LMFDB - Embedded newform 4320.2.h.b.2591.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(2591,4320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4320.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 523 x^{12} - 1120 x^{11} + 2214 x^{10} - 3524 x^{9} + \cdots + 2116 $ x^16 - 8x^15 + 44x^14 - 168x^13 + 523x^12 - 1120x^11 + 2214x^10 - 3524x^9 + 6063x^8 - 18332x^7 + 43638x^6 - 65248x^5 + 6502x^4 - 2712x^3 + 160676x^2 + 36616*x + 2116
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.15
Root			$0.425176 + 3.22953i$ of defining polynomial
Character	$\chi$	$=$	4320.2591
Dual form			4320.2.h.b.2591.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +2.44295i q^{7} -4.23668 q^{11} +6.38948 q^{13} -5.57997i q^{17} -2.14365i q^{19} +2.71090 q^{23} -1.00000 q^{25} +0.719614i q^{29} +6.30723i q^{31} -2.44295 q^{35} +7.25427 q^{37} -1.13558i q^{41} -0.721630i q^{43} +4.62923 q^{47} +1.03199 q^{49} -5.99323i q^{53} -4.23668i q^{55} +12.3247 q^{59} -4.24205 q^{61} +6.38948i q^{65} -2.37240i q^{67} +1.19306 q^{71} +12.3341 q^{73} -10.3500i q^{77} +0.170278i q^{79} -9.42811 q^{83} +5.57997 q^{85} +8.07451i q^{89} +15.6092i q^{91} +2.14365 q^{95} -12.0803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{11} - 8 q^{13} + 32 q^{23} - 16 q^{25} + 8 q^{37} + 16 q^{47} - 8 q^{49} - 32 q^{59} + 8 q^{61} - 32 q^{71} - 8 q^{73} - 32 q^{83} - 8 q^{97}+O(q^{100}) $ 16 * q - 8 * q^11 - 8 * q^13 + 32 * q^23 - 16 * q^25 + 8 * q^37 + 16 * q^47 - 8 * q^49 - 32 * q^59 + 8 * q^61 - 32 * q^71 - 8 * q^73 - 32 * q^83 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.44295i	0.923349i	0.887049	+	0.461674i	$0.152751\pi$
$7$	2.44295i	0.923349i	−0.887049	+	0.461674i	$0.847249\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.23668	−1.27741	−0.638704	−	0.769453i	$-0.720529\pi$
$11$	−4.23668	−1.27741	−0.638704	+	0.769453i	$0.720529\pi$
$12$	0	0
$13$	6.38948	1.77212	0.886062	−	0.463567i	$-0.153431\pi$
$13$	6.38948	1.77212	0.886062	+	0.463567i	$0.153431\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.57997i	− 1.35334i	−0.736286	−	0.676671i	$-0.763422\pi$
$17$	− 5.57997i	− 1.35334i	0.736286	−	0.676671i	$-0.236578\pi$
$18$	0	0
$19$	− 2.14365i	− 0.491788i	−0.969297	−	0.245894i	$-0.920919\pi$
$19$	− 2.14365i	− 0.491788i	0.969297	−	0.245894i	$-0.0790815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.71090	0.565262	0.282631	−	0.959229i	$-0.408793\pi$
$23$	2.71090	0.565262	0.282631	+	0.959229i	$0.408793\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.719614i	0.133629i	0.997765	+	0.0668145i	$0.0212836\pi$
$29$	0.719614i	0.133629i	−0.997765	+	0.0668145i	$0.978716\pi$
$30$	0	0
$31$	6.30723i	1.13281i	0.824126	+	0.566406i	$0.191667\pi$
$31$	6.30723i	1.13281i	−0.824126	+	0.566406i	$0.808333\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.44295	−0.412934
$36$	0	0
$37$	7.25427	1.19259	0.596297	−	0.802764i	$-0.296638\pi$
$37$	7.25427	1.19259	0.596297	+	0.802764i	$0.296638\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.13558i	− 0.177348i	−0.996061	−	0.0886739i	$-0.971737\pi$
$41$	− 1.13558i	− 0.177348i	0.996061	−	0.0886739i	$-0.0282629\pi$
$42$	0	0
$43$	− 0.721630i	− 0.110048i	−0.998485	−	0.0550238i	$-0.982477\pi$
$43$	− 0.721630i	− 0.110048i	0.998485	−	0.0550238i	$-0.0175235\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.62923	0.675242	0.337621	−	0.941282i	$-0.390378\pi$
$47$	4.62923	0.675242	0.337621	+	0.941282i	$0.390378\pi$
$48$	0	0
$49$	1.03199	0.147427
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.99323i	− 0.823234i	−0.911357	−	0.411617i	$-0.864964\pi$
$53$	− 5.99323i	− 0.823234i	0.911357	−	0.411617i	$-0.135036\pi$
$54$	0	0
$55$	− 4.23668i	− 0.571274i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.3247	1.60454	0.802272	−	0.596958i	$-0.203624\pi$
$59$	12.3247	1.60454	0.802272	+	0.596958i	$0.203624\pi$
$60$	0	0
$61$	−4.24205	−0.543138	−0.271569	−	0.962419i	$-0.587543\pi$
$61$	−4.24205	−0.543138	−0.271569	+	0.962419i	$0.587543\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.38948i	0.792518i
$66$	0	0
$67$	− 2.37240i	− 0.289835i	−0.989444	−	0.144917i	$-0.953708\pi$
$67$	− 2.37240i	− 0.289835i	0.989444	−	0.144917i	$-0.0462916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.19306	0.141590	0.0707948	−	0.997491i	$-0.477446\pi$
$71$	1.19306	0.141590	0.0707948	+	0.997491i	$0.477446\pi$
$72$	0	0
$73$	12.3341	1.44359	0.721797	−	0.692104i	$-0.243316\pi$
$73$	12.3341	1.44359	0.721797	+	0.692104i	$0.243316\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.3500i	− 1.17949i
$78$	0	0
$79$	0.170278i	0.0191577i	0.999954	+	0.00957887i	$0.00304909\pi$
$79$	0.170278i	0.0191577i	−0.999954	+	0.00957887i	$0.996951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.42811	−1.03487	−0.517435	−	0.855723i	$-0.673113\pi$
$83$	−9.42811	−1.03487	−0.517435	+	0.855723i	$0.673113\pi$
$84$	0	0
$85$	5.57997	0.605233
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.07451i	0.855897i	0.903803	+	0.427948i	$0.140763\pi$
$89$	8.07451i	0.855897i	−0.903803	+	0.427948i	$0.859237\pi$
$90$	0	0
$91$	15.6092i	1.63629i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.14365	0.219934
$96$	0	0
$97$	−12.0803	−1.22657	−0.613287	−	0.789860i	$-0.710153\pi$
$97$	−12.0803	−1.22657	−0.613287	+	0.789860i	$0.710153\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.24058i	0.819968i	0.912093	+	0.409984i	$0.134466\pi$
$101$	8.24058i	0.819968i	−0.912093	+	0.409984i	$0.865534\pi$
$102$	0	0
$103$	2.05087i	0.202078i	0.994882	+	0.101039i	$0.0322168\pi$
$103$	2.05087i	0.202078i	−0.994882	+	0.101039i	$0.967783\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.9672	1.25359	0.626793	−	0.779186i	$-0.284367\pi$
$107$	12.9672	1.25359	0.626793	+	0.779186i	$0.284367\pi$
$108$	0	0
$109$	−11.9126	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$109$	−11.9126	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.9830i	1.87984i	0.341388	+	0.939922i	$0.389103\pi$
$113$	19.9830i	1.87984i	−0.341388	+	0.939922i	$0.610897\pi$
$114$	0	0
$115$	2.71090i	0.252793i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.6316	1.24961
$120$	0	0
$121$	6.94946	0.631769
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	19.0423i	1.68973i	0.534981	+	0.844864i	$0.320319\pi$
$127$	19.0423i	1.68973i	−0.534981	+	0.844864i	$0.679681\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.8332	−1.38335	−0.691675	−	0.722209i	$-0.743127\pi$
$131$	−15.8332	−1.38335	−0.691675	+	0.722209i	$0.743127\pi$
$132$	0	0
$133$	5.23684	0.454092
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.88737i	0.332121i	0.986116	+	0.166060i	$0.0531047\pi$
$137$	3.88737i	0.332121i	−0.986116	+	0.166060i	$0.946895\pi$
$138$	0	0
$139$	13.1814i	1.11803i	0.829157	+	0.559015i	$0.188821\pi$
$139$	13.1814i	1.11803i	−0.829157	+	0.559015i	$0.811179\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−27.0702	−2.26372
$144$	0	0
$145$	−0.719614	−0.0597607
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 10.0046i	− 0.819608i	−0.912174	−	0.409804i	$-0.865597\pi$
$149$	− 10.0046i	− 0.819608i	0.912174	−	0.409804i	$-0.134403\pi$
$150$	0	0
$151$	12.8613i	1.04663i	0.852138	+	0.523317i	$0.175306\pi$
$151$	12.8613i	1.04663i	−0.852138	+	0.523317i	$0.824694\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.30723	−0.506609
$156$	0	0
$157$	23.8992	1.90736	0.953682	−	0.300817i	$-0.0972592\pi$
$157$	23.8992	1.90736	0.953682	+	0.300817i	$0.0972592\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.62260i	0.521934i
$162$	0	0
$163$	23.4319i	1.83533i	0.397360	+	0.917663i	$0.369926\pi$
$163$	23.4319i	1.83533i	−0.397360	+	0.917663i	$0.630074\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.7087	−0.906048	−0.453024	−	0.891498i	$-0.649655\pi$
$167$	−11.7087	−0.906048	−0.453024	+	0.891498i	$0.649655\pi$
$168$	0	0
$169$	27.8255	2.14042
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 23.8924i	− 1.81650i	−0.418426	−	0.908251i	$-0.637418\pi$
$173$	− 23.8924i	− 1.81650i	0.418426	−	0.908251i	$-0.362582\pi$
$174$	0	0
$175$	− 2.44295i	− 0.184670i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.03743	−0.301772	−0.150886	−	0.988551i	$-0.548213\pi$
$179$	−4.03743	−0.301772	−0.150886	+	0.988551i	$0.548213\pi$
$180$	0	0
$181$	3.45495	0.256804	0.128402	−	0.991722i	$-0.459015\pi$
$181$	3.45495	0.256804	0.128402	+	0.991722i	$0.459015\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.25427i	0.533344i
$186$	0	0
$187$	23.6405i	1.72877i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.9663	1.15528	0.577642	−	0.816291i	$-0.303973\pi$
$191$	15.9663	1.15528	0.577642	+	0.816291i	$0.303973\pi$
$192$	0	0
$193$	12.2807	0.883982	0.441991	−	0.897019i	$-0.354272\pi$
$193$	12.2807	0.883982	0.441991	+	0.897019i	$0.354272\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.82951i	− 0.415335i	−0.978199	−	0.207668i	$-0.933413\pi$
$197$	− 5.82951i	− 0.415335i	0.978199	−	0.207668i	$-0.0665872\pi$
$198$	0	0
$199$	15.3161i	1.08573i	0.839820	+	0.542865i	$0.182660\pi$
$199$	15.3161i	1.08573i	−0.839820	+	0.542865i	$0.817340\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.75798	−0.123386
$204$	0	0
$205$	1.13558	0.0793123
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.08197i	0.628213i
$210$	0	0
$211$	− 21.9307i	− 1.50977i	−0.655855	−	0.754887i	$-0.727692\pi$
$211$	− 21.9307i	− 1.50977i	0.655855	−	0.754887i	$-0.272308\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.721630	0.0492148
$216$	0	0
$217$	−15.4083	−1.04598
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 35.6531i	− 2.39829i
$222$	0	0
$223$	− 22.1825i	− 1.48545i	−0.669598	−	0.742724i	$-0.733533\pi$
$223$	− 22.1825i	− 1.48545i	0.669598	−	0.742724i	$-0.266467\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.09492	0.338161	0.169081	−	0.985602i	$-0.445920\pi$
$227$	5.09492	0.338161	0.169081	+	0.985602i	$0.445920\pi$
$228$	0	0
$229$	25.5983	1.69158	0.845792	−	0.533513i	$-0.179129\pi$
$229$	25.5983	1.69158	0.845792	+	0.533513i	$0.179129\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.13115i	0.139616i	0.997560	+	0.0698082i	$0.0222387\pi$
$233$	2.13115i	0.139616i	−0.997560	+	0.0698082i	$0.977761\pi$
$234$	0	0
$235$	4.62923i	0.301978i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.3611	−1.12300	−0.561498	−	0.827478i	$-0.689775\pi$
$239$	−17.3611	−1.12300	−0.561498	+	0.827478i	$0.689775\pi$
$240$	0	0
$241$	12.0800	0.778144	0.389072	−	0.921207i	$-0.372796\pi$
$241$	12.0800	0.778144	0.389072	+	0.921207i	$0.372796\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.03199i	0.0659314i
$246$	0	0
$247$	− 13.6968i	− 0.871509i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.2950	1.40725	0.703623	−	0.710574i	$-0.251565\pi$
$251$	22.2950	1.40725	0.703623	+	0.710574i	$0.251565\pi$
$252$	0	0
$253$	−11.4852	−0.722070
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.8460i	1.11320i	0.830781	+	0.556600i	$0.187894\pi$
$257$	17.8460i	1.11320i	−0.830781	+	0.556600i	$0.812106\pi$
$258$	0	0
$259$	17.7218i	1.10118i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.8048	−1.22122	−0.610609	−	0.791932i	$-0.709075\pi$
$263$	−19.8048	−1.22122	−0.610609	+	0.791932i	$0.709075\pi$
$264$	0	0
$265$	5.99323	0.368161
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.4166i	0.635112i	0.948240	+	0.317556i	$0.102862\pi$
$269$	10.4166i	0.635112i	−0.948240	+	0.317556i	$0.897138\pi$
$270$	0	0
$271$	13.5925i	0.825689i	0.910802	+	0.412844i	$0.135465\pi$
$271$	13.5925i	0.825689i	−0.910802	+	0.412844i	$0.864535\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.23668	0.255481
$276$	0	0
$277$	9.08674	0.545969	0.272985	−	0.962018i	$-0.411989\pi$
$277$	9.08674	0.545969	0.272985	+	0.962018i	$0.411989\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 25.8543i	− 1.54234i	−0.636630	−	0.771169i	$-0.719672\pi$
$281$	− 25.8543i	− 1.54234i	0.636630	−	0.771169i	$-0.280328\pi$
$282$	0	0
$283$	− 12.0976i	− 0.719125i	−0.933121	−	0.359562i	$-0.882926\pi$
$283$	− 12.0976i	− 0.719125i	0.933121	−	0.359562i	$-0.117074\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.77417	0.163754
$288$	0	0
$289$	−14.1361	−0.831532
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.3155i	1.59579i	0.602796	+	0.797895i	$0.294053\pi$
$293$	27.3155i	1.59579i	−0.602796	+	0.797895i	$0.705947\pi$
$294$	0	0
$295$	12.3247i	0.717574i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.3213	1.00171
$300$	0	0
$301$	1.76291	0.101612
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.24205i	− 0.242899i
$306$	0	0
$307$	− 18.8846i	− 1.07780i	−0.842370	−	0.538900i	$-0.818840\pi$
$307$	− 18.8846i	− 1.07780i	0.842370	−	0.538900i	$-0.181160\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.8447	1.29540	0.647701	−	0.761895i	$-0.275731\pi$
$311$	22.8447	1.29540	0.647701	+	0.761895i	$0.275731\pi$
$312$	0	0
$313$	−15.9648	−0.902383	−0.451191	−	0.892427i	$-0.649001\pi$
$313$	−15.9648	−0.902383	−0.451191	+	0.892427i	$0.649001\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.48494i	− 0.0834027i	−0.999130	−	0.0417013i	$-0.986722\pi$
$317$	− 1.48494i	− 0.0834027i	0.999130	−	0.0417013i	$-0.0132778\pi$
$318$	0	0
$319$	− 3.04878i	− 0.170699i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11.9615	−0.665557
$324$	0	0
$325$	−6.38948	−0.354425
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.3090i	0.623484i
$330$	0	0
$331$	− 20.1312i	− 1.10651i	−0.833012	−	0.553256i	$-0.813385\pi$
$331$	− 20.1312i	− 1.10651i	0.833012	−	0.553256i	$-0.186615\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.37240	0.129618
$336$	0	0
$337$	20.3897	1.11070	0.555349	−	0.831617i	$-0.312585\pi$
$337$	20.3897	1.11070	0.555349	+	0.831617i	$0.312585\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 26.7217i	− 1.44706i
$342$	0	0
$343$	19.6218i	1.05948i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.44008	−0.345722	−0.172861	−	0.984946i	$-0.555301\pi$
$347$	−6.44008	−0.345722	−0.172861	+	0.984946i	$0.555301\pi$
$348$	0	0
$349$	2.62208	0.140356	0.0701782	−	0.997534i	$-0.477643\pi$
$349$	2.62208	0.140356	0.0701782	+	0.997534i	$0.477643\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.9149i	1.69866i	0.527861	+	0.849331i	$0.322994\pi$
$353$	31.9149i	1.69866i	−0.527861	+	0.849331i	$0.677006\pi$
$354$	0	0
$355$	1.19306i	0.0633208i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.4748	−1.13340	−0.566699	−	0.823925i	$-0.691780\pi$
$359$	−21.4748	−1.13340	−0.566699	+	0.823925i	$0.691780\pi$
$360$	0	0
$361$	14.4048	0.758145
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.3341i	0.645595i
$366$	0	0
$367$	− 0.608609i	− 0.0317691i	−0.999874	−	0.0158846i	$-0.994944\pi$
$367$	− 0.608609i	− 0.0317691i	0.999874	−	0.0158846i	$-0.00505643\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.6412	0.760132
$372$	0	0
$373$	−13.3317	−0.690288	−0.345144	−	0.938550i	$-0.612170\pi$
$373$	−13.3317	−0.690288	−0.345144	+	0.938550i	$0.612170\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.59796i	0.236807i
$378$	0	0
$379$	− 13.2732i	− 0.681797i	−0.940100	−	0.340899i	$-0.889269\pi$
$379$	− 13.2732i	− 0.681797i	0.940100	−	0.340899i	$-0.110731\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.6834	1.77224	0.886119	−	0.463458i	$-0.153391\pi$
$383$	34.6834	1.77224	0.886119	+	0.463458i	$0.153391\pi$
$384$	0	0
$385$	10.3500	0.527485
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 12.8534i	− 0.651692i	−0.945423	−	0.325846i	$-0.894351\pi$
$389$	− 12.8534i	− 0.651692i	0.945423	−	0.325846i	$-0.105649\pi$
$390$	0	0
$391$	− 15.1267i	− 0.764992i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.170278	−0.00856760
$396$	0	0
$397$	16.4668	0.826447	0.413223	−	0.910630i	$-0.364403\pi$
$397$	16.4668	0.826447	0.413223	+	0.910630i	$0.364403\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.75317i	− 0.237362i	−0.992932	−	0.118681i	$-0.962133\pi$
$401$	− 4.75317i	− 0.237362i	0.992932	−	0.118681i	$-0.0378666\pi$
$402$	0	0
$403$	40.3000i	2.00748i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−30.7340	−1.52343
$408$	0	0
$409$	11.9615	0.591460	0.295730	−	0.955272i	$-0.404437\pi$
$409$	11.9615	0.591460	0.295730	+	0.955272i	$0.404437\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	30.1087i	1.48155i
$414$	0	0
$415$	− 9.42811i	− 0.462808i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9884	−1.02535	−0.512675	−	0.858583i	$-0.671345\pi$
$419$	−20.9884	−1.02535	−0.512675	+	0.858583i	$0.671345\pi$
$420$	0	0
$421$	−8.19926	−0.399607	−0.199804	−	0.979836i	$-0.564030\pi$
$421$	−8.19926	−0.399607	−0.199804	+	0.979836i	$0.564030\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.57997i	0.270668i
$426$	0	0
$427$	− 10.3631i	− 0.501506i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.8879	0.909800	0.454900	−	0.890542i	$-0.349675\pi$
$431$	18.8879	0.909800	0.454900	+	0.890542i	$0.349675\pi$
$432$	0	0
$433$	−5.26261	−0.252905	−0.126453	−	0.991973i	$-0.540359\pi$
$433$	−5.26261	−0.252905	−0.126453	+	0.991973i	$0.540359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.81123i	− 0.277989i
$438$	0	0
$439$	15.3700i	0.733570i	0.930306	+	0.366785i	$0.119541\pi$
$439$	15.3700i	0.733570i	−0.930306	+	0.366785i	$0.880459\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.7438	−0.843032	−0.421516	−	0.906821i	$-0.638502\pi$
$443$	−17.7438	−0.843032	−0.421516	+	0.906821i	$0.638502\pi$
$444$	0	0
$445$	−8.07451	−0.382769
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 3.03744i	− 0.143346i	−0.997428	−	0.0716728i	$-0.977166\pi$
$449$	− 3.03744i	− 0.143346i	0.997428	−	0.0716728i	$-0.0228337\pi$
$450$	0	0
$451$	4.81109i	0.226545i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.6092	−0.731770
$456$	0	0
$457$	−5.80647	−0.271615	−0.135808	−	0.990735i	$-0.543363\pi$
$457$	−5.80647	−0.271615	−0.135808	+	0.990735i	$0.543363\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 31.3510i	− 1.46016i	−0.683361	−	0.730081i	$-0.739482\pi$
$461$	− 31.3510i	− 1.46016i	0.683361	−	0.730081i	$-0.260518\pi$
$462$	0	0
$463$	− 14.6360i	− 0.680192i	−0.940391	−	0.340096i	$-0.889540\pi$
$463$	− 14.6360i	− 0.680192i	0.940391	−	0.340096i	$-0.110460\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.0783	1.66950	0.834752	−	0.550626i	$-0.185611\pi$
$467$	36.0783	1.66950	0.834752	+	0.550626i	$0.185611\pi$
$468$	0	0
$469$	5.79565	0.267618
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.05732i	0.140576i
$474$	0	0
$475$	2.14365i	0.0983576i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.3427	1.61485	0.807425	−	0.589970i	$-0.200860\pi$
$479$	35.3427	1.61485	0.807425	+	0.589970i	$0.200860\pi$
$480$	0	0
$481$	46.3510	2.11342
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 12.0803i	− 0.548540i
$486$	0	0
$487$	23.6551i	1.07192i	0.844244	+	0.535958i	$0.180050\pi$
$487$	23.6551i	1.07192i	−0.844244	+	0.535958i	$0.819950\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.9795	1.17244	0.586219	−	0.810153i	$-0.300616\pi$
$491$	25.9795	1.17244	0.586219	+	0.810153i	$0.300616\pi$
$492$	0	0
$493$	4.01543	0.180846
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.91458i	0.130737i
$498$	0	0
$499$	− 37.7530i	− 1.69006i	−0.534721	−	0.845028i	$-0.679583\pi$
$499$	− 37.7530i	− 1.69006i	0.534721	−	0.845028i	$-0.320417\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.0383	−1.29475	−0.647377	−	0.762170i	$-0.724134\pi$
$503$	−29.0383	−1.29475	−0.647377	+	0.762170i	$0.724134\pi$
$504$	0	0
$505$	−8.24058	−0.366701
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 35.3219i	− 1.56562i	−0.622264	−	0.782808i	$-0.713787\pi$
$509$	− 35.3219i	− 1.56562i	0.622264	−	0.782808i	$-0.286213\pi$
$510$	0	0
$511$	30.1316i	1.33294i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.05087	−0.0903722
$516$	0	0
$517$	−19.6126	−0.862559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.4787i	0.940997i	0.882401	+	0.470498i	$0.155926\pi$
$521$	21.4787i	0.940997i	−0.882401	+	0.470498i	$0.844074\pi$
$522$	0	0
$523$	10.7863i	0.471650i	0.971796	+	0.235825i	$0.0757793\pi$
$523$	10.7863i	0.471650i	−0.971796	+	0.235825i	$0.924221\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.1942	1.53308
$528$	0	0
$529$	−15.6510	−0.680479
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7.25577i	− 0.314282i
$534$	0	0
$535$	12.9672i	0.560620i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.37221	−0.188324
$540$	0	0
$541$	−38.6595	−1.66210	−0.831051	−	0.556196i	$-0.812260\pi$
$541$	−38.6595	−1.66210	−0.831051	+	0.556196i	$0.812260\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 11.9126i	− 0.510279i
$546$	0	0
$547$	− 0.596115i	− 0.0254880i	−0.999919	−	0.0127440i	$-0.995943\pi$
$547$	− 0.596115i	− 0.0254880i	0.999919	−	0.0127440i	$-0.00405666\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.54260	0.0657171
$552$	0	0
$553$	−0.415980	−0.0176893
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.2869i	− 1.62227i	−0.584860	−	0.811134i	$-0.698851\pi$
$557$	− 38.2869i	− 1.62227i	0.584860	−	0.811134i	$-0.301149\pi$
$558$	0	0
$559$	− 4.61085i	− 0.195018i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.6229	−1.41704	−0.708518	−	0.705693i	$-0.750636\pi$
$563$	−33.6229	−1.41704	−0.708518	+	0.705693i	$0.750636\pi$
$564$	0	0
$565$	−19.9830	−0.840692
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.2247i	0.722095i	0.932547	+	0.361048i	$0.117581\pi$
$569$	17.2247i	0.722095i	−0.932547	+	0.361048i	$0.882419\pi$
$570$	0	0
$571$	9.50895i	0.397937i	0.980006	+	0.198969i	$0.0637592\pi$
$571$	9.50895i	0.397937i	−0.980006	+	0.198969i	$0.936241\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.71090	−0.113052
$576$	0	0
$577$	−3.26046	−0.135735	−0.0678674	−	0.997694i	$-0.521619\pi$
$577$	−3.26046	−0.135735	−0.0678674	+	0.997694i	$0.521619\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 23.0324i	− 0.955545i
$582$	0	0
$583$	25.3914i	1.05160i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.3973	−1.58483	−0.792414	−	0.609984i	$-0.791176\pi$
$587$	−38.3973	−1.58483	−0.792414	+	0.609984i	$0.791176\pi$
$588$	0	0
$589$	13.5205	0.557103
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.4818i	1.70345i	0.523987	+	0.851726i	$0.324444\pi$
$593$	41.4818i	1.70345i	−0.523987	+	0.851726i	$0.675556\pi$
$594$	0	0
$595$	13.6316i	0.558841i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.83558	0.115858	0.0579292	−	0.998321i	$-0.481550\pi$
$599$	2.83558	0.115858	0.0579292	+	0.998321i	$0.481550\pi$
$600$	0	0
$601$	−14.3210	−0.584165	−0.292083	−	0.956393i	$-0.594348\pi$
$601$	−14.3210	−0.584165	−0.292083	+	0.956393i	$0.594348\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.94946i	0.282536i
$606$	0	0
$607$	− 18.7014i	− 0.759068i	−0.925178	−	0.379534i	$-0.876084\pi$
$607$	− 18.7014i	− 0.759068i	0.925178	−	0.379534i	$-0.123916\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.5784	1.19661
$612$	0	0
$613$	18.6007	0.751275	0.375638	−	0.926767i	$-0.377424\pi$
$613$	18.6007	0.751275	0.375638	+	0.926767i	$0.377424\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.01540i	− 0.282429i	−0.989979	−	0.141215i	$-0.954899\pi$
$617$	− 7.01540i	− 0.282429i	0.989979	−	0.141215i	$-0.0451008\pi$
$618$	0	0
$619$	13.7680i	0.553383i	0.960959	+	0.276691i	$0.0892380\pi$
$619$	13.7680i	0.553383i	−0.960959	+	0.276691i	$0.910762\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19.7256	−0.790291
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 40.4786i	− 1.61399i
$630$	0	0
$631$	− 43.1272i	− 1.71686i	−0.512927	−	0.858432i	$-0.671439\pi$
$631$	− 43.1272i	− 1.71686i	0.512927	−	0.858432i	$-0.328561\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.0423	−0.755670
$636$	0	0
$637$	6.59388	0.261259
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.90837i	0.114874i	0.998349	+	0.0574368i	$0.0182928\pi$
$641$	2.90837i	0.114874i	−0.998349	+	0.0574368i	$0.981707\pi$
$642$	0	0
$643$	− 13.1111i	− 0.517051i	−0.966004	−	0.258526i	$-0.916763\pi$
$643$	− 13.1111i	− 0.517051i	0.966004	−	0.258526i	$-0.0832367\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.03853	0.237399	0.118699	−	0.992930i	$-0.462127\pi$
$647$	6.03853	0.237399	0.118699	+	0.992930i	$0.462127\pi$
$648$	0	0
$649$	−52.2160	−2.04966
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.2209i	1.02610i	0.858358	+	0.513051i	$0.171485\pi$
$653$	26.2209i	1.02610i	−0.858358	+	0.513051i	$0.828515\pi$
$654$	0	0
$655$	− 15.8332i	− 0.618653i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.7946	−0.965859	−0.482929	−	0.875659i	$-0.660427\pi$
$659$	−24.7946	−0.965859	−0.482929	+	0.875659i	$0.660427\pi$
$660$	0	0
$661$	−18.2396	−0.709437	−0.354718	−	0.934973i	$-0.615423\pi$
$661$	−18.2396	−0.709437	−0.354718	+	0.934973i	$0.615423\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.23684i	0.203076i
$666$	0	0
$667$	1.95080i	0.0755354i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.9722	0.693809
$672$	0	0
$673$	−46.8651	−1.80652	−0.903259	−	0.429097i	$-0.858832\pi$
$673$	−46.8651	−1.80652	−0.903259	+	0.429097i	$0.858832\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.03293i	− 0.231864i	−0.993257	−	0.115932i	$-0.963014\pi$
$677$	− 6.03293i	− 0.231864i	0.993257	−	0.115932i	$-0.0369855\pi$
$678$	0	0
$679$	− 29.5117i	− 1.13256i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.8191	1.17926	0.589631	−	0.807673i	$-0.299273\pi$
$683$	30.8191	1.17926	0.589631	+	0.807673i	$0.299273\pi$
$684$	0	0
$685$	−3.88737	−0.148529
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 38.2936i	− 1.45887i
$690$	0	0
$691$	− 15.5475i	− 0.591456i	−0.955272	−	0.295728i	$-0.904438\pi$
$691$	− 15.5475i	− 0.591456i	0.955272	−	0.295728i	$-0.0955622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.1814	−0.499999
$696$	0	0
$697$	−6.33650	−0.240012
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 11.0905i	− 0.418884i	−0.977821	−	0.209442i	$-0.932835\pi$
$701$	− 11.0905i	− 0.418884i	0.977821	−	0.209442i	$-0.0671648\pi$
$702$	0	0
$703$	− 15.5506i	− 0.586503i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.1313	−0.757116
$708$	0	0
$709$	−32.5889	−1.22390	−0.611951	−	0.790896i	$-0.709615\pi$
$709$	−32.5889	−1.22390	−0.611951	+	0.790896i	$0.709615\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.0983i	0.640336i
$714$	0	0
$715$	− 27.0702i	− 1.01237i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.489793	−0.0182662	−0.00913309	−	0.999958i	$-0.502907\pi$
$719$	−0.489793	−0.0182662	−0.00913309	+	0.999958i	$0.502907\pi$
$720$	0	0
$721$	−5.01018	−0.186589
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 0.719614i	− 0.0267258i
$726$	0	0
$727$	− 30.6315i	− 1.13606i	−0.823008	−	0.568030i	$-0.807706\pi$
$727$	− 30.6315i	− 1.13606i	0.823008	−	0.568030i	$-0.192294\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.02668	−0.148932
$732$	0	0
$733$	30.3625	1.12146	0.560732	−	0.827998i	$-0.310520\pi$
$733$	30.3625	1.12146	0.560732	+	0.827998i	$0.310520\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0511i	0.370237i
$738$	0	0
$739$	− 6.82740i	− 0.251150i	−0.992084	−	0.125575i	$-0.959922\pi$
$739$	− 6.82740i	− 0.251150i	0.992084	−	0.125575i	$-0.0400775\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.5692	−0.791298	−0.395649	−	0.918402i	$-0.629480\pi$
$743$	−21.5692	−0.791298	−0.395649	+	0.918402i	$0.629480\pi$
$744$	0	0
$745$	10.0046	0.366540
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31.6782i	1.15750i
$750$	0	0
$751$	− 23.8218i	− 0.869271i	−0.900607	−	0.434635i	$-0.856877\pi$
$751$	− 23.8218i	− 0.869271i	0.900607	−	0.434635i	$-0.143123\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.8613	−0.468069
$756$	0	0
$757$	−47.7086	−1.73400	−0.866998	−	0.498311i	$-0.833954\pi$
$757$	−47.7086	−1.73400	−0.866998	+	0.498311i	$0.833954\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.5846i	1.50744i	0.657196	+	0.753720i	$0.271743\pi$
$761$	41.5846i	1.50744i	−0.657196	+	0.753720i	$0.728257\pi$
$762$	0	0
$763$	− 29.1019i	− 1.05356i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	78.7487	2.84345
$768$	0	0
$769$	−53.8617	−1.94230	−0.971151	−	0.238466i	$-0.923356\pi$
$769$	−53.8617	−1.94230	−0.971151	+	0.238466i	$0.923356\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	53.8369i	1.93638i	0.250218	+	0.968190i	$0.419498\pi$
$773$	53.8369i	1.93638i	−0.250218	+	0.968190i	$0.580502\pi$
$774$	0	0
$775$	− 6.30723i	− 0.226562i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.43429	−0.0872175
$780$	0	0
$781$	−5.05459	−0.180868
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23.8992i	0.852999i
$786$	0	0
$787$	7.22815i	0.257656i	0.991667	+	0.128828i	$0.0411215\pi$
$787$	7.22815i	0.257656i	−0.991667	+	0.128828i	$0.958879\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−48.8176	−1.73575
$792$	0	0
$793$	−27.1045	−0.962508
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.3075i	− 0.861015i	−0.902587	−	0.430507i	$-0.858335\pi$
$797$	− 24.3075i	− 0.861015i	0.902587	−	0.430507i	$-0.141665\pi$
$798$	0	0
$799$	− 25.8309i	− 0.913833i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−52.2556	−1.84406
$804$	0	0
$805$	−6.62260	−0.233416
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.9026i	0.629423i	0.949187	+	0.314711i	$0.101908\pi$
$809$	17.9026i	0.629423i	−0.949187	+	0.314711i	$0.898092\pi$
$810$	0	0
$811$	39.1739i	1.37558i	0.725909	+	0.687790i	$0.241419\pi$
$811$	39.1739i	1.37558i	−0.725909	+	0.687790i	$0.758581\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.4319	−0.820783
$816$	0	0
$817$	−1.54693	−0.0541201
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.28332i	− 0.0447880i	−0.999749	−	0.0223940i	$-0.992871\pi$
$821$	− 1.28332i	− 0.0447880i	0.999749	−	0.0223940i	$-0.00712883\pi$
$822$	0	0
$823$	− 27.1743i	− 0.947237i	−0.880730	−	0.473619i	$-0.842948\pi$
$823$	− 27.1743i	− 0.947237i	0.880730	−	0.473619i	$-0.157052\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.4825	1.26862	0.634311	−	0.773078i	$-0.281284\pi$
$827$	36.4825	1.26862	0.634311	+	0.773078i	$0.281284\pi$
$828$	0	0
$829$	28.2354	0.980657	0.490329	−	0.871538i	$-0.336877\pi$
$829$	28.2354	0.980657	0.490329	+	0.871538i	$0.336877\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.75847i	− 0.199519i
$834$	0	0
$835$	− 11.7087i	− 0.405197i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.3765	0.427284	0.213642	−	0.976912i	$-0.431467\pi$
$839$	12.3765	0.427284	0.213642	+	0.976912i	$0.431467\pi$
$840$	0	0
$841$	28.4822	0.982143
$842$	0	0
$843$	0	0
$844$	0	0
$845$	27.8255i	0.957226i
$846$	0	0
$847$	16.9772i	0.583343i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.6656	0.674128
$852$	0	0
$853$	14.0166	0.479919	0.239960	−	0.970783i	$-0.422866\pi$
$853$	14.0166	0.479919	0.239960	+	0.970783i	$0.422866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.29151i	− 0.249073i	−0.992215	−	0.124537i	$-0.960256\pi$
$857$	− 7.29151i	− 0.249073i	0.992215	−	0.124537i	$-0.0397444\pi$
$858$	0	0
$859$	− 9.55060i	− 0.325862i	−0.986637	−	0.162931i	$-0.947905\pi$
$859$	− 9.55060i	− 0.325862i	0.986637	−	0.162931i	$-0.0520948\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.5186	0.358058	0.179029	−	0.983844i	$-0.442704\pi$
$863$	10.5186	0.358058	0.179029	+	0.983844i	$0.442704\pi$
$864$	0	0
$865$	23.8924	0.812364
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 0.721412i	− 0.0244722i
$870$	0	0
$871$	− 15.1584i	− 0.513623i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.44295	0.0825868
$876$	0	0
$877$	25.3528	0.856103	0.428052	−	0.903754i	$-0.359200\pi$
$877$	25.3528	0.856103	0.428052	+	0.903754i	$0.359200\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 19.4161i	− 0.654146i	−0.944999	−	0.327073i	$-0.893938\pi$
$881$	− 19.4161i	− 0.654146i	0.944999	−	0.327073i	$-0.106062\pi$
$882$	0	0
$883$	6.25508i	0.210500i	0.994446	+	0.105250i	$0.0335643\pi$
$883$	6.25508i	0.210500i	−0.994446	+	0.105250i	$0.966436\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.4860	0.922889	0.461445	−	0.887169i	$-0.347331\pi$
$887$	27.4860	0.922889	0.461445	+	0.887169i	$0.347331\pi$
$888$	0	0
$889$	−46.5193	−1.56021
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.92346i	− 0.332076i
$894$	0	0
$895$	− 4.03743i	− 0.134957i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.53878	−0.151377
$900$	0	0
$901$	−33.4420	−1.11412
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.45495i	0.114846i
$906$	0	0
$907$	− 46.9987i	− 1.56057i	−0.625426	−	0.780283i	$-0.715075\pi$
$907$	− 46.9987i	− 1.56057i	0.625426	−	0.780283i	$-0.284925\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.3713	0.674933	0.337466	−	0.941338i	$-0.390430\pi$
$911$	20.3713	0.674933	0.337466	+	0.941338i	$0.390430\pi$
$912$	0	0
$913$	39.9439	1.32195
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 38.6796i	− 1.27731i
$918$	0	0
$919$	9.32861i	0.307722i	0.988092	+	0.153861i	$0.0491709\pi$
$919$	9.32861i	0.307722i	−0.988092	+	0.153861i	$0.950829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.62301	0.250914
$924$	0	0
$925$	−7.25427	−0.238519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25.0444i	− 0.821680i	−0.911707	−	0.410840i	$-0.865235\pi$
$929$	− 25.0444i	− 0.821680i	0.911707	−	0.410840i	$-0.134765\pi$
$930$	0	0
$931$	− 2.21223i	− 0.0725028i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−23.6405	−0.773129
$936$	0	0
$937$	−43.6231	−1.42510	−0.712552	−	0.701619i	$-0.752461\pi$
$937$	−43.6231	−1.42510	−0.712552	+	0.701619i	$0.752461\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	60.0770i	1.95845i	0.202768	+	0.979227i	$0.435006\pi$
$941$	60.0770i	1.95845i	−0.202768	+	0.979227i	$0.564994\pi$
$942$	0	0
$943$	− 3.07844i	− 0.100248i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.0289922	−0.000942121 0	−0.000471061	−	1.00000i	$-0.500150\pi$
$947$	−0.0289922	−0.000942121 0	−0.000471061		1.00000i	$0.500150\pi$
$948$	0	0
$949$	78.8084	2.55823
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.4249i	0.758806i	0.925232	+	0.379403i	$0.123871\pi$
$953$	23.4249i	0.758806i	−0.925232	+	0.379403i	$0.876129\pi$
$954$	0	0
$955$	15.9663i	0.516658i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.49666	−0.306663
$960$	0	0
$961$	−8.78119	−0.283264
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.2807i	0.395329i
$966$	0	0
$967$	− 13.7977i	− 0.443705i	−0.975080	−	0.221853i	$-0.928790\pi$
$967$	− 13.7977i	− 0.443705i	0.975080	−	0.221853i	$-0.0712104\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.5798	1.30227	0.651133	−	0.758963i	$-0.274294\pi$
$971$	40.5798	1.30227	0.651133	+	0.758963i	$0.274294\pi$
$972$	0	0
$973$	−32.2015	−1.03233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 22.6254i	− 0.723850i	−0.932207	−	0.361925i	$-0.882120\pi$
$977$	− 22.6254i	− 0.723850i	0.932207	−	0.361925i	$-0.117880\pi$
$978$	0	0
$979$	− 34.2091i	− 1.09333i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.91037	0.316092	0.158046	−	0.987432i	$-0.449481\pi$
$983$	9.91037	0.316092	0.158046	+	0.987432i	$0.449481\pi$
$984$	0	0
$985$	5.82951	0.185744
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.95627i	− 0.0622057i
$990$	0	0
$991$	26.7323i	0.849179i	0.905386	+	0.424589i	$0.139582\pi$
$991$	26.7323i	0.849179i	−0.905386	+	0.424589i	$0.860418\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.3161	−0.485553
$996$	0	0
$997$	3.46079	0.109604	0.0548022	−	0.998497i	$-0.482547\pi$
$997$	3.46079	0.109604	0.0548022	+	0.998497i	$0.482547\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.h.b.2591.15	yes	16
3.2	odd	2		4320.2.h.c.2591.7	yes	16
4.3	odd	2		4320.2.h.c.2591.10	yes	16
12.11	even	2	inner	4320.2.h.b.2591.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.h.b.2591.2	✓	16	12.11	even	2	inner
4320.2.h.b.2591.15	yes	16	1.1	even	1	trivial
4320.2.h.c.2591.7	yes	16	3.2	odd	2
4320.2.h.c.2591.10	yes	16	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +2.44295i q^{7} -4.23668 q^{11} +6.38948 q^{13} -5.57997i q^{17} -2.14365i q^{19} +2.71090 q^{23} -1.00000 q^{25} +0.719614i q^{29} +6.30723i q^{31} -2.44295 q^{35} +7.25427 q^{37} -1.13558i q^{41} -0.721630i q^{43} +4.62923 q^{47} +1.03199 q^{49} -5.99323i q^{53} -4.23668i q^{55} +12.3247 q^{59} -4.24205 q^{61} +6.38948i q^{65} -2.37240i q^{67} +1.19306 q^{71} +12.3341 q^{73} -10.3500i q^{77} +0.170278i q^{79} -9.42811 q^{83} +5.57997 q^{85} +8.07451i q^{89} +15.6092i q^{91} +2.14365 q^{95} -12.0803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{11} - 8 q^{13} + 32 q^{23} - 16 q^{25} + 8 q^{37} + 16 q^{47} - 8 q^{49} - 32 q^{59} + 8 q^{61} - 32 q^{71} - 8 q^{73} - 32 q^{83} - 8 q^{97}+O(q^{100}) \) 16 * q - 8 * q^11 - 8 * q^13 + 32 * q^23 - 16 * q^25 + 8 * q^37 + 16 * q^47 - 8 * q^49 - 32 * q^59 + 8 * q^61 - 32 * q^71 - 8 * q^73 - 32 * q^83 - 8 * q^97

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