LMFDB - Embedded newform 1216.3.e.n.1025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(1025,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.e (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34x^6 + 345x^4 + 1064*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.1
Root			$-4.27138i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.n.1025.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.27138i q^{3} +7.92033 q^{5} -5.75693 q^{7} -9.24469 q^{9} +O(q^{10})$	$q-4.27138i q^{3} +7.92033 q^{5} -5.75693 q^{7} -9.24469 q^{9} -16.5691 q^{11} +7.95976i q^{13} -33.8307i q^{15} -18.1650 q^{17} +(-0.595968 + 18.9907i) q^{19} +24.5900i q^{21} -7.02853 q^{23} +37.7316 q^{25} +1.04517i q^{27} +30.4768i q^{29} +34.9967i q^{31} +70.7727i q^{33} -45.5968 q^{35} +58.6635i q^{37} +33.9992 q^{39} -40.7686i q^{41} +12.5739 q^{43} -73.2210 q^{45} -88.8974 q^{47} -15.8578 q^{49} +77.5897i q^{51} -62.8130i q^{53} -131.232 q^{55} +(81.1163 + 2.54561i) q^{57} -51.2763i q^{59} +68.1970 q^{61} +53.2210 q^{63} +63.0439i q^{65} +82.5313i q^{67} +30.0215i q^{69} +21.9213i q^{71} -57.3956 q^{73} -161.166i q^{75} +95.3868 q^{77} -88.3089i q^{79} -78.7379 q^{81} -0.720460 q^{83} -143.873 q^{85} +130.178 q^{87} +30.1376i q^{89} -45.8238i q^{91} +149.484 q^{93} +(-4.72027 + 150.412i) q^{95} -169.956i q^{97} +153.176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	4.27138i		−	1.42379i	−0.702284	−	0.711897i	$-0.747836\pi$
$3$		−	4.27138i		−	1.42379i	0.702284	−	0.711897i	$-0.252164\pi$
$4$		0			0
$5$	7.92033			1.58407			0.792033	−	0.610478i	$-0.209023\pi$
$5$	7.92033			1.58407			0.792033	+	0.610478i	$0.209023\pi$
$6$		0			0
$7$	−5.75693			−0.822418			−0.411209	−	0.911541i	$-0.634893\pi$
$7$	−5.75693			−0.822418			−0.411209	+	0.911541i	$0.634893\pi$
$8$		0			0
$9$	−9.24469			−1.02719
$10$		0			0
$11$	−16.5691			−1.50628			−0.753139	−	0.657862i	$-0.771461\pi$
$11$	−16.5691			−1.50628			−0.753139	+	0.657862i	$0.771461\pi$
$12$		0			0
$13$			7.95976i			0.612289i	0.951985	+	0.306145i	$0.0990391\pi$
$13$			7.95976i			0.612289i	−0.951985	+	0.306145i	$0.900961\pi$
$14$		0			0
$15$		−	33.8307i		−	2.25538i
$16$		0			0
$17$	−18.1650			−1.06853			−0.534265	−	0.845317i	$-0.679412\pi$
$17$	−18.1650			−1.06853			−0.534265	+	0.845317i	$0.679412\pi$
$18$		0			0
$19$	−0.595968	+	18.9907i	−0.0313668	+	0.999508i
$19$	−0.595968	+	18.9907i	−0.0313668	+	0.999508i
$20$		0			0
$21$			24.5900i			1.17095i
$22$		0			0
$23$	−7.02853			−0.305588			−0.152794	−	0.988258i	$-0.548827\pi$
$23$	−7.02853			−0.305588			−0.152794	+	0.988258i	$0.548827\pi$
$24$		0			0
$25$	37.7316			1.50927
$26$		0			0
$27$			1.04517i			0.0387101i
$28$		0			0
$29$			30.4768i			1.05092i	0.850817	+	0.525462i	$0.176108\pi$
$29$			30.4768i			1.05092i	−0.850817	+	0.525462i	$0.823892\pi$
$30$		0			0
$31$			34.9967i			1.12893i	0.825458	+	0.564464i	$0.190917\pi$
$31$			34.9967i			1.12893i	−0.825458	+	0.564464i	$0.809083\pi$
$32$		0			0
$33$			70.7727i			2.14463i
$34$		0			0
$35$	−45.5968			−1.30276
$36$		0			0
$37$			58.6635i			1.58550i	0.609547	+	0.792750i	$0.291351\pi$
$37$			58.6635i			1.58550i	−0.609547	+	0.792750i	$0.708649\pi$
$38$		0			0
$39$	33.9992			0.871774
$40$		0			0
$41$		−	40.7686i		−	0.994355i	−0.867649	−	0.497178i	$-0.834370\pi$
$41$		−	40.7686i		−	0.994355i	0.867649	−	0.497178i	$-0.165630\pi$
$42$		0			0
$43$	12.5739			0.292417			0.146209	−	0.989254i	$-0.453293\pi$
$43$	12.5739			0.292417			0.146209	+	0.989254i	$0.453293\pi$
$44$		0			0
$45$	−73.2210			−1.62713
$46$		0			0
$47$	−88.8974			−1.89144			−0.945718	−	0.324990i	$-0.894639\pi$
$47$	−88.8974			−1.89144			−0.945718	+	0.324990i	$0.894639\pi$
$48$		0			0
$49$	−15.8578			−0.323628
$50$		0			0
$51$			77.5897i			1.52137i
$52$		0			0
$53$		−	62.8130i		−	1.18515i	−0.805515	−	0.592575i	$-0.798111\pi$
$53$		−	62.8130i		−	1.18515i	0.805515	−	0.592575i	$-0.201889\pi$
$54$		0			0
$55$	−131.232			−2.38604
$56$		0			0
$57$	81.1163	+	2.54561i	1.42309	+	0.0446598i
$58$		0			0
$59$		−	51.2763i		−	0.869089i	−0.900650	−	0.434545i	$-0.856909\pi$
$59$		−	51.2763i		−	0.869089i	0.900650	−	0.434545i	$-0.143091\pi$
$60$		0			0
$61$	68.1970			1.11798			0.558992	−	0.829173i	$-0.311188\pi$
$61$	68.1970			1.11798			0.558992	+	0.829173i	$0.311188\pi$
$62$		0			0
$63$	53.2210			0.844778
$64$		0			0
$65$			63.0439i			0.969907i
$66$		0			0
$67$			82.5313i			1.23181i	0.787820	+	0.615905i	$0.211209\pi$
$67$			82.5313i			1.23181i	−0.787820	+	0.615905i	$0.788791\pi$
$68$		0			0
$69$			30.0215i			0.435095i
$70$		0			0
$71$			21.9213i			0.308750i	0.988012	+	0.154375i	$0.0493364\pi$
$71$			21.9213i			0.308750i	−0.988012	+	0.154375i	$0.950664\pi$
$72$		0			0
$73$	−57.3956			−0.786241			−0.393121	−	0.919487i	$-0.628605\pi$
$73$	−57.3956			−0.786241			−0.393121	+	0.919487i	$0.628605\pi$
$74$		0			0
$75$		−	161.166i		−	2.14888i
$76$		0			0
$77$	95.3868			1.23879
$78$		0			0
$79$		−	88.3089i		−	1.11783i	−0.829223	−	0.558917i	$-0.811217\pi$
$79$		−	88.3089i		−	1.11783i	0.829223	−	0.558917i	$-0.188783\pi$
$80$		0			0
$81$	−78.7379			−0.972073
$82$		0			0
$83$	−0.720460			−0.00868024			−0.00434012	−	0.999991i	$-0.501382\pi$
$83$	−0.720460			−0.00868024			−0.00434012	+	0.999991i	$0.501382\pi$
$84$		0			0
$85$	−143.873			−1.69262
$86$		0			0
$87$	130.178			1.49630
$88$		0			0
$89$			30.1376i			0.338624i	0.985562	+	0.169312i	$0.0541546\pi$
$89$			30.1376i			0.338624i	−0.985562	+	0.169312i	$0.945845\pi$
$90$		0			0
$91$		−	45.8238i		−	0.503558i
$92$		0			0
$93$	149.484			1.60736
$94$		0			0
$95$	−4.72027	+	150.412i	−0.0496870	+	1.58329i
$96$		0			0
$97$		−	169.956i		−	1.75213i	−0.482195	−	0.876064i	$-0.660160\pi$
$97$		−	169.956i		−	1.75213i	0.482195	−	0.876064i	$-0.339840\pi$
$98$		0			0
$99$	153.176			1.54723
$100$		0			0
$101$	−111.727			−1.10621			−0.553104	−	0.833112i	$-0.686557\pi$
$101$	−111.727			−1.10621			−0.553104	+	0.833112i	$0.686557\pi$
$102$		0			0
$103$			23.7725i			0.230801i	0.993319	+	0.115401i	$0.0368152\pi$
$103$			23.7725i			0.230801i	−0.993319	+	0.115401i	$0.963185\pi$
$104$		0			0
$105$			194.761i			1.85487i
$106$		0			0
$107$		−	70.3801i		−	0.657758i	−0.944372	−	0.328879i	$-0.893329\pi$
$107$		−	70.3801i		−	0.657758i	0.944372	−	0.328879i	$-0.106671\pi$
$108$		0			0
$109$		−	21.0005i		−	0.192665i	−0.995349	−	0.0963327i	$-0.969289\pi$
$109$		−	21.0005i		−	0.192665i	0.995349	−	0.0963327i	$-0.0307113\pi$
$110$		0			0
$111$	250.574			2.25742
$112$		0			0
$113$			223.885i			1.98128i	0.136494	+	0.990641i	$0.456416\pi$
$113$			223.885i			1.98128i	−0.136494	+	0.990641i	$0.543584\pi$
$114$		0			0
$115$	−55.6683			−0.484072
$116$		0			0
$117$		−	73.5856i		−	0.628936i
$118$		0			0
$119$	104.575			0.878779
$120$		0			0
$121$	153.534			1.26887
$122$		0			0
$123$	−174.138			−1.41576
$124$		0			0
$125$	100.839			0.806710
$126$		0			0
$127$		−	37.2567i		−	0.293360i	−0.989184	−	0.146680i	$-0.953141\pi$
$127$		−	37.2567i		−	0.293360i	0.989184	−	0.146680i	$-0.0468588\pi$
$128$		0			0
$129$		−	53.7081i		−	0.416342i
$130$		0			0
$131$	33.3254			0.254393			0.127196	−	0.991878i	$-0.459402\pi$
$131$	33.3254			0.254393			0.127196	+	0.991878i	$0.459402\pi$
$132$		0			0
$133$	3.43095	−	109.328i	0.0257966	−	0.822013i
$134$		0			0
$135$			8.27812i			0.0613194i
$136$		0			0
$137$	−227.044			−1.65726			−0.828629	−	0.559799i	$-0.810878\pi$
$137$	−227.044			−1.65726			−0.828629	+	0.559799i	$0.810878\pi$
$138$		0			0
$139$	239.245			1.72119			0.860593	−	0.509293i	$-0.170093\pi$
$139$	239.245			1.72119			0.860593	+	0.509293i	$0.170093\pi$
$140$		0			0
$141$			379.715i			2.69301i
$142$		0			0
$143$		−	131.886i		−	0.922278i
$144$		0			0
$145$			241.386i			1.66473i
$146$		0			0
$147$			67.7347i			0.460780i
$148$		0			0
$149$	−142.214			−0.954459			−0.477229	−	0.878779i	$-0.658359\pi$
$149$	−142.214			−0.954459			−0.477229	+	0.878779i	$0.658359\pi$
$150$		0			0
$151$			88.4519i			0.585774i	0.956147	+	0.292887i	$0.0946160\pi$
$151$			88.4519i			0.585774i	−0.956147	+	0.292887i	$0.905384\pi$
$152$		0			0
$153$	167.930			1.09758
$154$		0			0
$155$			277.186i			1.78830i
$156$		0			0
$157$	−45.8364			−0.291952			−0.145976	−	0.989288i	$-0.546632\pi$
$157$	−45.8364			−0.291952			−0.145976	+	0.989288i	$0.546632\pi$
$158$		0			0
$159$	−268.298			−1.68741
$160$		0			0
$161$	40.4627			0.251321
$162$		0			0
$163$	−50.0387			−0.306986			−0.153493	−	0.988150i	$-0.549052\pi$
$163$	−50.0387			−0.306986			−0.153493	+	0.988150i	$0.549052\pi$
$164$		0			0
$165$			560.543i			3.39723i
$166$		0			0
$167$			147.739i			0.884663i	0.896852	+	0.442332i	$0.145849\pi$
$167$			147.739i			0.884663i	−0.896852	+	0.442332i	$0.854151\pi$
$168$		0			0
$169$	105.642			0.625102
$170$		0			0
$171$	5.50954	−	175.563i	0.0322196	−	1.02668i
$172$		0			0
$173$			221.272i			1.27903i	0.768778	+	0.639515i	$0.220865\pi$
$173$			221.272i			1.27903i	−0.768778	+	0.639515i	$0.779135\pi$
$174$		0			0
$175$	−217.218			−1.24125
$176$		0			0
$177$	−219.020			−1.23740
$178$		0			0
$179$		−	119.436i		−	0.667241i	−0.942707	−	0.333621i	$-0.891730\pi$
$179$		−	119.436i		−	0.667241i	0.942707	−	0.333621i	$-0.108270\pi$
$180$		0			0
$181$			3.14798i			0.0173921i	0.999962	+	0.00869606i	$0.00276808\pi$
$181$			3.14798i			0.0173921i	−0.999962	+	0.00869606i	$0.997232\pi$
$182$		0			0
$183$		−	291.296i		−	1.59178i
$184$		0			0
$185$			464.634i			2.51154i
$186$		0			0
$187$	300.977			1.60950
$188$		0			0
$189$		−	6.01699i		−	0.0318359i
$190$		0			0
$191$	213.295			1.11673			0.558364	−	0.829596i	$-0.311429\pi$
$191$	213.295			1.11673			0.558364	+	0.829596i	$0.311429\pi$
$192$		0			0
$193$		−	104.442i		−	0.541150i	−0.962699	−	0.270575i	$-0.912786\pi$
$193$		−	104.442i		−	0.541150i	0.962699	−	0.270575i	$-0.0872138\pi$
$194$		0			0
$195$	269.285			1.38095
$196$		0			0
$197$	268.231			1.36158			0.680789	−	0.732479i	$-0.261637\pi$
$197$	268.231			1.36158			0.680789	+	0.732479i	$0.261637\pi$
$198$		0			0
$199$	3.33565			0.0167621			0.00838103	−	0.999965i	$-0.497332\pi$
$199$	3.33565			0.0167621			0.00838103	+	0.999965i	$0.497332\pi$
$200$		0			0
$201$	352.522			1.75384
$202$		0			0
$203$		−	175.453i		−	0.864299i
$204$		0			0
$205$		−	322.900i		−	1.57512i
$206$		0			0
$207$	64.9766			0.313897
$208$		0			0
$209$	9.87463	−	314.657i	0.0472470	−	1.50554i
$210$		0			0
$211$			188.650i			0.894075i	0.894515	+	0.447037i	$0.147521\pi$
$211$			188.650i			0.894075i	−0.894515	+	0.447037i	$0.852479\pi$
$212$		0			0
$213$	93.6340			0.439596
$214$		0			0
$215$	99.5898			0.463208
$216$		0			0
$217$		−	201.474i		−	0.928450i
$218$		0			0
$219$			245.159i			1.11945i
$220$		0			0
$221$		−	144.589i		−	0.654250i
$222$		0			0
$223$			43.0515i			0.193056i	0.995330	+	0.0965281i	$0.0307738\pi$
$223$			43.0515i			0.193056i	−0.995330	+	0.0965281i	$0.969226\pi$
$224$		0			0
$225$	−348.817			−1.55030
$226$		0			0
$227$			107.923i			0.475433i	0.971335	+	0.237717i	$0.0763990\pi$
$227$			107.923i			0.475433i	−0.971335	+	0.237717i	$0.923601\pi$
$228$		0			0
$229$	−173.435			−0.757360			−0.378680	−	0.925528i	$-0.623622\pi$
$229$	−173.435			−0.757360			−0.378680	+	0.925528i	$0.623622\pi$
$230$		0			0
$231$		−	407.433i		−	1.76378i
$232$		0			0
$233$	−6.95848			−0.0298647			−0.0149324	−	0.999889i	$-0.504753\pi$
$233$	−6.95848			−0.0298647			−0.0149324	+	0.999889i	$0.504753\pi$
$234$		0			0
$235$	−704.097			−2.99616
$236$		0			0
$237$	−377.201			−1.59157
$238$		0			0
$239$	−55.8581			−0.233716			−0.116858	−	0.993149i	$-0.537282\pi$
$239$	−55.8581			−0.233716			−0.116858	+	0.993149i	$0.537282\pi$
$240$		0			0
$241$		−	156.862i		−	0.650880i	−0.945563	−	0.325440i	$-0.894488\pi$
$241$		−	156.862i		−	0.650880i	0.945563	−	0.325440i	$-0.105512\pi$
$242$		0			0
$243$			345.726i			1.42274i
$244$		0			0
$245$	−125.599			−0.512649
$246$		0			0
$247$	−151.161	−	4.74377i	−0.611988	−	0.0192055i
$248$		0			0
$249$			3.07736i			0.0123589i
$250$		0			0
$251$	−99.2555			−0.395440			−0.197720	−	0.980258i	$-0.563354\pi$
$251$	−99.2555			−0.395440			−0.197720	+	0.980258i	$0.563354\pi$
$252$		0			0
$253$	116.456			0.460301
$254$		0			0
$255$			614.536i			2.40995i
$256$		0			0
$257$		−	27.8601i		−	0.108405i	−0.998530	−	0.0542026i	$-0.982738\pi$
$257$		−	27.8601i		−	0.108405i	0.998530	−	0.0542026i	$-0.0172617\pi$
$258$		0			0
$259$		−	337.721i		−	1.30394i
$260$		0			0
$261$		−	281.749i		−	1.07950i
$262$		0			0
$263$	−378.378			−1.43870			−0.719349	−	0.694649i	$-0.755560\pi$
$263$	−378.378			−1.43870			−0.719349	+	0.694649i	$0.755560\pi$
$264$		0			0
$265$		−	497.500i		−	1.87736i
$266$		0			0
$267$	128.729			0.482131
$268$		0			0
$269$		−	132.873i		−	0.493950i	−0.969022	−	0.246975i	$-0.920563\pi$
$269$		−	132.873i		−	0.493950i	0.969022	−	0.246975i	$-0.0794366\pi$
$270$		0			0
$271$	186.215			0.687140			0.343570	−	0.939127i	$-0.388364\pi$
$271$	186.215			0.687140			0.343570	+	0.939127i	$0.388364\pi$
$272$		0			0
$273$	−195.731			−0.716962
$274$		0			0
$275$	−625.177			−2.27337
$276$		0			0
$277$	−44.9078			−0.162122			−0.0810611	−	0.996709i	$-0.525831\pi$
$277$	−44.9078			−0.162122			−0.0810611	+	0.996709i	$0.525831\pi$
$278$		0			0
$279$		−	323.534i		−	1.15962i
$280$		0			0
$281$		−	344.957i		−	1.22761i	−0.789459	−	0.613803i	$-0.789639\pi$
$281$		−	344.957i		−	1.22761i	0.789459	−	0.613803i	$-0.210361\pi$
$282$		0			0
$283$	−181.986			−0.643059			−0.321530	−	0.946900i	$-0.604197\pi$
$283$	−181.986			−0.643059			−0.321530	+	0.946900i	$0.604197\pi$
$284$		0			0
$285$	642.468	+	20.1621i	2.25427	+	0.0707440i
$286$		0			0
$287$			234.702i			0.817776i
$288$		0			0
$289$	40.9680			0.141758
$290$		0			0
$291$	−725.949			−2.49467
$292$		0			0
$293$		−	11.4268i		−	0.0389992i	−0.999810	−	0.0194996i	$-0.993793\pi$
$293$		−	11.4268i		−	0.0389992i	0.999810	−	0.0194996i	$-0.00620731\pi$
$294$		0			0
$295$		−	406.125i		−	1.37669i
$296$		0			0
$297$		−	17.3175i		−	0.0583082i
$298$		0			0
$299$		−	55.9455i		−	0.187109i
$300$		0			0
$301$	−72.3873			−0.240489
$302$		0			0
$303$			477.228i			1.57501i
$304$		0			0
$305$	540.143			1.77096
$306$		0			0
$307$			66.7163i			0.217317i	0.994079	+	0.108659i	$0.0346555\pi$
$307$			66.7163i			0.217317i	−0.994079	+	0.108659i	$0.965344\pi$
$308$		0			0
$309$	101.542			0.328613
$310$		0			0
$311$	−456.827			−1.46890			−0.734449	−	0.678664i	$-0.762559\pi$
$311$	−456.827			−1.46890			−0.734449	+	0.678664i	$0.762559\pi$
$312$		0			0
$313$	−571.983			−1.82742			−0.913711	−	0.406364i	$-0.866797\pi$
$313$	−571.983			−1.82742			−0.913711	+	0.406364i	$0.866797\pi$
$314$		0			0
$315$	421.528			1.33818
$316$		0			0
$317$		−	564.021i		−	1.77925i	−0.456695	−	0.889623i	$-0.650967\pi$
$317$		−	564.021i		−	1.77925i	0.456695	−	0.889623i	$-0.349033\pi$
$318$		0			0
$319$		−	504.972i		−	1.58298i
$320$		0			0
$321$	−300.620			−0.936512
$322$		0			0
$323$	10.8258	−	344.966i	0.0335163	−	1.06800i
$324$		0			0
$325$			300.335i			0.924107i
$326$		0			0
$327$	−89.7013			−0.274316
$328$		0			0
$329$	511.776			1.55555
$330$		0			0
$331$		−	101.354i		−	0.306206i	−0.988210	−	0.153103i	$-0.951073\pi$
$331$		−	101.354i		−	0.306206i	0.988210	−	0.153103i	$-0.0489267\pi$
$332$		0			0
$333$		−	542.326i		−	1.62861i
$334$		0			0
$335$			653.675i			1.95127i
$336$		0			0
$337$			251.262i			0.745583i	0.927915	+	0.372792i	$0.121599\pi$
$337$			251.262i			0.745583i	−0.927915	+	0.372792i	$0.878401\pi$
$338$		0			0
$339$	956.297			2.82094
$340$		0			0
$341$		−	579.863i		−	1.70048i
$342$		0			0
$343$	373.382			1.08858
$344$		0			0
$345$			237.781i			0.689219i
$346$		0			0
$347$	382.247			1.10158			0.550788	−	0.834645i	$-0.314327\pi$
$347$	382.247			1.10158			0.550788	+	0.834645i	$0.314327\pi$
$348$		0			0
$349$	25.1254			0.0719925			0.0359963	−	0.999352i	$-0.488540\pi$
$349$	25.1254			0.0719925			0.0359963	+	0.999352i	$0.488540\pi$
$350$		0			0
$351$	−8.31934			−0.0237018
$352$		0			0
$353$	140.407			0.397754			0.198877	−	0.980024i	$-0.436271\pi$
$353$	140.407			0.397754			0.198877	+	0.980024i	$0.436271\pi$
$354$		0			0
$355$			173.624i			0.489081i
$356$		0			0
$357$		−	446.678i		−	1.25120i
$358$		0			0
$359$	51.9710			0.144766			0.0723830	−	0.997377i	$-0.476940\pi$
$359$	51.9710			0.144766			0.0723830	+	0.997377i	$0.476940\pi$
$360$		0			0
$361$	−360.290	−	22.6357i	−0.998032	−	0.0627026i
$362$		0			0
$363$		−	655.800i		−	1.80661i
$364$		0			0
$365$	−454.592			−1.24546
$366$		0			0
$367$	−211.340			−0.575857			−0.287929	−	0.957652i	$-0.592967\pi$
$367$	−211.340			−0.575857			−0.287929	+	0.957652i	$0.592967\pi$
$368$		0			0
$369$			376.893i			1.02139i
$370$		0			0
$371$			361.610i			0.974689i
$372$		0			0
$373$			530.062i			1.42108i	0.703657	+	0.710539i	$0.251549\pi$
$373$			530.062i			1.42108i	−0.703657	+	0.710539i	$0.748451\pi$
$374$		0			0
$375$		−	430.721i		−	1.14859i
$376$		0			0
$377$	−242.588			−0.643470
$378$		0			0
$379$		−	498.205i		−	1.31453i	−0.753661	−	0.657263i	$-0.771714\pi$
$379$		−	498.205i		−	1.31453i	0.753661	−	0.657263i	$-0.228286\pi$
$380$		0			0
$381$	−159.138			−0.417684
$382$		0			0
$383$			626.130i			1.63480i	0.576067	+	0.817402i	$0.304587\pi$
$383$			626.130i			1.63480i	−0.576067	+	0.817402i	$0.695413\pi$
$384$		0			0
$385$	755.495			1.96233
$386$		0			0
$387$	−116.242			−0.300368
$388$		0			0
$389$	209.215			0.537828			0.268914	−	0.963164i	$-0.413335\pi$
$389$	209.215			0.537828			0.268914	+	0.963164i	$0.413335\pi$
$390$		0			0
$391$	127.673			0.326531
$392$		0			0
$393$		−	142.346i		−	0.362203i
$394$		0			0
$395$		−	699.436i		−	1.77072i
$396$		0			0
$397$	355.260			0.894861			0.447430	−	0.894319i	$-0.352339\pi$
$397$	355.260			0.894861			0.447430	+	0.894319i	$0.352339\pi$
$398$		0			0
$399$	−466.981	−	14.6549i	−1.17038	−	0.0367290i
$400$		0			0
$401$		−	335.760i		−	0.837308i	−0.908146	−	0.418654i	$-0.862502\pi$
$401$		−	335.760i		−	0.837308i	0.908146	−	0.418654i	$-0.137498\pi$
$402$		0			0
$403$	−278.566			−0.691230
$404$		0			0
$405$	−623.630			−1.53983
$406$		0			0
$407$		−	971.998i		−	2.38820i
$408$		0			0
$409$		−	267.604i		−	0.654289i	−0.944974	−	0.327144i	$-0.893914\pi$
$409$		−	267.604i		−	0.654289i	0.944974	−	0.327144i	$-0.106086\pi$
$410$		0			0
$411$			969.792i			2.35959i
$412$		0			0
$413$			295.194i			0.714755i
$414$		0			0
$415$	−5.70628			−0.0137501
$416$		0			0
$417$		−	1021.91i		−	2.45061i
$418$		0			0
$419$	460.072			1.09802			0.549012	−	0.835815i	$-0.315004\pi$
$419$	460.072			1.09802			0.549012	+	0.835815i	$0.315004\pi$
$420$		0			0
$421$			11.7047i			0.0278021i	0.999903	+	0.0139011i	$0.00442499\pi$
$421$			11.7047i			0.0278021i	−0.999903	+	0.0139011i	$0.995575\pi$
$422$		0			0
$423$	821.830			1.94286
$424$		0			0
$425$	−685.396			−1.61270
$426$		0			0
$427$	−392.605			−0.919450
$428$		0			0
$429$	−563.334			−1.31313
$430$		0			0
$431$			262.531i			0.609121i	0.952493	+	0.304560i	$0.0985095\pi$
$431$			262.531i			0.609121i	−0.952493	+	0.304560i	$0.901490\pi$
$432$		0			0
$433$			476.152i			1.09966i	0.835277	+	0.549829i	$0.185307\pi$
$433$			476.152i			1.09966i	−0.835277	+	0.549829i	$0.814693\pi$
$434$		0			0
$435$	1031.05			2.37024
$436$		0			0
$437$	4.18878	−	133.476i	0.00958531	−	0.305438i
$438$		0			0
$439$			712.061i			1.62201i	0.585042	+	0.811003i	$0.301078\pi$
$439$			712.061i			1.62201i	−0.585042	+	0.811003i	$0.698922\pi$
$440$		0			0
$441$	146.600			0.332427
$442$		0			0
$443$	−771.711			−1.74201			−0.871006	−	0.491272i	$-0.836532\pi$
$443$	−771.711			−1.74201			−0.871006	+	0.491272i	$0.836532\pi$
$444$		0			0
$445$			238.699i			0.536403i
$446$		0			0
$447$			607.452i			1.35895i
$448$		0			0
$449$			318.364i			0.709050i	0.935047	+	0.354525i	$0.115357\pi$
$449$			318.364i			0.709050i	−0.935047	+	0.354525i	$0.884643\pi$
$450$		0			0
$451$			675.496i			1.49777i
$452$		0			0
$453$	377.812			0.834022
$454$		0			0
$455$		−	362.939i		−	0.797669i
$456$		0			0
$457$	548.540			1.20031			0.600153	−	0.799885i	$-0.295106\pi$
$457$	548.540			1.20031			0.600153	+	0.799885i	$0.295106\pi$
$458$		0			0
$459$		−	18.9856i		−	0.0413630i
$460$		0			0
$461$	−233.459			−0.506419			−0.253209	−	0.967412i	$-0.581486\pi$
$461$	−233.459			−0.506419			−0.253209	+	0.967412i	$0.581486\pi$
$462$		0			0
$463$	−315.936			−0.682367			−0.341183	−	0.939997i	$-0.610828\pi$
$463$	−315.936			−0.682367			−0.341183	+	0.939997i	$0.610828\pi$
$464$		0			0
$465$	1183.97			2.54616
$466$		0			0
$467$	−340.599			−0.729335			−0.364667	−	0.931138i	$-0.618817\pi$
$467$	−340.599			−0.729335			−0.364667	+	0.931138i	$0.618817\pi$
$468$		0			0
$469$		−	475.126i		−	1.01306i
$470$		0			0
$471$			195.785i			0.415679i
$472$		0			0
$473$	−208.338			−0.440462
$474$		0			0
$475$	−22.4869	+	716.548i	−0.0473407	+	1.50852i
$476$		0			0
$477$			580.687i			1.21737i
$478$		0			0
$479$	414.927			0.866236			0.433118	−	0.901337i	$-0.357413\pi$
$479$	414.927			0.866236			0.433118	+	0.901337i	$0.357413\pi$
$480$		0			0
$481$	−466.947			−0.970784
$482$		0			0
$483$		−	172.832i		−	0.357830i
$484$		0			0
$485$		−	1346.11i		−	2.77549i
$486$		0			0
$487$		−	173.832i		−	0.356945i	−0.983945	−	0.178473i	$-0.942884\pi$
$487$		−	173.832i		−	0.356945i	0.983945	−	0.178473i	$-0.0571156\pi$
$488$		0			0
$489$			213.734i			0.437084i
$490$		0			0
$491$	436.406			0.888811			0.444406	−	0.895826i	$-0.353415\pi$
$491$	436.406			0.888811			0.444406	+	0.895826i	$0.353415\pi$
$492$		0			0
$493$		−	553.612i		−	1.12294i
$494$		0			0
$495$	1213.20			2.45092
$496$		0			0
$497$		−	126.199i		−	0.253922i
$498$		0			0
$499$	−32.6139			−0.0653585			−0.0326792	−	0.999466i	$-0.510404\pi$
$499$	−32.6139			−0.0653585			−0.0326792	+	0.999466i	$0.510404\pi$
$500$		0			0
$501$	631.048			1.25958
$502$		0			0
$503$	−464.682			−0.923820			−0.461910	−	0.886927i	$-0.652836\pi$
$503$	−464.682			−0.923820			−0.461910	+	0.886927i	$0.652836\pi$
$504$		0			0
$505$	−884.914			−1.75231
$506$		0			0
$507$		−	451.238i		−	0.890016i
$508$		0			0
$509$			428.140i			0.841140i	0.907260	+	0.420570i	$0.138170\pi$
$509$			428.140i			0.841140i	−0.907260	+	0.420570i	$0.861830\pi$
$510$		0			0
$511$	330.422			0.646619
$512$		0			0
$513$	−19.8485	−	0.622891i	−0.0386911	−	0.00121421i
$514$		0			0
$515$			188.286i			0.365604i
$516$		0			0
$517$	1472.95			2.84903
$518$		0			0
$519$	945.138			1.82108
$520$		0			0
$521$		−	336.826i		−	0.646499i	−0.946314	−	0.323250i	$-0.895225\pi$
$521$		−	336.826i		−	0.646499i	0.946314	−	0.323250i	$-0.104775\pi$
$522$		0			0
$523$		−	850.674i		−	1.62653i	−0.581895	−	0.813264i	$-0.697689\pi$
$523$		−	850.674i		−	1.62653i	0.581895	−	0.813264i	$-0.302311\pi$
$524$		0			0
$525$			927.822i			1.76728i
$526$		0			0
$527$		−	635.717i		−	1.20629i
$528$		0			0
$529$	−479.600			−0.906616
$530$		0			0
$531$			474.033i			0.892718i
$532$		0			0
$533$	324.508			0.608833
$534$		0			0
$535$		−	557.434i		−	1.04193i
$536$		0			0
$537$	−510.157			−0.950014
$538$		0			0
$539$	262.749			0.487474
$540$		0			0
$541$	−32.9805			−0.0609620			−0.0304810	−	0.999535i	$-0.509704\pi$
$541$	−32.9805			−0.0609620			−0.0304810	+	0.999535i	$0.509704\pi$
$542$		0			0
$543$	13.4462			0.0247628
$544$		0			0
$545$		−	166.331i		−	0.305195i
$546$		0			0
$547$			811.240i			1.48307i	0.670914	+	0.741535i	$0.265902\pi$
$547$			811.240i			1.48307i	−0.670914	+	0.741535i	$0.734098\pi$
$548$		0			0
$549$	−630.461			−1.14838
$550$		0			0
$551$	−578.774	−	18.1632i	−1.05041	−	0.0329641i
$552$		0			0
$553$			508.388i			0.919327i
$554$		0			0
$555$	1984.63			3.57591
$556$		0			0
$557$	−942.150			−1.69147			−0.845736	−	0.533602i	$-0.820838\pi$
$557$	−942.150			−1.69147			−0.845736	+	0.533602i	$0.820838\pi$
$558$		0			0
$559$			100.086i			0.179044i
$560$		0			0
$561$		−	1285.59i		−	2.29160i
$562$		0			0
$563$			193.631i			0.343927i	0.985103	+	0.171964i	$0.0550111\pi$
$563$			193.631i			0.343927i	−0.985103	+	0.171964i	$0.944989\pi$
$564$		0			0
$565$			1773.24i			3.13848i
$566$		0			0
$567$	453.288			0.799450
$568$		0			0
$569$		−	1072.23i		−	1.88440i	−0.335047	−	0.942202i	$-0.608752\pi$
$569$		−	1072.23i		−	1.88440i	0.335047	−	0.942202i	$-0.391248\pi$
$570$		0			0
$571$	−470.870			−0.824641			−0.412321	−	0.911039i	$-0.635282\pi$
$571$	−470.870			−0.824641			−0.412321	+	0.911039i	$0.635282\pi$
$572$		0			0
$573$		−	911.065i		−	1.58999i
$574$		0			0
$575$	−265.198			−0.461214
$576$		0			0
$577$	863.970			1.49735			0.748675	−	0.662938i	$-0.230691\pi$
$577$	863.970			1.49735			0.748675	+	0.662938i	$0.230691\pi$
$578$		0			0
$579$	−446.111			−0.770486
$580$		0			0
$581$	4.14764			0.00713879
$582$		0			0
$583$			1040.75i			1.78517i
$584$		0			0
$585$		−	582.822i		−	0.996277i
$586$		0			0
$587$	−478.667			−0.815446			−0.407723	−	0.913106i	$-0.633677\pi$
$587$	−478.667			−0.815446			−0.407723	+	0.913106i	$0.633677\pi$
$588$		0			0
$589$	−664.611	−	20.8569i	−1.12837	−	0.0354108i
$590$		0			0
$591$		−	1145.72i		−	1.93861i
$592$		0			0
$593$	−444.287			−0.749219			−0.374610	−	0.927183i	$-0.622223\pi$
$593$	−444.287			−0.749219			−0.374610	+	0.927183i	$0.622223\pi$
$594$		0			0
$595$	828.266			1.39204
$596$		0			0
$597$		−	14.2478i		−	0.0238657i
$598$		0			0
$599$		−	644.065i		−	1.07523i	−0.843189	−	0.537617i	$-0.819325\pi$
$599$		−	644.065i		−	1.07523i	0.843189	−	0.537617i	$-0.180675\pi$
$600$		0			0
$601$			858.767i			1.42890i	0.699688	+	0.714449i	$0.253322\pi$
$601$			858.767i			1.42890i	−0.699688	+	0.714449i	$0.746678\pi$
$602$		0			0
$603$		−	762.976i		−	1.26530i
$604$		0			0
$605$	1216.04			2.00998
$606$		0			0
$607$		−	570.384i		−	0.939677i	−0.882752	−	0.469839i	$-0.844312\pi$
$607$		−	570.384i		−	0.939677i	0.882752	−	0.469839i	$-0.155688\pi$
$608$		0			0
$609$	−749.425			−1.23058
$610$		0			0
$611$		−	707.603i		−	1.15811i
$612$		0			0
$613$	225.808			0.368366			0.184183	−	0.982892i	$-0.441036\pi$
$613$	225.808			0.368366			0.184183	+	0.982892i	$0.441036\pi$
$614$		0			0
$615$	−1379.23			−2.24265
$616$		0			0
$617$	484.705			0.785583			0.392792	−	0.919628i	$-0.371509\pi$
$617$	484.705			0.785583			0.392792	+	0.919628i	$0.371509\pi$
$618$		0			0
$619$	−4.30576			−0.00695599			−0.00347799	−	0.999994i	$-0.501107\pi$
$619$	−4.30576			−0.00695599			−0.00347799	+	0.999994i	$0.501107\pi$
$620$		0			0
$621$		−	7.34604i		−	0.0118294i
$622$		0			0
$623$		−	173.500i		−	0.278491i
$624$		0			0
$625$	−144.615			−0.231384
$626$		0			0
$627$	−1344.02	−	42.1783i	−2.14357	−	0.0672700i
$628$		0			0
$629$		−	1065.62i		−	1.69415i
$630$		0			0
$631$	−428.682			−0.679370			−0.339685	−	0.940539i	$-0.610320\pi$
$631$	−428.682			−0.679370			−0.339685	+	0.940539i	$0.610320\pi$
$632$		0			0
$633$	805.795			1.27298
$634$		0			0
$635$		−	295.086i		−	0.464702i
$636$		0			0
$637$		−	126.224i		−	0.198154i
$638$		0			0
$639$		−	202.655i		−	0.317144i
$640$		0			0
$641$		−	28.2734i		−	0.0441083i	−0.999757	−	0.0220542i	$-0.992979\pi$
$641$		−	28.2734i		−	0.0441083i	0.999757	−	0.0220542i	$-0.00702063\pi$
$642$		0			0
$643$	−555.039			−0.863202			−0.431601	−	0.902065i	$-0.642051\pi$
$643$	−555.039			−0.863202			−0.431601	+	0.902065i	$0.642051\pi$
$644$		0			0
$645$		−	425.386i		−	0.659513i
$646$		0			0
$647$	−258.218			−0.399101			−0.199550	−	0.979888i	$-0.563948\pi$
$647$	−258.218			−0.399101			−0.199550	+	0.979888i	$0.563948\pi$
$648$		0			0
$649$			849.599i			1.30909i
$650$		0			0
$651$	−860.571			−1.32192
$652$		0			0
$653$	−889.632			−1.36238			−0.681189	−	0.732108i	$-0.738537\pi$
$653$	−889.632			−1.36238			−0.681189	+	0.732108i	$0.738537\pi$
$654$		0			0
$655$	263.948			0.402975
$656$		0			0
$657$	530.605			0.807618
$658$		0			0
$659$			611.017i			0.927188i	0.886048	+	0.463594i	$0.153440\pi$
$659$			611.017i			0.927188i	−0.886048	+	0.463594i	$0.846560\pi$
$660$		0			0
$661$			114.885i			0.173805i	0.996217	+	0.0869025i	$0.0276969\pi$
$661$			114.885i			0.173805i	−0.996217	+	0.0869025i	$0.972303\pi$
$662$		0			0
$663$	−617.596			−0.931517
$664$		0			0
$665$	27.1742	−	865.912i	0.0408635	−	1.30212i
$666$		0			0
$667$		−	214.207i		−	0.321150i
$668$		0			0
$669$	183.889			0.274872
$670$		0			0
$671$	−1129.96			−1.68399
$672$		0			0
$673$			365.864i			0.543631i	0.962349	+	0.271815i	$0.0876240\pi$
$673$			365.864i			0.543631i	−0.962349	+	0.271815i	$0.912376\pi$
$674$		0			0
$675$			39.4361i			0.0584239i
$676$		0			0
$677$			42.7480i			0.0631433i	0.999501	+	0.0315716i	$0.0100512\pi$
$677$			42.7480i			0.0631433i	−0.999501	+	0.0315716i	$0.989949\pi$
$678$		0			0
$679$			978.427i			1.44098i
$680$		0			0
$681$	460.982			0.676919
$682$		0			0
$683$			862.409i			1.26268i	0.775507	+	0.631339i	$0.217494\pi$
$683$			862.409i			1.26268i	−0.775507	+	0.631339i	$0.782506\pi$
$684$		0			0
$685$	−1798.27			−2.62521
$686$		0			0
$687$			740.808i			1.07832i
$688$		0			0
$689$	499.976			0.725655
$690$		0			0
$691$	−84.3258			−0.122034			−0.0610172	−	0.998137i	$-0.519434\pi$
$691$	−84.3258			−0.122034			−0.0610172	+	0.998137i	$0.519434\pi$
$692$		0			0
$693$	−881.822			−1.27247
$694$		0			0
$695$	1894.90			2.72647
$696$		0			0
$697$			740.562i			1.06250i
$698$		0			0
$699$			29.7223i			0.0425212i
$700$		0			0
$701$	87.4458			0.124744			0.0623722	−	0.998053i	$-0.480133\pi$
$701$	87.4458			0.124744			0.0623722	+	0.998053i	$0.480133\pi$
$702$		0			0
$703$	−1114.06	−	34.9616i	−1.58472	−	0.0497320i
$704$		0			0
$705$			3007.47i			4.26591i
$706$		0			0
$707$	643.204			0.909765
$708$		0			0
$709$	588.263			0.829709			0.414854	−	0.909888i	$-0.363833\pi$
$709$	588.263			0.829709			0.414854	+	0.909888i	$0.363833\pi$
$710$		0			0
$711$			816.389i			1.14823i
$712$		0			0
$713$		−	245.976i		−	0.344987i
$714$		0			0
$715$		−	1044.58i		−	1.46095i
$716$		0			0
$717$			238.591i			0.332763i
$718$		0			0
$719$	−356.156			−0.495349			−0.247674	−	0.968843i	$-0.579666\pi$
$719$	−356.156			−0.495349			−0.247674	+	0.968843i	$0.579666\pi$
$720$		0			0
$721$		−	136.857i		−	0.189815i
$722$		0			0
$723$	−670.018			−0.926719
$724$		0			0
$725$			1149.94i			1.58612i
$726$		0			0
$727$	−177.681			−0.244403			−0.122202	−	0.992505i	$-0.538995\pi$
$727$	−177.681			−0.244403			−0.122202	+	0.992505i	$0.538995\pi$
$728$		0			0
$729$	768.087			1.05362
$730$		0			0
$731$	−228.406			−0.312457
$732$		0			0
$733$	942.078			1.28524			0.642618	−	0.766187i	$-0.277848\pi$
$733$	942.078			1.28524			0.642618	+	0.766187i	$0.277848\pi$
$734$		0			0
$735$			536.481i			0.729906i
$736$		0			0
$737$		−	1367.46i		−	1.85545i
$738$		0			0
$739$	578.070			0.782233			0.391116	−	0.920341i	$-0.372089\pi$
$739$	578.070			0.782233			0.391116	+	0.920341i	$0.372089\pi$
$740$		0			0
$741$	−20.2624	+	645.666i	−0.0273447	+	0.871345i
$742$		0			0
$743$			124.430i			0.167470i	0.996488	+	0.0837349i	$0.0266849\pi$
$743$			124.430i			0.167470i	−0.996488	+	0.0837349i	$0.973315\pi$
$744$		0			0
$745$	−1126.38			−1.51193
$746$		0			0
$747$	6.66043			0.00891624
$748$		0			0
$749$			405.173i			0.540952i
$750$		0			0
$751$			774.593i			1.03141i	0.856765	+	0.515707i	$0.172471\pi$
$751$			774.593i			1.03141i	−0.856765	+	0.515707i	$0.827529\pi$
$752$		0			0
$753$			423.958i			0.563025i
$754$		0			0
$755$			700.569i			0.927905i
$756$		0			0
$757$	−551.096			−0.727999			−0.364000	−	0.931399i	$-0.618589\pi$
$757$	−551.096			−0.727999			−0.364000	+	0.931399i	$0.618589\pi$
$758$		0			0
$759$		−	497.429i		−	0.655374i
$760$		0			0
$761$	−453.591			−0.596046			−0.298023	−	0.954559i	$-0.596327\pi$
$761$	−453.591			−0.596046			−0.298023	+	0.954559i	$0.596327\pi$
$762$		0			0
$763$			120.899i			0.158452i
$764$		0			0
$765$	1330.06			1.73864
$766$		0			0
$767$	408.147			0.532134
$768$		0			0
$769$	1000.00			1.30039			0.650197	−	0.759766i	$-0.274686\pi$
$769$	1000.00			1.30039			0.650197	+	0.759766i	$0.274686\pi$
$770$		0			0
$771$	−119.001			−0.154347
$772$		0			0
$773$		−	757.916i		−	0.980486i	−0.871586	−	0.490243i	$-0.836908\pi$
$773$		−	757.916i		−	0.980486i	0.871586	−	0.490243i	$-0.163092\pi$
$774$		0			0
$775$			1320.48i			1.70385i
$776$		0			0
$777$	−1442.54			−1.85655
$778$		0			0
$779$	774.221	+	24.2968i	0.993866	+	0.0311897i
$780$		0			0
$781$		−	363.215i		−	0.465063i
$782$		0			0
$783$	−31.8536			−0.0406814
$784$		0			0
$785$	−363.040			−0.462471
$786$		0			0
$787$			384.175i			0.488151i	0.969756	+	0.244075i	$0.0784845\pi$
$787$			384.175i			0.488151i	−0.969756	+	0.244075i	$0.921516\pi$
$788$		0			0
$789$			1616.19i			2.04841i
$790$		0			0
$791$		−	1288.89i		−	1.62944i
$792$		0			0
$793$			542.832i			0.684530i
$794$		0			0
$795$	−2125.01			−2.67297
$796$		0			0
$797$		−	566.234i		−	0.710457i	−0.934780	−	0.355228i	$-0.884403\pi$
$797$		−	566.234i		−	0.710457i	0.934780	−	0.355228i	$-0.115597\pi$
$798$		0			0
$799$	1614.82			2.02106
$800$		0			0
$801$		−	278.612i		−	0.347831i
$802$		0			0
$803$	950.991			1.18430
$804$		0			0
$805$	320.478			0.398110
$806$		0			0
$807$	−567.550			−0.703283
$808$		0			0
$809$	−357.919			−0.442422			−0.221211	−	0.975226i	$-0.571001\pi$
$809$	−357.919			−0.442422			−0.221211	+	0.975226i	$0.571001\pi$
$810$		0			0
$811$		−	903.689i		−	1.11429i	−0.830415	−	0.557145i	$-0.811897\pi$
$811$		−	903.689i		−	1.11429i	0.830415	−	0.557145i	$-0.188103\pi$
$812$		0			0
$813$		−	795.395i		−	0.978346i
$814$		0			0
$815$	−396.323			−0.486286
$816$		0			0
$817$	−7.49367	+	238.787i	−0.00917218	+	0.292273i
$818$		0			0
$819$			423.627i			0.517249i
$820$		0			0
$821$	853.605			1.03971			0.519857	−	0.854253i	$-0.325985\pi$
$821$	853.605			1.03971			0.519857	+	0.854253i	$0.325985\pi$
$822$		0			0
$823$	−173.599			−0.210934			−0.105467	−	0.994423i	$-0.533634\pi$
$823$	−173.599			−0.210934			−0.105467	+	0.994423i	$0.533634\pi$
$824$		0			0
$825$			2670.37i			3.23681i
$826$		0			0
$827$		−	1114.69i		−	1.34787i	−0.738789	−	0.673937i	$-0.764602\pi$
$827$		−	1114.69i		−	1.34787i	0.738789	−	0.673937i	$-0.235398\pi$
$828$		0			0
$829$			1382.86i			1.66810i	0.551686	+	0.834052i	$0.313985\pi$
$829$			1382.86i			1.66810i	−0.551686	+	0.834052i	$0.686015\pi$
$830$		0			0
$831$			191.818i			0.230828i
$832$		0			0
$833$	288.057			0.345807
$834$		0			0
$835$			1170.14i			1.40136i
$836$		0			0
$837$	−36.5777			−0.0437009
$838$		0			0
$839$			1132.53i			1.34986i	0.737882	+	0.674929i	$0.235826\pi$
$839$			1132.53i			1.34986i	−0.737882	+	0.674929i	$0.764174\pi$
$840$		0			0
$841$	−87.8353			−0.104441
$842$		0			0
$843$	−1473.44			−1.74786
$844$		0			0
$845$	836.721			0.990202
$846$		0			0
$847$	−883.881			−1.04354
$848$		0			0
$849$			777.331i			0.915584i
$850$		0			0
$851$		−	412.318i		−	0.484510i
$852$		0			0
$853$	−403.013			−0.472465			−0.236232	−	0.971697i	$-0.575913\pi$
$853$	−403.013			−0.472465			−0.236232	+	0.971697i	$0.575913\pi$
$854$		0			0
$855$	43.6374	−	1390.51i	0.0510379	−	1.62633i
$856$		0			0
$857$		−	873.926i		−	1.01975i	−0.860248	−	0.509875i	$-0.829692\pi$
$857$		−	873.926i		−	1.01975i	0.860248	−	0.509875i	$-0.170308\pi$
$858$		0			0
$859$	1159.36			1.34966			0.674828	−	0.737975i	$-0.264218\pi$
$859$	1159.36			1.34966			0.674828	+	0.737975i	$0.264218\pi$
$860$		0			0
$861$	1002.50			1.16434
$862$		0			0
$863$		−	36.7815i		−	0.0426206i	−0.999773	−	0.0213103i	$-0.993216\pi$
$863$		−	36.7815i		−	0.0426206i	0.999773	−	0.0213103i	$-0.00678379\pi$
$864$		0			0
$865$			1752.55i			2.02607i
$866$		0			0
$867$		−	174.990i		−	0.201834i
$868$		0			0
$869$			1463.20i			1.68377i
$870$		0			0
$871$	−656.929			−0.754224
$872$		0			0
$873$			1571.19i			1.79977i
$874$		0			0
$875$	−580.521			−0.663453
$876$		0			0
$877$			247.696i			0.282435i	0.989979	+	0.141218i	$0.0451017\pi$
$877$			247.696i			0.282435i	−0.989979	+	0.141218i	$0.954898\pi$
$878$		0			0
$879$	−48.8081			−0.0555269
$880$		0			0
$881$	−882.282			−1.00146			−0.500728	−	0.865605i	$-0.666934\pi$
$881$	−882.282			−1.00146			−0.500728	+	0.865605i	$0.666934\pi$
$882$		0			0
$883$	−241.560			−0.273567			−0.136784	−	0.990601i	$-0.543677\pi$
$883$	−241.560			−0.273567			−0.136784	+	0.990601i	$0.543677\pi$
$884$		0			0
$885$	−1734.71			−1.96013
$886$		0			0
$887$		−	1060.17i		−	1.19523i	−0.801782	−	0.597617i	$-0.796114\pi$
$887$		−	1060.17i		−	1.19523i	0.801782	−	0.597617i	$-0.203886\pi$
$888$		0			0
$889$			214.484i			0.241265i
$890$		0			0
$891$	1304.61			1.46421
$892$		0			0
$893$	52.9801	−	1688.22i	0.0593282	−	1.89050i
$894$		0			0
$895$		−	945.974i		−	1.05695i
$896$		0			0
$897$	−238.964			−0.266404
$898$		0			0
$899$	−1066.59			−1.18642
$900$		0			0
$901$			1141.00i			1.26637i
$902$		0			0
$903$			309.194i			0.342407i
$904$		0			0
$905$			24.9330i			0.0275503i
$906$		0			0
$907$		−	590.873i		−	0.651459i	−0.945463	−	0.325729i	$-0.894390\pi$
$907$		−	590.873i		−	0.651459i	0.945463	−	0.325729i	$-0.105610\pi$
$908$		0			0
$909$	1032.88			1.13628
$910$		0			0
$911$			442.176i			0.485374i	0.970105	+	0.242687i	$0.0780288\pi$
$911$			442.176i			0.485374i	−0.970105	+	0.242687i	$0.921971\pi$
$912$		0			0
$913$	11.9373			0.0130749
$914$		0			0
$915$		−	2307.16i		−	2.52148i
$916$		0			0
$917$	−191.852			−0.209217
$918$		0			0
$919$	951.533			1.03540			0.517700	−	0.855562i	$-0.326788\pi$
$919$	951.533			1.03540			0.517700	+	0.855562i	$0.326788\pi$
$920$		0			0
$921$	284.971			0.309415
$922$		0			0
$923$	−174.488			−0.189044
$924$		0			0
$925$			2213.47i			2.39294i
$926$		0			0
$927$		−	219.770i		−	0.237076i
$928$		0			0
$929$	518.679			0.558320			0.279160	−	0.960245i	$-0.409944\pi$
$929$	518.679			0.558320			0.279160	+	0.960245i	$0.409944\pi$
$930$		0			0
$931$	9.45074	−	301.150i	0.0101512	−	0.323469i
$932$		0			0
$933$			1951.28i			2.09141i
$934$		0			0
$935$	2383.84			2.54956
$936$		0			0
$937$	1238.36			1.32162			0.660809	−	0.750554i	$-0.270213\pi$
$937$	1238.36			1.32162			0.660809	+	0.750554i	$0.270213\pi$
$938$		0			0
$939$			2443.16i			2.60187i
$940$		0			0
$941$			492.751i			0.523646i	0.965116	+	0.261823i	$0.0843236\pi$
$941$			492.751i			0.523646i	−0.965116	+	0.261823i	$0.915676\pi$
$942$		0			0
$943$			286.543i			0.303863i
$944$		0			0
$945$		−	47.6565i		−	0.0504302i
$946$		0			0
$947$	−270.266			−0.285392			−0.142696	−	0.989767i	$-0.545577\pi$
$947$	−270.266			−0.285392			−0.142696	+	0.989767i	$0.545577\pi$
$948$		0			0
$949$		−	456.856i		−	0.481407i
$950$		0			0
$951$	−2409.15			−2.53328
$952$		0			0
$953$		−	1576.70i		−	1.65446i	−0.561860	−	0.827232i	$-0.689914\pi$
$953$		−	1576.70i		−	1.65446i	0.561860	−	0.827232i	$-0.310086\pi$
$954$		0			0
$955$	1689.37			1.76897
$956$		0			0
$957$	−2156.93			−2.25384
$958$		0			0
$959$	1307.08			1.36296
$960$		0			0
$961$	−263.772			−0.274477
$962$		0			0
$963$			650.643i			0.675641i
$964$		0			0
$965$		−	827.215i		−	0.857218i
$966$		0			0
$967$	533.968			0.552190			0.276095	−	0.961130i	$-0.410960\pi$
$967$	533.968			0.552190			0.276095	+	0.961130i	$0.410960\pi$
$968$		0			0
$969$	−1473.48	−	46.2410i	−1.52062	−	0.0477203i
$970$		0			0
$971$			431.827i			0.444724i	0.974964	+	0.222362i	$0.0713767\pi$
$971$			431.827i			0.444724i	−0.974964	+	0.222362i	$0.928623\pi$
$972$		0			0
$973$	−1377.32			−1.41553
$974$		0			0
$975$	1282.84			1.31574
$976$		0			0
$977$			666.524i			0.682215i	0.940024	+	0.341107i	$0.110802\pi$
$977$			666.524i			0.682215i	−0.940024	+	0.341107i	$0.889198\pi$
$978$		0			0
$979$		−	499.351i		−	0.510062i
$980$		0			0
$981$			194.143i			0.197904i
$982$		0			0
$983$			16.4228i			0.0167068i	0.999965	+	0.00835338i	$0.00265900\pi$
$983$			16.4228i			0.0167068i	−0.999965	+	0.00835338i	$0.997341\pi$
$984$		0			0
$985$	2124.48			2.15683
$986$		0			0
$987$		−	2185.99i		−	2.21478i
$988$		0			0
$989$	−88.3764			−0.0893593
$990$		0			0
$991$		−	505.687i		−	0.510280i	−0.966904	−	0.255140i	$-0.917879\pi$
$991$		−	505.687i		−	0.510280i	0.966904	−	0.255140i	$-0.0821215\pi$
$992$		0			0
$993$	−432.923			−0.435974
$994$		0			0
$995$	26.4195			0.0265522
$996$		0			0
$997$	−682.491			−0.684545			−0.342272	−	0.939601i	$-0.611197\pi$
$997$	−682.491			−0.684545			−0.342272	+	0.939601i	$0.611197\pi$
$998$		0			0
$999$	−61.3135			−0.0613749

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.n.1025.1		8
4.3	odd	2		1216.3.e.m.1025.8		8
8.3	odd	2		152.3.e.b.113.1	✓	8
8.5	even	2		304.3.e.g.113.8		8
19.18	odd	2	inner	1216.3.e.n.1025.8		8
24.5	odd	2		2736.3.o.p.721.7		8
24.11	even	2		1368.3.o.b.721.7		8
76.75	even	2		1216.3.e.m.1025.1		8
152.37	odd	2		304.3.e.g.113.1		8
152.75	even	2		152.3.e.b.113.8	yes	8
456.227	odd	2		1368.3.o.b.721.8		8
456.341	even	2		2736.3.o.p.721.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.b.113.1	✓	8	8.3	odd	2
152.3.e.b.113.8	yes	8	152.75	even	2
304.3.e.g.113.1		8	152.37	odd	2
304.3.e.g.113.8		8	8.5	even	2
1216.3.e.m.1025.1		8	76.75	even	2
1216.3.e.m.1025.8		8	4.3	odd	2
1216.3.e.n.1025.1		8	1.1	even	1	trivial
1216.3.e.n.1025.8		8	19.18	odd	2	inner
1368.3.o.b.721.7		8	24.11	even	2
1368.3.o.b.721.8		8	456.227	odd	2
2736.3.o.p.721.7		8	24.5	odd	2
2736.3.o.p.721.8		8	456.341	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.27138i q^{3} +7.92033 q^{5} -5.75693 q^{7} -9.24469 q^{9} +O(q^{10})\)	\(q-4.27138i q^{3} +7.92033 q^{5} -5.75693 q^{7} -9.24469 q^{9} -16.5691 q^{11} +7.95976i q^{13} -33.8307i q^{15} -18.1650 q^{17} +(-0.595968 + 18.9907i) q^{19} +24.5900i q^{21} -7.02853 q^{23} +37.7316 q^{25} +1.04517i q^{27} +30.4768i q^{29} +34.9967i q^{31} +70.7727i q^{33} -45.5968 q^{35} +58.6635i q^{37} +33.9992 q^{39} -40.7686i q^{41} +12.5739 q^{43} -73.2210 q^{45} -88.8974 q^{47} -15.8578 q^{49} +77.5897i q^{51} -62.8130i q^{53} -131.232 q^{55} +(81.1163 + 2.54561i) q^{57} -51.2763i q^{59} +68.1970 q^{61} +53.2210 q^{63} +63.0439i q^{65} +82.5313i q^{67} +30.0215i q^{69} +21.9213i q^{71} -57.3956 q^{73} -161.166i q^{75} +95.3868 q^{77} -88.3089i q^{79} -78.7379 q^{81} -0.720460 q^{83} -143.873 q^{85} +130.178 q^{87} +30.1376i q^{89} -45.8238i q^{91} +149.484 q^{93} +(-4.72027 + 150.412i) q^{95} -169.956i q^{97} +153.176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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