LMFDB - Embedded newform 2100.2.d.h.1301.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1301,2100)]

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, 0, 1]))

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

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

chi := DirichletCharacter("2100.1301");

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.d (of order $2$, degree $1$, 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			1301.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.h.1301.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})$	$q+(1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} -0.763932i q^{11} -1.23607i q^{13} +4.47214 q^{17} +7.23607i q^{19} +(-1.00000 - 4.47214i) q^{21} -7.70820i q^{23} +(2.38197 - 4.61803i) q^{27} +4.00000i q^{29} -3.23607i q^{31} +(-0.472136 - 1.23607i) q^{33} -4.47214 q^{37} +(-0.763932 - 2.00000i) q^{39} +3.52786 q^{41} +7.23607 q^{43} -3.23607 q^{47} +(-6.70820 - 2.00000i) q^{49} +(7.23607 - 2.76393i) q^{51} -9.23607i q^{53} +(4.47214 + 11.7082i) q^{57} -8.94427 q^{59} -4.94427i q^{61} +(-4.38197 - 6.61803i) q^{63} -9.70820 q^{67} +(-4.76393 - 12.4721i) q^{69} -12.1803i q^{71} +6.76393i q^{73} +(-2.00000 - 0.291796i) q^{77} +10.4721 q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607 q^{83} +(2.47214 + 6.47214i) q^{87} +12.4721 q^{89} +(-3.23607 - 0.472136i) q^{91} +(-2.00000 - 5.23607i) q^{93} +14.1803i q^{97} +(-1.52786 - 1.70820i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{21} + 14 q^{27} + 16 q^{33} - 12 q^{39} + 32 q^{41} + 20 q^{43} - 4 q^{47} + 20 q^{51} - 22 q^{63} - 12 q^{67} - 28 q^{69} - 8 q^{77} + 24 q^{79} + 4 q^{81} - 20 q^{83} - 8 q^{87} + 32 q^{89} - 4 q^{91} - 8 q^{93} - 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^7 - 4 * q^21 + 14 * q^27 + 16 * q^33 - 12 * q^39 + 32 * q^41 + 20 * q^43 - 4 * q^47 + 20 * q^51 - 22 * q^63 - 12 * q^67 - 28 * q^69 - 8 * q^77 + 24 * q^79 + 4 * q^81 - 20 * q^83 - 8 * q^87 + 32 * q^89 - 4 * q^91 - 8 * q^93 - 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
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.381966	−	2.61803i	0.144370	−	0.989524i
$7$	0.381966	−	2.61803i	0.144370	−	0.989524i
$8$		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
$13$		−	1.23607i		−	0.342824i	−0.985199	−	0.171412i	$-0.945167\pi$
$13$		−	1.23607i		−	0.342824i	0.985199	−	0.171412i	$-0.0548329\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$			7.23607i			1.66007i	0.557713	+	0.830034i	$0.311679\pi$
$19$			7.23607i			1.66007i	−0.557713	+	0.830034i	$0.688321\pi$
$20$		0			0
$21$	−1.00000	−	4.47214i	−0.218218	−	0.975900i
$22$		0			0
$23$		−	7.70820i		−	1.60727i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$		−	7.70820i		−	1.60727i	0.595121	−	0.803636i	$-0.297104\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.38197	−	4.61803i	0.458410	−	0.888741i
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		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
$33$	−0.472136	−	1.23607i	−0.0821883	−	0.215172i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.47214			−0.735215			−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214			−0.735215			−0.367607	+	0.929981i	$0.619823\pi$
$38$		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
$43$	7.23607			1.10349			0.551745	−	0.834013i	$-0.313962\pi$
$43$	7.23607			1.10349			0.551745	+	0.834013i	$0.313962\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.23607			−0.472029			−0.236015	−	0.971750i	$-0.575841\pi$
$47$	−3.23607			−0.472029			−0.236015	+	0.971750i	$0.575841\pi$
$48$		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
$53$		−	9.23607i		−	1.26867i	−0.773058	−	0.634336i	$-0.781274\pi$
$53$		−	9.23607i		−	1.26867i	0.773058	−	0.634336i	$-0.218726\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.47214	+	11.7082i	0.592349	+	1.55079i
$58$		0			0
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		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
$63$	−4.38197	−	6.61803i	−0.552076	−	0.833794i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.70820			−1.18605			−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820			−1.18605			−0.593023	+	0.805186i	$0.702066\pi$
$68$		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
$73$			6.76393i			0.791658i	0.918324	+	0.395829i	$0.129543\pi$
$73$			6.76393i			0.791658i	−0.918324	+	0.395829i	$0.870457\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	−	0.291796i	−0.227921	−	0.0332532i
$78$		0			0
$79$	10.4721			1.17821			0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721			1.17821			0.589104	+	0.808057i	$0.299481\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−7.23607			−0.794262			−0.397131	−	0.917762i	$-0.629994\pi$
$83$	−7.23607			−0.794262			−0.397131	+	0.917762i	$0.629994\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.47214	+	6.47214i	0.265041	+	0.693886i
$88$		0			0
$89$	12.4721			1.32204			0.661022	−	0.750367i	$-0.270123\pi$
$89$	12.4721			1.32204			0.661022	+	0.750367i	$0.270123\pi$
$90$		0			0
$91$	−3.23607	−	0.472136i	−0.339232	−	0.0494933i
$92$		0			0
$93$	−2.00000	−	5.23607i	−0.207390	−	0.542955i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			14.1803i			1.43980i	0.694080	+	0.719898i	$0.255811\pi$
$97$			14.1803i			1.43980i	−0.694080	+	0.719898i	$0.744189\pi$
$98$		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
$103$			10.7639i			1.06060i	0.847810	+	0.530301i	$0.177921\pi$
$103$			10.7639i			1.06060i	−0.847810	+	0.530301i	$0.822079\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			0.291796i			0.0282090i	0.999901	+	0.0141045i	$0.00448975\pi$
$107$			0.291796i			0.0282090i	−0.999901	+	0.0141045i	$0.995510\pi$
$108$		0			0
$109$	−2.94427			−0.282010			−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427			−0.282010			−0.141005	+	0.990009i	$0.545033\pi$
$110$		0			0
$111$	−7.23607	+	2.76393i	−0.686817	+	0.262341i
$112$		0			0
$113$		−	17.2361i		−	1.62143i	−0.585439	−	0.810716i	$-0.699078\pi$
$113$		−	17.2361i		−	1.62143i	0.585439	−	0.810716i	$-0.300922\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.47214	−	2.76393i	−0.228549	−	0.255526i
$118$		0			0
$119$	1.70820	−	11.7082i	0.156591	−	1.07329i
$120$		0			0
$121$	10.4164			0.946946
$122$		0			0
$123$	5.70820	−	2.18034i	0.514691	−	0.196595i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	21.1246			1.87451			0.937253	−	0.348650i	$-0.113360\pi$
$127$	21.1246			1.87451			0.937253	+	0.348650i	$0.113360\pi$
$128$		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
$133$	18.9443	+	2.76393i	1.64268	+	0.239663i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.23607i			0.105604i	0.998605	+	0.0528022i	$0.0168153\pi$
$137$			1.23607i			0.105604i	−0.998605	+	0.0528022i	$0.983185\pi$
$138$		0			0
$139$			15.2361i			1.29231i	0.763208	+	0.646153i	$0.223623\pi$
$139$			15.2361i			1.29231i	−0.763208	+	0.646153i	$0.776377\pi$
$140$		0			0
$141$	−5.23607	+	2.00000i	−0.440956	+	0.168430i
$142$		0			0
$143$	−0.944272			−0.0789640
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−12.0902	+	0.909830i	−0.997180	+	0.0750415i
$148$		0			0
$149$			15.4164i			1.26296i	0.775392	+	0.631481i	$0.217552\pi$
$149$			15.4164i			1.26296i	−0.775392	+	0.631481i	$0.782448\pi$
$150$		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
$153$	10.0000	−	8.94427i	0.808452	−	0.723102i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	1.81966i		−	0.145225i	−0.997360	−	0.0726123i	$-0.976866\pi$
$157$		−	1.81966i		−	0.145225i	0.997360	−	0.0726123i	$-0.0231336\pi$
$158$		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
$163$	−18.6525			−1.46097			−0.730487	−	0.682926i	$-0.760707\pi$
$163$	−18.6525			−1.46097			−0.730487	+	0.682926i	$0.760707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.76393			−0.678173			−0.339087	−	0.940755i	$-0.610118\pi$
$167$	−8.76393			−0.678173			−0.339087	+	0.940755i	$0.610118\pi$
$168$		0			0
$169$	11.4721			0.882472
$170$		0			0
$171$	14.4721	+	16.1803i	1.10671	+	1.23734i
$172$		0			0
$173$	2.00000			0.152057			0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000			0.152057			0.0760286	+	0.997106i	$0.475776\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−14.4721	+	5.52786i	−1.08779	+	0.415500i
$178$		0			0
$179$		−	12.7639i		−	0.954021i	−0.878898	−	0.477011i	$-0.841720\pi$
$179$		−	12.7639i		−	0.954021i	0.878898	−	0.477011i	$-0.158280\pi$
$180$		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
$183$	−3.05573	−	8.00000i	−0.225886	−	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	3.41641i		−	0.249832i
$188$		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
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			25.2361i			1.79800i	0.437953	+	0.898998i	$0.355704\pi$
$197$			25.2361i			1.79800i	−0.437953	+	0.898998i	$0.644296\pi$
$198$		0			0
$199$			8.76393i			0.621259i	0.950531	+	0.310629i	$0.100540\pi$
$199$			8.76393i			0.621259i	−0.950531	+	0.310629i	$0.899460\pi$
$200$		0			0
$201$	−15.7082	+	6.00000i	−1.10797	+	0.423207i
$202$		0			0
$203$	10.4721	+	1.52786i	0.735000	+	0.107235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−15.4164	−	17.2361i	−1.07151	−	1.19799i
$208$		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
$213$	−7.52786	−	19.7082i	−0.515801	−	1.35038i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.47214	−	1.23607i	−0.575126	−	0.0839098i
$218$		0			0
$219$	4.18034	+	10.9443i	0.282481	+	0.739545i
$220$		0			0
$221$		−	5.52786i		−	0.371844i
$222$		0			0
$223$			11.7082i			0.784039i	0.919957	+	0.392020i	$0.128223\pi$
$223$			11.7082i			0.784039i	−0.919957	+	0.392020i	$0.871777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	23.2361			1.54223			0.771116	−	0.636695i	$-0.219699\pi$
$227$	23.2361			1.54223			0.771116	+	0.636695i	$0.219699\pi$
$228$		0			0
$229$			13.8885i			0.917781i	0.888493	+	0.458890i	$0.151753\pi$
$229$			13.8885i			0.917781i	−0.888493	+	0.458890i	$0.848247\pi$
$230$		0			0
$231$	−3.41641	+	0.763932i	−0.224783	+	0.0502630i
$232$		0			0
$233$			1.81966i			0.119210i	0.998222	+	0.0596049i	$0.0189841\pi$
$233$			1.81966i			0.119210i	−0.998222	+	0.0596049i	$0.981016\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	16.9443	−	6.47214i	1.10065	−	0.420410i
$238$		0			0
$239$		−	0.180340i		−	0.0116652i	−0.999983	−	0.00583261i	$-0.998143\pi$
$239$		−	0.180340i		−	0.0116652i	0.999983	−	0.00583261i	$-0.00185659\pi$
$240$		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
$243$	−3.90983	−	15.0902i	−0.250816	−	0.968035i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.94427			0.569110
$248$		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
$253$	−5.88854			−0.370210
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.4721			−1.02750			−0.513752	−	0.857939i	$-0.671745\pi$
$257$	−16.4721			−1.02750			−0.513752	+	0.857939i	$0.671745\pi$
$258$		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
$263$			22.1803i			1.36770i	0.729623	+	0.683849i	$0.239695\pi$
$263$			22.1803i			1.36770i	−0.729623	+	0.683849i	$0.760305\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	20.1803	−	7.70820i	1.23502	−	0.471734i
$268$		0			0
$269$	−19.5279			−1.19063			−0.595317	−	0.803491i	$-0.702974\pi$
$269$	−19.5279			−1.19063			−0.595317	+	0.803491i	$0.702974\pi$
$270$		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
$273$	−5.52786	+	1.23607i	−0.334562	+	0.0748102i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.41641			−0.0851037			−0.0425519	−	0.999094i	$-0.513549\pi$
$277$	−1.41641			−0.0851037			−0.0425519	+	0.999094i	$0.513549\pi$
$278$		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
$283$			8.65248i			0.514336i	0.966367	+	0.257168i	$0.0827894\pi$
$283$			8.65248i			0.514336i	−0.966367	+	0.257168i	$0.917211\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.34752	−	9.23607i	0.0795418	−	0.545188i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$	8.76393	+	22.9443i	0.513751	+	1.34502i
$292$		0			0
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.52786	−	1.81966i	−0.204707	−	0.105587i
$298$		0			0
$299$	−9.52786			−0.551011
$300$		0			0
$301$	2.76393	−	18.9443i	0.159310	−	1.09193i
$302$		0			0
$303$	11.2361	−	4.29180i	0.645495	−	0.246557i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	18.1803i		−	1.03761i	−0.854894	−	0.518803i	$-0.826378\pi$
$307$		−	18.1803i		−	1.03761i	0.854894	−	0.518803i	$-0.173622\pi$
$308$		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
$313$		−	4.29180i		−	0.242587i	−0.992617	−	0.121293i	$-0.961296\pi$
$313$		−	4.29180i		−	0.242587i	0.992617	−	0.121293i	$-0.0387042\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.18034i			0.347122i	0.984823	+	0.173561i	$0.0555275\pi$
$317$			6.18034i			0.347122i	−0.984823	+	0.173561i	$0.944473\pi$
$318$		0			0
$319$	3.05573			0.171088
$320$		0			0
$321$	0.180340	+	0.472136i	0.0100656	+	0.0263521i
$322$		0			0
$323$			32.3607i			1.80060i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.76393	+	1.81966i	−0.263446	+	0.100627i
$328$		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
$333$	−10.0000	+	8.94427i	−0.547997	+	0.490143i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	−10.6525	−	27.8885i	−0.578563	−	1.51470i
$340$		0			0
$341$	−2.47214			−0.133874
$342$		0			0
$343$	−7.79837	+	16.7984i	−0.421073	+	0.907027i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	32.6525i		−	1.75288i	−0.481514	−	0.876438i	$-0.659913\pi$
$347$		−	32.6525i		−	1.75288i	0.481514	−	0.876438i	$-0.340087\pi$
$348$		0			0
$349$			8.94427i			0.478776i	0.970924	+	0.239388i	$0.0769468\pi$
$349$			8.94427i			0.478776i	−0.970924	+	0.239388i	$0.923053\pi$
$350$		0			0
$351$	−5.70820	−	2.94427i	−0.304681	−	0.157154i
$352$		0			0
$353$	25.4164			1.35278			0.676389	−	0.736544i	$-0.263544\pi$
$353$	25.4164			1.35278			0.676389	+	0.736544i	$0.263544\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−4.47214	−	20.0000i	−0.236691	−	1.05851i
$358$		0			0
$359$		−	34.0689i		−	1.79809i	−0.437859	−	0.899043i	$-0.644263\pi$
$359$		−	34.0689i		−	1.79809i	0.437859	−	0.899043i	$-0.355737\pi$
$360$		0			0
$361$	−33.3607			−1.75583
$362$		0			0
$363$	16.8541	−	6.43769i	0.884611	−	0.337891i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			24.6525i			1.28685i	0.765510	+	0.643424i	$0.222487\pi$
$367$			24.6525i			1.28685i	−0.765510	+	0.643424i	$0.777513\pi$
$368$		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
$373$	−7.52786			−0.389778			−0.194889	−	0.980825i	$-0.562435\pi$
$373$	−7.52786			−0.389778			−0.194889	+	0.980825i	$0.562435\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.94427			0.254643
$378$		0			0
$379$	1.52786			0.0784811			0.0392406	−	0.999230i	$-0.487506\pi$
$379$	1.52786			0.0784811			0.0392406	+	0.999230i	$0.487506\pi$
$380$		0			0
$381$	34.1803	−	13.0557i	1.75111	−	0.668865i
$382$		0			0
$383$	−15.2361			−0.778527			−0.389263	−	0.921127i	$-0.627270\pi$
$383$	−15.2361			−0.778527			−0.389263	+	0.921127i	$0.627270\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	16.1803	−	14.4721i	0.822493	−	0.735660i
$388$		0			0
$389$		−	2.47214i		−	0.125342i	−0.998034	−	0.0626711i	$-0.980038\pi$
$389$		−	2.47214i		−	0.125342i	0.998034	−	0.0626711i	$-0.0199619\pi$
$390$		0			0
$391$		−	34.4721i		−	1.74333i
$392$		0			0
$393$	−24.9443	+	9.52786i	−1.25827	+	0.480617i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	3.70820i		−	0.186109i	−0.995661	−	0.0930547i	$-0.970337\pi$
$397$		−	3.70820i		−	0.186109i	0.995661	−	0.0930547i	$-0.0296631\pi$
$398$		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
$403$	−4.00000			−0.199254
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.41641i			0.169345i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$	0.763932	+	2.00000i	0.0376820	+	0.0986527i
$412$		0			0
$413$	−3.41641	+	23.4164i	−0.168110	+	1.15225i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9.41641	+	24.6525i	0.461123	+	1.20724i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		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
$423$	−7.23607	+	6.47214i	−0.351830	+	0.314686i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.9443	−	1.88854i	−0.626417	−	0.0913930i
$428$		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
$433$			32.6525i			1.56918i	0.620016	+	0.784589i	$0.287126\pi$
$433$			32.6525i			1.56918i	−0.620016	+	0.784589i	$0.712874\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	55.7771			2.66818
$438$		0			0
$439$		−	4.18034i		−	0.199517i	−0.995012	−	0.0997584i	$-0.968193\pi$
$439$		−	4.18034i		−	0.199517i	0.995012	−	0.0997584i	$-0.0318070\pi$
$440$		0			0
$441$	−19.0000	+	8.94427i	−0.904762	+	0.425918i
$442$		0			0
$443$			24.0689i			1.14355i	0.820411	+	0.571774i	$0.193744\pi$
$443$			24.0689i			1.14355i	−0.820411	+	0.571774i	$0.806256\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	9.52786	+	24.9443i	0.450653	+	1.17982i
$448$		0			0
$449$			29.8885i			1.41053i	0.708945	+	0.705264i	$0.249171\pi$
$449$			29.8885i			1.41053i	−0.708945	+	0.705264i	$0.750829\pi$
$450$		0			0
$451$		−	2.69505i		−	0.126905i
$452$		0			0
$453$	14.4721	−	5.52786i	0.679960	−	0.259722i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.05573			0.423609			0.211805	−	0.977312i	$-0.432066\pi$
$457$	9.05573			0.423609			0.211805	+	0.977312i	$0.432066\pi$
$458$		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
$463$	20.7639			0.964982			0.482491	−	0.875901i	$-0.339732\pi$
$463$	20.7639			0.964982			0.482491	+	0.875901i	$0.339732\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.5410			−1.69092			−0.845458	−	0.534041i	$-0.820673\pi$
$467$	−36.5410			−1.69092			−0.845458	+	0.534041i	$0.820673\pi$
$468$		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
$473$		−	5.52786i		−	0.254171i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−18.4721	−	20.6525i	−0.845781	−	0.945612i
$478$		0			0
$479$	−12.9443			−0.591439			−0.295719	−	0.955275i	$-0.595559\pi$
$479$	−12.9443			−0.591439			−0.295719	+	0.955275i	$0.595559\pi$
$480$		0			0
$481$			5.52786i			0.252049i
$482$		0			0
$483$	−34.4721	+	7.70820i	−1.56854	+	0.350735i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.23607			0.146640			0.0733201	−	0.997308i	$-0.476641\pi$
$487$	3.23607			0.146640			0.0733201	+	0.997308i	$0.476641\pi$
$488$		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
$493$			17.8885i			0.805659i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.8885	−	4.65248i	−1.43040	−	0.208692i
$498$		0			0
$499$	−16.3607			−0.732405			−0.366202	−	0.930535i	$-0.619342\pi$
$499$	−16.3607			−0.732405			−0.366202	+	0.930535i	$0.619342\pi$
$500$		0			0
$501$	−14.1803	+	5.41641i	−0.633531	+	0.241987i
$502$		0			0
$503$	18.6525			0.831673			0.415836	−	0.909439i	$-0.363489\pi$
$503$	18.6525			0.831673			0.415836	+	0.909439i	$0.363489\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	18.5623	−	7.09017i	0.824381	−	0.314886i
$508$		0			0
$509$	22.9443			1.01699			0.508493	−	0.861066i	$-0.330203\pi$
$509$	22.9443			1.01699			0.508493	+	0.861066i	$0.330203\pi$
$510$		0			0
$511$	17.7082	+	2.58359i	0.783365	+	0.114291i
$512$		0			0
$513$	33.4164	+	17.2361i	1.47537	+	0.760991i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			2.47214i			0.108724i
$518$		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
$523$		−	8.29180i		−	0.362575i	−0.983430	−	0.181287i	$-0.941974\pi$
$523$		−	8.29180i		−	0.362575i	0.983430	−	0.181287i	$-0.0580264\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	14.4721i		−	0.630416i
$528$		0			0
$529$	−36.4164			−1.58332
$530$		0			0
$531$	−20.0000	+	17.8885i	−0.867926	+	0.776297i
$532$		0			0
$533$		−	4.36068i		−	0.188882i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.88854	−	20.6525i	−0.340416	−	0.891220i
$538$		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
$543$	4.00000	+	10.4721i	0.171656	+	0.449402i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.7082			−0.757148			−0.378574	−	0.925571i	$-0.623585\pi$
$547$	−17.7082			−0.757148			−0.378574	+	0.925571i	$0.623585\pi$
$548$		0			0
$549$	−9.88854	−	11.0557i	−0.422033	−	0.471847i
$550$		0			0
$551$	−28.9443			−1.23307
$552$		0			0
$553$	4.00000	−	27.4164i	0.170097	−	1.16586i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			14.1803i			0.600840i	0.953807	+	0.300420i	$0.0971269\pi$
$557$			14.1803i			0.600840i	−0.953807	+	0.300420i	$0.902873\pi$
$558$		0			0
$559$		−	8.94427i		−	0.378302i
$560$		0			0
$561$	−2.11146	−	5.52786i	−0.0891457	−	0.233387i
$562$		0			0
$563$	18.0689			0.761513			0.380756	−	0.924675i	$-0.375664\pi$
$563$	18.0689			0.761513			0.380756	+	0.924675i	$0.375664\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.0344	−	6.03444i	−0.967356	−	0.253423i
$568$		0			0
$569$			21.8885i			0.917615i	0.888536	+	0.458808i	$0.151723\pi$
$569$			21.8885i			0.917615i	−0.888536	+	0.458808i	$0.848277\pi$
$570$		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
$573$	−4.47214	−	11.7082i	−0.186826	−	0.489117i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			32.0689i			1.33505i	0.744590	+	0.667523i	$0.232645\pi$
$577$			32.0689i			1.33505i	−0.744590	+	0.667523i	$0.767355\pi$
$578$		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
$583$	−7.05573			−0.292218
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.29180			−0.259690			−0.129845	−	0.991534i	$-0.541448\pi$
$587$	−6.29180			−0.259690			−0.129845	+	0.991534i	$0.541448\pi$
$588$		0			0
$589$	23.4164			0.964856
$590$		0			0
$591$	15.5967	+	40.8328i	0.641564	+	1.67964i
$592$		0			0
$593$	−19.5279			−0.801913			−0.400957	−	0.916097i	$-0.631322\pi$
$593$	−19.5279			−0.801913			−0.400957	+	0.916097i	$0.631322\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.41641	+	14.1803i	0.221679	+	0.580363i
$598$		0			0
$599$			30.6525i			1.25243i	0.779652	+	0.626213i	$0.215396\pi$
$599$			30.6525i			1.25243i	−0.779652	+	0.626213i	$0.784604\pi$
$600$		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
$603$	−21.7082	+	19.4164i	−0.884026	+	0.790697i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	40.0689i		−	1.62635i	−0.582022	−	0.813173i	$-0.697738\pi$
$607$		−	40.0689i		−	1.62635i	0.582022	−	0.813173i	$-0.302262\pi$
$608$		0			0
$609$	17.8885	−	4.00000i	0.724880	−	0.162088i
$610$		0			0
$611$			4.00000i			0.161823i
$612$		0			0
$613$	−22.3607			−0.903139			−0.451570	−	0.892236i	$-0.649136\pi$
$613$	−22.3607			−0.903139			−0.451570	+	0.892236i	$0.649136\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.5410i		−	1.06850i	−0.845326	−	0.534251i	$-0.820594\pi$
$617$		−	26.5410i		−	1.06850i	0.845326	−	0.534251i	$-0.179406\pi$
$618$		0			0
$619$		−	5.70820i		−	0.229432i	−0.993398	−	0.114716i	$-0.963404\pi$
$619$		−	5.70820i		−	0.229432i	0.993398	−	0.114716i	$-0.0365958\pi$
$620$		0			0
$621$	−35.5967	−	18.3607i	−1.42845	−	0.736789i
$622$		0			0
$623$	4.76393	−	32.6525i	0.190863	−	1.30819i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	8.94427	−	3.41641i	0.357200	−	0.136438i
$628$		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
$633$	−12.9443	+	4.94427i	−0.514489	+	0.196517i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.47214	+	8.29180i	−0.0979496	+	0.328533i
$638$		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
$643$		−	24.2918i		−	0.957975i	−0.877822	−	0.478987i	$-0.841004\pi$
$643$		−	24.2918i		−	0.957975i	0.877822	−	0.478987i	$-0.158996\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.12461			0.358726			0.179363	−	0.983783i	$-0.442596\pi$
$647$	9.12461			0.358726			0.179363	+	0.983783i	$0.442596\pi$
$648$		0			0
$649$			6.83282i			0.268211i
$650$		0			0
$651$	−14.4721	+	3.23607i	−0.567208	+	0.126832i
$652$		0			0
$653$			21.5967i			0.845146i	0.906329	+	0.422573i	$0.138873\pi$
$653$			21.5967i			0.845146i	−0.906329	+	0.422573i	$0.861127\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	13.5279	+	15.1246i	0.527772	+	0.590067i
$658$		0			0
$659$		−	36.7639i		−	1.43212i	−0.698039	−	0.716060i	$-0.745944\pi$
$659$		−	36.7639i		−	1.43212i	0.698039	−	0.716060i	$-0.254056\pi$
$660$		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
$663$	−3.41641	−	8.94427i	−0.132682	−	0.347367i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	30.8328			1.19385
$668$		0			0
$669$	7.23607	+	18.9443i	0.279763	+	0.732428i
$670$		0			0
$671$	−3.77709			−0.145813
$672$		0			0
$673$	−14.9443			−0.576059			−0.288030	−	0.957621i	$-0.593000\pi$
$673$	−14.9443			−0.576059			−0.288030	+	0.957621i	$0.593000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		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
$683$			20.2918i			0.776444i	0.921566	+	0.388222i	$0.126911\pi$
$683$			20.2918i			0.776444i	−0.921566	+	0.388222i	$0.873089\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	8.58359	+	22.4721i	0.327484	+	0.857365i
$688$		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
$693$	−5.05573	+	3.34752i	−0.192051	+	0.127162i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	15.7771			0.597600
$698$		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
$703$		−	32.3607i		−	1.22051i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.65248	−	18.1803i	0.0997566	−	0.683742i
$708$		0			0
$709$	−22.9443			−0.861690			−0.430845	−	0.902426i	$-0.641784\pi$
$709$	−22.9443			−0.861690			−0.430845	+	0.902426i	$0.641784\pi$
$710$		0			0
$711$	23.4164	−	20.9443i	0.878184	−	0.785472i
$712$		0			0
$713$	−24.9443			−0.934170
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.111456	−	0.291796i	−0.00416241	−	0.0108973i
$718$		0			0
$719$	6.11146			0.227919			0.113959	−	0.993485i	$-0.463647\pi$
$719$	6.11146			0.227919			0.113959	+	0.993485i	$0.463647\pi$
$720$		0			0
$721$	28.1803	+	4.11146i	1.04949	+	0.153119i
$722$		0			0
$723$	10.1115	+	26.4721i	0.376049	+	0.984509i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			3.70820i			0.137530i	0.997633	+	0.0687648i	$0.0219058\pi$
$727$			3.70820i			0.137530i	−0.997633	+	0.0687648i	$0.978094\pi$
$728$		0			0
$729$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$730$		0			0
$731$	32.3607			1.19690
$732$		0			0
$733$			48.6525i			1.79702i	0.438953	+	0.898510i	$0.355350\pi$
$733$			48.6525i			1.79702i	−0.438953	+	0.898510i	$0.644650\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.41641i			0.273187i
$738$		0			0
$739$	40.3607			1.48469			0.742346	−	0.670017i	$-0.233713\pi$
$739$	40.3607			1.48469			0.742346	+	0.670017i	$0.233713\pi$
$740$		0			0
$741$	14.4721	−	5.52786i	0.531647	−	0.203071i
$742$		0			0
$743$		−	24.6525i		−	0.904412i	−0.891914	−	0.452206i	$-0.850637\pi$
$743$		−	24.6525i		−	0.904412i	0.891914	−	0.452206i	$-0.149363\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−16.1803	+	14.4721i	−0.592008	+	0.529508i
$748$		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
$753$	32.9443	−	12.5836i	1.20056	−	0.458572i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−35.3050			−1.28318			−0.641590	−	0.767048i	$-0.721725\pi$
$757$	−35.3050			−1.28318			−0.641590	+	0.767048i	$0.721725\pi$
$758$		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
$763$	−1.12461	+	7.70820i	−0.0407137	+	0.279056i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.0557i			0.399199i
$768$		0			0
$769$		−	31.4164i		−	1.13290i	−0.824095	−	0.566452i	$-0.808316\pi$
$769$		−	31.4164i		−	1.13290i	0.824095	−	0.566452i	$-0.191684\pi$
$770$		0			0
$771$	−26.6525	+	10.1803i	−0.959865	+	0.366636i
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	4.47214	+	20.0000i	0.160437	+	0.717496i
$778$		0			0
$779$			25.5279i			0.914631i
$780$		0			0
$781$	−9.30495			−0.332957
$782$		0			0
$783$	18.4721	+	9.52786i	0.660140	+	0.340498i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	16.2918i		−	0.580740i	−0.956914	−	0.290370i	$-0.906222\pi$
$787$		−	16.2918i		−	0.580740i	0.956914	−	0.290370i	$-0.0937784\pi$
$788$		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
$793$	−6.11146			−0.217024
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.8328			−1.02131			−0.510655	−	0.859785i	$-0.670597\pi$
$797$	−28.8328			−1.02131			−0.510655	+	0.859785i	$0.670597\pi$
$798$		0			0
$799$	−14.4721			−0.511987
$800$		0			0
$801$	27.8885	−	24.9443i	0.985393	−	0.881363i
$802$		0			0
$803$	5.16718			0.182346
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−31.5967	+	12.0689i	−1.11226	+	0.424845i
$808$		0			0
$809$		−	12.3607i		−	0.434578i	−0.976107	−	0.217289i	$-0.930279\pi$
$809$		−	12.3607i		−	0.434578i	0.976107	−	0.217289i	$-0.0697215\pi$
$810$		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
$813$	−13.0557	−	34.1803i	−0.457884	−	1.19876i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			52.3607i			1.83187i
$818$		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
$823$	11.5967			0.404237			0.202119	−	0.979361i	$-0.435217\pi$
$823$	11.5967			0.404237			0.202119	+	0.979361i	$0.435217\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	39.7082i		−	1.38079i	−0.723433	−	0.690395i	$-0.757437\pi$
$827$		−	39.7082i		−	1.38079i	0.723433	−	0.690395i	$-0.242563\pi$
$828$		0			0
$829$		−	20.3607i		−	0.707156i	−0.935405	−	0.353578i	$-0.884965\pi$
$829$		−	20.3607i		−	0.707156i	0.935405	−	0.353578i	$-0.115035\pi$
$830$		0			0
$831$	−2.29180	+	0.875388i	−0.0795015	+	0.0303669i
$832$		0			0
$833$	−30.0000	−	8.94427i	−1.03944	−	0.309901i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−14.9443	−	7.70820i	−0.516550	−	0.266435i
$838$		0			0
$839$	12.9443			0.446886			0.223443	−	0.974717i	$-0.428270\pi$
$839$	12.9443			0.446886			0.223443	+	0.974717i	$0.428270\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	7.05573	+	18.4721i	0.243012	+	0.636214i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3.97871	−	27.2705i	0.136710	−	0.937026i
$848$		0			0
$849$	5.34752	+	14.0000i	0.183527	+	0.480479i
$850$		0			0
$851$			34.4721i			1.18169i
$852$		0			0
$853$		−	19.1246i		−	0.654814i	−0.944883	−	0.327407i	$-0.893825\pi$
$853$		−	19.1246i		−	0.654814i	0.944883	−	0.327407i	$-0.106175\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.4721			0.972590			0.486295	−	0.873795i	$-0.338348\pi$
$857$	28.4721			0.972590			0.486295	+	0.873795i	$0.338348\pi$
$858$		0			0
$859$			13.3475i			0.455412i	0.973730	+	0.227706i	$0.0731225\pi$
$859$			13.3475i			0.455412i	−0.973730	+	0.227706i	$0.926878\pi$
$860$		0			0
$861$	−3.52786	−	15.7771i	−0.120229	−	0.537682i
$862$		0			0
$863$		−	52.6525i		−	1.79231i	−0.443740	−	0.896156i	$-0.646349\pi$
$863$		−	52.6525i		−	1.79231i	0.443740	−	0.896156i	$-0.353651\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4.85410	−	1.85410i	0.164854	−	0.0629686i
$868$		0			0
$869$		−	8.00000i		−	0.271381i
$870$		0			0
$871$			12.0000i			0.406604i
$872$		0			0
$873$	28.3607	+	31.7082i	0.959864	+	1.07316i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	32.4721			1.09651			0.548253	−	0.836312i	$-0.315293\pi$
$877$	32.4721			1.09651			0.548253	+	0.836312i	$0.315293\pi$
$878$		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
$883$	35.5967			1.19793			0.598963	−	0.800777i	$-0.295580\pi$
$883$	35.5967			1.19793			0.598963	+	0.800777i	$0.295580\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.1803			−0.543283			−0.271641	−	0.962399i	$-0.587566\pi$
$887$	−16.1803			−0.543283			−0.271641	+	0.962399i	$0.587566\pi$
$888$		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
$893$		−	23.4164i		−	0.783600i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−15.4164	+	5.88854i	−0.514739	+	0.196613i
$898$		0			0
$899$	12.9443			0.431716
$900$		0			0
$901$		−	41.3050i		−	1.37607i
$902$		0			0
$903$	−7.23607	−	32.3607i	−0.240801	−	1.07690i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	27.2361			0.904359			0.452179	−	0.891927i	$-0.350647\pi$
$907$	27.2361			0.904359			0.452179	+	0.891927i	$0.350647\pi$
$908$		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
$913$			5.52786i			0.182946i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.88854	+	40.3607i	−0.194457	+	1.33283i
$918$		0			0
$919$	−34.4721			−1.13713			−0.568565	−	0.822638i	$-0.692501\pi$
$919$	−34.4721			−1.13713			−0.568565	+	0.822638i	$0.692501\pi$
$920$		0			0
$921$	−11.2361	−	29.4164i	−0.370241	−	0.969304i
$922$		0			0
$923$	−15.0557			−0.495565
$924$		0			0
$925$		0			0
$926$		0			0
$927$	21.5279	+	24.0689i	0.707068	+	0.790526i
$928$		0			0
$929$	19.8885			0.652522			0.326261	−	0.945280i	$-0.394211\pi$
$929$	19.8885			0.652522			0.326261	+	0.945280i	$0.394211\pi$
$930$		0			0
$931$	14.4721	−	48.5410i	0.474305	−	1.59087i
$932$		0			0
$933$	52.3607	−	20.0000i	1.71421	−	0.654771i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	14.7639i		−	0.482317i	−0.970486	−	0.241158i	$-0.922473\pi$
$937$		−	14.7639i		−	0.482317i	0.970486	−	0.241158i	$-0.0775273\pi$
$938$		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
$943$		−	27.1935i		−	0.885542i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			25.2361i			0.820062i	0.912072	+	0.410031i	$0.134482\pi$
$947$			25.2361i			0.820062i	−0.912072	+	0.410031i	$0.865518\pi$
$948$		0			0
$949$	8.36068			0.271399
$950$		0			0
$951$	3.81966	+	10.0000i	0.123861	+	0.324272i
$952$		0			0
$953$			3.70820i			0.120121i	0.998195	+	0.0600603i	$0.0191293\pi$
$953$			3.70820i			0.120121i	−0.998195	+	0.0600603i	$0.980871\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	4.94427	−	1.88854i	0.159826	−	0.0610480i
$958$		0			0
$959$	3.23607	+	0.472136i	0.104498	+	0.0152461i
$960$		0			0
$961$	20.5279			0.662189
$962$		0			0
$963$	0.583592	+	0.652476i	0.0188060	+	0.0210257i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−44.1803			−1.42074			−0.710372	−	0.703826i	$-0.751473\pi$
$967$	−44.1803			−1.42074			−0.710372	+	0.703826i	$0.751473\pi$
$968$		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
$973$	39.8885	+	5.81966i	1.27877	+	0.186570i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			49.9574i			1.59828i	0.601145	+	0.799140i	$0.294711\pi$
$977$			49.9574i			1.59828i	−0.601145	+	0.799140i	$0.705289\pi$
$978$		0			0
$979$		−	9.52786i		−	0.304512i
$980$		0			0
$981$	−6.58359	+	5.88854i	−0.210198	+	0.188007i
$982$		0			0
$983$	−40.5410			−1.29306			−0.646529	−	0.762890i	$-0.723780\pi$
$983$	−40.5410			−1.29306			−0.646529	+	0.762890i	$0.723780\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.23607	+	14.4721i	0.103005	+	0.460653i
$988$		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
$993$	−33.8885	+	12.9443i	−1.07542	+	0.410774i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			16.0689i			0.508907i	0.967085	+	0.254453i	$0.0818955\pi$
$997$			16.0689i			0.508907i	−0.967085	+	0.254453i	$0.918104\pi$
$998$		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.d.h.1301.3		4
3.2	odd	2		2100.2.d.g.1301.1		4
5.2	odd	4		2100.2.f.g.1049.1		4
5.3	odd	4		2100.2.f.a.1049.4		4
5.4	even	2		420.2.d.c.41.2	yes	4
7.6	odd	2		2100.2.d.g.1301.2		4
15.2	even	4		2100.2.f.h.1049.2		4
15.8	even	4		2100.2.f.b.1049.3		4
15.14	odd	2		420.2.d.d.41.4	yes	4
20.19	odd	2		1680.2.f.j.881.3		4
21.20	even	2	inner	2100.2.d.h.1301.4		4
35.13	even	4		2100.2.f.h.1049.1		4
35.27	even	4		2100.2.f.b.1049.4		4
35.34	odd	2		420.2.d.d.41.3	yes	4
60.59	even	2		1680.2.f.f.881.1		4
105.62	odd	4		2100.2.f.a.1049.3		4
105.83	odd	4		2100.2.f.g.1049.2		4
105.104	even	2		420.2.d.c.41.1	✓	4
140.139	even	2		1680.2.f.f.881.2		4
420.419	odd	2		1680.2.f.j.881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.d.c.41.1	✓	4	105.104	even	2
420.2.d.c.41.2	yes	4	5.4	even	2
420.2.d.d.41.3	yes	4	35.34	odd	2
420.2.d.d.41.4	yes	4	15.14	odd	2
1680.2.f.f.881.1		4	60.59	even	2
1680.2.f.f.881.2		4	140.139	even	2
1680.2.f.j.881.3		4	20.19	odd	2
1680.2.f.j.881.4		4	420.419	odd	2
2100.2.d.g.1301.1		4	3.2	odd	2
2100.2.d.g.1301.2		4	7.6	odd	2
2100.2.d.h.1301.3		4	1.1	even	1	trivial
2100.2.d.h.1301.4		4	21.20	even	2	inner
2100.2.f.a.1049.3		4	105.62	odd	4
2100.2.f.a.1049.4		4	5.3	odd	4
2100.2.f.b.1049.3		4	15.8	even	4
2100.2.f.b.1049.4		4	35.27	even	4
2100.2.f.g.1049.1		4	5.2	odd	4
2100.2.f.g.1049.2		4	105.83	odd	4
2100.2.f.h.1049.1		4	35.13	even	4
2100.2.f.h.1049.2		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} -0.763932i q^{11} -1.23607i q^{13} +4.47214 q^{17} +7.23607i q^{19} +(-1.00000 - 4.47214i) q^{21} -7.70820i q^{23} +(2.38197 - 4.61803i) q^{27} +4.00000i q^{29} -3.23607i q^{31} +(-0.472136 - 1.23607i) q^{33} -4.47214 q^{37} +(-0.763932 - 2.00000i) q^{39} +3.52786 q^{41} +7.23607 q^{43} -3.23607 q^{47} +(-6.70820 - 2.00000i) q^{49} +(7.23607 - 2.76393i) q^{51} -9.23607i q^{53} +(4.47214 + 11.7082i) q^{57} -8.94427 q^{59} -4.94427i q^{61} +(-4.38197 - 6.61803i) q^{63} -9.70820 q^{67} +(-4.76393 - 12.4721i) q^{69} -12.1803i q^{71} +6.76393i q^{73} +(-2.00000 - 0.291796i) q^{77} +10.4721 q^{79} +(1.00000 - 8.94427i) q^{81} -7.23607 q^{83} +(2.47214 + 6.47214i) q^{87} +12.4721 q^{89} +(-3.23607 - 0.472136i) q^{91} +(-2.00000 - 5.23607i) q^{93} +14.1803i q^{97} +(-1.52786 - 1.70820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) 4 * q + 2 * q^3 + 6 * q^7	\( 4 q + 2 q^{3} + 6 q^{7} - 4 q^{21} + 14 q^{27} + 16 q^{33} - 12 q^{39} + 32 q^{41} + 20 q^{43} - 4 q^{47} + 20 q^{51} - 22 q^{63} - 12 q^{67} - 28 q^{69} - 8 q^{77} + 24 q^{79} + 4 q^{81} - 20 q^{83} - 8 q^{87} + 32 q^{89} - 4 q^{91} - 8 q^{93} - 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^7 - 4 * q^21 + 14 * q^27 + 16 * q^33 - 12 * q^39 + 32 * q^41 + 20 * q^43 - 4 * q^47 + 20 * q^51 - 22 * q^63 - 12 * q^67 - 28 * q^69 - 8 * q^77 + 24 * q^79 + 4 * q^81 - 20 * q^83 - 8 * q^87 + 32 * q^89 - 4 * q^91 - 8 * q^93 - 24 * q^99

Introduction

L-functions

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Database

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