LMFDB - Embedded newform 2400.2.d.g.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(49,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (of order $2$, degree $1$, not 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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.41216 - 0.0762223i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.g.49.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.97676i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.97676i q^{7} +1.00000 q^{9} -1.43055i q^{11} +0.241319 q^{13} +7.38407i q^{17} -3.04033i q^{19} +1.97676i q^{21} -0.874337i q^{23} -1.00000 q^{27} -9.07918i q^{29} +7.44764 q^{31} +1.43055i q^{33} +8.81463 q^{37} -0.241319 q^{39} -1.91319 q^{41} -11.2452 q^{43} -3.34374i q^{47} +3.09242 q^{49} -7.38407i q^{51} -9.20632 q^{53} +3.04033i q^{57} +6.43616i q^{59} -4.57331i q^{61} -1.97676i q^{63} -4.86671 q^{67} +0.874337i q^{69} +8.21808 q^{71} -4.12714i q^{73} -2.82786 q^{77} -13.6757 q^{79} +1.00000 q^{81} -12.3320 q^{83} +9.07918i q^{87} +8.08066 q^{89} -0.477031i q^{91} -7.44764 q^{93} -10.6757i q^{97} -1.43055i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 8 * q^9	$ 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.97676i	− 0.747145i	−0.927601	−	0.373573i	$-0.878133\pi$
$7$	− 1.97676i	− 0.747145i	0.927601	−	0.373573i	$-0.121867\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.43055i	− 0.431328i	−0.976468	−	0.215664i	$-0.930808\pi$
$11$	− 1.43055i	− 0.431328i	0.976468	−	0.215664i	$-0.0691915\pi$
$12$	0	0
$13$	0.241319	0.0669300	0.0334650	−	0.999440i	$-0.489346\pi$
$13$	0.241319	0.0669300	0.0334650	+	0.999440i	$0.489346\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.38407i	1.79090i	0.445161	+	0.895450i	$0.353146\pi$
$17$	7.38407i	1.79090i	−0.445161	+	0.895450i	$0.646854\pi$
$18$	0	0
$19$	− 3.04033i	− 0.697500i	−0.937216	−	0.348750i	$-0.886606\pi$
$19$	− 3.04033i	− 0.697500i	0.937216	−	0.348750i	$-0.113394\pi$
$20$	0	0
$21$	1.97676i	0.431365i
$22$	0	0
$23$	− 0.874337i	− 0.182312i	−0.995837	−	0.0911560i	$-0.970944\pi$
$23$	− 0.874337i	− 0.182312i	0.995837	−	0.0911560i	$-0.0290562\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	− 9.07918i	− 1.68596i	−0.537943	−	0.842981i	$-0.680799\pi$
$29$	− 9.07918i	− 1.68596i	0.537943	−	0.842981i	$-0.319201\pi$
$30$	0	0
$31$	7.44764	1.33764	0.668818	−	0.743426i	$-0.266800\pi$
$31$	7.44764	1.33764	0.668818	+	0.743426i	$0.266800\pi$
$32$	0	0
$33$	1.43055i	0.249027i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.81463	1.44912	0.724558	−	0.689214i	$-0.242044\pi$
$37$	8.81463	1.44912	0.724558	+	0.689214i	$0.242044\pi$
$38$	0	0
$39$	−0.241319	−0.0386420
$40$	0	0
$41$	−1.91319	−0.298790	−0.149395	−	0.988778i	$-0.547733\pi$
$41$	−1.91319	−0.298790	−0.149395	+	0.988778i	$0.547733\pi$
$42$	0	0
$43$	−11.2452	−1.71487	−0.857437	−	0.514589i	$-0.827944\pi$
$43$	−11.2452	−1.71487	−0.857437	+	0.514589i	$0.827944\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.34374i	− 0.487735i	−0.969809	−	0.243867i	$-0.921584\pi$
$47$	− 3.34374i	− 0.487735i	0.969809	−	0.243867i	$-0.0784162\pi$
$48$	0	0
$49$	3.09242	0.441774
$50$	0	0
$51$	− 7.38407i	− 1.03398i
$52$	0	0
$53$	−9.20632	−1.26459	−0.632293	−	0.774729i	$-0.717886\pi$
$53$	−9.20632	−1.26459	−0.632293	+	0.774729i	$0.717886\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.04033i	0.402702i
$58$	0	0
$59$	6.43616i	0.837917i	0.908005	+	0.418958i	$0.137605\pi$
$59$	6.43616i	0.837917i	−0.908005	+	0.418958i	$0.862395\pi$
$60$	0	0
$61$	− 4.57331i	− 0.585552i	−0.956181	−	0.292776i	$-0.905421\pi$
$61$	− 4.57331i	− 0.585552i	0.956181	−	0.292776i	$-0.0945790\pi$
$62$	0	0
$63$	− 1.97676i	− 0.249048i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.86671	−0.594563	−0.297282	−	0.954790i	$-0.596080\pi$
$67$	−4.86671	−0.594563	−0.297282	+	0.954790i	$0.596080\pi$
$68$	0	0
$69$	0.874337i	0.105258i
$70$	0	0
$71$	8.21808	0.975307	0.487653	−	0.873037i	$-0.337853\pi$
$71$	8.21808	0.975307	0.487653	+	0.873037i	$0.337853\pi$
$72$	0	0
$73$	− 4.12714i	− 0.483045i	−0.970395	−	0.241523i	$-0.922353\pi$
$73$	− 4.12714i	− 0.483045i	0.970395	−	0.241523i	$-0.0776468\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82786	−0.322264
$78$	0	0
$79$	−13.6757	−1.53864	−0.769320	−	0.638864i	$-0.779405\pi$
$79$	−13.6757	−1.53864	−0.769320	+	0.638864i	$0.779405\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.3320	−1.35361	−0.676806	−	0.736162i	$-0.736636\pi$
$83$	−12.3320	−1.35361	−0.676806	+	0.736162i	$0.736636\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.07918i	0.973391i
$88$	0	0
$89$	8.08066	0.856548	0.428274	−	0.903649i	$-0.359122\pi$
$89$	8.08066	0.856548	0.428274	+	0.903649i	$0.359122\pi$
$90$	0	0
$91$	− 0.477031i	− 0.0500064i
$92$	0	0
$93$	−7.44764	−0.772285
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.6757i	− 1.08396i	−0.840393	−	0.541978i	$-0.817676\pi$
$97$	− 10.6757i	− 1.08396i	0.840393	−	0.541978i	$-0.182324\pi$
$98$	0	0
$99$	− 1.43055i	− 0.143776i
$100$	0	0
$101$	− 13.2063i	− 1.31408i	−0.753856	−	0.657039i	$-0.771809\pi$
$101$	− 13.2063i	− 1.31408i	0.753856	−	0.657039i	$-0.228191\pi$
$102$	0	0
$103$	− 19.4244i	− 1.91394i	−0.290181	−	0.956972i	$-0.593716\pi$
$103$	− 19.4244i	− 1.91394i	0.290181	−	0.956972i	$-0.406284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.8085	1.43159	0.715795	−	0.698311i	$-0.246065\pi$
$107$	14.8085	1.43159	0.715795	+	0.698311i	$0.246065\pi$
$108$	0	0
$109$	− 15.2296i	− 1.45873i	−0.684126	−	0.729364i	$-0.739816\pi$
$109$	− 15.2296i	− 1.45873i	0.684126	−	0.729364i	$-0.260184\pi$
$110$	0	0
$111$	−8.81463	−0.836647
$112$	0	0
$113$	− 1.13890i	− 0.107138i	−0.998564	−	0.0535692i	$-0.982940\pi$
$113$	− 1.13890i	− 0.107138i	0.998564	−	0.0535692i	$-0.0170598\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.241319	0.0223100
$118$	0	0
$119$	14.5965	1.33806
$120$	0	0
$121$	8.95352	0.813956
$122$	0	0
$123$	1.91319	0.172507
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.43616i	0.216174i	0.994141	+	0.108087i	$0.0344725\pi$
$127$	2.43616i	0.216174i	−0.994141	+	0.108087i	$0.965527\pi$
$128$	0	0
$129$	11.2452	0.990083
$130$	0	0
$131$	− 6.90143i	− 0.602981i	−0.953469	−	0.301491i	$-0.902516\pi$
$131$	− 6.90143i	− 0.602981i	0.953469	−	0.301491i	$-0.0974842\pi$
$132$	0	0
$133$	−6.01001	−0.521134
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.39022i	0.460518i	0.973129	+	0.230259i	$0.0739573\pi$
$137$	5.39022i	0.460518i	−0.973129	+	0.230259i	$0.926043\pi$
$138$	0	0
$139$	− 17.4244i	− 1.47792i	−0.673750	−	0.738959i	$-0.735318\pi$
$139$	− 17.4244i	− 1.47792i	0.673750	−	0.738959i	$-0.264682\pi$
$140$	0	0
$141$	3.34374i	0.281594i
$142$	0	0
$143$	− 0.345220i	− 0.0288687i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.09242	−0.255058
$148$	0	0
$149$	− 2.28551i	− 0.187236i	−0.995608	−	0.0936180i	$-0.970157\pi$
$149$	− 2.28551i	− 0.187236i	0.995608	−	0.0936180i	$-0.0298432\pi$
$150$	0	0
$151$	−6.66425	−0.542329	−0.271164	−	0.962533i	$-0.587409\pi$
$151$	−6.66425	−0.542329	−0.271164	+	0.962533i	$0.587409\pi$
$152$	0	0
$153$	7.38407i	0.596967i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.4144	1.38982	0.694910	−	0.719097i	$-0.255444\pi$
$157$	17.4144	1.38982	0.694910	+	0.719097i	$0.255444\pi$
$158$	0	0
$159$	9.20632	0.730109
$160$	0	0
$161$	−1.72836	−0.136214
$162$	0	0
$163$	4.66187	0.365145	0.182573	−	0.983192i	$-0.441557\pi$
$163$	4.66187	0.365145	0.182573	+	0.983192i	$0.441557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 0.137419i	− 0.0106338i	−0.999986	−	0.00531690i	$-0.998308\pi$
$167$	− 0.137419i	− 0.0106338i	0.999986	−	0.00531690i	$-0.00169243\pi$
$168$	0	0
$169$	−12.9418	−0.995520
$170$	0	0
$171$	− 3.04033i	− 0.232500i
$172$	0	0
$173$	3.96675	0.301587	0.150793	−	0.988565i	$-0.451817\pi$
$173$	3.96675	0.301587	0.150793	+	0.988565i	$0.451817\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.43616i	− 0.483771i
$178$	0	0
$179$	− 4.68749i	− 0.350359i	−0.984537	−	0.175180i	$-0.943949\pi$
$179$	− 4.68749i	− 0.350359i	0.984537	−	0.175180i	$-0.0560506\pi$
$180$	0	0
$181$	− 9.10242i	− 0.676578i	−0.941042	−	0.338289i	$-0.890152\pi$
$181$	− 9.10242i	− 0.676578i	0.941042	−	0.338289i	$-0.109848\pi$
$182$	0	0
$183$	4.57331i	0.338068i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.5633	0.772465
$188$	0	0
$189$	1.97676i	0.143788i
$190$	0	0
$191$	−15.2063	−1.10029	−0.550145	−	0.835069i	$-0.685428\pi$
$191$	−15.2063	−1.10029	−0.550145	+	0.835069i	$0.685428\pi$
$192$	0	0
$193$	− 20.7564i	− 1.49408i	−0.664780	−	0.747039i	$-0.731475\pi$
$193$	− 20.7564i	− 1.49408i	0.664780	−	0.747039i	$-0.268525\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.2508	−1.65655	−0.828275	−	0.560322i	$-0.810677\pi$
$197$	−23.2508	−1.65655	−0.828275	+	0.560322i	$0.810677\pi$
$198$	0	0
$199$	−7.21633	−0.511552	−0.255776	−	0.966736i	$-0.582331\pi$
$199$	−7.21633	−0.511552	−0.255776	+	0.966736i	$0.582331\pi$
$200$	0	0
$201$	4.86671	0.343271
$202$	0	0
$203$	−17.9474	−1.25966
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.874337i	− 0.0607707i
$208$	0	0
$209$	−4.34935	−0.300851
$210$	0	0
$211$	− 4.38407i	− 0.301812i	−0.988548	−	0.150906i	$-0.951781\pi$
$211$	− 4.38407i	− 0.301812i	0.988548	−	0.150906i	$-0.0482191\pi$
$212$	0	0
$213$	−8.21808	−0.563094
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 14.7222i	− 0.999409i
$218$	0	0
$219$	4.12714i	0.278886i
$220$	0	0
$221$	1.78192i	0.119865i
$222$	0	0
$223$	4.98852i	0.334056i	0.985952	+	0.167028i	$0.0534170\pi$
$223$	4.98852i	0.334056i	−0.985952	+	0.167028i	$0.946583\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2569	0.747149	0.373574	−	0.927600i	$-0.378132\pi$
$227$	11.2569	0.747149	0.373574	+	0.927600i	$0.378132\pi$
$228$	0	0
$229$	15.8364i	1.04650i	0.852180	+	0.523249i	$0.175280\pi$
$229$	15.8364i	1.04650i	−0.852180	+	0.523249i	$0.824720\pi$
$230$	0	0
$231$	2.82786	0.186059
$232$	0	0
$233$	− 10.9591i	− 0.717956i	−0.933346	−	0.358978i	$-0.883125\pi$
$233$	− 10.9591i	− 0.717956i	0.933346	−	0.358978i	$-0.116875\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.6757	0.888334
$238$	0	0
$239$	17.3182	1.12022	0.560111	−	0.828418i	$-0.310758\pi$
$239$	17.3182	1.12022	0.560111	+	0.828418i	$0.310758\pi$
$240$	0	0
$241$	4.76869	0.307178	0.153589	−	0.988135i	$-0.450917\pi$
$241$	4.76869	0.307178	0.153589	+	0.988135i	$0.450917\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.733691i	− 0.0466836i
$248$	0	0
$249$	12.3320	0.781508
$250$	0	0
$251$	6.15837i	0.388713i	0.980931	+	0.194356i	$0.0622618\pi$
$251$	6.15837i	0.388713i	−0.980931	+	0.194356i	$0.937738\pi$
$252$	0	0
$253$	−1.25079	−0.0786362
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1584i	0.883175i	0.897218	+	0.441587i	$0.145584\pi$
$257$	14.1584i	0.883175i	−0.897218	+	0.441587i	$0.854416\pi$
$258$	0	0
$259$	− 17.4244i	− 1.08270i
$260$	0	0
$261$	− 9.07918i	− 0.561987i
$262$	0	0
$263$	15.5960i	0.961691i	0.876805	+	0.480845i	$0.159670\pi$
$263$	15.5960i	0.961691i	−0.876805	+	0.480845i	$0.840330\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.08066	−0.494528
$268$	0	0
$269$	11.3182i	0.690084i	0.938587	+	0.345042i	$0.112135\pi$
$269$	11.3182i	0.690084i	−0.938587	+	0.345042i	$0.887865\pi$
$270$	0	0
$271$	6.20485	0.376918	0.188459	−	0.982081i	$-0.439651\pi$
$271$	6.20485	0.376918	0.188459	+	0.982081i	$0.439651\pi$
$272$	0	0
$273$	0.477031i	0.0288712i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.9288	−1.13732	−0.568661	−	0.822572i	$-0.692538\pi$
$277$	−18.9288	−1.13732	−0.568661	+	0.822572i	$0.692538\pi$
$278$	0	0
$279$	7.44764	0.445879
$280$	0	0
$281$	−21.6231	−1.28993	−0.644963	−	0.764214i	$-0.723127\pi$
$281$	−21.6231	−1.28993	−0.644963	+	0.764214i	$0.723127\pi$
$282$	0	0
$283$	29.1522	1.73292	0.866460	−	0.499247i	$-0.166390\pi$
$283$	29.1522	1.73292	0.866460	+	0.499247i	$0.166390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.78192i	0.223240i
$288$	0	0
$289$	−37.5245	−2.20733
$290$	0	0
$291$	10.6757i	0.625822i
$292$	0	0
$293$	−2.32427	−0.135785	−0.0678927	−	0.997693i	$-0.521628\pi$
$293$	−2.32427	−0.135785	−0.0678927	+	0.997693i	$0.521628\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43055i	0.0830090i
$298$	0	0
$299$	− 0.210995i	− 0.0122021i
$300$	0	0
$301$	22.2290i	1.28126i
$302$	0	0
$303$	13.2063i	0.758683i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.52297	0.201066	0.100533	−	0.994934i	$-0.467945\pi$
$307$	3.52297	0.201066	0.100533	+	0.994934i	$0.467945\pi$
$308$	0	0
$309$	19.4244i	1.10502i
$310$	0	0
$311$	21.6757	1.22912	0.614559	−	0.788871i	$-0.289334\pi$
$311$	21.6757	1.22912	0.614559	+	0.788871i	$0.289334\pi$
$312$	0	0
$313$	− 12.5486i	− 0.709288i	−0.935001	−	0.354644i	$-0.884602\pi$
$313$	− 12.5486i	− 0.709288i	0.935001	−	0.354644i	$-0.115398\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.8611	0.610020	0.305010	−	0.952349i	$-0.401340\pi$
$317$	10.8611	0.610020	0.305010	+	0.952349i	$0.401340\pi$
$318$	0	0
$319$	−12.9882	−0.727202
$320$	0	0
$321$	−14.8085	−0.826529
$322$	0	0
$323$	22.4500	1.24915
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.2296i	0.842197i
$328$	0	0
$329$	−6.60978	−0.364409
$330$	0	0
$331$	1.23185i	0.0677088i	0.999427	+	0.0338544i	$0.0107783\pi$
$331$	1.23185i	0.0677088i	−0.999427	+	0.0338544i	$0.989222\pi$
$332$	0	0
$333$	8.81463	0.483038
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.13890i	− 0.225460i	−0.993626	−	0.112730i	$-0.964040\pi$
$337$	− 4.13890i	− 0.225460i	0.993626	−	0.112730i	$-0.0359595\pi$
$338$	0	0
$339$	1.13890i	0.0618564i
$340$	0	0
$341$	− 10.6542i	− 0.576959i
$342$	0	0
$343$	− 19.9503i	− 1.07721i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.4586	−0.937226	−0.468613	−	0.883404i	$-0.655246\pi$
$347$	−17.4586	−0.937226	−0.468613	+	0.883404i	$0.655246\pi$
$348$	0	0
$349$	− 21.2196i	− 1.13586i	−0.823078	−	0.567928i	$-0.807745\pi$
$349$	− 21.2196i	− 1.13586i	0.823078	−	0.567928i	$-0.192255\pi$
$350$	0	0
$351$	−0.241319	−0.0128807
$352$	0	0
$353$	21.0398i	1.11984i	0.828548	+	0.559918i	$0.189167\pi$
$353$	21.0398i	1.11984i	−0.828548	+	0.559918i	$0.810833\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.5965	−0.772531
$358$	0	0
$359$	23.5153	1.24109	0.620546	−	0.784170i	$-0.286911\pi$
$359$	23.5153	1.24109	0.620546	+	0.784170i	$0.286911\pi$
$360$	0	0
$361$	9.75639	0.513494
$362$	0	0
$363$	−8.95352	−0.469938
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 25.4012i	− 1.32593i	−0.748650	−	0.662965i	$-0.769298\pi$
$367$	− 25.4012i	− 1.32593i	0.748650	−	0.662965i	$-0.230702\pi$
$368$	0	0
$369$	−1.91319	−0.0995967
$370$	0	0
$371$	18.1987i	0.944829i
$372$	0	0
$373$	−10.0677	−0.521286	−0.260643	−	0.965435i	$-0.583935\pi$
$373$	−10.0677	−0.521286	−0.260643	+	0.965435i	$0.583935\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.19098i	− 0.112841i
$378$	0	0
$379$	18.9674i	0.974289i	0.873321	+	0.487145i	$0.161962\pi$
$379$	18.9674i	0.974289i	−0.873321	+	0.487145i	$0.838038\pi$
$380$	0	0
$381$	− 2.43616i	− 0.124808i
$382$	0	0
$383$	28.7446i	1.46878i	0.678727	+	0.734391i	$0.262532\pi$
$383$	28.7446i	1.46878i	−0.678727	+	0.734391i	$0.737468\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.2452	−0.571624
$388$	0	0
$389$	29.8161i	1.51174i	0.654724	+	0.755868i	$0.272785\pi$
$389$	29.8161i	1.51174i	−0.654724	+	0.755868i	$0.727215\pi$
$390$	0	0
$391$	6.45617	0.326503
$392$	0	0
$393$	6.90143i	0.348131i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.73167	−0.137099	−0.0685494	−	0.997648i	$-0.521837\pi$
$397$	−2.73167	−0.137099	−0.0685494	+	0.997648i	$0.521837\pi$
$398$	0	0
$399$	6.01001	0.300877
$400$	0	0
$401$	25.8744	1.29211	0.646054	−	0.763292i	$-0.276418\pi$
$401$	25.8744	1.29211	0.646054	+	0.763292i	$0.276418\pi$
$402$	0	0
$403$	1.79726	0.0895279
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.6098i	− 0.625044i
$408$	0	0
$409$	−22.4786	−1.11150	−0.555748	−	0.831351i	$-0.687568\pi$
$409$	−22.4786	−1.11150	−0.555748	+	0.831351i	$0.687568\pi$
$410$	0	0
$411$	− 5.39022i	− 0.265880i
$412$	0	0
$413$	12.7227	0.626045
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.4244i	0.853277i
$418$	0	0
$419$	24.5307i	1.19840i	0.800598	+	0.599201i	$0.204515\pi$
$419$	24.5307i	1.19840i	−0.800598	+	0.599201i	$0.795485\pi$
$420$	0	0
$421$	33.3856i	1.62712i	0.581483	+	0.813558i	$0.302473\pi$
$421$	33.3856i	1.62712i	−0.581483	+	0.813558i	$0.697527\pi$
$422$	0	0
$423$	− 3.34374i	− 0.162578i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9.04033	−0.437492
$428$	0	0
$429$	0.345220i	0.0166674i
$430$	0	0
$431$	−11.6548	−0.561391	−0.280696	−	0.959797i	$-0.590565\pi$
$431$	−11.6548	−0.561391	−0.280696	+	0.959797i	$0.590565\pi$
$432$	0	0
$433$	19.7681i	0.949996i	0.879987	+	0.474998i	$0.157551\pi$
$433$	19.7681i	0.949996i	−0.879987	+	0.474998i	$0.842449\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.65827	−0.127163
$438$	0	0
$439$	−25.1699	−1.20129	−0.600646	−	0.799515i	$-0.705090\pi$
$439$	−25.1699	−1.20129	−0.600646	+	0.799515i	$0.705090\pi$
$440$	0	0
$441$	3.09242	0.147258
$442$	0	0
$443$	19.5515	0.928922	0.464461	−	0.885594i	$-0.346248\pi$
$443$	19.5515	0.928922	0.464461	+	0.885594i	$0.346248\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.28551i	0.108101i
$448$	0	0
$449$	19.9612	0.942029	0.471014	−	0.882125i	$-0.343888\pi$
$449$	19.9612	0.942029	0.471014	+	0.882125i	$0.343888\pi$
$450$	0	0
$451$	2.73692i	0.128876i
$452$	0	0
$453$	6.66425	0.313114
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.01176i	0.234440i	0.993106	+	0.117220i	$0.0373983\pi$
$457$	5.01176i	0.234440i	−0.993106	+	0.117220i	$0.962602\pi$
$458$	0	0
$459$	− 7.38407i	− 0.344659i
$460$	0	0
$461$	5.12566i	0.238726i	0.992851	+	0.119363i	$0.0380852\pi$
$461$	5.12566i	0.238726i	−0.992851	+	0.119363i	$0.961915\pi$
$462$	0	0
$463$	5.79515i	0.269324i	0.990892	+	0.134662i	$0.0429948\pi$
$463$	5.79515i	0.269324i	−0.990892	+	0.134662i	$0.957005\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.8208	−0.778373	−0.389186	−	0.921159i	$-0.627244\pi$
$467$	−16.8208	−0.778373	−0.389186	+	0.921159i	$0.627244\pi$
$468$	0	0
$469$	9.62032i	0.444225i
$470$	0	0
$471$	−17.4144	−0.802413
$472$	0	0
$473$	16.0868i	0.739672i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.20632	−0.421529
$478$	0	0
$479$	−36.9065	−1.68630	−0.843151	−	0.537678i	$-0.819302\pi$
$479$	−36.9065	−1.68630	−0.843151	+	0.537678i	$0.819302\pi$
$480$	0	0
$481$	2.12714	0.0969892
$482$	0	0
$483$	1.72836	0.0786429
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.14984i	− 0.233361i	−0.993169	−	0.116681i	$-0.962775\pi$
$487$	− 5.14984i	− 0.233361i	0.993169	−	0.116681i	$-0.0372254\pi$
$488$	0	0
$489$	−4.66187	−0.210817
$490$	0	0
$491$	23.1154i	1.04318i	0.853195	+	0.521591i	$0.174661\pi$
$491$	23.1154i	1.04318i	−0.853195	+	0.521591i	$0.825339\pi$
$492$	0	0
$493$	67.0414	3.01939
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.2452i	− 0.728696i
$498$	0	0
$499$	14.3111i	0.640654i	0.947307	+	0.320327i	$0.103793\pi$
$499$	14.3111i	0.640654i	−0.947307	+	0.320327i	$0.896207\pi$
$500$	0	0
$501$	0.137419i	0.00613942i
$502$	0	0
$503$	− 15.4224i	− 0.687650i	−0.939034	−	0.343825i	$-0.888277\pi$
$503$	− 15.4224i	− 0.687650i	0.939034	−	0.343825i	$-0.111723\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.9418	0.574764
$508$	0	0
$509$	− 43.1578i	− 1.91294i	−0.291835	−	0.956469i	$-0.594266\pi$
$509$	− 43.1578i	− 1.91294i	0.291835	−	0.956469i	$-0.405734\pi$
$510$	0	0
$511$	−8.15837	−0.360905
$512$	0	0
$513$	3.04033i	0.134234i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.78340	−0.210374
$518$	0	0
$519$	−3.96675	−0.174121
$520$	0	0
$521$	−17.8232	−0.780848	−0.390424	−	0.920635i	$-0.627672\pi$
$521$	−17.8232	−0.780848	−0.390424	+	0.920635i	$0.627672\pi$
$522$	0	0
$523$	24.7502	1.08225	0.541126	−	0.840941i	$-0.317998\pi$
$523$	24.7502	1.08225	0.541126	+	0.840941i	$0.317998\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	54.9939i	2.39557i
$528$	0	0
$529$	22.2355	0.966762
$530$	0	0
$531$	6.43616i	0.279306i
$532$	0	0
$533$	−0.461690	−0.0199980
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.68749i	0.202280i
$538$	0	0
$539$	− 4.42386i	− 0.190549i
$540$	0	0
$541$	16.9982i	0.730812i	0.930848	+	0.365406i	$0.119070\pi$
$541$	16.9982i	0.730812i	−0.930848	+	0.365406i	$0.880930\pi$
$542$	0	0
$543$	9.10242i	0.390622i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.2385	1.59220	0.796101	−	0.605163i	$-0.206892\pi$
$547$	37.2385	1.59220	0.796101	+	0.605163i	$0.206892\pi$
$548$	0	0
$549$	− 4.57331i	− 0.195184i
$550$	0	0
$551$	−27.6037	−1.17596
$552$	0	0
$553$	27.0336i	1.14959i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7604	0.625420	0.312710	−	0.949849i	$-0.398763\pi$
$557$	14.7604	0.625420	0.312710	+	0.949849i	$0.398763\pi$
$558$	0	0
$559$	−2.71368	−0.114776
$560$	0	0
$561$	−10.5633	−0.445983
$562$	0	0
$563$	−3.00561	−0.126671	−0.0633356	−	0.997992i	$-0.520174\pi$
$563$	−3.00561	−0.126671	−0.0633356	+	0.997992i	$0.520174\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.97676i	− 0.0830161i
$568$	0	0
$569$	23.1840	0.971923	0.485962	−	0.873980i	$-0.338469\pi$
$569$	23.1840	0.971923	0.485962	+	0.873980i	$0.338469\pi$
$570$	0	0
$571$	− 0.202739i	− 0.00848438i	−0.999991	−	0.00424219i	$-0.998650\pi$
$571$	− 0.202739i	− 0.00848438i	0.999991	−	0.00424219i	$-0.00135033\pi$
$572$	0	0
$573$	15.2063	0.635253
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 21.8023i	− 0.907643i	−0.891093	−	0.453821i	$-0.850060\pi$
$577$	− 21.8023i	− 0.907643i	0.891093	−	0.453821i	$-0.149940\pi$
$578$	0	0
$579$	20.7564i	0.862606i
$580$	0	0
$581$	24.3774i	1.01134i
$582$	0	0
$583$	13.1701i	0.545451i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.7126	−1.51529	−0.757645	−	0.652667i	$-0.773650\pi$
$587$	−36.7126	−1.51529	−0.757645	+	0.652667i	$0.773650\pi$
$588$	0	0
$589$	− 22.6433i	− 0.933001i
$590$	0	0
$591$	23.2508	0.956409
$592$	0	0
$593$	10.6036i	0.435439i	0.976011	+	0.217719i	$0.0698618\pi$
$593$	10.6036i	0.435439i	−0.976011	+	0.217719i	$0.930138\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.21633	0.295345
$598$	0	0
$599$	−25.7988	−1.05411	−0.527056	−	0.849831i	$-0.676704\pi$
$599$	−25.7988	−1.05411	−0.527056	+	0.849831i	$0.676704\pi$
$600$	0	0
$601$	18.5021	0.754717	0.377358	−	0.926067i	$-0.376832\pi$
$601$	18.5021	0.754717	0.377358	+	0.926067i	$0.376832\pi$
$602$	0	0
$603$	−4.86671	−0.198188
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 37.5828i	− 1.52544i	−0.646730	−	0.762719i	$-0.723864\pi$
$607$	− 37.5828i	− 1.52544i	0.646730	−	0.762719i	$-0.276136\pi$
$608$	0	0
$609$	17.9474	0.727264
$610$	0	0
$611$	− 0.806910i	− 0.0326441i
$612$	0	0
$613$	−4.93405	−0.199284	−0.0996422	−	0.995023i	$-0.531770\pi$
$613$	−4.93405	−0.199284	−0.0996422	+	0.995023i	$0.531770\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.26043i	0.252035i	0.992028	+	0.126018i	$0.0402196\pi$
$617$	6.26043i	0.252035i	−0.992028	+	0.126018i	$0.959780\pi$
$618$	0	0
$619$	8.02562i	0.322577i	0.986907	+	0.161288i	$0.0515649\pi$
$619$	8.02562i	0.322577i	−0.986907	+	0.161288i	$0.948435\pi$
$620$	0	0
$621$	0.874337i	0.0350860i
$622$	0	0
$623$	− 15.9735i	− 0.639966i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.34935	0.173696
$628$	0	0
$629$	65.0878i	2.59522i
$630$	0	0
$631$	26.5248	1.05594	0.527968	−	0.849264i	$-0.322954\pi$
$631$	26.5248	1.05594	0.527968	+	0.849264i	$0.322954\pi$
$632$	0	0
$633$	4.38407i	0.174251i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.746260	0.0295679
$638$	0	0
$639$	8.21808	0.325102
$640$	0	0
$641$	−26.5863	−1.05009	−0.525047	−	0.851073i	$-0.675952\pi$
$641$	−26.5863	−1.05009	−0.525047	+	0.851073i	$0.675952\pi$
$642$	0	0
$643$	−2.89233	−0.114062	−0.0570312	−	0.998372i	$-0.518163\pi$
$643$	−2.89233	−0.114062	−0.0570312	+	0.998372i	$0.518163\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.3472i	− 0.485420i	−0.970099	−	0.242710i	$-0.921964\pi$
$647$	− 12.3472i	− 0.485420i	0.970099	−	0.242710i	$-0.0780363\pi$
$648$	0	0
$649$	9.20726	0.361417
$650$	0	0
$651$	14.7222i	0.577009i
$652$	0	0
$653$	39.0507	1.52817	0.764086	−	0.645114i	$-0.223190\pi$
$653$	39.0507	1.52817	0.764086	+	0.645114i	$0.223190\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.12714i	− 0.161015i
$658$	0	0
$659$	23.7738i	0.926094i	0.886334	+	0.463047i	$0.153244\pi$
$659$	23.7738i	0.926094i	−0.886334	+	0.463047i	$0.846756\pi$
$660$	0	0
$661$	− 21.5051i	− 0.836450i	−0.908343	−	0.418225i	$-0.862652\pi$
$661$	− 21.5051i	− 0.836450i	0.908343	−	0.418225i	$-0.137348\pi$
$662$	0	0
$663$	− 1.78192i	− 0.0692040i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.93827	−0.307371
$668$	0	0
$669$	− 4.98852i	− 0.192867i
$670$	0	0
$671$	−6.54235	−0.252565
$672$	0	0
$673$	− 36.1896i	− 1.39501i	−0.716582	−	0.697503i	$-0.754294\pi$
$673$	− 36.1896i	− 1.39501i	0.716582	−	0.697503i	$-0.245706\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.17214	−0.352514	−0.176257	−	0.984344i	$-0.556399\pi$
$677$	−9.17214	−0.352514	−0.176257	+	0.984344i	$0.556399\pi$
$678$	0	0
$679$	−21.1034	−0.809873
$680$	0	0
$681$	−11.2569	−0.431367
$682$	0	0
$683$	16.3974	0.627429	0.313714	−	0.949517i	$-0.398427\pi$
$683$	16.3974	0.627429	0.313714	+	0.949517i	$0.398427\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15.8364i	− 0.604196i
$688$	0	0
$689$	−2.22166	−0.0846387
$690$	0	0
$691$	24.4904i	0.931657i	0.884875	+	0.465828i	$0.154244\pi$
$691$	24.4904i	0.931657i	−0.884875	+	0.465828i	$0.845756\pi$
$692$	0	0
$693$	−2.82786	−0.107421
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 14.1271i	− 0.535104i
$698$	0	0
$699$	10.9591i	0.414512i
$700$	0	0
$701$	12.3887i	0.467916i	0.972247	+	0.233958i	$0.0751679\pi$
$701$	12.3887i	0.467916i	−0.972247	+	0.233958i	$0.924832\pi$
$702$	0	0
$703$	− 26.7994i	− 1.01076i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.1057	−0.981807
$708$	0	0
$709$	33.4144i	1.25490i	0.778655	+	0.627452i	$0.215902\pi$
$709$	33.4144i	1.25490i	−0.778655	+	0.627452i	$0.784098\pi$
$710$	0	0
$711$	−13.6757	−0.512880
$712$	0	0
$713$	− 6.51175i	− 0.243867i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.3182	−0.646761
$718$	0	0
$719$	−33.8938	−1.26403	−0.632013	−	0.774958i	$-0.717771\pi$
$719$	−33.8938	−1.26403	−0.632013	+	0.774958i	$0.717771\pi$
$720$	0	0
$721$	−38.3974	−1.42999
$722$	0	0
$723$	−4.76869	−0.177349
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.1846i	0.526076i	0.964785	+	0.263038i	$0.0847245\pi$
$727$	14.1846i	0.526076i	−0.964785	+	0.263038i	$0.915275\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	− 83.0352i	− 3.07117i
$732$	0	0
$733$	8.09296	0.298920	0.149460	−	0.988768i	$-0.452246\pi$
$733$	8.09296	0.298920	0.149460	+	0.988768i	$0.452246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.96208i	0.256452i
$738$	0	0
$739$	22.0919i	0.812663i	0.913726	+	0.406331i	$0.133192\pi$
$739$	22.0919i	0.812663i	−0.913726	+	0.406331i	$0.866808\pi$
$740$	0	0
$741$	0.733691i	0.0269528i
$742$	0	0
$743$	8.78340i	0.322232i	0.986936	+	0.161116i	$0.0515093\pi$
$743$	8.78340i	0.322232i	−0.986936	+	0.161116i	$0.948491\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.3320	−0.451204
$748$	0	0
$749$	− 29.2728i	− 1.06961i
$750$	0	0
$751$	−13.3779	−0.488167	−0.244084	−	0.969754i	$-0.578487\pi$
$751$	−13.3779	−0.488167	−0.244084	+	0.969754i	$0.578487\pi$
$752$	0	0
$753$	− 6.15837i	− 0.224423i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.72450	−0.353443	−0.176721	−	0.984261i	$-0.556549\pi$
$757$	−9.72450	−0.353443	−0.176721	+	0.984261i	$0.556549\pi$
$758$	0	0
$759$	1.25079	0.0454006
$760$	0	0
$761$	33.8835	1.22828	0.614138	−	0.789198i	$-0.289504\pi$
$761$	33.8835	1.22828	0.614138	+	0.789198i	$0.289504\pi$
$762$	0	0
$763$	−30.1052	−1.08988
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.55317i	0.0560817i
$768$	0	0
$769$	−18.7334	−0.675545	−0.337772	−	0.941228i	$-0.609673\pi$
$769$	−18.7334	−0.675545	−0.337772	+	0.941228i	$0.609673\pi$
$770$	0	0
$771$	− 14.1584i	− 0.509901i
$772$	0	0
$773$	−22.5006	−0.809292	−0.404646	−	0.914474i	$-0.632605\pi$
$773$	−22.5006	−0.809292	−0.404646	+	0.914474i	$0.632605\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17.4244i	0.625097i
$778$	0	0
$779$	5.81673i	0.208406i
$780$	0	0
$781$	− 11.7564i	− 0.420677i
$782$	0	0
$783$	9.07918i	0.324464i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−28.1063	−1.00188	−0.500940	−	0.865482i	$-0.667012\pi$
$787$	−28.1063	−1.00188	−0.500940	+	0.865482i	$0.667012\pi$
$788$	0	0
$789$	− 15.5960i	− 0.555232i
$790$	0	0
$791$	−2.25133	−0.0800479
$792$	0	0
$793$	− 1.10363i	− 0.0391910i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.1461	1.35120	0.675602	−	0.737267i	$-0.263884\pi$
$797$	38.1461	1.35120	0.675602	+	0.737267i	$0.263884\pi$
$798$	0	0
$799$	24.6904	0.873485
$800$	0	0
$801$	8.08066	0.285516
$802$	0	0
$803$	−5.90409	−0.208351
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11.3182i	− 0.398420i
$808$	0	0
$809$	10.0745	0.354201	0.177101	−	0.984193i	$-0.443328\pi$
$809$	10.0745	0.354201	0.177101	+	0.984193i	$0.443328\pi$
$810$	0	0
$811$	40.6001i	1.42566i	0.701335	+	0.712832i	$0.252588\pi$
$811$	40.6001i	1.42566i	−0.701335	+	0.712832i	$0.747412\pi$
$812$	0	0
$813$	−6.20485	−0.217613
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34.1891i	1.19612i
$818$	0	0
$819$	− 0.477031i	− 0.0166688i
$820$	0	0
$821$	10.3397i	0.360858i	0.983588	+	0.180429i	$0.0577486\pi$
$821$	10.3397i	0.360858i	−0.983588	+	0.180429i	$0.942251\pi$
$822$	0	0
$823$	28.5013i	0.993493i	0.867896	+	0.496746i	$0.165472\pi$
$823$	28.5013i	0.993493i	−0.867896	+	0.496746i	$0.834528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.1957	1.11956	0.559778	−	0.828643i	$-0.310886\pi$
$827$	32.1957	1.11956	0.559778	+	0.828643i	$0.310886\pi$
$828$	0	0
$829$	− 31.5286i	− 1.09503i	−0.836795	−	0.547516i	$-0.815573\pi$
$829$	− 31.5286i	− 1.09503i	0.836795	−	0.547516i	$-0.184427\pi$
$830$	0	0
$831$	18.9288	0.656633
$832$	0	0
$833$	22.8346i	0.791173i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.44764	−0.257428
$838$	0	0
$839$	54.9816	1.89818	0.949089	−	0.315009i	$-0.102008\pi$
$839$	54.9816	1.89818	0.949089	+	0.315009i	$0.102008\pi$
$840$	0	0
$841$	−53.4316	−1.84247
$842$	0	0
$843$	21.6231	0.744739
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 17.6990i	− 0.608144i
$848$	0	0
$849$	−29.1522	−1.00050
$850$	0	0
$851$	− 7.70696i	− 0.264191i
$852$	0	0
$853$	4.22607	0.144698	0.0723489	−	0.997379i	$-0.476950\pi$
$853$	4.22607	0.144698	0.0723489	+	0.997379i	$0.476950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 54.8223i	− 1.87270i	−0.351074	−	0.936348i	$-0.614183\pi$
$857$	− 54.8223i	− 1.87270i	0.351074	−	0.936348i	$-0.385817\pi$
$858$	0	0
$859$	− 14.2126i	− 0.484926i	−0.970161	−	0.242463i	$-0.922045\pi$
$859$	− 14.2126i	− 0.484926i	0.970161	−	0.242463i	$-0.0779554\pi$
$860$	0	0
$861$	− 3.78192i	− 0.128887i
$862$	0	0
$863$	− 40.7446i	− 1.38696i	−0.720474	−	0.693482i	$-0.756076\pi$
$863$	− 40.7446i	− 1.38696i	0.720474	−	0.693482i	$-0.243924\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37.5245	1.27440
$868$	0	0
$869$	19.5638i	0.663658i
$870$	0	0
$871$	−1.17443	−0.0397941
$872$	0	0
$873$	− 10.6757i	− 0.361319i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.4568	1.16352	0.581762	−	0.813359i	$-0.302363\pi$
$877$	34.4568	1.16352	0.581762	+	0.813359i	$0.302363\pi$
$878$	0	0
$879$	2.32427	0.0783957
$880$	0	0
$881$	23.8528	0.803623	0.401811	−	0.915722i	$-0.368381\pi$
$881$	23.8528	0.803623	0.401811	+	0.915722i	$0.368381\pi$
$882$	0	0
$883$	24.2417	0.815798	0.407899	−	0.913027i	$-0.366262\pi$
$883$	24.2417	0.815798	0.407899	+	0.913027i	$0.366262\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 39.1843i	− 1.31568i	−0.753158	−	0.657840i	$-0.771470\pi$
$887$	− 39.1843i	− 1.31568i	0.753158	−	0.657840i	$-0.228530\pi$
$888$	0	0
$889$	4.81571	0.161514
$890$	0	0
$891$	− 1.43055i	− 0.0479253i
$892$	0	0
$893$	−10.1661	−0.340195
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.210995i	0.00704490i
$898$	0	0
$899$	− 67.6185i	− 2.25520i
$900$	0	0
$901$	− 67.9802i	− 2.26475i
$902$	0	0
$903$	− 22.2290i	− 0.739736i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−18.9418	−0.628951	−0.314475	−	0.949266i	$-0.601829\pi$
$907$	−18.9418	−0.628951	−0.314475	+	0.949266i	$0.601829\pi$
$908$	0	0
$909$	− 13.2063i	− 0.438026i
$910$	0	0
$911$	−17.8493	−0.591375	−0.295688	−	0.955285i	$-0.595549\pi$
$911$	−17.8493	−0.591375	−0.295688	+	0.955285i	$0.595549\pi$
$912$	0	0
$913$	17.6415i	0.583850i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.6425	−0.450515
$918$	0	0
$919$	24.4983	0.808126	0.404063	−	0.914731i	$-0.367598\pi$
$919$	24.4983	0.808126	0.404063	+	0.914731i	$0.367598\pi$
$920$	0	0
$921$	−3.52297	−0.116086
$922$	0	0
$923$	1.98318	0.0652772
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 19.4244i	− 0.637981i
$928$	0	0
$929$	−11.5577	−0.379196	−0.189598	−	0.981862i	$-0.560718\pi$
$929$	−11.5577	−0.379196	−0.189598	+	0.981862i	$0.560718\pi$
$930$	0	0
$931$	− 9.40197i	− 0.308137i
$932$	0	0
$933$	−21.6757	−0.709632
$934$	0	0
$935$	0	0
$936$	0	0
$937$	59.5587i	1.94570i	0.231440	+	0.972849i	$0.425656\pi$
$937$	59.5587i	1.94570i	−0.231440	+	0.972849i	$0.574344\pi$
$938$	0	0
$939$	12.5486i	0.409508i
$940$	0	0
$941$	55.9289i	1.82323i	0.411044	+	0.911615i	$0.365164\pi$
$941$	55.9289i	1.82323i	−0.411044	+	0.911615i	$0.634836\pi$
$942$	0	0
$943$	1.67277i	0.0544730i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0945	0.977941	0.488970	−	0.872300i	$-0.337373\pi$
$947$	30.0945	0.977941	0.488970	+	0.872300i	$0.337373\pi$
$948$	0	0
$949$	− 0.995959i	− 0.0323302i
$950$	0	0
$951$	−10.8611	−0.352195
$952$	0	0
$953$	12.4097i	0.401989i	0.979592	+	0.200995i	$0.0644174\pi$
$953$	12.4097i	0.401989i	−0.979592	+	0.200995i	$0.935583\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.9882	0.419850
$958$	0	0
$959$	10.6552	0.344074
$960$	0	0
$961$	24.4674	0.789271
$962$	0	0
$963$	14.8085	0.477196
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.8371i	1.34539i	0.739920	+	0.672694i	$0.234863\pi$
$967$	41.8371i	1.34539i	−0.739920	+	0.672694i	$0.765137\pi$
$968$	0	0
$969$	−22.4500	−0.721199
$970$	0	0
$971$	40.6875i	1.30572i	0.757477	+	0.652862i	$0.226432\pi$
$971$	40.6875i	1.30572i	−0.757477	+	0.652862i	$0.773568\pi$
$972$	0	0
$973$	−34.4439	−1.10422
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.67923i	− 0.0537233i	−0.999639	−	0.0268617i	$-0.991449\pi$
$977$	− 1.67923i	− 0.0537233i	0.999639	−	0.0268617i	$-0.00855136\pi$
$978$	0	0
$979$	− 11.5598i	− 0.369453i
$980$	0	0
$981$	− 15.2296i	− 0.486243i
$982$	0	0
$983$	13.4944i	0.430404i	0.976570	+	0.215202i	$0.0690411\pi$
$983$	13.4944i	0.430404i	−0.976570	+	0.215202i	$0.930959\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.60978	0.210392
$988$	0	0
$989$	9.83208i	0.312642i
$990$	0	0
$991$	10.8502	0.344667	0.172333	−	0.985039i	$-0.444869\pi$
$991$	10.8502	0.344667	0.172333	+	0.985039i	$0.444869\pi$
$992$	0	0
$993$	− 1.23185i	− 0.0390917i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.0972	−1.23822	−0.619110	−	0.785304i	$-0.712507\pi$
$997$	−39.0972	−1.23822	−0.619110	+	0.785304i	$0.712507\pi$
$998$	0	0
$999$	−8.81463	−0.278882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.g.49.2		8
3.2	odd	2		7200.2.d.s.2449.2		8
4.3	odd	2		600.2.d.h.349.6		8
5.2	odd	4		2400.2.k.d.1201.8		8
5.3	odd	4		2400.2.k.e.1201.1		8
5.4	even	2		2400.2.d.h.49.7		8
8.3	odd	2		600.2.d.g.349.4		8
8.5	even	2		2400.2.d.h.49.2		8
12.11	even	2		1800.2.d.s.1549.3		8
15.2	even	4		7200.2.k.r.3601.8		8
15.8	even	4		7200.2.k.s.3601.2		8
15.14	odd	2		7200.2.d.t.2449.7		8
20.3	even	4		600.2.k.e.301.7	yes	8
20.7	even	4		600.2.k.d.301.2	yes	8
20.19	odd	2		600.2.d.g.349.3		8
24.5	odd	2		7200.2.d.t.2449.2		8
24.11	even	2		1800.2.d.t.1549.5		8
40.3	even	4		600.2.k.e.301.8	yes	8
40.13	odd	4		2400.2.k.e.1201.5		8
40.19	odd	2		600.2.d.h.349.5		8
40.27	even	4		600.2.k.d.301.1	✓	8
40.29	even	2	inner	2400.2.d.g.49.7		8
40.37	odd	4		2400.2.k.d.1201.4		8
60.23	odd	4		1800.2.k.q.901.2		8
60.47	odd	4		1800.2.k.t.901.7		8
60.59	even	2		1800.2.d.t.1549.6		8
120.29	odd	2		7200.2.d.s.2449.7		8
120.53	even	4		7200.2.k.s.3601.1		8
120.59	even	2		1800.2.d.s.1549.4		8
120.77	even	4		7200.2.k.r.3601.7		8
120.83	odd	4		1800.2.k.q.901.1		8
120.107	odd	4		1800.2.k.t.901.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.3		8	20.19	odd	2
600.2.d.g.349.4		8	8.3	odd	2
600.2.d.h.349.5		8	40.19	odd	2
600.2.d.h.349.6		8	4.3	odd	2
600.2.k.d.301.1	✓	8	40.27	even	4
600.2.k.d.301.2	yes	8	20.7	even	4
600.2.k.e.301.7	yes	8	20.3	even	4
600.2.k.e.301.8	yes	8	40.3	even	4
1800.2.d.s.1549.3		8	12.11	even	2
1800.2.d.s.1549.4		8	120.59	even	2
1800.2.d.t.1549.5		8	24.11	even	2
1800.2.d.t.1549.6		8	60.59	even	2
1800.2.k.q.901.1		8	120.83	odd	4
1800.2.k.q.901.2		8	60.23	odd	4
1800.2.k.t.901.7		8	60.47	odd	4
1800.2.k.t.901.8		8	120.107	odd	4
2400.2.d.g.49.2		8	1.1	even	1	trivial
2400.2.d.g.49.7		8	40.29	even	2	inner
2400.2.d.h.49.2		8	8.5	even	2
2400.2.d.h.49.7		8	5.4	even	2
2400.2.k.d.1201.4		8	40.37	odd	4
2400.2.k.d.1201.8		8	5.2	odd	4
2400.2.k.e.1201.1		8	5.3	odd	4
2400.2.k.e.1201.5		8	40.13	odd	4
7200.2.d.s.2449.2		8	3.2	odd	2
7200.2.d.s.2449.7		8	120.29	odd	2
7200.2.d.t.2449.2		8	24.5	odd	2
7200.2.d.t.2449.7		8	15.14	odd	2
7200.2.k.r.3601.7		8	120.77	even	4
7200.2.k.r.3601.8		8	15.2	even	4
7200.2.k.s.3601.1		8	120.53	even	4
7200.2.k.s.3601.2		8	15.8	even	4

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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.97676i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.97676i q^{7} +1.00000 q^{9} -1.43055i q^{11} +0.241319 q^{13} +7.38407i q^{17} -3.04033i q^{19} +1.97676i q^{21} -0.874337i q^{23} -1.00000 q^{27} -9.07918i q^{29} +7.44764 q^{31} +1.43055i q^{33} +8.81463 q^{37} -0.241319 q^{39} -1.91319 q^{41} -11.2452 q^{43} -3.34374i q^{47} +3.09242 q^{49} -7.38407i q^{51} -9.20632 q^{53} +3.04033i q^{57} +6.43616i q^{59} -4.57331i q^{61} -1.97676i q^{63} -4.86671 q^{67} +0.874337i q^{69} +8.21808 q^{71} -4.12714i q^{73} -2.82786 q^{77} -13.6757 q^{79} +1.00000 q^{81} -12.3320 q^{83} +9.07918i q^{87} +8.08066 q^{89} -0.477031i q^{91} -7.44764 q^{93} -10.6757i q^{97} -1.43055i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 8 * q^9	\( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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