LMFDB - Embedded newform 2340.2.y.b.1457.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(53,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3, 0]))

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

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

chi := DirichletCharacter("2340.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.y (of order $4$, degree $2$, 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:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.1
Character	$\chi$	$=$	2340.1457
Dual form			2340.2.y.b.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.15870 - 0.583115i) q^{5} +(-1.95569 - 1.95569i) q^{7} +O(q^{10})$	$q+(-2.15870 - 0.583115i) q^{5} +(-1.95569 - 1.95569i) q^{7} +5.98275i q^{11} +(-0.707107 + 0.707107i) q^{13} +(3.87703 - 3.87703i) q^{17} -0.340690i q^{19} +(5.93175 + 5.93175i) q^{23} +(4.31995 + 2.51754i) q^{25} -1.93115 q^{29} -5.01279 q^{31} +(3.08135 + 5.36214i) q^{35} +(-5.36873 - 5.36873i) q^{37} -8.26967i q^{41} +(-1.30379 + 1.30379i) q^{43} +(9.14231 - 9.14231i) q^{47} +0.649454i q^{49} +(0.0960951 + 0.0960951i) q^{53} +(3.48863 - 12.9150i) q^{55} -0.114314 q^{59} -6.89446 q^{61} +(1.93875 - 1.11411i) q^{65} +(-5.13411 - 5.13411i) q^{67} -15.8186i q^{71} +(5.50572 - 5.50572i) q^{73} +(11.7004 - 11.7004i) q^{77} -6.60341i q^{79} +(0.738605 + 0.738605i) q^{83} +(-10.6301 + 6.10858i) q^{85} -10.0964 q^{89} +2.76576 q^{91} +(-0.198662 + 0.735448i) q^{95} +(5.79504 + 5.79504i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} - 8 q^{7}+O(q^{10}) $ 24 * q + 4 * q^5 - 8 * q^7	$ 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83} - 32 q^{85} - 128 q^{89} - 8 q^{91} - 16 q^{95} + 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.15870	−	0.583115i	−0.965399	−	0.260777i
$5$	−2.15870	−	0.583115i	−0.965399	−	0.260777i
$6$		0			0
$7$	−1.95569	−	1.95569i	−0.739182	−	0.739182i	0.233238	−	0.972420i	$-0.425068\pi$
$7$	−1.95569	−	1.95569i	−0.739182	−	0.739182i	−0.972420	+	0.233238i	$0.925068\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.98275i			1.80387i	0.431874	+	0.901934i	$0.357852\pi$
$11$			5.98275i			1.80387i	−0.431874	+	0.901934i	$0.642148\pi$
$12$		0			0
$13$	−0.707107	+	0.707107i	−0.196116	+	0.196116i
$13$	−0.707107	+	0.707107i	−0.196116	+	0.196116i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.87703	−	3.87703i	0.940317	−	0.940317i	−0.0579997	−	0.998317i	$-0.518472\pi$
$17$	3.87703	−	3.87703i	0.940317	−	0.940317i	0.998317	+	0.0579997i	$0.0184722\pi$
$18$		0			0
$19$		−	0.340690i		−	0.0781598i	−0.999236	−	0.0390799i	$-0.987557\pi$
$19$		−	0.340690i		−	0.0781598i	0.999236	−	0.0390799i	$-0.0124427\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.93175	+	5.93175i	1.23686	+	1.23686i	0.961279	+	0.275577i	$0.0888689\pi$
$23$	5.93175	+	5.93175i	1.23686	+	1.23686i	0.275577	+	0.961279i	$0.411131\pi$
$24$		0			0
$25$	4.31995	+	2.51754i	0.863991	+	0.503508i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.93115			−0.358606			−0.179303	−	0.983794i	$-0.557384\pi$
$29$	−1.93115			−0.358606			−0.179303	+	0.983794i	$0.557384\pi$
$30$		0			0
$31$	−5.01279			−0.900324			−0.450162	−	0.892947i	$-0.648634\pi$
$31$	−5.01279			−0.900324			−0.450162	+	0.892947i	$0.648634\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.08135	+	5.36214i	0.520844	+	0.906367i
$36$		0			0
$37$	−5.36873	−	5.36873i	−0.882614	−	0.882614i	0.111186	−	0.993800i	$-0.464535\pi$
$37$	−5.36873	−	5.36873i	−0.882614	−	0.882614i	−0.993800	+	0.111186i	$0.964535\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.26967i		−	1.29151i	−0.763547	−	0.645753i	$-0.776544\pi$
$41$		−	8.26967i		−	1.29151i	0.763547	−	0.645753i	$-0.223456\pi$
$42$		0			0
$43$	−1.30379	+	1.30379i	−0.198827	+	0.198827i	−0.799497	−	0.600670i	$-0.794900\pi$
$43$	−1.30379	+	1.30379i	−0.198827	+	0.198827i	0.600670	+	0.799497i	$0.294900\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.14231	−	9.14231i	1.33354	−	1.33354i	0.431366	−	0.902177i	$-0.358032\pi$
$47$	9.14231	−	9.14231i	1.33354	−	1.33354i	0.902177	−	0.431366i	$-0.141968\pi$
$48$		0			0
$49$			0.649454i			0.0927791i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.0960951	+	0.0960951i	0.0131997	+	0.0131997i	0.713676	−	0.700476i	$-0.247029\pi$
$53$	0.0960951	+	0.0960951i	0.0131997	+	0.0131997i	−0.700476	+	0.713676i	$0.747029\pi$
$54$		0			0
$55$	3.48863	−	12.9150i	0.470407	−	1.74145i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.114314			−0.0148824			−0.00744120	−	0.999972i	$-0.502369\pi$
$59$	−0.114314			−0.0148824			−0.00744120	+	0.999972i	$0.502369\pi$
$60$		0			0
$61$	−6.89446			−0.882745			−0.441372	−	0.897324i	$-0.645508\pi$
$61$	−6.89446			−0.882745			−0.441372	+	0.897324i	$0.645508\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.93875	−	1.11411i	0.240473	−	0.138188i
$66$		0			0
$67$	−5.13411	−	5.13411i	−0.627232	−	0.627232i	0.320139	−	0.947371i	$-0.396270\pi$
$67$	−5.13411	−	5.13411i	−0.627232	−	0.627232i	−0.947371	+	0.320139i	$0.896270\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	15.8186i		−	1.87732i	−0.344849	−	0.938658i	$-0.612070\pi$
$71$		−	15.8186i		−	1.87732i	0.344849	−	0.938658i	$-0.387930\pi$
$72$		0			0
$73$	5.50572	−	5.50572i	0.644395	−	0.644395i	−0.307238	−	0.951633i	$-0.599405\pi$
$73$	5.50572	−	5.50572i	0.644395	−	0.644395i	0.951633	+	0.307238i	$0.0994046\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.7004	−	11.7004i	1.33339	−	1.33339i
$78$		0			0
$79$		−	6.60341i		−	0.742942i	−0.928445	−	0.371471i	$-0.878854\pi$
$79$		−	6.60341i		−	0.742942i	0.928445	−	0.371471i	$-0.121146\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.738605	+	0.738605i	0.0810725	+	0.0810725i	0.746480	−	0.665408i	$-0.231742\pi$
$83$	0.738605	+	0.738605i	0.0810725	+	0.0810725i	−0.665408	+	0.746480i	$0.731742\pi$
$84$		0			0
$85$	−10.6301	+	6.10858i	−1.15299	+	0.662568i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0964			−1.07022			−0.535110	−	0.844782i	$-0.679730\pi$
$89$	−10.0964			−1.07022			−0.535110	+	0.844782i	$0.679730\pi$
$90$		0			0
$91$	2.76576			0.289931
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.198662	+	0.735448i	−0.0203823	+	0.0754554i
$96$		0			0
$97$	5.79504	+	5.79504i	0.588398	+	0.588398i	0.937197	−	0.348800i	$-0.113411\pi$
$97$	5.79504	+	5.79504i	0.588398	+	0.588398i	−0.348800	+	0.937197i	$0.613411\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			4.74659i			0.472304i	0.971716	+	0.236152i	$0.0758862\pi$
$101$			4.74659i			0.472304i	−0.971716	+	0.236152i	$0.924114\pi$
$102$		0			0
$103$	4.38950	−	4.38950i	0.432510	−	0.432510i	−0.456971	−	0.889481i	$-0.651066\pi$
$103$	4.38950	−	4.38950i	0.432510	−	0.432510i	0.889481	+	0.456971i	$0.151066\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.75740	+	7.75740i	−0.749936	+	0.749936i	−0.974467	−	0.224531i	$-0.927915\pi$
$107$	−7.75740	+	7.75740i	−0.749936	+	0.749936i	0.224531	+	0.974467i	$0.427915\pi$
$108$		0			0
$109$		−	19.6851i		−	1.88549i	−0.333514	−	0.942745i	$-0.608235\pi$
$109$		−	19.6851i		−	1.88549i	0.333514	−	0.942745i	$-0.391765\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.87193	+	8.87193i	0.834601	+	0.834601i	0.988142	−	0.153541i	$-0.0490678\pi$
$113$	8.87193	+	8.87193i	0.834601	+	0.834601i	−0.153541	+	0.988142i	$0.549068\pi$
$114$		0			0
$115$	−9.34597	−	16.2638i	−0.871516	−	1.51660i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−15.1645			−1.39013
$120$		0			0
$121$	−24.7933			−2.25394
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.85746	−	7.95363i	−0.702793	−	0.711395i
$126$		0			0
$127$	−0.0699012	−	0.0699012i	−0.00620273	−	0.00620273i	0.703999	−	0.710201i	$-0.251396\pi$
$127$	−0.0699012	−	0.0699012i	−0.00620273	−	0.00620273i	−0.710201	+	0.703999i	$0.751396\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.60143i			0.664140i	0.943255	+	0.332070i	$0.107747\pi$
$131$			7.60143i			0.664140i	−0.943255	+	0.332070i	$0.892253\pi$
$132$		0			0
$133$	−0.666285	+	0.666285i	−0.0577743	+	0.0577743i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.90396	−	4.90396i	0.418973	−	0.418973i	−0.465877	−	0.884850i	$-0.654261\pi$
$137$	4.90396	−	4.90396i	0.418973	−	0.418973i	0.884850	+	0.465877i	$0.154261\pi$
$138$		0			0
$139$		−	13.4210i		−	1.13835i	−0.822216	−	0.569176i	$-0.807262\pi$
$139$		−	13.4210i		−	1.13835i	0.822216	−	0.569176i	$-0.192738\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.23044	−	4.23044i	−0.353768	−	0.353768i
$144$		0			0
$145$	4.16878	+	1.12609i	0.346198	+	0.0935163i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.8931			1.13817			0.569084	−	0.822279i	$-0.307298\pi$
$149$	13.8931			1.13817			0.569084	+	0.822279i	$0.307298\pi$
$150$		0			0
$151$	20.6292			1.67878			0.839389	−	0.543531i	$-0.182913\pi$
$151$	20.6292			1.67878			0.839389	+	0.543531i	$0.182913\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.8211	+	2.92304i	0.869172	+	0.234784i
$156$		0			0
$157$	−1.74709	−	1.74709i	−0.139433	−	0.139433i	0.633945	−	0.773378i	$-0.281435\pi$
$157$	−1.74709	−	1.74709i	−0.139433	−	0.139433i	−0.773378	+	0.633945i	$0.781435\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	23.2014i		−	1.82852i
$162$		0			0
$163$	−2.38679	+	2.38679i	−0.186948	+	0.186948i	−0.794375	−	0.607428i	$-0.792202\pi$
$163$	−2.38679	+	2.38679i	−0.186948	+	0.186948i	0.607428	+	0.794375i	$0.292202\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.46375	−	1.46375i	0.113268	−	0.113268i	−0.648201	−	0.761469i	$-0.724478\pi$
$167$	1.46375	−	1.46375i	0.113268	−	0.113268i	0.761469	+	0.648201i	$0.224478\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.08237	−	1.08237i	−0.0822914	−	0.0822914i	0.664763	−	0.747054i	$-0.268533\pi$
$173$	−1.08237	−	1.08237i	−0.0822914	−	0.0822914i	−0.747054	+	0.664763i	$0.768533\pi$
$174$		0			0
$175$	−3.52497	−	13.3720i	−0.266462	−	1.01083i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.05289			0.527158			0.263579	−	0.964638i	$-0.415097\pi$
$179$	7.05289			0.527158			0.263579	+	0.964638i	$0.415097\pi$
$180$		0			0
$181$	6.88898			0.512054			0.256027	−	0.966670i	$-0.417586\pi$
$181$	6.88898			0.512054			0.256027	+	0.966670i	$0.417586\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.45888	+	14.7201i	0.621909	+	1.08224i
$186$		0			0
$187$	23.1953	+	23.1953i	1.69621	+	1.69621i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.5802i			1.12735i	0.825998	+	0.563673i	$0.190612\pi$
$191$			15.5802i			1.12735i	−0.825998	+	0.563673i	$0.809388\pi$
$192$		0			0
$193$	−16.0875	+	16.0875i	−1.15800	+	1.15800i	−0.173097	+	0.984905i	$0.555377\pi$
$193$	−16.0875	+	16.0875i	−1.15800	+	1.15800i	−0.984905	+	0.173097i	$0.944623\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.942431	+	0.942431i	−0.0671454	+	0.0671454i	−0.739882	−	0.672737i	$-0.765119\pi$
$197$	−0.942431	+	0.942431i	−0.0671454	+	0.0671454i	0.672737	+	0.739882i	$0.265119\pi$
$198$		0			0
$199$		−	4.98784i		−	0.353578i	−0.984249	−	0.176789i	$-0.943429\pi$
$199$		−	4.98784i		−	0.353578i	0.984249	−	0.176789i	$-0.0565711\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.77674	+	3.77674i	0.265075	+	0.265075i
$204$		0			0
$205$	−4.82217	+	17.8517i	−0.336795	+	1.24682i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.03827			0.140990
$210$		0			0
$211$	15.0458			1.03580			0.517899	−	0.855442i	$-0.326714\pi$
$211$	15.0458			1.03580			0.517899	+	0.855442i	$0.326714\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.57476	−	2.05423i	0.243796	−	0.140098i
$216$		0			0
$217$	9.80348	+	9.80348i	0.665503	+	0.665503i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5.48294i			0.368823i
$222$		0			0
$223$	11.6223	−	11.6223i	0.778290	−	0.778290i	−0.201250	−	0.979540i	$-0.564500\pi$
$223$	11.6223	−	11.6223i	0.778290	−	0.778290i	0.979540	+	0.201250i	$0.0645004\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.0320	−	13.0320i	0.864962	−	0.864962i	−0.126947	−	0.991909i	$-0.540518\pi$
$227$	13.0320	−	13.0320i	0.864962	−	0.864962i	0.991909	+	0.126947i	$0.0405179\pi$
$228$		0			0
$229$		−	28.0271i		−	1.85209i	−0.377419	−	0.926043i	$-0.623188\pi$
$229$		−	28.0271i		−	1.85209i	0.377419	−	0.926043i	$-0.376812\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.52241	+	5.52241i	0.361785	+	0.361785i	0.864470	−	0.502685i	$-0.167654\pi$
$233$	5.52241	+	5.52241i	0.361785	+	0.361785i	−0.502685	+	0.864470i	$0.667654\pi$
$234$		0			0
$235$	−25.0665	+	14.4045i	−1.63516	+	0.939644i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	27.0935			1.75253			0.876265	−	0.481830i	$-0.160028\pi$
$239$	27.0935			1.75253			0.876265	+	0.481830i	$0.160028\pi$
$240$		0			0
$241$	8.43396			0.543279			0.271639	−	0.962399i	$-0.412434\pi$
$241$	8.43396			0.543279			0.271639	+	0.962399i	$0.412434\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.378706	−	1.40197i	0.0241947	−	0.0895689i
$246$		0			0
$247$	0.240905	+	0.240905i	0.0153284	+	0.0153284i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	18.4081i		−	1.16191i	−0.813937	−	0.580954i	$-0.802680\pi$
$251$		−	18.4081i		−	1.16191i	0.813937	−	0.580954i	$-0.197320\pi$
$252$		0			0
$253$	−35.4882	+	35.4882i	−2.23112	+	2.23112i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.18921	−	4.18921i	0.261316	−	0.261316i	−0.564273	−	0.825588i	$-0.690843\pi$
$257$	4.18921	−	4.18921i	0.261316	−	0.261316i	0.825588	+	0.564273i	$0.190843\pi$
$258$		0			0
$259$			20.9992i			1.30482i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.55167	−	6.55167i	−0.403993	−	0.403993i	0.475644	−	0.879638i	$-0.342215\pi$
$263$	−6.55167	−	6.55167i	−0.403993	−	0.403993i	−0.879638	+	0.475644i	$0.842215\pi$
$264$		0			0
$265$	−0.151406	−	0.263475i	−0.00930079	−	0.0161851i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.9756			−0.669192			−0.334596	−	0.942362i	$-0.608600\pi$
$269$	−10.9756			−0.669192			−0.334596	+	0.942362i	$0.608600\pi$
$270$		0			0
$271$	−13.7211			−0.833498			−0.416749	−	0.909022i	$-0.636831\pi$
$271$	−13.7211			−0.833498			−0.416749	+	0.909022i	$0.636831\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.0618	+	25.8452i	−0.908261	+	1.55852i
$276$		0			0
$277$	−20.3866	−	20.3866i	−1.22491	−	1.22491i	−0.965864	−	0.259050i	$-0.916591\pi$
$277$	−20.3866	−	20.3866i	−1.22491	−	1.22491i	−0.259050	−	0.965864i	$-0.583409\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.3982i			0.978232i	0.872219	+	0.489116i	$0.162681\pi$
$281$			16.3982i			0.978232i	−0.872219	+	0.489116i	$0.837319\pi$
$282$		0			0
$283$	9.18541	−	9.18541i	0.546016	−	0.546016i	−0.379270	−	0.925286i	$-0.623825\pi$
$283$	9.18541	−	9.18541i	0.546016	−	0.546016i	0.925286	+	0.379270i	$0.123825\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.1729	+	16.1729i	−0.954657	+	0.954657i
$288$		0			0
$289$		−	13.0627i		−	0.768392i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.5035	−	15.5035i	−0.905725	−	0.905725i	0.0901992	−	0.995924i	$-0.471250\pi$
$293$	−15.5035	−	15.5035i	−0.905725	−	0.905725i	−0.995924	+	0.0901992i	$0.971250\pi$
$294$		0			0
$295$	0.246769	+	0.0666582i	0.0143675	+	0.00388099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.38877			−0.485135
$300$		0			0
$301$	5.09963			0.293938
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.8831	+	4.02026i	0.852201	+	0.230200i
$306$		0			0
$307$	−14.7219	−	14.7219i	−0.840221	−	0.840221i	0.148666	−	0.988887i	$-0.452502\pi$
$307$	−14.7219	−	14.7219i	−0.840221	−	0.840221i	−0.988887	+	0.148666i	$0.952502\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.18137i		−	0.123694i	−0.998086	−	0.0618471i	$-0.980301\pi$
$311$		−	2.18137i		−	0.123694i	0.998086	−	0.0618471i	$-0.0196991\pi$
$312$		0			0
$313$	9.34208	−	9.34208i	0.528046	−	0.528046i	−0.391944	−	0.919989i	$-0.628197\pi$
$313$	9.34208	−	9.34208i	0.528046	−	0.528046i	0.919989	+	0.391944i	$0.128197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.5890	+	20.5890i	−1.15639	+	1.15639i	−0.171145	+	0.985246i	$0.554747\pi$
$317$	−20.5890	+	20.5890i	−1.15639	+	1.15639i	−0.985246	+	0.171145i	$0.945253\pi$
$318$		0			0
$319$		−	11.5536i		−	0.646878i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.32087	−	1.32087i	−0.0734949	−	0.0734949i
$324$		0			0
$325$	−4.83484	+	1.27450i	−0.268189	+	0.0706965i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−35.7591			−1.97146
$330$		0			0
$331$	14.3245			0.787347			0.393674	−	0.919250i	$-0.371204\pi$
$331$	14.3245			0.787347			0.393674	+	0.919250i	$0.371204\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.08922	+	14.0768i	0.441961	+	0.769097i
$336$		0			0
$337$	−6.11701	−	6.11701i	−0.333215	−	0.333215i	0.520591	−	0.853806i	$-0.325711\pi$
$337$	−6.11701	−	6.11701i	−0.333215	−	0.333215i	−0.853806	+	0.520591i	$0.825711\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	29.9903i		−	1.62407i
$342$		0			0
$343$	−12.4197	+	12.4197i	−0.670601	+	0.670601i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.3070	−	10.3070i	0.553308	−	0.553308i	−0.374086	−	0.927394i	$-0.622044\pi$
$347$	10.3070	−	10.3070i	0.553308	−	0.553308i	0.927394	+	0.374086i	$0.122044\pi$
$348$		0			0
$349$		−	20.6641i		−	1.10612i	−0.833140	−	0.553061i	$-0.813459\pi$
$349$		−	20.6641i		−	1.10612i	0.833140	−	0.553061i	$-0.186541\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.99930	+	7.99930i	0.425760	+	0.425760i	0.887181	−	0.461421i	$-0.152660\pi$
$353$	7.99930	+	7.99930i	0.425760	+	0.425760i	−0.461421	+	0.887181i	$0.652660\pi$
$354$		0			0
$355$	−9.22403	+	34.1475i	−0.489561	+	1.81236i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.6813			−1.19707			−0.598537	−	0.801095i	$-0.704251\pi$
$359$	−22.6813			−1.19707			−0.598537	+	0.801095i	$0.704251\pi$
$360$		0			0
$361$	18.8839			0.993891
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.0956	+	8.67471i	−0.790142	+	0.454055i
$366$		0			0
$367$	20.3206	+	20.3206i	1.06073	+	1.06073i	0.998033	+	0.0626926i	$0.0199688\pi$
$367$	20.3206	+	20.3206i	1.06073	+	1.06073i	0.0626926	+	0.998033i	$0.480031\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.375865i		−	0.0195139i
$372$		0			0
$373$	12.4028	−	12.4028i	0.642194	−	0.642194i	−0.308900	−	0.951094i	$-0.599961\pi$
$373$	12.4028	−	12.4028i	0.642194	−	0.642194i	0.951094	+	0.308900i	$0.0999609\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.36553	−	1.36553i	0.0703285	−	0.0703285i
$378$		0			0
$379$			26.9361i			1.38361i	0.722083	+	0.691806i	$0.243185\pi$
$379$			26.9361i			1.38361i	−0.722083	+	0.691806i	$0.756815\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.60208	+	9.60208i	0.490643	+	0.490643i	0.908509	−	0.417866i	$-0.137222\pi$
$383$	9.60208	+	9.60208i	0.490643	+	0.490643i	−0.417866	+	0.908509i	$0.637222\pi$
$384$		0			0
$385$	−32.0803	+	18.4350i	−1.63497	+	0.939533i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.5559			−1.04223			−0.521114	−	0.853487i	$-0.674483\pi$
$389$	−20.5559			−1.04223			−0.521114	+	0.853487i	$0.674483\pi$
$390$		0			0
$391$	45.9951			2.32607
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.85055	+	14.2548i	−0.193742	+	0.717235i
$396$		0			0
$397$	9.75468	+	9.75468i	0.489574	+	0.489574i	0.908172	−	0.418598i	$-0.137478\pi$
$397$	9.75468	+	9.75468i	0.489574	+	0.489574i	−0.418598	+	0.908172i	$0.637478\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	33.1920i		−	1.65753i	−0.559597	−	0.828765i	$-0.689044\pi$
$401$		−	33.1920i		−	1.65753i	0.559597	−	0.828765i	$-0.310956\pi$
$402$		0			0
$403$	3.54458	−	3.54458i	0.176568	−	0.176568i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	32.1198	−	32.1198i	1.59212	−	1.59212i
$408$		0			0
$409$			27.8800i			1.37858i	0.724487	+	0.689289i	$0.242077\pi$
$409$			27.8800i			1.37858i	−0.724487	+	0.689289i	$0.757923\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.223563	+	0.223563i	0.0110008	+	0.0110008i
$414$		0			0
$415$	−1.16373	−	2.02512i	−0.0571255	−	0.0994091i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.61678			−0.420957			−0.210479	−	0.977598i	$-0.567502\pi$
$419$	−8.61678			−0.420957			−0.210479	+	0.977598i	$0.567502\pi$
$420$		0			0
$421$	−27.0971			−1.32063			−0.660316	−	0.750988i	$-0.729578\pi$
$421$	−27.0971			−1.32063			−0.660316	+	0.750988i	$0.729578\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	26.5091	−	6.98801i	1.28588	−	0.338968i
$426$		0			0
$427$	13.4834	+	13.4834i	0.652509	+	0.652509i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.05423i		−	0.243454i	−0.992564	−	0.121727i	$-0.961157\pi$
$431$		−	5.05423i		−	0.243454i	0.992564	−	0.121727i	$-0.0388432\pi$
$432$		0			0
$433$	18.9340	−	18.9340i	0.909909	−	0.909909i	−0.0863558	−	0.996264i	$-0.527522\pi$
$433$	18.9340	−	18.9340i	0.909909	−	0.909909i	0.996264	+	0.0863558i	$0.0275222\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.02089	−	2.02089i	0.0966724	−	0.0966724i
$438$		0			0
$439$			6.06800i			0.289610i	0.989460	+	0.144805i	$0.0462555\pi$
$439$			6.06800i			0.289610i	−0.989460	+	0.144805i	$0.953745\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.5350	+	29.5350i	1.40325	+	1.40325i	0.789510	+	0.613738i	$0.210335\pi$
$443$	29.5350	+	29.5350i	1.40325	+	1.40325i	0.613738	+	0.789510i	$0.289665\pi$
$444$		0			0
$445$	21.7952	+	5.88738i	1.03319	+	0.279089i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.6092			−1.16138			−0.580691	−	0.814124i	$-0.697217\pi$
$449$	−24.6092			−1.16138			−0.580691	+	0.814124i	$0.697217\pi$
$450$		0			0
$451$	49.4754			2.32970
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.97045	−	1.61276i	−0.279899	−	0.0756073i
$456$		0			0
$457$	1.83165	+	1.83165i	0.0856811	+	0.0856811i	0.748648	−	0.662967i	$-0.230703\pi$
$457$	1.83165	+	1.83165i	0.0856811	+	0.0856811i	−0.662967	+	0.748648i	$0.730703\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			10.6817i			0.497494i	0.968568	+	0.248747i	$0.0800188\pi$
$461$			10.6817i			0.497494i	−0.968568	+	0.248747i	$0.919981\pi$
$462$		0			0
$463$	−1.86660	+	1.86660i	−0.0867485	+	0.0867485i	−0.749149	−	0.662401i	$-0.769538\pi$
$463$	−1.86660	+	1.86660i	−0.0867485	+	0.0867485i	0.662401	+	0.749149i	$0.269538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.856714	+	0.856714i	−0.0396440	+	0.0396440i	−0.726651	−	0.687007i	$-0.758924\pi$
$467$	−0.856714	+	0.856714i	−0.0396440	+	0.0396440i	0.687007	+	0.726651i	$0.258924\pi$
$468$		0			0
$469$			20.0815i			0.927276i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.80027	−	7.80027i	−0.358657	−	0.358657i
$474$		0			0
$475$	0.857701	−	1.47177i	0.0393540	−	0.0675293i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.0760			−0.506077			−0.253038	−	0.967456i	$-0.581430\pi$
$479$	−11.0760			−0.506077			−0.253038	+	0.967456i	$0.581430\pi$
$480$		0			0
$481$	7.59253			0.346190
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.13057	−	15.8889i	−0.414598	−	0.721479i
$486$		0			0
$487$	−3.98039	−	3.98039i	−0.180369	−	0.180369i	0.611148	−	0.791516i	$-0.290708\pi$
$487$	−3.98039	−	3.98039i	−0.180369	−	0.180369i	−0.791516	+	0.611148i	$0.790708\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.4852i			0.653707i	0.945075	+	0.326853i	$0.105988\pi$
$491$			14.4852i			0.653707i	−0.945075	+	0.326853i	$0.894012\pi$
$492$		0			0
$493$	−7.48714	+	7.48714i	−0.337204	+	0.337204i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−30.9362	+	30.9362i	−1.38768	+	1.38768i
$498$		0			0
$499$		−	35.6999i		−	1.59815i	−0.601235	−	0.799073i	$-0.705324\pi$
$499$		−	35.6999i		−	1.59815i	0.601235	−	0.799073i	$-0.294676\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.7161	−	28.7161i	−1.28039	−	1.28039i	−0.940448	−	0.339938i	$-0.889594\pi$
$503$	−28.7161	−	28.7161i	−1.28039	−	1.28039i	−0.339938	−	0.940448i	$-0.610406\pi$
$504$		0			0
$505$	2.76781	−	10.2465i	0.123166	−	0.455961i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.7549			−1.09724			−0.548621	−	0.836071i	$-0.684847\pi$
$509$	−24.7549			−1.09724			−0.548621	+	0.836071i	$0.684847\pi$
$510$		0			0
$511$	−21.5350			−0.952650
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.0352	+	6.91602i	−0.530333	+	0.304756i
$516$		0			0
$517$	54.6962	+	54.6962i	2.40553	+	2.40553i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.3540i		−	1.37364i	−0.726826	−	0.686822i	$-0.759005\pi$
$521$		−	31.3540i		−	1.37364i	0.726826	−	0.686822i	$-0.240995\pi$
$522$		0			0
$523$	5.14628	−	5.14628i	0.225031	−	0.225031i	−0.585582	−	0.810613i	$-0.699134\pi$
$523$	5.14628	−	5.14628i	0.225031	−	0.225031i	0.810613	+	0.585582i	$0.199134\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−19.4347	+	19.4347i	−0.846590	+	0.846590i
$528$		0			0
$529$			47.3714i			2.05963i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.84754	+	5.84754i	0.253285	+	0.253285i
$534$		0			0
$535$	21.2693	−	12.2224i	0.919554	−	0.528422i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.88552			−0.167361
$540$		0			0
$541$	−26.0336			−1.11927			−0.559636	−	0.828738i	$-0.689059\pi$
$541$	−26.0336			−1.11927			−0.559636	+	0.828738i	$0.689059\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.4787	+	42.4942i	−0.491692	+	1.82025i
$546$		0			0
$547$	9.95938	+	9.95938i	0.425832	+	0.425832i	0.887206	−	0.461374i	$-0.152643\pi$
$547$	9.95938	+	9.95938i	0.425832	+	0.425832i	−0.461374	+	0.887206i	$0.652643\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			0.657926i			0.0280286i
$552$		0			0
$553$	−12.9142	+	12.9142i	−0.549169	+	0.549169i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.7910	−	31.7910i	1.34703	−	1.34703i	0.458153	−	0.888873i	$-0.348511\pi$
$557$	31.7910	−	31.7910i	1.34703	−	1.34703i	0.888873	−	0.458153i	$-0.151489\pi$
$558$		0			0
$559$		−	1.84384i		−	0.0779862i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.5912	+	13.5912i	0.572803	+	0.572803i	0.932911	−	0.360108i	$-0.117260\pi$
$563$	13.5912	+	13.5912i	0.572803	+	0.572803i	−0.360108	+	0.932911i	$0.617260\pi$
$564$		0			0
$565$	−13.9785	−	24.3252i	−0.588078	−	1.02337i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.1544			−1.51567			−0.757835	−	0.652446i	$-0.773743\pi$
$569$	−36.1544			−1.51567			−0.757835	+	0.652446i	$0.773743\pi$
$570$		0			0
$571$	9.71908			0.406731			0.203365	−	0.979103i	$-0.434812\pi$
$571$	9.71908			0.406731			0.203365	+	0.979103i	$0.434812\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.6915	+	40.5583i	0.445866	+	1.69140i
$576$		0			0
$577$	17.0114	+	17.0114i	0.708194	+	0.708194i	0.966155	−	0.257962i	$-0.0830508\pi$
$577$	17.0114	+	17.0114i	0.708194	+	0.708194i	−0.257962	+	0.966155i	$0.583051\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	2.88897i		−	0.119855i
$582$		0			0
$583$	−0.574913	+	0.574913i	−0.0238105	+	0.0238105i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.8814	+	13.8814i	−0.572947	+	0.572947i	−0.932951	−	0.360004i	$-0.882775\pi$
$587$	−13.8814	+	13.8814i	−0.572947	+	0.572947i	0.360004	+	0.932951i	$0.382775\pi$
$588$		0			0
$589$			1.70781i			0.0703691i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.92495	+	6.92495i	0.284373	+	0.284373i	0.834850	−	0.550477i	$-0.185554\pi$
$593$	6.92495	+	6.92495i	0.284373	+	0.284373i	−0.550477	+	0.834850i	$0.685554\pi$
$594$		0			0
$595$	32.7356	+	8.84266i	1.34203	+	0.362514i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.1493			0.496409			0.248204	−	0.968708i	$-0.420160\pi$
$599$	12.1493			0.496409			0.248204	+	0.968708i	$0.420160\pi$
$600$		0			0
$601$	−43.8224			−1.78755			−0.893777	−	0.448511i	$-0.851955\pi$
$601$	−43.8224			−1.78755			−0.893777	+	0.448511i	$0.851955\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	53.5213	+	14.4574i	2.17595	+	0.587775i
$606$		0			0
$607$	−23.8953	−	23.8953i	−0.969880	−	0.969880i	0.0296796	−	0.999559i	$-0.490551\pi$
$607$	−23.8953	−	23.8953i	−0.969880	−	0.969880i	−0.999559	+	0.0296796i	$0.990551\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.9292i			0.523059i
$612$		0			0
$613$	−14.2369	+	14.2369i	−0.575024	+	0.575024i	−0.933528	−	0.358504i	$-0.883287\pi$
$613$	−14.2369	+	14.2369i	−0.575024	+	0.575024i	0.358504	+	0.933528i	$0.383287\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.3924	+	27.3924i	−1.10278	+	1.10278i	−0.108700	+	0.994075i	$0.534669\pi$
$617$	−27.3924	+	27.3924i	−1.10278	+	1.10278i	−0.994075	+	0.108700i	$0.965331\pi$
$618$		0			0
$619$			39.6515i			1.59373i	0.604158	+	0.796864i	$0.293510\pi$
$619$			39.6515i			1.59373i	−0.604158	+	0.796864i	$0.706490\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	19.7455	+	19.7455i	0.791087	+	0.791087i
$624$		0			0
$625$	12.3240	+	21.7513i	0.492960	+	0.870052i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41.6294			−1.65987
$630$		0			0
$631$	2.62125			0.104350			0.0521751	−	0.998638i	$-0.483385\pi$
$631$	2.62125			0.104350			0.0521751	+	0.998638i	$0.483385\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.110135	+	0.191656i	0.00437058	+	0.00760563i
$636$		0			0
$637$	−0.459233	−	0.459233i	−0.0181955	−	0.0181955i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	11.5348i		−	0.455598i	−0.973708	−	0.227799i	$-0.926847\pi$
$641$		−	11.5348i		−	0.455598i	0.973708	−	0.227799i	$-0.0731529\pi$
$642$		0			0
$643$	−7.44660	+	7.44660i	−0.293665	+	0.293665i	−0.838526	−	0.544861i	$-0.816582\pi$
$643$	−7.44660	+	7.44660i	−0.293665	+	0.293665i	0.544861	+	0.838526i	$0.316582\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.37088	−	9.37088i	0.368407	−	0.368407i	−0.498489	−	0.866896i	$-0.666112\pi$
$647$	9.37088	−	9.37088i	0.368407	−	0.368407i	0.866896	+	0.498489i	$0.166112\pi$
$648$		0			0
$649$		−	0.683912i		−	0.0268459i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.77000	−	1.77000i	−0.0692654	−	0.0692654i	0.671625	−	0.740891i	$-0.265596\pi$
$653$	−1.77000	−	1.77000i	−0.0692654	−	0.0692654i	−0.740891	+	0.671625i	$0.765596\pi$
$654$		0			0
$655$	4.43251	−	16.4092i	0.173192	−	0.641160i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.8987			−1.28155			−0.640775	−	0.767729i	$-0.721387\pi$
$659$	−32.8987			−1.28155			−0.640775	+	0.767729i	$0.721387\pi$
$660$		0			0
$661$	−16.1835			−0.629464			−0.314732	−	0.949181i	$-0.601915\pi$
$661$	−16.1835			−0.629464			−0.314732	+	0.949181i	$0.601915\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.82683	−	1.04979i	0.0708414	−	0.0407090i
$666$		0			0
$667$	−11.4551	−	11.4551i	−0.443544	−	0.443544i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	41.2478i		−	1.59235i
$672$		0			0
$673$	11.0802	−	11.0802i	0.427112	−	0.427112i	−0.460531	−	0.887643i	$-0.652341\pi$
$673$	11.0802	−	11.0802i	0.427112	−	0.427112i	0.887643	+	0.460531i	$0.152341\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.2151	−	32.2151i	1.23813	−	1.23813i	0.277360	−	0.960766i	$-0.410540\pi$
$677$	32.2151	−	32.2151i	1.23813	−	1.23813i	0.960766	−	0.277360i	$-0.0894595\pi$
$678$		0			0
$679$		−	22.6666i		−	0.869866i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.5078	−	11.5078i	−0.440334	−	0.440334i	0.451790	−	0.892124i	$-0.350786\pi$
$683$	−11.5078	−	11.5078i	−0.440334	−	0.440334i	−0.892124	+	0.451790i	$0.850786\pi$
$684$		0			0
$685$	−13.4457	+	7.72659i	−0.513735	+	0.295218i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.135899			−0.00517734
$690$		0			0
$691$	47.2543			1.79764			0.898819	−	0.438320i	$-0.144426\pi$
$691$	47.2543			1.79764			0.898819	+	0.438320i	$0.144426\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.82597	+	28.9718i	−0.296856	+	1.09896i
$696$		0			0
$697$	−32.0617	−	32.0617i	−1.21442	−	1.21442i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			17.5099i			0.661341i	0.943746	+	0.330670i	$0.107275\pi$
$701$			17.5099i			0.661341i	−0.943746	+	0.330670i	$0.892725\pi$
$702$		0			0
$703$	−1.82908	+	1.82908i	−0.0689849	+	0.0689849i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.28287	−	9.28287i	0.349118	−	0.349118i
$708$		0			0
$709$		−	8.16282i		−	0.306561i	−0.988183	−	0.153281i	$-0.951016\pi$
$709$		−	8.16282i		−	0.306561i	0.988183	−	0.153281i	$-0.0489838\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−29.7347	−	29.7347i	−1.11357	−	1.11357i
$714$		0			0
$715$	6.66542	+	11.5991i	0.249272	+	0.433781i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	31.0418			1.15766			0.578832	−	0.815447i	$-0.303509\pi$
$719$	31.0418			1.15766			0.578832	+	0.815447i	$0.303509\pi$
$720$		0			0
$721$	−17.1690			−0.639407
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.34250	−	4.86176i	−0.309833	−	0.180561i
$726$		0			0
$727$	−7.42114	−	7.42114i	−0.275235	−	0.275235i	0.555969	−	0.831203i	$-0.312347\pi$
$727$	−7.42114	−	7.42114i	−0.275235	−	0.275235i	−0.831203	+	0.555969i	$0.812347\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			10.1097i			0.373920i
$732$		0			0
$733$	−16.3026	+	16.3026i	−0.602149	+	0.602149i	−0.940882	−	0.338733i	$-0.890002\pi$
$733$	−16.3026	+	16.3026i	−0.602149	+	0.602149i	0.338733	+	0.940882i	$0.390002\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	30.7161	−	30.7161i	1.13144	−	1.13144i
$738$		0			0
$739$		−	1.01905i		−	0.0374863i	−0.999824	−	0.0187432i	$-0.994034\pi$
$739$		−	1.01905i		−	0.0374863i	0.999824	−	0.0187432i	$-0.00596648\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.9082	+	21.9082i	0.803733	+	0.803733i	0.983677	−	0.179944i	$-0.0575917\pi$
$743$	21.9082	+	21.9082i	0.803733	+	0.803733i	−0.179944	+	0.983677i	$0.557592\pi$
$744$		0			0
$745$	−29.9911	−	8.10129i	−1.09879	−	0.296808i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.3422			1.10868
$750$		0			0
$751$	−23.7471			−0.866543			−0.433272	−	0.901263i	$-0.642641\pi$
$751$	−23.7471			−0.866543			−0.433272	+	0.901263i	$0.642641\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−44.5321	−	12.0292i	−1.62069	−	0.437787i
$756$		0			0
$757$	31.4678	+	31.4678i	1.14372	+	1.14372i	0.987765	+	0.155953i	$0.0498448\pi$
$757$	31.4678	+	31.4678i	1.14372	+	1.14372i	0.155953	+	0.987765i	$0.450155\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			39.5960i			1.43535i	0.696376	+	0.717677i	$0.254795\pi$
$761$			39.5960i			1.43535i	−0.696376	+	0.717677i	$0.745205\pi$
$762$		0			0
$763$	−38.4980	+	38.4980i	−1.39372	+	1.39372i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.0808322	−	0.0808322i	0.00291868	−	0.00291868i
$768$		0			0
$769$		−	11.9990i		−	0.432695i	−0.976316	−	0.216348i	$-0.930586\pi$
$769$		−	11.9990i		−	0.432695i	0.976316	−	0.216348i	$-0.0694144\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.81832	−	7.81832i	−0.281205	−	0.281205i	0.552384	−	0.833590i	$-0.313718\pi$
$773$	−7.81832	−	7.81832i	−0.281205	−	0.281205i	−0.833590	+	0.552384i	$0.813718\pi$
$774$		0			0
$775$	−21.6550	−	12.6199i	−0.777872	−	0.453320i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.81740			−0.100944
$780$		0			0
$781$	94.6385			3.38643
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.75269	+	4.79021i	0.0982478	+	0.170970i
$786$		0			0
$787$	17.0991	+	17.0991i	0.609516	+	0.609516i	0.942820	−	0.333304i	$-0.108163\pi$
$787$	17.0991	+	17.0991i	0.609516	+	0.609516i	−0.333304	+	0.942820i	$0.608163\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	34.7015i		−	1.23384i
$792$		0			0
$793$	4.87512	−	4.87512i	0.173121	−	0.173121i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.5551	−	14.5551i	0.515568	−	0.515568i	−0.400659	−	0.916227i	$-0.631219\pi$
$797$	14.5551	−	14.5551i	0.515568	−	0.515568i	0.916227	+	0.400659i	$0.131219\pi$
$798$		0			0
$799$		−	70.8899i		−	2.50791i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.9393	+	32.9393i	1.16240	+	1.16240i
$804$		0			0
$805$	−13.5291	+	50.0847i	−0.476837	+	1.76525i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.4833			−1.07173			−0.535867	−	0.844302i	$-0.680015\pi$
$809$	−30.4833			−1.07173			−0.535867	+	0.844302i	$0.680015\pi$
$810$		0			0
$811$	−24.1209			−0.847000			−0.423500	−	0.905896i	$-0.639199\pi$
$811$	−24.1209			−0.847000			−0.423500	+	0.905896i	$0.639199\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.54412	−	3.76058i	0.229231	−	0.131727i
$816$		0			0
$817$	0.444190	+	0.444190i	0.0155402	+	0.0155402i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			49.4675i			1.72643i	0.504837	+	0.863215i	$0.331553\pi$
$821$			49.4675i			1.72643i	−0.504837	+	0.863215i	$0.668447\pi$
$822$		0			0
$823$	−22.3679	+	22.3679i	−0.779695	+	0.779695i	−0.979779	−	0.200084i	$-0.935878\pi$
$823$	−22.3679	+	22.3679i	−0.779695	+	0.779695i	0.200084	+	0.979779i	$0.435878\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.4785	−	19.4785i	0.677334	−	0.677334i	−0.282062	−	0.959396i	$-0.591018\pi$
$827$	19.4785	−	19.4785i	0.677334	−	0.677334i	0.959396	+	0.282062i	$0.0910185\pi$
$828$		0			0
$829$		−	52.3075i		−	1.81671i	−0.418197	−	0.908356i	$-0.637338\pi$
$829$		−	52.3075i		−	1.81671i	0.418197	−	0.908356i	$-0.362662\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.51795	+	2.51795i	0.0872418	+	0.0872418i
$834$		0			0
$835$	−4.01332	+	2.30626i	−0.138887	+	0.0798113i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.3487			0.426325			0.213163	−	0.977017i	$-0.431624\pi$
$839$	12.3487			0.426325			0.213163	+	0.977017i	$0.431624\pi$
$840$		0			0
$841$	−25.2706			−0.871401
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.583115	+	2.15870i	−0.0200598	+	0.0742615i
$846$		0			0
$847$	48.4881	+	48.4881i	1.66607	+	1.66607i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	63.6920i		−	2.18333i
$852$		0			0
$853$	−26.5718	+	26.5718i	−0.909801	+	0.909801i	−0.996256	−	0.0864548i	$-0.972446\pi$
$853$	−26.5718	+	26.5718i	−0.909801	+	0.909801i	0.0864548	+	0.996256i	$0.472446\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.63509	−	5.63509i	0.192491	−	0.192491i	−0.604281	−	0.796772i	$-0.706539\pi$
$857$	5.63509	−	5.63509i	0.192491	−	0.192491i	0.796772	+	0.604281i	$0.206539\pi$
$858$		0			0
$859$			27.3841i			0.934333i	0.884169	+	0.467166i	$0.154725\pi$
$859$			27.3841i			0.934333i	−0.884169	+	0.467166i	$0.845275\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.5625	+	17.5625i	0.597835	+	0.597835i	0.939736	−	0.341901i	$-0.111071\pi$
$863$	17.5625	+	17.5625i	0.597835	+	0.597835i	−0.341901	+	0.939736i	$0.611071\pi$
$864$		0			0
$865$	1.70537	+	2.96767i	0.0579844	+	0.100904i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	39.5066			1.34017
$870$		0			0
$871$	7.26073			0.246021
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.188087	+	30.9216i	−0.00635848	+	1.04534i
$876$		0			0
$877$	12.8894	+	12.8894i	0.435245	+	0.435245i	0.890408	−	0.455163i	$-0.150419\pi$
$877$	12.8894	+	12.8894i	0.435245	+	0.435245i	−0.455163	+	0.890408i	$0.650419\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			22.5565i			0.759948i	0.924997	+	0.379974i	$0.124067\pi$
$881$			22.5565i			0.759948i	−0.924997	+	0.379974i	$0.875933\pi$
$882$		0			0
$883$	29.9948	−	29.9948i	1.00940	−	1.00940i	0.00944911	−	0.999955i	$-0.496992\pi$
$883$	29.9948	−	29.9948i	1.00940	−	1.00940i	0.999955	−	0.00944911i	$-0.00300779\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.4105	+	17.4105i	−0.584587	+	0.584587i	−0.936160	−	0.351573i	$-0.885647\pi$
$887$	−17.4105	+	17.4105i	−0.584587	+	0.584587i	0.351573	+	0.936160i	$0.385647\pi$
$888$		0			0
$889$			0.273410i			0.00916988i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.11470	−	3.11470i	−0.104229	−	0.104229i
$894$		0			0
$895$	−15.2251	−	4.11265i	−0.508918	−	0.137471i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.68048			0.322862
$900$		0			0
$901$	0.745127			0.0248238
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.8712	−	4.01707i	−0.494337	−	0.133532i
$906$		0			0
$907$	12.8922	+	12.8922i	0.428080	+	0.428080i	0.887974	−	0.459894i	$-0.152113\pi$
$907$	12.8922	+	12.8922i	0.428080	+	0.428080i	−0.459894	+	0.887974i	$0.652113\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.9664i		−	0.827173i	−0.910465	−	0.413587i	$-0.864276\pi$
$911$		−	24.9664i		−	0.827173i	0.910465	−	0.413587i	$-0.135724\pi$
$912$		0			0
$913$	−4.41889	+	4.41889i	−0.146244	+	0.146244i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.8661	−	14.8661i	0.490920	−	0.490920i
$918$		0			0
$919$		−	29.7222i		−	0.980446i	−0.871597	−	0.490223i	$-0.836915\pi$
$919$		−	29.7222i		−	0.980446i	0.871597	−	0.490223i	$-0.163085\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.1854	+	11.1854i	0.368172	+	0.368172i
$924$		0			0
$925$	−9.67669	−	36.7087i	−0.318167	−	1.20697i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.7752			−0.386332			−0.193166	−	0.981166i	$-0.561876\pi$
$929$	−11.7752			−0.386332			−0.193166	+	0.981166i	$0.561876\pi$
$930$		0			0
$931$	0.221263			0.00725159
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−36.5461	−	63.5971i	−1.19519	−	2.07985i
$936$		0			0
$937$	−32.9567	−	32.9567i	−1.07665	−	1.07665i	−0.996808	−	0.0798391i	$-0.974559\pi$
$937$	−32.9567	−	32.9567i	−1.07665	−	1.07665i	−0.0798391	−	0.996808i	$-0.525441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			43.3995i			1.41478i	0.706822	+	0.707391i	$0.250128\pi$
$941$			43.3995i			1.41478i	−0.706822	+	0.707391i	$0.749872\pi$
$942$		0			0
$943$	49.0536	−	49.0536i	1.59741	−	1.59741i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.82419	−	7.82419i	0.254252	−	0.254252i	−0.568459	−	0.822711i	$-0.692460\pi$
$947$	7.82419	−	7.82419i	0.254252	−	0.254252i	0.822711	+	0.568459i	$0.192460\pi$
$948$		0			0
$949$			7.78626i			0.252753i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.7889	−	27.7889i	−0.900170	−	0.900170i	0.0952808	−	0.995450i	$-0.469625\pi$
$953$	−27.7889	−	27.7889i	−0.900170	−	0.900170i	−0.995450	+	0.0952808i	$0.969625\pi$
$954$		0			0
$955$	9.08507	−	33.6330i	0.293986	−	1.08834i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.1812			−0.619395
$960$		0			0
$961$	−5.87189			−0.189416
$962$		0			0
$963$		0			0
$964$		0			0
$965$	44.1088	−	25.3471i	1.41991	−	0.815954i
$966$		0			0
$967$	1.94087	+	1.94087i	0.0624143	+	0.0624143i	0.737625	−	0.675211i	$-0.235947\pi$
$967$	1.94087	+	1.94087i	0.0624143	+	0.0624143i	−0.675211	+	0.737625i	$0.735947\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			47.4360i			1.52229i	0.648580	+	0.761146i	$0.275363\pi$
$971$			47.4360i			1.52229i	−0.648580	+	0.761146i	$0.724637\pi$
$972$		0			0
$973$	−26.2473	+	26.2473i	−0.841449	+	0.841449i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.9589	−	22.9589i	0.734520	−	0.734520i	−0.236991	−	0.971512i	$-0.576161\pi$
$977$	22.9589	−	22.9589i	0.734520	−	0.734520i	0.971512	+	0.236991i	$0.0761613\pi$
$978$		0			0
$979$		−	60.4045i		−	1.93054i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.28154	−	6.28154i	−0.200350	−	0.200350i	0.599800	−	0.800150i	$-0.295247\pi$
$983$	−6.28154	−	6.28154i	−0.200350	−	0.200350i	−0.800150	+	0.599800i	$0.795247\pi$
$984$		0			0
$985$	2.58397	−	1.48488i	0.0823321	−	0.0473122i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.4676			−0.491840
$990$		0			0
$991$	19.7292			0.626719			0.313359	−	0.949635i	$-0.398546\pi$
$991$	19.7292			0.626719			0.313359	+	0.949635i	$0.398546\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.90848	+	10.7672i	−0.0922051	+	0.341344i
$996$		0			0
$997$	22.7904	+	22.7904i	0.721779	+	0.721779i	0.968967	−	0.247189i	$-0.0795067\pi$
$997$	22.7904	+	22.7904i	0.721779	+	0.721779i	−0.247189	+	0.968967i	$0.579507\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.y.b.1457.1	yes	24
3.2	odd	2		2340.2.y.a.1457.12	yes	24
5.3	odd	4		2340.2.y.a.53.12	✓	24
15.8	even	4	inner	2340.2.y.b.53.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.12	✓	24	5.3	odd	4
2340.2.y.a.1457.12	yes	24	3.2	odd	2
2340.2.y.b.53.1	yes	24	15.8	even	4	inner
2340.2.y.b.1457.1	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.15870 - 0.583115i) q^{5} +(-1.95569 - 1.95569i) q^{7} +O(q^{10})\)	\(q+(-2.15870 - 0.583115i) q^{5} +(-1.95569 - 1.95569i) q^{7} +5.98275i q^{11} +(-0.707107 + 0.707107i) q^{13} +(3.87703 - 3.87703i) q^{17} -0.340690i q^{19} +(5.93175 + 5.93175i) q^{23} +(4.31995 + 2.51754i) q^{25} -1.93115 q^{29} -5.01279 q^{31} +(3.08135 + 5.36214i) q^{35} +(-5.36873 - 5.36873i) q^{37} -8.26967i q^{41} +(-1.30379 + 1.30379i) q^{43} +(9.14231 - 9.14231i) q^{47} +0.649454i q^{49} +(0.0960951 + 0.0960951i) q^{53} +(3.48863 - 12.9150i) q^{55} -0.114314 q^{59} -6.89446 q^{61} +(1.93875 - 1.11411i) q^{65} +(-5.13411 - 5.13411i) q^{67} -15.8186i q^{71} +(5.50572 - 5.50572i) q^{73} +(11.7004 - 11.7004i) q^{77} -6.60341i q^{79} +(0.738605 + 0.738605i) q^{83} +(-10.6301 + 6.10858i) q^{85} -10.0964 q^{89} +2.76576 q^{91} +(-0.198662 + 0.735448i) q^{95} +(5.79504 + 5.79504i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7}+O(q^{10}) \) 24 * q + 4 * q^5 - 8 * q^7	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83} - 32 q^{85} - 128 q^{89} - 8 q^{91} - 16 q^{95} + 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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