LMFDB - Embedded newform 1560.4.g.c.961.10

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			961.10
Root			$-7.94352i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.4.g.c.961.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000i q^{5} +24.8312i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000i q^{5} +24.8312i q^{7} +9.00000 q^{9} -40.7182i q^{11} +(-34.2400 - 32.0097i) q^{13} +15.0000i q^{15} -27.4391 q^{17} +13.7504i q^{19} -74.4936i q^{21} +131.732 q^{23} -25.0000 q^{25} -27.0000 q^{27} -8.90679 q^{29} -95.6531i q^{31} +122.155i q^{33} +124.156 q^{35} +100.802i q^{37} +(102.720 + 96.0290i) q^{39} +473.938i q^{41} -15.3147 q^{43} -45.0000i q^{45} -502.427i q^{47} -273.588 q^{49} +82.3172 q^{51} +160.498 q^{53} -203.591 q^{55} -41.2511i q^{57} +612.200i q^{59} +397.485 q^{61} +223.481i q^{63} +(-160.048 + 171.200i) q^{65} -330.989i q^{67} -395.196 q^{69} -313.623i q^{71} -671.046i q^{73} +75.0000 q^{75} +1011.08 q^{77} -1188.72 q^{79} +81.0000 q^{81} -391.049i q^{83} +137.195i q^{85} +26.7204 q^{87} +1483.30i q^{89} +(794.838 - 850.221i) q^{91} +286.959i q^{93} +68.7518 q^{95} -389.415i q^{97} -366.464i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			24.8312i			1.34076i	0.742019	+	0.670379i	$0.233869\pi$
$7$			24.8312i			1.34076i	−0.742019	+	0.670379i	$0.766131\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	40.7182i		−	1.11609i	−0.829810	−	0.558046i	$-0.811551\pi$
$11$		−	40.7182i		−	1.11609i	0.829810	−	0.558046i	$-0.188449\pi$
$12$		0			0
$13$	−34.2400	−	32.0097i	−0.730498	−	0.682914i
$13$	−34.2400	−	32.0097i	−0.730498	−	0.682914i
$14$		0			0
$15$			15.0000i			0.258199i
$16$		0			0
$17$	−27.4391			−0.391468			−0.195734	−	0.980657i	$-0.562709\pi$
$17$	−27.4391			−0.391468			−0.195734	+	0.980657i	$0.562709\pi$
$18$		0			0
$19$			13.7504i			0.166029i	0.996548	+	0.0830145i	$0.0264548\pi$
$19$			13.7504i			0.166029i	−0.996548	+	0.0830145i	$0.973545\pi$
$20$		0			0
$21$		−	74.4936i		−	0.774087i
$22$		0			0
$23$	131.732			1.19426			0.597130	−	0.802144i	$-0.296307\pi$
$23$	131.732			1.19426			0.597130	+	0.802144i	$0.296307\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−8.90679			−0.0570328			−0.0285164	−	0.999593i	$-0.509078\pi$
$29$	−8.90679			−0.0570328			−0.0285164	+	0.999593i	$0.509078\pi$
$30$		0			0
$31$		−	95.6531i		−	0.554187i	−0.960843	−	0.277094i	$-0.910629\pi$
$31$		−	95.6531i		−	0.554187i	0.960843	−	0.277094i	$-0.0893712\pi$
$32$		0			0
$33$			122.155i			0.644376i
$34$		0			0
$35$	124.156			0.599605
$36$		0			0
$37$			100.802i			0.447887i	0.974602	+	0.223943i	$0.0718931\pi$
$37$			100.802i			0.447887i	−0.974602	+	0.223943i	$0.928107\pi$
$38$		0			0
$39$	102.720	+	96.0290i	0.421754	+	0.394281i
$40$		0			0
$41$			473.938i			1.80528i	0.430391	+	0.902642i	$0.358376\pi$
$41$			473.938i			1.80528i	−0.430391	+	0.902642i	$0.641624\pi$
$42$		0			0
$43$	−15.3147			−0.0543131			−0.0271566	−	0.999631i	$-0.508645\pi$
$43$	−15.3147			−0.0543131			−0.0271566	+	0.999631i	$0.508645\pi$
$44$		0			0
$45$		−	45.0000i		−	0.149071i
$46$		0			0
$47$		−	502.427i		−	1.55929i	−0.626224	−	0.779643i	$-0.715400\pi$
$47$		−	502.427i		−	1.55929i	0.626224	−	0.779643i	$-0.284600\pi$
$48$		0			0
$49$	−273.588			−0.797633
$50$		0			0
$51$	82.3172			0.226014
$52$		0			0
$53$	160.498			0.415963			0.207982	−	0.978133i	$-0.433311\pi$
$53$	160.498			0.415963			0.207982	+	0.978133i	$0.433311\pi$
$54$		0			0
$55$	−203.591			−0.499131
$56$		0			0
$57$		−	41.2511i		−	0.0958568i
$58$		0			0
$59$			612.200i			1.35087i	0.737417	+	0.675437i	$0.236045\pi$
$59$			612.200i			1.35087i	−0.737417	+	0.675437i	$0.763955\pi$
$60$		0			0
$61$	397.485			0.834307			0.417154	−	0.908836i	$-0.363028\pi$
$61$	397.485			0.834307			0.417154	+	0.908836i	$0.363028\pi$
$62$		0			0
$63$			223.481i			0.446920i
$64$		0			0
$65$	−160.048	+	171.200i	−0.305409	+	0.326689i
$66$		0			0
$67$		−	330.989i		−	0.603534i	−0.953382	−	0.301767i	$-0.902424\pi$
$67$		−	330.989i		−	0.603534i	0.953382	−	0.301767i	$-0.0975765\pi$
$68$		0			0
$69$	−395.196			−0.689507
$70$		0			0
$71$		−	313.623i		−	0.524228i	−0.965037	−	0.262114i	$-0.915580\pi$
$71$		−	313.623i		−	0.524228i	0.965037	−	0.262114i	$-0.0844197\pi$
$72$		0			0
$73$		−	671.046i		−	1.07589i	−0.842980	−	0.537946i	$-0.819201\pi$
$73$		−	671.046i		−	1.07589i	0.842980	−	0.537946i	$-0.180799\pi$
$74$		0			0
$75$	75.0000			0.115470
$76$		0			0
$77$	1011.08			1.49641
$78$		0			0
$79$	−1188.72			−1.69293			−0.846465	−	0.532444i	$-0.821274\pi$
$79$	−1188.72			−1.69293			−0.846465	+	0.532444i	$0.821274\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	391.049i		−	0.517147i	−0.965992	−	0.258574i	$-0.916748\pi$
$83$		−	391.049i		−	0.517147i	0.965992	−	0.258574i	$-0.0832525\pi$
$84$		0			0
$85$			137.195i			0.175070i
$86$		0			0
$87$	26.7204			0.0329279
$88$		0			0
$89$			1483.30i			1.76663i	0.468783	+	0.883314i	$0.344693\pi$
$89$			1483.30i			1.76663i	−0.468783	+	0.883314i	$0.655307\pi$
$90$		0			0
$91$	794.838	−	850.221i	0.915623	−	0.979422i
$92$		0			0
$93$			286.959i			0.319960i
$94$		0			0
$95$	68.7518			0.0742504
$96$		0			0
$97$		−	389.415i		−	0.407620i	−0.979010	−	0.203810i	$-0.934668\pi$
$97$		−	389.415i		−	0.407620i	0.979010	−	0.203810i	$-0.0653325\pi$
$98$		0			0
$99$		−	366.464i		−	0.372031i
$100$		0			0
$101$	−360.311			−0.354973			−0.177487	−	0.984123i	$-0.556797\pi$
$101$	−360.311			−0.354973			−0.177487	+	0.984123i	$0.556797\pi$
$102$		0			0
$103$	−512.910			−0.490666			−0.245333	−	0.969439i	$-0.578897\pi$
$103$	−512.910			−0.490666			−0.245333	+	0.969439i	$0.578897\pi$
$104$		0			0
$105$	−372.468			−0.346182
$106$		0			0
$107$	784.881			0.709134			0.354567	−	0.935031i	$-0.384628\pi$
$107$	784.881			0.709134			0.354567	+	0.935031i	$0.384628\pi$
$108$		0			0
$109$			58.5862i			0.0514820i	0.999669	+	0.0257410i	$0.00819452\pi$
$109$			58.5862i			0.0514820i	−0.999669	+	0.0257410i	$0.991805\pi$
$110$		0			0
$111$		−	302.407i		−	0.258588i
$112$		0			0
$113$	−1744.75			−1.45250			−0.726249	−	0.687432i	$-0.758738\pi$
$113$	−1744.75			−1.45250			−0.726249	+	0.687432i	$0.758738\pi$
$114$		0			0
$115$		−	658.660i		−	0.534090i
$116$		0			0
$117$	−308.160	−	288.087i	−0.243499	−	0.227638i
$118$		0			0
$119$		−	681.345i		−	0.524864i
$120$		0			0
$121$	−326.974			−0.245661
$122$		0			0
$123$		−	1421.81i		−	1.04228i
$124$		0			0
$125$			125.000i			0.0894427i
$126$		0			0
$127$	−2141.81			−1.49649			−0.748246	−	0.663421i	$-0.769104\pi$
$127$	−2141.81			−1.49649			−0.748246	+	0.663421i	$0.769104\pi$
$128$		0			0
$129$	45.9440			0.0313577
$130$		0			0
$131$	316.215			0.210899			0.105450	−	0.994425i	$-0.466372\pi$
$131$	316.215			0.210899			0.105450	+	0.994425i	$0.466372\pi$
$132$		0			0
$133$	−341.438			−0.222605
$134$		0			0
$135$			135.000i			0.0860663i
$136$		0			0
$137$		−	597.779i		−	0.372786i	−0.982475	−	0.186393i	$-0.940320\pi$
$137$		−	597.779i		−	0.372786i	0.982475	−	0.186393i	$-0.0596798\pi$
$138$		0			0
$139$	−1390.97			−0.848779			−0.424390	−	0.905480i	$-0.639511\pi$
$139$	−1390.97			−0.848779			−0.424390	+	0.905480i	$0.639511\pi$
$140$		0			0
$141$			1507.28i			0.900255i
$142$		0			0
$143$	−1303.38	+	1394.19i	−0.762195	+	0.815303i
$144$		0			0
$145$			44.5340i			0.0255058i
$146$		0			0
$147$	820.765			0.460514
$148$		0			0
$149$			2512.06i			1.38118i	0.723245	+	0.690592i	$0.242650\pi$
$149$			2512.06i			1.38118i	−0.723245	+	0.690592i	$0.757350\pi$
$150$		0			0
$151$			2476.88i			1.33487i	0.744667	+	0.667436i	$0.232608\pi$
$151$			2476.88i			1.33487i	−0.744667	+	0.667436i	$0.767392\pi$
$152$		0			0
$153$	−246.952			−0.130489
$154$		0			0
$155$	−478.265			−0.247840
$156$		0			0
$157$	1467.37			0.745918			0.372959	−	0.927848i	$-0.378343\pi$
$157$	1467.37			0.745918			0.372959	+	0.927848i	$0.378343\pi$
$158$		0			0
$159$	−481.493			−0.240157
$160$		0			0
$161$			3271.06i			1.60122i
$162$		0			0
$163$			1302.17i			0.625728i	0.949798	+	0.312864i	$0.101288\pi$
$163$			1302.17i			0.625728i	−0.949798	+	0.312864i	$0.898712\pi$
$164$		0			0
$165$	610.773			0.288174
$166$		0			0
$167$			2961.61i			1.37231i	0.727454	+	0.686157i	$0.240704\pi$
$167$			2961.61i			1.37231i	−0.727454	+	0.686157i	$0.759296\pi$
$168$		0			0
$169$	147.762	+	2192.03i	0.0672561	+	0.997736i
$170$		0			0
$171$			123.753i			0.0553430i
$172$		0			0
$173$	−2413.88			−1.06083			−0.530415	−	0.847738i	$-0.677964\pi$
$173$	−2413.88			−1.06083			−0.530415	+	0.847738i	$0.677964\pi$
$174$		0			0
$175$		−	620.780i		−	0.268152i
$176$		0			0
$177$		−	1836.60i		−	0.779928i
$178$		0			0
$179$	750.448			0.313358			0.156679	−	0.987650i	$-0.449921\pi$
$179$	750.448			0.313358			0.156679	+	0.987650i	$0.449921\pi$
$180$		0			0
$181$	605.971			0.248848			0.124424	−	0.992229i	$-0.460292\pi$
$181$	605.971			0.248848			0.124424	+	0.992229i	$0.460292\pi$
$182$		0			0
$183$	−1192.45			−0.481687
$184$		0			0
$185$	504.012			0.200301
$186$		0			0
$187$			1117.27i			0.436914i
$188$		0			0
$189$		−	670.442i		−	0.258029i
$190$		0			0
$191$	−1130.33			−0.428207			−0.214104	−	0.976811i	$-0.568683\pi$
$191$	−1130.33			−0.428207			−0.214104	+	0.976811i	$0.568683\pi$
$192$		0			0
$193$			229.343i			0.0855360i	0.999085	+	0.0427680i	$0.0136176\pi$
$193$			229.343i			0.0855360i	−0.999085	+	0.0427680i	$0.986382\pi$
$194$		0			0
$195$	480.145	−	513.601i	0.176328	−	0.188614i
$196$		0			0
$197$			2321.81i			0.839706i	0.907592	+	0.419853i	$0.137918\pi$
$197$			2321.81i			0.839706i	−0.907592	+	0.419853i	$0.862082\pi$
$198$		0			0
$199$	−3991.49			−1.42185			−0.710927	−	0.703266i	$-0.751724\pi$
$199$	−3991.49			−1.42185			−0.710927	+	0.703266i	$0.751724\pi$
$200$		0			0
$201$			992.968i			0.348451i
$202$		0			0
$203$		−	221.166i		−	0.0764672i
$204$		0			0
$205$	2369.69			0.807348
$206$		0			0
$207$	1185.59			0.398087
$208$		0			0
$209$	559.890			0.185303
$210$		0			0
$211$	−560.820			−0.182979			−0.0914893	−	0.995806i	$-0.529163\pi$
$211$	−560.820			−0.182979			−0.0914893	+	0.995806i	$0.529163\pi$
$212$		0			0
$213$			940.869i			0.302663i
$214$		0			0
$215$			76.5733i			0.0242896i
$216$		0			0
$217$	2375.18			0.743031
$218$		0			0
$219$			2013.14i			0.621166i
$220$		0			0
$221$	939.515	+	878.316i	0.285967	+	0.267339i
$222$		0			0
$223$			4422.27i			1.32797i	0.747746	+	0.663985i	$0.231136\pi$
$223$			4422.27i			1.32797i	−0.747746	+	0.663985i	$0.768864\pi$
$224$		0			0
$225$	−225.000			−0.0666667
$226$		0			0
$227$		−	955.237i		−	0.279301i	−0.990201	−	0.139651i	$-0.955402\pi$
$227$		−	955.237i		−	0.279301i	0.990201	−	0.139651i	$-0.0445979\pi$
$228$		0			0
$229$			3246.86i			0.936936i	0.883480	+	0.468468i	$0.155194\pi$
$229$			3246.86i			0.936936i	−0.883480	+	0.468468i	$0.844806\pi$
$230$		0			0
$231$	−3033.25			−0.863952
$232$		0			0
$233$	1597.19			0.449078			0.224539	−	0.974465i	$-0.427912\pi$
$233$	1597.19			0.449078			0.224539	+	0.974465i	$0.427912\pi$
$234$		0			0
$235$	−2512.13			−0.697334
$236$		0			0
$237$	3566.16			0.977414
$238$		0			0
$239$			3240.85i			0.877126i	0.898701	+	0.438563i	$0.144512\pi$
$239$			3240.85i			0.877126i	−0.898701	+	0.438563i	$0.855488\pi$
$240$		0			0
$241$			2579.69i			0.689513i	0.938692	+	0.344756i	$0.112038\pi$
$241$			2579.69i			0.689513i	−0.938692	+	0.344756i	$0.887962\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$			1367.94i			0.356713i
$246$		0			0
$247$	440.145	−	470.813i	0.113384	−	0.121284i
$248$		0			0
$249$			1173.15i			0.298575i
$250$		0			0
$251$	−5461.60			−1.37344			−0.686720	−	0.726922i	$-0.740950\pi$
$251$	−5461.60			−1.37344			−0.686720	+	0.726922i	$0.740950\pi$
$252$		0			0
$253$		−	5363.89i		−	1.33290i
$254$		0			0
$255$		−	411.586i		−	0.101077i
$256$		0			0
$257$	−7708.93			−1.87109			−0.935545	−	0.353208i	$-0.885091\pi$
$257$	−7708.93			−1.87109			−0.935545	+	0.353208i	$0.885091\pi$
$258$		0			0
$259$	−2503.04			−0.600508
$260$		0			0
$261$	−80.1612			−0.0190109
$262$		0			0
$263$	−5692.99			−1.33477			−0.667385	−	0.744713i	$-0.732587\pi$
$263$	−5692.99			−1.33477			−0.667385	+	0.744713i	$0.732587\pi$
$264$		0			0
$265$		−	802.488i		−	0.186024i
$266$		0			0
$267$		−	4449.91i		−	1.01996i
$268$		0			0
$269$	−2288.88			−0.518794			−0.259397	−	0.965771i	$-0.583524\pi$
$269$	−2288.88			−0.518794			−0.259397	+	0.965771i	$0.583524\pi$
$270$		0			0
$271$			468.413i			0.104996i	0.998621	+	0.0524982i	$0.0167184\pi$
$271$			468.413i			0.104996i	−0.998621	+	0.0524982i	$0.983282\pi$
$272$		0			0
$273$	−2384.52	+	2550.66i	−0.528635	+	0.565470i
$274$		0			0
$275$			1017.96i			0.223218i
$276$		0			0
$277$	1726.51			0.374498			0.187249	−	0.982312i	$-0.440043\pi$
$277$	1726.51			0.374498			0.187249	+	0.982312i	$0.440043\pi$
$278$		0			0
$279$		−	860.878i		−	0.184729i
$280$		0			0
$281$		−	3742.59i		−	0.794534i	−0.917703	−	0.397267i	$-0.869959\pi$
$281$		−	3742.59i		−	0.794534i	0.917703	−	0.397267i	$-0.130041\pi$
$282$		0			0
$283$	2814.18			0.591115			0.295557	−	0.955325i	$-0.404495\pi$
$283$	2814.18			0.591115			0.295557	+	0.955325i	$0.404495\pi$
$284$		0			0
$285$	−206.255			−0.0428685
$286$		0			0
$287$	−11768.4			−2.42045
$288$		0			0
$289$	−4160.10			−0.846753
$290$		0			0
$291$			1168.25i			0.235340i
$292$		0			0
$293$		−	6288.59i		−	1.25387i	−0.779072	−	0.626934i	$-0.784309\pi$
$293$		−	6288.59i		−	1.25387i	0.779072	−	0.626934i	$-0.215691\pi$
$294$		0			0
$295$	3061.00			0.604130
$296$		0			0
$297$			1099.39i			0.214792i
$298$		0			0
$299$	−4510.51	−	4216.70i	−0.872406	−	0.815578i
$300$		0			0
$301$		−	380.282i		−	0.0728208i
$302$		0			0
$303$	1080.93			0.204944
$304$		0			0
$305$		−	1987.42i		−	0.373113i
$306$		0			0
$307$		−	673.340i		−	0.125178i	−0.998039	−	0.0625888i	$-0.980064\pi$
$307$		−	673.340i		−	0.125178i	0.998039	−	0.0625888i	$-0.0199357\pi$
$308$		0			0
$309$	1538.73			0.283286
$310$		0			0
$311$	−9391.96			−1.71244			−0.856221	−	0.516610i	$-0.827194\pi$
$311$	−9391.96			−1.71244			−0.856221	+	0.516610i	$0.827194\pi$
$312$		0			0
$313$	−4445.30			−0.802758			−0.401379	−	0.915912i	$-0.631469\pi$
$313$	−4445.30			−0.802758			−0.401379	+	0.915912i	$0.631469\pi$
$314$		0			0
$315$	1117.40			0.199868
$316$		0			0
$317$			8650.21i			1.53263i	0.642464	+	0.766316i	$0.277912\pi$
$317$			8650.21i			1.53263i	−0.642464	+	0.766316i	$0.722088\pi$
$318$		0			0
$319$			362.669i			0.0636538i
$320$		0			0
$321$	−2354.64			−0.409419
$322$		0			0
$323$		−	377.297i		−	0.0649950i
$324$		0			0
$325$	856.001	+	800.242i	0.146100	+	0.136583i
$326$		0			0
$327$		−	175.758i		−	0.0297231i
$328$		0			0
$329$	12475.9			2.09063
$330$		0			0
$331$		−	8485.12i		−	1.40902i	−0.709696	−	0.704508i	$-0.751168\pi$
$331$		−	8485.12i		−	1.40902i	0.709696	−	0.704508i	$-0.248832\pi$
$332$		0			0
$333$			907.221i			0.149296i
$334$		0			0
$335$	−1654.95			−0.269909
$336$		0			0
$337$	−6496.68			−1.05014			−0.525069	−	0.851059i	$-0.675961\pi$
$337$	−6496.68			−1.05014			−0.525069	+	0.851059i	$0.675961\pi$
$338$		0			0
$339$	5234.25			0.838600
$340$		0			0
$341$	−3894.82			−0.618524
$342$		0			0
$343$			1723.58i			0.271325i
$344$		0			0
$345$			1975.98i			0.308357i
$346$		0			0
$347$	1293.95			0.200181			0.100090	−	0.994978i	$-0.468087\pi$
$347$	1293.95			0.200181			0.100090	+	0.994978i	$0.468087\pi$
$348$		0			0
$349$			12324.5i			1.89031i	0.326625	+	0.945154i	$0.394089\pi$
$349$			12324.5i			1.89031i	−0.326625	+	0.945154i	$0.605911\pi$
$350$		0			0
$351$	924.481	+	864.261i	0.140585	+	0.131427i
$352$		0			0
$353$			3723.05i			0.561354i	0.959802	+	0.280677i	$0.0905590\pi$
$353$			3723.05i			0.561354i	−0.959802	+	0.280677i	$0.909441\pi$
$354$		0			0
$355$	−1568.12			−0.234442
$356$		0			0
$357$			2044.04i			0.303030i
$358$		0			0
$359$		−	9596.37i		−	1.41080i	−0.708810	−	0.705400i	$-0.750768\pi$
$359$		−	9596.37i		−	1.41080i	0.708810	−	0.705400i	$-0.249232\pi$
$360$		0			0
$361$	6669.93			0.972434
$362$		0			0
$363$	980.923			0.141832
$364$		0			0
$365$	−3355.23			−0.481153
$366$		0			0
$367$	2874.13			0.408796			0.204398	−	0.978888i	$-0.434476\pi$
$367$	2874.13			0.408796			0.204398	+	0.978888i	$0.434476\pi$
$368$		0			0
$369$			4265.44i			0.601762i
$370$		0			0
$371$			3985.35i			0.557706i
$372$		0			0
$373$	−6629.98			−0.920341			−0.460171	−	0.887830i	$-0.652212\pi$
$373$	−6629.98			−0.920341			−0.460171	+	0.887830i	$0.652212\pi$
$374$		0			0
$375$		−	375.000i		−	0.0516398i
$376$		0			0
$377$	304.969	+	285.104i	0.0416624	+	0.0389485i
$378$		0			0
$379$		−	6862.04i		−	0.930024i	−0.885304	−	0.465012i	$-0.846050\pi$
$379$		−	6862.04i		−	0.930024i	0.885304	−	0.465012i	$-0.153950\pi$
$380$		0			0
$381$	6425.42			0.864000
$382$		0			0
$383$			854.055i			0.113943i	0.998376	+	0.0569715i	$0.0181444\pi$
$383$			854.055i			0.113943i	−0.998376	+	0.0569715i	$0.981856\pi$
$384$		0			0
$385$		−	5055.41i		−	0.669215i
$386$		0			0
$387$	−137.832			−0.0181044
$388$		0			0
$389$	11159.5			1.45452			0.727258	−	0.686364i	$-0.240794\pi$
$389$	11159.5			1.45452			0.727258	+	0.686364i	$0.240794\pi$
$390$		0			0
$391$	−3614.60			−0.467515
$392$		0			0
$393$	−948.645			−0.121763
$394$		0			0
$395$			5943.60i			0.757101i
$396$		0			0
$397$			4731.66i			0.598175i	0.954226	+	0.299087i	$0.0966822\pi$
$397$			4731.66i			0.598175i	−0.954226	+	0.299087i	$0.903318\pi$
$398$		0			0
$399$	1024.31			0.128521
$400$		0			0
$401$			6927.61i			0.862714i	0.902181	+	0.431357i	$0.141965\pi$
$401$			6927.61i			0.862714i	−0.902181	+	0.431357i	$0.858035\pi$
$402$		0			0
$403$	−3061.82	+	3275.17i	−0.378462	+	0.404833i
$404$		0			0
$405$		−	405.000i		−	0.0496904i
$406$		0			0
$407$	4104.49			0.499883
$408$		0			0
$409$		−	1947.96i		−	0.235502i	−0.993043	−	0.117751i	$-0.962431\pi$
$409$		−	1947.96i		−	0.235502i	0.993043	−	0.117751i	$-0.0375685\pi$
$410$		0			0
$411$			1793.34i			0.215228i
$412$		0			0
$413$	−15201.6			−1.81120
$414$		0			0
$415$	−1955.25			−0.231275
$416$		0			0
$417$	4172.90			0.490043
$418$		0			0
$419$	−2923.34			−0.340846			−0.170423	−	0.985371i	$-0.554513\pi$
$419$	−2923.34			−0.340846			−0.170423	+	0.985371i	$0.554513\pi$
$420$		0			0
$421$		−	921.421i		−	0.106668i	−0.998577	−	0.0533341i	$-0.983015\pi$
$421$		−	921.421i		−	0.106668i	0.998577	−	0.0533341i	$-0.0169848\pi$
$422$		0			0
$423$		−	4521.84i		−	0.519762i
$424$		0			0
$425$	685.977			0.0782936
$426$		0			0
$427$			9870.03i			1.11860i
$428$		0			0
$429$	3910.13	−	4182.58i	0.440053	−	0.470716i
$430$		0			0
$431$			2497.26i			0.279092i	0.990216	+	0.139546i	$0.0445643\pi$
$431$			2497.26i			0.279092i	−0.990216	+	0.139546i	$0.955436\pi$
$432$		0			0
$433$	3585.23			0.397910			0.198955	−	0.980009i	$-0.436245\pi$
$433$	3585.23			0.397910			0.198955	+	0.980009i	$0.436245\pi$
$434$		0			0
$435$		−	133.602i		−	0.0147258i
$436$		0			0
$437$			1811.36i			0.198282i
$438$		0			0
$439$	14262.1			1.55055			0.775275	−	0.631624i	$-0.217612\pi$
$439$	14262.1			1.55055			0.775275	+	0.631624i	$0.217612\pi$
$440$		0			0
$441$	−2462.29			−0.265878
$442$		0			0
$443$	2354.07			0.252473			0.126236	−	0.992000i	$-0.459710\pi$
$443$	2354.07			0.252473			0.126236	+	0.992000i	$0.459710\pi$
$444$		0			0
$445$	7416.52			0.790060
$446$		0			0
$447$		−	7536.19i		−	0.797426i
$448$		0			0
$449$			7311.35i			0.768472i	0.923235	+	0.384236i	$0.125535\pi$
$449$			7311.35i			0.768472i	−0.923235	+	0.384236i	$0.874465\pi$
$450$		0			0
$451$	19297.9			2.01486
$452$		0			0
$453$		−	7430.64i		−	0.770688i
$454$		0			0
$455$	−4251.11	−	3974.19i	−0.438011	−	0.409479i
$456$		0			0
$457$			10084.0i			1.03219i	0.856532	+	0.516095i	$0.172615\pi$
$457$			10084.0i			1.03219i	−0.856532	+	0.516095i	$0.827385\pi$
$458$		0			0
$459$	740.855			0.0753380
$460$		0			0
$461$			13219.0i			1.33551i	0.744381	+	0.667755i	$0.232745\pi$
$461$			13219.0i			1.33551i	−0.744381	+	0.667755i	$0.767255\pi$
$462$		0			0
$463$		−	19107.4i		−	1.91792i	−0.283541	−	0.958960i	$-0.591509\pi$
$463$		−	19107.4i		−	1.91792i	0.283541	−	0.958960i	$-0.408491\pi$
$464$		0			0
$465$	1434.80			0.143091
$466$		0			0
$467$	494.249			0.0489746			0.0244873	−	0.999700i	$-0.492205\pi$
$467$	494.249			0.0489746			0.0244873	+	0.999700i	$0.492205\pi$
$468$		0			0
$469$	8218.86			0.809194
$470$		0			0
$471$	−4402.12			−0.430656
$472$		0			0
$473$			623.586i			0.0606184i
$474$		0			0
$475$		−	343.759i		−	0.0332058i
$476$		0			0
$477$	1444.48			0.138654
$478$		0			0
$479$		−	4644.44i		−	0.443027i	−0.975157	−	0.221514i	$-0.928900\pi$
$479$		−	4644.44i		−	0.443027i	0.975157	−	0.221514i	$-0.0710997\pi$
$480$		0			0
$481$	3226.65	−	3451.48i	0.305868	−	0.327181i
$482$		0			0
$483$		−	9813.18i		−	0.924462i
$484$		0			0
$485$	−1947.08			−0.182293
$486$		0			0
$487$		−	5946.96i		−	0.553352i	−0.960963	−	0.276676i	$-0.910767\pi$
$487$		−	5946.96i		−	0.553352i	0.960963	−	0.276676i	$-0.0892329\pi$
$488$		0			0
$489$		−	3906.50i		−	0.361264i
$490$		0			0
$491$	12687.0			1.16611			0.583053	−	0.812434i	$-0.301858\pi$
$491$	12687.0			1.16611			0.583053	+	0.812434i	$0.301858\pi$
$492$		0			0
$493$	244.394			0.0223265
$494$		0			0
$495$	−1832.32			−0.166377
$496$		0			0
$497$	7787.64			0.702864
$498$		0			0
$499$			5679.62i			0.509529i	0.967003	+	0.254764i	$0.0819979\pi$
$499$			5679.62i			0.509529i	−0.967003	+	0.254764i	$0.918002\pi$
$500$		0			0
$501$		−	8884.83i		−	0.792306i
$502$		0			0
$503$	458.723			0.0406629			0.0203315	−	0.999793i	$-0.493528\pi$
$503$	458.723			0.0406629			0.0203315	+	0.999793i	$0.493528\pi$
$504$		0			0
$505$			1801.56i			0.158749i
$506$		0			0
$507$	−443.285	−	6576.08i	−0.0388303	−	0.576043i
$508$		0			0
$509$			16527.2i			1.43921i	0.694385	+	0.719604i	$0.255676\pi$
$509$			16527.2i			1.43921i	−0.694385	+	0.719604i	$0.744324\pi$
$510$		0			0
$511$	16662.9			1.44251
$512$		0			0
$513$		−	371.260i		−	0.0319523i
$514$		0			0
$515$			2564.55i			0.219432i
$516$		0			0
$517$	−20457.9			−1.74031
$518$		0			0
$519$	7241.63			0.612471
$520$		0			0
$521$	−6487.40			−0.545524			−0.272762	−	0.962082i	$-0.587937\pi$
$521$	−6487.40			−0.545524			−0.272762	+	0.962082i	$0.587937\pi$
$522$		0			0
$523$	−6829.73			−0.571019			−0.285510	−	0.958376i	$-0.592163\pi$
$523$	−6829.73			−0.571019			−0.285510	+	0.958376i	$0.592163\pi$
$524$		0			0
$525$			1862.34i			0.154817i
$526$		0			0
$527$			2624.63i			0.216946i
$528$		0			0
$529$	5186.30			0.426259
$530$		0			0
$531$			5509.80i			0.450292i
$532$		0			0
$533$	15170.6	−	16227.7i	1.23285	−	1.31876i
$534$		0			0
$535$		−	3924.41i		−	0.317134i
$536$		0			0
$537$	−2251.35			−0.180918
$538$		0			0
$539$			11140.0i			0.890232i
$540$		0			0
$541$		−	24892.2i		−	1.97819i	−0.147289	−	0.989093i	$-0.547055\pi$
$541$		−	24892.2i		−	1.97819i	0.147289	−	0.989093i	$-0.452945\pi$
$542$		0			0
$543$	−1817.91			−0.143672
$544$		0			0
$545$	292.931			0.0230234
$546$		0			0
$547$	2093.73			0.163659			0.0818294	−	0.996646i	$-0.473924\pi$
$547$	2093.73			0.163659			0.0818294	+	0.996646i	$0.473924\pi$
$548$		0			0
$549$	3577.36			0.278102
$550$		0			0
$551$		−	122.472i		−	0.00946909i
$552$		0			0
$553$		−	29517.3i		−	2.26981i
$554$		0			0
$555$	−1512.04			−0.115644
$556$		0			0
$557$		−	7167.43i		−	0.545231i	−0.962123	−	0.272615i	$-0.912111\pi$
$557$		−	7167.43i		−	0.545231i	0.962123	−	0.272615i	$-0.0878887\pi$
$558$		0			0
$559$	524.375	+	490.218i	0.0396757	+	0.0370912i
$560$		0			0
$561$		−	3351.81i		−	0.252252i
$562$		0			0
$563$	−8743.45			−0.654516			−0.327258	−	0.944935i	$-0.606125\pi$
$563$	−8743.45			−0.654516			−0.327258	+	0.944935i	$0.606125\pi$
$564$		0			0
$565$			8723.75i			0.649577i
$566$		0			0
$567$			2011.33i			0.148973i
$568$		0			0
$569$	−535.459			−0.0394510			−0.0197255	−	0.999805i	$-0.506279\pi$
$569$	−535.459			−0.0394510			−0.0197255	+	0.999805i	$0.506279\pi$
$570$		0			0
$571$	−1557.99			−0.114185			−0.0570926	−	0.998369i	$-0.518183\pi$
$571$	−1557.99			−0.114185			−0.0570926	+	0.998369i	$0.518183\pi$
$572$		0			0
$573$	3390.98			0.247225
$574$		0			0
$575$	−3293.30			−0.238852
$576$		0			0
$577$			6909.60i			0.498528i	0.968436	+	0.249264i	$0.0801886\pi$
$577$			6909.60i			0.498528i	−0.968436	+	0.249264i	$0.919811\pi$
$578$		0			0
$579$		−	688.028i		−	0.0493842i
$580$		0			0
$581$	9710.22			0.693370
$582$		0			0
$583$		−	6535.18i		−	0.464253i
$584$		0			0
$585$	−1440.44	+	1540.80i	−0.101803	+	0.108896i
$586$		0			0
$587$			4350.60i			0.305909i	0.988233	+	0.152955i	$0.0488788\pi$
$587$			4350.60i			0.305909i	−0.988233	+	0.152955i	$0.951121\pi$
$588$		0			0
$589$	1315.26			0.0920111
$590$		0			0
$591$		−	6965.43i		−	0.484804i
$592$		0			0
$593$			12591.6i			0.871963i	0.899956	+	0.435982i	$0.143599\pi$
$593$			12591.6i			0.871963i	−0.899956	+	0.435982i	$0.856401\pi$
$594$		0			0
$595$	−3406.73			−0.234726
$596$		0			0
$597$	11974.5			0.820908
$598$		0			0
$599$	126.235			0.00861075			0.00430537	−	0.999991i	$-0.498630\pi$
$599$	126.235			0.00861075			0.00430537	+	0.999991i	$0.498630\pi$
$600$		0			0
$601$	8540.69			0.579670			0.289835	−	0.957077i	$-0.406400\pi$
$601$	8540.69			0.579670			0.289835	+	0.957077i	$0.406400\pi$
$602$		0			0
$603$		−	2978.90i		−	0.201178i
$604$		0			0
$605$			1634.87i			0.109863i
$606$		0			0
$607$	12911.1			0.863334			0.431667	−	0.902033i	$-0.357926\pi$
$607$	12911.1			0.863334			0.431667	+	0.902033i	$0.357926\pi$
$608$		0			0
$609$			663.499i			0.0441483i
$610$		0			0
$611$	−16082.5	+	17203.1i	−1.06486	+	1.13906i
$612$		0			0
$613$			11303.8i			0.744789i	0.928075	+	0.372394i	$0.121463\pi$
$613$			11303.8i			0.744789i	−0.928075	+	0.372394i	$0.878537\pi$
$614$		0			0
$615$	−7109.07			−0.466123
$616$		0			0
$617$			6233.55i			0.406732i	0.979103	+	0.203366i	$0.0651881\pi$
$617$			6233.55i			0.406732i	−0.979103	+	0.203366i	$0.934812\pi$
$618$		0			0
$619$		−	17638.0i		−	1.14529i	−0.819805	−	0.572643i	$-0.805918\pi$
$619$		−	17638.0i		−	1.14529i	0.819805	−	0.572643i	$-0.194082\pi$
$620$		0			0
$621$	−3556.76			−0.229836
$622$		0			0
$623$	−36832.2			−2.36862
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−1679.67			−0.106985
$628$		0			0
$629$		−	2765.92i		−	0.175333i
$630$		0			0
$631$		−	19695.0i		−	1.24254i	−0.783595	−	0.621272i	$-0.786616\pi$
$631$		−	19695.0i		−	1.24254i	0.783595	−	0.621272i	$-0.213384\pi$
$632$		0			0
$633$	1682.46			0.105643
$634$		0			0
$635$			10709.0i			0.669252i
$636$		0			0
$637$	9367.68	+	8757.47i	0.582670	+	0.544715i
$638$		0			0
$639$		−	2822.61i		−	0.174743i
$640$		0			0
$641$	−19227.4			−1.18477			−0.592386	−	0.805655i	$-0.701814\pi$
$641$	−19227.4			−1.18477			−0.592386	+	0.805655i	$0.701814\pi$
$642$		0			0
$643$		−	24392.3i		−	1.49602i	−0.663689	−	0.748008i	$-0.731010\pi$
$643$		−	24392.3i		−	1.49602i	0.663689	−	0.748008i	$-0.268990\pi$
$644$		0			0
$645$		−	229.720i		−	0.0140236i
$646$		0			0
$647$	−2828.17			−0.171850			−0.0859250	−	0.996302i	$-0.527385\pi$
$647$	−2828.17			−0.171850			−0.0859250	+	0.996302i	$0.527385\pi$
$648$		0			0
$649$	24927.7			1.50770
$650$		0			0
$651$	−7125.54			−0.428989
$652$		0			0
$653$	24527.3			1.46987			0.734936	−	0.678136i	$-0.237212\pi$
$653$	24527.3			1.46987			0.734936	+	0.678136i	$0.237212\pi$
$654$		0			0
$655$		−	1581.07i		−	0.0943171i
$656$		0			0
$657$		−	6039.42i		−	0.358630i
$658$		0			0
$659$	−13560.0			−0.801552			−0.400776	−	0.916176i	$-0.631260\pi$
$659$	−13560.0			−0.801552			−0.400776	+	0.916176i	$0.631260\pi$
$660$		0			0
$661$		−	27158.6i		−	1.59810i	−0.601262	−	0.799052i	$-0.705335\pi$
$661$		−	27158.6i		−	1.59810i	0.601262	−	0.799052i	$-0.294665\pi$
$662$		0			0
$663$	−2818.55	−	2634.95i	−0.165103	−	0.154348i
$664$		0			0
$665$			1707.19i			0.0995518i
$666$		0			0
$667$	−1173.31			−0.0681120
$668$		0			0
$669$		−	13266.8i		−	0.766704i
$670$		0			0
$671$		−	16184.9i		−	0.931163i
$672$		0			0
$673$	−589.635			−0.0337723			−0.0168862	−	0.999857i	$-0.505375\pi$
$673$	−589.635			−0.0337723			−0.0168862	+	0.999857i	$0.505375\pi$
$674$		0			0
$675$	675.000			0.0384900
$676$		0			0
$677$	−19189.9			−1.08940			−0.544702	−	0.838630i	$-0.683357\pi$
$677$	−19189.9			−1.08940			−0.544702	+	0.838630i	$0.683357\pi$
$678$		0			0
$679$	9669.65			0.546520
$680$		0			0
$681$			2865.71i			0.161255i
$682$		0			0
$683$		−	24872.6i		−	1.39344i	−0.717341	−	0.696722i	$-0.754641\pi$
$683$		−	24872.6i		−	1.39344i	0.717341	−	0.696722i	$-0.245359\pi$
$684$		0			0
$685$	−2988.90			−0.166715
$686$		0			0
$687$		−	9740.57i		−	0.540940i
$688$		0			0
$689$	−5495.45	−	5137.48i	−0.303861	−	0.284067i
$690$		0			0
$691$			31092.3i			1.71173i	0.517196	+	0.855867i	$0.326976\pi$
$691$			31092.3i			1.71173i	−0.517196	+	0.855867i	$0.673024\pi$
$692$		0			0
$693$	9099.74			0.498803
$694$		0			0
$695$			6954.84i			0.379586i
$696$		0			0
$697$		−	13004.4i		−	0.706711i
$698$		0			0
$699$	−4791.56			−0.259276
$700$		0			0
$701$	32400.0			1.74569			0.872847	−	0.487994i	$-0.162271\pi$
$701$	32400.0			1.74569			0.872847	+	0.487994i	$0.162271\pi$
$702$		0			0
$703$	−1386.07			−0.0743622
$704$		0			0
$705$	7536.40			0.402606
$706$		0			0
$707$		−	8946.96i		−	0.475934i
$708$		0			0
$709$			16999.8i			0.900481i	0.892907	+	0.450240i	$0.148662\pi$
$709$			16999.8i			0.900481i	−0.892907	+	0.450240i	$0.851338\pi$
$710$		0			0
$711$	−10698.5			−0.564310
$712$		0			0
$713$		−	12600.6i		−	0.661844i
$714$		0			0
$715$	6970.97	+	6516.89i	0.364615	+	0.340864i
$716$		0			0
$717$		−	9722.54i		−	0.506409i
$718$		0			0
$719$	35614.5			1.84728			0.923641	−	0.383258i	$-0.125198\pi$
$719$	35614.5			1.84728			0.923641	+	0.383258i	$0.125198\pi$
$720$		0			0
$721$		−	12736.2i		−	0.657864i
$722$		0			0
$723$		−	7739.07i		−	0.398090i
$724$		0			0
$725$	222.670			0.0114066
$726$		0			0
$727$	−29844.1			−1.52250			−0.761248	−	0.648461i	$-0.775413\pi$
$727$	−29844.1			−1.52250			−0.761248	+	0.648461i	$0.775413\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	420.220			0.0212618
$732$		0			0
$733$			7919.64i			0.399070i	0.979891	+	0.199535i	$0.0639432\pi$
$733$			7919.64i			0.399070i	−0.979891	+	0.199535i	$0.936057\pi$
$734$		0			0
$735$		−	4103.82i		−	0.205948i
$736$		0			0
$737$	−13477.3			−0.673599
$738$		0			0
$739$			18723.9i			0.932028i	0.884777	+	0.466014i	$0.154310\pi$
$739$			18723.9i			0.932028i	−0.884777	+	0.466014i	$0.845690\pi$
$740$		0			0
$741$	−1320.43	+	1412.44i	−0.0654620	+	0.0700233i
$742$		0			0
$743$			20668.4i			1.02053i	0.860018	+	0.510263i	$0.170452\pi$
$743$			20668.4i			1.02053i	−0.860018	+	0.510263i	$0.829548\pi$
$744$		0			0
$745$	12560.3			0.617684
$746$		0			0
$747$		−	3519.44i		−	0.172382i
$748$		0			0
$749$			19489.5i			0.950777i
$750$		0			0
$751$	−3659.15			−0.177795			−0.0888976	−	0.996041i	$-0.528334\pi$
$751$	−3659.15			−0.177795			−0.0888976	+	0.996041i	$0.528334\pi$
$752$		0			0
$753$	16384.8			0.792956
$754$		0			0
$755$	12384.4			0.596973
$756$		0			0
$757$	21530.8			1.03375			0.516876	−	0.856060i	$-0.327095\pi$
$757$	21530.8			1.03375			0.516876	+	0.856060i	$0.327095\pi$
$758$		0			0
$759$			16091.7i			0.769553i
$760$		0			0
$761$			4952.04i			0.235889i	0.993020	+	0.117944i	$0.0376304\pi$
$761$			4952.04i			0.235889i	−0.993020	+	0.117944i	$0.962370\pi$
$762$		0			0
$763$	−1454.76			−0.0690249
$764$		0			0
$765$			1234.76i			0.0583566i
$766$		0			0
$767$	19596.3	−	20961.7i	0.922532	−	0.986812i
$768$		0			0
$769$		−	5333.22i		−	0.250092i	−0.992151	−	0.125046i	$-0.960092\pi$
$769$		−	5333.22i		−	0.250092i	0.992151	−	0.125046i	$-0.0399079\pi$
$770$		0			0
$771$	23126.8			1.08027
$772$		0			0
$773$			19036.7i			0.885773i	0.896578	+	0.442887i	$0.146046\pi$
$773$			19036.7i			0.885773i	−0.896578	+	0.442887i	$0.853954\pi$
$774$		0			0
$775$			2391.33i			0.110837i
$776$		0			0
$777$	7509.13			0.346703
$778$		0			0
$779$	−6516.82			−0.299730
$780$		0			0
$781$	−12770.2			−0.585087
$782$		0			0
$783$	240.483			0.0109760
$784$		0			0
$785$		−	7336.86i		−	0.333585i
$786$		0			0
$787$		−	14829.2i		−	0.671669i	−0.941921	−	0.335835i	$-0.890982\pi$
$787$		−	14829.2i		−	0.671669i	0.941921	−	0.335835i	$-0.109018\pi$
$788$		0			0
$789$	17079.0			0.770630
$790$		0			0
$791$		−	43324.2i		−	1.94745i
$792$		0			0
$793$	−13609.9	−	12723.4i	−0.609460	−	0.569760i
$794$		0			0
$795$			2407.47i			0.107401i
$796$		0			0
$797$	−19003.4			−0.844585			−0.422293	−	0.906460i	$-0.638775\pi$
$797$	−19003.4			−0.844585			−0.422293	+	0.906460i	$0.638775\pi$
$798$		0			0
$799$			13786.1i			0.610411i
$800$		0			0
$801$			13349.7i			0.588876i
$802$		0			0
$803$	−27323.8			−1.20079
$804$		0			0
$805$	16355.3			0.716085
$806$		0			0
$807$	6866.64			0.299526
$808$		0			0
$809$	−8842.52			−0.384285			−0.192142	−	0.981367i	$-0.561544\pi$
$809$	−8842.52			−0.384285			−0.192142	+	0.981367i	$0.561544\pi$
$810$		0			0
$811$		−	30565.8i		−	1.32344i	−0.749752	−	0.661719i	$-0.769827\pi$
$811$		−	30565.8i		−	1.32344i	0.749752	−	0.661719i	$-0.230173\pi$
$812$		0			0
$813$		−	1405.24i		−	0.0606197i
$814$		0			0
$815$	6510.84			0.279834
$816$		0			0
$817$		−	210.582i		−	0.00901755i
$818$		0			0
$819$	7153.55	−	7651.99i	0.305208	−	0.326474i
$820$		0			0
$821$			19276.1i			0.819418i	0.912216	+	0.409709i	$0.134370\pi$
$821$			19276.1i			0.819418i	−0.912216	+	0.409709i	$0.865630\pi$
$822$		0			0
$823$	22270.7			0.943265			0.471633	−	0.881795i	$-0.343665\pi$
$823$	22270.7			0.943265			0.471633	+	0.881795i	$0.343665\pi$
$824$		0			0
$825$		−	3053.87i		−	0.128875i
$826$		0			0
$827$			3433.88i			0.144386i	0.997391	+	0.0721932i	$0.0229998\pi$
$827$			3433.88i			0.144386i	−0.997391	+	0.0721932i	$0.977000\pi$
$828$		0			0
$829$	−18614.6			−0.779868			−0.389934	−	0.920843i	$-0.627502\pi$
$829$	−18614.6			−0.779868			−0.389934	+	0.920843i	$0.627502\pi$
$830$		0			0
$831$	−5179.54			−0.216217
$832$		0			0
$833$	7507.01			0.312248
$834$		0			0
$835$	14808.1			0.613717
$836$		0			0
$837$			2582.63i			0.106653i
$838$		0			0
$839$			25457.8i			1.04756i	0.851854	+	0.523779i	$0.175478\pi$
$839$			25457.8i			1.04756i	−0.851854	+	0.523779i	$0.824522\pi$
$840$		0			0
$841$	−24309.7			−0.996747
$842$		0			0
$843$			11227.8i			0.458724i
$844$		0			0
$845$	10960.1	−	738.808i	0.446201	−	0.0300778i
$846$		0			0
$847$		−	8119.16i		−	0.329372i
$848$		0			0
$849$	−8442.53			−0.341280
$850$		0			0
$851$			13278.9i			0.534894i
$852$		0			0
$853$		−	16549.4i		−	0.664292i	−0.943228	−	0.332146i	$-0.892227\pi$
$853$		−	16549.4i		−	0.664292i	0.943228	−	0.332146i	$-0.107773\pi$
$854$		0			0
$855$	618.766			0.0247501
$856$		0			0
$857$	−15407.7			−0.614139			−0.307070	−	0.951687i	$-0.599348\pi$
$857$	−15407.7			−0.614139			−0.307070	+	0.951687i	$0.599348\pi$
$858$		0			0
$859$	−4020.45			−0.159692			−0.0798462	−	0.996807i	$-0.525443\pi$
$859$	−4020.45			−0.159692			−0.0798462	+	0.996807i	$0.525443\pi$
$860$		0			0
$861$	35305.3			1.39745
$862$		0			0
$863$		−	46445.2i		−	1.83200i	−0.401182	−	0.915999i	$-0.631400\pi$
$863$		−	46445.2i		−	1.83200i	0.401182	−	0.915999i	$-0.368600\pi$
$864$		0			0
$865$			12069.4i			0.474418i
$866$		0			0
$867$	12480.3			0.488873
$868$		0			0
$869$			48402.6i			1.88947i
$870$		0			0
$871$	−10594.9	+	11333.1i	−0.412162	+	0.440881i
$872$		0			0
$873$		−	3504.74i		−	0.135873i
$874$		0			0
$875$	−3103.90			−0.119921
$876$		0			0
$877$			3863.65i			0.148764i	0.997230	+	0.0743821i	$0.0236984\pi$
$877$			3863.65i			0.148764i	−0.997230	+	0.0743821i	$0.976302\pi$
$878$		0			0
$879$			18865.8i			0.723921i
$880$		0			0
$881$	−16256.8			−0.621686			−0.310843	−	0.950461i	$-0.600611\pi$
$881$	−16256.8			−0.621686			−0.310843	+	0.950461i	$0.600611\pi$
$882$		0			0
$883$	−12872.5			−0.490592			−0.245296	−	0.969448i	$-0.578885\pi$
$883$	−12872.5			−0.490592			−0.245296	+	0.969448i	$0.578885\pi$
$884$		0			0
$885$	−9182.99			−0.348794
$886$		0			0
$887$	46228.9			1.74996			0.874981	−	0.484158i	$-0.160874\pi$
$887$	46228.9			1.74996			0.874981	+	0.484158i	$0.160874\pi$
$888$		0			0
$889$		−	53183.6i		−	2.00643i
$890$		0			0
$891$		−	3298.18i		−	0.124010i
$892$		0			0
$893$	6908.55			0.258887
$894$		0			0
$895$		−	3752.24i		−	0.140138i
$896$		0			0
$897$	13531.5	+	12650.1i	0.503684	+	0.470874i
$898$		0			0
$899$			851.962i			0.0316068i
$900$		0			0
$901$	−4403.91			−0.162836
$902$		0			0
$903$			1140.84i			0.0420431i
$904$		0			0
$905$		−	3029.86i		−	0.111288i
$906$		0			0
$907$	20900.6			0.765153			0.382576	−	0.923924i	$-0.375037\pi$
$907$	20900.6			0.765153			0.382576	+	0.923924i	$0.375037\pi$
$908$		0			0
$909$	−3242.80			−0.118324
$910$		0			0
$911$	−38443.5			−1.39812			−0.699061	−	0.715062i	$-0.746399\pi$
$911$	−38443.5			−1.39812			−0.699061	+	0.715062i	$0.746399\pi$
$912$		0			0
$913$	−15922.8			−0.577184
$914$		0			0
$915$			5962.27i			0.215417i
$916$		0			0
$917$			7851.99i			0.282765i
$918$		0			0
$919$	47651.1			1.71041			0.855204	−	0.518291i	$-0.173432\pi$
$919$	47651.1			1.71041			0.855204	+	0.518291i	$0.173432\pi$
$920$		0			0
$921$			2020.02i			0.0722714i
$922$		0			0
$923$	−10039.0	+	10738.5i	−0.358003	+	0.382948i
$924$		0			0
$925$		−	2520.06i		−	0.0895774i
$926$		0			0
$927$	−4616.19			−0.163555
$928$		0			0
$929$			48545.6i			1.71446i	0.514937	+	0.857228i	$0.327815\pi$
$929$			48545.6i			1.71446i	−0.514937	+	0.857228i	$0.672185\pi$
$930$		0			0
$931$		−	3761.94i		−	0.132430i
$932$		0			0
$933$	28175.9			0.988678
$934$		0			0
$935$	5586.35			0.195394
$936$		0			0
$937$	25915.2			0.903534			0.451767	−	0.892136i	$-0.350794\pi$
$937$	25915.2			0.903534			0.451767	+	0.892136i	$0.350794\pi$
$938$		0			0
$939$	13335.9			0.463472
$940$		0			0
$941$			5145.78i			0.178265i	0.996020	+	0.0891326i	$0.0284095\pi$
$941$			5145.78i			0.178265i	−0.996020	+	0.0891326i	$0.971591\pi$
$942$		0			0
$943$			62432.8i			2.15598i
$944$		0			0
$945$	−3352.21			−0.115394
$946$		0			0
$947$		−	31949.6i		−	1.09633i	−0.836371	−	0.548163i	$-0.815327\pi$
$947$		−	31949.6i		−	1.09633i	0.836371	−	0.548163i	$-0.184673\pi$
$948$		0			0
$949$	−21480.0	+	22976.7i	−0.734741	+	0.785937i
$950$		0			0
$951$		−	25950.6i		−	0.884865i
$952$		0			0
$953$	24344.6			0.827489			0.413745	−	0.910393i	$-0.364221\pi$
$953$	24344.6			0.827489			0.413745	+	0.910393i	$0.364221\pi$
$954$		0			0
$955$			5651.63i			0.191500i
$956$		0			0
$957$		−	1088.01i		−	0.0367505i
$958$		0			0
$959$	14843.6			0.499817
$960$		0			0
$961$	20641.5			0.692877
$962$		0			0
$963$	7063.93			0.236378
$964$		0			0
$965$	1146.71			0.0382529
$966$		0			0
$967$		−	49601.9i		−	1.64953i	−0.565479	−	0.824763i	$-0.691309\pi$
$967$		−	49601.9i		−	1.64953i	0.565479	−	0.824763i	$-0.308691\pi$
$968$		0			0
$969$			1131.89i			0.0375249i
$970$		0			0
$971$	−28770.9			−0.950877			−0.475438	−	0.879749i	$-0.657711\pi$
$971$	−28770.9			−0.950877			−0.475438	+	0.879749i	$0.657711\pi$
$972$		0			0
$973$		−	34539.4i		−	1.13801i
$974$		0			0
$975$	−2568.00	−	2400.73i	−0.0843507	−	0.0788562i
$976$		0			0
$977$		−	8269.27i		−	0.270785i	−0.990792	−	0.135393i	$-0.956770\pi$
$977$		−	8269.27i		−	0.270785i	0.990792	−	0.135393i	$-0.0432296\pi$
$978$		0			0
$979$	60397.5			1.97172
$980$		0			0
$981$			527.275i			0.0171607i
$982$		0			0
$983$		−	17887.3i		−	0.580383i	−0.956969	−	0.290191i	$-0.906281\pi$
$983$		−	17887.3i		−	0.580383i	0.956969	−	0.290191i	$-0.0937190\pi$
$984$		0			0
$985$	11609.0			0.375528
$986$		0			0
$987$	−37427.6			−1.20702
$988$		0			0
$989$	−2017.43			−0.0648641
$990$		0			0
$991$	24849.0			0.796524			0.398262	−	0.917272i	$-0.369614\pi$
$991$	24849.0			0.796524			0.398262	+	0.917272i	$0.369614\pi$
$992$		0			0
$993$			25455.4i			0.813496i
$994$		0			0
$995$			19957.4i			0.635873i
$996$		0			0
$997$	−3630.69			−0.115331			−0.0576656	−	0.998336i	$-0.518366\pi$
$997$	−3630.69			−0.115331			−0.0576656	+	0.998336i	$0.518366\pi$
$998$		0			0
$999$		−	2721.66i		−	0.0861958i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.g.c.961.10	✓	20
13.12	even	2	inner	1560.4.g.c.961.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.g.c.961.10	✓	20	1.1	even	1	trivial
1560.4.g.c.961.11	yes	20	13.12	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000i q^{5} +24.8312i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000i q^{5} +24.8312i q^{7} +9.00000 q^{9} -40.7182i q^{11} +(-34.2400 - 32.0097i) q^{13} +15.0000i q^{15} -27.4391 q^{17} +13.7504i q^{19} -74.4936i q^{21} +131.732 q^{23} -25.0000 q^{25} -27.0000 q^{27} -8.90679 q^{29} -95.6531i q^{31} +122.155i q^{33} +124.156 q^{35} +100.802i q^{37} +(102.720 + 96.0290i) q^{39} +473.938i q^{41} -15.3147 q^{43} -45.0000i q^{45} -502.427i q^{47} -273.588 q^{49} +82.3172 q^{51} +160.498 q^{53} -203.591 q^{55} -41.2511i q^{57} +612.200i q^{59} +397.485 q^{61} +223.481i q^{63} +(-160.048 + 171.200i) q^{65} -330.989i q^{67} -395.196 q^{69} -313.623i q^{71} -671.046i q^{73} +75.0000 q^{75} +1011.08 q^{77} -1188.72 q^{79} +81.0000 q^{81} -391.049i q^{83} +137.195i q^{85} +26.7204 q^{87} +1483.30i q^{89} +(794.838 - 850.221i) q^{91} +286.959i q^{93} +68.7518 q^{95} -389.415i q^{97} -366.464i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{3} + 180 q^{9}+O(q^{10}) \) 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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