LMFDB - Embedded newform 1176.3.d.h.785.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(785,1176)] 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("1176.785"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.18
Character	$\chi$	$=$	1176.785
Dual form			1176.3.d.h.785.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.80925 + 2.39304i) q^{3} -2.81249i q^{5} +(-2.45325 + 8.65919i) q^{9} -7.08791i q^{11} -8.30821 q^{13} +(6.73039 - 5.08849i) q^{15} -1.76276i q^{17} +27.3699 q^{19} -1.97739i q^{23} +17.0899 q^{25} +(-25.1603 + 9.79590i) q^{27} +22.7057i q^{29} +31.4404 q^{31} +(16.9616 - 12.8238i) q^{33} +49.5354 q^{37} +(-15.0316 - 19.8819i) q^{39} +34.3551i q^{41} +30.0894 q^{43} +(24.3539 + 6.89974i) q^{45} +71.4067i q^{47} +(4.21835 - 3.18927i) q^{51} -52.7048i q^{53} -19.9347 q^{55} +(49.5189 + 65.4972i) q^{57} +36.7457i q^{59} +63.8926 q^{61} +23.3668i q^{65} -81.7727 q^{67} +(4.73197 - 3.57759i) q^{69} -124.811i q^{71} +97.3274 q^{73} +(30.9198 + 40.8968i) q^{75} +15.7828 q^{79} +(-68.9631 - 42.4863i) q^{81} -86.0514i q^{83} -4.95774 q^{85} +(-54.3355 + 41.0802i) q^{87} +71.8968i q^{89} +(56.8835 + 75.2381i) q^{93} -76.9776i q^{95} +162.154 q^{97} +(61.3756 + 17.3884i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 16 q^{15} - 152 q^{25} + 64 q^{37} - 16 q^{39} + 16 q^{43} + 248 q^{51} - 120 q^{57} - 416 q^{67} - 224 q^{79} - 344 q^{81} + 496 q^{85} - 464 q^{93}+O(q^{100}) $ 24 * q - 16 * q^15 - 152 * q^25 + 64 * q^37 - 16 * q^39 + 16 * q^43 + 248 * q^51 - 120 * q^57 - 416 * q^67 - 224 * q^79 - 344 * q^81 + 496 * q^85 - 464 * q^93

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
$3$	1.80925	+	2.39304i	0.603082	+	0.797679i
$3$	1.80925	+	2.39304i	0.603082	+	0.797679i
$4$		0			0
$5$		−	2.81249i		−	0.562498i	−0.959635	−	0.281249i	$-0.909251\pi$
$5$		−	2.81249i		−	0.562498i	0.959635	−	0.281249i	$-0.0907487\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.45325	+	8.65919i	−0.272583	+	0.962132i
$10$		0			0
$11$		−	7.08791i		−	0.644356i	−0.946679	−	0.322178i	$-0.895585\pi$
$11$		−	7.08791i		−	0.644356i	0.946679	−	0.322178i	$-0.104415\pi$
$12$		0			0
$13$	−8.30821			−0.639093			−0.319547	−	0.947571i	$-0.603531\pi$
$13$	−8.30821			−0.639093			−0.319547	+	0.947571i	$0.603531\pi$
$14$		0			0
$15$	6.73039	−	5.08849i	0.448693	−	0.339233i
$16$		0			0
$17$		−	1.76276i		−	0.103692i	−0.998655	−	0.0518458i	$-0.983490\pi$
$17$		−	1.76276i		−	0.103692i	0.998655	−	0.0518458i	$-0.0165104\pi$
$18$		0			0
$19$	27.3699			1.44052			0.720260	−	0.693704i	$-0.244022\pi$
$19$	27.3699			1.44052			0.720260	+	0.693704i	$0.244022\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.97739i		−	0.0859735i	−0.999076	−	0.0429868i	$-0.986313\pi$
$23$		−	1.97739i		−	0.0859735i	0.999076	−	0.0429868i	$-0.0136873\pi$
$24$		0			0
$25$	17.0899			0.683596
$26$		0			0
$27$	−25.1603	+	9.79590i	−0.931863	+	0.362811i
$28$		0			0
$29$			22.7057i			0.782954i	0.920188	+	0.391477i	$0.128036\pi$
$29$			22.7057i			0.782954i	−0.920188	+	0.391477i	$0.871964\pi$
$30$		0			0
$31$	31.4404			1.01421			0.507104	−	0.861885i	$-0.330716\pi$
$31$	31.4404			1.01421			0.507104	+	0.861885i	$0.330716\pi$
$32$		0			0
$33$	16.9616	−	12.8238i	0.513989	−	0.388600i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	49.5354			1.33879			0.669397	−	0.742905i	$-0.266552\pi$
$37$	49.5354			1.33879			0.669397	+	0.742905i	$0.266552\pi$
$38$		0			0
$39$	−15.0316	−	19.8819i	−0.385426	−	0.509791i
$40$		0			0
$41$			34.3551i			0.837930i	0.908002	+	0.418965i	$0.137607\pi$
$41$			34.3551i			0.837930i	−0.908002	+	0.418965i	$0.862393\pi$
$42$		0			0
$43$	30.0894			0.699753			0.349877	−	0.936796i	$-0.386224\pi$
$43$	30.0894			0.699753			0.349877	+	0.936796i	$0.386224\pi$
$44$		0			0
$45$	24.3539	+	6.89974i	0.541198	+	0.153328i
$46$		0			0
$47$			71.4067i			1.51929i	0.650338	+	0.759645i	$0.274627\pi$
$47$			71.4067i			1.51929i	−0.650338	+	0.759645i	$0.725373\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	4.21835	−	3.18927i	0.0827127	−	0.0625346i
$52$		0			0
$53$		−	52.7048i		−	0.994429i	−0.867628	−	0.497215i	$-0.834356\pi$
$53$		−	52.7048i		−	0.994429i	0.867628	−	0.497215i	$-0.165644\pi$
$54$		0			0
$55$	−19.9347			−0.362449
$56$		0			0
$57$	49.5189	+	65.4972i	0.868753	+	1.14907i
$58$		0			0
$59$			36.7457i			0.622808i	0.950278	+	0.311404i	$0.100799\pi$
$59$			36.7457i			0.622808i	−0.950278	+	0.311404i	$0.899201\pi$
$60$		0			0
$61$	63.8926			1.04742			0.523710	−	0.851897i	$-0.324548\pi$
$61$	63.8926			1.04742			0.523710	+	0.851897i	$0.324548\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			23.3668i			0.359489i
$66$		0			0
$67$	−81.7727			−1.22049			−0.610244	−	0.792214i	$-0.708929\pi$
$67$	−81.7727			−1.22049			−0.610244	+	0.792214i	$0.708929\pi$
$68$		0			0
$69$	4.73197	−	3.57759i	0.0685793	−	0.0518491i
$70$		0			0
$71$		−	124.811i		−	1.75791i	−0.476906	−	0.878954i	$-0.658242\pi$
$71$		−	124.811i		−	1.75791i	0.476906	−	0.878954i	$-0.341758\pi$
$72$		0			0
$73$	97.3274			1.33325			0.666626	−	0.745393i	$-0.267738\pi$
$73$	97.3274			1.33325			0.666626	+	0.745393i	$0.267738\pi$
$74$		0			0
$75$	30.9198	+	40.8968i	0.412265	+	0.545290i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	15.7828			0.199783			0.0998914	−	0.994998i	$-0.468150\pi$
$79$	15.7828			0.199783			0.0998914	+	0.994998i	$0.468150\pi$
$80$		0			0
$81$	−68.9631	−	42.4863i	−0.851397	−	0.524522i
$82$		0			0
$83$		−	86.0514i		−	1.03676i	−0.855149	−	0.518382i	$-0.826535\pi$
$83$		−	86.0514i		−	1.03676i	0.855149	−	0.518382i	$-0.173465\pi$
$84$		0			0
$85$	−4.95774			−0.0583264
$86$		0			0
$87$	−54.3355	+	41.0802i	−0.624546	+	0.472186i
$88$		0			0
$89$			71.8968i			0.807829i	0.914797	+	0.403914i	$0.132351\pi$
$89$			71.8968i			0.807829i	−0.914797	+	0.403914i	$0.867649\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	56.8835	+	75.2381i	0.611651	+	0.809012i
$94$		0			0
$95$		−	76.9776i		−	0.810290i
$96$		0			0
$97$	162.154			1.67169			0.835846	−	0.548964i	$-0.184977\pi$
$97$	162.154			1.67169			0.835846	+	0.548964i	$0.184977\pi$
$98$		0			0
$99$	61.3756	+	17.3884i	0.619955	+	0.175641i
$100$		0			0
$101$		−	21.8587i		−	0.216423i	−0.994128	−	0.108211i	$-0.965488\pi$
$101$		−	21.8587i		−	0.216423i	0.994128	−	0.108211i	$-0.0345123\pi$
$102$		0			0
$103$	41.8428			0.406241			0.203121	−	0.979154i	$-0.434892\pi$
$103$	41.8428			0.406241			0.203121	+	0.979154i	$0.434892\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			68.2464i			0.637817i	0.947786	+	0.318908i	$0.103316\pi$
$107$			68.2464i			0.637817i	−0.947786	+	0.318908i	$0.896684\pi$
$108$		0			0
$109$	−162.614			−1.49188			−0.745938	−	0.666016i	$-0.767998\pi$
$109$	−162.614			−1.49188			−0.745938	+	0.666016i	$0.767998\pi$
$110$		0			0
$111$	89.6218	+	118.540i	0.807403	+	1.06793i
$112$		0			0
$113$			170.565i			1.50942i	0.656058	+	0.754711i	$0.272223\pi$
$113$			170.565i			1.50942i	−0.656058	+	0.754711i	$0.727777\pi$
$114$		0			0
$115$	−5.56139			−0.0483600
$116$		0			0
$117$	20.3821	−	71.9424i	0.174206	−	0.614892i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	70.7615			0.584806
$122$		0			0
$123$	−82.2131	+	62.1569i	−0.668399	+	0.505341i
$124$		0			0
$125$		−	118.377i		−	0.947019i
$126$		0			0
$127$	18.2677			0.143840			0.0719199	−	0.997410i	$-0.477087\pi$
$127$	18.2677			0.143840			0.0719199	+	0.997410i	$0.477087\pi$
$128$		0			0
$129$	54.4391	+	72.0050i	0.422009	+	0.558178i
$130$		0			0
$131$			230.591i			1.76023i	0.474758	+	0.880117i	$0.342536\pi$
$131$			230.591i			1.76023i	−0.474758	+	0.880117i	$0.657464\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	27.5509	+	70.7631i	0.204080	+	0.524171i
$136$		0			0
$137$		−	155.061i		−	1.13183i	−0.824464	−	0.565915i	$-0.808523\pi$
$137$		−	155.061i		−	1.13183i	0.824464	−	0.565915i	$-0.191477\pi$
$138$		0			0
$139$	4.64105			0.0333888			0.0166944	−	0.999861i	$-0.494686\pi$
$139$	4.64105			0.0333888			0.0166944	+	0.999861i	$0.494686\pi$
$140$		0			0
$141$	−170.879	+	129.192i	−1.21191	+	0.916257i
$142$		0			0
$143$			58.8879i			0.411803i
$144$		0			0
$145$	63.8595			0.440410
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	191.548i		−	1.28555i	−0.766053	−	0.642777i	$-0.777782\pi$
$149$		−	191.548i		−	1.28555i	0.766053	−	0.642777i	$-0.222218\pi$
$150$		0			0
$151$	−294.021			−1.94716			−0.973580	−	0.228346i	$-0.926668\pi$
$151$	−294.021			−1.94716			−0.973580	+	0.228346i	$0.926668\pi$
$152$		0			0
$153$	15.2641	+	4.32449i	0.0997651	+	0.0282646i
$154$		0			0
$155$		−	88.4259i		−	0.570490i
$156$		0			0
$157$	27.4748			0.174999			0.0874995	−	0.996165i	$-0.472112\pi$
$157$	27.4748			0.174999			0.0874995	+	0.996165i	$0.472112\pi$
$158$		0			0
$159$	126.124	−	95.3559i	0.793235	−	0.599723i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−38.7573			−0.237775			−0.118887	−	0.992908i	$-0.537933\pi$
$163$	−38.7573			−0.237775			−0.118887	+	0.992908i	$0.537933\pi$
$164$		0			0
$165$	−36.0668	−	47.7044i	−0.218587	−	0.289118i
$166$		0			0
$167$			207.180i			1.24060i	0.784364	+	0.620301i	$0.212989\pi$
$167$			207.180i			1.24060i	−0.784364	+	0.620301i	$0.787011\pi$
$168$		0			0
$169$	−99.9737			−0.591560
$170$		0			0
$171$	−67.1452	+	237.001i	−0.392662	+	1.38597i
$172$		0			0
$173$			165.438i			0.956288i	0.878282	+	0.478144i	$0.158690\pi$
$173$			165.438i			0.956288i	−0.878282	+	0.478144i	$0.841310\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−87.9337	+	66.4820i	−0.496801	+	0.375605i
$178$		0			0
$179$			123.667i			0.690879i	0.938441	+	0.345440i	$0.112270\pi$
$179$			123.667i			0.690879i	−0.938441	+	0.345440i	$0.887730\pi$
$180$		0			0
$181$	−2.95926			−0.0163495			−0.00817474	−	0.999967i	$-0.502602\pi$
$181$	−2.95926			−0.0163495			−0.00817474	+	0.999967i	$0.502602\pi$
$182$		0			0
$183$	115.598	+	152.897i	0.631681	+	0.835505i
$184$		0			0
$185$		−	139.318i		−	0.753069i
$186$		0			0
$187$	−12.4943			−0.0668143
$188$		0			0
$189$		0			0
$190$		0			0
$191$			95.4544i			0.499761i	0.968277	+	0.249881i	$0.0803914\pi$
$191$			95.4544i			0.499761i	−0.968277	+	0.249881i	$0.919609\pi$
$192$		0			0
$193$	177.002			0.917108			0.458554	−	0.888667i	$-0.348368\pi$
$193$	177.002			0.917108			0.458554	+	0.888667i	$0.348368\pi$
$194$		0			0
$195$	−55.9175	+	42.2762i	−0.286756	+	0.216801i
$196$		0			0
$197$		−	137.448i		−	0.697706i	−0.937177	−	0.348853i	$-0.886571\pi$
$197$		−	137.448i		−	0.697706i	0.937177	−	0.348853i	$-0.113429\pi$
$198$		0			0
$199$	−140.795			−0.707511			−0.353756	−	0.935338i	$-0.615096\pi$
$199$	−140.795			−0.707511			−0.353756	+	0.935338i	$0.615096\pi$
$200$		0			0
$201$	−147.947	−	195.685i	−0.736055	−	0.973557i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	96.6235			0.471334
$206$		0			0
$207$	17.1226	+	4.85103i	0.0827179	+	0.0234350i
$208$		0			0
$209$		−	193.995i		−	0.928208i
$210$		0			0
$211$	324.014			1.53561			0.767806	−	0.640682i	$-0.221348\pi$
$211$	324.014			1.53561			0.767806	+	0.640682i	$0.221348\pi$
$212$		0			0
$213$	298.678	−	225.815i	1.40225	−	1.06016i
$214$		0			0
$215$		−	84.6261i		−	0.393610i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	176.089	+	232.908i	0.804061	+	1.06351i
$220$		0			0
$221$			14.6454i			0.0662686i
$222$		0			0
$223$	−126.830			−0.568745			−0.284372	−	0.958714i	$-0.591785\pi$
$223$	−126.830			−0.568745			−0.284372	+	0.958714i	$0.591785\pi$
$224$		0			0
$225$	−41.9258	+	147.985i	−0.186337	+	0.657710i
$226$		0			0
$227$			169.535i			0.746852i	0.927660	+	0.373426i	$0.121817\pi$
$227$			169.535i			0.746852i	−0.927660	+	0.373426i	$0.878183\pi$
$228$		0			0
$229$	232.555			1.01552			0.507762	−	0.861497i	$-0.330473\pi$
$229$	232.555			1.01552			0.507762	+	0.861497i	$0.330473\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	198.859i		−	0.853470i	−0.904377	−	0.426735i	$-0.859664\pi$
$233$		−	198.859i		−	0.853470i	0.904377	−	0.426735i	$-0.140336\pi$
$234$		0			0
$235$	200.831			0.854598
$236$		0			0
$237$	28.5551	+	37.7689i	0.120486	+	0.159363i
$238$		0			0
$239$		−	463.599i		−	1.93974i	−0.243615	−	0.969872i	$-0.578333\pi$
$239$		−	463.599i		−	1.93974i	0.243615	−	0.969872i	$-0.421667\pi$
$240$		0			0
$241$	−95.3635			−0.395699			−0.197850	−	0.980232i	$-0.563396\pi$
$241$	−95.3635			−0.395699			−0.197850	+	0.980232i	$0.563396\pi$
$242$		0			0
$243$	−23.1000	−	241.900i	−0.0950619	−	0.995471i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−227.395			−0.920627
$248$		0			0
$249$	205.924	−	155.688i	0.827005	−	0.625254i
$250$		0			0
$251$		−	154.108i		−	0.613975i	−0.951714	−	0.306988i	$-0.900679\pi$
$251$		−	154.108i		−	0.613975i	0.951714	−	0.306988i	$-0.0993210\pi$
$252$		0			0
$253$	−14.0156			−0.0553975
$254$		0			0
$255$	−8.96978	−	11.8641i	−0.0351756	−	0.0465257i
$256$		0			0
$257$			426.400i			1.65915i	0.558398	+	0.829573i	$0.311416\pi$
$257$			426.400i			1.65915i	−0.558398	+	0.829573i	$0.688584\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−196.613	−	55.7027i	−0.753305	−	0.213420i
$262$		0			0
$263$		−	101.054i		−	0.384235i	−0.981372	−	0.192118i	$-0.938465\pi$
$263$		−	101.054i		−	0.384235i	0.981372	−	0.192118i	$-0.0615355\pi$
$264$		0			0
$265$	−148.232			−0.559365
$266$		0			0
$267$	−172.052	+	130.079i	−0.644388	+	0.487187i
$268$		0			0
$269$		−	482.835i		−	1.79492i	−0.441091	−	0.897462i	$-0.645408\pi$
$269$		−	482.835i		−	1.79492i	0.441091	−	0.897462i	$-0.354592\pi$
$270$		0			0
$271$	−322.665			−1.19065			−0.595324	−	0.803486i	$-0.702976\pi$
$271$	−322.665			−1.19065			−0.595324	+	0.803486i	$0.702976\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	121.132i		−	0.440479i
$276$		0			0
$277$	348.095			1.25666			0.628331	−	0.777946i	$-0.283738\pi$
$277$	348.095			1.25666			0.628331	+	0.777946i	$0.283738\pi$
$278$		0			0
$279$	−77.1312	+	272.249i	−0.276456	+	0.975801i
$280$		0			0
$281$		−	198.736i		−	0.707244i	−0.935388	−	0.353622i	$-0.884950\pi$
$281$		−	198.736i		−	0.707244i	0.935388	−	0.353622i	$-0.115050\pi$
$282$		0			0
$283$	204.328			0.722007			0.361004	−	0.932564i	$-0.382434\pi$
$283$	204.328			0.722007			0.361004	+	0.932564i	$0.382434\pi$
$284$		0			0
$285$	184.210	−	139.271i	0.646351	−	0.488672i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	285.893			0.989248
$290$		0			0
$291$	293.377	+	388.041i	1.00817	+	1.33347i
$292$		0			0
$293$		−	326.520i		−	1.11440i	−0.830377	−	0.557202i	$-0.811875\pi$
$293$		−	326.520i		−	1.11440i	0.830377	−	0.557202i	$-0.188125\pi$
$294$		0			0
$295$	103.347			0.350328
$296$		0			0
$297$	69.4325	+	178.334i	0.233779	+	0.600451i
$298$		0			0
$299$			16.4286i			0.0549451i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	52.3087	−	39.5478i	0.172636	−	0.130521i
$304$		0			0
$305$		−	179.697i		−	0.589172i
$306$		0			0
$307$	−435.182			−1.41753			−0.708766	−	0.705444i	$-0.750748\pi$
$307$	−435.182			−1.41753			−0.708766	+	0.705444i	$0.750748\pi$
$308$		0			0
$309$	75.7040	+	100.131i	0.244997	+	0.324050i
$310$		0			0
$311$		−	218.242i		−	0.701744i	−0.936423	−	0.350872i	$-0.885885\pi$
$311$		−	218.242i		−	0.701744i	0.936423	−	0.350872i	$-0.114115\pi$
$312$		0			0
$313$	−276.065			−0.881997			−0.440998	−	0.897508i	$-0.645376\pi$
$313$	−276.065			−0.881997			−0.440998	+	0.897508i	$0.645376\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	442.278i		−	1.39520i	−0.716488	−	0.697600i	$-0.754251\pi$
$317$		−	442.278i		−	1.39520i	0.716488	−	0.697600i	$-0.245749\pi$
$318$		0			0
$319$	160.936			0.504501
$320$		0			0
$321$	−163.316	+	123.475i	−0.508773	+	0.384656i
$322$		0			0
$323$		−	48.2465i		−	0.149370i
$324$		0			0
$325$	−141.986			−0.436881
$326$		0			0
$327$	−294.210	−	389.142i	−0.899724	−	1.19004i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−483.615			−1.46107			−0.730536	−	0.682874i	$-0.760730\pi$
$331$	−483.615			−1.46107			−0.730536	+	0.682874i	$0.760730\pi$
$332$		0			0
$333$	−121.523	+	428.936i	−0.364933	+	1.28810i
$334$		0			0
$335$			229.985i			0.686522i
$336$		0			0
$337$	−174.082			−0.516563			−0.258282	−	0.966070i	$-0.583156\pi$
$337$	−174.082			−0.516563			−0.258282	+	0.966070i	$0.583156\pi$
$338$		0			0
$339$	−408.167	+	308.594i	−1.20403	+	0.910306i
$340$		0			0
$341$		−	222.847i		−	0.653510i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−10.0619	−	13.3086i	−0.0291650	−	0.0385757i
$346$		0			0
$347$			464.621i			1.33897i	0.742827	+	0.669483i	$0.233484\pi$
$347$			464.621i			1.33897i	−0.742827	+	0.669483i	$0.766516\pi$
$348$		0			0
$349$	324.129			0.928735			0.464368	−	0.885643i	$-0.346282\pi$
$349$	324.129			0.928735			0.464368	+	0.885643i	$0.346282\pi$
$350$		0			0
$351$	209.037	−	81.3864i	0.595547	−	0.231870i
$352$		0			0
$353$		−	311.534i		−	0.882531i	−0.897377	−	0.441266i	$-0.854530\pi$
$353$		−	311.534i		−	0.882531i	0.897377	−	0.441266i	$-0.145470\pi$
$354$		0			0
$355$	−351.031			−0.988820
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	386.977i		−	1.07793i	−0.842328	−	0.538965i	$-0.818816\pi$
$359$		−	386.977i		−	1.07793i	0.842328	−	0.538965i	$-0.181184\pi$
$360$		0			0
$361$	388.111			1.07510
$362$		0			0
$363$	128.025	+	169.335i	0.352686	+	0.466487i
$364$		0			0
$365$		−	273.732i		−	0.749951i
$366$		0			0
$367$	−16.3898			−0.0446589			−0.0223294	−	0.999751i	$-0.507108\pi$
$367$	−16.3898			−0.0446589			−0.0223294	+	0.999751i	$0.507108\pi$
$368$		0			0
$369$	−297.488	−	84.2817i	−0.806199	−	0.228406i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−73.4946			−0.197037			−0.0985183	−	0.995135i	$-0.531410\pi$
$373$	−73.4946			−0.197037			−0.0985183	+	0.995135i	$0.531410\pi$
$374$		0			0
$375$	283.282	−	214.174i	0.755417	−	0.571131i
$376$		0			0
$377$		−	188.643i		−	0.500380i
$378$		0			0
$379$	−355.783			−0.938742			−0.469371	−	0.883001i	$-0.655519\pi$
$379$	−355.783			−0.938742			−0.469371	+	0.883001i	$0.655519\pi$
$380$		0			0
$381$	33.0507	+	43.7152i	0.0867472	+	0.114738i
$382$		0			0
$383$			82.1504i			0.214492i	0.994233	+	0.107246i	$0.0342032\pi$
$383$			82.1504i			0.214492i	−0.994233	+	0.107246i	$0.965797\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−73.8168	+	260.550i	−0.190741	+	0.673255i
$388$		0			0
$389$			416.878i			1.07167i	0.844324	+	0.535833i	$0.180002\pi$
$389$			416.878i			1.07167i	−0.844324	+	0.535833i	$0.819998\pi$
$390$		0			0
$391$	−3.48566			−0.00891474
$392$		0			0
$393$	−551.812	+	417.195i	−1.40410	+	1.06157i
$394$		0			0
$395$		−	44.3891i		−	0.112377i
$396$		0			0
$397$	23.2785			0.0586361			0.0293180	−	0.999570i	$-0.490666\pi$
$397$	23.2785			0.0586361			0.0293180	+	0.999570i	$0.490666\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			473.312i			1.18033i	0.807283	+	0.590165i	$0.200937\pi$
$401$			473.312i			1.18033i	−0.807283	+	0.590165i	$0.799063\pi$
$402$		0			0
$403$	−261.214			−0.648173
$404$		0			0
$405$	−119.492	+	193.958i	−0.295043	+	0.478909i
$406$		0			0
$407$		−	351.103i		−	0.862660i
$408$		0			0
$409$	−747.729			−1.82819			−0.914095	−	0.405501i	$-0.867097\pi$
$409$	−747.729			−1.82819			−0.914095	+	0.405501i	$0.867097\pi$
$410$		0			0
$411$	371.066	−	280.543i	0.902837	−	0.682587i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−242.019			−0.583178
$416$		0			0
$417$	8.39681	+	11.1062i	0.0201362	+	0.0266336i
$418$		0			0
$419$		−	614.107i		−	1.46565i	−0.680418	−	0.732824i	$-0.738202\pi$
$419$		−	614.107i		−	1.46565i	0.680418	−	0.732824i	$-0.261798\pi$
$420$		0			0
$421$	−255.585			−0.607090			−0.303545	−	0.952817i	$-0.598170\pi$
$421$	−255.585			−0.607090			−0.303545	+	0.952817i	$0.598170\pi$
$422$		0			0
$423$	−618.324	−	175.178i	−1.46176	−	0.414133i
$424$		0			0
$425$		−	30.1254i		−	0.0708832i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−140.921	+	106.543i	−0.328487	+	0.248351i
$430$		0			0
$431$			223.697i			0.519018i	0.965741	+	0.259509i	$0.0835607\pi$
$431$			223.697i			0.519018i	−0.965741	+	0.259509i	$0.916439\pi$
$432$		0			0
$433$	90.0321			0.207926			0.103963	−	0.994581i	$-0.466848\pi$
$433$	90.0321			0.207926			0.103963	+	0.994581i	$0.466848\pi$
$434$		0			0
$435$	115.538	+	152.818i	0.265604	+	0.351306i
$436$		0			0
$437$		−	54.1210i		−	0.123847i
$438$		0			0
$439$	−100.225			−0.228304			−0.114152	−	0.993463i	$-0.536415\pi$
$439$	−100.225			−0.228304			−0.114152	+	0.993463i	$0.536415\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	261.472i		−	0.590229i	−0.955462	−	0.295115i	$-0.904642\pi$
$443$		−	261.472i		−	0.590229i	0.955462	−	0.295115i	$-0.0953579\pi$
$444$		0			0
$445$	202.209			0.454402
$446$		0			0
$447$	458.380	−	346.557i	1.02546	−	0.775295i
$448$		0			0
$449$		−	92.5691i		−	0.206167i	−0.994673	−	0.103084i	$-0.967129\pi$
$449$		−	92.5691i		−	0.206167i	0.994673	−	0.103084i	$-0.0328709\pi$
$450$		0			0
$451$	243.506			0.539925
$452$		0			0
$453$	−531.957	−	703.604i	−1.17430	−	1.55321i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−525.494			−1.14988			−0.574939	−	0.818196i	$-0.694974\pi$
$457$	−525.494			−1.14988			−0.574939	+	0.818196i	$0.694974\pi$
$458$		0			0
$459$	17.2678	+	44.3515i	0.0376205	+	0.0966264i
$460$		0			0
$461$			60.1512i			0.130480i	0.997870	+	0.0652399i	$0.0207813\pi$
$461$			60.1512i			0.130480i	−0.997870	+	0.0652399i	$0.979219\pi$
$462$		0			0
$463$	−731.890			−1.58076			−0.790379	−	0.612619i	$-0.790116\pi$
$463$	−731.890			−1.58076			−0.790379	+	0.612619i	$0.790116\pi$
$464$		0			0
$465$	211.606	−	159.984i	0.455068	−	0.344052i
$466$		0			0
$467$			423.996i			0.907914i	0.891024	+	0.453957i	$0.149988\pi$
$467$			423.996i			0.907914i	−0.891024	+	0.453957i	$0.850012\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	49.7088	+	65.7483i	0.105539	+	0.139593i
$472$		0			0
$473$		−	213.271i		−	0.450890i
$474$		0			0
$475$	467.749			0.984734
$476$		0			0
$477$	456.381	+	129.298i	0.956773	+	0.271065i
$478$		0			0
$479$		−	10.0436i		−	0.0209679i	−0.999945	−	0.0104840i	$-0.996663\pi$
$479$		−	10.0436i		−	0.0209679i	0.999945	−	0.0104840i	$-0.00333721\pi$
$480$		0			0
$481$	−411.550			−0.855614
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	456.057i		−	0.940324i
$486$		0			0
$487$	−216.513			−0.444585			−0.222292	−	0.974980i	$-0.571354\pi$
$487$	−216.513			−0.444585			−0.222292	+	0.974980i	$0.571354\pi$
$488$		0			0
$489$	−70.1215	−	92.7476i	−0.143398	−	0.189668i
$490$		0			0
$491$		−	26.7752i		−	0.0545320i	−0.999628	−	0.0272660i	$-0.991320\pi$
$491$		−	26.7752i		−	0.0545320i	0.999628	−	0.0272660i	$-0.00868012\pi$
$492$		0			0
$493$	40.0246			0.0811858
$494$		0			0
$495$	48.9048	−	172.618i	0.0987975	−	0.348724i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	242.705			0.486382			0.243191	−	0.969978i	$-0.421806\pi$
$499$	242.705			0.486382			0.243191	+	0.969978i	$0.421806\pi$
$500$		0			0
$501$	−495.790	+	374.841i	−0.989602	+	0.748185i
$502$		0			0
$503$			22.2499i			0.0442343i	0.999755	+	0.0221171i	$0.00704068\pi$
$503$			22.2499i			0.0442343i	−0.999755	+	0.0221171i	$0.992959\pi$
$504$		0			0
$505$	−61.4774			−0.121737
$506$		0			0
$507$	−180.877	−	239.241i	−0.356759	−	0.471875i
$508$		0			0
$509$		−	683.092i		−	1.34203i	−0.741445	−	0.671014i	$-0.765859\pi$
$509$		−	683.092i		−	1.34203i	0.741445	−	0.671014i	$-0.234141\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−688.635	+	268.113i	−1.34237	+	0.522637i
$514$		0			0
$515$		−	117.683i		−	0.228510i
$516$		0			0
$517$	506.124			0.978964
$518$		0			0
$519$	−395.899	+	299.318i	−0.762811	+	0.576720i
$520$		0			0
$521$			791.821i			1.51981i	0.650034	+	0.759905i	$0.274755\pi$
$521$			791.821i			1.51981i	−0.650034	+	0.759905i	$0.725245\pi$
$522$		0			0
$523$	−556.281			−1.06363			−0.531817	−	0.846859i	$-0.678491\pi$
$523$	−556.281			−1.06363			−0.531817	+	0.846859i	$0.678491\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	55.4219i		−	0.105165i
$528$		0			0
$529$	525.090			0.992609
$530$		0			0
$531$	−318.188	−	90.1463i	−0.599224	−	0.169767i
$532$		0			0
$533$		−	285.430i		−	0.535515i
$534$		0			0
$535$	191.942			0.358771
$536$		0			0
$537$	−295.941	+	223.745i	−0.551100	+	0.416657i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	936.780			1.73157			0.865785	−	0.500416i	$-0.166820\pi$
$541$	936.780			1.73157			0.865785	+	0.500416i	$0.166820\pi$
$542$		0			0
$543$	−5.35403	−	7.08161i	−0.00986009	−	0.0130416i
$544$		0			0
$545$			457.352i			0.839177i
$546$		0			0
$547$	99.8090			0.182466			0.0912331	−	0.995830i	$-0.470919\pi$
$547$	99.8090			0.182466			0.0912331	+	0.995830i	$0.470919\pi$
$548$		0			0
$549$	−156.745	+	553.258i	−0.285509	+	1.00776i
$550$		0			0
$551$			621.452i			1.12786i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	333.393	−	252.060i	0.600708	−	0.454163i
$556$		0			0
$557$			604.501i			1.08528i	0.839965	+	0.542640i	$0.182575\pi$
$557$			604.501i			1.08528i	−0.839965	+	0.542640i	$0.817425\pi$
$558$		0			0
$559$	−249.989			−0.447207
$560$		0			0
$561$	−22.6052	−	29.8993i	−0.0402946	−	0.0532964i
$562$		0			0
$563$		−	1011.08i		−	1.79587i	−0.440127	−	0.897935i	$-0.645067\pi$
$563$		−	1011.08i		−	1.79587i	0.440127	−	0.897935i	$-0.354933\pi$
$564$		0			0
$565$	479.711			0.849047
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	15.8402i		−	0.0278387i	−0.999903	−	0.0139193i	$-0.995569\pi$
$569$		−	15.8402i		−	0.0278387i	0.999903	−	0.0139193i	$-0.00443080\pi$
$570$		0			0
$571$	124.057			0.217262			0.108631	−	0.994082i	$-0.465353\pi$
$571$	124.057			0.217262			0.108631	+	0.994082i	$0.465353\pi$
$572$		0			0
$573$	−228.426	+	172.701i	−0.398649	+	0.301397i
$574$		0			0
$575$		−	33.7934i		−	0.0587712i
$576$		0			0
$577$	−655.653			−1.13631			−0.568156	−	0.822921i	$-0.692343\pi$
$577$	−655.653			−1.13631			−0.568156	+	0.822921i	$0.692343\pi$
$578$		0			0
$579$	320.240	+	423.572i	0.553092	+	0.731558i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−373.567			−0.640766
$584$		0			0
$585$	−202.337	−	57.3245i	−0.345876	−	0.0979906i
$586$		0			0
$587$		−	220.023i		−	0.374826i	−0.982281	−	0.187413i	$-0.939990\pi$
$587$		−	220.023i		−	0.374826i	0.982281	−	0.187413i	$-0.0600103\pi$
$588$		0			0
$589$	860.521			1.46099
$590$		0			0
$591$	328.919	−	248.678i	0.556546	−	0.420774i
$592$		0			0
$593$			910.052i			1.53466i	0.641254	+	0.767329i	$0.278414\pi$
$593$			910.052i			1.53466i	−0.641254	+	0.767329i	$0.721586\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−254.733	−	336.927i	−0.426688	−	0.564367i
$598$		0			0
$599$		−	159.533i		−	0.266332i	−0.991094	−	0.133166i	$-0.957486\pi$
$599$		−	159.533i		−	0.266332i	0.991094	−	0.133166i	$-0.0425144\pi$
$600$		0			0
$601$	−283.623			−0.471918			−0.235959	−	0.971763i	$-0.575823\pi$
$601$	−283.623			−0.471918			−0.235959	+	0.971763i	$0.575823\pi$
$602$		0			0
$603$	200.609	−	708.085i	0.332685	−	1.17427i
$604$		0			0
$605$		−	199.016i		−	0.328952i
$606$		0			0
$607$	664.160			1.09417			0.547084	−	0.837078i	$-0.315738\pi$
$607$	664.160			1.09417			0.547084	+	0.837078i	$0.315738\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	593.261i		−	0.970968i
$612$		0			0
$613$	−993.227			−1.62027			−0.810136	−	0.586242i	$-0.800607\pi$
$613$	−993.227			−1.62027			−0.810136	+	0.586242i	$0.800607\pi$
$614$		0			0
$615$	174.816	+	231.223i	0.284253	+	0.375973i
$616$		0			0
$617$			232.977i			0.377596i	0.982016	+	0.188798i	$0.0604592\pi$
$617$			232.977i			0.377596i	−0.982016	+	0.188798i	$0.939541\pi$
$618$		0			0
$619$	255.888			0.413389			0.206694	−	0.978406i	$-0.433729\pi$
$619$	255.888			0.413389			0.206694	+	0.978406i	$0.433729\pi$
$620$		0			0
$621$	19.3703	+	49.7517i	0.0311921	+	0.0801155i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	94.3120			0.150899
$626$		0			0
$627$	464.238	−	350.986i	0.740412	−	0.559786i
$628$		0			0
$629$		−	87.3190i		−	0.138822i
$630$		0			0
$631$	−184.423			−0.292270			−0.146135	−	0.989265i	$-0.546683\pi$
$631$	−184.423			−0.292270			−0.146135	+	0.989265i	$0.546683\pi$
$632$		0			0
$633$	586.222	+	775.378i	0.926101	+	1.22493i
$634$		0			0
$635$		−	51.3776i		−	0.0809096i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1080.77	+	306.194i	1.69134	+	0.479176i
$640$		0			0
$641$		−	601.015i		−	0.937621i	−0.883299	−	0.468810i	$-0.844683\pi$
$641$		−	601.015i		−	0.937621i	0.883299	−	0.468810i	$-0.155317\pi$
$642$		0			0
$643$	−191.858			−0.298379			−0.149189	−	0.988809i	$-0.547666\pi$
$643$	−191.858			−0.298379			−0.149189	+	0.988809i	$0.547666\pi$
$644$		0			0
$645$	202.513	−	153.110i	0.313974	−	0.237379i
$646$		0			0
$647$			63.0112i			0.0973898i	0.998814	+	0.0486949i	$0.0155062\pi$
$647$			63.0112i			0.0973898i	−0.998814	+	0.0486949i	$0.984494\pi$
$648$		0			0
$649$	260.450			0.401310
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1222.44i			1.87203i	0.351954	+	0.936017i	$0.385517\pi$
$653$			1222.44i			1.87203i	−0.351954	+	0.936017i	$0.614483\pi$
$654$		0			0
$655$	648.534			0.990128
$656$		0			0
$657$	−238.768	+	842.776i	−0.363422	+	1.28276i
$658$		0			0
$659$			640.698i			0.972227i	0.873896	+	0.486114i	$0.161586\pi$
$659$			640.698i			0.972227i	−0.873896	+	0.486114i	$0.838414\pi$
$660$		0			0
$661$	520.336			0.787196			0.393598	−	0.919283i	$-0.371230\pi$
$661$	520.336			0.787196			0.393598	+	0.919283i	$0.371230\pi$
$662$		0			0
$663$	−35.0469	+	26.4971i	−0.0528611	+	0.0399654i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	44.8980			0.0673133
$668$		0			0
$669$	−229.467	−	303.509i	−0.343000	−	0.453676i
$670$		0			0
$671$		−	452.865i		−	0.674911i
$672$		0			0
$673$	517.231			0.768546			0.384273	−	0.923220i	$-0.374452\pi$
$673$	517.231			0.768546			0.384273	+	0.923220i	$0.374452\pi$
$674$		0			0
$675$	−429.987	+	167.411i	−0.637018	+	0.248016i
$676$		0			0
$677$		−	637.100i		−	0.941064i	−0.882383	−	0.470532i	$-0.844062\pi$
$677$		−	637.100i		−	0.941064i	0.882383	−	0.470532i	$-0.155938\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−405.704	+	306.731i	−0.595748	+	0.450413i
$682$		0			0
$683$		−	1289.75i		−	1.88836i	−0.329430	−	0.944180i	$-0.606857\pi$
$683$		−	1289.75i		−	1.88836i	0.329430	−	0.944180i	$-0.393143\pi$
$684$		0			0
$685$	−436.107			−0.636652
$686$		0			0
$687$	420.749	+	556.512i	0.612444	+	0.810062i
$688$		0			0
$689$			437.882i			0.635533i
$690$		0			0
$691$	152.458			0.220634			0.110317	−	0.993896i	$-0.464813\pi$
$691$	152.458			0.220634			0.110317	+	0.993896i	$0.464813\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	13.0529i		−	0.0187812i
$696$		0			0
$697$	60.5598			0.0868864
$698$		0			0
$699$	475.876	−	359.784i	0.680795	−	0.514713i
$700$		0			0
$701$		−	444.350i		−	0.633880i	−0.948446	−	0.316940i	$-0.897345\pi$
$701$		−	444.350i		−	0.633880i	0.948446	−	0.316940i	$-0.102655\pi$
$702$		0			0
$703$	1355.78			1.92856
$704$		0			0
$705$	363.352	+	480.595i	0.515393	+	0.681695i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1368.53			1.93022			0.965110	−	0.261845i	$-0.0843309\pi$
$709$	1368.53			1.93022			0.965110	+	0.261845i	$0.0843309\pi$
$710$		0			0
$711$	−38.7193	+	136.667i	−0.0544575	+	0.192218i
$712$		0			0
$713$		−	62.1700i		−	0.0871950i
$714$		0			0
$715$	165.622			0.231639
$716$		0			0
$717$	1109.41	−	838.765i	1.54729	−	1.16983i
$718$		0			0
$719$		−	581.443i		−	0.808683i	−0.914608	−	0.404342i	$-0.867501\pi$
$719$		−	581.443i		−	0.808683i	0.914608	−	0.404342i	$-0.132499\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−172.536	−	228.208i	−0.238639	−	0.315641i
$724$		0			0
$725$			388.038i			0.535224i
$726$		0			0
$727$	−167.092			−0.229838			−0.114919	−	0.993375i	$-0.536661\pi$
$727$	−167.092			−0.229838			−0.114919	+	0.993375i	$0.536661\pi$
$728$		0			0
$729$	537.081	−	492.935i	0.736736	−	0.676180i
$730$		0			0
$731$		−	53.0403i		−	0.0725586i
$732$		0			0
$733$	879.205			1.19946			0.599731	−	0.800202i	$-0.295274\pi$
$733$	879.205			1.19946			0.599731	+	0.800202i	$0.295274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			579.598i			0.786428i
$738$		0			0
$739$	−292.338			−0.395586			−0.197793	−	0.980244i	$-0.563377\pi$
$739$	−292.338			−0.395586			−0.197793	+	0.980244i	$0.563377\pi$
$740$		0			0
$741$	−411.413	−	544.164i	−0.555214	−	0.734365i
$742$		0			0
$743$			540.525i			0.727490i	0.931499	+	0.363745i	$0.118502\pi$
$743$			540.525i			0.727490i	−0.931499	+	0.363745i	$0.881498\pi$
$744$		0			0
$745$	−538.726			−0.723122
$746$		0			0
$747$	745.136	+	211.106i	0.997504	+	0.282605i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−786.662			−1.04749			−0.523743	−	0.851876i	$-0.675465\pi$
$751$	−786.662			−1.04749			−0.523743	+	0.851876i	$0.675465\pi$
$752$		0			0
$753$	368.786	−	278.819i	0.489755	−	0.370278i
$754$		0			0
$755$			826.932i			1.09527i
$756$		0			0
$757$	703.078			0.928769			0.464384	−	0.885634i	$-0.346276\pi$
$757$	703.078			0.928769			0.464384	+	0.885634i	$0.346276\pi$
$758$		0			0
$759$	−25.3576	−	33.5398i	−0.0334093	−	0.0441895i
$760$		0			0
$761$			778.050i			1.02240i	0.859460	+	0.511202i	$0.170800\pi$
$761$			778.050i			1.02240i	−0.859460	+	0.511202i	$0.829200\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	12.1626	−	42.9300i	0.0158988	−	0.0561177i
$766$		0			0
$767$		−	305.291i		−	0.398032i
$768$		0			0
$769$	−197.969			−0.257437			−0.128718	−	0.991681i	$-0.541086\pi$
$769$	−197.969			−0.257437			−0.128718	+	0.991681i	$0.541086\pi$
$770$		0			0
$771$	−1020.39	+	771.464i	−1.32347	+	1.00060i
$772$		0			0
$773$			1005.50i			1.30078i	0.759600	+	0.650390i	$0.225395\pi$
$773$			1005.50i			1.30078i	−0.759600	+	0.650390i	$0.774605\pi$
$774$		0			0
$775$	537.314			0.693308
$776$		0			0
$777$		0			0
$778$		0			0
$779$			940.296i			1.20706i
$780$		0			0
$781$	−884.653			−1.13272
$782$		0			0
$783$	−222.422	−	571.281i	−0.284064	−	0.729606i
$784$		0			0
$785$		−	77.2727i		−	0.0984366i
$786$		0			0
$787$	1196.49			1.52032			0.760161	−	0.649735i	$-0.225120\pi$
$787$	1196.49			1.52032			0.760161	+	0.649735i	$0.225120\pi$
$788$		0			0
$789$	241.825	−	182.831i	0.306496	−	0.231725i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−530.833			−0.669399
$794$		0			0
$795$	−268.188	−	354.724i	−0.337343	−	0.446193i
$796$		0			0
$797$			1142.13i			1.43303i	0.697570	+	0.716517i	$0.254265\pi$
$797$			1142.13i			1.43303i	−0.697570	+	0.716517i	$0.745735\pi$
$798$		0			0
$799$	125.873			0.157538
$800$		0			0
$801$	−622.568	−	176.381i	−0.777238	−	0.220201i
$802$		0			0
$803$		−	689.848i		−	0.859088i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1155.44	−	873.567i	1.43177	−	1.08249i
$808$		0			0
$809$		−	1157.22i		−	1.43043i	−0.698905	−	0.715215i	$-0.746329\pi$
$809$		−	1157.22i		−	1.43043i	0.698905	−	0.715215i	$-0.253671\pi$
$810$		0			0
$811$	−3.86014			−0.00475973			−0.00237986	−	0.999997i	$-0.500758\pi$
$811$	−3.86014			−0.00475973			−0.00237986	+	0.999997i	$0.500758\pi$
$812$		0			0
$813$	−583.782	−	772.150i	−0.718058	−	0.949754i
$814$		0			0
$815$			109.005i			0.133748i
$816$		0			0
$817$	823.543			1.00801
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	462.987i		−	0.563930i	−0.959425	−	0.281965i	$-0.909014\pi$
$821$		−	462.987i		−	0.563930i	0.959425	−	0.281965i	$-0.0909862\pi$
$822$		0			0
$823$	−1566.99			−1.90399			−0.951996	−	0.306110i	$-0.900972\pi$
$823$	−1566.99			−1.90399			−0.951996	+	0.306110i	$0.900972\pi$
$824$		0			0
$825$	289.873	−	219.157i	0.351361	−	0.265645i
$826$		0			0
$827$			926.069i			1.11979i	0.828563	+	0.559897i	$0.189159\pi$
$827$			926.069i			1.11979i	−0.828563	+	0.559897i	$0.810841\pi$
$828$		0			0
$829$	−1300.80			−1.56912			−0.784559	−	0.620054i	$-0.787111\pi$
$829$	−1300.80			−1.56912			−0.784559	+	0.620054i	$0.787111\pi$
$830$		0			0
$831$	629.790	+	833.005i	0.757871	+	1.00241i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	582.693			0.697836
$836$		0			0
$837$	−791.050	+	307.987i	−0.945102	+	0.367966i
$838$		0			0
$839$		−	278.357i		−	0.331773i	−0.986145	−	0.165886i	$-0.946951\pi$
$839$		−	278.357i		−	0.331773i	0.986145	−	0.165886i	$-0.0530485\pi$
$840$		0			0
$841$	325.453			0.386983
$842$		0			0
$843$	475.581	−	359.562i	0.564153	−	0.426526i
$844$		0			0
$845$			281.175i			0.332751i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	369.680	+	488.965i	0.435430	+	0.575930i
$850$		0			0
$851$		−	97.9509i		−	0.115101i
$852$		0			0
$853$	−483.798			−0.567172			−0.283586	−	0.958947i	$-0.591524\pi$
$853$	−483.798			−0.567172			−0.283586	+	0.958947i	$0.591524\pi$
$854$		0			0
$855$	666.563	+	188.845i	0.779606	+	0.220872i
$856$		0			0
$857$			1599.87i			1.86683i	0.358804	+	0.933413i	$0.383185\pi$
$857$			1599.87i			1.86683i	−0.358804	+	0.933413i	$0.616815\pi$
$858$		0			0
$859$	−572.258			−0.666191			−0.333096	−	0.942893i	$-0.608093\pi$
$859$	−572.258			−0.666191			−0.333096	+	0.942893i	$0.608093\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			863.459i			1.00053i	0.865872	+	0.500266i	$0.166765\pi$
$863$			863.459i			1.00053i	−0.865872	+	0.500266i	$0.833235\pi$
$864$		0			0
$865$	465.292			0.537910
$866$		0			0
$867$	517.251	+	684.152i	0.596598	+	0.789102i
$868$		0			0
$869$		−	111.867i		−	0.128731i
$870$		0			0
$871$	679.385			0.780005
$872$		0			0
$873$	−397.805	+	1404.12i	−0.455675	+	1.60839i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−319.604			−0.364429			−0.182214	−	0.983259i	$-0.558326\pi$
$877$	−319.604			−0.364429			−0.182214	+	0.983259i	$0.558326\pi$
$878$		0			0
$879$	781.375	−	590.756i	0.888936	−	0.672077i
$880$		0			0
$881$			698.076i			0.792368i	0.918171	+	0.396184i	$0.129666\pi$
$881$			698.076i			0.792368i	−0.918171	+	0.396184i	$0.870334\pi$
$882$		0			0
$883$	−3.04056			−0.00344344			−0.00172172	−	0.999999i	$-0.500548\pi$
$883$	−3.04056			−0.00344344			−0.00172172	+	0.999999i	$0.500548\pi$
$884$		0			0
$885$	186.980	+	247.313i	0.211277	+	0.279449i
$886$		0			0
$887$			499.307i			0.562917i	0.959573	+	0.281458i	$0.0908181\pi$
$887$			499.307i			0.562917i	−0.959573	+	0.281458i	$0.909182\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−301.139	+	488.805i	−0.337979	+	0.548602i
$892$		0			0
$893$			1954.39i			2.18857i
$894$		0			0
$895$	347.813			0.388618
$896$		0			0
$897$	−39.3142	+	29.7234i	−0.0438285	+	0.0331364i
$898$		0			0
$899$			713.876i			0.794078i
$900$		0			0
$901$	−92.9058			−0.103114
$902$		0			0
$903$		0			0
$904$		0			0
$905$			8.32288i			0.00919655i
$906$		0			0
$907$	−989.615			−1.09109			−0.545543	−	0.838083i	$-0.683677\pi$
$907$	−989.615			−1.09109			−0.545543	+	0.838083i	$0.683677\pi$
$908$		0			0
$909$	189.279	+	53.6249i	0.208227	+	0.0589933i
$910$		0			0
$911$			1772.23i			1.94537i	0.232135	+	0.972683i	$0.425429\pi$
$911$			1772.23i			1.94537i	−0.232135	+	0.972683i	$0.574571\pi$
$912$		0			0
$913$	−609.925			−0.668045
$914$		0			0
$915$	430.022	−	325.117i	0.469970	−	0.355319i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1103.89			1.20119			0.600593	−	0.799555i	$-0.294931\pi$
$919$	1103.89			1.20119			0.600593	+	0.799555i	$0.294931\pi$
$920$		0			0
$921$	−787.352	−	1041.41i	−0.854889	−	1.13074i
$922$		0			0
$923$			1036.96i			1.12347i
$924$		0			0
$925$	846.555			0.915195
$926$		0			0
$927$	−102.651	+	362.325i	−0.110735	+	0.390858i
$928$		0			0
$929$			521.843i			0.561725i	0.959748	+	0.280863i	$0.0906205\pi$
$929$			521.843i			0.561725i	−0.959748	+	0.280863i	$0.909380\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	522.262	−	394.854i	0.559766	−	0.423209i
$934$		0			0
$935$			35.1400i			0.0375829i
$936$		0			0
$937$	1176.86			1.25599			0.627993	−	0.778219i	$-0.283877\pi$
$937$	1176.86			1.25599			0.627993	+	0.778219i	$0.283877\pi$
$938$		0			0
$939$	−499.470	−	660.634i	−0.531917	−	0.703550i
$940$		0			0
$941$		−	1799.80i		−	1.91265i	−0.292308	−	0.956324i	$-0.594423\pi$
$941$		−	1799.80i		−	1.91265i	0.292308	−	0.956324i	$-0.405577\pi$
$942$		0			0
$943$	67.9335			0.0720398
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	163.647i		−	0.172806i	−0.996260	−	0.0864029i	$-0.972463\pi$
$947$		−	163.647i		−	0.172806i	0.996260	−	0.0864029i	$-0.0275372\pi$
$948$		0			0
$949$	−808.616			−0.852072
$950$		0			0
$951$	1058.39	−	800.191i	1.11292	−	0.841420i
$952$		0			0
$953$			447.912i			0.470002i	0.971995	+	0.235001i	$0.0755094\pi$
$953$			447.912i			0.470002i	−0.971995	+	0.235001i	$0.924491\pi$
$954$		0			0
$955$	268.465			0.281115
$956$		0			0
$957$	291.173	+	385.125i	0.304256	+	0.402430i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	27.5004			0.0286165
$962$		0			0
$963$	−590.959	−	167.426i	−0.613664	−	0.173858i
$964$		0			0
$965$		−	497.816i		−	0.515872i
$966$		0			0
$967$	−860.084			−0.889435			−0.444718	−	0.895671i	$-0.646696\pi$
$967$	−860.084			−0.889435			−0.444718	+	0.895671i	$0.646696\pi$
$968$		0			0
$969$	115.456	−	87.2899i	0.119149	−	0.0900824i
$970$		0			0
$971$		−	1602.86i		−	1.65074i	−0.564595	−	0.825368i	$-0.690968\pi$
$971$		−	1602.86i		−	1.65074i	0.564595	−	0.825368i	$-0.309032\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−256.889	−	339.779i	−0.263475	−	0.348491i
$976$		0			0
$977$			45.9086i			0.0469894i	0.999724	+	0.0234947i	$0.00747928\pi$
$977$			45.9086i			0.0469894i	−0.999724	+	0.0234947i	$0.992521\pi$
$978$		0			0
$979$	509.598			0.520529
$980$		0			0
$981$	398.934	−	1408.11i	0.406660	−	1.43538i
$982$		0			0
$983$			576.361i			0.586328i	0.956062	+	0.293164i	$0.0947082\pi$
$983$			576.361i			0.586328i	−0.956062	+	0.293164i	$0.905292\pi$
$984$		0			0
$985$	−386.572			−0.392459
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	59.4985i		−	0.0601603i
$990$		0			0
$991$	−754.773			−0.761628			−0.380814	−	0.924652i	$-0.624356\pi$
$991$	−754.773			−0.761628			−0.380814	+	0.924652i	$0.624356\pi$
$992$		0			0
$993$	−874.978	−	1157.31i	−0.881146	−	1.16547i
$994$		0			0
$995$			395.984i			0.397974i
$996$		0			0
$997$	315.285			0.316233			0.158117	−	0.987420i	$-0.449458\pi$
$997$	315.285			0.316233			0.158117	+	0.987420i	$0.449458\pi$
$998$		0			0
$999$	−1246.33	+	485.244i	−1.24757	+	0.485729i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.d.h.785.18	yes	24
3.2	odd	2	inner	1176.3.d.h.785.17	yes	24
7.6	odd	2	inner	1176.3.d.h.785.7	✓	24
21.20	even	2	inner	1176.3.d.h.785.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.d.h.785.7	✓	24	7.6	odd	2	inner
1176.3.d.h.785.8	yes	24	21.20	even	2	inner
1176.3.d.h.785.17	yes	24	3.2	odd	2	inner
1176.3.d.h.785.18	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.80925 + 2.39304i) q^{3} -2.81249i q^{5} +(-2.45325 + 8.65919i) q^{9} -7.08791i q^{11} -8.30821 q^{13} +(6.73039 - 5.08849i) q^{15} -1.76276i q^{17} +27.3699 q^{19} -1.97739i q^{23} +17.0899 q^{25} +(-25.1603 + 9.79590i) q^{27} +22.7057i q^{29} +31.4404 q^{31} +(16.9616 - 12.8238i) q^{33} +49.5354 q^{37} +(-15.0316 - 19.8819i) q^{39} +34.3551i q^{41} +30.0894 q^{43} +(24.3539 + 6.89974i) q^{45} +71.4067i q^{47} +(4.21835 - 3.18927i) q^{51} -52.7048i q^{53} -19.9347 q^{55} +(49.5189 + 65.4972i) q^{57} +36.7457i q^{59} +63.8926 q^{61} +23.3668i q^{65} -81.7727 q^{67} +(4.73197 - 3.57759i) q^{69} -124.811i q^{71} +97.3274 q^{73} +(30.9198 + 40.8968i) q^{75} +15.7828 q^{79} +(-68.9631 - 42.4863i) q^{81} -86.0514i q^{83} -4.95774 q^{85} +(-54.3355 + 41.0802i) q^{87} +71.8968i q^{89} +(56.8835 + 75.2381i) q^{93} -76.9776i q^{95} +162.154 q^{97} +(61.3756 + 17.3884i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 16 q^{15} - 152 q^{25} + 64 q^{37} - 16 q^{39} + 16 q^{43} + 248 q^{51} - 120 q^{57} - 416 q^{67} - 224 q^{79} - 344 q^{81} + 496 q^{85} - 464 q^{93}+O(q^{100}) \) 24 * q - 16 * q^15 - 152 * q^25 + 64 * q^37 - 16 * q^39 + 16 * q^43 + 248 * q^51 - 120 * q^57 - 416 * q^67 - 224 * q^79 - 344 * q^81 + 496 * q^85 - 464 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more