LMFDB - Embedded newform 1400.2.x.a.993.4

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			993.4
Root			$-1.32968 + 0.481610i$ of defining polynomial
Character	$\chi$	$=$	1400.993
Dual form			1400.2.x.a.657.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.848071 + 0.848071i) q^{3} +(2.61441 + 0.406039i) q^{7} +1.56155i q^{9} +O(q^{10})$	$q+(-0.848071 + 0.848071i) q^{3} +(2.61441 + 0.406039i) q^{7} +1.56155i q^{9} -2.56155 q^{11} +(2.17238 - 2.17238i) q^{13} +(2.17238 + 2.17238i) q^{17} +8.54312 q^{19} +(-2.56155 + 1.87285i) q^{21} +(-5.03680 - 5.03680i) q^{23} +(-3.86852 - 3.86852i) q^{27} +5.68466i q^{29} +4.79741i q^{31} +(2.17238 - 2.17238i) q^{33} +(2.82843 - 2.82843i) q^{37} +3.68466i q^{39} +8.54312i q^{41} +(0.620058 + 0.620058i) q^{43} +(-0.848071 - 0.848071i) q^{47} +(6.67026 + 2.12311i) q^{49} -3.68466 q^{51} +(4.41674 + 4.41674i) q^{53} +(-7.24517 + 7.24517i) q^{57} +3.74571 q^{59} -4.79741i q^{61} +(-0.634052 + 4.08254i) q^{63} +(2.20837 - 2.20837i) q^{67} +8.54312 q^{69} -10.2462 q^{71} +(3.39228 - 3.39228i) q^{73} +(-6.69695 - 1.04009i) q^{77} +8.80776i q^{79} +1.87689 q^{81} +(-5.66906 + 5.66906i) q^{83} +(-4.82099 - 4.82099i) q^{87} +4.79741 q^{89} +(6.56155 - 4.79741i) q^{91} +(-4.06854 - 4.06854i) q^{93} +(13.3017 + 13.3017i) q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.848071	+	0.848071i	−0.489634	+	0.489634i	−0.908191	−	0.418557i	$-0.862536\pi$
$3$	−0.848071	+	0.848071i	−0.489634	+	0.489634i	0.418557	+	0.908191i	$0.362536\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.61441	+	0.406039i	0.988154	+	0.153468i
$7$	2.61441	+	0.406039i	0.988154	+	0.153468i
$8$		0			0
$9$			1.56155i			0.520518i
$10$		0			0
$11$	−2.56155			−0.772337			−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155			−0.772337			−0.386169	+	0.922428i	$0.626202\pi$
$12$		0			0
$13$	2.17238	−	2.17238i	0.602509	−	0.602509i	−0.338469	−	0.940978i	$-0.609909\pi$
$13$	2.17238	−	2.17238i	0.602509	−	0.602509i	0.940978	+	0.338469i	$0.109909\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.17238	+	2.17238i	0.526879	+	0.526879i	0.919640	−	0.392761i	$-0.128480\pi$
$17$	2.17238	+	2.17238i	0.526879	+	0.526879i	−0.392761	+	0.919640i	$0.628480\pi$
$18$		0			0
$19$	8.54312			1.95993			0.979963	−	0.199181i	$-0.0638282\pi$
$19$	8.54312			1.95993			0.979963	+	0.199181i	$0.0638282\pi$
$20$		0			0
$21$	−2.56155	+	1.87285i	−0.558977	+	0.408690i
$22$		0			0
$23$	−5.03680	−	5.03680i	−1.05024	−	1.05024i	−0.998669	−	0.0515755i	$-0.983576\pi$
$23$	−5.03680	−	5.03680i	−1.05024	−	1.05024i	−0.0515755	−	0.998669i	$-0.516424\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.86852	−	3.86852i	−0.744497	−	0.744497i
$28$		0			0
$29$			5.68466i			1.05561i	0.849364	+	0.527807i	$0.176986\pi$
$29$			5.68466i			1.05561i	−0.849364	+	0.527807i	$0.823014\pi$
$30$		0			0
$31$			4.79741i			0.861641i	0.902438	+	0.430820i	$0.141776\pi$
$31$			4.79741i			0.861641i	−0.902438	+	0.430820i	$0.858224\pi$
$32$		0			0
$33$	2.17238	−	2.17238i	0.378162	−	0.378162i
$34$		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
$39$			3.68466i			0.590018i
$40$		0			0
$41$			8.54312i			1.33421i	0.744963	+	0.667105i	$0.232467\pi$
$41$			8.54312i			1.33421i	−0.744963	+	0.667105i	$0.767533\pi$
$42$		0			0
$43$	0.620058	+	0.620058i	0.0945580	+	0.0945580i	0.752803	−	0.658245i	$-0.228701\pi$
$43$	0.620058	+	0.620058i	0.0945580	+	0.0945580i	−0.658245	+	0.752803i	$0.728701\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.848071	−	0.848071i	−0.123704	−	0.123704i	0.642545	−	0.766248i	$-0.277879\pi$
$47$	−0.848071	−	0.848071i	−0.123704	−	0.123704i	−0.766248	+	0.642545i	$0.777879\pi$
$48$		0			0
$49$	6.67026	+	2.12311i	0.952895	+	0.303301i
$50$		0			0
$51$	−3.68466			−0.515955
$52$		0			0
$53$	4.41674	+	4.41674i	0.606686	+	0.606686i	0.942078	−	0.335393i	$-0.108869\pi$
$53$	4.41674	+	4.41674i	0.606686	+	0.606686i	−0.335393	+	0.942078i	$0.608869\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.24517	+	7.24517i	−0.959646	+	0.959646i
$58$		0			0
$59$	3.74571			0.487649			0.243825	−	0.969819i	$-0.421598\pi$
$59$	3.74571			0.487649			0.243825	+	0.969819i	$0.421598\pi$
$60$		0			0
$61$		−	4.79741i		−	0.614246i	−0.951670	−	0.307123i	$-0.900634\pi$
$61$		−	4.79741i		−	0.614246i	0.951670	−	0.307123i	$-0.0993662\pi$
$62$		0			0
$63$	−0.634052	+	4.08254i	−0.0798830	+	0.514351i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.20837	−	2.20837i	0.269795	−	0.269795i	−0.559222	−	0.829018i	$-0.688900\pi$
$67$	2.20837	−	2.20837i	0.269795	−	0.269795i	0.829018	+	0.559222i	$0.188900\pi$
$68$		0			0
$69$	8.54312			1.02847
$70$		0			0
$71$	−10.2462			−1.21600			−0.608001	−	0.793936i	$-0.708028\pi$
$71$	−10.2462			−1.21600			−0.608001	+	0.793936i	$0.708028\pi$
$72$		0			0
$73$	3.39228	−	3.39228i	0.397037	−	0.397037i	−0.480150	−	0.877186i	$-0.659418\pi$
$73$	3.39228	−	3.39228i	0.397037	−	0.397037i	0.877186	+	0.480150i	$0.159418\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.69695	−	1.04009i	−0.763188	−	0.118529i
$78$		0			0
$79$			8.80776i			0.990951i	0.868622	+	0.495475i	$0.165006\pi$
$79$			8.80776i			0.990951i	−0.868622	+	0.495475i	$0.834994\pi$
$80$		0			0
$81$	1.87689			0.208544
$82$		0			0
$83$	−5.66906	+	5.66906i	−0.622260	+	0.622260i	−0.946109	−	0.323849i	$-0.895023\pi$
$83$	−5.66906	+	5.66906i	−0.622260	+	0.622260i	0.323849	+	0.946109i	$0.395023\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.82099	−	4.82099i	−0.516865	−	0.516865i
$88$		0			0
$89$	4.79741			0.508525			0.254262	−	0.967135i	$-0.418167\pi$
$89$	4.79741			0.508525			0.254262	+	0.967135i	$0.418167\pi$
$90$		0			0
$91$	6.56155	−	4.79741i	0.687838	−	0.502905i
$92$		0			0
$93$	−4.06854	−	4.06854i	−0.421888	−	0.421888i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.3017	+	13.3017i	1.35058	+	1.35058i	0.885009	+	0.465573i	$0.154152\pi$
$97$	13.3017	+	13.3017i	1.35058	+	1.35058i	0.465573	+	0.885009i	$0.345848\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$			18.1379i			1.80479i	0.430907	+	0.902396i	$0.358194\pi$
$101$			18.1379i			1.80479i	−0.430907	+	0.902396i	$0.641806\pi$
$102$		0			0
$103$	−9.53758	+	9.53758i	−0.939766	+	0.939766i	−0.998286	−	0.0585205i	$-0.981362\pi$
$103$	−9.53758	+	9.53758i	−0.939766	+	0.939766i	0.0585205	+	0.998286i	$0.481362\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.86522	−	7.86522i	0.760360	−	0.760360i	−0.216027	−	0.976387i	$-0.569310\pi$
$107$	7.86522	−	7.86522i	0.760360	−	0.760360i	0.976387	+	0.216027i	$0.0693101\pi$
$108$		0			0
$109$			9.68466i			0.927622i	0.885934	+	0.463811i	$0.153518\pi$
$109$			9.68466i			0.927622i	−0.885934	+	0.463811i	$0.846482\pi$
$110$		0			0
$111$			4.79741i			0.455350i
$112$		0			0
$113$	12.9020	+	12.9020i	1.21372	+	1.21372i	0.969794	+	0.243926i	$0.0784354\pi$
$113$	12.9020	+	12.9020i	1.21372	+	1.21372i	0.243926	+	0.969794i	$0.421565\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.39228	+	3.39228i	0.313617	+	0.313617i
$118$		0			0
$119$	4.79741	+	6.56155i	0.439778	+	0.601497i
$120$		0			0
$121$	−4.43845			−0.403495
$122$		0			0
$123$	−7.24517	−	7.24517i	−0.653275	−	0.653275i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.44849	+	3.44849i	−0.306004	+	0.306004i	−0.843357	−	0.537354i	$-0.819424\pi$
$127$	−3.44849	+	3.44849i	−0.306004	+	0.306004i	0.537354	+	0.843357i	$0.319424\pi$
$128$		0			0
$129$	−1.05171			−0.0925975
$130$		0			0
$131$		−	1.05171i		−	0.0918880i	−0.998944	−	0.0459440i	$-0.985370\pi$
$131$		−	1.05171i		−	0.0918880i	0.998944	−	0.0459440i	$-0.0146296\pi$
$132$		0			0
$133$	22.3352	+	3.46884i	1.93671	+	0.300787i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.58831	+	1.58831i	−0.135699	+	0.135699i	−0.771693	−	0.635995i	$-0.780590\pi$
$137$	−1.58831	+	1.58831i	−0.135699	+	0.135699i	0.635995	+	0.771693i	$0.280590\pi$
$138$		0			0
$139$	3.74571			0.317707			0.158853	−	0.987302i	$-0.449220\pi$
$139$	3.74571			0.317707			0.158853	+	0.987302i	$0.449220\pi$
$140$		0			0
$141$	1.43845			0.121139
$142$		0			0
$143$	−5.56466	+	5.56466i	−0.465340	+	0.465340i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.45740	+	3.85631i	−0.615076	+	0.318063i
$148$		0			0
$149$		−	12.2462i		−	1.00325i	−0.865086	−	0.501624i	$-0.832736\pi$
$149$		−	12.2462i		−	1.00325i	0.865086	−	0.501624i	$-0.167264\pi$
$150$		0			0
$151$	−4.31534			−0.351178			−0.175589	−	0.984464i	$-0.556183\pi$
$151$	−4.31534			−0.351178			−0.175589	+	0.984464i	$0.556183\pi$
$152$		0			0
$153$	−3.39228	+	3.39228i	−0.274250	+	0.274250i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.78456	−	6.78456i	−0.541467	−	0.541467i	0.382492	−	0.923959i	$-0.375066\pi$
$157$	−6.78456	−	6.78456i	−0.541467	−	0.541467i	−0.923959	+	0.382492i	$0.875066\pi$
$158$		0			0
$159$	−7.49141			−0.594108
$160$		0			0
$161$	−11.1231	−	15.2134i	−0.876624	−	1.19898i
$162$		0			0
$163$	−13.5221	−	13.5221i	−1.05913	−	1.05913i	−0.998138	−	0.0609926i	$-0.980573\pi$
$163$	−13.5221	−	13.5221i	−1.05913	−	1.05913i	−0.0609926	−	0.998138i	$-0.519427\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.63263	−	7.63263i	−0.590631	−	0.590631i	0.347171	−	0.937802i	$-0.387142\pi$
$167$	−7.63263	−	7.63263i	−0.590631	−	0.590631i	−0.937802	+	0.347171i	$0.887142\pi$
$168$		0			0
$169$			3.56155i			0.273966i
$170$		0			0
$171$			13.3405i			1.02018i
$172$		0			0
$173$	9.90941	−	9.90941i	0.753399	−	0.753399i	−0.221713	−	0.975112i	$-0.571165\pi$
$173$	9.90941	−	9.90941i	0.753399	−	0.753399i	0.975112	+	0.221713i	$0.0711648\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.17662	+	3.17662i	−0.238770	+	0.238770i
$178$		0			0
$179$		−	14.2462i		−	1.06481i	−0.846489	−	0.532406i	$-0.821288\pi$
$179$		−	14.2462i		−	1.06481i	0.846489	−	0.532406i	$-0.178712\pi$
$180$		0			0
$181$		−	20.8319i		−	1.54843i	−0.632925	−	0.774213i	$-0.718146\pi$
$181$		−	20.8319i		−	1.54843i	0.632925	−	0.774213i	$-0.281854\pi$
$182$		0			0
$183$	4.06854	+	4.06854i	0.300755	+	0.300755i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−5.56466	−	5.56466i	−0.406928	−	0.406928i
$188$		0			0
$189$	−8.54312	−	11.6847i	−0.621420	−	0.849934i
$190$		0			0
$191$	14.5616			1.05364			0.526818	−	0.849978i	$-0.323385\pi$
$191$	14.5616			1.05364			0.526818	+	0.849978i	$0.323385\pi$
$192$		0			0
$193$	−12.9020	−	12.9020i	−0.928708	−	0.928708i	0.0689148	−	0.997623i	$-0.478046\pi$
$193$	−12.9020	−	12.9020i	−0.928708	−	0.928708i	−0.997623	+	0.0689148i	$0.978046\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00505	−	6.00505i	0.427842	−	0.427842i	−0.460051	−	0.887893i	$-0.652169\pi$
$197$	6.00505	−	6.00505i	0.427842	−	0.427842i	0.887893	+	0.460051i	$0.152169\pi$
$198$		0			0
$199$	−17.0862			−1.21121			−0.605605	−	0.795765i	$-0.707069\pi$
$199$	−17.0862			−1.21121			−0.605605	+	0.795765i	$0.707069\pi$
$200$		0			0
$201$			3.74571i			0.264202i
$202$		0			0
$203$	−2.30820	+	14.8620i	−0.162004	+	1.04311i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.86522	−	7.86522i	0.546671	−	0.546671i
$208$		0			0
$209$	−21.8836			−1.51372
$210$		0			0
$211$	20.1771			1.38905			0.694524	−	0.719470i	$-0.255615\pi$
$211$	20.1771			1.38905			0.694524	+	0.719470i	$0.255615\pi$
$212$		0			0
$213$	8.68951	−	8.68951i	0.595395	−	0.595395i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.94794	+	12.5424i	−0.132235	+	0.851433i
$218$		0			0
$219$			5.75379i			0.388805i
$220$		0			0
$221$	9.43845			0.634899
$222$		0			0
$223$	12.9299	−	12.9299i	0.865848	−	0.865848i	−0.126162	−	0.992010i	$-0.540266\pi$
$223$	12.9299	−	12.9299i	0.865848	−	0.865848i	0.992010	+	0.126162i	$0.0402659\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.6260	+	14.6260i	0.970762	+	0.970762i	0.999585	−	0.0288226i	$-0.00917579\pi$
$227$	14.6260	+	14.6260i	0.970762	+	0.970762i	−0.0288226	+	0.999585i	$0.509176\pi$
$228$		0			0
$229$	8.54312			0.564545			0.282273	−	0.959334i	$-0.408912\pi$
$229$	8.54312			0.564545			0.282273	+	0.959334i	$0.408912\pi$
$230$		0			0
$231$	6.56155	−	4.79741i	0.431718	−	0.315646i
$232$		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
$237$	−7.46960	−	7.46960i	−0.485203	−	0.485203i
$238$		0			0
$239$		−	0.807764i		−	0.0522499i	−0.999659	−	0.0261250i	$-0.991683\pi$
$239$		−	0.807764i		−	0.0522499i	0.999659	−	0.0261250i	$-0.00831678\pi$
$240$		0			0
$241$		−	16.0345i		−	1.03287i	−0.856325	−	0.516437i	$-0.827258\pi$
$241$		−	16.0345i		−	1.03287i	0.856325	−	0.516437i	$-0.172742\pi$
$242$		0			0
$243$	10.0138	−	10.0138i	0.642387	−	0.642387i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	18.5589	−	18.5589i	1.18087	−	1.18087i
$248$		0			0
$249$		−	9.61553i		−	0.609359i
$250$		0			0
$251$			3.74571i			0.236427i	0.992988	+	0.118213i	$0.0377167\pi$
$251$			3.74571i			0.236427i	−0.992988	+	0.118213i	$0.962283\pi$
$252$		0			0
$253$	12.9020	+	12.9020i	0.811143	+	0.811143i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.68951	+	8.68951i	0.542037	+	0.542037i	0.924126	−	0.382089i	$-0.124795\pi$
$257$	8.68951	+	8.68951i	0.542037	+	0.542037i	−0.382089	+	0.924126i	$0.624795\pi$
$258$		0			0
$259$	8.54312	−	6.24621i	0.530843	−	0.388121i
$260$		0			0
$261$	−8.87689			−0.549466
$262$		0			0
$263$	−12.2820	−	12.2820i	−0.757338	−	0.757338i	0.218499	−	0.975837i	$-0.429884\pi$
$263$	−12.2820	−	12.2820i	−0.757338	−	0.757338i	−0.975837	+	0.218499i	$0.929884\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.06854	+	4.06854i	−0.248991	+	0.248991i
$268$		0			0
$269$	−17.0862			−1.04177			−0.520883	−	0.853628i	$-0.674397\pi$
$269$	−17.0862			−1.04177			−0.520883	+	0.853628i	$0.674397\pi$
$270$		0			0
$271$		−	21.8836i		−	1.32934i	−0.747139	−	0.664668i	$-0.768573\pi$
$271$		−	21.8836i		−	1.32934i	0.747139	−	0.664668i	$-0.231427\pi$
$272$		0			0
$273$	−1.49612	+	9.63320i	−0.0905491	+	0.583028i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.41674	−	4.41674i	0.265376	−	0.265376i	−0.561858	−	0.827234i	$-0.689913\pi$
$277$	4.41674	−	4.41674i	0.265376	−	0.265376i	0.827234	+	0.561858i	$0.189913\pi$
$278$		0			0
$279$	−7.49141			−0.448499
$280$		0			0
$281$	4.56155			0.272119			0.136060	−	0.990701i	$-0.456556\pi$
$281$	4.56155			0.272119			0.136060	+	0.990701i	$0.456556\pi$
$282$		0			0
$283$	−12.9299	+	12.9299i	−0.768601	+	0.768601i	−0.977860	−	0.209260i	$-0.932895\pi$
$283$	−12.9299	+	12.9299i	−0.768601	+	0.768601i	0.209260	+	0.977860i	$0.432895\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46884	+	22.3352i	−0.204759	+	1.31841i
$288$		0			0
$289$		−	7.56155i		−	0.444797i
$290$		0			0
$291$	−22.5616			−1.32258
$292$		0			0
$293$	3.12485	−	3.12485i	0.182556	−	0.182556i	−0.609913	−	0.792469i	$-0.708796\pi$
$293$	3.12485	−	3.12485i	0.182556	−	0.182556i	0.792469	+	0.609913i	$0.208796\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.90941	+	9.90941i	0.575003	+	0.575003i
$298$		0			0
$299$	−21.8836			−1.26556
$300$		0			0
$301$	1.36932	+	1.87285i	0.0789261	+	0.107949i
$302$		0			0
$303$	−15.3823	−	15.3823i	−0.883687	−	0.883687i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−9.32878	−	9.32878i	−0.532421	−	0.532421i	0.388871	−	0.921292i	$-0.372865\pi$
$307$	−9.32878	−	9.32878i	−0.532421	−	0.532421i	−0.921292	+	0.388871i	$0.872865\pi$
$308$		0			0
$309$		−	16.1771i		−	0.920282i
$310$		0			0
$311$		−	12.2888i		−	0.696835i	−0.937339	−	0.348418i	$-0.886719\pi$
$311$		−	12.2888i		−	0.696835i	0.937339	−	0.348418i	$-0.113281\pi$
$312$		0			0
$313$	−21.0387	+	21.0387i	−1.18918	+	1.18918i	−0.211885	+	0.977295i	$0.567960\pi$
$313$	−21.0387	+	21.0387i	−1.18918	+	1.18918i	−0.977295	+	0.211885i	$0.932040\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.5639	−	24.5639i	1.37965	−	1.37965i	0.534443	−	0.845205i	$-0.320522\pi$
$317$	24.5639	−	24.5639i	1.37965	−	1.37965i	0.845205	−	0.534443i	$-0.179478\pi$
$318$		0			0
$319$		−	14.5616i		−	0.815290i
$320$		0			0
$321$			13.3405i			0.744596i
$322$		0			0
$323$	18.5589	+	18.5589i	1.03264	+	1.03264i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.21327	−	8.21327i	−0.454195	−	0.454195i
$328$		0			0
$329$	−1.87285	−	2.56155i	−0.103254	−	0.141223i
$330$		0			0
$331$	26.7386			1.46969			0.734844	−	0.678236i	$-0.237255\pi$
$331$	26.7386			1.46969			0.734844	+	0.678236i	$0.237255\pi$
$332$		0			0
$333$	4.41674	+	4.41674i	0.242036	+	0.242036i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.1472	+	20.1472i	−1.09749	+	1.09749i	−0.102783	+	0.994704i	$0.532775\pi$
$337$	−20.1472	+	20.1472i	−1.09749	+	1.09749i	−0.994704	+	0.102783i	$0.967225\pi$
$338$		0			0
$339$	−21.8836			−1.18856
$340$		0			0
$341$		−	12.2888i		−	0.665477i
$342$		0			0
$343$	16.5767	+	8.25906i	0.895059	+	0.445947i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.79668	+	3.79668i	−0.203816	+	0.203816i	−0.801633	−	0.597816i	$-0.796035\pi$
$347$	−3.79668	+	3.79668i	−0.203816	+	0.203816i	0.597816	+	0.801633i	$0.296035\pi$
$348$		0			0
$349$	−8.54312			−0.457303			−0.228651	−	0.973508i	$-0.573432\pi$
$349$	−8.54312			−0.457303			−0.228651	+	0.973508i	$0.573432\pi$
$350$		0			0
$351$	−16.8078			−0.897132
$352$		0			0
$353$	8.00447	−	8.00447i	0.426035	−	0.426035i	−0.461240	−	0.887275i	$-0.652595\pi$
$353$	8.00447	−	8.00447i	0.426035	−	0.426035i	0.887275	+	0.461240i	$0.152595\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−9.63320	−	1.49612i	−0.509843	−	0.0791829i
$358$		0			0
$359$		−	22.7386i		−	1.20010i	−0.799963	−	0.600050i	$-0.795148\pi$
$359$		−	22.7386i		−	1.20010i	0.799963	−	0.600050i	$-0.204852\pi$
$360$		0			0
$361$	53.9848			2.84131
$362$		0			0
$363$	3.76412	−	3.76412i	0.197565	−	0.197565i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.24035	+	4.24035i	0.221345	+	0.221345i	0.809064	−	0.587720i	$-0.199974\pi$
$367$	4.24035	+	4.24035i	0.221345	+	0.221345i	−0.587720	+	0.809064i	$0.699974\pi$
$368$		0			0
$369$	−13.3405			−0.694480
$370$		0			0
$371$	9.75379	+	13.3405i	0.506391	+	0.692606i
$372$		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
$377$	12.3492	+	12.3492i	0.636017	+	0.636017i
$378$		0			0
$379$		−	9.75379i		−	0.501018i	−0.968114	−	0.250509i	$-0.919402\pi$
$379$		−	9.75379i		−	0.501018i	0.968114	−	0.250509i	$-0.0805980\pi$
$380$		0			0
$381$		−	5.84912i		−	0.299659i
$382$		0			0
$383$	3.76412	−	3.76412i	0.192337	−	0.192337i	−0.604368	−	0.796705i	$-0.706574\pi$
$383$	3.76412	−	3.76412i	0.192337	−	0.192337i	0.796705	+	0.604368i	$0.206574\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.968253	+	0.968253i	−0.0492191	+	0.0492191i
$388$		0			0
$389$			24.5616i			1.24532i	0.782492	+	0.622660i	$0.213948\pi$
$389$			24.5616i			1.24532i	−0.782492	+	0.622660i	$0.786052\pi$
$390$		0			0
$391$		−	21.8836i		−	1.10670i
$392$		0			0
$393$	0.891921	+	0.891921i	0.0449914	+	0.0449914i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−10.8619	−	10.8619i	−0.545142	−	0.545142i	0.379889	−	0.925032i	$-0.375962\pi$
$397$	−10.8619	−	10.8619i	−0.545142	−	0.545142i	−0.925032	+	0.379889i	$0.875962\pi$
$398$		0			0
$399$	−21.8836	+	16.0000i	−1.09555	+	0.801002i
$400$		0			0
$401$	−27.9309			−1.39480			−0.697401	−	0.716682i	$-0.745660\pi$
$401$	−27.9309			−1.39480			−0.697401	+	0.716682i	$0.745660\pi$
$402$		0			0
$403$	10.4218	+	10.4218i	0.519146	+	0.519146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.24517	+	7.24517i	−0.359130	+	0.359130i
$408$		0			0
$409$	5.84912			0.289220			0.144610	−	0.989489i	$-0.453807\pi$
$409$	5.84912			0.289220			0.144610	+	0.989489i	$0.453807\pi$
$410$		0			0
$411$		−	2.69400i		−	0.132885i
$412$		0			0
$413$	9.79280	+	1.52090i	0.481872	+	0.0748388i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.17662	+	3.17662i	−0.155560	+	0.155560i
$418$		0			0
$419$	−13.3405			−0.651727			−0.325864	−	0.945417i	$-0.605655\pi$
$419$	−13.3405			−0.651727			−0.325864	+	0.945417i	$0.605655\pi$
$420$		0			0
$421$	1.68466			0.0821052			0.0410526	−	0.999157i	$-0.486929\pi$
$421$	1.68466			0.0821052			0.0410526	+	0.999157i	$0.486929\pi$
$422$		0			0
$423$	1.32431	−	1.32431i	0.0643900	−	0.0643900i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.94794	−	12.5424i	0.0942673	−	0.606969i
$428$		0			0
$429$		−	9.43845i		−	0.455693i
$430$		0			0
$431$	−22.5616			−1.08675			−0.543376	−	0.839490i	$-0.682854\pi$
$431$	−22.5616			−1.08675			−0.543376	+	0.839490i	$0.682854\pi$
$432$		0			0
$433$	−20.7713	+	20.7713i	−0.998205	+	0.998205i	−0.999998	−	0.00179335i	$-0.999429\pi$
$433$	−20.7713	+	20.7713i	−0.998205	+	0.998205i	0.00179335	+	0.999998i	$0.499429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−43.0299	−	43.0299i	−2.05840	−	2.05840i
$438$		0			0
$439$	24.5776			1.17303			0.586513	−	0.809939i	$-0.300500\pi$
$439$	24.5776			1.17303			0.586513	+	0.809939i	$0.300500\pi$
$440$		0			0
$441$	−3.31534	+	10.4160i	−0.157873	+	0.495999i
$442$		0			0
$443$	−22.0074	−	22.0074i	−1.04560	−	1.04560i	−0.998909	−	0.0466918i	$-0.985132\pi$
$443$	−22.0074	−	22.0074i	−1.04560	−	1.04560i	−0.0466918	−	0.998909i	$-0.514868\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	10.3857	+	10.3857i	0.491224	+	0.491224i
$448$		0			0
$449$		−	17.0540i		−	0.804827i	−0.915458	−	0.402413i	$-0.868171\pi$
$449$		−	17.0540i		−	0.804827i	0.915458	−	0.402413i	$-0.131829\pi$
$450$		0			0
$451$		−	21.8836i		−	1.03046i
$452$		0			0
$453$	3.65971	−	3.65971i	0.171948	−	0.171948i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.3823	−	15.3823i	0.719551	−	0.719551i	−0.248962	−	0.968513i	$-0.580089\pi$
$457$	15.3823	−	15.3823i	0.719551	−	0.719551i	0.968513	+	0.248962i	$0.0800894\pi$
$458$		0			0
$459$		−	16.8078i		−	0.784519i
$460$		0			0
$461$			11.2371i			0.523365i	0.965154	+	0.261682i	$0.0842773\pi$
$461$			11.2371i			0.523365i	−0.965154	+	0.261682i	$0.915723\pi$
$462$		0			0
$463$	25.1840	+	25.1840i	1.17040	+	1.17040i	0.982115	+	0.188284i	$0.0602926\pi$
$463$	25.1840	+	25.1840i	1.17040	+	1.17040i	0.188284	+	0.982115i	$0.439707\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.6260	+	14.6260i	0.676811	+	0.676811i	0.959277	−	0.282466i	$-0.0911526\pi$
$467$	14.6260	+	14.6260i	0.676811	+	0.676811i	−0.282466	+	0.959277i	$0.591153\pi$
$468$		0			0
$469$	6.67026	−	4.87689i	0.308004	−	0.225194i
$470$		0			0
$471$	11.5076			0.530241
$472$		0			0
$473$	−1.58831	−	1.58831i	−0.0730306	−	0.0730306i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.89697	+	6.89697i	−0.315791	+	0.315791i
$478$		0			0
$479$	38.9699			1.78058			0.890290	−	0.455395i	$-0.150502\pi$
$479$	38.9699			1.78058			0.890290	+	0.455395i	$0.150502\pi$
$480$		0			0
$481$		−	12.2888i		−	0.560322i
$482$		0			0
$483$	22.3352	+	3.46884i	1.01629	+	0.157838i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−13.5221	+	13.5221i	−0.612744	+	0.612744i	−0.943660	−	0.330916i	$-0.892642\pi$
$487$	−13.5221	+	13.5221i	−0.612744	+	0.612744i	0.330916	+	0.943660i	$0.392642\pi$
$488$		0			0
$489$	22.9354			1.03717
$490$		0			0
$491$	−7.68466			−0.346804			−0.173402	−	0.984851i	$-0.555476\pi$
$491$	−7.68466			−0.346804			−0.173402	+	0.984851i	$0.555476\pi$
$492$		0			0
$493$	−12.3492	+	12.3492i	−0.556181	+	0.556181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−26.7878	−	4.16037i	−1.20160	−	0.186618i
$498$		0			0
$499$			36.8078i			1.64774i	0.566777	+	0.823871i	$0.308190\pi$
$499$			36.8078i			1.64774i	−0.566777	+	0.823871i	$0.691810\pi$
$500$		0			0
$501$	12.9460			0.578386
$502$		0			0
$503$	−7.84144	+	7.84144i	−0.349632	+	0.349632i	−0.859973	−	0.510340i	$-0.829520\pi$
$503$	−7.84144	+	7.84144i	−0.349632	+	0.349632i	0.510340	+	0.859973i	$0.329520\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.02045	−	3.02045i	−0.134143	−	0.134143i
$508$		0			0
$509$	−20.8319			−0.923359			−0.461680	−	0.887047i	$-0.652753\pi$
$509$	−20.8319			−0.923359			−0.461680	+	0.887047i	$0.652753\pi$
$510$		0			0
$511$	10.2462	−	7.49141i	0.453266	−	0.331401i
$512$		0			0
$513$	−33.0492	−	33.0492i	−1.45916	−	1.45916i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.17238	+	2.17238i	0.0955410	+	0.0955410i
$518$		0			0
$519$			16.8078i			0.737779i
$520$		0			0
$521$			19.7802i			0.866588i	0.901253	+	0.433294i	$0.142649\pi$
$521$			19.7802i			0.866588i	−0.901253	+	0.433294i	$0.857351\pi$
$522$		0			0
$523$	12.4536	−	12.4536i	0.544559	−	0.544559i	−0.380303	−	0.924862i	$-0.624180\pi$
$523$	12.4536	−	12.4536i	0.544559	−	0.544559i	0.924862	+	0.380303i	$0.124180\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.4218	+	10.4218i	−0.453980	+	0.453980i
$528$		0			0
$529$			27.7386i			1.20603i
$530$		0			0
$531$			5.84912i			0.253830i
$532$		0			0
$533$	18.5589	+	18.5589i	0.803874	+	0.803874i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0818	+	12.0818i	0.521368	+	0.521368i
$538$		0			0
$539$	−17.0862	−	5.43845i	−0.735956	−	0.234251i
$540$		0			0
$541$	−0.0691303			−0.00297214			−0.00148607	−	0.999999i	$-0.500473\pi$
$541$	−0.0691303			−0.00297214			−0.00148607	+	0.999999i	$0.500473\pi$
$542$		0			0
$543$	17.6670	+	17.6670i	0.758162	+	0.758162i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−19.5271	+	19.5271i	−0.834920	+	0.834920i	−0.988185	−	0.153265i	$-0.951021\pi$
$547$	−19.5271	+	19.5271i	−0.834920	+	0.834920i	0.153265	+	0.988185i	$0.451021\pi$
$548$		0			0
$549$	7.49141			0.319726
$550$		0			0
$551$			48.5647i			2.06893i
$552$		0			0
$553$	−3.57630	+	23.0271i	−0.152080	+	0.979212i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	27.7405	−	27.7405i	1.17540	−	1.17540i	0.194503	−	0.980902i	$-0.437691\pi$
$557$	27.7405	−	27.7405i	1.17540	−	1.17540i	0.980902	−	0.194503i	$-0.0623093\pi$
$558$		0			0
$559$	2.69400			0.113944
$560$		0			0
$561$	9.43845			0.398492
$562$		0			0
$563$	23.7917	−	23.7917i	1.00270	−	1.00270i	0.00270634	−	0.999996i	$-0.499139\pi$
$563$	23.7917	−	23.7917i	1.00270	−	1.00270i	0.999996	−	0.00270634i	$-0.000861457\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.90697	+	0.762093i	0.206073	+	0.0320049i
$568$		0			0
$569$			12.7386i			0.534031i	0.963692	+	0.267016i	$0.0860375\pi$
$569$			12.7386i			0.534031i	−0.963692	+	0.267016i	$0.913962\pi$
$570$		0			0
$571$	−38.2462			−1.60055			−0.800277	−	0.599630i	$-0.795314\pi$
$571$	−38.2462			−1.60055			−0.800277	+	0.599630i	$0.795314\pi$
$572$		0			0
$573$	−12.3492	+	12.3492i	−0.515896	+	0.515896i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.6940	+	16.6940i	0.694979	+	0.694979i	0.963323	−	0.268344i	$-0.0864763\pi$
$577$	16.6940	+	16.6940i	0.694979	+	0.694979i	−0.268344	+	0.963323i	$0.586476\pi$
$578$		0			0
$579$	21.8836			0.909453
$580$		0			0
$581$	−17.1231	+	12.5194i	−0.710386	+	0.519391i
$582$		0			0
$583$	−11.3137	−	11.3137i	−0.468566	−	0.468566i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.41273	+	6.41273i	0.264682	+	0.264682i	0.826953	−	0.562271i	$-0.190072\pi$
$587$	6.41273	+	6.41273i	0.264682	+	0.264682i	−0.562271	+	0.826953i	$0.690072\pi$
$588$		0			0
$589$			40.9848i			1.68875i
$590$		0			0
$591$			10.1854i			0.418972i
$592$		0			0
$593$	−19.1338	+	19.1338i	−0.785730	+	0.785730i	−0.980791	−	0.195061i	$-0.937510\pi$
$593$	−19.1338	+	19.1338i	−0.785730	+	0.785730i	0.195061	+	0.980791i	$0.437510\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	14.4903	−	14.4903i	0.593050	−	0.593050i
$598$		0			0
$599$		−	35.6847i		−	1.45804i	−0.684495	−	0.729018i	$-0.739977\pi$
$599$		−	35.6847i		−	1.45804i	0.684495	−	0.729018i	$-0.260023\pi$
$600$		0			0
$601$		−	22.9354i		−	0.935552i	−0.883847	−	0.467776i	$-0.845055\pi$
$601$		−	22.9354i		−	0.935552i	0.883847	−	0.467776i	$-0.154945\pi$
$602$		0			0
$603$	3.44849	+	3.44849i	0.140433	+	0.140433i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	7.84144	+	7.84144i	0.318274	+	0.318274i	0.848104	−	0.529830i	$-0.177744\pi$
$607$	7.84144	+	7.84144i	0.318274	+	0.318274i	−0.529830	+	0.848104i	$0.677744\pi$
$608$		0			0
$609$	−10.6465	−	14.5616i	−0.431419	−	0.590064i
$610$		0			0
$611$	−3.68466			−0.149065
$612$		0			0
$613$	−21.3873	−	21.3873i	−0.863825	−	0.863825i	0.127955	−	0.991780i	$-0.459159\pi$
$613$	−21.3873	−	21.3873i	−0.863825	−	0.863825i	−0.991780	+	0.127955i	$0.959159\pi$
$614$		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
$619$	44.8190			1.80143			0.900714	−	0.434413i	$-0.143044\pi$
$619$	44.8190			1.80143			0.900714	+	0.434413i	$0.143044\pi$
$620$		0			0
$621$			38.9699i			1.56381i
$622$		0			0
$623$	12.5424	+	1.94794i	0.502500	+	0.0780425i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	18.5589	−	18.5589i	0.741170	−	0.741170i
$628$		0			0
$629$	12.2888			0.489987
$630$		0			0
$631$	0.807764			0.0321566			0.0160783	−	0.999871i	$-0.494882\pi$
$631$	0.807764			0.0321566			0.0160783	+	0.999871i	$0.494882\pi$
$632$		0			0
$633$	−17.1116	+	17.1116i	−0.680125	+	0.680125i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19.1025	−	9.87814i	0.756869	−	0.391386i
$638$		0			0
$639$		−	16.0000i		−	0.632950i
$640$		0			0
$641$	−4.73863			−0.187165			−0.0935824	−	0.995612i	$-0.529832\pi$
$641$	−4.73863			−0.187165			−0.0935824	+	0.995612i	$0.529832\pi$
$642$		0			0
$643$	19.9232	−	19.9232i	0.785696	−	0.785696i	−0.195090	−	0.980785i	$-0.562500\pi$
$643$	19.9232	−	19.9232i	0.785696	−	0.785696i	0.980785	+	0.195090i	$0.0624998\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.1431	+	21.1431i	0.831222	+	0.831222i	0.987684	−	0.156462i	$-0.0500087\pi$
$647$	21.1431	+	21.1431i	0.831222	+	0.831222i	−0.156462	+	0.987684i	$0.550009\pi$
$648$		0			0
$649$	−9.59482			−0.376630
$650$		0			0
$651$	−8.98485	−	12.2888i	−0.352144	−	0.481637i
$652$		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
$657$	5.29723	+	5.29723i	0.206665	+	0.206665i
$658$		0			0
$659$			1.93087i			0.0752160i	0.999293	+	0.0376080i	$0.0119738\pi$
$659$			1.93087i			0.0752160i	−0.999293	+	0.0376080i	$0.988026\pi$
$660$		0			0
$661$		−	43.7673i		−	1.70235i	−0.524882	−	0.851175i	$-0.675890\pi$
$661$		−	43.7673i		−	1.70235i	0.524882	−	0.851175i	$-0.324110\pi$
$662$		0			0
$663$	−8.00447	+	8.00447i	−0.310868	+	0.310868i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	28.6325	−	28.6325i	1.10865	−	1.10865i
$668$		0			0
$669$			21.9309i			0.847896i
$670$		0			0
$671$			12.2888i			0.474405i
$672$		0			0
$673$	−24.9121	−	24.9121i	−0.960292	−	0.960292i	0.0389496	−	0.999241i	$-0.487599\pi$
$673$	−24.9121	−	24.9121i	−0.960292	−	0.960292i	−0.999241	+	0.0389496i	$0.987599\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.6807	−	30.6807i	−1.17916	−	1.17916i	−0.979960	−	0.199196i	$-0.936167\pi$
$677$	−30.6807	−	30.6807i	−1.17916	−	1.17916i	−0.199196	−	0.979960i	$-0.563833\pi$
$678$		0			0
$679$	29.3751	+	40.1771i	1.12731	+	1.54185i
$680$		0			0
$681$	−24.8078			−0.950636
$682$		0			0
$683$	−13.8703	−	13.8703i	−0.530731	−	0.530731i	0.390059	−	0.920790i	$-0.372455\pi$
$683$	−13.8703	−	13.8703i	−0.530731	−	0.530731i	−0.920790	+	0.390059i	$0.872455\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−7.24517	+	7.24517i	−0.276420	+	0.276420i
$688$		0			0
$689$	19.1896			0.731067
$690$		0			0
$691$			13.3405i			0.507498i	0.967270	+	0.253749i	$0.0816637\pi$
$691$			13.3405i			0.507498i	−0.967270	+	0.253749i	$0.918336\pi$
$692$		0			0
$693$	1.62416	−	10.4576i	0.0616966	−	0.397253i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−18.5589	+	18.5589i	−0.702967	+	0.702967i
$698$		0			0
$699$	28.7845			1.08873
$700$		0			0
$701$	12.0691			0.455845			0.227922	−	0.973679i	$-0.426807\pi$
$701$	12.0691			0.455845			0.227922	+	0.973679i	$0.426807\pi$
$702$		0			0
$703$	24.1636	−	24.1636i	0.911347	−	0.911347i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.36472	+	47.4200i	−0.276979	+	1.78341i
$708$		0			0
$709$		−	9.19224i		−	0.345222i	−0.984990	−	0.172611i	$-0.944780\pi$
$709$		−	9.19224i		−	0.345222i	0.984990	−	0.172611i	$-0.0552203\pi$
$710$		0			0
$711$	−13.7538			−0.515807
$712$		0			0
$713$	24.1636	−	24.1636i	0.904933	−	0.904933i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.685041	+	0.685041i	0.0255833	+	0.0255833i
$718$		0			0
$719$	−14.3922			−0.536740			−0.268370	−	0.963316i	$-0.586485\pi$
$719$	−14.3922			−0.536740			−0.268370	+	0.963316i	$0.586485\pi$
$720$		0			0
$721$	−28.8078	+	21.0625i	−1.07286	+	0.784408i
$722$		0			0
$723$	13.5984	+	13.5984i	0.505730	+	0.505730i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−21.1431	−	21.1431i	−0.784155	−	0.784155i	0.196374	−	0.980529i	$-0.437083\pi$
$727$	−21.1431	−	21.1431i	−0.784155	−	0.784155i	−0.980529	+	0.196374i	$0.937083\pi$
$728$		0			0
$729$			22.6155i			0.837612i
$730$		0			0
$731$			2.69400i			0.0996412i
$732$		0			0
$733$	29.7282	−	29.7282i	1.09804	−	1.09804i	0.103398	−	0.994640i	$-0.467029\pi$
$733$	29.7282	−	29.7282i	1.09804	−	1.09804i	0.994640	−	0.103398i	$-0.0329714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.65685	+	5.65685i	−0.208373	+	0.208373i
$738$		0			0
$739$			22.4233i			0.824854i	0.910991	+	0.412427i	$0.135319\pi$
$739$			22.4233i			0.824854i	−0.910991	+	0.412427i	$0.864681\pi$
$740$		0			0
$741$			31.4785i			1.15639i
$742$		0			0
$743$	−0.968253	−	0.968253i	−0.0355218	−	0.0355218i	0.689123	−	0.724645i	$-0.257996\pi$
$743$	−0.968253	−	0.968253i	−0.0355218	−	0.0355218i	−0.724645	+	0.689123i	$0.757996\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−8.85254	−	8.85254i	−0.323897	−	0.323897i
$748$		0			0
$749$	23.7565	−	17.3693i	0.868044	−	0.634661i
$750$		0			0
$751$	38.5616			1.40713			0.703566	−	0.710630i	$-0.251590\pi$
$751$	38.5616			1.40713			0.703566	+	0.710630i	$0.251590\pi$
$752$		0			0
$753$	−3.17662	−	3.17662i	−0.115763	−	0.115763i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.24012	+	1.24012i	−0.0450728	+	0.0450728i	−0.729284	−	0.684211i	$-0.760147\pi$
$757$	−1.24012	+	1.24012i	−0.0450728	+	0.0450728i	0.684211	+	0.729284i	$0.260147\pi$
$758$		0			0
$759$	−21.8836			−0.794326
$760$		0			0
$761$			21.8836i			0.793282i	0.917974	+	0.396641i	$0.129824\pi$
$761$			21.8836i			0.793282i	−0.917974	+	0.396641i	$0.870176\pi$
$762$		0			0
$763$	−3.93235	+	25.3197i	−0.142361	+	0.916633i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.13709	−	8.13709i	0.293813	−	0.293813i
$768$		0			0
$769$	41.6639			1.50244			0.751219	−	0.660053i	$-0.229466\pi$
$769$	41.6639			1.50244			0.751219	+	0.660053i	$0.229466\pi$
$770$		0			0
$771$	−14.7386			−0.530799
$772$		0			0
$773$	−26.8708	+	26.8708i	−0.966476	+	0.966476i	−0.999456	−	0.0329796i	$-0.989500\pi$
$773$	−26.8708	+	26.8708i	−0.966476	+	0.966476i	0.0329796	+	0.999456i	$0.489500\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.94794	+	12.5424i	−0.0698819	+	0.449956i
$778$		0			0
$779$			72.9848i			2.61495i
$780$		0			0
$781$	26.2462			0.939163
$782$		0			0
$783$	21.9912	−	21.9912i	0.785902	−	0.785902i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.14530	−	6.14530i	−0.219056	−	0.219056i	0.589044	−	0.808101i	$-0.299504\pi$
$787$	−6.14530	−	6.14530i	−0.219056	−	0.219056i	−0.808101	+	0.589044i	$0.799504\pi$
$788$		0			0
$789$	20.8319			0.741637
$790$		0			0
$791$	28.4924	+	38.9699i	1.01307	+	1.38561i
$792$		0			0
$793$	−10.4218	−	10.4218i	−0.370089	−	0.370089i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.4653	−	37.4653i	−1.32709	−	1.32709i	−0.907899	−	0.419190i	$-0.862314\pi$
$797$	−37.4653	−	37.4653i	−1.32709	−	1.32709i	−0.419190	−	0.907899i	$-0.637686\pi$
$798$		0			0
$799$		−	3.68466i		−	0.130354i
$800$		0			0
$801$			7.49141i			0.264696i
$802$		0			0
$803$	−8.68951	+	8.68951i	−0.306646	+	0.306646i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.4903	−	14.4903i	0.510084	−	0.510084i
$808$		0			0
$809$			22.3153i			0.784566i	0.919845	+	0.392283i	$0.128315\pi$
$809$			22.3153i			0.784566i	−0.919845	+	0.392283i	$0.871685\pi$
$810$		0			0
$811$		−	22.9354i		−	0.805369i	−0.915339	−	0.402685i	$-0.868077\pi$
$811$		−	22.9354i		−	0.805369i	0.915339	−	0.402685i	$-0.131923\pi$
$812$		0			0
$813$	18.5589	+	18.5589i	0.650888	+	0.650888i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	5.29723	+	5.29723i	0.185327	+	0.185327i
$818$		0			0
$819$	7.49141	+	10.2462i	0.261771	+	0.358032i
$820$		0			0
$821$	−49.5464			−1.72918			−0.864591	−	0.502477i	$-0.832422\pi$
$821$	−49.5464			−1.72918			−0.864591	+	0.502477i	$0.832422\pi$
$822$		0			0
$823$	−20.7672	−	20.7672i	−0.723901	−	0.723901i	0.245497	−	0.969397i	$-0.421049\pi$
$823$	−20.7672	−	20.7672i	−0.723901	−	0.723901i	−0.969397	+	0.245497i	$0.921049\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.86522	+	7.86522i	−0.273501	+	0.273501i	−0.830508	−	0.557007i	$-0.811950\pi$
$827$	−7.86522	+	7.86522i	−0.273501	+	0.273501i	0.557007	+	0.830508i	$0.311950\pi$
$828$		0			0
$829$	48.5647			1.68672			0.843361	−	0.537348i	$-0.180574\pi$
$829$	48.5647			1.68672			0.843361	+	0.537348i	$0.180574\pi$
$830$		0			0
$831$			7.49141i			0.259874i
$832$		0			0
$833$	9.87814	+	19.1025i	0.342257	+	0.661863i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	18.5589	−	18.5589i	0.641489	−	0.641489i
$838$		0			0
$839$	19.7802			0.682889			0.341445	−	0.939902i	$-0.389084\pi$
$839$	19.7802			0.682889			0.341445	+	0.939902i	$0.389084\pi$
$840$		0			0
$841$	−3.31534			−0.114322
$842$		0			0
$843$	−3.86852	+	3.86852i	−0.133239	+	0.133239i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−11.6039	−	1.80218i	−0.398715	−	0.0619238i
$848$		0			0
$849$		−	21.9309i		−	0.752665i
$850$		0			0
$851$	−28.4924			−0.976708
$852$		0			0
$853$	−5.29723	+	5.29723i	−0.181374	+	0.181374i	−0.791954	−	0.610581i	$-0.790936\pi$
$853$	−5.29723	+	5.29723i	−0.181374	+	0.181374i	0.610581	+	0.791954i	$0.290936\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.2586	+	22.2586i	0.760341	+	0.760341i	0.976384	−	0.216043i	$-0.0693152\pi$
$857$	22.2586	+	22.2586i	0.760341	+	0.760341i	−0.216043	+	0.976384i	$0.569315\pi$
$858$		0			0
$859$	−20.8319			−0.710777			−0.355388	−	0.934719i	$-0.615651\pi$
$859$	−20.8319			−0.710777			−0.355388	+	0.934719i	$0.615651\pi$
$860$		0			0
$861$	−16.0000	−	21.8836i	−0.545279	−	0.745793i
$862$		0			0
$863$	−0.620058	−	0.620058i	−0.0211070	−	0.0211070i	0.696475	−	0.717582i	$-0.254751\pi$
$863$	−0.620058	−	0.620058i	−0.0211070	−	0.0211070i	−0.717582	+	0.696475i	$0.754751\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	6.41273	+	6.41273i	0.217788	+	0.217788i
$868$		0			0
$869$		−	22.5616i		−	0.765348i
$870$		0			0
$871$		−	9.59482i		−	0.325108i
$872$		0			0
$873$	−20.7713	+	20.7713i	−0.703002	+	0.703002i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.00505	+	6.00505i	−0.202776	+	0.202776i	−0.801188	−	0.598412i	$-0.795799\pi$
$877$	−6.00505	+	6.00505i	−0.202776	+	0.202776i	0.598412	+	0.801188i	$0.295799\pi$
$878$		0			0
$879$			5.30019i			0.178771i
$880$		0			0
$881$		−	45.4096i		−	1.52989i	−0.644097	−	0.764944i	$-0.722767\pi$
$881$		−	45.4096i		−	1.52989i	0.644097	−	0.764944i	$-0.277233\pi$
$882$		0			0
$883$	8.21342	+	8.21342i	0.276404	+	0.276404i	0.831672	−	0.555268i	$-0.187384\pi$
$883$	8.21342	+	8.21342i	0.276404	+	0.276404i	−0.555268	+	0.831672i	$0.687384\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.5487	+	10.5487i	0.354190	+	0.354190i	0.861666	−	0.507476i	$-0.169421\pi$
$887$	10.5487	+	10.5487i	0.354190	+	0.354190i	−0.507476	+	0.861666i	$0.669421\pi$
$888$		0			0
$889$	−10.4160	+	7.61553i	−0.349340	+	0.255417i
$890$		0			0
$891$	−4.80776			−0.161066
$892$		0			0
$893$	−7.24517	−	7.24517i	−0.242450	−	0.242450i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	18.5589	−	18.5589i	0.619663	−	0.619663i
$898$		0			0
$899$	−27.2716			−0.909560
$900$		0			0
$901$			19.1896i			0.639300i
$902$		0			0
$903$	−2.74959	−	0.427034i	−0.0915006	−	0.0142108i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−18.8307	+	18.8307i	−0.625264	+	0.625264i	−0.946873	−	0.321608i	$-0.895777\pi$
$907$	−18.8307	+	18.8307i	−0.625264	+	0.625264i	0.321608	+	0.946873i	$0.395777\pi$
$908$		0			0
$909$	−28.3234			−0.939426
$910$		0			0
$911$	−42.2462			−1.39968			−0.699840	−	0.714300i	$-0.746745\pi$
$911$	−42.2462			−1.39968			−0.699840	+	0.714300i	$0.746745\pi$
$912$		0			0
$913$	14.5216	−	14.5216i	0.480595	−	0.480595i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.427034	−	2.74959i	0.0141019	−	0.0907994i
$918$		0			0
$919$		−	18.0691i		−	0.596046i	−0.954559	−	0.298023i	$-0.903673\pi$
$919$		−	18.0691i		−	0.596046i	0.954559	−	0.298023i	$-0.0963271\pi$
$920$		0			0
$921$	15.8229			0.521383
$922$		0			0
$923$	−22.2586	+	22.2586i	−0.732652	+	0.732652i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−14.8934	−	14.8934i	−0.489165	−	0.489165i
$928$		0			0
$929$	45.4096			1.48984			0.744920	−	0.667154i	$-0.232488\pi$
$929$	45.4096			1.48984			0.744920	+	0.667154i	$0.232488\pi$
$930$		0			0
$931$	56.9848	+	18.1379i	1.86760	+	0.594447i
$932$		0			0
$933$	10.4218	+	10.4218i	0.341194	+	0.341194i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−6.51713	−	6.51713i	−0.212905	−	0.212905i	0.592595	−	0.805501i	$-0.298103\pi$
$937$	−6.51713	−	6.51713i	−0.212905	−	0.212905i	−0.805501	+	0.592595i	$0.798103\pi$
$938$		0			0
$939$		−	35.6847i		−	1.16452i
$940$		0			0
$941$		−	8.54312i		−	0.278498i	−0.990257	−	0.139249i	$-0.955531\pi$
$941$		−	8.54312i		−	0.278498i	0.990257	−	0.139249i	$-0.0444688\pi$
$942$		0			0
$943$	43.0299	−	43.0299i	1.40125	−	1.40125i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.9439	+	23.9439i	−0.778071	+	0.778071i	−0.979503	−	0.201432i	$-0.935441\pi$
$947$	−23.9439	+	23.9439i	−0.778071	+	0.778071i	0.201432	+	0.979503i	$0.435441\pi$
$948$		0			0
$949$		−	14.7386i		−	0.478436i
$950$		0			0
$951$			41.6639i			1.35104i
$952$		0			0
$953$	4.76493	+	4.76493i	0.154351	+	0.154351i	0.780058	−	0.625707i	$-0.215189\pi$
$953$	4.76493	+	4.76493i	0.154351	+	0.154351i	−0.625707	+	0.780058i	$0.715189\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	12.3492	+	12.3492i	0.399194	+	0.399194i
$958$		0			0
$959$	−4.79741	+	3.50758i	−0.154916	+	0.113266i
$960$		0			0
$961$	7.98485			0.257576
$962$		0			0
$963$	12.2820	+	12.2820i	0.395781	+	0.395781i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.14488	−	4.14488i	0.133290	−	0.133290i	−0.637314	−	0.770604i	$-0.719955\pi$
$967$	4.14488	−	4.14488i	0.133290	−	0.133290i	0.770604	+	0.637314i	$0.219955\pi$
$968$		0			0
$969$	−31.4785			−1.01123
$970$		0			0
$971$		−	8.54312i		−	0.274162i	−0.990560	−	0.137081i	$-0.956228\pi$
$971$		−	8.54312i		−	0.274162i	0.990560	−	0.137081i	$-0.0437720\pi$
$972$		0			0
$973$	9.79280	+	1.52090i	0.313943	+	0.0487579i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.06854	−	4.06854i	0.130164	−	0.130164i	−0.639023	−	0.769187i	$-0.720661\pi$
$977$	4.06854	−	4.06854i	0.130164	−	0.130164i	0.769187	+	0.639023i	$0.220661\pi$
$978$		0			0
$979$	−12.2888			−0.392752
$980$		0			0
$981$	−15.1231			−0.482844
$982$		0			0
$983$	−30.3089	+	30.3089i	−0.966703	+	0.966703i	−0.999463	−	0.0327601i	$-0.989570\pi$
$983$	−30.3089	+	30.3089i	−0.966703	+	0.966703i	0.0327601	+	0.999463i	$0.489570\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.76069	+	0.584066i	0.119704	+	0.0185910i
$988$		0			0
$989$		−	6.24621i		−	0.198618i
$990$		0			0
$991$	6.73863			0.214060			0.107030	−	0.994256i	$-0.465866\pi$
$991$	6.73863			0.214060			0.107030	+	0.994256i	$0.465866\pi$
$992$		0			0
$993$	−22.6762	+	22.6762i	−0.719609	+	0.719609i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.17238	−	2.17238i	−0.0687999	−	0.0687999i	0.671870	−	0.740669i	$-0.265492\pi$
$997$	−2.17238	−	2.17238i	−0.0687999	−	0.0687999i	−0.740669	+	0.671870i	$0.765492\pi$
$998$		0			0
$999$	−21.8836			−0.692368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.x.a.993.4	yes	16
5.2	odd	4	inner	1400.2.x.a.657.6	yes	16
5.3	odd	4	inner	1400.2.x.a.657.3	✓	16
5.4	even	2	inner	1400.2.x.a.993.5	yes	16
7.6	odd	2	inner	1400.2.x.a.993.6	yes	16
35.13	even	4	inner	1400.2.x.a.657.5	yes	16
35.27	even	4	inner	1400.2.x.a.657.4	yes	16
35.34	odd	2	inner	1400.2.x.a.993.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.x.a.657.3	✓	16	5.3	odd	4	inner
1400.2.x.a.657.4	yes	16	35.27	even	4	inner
1400.2.x.a.657.5	yes	16	35.13	even	4	inner
1400.2.x.a.657.6	yes	16	5.2	odd	4	inner
1400.2.x.a.993.3	yes	16	35.34	odd	2	inner
1400.2.x.a.993.4	yes	16	1.1	even	1	trivial
1400.2.x.a.993.5	yes	16	5.4	even	2	inner
1400.2.x.a.993.6	yes	16	7.6	odd	2	inner

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+(-0.848071 + 0.848071i) q^{3} +(2.61441 + 0.406039i) q^{7} +1.56155i q^{9} +O(q^{10})\)	\(q+(-0.848071 + 0.848071i) q^{3} +(2.61441 + 0.406039i) q^{7} +1.56155i q^{9} -2.56155 q^{11} +(2.17238 - 2.17238i) q^{13} +(2.17238 + 2.17238i) q^{17} +8.54312 q^{19} +(-2.56155 + 1.87285i) q^{21} +(-5.03680 - 5.03680i) q^{23} +(-3.86852 - 3.86852i) q^{27} +5.68466i q^{29} +4.79741i q^{31} +(2.17238 - 2.17238i) q^{33} +(2.82843 - 2.82843i) q^{37} +3.68466i q^{39} +8.54312i q^{41} +(0.620058 + 0.620058i) q^{43} +(-0.848071 - 0.848071i) q^{47} +(6.67026 + 2.12311i) q^{49} -3.68466 q^{51} +(4.41674 + 4.41674i) q^{53} +(-7.24517 + 7.24517i) q^{57} +3.74571 q^{59} -4.79741i q^{61} +(-0.634052 + 4.08254i) q^{63} +(2.20837 - 2.20837i) q^{67} +8.54312 q^{69} -10.2462 q^{71} +(3.39228 - 3.39228i) q^{73} +(-6.69695 - 1.04009i) q^{77} +8.80776i q^{79} +1.87689 q^{81} +(-5.66906 + 5.66906i) q^{83} +(-4.82099 - 4.82099i) q^{87} +4.79741 q^{89} +(6.56155 - 4.79741i) q^{91} +(-4.06854 - 4.06854i) q^{93} +(13.3017 + 13.3017i) q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

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