LMFDB - Embedded newform 1176.2.k.b.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(881,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1176.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.k (of order $2$, degree $1$, 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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Character	$\chi$	$=$	1176.881
Dual form			1176.2.k.b.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72171 + 0.188994i) q^{3} +4.01535 q^{5} +(2.92856 - 0.650785i) q^{9} +3.77164i q^{11} +4.48235i q^{13} +(-6.91326 + 0.758876i) q^{15} -4.11977 q^{17} +6.57031i q^{19} -5.01026i q^{23} +11.1230 q^{25} +(-4.91914 + 1.67394i) q^{27} -1.23251i q^{29} +0.934715i q^{31} +(-0.712817 - 6.49367i) q^{33} +2.02362 q^{37} +(-0.847136 - 7.71730i) q^{39} -2.21527 q^{41} -3.14096 q^{43} +(11.7592 - 2.61313i) q^{45} -5.85659 q^{47} +(7.09304 - 0.778611i) q^{51} -4.56288i q^{53} +15.1444i q^{55} +(-1.24175 - 11.3122i) q^{57} +1.96554 q^{59} +9.24698i q^{61} +17.9982i q^{65} +7.76053 q^{67} +(0.946908 + 8.62621i) q^{69} +7.97495i q^{71} +14.8218i q^{73} +(-19.1506 + 2.10218i) q^{75} +3.70259 q^{79} +(8.15296 - 3.81173i) q^{81} +9.15704 q^{83} -16.5423 q^{85} +(0.232938 + 2.12203i) q^{87} +13.0719 q^{89} +(-0.176655 - 1.60931i) q^{93} +26.3821i q^{95} -7.06073i q^{97} +(2.45453 + 11.0455i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 16 q^{15} + 8 q^{25} + 16 q^{37} - 64 q^{39} + 16 q^{43} + 48 q^{51} + 48 q^{57} + 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} - 56 q^{99}+O(q^{100}) $ 24 * q - 16 * q^15 + 8 * q^25 + 16 * q^37 - 64 * q^39 + 16 * q^43 + 48 * q^51 + 48 * q^57 + 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 - 56 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−1.72171	+	0.188994i	−0.994029	+	0.109116i
$3$	−1.72171	+	0.188994i	−0.994029	+	0.109116i
$4$		0			0
$5$	4.01535			1.79572			0.897859	−	0.440284i	$-0.145122\pi$
$5$	4.01535			1.79572			0.897859	+	0.440284i	$0.145122\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.92856	−	0.650785i	0.976188	−	0.216928i
$10$		0			0
$11$			3.77164i			1.13719i	0.822617	+	0.568597i	$0.192513\pi$
$11$			3.77164i			1.13719i	−0.822617	+	0.568597i	$0.807487\pi$
$12$		0			0
$13$			4.48235i			1.24318i	0.783343	+	0.621590i	$0.213513\pi$
$13$			4.48235i			1.24318i	−0.783343	+	0.621590i	$0.786487\pi$
$14$		0			0
$15$	−6.91326	+	0.758876i	−1.78499	+	0.195941i
$16$		0			0
$17$	−4.11977			−0.999191			−0.499595	−	0.866259i	$-0.666518\pi$
$17$	−4.11977			−0.999191			−0.499595	+	0.866259i	$0.666518\pi$
$18$		0			0
$19$			6.57031i			1.50733i	0.657257	+	0.753666i	$0.271717\pi$
$19$			6.57031i			1.50733i	−0.657257	+	0.753666i	$0.728283\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.01026i		−	1.04471i	−0.852728	−	0.522356i	$-0.825053\pi$
$23$		−	5.01026i		−	1.04471i	0.852728	−	0.522356i	$-0.174947\pi$
$24$		0			0
$25$	11.1230			2.22460
$26$		0			0
$27$	−4.91914	+	1.67394i	−0.946689	+	0.322150i
$28$		0			0
$29$		−	1.23251i		−	0.228872i	−0.993431	−	0.114436i	$-0.963494\pi$
$29$		−	1.23251i		−	0.228872i	0.993431	−	0.114436i	$-0.0365061\pi$
$30$		0			0
$31$			0.934715i			0.167880i	0.996471	+	0.0839399i	$0.0267504\pi$
$31$			0.934715i			0.167880i	−0.996471	+	0.0839399i	$0.973250\pi$
$32$		0			0
$33$	−0.712817	−	6.49367i	−0.124086	−	1.13040i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.02362			0.332680			0.166340	−	0.986068i	$-0.446805\pi$
$37$	2.02362			0.332680			0.166340	+	0.986068i	$0.446805\pi$
$38$		0			0
$39$	−0.847136	−	7.71730i	−0.135650	−	1.23576i
$40$		0			0
$41$	−2.21527			−0.345967			−0.172983	−	0.984925i	$-0.555341\pi$
$41$	−2.21527			−0.345967			−0.172983	+	0.984925i	$0.555341\pi$
$42$		0			0
$43$	−3.14096			−0.478992			−0.239496	−	0.970897i	$-0.576982\pi$
$43$	−3.14096			−0.478992			−0.239496	+	0.970897i	$0.576982\pi$
$44$		0			0
$45$	11.7592	−	2.61313i	1.75296	−	0.389542i
$46$		0			0
$47$	−5.85659			−0.854271			−0.427136	−	0.904188i	$-0.640477\pi$
$47$	−5.85659			−0.854271			−0.427136	+	0.904188i	$0.640477\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	7.09304	−	0.778611i	0.993225	−	0.109027i
$52$		0			0
$53$		−	4.56288i		−	0.626759i	−0.949628	−	0.313380i	$-0.898539\pi$
$53$		−	4.56288i		−	0.626759i	0.949628	−	0.313380i	$-0.101461\pi$
$54$		0			0
$55$			15.1444i			2.04208i
$56$		0			0
$57$	−1.24175	−	11.3122i	−0.164474	−	1.49833i
$58$		0			0
$59$	1.96554			0.255891			0.127946	−	0.991781i	$-0.459162\pi$
$59$	1.96554			0.255891			0.127946	+	0.991781i	$0.459162\pi$
$60$		0			0
$61$			9.24698i			1.18395i	0.805955	+	0.591977i	$0.201652\pi$
$61$			9.24698i			1.18395i	−0.805955	+	0.591977i	$0.798348\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			17.9982i			2.23240i
$66$		0			0
$67$	7.76053			0.948099			0.474050	−	0.880498i	$-0.342792\pi$
$67$	7.76053			0.948099			0.474050	+	0.880498i	$0.342792\pi$
$68$		0			0
$69$	0.946908	+	8.62621i	0.113994	+	1.03847i
$70$		0			0
$71$			7.97495i			0.946452i	0.880941	+	0.473226i	$0.156911\pi$
$71$			7.97495i			0.946452i	−0.880941	+	0.473226i	$0.843089\pi$
$72$		0			0
$73$			14.8218i			1.73476i	0.497644	+	0.867381i	$0.334199\pi$
$73$			14.8218i			1.73476i	−0.497644	+	0.867381i	$0.665801\pi$
$74$		0			0
$75$	−19.1506	+	2.10218i	−2.21132	+	0.242739i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.70259			0.416574			0.208287	−	0.978068i	$-0.433211\pi$
$79$	3.70259			0.416574			0.208287	+	0.978068i	$0.433211\pi$
$80$		0			0
$81$	8.15296	−	3.81173i	0.905884	−	0.423525i
$82$		0			0
$83$	9.15704			1.00512			0.502558	−	0.864543i	$-0.332392\pi$
$83$	9.15704			1.00512			0.502558	+	0.864543i	$0.332392\pi$
$84$		0			0
$85$	−16.5423			−1.79426
$86$		0			0
$87$	0.232938	+	2.12203i	0.0249735	+	0.227506i
$88$		0			0
$89$	13.0719			1.38562			0.692809	−	0.721121i	$-0.256373\pi$
$89$	13.0719			1.38562			0.692809	+	0.721121i	$0.256373\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.176655	−	1.60931i	−0.0183183	−	0.166877i
$94$		0			0
$95$			26.3821i			2.70674i
$96$		0			0
$97$		−	7.06073i		−	0.716908i	−0.933547	−	0.358454i	$-0.883304\pi$
$97$		−	7.06073i		−	0.716908i	0.933547	−	0.358454i	$-0.116696\pi$
$98$		0			0
$99$	2.45453	+	11.0455i	0.246689	+	1.11011i
$100$		0			0
$101$	−8.78929			−0.874567			−0.437284	−	0.899324i	$-0.644059\pi$
$101$	−8.78929			−0.874567			−0.437284	+	0.899324i	$0.644059\pi$
$102$		0			0
$103$		−	17.0387i		−	1.67887i	−0.543460	−	0.839435i	$-0.682886\pi$
$103$		−	17.0387i		−	1.67887i	0.543460	−	0.839435i	$-0.317114\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.74103i		−	0.361659i	−0.983514	−	0.180829i	$-0.942122\pi$
$107$		−	3.74103i		−	0.361659i	0.983514	−	0.180829i	$-0.0578782\pi$
$108$		0			0
$109$	−14.3170			−1.37132			−0.685660	−	0.727922i	$-0.740486\pi$
$109$	−14.3170			−1.37132			−0.685660	+	0.727922i	$0.740486\pi$
$110$		0			0
$111$	−3.48408	+	0.382451i	−0.330694	+	0.0363006i
$112$		0			0
$113$		−	3.14828i		−	0.296165i	−0.988975	−	0.148083i	$-0.952690\pi$
$113$		−	3.14828i		−	0.296165i	0.988975	−	0.148083i	$-0.0473102\pi$
$114$		0			0
$115$		−	20.1179i		−	1.87601i
$116$		0			0
$117$	2.91704	+	13.1268i	0.269681	+	1.21358i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.22529			−0.293208
$122$		0			0
$123$	3.81405	−	0.418672i	0.343901	−	0.0377504i
$124$		0			0
$125$	24.5860			2.19903
$126$		0			0
$127$	9.79366			0.869046			0.434523	−	0.900661i	$-0.356917\pi$
$127$	9.79366			0.869046			0.434523	+	0.900661i	$0.356917\pi$
$128$		0			0
$129$	5.40782	−	0.593622i	0.476132	−	0.0522655i
$130$		0			0
$131$	11.3387			0.990663			0.495331	−	0.868704i	$-0.335047\pi$
$131$	11.3387			0.990663			0.495331	+	0.868704i	$0.335047\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−19.7520	+	6.72146i	−1.69998	+	0.578491i
$136$		0			0
$137$			16.6995i			1.42673i	0.700792	+	0.713365i	$0.252830\pi$
$137$			16.6995i			1.42673i	−0.700792	+	0.713365i	$0.747170\pi$
$138$		0			0
$139$		−	1.74422i		−	0.147943i	−0.997260	−	0.0739713i	$-0.976433\pi$
$139$		−	1.74422i		−	0.147943i	0.997260	−	0.0739713i	$-0.0235673\pi$
$140$		0			0
$141$	10.0833	−	1.10686i	0.849170	−	0.0932144i
$142$		0			0
$143$	−16.9058			−1.41374
$144$		0			0
$145$		−	4.94897i		−	0.410990i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1.19765i			0.0981152i	0.998796	+	0.0490576i	$0.0156218\pi$
$149$			1.19765i			0.0981152i	−0.998796	+	0.0490576i	$0.984378\pi$
$150$		0			0
$151$	0.739210			0.0601560			0.0300780	−	0.999548i	$-0.490424\pi$
$151$	0.739210			0.0601560			0.0300780	+	0.999548i	$0.490424\pi$
$152$		0			0
$153$	−12.0650	+	2.68108i	−0.975398	+	0.216753i
$154$		0			0
$155$			3.75320i			0.301465i
$156$		0			0
$157$		−	10.4051i		−	0.830415i	−0.909727	−	0.415207i	$-0.863709\pi$
$157$		−	10.4051i		−	0.830415i	0.909727	−	0.415207i	$-0.136291\pi$
$158$		0			0
$159$	0.862356	+	7.85595i	0.0683893	+	0.623017i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	23.0449			1.80502			0.902509	−	0.430671i	$-0.141723\pi$
$163$	23.0449			1.80502			0.902509	+	0.430671i	$0.141723\pi$
$164$		0			0
$165$	−2.86221	−	26.0743i	−0.222823	−	2.02988i
$166$		0			0
$167$	−9.84130			−0.761543			−0.380771	−	0.924669i	$-0.624342\pi$
$167$	−9.84130			−0.761543			−0.380771	+	0.924669i	$0.624342\pi$
$168$		0			0
$169$	−7.09145			−0.545496
$170$		0			0
$171$	4.27586	+	19.2416i	0.326983	+	1.47144i
$172$		0			0
$173$	15.9678			1.21401			0.607003	−	0.794699i	$-0.292371\pi$
$173$	15.9678			1.21401			0.607003	+	0.794699i	$0.292371\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.38408	+	0.371474i	−0.254363	+	0.0279217i
$178$		0			0
$179$		−	2.03410i		−	0.152036i	−0.997106	−	0.0760179i	$-0.975779\pi$
$179$		−	2.03410i		−	0.152036i	0.997106	−	0.0760179i	$-0.0242206\pi$
$180$		0			0
$181$		−	7.58845i		−	0.564045i	−0.959408	−	0.282023i	$-0.908995\pi$
$181$		−	7.58845i		−	0.564045i	0.959408	−	0.282023i	$-0.0910053\pi$
$182$		0			0
$183$	−1.74762	−	15.9206i	−0.129188	−	1.17688i
$184$		0			0
$185$	8.12552			0.597400
$186$		0			0
$187$		−	15.5383i		−	1.13627i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.3242i		−	1.61532i	−0.589649	−	0.807660i	$-0.700734\pi$
$191$		−	22.3242i		−	1.61532i	0.589649	−	0.807660i	$-0.299266\pi$
$192$		0			0
$193$	−17.3106			−1.24605			−0.623023	−	0.782204i	$-0.714096\pi$
$193$	−17.3106			−1.24605			−0.623023	+	0.782204i	$0.714096\pi$
$194$		0			0
$195$	−3.40155	−	30.9876i	−0.243590	−	2.21907i
$196$		0			0
$197$			0.00885665i			0.000631010i	1.00000		0.000315505i	$0.000100428\pi$
$197$			0.00885665i			0.000631010i	−1.00000		0.000315505i	$0.999900\pi$
$198$		0			0
$199$			19.3934i			1.37476i	0.726297	+	0.687381i	$0.241240\pi$
$199$			19.3934i			1.37476i	−0.726297	+	0.687381i	$0.758760\pi$
$200$		0			0
$201$	−13.3614	+	1.46669i	−0.942438	+	0.103452i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−8.89506			−0.621258
$206$		0			0
$207$	−3.26060	−	14.6729i	−0.226627	−	1.01983i
$208$		0			0
$209$	−24.7809			−1.71413
$210$		0			0
$211$	13.4174			0.923691			0.461845	−	0.886961i	$-0.347188\pi$
$211$	13.4174			0.923691			0.461845	+	0.886961i	$0.347188\pi$
$212$		0			0
$213$	−1.50722	−	13.7305i	−0.103273	−	0.940801i
$214$		0			0
$215$	−12.6120			−0.860134
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.80123	−	25.5189i	−0.189290	−	1.72440i
$220$		0			0
$221$		−	18.4662i		−	1.24217i
$222$		0			0
$223$		−	22.6164i		−	1.51451i	−0.653120	−	0.757255i	$-0.726540\pi$
$223$		−	22.6164i		−	1.51451i	0.653120	−	0.757255i	$-0.273460\pi$
$224$		0			0
$225$	32.5744	−	7.23868i	2.17163	−	0.482579i
$226$		0			0
$227$	−12.9582			−0.860066			−0.430033	−	0.902813i	$-0.641498\pi$
$227$	−12.9582			−0.860066			−0.430033	+	0.902813i	$0.641498\pi$
$228$		0			0
$229$		−	0.613695i		−	0.0405541i	−0.999794	−	0.0202770i	$-0.993545\pi$
$229$		−	0.613695i		−	0.0405541i	0.999794	−	0.0202770i	$-0.00645483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.08800i		−	0.529862i	−0.964267	−	0.264931i	$-0.914651\pi$
$233$		−	8.08800i		−	0.529862i	0.964267	−	0.264931i	$-0.0853492\pi$
$234$		0			0
$235$	−23.5162			−1.53403
$236$		0			0
$237$	−6.37478	+	0.699766i	−0.414086	+	0.0454547i
$238$		0			0
$239$			24.8454i			1.60711i	0.595229	+	0.803556i	$0.297061\pi$
$239$			24.8454i			1.60711i	−0.595229	+	0.803556i	$0.702939\pi$
$240$		0			0
$241$		−	16.7171i		−	1.07684i	−0.842676	−	0.538422i	$-0.819021\pi$
$241$		−	16.7171i		−	1.07684i	0.842676	−	0.538422i	$-0.180979\pi$
$242$		0			0
$243$	−13.3166	+	8.10355i	−0.854262	+	0.519843i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−29.4504			−1.87389
$248$		0			0
$249$	−15.7658	+	1.73063i	−0.999115	+	0.109674i
$250$		0			0
$251$	4.42932			0.279576			0.139788	−	0.990181i	$-0.455358\pi$
$251$	4.42932			0.279576			0.139788	+	0.990181i	$0.455358\pi$
$252$		0			0
$253$	18.8969			1.18804
$254$		0			0
$255$	28.4810	−	3.12639i	1.78355	−	0.195782i
$256$		0			0
$257$	3.53994			0.220815			0.110408	−	0.993886i	$-0.464784\pi$
$257$	3.53994			0.220815			0.110408	+	0.993886i	$0.464784\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.802102	−	3.60950i	−0.0496489	−	0.223422i
$262$		0			0
$263$		−	14.4400i		−	0.890407i	−0.895429	−	0.445204i	$-0.853131\pi$
$263$		−	14.4400i		−	0.890407i	0.895429	−	0.445204i	$-0.146869\pi$
$264$		0			0
$265$		−	18.3215i		−	1.12548i
$266$		0			0
$267$	−22.5060	+	2.47051i	−1.37734	+	0.151193i
$268$		0			0
$269$	3.35036			0.204275			0.102137	−	0.994770i	$-0.467432\pi$
$269$	3.35036			0.204275			0.102137	+	0.994770i	$0.467432\pi$
$270$		0			0
$271$		−	2.63888i		−	0.160300i	−0.996783	−	0.0801502i	$-0.974460\pi$
$271$		−	2.63888i		−	0.160300i	0.996783	−	0.0801502i	$-0.0255400\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			41.9520i			2.52980i
$276$		0			0
$277$	11.9155			0.715935			0.357968	−	0.933734i	$-0.383470\pi$
$277$	11.9155			0.715935			0.357968	+	0.933734i	$0.383470\pi$
$278$		0			0
$279$	0.608298	+	2.73737i	0.0364179	+	0.163882i
$280$		0			0
$281$		−	23.9059i		−	1.42611i	−0.701109	−	0.713054i	$-0.747312\pi$
$281$		−	23.9059i		−	1.42611i	0.701109	−	0.713054i	$-0.252688\pi$
$282$		0			0
$283$		−	14.6318i		−	0.869771i	−0.900486	−	0.434886i	$-0.856789\pi$
$283$		−	14.6318i		−	0.869771i	0.900486	−	0.434886i	$-0.143211\pi$
$284$		0			0
$285$	−4.98605	−	45.4222i	−0.295348	−	2.69058i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.0275052			−0.00161795
$290$		0			0
$291$	1.33443	+	12.1565i	0.0782259	+	0.712628i
$292$		0			0
$293$	23.0499			1.34659			0.673294	−	0.739375i	$-0.264879\pi$
$293$	23.0499			1.34659			0.673294	+	0.739375i	$0.264879\pi$
$294$		0			0
$295$	7.89231			0.459508
$296$		0			0
$297$	−6.31351	−	18.5532i	−0.366347	−	1.07657i
$298$		0			0
$299$	22.4577			1.29876
$300$		0			0
$301$		0			0
$302$		0			0
$303$	15.1326	−	1.66112i	0.869345	−	0.0954290i
$304$		0			0
$305$			37.1298i			2.12605i
$306$		0			0
$307$		−	6.05212i		−	0.345413i	−0.984973	−	0.172706i	$-0.944749\pi$
$307$		−	6.05212i		−	0.345413i	0.984973	−	0.172706i	$-0.0552512\pi$
$308$		0			0
$309$	3.22021	+	29.3356i	0.183191	+	1.66885i
$310$		0			0
$311$	−10.0698			−0.571008			−0.285504	−	0.958378i	$-0.592161\pi$
$311$	−10.0698			−0.571008			−0.285504	+	0.958378i	$0.592161\pi$
$312$		0			0
$313$			18.5429i			1.04811i	0.851685	+	0.524053i	$0.175581\pi$
$313$			18.5429i			1.04811i	−0.851685	+	0.524053i	$0.824419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	21.3925i		−	1.20152i	−0.799429	−	0.600761i	$-0.794864\pi$
$317$		−	21.3925i		−	1.20152i	0.799429	−	0.600761i	$-0.205136\pi$
$318$		0			0
$319$	4.64861			0.260272
$320$		0			0
$321$	0.707031	+	6.44096i	0.0394626	+	0.359499i
$322$		0			0
$323$		−	27.0682i		−	1.50611i
$324$		0			0
$325$			49.8572i			2.76558i
$326$		0			0
$327$	24.6497	−	2.70583i	1.36313	−	0.149633i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−31.1855			−1.71411			−0.857054	−	0.515226i	$-0.827708\pi$
$331$	−31.1855			−1.71411			−0.857054	+	0.515226i	$0.827708\pi$
$332$		0			0
$333$	5.92628	−	1.31694i	0.324758	−	0.0721678i
$334$		0			0
$335$	31.1612			1.70252
$336$		0			0
$337$	−1.80637			−0.0983995			−0.0491998	−	0.998789i	$-0.515667\pi$
$337$	−1.80637			−0.0983995			−0.0491998	+	0.998789i	$0.515667\pi$
$338$		0			0
$339$	0.595006	+	5.42042i	0.0323163	+	0.294397i
$340$		0			0
$341$	−3.52541			−0.190912
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.80216	+	34.6372i	0.204702	+	1.86480i
$346$		0			0
$347$			7.42470i			0.398579i	0.979941	+	0.199289i	$0.0638634\pi$
$347$			7.42470i			0.398579i	−0.979941	+	0.199289i	$0.936137\pi$
$348$		0			0
$349$		−	19.1105i		−	1.02296i	−0.859295	−	0.511480i	$-0.829097\pi$
$349$		−	19.1105i		−	1.02296i	0.859295	−	0.511480i	$-0.170903\pi$
$350$		0			0
$351$	−7.50319	−	22.0493i	−0.400491	−	1.17690i
$352$		0			0
$353$	−11.2491			−0.598729			−0.299365	−	0.954139i	$-0.596775\pi$
$353$	−11.2491			−0.598729			−0.299365	+	0.954139i	$0.596775\pi$
$354$		0			0
$355$			32.0222i			1.69956i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			23.9845i			1.26586i	0.774211	+	0.632928i	$0.218147\pi$
$359$			23.9845i			1.26586i	−0.774211	+	0.632928i	$0.781853\pi$
$360$		0			0
$361$	−24.1690			−1.27205
$362$		0			0
$363$	5.55301	−	0.609560i	0.291457	−	0.0319936i
$364$		0			0
$365$			59.5147i			3.11514i
$366$		0			0
$367$		−	27.9230i		−	1.45757i	−0.684742	−	0.728786i	$-0.740085\pi$
$367$		−	27.9230i		−	1.45757i	0.684742	−	0.728786i	$-0.259915\pi$
$368$		0			0
$369$	−6.48755	+	1.44166i	−0.337728	+	0.0750499i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.7495			−0.711922			−0.355961	−	0.934501i	$-0.615846\pi$
$373$	−13.7495			−0.711922			−0.355961	+	0.934501i	$0.615846\pi$
$374$		0			0
$375$	−42.3299	+	4.64659i	−2.18590	+	0.239949i
$376$		0			0
$377$	5.52456			0.284529
$378$		0			0
$379$	37.2400			1.91289			0.956445	−	0.291913i	$-0.0942918\pi$
$379$	37.2400			1.91289			0.956445	+	0.291913i	$0.0942918\pi$
$380$		0			0
$381$	−16.8618	+	1.85094i	−0.863857	+	0.0948266i
$382$		0			0
$383$	3.25244			0.166192			0.0830959	−	0.996542i	$-0.473519\pi$
$383$	3.25244			0.166192			0.0830959	+	0.996542i	$0.473519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.19849	+	2.04409i	−0.467586	+	0.103907i
$388$		0			0
$389$			22.1960i			1.12538i	0.826668	+	0.562691i	$0.190234\pi$
$389$			22.1960i			1.12538i	−0.826668	+	0.562691i	$0.809766\pi$
$390$		0			0
$391$			20.6411i			1.04387i
$392$		0			0
$393$	−19.5219	+	2.14294i	−0.984748	+	0.108097i
$394$		0			0
$395$	14.8672			0.748048
$396$		0			0
$397$		−	2.73526i		−	0.137279i	−0.997642	−	0.0686393i	$-0.978134\pi$
$397$		−	2.73526i		−	0.137279i	0.997642	−	0.0686393i	$-0.0218658\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.24402i		−	0.211936i	−0.994370	−	0.105968i	$-0.966206\pi$
$401$		−	4.24402i		−	0.211936i	0.994370	−	0.105968i	$-0.0337941\pi$
$402$		0			0
$403$	−4.18972			−0.208705
$404$		0			0
$405$	32.7369	−	15.3054i	1.62671	−	0.760532i
$406$		0			0
$407$			7.63235i			0.378322i
$408$		0			0
$409$			18.2543i			0.902619i	0.892367	+	0.451310i	$0.149043\pi$
$409$			18.2543i			0.902619i	−0.892367	+	0.451310i	$0.850957\pi$
$410$		0			0
$411$	−3.15609	−	28.7516i	−0.155679	−	1.41821i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	36.7687			1.80490
$416$		0			0
$417$	0.329647	+	3.00304i	0.0161429	+	0.147059i
$418$		0			0
$419$	−27.1802			−1.32784			−0.663919	−	0.747805i	$-0.731108\pi$
$419$	−27.1802			−1.32784			−0.663919	+	0.747805i	$0.731108\pi$
$420$		0			0
$421$	−17.8910			−0.871954			−0.435977	−	0.899958i	$-0.643597\pi$
$421$	−17.8910			−0.871954			−0.435977	+	0.899958i	$0.643597\pi$
$422$		0			0
$423$	−17.1514	+	3.81138i	−0.833929	+	0.185316i
$424$		0			0
$425$	−45.8242			−2.22280
$426$		0			0
$427$		0			0
$428$		0			0
$429$	29.1069	−	3.19510i	1.40529	−	0.154261i
$430$		0			0
$431$		−	6.51842i		−	0.313981i	−0.987600	−	0.156991i	$-0.949821\pi$
$431$		−	6.51842i		−	0.313981i	0.987600	−	0.156991i	$-0.0501793\pi$
$432$		0			0
$433$			16.4285i			0.789506i	0.918787	+	0.394753i	$0.129170\pi$
$433$			16.4285i			0.789506i	−0.918787	+	0.394753i	$0.870830\pi$
$434$		0			0
$435$	0.935325	+	8.52069i	0.0448454	+	0.408536i
$436$		0			0
$437$	32.9190			1.57473
$438$		0			0
$439$		−	18.2864i		−	0.872761i	−0.899762	−	0.436381i	$-0.856260\pi$
$439$		−	18.2864i		−	0.872761i	0.899762	−	0.436381i	$-0.143740\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			0.679450i			0.0322816i	0.999870	+	0.0161408i	$0.00513801\pi$
$443$			0.679450i			0.0322816i	−0.999870	+	0.0161408i	$0.994862\pi$
$444$		0			0
$445$	52.4882			2.48818
$446$		0			0
$447$	−0.226348	−	2.06200i	−0.0107059	−	0.0975294i
$448$		0			0
$449$		−	23.7906i		−	1.12275i	−0.827562	−	0.561375i	$-0.810273\pi$
$449$		−	23.7906i		−	1.12275i	0.827562	−	0.561375i	$-0.189727\pi$
$450$		0			0
$451$		−	8.35520i		−	0.393431i
$452$		0			0
$453$	−1.27270	+	0.139706i	−0.0597968	+	0.00656396i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−24.3188			−1.13758			−0.568792	−	0.822481i	$-0.692589\pi$
$457$	−24.3188			−1.13758			−0.568792	+	0.822481i	$0.692589\pi$
$458$		0			0
$459$	20.2657	−	6.89626i	0.945922	−	0.321890i
$460$		0			0
$461$	4.08266			0.190148			0.0950742	−	0.995470i	$-0.469691\pi$
$461$	4.08266			0.190148			0.0950742	+	0.995470i	$0.469691\pi$
$462$		0			0
$463$	17.5152			0.814001			0.407001	−	0.913428i	$-0.366575\pi$
$463$	17.5152			0.814001			0.407001	+	0.913428i	$0.366575\pi$
$464$		0			0
$465$	−0.709333	−	6.46192i	−0.0328945	−	0.299665i
$466$		0			0
$467$	−6.34194			−0.293470			−0.146735	−	0.989176i	$-0.546876\pi$
$467$	−6.34194			−0.293470			−0.146735	+	0.989176i	$0.546876\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.96649	+	17.9145i	0.0906113	+	0.825457i
$472$		0			0
$473$		−	11.8466i		−	0.544706i
$474$		0			0
$475$			73.0816i			3.35321i
$476$		0			0
$477$	−2.96945	−	13.3627i	−0.135962	−	0.611835i
$478$		0			0
$479$	30.9762			1.41534			0.707669	−	0.706544i	$-0.249747\pi$
$479$	30.9762			1.41534			0.707669	+	0.706544i	$0.249747\pi$
$480$		0			0
$481$			9.07055i			0.413582i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	28.3513i		−	1.28736i
$486$		0			0
$487$	35.9430			1.62873			0.814366	−	0.580351i	$-0.197085\pi$
$487$	35.9430			1.62873			0.814366	+	0.580351i	$0.197085\pi$
$488$		0			0
$489$	−39.6767	+	4.35535i	−1.79424	+	0.196956i
$490$		0			0
$491$		−	23.7756i		−	1.07298i	−0.843907	−	0.536490i	$-0.819750\pi$
$491$		−	23.7756i		−	1.07298i	0.843907	−	0.536490i	$-0.180250\pi$
$492$		0			0
$493$			5.07768i			0.228687i
$494$		0			0
$495$	9.85578	+	44.3515i	0.442984	+	1.99345i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.2810			0.773602			0.386801	−	0.922163i	$-0.373580\pi$
$499$	17.2810			0.773602			0.386801	+	0.922163i	$0.373580\pi$
$500$		0			0
$501$	16.9439	−	1.85995i	0.756996	−	0.0830962i
$502$		0			0
$503$	−22.4974			−1.00311			−0.501554	−	0.865126i	$-0.667238\pi$
$503$	−22.4974			−1.00311			−0.501554	+	0.865126i	$0.667238\pi$
$504$		0			0
$505$	−35.2920			−1.57048
$506$		0			0
$507$	12.2094	−	1.34024i	0.542239	−	0.0595222i
$508$		0			0
$509$	3.37671			0.149670			0.0748351	−	0.997196i	$-0.476157\pi$
$509$	3.37671			0.149670			0.0748351	+	0.997196i	$0.476157\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−10.9983	−	32.3203i	−0.485588	−	1.42697i
$514$		0			0
$515$		−	68.4162i		−	3.01478i
$516$		0			0
$517$		−	22.0890i		−	0.971471i
$518$		0			0
$519$	−27.4918	+	3.01781i	−1.20676	+	0.132467i
$520$		0			0
$521$	−30.5574			−1.33874			−0.669371	−	0.742928i	$-0.733436\pi$
$521$	−30.5574			−1.33874			−0.669371	+	0.742928i	$0.733436\pi$
$522$		0			0
$523$			9.96089i			0.435559i	0.975998	+	0.217780i	$0.0698814\pi$
$523$			9.96089i			0.435559i	−0.975998	+	0.217780i	$0.930119\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.85081i		−	0.167744i
$528$		0			0
$529$	−2.10270			−0.0914219
$530$		0			0
$531$	5.75620	−	1.27914i	0.249798	−	0.0555100i
$532$		0			0
$533$		−	9.92960i		−	0.430099i
$534$		0			0
$535$		−	15.0215i		−	0.649437i
$536$		0			0
$537$	0.384432	+	3.50213i	0.0165895	+	0.151128i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	30.8718			1.32728			0.663642	−	0.748051i	$-0.269010\pi$
$541$	30.8718			1.32728			0.663642	+	0.748051i	$0.269010\pi$
$542$		0			0
$543$	1.43417	+	13.0651i	0.0615462	+	0.560677i
$544$		0			0
$545$	−57.4877			−2.46250
$546$		0			0
$547$	−38.9414			−1.66501			−0.832506	−	0.554016i	$-0.813095\pi$
$547$	−38.9414			−1.66501			−0.832506	+	0.554016i	$0.813095\pi$
$548$		0			0
$549$	6.01779	+	27.0804i	0.256833	+	1.15576i
$550$		0			0
$551$	8.09800			0.344987
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−13.9898	+	1.53567i	−0.593833	+	0.0651857i
$556$		0			0
$557$		−	45.4169i		−	1.92438i	−0.272387	−	0.962188i	$-0.587813\pi$
$557$		−	45.4169i		−	1.92438i	0.272387	−	0.962188i	$-0.412187\pi$
$558$		0			0
$559$		−	14.0789i		−	0.595473i
$560$		0			0
$561$	2.93664	+	26.7524i	0.123985	+	1.12949i
$562$		0			0
$563$	10.0487			0.423503			0.211751	−	0.977324i	$-0.432083\pi$
$563$	10.0487			0.423503			0.211751	+	0.977324i	$0.432083\pi$
$564$		0			0
$565$		−	12.6414i		−	0.531829i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			24.0130i			1.00668i	0.864089	+	0.503339i	$0.167895\pi$
$569$			24.0130i			1.00668i	−0.864089	+	0.503339i	$0.832105\pi$
$570$		0			0
$571$	−15.7396			−0.658683			−0.329341	−	0.944211i	$-0.606827\pi$
$571$	−15.7396			−0.658683			−0.329341	+	0.944211i	$0.606827\pi$
$572$		0			0
$573$	4.21913	+	38.4357i	0.176257	+	1.60567i
$574$		0			0
$575$		−	55.7291i		−	2.32406i
$576$		0			0
$577$		−	24.7582i		−	1.03070i	−0.856981	−	0.515348i	$-0.827662\pi$
$577$		−	24.7582i		−	1.03070i	0.856981	−	0.515348i	$-0.172338\pi$
$578$		0			0
$579$	29.8038	−	3.27160i	1.23861	−	0.135963i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	17.2095			0.712746
$584$		0			0
$585$	11.7129	+	52.7088i	0.484271	+	2.17924i
$586$		0			0
$587$	23.8593			0.984778			0.492389	−	0.870375i	$-0.336124\pi$
$587$	23.8593			0.984778			0.492389	+	0.870375i	$0.336124\pi$
$588$		0			0
$589$	−6.14137			−0.253051
$590$		0			0
$591$	−0.00167385	−	0.0152486i	−6.88531e−5	−	0.000627242i
$592$		0			0
$593$	−40.4394			−1.66065			−0.830324	−	0.557280i	$-0.811845\pi$
$593$	−40.4394			−1.66065			−0.830324	+	0.557280i	$0.811845\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.66523	−	33.3898i	−0.150008	−	1.36655i
$598$		0			0
$599$		−	4.77807i		−	0.195226i	−0.995224	−	0.0976132i	$-0.968879\pi$
$599$		−	4.77807i		−	0.195226i	0.995224	−	0.0976132i	$-0.0311208\pi$
$600$		0			0
$601$			4.52159i			0.184440i	0.995739	+	0.0922198i	$0.0293963\pi$
$601$			4.52159i			0.184440i	−0.995739	+	0.0922198i	$0.970604\pi$
$602$		0			0
$603$	22.7272	−	5.05043i	0.925523	−	0.205670i
$604$		0			0
$605$	−12.9506			−0.526519
$606$		0			0
$607$		−	10.6263i		−	0.431310i	−0.976470	−	0.215655i	$-0.930811\pi$
$607$		−	10.6263i		−	0.431310i	0.976470	−	0.215655i	$-0.0691887\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	26.2513i		−	1.06201i
$612$		0			0
$613$	−23.3019			−0.941157			−0.470578	−	0.882358i	$-0.655955\pi$
$613$	−23.3019			−0.941157			−0.470578	+	0.882358i	$0.655955\pi$
$614$		0			0
$615$	15.3147	−	1.68111i	0.617549	−	0.0677890i
$616$		0			0
$617$			38.6616i			1.55646i	0.627982	+	0.778228i	$0.283881\pi$
$617$			38.6616i			1.55646i	−0.627982	+	0.778228i	$0.716119\pi$
$618$		0			0
$619$			49.1755i			1.97653i	0.152743	+	0.988266i	$0.451189\pi$
$619$			49.1755i			1.97653i	−0.152743	+	0.988266i	$0.548811\pi$
$620$		0			0
$621$	8.38689	+	24.6462i	0.336554	+	0.989016i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	43.1061			1.72424
$626$		0			0
$627$	42.6654	−	4.68343i	1.70389	−	0.187038i
$628$		0			0
$629$	−8.33683			−0.332411
$630$		0			0
$631$	−18.6538			−0.742597			−0.371298	−	0.928514i	$-0.621087\pi$
$631$	−18.6538			−0.742597			−0.371298	+	0.928514i	$0.621087\pi$
$632$		0			0
$633$	−23.1008	+	2.53580i	−0.918175	+	0.100789i
$634$		0			0
$635$	39.3249			1.56056
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5.18997	+	23.3551i	0.205312	+	0.923915i
$640$		0			0
$641$			18.0042i			0.711123i	0.934653	+	0.355561i	$0.115710\pi$
$641$			18.0042i			0.711123i	−0.934653	+	0.355561i	$0.884290\pi$
$642$		0			0
$643$			8.31261i			0.327817i	0.986476	+	0.163909i	$0.0524102\pi$
$643$			8.31261i			0.327817i	−0.986476	+	0.163909i	$0.947590\pi$
$644$		0			0
$645$	21.7143	−	2.38360i	0.854998	−	0.0938540i
$646$		0			0
$647$	−20.9963			−0.825451			−0.412726	−	0.910855i	$-0.635423\pi$
$647$	−20.9963			−0.825451			−0.412726	+	0.910855i	$0.635423\pi$
$648$		0			0
$649$			7.41330i			0.290997i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.2535i		−	0.440383i	−0.975457	−	0.220191i	$-0.929332\pi$
$653$		−	11.2535i		−	0.440383i	0.975457	−	0.220191i	$-0.0706682\pi$
$654$		0			0
$655$	45.5286			1.77895
$656$		0			0
$657$	9.64581	+	43.4066i	0.376319	+	1.69345i
$658$		0			0
$659$			30.5790i			1.19119i	0.803286	+	0.595594i	$0.203083\pi$
$659$			30.5790i			1.19119i	−0.803286	+	0.595594i	$0.796917\pi$
$660$		0			0
$661$		−	21.9455i		−	0.853580i	−0.904351	−	0.426790i	$-0.859644\pi$
$661$		−	21.9455i		−	0.853580i	0.904351	−	0.426790i	$-0.140356\pi$
$662$		0			0
$663$	3.49001	+	31.7935i	0.135541	+	1.23476i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.17522			−0.239105
$668$		0			0
$669$	4.27437	+	38.9389i	0.165257	+	1.50547i
$670$		0			0
$671$	−34.8763			−1.34638
$672$		0			0
$673$	2.10181			0.0810190			0.0405095	−	0.999179i	$-0.487102\pi$
$673$	2.10181			0.0810190			0.0405095	+	0.999179i	$0.487102\pi$
$674$		0			0
$675$	−54.7156	+	18.6193i	−2.10600	+	0.716656i
$676$		0			0
$677$	−22.9306			−0.881293			−0.440646	−	0.897681i	$-0.645251\pi$
$677$	−22.9306			−0.881293			−0.440646	+	0.897681i	$0.645251\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	22.3102	−	2.44902i	0.854930	−	0.0938467i
$682$		0			0
$683$			48.2273i			1.84537i	0.385560	+	0.922683i	$0.374008\pi$
$683$			48.2273i			1.84537i	−0.385560	+	0.922683i	$0.625992\pi$
$684$		0			0
$685$			67.0541i			2.56200i
$686$		0			0
$687$	0.115985	+	1.05660i	0.00442509	+	0.0403119i
$688$		0			0
$689$	20.4524			0.779175
$690$		0			0
$691$		−	19.3373i		−	0.735627i	−0.929900	−	0.367813i	$-0.880106\pi$
$691$		−	19.3373i		−	0.735627i	0.929900	−	0.367813i	$-0.119894\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.00364i		−	0.265663i
$696$		0			0
$697$	9.12639			0.345687
$698$		0			0
$699$	1.52858	+	13.9252i	0.0578163	+	0.526699i
$700$		0			0
$701$			22.3461i			0.844001i	0.906596	+	0.422001i	$0.138672\pi$
$701$			22.3461i			0.844001i	−0.906596	+	0.422001i	$0.861328\pi$
$702$		0			0
$703$			13.2958i			0.501460i
$704$		0			0
$705$	40.4881	−	4.44442i	1.52487	−	0.167387i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.0294			0.977555			0.488777	−	0.872409i	$-0.337443\pi$
$709$	26.0294			0.977555			0.488777	+	0.872409i	$0.337443\pi$
$710$		0			0
$711$	10.8433	−	2.40959i	0.406654	−	0.0903666i
$712$		0			0
$713$	4.68317			0.175386
$714$		0			0
$715$	−67.8827			−2.53867
$716$		0			0
$717$	−4.69562	−	42.7765i	−0.175361	−	1.59752i
$718$		0			0
$719$	−1.25557			−0.0468250			−0.0234125	−	0.999726i	$-0.507453\pi$
$719$	−1.25557			−0.0468250			−0.0234125	+	0.999726i	$0.507453\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	3.15943	+	28.7820i	0.117500	+	1.07041i
$724$		0			0
$725$		−	13.7093i		−	0.509149i
$726$		0			0
$727$		−	33.5608i		−	1.24470i	−0.782738	−	0.622351i	$-0.786178\pi$
$727$		−	33.5608i		−	1.24470i	0.782738	−	0.622351i	$-0.213822\pi$
$728$		0			0
$729$	21.3958	−	16.4687i	0.792438	−	0.609952i
$730$		0			0
$731$	12.9400			0.478604
$732$		0			0
$733$			0.0637987i			0.00235646i	0.999999	+	0.00117823i	$0.000375042\pi$
$733$			0.0637987i			0.00235646i	−0.999999	+	0.00117823i	$0.999625\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			29.2699i			1.07817i
$738$		0			0
$739$	−24.8559			−0.914341			−0.457170	−	0.889379i	$-0.651137\pi$
$739$	−24.8559			−0.914341			−0.457170	+	0.889379i	$0.651137\pi$
$740$		0			0
$741$	50.7051	−	5.56595i	1.86270	−	0.204470i
$742$		0			0
$743$		−	1.14074i		−	0.0418496i	−0.999781	−	0.0209248i	$-0.993339\pi$
$743$		−	1.14074i		−	0.0418496i	0.999781	−	0.0209248i	$-0.00666106\pi$
$744$		0			0
$745$			4.80897i			0.176187i
$746$		0			0
$747$	26.8170	−	5.95927i	0.981182	−	0.218038i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.3386			0.669184			0.334592	−	0.942363i	$-0.391401\pi$
$751$	18.3386			0.669184			0.334592	+	0.942363i	$0.391401\pi$
$752$		0			0
$753$	−7.62600	+	0.837115i	−0.277907	+	0.0305062i
$754$		0			0
$755$	2.96818			0.108023
$756$		0			0
$757$	−41.6253			−1.51290			−0.756449	−	0.654053i	$-0.773067\pi$
$757$	−41.6253			−1.51290			−0.756449	+	0.654053i	$0.773067\pi$
$758$		0			0
$759$	−32.5350	+	3.57140i	−1.18094	+	0.129634i
$760$		0			0
$761$	17.3337			0.628346			0.314173	−	0.949366i	$-0.398273\pi$
$761$	17.3337			0.628346			0.314173	+	0.949366i	$0.398273\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−48.4451	+	10.7655i	−1.75154	+	0.389227i
$766$		0			0
$767$			8.81022i			0.318119i
$768$		0			0
$769$			15.7285i			0.567184i	0.958945	+	0.283592i	$0.0915261\pi$
$769$			15.7285i			0.567184i	−0.958945	+	0.283592i	$0.908474\pi$
$770$		0			0
$771$	−6.09474	+	0.669026i	−0.219497	+	0.0240944i
$772$		0			0
$773$	−14.2544			−0.512695			−0.256348	−	0.966585i	$-0.582519\pi$
$773$	−14.2544			−0.512695			−0.256348	+	0.966585i	$0.582519\pi$
$774$		0			0
$775$			10.3968i			0.373465i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	14.5550i		−	0.521487i
$780$		0			0
$781$	−30.0786			−1.07630
$782$		0			0
$783$	2.06316	+	6.06291i	0.0737313	+	0.216671i
$784$		0			0
$785$		−	41.7799i		−	1.49119i
$786$		0			0
$787$		−	12.9471i		−	0.461516i	−0.973011	−	0.230758i	$-0.925879\pi$
$787$		−	12.9471i		−	0.461516i	0.973011	−	0.230758i	$-0.0741206\pi$
$788$		0			0
$789$	2.72907	+	24.8614i	0.0971574	+	0.885090i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−41.4482			−1.47187
$794$		0			0
$795$	3.46266	+	31.5443i	0.122808	+	1.11876i
$796$		0			0
$797$	21.1678			0.749803			0.374901	−	0.927065i	$-0.377677\pi$
$797$	21.1678			0.749803			0.374901	+	0.927065i	$0.377677\pi$
$798$		0			0
$799$	24.1278			0.853580
$800$		0			0
$801$	38.2819	−	8.50699i	1.35262	−	0.300580i
$802$		0			0
$803$	−55.9026			−1.97276
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.76834	+	0.633197i	−0.203055	+	0.0222896i
$808$		0			0
$809$		−	41.6834i		−	1.46551i	−0.680491	−	0.732756i	$-0.738234\pi$
$809$		−	41.6834i		−	1.46551i	0.680491	−	0.732756i	$-0.261766\pi$
$810$		0			0
$811$		−	31.7652i		−	1.11543i	−0.830033	−	0.557714i	$-0.811679\pi$
$811$		−	31.7652i		−	1.11543i	0.830033	−	0.557714i	$-0.188321\pi$
$812$		0			0
$813$	0.498732	+	4.54338i	0.0174913	+	0.159343i
$814$		0			0
$815$	92.5333			3.24130
$816$		0			0
$817$		−	20.6371i		−	0.722000i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			14.1922i			0.495312i	0.968848	+	0.247656i	$0.0796602\pi$
$821$			14.1922i			0.495312i	−0.968848	+	0.247656i	$0.920340\pi$
$822$		0			0
$823$	14.2603			0.497084			0.248542	−	0.968621i	$-0.420049\pi$
$823$	14.2603			0.497084			0.248542	+	0.968621i	$0.420049\pi$
$824$		0			0
$825$	−7.92867	−	72.2291i	−0.276041	−	2.51469i
$826$		0			0
$827$		−	43.8269i		−	1.52401i	−0.647571	−	0.762006i	$-0.724215\pi$
$827$		−	43.8269i		−	1.52401i	0.647571	−	0.762006i	$-0.275785\pi$
$828$		0			0
$829$		−	19.5526i		−	0.679090i	−0.940590	−	0.339545i	$-0.889727\pi$
$829$		−	19.5526i		−	0.679090i	0.940590	−	0.339545i	$-0.110273\pi$
$830$		0			0
$831$	−20.5151	+	2.25196i	−0.711661	+	0.0781198i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−39.5162			−1.36752
$836$		0			0
$837$	−1.56466	−	4.59799i	−0.0540825	−	0.158930i
$838$		0			0
$839$	41.5510			1.43450			0.717251	−	0.696815i	$-0.245400\pi$
$839$	41.5510			1.43450			0.717251	+	0.696815i	$0.245400\pi$
$840$		0			0
$841$	27.4809			0.947617
$842$		0			0
$843$	4.51807	+	41.1590i	0.155611	+	1.41759i
$844$		0			0
$845$	−28.4746			−0.979557
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.76532	+	25.1917i	0.0949057	+	0.864578i
$850$		0			0
$851$		−	10.1388i		−	0.347555i
$852$		0			0
$853$			17.5344i			0.600367i	0.953882	+	0.300183i	$0.0970478\pi$
$853$			17.5344i			0.600367i	−0.953882	+	0.300183i	$0.902952\pi$
$854$		0			0
$855$	17.1690	+	77.2615i	0.587169	+	2.64229i
$856$		0			0
$857$	19.6428			0.670986			0.335493	−	0.942043i	$-0.391097\pi$
$857$	19.6428			0.670986			0.335493	+	0.942043i	$0.391097\pi$
$858$		0			0
$859$		−	13.4428i		−	0.458662i	−0.973348	−	0.229331i	$-0.926346\pi$
$859$		−	13.4428i		−	0.458662i	0.973348	−	0.229331i	$-0.0736538\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			22.5635i			0.768070i	0.923319	+	0.384035i	$0.125466\pi$
$863$			22.5635i			0.768070i	−0.923319	+	0.384035i	$0.874534\pi$
$864$		0			0
$865$	64.1161			2.18001
$866$		0			0
$867$	0.0473560	−	0.00519832i	0.00160829	−	0.000176544i
$868$		0			0
$869$			13.9648i			0.473725i
$870$		0			0
$871$			34.7854i			1.17866i
$872$		0			0
$873$	−4.59502	−	20.6778i	−0.155518	−	0.699837i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.57698			−0.323392			−0.161696	−	0.986841i	$-0.551696\pi$
$877$	−9.57698			−0.323392			−0.161696	+	0.986841i	$0.551696\pi$
$878$		0			0
$879$	−39.6852	+	4.35629i	−1.33855	+	0.146934i
$880$		0			0
$881$	−4.74822			−0.159971			−0.0799857	−	0.996796i	$-0.525487\pi$
$881$	−4.74822			−0.159971			−0.0799857	+	0.996796i	$0.525487\pi$
$882$		0			0
$883$	36.7930			1.23818			0.619092	−	0.785318i	$-0.287501\pi$
$883$	36.7930			1.23818			0.619092	+	0.785318i	$0.287501\pi$
$884$		0			0
$885$	−13.5883	+	1.49160i	−0.456764	+	0.0501395i
$886$		0			0
$887$	−34.4816			−1.15778			−0.578889	−	0.815406i	$-0.696514\pi$
$887$	−34.4816			−1.15778			−0.578889	+	0.815406i	$0.696514\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	14.3765	+	30.7500i	0.481630	+	1.03017i
$892$		0			0
$893$		−	38.4796i		−	1.28767i
$894$		0			0
$895$		−	8.16761i		−	0.273013i
$896$		0			0
$897$	−38.6657	+	4.24437i	−1.29101	+	0.141716i
$898$		0			0
$899$	1.15205			0.0384230
$900$		0			0
$901$			18.7980i			0.626252i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	30.4703i		−	1.01287i
$906$		0			0
$907$	−18.6067			−0.617826			−0.308913	−	0.951090i	$-0.599965\pi$
$907$	−18.6067			−0.617826			−0.308913	+	0.951090i	$0.599965\pi$
$908$		0			0
$909$	−25.7400	+	5.71994i	−0.853742	+	0.189718i
$910$		0			0
$911$		−	27.0277i		−	0.895467i	−0.894167	−	0.447733i	$-0.852231\pi$
$911$		−	27.0277i		−	0.895467i	0.894167	−	0.447733i	$-0.147769\pi$
$912$		0			0
$913$			34.5371i			1.14301i
$914$		0			0
$915$	−7.01731	−	63.9267i	−0.231985	−	2.11335i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	17.6917			0.583594			0.291797	−	0.956480i	$-0.405747\pi$
$919$	17.6917			0.583594			0.291797	+	0.956480i	$0.405747\pi$
$920$		0			0
$921$	1.14381	+	10.4200i	0.0376899	+	0.343350i
$922$		0			0
$923$	−35.7465			−1.17661
$924$		0			0
$925$	22.5087			0.740081
$926$		0			0
$927$	−11.0885	−	49.8988i	−0.364194	−	1.63889i
$928$		0			0
$929$	−22.7015			−0.744812			−0.372406	−	0.928070i	$-0.621467\pi$
$929$	−22.7015			−0.744812			−0.372406	+	0.928070i	$0.621467\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	17.3373	−	1.90314i	0.567599	−	0.0623059i
$934$		0			0
$935$		−	62.3916i		−	2.04042i
$936$		0			0
$937$		−	29.4147i		−	0.960935i	−0.877012	−	0.480468i	$-0.840467\pi$
$937$		−	29.4147i		−	0.960935i	0.877012	−	0.480468i	$-0.159533\pi$
$938$		0			0
$939$	−3.50449	−	31.9255i	−0.114365	−	1.04185i
$940$		0			0
$941$	41.1024			1.33990			0.669949	−	0.742407i	$-0.266316\pi$
$941$	41.1024			1.33990			0.669949	+	0.742407i	$0.266316\pi$
$942$		0			0
$943$			11.0991i			0.361435i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.960702i			0.0312186i	0.999878	+	0.0156093i	$0.00496880\pi$
$947$			0.960702i			0.0312186i	−0.999878	+	0.0156093i	$0.995031\pi$
$948$		0			0
$949$	−66.4366			−2.15662
$950$		0			0
$951$	4.04305	+	36.8316i	0.131105	+	1.19435i
$952$		0			0
$953$		−	14.0678i		−	0.455702i	−0.973696	−	0.227851i	$-0.926830\pi$
$953$		−	14.0678i		−	0.455702i	0.973696	−	0.227851i	$-0.0731699\pi$
$954$		0			0
$955$		−	89.6392i		−	2.90066i
$956$		0			0
$957$	−8.00354	+	0.878558i	−0.258718	+	0.0283997i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.1263			0.971816
$962$		0			0
$963$	−2.43460	−	10.9558i	−0.0784540	−	0.353047i
$964$		0			0
$965$	−69.5081			−2.23754
$966$		0			0
$967$	40.1695			1.29177			0.645883	−	0.763437i	$-0.276490\pi$
$967$	40.1695			1.29177			0.645883	+	0.763437i	$0.276490\pi$
$968$		0			0
$969$	5.11572	+	46.6035i	0.164340	+	1.49712i
$970$		0			0
$971$	−49.5568			−1.59035			−0.795177	−	0.606377i	$-0.792622\pi$
$971$	−49.5568			−1.59035			−0.795177	+	0.606377i	$0.792622\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−9.42270	−	85.8395i	−0.301768	−	2.74906i
$976$		0			0
$977$			15.1242i			0.483865i	0.970293	+	0.241932i	$0.0777812\pi$
$977$			15.1242i			0.483865i	−0.970293	+	0.241932i	$0.922219\pi$
$978$		0			0
$979$			49.3025i			1.57572i
$980$		0			0
$981$	−41.9283	+	9.31729i	−1.33867	+	0.297478i
$982$		0			0
$983$	−28.4908			−0.908715			−0.454358	−	0.890819i	$-0.650131\pi$
$983$	−28.4908			−0.908715			−0.454358	+	0.890819i	$0.650131\pi$
$984$		0			0
$985$			0.0355625i			0.00113312i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			15.7370i			0.500408i
$990$		0			0
$991$	10.2998			0.327183			0.163591	−	0.986528i	$-0.447692\pi$
$991$	10.2998			0.327183			0.163591	+	0.986528i	$0.447692\pi$
$992$		0			0
$993$	53.6923	−	5.89386i	1.70387	−	0.187036i
$994$		0			0
$995$			77.8712i			2.46868i
$996$		0			0
$997$		−	20.4447i		−	0.647489i	−0.946145	−	0.323744i	$-0.895058\pi$
$997$		−	20.4447i		−	0.647489i	0.946145	−	0.323744i	$-0.104942\pi$
$998$		0			0
$999$	−9.95444	+	3.38742i	−0.314945	+	0.107173i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.k.b.881.2	yes	24
3.2	odd	2	inner	1176.2.k.b.881.24	yes	24
4.3	odd	2		2352.2.k.j.881.23		24
7.2	even	3		1176.2.u.c.521.15		48
7.3	odd	6		1176.2.u.c.1097.8		48
7.4	even	3		1176.2.u.c.1097.17		48
7.5	odd	6		1176.2.u.c.521.10		48
7.6	odd	2	inner	1176.2.k.b.881.23	yes	24
12.11	even	2		2352.2.k.j.881.1		24
21.2	odd	6		1176.2.u.c.521.8		48
21.5	even	6		1176.2.u.c.521.17		48
21.11	odd	6		1176.2.u.c.1097.10		48
21.17	even	6		1176.2.u.c.1097.15		48
21.20	even	2	inner	1176.2.k.b.881.1	✓	24
28.27	even	2		2352.2.k.j.881.2		24
84.83	odd	2		2352.2.k.j.881.24		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.k.b.881.1	✓	24	21.20	even	2	inner
1176.2.k.b.881.2	yes	24	1.1	even	1	trivial
1176.2.k.b.881.23	yes	24	7.6	odd	2	inner
1176.2.k.b.881.24	yes	24	3.2	odd	2	inner
1176.2.u.c.521.8		48	21.2	odd	6
1176.2.u.c.521.10		48	7.5	odd	6
1176.2.u.c.521.15		48	7.2	even	3
1176.2.u.c.521.17		48	21.5	even	6
1176.2.u.c.1097.8		48	7.3	odd	6
1176.2.u.c.1097.10		48	21.11	odd	6
1176.2.u.c.1097.15		48	21.17	even	6
1176.2.u.c.1097.17		48	7.4	even	3
2352.2.k.j.881.1		24	12.11	even	2
2352.2.k.j.881.2		24	28.27	even	2
2352.2.k.j.881.23		24	4.3	odd	2
2352.2.k.j.881.24		24	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72171 + 0.188994i) q^{3} +4.01535 q^{5} +(2.92856 - 0.650785i) q^{9} +3.77164i q^{11} +4.48235i q^{13} +(-6.91326 + 0.758876i) q^{15} -4.11977 q^{17} +6.57031i q^{19} -5.01026i q^{23} +11.1230 q^{25} +(-4.91914 + 1.67394i) q^{27} -1.23251i q^{29} +0.934715i q^{31} +(-0.712817 - 6.49367i) q^{33} +2.02362 q^{37} +(-0.847136 - 7.71730i) q^{39} -2.21527 q^{41} -3.14096 q^{43} +(11.7592 - 2.61313i) q^{45} -5.85659 q^{47} +(7.09304 - 0.778611i) q^{51} -4.56288i q^{53} +15.1444i q^{55} +(-1.24175 - 11.3122i) q^{57} +1.96554 q^{59} +9.24698i q^{61} +17.9982i q^{65} +7.76053 q^{67} +(0.946908 + 8.62621i) q^{69} +7.97495i q^{71} +14.8218i q^{73} +(-19.1506 + 2.10218i) q^{75} +3.70259 q^{79} +(8.15296 - 3.81173i) q^{81} +9.15704 q^{83} -16.5423 q^{85} +(0.232938 + 2.12203i) q^{87} +13.0719 q^{89} +(-0.176655 - 1.60931i) q^{93} +26.3821i q^{95} -7.06073i q^{97} +(2.45453 + 11.0455i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 16 q^{15} + 8 q^{25} + 16 q^{37} - 64 q^{39} + 16 q^{43} + 48 q^{51} + 48 q^{57} + 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} - 56 q^{99}+O(q^{100}) \) 24 * q - 16 * q^15 + 8 * q^25 + 16 * q^37 - 64 * q^39 + 16 * q^43 + 48 * q^51 + 48 * q^57 + 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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