LMFDB - Embedded newform 252.9.d.b.181.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,9,Mod(181,252)] 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("252.181"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 2160x^{4} + 976392x^{2} + 85162752 $ x^6 + 2160x^4 + 976392x^2 + 85162752
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.6
Root			$-39.6746i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.9.d.b.181.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+964.066i q^{5} +(2367.93 - 397.129i) q^{7} -11856.2 q^{11} +28897.2i q^{13} +46018.1i q^{17} -19494.9i q^{19} +131645. q^{23} -538799. q^{25} -1.22466e6 q^{29} +1.08710e6i q^{31} +(382858. + 2.28284e6i) q^{35} +2.48906e6 q^{37} +26466.8i q^{41} -4.08031e6 q^{43} +6.95060e6i q^{47} +(5.44938e6 - 1.88075e6i) q^{49} +8.77983e6 q^{53} -1.14302e7i q^{55} -3.74308e6i q^{59} -1.40801e7i q^{61} -2.78588e7 q^{65} -7.62251e6 q^{67} -158650. q^{71} -3.88616e7i q^{73} +(-2.80746e7 + 4.70843e6i) q^{77} +1.13017e7 q^{79} -5.38238e6i q^{83} -4.43645e7 q^{85} -1.08864e7i q^{89} +(1.14759e7 + 6.84265e7i) q^{91} +1.87944e7 q^{95} +5.55681e7i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71}+ \cdots - 85912896 q^{95}+O(q^{100}) $ 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			964.066i			1.54251i	0.636529	+	0.771253i	$0.280370\pi$
$5$			964.066i			1.54251i	−0.636529	+	0.771253i	$0.719630\pi$
$6$		0			0
$7$	2367.93	−	397.129i	0.986226	−	0.165401i
$7$	2367.93	−	397.129i	0.986226	−	0.165401i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11856.2			−0.809794			−0.404897	−	0.914362i	$-0.632693\pi$
$11$	−11856.2			−0.809794			−0.404897	+	0.914362i	$0.632693\pi$
$12$		0			0
$13$			28897.2i			1.01177i	0.862600	+	0.505886i	$0.168834\pi$
$13$			28897.2i			1.01177i	−0.862600	+	0.505886i	$0.831166\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			46018.1i			0.550976i	0.961305	+	0.275488i	$0.0888394\pi$
$17$			46018.1i			0.550976i	−0.961305	+	0.275488i	$0.911161\pi$
$18$		0			0
$19$		−	19494.9i		−	0.149592i	−0.997199	−	0.0747959i	$-0.976169\pi$
$19$		−	19494.9i		−	0.149592i	0.997199	−	0.0747959i	$-0.0238305\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	131645.			0.470428			0.235214	−	0.971944i	$-0.424421\pi$
$23$	131645.			0.470428			0.235214	+	0.971944i	$0.424421\pi$
$24$		0			0
$25$	−538799.			−1.37933
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.22466e6			−1.73151			−0.865755	−	0.500468i	$-0.833161\pi$
$29$	−1.22466e6			−1.73151			−0.865755	+	0.500468i	$0.833161\pi$
$30$		0			0
$31$			1.08710e6i			1.17712i	0.808452	+	0.588562i	$0.200306\pi$
$31$			1.08710e6i			1.17712i	−0.808452	+	0.588562i	$0.799694\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	382858.	+	2.28284e6i	0.255133	+	1.52126i
$36$		0			0
$37$	2.48906e6			1.32809			0.664047	−	0.747691i	$-0.268837\pi$
$37$	2.48906e6			1.32809			0.664047	+	0.747691i	$0.268837\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			26466.8i			0.00936627i	0.999989	+	0.00468314i	$0.00149069\pi$
$41$			26466.8i			0.00936627i	−0.999989	+	0.00468314i	$0.998509\pi$
$42$		0			0
$43$	−4.08031e6			−1.19349			−0.596746	−	0.802430i	$-0.703540\pi$
$43$	−4.08031e6			−1.19349			−0.596746	+	0.802430i	$0.703540\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.95060e6i			1.42440i	0.701979	+	0.712198i	$0.252300\pi$
$47$			6.95060e6i			1.42440i	−0.701979	+	0.712198i	$0.747700\pi$
$48$		0			0
$49$	5.44938e6	−	1.88075e6i	0.945285	−	0.326246i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.77983e6			1.11271			0.556356	−	0.830944i	$-0.312199\pi$
$53$	8.77983e6			1.11271			0.556356	+	0.830944i	$0.312199\pi$
$54$		0			0
$55$		−	1.14302e7i		−	1.24911i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.74308e6i		−	0.308903i	−0.988000	−	0.154451i	$-0.950639\pi$
$59$		−	3.74308e6i		−	0.308903i	0.988000	−	0.154451i	$-0.0493610\pi$
$60$		0			0
$61$		−	1.40801e7i		−	1.01692i	−0.861085	−	0.508461i	$-0.830215\pi$
$61$		−	1.40801e7i		−	1.01692i	0.861085	−	0.508461i	$-0.169785\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.78588e7			−1.56066
$66$		0			0
$67$	−7.62251e6			−0.378267			−0.189134	−	0.981951i	$-0.560568\pi$
$67$	−7.62251e6			−0.378267			−0.189134	+	0.981951i	$0.560568\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−158650.			−0.00624319			−0.00312159	−	0.999995i	$-0.500994\pi$
$71$	−158650.			−0.00624319			−0.00312159	+	0.999995i	$0.500994\pi$
$72$		0			0
$73$		−	3.88616e7i		−	1.36845i	−0.729271	−	0.684225i	$-0.760141\pi$
$73$		−	3.88616e7i		−	1.36845i	0.729271	−	0.684225i	$-0.239859\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.80746e7	+	4.70843e6i	−0.798640	+	0.133941i
$78$		0			0
$79$	1.13017e7			0.290159			0.145080	−	0.989420i	$-0.453656\pi$
$79$	1.13017e7			0.290159			0.145080	+	0.989420i	$0.453656\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.38238e6i		−	0.113413i	−0.998391	−	0.0567064i	$-0.981940\pi$
$83$		−	5.38238e6i		−	0.113413i	0.998391	−	0.0567064i	$-0.0180599\pi$
$84$		0			0
$85$	−4.43645e7			−0.849884
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.08864e7i		−	0.173510i	−0.996230	−	0.0867552i	$-0.972350\pi$
$89$		−	1.08864e7i		−	0.173510i	0.996230	−	0.0867552i	$-0.0276498\pi$
$90$		0			0
$91$	1.14759e7	+	6.84265e7i	0.167348	+	0.997836i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.87944e7			0.230746
$96$		0			0
$97$			5.55681e7i			0.627681i	0.949476	+	0.313840i	$0.101616\pi$
$97$			5.55681e7i			0.627681i	−0.949476	+	0.313840i	$0.898384\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.65688e7i		−	0.735811i	−0.929863	−	0.367905i	$-0.880075\pi$
$101$		−	7.65688e7i		−	0.735811i	0.929863	−	0.367905i	$-0.119925\pi$
$102$		0			0
$103$			1.27219e8i			1.13033i	0.824979	+	0.565163i	$0.191187\pi$
$103$			1.27219e8i			1.13033i	−0.824979	+	0.565163i	$0.808813\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.35004e8			−1.02994			−0.514969	−	0.857209i	$-0.672197\pi$
$107$	−1.35004e8			−1.02994			−0.514969	+	0.857209i	$0.672197\pi$
$108$		0			0
$109$	−7.70178e7			−0.545613			−0.272807	−	0.962069i	$-0.587952\pi$
$109$	−7.70178e7			−0.545613			−0.272807	+	0.962069i	$0.587952\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.20801e7			−0.564745			−0.282372	−	0.959305i	$-0.591121\pi$
$113$	−9.20801e7			−0.564745			−0.282372	+	0.959305i	$0.591121\pi$
$114$		0			0
$115$			1.26915e8i			0.725639i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.82751e7	+	1.08968e8i	0.0911322	+	0.543387i
$120$		0			0
$121$	−7.37896e7			−0.344234
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.42850e8i		−	0.585112i
$126$		0			0
$127$	−3.72776e8			−1.43296			−0.716479	−	0.697609i	$-0.754247\pi$
$127$	−3.72776e8			−1.43296			−0.716479	+	0.697609i	$0.754247\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.33849e8i		−	0.794054i	−0.917807	−	0.397027i	$-0.870042\pi$
$131$		−	2.33849e8i		−	0.794054i	0.917807	−	0.397027i	$-0.129958\pi$
$132$		0			0
$133$	−7.74200e6	−	4.61627e7i	−0.0247427	−	0.147531i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.78252e8			−1.92535			−0.962673	−	0.270666i	$-0.912756\pi$
$137$	−6.78252e8			−1.92535			−0.962673	+	0.270666i	$0.912756\pi$
$138$		0			0
$139$			4.09826e8i			1.09784i	0.835874	+	0.548921i	$0.184961\pi$
$139$			4.09826e8i			1.09784i	−0.835874	+	0.548921i	$0.815039\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3.42611e8i		−	0.819326i
$144$		0			0
$145$		−	1.18066e9i		−	2.67087i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.64958e8			−0.943342			−0.471671	−	0.881775i	$-0.656349\pi$
$149$	−4.64958e8			−0.943342			−0.471671	+	0.881775i	$0.656349\pi$
$150$		0			0
$151$	4.25253e8			0.817974			0.408987	−	0.912540i	$-0.365882\pi$
$151$	4.25253e8			0.817974			0.408987	+	0.912540i	$0.365882\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.04804e9			−1.81572
$156$		0			0
$157$		−	9.13759e8i		−	1.50395i	−0.659192	−	0.751974i	$-0.729102\pi$
$157$		−	9.13759e8i		−	1.50395i	0.659192	−	0.751974i	$-0.270898\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.11726e8	−	5.22801e7i	0.463949	−	0.0778095i
$162$		0			0
$163$	−5.93705e8			−0.841047			−0.420524	−	0.907282i	$-0.638154\pi$
$163$	−5.93705e8			−0.841047			−0.420524	+	0.907282i	$0.638154\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.97081e8i			0.896225i	0.893977	+	0.448113i	$0.147904\pi$
$167$			6.97081e8i			0.896225i	−0.893977	+	0.448113i	$0.852096\pi$
$168$		0			0
$169$	−1.93178e7			−0.0236816
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.66618e8i		−	0.520927i	−0.965484	−	0.260463i	$-0.916125\pi$
$173$		−	4.66618e8i		−	0.520927i	0.965484	−	0.260463i	$-0.0838754\pi$
$174$		0			0
$175$	−1.27584e9	+	2.13973e8i	−1.36033	+	0.228142i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.52236e8			−0.830133			−0.415066	−	0.909791i	$-0.636242\pi$
$179$	−8.52236e8			−0.830133			−0.415066	+	0.909791i	$0.636242\pi$
$180$		0			0
$181$			7.41085e8i			0.690484i	0.938514	+	0.345242i	$0.112203\pi$
$181$			7.41085e8i			0.690484i	−0.938514	+	0.345242i	$0.887797\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.39962e9i			2.04859i
$186$		0			0
$187$		−	5.45599e8i		−	0.446177i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.47730e8			−0.261282			−0.130641	−	0.991430i	$-0.541703\pi$
$191$	−3.47730e8			−0.261282			−0.130641	+	0.991430i	$0.541703\pi$
$192$		0			0
$193$	−4.97355e8			−0.358457			−0.179228	−	0.983807i	$-0.557360\pi$
$193$	−4.97355e8			−0.358457			−0.179228	+	0.983807i	$0.557360\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.99473e8			0.198835			0.0994175	−	0.995046i	$-0.468302\pi$
$197$	2.99473e8			0.198835			0.0994175	+	0.995046i	$0.468302\pi$
$198$		0			0
$199$			6.90200e7i			0.0440112i	0.999758	+	0.0220056i	$0.00700516\pi$
$199$			6.90200e7i			0.0440112i	−0.999758	+	0.0220056i	$0.992995\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.89992e9	+	4.86349e8i	−1.70766	+	0.286394i
$204$		0			0
$205$	−2.55158e7			−0.0144475
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.31136e8i			0.121138i
$210$		0			0
$211$	1.38118e9			0.696820			0.348410	−	0.937342i	$-0.386722\pi$
$211$	1.38118e9			0.696820			0.348410	+	0.937342i	$0.386722\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.93369e9i		−	1.84097i
$216$		0			0
$217$	4.31718e8	+	2.57417e9i	0.194698	+	1.16091i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.32979e9			−0.557462
$222$		0			0
$223$			3.59070e9i			1.45198i	0.687707	+	0.725989i	$0.258617\pi$
$223$			3.59070e9i			1.45198i	−0.687707	+	0.725989i	$0.741383\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.45220e9i		−	0.546919i	−0.961884	−	0.273460i	$-0.911832\pi$
$227$		−	1.45220e9i		−	0.546919i	0.961884	−	0.273460i	$-0.0881680\pi$
$228$		0			0
$229$		−	4.39121e9i		−	1.59677i	−0.602148	−	0.798384i	$-0.705688\pi$
$229$		−	4.39121e9i		−	1.59677i	0.602148	−	0.798384i	$-0.294312\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.51324e9			−0.852729			−0.426364	−	0.904551i	$-0.640206\pi$
$233$	−2.51324e9			−0.852729			−0.426364	+	0.904551i	$0.640206\pi$
$234$		0			0
$235$	−6.70084e9			−2.19714
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.38734e9			0.425198			0.212599	−	0.977140i	$-0.431807\pi$
$239$	1.38734e9			0.425198			0.212599	+	0.977140i	$0.431807\pi$
$240$		0			0
$241$		−	2.00561e9i		−	0.594535i	−0.954794	−	0.297267i	$-0.903925\pi$
$241$		−	2.00561e9i		−	0.594535i	0.954794	−	0.297267i	$-0.0960753\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.81316e9	+	5.25356e9i	0.503237	+	1.45811i
$246$		0			0
$247$	5.63349e8			0.151353
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.22678e9i		−	1.82075i	−0.413786	−	0.910374i	$-0.635794\pi$
$251$		−	7.22678e9i		−	1.82075i	0.413786	−	0.910374i	$-0.364206\pi$
$252$		0			0
$253$	−1.56081e9			−0.380950
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	7.65025e8i		−	0.175365i	−0.996148	−	0.0876825i	$-0.972054\pi$
$257$		−	7.65025e8i		−	0.175365i	0.996148	−	0.0876825i	$-0.0279461\pi$
$258$		0			0
$259$	5.89393e9	−	9.88479e8i	1.30980	−	0.219669i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.17529e9			1.29073			0.645363	−	0.763876i	$-0.276706\pi$
$263$	6.17529e9			1.29073			0.645363	+	0.763876i	$0.276706\pi$
$264$		0			0
$265$			8.46434e9i			1.71636i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.41507e9i		−	1.22516i	−0.790409	−	0.612579i	$-0.790132\pi$
$269$		−	6.41507e9i		−	1.22516i	0.790409	−	0.612579i	$-0.209868\pi$
$270$		0			0
$271$		−	5.16209e9i		−	0.957080i	−0.878066	−	0.478540i	$-0.841166\pi$
$271$		−	5.16209e9i		−	0.957080i	0.878066	−	0.478540i	$-0.158834\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.38810e9			1.11697
$276$		0			0
$277$	6.33607e9			1.07622			0.538110	−	0.842875i	$-0.319138\pi$
$277$	6.33607e9			1.07622			0.538110	+	0.842875i	$0.319138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.11288e9			−0.338883			−0.169441	−	0.985540i	$-0.554196\pi$
$281$	−2.11288e9			−0.338883			−0.169441	+	0.985540i	$0.554196\pi$
$282$		0			0
$283$		−	5.14406e9i		−	0.801974i	−0.916084	−	0.400987i	$-0.868667\pi$
$283$		−	5.14406e9i		−	0.801974i	0.916084	−	0.400987i	$-0.131333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.05107e7	+	6.26716e7i	0.00154919	+	0.00923726i
$288$		0			0
$289$	4.85810e9			0.696426
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7.60278e9i			1.03158i	0.856716	+	0.515789i	$0.172501\pi$
$293$			7.60278e9i			1.03158i	−0.856716	+	0.515789i	$0.827499\pi$
$294$		0			0
$295$	3.60858e9			0.476484
$296$		0			0
$297$		0			0
$298$		0			0
$299$			3.80418e9i			0.475966i
$300$		0			0
$301$	−9.66189e9	+	1.62041e9i	−1.17705	+	0.197405i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.35742e10			1.56861
$306$		0			0
$307$			1.31307e10i			1.47820i	0.673593	+	0.739102i	$0.264750\pi$
$307$			1.31307e10i			1.47820i	−0.673593	+	0.739102i	$0.735250\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.50081e10i			1.60429i	0.597128	+	0.802146i	$0.296309\pi$
$311$			1.50081e10i			1.60429i	−0.597128	+	0.802146i	$0.703691\pi$
$312$		0			0
$313$			8.16769e9i			0.850985i	0.904962	+	0.425492i	$0.139899\pi$
$313$			8.16769e9i			0.850985i	−0.904962	+	0.425492i	$0.860101\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.60016e9			0.356520			0.178260	−	0.983983i	$-0.442953\pi$
$317$	3.60016e9			0.356520			0.178260	+	0.983983i	$0.442953\pi$
$318$		0			0
$319$	1.45199e10			1.40217
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.97120e8			0.0824214
$324$		0			0
$325$		−	1.55698e10i		−	1.39556i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.76028e9	+	1.64585e10i	0.235597	+	1.40478i
$330$		0			0
$331$	−1.70539e10			−1.42073			−0.710367	−	0.703831i	$-0.751471\pi$
$331$	−1.70539e10			−1.42073			−0.710367	+	0.703831i	$0.751471\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	7.34861e9i		−	0.583480i
$336$		0			0
$337$	−2.14873e9			−0.166595			−0.0832976	−	0.996525i	$-0.526545\pi$
$337$	−2.14873e9			−0.166595			−0.0832976	+	0.996525i	$0.526545\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.28889e10i		−	0.953228i
$342$		0			0
$343$	1.21568e10	−	6.61758e9i	0.878303	−	0.478104i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.21427e10			−1.52726			−0.763631	−	0.645653i	$-0.776585\pi$
$347$	−2.21427e10			−1.52726			−0.763631	+	0.645653i	$0.776585\pi$
$348$		0			0
$349$		−	5.06835e8i		−	0.0341637i	−0.999854	−	0.0170818i	$-0.994562\pi$
$349$		−	5.06835e8i		−	0.0341637i	0.999854	−	0.0170818i	$-0.00543758\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.18664e10i			1.40825i	0.710077	+	0.704124i	$0.248660\pi$
$353$			2.18664e10i			1.40825i	−0.710077	+	0.704124i	$0.751340\pi$
$354$		0			0
$355$		−	1.52949e8i		−	0.00963015i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.65236e10			0.994780			0.497390	−	0.867527i	$-0.334292\pi$
$359$	1.65236e10			0.994780			0.497390	+	0.867527i	$0.334292\pi$
$360$		0			0
$361$	1.66035e10			0.977622
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.74651e10			2.11084
$366$		0			0
$367$		−	8.46251e9i		−	0.466482i	−0.972419	−	0.233241i	$-0.925067\pi$
$367$		−	8.46251e9i		−	0.466482i	0.972419	−	0.233241i	$-0.0749331\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.07900e10	−	3.48672e9i	1.09739	−	0.184044i
$372$		0			0
$373$	2.54227e10			1.31337			0.656684	−	0.754166i	$-0.271959\pi$
$373$	2.54227e10			1.31337			0.656684	+	0.754166i	$0.271959\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.53894e10i		−	1.75189i
$378$		0			0
$379$	3.76359e9			0.182409			0.0912044	−	0.995832i	$-0.470928\pi$
$379$	3.76359e9			0.182409			0.0912044	+	0.995832i	$0.470928\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			1.59784e10i			0.742571i	0.928519	+	0.371285i	$0.121083\pi$
$383$			1.59784e10i			0.742571i	−0.928519	+	0.371285i	$0.878917\pi$
$384$		0			0
$385$	−4.53924e9	−	2.70658e10i	−0.206605	−	1.23191i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.04578e10			0.893432			0.446716	−	0.894676i	$-0.352594\pi$
$389$	2.04578e10			0.893432			0.446716	+	0.894676i	$0.352594\pi$
$390$		0			0
$391$			6.05805e9i			0.259195i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.08956e10i			0.447572i
$396$		0			0
$397$			7.26355e9i			0.292406i	0.989255	+	0.146203i	$0.0467053\pi$
$397$			7.26355e9i			0.292406i	−0.989255	+	0.146203i	$0.953295\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.21706e9			−0.0857432			−0.0428716	−	0.999081i	$-0.513651\pi$
$401$	−2.21706e9			−0.0857432			−0.0428716	+	0.999081i	$0.513651\pi$
$402$		0			0
$403$	−3.14141e10			−1.19098
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.95108e10			−1.07548
$408$		0			0
$409$			3.38513e10i			1.20971i	0.796335	+	0.604856i	$0.206769\pi$
$409$			3.38513e10i			1.20971i	−0.796335	+	0.604856i	$0.793231\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.48649e9	−	8.86336e9i	−0.0510929	−	0.304648i
$414$		0			0
$415$	5.18897e9			0.174940
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	3.14074e10i		−	1.01900i	−0.860469	−	0.509502i	$-0.829830\pi$
$419$		−	3.14074e10i		−	1.01900i	0.860469	−	0.509502i	$-0.170170\pi$
$420$		0			0
$421$	1.95780e10			0.623217			0.311608	−	0.950211i	$-0.399132\pi$
$421$	1.95780e10			0.623217			0.311608	+	0.950211i	$0.399132\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.47945e10i		−	0.759975i
$426$		0			0
$427$	−5.59162e9	−	3.33408e10i	−0.168200	−	1.00291i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.98097e10			0.574076			0.287038	−	0.957919i	$-0.407329\pi$
$431$	1.98097e10			0.574076			0.287038	+	0.957919i	$0.407329\pi$
$432$		0			0
$433$			1.98951e10i			0.565972i	0.959124	+	0.282986i	$0.0913250\pi$
$433$			1.98951e10i			0.565972i	−0.959124	+	0.282986i	$0.908675\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.56641e9i		−	0.0703722i
$438$		0			0
$439$			6.74308e10i			1.81552i	0.419493	+	0.907759i	$0.362208\pi$
$439$			6.74308e10i			1.81552i	−0.419493	+	0.907759i	$0.637792\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.14164e10			−1.33502			−0.667508	−	0.744603i	$-0.732639\pi$
$443$	−5.14164e10			−1.33502			−0.667508	+	0.744603i	$0.732639\pi$
$444$		0			0
$445$	1.04952e10			0.267641
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.77031e9			0.0681620			0.0340810	−	0.999419i	$-0.489150\pi$
$449$	2.77031e9			0.0681620			0.0340810	+	0.999419i	$0.489150\pi$
$450$		0			0
$451$		−	3.13796e8i		−	0.00758475i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.59677e10	+	1.10635e10i	−1.53917	+	0.258136i
$456$		0			0
$457$	−3.73325e10			−0.855898			−0.427949	−	0.903803i	$-0.640764\pi$
$457$	−3.73325e10			−0.855898			−0.427949	+	0.903803i	$0.640764\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.50363e10i		−	0.775736i	−0.921715	−	0.387868i	$-0.873212\pi$
$461$		−	3.50363e10i		−	0.775736i	0.921715	−	0.387868i	$-0.126788\pi$
$462$		0			0
$463$	−7.33892e10			−1.59701			−0.798506	−	0.601986i	$-0.794376\pi$
$463$	−7.33892e10			−1.59701			−0.798506	+	0.601986i	$0.794376\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.81052e10i			1.01140i	0.862708	+	0.505702i	$0.168766\pi$
$467$			4.81052e10i			1.01140i	−0.862708	+	0.505702i	$0.831234\pi$
$468$		0			0
$469$	−1.80496e10	+	3.02712e9i	−0.373057	+	0.0625660i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.83769e10			0.966482
$474$		0			0
$475$			1.05039e10i			0.206336i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			3.16459e10i			0.601139i	0.953760	+	0.300570i	$0.0971768\pi$
$479$			3.16459e10i			0.601139i	−0.953760	+	0.300570i	$0.902823\pi$
$480$		0			0
$481$			7.19270e10i			1.34373i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.35714e10			−0.968202
$486$		0			0
$487$	−2.64463e10			−0.470163			−0.235082	−	0.971976i	$-0.575536\pi$
$487$	−2.64463e10			−0.470163			−0.235082	+	0.971976i	$0.575536\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.51063e9			−0.146432			−0.0732159	−	0.997316i	$-0.523326\pi$
$491$	−8.51063e9			−0.146432			−0.0732159	+	0.997316i	$0.523326\pi$
$492$		0			0
$493$		−	5.63567e10i		−	0.954021i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.75672e8	+	6.30044e7i	−0.00615719	+	0.00103263i
$498$		0			0
$499$	6.43265e10			1.03750			0.518750	−	0.854926i	$-0.326398\pi$
$499$	6.43265e10			1.03750			0.518750	+	0.854926i	$0.326398\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1.04123e10i		−	0.162658i	−0.996687	−	0.0813292i	$-0.974083\pi$
$503$		−	1.04123e10i		−	0.162658i	0.996687	−	0.0813292i	$-0.0259165\pi$
$504$		0			0
$505$	7.38174e10			1.13499
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	1.37157e10i		−	0.204337i	−0.994767	−	0.102168i	$-0.967422\pi$
$509$		−	1.37157e10i		−	0.204337i	0.994767	−	0.102168i	$-0.0325781\pi$
$510$		0			0
$511$	−1.54330e10	−	9.20214e10i	−0.226343	−	1.34960i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.22648e11			−1.74354
$516$		0			0
$517$		−	8.24076e10i		−	1.15347i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.00567e10i			1.22226i	0.791528	+	0.611132i	$0.209286\pi$
$521$			9.00567e10i			1.22226i	−0.791528	+	0.611132i	$0.790714\pi$
$522$		0			0
$523$			4.39983e10i			0.588070i	0.955795	+	0.294035i	$0.0949983\pi$
$523$			4.39983e10i			0.588070i	−0.955795	+	0.294035i	$0.905002\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.00262e10			−0.648567
$528$		0			0
$529$	−6.09805e10			−0.778697
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.64818e8			−0.00947653
$534$		0			0
$535$		−	1.30153e11i		−	1.58869i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.46089e10	+	2.22985e10i	−0.765486	+	0.264192i
$540$		0			0
$541$	−9.81531e10			−1.14582			−0.572908	−	0.819619i	$-0.694185\pi$
$541$	−9.81531e10			−1.14582			−0.572908	+	0.819619i	$0.694185\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	7.42503e10i		−	0.841612i
$546$		0			0
$547$	−9.97101e8			−0.0111376			−0.00556878	−	0.999984i	$-0.501773\pi$
$547$	−9.97101e8			−0.0111376			−0.00556878	+	0.999984i	$0.501773\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2.38748e10i			0.259020i
$552$		0			0
$553$	2.67617e10	−	4.48824e9i	0.286163	−	0.0479927i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.53863e10			−0.367632			−0.183816	−	0.982961i	$-0.558845\pi$
$557$	−3.53863e10			−0.367632			−0.183816	+	0.982961i	$0.558845\pi$
$558$		0			0
$559$		−	1.17910e11i		−	1.20754i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.78696e11i		−	1.77862i	−0.457308	−	0.889308i	$-0.651186\pi$
$563$		−	1.78696e11i		−	1.77862i	0.457308	−	0.889308i	$-0.348814\pi$
$564$		0			0
$565$		−	8.87714e10i		−	0.871122i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.16763e10			0.874597			0.437298	−	0.899316i	$-0.355935\pi$
$569$	9.16763e10			0.874597			0.437298	+	0.899316i	$0.355935\pi$
$570$		0			0
$571$	1.66178e10			0.156325			0.0781626	−	0.996941i	$-0.475095\pi$
$571$	1.66178e10			0.156325			0.0781626	+	0.996941i	$0.475095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.09303e10			−0.648874
$576$		0			0
$577$			1.35959e11i			1.22661i	0.789848	+	0.613303i	$0.210159\pi$
$577$			1.35959e11i			1.22661i	−0.789848	+	0.613303i	$0.789841\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.13750e9	−	1.27451e10i	−0.0187586	−	0.111851i
$582$		0			0
$583$	−1.04095e11			−0.901067
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.24512e10i			0.189098i	0.995520	+	0.0945491i	$0.0301409\pi$
$587$			2.24512e10i			0.189098i	−0.995520	+	0.0945491i	$0.969859\pi$
$588$		0			0
$589$	2.11929e10			0.176088
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.40438e11i			1.94439i	0.234170	+	0.972196i	$0.424763\pi$
$593$			2.40438e11i			1.94439i	−0.234170	+	0.972196i	$0.575237\pi$
$594$		0			0
$595$	−1.05052e11	+	1.76184e10i	−0.838178	+	0.140572i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.62590e10			0.281649			0.140825	−	0.990035i	$-0.455025\pi$
$599$	3.62590e10			0.281649			0.140825	+	0.990035i	$0.455025\pi$
$600$		0			0
$601$			1.88260e11i			1.44298i	0.692426	+	0.721488i	$0.256542\pi$
$601$			1.88260e11i			1.44298i	−0.692426	+	0.721488i	$0.743458\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	7.11381e10i		−	0.530983i
$606$		0			0
$607$		−	2.96089e10i		−	0.218106i	−0.994036	−	0.109053i	$-0.965218\pi$
$607$		−	2.96089e10i		−	0.218106i	0.994036	−	0.109053i	$-0.0347819\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.00853e11			−1.44116
$612$		0			0
$613$	2.29090e11			1.62242			0.811211	−	0.584754i	$-0.198809\pi$
$613$	2.29090e11			1.62242			0.811211	+	0.584754i	$0.198809\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.19299e11			−1.51320			−0.756598	−	0.653880i	$-0.773140\pi$
$617$	−2.19299e11			−1.51320			−0.756598	+	0.653880i	$0.773140\pi$
$618$		0			0
$619$			6.01757e10i			0.409882i	0.978774	+	0.204941i	$0.0657002\pi$
$619$			6.01757e10i			0.409882i	−0.978774	+	0.204941i	$0.934300\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.32332e9	−	2.57783e10i	−0.0286989	−	0.171121i
$624$		0			0
$625$	−7.27519e10			−0.476787
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1.14542e11i			0.731748i
$630$		0			0
$631$	−1.82301e11			−1.14993			−0.574965	−	0.818178i	$-0.694984\pi$
$631$	−1.82301e11			−1.14993			−0.574965	+	0.818178i	$0.694984\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	3.59381e11i		−	2.21035i
$636$		0			0
$637$	5.43483e10	+	1.57472e11i	0.330087	+	0.956412i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.38968e11			1.41549			0.707746	−	0.706467i	$-0.249712\pi$
$641$	2.38968e11			1.41549			0.707746	+	0.706467i	$0.249712\pi$
$642$		0			0
$643$			2.67685e11i			1.56596i	0.622046	+	0.782980i	$0.286302\pi$
$643$			2.67685e11i			1.56596i	−0.622046	+	0.782980i	$0.713698\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.95204e9i			0.0111396i	0.999984	+	0.00556981i	$0.00177294\pi$
$647$			1.95204e9i			0.0111396i	−0.999984	+	0.00556981i	$0.998227\pi$
$648$		0			0
$649$			4.43787e10i			0.250147i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.52280e11			1.38749			0.693744	−	0.720221i	$-0.255960\pi$
$653$	2.52280e11			1.38749			0.693744	+	0.720221i	$0.255960\pi$
$654$		0			0
$655$	2.25446e11			1.22483
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.72053e11			1.44249			0.721244	−	0.692681i	$-0.243571\pi$
$659$	2.72053e11			1.44249			0.721244	+	0.692681i	$0.243571\pi$
$660$		0			0
$661$			2.43044e11i			1.27315i	0.771215	+	0.636574i	$0.219649\pi$
$661$			2.43044e11i			1.27315i	−0.771215	+	0.636574i	$0.780351\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.45039e10	−	7.46380e9i	0.227568	−	0.0381657i
$666$		0			0
$667$	−1.61221e11			−0.814552
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.66937e11i			0.823496i
$672$		0			0
$673$	4.27524e9			0.0208401			0.0104201	−	0.999946i	$-0.496683\pi$
$673$	4.27524e9			0.0208401			0.0104201	+	0.999946i	$0.496683\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.98728e11i		−	0.946029i	−0.881055	−	0.473014i	$-0.843166\pi$
$677$		−	1.98728e11i		−	0.946029i	0.881055	−	0.473014i	$-0.156834\pi$
$678$		0			0
$679$	2.20677e10	+	1.31581e11i	0.103819	+	0.619035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.67236e11			0.768504			0.384252	−	0.923228i	$-0.374459\pi$
$683$	1.67236e11			0.768504			0.384252	+	0.923228i	$0.374459\pi$
$684$		0			0
$685$		−	6.53880e11i		−	2.96986i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2.53713e11i			1.12581i
$690$		0			0
$691$		−	2.41970e11i		−	1.06133i	−0.847583	−	0.530663i	$-0.821943\pi$
$691$		−	2.41970e11i		−	1.06133i	0.847583	−	0.530663i	$-0.178057\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.95099e11			−1.69343
$696$		0			0
$697$	−1.21795e9			−0.00516059
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.65999e11			1.51568			0.757840	−	0.652441i	$-0.226255\pi$
$701$	3.65999e11			1.51568			0.757840	+	0.652441i	$0.226255\pi$
$702$		0			0
$703$		−	4.85242e10i		−	0.198672i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.04077e10	−	1.81309e11i	−0.121704	−	0.725676i
$708$		0			0
$709$	3.74001e11			1.48009			0.740045	−	0.672557i	$-0.234804\pi$
$709$	3.74001e11			1.48009			0.740045	+	0.672557i	$0.234804\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.43111e11i			0.553753i
$714$		0			0
$715$	3.30300e11			1.26382
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.04752e10i		−	0.0391965i	−0.999808	−	0.0195982i	$-0.993761\pi$
$719$		−	1.04752e10i		−	0.0391965i	0.999808	−	0.0195982i	$-0.00623871\pi$
$720$		0			0
$721$	5.05224e10	+	3.01246e11i	0.186958	+	1.11476i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.59848e11			2.38832
$726$		0			0
$727$			4.11284e11i			1.47233i	0.676804	+	0.736163i	$0.263364\pi$
$727$			4.11284e11i			1.47233i	−0.676804	+	0.736163i	$0.736636\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.87768e11i		−	0.657585i
$732$		0			0
$733$		−	4.88417e11i		−	1.69190i	−0.533263	−	0.845950i	$-0.679034\pi$
$733$		−	4.88417e11i		−	1.69190i	0.533263	−	0.845950i	$-0.320966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.03740e10			0.306319
$738$		0			0
$739$	−2.88067e11			−0.965865			−0.482932	−	0.875658i	$-0.660428\pi$
$739$	−2.88067e11			−0.965865			−0.482932	+	0.875658i	$0.660428\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.14288e11			−0.703142			−0.351571	−	0.936161i	$-0.614352\pi$
$743$	−2.14288e11			−0.703142			−0.351571	+	0.936161i	$0.614352\pi$
$744$		0			0
$745$		−	4.48251e11i		−	1.45511i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.19680e11	+	5.36140e10i	−1.01575	+	0.170353i
$750$		0			0
$751$	1.85479e11			0.583088			0.291544	−	0.956557i	$-0.405831\pi$
$751$	1.85479e11			0.583088			0.291544	+	0.956557i	$0.405831\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			4.09972e11i			1.26173i
$756$		0			0
$757$	6.69254e10			0.203802			0.101901	−	0.994795i	$-0.467508\pi$
$757$	6.69254e10			0.203802			0.101901	+	0.994795i	$0.467508\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.57239e10i			0.136334i	0.997674	+	0.0681670i	$0.0217151\pi$
$761$			4.57239e10i			0.136334i	−0.997674	+	0.0681670i	$0.978285\pi$
$762$		0			0
$763$	−1.82373e11	+	3.05860e10i	−0.538098	+	0.0902452i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.08165e11			0.312539
$768$		0			0
$769$		−	4.25562e11i		−	1.21691i	−0.793589	−	0.608454i	$-0.791790\pi$
$769$		−	4.25562e11i		−	1.21691i	0.793589	−	0.608454i	$-0.208210\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.61674e11i		−	0.452816i	−0.974032	−	0.226408i	$-0.927302\pi$
$773$		−	1.61674e11i		−	0.452816i	0.974032	−	0.226408i	$-0.0726983\pi$
$774$		0			0
$775$		−	5.85728e11i		−	1.62364i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.15970e8			0.00140112
$780$		0			0
$781$	1.88098e9			0.00505569
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.80924e11			2.31985
$786$		0			0
$787$			3.11384e11i			0.811705i	0.913939	+	0.405852i	$0.133025\pi$
$787$			3.11384e11i			0.811705i	−0.913939	+	0.405852i	$0.866975\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.18039e11	+	3.65677e10i	−0.556966	+	0.0934096i
$792$		0			0
$793$	4.06876e11			1.02889
$794$		0			0
$795$		0			0
$796$		0			0
$797$			5.39201e11i			1.33634i	0.744008	+	0.668171i	$0.232922\pi$
$797$			5.39201e11i			1.33634i	−0.744008	+	0.668171i	$0.767078\pi$
$798$		0			0
$799$	−3.19853e11			−0.784808
$800$		0			0
$801$		0			0
$802$		0			0
$803$			4.60750e11i			1.10816i
$804$		0			0
$805$	5.04014e10	+	3.00525e11i	0.120022	+	0.715644i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.72257e11			0.869059			0.434530	−	0.900658i	$-0.356915\pi$
$809$	3.72257e11			0.869059			0.434530	+	0.900658i	$0.356915\pi$
$810$		0			0
$811$			6.84823e11i			1.58305i	0.611137	+	0.791525i	$0.290713\pi$
$811$			6.84823e11i			1.58305i	−0.611137	+	0.791525i	$0.709287\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	5.72371e11i		−	1.29732i
$816$		0			0
$817$			7.95454e10i			0.178537i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.06763e11			−0.895300			−0.447650	−	0.894209i	$-0.647739\pi$
$821$	−4.06763e11			−0.895300			−0.447650	+	0.894209i	$0.647739\pi$
$822$		0			0
$823$	5.61961e11			1.22492			0.612459	−	0.790503i	$-0.290181\pi$
$823$	5.61961e11			1.22492			0.612459	+	0.790503i	$0.290181\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.56518e11			0.334612			0.167306	−	0.985905i	$-0.446493\pi$
$827$	1.56518e11			0.334612			0.167306	+	0.985905i	$0.446493\pi$
$828$		0			0
$829$			6.83001e11i			1.44612i	0.690787	+	0.723058i	$0.257264\pi$
$829$			6.83001e11i			1.44612i	−0.690787	+	0.723058i	$0.742736\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.65483e10	+	2.50770e11i	0.179754	+	0.520829i
$834$		0			0
$835$	−6.72032e11			−1.38243
$836$		0			0
$837$		0			0
$838$		0			0
$839$			6.02058e11i			1.21504i	0.794304	+	0.607520i	$0.207836\pi$
$839$			6.02058e11i			1.21504i	−0.794304	+	0.607520i	$0.792164\pi$
$840$		0			0
$841$	9.99557e11			1.99813
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.86237e10i		−	0.0365291i
$846$		0			0
$847$	−1.74729e11	+	2.93040e10i	−0.339493	+	0.0569368i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.27673e11			0.624773
$852$		0			0
$853$		−	2.62751e11i		−	0.496305i	−0.968721	−	0.248153i	$-0.920177\pi$
$853$		−	2.62751e11i		−	0.496305i	0.968721	−	0.248153i	$-0.0798235\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.71804e10i			0.0874658i	0.999043	+	0.0437329i	$0.0139251\pi$
$857$			4.71804e10i			0.0874658i	−0.999043	+	0.0437329i	$0.986075\pi$
$858$		0			0
$859$			5.10713e11i			0.938003i	0.883197	+	0.469001i	$0.155386\pi$
$859$			5.10713e11i			0.938003i	−0.883197	+	0.469001i	$0.844614\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.69496e10			−0.0666142			−0.0333071	−	0.999445i	$-0.510604\pi$
$863$	−3.69496e10			−0.0666142			−0.0333071	+	0.999445i	$0.510604\pi$
$864$		0			0
$865$	4.49850e11			0.803533
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.33995e11			−0.234969
$870$		0			0
$871$		−	2.20269e11i		−	0.382720i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.67297e10	−	3.38258e11i	−0.0967783	−	0.577053i
$876$		0			0
$877$	−9.68347e11			−1.63694			−0.818470	−	0.574549i	$-0.805177\pi$
$877$	−9.68347e11			−1.63694			−0.818470	+	0.574549i	$0.805177\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	9.95652e11i		−	1.65274i	−0.563129	−	0.826369i	$-0.690403\pi$
$881$		−	9.95652e11i		−	1.65274i	0.563129	−	0.826369i	$-0.309597\pi$
$882$		0			0
$883$	5.84931e10			0.0962192			0.0481096	−	0.998842i	$-0.484680\pi$
$883$	5.84931e10			0.0962192			0.0481096	+	0.998842i	$0.484680\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.91546e11i			0.794090i	0.917799	+	0.397045i	$0.129964\pi$
$887$			4.91546e11i			0.794090i	−0.917799	+	0.397045i	$0.870036\pi$
$888$		0			0
$889$	−8.82708e11	+	1.48040e11i	−1.41322	+	0.237013i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.35502e11			0.213078
$894$		0			0
$895$		−	8.21612e11i		−	1.28049i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1.33133e12i		−	2.03820i
$900$		0			0
$901$			4.04031e11i			0.613077i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.14455e11			−1.06508
$906$		0			0
$907$	3.70620e11			0.547645			0.273823	−	0.961780i	$-0.411712\pi$
$907$	3.70620e11			0.547645			0.273823	+	0.961780i	$0.411712\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.13289e11			−1.03560			−0.517800	−	0.855502i	$-0.673249\pi$
$911$	−7.13289e11			−1.03560			−0.517800	+	0.855502i	$0.673249\pi$
$912$		0			0
$913$			6.38145e10i			0.0918410i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.28681e10	−	5.53737e11i	−0.131338	−	0.783117i
$918$		0			0
$919$	−3.56109e11			−0.499253			−0.249626	−	0.968342i	$-0.580308\pi$
$919$	−3.56109e11			−0.499253			−0.249626	+	0.968342i	$0.580308\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.58454e9i		−	0.00631668i
$924$		0			0
$925$	−1.34110e12			−1.83187
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.23321e12i		−	1.65567i	−0.560971	−	0.827836i	$-0.689572\pi$
$929$		−	1.23321e12i		−	1.65567i	0.560971	−	0.827836i	$-0.310428\pi$
$930$		0			0
$931$	−3.66650e10	−	1.06235e11i	−0.0488038	−	0.141407i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.25993e11			0.688231
$936$		0			0
$937$		−	1.10549e12i		−	1.43415i	−0.696994	−	0.717077i	$-0.745479\pi$
$937$		−	1.10549e12i		−	1.43415i	0.696994	−	0.717077i	$-0.254521\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			7.52243e11i			0.959400i	0.877433	+	0.479700i	$0.159254\pi$
$941$			7.52243e11i			0.959400i	−0.877433	+	0.479700i	$0.840746\pi$
$942$		0			0
$943$			3.48423e9i			0.00440616i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.01671e11			−1.12111			−0.560555	−	0.828117i	$-0.689412\pi$
$947$	−9.01671e11			−1.12111			−0.560555	+	0.828117i	$0.689412\pi$
$948$		0			0
$949$	1.12299e12			1.38456
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.66082e11			−0.686290			−0.343145	−	0.939282i	$-0.611492\pi$
$953$	−5.66082e11			−0.686290			−0.343145	+	0.939282i	$0.611492\pi$
$954$		0			0
$955$		−	3.35235e11i		−	0.403029i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.60605e12	+	2.69353e11i	−1.89883	+	0.318455i
$960$		0			0
$961$	−3.28893e11			−0.385622
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	4.79483e11i		−	0.552922i
$966$		0			0
$967$	8.41824e11			0.962754			0.481377	−	0.876514i	$-0.340137\pi$
$967$	8.41824e11			0.962754			0.481377	+	0.876514i	$0.340137\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.22117e11i		−	0.474849i	−0.971406	−	0.237425i	$-0.923697\pi$
$971$		−	4.22117e11i		−	0.474849i	0.971406	−	0.237425i	$-0.0763033\pi$
$972$		0			0
$973$	1.62754e11	+	9.70438e11i	0.181585	+	1.08272i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.43992e11			0.158037			0.0790187	−	0.996873i	$-0.474821\pi$
$977$	1.43992e11			0.158037			0.0790187	+	0.996873i	$0.474821\pi$
$978$		0			0
$979$			1.29072e11i			0.140508i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.68786e12i			1.80769i	0.427863	+	0.903844i	$0.359267\pi$
$983$			1.68786e12i			1.80769i	−0.427863	+	0.903844i	$0.640733\pi$
$984$		0			0
$985$			2.88712e11i			0.306704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.37153e11			−0.561452
$990$		0			0
$991$	−6.08937e11			−0.631361			−0.315681	−	0.948866i	$-0.602233\pi$
$991$	−6.08937e11			−0.631361			−0.315681	+	0.948866i	$0.602233\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.65399e10			−0.0678875
$996$		0			0
$997$			6.40936e11i			0.648686i	0.945940	+	0.324343i	$0.105143\pi$
$997$			6.40936e11i			0.648686i	−0.945940	+	0.324343i	$0.894857\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.d.b.181.6		6
3.2	odd	2		28.9.b.a.13.1	✓	6
7.6	odd	2	inner	252.9.d.b.181.1		6
12.11	even	2		112.9.c.d.97.6		6
21.2	odd	6		196.9.h.b.129.6		12
21.5	even	6		196.9.h.b.129.1		12
21.11	odd	6		196.9.h.b.117.1		12
21.17	even	6		196.9.h.b.117.6		12
21.20	even	2		28.9.b.a.13.6	yes	6
84.83	odd	2		112.9.c.d.97.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.9.b.a.13.1	✓	6	3.2	odd	2
28.9.b.a.13.6	yes	6	21.20	even	2
112.9.c.d.97.1		6	84.83	odd	2
112.9.c.d.97.6		6	12.11	even	2
196.9.h.b.117.1		12	21.11	odd	6
196.9.h.b.117.6		12	21.17	even	6
196.9.h.b.129.1		12	21.5	even	6
196.9.h.b.129.6		12	21.2	odd	6
252.9.d.b.181.1		6	7.6	odd	2	inner
252.9.d.b.181.6		6	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+964.066i q^{5} +(2367.93 - 397.129i) q^{7} -11856.2 q^{11} +28897.2i q^{13} +46018.1i q^{17} -19494.9i q^{19} +131645. q^{23} -538799. q^{25} -1.22466e6 q^{29} +1.08710e6i q^{31} +(382858. + 2.28284e6i) q^{35} +2.48906e6 q^{37} +26466.8i q^{41} -4.08031e6 q^{43} +6.95060e6i q^{47} +(5.44938e6 - 1.88075e6i) q^{49} +8.77983e6 q^{53} -1.14302e7i q^{55} -3.74308e6i q^{59} -1.40801e7i q^{61} -2.78588e7 q^{65} -7.62251e6 q^{67} -158650. q^{71} -3.88616e7i q^{73} +(-2.80746e7 + 4.70843e6i) q^{77} +1.13017e7 q^{79} -5.38238e6i q^{83} -4.43645e7 q^{85} -1.08864e7i q^{89} +(1.14759e7 + 6.84265e7i) q^{91} +1.87944e7 q^{95} +5.55681e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71}+ \cdots - 85912896 q^{95}+O(q^{100}) \) 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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