LMFDB - Embedded newform 2100.2.d.h.1301.1

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			1301.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.h.1301.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.618034 - 1.61803i) q^{3} +(2.61803 + 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	$q+(-0.618034 - 1.61803i) q^{3} +(2.61803 + 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +5.23607i q^{11} -3.23607i q^{13} -4.47214 q^{17} -2.76393i q^{19} +(-1.00000 - 4.47214i) q^{21} -5.70820i q^{23} +(4.61803 + 2.38197i) q^{27} -4.00000i q^{29} -1.23607i q^{31} +(8.47214 - 3.23607i) q^{33} +4.47214 q^{37} +(-5.23607 + 2.00000i) q^{39} +12.4721 q^{41} +2.76393 q^{43} +1.23607 q^{47} +(6.70820 + 2.00000i) q^{49} +(2.76393 + 7.23607i) q^{51} +4.76393i q^{53} +(-4.47214 + 1.70820i) q^{57} +8.94427 q^{59} -12.9443i q^{61} +(-6.61803 + 4.38197i) q^{63} +3.70820 q^{67} +(-9.23607 + 3.52786i) q^{69} -10.1803i q^{71} -11.2361i q^{73} +(-2.00000 + 13.7082i) q^{77} +1.52786 q^{79} +(1.00000 - 8.94427i) q^{81} -2.76393 q^{83} +(-6.47214 + 2.47214i) q^{87} +3.52786 q^{89} +(1.23607 - 8.47214i) q^{91} +(-2.00000 + 0.763932i) q^{93} +8.18034i q^{97} +(-10.4721 - 11.7082i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−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
$11$			5.23607i			1.57873i	0.613922	+	0.789367i	$0.289591\pi$
$11$			5.23607i			1.57873i	−0.613922	+	0.789367i	$0.710409\pi$
$12$		0			0
$13$		−	3.23607i		−	0.897524i	−0.893651	−	0.448762i	$-0.851865\pi$
$13$		−	3.23607i		−	0.897524i	0.893651	−	0.448762i	$-0.148135\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.47214			−1.08465			−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214			−1.08465			−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$		−	2.76393i		−	0.634089i	−0.948411	−	0.317045i	$-0.897309\pi$
$19$		−	2.76393i		−	0.634089i	0.948411	−	0.317045i	$-0.102691\pi$
$20$		0			0
$21$	−1.00000	−	4.47214i	−0.218218	−	0.975900i
$22$		0			0
$23$		−	5.70820i		−	1.19024i	−0.803636	−	0.595121i	$-0.797104\pi$
$23$		−	5.70820i		−	1.19024i	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
$31$		−	1.23607i		−	0.222004i	−0.993820	−	0.111002i	$-0.964594\pi$
$31$		−	1.23607i		−	0.222004i	0.993820	−	0.111002i	$-0.0354061\pi$
$32$		0			0
$33$	8.47214	−	3.23607i	1.47481	−	0.563327i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.47214			0.735215			0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214			0.735215			0.367607	+	0.929981i	$0.380177\pi$
$38$		0			0
$39$	−5.23607	+	2.00000i	−0.838442	+	0.320256i
$40$		0			0
$41$	12.4721			1.94782			0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721			1.94782			0.973910	+	0.226934i	$0.0728701\pi$
$42$		0			0
$43$	2.76393			0.421496			0.210748	−	0.977540i	$-0.432410\pi$
$43$	2.76393			0.421496			0.210748	+	0.977540i	$0.432410\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.23607			0.180299			0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607			0.180299			0.0901495	+	0.995928i	$0.471266\pi$
$48$		0			0
$49$	6.70820	+	2.00000i	0.958315	+	0.285714i
$50$		0			0
$51$	2.76393	+	7.23607i	0.387028	+	1.01325i
$52$		0			0
$53$			4.76393i			0.654376i	0.944959	+	0.327188i	$0.106101\pi$
$53$			4.76393i			0.654376i	−0.944959	+	0.327188i	$0.893899\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.47214	+	1.70820i	−0.592349	+	0.226257i
$58$		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
$61$		−	12.9443i		−	1.65734i	−0.559734	−	0.828672i	$-0.689097\pi$
$61$		−	12.9443i		−	1.65734i	0.559734	−	0.828672i	$-0.310903\pi$
$62$		0			0
$63$	−6.61803	+	4.38197i	−0.833794	+	0.552076i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.70820			0.453029			0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820			0.453029			0.226515	+	0.974008i	$0.427267\pi$
$68$		0			0
$69$	−9.23607	+	3.52786i	−1.11189	+	0.424705i
$70$		0			0
$71$		−	10.1803i		−	1.20818i	−0.796915	−	0.604092i	$-0.793536\pi$
$71$		−	10.1803i		−	1.20818i	0.796915	−	0.604092i	$-0.206464\pi$
$72$		0			0
$73$		−	11.2361i		−	1.31508i	−0.753419	−	0.657541i	$-0.771597\pi$
$73$		−	11.2361i		−	1.31508i	0.753419	−	0.657541i	$-0.228403\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	+	13.7082i	−0.227921	+	1.56219i
$78$		0			0
$79$	1.52786			0.171898			0.0859491	−	0.996300i	$-0.472608\pi$
$79$	1.52786			0.171898			0.0859491	+	0.996300i	$0.472608\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−2.76393			−0.303381			−0.151690	−	0.988428i	$-0.548472\pi$
$83$	−2.76393			−0.303381			−0.151690	+	0.988428i	$0.548472\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.47214	+	2.47214i	−0.693886	+	0.265041i
$88$		0			0
$89$	3.52786			0.373953			0.186976	−	0.982364i	$-0.440131\pi$
$89$	3.52786			0.373953			0.186976	+	0.982364i	$0.440131\pi$
$90$		0			0
$91$	1.23607	−	8.47214i	0.129575	−	0.888121i
$92$		0			0
$93$	−2.00000	+	0.763932i	−0.207390	+	0.0792161i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.18034i			0.830588i	0.909687	+	0.415294i	$0.136321\pi$
$97$			8.18034i			0.830588i	−0.909687	+	0.415294i	$0.863679\pi$
$98$		0			0
$99$	−10.4721	−	11.7082i	−1.05249	−	1.17672i
$100$		0			0
$101$	−10.9443			−1.08900			−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443			−1.08900			−0.544498	+	0.838762i	$0.683280\pi$
$102$		0			0
$103$		−	15.2361i		−	1.50125i	−0.660726	−	0.750627i	$-0.729751\pi$
$103$		−	15.2361i		−	1.50125i	0.660726	−	0.750627i	$-0.270249\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.7082i		−	1.32522i	−0.748964	−	0.662611i	$-0.769448\pi$
$107$		−	13.7082i		−	1.32522i	0.748964	−	0.662611i	$-0.230552\pi$
$108$		0			0
$109$	14.9443			1.43140			0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443			1.43140			0.715701	+	0.698407i	$0.246107\pi$
$110$		0			0
$111$	−2.76393	−	7.23607i	−0.262341	−	0.686817i
$112$		0			0
$113$			12.7639i			1.20073i	0.799726	+	0.600365i	$0.204978\pi$
$113$			12.7639i			1.20073i	−0.799726	+	0.600365i	$0.795022\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.47214	+	7.23607i	0.598349	+	0.668975i
$118$		0			0
$119$	−11.7082	−	1.70820i	−1.07329	−	0.156591i
$120$		0			0
$121$	−16.4164			−1.49240
$122$		0			0
$123$	−7.70820	−	20.1803i	−0.695025	−	1.81960i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−19.1246			−1.69703			−0.848517	−	0.529167i	$-0.822504\pi$
$127$	−19.1246			−1.69703			−0.848517	+	0.529167i	$0.822504\pi$
$128$		0			0
$129$	−1.70820	−	4.47214i	−0.150399	−	0.393750i
$130$		0			0
$131$	11.4164			0.997456			0.498728	−	0.866758i	$-0.333801\pi$
$131$	11.4164			0.997456			0.498728	+	0.866758i	$0.333801\pi$
$132$		0			0
$133$	1.05573	−	7.23607i	0.0915432	−	0.627447i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.23607i			0.276476i	0.990399	+	0.138238i	$0.0441439\pi$
$137$			3.23607i			0.276476i	−0.990399	+	0.138238i	$0.955856\pi$
$138$		0			0
$139$		−	10.7639i		−	0.912985i	−0.889727	−	0.456492i	$-0.849106\pi$
$139$		−	10.7639i		−	0.912985i	0.889727	−	0.456492i	$-0.150894\pi$
$140$		0			0
$141$	−0.763932	−	2.00000i	−0.0643347	−	0.168430i
$142$		0			0
$143$	16.9443			1.41695
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.909830	−	12.0902i	−0.0750415	−	0.997180i
$148$		0			0
$149$			11.4164i			0.935269i	0.883922	+	0.467634i	$0.154894\pi$
$149$			11.4164i			0.935269i	−0.883922	+	0.467634i	$0.845106\pi$
$150$		0			0
$151$	−8.94427			−0.727875			−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427			−0.727875			−0.363937	+	0.931423i	$0.618568\pi$
$152$		0			0
$153$	10.0000	−	8.94427i	0.808452	−	0.723102i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			24.1803i			1.92980i	0.262615	+	0.964901i	$0.415415\pi$
$157$			24.1803i			1.92980i	−0.262615	+	0.964901i	$0.584585\pi$
$158$		0			0
$159$	7.70820	−	2.94427i	0.611300	−	0.233496i
$160$		0			0
$161$	2.18034	−	14.9443i	0.171835	−	1.17777i
$162$		0			0
$163$	12.6525			0.991018			0.495509	−	0.868603i	$-0.334981\pi$
$163$	12.6525			0.991018			0.495509	+	0.868603i	$0.334981\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.2361			−1.02424			−0.512119	−	0.858915i	$-0.671139\pi$
$167$	−13.2361			−1.02424			−0.512119	+	0.858915i	$0.671139\pi$
$168$		0			0
$169$	2.52786			0.194451
$170$		0			0
$171$	5.52786	+	6.18034i	0.422726	+	0.472622i
$172$		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
$177$	−5.52786	−	14.4721i	−0.415500	−	1.08779i
$178$		0			0
$179$			17.2361i			1.28828i	0.764906	+	0.644142i	$0.222785\pi$
$179$			17.2361i			1.28828i	−0.764906	+	0.644142i	$0.777215\pi$
$180$		0			0
$181$			2.47214i			0.183752i	0.995770	+	0.0918762i	$0.0292864\pi$
$181$			2.47214i			0.183752i	−0.995770	+	0.0918762i	$0.970714\pi$
$182$		0			0
$183$	−20.9443	+	8.00000i	−1.54825	+	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	23.4164i		−	1.71238i
$188$		0			0
$189$	11.1803	+	8.00000i	0.813250	+	0.581914i
$190$		0			0
$191$			2.76393i			0.199991i	0.994988	+	0.0999956i	$0.0318829\pi$
$191$			2.76393i			0.199991i	−0.994988	+	0.0999956i	$0.968117\pi$
$192$		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
$197$		−	20.7639i		−	1.47937i	−0.672954	−	0.739684i	$-0.734975\pi$
$197$		−	20.7639i		−	1.47937i	0.672954	−	0.739684i	$-0.265025\pi$
$198$		0			0
$199$		−	13.2361i		−	0.938280i	−0.883124	−	0.469140i	$-0.844564\pi$
$199$		−	13.2361i		−	0.938280i	0.883124	−	0.469140i	$-0.155436\pi$
$200$		0			0
$201$	−2.29180	−	6.00000i	−0.161651	−	0.423207i
$202$		0			0
$203$	1.52786	−	10.4721i	0.107235	−	0.735000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	11.4164	+	12.7639i	0.793495	+	0.887155i
$208$		0			0
$209$	14.4721			1.00106
$210$		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
$213$	−16.4721	+	6.29180i	−1.12865	+	0.431107i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.472136	−	3.23607i	0.0320507	−	0.219679i
$218$		0			0
$219$	−18.1803	+	6.94427i	−1.22851	+	0.469250i
$220$		0			0
$221$			14.4721i			0.973501i
$222$		0			0
$223$			1.70820i			0.114390i	0.998363	+	0.0571949i	$0.0182156\pi$
$223$			1.70820i			0.114390i	−0.998363	+	0.0571949i	$0.981784\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.7639			1.24541			0.622703	−	0.782458i	$-0.286035\pi$
$227$	18.7639			1.24541			0.622703	+	0.782458i	$0.286035\pi$
$228$		0			0
$229$			21.8885i			1.44644i	0.690620	+	0.723218i	$0.257338\pi$
$229$			21.8885i			1.44644i	−0.690620	+	0.723218i	$0.742662\pi$
$230$		0			0
$231$	23.4164	−	5.23607i	1.54069	−	0.344508i
$232$		0			0
$233$		−	24.1803i		−	1.58411i	−0.610452	−	0.792053i	$-0.709012\pi$
$233$		−	24.1803i		−	1.58411i	0.610452	−	0.792053i	$-0.290988\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.944272	−	2.47214i	−0.0613371	−	0.160582i
$238$		0			0
$239$		−	22.1803i		−	1.43473i	−0.696699	−	0.717363i	$-0.745349\pi$
$239$		−	22.1803i		−	1.43473i	0.696699	−	0.717363i	$-0.254651\pi$
$240$		0			0
$241$			28.3607i			1.82687i	0.406982	+	0.913436i	$0.366581\pi$
$241$			28.3607i			1.82687i	−0.406982	+	0.913436i	$0.633419\pi$
$242$		0			0
$243$	−15.0902	+	3.90983i	−0.968035	+	0.250816i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.94427			−0.569110
$248$		0			0
$249$	1.70820	+	4.47214i	0.108253	+	0.283410i
$250$		0			0
$251$	−24.3607			−1.53763			−0.768816	−	0.639470i	$-0.779154\pi$
$251$	−24.3607			−1.53763			−0.768816	+	0.639470i	$0.779154\pi$
$252$		0			0
$253$	29.8885			1.87908
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.52786			−0.469575			−0.234788	−	0.972047i	$-0.575439\pi$
$257$	−7.52786			−0.469575			−0.234788	+	0.972047i	$0.575439\pi$
$258$		0			0
$259$	11.7082	+	1.70820i	0.727512	+	0.106143i
$260$		0			0
$261$	8.00000	+	8.94427i	0.495188	+	0.553637i
$262$		0			0
$263$			0.180340i			0.0111202i	0.999985	+	0.00556012i	$0.00176985\pi$
$263$			0.180340i			0.0111202i	−0.999985	+	0.00556012i	$0.998230\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.18034	−	5.70820i	−0.133435	−	0.349336i
$268$		0			0
$269$	−28.4721			−1.73598			−0.867988	−	0.496584i	$-0.834587\pi$
$269$	−28.4721			−1.73598			−0.867988	+	0.496584i	$0.834587\pi$
$270$		0			0
$271$		−	19.1246i		−	1.16174i	−0.813997	−	0.580869i	$-0.802713\pi$
$271$		−	19.1246i		−	1.16174i	0.813997	−	0.580869i	$-0.197287\pi$
$272$		0			0
$273$	−14.4721	+	3.23607i	−0.875894	+	0.195856i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	25.4164			1.52712			0.763562	−	0.645735i	$-0.223449\pi$
$277$	25.4164			1.52712			0.763562	+	0.645735i	$0.223449\pi$
$278$		0			0
$279$	2.47214	+	2.76393i	0.148003	+	0.165472i
$280$		0			0
$281$			15.4164i			0.919666i	0.888005	+	0.459833i	$0.152091\pi$
$281$			15.4164i			0.919666i	−0.888005	+	0.459833i	$0.847909\pi$
$282$		0			0
$283$			22.6525i			1.34655i	0.739392	+	0.673275i	$0.235113\pi$
$283$			22.6525i			1.34655i	−0.739392	+	0.673275i	$0.764887\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	32.6525	+	4.76393i	1.92741	+	0.281206i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$	13.2361	−	5.05573i	0.775912	−	0.296372i
$292$		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
$297$	−12.4721	+	24.1803i	−0.723707	+	1.40309i
$298$		0			0
$299$	−18.4721			−1.06827
$300$		0			0
$301$	7.23607	+	1.05573i	0.417080	+	0.0608512i
$302$		0			0
$303$	6.76393	+	17.7082i	0.388578	+	1.01731i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	4.18034i		−	0.238585i	−0.992859	−	0.119292i	$-0.961937\pi$
$307$		−	4.18034i		−	0.238585i	0.992859	−	0.119292i	$-0.0380626\pi$
$308$		0			0
$309$	−24.6525	+	9.41641i	−1.40243	+	0.535681i
$310$		0			0
$311$	−12.3607			−0.700910			−0.350455	−	0.936580i	$-0.613973\pi$
$311$	−12.3607			−0.700910			−0.350455	+	0.936580i	$0.613973\pi$
$312$		0			0
$313$			17.7082i			1.00093i	0.865758	+	0.500463i	$0.166837\pi$
$313$			17.7082i			1.00093i	−0.865758	+	0.500463i	$0.833163\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.1803i			0.908778i	0.890803	+	0.454389i	$0.150142\pi$
$317$			16.1803i			0.908778i	−0.890803	+	0.454389i	$0.849858\pi$
$318$		0			0
$319$	20.9443			1.17265
$320$		0			0
$321$	−22.1803	+	8.47214i	−1.23799	+	0.472869i
$322$		0			0
$323$			12.3607i			0.687767i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−9.23607	−	24.1803i	−0.510756	−	1.33718i
$328$		0			0
$329$	3.23607	+	0.472136i	0.178410	+	0.0260297i
$330$		0			0
$331$	−3.05573			−0.167958			−0.0839790	−	0.996468i	$-0.526763\pi$
$331$	−3.05573			−0.167958			−0.0839790	+	0.996468i	$0.526763\pi$
$332$		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
$339$	20.6525	−	7.88854i	1.12169	−	0.428447i
$340$		0			0
$341$	6.47214			0.350486
$342$		0			0
$343$	16.7984	+	7.79837i	0.907027	+	0.421073i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.34752i			0.0723389i	0.999346	+	0.0361694i	$0.0115156\pi$
$347$			1.34752i			0.0723389i	−0.999346	+	0.0361694i	$0.988484\pi$
$348$		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
$351$	7.70820	−	14.9443i	0.411433	−	0.797666i
$352$		0			0
$353$	−1.41641			−0.0753878			−0.0376939	−	0.999289i	$-0.512001\pi$
$353$	−1.41641			−0.0753878			−0.0376939	+	0.999289i	$0.512001\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.47214	+	20.0000i	0.236691	+	1.05851i
$358$		0			0
$359$		−	24.0689i		−	1.27031i	−0.772386	−	0.635154i	$-0.780937\pi$
$359$		−	24.0689i		−	1.27031i	0.772386	−	0.635154i	$-0.219063\pi$
$360$		0			0
$361$	11.3607			0.597931
$362$		0			0
$363$	10.1459	+	26.5623i	0.532522	+	1.39416i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			6.65248i			0.347256i	0.984811	+	0.173628i	$0.0555491\pi$
$367$			6.65248i			0.347256i	−0.984811	+	0.173628i	$0.944451\pi$
$368$		0			0
$369$	−27.8885	+	24.9443i	−1.45182	+	1.29855i
$370$		0			0
$371$	−1.81966	+	12.4721i	−0.0944720	+	0.647521i
$372$		0			0
$373$	−16.4721			−0.852895			−0.426447	−	0.904512i	$-0.640235\pi$
$373$	−16.4721			−0.852895			−0.426447	+	0.904512i	$0.640235\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.9443			−0.666664
$378$		0			0
$379$	10.4721			0.537917			0.268959	−	0.963152i	$-0.413320\pi$
$379$	10.4721			0.537917			0.268959	+	0.963152i	$0.413320\pi$
$380$		0			0
$381$	11.8197	+	30.9443i	0.605540	+	1.58532i
$382$		0			0
$383$	−10.7639			−0.550011			−0.275006	−	0.961443i	$-0.588680\pi$
$383$	−10.7639			−0.550011			−0.275006	+	0.961443i	$0.588680\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−6.18034	+	5.52786i	−0.314164	+	0.280997i
$388$		0			0
$389$		−	6.47214i		−	0.328150i	−0.986448	−	0.164075i	$-0.947536\pi$
$389$		−	6.47214i		−	0.328150i	0.986448	−	0.164075i	$-0.0524640\pi$
$390$		0			0
$391$			25.5279i			1.29100i
$392$		0			0
$393$	−7.05573	−	18.4721i	−0.355914	−	0.931796i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	9.70820i		−	0.487241i	−0.969871	−	0.243620i	$-0.921665\pi$
$397$		−	9.70820i		−	0.487241i	0.969871	−	0.243620i	$-0.0783351\pi$
$398$		0			0
$399$	−12.3607	+	2.76393i	−0.618808	+	0.138370i
$400$		0			0
$401$		−	11.0557i		−	0.552097i	−0.961144	−	0.276048i	$-0.910975\pi$
$401$		−	11.0557i		−	0.552097i	0.961144	−	0.276048i	$-0.0890250\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		0			0
$405$		0			0
$406$		0			0
$407$			23.4164i			1.16071i
$408$		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
$411$	5.23607	−	2.00000i	0.258276	−	0.0986527i
$412$		0			0
$413$	23.4164	+	3.41641i	1.15225	+	0.168110i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−17.4164	+	6.65248i	−0.852885	+	0.325773i
$418$		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
$421$	23.5279			1.14668			0.573339	−	0.819318i	$-0.305648\pi$
$421$	23.5279			1.14668			0.573339	+	0.819318i	$0.305648\pi$
$422$		0			0
$423$	−2.76393	+	2.47214i	−0.134387	+	0.120199i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.94427	−	33.8885i	0.239270	−	1.63998i
$428$		0			0
$429$	−10.4721	−	27.4164i	−0.505599	−	1.32368i
$430$		0			0
$431$		−	5.23607i		−	0.252213i	−0.992017	−	0.126106i	$-0.959752\pi$
$431$		−	5.23607i		−	0.252213i	0.992017	−	0.126106i	$-0.0402480\pi$
$432$		0			0
$433$		−	1.34752i		−	0.0647579i	−0.999476	−	0.0323789i	$-0.989692\pi$
$433$		−	1.34752i		−	0.0647579i	0.999476	−	0.0323789i	$-0.0103083\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.7771			−0.754720
$438$		0			0
$439$		−	18.1803i		−	0.867700i	−0.900985	−	0.433850i	$-0.857155\pi$
$439$		−	18.1803i		−	0.867700i	0.900985	−	0.433850i	$-0.142845\pi$
$440$		0			0
$441$	−19.0000	+	8.94427i	−0.904762	+	0.425918i
$442$		0			0
$443$			34.0689i			1.61866i	0.587353	+	0.809331i	$0.300170\pi$
$443$			34.0689i			1.61866i	−0.587353	+	0.809331i	$0.699830\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	18.4721	−	7.05573i	0.873702	−	0.333724i
$448$		0			0
$449$			5.88854i			0.277898i	0.990300	+	0.138949i	$0.0443723\pi$
$449$			5.88854i			0.277898i	−0.990300	+	0.138949i	$0.955628\pi$
$450$		0			0
$451$			65.3050i			3.07509i
$452$		0			0
$453$	5.52786	+	14.4721i	0.259722	+	0.679960i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.9443			1.26040			0.630200	−	0.776433i	$-0.282973\pi$
$457$	26.9443			1.26040			0.630200	+	0.776433i	$0.282973\pi$
$458$		0			0
$459$	−20.6525	−	10.6525i	−0.963975	−	0.497215i
$460$		0			0
$461$	15.5279			0.723205			0.361602	−	0.932332i	$-0.382230\pi$
$461$	15.5279			0.723205			0.361602	+	0.932332i	$0.382230\pi$
$462$		0			0
$463$	25.2361			1.17282			0.586410	−	0.810015i	$-0.300541\pi$
$463$	25.2361			1.17282			0.586410	+	0.810015i	$0.300541\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.5410			1.41327			0.706635	−	0.707578i	$-0.250212\pi$
$467$	30.5410			1.41327			0.706635	+	0.707578i	$0.250212\pi$
$468$		0			0
$469$	9.70820	+	1.41641i	0.448283	+	0.0654036i
$470$		0			0
$471$	39.1246	−	14.9443i	1.80277	−	0.688596i
$472$		0			0
$473$			14.4721i			0.665430i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−9.52786	−	10.6525i	−0.436251	−	0.487743i
$478$		0			0
$479$	4.94427			0.225910			0.112955	−	0.993600i	$-0.463968\pi$
$479$	4.94427			0.225910			0.112955	+	0.993600i	$0.463968\pi$
$480$		0			0
$481$		−	14.4721i		−	0.659873i
$482$		0			0
$483$	−25.5279	+	5.70820i	−1.16156	+	0.259732i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.23607			−0.0560116			−0.0280058	−	0.999608i	$-0.508916\pi$
$487$	−1.23607			−0.0560116			−0.0280058	+	0.999608i	$0.508916\pi$
$488$		0			0
$489$	−7.81966	−	20.4721i	−0.353617	−	0.925782i
$490$		0			0
$491$		−	0.875388i		−	0.0395057i	−0.999805	−	0.0197529i	$-0.993712\pi$
$491$		−	0.875388i		−	0.0395057i	0.999805	−	0.0197529i	$-0.00628794\pi$
$492$		0			0
$493$			17.8885i			0.805659i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.88854	−	26.6525i	0.174425	−	1.19553i
$498$		0			0
$499$	28.3607			1.26960			0.634799	−	0.772677i	$-0.281083\pi$
$499$	28.3607			1.26960			0.634799	+	0.772677i	$0.281083\pi$
$500$		0			0
$501$	8.18034	+	21.4164i	0.365471	+	0.956815i
$502$		0			0
$503$	−12.6525			−0.564146			−0.282073	−	0.959393i	$-0.591022\pi$
$503$	−12.6525			−0.564146			−0.282073	+	0.959393i	$0.591022\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.56231	−	4.09017i	−0.0693844	−	0.181651i
$508$		0			0
$509$	5.05573			0.224091			0.112046	−	0.993703i	$-0.464260\pi$
$509$	5.05573			0.224091			0.112046	+	0.993703i	$0.464260\pi$
$510$		0			0
$511$	4.29180	−	29.4164i	0.189858	−	1.30131i
$512$		0			0
$513$	6.58359	−	12.7639i	0.290673	−	0.563541i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			6.47214i			0.284644i
$518$		0			0
$519$	−1.23607	−	3.23607i	−0.0542574	−	0.142048i
$520$		0			0
$521$	−27.8885			−1.22182			−0.610910	−	0.791700i	$-0.709196\pi$
$521$	−27.8885			−1.22182			−0.610910	+	0.791700i	$0.709196\pi$
$522$		0			0
$523$			21.7082i			0.949233i	0.880193	+	0.474617i	$0.157413\pi$
$523$			21.7082i			0.949233i	−0.880193	+	0.474617i	$0.842587\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.52786i			0.240798i
$528$		0			0
$529$	−9.58359			−0.416678
$530$		0			0
$531$	−20.0000	+	17.8885i	−0.867926	+	0.776297i
$532$		0			0
$533$		−	40.3607i		−	1.74822i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	27.8885	−	10.6525i	1.20348	−	0.459688i
$538$		0			0
$539$	−10.4721	+	35.1246i	−0.451067	+	1.51292i
$540$		0			0
$541$	−8.11146			−0.348739			−0.174369	−	0.984680i	$-0.555789\pi$
$541$	−8.11146			−0.348739			−0.174369	+	0.984680i	$0.555789\pi$
$542$		0			0
$543$	4.00000	−	1.52786i	0.171656	−	0.0655669i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.29180			−0.183504			−0.0917520	−	0.995782i	$-0.529247\pi$
$547$	−4.29180			−0.183504			−0.0917520	+	0.995782i	$0.529247\pi$
$548$		0			0
$549$	25.8885	+	28.9443i	1.10490	+	1.23531i
$550$		0			0
$551$	−11.0557			−0.470990
$552$		0			0
$553$	4.00000	+	0.583592i	0.170097	+	0.0248169i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.18034i			0.346612i	0.984868	+	0.173306i	$0.0554450\pi$
$557$			8.18034i			0.346612i	−0.984868	+	0.173306i	$0.944555\pi$
$558$		0			0
$559$		−	8.94427i		−	0.378302i
$560$		0			0
$561$	−37.8885	+	14.4721i	−1.59966	+	0.611014i
$562$		0			0
$563$	−40.0689			−1.68870			−0.844351	−	0.535790i	$-0.820014\pi$
$563$	−40.0689			−1.68870			−0.844351	+	0.535790i	$0.820014\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.03444	−	23.0344i	0.253423	−	0.967356i
$568$		0			0
$569$			13.8885i			0.582238i	0.956687	+	0.291119i	$0.0940276\pi$
$569$			13.8885i			0.582238i	−0.956687	+	0.291119i	$0.905972\pi$
$570$		0			0
$571$	−46.8328			−1.95989			−0.979946	−	0.199262i	$-0.936145\pi$
$571$	−46.8328			−1.95989			−0.979946	+	0.199262i	$0.936145\pi$
$572$		0			0
$573$	4.47214	−	1.70820i	0.186826	−	0.0713612i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			26.0689i			1.08526i	0.839971	+	0.542631i	$0.182572\pi$
$577$			26.0689i			1.08526i	−0.839971	+	0.542631i	$0.817428\pi$
$578$		0			0
$579$	−3.70820	−	9.70820i	−0.154108	−	0.403459i
$580$		0			0
$581$	−7.23607	−	1.05573i	−0.300203	−	0.0437990i
$582$		0			0
$583$	−24.9443			−1.03309
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.7082			−0.813445			−0.406722	−	0.913552i	$-0.633328\pi$
$587$	−19.7082			−0.813445			−0.406722	+	0.913552i	$0.633328\pi$
$588$		0			0
$589$	−3.41641			−0.140771
$590$		0			0
$591$	−33.5967	+	12.8328i	−1.38199	+	0.527872i
$592$		0			0
$593$	−28.4721			−1.16921			−0.584605	−	0.811318i	$-0.698751\pi$
$593$	−28.4721			−1.16921			−0.584605	+	0.811318i	$0.698751\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−21.4164	+	8.18034i	−0.876515	+	0.334799i
$598$		0			0
$599$			0.652476i			0.0266594i	0.999911	+	0.0133297i	$0.00424311\pi$
$599$			0.652476i			0.0266594i	−0.999911	+	0.0133297i	$0.995757\pi$
$600$		0			0
$601$			17.3050i			0.705884i	0.935645	+	0.352942i	$0.114819\pi$
$601$			17.3050i			0.705884i	−0.935645	+	0.352942i	$0.885181\pi$
$602$		0			0
$603$	−8.29180	+	7.41641i	−0.337668	+	0.302019i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	18.0689i		−	0.733393i	−0.930341	−	0.366697i	$-0.880489\pi$
$607$		−	18.0689i		−	0.733393i	0.930341	−	0.366697i	$-0.119511\pi$
$608$		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	$-0.350864\pi$
$613$	22.3607			0.903139			0.451570	+	0.892236i	$0.350864\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	40.5410i		−	1.63212i	−0.577967	−	0.816060i	$-0.696154\pi$
$617$		−	40.5410i		−	1.63212i	0.577967	−	0.816060i	$-0.303846\pi$
$618$		0			0
$619$		−	7.70820i		−	0.309819i	−0.987929	−	0.154909i	$-0.950491\pi$
$619$		−	7.70820i		−	0.309819i	0.987929	−	0.154909i	$-0.0495086\pi$
$620$		0			0
$621$	13.5967	−	26.3607i	0.545619	−	1.05782i
$622$		0			0
$623$	9.23607	+	1.34752i	0.370035	+	0.0539874i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.94427	−	23.4164i	−0.357200	−	0.935161i
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−8.94427			−0.356066			−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427			−0.356066			−0.178033	+	0.984025i	$0.556973\pi$
$632$		0			0
$633$	4.94427	+	12.9443i	0.196517	+	0.514489i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.47214	−	21.7082i	0.256435	−	0.860110i
$638$		0			0
$639$	20.3607	+	22.7639i	0.805456	+	0.900527i
$640$		0			0
$641$		−	39.4164i		−	1.55685i	−0.627735	−	0.778427i	$-0.716018\pi$
$641$		−	39.4164i		−	1.55685i	0.627735	−	0.778427i	$-0.283982\pi$
$642$		0			0
$643$			37.7082i			1.48707i	0.668699	+	0.743533i	$0.266851\pi$
$643$			37.7082i			1.48707i	−0.668699	+	0.743533i	$0.733149\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−31.1246			−1.22363			−0.611817	−	0.790999i	$-0.709561\pi$
$647$	−31.1246			−1.22363			−0.611817	+	0.790999i	$0.709561\pi$
$648$		0			0
$649$			46.8328i			1.83835i
$650$		0			0
$651$	−5.52786	+	1.23607i	−0.216654	+	0.0484453i
$652$		0			0
$653$			27.5967i			1.07994i	0.841683	+	0.539972i	$0.181565\pi$
$653$			27.5967i			1.07994i	−0.841683	+	0.539972i	$0.818435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	22.4721	+	25.1246i	0.876722	+	0.980204i
$658$		0			0
$659$			41.2361i			1.60633i	0.595757	+	0.803165i	$0.296852\pi$
$659$			41.2361i			1.60633i	−0.595757	+	0.803165i	$0.703148\pi$
$660$		0			0
$661$			30.8328i			1.19926i	0.800278	+	0.599629i	$0.204685\pi$
$661$			30.8328i			1.19926i	−0.800278	+	0.599629i	$0.795315\pi$
$662$		0			0
$663$	23.4164	−	8.94427i	0.909418	−	0.347367i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−22.8328			−0.884090
$668$		0			0
$669$	2.76393	−	1.05573i	0.106860	−	0.0408168i
$670$		0			0
$671$	67.7771			2.61651
$672$		0			0
$673$	2.94427			0.113493			0.0567467	−	0.998389i	$-0.481927\pi$
$673$	2.94427			0.113493			0.0567467	+	0.998389i	$0.481927\pi$
$674$		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
$679$	−3.12461	+	21.4164i	−0.119912	+	0.821886i
$680$		0			0
$681$	−11.5967	−	30.3607i	−0.444388	−	1.16342i
$682$		0			0
$683$		−	33.7082i		−	1.28981i	−0.764263	−	0.644904i	$-0.776897\pi$
$683$		−	33.7082i		−	1.28981i	0.764263	−	0.644904i	$-0.223103\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	35.4164	−	13.5279i	1.35122	−	0.516120i
$688$		0			0
$689$	15.4164			0.587318
$690$		0			0
$691$			14.1803i			0.539446i	0.962938	+	0.269723i	$0.0869321\pi$
$691$			14.1803i			0.539446i	−0.962938	+	0.269723i	$0.913068\pi$
$692$		0			0
$693$	−22.9443	−	34.6525i	−0.871581	−	1.31634i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−55.7771			−2.11271
$698$		0			0
$699$	−39.1246	+	14.9443i	−1.47983	+	0.565244i
$700$		0			0
$701$			20.9443i			0.791054i	0.918454	+	0.395527i	$0.129438\pi$
$701$			20.9443i			0.791054i	−0.918454	+	0.395527i	$0.870562\pi$
$702$		0			0
$703$		−	12.3607i		−	0.466192i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.6525	−	4.18034i	−1.07759	−	0.157218i
$708$		0			0
$709$	−5.05573			−0.189872			−0.0949359	−	0.995483i	$-0.530265\pi$
$709$	−5.05573			−0.189872			−0.0949359	+	0.995483i	$0.530265\pi$
$710$		0			0
$711$	−3.41641	+	3.05573i	−0.128125	+	0.114599i
$712$		0			0
$713$	−7.05573			−0.264239
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−35.8885	+	13.7082i	−1.34028	+	0.511942i
$718$		0			0
$719$	41.8885			1.56218			0.781090	−	0.624419i	$-0.214664\pi$
$719$	41.8885			1.56218			0.781090	+	0.624419i	$0.214664\pi$
$720$		0			0
$721$	5.81966	−	39.8885i	0.216735	−	1.48553i
$722$		0			0
$723$	45.8885	−	17.5279i	1.70661	−	0.651868i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			9.70820i			0.360057i	0.983661	+	0.180029i	$0.0576191\pi$
$727$			9.70820i			0.360057i	−0.983661	+	0.180029i	$0.942381\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	−12.3607			−0.457176
$732$		0			0
$733$		−	17.3475i		−	0.640745i	−0.947292	−	0.320373i	$-0.896192\pi$
$733$		−	17.3475i		−	0.640745i	0.947292	−	0.320373i	$-0.103808\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			19.4164i			0.715213i
$738$		0			0
$739$	−4.36068			−0.160410			−0.0802051	−	0.996778i	$-0.525558\pi$
$739$	−4.36068			−0.160410			−0.0802051	+	0.996778i	$0.525558\pi$
$740$		0			0
$741$	5.52786	+	14.4721i	0.203071	+	0.531647i
$742$		0			0
$743$		−	6.65248i		−	0.244056i	−0.992527	−	0.122028i	$-0.961060\pi$
$743$		−	6.65248i		−	0.244056i	0.992527	−	0.122028i	$-0.0389397\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.18034	−	5.52786i	0.226127	−	0.202254i
$748$		0			0
$749$	5.23607	−	35.8885i	0.191322	−	1.31134i
$750$		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
$753$	15.0557	+	39.4164i	0.548661	+	1.43641i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	27.3050			0.992415			0.496208	−	0.868204i	$-0.334725\pi$
$757$	27.3050			0.992415			0.496208	+	0.868204i	$0.334725\pi$
$758$		0			0
$759$	−18.4721	−	48.3607i	−0.670496	−	1.75538i
$760$		0			0
$761$	31.3050			1.13480			0.567402	−	0.823441i	$-0.307949\pi$
$761$	31.3050			1.13480			0.567402	+	0.823441i	$0.307949\pi$
$762$		0			0
$763$	39.1246	+	5.70820i	1.41641	+	0.206651i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28.9443i		−	1.04512i
$768$		0			0
$769$			4.58359i			0.165289i	0.996579	+	0.0826443i	$0.0263365\pi$
$769$			4.58359i			0.165289i	−0.996579	+	0.0826443i	$0.973663\pi$
$770$		0			0
$771$	4.65248	+	12.1803i	0.167555	+	0.438664i
$772$		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
$779$		−	34.4721i		−	1.23509i
$780$		0			0
$781$	53.3050			1.90740
$782$		0			0
$783$	9.52786	−	18.4721i	0.340498	−	0.660140i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			29.7082i			1.05898i	0.848315	+	0.529492i	$0.177617\pi$
$787$			29.7082i			1.05898i	−0.848315	+	0.529492i	$0.822383\pi$
$788$		0			0
$789$	0.291796	−	0.111456i	0.0103882	−	0.00396795i
$790$		0			0
$791$	−4.87539	+	33.4164i	−0.173349	+	1.18815i
$792$		0			0
$793$	−41.8885			−1.48751
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24.8328			0.879623			0.439812	−	0.898090i	$-0.355045\pi$
$797$	24.8328			0.879623			0.439812	+	0.898090i	$0.355045\pi$
$798$		0			0
$799$	−5.52786			−0.195562
$800$		0			0
$801$	−7.88854	+	7.05573i	−0.278728	+	0.249302i
$802$		0			0
$803$	58.8328			2.07616
$804$		0			0
$805$		0			0
$806$		0			0
$807$	17.5967	+	46.0689i	0.619435	+	1.62170i
$808$		0			0
$809$		−	32.3607i		−	1.13774i	−0.822427	−	0.568870i	$-0.807381\pi$
$809$		−	32.3607i		−	1.13774i	0.822427	−	0.568870i	$-0.192619\pi$
$810$		0			0
$811$		−	11.7082i		−	0.411131i	−0.978643	−	0.205565i	$-0.934097\pi$
$811$		−	11.7082i		−	0.411131i	0.978643	−	0.205565i	$-0.0659033\pi$
$812$		0			0
$813$	−30.9443	+	11.8197i	−1.08526	+	0.414534i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	7.63932i		−	0.267266i
$818$		0			0
$819$	14.1803	+	21.4164i	0.495501	+	0.748350i
$820$		0			0
$821$		−	6.83282i		−	0.238467i	−0.992866	−	0.119233i	$-0.961956\pi$
$821$		−	6.83282i		−	0.238467i	0.992866	−	0.119233i	$-0.0380437\pi$
$822$		0			0
$823$	−37.5967			−1.31054			−0.655270	−	0.755395i	$-0.727445\pi$
$823$	−37.5967			−1.31054			−0.655270	+	0.755395i	$0.727445\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			26.2918i			0.914255i	0.889401	+	0.457128i	$0.151122\pi$
$827$			26.2918i			0.914255i	−0.889401	+	0.457128i	$0.848878\pi$
$828$		0			0
$829$		−	24.3607i		−	0.846081i	−0.906111	−	0.423041i	$-0.860963\pi$
$829$		−	24.3607i		−	0.846081i	0.906111	−	0.423041i	$-0.139037\pi$
$830$		0			0
$831$	−15.7082	−	41.1246i	−0.544912	−	1.42660i
$832$		0			0
$833$	−30.0000	−	8.94427i	−1.03944	−	0.309901i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.94427	−	5.70820i	0.101769	−	0.197304i
$838$		0			0
$839$	−4.94427			−0.170695			−0.0853476	−	0.996351i	$-0.527200\pi$
$839$	−4.94427			−0.170695			−0.0853476	+	0.996351i	$0.527200\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	24.9443	−	9.52786i	0.859126	−	0.328157i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−42.9787	−	6.27051i	−1.47677	−	0.215457i
$848$		0			0
$849$	36.6525	−	14.0000i	1.25791	−	0.480479i
$850$		0			0
$851$		−	25.5279i		−	0.875084i
$852$		0			0
$853$		−	21.1246i		−	0.723293i	−0.932315	−	0.361646i	$-0.882215\pi$
$853$		−	21.1246i		−	0.723293i	0.932315	−	0.361646i	$-0.117785\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.5279			0.667059			0.333530	−	0.942740i	$-0.391760\pi$
$857$	19.5279			0.667059			0.333530	+	0.942740i	$0.391760\pi$
$858$		0			0
$859$		−	44.6525i		−	1.52352i	−0.647858	−	0.761761i	$-0.724335\pi$
$859$		−	44.6525i		−	1.52352i	0.647858	−	0.761761i	$-0.275665\pi$
$860$		0			0
$861$	−12.4721	−	55.7771i	−0.425049	−	1.90088i
$862$		0			0
$863$			21.3475i			0.726678i	0.931657	+	0.363339i	$0.118363\pi$
$863$			21.3475i			0.726678i	−0.931657	+	0.363339i	$0.881637\pi$
$864$		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
$873$	−16.3607	−	18.2918i	−0.553725	−	0.619083i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.5279			0.794480			0.397240	−	0.917715i	$-0.369968\pi$
$877$	23.5279			0.794480			0.397240	+	0.917715i	$0.369968\pi$
$878$		0			0
$879$	8.65248	+	22.6525i	0.291841	+	0.764049i
$880$		0			0
$881$	−36.8328			−1.24093			−0.620465	−	0.784234i	$-0.713056\pi$
$881$	−36.8328			−1.24093			−0.620465	+	0.784234i	$0.713056\pi$
$882$		0			0
$883$	−13.5967			−0.457567			−0.228783	−	0.973477i	$-0.573475\pi$
$883$	−13.5967			−0.457567			−0.228783	+	0.973477i	$0.573475\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.18034			0.207516			0.103758	−	0.994603i	$-0.466913\pi$
$887$	6.18034			0.207516			0.103758	+	0.994603i	$0.466913\pi$
$888$		0			0
$889$	−50.0689	−	7.30495i	−1.67926	−	0.245000i
$890$		0			0
$891$	46.8328	+	5.23607i	1.56896	+	0.175415i
$892$		0			0
$893$		−	3.41641i		−	0.114326i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	11.4164	+	29.8885i	0.381183	+	0.997949i
$898$		0			0
$899$	−4.94427			−0.164901
$900$		0			0
$901$		−	21.3050i		−	0.709771i
$902$		0			0
$903$	−2.76393	−	12.3607i	−0.0919779	−	0.411338i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	22.7639			0.755864			0.377932	−	0.925833i	$-0.376635\pi$
$907$	22.7639			0.755864			0.377932	+	0.925833i	$0.376635\pi$
$908$		0			0
$909$	24.4721	−	21.8885i	0.811690	−	0.725997i
$910$		0			0
$911$		−	12.0689i		−	0.399860i	−0.979810	−	0.199930i	$-0.935929\pi$
$911$		−	12.0689i		−	0.399860i	0.979810	−	0.199930i	$-0.0640715\pi$
$912$		0			0
$913$		−	14.4721i		−	0.478958i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	29.8885	+	4.36068i	0.987007	+	0.144002i
$918$		0			0
$919$	−25.5279			−0.842087			−0.421043	−	0.907041i	$-0.638336\pi$
$919$	−25.5279			−0.842087			−0.421043	+	0.907041i	$0.638336\pi$
$920$		0			0
$921$	−6.76393	+	2.58359i	−0.222879	+	0.0851323i
$922$		0			0
$923$	−32.9443			−1.08437
$924$		0			0
$925$		0			0
$926$		0			0
$927$	30.4721	+	34.0689i	1.00084	+	1.11897i
$928$		0			0
$929$	−15.8885			−0.521286			−0.260643	−	0.965435i	$-0.583935\pi$
$929$	−15.8885			−0.521286			−0.260643	+	0.965435i	$0.583935\pi$
$930$		0			0
$931$	5.52786	−	18.5410i	0.181168	−	0.607657i
$932$		0			0
$933$	7.63932	+	20.0000i	0.250100	+	0.654771i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			19.2361i			0.628415i	0.949354	+	0.314207i	$0.101739\pi$
$937$			19.2361i			0.628415i	−0.949354	+	0.314207i	$0.898261\pi$
$938$		0			0
$939$	28.6525	−	10.9443i	0.935038	−	0.357153i
$940$		0			0
$941$	−36.2492			−1.18169			−0.590845	−	0.806785i	$-0.701206\pi$
$941$	−36.2492			−1.18169			−0.590845	+	0.806785i	$0.701206\pi$
$942$		0			0
$943$		−	71.1935i		−	2.31838i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.7639i		−	0.674737i	−0.941373	−	0.337369i	$-0.890463\pi$
$947$		−	20.7639i		−	0.674737i	0.941373	−	0.337369i	$-0.109537\pi$
$948$		0			0
$949$	−36.3607			−1.18032
$950$		0			0
$951$	26.1803	−	10.0000i	0.848956	−	0.324272i
$952$		0			0
$953$			9.70820i			0.314480i	0.987560	+	0.157240i	$0.0502596\pi$
$953$			9.70820i			0.314480i	−0.987560	+	0.157240i	$0.949740\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−12.9443	−	33.8885i	−0.418429	−	1.09546i
$958$		0			0
$959$	−1.23607	+	8.47214i	−0.0399147	+	0.273580i
$960$		0			0
$961$	29.4721			0.950714
$962$		0			0
$963$	27.4164	+	30.6525i	0.883481	+	0.987762i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−21.8197			−0.701673			−0.350836	−	0.936437i	$-0.614103\pi$
$967$	−21.8197			−0.701673			−0.350836	+	0.936437i	$0.614103\pi$
$968$		0			0
$969$	20.0000	−	7.63932i	0.642493	−	0.245410i
$970$		0			0
$971$	27.4164			0.879834			0.439917	−	0.898038i	$-0.355008\pi$
$971$	27.4164			0.879834			0.439917	+	0.898038i	$0.355008\pi$
$972$		0			0
$973$	4.11146	−	28.1803i	0.131807	−	0.903420i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			43.9574i			1.40632i	0.711030	+	0.703161i	$0.248229\pi$
$977$			43.9574i			1.40632i	−0.711030	+	0.703161i	$0.751771\pi$
$978$		0			0
$979$			18.4721i			0.590372i
$980$		0			0
$981$	−33.4164	+	29.8885i	−1.06690	+	0.954268i
$982$		0			0
$983$	26.5410			0.846527			0.423264	−	0.906007i	$-0.360884\pi$
$983$	26.5410			0.846527			0.423264	+	0.906007i	$0.360884\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.23607	−	5.52786i	−0.0393445	−	0.175954i
$988$		0			0
$989$		−	15.7771i		−	0.501682i
$990$		0			0
$991$	−31.7771			−1.00943			−0.504716	−	0.863285i	$-0.668403\pi$
$991$	−31.7771			−1.00943			−0.504716	+	0.863285i	$0.668403\pi$
$992$		0			0
$993$	1.88854	+	4.94427i	0.0599311	+	0.156902i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			42.0689i			1.33233i	0.745802	+	0.666167i	$0.232066\pi$
$997$			42.0689i			1.33233i	−0.745802	+	0.666167i	$0.767934\pi$
$998$		0			0
$999$	20.6525	+	10.6525i	0.653415	+	0.337029i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-0.618034 - 1.61803i) q^{3} +(2.61803 + 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(-0.618034 - 1.61803i) q^{3} +(2.61803 + 0.381966i) q^{7} +(-2.23607 + 2.00000i) q^{9} +5.23607i q^{11} -3.23607i q^{13} -4.47214 q^{17} -2.76393i q^{19} +(-1.00000 - 4.47214i) q^{21} -5.70820i q^{23} +(4.61803 + 2.38197i) q^{27} -4.00000i q^{29} -1.23607i q^{31} +(8.47214 - 3.23607i) q^{33} +4.47214 q^{37} +(-5.23607 + 2.00000i) q^{39} +12.4721 q^{41} +2.76393 q^{43} +1.23607 q^{47} +(6.70820 + 2.00000i) q^{49} +(2.76393 + 7.23607i) q^{51} +4.76393i q^{53} +(-4.47214 + 1.70820i) q^{57} +8.94427 q^{59} -12.9443i q^{61} +(-6.61803 + 4.38197i) q^{63} +3.70820 q^{67} +(-9.23607 + 3.52786i) q^{69} -10.1803i q^{71} -11.2361i q^{73} +(-2.00000 + 13.7082i) q^{77} +1.52786 q^{79} +(1.00000 - 8.94427i) q^{81} -2.76393 q^{83} +(-6.47214 + 2.47214i) q^{87} +3.52786 q^{89} +(1.23607 - 8.47214i) q^{91} +(-2.00000 + 0.763932i) q^{93} +8.18034i q^{97} +(-10.4721 - 11.7082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) 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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