LMFDB - Embedded newform 2100.2.x.b.1693.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, 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("2100.1357");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1693.1
Root			$0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	2100.1693
Dual form			2100.2.x.b.1357.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} -2.00000 q^{11} +(-1.41421 + 1.41421i) q^{13} +(-2.82843 - 2.82843i) q^{17} +5.29150 q^{19} -2.64575i q^{21} +(-3.74166 - 3.74166i) q^{23} +(0.707107 + 0.707107i) q^{27} +6.00000i q^{29} -5.29150i q^{31} +(1.41421 - 1.41421i) q^{33} -2.00000i q^{39} -10.5830i q^{41} +(3.74166 + 3.74166i) q^{43} +(5.65685 + 5.65685i) q^{47} -7.00000i q^{49} +4.00000 q^{51} +(-3.74166 - 3.74166i) q^{53} +(-3.74166 + 3.74166i) q^{57} +10.5830 q^{59} -10.5830i q^{61} +(1.87083 + 1.87083i) q^{63} +(-3.74166 + 3.74166i) q^{67} +5.29150 q^{69} -2.00000 q^{71} +(9.89949 - 9.89949i) q^{73} +(3.74166 - 3.74166i) q^{77} +4.00000i q^{79} -1.00000 q^{81} +(-4.24264 - 4.24264i) q^{87} -5.29150i q^{91} +(3.74166 + 3.74166i) q^{93} +(1.41421 + 1.41421i) q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{11} + 32 q^{51} - 16 q^{71} - 8 q^{81}+O(q^{100}) $ 8 * q - 16 * q^11 + 32 * q^51 - 16 * q^71 - 8 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.87083	+	1.87083i	−0.707107	+	0.707107i
$7$	−1.87083	+	1.87083i	−0.707107	+	0.707107i
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	−1.41421	+	1.41421i	−0.392232	+	0.392232i	−0.875482	−	0.483250i	$-0.839456\pi$
$13$	−1.41421	+	1.41421i	−0.392232	+	0.392232i	0.483250	+	0.875482i	$0.339456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.82843	−	2.82843i	−0.685994	−	0.685994i	0.275350	−	0.961344i	$-0.411206\pi$
$17$	−2.82843	−	2.82843i	−0.685994	−	0.685994i	−0.961344	+	0.275350i	$0.911206\pi$
$18$		0			0
$19$	5.29150			1.21395			0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150			1.21395			0.606977	+	0.794719i	$0.292382\pi$
$20$		0			0
$21$		−	2.64575i		−	0.577350i
$22$		0			0
$23$	−3.74166	−	3.74166i	−0.780189	−	0.780189i	0.199673	−	0.979863i	$-0.436012\pi$
$23$	−3.74166	−	3.74166i	−0.780189	−	0.780189i	−0.979863	+	0.199673i	$0.936012\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		−	5.29150i		−	0.950382i	−0.879883	−	0.475191i	$-0.842379\pi$
$31$		−	5.29150i		−	0.950382i	0.879883	−	0.475191i	$-0.157621\pi$
$32$		0			0
$33$	1.41421	−	1.41421i	0.246183	−	0.246183i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		−	2.00000i		−	0.320256i
$40$		0			0
$41$		−	10.5830i		−	1.65279i	−0.563093	−	0.826394i	$-0.690389\pi$
$41$		−	10.5830i		−	1.65279i	0.563093	−	0.826394i	$-0.309611\pi$
$42$		0			0
$43$	3.74166	+	3.74166i	0.570597	+	0.570597i	0.932295	−	0.361698i	$-0.117803\pi$
$43$	3.74166	+	3.74166i	0.570597	+	0.570597i	−0.361698	+	0.932295i	$0.617803\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	0.986840	−	0.161703i	$-0.0516985\pi$
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	−0.161703	+	0.986840i	$0.551699\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$	−3.74166	−	3.74166i	−0.513956	−	0.513956i	0.401780	−	0.915736i	$-0.368392\pi$
$53$	−3.74166	−	3.74166i	−0.513956	−	0.513956i	−0.915736	+	0.401780i	$0.868392\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.74166	+	3.74166i	−0.495595	+	0.495595i
$58$		0			0
$59$	10.5830			1.37779			0.688895	−	0.724861i	$-0.258096\pi$
$59$	10.5830			1.37779			0.688895	+	0.724861i	$0.258096\pi$
$60$		0			0
$61$		−	10.5830i		−	1.35501i	−0.735516	−	0.677507i	$-0.763060\pi$
$61$		−	10.5830i		−	1.35501i	0.735516	−	0.677507i	$-0.236940\pi$
$62$		0			0
$63$	1.87083	+	1.87083i	0.235702	+	0.235702i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.74166	+	3.74166i	−0.457116	+	0.457116i	−0.897708	−	0.440592i	$-0.854769\pi$
$67$	−3.74166	+	3.74166i	−0.457116	+	0.457116i	0.440592	+	0.897708i	$0.354769\pi$
$68$		0			0
$69$	5.29150			0.637022
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	9.89949	−	9.89949i	1.15865	−	1.15865i	0.173882	−	0.984767i	$-0.444369\pi$
$73$	9.89949	−	9.89949i	1.15865	−	1.15865i	0.984767	−	0.173882i	$-0.0556310\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.74166	−	3.74166i	0.426401	−	0.426401i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.24264	−	4.24264i	−0.454859	−	0.454859i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		−	5.29150i		−	0.554700i
$92$		0			0
$93$	3.74166	+	3.74166i	0.387992	+	0.387992i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.41421	+	1.41421i	0.143592	+	0.143592i	0.775248	−	0.631657i	$-0.217625\pi$
$97$	1.41421	+	1.41421i	0.143592	+	0.143592i	−0.631657	+	0.775248i	$0.717625\pi$
$98$		0			0
$99$			2.00000i			0.201008i
$100$		0			0
$101$		−	10.5830i		−	1.05305i	−0.850160	−	0.526524i	$-0.823495\pi$
$101$		−	10.5830i		−	1.05305i	0.850160	−	0.526524i	$-0.176505\pi$
$102$		0			0
$103$	−5.65685	+	5.65685i	−0.557386	+	0.557386i	−0.928562	−	0.371176i	$-0.878955\pi$
$103$	−5.65685	+	5.65685i	−0.557386	+	0.557386i	0.371176	+	0.928562i	$0.378955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	−0.502726	−	0.864446i	$-0.667670\pi$
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	0.864446	+	0.502726i	$0.167670\pi$
$108$		0			0
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.2250	−	11.2250i	−1.05596	−	1.05596i	−0.998339	−	0.0576178i	$-0.981650\pi$
$113$	−11.2250	−	11.2250i	−1.05596	−	1.05596i	−0.0576178	−	0.998339i	$-0.518350\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.41421	+	1.41421i	0.130744	+	0.130744i
$118$		0			0
$119$	10.5830			0.970143
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	7.48331	+	7.48331i	0.674748	+	0.674748i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.74166	−	3.74166i	0.332018	−	0.332018i	−0.521334	−	0.853353i	$-0.674565\pi$
$127$	3.74166	−	3.74166i	0.332018	−	0.332018i	0.853353	+	0.521334i	$0.174565\pi$
$128$		0			0
$129$	−5.29150			−0.465891
$130$		0			0
$131$		−	10.5830i		−	0.924641i	−0.886713	−	0.462321i	$-0.847017\pi$
$131$		−	10.5830i		−	0.924641i	0.886713	−	0.462321i	$-0.152983\pi$
$132$		0			0
$133$	−9.89949	+	9.89949i	−0.858395	+	0.858395i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.2250	−	11.2250i	0.959014	−	0.959014i	−0.0401784	−	0.999193i	$-0.512793\pi$
$137$	11.2250	−	11.2250i	0.959014	−	0.959014i	0.999193	+	0.0401784i	$0.0127926\pi$
$138$		0			0
$139$	15.8745			1.34646			0.673229	−	0.739434i	$-0.264907\pi$
$139$	15.8745			1.34646			0.673229	+	0.739434i	$0.264907\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$	2.82843	−	2.82843i	0.236525	−	0.236525i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.94975	+	4.94975i	0.408248	+	0.408248i
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$	−2.82843	+	2.82843i	−0.228665	+	0.228665i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.89949	−	9.89949i	−0.790066	−	0.790066i	0.191439	−	0.981505i	$-0.438685\pi$
$157$	−9.89949	−	9.89949i	−0.790066	−	0.790066i	−0.981505	+	0.191439i	$0.938685\pi$
$158$		0			0
$159$	5.29150			0.419643
$160$		0			0
$161$	14.0000			1.10335
$162$		0			0
$163$	11.2250	+	11.2250i	0.879208	+	0.879208i	0.993453	−	0.114245i	$-0.0364449\pi$
$163$	11.2250	+	11.2250i	0.879208	+	0.879208i	−0.114245	+	0.993453i	$0.536445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.48528	−	8.48528i	−0.656611	−	0.656611i	0.297966	−	0.954577i	$-0.403692\pi$
$167$	−8.48528	−	8.48528i	−0.656611	−	0.656611i	−0.954577	+	0.297966i	$0.903692\pi$
$168$		0			0
$169$			9.00000i			0.692308i
$170$		0			0
$171$		−	5.29150i		−	0.404651i
$172$		0			0
$173$	−5.65685	+	5.65685i	−0.430083	+	0.430083i	−0.888656	−	0.458574i	$-0.848361\pi$
$173$	−5.65685	+	5.65685i	−0.430083	+	0.430083i	0.458574	+	0.888656i	$0.348361\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−7.48331	+	7.48331i	−0.562480	+	0.562480i
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$		−	10.5830i		−	0.786629i	−0.919404	−	0.393314i	$-0.871328\pi$
$181$		−	10.5830i		−	0.786629i	0.919404	−	0.393314i	$-0.128672\pi$
$182$		0			0
$183$	7.48331	+	7.48331i	0.553183	+	0.553183i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	5.65685	+	5.65685i	0.413670	+	0.413670i
$188$		0			0
$189$	−2.64575			−0.192450
$190$		0			0
$191$	−10.0000			−0.723575			−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000			−0.723575			−0.361787	+	0.932261i	$0.617833\pi$
$192$		0			0
$193$	7.48331	+	7.48331i	0.538661	+	0.538661i	0.923136	−	0.384475i	$-0.125617\pi$
$193$	7.48331	+	7.48331i	0.538661	+	0.538661i	−0.384475	+	0.923136i	$0.625617\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.2250	−	11.2250i	0.799746	−	0.799746i	−0.183309	−	0.983055i	$-0.558681\pi$
$197$	11.2250	−	11.2250i	0.799746	−	0.799746i	0.983055	+	0.183309i	$0.0586809\pi$
$198$		0			0
$199$	5.29150			0.375105			0.187552	−	0.982255i	$-0.439945\pi$
$199$	5.29150			0.375105			0.187552	+	0.982255i	$0.439945\pi$
$200$		0			0
$201$		−	5.29150i		−	0.373234i
$202$		0			0
$203$	−11.2250	−	11.2250i	−0.787839	−	0.787839i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.74166	+	3.74166i	−0.260063	+	0.260063i
$208$		0			0
$209$	−10.5830			−0.732042
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$	1.41421	−	1.41421i	0.0969003	−	0.0969003i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.89949	+	9.89949i	0.672022	+	0.672022i
$218$		0			0
$219$			14.0000i			0.946032i
$220$		0			0
$221$	8.00000			0.538138
$222$		0			0
$223$	11.3137	−	11.3137i	0.757622	−	0.757622i	−0.218267	−	0.975889i	$-0.570040\pi$
$223$	11.3137	−	11.3137i	0.757622	−	0.757622i	0.975889	+	0.218267i	$0.0700404\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.9706	−	16.9706i	−1.12638	−	1.12638i	−0.990762	−	0.135614i	$-0.956699\pi$
$227$	−16.9706	−	16.9706i	−1.12638	−	1.12638i	−0.135614	−	0.990762i	$-0.543301\pi$
$228$		0			0
$229$	−10.5830			−0.699345			−0.349672	−	0.936872i	$-0.613707\pi$
$229$	−10.5830			−0.699345			−0.349672	+	0.936872i	$0.613707\pi$
$230$		0			0
$231$			5.29150i			0.348155i
$232$		0			0
$233$	−11.2250	−	11.2250i	−0.735372	−	0.735372i	0.236306	−	0.971679i	$-0.424063\pi$
$233$	−11.2250	−	11.2250i	−0.735372	−	0.735372i	−0.971679	+	0.236306i	$0.924063\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.82843	−	2.82843i	−0.183726	−	0.183726i
$238$		0			0
$239$			6.00000i			0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$			6.00000i			0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.48331	+	7.48331i	−0.476152	+	0.476152i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.5830i			0.667993i	0.942574	+	0.333997i	$0.108397\pi$
$251$			10.5830i			0.667993i	−0.942574	+	0.333997i	$0.891603\pi$
$252$		0			0
$253$	7.48331	+	7.48331i	0.470472	+	0.470472i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.7990	+	19.7990i	1.23503	+	1.23503i	0.962009	+	0.273018i	$0.0880219\pi$
$257$	19.7990	+	19.7990i	1.23503	+	1.23503i	0.273018	+	0.962009i	$0.411978\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	−18.7083	−	18.7083i	−1.15360	−	1.15360i	−0.985825	−	0.167778i	$-0.946341\pi$
$263$	−18.7083	−	18.7083i	−1.15360	−	1.15360i	−0.167778	−	0.985825i	$-0.553659\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.5830			−0.645257			−0.322629	−	0.946526i	$-0.604566\pi$
$269$	−10.5830			−0.645257			−0.322629	+	0.946526i	$0.604566\pi$
$270$		0			0
$271$			26.4575i			1.60718i	0.595184	+	0.803590i	$0.297079\pi$
$271$			26.4575i			1.60718i	−0.595184	+	0.803590i	$0.702921\pi$
$272$		0			0
$273$	3.74166	+	3.74166i	0.226455	+	0.226455i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.48331	+	7.48331i	−0.449629	+	0.449629i	−0.895231	−	0.445602i	$-0.852990\pi$
$277$	−7.48331	+	7.48331i	−0.449629	+	0.449629i	0.445602	+	0.895231i	$0.352990\pi$
$278$		0			0
$279$	−5.29150			−0.316794
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	2.82843	−	2.82843i	0.168133	−	0.168133i	−0.618026	−	0.786158i	$-0.712067\pi$
$283$	2.82843	−	2.82843i	0.168133	−	0.168133i	0.786158	+	0.618026i	$0.212067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	19.7990	+	19.7990i	1.16870	+	1.16870i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$	−2.00000			−0.117242
$292$		0			0
$293$	8.48528	−	8.48528i	0.495715	−	0.495715i	−0.414386	−	0.910101i	$-0.636004\pi$
$293$	8.48528	−	8.48528i	0.495715	−	0.495715i	0.910101	+	0.414386i	$0.136004\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.41421	−	1.41421i	−0.0820610	−	0.0820610i
$298$		0			0
$299$	10.5830			0.612031
$300$		0			0
$301$	−14.0000			−0.806947
$302$		0			0
$303$	7.48331	+	7.48331i	0.429905	+	0.429905i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.48528	−	8.48528i	−0.484281	−	0.484281i	0.422215	−	0.906496i	$-0.361253\pi$
$307$	−8.48528	−	8.48528i	−0.484281	−	0.484281i	−0.906496	+	0.422215i	$0.861253\pi$
$308$		0			0
$309$		−	8.00000i		−	0.455104i
$310$		0			0
$311$			31.7490i			1.80032i	0.435558	+	0.900161i	$0.356551\pi$
$311$			31.7490i			1.80032i	−0.435558	+	0.900161i	$0.643449\pi$
$312$		0			0
$313$	−4.24264	+	4.24264i	−0.239808	+	0.239808i	−0.816771	−	0.576962i	$-0.804238\pi$
$313$	−4.24264	+	4.24264i	−0.239808	+	0.239808i	0.576962	+	0.816771i	$0.304238\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.7083	−	18.7083i	1.05076	−	1.05076i	0.0521213	−	0.998641i	$-0.483402\pi$
$317$	18.7083	−	18.7083i	1.05076	−	1.05076i	0.998641	−	0.0521213i	$-0.0165983\pi$
$318$		0			0
$319$		−	12.0000i		−	0.671871i
$320$		0			0
$321$			5.29150i			0.295343i
$322$		0			0
$323$	−14.9666	−	14.9666i	−0.832766	−	0.832766i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.41421	−	1.41421i	−0.0782062	−	0.0782062i
$328$		0			0
$329$	−21.1660			−1.16692
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.4499	+	22.4499i	−1.22293	+	1.22293i	−0.256340	+	0.966587i	$0.582517\pi$
$337$	−22.4499	+	22.4499i	−1.22293	+	1.22293i	−0.966587	+	0.256340i	$0.917483\pi$
$338$		0			0
$339$	15.8745			0.862185
$340$		0			0
$341$			10.5830i			0.573102i
$342$		0			0
$343$	13.0958	+	13.0958i	0.707107	+	0.707107i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.74166	+	3.74166i	−0.200863	+	0.200863i	−0.800370	−	0.599507i	$-0.795363\pi$
$347$	−3.74166	+	3.74166i	−0.200863	+	0.200863i	0.599507	+	0.800370i	$0.295363\pi$
$348$		0			0
$349$	−31.7490			−1.69949			−0.849743	−	0.527197i	$-0.823243\pi$
$349$	−31.7490			−1.69949			−0.849743	+	0.527197i	$0.823243\pi$
$350$		0			0
$351$	−2.00000			−0.106752
$352$		0			0
$353$	16.9706	−	16.9706i	0.903252	−	0.903252i	−0.0924641	−	0.995716i	$-0.529474\pi$
$353$	16.9706	−	16.9706i	0.903252	−	0.903252i	0.995716	+	0.0924641i	$0.0294743\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−7.48331	+	7.48331i	−0.396059	+	0.396059i
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$	4.94975	−	4.94975i	0.259794	−	0.259794i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−22.6274	−	22.6274i	−1.18114	−	1.18114i	−0.979449	−	0.201693i	$-0.935356\pi$
$367$	−22.6274	−	22.6274i	−1.18114	−	1.18114i	−0.201693	−	0.979449i	$-0.564644\pi$
$368$		0			0
$369$	−10.5830			−0.550929
$370$		0			0
$371$	14.0000			0.726844
$372$		0			0
$373$	−7.48331	−	7.48331i	−0.387471	−	0.387471i	0.486313	−	0.873785i	$-0.338341\pi$
$373$	−7.48331	−	7.48331i	−0.387471	−	0.387471i	−0.873785	+	0.486313i	$0.838341\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.48528	−	8.48528i	−0.437014	−	0.437014i
$378$		0			0
$379$			12.0000i			0.616399i	0.951322	+	0.308199i	$0.0997264\pi$
$379$			12.0000i			0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$			5.29150i			0.271092i
$382$		0			0
$383$	−19.7990	+	19.7990i	−1.01168	+	1.01168i	−0.0117502	+	0.999931i	$0.503740\pi$
$383$	−19.7990	+	19.7990i	−1.01168	+	1.01168i	−0.999931	+	0.0117502i	$0.996260\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.74166	−	3.74166i	0.190199	−	0.190199i
$388$		0			0
$389$		−	2.00000i		−	0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$		−	2.00000i		−	0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$		0			0
$391$			21.1660i			1.07041i
$392$		0			0
$393$	7.48331	+	7.48331i	0.377483	+	0.377483i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	15.5563	+	15.5563i	0.780751	+	0.780751i	0.979957	−	0.199207i	$-0.0638365\pi$
$397$	15.5563	+	15.5563i	0.780751	+	0.780751i	−0.199207	+	0.979957i	$0.563837\pi$
$398$		0			0
$399$		−	14.0000i		−	0.700877i
$400$		0			0
$401$	−38.0000			−1.89763			−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000			−1.89763			−0.948815	+	0.315833i	$0.897716\pi$
$402$		0			0
$403$	7.48331	+	7.48331i	0.372770	+	0.372770i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−21.1660			−1.04659			−0.523296	−	0.852151i	$-0.675298\pi$
$409$	−21.1660			−1.04659			−0.523296	+	0.852151i	$0.675298\pi$
$410$		0			0
$411$			15.8745i			0.783032i
$412$		0			0
$413$	−19.7990	+	19.7990i	−0.974245	+	0.974245i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11.2250	+	11.2250i	−0.549689	+	0.549689i
$418$		0			0
$419$	21.1660			1.03403			0.517014	−	0.855977i	$-0.327044\pi$
$419$	21.1660			1.03403			0.517014	+	0.855977i	$0.327044\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	5.65685	−	5.65685i	0.275046	−	0.275046i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	19.7990	+	19.7990i	0.958140	+	0.958140i
$428$		0			0
$429$			4.00000i			0.193122i
$430$		0			0
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$		0			0
$433$	−9.89949	+	9.89949i	−0.475739	+	0.475739i	−0.903766	−	0.428027i	$-0.859209\pi$
$433$	−9.89949	+	9.89949i	−0.475739	+	0.475739i	0.428027	+	0.903766i	$0.359209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.7990	−	19.7990i	−0.947114	−	0.947114i
$438$		0			0
$439$	5.29150			0.252550			0.126275	−	0.991995i	$-0.459698\pi$
$439$	5.29150			0.252550			0.126275	+	0.991995i	$0.459698\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−26.1916	−	26.1916i	−1.24440	−	1.24440i	−0.958157	−	0.286244i	$-0.907593\pi$
$443$	−26.1916	−	26.1916i	−1.24440	−	1.24440i	−0.286244	−	0.958157i	$-0.592407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	4.24264	+	4.24264i	0.200670	+	0.200670i
$448$		0			0
$449$			34.0000i			1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$			34.0000i			1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$		0			0
$451$			21.1660i			0.996669i
$452$		0			0
$453$	−11.3137	+	11.3137i	−0.531564	+	0.531564i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.48331	+	7.48331i	−0.350055	+	0.350055i	−0.860130	−	0.510075i	$-0.829618\pi$
$457$	−7.48331	+	7.48331i	−0.350055	+	0.350055i	0.510075	+	0.860130i	$0.329618\pi$
$458$		0			0
$459$		−	4.00000i		−	0.186704i
$460$		0			0
$461$			21.1660i			0.985799i	0.870086	+	0.492900i	$0.164063\pi$
$461$			21.1660i			0.985799i	−0.870086	+	0.492900i	$0.835937\pi$
$462$		0			0
$463$	−3.74166	−	3.74166i	−0.173890	−	0.173890i	0.614796	−	0.788686i	$-0.289238\pi$
$463$	−3.74166	−	3.74166i	−0.173890	−	0.173890i	−0.788686	+	0.614796i	$0.789238\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.65685	−	5.65685i	−0.261768	−	0.261768i	0.564004	−	0.825772i	$-0.309260\pi$
$467$	−5.65685	−	5.65685i	−0.261768	−	0.261768i	−0.825772	+	0.564004i	$0.809260\pi$
$468$		0			0
$469$		−	14.0000i		−	0.646460i
$470$		0			0
$471$	14.0000			0.645086
$472$		0			0
$473$	−7.48331	−	7.48331i	−0.344083	−	0.344083i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.74166	+	3.74166i	−0.171319	+	0.171319i
$478$		0			0
$479$	−31.7490			−1.45065			−0.725325	−	0.688407i	$-0.758310\pi$
$479$	−31.7490			−1.45065			−0.725325	+	0.688407i	$0.758310\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−9.89949	+	9.89949i	−0.450443	+	0.450443i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	26.1916	−	26.1916i	1.18685	−	1.18685i	0.208923	−	0.977932i	$-0.433004\pi$
$487$	26.1916	−	26.1916i	1.18685	−	1.18685i	0.977932	−	0.208923i	$-0.0669957\pi$
$488$		0			0
$489$	−15.8745			−0.717870
$490$		0			0
$491$	34.0000			1.53440			0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000			1.53440			0.767199	+	0.641409i	$0.221650\pi$
$492$		0			0
$493$	16.9706	−	16.9706i	0.764316	−	0.764316i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.74166	−	3.74166i	0.167836	−	0.167836i
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$	8.48528	−	8.48528i	0.378340	−	0.378340i	−0.492163	−	0.870503i	$-0.663794\pi$
$503$	8.48528	−	8.48528i	0.378340	−	0.378340i	0.870503	+	0.492163i	$0.163794\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6.36396	−	6.36396i	−0.282633	−	0.282633i
$508$		0			0
$509$	−31.7490			−1.40725			−0.703625	−	0.710571i	$-0.748437\pi$
$509$	−31.7490			−1.40725			−0.703625	+	0.710571i	$0.748437\pi$
$510$		0			0
$511$			37.0405i			1.63858i
$512$		0			0
$513$	3.74166	+	3.74166i	0.165198	+	0.165198i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−11.3137	−	11.3137i	−0.497576	−	0.497576i
$518$		0			0
$519$		−	8.00000i		−	0.351161i
$520$		0			0
$521$		−	10.5830i		−	0.463650i	−0.972758	−	0.231825i	$-0.925530\pi$
$521$		−	10.5830i		−	0.463650i	0.972758	−	0.231825i	$-0.0744697\pi$
$522$		0			0
$523$	−19.7990	+	19.7990i	−0.865749	+	0.865749i	−0.991999	−	0.126249i	$-0.959706\pi$
$523$	−19.7990	+	19.7990i	−0.865749	+	0.865749i	0.126249	+	0.991999i	$0.459706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.9666	+	14.9666i	−0.651957	+	0.651957i
$528$		0			0
$529$			5.00000i			0.217391i
$530$		0			0
$531$		−	10.5830i		−	0.459263i
$532$		0			0
$533$	14.9666	+	14.9666i	0.648277	+	0.648277i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.7279	+	12.7279i	0.549250	+	0.549250i
$538$		0			0
$539$			14.0000i			0.603023i
$540$		0			0
$541$	18.0000			0.773880			0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000			0.773880			0.386940	+	0.922105i	$0.373532\pi$
$542$		0			0
$543$	7.48331	+	7.48331i	0.321140	+	0.321140i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.74166	+	3.74166i	−0.159982	+	0.159982i	−0.782559	−	0.622577i	$-0.786086\pi$
$547$	−3.74166	+	3.74166i	−0.159982	+	0.159982i	0.622577	+	0.782559i	$0.286086\pi$
$548$		0			0
$549$	−10.5830			−0.451672
$550$		0			0
$551$			31.7490i			1.35255i
$552$		0			0
$553$	−7.48331	−	7.48331i	−0.318223	−	0.318223i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.2250	−	11.2250i	0.475617	−	0.475617i	−0.428110	−	0.903727i	$-0.640820\pi$
$557$	11.2250	−	11.2250i	0.475617	−	0.475617i	0.903727	+	0.428110i	$0.140820\pi$
$558$		0			0
$559$	−10.5830			−0.447613
$560$		0			0
$561$	−8.00000			−0.337760
$562$		0			0
$563$	19.7990	−	19.7990i	0.834428	−	0.834428i	−0.153691	−	0.988119i	$-0.549116\pi$
$563$	19.7990	−	19.7990i	0.834428	−	0.834428i	0.988119	+	0.153691i	$0.0491160\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.87083	−	1.87083i	0.0785674	−	0.0785674i
$568$		0			0
$569$		−	34.0000i		−	1.42535i	−0.701492	−	0.712677i	$-0.747483\pi$
$569$		−	34.0000i		−	1.42535i	0.701492	−	0.712677i	$-0.252517\pi$
$570$		0			0
$571$	−24.0000			−1.00437			−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000			−1.00437			−0.502184	+	0.864761i	$0.667470\pi$
$572$		0			0
$573$	7.07107	−	7.07107i	0.295398	−	0.295398i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−32.5269	−	32.5269i	−1.35411	−	1.35411i	−0.881004	−	0.473109i	$-0.843132\pi$
$577$	−32.5269	−	32.5269i	−1.35411	−	1.35411i	−0.473109	−	0.881004i	$-0.656868\pi$
$578$		0			0
$579$	−10.5830			−0.439815
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7.48331	+	7.48331i	0.309927	+	0.309927i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.7990	+	19.7990i	0.817192	+	0.817192i	0.985700	−	0.168508i	$-0.0538950\pi$
$587$	19.7990	+	19.7990i	0.817192	+	0.817192i	−0.168508	+	0.985700i	$0.553895\pi$
$588$		0			0
$589$		−	28.0000i		−	1.15372i
$590$		0			0
$591$			15.8745i			0.652990i
$592$		0			0
$593$	5.65685	−	5.65685i	0.232299	−	0.232299i	−0.581353	−	0.813652i	$-0.697476\pi$
$593$	5.65685	−	5.65685i	0.232299	−	0.232299i	0.813652	+	0.581353i	$0.197476\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.74166	+	3.74166i	−0.153136	+	0.153136i
$598$		0			0
$599$			26.0000i			1.06233i	0.847268	+	0.531166i	$0.178246\pi$
$599$			26.0000i			1.06233i	−0.847268	+	0.531166i	$0.821754\pi$
$600$		0			0
$601$			21.1660i			0.863380i	0.902022	+	0.431690i	$0.142082\pi$
$601$			21.1660i			0.863380i	−0.902022	+	0.431690i	$0.857918\pi$
$602$		0			0
$603$	3.74166	+	3.74166i	0.152372	+	0.152372i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	33.9411	+	33.9411i	1.37763	+	1.37763i	0.848612	+	0.529016i	$0.177439\pi$
$607$	33.9411	+	33.9411i	1.37763	+	1.37763i	0.529016	+	0.848612i	$0.322561\pi$
$608$		0			0
$609$	15.8745			0.643268
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$	7.48331	+	7.48331i	0.302248	+	0.302248i	0.841893	−	0.539645i	$-0.181441\pi$
$613$	7.48331	+	7.48331i	0.302248	+	0.302248i	−0.539645	+	0.841893i	$0.681441\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	26.1916	−	26.1916i	1.05443	−	1.05443i	0.0560036	−	0.998431i	$-0.482164\pi$
$617$	26.1916	−	26.1916i	1.05443	−	1.05443i	0.998431	−	0.0560036i	$-0.0178358\pi$
$618$		0			0
$619$	26.4575			1.06342			0.531709	−	0.846927i	$-0.321550\pi$
$619$	26.4575			1.06342			0.531709	+	0.846927i	$0.321550\pi$
$620$		0			0
$621$		−	5.29150i		−	0.212341i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	7.48331	−	7.48331i	0.298855	−	0.298855i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$	14.1421	−	14.1421i	0.562099	−	0.562099i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	9.89949	+	9.89949i	0.392232	+	0.392232i
$638$		0			0
$639$			2.00000i			0.0791188i
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$	8.48528	−	8.48528i	0.334627	−	0.334627i	−0.519714	−	0.854341i	$-0.673961\pi$
$643$	8.48528	−	8.48528i	0.334627	−	0.334627i	0.854341	+	0.519714i	$0.173961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.7990	−	19.7990i	−0.778379	−	0.778379i	0.201176	−	0.979555i	$-0.435524\pi$
$647$	−19.7990	−	19.7990i	−0.778379	−	0.778379i	−0.979555	+	0.201176i	$0.935524\pi$
$648$		0			0
$649$	−21.1660			−0.830839
$650$		0			0
$651$	−14.0000			−0.548703
$652$		0			0
$653$	33.6749	+	33.6749i	1.31780	+	1.31780i	0.915513	+	0.402288i	$0.131785\pi$
$653$	33.6749	+	33.6749i	1.31780	+	1.31780i	0.402288	+	0.915513i	$0.368215\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−9.89949	−	9.89949i	−0.386216	−	0.386216i
$658$		0			0
$659$		−	6.00000i		−	0.233727i	−0.993148	−	0.116863i	$-0.962716\pi$
$659$		−	6.00000i		−	0.233727i	0.993148	−	0.116863i	$-0.0372840\pi$
$660$		0			0
$661$			10.5830i			0.411631i	0.978591	+	0.205816i	$0.0659847\pi$
$661$			10.5830i			0.411631i	−0.978591	+	0.205816i	$0.934015\pi$
$662$		0			0
$663$	−5.65685	+	5.65685i	−0.219694	+	0.219694i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	22.4499	−	22.4499i	0.869265	−	0.869265i
$668$		0			0
$669$			16.0000i			0.618596i
$670$		0			0
$671$			21.1660i			0.817105i
$672$		0			0
$673$	−7.48331	−	7.48331i	−0.288461	−	0.288461i	0.548011	−	0.836471i	$-0.315385\pi$
$673$	−7.48331	−	7.48331i	−0.288461	−	0.288461i	−0.836471	+	0.548011i	$0.815385\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.4558	+	25.4558i	0.978348	+	0.978348i	0.999771	−	0.0214229i	$-0.00681965\pi$
$677$	25.4558	+	25.4558i	0.978348	+	0.978348i	−0.0214229	+	0.999771i	$0.506820\pi$
$678$		0			0
$679$	−5.29150			−0.203069
$680$		0			0
$681$	24.0000			0.919682
$682$		0			0
$683$	3.74166	+	3.74166i	0.143171	+	0.143171i	0.775059	−	0.631889i	$-0.217720\pi$
$683$	3.74166	+	3.74166i	0.143171	+	0.143171i	−0.631889	+	0.775059i	$0.717720\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	7.48331	−	7.48331i	0.285506	−	0.285506i
$688$		0			0
$689$	10.5830			0.403180
$690$		0			0
$691$		−	37.0405i		−	1.40909i	−0.709660	−	0.704544i	$-0.751152\pi$
$691$		−	37.0405i		−	1.40909i	0.709660	−	0.704544i	$-0.248848\pi$
$692$		0			0
$693$	−3.74166	−	3.74166i	−0.142134	−	0.142134i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−29.9333	+	29.9333i	−1.13380	+	1.13380i
$698$		0			0
$699$	15.8745			0.600429
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.7990	+	19.7990i	0.744618	+	0.744618i
$708$		0			0
$709$			46.0000i			1.72757i	0.503864	+	0.863783i	$0.331911\pi$
$709$			46.0000i			1.72757i	−0.503864	+	0.863783i	$0.668089\pi$
$710$		0			0
$711$	4.00000			0.150012
$712$		0			0
$713$	−19.7990	+	19.7990i	−0.741478	+	0.741478i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−4.24264	−	4.24264i	−0.158444	−	0.158444i
$718$		0			0
$719$	10.5830			0.394679			0.197340	−	0.980335i	$-0.436770\pi$
$719$	10.5830			0.394679			0.197340	+	0.980335i	$0.436770\pi$
$720$		0			0
$721$		−	21.1660i		−	0.788263i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$		−	21.1660i		−	0.782853i
$732$		0			0
$733$	24.0416	−	24.0416i	0.887998	−	0.887998i	−0.106333	−	0.994331i	$-0.533911\pi$
$733$	24.0416	−	24.0416i	0.887998	−	0.887998i	0.994331	+	0.106333i	$0.0339109\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.48331	−	7.48331i	0.275651	−	0.275651i
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$		−	10.5830i		−	0.388776i
$742$		0			0
$743$	−18.7083	−	18.7083i	−0.686340	−	0.686340i	0.275081	−	0.961421i	$-0.411295\pi$
$743$	−18.7083	−	18.7083i	−0.686340	−	0.686340i	−0.961421	+	0.275081i	$0.911295\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			14.0000i			0.511549i
$750$		0			0
$751$	24.0000			0.875772			0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000			0.875772			0.437886	+	0.899030i	$0.355727\pi$
$752$		0			0
$753$	−7.48331	−	7.48331i	−0.272707	−	0.272707i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	29.9333	−	29.9333i	1.08794	−	1.08794i	0.0922025	−	0.995740i	$-0.470609\pi$
$757$	29.9333	−	29.9333i	1.08794	−	1.08794i	0.995740	−	0.0922025i	$-0.0293907\pi$
$758$		0			0
$759$	−10.5830			−0.384139
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−3.74166	−	3.74166i	−0.135457	−	0.135457i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.9666	+	14.9666i	−0.540414	+	0.540414i
$768$		0			0
$769$	21.1660			0.763266			0.381633	−	0.924314i	$-0.375362\pi$
$769$	21.1660			0.763266			0.381633	+	0.924314i	$0.375362\pi$
$770$		0			0
$771$	−28.0000			−1.00840
$772$		0			0
$773$	16.9706	−	16.9706i	0.610389	−	0.610389i	−0.332659	−	0.943047i	$-0.607946\pi$
$773$	16.9706	−	16.9706i	0.610389	−	0.610389i	0.943047	+	0.332659i	$0.107946\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	56.0000i		−	2.00641i
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$	−4.24264	+	4.24264i	−0.151620	+	0.151620i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−36.7696	−	36.7696i	−1.31069	−	1.31069i	−0.920904	−	0.389789i	$-0.872548\pi$
$787$	−36.7696	−	36.7696i	−1.31069	−	1.31069i	−0.389789	−	0.920904i	$-0.627452\pi$
$788$		0			0
$789$	26.4575			0.941912
$790$		0			0
$791$	42.0000			1.49335
$792$		0			0
$793$	14.9666	+	14.9666i	0.531481	+	0.531481i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.3137	−	11.3137i	−0.400752	−	0.400752i	0.477746	−	0.878498i	$-0.341454\pi$
$797$	−11.3137	−	11.3137i	−0.400752	−	0.400752i	−0.878498	+	0.477746i	$0.841454\pi$
$798$		0			0
$799$		−	32.0000i		−	1.13208i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.7990	+	19.7990i	−0.698691	+	0.698691i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	7.48331	−	7.48331i	0.263425	−	0.263425i
$808$		0			0
$809$		−	30.0000i		−	1.05474i	−0.849635	−	0.527372i	$-0.823177\pi$
$809$		−	30.0000i		−	1.05474i	0.849635	−	0.527372i	$-0.176823\pi$
$810$		0			0
$811$		−	37.0405i		−	1.30067i	−0.759648	−	0.650334i	$-0.774629\pi$
$811$		−	37.0405i		−	1.30067i	0.759648	−	0.650334i	$-0.225371\pi$
$812$		0			0
$813$	−18.7083	−	18.7083i	−0.656128	−	0.656128i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	19.7990	+	19.7990i	0.692679	+	0.692679i
$818$		0			0
$819$	−5.29150			−0.184900
$820$		0			0
$821$	−46.0000			−1.60541			−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000			−1.60541			−0.802706	+	0.596376i	$0.796607\pi$
$822$		0			0
$823$	−18.7083	−	18.7083i	−0.652130	−	0.652130i	0.301376	−	0.953506i	$-0.402554\pi$
$823$	−18.7083	−	18.7083i	−0.652130	−	0.652130i	−0.953506	+	0.301376i	$0.902554\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33.6749	+	33.6749i	−1.17099	+	1.17099i	−0.189018	+	0.981974i	$0.560530\pi$
$827$	−33.6749	+	33.6749i	−1.17099	+	1.17099i	−0.981974	+	0.189018i	$0.939470\pi$
$828$		0			0
$829$	−10.5830			−0.367563			−0.183781	−	0.982967i	$-0.558834\pi$
$829$	−10.5830			−0.367563			−0.183781	+	0.982967i	$0.558834\pi$
$830$		0			0
$831$		−	10.5830i		−	0.367120i
$832$		0			0
$833$	−19.7990	+	19.7990i	−0.685994	+	0.685994i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.74166	−	3.74166i	0.129331	−	0.129331i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−4.24264	+	4.24264i	−0.146124	+	0.146124i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.0958	−	13.0958i	0.449977	−	0.449977i
$848$		0			0
$849$			4.00000i			0.137280i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−9.89949	+	9.89949i	−0.338952	+	0.338952i	−0.855973	−	0.517021i	$-0.827041\pi$
$853$	−9.89949	+	9.89949i	−0.338952	+	0.338952i	0.517021	+	0.855973i	$0.327041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−39.5980	−	39.5980i	−1.35264	−	1.35264i	−0.882695	−	0.469946i	$-0.844273\pi$
$857$	−39.5980	−	39.5980i	−1.35264	−	1.35264i	−0.469946	−	0.882695i	$-0.655727\pi$
$858$		0			0
$859$	15.8745			0.541631			0.270816	−	0.962631i	$-0.412707\pi$
$859$	15.8745			0.541631			0.270816	+	0.962631i	$0.412707\pi$
$860$		0			0
$861$	−28.0000			−0.954237
$862$		0			0
$863$	3.74166	+	3.74166i	0.127367	+	0.127367i	0.767917	−	0.640549i	$-0.221293\pi$
$863$	3.74166	+	3.74166i	0.127367	+	0.127367i	−0.640549	+	0.767917i	$0.721293\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.707107	+	0.707107i	0.0240146	+	0.0240146i
$868$		0			0
$869$		−	8.00000i		−	0.271381i
$870$		0			0
$871$		−	10.5830i		−	0.358591i
$872$		0			0
$873$	1.41421	−	1.41421i	0.0478639	−	0.0478639i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−22.4499	+	22.4499i	−0.758081	+	0.758081i	−0.975973	−	0.217892i	$-0.930082\pi$
$877$	−22.4499	+	22.4499i	−0.758081	+	0.758081i	0.217892	+	0.975973i	$0.430082\pi$
$878$		0			0
$879$			12.0000i			0.404750i
$880$		0			0
$881$		−	42.3320i		−	1.42620i	−0.701061	−	0.713101i	$-0.747290\pi$
$881$		−	42.3320i		−	1.42620i	0.701061	−	0.713101i	$-0.252710\pi$
$882$		0			0
$883$	18.7083	+	18.7083i	0.629584	+	0.629584i	0.947963	−	0.318379i	$-0.103139\pi$
$883$	18.7083	+	18.7083i	0.629584	+	0.629584i	−0.318379	+	0.947963i	$0.603139\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$			14.0000i			0.469545i
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$	29.9333	+	29.9333i	1.00168	+	1.00168i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−7.48331	+	7.48331i	−0.249861	+	0.249861i
$898$		0			0
$899$	31.7490			1.05889
$900$		0			0
$901$			21.1660i			0.705142i
$902$		0			0
$903$	9.89949	−	9.89949i	0.329435	−	0.329435i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.74166	+	3.74166i	−0.124240	+	0.124240i	−0.766493	−	0.642253i	$-0.778000\pi$
$907$	−3.74166	+	3.74166i	−0.124240	+	0.124240i	0.642253	+	0.766493i	$0.278000\pi$
$908$		0			0
$909$	−10.5830			−0.351016
$910$		0			0
$911$	54.0000			1.78910			0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000			1.78910			0.894550	+	0.446968i	$0.147496\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19.7990	+	19.7990i	0.653820	+	0.653820i
$918$		0			0
$919$			24.0000i			0.791687i	0.918318	+	0.395843i	$0.129548\pi$
$919$			24.0000i			0.791687i	−0.918318	+	0.395843i	$0.870452\pi$
$920$		0			0
$921$	12.0000			0.395413
$922$		0			0
$923$	2.82843	−	2.82843i	0.0930988	−	0.0930988i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	5.65685	+	5.65685i	0.185795	+	0.185795i
$928$		0			0
$929$	21.1660			0.694434			0.347217	−	0.937785i	$-0.387127\pi$
$929$	21.1660			0.694434			0.347217	+	0.937785i	$0.387127\pi$
$930$		0			0
$931$		−	37.0405i		−	1.21395i
$932$		0			0
$933$	−22.4499	−	22.4499i	−0.734978	−	0.734978i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.41421	−	1.41421i	−0.0462003	−	0.0462003i	0.683629	−	0.729830i	$-0.260401\pi$
$937$	−1.41421	−	1.41421i	−0.0462003	−	0.0462003i	−0.729830	+	0.683629i	$0.760401\pi$
$938$		0			0
$939$		−	6.00000i		−	0.195803i
$940$		0			0
$941$		−	21.1660i		−	0.689992i	−0.938604	−	0.344996i	$-0.887880\pi$
$941$		−	21.1660i		−	0.689992i	0.938604	−	0.344996i	$-0.112120\pi$
$942$		0			0
$943$	−39.5980	+	39.5980i	−1.28949	+	1.28949i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.7083	+	18.7083i	−0.607938	+	0.607938i	−0.942407	−	0.334469i	$-0.891443\pi$
$947$	−18.7083	+	18.7083i	−0.607938	+	0.607938i	0.334469	+	0.942407i	$0.391443\pi$
$948$		0			0
$949$			28.0000i			0.908918i
$950$		0			0
$951$			26.4575i			0.857944i
$952$		0			0
$953$	26.1916	+	26.1916i	0.848429	+	0.848429i	0.989937	−	0.141508i	$-0.0451951\pi$
$953$	26.1916	+	26.1916i	0.848429	+	0.848429i	−0.141508	+	0.989937i	$0.545195\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	8.48528	+	8.48528i	0.274290	+	0.274290i
$958$		0			0
$959$			42.0000i			1.35625i
$960$		0			0
$961$	3.00000			0.0967742
$962$		0			0
$963$	−3.74166	−	3.74166i	−0.120573	−	0.120573i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	3.74166	−	3.74166i	0.120324	−	0.120324i	−0.644381	−	0.764705i	$-0.722885\pi$
$967$	3.74166	−	3.74166i	0.120324	−	0.120324i	0.764705	+	0.644381i	$0.222885\pi$
$968$		0			0
$969$	21.1660			0.679950
$970$		0			0
$971$		−	42.3320i		−	1.35850i	−0.733907	−	0.679250i	$-0.762305\pi$
$971$		−	42.3320i		−	1.35850i	0.733907	−	0.679250i	$-0.237695\pi$
$972$		0			0
$973$	−29.6985	+	29.6985i	−0.952090	+	0.952090i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.2250	−	11.2250i	0.359119	−	0.359119i	−0.504369	−	0.863488i	$-0.668275\pi$
$977$	11.2250	−	11.2250i	0.359119	−	0.359119i	0.863488	+	0.504369i	$0.168275\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$	22.6274	−	22.6274i	0.721703	−	0.721703i	−0.247249	−	0.968952i	$-0.579527\pi$
$983$	22.6274	−	22.6274i	0.721703	−	0.721703i	0.968952	+	0.247249i	$0.0795267\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	14.9666	−	14.9666i	0.476393	−	0.476393i
$988$		0			0
$989$		−	28.0000i		−	0.890348i
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$	−14.1421	+	14.1421i	−0.448787	+	0.448787i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	12.7279	+	12.7279i	0.403097	+	0.403097i	0.879323	−	0.476226i	$-0.157995\pi$
$997$	12.7279	+	12.7279i	0.403097	+	0.403097i	−0.476226	+	0.879323i	$0.657995\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.x.b.1693.1	yes	8
5.2	odd	4	inner	2100.2.x.b.1357.3	yes	8
5.3	odd	4	inner	2100.2.x.b.1357.2	yes	8
5.4	even	2	inner	2100.2.x.b.1693.4	yes	8
7.6	odd	2	inner	2100.2.x.b.1693.3	yes	8
35.13	even	4	inner	2100.2.x.b.1357.4	yes	8
35.27	even	4	inner	2100.2.x.b.1357.1	✓	8
35.34	odd	2	inner	2100.2.x.b.1693.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.x.b.1357.1	✓	8	35.27	even	4	inner
2100.2.x.b.1357.2	yes	8	5.3	odd	4	inner
2100.2.x.b.1357.3	yes	8	5.2	odd	4	inner
2100.2.x.b.1357.4	yes	8	35.13	even	4	inner
2100.2.x.b.1693.1	yes	8	1.1	even	1	trivial
2100.2.x.b.1693.2	yes	8	35.34	odd	2	inner
2100.2.x.b.1693.3	yes	8	7.6	odd	2	inner
2100.2.x.b.1693.4	yes	8	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} -2.00000 q^{11} +(-1.41421 + 1.41421i) q^{13} +(-2.82843 - 2.82843i) q^{17} +5.29150 q^{19} -2.64575i q^{21} +(-3.74166 - 3.74166i) q^{23} +(0.707107 + 0.707107i) q^{27} +6.00000i q^{29} -5.29150i q^{31} +(1.41421 - 1.41421i) q^{33} -2.00000i q^{39} -10.5830i q^{41} +(3.74166 + 3.74166i) q^{43} +(5.65685 + 5.65685i) q^{47} -7.00000i q^{49} +4.00000 q^{51} +(-3.74166 - 3.74166i) q^{53} +(-3.74166 + 3.74166i) q^{57} +10.5830 q^{59} -10.5830i q^{61} +(1.87083 + 1.87083i) q^{63} +(-3.74166 + 3.74166i) q^{67} +5.29150 q^{69} -2.00000 q^{71} +(9.89949 - 9.89949i) q^{73} +(3.74166 - 3.74166i) q^{77} +4.00000i q^{79} -1.00000 q^{81} +(-4.24264 - 4.24264i) q^{87} -5.29150i q^{91} +(3.74166 + 3.74166i) q^{93} +(1.41421 + 1.41421i) q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{11} + 32 q^{51} - 16 q^{71} - 8 q^{81}+O(q^{100}) \) 8 * q - 16 * q^11 + 32 * q^51 - 16 * q^71 - 8 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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