LMFDB - Embedded newform 1960.2.g.c.1569.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1569,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, 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("1960.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.g (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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.5
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.c.1569.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76156i q^{3} +(-0.432320 + 2.19388i) q^{5} -0.103084 q^{9} +O(q^{10})$	$q+1.76156i q^{3} +(-0.432320 + 2.19388i) q^{5} -0.103084 q^{9} -0.626198 q^{11} +5.49084i q^{13} +(-3.86464 - 0.761557i) q^{15} -0.896916i q^{17} +6.38776 q^{19} +3.72928i q^{23} +(-4.62620 - 1.89692i) q^{25} +5.10308i q^{27} +7.87859 q^{29} -7.52311 q^{31} -1.10308i q^{33} +6.00000i q^{37} -9.67243 q^{39} -7.72928 q^{41} -1.72928i q^{43} +(0.0445652 - 0.226153i) q^{45} -5.87859i q^{47} +1.57997 q^{51} +6.77551i q^{53} +(0.270718 - 1.37380i) q^{55} +11.2524i q^{57} -0.593923 q^{59} -7.13536 q^{61} +(-12.0462 - 2.37380i) q^{65} -5.79383i q^{67} -6.56934 q^{69} +5.52311 q^{71} -3.72928i q^{73} +(3.34153 - 8.14931i) q^{75} +5.67243 q^{79} -9.29862 q^{81} -17.4340i q^{83} +(1.96772 + 0.387755i) q^{85} +13.8786i q^{87} +14.2986 q^{89} -13.2524i q^{93} +(-2.76156 + 14.0140i) q^{95} +10.1493i q^{97} +0.0645508 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} + 14 q^{11} - 18 q^{15} + 8 q^{19} - 10 q^{25} - 6 q^{29} - 20 q^{31} + 10 q^{39} - 36 q^{41} + 28 q^{45} + 42 q^{51} + 12 q^{55} + 12 q^{59} - 48 q^{61} - 22 q^{65} + 36 q^{69} + 8 q^{71} + 40 q^{75} - 34 q^{79} + 30 q^{81} + 14 q^{85} - 4 q^{95} - 4 q^{99}+O(q^{100}) $ 6 * q - 8 * q^9 + 14 * q^11 - 18 * q^15 + 8 * q^19 - 10 * q^25 - 6 * q^29 - 20 * q^31 + 10 * q^39 - 36 * q^41 + 28 * q^45 + 42 * q^51 + 12 * q^55 + 12 * q^59 - 48 * q^61 - 22 * q^65 + 36 * q^69 + 8 * q^71 + 40 * q^75 - 34 * q^79 + 30 * q^81 + 14 * q^85 - 4 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			1.76156i			1.01704i	0.861052	+	0.508518i	$0.169806\pi$
$3$			1.76156i			1.01704i	−0.861052	+	0.508518i	$0.830194\pi$
$4$		0			0
$5$	−0.432320	+	2.19388i	−0.193340	+	0.981132i
$5$	−0.432320	+	2.19388i	−0.193340	+	0.981132i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.103084			−0.0343612
$10$		0			0
$11$	−0.626198			−0.188806			−0.0944029	−	0.995534i	$-0.530094\pi$
$11$	−0.626198			−0.188806			−0.0944029	+	0.995534i	$0.530094\pi$
$12$		0			0
$13$			5.49084i			1.52288i	0.648233	+	0.761442i	$0.275508\pi$
$13$			5.49084i			1.52288i	−0.648233	+	0.761442i	$0.724492\pi$
$14$		0			0
$15$	−3.86464	−	0.761557i	−0.997846	−	0.196633i
$16$		0			0
$17$		−	0.896916i		−	0.217534i	−0.994067	−	0.108767i	$-0.965310\pi$
$17$		−	0.896916i		−	0.217534i	0.994067	−	0.108767i	$-0.0346903\pi$
$18$		0			0
$19$	6.38776			1.46545			0.732726	−	0.680524i	$-0.238248\pi$
$19$	6.38776			1.46545			0.732726	+	0.680524i	$0.238248\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.72928i			0.777609i	0.921320	+	0.388805i	$0.127112\pi$
$23$			3.72928i			0.777609i	−0.921320	+	0.388805i	$0.872888\pi$
$24$		0			0
$25$	−4.62620	−	1.89692i	−0.925240	−	0.379383i
$26$		0			0
$27$			5.10308i			0.982089i
$28$		0			0
$29$	7.87859			1.46302			0.731509	−	0.681832i	$-0.238816\pi$
$29$	7.87859			1.46302			0.731509	+	0.681832i	$0.238816\pi$
$30$		0			0
$31$	−7.52311			−1.35119			−0.675596	−	0.737272i	$-0.736113\pi$
$31$	−7.52311			−1.35119			−0.675596	+	0.737272i	$0.736113\pi$
$32$		0			0
$33$		−	1.10308i		−	0.192022i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$	−9.67243			−1.54883
$40$		0			0
$41$	−7.72928			−1.20711			−0.603556	−	0.797321i	$-0.706250\pi$
$41$	−7.72928			−1.20711			−0.603556	+	0.797321i	$0.706250\pi$
$42$		0			0
$43$		−	1.72928i		−	0.263713i	−0.991269	−	0.131856i	$-0.957906\pi$
$43$		−	1.72928i		−	0.263713i	0.991269	−	0.131856i	$-0.0420938\pi$
$44$		0			0
$45$	0.0445652	−	0.226153i	0.00664339	−	0.0337129i
$46$		0			0
$47$		−	5.87859i		−	0.857481i	−0.903428	−	0.428741i	$-0.858957\pi$
$47$		−	5.87859i		−	0.857481i	0.903428	−	0.428741i	$-0.141043\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.57997			0.221240
$52$		0			0
$53$			6.77551i			0.930688i	0.885130	+	0.465344i	$0.154069\pi$
$53$			6.77551i			0.930688i	−0.885130	+	0.465344i	$0.845931\pi$
$54$		0			0
$55$	0.270718	−	1.37380i	0.0365036	−	0.185243i
$56$		0			0
$57$			11.2524i			1.49042i
$58$		0			0
$59$	−0.593923			−0.0773221			−0.0386611	−	0.999252i	$-0.512309\pi$
$59$	−0.593923			−0.0773221			−0.0386611	+	0.999252i	$0.512309\pi$
$60$		0			0
$61$	−7.13536			−0.913589			−0.456795	−	0.889572i	$-0.651003\pi$
$61$	−7.13536			−0.913589			−0.456795	+	0.889572i	$0.651003\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0462	−	2.37380i	−1.49415	−	0.294434i
$66$		0			0
$67$		−	5.79383i		−	0.707829i	−0.935278	−	0.353915i	$-0.884850\pi$
$67$		−	5.79383i		−	0.707829i	0.935278	−	0.353915i	$-0.115150\pi$
$68$		0			0
$69$	−6.56934			−0.790856
$70$		0			0
$71$	5.52311			0.655473			0.327737	−	0.944769i	$-0.393714\pi$
$71$	5.52311			0.655473			0.327737	+	0.944769i	$0.393714\pi$
$72$		0			0
$73$		−	3.72928i		−	0.436479i	−0.975895	−	0.218240i	$-0.929969\pi$
$73$		−	3.72928i		−	0.436479i	0.975895	−	0.218240i	$-0.0700315\pi$
$74$		0			0
$75$	3.34153	−	8.14931i	0.385846	−	0.941002i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.67243			0.638198			0.319099	−	0.947721i	$-0.396620\pi$
$79$	5.67243			0.638198			0.319099	+	0.947721i	$0.396620\pi$
$80$		0			0
$81$	−9.29862			−1.03318
$82$		0			0
$83$		−	17.4340i		−	1.91363i	−0.290700	−	0.956814i	$-0.593888\pi$
$83$		−	17.4340i		−	1.91363i	0.290700	−	0.956814i	$-0.406112\pi$
$84$		0			0
$85$	1.96772	+	0.387755i	0.213430	+	0.0420580i
$86$		0			0
$87$			13.8786i			1.48794i
$88$		0			0
$89$	14.2986			1.51565			0.757826	−	0.652457i	$-0.226262\pi$
$89$	14.2986			1.51565			0.757826	+	0.652457i	$0.226262\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	13.2524i		−	1.37421i
$94$		0			0
$95$	−2.76156	+	14.0140i	−0.283330	+	1.43780i
$96$		0			0
$97$			10.1493i			1.03051i	0.857038	+	0.515253i	$0.172302\pi$
$97$			10.1493i			1.03051i	−0.857038	+	0.515253i	$0.827698\pi$
$98$		0			0
$99$	0.0645508			0.00648760
$100$		0			0
$101$	−9.64015			−0.959231			−0.479615	−	0.877479i	$-0.659224\pi$
$101$	−9.64015			−0.959231			−0.479615	+	0.877479i	$0.659224\pi$
$102$		0			0
$103$		−	0.626198i		−	0.0617011i	−0.999524	−	0.0308506i	$-0.990178\pi$
$103$		−	0.626198i		−	0.0617011i	0.999524	−	0.0308506i	$-0.00982160\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−18.9248			−1.81267			−0.906335	−	0.422561i	$-0.861131\pi$
$109$	−18.9248			−1.81267			−0.906335	+	0.422561i	$0.861131\pi$
$110$		0			0
$111$	−10.5693			−1.00320
$112$		0			0
$113$			1.04623i			0.0984209i	0.998788	+	0.0492105i	$0.0156705\pi$
$113$			1.04623i			0.0984209i	−0.998788	+	0.0492105i	$0.984329\pi$
$114$		0			0
$115$	−8.18159	−	1.61224i	−0.762937	−	0.150343i
$116$		0			0
$117$		−	0.566016i		−	0.0523282i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.6079			−0.964352
$122$		0			0
$123$		−	13.6156i		−	1.22767i
$124$		0			0
$125$	6.16160	−	9.32924i	0.551110	−	0.834432i
$126$		0			0
$127$		−	21.2803i		−	1.88832i	−0.329485	−	0.944161i	$-0.606875\pi$
$127$		−	21.2803i		−	1.88832i	0.329485	−	0.944161i	$-0.393125\pi$
$128$		0			0
$129$	3.04623			0.268205
$130$		0			0
$131$	9.91087			0.865917			0.432958	−	0.901414i	$-0.357470\pi$
$131$	9.91087			0.865917			0.432958	+	0.901414i	$0.357470\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−11.1955	−	2.20617i	−0.963559	−	0.189877i
$136$		0			0
$137$			3.45856i			0.295485i	0.989026	+	0.147743i	$0.0472007\pi$
$137$			3.45856i			0.295485i	−0.989026	+	0.147743i	$0.952799\pi$
$138$		0			0
$139$	−20.4157			−1.73163			−0.865817	−	0.500361i	$-0.833201\pi$
$139$	−20.4157			−1.73163			−0.865817	+	0.500361i	$0.833201\pi$
$140$		0			0
$141$	10.3555			0.872089
$142$		0			0
$143$		−	3.43835i		−	0.287530i
$144$		0			0
$145$	−3.40608	+	17.2847i	−0.282859	+	1.43541i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.0462			1.39648			0.698241	−	0.715863i	$-0.253967\pi$
$149$	17.0462			1.39648			0.698241	+	0.715863i	$0.253967\pi$
$150$		0			0
$151$	2.89692			0.235748			0.117874	−	0.993029i	$-0.462392\pi$
$151$	2.89692			0.235748			0.117874	+	0.993029i	$0.462392\pi$
$152$		0			0
$153$			0.0924575i			0.00747474i
$154$		0			0
$155$	3.25240	−	16.5048i	0.261239	−	1.32570i
$156$		0			0
$157$		−	10.1170i		−	0.807427i	−0.914885	−	0.403714i	$-0.867719\pi$
$157$		−	10.1170i		−	0.807427i	0.914885	−	0.403714i	$-0.132281\pi$
$158$		0			0
$159$	−11.9354			−0.946543
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	0.476886i		−	0.0373526i	−0.999826	−	0.0186763i	$-0.994055\pi$
$163$		−	0.476886i		−	0.0373526i	0.999826	−	0.0186763i	$-0.00594519\pi$
$164$		0			0
$165$	2.42003	+	0.476886i	0.188399	+	0.0371255i
$166$		0			0
$167$			13.1676i			1.01894i	0.860488	+	0.509471i	$0.170159\pi$
$167$			13.1676i			1.01894i	−0.860488	+	0.509471i	$0.829841\pi$
$168$		0			0
$169$	−17.1493			−1.31918
$170$		0			0
$171$	−0.658473			−0.0503547
$172$		0			0
$173$			13.9677i			1.06195i	0.847389	+	0.530973i	$0.178174\pi$
$173$			13.9677i			1.06195i	−0.847389	+	0.530973i	$0.821826\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	1.04623i		−	0.0786394i
$178$		0			0
$179$	−7.45856			−0.557479			−0.278740	−	0.960367i	$-0.589917\pi$
$179$	−7.45856			−0.557479			−0.278740	+	0.960367i	$0.589917\pi$
$180$		0			0
$181$	14.1170			1.04931			0.524656	−	0.851315i	$-0.324194\pi$
$181$	14.1170			1.04931			0.524656	+	0.851315i	$0.324194\pi$
$182$		0			0
$183$		−	12.5693i		−	0.929153i
$184$		0			0
$185$	−13.1633	−	2.59392i	−0.967783	−	0.190709i
$186$		0			0
$187$			0.561647i			0.0410717i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.42003			−0.609252			−0.304626	−	0.952472i	$-0.598531\pi$
$191$	−8.42003			−0.609252			−0.304626	+	0.952472i	$0.598531\pi$
$192$		0			0
$193$			22.2986i			1.60509i	0.596591	+	0.802545i	$0.296521\pi$
$193$			22.2986i			1.60509i	−0.596591	+	0.802545i	$0.703479\pi$
$194$		0			0
$195$	4.18159	−	21.2201i	0.299450	−	1.51960i
$196$		0			0
$197$			26.3632i			1.87830i	0.343509	+	0.939149i	$0.388384\pi$
$197$			26.3632i			1.87830i	−0.343509	+	0.939149i	$0.611616\pi$
$198$		0			0
$199$	22.4402			1.59075			0.795373	−	0.606120i	$-0.207275\pi$
$199$	22.4402			1.59075			0.795373	+	0.606120i	$0.207275\pi$
$200$		0			0
$201$	10.2062			0.719888
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.34153	−	16.9571i	0.233382	−	1.18434i
$206$		0			0
$207$		−	0.384428i		−	0.0267196i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−15.1955			−1.04610			−0.523052	−	0.852301i	$-0.675207\pi$
$211$	−15.1955			−1.04610			−0.523052	+	0.852301i	$0.675207\pi$
$212$		0			0
$213$			9.72928i			0.666639i
$214$		0			0
$215$	3.79383	+	0.747604i	0.258737	+	0.0509862i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.56934			0.443915
$220$		0			0
$221$	4.92482			0.331279
$222$		0			0
$223$		−	6.89692i		−	0.461852i	−0.972971	−	0.230926i	$-0.925825\pi$
$223$		−	6.89692i		−	0.461852i	0.972971	−	0.230926i	$-0.0741755\pi$
$224$		0			0
$225$	0.476886	+	0.195541i	0.0317924	+	0.0130361i
$226$		0			0
$227$			2.77988i			0.184507i	0.995736	+	0.0922535i	$0.0294070\pi$
$227$			2.77988i			0.184507i	−0.995736	+	0.0922535i	$0.970593\pi$
$228$		0			0
$229$	18.6585			1.23299			0.616493	−	0.787360i	$-0.288553\pi$
$229$	18.6585			1.23299			0.616493	+	0.787360i	$0.288553\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			11.0462i			0.723663i	0.932244	+	0.361831i	$0.117848\pi$
$233$			11.0462i			0.723663i	−0.932244	+	0.361831i	$0.882152\pi$
$234$		0			0
$235$	12.8969	+	2.54144i	0.841302	+	0.165785i
$236$		0			0
$237$			9.99230i			0.649070i
$238$		0			0
$239$	20.1493			1.30335			0.651675	−	0.758498i	$-0.274066\pi$
$239$	20.1493			1.30335			0.651675	+	0.758498i	$0.274066\pi$
$240$		0			0
$241$	−3.72928			−0.240224			−0.120112	−	0.992760i	$-0.538325\pi$
$241$	−3.72928			−0.240224			−0.120112	+	0.992760i	$0.538325\pi$
$242$		0			0
$243$		−	1.07081i		−	0.0686924i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			35.0741i			2.23171i
$248$		0			0
$249$	30.7110			1.94623
$250$		0			0
$251$	21.4985			1.35698			0.678488	−	0.734612i	$-0.262636\pi$
$251$	21.4985			1.35698			0.678488	+	0.734612i	$0.262636\pi$
$252$		0			0
$253$		−	2.33527i		−	0.146817i
$254$		0			0
$255$	−0.683053	+	3.46626i	−0.0427744	+	0.217066i
$256$		0			0
$257$		−	4.27072i		−	0.266400i	−0.991089	−	0.133200i	$-0.957475\pi$
$257$		−	4.27072i		−	0.266400i	0.991089	−	0.133200i	$-0.0425253\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.812155			−0.0502711
$262$		0			0
$263$			11.4586i			0.706565i	0.935517	+	0.353283i	$0.114935\pi$
$263$			11.4586i			0.706565i	−0.935517	+	0.353283i	$0.885065\pi$
$264$		0			0
$265$	−14.8646	−	2.92919i	−0.913128	−	0.179939i
$266$		0			0
$267$			25.1878i			1.54147i
$268$		0			0
$269$	−7.07081			−0.431115			−0.215557	−	0.976491i	$-0.569157\pi$
$269$	−7.07081			−0.431115			−0.215557	+	0.976491i	$0.569157\pi$
$270$		0			0
$271$	−17.0096			−1.03326			−0.516629	−	0.856209i	$-0.672813\pi$
$271$	−17.0096			−1.03326			−0.516629	+	0.856209i	$0.672813\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.89692	+	1.18785i	0.174691	+	0.0716298i
$276$		0			0
$277$		−	18.7755i		−	1.12811i	−0.825737	−	0.564056i	$-0.809240\pi$
$277$		−	18.7755i		−	1.12811i	0.825737	−	0.564056i	$-0.190760\pi$
$278$		0			0
$279$	0.775511			0.0464286
$280$		0			0
$281$	−4.59829			−0.274311			−0.137156	−	0.990550i	$-0.543796\pi$
$281$	−4.59829			−0.274311			−0.137156	+	0.990550i	$0.543796\pi$
$282$		0			0
$283$		−	13.5833i		−	0.807443i	−0.914882	−	0.403722i	$-0.867716\pi$
$283$		−	13.5833i		−	0.807443i	0.914882	−	0.403722i	$-0.132284\pi$
$284$		0			0
$285$	−24.6864	−	4.86464i	−1.46229	−	0.288156i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.1955			0.952679
$290$		0			0
$291$	−17.8786			−1.04806
$292$		0			0
$293$			11.8261i			0.690889i	0.938439	+	0.345444i	$0.112272\pi$
$293$			11.8261i			0.690889i	−0.938439	+	0.345444i	$0.887728\pi$
$294$		0			0
$295$	0.256765	−	1.30299i	0.0149494	−	0.0758632i
$296$		0			0
$297$		−	3.19554i		−	0.185424i
$298$		0			0
$299$	−20.4769			−1.18421
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	16.9817i		−	0.975572i
$304$		0			0
$305$	3.08476	−	15.6541i	0.176633	−	0.896351i
$306$		0			0
$307$			20.9860i			1.19774i	0.800847	+	0.598868i	$0.204383\pi$
$307$			20.9860i			1.19774i	−0.800847	+	0.598868i	$0.795617\pi$
$308$		0			0
$309$	1.10308			0.0627522
$310$		0			0
$311$	22.5048			1.27613			0.638065	−	0.769983i	$-0.279735\pi$
$311$	22.5048			1.27613			0.638065	+	0.769983i	$0.279735\pi$
$312$		0			0
$313$		−	12.4846i		−	0.705670i	−0.935686	−	0.352835i	$-0.885218\pi$
$313$		−	12.4846i		−	0.705670i	0.935686	−	0.352835i	$-0.114782\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.5231i			1.43352i	0.697319	+	0.716760i	$0.254376\pi$
$317$			25.5231i			1.43352i	−0.697319	+	0.716760i	$0.745624\pi$
$318$		0			0
$319$	−4.93356			−0.276226
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.72928i		−	0.318786i
$324$		0			0
$325$	10.4157	−	25.4017i	0.577757	−	1.40903i
$326$		0			0
$327$		−	33.3372i		−	1.84355i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.68305			−0.147474			−0.0737370	−	0.997278i	$-0.523493\pi$
$331$	−2.68305			−0.147474			−0.0737370	+	0.997278i	$0.523493\pi$
$332$		0			0
$333$		−	0.618502i		−	0.0338937i
$334$		0			0
$335$	12.7110	+	2.50479i	0.694474	+	0.136851i
$336$		0			0
$337$			12.5048i			0.681179i	0.940212	+	0.340590i	$0.110627\pi$
$337$			12.5048i			0.681179i	−0.940212	+	0.340590i	$0.889373\pi$
$338$		0			0
$339$	−1.84299			−0.100098
$340$		0			0
$341$	4.71096			0.255113
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.84006	−	14.4123i	0.152904	−	0.775934i
$346$		0			0
$347$		−	28.9450i		−	1.55385i	−0.629593	−	0.776925i	$-0.716778\pi$
$347$		−	28.9450i		−	1.55385i	0.629593	−	0.776925i	$-0.283222\pi$
$348$		0			0
$349$	32.1449			1.72068			0.860340	−	0.509721i	$-0.170251\pi$
$349$	32.1449			1.72068			0.860340	+	0.509721i	$0.170251\pi$
$350$		0			0
$351$	−28.0202			−1.49561
$352$		0			0
$353$			8.17722i			0.435229i	0.976035	+	0.217615i	$0.0698276\pi$
$353$			8.17722i			0.435229i	−0.976035	+	0.217615i	$0.930172\pi$
$354$		0			0
$355$	−2.38776	+	12.1170i	−0.126729	+	0.643106i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.5048			−0.976646			−0.488323	−	0.872663i	$-0.662391\pi$
$359$	−18.5048			−0.976646			−0.488323	+	0.872663i	$0.662391\pi$
$360$		0			0
$361$	21.8034			1.14755
$362$		0			0
$363$		−	18.6864i		−	0.980781i
$364$		0			0
$365$	8.18159	+	1.61224i	0.428244	+	0.0843887i
$366$		0			0
$367$			27.4942i			1.43518i	0.696464	+	0.717592i	$0.254756\pi$
$367$			27.4942i			1.43518i	−0.696464	+	0.717592i	$0.745244\pi$
$368$		0			0
$369$	0.796763			0.0414778
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	6.06455i		−	0.314011i	−0.987598	−	0.157005i	$-0.949816\pi$
$373$		−	6.06455i		−	0.314011i	0.987598	−	0.157005i	$-0.0501840\pi$
$374$		0			0
$375$	16.4340	+	10.8540i	0.848647	+	0.560499i
$376$		0			0
$377$			43.2601i			2.22801i
$378$		0			0
$379$	5.72928			0.294293			0.147147	−	0.989115i	$-0.452991\pi$
$379$	5.72928			0.294293			0.147147	+	0.989115i	$0.452991\pi$
$380$		0			0
$381$	37.4865			1.92049
$382$		0			0
$383$		−	1.72928i		−	0.0883622i	−0.999024	−	0.0441811i	$-0.985932\pi$
$383$		−	1.72928i		−	0.0883622i	0.999024	−	0.0441811i	$-0.0140679\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			0.178261i			0.00906150i
$388$		0			0
$389$	36.7187			1.86171			0.930855	−	0.365389i	$-0.119064\pi$
$389$	36.7187			1.86171			0.930855	+	0.365389i	$0.119064\pi$
$390$		0			0
$391$	3.34485			0.169157
$392$		0			0
$393$			17.4586i			0.880668i
$394$		0			0
$395$	−2.45231	+	12.4446i	−0.123389	+	0.626156i
$396$		0			0
$397$			2.03228i			0.101997i	0.998699	+	0.0509985i	$0.0162404\pi$
$397$			2.03228i			0.101997i	−0.998699	+	0.0509985i	$0.983760\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.64452			−0.0821234			−0.0410617	−	0.999157i	$-0.513074\pi$
$401$	−1.64452			−0.0821234			−0.0410617	+	0.999157i	$0.513074\pi$
$402$		0			0
$403$		−	41.3082i		−	2.05771i
$404$		0			0
$405$	4.01999	−	20.4000i	0.199755	−	1.01369i
$406$		0			0
$407$		−	3.75719i		−	0.186237i
$408$		0			0
$409$	−4.98168			−0.246328			−0.123164	−	0.992386i	$-0.539304\pi$
$409$	−4.98168			−0.246328			−0.123164	+	0.992386i	$0.539304\pi$
$410$		0			0
$411$	−6.09246			−0.300519
$412$		0			0
$413$		0			0
$414$		0			0
$415$	38.2480	+	7.53707i	1.87752	+	0.369980i
$416$		0			0
$417$		−	35.9634i		−	1.76113i
$418$		0			0
$419$	22.9205			1.11974			0.559869	−	0.828581i	$-0.310852\pi$
$419$	22.9205			1.11974			0.559869	+	0.828581i	$0.310852\pi$
$420$		0			0
$421$	−20.9527			−1.02117			−0.510587	−	0.859826i	$-0.670572\pi$
$421$	−20.9527			−1.02117			−0.510587	+	0.859826i	$0.670572\pi$
$422$		0			0
$423$			0.605987i			0.0294641i
$424$		0			0
$425$	−1.70138	+	4.14931i	−0.0825288	+	0.201271i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	6.05685			0.292428
$430$		0			0
$431$	33.1589			1.59721			0.798604	−	0.601857i	$-0.205572\pi$
$431$	33.1589			1.59721			0.798604	+	0.601857i	$0.205572\pi$
$432$		0			0
$433$		−	18.5414i		−	0.891045i	−0.895271	−	0.445522i	$-0.853018\pi$
$433$		−	18.5414i		−	0.891045i	0.895271	−	0.445522i	$-0.146982\pi$
$434$		0			0
$435$	−30.4479	−	6.00000i	−1.45987	−	0.287678i
$436$		0			0
$437$			23.8217i			1.13955i
$438$		0			0
$439$	−20.8401			−0.994642			−0.497321	−	0.867567i	$-0.665683\pi$
$439$	−20.8401			−0.994642			−0.497321	+	0.867567i	$0.665683\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.5510i		−	0.833874i	−0.908935	−	0.416937i	$-0.863104\pi$
$443$		−	17.5510i		−	0.833874i	0.908935	−	0.416937i	$-0.136896\pi$
$444$		0			0
$445$	−6.18159	+	31.3694i	−0.293035	+	1.48705i
$446$		0			0
$447$			30.0279i			1.42027i
$448$		0			0
$449$	13.1955			0.622736			0.311368	−	0.950289i	$-0.399213\pi$
$449$	13.1955			0.622736			0.311368	+	0.950289i	$0.399213\pi$
$450$		0			0
$451$	4.84006			0.227910
$452$		0			0
$453$			5.10308i			0.239764i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	29.1020i		−	1.36134i	−0.732592	−	0.680668i	$-0.761690\pi$
$457$		−	29.1020i		−	1.36134i	0.732592	−	0.680668i	$-0.238310\pi$
$458$		0			0
$459$	4.57704			0.213638
$460$		0			0
$461$	18.8280			0.876907			0.438454	−	0.898754i	$-0.355526\pi$
$461$	18.8280			0.876907			0.438454	+	0.898754i	$0.355526\pi$
$462$		0			0
$463$			14.0925i			0.654932i	0.944863	+	0.327466i	$0.106195\pi$
$463$			14.0925i			0.654932i	−0.944863	+	0.327466i	$0.893805\pi$
$464$		0			0
$465$	29.0741	+	5.72928i	1.34828	+	0.265689i
$466$		0			0
$467$			12.2663i			0.567619i	0.958881	+	0.283809i	$0.0915983\pi$
$467$			12.2663i			0.567619i	−0.958881	+	0.283809i	$0.908402\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	17.8217			0.821182
$472$		0			0
$473$			1.08287i			0.0497905i
$474$		0			0
$475$	−29.5510	−	12.1170i	−1.35589	−	0.555968i
$476$		0			0
$477$		−	0.698445i		−	0.0319796i
$478$		0			0
$479$	−14.1570			−0.646850			−0.323425	−	0.946254i	$-0.604834\pi$
$479$	−14.1570			−0.646850			−0.323425	+	0.946254i	$0.604834\pi$
$480$		0			0
$481$	−32.9450			−1.50216
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.2663	−	4.38776i	−1.01106	−	0.199238i
$486$		0			0
$487$			30.2341i			1.37004i	0.728526	+	0.685018i	$0.240206\pi$
$487$			30.2341i			1.37004i	−0.728526	+	0.685018i	$0.759794\pi$
$488$		0			0
$489$	0.840061			0.0379889
$490$		0			0
$491$	39.9065			1.80096			0.900478	−	0.434902i	$-0.143217\pi$
$491$	39.9065			1.80096			0.900478	+	0.434902i	$0.143217\pi$
$492$		0			0
$493$		−	7.06644i		−	0.318256i
$494$		0			0
$495$	−0.0279066	+	0.141617i	−0.00125431	+	0.00636519i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−27.3246			−1.22322			−0.611610	−	0.791160i	$-0.709478\pi$
$499$	−27.3246			−1.22322			−0.611610	+	0.791160i	$0.709478\pi$
$500$		0			0
$501$	−23.1955			−1.03630
$502$		0			0
$503$			1.40171i			0.0624991i	0.999512	+	0.0312495i	$0.00994866\pi$
$503$			1.40171i			0.0624991i	−0.999512	+	0.0312495i	$0.990051\pi$
$504$		0			0
$505$	4.16763	−	21.1493i	0.185457	−	0.941132i
$506$		0			0
$507$		−	30.2095i		−	1.34165i
$508$		0			0
$509$	18.6585			0.827022			0.413511	−	0.910499i	$-0.364302\pi$
$509$	18.6585			0.827022			0.413511	+	0.910499i	$0.364302\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			32.5972i			1.43920i
$514$		0			0
$515$	1.37380	+	0.270718i	0.0605369	+	0.0119293i
$516$		0			0
$517$			3.68116i			0.161897i
$518$		0			0
$519$	−24.6049			−1.08004
$520$		0			0
$521$	4.02791			0.176466			0.0882329	−	0.996100i	$-0.471878\pi$
$521$	4.02791			0.176466			0.0882329	+	0.996100i	$0.471878\pi$
$522$		0			0
$523$			38.4436i			1.68102i	0.541796	+	0.840510i	$0.317744\pi$
$523$			38.4436i			1.68102i	−0.541796	+	0.840510i	$0.682256\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			6.74760i			0.293930i
$528$		0			0
$529$	9.09246			0.395324
$530$		0			0
$531$	0.0612237			0.00265688
$532$		0			0
$533$		−	42.4402i		−	1.83829i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	13.1387i		−	0.566976i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.4846			0.536754			0.268377	−	0.963314i	$-0.413513\pi$
$541$	12.4846			0.536754			0.268377	+	0.963314i	$0.413513\pi$
$542$		0			0
$543$			24.8680i			1.06719i
$544$		0			0
$545$	8.18159	−	41.5187i	0.350461	−	1.77847i
$546$		0			0
$547$			37.7851i			1.61557i	0.589474	+	0.807787i	$0.299335\pi$
$547$			37.7851i			1.61557i	−0.589474	+	0.807787i	$0.700665\pi$
$548$		0			0
$549$	0.735539			0.0313921
$550$		0			0
$551$	50.3265			2.14398
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.56934	−	23.1878i	0.193958	−	0.984269i
$556$		0			0
$557$			1.93545i			0.0820076i	0.999159	+	0.0410038i	$0.0130556\pi$
$557$			1.93545i			0.0820076i	−0.999159	+	0.0410038i	$0.986944\pi$
$558$		0			0
$559$	9.49521			0.401604
$560$		0			0
$561$	−0.989374			−0.0417714
$562$		0			0
$563$		−	29.8463i		−	1.25787i	−0.777457	−	0.628936i	$-0.783491\pi$
$563$		−	29.8463i		−	1.25787i	0.777457	−	0.628936i	$-0.216509\pi$
$564$		0			0
$565$	−2.29530	−	0.452306i	−0.0965639	−	0.0190287i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.3449			−0.643290			−0.321645	−	0.946860i	$-0.604236\pi$
$569$	−15.3449			−0.643290			−0.321645	+	0.946860i	$0.604236\pi$
$570$		0			0
$571$	35.8217			1.49909			0.749547	−	0.661952i	$-0.230272\pi$
$571$	35.8217			1.49909			0.749547	+	0.661952i	$0.230272\pi$
$572$		0			0
$573$		−	14.8324i		−	0.619631i
$574$		0			0
$575$	7.07414	−	17.2524i	0.295012	−	0.719475i
$576$		0			0
$577$		−	2.92482i		−	0.121762i	−0.998145	−	0.0608810i	$-0.980609\pi$
$577$		−	2.92482i		−	0.121762i	0.998145	−	0.0608810i	$-0.0193910\pi$
$578$		0			0
$579$	−39.2803			−1.63243
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	4.24281i		−	0.175719i
$584$		0			0
$585$	1.24177	+	0.244700i	0.0513409	+	0.0101171i
$586$		0			0
$587$		−	19.5756i		−	0.807972i	−0.914765	−	0.403986i	$-0.867625\pi$
$587$		−	19.5756i		−	0.807972i	0.914765	−	0.403986i	$-0.132375\pi$
$588$		0			0
$589$	−48.0558			−1.98011
$590$		0			0
$591$	−46.4402			−1.91030
$592$		0			0
$593$			8.00770i			0.328837i	0.986391	+	0.164418i	$0.0525747\pi$
$593$			8.00770i			0.328837i	−0.986391	+	0.164418i	$0.947425\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			39.5298i			1.61785i
$598$		0			0
$599$	29.3005			1.19719			0.598593	−	0.801053i	$-0.295727\pi$
$599$	29.3005			1.19719			0.598593	+	0.801053i	$0.295727\pi$
$600$		0			0
$601$	−21.3449			−0.870675			−0.435337	−	0.900267i	$-0.643371\pi$
$601$	−21.3449			−0.870675			−0.435337	+	0.900267i	$0.643371\pi$
$602$		0			0
$603$			0.597250i			0.0243219i
$604$		0			0
$605$	4.58600	−	23.2724i	0.186447	−	0.946157i
$606$		0			0
$607$		−	9.53081i		−	0.386844i	−0.981116	−	0.193422i	$-0.938041\pi$
$607$		−	9.53081i		−	0.386844i	0.981116	−	0.193422i	$-0.0619586\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.2784			1.30584
$612$		0			0
$613$			9.75719i			0.394089i	0.980395	+	0.197045i	$0.0631344\pi$
$613$			9.75719i			0.394089i	−0.980395	+	0.197045i	$0.936866\pi$
$614$		0			0
$615$	29.8709	+	5.88629i	1.20451	+	0.237358i
$616$		0			0
$617$		−	37.9267i		−	1.52687i	−0.645884	−	0.763436i	$-0.723511\pi$
$617$		−	37.9267i		−	1.52687i	0.645884	−	0.763436i	$-0.276489\pi$
$618$		0			0
$619$	41.9109			1.68454			0.842270	−	0.539056i	$-0.181219\pi$
$619$	41.9109			1.68454			0.842270	+	0.539056i	$0.181219\pi$
$620$		0			0
$621$	−19.0308			−0.763681
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.8034	+	17.5510i	0.712137	+	0.702041i
$626$		0			0
$627$		−	7.04623i		−	0.281399i
$628$		0			0
$629$	5.38150			0.214574
$630$		0			0
$631$	−8.42003			−0.335196			−0.167598	−	0.985855i	$-0.553601\pi$
$631$	−8.42003			−0.335196			−0.167598	+	0.985855i	$0.553601\pi$
$632$		0			0
$633$		−	26.7678i		−	1.06393i
$634$		0			0
$635$	46.6864	+	9.19991i	1.85269	+	0.365087i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.569343			−0.0225229
$640$		0			0
$641$	−2.07707			−0.0820392			−0.0410196	−	0.999158i	$-0.513061\pi$
$641$	−2.07707			−0.0820392			−0.0410196	+	0.999158i	$0.513061\pi$
$642$		0			0
$643$		−	27.2480i		−	1.07456i	−0.843405	−	0.537279i	$-0.819452\pi$
$643$		−	27.2480i		−	1.07456i	0.843405	−	0.537279i	$-0.180548\pi$
$644$		0			0
$645$	−1.31695	+	6.68305i	−0.0518547	+	0.263145i
$646$		0			0
$647$		−	12.3632i		−	0.486047i	−0.970020	−	0.243023i	$-0.921861\pi$
$647$		−	12.3632i		−	0.486047i	0.970020	−	0.243023i	$-0.0781391\pi$
$648$		0			0
$649$	0.371913			0.0145989
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	37.3449i		−	1.46142i	−0.682690	−	0.730709i	$-0.739190\pi$
$653$		−	37.3449i		−	1.46142i	0.682690	−	0.730709i	$-0.260810\pi$
$654$		0			0
$655$	−4.28467	+	21.7432i	−0.167416	+	0.849578i
$656$		0			0
$657$			0.384428i			0.0149980i
$658$		0			0
$659$	14.1772			0.552266			0.276133	−	0.961119i	$-0.410947\pi$
$659$	14.1772			0.552266			0.276133	+	0.961119i	$0.410947\pi$
$660$		0			0
$661$	1.76925			0.0688160			0.0344080	−	0.999408i	$-0.489045\pi$
$661$	1.76925			0.0688160			0.0344080	+	0.999408i	$0.489045\pi$
$662$		0			0
$663$			8.67536i			0.336923i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			29.3815i			1.13766i
$668$		0			0
$669$	12.1493			0.469720
$670$		0			0
$671$	4.46815			0.172491
$672$		0			0
$673$		−	4.05581i		−	0.156340i	−0.996940	−	0.0781701i	$-0.975092\pi$
$673$		−	4.05581i		−	0.156340i	0.996940	−	0.0781701i	$-0.0249077\pi$
$674$		0			0
$675$	9.68012	−	23.6079i	0.372588	−	0.908668i
$676$		0			0
$677$			9.37713i			0.360392i	0.983631	+	0.180196i	$0.0576733\pi$
$677$			9.37713i			0.360392i	−0.983631	+	0.180196i	$0.942327\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−4.89692			−0.187650
$682$		0			0
$683$			5.49521i			0.210268i	0.994458	+	0.105134i	$0.0335272\pi$
$683$			5.49521i			0.210268i	−0.994458	+	0.105134i	$0.966473\pi$
$684$		0			0
$685$	−7.58767	−	1.49521i	−0.289910	−	0.0571290i
$686$		0			0
$687$			32.8680i			1.25399i
$688$		0			0
$689$	−37.2032			−1.41733
$690$		0			0
$691$	3.03416			0.115425			0.0577125	−	0.998333i	$-0.481619\pi$
$691$	3.03416			0.115425			0.0577125	+	0.998333i	$0.481619\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.82611	−	44.7895i	0.334793	−	1.69896i
$696$		0			0
$697$			6.93252i			0.262588i
$698$		0			0
$699$	−19.4586			−0.735990
$700$		0			0
$701$	35.2234			1.33037			0.665186	−	0.746678i	$-0.268352\pi$
$701$	35.2234			1.33037			0.665186	+	0.746678i	$0.268352\pi$
$702$		0			0
$703$			38.3265i			1.44551i
$704$		0			0
$705$	−4.47689	+	22.7187i	−0.168609	+	0.855634i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−28.7187			−1.07855			−0.539276	−	0.842129i	$-0.681302\pi$
$709$	−28.7187			−1.07855			−0.539276	+	0.842129i	$0.681302\pi$
$710$		0			0
$711$	−0.584735			−0.0219293
$712$		0			0
$713$		−	28.0558i		−	1.05070i
$714$		0			0
$715$	7.54333	+	1.48647i	0.282104	+	0.0555908i
$716$		0			0
$717$			35.4942i			1.32555i
$718$		0			0
$719$	−8.60599			−0.320949			−0.160475	−	0.987040i	$-0.551302\pi$
$719$	−8.60599			−0.320949			−0.160475	+	0.987040i	$0.551302\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	6.56934i		−	0.244316i
$724$		0			0
$725$	−36.4479	−	14.9450i	−1.35364	−	0.555045i
$726$		0			0
$727$			15.2803i			0.566715i	0.959014	+	0.283358i	$0.0914483\pi$
$727$			15.2803i			0.566715i	−0.959014	+	0.283358i	$0.908552\pi$
$728$		0			0
$729$	−26.0096			−0.963318
$730$		0			0
$731$	−1.55102			−0.0573666
$732$		0			0
$733$		−	46.0235i		−	1.69992i	−0.526849	−	0.849959i	$-0.676627\pi$
$733$		−	46.0235i		−	1.69992i	0.526849	−	0.849959i	$-0.323373\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.62809i			0.133642i
$738$		0			0
$739$	−46.3467			−1.70489			−0.852446	−	0.522815i	$-0.824882\pi$
$739$	−46.3467			−1.70489			−0.852446	+	0.522815i	$0.824882\pi$
$740$		0			0
$741$	−61.7851			−2.26973
$742$		0			0
$743$		−	10.7755i		−	0.395315i	−0.980271	−	0.197658i	$-0.936667\pi$
$743$		−	10.7755i		−	0.395315i	0.980271	−	0.197658i	$-0.0633334\pi$
$744$		0			0
$745$	−7.36943	+	37.3973i	−0.269995	+	1.37013i
$746$		0			0
$747$			1.79716i			0.0657546i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.76781			0.100999			0.0504995	−	0.998724i	$-0.483919\pi$
$751$	2.76781			0.100999			0.0504995	+	0.998724i	$0.483919\pi$
$752$		0			0
$753$			37.8709i			1.38009i
$754$		0			0
$755$	−1.25240	+	6.35548i	−0.0455794	+	0.231300i
$756$		0			0
$757$			14.8401i			0.539371i	0.962948	+	0.269686i	$0.0869198\pi$
$757$			14.8401i			0.539371i	−0.962948	+	0.269686i	$0.913080\pi$
$758$		0			0
$759$	4.11371			0.149318
$760$		0			0
$761$	−31.3169			−1.13524			−0.567619	−	0.823291i	$-0.692135\pi$
$761$	−31.3169			−1.13524			−0.567619	+	0.823291i	$0.692135\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.202840	−	0.0399712i	−0.00733371	−	0.00144516i
$766$		0			0
$767$		−	3.26113i		−	0.117753i
$768$		0			0
$769$	8.92586			0.321875			0.160937	−	0.986965i	$-0.448548\pi$
$769$	8.92586			0.321875			0.160937	+	0.986965i	$0.448548\pi$
$770$		0			0
$771$	7.52311			0.270938
$772$		0			0
$773$			36.7711i			1.32257i	0.750136	+	0.661283i	$0.229988\pi$
$773$			36.7711i			1.32257i	−0.750136	+	0.661283i	$0.770012\pi$
$774$		0			0
$775$	34.8034	+	14.2707i	1.25018	+	0.512619i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−49.3728			−1.76896
$780$		0			0
$781$	−3.45856			−0.123757
$782$		0			0
$783$			40.2051i			1.43681i
$784$		0			0
$785$	22.1955	+	4.37380i	0.792193	+	0.156108i
$786$		0			0
$787$		−	6.53707i		−	0.233021i	−0.993189	−	0.116511i	$-0.962829\pi$
$787$		−	6.53707i		−	0.233021i	0.993189	−	0.116511i	$-0.0371709\pi$
$788$		0			0
$789$	−20.1849			−0.718602
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	39.1791i		−	1.39129i
$794$		0			0
$795$	5.15994	−	26.1849i	0.183004	−	0.928683i
$796$		0			0
$797$		−	7.82611i		−	0.277215i	−0.990347	−	0.138607i	$-0.955737\pi$
$797$		−	7.82611i		−	0.277215i	0.990347	−	0.138607i	$-0.0442626\pi$
$798$		0			0
$799$	−5.27261			−0.186531
$800$		0			0
$801$	−1.47396			−0.0520796
$802$		0			0
$803$			2.33527i			0.0824099i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	12.4556i		−	0.438459i
$808$		0			0
$809$	−38.4113			−1.35047			−0.675235	−	0.737603i	$-0.735958\pi$
$809$	−38.4113			−1.35047			−0.675235	+	0.737603i	$0.735958\pi$
$810$		0			0
$811$	−7.64015			−0.268282			−0.134141	−	0.990962i	$-0.542828\pi$
$811$	−7.64015			−0.268282			−0.134141	+	0.990962i	$0.542828\pi$
$812$		0			0
$813$		−	29.9634i		−	1.05086i
$814$		0			0
$815$	1.04623	+	0.206167i	0.0366478	+	0.00722173i
$816$		0			0
$817$		−	11.0462i		−	0.386459i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.56165			0.0894021			0.0447011	−	0.999000i	$-0.485766\pi$
$821$	2.56165			0.0894021			0.0447011	+	0.999000i	$0.485766\pi$
$822$		0			0
$823$		−	0.784248i		−	0.0273372i	−0.999907	−	0.0136686i	$-0.995649\pi$
$823$		−	0.784248i		−	0.0273372i	0.999907	−	0.0136686i	$-0.00435098\pi$
$824$		0			0
$825$	−2.09246	+	5.10308i	−0.0728500	+	0.177667i
$826$		0			0
$827$			7.28904i			0.253465i	0.991937	+	0.126732i	$0.0404489\pi$
$827$			7.28904i			0.253465i	−0.991937	+	0.126732i	$0.959551\pi$
$828$		0			0
$829$	−11.0708			−0.384505			−0.192253	−	0.981345i	$-0.561579\pi$
$829$	−11.0708			−0.384505			−0.192253	+	0.981345i	$0.561579\pi$
$830$		0			0
$831$	33.0741			1.14733
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−28.8882	−	5.69264i	−0.999717	−	0.197002i
$836$		0			0
$837$		−	38.3911i		−	1.32699i
$838$		0			0
$839$	−5.55976			−0.191944			−0.0959721	−	0.995384i	$-0.530596\pi$
$839$	−5.55976			−0.191944			−0.0959721	+	0.995384i	$0.530596\pi$
$840$		0			0
$841$	33.0722			1.14042
$842$		0			0
$843$		−	8.10015i		−	0.278984i
$844$		0			0
$845$	7.41400	−	37.6235i	0.255049	−	1.29429i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	23.9278			0.821198
$850$		0			0
$851$	−22.3757			−0.767029
$852$		0			0
$853$			16.0804i			0.550582i	0.961361	+	0.275291i	$0.0887742\pi$
$853$			16.0804i			0.550582i	−0.961361	+	0.275291i	$0.911226\pi$
$854$		0			0
$855$	0.284672	−	1.44461i	0.00973556	−	0.0494046i
$856$		0			0
$857$		−	1.58767i		−	0.0542336i	−0.999632	−	0.0271168i	$-0.991367\pi$
$857$		−	1.58767i		−	0.0542336i	0.999632	−	0.0271168i	$-0.00863261\pi$
$858$		0			0
$859$	4.94171			0.168609			0.0843044	−	0.996440i	$-0.473133\pi$
$859$	4.94171			0.168609			0.0843044	+	0.996440i	$0.473133\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			37.6647i			1.28212i	0.767490	+	0.641061i	$0.221506\pi$
$863$			37.6647i			1.28212i	−0.767490	+	0.641061i	$0.778494\pi$
$864$		0			0
$865$	−30.6435	−	6.03853i	−1.04191	−	0.205316i
$866$		0			0
$867$			28.5294i			0.968908i
$868$		0			0
$869$	−3.55206			−0.120495
$870$		0			0
$871$	31.8130			1.07794
$872$		0			0
$873$		−	1.04623i		−	0.0354095i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	3.66473i		−	0.123749i	−0.998084	−	0.0618746i	$-0.980292\pi$
$877$		−	3.66473i		−	0.123749i	0.998084	−	0.0618746i	$-0.0197079\pi$
$878$		0			0
$879$	−20.8324			−0.702658
$880$		0			0
$881$	43.4373			1.46344			0.731720	−	0.681605i	$-0.238718\pi$
$881$	43.4373			1.46344			0.731720	+	0.681605i	$0.238718\pi$
$882$		0			0
$883$			47.3853i			1.59464i	0.603556	+	0.797321i	$0.293750\pi$
$883$			47.3853i			1.59464i	−0.603556	+	0.797321i	$0.706250\pi$
$884$		0			0
$885$	2.29530	+	0.452306i	0.0771556	+	0.0152041i
$886$		0			0
$887$		−	21.3169i		−	0.715753i	−0.933769	−	0.357877i	$-0.883501\pi$
$887$		−	21.3169i		−	0.715753i	0.933769	−	0.357877i	$-0.116499\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	5.82278			0.195071
$892$		0			0
$893$		−	37.5510i		−	1.25660i
$894$		0			0
$895$	3.22449	−	16.3632i	0.107783	−	0.546961i
$896$		0			0
$897$		−	36.0712i		−	1.20438i
$898$		0			0
$899$	−59.2716			−1.97682
$900$		0			0
$901$	6.07707			0.202456
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.10308	+	30.9711i	−0.202873	+	1.02951i
$906$		0			0
$907$		−	18.9325i		−	0.628644i	−0.949316	−	0.314322i	$-0.898223\pi$
$907$		−	18.9325i		−	0.628644i	0.949316	−	0.314322i	$-0.101777\pi$
$908$		0			0
$909$	0.993743			0.0329604
$910$		0			0
$911$	−12.6705			−0.419794			−0.209897	−	0.977724i	$-0.567313\pi$
$911$	−12.6705			−0.419794			−0.209897	+	0.977724i	$0.567313\pi$
$912$		0			0
$913$			10.9171i			0.361304i
$914$		0			0
$915$	27.5756	+	5.43398i	0.911621	+	0.179642i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−22.8690			−0.754379			−0.377190	−	0.926136i	$-0.623109\pi$
$919$	−22.8690			−0.754379			−0.377190	+	0.926136i	$0.623109\pi$
$920$		0			0
$921$	−36.9681			−1.21814
$922$		0			0
$923$			30.3265i			0.998210i
$924$		0			0
$925$	11.3815	−	27.7572i	0.374221	−	0.912651i
$926$		0			0
$927$			0.0645508i			0.00212013i
$928$		0			0
$929$	50.4190			1.65419			0.827097	−	0.562060i	$-0.189991\pi$
$929$	50.4190			1.65419			0.827097	+	0.562060i	$0.189991\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			39.6435i			1.29787i
$934$		0			0
$935$	−1.23219	−	0.242812i	−0.0402968	−	0.00794079i
$936$		0			0
$937$			47.9344i			1.56595i	0.622054	+	0.782974i	$0.286298\pi$
$937$			47.9344i			1.56595i	−0.622054	+	0.782974i	$0.713702\pi$
$938$		0			0
$939$	21.9923			0.717692
$940$		0			0
$941$	−59.9667			−1.95486			−0.977429	−	0.211264i	$-0.932242\pi$
$941$	−59.9667			−1.95486			−0.977429	+	0.211264i	$0.932242\pi$
$942$		0			0
$943$		−	28.8247i		−	0.938660i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	27.5231i		−	0.894381i	−0.894439	−	0.447191i	$-0.852425\pi$
$947$		−	27.5231i		−	0.894381i	0.894439	−	0.447191i	$-0.147575\pi$
$948$		0			0
$949$	20.4769			0.664708
$950$		0			0
$951$	−44.9604			−1.45794
$952$		0			0
$953$		−	0.840061i		−	0.0272123i	−0.999907	−	0.0136061i	$-0.995669\pi$
$953$		−	0.840061i		−	0.0272123i	0.999907	−	0.0136061i	$-0.00433110\pi$
$954$		0			0
$955$	3.64015	−	18.4725i	0.117793	−	0.597757i
$956$		0			0
$957$		−	8.69075i		−	0.280932i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	25.5972			0.825718
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−48.9205	−	9.64015i	−1.57481	−	0.310327i
$966$		0			0
$967$			31.8988i			1.02580i	0.858449	+	0.512898i	$0.171428\pi$
$967$			31.8988i			1.02580i	−0.858449	+	0.512898i	$0.828572\pi$
$968$		0			0
$969$	10.0925			0.324216
$970$		0			0
$971$	−28.4157			−0.911902			−0.455951	−	0.890005i	$-0.650701\pi$
$971$	−28.4157			−0.911902			−0.455951	+	0.890005i	$0.650701\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	44.7466	+	18.3478i	1.43304	+	0.587599i
$976$		0			0
$977$			9.55102i			0.305564i	0.988260	+	0.152782i	$0.0488233\pi$
$977$			9.55102i			0.305564i	−0.988260	+	0.152782i	$0.951177\pi$
$978$		0			0
$979$	−8.95377			−0.286164
$980$		0			0
$981$	1.95084			0.0622856
$982$		0			0
$983$		−	0.0443400i		−	0.00141423i	−1.00000		0.000707113i	$-0.999775\pi$
$983$		−	0.0443400i		−	0.00141423i	1.00000		0.000707113i	$-0.000225081\pi$
$984$		0			0
$985$	−57.8376	−	11.3973i	−1.84286	−	0.363149i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.44898			0.205066
$990$		0			0
$991$	−13.9913			−0.444447			−0.222224	−	0.974996i	$-0.571331\pi$
$991$	−13.9913			−0.444447			−0.222224	+	0.974996i	$0.571331\pi$
$992$		0			0
$993$		−	4.72635i		−	0.149986i
$994$		0			0
$995$	−9.70138	+	49.2311i	−0.307554	+	1.56073i
$996$		0			0
$997$			19.0419i			0.603062i	0.953456	+	0.301531i	$0.0974976\pi$
$997$			19.0419i			0.603062i	−0.953456	+	0.301531i	$0.902502\pi$
$998$		0			0
$999$	−30.6185			−0.968727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.c.1569.5		6
5.2	odd	4		9800.2.a.cd.1.3		3
5.3	odd	4		9800.2.a.cg.1.1		3
5.4	even	2	inner	1960.2.g.c.1569.2		6
7.6	odd	2		280.2.g.b.169.2	✓	6
21.20	even	2		2520.2.t.g.1009.4		6
28.27	even	2		560.2.g.f.449.5		6
35.13	even	4		1400.2.a.s.1.3		3
35.27	even	4		1400.2.a.t.1.1		3
35.34	odd	2		280.2.g.b.169.5	yes	6
56.13	odd	2		2240.2.g.l.449.5		6
56.27	even	2		2240.2.g.m.449.2		6
84.83	odd	2		5040.2.t.y.1009.4		6
105.104	even	2		2520.2.t.g.1009.3		6
140.27	odd	4		2800.2.a.bq.1.3		3
140.83	odd	4		2800.2.a.br.1.1		3
140.139	even	2		560.2.g.f.449.2		6
280.69	odd	2		2240.2.g.l.449.2		6
280.139	even	2		2240.2.g.m.449.5		6
420.419	odd	2		5040.2.t.y.1009.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.g.b.169.2	✓	6	7.6	odd	2
280.2.g.b.169.5	yes	6	35.34	odd	2
560.2.g.f.449.2		6	140.139	even	2
560.2.g.f.449.5		6	28.27	even	2
1400.2.a.s.1.3		3	35.13	even	4
1400.2.a.t.1.1		3	35.27	even	4
1960.2.g.c.1569.2		6	5.4	even	2	inner
1960.2.g.c.1569.5		6	1.1	even	1	trivial
2240.2.g.l.449.2		6	280.69	odd	2
2240.2.g.l.449.5		6	56.13	odd	2
2240.2.g.m.449.2		6	56.27	even	2
2240.2.g.m.449.5		6	280.139	even	2
2520.2.t.g.1009.3		6	105.104	even	2
2520.2.t.g.1009.4		6	21.20	even	2
2800.2.a.bq.1.3		3	140.27	odd	4
2800.2.a.br.1.1		3	140.83	odd	4
5040.2.t.y.1009.3		6	420.419	odd	2
5040.2.t.y.1009.4		6	84.83	odd	2
9800.2.a.cd.1.3		3	5.2	odd	4
9800.2.a.cg.1.1		3	5.3	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.76156i q^{3} +(-0.432320 + 2.19388i) q^{5} -0.103084 q^{9} +O(q^{10})\)	\(q+1.76156i q^{3} +(-0.432320 + 2.19388i) q^{5} -0.103084 q^{9} -0.626198 q^{11} +5.49084i q^{13} +(-3.86464 - 0.761557i) q^{15} -0.896916i q^{17} +6.38776 q^{19} +3.72928i q^{23} +(-4.62620 - 1.89692i) q^{25} +5.10308i q^{27} +7.87859 q^{29} -7.52311 q^{31} -1.10308i q^{33} +6.00000i q^{37} -9.67243 q^{39} -7.72928 q^{41} -1.72928i q^{43} +(0.0445652 - 0.226153i) q^{45} -5.87859i q^{47} +1.57997 q^{51} +6.77551i q^{53} +(0.270718 - 1.37380i) q^{55} +11.2524i q^{57} -0.593923 q^{59} -7.13536 q^{61} +(-12.0462 - 2.37380i) q^{65} -5.79383i q^{67} -6.56934 q^{69} +5.52311 q^{71} -3.72928i q^{73} +(3.34153 - 8.14931i) q^{75} +5.67243 q^{79} -9.29862 q^{81} -17.4340i q^{83} +(1.96772 + 0.387755i) q^{85} +13.8786i q^{87} +14.2986 q^{89} -13.2524i q^{93} +(-2.76156 + 14.0140i) q^{95} +10.1493i q^{97} +0.0645508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} + 14 q^{11} - 18 q^{15} + 8 q^{19} - 10 q^{25} - 6 q^{29} - 20 q^{31} + 10 q^{39} - 36 q^{41} + 28 q^{45} + 42 q^{51} + 12 q^{55} + 12 q^{59} - 48 q^{61} - 22 q^{65} + 36 q^{69} + 8 q^{71} + 40 q^{75} - 34 q^{79} + 30 q^{81} + 14 q^{85} - 4 q^{95} - 4 q^{99}+O(q^{100}) \) 6 * q - 8 * q^9 + 14 * q^11 - 18 * q^15 + 8 * q^19 - 10 * q^25 - 6 * q^29 - 20 * q^31 + 10 * q^39 - 36 * q^41 + 28 * q^45 + 42 * q^51 + 12 * q^55 + 12 * q^59 - 48 * q^61 - 22 * q^65 + 36 * q^69 + 8 * q^71 + 40 * q^75 - 34 * q^79 + 30 * q^81 + 14 * q^85 - 4 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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