LMFDB - Embedded newform 2400.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$141.604584014$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +20.0000 q^{11} -70.0000 q^{13} -90.0000 q^{17} +140.000 q^{19} +12.0000 q^{21} +192.000 q^{23} +27.0000 q^{27} -134.000 q^{29} +100.000 q^{31} +60.0000 q^{33} +170.000 q^{37} -210.000 q^{39} -110.000 q^{41} -532.000 q^{43} +56.0000 q^{47} -327.000 q^{49} -270.000 q^{51} +430.000 q^{53} +420.000 q^{57} -20.0000 q^{59} +270.000 q^{61} +36.0000 q^{63} +524.000 q^{67} +576.000 q^{69} -80.0000 q^{71} -330.000 q^{73} +80.0000 q^{77} +1060.00 q^{79} +81.0000 q^{81} +1188.00 q^{83} -402.000 q^{87} +1274.00 q^{89} -280.000 q^{91} +300.000 q^{93} +590.000 q^{97} +180.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−90.0000	−1.28401	−0.642006	−	0.766700i	$-0.721898\pi$
$17$	−90.0000	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	12.0000	0.124696
$22$	0	0
$23$	192.000	1.74064	0.870321	−	0.492485i	$-0.163911\pi$
$23$	192.000	1.74064	0.870321	+	0.492485i	$0.163911\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−134.000	−0.858041	−0.429020	−	0.903295i	$-0.641141\pi$
$29$	−134.000	−0.858041	−0.429020	+	0.903295i	$0.641141\pi$
$30$	0	0
$31$	100.000	0.579372	0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000	0.579372	0.289686	+	0.957122i	$0.406449\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	0	0
$36$	0	0
$37$	170.000	0.755347	0.377673	−	0.925939i	$-0.376724\pi$
$37$	170.000	0.755347	0.377673	+	0.925939i	$0.376724\pi$
$38$	0	0
$39$	−210.000	−0.862229
$40$	0	0
$41$	−110.000	−0.419003	−0.209501	−	0.977808i	$-0.567184\pi$
$41$	−110.000	−0.419003	−0.209501	+	0.977808i	$0.567184\pi$
$42$	0	0
$43$	−532.000	−1.88673	−0.943363	−	0.331762i	$-0.892357\pi$
$43$	−532.000	−1.88673	−0.943363	+	0.331762i	$0.892357\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	56.0000	0.173797	0.0868983	−	0.996217i	$-0.472304\pi$
$47$	56.0000	0.173797	0.0868983	+	0.996217i	$0.472304\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	−270.000	−0.741325
$52$	0	0
$53$	430.000	1.11443	0.557217	−	0.830367i	$-0.311869\pi$
$53$	430.000	1.11443	0.557217	+	0.830367i	$0.311869\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	420.000	0.975971
$58$	0	0
$59$	−20.0000	−0.0441318	−0.0220659	−	0.999757i	$-0.507024\pi$
$59$	−20.0000	−0.0441318	−0.0220659	+	0.999757i	$0.507024\pi$
$60$	0	0
$61$	270.000	0.566721	0.283360	−	0.959014i	$-0.408551\pi$
$61$	270.000	0.566721	0.283360	+	0.959014i	$0.408551\pi$
$62$	0	0
$63$	36.0000	0.0719932
$64$	0	0
$65$	0	0
$66$	0	0
$67$	524.000	0.955474	0.477737	−	0.878503i	$-0.341457\pi$
$67$	524.000	0.955474	0.477737	+	0.878503i	$0.341457\pi$
$68$	0	0
$69$	576.000	1.00496
$70$	0	0
$71$	−80.0000	−0.133722	−0.0668609	−	0.997762i	$-0.521298\pi$
$71$	−80.0000	−0.133722	−0.0668609	+	0.997762i	$0.521298\pi$
$72$	0	0
$73$	−330.000	−0.529090	−0.264545	−	0.964373i	$-0.585222\pi$
$73$	−330.000	−0.529090	−0.264545	+	0.964373i	$0.585222\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	80.0000	0.118401
$78$	0	0
$79$	1060.00	1.50961	0.754806	−	0.655948i	$-0.227731\pi$
$79$	1060.00	1.50961	0.754806	+	0.655948i	$0.227731\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1188.00	1.57108	0.785542	−	0.618809i	$-0.212384\pi$
$83$	1188.00	1.57108	0.785542	+	0.618809i	$0.212384\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−402.000	−0.495390
$88$	0	0
$89$	1274.00	1.51735	0.758673	−	0.651472i	$-0.225848\pi$
$89$	1274.00	1.51735	0.758673	+	0.651472i	$0.225848\pi$
$90$	0	0
$91$	−280.000	−0.322549
$92$	0	0
$93$	300.000	0.334501
$94$	0	0
$95$	0	0
$96$	0	0
$97$	590.000	0.617582	0.308791	−	0.951130i	$-0.400076\pi$
$97$	590.000	0.617582	0.308791	+	0.951130i	$0.400076\pi$
$98$	0	0
$99$	180.000	0.182734
$100$	0	0
$101$	−798.000	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$101$	−798.000	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$102$	0	0
$103$	1548.00	1.48086	0.740432	−	0.672131i	$-0.234621\pi$
$103$	1548.00	1.48086	0.740432	+	0.672131i	$0.234621\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1324.00	−1.19622	−0.598112	−	0.801413i	$-0.704082\pi$
$107$	−1324.00	−1.19622	−0.598112	+	0.801413i	$0.704082\pi$
$108$	0	0
$109$	470.000	0.413008	0.206504	−	0.978446i	$-0.433791\pi$
$109$	470.000	0.413008	0.206504	+	0.978446i	$0.433791\pi$
$110$	0	0
$111$	510.000	0.436100
$112$	0	0
$113$	−610.000	−0.507823	−0.253911	−	0.967227i	$-0.581717\pi$
$113$	−610.000	−0.507823	−0.253911	+	0.967227i	$0.581717\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−630.000	−0.497808
$118$	0	0
$119$	−360.000	−0.277321
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−330.000	−0.241911
$124$	0	0
$125$	0	0
$126$	0	0
$127$	884.000	0.617656	0.308828	−	0.951118i	$-0.400063\pi$
$127$	884.000	0.617656	0.308828	+	0.951118i	$0.400063\pi$
$128$	0	0
$129$	−1596.00	−1.08930
$130$	0	0
$131$	500.000	0.333475	0.166737	−	0.986001i	$-0.446677\pi$
$131$	500.000	0.333475	0.166737	+	0.986001i	$0.446677\pi$
$132$	0	0
$133$	560.000	0.365099
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1950.00	1.21606	0.608028	−	0.793915i	$-0.291961\pi$
$137$	1950.00	1.21606	0.608028	+	0.793915i	$0.291961\pi$
$138$	0	0
$139$	220.000	0.134246	0.0671229	−	0.997745i	$-0.478618\pi$
$139$	220.000	0.134246	0.0671229	+	0.997745i	$0.478618\pi$
$140$	0	0
$141$	168.000	0.100342
$142$	0	0
$143$	−1400.00	−0.818698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−981.000	−0.550418
$148$	0	0
$149$	−390.000	−0.214430	−0.107215	−	0.994236i	$-0.534193\pi$
$149$	−390.000	−0.214430	−0.107215	+	0.994236i	$0.534193\pi$
$150$	0	0
$151$	−2100.00	−1.13176	−0.565879	−	0.824488i	$-0.691463\pi$
$151$	−2100.00	−1.13176	−0.565879	+	0.824488i	$0.691463\pi$
$152$	0	0
$153$	−810.000	−0.428004
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2050.00	1.04209	0.521044	−	0.853530i	$-0.325543\pi$
$157$	2050.00	1.04209	0.521044	+	0.853530i	$0.325543\pi$
$158$	0	0
$159$	1290.00	0.643419
$160$	0	0
$161$	768.000	0.375943
$162$	0	0
$163$	−332.000	−0.159535	−0.0797676	−	0.996813i	$-0.525418\pi$
$163$	−332.000	−0.159535	−0.0797676	+	0.996813i	$0.525418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2744.00	1.27148	0.635740	−	0.771903i	$-0.280695\pi$
$167$	2744.00	1.27148	0.635740	+	0.771903i	$0.280695\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	0	0
$171$	1260.00	0.563477
$172$	0	0
$173$	−3570.00	−1.56891	−0.784457	−	0.620183i	$-0.787058\pi$
$173$	−3570.00	−1.56891	−0.784457	+	0.620183i	$0.787058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−60.0000	−0.0254795
$178$	0	0
$179$	1380.00	0.576235	0.288117	−	0.957595i	$-0.406971\pi$
$179$	1380.00	0.576235	0.288117	+	0.957595i	$0.406971\pi$
$180$	0	0
$181$	1358.00	0.557676	0.278838	−	0.960338i	$-0.410051\pi$
$181$	1358.00	0.557676	0.278838	+	0.960338i	$0.410051\pi$
$182$	0	0
$183$	810.000	0.327196
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1800.00	−0.703899
$188$	0	0
$189$	108.000	0.0415653
$190$	0	0
$191$	−3840.00	−1.45473	−0.727363	−	0.686253i	$-0.759254\pi$
$191$	−3840.00	−1.45473	−0.727363	+	0.686253i	$0.759254\pi$
$192$	0	0
$193$	−3090.00	−1.15245	−0.576226	−	0.817291i	$-0.695475\pi$
$193$	−3090.00	−1.15245	−0.576226	+	0.817291i	$0.695475\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1070.00	0.386976	0.193488	−	0.981103i	$-0.438020\pi$
$197$	1070.00	0.386976	0.193488	+	0.981103i	$0.438020\pi$
$198$	0	0
$199$	−380.000	−0.135364	−0.0676821	−	0.997707i	$-0.521560\pi$
$199$	−380.000	−0.135364	−0.0676821	+	0.997707i	$0.521560\pi$
$200$	0	0
$201$	1572.00	0.551643
$202$	0	0
$203$	−536.000	−0.185319
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1728.00	0.580214
$208$	0	0
$209$	2800.00	0.926699
$210$	0	0
$211$	2180.00	0.711267	0.355634	−	0.934625i	$-0.384265\pi$
$211$	2180.00	0.711267	0.355634	+	0.934625i	$0.384265\pi$
$212$	0	0
$213$	−240.000	−0.0772044
$214$	0	0
$215$	0	0
$216$	0	0
$217$	400.000	0.125133
$218$	0	0
$219$	−990.000	−0.305470
$220$	0	0
$221$	6300.00	1.91757
$222$	0	0
$223$	−668.000	−0.200595	−0.100297	−	0.994958i	$-0.531979\pi$
$223$	−668.000	−0.200595	−0.100297	+	0.994958i	$0.531979\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4836.00	1.41399	0.706997	−	0.707217i	$-0.250049\pi$
$227$	4836.00	1.41399	0.706997	+	0.707217i	$0.250049\pi$
$228$	0	0
$229$	4334.00	1.25065	0.625325	−	0.780365i	$-0.284966\pi$
$229$	4334.00	1.25065	0.625325	+	0.780365i	$0.284966\pi$
$230$	0	0
$231$	240.000	0.0683586
$232$	0	0
$233$	2550.00	0.716979	0.358489	−	0.933534i	$-0.383292\pi$
$233$	2550.00	0.716979	0.358489	+	0.933534i	$0.383292\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3180.00	0.871575
$238$	0	0
$239$	1920.00	0.519642	0.259821	−	0.965657i	$-0.416336\pi$
$239$	1920.00	0.519642	0.259821	+	0.965657i	$0.416336\pi$
$240$	0	0
$241$	−1070.00	−0.285995	−0.142997	−	0.989723i	$-0.545674\pi$
$241$	−1070.00	−0.285995	−0.142997	+	0.989723i	$0.545674\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9800.00	−2.52453
$248$	0	0
$249$	3564.00	0.907066
$250$	0	0
$251$	−1020.00	−0.256501	−0.128251	−	0.991742i	$-0.540936\pi$
$251$	−1020.00	−0.256501	−0.128251	+	0.991742i	$0.540936\pi$
$252$	0	0
$253$	3840.00	0.954224
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5630.00	1.36650	0.683249	−	0.730186i	$-0.260567\pi$
$257$	5630.00	1.36650	0.683249	+	0.730186i	$0.260567\pi$
$258$	0	0
$259$	680.000	0.163140
$260$	0	0
$261$	−1206.00	−0.286014
$262$	0	0
$263$	7128.00	1.67122	0.835611	−	0.549322i	$-0.185114\pi$
$263$	7128.00	1.67122	0.835611	+	0.549322i	$0.185114\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3822.00	0.876040
$268$	0	0
$269$	4650.00	1.05396	0.526980	−	0.849877i	$-0.323324\pi$
$269$	4650.00	1.05396	0.526980	+	0.849877i	$0.323324\pi$
$270$	0	0
$271$	3180.00	0.712809	0.356405	−	0.934332i	$-0.384003\pi$
$271$	3180.00	0.712809	0.356405	+	0.934332i	$0.384003\pi$
$272$	0	0
$273$	−840.000	−0.186224
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5330.00	1.15613	0.578066	−	0.815990i	$-0.303808\pi$
$277$	5330.00	1.15613	0.578066	+	0.815990i	$0.303808\pi$
$278$	0	0
$279$	900.000	0.193124
$280$	0	0
$281$	−7830.00	−1.66227	−0.831136	−	0.556069i	$-0.812309\pi$
$281$	−7830.00	−1.66227	−0.831136	+	0.556069i	$0.812309\pi$
$282$	0	0
$283$	−268.000	−0.0562931	−0.0281465	−	0.999604i	$-0.508961\pi$
$283$	−268.000	−0.0562931	−0.0281465	+	0.999604i	$0.508961\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−440.000	−0.0904961
$288$	0	0
$289$	3187.00	0.648687
$290$	0	0
$291$	1770.00	0.356561
$292$	0	0
$293$	1950.00	0.388806	0.194403	−	0.980922i	$-0.437723\pi$
$293$	1950.00	0.388806	0.194403	+	0.980922i	$0.437723\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	−13440.0	−2.59952
$300$	0	0
$301$	−2128.00	−0.407495
$302$	0	0
$303$	−2394.00	−0.453900
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2916.00	−0.542101	−0.271050	−	0.962565i	$-0.587371\pi$
$307$	−2916.00	−0.542101	−0.271050	+	0.962565i	$0.587371\pi$
$308$	0	0
$309$	4644.00	0.854977
$310$	0	0
$311$	9000.00	1.64097	0.820487	−	0.571665i	$-0.193702\pi$
$311$	9000.00	1.64097	0.820487	+	0.571665i	$0.193702\pi$
$312$	0	0
$313$	−8890.00	−1.60541	−0.802704	−	0.596378i	$-0.796606\pi$
$313$	−8890.00	−1.60541	−0.802704	+	0.596378i	$0.796606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1290.00	−0.228560	−0.114280	−	0.993449i	$-0.536456\pi$
$317$	−1290.00	−0.228560	−0.114280	+	0.993449i	$0.536456\pi$
$318$	0	0
$319$	−2680.00	−0.470380
$320$	0	0
$321$	−3972.00	−0.690640
$322$	0	0
$323$	−12600.0	−2.17053
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1410.00	0.238450
$328$	0	0
$329$	224.000	0.0375365
$330$	0	0
$331$	−5300.00	−0.880104	−0.440052	−	0.897972i	$-0.645040\pi$
$331$	−5300.00	−0.880104	−0.440052	+	0.897972i	$0.645040\pi$
$332$	0	0
$333$	1530.00	0.251782
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9310.00	1.50489	0.752445	−	0.658655i	$-0.228874\pi$
$337$	9310.00	1.50489	0.752445	+	0.658655i	$0.228874\pi$
$338$	0	0
$339$	−1830.00	−0.293192
$340$	0	0
$341$	2000.00	0.317613
$342$	0	0
$343$	−2680.00	−0.421885
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2596.00	−0.401616	−0.200808	−	0.979631i	$-0.564357\pi$
$347$	−2596.00	−0.401616	−0.200808	+	0.979631i	$0.564357\pi$
$348$	0	0
$349$	814.000	0.124849	0.0624247	−	0.998050i	$-0.480117\pi$
$349$	814.000	0.124849	0.0624247	+	0.998050i	$0.480117\pi$
$350$	0	0
$351$	−1890.00	−0.287410
$352$	0	0
$353$	−7730.00	−1.16551	−0.582757	−	0.812647i	$-0.698026\pi$
$353$	−7730.00	−1.16551	−0.582757	+	0.812647i	$0.698026\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1080.00	−0.160111
$358$	0	0
$359$	5840.00	0.858561	0.429281	−	0.903171i	$-0.358767\pi$
$359$	5840.00	0.858561	0.429281	+	0.903171i	$0.358767\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6636.00	0.943859	0.471930	−	0.881636i	$-0.343558\pi$
$367$	6636.00	0.943859	0.471930	+	0.881636i	$0.343558\pi$
$368$	0	0
$369$	−990.000	−0.139668
$370$	0	0
$371$	1720.00	0.240695
$372$	0	0
$373$	−12950.0	−1.79766	−0.898828	−	0.438302i	$-0.855580\pi$
$373$	−12950.0	−1.79766	−0.898828	+	0.438302i	$0.855580\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9380.00	1.28142
$378$	0	0
$379$	−6220.00	−0.843008	−0.421504	−	0.906827i	$-0.638498\pi$
$379$	−6220.00	−0.843008	−0.421504	+	0.906827i	$0.638498\pi$
$380$	0	0
$381$	2652.00	0.356604
$382$	0	0
$383$	8672.00	1.15697	0.578484	−	0.815694i	$-0.303645\pi$
$383$	8672.00	1.15697	0.578484	+	0.815694i	$0.303645\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4788.00	−0.628909
$388$	0	0
$389$	7530.00	0.981455	0.490728	−	0.871313i	$-0.336731\pi$
$389$	7530.00	0.981455	0.490728	+	0.871313i	$0.336731\pi$
$390$	0	0
$391$	−17280.0	−2.23501
$392$	0	0
$393$	1500.00	0.192532
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3650.00	0.461431	0.230716	−	0.973021i	$-0.425893\pi$
$397$	3650.00	0.461431	0.230716	+	0.973021i	$0.425893\pi$
$398$	0	0
$399$	1680.00	0.210790
$400$	0	0
$401$	11498.0	1.43188	0.715939	−	0.698163i	$-0.245999\pi$
$401$	11498.0	1.43188	0.715939	+	0.698163i	$0.245999\pi$
$402$	0	0
$403$	−7000.00	−0.865248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3400.00	0.414083
$408$	0	0
$409$	−3590.00	−0.434020	−0.217010	−	0.976169i	$-0.569630\pi$
$409$	−3590.00	−0.434020	−0.217010	+	0.976169i	$0.569630\pi$
$410$	0	0
$411$	5850.00	0.702091
$412$	0	0
$413$	−80.0000	−0.00953158
$414$	0	0
$415$	0	0
$416$	0	0
$417$	660.000	0.0775068
$418$	0	0
$419$	380.000	0.0443060	0.0221530	−	0.999755i	$-0.492948\pi$
$419$	380.000	0.0443060	0.0221530	+	0.999755i	$0.492948\pi$
$420$	0	0
$421$	−5410.00	−0.626288	−0.313144	−	0.949706i	$-0.601382\pi$
$421$	−5410.00	−0.626288	−0.313144	+	0.949706i	$0.601382\pi$
$422$	0	0
$423$	504.000	0.0579322
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1080.00	0.122400
$428$	0	0
$429$	−4200.00	−0.472676
$430$	0	0
$431$	8280.00	0.925368	0.462684	−	0.886523i	$-0.346886\pi$
$431$	8280.00	0.925368	0.462684	+	0.886523i	$0.346886\pi$
$432$	0	0
$433$	12590.0	1.39731	0.698657	−	0.715457i	$-0.253781\pi$
$433$	12590.0	1.39731	0.698657	+	0.715457i	$0.253781\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26880.0	2.94244
$438$	0	0
$439$	15140.0	1.64600	0.822999	−	0.568043i	$-0.192299\pi$
$439$	15140.0	1.64600	0.822999	+	0.568043i	$0.192299\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	5628.00	0.603599	0.301799	−	0.953371i	$-0.402413\pi$
$443$	5628.00	0.603599	0.301799	+	0.953371i	$0.402413\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1170.00	−0.123801
$448$	0	0
$449$	5450.00	0.572832	0.286416	−	0.958105i	$-0.407536\pi$
$449$	5450.00	0.572832	0.286416	+	0.958105i	$0.407536\pi$
$450$	0	0
$451$	−2200.00	−0.229698
$452$	0	0
$453$	−6300.00	−0.653421
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2710.00	0.277393	0.138696	−	0.990335i	$-0.455709\pi$
$457$	2710.00	0.277393	0.138696	+	0.990335i	$0.455709\pi$
$458$	0	0
$459$	−2430.00	−0.247108
$460$	0	0
$461$	−11982.0	−1.21054	−0.605268	−	0.796022i	$-0.706934\pi$
$461$	−11982.0	−1.21054	−0.605268	+	0.796022i	$0.706934\pi$
$462$	0	0
$463$	6068.00	0.609080	0.304540	−	0.952500i	$-0.401497\pi$
$463$	6068.00	0.609080	0.304540	+	0.952500i	$0.401497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5036.00	−0.499011	−0.249506	−	0.968373i	$-0.580268\pi$
$467$	−5036.00	−0.499011	−0.249506	+	0.968373i	$0.580268\pi$
$468$	0	0
$469$	2096.00	0.206363
$470$	0	0
$471$	6150.00	0.601650
$472$	0	0
$473$	−10640.0	−1.03431
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3870.00	0.371478
$478$	0	0
$479$	−4440.00	−0.423526	−0.211763	−	0.977321i	$-0.567920\pi$
$479$	−4440.00	−0.423526	−0.211763	+	0.977321i	$0.567920\pi$
$480$	0	0
$481$	−11900.0	−1.12805
$482$	0	0
$483$	2304.00	0.217051
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4996.00	0.464867	0.232434	−	0.972612i	$-0.425331\pi$
$487$	4996.00	0.464867	0.232434	+	0.972612i	$0.425331\pi$
$488$	0	0
$489$	−996.000	−0.0921077
$490$	0	0
$491$	10700.0	0.983471	0.491735	−	0.870745i	$-0.336363\pi$
$491$	10700.0	0.983471	0.491735	+	0.870745i	$0.336363\pi$
$492$	0	0
$493$	12060.0	1.10173
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−320.000	−0.0288812
$498$	0	0
$499$	−13980.0	−1.25417	−0.627085	−	0.778951i	$-0.715752\pi$
$499$	−13980.0	−1.25417	−0.627085	+	0.778951i	$0.715752\pi$
$500$	0	0
$501$	8232.00	0.734089
$502$	0	0
$503$	−13632.0	−1.20839	−0.604196	−	0.796836i	$-0.706505\pi$
$503$	−13632.0	−1.20839	−0.604196	+	0.796836i	$0.706505\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8109.00	0.710322
$508$	0	0
$509$	4746.00	0.413286	0.206643	−	0.978416i	$-0.433746\pi$
$509$	4746.00	0.413286	0.206643	+	0.978416i	$0.433746\pi$
$510$	0	0
$511$	−1320.00	−0.114273
$512$	0	0
$513$	3780.00	0.325324
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1120.00	0.0952757
$518$	0	0
$519$	−10710.0	−0.905813
$520$	0	0
$521$	−5838.00	−0.490916	−0.245458	−	0.969407i	$-0.578938\pi$
$521$	−5838.00	−0.490916	−0.245458	+	0.969407i	$0.578938\pi$
$522$	0	0
$523$	−8388.00	−0.701303	−0.350652	−	0.936506i	$-0.614040\pi$
$523$	−8388.00	−0.701303	−0.350652	+	0.936506i	$0.614040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9000.00	−0.743921
$528$	0	0
$529$	24697.0	2.02983
$530$	0	0
$531$	−180.000	−0.0147106
$532$	0	0
$533$	7700.00	0.625749
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4140.00	0.332689
$538$	0	0
$539$	−6540.00	−0.522630
$540$	0	0
$541$	7078.00	0.562490	0.281245	−	0.959636i	$-0.409253\pi$
$541$	7078.00	0.562490	0.281245	+	0.959636i	$0.409253\pi$
$542$	0	0
$543$	4074.00	0.321974
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8604.00	−0.672542	−0.336271	−	0.941765i	$-0.609166\pi$
$547$	−8604.00	−0.672542	−0.336271	+	0.941765i	$0.609166\pi$
$548$	0	0
$549$	2430.00	0.188907
$550$	0	0
$551$	−18760.0	−1.45046
$552$	0	0
$553$	4240.00	0.326045
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18850.0	−1.43393	−0.716966	−	0.697108i	$-0.754470\pi$
$557$	−18850.0	−1.43393	−0.716966	+	0.697108i	$0.754470\pi$
$558$	0	0
$559$	37240.0	2.81768
$560$	0	0
$561$	−5400.00	−0.406396
$562$	0	0
$563$	5412.00	0.405131	0.202565	−	0.979269i	$-0.435072\pi$
$563$	5412.00	0.405131	0.202565	+	0.979269i	$0.435072\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	324.000	0.0239977
$568$	0	0
$569$	13570.0	0.999796	0.499898	−	0.866084i	$-0.333371\pi$
$569$	13570.0	0.999796	0.499898	+	0.866084i	$0.333371\pi$
$570$	0	0
$571$	860.000	0.0630296	0.0315148	−	0.999503i	$-0.489967\pi$
$571$	860.000	0.0630296	0.0315148	+	0.999503i	$0.489967\pi$
$572$	0	0
$573$	−11520.0	−0.839886
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5010.00	−0.361471	−0.180736	−	0.983532i	$-0.557848\pi$
$577$	−5010.00	−0.361471	−0.180736	+	0.983532i	$0.557848\pi$
$578$	0	0
$579$	−9270.00	−0.665368
$580$	0	0
$581$	4752.00	0.339322
$582$	0	0
$583$	8600.00	0.610936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16116.0	1.13318	0.566592	−	0.823999i	$-0.308262\pi$
$587$	16116.0	1.13318	0.566592	+	0.823999i	$0.308262\pi$
$588$	0	0
$589$	14000.0	0.979389
$590$	0	0
$591$	3210.00	0.223421
$592$	0	0
$593$	−7170.00	−0.496520	−0.248260	−	0.968693i	$-0.579859\pi$
$593$	−7170.00	−0.496520	−0.248260	+	0.968693i	$0.579859\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1140.00	−0.0781526
$598$	0	0
$599$	9520.00	0.649377	0.324688	−	0.945821i	$-0.394741\pi$
$599$	9520.00	0.649377	0.324688	+	0.945821i	$0.394741\pi$
$600$	0	0
$601$	6010.00	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	6010.00	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	4716.00	0.318491
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24844.0	−1.66126	−0.830632	−	0.556822i	$-0.812020\pi$
$607$	−24844.0	−1.66126	−0.830632	+	0.556822i	$0.812020\pi$
$608$	0	0
$609$	−1608.00	−0.106994
$610$	0	0
$611$	−3920.00	−0.259552
$612$	0	0
$613$	−9990.00	−0.658226	−0.329113	−	0.944291i	$-0.606750\pi$
$613$	−9990.00	−0.658226	−0.329113	+	0.944291i	$0.606750\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−570.000	−0.0371918	−0.0185959	−	0.999827i	$-0.505920\pi$
$617$	−570.000	−0.0371918	−0.0185959	+	0.999827i	$0.505920\pi$
$618$	0	0
$619$	8700.00	0.564915	0.282458	−	0.959280i	$-0.408850\pi$
$619$	8700.00	0.564915	0.282458	+	0.959280i	$0.408850\pi$
$620$	0	0
$621$	5184.00	0.334987
$622$	0	0
$623$	5096.00	0.327716
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8400.00	0.535030
$628$	0	0
$629$	−15300.0	−0.969874
$630$	0	0
$631$	−16340.0	−1.03088	−0.515440	−	0.856926i	$-0.672371\pi$
$631$	−16340.0	−1.03088	−0.515440	+	0.856926i	$0.672371\pi$
$632$	0	0
$633$	6540.00	0.410650
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22890.0	1.42376
$638$	0	0
$639$	−720.000	−0.0445740
$640$	0	0
$641$	11210.0	0.690746	0.345373	−	0.938465i	$-0.387752\pi$
$641$	11210.0	0.690746	0.345373	+	0.938465i	$0.387752\pi$
$642$	0	0
$643$	−12828.0	−0.786760	−0.393380	−	0.919376i	$-0.628694\pi$
$643$	−12828.0	−0.786760	−0.393380	+	0.919376i	$0.628694\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24336.0	1.47874	0.739372	−	0.673298i	$-0.235123\pi$
$647$	24336.0	1.47874	0.739372	+	0.673298i	$0.235123\pi$
$648$	0	0
$649$	−400.000	−0.0241932
$650$	0	0
$651$	1200.00	0.0722453
$652$	0	0
$653$	−770.000	−0.0461446	−0.0230723	−	0.999734i	$-0.507345\pi$
$653$	−770.000	−0.0461446	−0.0230723	+	0.999734i	$0.507345\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2970.00	−0.176363
$658$	0	0
$659$	−4140.00	−0.244722	−0.122361	−	0.992486i	$-0.539046\pi$
$659$	−4140.00	−0.244722	−0.122361	+	0.992486i	$0.539046\pi$
$660$	0	0
$661$	8390.00	0.493696	0.246848	−	0.969054i	$-0.420605\pi$
$661$	8390.00	0.493696	0.246848	+	0.969054i	$0.420605\pi$
$662$	0	0
$663$	18900.0	1.10711
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−25728.0	−1.49354
$668$	0	0
$669$	−2004.00	−0.115813
$670$	0	0
$671$	5400.00	0.310678
$672$	0	0
$673$	−1730.00	−0.0990886	−0.0495443	−	0.998772i	$-0.515777\pi$
$673$	−1730.00	−0.0990886	−0.0495443	+	0.998772i	$0.515777\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23930.0	−1.35850	−0.679250	−	0.733907i	$-0.737695\pi$
$677$	−23930.0	−1.35850	−0.679250	+	0.733907i	$0.737695\pi$
$678$	0	0
$679$	2360.00	0.133385
$680$	0	0
$681$	14508.0	0.816370
$682$	0	0
$683$	−29892.0	−1.67465	−0.837325	−	0.546706i	$-0.815882\pi$
$683$	−29892.0	−1.67465	−0.837325	+	0.546706i	$0.815882\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13002.0	0.722063
$688$	0	0
$689$	−30100.0	−1.66432
$690$	0	0
$691$	−21220.0	−1.16823	−0.584115	−	0.811671i	$-0.698558\pi$
$691$	−21220.0	−1.16823	−0.584115	+	0.811671i	$0.698558\pi$
$692$	0	0
$693$	720.000	0.0394669
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9900.00	0.538005
$698$	0	0
$699$	7650.00	0.413948
$700$	0	0
$701$	−21750.0	−1.17188	−0.585939	−	0.810355i	$-0.699274\pi$
$701$	−21750.0	−1.17188	−0.585939	+	0.810355i	$0.699274\pi$
$702$	0	0
$703$	23800.0	1.27686
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3192.00	−0.169798
$708$	0	0
$709$	−21554.0	−1.14172	−0.570859	−	0.821048i	$-0.693390\pi$
$709$	−21554.0	−1.14172	−0.570859	+	0.821048i	$0.693390\pi$
$710$	0	0
$711$	9540.00	0.503204
$712$	0	0
$713$	19200.0	1.00848
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5760.00	0.300016
$718$	0	0
$719$	13000.0	0.674295	0.337148	−	0.941452i	$-0.390538\pi$
$719$	13000.0	0.674295	0.337148	+	0.941452i	$0.390538\pi$
$720$	0	0
$721$	6192.00	0.319837
$722$	0	0
$723$	−3210.00	−0.165119
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3276.00	−0.167125	−0.0835627	−	0.996503i	$-0.526630\pi$
$727$	−3276.00	−0.167125	−0.0835627	+	0.996503i	$0.526630\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	47880.0	2.42258
$732$	0	0
$733$	−21110.0	−1.06373	−0.531866	−	0.846828i	$-0.678509\pi$
$733$	−21110.0	−1.06373	−0.531866	+	0.846828i	$0.678509\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10480.0	0.523793
$738$	0	0
$739$	−17980.0	−0.895000	−0.447500	−	0.894284i	$-0.647686\pi$
$739$	−17980.0	−0.895000	−0.447500	+	0.894284i	$0.647686\pi$
$740$	0	0
$741$	−29400.0	−1.45754
$742$	0	0
$743$	33192.0	1.63889	0.819446	−	0.573157i	$-0.194281\pi$
$743$	33192.0	1.63889	0.819446	+	0.573157i	$0.194281\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10692.0	0.523695
$748$	0	0
$749$	−5296.00	−0.258360
$750$	0	0
$751$	−23900.0	−1.16128	−0.580641	−	0.814159i	$-0.697198\pi$
$751$	−23900.0	−1.16128	−0.580641	+	0.814159i	$0.697198\pi$
$752$	0	0
$753$	−3060.00	−0.148091
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1950.00	−0.0936248	−0.0468124	−	0.998904i	$-0.514906\pi$
$757$	−1950.00	−0.0936248	−0.0468124	+	0.998904i	$0.514906\pi$
$758$	0	0
$759$	11520.0	0.550922
$760$	0	0
$761$	8082.00	0.384983	0.192492	−	0.981299i	$-0.438343\pi$
$761$	8082.00	0.384983	0.192492	+	0.981299i	$0.438343\pi$
$762$	0	0
$763$	1880.00	0.0892013
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1400.00	0.0659075
$768$	0	0
$769$	−36494.0	−1.71132	−0.855661	−	0.517536i	$-0.826849\pi$
$769$	−36494.0	−1.71132	−0.855661	+	0.517536i	$0.826849\pi$
$770$	0	0
$771$	16890.0	0.788947
$772$	0	0
$773$	−28050.0	−1.30516	−0.652580	−	0.757720i	$-0.726313\pi$
$773$	−28050.0	−1.30516	−0.652580	+	0.757720i	$0.726313\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2040.00	0.0941887
$778$	0	0
$779$	−15400.0	−0.708296
$780$	0	0
$781$	−1600.00	−0.0733067
$782$	0	0
$783$	−3618.00	−0.165130
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23004.0	−1.04194	−0.520968	−	0.853576i	$-0.674429\pi$
$787$	−23004.0	−1.04194	−0.520968	+	0.853576i	$0.674429\pi$
$788$	0	0
$789$	21384.0	0.964880
$790$	0	0
$791$	−2440.00	−0.109679
$792$	0	0
$793$	−18900.0	−0.846354
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19410.0	−0.862657	−0.431328	−	0.902195i	$-0.641955\pi$
$797$	−19410.0	−0.862657	−0.431328	+	0.902195i	$0.641955\pi$
$798$	0	0
$799$	−5040.00	−0.223157
$800$	0	0
$801$	11466.0	0.505782
$802$	0	0
$803$	−6600.00	−0.290048
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13950.0	0.608505
$808$	0	0
$809$	34946.0	1.51871	0.759355	−	0.650677i	$-0.225515\pi$
$809$	34946.0	1.51871	0.759355	+	0.650677i	$0.225515\pi$
$810$	0	0
$811$	−28300.0	−1.22534	−0.612668	−	0.790340i	$-0.709904\pi$
$811$	−28300.0	−1.22534	−0.612668	+	0.790340i	$0.709904\pi$
$812$	0	0
$813$	9540.00	0.411540
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−74480.0	−3.18938
$818$	0	0
$819$	−2520.00	−0.107516
$820$	0	0
$821$	16930.0	0.719685	0.359842	−	0.933013i	$-0.382830\pi$
$821$	16930.0	0.719685	0.359842	+	0.933013i	$0.382830\pi$
$822$	0	0
$823$	24772.0	1.04921	0.524604	−	0.851347i	$-0.324214\pi$
$823$	24772.0	1.04921	0.524604	+	0.851347i	$0.324214\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12004.0	0.504740	0.252370	−	0.967631i	$-0.418790\pi$
$827$	12004.0	0.504740	0.252370	+	0.967631i	$0.418790\pi$
$828$	0	0
$829$	−22330.0	−0.935528	−0.467764	−	0.883853i	$-0.654940\pi$
$829$	−22330.0	−0.935528	−0.467764	+	0.883853i	$0.654940\pi$
$830$	0	0
$831$	15990.0	0.667493
$832$	0	0
$833$	29430.0	1.22412
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2700.00	0.111500
$838$	0	0
$839$	45280.0	1.86322	0.931609	−	0.363463i	$-0.118406\pi$
$839$	45280.0	1.86322	0.931609	+	0.363463i	$0.118406\pi$
$840$	0	0
$841$	−6433.00	−0.263766
$842$	0	0
$843$	−23490.0	−0.959714
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3724.00	−0.151072
$848$	0	0
$849$	−804.000	−0.0325008
$850$	0	0
$851$	32640.0	1.31479
$852$	0	0
$853$	26970.0	1.08257	0.541287	−	0.840838i	$-0.317937\pi$
$853$	26970.0	1.08257	0.541287	+	0.840838i	$0.317937\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12190.0	0.485884	0.242942	−	0.970041i	$-0.421888\pi$
$857$	12190.0	0.485884	0.242942	+	0.970041i	$0.421888\pi$
$858$	0	0
$859$	−9260.00	−0.367808	−0.183904	−	0.982944i	$-0.558874\pi$
$859$	−9260.00	−0.367808	−0.183904	+	0.982944i	$0.558874\pi$
$860$	0	0
$861$	−1320.00	−0.0522479
$862$	0	0
$863$	4832.00	0.190595	0.0952973	−	0.995449i	$-0.469620\pi$
$863$	4832.00	0.190595	0.0952973	+	0.995449i	$0.469620\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9561.00	0.374520
$868$	0	0
$869$	21200.0	0.827573
$870$	0	0
$871$	−36680.0	−1.42693
$872$	0	0
$873$	5310.00	0.205861
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37630.0	−1.44889	−0.724444	−	0.689334i	$-0.757903\pi$
$877$	−37630.0	−1.44889	−0.724444	+	0.689334i	$0.757903\pi$
$878$	0	0
$879$	5850.00	0.224477
$880$	0	0
$881$	−15630.0	−0.597716	−0.298858	−	0.954298i	$-0.596606\pi$
$881$	−15630.0	−0.597716	−0.298858	+	0.954298i	$0.596606\pi$
$882$	0	0
$883$	−12988.0	−0.494995	−0.247498	−	0.968888i	$-0.579608\pi$
$883$	−12988.0	−0.494995	−0.247498	+	0.968888i	$0.579608\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5976.00	−0.226217	−0.113108	−	0.993583i	$-0.536081\pi$
$887$	−5976.00	−0.226217	−0.113108	+	0.993583i	$0.536081\pi$
$888$	0	0
$889$	3536.00	0.133401
$890$	0	0
$891$	1620.00	0.0609114
$892$	0	0
$893$	7840.00	0.293791
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−40320.0	−1.50083
$898$	0	0
$899$	−13400.0	−0.497125
$900$	0	0
$901$	−38700.0	−1.43095
$902$	0	0
$903$	−6384.00	−0.235267
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15004.0	0.549283	0.274641	−	0.961547i	$-0.411441\pi$
$907$	15004.0	0.549283	0.274641	+	0.961547i	$0.411441\pi$
$908$	0	0
$909$	−7182.00	−0.262059
$910$	0	0
$911$	21120.0	0.768097	0.384049	−	0.923313i	$-0.374529\pi$
$911$	21120.0	0.768097	0.384049	+	0.923313i	$0.374529\pi$
$912$	0	0
$913$	23760.0	0.861272
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2000.00	0.0720238
$918$	0	0
$919$	25500.0	0.915307	0.457654	−	0.889131i	$-0.348690\pi$
$919$	25500.0	0.915307	0.457654	+	0.889131i	$0.348690\pi$
$920$	0	0
$921$	−8748.00	−0.312982
$922$	0	0
$923$	5600.00	0.199703
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13932.0	0.493621
$928$	0	0
$929$	−390.000	−0.0137734	−0.00688670	−	0.999976i	$-0.502192\pi$
$929$	−390.000	−0.0137734	−0.00688670	+	0.999976i	$0.502192\pi$
$930$	0	0
$931$	−45780.0	−1.61158
$932$	0	0
$933$	27000.0	0.947417
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−890.000	−0.0310299	−0.0155150	−	0.999880i	$-0.504939\pi$
$937$	−890.000	−0.0310299	−0.0155150	+	0.999880i	$0.504939\pi$
$938$	0	0
$939$	−26670.0	−0.926882
$940$	0	0
$941$	−3022.00	−0.104691	−0.0523456	−	0.998629i	$-0.516670\pi$
$941$	−3022.00	−0.104691	−0.0523456	+	0.998629i	$0.516670\pi$
$942$	0	0
$943$	−21120.0	−0.729334
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2916.00	−0.100060	−0.0500302	−	0.998748i	$-0.515932\pi$
$947$	−2916.00	−0.100060	−0.0500302	+	0.998748i	$0.515932\pi$
$948$	0	0
$949$	23100.0	0.790156
$950$	0	0
$951$	−3870.00	−0.131959
$952$	0	0
$953$	12990.0	0.441540	0.220770	−	0.975326i	$-0.429143\pi$
$953$	12990.0	0.441540	0.220770	+	0.975326i	$0.429143\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8040.00	−0.271574
$958$	0	0
$959$	7800.00	0.262644
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	−11916.0	−0.398741
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41844.0	−1.39153	−0.695766	−	0.718268i	$-0.744935\pi$
$967$	−41844.0	−1.39153	−0.695766	+	0.718268i	$0.744935\pi$
$968$	0	0
$969$	−37800.0	−1.25316
$970$	0	0
$971$	−43900.0	−1.45089	−0.725447	−	0.688278i	$-0.758367\pi$
$971$	−43900.0	−1.45089	−0.725447	+	0.688278i	$0.758367\pi$
$972$	0	0
$973$	880.000	0.0289944
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8630.00	0.282598	0.141299	−	0.989967i	$-0.454872\pi$
$977$	8630.00	0.282598	0.141299	+	0.989967i	$0.454872\pi$
$978$	0	0
$979$	25480.0	0.831812
$980$	0	0
$981$	4230.00	0.137669
$982$	0	0
$983$	168.000	0.00545104	0.00272552	−	0.999996i	$-0.499132\pi$
$983$	168.000	0.00545104	0.00272552	+	0.999996i	$0.499132\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	672.000	0.0216717
$988$	0	0
$989$	−102144.	−3.28412
$990$	0	0
$991$	−25580.0	−0.819955	−0.409978	−	0.912096i	$-0.634463\pi$
$991$	−25580.0	−0.819955	−0.409978	+	0.912096i	$0.634463\pi$
$992$	0	0
$993$	−15900.0	−0.508128
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47610.0	1.51236	0.756180	−	0.654363i	$-0.227063\pi$
$997$	47610.0	1.51236	0.756180	+	0.654363i	$0.227063\pi$
$998$	0	0
$999$	4590.00	0.145367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.r.1.1		1
4.3	odd	2		2400.4.a.e.1.1		1
5.4	even	2		96.4.a.c.1.1	✓	1
15.14	odd	2		288.4.a.b.1.1		1
20.19	odd	2		96.4.a.f.1.1	yes	1
40.19	odd	2		192.4.a.b.1.1		1
40.29	even	2		192.4.a.h.1.1		1
60.59	even	2		288.4.a.c.1.1		1
80.19	odd	4		768.4.d.h.385.2		2
80.29	even	4		768.4.d.i.385.1		2
80.59	odd	4		768.4.d.h.385.1		2
80.69	even	4		768.4.d.i.385.2		2
120.29	odd	2		576.4.a.s.1.1		1
120.59	even	2		576.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.4.a.c.1.1	✓	1	5.4	even	2
96.4.a.f.1.1	yes	1	20.19	odd	2
192.4.a.b.1.1		1	40.19	odd	2
192.4.a.h.1.1		1	40.29	even	2
288.4.a.b.1.1		1	15.14	odd	2
288.4.a.c.1.1		1	60.59	even	2
576.4.a.s.1.1		1	120.29	odd	2
576.4.a.t.1.1		1	120.59	even	2
768.4.d.h.385.1		2	80.59	odd	4
768.4.d.h.385.2		2	80.19	odd	4
768.4.d.i.385.1		2	80.29	even	4
768.4.d.i.385.2		2	80.69	even	4
2400.4.a.e.1.1		1	4.3	odd	2
2400.4.a.r.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more