LMFDB - Embedded newform 96.3.b.a.79.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,3,Mod(79,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("96.79");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	96.b (of order $2$, degree $1$, not 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:	$2.61581053786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} - 6x + 6 $ x^4 + 3x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			79.1
Root			$0.866025 + 0.719687i$ of defining polynomial
Character	$\chi$	$=$	96.79
Dual form			96.3.b.a.79.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -2.87875i q^{5} -10.7436i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} -2.87875i q^{5} -10.7436i q^{7} +3.00000 q^{9} +8.00000 q^{11} -15.7298i q^{13} +4.98614i q^{15} -15.8564 q^{17} -1.07180 q^{19} +18.6085i q^{21} +21.4873i q^{23} +16.7128 q^{25} -5.19615 q^{27} +40.0958i q^{29} -9.20092i q^{31} -13.8564 q^{33} -30.9282 q^{35} -9.97227i q^{37} +27.2448i q^{39} +51.5692 q^{41} +12.7846 q^{43} -8.63624i q^{45} -1.54272i q^{47} -66.4256 q^{49} +27.4641 q^{51} -28.5808i q^{53} -23.0300i q^{55} +1.85641 q^{57} +11.2154 q^{59} -1.54272i q^{61} -32.2309i q^{63} -45.2820 q^{65} +43.2154 q^{67} -37.2170i q^{69} -84.4063i q^{71} +105.426 q^{73} -28.9474 q^{75} -85.9491i q^{77} +73.6627i q^{79} +9.00000 q^{81} -12.2872 q^{83} +45.6466i q^{85} -69.4479i q^{87} +33.1384 q^{89} -168.995 q^{91} +15.9365i q^{93} +3.08543i q^{95} -69.1384 q^{97} +24.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 32 q^{11} - 8 q^{17} - 32 q^{19} - 44 q^{25} - 96 q^{35} + 40 q^{41} - 32 q^{43} - 44 q^{49} + 96 q^{51} - 48 q^{57} + 128 q^{59} + 96 q^{65} + 256 q^{67} + 200 q^{73} - 192 q^{75} + 36 q^{81} - 160 q^{83} - 200 q^{89} - 288 q^{91} + 56 q^{97} + 96 q^{99}+O(q^{100}) $ 4 * q + 12 * q^9 + 32 * q^11 - 8 * q^17 - 32 * q^19 - 44 * q^25 - 96 * q^35 + 40 * q^41 - 32 * q^43 - 44 * q^49 + 96 * q^51 - 48 * q^57 + 128 * q^59 + 96 * q^65 + 256 * q^67 + 200 * q^73 - 192 * q^75 + 36 * q^81 - 160 * q^83 - 200 * q^89 - 288 * q^91 + 56 * q^97 + 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	− 2.87875i	− 0.575749i	−0.957668	−	0.287875i	$-0.907051\pi$
$5$	− 2.87875i	− 0.575749i	0.957668	−	0.287875i	$-0.0929487\pi$
$6$	0	0
$7$	− 10.7436i	− 1.53480i	−0.641166	−	0.767402i	$-0.721549\pi$
$7$	− 10.7436i	− 1.53480i	0.641166	−	0.767402i	$-0.278451\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	8.00000	0.727273	0.363636	−	0.931541i	$-0.381535\pi$
$11$	8.00000	0.727273	0.363636	+	0.931541i	$0.381535\pi$
$12$	0	0
$13$	− 15.7298i	− 1.20998i	−0.796232	−	0.604991i	$-0.793177\pi$
$13$	− 15.7298i	− 1.20998i	0.796232	−	0.604991i	$-0.206823\pi$
$14$	0	0
$15$	4.98614i	0.332409i
$16$	0	0
$17$	−15.8564	−0.932730	−0.466365	−	0.884592i	$-0.654437\pi$
$17$	−15.8564	−0.932730	−0.466365	+	0.884592i	$0.654437\pi$
$18$	0	0
$19$	−1.07180	−0.0564104	−0.0282052	−	0.999602i	$-0.508979\pi$
$19$	−1.07180	−0.0564104	−0.0282052	+	0.999602i	$0.508979\pi$
$20$	0	0
$21$	18.6085i	0.886120i
$22$	0	0
$23$	21.4873i	0.934229i	0.884197	+	0.467114i	$0.154706\pi$
$23$	21.4873i	0.934229i	−0.884197	+	0.467114i	$0.845294\pi$
$24$	0	0
$25$	16.7128	0.668513
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	40.0958i	1.38261i	0.722562	+	0.691307i	$0.242965\pi$
$29$	40.0958i	1.38261i	−0.722562	+	0.691307i	$0.757035\pi$
$30$	0	0
$31$	− 9.20092i	− 0.296804i	−0.988927	−	0.148402i	$-0.952587\pi$
$31$	− 9.20092i	− 0.296804i	0.988927	−	0.148402i	$-0.0474129\pi$
$32$	0	0
$33$	−13.8564	−0.419891
$34$	0	0
$35$	−30.9282	−0.883663
$36$	0	0
$37$	− 9.97227i	− 0.269521i	−0.990878	−	0.134760i	$-0.956974\pi$
$37$	− 9.97227i	− 0.269521i	0.990878	−	0.134760i	$-0.0430265\pi$
$38$	0	0
$39$	27.2448i	0.698584i
$40$	0	0
$41$	51.5692	1.25779	0.628893	−	0.777492i	$-0.283508\pi$
$41$	51.5692	1.25779	0.628893	+	0.777492i	$0.283508\pi$
$42$	0	0
$43$	12.7846	0.297317	0.148658	−	0.988889i	$-0.452505\pi$
$43$	12.7846	0.297317	0.148658	+	0.988889i	$0.452505\pi$
$44$	0	0
$45$	− 8.63624i	− 0.191916i
$46$	0	0
$47$	− 1.54272i	− 0.0328237i	−0.999865	−	0.0164119i	$-0.994776\pi$
$47$	− 1.54272i	− 0.0328237i	0.999865	−	0.0164119i	$-0.00522430\pi$
$48$	0	0
$49$	−66.4256	−1.35563
$50$	0	0
$51$	27.4641	0.538512
$52$	0	0
$53$	− 28.5808i	− 0.539260i	−0.962964	−	0.269630i	$-0.913099\pi$
$53$	− 28.5808i	− 0.539260i	0.962964	−	0.269630i	$-0.0869014\pi$
$54$	0	0
$55$	− 23.0300i	− 0.418727i
$56$	0	0
$57$	1.85641	0.0325685
$58$	0	0
$59$	11.2154	0.190091	0.0950457	−	0.995473i	$-0.469700\pi$
$59$	11.2154	0.190091	0.0950457	+	0.995473i	$0.469700\pi$
$60$	0	0
$61$	− 1.54272i	− 0.0252904i	−0.999920	−	0.0126452i	$-0.995975\pi$
$61$	− 1.54272i	− 0.0252904i	0.999920	−	0.0126452i	$-0.00402520\pi$
$62$	0	0
$63$	− 32.2309i	− 0.511602i
$64$	0	0
$65$	−45.2820	−0.696647
$66$	0	0
$67$	43.2154	0.645006	0.322503	−	0.946568i	$-0.395476\pi$
$67$	43.2154	0.645006	0.322503	+	0.946568i	$0.395476\pi$
$68$	0	0
$69$	− 37.2170i	− 0.539377i
$70$	0	0
$71$	− 84.4063i	− 1.18882i	−0.804162	−	0.594411i	$-0.797385\pi$
$71$	− 84.4063i	− 1.18882i	0.804162	−	0.594411i	$-0.202615\pi$
$72$	0	0
$73$	105.426	1.44419	0.722093	−	0.691796i	$-0.243180\pi$
$73$	105.426	1.44419	0.722093	+	0.691796i	$0.243180\pi$
$74$	0	0
$75$	−28.9474	−0.385966
$76$	0	0
$77$	− 85.9491i	− 1.11622i
$78$	0	0
$79$	73.6627i	0.932439i	0.884669	+	0.466220i	$0.154384\pi$
$79$	73.6627i	0.932439i	−0.884669	+	0.466220i	$0.845616\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−12.2872	−0.148038	−0.0740192	−	0.997257i	$-0.523583\pi$
$83$	−12.2872	−0.148038	−0.0740192	+	0.997257i	$0.523583\pi$
$84$	0	0
$85$	45.6466i	0.537019i
$86$	0	0
$87$	− 69.4479i	− 0.798252i
$88$	0	0
$89$	33.1384	0.372342	0.186171	−	0.982517i	$-0.440392\pi$
$89$	33.1384	0.372342	0.186171	+	0.982517i	$0.440392\pi$
$90$	0	0
$91$	−168.995	−1.85709
$92$	0	0
$93$	15.9365i	0.171360i
$94$	0	0
$95$	3.08543i	0.0324782i
$96$	0	0
$97$	−69.1384	−0.712767	−0.356384	−	0.934340i	$-0.615990\pi$
$97$	−69.1384	−0.712767	−0.356384	+	0.934340i	$0.615990\pi$
$98$	0	0
$99$	24.0000	0.242424
$100$	0	0
$101$	97.2574i	0.962944i	0.876461	+	0.481472i	$0.159898\pi$
$101$	97.2574i	0.962944i	−0.876461	+	0.481472i	$0.840102\pi$
$102$	0	0
$103$	139.667i	1.35599i	0.735065	+	0.677996i	$0.237151\pi$
$103$	139.667i	1.35599i	−0.735065	+	0.677996i	$0.762849\pi$
$104$	0	0
$105$	53.5692	0.510183
$106$	0	0
$107$	197.779	1.84841	0.924203	−	0.381901i	$-0.124731\pi$
$107$	197.779	1.84841	0.924203	+	0.381901i	$0.124731\pi$
$108$	0	0
$109$	190.713i	1.74966i	0.484427	+	0.874832i	$0.339028\pi$
$109$	190.713i	1.74966i	−0.484427	+	0.874832i	$0.660972\pi$
$110$	0	0
$111$	17.2725i	0.155608i
$112$	0	0
$113$	−113.713	−1.00631	−0.503154	−	0.864197i	$-0.667827\pi$
$113$	−113.713	−1.00631	−0.503154	+	0.864197i	$0.667827\pi$
$114$	0	0
$115$	61.8564	0.537882
$116$	0	0
$117$	− 47.1893i	− 0.403327i
$118$	0	0
$119$	170.355i	1.43156i
$120$	0	0
$121$	−57.0000	−0.471074
$122$	0	0
$123$	−89.3205	−0.726183
$124$	0	0
$125$	− 120.081i	− 0.960645i
$126$	0	0
$127$	− 36.8590i	− 0.290229i	−0.989415	−	0.145114i	$-0.953645\pi$
$127$	− 36.8590i	− 0.290229i	0.989415	−	0.145114i	$-0.0463550\pi$
$128$	0	0
$129$	−22.1436	−0.171656
$130$	0	0
$131$	−194.641	−1.48581	−0.742905	−	0.669397i	$-0.766552\pi$
$131$	−194.641	−1.48581	−0.742905	+	0.669397i	$0.766552\pi$
$132$	0	0
$133$	11.5150i	0.0865789i
$134$	0	0
$135$	14.9584i	0.110803i
$136$	0	0
$137$	16.4308	0.119933	0.0599664	−	0.998200i	$-0.480901\pi$
$137$	16.4308	0.119933	0.0599664	+	0.998200i	$0.480901\pi$
$138$	0	0
$139$	113.492	0.816491	0.408246	−	0.912872i	$-0.366141\pi$
$139$	113.492	0.816491	0.408246	+	0.912872i	$0.366141\pi$
$140$	0	0
$141$	2.67206i	0.0189508i
$142$	0	0
$143$	− 125.838i	− 0.879987i
$144$	0	0
$145$	115.426	0.796039
$146$	0	0
$147$	115.053	0.782670
$148$	0	0
$149$	31.6662i	0.212525i	0.994338	+	0.106262i	$0.0338884\pi$
$149$	31.6662i	0.212525i	−0.994338	+	0.106262i	$0.966112\pi$
$150$	0	0
$151$	− 90.5218i	− 0.599482i	−0.954021	−	0.299741i	$-0.903100\pi$
$151$	− 90.5218i	− 0.599482i	0.954021	−	0.299741i	$-0.0969003\pi$
$152$	0	0
$153$	−47.5692	−0.310910
$154$	0	0
$155$	−26.4871	−0.170885
$156$	0	0
$157$	− 231.016i	− 1.47144i	−0.677287	−	0.735719i	$-0.736844\pi$
$157$	− 231.016i	− 1.47144i	0.677287	−	0.735719i	$-0.263156\pi$
$158$	0	0
$159$	49.5034i	0.311342i
$160$	0	0
$161$	230.851	1.43386
$162$	0	0
$163$	−22.3538	−0.137140	−0.0685700	−	0.997646i	$-0.521844\pi$
$163$	−22.3538	−0.137140	−0.0685700	+	0.997646i	$0.521844\pi$
$164$	0	0
$165$	39.8891i	0.241752i
$166$	0	0
$167$	− 221.044i	− 1.32361i	−0.749674	−	0.661807i	$-0.769790\pi$
$167$	− 221.044i	− 1.32361i	0.749674	−	0.661807i	$-0.230210\pi$
$168$	0	0
$169$	−78.4256	−0.464057
$170$	0	0
$171$	−3.21539	−0.0188035
$172$	0	0
$173$	231.525i	1.33830i	0.743130	+	0.669148i	$0.233341\pi$
$173$	231.525i	1.33830i	−0.743130	+	0.669148i	$0.766659\pi$
$174$	0	0
$175$	− 179.556i	− 1.02604i
$176$	0	0
$177$	−19.4256	−0.109749
$178$	0	0
$179$	−193.646	−1.08182	−0.540911	−	0.841080i	$-0.681920\pi$
$179$	−193.646	−1.08182	−0.540911	+	0.841080i	$0.681920\pi$
$180$	0	0
$181$	− 270.492i	− 1.49443i	−0.664583	−	0.747214i	$-0.731391\pi$
$181$	− 270.492i	− 1.49443i	0.664583	−	0.747214i	$-0.268609\pi$
$182$	0	0
$183$	2.67206i	0.0146014i
$184$	0	0
$185$	−28.7077	−0.155177
$186$	0	0
$187$	−126.851	−0.678349
$188$	0	0
$189$	55.8255i	0.295373i
$190$	0	0
$191$	311.510i	1.63094i	0.578798	+	0.815471i	$0.303522\pi$
$191$	311.510i	1.63094i	−0.578798	+	0.815471i	$0.696478\pi$
$192$	0	0
$193$	48.2769	0.250139	0.125070	−	0.992148i	$-0.460085\pi$
$193$	48.2769	0.250139	0.125070	+	0.992148i	$0.460085\pi$
$194$	0	0
$195$	78.4308	0.402209
$196$	0	0
$197$	− 251.883i	− 1.27859i	−0.768960	−	0.639297i	$-0.779225\pi$
$197$	− 251.883i	− 1.27859i	0.768960	−	0.639297i	$-0.220775\pi$
$198$	0	0
$199$	− 72.1200i	− 0.362412i	−0.983445	−	0.181206i	$-0.942000\pi$
$199$	− 72.1200i	− 0.362412i	0.983445	−	0.181206i	$-0.0580001\pi$
$200$	0	0
$201$	−74.8513	−0.372394
$202$	0	0
$203$	430.774	2.12204
$204$	0	0
$205$	− 148.455i	− 0.724170i
$206$	0	0
$207$	64.4618i	0.311410i
$208$	0	0
$209$	−8.57437	−0.0410257
$210$	0	0
$211$	264.918	1.25554	0.627768	−	0.778401i	$-0.283969\pi$
$211$	264.918	1.25554	0.627768	+	0.778401i	$0.283969\pi$
$212$	0	0
$213$	146.196i	0.686367i
$214$	0	0
$215$	− 36.8037i	− 0.171180i
$216$	0	0
$217$	−98.8513	−0.455536
$218$	0	0
$219$	−182.603	−0.833802
$220$	0	0
$221$	249.418i	1.12859i
$222$	0	0
$223$	− 30.6882i	− 0.137615i	−0.997630	−	0.0688076i	$-0.978081\pi$
$223$	− 30.6882i	− 0.137615i	0.997630	−	0.0688076i	$-0.0219195\pi$
$224$	0	0
$225$	50.1384	0.222838
$226$	0	0
$227$	−295.846	−1.30329	−0.651643	−	0.758525i	$-0.725920\pi$
$227$	−295.846	−1.30329	−0.651643	+	0.758525i	$0.725920\pi$
$228$	0	0
$229$	256.718i	1.12104i	0.828141	+	0.560519i	$0.189398\pi$
$229$	256.718i	1.12104i	−0.828141	+	0.560519i	$0.810602\pi$
$230$	0	0
$231$	148.868i	0.644451i
$232$	0	0
$233$	−404.564	−1.73633	−0.868163	−	0.496279i	$-0.834699\pi$
$233$	−404.564	−1.73633	−0.868163	+	0.496279i	$0.834699\pi$
$234$	0	0
$235$	−4.44109	−0.0188983
$236$	0	0
$237$	− 127.588i	− 0.538344i
$238$	0	0
$239$	115.150i	0.481799i	0.970550	+	0.240899i	$0.0774424\pi$
$239$	115.150i	0.481799i	−0.970550	+	0.240899i	$0.922558\pi$
$240$	0	0
$241$	251.415	1.04322	0.521609	−	0.853185i	$-0.325332\pi$
$241$	251.415	1.04322	0.521609	+	0.853185i	$0.325332\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	191.223i	0.780500i
$246$	0	0
$247$	16.8591i	0.0682555i
$248$	0	0
$249$	21.2820	0.0854700
$250$	0	0
$251$	−6.85125	−0.0272958	−0.0136479	−	0.999907i	$-0.504344\pi$
$251$	−6.85125	−0.0272958	−0.0136479	+	0.999907i	$0.504344\pi$
$252$	0	0
$253$	171.898i	0.679439i
$254$	0	0
$255$	− 79.0622i	− 0.310048i
$256$	0	0
$257$	248.277	0.966058	0.483029	−	0.875604i	$-0.339537\pi$
$257$	248.277	0.966058	0.483029	+	0.875604i	$0.339537\pi$
$258$	0	0
$259$	−107.138	−0.413662
$260$	0	0
$261$	120.287i	0.460871i
$262$	0	0
$263$	498.835i	1.89671i	0.317208	+	0.948356i	$0.397255\pi$
$263$	498.835i	1.89671i	−0.317208	+	0.948356i	$0.602745\pi$
$264$	0	0
$265$	−82.2769	−0.310479
$266$	0	0
$267$	−57.3975	−0.214972
$268$	0	0
$269$	− 234.611i	− 0.872158i	−0.899908	−	0.436079i	$-0.856367\pi$
$269$	− 234.611i	− 0.872158i	0.899908	−	0.436079i	$-0.143633\pi$
$270$	0	0
$271$	101.321i	0.373878i	0.982372	+	0.186939i	$0.0598566\pi$
$271$	101.321i	0.373878i	−0.982372	+	0.186939i	$0.940143\pi$
$272$	0	0
$273$	292.708	1.07219
$274$	0	0
$275$	133.703	0.486191
$276$	0	0
$277$	169.942i	0.613509i	0.951789	+	0.306755i	$0.0992431\pi$
$277$	169.942i	0.613509i	−0.951789	+	0.306755i	$0.900757\pi$
$278$	0	0
$279$	− 27.6027i	− 0.0989346i
$280$	0	0
$281$	−111.128	−0.395474	−0.197737	−	0.980255i	$-0.563359\pi$
$281$	−111.128	−0.395474	−0.197737	+	0.980255i	$0.563359\pi$
$282$	0	0
$283$	−550.620	−1.94566	−0.972828	−	0.231531i	$-0.925627\pi$
$283$	−550.620	−1.94566	−0.972828	+	0.231531i	$0.925627\pi$
$284$	0	0
$285$	− 5.34413i	− 0.0187513i
$286$	0	0
$287$	− 554.041i	− 1.93046i
$288$	0	0
$289$	−37.5744	−0.130015
$290$	0	0
$291$	119.751	0.411516
$292$	0	0
$293$	− 223.509i	− 0.762829i	−0.924404	−	0.381414i	$-0.875437\pi$
$293$	− 223.509i	− 0.762829i	0.924404	−	0.381414i	$-0.124563\pi$
$294$	0	0
$295$	− 32.2863i	− 0.109445i
$296$	0	0
$297$	−41.5692	−0.139964
$298$	0	0
$299$	337.990	1.13040
$300$	0	0
$301$	− 137.353i	− 0.456323i
$302$	0	0
$303$	− 168.455i	− 0.555956i
$304$	0	0
$305$	−4.44109	−0.0145610
$306$	0	0
$307$	371.790	1.21104	0.605521	−	0.795829i	$-0.292965\pi$
$307$	371.790	1.21104	0.605521	+	0.795829i	$0.292965\pi$
$308$	0	0
$309$	− 241.911i	− 0.782883i
$310$	0	0
$311$	− 330.023i	− 1.06117i	−0.847633	−	0.530583i	$-0.821973\pi$
$311$	− 330.023i	− 1.06117i	0.847633	−	0.530583i	$-0.178027\pi$
$312$	0	0
$313$	−80.2769	−0.256476	−0.128238	−	0.991743i	$-0.540932\pi$
$313$	−80.2769	−0.256476	−0.128238	+	0.991743i	$0.540932\pi$
$314$	0	0
$315$	−92.7846	−0.294554
$316$	0	0
$317$	− 68.8833i	− 0.217297i	−0.994080	−	0.108649i	$-0.965348\pi$
$317$	− 68.8833i	− 0.217297i	0.994080	−	0.108649i	$-0.0346524\pi$
$318$	0	0
$319$	320.766i	1.00554i
$320$	0	0
$321$	−342.564	−1.06718
$322$	0	0
$323$	16.9948	0.0526156
$324$	0	0
$325$	− 262.889i	− 0.808888i
$326$	0	0
$327$	− 330.325i	− 1.01017i
$328$	0	0
$329$	−16.5744	−0.0503780
$330$	0	0
$331$	396.056	1.19654	0.598272	−	0.801293i	$-0.295854\pi$
$331$	396.056	1.19654	0.598272	+	0.801293i	$0.295854\pi$
$332$	0	0
$333$	− 29.9168i	− 0.0898403i
$334$	0	0
$335$	− 124.406i	− 0.371362i
$336$	0	0
$337$	231.723	0.687606	0.343803	−	0.939042i	$-0.388285\pi$
$337$	231.723	0.687606	0.343803	+	0.939042i	$0.388285\pi$
$338$	0	0
$339$	196.956	0.580992
$340$	0	0
$341$	− 73.6073i	− 0.215857i
$342$	0	0
$343$	187.215i	0.545815i
$344$	0	0
$345$	−107.138	−0.310546
$346$	0	0
$347$	−462.123	−1.33177	−0.665883	−	0.746056i	$-0.731945\pi$
$347$	−462.123	−1.33177	−0.665883	+	0.746056i	$0.731945\pi$
$348$	0	0
$349$	266.993i	0.765022i	0.923951	+	0.382511i	$0.124941\pi$
$349$	266.993i	0.765022i	−0.923951	+	0.382511i	$0.875059\pi$
$350$	0	0
$351$	81.7343i	0.232861i
$352$	0	0
$353$	262.862	0.744650	0.372325	−	0.928102i	$-0.378561\pi$
$353$	262.862	0.744650	0.372325	+	0.928102i	$0.378561\pi$
$354$	0	0
$355$	−242.985	−0.684463
$356$	0	0
$357$	− 295.064i	− 0.826510i
$358$	0	0
$359$	− 164.185i	− 0.457339i	−0.973504	−	0.228669i	$-0.926563\pi$
$359$	− 164.185i	− 0.457339i	0.973504	−	0.228669i	$-0.0734374\pi$
$360$	0	0
$361$	−359.851	−0.996818
$362$	0	0
$363$	98.7269	0.271975
$364$	0	0
$365$	− 303.494i	− 0.831490i
$366$	0	0
$367$	242.475i	0.660696i	0.943859	+	0.330348i	$0.107166\pi$
$367$	242.475i	0.660696i	−0.943859	+	0.330348i	$0.892834\pi$
$368$	0	0
$369$	154.708	0.419262
$370$	0	0
$371$	−307.061	−0.827659
$372$	0	0
$373$	328.480i	0.880643i	0.897840	+	0.440321i	$0.145136\pi$
$373$	328.480i	0.880643i	−0.897840	+	0.440321i	$0.854864\pi$
$374$	0	0
$375$	207.986i	0.554629i
$376$	0	0
$377$	630.697	1.67294
$378$	0	0
$379$	−36.2102	−0.0955415	−0.0477708	−	0.998858i	$-0.515212\pi$
$379$	−36.2102	−0.0955415	−0.0477708	+	0.998858i	$0.515212\pi$
$380$	0	0
$381$	63.8417i	0.167564i
$382$	0	0
$383$	164.295i	0.428969i	0.976727	+	0.214485i	$0.0688072\pi$
$383$	164.295i	0.428969i	−0.976727	+	0.214485i	$0.931193\pi$
$384$	0	0
$385$	−247.426	−0.642664
$386$	0	0
$387$	38.3538	0.0991055
$388$	0	0
$389$	604.936i	1.55510i	0.628819	+	0.777552i	$0.283539\pi$
$389$	604.936i	1.55510i	−0.628819	+	0.777552i	$0.716461\pi$
$390$	0	0
$391$	− 340.711i	− 0.871383i
$392$	0	0
$393$	337.128	0.857832
$394$	0	0
$395$	212.056	0.536851
$396$	0	0
$397$	541.699i	1.36448i	0.731128	+	0.682241i	$0.238994\pi$
$397$	541.699i	1.36448i	−0.731128	+	0.682241i	$0.761006\pi$
$398$	0	0
$399$	− 19.9445i	− 0.0499863i
$400$	0	0
$401$	−379.569	−0.946557	−0.473278	−	0.880913i	$-0.656930\pi$
$401$	−379.569	−0.946557	−0.473278	+	0.880913i	$0.656930\pi$
$402$	0	0
$403$	−144.728	−0.359127
$404$	0	0
$405$	− 25.9087i	− 0.0639722i
$406$	0	0
$407$	− 79.7782i	− 0.196015i
$408$	0	0
$409$	251.415	0.614707	0.307354	−	0.951595i	$-0.400557\pi$
$409$	251.415	0.614707	0.307354	+	0.951595i	$0.400557\pi$
$410$	0	0
$411$	−28.4589	−0.0692432
$412$	0	0
$413$	− 120.494i	− 0.291753i
$414$	0	0
$415$	35.3717i	0.0852330i
$416$	0	0
$417$	−196.574	−0.471401
$418$	0	0
$419$	−268.133	−0.639936	−0.319968	−	0.947428i	$-0.603672\pi$
$419$	−268.133	−0.639936	−0.319968	+	0.947428i	$0.603672\pi$
$420$	0	0
$421$	− 218.261i	− 0.518434i	−0.965819	−	0.259217i	$-0.916536\pi$
$421$	− 218.261i	− 0.518434i	0.965819	−	0.259217i	$-0.0834645\pi$
$422$	0	0
$423$	− 4.62815i	− 0.0109412i
$424$	0	0
$425$	−265.005	−0.623542
$426$	0	0
$427$	−16.5744	−0.0388159
$428$	0	0
$429$	217.958i	0.508061i
$430$	0	0
$431$	550.955i	1.27832i	0.769074	+	0.639159i	$0.220718\pi$
$431$	550.955i	1.27832i	−0.769074	+	0.639159i	$0.779282\pi$
$432$	0	0
$433$	−263.128	−0.607686	−0.303843	−	0.952722i	$-0.598270\pi$
$433$	−263.128	−0.607686	−0.303843	+	0.952722i	$0.598270\pi$
$434$	0	0
$435$	−199.923	−0.459593
$436$	0	0
$437$	− 23.0300i	− 0.0527002i
$438$	0	0
$439$	440.489i	1.00339i	0.865044	+	0.501696i	$0.167290\pi$
$439$	440.489i	1.00339i	−0.865044	+	0.501696i	$0.832710\pi$
$440$	0	0
$441$	−199.277	−0.451875
$442$	0	0
$443$	228.708	0.516270	0.258135	−	0.966109i	$-0.416892\pi$
$443$	228.708	0.516270	0.258135	+	0.966109i	$0.416892\pi$
$444$	0	0
$445$	− 95.3972i	− 0.214376i
$446$	0	0
$447$	− 54.8475i	− 0.122701i
$448$	0	0
$449$	−108.410	−0.241448	−0.120724	−	0.992686i	$-0.538522\pi$
$449$	−108.410	−0.241448	−0.120724	+	0.992686i	$0.538522\pi$
$450$	0	0
$451$	412.554	0.914753
$452$	0	0
$453$	156.788i	0.346111i
$454$	0	0
$455$	486.493i	1.06922i
$456$	0	0
$457$	561.692	1.22909	0.614543	−	0.788883i	$-0.289340\pi$
$457$	561.692	1.22909	0.614543	+	0.788883i	$0.289340\pi$
$458$	0	0
$459$	82.3923	0.179504
$460$	0	0
$461$	335.160i	0.727028i	0.931589	+	0.363514i	$0.118423\pi$
$461$	335.160i	0.727028i	−0.931589	+	0.363514i	$0.881577\pi$
$462$	0	0
$463$	− 389.912i	− 0.842142i	−0.907028	−	0.421071i	$-0.861654\pi$
$463$	− 389.912i	− 0.842142i	0.907028	−	0.421071i	$-0.138346\pi$
$464$	0	0
$465$	45.8770	0.0986603
$466$	0	0
$467$	546.410	1.17004	0.585022	−	0.811018i	$-0.301086\pi$
$467$	546.410	1.17004	0.585022	+	0.811018i	$0.301086\pi$
$468$	0	0
$469$	− 464.290i	− 0.989958i
$470$	0	0
$471$	400.131i	0.849535i
$472$	0	0
$473$	102.277	0.216230
$474$	0	0
$475$	−17.9127	−0.0377110
$476$	0	0
$477$	− 85.7424i	− 0.179753i
$478$	0	0
$479$	− 368.369i	− 0.769037i	−0.923117	−	0.384519i	$-0.874367\pi$
$479$	− 368.369i	− 0.769037i	0.923117	−	0.384519i	$-0.125633\pi$
$480$	0	0
$481$	−156.862	−0.326116
$482$	0	0
$483$	−399.846	−0.827839
$484$	0	0
$485$	199.032i	0.410375i
$486$	0	0
$487$	90.6326i	0.186104i	0.995661	+	0.0930519i	$0.0296623\pi$
$487$	90.6326i	0.186104i	−0.995661	+	0.0930519i	$0.970338\pi$
$488$	0	0
$489$	38.7180	0.0791778
$490$	0	0
$491$	17.6462	0.0359392	0.0179696	−	0.999839i	$-0.494280\pi$
$491$	17.6462	0.0359392	0.0179696	+	0.999839i	$0.494280\pi$
$492$	0	0
$493$	− 635.775i	− 1.28960i
$494$	0	0
$495$	− 69.0899i	− 0.139576i
$496$	0	0
$497$	−906.831	−1.82461
$498$	0	0
$499$	−548.631	−1.09946	−0.549730	−	0.835342i	$-0.685269\pi$
$499$	−548.631	−1.09946	−0.549730	+	0.835342i	$0.685269\pi$
$500$	0	0
$501$	382.859i	0.764189i
$502$	0	0
$503$	− 262.475i	− 0.521820i	−0.965363	−	0.260910i	$-0.915977\pi$
$503$	− 262.475i	− 0.521820i	0.965363	−	0.260910i	$-0.0840225\pi$
$504$	0	0
$505$	279.979	0.554415
$506$	0	0
$507$	135.837	0.267923
$508$	0	0
$509$	− 230.093i	− 0.452049i	−0.974122	−	0.226025i	$-0.927427\pi$
$509$	− 230.093i	− 0.452049i	0.974122	−	0.226025i	$-0.0725730\pi$
$510$	0	0
$511$	− 1132.65i	− 2.21654i
$512$	0	0
$513$	5.56922	0.0108562
$514$	0	0
$515$	402.067	0.780712
$516$	0	0
$517$	− 12.3417i	− 0.0238718i
$518$	0	0
$519$	− 401.013i	− 0.772665i
$520$	0	0
$521$	164.144	0.315055	0.157527	−	0.987515i	$-0.449648\pi$
$521$	164.144	0.315055	0.157527	+	0.987515i	$0.449648\pi$
$522$	0	0
$523$	185.492	0.354670	0.177335	−	0.984151i	$-0.443252\pi$
$523$	185.492	0.354670	0.177335	+	0.984151i	$0.443252\pi$
$524$	0	0
$525$	311.001i	0.592382i
$526$	0	0
$527$	145.893i	0.276838i
$528$	0	0
$529$	67.2975	0.127216
$530$	0	0
$531$	33.6462	0.0633638
$532$	0	0
$533$	− 811.172i	− 1.52190i
$534$	0	0
$535$	− 569.357i	− 1.06422i
$536$	0	0
$537$	335.405	0.624590
$538$	0	0
$539$	−531.405	−0.985909
$540$	0	0
$541$	− 891.253i	− 1.64742i	−0.567013	−	0.823709i	$-0.691901\pi$
$541$	− 891.253i	− 1.64742i	0.567013	−	0.823709i	$-0.308099\pi$
$542$	0	0
$543$	468.505i	0.862809i
$544$	0	0
$545$	549.015	1.00737
$546$	0	0
$547$	−524.631	−0.959105	−0.479553	−	0.877513i	$-0.659201\pi$
$547$	−524.631	−0.959105	−0.479553	+	0.877513i	$0.659201\pi$
$548$	0	0
$549$	− 4.62815i	− 0.00843014i
$550$	0	0
$551$	− 42.9745i	− 0.0779937i
$552$	0	0
$553$	791.405	1.43111
$554$	0	0
$555$	49.7231	0.0895912
$556$	0	0
$557$	570.804i	1.02478i	0.858752	+	0.512391i	$0.171240\pi$
$557$	570.804i	1.02478i	−0.858752	+	0.512391i	$0.828760\pi$
$558$	0	0
$559$	− 201.099i	− 0.359748i
$560$	0	0
$561$	219.713	0.391645
$562$	0	0
$563$	161.877	0.287526	0.143763	−	0.989612i	$-0.454080\pi$
$563$	161.877	0.287526	0.143763	+	0.989612i	$0.454080\pi$
$564$	0	0
$565$	327.350i	0.579381i
$566$	0	0
$567$	− 96.6927i	− 0.170534i
$568$	0	0
$569$	624.123	1.09688	0.548438	−	0.836191i	$-0.315222\pi$
$569$	624.123	1.09688	0.548438	+	0.836191i	$0.315222\pi$
$570$	0	0
$571$	593.031	1.03858	0.519291	−	0.854597i	$-0.326196\pi$
$571$	593.031	1.03858	0.519291	+	0.854597i	$0.326196\pi$
$572$	0	0
$573$	− 539.551i	− 0.941625i
$574$	0	0
$575$	359.113i	0.624544i
$576$	0	0
$577$	−1003.68	−1.73948	−0.869742	−	0.493507i	$-0.835715\pi$
$577$	−1003.68	−1.73948	−0.869742	+	0.493507i	$0.835715\pi$
$578$	0	0
$579$	−83.6180	−0.144418
$580$	0	0
$581$	132.009i	0.227210i
$582$	0	0
$583$	− 228.646i	− 0.392189i
$584$	0	0
$585$	−135.846	−0.232216
$586$	0	0
$587$	−859.215	−1.46374	−0.731870	−	0.681444i	$-0.761352\pi$
$587$	−859.215	−1.46374	−0.731870	+	0.681444i	$0.761352\pi$
$588$	0	0
$589$	9.86151i	0.0167428i
$590$	0	0
$591$	436.274i	0.738196i
$592$	0	0
$593$	−1007.42	−1.69885	−0.849423	−	0.527713i	$-0.823050\pi$
$593$	−1007.42	−1.69885	−0.849423	+	0.527713i	$0.823050\pi$
$594$	0	0
$595$	490.410	0.824219
$596$	0	0
$597$	124.915i	0.209239i
$598$	0	0
$599$	− 86.0598i	− 0.143672i	−0.997416	−	0.0718362i	$-0.977114\pi$
$599$	− 86.0598i	− 0.143672i	0.997416	−	0.0718362i	$-0.0228859\pi$
$600$	0	0
$601$	−406.000	−0.675541	−0.337770	−	0.941229i	$-0.609673\pi$
$601$	−406.000	−0.675541	−0.337770	+	0.941229i	$0.609673\pi$
$602$	0	0
$603$	129.646	0.215002
$604$	0	0
$605$	164.089i	0.271221i
$606$	0	0
$607$	1187.80i	1.95684i	0.206617	+	0.978422i	$0.433755\pi$
$607$	1187.80i	1.95684i	−0.206617	+	0.978422i	$0.566245\pi$
$608$	0	0
$609$	−746.123	−1.22516
$610$	0	0
$611$	−24.2666	−0.0397162
$612$	0	0
$613$	500.378i	0.816277i	0.912920	+	0.408139i	$0.133822\pi$
$613$	500.378i	0.816277i	−0.912920	+	0.408139i	$0.866178\pi$
$614$	0	0
$615$	257.131i	0.418099i
$616$	0	0
$617$	88.8306	0.143972	0.0719859	−	0.997406i	$-0.477066\pi$
$617$	88.8306	0.143972	0.0719859	+	0.997406i	$0.477066\pi$
$618$	0	0
$619$	424.231	0.685349	0.342674	−	0.939454i	$-0.388667\pi$
$619$	424.231	0.685349	0.342674	+	0.939454i	$0.388667\pi$
$620$	0	0
$621$	− 111.651i	− 0.179792i
$622$	0	0
$623$	− 356.027i	− 0.571472i
$624$	0	0
$625$	72.1384	0.115422
$626$	0	0
$627$	14.8513	0.0236862
$628$	0	0
$629$	158.124i	0.251390i
$630$	0	0
$631$	90.6326i	0.143633i	0.997418	+	0.0718166i	$0.0228796\pi$
$631$	90.6326i	0.143633i	−0.997418	+	0.0718166i	$0.977120\pi$
$632$	0	0
$633$	−458.851	−0.724883
$634$	0	0
$635$	−106.108	−0.167099
$636$	0	0
$637$	1044.86i	1.64028i
$638$	0	0
$639$	− 253.219i	− 0.396274i
$640$	0	0
$641$	1090.40	1.70109	0.850546	−	0.525901i	$-0.176272\pi$
$641$	1090.40	1.70109	0.850546	+	0.525901i	$0.176272\pi$
$642$	0	0
$643$	454.200	0.706376	0.353188	−	0.935552i	$-0.385098\pi$
$643$	454.200	0.706376	0.353188	+	0.935552i	$0.385098\pi$
$644$	0	0
$645$	63.7458i	0.0988307i
$646$	0	0
$647$	610.789i	0.944032i	0.881590	+	0.472016i	$0.156474\pi$
$647$	610.789i	0.944032i	−0.881590	+	0.472016i	$0.843526\pi$
$648$	0	0
$649$	89.7231	0.138248
$650$	0	0
$651$	171.215	0.263004
$652$	0	0
$653$	673.612i	1.03157i	0.856720	+	0.515783i	$0.172499\pi$
$653$	673.612i	1.03157i	−0.856720	+	0.515783i	$0.827501\pi$
$654$	0	0
$655$	560.322i	0.855454i
$656$	0	0
$657$	316.277	0.481396
$658$	0	0
$659$	819.328	1.24329	0.621645	−	0.783299i	$-0.286465\pi$
$659$	819.328	1.24329	0.621645	+	0.783299i	$0.286465\pi$
$660$	0	0
$661$	− 370.628i	− 0.560707i	−0.959897	−	0.280354i	$-0.909548\pi$
$661$	− 370.628i	− 0.560707i	0.959897	−	0.280354i	$-0.0904518\pi$
$662$	0	0
$663$	− 432.004i	− 0.651590i
$664$	0	0
$665$	33.1487	0.0498477
$666$	0	0
$667$	−861.549	−1.29168
$668$	0	0
$669$	53.1535i	0.0794521i
$670$	0	0
$671$	− 12.3417i	− 0.0183930i
$672$	0	0
$673$	−255.703	−0.379944	−0.189972	−	0.981789i	$-0.560840\pi$
$673$	−255.703	−0.379944	−0.189972	+	0.981789i	$0.560840\pi$
$674$	0	0
$675$	−86.8423	−0.128655
$676$	0	0
$677$	− 934.323i	− 1.38009i	−0.723765	−	0.690047i	$-0.757590\pi$
$677$	− 934.323i	− 1.38009i	0.723765	−	0.690047i	$-0.242410\pi$
$678$	0	0
$679$	742.798i	1.09396i
$680$	0	0
$681$	512.420	0.752453
$682$	0	0
$683$	−1142.54	−1.67283	−0.836415	−	0.548096i	$-0.815353\pi$
$683$	−1142.54	−1.67283	−0.836415	+	0.548096i	$0.815353\pi$
$684$	0	0
$685$	− 47.3001i	− 0.0690512i
$686$	0	0
$687$	− 444.648i	− 0.647232i
$688$	0	0
$689$	−449.569	−0.652495
$690$	0	0
$691$	−1316.90	−1.90578	−0.952892	−	0.303309i	$-0.901909\pi$
$691$	−1316.90	−1.90578	−0.952892	+	0.303309i	$0.901909\pi$
$692$	0	0
$693$	− 257.847i	− 0.372074i
$694$	0	0
$695$	− 326.716i	− 0.470094i
$696$	0	0
$697$	−817.703	−1.17317
$698$	0	0
$699$	700.726	1.00247
$700$	0	0
$701$	318.493i	0.454341i	0.973855	+	0.227170i	$0.0729474\pi$
$701$	318.493i	0.454341i	−0.973855	+	0.227170i	$0.927053\pi$
$702$	0	0
$703$	10.6883i	0.0152038i
$704$	0	0
$705$	7.69219	0.0109109
$706$	0	0
$707$	1044.90	1.47793
$708$	0	0
$709$	− 289.831i	− 0.408788i	−0.978889	−	0.204394i	$-0.934478\pi$
$709$	− 289.831i	− 0.408788i	0.978889	−	0.204394i	$-0.0655224\pi$
$710$	0	0
$711$	220.988i	0.310813i
$712$	0	0
$713$	197.703	0.277283
$714$	0	0
$715$	−362.256	−0.506652
$716$	0	0
$717$	− 199.445i	− 0.278167i
$718$	0	0
$719$	− 491.122i	− 0.683062i	−0.939870	−	0.341531i	$-0.889055\pi$
$719$	− 491.122i	− 0.683062i	0.939870	−	0.341531i	$-0.110945\pi$
$720$	0	0
$721$	1500.53	2.08118
$722$	0	0
$723$	−435.464	−0.602302
$724$	0	0
$725$	670.113i	0.924294i
$726$	0	0
$727$	774.918i	1.06591i	0.846143	+	0.532956i	$0.178919\pi$
$727$	774.918i	1.06591i	−0.846143	+	0.532956i	$0.821081\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−202.718	−0.277316
$732$	0	0
$733$	858.966i	1.17185i	0.810365	+	0.585925i	$0.199269\pi$
$733$	858.966i	1.17185i	−0.810365	+	0.585925i	$0.800731\pi$
$734$	0	0
$735$	− 331.207i	− 0.450622i
$736$	0	0
$737$	345.723	0.469095
$738$	0	0
$739$	−63.1948	−0.0855139	−0.0427569	−	0.999086i	$-0.513614\pi$
$739$	−63.1948	−0.0855139	−0.0427569	+	0.999086i	$0.513614\pi$
$740$	0	0
$741$	− 29.2008i	− 0.0394073i
$742$	0	0
$743$	− 1105.00i	− 1.48721i	−0.668620	−	0.743604i	$-0.733115\pi$
$743$	− 1105.00i	− 1.48721i	0.668620	−	0.743604i	$-0.266885\pi$
$744$	0	0
$745$	91.1591	0.122361
$746$	0	0
$747$	−36.8616	−0.0493461
$748$	0	0
$749$	− 2124.87i	− 2.83694i
$750$	0	0
$751$	− 804.119i	− 1.07073i	−0.844621	−	0.535365i	$-0.820174\pi$
$751$	− 804.119i	− 1.07073i	0.844621	−	0.535365i	$-0.179826\pi$
$752$	0	0
$753$	11.8667	0.0157593
$754$	0	0
$755$	−260.589	−0.345152
$756$	0	0
$757$	1351.03i	1.78471i	0.451334	+	0.892355i	$0.350948\pi$
$757$	1351.03i	1.78471i	−0.451334	+	0.892355i	$0.649052\pi$
$758$	0	0
$759$	− 297.736i	− 0.392274i
$760$	0	0
$761$	−709.805	−0.932726	−0.466363	−	0.884593i	$-0.654436\pi$
$761$	−709.805	−0.932726	−0.466363	+	0.884593i	$0.654436\pi$
$762$	0	0
$763$	2048.95	2.68539
$764$	0	0
$765$	136.940i	0.179006i
$766$	0	0
$767$	− 176.415i	− 0.230007i
$768$	0	0
$769$	−195.703	−0.254490	−0.127245	−	0.991871i	$-0.540613\pi$
$769$	−195.703	−0.254490	−0.127245	+	0.991871i	$0.540613\pi$
$770$	0	0
$771$	−430.028	−0.557754
$772$	0	0
$773$	− 12.5484i	− 0.0162334i	−0.999967	−	0.00811670i	$-0.997416\pi$
$773$	− 12.5484i	− 0.0162334i	0.999967	−	0.00811670i	$-0.00258365\pi$
$774$	0	0
$775$	− 153.773i	− 0.198417i
$776$	0	0
$777$	185.569	0.238828
$778$	0	0
$779$	−55.2717	−0.0709521
$780$	0	0
$781$	− 675.251i	− 0.864598i
$782$	0	0
$783$	− 208.344i	− 0.266084i
$784$	0	0
$785$	−665.036	−0.847180
$786$	0	0
$787$	432.918	0.550086	0.275043	−	0.961432i	$-0.411308\pi$
$787$	432.918	0.550086	0.275043	+	0.961432i	$0.411308\pi$
$788$	0	0
$789$	− 864.008i	− 1.09507i
$790$	0	0
$791$	1221.69i	1.54449i
$792$	0	0
$793$	−24.2666	−0.0306010
$794$	0	0
$795$	142.508	0.179255
$796$	0	0
$797$	− 700.746i	− 0.879230i	−0.898186	−	0.439615i	$-0.855115\pi$
$797$	− 700.746i	− 0.879230i	0.898186	−	0.439615i	$-0.144885\pi$
$798$	0	0
$799$	24.4619i	0.0306157i
$800$	0	0
$801$	99.4153	0.124114
$802$	0	0
$803$	843.405	1.05032
$804$	0	0
$805$	− 664.562i	− 0.825543i
$806$	0	0
$807$	406.357i	0.503541i
$808$	0	0
$809$	−848.102	−1.04833	−0.524167	−	0.851615i	$-0.675623\pi$
$809$	−848.102	−1.04833	−0.524167	+	0.851615i	$0.675623\pi$
$810$	0	0
$811$	−1242.18	−1.53166	−0.765832	−	0.643041i	$-0.777673\pi$
$811$	−1242.18	−1.53166	−0.765832	+	0.643041i	$0.777673\pi$
$812$	0	0
$813$	− 175.493i	− 0.215858i
$814$	0	0
$815$	64.3510i	0.0789583i
$816$	0	0
$817$	−13.7025	−0.0167717
$818$	0	0
$819$	−506.985	−0.619029
$820$	0	0
$821$	− 208.303i	− 0.253719i	−0.991921	−	0.126859i	$-0.959510\pi$
$821$	− 208.303i	− 0.253719i	0.991921	−	0.126859i	$-0.0404897\pi$
$822$	0	0
$823$	− 778.114i	− 0.945461i	−0.881207	−	0.472730i	$-0.843268\pi$
$823$	− 778.114i	− 0.945461i	0.881207	−	0.472730i	$-0.156732\pi$
$824$	0	0
$825$	−231.580	−0.280702
$826$	0	0
$827$	1280.57	1.54845	0.774225	−	0.632910i	$-0.218140\pi$
$827$	1280.57	1.54845	0.774225	+	0.632910i	$0.218140\pi$
$828$	0	0
$829$	9.78043i	0.0117979i	0.999983	+	0.00589893i	$0.00187770\pi$
$829$	9.78043i	0.0117979i	−0.999983	+	0.00589893i	$0.998122\pi$
$830$	0	0
$831$	− 294.348i	− 0.354210i
$832$	0	0
$833$	1053.27	1.26443
$834$	0	0
$835$	−636.328	−0.762070
$836$	0	0
$837$	47.8094i	0.0571199i
$838$	0	0
$839$	− 1232.38i	− 1.46886i	−0.678682	−	0.734432i	$-0.737449\pi$
$839$	− 1232.38i	− 1.46886i	0.678682	−	0.734432i	$-0.262551\pi$
$840$	0	0
$841$	−766.672	−0.911619
$842$	0	0
$843$	192.480	0.228327
$844$	0	0
$845$	225.768i	0.267181i
$846$	0	0
$847$	612.387i	0.723007i
$848$	0	0
$849$	953.703	1.12332
$850$	0	0
$851$	214.277	0.251794
$852$	0	0
$853$	− 412.170i	− 0.483201i	−0.970376	−	0.241600i	$-0.922328\pi$
$853$	− 412.170i	− 0.483201i	0.970376	−	0.241600i	$-0.0776723\pi$
$854$	0	0
$855$	9.25630i	0.0108261i
$856$	0	0
$857$	−136.451	−0.159220	−0.0796099	−	0.996826i	$-0.525367\pi$
$857$	−136.451	−0.159220	−0.0796099	+	0.996826i	$0.525367\pi$
$858$	0	0
$859$	649.646	0.756282	0.378141	−	0.925748i	$-0.376563\pi$
$859$	649.646	0.756282	0.378141	+	0.925748i	$0.376563\pi$
$860$	0	0
$861$	959.627i	1.11455i
$862$	0	0
$863$	1192.38i	1.38167i	0.723015	+	0.690833i	$0.242756\pi$
$863$	1192.38i	1.38167i	−0.723015	+	0.690833i	$0.757244\pi$
$864$	0	0
$865$	666.502	0.770523
$866$	0	0
$867$	65.0807	0.0750643
$868$	0	0
$869$	589.302i	0.678138i
$870$	0	0
$871$	− 679.768i	− 0.780446i
$872$	0	0
$873$	−207.415	−0.237589
$874$	0	0
$875$	−1290.10	−1.47440
$876$	0	0
$877$	744.451i	0.848861i	0.905460	+	0.424431i	$0.139526\pi$
$877$	744.451i	0.848861i	−0.905460	+	0.424431i	$0.860474\pi$
$878$	0	0
$879$	387.129i	0.440419i
$880$	0	0
$881$	651.108	0.739055	0.369528	−	0.929220i	$-0.379520\pi$
$881$	651.108	0.739055	0.369528	+	0.929220i	$0.379520\pi$
$882$	0	0
$883$	−1171.44	−1.32666	−0.663330	−	0.748327i	$-0.730857\pi$
$883$	−1171.44	−1.32666	−0.663330	+	0.748327i	$0.730857\pi$
$884$	0	0
$885$	55.9215i	0.0631881i
$886$	0	0
$887$	1026.87i	1.15769i	0.815437	+	0.578845i	$0.196496\pi$
$887$	1026.87i	1.15769i	−0.815437	+	0.578845i	$0.803504\pi$
$888$	0	0
$889$	−396.000	−0.445444
$890$	0	0
$891$	72.0000	0.0808081
$892$	0	0
$893$	1.65348i	0.00185160i
$894$	0	0
$895$	557.458i	0.622859i
$896$	0	0
$897$	−585.415	−0.652637
$898$	0	0
$899$	368.918	0.410365
$900$	0	0
$901$	453.189i	0.502984i
$902$	0	0
$903$	237.903i	0.263458i
$904$	0	0
$905$	−778.677	−0.860416
$906$	0	0
$907$	470.508	0.518752	0.259376	−	0.965776i	$-0.416483\pi$
$907$	470.508	0.518752	0.259376	+	0.965776i	$0.416483\pi$
$908$	0	0
$909$	291.772i	0.320981i
$910$	0	0
$911$	− 553.930i	− 0.608046i	−0.952665	−	0.304023i	$-0.901670\pi$
$911$	− 553.930i	− 0.608046i	0.952665	−	0.304023i	$-0.0983300\pi$
$912$	0	0
$913$	−98.2975	−0.107664
$914$	0	0
$915$	7.69219	0.00840677
$916$	0	0
$917$	2091.15i	2.28043i
$918$	0	0
$919$	− 910.123i	− 0.990341i	−0.868796	−	0.495170i	$-0.835106\pi$
$919$	− 910.123i	− 0.990341i	0.868796	−	0.495170i	$-0.164894\pi$
$920$	0	0
$921$	−643.959	−0.699195
$922$	0	0
$923$	−1327.69	−1.43845
$924$	0	0
$925$	− 166.665i	− 0.180178i
$926$	0	0
$927$	419.002i	0.451997i
$928$	0	0
$929$	1278.69	1.37641	0.688206	−	0.725515i	$-0.258398\pi$
$929$	1278.69	1.37641	0.688206	+	0.725515i	$0.258398\pi$
$930$	0	0
$931$	71.1948	0.0764713
$932$	0	0
$933$	571.616i	0.612664i
$934$	0	0
$935$	365.173i	0.390559i
$936$	0	0
$937$	634.554	0.677219	0.338609	−	0.940927i	$-0.390044\pi$
$937$	634.554	0.677219	0.338609	+	0.940927i	$0.390044\pi$
$938$	0	0
$939$	139.044	0.148076
$940$	0	0
$941$	1136.25i	1.20749i	0.797177	+	0.603745i	$0.206326\pi$
$941$	1136.25i	1.20749i	−0.797177	+	0.603745i	$0.793674\pi$
$942$	0	0
$943$	1108.08i	1.17506i
$944$	0	0
$945$	160.708	0.170061
$946$	0	0
$947$	−775.615	−0.819023	−0.409512	−	0.912305i	$-0.634301\pi$
$947$	−775.615	−0.819023	−0.409512	+	0.912305i	$0.634301\pi$
$948$	0	0
$949$	− 1658.32i	− 1.74744i
$950$	0	0
$951$	119.309i	0.125457i
$952$	0	0
$953$	599.590	0.629160	0.314580	−	0.949231i	$-0.398136\pi$
$953$	599.590	0.629160	0.314580	+	0.949231i	$0.398136\pi$
$954$	0	0
$955$	896.758	0.939014
$956$	0	0
$957$	− 555.583i	− 0.580547i
$958$	0	0
$959$	− 176.526i	− 0.184073i
$960$	0	0
$961$	876.343	0.911908
$962$	0	0
$963$	593.338	0.616135
$964$	0	0
$965$	− 138.977i	− 0.144018i
$966$	0	0
$967$	1407.53i	1.45556i	0.685811	+	0.727780i	$0.259448\pi$
$967$	1407.53i	1.45556i	−0.685811	+	0.727780i	$0.740552\pi$
$968$	0	0
$969$	−29.4359	−0.0303776
$970$	0	0
$971$	−653.661	−0.673184	−0.336592	−	0.941651i	$-0.609274\pi$
$971$	−653.661	−0.673184	−0.336592	+	0.941651i	$0.609274\pi$
$972$	0	0
$973$	− 1219.32i	− 1.25315i
$974$	0	0
$975$	455.337i	0.467012i
$976$	0	0
$977$	−1003.57	−1.02719	−0.513597	−	0.858031i	$-0.671687\pi$
$977$	−1003.57	−1.02719	−0.513597	+	0.858031i	$0.671687\pi$
$978$	0	0
$979$	265.108	0.270794
$980$	0	0
$981$	572.140i	0.583221i
$982$	0	0
$983$	105.672i	0.107500i	0.998554	+	0.0537498i	$0.0171173\pi$
$983$	105.672i	0.107500i	−0.998554	+	0.0537498i	$0.982883\pi$
$984$	0	0
$985$	−725.108	−0.736150
$986$	0	0
$987$	28.7077	0.0290858
$988$	0	0
$989$	274.706i	0.277762i
$990$	0	0
$991$	− 1728.18i	− 1.74388i	−0.489615	−	0.871938i	$-0.662863\pi$
$991$	− 1728.18i	− 1.74388i	0.489615	−	0.871938i	$-0.337137\pi$
$992$	0	0
$993$	−685.990	−0.690825
$994$	0	0
$995$	−207.615	−0.208659
$996$	0	0
$997$	135.205i	0.135612i	0.997699	+	0.0678060i	$0.0215999\pi$
$997$	135.205i	0.135612i	−0.997699	+	0.0678060i	$0.978400\pi$
$998$	0	0
$999$	51.8175i	0.0518693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.b.a.79.1		4
3.2	odd	2		288.3.b.b.271.3		4
4.3	odd	2		24.3.b.a.19.1	✓	4
5.2	odd	4		2400.3.p.a.1999.8		8
5.3	odd	4		2400.3.p.a.1999.1		8
5.4	even	2		2400.3.g.a.751.4		4
8.3	odd	2	inner	96.3.b.a.79.2		4
8.5	even	2		24.3.b.a.19.2	yes	4
12.11	even	2		72.3.b.b.19.4		4
16.3	odd	4		768.3.g.h.511.6		8
16.5	even	4		768.3.g.h.511.7		8
16.11	odd	4		768.3.g.h.511.3		8
16.13	even	4		768.3.g.h.511.2		8
20.3	even	4		600.3.p.a.499.2		8
20.7	even	4		600.3.p.a.499.7		8
20.19	odd	2		600.3.g.a.451.4		4
24.5	odd	2		72.3.b.b.19.3		4
24.11	even	2		288.3.b.b.271.2		4
40.3	even	4		2400.3.p.a.1999.4		8
40.13	odd	4		600.3.p.a.499.8		8
40.19	odd	2		2400.3.g.a.751.3		4
40.27	even	4		2400.3.p.a.1999.5		8
40.29	even	2		600.3.g.a.451.3		4
40.37	odd	4		600.3.p.a.499.1		8
48.5	odd	4		2304.3.g.z.1279.4		8
48.11	even	4		2304.3.g.z.1279.3		8
48.29	odd	4		2304.3.g.z.1279.6		8
48.35	even	4		2304.3.g.z.1279.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.1	✓	4	4.3	odd	2
24.3.b.a.19.2	yes	4	8.5	even	2
72.3.b.b.19.3		4	24.5	odd	2
72.3.b.b.19.4		4	12.11	even	2
96.3.b.a.79.1		4	1.1	even	1	trivial
96.3.b.a.79.2		4	8.3	odd	2	inner
288.3.b.b.271.2		4	24.11	even	2
288.3.b.b.271.3		4	3.2	odd	2
600.3.g.a.451.3		4	40.29	even	2
600.3.g.a.451.4		4	20.19	odd	2
600.3.p.a.499.1		8	40.37	odd	4
600.3.p.a.499.2		8	20.3	even	4
600.3.p.a.499.7		8	20.7	even	4
600.3.p.a.499.8		8	40.13	odd	4
768.3.g.h.511.2		8	16.13	even	4
768.3.g.h.511.3		8	16.11	odd	4
768.3.g.h.511.6		8	16.3	odd	4
768.3.g.h.511.7		8	16.5	even	4
2304.3.g.z.1279.3		8	48.11	even	4
2304.3.g.z.1279.4		8	48.5	odd	4
2304.3.g.z.1279.5		8	48.35	even	4
2304.3.g.z.1279.6		8	48.29	odd	4
2400.3.g.a.751.3		4	40.19	odd	2
2400.3.g.a.751.4		4	5.4	even	2
2400.3.p.a.1999.1		8	5.3	odd	4
2400.3.p.a.1999.4		8	40.3	even	4
2400.3.p.a.1999.5		8	40.27	even	4
2400.3.p.a.1999.8		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -2.87875i q^{5} -10.7436i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} -2.87875i q^{5} -10.7436i q^{7} +3.00000 q^{9} +8.00000 q^{11} -15.7298i q^{13} +4.98614i q^{15} -15.8564 q^{17} -1.07180 q^{19} +18.6085i q^{21} +21.4873i q^{23} +16.7128 q^{25} -5.19615 q^{27} +40.0958i q^{29} -9.20092i q^{31} -13.8564 q^{33} -30.9282 q^{35} -9.97227i q^{37} +27.2448i q^{39} +51.5692 q^{41} +12.7846 q^{43} -8.63624i q^{45} -1.54272i q^{47} -66.4256 q^{49} +27.4641 q^{51} -28.5808i q^{53} -23.0300i q^{55} +1.85641 q^{57} +11.2154 q^{59} -1.54272i q^{61} -32.2309i q^{63} -45.2820 q^{65} +43.2154 q^{67} -37.2170i q^{69} -84.4063i q^{71} +105.426 q^{73} -28.9474 q^{75} -85.9491i q^{77} +73.6627i q^{79} +9.00000 q^{81} -12.2872 q^{83} +45.6466i q^{85} -69.4479i q^{87} +33.1384 q^{89} -168.995 q^{91} +15.9365i q^{93} +3.08543i q^{95} -69.1384 q^{97} +24.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 32 q^{11} - 8 q^{17} - 32 q^{19} - 44 q^{25} - 96 q^{35} + 40 q^{41} - 32 q^{43} - 44 q^{49} + 96 q^{51} - 48 q^{57} + 128 q^{59} + 96 q^{65} + 256 q^{67} + 200 q^{73} - 192 q^{75} + 36 q^{81} - 160 q^{83} - 200 q^{89} - 288 q^{91} + 56 q^{97} + 96 q^{99}+O(q^{100}) \) 4 * q + 12 * q^9 + 32 * q^11 - 8 * q^17 - 32 * q^19 - 44 * q^25 - 96 * q^35 + 40 * q^41 - 32 * q^43 - 44 * q^49 + 96 * q^51 - 48 * q^57 + 128 * q^59 + 96 * q^65 + 256 * q^67 + 200 * q^73 - 192 * q^75 + 36 * q^81 - 160 * q^83 - 200 * q^89 - 288 * q^91 + 56 * q^97 + 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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