LMFDB - Embedded newform 4000.2.f.c.3249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(3249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.3249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.f (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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 $ x^20 - 3x^19 + 6x^18 - 5x^17 - 3x^16 + 20x^15 - 28x^14 + 24x^13 + 16x^12 - 60x^11 + 116x^10 - 120x^9 + 64x^8 + 192x^7 - 448x^6 + 640x^5 - 192x^4 - 640x^3 + 1536x^2 - 1536*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3249.6
Root			$1.32522 + 0.493756i$ of defining polynomial
Character	$\chi$	$=$	4000.3249
Dual form			4000.2.f.c.3249.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83526 q^{3} +1.31564i q^{7} +0.368171 q^{9} +O(q^{10})$	$q-1.83526 q^{3} +1.31564i q^{7} +0.368171 q^{9} -6.60274i q^{11} +2.96705 q^{13} -2.69502i q^{17} -4.97013i q^{19} -2.41454i q^{21} -3.73739i q^{23} +4.83008 q^{27} +5.52846i q^{29} -8.36121 q^{31} +12.1177i q^{33} +7.19517 q^{37} -5.44530 q^{39} -3.77169 q^{41} -3.07951 q^{43} +8.77645i q^{47} +5.26908 q^{49} +4.94606i q^{51} -0.0464228 q^{53} +9.12147i q^{57} +1.02084i q^{59} +5.23191i q^{61} +0.484382i q^{63} +10.8187 q^{67} +6.85908i q^{69} -9.35643 q^{71} -12.4143i q^{73} +8.68684 q^{77} +1.43842 q^{79} -9.96896 q^{81} -13.0280 q^{83} -10.1462i q^{87} -3.94185 q^{89} +3.90357i q^{91} +15.3450 q^{93} +1.00691i q^{97} -2.43094i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 12 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 + 12 * q^9	$ 20 q - 4 q^{3} + 12 q^{9} + 6 q^{13} + 8 q^{27} - 24 q^{31} + 18 q^{37} + 4 q^{39} + 22 q^{41} + 60 q^{43} - 6 q^{49} + 10 q^{53} + 40 q^{67} - 48 q^{71} + 24 q^{77} + 48 q^{79} - 28 q^{81} - 40 q^{83} + 22 q^{89} + 8 q^{93}+O(q^{100}) $ 20 * q - 4 * q^3 + 12 * q^9 + 6 * q^13 + 8 * q^27 - 24 * q^31 + 18 * q^37 + 4 * q^39 + 22 * q^41 + 60 * q^43 - 6 * q^49 + 10 * q^53 + 40 * q^67 - 48 * q^71 + 24 * q^77 + 48 * q^79 - 28 * q^81 - 40 * q^83 + 22 * q^89 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.83526	−1.05959	−0.529793	−	0.848127i	$-0.677730\pi$
$3$	−1.83526	−1.05959	−0.529793	+	0.848127i	$0.677730\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.31564i	0.497266i	0.968598	+	0.248633i	$0.0799813\pi$
$7$	1.31564i	0.497266i	−0.968598	+	0.248633i	$0.920019\pi$
$8$	0	0
$9$	0.368171	0.122724
$10$	0	0
$11$	− 6.60274i	− 1.99080i	−0.0958055	−	0.995400i	$-0.530543\pi$
$11$	− 6.60274i	− 1.99080i	0.0958055	−	0.995400i	$-0.469457\pi$
$12$	0	0
$13$	2.96705	0.822911	0.411456	−	0.911430i	$-0.365021\pi$
$13$	2.96705	0.822911	0.411456	+	0.911430i	$0.365021\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.69502i	− 0.653639i	−0.945087	−	0.326819i	$-0.894023\pi$
$17$	− 2.69502i	− 0.653639i	0.945087	−	0.326819i	$-0.105977\pi$
$18$	0	0
$19$	− 4.97013i	− 1.14023i	−0.821566	−	0.570113i	$-0.806899\pi$
$19$	− 4.97013i	− 1.14023i	0.821566	−	0.570113i	$-0.193101\pi$
$20$	0	0
$21$	− 2.41454i	− 0.526897i
$22$	0	0
$23$	− 3.73739i	− 0.779301i	−0.920963	−	0.389650i	$-0.872596\pi$
$23$	− 3.73739i	− 0.779301i	0.920963	−	0.389650i	$-0.127404\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.83008	0.929550
$28$	0	0
$29$	5.52846i	1.02661i	0.858206	+	0.513305i	$0.171579\pi$
$29$	5.52846i	1.02661i	−0.858206	+	0.513305i	$0.828421\pi$
$30$	0	0
$31$	−8.36121	−1.50172	−0.750859	−	0.660462i	$-0.770360\pi$
$31$	−8.36121	−1.50172	−0.750859	+	0.660462i	$0.770360\pi$
$32$	0	0
$33$	12.1177i	2.10943i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.19517	1.18288	0.591439	−	0.806349i	$-0.298560\pi$
$37$	7.19517	1.18288	0.591439	+	0.806349i	$0.298560\pi$
$38$	0	0
$39$	−5.44530	−0.871945
$40$	0	0
$41$	−3.77169	−0.589040	−0.294520	−	0.955645i	$-0.595160\pi$
$41$	−3.77169	−0.589040	−0.294520	+	0.955645i	$0.595160\pi$
$42$	0	0
$43$	−3.07951	−0.469621	−0.234810	−	0.972041i	$-0.575447\pi$
$43$	−3.07951	−0.469621	−0.234810	+	0.972041i	$0.575447\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.77645i	1.28018i	0.768301	+	0.640088i	$0.221102\pi$
$47$	8.77645i	1.28018i	−0.768301	+	0.640088i	$0.778898\pi$
$48$	0	0
$49$	5.26908	0.752726
$50$	0	0
$51$	4.94606i	0.692587i
$52$	0	0
$53$	−0.0464228	−0.00637666	−0.00318833	−	0.999995i	$-0.501015\pi$
$53$	−0.0464228	−0.00637666	−0.00318833	+	0.999995i	$0.501015\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.12147i	1.20817i
$58$	0	0
$59$	1.02084i	0.132902i	0.997790	+	0.0664510i	$0.0211676\pi$
$59$	1.02084i	0.132902i	−0.997790	+	0.0664510i	$0.978832\pi$
$60$	0	0
$61$	5.23191i	0.669877i	0.942240	+	0.334938i	$0.108716\pi$
$61$	5.23191i	0.669877i	−0.942240	+	0.334938i	$0.891284\pi$
$62$	0	0
$63$	0.484382i	0.0610264i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8187	1.32172	0.660858	−	0.750511i	$-0.270193\pi$
$67$	10.8187	1.32172	0.660858	+	0.750511i	$0.270193\pi$
$68$	0	0
$69$	6.85908i	0.825737i
$70$	0	0
$71$	−9.35643	−1.11040	−0.555202	−	0.831715i	$-0.687359\pi$
$71$	−9.35643	−1.11040	−0.555202	+	0.831715i	$0.687359\pi$
$72$	0	0
$73$	− 12.4143i	− 1.45298i	−0.687177	−	0.726490i	$-0.741150\pi$
$73$	− 12.4143i	− 1.45298i	0.687177	−	0.726490i	$-0.258850\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.68684	0.989958
$78$	0	0
$79$	1.43842	0.161835	0.0809174	−	0.996721i	$-0.474215\pi$
$79$	1.43842	0.161835	0.0809174	+	0.996721i	$0.474215\pi$
$80$	0	0
$81$	−9.96896	−1.10766
$82$	0	0
$83$	−13.0280	−1.43001	−0.715005	−	0.699120i	$-0.753575\pi$
$83$	−13.0280	−1.43001	−0.715005	+	0.699120i	$0.753575\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 10.1462i	− 1.08778i
$88$	0	0
$89$	−3.94185	−0.417835	−0.208917	−	0.977933i	$-0.566994\pi$
$89$	−3.94185	−0.417835	−0.208917	+	0.977933i	$0.566994\pi$
$90$	0	0
$91$	3.90357i	0.409206i
$92$	0	0
$93$	15.3450	1.59120
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00691i	0.102236i	0.998693	+	0.0511181i	$0.0162785\pi$
$97$	1.00691i	0.102236i	−0.998693	+	0.0511181i	$0.983722\pi$
$98$	0	0
$99$	− 2.43094i	− 0.244318i
$100$	0	0
$101$	− 2.50175i	− 0.248933i	−0.992224	−	0.124466i	$-0.960278\pi$
$101$	− 2.50175i	− 0.248933i	0.992224	−	0.124466i	$-0.0397219\pi$
$102$	0	0
$103$	− 5.34336i	− 0.526497i	−0.964728	−	0.263248i	$-0.915206\pi$
$103$	− 5.34336i	− 0.526497i	0.964728	−	0.263248i	$-0.0847938\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.971619	0.0939299	0.0469650	−	0.998897i	$-0.485045\pi$
$107$	0.971619	0.0939299	0.0469650	+	0.998897i	$0.485045\pi$
$108$	0	0
$109$	− 18.8454i	− 1.80506i	−0.430626	−	0.902531i	$-0.641707\pi$
$109$	− 18.8454i	− 1.80506i	0.430626	−	0.902531i	$-0.358293\pi$
$110$	0	0
$111$	−13.2050	−1.25336
$112$	0	0
$113$	12.3812i	1.16472i	0.812930	+	0.582361i	$0.197871\pi$
$113$	12.3812i	1.16472i	−0.812930	+	0.582361i	$0.802129\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.09238	0.100991
$118$	0	0
$119$	3.54568	0.325032
$120$	0	0
$121$	−32.5961	−2.96329
$122$	0	0
$123$	6.92203	0.624139
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.8083i	− 1.04782i	−0.851774	−	0.523909i	$-0.824473\pi$
$127$	− 11.8083i	− 1.04782i	0.851774	−	0.523909i	$-0.175527\pi$
$128$	0	0
$129$	5.65169	0.497604
$130$	0	0
$131$	− 12.6735i	− 1.10729i	−0.832752	−	0.553646i	$-0.813236\pi$
$131$	− 12.6735i	− 1.10729i	0.832752	−	0.553646i	$-0.186764\pi$
$132$	0	0
$133$	6.53892	0.566996
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.33562i	0.370417i	0.982699	+	0.185209i	$0.0592961\pi$
$137$	4.33562i	0.370417i	−0.982699	+	0.185209i	$0.940704\pi$
$138$	0	0
$139$	− 3.48791i	− 0.295841i	−0.988999	−	0.147920i	$-0.952742\pi$
$139$	− 3.48791i	− 0.295841i	0.988999	−	0.147920i	$-0.0472579\pi$
$140$	0	0
$141$	− 16.1070i	− 1.35646i
$142$	0	0
$143$	− 19.5906i	− 1.63825i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−9.67013	−0.797579
$148$	0	0
$149$	9.11838i	0.747007i	0.927629	+	0.373503i	$0.121844\pi$
$149$	9.11838i	0.747007i	−0.927629	+	0.373503i	$0.878156\pi$
$150$	0	0
$151$	10.5180	0.855944	0.427972	−	0.903792i	$-0.359228\pi$
$151$	10.5180	0.855944	0.427972	+	0.903792i	$0.359228\pi$
$152$	0	0
$153$	− 0.992229i	− 0.0802170i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.59017	−0.446144	−0.223072	−	0.974802i	$-0.571608\pi$
$157$	−5.59017	−0.446144	−0.223072	+	0.974802i	$0.571608\pi$
$158$	0	0
$159$	0.0851977	0.00675662
$160$	0	0
$161$	4.91708	0.387520
$162$	0	0
$163$	−5.48367	−0.429514	−0.214757	−	0.976667i	$-0.568896\pi$
$163$	−5.48367	−0.429514	−0.214757	+	0.976667i	$0.568896\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.4503i	1.50511i	0.658528	+	0.752556i	$0.271179\pi$
$167$	19.4503i	1.50511i	−0.658528	+	0.752556i	$0.728821\pi$
$168$	0	0
$169$	−4.19663	−0.322817
$170$	0	0
$171$	− 1.82986i	− 0.139933i
$172$	0	0
$173$	−11.9677	−0.909885	−0.454942	−	0.890521i	$-0.650340\pi$
$173$	−11.9677	−0.909885	−0.454942	+	0.890521i	$0.650340\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.87350i	− 0.140821i
$178$	0	0
$179$	11.0142i	0.823244i	0.911355	+	0.411622i	$0.135038\pi$
$179$	11.0142i	0.823244i	−0.911355	+	0.411622i	$0.864962\pi$
$180$	0	0
$181$	− 18.6306i	− 1.38480i	−0.721511	−	0.692402i	$-0.756552\pi$
$181$	− 18.6306i	− 1.38480i	0.721511	−	0.692402i	$-0.243448\pi$
$182$	0	0
$183$	− 9.60190i	− 0.709793i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.7945	−1.30126
$188$	0	0
$189$	6.35466i	0.462234i
$190$	0	0
$191$	−10.6266	−0.768912	−0.384456	−	0.923143i	$-0.625611\pi$
$191$	−10.6266	−0.768912	−0.384456	+	0.923143i	$0.625611\pi$
$192$	0	0
$193$	17.9777i	1.29406i	0.762464	+	0.647031i	$0.223989\pi$
$193$	17.9777i	1.29406i	−0.762464	+	0.647031i	$0.776011\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.6813	−1.68722	−0.843610	−	0.536956i	$-0.819574\pi$
$197$	−23.6813	−1.68722	−0.843610	+	0.536956i	$0.819574\pi$
$198$	0	0
$199$	17.9303	1.27104	0.635522	−	0.772082i	$-0.280785\pi$
$199$	17.9303	1.27104	0.635522	+	0.772082i	$0.280785\pi$
$200$	0	0
$201$	−19.8551	−1.40047
$202$	0	0
$203$	−7.27348	−0.510498
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.37600i	− 0.0956387i
$208$	0	0
$209$	−32.8165	−2.26996
$210$	0	0
$211$	− 15.0943i	− 1.03913i	−0.854430	−	0.519566i	$-0.826094\pi$
$211$	− 15.0943i	− 1.03913i	0.854430	−	0.519566i	$-0.173906\pi$
$212$	0	0
$213$	17.1715	1.17657
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 11.0004i	− 0.746754i
$218$	0	0
$219$	22.7834i	1.53956i
$220$	0	0
$221$	− 7.99626i	− 0.537886i
$222$	0	0
$223$	21.0583i	1.41017i	0.709125	+	0.705083i	$0.249090\pi$
$223$	21.0583i	1.41017i	−0.709125	+	0.705083i	$0.750910\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.94442	0.460917	0.230459	−	0.973082i	$-0.425977\pi$
$227$	6.94442	0.460917	0.230459	+	0.973082i	$0.425977\pi$
$228$	0	0
$229$	− 19.8805i	− 1.31374i	−0.754003	−	0.656871i	$-0.771879\pi$
$229$	− 19.8805i	− 1.31374i	0.754003	−	0.656871i	$-0.228121\pi$
$230$	0	0
$231$	−15.9426	−1.04895
$232$	0	0
$233$	− 9.43816i	− 0.618315i	−0.951011	−	0.309157i	$-0.899953\pi$
$233$	− 9.43816i	− 0.618315i	0.951011	−	0.309157i	$-0.100047\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.63987	−0.171478
$238$	0	0
$239$	−21.3865	−1.38338	−0.691688	−	0.722197i	$-0.743132\pi$
$239$	−21.3865	−1.38338	−0.691688	+	0.722197i	$0.743132\pi$
$240$	0	0
$241$	11.4117	0.735094	0.367547	−	0.930005i	$-0.380198\pi$
$241$	11.4117	0.735094	0.367547	+	0.930005i	$0.380198\pi$
$242$	0	0
$243$	3.80536	0.244114
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 14.7466i	− 0.938305i
$248$	0	0
$249$	23.9097	1.51522
$250$	0	0
$251$	2.76130i	0.174292i	0.996196	+	0.0871459i	$0.0277746\pi$
$251$	2.76130i	0.174292i	−0.996196	+	0.0871459i	$0.972225\pi$
$252$	0	0
$253$	−24.6770	−1.55143
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.1685i	− 0.759052i	−0.925181	−	0.379526i	$-0.876087\pi$
$257$	− 12.1685i	− 0.759052i	0.925181	−	0.379526i	$-0.123913\pi$
$258$	0	0
$259$	9.46627i	0.588206i
$260$	0	0
$261$	2.03542i	0.125989i
$262$	0	0
$263$	22.4088i	1.38179i	0.722956	+	0.690894i	$0.242783\pi$
$263$	22.4088i	1.38179i	−0.722956	+	0.690894i	$0.757217\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.23430	0.442732
$268$	0	0
$269$	− 29.7546i	− 1.81417i	−0.420946	−	0.907086i	$-0.638302\pi$
$269$	− 29.7546i	− 1.81417i	0.420946	−	0.907086i	$-0.361698\pi$
$270$	0	0
$271$	−15.1173	−0.918308	−0.459154	−	0.888357i	$-0.651847\pi$
$271$	−15.1173	−0.918308	−0.459154	+	0.888357i	$0.651847\pi$
$272$	0	0
$273$	− 7.16407i	− 0.433589i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.51304	0.571583	0.285792	−	0.958292i	$-0.407743\pi$
$277$	9.51304	0.571583	0.285792	+	0.958292i	$0.407743\pi$
$278$	0	0
$279$	−3.07836	−0.184296
$280$	0	0
$281$	−17.5890	−1.04927	−0.524637	−	0.851326i	$-0.675799\pi$
$281$	−17.5890	−1.04927	−0.524637	+	0.851326i	$0.675799\pi$
$282$	0	0
$283$	15.3245	0.910944	0.455472	−	0.890250i	$-0.349471\pi$
$283$	15.3245	0.910944	0.455472	+	0.890250i	$0.349471\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.96220i	− 0.292909i
$288$	0	0
$289$	9.73686	0.572756
$290$	0	0
$291$	− 1.84794i	− 0.108328i
$292$	0	0
$293$	−3.18687	−0.186179	−0.0930894	−	0.995658i	$-0.529674\pi$
$293$	−3.18687	−0.186179	−0.0930894	+	0.995658i	$0.529674\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 31.8918i	− 1.85055i
$298$	0	0
$299$	− 11.0890i	− 0.641295i
$300$	0	0
$301$	− 4.05153i	− 0.233526i
$302$	0	0
$303$	4.59135i	0.263766i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.0569	−1.60129	−0.800645	−	0.599138i	$-0.795510\pi$
$307$	−28.0569	−1.60129	−0.800645	+	0.599138i	$0.795510\pi$
$308$	0	0
$309$	9.80644i	0.557869i
$310$	0	0
$311$	−27.7970	−1.57622	−0.788112	−	0.615532i	$-0.788941\pi$
$311$	−27.7970	−1.57622	−0.788112	+	0.615532i	$0.788941\pi$
$312$	0	0
$313$	− 21.6443i	− 1.22341i	−0.791088	−	0.611703i	$-0.790485\pi$
$313$	− 21.6443i	− 1.22341i	0.791088	−	0.611703i	$-0.209515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.0032	−0.898827	−0.449414	−	0.893324i	$-0.648367\pi$
$317$	−16.0032	−0.898827	−0.449414	+	0.893324i	$0.648367\pi$
$318$	0	0
$319$	36.5030	2.04377
$320$	0	0
$321$	−1.78317	−0.0995269
$322$	0	0
$323$	−13.3946	−0.745296
$324$	0	0
$325$	0	0
$326$	0	0
$327$	34.5862i	1.91262i
$328$	0	0
$329$	−11.5467	−0.636589
$330$	0	0
$331$	5.39927i	0.296771i	0.988930	+	0.148385i	$0.0474076\pi$
$331$	5.39927i	0.296771i	−0.988930	+	0.148385i	$0.952592\pi$
$332$	0	0
$333$	2.64905	0.145167
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.3679i	1.43635i	0.695861	+	0.718176i	$0.255023\pi$
$337$	26.3679i	1.43635i	−0.695861	+	0.718176i	$0.744977\pi$
$338$	0	0
$339$	− 22.7226i	− 1.23412i
$340$	0	0
$341$	55.2069i	2.98962i
$342$	0	0
$343$	16.1417i	0.871571i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.70069	−0.467077	−0.233539	−	0.972348i	$-0.575031\pi$
$347$	−8.70069	−0.467077	−0.233539	+	0.972348i	$0.575031\pi$
$348$	0	0
$349$	14.7757i	0.790923i	0.918483	+	0.395462i	$0.129415\pi$
$349$	14.7757i	0.790923i	−0.918483	+	0.395462i	$0.870585\pi$
$350$	0	0
$351$	14.3311	0.764937
$352$	0	0
$353$	− 17.4259i	− 0.927485i	−0.885970	−	0.463743i	$-0.846506\pi$
$353$	− 17.4259i	− 0.927485i	0.885970	−	0.463743i	$-0.153494\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.50725	−0.344400
$358$	0	0
$359$	−12.3632	−0.652506	−0.326253	−	0.945282i	$-0.605786\pi$
$359$	−12.3632	−0.652506	−0.326253	+	0.945282i	$0.605786\pi$
$360$	0	0
$361$	−5.70221	−0.300116
$362$	0	0
$363$	59.8223	3.13986
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.15969i	0.217134i	0.994089	+	0.108567i	$0.0346262\pi$
$367$	4.15969i	0.217134i	−0.994089	+	0.108567i	$0.965374\pi$
$368$	0	0
$369$	−1.38863	−0.0722891
$370$	0	0
$371$	− 0.0610758i	− 0.00317090i
$372$	0	0
$373$	−33.9114	−1.75586	−0.877932	−	0.478786i	$-0.841077\pi$
$373$	−33.9114	−1.75586	−0.877932	+	0.478786i	$0.841077\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.4032i	0.844808i
$378$	0	0
$379$	− 8.17397i	− 0.419869i	−0.977715	−	0.209934i	$-0.932675\pi$
$379$	− 8.17397i	− 0.419869i	0.977715	−	0.209934i	$-0.0673250\pi$
$380$	0	0
$381$	21.6713i	1.11025i
$382$	0	0
$383$	− 3.27290i	− 0.167238i	−0.996498	−	0.0836188i	$-0.973352\pi$
$383$	− 3.27290i	− 0.167238i	0.996498	−	0.0836188i	$-0.0266478\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.13379	−0.0576336
$388$	0	0
$389$	2.07560i	0.105237i	0.998615	+	0.0526186i	$0.0167568\pi$
$389$	2.07560i	0.105237i	−0.998615	+	0.0526186i	$0.983243\pi$
$390$	0	0
$391$	−10.0724	−0.509381
$392$	0	0
$393$	23.2592i	1.17327i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.23114	−0.312732	−0.156366	−	0.987699i	$-0.549978\pi$
$397$	−6.23114	−0.312732	−0.156366	+	0.987699i	$0.549978\pi$
$398$	0	0
$399$	−12.0006	−0.600781
$400$	0	0
$401$	20.9959	1.04849	0.524243	−	0.851569i	$-0.324348\pi$
$401$	20.9959	1.04849	0.524243	+	0.851569i	$0.324348\pi$
$402$	0	0
$403$	−24.8081	−1.23578
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 47.5078i	− 2.35488i
$408$	0	0
$409$	−6.61521	−0.327101	−0.163551	−	0.986535i	$-0.552295\pi$
$409$	−6.61521	−0.327101	−0.163551	+	0.986535i	$0.552295\pi$
$410$	0	0
$411$	− 7.95698i	− 0.392489i
$412$	0	0
$413$	−1.34306	−0.0660876
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.40121i	0.313469i
$418$	0	0
$419$	17.6296i	0.861262i	0.902528	+	0.430631i	$0.141709\pi$
$419$	17.6296i	0.861262i	−0.902528	+	0.430631i	$0.858291\pi$
$420$	0	0
$421$	− 18.0508i	− 0.879742i	−0.898061	−	0.439871i	$-0.855024\pi$
$421$	− 18.0508i	− 0.879742i	0.898061	−	0.439871i	$-0.144976\pi$
$422$	0	0
$423$	3.23124i	0.157108i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.88332	−0.333107
$428$	0	0
$429$	35.9539i	1.73587i
$430$	0	0
$431$	−4.44104	−0.213917	−0.106959	−	0.994263i	$-0.534111\pi$
$431$	−4.44104	−0.213917	−0.106959	+	0.994263i	$0.534111\pi$
$432$	0	0
$433$	− 24.5345i	− 1.17905i	−0.807749	−	0.589526i	$-0.799315\pi$
$433$	− 24.5345i	− 1.17905i	0.807749	−	0.589526i	$-0.200685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−18.5753	−0.888579
$438$	0	0
$439$	−26.1063	−1.24598	−0.622992	−	0.782228i	$-0.714083\pi$
$439$	−26.1063	−1.24598	−0.622992	+	0.782228i	$0.714083\pi$
$440$	0	0
$441$	1.93993	0.0923774
$442$	0	0
$443$	29.4964	1.40141	0.700707	−	0.713449i	$-0.252868\pi$
$443$	29.4964	1.40141	0.700707	+	0.713449i	$0.252868\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 16.7346i	− 0.791518i
$448$	0	0
$449$	18.7321	0.884021	0.442010	−	0.897010i	$-0.354265\pi$
$449$	18.7321	0.884021	0.442010	+	0.897010i	$0.354265\pi$
$450$	0	0
$451$	24.9035i	1.17266i
$452$	0	0
$453$	−19.3033	−0.906947
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.111216i	− 0.00520245i	−0.999997	−	0.00260123i	$-0.999172\pi$
$457$	− 0.111216i	− 0.00520245i	0.999997	−	0.00260123i	$-0.000827997\pi$
$458$	0	0
$459$	− 13.0172i	− 0.607590i
$460$	0	0
$461$	3.71668i	0.173103i	0.996247	+	0.0865516i	$0.0275847\pi$
$461$	3.71668i	0.173103i	−0.996247	+	0.0865516i	$0.972415\pi$
$462$	0	0
$463$	− 11.9601i	− 0.555831i	−0.960606	−	0.277915i	$-0.910356\pi$
$463$	− 11.9601i	− 0.555831i	0.960606	−	0.277915i	$-0.0896435\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.1232	0.792365	0.396183	−	0.918172i	$-0.370335\pi$
$467$	17.1232	0.792365	0.396183	+	0.918172i	$0.370335\pi$
$468$	0	0
$469$	14.2336i	0.657244i
$470$	0	0
$471$	10.2594	0.472728
$472$	0	0
$473$	20.3332i	0.934921i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.0170915	−0.000782567 0
$478$	0	0
$479$	7.70565	0.352080	0.176040	−	0.984383i	$-0.443671\pi$
$479$	7.70565	0.352080	0.176040	+	0.984383i	$0.443671\pi$
$480$	0	0
$481$	21.3484	0.973404
$482$	0	0
$483$	−9.02410	−0.410611
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 0.471068i	− 0.0213461i	−0.999943	−	0.0106731i	$-0.996603\pi$
$487$	− 0.471068i	− 0.0213461i	0.999943	−	0.0106731i	$-0.00339740\pi$
$488$	0	0
$489$	10.0639	0.455107
$490$	0	0
$491$	− 42.9128i	− 1.93663i	−0.249735	−	0.968314i	$-0.580343\pi$
$491$	− 42.9128i	− 1.93663i	0.249735	−	0.968314i	$-0.419657\pi$
$492$	0	0
$493$	14.8993	0.671032
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.3097i	− 0.552167i
$498$	0	0
$499$	20.7158i	0.927365i	0.886001	+	0.463683i	$0.153472\pi$
$499$	20.7158i	0.927365i	−0.886001	+	0.463683i	$0.846528\pi$
$500$	0	0
$501$	− 35.6964i	− 1.59480i
$502$	0	0
$503$	− 6.72612i	− 0.299903i	−0.988693	−	0.149951i	$-0.952088\pi$
$503$	− 6.72612i	− 0.299903i	0.988693	−	0.149951i	$-0.0479117\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.70189	0.342053
$508$	0	0
$509$	− 28.6335i	− 1.26916i	−0.772858	−	0.634579i	$-0.781174\pi$
$509$	− 28.6335i	− 1.26916i	0.772858	−	0.634579i	$-0.218826\pi$
$510$	0	0
$511$	16.3327	0.722518
$512$	0	0
$513$	− 24.0062i	− 1.05990i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	57.9486	2.54858
$518$	0	0
$519$	21.9637	0.964102
$520$	0	0
$521$	−23.0885	−1.01152	−0.505762	−	0.862673i	$-0.668789\pi$
$521$	−23.0885	−1.01152	−0.505762	+	0.862673i	$0.668789\pi$
$522$	0	0
$523$	−11.3256	−0.495236	−0.247618	−	0.968858i	$-0.579648\pi$
$523$	−11.3256	−0.495236	−0.247618	+	0.968858i	$0.579648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.5337i	0.981581i
$528$	0	0
$529$	9.03188	0.392690
$530$	0	0
$531$	0.375844i	0.0163102i
$532$	0	0
$533$	−11.1908	−0.484727
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20.2140i	− 0.872298i
$538$	0	0
$539$	− 34.7904i	− 1.49853i
$540$	0	0
$541$	− 28.1159i	− 1.20880i	−0.796683	−	0.604398i	$-0.793414\pi$
$541$	− 28.1159i	− 1.20880i	0.796683	−	0.604398i	$-0.206586\pi$
$542$	0	0
$543$	34.1920i	1.46732i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.7268	−1.10000	−0.549998	−	0.835166i	$-0.685372\pi$
$547$	−25.7268	−1.10000	−0.549998	+	0.835166i	$0.685372\pi$
$548$	0	0
$549$	1.92624i	0.0822098i
$550$	0	0
$551$	27.4772	1.17057
$552$	0	0
$553$	1.89245i	0.0804750i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.3682	0.481685	0.240842	−	0.970564i	$-0.422576\pi$
$557$	11.3682	0.481685	0.240842	+	0.970564i	$0.422576\pi$
$558$	0	0
$559$	−9.13705	−0.386456
$560$	0	0
$561$	32.6575	1.37880
$562$	0	0
$563$	6.10463	0.257279	0.128640	−	0.991691i	$-0.458939\pi$
$563$	6.10463	0.257279	0.128640	+	0.991691i	$0.458939\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 13.1156i	− 0.550803i
$568$	0	0
$569$	−20.0788	−0.841749	−0.420875	−	0.907119i	$-0.638277\pi$
$569$	−20.0788	−0.841749	−0.420875	+	0.907119i	$0.638277\pi$
$570$	0	0
$571$	12.1145i	0.506975i	0.967339	+	0.253487i	$0.0815776\pi$
$571$	12.1145i	0.506975i	−0.967339	+	0.253487i	$0.918422\pi$
$572$	0	0
$573$	19.5025	0.814729
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.1530i	1.54670i	0.633979	+	0.773351i	$0.281421\pi$
$577$	37.1530i	1.54670i	−0.633979	+	0.773351i	$0.718579\pi$
$578$	0	0
$579$	− 32.9937i	− 1.37117i
$580$	0	0
$581$	− 17.1402i	− 0.711095i
$582$	0	0
$583$	0.306517i	0.0126946i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6112	0.603071	0.301535	−	0.953455i	$-0.402501\pi$
$587$	14.6112	0.603071	0.301535	+	0.953455i	$0.402501\pi$
$588$	0	0
$589$	41.5563i	1.71230i
$590$	0	0
$591$	43.4612	1.78776
$592$	0	0
$593$	− 43.9414i	− 1.80446i	−0.431257	−	0.902229i	$-0.641930\pi$
$593$	− 43.9414i	− 1.80446i	0.431257	−	0.902229i	$-0.358070\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.9067	−1.34678
$598$	0	0
$599$	−19.8918	−0.812756	−0.406378	−	0.913705i	$-0.633208\pi$
$599$	−19.8918	−0.812756	−0.406378	+	0.913705i	$0.633208\pi$
$600$	0	0
$601$	−7.20689	−0.293975	−0.146988	−	0.989138i	$-0.546958\pi$
$601$	−7.20689	−0.293975	−0.146988	+	0.989138i	$0.546958\pi$
$602$	0	0
$603$	3.98314	0.162206
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.17988i	0.250834i	0.992104	+	0.125417i	$0.0400268\pi$
$607$	6.17988i	0.250834i	−0.992104	+	0.125417i	$0.959973\pi$
$608$	0	0
$609$	13.3487	0.540917
$610$	0	0
$611$	26.0401i	1.05347i
$612$	0	0
$613$	14.3999	0.581605	0.290803	−	0.956783i	$-0.406078\pi$
$613$	14.3999	0.581605	0.290803	+	0.956783i	$0.406078\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.5229i	1.55087i	0.631425	+	0.775437i	$0.282470\pi$
$617$	38.5229i	1.55087i	−0.631425	+	0.775437i	$0.717530\pi$
$618$	0	0
$619$	4.77140i	0.191779i	0.995392	+	0.0958894i	$0.0305695\pi$
$619$	4.77140i	0.191779i	−0.995392	+	0.0958894i	$0.969430\pi$
$620$	0	0
$621$	− 18.0519i	− 0.724399i
$622$	0	0
$623$	− 5.18606i	− 0.207775i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	60.2267	2.40522
$628$	0	0
$629$	− 19.3911i	− 0.773175i
$630$	0	0
$631$	16.4566	0.655127	0.327563	−	0.944829i	$-0.393772\pi$
$631$	16.4566	0.655127	0.327563	+	0.944829i	$0.393772\pi$
$632$	0	0
$633$	27.7019i	1.10105i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.6336	0.619427
$638$	0	0
$639$	−3.44477	−0.136273
$640$	0	0
$641$	37.6381	1.48662	0.743309	−	0.668949i	$-0.233255\pi$
$641$	37.6381	1.48662	0.743309	+	0.668949i	$0.233255\pi$
$642$	0	0
$643$	−0.647618	−0.0255396	−0.0127698	−	0.999918i	$-0.504065\pi$
$643$	−0.647618	−0.0255396	−0.0127698	+	0.999918i	$0.504065\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.6274i	1.59723i	0.601845	+	0.798613i	$0.294433\pi$
$647$	40.6274i	1.59723i	−0.601845	+	0.798613i	$0.705567\pi$
$648$	0	0
$649$	6.74033	0.264581
$650$	0	0
$651$	20.1885i	0.791250i
$652$	0	0
$653$	−21.7949	−0.852900	−0.426450	−	0.904511i	$-0.640236\pi$
$653$	−21.7949	−0.852900	−0.426450	+	0.904511i	$0.640236\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.57057i	− 0.178315i
$658$	0	0
$659$	− 10.4657i	− 0.407684i	−0.979004	−	0.203842i	$-0.934657\pi$
$659$	− 10.4657i	− 0.407684i	0.979004	−	0.203842i	$-0.0653429\pi$
$660$	0	0
$661$	23.8297i	0.926870i	0.886131	+	0.463435i	$0.153383\pi$
$661$	23.8297i	0.926870i	−0.886131	+	0.463435i	$0.846617\pi$
$662$	0	0
$663$	14.6752i	0.569937i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.6620	0.800037
$668$	0	0
$669$	− 38.6474i	− 1.49419i
$670$	0	0
$671$	34.5449	1.33359
$672$	0	0
$673$	− 18.0022i	− 0.693935i	−0.937877	−	0.346968i	$-0.887211\pi$
$673$	− 18.0022i	− 0.693935i	0.937877	−	0.346968i	$-0.112789\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0024	1.46055	0.730276	−	0.683152i	$-0.239391\pi$
$677$	38.0024	1.46055	0.730276	+	0.683152i	$0.239391\pi$
$678$	0	0
$679$	−1.32473	−0.0508386
$680$	0	0
$681$	−12.7448	−0.488382
$682$	0	0
$683$	44.0579	1.68583	0.842915	−	0.538046i	$-0.180837\pi$
$683$	44.0579	1.68583	0.842915	+	0.538046i	$0.180837\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	36.4859i	1.39202i
$688$	0	0
$689$	−0.137739	−0.00524742
$690$	0	0
$691$	6.13682i	0.233456i	0.993164	+	0.116728i	$0.0372405\pi$
$691$	6.13682i	0.233456i	−0.993164	+	0.116728i	$0.962759\pi$
$692$	0	0
$693$	3.19824	0.121491
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.1648i	0.385019i
$698$	0	0
$699$	17.3215i	0.655158i
$700$	0	0
$701$	41.6486i	1.57305i	0.617561	+	0.786523i	$0.288121\pi$
$701$	41.6486i	1.57305i	−0.617561	+	0.786523i	$0.711879\pi$
$702$	0	0
$703$	− 35.7609i	− 1.34875i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.29140	0.123786
$708$	0	0
$709$	46.6032i	1.75022i	0.483925	+	0.875109i	$0.339211\pi$
$709$	46.6032i	1.75022i	−0.483925	+	0.875109i	$0.660789\pi$
$710$	0	0
$711$	0.529584	0.0198610
$712$	0	0
$713$	31.2492i	1.17029i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.2497	1.46581
$718$	0	0
$719$	−33.3634	−1.24424	−0.622122	−	0.782920i	$-0.713729\pi$
$719$	−33.3634	−1.24424	−0.622122	+	0.782920i	$0.713729\pi$
$720$	0	0
$721$	7.02995	0.261809
$722$	0	0
$723$	−20.9435	−0.778895
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 34.3672i	− 1.27461i	−0.770612	−	0.637305i	$-0.780049\pi$
$727$	− 34.3672i	− 1.27461i	0.770612	−	0.637305i	$-0.219951\pi$
$728$	0	0
$729$	22.9231	0.849002
$730$	0	0
$731$	8.29934i	0.306962i
$732$	0	0
$733$	−40.7127	−1.50376	−0.751878	−	0.659302i	$-0.770852\pi$
$733$	−40.7127	−1.50376	−0.751878	+	0.659302i	$0.770852\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 71.4331i	− 2.63127i
$738$	0	0
$739$	22.9998i	0.846061i	0.906115	+	0.423030i	$0.139034\pi$
$739$	22.9998i	0.846061i	−0.906115	+	0.423030i	$0.860966\pi$
$740$	0	0
$741$	27.0638i	0.994215i
$742$	0	0
$743$	− 14.7518i	− 0.541192i	−0.962693	−	0.270596i	$-0.912779\pi$
$743$	− 14.7518i	− 0.541192i	0.962693	−	0.270596i	$-0.0872208\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.79654	−0.175496
$748$	0	0
$749$	1.27830i	0.0467082i
$750$	0	0
$751$	19.6790	0.718099	0.359049	−	0.933319i	$-0.383101\pi$
$751$	19.6790	0.718099	0.359049	+	0.933319i	$0.383101\pi$
$752$	0	0
$753$	− 5.06770i	− 0.184677i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.22189	0.117102	0.0585508	−	0.998284i	$-0.481352\pi$
$757$	3.22189	0.117102	0.0585508	+	0.998284i	$0.481352\pi$
$758$	0	0
$759$	45.2887	1.64388
$760$	0	0
$761$	21.5082	0.779672	0.389836	−	0.920884i	$-0.372532\pi$
$761$	21.5082	0.779672	0.389836	+	0.920884i	$0.372532\pi$
$762$	0	0
$763$	24.7938	0.897596
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.02888i	0.109366i
$768$	0	0
$769$	−16.0974	−0.580487	−0.290243	−	0.956953i	$-0.593736\pi$
$769$	−16.0974	−0.580487	−0.290243	+	0.956953i	$0.593736\pi$
$770$	0	0
$771$	22.3324i	0.804281i
$772$	0	0
$773$	−2.13036	−0.0766237	−0.0383118	−	0.999266i	$-0.512198\pi$
$773$	−2.13036	−0.0766237	−0.0383118	+	0.999266i	$0.512198\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 17.3731i	− 0.623255i
$778$	0	0
$779$	18.7458i	0.671639i
$780$	0	0
$781$	61.7781i	2.21059i
$782$	0	0
$783$	26.7029i	0.954285i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.4913	0.409619	0.204810	−	0.978802i	$-0.434342\pi$
$787$	11.4913	0.409619	0.204810	+	0.978802i	$0.434342\pi$
$788$	0	0
$789$	− 41.1260i	− 1.46412i
$790$	0	0
$791$	−16.2892	−0.579177
$792$	0	0
$793$	15.5233i	0.551249i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.3363	−0.507818	−0.253909	−	0.967228i	$-0.581716\pi$
$797$	−14.3363	−0.507818	−0.253909	+	0.967228i	$0.581716\pi$
$798$	0	0
$799$	23.6527	0.836773
$800$	0	0
$801$	−1.45127	−0.0512782
$802$	0	0
$803$	−81.9681	−2.89259
$804$	0	0
$805$	0	0
$806$	0	0
$807$	54.6074i	1.92227i
$808$	0	0
$809$	−43.6620	−1.53508	−0.767538	−	0.641004i	$-0.778518\pi$
$809$	−43.6620	−1.53508	−0.767538	+	0.641004i	$0.778518\pi$
$810$	0	0
$811$	18.8353i	0.661396i	0.943737	+	0.330698i	$0.107284\pi$
$811$	18.8353i	0.661396i	−0.943737	+	0.330698i	$0.892716\pi$
$812$	0	0
$813$	27.7441	0.973027
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.3056i	0.535474i
$818$	0	0
$819$	1.43718i	0.0502193i
$820$	0	0
$821$	− 1.64016i	− 0.0572421i	−0.999590	−	0.0286210i	$-0.990888\pi$
$821$	− 1.64016i	− 0.0572421i	0.999590	−	0.0286210i	$-0.00911160\pi$
$822$	0	0
$823$	− 43.7219i	− 1.52405i	−0.647549	−	0.762024i	$-0.724206\pi$
$823$	− 43.7219i	− 1.52405i	0.647549	−	0.762024i	$-0.275794\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.2651	1.12197	0.560983	−	0.827827i	$-0.310423\pi$
$827$	32.2651	1.12197	0.560983	+	0.827827i	$0.310423\pi$
$828$	0	0
$829$	6.83658i	0.237444i	0.992928	+	0.118722i	$0.0378798\pi$
$829$	6.83658i	0.237444i	−0.992928	+	0.118722i	$0.962120\pi$
$830$	0	0
$831$	−17.4589	−0.605642
$832$	0	0
$833$	− 14.2003i	− 0.492011i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−40.3854	−1.39592
$838$	0	0
$839$	12.3983	0.428037	0.214018	−	0.976830i	$-0.431345\pi$
$839$	12.3983	0.428037	0.214018	+	0.976830i	$0.431345\pi$
$840$	0	0
$841$	−1.56388	−0.0539268
$842$	0	0
$843$	32.2804	1.11180
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 42.8849i	− 1.47354i
$848$	0	0
$849$	−28.1243	−0.965224
$850$	0	0
$851$	− 26.8912i	− 0.921818i
$852$	0	0
$853$	10.2780	0.351911	0.175955	−	0.984398i	$-0.443699\pi$
$853$	10.2780	0.351911	0.175955	+	0.984398i	$0.443699\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1493i	0.380852i	0.981702	+	0.190426i	$0.0609870\pi$
$857$	11.1493i	0.380852i	−0.981702	+	0.190426i	$0.939013\pi$
$858$	0	0
$859$	30.3826i	1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$859$	30.3826i	1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$860$	0	0
$861$	9.10692i	0.310363i
$862$	0	0
$863$	46.2899i	1.57573i	0.615851	+	0.787863i	$0.288812\pi$
$863$	46.2899i	1.57573i	−0.615851	+	0.787863i	$0.711188\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−17.8696	−0.606885
$868$	0	0
$869$	− 9.49750i	− 0.322181i
$870$	0	0
$871$	32.0996	1.08765
$872$	0	0
$873$	0.370715i	0.0125468i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.8781	1.34659	0.673293	−	0.739376i	$-0.264879\pi$
$877$	39.8781	1.34659	0.673293	+	0.739376i	$0.264879\pi$
$878$	0	0
$879$	5.84872	0.197273
$880$	0	0
$881$	38.6037	1.30059	0.650297	−	0.759680i	$-0.274645\pi$
$881$	38.6037	1.30059	0.650297	+	0.759680i	$0.274645\pi$
$882$	0	0
$883$	−57.2207	−1.92563	−0.962814	−	0.270164i	$-0.912922\pi$
$883$	−57.2207	−1.92563	−0.962814	+	0.270164i	$0.912922\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.8099i	0.396537i	0.980148	+	0.198269i	$0.0635319\pi$
$887$	11.8099i	0.396537i	−0.980148	+	0.198269i	$0.936468\pi$
$888$	0	0
$889$	15.5355	0.521045
$890$	0	0
$891$	65.8224i	2.20513i
$892$	0	0
$893$	43.6201	1.45969
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.3512i	0.679508i
$898$	0	0
$899$	− 46.2246i	− 1.54168i
$900$	0	0
$901$	0.125110i	0.00416803i
$902$	0	0
$903$	7.43561i	0.247441i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.7370	−1.21983	−0.609917	−	0.792466i	$-0.708797\pi$
$907$	−36.7370	−1.21983	−0.609917	+	0.792466i	$0.708797\pi$
$908$	0	0
$909$	− 0.921071i	− 0.0305500i
$910$	0	0
$911$	16.4318	0.544409	0.272205	−	0.962239i	$-0.412247\pi$
$911$	16.4318	0.544409	0.272205	+	0.962239i	$0.412247\pi$
$912$	0	0
$913$	86.0205i	2.84686i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.6738	0.550618
$918$	0	0
$919$	−30.9480	−1.02088	−0.510440	−	0.859913i	$-0.670518\pi$
$919$	−30.9480	−1.02088	−0.510440	+	0.859913i	$0.670518\pi$
$920$	0	0
$921$	51.4916	1.69671
$922$	0	0
$923$	−27.7610	−0.913764
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.96727i	− 0.0646136i
$928$	0	0
$929$	−11.4537	−0.375784	−0.187892	−	0.982190i	$-0.560165\pi$
$929$	−11.4537	−0.375784	−0.187892	+	0.982190i	$0.560165\pi$
$930$	0	0
$931$	− 26.1880i	− 0.858278i
$932$	0	0
$933$	51.0147	1.67015
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.7274i	0.971151i	0.874195	+	0.485575i	$0.161390\pi$
$937$	29.7274i	0.971151i	−0.874195	+	0.485575i	$0.838610\pi$
$938$	0	0
$939$	39.7228i	1.29630i
$940$	0	0
$941$	− 22.9601i	− 0.748477i	−0.927332	−	0.374239i	$-0.877904\pi$
$941$	− 22.9601i	− 0.748477i	0.927332	−	0.374239i	$-0.122096\pi$
$942$	0	0
$943$	14.0963i	0.459039i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.4333	−1.15143	−0.575714	−	0.817651i	$-0.695276\pi$
$947$	−35.4333	−1.15143	−0.575714	+	0.817651i	$0.695276\pi$
$948$	0	0
$949$	− 36.8337i	− 1.19567i
$950$	0	0
$951$	29.3699	0.952385
$952$	0	0
$953$	1.35496i	0.0438914i	0.999759	+	0.0219457i	$0.00698610\pi$
$953$	1.35496i	0.0438914i	−0.999759	+	0.0219457i	$0.993014\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−66.9924	−2.16556
$958$	0	0
$959$	−5.70413	−0.184196
$960$	0	0
$961$	38.9099	1.25516
$962$	0	0
$963$	0.357722	0.0115274
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 19.6245i	− 0.631080i	−0.948912	−	0.315540i	$-0.897814\pi$
$967$	− 19.6245i	− 0.631080i	0.948912	−	0.315540i	$-0.102186\pi$
$968$	0	0
$969$	24.5826	0.789706
$970$	0	0
$971$	− 1.55749i	− 0.0499822i	−0.999688	−	0.0249911i	$-0.992044\pi$
$971$	− 1.55749i	− 0.0499822i	0.999688	−	0.0249911i	$-0.00795575\pi$
$972$	0	0
$973$	4.58884	0.147112
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 49.4288i	− 1.58137i	−0.612226	−	0.790683i	$-0.709726\pi$
$977$	− 49.4288i	− 1.58137i	0.612226	−	0.790683i	$-0.290274\pi$
$978$	0	0
$979$	26.0270i	0.831825i
$980$	0	0
$981$	− 6.93833i	− 0.221524i
$982$	0	0
$983$	18.1091i	0.577591i	0.957391	+	0.288795i	$0.0932547\pi$
$983$	18.1091i	0.577591i	−0.957391	+	0.288795i	$0.906745\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.1911	0.674521
$988$	0	0
$989$	11.5093i	0.365976i
$990$	0	0
$991$	−4.82935	−0.153409	−0.0767046	−	0.997054i	$-0.524440\pi$
$991$	−4.82935	−0.153409	−0.0767046	+	0.997054i	$0.524440\pi$
$992$	0	0
$993$	− 9.90905i	− 0.314454i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18.3807	−0.582122	−0.291061	−	0.956705i	$-0.594008\pi$
$997$	−18.3807	−0.582122	−0.291061	+	0.956705i	$0.594008\pi$
$998$	0	0
$999$	34.7533	1.09955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.f.c.3249.6		20
4.3	odd	2		1000.2.f.c.749.2		20
5.2	odd	4		4000.2.d.c.2001.26		40
5.3	odd	4		4000.2.d.c.2001.15		40
5.4	even	2		4000.2.f.d.3249.15		20
8.3	odd	2		1000.2.f.d.749.20		20
8.5	even	2		4000.2.f.d.3249.16		20
20.3	even	4		1000.2.d.c.501.24	yes	40
20.7	even	4		1000.2.d.c.501.17	✓	40
20.19	odd	2		1000.2.f.d.749.19		20
40.3	even	4		1000.2.d.c.501.23	yes	40
40.13	odd	4		4000.2.d.c.2001.16		40
40.19	odd	2		1000.2.f.c.749.1		20
40.27	even	4		1000.2.d.c.501.18	yes	40
40.29	even	2	inner	4000.2.f.c.3249.5		20
40.37	odd	4		4000.2.d.c.2001.25		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.17	✓	40	20.7	even	4
1000.2.d.c.501.18	yes	40	40.27	even	4
1000.2.d.c.501.23	yes	40	40.3	even	4
1000.2.d.c.501.24	yes	40	20.3	even	4
1000.2.f.c.749.1		20	40.19	odd	2
1000.2.f.c.749.2		20	4.3	odd	2
1000.2.f.d.749.19		20	20.19	odd	2
1000.2.f.d.749.20		20	8.3	odd	2
4000.2.d.c.2001.15		40	5.3	odd	4
4000.2.d.c.2001.16		40	40.13	odd	4
4000.2.d.c.2001.25		40	40.37	odd	4
4000.2.d.c.2001.26		40	5.2	odd	4
4000.2.f.c.3249.5		20	40.29	even	2	inner
4000.2.f.c.3249.6		20	1.1	even	1	trivial
4000.2.f.d.3249.15		20	5.4	even	2
4000.2.f.d.3249.16		20	8.5	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.83526 q^{3} +1.31564i q^{7} +0.368171 q^{9} +O(q^{10})\)	\(q-1.83526 q^{3} +1.31564i q^{7} +0.368171 q^{9} -6.60274i q^{11} +2.96705 q^{13} -2.69502i q^{17} -4.97013i q^{19} -2.41454i q^{21} -3.73739i q^{23} +4.83008 q^{27} +5.52846i q^{29} -8.36121 q^{31} +12.1177i q^{33} +7.19517 q^{37} -5.44530 q^{39} -3.77169 q^{41} -3.07951 q^{43} +8.77645i q^{47} +5.26908 q^{49} +4.94606i q^{51} -0.0464228 q^{53} +9.12147i q^{57} +1.02084i q^{59} +5.23191i q^{61} +0.484382i q^{63} +10.8187 q^{67} +6.85908i q^{69} -9.35643 q^{71} -12.4143i q^{73} +8.68684 q^{77} +1.43842 q^{79} -9.96896 q^{81} -13.0280 q^{83} -10.1462i q^{87} -3.94185 q^{89} +3.90357i q^{91} +15.3450 q^{93} +1.00691i q^{97} -2.43094i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 12 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 + 12 * q^9	\( 20 q - 4 q^{3} + 12 q^{9} + 6 q^{13} + 8 q^{27} - 24 q^{31} + 18 q^{37} + 4 q^{39} + 22 q^{41} + 60 q^{43} - 6 q^{49} + 10 q^{53} + 40 q^{67} - 48 q^{71} + 24 q^{77} + 48 q^{79} - 28 q^{81} - 40 q^{83} + 22 q^{89} + 8 q^{93}+O(q^{100}) \) 20 * q - 4 * q^3 + 12 * q^9 + 6 * q^13 + 8 * q^27 - 24 * q^31 + 18 * q^37 + 4 * q^39 + 22 * q^41 + 60 * q^43 - 6 * q^49 + 10 * q^53 + 40 * q^67 - 48 * q^71 + 24 * q^77 + 48 * q^79 - 28 * q^81 - 40 * q^83 + 22 * q^89 + 8 * q^93

Introduction

L-functions

Modular forms

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