LMFDB - Embedded newform 912.3.o.d.721.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.184143974400.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 215x^4 - 568x^3 - 1022x^2 + 1320*x + 2628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.3
Root			$-2.87998 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	912.721
Dual form			912.3.o.d.721.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +2.94975 q^{5} -5.84873 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +2.94975 q^{5} -5.84873 q^{7} -3.00000 q^{9} -18.6246 q^{11} -1.55708i q^{13} -5.10912i q^{15} +12.7477 q^{17} +(17.7256 + 6.84117i) q^{19} +10.1303i q^{21} +15.8205 q^{23} -16.2990 q^{25} +5.19615i q^{27} +46.0273i q^{29} +37.5811i q^{31} +32.2588i q^{33} -17.2523 q^{35} +17.6705i q^{37} -2.69694 q^{39} +33.3798i q^{41} +50.6939 q^{43} -8.84925 q^{45} +15.1733 q^{47} -14.7924 q^{49} -22.0797i q^{51} -75.3754i q^{53} -54.9380 q^{55} +(11.8493 - 30.7017i) q^{57} +53.0940i q^{59} +11.3011 q^{61} +17.5462 q^{63} -4.59299i q^{65} -12.1999i q^{67} -27.4020i q^{69} +20.8673i q^{71} -34.2076 q^{73} +28.2307i q^{75} +108.930 q^{77} +128.601i q^{79} +9.00000 q^{81} -66.9672 q^{83} +37.6026 q^{85} +79.7216 q^{87} +25.1600i q^{89} +9.10693i q^{91} +65.0924 q^{93} +(52.2862 + 20.1797i) q^{95} +117.436i q^{97} +55.8739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9	$ 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9} - 4 q^{11} + 4 q^{17} + 36 q^{19} + 56 q^{23} + 140 q^{25} - 236 q^{35} + 96 q^{39} - 100 q^{43} - 12 q^{45} + 188 q^{47} - 36 q^{49} - 28 q^{55} + 36 q^{57} - 180 q^{61} - 36 q^{63} - 356 q^{73} + 68 q^{77} + 72 q^{81} - 136 q^{83} + 148 q^{85} + 144 q^{87} + 168 q^{93} + 140 q^{95} + 12 q^{99}+O(q^{100}) $ 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9 - 4 * q^11 + 4 * q^17 + 36 * q^19 + 56 * q^23 + 140 * q^25 - 236 * q^35 + 96 * q^39 - 100 * q^43 - 12 * q^45 + 188 * q^47 - 36 * q^49 - 28 * q^55 + 36 * q^57 - 180 * q^61 - 36 * q^63 - 356 * q^73 + 68 * q^77 + 72 * q^81 - 136 * q^83 + 148 * q^85 + 144 * q^87 + 168 * q^93 + 140 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$	2.94975			0.589950			0.294975	−	0.955505i	$-0.404689\pi$
$5$	2.94975			0.589950			0.294975	+	0.955505i	$0.404689\pi$
$6$		0			0
$7$	−5.84873			−0.835533			−0.417766	−	0.908554i	$-0.637187\pi$
$7$	−5.84873			−0.835533			−0.417766	+	0.908554i	$0.637187\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−18.6246			−1.69315			−0.846574	−	0.532271i	$-0.821339\pi$
$11$	−18.6246			−1.69315			−0.846574	+	0.532271i	$0.821339\pi$
$12$		0			0
$13$		−	1.55708i		−	0.119775i	−0.998205	−	0.0598876i	$-0.980926\pi$
$13$		−	1.55708i		−	0.119775i	0.998205	−	0.0598876i	$-0.0190742\pi$
$14$		0			0
$15$		−	5.10912i		−	0.340608i
$16$		0			0
$17$	12.7477			0.749865			0.374933	−	0.927052i	$-0.377666\pi$
$17$	12.7477			0.749865			0.374933	+	0.927052i	$0.377666\pi$
$18$		0			0
$19$	17.7256	+	6.84117i	0.932929	+	0.360061i
$19$	17.7256	+	6.84117i	0.932929	+	0.360061i
$20$		0			0
$21$			10.1303i			0.482395i
$22$		0			0
$23$	15.8205			0.687850			0.343925	−	0.938997i	$-0.388243\pi$
$23$	15.8205			0.687850			0.343925	+	0.938997i	$0.388243\pi$
$24$		0			0
$25$	−16.2990			−0.651959
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$			46.0273i			1.58715i	0.608474	+	0.793573i	$0.291782\pi$
$29$			46.0273i			1.58715i	−0.608474	+	0.793573i	$0.708218\pi$
$30$		0			0
$31$			37.5811i			1.21229i	0.795353	+	0.606147i	$0.207286\pi$
$31$			37.5811i			1.21229i	−0.795353	+	0.606147i	$0.792714\pi$
$32$		0			0
$33$			32.2588i			0.977539i
$34$		0			0
$35$	−17.2523			−0.492923
$36$		0			0
$37$			17.6705i			0.477580i	0.971071	+	0.238790i	$0.0767507\pi$
$37$			17.6705i			0.477580i	−0.971071	+	0.238790i	$0.923249\pi$
$38$		0			0
$39$	−2.69694			−0.0691523
$40$		0			0
$41$			33.3798i			0.814141i	0.913397	+	0.407071i	$0.133450\pi$
$41$			33.3798i			0.814141i	−0.913397	+	0.407071i	$0.866550\pi$
$42$		0			0
$43$	50.6939			1.17893			0.589464	−	0.807795i	$-0.299339\pi$
$43$	50.6939			1.17893			0.589464	+	0.807795i	$0.299339\pi$
$44$		0			0
$45$	−8.84925			−0.196650
$46$		0			0
$47$	15.1733			0.322837			0.161418	−	0.986886i	$-0.448393\pi$
$47$	15.1733			0.322837			0.161418	+	0.986886i	$0.448393\pi$
$48$		0			0
$49$	−14.7924			−0.301885
$50$		0			0
$51$		−	22.0797i		−	0.432935i
$52$		0			0
$53$		−	75.3754i		−	1.42218i	−0.703102	−	0.711089i	$-0.748202\pi$
$53$		−	75.3754i		−	1.42218i	0.703102	−	0.711089i	$-0.251798\pi$
$54$		0			0
$55$	−54.9380			−0.998872
$56$		0			0
$57$	11.8493	−	30.7017i	0.207882	−	0.538627i
$58$		0			0
$59$			53.0940i			0.899899i	0.893054	+	0.449949i	$0.148558\pi$
$59$			53.0940i			0.899899i	−0.893054	+	0.449949i	$0.851442\pi$
$60$		0			0
$61$	11.3011			0.185263			0.0926316	−	0.995700i	$-0.470472\pi$
$61$	11.3011			0.185263			0.0926316	+	0.995700i	$0.470472\pi$
$62$		0			0
$63$	17.5462			0.278511
$64$		0			0
$65$		−	4.59299i		−	0.0706614i
$66$		0			0
$67$		−	12.1999i		−	0.182088i	−0.995847	−	0.0910440i	$-0.970980\pi$
$67$		−	12.1999i		−	0.182088i	0.995847	−	0.0910440i	$-0.0290204\pi$
$68$		0			0
$69$		−	27.4020i		−	0.397130i
$70$		0			0
$71$			20.8673i			0.293905i	0.989144	+	0.146953i	$0.0469465\pi$
$71$			20.8673i			0.293905i	−0.989144	+	0.146953i	$0.953054\pi$
$72$		0			0
$73$	−34.2076			−0.468598			−0.234299	−	0.972165i	$-0.575279\pi$
$73$	−34.2076			−0.468598			−0.234299	+	0.972165i	$0.575279\pi$
$74$		0			0
$75$			28.2307i			0.376409i
$76$		0			0
$77$	108.930			1.41468
$78$		0			0
$79$			128.601i			1.62786i	0.580965	+	0.813929i	$0.302675\pi$
$79$			128.601i			1.62786i	−0.580965	+	0.813929i	$0.697325\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−66.9672			−0.806834			−0.403417	−	0.915016i	$-0.632178\pi$
$83$	−66.9672			−0.806834			−0.403417	+	0.915016i	$0.632178\pi$
$84$		0			0
$85$	37.6026			0.442383
$86$		0			0
$87$	79.7216			0.916340
$88$		0			0
$89$			25.1600i			0.282697i	0.989960	+	0.141348i	$0.0451437\pi$
$89$			25.1600i			0.282697i	−0.989960	+	0.141348i	$0.954856\pi$
$90$		0			0
$91$			9.10693i			0.100076i
$92$		0			0
$93$	65.0924			0.699918
$94$		0			0
$95$	52.2862	+	20.1797i	0.550381	+	0.212418i
$96$		0			0
$97$			117.436i			1.21068i	0.795968	+	0.605339i	$0.206962\pi$
$97$			117.436i			1.21068i	−0.795968	+	0.605339i	$0.793038\pi$
$98$		0			0
$99$	55.8739			0.564383
$100$		0			0
$101$	−45.8852			−0.454309			−0.227154	−	0.973859i	$-0.572942\pi$
$101$	−45.8852			−0.454309			−0.227154	+	0.973859i	$0.572942\pi$
$102$		0			0
$103$		−	169.904i		−	1.64956i	−0.565455	−	0.824779i	$-0.691299\pi$
$103$		−	169.904i		−	1.64956i	0.565455	−	0.824779i	$-0.308701\pi$
$104$		0			0
$105$			29.8818i			0.284589i
$106$		0			0
$107$			183.761i			1.71739i	0.512486	+	0.858696i	$0.328725\pi$
$107$			183.761i			1.71739i	−0.512486	+	0.858696i	$0.671275\pi$
$108$		0			0
$109$			59.0488i			0.541732i	0.962617	+	0.270866i	$0.0873101\pi$
$109$			59.0488i			0.541732i	−0.962617	+	0.270866i	$0.912690\pi$
$110$		0			0
$111$	30.6061			0.275731
$112$		0			0
$113$			103.714i			0.917825i	0.888481	+	0.458913i	$0.151761\pi$
$113$			103.714i			0.917825i	−0.888481	+	0.458913i	$0.848239\pi$
$114$		0			0
$115$	46.6667			0.405797
$116$		0			0
$117$			4.67123i			0.0399251i
$118$		0			0
$119$	−74.5579			−0.626537
$120$		0			0
$121$	225.877			1.86675
$122$		0			0
$123$	57.8155			0.470045
$124$		0			0
$125$	−121.822			−0.974573
$126$		0			0
$127$			27.7707i			0.218667i	0.994005	+	0.109333i	$0.0348716\pi$
$127$			27.7707i			0.218667i	−0.994005	+	0.109333i	$0.965128\pi$
$128$		0			0
$129$		−	87.8044i		−	0.680654i
$130$		0			0
$131$	−102.402			−0.781696			−0.390848	−	0.920455i	$-0.627818\pi$
$131$	−102.402			−0.781696			−0.390848	+	0.920455i	$0.627818\pi$
$132$		0			0
$133$	−103.672	−	40.0121i	−0.779492	−	0.300843i
$134$		0			0
$135$			15.3274i			0.113536i
$136$		0			0
$137$	−243.826			−1.77975			−0.889875	−	0.456204i	$-0.849209\pi$
$137$	−243.826			−1.77975			−0.889875	+	0.456204i	$0.849209\pi$
$138$		0			0
$139$	183.616			1.32098			0.660490	−	0.750835i	$-0.270349\pi$
$139$	183.616			1.32098			0.660490	+	0.750835i	$0.270349\pi$
$140$		0			0
$141$		−	26.2810i		−	0.186390i
$142$		0			0
$143$			29.0000i			0.202797i
$144$		0			0
$145$			135.769i			0.936337i
$146$		0			0
$147$			25.6211i			0.174293i
$148$		0			0
$149$	154.434			1.03647			0.518234	−	0.855239i	$-0.326589\pi$
$149$	154.434			1.03647			0.518234	+	0.855239i	$0.326589\pi$
$150$		0			0
$151$		−	193.877i		−	1.28395i	−0.766724	−	0.641977i	$-0.778114\pi$
$151$		−	193.877i		−	1.28395i	0.766724	−	0.641977i	$-0.221886\pi$
$152$		0			0
$153$	−38.2431			−0.249955
$154$		0			0
$155$			110.855i			0.715193i
$156$		0			0
$157$	129.392			0.824153			0.412076	−	0.911149i	$-0.364804\pi$
$157$	129.392			0.824153			0.412076	+	0.911149i	$0.364804\pi$
$158$		0			0
$159$	−130.554			−0.821094
$160$		0			0
$161$	−92.5301			−0.574721
$162$		0			0
$163$	211.815			1.29948			0.649738	−	0.760158i	$-0.274879\pi$
$163$	211.815			1.29948			0.649738	+	0.760158i	$0.274879\pi$
$164$		0			0
$165$			95.1554i			0.576699i
$166$		0			0
$167$		−	315.483i		−	1.88912i	−0.328339	−	0.944560i	$-0.606489\pi$
$167$		−	315.483i		−	1.88912i	0.328339	−	0.944560i	$-0.393511\pi$
$168$		0			0
$169$	166.576			0.985654
$170$		0			0
$171$	−53.1769	−	20.5235i	−0.310976	−	0.120020i
$172$		0			0
$173$			236.367i			1.36629i	0.730285	+	0.683143i	$0.239387\pi$
$173$			236.367i			1.36629i	−0.730285	+	0.683143i	$0.760613\pi$
$174$		0			0
$175$	95.3283			0.544733
$176$		0			0
$177$	91.9615			0.519557
$178$		0			0
$179$			184.348i			1.02988i	0.857227	+	0.514939i	$0.172185\pi$
$179$			184.348i			1.02988i	−0.857227	+	0.514939i	$0.827815\pi$
$180$		0			0
$181$		−	80.2606i		−	0.443429i	−0.975112	−	0.221714i	$-0.928835\pi$
$181$		−	80.2606i		−	0.443429i	0.975112	−	0.221714i	$-0.0711653\pi$
$182$		0			0
$183$		−	19.5740i		−	0.106962i
$184$		0			0
$185$			52.1234i			0.281748i
$186$		0			0
$187$	−237.421			−1.26963
$188$		0			0
$189$		−	30.3909i		−	0.160798i
$190$		0			0
$191$	−362.279			−1.89675			−0.948375	−	0.317150i	$-0.897274\pi$
$191$	−362.279			−1.89675			−0.948375	+	0.317150i	$0.897274\pi$
$192$		0			0
$193$		−	271.802i		−	1.40830i	−0.710052	−	0.704149i	$-0.751328\pi$
$193$		−	271.802i		−	1.40830i	0.710052	−	0.704149i	$-0.248672\pi$
$194$		0			0
$195$	−7.95529			−0.0407964
$196$		0			0
$197$	−118.230			−0.600150			−0.300075	−	0.953916i	$-0.597012\pi$
$197$	−118.230			−0.600150			−0.300075	+	0.953916i	$0.597012\pi$
$198$		0			0
$199$	−185.073			−0.930013			−0.465007	−	0.885307i	$-0.653948\pi$
$199$	−185.073			−0.930013			−0.465007	+	0.885307i	$0.653948\pi$
$200$		0			0
$201$	−21.1308			−0.105129
$202$		0			0
$203$		−	269.201i		−	1.32611i
$204$		0			0
$205$			98.4620i			0.480303i
$206$		0			0
$207$	−47.4616			−0.229283
$208$		0			0
$209$	−330.133	−	127.414i	−1.57959	−	0.609637i
$210$		0			0
$211$			417.799i			1.98009i	0.140756	+	0.990044i	$0.455047\pi$
$211$			417.799i			1.98009i	−0.140756	+	0.990044i	$0.544953\pi$
$212$		0			0
$213$	36.1432			0.169686
$214$		0			0
$215$	149.534			0.695508
$216$		0			0
$217$		−	219.802i		−	1.01291i
$218$		0			0
$219$			59.2494i			0.270545i
$220$		0			0
$221$		−	19.8492i		−	0.0898153i
$222$		0			0
$223$		−	172.345i		−	0.772848i	−0.922321	−	0.386424i	$-0.873710\pi$
$223$		−	172.345i		−	0.772848i	0.922321	−	0.386424i	$-0.126290\pi$
$224$		0			0
$225$	48.8969			0.217320
$226$		0			0
$227$			12.5692i			0.0553710i	0.999617	+	0.0276855i	$0.00881370\pi$
$227$			12.5692i			0.0553710i	−0.999617	+	0.0276855i	$0.991186\pi$
$228$		0			0
$229$	240.867			1.05182			0.525911	−	0.850539i	$-0.323724\pi$
$229$	240.867			1.05182			0.525911	+	0.850539i	$0.323724\pi$
$230$		0			0
$231$		−	188.673i		−	0.816766i
$232$		0			0
$233$	−222.034			−0.952936			−0.476468	−	0.879192i	$-0.658083\pi$
$233$	−222.034			−0.952936			−0.476468	+	0.879192i	$0.658083\pi$
$234$		0			0
$235$	44.7576			0.190458
$236$		0			0
$237$	222.743			0.939844
$238$		0			0
$239$	−10.0203			−0.0419259			−0.0209630	−	0.999780i	$-0.506673\pi$
$239$	−10.0203			−0.0419259			−0.0209630	+	0.999780i	$0.506673\pi$
$240$		0			0
$241$		−	420.230i		−	1.74369i	−0.489780	−	0.871846i	$-0.662923\pi$
$241$		−	420.230i		−	1.74369i	0.489780	−	0.871846i	$-0.337077\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	−43.6338			−0.178097
$246$		0			0
$247$	10.6522	−	27.6002i	0.0431264	−	0.111742i
$248$		0			0
$249$			115.991i			0.465826i
$250$		0			0
$251$	−185.394			−0.738621			−0.369310	−	0.929306i	$-0.620406\pi$
$251$	−185.394			−0.738621			−0.369310	+	0.929306i	$0.620406\pi$
$252$		0			0
$253$	−294.652			−1.16463
$254$		0			0
$255$		−	65.1295i		−	0.255410i
$256$		0			0
$257$			26.5370i			0.103257i	0.998666	+	0.0516285i	$0.0164412\pi$
$257$			26.5370i			0.103257i	−0.998666	+	0.0516285i	$0.983559\pi$
$258$		0			0
$259$		−	103.350i		−	0.399034i
$260$		0			0
$261$		−	138.082i		−	0.529049i
$262$		0			0
$263$	−71.2398			−0.270874			−0.135437	−	0.990786i	$-0.543244\pi$
$263$	−71.2398			−0.270874			−0.135437	+	0.990786i	$0.543244\pi$
$264$		0			0
$265$		−	222.339i		−	0.839014i
$266$		0			0
$267$	43.5784			0.163215
$268$		0			0
$269$		−	164.593i		−	0.611869i	−0.952052	−	0.305935i	$-0.901031\pi$
$269$		−	164.593i		−	0.611869i	0.952052	−	0.305935i	$-0.0989689\pi$
$270$		0			0
$271$	156.397			0.577110			0.288555	−	0.957463i	$-0.406825\pi$
$271$	156.397			0.577110			0.288555	+	0.957463i	$0.406825\pi$
$272$		0			0
$273$	15.7737			0.0577790
$274$		0			0
$275$	303.562			1.10386
$276$		0			0
$277$	−278.050			−1.00379			−0.501896	−	0.864928i	$-0.667364\pi$
$277$	−278.050			−1.00379			−0.501896	+	0.864928i	$0.667364\pi$
$278$		0			0
$279$		−	112.743i		−	0.404098i
$280$		0			0
$281$			64.5425i			0.229688i	0.993384	+	0.114844i	$0.0366369\pi$
$281$			64.5425i			0.229688i	−0.993384	+	0.114844i	$0.963363\pi$
$282$		0			0
$283$	−209.171			−0.739119			−0.369560	−	0.929207i	$-0.620491\pi$
$283$	−209.171			−0.739119			−0.369560	+	0.929207i	$0.620491\pi$
$284$		0			0
$285$	34.9523	−	90.5624i	0.122640	−	0.317763i
$286$		0			0
$287$		−	195.229i		−	0.680242i
$288$		0			0
$289$	−126.496			−0.437702
$290$		0			0
$291$	203.405			0.698985
$292$		0			0
$293$		−	75.5171i		−	0.257738i	−0.991662	−	0.128869i	$-0.958865\pi$
$293$		−	75.5171i		−	0.257738i	0.991662	−	0.128869i	$-0.0411346\pi$
$294$		0			0
$295$			156.614i			0.530895i
$296$		0			0
$297$		−	96.7764i		−	0.325846i
$298$		0			0
$299$		−	24.6338i		−	0.0823874i
$300$		0			0
$301$	−296.495			−0.985033
$302$		0			0
$303$			79.4755i			0.262295i
$304$		0			0
$305$	33.3353			0.109296
$306$		0			0
$307$			46.7862i			0.152398i	0.997093	+	0.0761990i	$0.0242784\pi$
$307$			46.7862i			0.152398i	−0.997093	+	0.0761990i	$0.975722\pi$
$308$		0			0
$309$	−294.283			−0.952373
$310$		0			0
$311$	−230.261			−0.740388			−0.370194	−	0.928954i	$-0.620709\pi$
$311$	−230.261			−0.740388			−0.370194	+	0.928954i	$0.620709\pi$
$312$		0			0
$313$	63.4469			0.202706			0.101353	−	0.994851i	$-0.467683\pi$
$313$	63.4469			0.202706			0.101353	+	0.994851i	$0.467683\pi$
$314$		0			0
$315$	51.7569			0.164308
$316$		0			0
$317$			261.464i			0.824809i	0.911001	+	0.412404i	$0.135311\pi$
$317$			261.464i			0.824809i	−0.911001	+	0.412404i	$0.864689\pi$
$318$		0			0
$319$		−	857.240i		−	2.68727i
$320$		0			0
$321$	318.283			0.991537
$322$		0			0
$323$	225.961	+	87.2092i	0.699571	+	0.269998i
$324$		0			0
$325$			25.3788i			0.0780885i
$326$		0			0
$327$	102.276			0.312769
$328$		0			0
$329$	−88.7447			−0.269741
$330$		0			0
$331$			587.914i			1.77618i	0.459673	+	0.888088i	$0.347967\pi$
$331$			587.914i			1.77618i	−0.459673	+	0.888088i	$0.652033\pi$
$332$		0			0
$333$		−	53.0114i		−	0.159193i
$334$		0			0
$335$		−	35.9866i		−	0.107423i
$336$		0			0
$337$			628.812i			1.86591i	0.359990	+	0.932956i	$0.382780\pi$
$337$			628.812i			1.86591i	−0.359990	+	0.932956i	$0.617220\pi$
$338$		0			0
$339$	179.638			0.529907
$340$		0			0
$341$		−	699.934i		−	2.05259i
$342$		0			0
$343$	373.104			1.08777
$344$		0			0
$345$		−	80.8290i		−	0.234287i
$346$		0			0
$347$	−113.695			−0.327651			−0.163825	−	0.986489i	$-0.552383\pi$
$347$	−113.695			−0.327651			−0.163825	+	0.986489i	$0.552383\pi$
$348$		0			0
$349$	359.962			1.03141			0.515705	−	0.856766i	$-0.327530\pi$
$349$	359.962			1.03141			0.515705	+	0.856766i	$0.327530\pi$
$350$		0			0
$351$	8.09082			0.0230508
$352$		0			0
$353$	353.032			1.00009			0.500046	−	0.865999i	$-0.333316\pi$
$353$	353.032			1.00009			0.500046	+	0.865999i	$0.333316\pi$
$354$		0			0
$355$			61.5532i			0.173389i
$356$		0			0
$357$			129.138i			0.361731i
$358$		0			0
$359$	−386.538			−1.07671			−0.538354	−	0.842719i	$-0.680954\pi$
$359$	−386.538			−1.07671			−0.538354	+	0.842719i	$0.680954\pi$
$360$		0			0
$361$	267.397	+	242.528i	0.740711	+	0.671823i
$362$		0			0
$363$		−	391.230i		−	1.07777i
$364$		0			0
$365$	−100.904			−0.276449
$366$		0			0
$367$	−584.274			−1.59203			−0.796014	−	0.605278i	$-0.793062\pi$
$367$	−584.274			−1.59203			−0.796014	+	0.605278i	$0.793062\pi$
$368$		0			0
$369$		−	100.139i		−	0.271380i
$370$		0			0
$371$			440.850i			1.18828i
$372$		0			0
$373$		−	199.190i		−	0.534020i	−0.963694	−	0.267010i	$-0.913964\pi$
$373$		−	199.190i		−	0.534020i	0.963694	−	0.267010i	$-0.0860358\pi$
$374$		0			0
$375$			211.001i			0.562670i
$376$		0			0
$377$	71.6680			0.190101
$378$		0			0
$379$		−	519.527i		−	1.37078i	−0.728174	−	0.685392i	$-0.759631\pi$
$379$		−	519.527i		−	1.37078i	0.728174	−	0.685392i	$-0.240369\pi$
$380$		0			0
$381$	48.1002			0.126247
$382$		0			0
$383$		−	575.644i		−	1.50299i	−0.659741	−	0.751493i	$-0.729334\pi$
$383$		−	575.644i		−	1.50299i	0.659741	−	0.751493i	$-0.270666\pi$
$384$		0			0
$385$	321.317			0.834591
$386$		0			0
$387$	−152.082			−0.392976
$388$		0			0
$389$	13.5845			0.0349216			0.0174608	−	0.999848i	$-0.494442\pi$
$389$	13.5845			0.0349216			0.0174608	+	0.999848i	$0.494442\pi$
$390$		0			0
$391$	201.676			0.515795
$392$		0			0
$393$			177.366i			0.451312i
$394$		0			0
$395$			379.340i			0.960355i
$396$		0			0
$397$	−220.020			−0.554206			−0.277103	−	0.960840i	$-0.589374\pi$
$397$	−220.020			−0.554206			−0.277103	+	0.960840i	$0.589374\pi$
$398$		0			0
$399$	−69.3031	+	179.566i	−0.173692	+	0.450040i
$400$		0			0
$401$			331.524i			0.826744i	0.910562	+	0.413372i	$0.135649\pi$
$401$			331.524i			0.826744i	−0.910562	+	0.413372i	$0.864351\pi$
$402$		0			0
$403$	58.5167			0.145203
$404$		0			0
$405$	26.5478			0.0655500
$406$		0			0
$407$		−	329.106i		−	0.808613i
$408$		0			0
$409$			13.7890i			0.0337139i	0.999858	+	0.0168570i	$0.00536599\pi$
$409$			13.7890i			0.0337139i	−0.999858	+	0.0168570i	$0.994634\pi$
$410$		0			0
$411$			422.319i			1.02754i
$412$		0			0
$413$		−	310.533i		−	0.751895i
$414$		0			0
$415$	−197.537			−0.475992
$416$		0			0
$417$		−	318.032i		−	0.762668i
$418$		0			0
$419$	−283.447			−0.676485			−0.338242	−	0.941059i	$-0.609832\pi$
$419$	−283.447			−0.676485			−0.338242	+	0.941059i	$0.609832\pi$
$420$		0			0
$421$		−	595.014i		−	1.41333i	−0.707546	−	0.706667i	$-0.750198\pi$
$421$		−	595.014i		−	1.41333i	0.707546	−	0.706667i	$-0.249802\pi$
$422$		0			0
$423$	−45.5200			−0.107612
$424$		0			0
$425$	−207.775			−0.488881
$426$		0			0
$427$	−66.0968			−0.154794
$428$		0			0
$429$	50.2295			0.117085
$430$		0			0
$431$			318.279i			0.738466i	0.929337	+	0.369233i	$0.120380\pi$
$431$			318.279i			0.738466i	−0.929337	+	0.369233i	$0.879620\pi$
$432$		0			0
$433$			180.258i			0.416301i	0.978097	+	0.208150i	$0.0667444\pi$
$433$			180.258i			0.416301i	−0.978097	+	0.208150i	$0.933256\pi$
$434$		0			0
$435$	235.159			0.540595
$436$		0			0
$437$	280.429	+	108.231i	0.641715	+	0.247668i
$438$		0			0
$439$		−	732.716i		−	1.66906i	−0.550965	−	0.834528i	$-0.685740\pi$
$439$		−	732.716i		−	1.66906i	0.550965	−	0.834528i	$-0.314260\pi$
$440$		0			0
$441$	44.3771			0.100628
$442$		0			0
$443$	112.171			0.253207			0.126603	−	0.991953i	$-0.459592\pi$
$443$	112.171			0.253207			0.126603	+	0.991953i	$0.459592\pi$
$444$		0			0
$445$			74.2157i			0.166777i
$446$		0			0
$447$		−	267.487i		−	0.598406i
$448$		0			0
$449$		−	104.449i		−	0.232626i	−0.993213	−	0.116313i	$-0.962892\pi$
$449$		−	104.449i		−	0.232626i	0.993213	−	0.116313i	$-0.0371076\pi$
$450$		0			0
$451$		−	621.686i		−	1.37846i
$452$		0			0
$453$	−335.805			−0.741291
$454$		0			0
$455$			26.8632i			0.0590399i
$456$		0			0
$457$	641.023			1.40268			0.701338	−	0.712828i	$-0.252586\pi$
$457$	641.023			1.40268			0.701338	+	0.712828i	$0.252586\pi$
$458$		0			0
$459$			66.2390i			0.144312i
$460$		0			0
$461$	−351.382			−0.762217			−0.381109	−	0.924530i	$-0.624458\pi$
$461$	−351.382			−0.762217			−0.381109	+	0.924530i	$0.624458\pi$
$462$		0			0
$463$	509.568			1.10058			0.550289	−	0.834974i	$-0.314517\pi$
$463$	509.568			1.10058			0.550289	+	0.834974i	$0.314517\pi$
$464$		0			0
$465$	192.006			0.412917
$466$		0			0
$467$	−24.6920			−0.0528736			−0.0264368	−	0.999650i	$-0.508416\pi$
$467$	−24.6920			−0.0528736			−0.0264368	+	0.999650i	$0.508416\pi$
$468$		0			0
$469$			71.3539i			0.152140i
$470$		0			0
$471$		−	224.113i		−	0.475825i
$472$		0			0
$473$	−944.155			−1.99610
$474$		0			0
$475$	−288.910	−	111.504i	−0.608231	−	0.234745i
$476$		0			0
$477$			226.126i			0.474059i
$478$		0			0
$479$	−872.814			−1.82216			−0.911080	−	0.412230i	$-0.864750\pi$
$479$	−872.814			−1.82216			−0.911080	+	0.412230i	$0.864750\pi$
$480$		0			0
$481$	27.5143			0.0572022
$482$		0			0
$483$			160.267i			0.331815i
$484$		0			0
$485$			346.406i			0.714239i
$486$		0			0
$487$			64.1843i			0.131795i	0.997826	+	0.0658977i	$0.0209911\pi$
$487$			64.1843i			0.131795i	−0.997826	+	0.0658977i	$0.979009\pi$
$488$		0			0
$489$		−	366.874i		−	0.750253i
$490$		0			0
$491$	−223.170			−0.454521			−0.227261	−	0.973834i	$-0.572977\pi$
$491$	−223.170			−0.454521			−0.227261	+	0.973834i	$0.572977\pi$
$492$		0			0
$493$			586.742i			1.19015i
$494$		0			0
$495$	164.814			0.332957
$496$		0			0
$497$		−	122.047i		−	0.245567i
$498$		0			0
$499$	540.219			1.08260			0.541302	−	0.840828i	$-0.317932\pi$
$499$	540.219			1.08260			0.541302	+	0.840828i	$0.317932\pi$
$500$		0			0
$501$	−546.433			−1.09068
$502$		0			0
$503$	575.174			1.14349			0.571744	−	0.820432i	$-0.306267\pi$
$503$	575.174			1.14349			0.571744	+	0.820432i	$0.306267\pi$
$504$		0			0
$505$	−135.350			−0.268019
$506$		0			0
$507$		−	288.517i		−	0.569068i
$508$		0			0
$509$			467.013i			0.917511i	0.888563	+	0.458756i	$0.151705\pi$
$509$			467.013i			0.917511i	−0.888563	+	0.458756i	$0.848295\pi$
$510$		0			0
$511$	200.071			0.391529
$512$		0			0
$513$	−35.5478	+	92.1051i	−0.0692939	+	0.179542i
$514$		0			0
$515$		−	501.176i		−	0.973157i
$516$		0			0
$517$	−282.598			−0.546611
$518$		0			0
$519$	409.400			0.788825
$520$		0			0
$521$			90.3378i			0.173393i	0.996235	+	0.0866965i	$0.0276311\pi$
$521$			90.3378i			0.173393i	−0.996235	+	0.0866965i	$0.972369\pi$
$522$		0			0
$523$			872.076i			1.66745i	0.552180	+	0.833725i	$0.313796\pi$
$523$			872.076i			1.66745i	−0.552180	+	0.833725i	$0.686204\pi$
$524$		0			0
$525$		−	165.113i		−	0.314502i
$526$		0			0
$527$			479.073i			0.909057i
$528$		0			0
$529$	−278.710			−0.526863
$530$		0			0
$531$		−	159.282i		−	0.299966i
$532$		0			0
$533$	51.9749			0.0975140
$534$		0			0
$535$			542.049i			1.01318i
$536$		0			0
$537$	319.300			0.594600
$538$		0			0
$539$	275.502			0.511136
$540$		0			0
$541$	−149.284			−0.275941			−0.137971	−	0.990436i	$-0.544058\pi$
$541$	−149.284			−0.275941			−0.137971	+	0.990436i	$0.544058\pi$
$542$		0			0
$543$	−139.015			−0.256014
$544$		0			0
$545$			174.179i			0.319595i
$546$		0			0
$547$			750.900i			1.37276i	0.727243	+	0.686380i	$0.240801\pi$
$547$			750.900i			1.37276i	−0.727243	+	0.686380i	$0.759199\pi$
$548$		0			0
$549$	−33.9032			−0.0617544
$550$		0			0
$551$	−314.880	+	815.863i	−0.571470	+	1.48069i
$552$		0			0
$553$		−	752.151i		−	1.36013i
$554$		0			0
$555$	90.2804			0.162667
$556$		0			0
$557$	1024.62			1.83954			0.919770	−	0.392457i	$-0.128375\pi$
$557$	1024.62			1.83954			0.919770	+	0.392457i	$0.128375\pi$
$558$		0			0
$559$		−	78.9344i		−	0.141206i
$560$		0			0
$561$			411.226i			0.733023i
$562$		0			0
$563$			879.509i			1.56218i	0.624417	+	0.781091i	$0.285337\pi$
$563$			879.509i			1.56218i	−0.624417	+	0.781091i	$0.714663\pi$
$564$		0			0
$565$			305.931i			0.541471i
$566$		0			0
$567$	−52.6386			−0.0928370
$568$		0			0
$569$		−	583.249i		−	1.02504i	−0.858675	−	0.512521i	$-0.828712\pi$
$569$		−	583.249i		−	1.02504i	0.858675	−	0.512521i	$-0.171288\pi$
$570$		0			0
$571$	927.321			1.62403			0.812015	−	0.583637i	$-0.198371\pi$
$571$	927.321			1.62403			0.812015	+	0.583637i	$0.198371\pi$
$572$		0			0
$573$			627.486i			1.09509i
$574$		0			0
$575$	−257.859			−0.448450
$576$		0			0
$577$	−983.592			−1.70467			−0.852333	−	0.523000i	$-0.824813\pi$
$577$	−983.592			−1.70467			−0.852333	+	0.523000i	$0.824813\pi$
$578$		0			0
$579$	−470.774			−0.813082
$580$		0			0
$581$	391.673			0.674136
$582$		0			0
$583$			1403.84i			2.40796i
$584$		0			0
$585$			13.7790i			0.0235538i
$586$		0			0
$587$	566.698			0.965415			0.482707	−	0.875782i	$-0.339653\pi$
$587$	566.698			0.965415			0.482707	+	0.875782i	$0.339653\pi$
$588$		0			0
$589$	−257.099	+	666.149i	−0.436500	+	1.13098i
$590$		0			0
$591$			204.780i			0.346497i
$592$		0			0
$593$	553.999			0.934230			0.467115	−	0.884196i	$-0.345293\pi$
$593$	553.999			0.934230			0.467115	+	0.884196i	$0.345293\pi$
$594$		0			0
$595$	−219.927			−0.369626
$596$		0			0
$597$			320.555i			0.536943i
$598$		0			0
$599$		−	324.382i		−	0.541540i	−0.962644	−	0.270770i	$-0.912722\pi$
$599$		−	324.382i		−	0.541540i	0.962644	−	0.270770i	$-0.0872783\pi$
$600$		0			0
$601$			166.488i			0.277018i	0.990361	+	0.138509i	$0.0442309\pi$
$601$			166.488i			0.277018i	−0.990361	+	0.138509i	$0.955769\pi$
$602$		0			0
$603$			36.5997i			0.0606960i
$604$		0			0
$605$	666.279			1.10129
$606$		0			0
$607$		−	130.347i		−	0.214739i	−0.994219	−	0.107370i	$-0.965757\pi$
$607$		−	130.347i		−	0.214739i	0.994219	−	0.107370i	$-0.0342428\pi$
$608$		0			0
$609$	−466.270			−0.765632
$610$		0			0
$611$		−	23.6261i		−	0.0386679i
$612$		0			0
$613$	776.760			1.26715			0.633573	−	0.773683i	$-0.281588\pi$
$613$	776.760			1.26715			0.633573	+	0.773683i	$0.281588\pi$
$614$		0			0
$615$	170.541			0.277303
$616$		0			0
$617$	121.953			0.197654			0.0988270	−	0.995105i	$-0.468491\pi$
$617$	121.953			0.197654			0.0988270	+	0.995105i	$0.468491\pi$
$618$		0			0
$619$	−899.812			−1.45365			−0.726827	−	0.686821i	$-0.759006\pi$
$619$	−899.812			−1.45365			−0.726827	+	0.686821i	$0.759006\pi$
$620$		0			0
$621$			82.2060i			0.132377i
$622$		0			0
$623$		−	147.154i		−	0.236202i
$624$		0			0
$625$	48.1309			0.0770095
$626$		0			0
$627$	−220.688	+	571.808i	−0.351974	+	0.911974i
$628$		0			0
$629$			225.258i			0.358121i
$630$		0			0
$631$	−471.639			−0.747447			−0.373724	−	0.927540i	$-0.621919\pi$
$631$	−471.639			−0.747447			−0.373724	+	0.927540i	$0.621919\pi$
$632$		0			0
$633$	723.649			1.14320
$634$		0			0
$635$			81.9165i			0.129002i
$636$		0			0
$637$			23.0329i			0.0361583i
$638$		0			0
$639$		−	62.6018i		−	0.0979684i
$640$		0			0
$641$			989.672i			1.54395i	0.635653	+	0.771975i	$0.280731\pi$
$641$			989.672i			1.54395i	−0.635653	+	0.771975i	$0.719269\pi$
$642$		0			0
$643$	−98.4136			−0.153054			−0.0765269	−	0.997068i	$-0.524383\pi$
$643$	−98.4136			−0.153054			−0.0765269	+	0.997068i	$0.524383\pi$
$644$		0			0
$645$		−	259.001i		−	0.401552i
$646$		0			0
$647$	713.056			1.10210			0.551048	−	0.834473i	$-0.314228\pi$
$647$	713.056			1.10210			0.551048	+	0.834473i	$0.314228\pi$
$648$		0			0
$649$		−	988.856i		−	1.52366i
$650$		0			0
$651$	−380.708			−0.584804
$652$		0			0
$653$	891.424			1.36512			0.682561	−	0.730829i	$-0.260866\pi$
$653$	891.424			1.36512			0.682561	+	0.730829i	$0.260866\pi$
$654$		0			0
$655$	−302.061			−0.461162
$656$		0			0
$657$	102.623			0.156199
$658$		0			0
$659$		−	706.071i		−	1.07143i	−0.844399	−	0.535714i	$-0.820042\pi$
$659$		−	706.071i		−	1.07143i	0.844399	−	0.535714i	$-0.179958\pi$
$660$		0			0
$661$		−	31.7149i		−	0.0479802i	−0.999712	−	0.0239901i	$-0.992363\pi$
$661$		−	31.7149i		−	0.0479802i	0.999712	−	0.0239901i	$-0.00763702\pi$
$662$		0			0
$663$	−34.3798			−0.0518549
$664$		0			0
$665$	−305.808	−	118.026i	−0.459862	−	0.177482i
$666$		0			0
$667$			728.176i			1.09172i
$668$		0			0
$669$	−298.510			−0.446204
$670$		0			0
$671$	−210.478			−0.313678
$672$		0			0
$673$		−	261.341i		−	0.388322i	−0.980970	−	0.194161i	$-0.937802\pi$
$673$		−	261.341i		−	0.388322i	0.980970	−	0.194161i	$-0.0621985\pi$
$674$		0			0
$675$		−	84.6920i		−	0.125470i
$676$		0			0
$677$		−	316.111i		−	0.466929i	−0.972365	−	0.233465i	$-0.924994\pi$
$677$		−	316.111i		−	0.466929i	0.972365	−	0.233465i	$-0.0750063\pi$
$678$		0			0
$679$		−	686.850i		−	1.01156i
$680$		0			0
$681$	21.7705			0.0319685
$682$		0			0
$683$		−	482.119i		−	0.705884i	−0.935645	−	0.352942i	$-0.885181\pi$
$683$		−	482.119i		−	0.705884i	0.935645	−	0.352942i	$-0.114819\pi$
$684$		0			0
$685$	−719.225			−1.04996
$686$		0			0
$687$		−	417.194i		−	0.607270i
$688$		0			0
$689$	−117.365			−0.170342
$690$		0			0
$691$	185.162			0.267962			0.133981	−	0.990984i	$-0.457224\pi$
$691$	185.162			0.267962			0.133981	+	0.990984i	$0.457224\pi$
$692$		0			0
$693$	−326.791			−0.471560
$694$		0			0
$695$	541.622			0.779312
$696$		0			0
$697$			425.516i			0.610496i
$698$		0			0
$699$			384.574i			0.550178i
$700$		0			0
$701$	−463.049			−0.660555			−0.330278	−	0.943884i	$-0.607142\pi$
$701$	−463.049			−0.660555			−0.330278	+	0.943884i	$0.607142\pi$
$702$		0			0
$703$	−120.887	+	313.220i	−0.171958	+	0.445548i
$704$		0			0
$705$		−	77.5224i		−	0.109961i
$706$		0			0
$707$	268.370			0.379590
$708$		0			0
$709$	11.5719			0.0163215			0.00816073	−	0.999967i	$-0.497402\pi$
$709$	11.5719			0.0163215			0.00816073	+	0.999967i	$0.497402\pi$
$710$		0			0
$711$		−	385.802i		−	0.542619i
$712$		0			0
$713$			594.554i			0.833876i
$714$		0			0
$715$			85.5427i			0.119640i
$716$		0			0
$717$			17.3557i			0.0242059i
$718$		0			0
$719$	1245.89			1.73280			0.866402	−	0.499347i	$-0.166427\pi$
$719$	1245.89			1.73280			0.866402	+	0.499347i	$0.166427\pi$
$720$		0			0
$721$			993.725i			1.37826i
$722$		0			0
$723$	−727.859			−1.00672
$724$		0			0
$725$		−	750.197i		−	1.03475i
$726$		0			0
$727$	−246.205			−0.338658			−0.169329	−	0.985560i	$-0.554160\pi$
$727$	−246.205			−0.338658			−0.169329	+	0.985560i	$0.554160\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$	646.231			0.884037
$732$		0			0
$733$	1390.30			1.89672			0.948362	−	0.317189i	$-0.102739\pi$
$733$	1390.30			1.89672			0.948362	+	0.317189i	$0.102739\pi$
$734$		0			0
$735$			75.5759i			0.102824i
$736$		0			0
$737$			227.218i			0.308302i
$738$		0			0
$739$	925.346			1.25216			0.626080	−	0.779759i	$-0.284658\pi$
$739$	925.346			1.25216			0.626080	+	0.779759i	$0.284658\pi$
$740$		0			0
$741$	−47.8050	−	18.4502i	−0.0645141	−	0.0248991i
$742$		0			0
$743$			379.097i			0.510225i	0.966911	+	0.255113i	$0.0821125\pi$
$743$			379.097i			0.510225i	−0.966911	+	0.255113i	$0.917887\pi$
$744$		0			0
$745$	455.541			0.611465
$746$		0			0
$747$	200.902			0.268945
$748$		0			0
$749$		−	1074.77i		−	1.43494i
$750$		0			0
$751$			657.073i			0.874931i	0.899235	+	0.437465i	$0.144124\pi$
$751$			657.073i			0.874931i	−0.899235	+	0.437465i	$0.855876\pi$
$752$		0			0
$753$			321.111i			0.426443i
$754$		0			0
$755$		−	571.889i		−	0.757469i
$756$		0			0
$757$	−965.336			−1.27521			−0.637606	−	0.770362i	$-0.720075\pi$
$757$	−965.336			−1.27521			−0.637606	+	0.770362i	$0.720075\pi$
$758$		0			0
$759$			510.352i			0.672400i
$760$		0			0
$761$	−1199.83			−1.57664			−0.788322	−	0.615263i	$-0.789050\pi$
$761$	−1199.83			−1.57664			−0.788322	+	0.615263i	$0.789050\pi$
$762$		0			0
$763$		−	345.361i		−	0.452635i
$764$		0			0
$765$	−112.808			−0.147461
$766$		0			0
$767$	82.6715			0.107786
$768$		0			0
$769$	771.953			1.00384			0.501920	−	0.864914i	$-0.332627\pi$
$769$	771.953			1.00384			0.501920	+	0.864914i	$0.332627\pi$
$770$		0			0
$771$	45.9635			0.0596155
$772$		0			0
$773$			1054.17i			1.36374i	0.731475	+	0.681868i	$0.238832\pi$
$773$			1054.17i			1.36374i	−0.731475	+	0.681868i	$0.761168\pi$
$774$		0			0
$775$		−	612.533i		−	0.790366i
$776$		0			0
$777$	−179.007			−0.230382
$778$		0			0
$779$	−228.357	+	591.678i	−0.293141	+	0.759536i
$780$		0			0
$781$		−	388.645i		−	0.497625i
$782$		0			0
$783$	−239.165			−0.305447
$784$		0			0
$785$	381.674			0.486209
$786$		0			0
$787$		−	1088.65i		−	1.38329i	−0.722239	−	0.691644i	$-0.756887\pi$
$787$		−	1088.65i		−	1.38329i	0.722239	−	0.691644i	$-0.243113\pi$
$788$		0			0
$789$			123.391i			0.156389i
$790$		0			0
$791$		−	606.597i		−	0.766873i
$792$		0			0
$793$		−	17.5966i		−	0.0221899i
$794$		0			0
$795$	−385.102			−0.484405
$796$		0			0
$797$			1096.14i			1.37533i	0.726026	+	0.687667i	$0.241365\pi$
$797$			1096.14i			1.37533i	−0.726026	+	0.687667i	$0.758635\pi$
$798$		0			0
$799$	193.425			0.242084
$800$		0			0
$801$		−	75.4800i		−	0.0942322i
$802$		0			0
$803$	637.104			0.793405
$804$		0			0
$805$	−272.941			−0.339057
$806$		0			0
$807$	−285.083			−0.353263
$808$		0			0
$809$	−216.962			−0.268186			−0.134093	−	0.990969i	$-0.542812\pi$
$809$	−216.962			−0.268186			−0.134093	+	0.990969i	$0.542812\pi$
$810$		0			0
$811$			133.364i			0.164444i	0.996614	+	0.0822219i	$0.0262016\pi$
$811$			133.364i			0.164444i	−0.996614	+	0.0822219i	$0.973798\pi$
$812$		0			0
$813$		−	270.887i		−	0.333195i
$814$		0			0
$815$	624.800			0.766626
$816$		0			0
$817$	898.582	+	346.805i	1.09986	+	0.424486i
$818$		0			0
$819$		−	27.3208i		−	0.0333587i
$820$		0			0
$821$	703.990			0.857478			0.428739	−	0.903428i	$-0.358958\pi$
$821$	703.990			0.857478			0.428739	+	0.903428i	$0.358958\pi$
$822$		0			0
$823$	381.732			0.463830			0.231915	−	0.972736i	$-0.425501\pi$
$823$	381.732			0.463830			0.231915	+	0.972736i	$0.425501\pi$
$824$		0			0
$825$		−	525.785i		−	0.637315i
$826$		0			0
$827$		−	385.027i		−	0.465571i	−0.972528	−	0.232785i	$-0.925216\pi$
$827$		−	385.027i		−	0.465571i	0.972528	−	0.232785i	$-0.0747840\pi$
$828$		0			0
$829$			1648.22i			1.98820i	0.108462	+	0.994101i	$0.465407\pi$
$829$			1648.22i			1.98820i	−0.108462	+	0.994101i	$0.534593\pi$
$830$		0			0
$831$			481.597i			0.579539i
$832$		0			0
$833$	−188.569			−0.226373
$834$		0			0
$835$		−	930.596i		−	1.11449i
$836$		0			0
$837$	−195.277			−0.233306
$838$		0			0
$839$		−	234.921i		−	0.280001i	−0.990151	−	0.140001i	$-0.955290\pi$
$839$		−	234.921i		−	0.280001i	0.990151	−	0.140001i	$-0.0447104\pi$
$840$		0			0
$841$	−1277.51			−1.51904
$842$		0			0
$843$	111.791			0.132611
$844$		0			0
$845$	491.356			0.581487
$846$		0			0
$847$	−1321.09			−1.55973
$848$		0			0
$849$			362.294i			0.426731i
$850$		0			0
$851$			279.556i			0.328503i
$852$		0			0
$853$	567.207			0.664955			0.332478	−	0.943111i	$-0.392115\pi$
$853$	567.207			0.664955			0.332478	+	0.943111i	$0.392115\pi$
$854$		0			0
$855$	−156.859	−	60.5392i	−0.183460	−	0.0708061i
$856$		0			0
$857$		−	111.109i		−	0.129649i	−0.997897	−	0.0648243i	$-0.979351\pi$
$857$		−	111.109i		−	0.129649i	0.997897	−	0.0648243i	$-0.0206487\pi$
$858$		0			0
$859$	−635.979			−0.740371			−0.370185	−	0.928958i	$-0.620706\pi$
$859$	−635.979			−0.740371			−0.370185	+	0.928958i	$0.620706\pi$
$860$		0			0
$861$	−338.147			−0.392738
$862$		0			0
$863$		−	824.629i		−	0.955537i	−0.878486	−	0.477769i	$-0.841446\pi$
$863$		−	824.629i		−	0.955537i	0.878486	−	0.477769i	$-0.158554\pi$
$864$		0			0
$865$			697.225i			0.806040i
$866$		0			0
$867$			219.097i			0.252707i
$868$		0			0
$869$		−	2395.14i		−	2.75620i
$870$		0			0
$871$	−18.9962			−0.0218096
$872$		0			0
$873$		−	352.307i		−	0.403559i
$874$		0			0
$875$	712.502			0.814288
$876$		0			0
$877$		−	952.668i		−	1.08628i	−0.839642	−	0.543140i	$-0.817235\pi$
$877$		−	952.668i		−	1.08628i	0.839642	−	0.543140i	$-0.182765\pi$
$878$		0			0
$879$	−130.799			−0.148805
$880$		0			0
$881$	642.008			0.728727			0.364363	−	0.931257i	$-0.381287\pi$
$881$	642.008			0.728727			0.364363	+	0.931257i	$0.381287\pi$
$882$		0			0
$883$	87.6409			0.0992535			0.0496268	−	0.998768i	$-0.484197\pi$
$883$	87.6409			0.0992535			0.0496268	+	0.998768i	$0.484197\pi$
$884$		0			0
$885$	271.264			0.306513
$886$		0			0
$887$			522.256i			0.588790i	0.955684	+	0.294395i	$0.0951181\pi$
$887$			522.256i			0.588790i	−0.955684	+	0.294395i	$0.904882\pi$
$888$		0			0
$889$		−	162.423i		−	0.182703i
$890$		0			0
$891$	−167.622			−0.188128
$892$		0			0
$893$	268.957	+	103.803i	0.301184	+	0.116241i
$894$		0			0
$895$			543.781i			0.607577i
$896$		0			0
$897$	−42.6670			−0.0475664
$898$		0			0
$899$	−1729.76			−1.92409
$900$		0			0
$901$		−	960.864i		−	1.06644i
$902$		0			0
$903$			513.544i			0.568709i
$904$		0			0
$905$		−	236.749i		−	0.261601i
$906$		0			0
$907$		−	734.866i		−	0.810216i	−0.914269	−	0.405108i	$-0.867234\pi$
$907$		−	734.866i		−	0.810216i	0.914269	−	0.405108i	$-0.132766\pi$
$908$		0			0
$909$	137.656			0.151436
$910$		0			0
$911$			446.769i			0.490416i	0.969470	+	0.245208i	$0.0788563\pi$
$911$			446.769i			0.490416i	−0.969470	+	0.245208i	$0.921144\pi$
$912$		0			0
$913$	1247.24			1.36609
$914$		0			0
$915$		−	57.7384i		−	0.0631021i
$916$		0			0
$917$	598.923			0.653133
$918$		0			0
$919$	−259.922			−0.282831			−0.141416	−	0.989950i	$-0.545165\pi$
$919$	−259.922			−0.282831			−0.141416	+	0.989950i	$0.545165\pi$
$920$		0			0
$921$	81.0360			0.0879870
$922$		0			0
$923$	32.4920			0.0352026
$924$		0			0
$925$		−	288.010i		−	0.311362i
$926$		0			0
$927$			509.713i			0.549853i
$928$		0			0
$929$	−1049.29			−1.12948			−0.564741	−	0.825268i	$-0.691024\pi$
$929$	−1049.29			−1.12948			−0.564741	+	0.825268i	$0.691024\pi$
$930$		0			0
$931$	−262.204	−	101.197i	−0.281637	−	0.108697i
$932$		0			0
$933$			398.823i			0.427463i
$934$		0			0
$935$	−700.333			−0.749020
$936$		0			0
$937$	−715.553			−0.763664			−0.381832	−	0.924232i	$-0.624707\pi$
$937$	−715.553			−0.763664			−0.381832	+	0.924232i	$0.624707\pi$
$938$		0			0
$939$		−	109.893i		−	0.117032i
$940$		0			0
$941$		−	703.553i		−	0.747666i	−0.927496	−	0.373833i	$-0.878043\pi$
$941$		−	703.553i		−	0.747666i	0.927496	−	0.373833i	$-0.121957\pi$
$942$		0			0
$943$			528.087i			0.560007i
$944$		0			0
$945$		−	89.6455i		−	0.0948630i
$946$		0			0
$947$	698.867			0.737980			0.368990	−	0.929433i	$-0.379704\pi$
$947$	698.867			0.737980			0.368990	+	0.929433i	$0.379704\pi$
$948$		0			0
$949$			53.2640i			0.0561264i
$950$		0			0
$951$	452.869			0.476203
$952$		0			0
$953$			826.216i			0.866963i	0.901162	+	0.433482i	$0.142715\pi$
$953$			826.216i			0.866963i	−0.901162	+	0.433482i	$0.857285\pi$
$954$		0			0
$955$	−1068.63			−1.11899
$956$		0			0
$957$	−1484.78			−1.55150
$958$		0			0
$959$	1426.07			1.48704
$960$		0			0
$961$	−451.339			−0.469656
$962$		0			0
$963$		−	551.283i		−	0.572464i
$964$		0			0
$965$		−	801.747i		−	0.830826i
$966$		0			0
$967$	−677.220			−0.700331			−0.350165	−	0.936688i	$-0.613875\pi$
$967$	−677.220			−0.700331			−0.350165	+	0.936688i	$0.613875\pi$
$968$		0			0
$969$	151.051	−	391.377i	0.155883	−	0.403897i
$970$		0			0
$971$			439.573i			0.452701i	0.974046	+	0.226351i	$0.0726795\pi$
$971$			439.573i			0.452701i	−0.974046	+	0.226351i	$0.927320\pi$
$972$		0			0
$973$	−1073.92			−1.10372
$974$		0			0
$975$	43.9573			0.0450844
$976$		0			0
$977$		−	305.179i		−	0.312363i	−0.987728	−	0.156182i	$-0.950081\pi$
$977$		−	305.179i		−	0.312363i	0.987728	−	0.156182i	$-0.0499185\pi$
$978$		0			0
$979$		−	468.595i		−	0.478647i
$980$		0			0
$981$		−	177.146i		−	0.180577i
$982$		0			0
$983$			141.357i			0.143801i	0.997412	+	0.0719006i	$0.0229064\pi$
$983$			141.357i			0.143801i	−0.997412	+	0.0719006i	$0.977094\pi$
$984$		0			0
$985$	−348.748			−0.354059
$986$		0			0
$987$			153.710i			0.155735i
$988$		0			0
$989$	802.005			0.810925
$990$		0			0
$991$			29.1163i			0.0293807i	0.999892	+	0.0146904i	$0.00467625\pi$
$991$			29.1163i			0.0293807i	−0.999892	+	0.0146904i	$0.995324\pi$
$992$		0			0
$993$	1018.30			1.02548
$994$		0			0
$995$	−545.918			−0.548661
$996$		0			0
$997$	−897.567			−0.900267			−0.450134	−	0.892961i	$-0.648624\pi$
$997$	−897.567			−0.900267			−0.450134	+	0.892961i	$0.648624\pi$
$998$		0			0
$999$	−91.8184			−0.0919103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.o.d.721.3		8
3.2	odd	2		2736.3.o.n.721.3		8
4.3	odd	2		114.3.d.a.37.4	✓	8
12.11	even	2		342.3.d.b.37.6		8
19.18	odd	2	inner	912.3.o.d.721.7		8
57.56	even	2		2736.3.o.n.721.4		8
76.75	even	2		114.3.d.a.37.6	yes	8
228.227	odd	2		342.3.d.b.37.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.d.a.37.4	✓	8	4.3	odd	2
114.3.d.a.37.6	yes	8	76.75	even	2
342.3.d.b.37.2		8	228.227	odd	2
342.3.d.b.37.6		8	12.11	even	2
912.3.o.d.721.3		8	1.1	even	1	trivial
912.3.o.d.721.7		8	19.18	odd	2	inner
2736.3.o.n.721.3		8	3.2	odd	2
2736.3.o.n.721.4		8	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +2.94975 q^{5} -5.84873 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +2.94975 q^{5} -5.84873 q^{7} -3.00000 q^{9} -18.6246 q^{11} -1.55708i q^{13} -5.10912i q^{15} +12.7477 q^{17} +(17.7256 + 6.84117i) q^{19} +10.1303i q^{21} +15.8205 q^{23} -16.2990 q^{25} +5.19615i q^{27} +46.0273i q^{29} +37.5811i q^{31} +32.2588i q^{33} -17.2523 q^{35} +17.6705i q^{37} -2.69694 q^{39} +33.3798i q^{41} +50.6939 q^{43} -8.84925 q^{45} +15.1733 q^{47} -14.7924 q^{49} -22.0797i q^{51} -75.3754i q^{53} -54.9380 q^{55} +(11.8493 - 30.7017i) q^{57} +53.0940i q^{59} +11.3011 q^{61} +17.5462 q^{63} -4.59299i q^{65} -12.1999i q^{67} -27.4020i q^{69} +20.8673i q^{71} -34.2076 q^{73} +28.2307i q^{75} +108.930 q^{77} +128.601i q^{79} +9.00000 q^{81} -66.9672 q^{83} +37.6026 q^{85} +79.7216 q^{87} +25.1600i q^{89} +9.10693i q^{91} +65.0924 q^{93} +(52.2862 + 20.1797i) q^{95} +117.436i q^{97} +55.8739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9	\( 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9} - 4 q^{11} + 4 q^{17} + 36 q^{19} + 56 q^{23} + 140 q^{25} - 236 q^{35} + 96 q^{39} - 100 q^{43} - 12 q^{45} + 188 q^{47} - 36 q^{49} - 28 q^{55} + 36 q^{57} - 180 q^{61} - 36 q^{63} - 356 q^{73} + 68 q^{77} + 72 q^{81} - 136 q^{83} + 148 q^{85} + 144 q^{87} + 168 q^{93} + 140 q^{95} + 12 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9 - 4 * q^11 + 4 * q^17 + 36 * q^19 + 56 * q^23 + 140 * q^25 - 236 * q^35 + 96 * q^39 - 100 * q^43 - 12 * q^45 + 188 * q^47 - 36 * q^49 - 28 * q^55 + 36 * q^57 - 180 * q^61 - 36 * q^63 - 356 * q^73 + 68 * q^77 + 72 * q^81 - 136 * q^83 + 148 * q^85 + 144 * q^87 + 168 * q^93 + 140 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

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Database

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