LMFDB - Embedded newform 1520.2.d.c.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1372611072$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.c.609.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607i q^{3} +(-1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +1.47214 q^{9} +O(q^{10})$	$q-1.23607i q^{3} +(-1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +1.47214 q^{9} +3.23607i q^{13} +(2.47214 + 1.23607i) q^{15} -6.47214i q^{17} +1.00000 q^{19} -5.52786 q^{21} +2.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -5.52786i q^{27} -2.00000 q^{29} +1.52786 q^{31} +(8.94427 + 4.47214i) q^{35} +4.76393i q^{37} +4.00000 q^{39} +3.52786 q^{41} -0.472136i q^{43} +(-1.47214 + 2.94427i) q^{45} -12.4721i q^{47} -13.0000 q^{49} -8.00000 q^{51} -11.2361i q^{53} -1.23607i q^{57} -10.4721 q^{59} -4.47214 q^{61} -6.58359i q^{63} +(-6.47214 - 3.23607i) q^{65} -1.23607i q^{67} +2.47214 q^{69} +1.52786 q^{71} -6.47214i q^{73} +(-4.94427 + 3.70820i) q^{75} -6.47214 q^{79} -2.41641 q^{81} -14.9443i q^{83} +(12.9443 + 6.47214i) q^{85} +2.47214i q^{87} +6.94427 q^{89} +14.4721 q^{91} -1.88854i q^{93} +(-1.00000 + 2.00000i) q^{95} +4.18034i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^5 - 12 * q^9	$ 4 q - 4 q^{5} - 12 q^{9} - 8 q^{15} + 4 q^{19} - 40 q^{21} - 12 q^{25} - 8 q^{29} + 24 q^{31} + 16 q^{39} + 32 q^{41} + 12 q^{45} - 52 q^{49} - 32 q^{51} - 24 q^{59} - 8 q^{65} - 8 q^{69} + 24 q^{71} + 16 q^{75} - 8 q^{79} + 44 q^{81} + 16 q^{85} - 8 q^{89} + 40 q^{91} - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^5 - 12 * q^9 - 8 * q^15 + 4 * q^19 - 40 * q^21 - 12 * q^25 - 8 * q^29 + 24 * q^31 + 16 * q^39 + 32 * q^41 + 12 * q^45 - 52 * q^49 - 32 * q^51 - 24 * q^59 - 8 * q^65 - 8 * q^69 + 24 * q^71 + 16 * q^75 - 8 * q^79 + 44 * q^81 + 16 * q^85 - 8 * q^89 + 40 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.23607i		−	0.713644i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$		−	1.23607i		−	0.713644i	0.934172	−	0.356822i	$-0.116140\pi$
$4$		0			0
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$		−	4.47214i		−	1.69031i	−0.534522	−	0.845154i	$-0.679509\pi$
$7$		−	4.47214i		−	1.69031i	0.534522	−	0.845154i	$-0.320491\pi$
$8$		0			0
$9$	1.47214			0.490712
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			3.23607i			0.897524i	0.893651	+	0.448762i	$0.148135\pi$
$13$			3.23607i			0.897524i	−0.893651	+	0.448762i	$0.851865\pi$
$14$		0			0
$15$	2.47214	+	1.23607i	0.638303	+	0.319151i
$16$		0			0
$17$		−	6.47214i		−	1.56972i	−0.619671	−	0.784862i	$-0.712734\pi$
$17$		−	6.47214i		−	1.56972i	0.619671	−	0.784862i	$-0.287266\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	−5.52786			−1.20628
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		−	5.52786i		−	1.06384i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	1.52786			0.274412			0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786			0.274412			0.137206	+	0.990543i	$0.456188\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.94427	+	4.47214i	1.51186	+	0.755929i
$36$		0			0
$37$			4.76393i			0.783186i	0.920139	+	0.391593i	$0.128076\pi$
$37$			4.76393i			0.783186i	−0.920139	+	0.391593i	$0.871924\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		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
$43$		−	0.472136i		−	0.0720001i	−0.999352	−	0.0360000i	$-0.988538\pi$
$43$		−	0.472136i		−	0.0720001i	0.999352	−	0.0360000i	$-0.0114616\pi$
$44$		0			0
$45$	−1.47214	+	2.94427i	−0.219453	+	0.438906i
$46$		0			0
$47$		−	12.4721i		−	1.81925i	−0.415433	−	0.909624i	$-0.636370\pi$
$47$		−	12.4721i		−	1.81925i	0.415433	−	0.909624i	$-0.363630\pi$
$48$		0			0
$49$	−13.0000			−1.85714
$50$		0			0
$51$	−8.00000			−1.12022
$52$		0			0
$53$		−	11.2361i		−	1.54339i	−0.635991	−	0.771696i	$-0.719409\pi$
$53$		−	11.2361i		−	1.54339i	0.635991	−	0.771696i	$-0.280591\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	1.23607i		−	0.163721i
$58$		0			0
$59$	−10.4721			−1.36336			−0.681678	−	0.731652i	$-0.738749\pi$
$59$	−10.4721			−1.36336			−0.681678	+	0.731652i	$0.738749\pi$
$60$		0			0
$61$	−4.47214			−0.572598			−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214			−0.572598			−0.286299	+	0.958140i	$0.592425\pi$
$62$		0			0
$63$		−	6.58359i		−	0.829455i
$64$		0			0
$65$	−6.47214	−	3.23607i	−0.802770	−	0.401385i
$66$		0			0
$67$		−	1.23607i		−	0.151010i	−0.997145	−	0.0755049i	$-0.975943\pi$
$67$		−	1.23607i		−	0.151010i	0.997145	−	0.0755049i	$-0.0240568\pi$
$68$		0			0
$69$	2.47214			0.297610
$70$		0			0
$71$	1.52786			0.181324			0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786			0.181324			0.0906621	+	0.995882i	$0.471102\pi$
$72$		0			0
$73$		−	6.47214i		−	0.757506i	−0.925498	−	0.378753i	$-0.876353\pi$
$73$		−	6.47214i		−	0.757506i	0.925498	−	0.378753i	$-0.123647\pi$
$74$		0			0
$75$	−4.94427	+	3.70820i	−0.570915	+	0.428187i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.47214			−0.728172			−0.364086	−	0.931365i	$-0.618619\pi$
$79$	−6.47214			−0.728172			−0.364086	+	0.931365i	$0.618619\pi$
$80$		0			0
$81$	−2.41641			−0.268490
$82$		0			0
$83$		−	14.9443i		−	1.64035i	−0.572115	−	0.820173i	$-0.693877\pi$
$83$		−	14.9443i		−	1.64035i	0.572115	−	0.820173i	$-0.306123\pi$
$84$		0			0
$85$	12.9443	+	6.47214i	1.40400	+	0.702002i
$86$		0			0
$87$			2.47214i			0.265041i
$88$		0			0
$89$	6.94427			0.736091			0.368046	−	0.929808i	$-0.380027\pi$
$89$	6.94427			0.736091			0.368046	+	0.929808i	$0.380027\pi$
$90$		0			0
$91$	14.4721			1.51709
$92$		0			0
$93$		−	1.88854i		−	0.195833i
$94$		0			0
$95$	−1.00000	+	2.00000i	−0.102598	+	0.205196i
$96$		0			0
$97$			4.18034i			0.424449i	0.977221	+	0.212225i	$0.0680708\pi$
$97$			4.18034i			0.424449i	−0.977221	+	0.212225i	$0.931929\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.47214			0.843009			0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214			0.843009			0.421505	+	0.906826i	$0.361502\pi$
$102$		0			0
$103$			17.2361i			1.69832i	0.528135	+	0.849160i	$0.322891\pi$
$103$			17.2361i			1.69832i	−0.528135	+	0.849160i	$0.677109\pi$
$104$		0			0
$105$	5.52786	−	11.0557i	0.539464	−	1.07893i
$106$		0			0
$107$		−	7.70820i		−	0.745180i	−0.927996	−	0.372590i	$-0.878470\pi$
$107$		−	7.70820i		−	0.745180i	0.927996	−	0.372590i	$-0.121530\pi$
$108$		0			0
$109$	−13.4164			−1.28506			−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164			−1.28506			−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$	5.88854			0.558916
$112$		0			0
$113$		−	15.2361i		−	1.43329i	−0.697439	−	0.716644i	$-0.745677\pi$
$113$		−	15.2361i		−	1.43329i	0.697439	−	0.716644i	$-0.254323\pi$
$114$		0			0
$115$	−4.00000	−	2.00000i	−0.373002	−	0.186501i
$116$		0			0
$117$			4.76393i			0.440426i
$118$		0			0
$119$	−28.9443			−2.65332
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	4.36068i		−	0.393189i
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$		−	6.76393i		−	0.600202i	−0.953907	−	0.300101i	$-0.902980\pi$
$127$		−	6.76393i		−	0.600202i	0.953907	−	0.300101i	$-0.0970204\pi$
$128$		0			0
$129$	−0.583592			−0.0513824
$130$		0			0
$131$	8.94427			0.781465			0.390732	−	0.920504i	$-0.372222\pi$
$131$	8.94427			0.781465			0.390732	+	0.920504i	$0.372222\pi$
$132$		0			0
$133$		−	4.47214i		−	0.387783i
$134$		0			0
$135$	11.0557	+	5.52786i	0.951526	+	0.475763i
$136$		0			0
$137$			11.4164i			0.975370i	0.873020	+	0.487685i	$0.162158\pi$
$137$			11.4164i			0.975370i	−0.873020	+	0.487685i	$0.837842\pi$
$138$		0			0
$139$	4.94427			0.419368			0.209684	−	0.977769i	$-0.432757\pi$
$139$	4.94427			0.419368			0.209684	+	0.977769i	$0.432757\pi$
$140$		0			0
$141$	−15.4164			−1.29830
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000	−	4.00000i	0.166091	−	0.332182i
$146$		0			0
$147$			16.0689i			1.32534i
$148$		0			0
$149$	21.4164			1.75450			0.877250	−	0.480033i	$-0.159375\pi$
$149$	21.4164			1.75450			0.877250	+	0.480033i	$0.159375\pi$
$150$		0			0
$151$	−17.8885			−1.45575			−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885			−1.45575			−0.727875	+	0.685710i	$0.759492\pi$
$152$		0			0
$153$		−	9.52786i		−	0.770282i
$154$		0			0
$155$	−1.52786	+	3.05573i	−0.122721	+	0.245442i
$156$		0			0
$157$			15.4164i			1.23036i	0.788385	+	0.615182i	$0.210917\pi$
$157$			15.4164i			1.23036i	−0.788385	+	0.615182i	$0.789083\pi$
$158$		0			0
$159$	−13.8885			−1.10143
$160$		0			0
$161$	8.94427			0.704907
$162$		0			0
$163$		−	8.47214i		−	0.663589i	−0.943352	−	0.331794i	$-0.892346\pi$
$163$		−	8.47214i		−	0.663589i	0.943352	−	0.331794i	$-0.107654\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.1803i		−	0.787778i	−0.919158	−	0.393889i	$-0.871129\pi$
$167$		−	10.1803i		−	0.787778i	0.919158	−	0.393889i	$-0.128871\pi$
$168$		0			0
$169$	2.52786			0.194451
$170$		0			0
$171$	1.47214			0.112577
$172$		0			0
$173$			14.6525i			1.11401i	0.830510	+	0.557004i	$0.188049\pi$
$173$			14.6525i			1.11401i	−0.830510	+	0.557004i	$0.811951\pi$
$174$		0			0
$175$	−17.8885	+	13.4164i	−1.35225	+	1.01419i
$176$		0			0
$177$			12.9443i			0.972951i
$178$		0			0
$179$	−5.52786			−0.413172			−0.206586	−	0.978428i	$-0.566235\pi$
$179$	−5.52786			−0.413172			−0.206586	+	0.978428i	$0.566235\pi$
$180$		0			0
$181$	23.8885			1.77562			0.887811	−	0.460209i	$-0.152225\pi$
$181$	23.8885			1.77562			0.887811	+	0.460209i	$0.152225\pi$
$182$		0			0
$183$			5.52786i			0.408631i
$184$		0			0
$185$	−9.52786	−	4.76393i	−0.700502	−	0.350251i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−24.7214			−1.79821
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			5.70820i			0.410886i	0.978669	+	0.205443i	$0.0658634\pi$
$193$			5.70820i			0.410886i	−0.978669	+	0.205443i	$0.934137\pi$
$194$		0			0
$195$	−4.00000	+	8.00000i	−0.286446	+	0.572892i
$196$		0			0
$197$		−	0.944272i		−	0.0672766i	−0.999434	−	0.0336383i	$-0.989291\pi$
$197$		−	0.944272i		−	0.0672766i	0.999434	−	0.0336383i	$-0.0107094\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−1.52786			−0.107767
$202$		0			0
$203$			8.94427i			0.627765i
$204$		0			0
$205$	−3.52786	+	7.05573i	−0.246397	+	0.492793i
$206$		0			0
$207$			2.94427i			0.204641i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−7.05573			−0.485736			−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573			−0.485736			−0.242868	+	0.970059i	$0.578088\pi$
$212$		0			0
$213$		−	1.88854i		−	0.129401i
$214$		0			0
$215$	0.944272	+	0.472136i	0.0643988	+	0.0321994i
$216$		0			0
$217$		−	6.83282i		−	0.463842i
$218$		0			0
$219$	−8.00000			−0.540590
$220$		0			0
$221$	20.9443			1.40886
$222$		0			0
$223$			25.2361i			1.68993i	0.534820	+	0.844966i	$0.320379\pi$
$223$			25.2361i			1.68993i	−0.534820	+	0.844966i	$0.679621\pi$
$224$		0			0
$225$	−4.41641	−	5.88854i	−0.294427	−	0.392570i
$226$		0			0
$227$			10.1803i			0.675693i	0.941201	+	0.337846i	$0.109698\pi$
$227$			10.1803i			0.675693i	−0.941201	+	0.337846i	$0.890302\pi$
$228$		0			0
$229$	−17.4164			−1.15091			−0.575454	−	0.817834i	$-0.695175\pi$
$229$	−17.4164			−1.15091			−0.575454	+	0.817834i	$0.695175\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.9443i		−	0.848007i	−0.905660	−	0.424004i	$-0.860624\pi$
$233$		−	12.9443i		−	0.848007i	0.905660	−	0.424004i	$-0.139376\pi$
$234$		0			0
$235$	24.9443	+	12.4721i	1.62718	+	0.813592i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	28.9443			1.87225			0.936125	−	0.351668i	$-0.114386\pi$
$239$	28.9443			1.87225			0.936125	+	0.351668i	$0.114386\pi$
$240$		0			0
$241$	21.4164			1.37955			0.689776	−	0.724023i	$-0.257709\pi$
$241$	21.4164			1.37955			0.689776	+	0.724023i	$0.257709\pi$
$242$		0			0
$243$		−	13.5967i		−	0.872232i
$244$		0			0
$245$	13.0000	−	26.0000i	0.830540	−	1.66108i
$246$		0			0
$247$			3.23607i			0.205906i
$248$		0			0
$249$	−18.4721			−1.17062
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	8.00000	−	16.0000i	0.500979	−	1.00196i
$256$		0			0
$257$		−	10.2918i		−	0.641985i	−0.947082	−	0.320992i	$-0.895984\pi$
$257$		−	10.2918i		−	0.641985i	0.947082	−	0.320992i	$-0.104016\pi$
$258$		0			0
$259$	21.3050			1.32383
$260$		0			0
$261$	−2.94427			−0.182246
$262$		0			0
$263$			11.8885i			0.733079i	0.930402	+	0.366540i	$0.119458\pi$
$263$			11.8885i			0.733079i	−0.930402	+	0.366540i	$0.880542\pi$
$264$		0			0
$265$	22.4721	+	11.2361i	1.38045	+	0.690226i
$266$		0			0
$267$		−	8.58359i		−	0.525307i
$268$		0			0
$269$	25.4164			1.54967			0.774833	−	0.632166i	$-0.217834\pi$
$269$	25.4164			1.54967			0.774833	+	0.632166i	$0.217834\pi$
$270$		0			0
$271$	−0.944272			−0.0573604			−0.0286802	−	0.999589i	$-0.509130\pi$
$271$	−0.944272			−0.0573604			−0.0286802	+	0.999589i	$0.509130\pi$
$272$		0			0
$273$		−	17.8885i		−	1.08266i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			26.4721i			1.59056i	0.606245	+	0.795278i	$0.292675\pi$
$277$			26.4721i			1.59056i	−0.606245	+	0.795278i	$0.707325\pi$
$278$		0			0
$279$	2.24922			0.134657
$280$		0			0
$281$	21.4164			1.27760			0.638798	−	0.769375i	$-0.279432\pi$
$281$	21.4164			1.27760			0.638798	+	0.769375i	$0.279432\pi$
$282$		0			0
$283$		−	19.8885i		−	1.18225i	−0.806580	−	0.591126i	$-0.798684\pi$
$283$		−	19.8885i		−	1.18225i	0.806580	−	0.591126i	$-0.201316\pi$
$284$		0			0
$285$	2.47214	+	1.23607i	0.146437	+	0.0732183i
$286$		0			0
$287$		−	15.7771i		−	0.931292i
$288$		0			0
$289$	−24.8885			−1.46403
$290$		0			0
$291$	5.16718			0.302906
$292$		0			0
$293$			17.7082i			1.03452i	0.855827	+	0.517262i	$0.173049\pi$
$293$			17.7082i			1.03452i	−0.855827	+	0.517262i	$0.826951\pi$
$294$		0			0
$295$	10.4721	−	20.9443i	0.609711	−	1.21942i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.47214			−0.374293
$300$		0			0
$301$	−2.11146			−0.121702
$302$		0			0
$303$		−	10.4721i		−	0.601608i
$304$		0			0
$305$	4.47214	−	8.94427i	0.256074	−	0.512148i
$306$		0			0
$307$			13.2361i			0.755422i	0.925923	+	0.377711i	$0.123289\pi$
$307$			13.2361i			0.755422i	−0.925923	+	0.377711i	$0.876711\pi$
$308$		0			0
$309$	21.3050			1.21200
$310$		0			0
$311$	20.0000			1.13410			0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000			1.13410			0.567048	+	0.823685i	$0.308085\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$	13.1672	+	6.58359i	0.741887	+	0.370943i
$316$		0			0
$317$		−	21.1246i		−	1.18648i	−0.805027	−	0.593238i	$-0.797849\pi$
$317$		−	21.1246i		−	1.18648i	0.805027	−	0.593238i	$-0.202151\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−9.52786			−0.531794
$322$		0			0
$323$		−	6.47214i		−	0.360119i
$324$		0			0
$325$	12.9443	−	9.70820i	0.718019	−	0.538514i
$326$		0			0
$327$			16.5836i			0.917075i
$328$		0			0
$329$	−55.7771			−3.07509
$330$		0			0
$331$	32.9443			1.81078			0.905390	−	0.424580i	$-0.139578\pi$
$331$	32.9443			1.81078			0.905390	+	0.424580i	$0.139578\pi$
$332$		0			0
$333$			7.01316i			0.384319i
$334$		0			0
$335$	2.47214	+	1.23607i	0.135067	+	0.0675336i
$336$		0			0
$337$			31.2361i			1.70154i	0.525541	+	0.850769i	$0.323863\pi$
$337$			31.2361i			1.70154i	−0.525541	+	0.850769i	$0.676137\pi$
$338$		0			0
$339$	−18.8328			−1.02286
$340$		0			0
$341$		0			0
$342$		0			0
$343$			26.8328i			1.44884i
$344$		0			0
$345$	−2.47214	+	4.94427i	−0.133095	+	0.266191i
$346$		0			0
$347$			10.9443i			0.587519i	0.955879	+	0.293760i	$0.0949065\pi$
$347$			10.9443i			0.587519i	−0.955879	+	0.293760i	$0.905093\pi$
$348$		0			0
$349$	7.88854			0.422264			0.211132	−	0.977458i	$-0.432285\pi$
$349$	7.88854			0.422264			0.211132	+	0.977458i	$0.432285\pi$
$350$		0			0
$351$	17.8885			0.954820
$352$		0			0
$353$			22.4721i			1.19607i	0.801470	+	0.598036i	$0.204052\pi$
$353$			22.4721i			1.19607i	−0.801470	+	0.598036i	$0.795948\pi$
$354$		0			0
$355$	−1.52786	+	3.05573i	−0.0810906	+	0.162181i
$356$		0			0
$357$			35.7771i			1.89352i
$358$		0			0
$359$	5.88854			0.310785			0.155393	−	0.987853i	$-0.450336\pi$
$359$	5.88854			0.310785			0.155393	+	0.987853i	$0.450336\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			13.5967i			0.713644i
$364$		0			0
$365$	12.9443	+	6.47214i	0.677534	+	0.338767i
$366$		0			0
$367$		−	30.3607i		−	1.58481i	−0.609992	−	0.792407i	$-0.708828\pi$
$367$		−	30.3607i		−	1.58481i	0.609992	−	0.792407i	$-0.291172\pi$
$368$		0			0
$369$	5.19350			0.270363
$370$		0			0
$371$	−50.2492			−2.60881
$372$		0			0
$373$		−	21.1246i		−	1.09379i	−0.837201	−	0.546895i	$-0.815810\pi$
$373$		−	21.1246i		−	1.09379i	0.837201	−	0.546895i	$-0.184190\pi$
$374$		0			0
$375$	−2.47214	−	13.5967i	−0.127661	−	0.702133i
$376$		0			0
$377$		−	6.47214i		−	0.333332i
$378$		0			0
$379$	21.8885			1.12434			0.562169	−	0.827022i	$-0.309967\pi$
$379$	21.8885			1.12434			0.562169	+	0.827022i	$0.309967\pi$
$380$		0			0
$381$	−8.36068			−0.428331
$382$		0			0
$383$			20.6525i			1.05529i	0.849464	+	0.527646i	$0.176925\pi$
$383$			20.6525i			1.05529i	−0.849464	+	0.527646i	$0.823075\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	0.695048i		−	0.0353313i
$388$		0			0
$389$	−37.7771			−1.91537			−0.957687	−	0.287811i	$-0.907072\pi$
$389$	−37.7771			−1.91537			−0.957687	+	0.287811i	$0.907072\pi$
$390$		0			0
$391$	12.9443			0.654620
$392$		0			0
$393$		−	11.0557i		−	0.557688i
$394$		0			0
$395$	6.47214	−	12.9443i	0.325649	−	0.651297i
$396$		0			0
$397$			23.4164i			1.17524i	0.809139	+	0.587618i	$0.199934\pi$
$397$			23.4164i			1.17524i	−0.809139	+	0.587618i	$0.800066\pi$
$398$		0			0
$399$	−5.52786			−0.276739
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$			4.94427i			0.246292i
$404$		0			0
$405$	2.41641	−	4.83282i	0.120072	−	0.240145i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	24.4721			1.21007			0.605035	−	0.796199i	$-0.293159\pi$
$409$	24.4721			1.21007			0.605035	+	0.796199i	$0.293159\pi$
$410$		0			0
$411$	14.1115			0.696067
$412$		0			0
$413$			46.8328i			2.30449i
$414$		0			0
$415$	29.8885	+	14.9443i	1.46717	+	0.733585i
$416$		0			0
$417$		−	6.11146i		−	0.299279i
$418$		0			0
$419$	−29.8885			−1.46015			−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885			−1.46015			−0.730075	+	0.683367i	$0.760515\pi$
$420$		0			0
$421$	20.4721			0.997751			0.498875	−	0.866674i	$-0.333747\pi$
$421$	20.4721			0.997751			0.498875	+	0.866674i	$0.333747\pi$
$422$		0			0
$423$		−	18.3607i		−	0.892727i
$424$		0			0
$425$	−25.8885	+	19.4164i	−1.25578	+	0.941834i
$426$		0			0
$427$			20.0000i			0.967868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.8885			0.861661			0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885			0.861661			0.430830	+	0.902433i	$0.358221\pi$
$432$		0			0
$433$		−	12.1803i		−	0.585350i	−0.956212	−	0.292675i	$-0.905455\pi$
$433$		−	12.1803i		−	0.585350i	0.956212	−	0.292675i	$-0.0945454\pi$
$434$		0			0
$435$	−4.94427	−	2.47214i	−0.237060	−	0.118530i
$436$		0			0
$437$			2.00000i			0.0956730i
$438$		0			0
$439$	22.4721			1.07254			0.536268	−	0.844048i	$-0.319834\pi$
$439$	22.4721			1.07254			0.536268	+	0.844048i	$0.319834\pi$
$440$		0			0
$441$	−19.1378			−0.911322
$442$		0			0
$443$			23.8885i			1.13498i	0.823381	+	0.567489i	$0.192085\pi$
$443$			23.8885i			1.13498i	−0.823381	+	0.567489i	$0.807915\pi$
$444$		0			0
$445$	−6.94427	+	13.8885i	−0.329190	+	0.658380i
$446$		0			0
$447$		−	26.4721i		−	1.25209i
$448$		0			0
$449$	−9.41641			−0.444388			−0.222194	−	0.975003i	$-0.571322\pi$
$449$	−9.41641			−0.444388			−0.222194	+	0.975003i	$0.571322\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			22.1115i			1.03889i
$454$		0			0
$455$	−14.4721	+	28.9443i	−0.678464	+	1.35693i
$456$		0			0
$457$		−	33.8885i		−	1.58524i	−0.609716	−	0.792620i	$-0.708717\pi$
$457$		−	33.8885i		−	1.58524i	0.609716	−	0.792620i	$-0.291283\pi$
$458$		0			0
$459$	−35.7771			−1.66993
$460$		0			0
$461$	−11.8885			−0.553705			−0.276852	−	0.960912i	$-0.589291\pi$
$461$	−11.8885			−0.553705			−0.276852	+	0.960912i	$0.589291\pi$
$462$		0			0
$463$		−	31.8885i		−	1.48199i	−0.671513	−	0.740993i	$-0.734355\pi$
$463$		−	31.8885i		−	1.48199i	0.671513	−	0.740993i	$-0.265645\pi$
$464$		0			0
$465$	3.77709	+	1.88854i	0.175158	+	0.0875791i
$466$		0			0
$467$			6.00000i			0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$			6.00000i			0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$		0			0
$469$	−5.52786			−0.255253
$470$		0			0
$471$	19.0557			0.878042
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−3.00000	−	4.00000i	−0.137649	−	0.183533i
$476$		0			0
$477$		−	16.5410i		−	0.757361i
$478$		0			0
$479$	−24.9443			−1.13973			−0.569866	−	0.821737i	$-0.693005\pi$
$479$	−24.9443			−1.13973			−0.569866	+	0.821737i	$0.693005\pi$
$480$		0			0
$481$	−15.4164			−0.702928
$482$		0			0
$483$		−	11.0557i		−	0.503053i
$484$		0			0
$485$	−8.36068	−	4.18034i	−0.379639	−	0.189819i
$486$		0			0
$487$			12.6525i			0.573338i	0.958030	+	0.286669i	$0.0925481\pi$
$487$			12.6525i			0.573338i	−0.958030	+	0.286669i	$0.907452\pi$
$488$		0			0
$489$	−10.4721			−0.473566
$490$		0			0
$491$	−7.05573			−0.318421			−0.159210	−	0.987245i	$-0.550895\pi$
$491$	−7.05573			−0.318421			−0.159210	+	0.987245i	$0.550895\pi$
$492$		0			0
$493$			12.9443i			0.582981i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.83282i		−	0.306494i
$498$		0			0
$499$	−14.8328			−0.664008			−0.332004	−	0.943278i	$-0.607725\pi$
$499$	−14.8328			−0.664008			−0.332004	+	0.943278i	$0.607725\pi$
$500$		0			0
$501$	−12.5836			−0.562193
$502$		0			0
$503$			37.7771i			1.68440i	0.539168	+	0.842199i	$0.318739\pi$
$503$			37.7771i			1.68440i	−0.539168	+	0.842199i	$0.681261\pi$
$504$		0			0
$505$	−8.47214	+	16.9443i	−0.377005	+	0.754010i
$506$		0			0
$507$		−	3.12461i		−	0.138769i
$508$		0			0
$509$	−2.00000			−0.0886484			−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000			−0.0886484			−0.0443242	+	0.999017i	$0.514113\pi$
$510$		0			0
$511$	−28.9443			−1.28042
$512$		0			0
$513$		−	5.52786i		−	0.244061i
$514$		0			0
$515$	−34.4721	−	17.2361i	−1.51902	−	0.759512i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	18.1115			0.795005
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$			21.2361i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$523$			21.2361i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$524$		0			0
$525$	16.5836	+	22.1115i	0.723767	+	0.965023i
$526$		0			0
$527$		−	9.88854i		−	0.430752i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	−15.4164			−0.669015
$532$		0			0
$533$			11.4164i			0.494500i
$534$		0			0
$535$	15.4164	+	7.70820i	0.666509	+	0.333255i
$536$		0			0
$537$			6.83282i			0.294858i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.3607			−1.30531			−0.652654	−	0.757656i	$-0.726344\pi$
$541$	−30.3607			−1.30531			−0.652654	+	0.757656i	$0.726344\pi$
$542$		0			0
$543$		−	29.5279i		−	1.26716i
$544$		0			0
$545$	13.4164	−	26.8328i	0.574696	−	1.14939i
$546$		0			0
$547$			0.291796i			0.0124763i	0.999981	+	0.00623815i	$0.00198568\pi$
$547$			0.291796i			0.0124763i	−0.999981	+	0.00623815i	$0.998014\pi$
$548$		0			0
$549$	−6.58359			−0.280981
$550$		0			0
$551$	−2.00000			−0.0852029
$552$		0			0
$553$			28.9443i			1.23084i
$554$		0			0
$555$	−5.88854	+	11.7771i	−0.249955	+	0.499910i
$556$		0			0
$557$		−	39.4164i		−	1.67013i	−0.550154	−	0.835063i	$-0.685431\pi$
$557$		−	39.4164i		−	1.67013i	0.550154	−	0.835063i	$-0.314569\pi$
$558$		0			0
$559$	1.52786			0.0646218
$560$		0			0
$561$		0			0
$562$		0			0
$563$			29.2361i			1.23215i	0.787686	+	0.616077i	$0.211279\pi$
$563$			29.2361i			1.23215i	−0.787686	+	0.616077i	$0.788721\pi$
$564$		0			0
$565$	30.4721	+	15.2361i	1.28197	+	0.640986i
$566$		0			0
$567$			10.8065i			0.453831i
$568$		0			0
$569$	−12.4721			−0.522859			−0.261430	−	0.965223i	$-0.584194\pi$
$569$	−12.4721			−0.522859			−0.261430	+	0.965223i	$0.584194\pi$
$570$		0			0
$571$	3.05573			0.127878			0.0639391	−	0.997954i	$-0.479634\pi$
$571$	3.05573			0.127878			0.0639391	+	0.997954i	$0.479634\pi$
$572$		0			0
$573$			9.88854i			0.413100i
$574$		0			0
$575$	8.00000	−	6.00000i	0.333623	−	0.250217i
$576$		0			0
$577$		−	20.9443i		−	0.871921i	−0.899966	−	0.435961i	$-0.856409\pi$
$577$		−	20.9443i		−	0.871921i	0.899966	−	0.435961i	$-0.143591\pi$
$578$		0			0
$579$	7.05573			0.293226
$580$		0			0
$581$	−66.8328			−2.77269
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−9.52786	−	4.76393i	−0.393929	−	0.196964i
$586$		0			0
$587$			25.7771i			1.06393i	0.846765	+	0.531967i	$0.178547\pi$
$587$			25.7771i			1.06393i	−0.846765	+	0.531967i	$0.821453\pi$
$588$		0			0
$589$	1.52786			0.0629545
$590$		0			0
$591$	−1.16718			−0.0480115
$592$		0			0
$593$		−	19.0557i		−	0.782525i	−0.920279	−	0.391262i	$-0.872038\pi$
$593$		−	19.0557i		−	0.782525i	0.920279	−	0.391262i	$-0.127962\pi$
$594$		0			0
$595$	28.9443	−	57.8885i	1.18660	−	2.37320i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−38.4721			−1.57193			−0.785964	−	0.618272i	$-0.787833\pi$
$599$	−38.4721			−1.57193			−0.785964	+	0.618272i	$0.787833\pi$
$600$		0			0
$601$	42.3607			1.72793			0.863964	−	0.503553i	$-0.167974\pi$
$601$	42.3607			1.72793			0.863964	+	0.503553i	$0.167974\pi$
$602$		0			0
$603$		−	1.81966i		−	0.0741023i
$604$		0			0
$605$	11.0000	−	22.0000i	0.447214	−	0.894427i
$606$		0			0
$607$		−	15.1246i		−	0.613889i	−0.951727	−	0.306945i	$-0.900693\pi$
$607$		−	15.1246i		−	0.613889i	0.951727	−	0.306945i	$-0.0993066\pi$
$608$		0			0
$609$	11.0557			0.448001
$610$		0			0
$611$	40.3607			1.63282
$612$		0			0
$613$		−	39.4164i		−	1.59201i	−0.605288	−	0.796007i	$-0.706942\pi$
$613$		−	39.4164i		−	1.59201i	0.605288	−	0.796007i	$-0.293058\pi$
$614$		0			0
$615$	8.72136	+	4.36068i	0.351679	+	0.175840i
$616$		0			0
$617$		−	42.2492i		−	1.70089i	−0.526064	−	0.850445i	$-0.676333\pi$
$617$		−	42.2492i		−	1.70089i	0.526064	−	0.850445i	$-0.323667\pi$
$618$		0			0
$619$	−11.0557			−0.444367			−0.222184	−	0.975005i	$-0.571318\pi$
$619$	−11.0557			−0.444367			−0.222184	+	0.975005i	$0.571318\pi$
$620$		0			0
$621$	11.0557			0.443651
$622$		0			0
$623$		−	31.0557i		−	1.24422i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	30.8328			1.22938
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$			8.72136i			0.346643i
$634$		0			0
$635$	13.5279	+	6.76393i	0.536837	+	0.268418i
$636$		0			0
$637$		−	42.0689i		−	1.66683i
$638$		0			0
$639$	2.24922			0.0889779
$640$		0			0
$641$	34.3607			1.35717			0.678583	−	0.734524i	$-0.262595\pi$
$641$	34.3607			1.35717			0.678583	+	0.734524i	$0.262595\pi$
$642$		0			0
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$		0			0
$645$	0.583592	−	1.16718i	0.0229789	−	0.0459578i
$646$		0			0
$647$		−	9.05573i		−	0.356017i	−0.984029	−	0.178009i	$-0.943034\pi$
$647$		−	9.05573i		−	0.356017i	0.984029	−	0.178009i	$-0.0569655\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−8.44582			−0.331018
$652$		0			0
$653$		−	21.5279i		−	0.842450i	−0.906956	−	0.421225i	$-0.861600\pi$
$653$		−	21.5279i		−	0.842450i	0.906956	−	0.421225i	$-0.138400\pi$
$654$		0			0
$655$	−8.94427	+	17.8885i	−0.349482	+	0.698963i
$656$		0			0
$657$		−	9.52786i		−	0.371717i
$658$		0			0
$659$	28.3607			1.10478			0.552388	−	0.833587i	$-0.313717\pi$
$659$	28.3607			1.10478			0.552388	+	0.833587i	$0.313717\pi$
$660$		0			0
$661$	−27.5279			−1.07071			−0.535355	−	0.844627i	$-0.679822\pi$
$661$	−27.5279			−1.07071			−0.535355	+	0.844627i	$0.679822\pi$
$662$		0			0
$663$		−	25.8885i		−	1.00543i
$664$		0			0
$665$	8.94427	+	4.47214i	0.346844	+	0.173422i
$666$		0			0
$667$		−	4.00000i		−	0.154881i
$668$		0			0
$669$	31.1935			1.20601
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	19.8197i		−	0.763992i	−0.924164	−	0.381996i	$-0.875237\pi$
$673$		−	19.8197i		−	0.763992i	0.924164	−	0.381996i	$-0.124763\pi$
$674$		0			0
$675$	−22.1115	+	16.5836i	−0.851070	+	0.638303i
$676$		0			0
$677$			13.1246i			0.504420i	0.967673	+	0.252210i	$0.0811573\pi$
$677$			13.1246i			0.504420i	−0.967673	+	0.252210i	$0.918843\pi$
$678$		0			0
$679$	18.6950			0.717450
$680$		0			0
$681$	12.5836			0.482204
$682$		0			0
$683$		−	13.8197i		−	0.528795i	−0.964414	−	0.264397i	$-0.914827\pi$
$683$		−	13.8197i		−	0.528795i	0.964414	−	0.264397i	$-0.0851730\pi$
$684$		0			0
$685$	−22.8328	−	11.4164i	−0.872397	−	0.436199i
$686$		0			0
$687$			21.5279i			0.821339i
$688$		0			0
$689$	36.3607			1.38523
$690$		0			0
$691$	17.8885			0.680512			0.340256	−	0.940333i	$-0.389486\pi$
$691$	17.8885			0.680512			0.340256	+	0.940333i	$0.389486\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.94427	+	9.88854i	−0.187547	+	0.375094i
$696$		0			0
$697$		−	22.8328i		−	0.864855i
$698$		0			0
$699$	−16.0000			−0.605176
$700$		0			0
$701$	23.3050			0.880216			0.440108	−	0.897945i	$-0.354940\pi$
$701$	23.3050			0.880216			0.440108	+	0.897945i	$0.354940\pi$
$702$		0			0
$703$			4.76393i			0.179675i
$704$		0			0
$705$	15.4164	−	30.8328i	0.580616	−	1.16123i
$706$		0			0
$707$		−	37.8885i		−	1.42495i
$708$		0			0
$709$	−0.111456			−0.00418582			−0.00209291	−	0.999998i	$-0.500666\pi$
$709$	−0.111456			−0.00418582			−0.00209291	+	0.999998i	$0.500666\pi$
$710$		0			0
$711$	−9.52786			−0.357323
$712$		0			0
$713$			3.05573i			0.114438i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	35.7771i		−	1.33612i
$718$		0			0
$719$	−45.8885			−1.71135			−0.855677	−	0.517510i	$-0.826859\pi$
$719$	−45.8885			−1.71135			−0.855677	+	0.517510i	$0.826859\pi$
$720$		0			0
$721$	77.0820			2.87069
$722$		0			0
$723$		−	26.4721i		−	0.984509i
$724$		0			0
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$		0			0
$727$			45.4164i			1.68440i	0.539164	+	0.842201i	$0.318740\pi$
$727$			45.4164i			1.68440i	−0.539164	+	0.842201i	$0.681260\pi$
$728$		0			0
$729$	−24.0557			−0.890953
$730$		0			0
$731$	−3.05573			−0.113020
$732$		0			0
$733$			16.9443i			0.625851i	0.949778	+	0.312925i	$0.101309\pi$
$733$			16.9443i			0.625851i	−0.949778	+	0.312925i	$0.898691\pi$
$734$		0			0
$735$	−32.1378	−	16.0689i	−1.18542	−	0.592710i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$	4.00000			0.146944
$742$		0			0
$743$		−	39.1246i		−	1.43534i	−0.696382	−	0.717671i	$-0.745208\pi$
$743$		−	39.1246i		−	1.43534i	0.696382	−	0.717671i	$-0.254792\pi$
$744$		0			0
$745$	−21.4164	+	42.8328i	−0.784636	+	1.56927i
$746$		0			0
$747$		−	22.0000i		−	0.804938i
$748$		0			0
$749$	−34.4721			−1.25958
$750$		0			0
$751$	−20.9443			−0.764267			−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443			−0.764267			−0.382134	+	0.924107i	$0.624811\pi$
$752$		0			0
$753$			4.94427i			0.180179i
$754$		0			0
$755$	17.8885	−	35.7771i	0.651031	−	1.30206i
$756$		0			0
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.472136			−0.0171149			−0.00855746	−	0.999963i	$-0.502724\pi$
$761$	−0.472136			−0.0171149			−0.00855746	+	0.999963i	$0.502724\pi$
$762$		0			0
$763$			60.0000i			2.17215i
$764$		0			0
$765$	19.0557	+	9.52786i	0.688961	+	0.344481i
$766$		0			0
$767$		−	33.8885i		−	1.22364i
$768$		0			0
$769$	2.58359			0.0931667			0.0465834	−	0.998914i	$-0.485167\pi$
$769$	2.58359			0.0931667			0.0465834	+	0.998914i	$0.485167\pi$
$770$		0			0
$771$	−12.7214			−0.458149
$772$		0			0
$773$			14.2918i			0.514040i	0.966406	+	0.257020i	$0.0827407\pi$
$773$			14.2918i			0.514040i	−0.966406	+	0.257020i	$0.917259\pi$
$774$		0			0
$775$	−4.58359	−	6.11146i	−0.164647	−	0.219530i
$776$		0			0
$777$		−	26.3344i		−	0.944740i
$778$		0			0
$779$	3.52786			0.126399
$780$		0			0
$781$		0			0
$782$		0			0
$783$			11.0557i			0.395099i
$784$		0			0
$785$	−30.8328	−	15.4164i	−1.10047	−	0.550235i
$786$		0			0
$787$		−	54.5410i		−	1.94418i	−0.234614	−	0.972089i	$-0.575383\pi$
$787$		−	54.5410i		−	1.94418i	0.234614	−	0.972089i	$-0.424617\pi$
$788$		0			0
$789$	14.6950			0.523158
$790$		0			0
$791$	−68.1378			−2.42270
$792$		0			0
$793$		−	14.4721i		−	0.513921i
$794$		0			0
$795$	13.8885	−	27.7771i	0.492576	−	0.985152i
$796$		0			0
$797$		−	14.6525i		−	0.519017i	−0.965741	−	0.259509i	$-0.916439\pi$
$797$		−	14.6525i		−	0.519017i	0.965741	−	0.259509i	$-0.0835606\pi$
$798$		0			0
$799$	−80.7214			−2.85572
$800$		0			0
$801$	10.2229			0.361209
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−8.94427	+	17.8885i	−0.315244	+	0.630488i
$806$		0			0
$807$		−	31.4164i		−	1.10591i
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−41.3050			−1.45041			−0.725207	−	0.688531i	$-0.758256\pi$
$811$	−41.3050			−1.45041			−0.725207	+	0.688531i	$0.758256\pi$
$812$		0			0
$813$			1.16718i			0.0409349i
$814$		0			0
$815$	16.9443	+	8.47214i	0.593532	+	0.296766i
$816$		0			0
$817$		−	0.472136i		−	0.0165179i
$818$		0			0
$819$	21.3050			0.744455
$820$		0			0
$821$	42.9443			1.49877			0.749383	−	0.662137i	$-0.230350\pi$
$821$	42.9443			1.49877			0.749383	+	0.662137i	$0.230350\pi$
$822$		0			0
$823$			22.5836i			0.787215i	0.919279	+	0.393607i	$0.128773\pi$
$823$			22.5836i			0.787215i	−0.919279	+	0.393607i	$0.871227\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			38.7639i			1.34795i	0.738752	+	0.673977i	$0.235415\pi$
$827$			38.7639i			1.34795i	−0.738752	+	0.673977i	$0.764585\pi$
$828$		0			0
$829$	9.41641			0.327045			0.163523	−	0.986540i	$-0.447714\pi$
$829$	9.41641			0.327045			0.163523	+	0.986540i	$0.447714\pi$
$830$		0			0
$831$	32.7214			1.13509
$832$		0			0
$833$			84.1378i			2.91520i
$834$		0			0
$835$	20.3607	+	10.1803i	0.704610	+	0.352305i
$836$		0			0
$837$		−	8.44582i		−	0.291930i
$838$		0			0
$839$	−17.5279			−0.605129			−0.302565	−	0.953129i	$-0.597843\pi$
$839$	−17.5279			−0.605129			−0.302565	+	0.953129i	$0.597843\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	26.4721i		−	0.911749i
$844$		0			0
$845$	−2.52786	+	5.05573i	−0.0869612	+	0.173922i
$846$		0			0
$847$			49.1935i			1.69031i
$848$		0			0
$849$	−24.5836			−0.843707
$850$		0			0
$851$	−9.52786			−0.326611
$852$		0			0
$853$		−	40.9443i		−	1.40191i	−0.713208	−	0.700953i	$-0.752758\pi$
$853$		−	40.9443i		−	1.40191i	0.713208	−	0.700953i	$-0.247242\pi$
$854$		0			0
$855$	−1.47214	+	2.94427i	−0.0503460	+	0.100692i
$856$		0			0
$857$		−	21.3475i		−	0.729218i	−0.931161	−	0.364609i	$-0.881203\pi$
$857$		−	21.3475i		−	0.729218i	0.931161	−	0.364609i	$-0.118797\pi$
$858$		0			0
$859$	−37.8885			−1.29274			−0.646370	−	0.763024i	$-0.723714\pi$
$859$	−37.8885			−1.29274			−0.646370	+	0.763024i	$0.723714\pi$
$860$		0			0
$861$	−19.5016			−0.664611
$862$		0			0
$863$		−	21.2361i		−	0.722884i	−0.932395	−	0.361442i	$-0.882285\pi$
$863$		−	21.2361i		−	0.722884i	0.932395	−	0.361442i	$-0.117715\pi$
$864$		0			0
$865$	−29.3050	−	14.6525i	−0.996398	−	0.498199i
$866$		0			0
$867$			30.7639i			1.04480i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	4.00000			0.135535
$872$		0			0
$873$			6.15403i			0.208282i
$874$		0			0
$875$	−8.94427	−	49.1935i	−0.302372	−	1.66304i
$876$		0			0
$877$			8.18034i			0.276230i	0.990416	+	0.138115i	$0.0441044\pi$
$877$			8.18034i			0.276230i	−0.990416	+	0.138115i	$0.955896\pi$
$878$		0			0
$879$	21.8885			0.738282
$880$		0			0
$881$	40.2492			1.35603			0.678015	−	0.735048i	$-0.262840\pi$
$881$	40.2492			1.35603			0.678015	+	0.735048i	$0.262840\pi$
$882$		0			0
$883$		−	35.5279i		−	1.19561i	−0.801642	−	0.597804i	$-0.796040\pi$
$883$		−	35.5279i		−	1.19561i	0.801642	−	0.597804i	$-0.203960\pi$
$884$		0			0
$885$	−25.8885	−	12.9443i	−0.870234	−	0.435117i
$886$		0			0
$887$			36.2918i			1.21856i	0.792955	+	0.609280i	$0.208541\pi$
$887$			36.2918i			1.21856i	−0.792955	+	0.609280i	$0.791459\pi$
$888$		0			0
$889$	−30.2492			−1.01453
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	12.4721i		−	0.417364i
$894$		0			0
$895$	5.52786	−	11.0557i	0.184776	−	0.369552i
$896$		0			0
$897$			8.00000i			0.267112i
$898$		0			0
$899$	−3.05573			−0.101914
$900$		0			0
$901$	−72.7214			−2.42270
$902$		0			0
$903$			2.60990i			0.0868521i
$904$		0			0
$905$	−23.8885	+	47.7771i	−0.794082	+	1.58816i
$906$		0			0
$907$			34.1803i			1.13494i	0.823394	+	0.567470i	$0.192078\pi$
$907$			34.1803i			1.13494i	−0.823394	+	0.567470i	$0.807922\pi$
$908$		0			0
$909$	12.4721			0.413675
$910$		0			0
$911$	1.16718			0.0386705			0.0193353	−	0.999813i	$-0.493845\pi$
$911$	1.16718			0.0386705			0.0193353	+	0.999813i	$0.493845\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−11.0557	−	5.52786i	−0.365491	−	0.182746i
$916$		0			0
$917$		−	40.0000i		−	1.32092i
$918$		0			0
$919$	33.8885			1.11788			0.558940	−	0.829208i	$-0.311208\pi$
$919$	33.8885			1.11788			0.558940	+	0.829208i	$0.311208\pi$
$920$		0			0
$921$	16.3607			0.539103
$922$		0			0
$923$			4.94427i			0.162743i
$924$		0			0
$925$	19.0557	−	14.2918i	0.626548	−	0.469911i
$926$		0			0
$927$			25.3738i			0.833386i
$928$		0			0
$929$	−41.7771			−1.37066			−0.685331	−	0.728232i	$-0.740342\pi$
$929$	−41.7771			−1.37066			−0.685331	+	0.728232i	$0.740342\pi$
$930$		0			0
$931$	−13.0000			−0.426058
$932$		0			0
$933$		−	24.7214i		−	0.809341i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			35.4164i			1.15700i	0.815681	+	0.578502i	$0.196362\pi$
$937$			35.4164i			1.15700i	−0.815681	+	0.578502i	$0.803638\pi$
$938$		0			0
$939$	19.7771			0.645401
$940$		0			0
$941$	−16.8328			−0.548734			−0.274367	−	0.961625i	$-0.588468\pi$
$941$	−16.8328			−0.548734			−0.274367	+	0.961625i	$0.588468\pi$
$942$		0			0
$943$			7.05573i			0.229766i
$944$		0			0
$945$	24.7214	−	49.4427i	0.804186	−	1.60837i
$946$		0			0
$947$			9.05573i			0.294272i	0.989116	+	0.147136i	$0.0470054\pi$
$947$			9.05573i			0.294272i	−0.989116	+	0.147136i	$0.952995\pi$
$948$		0			0
$949$	20.9443			0.679880
$950$		0			0
$951$	−26.1115			−0.846722
$952$		0			0
$953$			2.29180i			0.0742386i	0.999311	+	0.0371193i	$0.0118181\pi$
$953$			2.29180i			0.0742386i	−0.999311	+	0.0371193i	$0.988182\pi$
$954$		0			0
$955$	8.00000	−	16.0000i	0.258874	−	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	51.0557			1.64868
$960$		0			0
$961$	−28.6656			−0.924698
$962$		0			0
$963$		−	11.3475i		−	0.365669i
$964$		0			0
$965$	−11.4164	−	5.70820i	−0.367507	−	0.183754i
$966$		0			0
$967$			32.8328i			1.05583i	0.849297	+	0.527916i	$0.177026\pi$
$967$			32.8328i			1.05583i	−0.849297	+	0.527916i	$0.822974\pi$
$968$		0			0
$969$	−8.00000			−0.256997
$970$		0			0
$971$	44.3607			1.42360			0.711801	−	0.702381i	$-0.247880\pi$
$971$	44.3607			1.42360			0.711801	+	0.702381i	$0.247880\pi$
$972$		0			0
$973$		−	22.1115i		−	0.708861i
$974$		0			0
$975$	−12.0000	−	16.0000i	−0.384308	−	0.512410i
$976$		0			0
$977$			3.81966i			0.122202i	0.998132	+	0.0611009i	$0.0194611\pi$
$977$			3.81966i			0.122202i	−0.998132	+	0.0611009i	$0.980539\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−19.7508			−0.630594
$982$		0			0
$983$			34.7639i			1.10880i	0.832251	+	0.554399i	$0.187052\pi$
$983$			34.7639i			1.10880i	−0.832251	+	0.554399i	$0.812948\pi$
$984$		0			0
$985$	1.88854	+	0.944272i	0.0601740	+	0.0300870i
$986$		0			0
$987$			68.9443i			2.19452i
$988$		0			0
$989$	0.944272			0.0300261
$990$		0			0
$991$	−9.88854			−0.314120			−0.157060	−	0.987589i	$-0.550202\pi$
$991$	−9.88854			−0.314120			−0.157060	+	0.987589i	$0.550202\pi$
$992$		0			0
$993$		−	40.7214i		−	1.29225i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			33.3050i			1.05478i	0.849624	+	0.527389i	$0.176829\pi$
$997$			33.3050i			1.05478i	−0.849624	+	0.527389i	$0.823171\pi$
$998$		0			0
$999$	26.3344			0.833183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.c.609.2		4
4.3	odd	2		760.2.d.b.609.3	yes	4
5.2	odd	4		7600.2.a.be.1.1		2
5.3	odd	4		7600.2.a.w.1.2		2
5.4	even	2	inner	1520.2.d.c.609.3		4
20.3	even	4		3800.2.a.q.1.1		2
20.7	even	4		3800.2.a.k.1.2		2
20.19	odd	2		760.2.d.b.609.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.b.609.2	✓	4	20.19	odd	2
760.2.d.b.609.3	yes	4	4.3	odd	2
1520.2.d.c.609.2		4	1.1	even	1	trivial
1520.2.d.c.609.3		4	5.4	even	2	inner
3800.2.a.k.1.2		2	20.7	even	4
3800.2.a.q.1.1		2	20.3	even	4
7600.2.a.w.1.2		2	5.3	odd	4
7600.2.a.be.1.1		2	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23607i q^{3} +(-1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q-1.23607i q^{3} +(-1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +1.47214 q^{9} +3.23607i q^{13} +(2.47214 + 1.23607i) q^{15} -6.47214i q^{17} +1.00000 q^{19} -5.52786 q^{21} +2.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -5.52786i q^{27} -2.00000 q^{29} +1.52786 q^{31} +(8.94427 + 4.47214i) q^{35} +4.76393i q^{37} +4.00000 q^{39} +3.52786 q^{41} -0.472136i q^{43} +(-1.47214 + 2.94427i) q^{45} -12.4721i q^{47} -13.0000 q^{49} -8.00000 q^{51} -11.2361i q^{53} -1.23607i q^{57} -10.4721 q^{59} -4.47214 q^{61} -6.58359i q^{63} +(-6.47214 - 3.23607i) q^{65} -1.23607i q^{67} +2.47214 q^{69} +1.52786 q^{71} -6.47214i q^{73} +(-4.94427 + 3.70820i) q^{75} -6.47214 q^{79} -2.41641 q^{81} -14.9443i q^{83} +(12.9443 + 6.47214i) q^{85} +2.47214i q^{87} +6.94427 q^{89} +14.4721 q^{91} -1.88854i q^{93} +(-1.00000 + 2.00000i) q^{95} +4.18034i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^5 - 12 * q^9	\( 4 q - 4 q^{5} - 12 q^{9} - 8 q^{15} + 4 q^{19} - 40 q^{21} - 12 q^{25} - 8 q^{29} + 24 q^{31} + 16 q^{39} + 32 q^{41} + 12 q^{45} - 52 q^{49} - 32 q^{51} - 24 q^{59} - 8 q^{65} - 8 q^{69} + 24 q^{71} + 16 q^{75} - 8 q^{79} + 44 q^{81} + 16 q^{85} - 8 q^{89} + 40 q^{91} - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^5 - 12 * q^9 - 8 * q^15 + 4 * q^19 - 40 * q^21 - 12 * q^25 - 8 * q^29 + 24 * q^31 + 16 * q^39 + 32 * q^41 + 12 * q^45 - 52 * q^49 - 32 * q^51 - 24 * q^59 - 8 * q^65 - 8 * q^69 + 24 * q^71 + 16 * q^75 - 8 * q^79 + 44 * q^81 + 16 * q^85 - 8 * q^89 + 40 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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