LMFDB - Embedded newform 6300.2.v.c.1457.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1457,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("6300.1457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.v (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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	6300.1457
Dual form			6300.2.v.c.5993.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{7} -5.41421i q^{11} +(3.41421 - 3.41421i) q^{13} +(2.82843 - 2.82843i) q^{17} -2.82843i q^{19} +(1.82843 + 1.82843i) q^{23} +1.41421 q^{29} -2.82843 q^{31} +(-5.41421 - 5.41421i) q^{37} +1.17157i q^{41} +(6.82843 - 6.82843i) q^{43} +(-6.82843 + 6.82843i) q^{47} +1.00000i q^{49} +(-7.82843 - 7.82843i) q^{53} -4.00000 q^{59} +13.3137 q^{61} +(2.58579 + 2.58579i) q^{67} +8.24264i q^{71} +(-1.75736 + 1.75736i) q^{73} +(-3.82843 + 3.82843i) q^{77} +9.65685i q^{79} +(-1.17157 - 1.17157i) q^{83} +10.8284 q^{89} -4.82843 q^{91} +(-6.24264 - 6.24264i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{13} - 4 q^{23} - 16 q^{37} + 16 q^{43} - 16 q^{47} - 20 q^{53} - 16 q^{59} + 8 q^{61} + 16 q^{67} - 24 q^{73} - 4 q^{77} - 16 q^{83} + 32 q^{89} - 8 q^{91} - 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^13 - 4 * q^23 - 16 * q^37 + 16 * q^43 - 16 * q^47 - 20 * q^53 - 16 * q^59 + 8 * q^61 + 16 * q^67 - 24 * q^73 - 4 * q^77 - 16 * q^83 + 32 * q^89 - 8 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.707107	−	0.707107i	−0.267261	−	0.267261i
$7$	−0.707107	−	0.707107i	−0.267261	−	0.267261i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.41421i		−	1.63245i	−0.577736	−	0.816223i	$-0.696064\pi$
$11$		−	5.41421i		−	1.63245i	0.577736	−	0.816223i	$-0.303936\pi$
$12$		0			0
$13$	3.41421	−	3.41421i	0.946932	−	0.946932i	−0.0517287	−	0.998661i	$-0.516473\pi$
$13$	3.41421	−	3.41421i	0.946932	−	0.946932i	0.998661	+	0.0517287i	$0.0164731\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843	−	2.82843i	0.685994	−	0.685994i	−0.275350	−	0.961344i	$-0.588794\pi$
$17$	2.82843	−	2.82843i	0.685994	−	0.685994i	0.961344	+	0.275350i	$0.0887937\pi$
$18$		0			0
$19$		−	2.82843i		−	0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$		−	2.82843i		−	0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.82843	+	1.82843i	0.381253	+	0.381253i	0.871554	−	0.490300i	$-0.163113\pi$
$23$	1.82843	+	1.82843i	0.381253	+	0.381253i	−0.490300	+	0.871554i	$0.663113\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.41421			0.262613			0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421			0.262613			0.131306	+	0.991342i	$0.458083\pi$
$30$		0			0
$31$	−2.82843			−0.508001			−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843			−0.508001			−0.254000	+	0.967204i	$0.581746\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.41421	−	5.41421i	−0.890091	−	0.890091i	0.104440	−	0.994531i	$-0.466695\pi$
$37$	−5.41421	−	5.41421i	−0.890091	−	0.890091i	−0.994531	+	0.104440i	$0.966695\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.17157i			0.182969i	0.995807	+	0.0914845i	$0.0291612\pi$
$41$			1.17157i			0.182969i	−0.995807	+	0.0914845i	$0.970839\pi$
$42$		0			0
$43$	6.82843	−	6.82843i	1.04133	−	1.04133i	0.0422169	−	0.999108i	$-0.486558\pi$
$43$	6.82843	−	6.82843i	1.04133	−	1.04133i	0.999108	−	0.0422169i	$-0.0134421\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.82843	+	6.82843i	−0.996028	+	0.996028i	−0.999992	−	0.00396368i	$-0.998738\pi$
$47$	−6.82843	+	6.82843i	−0.996028	+	0.996028i	0.00396368	+	0.999992i	$0.498738\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.82843	−	7.82843i	−1.07532	−	1.07532i	−0.996922	−	0.0783948i	$-0.975021\pi$
$53$	−7.82843	−	7.82843i	−1.07532	−	1.07532i	−0.0783948	−	0.996922i	$-0.524979\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	13.3137			1.70465			0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137			1.70465			0.852323	+	0.523016i	$0.175193\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.58579	+	2.58579i	0.315904	+	0.315904i	0.847192	−	0.531287i	$-0.178292\pi$
$67$	2.58579	+	2.58579i	0.315904	+	0.315904i	−0.531287	+	0.847192i	$0.678292\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.24264i			0.978221i	0.872222	+	0.489111i	$0.162679\pi$
$71$			8.24264i			0.978221i	−0.872222	+	0.489111i	$0.837321\pi$
$72$		0			0
$73$	−1.75736	+	1.75736i	−0.205683	+	0.205683i	−0.802430	−	0.596746i	$-0.796460\pi$
$73$	−1.75736	+	1.75736i	−0.205683	+	0.205683i	0.596746	+	0.802430i	$0.296460\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.82843	+	3.82843i	−0.436290	+	0.436290i
$78$		0			0
$79$			9.65685i			1.08648i	0.839577	+	0.543240i	$0.182803\pi$
$79$			9.65685i			1.08648i	−0.839577	+	0.543240i	$0.817197\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.17157	−	1.17157i	−0.128597	−	0.128597i	0.639879	−	0.768476i	$-0.278984\pi$
$83$	−1.17157	−	1.17157i	−0.128597	−	0.128597i	−0.768476	+	0.639879i	$0.778984\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.8284			1.14781			0.573905	−	0.818922i	$-0.305428\pi$
$89$	10.8284			1.14781			0.573905	+	0.818922i	$0.305428\pi$
$90$		0			0
$91$	−4.82843			−0.506157
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.24264	−	6.24264i	−0.633844	−	0.633844i	0.315186	−	0.949030i	$-0.397933\pi$
$97$	−6.24264	−	6.24264i	−0.633844	−	0.633844i	−0.949030	+	0.315186i	$0.897933\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.82843i		−	0.679454i	−0.940524	−	0.339727i	$-0.889665\pi$
$101$		−	6.82843i		−	0.679454i	0.940524	−	0.339727i	$-0.110335\pi$
$102$		0			0
$103$	−5.17157	+	5.17157i	−0.509570	+	0.509570i	−0.914394	−	0.404824i	$-0.867333\pi$
$103$	−5.17157	+	5.17157i	−0.509570	+	0.509570i	0.404824	+	0.914394i	$0.367333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.17157	+	4.17157i	−0.403281	+	0.403281i	−0.879388	−	0.476106i	$-0.842048\pi$
$107$	−4.17157	+	4.17157i	−0.403281	+	0.403281i	0.476106	+	0.879388i	$0.342048\pi$
$108$		0			0
$109$			11.3137i			1.08366i	0.840489	+	0.541828i	$0.182268\pi$
$109$			11.3137i			1.08366i	−0.840489	+	0.541828i	$0.817732\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.8284	−	13.8284i	−1.30087	−	1.30087i	−0.927806	−	0.373063i	$-0.878308\pi$
$113$	−13.8284	−	13.8284i	−1.30087	−	1.30087i	−0.373063	−	0.927806i	$-0.621692\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−18.3137			−1.66488
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.58579	+	6.58579i	0.584394	+	0.584394i	0.936108	−	0.351714i	$-0.114401\pi$
$127$	6.58579	+	6.58579i	0.584394	+	0.584394i	−0.351714	+	0.936108i	$0.614401\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			20.4853i			1.78981i	0.446259	+	0.894904i	$0.352756\pi$
$131$			20.4853i			1.78981i	−0.446259	+	0.894904i	$0.647244\pi$
$132$		0			0
$133$	−2.00000	+	2.00000i	−0.173422	+	0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.82843	+	5.82843i	−0.497956	+	0.497956i	−0.910801	−	0.412845i	$-0.864535\pi$
$137$	−5.82843	+	5.82843i	−0.497956	+	0.497956i	0.412845	+	0.910801i	$0.364535\pi$
$138$		0			0
$139$		−	16.4853i		−	1.39826i	−0.714993	−	0.699132i	$-0.753570\pi$
$139$		−	16.4853i		−	1.39826i	0.714993	−	0.699132i	$-0.246430\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.4853	−	18.4853i	−1.54582	−	1.54582i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.41421			0.115857			0.0579284	−	0.998321i	$-0.481550\pi$
$149$	1.41421			0.115857			0.0579284	+	0.998321i	$0.481550\pi$
$150$		0			0
$151$	−13.3137			−1.08345			−0.541727	−	0.840554i	$-0.682229\pi$
$151$	−13.3137			−1.08345			−0.541727	+	0.840554i	$0.682229\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.41421	+	3.41421i	0.272484	+	0.272484i	0.830099	−	0.557615i	$-0.188284\pi$
$157$	3.41421	+	3.41421i	0.272484	+	0.272484i	−0.557615	+	0.830099i	$0.688284\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.58579i		−	0.203789i
$162$		0			0
$163$	15.0711	−	15.0711i	1.18046	−	1.18046i	0.200831	−	0.979626i	$-0.435636\pi$
$163$	15.0711	−	15.0711i	1.18046	−	1.18046i	0.979626	−	0.200831i	$-0.0643643\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.6569	+	17.6569i	−1.36633	+	1.36633i	−0.500718	+	0.865611i	$0.666931\pi$
$167$	−17.6569	+	17.6569i	−1.36633	+	1.36633i	−0.865611	+	0.500718i	$0.833069\pi$
$168$		0			0
$169$		−	10.3137i		−	0.793362i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.0000	−	10.0000i	−0.760286	−	0.760286i	0.216088	−	0.976374i	$-0.430670\pi$
$173$	−10.0000	−	10.0000i	−0.760286	−	0.760286i	−0.976374	+	0.216088i	$0.930670\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.07107			−0.229542			−0.114771	−	0.993392i	$-0.536613\pi$
$179$	−3.07107			−0.229542			−0.114771	+	0.993392i	$0.536613\pi$
$180$		0			0
$181$	19.6569			1.46108			0.730541	−	0.682869i	$-0.239268\pi$
$181$	19.6569			1.46108			0.730541	+	0.682869i	$0.239268\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−15.3137	−	15.3137i	−1.11985	−	1.11985i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4.92893i		−	0.356645i	−0.983972	−	0.178323i	$-0.942933\pi$
$191$		−	4.92893i		−	0.356645i	0.983972	−	0.178323i	$-0.0570670\pi$
$192$		0			0
$193$	12.7279	−	12.7279i	0.916176	−	0.916176i	−0.0805728	−	0.996749i	$-0.525675\pi$
$193$	12.7279	−	12.7279i	0.916176	−	0.916176i	0.996749	+	0.0805728i	$0.0256750\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.1421	−	17.1421i	1.22133	−	1.22133i	0.254166	−	0.967161i	$-0.418199\pi$
$197$	17.1421	−	17.1421i	1.22133	−	1.22133i	0.967161	−	0.254166i	$-0.0818010\pi$
$198$		0			0
$199$			2.82843i			0.200502i	0.994962	+	0.100251i	$0.0319646\pi$
$199$			2.82843i			0.200502i	−0.994962	+	0.100251i	$0.968035\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.00000	−	1.00000i	−0.0701862	−	0.0701862i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−15.3137			−1.05927
$210$		0			0
$211$	10.9706			0.755245			0.377622	−	0.925960i	$-0.376742\pi$
$211$	10.9706			0.755245			0.377622	+	0.925960i	$0.376742\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.00000	+	2.00000i	0.135769	+	0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	19.3137i		−	1.29918i
$222$		0			0
$223$	−1.17157	+	1.17157i	−0.0784543	+	0.0784543i	−0.745245	−	0.666791i	$-0.767667\pi$
$223$	−1.17157	+	1.17157i	−0.0784543	+	0.0784543i	0.666791	+	0.745245i	$0.267667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.8284	+	12.8284i	−0.851453	+	0.851453i	−0.990312	−	0.138859i	$-0.955656\pi$
$227$	−12.8284	+	12.8284i	−0.851453	+	0.851453i	0.138859	+	0.990312i	$0.455656\pi$
$228$		0			0
$229$		−	17.3137i		−	1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$229$		−	17.3137i		−	1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.1716	−	10.1716i	−0.666362	−	0.666362i	0.290510	−	0.956872i	$-0.406175\pi$
$233$	−10.1716	−	10.1716i	−0.666362	−	0.666362i	−0.956872	+	0.290510i	$0.906175\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.24264			0.533172			0.266586	−	0.963811i	$-0.414104\pi$
$239$	8.24264			0.533172			0.266586	+	0.963811i	$0.414104\pi$
$240$		0			0
$241$	−7.65685			−0.493221			−0.246611	−	0.969115i	$-0.579317\pi$
$241$	−7.65685			−0.493221			−0.246611	+	0.969115i	$0.579317\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.65685	−	9.65685i	−0.614451	−	0.614451i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	20.9706i		−	1.32365i	−0.749658	−	0.661825i	$-0.769782\pi$
$251$		−	20.9706i		−	1.32365i	0.749658	−	0.661825i	$-0.230218\pi$
$252$		0			0
$253$	9.89949	−	9.89949i	0.622376	−	0.622376i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.82843	+	8.82843i	−0.550702	+	0.550702i	−0.926643	−	0.375941i	$-0.877319\pi$
$257$	−8.82843	+	8.82843i	−0.550702	+	0.550702i	0.375941	+	0.926643i	$0.377319\pi$
$258$		0			0
$259$			7.65685i			0.475774i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.48528	+	3.48528i	0.214912	+	0.214912i	0.806350	−	0.591438i	$-0.201440\pi$
$263$	3.48528	+	3.48528i	0.214912	+	0.214912i	−0.591438	+	0.806350i	$0.701440\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.3137			−0.933693			−0.466847	−	0.884338i	$-0.654610\pi$
$269$	−15.3137			−0.933693			−0.466847	+	0.884338i	$0.654610\pi$
$270$		0			0
$271$	18.1421			1.10206			0.551028	−	0.834487i	$-0.314236\pi$
$271$	18.1421			1.10206			0.551028	+	0.834487i	$0.314236\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.65685	−	9.65685i	−0.580224	−	0.580224i	0.354741	−	0.934965i	$-0.384569\pi$
$277$	−9.65685	−	9.65685i	−0.580224	−	0.580224i	−0.934965	+	0.354741i	$0.884569\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			17.8995i			1.06779i	0.845549	+	0.533897i	$0.179273\pi$
$281$			17.8995i			1.06779i	−0.845549	+	0.533897i	$0.820727\pi$
$282$		0			0
$283$	20.4853	−	20.4853i	1.21772	−	1.21772i	0.249296	−	0.968427i	$-0.419801\pi$
$283$	20.4853	−	20.4853i	1.21772	−	1.21772i	0.968427	−	0.249296i	$-0.0801993\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.828427	−	0.828427i	0.0489005	−	0.0489005i
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.51472	−	1.51472i	−0.0884908	−	0.0884908i	0.661476	−	0.749967i	$-0.269930\pi$
$293$	−1.51472	−	1.51472i	−0.0884908	−	0.0884908i	−0.749967	+	0.661476i	$0.769930\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.4853			0.722042
$300$		0			0
$301$	−9.65685			−0.556612
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−16.4853	−	16.4853i	−0.940865	−	0.940865i	0.0574818	−	0.998347i	$-0.481693\pi$
$307$	−16.4853	−	16.4853i	−0.940865	−	0.940865i	−0.998347	+	0.0574818i	$0.981693\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.48528i		−	0.254337i	−0.991881	−	0.127168i	$-0.959411\pi$
$311$		−	4.48528i		−	0.254337i	0.991881	−	0.127168i	$-0.0405889\pi$
$312$		0			0
$313$	23.8995	−	23.8995i	1.35088	−	1.35088i	0.466200	−	0.884679i	$-0.345623\pi$
$313$	23.8995	−	23.8995i	1.35088	−	1.35088i	0.884679	−	0.466200i	$-0.154377\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.48528	+	9.48528i	−0.532746	+	0.532746i	−0.921389	−	0.388642i	$-0.872944\pi$
$317$	−9.48528	+	9.48528i	−0.532746	+	0.532746i	0.388642	+	0.921389i	$0.372944\pi$
$318$		0			0
$319$		−	7.65685i		−	0.428702i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.00000	−	8.00000i	−0.445132	−	0.445132i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.65685			0.532400
$330$		0			0
$331$	−4.34315			−0.238721			−0.119360	−	0.992851i	$-0.538084\pi$
$331$	−4.34315			−0.238721			−0.119360	+	0.992851i	$0.538084\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.75736	−	3.75736i	−0.204676	−	0.204676i	0.597324	−	0.802000i	$-0.296231\pi$
$337$	−3.75736	−	3.75736i	−0.204676	−	0.204676i	−0.802000	+	0.597324i	$0.796231\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			15.3137i			0.829284i
$342$		0			0
$343$	0.707107	−	0.707107i	0.0381802	−	0.0381802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.14214	−	5.14214i	0.276044	−	0.276044i	−0.555483	−	0.831528i	$-0.687467\pi$
$347$	5.14214	−	5.14214i	0.276044	−	0.276044i	0.831528	+	0.555483i	$0.187467\pi$
$348$		0			0
$349$			14.9706i			0.801356i	0.916219	+	0.400678i	$0.131225\pi$
$349$			14.9706i			0.801356i	−0.916219	+	0.400678i	$0.868775\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.34315	−	6.34315i	−0.337612	−	0.337612i	0.517856	−	0.855468i	$-0.326730\pi$
$353$	−6.34315	−	6.34315i	−0.337612	−	0.337612i	−0.855468	+	0.517856i	$0.826730\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.27208			−0.383806			−0.191903	−	0.981414i	$-0.561466\pi$
$359$	−7.27208			−0.383806			−0.191903	+	0.981414i	$0.561466\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.9706	−	12.9706i	−0.677058	−	0.677058i	0.282276	−	0.959333i	$-0.408911\pi$
$367$	−12.9706	−	12.9706i	−0.677058	−	0.677058i	−0.959333	+	0.282276i	$0.908911\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			11.0711i			0.574781i
$372$		0			0
$373$	0.485281	−	0.485281i	0.0251269	−	0.0251269i	−0.694432	−	0.719559i	$-0.744344\pi$
$373$	0.485281	−	0.485281i	0.0251269	−	0.0251269i	0.719559	+	0.694432i	$0.244344\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.82843	−	4.82843i	0.248677	−	0.248677i
$378$		0			0
$379$			17.3137i			0.889345i	0.895693	+	0.444673i	$0.146680\pi$
$379$			17.3137i			0.889345i	−0.895693	+	0.444673i	$0.853320\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.00000	−	2.00000i	−0.102195	−	0.102195i	0.654161	−	0.756356i	$-0.273022\pi$
$383$	−2.00000	−	2.00000i	−0.102195	−	0.102195i	−0.756356	+	0.654161i	$0.773022\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−31.0711			−1.57537			−0.787683	−	0.616081i	$-0.788719\pi$
$389$	−31.0711			−1.57537			−0.787683	+	0.616081i	$0.788719\pi$
$390$		0			0
$391$	10.3431			0.523075
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.0711	+	13.0711i	0.656018	+	0.656018i	0.954435	−	0.298417i	$-0.0964587\pi$
$397$	13.0711	+	13.0711i	0.656018	+	0.656018i	−0.298417	+	0.954435i	$0.596459\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			15.5563i			0.776847i	0.921481	+	0.388424i	$0.126980\pi$
$401$			15.5563i			0.776847i	−0.921481	+	0.388424i	$0.873020\pi$
$402$		0			0
$403$	−9.65685	+	9.65685i	−0.481042	+	0.481042i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−29.3137	+	29.3137i	−1.45303	+	1.45303i
$408$		0			0
$409$			3.65685i			0.180820i	0.995905	+	0.0904099i	$0.0288177\pi$
$409$			3.65685i			0.180820i	−0.995905	+	0.0904099i	$0.971182\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.82843	+	2.82843i	0.139178	+	0.139178i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.970563			−0.0474151			−0.0237075	−	0.999719i	$-0.507547\pi$
$419$	−0.970563			−0.0474151			−0.0237075	+	0.999719i	$0.507547\pi$
$420$		0			0
$421$	−21.3137			−1.03877			−0.519383	−	0.854541i	$-0.673838\pi$
$421$	−21.3137			−1.03877			−0.519383	+	0.854541i	$0.673838\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−9.41421	−	9.41421i	−0.455586	−	0.455586i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.72792i		−	0.420409i	−0.977657	−	0.210205i	$-0.932587\pi$
$431$		−	8.72792i		−	0.420409i	0.977657	−	0.210205i	$-0.0674130\pi$
$432$		0			0
$433$	10.7279	−	10.7279i	0.515551	−	0.515551i	−0.400671	−	0.916222i	$-0.631223\pi$
$433$	10.7279	−	10.7279i	0.515551	−	0.515551i	0.916222	+	0.400671i	$0.131223\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.17157	−	5.17157i	0.247390	−	0.247390i
$438$		0			0
$439$			14.8284i			0.707722i	0.935298	+	0.353861i	$0.115131\pi$
$439$			14.8284i			0.707722i	−0.935298	+	0.353861i	$0.884869\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.4853	−	13.4853i	−0.640705	−	0.640705i	0.310024	−	0.950729i	$-0.399663\pi$
$443$	−13.4853	−	13.4853i	−0.640705	−	0.640705i	−0.950729	+	0.310024i	$0.899663\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	29.2132			1.37866			0.689328	−	0.724449i	$-0.257906\pi$
$449$	29.2132			1.37866			0.689328	+	0.724449i	$0.257906\pi$
$450$		0			0
$451$	6.34315			0.298687
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.7574	+	15.7574i	0.737098	+	0.737098i	0.972015	−	0.234917i	$-0.0754819\pi$
$457$	15.7574	+	15.7574i	0.737098	+	0.737098i	−0.234917	+	0.972015i	$0.575482\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	23.1127i		−	1.07647i	−0.842796	−	0.538233i	$-0.819092\pi$
$461$		−	23.1127i		−	1.07647i	0.842796	−	0.538233i	$-0.180908\pi$
$462$		0			0
$463$	−3.75736	+	3.75736i	−0.174619	+	0.174619i	−0.789005	−	0.614386i	$-0.789404\pi$
$463$	−3.75736	+	3.75736i	−0.174619	+	0.174619i	0.614386	+	0.789005i	$0.289404\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.17157	+	7.17157i	−0.331861	+	0.331861i	−0.853293	−	0.521432i	$-0.825398\pi$
$467$	−7.17157	+	7.17157i	−0.331861	+	0.331861i	0.521432	+	0.853293i	$0.325398\pi$
$468$		0			0
$469$		−	3.65685i		−	0.168858i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−36.9706	−	36.9706i	−1.69991	−	1.69991i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.1421			1.92552			0.962762	−	0.270352i	$-0.0871399\pi$
$479$	42.1421			1.92552			0.962762	+	0.270352i	$0.0871399\pi$
$480$		0			0
$481$	−36.9706			−1.68571
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.00000	−	4.00000i	−0.181257	−	0.181257i	0.610646	−	0.791904i	$-0.290910\pi$
$487$	−4.00000	−	4.00000i	−0.181257	−	0.181257i	−0.791904	+	0.610646i	$0.790910\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9.21320i			0.415786i	0.978152	+	0.207893i	$0.0666606\pi$
$491$			9.21320i			0.415786i	−0.978152	+	0.207893i	$0.933339\pi$
$492$		0			0
$493$	4.00000	−	4.00000i	0.180151	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.82843	−	5.82843i	0.261441	−	0.261441i
$498$		0			0
$499$			18.0000i			0.805791i	0.915246	+	0.402895i	$0.131996\pi$
$499$			18.0000i			0.805791i	−0.915246	+	0.402895i	$0.868004\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−30.2843	−	30.2843i	−1.35031	−	1.35031i	−0.885313	−	0.464996i	$-0.846056\pi$
$503$	−30.2843	−	30.2843i	−1.35031	−	1.35031i	−0.464996	−	0.885313i	$-0.653944\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.34315			−0.281155			−0.140577	−	0.990070i	$-0.544896\pi$
$509$	−6.34315			−0.281155			−0.140577	+	0.990070i	$0.544896\pi$
$510$		0			0
$511$	2.48528			0.109942
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	36.9706	+	36.9706i	1.62596	+	1.62596i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	4.00000i		−	0.175243i	−0.996154	−	0.0876216i	$-0.972073\pi$
$521$		−	4.00000i		−	0.175243i	0.996154	−	0.0876216i	$-0.0279266\pi$
$522$		0			0
$523$	−10.8284	+	10.8284i	−0.473494	+	0.473494i	−0.903043	−	0.429549i	$-0.858672\pi$
$523$	−10.8284	+	10.8284i	−0.473494	+	0.473494i	0.429549	+	0.903043i	$0.358672\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.00000	+	8.00000i	−0.348485	+	0.348485i
$528$		0			0
$529$		−	16.3137i		−	0.709292i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.00000	+	4.00000i	0.173259	+	0.173259i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.41421			0.233207
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−25.2132	−	25.2132i	−1.07804	−	1.07804i	−0.996685	−	0.0813534i	$-0.974076\pi$
$547$	−25.2132	−	25.2132i	−1.07804	−	1.07804i	−0.0813534	−	0.996685i	$-0.525924\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.00000i		−	0.170406i
$552$		0			0
$553$	6.82843	−	6.82843i	0.290374	−	0.290374i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.1421	+	11.1421i	−0.472107	+	0.472107i	−0.902596	−	0.430489i	$-0.858341\pi$
$557$	−11.1421	+	11.1421i	−0.472107	+	0.472107i	0.430489	+	0.902596i	$0.358341\pi$
$558$		0			0
$559$		−	46.6274i		−	1.97213i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.6274	+	28.6274i	1.20650	+	1.20650i	0.972152	+	0.234349i	$0.0752959\pi$
$563$	28.6274	+	28.6274i	1.20650	+	1.20650i	0.234349	+	0.972152i	$0.424704\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	47.5563			1.99367			0.996833	−	0.0795242i	$-0.0253401\pi$
$569$	47.5563			1.99367			0.996833	+	0.0795242i	$0.0253401\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	5.07107	+	5.07107i	0.211111	+	0.211111i	0.804739	−	0.593628i	$-0.202305\pi$
$577$	5.07107	+	5.07107i	0.211111	+	0.211111i	−0.593628	+	0.804739i	$0.702305\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.65685i			0.0687379i
$582$		0			0
$583$	−42.3848	+	42.3848i	−1.75540	+	1.75540i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.9706	−	14.9706i	0.617901	−	0.617901i	−0.327091	−	0.944993i	$-0.606069\pi$
$587$	14.9706	−	14.9706i	0.617901	−	0.617901i	0.944993	+	0.327091i	$0.106069\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.68629	−	4.68629i	−0.192443	−	0.192443i	0.604308	−	0.796751i	$-0.293450\pi$
$593$	−4.68629	−	4.68629i	−0.192443	−	0.192443i	−0.796751	+	0.604308i	$0.793450\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.92893			0.201391			0.100695	−	0.994917i	$-0.467893\pi$
$599$	4.92893			0.201391			0.100695	+	0.994917i	$0.467893\pi$
$600$		0			0
$601$	46.9706			1.91597			0.957985	−	0.286820i	$-0.0925980\pi$
$601$	46.9706			1.91597			0.957985	+	0.286820i	$0.0925980\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.82843	+	2.82843i	0.114802	+	0.114802i	0.762174	−	0.647372i	$-0.224132\pi$
$607$	2.82843	+	2.82843i	0.114802	+	0.114802i	−0.647372	+	0.762174i	$0.724132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			46.6274i			1.88634i
$612$		0			0
$613$	9.65685	−	9.65685i	0.390037	−	0.390037i	−0.484664	−	0.874701i	$-0.661058\pi$
$613$	9.65685	−	9.65685i	0.390037	−	0.390037i	0.874701	+	0.484664i	$0.161058\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.1421	+	15.1421i	−0.609599	+	0.609599i	−0.942841	−	0.333242i	$-0.891857\pi$
$617$	−15.1421	+	15.1421i	−0.609599	+	0.609599i	0.333242	+	0.942841i	$0.391857\pi$
$618$		0			0
$619$		−	6.82843i		−	0.274458i	−0.990539	−	0.137229i	$-0.956180\pi$
$619$		−	6.82843i		−	0.274458i	0.990539	−	0.137229i	$-0.0438196\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.65685	−	7.65685i	−0.306765	−	0.306765i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−30.6274			−1.22120
$630$		0			0
$631$	−28.9706			−1.15330			−0.576650	−	0.816991i	$-0.695640\pi$
$631$	−28.9706			−1.15330			−0.576650	+	0.816991i	$0.695640\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.41421	+	3.41421i	0.135276	+	0.135276i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			22.1005i			0.872917i	0.899724	+	0.436459i	$0.143768\pi$
$641$			22.1005i			0.872917i	−0.899724	+	0.436459i	$0.856232\pi$
$642$		0			0
$643$	9.65685	−	9.65685i	0.380829	−	0.380829i	−0.490572	−	0.871401i	$-0.663212\pi$
$643$	9.65685	−	9.65685i	0.380829	−	0.380829i	0.871401	+	0.490572i	$0.163212\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.0000	+	12.0000i	−0.471769	+	0.471769i	−0.902487	−	0.430718i	$-0.858260\pi$
$647$	−12.0000	+	12.0000i	−0.471769	+	0.471769i	0.430718	+	0.902487i	$0.358260\pi$
$648$		0			0
$649$			21.6569i			0.850106i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.1421	−	13.1421i	−0.514292	−	0.514292i	0.401547	−	0.915838i	$-0.368473\pi$
$653$	−13.1421	−	13.1421i	−0.514292	−	0.514292i	−0.915838	+	0.401547i	$0.868473\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−35.0711			−1.36617			−0.683087	−	0.730337i	$-0.739363\pi$
$659$	−35.0711			−1.36617			−0.683087	+	0.730337i	$0.739363\pi$
$660$		0			0
$661$	43.9411			1.70911			0.854556	−	0.519360i	$-0.173829\pi$
$661$	43.9411			1.70911			0.854556	+	0.519360i	$0.173829\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.58579	+	2.58579i	0.100122	+	0.100122i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	72.0833i		−	2.78274i
$672$		0			0
$673$	−21.8995	+	21.8995i	−0.844163	+	0.844163i	−0.989397	−	0.145234i	$-0.953607\pi$
$673$	−21.8995	+	21.8995i	−0.844163	+	0.844163i	0.145234	+	0.989397i	$0.453607\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30.4853	−	30.4853i	1.17164	−	1.17164i	0.189827	−	0.981818i	$-0.439207\pi$
$677$	30.4853	−	30.4853i	1.17164	−	1.17164i	0.981818	−	0.189827i	$-0.0607927\pi$
$678$		0			0
$679$			8.82843i			0.338804i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.1421	+	35.1421i	1.34468	+	1.34468i	0.891339	+	0.453338i	$0.149767\pi$
$683$	35.1421	+	35.1421i	1.34468	+	1.34468i	0.453338	+	0.891339i	$0.350233\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−53.4558			−2.03651
$690$		0			0
$691$	−45.4558			−1.72922			−0.864611	−	0.502442i	$-0.832435\pi$
$691$	−45.4558			−1.72922			−0.864611	+	0.502442i	$0.832435\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.31371	+	3.31371i	0.125516	+	0.125516i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	11.2721i		−	0.425741i	−0.977080	−	0.212870i	$-0.931719\pi$
$701$		−	11.2721i		−	0.425741i	0.977080	−	0.212870i	$-0.0682812\pi$
$702$		0			0
$703$	−15.3137	+	15.3137i	−0.577567	+	0.577567i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.82843	+	4.82843i	−0.181592	+	0.181592i
$708$		0			0
$709$		−	22.6274i		−	0.849790i	−0.905243	−	0.424895i	$-0.860311\pi$
$709$		−	22.6274i		−	0.849790i	0.905243	−	0.424895i	$-0.139689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.17157	−	5.17157i	−0.193677	−	0.193677i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	7.31371			0.272377
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−16.9706	−	16.9706i	−0.629403	−	0.629403i	0.318515	−	0.947918i	$-0.396816\pi$
$727$	−16.9706	−	16.9706i	−0.629403	−	0.629403i	−0.947918	+	0.318515i	$0.896816\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	38.6274i		−	1.42869i
$732$		0			0
$733$	18.2426	−	18.2426i	0.673807	−	0.673807i	−0.284784	−	0.958592i	$-0.591922\pi$
$733$	18.2426	−	18.2426i	0.673807	−	0.673807i	0.958592	+	0.284784i	$0.0919220\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.0000	−	14.0000i	0.515697	−	0.515697i
$738$		0			0
$739$		−	43.3137i		−	1.59332i	−0.604427	−	0.796660i	$-0.706598\pi$
$739$		−	43.3137i		−	1.59332i	0.604427	−	0.796660i	$-0.293402\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.8284	+	13.8284i	0.507316	+	0.507316i	0.913702	−	0.406386i	$-0.133211\pi$
$743$	13.8284	+	13.8284i	0.507316	+	0.507316i	−0.406386	+	0.913702i	$0.633211\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.89949			0.215563
$750$		0			0
$751$	14.9706			0.546284			0.273142	−	0.961974i	$-0.411937\pi$
$751$	14.9706			0.546284			0.273142	+	0.961974i	$0.411937\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−8.92893	−	8.92893i	−0.324528	−	0.324528i	0.525973	−	0.850501i	$-0.323701\pi$
$757$	−8.92893	−	8.92893i	−0.324528	−	0.324528i	−0.850501	+	0.525973i	$0.823701\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17.8579i		−	0.647347i	−0.946169	−	0.323674i	$-0.895082\pi$
$761$		−	17.8579i		−	0.647347i	0.946169	−	0.323674i	$-0.104918\pi$
$762$		0			0
$763$	8.00000	−	8.00000i	0.289619	−	0.289619i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.6569	+	13.6569i	−0.493120	+	0.493120i
$768$		0			0
$769$			25.5980i			0.923087i	0.887118	+	0.461543i	$0.152704\pi$
$769$			25.5980i			0.923087i	−0.887118	+	0.461543i	$0.847296\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.0000	+	10.0000i	0.359675	+	0.359675i	0.863693	−	0.504018i	$-0.168146\pi$
$773$	10.0000	+	10.0000i	0.359675	+	0.359675i	−0.504018	+	0.863693i	$0.668146\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.31371			0.118726
$780$		0			0
$781$	44.6274			1.59689
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.34315	−	6.34315i	−0.226109	−	0.226109i	0.584956	−	0.811065i	$-0.301112\pi$
$787$	−6.34315	−	6.34315i	−0.226109	−	0.226109i	−0.811065	+	0.584956i	$0.801112\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			19.5563i			0.695344i
$792$		0			0
$793$	45.4558	−	45.4558i	1.61418	−	1.61418i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.6863	−	10.6863i	0.378528	−	0.378528i	−0.492043	−	0.870571i	$-0.663750\pi$
$797$	10.6863	−	10.6863i	0.378528	−	0.378528i	0.870571	+	0.492043i	$0.163750\pi$
$798$		0			0
$799$			38.6274i			1.36654i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.51472	+	9.51472i	0.335767	+	0.335767i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.7574			0.413367			0.206683	−	0.978408i	$-0.433733\pi$
$809$	11.7574			0.413367			0.206683	+	0.978408i	$0.433733\pi$
$810$		0			0
$811$	−5.45584			−0.191581			−0.0957903	−	0.995402i	$-0.530538\pi$
$811$	−5.45584			−0.191581			−0.0957903	+	0.995402i	$0.530538\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−19.3137	−	19.3137i	−0.675701	−	0.675701i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			49.0122i			1.71054i	0.518185	+	0.855269i	$0.326608\pi$
$821$			49.0122i			1.71054i	−0.518185	+	0.855269i	$0.673392\pi$
$822$		0			0
$823$	14.5858	−	14.5858i	0.508429	−	0.508429i	−0.405615	−	0.914044i	$-0.632943\pi$
$823$	14.5858	−	14.5858i	0.508429	−	0.508429i	0.914044	+	0.405615i	$0.132943\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.1421	+	25.1421i	−0.874278	+	0.874278i	−0.992935	−	0.118658i	$-0.962141\pi$
$827$	−25.1421	+	25.1421i	−0.874278	+	0.874278i	0.118658	+	0.992935i	$0.462141\pi$
$828$		0			0
$829$		−	37.3137i		−	1.29596i	−0.761658	−	0.647979i	$-0.775614\pi$
$829$		−	37.3137i		−	1.29596i	0.761658	−	0.647979i	$-0.224386\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.82843	+	2.82843i	0.0979992	+	0.0979992i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.7696			−1.26943			−0.634713	−	0.772748i	$-0.718882\pi$
$839$	−36.7696			−1.26943			−0.634713	+	0.772748i	$0.718882\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	12.9497	+	12.9497i	0.444959	+	0.444959i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	19.7990i		−	0.678701i
$852$		0			0
$853$	−32.8701	+	32.8701i	−1.12545	+	1.12545i	−0.134541	+	0.990908i	$0.542956\pi$
$853$	−32.8701	+	32.8701i	−1.12545	+	1.12545i	−0.990908	+	0.134541i	$0.957044\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.3137	−	17.3137i	0.591425	−	0.591425i	−0.346591	−	0.938016i	$-0.612661\pi$
$857$	17.3137	−	17.3137i	0.591425	−	0.591425i	0.938016	+	0.346591i	$0.112661\pi$
$858$		0			0
$859$			46.4264i			1.58405i	0.610490	+	0.792024i	$0.290973\pi$
$859$			46.4264i			1.58405i	−0.610490	+	0.792024i	$0.709027\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31.1421	+	31.1421i	1.06009	+	1.06009i	0.998075	+	0.0620154i	$0.0197528\pi$
$863$	31.1421	+	31.1421i	1.06009	+	1.06009i	0.0620154	+	0.998075i	$0.480247\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	52.2843			1.77362
$870$		0			0
$871$	17.6569			0.598280
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.9411	+	29.9411i	1.01104	+	1.01104i	0.999938	+	0.0111016i	$0.00353382\pi$
$877$	29.9411	+	29.9411i	1.01104	+	1.01104i	0.0111016	+	0.999938i	$0.496466\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42.9117i		−	1.44573i	−0.690988	−	0.722866i	$-0.742824\pi$
$881$		−	42.9117i		−	1.44573i	0.690988	−	0.722866i	$-0.257176\pi$
$882$		0			0
$883$	1.85786	−	1.85786i	0.0625221	−	0.0625221i	−0.675154	−	0.737676i	$-0.735923\pi$
$883$	1.85786	−	1.85786i	0.0625221	−	0.0625221i	0.737676	+	0.675154i	$0.235923\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	39.1127	−	39.1127i	1.31328	−	1.31328i	0.394290	−	0.918986i	$-0.370991\pi$
$887$	39.1127	−	39.1127i	1.31328	−	1.31328i	0.918986	−	0.394290i	$-0.129009\pi$
$888$		0			0
$889$		−	9.31371i		−	0.312372i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19.3137	+	19.3137i	0.646309	+	0.646309i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.00000			−0.133407
$900$		0			0
$901$	−44.2843			−1.47532
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−17.4558	−	17.4558i	−0.579612	−	0.579612i	0.355184	−	0.934796i	$-0.384418\pi$
$907$	−17.4558	−	17.4558i	−0.579612	−	0.579612i	−0.934796	+	0.355184i	$0.884418\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			52.5269i			1.74029i	0.492792	+	0.870147i	$0.335976\pi$
$911$			52.5269i			1.74029i	−0.492792	+	0.870147i	$0.664024\pi$
$912$		0			0
$913$	−6.34315	+	6.34315i	−0.209927	+	0.209927i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.4853	−	14.4853i	0.478346	−	0.478346i
$918$		0			0
$919$			15.6569i			0.516472i	0.966082	+	0.258236i	$0.0831412\pi$
$919$			15.6569i			0.516472i	−0.966082	+	0.258236i	$0.916859\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28.1421	+	28.1421i	0.926310	+	0.926310i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.82843			0.0927977			0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843			0.0927977			0.0463988	+	0.998923i	$0.485225\pi$
$930$		0			0
$931$	2.82843			0.0926980
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	12.3848	+	12.3848i	0.404593	+	0.404593i	0.879848	−	0.475255i	$-0.157644\pi$
$937$	12.3848	+	12.3848i	0.404593	+	0.404593i	−0.475255	+	0.879848i	$0.657644\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			51.3137i			1.67278i	0.548136	+	0.836390i	$0.315338\pi$
$941$			51.3137i			1.67278i	−0.548136	+	0.836390i	$0.684662\pi$
$942$		0			0
$943$	−2.14214	+	2.14214i	−0.0697575	+	0.0697575i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.1421	−	21.1421i	0.687027	−	0.687027i	−0.274547	−	0.961574i	$-0.588528\pi$
$947$	21.1421	−	21.1421i	0.687027	−	0.687027i	0.961574	+	0.274547i	$0.0885278\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.7990	+	24.7990i	0.803318	+	0.803318i	0.983613	−	0.180295i	$-0.0577051\pi$
$953$	24.7990	+	24.7990i	0.803318	+	0.803318i	−0.180295	+	0.983613i	$0.557705\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.24264			0.266169
$960$		0			0
$961$	−23.0000			−0.741935
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	16.2426	+	16.2426i	0.522328	+	0.522328i	0.918274	−	0.395946i	$-0.129583\pi$
$967$	16.2426	+	16.2426i	0.522328	+	0.522328i	−0.395946	+	0.918274i	$0.629583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.20101i		−	0.134817i	−0.997725	−	0.0674084i	$-0.978527\pi$
$971$		−	4.20101i		−	0.134817i	0.997725	−	0.0674084i	$-0.0214730\pi$
$972$		0			0
$973$	−11.6569	+	11.6569i	−0.373702	+	0.373702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	38.7990	−	38.7990i	1.24129	−	1.24129i	0.281823	−	0.959466i	$-0.409061\pi$
$977$	38.7990	−	38.7990i	1.24129	−	1.24129i	0.959466	−	0.281823i	$-0.0909392\pi$
$978$		0			0
$979$		−	58.6274i		−	1.87374i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.20101	+	2.20101i	0.0702013	+	0.0702013i	0.741336	−	0.671134i	$-0.234193\pi$
$983$	2.20101	+	2.20101i	0.0702013	+	0.0702013i	−0.671134	+	0.741336i	$0.734193\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.9706			0.794018
$990$		0			0
$991$	34.2843			1.08908			0.544538	−	0.838736i	$-0.316705\pi$
$991$	34.2843			1.08908			0.544538	+	0.838736i	$0.316705\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−23.4142	−	23.4142i	−0.741536	−	0.741536i	0.231338	−	0.972873i	$-0.425690\pi$
$997$	−23.4142	−	23.4142i	−0.741536	−	0.741536i	−0.972873	+	0.231338i	$0.925690\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.v.c.1457.1	yes	4
3.2	odd	2		6300.2.v.d.1457.1	yes	4
5.2	odd	4		6300.2.v.a.5993.2	yes	4
5.3	odd	4		6300.2.v.d.5993.1	yes	4
5.4	even	2		6300.2.v.b.1457.2	yes	4
15.2	even	4		6300.2.v.b.5993.2	yes	4
15.8	even	4	inner	6300.2.v.c.5993.1	yes	4
15.14	odd	2		6300.2.v.a.1457.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.v.a.1457.2	✓	4	15.14	odd	2
6300.2.v.a.5993.2	yes	4	5.2	odd	4
6300.2.v.b.1457.2	yes	4	5.4	even	2
6300.2.v.b.5993.2	yes	4	15.2	even	4
6300.2.v.c.1457.1	yes	4	1.1	even	1	trivial
6300.2.v.c.5993.1	yes	4	15.8	even	4	inner
6300.2.v.d.1457.1	yes	4	3.2	odd	2
6300.2.v.d.5993.1	yes	4	5.3	odd	4

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{7} -5.41421i q^{11} +(3.41421 - 3.41421i) q^{13} +(2.82843 - 2.82843i) q^{17} -2.82843i q^{19} +(1.82843 + 1.82843i) q^{23} +1.41421 q^{29} -2.82843 q^{31} +(-5.41421 - 5.41421i) q^{37} +1.17157i q^{41} +(6.82843 - 6.82843i) q^{43} +(-6.82843 + 6.82843i) q^{47} +1.00000i q^{49} +(-7.82843 - 7.82843i) q^{53} -4.00000 q^{59} +13.3137 q^{61} +(2.58579 + 2.58579i) q^{67} +8.24264i q^{71} +(-1.75736 + 1.75736i) q^{73} +(-3.82843 + 3.82843i) q^{77} +9.65685i q^{79} +(-1.17157 - 1.17157i) q^{83} +10.8284 q^{89} -4.82843 q^{91} +(-6.24264 - 6.24264i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{13} - 4 q^{23} - 16 q^{37} + 16 q^{43} - 16 q^{47} - 20 q^{53} - 16 q^{59} + 8 q^{61} + 16 q^{67} - 24 q^{73} - 4 q^{77} - 16 q^{83} + 32 q^{89} - 8 q^{91} - 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^13 - 4 * q^23 - 16 * q^37 + 16 * q^43 - 16 * q^47 - 20 * q^53 - 16 * q^59 + 8 * q^61 + 16 * q^67 - 24 * q^73 - 4 * q^77 - 16 * q^83 + 32 * q^89 - 8 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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