LMFDB - Embedded newform 1560.4.g.c.961.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(961,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1560.961");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$92.0429796090$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 2831 x^{18} + 3052845 x^{16} + 1573840355 x^{14} + 402678072963 x^{12} + 50151013456349 x^{10} + \cdots + 17\!\cdots\!36 $ x^20 + 2831x^18 + 3052845x^16 + 1573840355x^14 + 402678072963x^12 + 50151013456349x^10 + 3182050366505631x^8 + 102921826733622513x^6 + 1575149188769282224x^4 + 9208618475376250752*x^2 + 17299064890999375936
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.14
Root			$5.44159i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.4.g.c.961.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000i q^{5} -15.1033i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.00000i q^{5} -15.1033i q^{7} +9.00000 q^{9} +25.9865i q^{11} +(43.0548 + 18.5278i) q^{13} -15.0000i q^{15} +106.624 q^{17} -126.133i q^{19} +45.3099i q^{21} -117.926 q^{23} -25.0000 q^{25} -27.0000 q^{27} +68.1290 q^{29} -40.3705i q^{31} -77.9595i q^{33} +75.5165 q^{35} +41.2204i q^{37} +(-129.165 - 55.5834i) q^{39} +100.213i q^{41} -124.780 q^{43} +45.0000i q^{45} +180.097i q^{47} +114.890 q^{49} -319.871 q^{51} +84.6212 q^{53} -129.932 q^{55} +378.399i q^{57} +97.5782i q^{59} -747.336 q^{61} -135.930i q^{63} +(-92.6391 + 215.274i) q^{65} -425.842i q^{67} +353.777 q^{69} +566.618i q^{71} -481.492i q^{73} +75.0000 q^{75} +392.482 q^{77} +862.592 q^{79} +81.0000 q^{81} +271.088i q^{83} +533.118i q^{85} -204.387 q^{87} -1526.25i q^{89} +(279.831 - 650.271i) q^{91} +121.112i q^{93} +630.665 q^{95} +290.785i q^{97} +233.878i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 60 q^{3} + 180 q^{9}+O(q^{10}) $ 20 * q - 60 * q^3 + 180 * q^9	$ 20 q - 60 q^{3} + 180 q^{9} - 38 q^{13} - 58 q^{17} + 86 q^{23} - 500 q^{25} - 540 q^{27} - 220 q^{29} - 90 q^{35} + 114 q^{39} - 340 q^{43} - 482 q^{49} + 174 q^{51} - 106 q^{53} + 110 q^{55} - 70 q^{61} - 150 q^{65} - 258 q^{69} + 1500 q^{75} + 4058 q^{77} + 2054 q^{79} + 1620 q^{81} + 660 q^{87} - 2366 q^{91} - 20 q^{95}+O(q^{100}) $ 20 * q - 60 * q^3 + 180 * q^9 - 38 * q^13 - 58 * q^17 + 86 * q^23 - 500 * q^25 - 540 * q^27 - 220 * q^29 - 90 * q^35 + 114 * q^39 - 340 * q^43 - 482 * q^49 + 174 * q^51 - 106 * q^53 + 110 * q^55 - 70 * q^61 - 150 * q^65 - 258 * q^69 + 1500 * q^75 + 4058 * q^77 + 2054 * q^79 + 1620 * q^81 + 660 * q^87 - 2366 * q^91 - 20 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$			5.00000i			0.447214i
$5$			5.00000i			0.447214i
$6$		0			0
$7$		−	15.1033i		−	0.815502i	−0.913093	−	0.407751i	$-0.866313\pi$
$7$		−	15.1033i		−	0.815502i	0.913093	−	0.407751i	$-0.133687\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			25.9865i			0.712293i	0.934430	+	0.356146i	$0.115910\pi$
$11$			25.9865i			0.712293i	−0.934430	+	0.356146i	$0.884090\pi$
$12$		0			0
$13$	43.0548	+	18.5278i	0.918559	+	0.395284i
$13$	43.0548	+	18.5278i	0.918559	+	0.395284i
$14$		0			0
$15$		−	15.0000i		−	0.258199i
$16$		0			0
$17$	106.624			1.52118			0.760590	−	0.649233i	$-0.224910\pi$
$17$	106.624			1.52118			0.760590	+	0.649233i	$0.224910\pi$
$18$		0			0
$19$		−	126.133i		−	1.52300i	−0.648167	−	0.761498i	$-0.724464\pi$
$19$		−	126.133i		−	1.52300i	0.648167	−	0.761498i	$-0.275536\pi$
$20$		0			0
$21$			45.3099i			0.470830i
$22$		0			0
$23$	−117.926			−1.06910			−0.534548	−	0.845138i	$-0.679518\pi$
$23$	−117.926			−1.06910			−0.534548	+	0.845138i	$0.679518\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	68.1290			0.436249			0.218125	−	0.975921i	$-0.430006\pi$
$29$	68.1290			0.436249			0.218125	+	0.975921i	$0.430006\pi$
$30$		0			0
$31$		−	40.3705i		−	0.233895i	−0.993138	−	0.116948i	$-0.962689\pi$
$31$		−	40.3705i		−	0.233895i	0.993138	−	0.116948i	$-0.0373110\pi$
$32$		0			0
$33$		−	77.9595i		−	0.411242i
$34$		0			0
$35$	75.5165			0.364704
$36$		0			0
$37$			41.2204i			0.183151i	0.995798	+	0.0915755i	$0.0291903\pi$
$37$			41.2204i			0.183151i	−0.995798	+	0.0915755i	$0.970810\pi$
$38$		0			0
$39$	−129.165	−	55.5834i	−0.530330	−	0.228217i
$40$		0			0
$41$			100.213i			0.381724i	0.981617	+	0.190862i	$0.0611284\pi$
$41$			100.213i			0.381724i	−0.981617	+	0.190862i	$0.938872\pi$
$42$		0			0
$43$	−124.780			−0.442530			−0.221265	−	0.975214i	$-0.571019\pi$
$43$	−124.780			−0.442530			−0.221265	+	0.975214i	$0.571019\pi$
$44$		0			0
$45$			45.0000i			0.149071i
$46$		0			0
$47$			180.097i			0.558934i	0.960155	+	0.279467i	$0.0901578\pi$
$47$			180.097i			0.558934i	−0.960155	+	0.279467i	$0.909842\pi$
$48$		0			0
$49$	114.890			0.334957
$50$		0			0
$51$	−319.871			−0.878253
$52$		0			0
$53$	84.6212			0.219314			0.109657	−	0.993970i	$-0.465025\pi$
$53$	84.6212			0.219314			0.109657	+	0.993970i	$0.465025\pi$
$54$		0			0
$55$	−129.932			−0.318547
$56$		0			0
$57$			378.399i			0.879302i
$58$		0			0
$59$			97.5782i			0.215315i	0.994188	+	0.107658i	$0.0343350\pi$
$59$			97.5782i			0.215315i	−0.994188	+	0.107658i	$0.965665\pi$
$60$		0			0
$61$	−747.336			−1.56863			−0.784316	−	0.620362i	$-0.786986\pi$
$61$	−747.336			−1.56863			−0.784316	+	0.620362i	$0.786986\pi$
$62$		0			0
$63$		−	135.930i		−	0.271834i
$64$		0			0
$65$	−92.6391	+	215.274i	−0.176776	+	0.410792i
$66$		0			0
$67$		−	425.842i		−	0.776491i	−0.921556	−	0.388246i	$-0.873081\pi$
$67$		−	425.842i		−	0.776491i	0.921556	−	0.388246i	$-0.126919\pi$
$68$		0			0
$69$	353.777			0.617243
$70$		0			0
$71$			566.618i			0.947115i	0.880763	+	0.473557i	$0.157030\pi$
$71$			566.618i			0.947115i	−0.880763	+	0.473557i	$0.842970\pi$
$72$		0			0
$73$		−	481.492i		−	0.771978i	−0.922503	−	0.385989i	$-0.873860\pi$
$73$		−	481.492i		−	0.771978i	0.922503	−	0.385989i	$-0.126140\pi$
$74$		0			0
$75$	75.0000			0.115470
$76$		0			0
$77$	392.482			0.580876
$78$		0			0
$79$	862.592			1.22847			0.614235	−	0.789123i	$-0.289465\pi$
$79$	862.592			1.22847			0.614235	+	0.789123i	$0.289465\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			271.088i			0.358503i	0.983803	+	0.179252i	$0.0573676\pi$
$83$			271.088i			0.358503i	−0.983803	+	0.179252i	$0.942632\pi$
$84$		0			0
$85$			533.118i			0.680292i
$86$		0			0
$87$	−204.387			−0.251869
$88$		0			0
$89$		−	1526.25i		−	1.81777i	−0.417044	−	0.908886i	$-0.636934\pi$
$89$		−	1526.25i		−	1.81777i	0.417044	−	0.908886i	$-0.363066\pi$
$90$		0			0
$91$	279.831	−	650.271i	0.322355	−	0.749087i
$92$		0			0
$93$			121.112i			0.135040i
$94$		0			0
$95$	630.665			0.681104
$96$		0			0
$97$			290.785i			0.304379i	0.988351	+	0.152190i	$0.0486324\pi$
$97$			290.785i			0.304379i	−0.988351	+	0.152190i	$0.951368\pi$
$98$		0			0
$99$			233.878i			0.237431i
$100$		0			0
$101$	100.790			0.0992966			0.0496483	−	0.998767i	$-0.484190\pi$
$101$	100.790			0.0992966			0.0496483	+	0.998767i	$0.484190\pi$
$102$		0			0
$103$	−283.478			−0.271184			−0.135592	−	0.990765i	$-0.543294\pi$
$103$	−283.478			−0.271184			−0.135592	+	0.990765i	$0.543294\pi$
$104$		0			0
$105$	−226.550			−0.210562
$106$		0			0
$107$	−991.736			−0.896025			−0.448013	−	0.894027i	$-0.647868\pi$
$107$	−991.736			−0.896025			−0.448013	+	0.894027i	$0.647868\pi$
$108$		0			0
$109$		−	930.670i		−	0.817817i	−0.912575	−	0.408908i	$-0.865910\pi$
$109$		−	930.670i		−	0.817817i	0.912575	−	0.408908i	$-0.134090\pi$
$110$		0			0
$111$		−	123.661i		−	0.105742i
$112$		0			0
$113$	2237.67			1.86286			0.931428	−	0.363926i	$-0.118564\pi$
$113$	2237.67			1.86286			0.931428	+	0.363926i	$0.118564\pi$
$114$		0			0
$115$		−	589.629i		−	0.478114i
$116$		0			0
$117$	387.494	+	166.750i	0.306186	+	0.131761i
$118$		0			0
$119$		−	1610.37i		−	1.24052i
$120$		0			0
$121$	655.702			0.492639
$122$		0			0
$123$		−	300.640i		−	0.220389i
$124$		0			0
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$	−1128.32			−0.788362			−0.394181	−	0.919033i	$-0.628972\pi$
$127$	−1128.32			−0.788362			−0.394181	+	0.919033i	$0.628972\pi$
$128$		0			0
$129$	374.341			0.255495
$130$		0			0
$131$	−373.153			−0.248875			−0.124437	−	0.992227i	$-0.539713\pi$
$131$	−373.153			−0.248875			−0.124437	+	0.992227i	$0.539713\pi$
$132$		0			0
$133$	−1905.03			−1.24201
$134$		0			0
$135$		−	135.000i		−	0.0860663i
$136$		0			0
$137$		−	693.912i		−	0.432736i	−0.976312	−	0.216368i	$-0.930579\pi$
$137$		−	693.912i		−	0.432736i	0.976312	−	0.216368i	$-0.0694211\pi$
$138$		0			0
$139$	625.796			0.381866			0.190933	−	0.981603i	$-0.438849\pi$
$139$	625.796			0.381866			0.190933	+	0.981603i	$0.438849\pi$
$140$		0			0
$141$		−	540.292i		−	0.322701i
$142$		0			0
$143$	−481.473	+	1118.84i	−0.281558	+	0.654283i
$144$		0			0
$145$			340.645i			0.195097i
$146$		0			0
$147$	−344.670			−0.193387
$148$		0			0
$149$			3037.96i			1.67033i	0.549999	+	0.835165i	$0.314628\pi$
$149$			3037.96i			1.67033i	−0.549999	+	0.835165i	$0.685372\pi$
$150$		0			0
$151$		−	1726.37i		−	0.930398i	−0.885206	−	0.465199i	$-0.845983\pi$
$151$		−	1726.37i		−	0.930398i	0.885206	−	0.465199i	$-0.154017\pi$
$152$		0			0
$153$	959.613			0.507060
$154$		0			0
$155$	201.853			0.104601
$156$		0			0
$157$	1642.97			0.835179			0.417590	−	0.908636i	$-0.362875\pi$
$157$	1642.97			0.835179			0.417590	+	0.908636i	$0.362875\pi$
$158$		0			0
$159$	−253.864			−0.126621
$160$		0			0
$161$			1781.07i			0.871850i
$162$		0			0
$163$			685.409i			0.329358i	0.986347	+	0.164679i	$0.0526588\pi$
$163$			685.409i			0.329358i	−0.986347	+	0.164679i	$0.947341\pi$
$164$		0			0
$165$	389.797			0.183913
$166$		0			0
$167$		−	2121.97i		−	0.983253i	−0.870806	−	0.491626i	$-0.836403\pi$
$167$		−	2121.97i		−	0.983253i	0.870806	−	0.491626i	$-0.163597\pi$
$168$		0			0
$169$	1510.44	+	1595.42i	0.687501	+	0.726183i
$170$		0			0
$171$		−	1135.20i		−	0.507665i
$172$		0			0
$173$	3694.85			1.62378			0.811891	−	0.583809i	$-0.198438\pi$
$173$	3694.85			1.62378			0.811891	+	0.583809i	$0.198438\pi$
$174$		0			0
$175$			377.583i			0.163100i
$176$		0			0
$177$		−	292.735i		−	0.124312i
$178$		0			0
$179$	684.192			0.285692			0.142846	−	0.989745i	$-0.454375\pi$
$179$	684.192			0.285692			0.142846	+	0.989745i	$0.454375\pi$
$180$		0			0
$181$	1761.25			0.723275			0.361637	−	0.932319i	$-0.382218\pi$
$181$	1761.25			0.723275			0.361637	+	0.932319i	$0.382218\pi$
$182$		0			0
$183$	2242.01			0.905650
$184$		0			0
$185$	−206.102			−0.0819077
$186$		0			0
$187$			2770.78i			1.08352i
$188$		0			0
$189$			407.789i			0.156943i
$190$		0			0
$191$	4891.15			1.85294			0.926469	−	0.376371i	$-0.122828\pi$
$191$	4891.15			1.85294			0.926469	+	0.376371i	$0.122828\pi$
$192$		0			0
$193$		−	2955.47i		−	1.10227i	−0.834415	−	0.551137i	$-0.814194\pi$
$193$		−	2955.47i		−	1.10227i	0.834415	−	0.551137i	$-0.185806\pi$
$194$		0			0
$195$	277.917	−	645.823i	0.102062	−	0.237171i
$196$		0			0
$197$			2505.45i			0.906120i	0.891480	+	0.453060i	$0.149668\pi$
$197$			2505.45i			0.906120i	−0.891480	+	0.453060i	$0.850332\pi$
$198$		0			0
$199$	4394.57			1.56544			0.782721	−	0.622373i	$-0.213831\pi$
$199$	4394.57			1.56544			0.782721	+	0.622373i	$0.213831\pi$
$200$		0			0
$201$			1277.53i			0.448307i
$202$		0			0
$203$		−	1028.97i		−	0.355762i
$204$		0			0
$205$	−501.067			−0.170712
$206$		0			0
$207$	−1061.33			−0.356365
$208$		0			0
$209$	3277.76			1.08482
$210$		0			0
$211$	3211.01			1.04765			0.523827	−	0.851825i	$-0.324504\pi$
$211$	3211.01			1.04765			0.523827	+	0.851825i	$0.324504\pi$
$212$		0			0
$213$		−	1699.85i		−	0.546817i
$214$		0			0
$215$		−	623.901i		−	0.197906i
$216$		0			0
$217$	−609.728			−0.190742
$218$		0			0
$219$			1444.48i			0.445702i
$220$		0			0
$221$	4590.67	+	1975.50i	1.39729	+	0.601298i
$222$		0			0
$223$		−	2649.83i		−	0.795721i	−0.917446	−	0.397861i	$-0.869753\pi$
$223$		−	2649.83i		−	0.795721i	0.917446	−	0.397861i	$-0.130247\pi$
$224$		0			0
$225$	−225.000			−0.0666667
$226$		0			0
$227$		−	5559.56i		−	1.62556i	−0.582574	−	0.812778i	$-0.697954\pi$
$227$		−	5559.56i		−	1.62556i	0.582574	−	0.812778i	$-0.302046\pi$
$228$		0			0
$229$			3681.04i			1.06223i	0.847301	+	0.531113i	$0.178226\pi$
$229$			3681.04i			1.06223i	−0.847301	+	0.531113i	$0.821774\pi$
$230$		0			0
$231$	−1177.45			−0.335369
$232$		0			0
$233$	−5204.47			−1.46333			−0.731665	−	0.681664i	$-0.761257\pi$
$233$	−5204.47			−1.46333			−0.731665	+	0.681664i	$0.761257\pi$
$234$		0			0
$235$	−900.487			−0.249963
$236$		0			0
$237$	−2587.78			−0.709258
$238$		0			0
$239$		−	4677.97i		−	1.26608i	−0.774120	−	0.633039i	$-0.781807\pi$
$239$		−	4677.97i		−	1.26608i	0.774120	−	0.633039i	$-0.218193\pi$
$240$		0			0
$241$			592.535i			0.158376i	0.996860	+	0.0791879i	$0.0252327\pi$
$241$			592.535i			0.158376i	−0.996860	+	0.0791879i	$0.974767\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$			574.450i			0.149797i
$246$		0			0
$247$	2336.97	−	5430.64i	0.602016	−	1.39896i
$248$		0			0
$249$		−	813.264i		−	0.206982i
$250$		0			0
$251$	1170.17			0.294264			0.147132	−	0.989117i	$-0.452996\pi$
$251$	1170.17			0.294264			0.147132	+	0.989117i	$0.452996\pi$
$252$		0			0
$253$		−	3064.48i		−	0.761510i
$254$		0			0
$255$		−	1599.36i		−	0.392767i
$256$		0			0
$257$	1977.88			0.480065			0.240032	−	0.970765i	$-0.422842\pi$
$257$	1977.88			0.480065			0.240032	+	0.970765i	$0.422842\pi$
$258$		0			0
$259$	622.564			0.149360
$260$		0			0
$261$	613.161			0.145416
$262$		0			0
$263$	6484.73			1.52040			0.760200	−	0.649689i	$-0.225101\pi$
$263$	6484.73			1.52040			0.760200	+	0.649689i	$0.225101\pi$
$264$		0			0
$265$			423.106i			0.0980800i
$266$		0			0
$267$			4578.74i			1.04949i
$268$		0			0
$269$	−2153.25			−0.488052			−0.244026	−	0.969769i	$-0.578468\pi$
$269$	−2153.25			−0.488052			−0.244026	+	0.969769i	$0.578468\pi$
$270$		0			0
$271$		−	6845.11i		−	1.53436i	−0.641434	−	0.767178i	$-0.721660\pi$
$271$		−	6845.11i		−	1.53436i	0.641434	−	0.767178i	$-0.278340\pi$
$272$		0			0
$273$	−839.494	+	1950.81i	−0.186112	+	0.432485i
$274$		0			0
$275$		−	649.662i		−	0.142459i
$276$		0			0
$277$	−76.1290			−0.0165132			−0.00825659	−	0.999966i	$-0.502628\pi$
$277$	−76.1290			−0.0165132			−0.00825659	+	0.999966i	$0.502628\pi$
$278$		0			0
$279$		−	363.335i		−	0.0779651i
$280$		0			0
$281$			2912.02i			0.618208i	0.951028	+	0.309104i	$0.100029\pi$
$281$			2912.02i			0.618208i	−0.951028	+	0.309104i	$0.899971\pi$
$282$		0			0
$283$	−2712.74			−0.569808			−0.284904	−	0.958556i	$-0.591962\pi$
$283$	−2712.74			−0.569808			−0.284904	+	0.958556i	$0.591962\pi$
$284$		0			0
$285$	−1892.00			−0.393236
$286$		0			0
$287$	1513.55			0.311297
$288$		0			0
$289$	6455.61			1.31399
$290$		0			0
$291$		−	872.356i		−	0.175733i
$292$		0			0
$293$		−	6983.91i		−	1.39251i	−0.717796	−	0.696253i	$-0.754849\pi$
$293$		−	6983.91i		−	1.39251i	0.717796	−	0.696253i	$-0.245151\pi$
$294$		0			0
$295$	−487.891			−0.0962919
$296$		0			0
$297$		−	701.635i		−	0.137081i
$298$		0			0
$299$	−5077.27	−	2184.91i	−0.982028	−	0.422597i
$300$		0			0
$301$			1884.59i			0.360884i
$302$		0			0
$303$	−302.369			−0.0573289
$304$		0			0
$305$		−	3736.68i		−	0.701513i
$306$		0			0
$307$			2684.16i			0.499000i	0.968375	+	0.249500i	$0.0802663\pi$
$307$			2684.16i			0.499000i	−0.968375	+	0.249500i	$0.919734\pi$
$308$		0			0
$309$	850.435			0.156568
$310$		0			0
$311$	1702.10			0.310345			0.155173	−	0.987887i	$-0.450407\pi$
$311$	1702.10			0.310345			0.155173	+	0.987887i	$0.450407\pi$
$312$		0			0
$313$	4296.18			0.775829			0.387915	−	0.921695i	$-0.373196\pi$
$313$	4296.18			0.775829			0.387915	+	0.921695i	$0.373196\pi$
$314$		0			0
$315$	679.649			0.121568
$316$		0			0
$317$		−	5385.49i		−	0.954194i	−0.878851	−	0.477097i	$-0.841689\pi$
$317$		−	5385.49i		−	0.954194i	0.878851	−	0.477097i	$-0.158311\pi$
$318$		0			0
$319$			1770.43i			0.310737i
$320$		0			0
$321$	2975.21			0.517321
$322$		0			0
$323$		−	13448.8i		−	2.31675i
$324$		0			0
$325$	−1076.37	−	463.195i	−0.183712	−	0.0790568i
$326$		0			0
$327$			2792.01i			0.472167i
$328$		0			0
$329$	2720.07			0.455812
$330$		0			0
$331$			9546.74i			1.58531i	0.609673	+	0.792653i	$0.291301\pi$
$331$			9546.74i			1.58531i	−0.609673	+	0.792653i	$0.708699\pi$
$332$		0			0
$333$			370.984i			0.0610504i
$334$		0			0
$335$	2129.21			0.347257
$336$		0			0
$337$	10984.4			1.77554			0.887770	−	0.460287i	$-0.152254\pi$
$337$	10984.4			1.77554			0.887770	+	0.460287i	$0.152254\pi$
$338$		0			0
$339$	−6713.02			−1.07552
$340$		0			0
$341$	1049.09			0.166602
$342$		0			0
$343$		−	6915.66i		−	1.08866i
$344$		0			0
$345$			1768.89i			0.276039i
$346$		0			0
$347$	11349.9			1.75589			0.877943	−	0.478766i	$-0.158916\pi$
$347$	11349.9			1.75589			0.877943	+	0.478766i	$0.158916\pi$
$348$		0			0
$349$			6749.06i			1.03515i	0.855637	+	0.517577i	$0.173166\pi$
$349$			6749.06i			1.03515i	−0.855637	+	0.517577i	$0.826834\pi$
$350$		0			0
$351$	−1162.48	−	500.251i	−0.176777	−	0.0760724i
$352$		0			0
$353$		−	2465.54i		−	0.371749i	−0.982573	−	0.185875i	$-0.940488\pi$
$353$		−	2465.54i		−	0.371749i	0.982573	−	0.185875i	$-0.0595119\pi$
$354$		0			0
$355$	−2833.09			−0.423563
$356$		0			0
$357$			4831.11i			0.716217i
$358$		0			0
$359$			668.417i			0.0982666i	0.998792	+	0.0491333i	$0.0156459\pi$
$359$			668.417i			0.0982666i	−0.998792	+	0.0491333i	$0.984354\pi$
$360$		0			0
$361$	−9050.56			−1.31952
$362$		0			0
$363$	−1967.11			−0.284425
$364$		0			0
$365$	2407.46			0.345239
$366$		0			0
$367$	−331.200			−0.0471077			−0.0235538	−	0.999723i	$-0.507498\pi$
$367$	−331.200			−0.0471077			−0.0235538	+	0.999723i	$0.507498\pi$
$368$		0			0
$369$			901.920i			0.127241i
$370$		0			0
$371$		−	1278.06i		−	0.178851i
$372$		0			0
$373$	−8875.84			−1.23210			−0.616051	−	0.787707i	$-0.711268\pi$
$373$	−8875.84			−1.23210			−0.616051	+	0.787707i	$0.711268\pi$
$374$		0			0
$375$			375.000i			0.0516398i
$376$		0			0
$377$	2933.28	+	1262.28i	0.400721	+	0.172442i
$378$		0			0
$379$		−	6422.25i		−	0.870419i	−0.900329	−	0.435210i	$-0.856674\pi$
$379$		−	6422.25i		−	0.870419i	0.900329	−	0.435210i	$-0.143326\pi$
$380$		0			0
$381$	3384.95			0.455161
$382$		0			0
$383$		−	300.306i		−	0.0400651i	−0.999799	−	0.0200325i	$-0.993623\pi$
$383$		−	300.306i		−	0.0400651i	0.999799	−	0.0200325i	$-0.00637698\pi$
$384$		0			0
$385$			1962.41i			0.259776i
$386$		0			0
$387$	−1123.02			−0.147510
$388$		0			0
$389$	12565.6			1.63780			0.818898	−	0.573938i	$-0.194585\pi$
$389$	12565.6			1.63780			0.818898	+	0.573938i	$0.194585\pi$
$390$		0			0
$391$	−12573.7			−1.62629
$392$		0			0
$393$	1119.46			0.143688
$394$		0			0
$395$			4312.96i			0.549389i
$396$		0			0
$397$		−	154.649i		−	0.0195506i	−0.999952	−	0.00977531i	$-0.996888\pi$
$397$		−	154.649i		−	0.0195506i	0.999952	−	0.00977531i	$-0.00311163\pi$
$398$		0			0
$399$	5715.08			0.717072
$400$		0			0
$401$		−	4604.86i		−	0.573456i	−0.958012	−	0.286728i	$-0.907432\pi$
$401$		−	4604.86i		−	0.573456i	0.958012	−	0.286728i	$-0.0925676\pi$
$402$		0			0
$403$	747.977	−	1738.15i	0.0924551	−	0.214847i
$404$		0			0
$405$			405.000i			0.0496904i
$406$		0			0
$407$	−1071.17			−0.130457
$408$		0			0
$409$			6032.78i			0.729344i	0.931136	+	0.364672i	$0.118819\pi$
$409$			6032.78i			0.729344i	−0.931136	+	0.364672i	$0.881181\pi$
$410$		0			0
$411$			2081.73i			0.249840i
$412$		0			0
$413$	1473.75			0.175590
$414$		0			0
$415$	−1355.44			−0.160327
$416$		0			0
$417$	−1877.39			−0.220470
$418$		0			0
$419$	−6542.01			−0.762764			−0.381382	−	0.924418i	$-0.624552\pi$
$419$	−6542.01			−0.762764			−0.381382	+	0.924418i	$0.624552\pi$
$420$		0			0
$421$			3826.68i			0.442995i	0.975161	+	0.221497i	$0.0710944\pi$
$421$			3826.68i			0.442995i	−0.975161	+	0.221497i	$0.928906\pi$
$422$		0			0
$423$			1620.88i			0.186311i
$424$		0			0
$425$	−2665.59			−0.304236
$426$		0			0
$427$			11287.2i			1.27922i
$428$		0			0
$429$	1444.42	−	3356.53i	0.162558	−	0.377750i
$430$		0			0
$431$		−	5266.37i		−	0.588567i	−0.955718	−	0.294283i	$-0.904919\pi$
$431$		−	5266.37i		−	0.588567i	0.955718	−	0.294283i	$-0.0950809\pi$
$432$		0			0
$433$	−9046.55			−1.00404			−0.502020	−	0.864856i	$-0.667410\pi$
$433$	−9046.55			−1.00404			−0.502020	+	0.864856i	$0.667410\pi$
$434$		0			0
$435$		−	1021.93i		−	0.112639i
$436$		0			0
$437$			14874.3i			1.62823i
$438$		0			0
$439$	−6559.10			−0.713096			−0.356548	−	0.934277i	$-0.616046\pi$
$439$	−6559.10			−0.713096			−0.356548	+	0.934277i	$0.616046\pi$
$440$		0			0
$441$	1034.01			0.111652
$442$		0			0
$443$	−4111.79			−0.440987			−0.220493	−	0.975388i	$-0.570767\pi$
$443$	−4111.79			−0.440987			−0.220493	+	0.975388i	$0.570767\pi$
$444$		0			0
$445$	7631.23			0.812933
$446$		0			0
$447$		−	9113.87i		−	0.964366i
$448$		0			0
$449$			9209.37i			0.967967i	0.875077	+	0.483984i	$0.160811\pi$
$449$			9209.37i			0.967967i	−0.875077	+	0.483984i	$0.839189\pi$
$450$		0			0
$451$	−2604.19			−0.271900
$452$		0			0
$453$			5179.11i			0.537165i
$454$		0			0
$455$	3251.35	+	1399.16i	0.335002	+	0.144161i
$456$		0			0
$457$		−	875.055i		−	0.0895697i	−0.998997	−	0.0447848i	$-0.985740\pi$
$457$		−	875.055i		−	0.0895697i	0.998997	−	0.0447848i	$-0.0142602\pi$
$458$		0			0
$459$	−2878.84			−0.292751
$460$		0			0
$461$		−	12219.7i		−	1.23455i	−0.786749	−	0.617273i	$-0.788237\pi$
$461$		−	12219.7i		−	1.23455i	0.786749	−	0.617273i	$-0.211763\pi$
$462$		0			0
$463$			8726.91i			0.875969i	0.898982	+	0.437985i	$0.144308\pi$
$463$			8726.91i			0.875969i	−0.898982	+	0.437985i	$0.855692\pi$
$464$		0			0
$465$	−605.558			−0.0603915
$466$		0			0
$467$	1963.77			0.194588			0.0972940	−	0.995256i	$-0.468981\pi$
$467$	1963.77			0.194588			0.0972940	+	0.995256i	$0.468981\pi$
$468$		0			0
$469$	−6431.63			−0.633230
$470$		0			0
$471$	−4928.90			−0.482191
$472$		0			0
$473$		−	3242.60i		−	0.315211i
$474$		0			0
$475$			3153.33i			0.304599i
$476$		0			0
$477$	761.591			0.0731045
$478$		0			0
$479$			12185.2i			1.16233i	0.813787	+	0.581163i	$0.197402\pi$
$479$			12185.2i			1.16233i	−0.813787	+	0.581163i	$0.802598\pi$
$480$		0			0
$481$	−763.724	+	1774.74i	−0.0723967	+	0.168235i
$482$		0			0
$483$		−	5343.21i		−	0.503363i
$484$		0			0
$485$	−1453.93			−0.136123
$486$		0			0
$487$			4548.62i			0.423240i	0.977352	+	0.211620i	$0.0678738\pi$
$487$			4548.62i			0.423240i	−0.977352	+	0.211620i	$0.932126\pi$
$488$		0			0
$489$		−	2056.23i		−	0.190155i
$490$		0			0
$491$	18166.3			1.66972			0.834861	−	0.550461i	$-0.185548\pi$
$491$	18166.3			1.66972			0.834861	+	0.550461i	$0.185548\pi$
$492$		0			0
$493$	7264.16			0.663614
$494$		0			0
$495$	−1169.39			−0.106182
$496$		0			0
$497$	8557.80			0.772374
$498$		0			0
$499$		−	4747.85i		−	0.425938i	−0.977059	−	0.212969i	$-0.931687\pi$
$499$		−	4747.85i		−	0.425938i	0.977059	−	0.212969i	$-0.0683134\pi$
$500$		0			0
$501$			6365.92i			0.567681i
$502$		0			0
$503$	−6558.06			−0.581331			−0.290665	−	0.956825i	$-0.593877\pi$
$503$	−6558.06			−0.581331			−0.290665	+	0.956825i	$0.593877\pi$
$504$		0			0
$505$			503.949i			0.0444068i
$506$		0			0
$507$	−4531.32	−	4786.27i	−0.396929	−	0.419262i
$508$		0			0
$509$			21449.9i			1.86788i	0.357435	+	0.933938i	$0.383651\pi$
$509$			21449.9i			1.86788i	−0.357435	+	0.933938i	$0.616349\pi$
$510$		0			0
$511$	−7272.12			−0.629549
$512$		0			0
$513$			3405.59i			0.293101i
$514$		0			0
$515$		−	1417.39i		−	0.121277i
$516$		0			0
$517$	−4680.10			−0.398125
$518$		0			0
$519$	−11084.6			−0.937491
$520$		0			0
$521$	−13266.5			−1.11557			−0.557787	−	0.829984i	$-0.688349\pi$
$521$	−13266.5			−1.11557			−0.557787	+	0.829984i	$0.688349\pi$
$522$		0			0
$523$	10765.2			0.900053			0.450026	−	0.893015i	$-0.351415\pi$
$523$	10765.2			0.900053			0.450026	+	0.893015i	$0.351415\pi$
$524$		0			0
$525$		−	1132.75i		−	0.0941661i
$526$		0			0
$527$		−	4304.45i		−	0.355797i
$528$		0			0
$529$	1739.48			0.142967
$530$		0			0
$531$			878.204i			0.0717717i
$532$		0			0
$533$	−1856.73	+	4314.67i	−0.150890	+	0.350636i
$534$		0			0
$535$		−	4958.68i		−	0.400715i
$536$		0			0
$537$	−2052.58			−0.164944
$538$		0			0
$539$			2985.59i			0.238587i
$540$		0			0
$541$			4409.95i			0.350459i	0.984528	+	0.175230i	$0.0560668\pi$
$541$			4409.95i			0.350459i	−0.984528	+	0.175230i	$0.943933\pi$
$542$		0			0
$543$	−5283.75			−0.417583
$544$		0			0
$545$	4653.35			0.365739
$546$		0			0
$547$	−8020.49			−0.626931			−0.313466	−	0.949600i	$-0.601490\pi$
$547$	−8020.49			−0.626931			−0.313466	+	0.949600i	$0.601490\pi$
$548$		0			0
$549$	−6726.02			−0.522877
$550$		0			0
$551$		−	8593.32i		−	0.664406i
$552$		0			0
$553$		−	13028.0i		−	1.00182i
$554$		0			0
$555$	618.306			0.0472894
$556$		0			0
$557$			1183.79i			0.0900515i	0.998986	+	0.0450258i	$0.0143370\pi$
$557$			1183.79i			0.0900515i	−0.998986	+	0.0450258i	$0.985663\pi$
$558$		0			0
$559$	−5372.39	−	2311.91i	−0.406490	−	0.174925i
$560$		0			0
$561$		−	8312.33i		−	0.625573i
$562$		0			0
$563$	2400.42			0.179690			0.0898450	−	0.995956i	$-0.471363\pi$
$563$	2400.42			0.179690			0.0898450	+	0.995956i	$0.471363\pi$
$564$		0			0
$565$			11188.4i			0.833094i
$566$		0			0
$567$		−	1223.37i		−	0.0906113i
$568$		0			0
$569$	5964.38			0.439438			0.219719	−	0.975563i	$-0.429486\pi$
$569$	5964.38			0.439438			0.219719	+	0.975563i	$0.429486\pi$
$570$		0			0
$571$	3227.67			0.236556			0.118278	−	0.992980i	$-0.462263\pi$
$571$	3227.67			0.236556			0.118278	+	0.992980i	$0.462263\pi$
$572$		0			0
$573$	−14673.4			−1.06979
$574$		0			0
$575$	2948.14			0.213819
$576$		0			0
$577$		−	19442.9i		−	1.40281i	−0.712764	−	0.701404i	$-0.752557\pi$
$577$		−	19442.9i		−	1.40281i	0.712764	−	0.701404i	$-0.247443\pi$
$578$		0			0
$579$			8866.40i			0.636399i
$580$		0			0
$581$	4094.32			0.292360
$582$		0			0
$583$			2199.01i			0.156216i
$584$		0			0
$585$	−833.752	+	1937.47i	−0.0589255	+	0.136931i
$586$		0			0
$587$		−	6555.72i		−	0.460960i	−0.973077	−	0.230480i	$-0.925970\pi$
$587$		−	6555.72i		−	0.460960i	0.973077	−	0.230480i	$-0.0740296\pi$
$588$		0			0
$589$	−5092.06			−0.356222
$590$		0			0
$591$		−	7516.34i		−	0.523148i
$592$		0			0
$593$			17793.4i			1.23218i	0.787674	+	0.616092i	$0.211285\pi$
$593$			17793.4i			1.23218i	−0.787674	+	0.616092i	$0.788715\pi$
$594$		0			0
$595$	8051.85			0.554779
$596$		0			0
$597$	−13183.7			−0.903808
$598$		0			0
$599$	−7633.24			−0.520677			−0.260339	−	0.965517i	$-0.583834\pi$
$599$	−7633.24			−0.520677			−0.260339	+	0.965517i	$0.583834\pi$
$600$		0			0
$601$	−19338.8			−1.31256			−0.656279	−	0.754518i	$-0.727871\pi$
$601$	−19338.8			−1.31256			−0.656279	+	0.754518i	$0.727871\pi$
$602$		0			0
$603$		−	3832.58i		−	0.258830i
$604$		0			0
$605$			3278.51i			0.220315i
$606$		0			0
$607$	−12237.6			−0.818298			−0.409149	−	0.912468i	$-0.634174\pi$
$607$	−12237.6			−0.818298			−0.409149	+	0.912468i	$0.634174\pi$
$608$		0			0
$609$			3086.92i			0.205399i
$610$		0			0
$611$	−3336.81	+	7754.07i	−0.220938	+	0.513414i
$612$		0			0
$613$		−	3579.61i		−	0.235855i	−0.993022	−	0.117927i	$-0.962375\pi$
$613$		−	3579.61i		−	0.235855i	0.993022	−	0.117927i	$-0.0376250\pi$
$614$		0			0
$615$	1503.20			0.0985608
$616$		0			0
$617$		−	10907.0i		−	0.711669i	−0.934549	−	0.355835i	$-0.884197\pi$
$617$		−	10907.0i		−	0.711669i	0.934549	−	0.355835i	$-0.115803\pi$
$618$		0			0
$619$			14027.2i			0.910823i	0.890281	+	0.455411i	$0.150508\pi$
$619$			14027.2i			0.910823i	−0.890281	+	0.455411i	$0.849492\pi$
$620$		0			0
$621$	3183.99			0.205748
$622$		0			0
$623$	−23051.4			−1.48240
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−9833.27			−0.626320
$628$		0			0
$629$			4395.07i			0.278606i
$630$		0			0
$631$			16789.7i			1.05925i	0.848231	+	0.529626i	$0.177668\pi$
$631$			16789.7i			1.05925i	−0.848231	+	0.529626i	$0.822332\pi$
$632$		0			0
$633$	−9633.03			−0.604863
$634$		0			0
$635$		−	5641.59i		−	0.352566i
$636$		0			0
$637$	4946.58	+	2128.66i	0.307677	+	0.132403i
$638$		0			0
$639$			5099.56i			0.315705i
$640$		0			0
$641$	5868.46			0.361607			0.180803	−	0.983519i	$-0.442130\pi$
$641$	5868.46			0.361607			0.180803	+	0.983519i	$0.442130\pi$
$642$		0			0
$643$		−	21807.5i		−	1.33749i	−0.743492	−	0.668745i	$-0.766832\pi$
$643$		−	21807.5i		−	1.33749i	0.743492	−	0.668745i	$-0.233168\pi$
$644$		0			0
$645$			1871.70i			0.114261i
$646$		0			0
$647$	−9598.18			−0.583220			−0.291610	−	0.956537i	$-0.594191\pi$
$647$	−9598.18			−0.583220			−0.291610	+	0.956537i	$0.594191\pi$
$648$		0			0
$649$	−2535.71			−0.153367
$650$		0			0
$651$	1829.18			0.110125
$652$		0			0
$653$	4653.63			0.278883			0.139441	−	0.990230i	$-0.455469\pi$
$653$	4653.63			0.278883			0.139441	+	0.990230i	$0.455469\pi$
$654$		0			0
$655$		−	1865.77i		−	0.111300i
$656$		0			0
$657$		−	4333.43i		−	0.257326i
$658$		0			0
$659$	14617.5			0.864064			0.432032	−	0.901858i	$-0.357797\pi$
$659$	14617.5			0.864064			0.432032	+	0.901858i	$0.357797\pi$
$660$		0			0
$661$		−	18792.2i		−	1.10580i	−0.833248	−	0.552900i	$-0.813521\pi$
$661$		−	18792.2i		−	1.10580i	0.833248	−	0.552900i	$-0.186479\pi$
$662$		0			0
$663$	−13772.0	−	5926.51i	−0.806727	−	0.347159i
$664$		0			0
$665$		−	9525.13i		−	0.555442i
$666$		0			0
$667$	−8034.16			−0.466393
$668$		0			0
$669$			7949.49i			0.459410i
$670$		0			0
$671$		−	19420.6i		−	1.11733i
$672$		0			0
$673$	−21164.1			−1.21221			−0.606103	−	0.795386i	$-0.707268\pi$
$673$	−21164.1			−1.21221			−0.606103	+	0.795386i	$0.707268\pi$
$674$		0			0
$675$	675.000			0.0384900
$676$		0			0
$677$	−10265.5			−0.582771			−0.291386	−	0.956606i	$-0.594116\pi$
$677$	−10265.5			−0.582771			−0.291386	+	0.956606i	$0.594116\pi$
$678$		0			0
$679$	4391.82			0.248222
$680$		0			0
$681$			16678.7i			0.938515i
$682$		0			0
$683$		−	34773.9i		−	1.94815i	−0.226221	−	0.974076i	$-0.572637\pi$
$683$		−	34773.9i		−	1.94815i	0.226221	−	0.974076i	$-0.427363\pi$
$684$		0			0
$685$	3469.56			0.193526
$686$		0			0
$687$		−	11043.1i		−	0.613276i
$688$		0			0
$689$	3643.35	+	1567.85i	0.201452	+	0.0866912i
$690$		0			0
$691$			15021.9i			0.827005i	0.910503	+	0.413502i	$0.135695\pi$
$691$			15021.9i			0.827005i	−0.910503	+	0.413502i	$0.864305\pi$
$692$		0			0
$693$	3532.34			0.193625
$694$		0			0
$695$			3128.98i			0.170776i
$696$		0			0
$697$			10685.1i			0.580671i
$698$		0			0
$699$	15613.4			0.844854
$700$		0			0
$701$	15982.4			0.861121			0.430561	−	0.902562i	$-0.358316\pi$
$701$	15982.4			0.861121			0.430561	+	0.902562i	$0.358316\pi$
$702$		0			0
$703$	5199.26			0.278938
$704$		0			0
$705$	2701.46			0.144316
$706$		0			0
$707$		−	1522.26i		−	0.0809766i
$708$		0			0
$709$			18323.0i			0.970572i	0.874356	+	0.485286i	$0.161284\pi$
$709$			18323.0i			0.970572i	−0.874356	+	0.485286i	$0.838716\pi$
$710$		0			0
$711$	7763.33			0.409490
$712$		0			0
$713$			4760.72i			0.250057i
$714$		0			0
$715$	−5594.22	−	2407.36i	−0.292604	−	0.125917i
$716$		0			0
$717$			14033.9i			0.730971i
$718$		0			0
$719$	31470.0			1.63231			0.816156	−	0.577831i	$-0.196101\pi$
$719$	31470.0			1.63231			0.816156	+	0.577831i	$0.196101\pi$
$720$		0			0
$721$			4281.46i			0.221151i
$722$		0			0
$723$		−	1777.61i		−	0.0914383i
$724$		0			0
$725$	−1703.22			−0.0872499
$726$		0			0
$727$	−19617.7			−1.00080			−0.500398	−	0.865796i	$-0.666813\pi$
$727$	−19617.7			−1.00080			−0.500398	+	0.865796i	$0.666813\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−13304.5			−0.673168
$732$		0			0
$733$			17732.1i			0.893519i	0.894654	+	0.446759i	$0.147422\pi$
$733$			17732.1i			0.893519i	−0.894654	+	0.446759i	$0.852578\pi$
$734$		0			0
$735$		−	1723.35i		−	0.0864854i
$736$		0			0
$737$	11066.1			0.553089
$738$		0			0
$739$			22747.7i			1.13232i	0.824295	+	0.566161i	$0.191572\pi$
$739$			22747.7i			1.13232i	−0.824295	+	0.566161i	$0.808428\pi$
$740$		0			0
$741$	−7010.91	+	16291.9i	−0.347574	+	0.807691i
$742$		0			0
$743$		−	9262.33i		−	0.457337i	−0.973504	−	0.228669i	$-0.926563\pi$
$743$		−	9262.33i		−	0.457337i	0.973504	−	0.228669i	$-0.0734373\pi$
$744$		0			0
$745$	−15189.8			−0.746994
$746$		0			0
$747$			2439.79i			0.119501i
$748$		0			0
$749$			14978.5i			0.730710i
$750$		0			0
$751$	−25606.1			−1.24418			−0.622091	−	0.782945i	$-0.713716\pi$
$751$	−25606.1			−1.24418			−0.622091	+	0.782945i	$0.713716\pi$
$752$		0			0
$753$	−3510.50			−0.169893
$754$		0			0
$755$	8631.85			0.416087
$756$		0			0
$757$	26684.9			1.28121			0.640607	−	0.767869i	$-0.278683\pi$
$757$	26684.9			1.28121			0.640607	+	0.767869i	$0.278683\pi$
$758$		0			0
$759$			9193.43i			0.439658i
$760$		0			0
$761$			9488.16i			0.451965i	0.974131	+	0.225983i	$0.0725593\pi$
$761$			9488.16i			0.451965i	−0.974131	+	0.225983i	$0.927441\pi$
$762$		0			0
$763$	−14056.2			−0.666931
$764$		0			0
$765$			4798.07i			0.226764i
$766$		0			0
$767$	−1807.91	+	4201.21i	−0.0851106	+	0.197780i
$768$		0			0
$769$		−	30349.9i		−	1.42320i	−0.702582	−	0.711602i	$-0.747970\pi$
$769$		−	30349.9i		−	1.42320i	0.702582	−	0.711602i	$-0.252030\pi$
$770$		0			0
$771$	−5933.63			−0.277166
$772$		0			0
$773$		−	28899.2i		−	1.34467i	−0.740246	−	0.672337i	$-0.765291\pi$
$773$		−	28899.2i		−	1.34467i	0.740246	−	0.672337i	$-0.234709\pi$
$774$		0			0
$775$			1009.26i			0.0467791i
$776$		0			0
$777$	−1867.69			−0.0862331
$778$		0			0
$779$	12640.2			0.581365
$780$		0			0
$781$	−14724.4			−0.674623
$782$		0			0
$783$	−1839.48			−0.0839562
$784$		0			0
$785$			8214.84i			0.373503i
$786$		0			0
$787$		−	12589.1i		−	0.570207i	−0.958497	−	0.285103i	$-0.907972\pi$
$787$		−	12589.1i		−	0.570207i	0.958497	−	0.285103i	$-0.0920279\pi$
$788$		0			0
$789$	−19454.2			−0.877804
$790$		0			0
$791$		−	33796.3i		−	1.51916i
$792$		0			0
$793$	−32176.4	−	13846.5i	−1.44088	−	0.620055i
$794$		0			0
$795$		−	1269.32i		−	0.0566265i
$796$		0			0
$797$	23427.7			1.04122			0.520611	−	0.853794i	$-0.325704\pi$
$797$	23427.7			1.04122			0.520611	+	0.853794i	$0.325704\pi$
$798$		0			0
$799$			19202.7i			0.850239i
$800$		0			0
$801$		−	13736.2i		−	0.605924i
$802$		0			0
$803$	12512.3			0.549874
$804$		0			0
$805$	−8905.34			−0.389903
$806$		0			0
$807$	6459.75			0.281777
$808$		0			0
$809$	28409.3			1.23463			0.617316	−	0.786716i	$-0.288220\pi$
$809$	28409.3			1.23463			0.617316	+	0.786716i	$0.288220\pi$
$810$		0			0
$811$			38544.2i			1.66889i	0.551092	+	0.834445i	$0.314211\pi$
$811$			38544.2i			1.66889i	−0.551092	+	0.834445i	$0.685789\pi$
$812$		0			0
$813$			20535.3i			0.885861i
$814$		0			0
$815$	−3427.05			−0.147293
$816$		0			0
$817$			15738.9i			0.673972i
$818$		0			0
$819$	2518.48	−	5852.44i	0.107452	−	0.249696i
$820$		0			0
$821$		−	15535.4i		−	0.660400i	−0.943911	−	0.330200i	$-0.892884\pi$
$821$		−	15535.4i		−	0.660400i	0.943911	−	0.330200i	$-0.107116\pi$
$822$		0			0
$823$	16358.2			0.692846			0.346423	−	0.938078i	$-0.387396\pi$
$823$	16358.2			0.692846			0.346423	+	0.938078i	$0.387396\pi$
$824$		0			0
$825$			1948.99i			0.0822485i
$826$		0			0
$827$			6574.18i			0.276429i	0.990402	+	0.138214i	$0.0441363\pi$
$827$			6574.18i			0.276429i	−0.990402	+	0.138214i	$0.955864\pi$
$828$		0			0
$829$	−22002.9			−0.921826			−0.460913	−	0.887445i	$-0.652478\pi$
$829$	−22002.9			−0.921826			−0.460913	+	0.887445i	$0.652478\pi$
$830$		0			0
$831$	228.387			0.00953389
$832$		0			0
$833$	12250.0			0.509529
$834$		0			0
$835$	10609.9			0.439724
$836$		0			0
$837$			1090.00i			0.0450132i
$838$		0			0
$839$		−	1963.07i		−	0.0807779i	−0.999184	−	0.0403889i	$-0.987140\pi$
$839$		−	1963.07i		−	0.0807779i	0.999184	−	0.0403889i	$-0.0128597\pi$
$840$		0			0
$841$	−19747.4			−0.809686
$842$		0			0
$843$		−	8736.05i		−	0.356922i
$844$		0			0
$845$	−7977.12	+	7552.20i	−0.324759	+	0.307460i
$846$		0			0
$847$		−	9903.28i		−	0.401748i
$848$		0			0
$849$	8138.21			0.328979
$850$		0			0
$851$		−	4860.94i		−	0.195806i
$852$		0			0
$853$		−	22583.0i		−	0.906479i	−0.891389	−	0.453239i	$-0.850268\pi$
$853$		−	22583.0i		−	0.906479i	0.891389	−	0.453239i	$-0.149732\pi$
$854$		0			0
$855$	5675.99			0.227035
$856$		0			0
$857$	−1838.44			−0.0732787			−0.0366393	−	0.999329i	$-0.511665\pi$
$857$	−1838.44			−0.0732787			−0.0366393	+	0.999329i	$0.511665\pi$
$858$		0			0
$859$	−19743.7			−0.784221			−0.392111	−	0.919918i	$-0.628255\pi$
$859$	−19743.7			−0.784221			−0.392111	+	0.919918i	$0.628255\pi$
$860$		0			0
$861$	−4540.66			−0.179727
$862$		0			0
$863$		−	1733.38i		−	0.0683720i	−0.999415	−	0.0341860i	$-0.989116\pi$
$863$		−	1733.38i		−	0.0683720i	0.999415	−	0.0341860i	$-0.0108839\pi$
$864$		0			0
$865$			18474.3i			0.726178i
$866$		0			0
$867$	−19366.8			−0.758630
$868$		0			0
$869$			22415.7i			0.875031i
$870$		0			0
$871$	7889.93	−	18334.6i	0.306935	−	0.713253i
$872$		0			0
$873$			2617.07i			0.101460i
$874$		0			0
$875$	−1887.91			−0.0729407
$876$		0			0
$877$		−	5602.78i		−	0.215727i	−0.994166	−	0.107863i	$-0.965599\pi$
$877$		−	5602.78i		−	0.215727i	0.994166	−	0.107863i	$-0.0344009\pi$
$878$		0			0
$879$			20951.7i			0.803964i
$880$		0			0
$881$	−1662.99			−0.0635952			−0.0317976	−	0.999494i	$-0.510123\pi$
$881$	−1662.99			−0.0635952			−0.0317976	+	0.999494i	$0.510123\pi$
$882$		0			0
$883$	−52366.4			−1.99577			−0.997887	−	0.0649723i	$-0.979304\pi$
$883$	−52366.4			−1.99577			−0.997887	+	0.0649723i	$0.979304\pi$
$884$		0			0
$885$	1463.67			0.0555941
$886$		0			0
$887$	−15807.8			−0.598391			−0.299195	−	0.954192i	$-0.596718\pi$
$887$	−15807.8			−0.598391			−0.299195	+	0.954192i	$0.596718\pi$
$888$		0			0
$889$			17041.3i			0.642911i
$890$		0			0
$891$			2104.91i			0.0791436i
$892$		0			0
$893$	22716.2			0.851254
$894$		0			0
$895$			3420.96i			0.127765i
$896$		0			0
$897$	15231.8	+	6554.72i	0.566974	+	0.243986i
$898$		0			0
$899$		−	2750.40i		−	0.102037i
$900$		0			0
$901$	9022.63			0.333615
$902$		0			0
$903$		−	5653.78i		−	0.208357i
$904$		0			0
$905$			8806.25i			0.323458i
$906$		0			0
$907$	−4819.04			−0.176421			−0.0882103	−	0.996102i	$-0.528115\pi$
$907$	−4819.04			−0.176421			−0.0882103	+	0.996102i	$0.528115\pi$
$908$		0			0
$909$	907.108			0.0330989
$910$		0			0
$911$	−15711.9			−0.571414			−0.285707	−	0.958317i	$-0.592228\pi$
$911$	−15711.9			−0.571414			−0.285707	+	0.958317i	$0.592228\pi$
$912$		0			0
$913$	−7044.62			−0.255359
$914$		0			0
$915$			11210.0i			0.405019i
$916$		0			0
$917$			5635.85i			0.202958i
$918$		0			0
$919$	10755.9			0.386077			0.193039	−	0.981191i	$-0.438166\pi$
$919$	10755.9			0.386077			0.193039	+	0.981191i	$0.438166\pi$
$920$		0			0
$921$		−	8052.48i		−	0.288098i
$922$		0			0
$923$	−10498.2	+	24395.6i	−0.374379	+	0.869981i
$924$		0			0
$925$		−	1030.51i		−	0.0366302i
$926$		0			0
$927$	−2551.31			−0.0903947
$928$		0			0
$929$		−	27896.7i		−	0.985211i	−0.870253	−	0.492605i	$-0.836045\pi$
$929$		−	27896.7i		−	0.985211i	0.870253	−	0.492605i	$-0.163955\pi$
$930$		0			0
$931$		−	14491.4i		−	0.510137i
$932$		0			0
$933$	−5106.31			−0.179178
$934$		0			0
$935$	−13853.9			−0.484567
$936$		0			0
$937$	−37527.3			−1.30839			−0.654196	−	0.756325i	$-0.726993\pi$
$937$	−37527.3			−1.30839			−0.654196	+	0.756325i	$0.726993\pi$
$938$		0			0
$939$	−12888.5			−0.447925
$940$		0			0
$941$			49626.8i			1.71922i	0.510949	+	0.859611i	$0.329294\pi$
$941$			49626.8i			1.71922i	−0.510949	+	0.859611i	$0.670706\pi$
$942$		0			0
$943$		−	11817.7i		−	0.408100i
$944$		0			0
$945$	−2038.95			−0.0701872
$946$		0			0
$947$			35179.9i			1.20717i	0.797297	+	0.603587i	$0.206262\pi$
$947$			35179.9i			1.20717i	−0.797297	+	0.603587i	$0.793738\pi$
$948$		0			0
$949$	8920.99	−	20730.6i	0.305150	−	0.709107i
$950$		0			0
$951$			16156.5i			0.550904i
$952$		0			0
$953$	13422.7			0.456247			0.228124	−	0.973632i	$-0.426741\pi$
$953$	13422.7			0.456247			0.228124	+	0.973632i	$0.426741\pi$
$954$		0			0
$955$			24455.7i			0.828659i
$956$		0			0
$957$		−	5311.30i		−	0.179404i
$958$		0			0
$959$	−10480.4			−0.352897
$960$		0			0
$961$	28161.2			0.945293
$962$		0			0
$963$	−8925.62			−0.298675
$964$		0			0
$965$	14777.3			0.492952
$966$		0			0
$967$			25510.3i			0.848352i	0.905580	+	0.424176i	$0.139436\pi$
$967$			25510.3i			0.848352i	−0.905580	+	0.424176i	$0.860564\pi$
$968$		0			0
$969$			40346.3i			1.33758i
$970$		0			0
$971$	−1526.15			−0.0504393			−0.0252197	−	0.999682i	$-0.508029\pi$
$971$	−1526.15			−0.0504393			−0.0252197	+	0.999682i	$0.508029\pi$
$972$		0			0
$973$		−	9451.59i		−	0.311412i
$974$		0			0
$975$	3229.11	+	1389.59i	0.106066	+	0.0456435i
$976$		0			0
$977$		−	12413.0i		−	0.406477i	−0.979129	−	0.203238i	$-0.934853\pi$
$977$		−	12413.0i		−	0.406477i	0.979129	−	0.203238i	$-0.0651467\pi$
$978$		0			0
$979$	39661.8			1.29479
$980$		0			0
$981$		−	8376.03i		−	0.272606i
$982$		0			0
$983$		−	30187.4i		−	0.979481i	−0.871868	−	0.489740i	$-0.837092\pi$
$983$		−	30187.4i		−	0.979481i	0.871868	−	0.489740i	$-0.162908\pi$
$984$		0			0
$985$	−12527.2			−0.405229
$986$		0			0
$987$	−8160.20			−0.263163
$988$		0			0
$989$	14714.8			0.473108
$990$		0			0
$991$	7702.34			0.246895			0.123447	−	0.992351i	$-0.460605\pi$
$991$	7702.34			0.246895			0.123447	+	0.992351i	$0.460605\pi$
$992$		0			0
$993$		−	28640.2i		−	0.915277i
$994$		0			0
$995$			21972.9i			0.700087i
$996$		0			0
$997$	51145.6			1.62467			0.812336	−	0.583189i	$-0.198195\pi$
$997$	51145.6			1.62467			0.812336	+	0.583189i	$0.198195\pi$
$998$		0			0
$999$		−	1112.95i		−	0.0352474i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.g.c.961.14	yes	20
13.12	even	2	inner	1560.4.g.c.961.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.g.c.961.7	✓	20	13.12	even	2	inner
1560.4.g.c.961.14	yes	20	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000i q^{5} -15.1033i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.00000i q^{5} -15.1033i q^{7} +9.00000 q^{9} +25.9865i q^{11} +(43.0548 + 18.5278i) q^{13} -15.0000i q^{15} +106.624 q^{17} -126.133i q^{19} +45.3099i q^{21} -117.926 q^{23} -25.0000 q^{25} -27.0000 q^{27} +68.1290 q^{29} -40.3705i q^{31} -77.9595i q^{33} +75.5165 q^{35} +41.2204i q^{37} +(-129.165 - 55.5834i) q^{39} +100.213i q^{41} -124.780 q^{43} +45.0000i q^{45} +180.097i q^{47} +114.890 q^{49} -319.871 q^{51} +84.6212 q^{53} -129.932 q^{55} +378.399i q^{57} +97.5782i q^{59} -747.336 q^{61} -135.930i q^{63} +(-92.6391 + 215.274i) q^{65} -425.842i q^{67} +353.777 q^{69} +566.618i q^{71} -481.492i q^{73} +75.0000 q^{75} +392.482 q^{77} +862.592 q^{79} +81.0000 q^{81} +271.088i q^{83} +533.118i q^{85} -204.387 q^{87} -1526.25i q^{89} +(279.831 - 650.271i) q^{91} +121.112i q^{93} +630.665 q^{95} +290.785i q^{97} +233.878i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{3} + 180 q^{9}+O(q^{10}) \) 20 * q - 60 * q^3 + 180 * q^9	\( 20 q - 60 q^{3} + 180 q^{9} - 38 q^{13} - 58 q^{17} + 86 q^{23} - 500 q^{25} - 540 q^{27} - 220 q^{29} - 90 q^{35} + 114 q^{39} - 340 q^{43} - 482 q^{49} + 174 q^{51} - 106 q^{53} + 110 q^{55} - 70 q^{61} - 150 q^{65} - 258 q^{69} + 1500 q^{75} + 4058 q^{77} + 2054 q^{79} + 1620 q^{81} + 660 q^{87} - 2366 q^{91} - 20 q^{95}+O(q^{100}) \) 20 * q - 60 * q^3 + 180 * q^9 - 38 * q^13 - 58 * q^17 + 86 * q^23 - 500 * q^25 - 540 * q^27 - 220 * q^29 - 90 * q^35 + 114 * q^39 - 340 * q^43 - 482 * q^49 + 174 * q^51 - 106 * q^53 + 110 * q^55 - 70 * q^61 - 150 * q^65 - 258 * q^69 + 1500 * q^75 + 4058 * q^77 + 2054 * q^79 + 1620 * q^81 + 660 * q^87 - 2366 * q^91 - 20 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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