154	−0.121743
$105$	0	0
$106$	−0.221411	−0.0215054
$107$	−2.44000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$107$	−2.44000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$108$	−1.89858	−0.182691
$109$	7.95587	0.762034	0.381017	−	0.924568i	$-0.375574\pi$
$109$	7.95587	0.762034	0.381017	+	0.924568i	$0.375574\pi$
$110$	0	0
$111$	−4.73397	−0.449328
$112$	3.40179	0.321439
$113$	17.1024	1.60886	0.804429	−	0.594048i	$-0.202471\pi$
$113$	17.1024	1.60886	0.804429	+	0.594048i	$0.202471\pi$
$114$	0.330785	0.0309809
$115$	0	0
$116$	17.4650	1.62158
$117$	1.00000	0.0924500
$118$	1.58217	0.145650
$119$	−4.38166	−0.401666
$120$	0	0
$121$	−10.6864	−0.971491
$122$	−2.04651	−0.185282
$123$	4.44486	0.400780
$124$	8.05292	0.723173
$125$	0	0
$126$	0.318459	0.0283706
$127$	−18.4374	−1.63605	−0.818027	−	0.575179i	$-0.804932\pi$
$127$	−18.4374	−1.63605	−0.818027	+	0.575179i	$0.804932\pi$
$128$	−8.93778	−0.789996
$129$	3.46499	0.305075
$130$	0	0
$131$	19.4934	1.70315	0.851574	−	0.524235i	$-0.175649\pi$
$131$	19.4934	1.70315	0.851574	+	0.524235i	$0.175649\pi$
$132$	1.06320	0.0925399
$133$	1.03871	0.0900673
$134$	−2.85347	−0.246502
$135$	0	0
$136$	5.44000	0.466476
$137$	−13.9883	−1.19510	−0.597552	−	0.801830i	$-0.703860\pi$
$137$	−13.9883	−1.19510	−0.597552	+	0.801830i	$0.703860\pi$
$138$	1.04462	0.0889242
$139$	−5.97987	−0.507206	−0.253603	−	0.967308i	$-0.581616\pi$
$139$	−5.97987	−0.507206	−0.253603	+	0.967308i	$0.581616\pi$
$140$	0	0
$141$	0.769240	0.0647817
$142$	3.60511	0.302534
$143$	−0.559998	−0.0468294
$144$	3.40179	0.283482
$145$	0	0
$146$	−3.23329	−0.267589
$147$	1.00000	0.0824786
$148$	8.98783	0.738795
$149$	−20.5429	−1.68294	−0.841470	−	0.540304i	$-0.818309\pi$
$149$	−20.5429	−1.68294	−0.841470	+	0.540304i	$0.818309\pi$
$150$	0	0
$151$	−10.4988	−0.854383	−0.427192	−	0.904161i	$-0.640497\pi$
$151$	−10.4988	−0.854383	−0.427192	+	0.904161i	$0.640497\pi$
$152$	−1.28959	−0.104600
$153$	−4.38166	−0.354236
$154$	−0.178336	−0.0143707
$155$	0	0
$156$	−1.89858	−0.152008
$157$	−3.88576	−0.310118	−0.155059	−	0.987905i	$-0.549557\pi$
$157$	−3.88576	−0.310118	−0.155059	+	0.987905i	$0.549557\pi$
$158$	−0.996125	−0.0792474
$159$	−0.695259	−0.0551376
$160$	0	0
$161$	3.28025	0.258520
$162$	0.318459	0.0250205
$163$	25.2567	1.97826	0.989130	−	0.147043i	$-0.0469756\pi$
$163$	25.2567	1.97826	0.989130	+	0.147043i	$0.0469756\pi$
$164$	−8.43895	−0.658971
$165$	0	0
$166$	−3.58997	−0.278635
$167$	2.22782	0.172394	0.0861970	−	0.996278i	$-0.472529\pi$
$167$	2.22782	0.172394	0.0861970	+	0.996278i	$0.472529\pi$
$168$	−1.24154	−0.0957868
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.03871	0.0794319
$172$	−6.57858	−0.501612
$173$	−17.2547	−1.31185	−0.655925	−	0.754826i	$-0.727721\pi$
$173$	−17.2547	−1.31185	−0.655925	+	0.754826i	$0.727721\pi$
$174$	−2.92949	−0.222084
$175$	0	0
$176$	−1.90499	−0.143594
$177$	4.96820	0.373432
$178$	1.21655	0.0911843
$179$	−14.4082	−1.07692	−0.538460	−	0.842651i	$-0.680994\pi$
$179$	−14.4082	−1.07692	−0.538460	+	0.842651i	$0.680994\pi$
$180$	0	0
$181$	4.62165	0.343525	0.171762	−	0.985138i	$-0.445054\pi$
$181$	4.62165	0.343525	0.171762	+	0.985138i	$0.445054\pi$
$182$	0.318459	0.0236057
$183$	−6.42628	−0.475045
$184$	−4.07255	−0.300232
$185$	0	0
$186$	−1.35076	−0.0990422
$187$	2.45372	0.179434
$188$	−1.46047	−0.106515
$189$	1.00000	0.0727393
$190$	0	0
$191$	−2.47667	−0.179205	−0.0896027	−	0.995978i	$-0.528560\pi$
$191$	−2.47667	−0.179205	−0.0896027	+	0.995978i	$0.528560\pi$
$192$	−5.66782	−0.409040
$193$	−24.1274	−1.73673	−0.868364	−	0.495928i	$-0.834828\pi$
$193$	−24.1274	−1.73673	−0.868364	+	0.495928i	$0.834828\pi$
$194$	−2.71632	−0.195021
$195$	0	0
$196$	−1.89858	−0.135613
$197$	4.52472	0.322373	0.161187	−	0.986924i	$-0.448468\pi$
$197$	4.52472	0.322373	0.161187	+	0.986924i	$0.448468\pi$
$198$	−0.178336	−0.0126738
$199$	−26.2513	−1.86090	−0.930451	−	0.366416i	$-0.880585\pi$
$199$	−26.2513	−1.86090	−0.930451	+	0.366416i	$0.880585\pi$
$200$	0	0
$201$	−8.96024	−0.632006
$202$	−1.96129	−0.137996
$203$	−9.19896	−0.645640
$204$	8.31895	0.582443
$205$	0	0
$206$	4.05242	0.282346
$207$	3.28025	0.227993
$208$	3.40179	0.235872
$209$	−0.581673	−0.0402352
$210$	0	0
$211$	5.82012	0.400673	0.200337	−	0.979727i	$-0.435796\pi$
$211$	5.82012	0.400673	0.200337	+	0.979727i	$0.435796\pi$
$212$	1.32001	0.0906585
$213$	11.3205	0.775668
$214$	−0.777040	−0.0531174
$215$	0	0
$216$	−1.24154	−0.0844760
$217$	−4.24154	−0.287934
$218$	2.53362	0.171598
$219$	−10.1529	−0.686072
$220$	0	0
$221$	−4.38166	−0.294742
$222$	−1.50757	−0.101182
$223$	21.0182	1.40748	0.703741	−	0.710457i	$-0.251512\pi$
$223$	21.0182	1.40748	0.703741	+	0.710457i	$0.251512\pi$
$224$	3.56641	0.238291
$225$	0	0
$226$	5.44641	0.362290
$227$	3.56000	0.236285	0.118143	−	0.992997i	$-0.462306\pi$
$227$	3.56000	0.236285	0.118143	+	0.992997i	$0.462306\pi$
$228$	−1.97207	−0.130604
$229$	−22.2275	−1.46884	−0.734418	−	0.678697i	$-0.762545\pi$
$229$	−22.2275	−1.46884	−0.734418	+	0.678697i	$0.762545\pi$
$230$	0	0
$231$	−0.559998	−0.0368451
$232$	11.4209	0.749816
$233$	−4.48998	−0.294148	−0.147074	−	0.989125i	$-0.546986\pi$
$233$	−4.48998	−0.294148	−0.147074	+	0.989125i	$0.546986\pi$
$234$	0.318459	0.0208183
$235$	0	0
$236$	−9.43254	−0.614006
$237$	−3.12795	−0.203182
$238$	−1.39538	−0.0904490
$239$	−28.0085	−1.81172	−0.905858	−	0.423581i	$-0.860773\pi$
$239$	−28.0085	−1.81172	−0.905858	+	0.423581i	$0.860773\pi$
$240$	0	0
$241$	−14.6468	−0.943481	−0.471740	−	0.881738i	$-0.656374\pi$
$241$	−14.6468	−0.943481	−0.471740	+	0.881738i	$0.656374\pi$
$242$	−3.40318	−0.218765
$243$	1.00000	0.0641500
$244$	12.2008	0.781079
$245$	0	0
$246$	1.41551	0.0902494
$247$	1.03871	0.0660913
$248$	5.26603	0.334394
$249$	−11.2729	−0.714393
$250$	0	0
$251$	−11.5619	−0.729780	−0.364890	−	0.931051i	$-0.618893\pi$
$251$	−11.5619	−0.729780	−0.364890	+	0.931051i	$0.618893\pi$
$252$	−1.89858	−0.119600
$253$	−1.83693	−0.115487
$254$	−5.87155	−0.368414
$255$	0	0
$256$	8.48933	0.530583
$257$	21.8506	1.36300	0.681501	−	0.731817i	$-0.261328\pi$
$257$	21.8506	1.36300	0.681501	+	0.731817i	$0.261328\pi$
$258$	1.10346	0.0686983
$259$	−4.73397	−0.294154
$260$	0	0
$261$	−9.19896	−0.569401
$262$	6.20785	0.383522
$263$	20.8055	1.28292	0.641460	−	0.767157i	$-0.278329\pi$
$263$	20.8055	1.28292	0.641460	+	0.767157i	$0.278329\pi$
$264$	0.695259	0.0427902
$265$	0	0
$266$	0.330785	0.0202818
$267$	3.82012	0.233787
$268$	17.0118	1.03916
$269$	9.95040	0.606686	0.303343	−	0.952881i	$-0.401897\pi$
$269$	9.95040	0.606686	0.303343	+	0.952881i	$0.401897\pi$
$270$	0	0
$271$	−9.26653	−0.562901	−0.281451	−	0.959576i	$-0.590816\pi$
$271$	−9.26653	−0.562901	−0.281451	+	0.959576i	$0.590816\pi$
$272$	−14.9055	−0.903778
$273$	1.00000	0.0605228
$274$	−4.45471	−0.269119
$275$	0	0
$276$	−6.22782	−0.374871
$277$	−24.8869	−1.49531	−0.747655	−	0.664088i	$-0.768820\pi$
$277$	−24.8869	−1.49531	−0.747655	+	0.664088i	$0.768820\pi$
$278$	−1.90434	−0.114215
$279$	−4.24154	−0.253934
$280$	0	0
$281$	3.67358	0.219148	0.109574	−	0.993979i	$-0.465051\pi$
$281$	3.67358	0.219148	0.109574	+	0.993979i	$0.465051\pi$
$282$	0.244971	0.0145878
$283$	12.4801	0.741865	0.370932	−	0.928660i	$-0.379038\pi$
$283$	12.4801	0.741865	0.370932	+	0.928660i	$0.379038\pi$
$284$	−21.4929	−1.27537
$285$	0	0
$286$	−0.178336	−0.0105452
$287$	4.44486	0.262372
$288$	3.56641	0.210153
$289$	2.19896	0.129350
$290$	0	0
$291$	−8.52959	−0.500013
$292$	19.2762	1.12806
$293$	−19.0921	−1.11537	−0.557687	−	0.830052i	$-0.688311\pi$
$293$	−19.0921	−1.11537	−0.557687	+	0.830052i	$0.688311\pi$
$294$	0.318459	0.0185729
$295$	0	0
$296$	5.87740	0.341617
$297$	−0.559998	−0.0324944
$298$	−6.54207	−0.378972
$299$	3.28025	0.189701
$300$	0	0
$301$	3.46499	0.199719
$302$	−3.34345	−0.192394
$303$	−6.15870	−0.353808
$304$	3.53346	0.202658
$305$	0	0
$306$	−1.39538	−0.0797685
$307$	−21.2812	−1.21458	−0.607292	−	0.794479i	$-0.707744\pi$
$307$	−21.2812	−1.21458	−0.607292	+	0.794479i	$0.707744\pi$
$308$	1.06320	0.0605816
$309$	12.7251	0.723906
$310$	0	0
$311$	−17.8373	−1.01146	−0.505730	−	0.862692i	$-0.668777\pi$
$311$	−17.8373	−1.01146	−0.505730	+	0.862692i	$0.668777\pi$
$312$	−1.24154	−0.0702883
$313$	−10.0969	−0.570712	−0.285356	−	0.958422i	$-0.592112\pi$
$313$	−10.0969	−0.570712	−0.285356	+	0.958422i	$0.592112\pi$
$314$	−1.23746	−0.0698337
$315$	0	0
$316$	5.93868	0.334077
$317$	6.22451	0.349603	0.174802	−	0.984604i	$-0.444072\pi$
$317$	6.22451	0.349603	0.174802	+	0.984604i	$0.444072\pi$
$318$	−0.221411	−0.0124161
$319$	5.15139	0.288423
$320$	0	0
$321$	−2.44000	−0.136188
$322$	1.04462	0.0582146
$323$	−4.55126	−0.253239
$324$	−1.89858	−0.105477
$325$	0	0
$326$	8.04323	0.445473
$327$	7.95587	0.439961
$328$	−5.51847	−0.304707
$329$	0.769240	0.0424095
$330$	0	0
$331$	16.7011	0.917976	0.458988	−	0.888443i	$-0.348212\pi$
$331$	16.7011	0.917976	0.458988	+	0.888443i	$0.348212\pi$
$332$	21.4026	1.17462
$333$	−4.73397	−0.259420
$334$	0.709469	0.0388204
$335$	0	0
$336$	3.40179	0.185583
$337$	−24.3002	−1.32372	−0.661859	−	0.749629i	$-0.730232\pi$
$337$	−24.3002	−1.32372	−0.661859	+	0.749629i	$0.730232\pi$
$338$	0.318459	0.0173219
$339$	17.1024	0.928875
$340$	0	0
$341$	2.37525	0.128627
$342$	0.330785	0.0178868
$343$	1.00000	0.0539949
$344$	−4.30192	−0.231944
$345$	0	0
$346$	−5.49491	−0.295408
$347$	−0.923080	−0.0495535	−0.0247768	−	0.999693i	$-0.507887\pi$
$347$	−0.923080	−0.0495535	−0.0247768	+	0.999693i	$0.507887\pi$
$348$	17.4650	0.936222
$349$	7.18475	0.384591	0.192295	−	0.981337i	$-0.438407\pi$
$349$	7.18475	0.384591	0.192295	+	0.981337i	$0.438407\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−1.99718	−0.106450
$353$	11.3220	0.602612	0.301306	−	0.953528i	$-0.402577\pi$
$353$	11.3220	0.602612	0.301306	+	0.953528i	$0.402577\pi$
$354$	1.58217	0.0840912
$355$	0	0
$356$	−7.25281	−0.384398
$357$	−4.38166	−0.231902
$358$	−4.58842	−0.242505
$359$	1.03373	0.0545580	0.0272790	−	0.999628i	$-0.491316\pi$
$359$	1.03373	0.0545580	0.0272790	+	0.999628i	$0.491316\pi$
$360$	0	0
$361$	−17.9211	−0.943215
$362$	1.47181	0.0773564
$363$	−10.6864	−0.560891
$364$	−1.89858	−0.0995128
$365$	0	0
$366$	−2.04651	−0.106973
$367$	18.1966	0.949856	0.474928	−	0.880025i	$-0.342474\pi$
$367$	18.1966	0.949856	0.474928	+	0.880025i	$0.342474\pi$
$368$	11.1587	0.581688
$369$	4.44486	0.231390
$370$	0	0
$371$	−0.695259	−0.0360960
$372$	8.05292	0.417524
$373$	7.09048	0.367131	0.183566	−	0.983007i	$-0.441236\pi$
$373$	7.09048	0.367131	0.183566	+	0.983007i	$0.441236\pi$
$374$	0.781409	0.0404057
$375$	0	0
$376$	−0.955041	−0.0492525
$377$	−9.19896	−0.473770
$378$	0.318459	0.0163798
$379$	17.3071	0.889007	0.444503	−	0.895777i	$-0.353380\pi$
$379$	17.3071	0.889007	0.444503	+	0.895777i	$0.353380\pi$
$380$	0	0
$381$	−18.4374	−0.944577
$382$	−0.788717	−0.0403543
$383$	7.62896	0.389822	0.194911	−	0.980821i	$-0.437558\pi$
$383$	7.62896	0.389822	0.194911	+	0.980821i	$0.437558\pi$
$384$	−8.93778	−0.456104
$385$	0	0
$386$	−7.68358	−0.391084
$387$	3.46499	0.176135
$388$	16.1941	0.822133
$389$	15.0423	0.762675	0.381337	−	0.924436i	$-0.375464\pi$
$389$	15.0423	0.762675	0.381337	+	0.924436i	$0.375464\pi$
$390$	0	0
$391$	−14.3729	−0.726870
$392$	−1.24154	−0.0627072
$393$	19.4934	0.983313
$394$	1.44094	0.0725934
$395$	0	0
$396$	1.06320	0.0534279
$397$	8.11690	0.407375	0.203688	−	0.979036i	$-0.434707\pi$
$397$	8.11690	0.407375	0.203688	+	0.979036i	$0.434707\pi$
$398$	−8.35995	−0.419046
$399$	1.03871	0.0520004
$400$	0	0
$401$	−18.7107	−0.934370	−0.467185	−	0.884160i	$-0.654732\pi$
$401$	−18.7107	−0.934370	−0.467185	+	0.884160i	$0.654732\pi$
$402$	−2.85347	−0.142318
$403$	−4.24154	−0.211286
$404$	11.6928	0.581739
$405$	0	0
$406$	−2.92949	−0.145388
$407$	2.65101	0.131406
$408$	5.44000	0.269320
$409$	−0.958197	−0.0473798	−0.0236899	−	0.999719i	$-0.507541\pi$
$409$	−0.958197	−0.0473798	−0.0236899	+	0.999719i	$0.507541\pi$
$410$	0	0
$411$	−13.9883	−0.689993
$412$	−24.1597	−1.19026
$413$	4.96820	0.244469
$414$	1.04462	0.0513404
$415$	0	0
$416$	3.56641	0.174858
$417$	−5.97987	−0.292836
$418$	−0.185239	−0.00906034
$419$	−31.2929	−1.52876	−0.764379	−	0.644767i	$-0.776954\pi$
$419$	−31.2929	−1.52876	−0.764379	+	0.644767i	$0.776954\pi$
$420$	0	0
$421$	20.9643	1.02174	0.510869	−	0.859658i	$-0.329324\pi$
$421$	20.9643	1.02174	0.510869	+	0.859658i	$0.329324\pi$
$422$	1.85347	0.0902254
$423$	0.769240	0.0374017
$424$	0.863191	0.0419202
$425$	0	0
$426$	3.60511	0.174668
$427$	−6.42628	−0.310990
$428$	4.63255	0.223923
$429$	−0.559998	−0.0270369
$430$	0	0
$431$	−16.9304	−0.815508	−0.407754	−	0.913092i	$-0.633688\pi$
$431$	−16.9304	−0.815508	−0.407754	+	0.913092i	$0.633688\pi$
$432$	3.40179	0.163669
$433$	−24.6154	−1.18294	−0.591471	−	0.806326i	$-0.701453\pi$
$433$	−24.6154	−1.18294	−0.591471	+	0.806326i	$0.701453\pi$
$434$	−1.35076	−0.0648384
$435$	0	0
$436$	−15.1049	−0.723393
$437$	3.40721	0.162989
$438$	−3.23329	−0.154493
$439$	−17.3333	−0.827274	−0.413637	−	0.910442i	$-0.635742\pi$
$439$	−17.3333	−0.827274	−0.413637	+	0.910442i	$0.635742\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−1.39538	−0.0663714
$443$	12.0744	0.573674	0.286837	−	0.957979i	$-0.407396\pi$
$443$	12.0744	0.573674	0.286837	+	0.957979i	$0.407396\pi$
$444$	8.98783	0.426544
$445$	0	0
$446$	6.69342	0.316943
$447$	−20.5429	−0.971646
$448$	−5.66782	−0.267779
$449$	−17.2235	−0.812825	−0.406412	−	0.913690i	$-0.633220\pi$
$449$	−17.2235	−0.812825	−0.406412	+	0.913690i	$0.633220\pi$
$450$	0	0
$451$	−2.48911	−0.117208
$452$	−32.4703	−1.52728
$453$	−10.4988	−0.493278
$454$	1.13371	0.0532078
$455$	0	0
$456$	−1.28959	−0.0603908
$457$	28.4970	1.33303	0.666517	−	0.745490i	$-0.267784\pi$
$457$	28.4970	1.33303	0.666517	+	0.745490i	$0.267784\pi$
$458$	−7.07856	−0.330759
$459$	−4.38166	−0.204518
$460$	0	0
$461$	−27.4323	−1.27765	−0.638824	−	0.769353i	$-0.720579\pi$
$461$	−27.4323	−1.27765	−0.638824	+	0.769353i	$0.720579\pi$
$462$	−0.178336	−0.00829695
$463$	−25.5851	−1.18904	−0.594520	−	0.804081i	$-0.702658\pi$
$463$	−25.5851	−1.18904	−0.594520	+	0.804081i	$0.702658\pi$
$464$	−31.2929	−1.45274
$465$	0	0
$466$	−1.42987	−0.0662376
$467$	−5.58155	−0.258284	−0.129142	−	0.991626i	$-0.541222\pi$
$467$	−5.58155	−0.258284	−0.129142	+	0.991626i	$0.541222\pi$
$468$	−1.89858	−0.0877621
$469$	−8.96024	−0.413745
$470$	0	0
$471$	−3.88576	−0.179047
$472$	−6.16821	−0.283915
$473$	−1.94039	−0.0892191
$474$	−0.996125	−0.0457535
$475$	0	0
$476$	8.31895	0.381299
$477$	−0.695259	−0.0318337
$478$	−8.91954	−0.407970
$479$	−21.4429	−0.979752	−0.489876	−	0.871792i	$-0.662958\pi$
$479$	−21.4429	−0.979752	−0.489876	+	0.871792i	$0.662958\pi$
$480$	0	0
$481$	−4.73397	−0.215850
$482$	−4.66439	−0.212457
$483$	3.28025	0.149256
$484$	20.2890	0.922229
$485$	0	0
$486$	0.318459	0.0144456
$487$	14.9191	0.676048	0.338024	−	0.941137i	$-0.390241\pi$
$487$	14.9191	0.676048	0.338024	+	0.941137i	$0.390241\pi$
$488$	7.97848	0.361169
$489$	25.2567	1.14215
$490$	0	0
$491$	3.73298	0.168467	0.0842335	−	0.996446i	$-0.473156\pi$
$491$	3.73298	0.168467	0.0842335	+	0.996446i	$0.473156\pi$
$492$	−8.43895	−0.380457
$493$	40.3067	1.81532
$494$	0.330785	0.0148827
$495$	0	0
$496$	−14.4288	−0.647873
$497$	11.3205	0.507794
$498$	−3.58997	−0.160870
$499$	16.2285	0.726486	0.363243	−	0.931694i	$-0.381669\pi$
$499$	16.2285	0.726486	0.363243	+	0.931694i	$0.381669\pi$
$500$	0	0
$501$	2.22782	0.0995317
$502$	−3.68198	−0.164335
$503$	11.1999	0.499379	0.249689	−	0.968326i	$-0.419672\pi$
$503$	11.1999	0.499379	0.249689	+	0.968326i	$0.419672\pi$
$504$	−1.24154	−0.0553025
$505$	0	0
$506$	−0.584987	−0.0260058
$507$	1.00000	0.0444116
$508$	35.0049	1.55309
$509$	−14.2965	−0.633683	−0.316841	−	0.948479i	$-0.602622\pi$
$509$	−14.2965	−0.633683	−0.316841	+	0.948479i	$0.602622\pi$
$510$	0	0
$511$	−10.1529	−0.449140
$512$	20.5791	0.909475
$513$	1.03871	0.0458600
$514$	6.95851	0.306927
$515$	0	0
$516$	−6.57858	−0.289606
$517$	−0.430772	−0.0189453
$518$	−1.50757	−0.0662390
$519$	−17.2547	−0.757397
$520$	0	0
$521$	−6.40522	−0.280618	−0.140309	−	0.990108i	$-0.544810\pi$
$521$	−6.40522	−0.280618	−0.140309	+	0.990108i	$0.544810\pi$
$522$	−2.92949	−0.128220
$523$	−24.9957	−1.09299	−0.546493	−	0.837464i	$-0.684038\pi$
$523$	−24.9957	−1.09299	−0.546493	+	0.837464i	$0.684038\pi$
$524$	−37.0099	−1.61678
$525$	0	0
$526$	6.62568	0.288894
$527$	18.5850	0.809575
$528$	−1.90499	−0.0829042
$529$	−12.2400	−0.532174
$530$	0	0
$531$	4.96820	0.215601
$532$	−1.97207	−0.0855001
$533$	4.44486	0.192528
$534$	1.21655	0.0526453
$535$	0	0
$536$	11.1245	0.480504
$537$	−14.4082	−0.621760
$538$	3.16879	0.136616
$539$	−0.559998	−0.0241208
$540$	0	0
$541$	−12.6561	−0.544130	−0.272065	−	0.962279i	$-0.587707\pi$
$541$	−12.6561	−0.544130	−0.272065	+	0.962279i	$0.587707\pi$
$542$	−2.95101	−0.126757
$543$	4.62165	0.198334
$544$	−15.6268	−0.669993
$545$	0	0
$546$	0.318459	0.0136288
$547$	29.2510	1.25068	0.625341	−	0.780352i	$-0.284960\pi$
$547$	29.2510	1.25068	0.625341	+	0.780352i	$0.284960\pi$
$548$	26.5580	1.13450
$549$	−6.42628	−0.274267
$550$	0	0
$551$	−9.55502	−0.407057
$552$	−4.07255	−0.173339
$553$	−3.12795	−0.133014
$554$	−7.92546	−0.336720
$555$	0	0
$556$	11.3533	0.481487
$557$	20.9412	0.887306	0.443653	−	0.896199i	$-0.353682\pi$
$557$	20.9412	0.887306	0.443653	+	0.896199i	$0.353682\pi$
$558$	−1.35076	−0.0571821
$559$	3.46499	0.146554
$560$	0	0
$561$	2.45372	0.103596
$562$	1.16989	0.0493486
$563$	18.3282	0.772441	0.386220	−	0.922407i	$-0.373780\pi$
$563$	18.3282	0.772441	0.386220	+	0.922407i	$0.373780\pi$
$564$	−1.46047	−0.0614967
$565$	0	0
$566$	3.97440	0.167056
$567$	1.00000	0.0419961
$568$	−14.0548	−0.589728
$569$	9.01535	0.377943	0.188972	−	0.981983i	$-0.439485\pi$
$569$	9.01535	0.377943	0.188972	+	0.981983i	$0.439485\pi$
$570$	0	0
$571$	−16.6491	−0.696743	−0.348371	−	0.937357i	$-0.613265\pi$
$571$	−16.6491	−0.696743	−0.348371	+	0.937357i	$0.613265\pi$
$572$	1.06320	0.0444547
$573$	−2.47667	−0.103464
$574$	1.41551	0.0590821
$575$	0	0
$576$	−5.66782	−0.236159
$577$	−11.9329	−0.496774	−0.248387	−	0.968661i	$-0.579900\pi$
$577$	−11.9329	−0.496774	−0.248387	+	0.968661i	$0.579900\pi$
$578$	0.700277	0.0291277
$579$	−24.1274	−1.00270
$580$	0	0
$581$	−11.2729	−0.467680
$582$	−2.71632	−0.112595
$583$	0.389343	0.0161250
$584$	12.6053	0.521610
$585$	0	0
$586$	−6.08005	−0.251165
$587$	41.8937	1.72914	0.864568	−	0.502516i	$-0.167592\pi$
$587$	41.8937	1.72914	0.864568	+	0.502516i	$0.167592\pi$
$588$	−1.89858	−0.0782963
$589$	−4.40571	−0.181534
$590$	0	0
$591$	4.52472	0.186122
$592$	−16.1040	−0.661868
$593$	0.756913	0.0310827	0.0155413	−	0.999879i	$-0.495053\pi$
$593$	0.756913	0.0310827	0.0155413	+	0.999879i	$0.495053\pi$
$594$	−0.178336	−0.00731722
$595$	0	0
$596$	39.0024	1.59760
$597$	−26.2513	−1.07439
$598$	1.04462	0.0427178
$599$	43.8116	1.79009	0.895047	−	0.445971i	$-0.147142\pi$
$599$	43.8116	1.79009	0.895047	+	0.445971i	$0.147142\pi$
$600$	0	0
$601$	−32.6738	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$601$	−32.6738	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$602$	1.10346	0.0449736
$603$	−8.96024	−0.364889
$604$	19.9329	0.811059
$605$	0	0
$606$	−1.96129	−0.0796721
$607$	−5.45870	−0.221562	−0.110781	−	0.993845i	$-0.535335\pi$
$607$	−5.45870	−0.221562	−0.110781	+	0.993845i	$0.535335\pi$
$608$	3.70445	0.150235
$609$	−9.19896	−0.372761
$610$	0	0
$611$	0.769240	0.0311201
$612$	8.31895	0.336274
$613$	9.19839	0.371520	0.185760	−	0.982595i	$-0.440525\pi$
$613$	9.19839	0.371520	0.185760	+	0.982595i	$0.440525\pi$
$614$	−6.77720	−0.273505
$615$	0	0
$616$	0.695259	0.0280128
$617$	16.5466	0.666142	0.333071	−	0.942902i	$-0.391915\pi$
$617$	16.5466	0.666142	0.333071	+	0.942902i	$0.391915\pi$
$618$	4.05242	0.163012
$619$	40.2010	1.61581	0.807907	−	0.589310i	$-0.200600\pi$
$619$	40.2010	1.61581	0.807907	+	0.589310i	$0.200600\pi$
$620$	0	0
$621$	3.28025	0.131632
$622$	−5.68045	−0.227765
$623$	3.82012	0.153050
$624$	3.40179	0.136181
$625$	0	0
$626$	−3.21546	−0.128515
$627$	−0.581673	−0.0232298
$628$	7.37745	0.294392
$629$	20.7426	0.827063
$630$	0	0
$631$	−16.5228	−0.657761	−0.328881	−	0.944372i	$-0.606671\pi$
$631$	−16.5228	−0.657761	−0.328881	+	0.944372i	$0.606671\pi$
$632$	3.88348	0.154476
$633$	5.82012	0.231329
$634$	1.98225	0.0787252
$635$	0	0
$636$	1.32001	0.0523417
$637$	1.00000	0.0396214
$638$	1.64051	0.0649483
$639$	11.3205	0.447832
$640$	0	0
$641$	39.1790	1.54748	0.773739	−	0.633504i	$-0.218384\pi$
$641$	39.1790	1.54748	0.773739	+	0.633504i	$0.218384\pi$
$642$	−0.777040	−0.0306673
$643$	−1.98613	−0.0783251	−0.0391626	−	0.999233i	$-0.512469\pi$
$643$	−1.98613	−0.0783251	−0.0391626	+	0.999233i	$0.512469\pi$
$644$	−6.22782	−0.245411
$645$	0	0
$646$	−1.44939	−0.0570255
$647$	−6.04859	−0.237794	−0.118897	−	0.992907i	$-0.537936\pi$
$647$	−6.04859	−0.237794	−0.118897	+	0.992907i	$0.537936\pi$
$648$	−1.24154	−0.0487722
$649$	−2.78218	−0.109210
$650$	0	0
$651$	−4.24154	−0.166239
$652$	−47.9520	−1.87795
$653$	38.2254	1.49588	0.747938	−	0.663768i	$-0.231044\pi$
$653$	38.2254	1.49588	0.747938	+	0.663768i	$0.231044\pi$
$654$	2.53362	0.0990723
$655$	0	0
$656$	15.1205	0.590356
$657$	−10.1529	−0.396104
$658$	0.244971	0.00954997
$659$	−8.38962	−0.326813	−0.163407	−	0.986559i	$-0.552248\pi$
$659$	−8.38962	−0.326813	−0.163407	+	0.986559i	$0.552248\pi$
$660$	0	0
$661$	23.1388	0.899994	0.449997	−	0.893030i	$-0.351425\pi$
$661$	23.1388	0.899994	0.449997	+	0.893030i	$0.351425\pi$
$662$	5.31862	0.206714
$663$	−4.38166	−0.170170
$664$	13.9958	0.543142
$665$	0	0
$666$	−1.50757	−0.0584173
$667$	−30.1748	−1.16837
$668$	−4.22971	−0.163652
$669$	21.0182	0.812610
$670$	0	0
$671$	3.59870	0.138926
$672$	3.56641	0.137577
$673$	6.62137	0.255235	0.127617	−	0.991823i	$-0.459267\pi$
$673$	6.62137	0.255235	0.127617	+	0.991823i	$0.459267\pi$
$674$	−7.73862	−0.298081
$675$	0	0
$676$	−1.89858	−0.0730225
$677$	31.0552	1.19355	0.596775	−	0.802409i	$-0.296448\pi$
$677$	31.0552	1.19355	0.596775	+	0.802409i	$0.296448\pi$
$678$	5.44641	0.209168
$679$	−8.52959	−0.327335
$680$	0	0
$681$	3.56000	0.136419
$682$	0.756420	0.0289648
$683$	−1.51194	−0.0578528	−0.0289264	−	0.999582i	$-0.509209\pi$
$683$	−1.51194	−0.0578528	−0.0289264	+	0.999582i	$0.509209\pi$
$684$	−1.97207	−0.0754040
$685$	0	0
$686$	0.318459	0.0121588
$687$	−22.2275	−0.848033
$688$	11.7872	0.449382
$689$	−0.695259	−0.0264872
$690$	0	0
$691$	−37.3647	−1.42142	−0.710710	−	0.703485i	$-0.751626\pi$
$691$	−37.3647	−1.42142	−0.710710	+	0.703485i	$0.751626\pi$
$692$	32.7595	1.24533
$693$	−0.559998	−0.0212726
$694$	−0.293963	−0.0111587
$695$	0	0
$696$	11.4209	0.432907
$697$	−19.4759	−0.737702
$698$	2.28805	0.0866038
$699$	−4.48998	−0.169827
$700$	0	0
$701$	38.5512	1.45606	0.728029	−	0.685546i	$-0.240437\pi$
$701$	38.5512	1.45606	0.728029	+	0.685546i	$0.240437\pi$
$702$	0.318459	0.0120195
$703$	−4.91720	−0.185456
$704$	3.17397	0.119623
$705$	0	0
$706$	3.60561	0.135699
$707$	−6.15870	−0.231622
$708$	−9.43254	−0.354496
$709$	−41.3214	−1.55186	−0.775929	−	0.630820i	$-0.782719\pi$
$709$	−41.3214	−1.55186	−0.775929	+	0.630820i	$0.782719\pi$
$710$	0	0
$711$	−3.12795	−0.117307
$712$	−4.74282	−0.177745
$713$	−13.9133	−0.521057
$714$	−1.39538	−0.0522207
$715$	0	0
$716$	27.3552	1.02231
$717$	−28.0085	−1.04599
$718$	0.329199	0.0122856
$719$	−25.1876	−0.939340	−0.469670	−	0.882842i	$-0.655627\pi$
$719$	−25.1876	−0.939340	−0.469670	+	0.882842i	$0.655627\pi$
$720$	0	0
$721$	12.7251	0.473908
$722$	−5.70713	−0.212397
$723$	−14.6468	−0.544719
$724$	−8.77459	−0.326105
$725$	0	0
$726$	−3.40318	−0.126304
$727$	12.9615	0.480716	0.240358	−	0.970684i	$-0.422735\pi$
$727$	12.9615	0.480716	0.240358	+	0.970684i	$0.422735\pi$
$728$	−1.24154	−0.0460145
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.1824	−0.561542
$732$	12.2008	0.450956
$733$	41.3640	1.52781	0.763907	−	0.645326i	$-0.223279\pi$
$733$	41.3640	1.52781	0.763907	+	0.645326i	$0.223279\pi$
$734$	5.79488	0.213893
$735$	0	0
$736$	11.6987	0.431219
$737$	5.01771	0.184830
$738$	1.41551	0.0521055
$739$	−10.9593	−0.403144	−0.201572	−	0.979474i	$-0.564605\pi$
$739$	−10.9593	−0.403144	−0.201572	+	0.979474i	$0.564605\pi$
$740$	0	0
$741$	1.03871	0.0381578
$742$	−0.221411	−0.00812827
$743$	−3.01156	−0.110483	−0.0552417	−	0.998473i	$-0.517593\pi$
$743$	−3.01156	−0.110483	−0.0552417	+	0.998473i	$0.517593\pi$
$744$	5.26603	0.193062
$745$	0	0
$746$	2.25803	0.0826722
$747$	−11.2729	−0.412455
$748$	−4.65859	−0.170335
$749$	−2.44000	−0.0891557
$750$	0	0
$751$	−12.0137	−0.438387	−0.219193	−	0.975681i	$-0.570343\pi$
$751$	−12.0137	−0.438387	−0.219193	+	0.975681i	$0.570343\pi$
$752$	2.61679	0.0954245
$753$	−11.5619	−0.421338
$754$	−2.92949	−0.106686
$755$	0	0
$756$	−1.89858	−0.0690508
$757$	2.42625	0.0881834	0.0440917	−	0.999027i	$-0.485961\pi$
$757$	2.42625	0.0881834	0.0440917	+	0.999027i	$0.485961\pi$
$758$	5.51161	0.200191
$759$	−1.83693	−0.0666763
$760$	0	0
$761$	21.3265	0.773085	0.386543	−	0.922272i	$-0.373669\pi$
$761$	21.3265	0.773085	0.386543	+	0.922272i	$0.373669\pi$
$762$	−5.87155	−0.212704
$763$	7.95587	0.288022
$764$	4.70216	0.170118
$765$	0	0
$766$	2.42951	0.0877818
$767$	4.96820	0.179391
$768$	8.48933	0.306332
$769$	5.24166	0.189019	0.0945095	−	0.995524i	$-0.469872\pi$
$769$	5.24166	0.189019	0.0945095	+	0.995524i	$0.469872\pi$
$770$	0	0
$771$	21.8506	0.786929
$772$	45.8079	1.64866
$773$	21.7015	0.780549	0.390275	−	0.920699i	$-0.372380\pi$
$773$	21.7015	0.780549	0.390275	+	0.920699i	$0.372380\pi$
$774$	1.10346	0.0396630
$775$	0	0
$776$	10.5898	0.380152
$777$	−4.73397	−0.169830
$778$	4.79035	0.171742
$779$	4.61691	0.165418
$780$	0	0
$781$	−6.33945	−0.226843
$782$	−4.57719	−0.163680
$783$	−9.19896	−0.328744
$784$	3.40179	0.121492
$785$	0	0
$786$	6.20785	0.221427
$787$	−13.0567	−0.465422	−0.232711	−	0.972546i	$-0.574760\pi$
$787$	−13.0567	−0.465422	−0.232711	+	0.972546i	$0.574760\pi$
$788$	−8.59057	−0.306026
$789$	20.8055	0.740694
$790$	0	0
$791$	17.1024	0.608091
$792$	0.695259	0.0247049
$793$	−6.42628	−0.228204
$794$	2.58490	0.0917346
$795$	0	0
$796$	49.8402	1.76654
$797$	−31.2399	−1.10657	−0.553286	−	0.832991i	$-0.686626\pi$
$797$	−31.2399	−1.10657	−0.553286	+	0.832991i	$0.686626\pi$
$798$	0.330785	0.0117097
$799$	−3.37055	−0.119241
$800$	0	0
$801$	3.82012	0.134977
$802$	−5.95860	−0.210406
$803$	5.68562	0.200641
$804$	17.0118	0.599959
$805$	0	0
$806$	−1.35076	−0.0475783
$807$	9.95040	0.350271
$808$	7.64627	0.268995
$809$	−41.5198	−1.45976	−0.729880	−	0.683576i	$-0.760424\pi$
$809$	−41.5198	−1.45976	−0.729880	+	0.683576i	$0.760424\pi$
$810$	0	0
$811$	27.1895	0.954752	0.477376	−	0.878699i	$-0.341588\pi$
$811$	27.1895	0.954752	0.477376	+	0.878699i	$0.341588\pi$
$812$	17.4650	0.612901
$813$	−9.26653	−0.324991
$814$	0.844238	0.0295905
$815$	0	0
$816$	−14.9055	−0.521796
$817$	3.59911	0.125917
$818$	−0.305146	−0.0106692
$819$	1.00000	0.0349428
$820$	0	0
$821$	−33.2419	−1.16015	−0.580075	−	0.814563i	$-0.696977\pi$
$821$	−33.2419	−1.16015	−0.580075	+	0.814563i	$0.696977\pi$
$822$	−4.45471	−0.155376
$823$	−45.2516	−1.57737	−0.788686	−	0.614796i	$-0.789238\pi$
$823$	−45.2516	−1.57737	−0.788686	+	0.614796i	$0.789238\pi$
$824$	−15.7987	−0.550374
$825$	0	0
$826$	1.58217	0.0550506
$827$	−8.83950	−0.307380	−0.153690	−	0.988119i	$-0.549116\pi$
$827$	−8.83950	−0.307380	−0.153690	+	0.988119i	$0.549116\pi$
$828$	−6.22782	−0.216432
$829$	26.1667	0.908809	0.454404	−	0.890796i	$-0.349852\pi$
$829$	26.1667	0.908809	0.454404	+	0.890796i	$0.349852\pi$
$830$	0	0
$831$	−24.8869	−0.863317
$832$	−5.66782	−0.196496
$833$	−4.38166	−0.151816
$834$	−1.90434	−0.0659420
$835$	0	0
$836$	1.10436	0.0381949
$837$	−4.24154	−0.146609
$838$	−9.96550	−0.344253
$839$	−9.98579	−0.344748	−0.172374	−	0.985032i	$-0.555144\pi$
$839$	−9.98579	−0.344748	−0.172374	+	0.985032i	$0.555144\pi$
$840$	0	0
$841$	55.6208	1.91796
$842$	6.67627	0.230080
$843$	3.67358	0.126525
$844$	−11.0500	−0.380356
$845$	0	0
$846$	0.244971	0.00842228
$847$	−10.6864	−0.367189
$848$	−2.36512	−0.0812187
$849$	12.4801	0.428316
$850$	0	0
$851$	−15.5286	−0.532312
$852$	−21.4929	−0.736335
$853$	−27.9640	−0.957469	−0.478734	−	0.877960i	$-0.658904\pi$
$853$	−27.9640	−0.957469	−0.478734	+	0.877960i	$0.658904\pi$
$854$	−2.04651	−0.0700300
$855$	0	0
$856$	3.02936	0.103541
$857$	27.9612	0.955137	0.477569	−	0.878594i	$-0.341518\pi$
$857$	27.9612	0.955137	0.477569	+	0.878594i	$0.341518\pi$
$858$	−0.178336	−0.00608830
$859$	48.6972	1.66153	0.830764	−	0.556625i	$-0.187904\pi$
$859$	48.6972	1.66153	0.830764	+	0.556625i	$0.187904\pi$
$860$	0	0
$861$	4.44486	0.151481
$862$	−5.39163	−0.183640
$863$	2.45964	0.0837270	0.0418635	−	0.999123i	$-0.486671\pi$
$863$	2.45964	0.0837270	0.0418635	+	0.999123i	$0.486671\pi$
$864$	3.56641	0.121332
$865$	0	0
$866$	−7.83901	−0.266380
$867$	2.19896	0.0746805
$868$	8.05292	0.273334
$869$	1.75165	0.0594205
$870$	0	0
$871$	−8.96024	−0.303606
$872$	−9.87752	−0.334495
$873$	−8.52959	−0.288683
$874$	1.08506	0.0367026
$875$	0	0
$876$	19.2762	0.651283
$877$	24.2250	0.818020	0.409010	−	0.912530i	$-0.365874\pi$
$877$	24.2250	0.818020	0.409010	+	0.912530i	$0.365874\pi$
$878$	−5.51995	−0.186289
$879$	−19.0921	−0.643961
$880$	0	0
$881$	9.06491	0.305405	0.152702	−	0.988272i	$-0.451202\pi$
$881$	9.06491	0.305405	0.152702	+	0.988272i	$0.451202\pi$
$882$	0.318459	0.0107231
$883$	9.14548	0.307770	0.153885	−	0.988089i	$-0.450821\pi$
$883$	9.14548	0.307770	0.153885	+	0.988089i	$0.450821\pi$
$884$	8.31895	0.279797
$885$	0	0
$886$	3.84521	0.129182
$887$	30.0582	1.00926	0.504628	−	0.863337i	$-0.331630\pi$
$887$	30.0582	1.00926	0.504628	+	0.863337i	$0.331630\pi$
$888$	5.87740	0.197233
$889$	−18.4374	−0.618371
$890$	0	0
$891$	−0.559998	−0.0187606
$892$	−39.9048	−1.33611
$893$	0.799014	0.0267380
$894$	−6.54207	−0.218800
$895$	0	0
$896$	−8.93778	−0.298590
$897$	3.28025	0.109524
$898$	−5.48496	−0.183036
$899$	39.0177	1.30131
$900$	0	0
$901$	3.04639	0.101490
$902$	−0.792680	−0.0263934
$903$	3.46499	0.115308
$904$	−21.2333	−0.706209
$905$	0	0
$906$	−3.34345	−0.111079
$907$	26.2932	0.873051	0.436525	−	0.899692i	$-0.356209\pi$
$907$	26.2932	0.873051	0.436525	+	0.899692i	$0.356209\pi$
$908$	−6.75895	−0.224304
$909$	−6.15870	−0.204271
$910$	0	0
$911$	−15.2163	−0.504137	−0.252069	−	0.967709i	$-0.581111\pi$
$911$	−15.2163	−0.504137	−0.252069	+	0.967709i	$0.581111\pi$
$912$	3.53346	0.117005
$913$	6.31282	0.208924
$914$	9.07512	0.300178
$915$	0	0
$916$	42.2008	1.39436
$917$	19.4934	0.643729
$918$	−1.39538	−0.0460544
$919$	11.7100	0.386277	0.193139	−	0.981171i	$-0.438133\pi$
$919$	11.7100	0.386277	0.193139	+	0.981171i	$0.438133\pi$
$920$	0	0
$921$	−21.2812	−0.701240
$922$	−8.73604	−0.287706
$923$	11.3205	0.372619
$924$	1.06320	0.0349768
$925$	0	0
$926$	−8.14779	−0.267753
$927$	12.7251	0.417947
$928$	−32.8072	−1.07695
$929$	28.1037	0.922052	0.461026	−	0.887387i	$-0.347482\pi$
$929$	28.1037	0.922052	0.461026	+	0.887387i	$0.347482\pi$
$930$	0	0
$931$	1.03871	0.0340422
$932$	8.52460	0.279233
$933$	−17.8373	−0.583967
$934$	−1.77750	−0.0581614
$935$	0	0
$936$	−1.24154	−0.0405810
$937$	−32.3947	−1.05829	−0.529144	−	0.848532i	$-0.677487\pi$
$937$	−32.3947	−1.05829	−0.529144	+	0.848532i	$0.677487\pi$
$938$	−2.85347	−0.0931690
$939$	−10.0969	−0.329501
$940$	0	0
$941$	−7.71494	−0.251500	−0.125750	−	0.992062i	$-0.540134\pi$
$941$	−7.71494	−0.251500	−0.125750	+	0.992062i	$0.540134\pi$
$942$	−1.23746	−0.0403185
$943$	14.5802	0.474798
$944$	16.9008	0.550073
$945$	0	0
$946$	−0.617933	−0.0200908
$947$	26.8915	0.873856	0.436928	−	0.899497i	$-0.356066\pi$
$947$	26.8915	0.873856	0.436928	+	0.899497i	$0.356066\pi$
$948$	5.93868	0.192879
$949$	−10.1529	−0.329578
$950$	0	0
$951$	6.22451	0.201843
$952$	5.44000	0.176311
$953$	−33.0003	−1.06899	−0.534493	−	0.845173i	$-0.679497\pi$
$953$	−33.0003	−1.06899	−0.534493	+	0.845173i	$0.679497\pi$
$954$	−0.221411	−0.00716846
$955$	0	0
$956$	53.1764	1.71985
$957$	5.15139	0.166521
$958$	−6.82869	−0.220625
$959$	−13.9883	−0.451707
$960$	0	0
$961$	−13.0093	−0.419656
$962$	−1.50757	−0.0486061
$963$	−2.44000	−0.0786280
$964$	27.8081	0.895639
$965$	0	0
$966$	1.04462	0.0336102
$967$	27.3346	0.879021	0.439511	−	0.898237i	$-0.355152\pi$
$967$	27.3346	0.879021	0.439511	+	0.898237i	$0.355152\pi$
$968$	13.2676	0.426436
$969$	−4.55126	−0.146208
$970$	0	0
$971$	−0.338862	−0.0108746	−0.00543731	−	0.999985i	$-0.501731\pi$
$971$	−0.338862	−0.0108746	−0.00543731	+	0.999985i	$0.501731\pi$
$972$	−1.89858	−0.0608971
$973$	−5.97987	−0.191706
$974$	4.75112	0.152236
$975$	0	0
$976$	−21.8609	−0.699749
$977$	36.6151	1.17142	0.585711	−	0.810520i	$-0.300815\pi$
$977$	36.6151	1.17142	0.585711	+	0.810520i	$0.300815\pi$
$978$	8.04323	0.257194
$979$	−2.13926	−0.0683709
$980$	0	0
$981$	7.95587	0.254011
$982$	1.18880	0.0379361
$983$	36.6089	1.16764	0.583822	−	0.811882i	$-0.301557\pi$
$983$	36.6089	1.16764	0.583822	+	0.811882i	$0.301557\pi$
$984$	−5.51847	−0.175922
$985$	0	0
$986$	12.8360	0.408783
$987$	0.769240	0.0244852
$988$	−1.97207	−0.0627400
$989$	11.3660	0.361418
$990$	0	0
$991$	54.6507	1.73604	0.868018	−	0.496533i	$-0.165394\pi$
$991$	54.6507	1.73604	0.868018	+	0.496533i	$0.165394\pi$
$992$	−15.1271	−0.480284
$993$	16.7011	0.529994
$994$	3.60511	0.114347
$995$	0	0
$996$	21.4026	0.678168
$997$	2.87263	0.0909770	0.0454885	−	0.998965i	$-0.485516\pi$
$997$	2.87263	0.0909770	0.0454885	+	0.998965i	$0.485516\pi$
$998$	5.16810	0.163593
$999$	−4.73397	−0.149776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bi.1.3	✓	4
5.4	even	2		6825.2.a.bj.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6825.2.a.bi.1.3	✓	4	1.1	even	1	trivial
6825.2.a.bj.1.2	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)

\(f(q)\)	\(=\)	\(q+0.318459 q^{2} +1.00000 q^{3} -1.89858 q^{4} +0.318459 q^{6} +1.00000 q^{7} -1.24154 q^{8} +1.00000 q^{9} -0.559998 q^{11} -1.89858 q^{12} +1.00000 q^{13} +0.318459 q^{14} +3.40179 q^{16} -4.38166 q^{17} +0.318459 q^{18} +1.03871 q^{19} +1.00000 q^{21} -0.178336 q^{22} +3.28025 q^{23} -1.24154 q^{24} +0.318459 q^{26} +1.00000 q^{27} -1.89858 q^{28} -9.19896 q^{29} -4.24154 q^{31} +3.56641 q^{32} -0.559998 q^{33} -1.39538 q^{34} -1.89858 q^{36} -4.73397 q^{37} +0.330785 q^{38} +1.00000 q^{39} +4.44486 q^{41} +0.318459 q^{42} +3.46499 q^{43} +1.06320 q^{44} +1.04462 q^{46} +0.769240 q^{47} +3.40179 q^{48} +1.00000 q^{49} -4.38166 q^{51} -1.89858 q^{52} -0.695259 q^{53} +0.318459 q^{54} -1.24154 q^{56} +1.03871 q^{57} -2.92949 q^{58} +4.96820 q^{59} -6.42628 q^{61} -1.35076 q^{62} +1.00000 q^{63} -5.66782 q^{64} -0.178336 q^{66} -8.96024 q^{67} +8.31895 q^{68} +3.28025 q^{69} +11.3205 q^{71} -1.24154 q^{72} -10.1529 q^{73} -1.50757 q^{74} -1.97207 q^{76} -0.559998 q^{77} +0.318459 q^{78} -3.12795 q^{79} +1.00000 q^{81} +1.41551 q^{82} -11.2729 q^{83} -1.89858 q^{84} +1.10346 q^{86} -9.19896 q^{87} +0.695259 q^{88} +3.82012 q^{89} +1.00000 q^{91} -6.22782 q^{92} -4.24154 q^{93} +0.244971 q^{94} +3.56641 q^{96} -8.52959 q^{97} +0.318459 q^{98} -0.559998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 4 q^{3} + q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{16} - 5 q^{17} - q^{18} - 15 q^{19} + 4 q^{21} - 9 q^{22} - 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 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

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