LMFDB - Embedded newform 3240.2.f.k.649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(649,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3240.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.f (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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - x^{14} + 26 x^{13} - 44 x^{12} + 6 x^{11} + 225 x^{10} - 174 x^{9} + 102 x^{8} + \cdots + 390625 $ x^16 - 2x^15 - x^14 + 26x^13 - 44x^12 + 6x^11 + 225x^10 - 174x^9 + 102x^8 - 870x^7 + 5625x^6 + 750x^5 - 27500x^4 + 81250x^3 - 15625x^2 - 156250x + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$-1.17326 + 1.90354i$ of defining polynomial
Character	$\chi$	$=$	3240.649
Dual form			3240.2.f.k.649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.17326 + 1.90354i) q^{5} +1.37527i q^{7} +O(q^{10})$	$q+(-1.17326 + 1.90354i) q^{5} +1.37527i q^{7} +0.0115956 q^{11} +1.06156i q^{13} -1.57910i q^{17} +6.38234 q^{19} +7.36519i q^{23} +(-2.24693 - 4.46669i) q^{25} -5.32462 q^{29} -2.31303 q^{31} +(-2.61787 - 1.61354i) q^{35} +10.8828i q^{37} +8.18265 q^{41} -3.38618i q^{43} +5.32277i q^{47} +5.10864 q^{49} -1.27370i q^{53} +(-0.0136046 + 0.0220726i) q^{55} -6.79425 q^{59} +8.58296 q^{61} +(-2.02071 - 1.24548i) q^{65} -9.43352i q^{67} -16.4442 q^{71} +8.32462i q^{73} +0.0159470i q^{77} -7.73325 q^{79} -5.39950i q^{83} +(3.00587 + 1.85269i) q^{85} -12.6835 q^{89} -1.45992 q^{91} +(-7.48814 + 12.1490i) q^{95} +0.0856468i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{5}+O(q^{10}) $ 16 * q + 2 * q^5	$ 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.17326	+	1.90354i	−0.524697	+	0.851289i
$5$	−1.17326	+	1.90354i	−0.524697	+	0.851289i
$6$		0			0
$7$			1.37527i			0.519802i	0.965635	+	0.259901i	$0.0836899\pi$
$7$			1.37527i			0.519802i	−0.965635	+	0.259901i	$0.916310\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.0115956			0.00349619			0.00174810	−	0.999998i	$-0.499444\pi$
$11$	0.0115956			0.00349619			0.00174810	+	0.999998i	$0.499444\pi$
$12$		0			0
$13$			1.06156i			0.294423i	0.989105	+	0.147211i	$0.0470297\pi$
$13$			1.06156i			0.294423i	−0.989105	+	0.147211i	$0.952970\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.57910i		−	0.382987i	−0.981494	−	0.191494i	$-0.938667\pi$
$17$		−	1.57910i		−	0.382987i	0.981494	−	0.191494i	$-0.0613331\pi$
$18$		0			0
$19$	6.38234			1.46421			0.732105	−	0.681192i	$-0.238538\pi$
$19$	6.38234			1.46421			0.732105	+	0.681192i	$0.238538\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.36519i			1.53575i	0.640601	+	0.767874i	$0.278685\pi$
$23$			7.36519i			1.53575i	−0.640601	+	0.767874i	$0.721315\pi$
$24$		0			0
$25$	−2.24693	−	4.46669i	−0.449386	−	0.893338i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.32462			−0.988757			−0.494379	−	0.869247i	$-0.664604\pi$
$29$	−5.32462			−0.988757			−0.494379	+	0.869247i	$0.664604\pi$
$30$		0			0
$31$	−2.31303			−0.415432			−0.207716	−	0.978189i	$-0.566603\pi$
$31$	−2.31303			−0.415432			−0.207716	+	0.978189i	$0.566603\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.61787	−	1.61354i	−0.442501	−	0.272738i
$36$		0			0
$37$			10.8828i			1.78912i	0.446948	+	0.894560i	$0.352511\pi$
$37$			10.8828i			1.78912i	−0.446948	+	0.894560i	$0.647489\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.18265			1.27792			0.638958	−	0.769242i	$-0.279366\pi$
$41$	8.18265			1.27792			0.638958	+	0.769242i	$0.279366\pi$
$42$		0			0
$43$		−	3.38618i		−	0.516387i	−0.966093	−	0.258194i	$-0.916873\pi$
$43$		−	3.38618i		−	0.516387i	0.966093	−	0.258194i	$-0.0831272\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.32277i			0.776406i	0.921574	+	0.388203i	$0.126904\pi$
$47$			5.32277i			0.776406i	−0.921574	+	0.388203i	$0.873096\pi$
$48$		0			0
$49$	5.10864			0.729806
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.27370i		−	0.174956i	−0.996166	−	0.0874782i	$-0.972119\pi$
$53$		−	1.27370i		−	0.174956i	0.996166	−	0.0874782i	$-0.0278808\pi$
$54$		0			0
$55$	−0.0136046	+	0.0220726i	−0.00183444	+	0.00297627i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.79425			−0.884536			−0.442268	−	0.896883i	$-0.645826\pi$
$59$	−6.79425			−0.884536			−0.442268	+	0.896883i	$0.645826\pi$
$60$		0			0
$61$	8.58296			1.09894			0.549468	−	0.835515i	$-0.314831\pi$
$61$	8.58296			1.09894			0.549468	+	0.835515i	$0.314831\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.02071	−	1.24548i	−0.250639	−	0.154483i
$66$		0			0
$67$		−	9.43352i		−	1.15249i	−0.817278	−	0.576244i	$-0.804518\pi$
$67$		−	9.43352i		−	1.15249i	0.817278	−	0.576244i	$-0.195482\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.4442			−1.95157			−0.975786	−	0.218727i	$-0.929809\pi$
$71$	−16.4442			−1.95157			−0.975786	+	0.218727i	$0.929809\pi$
$72$		0			0
$73$			8.32462i			0.974323i	0.873312	+	0.487162i	$0.161968\pi$
$73$			8.32462i			0.974323i	−0.873312	+	0.487162i	$0.838032\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.0159470i			0.00181733i
$78$		0			0
$79$	−7.73325			−0.870059			−0.435030	−	0.900416i	$-0.643262\pi$
$79$	−7.73325			−0.870059			−0.435030	+	0.900416i	$0.643262\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.39950i		−	0.592672i	−0.955084	−	0.296336i	$-0.904235\pi$
$83$		−	5.39950i		−	0.592672i	0.955084	−	0.296336i	$-0.0957648\pi$
$84$		0			0
$85$	3.00587	+	1.85269i	0.326033	+	0.200952i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.6835			−1.34445			−0.672223	−	0.740349i	$-0.734660\pi$
$89$	−12.6835			−1.34445			−0.672223	+	0.740349i	$0.734660\pi$
$90$		0			0
$91$	−1.45992			−0.153041
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.48814	+	12.1490i	−0.768267	+	1.24647i
$96$		0			0
$97$			0.0856468i			0.00869612i	0.999991	+	0.00434806i	$0.00138403\pi$
$97$			0.0856468i			0.00869612i	−0.999991	+	0.00434806i	$0.998616\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.4505			−1.43788			−0.718940	−	0.695072i	$-0.755373\pi$
$101$	−14.4505			−1.43788			−0.718940	+	0.695072i	$0.755373\pi$
$102$		0			0
$103$			4.93844i			0.486599i	0.969951	+	0.243300i	$0.0782298\pi$
$103$			4.93844i			0.486599i	−0.969951	+	0.243300i	$0.921770\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.5968i			1.31445i	0.753693	+	0.657227i	$0.228271\pi$
$107$			13.5968i			1.31445i	−0.753693	+	0.657227i	$0.771729\pi$
$108$		0			0
$109$	−9.92265			−0.950417			−0.475209	−	0.879873i	$-0.657627\pi$
$109$	−9.92265			−0.950417			−0.475209	+	0.879873i	$0.657627\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.53338i			0.238320i	0.992875	+	0.119160i	$0.0380202\pi$
$113$			2.53338i			0.238320i	−0.992875	+	0.119160i	$0.961980\pi$
$114$		0			0
$115$	−14.0199	−	8.64127i	−1.30737	−	0.805803i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.17168			0.199077
$120$		0			0
$121$	−10.9999			−0.999988
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1388	+	0.963456i	0.996280	+	0.0861742i
$126$		0			0
$127$		−	18.2790i		−	1.62200i	−0.585046	−	0.811000i	$-0.698924\pi$
$127$		−	18.2790i		−	1.62200i	0.585046	−	0.811000i	$-0.301076\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.29227			−0.287647			−0.143823	−	0.989603i	$-0.545940\pi$
$131$	−3.29227			−0.287647			−0.143823	+	0.989603i	$0.545940\pi$
$132$		0			0
$133$			8.77742i			0.761099i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.73489i			0.746272i	0.927777	+	0.373136i	$0.121718\pi$
$137$			8.73489i			0.746272i	−0.927777	+	0.373136i	$0.878282\pi$
$138$		0			0
$139$	−6.61501			−0.561078			−0.280539	−	0.959843i	$-0.590513\pi$
$139$	−6.61501			−0.561078			−0.280539	+	0.959843i	$0.590513\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.0123093i			0.00102936i
$144$		0			0
$145$	6.24716	−	10.1356i	0.518798	−	0.841718i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.22278			−0.509790			−0.254895	−	0.966969i	$-0.582041\pi$
$149$	−6.22278			−0.509790			−0.254895	+	0.966969i	$0.582041\pi$
$150$		0			0
$151$	4.92598			0.400871			0.200435	−	0.979707i	$-0.435764\pi$
$151$	4.92598			0.400871			0.200435	+	0.979707i	$0.435764\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.71378	−	4.40294i	0.217976	−	0.353652i
$156$		0			0
$157$		−	9.41010i		−	0.751008i	−0.926821	−	0.375504i	$-0.877470\pi$
$157$		−	9.41010i		−	0.751008i	0.926821	−	0.375504i	$-0.122530\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.1291			−0.798285
$162$		0			0
$163$			0.333348i			0.0261098i	0.999915	+	0.0130549i	$0.00415562\pi$
$163$			0.333348i			0.0261098i	−0.999915	+	0.0130549i	$0.995844\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.64253i		−	0.514014i	−0.966409	−	0.257007i	$-0.917264\pi$
$167$		−	6.64253i		−	0.514014i	0.966409	−	0.257007i	$-0.0827364\pi$
$168$		0			0
$169$	11.8731			0.913315
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.4910i			1.55790i	0.627084	+	0.778952i	$0.284248\pi$
$173$			20.4910i			1.55790i	−0.627084	+	0.778952i	$0.715752\pi$
$174$		0			0
$175$	6.14288	−	3.09013i	0.464358	−	0.233592i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.0795			1.20184			0.600919	−	0.799310i	$-0.294802\pi$
$179$	16.0795			1.20184			0.600919	+	0.799310i	$0.294802\pi$
$180$		0			0
$181$	−10.6942			−0.794891			−0.397445	−	0.917626i	$-0.630103\pi$
$181$	−10.6942			−0.794891			−0.397445	+	0.917626i	$0.630103\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.7158	−	12.7683i	−1.52306	−	0.938746i
$186$		0			0
$187$		−	0.0183105i		−	0.00133900i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.3235			−1.18113			−0.590563	−	0.806991i	$-0.701094\pi$
$191$	−16.3235			−1.18113			−0.590563	+	0.806991i	$0.701094\pi$
$192$		0			0
$193$			20.8453i			1.50048i	0.661166	+	0.750240i	$0.270062\pi$
$193$			20.8453i			1.50048i	−0.661166	+	0.750240i	$0.729938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.80666i		−	0.342460i	−0.985231	−	0.171230i	$-0.945226\pi$
$197$		−	4.80666i		−	0.342460i	0.985231	−	0.171230i	$-0.0547742\pi$
$198$		0			0
$199$	7.79500			0.552573			0.276286	−	0.961075i	$-0.410896\pi$
$199$	7.79500			0.552573			0.276286	+	0.961075i	$0.410896\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	7.32277i		−	0.513958i
$204$		0			0
$205$	−9.60036	+	15.5760i	−0.670518	+	1.08788i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0740068			0.00511916
$210$		0			0
$211$	−21.6069			−1.48748			−0.743740	−	0.668469i	$-0.766950\pi$
$211$	−21.6069			−1.48748			−0.743740	+	0.668469i	$0.766950\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.44572	+	3.97286i	0.439595	+	0.270947i
$216$		0			0
$217$		−	3.18102i		−	0.215942i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.67630			0.112760
$222$		0			0
$223$			9.92663i			0.664737i	0.943150	+	0.332368i	$0.107848\pi$
$223$			9.92663i			0.664737i	−0.943150	+	0.332368i	$0.892152\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.42093i			0.0943107i	0.998888	+	0.0471554i	$0.0150156\pi$
$227$			1.42093i			0.0943107i	−0.998888	+	0.0471554i	$0.984984\pi$
$228$		0			0
$229$	27.5073			1.81773			0.908866	−	0.417088i	$-0.136949\pi$
$229$	27.5073			1.81773			0.908866	+	0.417088i	$0.136949\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.1509i			1.77871i	0.457213	+	0.889357i	$0.348848\pi$
$233$			27.1509i			1.77871i	−0.457213	+	0.889357i	$0.651152\pi$
$234$		0			0
$235$	−10.1321	−	6.24499i	−0.660946	−	0.407378i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.8285			−1.54134			−0.770668	−	0.637237i	$-0.780077\pi$
$239$	−23.8285			−1.54134			−0.770668	+	0.637237i	$0.780077\pi$
$240$		0			0
$241$	24.3102			1.56596			0.782979	−	0.622048i	$-0.213699\pi$
$241$	24.3102			1.56596			0.782979	+	0.622048i	$0.213699\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.99376	+	9.72451i	−0.382927	+	0.621276i
$246$		0			0
$247$			6.77521i			0.431097i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.1181			−1.01737			−0.508683	−	0.860954i	$-0.669868\pi$
$251$	−16.1181			−1.01737			−0.508683	+	0.860954i	$0.669868\pi$
$252$		0			0
$253$			0.0854035i			0.00536927i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.2895i			0.891355i	0.895194	+	0.445678i	$0.147037\pi$
$257$			14.2895i			0.891355i	−0.895194	+	0.445678i	$0.852963\pi$
$258$		0			0
$259$	−14.9667			−0.929988
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.42949i			0.396460i	0.980156	+	0.198230i	$0.0635192\pi$
$263$			6.42949i			0.396460i	−0.980156	+	0.198230i	$0.936481\pi$
$264$		0			0
$265$	2.42454	+	1.49438i	0.148938	+	0.0917991i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.18395			0.498984			0.249492	−	0.968377i	$-0.419736\pi$
$269$	8.18395			0.498984			0.249492	+	0.968377i	$0.419736\pi$
$270$		0			0
$271$	−21.8758			−1.32886			−0.664430	−	0.747350i	$-0.731326\pi$
$271$	−21.8758			−1.32886			−0.664430	+	0.747350i	$0.731326\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0260544	−	0.0517937i	−0.00157114	−	0.00312328i
$276$		0			0
$277$		−	12.2897i		−	0.738416i	−0.929347	−	0.369208i	$-0.879629\pi$
$277$		−	12.2897i		−	0.738416i	0.929347	−	0.369208i	$-0.120371\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.5870			0.870186			0.435093	−	0.900385i	$-0.356715\pi$
$281$	14.5870			0.870186			0.435093	+	0.900385i	$0.356715\pi$
$282$		0			0
$283$		−	11.6492i		−	0.692473i	−0.938147	−	0.346236i	$-0.887460\pi$
$283$		−	11.6492i		−	0.692473i	0.938147	−	0.346236i	$-0.112540\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			11.2533i			0.664262i
$288$		0			0
$289$	14.5065			0.853321
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5.19641i		−	0.303578i	−0.988413	−	0.151789i	$-0.951497\pi$
$293$		−	5.19641i		−	0.303578i	0.988413	−	0.151789i	$-0.0485034\pi$
$294$		0			0
$295$	7.97141	−	12.9331i	0.464113	−	0.752996i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.81856			−0.452159
$300$		0			0
$301$	4.65689			0.268419
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0700	+	16.3380i	−0.576608	+	0.935512i
$306$		0			0
$307$		−	31.2676i		−	1.78453i	−0.451509	−	0.892267i	$-0.649114\pi$
$307$		−	31.2676i		−	1.78453i	0.451509	−	0.892267i	$-0.350886\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	27.0999			1.53670			0.768348	−	0.640032i	$-0.221079\pi$
$311$	27.0999			1.53670			0.768348	+	0.640032i	$0.221079\pi$
$312$		0			0
$313$			35.0007i			1.97836i	0.146714	+	0.989179i	$0.453130\pi$
$313$			35.0007i			1.97836i	−0.146714	+	0.989179i	$0.546870\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.4605i		−	0.980679i	−0.871532	−	0.490339i	$-0.836873\pi$
$317$		−	17.4605i		−	0.980679i	0.871532	−	0.490339i	$-0.163127\pi$
$318$		0			0
$319$	−0.0617419			−0.00345688
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10.0783i		−	0.560774i
$324$		0			0
$325$	4.74164	−	2.38524i	0.263019	−	0.132309i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.32023			−0.403577
$330$		0			0
$331$	29.1891			1.60438			0.802189	−	0.597070i	$-0.203669\pi$
$331$	29.1891			1.60438			0.802189	+	0.597070i	$0.203669\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.9571	+	11.0680i	0.981100	+	0.604707i
$336$		0			0
$337$			16.7487i			0.912361i	0.889887	+	0.456181i	$0.150783\pi$
$337$			16.7487i			0.912361i	−0.889887	+	0.456181i	$0.849217\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.0268208			−0.00145243
$342$		0			0
$343$			16.6526i			0.899156i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.9303i		−	1.12360i	−0.827275	−	0.561798i	$-0.810110\pi$
$347$		−	20.9303i		−	1.12360i	0.827275	−	0.561798i	$-0.189890\pi$
$348$		0			0
$349$	−12.1125			−0.648367			−0.324184	−	0.945994i	$-0.605090\pi$
$349$	−12.1125			−0.648367			−0.324184	+	0.945994i	$0.605090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	25.5325i		−	1.35896i	−0.733694	−	0.679480i	$-0.762205\pi$
$353$		−	25.5325i		−	1.35896i	0.733694	−	0.679480i	$-0.237795\pi$
$354$		0			0
$355$	19.2933	−	31.3023i	1.02398	−	1.66135i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.41790			−0.338724			−0.169362	−	0.985554i	$-0.554171\pi$
$359$	−6.41790			−0.338724			−0.169362	+	0.985554i	$0.554171\pi$
$360$		0			0
$361$	21.7343			1.14391
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.8463	−	9.76693i	−0.829431	−	0.511224i
$366$		0			0
$367$			4.64108i			0.242262i	0.992637	+	0.121131i	$0.0386522\pi$
$367$			4.64108i			0.242262i	−0.992637	+	0.121131i	$0.961348\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.75168			0.0909426
$372$		0			0
$373$			21.8249i			1.13005i	0.825074	+	0.565025i	$0.191133\pi$
$373$			21.8249i			1.13005i	−0.825074	+	0.565025i	$0.808867\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.65238i		−	0.291112i
$378$		0			0
$379$	−29.1095			−1.49526			−0.747628	−	0.664117i	$-0.768808\pi$
$379$	−29.1095			−1.49526			−0.747628	+	0.664117i	$0.768808\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	8.32451i		−	0.425363i	−0.977122	−	0.212681i	$-0.931780\pi$
$383$		−	8.32451i		−	0.425363i	0.977122	−	0.212681i	$-0.0682196\pi$
$384$		0			0
$385$	−0.0303557	−	0.0187099i	−0.00154707	−	0.000953545i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.85157			0.144580			0.0722902	−	0.997384i	$-0.476969\pi$
$389$	2.85157			0.144580			0.0722902	+	0.997384i	$0.476969\pi$
$390$		0			0
$391$	11.6303			0.588172
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.07310	−	14.7206i	0.456517	−	0.740672i
$396$		0			0
$397$			29.4781i			1.47946i	0.672903	+	0.739731i	$0.265047\pi$
$397$			29.4781i			1.47946i	−0.672903	+	0.739731i	$0.734953\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	25.8106			1.28892			0.644460	−	0.764638i	$-0.277082\pi$
$401$	25.8106			1.28892			0.644460	+	0.764638i	$0.277082\pi$
$402$		0			0
$403$		−	2.45541i		−	0.122312i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.126192i			0.00625511i
$408$		0			0
$409$	−27.0742			−1.33873			−0.669367	−	0.742932i	$-0.733435\pi$
$409$	−27.0742			−1.33873			−0.669367	+	0.742932i	$0.733435\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	9.34390i		−	0.459783i
$414$		0			0
$415$	10.2782	+	6.33501i	0.504535	+	0.310973i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0184			1.17338			0.586688	−	0.809813i	$-0.300432\pi$
$419$	24.0184			1.17338			0.586688	+	0.809813i	$0.300432\pi$
$420$		0			0
$421$	−8.18877			−0.399096			−0.199548	−	0.979888i	$-0.563947\pi$
$421$	−8.18877			−0.399096			−0.199548	+	0.979888i	$0.563947\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.05333	+	3.54812i	−0.342137	+	0.172109i
$426$		0			0
$427$			11.8039i			0.571228i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.20494			0.250713			0.125357	−	0.992112i	$-0.459992\pi$
$431$	5.20494			0.250713			0.125357	+	0.992112i	$0.459992\pi$
$432$		0			0
$433$			22.2500i			1.06927i	0.845084	+	0.534633i	$0.179550\pi$
$433$			22.2500i			1.06927i	−0.845084	+	0.534633i	$0.820450\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			47.0072i			2.24866i
$438$		0			0
$439$	13.9739			0.666936			0.333468	−	0.942761i	$-0.391781\pi$
$439$	13.9739			0.666936			0.333468	+	0.942761i	$0.391781\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.28238i			0.250973i	0.992095	+	0.125487i	$0.0400492\pi$
$443$			5.28238i			0.250973i	−0.992095	+	0.125487i	$0.959951\pi$
$444$		0			0
$445$	14.8810	−	24.1435i	0.705427	−	1.14451i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.87417			−0.0884474			−0.0442237	−	0.999022i	$-0.514081\pi$
$449$	−1.87417			−0.0884474			−0.0442237	+	0.999022i	$0.514081\pi$
$450$		0			0
$451$	0.0948824			0.00446784
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.71286	−	2.77902i	0.0803003	−	0.130282i
$456$		0			0
$457$		−	22.3159i		−	1.04390i	−0.852977	−	0.521948i	$-0.825206\pi$
$457$		−	22.3159i		−	1.04390i	0.852977	−	0.521948i	$-0.174794\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.7564			0.547548			0.273774	−	0.961794i	$-0.411728\pi$
$461$	11.7564			0.547548			0.273774	+	0.961794i	$0.411728\pi$
$462$		0			0
$463$			11.7955i			0.548184i	0.961703	+	0.274092i	$0.0883773\pi$
$463$			11.7955i			0.548184i	−0.961703	+	0.274092i	$0.911623\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	21.4898i		−	0.994431i	−0.867627	−	0.497215i	$-0.834356\pi$
$467$		−	21.4898i		−	0.994431i	0.867627	−	0.497215i	$-0.165644\pi$
$468$		0			0
$469$	12.9736			0.599065
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.0392646i		−	0.00180539i
$474$		0			0
$475$	−14.3407	−	28.5079i	−0.657996	−	1.30803i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.97091			−0.181436			−0.0907178	−	0.995877i	$-0.528916\pi$
$479$	−3.97091			−0.181436			−0.0907178	+	0.995877i	$0.528916\pi$
$480$		0			0
$481$	−11.5527			−0.526757
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.163032	−	0.100486i	−0.00740291	−	0.00456283i
$486$		0			0
$487$			12.7847i			0.579329i	0.957128	+	0.289664i	$0.0935437\pi$
$487$			12.7847i			0.579329i	−0.957128	+	0.289664i	$0.906456\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.00122			−0.315961			−0.157980	−	0.987442i	$-0.550498\pi$
$491$	−7.00122			−0.315961			−0.157980	+	0.987442i	$0.550498\pi$
$492$		0			0
$493$			8.40809i			0.378681i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	22.6152i		−	1.01443i
$498$		0			0
$499$	11.5247			0.515916			0.257958	−	0.966156i	$-0.416950\pi$
$499$	11.5247			0.515916			0.257958	+	0.966156i	$0.416950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.38880i		−	0.329450i	−0.986340	−	0.164725i	$-0.947326\pi$
$503$		−	7.38880i		−	0.329450i	0.986340	−	0.164725i	$-0.0526737\pi$
$504$		0			0
$505$	16.9542	−	27.5071i	0.754452	−	1.22405i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.22133			−0.0984587			−0.0492294	−	0.998787i	$-0.515677\pi$
$509$	−2.22133			−0.0984587			−0.0492294	+	0.998787i	$0.515677\pi$
$510$		0			0
$511$	−11.4486			−0.506455
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.40053	−	5.79407i	−0.414237	−	0.255317i
$516$		0			0
$517$			0.0617205i			0.00271446i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.09160			0.354499			0.177250	−	0.984166i	$-0.443280\pi$
$521$	8.09160			0.354499			0.177250	+	0.984166i	$0.443280\pi$
$522$		0			0
$523$			14.4193i			0.630512i	0.949007	+	0.315256i	$0.102090\pi$
$523$			14.4193i			0.630512i	−0.949007	+	0.315256i	$0.897910\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			3.65249i			0.159105i
$528$		0			0
$529$	−31.2460			−1.35852
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8.68634i			0.376247i
$534$		0			0
$535$	−25.8821	−	15.9526i	−1.11898	−	0.689690i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.0592376			0.00255154
$540$		0			0
$541$	5.37804			0.231220			0.115610	−	0.993295i	$-0.463118\pi$
$541$	5.37804			0.231220			0.115610	+	0.993295i	$0.463118\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.6418	−	18.8882i	0.498681	−	0.809080i
$546$		0			0
$547$		−	25.5753i		−	1.09352i	−0.837290	−	0.546760i	$-0.815861\pi$
$547$		−	25.5753i		−	1.09352i	0.837290	−	0.546760i	$-0.184139\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−33.9836			−1.44775
$552$		0			0
$553$		−	10.6353i		−	0.452258i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	36.1519i		−	1.53181i	−0.642956	−	0.765903i	$-0.722292\pi$
$557$		−	36.1519i		−	1.53181i	0.642956	−	0.765903i	$-0.277708\pi$
$558$		0			0
$559$	3.59462			0.152036
$560$		0			0
$561$		0			0
$562$		0			0
$563$			33.9013i			1.42877i	0.699752	+	0.714385i	$0.253294\pi$
$563$			33.9013i			1.42877i	−0.699752	+	0.714385i	$0.746706\pi$
$564$		0			0
$565$	−4.82239	−	2.97231i	−0.202879	−	0.125046i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.0114			1.17430			0.587149	−	0.809479i	$-0.300250\pi$
$569$	28.0114			1.17430			0.587149	+	0.809479i	$0.300250\pi$
$570$		0			0
$571$	−34.6080			−1.44830			−0.724150	−	0.689642i	$-0.757768\pi$
$571$	−34.6080			−1.44830			−0.724150	+	0.689642i	$0.757768\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.8980	−	16.5491i	1.37194	−	0.690144i
$576$		0			0
$577$		−	1.88098i		−	0.0783062i	−0.999233	−	0.0391531i	$-0.987534\pi$
$577$		−	1.88098i		−	0.0783062i	0.999233	−	0.0391531i	$-0.0124660\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.42575			0.308072
$582$		0			0
$583$		−	0.0147693i		−	0.000611681i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			5.68881i			0.234802i	0.993085	+	0.117401i	$0.0374563\pi$
$587$			5.68881i			0.234802i	−0.993085	+	0.117401i	$0.962544\pi$
$588$		0			0
$589$	−14.7625			−0.608279
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	23.6302i		−	0.970378i	−0.874409	−	0.485189i	$-0.838751\pi$
$593$		−	23.6302i		−	0.970378i	0.874409	−	0.485189i	$-0.161249\pi$
$594$		0			0
$595$	−2.54794	+	4.13388i	−0.104455	+	0.169472i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23.7847			0.971816			0.485908	−	0.874010i	$-0.338489\pi$
$599$	23.7847			0.971816			0.485908	+	0.874010i	$0.338489\pi$
$600$		0			0
$601$	−8.61803			−0.351537			−0.175768	−	0.984432i	$-0.556241\pi$
$601$	−8.61803			−0.351537			−0.175768	+	0.984432i	$0.556241\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.9057	−	20.9387i	0.524691	−	0.851279i
$606$		0			0
$607$		−	27.3863i		−	1.11157i	−0.831325	−	0.555787i	$-0.812417\pi$
$607$		−	27.3863i		−	1.11157i	0.831325	−	0.555787i	$-0.187583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.65042			−0.228592
$612$		0			0
$613$		−	1.61637i		−	0.0652845i	−0.999467	−	0.0326422i	$-0.989608\pi$
$613$		−	1.61637i		−	0.0652845i	0.999467	−	0.0326422i	$-0.0103922\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			9.40528i			0.378642i	0.981915	+	0.189321i	$0.0606287\pi$
$617$			9.40528i			0.378642i	−0.981915	+	0.189321i	$0.939371\pi$
$618$		0			0
$619$	−3.24858			−0.130572			−0.0652858	−	0.997867i	$-0.520796\pi$
$619$	−3.24858			−0.130572			−0.0652858	+	0.997867i	$0.520796\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	17.4432i		−	0.698845i
$624$		0			0
$625$	−14.9026	+	20.0727i	−0.596104	+	0.802907i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.1850			0.685210
$630$		0			0
$631$	22.0746			0.878777			0.439389	−	0.898297i	$-0.355195\pi$
$631$	22.0746			0.878777			0.439389	+	0.898297i	$0.355195\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	34.7948	+	21.4460i	1.38079	+	0.851059i
$636$		0			0
$637$			5.42311i			0.214871i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	43.2299			1.70748			0.853739	−	0.520700i	$-0.174329\pi$
$641$	43.2299			1.70748			0.853739	+	0.520700i	$0.174329\pi$
$642$		0			0
$643$			15.7309i			0.620365i	0.950677	+	0.310183i	$0.100390\pi$
$643$			15.7309i			0.620365i	−0.950677	+	0.310183i	$0.899610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.8513i			1.01632i	0.861263	+	0.508159i	$0.169674\pi$
$647$			25.8513i			1.01632i	−0.861263	+	0.508159i	$0.830326\pi$
$648$		0			0
$649$	−0.0787831			−0.00309251
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	25.3107i		−	0.990483i	−0.868755	−	0.495242i	$-0.835080\pi$
$653$		−	25.3107i		−	0.990483i	0.868755	−	0.495242i	$-0.164920\pi$
$654$		0			0
$655$	3.86268	−	6.26696i	0.150927	−	0.244870i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27.2853			1.06288			0.531442	−	0.847095i	$-0.321650\pi$
$659$	27.2853			1.06288			0.531442	+	0.847095i	$0.321650\pi$
$660$		0			0
$661$	22.2554			0.865634			0.432817	−	0.901482i	$-0.357520\pi$
$661$	22.2554			0.865634			0.432817	+	0.901482i	$0.357520\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.7082	−	10.2982i	−0.647915	−	0.399346i
$666$		0			0
$667$		−	39.2169i		−	1.51848i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.0995242			0.00384209
$672$		0			0
$673$			17.2437i			0.664694i	0.943157	+	0.332347i	$0.107841\pi$
$673$			17.2437i			0.664694i	−0.943157	+	0.332347i	$0.892159\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.08573i		−	0.0417278i	−0.999782	−	0.0208639i	$-0.993358\pi$
$677$		−	1.08573i		−	0.0417278i	0.999782	−	0.0208639i	$-0.00664167\pi$
$678$		0			0
$679$	−0.117787			−0.00452026
$680$		0			0
$681$		0			0
$682$		0			0
$683$			3.62926i			0.138870i	0.997586	+	0.0694349i	$0.0221196\pi$
$683$			3.62926i			0.138870i	−0.997586	+	0.0694349i	$0.977880\pi$
$684$		0			0
$685$	−16.6272	−	10.2483i	−0.635293	−	0.391567i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.35210			0.0515111
$690$		0			0
$691$	−28.9733			−1.10220			−0.551099	−	0.834440i	$-0.685791\pi$
$691$	−28.9733			−1.10220			−0.551099	+	0.834440i	$0.685791\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.76111	−	12.5919i	0.294396	−	0.477639i
$696$		0			0
$697$		−	12.9212i		−	0.489425i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28.4520			1.07462			0.537309	−	0.843385i	$-0.319441\pi$
$701$	28.4520			1.07462			0.537309	+	0.843385i	$0.319441\pi$
$702$		0			0
$703$			69.4577i			2.61965i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	19.8733i		−	0.747413i
$708$		0			0
$709$	18.8116			0.706486			0.353243	−	0.935532i	$-0.385079\pi$
$709$	18.8116			0.706486			0.353243	+	0.935532i	$0.385079\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	17.0359i		−	0.637999i
$714$		0			0
$715$	−0.0234313	−	0.0144420i	−0.000876281	−	0.000540101i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.22562			0.157589			0.0787946	−	0.996891i	$-0.474893\pi$
$719$	4.22562			0.157589			0.0787946	+	0.996891i	$0.474893\pi$
$720$		0			0
$721$	−6.79167			−0.252935
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.9641	+	23.7834i	0.444334	+	0.883294i
$726$		0			0
$727$			4.65400i			0.172607i	0.996269	+	0.0863036i	$0.0275055\pi$
$727$			4.65400i			0.172607i	−0.996269	+	0.0863036i	$0.972495\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.34710			−0.197770
$732$		0			0
$733$			25.0000i			0.923395i	0.887037	+	0.461697i	$0.152759\pi$
$733$			25.0000i			0.923395i	−0.887037	+	0.461697i	$0.847241\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.109387i		−	0.00402932i
$738$		0			0
$739$	−18.4824			−0.679887			−0.339944	−	0.940446i	$-0.610408\pi$
$739$	−18.4824			−0.679887			−0.339944	+	0.940446i	$0.610408\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.30761i			0.121345i	0.998158	+	0.0606723i	$0.0193245\pi$
$743$			3.30761i			0.121345i	−0.998158	+	0.0606723i	$0.980676\pi$
$744$		0			0
$745$	7.30093	−	11.8453i	0.267485	−	0.433979i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.6992			−0.683255
$750$		0			0
$751$	45.8466			1.67297			0.836483	−	0.547992i	$-0.184608\pi$
$751$	45.8466			1.67297			0.836483	+	0.547992i	$0.184608\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.77945	+	9.37680i	−0.210336	+	0.341257i
$756$		0			0
$757$			12.2649i			0.445775i	0.974844	+	0.222888i	$0.0715483\pi$
$757$			12.2649i			0.445775i	−0.974844	+	0.222888i	$0.928452\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.0884			0.691954			0.345977	−	0.938243i	$-0.387548\pi$
$761$	19.0884			0.691954			0.345977	+	0.938243i	$0.387548\pi$
$762$		0			0
$763$		−	13.6463i		−	0.494029i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	7.21247i		−	0.260427i
$768$		0			0
$769$	7.20065			0.259662			0.129831	−	0.991536i	$-0.458556\pi$
$769$	7.20065			0.259662			0.129831	+	0.991536i	$0.458556\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.69028i		−	0.0607950i	−0.999538	−	0.0303975i	$-0.990323\pi$
$773$		−	1.69028i		−	0.0607950i	0.999538	−	0.0303975i	$-0.00967732\pi$
$774$		0			0
$775$	5.19721	+	10.3316i	0.186689	+	0.371121i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	52.2245			1.87114
$780$		0			0
$781$	−0.190680			−0.00682307
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.9125	+	11.0405i	0.639325	+	0.394052i
$786$		0			0
$787$		−	33.2410i		−	1.18491i	−0.805602	−	0.592457i	$-0.798158\pi$
$787$		−	33.2410i		−	1.18491i	0.805602	−	0.592457i	$-0.201842\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.48407			−0.123879
$792$		0			0
$793$			9.11129i			0.323551i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	43.2722i		−	1.53278i	−0.642376	−	0.766390i	$-0.722051\pi$
$797$		−	43.2722i		−	1.53278i	0.642376	−	0.766390i	$-0.277949\pi$
$798$		0			0
$799$	8.40517			0.297354
$800$		0			0
$801$		0			0
$802$		0			0
$803$			0.0965286i			0.00340642i
$804$		0			0
$805$	11.8840	−	19.2811i	0.418858	−	0.679571i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.4664			−0.367981			−0.183990	−	0.982928i	$-0.558902\pi$
$809$	−10.4664			−0.367981			−0.183990	+	0.982928i	$0.558902\pi$
$810$		0			0
$811$	−21.4200			−0.752158			−0.376079	−	0.926588i	$-0.622728\pi$
$811$	−21.4200			−0.752158			−0.376079	+	0.926588i	$0.622728\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.634540	−	0.391103i	−0.0222270	−	0.0136997i
$816$		0			0
$817$		−	21.6117i		−	0.756099i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.04929			0.280922			0.140461	−	0.990086i	$-0.455142\pi$
$821$	8.04929			0.280922			0.140461	+	0.990086i	$0.455142\pi$
$822$		0			0
$823$			36.5613i			1.27445i	0.770679	+	0.637223i	$0.219917\pi$
$823$			36.5613i			1.27445i	−0.770679	+	0.637223i	$0.780083\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7.52915i			0.261814i	0.991395	+	0.130907i	$0.0417890\pi$
$827$			7.52915i			0.261814i	−0.991395	+	0.130907i	$0.958211\pi$
$828$		0			0
$829$	−0.188646			−0.00655195			−0.00327597	−	0.999995i	$-0.501043\pi$
$829$	−0.188646			−0.00655195			−0.00327597	+	0.999995i	$0.501043\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	8.06704i		−	0.279506i
$834$		0			0
$835$	12.6443	+	7.79340i	0.437575	+	0.269702i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.4500			0.567916			0.283958	−	0.958837i	$-0.408352\pi$
$839$	16.4500			0.567916			0.283958	+	0.958837i	$0.408352\pi$
$840$		0			0
$841$	−0.648411			−0.0223590
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.9302	+	22.6009i	−0.479214	+	0.777495i
$846$		0			0
$847$		−	15.1277i		−	0.519795i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−80.1539			−2.74764
$852$		0			0
$853$			23.4556i			0.803105i	0.915836	+	0.401552i	$0.131529\pi$
$853$			23.4556i			0.803105i	−0.915836	+	0.401552i	$0.868471\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			19.0205i			0.649727i	0.945761	+	0.324864i	$0.105318\pi$
$857$			19.0205i			0.649727i	−0.945761	+	0.324864i	$0.894682\pi$
$858$		0			0
$859$	−15.8182			−0.539711			−0.269856	−	0.962901i	$-0.586976\pi$
$859$	−15.8182			−0.539711			−0.269856	+	0.962901i	$0.586976\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7.81226i		−	0.265933i	−0.991121	−	0.132966i	$-0.957550\pi$
$863$		−	7.81226i		−	0.265933i	0.991121	−	0.132966i	$-0.0424502\pi$
$864$		0			0
$865$	−39.0055	−	24.0413i	−1.32623	−	0.817427i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.0896714			−0.00304189
$870$		0			0
$871$	10.0142			0.339319
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.32501	+	15.3187i	−0.0447935	+	0.517868i
$876$		0			0
$877$		−	54.8706i		−	1.85285i	−0.376483	−	0.926424i	$-0.622867\pi$
$877$		−	54.8706i		−	1.85285i	0.376483	−	0.926424i	$-0.377133\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.1017			1.04784			0.523922	−	0.851766i	$-0.324468\pi$
$881$	31.1017			1.04784			0.523922	+	0.851766i	$0.324468\pi$
$882$		0			0
$883$		−	20.5411i		−	0.691264i	−0.938370	−	0.345632i	$-0.887665\pi$
$883$		−	20.5411i		−	0.691264i	0.938370	−	0.345632i	$-0.112335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	52.1838i		−	1.75216i	−0.482165	−	0.876081i	$-0.660149\pi$
$887$		−	52.1838i		−	1.75216i	0.482165	−	0.876081i	$-0.339851\pi$
$888$		0			0
$889$	25.1385			0.843118
$890$		0			0
$891$		0			0
$892$		0			0
$893$			33.9718i			1.13682i
$894$		0			0
$895$	−18.8654	+	30.6079i	−0.630600	+	1.02311i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.3160			0.410761
$900$		0			0
$901$	−2.01130			−0.0670060
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.5470	−	20.3568i	0.417077	−	0.676682i
$906$		0			0
$907$			25.5965i			0.849918i	0.905212	+	0.424959i	$0.139712\pi$
$907$			25.5965i			0.849918i	−0.905212	+	0.424959i	$0.860288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	41.1950			1.36485			0.682426	−	0.730954i	$-0.260925\pi$
$911$	41.1950			1.36485			0.682426	+	0.730954i	$0.260925\pi$
$912$		0			0
$913$		−	0.0626102i		−	0.00207209i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.52774i		−	0.149519i
$918$		0			0
$919$	−4.84011			−0.159661			−0.0798303	−	0.996808i	$-0.525438\pi$
$919$	−4.84011			−0.159661			−0.0798303	+	0.996808i	$0.525438\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	17.4565i		−	0.574587i
$924$		0			0
$925$	48.6100	−	24.4529i	1.59829	−	0.804006i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−39.8639			−1.30789			−0.653945	−	0.756542i	$-0.726887\pi$
$929$	−39.8639			−1.30789			−0.653945	+	0.756542i	$0.726887\pi$
$930$		0			0
$931$	32.6051			1.06859
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.0348548	+	0.0214829i	0.00113987	+	0.000702567i
$936$		0			0
$937$			5.03028i			0.164332i	0.996619	+	0.0821661i	$0.0261838\pi$
$937$			5.03028i			0.164332i	−0.996619	+	0.0821661i	$0.973816\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.8748			0.680499			0.340250	−	0.940335i	$-0.389488\pi$
$941$	20.8748			0.680499			0.340250	+	0.940335i	$0.389488\pi$
$942$		0			0
$943$			60.2668i			1.96256i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.0754729i			0.00245254i	0.999999	+	0.00122627i	$0.000390334\pi$
$947$			0.0754729i			0.00245254i	−0.999999	+	0.00122627i	$0.999610\pi$
$948$		0			0
$949$	−8.83705			−0.286863
$950$		0			0
$951$		0			0
$952$		0			0
$953$			23.6892i			0.767370i	0.923464	+	0.383685i	$0.125345\pi$
$953$			23.6892i			0.767370i	−0.923464	+	0.383685i	$0.874655\pi$
$954$		0			0
$955$	19.1517	−	31.0724i	0.619733	−	1.00548i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.0128			−0.387913
$960$		0			0
$961$	−25.6499			−0.827417
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−39.6799	−	24.4570i	−1.27734	−	0.787297i
$966$		0			0
$967$			45.2881i			1.45637i	0.685383	+	0.728183i	$0.259635\pi$
$967$			45.2881i			1.45637i	−0.685383	+	0.728183i	$0.740365\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.4586			0.817004			0.408502	−	0.912757i	$-0.366051\pi$
$971$	25.4586			0.817004			0.408502	+	0.912757i	$0.366051\pi$
$972$		0			0
$973$		−	9.09739i		−	0.291649i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	8.18830i		−	0.261967i	−0.991385	−	0.130983i	$-0.958187\pi$
$977$		−	8.18830i		−	0.261967i	0.991385	−	0.130983i	$-0.0418135\pi$
$978$		0			0
$979$	−0.147072			−0.00470044
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	12.2400i		−	0.390394i	−0.980764	−	0.195197i	$-0.937465\pi$
$983$		−	12.2400i		−	0.390394i	0.980764	−	0.195197i	$-0.0625346\pi$
$984$		0			0
$985$	9.14967	+	5.63945i	0.291533	+	0.179688i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.9398			0.793041
$990$		0			0
$991$	48.0040			1.52490			0.762448	−	0.647049i	$-0.223997\pi$
$991$	48.0040			1.52490			0.762448	+	0.647049i	$0.223997\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.14554	+	14.8381i	−0.289933	+	0.470399i
$996$		0			0
$997$			6.19582i			0.196224i	0.995175	+	0.0981118i	$0.0312803\pi$
$997$			6.19582i			0.196224i	−0.995175	+	0.0981118i	$0.968720\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.f.k.649.6		16
3.2	odd	2		3240.2.f.i.649.11		16
5.4	even	2	inner	3240.2.f.k.649.5		16
9.2	odd	6		1080.2.bi.b.1009.11		32
9.4	even	3		360.2.bi.b.169.1	yes	32
9.5	odd	6		1080.2.bi.b.289.1		32
9.7	even	3		360.2.bi.b.49.16	yes	32
15.14	odd	2		3240.2.f.i.649.12		16
36.7	odd	6		720.2.by.f.49.1		32
36.11	even	6		2160.2.by.f.1009.11		32
36.23	even	6		2160.2.by.f.289.1		32
36.31	odd	6		720.2.by.f.529.16		32
45.4	even	6		360.2.bi.b.169.16	yes	32
45.14	odd	6		1080.2.bi.b.289.11		32
45.29	odd	6		1080.2.bi.b.1009.1		32
45.34	even	6		360.2.bi.b.49.1	✓	32
180.59	even	6		2160.2.by.f.289.11		32
180.79	odd	6		720.2.by.f.49.16		32
180.119	even	6		2160.2.by.f.1009.1		32
180.139	odd	6		720.2.by.f.529.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bi.b.49.1	✓	32	45.34	even	6
360.2.bi.b.49.16	yes	32	9.7	even	3
360.2.bi.b.169.1	yes	32	9.4	even	3
360.2.bi.b.169.16	yes	32	45.4	even	6
720.2.by.f.49.1		32	36.7	odd	6
720.2.by.f.49.16		32	180.79	odd	6
720.2.by.f.529.1		32	180.139	odd	6
720.2.by.f.529.16		32	36.31	odd	6
1080.2.bi.b.289.1		32	9.5	odd	6
1080.2.bi.b.289.11		32	45.14	odd	6
1080.2.bi.b.1009.1		32	45.29	odd	6
1080.2.bi.b.1009.11		32	9.2	odd	6
2160.2.by.f.289.1		32	36.23	even	6
2160.2.by.f.289.11		32	180.59	even	6
2160.2.by.f.1009.1		32	180.119	even	6
2160.2.by.f.1009.11		32	36.11	even	6
3240.2.f.i.649.11		16	3.2	odd	2
3240.2.f.i.649.12		16	15.14	odd	2
3240.2.f.k.649.5		16	5.4	even	2	inner
3240.2.f.k.649.6		16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.17326 + 1.90354i) q^{5} +1.37527i q^{7} +O(q^{10})\)	\(q+(-1.17326 + 1.90354i) q^{5} +1.37527i q^{7} +0.0115956 q^{11} +1.06156i q^{13} -1.57910i q^{17} +6.38234 q^{19} +7.36519i q^{23} +(-2.24693 - 4.46669i) q^{25} -5.32462 q^{29} -2.31303 q^{31} +(-2.61787 - 1.61354i) q^{35} +10.8828i q^{37} +8.18265 q^{41} -3.38618i q^{43} +5.32277i q^{47} +5.10864 q^{49} -1.27370i q^{53} +(-0.0136046 + 0.0220726i) q^{55} -6.79425 q^{59} +8.58296 q^{61} +(-2.02071 - 1.24548i) q^{65} -9.43352i q^{67} -16.4442 q^{71} +8.32462i q^{73} +0.0159470i q^{77} -7.73325 q^{79} -5.39950i q^{83} +(3.00587 + 1.85269i) q^{85} -12.6835 q^{89} -1.45992 q^{91} +(-7.48814 + 12.1490i) q^{95} +0.0856468i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{5}+O(q^{10}) \) 16 * q + 2 * q^5	\( 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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