LMFDB - Embedded newform 1400.2.x.c.657.11

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:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			657.11
Character	$\chi$	$=$	1400.657
Dual form			1400.2.x.c.993.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.923076 + 0.923076i) q^{3} +(0.915096 - 2.48246i) q^{7} -1.29586i q^{9} +O(q^{10})$	$q+(0.923076 + 0.923076i) q^{3} +(0.915096 - 2.48246i) q^{7} -1.29586i q^{9} +1.21658 q^{11} +(0.996254 + 0.996254i) q^{13} +(0.567852 - 0.567852i) q^{17} +0.103488 q^{19} +(3.13620 - 1.44679i) q^{21} +(-1.43395 + 1.43395i) q^{23} +(3.96541 - 3.96541i) q^{27} -4.97655i q^{29} -4.35159i q^{31} +(1.12300 + 1.12300i) q^{33} +(2.54187 + 2.54187i) q^{37} +1.83924i q^{39} -7.06593i q^{41} +(-3.11342 + 3.11342i) q^{43} +(7.23621 - 7.23621i) q^{47} +(-5.32520 - 4.54338i) q^{49} +1.04834 q^{51} +(-0.882128 + 0.882128i) q^{53} +(0.0955275 + 0.0955275i) q^{57} +8.28584 q^{59} -10.0630i q^{61} +(-3.21692 - 1.18584i) q^{63} +(7.07422 + 7.07422i) q^{67} -2.64729 q^{69} +0.329005 q^{71} +(11.3569 + 11.3569i) q^{73} +(1.11329 - 3.02012i) q^{77} +13.6882i q^{79} +3.43317 q^{81} +(4.61538 + 4.61538i) q^{83} +(4.59373 - 4.59373i) q^{87} -13.1942 q^{89} +(3.38483 - 1.56149i) q^{91} +(4.01685 - 4.01685i) q^{93} +(-2.40130 + 2.40130i) q^{97} -1.57652i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) $ 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	0.923076	+	0.923076i	0.532938	+	0.532938i	0.921446	−	0.388507i	$-0.127009\pi$
$3$	0.923076	+	0.923076i	0.532938	+	0.532938i	−0.388507	+	0.921446i	$0.627009\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.915096	−	2.48246i	0.345874	−	0.938281i
$7$	0.915096	−	2.48246i	0.345874	−	0.938281i
$8$		0			0
$9$		−	1.29586i		−	0.431953i
$10$		0			0
$11$	1.21658			0.366814			0.183407	−	0.983037i	$-0.441287\pi$
$11$	1.21658			0.366814			0.183407	+	0.983037i	$0.441287\pi$
$12$		0			0
$13$	0.996254	+	0.996254i	0.276311	+	0.276311i	0.831634	−	0.555323i	$-0.187406\pi$
$13$	0.996254	+	0.996254i	0.276311	+	0.276311i	−0.555323	+	0.831634i	$0.687406\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.567852	−	0.567852i	0.137724	−	0.137724i	−0.634883	−	0.772608i	$-0.718952\pi$
$17$	0.567852	−	0.567852i	0.137724	−	0.137724i	0.772608	+	0.634883i	$0.218952\pi$
$18$		0			0
$19$	0.103488			0.0237418			0.0118709	−	0.999930i	$-0.496221\pi$
$19$	0.103488			0.0237418			0.0118709	+	0.999930i	$0.496221\pi$
$20$		0			0
$21$	3.13620	−	1.44679i	0.684376	−	0.315717i
$22$		0			0
$23$	−1.43395	+	1.43395i	−0.298999	+	0.298999i	−0.840622	−	0.541623i	$-0.817810\pi$
$23$	−1.43395	+	1.43395i	−0.298999	+	0.298999i	0.541623	+	0.840622i	$0.317810\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.96541	−	3.96541i	0.763143	−	0.763143i
$28$		0			0
$29$		−	4.97655i		−	0.924121i	−0.886848	−	0.462061i	$-0.847110\pi$
$29$		−	4.97655i		−	0.924121i	0.886848	−	0.462061i	$-0.152890\pi$
$30$		0			0
$31$		−	4.35159i		−	0.781569i	−0.920482	−	0.390784i	$-0.872204\pi$
$31$		−	4.35159i		−	0.781569i	0.920482	−	0.390784i	$-0.127796\pi$
$32$		0			0
$33$	1.12300	+	1.12300i	0.195489	+	0.195489i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.54187	+	2.54187i	0.417880	+	0.417880i	0.884473	−	0.466592i	$-0.154518\pi$
$37$	2.54187	+	2.54187i	0.417880	+	0.417880i	−0.466592	+	0.884473i	$0.654518\pi$
$38$		0			0
$39$			1.83924i			0.294514i
$40$		0			0
$41$		−	7.06593i		−	1.10351i	−0.834005	−	0.551757i	$-0.813958\pi$
$41$		−	7.06593i		−	1.10351i	0.834005	−	0.551757i	$-0.186042\pi$
$42$		0			0
$43$	−3.11342	+	3.11342i	−0.474793	+	0.474793i	−0.903462	−	0.428669i	$-0.858983\pi$
$43$	−3.11342	+	3.11342i	−0.474793	+	0.474793i	0.428669	+	0.903462i	$0.358983\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.23621	−	7.23621i	1.05551	−	1.05551i	0.0571445	−	0.998366i	$-0.481800\pi$
$47$	7.23621	−	7.23621i	1.05551	−	1.05551i	0.998366	−	0.0571445i	$-0.0181996\pi$
$48$		0			0
$49$	−5.32520	−	4.54338i	−0.760742	−	0.649054i
$50$		0			0
$51$	1.04834			0.146797
$52$		0			0
$53$	−0.882128	+	0.882128i	−0.121170	+	0.121170i	−0.765091	−	0.643922i	$-0.777306\pi$
$53$	−0.882128	+	0.882128i	−0.121170	+	0.121170i	0.643922	+	0.765091i	$0.277306\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.0955275	+	0.0955275i	0.0126529	+	0.0126529i
$58$		0			0
$59$	8.28584			1.07872			0.539362	−	0.842074i	$-0.318665\pi$
$59$	8.28584			1.07872			0.539362	+	0.842074i	$0.318665\pi$
$60$		0			0
$61$		−	10.0630i		−	1.28844i	−0.764842	−	0.644218i	$-0.777183\pi$
$61$		−	10.0630i		−	1.28844i	0.764842	−	0.644218i	$-0.222817\pi$
$62$		0			0
$63$	−3.21692	−	1.18584i	−0.405294	−	0.149401i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.07422	+	7.07422i	0.864253	+	0.864253i	0.991829	−	0.127575i	$-0.0407195\pi$
$67$	7.07422	+	7.07422i	0.864253	+	0.864253i	−0.127575	+	0.991829i	$0.540720\pi$
$68$		0			0
$69$	−2.64729			−0.318696
$70$		0			0
$71$	0.329005			0.0390457			0.0195229	−	0.999809i	$-0.493785\pi$
$71$	0.329005			0.0390457			0.0195229	+	0.999809i	$0.493785\pi$
$72$		0			0
$73$	11.3569	+	11.3569i	1.32923	+	1.32923i	0.906044	+	0.423183i	$0.139087\pi$
$73$	11.3569	+	11.3569i	1.32923	+	1.32923i	0.423183	+	0.906044i	$0.360913\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.11329	−	3.02012i	0.126871	−	0.344175i
$78$		0			0
$79$			13.6882i			1.54004i	0.638020	+	0.770019i	$0.279754\pi$
$79$			13.6882i			1.54004i	−0.638020	+	0.770019i	$0.720246\pi$
$80$		0			0
$81$	3.43317			0.381463
$82$		0			0
$83$	4.61538	+	4.61538i	0.506604	+	0.506604i	0.913482	−	0.406878i	$-0.133383\pi$
$83$	4.61538	+	4.61538i	0.506604	+	0.506604i	−0.406878	+	0.913482i	$0.633383\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.59373	−	4.59373i	0.492500	−	0.492500i
$88$		0			0
$89$	−13.1942			−1.39858			−0.699291	−	0.714837i	$-0.746501\pi$
$89$	−13.1942			−1.39858			−0.699291	+	0.714837i	$0.746501\pi$
$90$		0			0
$91$	3.38483	−	1.56149i	0.354826	−	0.163689i
$92$		0			0
$93$	4.01685	−	4.01685i	0.416528	−	0.416528i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.40130	+	2.40130i	−0.243815	+	0.243815i	−0.818426	−	0.574611i	$-0.805153\pi$
$97$	−2.40130	+	2.40130i	−0.243815	+	0.243815i	0.574611	+	0.818426i	$0.305153\pi$
$98$		0			0
$99$		−	1.57652i		−	0.158446i
$100$		0			0
$101$		−	8.77555i		−	0.873200i	−0.899656	−	0.436600i	$-0.856183\pi$
$101$		−	8.77555i		−	0.873200i	0.899656	−	0.436600i	$-0.143817\pi$
$102$		0			0
$103$	−0.710448	−	0.710448i	−0.0700025	−	0.0700025i	0.671239	−	0.741241i	$-0.265763\pi$
$103$	−0.710448	−	0.710448i	−0.0700025	−	0.0700025i	−0.741241	+	0.671239i	$0.765763\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.2829	+	13.2829i	1.28411	+	1.28411i	0.938310	+	0.345796i	$0.112391\pi$
$107$	13.2829	+	13.2829i	1.28411	+	1.28411i	0.345796	+	0.938310i	$0.387609\pi$
$108$		0			0
$109$			7.13731i			0.683630i	0.939767	+	0.341815i	$0.111042\pi$
$109$			7.13731i			0.683630i	−0.939767	+	0.341815i	$0.888958\pi$
$110$		0			0
$111$			4.69267i			0.445409i
$112$		0			0
$113$	−4.35947	+	4.35947i	−0.410104	+	0.410104i	−0.881775	−	0.471671i	$-0.843651\pi$
$113$	−4.35947	+	4.35947i	−0.410104	+	0.410104i	0.471671	+	0.881775i	$0.343651\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.29100	−	1.29100i	0.119353	−	0.119353i
$118$		0			0
$119$	−0.890030	−	1.92931i	−0.0815889	−	0.176860i
$120$		0			0
$121$	−9.51992			−0.865448
$122$		0			0
$123$	6.52240	−	6.52240i	0.588105	−	0.588105i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.07395	−	3.07395i	−0.272769	−	0.272769i	0.557445	−	0.830214i	$-0.311782\pi$
$127$	−3.07395	−	3.07395i	−0.272769	−	0.272769i	−0.830214	+	0.557445i	$0.811782\pi$
$128$		0			0
$129$	−5.74786			−0.506070
$130$		0			0
$131$			6.12826i			0.535429i	0.963498	+	0.267714i	$0.0862684\pi$
$131$			6.12826i			0.535429i	−0.963498	+	0.267714i	$0.913732\pi$
$132$		0			0
$133$	0.0947017	−	0.256905i	0.00821168	−	0.0222765i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.69973	−	2.69973i	−0.230654	−	0.230654i	0.582312	−	0.812965i	$-0.302148\pi$
$137$	−2.69973	−	2.69973i	−0.230654	−	0.230654i	−0.812965	+	0.582312i	$0.802148\pi$
$138$		0			0
$139$	8.95480			0.759536			0.379768	−	0.925082i	$-0.376004\pi$
$139$	8.95480			0.759536			0.379768	+	0.925082i	$0.376004\pi$
$140$		0			0
$141$	13.3592			1.12504
$142$		0			0
$143$	1.21203	+	1.21203i	0.101355	+	0.101355i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.721680	−	9.10945i	−0.0595232	−	0.751335i
$148$		0			0
$149$			9.15227i			0.749783i	0.927069	+	0.374891i	$0.122320\pi$
$149$			9.15227i			0.749783i	−0.927069	+	0.374891i	$0.877680\pi$
$150$		0			0
$151$	−10.4406			−0.849648			−0.424824	−	0.905276i	$-0.639664\pi$
$151$	−10.4406			−0.849648			−0.424824	+	0.905276i	$0.639664\pi$
$152$		0			0
$153$	−0.735857	−	0.735857i	−0.0594905	−	0.0594905i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.3298	+	13.3298i	−1.06384	+	1.06384i	−0.0660171	+	0.997818i	$0.521029\pi$
$157$	−13.3298	+	13.3298i	−1.06384	+	1.06384i	−0.997818	+	0.0660171i	$0.978971\pi$
$158$		0			0
$159$	−1.62854			−0.129152
$160$		0			0
$161$	2.24752	+	4.87192i	0.177129	+	0.383961i
$162$		0			0
$163$	−3.31503	+	3.31503i	−0.259653	+	0.259653i	−0.824913	−	0.565260i	$-0.808776\pi$
$163$	−3.31503	+	3.31503i	−0.259653	+	0.259653i	0.565260	+	0.824913i	$0.308776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.75278	−	3.75278i	0.290399	−	0.290399i	−0.546839	−	0.837238i	$-0.684169\pi$
$167$	3.75278	−	3.75278i	0.290399	−	0.290399i	0.837238	+	0.546839i	$0.184169\pi$
$168$		0			0
$169$		−	11.0150i		−	0.847304i
$170$		0			0
$171$		−	0.134106i		−	0.0102554i
$172$		0			0
$173$	−13.8850	−	13.8850i	−1.05565	−	1.05565i	−0.998357	−	0.0572975i	$-0.981752\pi$
$173$	−13.8850	−	13.8850i	−1.05565	−	1.05565i	−0.0572975	−	0.998357i	$-0.518248\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.64846	+	7.64846i	0.574894	+	0.574894i
$178$		0			0
$179$		−	9.64006i		−	0.720532i	−0.932850	−	0.360266i	$-0.882686\pi$
$179$		−	9.64006i		−	0.720532i	0.932850	−	0.360266i	$-0.117314\pi$
$180$		0			0
$181$		−	15.6258i		−	1.16146i	−0.814097	−	0.580728i	$-0.802768\pi$
$181$		−	15.6258i		−	1.16146i	0.814097	−	0.580728i	$-0.197232\pi$
$182$		0			0
$183$	9.28893	−	9.28893i	0.686657	−	0.686657i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.690840	−	0.690840i	0.0505192	−	0.0505192i
$188$		0			0
$189$	−6.21523	−	13.4727i	−0.452091	−	0.979994i
$190$		0			0
$191$	−18.4481			−1.33486			−0.667430	−	0.744673i	$-0.732606\pi$
$191$	−18.4481			−1.33486			−0.667430	+	0.744673i	$0.732606\pi$
$192$		0			0
$193$	−15.3732	+	15.3732i	−1.10659	+	1.10659i	−0.112990	+	0.993596i	$0.536043\pi$
$193$	−15.3732	+	15.3732i	−1.10659	+	1.10659i	−0.993596	+	0.112990i	$0.963957\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.7687	−	15.7687i	−1.12347	−	1.12347i	−0.991215	−	0.132258i	$-0.957777\pi$
$197$	−15.7687	−	15.7687i	−1.12347	−	1.12347i	−0.132258	−	0.991215i	$-0.542223\pi$
$198$		0			0
$199$	24.6117			1.74468			0.872339	−	0.488901i	$-0.162602\pi$
$199$	24.6117			1.74468			0.872339	+	0.488901i	$0.162602\pi$
$200$		0			0
$201$			13.0601i			0.921188i
$202$		0			0
$203$	−12.3541	−	4.55402i	−0.867085	−	0.319629i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.85820	+	1.85820i	0.129154	+	0.129154i
$208$		0			0
$209$	0.125902			0.00870883
$210$		0			0
$211$	7.49647			0.516078			0.258039	−	0.966134i	$-0.416924\pi$
$211$	7.49647			0.516078			0.258039	+	0.966134i	$0.416924\pi$
$212$		0			0
$213$	0.303697	+	0.303697i	0.0208090	+	0.0208090i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−10.8026	−	3.98213i	−0.733331	−	0.270324i
$218$		0			0
$219$			20.9666i			1.41679i
$220$		0			0
$221$	1.13145			0.0761096
$222$		0			0
$223$	7.88619	+	7.88619i	0.528098	+	0.528098i	0.920005	−	0.391907i	$-0.128184\pi$
$223$	7.88619	+	7.88619i	0.528098	+	0.528098i	−0.391907	+	0.920005i	$0.628184\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.54162	+	6.54162i	−0.434182	+	0.434182i	−0.890048	−	0.455866i	$-0.849330\pi$
$227$	−6.54162	+	6.54162i	−0.434182	+	0.434182i	0.455866	+	0.890048i	$0.349330\pi$
$228$		0			0
$229$	6.01657			0.397586			0.198793	−	0.980041i	$-0.436298\pi$
$229$	6.01657			0.397586			0.198793	+	0.980041i	$0.436298\pi$
$230$		0			0
$231$	3.81545	−	1.76015i	0.251039	−	0.115809i
$232$		0			0
$233$	8.74138	−	8.74138i	0.572667	−	0.572667i	−0.360206	−	0.932873i	$-0.617294\pi$
$233$	8.74138	−	8.74138i	0.572667	−	0.572667i	0.932873	+	0.360206i	$0.117294\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.6352	+	12.6352i	−0.820746	+	0.820746i
$238$		0			0
$239$		−	23.0398i		−	1.49032i	−0.666884	−	0.745162i	$-0.732372\pi$
$239$		−	23.0398i		−	1.49032i	0.666884	−	0.745162i	$-0.267628\pi$
$240$		0			0
$241$			22.8987i			1.47504i	0.675328	+	0.737518i	$0.264002\pi$
$241$			22.8987i			1.47504i	−0.675328	+	0.737518i	$0.735998\pi$
$242$		0			0
$243$	−8.72714	−	8.72714i	−0.559847	−	0.559847i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.103101	+	0.103101i	0.00656013	+	0.00656013i
$248$		0			0
$249$			8.52070i			0.539978i
$250$		0			0
$251$			24.5447i			1.54925i	0.632423	+	0.774623i	$0.282060\pi$
$251$			24.5447i			1.54925i	−0.632423	+	0.774623i	$0.717940\pi$
$252$		0			0
$253$	−1.74452	+	1.74452i	−0.109677	+	0.109677i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.22931	+	7.22931i	−0.450952	+	0.450952i	−0.895670	−	0.444718i	$-0.853304\pi$
$257$	−7.22931	+	7.22931i	−0.450952	+	0.450952i	0.444718	+	0.895670i	$0.353304\pi$
$258$		0			0
$259$	8.63613	−	3.98402i	0.536623	−	0.247555i
$260$		0			0
$261$	−6.44890			−0.399177
$262$		0			0
$263$	12.0746	−	12.0746i	0.744550	−	0.744550i	−0.228900	−	0.973450i	$-0.573513\pi$
$263$	12.0746	−	12.0746i	0.744550	−	0.744550i	0.973450	+	0.228900i	$0.0735130\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−12.1793	−	12.1793i	−0.745358	−	0.745358i
$268$		0			0
$269$	0.430862			0.0262701			0.0131351	−	0.999914i	$-0.495819\pi$
$269$	0.430862			0.0262701			0.0131351	+	0.999914i	$0.495819\pi$
$270$		0			0
$271$			10.1612i			0.617248i	0.951184	+	0.308624i	$0.0998685\pi$
$271$			10.1612i			0.617248i	−0.951184	+	0.308624i	$0.900132\pi$
$272$		0			0
$273$	4.56583	+	1.68308i	0.276336	+	0.101865i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.9353	−	11.9353i	−0.717123	−	0.717123i	0.250892	−	0.968015i	$-0.419276\pi$
$277$	−11.9353	−	11.9353i	−0.717123	−	0.717123i	−0.968015	+	0.250892i	$0.919276\pi$
$278$		0			0
$279$	−5.63905			−0.337601
$280$		0			0
$281$	1.47379			0.0879191			0.0439596	−	0.999033i	$-0.486003\pi$
$281$	1.47379			0.0879191			0.0439596	+	0.999033i	$0.486003\pi$
$282$		0			0
$283$	−2.68335	−	2.68335i	−0.159508	−	0.159508i	0.622840	−	0.782349i	$-0.285979\pi$
$283$	−2.68335	−	2.68335i	−0.159508	−	0.159508i	−0.782349	+	0.622840i	$0.785979\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.5409	−	6.46601i	−1.03541	−	0.381677i
$288$		0			0
$289$			16.3551i			0.962064i
$290$		0			0
$291$	−4.43317			−0.259877
$292$		0			0
$293$	−20.8195	−	20.8195i	−1.21629	−	1.21629i	−0.968921	−	0.247368i	$-0.920434\pi$
$293$	−20.8195	−	20.8195i	−1.21629	−	1.21629i	−0.247368	−	0.968921i	$-0.579566\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.82425	−	4.82425i	0.279931	−	0.279931i
$298$		0			0
$299$	−2.85715			−0.165234
$300$		0			0
$301$	4.87986	+	10.5780i	0.281271	+	0.609707i
$302$		0			0
$303$	8.10050	−	8.10050i	0.465362	−	0.465362i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.0501	+	10.0501i	−0.573588	+	0.573588i	−0.933129	−	0.359541i	$-0.882933\pi$
$307$	−10.0501	+	10.0501i	−0.573588	+	0.573588i	0.359541	+	0.933129i	$0.382933\pi$
$308$		0			0
$309$		−	1.31160i		−	0.0746141i
$310$		0			0
$311$			6.48676i			0.367830i	0.982942	+	0.183915i	$0.0588771\pi$
$311$			6.48676i			0.367830i	−0.982942	+	0.183915i	$0.941123\pi$
$312$		0			0
$313$	11.7441	+	11.7441i	0.663814	+	0.663814i	0.956277	−	0.292463i	$-0.0944748\pi$
$313$	11.7441	+	11.7441i	0.663814	+	0.663814i	−0.292463	+	0.956277i	$0.594475\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.1787	+	18.1787i	1.02102	+	1.02102i	0.999774	+	0.0212445i	$0.00676284\pi$
$317$	18.1787	+	18.1787i	1.02102	+	1.02102i	0.0212445	+	0.999774i	$0.493237\pi$
$318$		0			0
$319$		−	6.05439i		−	0.338981i
$320$		0			0
$321$			24.5223i			1.36870i
$322$		0			0
$323$	0.0587660	−	0.0587660i	0.00326983	−	0.00326983i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.58828	+	6.58828i	−0.364333	+	0.364333i
$328$		0			0
$329$	−11.3418	−	24.5854i	−0.625292	−	1.35544i
$330$		0			0
$331$	−2.57796			−0.141697			−0.0708486	−	0.997487i	$-0.522571\pi$
$331$	−2.57796			−0.141697			−0.0708486	+	0.997487i	$0.522571\pi$
$332$		0			0
$333$	3.29390	−	3.29390i	0.180505	−	0.180505i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−16.8387	−	16.8387i	−0.917264	−	0.917264i	0.0795652	−	0.996830i	$-0.474647\pi$
$337$	−16.8387	−	16.8387i	−0.917264	−	0.917264i	−0.996830	+	0.0795652i	$0.974647\pi$
$338$		0			0
$339$	−8.04825			−0.437121
$340$		0			0
$341$		−	5.29408i		−	0.286690i
$342$		0			0
$343$	−16.1518	+	9.06195i	−0.872116	+	0.489299i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.2400	+	12.2400i	0.657079	+	0.657079i	0.954688	−	0.297609i	$-0.0961892\pi$
$347$	12.2400	+	12.2400i	0.657079	+	0.657079i	−0.297609	+	0.954688i	$0.596189\pi$
$348$		0			0
$349$	27.3348			1.46320			0.731598	−	0.681736i	$-0.238775\pi$
$349$	27.3348			1.46320			0.731598	+	0.681736i	$0.238775\pi$
$350$		0			0
$351$	7.90110			0.421730
$352$		0			0
$353$	−3.42987	−	3.42987i	−0.182553	−	0.182553i	0.609914	−	0.792468i	$-0.291204\pi$
$353$	−3.42987	−	3.42987i	−0.182553	−	0.182553i	−0.792468	+	0.609914i	$0.791204\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.959334	−	2.60247i	0.0507734	−	0.137737i
$358$		0			0
$359$			30.4578i			1.60750i	0.594966	+	0.803751i	$0.297166\pi$
$359$			30.4578i			1.60750i	−0.594966	+	0.803751i	$0.702834\pi$
$360$		0			0
$361$	−18.9893			−0.999436
$362$		0			0
$363$	−8.78762	−	8.78762i	−0.461230	−	0.461230i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−17.6219	+	17.6219i	−0.919856	+	0.919856i	−0.997019	−	0.0771629i	$-0.975414\pi$
$367$	−17.6219	+	17.6219i	−0.919856	+	0.919856i	0.0771629	+	0.997019i	$0.475414\pi$
$368$		0			0
$369$	−9.15646			−0.476666
$370$		0			0
$371$	1.38261	+	2.99708i	0.0717817	+	0.155601i
$372$		0			0
$373$	13.1403	−	13.1403i	0.680379	−	0.680379i	−0.279707	−	0.960086i	$-0.590237\pi$
$373$	13.1403	−	13.1403i	0.680379	−	0.680379i	0.960086	+	0.279707i	$0.0902372\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.95790	−	4.95790i	0.255345	−	0.255345i
$378$		0			0
$379$		−	9.88955i		−	0.507992i	−0.967205	−	0.253996i	$-0.918255\pi$
$379$		−	9.88955i		−	0.507992i	0.967205	−	0.253996i	$-0.0817450\pi$
$380$		0			0
$381$		−	5.67499i		−	0.290738i
$382$		0			0
$383$	1.61392	+	1.61392i	0.0824673	+	0.0824673i	0.747137	−	0.664670i	$-0.231428\pi$
$383$	1.61392	+	1.61392i	0.0824673	+	0.0824673i	−0.664670	+	0.747137i	$0.731428\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.03456	+	4.03456i	0.205088	+	0.205088i
$388$		0			0
$389$			33.9003i			1.71881i	0.511292	+	0.859407i	$0.329167\pi$
$389$			33.9003i			1.71881i	−0.511292	+	0.859407i	$0.670833\pi$
$390$		0			0
$391$			1.62854i			0.0823590i
$392$		0			0
$393$	−5.65685	+	5.65685i	−0.285351	+	0.285351i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.67961	+	4.67961i	−0.234863	+	0.234863i	−0.814719	−	0.579856i	$-0.803109\pi$
$397$	−4.67961	+	4.67961i	−0.234863	+	0.234863i	0.579856	+	0.814719i	$0.303109\pi$
$398$		0			0
$399$	0.324560	−	0.149726i	0.0162483	−	0.00749569i
$400$		0			0
$401$	−31.2427			−1.56018			−0.780092	−	0.625665i	$-0.784828\pi$
$401$	−31.2427			−1.56018			−0.780092	+	0.625665i	$0.784828\pi$
$402$		0			0
$403$	4.33529	−	4.33529i	0.215956	−	0.215956i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.09239	+	3.09239i	0.153284	+	0.153284i
$408$		0			0
$409$	15.5228			0.767554			0.383777	−	0.923426i	$-0.374623\pi$
$409$	15.5228			0.767554			0.383777	+	0.923426i	$0.374623\pi$
$410$		0			0
$411$		−	4.98411i		−	0.245848i
$412$		0			0
$413$	7.58234	−	20.5693i	0.373103	−	1.01215i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8.26596	+	8.26596i	0.404786	+	0.404786i
$418$		0			0
$419$	−29.0634			−1.41984			−0.709920	−	0.704282i	$-0.751269\pi$
$419$	−29.0634			−1.41984			−0.709920	+	0.704282i	$0.751269\pi$
$420$		0			0
$421$	2.76601			0.134807			0.0674034	−	0.997726i	$-0.478529\pi$
$421$	2.76601			0.134807			0.0674034	+	0.997726i	$0.478529\pi$
$422$		0			0
$423$	−9.37712	−	9.37712i	−0.455931	−	0.455931i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.9810	−	9.20863i	−1.20892	−	0.445637i
$428$		0			0
$429$			2.23759i			0.108032i
$430$		0			0
$431$	−6.31931			−0.304391			−0.152195	−	0.988350i	$-0.548634\pi$
$431$	−6.31931			−0.304391			−0.152195	+	0.988350i	$0.548634\pi$
$432$		0			0
$433$	12.4583	+	12.4583i	0.598706	+	0.598706i	0.939968	−	0.341262i	$-0.110854\pi$
$433$	12.4583	+	12.4583i	0.598706	+	0.598706i	−0.341262	+	0.939968i	$0.610854\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.148397	+	0.148397i	−0.00709878	+	0.00709878i
$438$		0			0
$439$	16.9890			0.810842			0.405421	−	0.914130i	$-0.367125\pi$
$439$	16.9890			0.810842			0.405421	+	0.914130i	$0.367125\pi$
$440$		0			0
$441$	−5.88758	+	6.90071i	−0.280361	+	0.328605i
$442$		0			0
$443$	−16.8935	+	16.8935i	−0.802633	+	0.802633i	−0.983506	−	0.180873i	$-0.942108\pi$
$443$	−16.8935	+	16.8935i	−0.802633	+	0.802633i	0.180873	+	0.983506i	$0.442108\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−8.44824	+	8.44824i	−0.399588	+	0.399588i
$448$		0			0
$449$		−	4.85641i		−	0.229188i	−0.993412	−	0.114594i	$-0.963443\pi$
$449$		−	4.85641i		−	0.229188i	0.993412	−	0.114594i	$-0.0365567\pi$
$450$		0			0
$451$		−	8.59630i		−	0.404784i
$452$		0			0
$453$	−9.63752	−	9.63752i	−0.452810	−	0.452810i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	27.2995	+	27.2995i	1.27702	+	1.27702i	0.942330	+	0.334686i	$0.108630\pi$
$457$	27.2995	+	27.2995i	1.27702	+	1.27702i	0.334686	+	0.942330i	$0.391370\pi$
$458$		0			0
$459$		−	4.50353i		−	0.210207i
$460$		0			0
$461$		−	33.0098i		−	1.53742i	−0.639599	−	0.768709i	$-0.720900\pi$
$461$		−	33.0098i		−	1.53742i	0.639599	−	0.768709i	$-0.279100\pi$
$462$		0			0
$463$	−11.5774	+	11.5774i	−0.538048	+	0.538048i	−0.922955	−	0.384908i	$-0.874233\pi$
$463$	−11.5774	+	11.5774i	−0.538048	+	0.538048i	0.384908	+	0.922955i	$0.374233\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.90541	+	7.90541i	−0.365819	+	0.365819i	−0.865950	−	0.500131i	$-0.833285\pi$
$467$	−7.90541	+	7.90541i	−0.365819	+	0.365819i	0.500131	+	0.865950i	$0.333285\pi$
$468$		0			0
$469$	24.0350	−	11.0879i	1.10984	−	0.511990i
$470$		0			0
$471$	−24.6089			−1.13392
$472$		0			0
$473$	−3.78774	+	3.78774i	−0.174161	+	0.174161i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.14311	+	1.14311i	0.0523396	+	0.0523396i
$478$		0			0
$479$	−39.6277			−1.81064			−0.905319	−	0.424733i	$-0.860368\pi$
$479$	−39.6277			−1.81064			−0.905319	+	0.424733i	$0.860368\pi$
$480$		0			0
$481$			5.06469i			0.230930i
$482$		0			0
$483$	−2.42253	+	6.57179i	−0.110229	+	0.299027i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	7.44241	+	7.44241i	0.337248	+	0.337248i	0.855331	−	0.518083i	$-0.173354\pi$
$487$	7.44241	+	7.44241i	0.337248	+	0.337248i	−0.518083	+	0.855331i	$0.673354\pi$
$488$		0			0
$489$	−6.12006			−0.276759
$490$		0			0
$491$	−0.423477			−0.0191113			−0.00955563	−	0.999954i	$-0.503042\pi$
$491$	−0.423477			−0.0191113			−0.00955563	+	0.999954i	$0.503042\pi$
$492$		0			0
$493$	−2.82594	−	2.82594i	−0.127274	−	0.127274i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.301072	−	0.816742i	0.0135049	−	0.0366359i
$498$		0			0
$499$			42.5173i			1.90334i	0.307130	+	0.951668i	$0.400631\pi$
$499$			42.5173i			1.90334i	−0.307130	+	0.951668i	$0.599369\pi$
$500$		0			0
$501$	6.92820			0.309529
$502$		0			0
$503$	6.94351	+	6.94351i	0.309596	+	0.309596i	0.844753	−	0.535157i	$-0.179748\pi$
$503$	6.94351	+	6.94351i	0.309596	+	0.309596i	−0.535157	+	0.844753i	$0.679748\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	10.1676	−	10.1676i	0.451561	−	0.451561i
$508$		0			0
$509$	38.8912			1.72382			0.861911	−	0.507059i	$-0.169267\pi$
$509$	38.8912			1.72382			0.861911	+	0.507059i	$0.169267\pi$
$510$		0			0
$511$	38.5858	−	17.8004i	1.70693	−	0.787444i
$512$		0			0
$513$	0.410373	−	0.410373i	0.0181184	−	0.0181184i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	8.80347	−	8.80347i	0.387176	−	0.387176i
$518$		0			0
$519$		−	25.6338i		−	1.12520i
$520$		0			0
$521$			31.1879i			1.36637i	0.730246	+	0.683184i	$0.239405\pi$
$521$			31.1879i			1.36637i	−0.730246	+	0.683184i	$0.760595\pi$
$522$		0			0
$523$	19.0273	+	19.0273i	0.832007	+	0.832007i	0.987791	−	0.155784i	$-0.0497903\pi$
$523$	19.0273	+	19.0273i	0.832007	+	0.832007i	−0.155784	+	0.987791i	$0.549790\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.47106	−	2.47106i	−0.107641	−	0.107641i
$528$		0			0
$529$			18.8876i			0.821199i
$530$		0			0
$531$		−	10.7373i		−	0.465959i
$532$		0			0
$533$	7.03946	−	7.03946i	0.304913	−	0.304913i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	8.89851	−	8.89851i	0.383999	−	0.383999i
$538$		0			0
$539$	−6.47855	−	5.52740i	−0.279051	−	0.238082i
$540$		0			0
$541$	2.48151			0.106688			0.0533442	−	0.998576i	$-0.483012\pi$
$541$	2.48151			0.106688			0.0533442	+	0.998576i	$0.483012\pi$
$542$		0			0
$543$	14.4238	−	14.4238i	0.618985	−	0.618985i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.6132	−	20.6132i	−0.881356	−	0.881356i	0.112316	−	0.993673i	$-0.464173\pi$
$547$	−20.6132	−	20.6132i	−0.881356	−	0.881356i	−0.993673	+	0.112316i	$0.964173\pi$
$548$		0			0
$549$	−13.0403			−0.556544
$550$		0			0
$551$		−	0.515014i		−	0.0219403i
$552$		0			0
$553$	33.9803	+	12.5260i	1.44499	+	0.532659i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.10076	+	5.10076i	0.216126	+	0.216126i	0.806864	−	0.590738i	$-0.201163\pi$
$557$	5.10076	+	5.10076i	0.216126	+	0.216126i	−0.590738	+	0.806864i	$0.701163\pi$
$558$		0			0
$559$	−6.20352			−0.262381
$560$		0			0
$561$	1.27540			0.0538473
$562$		0			0
$563$	−7.13112	−	7.13112i	−0.300541	−	0.300541i	0.540685	−	0.841225i	$-0.318165\pi$
$563$	−7.13112	−	7.13112i	−0.300541	−	0.300541i	−0.841225	+	0.540685i	$0.818165\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.14168	−	8.52270i	0.131938	−	0.357920i
$568$		0			0
$569$		−	8.16848i		−	0.342441i	−0.985233	−	0.171220i	$-0.945229\pi$
$569$		−	8.16848i		−	0.342441i	0.985233	−	0.171220i	$-0.0547710\pi$
$570$		0			0
$571$	−6.66328			−0.278849			−0.139425	−	0.990233i	$-0.544525\pi$
$571$	−6.66328			−0.278849			−0.139425	+	0.990233i	$0.544525\pi$
$572$		0			0
$573$	−17.0290	−	17.0290i	−0.711398	−	0.711398i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	22.2531	−	22.2531i	0.926410	−	0.926410i	−0.0710620	−	0.997472i	$-0.522639\pi$
$577$	22.2531	−	22.2531i	0.926410	−	0.926410i	0.997472	+	0.0710620i	$0.0226388\pi$
$578$		0			0
$579$	−28.3812			−1.17948
$580$		0			0
$581$	15.6810	−	7.23397i	0.650558	−	0.300116i
$582$		0			0
$583$	−1.07318	+	1.07318i	−0.0444467	+	0.0444467i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.6776	−	16.6776i	0.688357	−	0.688357i	−0.273512	−	0.961869i	$-0.588185\pi$
$587$	16.6776	−	16.6776i	0.688357	−	0.688357i	0.961869	+	0.273512i	$0.0881853\pi$
$588$		0			0
$589$		−	0.450338i		−	0.0185559i
$590$		0			0
$591$		−	29.1114i		−	1.19748i
$592$		0			0
$593$	6.56327	+	6.56327i	0.269521	+	0.269521i	0.828907	−	0.559386i	$-0.188963\pi$
$593$	6.56327	+	6.56327i	0.269521	+	0.269521i	−0.559386	+	0.828907i	$0.688963\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	22.7185	+	22.7185i	0.929806	+	0.929806i
$598$		0			0
$599$		−	9.55935i		−	0.390585i	−0.980745	−	0.195292i	$-0.937434\pi$
$599$		−	9.55935i		−	0.390585i	0.980745	−	0.195292i	$-0.0625655\pi$
$600$		0			0
$601$			20.1897i			0.823555i	0.911285	+	0.411777i	$0.135092\pi$
$601$			20.1897i			0.823555i	−0.911285	+	0.411777i	$0.864908\pi$
$602$		0			0
$603$	9.16719	−	9.16719i	0.373317	−	0.373317i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−9.55854	+	9.55854i	−0.387969	+	0.387969i	−0.873962	−	0.485993i	$-0.838458\pi$
$607$	−9.55854	+	9.55854i	−0.387969	+	0.387969i	0.485993	+	0.873962i	$0.338458\pi$
$608$		0			0
$609$	−7.20004	−	15.6075i	−0.291760	−	0.632446i
$610$		0			0
$611$	14.4182			0.583298
$612$		0			0
$613$	−27.0641	+	27.0641i	−1.09311	+	1.09311i	−0.0979135	+	0.995195i	$0.531217\pi$
$613$	−27.0641	+	27.0641i	−1.09311	+	1.09311i	−0.995195	+	0.0979135i	$0.968783\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.331871	−	0.331871i	−0.0133606	−	0.0133606i	0.700395	−	0.713756i	$-0.253007\pi$
$617$	−0.331871	−	0.331871i	−0.0133606	−	0.0133606i	−0.713756	+	0.700395i	$0.753007\pi$
$618$		0			0
$619$	−2.99126			−0.120229			−0.0601145	−	0.998191i	$-0.519147\pi$
$619$	−2.99126			−0.120229			−0.0601145	+	0.998191i	$0.519147\pi$
$620$		0			0
$621$			11.3724i			0.456358i
$622$		0			0
$623$	−12.0740	+	32.7540i	−0.483733	+	1.31226i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	0.116217	+	0.116217i	0.00464127	+	0.00464127i
$628$		0			0
$629$	2.88681			0.115105
$630$		0			0
$631$	28.0573			1.11694			0.558471	−	0.829524i	$-0.311388\pi$
$631$	28.0573			1.11694			0.558471	+	0.829524i	$0.311388\pi$
$632$		0			0
$633$	6.91981	+	6.91981i	0.275038	+	0.275038i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.778891	−	9.83160i	−0.0308608	−	0.389542i
$638$		0			0
$639$		−	0.426345i		−	0.0168659i
$640$		0			0
$641$	26.1153			1.03149			0.515746	−	0.856742i	$-0.327515\pi$
$641$	26.1153			1.03149			0.515746	+	0.856742i	$0.327515\pi$
$642$		0			0
$643$	20.4107	+	20.4107i	0.804921	+	0.804921i	0.983860	−	0.178939i	$-0.0572666\pi$
$643$	20.4107	+	20.4107i	0.804921	+	0.804921i	−0.178939	+	0.983860i	$0.557267\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.2719	−	18.2719i	0.718342	−	0.718342i	−0.249924	−	0.968265i	$-0.580406\pi$
$647$	18.2719	−	18.2719i	0.718342	−	0.718342i	0.968265	+	0.249924i	$0.0804056\pi$
$648$		0			0
$649$	10.0804			0.395691
$650$		0			0
$651$	−6.29586	−	13.6475i	−0.246754	−	0.534887i
$652$		0			0
$653$	24.0558	−	24.0558i	0.941376	−	0.941376i	−0.0569981	−	0.998374i	$-0.518153\pi$
$653$	24.0558	−	24.0558i	0.941376	−	0.941376i	0.998374	+	0.0569981i	$0.0181529\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	14.7170	−	14.7170i	0.574164	−	0.574164i
$658$		0			0
$659$		−	30.6949i		−	1.19570i	−0.801607	−	0.597851i	$-0.796021\pi$
$659$		−	30.6949i		−	1.19570i	0.801607	−	0.597851i	$-0.203979\pi$
$660$		0			0
$661$		−	30.4577i		−	1.18467i	−0.805692	−	0.592335i	$-0.798206\pi$
$661$		−	30.4577i		−	1.18467i	0.805692	−	0.592335i	$-0.201794\pi$
$662$		0			0
$663$	1.04441	+	1.04441i	0.0405617	+	0.0405617i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.13611	+	7.13611i	0.276311	+	0.276311i
$668$		0			0
$669$			14.5591i			0.562888i
$670$		0			0
$671$		−	12.2425i		−	0.472617i
$672$		0			0
$673$	−24.7437	+	24.7437i	−0.953801	+	0.953801i	−0.998979	−	0.0451778i	$-0.985615\pi$
$673$	−24.7437	+	24.7437i	−0.953801	+	0.953801i	0.0451778	+	0.998979i	$0.485615\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.6461	−	23.6461i	0.908793	−	0.908793i	−0.0873819	−	0.996175i	$-0.527850\pi$
$677$	23.6461	−	23.6461i	0.908793	−	0.908793i	0.996175	+	0.0873819i	$0.0278500\pi$
$678$		0			0
$679$	3.76371	+	8.15855i	0.144438	+	0.313096i
$680$		0			0
$681$	−12.0768			−0.462785
$682$		0			0
$683$	25.8548	−	25.8548i	0.989307	−	0.989307i	−0.0106366	−	0.999943i	$-0.503386\pi$
$683$	25.8548	−	25.8548i	0.989307	−	0.989307i	0.999943	+	0.0106366i	$0.00338580\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	5.55375	+	5.55375i	0.211889	+	0.211889i
$688$		0			0
$689$	−1.75765			−0.0669610
$690$		0			0
$691$		−	40.5649i		−	1.54316i	−0.636131	−	0.771581i	$-0.719466\pi$
$691$		−	40.5649i		−	1.54316i	0.636131	−	0.771581i	$-0.280534\pi$
$692$		0			0
$693$	−3.91365	−	1.44267i	−0.148667	−	0.0548025i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−4.01241	−	4.01241i	−0.151981	−	0.151981i
$698$		0			0
$699$	16.1379			0.610393
$700$		0			0
$701$	12.4481			0.470159			0.235080	−	0.971976i	$-0.424465\pi$
$701$	12.4481			0.470159			0.235080	+	0.971976i	$0.424465\pi$
$702$		0			0
$703$	0.263053	+	0.263053i	0.00992124	+	0.00992124i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.7849	−	8.03048i	−0.819307	−	0.302017i
$708$		0			0
$709$			23.9211i			0.898377i	0.893437	+	0.449189i	$0.148287\pi$
$709$			23.9211i			0.898377i	−0.893437	+	0.449189i	$0.851713\pi$
$710$		0			0
$711$	17.7379			0.665225
$712$		0			0
$713$	6.23996	+	6.23996i	0.233688	+	0.233688i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	21.2675	−	21.2675i	0.794251	−	0.794251i
$718$		0			0
$719$	−32.4190			−1.20902			−0.604512	−	0.796596i	$-0.706632\pi$
$719$	−32.4190			−1.20902			−0.604512	+	0.796596i	$0.706632\pi$
$720$		0			0
$721$	−2.41379	+	1.11353i	−0.0898941	+	0.0414700i
$722$		0			0
$723$	−21.1373	+	21.1373i	−0.786103	+	0.786103i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.14090	−	2.14090i	0.0794017	−	0.0794017i	−0.666291	−	0.745692i	$-0.732119\pi$
$727$	2.14090	−	2.14090i	0.0794017	−	0.0794017i	0.745692	+	0.666291i	$0.232119\pi$
$728$		0			0
$729$		−	26.4111i		−	0.978191i
$730$		0			0
$731$			3.53593i			0.130781i
$732$		0			0
$733$	18.1647	+	18.1647i	0.670930	+	0.670930i	0.957930	−	0.287001i	$-0.0926582\pi$
$733$	18.1647	+	18.1647i	0.670930	+	0.670930i	−0.287001	+	0.957930i	$0.592658\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.60638	+	8.60638i	0.317020	+	0.317020i
$738$		0			0
$739$		−	42.7086i		−	1.57106i	−0.618822	−	0.785531i	$-0.712390\pi$
$739$		−	42.7086i		−	1.57106i	0.618822	−	0.785531i	$-0.287610\pi$
$740$		0			0
$741$			0.190339i			0.00699229i
$742$		0			0
$743$	−15.2579	+	15.2579i	−0.559759	+	0.559759i	−0.929239	−	0.369480i	$-0.879536\pi$
$743$	−15.2579	+	15.2579i	−0.559759	+	0.559759i	0.369480	+	0.929239i	$0.379536\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.98089	−	5.98089i	0.218829	−	0.218829i
$748$		0			0
$749$	45.1294	−	20.8191i	1.64899	−	0.760713i
$750$		0			0
$751$	8.87908			0.324002			0.162001	−	0.986791i	$-0.448205\pi$
$751$	8.87908			0.324002			0.162001	+	0.986791i	$0.448205\pi$
$752$		0			0
$753$	−22.6566	+	22.6566i	−0.825653	+	0.825653i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−7.15917	−	7.15917i	−0.260204	−	0.260204i	0.564933	−	0.825137i	$-0.308902\pi$
$757$	−7.15917	−	7.15917i	−0.260204	−	0.260204i	−0.825137	+	0.564933i	$0.808902\pi$
$758$		0			0
$759$	−3.22065			−0.116902
$760$		0			0
$761$		−	36.7681i		−	1.33284i	−0.745575	−	0.666422i	$-0.767825\pi$
$761$		−	36.7681i		−	1.33284i	0.745575	−	0.666422i	$-0.232175\pi$
$762$		0			0
$763$	17.7181	+	6.53133i	0.641437	+	0.236450i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.25480	+	8.25480i	0.298064	+	0.298064i
$768$		0			0
$769$	0.372202			0.0134220			0.00671098	−	0.999977i	$-0.497864\pi$
$769$	0.372202			0.0134220			0.00671098	+	0.999977i	$0.497864\pi$
$770$		0			0
$771$	−13.3464			−0.480659
$772$		0			0
$773$	−28.1925	−	28.1925i	−1.01402	−	1.01402i	−0.999900	−	0.0141148i	$-0.995507\pi$
$773$	−28.1925	−	28.1925i	−1.01402	−	1.01402i	−0.0141148	−	0.999900i	$-0.504493\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	11.6494	+	4.29425i	0.417919	+	0.154055i
$778$		0			0
$779$		−	0.731241i		−	0.0261994i
$780$		0			0
$781$	0.400263			0.0143225
$782$		0			0
$783$	−19.7340	−	19.7340i	−0.705237	−	0.705237i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.61707	+	2.61707i	−0.0932886	+	0.0932886i	−0.752211	−	0.658922i	$-0.771013\pi$
$787$	−2.61707	+	2.61707i	−0.0932886	+	0.0932886i	0.658922	+	0.752211i	$0.271013\pi$
$788$		0			0
$789$	22.2915			0.793598
$790$		0			0
$791$	6.83286	+	14.8115i	0.242949	+	0.526637i
$792$		0			0
$793$	10.0253	−	10.0253i	0.356009	−	0.356009i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.2158	−	13.2158i	0.468127	−	0.468127i	−0.433180	−	0.901307i	$-0.642609\pi$
$797$	13.2158	−	13.2158i	0.468127	−	0.468127i	0.901307	+	0.433180i	$0.142609\pi$
$798$		0			0
$799$		−	8.21820i		−	0.290739i
$800$		0			0
$801$			17.0978i			0.604122i
$802$		0			0
$803$	13.8167	+	13.8167i	0.487579	+	0.487579i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.397719	+	0.397719i	0.0140004	+	0.0140004i
$808$		0			0
$809$			35.0904i			1.23371i	0.787076	+	0.616856i	$0.211594\pi$
$809$			35.0904i			1.23371i	−0.787076	+	0.616856i	$0.788406\pi$
$810$		0			0
$811$		−	41.9817i		−	1.47418i	−0.675797	−	0.737088i	$-0.736201\pi$
$811$		−	41.9817i		−	1.47418i	0.675797	−	0.737088i	$-0.263799\pi$
$812$		0			0
$813$	−9.37955	+	9.37955i	−0.328955	+	0.328955i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.322203	+	0.322203i	−0.0112724	+	0.0112724i
$818$		0			0
$819$	−2.02347	−	4.38626i	−0.0707058	−	0.153268i
$820$		0			0
$821$	−6.93042			−0.241873			−0.120937	−	0.992660i	$-0.538590\pi$
$821$	−6.93042			−0.241873			−0.120937	+	0.992660i	$0.538590\pi$
$822$		0			0
$823$	−0.193913	+	0.193913i	−0.00675937	+	0.00675937i	−0.710478	−	0.703719i	$-0.751521\pi$
$823$	−0.193913	+	0.193913i	−0.00675937	+	0.00675937i	0.703719	+	0.710478i	$0.251521\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.94789	−	1.94789i	−0.0677346	−	0.0677346i	0.672428	−	0.740163i	$-0.265251\pi$
$827$	−1.94789	−	1.94789i	−0.0677346	−	0.0677346i	−0.740163	+	0.672428i	$0.765251\pi$
$828$		0			0
$829$	−27.6983			−0.962001			−0.481000	−	0.876720i	$-0.659726\pi$
$829$	−27.6983			−0.962001			−0.481000	+	0.876720i	$0.659726\pi$
$830$		0			0
$831$		−	22.0344i		−	0.764365i
$832$		0			0
$833$	−5.60389	+	0.443959i	−0.194163	+	0.0153823i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−17.2558	−	17.2558i	−0.596449	−	0.596449i
$838$		0			0
$839$	−17.4254			−0.601591			−0.300796	−	0.953689i	$-0.597252\pi$
$839$	−17.4254			−0.601591			−0.300796	+	0.953689i	$0.597252\pi$
$840$		0			0
$841$	4.23399			0.146000
$842$		0			0
$843$	1.36042	+	1.36042i	0.0468555	+	0.0468555i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−8.71165	+	23.6328i	−0.299336	+	0.812033i
$848$		0			0
$849$		−	4.95387i		−	0.170016i
$850$		0			0
$851$	−7.28981			−0.249892
$852$		0			0
$853$	20.2898	+	20.2898i	0.694709	+	0.694709i	0.963264	−	0.268555i	$-0.0865461\pi$
$853$	20.2898	+	20.2898i	0.694709	+	0.694709i	−0.268555	+	0.963264i	$0.586546\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.9133	−	18.9133i	0.646066	−	0.646066i	−0.305974	−	0.952040i	$-0.598982\pi$
$857$	18.9133	−	18.9133i	0.646066	−	0.646066i	0.952040	+	0.305974i	$0.0989821\pi$
$858$		0			0
$859$	47.6031			1.62420			0.812098	−	0.583520i	$-0.198325\pi$
$859$	47.6031			1.62420			0.812098	+	0.583520i	$0.198325\pi$
$860$		0			0
$861$	−10.2230	−	22.1602i	−0.348397	−	0.755218i
$862$		0			0
$863$	−8.74024	+	8.74024i	−0.297521	+	0.297521i	−0.840042	−	0.542521i	$-0.817470\pi$
$863$	−8.74024	+	8.74024i	−0.297521	+	0.297521i	0.542521	+	0.840042i	$0.317470\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.0970	+	15.0970i	−0.512721	+	0.512721i
$868$		0			0
$869$			16.6528i			0.564908i
$870$		0			0
$871$			14.0954i			0.477606i
$872$		0			0
$873$	3.11175	+	3.11175i	0.105317	+	0.105317i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.95163	+	7.95163i	0.268508	+	0.268508i	0.828499	−	0.559991i	$-0.189195\pi$
$877$	7.95163	+	7.95163i	0.268508	+	0.268508i	−0.559991	+	0.828499i	$0.689195\pi$
$878$		0			0
$879$		−	38.4360i		−	1.29642i
$880$		0			0
$881$		−	8.18767i		−	0.275850i	−0.990443	−	0.137925i	$-0.955957\pi$
$881$		−	8.18767i		−	0.275850i	0.990443	−	0.137925i	$-0.0440432\pi$
$882$		0			0
$883$	21.3047	−	21.3047i	0.716961	−	0.716961i	−0.251021	−	0.967982i	$-0.580766\pi$
$883$	21.3047	−	21.3047i	0.716961	−	0.716961i	0.967982	+	0.251021i	$0.0807662\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15.4989	−	15.4989i	0.520402	−	0.520402i	−0.397291	−	0.917693i	$-0.630050\pi$
$887$	15.4989	−	15.4989i	0.520402	−	0.520402i	0.917693	+	0.397291i	$0.130050\pi$
$888$		0			0
$889$	−10.4439	+	4.81799i	−0.350278	+	0.161590i
$890$		0			0
$891$	4.17674			0.139926
$892$		0			0
$893$	0.748863	−	0.748863i	0.0250597	−	0.0250597i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2.63737	−	2.63737i	−0.0880593	−	0.0880593i
$898$		0			0
$899$	−21.6559			−0.722264
$900$		0			0
$901$			1.00184i			0.0333760i
$902$		0			0
$903$	−5.25984	+	14.2688i	−0.175037	+	0.474836i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	26.3564	+	26.3564i	0.875150	+	0.875150i	0.993028	−	0.117878i	$-0.0376091\pi$
$907$	26.3564	+	26.3564i	0.875150	+	0.875150i	−0.117878	+	0.993028i	$0.537609\pi$
$908$		0			0
$909$	−11.3719			−0.377182
$910$		0			0
$911$	41.4208			1.37233			0.686167	−	0.727444i	$-0.259292\pi$
$911$	41.4208			1.37233			0.686167	+	0.727444i	$0.259292\pi$
$912$		0			0
$913$	5.61500	+	5.61500i	0.185829	+	0.185829i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.2132	+	5.60795i	0.502383	+	0.185191i
$918$		0			0
$919$			44.8247i			1.47863i	0.673360	+	0.739315i	$0.264851\pi$
$919$			44.8247i			1.47863i	−0.673360	+	0.739315i	$0.735149\pi$
$920$		0			0
$921$	−18.5540			−0.611374
$922$		0			0
$923$	0.327773	+	0.327773i	0.0107888	+	0.0107888i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.920641	+	0.920641i	−0.0302378	+	0.0302378i
$928$		0			0
$929$	−36.7206			−1.20476			−0.602382	−	0.798208i	$-0.705782\pi$
$929$	−36.7206			−1.20476			−0.602382	+	0.798208i	$0.705782\pi$
$930$		0			0
$931$	−0.551095	−	0.470186i	−0.0180614	−	0.0154097i
$932$		0			0
$933$	−5.98777	+	5.98777i	−0.196031	+	0.196031i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	9.07171	−	9.07171i	0.296360	−	0.296360i	−0.543226	−	0.839586i	$-0.682798\pi$
$937$	9.07171	−	9.07171i	0.296360	−	0.296360i	0.839586	+	0.543226i	$0.182798\pi$
$938$		0			0
$939$			21.6813i			0.707544i
$940$		0			0
$941$		−	28.6996i		−	0.935580i	−0.883840	−	0.467790i	$-0.845050\pi$
$941$		−	28.6996i		−	0.935580i	0.883840	−	0.467790i	$-0.154950\pi$
$942$		0			0
$943$	10.1322	+	10.1322i	0.329950	+	0.329950i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.9306	+	14.9306i	0.485179	+	0.485179i	0.906781	−	0.421602i	$-0.138532\pi$
$947$	14.9306	+	14.9306i	0.485179	+	0.485179i	−0.421602	+	0.906781i	$0.638532\pi$
$948$		0			0
$949$			22.6288i			0.734560i
$950$		0			0
$951$			33.5607i			1.08828i
$952$		0			0
$953$	20.5690	−	20.5690i	0.666296	−	0.666296i	−0.290560	−	0.956857i	$-0.593842\pi$
$953$	20.5690	−	20.5690i	0.666296	−	0.666296i	0.956857	+	0.290560i	$0.0938417\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	5.58866	−	5.58866i	0.180656	−	0.180656i
$958$		0			0
$959$	−9.17248	+	4.23145i	−0.296195	+	0.136641i
$960$		0			0
$961$	12.0637			0.389150
$962$		0			0
$963$	17.2128	−	17.2128i	0.554674	−	0.554674i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	20.2690	+	20.2690i	0.651807	+	0.651807i	0.953428	−	0.301621i	$-0.0975276\pi$
$967$	20.2690	+	20.2690i	0.651807	+	0.651807i	−0.301621	+	0.953428i	$0.597528\pi$
$968$		0			0
$969$	0.108491			0.00348524
$970$		0			0
$971$			58.5227i			1.87808i	0.343803	+	0.939042i	$0.388285\pi$
$971$			58.5227i			1.87808i	−0.343803	+	0.939042i	$0.611715\pi$
$972$		0			0
$973$	8.19450	−	22.2299i	0.262704	−	0.712658i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.0405	−	17.0405i	−0.545174	−	0.545174i	0.379867	−	0.925041i	$-0.375970\pi$
$977$	−17.0405	−	17.0405i	−0.545174	−	0.545174i	−0.925041	+	0.379867i	$0.875970\pi$
$978$		0			0
$979$	−16.0519			−0.513019
$980$		0			0
$981$	9.24895			0.295296
$982$		0			0
$983$	−0.0763218	−	0.0763218i	−0.00243429	−	0.00243429i	0.705889	−	0.708323i	$-0.250548\pi$
$983$	−0.0763218	−	0.0763218i	−0.00243429	−	0.00243429i	−0.708323	+	0.705889i	$0.750548\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	12.2249	−	33.1636i	0.389123	−	1.05561i
$988$		0			0
$989$		−	8.92898i		−	0.283925i
$990$		0			0
$991$	−22.8932			−0.727226			−0.363613	−	0.931550i	$-0.618457\pi$
$991$	−22.8932			−0.727226			−0.363613	+	0.931550i	$0.618457\pi$
$992$		0			0
$993$	−2.37965	−	2.37965i	−0.0755159	−	0.0755159i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−36.3315	+	36.3315i	−1.15063	+	1.15063i	−0.164204	+	0.986426i	$0.552506\pi$
$997$	−36.3315	+	36.3315i	−1.15063	+	1.15063i	−0.986426	+	0.164204i	$0.947494\pi$
$998$		0			0
$999$	20.1591			0.637805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.x.c.657.11	yes	32
5.2	odd	4	inner	1400.2.x.c.993.12	yes	32
5.3	odd	4	inner	1400.2.x.c.993.6	yes	32
5.4	even	2	inner	1400.2.x.c.657.6	yes	32
7.6	odd	2	inner	1400.2.x.c.657.5	✓	32
35.13	even	4	inner	1400.2.x.c.993.11	yes	32
35.27	even	4	inner	1400.2.x.c.993.5	yes	32
35.34	odd	2	inner	1400.2.x.c.657.12	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.x.c.657.5	✓	32	7.6	odd	2	inner
1400.2.x.c.657.6	yes	32	5.4	even	2	inner
1400.2.x.c.657.11	yes	32	1.1	even	1	trivial
1400.2.x.c.657.12	yes	32	35.34	odd	2	inner
1400.2.x.c.993.5	yes	32	35.27	even	4	inner
1400.2.x.c.993.6	yes	32	5.3	odd	4	inner
1400.2.x.c.993.11	yes	32	35.13	even	4	inner
1400.2.x.c.993.12	yes	32	5.2	odd	4	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)

\(f(q)\)	\(=\)	\(q+(0.923076 + 0.923076i) q^{3} +(0.915096 - 2.48246i) q^{7} -1.29586i q^{9} +O(q^{10})\)	\(q+(0.923076 + 0.923076i) q^{3} +(0.915096 - 2.48246i) q^{7} -1.29586i q^{9} +1.21658 q^{11} +(0.996254 + 0.996254i) q^{13} +(0.567852 - 0.567852i) q^{17} +0.103488 q^{19} +(3.13620 - 1.44679i) q^{21} +(-1.43395 + 1.43395i) q^{23} +(3.96541 - 3.96541i) q^{27} -4.97655i q^{29} -4.35159i q^{31} +(1.12300 + 1.12300i) q^{33} +(2.54187 + 2.54187i) q^{37} +1.83924i q^{39} -7.06593i q^{41} +(-3.11342 + 3.11342i) q^{43} +(7.23621 - 7.23621i) q^{47} +(-5.32520 - 4.54338i) q^{49} +1.04834 q^{51} +(-0.882128 + 0.882128i) q^{53} +(0.0955275 + 0.0955275i) q^{57} +8.28584 q^{59} -10.0630i q^{61} +(-3.21692 - 1.18584i) q^{63} +(7.07422 + 7.07422i) q^{67} -2.64729 q^{69} +0.329005 q^{71} +(11.3569 + 11.3569i) q^{73} +(1.11329 - 3.02012i) q^{77} +13.6882i q^{79} +3.43317 q^{81} +(4.61538 + 4.61538i) q^{83} +(4.59373 - 4.59373i) q^{87} -13.1942 q^{89} +(3.38483 - 1.56149i) q^{91} +(4.01685 - 4.01685i) q^{93} +(-2.40130 + 2.40130i) q^{97} -1.57652i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) \) 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

Introduction

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