LMFDB - Embedded newform 2400.2.m.e.1199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1199,2400)]

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([1, 1, 1, 1]))

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

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

chi := DirichletCharacter("2400.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.m (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:	$24$
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1199.3
Character	$\chi$	$=$	2400.1199
Dual form			2400.2.m.e.1199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71500 + 0.242431i) q^{3} -3.08957 q^{7} +(2.88245 - 0.831539i) q^{9} +O(q^{10})$	$q+(-1.71500 + 0.242431i) q^{3} -3.08957 q^{7} +(2.88245 - 0.831539i) q^{9} +2.54654i q^{11} -5.06696 q^{13} +0.214179 q^{17} +2.60975 q^{19} +(5.29861 - 0.749006i) q^{21} +4.47647i q^{23} +(-4.74182 + 2.12489i) q^{27} -7.86770 q^{29} -4.58758i q^{31} +(-0.617360 - 4.36732i) q^{33} +7.67714 q^{37} +(8.68984 - 1.22839i) q^{39} +9.26946i q^{41} -11.4049i q^{43} +10.5972i q^{47} +2.54541 q^{49} +(-0.367316 + 0.0519235i) q^{51} -9.51198i q^{53} +(-4.47572 + 0.632684i) q^{57} +0.428357i q^{59} +1.11217i q^{61} +(-8.90553 + 2.56909i) q^{63} +2.35998i q^{67} +(-1.08523 - 7.67714i) q^{69} +6.12075 q^{71} -12.0147i q^{73} -7.86770i q^{77} -11.6319i q^{79} +(7.61709 - 4.79374i) q^{81} +2.29913 q^{83} +(13.4931 - 1.90737i) q^{87} -12.4853i q^{89} +15.6547 q^{91} +(1.11217 + 7.86770i) q^{93} +8.04496i q^{97} +(2.11755 + 7.34028i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) $ 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

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
$3$	−1.71500	+	0.242431i	−0.990156	+	0.139968i
$3$	−1.71500	+	0.242431i	−0.990156	+	0.139968i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.08957			−1.16775			−0.583873	−	0.811845i	$-0.698463\pi$
$7$	−3.08957			−1.16775			−0.583873	+	0.811845i	$0.698463\pi$
$8$		0			0
$9$	2.88245	−	0.831539i	0.960818	−	0.277180i
$10$		0			0
$11$			2.54654i			0.767810i	0.923373	+	0.383905i	$0.125421\pi$
$11$			2.54654i			0.767810i	−0.923373	+	0.383905i	$0.874579\pi$
$12$		0			0
$13$	−5.06696			−1.40532			−0.702661	−	0.711525i	$-0.748005\pi$
$13$	−5.06696			−1.40532			−0.702661	+	0.711525i	$0.748005\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.214179			0.0519459			0.0259730	−	0.999663i	$-0.491732\pi$
$17$	0.214179			0.0519459			0.0259730	+	0.999663i	$0.491732\pi$
$18$		0			0
$19$	2.60975			0.598717			0.299359	−	0.954141i	$-0.403227\pi$
$19$	2.60975			0.598717			0.299359	+	0.954141i	$0.403227\pi$
$20$		0			0
$21$	5.29861	−	0.749006i	1.15625	−	0.163447i
$22$		0			0
$23$			4.47647i			0.933408i	0.884414	+	0.466704i	$0.154559\pi$
$23$			4.47647i			0.933408i	−0.884414	+	0.466704i	$0.845441\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.74182	+	2.12489i	−0.912564	+	0.408934i
$28$		0			0
$29$	−7.86770			−1.46100			−0.730498	−	0.682915i	$-0.760712\pi$
$29$	−7.86770			−1.46100			−0.730498	+	0.682915i	$0.760712\pi$
$30$		0			0
$31$		−	4.58758i		−	0.823953i	−0.911194	−	0.411977i	$-0.864839\pi$
$31$		−	4.58758i		−	0.823953i	0.911194	−	0.411977i	$-0.135161\pi$
$32$		0			0
$33$	−0.617360	−	4.36732i	−0.107469	−	0.760252i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.67714			1.26211			0.631057	−	0.775736i	$-0.282621\pi$
$37$	7.67714			1.26211			0.631057	+	0.775736i	$0.282621\pi$
$38$		0			0
$39$	8.68984	−	1.22839i	1.39149	−	0.196700i
$40$		0			0
$41$			9.26946i			1.44765i	0.689985	+	0.723823i	$0.257617\pi$
$41$			9.26946i			1.44765i	−0.689985	+	0.723823i	$0.742383\pi$
$42$		0			0
$43$		−	11.4049i		−	1.73924i	−0.493725	−	0.869618i	$-0.664365\pi$
$43$		−	11.4049i		−	1.73924i	0.493725	−	0.869618i	$-0.335635\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.5972i			1.54576i	0.634551	+	0.772881i	$0.281185\pi$
$47$			10.5972i			1.54576i	−0.634551	+	0.772881i	$0.718815\pi$
$48$		0			0
$49$	2.54541			0.363631
$50$		0			0
$51$	−0.367316	+	0.0519235i	−0.0514346	+	0.00727075i
$52$		0			0
$53$		−	9.51198i		−	1.30657i	−0.757112	−	0.653285i	$-0.773390\pi$
$53$		−	9.51198i		−	1.30657i	0.757112	−	0.653285i	$-0.226610\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.47572	+	0.632684i	−0.592823	+	0.0838010i
$58$		0			0
$59$			0.428357i			0.0557674i	0.999611	+	0.0278837i	$0.00887680\pi$
$59$			0.428357i			0.0557674i	−0.999611	+	0.0278837i	$0.991123\pi$
$60$		0			0
$61$			1.11217i			0.142399i	0.997462	+	0.0711995i	$0.0226827\pi$
$61$			1.11217i			0.142399i	−0.997462	+	0.0711995i	$0.977317\pi$
$62$		0			0
$63$	−8.90553	+	2.56909i	−1.12199	+	0.323675i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			2.35998i			0.288317i	0.989555	+	0.144159i	$0.0460475\pi$
$67$			2.35998i			0.288317i	−0.989555	+	0.144159i	$0.953952\pi$
$68$		0			0
$69$	−1.08523	−	7.67714i	−0.130647	−	0.924219i
$70$		0			0
$71$	6.12075			0.726399			0.363199	−	0.931711i	$-0.381684\pi$
$71$	6.12075			0.726399			0.363199	+	0.931711i	$0.381684\pi$
$72$		0			0
$73$		−	12.0147i		−	1.40621i	−0.711085	−	0.703106i	$-0.751796\pi$
$73$		−	12.0147i		−	1.40621i	0.711085	−	0.703106i	$-0.248204\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	7.86770i		−	0.896608i
$78$		0			0
$79$		−	11.6319i		−	1.30869i	−0.756194	−	0.654347i	$-0.772943\pi$
$79$		−	11.6319i		−	1.30869i	0.756194	−	0.654347i	$-0.227057\pi$
$80$		0			0
$81$	7.61709	−	4.79374i	0.846343	−	0.532638i
$82$		0			0
$83$	2.29913			0.252362			0.126181	−	0.992007i	$-0.459728\pi$
$83$	2.29913			0.252362			0.126181	+	0.992007i	$0.459728\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	13.4931	−	1.90737i	1.44661	−	0.204492i
$88$		0			0
$89$		−	12.4853i		−	1.32344i	−0.749752	−	0.661719i	$-0.769827\pi$
$89$		−	12.4853i		−	1.32344i	0.749752	−	0.661719i	$-0.230173\pi$
$90$		0			0
$91$	15.6547			1.64106
$92$		0			0
$93$	1.11217	+	7.86770i	0.115327	+	0.815842i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.04496i			0.816842i	0.912794	+	0.408421i	$0.133920\pi$
$97$			8.04496i			0.816842i	−0.912794	+	0.408421i	$0.866080\pi$
$98$		0			0
$99$	2.11755	+	7.34028i	0.212821	+	0.737726i
$100$		0			0
$101$	−1.08523			−0.107985			−0.0539924	−	0.998541i	$-0.517195\pi$
$101$	−1.08523			−0.107985			−0.0539924	+	0.998541i	$0.517195\pi$
$102$		0			0
$103$	−11.6319			−1.14613			−0.573064	−	0.819511i	$-0.694245\pi$
$103$	−11.6319			−1.14613			−0.573064	+	0.819511i	$0.694245\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.50874			0.629224			0.314612	−	0.949220i	$-0.398126\pi$
$107$	6.50874			0.629224			0.314612	+	0.949220i	$0.398126\pi$
$108$		0			0
$109$		−	5.06696i		−	0.485327i	−0.970111	−	0.242663i	$-0.921979\pi$
$109$		−	5.06696i		−	0.485327i	0.970111	−	0.242663i	$-0.0780210\pi$
$110$		0			0
$111$	−13.1663	+	1.86118i	−1.24969	+	0.176655i
$112$		0			0
$113$	6.05364			0.569479			0.284739	−	0.958605i	$-0.408093\pi$
$113$	6.05364			0.569479			0.284739	+	0.958605i	$0.408093\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.6053	+	4.21337i	−1.35026	+	0.389526i
$118$		0			0
$119$	−0.661719			−0.0606597
$120$		0			0
$121$	4.51514			0.410467
$122$		0			0
$123$	−2.24720	−	15.8971i	−0.202624	−	1.43340i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.958763			0.0850765			0.0425382	−	0.999095i	$-0.486456\pi$
$127$	0.958763			0.0850765			0.0425382	+	0.999095i	$0.486456\pi$
$128$		0			0
$129$	2.76491	+	19.5595i	0.243437	+	1.72211i
$130$		0			0
$131$		−	3.78126i		−	0.330370i	−0.986263	−	0.165185i	$-0.947178\pi$
$131$		−	3.78126i		−	0.330370i	0.986263	−	0.165185i	$-0.0528221\pi$
$132$		0			0
$133$	−8.06299			−0.699149
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.1878			1.12671			0.563355	−	0.826215i	$-0.309510\pi$
$137$	13.1878			1.12671			0.563355	+	0.826215i	$0.309510\pi$
$138$		0			0
$139$	17.2947			1.46692			0.733460	−	0.679733i	$-0.237904\pi$
$139$	17.2947			1.46692			0.733460	+	0.679733i	$0.237904\pi$
$140$		0			0
$141$	−2.56909	−	18.1742i	−0.216357	−	1.53055i
$142$		0			0
$143$		−	12.9032i		−	1.07902i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.36539	+	0.617087i	−0.360051	+	0.0508965i
$148$		0			0
$149$	3.81475			0.312516			0.156258	−	0.987716i	$-0.450057\pi$
$149$	3.81475			0.312516			0.156258	+	0.987716i	$0.450057\pi$
$150$		0			0
$151$		−	13.2235i		−	1.07611i	−0.842909	−	0.538056i	$-0.819159\pi$
$151$		−	13.2235i		−	1.07611i	0.842909	−	0.538056i	$-0.180841\pi$
$152$		0			0
$153$	0.617360	−	0.178098i	0.0499106	−	0.0143983i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.56497			0.523942			0.261971	−	0.965076i	$-0.415628\pi$
$157$	6.56497			0.523942			0.261971	+	0.965076i	$0.415628\pi$
$158$		0			0
$159$	2.30600	+	16.3131i	0.182878	+	1.29371i
$160$		0			0
$161$		−	13.8303i		−	1.08998i
$162$		0			0
$163$			8.13957i			0.637540i	0.947832	+	0.318770i	$0.103270\pi$
$163$			8.13957i			0.637540i	−0.947832	+	0.318770i	$0.896730\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.8561i		−	1.69128i	−0.533754	−	0.845640i	$-0.679219\pi$
$167$		−	21.8561i		−	1.69128i	0.533754	−	0.845640i	$-0.320781\pi$
$168$		0			0
$169$	12.6741			0.974929
$170$		0			0
$171$	7.52248	−	2.17011i	0.575258	−	0.165952i
$172$		0			0
$173$		−	9.51198i		−	0.723182i	−0.932337	−	0.361591i	$-0.882234\pi$
$173$		−	9.51198i		−	0.723182i	0.932337	−	0.361591i	$-0.117766\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.103847	−	0.734633i	−0.00780562	−	0.0552184i
$178$		0			0
$179$		−	16.2398i		−	1.21382i	−0.794771	−	0.606910i	$-0.792409\pi$
$179$		−	16.2398i		−	1.21382i	0.794771	−	0.606910i	$-0.207591\pi$
$180$		0			0
$181$		−	9.74808i		−	0.724569i	−0.932068	−	0.362284i	$-0.881997\pi$
$181$		−	9.74808i		−	0.724569i	0.932068	−	0.362284i	$-0.118003\pi$
$182$		0			0
$183$	−0.269625	−	1.90737i	−0.0199312	−	0.140997i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			0.545414i			0.0398846i
$188$		0			0
$189$	14.6502	−	6.56497i	1.06564	−	0.477531i
$190$		0			0
$191$	2.30600			0.166856			0.0834281	−	0.996514i	$-0.473413\pi$
$191$	2.30600			0.166856			0.0834281	+	0.996514i	$0.473413\pi$
$192$		0			0
$193$		−	11.2498i		−	0.809776i	−0.914366	−	0.404888i	$-0.867310\pi$
$193$		−	11.2498i		−	0.809776i	0.914366	−	0.404888i	$-0.132690\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.78247i		−	0.483231i	−0.970372	−	0.241615i	$-0.922323\pi$
$197$		−	6.78247i		−	0.483231i	0.970372	−	0.241615i	$-0.0776772\pi$
$198$		0			0
$199$		−	0.632789i		−	0.0448573i	−0.999748	−	0.0224286i	$-0.992860\pi$
$199$		−	0.632789i		−	0.0448573i	0.999748	−	0.0224286i	$-0.00713985\pi$
$200$		0			0
$201$	−0.572131	−	4.04736i	−0.0403550	−	0.285479i
$202$		0			0
$203$	24.3078			1.70607
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.72235	+	12.9032i	0.258722	+	0.896835i
$208$		0			0
$209$			6.64582i			0.459701i
$210$		0			0
$211$	−11.6400			−0.801332			−0.400666	−	0.916224i	$-0.631221\pi$
$211$	−11.6400			−0.801332			−0.400666	+	0.916224i	$0.631221\pi$
$212$		0			0
$213$	−10.4971	+	1.48386i	−0.719248	+	0.101672i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			14.1736i			0.962168i
$218$		0			0
$219$	2.91273	+	20.6052i	0.196824	+	1.39237i
$220$		0			0
$221$	−1.08523			−0.0730008
$222$		0			0
$223$	10.7667			0.720992			0.360496	−	0.932761i	$-0.382607\pi$
$223$	10.7667			0.720992			0.360496	+	0.932761i	$0.382607\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.8365			0.851991			0.425996	−	0.904725i	$-0.359924\pi$
$227$	12.8365			0.851991			0.425996	+	0.904725i	$0.359924\pi$
$228$		0			0
$229$			14.9684i			0.989143i	0.869137	+	0.494571i	$0.164675\pi$
$229$			14.9684i			0.989143i	−0.869137	+	0.494571i	$0.835325\pi$
$230$		0			0
$231$	1.90737	+	13.4931i	0.125496	+	0.887781i
$232$		0			0
$233$	20.8980			1.36908			0.684538	−	0.728977i	$-0.260004\pi$
$233$	20.8980			1.36908			0.684538	+	0.728977i	$0.260004\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.81994	+	19.9488i	0.183175	+	1.29581i
$238$		0			0
$239$	−28.6386			−1.85248			−0.926239	−	0.376937i	$-0.876977\pi$
$239$	−28.6386			−1.85248			−0.926239	+	0.376937i	$0.876977\pi$
$240$		0			0
$241$	9.24977			0.595830			0.297915	−	0.954592i	$-0.403709\pi$
$241$	9.24977			0.595830			0.297915	+	0.954592i	$0.403709\pi$
$242$		0			0
$243$	−11.9012	+	10.0679i	−0.763460	+	0.645856i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−13.2235			−0.841390
$248$		0			0
$249$	−3.94301	+	0.557380i	−0.249878	+	0.0353225i
$250$		0			0
$251$			1.23472i			0.0779348i	0.999240	+	0.0389674i	$0.0124069\pi$
$251$			1.23472i			0.0779348i	−0.999240	+	0.0389674i	$0.987593\pi$
$252$		0			0
$253$	−11.3995			−0.716680
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.6054			−1.16057			−0.580286	−	0.814413i	$-0.697059\pi$
$257$	−18.6054			−1.16057			−0.580286	+	0.814413i	$0.697059\pi$
$258$		0			0
$259$	−23.7190			−1.47383
$260$		0			0
$261$	−22.6783	+	6.54230i	−1.40375	+	0.404958i
$262$		0			0
$263$		−	11.2589i		−	0.694255i	−0.937818	−	0.347128i	$-0.887157\pi$
$263$		−	11.2589i		−	0.694255i	0.937818	−	0.347128i	$-0.112843\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	3.02682	+	21.4123i	0.185238	+	1.31041i
$268$		0			0
$269$	11.6824			0.712291			0.356146	−	0.934430i	$-0.384091\pi$
$269$	11.6824			0.712291			0.356146	+	0.934430i	$0.384091\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−26.8478	+	3.79518i	−1.62490	+	0.229695i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	17.4252			1.04698			0.523490	−	0.852032i	$-0.324630\pi$
$277$	17.4252			1.04698			0.523490	+	0.852032i	$0.324630\pi$
$278$		0			0
$279$	−3.81475	−	13.2235i	−0.228383	−	0.791669i
$280$		0			0
$281$			12.9736i			0.773941i	0.922092	+	0.386971i	$0.126479\pi$
$281$			12.9736i			0.773941i	−0.922092	+	0.386971i	$0.873521\pi$
$282$		0			0
$283$		−	12.3893i		−	0.736470i	−0.929733	−	0.368235i	$-0.879962\pi$
$283$		−	12.3893i		−	0.736470i	0.929733	−	0.368235i	$-0.120038\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	28.6386i		−	1.69048i
$288$		0			0
$289$	−16.9541			−0.997302
$290$		0			0
$291$	−1.95035	−	13.7971i	−0.114331	−	0.808801i
$292$		0			0
$293$			31.7916i			1.85729i	0.370973	+	0.928644i	$0.379024\pi$
$293$			31.7916i			1.85729i	−0.370973	+	0.928644i	$0.620976\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−5.41110	−	12.0752i	−0.313984	−	0.700676i
$298$		0			0
$299$		−	22.6821i		−	1.31174i
$300$		0			0
$301$			35.2363i			2.03099i
$302$		0			0
$303$	1.86118	−	0.263094i	0.106922	−	0.0151144i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	10.3288i		−	0.589495i	−0.955575	−	0.294747i	$-0.904764\pi$
$307$		−	10.3288i		−	0.589495i	0.955575	−	0.294747i	$-0.0952355\pi$
$308$		0			0
$309$	19.9488	−	2.81994i	1.13485	−	0.160421i
$310$		0			0
$311$	5.13819			0.291360			0.145680	−	0.989332i	$-0.453463\pi$
$311$	5.13819			0.291360			0.145680	+	0.989332i	$0.453463\pi$
$312$		0			0
$313$			15.6741i			0.885951i	0.896534	+	0.442976i	$0.146077\pi$
$313$			15.6741i			0.885951i	−0.896534	+	0.442976i	$0.853923\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			8.95293i			0.502847i	0.967877	+	0.251423i	$0.0808987\pi$
$317$			8.95293i			0.502847i	−0.967877	+	0.251423i	$0.919101\pi$
$318$		0			0
$319$		−	20.0354i		−	1.12177i
$320$		0			0
$321$	−11.1625	+	1.57792i	−0.623030	+	0.0880710i
$322$		0			0
$323$	0.558952			0.0311009
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.22839	+	8.68984i	0.0679300	+	0.480549i
$328$		0			0
$329$		−	32.7408i		−	1.80506i
$330$		0			0
$331$	4.48486			0.246510			0.123255	−	0.992375i	$-0.460667\pi$
$331$	4.48486			0.246510			0.123255	+	0.992375i	$0.460667\pi$
$332$		0			0
$333$	22.1290	−	6.38384i	1.21266	−	0.349832i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	25.9991i		−	1.41626i	−0.706082	−	0.708130i	$-0.749539\pi$
$337$		−	25.9991i		−	1.41626i	0.706082	−	0.708130i	$-0.250461\pi$
$338$		0			0
$339$	−10.3820	+	1.46759i	−0.563873	+	0.0797085i
$340$		0			0
$341$	11.6824			0.632640
$342$		0			0
$343$	13.7627			0.743118
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.7184			−0.575392			−0.287696	−	0.957722i	$-0.592889\pi$
$347$	−10.7184			−0.575392			−0.287696	+	0.957722i	$0.592889\pi$
$348$		0			0
$349$			11.6319i			0.622643i	0.950305	+	0.311322i	$0.100772\pi$
$349$			11.6319i			0.622643i	−0.950305	+	0.311322i	$0.899228\pi$
$350$		0			0
$351$	24.0266	−	10.7667i	1.28245	−	0.574684i
$352$		0			0
$353$	−21.7547			−1.15789			−0.578944	−	0.815367i	$-0.696535\pi$
$353$	−21.7547			−1.15789			−0.578944	+	0.815367i	$0.696535\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.13485	−	0.160421i	0.0600625	−	0.00849039i
$358$		0			0
$359$	−12.9032			−0.681005			−0.340503	−	0.940244i	$-0.610597\pi$
$359$	−12.9032			−0.681005			−0.340503	+	0.940244i	$0.610597\pi$
$360$		0			0
$361$	−12.1892			−0.641538
$362$		0			0
$363$	−7.74346	+	1.09461i	−0.406426	+	0.0574521i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.7255			−0.612065			−0.306032	−	0.952021i	$-0.599002\pi$
$367$	−11.7255			−0.612065			−0.306032	+	0.952021i	$0.599002\pi$
$368$		0			0
$369$	7.70792	+	26.7188i	0.401258	+	1.39093i
$370$		0			0
$371$			29.3879i			1.52574i
$372$		0			0
$373$	31.0944			1.61001			0.805004	−	0.593270i	$-0.202163\pi$
$373$	31.0944			1.61001			0.805004	+	0.593270i	$0.202163\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	39.8653			2.05317
$378$		0			0
$379$	12.6097			0.647719			0.323860	−	0.946105i	$-0.395019\pi$
$379$	12.6097			0.647719			0.323860	+	0.946105i	$0.395019\pi$
$380$		0			0
$381$	−1.64428	+	0.232434i	−0.0842390	+	0.0119080i
$382$		0			0
$383$			1.64428i			0.0840188i	0.999117	+	0.0420094i	$0.0133759\pi$
$383$			1.64428i			0.0840188i	−0.999117	+	0.0420094i	$0.986624\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.48364	−	32.8742i	−0.482081	−	1.67109i
$388$		0			0
$389$	−37.4889			−1.90076			−0.950381	−	0.311090i	$-0.899306\pi$
$389$	−37.4889			−1.90076			−0.950381	+	0.311090i	$0.899306\pi$
$390$		0			0
$391$			0.958763i			0.0484867i
$392$		0			0
$393$	0.916694	+	6.48486i	0.0462411	+	0.327118i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.8820			−0.997849			−0.498924	−	0.866646i	$-0.666271\pi$
$397$	−19.8820			−0.997849			−0.498924	+	0.866646i	$0.666271\pi$
$398$		0			0
$399$	13.8280	−	1.95472i	0.692267	−	0.0978583i
$400$		0			0
$401$			11.2570i			0.562150i	0.959686	+	0.281075i	$0.0906910\pi$
$401$			11.2570i			0.562150i	−0.959686	+	0.281075i	$0.909309\pi$
$402$		0			0
$403$			23.2451i			1.15792i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			19.5501i			0.969065i
$408$		0			0
$409$	−16.9007			−0.835685			−0.417843	−	0.908519i	$-0.637214\pi$
$409$	−16.9007			−0.835685			−0.417843	+	0.908519i	$0.637214\pi$
$410$		0			0
$411$	−22.6171	+	3.19713i	−1.11562	+	0.157703i
$412$		0			0
$413$		−	1.32344i		−	0.0651221i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−29.6605	+	4.19278i	−1.45248	+	0.205321i
$418$		0			0
$419$		−	7.21126i		−	0.352293i	−0.984364	−	0.176147i	$-0.943637\pi$
$419$		−	7.21126i		−	0.352293i	0.984364	−	0.176147i	$-0.0563633\pi$
$420$		0			0
$421$		−	11.3995i		−	0.555578i	−0.960642	−	0.277789i	$-0.910398\pi$
$421$		−	11.3995i		−	0.555578i	0.960642	−	0.277789i	$-0.0896015\pi$
$422$		0			0
$423$	8.81199	+	30.5460i	0.428454	+	1.48520i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	3.43613i		−	0.166286i
$428$		0			0
$429$	3.12814	+	22.1290i	0.151028	+	1.06840i
$430$		0			0
$431$	−3.95028			−0.190278			−0.0951391	−	0.995464i	$-0.530330\pi$
$431$	−3.95028			−0.190278			−0.0951391	+	0.995464i	$0.530330\pi$
$432$		0			0
$433$			20.6509i			0.992420i	0.868203	+	0.496210i	$0.165275\pi$
$433$			20.6509i			0.992420i	−0.868203	+	0.496210i	$0.834725\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			11.6824i			0.558847i
$438$		0			0
$439$			28.5778i			1.36394i	0.731379	+	0.681971i	$0.238877\pi$
$439$			28.5778i			1.36394i	−0.731379	+	0.681971i	$0.761123\pi$
$440$		0			0
$441$	7.33704	−	2.11661i	0.349383	−	0.100791i
$442$		0			0
$443$	−21.9689			−1.04378			−0.521888	−	0.853014i	$-0.674772\pi$
$443$	−21.9689			−1.04378			−0.521888	+	0.853014i	$0.674772\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−6.54230	+	0.924813i	−0.309440	+	0.0437422i
$448$		0			0
$449$			9.00493i			0.424969i	0.977164	+	0.212485i	$0.0681555\pi$
$449$			9.00493i			0.424969i	−0.977164	+	0.212485i	$0.931844\pi$
$450$		0			0
$451$	−23.6050			−1.11152
$452$		0			0
$453$	3.20578	+	22.6783i	0.150621	+	1.06552i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	18.5748i		−	0.868891i	−0.900698	−	0.434446i	$-0.856944\pi$
$457$		−	18.5748i		−	0.868891i	0.900698	−	0.434446i	$-0.143056\pi$
$458$		0			0
$459$	−1.01560	+	0.455105i	−0.0474040	+	0.0212425i
$460$		0			0
$461$	34.2003			1.59287			0.796434	−	0.604726i	$-0.206717\pi$
$461$	34.2003			1.59287			0.796434	+	0.604726i	$0.206717\pi$
$462$		0			0
$463$	−7.44471			−0.345985			−0.172992	−	0.984923i	$-0.555344\pi$
$463$	−7.44471			−0.345985			−0.172992	+	0.984923i	$0.555344\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.26161			0.336027			0.168014	−	0.985785i	$-0.446265\pi$
$467$	7.26161			0.336027			0.168014	+	0.985785i	$0.446265\pi$
$468$		0			0
$469$		−	7.29130i		−	0.336681i
$470$		0			0
$471$	−11.2589	+	1.59155i	−0.518784	+	0.0733349i
$472$		0			0
$473$	29.0431			1.33540
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−7.90958	−	27.4178i	−0.362155	−	1.25538i
$478$		0			0
$479$	23.3649			1.06757			0.533785	−	0.845620i	$-0.320769\pi$
$479$	23.3649			1.06757			0.533785	+	0.845620i	$0.320769\pi$
$480$		0			0
$481$	−38.8998			−1.77368
$482$		0			0
$483$	3.35290	+	23.7190i	0.152562	+	1.07925i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	7.77068			0.352123			0.176062	−	0.984379i	$-0.443664\pi$
$487$	7.77068			0.352123			0.176062	+	0.984379i	$0.443664\pi$
$488$		0			0
$489$	−1.97328	−	13.9594i	−0.0892349	−	0.631264i
$490$		0			0
$491$			20.5265i			0.926349i	0.886267	+	0.463174i	$0.153290\pi$
$491$			20.5265i			0.926349i	−0.886267	+	0.463174i	$0.846710\pi$
$492$		0			0
$493$	−1.68509			−0.0758928
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.9104			−0.848249
$498$		0			0
$499$	−26.4958			−1.18611			−0.593057	−	0.805161i	$-0.702079\pi$
$499$	−26.4958			−1.18611			−0.593057	+	0.805161i	$0.702079\pi$
$500$		0			0
$501$	5.29861	+	37.4833i	0.236724	+	1.67463i
$502$		0			0
$503$		−	26.9943i		−	1.20362i	−0.798640	−	0.601809i	$-0.794447\pi$
$503$		−	26.9943i		−	1.20362i	0.798640	−	0.601809i	$-0.205553\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.7361	+	3.07259i	−0.965332	+	0.136459i
$508$		0			0
$509$	−25.8064			−1.14385			−0.571925	−	0.820306i	$-0.693803\pi$
$509$	−25.8064			−1.14385			−0.571925	+	0.820306i	$0.693803\pi$
$510$		0			0
$511$			37.1201i			1.64210i
$512$		0			0
$513$	−12.3750	+	5.54541i	−0.546368	+	0.244836i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−26.9862			−1.18685
$518$		0			0
$519$	2.30600	+	16.3131i	0.101222	+	0.716063i
$520$		0			0
$521$		−	1.70694i		−	0.0747826i	−0.999301	−	0.0373913i	$-0.988095\pi$
$521$		−	1.70694i		−	0.0747826i	0.999301	−	0.0373913i	$-0.0119048\pi$
$522$		0			0
$523$			24.0790i			1.05290i	0.850206	+	0.526451i	$0.176478\pi$
$523$			24.0790i			1.05290i	−0.850206	+	0.526451i	$0.823522\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	0.982561i		−	0.0428010i
$528$		0			0
$529$	2.96125			0.128750
$530$		0			0
$531$	0.356195	+	1.23472i	0.0154576	+	0.0535823i
$532$		0			0
$533$		−	46.9680i		−	2.03441i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.93703	+	27.8513i	0.169895	+	1.20187i
$538$		0			0
$539$			6.48200i			0.279199i
$540$		0			0
$541$		−	16.6989i		−	0.717941i	−0.933349	−	0.358971i	$-0.883128\pi$
$541$		−	16.6989i		−	0.717941i	0.933349	−	0.358971i	$-0.116872\pi$
$542$		0			0
$543$	2.36324	+	16.7180i	0.101416	+	0.717436i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			15.5833i			0.666292i	0.942875	+	0.333146i	$0.108110\pi$
$547$			15.5833i			0.666292i	−0.942875	+	0.333146i	$0.891890\pi$
$548$		0			0
$549$	0.924813	+	3.20578i	0.0394701	+	0.136819i
$550$		0			0
$551$	−20.5327			−0.874723
$552$		0			0
$553$			35.9376i			1.52822i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	2.96772i		−	0.125746i	−0.998022	−	0.0628731i	$-0.979974\pi$
$557$		−	2.96772i		−	0.125746i	0.998022	−	0.0628731i	$-0.0200263\pi$
$558$		0			0
$559$			57.7883i			2.44419i
$560$		0			0
$561$	−0.132225	−	0.935386i	−0.00558256	−	0.0394920i
$562$		0			0
$563$	−11.7057			−0.493335			−0.246668	−	0.969100i	$-0.579336\pi$
$563$	−11.7057			−0.493335			−0.246668	+	0.969100i	$0.579336\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.5335	+	14.8106i	−0.988314	+	0.621986i
$568$		0			0
$569$		−	43.3618i		−	1.81782i	−0.416992	−	0.908910i	$-0.636916\pi$
$569$		−	43.3618i		−	1.81782i	0.416992	−	0.908910i	$-0.363084\pi$
$570$		0			0
$571$	8.18544			0.342550			0.171275	−	0.985223i	$-0.445211\pi$
$571$	8.18544			0.342550			0.171275	+	0.985223i	$0.445211\pi$
$572$		0			0
$573$	−3.95479	+	0.559045i	−0.165214	+	0.0233545i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			18.9612i			0.789367i	0.918817	+	0.394683i	$0.129146\pi$
$577$			18.9612i			0.789367i	−0.918817	+	0.394683i	$0.870854\pi$
$578$		0			0
$579$	2.72729	+	19.2934i	0.113342	+	0.801805i
$580$		0			0
$581$	−7.10331			−0.294695
$582$		0			0
$583$	24.2226			1.00320
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.9546			0.823614			0.411807	−	0.911271i	$-0.364898\pi$
$587$	19.9546			0.823614			0.411807	+	0.911271i	$0.364898\pi$
$588$		0			0
$589$		−	11.9724i		−	0.493315i
$590$		0			0
$591$	1.64428	+	11.6319i	0.0676366	+	0.478474i
$592$		0			0
$593$	−27.4465			−1.12709			−0.563546	−	0.826085i	$-0.690563\pi$
$593$	−27.4465			−1.12709			−0.563546	+	0.826085i	$0.690563\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.153408	+	1.08523i	0.00627856	+	0.0444157i
$598$		0			0
$599$	−13.8858			−0.567357			−0.283679	−	0.958919i	$-0.591555\pi$
$599$	−13.8858			−0.567357			−0.283679	+	0.958919i	$0.591555\pi$
$600$		0			0
$601$	−10.7502			−0.438511			−0.219255	−	0.975667i	$-0.570363\pi$
$601$	−10.7502			−0.438511			−0.219255	+	0.975667i	$0.570363\pi$
$602$		0			0
$603$	1.96241	+	6.80252i	0.0799156	+	0.277020i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	30.7086			1.24642			0.623211	−	0.782054i	$-0.285828\pi$
$607$	30.7086			1.24642			0.623211	+	0.782054i	$0.285828\pi$
$608$		0			0
$609$	−41.6878	+	5.89296i	−1.68928	+	0.238795i
$610$		0			0
$611$		−	53.6957i		−	2.17229i
$612$		0			0
$613$	13.3170			0.537870			0.268935	−	0.963158i	$-0.413328\pi$
$613$	13.3170			0.537870			0.268935	+	0.963158i	$0.413328\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.3725			1.54482			0.772410	−	0.635124i	$-0.219051\pi$
$617$	38.3725			1.54482			0.772410	+	0.635124i	$0.219051\pi$
$618$		0			0
$619$	4.59507			0.184691			0.0923457	−	0.995727i	$-0.470564\pi$
$619$	4.59507			0.184691			0.0923457	+	0.995727i	$0.470564\pi$
$620$		0			0
$621$	−9.51198	−	21.2266i	−0.381703	−	0.851794i
$622$		0			0
$623$			38.5741i			1.54544i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.61115	−	11.3976i	−0.0643433	−	0.455176i
$628$		0			0
$629$	1.64428			0.0655617
$630$		0			0
$631$			40.2097i			1.60072i	0.599518	+	0.800362i	$0.295359\pi$
$631$			40.2097i			1.60072i	−0.599518	+	0.800362i	$0.704641\pi$
$632$		0			0
$633$	19.9626	−	2.82190i	0.793444	−	0.112161i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−12.8975			−0.511018
$638$		0			0
$639$	17.6428	−	5.08964i	0.697937	−	0.201343i
$640$		0			0
$641$		−	7.56252i		−	0.298702i	−0.988784	−	0.149351i	$-0.952282\pi$
$641$		−	7.56252i		−	0.298702i	0.988784	−	0.149351i	$-0.0477184\pi$
$642$		0			0
$643$			2.47018i			0.0974145i	0.998813	+	0.0487072i	$0.0155101\pi$
$643$			2.47018i			0.0974145i	−0.998813	+	0.0487072i	$0.984490\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.1474i			1.18522i	0.805491	+	0.592608i	$0.201901\pi$
$647$			30.1474i			1.18522i	−0.805491	+	0.592608i	$0.798099\pi$
$648$		0			0
$649$	−1.09083			−0.0428188
$650$		0			0
$651$	−3.43613	−	24.3078i	−0.134672	−	0.952697i
$652$		0			0
$653$			8.42674i			0.329764i	0.986313	+	0.164882i	$0.0527243\pi$
$653$			8.42674i			0.329764i	−0.986313	+	0.164882i	$0.947276\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−9.99067	−	34.6318i	−0.389773	−	1.35111i
$658$		0			0
$659$		−	13.5764i		−	0.528863i	−0.964404	−	0.264432i	$-0.914816\pi$
$659$		−	13.5764i		−	0.528863i	0.964404	−	0.264432i	$-0.0851843\pi$
$660$		0			0
$661$		−	1.68509i		−	0.0655425i	−0.999463	−	0.0327713i	$-0.989567\pi$
$661$		−	1.68509i		−	0.0655425i	0.999463	−	0.0327713i	$-0.0104333\pi$
$662$		0			0
$663$	1.86118	−	0.263094i	0.0722821	−	0.0102177i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	35.2195i		−	1.36370i
$668$		0			0
$669$	−18.4649	+	2.61018i	−0.713895	+	0.100916i
$670$		0			0
$671$	−2.83219			−0.109335
$672$		0			0
$673$		−	7.03784i		−	0.271289i	−0.990758	−	0.135644i	$-0.956690\pi$
$673$		−	7.03784i		−	0.271289i	0.990758	−	0.135644i	$-0.0433105\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	41.8298i		−	1.60765i	−0.594866	−	0.803825i	$-0.702795\pi$
$677$		−	41.8298i		−	1.60765i	0.594866	−	0.803825i	$-0.297205\pi$
$678$		0			0
$679$		−	24.8554i		−	0.953863i
$680$		0			0
$681$	−22.0147	+	3.11198i	−0.843604	+	0.119251i
$682$		0			0
$683$	28.2464			1.08082			0.540409	−	0.841403i	$-0.318270\pi$
$683$	28.2464			1.08082			0.540409	+	0.841403i	$0.318270\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−3.62881	−	25.6709i	−0.138448	−	0.979406i
$688$		0			0
$689$			48.1968i			1.83615i
$690$		0			0
$691$	−24.8633			−0.945844			−0.472922	−	0.881104i	$-0.656801\pi$
$691$	−24.8633			−0.945844			−0.472922	+	0.881104i	$0.656801\pi$
$692$		0			0
$693$	−6.54230	−	22.6783i	−0.248521	−	0.861477i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			1.98532i			0.0751994i
$698$		0			0
$699$	−35.8401	+	5.06633i	−1.35560	+	0.191626i
$700$		0			0
$701$	−42.6271			−1.61000			−0.805001	−	0.593274i	$-0.797835\pi$
$701$	−42.6271			−1.61000			−0.805001	+	0.593274i	$0.797835\pi$
$702$		0			0
$703$	20.0354			0.755650
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.35290			0.126099
$708$		0			0
$709$		−	17.0057i		−	0.638663i	−0.947643	−	0.319331i	$-0.896542\pi$
$709$		−	17.0057i		−	0.638663i	0.947643	−	0.319331i	$-0.103458\pi$
$710$		0			0
$711$	−9.67240	−	33.5285i	−0.362743	−	1.25742i
$712$		0			0
$713$	20.5361			0.769085
$714$		0			0
$715$		0			0
$716$		0			0
$717$	49.1152	−	6.94289i	1.83424	−	0.259287i
$718$		0			0
$719$	28.6386			1.06804			0.534020	−	0.845472i	$-0.320681\pi$
$719$	28.6386			1.06804			0.534020	+	0.845472i	$0.320681\pi$
$720$		0			0
$721$	35.9376			1.33839
$722$		0			0
$723$	−15.8634	+	2.24243i	−0.589965	+	0.0833969i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	14.1822			0.525990			0.262995	−	0.964797i	$-0.415290\pi$
$727$	14.1822			0.525990			0.262995	+	0.964797i	$0.415290\pi$
$728$		0			0
$729$	17.9697	−	20.1517i	0.665545	−	0.746357i
$730$		0			0
$731$		−	2.44269i		−	0.0903462i
$732$		0			0
$733$	51.7027			1.90968			0.954842	−	0.297113i	$-0.0960240\pi$
$733$	51.7027			1.90968			0.954842	+	0.297113i	$0.0960240\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.00977			−0.221373
$738$		0			0
$739$	−16.7493			−0.616133			−0.308067	−	0.951365i	$-0.599682\pi$
$739$	−16.7493			−0.616133			−0.308067	+	0.951365i	$0.599682\pi$
$740$		0			0
$741$	22.6783	−	3.20578i	0.833108	−	0.117767i
$742$		0			0
$743$			13.5649i			0.497649i	0.968549	+	0.248825i	$0.0800442\pi$
$743$			13.5649i			0.497649i	−0.968549	+	0.248825i	$0.919956\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.62713	−	1.91181i	0.242474	−	0.0699496i
$748$		0			0
$749$	−20.1092			−0.734774
$750$		0			0
$751$		−	37.3072i		−	1.36136i	−0.732581	−	0.680680i	$-0.761684\pi$
$751$		−	37.3072i		−	1.36136i	0.732581	−	0.680680i	$-0.238316\pi$
$752$		0			0
$753$	−0.299334	−	2.11755i	−0.0109084	−	0.0771676i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.385842			−0.0140237			−0.00701183	−	0.999975i	$-0.502232\pi$
$757$	−0.385842			−0.0140237			−0.00701183	+	0.999975i	$0.502232\pi$
$758$		0			0
$759$	19.5501	−	2.76359i	0.709625	−	0.100312i
$760$		0			0
$761$			10.2235i			0.370603i	0.982682	+	0.185302i	$0.0593262\pi$
$761$			10.2235i			0.370603i	−0.982682	+	0.185302i	$0.940674\pi$
$762$		0			0
$763$			15.6547i			0.566738i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2.17047i		−	0.0783711i
$768$		0			0
$769$	−4.34816			−0.156799			−0.0783994	−	0.996922i	$-0.524981\pi$
$769$	−4.34816			−0.156799			−0.0783994	+	0.996922i	$0.524981\pi$
$770$		0			0
$771$	31.9083	−	4.51052i	1.14915	−	0.162443i
$772$		0			0
$773$		−	21.4326i		−	0.770878i	−0.922733	−	0.385439i	$-0.874050\pi$
$773$		−	21.4326i		−	0.770878i	0.922733	−	0.385439i	$-0.125950\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	40.6782	−	5.75023i	1.45932	−	0.206288i
$778$		0			0
$779$			24.1910i			0.866731i
$780$		0			0
$781$			15.5867i			0.557737i
$782$		0			0
$783$	37.3072	−	16.7180i	1.33325	−	0.597451i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			32.2753i			1.15049i	0.817980	+	0.575246i	$0.195094\pi$
$787$			32.2753i			1.15049i	−0.817980	+	0.575246i	$0.804906\pi$
$788$		0			0
$789$	2.72951	+	19.3091i	0.0971733	+	0.687421i
$790$		0			0
$791$	−18.7031			−0.665006
$792$		0			0
$793$		−	5.63533i		−	0.200116i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	14.4120i		−	0.510498i	−0.966875	−	0.255249i	$-0.917843\pi$
$797$		−	14.4120i		−	0.510498i	0.966875	−	0.255249i	$-0.0821574\pi$
$798$		0			0
$799$			2.26970i			0.0802961i
$800$		0			0
$801$	−10.3820	−	35.9883i	−0.366830	−	1.27158i
$802$		0			0
$803$	30.5959			1.07970
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−20.0354	+	2.83219i	−0.705280	+	0.0996977i
$808$		0			0
$809$		−	27.3297i		−	0.960860i	−0.877033	−	0.480430i	$-0.840481\pi$
$809$		−	27.3297i		−	0.960860i	0.877033	−	0.480430i	$-0.159519\pi$
$810$		0			0
$811$	−1.18452			−0.0415942			−0.0207971	−	0.999784i	$-0.506620\pi$
$811$	−1.18452			−0.0415942			−0.0207971	+	0.999784i	$0.506620\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	29.7640i		−	1.04131i
$818$		0			0
$819$	45.1240	−	13.0175i	1.57676	−	0.454868i
$820$		0			0
$821$	16.8535			0.588191			0.294095	−	0.955776i	$-0.404982\pi$
$821$	16.8535			0.588191			0.294095	+	0.955776i	$0.404982\pi$
$822$		0			0
$823$	−47.6544			−1.66113			−0.830564	−	0.556923i	$-0.811982\pi$
$823$	−47.6544			−1.66113			−0.830564	+	0.556923i	$0.811982\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.4476			0.398073			0.199036	−	0.979992i	$-0.436219\pi$
$827$	11.4476			0.398073			0.199036	+	0.979992i	$0.436219\pi$
$828$		0			0
$829$		−	13.6692i		−	0.474751i	−0.971418	−	0.237375i	$-0.923713\pi$
$829$		−	13.6692i		−	0.474751i	0.971418	−	0.237375i	$-0.0762871\pi$
$830$		0			0
$831$	−29.8843	+	4.22441i	−1.03667	+	0.146543i
$832$		0			0
$833$	0.545173			0.0188891
$834$		0			0
$835$		0			0
$836$		0			0
$837$	9.74808	+	21.7535i	0.336943	+	0.751910i
$838$		0			0
$839$	16.7180			0.577168			0.288584	−	0.957455i	$-0.406816\pi$
$839$	16.7180			0.577168			0.288584	+	0.957455i	$0.406816\pi$
$840$		0			0
$841$	32.9007			1.13451
$842$		0			0
$843$	−3.14521	−	22.2498i	−0.108327	−	0.766323i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−13.9498			−0.479321
$848$		0			0
$849$	3.00356	+	21.2477i	0.103082	+	0.729220i
$850$		0			0
$851$			34.3665i			1.17807i
$852$		0			0
$853$	−7.83055			−0.268113			−0.134056	−	0.990974i	$-0.542800\pi$
$853$	−7.83055			−0.268113			−0.134056	+	0.990974i	$0.542800\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−25.7234			−0.878696			−0.439348	−	0.898317i	$-0.644790\pi$
$857$	−25.7234			−0.878696			−0.439348	+	0.898317i	$0.644790\pi$
$858$		0			0
$859$	−12.5142			−0.426980			−0.213490	−	0.976945i	$-0.568483\pi$
$859$	−12.5142			−0.426980			−0.213490	+	0.976945i	$0.568483\pi$
$860$		0			0
$861$	6.94289	+	49.1152i	0.236613	+	1.67384i
$862$		0			0
$863$		−	24.1621i		−	0.822489i	−0.911525	−	0.411244i	$-0.865094\pi$
$863$		−	24.1621i		−	0.822489i	0.911525	−	0.411244i	$-0.134906\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.0763	−	4.11021i	0.987484	−	0.139590i
$868$		0			0
$869$	29.6212			1.00483
$870$		0			0
$871$		−	11.9579i		−	0.405178i
$872$		0			0
$873$	6.68969	+	23.1892i	0.226412	+	0.784836i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−18.7362			−0.632675			−0.316337	−	0.948647i	$-0.602453\pi$
$877$	−18.7362			−0.632675			−0.316337	+	0.948647i	$0.602453\pi$
$878$		0			0
$879$	−7.70728	−	54.5227i	−0.259960	−	1.83900i
$880$		0			0
$881$		−	45.2764i		−	1.52540i	−0.646752	−	0.762701i	$-0.723873\pi$
$881$		−	45.2764i		−	1.52540i	0.646752	−	0.762701i	$-0.276127\pi$
$882$		0			0
$883$		−	32.6703i		−	1.09944i	−0.835348	−	0.549722i	$-0.814734\pi$
$883$		−	32.6703i		−	1.09944i	0.835348	−	0.549722i	$-0.185266\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	24.6883i		−	0.828953i	−0.910060	−	0.414477i	$-0.863965\pi$
$887$		−	24.6883i		−	0.828953i	0.910060	−	0.414477i	$-0.136035\pi$
$888$		0			0
$889$	−2.96216			−0.0993477
$890$		0			0
$891$	12.2075	+	19.3972i	0.408965	+	0.649831i
$892$		0			0
$893$			27.6560i			0.925474i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5.49884	+	38.8998i	0.183601	+	1.29883i
$898$		0			0
$899$			36.0937i			1.20379i
$900$		0			0
$901$		−	2.03726i		−	0.0678710i
$902$		0			0
$903$	−8.54237	−	60.4302i	−0.284272	−	2.01099i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	15.7502i		−	0.522978i	−0.965206	−	0.261489i	$-0.915787\pi$
$907$		−	15.7502i		−	0.522978i	0.965206	−	0.261489i	$-0.0842135\pi$
$908$		0			0
$909$	−3.12814	+	0.902414i	−0.103754	+	0.0299312i
$910$		0			0
$911$	50.4948			1.67297			0.836483	−	0.547993i	$-0.184608\pi$
$911$	50.4948			1.67297			0.836483	+	0.547993i	$0.184608\pi$
$912$		0			0
$913$			5.85482i			0.193766i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			11.6824i			0.385788i
$918$		0			0
$919$		−	22.6311i		−	0.746530i	−0.927725	−	0.373265i	$-0.878238\pi$
$919$		−	22.6311i		−	0.746530i	0.927725	−	0.373265i	$-0.121762\pi$
$920$		0			0
$921$	2.50402	+	17.7139i	0.0825102	+	0.583692i
$922$		0			0
$923$	−31.0136			−1.02082
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−33.5285	+	9.67240i	−1.10122	+	0.317683i
$928$		0			0
$929$			45.8021i			1.50272i	0.659893	+	0.751360i	$0.270602\pi$
$929$			45.8021i			1.50272i	−0.659893	+	0.751360i	$0.729398\pi$
$930$		0			0
$931$	6.64289			0.217712
$932$		0			0
$933$	−8.81199	+	1.24566i	−0.288492	+	0.0407809i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	4.77203i		−	0.155895i	−0.996957	−	0.0779477i	$-0.975163\pi$
$937$		−	4.77203i		−	0.155895i	0.996957	−	0.0779477i	$-0.0248367\pi$
$938$		0			0
$939$	−3.79988	−	26.8811i	−0.124004	−	0.877230i
$940$		0			0
$941$	−28.5031			−0.929174			−0.464587	−	0.885528i	$-0.653797\pi$
$941$	−28.5031			−0.929174			−0.464587	+	0.885528i	$0.653797\pi$
$942$		0			0
$943$	−41.4944			−1.35124
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−59.7387			−1.94125			−0.970623	−	0.240606i	$-0.922654\pi$
$947$	−59.7387			−1.94125			−0.970623	+	0.240606i	$0.922654\pi$
$948$		0			0
$949$			60.8779i			1.97618i
$950$		0			0
$951$	−2.17047	−	15.3543i	−0.0703823	−	0.497897i
$952$		0			0
$953$	32.0009			1.03661			0.518305	−	0.855196i	$-0.326563\pi$
$953$	32.0009			1.03661			0.518305	+	0.855196i	$0.326563\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	4.85720	+	34.3607i	0.157011	+	1.11072i
$958$		0			0
$959$	−40.7446			−1.31571
$960$		0			0
$961$	9.95413			0.321101
$962$		0			0
$963$	18.7612	−	5.41227i	0.604570	−	0.174408i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	30.7086			0.987521			0.493761	−	0.869598i	$-0.335622\pi$
$967$	30.7086			0.987521			0.493761	+	0.869598i	$0.335622\pi$
$968$		0			0
$969$	−0.958603	+	0.135507i	−0.0307948	+	0.00435312i
$970$		0			0
$971$		−	54.2279i		−	1.74025i	−0.492827	−	0.870127i	$-0.664036\pi$
$971$		−	54.2279i		−	1.74025i	0.492827	−	0.870127i	$-0.335964\pi$
$972$		0			0
$973$	−53.4332			−1.71299
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.5621			−0.369905			−0.184952	−	0.982748i	$-0.559213\pi$
$977$	−11.5621			−0.369905			−0.184952	+	0.982748i	$0.559213\pi$
$978$		0			0
$979$	31.7943			1.01615
$980$		0			0
$981$	−4.21337	−	14.6053i	−0.134523	−	0.466311i
$982$		0			0
$983$			48.8505i			1.55809i	0.626969	+	0.779044i	$0.284295\pi$
$983$			48.8505i			1.55809i	−0.626969	+	0.779044i	$0.715705\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.93738	+	56.1505i	0.252650	+	1.78729i
$988$		0			0
$989$	51.0538			1.62342
$990$		0			0
$991$		−	8.77480i		−	0.278741i	−0.990240	−	0.139370i	$-0.955492\pi$
$991$		−	8.77480i		−	0.278741i	0.990240	−	0.139370i	$-0.0445079\pi$
$992$		0			0
$993$	−7.69154	+	1.08727i	−0.244084	+	0.0345035i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.99205			0.189770			0.0948851	−	0.995488i	$-0.469752\pi$
$997$	5.99205			0.189770			0.0948851	+	0.995488i	$0.469752\pi$
$998$		0			0
$999$	−36.4036	+	16.3131i	−1.15176	+	0.516122i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.m.e.1199.3		24
3.2	odd	2	inner	2400.2.m.e.1199.23		24
4.3	odd	2		600.2.m.e.299.23		24
5.2	odd	4		2400.2.b.h.2351.7		12
5.3	odd	4		2400.2.b.g.2351.6		12
5.4	even	2	inner	2400.2.m.e.1199.22		24
8.3	odd	2	inner	2400.2.m.e.1199.4		24
8.5	even	2		600.2.m.e.299.21		24
12.11	even	2		600.2.m.e.299.1		24
15.2	even	4		2400.2.b.h.2351.5		12
15.8	even	4		2400.2.b.g.2351.8		12
15.14	odd	2	inner	2400.2.m.e.1199.2		24
20.3	even	4		600.2.b.h.251.7	yes	12
20.7	even	4		600.2.b.g.251.6	yes	12
20.19	odd	2		600.2.m.e.299.2		24
24.5	odd	2		600.2.m.e.299.3		24
24.11	even	2	inner	2400.2.m.e.1199.24		24
40.3	even	4		2400.2.b.g.2351.5		12
40.13	odd	4		600.2.b.h.251.5	yes	12
40.19	odd	2	inner	2400.2.m.e.1199.21		24
40.27	even	4		2400.2.b.h.2351.8		12
40.29	even	2		600.2.m.e.299.4		24
40.37	odd	4		600.2.b.g.251.8	yes	12
60.23	odd	4		600.2.b.h.251.6	yes	12
60.47	odd	4		600.2.b.g.251.7	yes	12
60.59	even	2		600.2.m.e.299.24		24
120.29	odd	2		600.2.m.e.299.22		24
120.53	even	4		600.2.b.h.251.8	yes	12
120.59	even	2	inner	2400.2.m.e.1199.1		24
120.77	even	4		600.2.b.g.251.5	✓	12
120.83	odd	4		2400.2.b.g.2351.7		12
120.107	odd	4		2400.2.b.h.2351.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.b.g.251.5	✓	12	120.77	even	4
600.2.b.g.251.6	yes	12	20.7	even	4
600.2.b.g.251.7	yes	12	60.47	odd	4
600.2.b.g.251.8	yes	12	40.37	odd	4
600.2.b.h.251.5	yes	12	40.13	odd	4
600.2.b.h.251.6	yes	12	60.23	odd	4
600.2.b.h.251.7	yes	12	20.3	even	4
600.2.b.h.251.8	yes	12	120.53	even	4
600.2.m.e.299.1		24	12.11	even	2
600.2.m.e.299.2		24	20.19	odd	2
600.2.m.e.299.3		24	24.5	odd	2
600.2.m.e.299.4		24	40.29	even	2
600.2.m.e.299.21		24	8.5	even	2
600.2.m.e.299.22		24	120.29	odd	2
600.2.m.e.299.23		24	4.3	odd	2
600.2.m.e.299.24		24	60.59	even	2
2400.2.b.g.2351.5		12	40.3	even	4
2400.2.b.g.2351.6		12	5.3	odd	4
2400.2.b.g.2351.7		12	120.83	odd	4
2400.2.b.g.2351.8		12	15.8	even	4
2400.2.b.h.2351.5		12	15.2	even	4
2400.2.b.h.2351.6		12	120.107	odd	4
2400.2.b.h.2351.7		12	5.2	odd	4
2400.2.b.h.2351.8		12	40.27	even	4
2400.2.m.e.1199.1		24	120.59	even	2	inner
2400.2.m.e.1199.2		24	15.14	odd	2	inner
2400.2.m.e.1199.3		24	1.1	even	1	trivial
2400.2.m.e.1199.4		24	8.3	odd	2	inner
2400.2.m.e.1199.21		24	40.19	odd	2	inner
2400.2.m.e.1199.22		24	5.4	even	2	inner
2400.2.m.e.1199.23		24	3.2	odd	2	inner
2400.2.m.e.1199.24		24	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.71500 + 0.242431i) q^{3} -3.08957 q^{7} +(2.88245 - 0.831539i) q^{9} +O(q^{10})\)	\(q+(-1.71500 + 0.242431i) q^{3} -3.08957 q^{7} +(2.88245 - 0.831539i) q^{9} +2.54654i q^{11} -5.06696 q^{13} +0.214179 q^{17} +2.60975 q^{19} +(5.29861 - 0.749006i) q^{21} +4.47647i q^{23} +(-4.74182 + 2.12489i) q^{27} -7.86770 q^{29} -4.58758i q^{31} +(-0.617360 - 4.36732i) q^{33} +7.67714 q^{37} +(8.68984 - 1.22839i) q^{39} +9.26946i q^{41} -11.4049i q^{43} +10.5972i q^{47} +2.54541 q^{49} +(-0.367316 + 0.0519235i) q^{51} -9.51198i q^{53} +(-4.47572 + 0.632684i) q^{57} +0.428357i q^{59} +1.11217i q^{61} +(-8.90553 + 2.56909i) q^{63} +2.35998i q^{67} +(-1.08523 - 7.67714i) q^{69} +6.12075 q^{71} -12.0147i q^{73} -7.86770i q^{77} -11.6319i q^{79} +(7.61709 - 4.79374i) q^{81} +2.29913 q^{83} +(13.4931 - 1.90737i) q^{87} -12.4853i q^{89} +15.6547 q^{91} +(1.11217 + 7.86770i) q^{93} +8.04496i q^{97} +(2.11755 + 7.34028i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) \) 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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