LMFDB - Embedded newform 1600.3.g.h.351.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(351,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1600.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.691798081536.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 $ x^8 - 4x^7 + 26x^6 - 64x^5 + 229x^4 - 356x^3 + 164x^2 + 4*x + 985
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.7
Root			$1.72474 - 0.954705i$ of defining polynomial
Character	$\chi$	$=$	1600.351
Dual form			1600.3.g.h.351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +10.6771i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} +10.6771i q^{7} -3.00000 q^{9} -8.71780 q^{11} -19.5959i q^{13} -21.3542 q^{17} +26.1534 q^{19} +26.1534i q^{21} +10.6771i q^{23} -29.3939 q^{27} -34.8712i q^{29} -4.00000i q^{31} -21.3542 q^{33} +14.6969i q^{37} -48.0000i q^{39} -24.0000 q^{41} -56.3383 q^{43} -10.6771i q^{47} -65.0000 q^{49} -52.3068 q^{51} -48.9898i q^{53} +64.0625 q^{57} -43.5890 q^{59} -26.1534i q^{61} -32.0312i q^{63} -7.34847 q^{67} +26.1534i q^{69} -84.0000i q^{71} +106.771 q^{73} -93.0806i q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464 q^{83} -85.4166i q^{87} -150.000 q^{89} +209.227 q^{91} -9.79796i q^{93} +21.3542 q^{97} +26.1534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} - 192 q^{41} - 520 q^{49} - 360 q^{81} - 1200 q^{89}+O(q^{100}) $ 8 * q - 24 * q^9 - 192 * q^41 - 520 * q^49 - 360 * q^81 - 1200 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$-1$

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	0
$2$	0	0
$3$	2.44949	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	2.44949	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.6771i	1.52530i	0.646813	+	0.762648i	$0.276101\pi$
$7$	10.6771i	1.52530i	−0.646813	+	0.762648i	$0.723899\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−8.71780	−0.792527	−0.396264	−	0.918137i	$-0.629693\pi$
$11$	−8.71780	−0.792527	−0.396264	+	0.918137i	$0.629693\pi$
$12$	0	0
$13$	− 19.5959i	− 1.50738i	−0.657231	−	0.753689i	$-0.728272\pi$
$13$	− 19.5959i	− 1.50738i	0.657231	−	0.753689i	$-0.271728\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−21.3542	−1.25613	−0.628063	−	0.778162i	$-0.716152\pi$
$17$	−21.3542	−1.25613	−0.628063	+	0.778162i	$0.716152\pi$
$18$	0	0
$19$	26.1534	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	26.1534	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	26.1534i	1.24540i
$22$	0	0
$23$	10.6771i	0.464221i	0.972689	+	0.232110i	$0.0745631\pi$
$23$	10.6771i	0.464221i	−0.972689	+	0.232110i	$0.925437\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−29.3939	−1.08866
$28$	0	0
$29$	− 34.8712i	− 1.20245i	−0.799078	−	0.601227i	$-0.794679\pi$
$29$	− 34.8712i	− 1.20245i	0.799078	−	0.601227i	$-0.205321\pi$
$30$	0	0
$31$	− 4.00000i	− 0.129032i	−0.997917	−	0.0645161i	$-0.979450\pi$
$31$	− 4.00000i	− 0.129032i	0.997917	−	0.0645161i	$-0.0205504\pi$
$32$	0	0
$33$	−21.3542	−0.647096
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14.6969i	0.397215i	0.980079	+	0.198607i	$0.0636418\pi$
$37$	14.6969i	0.397215i	−0.980079	+	0.198607i	$0.936358\pi$
$38$	0	0
$39$	− 48.0000i	− 1.23077i
$40$	0	0
$41$	−24.0000	−0.585366	−0.292683	−	0.956210i	$-0.594548\pi$
$41$	−24.0000	−0.585366	−0.292683	+	0.956210i	$0.594548\pi$
$42$	0	0
$43$	−56.3383	−1.31019	−0.655096	−	0.755546i	$-0.727372\pi$
$43$	−56.3383	−1.31019	−0.655096	+	0.755546i	$0.727372\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.6771i	− 0.227172i	−0.993528	−	0.113586i	$-0.963766\pi$
$47$	− 10.6771i	− 0.227172i	0.993528	−	0.113586i	$-0.0362337\pi$
$48$	0	0
$49$	−65.0000	−1.32653
$50$	0	0
$51$	−52.3068	−1.02562
$52$	0	0
$53$	− 48.9898i	− 0.924336i	−0.886792	−	0.462168i	$-0.847072\pi$
$53$	− 48.9898i	− 0.924336i	0.886792	−	0.462168i	$-0.152928\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	64.0625	1.12390
$58$	0	0
$59$	−43.5890	−0.738796	−0.369398	−	0.929271i	$-0.620436\pi$
$59$	−43.5890	−0.738796	−0.369398	+	0.929271i	$0.620436\pi$
$60$	0	0
$61$	− 26.1534i	− 0.428744i	−0.976752	−	0.214372i	$-0.931229\pi$
$61$	− 26.1534i	− 0.428744i	0.976752	−	0.214372i	$-0.0687705\pi$
$62$	0	0
$63$	− 32.0312i	− 0.508432i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.34847	−0.109679	−0.0548393	−	0.998495i	$-0.517465\pi$
$67$	−7.34847	−0.109679	−0.0548393	+	0.998495i	$0.517465\pi$
$68$	0	0
$69$	26.1534i	0.379035i
$70$	0	0
$71$	− 84.0000i	− 1.18310i	−0.806269	−	0.591549i	$-0.798517\pi$
$71$	− 84.0000i	− 1.18310i	0.806269	−	0.591549i	$-0.201483\pi$
$72$	0	0
$73$	106.771	1.46261	0.731307	−	0.682049i	$-0.238911\pi$
$73$	106.771	1.46261	0.731307	+	0.682049i	$0.238911\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 93.0806i	− 1.20884i
$78$	0	0
$79$	− 100.000i	− 1.26582i	−0.774224	−	0.632911i	$-0.781860\pi$
$79$	− 100.000i	− 1.26582i	0.774224	−	0.632911i	$-0.218140\pi$
$80$	0	0
$81$	−45.0000	−0.555556
$82$	0	0
$83$	17.1464	0.206583	0.103292	−	0.994651i	$-0.467062\pi$
$83$	17.1464	0.206583	0.103292	+	0.994651i	$0.467062\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 85.4166i	− 0.981800i
$88$	0	0
$89$	−150.000	−1.68539	−0.842697	−	0.538389i	$-0.819033\pi$
$89$	−150.000	−1.68539	−0.842697	+	0.538389i	$0.819033\pi$
$90$	0	0
$91$	209.227	2.29920
$92$	0	0
$93$	− 9.79796i	− 0.105354i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	21.3542	0.220146	0.110073	−	0.993924i	$-0.464892\pi$
$97$	21.3542	0.220146	0.110073	+	0.993924i	$0.464892\pi$
$98$	0	0
$99$	26.1534	0.264176
$100$	0	0
$101$	− 69.7424i	− 0.690519i	−0.938507	−	0.345259i	$-0.887791\pi$
$101$	− 69.7424i	− 0.690519i	0.938507	−	0.345259i	$-0.112209\pi$
$102$	0	0
$103$	− 32.0312i	− 0.310983i	−0.987837	−	0.155491i	$-0.950304\pi$
$103$	− 32.0312i	− 0.310983i	0.987837	−	0.155491i	$-0.0496961\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	149.419	1.39644	0.698219	−	0.715884i	$-0.253976\pi$
$107$	149.419	1.39644	0.698219	+	0.715884i	$0.253976\pi$
$108$	0	0
$109$	183.074i	1.67958i	0.542915	+	0.839788i	$0.317321\pi$
$109$	183.074i	1.67958i	−0.542915	+	0.839788i	$0.682679\pi$
$110$	0	0
$111$	36.0000i	0.324324i
$112$	0	0
$113$	−170.833	−1.51180	−0.755899	−	0.654688i	$-0.772800\pi$
$113$	−170.833	−1.51180	−0.755899	+	0.654688i	$0.772800\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	58.7878i	0.502459i
$118$	0	0
$119$	− 228.000i	− 1.91597i
$120$	0	0
$121$	−45.0000	−0.371901
$122$	0	0
$123$	−58.7878	−0.477949
$124$	0	0
$125$	0	0
$126$	0	0
$127$	53.3854i	0.420357i	0.977663	+	0.210179i	$0.0674046\pi$
$127$	53.3854i	0.420357i	−0.977663	+	0.210179i	$0.932595\pi$
$128$	0	0
$129$	−138.000	−1.06977
$130$	0	0
$131$	−95.8958	−0.732029	−0.366014	−	0.930609i	$-0.619278\pi$
$131$	−95.8958	−0.732029	−0.366014	+	0.930609i	$0.619278\pi$
$132$	0	0
$133$	279.242i	2.09956i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−128.125	−0.935219	−0.467609	−	0.883935i	$-0.654885\pi$
$137$	−128.125	−0.935219	−0.467609	+	0.883935i	$0.654885\pi$
$138$	0	0
$139$	−235.381	−1.69339	−0.846693	−	0.532082i	$-0.821410\pi$
$139$	−235.381	−1.69339	−0.846693	+	0.532082i	$0.821410\pi$
$140$	0	0
$141$	− 26.1534i	− 0.185485i
$142$	0	0
$143$	170.833i	1.19464i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−159.217	−1.08311
$148$	0	0
$149$	− 95.8958i	− 0.643596i	−0.946808	−	0.321798i	$-0.895713\pi$
$149$	− 95.8958i	− 0.643596i	0.946808	−	0.321798i	$-0.104287\pi$
$150$	0	0
$151$	− 160.000i	− 1.05960i	−0.848122	−	0.529801i	$-0.822266\pi$
$151$	− 160.000i	− 1.05960i	0.848122	−	0.529801i	$-0.177734\pi$
$152$	0	0
$153$	64.0625	0.418709
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 34.2929i	− 0.218426i	−0.994018	−	0.109213i	$-0.965167\pi$
$157$	− 34.2929i	− 0.218426i	0.994018	−	0.109213i	$-0.0348330\pi$
$158$	0	0
$159$	− 120.000i	− 0.754717i
$160$	0	0
$161$	−114.000	−0.708075
$162$	0	0
$163$	120.025	0.736350	0.368175	−	0.929757i	$-0.379983\pi$
$163$	120.025	0.736350	0.368175	+	0.929757i	$0.379983\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	288.281i	1.72623i	0.505004	+	0.863117i	$0.331491\pi$
$167$	288.281i	1.72623i	−0.505004	+	0.863117i	$0.668509\pi$
$168$	0	0
$169$	−215.000	−1.27219
$170$	0	0
$171$	−78.4602	−0.458831
$172$	0	0
$173$	24.4949i	0.141589i	0.997491	+	0.0707945i	$0.0225535\pi$
$173$	24.4949i	0.141589i	−0.997491	+	0.0707945i	$0.977447\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−106.771	−0.603225
$178$	0	0
$179$	252.816	1.41238	0.706190	−	0.708022i	$-0.250412\pi$
$179$	252.816	1.41238	0.706190	+	0.708022i	$0.250412\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	− 64.0625i	− 0.350068i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	186.161	0.995515
$188$	0	0
$189$	− 313.841i	− 1.66053i
$190$	0	0
$191$	− 24.0000i	− 0.125654i	−0.998024	−	0.0628272i	$-0.979988\pi$
$191$	− 24.0000i	− 0.125654i	0.998024	−	0.0628272i	$-0.0200117\pi$
$192$	0	0
$193$	106.771	0.553216	0.276608	−	0.960983i	$-0.410790\pi$
$193$	106.771	0.553216	0.276608	+	0.960983i	$0.410790\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	342.929i	1.74075i	0.492386	+	0.870377i	$0.336125\pi$
$197$	342.929i	1.74075i	−0.492386	+	0.870377i	$0.663875\pi$
$198$	0	0
$199$	− 188.000i	− 0.944724i	−0.881405	−	0.472362i	$-0.843402\pi$
$199$	− 188.000i	− 0.944724i	0.881405	−	0.472362i	$-0.156598\pi$
$200$	0	0
$201$	−18.0000	−0.0895522
$202$	0	0
$203$	372.322	1.83410
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 32.0312i	− 0.154740i
$208$	0	0
$209$	−228.000	−1.09091
$210$	0	0
$211$	−26.1534	−0.123950	−0.0619749	−	0.998078i	$-0.519740\pi$
$211$	−26.1534	−0.123950	−0.0619749	+	0.998078i	$0.519740\pi$
$212$	0	0
$213$	− 205.757i	− 0.965996i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	42.7083	0.196813
$218$	0	0
$219$	261.534	1.19422
$220$	0	0
$221$	418.454i	1.89346i
$222$	0	0
$223$	− 224.219i	− 1.00546i	−0.864442	−	0.502732i	$-0.832328\pi$
$223$	− 224.219i	− 1.00546i	0.864442	−	0.502732i	$-0.167672\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−80.8332	−0.356093	−0.178047	−	0.984022i	$-0.556978\pi$
$227$	−80.8332	−0.356093	−0.178047	+	0.984022i	$0.556978\pi$
$228$	0	0
$229$	104.614i	0.456828i	0.973564	+	0.228414i	$0.0733539\pi$
$229$	104.614i	0.456828i	−0.973564	+	0.228414i	$0.926646\pi$
$230$	0	0
$231$	− 228.000i	− 0.987013i
$232$	0	0
$233$	−320.312	−1.37473	−0.687366	−	0.726312i	$-0.741233\pi$
$233$	−320.312	−1.37473	−0.687366	+	0.726312i	$0.741233\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 244.949i	− 1.03354i
$238$	0	0
$239$	− 108.000i	− 0.451883i	−0.974141	−	0.225941i	$-0.927454\pi$
$239$	− 108.000i	− 0.451883i	0.974141	−	0.225941i	$-0.0725458\pi$
$240$	0	0
$241$	−28.0000	−0.116183	−0.0580913	−	0.998311i	$-0.518501\pi$
$241$	−28.0000	−0.116183	−0.0580913	+	0.998311i	$0.518501\pi$
$242$	0	0
$243$	154.318	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 512.500i	− 2.07490i
$248$	0	0
$249$	42.0000	0.168675
$250$	0	0
$251$	−252.816	−1.00724	−0.503618	−	0.863927i	$-0.667998\pi$
$251$	−252.816	−1.00724	−0.503618	+	0.863927i	$0.667998\pi$
$252$	0	0
$253$	− 93.0806i	− 0.367908i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	298.958	1.16326	0.581631	−	0.813453i	$-0.302415\pi$
$257$	298.958	1.16326	0.581631	+	0.813453i	$0.302415\pi$
$258$	0	0
$259$	−156.920	−0.605870
$260$	0	0
$261$	104.614i	0.400818i
$262$	0	0
$263$	288.281i	1.09613i	0.836437	+	0.548063i	$0.184635\pi$
$263$	288.281i	1.09613i	−0.836437	+	0.548063i	$0.815365\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−367.423	−1.37612
$268$	0	0
$269$	61.0246i	0.226857i	0.993546	+	0.113429i	$0.0361833\pi$
$269$	61.0246i	0.226857i	−0.993546	+	0.113429i	$0.963817\pi$
$270$	0	0
$271$	428.000i	1.57934i	0.613535	+	0.789668i	$0.289747\pi$
$271$	428.000i	1.57934i	−0.613535	+	0.789668i	$0.710253\pi$
$272$	0	0
$273$	512.500	1.87729
$274$	0	0
$275$	0	0
$276$	0	0
$277$	73.4847i	0.265288i	0.991164	+	0.132644i	$0.0423467\pi$
$277$	73.4847i	0.265288i	−0.991164	+	0.132644i	$0.957653\pi$
$278$	0	0
$279$	12.0000i	0.0430108i
$280$	0	0
$281$	−204.000	−0.725979	−0.362989	−	0.931793i	$-0.618244\pi$
$281$	−204.000	−0.725979	−0.362989	+	0.931793i	$0.618244\pi$
$282$	0	0
$283$	−404.166	−1.42815	−0.714074	−	0.700070i	$-0.753152\pi$
$283$	−404.166	−1.42815	−0.714074	+	0.700070i	$0.753152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 256.250i	− 0.892857i
$288$	0	0
$289$	167.000	0.577855
$290$	0	0
$291$	52.3068	0.179748
$292$	0	0
$293$	161.666i	0.551762i	0.961192	+	0.275881i	$0.0889696\pi$
$293$	161.666i	0.551762i	−0.961192	+	0.275881i	$0.911030\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	256.250	0.862794
$298$	0	0
$299$	209.227	0.699756
$300$	0	0
$301$	− 601.528i	− 1.99843i
$302$	0	0
$303$	− 170.833i	− 0.563806i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	502.145	1.63565	0.817826	−	0.575465i	$-0.195179\pi$
$307$	502.145	1.63565	0.817826	+	0.575465i	$0.195179\pi$
$308$	0	0
$309$	− 78.4602i	− 0.253916i
$310$	0	0
$311$	− 408.000i	− 1.31190i	−0.754806	−	0.655949i	$-0.772269\pi$
$311$	− 408.000i	− 1.31190i	0.754806	−	0.655949i	$-0.227731\pi$
$312$	0	0
$313$	−298.958	−0.955138	−0.477569	−	0.878594i	$-0.658482\pi$
$313$	−298.958	−0.955138	−0.477569	+	0.878594i	$0.658482\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	186.161i	0.587259i	0.955919	+	0.293630i	$0.0948633\pi$
$317$	186.161i	0.587259i	−0.955919	+	0.293630i	$0.905137\pi$
$318$	0	0
$319$	304.000i	0.952978i
$320$	0	0
$321$	366.000	1.14019
$322$	0	0
$323$	−558.484	−1.72905
$324$	0	0
$325$	0	0
$326$	0	0
$327$	448.437i	1.37137i
$328$	0	0
$329$	114.000	0.346505
$330$	0	0
$331$	287.687	0.869146	0.434573	−	0.900637i	$-0.356899\pi$
$331$	287.687	0.869146	0.434573	+	0.900637i	$0.356899\pi$
$332$	0	0
$333$	− 44.0908i	− 0.132405i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	−418.454	−1.23438
$340$	0	0
$341$	34.8712i	0.102262i
$342$	0	0
$343$	− 170.833i	− 0.498056i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−413.964	−1.19298	−0.596490	−	0.802621i	$-0.703438\pi$
$347$	−413.964	−1.19298	−0.596490	+	0.802621i	$0.703438\pi$
$348$	0	0
$349$	209.227i	0.599505i	0.954017	+	0.299752i	$0.0969041\pi$
$349$	209.227i	0.599505i	−0.954017	+	0.299752i	$0.903096\pi$
$350$	0	0
$351$	576.000i	1.64103i
$352$	0	0
$353$	213.542	0.604934	0.302467	−	0.953160i	$-0.402190\pi$
$353$	213.542	0.604934	0.302467	+	0.953160i	$0.402190\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 558.484i	− 1.56438i
$358$	0	0
$359$	72.0000i	0.200557i	0.994959	+	0.100279i	$0.0319734\pi$
$359$	72.0000i	0.200557i	−0.994959	+	0.100279i	$0.968027\pi$
$360$	0	0
$361$	323.000	0.894737
$362$	0	0
$363$	−110.227	−0.303656
$364$	0	0
$365$	0	0
$366$	0	0
$367$	523.177i	1.42555i	0.701393	+	0.712775i	$0.252562\pi$
$367$	523.177i	1.42555i	−0.701393	+	0.712775i	$0.747438\pi$
$368$	0	0
$369$	72.0000	0.195122
$370$	0	0
$371$	523.068	1.40989
$372$	0	0
$373$	− 93.0806i	− 0.249546i	−0.992185	−	0.124773i	$-0.960180\pi$
$373$	− 93.0806i	− 0.249546i	0.992185	−	0.124773i	$-0.0398202\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−683.333	−1.81255
$378$	0	0
$379$	−339.994	−0.897082	−0.448541	−	0.893762i	$-0.648056\pi$
$379$	−339.994	−0.897082	−0.448541	+	0.893762i	$0.648056\pi$
$380$	0	0
$381$	130.767i	0.343220i
$382$	0	0
$383$	245.573i	0.641182i	0.947218	+	0.320591i	$0.103882\pi$
$383$	245.573i	0.641182i	−0.947218	+	0.320591i	$0.896118\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	169.015	0.436731
$388$	0	0
$389$	148.203i	0.380983i	0.981689	+	0.190492i	$0.0610082\pi$
$389$	148.203i	0.380983i	−0.981689	+	0.190492i	$0.938992\pi$
$390$	0	0
$391$	− 228.000i	− 0.583120i
$392$	0	0
$393$	−234.896	−0.597699
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 460.504i	− 1.15996i	−0.814631	−	0.579980i	$-0.803060\pi$
$397$	− 460.504i	− 1.15996i	0.814631	−	0.579980i	$-0.196940\pi$
$398$	0	0
$399$	684.000i	1.71429i
$400$	0	0
$401$	−330.000	−0.822943	−0.411471	−	0.911423i	$-0.634985\pi$
$401$	−330.000	−0.822943	−0.411471	+	0.911423i	$0.634985\pi$
$402$	0	0
$403$	−78.3837	−0.194500
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 128.125i	− 0.314803i
$408$	0	0
$409$	536.000	1.31051	0.655257	−	0.755406i	$-0.272560\pi$
$409$	536.000	1.31051	0.655257	+	0.755406i	$0.272560\pi$
$410$	0	0
$411$	−313.841	−0.763603
$412$	0	0
$413$	− 465.403i	− 1.12688i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−576.562	−1.38264
$418$	0	0
$419$	−479.479	−1.14434	−0.572171	−	0.820135i	$-0.693898\pi$
$419$	−479.479	−1.14434	−0.572171	+	0.820135i	$0.693898\pi$
$420$	0	0
$421$	− 496.914i	− 1.18032i	−0.807287	−	0.590160i	$-0.799065\pi$
$421$	− 496.914i	− 1.18032i	0.807287	−	0.590160i	$-0.200935\pi$
$422$	0	0
$423$	32.0312i	0.0757240i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	279.242	0.653962
$428$	0	0
$429$	418.454i	0.975418i
$430$	0	0
$431$	180.000i	0.417633i	0.977955	+	0.208817i	$0.0669612\pi$
$431$	180.000i	0.417633i	−0.977955	+	0.208817i	$0.933039\pi$
$432$	0	0
$433$	−363.021	−0.838385	−0.419192	−	0.907897i	$-0.637687\pi$
$433$	−363.021	−0.838385	−0.419192	+	0.907897i	$0.637687\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	279.242i	0.638997i
$438$	0	0
$439$	136.000i	0.309795i	0.987931	+	0.154897i	$0.0495047\pi$
$439$	136.000i	0.309795i	−0.987931	+	0.154897i	$0.950495\pi$
$440$	0	0
$441$	195.000	0.442177
$442$	0	0
$443$	183.712	0.414699	0.207350	−	0.978267i	$-0.433516\pi$
$443$	183.712	0.414699	0.207350	+	0.978267i	$0.433516\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 234.896i	− 0.525494i
$448$	0	0
$449$	684.000	1.52339	0.761693	−	0.647939i	$-0.224369\pi$
$449$	684.000	1.52339	0.761693	+	0.647939i	$0.224369\pi$
$450$	0	0
$451$	209.227	0.463918
$452$	0	0
$453$	− 391.918i	− 0.865162i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	85.4166	0.186907	0.0934536	−	0.995624i	$-0.470209\pi$
$457$	85.4166	0.186907	0.0934536	+	0.995624i	$0.470209\pi$
$458$	0	0
$459$	627.681	1.36750
$460$	0	0
$461$	− 348.712i	− 0.756425i	−0.925719	−	0.378212i	$-0.876539\pi$
$461$	− 348.712i	− 0.756425i	0.925719	−	0.378212i	$-0.123461\pi$
$462$	0	0
$463$	565.885i	1.22221i	0.791548	+	0.611107i	$0.209276\pi$
$463$	565.885i	1.22221i	−0.791548	+	0.611107i	$0.790724\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	105.328	0.225542	0.112771	−	0.993621i	$-0.464027\pi$
$467$	105.328	0.225542	0.112771	+	0.993621i	$0.464027\pi$
$468$	0	0
$469$	− 78.4602i	− 0.167292i
$470$	0	0
$471$	− 84.0000i	− 0.178344i
$472$	0	0
$473$	491.146	1.03836
$474$	0	0
$475$	0	0
$476$	0	0
$477$	146.969i	0.308112i
$478$	0	0
$479$	552.000i	1.15240i	0.817308	+	0.576200i	$0.195465\pi$
$479$	552.000i	1.15240i	−0.817308	+	0.576200i	$0.804535\pi$
$480$	0	0
$481$	288.000	0.598753
$482$	0	0
$483$	−279.242	−0.578140
$484$	0	0
$485$	0	0
$486$	0	0
$487$	224.219i	0.460408i	0.973142	+	0.230204i	$0.0739393\pi$
$487$	224.219i	0.460408i	−0.973142	+	0.230204i	$0.926061\pi$
$488$	0	0
$489$	294.000	0.601227
$490$	0	0
$491$	−479.479	−0.976535	−0.488268	−	0.872694i	$-0.662371\pi$
$491$	−479.479	−0.976535	−0.488268	+	0.872694i	$0.662371\pi$
$492$	0	0
$493$	744.645i	1.51044i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	896.875	1.80458
$498$	0	0
$499$	287.687	0.576528	0.288264	−	0.957551i	$-0.406922\pi$
$499$	287.687	0.576528	0.288264	+	0.957551i	$0.406922\pi$
$500$	0	0
$501$	706.142i	1.40946i
$502$	0	0
$503$	− 715.364i	− 1.42220i	−0.703093	−	0.711098i	$-0.748198\pi$
$503$	− 715.364i	− 1.42220i	0.703093	−	0.711098i	$-0.251802\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−526.640	−1.03874
$508$	0	0
$509$	− 802.037i	− 1.57571i	−0.615860	−	0.787856i	$-0.711191\pi$
$509$	− 802.037i	− 1.57571i	0.615860	−	0.787856i	$-0.288809\pi$
$510$	0	0
$511$	1140.00i	2.23092i
$512$	0	0
$513$	−768.750	−1.49854
$514$	0	0
$515$	0	0
$516$	0	0
$517$	93.0806i	0.180040i
$518$	0	0
$519$	60.0000i	0.115607i
$520$	0	0
$521$	−102.000	−0.195777	−0.0978887	−	0.995197i	$-0.531209\pi$
$521$	−102.000	−0.195777	−0.0978887	+	0.995197i	$0.531209\pi$
$522$	0	0
$523$	575.630	1.10063	0.550316	−	0.834957i	$-0.314507\pi$
$523$	575.630	1.10063	0.550316	+	0.834957i	$0.314507\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	85.4166i	0.162081i
$528$	0	0
$529$	415.000	0.784499
$530$	0	0
$531$	130.767	0.246265
$532$	0	0
$533$	470.302i	0.882368i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	619.271	1.15320
$538$	0	0
$539$	566.657	1.05131
$540$	0	0
$541$	836.909i	1.54697i	0.633817	+	0.773483i	$0.281487\pi$
$541$	836.909i	1.54697i	−0.633817	+	0.773483i	$0.718513\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−560.933	−1.02547	−0.512736	−	0.858546i	$-0.671368\pi$
$547$	−560.933	−1.02547	−0.512736	+	0.858546i	$0.671368\pi$
$548$	0	0
$549$	78.4602i	0.142915i
$550$	0	0
$551$	− 912.000i	− 1.65517i
$552$	0	0
$553$	1067.71	1.93076
$554$	0	0
$555$	0	0
$556$	0	0
$557$	788.736i	1.41604i	0.706191	+	0.708021i	$0.250412\pi$
$557$	788.736i	1.41604i	−0.706191	+	0.708021i	$0.749588\pi$
$558$	0	0
$559$	1104.00i	1.97496i
$560$	0	0
$561$	456.000	0.812834
$562$	0	0
$563$	149.419	0.265398	0.132699	−	0.991156i	$-0.457636\pi$
$563$	149.419	0.265398	0.132699	+	0.991156i	$0.457636\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 480.469i	− 0.847387i
$568$	0	0
$569$	144.000	0.253076	0.126538	−	0.991962i	$-0.459614\pi$
$569$	144.000	0.253076	0.126538	+	0.991962i	$0.459614\pi$
$570$	0	0
$571$	−496.914	−0.870253	−0.435127	−	0.900369i	$-0.643296\pi$
$571$	−496.914	−0.870253	−0.435127	+	0.900369i	$0.643296\pi$
$572$	0	0
$573$	− 58.7878i	− 0.102596i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−384.375	−0.666161	−0.333080	−	0.942898i	$-0.608088\pi$
$577$	−384.375	−0.666161	−0.333080	+	0.942898i	$0.608088\pi$
$578$	0	0
$579$	261.534	0.451699
$580$	0	0
$581$	183.074i	0.315101i
$582$	0	0
$583$	427.083i	0.732561i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	869.569	1.48138	0.740689	−	0.671848i	$-0.234499\pi$
$587$	869.569	1.48138	0.740689	+	0.671848i	$0.234499\pi$
$588$	0	0
$589$	− 104.614i	− 0.177612i
$590$	0	0
$591$	840.000i	1.42132i
$592$	0	0
$593$	213.542	0.360104	0.180052	−	0.983657i	$-0.442373\pi$
$593$	213.542	0.360104	0.180052	+	0.983657i	$0.442373\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 460.504i	− 0.771364i
$598$	0	0
$599$	300.000i	0.500835i	0.968138	+	0.250417i	$0.0805678\pi$
$599$	300.000i	0.500835i	−0.968138	+	0.250417i	$0.919432\pi$
$600$	0	0
$601$	740.000	1.23128	0.615641	−	0.788027i	$-0.288897\pi$
$601$	740.000	1.23128	0.615641	+	0.788027i	$0.288897\pi$
$602$	0	0
$603$	22.0454	0.0365595
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 608.593i	− 1.00263i	−0.865266	−	0.501313i	$-0.832851\pi$
$607$	− 608.593i	− 1.00263i	0.865266	−	0.501313i	$-0.167149\pi$
$608$	0	0
$609$	912.000	1.49754
$610$	0	0
$611$	−209.227	−0.342434
$612$	0	0
$613$	− 431.110i	− 0.703279i	−0.936135	−	0.351640i	$-0.885624\pi$
$613$	− 431.110i	− 0.703279i	0.936135	−	0.351640i	$-0.114376\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−106.771	−0.173048	−0.0865241	−	0.996250i	$-0.527576\pi$
$617$	−106.771	−0.173048	−0.0865241	+	0.996250i	$0.527576\pi$
$618$	0	0
$619$	−183.074	−0.295757	−0.147879	−	0.989006i	$-0.547245\pi$
$619$	−183.074	−0.295757	−0.147879	+	0.989006i	$0.547245\pi$
$620$	0	0
$621$	− 313.841i	− 0.505380i
$622$	0	0
$623$	− 1601.56i	− 2.57073i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−558.484	−0.890724
$628$	0	0
$629$	− 313.841i	− 0.498952i
$630$	0	0
$631$	− 320.000i	− 0.507132i	−0.967318	−	0.253566i	$-0.918397\pi$
$631$	− 320.000i	− 0.507132i	0.967318	−	0.253566i	$-0.0816034\pi$
$632$	0	0
$633$	−64.0625	−0.101205
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1273.73i	1.99958i
$638$	0	0
$639$	252.000i	0.394366i
$640$	0	0
$641$	−420.000	−0.655226	−0.327613	−	0.944812i	$-0.606244\pi$
$641$	−420.000	−0.655226	−0.327613	+	0.944812i	$0.606244\pi$
$642$	0	0
$643$	1045.93	1.62664	0.813322	−	0.581814i	$-0.197657\pi$
$643$	1045.93	1.62664	0.813322	+	0.581814i	$0.197657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 544.531i	− 0.841624i	−0.907148	−	0.420812i	$-0.861745\pi$
$647$	− 544.531i	− 0.841624i	0.907148	−	0.420812i	$-0.138255\pi$
$648$	0	0
$649$	380.000	0.585516
$650$	0	0
$651$	104.614	0.160697
$652$	0	0
$653$	509.494i	0.780236i	0.920765	+	0.390118i	$0.127566\pi$
$653$	509.494i	0.780236i	−0.920765	+	0.390118i	$0.872434\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−320.312	−0.487538
$658$	0	0
$659$	−514.350	−0.780501	−0.390250	−	0.920709i	$-0.627612\pi$
$659$	−514.350	−0.780501	−0.390250	+	0.920709i	$0.627612\pi$
$660$	0	0
$661$	− 26.1534i	− 0.0395664i	−0.999804	−	0.0197832i	$-0.993702\pi$
$661$	− 26.1534i	− 0.0395664i	0.999804	−	0.0197832i	$-0.00629760\pi$
$662$	0	0
$663$	1025.00i	1.54600i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	372.322	0.558205
$668$	0	0
$669$	− 549.221i	− 0.820959i
$670$	0	0
$671$	228.000i	0.339791i
$672$	0	0
$673$	−533.854	−0.793245	−0.396623	−	0.917982i	$-0.629818\pi$
$673$	−533.854	−0.793245	−0.396623	+	0.917982i	$0.629818\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1116.97i	− 1.64988i	−0.565222	−	0.824939i	$-0.691209\pi$
$677$	− 1116.97i	− 1.64988i	0.565222	−	0.824939i	$-0.308791\pi$
$678$	0	0
$679$	228.000i	0.335788i
$680$	0	0
$681$	−198.000	−0.290749
$682$	0	0
$683$	1256.59	1.83981	0.919904	−	0.392145i	$-0.128267\pi$
$683$	1256.59	1.83981	0.919904	+	0.392145i	$0.128267\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	256.250i	0.372998i
$688$	0	0
$689$	−960.000	−1.39332
$690$	0	0
$691$	−1072.29	−1.55179	−0.775897	−	0.630860i	$-0.782702\pi$
$691$	−1072.29	−1.55179	−0.775897	+	0.630860i	$0.782702\pi$
$692$	0	0
$693$	279.242i	0.402946i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	512.500	0.735294
$698$	0	0
$699$	−784.602	−1.12246
$700$	0	0
$701$	− 43.5890i	− 0.0621812i	−0.999517	−	0.0310906i	$-0.990102\pi$
$701$	− 43.5890i	− 0.0621812i	0.999517	−	0.0310906i	$-0.00989803\pi$
$702$	0	0
$703$	384.375i	0.546764i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	744.645	1.05325
$708$	0	0
$709$	− 209.227i	− 0.295102i	−0.989054	−	0.147551i	$-0.952861\pi$
$709$	− 209.227i	− 0.295102i	0.989054	−	0.147551i	$-0.0471390\pi$
$710$	0	0
$711$	300.000i	0.421941i
$712$	0	0
$713$	42.7083	0.0598995
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 264.545i	− 0.368961i
$718$	0	0
$719$	− 612.000i	− 0.851182i	−0.904916	−	0.425591i	$-0.860066\pi$
$719$	− 612.000i	− 0.851182i	0.904916	−	0.425591i	$-0.139934\pi$
$720$	0	0
$721$	342.000	0.474341
$722$	0	0
$723$	−68.5857	−0.0948627
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1334.63i	1.83581i	0.396799	+	0.917906i	$0.370121\pi$
$727$	1334.63i	1.83581i	−0.396799	+	0.917906i	$0.629879\pi$
$728$	0	0
$729$	783.000	1.07407
$730$	0	0
$731$	1203.06	1.64577
$732$	0	0
$733$	− 220.454i	− 0.300756i	−0.988629	−	0.150378i	$-0.951951\pi$
$733$	− 220.454i	− 0.300756i	0.988629	−	0.150378i	$-0.0480491\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	64.0625	0.0869233
$738$	0	0
$739$	1438.44	1.94646	0.973232	−	0.229826i	$-0.0738156\pi$
$739$	1438.44	1.94646	0.973232	+	0.229826i	$0.0738156\pi$
$740$	0	0
$741$	− 1255.36i	− 1.69415i
$742$	0	0
$743$	− 138.802i	− 0.186813i	−0.995628	−	0.0934065i	$-0.970224\pi$
$743$	− 138.802i	− 0.186813i	0.995628	−	0.0934065i	$-0.0297756\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−51.4393	−0.0688612
$748$	0	0
$749$	1595.36i	2.12998i
$750$	0	0
$751$	608.000i	0.809587i	0.914408	+	0.404794i	$0.132657\pi$
$751$	608.000i	0.809587i	−0.914408	+	0.404794i	$0.867343\pi$
$752$	0	0
$753$	−619.271	−0.822404
$754$	0	0
$755$	0	0
$756$	0	0
$757$	592.777i	0.783060i	0.920165	+	0.391530i	$0.128054\pi$
$757$	592.777i	0.783060i	−0.920165	+	0.391530i	$0.871946\pi$
$758$	0	0
$759$	− 228.000i	− 0.300395i
$760$	0	0
$761$	−642.000	−0.843627	−0.421813	−	0.906683i	$-0.638606\pi$
$761$	−642.000	−0.843627	−0.421813	+	0.906683i	$0.638606\pi$
$762$	0	0
$763$	−1954.69	−2.56185
$764$	0	0
$765$	0	0
$766$	0	0
$767$	854.166i	1.11365i
$768$	0	0
$769$	230.000	0.299090	0.149545	−	0.988755i	$-0.452219\pi$
$769$	230.000	0.299090	0.149545	+	0.988755i	$0.452219\pi$
$770$	0	0
$771$	732.295	0.949799
$772$	0	0
$773$	− 88.1816i	− 0.114077i	−0.998372	−	0.0570386i	$-0.981834\pi$
$773$	− 88.1816i	− 0.114077i	0.998372	−	0.0570386i	$-0.0181658\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−384.375	−0.494691
$778$	0	0
$779$	−627.681	−0.805753
$780$	0	0
$781$	732.295i	0.937638i
$782$	0	0
$783$	1025.00i	1.30907i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−232.702	−0.295682	−0.147841	−	0.989011i	$-0.547232\pi$
$787$	−232.702	−0.295682	−0.147841	+	0.989011i	$0.547232\pi$
$788$	0	0
$789$	706.142i	0.894983i
$790$	0	0
$791$	− 1824.00i	− 2.30594i
$792$	0	0
$793$	−512.500	−0.646280
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 989.594i	− 1.24165i	−0.783950	−	0.620824i	$-0.786798\pi$
$797$	− 989.594i	− 1.24165i	0.783950	−	0.620824i	$-0.213202\pi$
$798$	0	0
$799$	228.000i	0.285357i
$800$	0	0
$801$	450.000	0.561798
$802$	0	0
$803$	−930.806	−1.15916
$804$	0	0
$805$	0	0
$806$	0	0
$807$	149.479i	0.185228i
$808$	0	0
$809$	−906.000	−1.11990	−0.559951	−	0.828526i	$-0.689180\pi$
$809$	−906.000	−1.11990	−0.559951	+	0.828526i	$0.689180\pi$
$810$	0	0
$811$	1281.52	1.58017	0.790084	−	0.612999i	$-0.210037\pi$
$811$	1281.52	1.58017	0.790084	+	0.612999i	$0.210037\pi$
$812$	0	0
$813$	1048.38i	1.28952i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1473.44	−1.80347
$818$	0	0
$819$	−627.681	−0.766400
$820$	0	0
$821$	− 409.737i	− 0.499070i	−0.968366	−	0.249535i	$-0.919722\pi$
$821$	− 409.737i	− 0.499070i	0.968366	−	0.249535i	$-0.0802778\pi$
$822$	0	0
$823$	− 373.698i	− 0.454068i	−0.973887	−	0.227034i	$-0.927097\pi$
$823$	− 373.698i	− 0.454068i	0.973887	−	0.227034i	$-0.0729028\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−169.015	−0.204371	−0.102185	−	0.994765i	$-0.532584\pi$
$827$	−169.015	−0.204371	−0.102185	+	0.994765i	$0.532584\pi$
$828$	0	0
$829$	130.767i	0.157741i	0.996885	+	0.0788703i	$0.0251313\pi$
$829$	130.767i	0.157741i	−0.996885	+	0.0788703i	$0.974869\pi$
$830$	0	0
$831$	180.000i	0.216606i
$832$	0	0
$833$	1388.02	1.66629
$834$	0	0
$835$	0	0
$836$	0	0
$837$	117.576i	0.140473i
$838$	0	0
$839$	− 816.000i	− 0.972586i	−0.873796	−	0.486293i	$-0.838349\pi$
$839$	− 816.000i	− 0.972586i	0.873796	−	0.486293i	$-0.161651\pi$
$840$	0	0
$841$	−375.000	−0.445898
$842$	0	0
$843$	−499.696	−0.592759
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 480.469i	− 0.567259i
$848$	0	0
$849$	−990.000	−1.16608
$850$	0	0
$851$	−156.920	−0.184395
$852$	0	0
$853$	450.706i	0.528378i	0.964471	+	0.264189i	$0.0851042\pi$
$853$	450.706i	0.528378i	−0.964471	+	0.264189i	$0.914896\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−512.500	−0.598016	−0.299008	−	0.954251i	$-0.596656\pi$
$857$	−512.500	−0.598016	−0.299008	+	0.954251i	$0.596656\pi$
$858$	0	0
$859$	−496.914	−0.578480	−0.289240	−	0.957257i	$-0.593403\pi$
$859$	−496.914	−0.578480	−0.289240	+	0.957257i	$0.593403\pi$
$860$	0	0
$861$	− 627.681i	− 0.729014i
$862$	0	0
$863$	53.3854i	0.0618602i	0.999522	+	0.0309301i	$0.00984693\pi$
$863$	53.3854i	0.0618602i	−0.999522	+	0.0309301i	$0.990153\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	409.065	0.471816
$868$	0	0
$869$	871.780i	1.00320i
$870$	0	0
$871$	144.000i	0.165327i
$872$	0	0
$873$	−64.0625	−0.0733820
$874$	0	0
$875$	0	0
$876$	0	0
$877$	230.252i	0.262545i	0.991346	+	0.131273i	$0.0419063\pi$
$877$	230.252i	0.262545i	−0.991346	+	0.131273i	$0.958094\pi$
$878$	0	0
$879$	396.000i	0.450512i
$880$	0	0
$881$	−1320.00	−1.49830	−0.749149	−	0.662402i	$-0.769537\pi$
$881$	−1320.00	−1.49830	−0.749149	+	0.662402i	$0.769537\pi$
$882$	0	0
$883$	1050.83	1.19007	0.595035	−	0.803700i	$-0.297138\pi$
$883$	1050.83	1.19007	0.595035	+	0.803700i	$0.297138\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 117.448i	− 0.132410i	−0.997806	−	0.0662051i	$-0.978911\pi$
$887$	− 117.448i	− 0.132410i	0.997806	−	0.0662051i	$-0.0210892\pi$
$888$	0	0
$889$	−570.000	−0.641170
$890$	0	0
$891$	392.301	0.440293
$892$	0	0
$893$	− 279.242i	− 0.312701i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	512.500	0.571349
$898$	0	0
$899$	−139.485	−0.155155
$900$	0	0
$901$	1046.14i	1.16108i
$902$	0	0
$903$	− 1473.44i	− 1.63171i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−585.428	−0.645455	−0.322728	−	0.946492i	$-0.604600\pi$
$907$	−585.428	−0.645455	−0.322728	+	0.946492i	$0.604600\pi$
$908$	0	0
$909$	209.227i	0.230173i
$910$	0	0
$911$	− 564.000i	− 0.619100i	−0.950883	−	0.309550i	$-0.899822\pi$
$911$	− 564.000i	− 0.619100i	0.950883	−	0.309550i	$-0.100178\pi$
$912$	0	0
$913$	−149.479	−0.163723
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1023.89i	− 1.11656i
$918$	0	0
$919$	− 1124.00i	− 1.22307i	−0.791218	−	0.611534i	$-0.790553\pi$
$919$	− 1124.00i	− 1.22307i	0.791218	−	0.611534i	$-0.209447\pi$
$920$	0	0
$921$	1230.00	1.33550
$922$	0	0
$923$	−1646.06	−1.78338
$924$	0	0
$925$	0	0
$926$	0	0
$927$	96.0937i	0.103661i
$928$	0	0
$929$	276.000	0.297094	0.148547	−	0.988905i	$-0.452540\pi$
$929$	276.000	0.297094	0.148547	+	0.988905i	$0.452540\pi$
$930$	0	0
$931$	−1699.97	−1.82596
$932$	0	0
$933$	− 999.392i	− 1.07116i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	619.271	0.660908	0.330454	−	0.943822i	$-0.392798\pi$
$937$	619.271	0.660908	0.330454	+	0.943822i	$0.392798\pi$
$938$	0	0
$939$	−732.295	−0.779867
$940$	0	0
$941$	− 1115.88i	− 1.18584i	−0.805260	−	0.592921i	$-0.797974\pi$
$941$	− 1115.88i	− 1.18584i	0.805260	−	0.592921i	$-0.202026\pi$
$942$	0	0
$943$	− 256.250i	− 0.271739i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	364.974	0.385400	0.192700	−	0.981258i	$-0.438276\pi$
$947$	364.974	0.385400	0.192700	+	0.981258i	$0.438276\pi$
$948$	0	0
$949$	− 2092.27i	− 2.20471i
$950$	0	0
$951$	456.000i	0.479495i
$952$	0	0
$953$	−341.667	−0.358517	−0.179258	−	0.983802i	$-0.557370\pi$
$953$	−341.667	−0.358517	−0.179258	+	0.983802i	$0.557370\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	744.645i	0.778103i
$958$	0	0
$959$	− 1368.00i	− 1.42649i
$960$	0	0
$961$	945.000	0.983351
$962$	0	0
$963$	−448.257	−0.465479
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1548.18i	− 1.60101i	−0.599326	−	0.800505i	$-0.704565\pi$
$967$	− 1548.18i	− 1.60101i	0.599326	−	0.800505i	$-0.295435\pi$
$968$	0	0
$969$	−1368.00	−1.41176
$970$	0	0
$971$	−828.191	−0.852926	−0.426463	−	0.904505i	$-0.640241\pi$
$971$	−828.191	−0.852926	−0.426463	+	0.904505i	$0.640241\pi$
$972$	0	0
$973$	− 2513.18i	− 2.58292i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1558.85	1.59555	0.797776	−	0.602955i	$-0.206010\pi$
$977$	1558.85	1.59555	0.797776	+	0.602955i	$0.206010\pi$
$978$	0	0
$979$	1307.67	1.33572
$980$	0	0
$981$	− 549.221i	− 0.559859i
$982$	0	0
$983$	843.489i	0.858076i	0.903286	+	0.429038i	$0.141147\pi$
$983$	843.489i	0.858076i	−0.903286	+	0.429038i	$0.858853\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	279.242	0.282920
$988$	0	0
$989$	− 601.528i	− 0.608218i
$990$	0	0
$991$	208.000i	0.209889i	0.994478	+	0.104945i	$0.0334665\pi$
$991$	208.000i	0.209889i	−0.994478	+	0.104945i	$0.966534\pi$
$992$	0	0
$993$	704.687	0.709655
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 950.402i	− 0.953262i	−0.879104	−	0.476631i	$-0.841858\pi$
$997$	− 950.402i	− 0.953262i	0.879104	−	0.476631i	$-0.158142\pi$
$998$	0	0
$999$	− 432.000i	− 0.432432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.g.h.351.7		8
4.3	odd	2	inner	1600.3.g.h.351.2		8
5.2	odd	4		320.3.e.a.159.4	yes	8
5.3	odd	4		320.3.e.a.159.6	yes	8
5.4	even	2	inner	1600.3.g.h.351.1		8
8.3	odd	2	inner	1600.3.g.h.351.5		8
8.5	even	2	inner	1600.3.g.h.351.4		8
20.3	even	4		320.3.e.a.159.2	yes	8
20.7	even	4		320.3.e.a.159.8	yes	8
20.19	odd	2	inner	1600.3.g.h.351.8		8
40.3	even	4		320.3.e.a.159.7	yes	8
40.13	odd	4		320.3.e.a.159.3	yes	8
40.19	odd	2	inner	1600.3.g.h.351.3		8
40.27	even	4		320.3.e.a.159.1	✓	8
40.29	even	2	inner	1600.3.g.h.351.6		8
40.37	odd	4		320.3.e.a.159.5	yes	8
80.3	even	4		1280.3.h.i.1279.8		8
80.13	odd	4		1280.3.h.i.1279.4		8
80.27	even	4		1280.3.h.i.1279.6		8
80.37	odd	4		1280.3.h.i.1279.2		8
80.43	even	4		1280.3.h.i.1279.1		8
80.53	odd	4		1280.3.h.i.1279.5		8
80.67	even	4		1280.3.h.i.1279.3		8
80.77	odd	4		1280.3.h.i.1279.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.a.159.1	✓	8	40.27	even	4
320.3.e.a.159.2	yes	8	20.3	even	4
320.3.e.a.159.3	yes	8	40.13	odd	4
320.3.e.a.159.4	yes	8	5.2	odd	4
320.3.e.a.159.5	yes	8	40.37	odd	4
320.3.e.a.159.6	yes	8	5.3	odd	4
320.3.e.a.159.7	yes	8	40.3	even	4
320.3.e.a.159.8	yes	8	20.7	even	4
1280.3.h.i.1279.1		8	80.43	even	4
1280.3.h.i.1279.2		8	80.37	odd	4
1280.3.h.i.1279.3		8	80.67	even	4
1280.3.h.i.1279.4		8	80.13	odd	4
1280.3.h.i.1279.5		8	80.53	odd	4
1280.3.h.i.1279.6		8	80.27	even	4
1280.3.h.i.1279.7		8	80.77	odd	4
1280.3.h.i.1279.8		8	80.3	even	4
1600.3.g.h.351.1		8	5.4	even	2	inner
1600.3.g.h.351.2		8	4.3	odd	2	inner
1600.3.g.h.351.3		8	40.19	odd	2	inner
1600.3.g.h.351.4		8	8.5	even	2	inner
1600.3.g.h.351.5		8	8.3	odd	2	inner
1600.3.g.h.351.6		8	40.29	even	2	inner
1600.3.g.h.351.7		8	1.1	even	1	trivial
1600.3.g.h.351.8		8	20.19	odd	2	inner

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +10.6771i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} +10.6771i q^{7} -3.00000 q^{9} -8.71780 q^{11} -19.5959i q^{13} -21.3542 q^{17} +26.1534 q^{19} +26.1534i q^{21} +10.6771i q^{23} -29.3939 q^{27} -34.8712i q^{29} -4.00000i q^{31} -21.3542 q^{33} +14.6969i q^{37} -48.0000i q^{39} -24.0000 q^{41} -56.3383 q^{43} -10.6771i q^{47} -65.0000 q^{49} -52.3068 q^{51} -48.9898i q^{53} +64.0625 q^{57} -43.5890 q^{59} -26.1534i q^{61} -32.0312i q^{63} -7.34847 q^{67} +26.1534i q^{69} -84.0000i q^{71} +106.771 q^{73} -93.0806i q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464 q^{83} -85.4166i q^{87} -150.000 q^{89} +209.227 q^{91} -9.79796i q^{93} +21.3542 q^{97} +26.1534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} - 192 q^{41} - 520 q^{49} - 360 q^{81} - 1200 q^{89}+O(q^{100}) \) 8 * q - 24 * q^9 - 192 * q^41 - 520 * q^49 - 360 * q^81 - 1200 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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