LMFDB - Embedded newform 1400.2.x.a.993.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(657,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("1400.657");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.x (of order $4$, degree $2$, 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			993.7
Root			$1.38588 + 0.281691i$ of defining polynomial
Character	$\chi$	$=$	1400.993
Dual form			1400.2.x.a.657.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66757 - 1.66757i) q^{3} +(-1.34308 + 2.27950i) q^{7} -2.56155i q^{9} +O(q^{10})$	$q+(1.66757 - 1.66757i) q^{3} +(-1.34308 + 2.27950i) q^{7} -2.56155i q^{9} +1.56155 q^{11} +(2.60399 - 2.60399i) q^{13} +(2.60399 + 2.60399i) q^{17} +2.64861 q^{19} +(1.56155 + 6.04090i) q^{21} +(0.794156 + 0.794156i) q^{23} +(0.731140 + 0.731140i) q^{27} -6.68466i q^{29} -9.43318i q^{31} +(2.60399 - 2.60399i) q^{33} +(2.82843 - 2.82843i) q^{37} -8.68466i q^{39} +2.64861i q^{41} +(6.45101 + 6.45101i) q^{43} +(1.66757 + 1.66757i) q^{47} +(-3.39228 - 6.12311i) q^{49} +8.68466 q^{51} +(-7.24517 - 7.24517i) q^{53} +(4.41674 - 4.41674i) q^{57} +12.0818 q^{59} +9.43318i q^{61} +(5.83907 + 3.44037i) q^{63} +(-3.62258 + 3.62258i) q^{67} +2.64861 q^{69} +6.24621 q^{71} +(-6.67026 + 6.67026i) q^{73} +(-2.09729 + 3.55957i) q^{77} -11.8078i q^{79} +10.1231 q^{81} +(-9.47954 + 9.47954i) q^{83} +(-11.1471 - 11.1471i) q^{87} -9.43318 q^{89} +(2.43845 + 9.43318i) q^{91} +(-15.7304 - 15.7304i) q^{93} +(-5.52855 - 5.52855i) q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} - 8 q^{21} + 40 q^{51} - 32 q^{71} + 96 q^{81} + 72 q^{91}+O(q^{100}) $ 16 * q - 8 * q^11 - 8 * q^21 + 40 * q^51 - 32 * q^71 + 96 * q^81 + 72 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	1.66757	−	1.66757i	0.962770	−	0.962770i	−0.0365617	−	0.999331i	$-0.511641\pi$
$3$	1.66757	−	1.66757i	0.962770	−	0.962770i	0.999331	+	0.0365617i	$0.0116405\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.34308	+	2.27950i	−0.507636	+	0.861572i
$7$	−1.34308	+	2.27950i	−0.507636	+	0.861572i
$8$		0			0
$9$		−	2.56155i		−	0.853851i
$10$		0			0
$11$	1.56155			0.470826			0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155			0.470826			0.235413	+	0.971895i	$0.424356\pi$
$12$		0			0
$13$	2.60399	−	2.60399i	0.722218	−	0.722218i	−0.246839	−	0.969057i	$-0.579392\pi$
$13$	2.60399	−	2.60399i	0.722218	−	0.722218i	0.969057	+	0.246839i	$0.0793919\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.60399	+	2.60399i	0.631561	+	0.631561i	0.948459	−	0.316899i	$-0.102641\pi$
$17$	2.60399	+	2.60399i	0.631561	+	0.631561i	−0.316899	+	0.948459i	$0.602641\pi$
$18$		0			0
$19$	2.64861			0.607634			0.303817	−	0.952730i	$-0.401739\pi$
$19$	2.64861			0.607634			0.303817	+	0.952730i	$0.401739\pi$
$20$		0			0
$21$	1.56155	+	6.04090i	0.340759	+	1.31823i
$22$		0			0
$23$	0.794156	+	0.794156i	0.165593	+	0.165593i	0.785039	−	0.619446i	$-0.212643\pi$
$23$	0.794156	+	0.794156i	0.165593	+	0.165593i	−0.619446	+	0.785039i	$0.712643\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.731140	+	0.731140i	0.140708	+	0.140708i
$28$		0			0
$29$		−	6.68466i		−	1.24131i	−0.784084	−	0.620655i	$-0.786867\pi$
$29$		−	6.68466i		−	1.24131i	0.784084	−	0.620655i	$-0.213133\pi$
$30$		0			0
$31$		−	9.43318i		−	1.69425i	−0.531395	−	0.847124i	$-0.678332\pi$
$31$		−	9.43318i		−	1.69425i	0.531395	−	0.847124i	$-0.321668\pi$
$32$		0			0
$33$	2.60399	−	2.60399i	0.453297	−	0.453297i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.82843	−	2.82843i	0.464991	−	0.464991i	−0.435297	−	0.900287i	$-0.643356\pi$
$37$	2.82843	−	2.82843i	0.464991	−	0.464991i	0.900287	+	0.435297i	$0.143356\pi$
$38$		0			0
$39$		−	8.68466i		−	1.39066i
$40$		0			0
$41$			2.64861i			0.413644i	0.978379	+	0.206822i	$0.0663121\pi$
$41$			2.64861i			0.413644i	−0.978379	+	0.206822i	$0.933688\pi$
$42$		0			0
$43$	6.45101	+	6.45101i	0.983770	+	0.983770i	0.999870	−	0.0161006i	$-0.00512520\pi$
$43$	6.45101	+	6.45101i	0.983770	+	0.983770i	−0.0161006	+	0.999870i	$0.505125\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.66757	+	1.66757i	0.243240	+	0.243240i	0.818189	−	0.574949i	$-0.194978\pi$
$47$	1.66757	+	1.66757i	0.243240	+	0.243240i	−0.574949	+	0.818189i	$0.694978\pi$
$48$		0			0
$49$	−3.39228	−	6.12311i	−0.484612	−	0.874729i
$50$		0			0
$51$	8.68466			1.21610
$52$		0			0
$53$	−7.24517	−	7.24517i	−0.995200	−	0.995200i	0.00478852	−	0.999989i	$-0.498476\pi$
$53$	−7.24517	−	7.24517i	−0.995200	−	0.995200i	−0.999989	+	0.00478852i	$0.998476\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.41674	−	4.41674i	0.585011	−	0.585011i
$58$		0			0
$59$	12.0818			1.57292			0.786458	−	0.617644i	$-0.211913\pi$
$59$	12.0818			1.57292			0.786458	+	0.617644i	$0.211913\pi$
$60$		0			0
$61$			9.43318i			1.20779i	0.797062	+	0.603897i	$0.206386\pi$
$61$			9.43318i			1.20779i	−0.797062	+	0.603897i	$0.793614\pi$
$62$		0			0
$63$	5.83907	+	3.44037i	0.735654	+	0.433445i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.62258	+	3.62258i	−0.442569	+	0.442569i	−0.892875	−	0.450306i	$-0.851315\pi$
$67$	−3.62258	+	3.62258i	−0.442569	+	0.442569i	0.450306	+	0.892875i	$0.351315\pi$
$68$		0			0
$69$	2.64861			0.318856
$70$		0			0
$71$	6.24621			0.741289			0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621			0.741289			0.370644	+	0.928775i	$0.379137\pi$
$72$		0			0
$73$	−6.67026	+	6.67026i	−0.780695	+	0.780695i	−0.979948	−	0.199253i	$-0.936149\pi$
$73$	−6.67026	+	6.67026i	−0.780695	+	0.780695i	0.199253	+	0.979948i	$0.436149\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.09729	+	3.55957i	−0.239008	+	0.405650i
$78$		0			0
$79$		−	11.8078i		−	1.32848i	−0.747521	−	0.664239i	$-0.768756\pi$
$79$		−	11.8078i		−	1.32848i	0.747521	−	0.664239i	$-0.231244\pi$
$80$		0			0
$81$	10.1231			1.12479
$82$		0			0
$83$	−9.47954	+	9.47954i	−1.04052	+	1.04052i	−0.0413712	+	0.999144i	$0.513173\pi$
$83$	−9.47954	+	9.47954i	−1.04052	+	1.04052i	−0.999144	+	0.0413712i	$0.986827\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−11.1471	−	11.1471i	−1.19510	−	1.19510i
$88$		0			0
$89$	−9.43318			−0.999915			−0.499957	−	0.866050i	$-0.666651\pi$
$89$	−9.43318			−0.999915			−0.499957	+	0.866050i	$0.666651\pi$
$90$		0			0
$91$	2.43845	+	9.43318i	0.255619	+	0.988866i
$92$		0			0
$93$	−15.7304	−	15.7304i	−1.63117	−	1.63117i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.52855	−	5.52855i	−0.561339	−	0.561339i	0.368348	−	0.929688i	$-0.379923\pi$
$97$	−5.52855	−	5.52855i	−0.561339	−	0.561339i	−0.929688	+	0.368348i	$0.879923\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$		−	16.2177i		−	1.61373i	−0.590739	−	0.806863i	$-0.701164\pi$
$101$		−	16.2177i		−	1.61373i	0.590739	−	0.806863i	$-0.298836\pi$
$102$		0			0
$103$	−8.74840	+	8.74840i	−0.862006	+	0.862006i	−0.991571	−	0.129565i	$-0.958642\pi$
$103$	−8.74840	+	8.74840i	−0.862006	+	0.862006i	0.129565	+	0.991571i	$0.458642\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.03427	−	2.03427i	0.196660	−	0.196660i	−0.601906	−	0.798567i	$-0.705592\pi$
$107$	2.03427	−	2.03427i	0.196660	−	0.196660i	0.798567	+	0.601906i	$0.205592\pi$
$108$		0			0
$109$		−	2.68466i		−	0.257144i	−0.991700	−	0.128572i	$-0.958961\pi$
$109$		−	2.68466i		−	0.257144i	0.991700	−	0.128572i	$-0.0410393\pi$
$110$		0			0
$111$		−	9.43318i		−	0.895358i
$112$		0			0
$113$	1.24012	+	1.24012i	0.116660	+	0.116660i	0.763027	−	0.646367i	$-0.223712\pi$
$113$	1.24012	+	1.24012i	0.116660	+	0.116660i	−0.646367	+	0.763027i	$0.723712\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−6.67026	−	6.67026i	−0.616666	−	0.616666i
$118$		0			0
$119$	−9.43318	+	2.43845i	−0.864738	+	0.223532i
$120$		0			0
$121$	−8.56155			−0.778323
$122$		0			0
$123$	4.41674	+	4.41674i	0.398244	+	0.398244i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.27944	+	9.27944i	−0.823417	+	0.823417i	−0.986596	−	0.163180i	$-0.947825\pi$
$127$	−9.27944	+	9.27944i	−0.823417	+	0.823417i	0.163180	+	0.986596i	$0.447825\pi$
$128$		0			0
$129$	21.5150			1.89429
$130$		0			0
$131$			21.5150i			1.87977i	0.341489	+	0.939886i	$0.389069\pi$
$131$			21.5150i			1.87977i	−0.341489	+	0.939886i	$0.610931\pi$
$132$		0			0
$133$	−3.55730	+	6.03753i	−0.308457	+	0.523520i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.0736	−	10.0736i	0.860645	−	0.860645i	−0.130768	−	0.991413i	$-0.541744\pi$
$137$	10.0736	−	10.0736i	0.860645	−	0.860645i	0.991413	+	0.130768i	$0.0417443\pi$
$138$		0			0
$139$	12.0818			1.02476			0.512382	−	0.858758i	$-0.328763\pi$
$139$	12.0818			1.02476			0.512382	+	0.858758i	$0.328763\pi$
$140$		0			0
$141$	5.56155			0.468367
$142$		0			0
$143$	4.06627	−	4.06627i	0.340039	−	0.340039i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−15.8675	−	4.55383i	−1.30873	−	0.375593i
$148$		0			0
$149$			4.24621i			0.347863i	0.984758	+	0.173932i	$0.0556472\pi$
$149$			4.24621i			0.347863i	−0.984758	+	0.173932i	$0.944353\pi$
$150$		0			0
$151$	−16.6847			−1.35778			−0.678889	−	0.734241i	$-0.737538\pi$
$151$	−16.6847			−1.35778			−0.678889	+	0.734241i	$0.737538\pi$
$152$		0			0
$153$	6.67026	−	6.67026i	0.539259	−	0.539259i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	13.3405	+	13.3405i	1.06469	+	1.06469i	0.997758	+	0.0669325i	$0.0213212\pi$
$157$	13.3405	+	13.3405i	1.06469	+	1.06469i	0.0669325	+	0.997758i	$0.478679\pi$
$158$		0			0
$159$	−24.1636			−1.91630
$160$		0			0
$161$	−2.87689	+	0.743668i	−0.226731	+	0.0586093i
$162$		0			0
$163$	−7.69113	−	7.69113i	−0.602415	−	0.602415i	0.338537	−	0.940953i	$-0.390068\pi$
$163$	−7.69113	−	7.69113i	−0.602415	−	0.602415i	−0.940953	+	0.338537i	$0.890068\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.0081	+	15.0081i	1.16136	+	1.16136i	0.984178	+	0.177183i	$0.0566986\pi$
$167$	15.0081	+	15.0081i	1.16136	+	1.16136i	0.177183	+	0.984178i	$0.443301\pi$
$168$		0			0
$169$		−	0.561553i		−	0.0431964i
$170$		0			0
$171$		−	6.78456i		−	0.518829i
$172$		0			0
$173$	1.14171	−	1.14171i	0.0868028	−	0.0868028i	−0.662372	−	0.749175i	$-0.730450\pi$
$173$	1.14171	−	1.14171i	0.0868028	−	0.0868028i	0.749175	+	0.662372i	$0.230450\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	20.1472	−	20.1472i	1.51436	−	1.51436i
$178$		0			0
$179$			2.24621i			0.167890i	0.996470	+	0.0839449i	$0.0267520\pi$
$179$			2.24621i			0.167890i	−0.996470	+	0.0839449i	$0.973248\pi$
$180$		0			0
$181$		−	17.3790i		−	1.29177i	−0.763434	−	0.645886i	$-0.776488\pi$
$181$		−	17.3790i		−	1.29177i	0.763434	−	0.645886i	$-0.223512\pi$
$182$		0			0
$183$	15.7304	+	15.7304i	1.16283	+	1.16283i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.06627	+	4.06627i	0.297355	+	0.297355i
$188$		0			0
$189$	−2.64861	+	0.684658i	−0.192658	+	0.0498016i
$190$		0			0
$191$	10.4384			0.755300			0.377650	−	0.925949i	$-0.376732\pi$
$191$	10.4384			0.755300			0.377650	+	0.925949i	$0.376732\pi$
$192$		0			0
$193$	−1.24012	−	1.24012i	−0.0892655	−	0.0892655i	0.661064	−	0.750330i	$-0.270105\pi$
$193$	−1.24012	−	1.24012i	−0.0892655	−	0.0892655i	−0.750330	+	0.661064i	$0.770105\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.3188	+	17.3188i	−1.23391	+	1.23391i	−0.271461	+	0.962449i	$0.587507\pi$
$197$	−17.3188	+	17.3188i	−1.23391	+	1.23391i	−0.962449	+	0.271461i	$0.912493\pi$
$198$		0			0
$199$	−5.29723			−0.375511			−0.187755	−	0.982216i	$-0.560121\pi$
$199$	−5.29723			−0.375511			−0.187755	+	0.982216i	$0.560121\pi$
$200$		0			0
$201$			12.0818i			0.852184i
$202$		0			0
$203$	15.2377	+	8.97802i	1.06948	+	0.630133i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.03427	−	2.03427i	0.141392	−	0.141392i
$208$		0			0
$209$	4.13595			0.286090
$210$		0			0
$211$	−25.1771			−1.73326			−0.866631	−	0.498950i	$-0.833719\pi$
$211$	−25.1771			−1.73326			−0.866631	+	0.498950i	$0.833719\pi$
$212$		0			0
$213$	10.4160	−	10.4160i	0.713690	−	0.713690i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	21.5030	+	12.6695i	1.45972	+	0.860061i
$218$		0			0
$219$			22.2462i			1.50326i
$220$		0			0
$221$	13.5616			0.912249
$222$		0			0
$223$	2.07814	−	2.07814i	0.139163	−	0.139163i	−0.634094	−	0.773256i	$-0.718627\pi$
$223$	2.07814	−	2.07814i	0.139163	−	0.139163i	0.773256	+	0.634094i	$0.218627\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.25699	−	1.25699i	−0.0834295	−	0.0834295i	0.664160	−	0.747590i	$-0.268789\pi$
$227$	−1.25699	−	1.25699i	−0.0834295	−	0.0834295i	−0.747590	+	0.664160i	$0.768789\pi$
$228$		0			0
$229$	2.64861			0.175025			0.0875127	−	0.996163i	$-0.472108\pi$
$229$	2.64861			0.175025			0.0875127	+	0.996163i	$0.472108\pi$
$230$		0			0
$231$	2.43845	+	9.43318i	0.160438	+	0.620658i
$232$		0			0
$233$	−16.9706	−	16.9706i	−1.11178	−	1.11178i	−0.992910	−	0.118869i	$-0.962073\pi$
$233$	−16.9706	−	16.9706i	−1.11178	−	1.11178i	−0.118869	−	0.992910i	$-0.537927\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.6902	−	19.6902i	−1.27902	−	1.27902i
$238$		0			0
$239$			19.8078i			1.28126i	0.767851	+	0.640629i	$0.221326\pi$
$239$			19.8078i			1.28126i	−0.767851	+	0.640629i	$0.778674\pi$
$240$		0			0
$241$		−	26.8122i		−	1.72713i	−0.504241	−	0.863563i	$-0.668228\pi$
$241$		−	26.8122i		−	1.72713i	0.504241	−	0.863563i	$-0.331772\pi$
$242$		0			0
$243$	14.6875	−	14.6875i	0.942205	−	0.942205i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.89697	−	6.89697i	0.438844	−	0.438844i
$248$		0			0
$249$			31.6155i			2.00355i
$250$		0			0
$251$			12.0818i			0.762596i	0.924452	+	0.381298i	$0.124523\pi$
$251$			12.0818i			0.762596i	−0.924452	+	0.381298i	$0.875477\pi$
$252$		0			0
$253$	1.24012	+	1.24012i	0.0779654	+	0.0779654i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.4160	+	10.4160i	0.649730	+	0.649730i	0.952928	−	0.303197i	$-0.0980542\pi$
$257$	10.4160	+	10.4160i	0.649730	+	0.649730i	−0.303197	+	0.952928i	$0.598054\pi$
$258$		0			0
$259$	2.64861	+	10.2462i	0.164577	+	0.636669i
$260$		0			0
$261$	−17.1231			−1.05989
$262$		0			0
$263$	5.21089	+	5.21089i	0.321317	+	0.321317i	0.849272	−	0.527955i	$-0.177041\pi$
$263$	5.21089	+	5.21089i	0.321317	+	0.321317i	−0.527955	+	0.849272i	$0.677041\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−15.7304	+	15.7304i	−0.962688	+	0.962688i
$268$		0			0
$269$	−5.29723			−0.322978			−0.161489	−	0.986875i	$-0.551630\pi$
$269$	−5.29723			−0.322978			−0.161489	+	0.986875i	$0.551630\pi$
$270$		0			0
$271$			4.13595i			0.251241i	0.992078	+	0.125621i	$0.0400922\pi$
$271$			4.13595i			0.251241i	−0.992078	+	0.125621i	$0.959908\pi$
$272$		0			0
$273$	19.7967	+	11.6642i	1.19815	+	0.705948i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.24517	+	7.24517i	−0.435320	+	0.435320i	−0.890433	−	0.455114i	$-0.849599\pi$
$277$	−7.24517	+	7.24517i	−0.435320	+	0.435320i	0.455114	+	0.890433i	$0.349599\pi$
$278$		0			0
$279$	−24.1636			−1.44664
$280$		0			0
$281$	0.438447			0.0261556			0.0130778	−	0.999914i	$-0.495837\pi$
$281$	0.438447			0.0261556			0.0130778	+	0.999914i	$0.495837\pi$
$282$		0			0
$283$	−2.07814	+	2.07814i	−0.123533	+	0.123533i	−0.766170	−	0.642638i	$-0.777840\pi$
$283$	−2.07814	+	2.07814i	−0.123533	+	0.123533i	0.642638	+	0.766170i	$0.277840\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.03753	−	3.55730i	−0.356384	−	0.209981i
$288$		0			0
$289$		−	3.43845i		−	0.202262i
$290$		0			0
$291$	−18.4384			−1.08088
$292$		0			0
$293$	14.4822	−	14.4822i	0.846062	−	0.846062i	−0.143578	−	0.989639i	$-0.545861\pi$
$293$	14.4822	−	14.4822i	0.846062	−	0.846062i	0.989639	+	0.143578i	$0.0458606\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.14171	+	1.14171i	0.0662489	+	0.0662489i
$298$		0			0
$299$	4.13595			0.239188
$300$		0			0
$301$	−23.3693	+	6.04090i	−1.34699	+	0.348191i
$302$		0			0
$303$	−27.0442	−	27.0442i	−1.55365	−	1.55365i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	18.3432	+	18.3432i	1.04690	+	1.04690i	0.998845	+	0.0480587i	$0.0153035\pi$
$307$	18.3432	+	18.3432i	1.04690	+	1.04690i	0.0480587	+	0.998845i	$0.484697\pi$
$308$		0			0
$309$			29.1771i			1.65983i
$310$		0			0
$311$		−	14.7304i		−	0.835285i	−0.908612	−	0.417642i	$-0.862857\pi$
$311$		−	14.7304i		−	0.835285i	0.908612	−	0.417642i	$-0.137143\pi$
$312$		0			0
$313$	6.99083	−	6.99083i	0.395145	−	0.395145i	−0.481372	−	0.876517i	$-0.659861\pi$
$313$	6.99083	−	6.99083i	0.395145	−	0.395145i	0.876517	+	0.481372i	$0.159861\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.4218	+	10.4218i	−0.585346	+	0.585346i	−0.936367	−	0.351021i	$-0.885834\pi$
$317$	−10.4218	+	10.4218i	−0.585346	+	0.585346i	0.351021	+	0.936367i	$0.385834\pi$
$318$		0			0
$319$		−	10.4384i		−	0.584441i
$320$		0			0
$321$		−	6.78456i		−	0.378677i
$322$		0			0
$323$	6.89697	+	6.89697i	0.383758	+	0.383758i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.47685	−	4.47685i	−0.247570	−	0.247570i
$328$		0			0
$329$	−6.04090	+	1.56155i	−0.333045	+	0.0860912i
$330$		0			0
$331$	−22.7386			−1.24983			−0.624914	−	0.780693i	$-0.714866\pi$
$331$	−22.7386			−1.24983			−0.624914	+	0.780693i	$0.714866\pi$
$332$		0			0
$333$	−7.24517	−	7.24517i	−0.397033	−	0.397033i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.17662	−	3.17662i	0.173042	−	0.173042i	−0.615273	−	0.788314i	$-0.710954\pi$
$337$	3.17662	−	3.17662i	0.173042	−	0.173042i	0.788314	+	0.615273i	$0.210954\pi$
$338$		0			0
$339$	4.13595			0.224634
$340$		0			0
$341$		−	14.7304i		−	0.797696i
$342$		0			0
$343$	18.5137	+	0.491087i	0.999648	+	0.0265162i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.6962	−	13.6962i	0.735249	−	0.735249i	−0.236405	−	0.971655i	$-0.575969\pi$
$347$	13.6962	−	13.6962i	0.735249	−	0.735249i	0.971655	+	0.236405i	$0.0759693\pi$
$348$		0			0
$349$	−2.64861			−0.141777			−0.0708885	−	0.997484i	$-0.522583\pi$
$349$	−2.64861			−0.141777			−0.0708885	+	0.997484i	$0.522583\pi$
$350$		0			0
$351$	3.80776			0.203243
$352$		0			0
$353$	−22.6148	+	22.6148i	−1.20366	+	1.20366i	−0.230620	+	0.973044i	$0.574075\pi$
$353$	−22.6148	+	22.6148i	−1.20366	+	1.20366i	−0.973044	+	0.230620i	$0.925925\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−11.6642	+	19.7967i	−0.617334	+	1.04775i
$358$		0			0
$359$			26.7386i			1.41121i	0.708605	+	0.705606i	$0.249325\pi$
$359$			26.7386i			1.41121i	−0.708605	+	0.705606i	$0.750675\pi$
$360$		0			0
$361$	−11.9848			−0.630781
$362$		0			0
$363$	−14.2770	+	14.2770i	−0.749346	+	0.749346i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−8.33783	−	8.33783i	−0.435231	−	0.435231i	0.455172	−	0.890403i	$-0.349578\pi$
$367$	−8.33783	−	8.33783i	−0.435231	−	0.435231i	−0.890403	+	0.455172i	$0.849578\pi$
$368$		0			0
$369$	6.78456			0.353190
$370$		0			0
$371$	26.2462	−	6.78456i	1.36264	−	0.352237i
$372$		0			0
$373$	−14.1421	−	14.1421i	−0.732252	−	0.732252i	0.238813	−	0.971065i	$-0.423242\pi$
$373$	−14.1421	−	14.1421i	−0.732252	−	0.732252i	−0.971065	+	0.238813i	$0.923242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−17.4068	−	17.4068i	−0.896496	−	0.896496i
$378$		0			0
$379$		−	26.2462i		−	1.34818i	−0.738650	−	0.674089i	$-0.764537\pi$
$379$		−	26.2462i		−	1.34818i	0.738650	−	0.674089i	$-0.235463\pi$
$380$		0			0
$381$			30.9481i			1.58552i
$382$		0			0
$383$	−14.2770	+	14.2770i	−0.729518	+	0.729518i	−0.970524	−	0.241005i	$-0.922523\pi$
$383$	−14.2770	+	14.2770i	−0.729518	+	0.729518i	0.241005	+	0.970524i	$0.422523\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	16.5246	−	16.5246i	0.839993	−	0.839993i
$388$		0			0
$389$			20.4384i			1.03627i	0.855299	+	0.518135i	$0.173374\pi$
$389$			20.4384i			1.03627i	−0.855299	+	0.518135i	$0.826626\pi$
$390$		0			0
$391$			4.13595i			0.209164i
$392$		0			0
$393$	35.8776	+	35.8776i	1.80979	+	1.80979i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13.0200	−	13.0200i	−0.653453	−	0.653453i	0.300370	−	0.953823i	$-0.402890\pi$
$397$	−13.0200	−	13.0200i	−0.653453	−	0.653453i	−0.953823	+	0.300370i	$0.902890\pi$
$398$		0			0
$399$	4.13595	+	16.0000i	0.207056	+	0.801002i
$400$		0			0
$401$	0.930870			0.0464854			0.0232427	−	0.999730i	$-0.492601\pi$
$401$	0.930870			0.0464854			0.0232427	+	0.999730i	$0.492601\pi$
$402$		0			0
$403$	−24.5639	−	24.5639i	−1.22362	−	1.22362i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.41674	−	4.41674i	0.218930	−	0.218930i
$408$		0			0
$409$	−30.9481			−1.53029			−0.765144	−	0.643860i	$-0.777332\pi$
$409$	−30.9481			−1.53029			−0.765144	+	0.643860i	$0.777332\pi$
$410$		0			0
$411$		−	33.5968i		−	1.65721i
$412$		0			0
$413$	−16.2268	+	27.5405i	−0.798468	+	1.35518i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	20.1472	−	20.1472i	0.986612	−	0.986612i
$418$		0			0
$419$	6.78456			0.331448			0.165724	−	0.986172i	$-0.447004\pi$
$419$	6.78456			0.331448			0.165724	+	0.986172i	$0.447004\pi$
$420$		0			0
$421$	−10.6847			−0.520738			−0.260369	−	0.965509i	$-0.583844\pi$
$421$	−10.6847			−0.520738			−0.260369	+	0.965509i	$0.583844\pi$
$422$		0			0
$423$	4.27156	−	4.27156i	0.207690	−	0.207690i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−21.5030	−	12.6695i	−1.04060	−	0.613120i
$428$		0			0
$429$		−	13.5616i		−	0.654758i
$430$		0			0
$431$	−18.4384			−0.888149			−0.444074	−	0.895990i	$-0.646467\pi$
$431$	−18.4384			−0.888149			−0.444074	+	0.895990i	$0.646467\pi$
$432$		0			0
$433$	−14.1617	+	14.1617i	−0.680567	+	0.680567i	−0.960128	−	0.279561i	$-0.909811\pi$
$433$	−14.1617	+	14.1617i	−0.680567	+	0.680567i	0.279561	+	0.960128i	$0.409811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.10341	+	2.10341i	0.100620	+	0.100620i
$438$		0			0
$439$	29.4608			1.40609			0.703044	−	0.711146i	$-0.251824\pi$
$439$	29.4608			1.40609			0.703044	+	0.711146i	$0.251824\pi$
$440$		0			0
$441$	−15.6847	+	8.68951i	−0.746888	+	0.413786i
$442$		0			0
$443$	−16.1764	−	16.1764i	−0.768564	−	0.768564i	0.209289	−	0.977854i	$-0.432885\pi$
$443$	−16.1764	−	16.1764i	−0.768564	−	0.768564i	−0.977854	+	0.209289i	$0.932885\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.08084	+	7.08084i	0.334912	+	0.334912i
$448$		0			0
$449$			20.0540i			0.946406i	0.880954	+	0.473203i	$0.156902\pi$
$449$			20.0540i			0.946406i	−0.880954	+	0.473203i	$0.843098\pi$
$450$		0			0
$451$			4.13595i			0.194754i
$452$		0			0
$453$	−27.8228	+	27.8228i	−1.30723	+	1.30723i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	27.0442	−	27.0442i	1.26507	−	1.26507i	0.316470	−	0.948603i	$-0.397502\pi$
$457$	27.0442	−	27.0442i	1.26507	−	1.26507i	0.948603	−	0.316470i	$-0.102498\pi$
$458$		0			0
$459$			3.80776i			0.177731i
$460$		0			0
$461$			36.2454i			1.68812i	0.536252	+	0.844058i	$0.319840\pi$
$461$			36.2454i			1.68812i	−0.536252	+	0.844058i	$0.680160\pi$
$462$		0			0
$463$	−3.97078	−	3.97078i	−0.184538	−	0.184538i	0.608792	−	0.793330i	$-0.291654\pi$
$463$	−3.97078	−	3.97078i	−0.184538	−	0.184538i	−0.793330	+	0.608792i	$0.791654\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.25699	−	1.25699i	−0.0581667	−	0.0581667i	0.677425	−	0.735592i	$-0.263096\pi$
$467$	−1.25699	−	1.25699i	−0.0581667	−	0.0581667i	−0.735592	+	0.677425i	$0.763096\pi$
$468$		0			0
$469$	−3.39228	−	13.1231i	−0.156641	−	0.605969i
$470$		0			0
$471$	44.4924			2.05010
$472$		0			0
$473$	10.0736	+	10.0736i	0.463184	+	0.463184i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−18.5589	+	18.5589i	−0.849752	+	0.849752i
$478$		0			0
$479$	1.16128			0.0530601			0.0265301	−	0.999648i	$-0.491554\pi$
$479$	1.16128			0.0530601			0.0265301	+	0.999648i	$0.491554\pi$
$480$		0			0
$481$		−	14.7304i		−	0.671649i
$482$		0			0
$483$	−3.55730	+	6.03753i	−0.161863	+	0.274717i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.69113	+	7.69113i	−0.348518	+	0.348518i	−0.859557	−	0.511039i	$-0.829261\pi$
$487$	−7.69113	+	7.69113i	−0.348518	+	0.348518i	0.511039	+	0.859557i	$0.329261\pi$
$488$		0			0
$489$	−25.6509			−1.15997
$490$		0			0
$491$	4.68466			0.211416			0.105708	−	0.994397i	$-0.466289\pi$
$491$	4.68466			0.211416			0.105708	+	0.994397i	$0.466289\pi$
$492$		0			0
$493$	17.4068	−	17.4068i	0.783963	−	0.783963i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.38915	+	14.2383i	−0.376305	+	0.638674i
$498$		0			0
$499$			16.1922i			0.724864i	0.932010	+	0.362432i	$0.118053\pi$
$499$			16.1922i			0.724864i	−0.932010	+	0.362432i	$0.881947\pi$
$500$		0			0
$501$	50.0540			2.23625
$502$		0			0
$503$	−12.0835	+	12.0835i	−0.538778	+	0.538778i	−0.923170	−	0.384392i	$-0.874411\pi$
$503$	−12.0835	+	12.0835i	−0.538778	+	0.538778i	0.384392	+	0.923170i	$0.374411\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.936426	−	0.936426i	−0.0415882	−	0.0415882i
$508$		0			0
$509$	−17.3790			−0.770311			−0.385156	−	0.922852i	$-0.625852\pi$
$509$	−17.3790			−0.770311			−0.385156	+	0.922852i	$0.625852\pi$
$510$		0			0
$511$	−6.24621	−	24.1636i	−0.276316	−	1.06893i
$512$		0			0
$513$	1.93651	+	1.93651i	0.0854989	+	0.0854989i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.60399	+	2.60399i	0.114523	+	0.114523i
$518$		0			0
$519$		−	3.80776i		−	0.167142i
$520$		0			0
$521$			38.8940i			1.70398i	0.523561	+	0.851988i	$0.324603\pi$
$521$			38.8940i			1.70398i	−0.523561	+	0.851988i	$0.675397\pi$
$522$		0			0
$523$	−3.86098	+	3.86098i	−0.168829	+	0.168829i	−0.786465	−	0.617635i	$-0.788091\pi$
$523$	−3.86098	+	3.86098i	−0.168829	+	0.168829i	0.617635	+	0.786465i	$0.288091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.5639	−	24.5639i	1.07002	−	1.07002i
$528$		0			0
$529$		−	21.7386i		−	0.945158i
$530$		0			0
$531$		−	30.9481i		−	1.34304i
$532$		0			0
$533$	6.89697	+	6.89697i	0.298741	+	0.298741i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.74571	+	3.74571i	0.161639	+	0.161639i
$538$		0			0
$539$	−5.29723	−	9.56155i	−0.228168	−	0.411845i
$540$		0			0
$541$	−28.9309			−1.24384			−0.621918	−	0.783083i	$-0.713646\pi$
$541$	−28.9309			−1.24384			−0.621918	+	0.783083i	$0.713646\pi$
$542$		0			0
$543$	−28.9807	−	28.9807i	−1.24368	−	1.24368i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	9.62763	−	9.62763i	0.411648	−	0.411648i	−0.470664	−	0.882312i	$-0.655986\pi$
$547$	9.62763	−	9.62763i	0.411648	−	0.411648i	0.882312	+	0.470664i	$0.155986\pi$
$548$		0			0
$549$	24.1636			1.03128
$550$		0			0
$551$		−	17.7051i		−	0.754262i
$552$		0			0
$553$	26.9159	+	15.8588i	1.14458	+	0.674383i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−30.5690	+	30.5690i	−1.29525	+	1.29525i	−0.363754	+	0.931495i	$0.618505\pi$
$557$	−30.5690	+	30.5690i	−1.29525	+	1.29525i	−0.931495	+	0.363754i	$0.881495\pi$
$558$		0			0
$559$	33.5968			1.42099
$560$		0			0
$561$	13.5616			0.572569
$562$		0			0
$563$	15.0981	−	15.0981i	0.636309	−	0.636309i	−0.313334	−	0.949643i	$-0.601446\pi$
$563$	15.0981	−	15.0981i	0.636309	−	0.636309i	0.949643	+	0.313334i	$0.101446\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−13.5961	+	23.0757i	−0.570983	+	0.969087i
$568$		0			0
$569$		−	36.7386i		−	1.54016i	−0.637945	−	0.770082i	$-0.720215\pi$
$569$		−	36.7386i		−	1.54016i	0.637945	−	0.770082i	$-0.279785\pi$
$570$		0			0
$571$	−21.7538			−0.910368			−0.455184	−	0.890397i	$-0.650426\pi$
$571$	−21.7538			−0.910368			−0.455184	+	0.890397i	$0.650426\pi$
$572$		0			0
$573$	17.4068	−	17.4068i	0.727179	−	0.727179i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.1988	−	12.1988i	−0.507843	−	0.507843i	0.406021	−	0.913864i	$-0.366916\pi$
$577$	−12.1988	−	12.1988i	−0.507843	−	0.507843i	−0.913864	+	0.406021i	$0.866916\pi$
$578$		0			0
$579$	−4.13595			−0.171884
$580$		0			0
$581$	−8.87689	−	34.3404i	−0.368276	−	1.42468i
$582$		0			0
$583$	−11.3137	−	11.3137i	−0.468566	−	0.468566i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.73384	−	5.73384i	−0.236661	−	0.236661i	0.578805	−	0.815466i	$-0.303519\pi$
$587$	−5.73384	−	5.73384i	−0.236661	−	0.236661i	−0.815466	+	0.578805i	$0.803519\pi$
$588$		0			0
$589$		−	24.9848i		−	1.02948i
$590$		0			0
$591$			57.7603i			2.37594i
$592$		0			0
$593$	30.7473	−	30.7473i	1.26264	−	1.26264i	0.312833	−	0.949808i	$-0.398722\pi$
$593$	30.7473	−	30.7473i	1.26264	−	1.26264i	0.949808	−	0.312833i	$-0.101278\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.83348	+	8.83348i	−0.361530	+	0.361530i
$598$		0			0
$599$		−	23.3153i		−	0.952639i	−0.879272	−	0.476320i	$-0.841971\pi$
$599$		−	23.3153i		−	0.952639i	0.879272	−	0.476320i	$-0.158029\pi$
$600$		0			0
$601$			25.6509i			1.04632i	0.852234	+	0.523161i	$0.175248\pi$
$601$			25.6509i			1.04632i	−0.852234	+	0.523161i	$0.824752\pi$
$602$		0			0
$603$	9.27944	+	9.27944i	0.377888	+	0.377888i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.0835	+	12.0835i	0.490456	+	0.490456i	0.908450	−	0.417994i	$-0.137267\pi$
$607$	12.0835	+	12.0835i	0.490456	+	0.490456i	−0.417994	+	0.908450i	$0.637267\pi$
$608$		0			0
$609$	40.3813	−	10.4384i	1.63633	−	0.422987i
$610$		0			0
$611$	8.68466			0.351344
$612$		0			0
$613$	−9.72540	−	9.72540i	−0.392805	−	0.392805i	0.482881	−	0.875686i	$-0.339590\pi$
$613$	−9.72540	−	9.72540i	−0.392805	−	0.392805i	−0.875686	+	0.482881i	$0.839590\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	−29.7869			−1.19724			−0.598618	−	0.801035i	$-0.704283\pi$
$619$	−29.7869			−1.19724			−0.598618	+	0.801035i	$0.704283\pi$
$620$		0			0
$621$			1.16128i			0.0466005i
$622$		0			0
$623$	12.6695	−	21.5030i	0.507593	−	0.861498i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	6.89697	−	6.89697i	0.275438	−	0.275438i
$628$		0			0
$629$	14.7304			0.587340
$630$		0			0
$631$	−19.8078			−0.788535			−0.394267	−	0.918996i	$-0.629002\pi$
$631$	−19.8078			−0.788535			−0.394267	+	0.918996i	$0.629002\pi$
$632$		0			0
$633$	−41.9844	+	41.9844i	−1.66873	+	1.66873i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−24.7780	−	7.11104i	−0.981740	−	0.281750i
$638$		0			0
$639$		−	16.0000i		−	0.632950i
$640$		0			0
$641$	44.7386			1.76707			0.883535	−	0.468365i	$-0.155157\pi$
$641$	44.7386			1.76707			0.883535	+	0.468365i	$0.155157\pi$
$642$		0			0
$643$	15.8292	−	15.8292i	0.624244	−	0.624244i	−0.322370	−	0.946614i	$-0.604479\pi$
$643$	15.8292	−	15.8292i	0.624244	−	0.624244i	0.946614	+	0.322370i	$0.104479\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.55498	+	6.55498i	0.257703	+	0.257703i	0.824119	−	0.566416i	$-0.191671\pi$
$647$	6.55498	+	6.55498i	0.257703	+	0.257703i	−0.566416	+	0.824119i	$0.691671\pi$
$648$		0			0
$649$	18.8664			0.740569
$650$		0			0
$651$	56.9848	−	14.7304i	2.23341	−	0.577330i
$652$		0			0
$653$	−14.1421	−	14.1421i	−0.553425	−	0.553425i	0.374003	−	0.927428i	$-0.377985\pi$
$653$	−14.1421	−	14.1421i	−0.553425	−	0.553425i	−0.927428	+	0.374003i	$0.877985\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	17.0862	+	17.0862i	0.666597	+	0.666597i
$658$		0			0
$659$		−	26.9309i		−	1.04908i	−0.851387	−	0.524539i	$-0.824238\pi$
$659$		−	26.9309i		−	1.04908i	0.851387	−	0.524539i	$-0.175762\pi$
$660$		0			0
$661$			8.27190i			0.321740i	0.986976	+	0.160870i	$0.0514299\pi$
$661$			8.27190i			0.321740i	−0.986976	+	0.160870i	$0.948570\pi$
$662$		0			0
$663$	22.6148	−	22.6148i	0.878285	−	0.878285i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	5.30866	−	5.30866i	0.205552	−	0.205552i
$668$		0			0
$669$		−	6.93087i		−	0.267963i
$670$		0			0
$671$			14.7304i			0.568661i
$672$		0			0
$673$	33.3974	+	33.3974i	1.28738	+	1.28738i	0.936376	+	0.351000i	$0.114158\pi$
$673$	33.3974	+	33.3974i	1.28738	+	1.28738i	0.351000	+	0.936376i	$0.385842\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.3034	−	15.3034i	−0.588157	−	0.588157i	0.348975	−	0.937132i	$-0.386530\pi$
$677$	−15.3034	−	15.3034i	−0.588157	−	0.588157i	−0.937132	+	0.348975i	$0.886530\pi$
$678$		0			0
$679$	20.0276	−	5.17708i	0.768590	−	0.198678i
$680$		0			0
$681$	−4.19224			−0.160647
$682$		0			0
$683$	15.2845	+	15.2845i	0.584845	+	0.584845i	0.936231	−	0.351386i	$-0.114290\pi$
$683$	15.2845	+	15.2845i	0.584845	+	0.584845i	−0.351386	+	0.936231i	$0.614290\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.41674	−	4.41674i	0.168509	−	0.168509i
$688$		0			0
$689$	−37.7327			−1.43750
$690$		0			0
$691$		−	6.78456i		−	0.258097i	−0.991638	−	0.129048i	$-0.958808\pi$
$691$		−	6.78456i		−	0.258097i	0.991638	−	0.129048i	$-0.0411923\pi$
$692$		0			0
$693$	9.11802	+	5.37231i	0.346365	+	0.204077i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−6.89697	+	6.89697i	−0.261241	+	0.261241i
$698$		0			0
$699$	−56.5991			−2.14077
$700$		0			0
$701$	40.9309			1.54594			0.772969	−	0.634444i	$-0.218771\pi$
$701$	40.9309			1.54594			0.772969	+	0.634444i	$0.218771\pi$
$702$		0			0
$703$	7.49141	−	7.49141i	0.282544	−	0.282544i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.9684	+	21.7817i	1.39034	+	0.819185i
$708$		0			0
$709$		−	29.8078i		−	1.11945i	−0.828677	−	0.559727i	$-0.810906\pi$
$709$		−	29.8078i		−	1.11945i	0.828677	−	0.559727i	$-0.189094\pi$
$710$		0			0
$711$	−30.2462			−1.13432
$712$		0			0
$713$	7.49141	−	7.49141i	0.280556	−	0.280556i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	33.0308	+	33.0308i	1.23356	+	1.23356i
$718$		0			0
$719$	28.2995			1.05539			0.527697	−	0.849433i	$-0.323056\pi$
$719$	28.2995			1.05539			0.527697	+	0.849433i	$0.323056\pi$
$720$		0			0
$721$	−8.19224	−	31.6918i	−0.305095	−	1.18026i
$722$		0			0
$723$	−44.7111	−	44.7111i	−1.66282	−	1.66282i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.55498	−	6.55498i	−0.243111	−	0.243111i	0.575025	−	0.818136i	$-0.304992\pi$
$727$	−6.55498	−	6.55498i	−0.243111	−	0.243111i	−0.818136	+	0.575025i	$0.804992\pi$
$728$		0			0
$729$		−	18.6155i		−	0.689464i
$730$		0			0
$731$			33.5968i			1.24262i
$732$		0			0
$733$	3.42514	−	3.42514i	0.126510	−	0.126510i	−0.641017	−	0.767527i	$-0.721487\pi$
$733$	3.42514	−	3.42514i	0.126510	−	0.126510i	0.767527	+	0.641017i	$0.221487\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.65685	+	5.65685i	−0.208373	+	0.208373i
$738$		0			0
$739$		−	39.4233i		−	1.45021i	−0.688639	−	0.725105i	$-0.741791\pi$
$739$		−	39.4233i		−	1.45021i	0.688639	−	0.725105i	$-0.258209\pi$
$740$		0			0
$741$		−	23.0023i		−	0.845011i
$742$		0			0
$743$	16.5246	+	16.5246i	0.606229	+	0.606229i	0.941958	−	0.335730i	$-0.108983\pi$
$743$	16.5246	+	16.5246i	0.606229	+	0.606229i	−0.335730	+	0.941958i	$0.608983\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	24.2824	+	24.2824i	0.888445	+	0.888445i
$748$		0			0
$749$	1.90495	+	7.36932i	0.0696052	+	0.269269i
$750$		0			0
$751$	34.4384			1.25668			0.628338	−	0.777940i	$-0.283735\pi$
$751$	34.4384			1.25668			0.628338	+	0.777940i	$0.283735\pi$
$752$		0			0
$753$	20.1472	+	20.1472i	0.734204	+	0.734204i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−12.9020	+	12.9020i	−0.468932	+	0.468932i	−0.901568	−	0.432637i	$-0.857583\pi$
$757$	−12.9020	+	12.9020i	−0.468932	+	0.468932i	0.432637	+	0.901568i	$0.357583\pi$
$758$		0			0
$759$	4.13595			0.150125
$760$		0			0
$761$		−	4.13595i		−	0.149928i	−0.997186	−	0.0749640i	$-0.976116\pi$
$761$		−	4.13595i		−	0.149928i	0.997186	−	0.0749640i	$-0.0238842\pi$
$762$		0			0
$763$	6.11969	+	3.60571i	0.221548	+	0.130535i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31.4609	−	31.4609i	1.13599	−	1.13599i
$768$		0			0
$769$	34.7580			1.25341			0.626703	−	0.779258i	$-0.284404\pi$
$769$	34.7580			1.25341			0.626703	+	0.779258i	$0.284404\pi$
$770$		0			0
$771$	34.7386			1.25108
$772$		0			0
$773$	32.2096	−	32.2096i	1.15850	−	1.15850i	0.173701	−	0.984799i	$-0.444428\pi$
$773$	32.2096	−	32.2096i	1.15850	−	1.15850i	0.984799	−	0.173701i	$-0.0555725\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	21.5030	+	12.6695i	0.771415	+	0.454516i
$778$		0			0
$779$			7.01515i			0.251344i
$780$		0			0
$781$	9.75379			0.349018
$782$		0			0
$783$	4.88742	−	4.88742i	0.174662	−	0.174662i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−15.4187	−	15.4187i	−0.549616	−	0.549616i	0.376714	−	0.926330i	$-0.377054\pi$
$787$	−15.4187	−	15.4187i	−0.549616	−	0.549616i	−0.926330	+	0.376714i	$0.877054\pi$
$788$		0			0
$789$	17.3790			0.618709
$790$		0			0
$791$	−4.49242	+	1.16128i	−0.159732	+	0.0412903i
$792$		0			0
$793$	24.5639	+	24.5639i	0.872290	+	0.872290i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.96286	−	1.96286i	−0.0695281	−	0.0695281i	0.671488	−	0.741016i	$-0.265656\pi$
$797$	−1.96286	−	1.96286i	−0.0695281	−	0.0695281i	−0.741016	+	0.671488i	$0.765656\pi$
$798$		0			0
$799$			8.68466i			0.307241i
$800$		0			0
$801$			24.1636i			0.853778i
$802$		0			0
$803$	−10.4160	+	10.4160i	−0.367572	+	0.367572i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−8.83348	+	8.83348i	−0.310953	+	0.310953i
$808$		0			0
$809$			34.6847i			1.21945i	0.792614	+	0.609724i	$0.208720\pi$
$809$			34.6847i			1.21945i	−0.792614	+	0.609724i	$0.791280\pi$
$810$		0			0
$811$			25.6509i			0.900726i	0.892846	+	0.450363i	$0.148705\pi$
$811$			25.6509i			0.900726i	−0.892846	+	0.450363i	$0.851295\pi$
$812$		0			0
$813$	6.89697	+	6.89697i	0.241887	+	0.241887i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	17.0862	+	17.0862i	0.597772	+	0.597772i
$818$		0			0
$819$	24.1636	−	6.24621i	0.844344	−	0.218260i
$820$		0			0
$821$	20.5464			0.717074			0.358537	−	0.933515i	$-0.383276\pi$
$821$	20.5464			0.717074			0.358537	+	0.933515i	$0.383276\pi$
$822$		0			0
$823$	−3.27439	−	3.27439i	−0.114138	−	0.114138i	0.647731	−	0.761869i	$-0.275718\pi$
$823$	−3.27439	−	3.27439i	−0.114138	−	0.114138i	−0.761869	+	0.647731i	$0.775718\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.03427	+	2.03427i	−0.0707386	+	0.0707386i	−0.741591	−	0.670852i	$-0.765928\pi$
$827$	−2.03427	+	2.03427i	−0.0707386	+	0.0707386i	0.670852	+	0.741591i	$0.265928\pi$
$828$		0			0
$829$	−17.7051			−0.614923			−0.307461	−	0.951561i	$-0.599479\pi$
$829$	−17.7051			−0.614923			−0.307461	+	0.951561i	$0.599479\pi$
$830$		0			0
$831$			24.1636i			0.838225i
$832$		0			0
$833$	7.11104	−	24.7780i	0.246383	−	0.858507i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.89697	−	6.89697i	0.238394	−	0.238394i
$838$		0			0
$839$	38.8940			1.34277			0.671385	−	0.741109i	$-0.265700\pi$
$839$	38.8940			1.34277			0.671385	+	0.741109i	$0.265700\pi$
$840$		0			0
$841$	−15.6847			−0.540850
$842$		0			0
$843$	0.731140	−	0.731140i	0.0251818	−	0.0251818i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	11.4988	−	19.5161i	0.395105	−	0.670581i
$848$		0			0
$849$			6.93087i			0.237867i
$850$		0			0
$851$	4.49242			0.153998
$852$		0			0
$853$	−17.0862	+	17.0862i	−0.585021	+	0.585021i	−0.936279	−	0.351257i	$-0.885754\pi$
$853$	−17.0862	+	17.0862i	−0.585021	+	0.585021i	0.351257	+	0.936279i	$0.385754\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.2651	−	16.2651i	−0.555605	−	0.555605i	0.372448	−	0.928053i	$-0.378518\pi$
$857$	−16.2651	−	16.2651i	−0.555605	−	0.555605i	−0.928053	+	0.372448i	$0.878518\pi$
$858$		0			0
$859$	−17.3790			−0.592964			−0.296482	−	0.955038i	$-0.595814\pi$
$859$	−17.3790			−0.592964			−0.296482	+	0.955038i	$0.595814\pi$
$860$		0			0
$861$	−16.0000	+	4.13595i	−0.545279	+	0.140953i
$862$		0			0
$863$	−6.45101	−	6.45101i	−0.219595	−	0.219595i	0.588733	−	0.808328i	$-0.299627\pi$
$863$	−6.45101	−	6.45101i	−0.219595	−	0.219595i	−0.808328	+	0.588733i	$0.799627\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.73384	−	5.73384i	−0.194731	−	0.194731i
$868$		0			0
$869$		−	18.4384i		−	0.625481i
$870$		0			0
$871$			18.8664i			0.639262i
$872$		0			0
$873$	−14.1617	+	14.1617i	−0.479300	+	0.479300i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17.3188	−	17.3188i	0.584813	−	0.584813i	−0.351409	−	0.936222i	$-0.614297\pi$
$877$	17.3188	−	17.3188i	0.584813	−	0.584813i	0.936222	+	0.351409i	$0.114297\pi$
$878$		0			0
$879$		−	48.3002i		−	1.62912i
$880$		0			0
$881$		−	46.8398i		−	1.57807i	−0.614346	−	0.789037i	$-0.710580\pi$
$881$		−	46.8398i		−	1.57807i	0.614346	−	0.789037i	$-0.289420\pi$
$882$		0			0
$883$	−20.9413	−	20.9413i	−0.704732	−	0.704732i	0.260690	−	0.965422i	$-0.416050\pi$
$883$	−20.9413	−	20.9413i	−0.704732	−	0.704732i	−0.965422	+	0.260690i	$0.916050\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−27.6175	−	27.6175i	−0.927304	−	0.927304i	0.0702267	−	0.997531i	$-0.477628\pi$
$887$	−27.6175	−	27.6175i	−0.927304	−	0.927304i	−0.997531	+	0.0702267i	$0.977628\pi$
$888$		0			0
$889$	−8.68951	−	33.6155i	−0.291437	−	1.12743i
$890$		0			0
$891$	15.8078			0.529580
$892$		0			0
$893$	4.41674	+	4.41674i	0.147801	+	0.147801i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.89697	−	6.89697i	0.230283	−	0.230283i
$898$		0			0
$899$	−63.0576			−2.10309
$900$		0			0
$901$		−	37.7327i		−	1.25706i
$902$		0			0
$903$	−28.8963	+	49.0435i	−0.961608	+	1.63206i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−36.3236	+	36.3236i	−1.20611	+	1.20611i	−0.233827	+	0.972278i	$0.575125\pi$
$907$	−36.3236	+	36.3236i	−1.20611	+	1.20611i	−0.972278	+	0.233827i	$0.924875\pi$
$908$		0			0
$909$	−41.5426			−1.37788
$910$		0			0
$911$	−25.7538			−0.853261			−0.426631	−	0.904426i	$-0.640300\pi$
$911$	−25.7538			−0.853261			−0.426631	+	0.904426i	$0.640300\pi$
$912$		0			0
$913$	−14.8028	+	14.8028i	−0.489901	+	0.489901i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−49.0435	−	28.8963i	−1.61956	−	0.954239i
$918$		0			0
$919$		−	46.9309i		−	1.54811i	−0.633120	−	0.774053i	$-0.718226\pi$
$919$		−	46.9309i		−	1.54811i	0.633120	−	0.774053i	$-0.281774\pi$
$920$		0			0
$921$	61.1771			2.01585
$922$		0			0
$923$	16.2651	−	16.2651i	0.535372	−	0.535372i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	22.4095	+	22.4095i	0.736024	+	0.736024i
$928$		0			0
$929$	46.8398			1.53676			0.768382	−	0.639991i	$-0.221062\pi$
$929$	46.8398			1.53676			0.768382	+	0.639991i	$0.221062\pi$
$930$		0			0
$931$	−8.98485	−	16.2177i	−0.294466	−	0.531515i
$932$		0			0
$933$	−24.5639	−	24.5639i	−0.804187	−	0.804187i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−7.81198	−	7.81198i	−0.255206	−	0.255206i	0.567895	−	0.823101i	$-0.307758\pi$
$937$	−7.81198	−	7.81198i	−0.255206	−	0.255206i	−0.823101	+	0.567895i	$0.807758\pi$
$938$		0			0
$939$		−	23.3153i		−	0.760867i
$940$		0			0
$941$		−	2.64861i		−	0.0863423i	−0.999068	−	0.0431712i	$-0.986254\pi$
$941$		−	2.64861i		−	0.0863423i	0.999068	−	0.0431712i	$-0.0137461\pi$
$942$		0			0
$943$	−2.10341	+	2.10341i	−0.0684965	+	0.0684965i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.8728	−	16.8728i	0.548292	−	0.548292i	−0.377654	−	0.925947i	$-0.623269\pi$
$947$	16.8728	−	16.8728i	0.548292	−	0.548292i	0.925947	+	0.377654i	$0.123269\pi$
$948$		0			0
$949$			34.7386i			1.12766i
$950$		0			0
$951$			34.7580i			1.12711i
$952$		0			0
$953$	−30.2208	−	30.2208i	−0.978947	−	0.978947i	0.0208359	−	0.999783i	$-0.493367\pi$
$953$	−30.2208	−	30.2208i	−0.978947	−	0.978947i	−0.999783	+	0.0208359i	$0.993367\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−17.4068	−	17.4068i	−0.562682	−	0.562682i
$958$		0			0
$959$	9.43318	+	36.4924i	0.304613	+	1.17840i
$960$		0			0
$961$	−57.9848			−1.87048
$962$		0			0
$963$	−5.21089	−	5.21089i	−0.167919	−	0.167919i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−36.6718	+	36.6718i	−1.17929	+	1.17929i	−0.199359	+	0.979927i	$0.563886\pi$
$967$	−36.6718	+	36.6718i	−1.17929	+	1.17929i	−0.979927	+	0.199359i	$0.936114\pi$
$968$		0			0
$969$	23.0023			0.738941
$970$		0			0
$971$		−	2.64861i		−	0.0849981i	−0.999097	−	0.0424990i	$-0.986468\pi$
$971$		−	2.64861i		−	0.0849981i	0.999097	−	0.0424990i	$-0.0135319\pi$
$972$		0			0
$973$	−16.2268	+	27.5405i	−0.520207	+	0.882908i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.7304	−	15.7304i	0.503262	−	0.503262i	−0.409188	−	0.912450i	$-0.634188\pi$
$977$	15.7304	−	15.7304i	0.503262	−	0.503262i	0.912450	+	0.409188i	$0.134188\pi$
$978$		0			0
$979$	−14.7304			−0.470786
$980$		0			0
$981$	−6.87689			−0.219562
$982$		0			0
$983$	−22.9101	+	22.9101i	−0.730718	+	0.730718i	−0.970762	−	0.240044i	$-0.922838\pi$
$983$	−22.9101	+	22.9101i	−0.730718	+	0.730718i	0.240044	+	0.970762i	$0.422838\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−7.46960	+	12.6776i	−0.237760	+	0.403532i
$988$		0			0
$989$			10.2462i			0.325811i
$990$		0			0
$991$	−42.7386			−1.35764			−0.678819	−	0.734306i	$-0.737508\pi$
$991$	−42.7386			−1.35764			−0.678819	+	0.734306i	$0.737508\pi$
$992$		0			0
$993$	−37.9182	+	37.9182i	−1.20330	+	1.20330i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.60399	−	2.60399i	−0.0824693	−	0.0824693i	0.664669	−	0.747138i	$-0.268573\pi$
$997$	−2.60399	−	2.60399i	−0.0824693	−	0.0824693i	−0.747138	+	0.664669i	$0.768573\pi$
$998$		0			0
$999$	4.13595			0.130856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.x.a.993.7	yes	16
5.2	odd	4	inner	1400.2.x.a.657.1	✓	16
5.3	odd	4	inner	1400.2.x.a.657.8	yes	16
5.4	even	2	inner	1400.2.x.a.993.2	yes	16
7.6	odd	2	inner	1400.2.x.a.993.1	yes	16
35.13	even	4	inner	1400.2.x.a.657.2	yes	16
35.27	even	4	inner	1400.2.x.a.657.7	yes	16
35.34	odd	2	inner	1400.2.x.a.993.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.x.a.657.1	✓	16	5.2	odd	4	inner
1400.2.x.a.657.2	yes	16	35.13	even	4	inner
1400.2.x.a.657.7	yes	16	35.27	even	4	inner
1400.2.x.a.657.8	yes	16	5.3	odd	4	inner
1400.2.x.a.993.1	yes	16	7.6	odd	2	inner
1400.2.x.a.993.2	yes	16	5.4	even	2	inner
1400.2.x.a.993.7	yes	16	1.1	even	1	trivial
1400.2.x.a.993.8	yes	16	35.34	odd	2	inner

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.66757 - 1.66757i) q^{3} +(-1.34308 + 2.27950i) q^{7} -2.56155i q^{9} +O(q^{10})\)	\(q+(1.66757 - 1.66757i) q^{3} +(-1.34308 + 2.27950i) q^{7} -2.56155i q^{9} +1.56155 q^{11} +(2.60399 - 2.60399i) q^{13} +(2.60399 + 2.60399i) q^{17} +2.64861 q^{19} +(1.56155 + 6.04090i) q^{21} +(0.794156 + 0.794156i) q^{23} +(0.731140 + 0.731140i) q^{27} -6.68466i q^{29} -9.43318i q^{31} +(2.60399 - 2.60399i) q^{33} +(2.82843 - 2.82843i) q^{37} -8.68466i q^{39} +2.64861i q^{41} +(6.45101 + 6.45101i) q^{43} +(1.66757 + 1.66757i) q^{47} +(-3.39228 - 6.12311i) q^{49} +8.68466 q^{51} +(-7.24517 - 7.24517i) q^{53} +(4.41674 - 4.41674i) q^{57} +12.0818 q^{59} +9.43318i q^{61} +(5.83907 + 3.44037i) q^{63} +(-3.62258 + 3.62258i) q^{67} +2.64861 q^{69} +6.24621 q^{71} +(-6.67026 + 6.67026i) q^{73} +(-2.09729 + 3.55957i) q^{77} -11.8078i q^{79} +10.1231 q^{81} +(-9.47954 + 9.47954i) q^{83} +(-11.1471 - 11.1471i) q^{87} -9.43318 q^{89} +(2.43845 + 9.43318i) q^{91} +(-15.7304 - 15.7304i) q^{93} +(-5.52855 - 5.52855i) q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} - 8 q^{21} + 40 q^{51} - 32 q^{71} + 96 q^{81} + 72 q^{91}+O(q^{100}) \) 16 * q - 8 * q^11 - 8 * q^21 + 40 * q^51 - 32 * q^71 + 96 * q^81 + 72 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more