LMFDB - Embedded newform 2100.2.f.a.1049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1049,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1049");

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.f (of order $2$, degree $1$, not 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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	2100.1049
Dual form			2100.2.f.a.1049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 + 1.61803i) q^{3} +(-2.61803 - 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	$q+(0.618034 + 1.61803i) q^{3} +(-2.61803 - 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} -0.763932i q^{11} +1.23607 q^{13} -4.47214i q^{17} -7.23607i q^{19} +(-1.00000 - 4.47214i) q^{21} +7.70820 q^{23} +(-4.61803 - 2.38197i) q^{27} -4.00000i q^{29} -3.23607i q^{31} +(1.23607 - 0.472136i) q^{33} +4.47214i q^{37} +(0.763932 + 2.00000i) q^{39} +3.52786 q^{41} +7.23607i q^{43} +3.23607i q^{47} +(6.70820 + 2.00000i) q^{49} +(7.23607 - 2.76393i) q^{51} +9.23607 q^{53} +(11.7082 - 4.47214i) q^{57} +8.94427 q^{59} -4.94427i q^{61} +(6.61803 - 4.38197i) q^{63} +9.70820i q^{67} +(4.76393 + 12.4721i) q^{69} -12.1803i q^{71} -6.76393 q^{73} +(-0.291796 + 2.00000i) q^{77} -10.4721 q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607i q^{83} +(6.47214 - 2.47214i) q^{87} -12.4721 q^{89} +(-3.23607 - 0.472136i) q^{91} +(5.23607 - 2.00000i) q^{93} +14.1803 q^{97} +(1.52786 + 1.70820i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 6 q^{7}+O(q^{10}) $ 4 * q - 2 * q^3 - 6 * q^7	$ 4 q - 2 q^{3} - 6 q^{7} - 4 q^{13} - 4 q^{21} + 4 q^{23} - 14 q^{27} - 4 q^{33} + 12 q^{39} + 32 q^{41} + 20 q^{51} + 28 q^{53} + 20 q^{57} + 22 q^{63} + 28 q^{69} - 36 q^{73} - 28 q^{77} - 24 q^{79} + 4 q^{81} + 8 q^{87} - 32 q^{89} - 4 q^{91} + 12 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 6 * q^7 - 4 * q^13 - 4 * q^21 + 4 * q^23 - 14 * q^27 - 4 * q^33 + 12 * q^39 + 32 * q^41 + 20 * q^51 + 28 * q^53 + 20 * q^57 + 22 * q^63 + 28 * q^69 - 36 * q^73 - 28 * q^77 - 24 * q^79 + 4 * q^81 + 8 * q^87 - 32 * q^89 - 4 * q^91 + 12 * q^93 + 12 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.61803	−	0.381966i	−0.989524	−	0.144370i
$7$	−2.61803	−	0.381966i	−0.989524	−	0.144370i
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$		−	0.763932i		−	0.230334i	−0.993346	−	0.115167i	$-0.963260\pi$
$11$		−	0.763932i		−	0.230334i	0.993346	−	0.115167i	$-0.0367403\pi$
$12$		0			0
$13$	1.23607			0.342824			0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607			0.342824			0.171412	+	0.985199i	$0.445167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.47214i		−	1.08465i	−0.840168	−	0.542326i	$-0.817544\pi$
$17$		−	4.47214i		−	1.08465i	0.840168	−	0.542326i	$-0.182456\pi$
$18$		0			0
$19$		−	7.23607i		−	1.66007i	−0.557713	−	0.830034i	$-0.688321\pi$
$19$		−	7.23607i		−	1.66007i	0.557713	−	0.830034i	$-0.311679\pi$
$20$		0			0
$21$	−1.00000	−	4.47214i	−0.218218	−	0.975900i
$22$		0			0
$23$	7.70820			1.60727			0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820			1.60727			0.803636	+	0.595121i	$0.202896\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.61803	−	2.38197i	−0.888741	−	0.458410i
$28$		0			0
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$		−	3.23607i		−	0.581215i	−0.956842	−	0.290607i	$-0.906143\pi$
$31$		−	3.23607i		−	0.581215i	0.956842	−	0.290607i	$-0.0938574\pi$
$32$		0			0
$33$	1.23607	−	0.472136i	0.215172	−	0.0821883i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			4.47214i			0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$37$			4.47214i			0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$38$		0			0
$39$	0.763932	+	2.00000i	0.122327	+	0.320256i
$40$		0			0
$41$	3.52786			0.550960			0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786			0.550960			0.275480	+	0.961307i	$0.411163\pi$
$42$		0			0
$43$			7.23607i			1.10349i	0.834013	+	0.551745i	$0.186038\pi$
$43$			7.23607i			1.10349i	−0.834013	+	0.551745i	$0.813962\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.23607i			0.472029i	0.971750	+	0.236015i	$0.0758413\pi$
$47$			3.23607i			0.472029i	−0.971750	+	0.236015i	$0.924159\pi$
$48$		0			0
$49$	6.70820	+	2.00000i	0.958315	+	0.285714i
$50$		0			0
$51$	7.23607	−	2.76393i	1.01325	−	0.387028i
$52$		0			0
$53$	9.23607			1.26867			0.634336	−	0.773058i	$-0.281274\pi$
$53$	9.23607			1.26867			0.634336	+	0.773058i	$0.281274\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	11.7082	−	4.47214i	1.55079	−	0.592349i
$58$		0			0
$59$	8.94427			1.16445			0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427			1.16445			0.582223	+	0.813029i	$0.302183\pi$
$60$		0			0
$61$		−	4.94427i		−	0.633049i	−0.948584	−	0.316525i	$-0.897484\pi$
$61$		−	4.94427i		−	0.633049i	0.948584	−	0.316525i	$-0.102516\pi$
$62$		0			0
$63$	6.61803	−	4.38197i	0.833794	−	0.552076i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			9.70820i			1.18605i	0.805186	+	0.593023i	$0.202066\pi$
$67$			9.70820i			1.18605i	−0.805186	+	0.593023i	$0.797934\pi$
$68$		0			0
$69$	4.76393	+	12.4721i	0.573510	+	1.50147i
$70$		0			0
$71$		−	12.1803i		−	1.44554i	−0.691088	−	0.722770i	$-0.742869\pi$
$71$		−	12.1803i		−	1.44554i	0.691088	−	0.722770i	$-0.257131\pi$
$72$		0			0
$73$	−6.76393			−0.791658			−0.395829	−	0.918324i	$-0.629543\pi$
$73$	−6.76393			−0.791658			−0.395829	+	0.918324i	$0.629543\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.291796	+	2.00000i	−0.0332532	+	0.227921i
$78$		0			0
$79$	−10.4721			−1.17821			−0.589104	−	0.808057i	$-0.700519\pi$
$79$	−10.4721			−1.17821			−0.589104	+	0.808057i	$0.700519\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$		−	7.23607i		−	0.794262i	−0.917762	−	0.397131i	$-0.870006\pi$
$83$		−	7.23607i		−	0.794262i	0.917762	−	0.397131i	$-0.129994\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	6.47214	−	2.47214i	0.693886	−	0.265041i
$88$		0			0
$89$	−12.4721			−1.32204			−0.661022	−	0.750367i	$-0.729877\pi$
$89$	−12.4721			−1.32204			−0.661022	+	0.750367i	$0.729877\pi$
$90$		0			0
$91$	−3.23607	−	0.472136i	−0.339232	−	0.0494933i
$92$		0			0
$93$	5.23607	−	2.00000i	0.542955	−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.1803			1.43980			0.719898	−	0.694080i	$-0.244189\pi$
$97$	14.1803			1.43980			0.719898	+	0.694080i	$0.244189\pi$
$98$		0			0
$99$	1.52786	+	1.70820i	0.153556	+	0.171681i
$100$		0			0
$101$	6.94427			0.690981			0.345490	−	0.938422i	$-0.387713\pi$
$101$	6.94427			0.690981			0.345490	+	0.938422i	$0.387713\pi$
$102$		0			0
$103$	−10.7639			−1.06060			−0.530301	−	0.847810i	$-0.677921\pi$
$103$	−10.7639			−1.06060			−0.530301	+	0.847810i	$0.677921\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.291796			0.0282090			0.0141045	−	0.999901i	$-0.495510\pi$
$107$	0.291796			0.0282090			0.0141045	+	0.999901i	$0.495510\pi$
$108$		0			0
$109$	2.94427			0.282010			0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427			0.282010			0.141005	+	0.990009i	$0.454967\pi$
$110$		0			0
$111$	−7.23607	+	2.76393i	−0.686817	+	0.262341i
$112$		0			0
$113$	17.2361			1.62143			0.810716	−	0.585439i	$-0.199078\pi$
$113$	17.2361			1.62143			0.810716	+	0.585439i	$0.199078\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.76393	+	2.47214i	−0.255526	+	0.228549i
$118$		0			0
$119$	−1.70820	+	11.7082i	−0.156591	+	1.07329i
$120$		0			0
$121$	10.4164			0.946946
$122$		0			0
$123$	2.18034	+	5.70820i	0.196595	+	0.514691i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	21.1246i		−	1.87451i	−0.348650	−	0.937253i	$-0.613360\pi$
$127$		−	21.1246i		−	1.87451i	0.348650	−	0.937253i	$-0.386640\pi$
$128$		0			0
$129$	−11.7082	+	4.47214i	−1.03085	+	0.393750i
$130$		0			0
$131$	−15.4164			−1.34694			−0.673469	−	0.739216i	$-0.735196\pi$
$131$	−15.4164			−1.34694			−0.673469	+	0.739216i	$0.735196\pi$
$132$		0			0
$133$	−2.76393	+	18.9443i	−0.239663	+	1.64268i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.23607			0.105604			0.0528022	−	0.998605i	$-0.483185\pi$
$137$	1.23607			0.105604			0.0528022	+	0.998605i	$0.483185\pi$
$138$		0			0
$139$		−	15.2361i		−	1.29231i	−0.763208	−	0.646153i	$-0.776377\pi$
$139$		−	15.2361i		−	1.29231i	0.763208	−	0.646153i	$-0.223623\pi$
$140$		0			0
$141$	−5.23607	+	2.00000i	−0.440956	+	0.168430i
$142$		0			0
$143$		−	0.944272i		−	0.0789640i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.909830	+	12.0902i	0.0750415	+	0.997180i
$148$		0			0
$149$		−	15.4164i		−	1.26296i	−0.775392	−	0.631481i	$-0.782448\pi$
$149$		−	15.4164i		−	1.26296i	0.775392	−	0.631481i	$-0.217552\pi$
$150$		0			0
$151$	8.94427			0.727875			0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427			0.727875			0.363937	+	0.931423i	$0.381432\pi$
$152$		0			0
$153$	8.94427	+	10.0000i	0.723102	+	0.808452i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.81966			−0.145225			−0.0726123	−	0.997360i	$-0.523134\pi$
$157$	−1.81966			−0.145225			−0.0726123	+	0.997360i	$0.523134\pi$
$158$		0			0
$159$	5.70820	+	14.9443i	0.452690	+	1.18516i
$160$		0			0
$161$	−20.1803	−	2.94427i	−1.59043	−	0.232041i
$162$		0			0
$163$		−	18.6525i		−	1.46097i	−0.682926	−	0.730487i	$-0.739293\pi$
$163$		−	18.6525i		−	1.46097i	0.682926	−	0.730487i	$-0.260707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.76393i			0.678173i	0.940755	+	0.339087i	$0.110118\pi$
$167$			8.76393i			0.678173i	−0.940755	+	0.339087i	$0.889882\pi$
$168$		0			0
$169$	−11.4721			−0.882472
$170$		0			0
$171$	14.4721	+	16.1803i	1.10671	+	1.23734i
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.52786	+	14.4721i	0.415500	+	1.08779i
$178$		0			0
$179$			12.7639i			0.954021i	0.878898	+	0.477011i	$0.158280\pi$
$179$			12.7639i			0.954021i	−0.878898	+	0.477011i	$0.841720\pi$
$180$		0			0
$181$			6.47214i			0.481070i	0.970640	+	0.240535i	$0.0773229\pi$
$181$			6.47214i			0.481070i	−0.970640	+	0.240535i	$0.922677\pi$
$182$		0			0
$183$	8.00000	−	3.05573i	0.591377	−	0.225886i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.41641			−0.249832
$188$		0			0
$189$	11.1803	+	8.00000i	0.813250	+	0.581914i
$190$		0			0
$191$		−	7.23607i		−	0.523584i	−0.965124	−	0.261792i	$-0.915687\pi$
$191$		−	7.23607i		−	0.523584i	0.965124	−	0.261792i	$-0.0843134\pi$
$192$		0			0
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	25.2361			1.79800			0.898998	−	0.437953i	$-0.144296\pi$
$197$	25.2361			1.79800			0.898998	+	0.437953i	$0.144296\pi$
$198$		0			0
$199$		−	8.76393i		−	0.621259i	−0.950531	−	0.310629i	$-0.899460\pi$
$199$		−	8.76393i		−	0.621259i	0.950531	−	0.310629i	$-0.100540\pi$
$200$		0			0
$201$	−15.7082	+	6.00000i	−1.10797	+	0.423207i
$202$		0			0
$203$	−1.52786	+	10.4721i	−0.107235	+	0.735000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−17.2361	+	15.4164i	−1.19799	+	1.07151i
$208$		0			0
$209$	−5.52786			−0.382370
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$	19.7082	−	7.52786i	1.35038	−	0.515801i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.23607	+	8.47214i	−0.0839098	+	0.575126i
$218$		0			0
$219$	−4.18034	−	10.9443i	−0.282481	−	0.739545i
$220$		0			0
$221$		−	5.52786i		−	0.371844i
$222$		0			0
$223$	−11.7082			−0.784039			−0.392020	−	0.919957i	$-0.628223\pi$
$223$	−11.7082			−0.784039			−0.392020	+	0.919957i	$0.628223\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	23.2361i		−	1.54223i	−0.636695	−	0.771116i	$-0.719699\pi$
$227$		−	23.2361i		−	1.54223i	0.636695	−	0.771116i	$-0.280301\pi$
$228$		0			0
$229$		−	13.8885i		−	0.917781i	−0.888493	−	0.458890i	$-0.848247\pi$
$229$		−	13.8885i		−	0.917781i	0.888493	−	0.458890i	$-0.151753\pi$
$230$		0			0
$231$	−3.41641	+	0.763932i	−0.224783	+	0.0502630i
$232$		0			0
$233$	−1.81966			−0.119210			−0.0596049	−	0.998222i	$-0.518984\pi$
$233$	−1.81966			−0.119210			−0.0596049	+	0.998222i	$0.518984\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.47214	−	16.9443i	−0.420410	−	1.10065i
$238$		0			0
$239$			0.180340i			0.0116652i	0.999983	+	0.00583261i	$0.00185659\pi$
$239$			0.180340i			0.0116652i	−0.999983	+	0.00583261i	$0.998143\pi$
$240$		0			0
$241$			16.3607i			1.05388i	0.849901	+	0.526942i	$0.176662\pi$
$241$			16.3607i			1.05388i	−0.849901	+	0.526942i	$0.823338\pi$
$242$		0			0
$243$	15.0902	−	3.90983i	0.968035	−	0.250816i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	8.94427i		−	0.569110i
$248$		0			0
$249$	11.7082	−	4.47214i	0.741977	−	0.283410i
$250$		0			0
$251$	20.3607			1.28515			0.642577	−	0.766221i	$-0.277865\pi$
$251$	20.3607			1.28515			0.642577	+	0.766221i	$0.277865\pi$
$252$		0			0
$253$		−	5.88854i		−	0.370210i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.4721i			1.02750i	0.857939	+	0.513752i	$0.171745\pi$
$257$			16.4721i			1.02750i	−0.857939	+	0.513752i	$0.828255\pi$
$258$		0			0
$259$	1.70820	−	11.7082i	0.106143	−	0.727512i
$260$		0			0
$261$	8.00000	+	8.94427i	0.495188	+	0.553637i
$262$		0			0
$263$	−22.1803			−1.36770			−0.683849	−	0.729623i	$-0.739695\pi$
$263$	−22.1803			−1.36770			−0.683849	+	0.729623i	$0.739695\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−7.70820	−	20.1803i	−0.471734	−	1.23502i
$268$		0			0
$269$	19.5279			1.19063			0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279			1.19063			0.595317	+	0.803491i	$0.297026\pi$
$270$		0			0
$271$		−	21.1246i		−	1.28323i	−0.767027	−	0.641614i	$-0.778265\pi$
$271$		−	21.1246i		−	1.28323i	0.767027	−	0.641614i	$-0.221735\pi$
$272$		0			0
$273$	−1.23607	−	5.52786i	−0.0748102	−	0.334562i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			1.41641i			0.0851037i	0.999094	+	0.0425519i	$0.0135488\pi$
$277$			1.41641i			0.0851037i	−0.999094	+	0.0425519i	$0.986451\pi$
$278$		0			0
$279$	6.47214	+	7.23607i	0.387477	+	0.433212i
$280$		0			0
$281$			11.4164i			0.681046i	0.940236	+	0.340523i	$0.110604\pi$
$281$			11.4164i			0.681046i	−0.940236	+	0.340523i	$0.889396\pi$
$282$		0			0
$283$	−8.65248			−0.514336			−0.257168	−	0.966367i	$-0.582789\pi$
$283$	−8.65248			−0.514336			−0.257168	+	0.966367i	$0.582789\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.23607	−	1.34752i	−0.545188	−	0.0795418i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	8.76393	+	22.9443i	0.513751	+	1.34502i
$292$		0			0
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.81966	+	3.52786i	−0.105587	+	0.204707i
$298$		0			0
$299$	9.52786			0.551011
$300$		0			0
$301$	2.76393	−	18.9443i	0.159310	−	1.09193i
$302$		0			0
$303$	4.29180	+	11.2361i	0.246557	+	0.645495i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−18.1803			−1.03761			−0.518803	−	0.854894i	$-0.673622\pi$
$307$	−18.1803			−1.03761			−0.518803	+	0.854894i	$0.673622\pi$
$308$		0			0
$309$	−6.65248	−	17.4164i	−0.378446	−	0.990785i
$310$		0			0
$311$	32.3607			1.83501			0.917503	−	0.397729i	$-0.130202\pi$
$311$	32.3607			1.83501			0.917503	+	0.397729i	$0.130202\pi$
$312$		0			0
$313$	4.29180			0.242587			0.121293	−	0.992617i	$-0.461296\pi$
$313$	4.29180			0.242587			0.121293	+	0.992617i	$0.461296\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.18034			0.347122			0.173561	−	0.984823i	$-0.444473\pi$
$317$	6.18034			0.347122			0.173561	+	0.984823i	$0.444473\pi$
$318$		0			0
$319$	−3.05573			−0.171088
$320$		0			0
$321$	0.180340	+	0.472136i	0.0100656	+	0.0263521i
$322$		0			0
$323$	−32.3607			−1.80060
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.81966	+	4.76393i	0.100627	+	0.263446i
$328$		0			0
$329$	1.23607	−	8.47214i	0.0681466	−	0.467084i
$330$		0			0
$331$	−20.9443			−1.15120			−0.575601	−	0.817731i	$-0.695232\pi$
$331$	−20.9443			−1.15120			−0.575601	+	0.817731i	$0.695232\pi$
$332$		0			0
$333$	−8.94427	−	10.0000i	−0.490143	−	0.547997i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			2.00000i			0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$			2.00000i			0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$		0			0
$339$	10.6525	+	27.8885i	0.578563	+	1.51470i
$340$		0			0
$341$	−2.47214			−0.133874
$342$		0			0
$343$	−16.7984	−	7.79837i	−0.907027	−	0.421073i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−32.6525			−1.75288			−0.876438	−	0.481514i	$-0.840087\pi$
$347$	−32.6525			−1.75288			−0.876438	+	0.481514i	$0.840087\pi$
$348$		0			0
$349$		−	8.94427i		−	0.478776i	−0.970924	−	0.239388i	$-0.923053\pi$
$349$		−	8.94427i		−	0.478776i	0.970924	−	0.239388i	$-0.0769468\pi$
$350$		0			0
$351$	−5.70820	−	2.94427i	−0.304681	−	0.157154i
$352$		0			0
$353$			25.4164i			1.35278i	0.736544	+	0.676389i	$0.236456\pi$
$353$			25.4164i			1.35278i	−0.736544	+	0.676389i	$0.763544\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−20.0000	+	4.47214i	−1.05851	+	0.236691i
$358$		0			0
$359$			34.0689i			1.79809i	0.437859	+	0.899043i	$0.355737\pi$
$359$			34.0689i			1.79809i	−0.437859	+	0.899043i	$0.644263\pi$
$360$		0			0
$361$	−33.3607			−1.75583
$362$		0			0
$363$	6.43769	+	16.8541i	0.337891	+	0.884611i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	24.6525			1.28685			0.643424	−	0.765510i	$-0.277513\pi$
$367$	24.6525			1.28685			0.643424	+	0.765510i	$0.277513\pi$
$368$		0			0
$369$	−7.88854	+	7.05573i	−0.410661	+	0.367307i
$370$		0			0
$371$	−24.1803	−	3.52786i	−1.25538	−	0.183158i
$372$		0			0
$373$		−	7.52786i		−	0.389778i	−0.980825	−	0.194889i	$-0.937565\pi$
$373$		−	7.52786i		−	0.389778i	0.980825	−	0.194889i	$-0.0624347\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.94427i		−	0.254643i
$378$		0			0
$379$	−1.52786			−0.0784811			−0.0392406	−	0.999230i	$-0.512494\pi$
$379$	−1.52786			−0.0784811			−0.0392406	+	0.999230i	$0.512494\pi$
$380$		0			0
$381$	34.1803	−	13.0557i	1.75111	−	0.668865i
$382$		0			0
$383$		−	15.2361i		−	0.778527i	−0.921127	−	0.389263i	$-0.872730\pi$
$383$		−	15.2361i		−	0.778527i	0.921127	−	0.389263i	$-0.127270\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−14.4721	−	16.1803i	−0.735660	−	0.822493i
$388$		0			0
$389$			2.47214i			0.125342i	0.998034	+	0.0626711i	$0.0199619\pi$
$389$			2.47214i			0.125342i	−0.998034	+	0.0626711i	$0.980038\pi$
$390$		0			0
$391$		−	34.4721i		−	1.74333i
$392$		0			0
$393$	−9.52786	−	24.9443i	−0.480617	−	1.25827i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.70820			−0.186109			−0.0930547	−	0.995661i	$-0.529663\pi$
$397$	−3.70820			−0.186109			−0.0930547	+	0.995661i	$0.529663\pi$
$398$		0			0
$399$	−32.3607	+	7.23607i	−1.62006	+	0.362257i
$400$		0			0
$401$			28.9443i			1.44541i	0.691158	+	0.722704i	$0.257101\pi$
$401$			28.9443i			1.44541i	−0.691158	+	0.722704i	$0.742899\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.41641			0.169345
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$	0.763932	+	2.00000i	0.0376820	+	0.0986527i
$412$		0			0
$413$	−23.4164	−	3.41641i	−1.15225	−	0.168110i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.6525	−	9.41641i	1.20724	−	0.461123i
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	32.4721			1.58260			0.791298	−	0.611431i	$-0.209406\pi$
$421$	32.4721			1.58260			0.791298	+	0.611431i	$0.209406\pi$
$422$		0			0
$423$	−6.47214	−	7.23607i	−0.314686	−	0.351830i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.88854	+	12.9443i	−0.0913930	+	0.626417i
$428$		0			0
$429$	1.52786	−	0.583592i	0.0737660	−	0.0281761i
$430$		0			0
$431$			0.763932i			0.0367973i	0.999831	+	0.0183987i	$0.00585680\pi$
$431$			0.763932i			0.0367973i	−0.999831	+	0.0183987i	$0.994143\pi$
$432$		0			0
$433$	−32.6525			−1.56918			−0.784589	−	0.620016i	$-0.787126\pi$
$433$	−32.6525			−1.56918			−0.784589	+	0.620016i	$0.787126\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	55.7771i		−	2.66818i
$438$		0			0
$439$			4.18034i			0.199517i	0.995012	+	0.0997584i	$0.0318070\pi$
$439$			4.18034i			0.199517i	−0.995012	+	0.0997584i	$0.968193\pi$
$440$		0			0
$441$	−19.0000	+	8.94427i	−0.904762	+	0.425918i
$442$		0			0
$443$	−24.0689			−1.14355			−0.571774	−	0.820411i	$-0.693744\pi$
$443$	−24.0689			−1.14355			−0.571774	+	0.820411i	$0.693744\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	24.9443	−	9.52786i	1.17982	−	0.450653i
$448$		0			0
$449$		−	29.8885i		−	1.41053i	−0.708945	−	0.705264i	$-0.750829\pi$
$449$		−	29.8885i		−	1.41053i	0.708945	−	0.705264i	$-0.249171\pi$
$450$		0			0
$451$		−	2.69505i		−	0.126905i
$452$		0			0
$453$	5.52786	+	14.4721i	0.259722	+	0.679960i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	9.05573i		−	0.423609i	−0.977312	−	0.211805i	$-0.932066\pi$
$457$		−	9.05573i		−	0.423609i	0.977312	−	0.211805i	$-0.0679340\pi$
$458$		0			0
$459$	−10.6525	+	20.6525i	−0.497215	+	0.963975i
$460$		0			0
$461$	24.4721			1.13978			0.569891	−	0.821721i	$-0.306986\pi$
$461$	24.4721			1.13978			0.569891	+	0.821721i	$0.306986\pi$
$462$		0			0
$463$			20.7639i			0.964982i	0.875901	+	0.482491i	$0.160268\pi$
$463$			20.7639i			0.964982i	−0.875901	+	0.482491i	$0.839732\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			36.5410i			1.69092i	0.534041	+	0.845458i	$0.320673\pi$
$467$			36.5410i			1.69092i	−0.534041	+	0.845458i	$0.679327\pi$
$468$		0			0
$469$	3.70820	−	25.4164i	0.171229	−	1.17362i
$470$		0			0
$471$	−1.12461	−	2.94427i	−0.0518194	−	0.135665i
$472$		0			0
$473$	5.52786			0.254171
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−20.6525	+	18.4721i	−0.945612	+	0.845781i
$478$		0			0
$479$	12.9443			0.591439			0.295719	−	0.955275i	$-0.404441\pi$
$479$	12.9443			0.591439			0.295719	+	0.955275i	$0.404441\pi$
$480$		0			0
$481$			5.52786i			0.252049i
$482$		0			0
$483$	−7.70820	−	34.4721i	−0.350735	−	1.56854i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	3.23607i		−	0.146640i	−0.997308	−	0.0733201i	$-0.976641\pi$
$487$		−	3.23607i		−	0.146640i	0.997308	−	0.0733201i	$-0.0233595\pi$
$488$		0			0
$489$	30.1803	−	11.5279i	1.36480	−	0.521308i
$490$		0			0
$491$			41.1246i			1.85593i	0.372670	+	0.927964i	$0.378442\pi$
$491$			41.1246i			1.85593i	−0.372670	+	0.927964i	$0.621558\pi$
$492$		0			0
$493$	−17.8885			−0.805659
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.65248	+	31.8885i	−0.208692	+	1.43040i
$498$		0			0
$499$	16.3607			0.732405			0.366202	−	0.930535i	$-0.380658\pi$
$499$	16.3607			0.732405			0.366202	+	0.930535i	$0.380658\pi$
$500$		0			0
$501$	−14.1803	+	5.41641i	−0.633531	+	0.241987i
$502$		0			0
$503$			18.6525i			0.831673i	0.909439	+	0.415836i	$0.136511\pi$
$503$			18.6525i			0.831673i	−0.909439	+	0.415836i	$0.863489\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.09017	−	18.5623i	−0.314886	−	0.824381i
$508$		0			0
$509$	−22.9443			−1.01699			−0.508493	−	0.861066i	$-0.669797\pi$
$509$	−22.9443			−1.01699			−0.508493	+	0.861066i	$0.669797\pi$
$510$		0			0
$511$	17.7082	+	2.58359i	0.783365	+	0.114291i
$512$		0			0
$513$	−17.2361	+	33.4164i	−0.760991	+	1.47537i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.47214			0.108724
$518$		0			0
$519$	−3.23607	+	1.23607i	−0.142048	+	0.0542574i
$520$		0			0
$521$	7.88854			0.345603			0.172802	−	0.984957i	$-0.444718\pi$
$521$	7.88854			0.345603			0.172802	+	0.984957i	$0.444718\pi$
$522$		0			0
$523$	8.29180			0.362575			0.181287	−	0.983430i	$-0.441974\pi$
$523$	8.29180			0.362575			0.181287	+	0.983430i	$0.441974\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.4721			−0.630416
$528$		0			0
$529$	36.4164			1.58332
$530$		0			0
$531$	−20.0000	+	17.8885i	−0.867926	+	0.776297i
$532$		0			0
$533$	4.36068			0.188882
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−20.6525	+	7.88854i	−0.891220	+	0.340416i
$538$		0			0
$539$	1.52786	−	5.12461i	0.0658098	−	0.220733i
$540$		0			0
$541$	−43.8885			−1.88692			−0.943458	−	0.331492i	$-0.892448\pi$
$541$	−43.8885			−1.88692			−0.943458	+	0.331492i	$0.892448\pi$
$542$		0			0
$543$	−10.4721	+	4.00000i	−0.449402	+	0.171656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			17.7082i			0.757148i	0.925571	+	0.378574i	$0.123585\pi$
$547$			17.7082i			0.757148i	−0.925571	+	0.378574i	$0.876415\pi$
$548$		0			0
$549$	9.88854	+	11.0557i	0.422033	+	0.471847i
$550$		0			0
$551$	−28.9443			−1.23307
$552$		0			0
$553$	27.4164	+	4.00000i	1.16586	+	0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.1803			0.600840			0.300420	−	0.953807i	$-0.402873\pi$
$557$	14.1803			0.600840			0.300420	+	0.953807i	$0.402873\pi$
$558$		0			0
$559$			8.94427i			0.378302i
$560$		0			0
$561$	−2.11146	−	5.52786i	−0.0891457	−	0.233387i
$562$		0			0
$563$			18.0689i			0.761513i	0.924675	+	0.380756i	$0.124336\pi$
$563$			18.0689i			0.761513i	−0.924675	+	0.380756i	$0.875664\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.03444	+	23.0344i	−0.253423	+	0.967356i
$568$		0			0
$569$		−	21.8885i		−	0.917615i	−0.888536	−	0.458808i	$-0.848277\pi$
$569$		−	21.8885i		−	0.917615i	0.888536	−	0.458808i	$-0.151723\pi$
$570$		0			0
$571$	6.83282			0.285944			0.142972	−	0.989727i	$-0.454334\pi$
$571$	6.83282			0.285944			0.142972	+	0.989727i	$0.454334\pi$
$572$		0			0
$573$	11.7082	−	4.47214i	0.489117	−	0.186826i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	32.0689			1.33505			0.667523	−	0.744590i	$-0.267355\pi$
$577$	32.0689			1.33505			0.667523	+	0.744590i	$0.267355\pi$
$578$		0			0
$579$	−9.70820	+	3.70820i	−0.403459	+	0.154108i
$580$		0			0
$581$	−2.76393	+	18.9443i	−0.114667	+	0.785941i
$582$		0			0
$583$		−	7.05573i		−	0.292218i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			6.29180i			0.259690i	0.991534	+	0.129845i	$0.0414480\pi$
$587$			6.29180i			0.259690i	−0.991534	+	0.129845i	$0.958552\pi$
$588$		0			0
$589$	−23.4164			−0.964856
$590$		0			0
$591$	15.5967	+	40.8328i	0.641564	+	1.67964i
$592$		0			0
$593$		−	19.5279i		−	0.801913i	−0.916097	−	0.400957i	$-0.868678\pi$
$593$		−	19.5279i		−	0.801913i	0.916097	−	0.400957i	$-0.131322\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	14.1803	−	5.41641i	0.580363	−	0.221679i
$598$		0			0
$599$		−	30.6525i		−	1.25243i	−0.779652	−	0.626213i	$-0.784604\pi$
$599$		−	30.6525i		−	1.25243i	0.779652	−	0.626213i	$-0.215396\pi$
$600$		0			0
$601$			45.3050i			1.84803i	0.382359	+	0.924014i	$0.375112\pi$
$601$			45.3050i			1.84803i	−0.382359	+	0.924014i	$0.624888\pi$
$602$		0			0
$603$	−19.4164	−	21.7082i	−0.790697	−	0.884026i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−40.0689			−1.62635			−0.813173	−	0.582022i	$-0.802262\pi$
$607$	−40.0689			−1.62635			−0.813173	+	0.582022i	$0.802262\pi$
$608$		0			0
$609$	−17.8885	+	4.00000i	−0.724880	+	0.162088i
$610$		0			0
$611$			4.00000i			0.161823i
$612$		0			0
$613$		−	22.3607i		−	0.903139i	−0.892236	−	0.451570i	$-0.850864\pi$
$613$		−	22.3607i		−	0.903139i	0.892236	−	0.451570i	$-0.149136\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.5410			−1.06850			−0.534251	−	0.845326i	$-0.679406\pi$
$617$	−26.5410			−1.06850			−0.534251	+	0.845326i	$0.679406\pi$
$618$		0			0
$619$			5.70820i			0.229432i	0.993398	+	0.114716i	$0.0365958\pi$
$619$			5.70820i			0.229432i	−0.993398	+	0.114716i	$0.963404\pi$
$620$		0			0
$621$	−35.5967	−	18.3607i	−1.42845	−	0.736789i
$622$		0			0
$623$	32.6525	+	4.76393i	1.30819	+	0.190863i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.41641	−	8.94427i	−0.136438	−	0.357200i
$628$		0			0
$629$	20.0000			0.797452
$630$		0			0
$631$	8.94427			0.356066			0.178033	−	0.984025i	$-0.443027\pi$
$631$	8.94427			0.356066			0.178033	+	0.984025i	$0.443027\pi$
$632$		0			0
$633$	−4.94427	−	12.9443i	−0.196517	−	0.514489i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	8.29180	+	2.47214i	0.328533	+	0.0979496i
$638$		0			0
$639$	24.3607	+	27.2361i	0.963694	+	1.07744i
$640$		0			0
$641$			12.5836i			0.497022i	0.968629	+	0.248511i	$0.0799412\pi$
$641$			12.5836i			0.497022i	−0.968629	+	0.248511i	$0.920059\pi$
$642$		0			0
$643$	24.2918			0.957975			0.478987	−	0.877822i	$-0.341004\pi$
$643$	24.2918			0.957975			0.478987	+	0.877822i	$0.341004\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.12461i		−	0.358726i	−0.983783	−	0.179363i	$-0.942596\pi$
$647$		−	9.12461i		−	0.358726i	0.983783	−	0.179363i	$-0.0574036\pi$
$648$		0			0
$649$		−	6.83282i		−	0.268211i
$650$		0			0
$651$	−14.4721	+	3.23607i	−0.567208	+	0.126832i
$652$		0			0
$653$	−21.5967			−0.845146			−0.422573	−	0.906329i	$-0.638873\pi$
$653$	−21.5967			−0.845146			−0.422573	+	0.906329i	$0.638873\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	15.1246	−	13.5279i	0.590067	−	0.527772i
$658$		0			0
$659$			36.7639i			1.43212i	0.698039	+	0.716060i	$0.254056\pi$
$659$			36.7639i			1.43212i	−0.698039	+	0.716060i	$0.745944\pi$
$660$		0			0
$661$			22.8328i			0.888094i	0.896004	+	0.444047i	$0.146458\pi$
$661$			22.8328i			0.888094i	−0.896004	+	0.444047i	$0.853542\pi$
$662$		0			0
$663$	8.94427	−	3.41641i	0.347367	−	0.132682i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	30.8328i		−	1.19385i
$668$		0			0
$669$	−7.23607	−	18.9443i	−0.279763	−	0.732428i
$670$		0			0
$671$	−3.77709			−0.145813
$672$		0			0
$673$		−	14.9443i		−	0.576059i	−0.957621	−	0.288030i	$-0.907000\pi$
$673$		−	14.9443i		−	0.576059i	0.957621	−	0.288030i	$-0.0930002\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$		0			0
$679$	−37.1246	−	5.41641i	−1.42471	−	0.207863i
$680$		0			0
$681$	37.5967	−	14.3607i	1.44071	−	0.550302i
$682$		0			0
$683$	−20.2918			−0.776444			−0.388222	−	0.921566i	$-0.626911\pi$
$683$	−20.2918			−0.776444			−0.388222	+	0.921566i	$0.626911\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	22.4721	−	8.58359i	0.857365	−	0.327484i
$688$		0			0
$689$	11.4164			0.434931
$690$		0			0
$691$			8.18034i			0.311195i	0.987821	+	0.155597i	$0.0497302\pi$
$691$			8.18034i			0.311195i	−0.987821	+	0.155597i	$0.950270\pi$
$692$		0			0
$693$	−3.34752	−	5.05573i	−0.127162	−	0.192051i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	15.7771i		−	0.597600i
$698$		0			0
$699$	−1.12461	−	2.94427i	−0.0425367	−	0.111363i
$700$		0			0
$701$		−	3.05573i		−	0.115413i	−0.998334	−	0.0577066i	$-0.981621\pi$
$701$		−	3.05573i		−	0.115413i	0.998334	−	0.0577066i	$-0.0183788\pi$
$702$		0			0
$703$	32.3607			1.22051
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.1803	−	2.65248i	−0.683742	−	0.0997566i
$708$		0			0
$709$	22.9443			0.861690			0.430845	−	0.902426i	$-0.358216\pi$
$709$	22.9443			0.861690			0.430845	+	0.902426i	$0.358216\pi$
$710$		0			0
$711$	23.4164	−	20.9443i	0.878184	−	0.785472i
$712$		0			0
$713$		−	24.9443i		−	0.934170i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.291796	+	0.111456i	−0.0108973	+	0.00416241i
$718$		0			0
$719$	−6.11146			−0.227919			−0.113959	−	0.993485i	$-0.536353\pi$
$719$	−6.11146			−0.227919			−0.113959	+	0.993485i	$0.536353\pi$
$720$		0			0
$721$	28.1803	+	4.11146i	1.04949	+	0.153119i
$722$		0			0
$723$	−26.4721	+	10.1115i	−0.984509	+	0.376049i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	3.70820			0.137530			0.0687648	−	0.997633i	$-0.478094\pi$
$727$	3.70820			0.137530			0.0687648	+	0.997633i	$0.478094\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	32.3607			1.19690
$732$		0			0
$733$	−48.6525			−1.79702			−0.898510	−	0.438953i	$-0.855350\pi$
$733$	−48.6525			−1.79702			−0.898510	+	0.438953i	$0.855350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.41641			0.273187
$738$		0			0
$739$	−40.3607			−1.48469			−0.742346	−	0.670017i	$-0.766287\pi$
$739$	−40.3607			−1.48469			−0.742346	+	0.670017i	$0.766287\pi$
$740$		0			0
$741$	14.4721	−	5.52786i	0.531647	−	0.203071i
$742$		0			0
$743$	24.6525			0.904412			0.452206	−	0.891914i	$-0.350637\pi$
$743$	24.6525			0.904412			0.452206	+	0.891914i	$0.350637\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	14.4721	+	16.1803i	0.529508	+	0.592008i
$748$		0			0
$749$	−0.763932	−	0.111456i	−0.0279135	−	0.00407252i
$750$		0			0
$751$	−44.0000			−1.60558			−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000			−1.60558			−0.802791	+	0.596260i	$0.796653\pi$
$752$		0			0
$753$	12.5836	+	32.9443i	0.458572	+	1.20056i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			35.3050i			1.28318i	0.767048	+	0.641590i	$0.221725\pi$
$757$			35.3050i			1.28318i	−0.767048	+	0.641590i	$0.778275\pi$
$758$		0			0
$759$	9.52786	−	3.63932i	0.345840	−	0.132099i
$760$		0			0
$761$	−31.3050			−1.13480			−0.567402	−	0.823441i	$-0.692051\pi$
$761$	−31.3050			−1.13480			−0.567402	+	0.823441i	$0.692051\pi$
$762$		0			0
$763$	−7.70820	−	1.12461i	−0.279056	−	0.0407137i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.0557			0.399199
$768$		0			0
$769$			31.4164i			1.13290i	0.824095	+	0.566452i	$0.191684\pi$
$769$			31.4164i			1.13290i	−0.824095	+	0.566452i	$0.808316\pi$
$770$		0			0
$771$	−26.6525	+	10.1803i	−0.959865	+	0.366636i
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	20.0000	−	4.47214i	0.717496	−	0.160437i
$778$		0			0
$779$		−	25.5279i		−	0.914631i
$780$		0			0
$781$	−9.30495			−0.332957
$782$		0			0
$783$	−9.52786	+	18.4721i	−0.340498	+	0.660140i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−16.2918			−0.580740			−0.290370	−	0.956914i	$-0.593778\pi$
$787$	−16.2918			−0.580740			−0.290370	+	0.956914i	$0.593778\pi$
$788$		0			0
$789$	−13.7082	−	35.8885i	−0.488025	−	1.27767i
$790$		0			0
$791$	−45.1246	−	6.58359i	−1.60445	−	0.234086i
$792$		0			0
$793$		−	6.11146i		−	0.217024i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			28.8328i			1.02131i	0.859785	+	0.510655i	$0.170597\pi$
$797$			28.8328i			1.02131i	−0.859785	+	0.510655i	$0.829403\pi$
$798$		0			0
$799$	14.4721			0.511987
$800$		0			0
$801$	27.8885	−	24.9443i	0.985393	−	0.881363i
$802$		0			0
$803$			5.16718i			0.182346i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	12.0689	+	31.5967i	0.424845	+	1.11226i
$808$		0			0
$809$			12.3607i			0.434578i	0.976107	+	0.217289i	$0.0697215\pi$
$809$			12.3607i			0.434578i	−0.976107	+	0.217289i	$0.930279\pi$
$810$		0			0
$811$		−	1.70820i		−	0.0599832i	−0.999550	−	0.0299916i	$-0.990452\pi$
$811$		−	1.70820i		−	0.0599832i	0.999550	−	0.0299916i	$-0.00954805\pi$
$812$		0			0
$813$	34.1803	−	13.0557i	1.19876	−	0.457884i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	52.3607			1.83187
$818$		0			0
$819$	8.18034	−	5.41641i	0.285844	−	0.189265i
$820$		0			0
$821$		−	46.8328i		−	1.63448i	−0.576300	−	0.817238i	$-0.695504\pi$
$821$		−	46.8328i		−	1.63448i	0.576300	−	0.817238i	$-0.304496\pi$
$822$		0			0
$823$			11.5967i			0.404237i	0.979361	+	0.202119i	$0.0647826\pi$
$823$			11.5967i			0.404237i	−0.979361	+	0.202119i	$0.935217\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.7082			−1.38079			−0.690395	−	0.723433i	$-0.742563\pi$
$827$	−39.7082			−1.38079			−0.690395	+	0.723433i	$0.742563\pi$
$828$		0			0
$829$			20.3607i			0.707156i	0.935405	+	0.353578i	$0.115035\pi$
$829$			20.3607i			0.707156i	−0.935405	+	0.353578i	$0.884965\pi$
$830$		0			0
$831$	−2.29180	+	0.875388i	−0.0795015	+	0.0303669i
$832$		0			0
$833$	8.94427	−	30.0000i	0.309901	−	1.03944i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−7.70820	+	14.9443i	−0.266435	+	0.516550i
$838$		0			0
$839$	−12.9443			−0.446886			−0.223443	−	0.974717i	$-0.571730\pi$
$839$	−12.9443			−0.446886			−0.223443	+	0.974717i	$0.571730\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	−18.4721	+	7.05573i	−0.636214	+	0.243012i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−27.2705	−	3.97871i	−0.937026	−	0.136710i
$848$		0			0
$849$	−5.34752	−	14.0000i	−0.183527	−	0.480479i
$850$		0			0
$851$			34.4721i			1.18169i
$852$		0			0
$853$	19.1246			0.654814			0.327407	−	0.944883i	$-0.393825\pi$
$853$	19.1246			0.654814			0.327407	+	0.944883i	$0.393825\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	28.4721i		−	0.972590i	−0.873795	−	0.486295i	$-0.838348\pi$
$857$		−	28.4721i		−	0.972590i	0.873795	−	0.486295i	$-0.161652\pi$
$858$		0			0
$859$		−	13.3475i		−	0.455412i	−0.973730	−	0.227706i	$-0.926878\pi$
$859$		−	13.3475i		−	0.455412i	0.973730	−	0.227706i	$-0.0731225\pi$
$860$		0			0
$861$	−3.52786	−	15.7771i	−0.120229	−	0.537682i
$862$		0			0
$863$	52.6525			1.79231			0.896156	−	0.443740i	$-0.146349\pi$
$863$	52.6525			1.79231			0.896156	+	0.443740i	$0.146349\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.85410	−	4.85410i	−0.0629686	−	0.164854i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$			12.0000i			0.406604i
$872$		0			0
$873$	−31.7082	+	28.3607i	−1.07316	+	0.959864i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	32.4721i		−	1.09651i	−0.836312	−	0.548253i	$-0.815293\pi$
$877$		−	32.4721i		−	1.09651i	0.836312	−	0.548253i	$-0.184707\pi$
$878$		0			0
$879$	22.6525	−	8.65248i	0.764049	−	0.291841i
$880$		0			0
$881$	16.8328			0.567112			0.283556	−	0.958956i	$-0.408486\pi$
$881$	16.8328			0.567112			0.283556	+	0.958956i	$0.408486\pi$
$882$		0			0
$883$			35.5967i			1.19793i	0.800777	+	0.598963i	$0.204420\pi$
$883$			35.5967i			1.19793i	−0.800777	+	0.598963i	$0.795580\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			16.1803i			0.543283i	0.962399	+	0.271641i	$0.0875664\pi$
$887$			16.1803i			0.543283i	−0.962399	+	0.271641i	$0.912434\pi$
$888$		0			0
$889$	−8.06888	+	55.3050i	−0.270622	+	1.85487i
$890$		0			0
$891$	−6.83282	−	0.763932i	−0.228908	−	0.0255927i
$892$		0			0
$893$	23.4164			0.783600
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5.88854	+	15.4164i	0.196613	+	0.514739i
$898$		0			0
$899$	−12.9443			−0.431716
$900$		0			0
$901$		−	41.3050i		−	1.37607i
$902$		0			0
$903$	32.3607	−	7.23607i	1.07690	−	0.240801i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	27.2361i		−	0.904359i	−0.891927	−	0.452179i	$-0.850647\pi$
$907$		−	27.2361i		−	0.904359i	0.891927	−	0.452179i	$-0.149353\pi$
$908$		0			0
$909$	−15.5279	+	13.8885i	−0.515027	+	0.460654i
$910$		0			0
$911$		−	46.0689i		−	1.52633i	−0.646204	−	0.763165i	$-0.723644\pi$
$911$		−	46.0689i		−	1.52633i	0.646204	−	0.763165i	$-0.276356\pi$
$912$		0			0
$913$	−5.52786			−0.182946
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.3607	+	5.88854i	1.33283	+	0.194457i
$918$		0			0
$919$	34.4721			1.13713			0.568565	−	0.822638i	$-0.307499\pi$
$919$	34.4721			1.13713			0.568565	+	0.822638i	$0.307499\pi$
$920$		0			0
$921$	−11.2361	−	29.4164i	−0.370241	−	0.969304i
$922$		0			0
$923$		−	15.0557i		−	0.495565i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	24.0689	−	21.5279i	0.790526	−	0.707068i
$928$		0			0
$929$	−19.8885			−0.652522			−0.326261	−	0.945280i	$-0.605789\pi$
$929$	−19.8885			−0.652522			−0.326261	+	0.945280i	$0.605789\pi$
$930$		0			0
$931$	14.4721	−	48.5410i	0.474305	−	1.59087i
$932$		0			0
$933$	20.0000	+	52.3607i	0.654771	+	1.71421i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−14.7639			−0.482317			−0.241158	−	0.970486i	$-0.577527\pi$
$937$	−14.7639			−0.482317			−0.241158	+	0.970486i	$0.577527\pi$
$938$		0			0
$939$	2.65248	+	6.94427i	0.0865603	+	0.226618i
$940$		0			0
$941$	44.2492			1.44248			0.721242	−	0.692683i	$-0.243572\pi$
$941$	44.2492			1.44248			0.721242	+	0.692683i	$0.243572\pi$
$942$		0			0
$943$	27.1935			0.885542
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.2361			0.820062			0.410031	−	0.912072i	$-0.365518\pi$
$947$	25.2361			0.820062			0.410031	+	0.912072i	$0.365518\pi$
$948$		0			0
$949$	−8.36068			−0.271399
$950$		0			0
$951$	3.81966	+	10.0000i	0.123861	+	0.324272i
$952$		0			0
$953$	−3.70820			−0.120121			−0.0600603	−	0.998195i	$-0.519129\pi$
$953$	−3.70820			−0.120121			−0.0600603	+	0.998195i	$0.519129\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1.88854	−	4.94427i	−0.0610480	−	0.159826i
$958$		0			0
$959$	−3.23607	−	0.472136i	−0.104498	−	0.0152461i
$960$		0			0
$961$	20.5279			0.662189
$962$		0			0
$963$	−0.652476	+	0.583592i	−0.0210257	+	0.0188060i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			44.1803i			1.42074i	0.703826	+	0.710372i	$0.251473\pi$
$967$			44.1803i			1.42074i	−0.703826	+	0.710372i	$0.748527\pi$
$968$		0			0
$969$	−20.0000	−	52.3607i	−0.642493	−	1.68207i
$970$		0			0
$971$	0.583592			0.0187284			0.00936418	−	0.999956i	$-0.497019\pi$
$971$	0.583592			0.0187284			0.00936418	+	0.999956i	$0.497019\pi$
$972$		0			0
$973$	−5.81966	+	39.8885i	−0.186570	+	1.27877i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.9574			1.59828			0.799140	−	0.601145i	$-0.205289\pi$
$977$	49.9574			1.59828			0.799140	+	0.601145i	$0.205289\pi$
$978$		0			0
$979$			9.52786i			0.304512i
$980$		0			0
$981$	−6.58359	+	5.88854i	−0.210198	+	0.188007i
$982$		0			0
$983$		−	40.5410i		−	1.29306i	−0.762890	−	0.646529i	$-0.776220\pi$
$983$		−	40.5410i		−	1.29306i	0.762890	−	0.646529i	$-0.223780\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	14.4721	−	3.23607i	0.460653	−	0.103005i
$988$		0			0
$989$			55.7771i			1.77361i
$990$		0			0
$991$	39.7771			1.26356			0.631780	−	0.775147i	$-0.282324\pi$
$991$	39.7771			1.26356			0.631780	+	0.775147i	$0.282324\pi$
$992$		0			0
$993$	−12.9443	−	33.8885i	−0.410774	−	1.07542i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.0689			0.508907			0.254453	−	0.967085i	$-0.418104\pi$
$997$	16.0689			0.508907			0.254453	+	0.967085i	$0.418104\pi$
$998$		0			0
$999$	10.6525	−	20.6525i	0.337029	−	0.653415i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.f.a.1049.4		4
3.2	odd	2		2100.2.f.b.1049.3		4
5.2	odd	4		2100.2.d.h.1301.3		4
5.3	odd	4		420.2.d.c.41.2	yes	4
5.4	even	2		2100.2.f.g.1049.1		4
7.6	odd	2		2100.2.f.h.1049.1		4
15.2	even	4		2100.2.d.g.1301.1		4
15.8	even	4		420.2.d.d.41.4	yes	4
15.14	odd	2		2100.2.f.h.1049.2		4
20.3	even	4		1680.2.f.j.881.3		4
21.20	even	2		2100.2.f.g.1049.2		4
35.13	even	4		420.2.d.d.41.3	yes	4
35.27	even	4		2100.2.d.g.1301.2		4
35.34	odd	2		2100.2.f.b.1049.4		4
60.23	odd	4		1680.2.f.f.881.1		4
105.62	odd	4		2100.2.d.h.1301.4		4
105.83	odd	4		420.2.d.c.41.1	✓	4
105.104	even	2	inner	2100.2.f.a.1049.3		4
140.83	odd	4		1680.2.f.f.881.2		4
420.83	even	4		1680.2.f.j.881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.d.c.41.1	✓	4	105.83	odd	4
420.2.d.c.41.2	yes	4	5.3	odd	4
420.2.d.d.41.3	yes	4	35.13	even	4
420.2.d.d.41.4	yes	4	15.8	even	4
1680.2.f.f.881.1		4	60.23	odd	4
1680.2.f.f.881.2		4	140.83	odd	4
1680.2.f.j.881.3		4	20.3	even	4
1680.2.f.j.881.4		4	420.83	even	4
2100.2.d.g.1301.1		4	15.2	even	4
2100.2.d.g.1301.2		4	35.27	even	4
2100.2.d.h.1301.3		4	5.2	odd	4
2100.2.d.h.1301.4		4	105.62	odd	4
2100.2.f.a.1049.3		4	105.104	even	2	inner
2100.2.f.a.1049.4		4	1.1	even	1	trivial
2100.2.f.b.1049.3		4	3.2	odd	2
2100.2.f.b.1049.4		4	35.34	odd	2
2100.2.f.g.1049.1		4	5.4	even	2
2100.2.f.g.1049.2		4	21.20	even	2
2100.2.f.h.1049.1		4	7.6	odd	2
2100.2.f.h.1049.2		4	15.14	odd	2

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 1.61803i) q^{3} +(-2.61803 - 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(0.618034 + 1.61803i) q^{3} +(-2.61803 - 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} -0.763932i q^{11} +1.23607 q^{13} -4.47214i q^{17} -7.23607i q^{19} +(-1.00000 - 4.47214i) q^{21} +7.70820 q^{23} +(-4.61803 - 2.38197i) q^{27} -4.00000i q^{29} -3.23607i q^{31} +(1.23607 - 0.472136i) q^{33} +4.47214i q^{37} +(0.763932 + 2.00000i) q^{39} +3.52786 q^{41} +7.23607i q^{43} +3.23607i q^{47} +(6.70820 + 2.00000i) q^{49} +(7.23607 - 2.76393i) q^{51} +9.23607 q^{53} +(11.7082 - 4.47214i) q^{57} +8.94427 q^{59} -4.94427i q^{61} +(6.61803 - 4.38197i) q^{63} +9.70820i q^{67} +(4.76393 + 12.4721i) q^{69} -12.1803i q^{71} -6.76393 q^{73} +(-0.291796 + 2.00000i) q^{77} -10.4721 q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607i q^{83} +(6.47214 - 2.47214i) q^{87} -12.4721 q^{89} +(-3.23607 - 0.472136i) q^{91} +(5.23607 - 2.00000i) q^{93} +14.1803 q^{97} +(1.52786 + 1.70820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) 4 * q - 2 * q^3 - 6 * q^7	\( 4 q - 2 q^{3} - 6 q^{7} - 4 q^{13} - 4 q^{21} + 4 q^{23} - 14 q^{27} - 4 q^{33} + 12 q^{39} + 32 q^{41} + 20 q^{51} + 28 q^{53} + 20 q^{57} + 22 q^{63} + 28 q^{69} - 36 q^{73} - 28 q^{77} - 24 q^{79} + 4 q^{81} + 8 q^{87} - 32 q^{89} - 4 q^{91} + 12 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 6 * q^7 - 4 * q^13 - 4 * q^21 + 4 * q^23 - 14 * q^27 - 4 * q^33 + 12 * q^39 + 32 * q^41 + 20 * q^51 + 28 * q^53 + 20 * q^57 + 22 * q^63 + 28 * q^69 - 36 * q^73 - 28 * q^77 - 24 * q^79 + 4 * q^81 + 8 * q^87 - 32 * q^89 - 4 * q^91 + 12 * q^93 + 12 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

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Database

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