LMFDB - Embedded newform 4056.2.c.p.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.649638144.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 $ x^8 - 14x^6 + 75x^4 - 170*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$1.42055 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.p.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.28657i q^{5} -1.96046i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.28657i q^{5} -1.96046i q^{7} +1.00000 q^{9} +5.51498i q^{11} +1.28657i q^{15} -1.82247 q^{17} -8.08812i q^{19} -1.96046i q^{21} +6.97908 q^{23} +3.34474 q^{25} +1.00000 q^{27} -2.22841 q^{29} -4.44548i q^{31} +5.51498i q^{33} +2.52227 q^{35} -2.80155i q^{37} -4.58347i q^{41} +5.39561 q^{43} +1.28657i q^{45} +5.05816i q^{47} +3.15661 q^{49} -1.82247 q^{51} -2.44649 q^{53} -7.09541 q^{55} -8.08812i q^{57} +10.0372i q^{59} +12.9073 q^{61} -1.96046i q^{63} +2.39460i q^{67} +6.97908 q^{69} +15.1181i q^{71} +2.62301i q^{73} +3.34474 q^{75} +10.8119 q^{77} +6.15661 q^{79} +1.00000 q^{81} -5.59406i q^{83} -2.34474i q^{85} -2.22841 q^{87} +15.0300i q^{89} -4.44548i q^{93} +10.4059 q^{95} +13.9468i q^{97} +5.51498i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.28657i	0.575372i	0.957725	+	0.287686i	$0.0928859\pi$
$5$	1.28657i	0.575372i	−0.957725	+	0.287686i	$0.907114\pi$
$6$	0	0
$7$	− 1.96046i	− 0.740983i	−0.928836	−	0.370492i	$-0.879189\pi$
$7$	− 1.96046i	− 0.740983i	0.928836	−	0.370492i	$-0.120811\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.51498i	1.66283i	0.555653	+	0.831414i	$0.312468\pi$
$11$	5.51498i	1.66283i	−0.555653	+	0.831414i	$0.687532\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.28657i	0.332191i
$16$	0	0
$17$	−1.82247	−0.442014	−0.221007	−	0.975272i	$-0.570934\pi$
$17$	−1.82247	−0.442014	−0.221007	+	0.975272i	$0.570934\pi$
$18$	0	0
$19$	− 8.08812i	− 1.85554i	−0.373151	−	0.927771i	$-0.621723\pi$
$19$	− 8.08812i	− 1.85554i	0.373151	−	0.927771i	$-0.378277\pi$
$20$	0	0
$21$	− 1.96046i	− 0.427807i
$22$	0	0
$23$	6.97908	1.45524	0.727619	−	0.685981i	$-0.240627\pi$
$23$	6.97908	1.45524	0.727619	+	0.685981i	$0.240627\pi$
$24$	0	0
$25$	3.34474	0.668947
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.22841	−0.413805	−0.206902	−	0.978362i	$-0.566338\pi$
$29$	−2.22841	−0.413805	−0.206902	+	0.978362i	$0.566338\pi$
$30$	0	0
$31$	− 4.44548i	− 0.798432i	−0.916857	−	0.399216i	$-0.869282\pi$
$31$	− 4.44548i	− 0.798432i	0.916857	−	0.399216i	$-0.130718\pi$
$32$	0	0
$33$	5.51498i	0.960034i
$34$	0	0
$35$	2.52227	0.426341
$36$	0	0
$37$	− 2.80155i	− 0.460572i	−0.973123	−	0.230286i	$-0.926034\pi$
$37$	− 2.80155i	− 0.460572i	0.973123	−	0.230286i	$-0.0739661\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.58347i	− 0.715817i	−0.933757	−	0.357909i	$-0.883490\pi$
$41$	− 4.58347i	− 0.715817i	0.933757	−	0.357909i	$-0.116510\pi$
$42$	0	0
$43$	5.39561	0.822823	0.411411	−	0.911450i	$-0.365036\pi$
$43$	5.39561	0.822823	0.411411	+	0.911450i	$0.365036\pi$
$44$	0	0
$45$	1.28657i	0.191791i
$46$	0	0
$47$	5.05816i	0.737809i	0.929467	+	0.368905i	$0.120267\pi$
$47$	5.05816i	0.737809i	−0.929467	+	0.368905i	$0.879733\pi$
$48$	0	0
$49$	3.15661	0.450944
$50$	0	0
$51$	−1.82247	−0.255197
$52$	0	0
$53$	−2.44649	−0.336051	−0.168025	−	0.985783i	$-0.553739\pi$
$53$	−2.44649	−0.336051	−0.168025	+	0.985783i	$0.553739\pi$
$54$	0	0
$55$	−7.09541	−0.956745
$56$	0	0
$57$	− 8.08812i	− 1.07130i
$58$	0	0
$59$	10.0372i	1.30674i	0.757040	+	0.653369i	$0.226645\pi$
$59$	10.0372i	1.30674i	−0.757040	+	0.653369i	$0.773355\pi$
$60$	0	0
$61$	12.9073	1.65261	0.826304	−	0.563224i	$-0.190439\pi$
$61$	12.9073	1.65261	0.826304	+	0.563224i	$0.190439\pi$
$62$	0	0
$63$	− 1.96046i	− 0.246994i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.39460i	0.292548i	0.989244	+	0.146274i	$0.0467280\pi$
$67$	2.39460i	0.292548i	−0.989244	+	0.146274i	$0.953272\pi$
$68$	0	0
$69$	6.97908	0.840182
$70$	0	0
$71$	15.1181i	1.79419i	0.441843	+	0.897093i	$0.354325\pi$
$71$	15.1181i	1.79419i	−0.441843	+	0.897093i	$0.645675\pi$
$72$	0	0
$73$	2.62301i	0.307000i	0.988149	+	0.153500i	$0.0490545\pi$
$73$	2.62301i	0.307000i	−0.988149	+	0.153500i	$0.950945\pi$
$74$	0	0
$75$	3.34474	0.386217
$76$	0	0
$77$	10.8119	1.23213
$78$	0	0
$79$	6.15661	0.692673	0.346336	−	0.938110i	$-0.387426\pi$
$79$	6.15661	0.692673	0.346336	+	0.938110i	$0.387426\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.59406i	− 0.614028i	−0.951705	−	0.307014i	$-0.900670\pi$
$83$	− 5.59406i	− 0.614028i	0.951705	−	0.307014i	$-0.0993299\pi$
$84$	0	0
$85$	− 2.34474i	− 0.254322i
$86$	0	0
$87$	−2.22841	−0.238910
$88$	0	0
$89$	15.0300i	1.59317i	0.604525	+	0.796586i	$0.293363\pi$
$89$	15.0300i	1.59317i	−0.604525	+	0.796586i	$0.706637\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 4.44548i	− 0.460975i
$94$	0	0
$95$	10.4059	1.06763
$96$	0	0
$97$	13.9468i	1.41609i	0.706170	+	0.708043i	$0.250422\pi$
$97$	13.9468i	1.41609i	−0.706170	+	0.708043i	$0.749578\pi$
$98$	0	0
$99$	5.51498i	0.554276i
$100$	0	0
$101$	−5.25836	−0.523226	−0.261613	−	0.965173i	$-0.584254\pi$
$101$	−5.25836	−0.523226	−0.261613	+	0.965173i	$0.584254\pi$
$102$	0	0
$103$	−7.39561	−0.728711	−0.364356	−	0.931260i	$-0.618711\pi$
$103$	−7.39561	−0.728711	−0.364356	+	0.931260i	$0.618711\pi$
$104$	0	0
$105$	2.52227	0.246148
$106$	0	0
$107$	7.77030	0.751183	0.375592	−	0.926785i	$-0.377440\pi$
$107$	7.77030	0.751183	0.375592	+	0.926785i	$0.377440\pi$
$108$	0	0
$109$	− 12.4455i	− 1.19206i	−0.802962	−	0.596030i	$-0.796744\pi$
$109$	− 12.4455i	− 1.19206i	0.802962	−	0.596030i	$-0.203256\pi$
$110$	0	0
$111$	− 2.80155i	− 0.265911i
$112$	0	0
$113$	4.17753	0.392989	0.196495	−	0.980505i	$-0.437044\pi$
$113$	4.17753	0.392989	0.196495	+	0.980505i	$0.437044\pi$
$114$	0	0
$115$	8.97908i	0.837303i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.57287i	0.327525i
$120$	0	0
$121$	−19.4150	−1.76500
$122$	0	0
$123$	− 4.58347i	− 0.413277i
$124$	0	0
$125$	10.7361i	0.960265i
$126$	0	0
$127$	4.19845	0.372552	0.186276	−	0.982497i	$-0.440358\pi$
$127$	4.19845	0.372552	0.186276	+	0.982497i	$0.440358\pi$
$128$	0	0
$129$	5.39561	0.475057
$130$	0	0
$131$	11.5013	1.00488	0.502439	−	0.864613i	$-0.332436\pi$
$131$	11.5013	1.00488	0.502439	+	0.864613i	$0.332436\pi$
$132$	0	0
$133$	−15.8564	−1.37492
$134$	0	0
$135$	1.28657i	0.110730i
$136$	0	0
$137$	− 8.75970i	− 0.748392i	−0.927350	−	0.374196i	$-0.877919\pi$
$137$	− 8.75970i	− 0.748392i	0.927350	−	0.374196i	$-0.122081\pi$
$138$	0	0
$139$	14.9685	1.26961	0.634805	−	0.772672i	$-0.281080\pi$
$139$	14.9685	1.26961	0.634805	+	0.772672i	$0.281080\pi$
$140$	0	0
$141$	5.05816i	0.425974i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 2.86700i	− 0.238092i
$146$	0	0
$147$	3.15661	0.260353
$148$	0	0
$149$	− 7.78063i	− 0.637414i	−0.947853	−	0.318707i	$-0.896751\pi$
$149$	− 7.78063i	− 0.637414i	0.947853	−	0.318707i	$-0.103249\pi$
$150$	0	0
$151$	− 5.51498i	− 0.448802i	−0.974497	−	0.224401i	$-0.927957\pi$
$151$	− 5.51498i	− 0.448802i	0.974497	−	0.224401i	$-0.0720426\pi$
$152$	0	0
$153$	−1.82247	−0.147338
$154$	0	0
$155$	5.71942	0.459395
$156$	0	0
$157$	20.5104	1.63691	0.818453	−	0.574573i	$-0.194832\pi$
$157$	20.5104	1.63691	0.818453	+	0.574573i	$0.194832\pi$
$158$	0	0
$159$	−2.44649	−0.194019
$160$	0	0
$161$	− 13.6822i	− 1.07831i
$162$	0	0
$163$	18.3246i	1.43529i	0.696409	+	0.717645i	$0.254780\pi$
$163$	18.3246i	1.43529i	−0.696409	+	0.717645i	$0.745220\pi$
$164$	0	0
$165$	−7.09541	−0.552377
$166$	0	0
$167$	− 8.17624i	− 0.632696i	−0.948643	−	0.316348i	$-0.897543\pi$
$167$	− 8.17624i	− 0.632696i	0.948643	−	0.316348i	$-0.102457\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 8.08812i	− 0.618514i
$172$	0	0
$173$	−8.57314	−0.651804	−0.325902	−	0.945404i	$-0.605668\pi$
$173$	−8.57314	−0.651804	−0.325902	+	0.945404i	$0.605668\pi$
$174$	0	0
$175$	− 6.55721i	− 0.495678i
$176$	0	0
$177$	10.0372i	0.754445i
$178$	0	0
$179$	3.73104	0.278871	0.139436	−	0.990231i	$-0.455471\pi$
$179$	3.73104	0.278871	0.139436	+	0.990231i	$0.455471\pi$
$180$	0	0
$181$	5.24933	0.390179	0.195090	−	0.980785i	$-0.437500\pi$
$181$	5.24933	0.390179	0.195090	+	0.980785i	$0.437500\pi$
$182$	0	0
$183$	12.9073	0.954134
$184$	0	0
$185$	3.60439	0.265000
$186$	0	0
$187$	− 10.0509i	− 0.734993i
$188$	0	0
$189$	− 1.96046i	− 0.142602i
$190$	0	0
$191$	−5.54319	−0.401091	−0.200546	−	0.979684i	$-0.564271\pi$
$191$	−5.54319	−0.401091	−0.200546	+	0.979684i	$0.564271\pi$
$192$	0	0
$193$	− 10.1889i	− 0.733410i	−0.930337	−	0.366705i	$-0.880486\pi$
$193$	− 10.1889i	− 0.733410i	0.930337	−	0.366705i	$-0.119514\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 25.8564i	− 1.84219i	−0.389335	−	0.921096i	$-0.627295\pi$
$197$	− 25.8564i	− 1.84219i	0.389335	−	0.921096i	$-0.372705\pi$
$198$	0	0
$199$	−12.5419	−0.889071	−0.444536	−	0.895761i	$-0.646631\pi$
$199$	−12.5419	−0.889071	−0.444536	+	0.895761i	$0.646631\pi$
$200$	0	0
$201$	2.39460i	0.168902i
$202$	0	0
$203$	4.36869i	0.306622i
$204$	0	0
$205$	5.89696	0.411861
$206$	0	0
$207$	6.97908	0.485079
$208$	0	0
$209$	44.6058	3.08545
$210$	0	0
$211$	−12.4099	−0.854334	−0.427167	−	0.904173i	$-0.640488\pi$
$211$	−12.4099	−0.854334	−0.427167	+	0.904173i	$0.640488\pi$
$212$	0	0
$213$	15.1181i	1.03587i
$214$	0	0
$215$	6.94184i	0.473429i
$216$	0	0
$217$	−8.71517	−0.591624
$218$	0	0
$219$	2.62301i	0.177247i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.1835i	1.28462i	0.766443	+	0.642312i	$0.222025\pi$
$223$	19.1835i	1.28462i	−0.766443	+	0.642312i	$0.777975\pi$
$224$	0	0
$225$	3.34474	0.222982
$226$	0	0
$227$	− 10.5822i	− 0.702364i	−0.936307	−	0.351182i	$-0.885780\pi$
$227$	− 10.5822i	− 0.702364i	0.936307	−	0.351182i	$-0.114220\pi$
$228$	0	0
$229$	− 19.4595i	− 1.28592i	−0.765900	−	0.642960i	$-0.777706\pi$
$229$	− 19.4595i	− 1.28592i	0.765900	−	0.642960i	$-0.222294\pi$
$230$	0	0
$231$	10.8119	0.711369
$232$	0	0
$233$	−27.4243	−1.79662	−0.898312	−	0.439358i	$-0.855206\pi$
$233$	−27.4243	−1.79662	−0.898312	+	0.439358i	$0.855206\pi$
$234$	0	0
$235$	−6.50769	−0.424515
$236$	0	0
$237$	6.15661	0.399915
$238$	0	0
$239$	− 2.76100i	− 0.178594i	−0.996005	−	0.0892971i	$-0.971538\pi$
$239$	− 2.76100i	− 0.178594i	0.996005	−	0.0892971i	$-0.0284621\pi$
$240$	0	0
$241$	− 6.17753i	− 0.397930i	−0.980007	−	0.198965i	$-0.936242\pi$
$241$	− 6.17753i	− 0.397930i	0.980007	−	0.198965i	$-0.0637580\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.06120i	0.259461i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 5.59406i	− 0.354509i
$250$	0	0
$251$	14.1163	0.891015	0.445507	−	0.895278i	$-0.353023\pi$
$251$	14.1163	0.891015	0.445507	+	0.895278i	$0.353023\pi$
$252$	0	0
$253$	38.4895i	2.41981i
$254$	0	0
$255$	− 2.34474i	− 0.146833i
$256$	0	0
$257$	11.6225	0.724989	0.362494	−	0.931986i	$-0.381925\pi$
$257$	11.6225	0.724989	0.362494	+	0.931986i	$0.381925\pi$
$258$	0	0
$259$	−5.49231	−0.341276
$260$	0	0
$261$	−2.22841	−0.137935
$262$	0	0
$263$	−8.62402	−0.531780	−0.265890	−	0.964003i	$-0.585666\pi$
$263$	−8.62402	−0.531780	−0.265890	+	0.964003i	$0.585666\pi$
$264$	0	0
$265$	− 3.14758i	− 0.193354i
$266$	0	0
$267$	15.0300i	0.919818i
$268$	0	0
$269$	25.2480	1.53940	0.769700	−	0.638406i	$-0.220406\pi$
$269$	25.2480	1.53940	0.769700	+	0.638406i	$0.220406\pi$
$270$	0	0
$271$	2.99797i	0.182114i	0.995846	+	0.0910568i	$0.0290245\pi$
$271$	2.99797i	0.182114i	−0.995846	+	0.0910568i	$0.970976\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.4461i	1.11234i
$276$	0	0
$277$	24.9123	1.49684	0.748418	−	0.663227i	$-0.230814\pi$
$277$	24.9123	1.49684	0.748418	+	0.663227i	$0.230814\pi$
$278$	0	0
$279$	− 4.44548i	− 0.266144i
$280$	0	0
$281$	8.38198i	0.500027i	0.968242	+	0.250013i	$0.0804351\pi$
$281$	8.38198i	0.500027i	−0.968242	+	0.250013i	$0.919565\pi$
$282$	0	0
$283$	25.8106	1.53428	0.767140	−	0.641480i	$-0.221679\pi$
$283$	25.8106	1.53428	0.767140	+	0.641480i	$0.221679\pi$
$284$	0	0
$285$	10.4059	0.616394
$286$	0	0
$287$	−8.98569	−0.530409
$288$	0	0
$289$	−13.6786	−0.804624
$290$	0	0
$291$	13.9468i	0.817577i
$292$	0	0
$293$	12.7527i	0.745020i	0.928028	+	0.372510i	$0.121503\pi$
$293$	12.7527i	0.745020i	−0.928028	+	0.372510i	$0.878497\pi$
$294$	0	0
$295$	−12.9136	−0.751860
$296$	0	0
$297$	5.51498i	0.320011i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 10.5779i	− 0.609698i
$302$	0	0
$303$	−5.25836	−0.302085
$304$	0	0
$305$	16.6061i	0.950864i
$306$	0	0
$307$	16.6307i	0.949167i	0.880210	+	0.474583i	$0.157401\pi$
$307$	16.6307i	0.949167i	−0.880210	+	0.474583i	$0.842599\pi$
$308$	0	0
$309$	−7.39561	−0.420722
$310$	0	0
$311$	−19.3940	−1.09974	−0.549868	−	0.835252i	$-0.685322\pi$
$311$	−19.3940	−1.09974	−0.549868	+	0.835252i	$0.685322\pi$
$312$	0	0
$313$	−26.5091	−1.49838	−0.749191	−	0.662354i	$-0.769557\pi$
$313$	−26.5091	−1.49838	−0.749191	+	0.662354i	$0.769557\pi$
$314$	0	0
$315$	2.52227	0.142114
$316$	0	0
$317$	− 8.13569i	− 0.456946i	−0.973550	−	0.228473i	$-0.926627\pi$
$317$	− 8.13569i	− 0.456946i	0.973550	−	0.228473i	$-0.0733732\pi$
$318$	0	0
$319$	− 12.2896i	− 0.688086i
$320$	0	0
$321$	7.77030	0.433696
$322$	0	0
$323$	14.7403i	0.820175i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12.4455i	− 0.688237i
$328$	0	0
$329$	9.91631	0.546704
$330$	0	0
$331$	31.9861i	1.75811i	0.476717	+	0.879057i	$0.341827\pi$
$331$	31.9861i	1.75811i	−0.476717	+	0.879057i	$0.658173\pi$
$332$	0	0
$333$	− 2.80155i	− 0.153524i
$334$	0	0
$335$	−3.08083	−0.168324
$336$	0	0
$337$	8.14628	0.443756	0.221878	−	0.975074i	$-0.428781\pi$
$337$	8.14628	0.443756	0.221878	+	0.975074i	$0.428781\pi$
$338$	0	0
$339$	4.17753	0.226892
$340$	0	0
$341$	24.5167	1.32765
$342$	0	0
$343$	− 19.9116i	− 1.07512i
$344$	0	0
$345$	8.97908i	0.483417i
$346$	0	0
$347$	−31.7491	−1.70438	−0.852191	−	0.523231i	$-0.824726\pi$
$347$	−31.7491	−1.70438	−0.852191	+	0.523231i	$0.824726\pi$
$348$	0	0
$349$	11.3963i	0.610032i	0.952347	+	0.305016i	$0.0986619\pi$
$349$	11.3963i	0.610032i	−0.952347	+	0.305016i	$0.901338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 22.9551i	− 1.22178i	−0.791716	−	0.610889i	$-0.790812\pi$
$353$	− 22.9551i	− 1.22178i	0.791716	−	0.610889i	$-0.209188\pi$
$354$	0	0
$355$	−19.4505	−1.03232
$356$	0	0
$357$	3.57287i	0.189096i
$358$	0	0
$359$	8.77558i	0.463157i	0.972816	+	0.231579i	$0.0743891\pi$
$359$	8.77558i	0.463157i	−0.972816	+	0.231579i	$0.925611\pi$
$360$	0	0
$361$	−46.4177	−2.44303
$362$	0	0
$363$	−19.4150	−1.01902
$364$	0	0
$365$	−3.37469	−0.176639
$366$	0	0
$367$	−33.6670	−1.75740	−0.878701	−	0.477372i	$-0.841589\pi$
$367$	−33.6670	−1.75740	−0.878701	+	0.477372i	$0.841589\pi$
$368$	0	0
$369$	− 4.58347i	− 0.238606i
$370$	0	0
$371$	4.79623i	0.249008i
$372$	0	0
$373$	6.49231	0.336159	0.168080	−	0.985773i	$-0.446243\pi$
$373$	6.49231	0.336159	0.168080	+	0.985773i	$0.446243\pi$
$374$	0	0
$375$	10.7361i	0.554409i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.5572i	1.56962i	0.619737	+	0.784809i	$0.287239\pi$
$379$	30.5572i	1.56962i	−0.619737	+	0.784809i	$0.712761\pi$
$380$	0	0
$381$	4.19845	0.215093
$382$	0	0
$383$	− 7.39163i	− 0.377694i	−0.982006	−	0.188847i	$-0.939525\pi$
$383$	− 7.39163i	− 0.377694i	0.982006	−	0.188847i	$-0.0604751\pi$
$384$	0	0
$385$	13.9102i	0.708932i
$386$	0	0
$387$	5.39561	0.274274
$388$	0	0
$389$	24.8408	1.25948	0.629739	−	0.776807i	$-0.283162\pi$
$389$	24.8408	1.25948	0.629739	+	0.776807i	$0.283162\pi$
$390$	0	0
$391$	−12.7192	−0.643235
$392$	0	0
$393$	11.5013	0.580166
$394$	0	0
$395$	7.92091i	0.398544i
$396$	0	0
$397$	− 27.0347i	− 1.35683i	−0.734678	−	0.678416i	$-0.762667\pi$
$397$	− 27.0347i	− 1.35683i	0.734678	−	0.678416i	$-0.237333\pi$
$398$	0	0
$399$	−15.8564	−0.793813
$400$	0	0
$401$	− 24.7804i	− 1.23747i	−0.785599	−	0.618736i	$-0.787645\pi$
$401$	− 24.7804i	− 1.23747i	0.785599	−	0.618736i	$-0.212355\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.28657i	0.0639302i
$406$	0	0
$407$	15.4505	0.765851
$408$	0	0
$409$	− 15.6338i	− 0.773041i	−0.922281	−	0.386520i	$-0.873677\pi$
$409$	− 15.6338i	− 0.773041i	0.922281	−	0.386520i	$-0.126323\pi$
$410$	0	0
$411$	− 8.75970i	− 0.432084i
$412$	0	0
$413$	19.6776	0.968270
$414$	0	0
$415$	7.19716	0.353295
$416$	0	0
$417$	14.9685	0.733010
$418$	0	0
$419$	−35.3079	−1.72491	−0.862453	−	0.506137i	$-0.831073\pi$
$419$	−35.3079	−1.72491	−0.862453	+	0.506137i	$0.831073\pi$
$420$	0	0
$421$	− 36.6693i	− 1.78715i	−0.448912	−	0.893576i	$-0.648188\pi$
$421$	− 36.6693i	− 1.78715i	0.448912	−	0.893576i	$-0.351812\pi$
$422$	0	0
$423$	5.05816i	0.245936i
$424$	0	0
$425$	−6.09568	−0.295684
$426$	0	0
$427$	− 25.3042i	− 1.22455i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.22442i	− 0.155315i	−0.996980	−	0.0776575i	$-0.975256\pi$
$431$	− 3.22442i	− 0.155315i	0.996980	−	0.0776575i	$-0.0247441\pi$
$432$	0	0
$433$	−6.79650	−0.326619	−0.163309	−	0.986575i	$-0.552217\pi$
$433$	−6.79650	−0.326619	−0.163309	+	0.986575i	$0.552217\pi$
$434$	0	0
$435$	− 2.86700i	− 0.137462i
$436$	0	0
$437$	− 56.4476i	− 2.70026i
$438$	0	0
$439$	8.83705	0.421769	0.210885	−	0.977511i	$-0.432366\pi$
$439$	8.83705	0.421769	0.210885	+	0.977511i	$0.432366\pi$
$440$	0	0
$441$	3.15661	0.150315
$442$	0	0
$443$	9.08986	0.431872	0.215936	−	0.976407i	$-0.430720\pi$
$443$	9.08986	0.431872	0.215936	+	0.976407i	$0.430720\pi$
$444$	0	0
$445$	−19.3371	−0.916666
$446$	0	0
$447$	− 7.78063i	− 0.368011i
$448$	0	0
$449$	3.92091i	0.185039i	0.995711	+	0.0925197i	$0.0294921\pi$
$449$	3.92091i	0.185039i	−0.995711	+	0.0925197i	$0.970508\pi$
$450$	0	0
$451$	25.2777	1.19028
$452$	0	0
$453$	− 5.51498i	− 0.259116i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.0944i	1.17387i	0.809635	+	0.586933i	$0.199665\pi$
$457$	25.0944i	1.17387i	−0.809635	+	0.586933i	$0.800335\pi$
$458$	0	0
$459$	−1.82247	−0.0850656
$460$	0	0
$461$	− 23.1863i	− 1.07989i	−0.841699	−	0.539947i	$-0.818444\pi$
$461$	− 23.1863i	− 1.07989i	0.841699	−	0.539947i	$-0.181556\pi$
$462$	0	0
$463$	− 8.88866i	− 0.413091i	−0.978437	−	0.206546i	$-0.933778\pi$
$463$	− 8.88866i	− 0.413091i	0.978437	−	0.206546i	$-0.0662222\pi$
$464$	0	0
$465$	5.71942	0.265232
$466$	0	0
$467$	23.1760	1.07246	0.536228	−	0.844073i	$-0.319849\pi$
$467$	23.1760	1.07246	0.536228	+	0.844073i	$0.319849\pi$
$468$	0	0
$469$	4.69452	0.216773
$470$	0	0
$471$	20.5104	0.945068
$472$	0	0
$473$	29.7567i	1.36821i
$474$	0	0
$475$	− 27.0526i	− 1.24126i
$476$	0	0
$477$	−2.44649	−0.112017
$478$	0	0
$479$	25.5965i	1.16953i	0.811202	+	0.584767i	$0.198814\pi$
$479$	25.5965i	1.16953i	−0.811202	+	0.584767i	$0.801186\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 13.6822i	− 0.622561i
$484$	0	0
$485$	−17.9436	−0.814776
$486$	0	0
$487$	− 23.1115i	− 1.04728i	−0.851940	−	0.523640i	$-0.824574\pi$
$487$	− 23.1115i	− 1.04728i	0.851940	−	0.523640i	$-0.175426\pi$
$488$	0	0
$489$	18.3246i	0.828665i
$490$	0	0
$491$	4.84210	0.218521	0.109260	−	0.994013i	$-0.465152\pi$
$491$	4.84210	0.218521	0.109260	+	0.994013i	$0.465152\pi$
$492$	0	0
$493$	4.06120	0.182907
$494$	0	0
$495$	−7.09541	−0.318915
$496$	0	0
$497$	29.6383	1.32946
$498$	0	0
$499$	40.8818i	1.83012i	0.403319	+	0.915059i	$0.367857\pi$
$499$	40.8818i	1.83012i	−0.403319	+	0.915059i	$0.632143\pi$
$500$	0	0
$501$	− 8.17624i	− 0.365287i
$502$	0	0
$503$	0.937235	0.0417892	0.0208946	−	0.999782i	$-0.493349\pi$
$503$	0.937235	0.0417892	0.0208946	+	0.999782i	$0.493349\pi$
$504$	0	0
$505$	− 6.76525i	− 0.301050i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 21.9694i	− 0.973778i	−0.873464	−	0.486889i	$-0.838132\pi$
$509$	− 21.9694i	− 0.973778i	0.873464	−	0.486889i	$-0.161868\pi$
$510$	0	0
$511$	5.14230	0.227482
$512$	0	0
$513$	− 8.08812i	− 0.357099i
$514$	0	0
$515$	− 9.51498i	− 0.419280i
$516$	0	0
$517$	−27.8957	−1.22685
$518$	0	0
$519$	−8.57314	−0.376319
$520$	0	0
$521$	22.1775	0.971615	0.485808	−	0.874066i	$-0.338526\pi$
$521$	22.1775	0.971615	0.485808	+	0.874066i	$0.338526\pi$
$522$	0	0
$523$	−14.3042	−0.625478	−0.312739	−	0.949839i	$-0.601247\pi$
$523$	−14.3042	−0.625478	−0.312739	+	0.949839i	$0.601247\pi$
$524$	0	0
$525$	− 6.55721i	− 0.286180i
$526$	0	0
$527$	8.10175i	0.352918i
$528$	0	0
$529$	25.7075	1.11772
$530$	0	0
$531$	10.0372i	0.435579i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.99704i	0.432210i
$536$	0	0
$537$	3.73104	0.161006
$538$	0	0
$539$	17.4086i	0.749843i
$540$	0	0
$541$	− 15.3779i	− 0.661149i	−0.943780	−	0.330575i	$-0.892757\pi$
$541$	− 15.3779i	− 0.661149i	0.943780	−	0.330575i	$-0.107243\pi$
$542$	0	0
$543$	5.24933	0.225270
$544$	0	0
$545$	16.0120	0.685878
$546$	0	0
$547$	−11.1503	−0.476751	−0.238375	−	0.971173i	$-0.576615\pi$
$547$	−11.1503	−0.476751	−0.238375	+	0.971173i	$0.576615\pi$
$548$	0	0
$549$	12.9073	0.550869
$550$	0	0
$551$	18.0236i	0.767832i
$552$	0	0
$553$	− 12.0698i	− 0.513259i
$554$	0	0
$555$	3.60439	0.152998
$556$	0	0
$557$	− 13.3092i	− 0.563930i	−0.959425	−	0.281965i	$-0.909014\pi$
$557$	− 13.3092i	− 0.563930i	0.959425	−	0.281965i	$-0.0909863\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 10.0509i	− 0.424348i
$562$	0	0
$563$	7.76127	0.327098	0.163549	−	0.986535i	$-0.447706\pi$
$563$	7.76127	0.327098	0.163549	+	0.986535i	$0.447706\pi$
$564$	0	0
$565$	5.37469i	0.226115i
$566$	0	0
$567$	− 1.96046i	− 0.0823314i
$568$	0	0
$569$	−14.2986	−0.599430	−0.299715	−	0.954029i	$-0.596892\pi$
$569$	−14.2986	−0.599430	−0.299715	+	0.954029i	$0.596892\pi$
$570$	0	0
$571$	−29.6121	−1.23923	−0.619614	−	0.784906i	$-0.712711\pi$
$571$	−29.6121	−1.23923	−0.619614	+	0.784906i	$0.712711\pi$
$572$	0	0
$573$	−5.54319	−0.231570
$574$	0	0
$575$	23.3432	0.973478
$576$	0	0
$577$	1.23475i	0.0514032i	0.999670	+	0.0257016i	$0.00818198\pi$
$577$	1.23475i	0.0514032i	−0.999670	+	0.0257016i	$0.991818\pi$
$578$	0	0
$579$	− 10.1889i	− 0.423435i
$580$	0	0
$581$	−10.9669	−0.454984
$582$	0	0
$583$	− 13.4923i	− 0.558795i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.9096i	− 0.450286i	−0.974326	−	0.225143i	$-0.927715\pi$
$587$	− 10.9096i	− 0.450286i	0.974326	−	0.225143i	$-0.0722849\pi$
$588$	0	0
$589$	−35.9556	−1.48152
$590$	0	0
$591$	− 25.8564i	− 1.06359i
$592$	0	0
$593$	17.2775i	0.709503i	0.934961	+	0.354752i	$0.115435\pi$
$593$	17.2775i	0.709503i	−0.934961	+	0.354752i	$0.884565\pi$
$594$	0	0
$595$	−4.59675	−0.188448
$596$	0	0
$597$	−12.5419	−0.513306
$598$	0	0
$599$	−44.0533	−1.79997	−0.899984	−	0.435922i	$-0.856422\pi$
$599$	−44.0533	−1.79997	−0.899984	+	0.435922i	$0.856422\pi$
$600$	0	0
$601$	−4.34474	−0.177225	−0.0886127	−	0.996066i	$-0.528243\pi$
$601$	−4.34474	−0.177225	−0.0886127	+	0.996066i	$0.528243\pi$
$602$	0	0
$603$	2.39460i	0.0975158i
$604$	0	0
$605$	− 24.9787i	− 1.01553i
$606$	0	0
$607$	30.2089	1.22614	0.613070	−	0.790028i	$-0.289934\pi$
$607$	30.2089	1.22614	0.613070	+	0.790028i	$0.289934\pi$
$608$	0	0
$609$	4.36869i	0.177028i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	6.63664i	0.268051i	0.990978	+	0.134026i	$0.0427905\pi$
$613$	6.63664i	0.268051i	−0.990978	+	0.134026i	$0.957210\pi$
$614$	0	0
$615$	5.89696	0.237788
$616$	0	0
$617$	− 24.5402i	− 0.987949i	−0.869476	−	0.493975i	$-0.835544\pi$
$617$	− 24.5402i	− 0.987949i	0.869476	−	0.493975i	$-0.164456\pi$
$618$	0	0
$619$	− 33.4563i	− 1.34472i	−0.740224	−	0.672360i	$-0.765281\pi$
$619$	− 33.4563i	− 1.34472i	0.740224	−	0.672360i	$-0.234719\pi$
$620$	0	0
$621$	6.97908	0.280061
$622$	0	0
$623$	29.4656	1.18051
$624$	0	0
$625$	2.91093	0.116437
$626$	0	0
$627$	44.6058	1.78138
$628$	0	0
$629$	5.10573i	0.203579i
$630$	0	0
$631$	14.4263i	0.574302i	0.957885	+	0.287151i	$0.0927081\pi$
$631$	14.4263i	0.574302i	−0.957885	+	0.287151i	$0.907292\pi$
$632$	0	0
$633$	−12.4099	−0.493250
$634$	0	0
$635$	5.40161i	0.214356i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.1181i	0.598062i
$640$	0	0
$641$	−41.6225	−1.64399	−0.821994	−	0.569496i	$-0.807138\pi$
$641$	−41.6225	−1.64399	−0.821994	+	0.569496i	$0.807138\pi$
$642$	0	0
$643$	− 11.6854i	− 0.460828i	−0.973093	−	0.230414i	$-0.925992\pi$
$643$	− 11.6854i	− 0.460828i	0.973093	−	0.230414i	$-0.0740081\pi$
$644$	0	0
$645$	6.94184i	0.273334i
$646$	0	0
$647$	10.9947	0.432247	0.216124	−	0.976366i	$-0.430659\pi$
$647$	10.9947	0.432247	0.216124	+	0.976366i	$0.430659\pi$
$648$	0	0
$649$	−55.3552	−2.17288
$650$	0	0
$651$	−8.71517	−0.341574
$652$	0	0
$653$	33.1190	1.29605	0.648024	−	0.761620i	$-0.275596\pi$
$653$	33.1190	1.29605	0.648024	+	0.761620i	$0.275596\pi$
$654$	0	0
$655$	14.7973i	0.578178i
$656$	0	0
$657$	2.62301i	0.102333i
$658$	0	0
$659$	−22.1908	−0.864431	−0.432216	−	0.901770i	$-0.642268\pi$
$659$	−22.1908	−0.864431	−0.432216	+	0.901770i	$0.642268\pi$
$660$	0	0
$661$	19.8368i	0.771561i	0.922591	+	0.385780i	$0.126068\pi$
$661$	19.8368i	0.771561i	−0.922591	+	0.385780i	$0.873932\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20.4004i	− 0.791093i
$666$	0	0
$667$	−15.5522	−0.602184
$668$	0	0
$669$	19.1835i	0.741678i
$670$	0	0
$671$	71.1834i	2.74800i
$672$	0	0
$673$	−38.8564	−1.49780	−0.748902	−	0.662681i	$-0.769419\pi$
$673$	−38.8564	−1.49780	−0.748902	+	0.662681i	$0.769419\pi$
$674$	0	0
$675$	3.34474	0.128739
$676$	0	0
$677$	−40.8864	−1.57139	−0.785695	−	0.618614i	$-0.787695\pi$
$677$	−40.8864	−1.57139	−0.785695	+	0.618614i	$0.787695\pi$
$678$	0	0
$679$	27.3421	1.04930
$680$	0	0
$681$	− 10.5822i	− 0.405510i
$682$	0	0
$683$	44.3298i	1.69623i	0.529810	+	0.848117i	$0.322263\pi$
$683$	44.3298i	1.69623i	−0.529810	+	0.848117i	$0.677737\pi$
$684$	0	0
$685$	11.2700	0.430604
$686$	0	0
$687$	− 19.4595i	− 0.742426i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.7144i	0.483680i	0.970316	+	0.241840i	$0.0777509\pi$
$691$	12.7144i	0.483680i	−0.970316	+	0.241840i	$0.922249\pi$
$692$	0	0
$693$	10.8119	0.410709
$694$	0	0
$695$	19.2580i	0.730498i
$696$	0	0
$697$	8.35323i	0.316401i
$698$	0	0
$699$	−27.4243	−1.03728
$700$	0	0
$701$	−13.9582	−0.527192	−0.263596	−	0.964633i	$-0.584909\pi$
$701$	−13.9582	−0.527192	−0.263596	+	0.964633i	$0.584909\pi$
$702$	0	0
$703$	−22.6592	−0.854610
$704$	0	0
$705$	−6.50769	−0.245094
$706$	0	0
$707$	10.3088i	0.387702i
$708$	0	0
$709$	9.38142i	0.352327i	0.984361	+	0.176163i	$0.0563687\pi$
$709$	9.38142i	0.352327i	−0.984361	+	0.176163i	$0.943631\pi$
$710$	0	0
$711$	6.15661	0.230891
$712$	0	0
$713$	− 31.0254i	− 1.16191i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.76100i	− 0.103111i
$718$	0	0
$719$	37.5613	1.40080	0.700399	−	0.713751i	$-0.253005\pi$
$719$	37.5613	1.40080	0.700399	+	0.713751i	$0.253005\pi$
$720$	0	0
$721$	14.4988i	0.539963i
$722$	0	0
$723$	− 6.17753i	− 0.229745i
$724$	0	0
$725$	−7.45343	−0.276813
$726$	0	0
$727$	25.5864	0.948948	0.474474	−	0.880270i	$-0.342638\pi$
$727$	25.5864	0.948948	0.474474	+	0.880270i	$0.342638\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.83333	−0.363699
$732$	0	0
$733$	− 27.1633i	− 1.00330i	−0.865070	−	0.501650i	$-0.832726\pi$
$733$	− 27.1633i	− 1.00330i	0.865070	−	0.501650i	$-0.167274\pi$
$734$	0	0
$735$	4.06120i	0.149800i
$736$	0	0
$737$	−13.2062	−0.486456
$738$	0	0
$739$	− 16.9363i	− 0.623011i	−0.950244	−	0.311506i	$-0.899167\pi$
$739$	− 16.9363i	− 0.623011i	0.950244	−	0.311506i	$-0.100833\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.8078i	0.689992i	0.938604	+	0.344996i	$0.112120\pi$
$743$	18.8078i	0.689992i	−0.938604	+	0.344996i	$0.887880\pi$
$744$	0	0
$745$	10.0103	0.366750
$746$	0	0
$747$	− 5.59406i	− 0.204676i
$748$	0	0
$749$	− 15.2333i	− 0.556614i
$750$	0	0
$751$	−3.47112	−0.126663	−0.0633315	−	0.997993i	$-0.520173\pi$
$751$	−3.47112	−0.126663	−0.0633315	+	0.997993i	$0.520173\pi$
$752$	0	0
$753$	14.1163	0.514428
$754$	0	0
$755$	7.09541	0.258228
$756$	0	0
$757$	−9.76734	−0.355000	−0.177500	−	0.984121i	$-0.556801\pi$
$757$	−9.76734	−0.355000	−0.177500	+	0.984121i	$0.556801\pi$
$758$	0	0
$759$	38.4895i	1.39708i
$760$	0	0
$761$	12.1762i	0.441388i	0.975343	+	0.220694i	$0.0708323\pi$
$761$	12.1762i	0.441388i	−0.975343	+	0.220694i	$0.929168\pi$
$762$	0	0
$763$	−24.3988	−0.883297
$764$	0	0
$765$	− 2.34474i	− 0.0847741i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	41.6776i	1.50293i	0.659772	+	0.751466i	$0.270653\pi$
$769$	41.6776i	1.50293i	−0.659772	+	0.751466i	$0.729347\pi$
$770$	0	0
$771$	11.6225	0.418572
$772$	0	0
$773$	− 47.0460i	− 1.69213i	−0.533082	−	0.846064i	$-0.678966\pi$
$773$	− 47.0460i	− 1.69213i	0.533082	−	0.846064i	$-0.321034\pi$
$774$	0	0
$775$	− 14.8690i	− 0.534109i
$776$	0	0
$777$	−5.49231	−0.197036
$778$	0	0
$779$	−37.0716	−1.32823
$780$	0	0
$781$	−83.3758	−2.98342
$782$	0	0
$783$	−2.22841	−0.0796367
$784$	0	0
$785$	26.3881i	0.941830i
$786$	0	0
$787$	− 4.26464i	− 0.152018i	−0.997107	−	0.0760091i	$-0.975782\pi$
$787$	− 4.26464i	− 0.152018i	0.997107	−	0.0760091i	$-0.0242178\pi$
$788$	0	0
$789$	−8.62402	−0.307023
$790$	0	0
$791$	− 8.18987i	− 0.291198i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 3.14758i	− 0.111633i
$796$	0	0
$797$	31.5048	1.11596	0.557979	−	0.829855i	$-0.311577\pi$
$797$	31.5048	1.11596	0.557979	+	0.829855i	$0.311577\pi$
$798$	0	0
$799$	− 9.21835i	− 0.326122i
$800$	0	0
$801$	15.0300i	0.531057i
$802$	0	0
$803$	−14.4658	−0.510489
$804$	0	0
$805$	17.6031	0.620428
$806$	0	0
$807$	25.2480	0.888773
$808$	0	0
$809$	−35.9004	−1.26219	−0.631096	−	0.775705i	$-0.717395\pi$
$809$	−35.9004	−1.26219	−0.631096	+	0.775705i	$0.717395\pi$
$810$	0	0
$811$	19.1515i	0.672502i	0.941772	+	0.336251i	$0.109159\pi$
$811$	19.1515i	0.672502i	−0.941772	+	0.336251i	$0.890841\pi$
$812$	0	0
$813$	2.99797i	0.105143i
$814$	0	0
$815$	−23.5758	−0.825825
$816$	0	0
$817$	− 43.6403i	− 1.52678i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.6139i	1.20803i	0.796972	+	0.604016i	$0.206434\pi$
$821$	34.6139i	1.20803i	−0.796972	+	0.604016i	$0.793566\pi$
$822$	0	0
$823$	−9.42025	−0.328369	−0.164185	−	0.986430i	$-0.552499\pi$
$823$	−9.42025	−0.328369	−0.164185	+	0.986430i	$0.552499\pi$
$824$	0	0
$825$	18.4461i	0.642212i
$826$	0	0
$827$	− 9.40621i	− 0.327086i	−0.986536	−	0.163543i	$-0.947708\pi$
$827$	− 9.40621i	− 0.327086i	0.986536	−	0.163543i	$-0.0522922\pi$
$828$	0	0
$829$	39.3615	1.36708	0.683541	−	0.729912i	$-0.260439\pi$
$829$	39.3615	1.36708	0.683541	+	0.729912i	$0.260439\pi$
$830$	0	0
$831$	24.9123	0.864199
$832$	0	0
$833$	−5.75282	−0.199324
$834$	0	0
$835$	10.5193	0.364036
$836$	0	0
$837$	− 4.44548i	− 0.153658i
$838$	0	0
$839$	− 30.5147i	− 1.05348i	−0.850025	−	0.526742i	$-0.823413\pi$
$839$	− 30.5147i	− 1.05348i	0.850025	−	0.526742i	$-0.176587\pi$
$840$	0	0
$841$	−24.0342	−0.828766
$842$	0	0
$843$	8.38198i	0.288691i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38.0622i	1.30783i
$848$	0	0
$849$	25.8106	0.885817
$850$	0	0
$851$	− 19.5522i	− 0.670241i
$852$	0	0
$853$	− 1.33219i	− 0.0456132i	−0.999740	−	0.0228066i	$-0.992740\pi$
$853$	− 1.33219i	− 0.0456132i	0.999740	−	0.0228066i	$-0.00726020\pi$
$854$	0	0
$855$	10.4059	0.355876
$856$	0	0
$857$	−0.771864	−0.0263664	−0.0131832	−	0.999913i	$-0.504196\pi$
$857$	−0.771864	−0.0263664	−0.0131832	+	0.999913i	$0.504196\pi$
$858$	0	0
$859$	−5.09036	−0.173681	−0.0868404	−	0.996222i	$-0.527677\pi$
$859$	−5.09036	−0.173681	−0.0868404	+	0.996222i	$0.527677\pi$
$860$	0	0
$861$	−8.98569	−0.306232
$862$	0	0
$863$	− 23.7264i	− 0.807658i	−0.914835	−	0.403829i	$-0.867679\pi$
$863$	− 23.7264i	− 0.807658i	0.914835	−	0.403829i	$-0.132321\pi$
$864$	0	0
$865$	− 11.0300i	− 0.375030i
$866$	0	0
$867$	−13.6786	−0.464550
$868$	0	0
$869$	33.9536i	1.15180i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.9468i	0.472029i
$874$	0	0
$875$	21.0476	0.711540
$876$	0	0
$877$	− 8.02893i	− 0.271118i	−0.990769	−	0.135559i	$-0.956717\pi$
$877$	− 8.02893i	− 0.271118i	0.990769	−	0.135559i	$-0.0432830\pi$
$878$	0	0
$879$	12.7527i	0.430137i
$880$	0	0
$881$	−21.7454	−0.732621	−0.366310	−	0.930493i	$-0.619379\pi$
$881$	−21.7454	−0.732621	−0.366310	+	0.930493i	$0.619379\pi$
$882$	0	0
$883$	0.00504815	0.000169884 0	8.49419e−5	−	1.00000i	$-0.499973\pi$
$883$	0.00504815	0.000169884 0	8.49419e−5		1.00000i	$0.499973\pi$
$884$	0	0
$885$	−12.9136	−0.434087
$886$	0	0
$887$	33.8503	1.13658	0.568291	−	0.822827i	$-0.307605\pi$
$887$	33.8503	1.13658	0.568291	+	0.822827i	$0.307605\pi$
$888$	0	0
$889$	− 8.23088i	− 0.276055i
$890$	0	0
$891$	5.51498i	0.184759i
$892$	0	0
$893$	40.9110	1.36904
$894$	0	0
$895$	4.80025i	0.160455i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.90633i	0.330395i
$900$	0	0
$901$	4.45865	0.148539
$902$	0	0
$903$	− 10.5779i	− 0.352009i
$904$	0	0
$905$	6.75363i	0.224498i
$906$	0	0
$907$	−25.2772	−0.839315	−0.419658	−	0.907682i	$-0.637850\pi$
$907$	−25.2772	−0.839315	−0.419658	+	0.907682i	$0.637850\pi$
$908$	0	0
$909$	−5.25836	−0.174409
$910$	0	0
$911$	−15.5366	−0.514750	−0.257375	−	0.966312i	$-0.582858\pi$
$911$	−15.5366	−0.514750	−0.257375	+	0.966312i	$0.582858\pi$
$912$	0	0
$913$	30.8511	1.02102
$914$	0	0
$915$	16.6061i	0.548982i
$916$	0	0
$917$	− 22.5479i	− 0.744597i
$918$	0	0
$919$	−50.5141	−1.66631	−0.833154	−	0.553041i	$-0.813467\pi$
$919$	−50.5141	−1.66631	−0.833154	+	0.553041i	$0.813467\pi$
$920$	0	0
$921$	16.6307i	0.548002i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 9.37044i	− 0.308098i
$926$	0	0
$927$	−7.39561	−0.242904
$928$	0	0
$929$	− 12.3800i	− 0.406174i	−0.979161	−	0.203087i	$-0.934903\pi$
$929$	− 12.3800i	− 0.406174i	0.979161	−	0.203087i	$-0.0650974\pi$
$930$	0	0
$931$	− 25.5310i	− 0.836746i
$932$	0	0
$933$	−19.3940	−0.634933
$934$	0	0
$935$	12.9312	0.422894
$936$	0	0
$937$	17.5721	0.574056	0.287028	−	0.957922i	$-0.407333\pi$
$937$	17.5721	0.574056	0.287028	+	0.957922i	$0.407333\pi$
$938$	0	0
$939$	−26.5091	−0.865091
$940$	0	0
$941$	51.8438i	1.69006i	0.534718	+	0.845030i	$0.320418\pi$
$941$	51.8438i	1.69006i	−0.534718	+	0.845030i	$0.679582\pi$
$942$	0	0
$943$	− 31.9884i	− 1.04168i
$944$	0	0
$945$	2.52227	0.0820493
$946$	0	0
$947$	− 22.0146i	− 0.715378i	−0.933841	−	0.357689i	$-0.883565\pi$
$947$	− 22.0146i	− 0.715378i	0.933841	−	0.357689i	$-0.116435\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 8.13569i	− 0.263818i
$952$	0	0
$953$	−25.5432	−0.827425	−0.413713	−	0.910408i	$-0.635768\pi$
$953$	−25.5432	−0.827425	−0.413713	+	0.910408i	$0.635768\pi$
$954$	0	0
$955$	− 7.13170i	− 0.230777i
$956$	0	0
$957$	− 12.2896i	− 0.397267i
$958$	0	0
$959$	−17.1730	−0.554546
$960$	0	0
$961$	11.2377	0.362507
$962$	0	0
$963$	7.77030	0.250394
$964$	0	0
$965$	13.1087	0.421984
$966$	0	0
$967$	14.8939i	0.478956i	0.970902	+	0.239478i	$0.0769763\pi$
$967$	14.8939i	0.478956i	−0.970902	+	0.239478i	$0.923024\pi$
$968$	0	0
$969$	14.7403i	0.473528i
$970$	0	0
$971$	−9.88367	−0.317182	−0.158591	−	0.987344i	$-0.550695\pi$
$971$	−9.88367	−0.317182	−0.158591	+	0.987344i	$0.550695\pi$
$972$	0	0
$973$	− 29.3451i	− 0.940760i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.6527i	1.58853i	0.607572	+	0.794265i	$0.292144\pi$
$977$	49.6527i	1.58853i	−0.607572	+	0.794265i	$0.707856\pi$
$978$	0	0
$979$	−82.8898	−2.64917
$980$	0	0
$981$	− 12.4455i	− 0.397354i
$982$	0	0
$983$	− 4.41350i	− 0.140769i	−0.997520	−	0.0703843i	$-0.977577\pi$
$983$	− 4.41350i	− 0.140769i	0.997520	−	0.0703843i	$-0.0224226\pi$
$984$	0	0
$985$	33.2661	1.05995
$986$	0	0
$987$	9.91631	0.315640
$988$	0	0
$989$	37.6564	1.19740
$990$	0	0
$991$	−0.802842	−0.0255031	−0.0127516	−	0.999919i	$-0.504059\pi$
$991$	−0.802842	−0.0255031	−0.0127516	+	0.999919i	$0.504059\pi$
$992$	0	0
$993$	31.9861i	1.01505i
$994$	0	0
$995$	− 16.1360i	− 0.511547i
$996$	0	0
$997$	7.94331	0.251567	0.125784	−	0.992058i	$-0.459856\pi$
$997$	7.94331	0.251567	0.125784	+	0.992058i	$0.459856\pi$
$998$	0	0
$999$	− 2.80155i	− 0.0886370i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.p.337.6		8
13.5	odd	4		4056.2.a.bd.1.4		4
13.8	odd	4		4056.2.a.be.1.1		4
13.9	even	3		312.2.bf.b.49.3	✓	8
13.10	even	6		312.2.bf.b.121.2	yes	8
13.12	even	2	inner	4056.2.c.p.337.3		8
39.23	odd	6		936.2.bi.c.433.3		8
39.35	odd	6		936.2.bi.c.361.2		8
52.23	odd	6		624.2.bv.g.433.2		8
52.31	even	4		8112.2.a.cq.1.4		4
52.35	odd	6		624.2.bv.g.49.3		8
52.47	even	4		8112.2.a.cs.1.1		4
156.23	even	6		1872.2.by.m.433.3		8
156.35	even	6		1872.2.by.m.1297.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.b.49.3	✓	8	13.9	even	3
312.2.bf.b.121.2	yes	8	13.10	even	6
624.2.bv.g.49.3		8	52.35	odd	6
624.2.bv.g.433.2		8	52.23	odd	6
936.2.bi.c.361.2		8	39.35	odd	6
936.2.bi.c.433.3		8	39.23	odd	6
1872.2.by.m.433.3		8	156.23	even	6
1872.2.by.m.1297.2		8	156.35	even	6
4056.2.a.bd.1.4		4	13.5	odd	4
4056.2.a.be.1.1		4	13.8	odd	4
4056.2.c.p.337.3		8	13.12	even	2	inner
4056.2.c.p.337.6		8	1.1	even	1	trivial
8112.2.a.cq.1.4		4	52.31	even	4
8112.2.a.cs.1.1		4	52.47	even	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.28657i q^{5} -1.96046i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.28657i q^{5} -1.96046i q^{7} +1.00000 q^{9} +5.51498i q^{11} +1.28657i q^{15} -1.82247 q^{17} -8.08812i q^{19} -1.96046i q^{21} +6.97908 q^{23} +3.34474 q^{25} +1.00000 q^{27} -2.22841 q^{29} -4.44548i q^{31} +5.51498i q^{33} +2.52227 q^{35} -2.80155i q^{37} -4.58347i q^{41} +5.39561 q^{43} +1.28657i q^{45} +5.05816i q^{47} +3.15661 q^{49} -1.82247 q^{51} -2.44649 q^{53} -7.09541 q^{55} -8.08812i q^{57} +10.0372i q^{59} +12.9073 q^{61} -1.96046i q^{63} +2.39460i q^{67} +6.97908 q^{69} +15.1181i q^{71} +2.62301i q^{73} +3.34474 q^{75} +10.8119 q^{77} +6.15661 q^{79} +1.00000 q^{81} -5.59406i q^{83} -2.34474i q^{85} -2.22841 q^{87} +15.0300i q^{89} -4.44548i q^{93} +10.4059 q^{95} +13.9468i q^{97} +5.51498i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Introduction

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