LMFDB - Embedded newform 1380.2.i.b.1241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1241,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, 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("1380.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.i (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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 $ x^16 - x^14 - 4x^13 - 2x^12 + 12x^11 + 17x^10 + 16x^9 - 110x^8 + 48x^7 + 153x^6 + 324x^5 - 162x^4 - 972x^3 - 729x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.1
Root			$1.68259 + 0.410981i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.b.1241.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.68259 - 0.410981i) q^{3} +1.00000 q^{5} +4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} +O(q^{10})$	\(q+(-1.68259 - 0.410981i) q^{3} +1.00000 q^{5} +4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} +1.77042 q^{11} +4.14699 q^{13} +(-1.68259 - 0.410981i) q^{15} -6.86944 q^{17} +1.46552i q^{19} +(1.75045 - 7.16646i) q^{21} +(1.73005 + 4.47291i) q^{23} +1.00000 q^{25} +(-3.91096 - 3.42116i) q^{27} -4.10332i q^{29} -5.05868 q^{31} +(-2.97888 - 0.727608i) q^{33} +4.25920i q^{35} +2.88245i q^{37} +(-6.97767 - 1.70434i) q^{39} -4.56378i q^{41} +3.27999i q^{43} +(2.66219 + 1.38302i) q^{45} +5.80230i q^{47} -11.1408 q^{49} +(11.5584 + 2.82321i) q^{51} -0.831510 q^{53} +1.77042 q^{55} +(0.602301 - 2.46586i) q^{57} -11.8329i q^{59} +12.0164i q^{61} +(-5.89057 + 11.3388i) q^{63} +4.14699 q^{65} +10.3996i q^{67} +(-1.07268 - 8.23707i) q^{69} +16.3046i q^{71} -1.69014 q^{73} +(-1.68259 - 0.410981i) q^{75} +7.54056i q^{77} +2.62553i q^{79} +(5.17450 + 7.36373i) q^{81} +5.22163 q^{83} -6.86944 q^{85} +(-1.68639 + 6.90419i) q^{87} +9.78113 q^{89} +17.6629i q^{91} +(8.51167 + 2.07902i) q^{93} +1.46552i q^{95} -9.26880i q^{97} +(4.71318 + 2.44853i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} + 2 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 + 2 * q^9	$ 16 q + 16 q^{5} + 2 q^{9} + 4 q^{21} - 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} - 2 q^{33} - 14 q^{39} + 2 q^{45} + 8 q^{49} + 2 q^{51} - 4 q^{53} + 2 q^{57} - 30 q^{63} + 8 q^{69} - 4 q^{73} + 10 q^{81} + 4 q^{83} - 10 q^{87} + 28 q^{89} - 10 q^{93} + 26 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 + 2 * q^9 + 4 * q^21 - 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 - 2 * q^33 - 14 * q^39 + 2 * q^45 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 2 * q^57 - 30 * q^63 + 8 * q^69 - 4 * q^73 + 10 * q^81 + 4 * q^83 - 10 * q^87 + 28 * q^89 - 10 * q^93 + 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−1.68259	−	0.410981i	−0.971441	−	0.237280i
$3$	−1.68259	−	0.410981i	−0.971441	−	0.237280i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			4.25920i			1.60983i	0.593393	+	0.804913i	$0.297788\pi$
$7$			4.25920i			1.60983i	−0.593393	+	0.804913i	$0.702212\pi$
$8$		0			0
$9$	2.66219	+	1.38302i	0.887396	+	0.461008i
$10$		0			0
$11$	1.77042			0.533801			0.266900	−	0.963724i	$-0.414001\pi$
$11$	1.77042			0.533801			0.266900	+	0.963724i	$0.414001\pi$
$12$		0			0
$13$	4.14699			1.15017			0.575085	−	0.818094i	$-0.304969\pi$
$13$	4.14699			1.15017			0.575085	+	0.818094i	$0.304969\pi$
$14$		0			0
$15$	−1.68259	−	0.410981i	−0.434442	−	0.106115i
$16$		0			0
$17$	−6.86944			−1.66609			−0.833043	−	0.553209i	$-0.813403\pi$
$17$	−6.86944			−1.66609			−0.833043	+	0.553209i	$0.813403\pi$
$18$		0			0
$19$			1.46552i			0.336213i	0.985769	+	0.168106i	$0.0537652\pi$
$19$			1.46552i			0.336213i	−0.985769	+	0.168106i	$0.946235\pi$
$20$		0			0
$21$	1.75045	−	7.16646i	0.381980	−	1.56385i
$22$		0			0
$23$	1.73005	+	4.47291i	0.360741	+	0.932666i
$23$	1.73005	+	4.47291i	0.360741	+	0.932666i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.91096	−	3.42116i	−0.752665	−	0.658403i
$28$		0			0
$29$		−	4.10332i		−	0.761968i	−0.924582	−	0.380984i	$-0.875585\pi$
$29$		−	4.10332i		−	0.761968i	0.924582	−	0.380984i	$-0.124415\pi$
$30$		0			0
$31$	−5.05868			−0.908566			−0.454283	−	0.890857i	$-0.650105\pi$
$31$	−5.05868			−0.908566			−0.454283	+	0.890857i	$0.650105\pi$
$32$		0			0
$33$	−2.97888	−	0.727608i	−0.518556	−	0.126660i
$34$		0			0
$35$			4.25920i			0.719936i
$36$		0			0
$37$			2.88245i			0.473873i	0.971525	+	0.236936i	$0.0761433\pi$
$37$			2.88245i			0.473873i	−0.971525	+	0.236936i	$0.923857\pi$
$38$		0			0
$39$	−6.97767	−	1.70434i	−1.11732	−	0.272912i
$40$		0			0
$41$		−	4.56378i		−	0.712742i	−0.934345	−	0.356371i	$-0.884014\pi$
$41$		−	4.56378i		−	0.712742i	0.934345	−	0.356371i	$-0.115986\pi$
$42$		0			0
$43$			3.27999i			0.500193i	0.968221	+	0.250097i	$0.0804624\pi$
$43$			3.27999i			0.500193i	−0.968221	+	0.250097i	$0.919538\pi$
$44$		0			0
$45$	2.66219	+	1.38302i	0.396856	+	0.206169i
$46$		0			0
$47$			5.80230i			0.846353i	0.906047	+	0.423177i	$0.139085\pi$
$47$			5.80230i			0.846353i	−0.906047	+	0.423177i	$0.860915\pi$
$48$		0			0
$49$	−11.1408			−1.59154
$50$		0			0
$51$	11.5584	+	2.82321i	1.61850	+	0.395329i
$52$		0			0
$53$	−0.831510			−0.114217			−0.0571083	−	0.998368i	$-0.518188\pi$
$53$	−0.831510			−0.114217			−0.0571083	+	0.998368i	$0.518188\pi$
$54$		0			0
$55$	1.77042			0.238723
$56$		0			0
$57$	0.602301	−	2.46586i	0.0797767	−	0.326611i
$58$		0			0
$59$		−	11.8329i		−	1.54052i	−0.637732	−	0.770258i	$-0.720127\pi$
$59$		−	11.8329i		−	1.54052i	0.637732	−	0.770258i	$-0.279873\pi$
$60$		0			0
$61$			12.0164i			1.53855i	0.638919	+	0.769274i	$0.279382\pi$
$61$			12.0164i			1.53855i	−0.638919	+	0.769274i	$0.720618\pi$
$62$		0			0
$63$	−5.89057	+	11.3388i	−0.742142	+	1.42855i
$64$		0			0
$65$	4.14699			0.514371
$66$		0			0
$67$			10.3996i			1.27051i	0.772303	+	0.635254i	$0.219105\pi$
$67$			10.3996i			1.27051i	−0.772303	+	0.635254i	$0.780895\pi$
$68$		0			0
$69$	−1.07268	−	8.23707i	−0.129136	−	0.991627i
$70$		0			0
$71$			16.3046i			1.93501i	0.252861	+	0.967503i	$0.418629\pi$
$71$			16.3046i			1.93501i	−0.252861	+	0.967503i	$0.581371\pi$
$72$		0			0
$73$	−1.69014			−0.197816			−0.0989080	−	0.995097i	$-0.531535\pi$
$73$	−1.69014			−0.197816			−0.0989080	+	0.995097i	$0.531535\pi$
$74$		0			0
$75$	−1.68259	−	0.410981i	−0.194288	−	0.0474560i
$76$		0			0
$77$			7.54056i			0.859326i
$78$		0			0
$79$			2.62553i			0.295395i	0.989033	+	0.147698i	$0.0471863\pi$
$79$			2.62553i			0.295395i	−0.989033	+	0.147698i	$0.952814\pi$
$80$		0			0
$81$	5.17450	+	7.36373i	0.574944	+	0.818193i
$82$		0			0
$83$	5.22163			0.573149			0.286574	−	0.958058i	$-0.407483\pi$
$83$	5.22163			0.573149			0.286574	+	0.958058i	$0.407483\pi$
$84$		0			0
$85$	−6.86944			−0.745096
$86$		0			0
$87$	−1.68639	+	6.90419i	−0.180800	+	0.740207i
$88$		0			0
$89$	9.78113			1.03680			0.518399	−	0.855139i	$-0.326528\pi$
$89$	9.78113			1.03680			0.518399	+	0.855139i	$0.326528\pi$
$90$		0			0
$91$			17.6629i			1.85157i
$92$		0			0
$93$	8.51167	+	2.07902i	0.882619	+	0.215585i
$94$		0			0
$95$			1.46552i			0.150359i
$96$		0			0
$97$		−	9.26880i		−	0.941104i	−0.882372	−	0.470552i	$-0.844055\pi$
$97$		−	9.26880i		−	0.941104i	0.882372	−	0.470552i	$-0.155945\pi$
$98$		0			0
$99$	4.71318	+	2.44853i	0.473693	+	0.246086i
$100$		0			0
$101$		−	10.1637i		−	1.01133i	−0.862731	−	0.505663i	$-0.831248\pi$
$101$		−	10.1637i		−	1.01133i	0.862731	−	0.505663i	$-0.168752\pi$
$102$		0			0
$103$			9.36818i			0.923074i	0.887121	+	0.461537i	$0.152702\pi$
$103$			9.36818i			0.923074i	−0.887121	+	0.461537i	$0.847298\pi$
$104$		0			0
$105$	1.75045	−	7.16646i	0.170827	−	0.699375i
$106$		0			0
$107$	−1.58889			−0.153604			−0.0768019	−	0.997046i	$-0.524471\pi$
$107$	−1.58889			−0.153604			−0.0768019	+	0.997046i	$0.524471\pi$
$108$		0			0
$109$			9.94643i			0.952695i	0.879257	+	0.476347i	$0.158040\pi$
$109$			9.94643i			0.952695i	−0.879257	+	0.476347i	$0.841960\pi$
$110$		0			0
$111$	1.18464	−	4.84998i	0.112441	−	0.460339i
$112$		0			0
$113$	4.69423			0.441596			0.220798	−	0.975320i	$-0.429134\pi$
$113$	4.69423			0.441596			0.220798	+	0.975320i	$0.429134\pi$
$114$		0			0
$115$	1.73005	+	4.47291i	0.161328	+	0.417101i
$116$		0			0
$117$	11.0401	+	5.73539i	1.02066	+	0.530237i
$118$		0			0
$119$		−	29.2583i		−	2.68211i
$120$		0			0
$121$	−7.86562			−0.715057
$122$		0			0
$123$	−1.87563	+	7.67894i	−0.169120	+	0.692387i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.49918			0.221767			0.110883	−	0.993833i	$-0.464632\pi$
$127$	2.49918			0.221767			0.110883	+	0.993833i	$0.464632\pi$
$128$		0			0
$129$	1.34801	−	5.51886i	0.118686	−	0.485908i
$130$		0			0
$131$			6.14095i			0.536537i	0.963344	+	0.268269i	$0.0864515\pi$
$131$			6.14095i			0.536537i	−0.963344	+	0.268269i	$0.913548\pi$
$132$		0			0
$133$	−6.24193			−0.541244
$134$		0			0
$135$	−3.91096	−	3.42116i	−0.336602	−	0.294447i
$136$		0			0
$137$	−11.5065			−0.983071			−0.491535	−	0.870858i	$-0.663564\pi$
$137$	−11.5065			−0.983071			−0.491535	+	0.870858i	$0.663564\pi$
$138$		0			0
$139$	−13.8249			−1.17262			−0.586308	−	0.810088i	$-0.699419\pi$
$139$	−13.8249			−1.17262			−0.586308	+	0.810088i	$0.699419\pi$
$140$		0			0
$141$	2.38464	−	9.76287i	0.200823	−	0.822182i
$142$		0			0
$143$	7.34191			0.613961
$144$		0			0
$145$		−	4.10332i		−	0.340762i
$146$		0			0
$147$	18.7453	+	4.57865i	1.54609	+	0.377640i
$148$		0			0
$149$	16.3047			1.33573			0.667867	−	0.744281i	$-0.267208\pi$
$149$	16.3047			1.33573			0.667867	+	0.744281i	$0.267208\pi$
$150$		0			0
$151$	4.81062			0.391483			0.195741	−	0.980656i	$-0.437289\pi$
$151$	4.81062			0.391483			0.195741	+	0.980656i	$0.437289\pi$
$152$		0			0
$153$	−18.2878	−	9.50060i	−1.47848	−	0.768078i
$154$		0			0
$155$	−5.05868			−0.406323
$156$		0			0
$157$			12.9122i			1.03050i	0.857039	+	0.515252i	$0.172302\pi$
$157$			12.9122i			1.03050i	−0.857039	+	0.515252i	$0.827698\pi$
$158$		0			0
$159$	1.39909	+	0.341735i	0.110955	+	0.0271014i
$160$		0			0
$161$	−19.0510	+	7.36864i	−1.50143	+	0.580730i
$162$		0			0
$163$	−11.1862			−0.876174			−0.438087	−	0.898933i	$-0.644344\pi$
$163$	−11.1862			−0.876174			−0.438087	+	0.898933i	$0.644344\pi$
$164$		0			0
$165$	−2.97888	−	0.727608i	−0.231905	−	0.0566442i
$166$		0			0
$167$		−	0.273329i		−	0.0211508i	−0.999944	−	0.0105754i	$-0.996634\pi$
$167$		−	0.273329i		−	0.0211508i	0.999944	−	0.0105754i	$-0.00336632\pi$
$168$		0			0
$169$	4.19756			0.322889
$170$		0			0
$171$	−2.02684	+	3.90149i	−0.154997	+	0.298354i
$172$		0			0
$173$			12.8456i			0.976631i	0.872667	+	0.488315i	$0.162388\pi$
$173$			12.8456i			0.976631i	−0.872667	+	0.488315i	$0.837612\pi$
$174$		0			0
$175$			4.25920i			0.321965i
$176$		0			0
$177$	−4.86311	+	19.9099i	−0.365534	+	1.49652i
$178$		0			0
$179$		−	5.54592i		−	0.414522i	−0.978286	−	0.207261i	$-0.933545\pi$
$179$		−	5.54592i		−	0.414522i	0.978286	−	0.207261i	$-0.0664549\pi$
$180$		0			0
$181$		−	9.79986i		−	0.728418i	−0.931317	−	0.364209i	$-0.881339\pi$
$181$		−	9.79986i		−	0.728418i	0.931317	−	0.364209i	$-0.118661\pi$
$182$		0			0
$183$	4.93853	−	20.2187i	0.365067	−	1.49461i
$184$		0			0
$185$			2.88245i			0.211922i
$186$		0			0
$187$	−12.1618			−0.889358
$188$		0			0
$189$	14.5714	−	16.6576i	1.05991	−	1.21166i
$190$		0			0
$191$	−18.3769			−1.32971			−0.664854	−	0.746973i	$-0.731506\pi$
$191$	−18.3769			−1.32971			−0.664854	+	0.746973i	$0.731506\pi$
$192$		0			0
$193$	−15.0944			−1.08652			−0.543260	−	0.839564i	$-0.682810\pi$
$193$	−15.0944			−1.08652			−0.543260	+	0.839564i	$0.682810\pi$
$194$		0			0
$195$	−6.97767	−	1.70434i	−0.499682	−	0.122050i
$196$		0			0
$197$			21.1615i			1.50769i	0.657050	+	0.753847i	$0.271804\pi$
$197$			21.1615i			1.50769i	−0.657050	+	0.753847i	$0.728196\pi$
$198$		0			0
$199$			3.16726i			0.224521i	0.993679	+	0.112261i	$0.0358092\pi$
$199$			3.16726i			0.224521i	−0.993679	+	0.112261i	$0.964191\pi$
$200$		0			0
$201$	4.27402	−	17.4981i	0.301466	−	1.23422i
$202$		0			0
$203$	17.4769			1.22663
$204$		0			0
$205$		−	4.56378i		−	0.318748i
$206$		0			0
$207$	−1.58040	+	14.3004i	−0.109846	+	0.993949i
$208$		0			0
$209$			2.59458i			0.179471i
$210$		0			0
$211$	5.16177			0.355351			0.177676	−	0.984089i	$-0.443142\pi$
$211$	5.16177			0.355351			0.177676	+	0.984089i	$0.443142\pi$
$212$		0			0
$213$	6.70091	−	27.4340i	0.459138	−	1.87974i
$214$		0			0
$215$			3.27999i			0.223693i
$216$		0			0
$217$		−	21.5459i		−	1.46263i
$218$		0			0
$219$	2.84381	+	0.694616i	0.192167	+	0.0469378i
$220$		0			0
$221$	−28.4875			−1.91628
$222$		0			0
$223$	29.1017			1.94879			0.974395	−	0.224842i	$-0.0721864\pi$
$223$	29.1017			1.94879			0.974395	+	0.224842i	$0.0721864\pi$
$224$		0			0
$225$	2.66219	+	1.38302i	0.177479	+	0.0922015i
$226$		0			0
$227$	5.42076			0.359789			0.179894	−	0.983686i	$-0.442424\pi$
$227$	5.42076			0.359789			0.179894	+	0.983686i	$0.442424\pi$
$228$		0			0
$229$		−	15.5263i		−	1.02601i	−0.858386	−	0.513004i	$-0.828533\pi$
$229$		−	15.5263i		−	1.02601i	0.858386	−	0.513004i	$-0.171467\pi$
$230$		0			0
$231$	3.09903	−	12.6876i	0.203901	−	0.834785i
$232$		0			0
$233$		−	29.1310i		−	1.90844i	−0.299113	−	0.954218i	$-0.596691\pi$
$233$		−	29.1310i		−	1.90844i	0.299113	−	0.954218i	$-0.403309\pi$
$234$		0			0
$235$			5.80230i			0.378501i
$236$		0			0
$237$	1.07904	−	4.41768i	0.0700915	−	0.286959i
$238$		0			0
$239$		−	25.1530i		−	1.62701i	−0.581555	−	0.813507i	$-0.697556\pi$
$239$		−	25.1530i		−	1.62701i	0.581555	−	0.813507i	$-0.302444\pi$
$240$		0			0
$241$		−	23.5388i		−	1.51627i	−0.652097	−	0.758135i	$-0.726111\pi$
$241$		−	23.5388i		−	1.51627i	0.652097	−	0.758135i	$-0.273889\pi$
$242$		0			0
$243$	−5.68018	−	14.5167i	−0.364383	−	0.931249i
$244$		0			0
$245$	−11.1408			−0.711757
$246$		0			0
$247$			6.07750i			0.386702i
$248$		0			0
$249$	−8.78584	−	2.14599i	−0.556780	−	0.135997i
$250$		0			0
$251$	−6.64673			−0.419538			−0.209769	−	0.977751i	$-0.567271\pi$
$251$	−6.64673			−0.419538			−0.209769	+	0.977751i	$0.567271\pi$
$252$		0			0
$253$	3.06292	+	7.91891i	0.192564	+	0.497858i
$254$		0			0
$255$	11.5584	+	2.82321i	0.723817	+	0.176797i
$256$		0			0
$257$			6.11601i			0.381506i	0.981638	+	0.190753i	$0.0610930\pi$
$257$			6.11601i			0.381506i	−0.981638	+	0.190753i	$0.938907\pi$
$258$		0			0
$259$	−12.2769			−0.762852
$260$		0			0
$261$	5.67499	−	10.9238i	0.351273	−	0.676167i
$262$		0			0
$263$	8.17715			0.504225			0.252112	−	0.967698i	$-0.418875\pi$
$263$	8.17715			0.504225			0.252112	+	0.967698i	$0.418875\pi$
$264$		0			0
$265$	−0.831510			−0.0510792
$266$		0			0
$267$	−16.4576	−	4.01986i	−1.00719	−	0.246012i
$268$		0			0
$269$			0.00864764i			0.000527256i	1.00000		0.000263628i	$8.39153e-5\pi$
$269$			0.00864764i			0.000527256i	−1.00000		0.000263628i	$0.999916\pi$
$270$		0			0
$271$	23.9068			1.45223			0.726117	−	0.687571i	$-0.241323\pi$
$271$	23.9068			1.45223			0.726117	+	0.687571i	$0.241323\pi$
$272$		0			0
$273$	7.25911	−	29.7193i	0.439341	−	1.79869i
$274$		0			0
$275$	1.77042			0.106760
$276$		0			0
$277$	22.6195			1.35907			0.679537	−	0.733641i	$-0.262181\pi$
$277$	22.6195			1.35907			0.679537	+	0.733641i	$0.262181\pi$
$278$		0			0
$279$	−13.4672	−	6.99627i	−0.806258	−	0.418856i
$280$		0			0
$281$	32.6193			1.94590			0.972951	−	0.231013i	$-0.0742038\pi$
$281$	32.6193			1.94590			0.972951	+	0.231013i	$0.0742038\pi$
$282$		0			0
$283$		−	12.2989i		−	0.731095i	−0.930793	−	0.365548i	$-0.880882\pi$
$283$		−	12.2989i		−	0.731095i	0.930793	−	0.365548i	$-0.119118\pi$
$284$		0			0
$285$	0.602301	−	2.46586i	0.0356772	−	0.146065i
$286$		0			0
$287$	19.4380			1.14739
$288$		0			0
$289$	30.1893			1.77584
$290$		0			0
$291$	−3.80930	+	15.5955i	−0.223305	+	0.914227i
$292$		0			0
$293$	−22.1635			−1.29481			−0.647403	−	0.762148i	$-0.724145\pi$
$293$	−22.1635			−1.29481			−0.647403	+	0.762148i	$0.724145\pi$
$294$		0			0
$295$		−	11.8329i		−	0.688940i
$296$		0			0
$297$	−6.92404	−	6.05689i	−0.401773	−	0.351456i
$298$		0			0
$299$	7.17452	+	18.5491i	0.414913	+	1.07272i
$300$		0			0
$301$	−13.9701			−0.805223
$302$		0			0
$303$	−4.17709	+	17.1013i	−0.239967	+	0.982443i
$304$		0			0
$305$			12.0164i			0.688059i
$306$		0			0
$307$	27.1627			1.55026			0.775129	−	0.631803i	$-0.217685\pi$
$307$	27.1627			1.55026			0.775129	+	0.631803i	$0.217685\pi$
$308$		0			0
$309$	3.85015	−	15.7628i	0.219027	−	0.896712i
$310$		0			0
$311$		−	13.8045i		−	0.782781i	−0.920225	−	0.391391i	$-0.871994\pi$
$311$		−	13.8045i		−	0.782781i	0.920225	−	0.391391i	$-0.128006\pi$
$312$		0			0
$313$			32.9723i			1.86371i	0.362834	+	0.931854i	$0.381809\pi$
$313$			32.9723i			1.86371i	−0.362834	+	0.931854i	$0.618191\pi$
$314$		0			0
$315$	−5.89057	+	11.3388i	−0.331896	+	0.638868i
$316$		0			0
$317$		−	10.9658i		−	0.615900i	−0.951402	−	0.307950i	$-0.900357\pi$
$317$		−	10.9658i		−	0.615900i	0.951402	−	0.307950i	$-0.0996430\pi$
$318$		0			0
$319$		−	7.26459i		−	0.406739i
$320$		0			0
$321$	2.67344	+	0.653004i	0.149217	+	0.0364471i
$322$		0			0
$323$		−	10.0673i		−	0.560159i
$324$		0			0
$325$	4.14699			0.230034
$326$		0			0
$327$	4.08780	−	16.7357i	0.226056	−	0.925487i
$328$		0			0
$329$	−24.7132			−1.36248
$330$		0			0
$331$	−18.0444			−0.991811			−0.495905	−	0.868377i	$-0.665164\pi$
$331$	−18.0444			−0.991811			−0.495905	+	0.868377i	$0.665164\pi$
$332$		0			0
$333$	−3.98650	+	7.67364i	−0.218459	+	0.420513i
$334$		0			0
$335$			10.3996i			0.568188i
$336$		0			0
$337$			0.830721i			0.0452523i	0.999744	+	0.0226261i	$0.00720274\pi$
$337$			0.830721i			0.0452523i	−0.999744	+	0.0226261i	$0.992797\pi$
$338$		0			0
$339$	−7.89845	−	1.92924i	−0.428985	−	0.104782i
$340$		0			0
$341$	−8.95598			−0.484993
$342$		0			0
$343$		−	17.6363i		−	0.952272i
$344$		0			0
$345$	−1.07268	−	8.23707i	−0.0577512	−	0.443469i
$346$		0			0
$347$			6.54648i			0.351434i	0.984441	+	0.175717i	$0.0562243\pi$
$347$			6.54648i			0.351434i	−0.984441	+	0.175717i	$0.943776\pi$
$348$		0			0
$349$	17.4584			0.934529			0.467264	−	0.884118i	$-0.345240\pi$
$349$	17.4584			0.934529			0.467264	+	0.884118i	$0.345240\pi$
$350$		0			0
$351$	−16.2187	−	14.1875i	−0.865693	−	0.757275i
$352$		0			0
$353$		−	11.3823i		−	0.605819i	−0.953019	−	0.302909i	$-0.902042\pi$
$353$		−	11.3823i		−	0.605819i	0.953019	−	0.302909i	$-0.0979579\pi$
$354$		0			0
$355$			16.3046i			0.865361i
$356$		0			0
$357$	−12.0246	+	49.2296i	−0.636411	+	2.60551i
$358$		0			0
$359$	−30.3119			−1.59980			−0.799900	−	0.600134i	$-0.795114\pi$
$359$	−30.3119			−1.59980			−0.799900	+	0.600134i	$0.795114\pi$
$360$		0			0
$361$	16.8523			0.886961
$362$		0			0
$363$	13.2346	+	3.23262i	0.694636	+	0.169669i
$364$		0			0
$365$	−1.69014			−0.0884660
$366$		0			0
$367$		−	11.1215i		−	0.580540i	−0.956945	−	0.290270i	$-0.906255\pi$
$367$		−	11.1215i		−	0.580540i	0.956945	−	0.290270i	$-0.0937451\pi$
$368$		0			0
$369$	6.31180	−	12.1496i	0.328579	−	0.632485i
$370$		0			0
$371$		−	3.54156i		−	0.183869i
$372$		0			0
$373$		−	7.32753i		−	0.379405i	−0.981842	−	0.189703i	$-0.939248\pi$
$373$		−	7.32753i		−	0.379405i	0.981842	−	0.189703i	$-0.0607524\pi$
$374$		0			0
$375$	−1.68259	−	0.410981i	−0.0868883	−	0.0212230i
$376$		0			0
$377$		−	17.0164i		−	0.876392i
$378$		0			0
$379$		−	1.54185i		−	0.0791997i	−0.999216	−	0.0395998i	$-0.987392\pi$
$379$		−	1.54185i		−	0.0791997i	0.999216	−	0.0395998i	$-0.0126083\pi$
$380$		0			0
$381$	−4.20509	−	1.02712i	−0.215433	−	0.0526208i
$382$		0			0
$383$	−33.1055			−1.69161			−0.845805	−	0.533492i	$-0.820879\pi$
$383$	−33.1055			−1.69161			−0.845805	+	0.533492i	$0.820879\pi$
$384$		0			0
$385$			7.54056i			0.384302i
$386$		0			0
$387$	−4.53629	+	8.73194i	−0.230593	+	0.443869i
$388$		0			0
$389$	27.5503			1.39686			0.698428	−	0.715680i	$-0.253883\pi$
$389$	27.5503			1.39686			0.698428	+	0.715680i	$0.253883\pi$
$390$		0			0
$391$	−11.8845	−	30.7264i	−0.601025	−	1.55390i
$392$		0			0
$393$	2.52382	−	10.3327i	0.127310	−	0.521215i
$394$		0			0
$395$			2.62553i			0.132105i
$396$		0			0
$397$	13.4334			0.674205			0.337102	−	0.941468i	$-0.390553\pi$
$397$	13.4334			0.674205			0.337102	+	0.941468i	$0.390553\pi$
$398$		0			0
$399$	10.5026	+	2.56532i	0.525787	+	0.128427i
$400$		0			0
$401$	17.8416			0.890965			0.445483	−	0.895291i	$-0.353032\pi$
$401$	17.8416			0.890965			0.445483	+	0.895291i	$0.353032\pi$
$402$		0			0
$403$	−20.9783			−1.04501
$404$		0			0
$405$	5.17450	+	7.36373i	0.257123	+	0.365907i
$406$		0			0
$407$			5.10315i			0.252954i
$408$		0			0
$409$	−7.54963			−0.373305			−0.186653	−	0.982426i	$-0.559764\pi$
$409$	−7.54963			−0.373305			−0.186653	+	0.982426i	$0.559764\pi$
$410$		0			0
$411$	19.3607	+	4.72898i	0.954995	+	0.233263i
$412$		0			0
$413$	50.3988			2.47996
$414$		0			0
$415$	5.22163			0.256320
$416$		0			0
$417$	23.2616	+	5.68179i	1.13913	+	0.278238i
$418$		0			0
$419$	6.14328			0.300119			0.150059	−	0.988677i	$-0.452054\pi$
$419$	6.14328			0.300119			0.150059	+	0.988677i	$0.452054\pi$
$420$		0			0
$421$			38.8307i			1.89249i	0.323449	+	0.946246i	$0.395158\pi$
$421$			38.8307i			1.89249i	−0.323449	+	0.946246i	$0.604842\pi$
$422$		0			0
$423$	−8.02472	+	15.4468i	−0.390175	+	0.751051i
$424$		0			0
$425$	−6.86944			−0.333217
$426$		0			0
$427$	−51.1804			−2.47679
$428$		0			0
$429$	−12.3534	−	3.01739i	−0.596427	−	0.145681i
$430$		0			0
$431$	25.7768			1.24163			0.620813	−	0.783959i	$-0.286803\pi$
$431$	25.7768			1.24163			0.620813	+	0.783959i	$0.286803\pi$
$432$		0			0
$433$		−	32.6236i		−	1.56779i	−0.620894	−	0.783894i	$-0.713230\pi$
$433$		−	32.6236i		−	1.56779i	0.620894	−	0.783894i	$-0.286770\pi$
$434$		0			0
$435$	−1.68639	+	6.90419i	−0.0808561	+	0.331031i
$436$		0			0
$437$	−6.55513	+	2.53543i	−0.313574	+	0.121286i
$438$		0			0
$439$	−19.8107			−0.945513			−0.472757	−	0.881193i	$-0.656741\pi$
$439$	−19.8107			−0.945513			−0.472757	+	0.881193i	$0.656741\pi$
$440$		0			0
$441$	−29.6588	−	15.4079i	−1.41232	−	0.733711i
$442$		0			0
$443$			1.51886i			0.0721631i	0.999349	+	0.0360816i	$0.0114876\pi$
$443$			1.51886i			0.0721631i	−0.999349	+	0.0360816i	$0.988512\pi$
$444$		0			0
$445$	9.78113			0.463670
$446$		0			0
$447$	−27.4341	−	6.70093i	−1.29759	−	0.316943i
$448$		0			0
$449$		−	12.0706i		−	0.569646i	−0.958580	−	0.284823i	$-0.908065\pi$
$449$		−	12.0706i		−	0.569646i	0.958580	−	0.284823i	$-0.0919348\pi$
$450$		0			0
$451$		−	8.07979i		−	0.380462i
$452$		0			0
$453$	−8.09428	−	1.97707i	−0.380302	−	0.0928911i
$454$		0			0
$455$			17.6629i			0.828048i
$456$		0			0
$457$			36.0870i			1.68808i	0.536281	+	0.844039i	$0.319829\pi$
$457$			36.0870i			1.68808i	−0.536281	+	0.844039i	$0.680171\pi$
$458$		0			0
$459$	26.8661	+	23.5015i	1.25400	+	1.09696i
$460$		0			0
$461$			39.9476i			1.86054i	0.366872	+	0.930271i	$0.380429\pi$
$461$			39.9476i			1.86054i	−0.366872	+	0.930271i	$0.619571\pi$
$462$		0			0
$463$	4.60520			0.214022			0.107011	−	0.994258i	$-0.465872\pi$
$463$	4.60520			0.214022			0.107011	+	0.994258i	$0.465872\pi$
$464$		0			0
$465$	8.51167	+	2.07902i	0.394719	+	0.0964125i
$466$		0			0
$467$	−30.4087			−1.40715			−0.703573	−	0.710623i	$-0.748413\pi$
$467$	−30.4087			−1.40715			−0.703573	+	0.710623i	$0.748413\pi$
$468$		0			0
$469$	−44.2938			−2.04530
$470$		0			0
$471$	5.30666	−	21.7258i	0.244518	−	1.00107i
$472$		0			0
$473$			5.80694i			0.267003i
$474$		0			0
$475$			1.46552i			0.0672426i
$476$		0			0
$477$	−2.21364	−	1.15000i	−0.101355	−	0.0526547i
$478$		0			0
$479$	31.0474			1.41859			0.709297	−	0.704910i	$-0.249013\pi$
$479$	31.0474			1.41859			0.709297	+	0.704910i	$0.249013\pi$
$480$		0			0
$481$			11.9535i			0.545034i
$482$		0			0
$483$	35.0833	−	4.56876i	1.59635	−	0.207886i
$484$		0			0
$485$		−	9.26880i		−	0.420875i
$486$		0			0
$487$	−11.7747			−0.533563			−0.266782	−	0.963757i	$-0.585960\pi$
$487$	−11.7747			−0.533563			−0.266782	+	0.963757i	$0.585960\pi$
$488$		0			0
$489$	18.8218	+	4.59733i	0.851151	+	0.207899i
$490$		0			0
$491$			23.6929i			1.06924i	0.845091	+	0.534622i	$0.179546\pi$
$491$			23.6929i			1.06924i	−0.845091	+	0.534622i	$0.820454\pi$
$492$		0			0
$493$			28.1875i			1.26950i
$494$		0			0
$495$	4.71318	+	2.44853i	0.211842	+	0.110053i
$496$		0			0
$497$	−69.4447			−3.11502
$498$		0			0
$499$	36.7997			1.64738			0.823689	−	0.567041i	$-0.191912\pi$
$499$	36.7997			1.64738			0.823689	+	0.567041i	$0.191912\pi$
$500$		0			0
$501$	−0.112333	+	0.459899i	−0.00501867	+	0.0205468i
$502$		0			0
$503$	−39.7986			−1.77453			−0.887266	−	0.461259i	$-0.847398\pi$
$503$	−39.7986			−1.77453			−0.887266	+	0.461259i	$0.847398\pi$
$504$		0			0
$505$		−	10.1637i		−	0.452278i
$506$		0			0
$507$	−7.06276	−	1.72512i	−0.313668	−	0.0766153i
$508$		0			0
$509$			10.3012i			0.456593i	0.973592	+	0.228297i	$0.0733156\pi$
$509$			10.3012i			0.456593i	−0.973592	+	0.228297i	$0.926684\pi$
$510$		0			0
$511$		−	7.19864i		−	0.318449i
$512$		0			0
$513$	5.01378	−	5.73159i	0.221364	−	0.253056i
$514$		0			0
$515$			9.36818i			0.412811i
$516$		0			0
$517$			10.2725i			0.451784i
$518$		0			0
$519$	5.27929	−	21.6138i	0.231735	−	0.948739i
$520$		0			0
$521$	24.8553			1.08893			0.544465	−	0.838784i	$-0.316733\pi$
$521$	24.8553			1.08893			0.544465	+	0.838784i	$0.316733\pi$
$522$		0			0
$523$			12.6351i			0.552496i	0.961086	+	0.276248i	$0.0890912\pi$
$523$			12.6351i			0.552496i	−0.961086	+	0.276248i	$0.910909\pi$
$524$		0			0
$525$	1.75045	−	7.16646i	0.0763959	−	0.312770i
$526$		0			0
$527$	34.7503			1.51375
$528$		0			0
$529$	−17.0138	+	15.4767i	−0.739732	+	0.672902i
$530$		0			0
$531$	16.3652	−	31.5015i	0.710190	−	1.36705i
$532$		0			0
$533$		−	18.9260i		−	0.819774i
$534$		0			0
$535$	−1.58889			−0.0686937
$536$		0			0
$537$	−2.27927	+	9.33149i	−0.0983578	+	0.402683i
$538$		0			0
$539$	−19.7238			−0.849564
$540$		0			0
$541$	−21.7627			−0.935653			−0.467826	−	0.883820i	$-0.654963\pi$
$541$	−21.7627			−0.935653			−0.467826	+	0.883820i	$0.654963\pi$
$542$		0			0
$543$	−4.02756	+	16.4891i	−0.172839	+	0.707615i
$544$		0			0
$545$			9.94643i			0.426058i
$546$		0			0
$547$	0.652614			0.0279038			0.0139519	−	0.999903i	$-0.495559\pi$
$547$	0.652614			0.0279038			0.0139519	+	0.999903i	$0.495559\pi$
$548$		0			0
$549$	−16.6190	+	31.9900i	−0.709282	+	1.36530i
$550$		0			0
$551$	6.01349			0.256183
$552$		0			0
$553$	−11.1827			−0.475535
$554$		0			0
$555$	1.18464	−	4.84998i	0.0502850	−	0.205870i
$556$		0			0
$557$	43.7961			1.85570			0.927850	−	0.372953i	$-0.121655\pi$
$557$	43.7961			1.85570			0.927850	+	0.372953i	$0.121655\pi$
$558$		0			0
$559$			13.6021i			0.575307i
$560$		0			0
$561$	20.4632	+	4.99827i	0.863959	+	0.211027i
$562$		0			0
$563$	−4.04662			−0.170545			−0.0852723	−	0.996358i	$-0.527176\pi$
$563$	−4.04662			−0.170545			−0.0852723	+	0.996358i	$0.527176\pi$
$564$		0			0
$565$	4.69423			0.197488
$566$		0			0
$567$	−31.3636	+	22.0392i	−1.31715	+	0.925560i
$568$		0			0
$569$	17.6778			0.741092			0.370546	−	0.928814i	$-0.379171\pi$
$569$	17.6778			0.741092			0.370546	+	0.928814i	$0.379171\pi$
$570$		0			0
$571$			9.47431i			0.396488i	0.980153	+	0.198244i	$0.0635238\pi$
$571$			9.47431i			0.396488i	−0.980153	+	0.198244i	$0.936476\pi$
$572$		0			0
$573$	30.9208	+	7.55257i	1.29173	+	0.315513i
$574$		0			0
$575$	1.73005	+	4.47291i	0.0721482	+	0.186533i
$576$		0			0
$577$	27.6920			1.15284			0.576418	−	0.817155i	$-0.304450\pi$
$577$	27.6920			1.15284			0.576418	+	0.817155i	$0.304450\pi$
$578$		0			0
$579$	25.3977	+	6.20353i	1.05549	+	0.257810i
$580$		0			0
$581$			22.2400i			0.922669i
$582$		0			0
$583$	−1.47212			−0.0609689
$584$		0			0
$585$	11.0401	+	5.73539i	0.456451	+	0.237129i
$586$		0			0
$587$		−	33.9922i		−	1.40301i	−0.712665	−	0.701504i	$-0.752512\pi$
$587$		−	33.9922i		−	1.40301i	0.712665	−	0.701504i	$-0.247488\pi$
$588$		0			0
$589$		−	7.41359i		−	0.305472i
$590$		0			0
$591$	8.69698	−	35.6060i	0.357746	−	1.46464i
$592$		0			0
$593$		−	45.4678i		−	1.86714i	−0.358398	−	0.933569i	$-0.616677\pi$
$593$		−	45.4678i		−	1.86714i	0.358398	−	0.933569i	$-0.383323\pi$
$594$		0			0
$595$		−	29.2583i		−	1.19947i
$596$		0			0
$597$	1.30169	−	5.32919i	0.0532745	−	0.218109i
$598$		0			0
$599$		−	23.7247i		−	0.969364i	−0.874691	−	0.484682i	$-0.838935\pi$
$599$		−	23.7247i		−	0.969364i	0.874691	−	0.484682i	$-0.161065\pi$
$600$		0			0
$601$	34.9900			1.42727			0.713636	−	0.700517i	$-0.247047\pi$
$601$	34.9900			1.42727			0.713636	+	0.700517i	$0.247047\pi$
$602$		0			0
$603$	−14.3828	+	27.6856i	−0.585714	+	1.12744i
$604$		0			0
$605$	−7.86562			−0.319783
$606$		0			0
$607$	26.2120			1.06391			0.531955	−	0.846773i	$-0.321457\pi$
$607$	26.2120			1.06391			0.531955	+	0.846773i	$0.321457\pi$
$608$		0			0
$609$	−29.4063	−	7.18266i	−1.19160	−	0.291056i
$610$		0			0
$611$			24.0621i			0.973449i
$612$		0			0
$613$		−	27.1765i		−	1.09765i	−0.835938	−	0.548824i	$-0.815076\pi$
$613$		−	27.1765i		−	1.09765i	0.835938	−	0.548824i	$-0.184924\pi$
$614$		0			0
$615$	−1.87563	+	7.67894i	−0.0756326	+	0.309645i
$616$		0			0
$617$	−0.364713			−0.0146828			−0.00734140	−	0.999973i	$-0.502337\pi$
$617$	−0.364713			−0.0146828			−0.00734140	+	0.999973i	$0.502337\pi$
$618$		0			0
$619$			0.854713i			0.0343538i	0.999852	+	0.0171769i	$0.00546785\pi$
$619$			0.854713i			0.0343538i	−0.999852	+	0.0171769i	$0.994532\pi$
$620$		0			0
$621$	8.53638	−	23.4122i	0.342553	−	0.939499i
$622$		0			0
$623$			41.6598i			1.66906i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.06632	−	4.36560i	0.0425848	−	0.174345i
$628$		0			0
$629$		−	19.8009i		−	0.789512i
$630$		0			0
$631$		−	43.6340i		−	1.73704i	−0.495654	−	0.868520i	$-0.665072\pi$
$631$		−	43.6340i		−	1.73704i	0.495654	−	0.868520i	$-0.334928\pi$
$632$		0			0
$633$	−8.68512	−	2.12139i	−0.345203	−	0.0843178i
$634$		0			0
$635$	2.49918			0.0991770
$636$		0			0
$637$	−46.2007			−1.83054
$638$		0			0
$639$	−22.5497	+	43.4060i	−0.892052	+	1.71712i
$640$		0			0
$641$	10.9445			0.432282			0.216141	−	0.976362i	$-0.430653\pi$
$641$	10.9445			0.432282			0.216141	+	0.976362i	$0.430653\pi$
$642$		0			0
$643$		−	27.3836i		−	1.07990i	−0.841696	−	0.539952i	$-0.818442\pi$
$643$		−	27.3836i		−	1.07990i	0.841696	−	0.539952i	$-0.181558\pi$
$644$		0			0
$645$	1.34801	−	5.51886i	0.0530780	−	0.217305i
$646$		0			0
$647$			11.8734i			0.466791i	0.972382	+	0.233395i	$0.0749836\pi$
$647$			11.8734i			0.466791i	−0.972382	+	0.233395i	$0.925016\pi$
$648$		0			0
$649$		−	20.9492i		−	0.822329i
$650$		0			0
$651$	−8.85498	+	36.2529i	−0.347054	+	1.42086i
$652$		0			0
$653$		−	15.0075i		−	0.587289i	−0.955915	−	0.293644i	$-0.905132\pi$
$653$		−	15.0075i		−	0.587289i	0.955915	−	0.293644i	$-0.0948681\pi$
$654$		0			0
$655$			6.14095i			0.239947i
$656$		0			0
$657$	−4.49947	−	2.33750i	−0.175541	−	0.0911946i
$658$		0			0
$659$	−25.1920			−0.981342			−0.490671	−	0.871345i	$-0.663248\pi$
$659$	−25.1920			−0.981342			−0.490671	+	0.871345i	$0.663248\pi$
$660$		0			0
$661$		−	11.3817i		−	0.442698i	−0.975195	−	0.221349i	$-0.928954\pi$
$661$		−	11.3817i		−	0.442698i	0.975195	−	0.221349i	$-0.0710459\pi$
$662$		0			0
$663$	47.9327	+	11.7079i	1.86155	+	0.454695i
$664$		0			0
$665$	−6.24193			−0.242052
$666$		0			0
$667$	18.3538	−	7.09897i	0.710661	−	0.274873i
$668$		0			0
$669$	−48.9660	−	11.9602i	−1.89314	−	0.462409i
$670$		0			0
$671$			21.2741i			0.821278i
$672$		0			0
$673$	7.04251			0.271469			0.135734	−	0.990745i	$-0.456661\pi$
$673$	7.04251			0.271469			0.135734	+	0.990745i	$0.456661\pi$
$674$		0			0
$675$	−3.91096	−	3.42116i	−0.150533	−	0.131681i
$676$		0			0
$677$	−4.06545			−0.156248			−0.0781239	−	0.996944i	$-0.524893\pi$
$677$	−4.06545			−0.156248			−0.0781239	+	0.996944i	$0.524893\pi$
$678$		0			0
$679$	39.4776			1.51501
$680$		0			0
$681$	−9.12090	−	2.22783i	−0.349514	−	0.0853707i
$682$		0			0
$683$		−	45.4211i		−	1.73799i	−0.494819	−	0.868996i	$-0.664766\pi$
$683$		−	45.4211i		−	1.73799i	0.494819	−	0.868996i	$-0.335234\pi$
$684$		0			0
$685$	−11.5065			−0.439643
$686$		0			0
$687$	−6.38102	+	26.1243i	−0.243451	+	0.996706i
$688$		0			0
$689$	−3.44827			−0.131369
$690$		0			0
$691$	−15.9715			−0.607586			−0.303793	−	0.952738i	$-0.598253\pi$
$691$	−15.9715			−0.607586			−0.303793	+	0.952738i	$0.598253\pi$
$692$		0			0
$693$	−10.4288	+	20.0744i	−0.396156	+	0.762563i
$694$		0			0
$695$	−13.8249			−0.524410
$696$		0			0
$697$			31.3506i			1.18749i
$698$		0			0
$699$	−11.9723	+	49.0154i	−0.452834	+	1.85393i
$700$		0			0
$701$	−19.3116			−0.729391			−0.364695	−	0.931127i	$-0.618827\pi$
$701$	−19.3116			−0.729391			−0.364695	+	0.931127i	$0.618827\pi$
$702$		0			0
$703$	−4.22429			−0.159322
$704$		0			0
$705$	2.38464	−	9.76287i	0.0898107	−	0.367691i
$706$		0			0
$707$	43.2892			1.62806
$708$		0			0
$709$			35.6708i			1.33964i	0.742521	+	0.669822i	$0.233630\pi$
$709$			35.6708i			1.33964i	−0.742521	+	0.669822i	$0.766370\pi$
$710$		0			0
$711$	−3.63117	+	6.98966i	−0.136180	+	0.262133i
$712$		0			0
$713$	−8.75179	−	22.6270i	−0.327757	−	0.847389i
$714$		0			0
$715$	7.34191			0.274572
$716$		0			0
$717$	−10.3374	+	42.3221i	−0.386058	+	1.58055i
$718$		0			0
$719$			8.77126i			0.327113i	0.986534	+	0.163556i	$0.0522966\pi$
$719$			8.77126i			0.327113i	−0.986534	+	0.163556i	$0.947703\pi$
$720$		0			0
$721$	−39.9009			−1.48599
$722$		0			0
$723$	−9.67403	+	39.6061i	−0.359781	+	1.47297i
$724$		0			0
$725$		−	4.10332i		−	0.152394i
$726$		0			0
$727$		−	26.4591i		−	0.981315i	−0.871353	−	0.490657i	$-0.836757\pi$
$727$		−	26.4591i		−	0.981315i	0.871353	−	0.490657i	$-0.163243\pi$
$728$		0			0
$729$	3.59127	+	26.7601i	0.133010	+	0.991115i
$730$		0			0
$731$		−	22.5317i		−	0.833364i
$732$		0			0
$733$		−	40.6758i		−	1.50240i	−0.660077	−	0.751198i	$-0.729477\pi$
$733$		−	40.6758i		−	1.50240i	0.660077	−	0.751198i	$-0.270523\pi$
$734$		0			0
$735$	18.7453	+	4.57865i	0.691430	+	0.168886i
$736$		0			0
$737$			18.4115i			0.678198i
$738$		0			0
$739$	−4.06596			−0.149569			−0.0747844	−	0.997200i	$-0.523827\pi$
$739$	−4.06596			−0.149569			−0.0747844	+	0.997200i	$0.523827\pi$
$740$		0			0
$741$	2.49774	−	10.2259i	0.0917567	−	0.375658i
$742$		0			0
$743$	−37.8267			−1.38773			−0.693863	−	0.720107i	$-0.744093\pi$
$743$	−37.8267			−1.38773			−0.693863	+	0.720107i	$0.744093\pi$
$744$		0			0
$745$	16.3047			0.597358
$746$		0			0
$747$	13.9010	+	7.22164i	0.508610	+	0.264226i
$748$		0			0
$749$		−	6.76739i		−	0.247275i
$750$		0			0
$751$			37.9249i			1.38390i	0.721946	+	0.691950i	$0.243248\pi$
$751$			37.9249i			1.38390i	−0.721946	+	0.691950i	$0.756752\pi$
$752$		0			0
$753$	11.1837	+	2.73168i	0.407556	+	0.0995480i
$754$		0			0
$755$	4.81062			0.175076
$756$		0			0
$757$		−	14.6543i		−	0.532618i	−0.963888	−	0.266309i	$-0.914196\pi$
$757$		−	14.6543i		−	0.532618i	0.963888	−	0.266309i	$-0.0858041\pi$
$758$		0			0
$759$	−1.89909	−	14.5831i	−0.0689327	−	0.529331i
$760$		0			0
$761$			7.11993i			0.258097i	0.991638	+	0.129049i	$0.0411923\pi$
$761$			7.11993i			0.258097i	−0.991638	+	0.129049i	$0.958808\pi$
$762$		0			0
$763$	−42.3638			−1.53367
$764$		0			0
$765$	−18.2878	−	9.50060i	−0.661195	−	0.343495i
$766$		0			0
$767$		−	49.0711i		−	1.77185i
$768$		0			0
$769$			32.0781i			1.15676i	0.815766	+	0.578382i	$0.196316\pi$
$769$			32.0781i			1.15676i	−0.815766	+	0.578382i	$0.803684\pi$
$770$		0			0
$771$	2.51356	−	10.2907i	0.0905238	−	0.370611i
$772$		0			0
$773$	−8.51413			−0.306232			−0.153116	−	0.988208i	$-0.548931\pi$
$773$	−8.51413			−0.306232			−0.153116	+	0.988208i	$0.548931\pi$
$774$		0			0
$775$	−5.05868			−0.181713
$776$		0			0
$777$	20.6570	+	5.04559i	0.741066	+	0.181010i
$778$		0			0
$779$	6.68830			0.239633
$780$		0			0
$781$			28.8660i			1.03291i
$782$		0			0
$783$	−14.0381	+	16.0479i	−0.501682	+	0.573507i
$784$		0			0
$785$			12.9122i			0.460855i
$786$		0			0
$787$		−	11.9810i		−	0.427078i	−0.976935	−	0.213539i	$-0.931501\pi$
$787$		−	11.9810i		−	0.427078i	0.976935	−	0.213539i	$-0.0684990\pi$
$788$		0			0
$789$	−13.7588	−	3.36066i	−0.489825	−	0.119643i
$790$		0			0
$791$			19.9937i			0.710893i
$792$		0			0
$793$			49.8321i			1.76959i
$794$		0			0
$795$	1.39909	+	0.341735i	0.0496205	+	0.0121201i
$796$		0			0
$797$	15.5987			0.552533			0.276266	−	0.961081i	$-0.410903\pi$
$797$	15.5987			0.552533			0.276266	+	0.961081i	$0.410903\pi$
$798$		0			0
$799$		−	39.8586i		−	1.41010i
$800$		0			0
$801$	26.0392	+	13.5275i	0.920051	+	0.477972i
$802$		0			0
$803$	−2.99225			−0.105594
$804$		0			0
$805$	−19.0510	+	7.36864i	−0.671460	+	0.259710i
$806$		0			0
$807$	0.00355402	−	0.0145504i	0.000125107	−	0.000512198i
$808$		0			0
$809$			12.8819i			0.452904i	0.974022	+	0.226452i	$0.0727127\pi$
$809$			12.8819i			0.452904i	−0.974022	+	0.226452i	$0.927287\pi$
$810$		0			0
$811$	18.9804			0.666491			0.333245	−	0.942840i	$-0.391856\pi$
$811$	18.9804			0.666491			0.333245	+	0.942840i	$0.391856\pi$
$812$		0			0
$813$	−40.2252	−	9.82525i	−1.41076	−	0.344587i
$814$		0			0
$815$	−11.1862			−0.391837
$816$		0			0
$817$	−4.80688			−0.168171
$818$		0			0
$819$	−24.4281	+	47.0219i	−0.853589	+	1.64308i
$820$		0			0
$821$		−	10.5143i		−	0.366952i	−0.983024	−	0.183476i	$-0.941265\pi$
$821$		−	10.5143i		−	0.366952i	0.983024	−	0.183476i	$-0.0587350\pi$
$822$		0			0
$823$	20.7205			0.722270			0.361135	−	0.932514i	$-0.382389\pi$
$823$	20.7205			0.722270			0.361135	+	0.932514i	$0.382389\pi$
$824$		0			0
$825$	−2.97888	−	0.727608i	−0.103711	−	0.0253321i
$826$		0			0
$827$	45.9796			1.59887			0.799434	−	0.600754i	$-0.205133\pi$
$827$	45.9796			1.59887			0.799434	+	0.600754i	$0.205133\pi$
$828$		0			0
$829$	46.1501			1.60286			0.801429	−	0.598089i	$-0.204073\pi$
$829$	46.1501			1.60286			0.801429	+	0.598089i	$0.204073\pi$
$830$		0			0
$831$	−38.0593	−	9.29619i	−1.32026	−	0.322481i
$832$		0			0
$833$	76.5309			2.65164
$834$		0			0
$835$		−	0.273329i		−	0.00945894i
$836$		0			0
$837$	19.7843	+	17.3066i	0.683846	+	0.598203i
$838$		0			0
$839$	53.3483			1.84179			0.920894	−	0.389812i	$-0.127460\pi$
$839$	53.3483			1.84179			0.920894	+	0.389812i	$0.127460\pi$
$840$		0			0
$841$	12.1628			0.419405
$842$		0			0
$843$	−54.8847	−	13.4059i	−1.89033	−	0.461724i
$844$		0			0
$845$	4.19756			0.144401
$846$		0			0
$847$		−	33.5012i		−	1.15112i
$848$		0			0
$849$	−5.05463	+	20.6940i	−0.173474	+	0.710216i
$850$		0			0
$851$	−12.8930	+	4.98680i	−0.441965	+	0.170945i
$852$		0			0
$853$	−20.9563			−0.717530			−0.358765	−	0.933428i	$-0.616802\pi$
$853$	−20.9563			−0.717530			−0.358765	+	0.933428i	$0.616802\pi$
$854$		0			0
$855$	−2.02684	+	3.90149i	−0.0693166	+	0.133428i
$856$		0			0
$857$			44.3899i			1.51633i	0.652063	+	0.758165i	$0.273904\pi$
$857$			44.3899i			1.51633i	−0.652063	+	0.758165i	$0.726096\pi$
$858$		0			0
$859$	−13.2571			−0.452326			−0.226163	−	0.974089i	$-0.572618\pi$
$859$	−13.2571			−0.452326			−0.226163	+	0.974089i	$0.572618\pi$
$860$		0			0
$861$	−32.7061	−	7.98866i	−1.11462	−	0.272253i
$862$		0			0
$863$		−	34.1455i		−	1.16233i	−0.813787	−	0.581164i	$-0.802598\pi$
$863$		−	34.1455i		−	1.16233i	0.813787	−	0.581164i	$-0.197402\pi$
$864$		0			0
$865$			12.8456i			0.436762i
$866$		0			0
$867$	−50.7960	−	12.4072i	−1.72512	−	0.421372i
$868$		0			0
$869$			4.64829i			0.157682i
$870$		0			0
$871$			43.1269i			1.46130i
$872$		0			0
$873$	12.8190	−	24.6753i	0.433856	−	0.835132i
$874$		0			0
$875$			4.25920i			0.143987i
$876$		0			0
$877$	−6.23583			−0.210569			−0.105284	−	0.994442i	$-0.533575\pi$
$877$	−6.23583			−0.210569			−0.105284	+	0.994442i	$0.533575\pi$
$878$		0			0
$879$	37.2920	+	9.10879i	1.25783	+	0.307232i
$880$		0			0
$881$	−32.4198			−1.09225			−0.546125	−	0.837704i	$-0.683898\pi$
$881$	−32.4198			−1.09225			−0.546125	+	0.837704i	$0.683898\pi$
$882$		0			0
$883$	47.9132			1.61241			0.806204	−	0.591637i	$-0.201518\pi$
$883$	47.9132			1.61241			0.806204	+	0.591637i	$0.201518\pi$
$884$		0			0
$885$	−4.86311	+	19.9099i	−0.163472	+	0.669265i
$886$		0			0
$887$			6.06146i			0.203524i	0.994809	+	0.101762i	$0.0324480\pi$
$887$			6.06146i			0.203524i	−0.994809	+	0.101762i	$0.967552\pi$
$888$		0			0
$889$			10.6445i			0.357005i
$890$		0			0
$891$	9.16102	+	13.0369i	0.306906	+	0.436752i
$892$		0			0
$893$	−8.50338			−0.284555
$894$		0			0
$895$		−	5.54592i		−	0.185380i
$896$		0			0
$897$	−4.44840	−	34.1591i	−0.148528	−	1.14054i
$898$		0			0
$899$			20.7574i			0.692298i
$900$		0			0
$901$	5.71201			0.190295
$902$		0			0
$903$	23.5059	+	5.74145i	0.782227	+	0.191064i
$904$		0			0
$905$		−	9.79986i		−	0.325758i
$906$		0			0
$907$			26.5492i			0.881552i	0.897617	+	0.440776i	$0.145297\pi$
$907$			26.5492i			0.881552i	−0.897617	+	0.440776i	$0.854703\pi$
$908$		0			0
$909$	14.0566	−	27.0577i	0.466229	−	0.897446i
$910$		0			0
$911$	−33.4165			−1.10714			−0.553569	−	0.832804i	$-0.686734\pi$
$911$	−33.4165			−1.10714			−0.553569	+	0.832804i	$0.686734\pi$
$912$		0			0
$913$	9.24447			0.305947
$914$		0			0
$915$	4.93853	−	20.2187i	0.163263	−	0.668409i
$916$		0			0
$917$	−26.1555			−0.863732
$918$		0			0
$919$		−	43.3201i		−	1.42900i	−0.699637	−	0.714498i	$-0.746655\pi$
$919$		−	43.3201i		−	1.42900i	0.699637	−	0.714498i	$-0.253345\pi$
$920$		0			0
$921$	−45.7036	−	11.1634i	−1.50599	−	0.367846i
$922$		0			0
$923$			67.6153i			2.22558i
$924$		0			0
$925$			2.88245i			0.0947745i
$926$		0			0
$927$	−12.9564	+	24.9399i	−0.425544	+	0.819132i
$928$		0			0
$929$			18.3726i			0.602785i	0.953500	+	0.301393i	$0.0974515\pi$
$929$			18.3726i			0.602785i	−0.953500	+	0.301393i	$0.902549\pi$
$930$		0			0
$931$		−	16.3270i		−	0.535095i
$932$		0			0
$933$	−5.67339	+	23.2273i	−0.185739	+	0.760426i
$934$		0			0
$935$	−12.1618			−0.397733
$936$		0			0
$937$		−	36.2126i		−	1.18301i	−0.806300	−	0.591507i	$-0.798533\pi$
$937$		−	36.2126i		−	1.18301i	0.806300	−	0.591507i	$-0.201467\pi$
$938$		0			0
$939$	13.5510	−	55.4788i	0.442221	−	1.81048i
$940$		0			0
$941$	−16.1480			−0.526409			−0.263204	−	0.964740i	$-0.584779\pi$
$941$	−16.1480			−0.526409			−0.263204	+	0.964740i	$0.584779\pi$
$942$		0			0
$943$	20.4134	−	7.89558i	0.664750	−	0.257115i
$944$		0			0
$945$	14.5714	−	16.6576i	0.474008	−	0.541871i
$946$		0			0
$947$		−	32.7857i		−	1.06539i	−0.846306	−	0.532697i	$-0.821179\pi$
$947$		−	32.7857i		−	1.06539i	0.846306	−	0.532697i	$-0.178821\pi$
$948$		0			0
$949$	−7.00900			−0.227522
$950$		0			0
$951$	−4.50674	+	18.4509i	−0.146141	+	0.598311i
$952$		0			0
$953$	−3.84790			−0.124646			−0.0623228	−	0.998056i	$-0.519851\pi$
$953$	−3.84790			−0.124646			−0.0623228	+	0.998056i	$0.519851\pi$
$954$		0			0
$955$	−18.3769			−0.594663
$956$		0			0
$957$	−2.98561	+	12.2233i	−0.0965111	+	0.395123i
$958$		0			0
$959$		−	49.0087i		−	1.58257i
$960$		0			0
$961$	−5.40973			−0.174507
$962$		0			0
$963$	−4.22992	−	2.19747i	−0.136307	−	0.0708125i
$964$		0			0
$965$	−15.0944			−0.485907
$966$		0			0
$967$	25.0017			0.804001			0.402001	−	0.915639i	$-0.368315\pi$
$967$	25.0017			0.804001			0.402001	+	0.915639i	$0.368315\pi$
$968$		0			0
$969$	−4.13747	+	16.9391i	−0.132915	+	0.544162i
$970$		0			0
$971$	26.2788			0.843326			0.421663	−	0.906753i	$-0.361447\pi$
$971$	26.2788			0.843326			0.421663	+	0.906753i	$0.361447\pi$
$972$		0			0
$973$		−	58.8831i		−	1.88771i
$974$		0			0
$975$	−6.97767	−	1.70434i	−0.223464	−	0.0545825i
$976$		0			0
$977$	21.4585			0.686517			0.343258	−	0.939241i	$-0.388469\pi$
$977$	21.4585			0.686517			0.343258	+	0.939241i	$0.388469\pi$
$978$		0			0
$979$	17.3167			0.553444
$980$		0			0
$981$	−13.7561	+	26.4793i	−0.439199	+	0.845418i
$982$		0			0
$983$	14.7143			0.469313			0.234656	−	0.972078i	$-0.424603\pi$
$983$	14.7143			0.469313			0.234656	+	0.972078i	$0.424603\pi$
$984$		0			0
$985$			21.1615i			0.674262i
$986$		0			0
$987$	41.5820	+	10.1566i	1.32357	+	0.323290i
$988$		0			0
$989$	−14.6711	+	5.67455i	−0.466513	+	0.180440i
$990$		0			0
$991$	38.7963			1.23240			0.616202	−	0.787588i	$-0.288670\pi$
$991$	38.7963			1.23240			0.616202	+	0.787588i	$0.288670\pi$
$992$		0			0
$993$	30.3613	+	7.41592i	0.963486	+	0.235337i
$994$		0			0
$995$			3.16726i			0.100409i
$996$		0			0
$997$	11.4649			0.363097			0.181549	−	0.983382i	$-0.441889\pi$
$997$	11.4649			0.363097			0.181549	+	0.983382i	$0.441889\pi$
$998$		0			0
$999$	9.86135	−	11.2732i	0.311999	−	0.356667i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.b.1241.1	yes	16
3.2	odd	2		1380.2.i.a.1241.2	yes	16
23.22	odd	2		1380.2.i.a.1241.1	✓	16
69.68	even	2	inner	1380.2.i.b.1241.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.1	✓	16	23.22	odd	2
1380.2.i.a.1241.2	yes	16	3.2	odd	2
1380.2.i.b.1241.1	yes	16	1.1	even	1	trivial
1380.2.i.b.1241.2	yes	16	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.68259 - 0.410981i) q^{3} +1.00000 q^{5} +4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} +O(q^{10})\)	\(q+(-1.68259 - 0.410981i) q^{3} +1.00000 q^{5} +4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} +1.77042 q^{11} +4.14699 q^{13} +(-1.68259 - 0.410981i) q^{15} -6.86944 q^{17} +1.46552i q^{19} +(1.75045 - 7.16646i) q^{21} +(1.73005 + 4.47291i) q^{23} +1.00000 q^{25} +(-3.91096 - 3.42116i) q^{27} -4.10332i q^{29} -5.05868 q^{31} +(-2.97888 - 0.727608i) q^{33} +4.25920i q^{35} +2.88245i q^{37} +(-6.97767 - 1.70434i) q^{39} -4.56378i q^{41} +3.27999i q^{43} +(2.66219 + 1.38302i) q^{45} +5.80230i q^{47} -11.1408 q^{49} +(11.5584 + 2.82321i) q^{51} -0.831510 q^{53} +1.77042 q^{55} +(0.602301 - 2.46586i) q^{57} -11.8329i q^{59} +12.0164i q^{61} +(-5.89057 + 11.3388i) q^{63} +4.14699 q^{65} +10.3996i q^{67} +(-1.07268 - 8.23707i) q^{69} +16.3046i q^{71} -1.69014 q^{73} +(-1.68259 - 0.410981i) q^{75} +7.54056i q^{77} +2.62553i q^{79} +(5.17450 + 7.36373i) q^{81} +5.22163 q^{83} -6.86944 q^{85} +(-1.68639 + 6.90419i) q^{87} +9.78113 q^{89} +17.6629i q^{91} +(8.51167 + 2.07902i) q^{93} +1.46552i q^{95} -9.26880i q^{97} +(4.71318 + 2.44853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} + 2 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 + 2 * q^9	\( 16 q + 16 q^{5} + 2 q^{9} + 4 q^{21} - 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} - 2 q^{33} - 14 q^{39} + 2 q^{45} + 8 q^{49} + 2 q^{51} - 4 q^{53} + 2 q^{57} - 30 q^{63} + 8 q^{69} - 4 q^{73} + 10 q^{81} + 4 q^{83} - 10 q^{87} + 28 q^{89} - 10 q^{93} + 26 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 + 2 * q^9 + 4 * q^21 - 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 - 2 * q^33 - 14 * q^39 + 2 * q^45 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 2 * q^57 - 30 * q^63 + 8 * q^69 - 4 * q^73 + 10 * q^81 + 4 * q^83 - 10 * q^87 + 28 * q^89 - 10 * q^93 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more