LMFDB - Embedded newform 2400.2.k.d.1201.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1201,2400)]

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, 0]))

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

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

chi := DirichletCharacter("2400.1201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (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:	$\Z[a_1, \ldots, a_{11}]$
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			1201.4
Root			$1.41216 - 0.0762223i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.d.1201.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.97676	0.747145	0.373573	−	0.927601i	$-0.378133\pi$
$7$	1.97676	0.747145	0.373573	+	0.927601i	$0.378133\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.43055i	0.431328i	0.976468	+	0.215664i	$0.0691915\pi$
$11$	1.43055i	0.431328i	−0.976468	+	0.215664i	$0.930808\pi$
$12$	0	0
$13$	0.241319i	0.0669300i	0.999440	+	0.0334650i	$0.0106542\pi$
$13$	0.241319i	0.0669300i	−0.999440	+	0.0334650i	$0.989346\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.38407	−1.79090	−0.895450	−	0.445161i	$-0.853146\pi$
$17$	−7.38407	−1.79090	−0.895450	+	0.445161i	$0.853146\pi$
$18$	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
$23$	−0.874337	−0.182312	−0.0911560	−	0.995837i	$-0.529056\pi$
$23$	−0.874337	−0.182312	−0.0911560	+	0.995837i	$0.529056\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	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
$33$	1.43055	0.249027
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.81463i	− 1.44912i	−0.689214	−	0.724558i	$-0.742044\pi$
$37$	− 8.81463i	− 1.44912i	0.689214	−	0.724558i	$-0.257956\pi$
$38$	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
$43$	− 11.2452i	− 1.71487i	−0.514589	−	0.857437i	$-0.672056\pi$
$43$	− 11.2452i	− 1.71487i	0.514589	−	0.857437i	$-0.327944\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.34374	0.487735	0.243867	−	0.969809i	$-0.421584\pi$
$47$	3.34374	0.487735	0.243867	+	0.969809i	$0.421584\pi$
$48$	0	0
$49$	−3.09242	−0.441774
$50$	0	0
$51$	7.38407i	1.03398i
$52$	0	0
$53$	− 9.20632i	− 1.26459i	−0.774729	−	0.632293i	$-0.782114\pi$
$53$	− 9.20632i	− 1.26459i	0.774729	−	0.632293i	$-0.217886\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.04033	−0.402702
$58$	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	$0.0945790\pi$
$61$	4.57331i	0.585552i	−0.956181	+	0.292776i	$0.905421\pi$
$62$	0	0
$63$	−1.97676	−0.249048
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.86671i	0.594563i	0.954790	+	0.297282i	$0.0960801\pi$
$67$	4.86671i	0.594563i	−0.954790	+	0.297282i	$0.903920\pi$
$68$	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
$73$	−4.12714	−0.483045	−0.241523	−	0.970395i	$-0.577647\pi$
$73$	−4.12714	−0.483045	−0.241523	+	0.970395i	$0.577647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82786i	0.322264i
$78$	0	0
$79$	13.6757	1.53864	0.769320	−	0.638864i	$-0.220595\pi$
$79$	13.6757	1.53864	0.769320	+	0.638864i	$0.220595\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.3320i	− 1.35361i	−0.736162	−	0.676806i	$-0.763364\pi$
$83$	− 12.3320i	− 1.35361i	0.736162	−	0.676806i	$-0.236636\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.07918	−0.973391
$88$	0	0
$89$	−8.08066	−0.856548	−0.428274	−	0.903649i	$-0.640878\pi$
$89$	−8.08066	−0.856548	−0.428274	+	0.903649i	$0.640878\pi$
$90$	0	0
$91$	0.477031i	0.0500064i
$92$	0	0
$93$	− 7.44764i	− 0.772285i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.6757	1.08396	0.541978	−	0.840393i	$-0.317676\pi$
$97$	10.6757	1.08396	0.541978	+	0.840393i	$0.317676\pi$
$98$	0	0
$99$	− 1.43055i	− 0.143776i
$100$	0	0
$101$	13.2063i	1.31408i	0.753856	+	0.657039i	$0.228191\pi$
$101$	13.2063i	1.31408i	−0.753856	+	0.657039i	$0.771809\pi$
$102$	0	0
$103$	−19.4244	−1.91394	−0.956972	−	0.290181i	$-0.906284\pi$
$103$	−19.4244	−1.91394	−0.956972	+	0.290181i	$0.906284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.8085i	− 1.43159i	−0.698311	−	0.715795i	$-0.746065\pi$
$107$	− 14.8085i	− 1.43159i	0.698311	−	0.715795i	$-0.253935\pi$
$108$	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
$113$	−1.13890	−0.107138	−0.0535692	−	0.998564i	$-0.517060\pi$
$113$	−1.13890	−0.107138	−0.0535692	+	0.998564i	$0.517060\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.241319i	− 0.0223100i
$118$	0	0
$119$	−14.5965	−1.33806
$120$	0	0
$121$	8.95352	0.813956
$122$	0	0
$123$	1.91319i	0.172507i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.43616	−0.216174	−0.108087	−	0.994141i	$-0.534473\pi$
$127$	−2.43616	−0.216174	−0.108087	+	0.994141i	$0.534473\pi$
$128$	0	0
$129$	−11.2452	−0.990083
$130$	0	0
$131$	6.90143i	0.602981i	0.953469	+	0.301491i	$0.0974842\pi$
$131$	6.90143i	0.602981i	−0.953469	+	0.301491i	$0.902516\pi$
$132$	0	0
$133$	− 6.01001i	− 0.521134i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.39022	−0.460518	−0.230259	−	0.973129i	$-0.573957\pi$
$137$	−5.39022	−0.460518	−0.230259	+	0.973129i	$0.573957\pi$
$138$	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
$143$	−0.345220	−0.0288687
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.09242i	0.255058i
$148$	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
$153$	7.38407	0.596967
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.4144i	− 1.38982i	−0.719097	−	0.694910i	$-0.755444\pi$
$157$	− 17.4144i	− 1.38982i	0.719097	−	0.694910i	$-0.244556\pi$
$158$	0	0
$159$	−9.20632	−0.730109
$160$	0	0
$161$	−1.72836	−0.136214
$162$	0	0
$163$	4.66187i	0.365145i	0.983192	+	0.182573i	$0.0584425\pi$
$163$	4.66187i	0.365145i	−0.983192	+	0.182573i	$0.941557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.137419	0.0106338	0.00531690	−	0.999986i	$-0.498308\pi$
$167$	0.137419	0.0106338	0.00531690	+	0.999986i	$0.498308\pi$
$168$	0	0
$169$	12.9418	0.995520
$170$	0	0
$171$	3.04033i	0.232500i
$172$	0	0
$173$	3.96675i	0.301587i	0.988565	+	0.150793i	$0.0481828\pi$
$173$	3.96675i	0.301587i	−0.988565	+	0.150793i	$0.951817\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.43616	0.483771
$178$	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	$0.109848\pi$
$181$	9.10242i	0.676578i	−0.941042	+	0.338289i	$0.890152\pi$
$182$	0	0
$183$	4.57331	0.338068
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 10.5633i	− 0.772465i
$188$	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
$193$	−20.7564	−1.49408	−0.747039	−	0.664780i	$-0.768525\pi$
$193$	−20.7564	−1.49408	−0.747039	+	0.664780i	$0.768525\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.2508i	1.65655i	0.560322	+	0.828275i	$0.310677\pi$
$197$	23.2508i	1.65655i	−0.560322	+	0.828275i	$0.689323\pi$
$198$	0	0
$199$	7.21633	0.511552	0.255776	−	0.966736i	$-0.417669\pi$
$199$	7.21633	0.511552	0.255776	+	0.966736i	$0.417669\pi$
$200$	0	0
$201$	4.86671	0.343271
$202$	0	0
$203$	− 17.9474i	− 1.25966i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.874337	0.0607707
$208$	0	0
$209$	4.34935	0.300851
$210$	0	0
$211$	4.38407i	0.301812i	0.988548	+	0.150906i	$0.0482191\pi$
$211$	4.38407i	0.301812i	−0.988548	+	0.150906i	$0.951781\pi$
$212$	0	0
$213$	− 8.21808i	− 0.563094i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.7222	0.999409
$218$	0	0
$219$	4.12714i	0.278886i
$220$	0	0
$221$	− 1.78192i	− 0.119865i
$222$	0	0
$223$	4.98852	0.334056	0.167028	−	0.985952i	$-0.446583\pi$
$223$	4.98852	0.334056	0.167028	+	0.985952i	$0.446583\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.2569i	− 0.747149i	−0.927600	−	0.373574i	$-0.878132\pi$
$227$	− 11.2569i	− 0.747149i	0.927600	−	0.373574i	$-0.121868\pi$
$228$	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
$233$	−10.9591	−0.717956	−0.358978	−	0.933346i	$-0.616875\pi$
$233$	−10.9591	−0.717956	−0.358978	+	0.933346i	$0.616875\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.6757i	− 0.888334i
$238$	0	0
$239$	−17.3182	−1.12022	−0.560111	−	0.828418i	$-0.689242\pi$
$239$	−17.3182	−1.12022	−0.560111	+	0.828418i	$0.689242\pi$
$240$	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
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.733691	0.0466836
$248$	0	0
$249$	−12.3320	−0.781508
$250$	0	0
$251$	− 6.15837i	− 0.388713i	−0.980931	−	0.194356i	$-0.937738\pi$
$251$	− 6.15837i	− 0.388713i	0.980931	−	0.194356i	$-0.0622618\pi$
$252$	0	0
$253$	− 1.25079i	− 0.0786362i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.1584	−0.883175	−0.441587	−	0.897218i	$-0.645584\pi$
$257$	−14.1584	−0.883175	−0.441587	+	0.897218i	$0.645584\pi$
$258$	0	0
$259$	− 17.4244i	− 1.08270i
$260$	0	0
$261$	9.07918i	0.561987i
$262$	0	0
$263$	15.5960	0.961691	0.480845	−	0.876805i	$-0.340330\pi$
$263$	15.5960	0.961691	0.480845	+	0.876805i	$0.340330\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.08066i	0.494528i
$268$	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
$273$	0.477031	0.0288712
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.9288i	1.13732i	0.822572	+	0.568661i	$0.192538\pi$
$277$	18.9288i	1.13732i	−0.822572	+	0.568661i	$0.807462\pi$
$278$	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
$283$	29.1522i	1.73292i	0.499247	+	0.866460i	$0.333610\pi$
$283$	29.1522i	1.73292i	−0.499247	+	0.866460i	$0.666390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.78192	−0.223240
$288$	0	0
$289$	37.5245	2.20733
$290$	0	0
$291$	− 10.6757i	− 0.625822i
$292$	0	0
$293$	− 2.32427i	− 0.135785i	−0.997693	−	0.0678927i	$-0.978372\pi$
$293$	− 2.32427i	− 0.135785i	0.997693	−	0.0678927i	$-0.0216275\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.43055	−0.0830090
$298$	0	0
$299$	− 0.210995i	− 0.0122021i
$300$	0	0
$301$	− 22.2290i	− 1.28126i
$302$	0	0
$303$	13.2063	0.758683
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.52297i	− 0.201066i	−0.994934	−	0.100533i	$-0.967945\pi$
$307$	− 3.52297i	− 0.201066i	0.994934	−	0.100533i	$-0.0320549\pi$
$308$	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
$313$	−12.5486	−0.709288	−0.354644	−	0.935001i	$-0.615398\pi$
$313$	−12.5486	−0.709288	−0.354644	+	0.935001i	$0.615398\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.8611i	− 0.610020i	−0.952349	−	0.305010i	$-0.901340\pi$
$317$	− 10.8611i	− 0.610020i	0.952349	−	0.305010i	$-0.0986599\pi$
$318$	0	0
$319$	12.9882	0.727202
$320$	0	0
$321$	−14.8085	−0.826529
$322$	0	0
$323$	22.4500i	1.24915i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.2296	−0.842197
$328$	0	0
$329$	6.60978	0.364409
$330$	0	0
$331$	− 1.23185i	− 0.0677088i	−0.999427	−	0.0338544i	$-0.989222\pi$
$331$	− 1.23185i	− 0.0677088i	0.999427	−	0.0338544i	$-0.0107783\pi$
$332$	0	0
$333$	8.81463i	0.483038i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.13890	0.225460	0.112730	−	0.993626i	$-0.464040\pi$
$337$	4.13890	0.225460	0.112730	+	0.993626i	$0.464040\pi$
$338$	0	0
$339$	1.13890i	0.0618564i
$340$	0	0
$341$	10.6542i	0.576959i
$342$	0	0
$343$	−19.9503	−1.07721
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.4586i	0.937226i	0.883404	+	0.468613i	$0.155246\pi$
$347$	17.4586i	0.937226i	−0.883404	+	0.468613i	$0.844754\pi$
$348$	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
$353$	21.0398	1.11984	0.559918	−	0.828548i	$-0.310833\pi$
$353$	21.0398	1.11984	0.559918	+	0.828548i	$0.310833\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.5965i	0.772531i
$358$	0	0
$359$	−23.5153	−1.24109	−0.620546	−	0.784170i	$-0.713089\pi$
$359$	−23.5153	−1.24109	−0.620546	+	0.784170i	$0.713089\pi$
$360$	0	0
$361$	9.75639	0.513494
$362$	0	0
$363$	− 8.95352i	− 0.469938i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.4012	1.32593	0.662965	−	0.748650i	$-0.269298\pi$
$367$	25.4012	1.32593	0.662965	+	0.748650i	$0.269298\pi$
$368$	0	0
$369$	1.91319	0.0995967
$370$	0	0
$371$	− 18.1987i	− 0.944829i
$372$	0	0
$373$	− 10.0677i	− 0.521286i	−0.965435	−	0.260643i	$-0.916065\pi$
$373$	− 10.0677i	− 0.521286i	0.965435	−	0.260643i	$-0.0839345\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.19098	0.112841
$378$	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
$383$	28.7446	1.46878	0.734391	−	0.678727i	$-0.237468\pi$
$383$	28.7446	1.46878	0.734391	+	0.678727i	$0.237468\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.2452i	0.571624i
$388$	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
$393$	6.90143	0.348131
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.73167i	0.137099i	0.997648	+	0.0685494i	$0.0218371\pi$
$397$	2.73167i	0.137099i	−0.997648	+	0.0685494i	$0.978163\pi$
$398$	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
$403$	1.79726i	0.0895279i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.6098	0.625044
$408$	0	0
$409$	22.4786	1.11150	0.555748	−	0.831351i	$-0.312432\pi$
$409$	22.4786	1.11150	0.555748	+	0.831351i	$0.312432\pi$
$410$	0	0
$411$	5.39022i	0.265880i
$412$	0	0
$413$	12.7227i	0.626045i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.4244	−0.853277
$418$	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	$-0.697527\pi$
$421$	− 33.3856i	− 1.62712i	0.581483	−	0.813558i	$-0.302473\pi$
$422$	0	0
$423$	−3.34374	−0.162578
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.04033i	0.437492i
$428$	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
$433$	19.7681	0.949996	0.474998	−	0.879987i	$-0.342449\pi$
$433$	19.7681	0.949996	0.474998	+	0.879987i	$0.342449\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.65827i	0.127163i
$438$	0	0
$439$	25.1699	1.20129	0.600646	−	0.799515i	$-0.294910\pi$
$439$	25.1699	1.20129	0.600646	+	0.799515i	$0.294910\pi$
$440$	0	0
$441$	3.09242	0.147258
$442$	0	0
$443$	19.5515i	0.928922i	0.885594	+	0.464461i	$0.153752\pi$
$443$	19.5515i	0.928922i	−0.885594	+	0.464461i	$0.846248\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.28551	−0.108101
$448$	0	0
$449$	−19.9612	−0.942029	−0.471014	−	0.882125i	$-0.656112\pi$
$449$	−19.9612	−0.942029	−0.471014	+	0.882125i	$0.656112\pi$
$450$	0	0
$451$	− 2.73692i	− 0.128876i
$452$	0	0
$453$	6.66425i	0.313114i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.01176	−0.234440	−0.117220	−	0.993106i	$-0.537398\pi$
$457$	−5.01176	−0.234440	−0.117220	+	0.993106i	$0.537398\pi$
$458$	0	0
$459$	− 7.38407i	− 0.344659i
$460$	0	0
$461$	− 5.12566i	− 0.238726i	−0.992851	−	0.119363i	$-0.961915\pi$
$461$	− 5.12566i	− 0.238726i	0.992851	−	0.119363i	$-0.0380852\pi$
$462$	0	0
$463$	5.79515	0.269324	0.134662	−	0.990892i	$-0.457005\pi$
$463$	5.79515	0.269324	0.134662	+	0.990892i	$0.457005\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.8208i	0.778373i	0.921159	+	0.389186i	$0.127244\pi$
$467$	16.8208i	0.778373i	−0.921159	+	0.389186i	$0.872756\pi$
$468$	0	0
$469$	9.62032i	0.444225i
$470$	0	0
$471$	−17.4144	−0.802413
$472$	0	0
$473$	16.0868	0.739672
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.20632i	0.421529i
$478$	0	0
$479$	36.9065	1.68630	0.843151	−	0.537678i	$-0.180698\pi$
$479$	36.9065	1.68630	0.843151	+	0.537678i	$0.180698\pi$
$480$	0	0
$481$	2.12714	0.0969892
$482$	0	0
$483$	1.72836i	0.0786429i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.14984	0.233361	0.116681	−	0.993169i	$-0.462775\pi$
$487$	5.14984	0.233361	0.116681	+	0.993169i	$0.462775\pi$
$488$	0	0
$489$	4.66187	0.210817
$490$	0	0
$491$	− 23.1154i	− 1.04318i	−0.853195	−	0.521591i	$-0.825339\pi$
$491$	− 23.1154i	− 1.04318i	0.853195	−	0.521591i	$-0.174661\pi$
$492$	0	0
$493$	67.0414i	3.01939i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.2452	0.728696
$498$	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
$503$	−15.4224	−0.687650	−0.343825	−	0.939034i	$-0.611723\pi$
$503$	−15.4224	−0.687650	−0.343825	+	0.939034i	$0.611723\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.9418i	− 0.574764i
$508$	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
$513$	3.04033	0.134234
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.78340i	0.210374i
$518$	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
$523$	24.7502i	1.08225i	0.840941	+	0.541126i	$0.182002\pi$
$523$	24.7502i	1.08225i	−0.840941	+	0.541126i	$0.817998\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−54.9939	−2.39557
$528$	0	0
$529$	−22.2355	−0.966762
$530$	0	0
$531$	− 6.43616i	− 0.279306i
$532$	0	0
$533$	− 0.461690i	− 0.0199980i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.68749	−0.202280
$538$	0	0
$539$	− 4.42386i	− 0.190549i
$540$	0	0
$541$	− 16.9982i	− 0.730812i	−0.930848	−	0.365406i	$-0.880930\pi$
$541$	− 16.9982i	− 0.730812i	0.930848	−	0.365406i	$-0.119070\pi$
$542$	0	0
$543$	9.10242	0.390622
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 37.2385i	− 1.59220i	−0.605163	−	0.796101i	$-0.706892\pi$
$547$	− 37.2385i	− 1.59220i	0.605163	−	0.796101i	$-0.293108\pi$
$548$	0	0
$549$	− 4.57331i	− 0.195184i
$550$	0	0
$551$	−27.6037	−1.17596
$552$	0	0
$553$	27.0336	1.14959
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.7604i	− 0.625420i	−0.949849	−	0.312710i	$-0.898763\pi$
$557$	− 14.7604i	− 0.625420i	0.949849	−	0.312710i	$-0.101237\pi$
$558$	0	0
$559$	2.71368	0.114776
$560$	0	0
$561$	−10.5633	−0.445983
$562$	0	0
$563$	− 3.00561i	− 0.126671i	−0.997992	−	0.0633356i	$-0.979826\pi$
$563$	− 3.00561i	− 0.126671i	0.997992	−	0.0633356i	$-0.0201739\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.97676	0.0830161
$568$	0	0
$569$	−23.1840	−0.971923	−0.485962	−	0.873980i	$-0.661531\pi$
$569$	−23.1840	−0.971923	−0.485962	+	0.873980i	$0.661531\pi$
$570$	0	0
$571$	0.202739i	0.00848438i	0.999991	+	0.00424219i	$0.00135033\pi$
$571$	0.202739i	0.00848438i	−0.999991	+	0.00424219i	$0.998650\pi$
$572$	0	0
$573$	15.2063i	0.635253i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.8023	0.907643	0.453821	−	0.891093i	$-0.350060\pi$
$577$	21.8023	0.907643	0.453821	+	0.891093i	$0.350060\pi$
$578$	0	0
$579$	20.7564i	0.862606i
$580$	0	0
$581$	− 24.3774i	− 1.01134i
$582$	0	0
$583$	13.1701	0.545451
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.7126i	1.51529i	0.652667	+	0.757645i	$0.273650\pi$
$587$	36.7126i	1.51529i	−0.652667	+	0.757645i	$0.726350\pi$
$588$	0	0
$589$	− 22.6433i	− 0.933001i
$590$	0	0
$591$	23.2508	0.956409
$592$	0	0
$593$	10.6036	0.435439	0.217719	−	0.976011i	$-0.430138\pi$
$593$	10.6036	0.435439	0.217719	+	0.976011i	$0.430138\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7.21633i	− 0.295345i
$598$	0	0
$599$	25.7988	1.05411	0.527056	−	0.849831i	$-0.323296\pi$
$599$	25.7988	1.05411	0.527056	+	0.849831i	$0.323296\pi$
$600$	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
$603$	− 4.86671i	− 0.198188i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.5828	1.52544	0.762719	−	0.646730i	$-0.223864\pi$
$607$	37.5828	1.52544	0.762719	+	0.646730i	$0.223864\pi$
$608$	0	0
$609$	−17.9474	−0.727264
$610$	0	0
$611$	0.806910i	0.0326441i
$612$	0	0
$613$	− 4.93405i	− 0.199284i	−0.995023	−	0.0996422i	$-0.968230\pi$
$613$	− 4.93405i	− 0.199284i	0.995023	−	0.0996422i	$-0.0317698\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.26043	−0.252035	−0.126018	−	0.992028i	$-0.540220\pi$
$617$	−6.26043	−0.252035	−0.126018	+	0.992028i	$0.540220\pi$
$618$	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
$623$	−15.9735	−0.639966
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.34935i	− 0.173696i
$628$	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
$633$	4.38407	0.174251
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.746260i	− 0.0295679i
$638$	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
$643$	− 2.89233i	− 0.114062i	−0.998372	−	0.0570312i	$-0.981837\pi$
$643$	− 2.89233i	− 0.114062i	0.998372	−	0.0570312i	$-0.0181635\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.3472	0.485420	0.242710	−	0.970099i	$-0.421964\pi$
$647$	12.3472	0.485420	0.242710	+	0.970099i	$0.421964\pi$
$648$	0	0
$649$	−9.20726	−0.361417
$650$	0	0
$651$	− 14.7222i	− 0.577009i
$652$	0	0
$653$	39.0507i	1.52817i	0.645114	+	0.764086i	$0.276810\pi$
$653$	39.0507i	1.52817i	−0.645114	+	0.764086i	$0.723190\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.12714	0.161015
$658$	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	$0.137348\pi$
$661$	21.5051i	0.836450i	−0.908343	+	0.418225i	$0.862652\pi$
$662$	0	0
$663$	−1.78192	−0.0692040
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.93827i	0.307371i
$668$	0	0
$669$	− 4.98852i	− 0.192867i
$670$	0	0
$671$	−6.54235	−0.252565
$672$	0	0
$673$	−36.1896	−1.39501	−0.697503	−	0.716582i	$-0.745706\pi$
$673$	−36.1896	−1.39501	−0.697503	+	0.716582i	$0.745706\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.17214i	0.352514i	0.984344	+	0.176257i	$0.0563990\pi$
$677$	9.17214i	0.352514i	−0.984344	+	0.176257i	$0.943601\pi$
$678$	0	0
$679$	21.1034	0.809873
$680$	0	0
$681$	−11.2569	−0.431367
$682$	0	0
$683$	16.3974i	0.627429i	0.949517	+	0.313714i	$0.101573\pi$
$683$	16.3974i	0.627429i	−0.949517	+	0.313714i	$0.898427\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.8364	0.604196
$688$	0	0
$689$	2.22166	0.0846387
$690$	0	0
$691$	− 24.4904i	− 0.931657i	−0.884875	−	0.465828i	$-0.845756\pi$
$691$	− 24.4904i	− 0.931657i	0.884875	−	0.465828i	$-0.154244\pi$
$692$	0	0
$693$	− 2.82786i	− 0.107421i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	14.1271	0.535104
$698$	0	0
$699$	10.9591i	0.414512i
$700$	0	0
$701$	− 12.3887i	− 0.467916i	−0.972247	−	0.233958i	$-0.924832\pi$
$701$	− 12.3887i	− 0.467916i	0.972247	−	0.233958i	$-0.0751679\pi$
$702$	0	0
$703$	−26.7994	−1.01076
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.1057i	0.981807i
$708$	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
$713$	−6.51175	−0.243867
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.3182i	0.646761i
$718$	0	0
$719$	33.8938	1.26403	0.632013	−	0.774958i	$-0.282229\pi$
$719$	33.8938	1.26403	0.632013	+	0.774958i	$0.282229\pi$
$720$	0	0
$721$	−38.3974	−1.42999
$722$	0	0
$723$	− 4.76869i	− 0.177349i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.1846	−0.526076	−0.263038	−	0.964785i	$-0.584725\pi$
$727$	−14.1846	−0.526076	−0.263038	+	0.964785i	$0.584725\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	83.0352i	3.07117i
$732$	0	0
$733$	8.09296i	0.298920i	0.988768	+	0.149460i	$0.0477536\pi$
$733$	8.09296i	0.298920i	−0.988768	+	0.149460i	$0.952246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.96208	−0.256452
$738$	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
$743$	8.78340	0.322232	0.161116	−	0.986936i	$-0.448491\pi$
$743$	8.78340	0.322232	0.161116	+	0.986936i	$0.448491\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.3320i	0.451204i
$748$	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
$753$	−6.15837	−0.224423
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.72450i	0.353443i	0.984261	+	0.176721i	$0.0565492\pi$
$757$	9.72450i	0.353443i	−0.984261	+	0.176721i	$0.943451\pi$
$758$	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
$763$	− 30.1052i	− 1.08988i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.55317	−0.0560817
$768$	0	0
$769$	18.7334	0.675545	0.337772	−	0.941228i	$-0.390327\pi$
$769$	18.7334	0.675545	0.337772	+	0.941228i	$0.390327\pi$
$770$	0	0
$771$	14.1584i	0.509901i
$772$	0	0
$773$	− 22.5006i	− 0.809292i	−0.914474	−	0.404646i	$-0.867395\pi$
$773$	− 22.5006i	− 0.809292i	0.914474	−	0.404646i	$-0.132605\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.4244	−0.625097
$778$	0	0
$779$	5.81673i	0.208406i
$780$	0	0
$781$	11.7564i	0.420677i
$782$	0	0
$783$	9.07918	0.324464
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.1063i	1.00188i	0.865482	+	0.500940i	$0.167012\pi$
$787$	28.1063i	1.00188i	−0.865482	+	0.500940i	$0.832988\pi$
$788$	0	0
$789$	− 15.5960i	− 0.555232i
$790$	0	0
$791$	−2.25133	−0.0800479
$792$	0	0
$793$	−1.10363	−0.0391910
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 38.1461i	− 1.35120i	−0.737267	−	0.675602i	$-0.763884\pi$
$797$	− 38.1461i	− 1.35120i	0.737267	−	0.675602i	$-0.236116\pi$
$798$	0	0
$799$	−24.6904	−0.873485
$800$	0	0
$801$	8.08066	0.285516
$802$	0	0
$803$	− 5.90409i	− 0.208351i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.3182	0.398420
$808$	0	0
$809$	−10.0745	−0.354201	−0.177101	−	0.984193i	$-0.556672\pi$
$809$	−10.0745	−0.354201	−0.177101	+	0.984193i	$0.556672\pi$
$810$	0	0
$811$	− 40.6001i	− 1.42566i	−0.701335	−	0.712832i	$-0.747412\pi$
$811$	− 40.6001i	− 1.42566i	0.701335	−	0.712832i	$-0.252588\pi$
$812$	0	0
$813$	− 6.20485i	− 0.217613i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−34.1891	−1.19612
$818$	0	0
$819$	− 0.477031i	− 0.0166688i
$820$	0	0
$821$	− 10.3397i	− 0.360858i	−0.983588	−	0.180429i	$-0.942251\pi$
$821$	− 10.3397i	− 0.360858i	0.983588	−	0.180429i	$-0.0577486\pi$
$822$	0	0
$823$	28.5013	0.993493	0.496746	−	0.867896i	$-0.334528\pi$
$823$	28.5013	0.993493	0.496746	+	0.867896i	$0.334528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32.1957i	− 1.11956i	−0.828643	−	0.559778i	$-0.810886\pi$
$827$	− 32.1957i	− 1.11956i	0.828643	−	0.559778i	$-0.189114\pi$
$828$	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
$833$	22.8346	0.791173
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.44764i	0.257428i
$838$	0	0
$839$	−54.9816	−1.89818	−0.949089	−	0.315009i	$-0.897992\pi$
$839$	−54.9816	−1.89818	−0.949089	+	0.315009i	$0.897992\pi$
$840$	0	0
$841$	−53.4316	−1.84247
$842$	0	0
$843$	21.6231i	0.744739i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.6990	0.608144
$848$	0	0
$849$	29.1522	1.00050
$850$	0	0
$851$	7.70696i	0.264191i
$852$	0	0
$853$	4.22607i	0.144698i	0.997379	+	0.0723489i	$0.0230495\pi$
$853$	4.22607i	0.144698i	−0.997379	+	0.0723489i	$0.976950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.8223	1.87270	0.936348	−	0.351074i	$-0.114183\pi$
$857$	54.8223	1.87270	0.936348	+	0.351074i	$0.114183\pi$
$858$	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
$863$	−40.7446	−1.38696	−0.693482	−	0.720474i	$-0.743924\pi$
$863$	−40.7446	−1.38696	−0.693482	+	0.720474i	$0.743924\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 37.5245i	− 1.27440i
$868$	0	0
$869$	19.5638i	0.663658i
$870$	0	0
$871$	−1.17443	−0.0397941
$872$	0	0
$873$	−10.6757	−0.361319
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 34.4568i	− 1.16352i	−0.813359	−	0.581762i	$-0.802363\pi$
$877$	− 34.4568i	− 1.16352i	0.813359	−	0.581762i	$-0.197637\pi$
$878$	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
$883$	24.2417i	0.815798i	0.913027	+	0.407899i	$0.133738\pi$
$883$	24.2417i	0.815798i	−0.913027	+	0.407899i	$0.866262\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.1843	1.31568	0.657840	−	0.753158i	$-0.271470\pi$
$887$	39.1843	1.31568	0.657840	+	0.753158i	$0.271470\pi$
$888$	0	0
$889$	−4.81571	−0.161514
$890$	0	0
$891$	1.43055i	0.0479253i
$892$	0	0
$893$	− 10.1661i	− 0.340195i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.210995	−0.00704490
$898$	0	0
$899$	− 67.6185i	− 2.25520i
$900$	0	0
$901$	67.9802i	2.26475i
$902$	0	0
$903$	−22.2290	−0.739736
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.9418i	0.628951i	0.949266	+	0.314475i	$0.101829\pi$
$907$	18.9418i	0.628951i	−0.949266	+	0.314475i	$0.898171\pi$
$908$	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
$913$	17.6415	0.583850
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.6425i	0.450515i
$918$	0	0
$919$	−24.4983	−0.808126	−0.404063	−	0.914731i	$-0.632402\pi$
$919$	−24.4983	−0.808126	−0.404063	+	0.914731i	$0.632402\pi$
$920$	0	0
$921$	−3.52297	−0.116086
$922$	0	0
$923$	1.98318i	0.0652772i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	19.4244	0.637981
$928$	0	0
$929$	11.5577	0.379196	0.189598	−	0.981862i	$-0.439282\pi$
$929$	11.5577	0.379196	0.189598	+	0.981862i	$0.439282\pi$
$930$	0	0
$931$	9.40197i	0.308137i
$932$	0	0
$933$	− 21.6757i	− 0.709632i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.5587	−1.94570	−0.972849	−	0.231440i	$-0.925656\pi$
$937$	−59.5587	−1.94570	−0.972849	+	0.231440i	$0.925656\pi$
$938$	0	0
$939$	12.5486i	0.409508i
$940$	0	0
$941$	− 55.9289i	− 1.82323i	−0.411044	−	0.911615i	$-0.634836\pi$
$941$	− 55.9289i	− 1.82323i	0.411044	−	0.911615i	$-0.365164\pi$
$942$	0	0
$943$	1.67277	0.0544730
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 30.0945i	− 0.977941i	−0.872300	−	0.488970i	$-0.837373\pi$
$947$	− 30.0945i	− 0.977941i	0.872300	−	0.488970i	$-0.162627\pi$
$948$	0	0
$949$	− 0.995959i	− 0.0323302i
$950$	0	0
$951$	−10.8611	−0.352195
$952$	0	0
$953$	12.4097	0.401989	0.200995	−	0.979592i	$-0.435583\pi$
$953$	12.4097	0.401989	0.200995	+	0.979592i	$0.435583\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.9882i	− 0.419850i
$958$	0	0
$959$	−10.6552	−0.344074
$960$	0	0
$961$	24.4674	0.789271
$962$	0	0
$963$	14.8085i	0.477196i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.8371	−1.34539	−0.672694	−	0.739920i	$-0.734863\pi$
$967$	−41.8371	−1.34539	−0.672694	+	0.739920i	$0.734863\pi$
$968$	0	0
$969$	22.4500	0.721199
$970$	0	0
$971$	− 40.6875i	− 1.30572i	−0.757477	−	0.652862i	$-0.773568\pi$
$971$	− 40.6875i	− 1.30572i	0.757477	−	0.652862i	$-0.226432\pi$
$972$	0	0
$973$	− 34.4439i	− 1.10422i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.67923	0.0537233	0.0268617	−	0.999639i	$-0.491449\pi$
$977$	1.67923	0.0537233	0.0268617	+	0.999639i	$0.491449\pi$
$978$	0	0
$979$	− 11.5598i	− 0.369453i
$980$	0	0
$981$	15.2296i	0.486243i
$982$	0	0
$983$	13.4944	0.430404	0.215202	−	0.976570i	$-0.430959\pi$
$983$	13.4944	0.430404	0.215202	+	0.976570i	$0.430959\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.60978i	− 0.210392i
$988$	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
$993$	−1.23185	−0.0390917
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.0972i	1.23822i	0.785304	+	0.619110i	$0.212507\pi$
$997$	39.0972i	1.23822i	−0.785304	+	0.619110i	$0.787493\pi$
$998$	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.k.d.1201.4		8
3.2	odd	2		7200.2.k.r.3601.7		8
4.3	odd	2		600.2.k.d.301.1	✓	8
5.2	odd	4		2400.2.d.g.49.7		8
5.3	odd	4		2400.2.d.h.49.2		8
5.4	even	2		2400.2.k.e.1201.5		8
8.3	odd	2		600.2.k.d.301.2	yes	8
8.5	even	2	inner	2400.2.k.d.1201.8		8
12.11	even	2		1800.2.k.t.901.8		8
15.2	even	4		7200.2.d.s.2449.7		8
15.8	even	4		7200.2.d.t.2449.2		8
15.14	odd	2		7200.2.k.s.3601.1		8
20.3	even	4		600.2.d.g.349.4		8
20.7	even	4		600.2.d.h.349.5		8
20.19	odd	2		600.2.k.e.301.8	yes	8
24.5	odd	2		7200.2.k.r.3601.8		8
24.11	even	2		1800.2.k.t.901.7		8
40.3	even	4		600.2.d.h.349.6		8
40.13	odd	4		2400.2.d.g.49.2		8
40.19	odd	2		600.2.k.e.301.7	yes	8
40.27	even	4		600.2.d.g.349.3		8
40.29	even	2		2400.2.k.e.1201.1		8
40.37	odd	4		2400.2.d.h.49.7		8
60.23	odd	4		1800.2.d.t.1549.5		8
60.47	odd	4		1800.2.d.s.1549.4		8
60.59	even	2		1800.2.k.q.901.1		8
120.29	odd	2		7200.2.k.s.3601.2		8
120.53	even	4		7200.2.d.s.2449.2		8
120.59	even	2		1800.2.k.q.901.2		8
120.77	even	4		7200.2.d.t.2449.7		8
120.83	odd	4		1800.2.d.s.1549.3		8
120.107	odd	4		1800.2.d.t.1549.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.3		8	40.27	even	4
600.2.d.g.349.4		8	20.3	even	4
600.2.d.h.349.5		8	20.7	even	4
600.2.d.h.349.6		8	40.3	even	4
600.2.k.d.301.1	✓	8	4.3	odd	2
600.2.k.d.301.2	yes	8	8.3	odd	2
600.2.k.e.301.7	yes	8	40.19	odd	2
600.2.k.e.301.8	yes	8	20.19	odd	2
1800.2.d.s.1549.3		8	120.83	odd	4
1800.2.d.s.1549.4		8	60.47	odd	4
1800.2.d.t.1549.5		8	60.23	odd	4
1800.2.d.t.1549.6		8	120.107	odd	4
1800.2.k.q.901.1		8	60.59	even	2
1800.2.k.q.901.2		8	120.59	even	2
1800.2.k.t.901.7		8	24.11	even	2
1800.2.k.t.901.8		8	12.11	even	2
2400.2.d.g.49.2		8	40.13	odd	4
2400.2.d.g.49.7		8	5.2	odd	4
2400.2.d.h.49.2		8	5.3	odd	4
2400.2.d.h.49.7		8	40.37	odd	4
2400.2.k.d.1201.4		8	1.1	even	1	trivial
2400.2.k.d.1201.8		8	8.5	even	2	inner
2400.2.k.e.1201.1		8	40.29	even	2
2400.2.k.e.1201.5		8	5.4	even	2
7200.2.d.s.2449.2		8	120.53	even	4
7200.2.d.s.2449.7		8	15.2	even	4
7200.2.d.t.2449.2		8	15.8	even	4
7200.2.d.t.2449.7		8	120.77	even	4
7200.2.k.r.3601.7		8	3.2	odd	2
7200.2.k.r.3601.8		8	24.5	odd	2
7200.2.k.s.3601.1		8	15.14	odd	2
7200.2.k.s.3601.2		8	120.29	odd	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.k (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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