LMFDB - Embedded newform 975.2.h.c.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(649,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("975.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.h (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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	975.649
Dual form			975.2.h.c.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000 q^{4} +3.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000 q^{4} +3.00000 q^{7} -1.00000 q^{9} +3.00000i q^{11} -2.00000i q^{12} +(-3.00000 - 2.00000i) q^{13} +4.00000 q^{16} +3.00000i q^{17} +3.00000i q^{21} +3.00000i q^{23} -1.00000i q^{27} -6.00000 q^{28} +6.00000 q^{29} +6.00000i q^{31} -3.00000 q^{33} +2.00000 q^{36} -9.00000 q^{37} +(2.00000 - 3.00000i) q^{39} -3.00000i q^{41} +10.0000i q^{43} -6.00000i q^{44} -12.0000 q^{47} +4.00000i q^{48} +2.00000 q^{49} -3.00000 q^{51} +(6.00000 + 4.00000i) q^{52} +3.00000i q^{53} +12.0000i q^{59} +1.00000 q^{61} -3.00000 q^{63} -8.00000 q^{64} -6.00000i q^{68} -3.00000 q^{69} +9.00000i q^{71} -6.00000 q^{73} +9.00000i q^{77} -1.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -6.00000i q^{84} +6.00000i q^{87} +15.0000i q^{89} +(-9.00000 - 6.00000i) q^{91} -6.00000i q^{92} -6.00000 q^{93} -9.00000 q^{97} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 6 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 6 * q^7 - 2 * q^9	$ 2 q - 4 q^{4} + 6 q^{7} - 2 q^{9} - 6 q^{13} + 8 q^{16} - 12 q^{28} + 12 q^{29} - 6 q^{33} + 4 q^{36} - 18 q^{37} + 4 q^{39} - 24 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{52} + 2 q^{61} - 6 q^{63} - 16 q^{64} - 6 q^{69} - 12 q^{73} - 2 q^{79} + 2 q^{81} + 12 q^{83} - 18 q^{91} - 12 q^{93} - 18 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 6 * q^7 - 2 * q^9 - 6 * q^13 + 8 * q^16 - 12 * q^28 + 12 * q^29 - 6 * q^33 + 4 * q^36 - 18 * q^37 + 4 * q^39 - 24 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^52 + 2 * q^61 - 6 * q^63 - 16 * q^64 - 6 * q^69 - 12 * q^73 - 2 * q^79 + 2 * q^81 + 12 * q^83 - 18 * q^91 - 12 * q^93 - 18 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$-1$	$1$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		−	2.00000i		−	0.577350i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$			3.00000i			0.654654i
$22$		0			0
$23$			3.00000i			0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$			3.00000i			0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$	−6.00000			−1.13389
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			6.00000i			1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$			6.00000i			1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$		0			0
$33$	−3.00000			−0.522233
$34$		0			0
$35$		0			0
$36$	2.00000			0.333333
$37$	−9.00000			−1.47959			−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000			−1.47959			−0.739795	+	0.672832i	$0.765078\pi$
$38$		0			0
$39$	2.00000	−	3.00000i	0.320256	−	0.480384i
$40$		0			0
$41$		−	3.00000i		−	0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$		−	3.00000i		−	0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$		0			0
$43$			10.0000i			1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$			10.0000i			1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$		−	6.00000i		−	0.904534i
$45$		0			0
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$			4.00000i			0.577350i
$49$	2.00000			0.285714
$50$		0			0
$51$	−3.00000			−0.420084
$52$	6.00000	+	4.00000i	0.832050	+	0.554700i
$53$			3.00000i			0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$			3.00000i			0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$		0			0
$61$	1.00000			0.128037			0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000			0.128037			0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$	−3.00000			−0.377964
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	6.00000i		−	0.727607i
$69$	−3.00000			−0.361158
$70$		0			0
$71$			9.00000i			1.06810i	0.845452	+	0.534052i	$0.179331\pi$
$71$			9.00000i			1.06810i	−0.845452	+	0.534052i	$0.820669\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			9.00000i			1.02565i
$78$		0			0
$79$	−1.00000			−0.112509			−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000			−0.112509			−0.0562544	+	0.998416i	$0.517916\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		−	6.00000i		−	0.654654i
$85$		0			0
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$			15.0000i			1.59000i	0.606612	+	0.794998i	$0.292528\pi$
$89$			15.0000i			1.59000i	−0.606612	+	0.794998i	$0.707472\pi$
$90$		0			0
$91$	−9.00000	−	6.00000i	−0.943456	−	0.628971i
$92$		−	6.00000i		−	0.625543i
$93$	−6.00000			−0.622171
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.00000			−0.913812			−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000			−0.913812			−0.456906	+	0.889515i	$0.651042\pi$
$98$		0			0
$99$		−	3.00000i		−	0.301511i
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.00000i		−	0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$		−	9.00000i		−	0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$			2.00000i			0.192450i
$109$		−	18.0000i		−	1.72409i	−0.506834	−	0.862044i	$-0.669184\pi$
$109$		−	18.0000i		−	1.72409i	0.506834	−	0.862044i	$-0.330816\pi$
$110$		0			0
$111$		−	9.00000i		−	0.854242i
$112$	12.0000			1.13389
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$		0			0
$116$	−12.0000			−1.11417
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$		0			0
$119$			9.00000i			0.825029i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$	3.00000			0.270501
$124$		−	12.0000i		−	1.07763i
$125$		0			0
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	−10.0000			−0.880451
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000			0.522233
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	13.0000			1.10265			0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000			1.10265			0.551323	+	0.834292i	$0.314123\pi$
$140$		0			0
$141$		−	12.0000i		−	1.01058i
$142$		0			0
$143$	6.00000	−	9.00000i	0.501745	−	0.752618i
$144$	−4.00000			−0.333333
$145$		0			0
$146$		0			0
$147$			2.00000i			0.164957i
$148$	18.0000			1.47959
$149$		−	3.00000i		−	0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$		−	3.00000i		−	0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$		0			0
$151$		−	6.00000i		−	0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$		−	6.00000i		−	0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$		0			0
$153$		−	3.00000i		−	0.242536i
$154$		0			0
$155$		0			0
$156$	−4.00000	+	6.00000i	−0.320256	+	0.480384i
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$	−3.00000			−0.237915
$160$		0			0
$161$			9.00000i			0.709299i
$162$		0			0
$163$	−3.00000			−0.234978			−0.117489	−	0.993074i	$-0.537485\pi$
$163$	−3.00000			−0.234978			−0.117489	+	0.993074i	$0.537485\pi$
$164$			6.00000i			0.468521i
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		−	20.0000i		−	1.52499i
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$		0			0
$176$			12.0000i			0.904534i
$177$	−12.0000			−0.901975
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$		0			0
$183$			1.00000i			0.0739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−9.00000			−0.658145
$188$	24.0000			1.75038
$189$		−	3.00000i		−	0.218218i
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		−	8.00000i		−	0.577350i
$193$	−3.00000			−0.215945			−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000			−0.215945			−0.107972	+	0.994154i	$0.534436\pi$
$194$		0			0
$195$		0			0
$196$	−4.00000			−0.285714
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.0000			1.26335
$204$	6.00000			0.420084
$205$		0			0
$206$		0			0
$207$		−	3.00000i		−	0.208514i
$208$	−12.0000	−	8.00000i	−0.832050	−	0.554700i
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		−	6.00000i		−	0.412082i
$213$	−9.00000			−0.616670
$214$		0			0
$215$		0			0
$216$		0			0
$217$			18.0000i			1.22192i
$218$		0			0
$219$		−	6.00000i		−	0.405442i
$220$		0			0
$221$	6.00000	−	9.00000i	0.403604	−	0.605406i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$	−9.00000			−0.592157
$232$		0			0
$233$			15.0000i			0.982683i	0.870967	+	0.491341i	$0.163493\pi$
$233$			15.0000i			0.982683i	−0.870967	+	0.491341i	$0.836507\pi$
$234$		0			0
$235$		0			0
$236$		−	24.0000i		−	1.56227i
$237$		−	1.00000i		−	0.0649570i
$238$		0			0
$239$			27.0000i			1.74648i	0.487286	+	0.873242i	$0.337987\pi$
$239$			27.0000i			1.74648i	−0.487286	+	0.873242i	$0.662013\pi$
$240$		0			0
$241$		−	30.0000i		−	1.93247i	−0.257663	−	0.966235i	$-0.582952\pi$
$241$		−	30.0000i		−	1.93247i	0.257663	−	0.966235i	$-0.417048\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$	−2.00000			−0.128037
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$	6.00000			0.377964
$253$	−9.00000			−0.565825
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	−27.0000			−1.67770
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−15.0000			−0.917985
$268$		0			0
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$		−	6.00000i		−	0.364474i	−0.983255	−	0.182237i	$-0.941666\pi$
$271$		−	6.00000i		−	0.364474i	0.983255	−	0.182237i	$-0.0583338\pi$
$272$			12.0000i			0.727607i
$273$	6.00000	−	9.00000i	0.363137	−	0.544705i
$274$		0			0
$275$		0			0
$276$	6.00000			0.361158
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$		0			0
$279$		−	6.00000i		−	0.359211i
$280$		0			0
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$		−	22.0000i		−	1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$		−	22.0000i		−	1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$		−	18.0000i		−	1.06810i
$285$		0			0
$286$		0			0
$287$		−	9.00000i		−	0.531253i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		−	9.00000i		−	0.527589i
$292$	12.0000			0.702247
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.00000			0.174078
$298$		0			0
$299$	6.00000	−	9.00000i	0.346989	−	0.520483i
$300$		0			0
$301$			30.0000i			1.72917i
$302$		0			0
$303$			18.0000i			1.03407i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	15.0000			0.856095			0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000			0.856095			0.428048	+	0.903756i	$0.359202\pi$
$308$		−	18.0000i		−	1.02565i
$309$	−4.00000			−0.227552
$310$		0			0
$311$	−6.00000			−0.340229			−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000			−0.340229			−0.170114	+	0.985424i	$0.554414\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$		0			0
$315$		0			0
$316$	2.00000			0.112509
$317$	−6.00000			−0.336994			−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000			−0.336994			−0.168497	+	0.985702i	$0.553891\pi$
$318$		0			0
$319$			18.0000i			1.00781i
$320$		0			0
$321$	9.00000			0.502331
$322$		0			0
$323$		0			0
$324$	−2.00000			−0.111111
$325$		0			0
$326$		0			0
$327$	18.0000			0.995402
$328$		0			0
$329$	−36.0000			−1.98474
$330$		0			0
$331$		−	24.0000i		−	1.31916i	−0.751635	−	0.659580i	$-0.770734\pi$
$331$		−	24.0000i		−	1.31916i	0.751635	−	0.659580i	$-0.229266\pi$
$332$	−12.0000			−0.658586
$333$	9.00000			0.493197
$334$		0			0
$335$		0			0
$336$			12.0000i			0.654654i
$337$		−	14.0000i		−	0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$		−	14.0000i		−	0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	−18.0000			−0.974755
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.00000i		−	0.483145i	−0.970383	−	0.241573i	$-0.922337\pi$
$347$		−	9.00000i		−	0.483145i	0.970383	−	0.241573i	$-0.0776632\pi$
$348$		−	12.0000i		−	0.643268i
$349$		−	18.0000i		−	0.963518i	−0.876304	−	0.481759i	$-0.839998\pi$
$349$		−	18.0000i		−	0.963518i	0.876304	−	0.481759i	$-0.160002\pi$
$350$		0			0
$351$	−2.00000	+	3.00000i	−0.106752	+	0.160128i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		−	30.0000i		−	1.59000i
$357$	−9.00000			−0.476331
$358$		0			0
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$			2.00000i			0.104973i
$364$	18.0000	+	12.0000i	0.943456	+	0.628971i
$365$		0			0
$366$		0			0
$367$			26.0000i			1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$			26.0000i			1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$			12.0000i			0.625543i
$369$			3.00000i			0.156174i
$370$		0			0
$371$			9.00000i			0.467257i
$372$	12.0000			0.622171
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.0000	−	12.0000i	−0.927047	−	0.618031i
$378$		0			0
$379$		−	30.0000i		−	1.54100i	−0.637442	−	0.770498i	$-0.720007\pi$
$379$		−	30.0000i		−	1.54100i	0.637442	−	0.770498i	$-0.279993\pi$
$380$		0			0
$381$	2.00000			0.102463
$382$		0			0
$383$	−30.0000			−1.53293			−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000			−1.53293			−0.766464	+	0.642287i	$0.777986\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	10.0000i		−	0.508329i
$388$	18.0000			0.913812
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$		0			0
$396$			6.00000i			0.301511i
$397$	3.00000			0.150566			0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000			0.150566			0.0752828	+	0.997162i	$0.476014\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.0000i			0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$			18.0000i			0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$		0			0
$403$	12.0000	−	18.0000i	0.597763	−	0.896644i
$404$	−36.0000			−1.79107
$405$		0			0
$406$		0			0
$407$		−	27.0000i		−	1.33834i
$408$		0			0
$409$			24.0000i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$409$			24.0000i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$410$		0			0
$411$			18.0000i			0.887875i
$412$		−	8.00000i		−	0.394132i
$413$			36.0000i			1.77144i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			13.0000i			0.636613i
$418$		0			0
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$			18.0000i			0.877266i	0.898666	+	0.438633i	$0.144537\pi$
$421$			18.0000i			0.877266i	−0.898666	+	0.438633i	$0.855463\pi$
$422$		0			0
$423$	12.0000			0.583460
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.00000			0.145180
$428$			18.0000i			0.870063i
$429$	9.00000	+	6.00000i	0.434524	+	0.289683i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	4.00000i		−	0.192450i
$433$			38.0000i			1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$			38.0000i			1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$		0			0
$435$		0			0
$436$			36.0000i			1.72409i
$437$		0			0
$438$		0			0
$439$	19.0000			0.906821			0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000			0.906821			0.453410	+	0.891302i	$0.350207\pi$
$440$		0			0
$441$	−2.00000			−0.0952381
$442$		0			0
$443$			21.0000i			0.997740i	0.866677	+	0.498870i	$0.166252\pi$
$443$			21.0000i			0.997740i	−0.866677	+	0.498870i	$0.833748\pi$
$444$			18.0000i			0.854242i
$445$		0			0
$446$		0			0
$447$	3.00000			0.141895
$448$	−24.0000			−1.13389
$449$		−	15.0000i		−	0.707894i	−0.935266	−	0.353947i	$-0.884839\pi$
$449$		−	15.0000i		−	0.707894i	0.935266	−	0.353947i	$-0.115161\pi$
$450$		0			0
$451$	9.00000			0.423793
$452$			36.0000i			1.69330i
$453$	6.00000			0.281905
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−33.0000			−1.54367			−0.771837	−	0.635820i	$-0.780662\pi$
$457$	−33.0000			−1.54367			−0.771837	+	0.635820i	$0.780662\pi$
$458$		0			0
$459$	3.00000			0.140028
$460$		0			0
$461$			9.00000i			0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$			9.00000i			0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$		0			0
$463$	−15.0000			−0.697109			−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000			−0.697109			−0.348555	+	0.937288i	$0.613327\pi$
$464$	24.0000			1.11417
$465$		0			0
$466$		0			0
$467$		−	15.0000i		−	0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$		−	15.0000i		−	0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	−6.00000	−	4.00000i	−0.277350	−	0.184900i
$469$		0			0
$470$		0			0
$471$	4.00000			0.184310
$472$		0			0
$473$	−30.0000			−1.37940
$474$		0			0
$475$		0			0
$476$		−	18.0000i		−	0.825029i
$477$		−	3.00000i		−	0.137361i
$478$		0			0
$479$			3.00000i			0.137073i	0.997649	+	0.0685367i	$0.0218330\pi$
$479$			3.00000i			0.137073i	−0.997649	+	0.0685367i	$0.978167\pi$
$480$		0			0
$481$	27.0000	+	18.0000i	1.23109	+	0.820729i
$482$		0			0
$483$	−9.00000			−0.409514
$484$	−4.00000			−0.181818
$485$		0			0
$486$		0			0
$487$	15.0000			0.679715			0.339857	−	0.940477i	$-0.389621\pi$
$487$	15.0000			0.679715			0.339857	+	0.940477i	$0.389621\pi$
$488$		0			0
$489$		−	3.00000i		−	0.135665i
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000			−0.270501
$493$			18.0000i			0.810679i
$494$		0			0
$495$		0			0
$496$			24.0000i			1.07763i
$497$			27.0000i			1.21112i
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−12.0000	+	5.00000i	−0.532939	+	0.222058i
$508$			4.00000i			0.177471i
$509$		−	15.0000i		−	0.664863i	−0.943127	−	0.332432i	$-0.892131\pi$
$509$		−	15.0000i		−	0.664863i	0.943127	−	0.332432i	$-0.107869\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$	20.0000			0.880451
$517$		−	36.0000i		−	1.58328i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$		−	2.00000i		−	0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$		−	2.00000i		−	0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$	−24.0000			−1.04844
$525$		0			0
$526$		0			0
$527$	−18.0000			−0.784092
$528$	−12.0000			−0.522233
$529$	14.0000			0.608696
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$	−6.00000	+	9.00000i	−0.259889	+	0.389833i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			24.0000i			1.03568i
$538$		0			0
$539$			6.00000i			0.258438i
$540$		0			0
$541$		−	12.0000i		−	0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$		−	12.0000i		−	0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$		0			0
$543$		−	25.0000i		−	1.07285i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			26.0000i			1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$			26.0000i			1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	−36.0000			−1.53784
$549$	−1.00000			−0.0426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.00000			−0.127573
$554$		0			0
$555$		0			0
$556$	−26.0000			−1.10265
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	20.0000	−	30.0000i	0.845910	−	1.26886i
$560$		0			0
$561$		−	9.00000i		−	0.379980i
$562$		0			0
$563$		−	9.00000i		−	0.379305i	−0.981851	−	0.189652i	$-0.939264\pi$
$563$		−	9.00000i		−	0.379305i	0.981851	−	0.189652i	$-0.0607361\pi$
$564$			24.0000i			1.01058i
$565$		0			0
$566$		0			0
$567$	3.00000			0.125988
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	31.0000			1.29731			0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000			1.29731			0.648655	+	0.761083i	$0.275332\pi$
$572$	−12.0000	+	18.0000i	−0.501745	+	0.752618i
$573$		−	12.0000i		−	0.501307i
$574$		0			0
$575$		0			0
$576$	8.00000			0.333333
$577$	−21.0000			−0.874241			−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000			−0.874241			−0.437121	+	0.899403i	$0.644002\pi$
$578$		0			0
$579$		−	3.00000i		−	0.124676i
$580$		0			0
$581$	18.0000			0.746766
$582$		0			0
$583$	−9.00000			−0.372742
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		−	4.00000i		−	0.164957i
$589$		0			0
$590$		0			0
$591$		−	6.00000i		−	0.246807i
$592$	−36.0000			−1.47959
$593$	−42.0000			−1.72473			−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000			−1.72473			−0.862367	+	0.506284i	$0.831019\pi$
$594$		0			0
$595$		0			0
$596$			6.00000i			0.245770i
$597$		−	16.0000i		−	0.654836i
$598$		0			0
$599$	−48.0000			−1.96123			−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000			−1.96123			−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	−17.0000			−0.693444			−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000			−0.693444			−0.346722	+	0.937968i	$0.612705\pi$
$602$		0			0
$603$		0			0
$604$			12.0000i			0.488273i
$605$		0			0
$606$		0			0
$607$		−	4.00000i		−	0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$		−	4.00000i		−	0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$		0			0
$609$			18.0000i			0.729397i
$610$		0			0
$611$	36.0000	+	24.0000i	1.45640	+	0.970936i
$612$			6.00000i			0.242536i
$613$	21.0000			0.848182			0.424091	−	0.905620i	$-0.360594\pi$
$613$	21.0000			0.848182			0.424091	+	0.905620i	$0.360594\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$			30.0000i			1.20580i	0.797816	+	0.602901i	$0.205989\pi$
$619$			30.0000i			1.20580i	−0.797816	+	0.602901i	$0.794011\pi$
$620$		0			0
$621$	3.00000			0.120386
$622$		0			0
$623$			45.0000i			1.80289i
$624$	8.00000	−	12.0000i	0.320256	−	0.480384i
$625$		0			0
$626$		0			0
$627$		0			0
$628$			8.00000i			0.319235i
$629$		−	27.0000i		−	1.07656i
$630$		0			0
$631$		−	6.00000i		−	0.238856i	−0.992843	−	0.119428i	$-0.961894\pi$
$631$		−	6.00000i		−	0.238856i	0.992843	−	0.119428i	$-0.0381061\pi$
$632$		0			0
$633$			4.00000i			0.158986i
$634$		0			0
$635$		0			0
$636$	6.00000			0.237915
$637$	−6.00000	−	4.00000i	−0.237729	−	0.158486i
$638$		0			0
$639$		−	9.00000i		−	0.356034i
$640$		0			0
$641$	−24.0000			−0.947943			−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000			−0.947943			−0.473972	+	0.880540i	$0.657180\pi$
$642$		0			0
$643$	27.0000			1.06478			0.532388	−	0.846500i	$-0.321295\pi$
$643$	27.0000			1.06478			0.532388	+	0.846500i	$0.321295\pi$
$644$		−	18.0000i		−	0.709299i
$645$		0			0
$646$		0			0
$647$			39.0000i			1.53325i	0.642096	+	0.766624i	$0.278065\pi$
$647$			39.0000i			1.53325i	−0.642096	+	0.766624i	$0.721935\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$		0			0
$651$	−18.0000			−0.705476
$652$	6.00000			0.234978
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$		0			0
$656$		−	12.0000i		−	0.468521i
$657$	6.00000			0.234082
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$			24.0000i			0.933492i	0.884391	+	0.466746i	$0.154574\pi$
$661$			24.0000i			0.933492i	−0.884391	+	0.466746i	$0.845426\pi$
$662$		0			0
$663$	9.00000	+	6.00000i	0.349531	+	0.233021i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			18.0000i			0.696963i
$668$	−24.0000			−0.928588
$669$			24.0000i			0.927894i
$670$		0			0
$671$			3.00000i			0.115814i
$672$		0			0
$673$		−	44.0000i		−	1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$		−	44.0000i		−	1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$		0			0
$675$		0			0
$676$	−10.0000	−	24.0000i	−0.384615	−	0.923077i
$677$		−	39.0000i		−	1.49889i	−0.662066	−	0.749446i	$-0.730320\pi$
$677$		−	39.0000i		−	1.49889i	0.662066	−	0.749446i	$-0.269680\pi$
$678$		0			0
$679$	−27.0000			−1.03616
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.0000			1.14792			0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000			1.14792			0.573959	+	0.818884i	$0.305407\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−6.00000			−0.228914
$688$			40.0000i			1.52499i
$689$	6.00000	−	9.00000i	0.228582	−	0.342873i
$690$		0			0
$691$		−	30.0000i		−	1.14125i	−0.821209	−	0.570627i	$-0.806700\pi$
$691$		−	30.0000i		−	1.14125i	0.821209	−	0.570627i	$-0.193300\pi$
$692$		−	12.0000i		−	0.456172i
$693$		−	9.00000i		−	0.341882i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.00000			0.340899
$698$		0			0
$699$	−15.0000			−0.567352
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$		0			0
$704$		−	24.0000i		−	0.904534i
$705$		0			0
$706$		0			0
$707$	54.0000			2.03088
$708$	24.0000			0.901975
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$	1.00000			0.0375029
$712$		0			0
$713$	−18.0000			−0.674105
$714$		0			0
$715$		0			0
$716$	−48.0000			−1.79384
$717$	−27.0000			−1.00833
$718$		0			0
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$			12.0000i			0.446903i
$722$		0			0
$723$	30.0000			1.11571
$724$	50.0000			1.85824
$725$		0			0
$726$		0			0
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−30.0000			−1.10959
$732$		−	2.00000i		−	0.0739221i
$733$	15.0000			0.554038			0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000			0.554038			0.277019	+	0.960864i	$0.410654\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			30.0000i			1.10357i	0.833987	+	0.551784i	$0.186053\pi$
$739$			30.0000i			1.10357i	−0.833987	+	0.551784i	$0.813947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−6.00000			−0.219529
$748$	18.0000			0.658145
$749$		−	27.0000i		−	0.986559i
$750$		0			0
$751$	5.00000			0.182453			0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000			0.182453			0.0912263	+	0.995830i	$0.470921\pi$
$752$	−48.0000			−1.75038
$753$		−	6.00000i		−	0.218652i
$754$		0			0
$755$		0			0
$756$			6.00000i			0.218218i
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$		0			0
$759$		−	9.00000i		−	0.326679i
$760$		0			0
$761$			18.0000i			0.652499i	0.945284	+	0.326250i	$0.105785\pi$
$761$			18.0000i			0.652499i	−0.945284	+	0.326250i	$0.894215\pi$
$762$		0			0
$763$		−	54.0000i		−	1.95493i
$764$	24.0000			0.868290
$765$		0			0
$766$		0			0
$767$	24.0000	−	36.0000i	0.866590	−	1.29988i
$768$			16.0000i			0.577350i
$769$		−	12.0000i		−	0.432731i	−0.976312	−	0.216366i	$-0.930580\pi$
$769$		−	12.0000i		−	0.432731i	0.976312	−	0.216366i	$-0.0694203\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$	6.00000			0.215945
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	27.0000i		−	0.968620i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−27.0000			−0.966136
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$	8.00000			0.285714
$785$		0			0
$786$		0			0
$787$	12.0000			0.427754			0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000			0.427754			0.213877	+	0.976861i	$0.431391\pi$
$788$	12.0000			0.427482
$789$	24.0000			0.854423
$790$		0			0
$791$		−	54.0000i		−	1.92002i
$792$		0			0
$793$	−3.00000	−	2.00000i	−0.106533	−	0.0710221i
$794$		0			0
$795$		0			0
$796$	32.0000			1.13421
$797$		−	51.0000i		−	1.80651i	−0.429101	−	0.903256i	$-0.641170\pi$
$797$		−	51.0000i		−	1.80651i	0.429101	−	0.903256i	$-0.358830\pi$
$798$		0			0
$799$		−	36.0000i		−	1.27359i
$800$		0			0
$801$		−	15.0000i		−	0.529999i
$802$		0			0
$803$		−	18.0000i		−	0.635206i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			24.0000i			0.844840i
$808$		0			0
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$		−	42.0000i		−	1.47482i	−0.675446	−	0.737410i	$-0.736049\pi$
$811$		−	42.0000i		−	1.47482i	0.675446	−	0.737410i	$-0.263951\pi$
$812$	−36.0000			−1.26335
$813$	6.00000			0.210429
$814$		0			0
$815$		0			0
$816$	−12.0000			−0.420084
$817$		0			0
$818$		0			0
$819$	9.00000	+	6.00000i	0.314485	+	0.209657i
$820$		0			0
$821$			33.0000i			1.15171i	0.817553	+	0.575854i	$0.195330\pi$
$821$			33.0000i			1.15171i	−0.817553	+	0.575854i	$0.804670\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$			6.00000i			0.208514i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$	24.0000	+	16.0000i	0.832050	+	0.554700i
$833$			6.00000i			0.207888i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.00000			0.207390
$838$		0			0
$839$			9.00000i			0.310715i	0.987858	+	0.155357i	$0.0496529\pi$
$839$			9.00000i			0.310715i	−0.987858	+	0.155357i	$0.950347\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−6.00000			−0.206651
$844$	−8.00000			−0.275371
$845$		0			0
$846$		0			0
$847$	6.00000			0.206162
$848$			12.0000i			0.412082i
$849$	22.0000			0.755038
$850$		0			0
$851$		−	27.0000i		−	0.925548i
$852$	18.0000			0.616670
$853$	−9.00000			−0.308154			−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000			−0.308154			−0.154077	+	0.988059i	$0.549240\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			21.0000i			0.717346i	0.933463	+	0.358673i	$0.116771\pi$
$857$			21.0000i			0.717346i	−0.933463	+	0.358673i	$0.883229\pi$
$858$		0			0
$859$	5.00000			0.170598			0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000			0.170598			0.0852989	+	0.996355i	$0.472815\pi$
$860$		0			0
$861$	9.00000			0.306719
$862$		0			0
$863$	30.0000			1.02121			0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000			1.02121			0.510606	+	0.859815i	$0.329421\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			8.00000i			0.271694i
$868$		−	36.0000i		−	1.22192i
$869$		−	3.00000i		−	0.101768i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	9.00000			0.304604
$874$		0			0
$875$		0			0
$876$			12.0000i			0.405442i
$877$	30.0000			1.01303			0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000			1.01303			0.506514	+	0.862232i	$0.330934\pi$
$878$		0			0
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	16.0000i		−	0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$		−	16.0000i		−	0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−12.0000	+	18.0000i	−0.403604	+	0.605406i
$885$		0			0
$886$		0			0
$887$			33.0000i			1.10803i	0.832506	+	0.554016i	$0.186905\pi$
$887$			33.0000i			1.10803i	−0.832506	+	0.554016i	$0.813095\pi$
$888$		0			0
$889$		−	6.00000i		−	0.201234i
$890$		0			0
$891$			3.00000i			0.100504i
$892$	−48.0000			−1.60716
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	9.00000	+	6.00000i	0.300501	+	0.200334i
$898$		0			0
$899$			36.0000i			1.20067i
$900$		0			0
$901$	−9.00000			−0.299833
$902$		0			0
$903$	−30.0000			−0.998337
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	8.00000i		−	0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$		−	8.00000i		−	0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$		0			0
$909$	−18.0000			−0.597022
$910$		0			0
$911$	18.0000			0.596367			0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000			0.596367			0.298183	+	0.954509i	$0.403619\pi$
$912$		0			0
$913$			18.0000i			0.595713i
$914$		0			0
$915$		0			0
$916$		−	12.0000i		−	0.396491i
$917$	36.0000			1.18882
$918$		0			0
$919$	11.0000			0.362857			0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000			0.362857			0.181428	+	0.983404i	$0.441928\pi$
$920$		0			0
$921$			15.0000i			0.494267i
$922$		0			0
$923$	18.0000	−	27.0000i	0.592477	−	0.888716i
$924$	18.0000			0.592157
$925$		0			0
$926$		0			0
$927$		−	4.00000i		−	0.131377i
$928$		0			0
$929$		−	27.0000i		−	0.885841i	−0.896561	−	0.442921i	$-0.853942\pi$
$929$		−	27.0000i		−	0.885841i	0.896561	−	0.442921i	$-0.146058\pi$
$930$		0			0
$931$		0			0
$932$		−	30.0000i		−	0.982683i
$933$		−	6.00000i		−	0.196431i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			34.0000i			1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$			34.0000i			1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$		0			0
$939$	−8.00000			−0.261070
$940$		0			0
$941$		−	9.00000i		−	0.293392i	−0.989182	−	0.146696i	$-0.953136\pi$
$941$		−	9.00000i		−	0.293392i	0.989182	−	0.146696i	$-0.0468638\pi$
$942$		0			0
$943$	9.00000			0.293080
$944$			48.0000i			1.56227i
$945$		0			0
$946$		0			0
$947$	−24.0000			−0.779895			−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000			−0.779895			−0.389948	+	0.920837i	$0.627507\pi$
$948$			2.00000i			0.0649570i
$949$	18.0000	+	12.0000i	0.584305	+	0.389536i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$		−	27.0000i		−	0.874616i	−0.899312	−	0.437308i	$-0.855932\pi$
$953$		−	27.0000i		−	0.874616i	0.899312	−	0.437308i	$-0.144068\pi$
$954$		0			0
$955$		0			0
$956$		−	54.0000i		−	1.74648i
$957$	−18.0000			−0.581857
$958$		0			0
$959$	54.0000			1.74375
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$			9.00000i			0.290021i
$964$			60.0000i			1.93247i
$965$		0			0
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000			−0.577647			−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000			−0.577647			−0.288824	+	0.957382i	$0.593264\pi$
$972$		−	2.00000i		−	0.0641500i
$973$	39.0000			1.25028
$974$		0			0
$975$		0			0
$976$	4.00000			0.128037
$977$	−24.0000			−0.767828			−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000			−0.767828			−0.383914	+	0.923369i	$0.625424\pi$
$978$		0			0
$979$	−45.0000			−1.43821
$980$		0			0
$981$			18.0000i			0.574696i
$982$		0			0
$983$	36.0000			1.14822			0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000			1.14822			0.574111	+	0.818778i	$0.305348\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	36.0000i		−	1.14589i
$988$		0			0
$989$	−30.0000			−0.953945
$990$		0			0
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		0			0
$993$	24.0000			0.761617
$994$		0			0
$995$		0			0
$996$		−	12.0000i		−	0.380235i
$997$			8.00000i			0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$			8.00000i			0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$		0			0
$999$			9.00000i			0.284747i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.h.c.649.2		2
5.2	odd	4		975.2.b.e.376.2		2
5.3	odd	4		195.2.b.a.181.1	✓	2
5.4	even	2		975.2.h.b.649.1		2
13.12	even	2		975.2.h.b.649.2		2
15.8	even	4		585.2.b.d.181.2		2
20.3	even	4		3120.2.g.j.961.1		2
65.8	even	4		2535.2.a.g.1.1		1
65.12	odd	4		975.2.b.e.376.1		2
65.18	even	4		2535.2.a.f.1.1		1
65.38	odd	4		195.2.b.a.181.2	yes	2
65.64	even	2	inner	975.2.h.c.649.1		2
195.8	odd	4		7605.2.a.i.1.1		1
195.38	even	4		585.2.b.d.181.1		2
195.83	odd	4		7605.2.a.n.1.1		1
260.103	even	4		3120.2.g.j.961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.b.a.181.1	✓	2	5.3	odd	4
195.2.b.a.181.2	yes	2	65.38	odd	4
585.2.b.d.181.1		2	195.38	even	4
585.2.b.d.181.2		2	15.8	even	4
975.2.b.e.376.1		2	65.12	odd	4
975.2.b.e.376.2		2	5.2	odd	4
975.2.h.b.649.1		2	5.4	even	2
975.2.h.b.649.2		2	13.12	even	2
975.2.h.c.649.1		2	65.64	even	2	inner
975.2.h.c.649.2		2	1.1	even	1	trivial
2535.2.a.f.1.1		1	65.18	even	4
2535.2.a.g.1.1		1	65.8	even	4
3120.2.g.j.961.1		2	20.3	even	4
3120.2.g.j.961.2		2	260.103	even	4
7605.2.a.i.1.1		1	195.8	odd	4
7605.2.a.n.1.1		1	195.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000 q^{4} +3.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000 q^{4} +3.00000 q^{7} -1.00000 q^{9} +3.00000i q^{11} -2.00000i q^{12} +(-3.00000 - 2.00000i) q^{13} +4.00000 q^{16} +3.00000i q^{17} +3.00000i q^{21} +3.00000i q^{23} -1.00000i q^{27} -6.00000 q^{28} +6.00000 q^{29} +6.00000i q^{31} -3.00000 q^{33} +2.00000 q^{36} -9.00000 q^{37} +(2.00000 - 3.00000i) q^{39} -3.00000i q^{41} +10.0000i q^{43} -6.00000i q^{44} -12.0000 q^{47} +4.00000i q^{48} +2.00000 q^{49} -3.00000 q^{51} +(6.00000 + 4.00000i) q^{52} +3.00000i q^{53} +12.0000i q^{59} +1.00000 q^{61} -3.00000 q^{63} -8.00000 q^{64} -6.00000i q^{68} -3.00000 q^{69} +9.00000i q^{71} -6.00000 q^{73} +9.00000i q^{77} -1.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -6.00000i q^{84} +6.00000i q^{87} +15.0000i q^{89} +(-9.00000 - 6.00000i) q^{91} -6.00000i q^{92} -6.00000 q^{93} -9.00000 q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 6 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 6 * q^7 - 2 * q^9	\( 2 q - 4 q^{4} + 6 q^{7} - 2 q^{9} - 6 q^{13} + 8 q^{16} - 12 q^{28} + 12 q^{29} - 6 q^{33} + 4 q^{36} - 18 q^{37} + 4 q^{39} - 24 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{52} + 2 q^{61} - 6 q^{63} - 16 q^{64} - 6 q^{69} - 12 q^{73} - 2 q^{79} + 2 q^{81} + 12 q^{83} - 18 q^{91} - 12 q^{93} - 18 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 6 * q^7 - 2 * q^9 - 6 * q^13 + 8 * q^16 - 12 * q^28 + 12 * q^29 - 6 * q^33 + 4 * q^36 - 18 * q^37 + 4 * q^39 - 24 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^52 + 2 * q^61 - 6 * q^63 - 16 * q^64 - 6 * q^69 - 12 * q^73 - 2 * q^79 + 2 * q^81 + 12 * q^83 - 18 * q^91 - 12 * q^93 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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