LMFDB - Embedded newform 2400.2.b.g.2351.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(2351,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.2351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [12,0,-2,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.537291533250985984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 14x^{8} - 30x^{6} + 56x^{4} - 80x^{2} + 64 $ x^12 - 5x^10 + 14x^8 - 30x^6 + 56x^4 - 80*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.6
Root			$-0.847808 - 1.13191i$ of defining polynomial
Character	$\chi$	$=$	2400.2351
Dual form			2400.2.b.g.2351.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.242431 - 1.71500i) q^{3} +3.08957i q^{7} +(-2.88245 + 0.831539i) q^{9} +2.54654i q^{11} -5.06696i q^{13} -0.214179i q^{17} -2.60975 q^{19} +(5.29861 - 0.749006i) q^{21} -4.47647 q^{23} +(2.12489 + 4.74182i) q^{27} +7.86770 q^{29} -4.58758i q^{31} +(4.36732 - 0.617360i) q^{33} -7.67714i q^{37} +(-8.68984 + 1.22839i) q^{39} +9.26946i q^{41} +11.4049 q^{43} +10.5972 q^{47} -2.54541 q^{49} +(-0.367316 + 0.0519235i) q^{51} +9.51198 q^{53} +(0.632684 + 4.47572i) q^{57} -0.428357i q^{59} +1.11217i q^{61} +(-2.56909 - 8.90553i) q^{63} +2.35998 q^{67} +(1.08523 + 7.67714i) q^{69} +6.12075 q^{71} +12.0147 q^{73} -7.86770 q^{77} +11.6319i q^{79} +(7.61709 - 4.79374i) q^{81} +2.29913i q^{83} +(-1.90737 - 13.4931i) q^{87} +12.4853i q^{89} +15.6547 q^{91} +(-7.86770 + 1.11217i) q^{93} +8.04496 q^{97} +(-2.11755 - 7.34028i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} - 2 q^{9} + 4 q^{19} - 8 q^{27} + 18 q^{33} + 40 q^{43} - 36 q^{49} + 30 q^{51} + 42 q^{57} + 60 q^{67} + 12 q^{73} - 10 q^{81} + 24 q^{91} - 32 q^{97} - 58 q^{99}+O(q^{100}) $ 12 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 - 8 * q^27 + 18 * q^33 + 40 * q^43 - 36 * q^49 + 30 * q^51 + 42 * q^57 + 60 * q^67 + 12 * q^73 - 10 * q^81 + 24 * q^91 - 32 * q^97 - 58 * q^99

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
$3$	−0.242431	−	1.71500i	−0.139968	−	0.990156i
$3$	−0.242431	−	1.71500i	−0.139968	−	0.990156i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			3.08957i			1.16775i	0.811845	+	0.583873i	$0.198463\pi$
$7$			3.08957i			1.16775i	−0.811845	+	0.583873i	$0.801537\pi$
$8$		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
$13$		−	5.06696i		−	1.40532i	−0.711525	−	0.702661i	$-0.751995\pi$
$13$		−	5.06696i		−	1.40532i	0.711525	−	0.702661i	$-0.248005\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.214179i		−	0.0519459i	−0.999663	−	0.0259730i	$-0.991732\pi$
$17$		−	0.214179i		−	0.0519459i	0.999663	−	0.0259730i	$-0.00826838\pi$
$18$		0			0
$19$	−2.60975			−0.598717			−0.299359	−	0.954141i	$-0.596773\pi$
$19$	−2.60975			−0.598717			−0.299359	+	0.954141i	$0.596773\pi$
$20$		0			0
$21$	5.29861	−	0.749006i	1.15625	−	0.163447i
$22$		0			0
$23$	−4.47647			−0.933408			−0.466704	−	0.884414i	$-0.654559\pi$
$23$	−4.47647			−0.933408			−0.466704	+	0.884414i	$0.654559\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.12489	+	4.74182i	0.408934	+	0.912564i
$28$		0			0
$29$	7.86770			1.46100			0.730498	−	0.682915i	$-0.239288\pi$
$29$	7.86770			1.46100			0.730498	+	0.682915i	$0.239288\pi$
$30$		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
$33$	4.36732	−	0.617360i	0.760252	−	0.107469i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	7.67714i		−	1.26211i	−0.775736	−	0.631057i	$-0.782621\pi$
$37$		−	7.67714i		−	1.26211i	0.775736	−	0.631057i	$-0.217379\pi$
$38$		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
$43$	11.4049			1.73924			0.869618	−	0.493725i	$-0.164365\pi$
$43$	11.4049			1.73924			0.869618	+	0.493725i	$0.164365\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.5972			1.54576			0.772881	−	0.634551i	$-0.218815\pi$
$47$	10.5972			1.54576			0.772881	+	0.634551i	$0.218815\pi$
$48$		0			0
$49$	−2.54541			−0.363631
$50$		0			0
$51$	−0.367316	+	0.0519235i	−0.0514346	+	0.00727075i
$52$		0			0
$53$	9.51198			1.30657			0.653285	−	0.757112i	$-0.273390\pi$
$53$	9.51198			1.30657			0.653285	+	0.757112i	$0.273390\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.632684	+	4.47572i	0.0838010	+	0.592823i
$58$		0			0
$59$		−	0.428357i		−	0.0557674i	−0.999611	−	0.0278837i	$-0.991123\pi$
$59$		−	0.428357i		−	0.0557674i	0.999611	−	0.0278837i	$-0.00887680\pi$
$60$		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
$63$	−2.56909	−	8.90553i	−0.323675	−	1.12199i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.35998			0.288317			0.144159	−	0.989555i	$-0.453952\pi$
$67$	2.35998			0.288317			0.144159	+	0.989555i	$0.453952\pi$
$68$		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
$73$	12.0147			1.40621			0.703106	−	0.711085i	$-0.251796\pi$
$73$	12.0147			1.40621			0.703106	+	0.711085i	$0.251796\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.86770			−0.896608
$78$		0			0
$79$			11.6319i			1.30869i	0.756194	+	0.654347i	$0.227057\pi$
$79$			11.6319i			1.30869i	−0.756194	+	0.654347i	$0.772943\pi$
$80$		0			0
$81$	7.61709	−	4.79374i	0.846343	−	0.532638i
$82$		0			0
$83$			2.29913i			0.252362i	0.992007	+	0.126181i	$0.0402720\pi$
$83$			2.29913i			0.252362i	−0.992007	+	0.126181i	$0.959728\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.90737	−	13.4931i	−0.204492	−	1.44661i
$88$		0			0
$89$			12.4853i			1.32344i	0.749752	+	0.661719i	$0.230173\pi$
$89$			12.4853i			1.32344i	−0.749752	+	0.661719i	$0.769827\pi$
$90$		0			0
$91$	15.6547			1.64106
$92$		0			0
$93$	−7.86770	+	1.11217i	−0.815842	+	0.115327i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.04496			0.816842			0.408421	−	0.912794i	$-0.366080\pi$
$97$	8.04496			0.816842			0.408421	+	0.912794i	$0.366080\pi$
$98$		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
$103$		−	11.6319i		−	1.14613i	−0.819511	−	0.573064i	$-0.805755\pi$
$103$		−	11.6319i		−	1.14613i	0.819511	−	0.573064i	$-0.194245\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.50874i		−	0.629224i	−0.949220	−	0.314612i	$-0.898126\pi$
$107$		−	6.50874i		−	0.629224i	0.949220	−	0.314612i	$-0.101874\pi$
$108$		0			0
$109$			5.06696i			0.485327i	0.970111	+	0.242663i	$0.0780210\pi$
$109$			5.06696i			0.485327i	−0.970111	+	0.242663i	$0.921979\pi$
$110$		0			0
$111$	−13.1663	+	1.86118i	−1.24969	+	0.176655i
$112$		0			0
$113$			6.05364i			0.569479i	0.958605	+	0.284739i	$0.0919070\pi$
$113$			6.05364i			0.569479i	−0.958605	+	0.284739i	$0.908093\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.21337	+	14.6053i	0.389526	+	1.35026i
$118$		0			0
$119$	0.661719			0.0606597
$120$		0			0
$121$	4.51514			0.410467
$122$		0			0
$123$	15.8971	−	2.24720i	1.43340	−	0.202624i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	0.958763i		−	0.0850765i	−0.999095	−	0.0425382i	$-0.986456\pi$
$127$		−	0.958763i		−	0.0850765i	0.999095	−	0.0425382i	$-0.0135444\pi$
$128$		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
$133$		−	8.06299i		−	0.699149i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	13.1878i		−	1.12671i	−0.826215	−	0.563355i	$-0.809510\pi$
$137$		−	13.1878i		−	1.12671i	0.826215	−	0.563355i	$-0.190490\pi$
$138$		0			0
$139$	−17.2947			−1.46692			−0.733460	−	0.679733i	$-0.762096\pi$
$139$	−17.2947			−1.46692			−0.733460	+	0.679733i	$0.762096\pi$
$140$		0			0
$141$	−2.56909	−	18.1742i	−0.216357	−	1.53055i
$142$		0			0
$143$	12.9032			1.07902
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.617087	+	4.36539i	0.0508965	+	0.360051i
$148$		0			0
$149$	−3.81475			−0.312516			−0.156258	−	0.987716i	$-0.549943\pi$
$149$	−3.81475			−0.312516			−0.156258	+	0.987716i	$0.549943\pi$
$150$		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
$153$	0.178098	+	0.617360i	0.0143983	+	0.0499106i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	6.56497i		−	0.523942i	−0.965076	−	0.261971i	$-0.915628\pi$
$157$		−	6.56497i		−	0.523942i	0.965076	−	0.261971i	$-0.0843725\pi$
$158$		0			0
$159$	−2.30600	−	16.3131i	−0.182878	−	1.29371i
$160$		0			0
$161$		−	13.8303i		−	1.08998i
$162$		0			0
$163$	−8.13957			−0.637540			−0.318770	−	0.947832i	$-0.603270\pi$
$163$	−8.13957			−0.637540			−0.318770	+	0.947832i	$0.603270\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.8561			−1.69128			−0.845640	−	0.533754i	$-0.820781\pi$
$167$	−21.8561			−1.69128			−0.845640	+	0.533754i	$0.820781\pi$
$168$		0			0
$169$	−12.6741			−0.974929
$170$		0			0
$171$	7.52248	−	2.17011i	0.575258	−	0.165952i
$172$		0			0
$173$	9.51198			0.723182			0.361591	−	0.932337i	$-0.382234\pi$
$173$	9.51198			0.723182			0.361591	+	0.932337i	$0.382234\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.734633	+	0.103847i	−0.0552184	+	0.00780562i
$178$		0			0
$179$			16.2398i			1.21382i	0.794771	+	0.606910i	$0.207591\pi$
$179$			16.2398i			1.21382i	−0.794771	+	0.606910i	$0.792409\pi$
$180$		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
$183$	1.90737	−	0.269625i	0.140997	−	0.0199312i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.545414			0.0398846
$188$		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
$193$	11.2498			0.809776			0.404888	−	0.914366i	$-0.367310\pi$
$193$	11.2498			0.809776			0.404888	+	0.914366i	$0.367310\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.78247			−0.483231			−0.241615	−	0.970372i	$-0.577677\pi$
$197$	−6.78247			−0.483231			−0.241615	+	0.970372i	$0.577677\pi$
$198$		0			0
$199$			0.632789i			0.0448573i	0.999748	+	0.0224286i	$0.00713985\pi$
$199$			0.632789i			0.0448573i	−0.999748	+	0.0224286i	$0.992860\pi$
$200$		0			0
$201$	−0.572131	−	4.04736i	−0.0403550	−	0.285479i
$202$		0			0
$203$			24.3078i			1.70607i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	12.9032	−	3.72235i	0.896835	−	0.258722i
$208$		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
$213$	−1.48386	−	10.4971i	−0.101672	−	0.719248i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14.1736			0.962168
$218$		0			0
$219$	−2.91273	−	20.6052i	−0.196824	−	1.39237i
$220$		0			0
$221$	−1.08523			−0.0730008
$222$		0			0
$223$			10.7667i			0.720992i	0.932761	+	0.360496i	$0.117393\pi$
$223$			10.7667i			0.720992i	−0.932761	+	0.360496i	$0.882607\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	12.8365i		−	0.851991i	−0.904725	−	0.425996i	$-0.859924\pi$
$227$		−	12.8365i		−	0.851991i	0.904725	−	0.425996i	$-0.140076\pi$
$228$		0			0
$229$		−	14.9684i		−	0.989143i	−0.869137	−	0.494571i	$-0.835325\pi$
$229$		−	14.9684i		−	0.989143i	0.869137	−	0.494571i	$-0.164675\pi$
$230$		0			0
$231$	1.90737	+	13.4931i	0.125496	+	0.887781i
$232$		0			0
$233$			20.8980i			1.36908i	0.728977	+	0.684538i	$0.239996\pi$
$233$			20.8980i			1.36908i	−0.728977	+	0.684538i	$0.760004\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	19.9488	−	2.81994i	1.29581	−	0.183175i
$238$		0			0
$239$	28.6386			1.85248			0.926239	−	0.376937i	$-0.123023\pi$
$239$	28.6386			1.85248			0.926239	+	0.376937i	$0.123023\pi$
$240$		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
$243$	−10.0679	−	11.9012i	−0.645856	−	0.763460i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			13.2235i			0.841390i
$248$		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
$253$		−	11.3995i		−	0.716680i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.6054i			1.16057i	0.814413	+	0.580286i	$0.197059\pi$
$257$			18.6054i			1.16057i	−0.814413	+	0.580286i	$0.802941\pi$
$258$		0			0
$259$	23.7190			1.47383
$260$		0			0
$261$	−22.6783	+	6.54230i	−1.40375	+	0.404958i
$262$		0			0
$263$	11.2589			0.694255			0.347128	−	0.937818i	$-0.387157\pi$
$263$	11.2589			0.694255			0.347128	+	0.937818i	$0.387157\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	21.4123	−	3.02682i	1.31041	−	0.185238i
$268$		0			0
$269$	−11.6824			−0.712291			−0.356146	−	0.934430i	$-0.615909\pi$
$269$	−11.6824			−0.712291			−0.356146	+	0.934430i	$0.615909\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−3.79518	−	26.8478i	−0.229695	−	1.62490i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	17.4252i		−	1.04698i	−0.852032	−	0.523490i	$-0.824630\pi$
$277$		−	17.4252i		−	1.04698i	0.852032	−	0.523490i	$-0.175370\pi$
$278$		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
$283$	12.3893			0.736470			0.368235	−	0.929733i	$-0.379962\pi$
$283$	12.3893			0.736470			0.368235	+	0.929733i	$0.379962\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−28.6386			−1.69048
$288$		0			0
$289$	16.9541			0.997302
$290$		0			0
$291$	−1.95035	−	13.7971i	−0.114331	−	0.808801i
$292$		0			0
$293$	−31.7916			−1.85729			−0.928644	−	0.370973i	$-0.879024\pi$
$293$	−31.7916			−1.85729			−0.928644	+	0.370973i	$0.879024\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.0752	+	5.41110i	−0.700676	+	0.313984i
$298$		0			0
$299$			22.6821i			1.31174i
$300$		0			0
$301$			35.2363i			2.03099i
$302$		0			0
$303$	0.263094	+	1.86118i	0.0151144	+	0.106922i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.3288			−0.589495			−0.294747	−	0.955575i	$-0.595236\pi$
$307$	−10.3288			−0.589495			−0.294747	+	0.955575i	$0.595236\pi$
$308$		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
$313$	−15.6741			−0.885951			−0.442976	−	0.896534i	$-0.646077\pi$
$313$	−15.6741			−0.885951			−0.442976	+	0.896534i	$0.646077\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.95293			0.502847			0.251423	−	0.967877i	$-0.419101\pi$
$317$	8.95293			0.502847			0.251423	+	0.967877i	$0.419101\pi$
$318$		0			0
$319$			20.0354i			1.12177i
$320$		0			0
$321$	−11.1625	+	1.57792i	−0.623030	+	0.0880710i
$322$		0			0
$323$			0.558952i			0.0311009i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.68984	−	1.22839i	0.480549	−	0.0679300i
$328$		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
$333$	6.38384	+	22.1290i	0.349832	+	1.21266i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.9991			−1.41626			−0.708130	−	0.706082i	$-0.750461\pi$
$337$	−25.9991			−1.41626			−0.708130	+	0.706082i	$0.750461\pi$
$338$		0			0
$339$	10.3820	−	1.46759i	0.563873	−	0.0797085i
$340$		0			0
$341$	11.6824			0.632640
$342$		0			0
$343$			13.7627i			0.743118i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.7184i			0.575392i	0.957722	+	0.287696i	$0.0928892\pi$
$347$			10.7184i			0.575392i	−0.957722	+	0.287696i	$0.907111\pi$
$348$		0			0
$349$		−	11.6319i		−	0.622643i	−0.950305	−	0.311322i	$-0.899228\pi$
$349$		−	11.6319i		−	0.622643i	0.950305	−	0.311322i	$-0.100772\pi$
$350$		0			0
$351$	24.0266	−	10.7667i	1.28245	−	0.574684i
$352$		0			0
$353$		−	21.7547i		−	1.15789i	−0.815367	−	0.578944i	$-0.803465\pi$
$353$		−	21.7547i		−	1.15789i	0.815367	−	0.578944i	$-0.196535\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.160421	−	1.13485i	−0.00849039	−	0.0600625i
$358$		0			0
$359$	12.9032			0.681005			0.340503	−	0.940244i	$-0.389403\pi$
$359$	12.9032			0.681005			0.340503	+	0.940244i	$0.389403\pi$
$360$		0			0
$361$	−12.1892			−0.641538
$362$		0			0
$363$	−1.09461	−	7.74346i	−0.0574521	−	0.406426i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			11.7255i			0.612065i	0.952021	+	0.306032i	$0.0990016\pi$
$367$			11.7255i			0.612065i	−0.952021	+	0.306032i	$0.900998\pi$
$368$		0			0
$369$	−7.70792	−	26.7188i	−0.401258	−	1.39093i
$370$		0			0
$371$			29.3879i			1.52574i
$372$		0			0
$373$			31.0944i			1.61001i	0.593270	+	0.805004i	$0.297837\pi$
$373$			31.0944i			1.61001i	−0.593270	+	0.805004i	$0.702163\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	39.8653i		−	2.05317i
$378$		0			0
$379$	−12.6097			−0.647719			−0.323860	−	0.946105i	$-0.604981\pi$
$379$	−12.6097			−0.647719			−0.323860	+	0.946105i	$0.604981\pi$
$380$		0			0
$381$	−1.64428	+	0.232434i	−0.0842390	+	0.0119080i
$382$		0			0
$383$	−1.64428			−0.0840188			−0.0420094	−	0.999117i	$-0.513376\pi$
$383$	−1.64428			−0.0840188			−0.0420094	+	0.999117i	$0.513376\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−32.8742	+	9.48364i	−1.67109	+	0.482081i
$388$		0			0
$389$	37.4889			1.90076			0.950381	−	0.311090i	$-0.100694\pi$
$389$	37.4889			1.90076			0.950381	+	0.311090i	$0.100694\pi$
$390$		0			0
$391$			0.958763i			0.0484867i
$392$		0			0
$393$	−6.48486	+	0.916694i	−0.327118	+	0.0462411i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			19.8820i			0.997849i	0.866646	+	0.498924i	$0.166271\pi$
$397$			19.8820i			0.997849i	−0.866646	+	0.498924i	$0.833729\pi$
$398$		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
$403$	−23.2451			−1.15792
$404$		0			0
$405$		0			0
$406$		0			0
$407$	19.5501			0.969065
$408$		0			0
$409$	16.9007			0.835685			0.417843	−	0.908519i	$-0.362786\pi$
$409$	16.9007			0.835685			0.417843	+	0.908519i	$0.362786\pi$
$410$		0			0
$411$	−22.6171	+	3.19713i	−1.11562	+	0.157703i
$412$		0			0
$413$	1.32344			0.0651221
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.19278	+	29.6605i	0.205321	+	1.45248i
$418$		0			0
$419$			7.21126i			0.352293i	0.984364	+	0.176147i	$0.0563633\pi$
$419$			7.21126i			0.352293i	−0.984364	+	0.176147i	$0.943637\pi$
$420$		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
$423$	−30.5460	+	8.81199i	−1.48520	+	0.428454i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.43613			−0.166286
$428$		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
$433$	−20.6509			−0.992420			−0.496210	−	0.868203i	$-0.665275\pi$
$433$	−20.6509			−0.992420			−0.496210	+	0.868203i	$0.665275\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.6824			0.558847
$438$		0			0
$439$		−	28.5778i		−	1.36394i	−0.731379	−	0.681971i	$-0.761123\pi$
$439$		−	28.5778i		−	1.36394i	0.731379	−	0.681971i	$-0.238877\pi$
$440$		0			0
$441$	7.33704	−	2.11661i	0.349383	−	0.100791i
$442$		0			0
$443$		−	21.9689i		−	1.04378i	−0.853014	−	0.521888i	$-0.825228\pi$
$443$		−	21.9689i		−	1.04378i	0.853014	−	0.521888i	$-0.174772\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.924813	+	6.54230i	0.0437422	+	0.309440i
$448$		0			0
$449$		−	9.00493i		−	0.424969i	−0.977164	−	0.212485i	$-0.931844\pi$
$449$		−	9.00493i		−	0.424969i	0.977164	−	0.212485i	$-0.0681555\pi$
$450$		0			0
$451$	−23.6050			−1.11152
$452$		0			0
$453$	−22.6783	+	3.20578i	−1.06552	+	0.150621i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.5748			−0.868891			−0.434446	−	0.900698i	$-0.643056\pi$
$457$	−18.5748			−0.868891			−0.434446	+	0.900698i	$0.643056\pi$
$458$		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
$463$		−	7.44471i		−	0.345985i	−0.984923	−	0.172992i	$-0.944656\pi$
$463$		−	7.44471i		−	0.345985i	0.984923	−	0.172992i	$-0.0553436\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	7.26161i		−	0.336027i	−0.985785	−	0.168014i	$-0.946265\pi$
$467$		−	7.26161i		−	0.336027i	0.985785	−	0.168014i	$-0.0537353\pi$
$468$		0			0
$469$			7.29130i			0.336681i
$470$		0			0
$471$	−11.2589	+	1.59155i	−0.518784	+	0.0733349i
$472$		0			0
$473$			29.0431i			1.33540i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.4178	+	7.90958i	−1.25538	+	0.362155i
$478$		0			0
$479$	−23.3649			−1.06757			−0.533785	−	0.845620i	$-0.679231\pi$
$479$	−23.3649			−1.06757			−0.533785	+	0.845620i	$0.679231\pi$
$480$		0			0
$481$	−38.8998			−1.77368
$482$		0			0
$483$	−23.7190	+	3.35290i	−1.07925	+	0.152562i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	7.77068i		−	0.352123i	−0.984379	−	0.176062i	$-0.943664\pi$
$487$		−	7.77068i		−	0.352123i	0.984379	−	0.176062i	$-0.0563358\pi$
$488$		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
$493$		−	1.68509i		−	0.0758928i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			18.9104i			0.848249i
$498$		0			0
$499$	26.4958			1.18611			0.593057	−	0.805161i	$-0.297921\pi$
$499$	26.4958			1.18611			0.593057	+	0.805161i	$0.297921\pi$
$500$		0			0
$501$	5.29861	+	37.4833i	0.236724	+	1.67463i
$502$		0			0
$503$	26.9943			1.20362			0.601809	−	0.798640i	$-0.294447\pi$
$503$	26.9943			1.20362			0.601809	+	0.798640i	$0.294447\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.07259	+	21.7361i	0.136459	+	0.965332i
$508$		0			0
$509$	25.8064			1.14385			0.571925	−	0.820306i	$-0.306197\pi$
$509$	25.8064			1.14385			0.571925	+	0.820306i	$0.306197\pi$
$510$		0			0
$511$			37.1201i			1.64210i
$512$		0			0
$513$	−5.54541	−	12.3750i	−0.244836	−	0.546368i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			26.9862i			1.18685i
$518$		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
$523$	−24.0790			−1.05290			−0.526451	−	0.850206i	$-0.676478\pi$
$523$	−24.0790			−1.05290			−0.526451	+	0.850206i	$0.676478\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.982561			−0.0428010
$528$		0			0
$529$	−2.96125			−0.128750
$530$		0			0
$531$	0.356195	+	1.23472i	0.0154576	+	0.0535823i
$532$		0			0
$533$	46.9680			2.03441
$534$		0			0
$535$		0			0
$536$		0			0
$537$	27.8513	−	3.93703i	1.20187	−	0.169895i
$538$		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
$543$	−16.7180	+	2.36324i	−0.717436	+	0.101416i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.5833			0.666292			0.333146	−	0.942875i	$-0.391890\pi$
$547$	15.5833			0.666292			0.333146	+	0.942875i	$0.391890\pi$
$548$		0			0
$549$	−0.924813	−	3.20578i	−0.0394701	−	0.136819i
$550$		0			0
$551$	−20.5327			−0.874723
$552$		0			0
$553$	−35.9376			−1.52822
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.96772			−0.125746			−0.0628731	−	0.998022i	$-0.520026\pi$
$557$	−2.96772			−0.125746			−0.0628731	+	0.998022i	$0.520026\pi$
$558$		0			0
$559$		−	57.7883i		−	2.44419i
$560$		0			0
$561$	−0.132225	−	0.935386i	−0.00558256	−	0.0394920i
$562$		0			0
$563$		−	11.7057i		−	0.493335i	−0.969100	−	0.246668i	$-0.920664\pi$
$563$		−	11.7057i		−	0.493335i	0.969100	−	0.246668i	$-0.0793356\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	14.8106	+	23.5335i	0.621986	+	0.988314i
$568$		0			0
$569$			43.3618i			1.81782i	0.416992	+	0.908910i	$0.363084\pi$
$569$			43.3618i			1.81782i	−0.416992	+	0.908910i	$0.636916\pi$
$570$		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
$573$	−0.559045	−	3.95479i	−0.0233545	−	0.165214i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	18.9612			0.789367			0.394683	−	0.918817i	$-0.370854\pi$
$577$	18.9612			0.789367			0.394683	+	0.918817i	$0.370854\pi$
$578$		0			0
$579$	−2.72729	−	19.2934i	−0.113342	−	0.801805i
$580$		0			0
$581$	−7.10331			−0.294695
$582$		0			0
$583$			24.2226i			1.00320i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	19.9546i		−	0.823614i	−0.911271	−	0.411807i	$-0.864898\pi$
$587$		−	19.9546i		−	0.823614i	0.911271	−	0.411807i	$-0.135102\pi$
$588$		0			0
$589$			11.9724i			0.493315i
$590$		0			0
$591$	1.64428	+	11.6319i	0.0676366	+	0.478474i
$592$		0			0
$593$		−	27.4465i		−	1.12709i	−0.826085	−	0.563546i	$-0.809437\pi$
$593$		−	27.4465i		−	1.12709i	0.826085	−	0.563546i	$-0.190563\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.08523	−	0.153408i	0.0444157	−	0.00627856i
$598$		0			0
$599$	13.8858			0.567357			0.283679	−	0.958919i	$-0.408445\pi$
$599$	13.8858			0.567357			0.283679	+	0.958919i	$0.408445\pi$
$600$		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
$603$	−6.80252	+	1.96241i	−0.277020	+	0.0799156i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	30.7086i		−	1.24642i	−0.782054	−	0.623211i	$-0.785828\pi$
$607$		−	30.7086i		−	1.24642i	0.782054	−	0.623211i	$-0.214172\pi$
$608$		0			0
$609$	41.6878	−	5.89296i	1.68928	−	0.238795i
$610$		0			0
$611$		−	53.6957i		−	2.17229i
$612$		0			0
$613$			13.3170i			0.537870i	0.963158	+	0.268935i	$0.0866716\pi$
$613$			13.3170i			0.537870i	−0.963158	+	0.268935i	$0.913328\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	38.3725i		−	1.54482i	−0.635124	−	0.772410i	$-0.719051\pi$
$617$		−	38.3725i		−	1.54482i	0.635124	−	0.772410i	$-0.280949\pi$
$618$		0			0
$619$	−4.59507			−0.184691			−0.0923457	−	0.995727i	$-0.529436\pi$
$619$	−4.59507			−0.184691			−0.0923457	+	0.995727i	$0.529436\pi$
$620$		0			0
$621$	−9.51198	−	21.2266i	−0.381703	−	0.851794i
$622$		0			0
$623$	−38.5741			−1.54544
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−11.3976	+	1.61115i	−0.455176	+	0.0643433i
$628$		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
$633$	2.82190	+	19.9626i	0.112161	+	0.793444i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			12.8975i			0.511018i
$638$		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
$643$	−2.47018			−0.0974145			−0.0487072	−	0.998813i	$-0.515510\pi$
$643$	−2.47018			−0.0974145			−0.0487072	+	0.998813i	$0.515510\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30.1474			1.18522			0.592608	−	0.805491i	$-0.298099\pi$
$647$	30.1474			1.18522			0.592608	+	0.805491i	$0.298099\pi$
$648$		0			0
$649$	1.09083			0.0428188
$650$		0			0
$651$	−3.43613	−	24.3078i	−0.134672	−	0.952697i
$652$		0			0
$653$	−8.42674			−0.329764			−0.164882	−	0.986313i	$-0.552724\pi$
$653$	−8.42674			−0.329764			−0.164882	+	0.986313i	$0.552724\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−34.6318	+	9.99067i	−1.35111	+	0.389773i
$658$		0			0
$659$			13.5764i			0.528863i	0.964404	+	0.264432i	$0.0851843\pi$
$659$			13.5764i			0.528863i	−0.964404	+	0.264432i	$0.914816\pi$
$660$		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
$663$	0.263094	+	1.86118i	0.0102177	+	0.0722821i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−35.2195			−1.36370
$668$		0			0
$669$	18.4649	−	2.61018i	0.713895	−	0.100916i
$670$		0			0
$671$	−2.83219			−0.109335
$672$		0			0
$673$	7.03784			0.271289			0.135644	−	0.990758i	$-0.456690\pi$
$673$	7.03784			0.271289			0.135644	+	0.990758i	$0.456690\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−41.8298			−1.60765			−0.803825	−	0.594866i	$-0.797205\pi$
$677$	−41.8298			−1.60765			−0.803825	+	0.594866i	$0.797205\pi$
$678$		0			0
$679$			24.8554i			0.953863i
$680$		0			0
$681$	−22.0147	+	3.11198i	−0.843604	+	0.119251i
$682$		0			0
$683$			28.2464i			1.08082i	0.841403	+	0.540409i	$0.181730\pi$
$683$			28.2464i			1.08082i	−0.841403	+	0.540409i	$0.818270\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−25.6709	+	3.62881i	−0.979406	+	0.138448i
$688$		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
$693$	22.6783	−	6.54230i	0.861477	−	0.248521i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.98532			0.0751994
$698$		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
$703$			20.0354i			0.755650i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.35290i		−	0.126099i
$708$		0			0
$709$			17.0057i			0.638663i	0.947643	+	0.319331i	$0.103458\pi$
$709$			17.0057i			0.638663i	−0.947643	+	0.319331i	$0.896542\pi$
$710$		0			0
$711$	−9.67240	−	33.5285i	−0.362743	−	1.25742i
$712$		0			0
$713$			20.5361i			0.769085i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−6.94289	−	49.1152i	−0.259287	−	1.83424i
$718$		0			0
$719$	−28.6386			−1.06804			−0.534020	−	0.845472i	$-0.679319\pi$
$719$	−28.6386			−1.06804			−0.534020	+	0.845472i	$0.679319\pi$
$720$		0			0
$721$	35.9376			1.33839
$722$		0			0
$723$	−2.24243	−	15.8634i	−0.0833969	−	0.589965i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	14.1822i		−	0.525990i	−0.964797	−	0.262995i	$-0.915290\pi$
$727$		−	14.1822i		−	0.525990i	0.964797	−	0.262995i	$-0.0847104\pi$
$728$		0			0
$729$	−17.9697	+	20.1517i	−0.665545	+	0.746357i
$730$		0			0
$731$		−	2.44269i		−	0.0903462i
$732$		0			0
$733$			51.7027i			1.90968i	0.297113	+	0.954842i	$0.403976\pi$
$733$			51.7027i			1.90968i	−0.297113	+	0.954842i	$0.596024\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.00977i			0.221373i
$738$		0			0
$739$	16.7493			0.616133			0.308067	−	0.951365i	$-0.400318\pi$
$739$	16.7493			0.616133			0.308067	+	0.951365i	$0.400318\pi$
$740$		0			0
$741$	22.6783	−	3.20578i	0.833108	−	0.117767i
$742$		0			0
$743$	−13.5649			−0.497649			−0.248825	−	0.968549i	$-0.580044\pi$
$743$	−13.5649			−0.497649			−0.248825	+	0.968549i	$0.580044\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.91181	−	6.62713i	−0.0699496	−	0.242474i
$748$		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
$753$	2.11755	−	0.299334i	0.0771676	−	0.0109084i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			0.385842i			0.0140237i	0.999975	+	0.00701183i	$0.00223195\pi$
$757$			0.385842i			0.0140237i	−0.999975	+	0.00701183i	$0.997768\pi$
$758$		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
$763$	−15.6547			−0.566738
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.17047			−0.0783711
$768$		0			0
$769$	4.34816			0.156799			0.0783994	−	0.996922i	$-0.475019\pi$
$769$	4.34816			0.156799			0.0783994	+	0.996922i	$0.475019\pi$
$770$		0			0
$771$	31.9083	−	4.51052i	1.14915	−	0.162443i
$772$		0			0
$773$	21.4326			0.770878			0.385439	−	0.922733i	$-0.374050\pi$
$773$	21.4326			0.770878			0.385439	+	0.922733i	$0.374050\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−5.75023	−	40.6782i	−0.206288	−	1.45932i
$778$		0			0
$779$		−	24.1910i		−	0.866731i
$780$		0			0
$781$			15.5867i			0.557737i
$782$		0			0
$783$	16.7180	+	37.3072i	0.597451	+	1.33325i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	32.2753			1.15049			0.575246	−	0.817980i	$-0.304906\pi$
$787$	32.2753			1.15049			0.575246	+	0.817980i	$0.304906\pi$
$788$		0			0
$789$	−2.72951	−	19.3091i	−0.0971733	−	0.687421i
$790$		0			0
$791$	−18.7031			−0.665006
$792$		0			0
$793$	5.63533			0.200116
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.4120			−0.510498			−0.255249	−	0.966875i	$-0.582157\pi$
$797$	−14.4120			−0.510498			−0.255249	+	0.966875i	$0.582157\pi$
$798$		0			0
$799$		−	2.26970i		−	0.0802961i
$800$		0			0
$801$	−10.3820	−	35.9883i	−0.366830	−	1.27158i
$802$		0			0
$803$			30.5959i			1.07970i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.83219	+	20.0354i	0.0996977	+	0.705280i
$808$		0			0
$809$			27.3297i			0.960860i	0.877033	+	0.480430i	$0.159519\pi$
$809$			27.3297i			0.960860i	−0.877033	+	0.480430i	$0.840481\pi$
$810$		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
$817$	−29.7640			−1.04131
$818$		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
$823$		−	47.6544i		−	1.66113i	−0.556923	−	0.830564i	$-0.688018\pi$
$823$		−	47.6544i		−	1.66113i	0.556923	−	0.830564i	$-0.311982\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11.4476i		−	0.398073i	−0.979992	−	0.199036i	$-0.936219\pi$
$827$		−	11.4476i		−	0.398073i	0.979992	−	0.199036i	$-0.0637812\pi$
$828$		0			0
$829$			13.6692i			0.474751i	0.971418	+	0.237375i	$0.0762871\pi$
$829$			13.6692i			0.474751i	−0.971418	+	0.237375i	$0.923713\pi$
$830$		0			0
$831$	−29.8843	+	4.22441i	−1.03667	+	0.146543i
$832$		0			0
$833$			0.545173i			0.0188891i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21.7535	−	9.74808i	0.751910	−	0.336943i
$838$		0			0
$839$	−16.7180			−0.577168			−0.288584	−	0.957455i	$-0.593184\pi$
$839$	−16.7180			−0.577168			−0.288584	+	0.957455i	$0.593184\pi$
$840$		0			0
$841$	32.9007			1.13451
$842$		0			0
$843$	22.2498	−	3.14521i	0.766323	−	0.108327i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			13.9498i			0.479321i
$848$		0			0
$849$	−3.00356	−	21.2477i	−0.103082	−	0.729220i
$850$		0			0
$851$			34.3665i			1.17807i
$852$		0			0
$853$		−	7.83055i		−	0.268113i	−0.990974	−	0.134056i	$-0.957200\pi$
$853$		−	7.83055i		−	0.268113i	0.990974	−	0.134056i	$-0.0428004\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			25.7234i			0.878696i	0.898317	+	0.439348i	$0.144790\pi$
$857$			25.7234i			0.878696i	−0.898317	+	0.439348i	$0.855210\pi$
$858$		0			0
$859$	12.5142			0.426980			0.213490	−	0.976945i	$-0.431517\pi$
$859$	12.5142			0.426980			0.213490	+	0.976945i	$0.431517\pi$
$860$		0			0
$861$	6.94289	+	49.1152i	0.236613	+	1.67384i
$862$		0			0
$863$	24.1621			0.822489			0.411244	−	0.911525i	$-0.365094\pi$
$863$	24.1621			0.822489			0.411244	+	0.911525i	$0.365094\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−4.11021	−	29.0763i	−0.139590	−	0.987484i
$868$		0			0
$869$	−29.6212			−1.00483
$870$		0			0
$871$		−	11.9579i		−	0.405178i
$872$		0			0
$873$	−23.1892	+	6.68969i	−0.784836	+	0.226412i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			18.7362i			0.632675i	0.948647	+	0.316337i	$0.102453\pi$
$877$			18.7362i			0.632675i	−0.948647	+	0.316337i	$0.897547\pi$
$878$		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
$883$	32.6703			1.09944			0.549722	−	0.835348i	$-0.314734\pi$
$883$	32.6703			1.09944			0.549722	+	0.835348i	$0.314734\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.6883			−0.828953			−0.414477	−	0.910060i	$-0.636035\pi$
$887$	−24.6883			−0.828953			−0.414477	+	0.910060i	$0.636035\pi$
$888$		0			0
$889$	2.96216			0.0993477
$890$		0			0
$891$	12.2075	+	19.3972i	0.408965	+	0.649831i
$892$		0			0
$893$	−27.6560			−0.925474
$894$		0			0
$895$		0			0
$896$		0			0
$897$	38.8998	−	5.49884i	1.29883	−	0.183601i
$898$		0			0
$899$		−	36.0937i		−	1.20379i
$900$		0			0
$901$		−	2.03726i		−	0.0678710i
$902$		0			0
$903$	60.4302	−	8.54237i	2.01099	−	0.284272i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−15.7502			−0.522978			−0.261489	−	0.965206i	$-0.584213\pi$
$907$	−15.7502			−0.522978			−0.261489	+	0.965206i	$0.584213\pi$
$908$		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
$913$	−5.85482			−0.193766
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11.6824			0.385788
$918$		0			0
$919$			22.6311i			0.746530i	0.927725	+	0.373265i	$0.121762\pi$
$919$			22.6311i			0.746530i	−0.927725	+	0.373265i	$0.878238\pi$
$920$		0			0
$921$	2.50402	+	17.7139i	0.0825102	+	0.583692i
$922$		0			0
$923$		−	31.0136i		−	1.02082i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	9.67240	+	33.5285i	0.317683	+	1.10122i
$928$		0			0
$929$		−	45.8021i		−	1.50272i	−0.659893	−	0.751360i	$-0.729398\pi$
$929$		−	45.8021i		−	1.50272i	0.659893	−	0.751360i	$-0.270602\pi$
$930$		0			0
$931$	6.64289			0.217712
$932$		0			0
$933$	−1.24566	−	8.81199i	−0.0407809	−	0.288492i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−4.77203			−0.155895			−0.0779477	−	0.996957i	$-0.524837\pi$
$937$	−4.77203			−0.155895			−0.0779477	+	0.996957i	$0.524837\pi$
$938$		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
$943$		−	41.4944i		−	1.35124i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			59.7387i			1.94125i	0.240606	+	0.970623i	$0.422654\pi$
$947$			59.7387i			1.94125i	−0.240606	+	0.970623i	$0.577346\pi$
$948$		0			0
$949$		−	60.8779i		−	1.97618i
$950$		0			0
$951$	−2.17047	−	15.3543i	−0.0703823	−	0.497897i
$952$		0			0
$953$			32.0009i			1.03661i	0.855196	+	0.518305i	$0.173437\pi$
$953$			32.0009i			1.03661i	−0.855196	+	0.518305i	$0.826563\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	34.3607	−	4.85720i	1.11072	−	0.157011i
$958$		0			0
$959$	40.7446			1.31571
$960$		0			0
$961$	9.95413			0.321101
$962$		0			0
$963$	5.41227	+	18.7612i	0.174408	+	0.604570i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	30.7086i		−	0.987521i	−0.869598	−	0.493761i	$-0.835622\pi$
$967$		−	30.7086i		−	0.987521i	0.869598	−	0.493761i	$-0.164378\pi$
$968$		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
$973$		−	53.4332i		−	1.71299i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			11.5621i			0.369905i	0.982748	+	0.184952i	$0.0592131\pi$
$977$			11.5621i			0.369905i	−0.982748	+	0.184952i	$0.940787\pi$
$978$		0			0
$979$	−31.7943			−1.01615
$980$		0			0
$981$	−4.21337	−	14.6053i	−0.134523	−	0.466311i
$982$		0			0
$983$	−48.8505			−1.55809			−0.779044	−	0.626969i	$-0.784295\pi$
$983$	−48.8505			−1.55809			−0.779044	+	0.626969i	$0.784295\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	56.1505	−	7.93738i	1.78729	−	0.252650i
$988$		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
$993$	−1.08727	−	7.69154i	−0.0345035	−	0.244084i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	5.99205i		−	0.189770i	−0.995488	−	0.0948851i	$-0.969752\pi$
$997$		−	5.99205i		−	0.189770i	0.995488	−	0.0948851i	$-0.0302484\pi$
$998$		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.b.g.2351.6		12
3.2	odd	2	inner	2400.2.b.g.2351.8		12
4.3	odd	2		600.2.b.h.251.7	yes	12
5.2	odd	4		2400.2.m.e.1199.3		24
5.3	odd	4		2400.2.m.e.1199.22		24
5.4	even	2		2400.2.b.h.2351.7		12
8.3	odd	2	inner	2400.2.b.g.2351.5		12
8.5	even	2		600.2.b.h.251.5	yes	12
12.11	even	2		600.2.b.h.251.6	yes	12
15.2	even	4		2400.2.m.e.1199.23		24
15.8	even	4		2400.2.m.e.1199.2		24
15.14	odd	2		2400.2.b.h.2351.5		12
20.3	even	4		600.2.m.e.299.2		24
20.7	even	4		600.2.m.e.299.23		24
20.19	odd	2		600.2.b.g.251.6	yes	12
24.5	odd	2		600.2.b.h.251.8	yes	12
24.11	even	2	inner	2400.2.b.g.2351.7		12
40.3	even	4		2400.2.m.e.1199.21		24
40.13	odd	4		600.2.m.e.299.4		24
40.19	odd	2		2400.2.b.h.2351.8		12
40.27	even	4		2400.2.m.e.1199.4		24
40.29	even	2		600.2.b.g.251.8	yes	12
40.37	odd	4		600.2.m.e.299.21		24
60.23	odd	4		600.2.m.e.299.24		24
60.47	odd	4		600.2.m.e.299.1		24
60.59	even	2		600.2.b.g.251.7	yes	12
120.29	odd	2		600.2.b.g.251.5	✓	12
120.53	even	4		600.2.m.e.299.22		24
120.59	even	2		2400.2.b.h.2351.6		12
120.77	even	4		600.2.m.e.299.3		24
120.83	odd	4		2400.2.m.e.1199.1		24
120.107	odd	4		2400.2.m.e.1199.24		24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.242431 - 1.71500i) q^{3} +3.08957i q^{7} +(-2.88245 + 0.831539i) q^{9} +2.54654i q^{11} -5.06696i q^{13} -0.214179i q^{17} -2.60975 q^{19} +(5.29861 - 0.749006i) q^{21} -4.47647 q^{23} +(2.12489 + 4.74182i) q^{27} +7.86770 q^{29} -4.58758i q^{31} +(4.36732 - 0.617360i) q^{33} -7.67714i q^{37} +(-8.68984 + 1.22839i) q^{39} +9.26946i q^{41} +11.4049 q^{43} +10.5972 q^{47} -2.54541 q^{49} +(-0.367316 + 0.0519235i) q^{51} +9.51198 q^{53} +(0.632684 + 4.47572i) q^{57} -0.428357i q^{59} +1.11217i q^{61} +(-2.56909 - 8.90553i) q^{63} +2.35998 q^{67} +(1.08523 + 7.67714i) q^{69} +6.12075 q^{71} +12.0147 q^{73} -7.86770 q^{77} +11.6319i q^{79} +(7.61709 - 4.79374i) q^{81} +2.29913i q^{83} +(-1.90737 - 13.4931i) q^{87} +12.4853i q^{89} +15.6547 q^{91} +(-7.86770 + 1.11217i) q^{93} +8.04496 q^{97} +(-2.11755 - 7.34028i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 2 q^{9} + 4 q^{19} - 8 q^{27} + 18 q^{33} + 40 q^{43} - 36 q^{49} + 30 q^{51} + 42 q^{57} + 60 q^{67} + 12 q^{73} - 10 q^{81} + 24 q^{91} - 32 q^{97} - 58 q^{99}+O(q^{100}) \) 12 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 - 8 * q^27 + 18 * q^33 + 40 * q^43 - 36 * q^49 + 30 * q^51 + 42 * q^57 + 60 * q^67 + 12 * q^73 - 10 * q^81 + 24 * q^91 - 32 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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