LMFDB - Embedded newform 2576.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.a (trivial)

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:	yes
Analytic conductor:	$20.5694635607$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.27841$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27841 q^{3} -0.477194 q^{5} -1.00000 q^{7} -1.36566 q^{9} +O(q^{10})$	$q-1.27841 q^{3} -0.477194 q^{5} -1.00000 q^{7} -1.36566 q^{9} +2.43556 q^{11} -7.18356 q^{13} +0.610051 q^{15} +0.302704 q^{17} +5.60244 q^{19} +1.27841 q^{21} +1.00000 q^{23} -4.77229 q^{25} +5.58112 q^{27} +0.0166785 q^{29} -7.66075 q^{31} -3.11365 q^{33} +0.477194 q^{35} -8.93916 q^{37} +9.18356 q^{39} +10.6115 q^{41} -5.03039 q^{43} +0.651684 q^{45} +3.27841 q^{47} +1.00000 q^{49} -0.386981 q^{51} +3.11365 q^{53} -1.16223 q^{55} -7.16223 q^{57} +7.90514 q^{59} +10.4620 q^{61} +1.36566 q^{63} +3.42795 q^{65} +2.78454 q^{67} -1.27841 q^{69} +0.641953 q^{71} +10.3124 q^{73} +6.10096 q^{75} -2.43556 q^{77} -2.43556 q^{79} -3.03800 q^{81} +11.7343 q^{83} -0.144448 q^{85} -0.0213220 q^{87} -7.59085 q^{89} +7.18356 q^{91} +9.79361 q^{93} -2.67345 q^{95} -1.12525 q^{97} -3.32615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 4 * q^7 + q^9	$ 4 q + 3 q^{3} - 4 q^{7} + q^{9} + 10 q^{11} - 3 q^{13} + 6 q^{15} - 4 q^{17} + 10 q^{19} - 3 q^{21} + 4 q^{23} + 4 q^{25} + 9 q^{27} - 13 q^{29} - 3 q^{31} + 20 q^{33} + 11 q^{39} + q^{41} + 8 q^{43} + 4 q^{45} + 5 q^{47} + 4 q^{49} + 4 q^{51} - 20 q^{53} + 22 q^{55} - 2 q^{57} + 14 q^{59} + 8 q^{61} - q^{63} - 2 q^{65} + 18 q^{67} + 3 q^{69} + 25 q^{71} + 15 q^{73} + 11 q^{75} - 10 q^{77} - 10 q^{79} + 36 q^{83} - 28 q^{85} - q^{87} + 4 q^{89} + 3 q^{91} + 17 q^{93} + 36 q^{95} + 6 q^{97} + 28 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 4 * q^7 + q^9 + 10 * q^11 - 3 * q^13 + 6 * q^15 - 4 * q^17 + 10 * q^19 - 3 * q^21 + 4 * q^23 + 4 * q^25 + 9 * q^27 - 13 * q^29 - 3 * q^31 + 20 * q^33 + 11 * q^39 + q^41 + 8 * q^43 + 4 * q^45 + 5 * q^47 + 4 * q^49 + 4 * q^51 - 20 * q^53 + 22 * q^55 - 2 * q^57 + 14 * q^59 + 8 * q^61 - q^63 - 2 * q^65 + 18 * q^67 + 3 * q^69 + 25 * q^71 + 15 * q^73 + 11 * q^75 - 10 * q^77 - 10 * q^79 + 36 * q^83 - 28 * q^85 - q^87 + 4 * q^89 + 3 * q^91 + 17 * q^93 + 36 * q^95 + 6 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.27841	−0.738092	−0.369046	−	0.929411i	$-0.620316\pi$
$3$	−1.27841	−0.738092	−0.369046	+	0.929411i	$0.620316\pi$
$4$	0	0
$5$	−0.477194	−0.213408	−0.106704	−	0.994291i	$-0.534030\pi$
$5$	−0.477194	−0.213408	−0.106704	+	0.994291i	$0.534030\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.36566	−0.455220
$10$	0	0
$11$	2.43556	0.734349	0.367175	−	0.930152i	$-0.380325\pi$
$11$	2.43556	0.734349	0.367175	+	0.930152i	$0.380325\pi$
$12$	0	0
$13$	−7.18356	−1.99236	−0.996180	−	0.0873224i	$-0.972169\pi$
$13$	−7.18356	−1.99236	−0.996180	+	0.0873224i	$0.972169\pi$
$14$	0	0
$15$	0.610051	0.157515
$16$	0	0
$17$	0.302704	0.0734165	0.0367082	−	0.999326i	$-0.488313\pi$
$17$	0.302704	0.0734165	0.0367082	+	0.999326i	$0.488313\pi$
$18$	0	0
$19$	5.60244	1.28529	0.642644	−	0.766165i	$-0.277837\pi$
$19$	5.60244	1.28529	0.642644	+	0.766165i	$0.277837\pi$
$20$	0	0
$21$	1.27841	0.278973
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.77229	−0.954457
$26$	0	0
$27$	5.58112	1.07409
$28$	0	0
$29$	0.0166785	0.00309712	0.00154856	−	0.999999i	$-0.499507\pi$
$29$	0.0166785	0.00309712	0.00154856	+	0.999999i	$0.499507\pi$
$30$	0	0
$31$	−7.66075	−1.37591	−0.687956	−	0.725753i	$-0.741492\pi$
$31$	−7.66075	−1.37591	−0.687956	+	0.725753i	$0.741492\pi$
$32$	0	0
$33$	−3.11365	−0.542018
$34$	0	0
$35$	0.477194	0.0806605
$36$	0	0
$37$	−8.93916	−1.46959	−0.734795	−	0.678289i	$-0.762722\pi$
$37$	−8.93916	−1.46959	−0.734795	+	0.678289i	$0.762722\pi$
$38$	0	0
$39$	9.18356	1.47055
$40$	0	0
$41$	10.6115	1.65724	0.828619	−	0.559812i	$-0.189127\pi$
$41$	10.6115	1.65724	0.828619	+	0.559812i	$0.189127\pi$
$42$	0	0
$43$	−5.03039	−0.767127	−0.383564	−	0.923514i	$-0.625303\pi$
$43$	−5.03039	−0.767127	−0.383564	+	0.923514i	$0.625303\pi$
$44$	0	0
$45$	0.651684	0.0971473
$46$	0	0
$47$	3.27841	0.478206	0.239103	−	0.970994i	$-0.423147\pi$
$47$	3.27841	0.478206	0.239103	+	0.970994i	$0.423147\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.386981	−0.0541881
$52$	0	0
$53$	3.11365	0.427693	0.213847	−	0.976867i	$-0.431401\pi$
$53$	3.11365	0.427693	0.213847	+	0.976867i	$0.431401\pi$
$54$	0	0
$55$	−1.16223	−0.156716
$56$	0	0
$57$	−7.16223	−0.948661
$58$	0	0
$59$	7.90514	1.02916	0.514581	−	0.857442i	$-0.327947\pi$
$59$	7.90514	1.02916	0.514581	+	0.857442i	$0.327947\pi$
$60$	0	0
$61$	10.4620	1.33952	0.669759	−	0.742579i	$-0.266397\pi$
$61$	10.4620	1.33952	0.669759	+	0.742579i	$0.266397\pi$
$62$	0	0
$63$	1.36566	0.172057
$64$	0	0
$65$	3.42795	0.425185
$66$	0	0
$67$	2.78454	0.340186	0.170093	−	0.985428i	$-0.445593\pi$
$67$	2.78454	0.340186	0.170093	+	0.985428i	$0.445593\pi$
$68$	0	0
$69$	−1.27841	−0.153903
$70$	0	0
$71$	0.641953	0.0761858	0.0380929	−	0.999274i	$-0.487872\pi$
$71$	0.641953	0.0761858	0.0380929	+	0.999274i	$0.487872\pi$
$72$	0	0
$73$	10.3124	1.20698	0.603490	−	0.797371i	$-0.293776\pi$
$73$	10.3124	1.20698	0.603490	+	0.797371i	$0.293776\pi$
$74$	0	0
$75$	6.10096	0.704478
$76$	0	0
$77$	−2.43556	−0.277558
$78$	0	0
$79$	−2.43556	−0.274022	−0.137011	−	0.990570i	$-0.543750\pi$
$79$	−2.43556	−0.274022	−0.137011	+	0.990570i	$0.543750\pi$
$80$	0	0
$81$	−3.03800	−0.337556
$82$	0	0
$83$	11.7343	1.28801	0.644003	−	0.765023i	$-0.277273\pi$
$83$	11.7343	1.28801	0.644003	+	0.765023i	$0.277273\pi$
$84$	0	0
$85$	−0.144448	−0.0156676
$86$	0	0
$87$	−0.0213220	−0.00228596
$88$	0	0
$89$	−7.59085	−0.804628	−0.402314	−	0.915502i	$-0.631794\pi$
$89$	−7.59085	−0.804628	−0.402314	+	0.915502i	$0.631794\pi$
$90$	0	0
$91$	7.18356	0.753041
$92$	0	0
$93$	9.79361	1.01555
$94$	0	0
$95$	−2.67345	−0.274290
$96$	0	0
$97$	−1.12525	−0.114251	−0.0571257	−	0.998367i	$-0.518194\pi$
$97$	−1.12525	−0.114251	−0.0571257	+	0.998367i	$0.518194\pi$
$98$	0	0
$99$	−3.32615	−0.334290
$100$	0	0
$101$	−9.46131	−0.941435	−0.470718	−	0.882284i	$-0.656005\pi$
$101$	−9.46131	−0.941435	−0.470718	+	0.882284i	$0.656005\pi$
$102$	0	0
$103$	−0.348980	−0.0343860	−0.0171930	−	0.999852i	$-0.505473\pi$
$103$	−0.348980	−0.0343860	−0.0171930	+	0.999852i	$0.505473\pi$
$104$	0	0
$105$	−0.610051	−0.0595349
$106$	0	0
$107$	0.223070	0.0215650	0.0107825	−	0.999942i	$-0.496568\pi$
$107$	0.223070	0.0215650	0.0107825	+	0.999942i	$0.496568\pi$
$108$	0	0
$109$	16.5264	1.58294	0.791470	−	0.611208i	$-0.209316\pi$
$109$	16.5264	1.58294	0.791470	+	0.611208i	$0.209316\pi$
$110$	0	0
$111$	11.4279	1.08469
$112$	0	0
$113$	−16.0181	−1.50686	−0.753430	−	0.657529i	$-0.771602\pi$
$113$	−16.0181	−1.50686	−0.753430	+	0.657529i	$0.771602\pi$
$114$	0	0
$115$	−0.477194	−0.0444986
$116$	0	0
$117$	9.81029	0.906961
$118$	0	0
$119$	−0.302704	−0.0277488
$120$	0	0
$121$	−5.06804	−0.460731
$122$	0	0
$123$	−13.5659	−1.22320
$124$	0	0
$125$	4.66328	0.417096
$126$	0	0
$127$	0.0699024	0.00620284	0.00310142	−	0.999995i	$-0.499013\pi$
$127$	0.0699024	0.00620284	0.00310142	+	0.999995i	$0.499013\pi$
$128$	0	0
$129$	6.43092	0.566211
$130$	0	0
$131$	8.14954	0.712028	0.356014	−	0.934481i	$-0.384135\pi$
$131$	8.14954	0.712028	0.356014	+	0.934481i	$0.384135\pi$
$132$	0	0
$133$	−5.60244	−0.485793
$134$	0	0
$135$	−2.66328	−0.229218
$136$	0	0
$137$	−9.42795	−0.805484	−0.402742	−	0.915314i	$-0.631943\pi$
$137$	−9.42795	−0.805484	−0.402742	+	0.915314i	$0.631943\pi$
$138$	0	0
$139$	13.5209	1.14683	0.573416	−	0.819264i	$-0.305618\pi$
$139$	13.5209	1.14683	0.573416	+	0.819264i	$0.305618\pi$
$140$	0	0
$141$	−4.19117	−0.352960
$142$	0	0
$143$	−17.4960	−1.46309
$144$	0	0
$145$	−0.00795888	−0.000660949 0
$146$	0	0
$147$	−1.27841	−0.105442
$148$	0	0
$149$	14.4157	1.18098	0.590490	−	0.807045i	$-0.298935\pi$
$149$	14.4157	1.18098	0.590490	+	0.807045i	$0.298935\pi$
$150$	0	0
$151$	−0.418883	−0.0340882	−0.0170441	−	0.999855i	$-0.505426\pi$
$151$	−0.418883	−0.0340882	−0.0170441	+	0.999855i	$0.505426\pi$
$152$	0	0
$153$	−0.413390	−0.0334206
$154$	0	0
$155$	3.65566	0.293630
$156$	0	0
$157$	24.3555	1.94378	0.971891	−	0.235431i	$-0.0756500\pi$
$157$	24.3555	1.94378	0.971891	+	0.235431i	$0.0756500\pi$
$158$	0	0
$159$	−3.98054	−0.315677
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−4.18653	−0.327914	−0.163957	−	0.986467i	$-0.552426\pi$
$163$	−4.18653	−0.327914	−0.163957	+	0.986467i	$0.552426\pi$
$164$	0	0
$165$	1.48582	0.115671
$166$	0	0
$167$	13.5756	1.05051	0.525257	−	0.850944i	$-0.323969\pi$
$167$	13.5756	1.05051	0.525257	+	0.850944i	$0.323969\pi$
$168$	0	0
$169$	38.6035	2.96950
$170$	0	0
$171$	−7.65102	−0.585088
$172$	0	0
$173$	15.5293	1.18067	0.590337	−	0.807157i	$-0.298995\pi$
$173$	15.5293	1.18067	0.590337	+	0.807157i	$0.298995\pi$
$174$	0	0
$175$	4.77229	0.360751
$176$	0	0
$177$	−10.1060	−0.759617
$178$	0	0
$179$	20.5550	1.53635	0.768175	−	0.640240i	$-0.221165\pi$
$179$	20.5550	1.53635	0.768175	+	0.640240i	$0.221165\pi$
$180$	0	0
$181$	−6.06441	−0.450764	−0.225382	−	0.974270i	$-0.572363\pi$
$181$	−6.06441	−0.450764	−0.225382	+	0.974270i	$0.572363\pi$
$182$	0	0
$183$	−13.3747	−0.988688
$184$	0	0
$185$	4.26571	0.313622
$186$	0	0
$187$	0.737254	0.0539133
$188$	0	0
$189$	−5.58112	−0.405967
$190$	0	0
$191$	14.2838	1.03354	0.516771	−	0.856123i	$-0.327134\pi$
$191$	14.2838	1.03354	0.516771	+	0.856123i	$0.327134\pi$
$192$	0	0
$193$	12.4037	0.892835	0.446417	−	0.894825i	$-0.352700\pi$
$193$	12.4037	0.892835	0.446417	+	0.894825i	$0.352700\pi$
$194$	0	0
$195$	−4.38234	−0.313826
$196$	0	0
$197$	−10.6420	−0.758208	−0.379104	−	0.925354i	$-0.623768\pi$
$197$	−10.6420	−0.758208	−0.379104	+	0.925354i	$0.623768\pi$
$198$	0	0
$199$	11.1318	0.789112	0.394556	−	0.918872i	$-0.370898\pi$
$199$	11.1318	0.789112	0.394556	+	0.918872i	$0.370898\pi$
$200$	0	0
$201$	−3.55980	−0.251089
$202$	0	0
$203$	−0.0166785	−0.00117060
$204$	0	0
$205$	−5.06375	−0.353667
$206$	0	0
$207$	−1.36566	−0.0949198
$208$	0	0
$209$	13.6451	0.943850
$210$	0	0
$211$	25.3489	1.74509	0.872546	−	0.488532i	$-0.162468\pi$
$211$	25.3489	1.74509	0.872546	+	0.488532i	$0.162468\pi$
$212$	0	0
$213$	−0.820682	−0.0562322
$214$	0	0
$215$	2.40047	0.163711
$216$	0	0
$217$	7.66075	0.520046
$218$	0	0
$219$	−13.1836	−0.890862
$220$	0	0
$221$	−2.17449	−0.146272
$222$	0	0
$223$	−3.45900	−0.231632	−0.115816	−	0.993271i	$-0.536948\pi$
$223$	−3.45900	−0.231632	−0.115816	+	0.993271i	$0.536948\pi$
$224$	0	0
$225$	6.51731	0.434488
$226$	0	0
$227$	4.24253	0.281587	0.140793	−	0.990039i	$-0.455035\pi$
$227$	4.24253	0.281587	0.140793	+	0.990039i	$0.455035\pi$
$228$	0	0
$229$	−14.0978	−0.931607	−0.465803	−	0.884888i	$-0.654235\pi$
$229$	−14.0978	−0.931607	−0.465803	+	0.884888i	$0.654235\pi$
$230$	0	0
$231$	3.11365	0.204863
$232$	0	0
$233$	−29.0239	−1.90142	−0.950709	−	0.310085i	$-0.899643\pi$
$233$	−29.0239	−1.90142	−0.950709	+	0.310085i	$0.899643\pi$
$234$	0	0
$235$	−1.56444	−0.102053
$236$	0	0
$237$	3.11365	0.202254
$238$	0	0
$239$	19.5507	1.26463	0.632314	−	0.774712i	$-0.282105\pi$
$239$	19.5507	1.26463	0.632314	+	0.774712i	$0.282105\pi$
$240$	0	0
$241$	10.6031	0.683006	0.341503	−	0.939881i	$-0.389064\pi$
$241$	10.6031	0.683006	0.341503	+	0.939881i	$0.389064\pi$
$242$	0	0
$243$	−12.8595	−0.824939
$244$	0	0
$245$	−0.477194	−0.0304868
$246$	0	0
$247$	−40.2454	−2.56076
$248$	0	0
$249$	−15.0013	−0.950667
$250$	0	0
$251$	−30.4765	−1.92366	−0.961829	−	0.273651i	$-0.911769\pi$
$251$	−30.4765	−1.92366	−0.961829	+	0.273651i	$0.911769\pi$
$252$	0	0
$253$	2.43556	0.153122
$254$	0	0
$255$	0.184665	0.0115642
$256$	0	0
$257$	13.6339	0.850462	0.425231	−	0.905085i	$-0.360193\pi$
$257$	13.6339	0.850462	0.425231	+	0.905085i	$0.360193\pi$
$258$	0	0
$259$	8.93916	0.555453
$260$	0	0
$261$	−0.0227771	−0.00140987
$262$	0	0
$263$	9.64044	0.594455	0.297227	−	0.954807i	$-0.403938\pi$
$263$	9.64044	0.594455	0.297227	+	0.954807i	$0.403938\pi$
$264$	0	0
$265$	−1.48582	−0.0912730
$266$	0	0
$267$	9.70424	0.593890
$268$	0	0
$269$	25.6766	1.56553	0.782764	−	0.622318i	$-0.213809\pi$
$269$	25.6766	1.56553	0.782764	+	0.622318i	$0.213809\pi$
$270$	0	0
$271$	−20.4801	−1.24408	−0.622039	−	0.782986i	$-0.713695\pi$
$271$	−20.4801	−1.24408	−0.622039	+	0.782986i	$0.713695\pi$
$272$	0	0
$273$	−9.18356	−0.555814
$274$	0	0
$275$	−11.6232	−0.700905
$276$	0	0
$277$	1.65247	0.0992876	0.0496438	−	0.998767i	$-0.484191\pi$
$277$	1.65247	0.0992876	0.0496438	+	0.998767i	$0.484191\pi$
$278$	0	0
$279$	10.4620	0.626342
$280$	0	0
$281$	−5.35195	−0.319270	−0.159635	−	0.987176i	$-0.551032\pi$
$281$	−5.35195	−0.319270	−0.159635	+	0.987176i	$0.551032\pi$
$282$	0	0
$283$	0.731317	0.0434723	0.0217362	−	0.999764i	$-0.493081\pi$
$283$	0.731317	0.0434723	0.0217362	+	0.999764i	$0.493081\pi$
$284$	0	0
$285$	3.41777	0.202451
$286$	0	0
$287$	−10.6115	−0.626377
$288$	0	0
$289$	−16.9084	−0.994610
$290$	0	0
$291$	1.43853	0.0843281
$292$	0	0
$293$	−10.2528	−0.598975	−0.299487	−	0.954100i	$-0.596816\pi$
$293$	−10.2528	−0.598975	−0.299487	+	0.954100i	$0.596816\pi$
$294$	0	0
$295$	−3.77229	−0.219631
$296$	0	0
$297$	13.5932	0.788755
$298$	0	0
$299$	−7.18356	−0.415436
$300$	0	0
$301$	5.03039	0.289947
$302$	0	0
$303$	12.0955	0.694866
$304$	0	0
$305$	−4.99239	−0.285863
$306$	0	0
$307$	16.9841	0.969335	0.484667	−	0.874699i	$-0.338941\pi$
$307$	16.9841	0.969335	0.484667	+	0.874699i	$0.338941\pi$
$308$	0	0
$309$	0.446141	0.0253801
$310$	0	0
$311$	−8.05831	−0.456945	−0.228472	−	0.973550i	$-0.573373\pi$
$311$	−8.05831	−0.456945	−0.228472	+	0.973550i	$0.573373\pi$
$312$	0	0
$313$	−3.68801	−0.208459	−0.104229	−	0.994553i	$-0.533238\pi$
$313$	−3.68801	−0.208459	−0.104229	+	0.994553i	$0.533238\pi$
$314$	0	0
$315$	−0.651684	−0.0367182
$316$	0	0
$317$	−4.30965	−0.242054	−0.121027	−	0.992649i	$-0.538619\pi$
$317$	−4.30965	−0.242054	−0.121027	+	0.992649i	$0.538619\pi$
$318$	0	0
$319$	0.0406215	0.00227437
$320$	0	0
$321$	−0.285176	−0.0159170
$322$	0	0
$323$	1.69588	0.0943613
$324$	0	0
$325$	34.2820	1.90162
$326$	0	0
$327$	−21.1275	−1.16836
$328$	0	0
$329$	−3.27841	−0.180745
$330$	0	0
$331$	−27.0285	−1.48562	−0.742811	−	0.669501i	$-0.766508\pi$
$331$	−27.0285	−1.48562	−0.742811	+	0.669501i	$0.766508\pi$
$332$	0	0
$333$	12.2078	0.668986
$334$	0	0
$335$	−1.32877	−0.0725982
$336$	0	0
$337$	−9.50395	−0.517713	−0.258857	−	0.965916i	$-0.583346\pi$
$337$	−9.50395	−0.517713	−0.258857	+	0.965916i	$0.583346\pi$
$338$	0	0
$339$	20.4778	1.11220
$340$	0	0
$341$	−18.6582	−1.01040
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.610051	0.0328440
$346$	0	0
$347$	19.8863	1.06755	0.533776	−	0.845626i	$-0.320773\pi$
$347$	19.8863	1.06755	0.533776	+	0.845626i	$0.320773\pi$
$348$	0	0
$349$	1.91064	0.102274	0.0511370	−	0.998692i	$-0.483715\pi$
$349$	1.91064	0.102274	0.0511370	+	0.998692i	$0.483715\pi$
$350$	0	0
$351$	−40.0923	−2.13997
$352$	0	0
$353$	25.4109	1.35248	0.676242	−	0.736680i	$-0.263607\pi$
$353$	25.4109	1.35248	0.676242	+	0.736680i	$0.263607\pi$
$354$	0	0
$355$	−0.306336	−0.0162586
$356$	0	0
$357$	0.386981	0.0204812
$358$	0	0
$359$	−33.9497	−1.79180	−0.895898	−	0.444260i	$-0.853467\pi$
$359$	−33.9497	−1.79180	−0.895898	+	0.444260i	$0.853467\pi$
$360$	0	0
$361$	12.3873	0.651965
$362$	0	0
$363$	6.47905	0.340062
$364$	0	0
$365$	−4.92103	−0.257579
$366$	0	0
$367$	−35.2577	−1.84044	−0.920218	−	0.391405i	$-0.871989\pi$
$367$	−35.2577	−1.84044	−0.920218	+	0.391405i	$0.871989\pi$
$368$	0	0
$369$	−14.4917	−0.754408
$370$	0	0
$371$	−3.11365	−0.161653
$372$	0	0
$373$	21.4381	1.11002	0.555012	−	0.831842i	$-0.312714\pi$
$373$	21.4381	1.11002	0.555012	+	0.831842i	$0.312714\pi$
$374$	0	0
$375$	−5.96159	−0.307855
$376$	0	0
$377$	−0.119811	−0.00617058
$378$	0	0
$379$	14.5754	0.748686	0.374343	−	0.927290i	$-0.377868\pi$
$379$	14.5754	0.748686	0.374343	+	0.927290i	$0.377868\pi$
$380$	0	0
$381$	−0.0893642	−0.00457827
$382$	0	0
$383$	−28.9999	−1.48183	−0.740914	−	0.671600i	$-0.765607\pi$
$383$	−28.9999	−1.48183	−0.740914	+	0.671600i	$0.765607\pi$
$384$	0	0
$385$	1.16223	0.0592330
$386$	0	0
$387$	6.86979	0.349211
$388$	0	0
$389$	4.72336	0.239484	0.119742	−	0.992805i	$-0.461793\pi$
$389$	4.72336	0.239484	0.119742	+	0.992805i	$0.461793\pi$
$390$	0	0
$391$	0.302704	0.0153084
$392$	0	0
$393$	−10.4185	−0.525543
$394$	0	0
$395$	1.16223	0.0584784
$396$	0	0
$397$	−32.2972	−1.62095	−0.810475	−	0.585773i	$-0.800791\pi$
$397$	−32.2972	−1.62095	−0.810475	+	0.585773i	$0.800791\pi$
$398$	0	0
$399$	7.16223	0.358560
$400$	0	0
$401$	22.6054	1.12886	0.564430	−	0.825481i	$-0.309096\pi$
$401$	22.6054	1.12886	0.564430	+	0.825481i	$0.309096\pi$
$402$	0	0
$403$	55.0314	2.74131
$404$	0	0
$405$	1.44972	0.0720369
$406$	0	0
$407$	−21.7719	−1.07919
$408$	0	0
$409$	−10.2097	−0.504838	−0.252419	−	0.967618i	$-0.581226\pi$
$409$	−10.2097	−0.504838	−0.252419	+	0.967618i	$0.581226\pi$
$410$	0	0
$411$	12.0528	0.594522
$412$	0	0
$413$	−7.90514	−0.388987
$414$	0	0
$415$	−5.59953	−0.274870
$416$	0	0
$417$	−17.2854	−0.846468
$418$	0	0
$419$	−18.2273	−0.890462	−0.445231	−	0.895416i	$-0.646878\pi$
$419$	−18.2273	−0.890462	−0.445231	+	0.895416i	$0.646878\pi$
$420$	0	0
$421$	−3.15630	−0.153829	−0.0769143	−	0.997038i	$-0.524507\pi$
$421$	−3.15630	−0.153829	−0.0769143	+	0.997038i	$0.524507\pi$
$422$	0	0
$423$	−4.47719	−0.217689
$424$	0	0
$425$	−1.44459	−0.0700729
$426$	0	0
$427$	−10.4620	−0.506290
$428$	0	0
$429$	22.3671	1.07989
$430$	0	0
$431$	5.39326	0.259784	0.129892	−	0.991528i	$-0.458537\pi$
$431$	5.39326	0.259784	0.129892	+	0.991528i	$0.458537\pi$
$432$	0	0
$433$	6.39526	0.307336	0.153668	−	0.988123i	$-0.450891\pi$
$433$	6.39526	0.307336	0.153668	+	0.988123i	$0.450891\pi$
$434$	0	0
$435$	0.0101747	0.000487842 0
$436$	0	0
$437$	5.60244	0.268001
$438$	0	0
$439$	37.1943	1.77519	0.887594	−	0.460626i	$-0.152375\pi$
$439$	37.1943	1.77519	0.887594	+	0.460626i	$0.152375\pi$
$440$	0	0
$441$	−1.36566	−0.0650314
$442$	0	0
$443$	36.2228	1.72100	0.860498	−	0.509453i	$-0.170152\pi$
$443$	36.2228	1.72100	0.860498	+	0.509453i	$0.170152\pi$
$444$	0	0
$445$	3.62231	0.171714
$446$	0	0
$447$	−18.4292	−0.871672
$448$	0	0
$449$	−30.8859	−1.45760	−0.728799	−	0.684728i	$-0.759921\pi$
$449$	−30.8859	−1.45760	−0.728799	+	0.684728i	$0.759921\pi$
$450$	0	0
$451$	25.8450	1.21699
$452$	0	0
$453$	0.535505	0.0251602
$454$	0	0
$455$	−3.42795	−0.160705
$456$	0	0
$457$	−14.2852	−0.668232	−0.334116	−	0.942532i	$-0.608438\pi$
$457$	−14.2852	−0.668232	−0.334116	+	0.942532i	$0.608438\pi$
$458$	0	0
$459$	1.68943	0.0788556
$460$	0	0
$461$	−9.40069	−0.437834	−0.218917	−	0.975744i	$-0.570252\pi$
$461$	−9.40069	−0.437834	−0.218917	+	0.975744i	$0.570252\pi$
$462$	0	0
$463$	−21.9015	−1.01785	−0.508925	−	0.860811i	$-0.669957\pi$
$463$	−21.9015	−1.01785	−0.508925	+	0.860811i	$0.669957\pi$
$464$	0	0
$465$	−4.67345	−0.216726
$466$	0	0
$467$	−29.9877	−1.38766	−0.693832	−	0.720137i	$-0.744079\pi$
$467$	−29.9877	−1.38766	−0.693832	+	0.720137i	$0.744079\pi$
$468$	0	0
$469$	−2.78454	−0.128578
$470$	0	0
$471$	−31.1364	−1.43469
$472$	0	0
$473$	−12.2518	−0.563339
$474$	0	0
$475$	−26.7364	−1.22675
$476$	0	0
$477$	−4.25219	−0.194694
$478$	0	0
$479$	−4.24253	−0.193846	−0.0969231	−	0.995292i	$-0.530900\pi$
$479$	−4.24253	−0.193846	−0.0969231	+	0.995292i	$0.530900\pi$
$480$	0	0
$481$	64.2150	2.92795
$482$	0	0
$483$	1.27841	0.0581698
$484$	0	0
$485$	0.536960	0.0243821
$486$	0	0
$487$	5.96478	0.270290	0.135145	−	0.990826i	$-0.456850\pi$
$487$	5.96478	0.270290	0.135145	+	0.990826i	$0.456850\pi$
$488$	0	0
$489$	5.35211	0.242031
$490$	0	0
$491$	−6.20175	−0.279881	−0.139940	−	0.990160i	$-0.544691\pi$
$491$	−6.20175	−0.279881	−0.139940	+	0.990160i	$0.544691\pi$
$492$	0	0
$493$	0.00504865	0.000227380 0
$494$	0	0
$495$	1.58722	0.0713401
$496$	0	0
$497$	−0.641953	−0.0287955
$498$	0	0
$499$	−17.6753	−0.791253	−0.395626	−	0.918412i	$-0.629473\pi$
$499$	−17.6753	−0.791253	−0.395626	+	0.918412i	$0.629473\pi$
$500$	0	0
$501$	−17.3553	−0.775376
$502$	0	0
$503$	34.1635	1.52328	0.761638	−	0.648003i	$-0.224396\pi$
$503$	34.1635	1.52328	0.761638	+	0.648003i	$0.224396\pi$
$504$	0	0
$505$	4.51488	0.200909
$506$	0	0
$507$	−49.3512	−2.19176
$508$	0	0
$509$	−18.4383	−0.817265	−0.408633	−	0.912699i	$-0.633994\pi$
$509$	−18.4383	−0.817265	−0.408633	+	0.912699i	$0.633994\pi$
$510$	0	0
$511$	−10.3124	−0.456195
$512$	0	0
$513$	31.2679	1.38051
$514$	0	0
$515$	0.166531	0.00733824
$516$	0	0
$517$	7.98478	0.351170
$518$	0	0
$519$	−19.8529	−0.871447
$520$	0	0
$521$	−11.8840	−0.520647	−0.260323	−	0.965521i	$-0.583829\pi$
$521$	−11.8840	−0.520647	−0.260323	+	0.965521i	$0.583829\pi$
$522$	0	0
$523$	12.8711	0.562815	0.281407	−	0.959588i	$-0.409199\pi$
$523$	12.8711	0.562815	0.281407	+	0.959588i	$0.409199\pi$
$524$	0	0
$525$	−6.10096	−0.266268
$526$	0	0
$527$	−2.31894	−0.101015
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.7957	−0.468495
$532$	0	0
$533$	−76.2284	−3.30182
$534$	0	0
$535$	−0.106448	−0.00460214
$536$	0	0
$537$	−26.2777	−1.13397
$538$	0	0
$539$	2.43556	0.104907
$540$	0	0
$541$	−30.3533	−1.30499	−0.652496	−	0.757792i	$-0.726278\pi$
$541$	−30.3533	−1.30499	−0.652496	+	0.757792i	$0.726278\pi$
$542$	0	0
$543$	7.75282	0.332706
$544$	0	0
$545$	−7.88629	−0.337811
$546$	0	0
$547$	34.2001	1.46229	0.731144	−	0.682223i	$-0.238987\pi$
$547$	34.2001	1.46229	0.731144	+	0.682223i	$0.238987\pi$
$548$	0	0
$549$	−14.2875	−0.609775
$550$	0	0
$551$	0.0934403	0.00398069
$552$	0	0
$553$	2.43556	0.103571
$554$	0	0
$555$	−5.45335	−0.231482
$556$	0	0
$557$	−29.2289	−1.23847	−0.619235	−	0.785206i	$-0.712557\pi$
$557$	−29.2289	−1.23847	−0.619235	+	0.785206i	$0.712557\pi$
$558$	0	0
$559$	36.1361	1.52839
$560$	0	0
$561$	−0.942515	−0.0397930
$562$	0	0
$563$	−23.4170	−0.986907	−0.493454	−	0.869772i	$-0.664266\pi$
$563$	−23.4170	−0.986907	−0.493454	+	0.869772i	$0.664266\pi$
$564$	0	0
$565$	7.64376	0.321575
$566$	0	0
$567$	3.03800	0.127584
$568$	0	0
$569$	−31.8255	−1.33419	−0.667097	−	0.744971i	$-0.732464\pi$
$569$	−31.8255	−1.33419	−0.667097	+	0.744971i	$0.732464\pi$
$570$	0	0
$571$	−12.8639	−0.538336	−0.269168	−	0.963093i	$-0.586749\pi$
$571$	−12.8639	−0.538336	−0.269168	+	0.963093i	$0.586749\pi$
$572$	0	0
$573$	−18.2607	−0.762850
$574$	0	0
$575$	−4.77229	−0.199018
$576$	0	0
$577$	−4.25457	−0.177120	−0.0885600	−	0.996071i	$-0.528226\pi$
$577$	−4.25457	−0.177120	−0.0885600	+	0.996071i	$0.528226\pi$
$578$	0	0
$579$	−15.8570	−0.658995
$580$	0	0
$581$	−11.7343	−0.486820
$582$	0	0
$583$	7.58350	0.314076
$584$	0	0
$585$	−4.68141	−0.193552
$586$	0	0
$587$	27.9185	1.15232	0.576160	−	0.817337i	$-0.304550\pi$
$587$	27.9185	1.15232	0.576160	+	0.817337i	$0.304550\pi$
$588$	0	0
$589$	−42.9189	−1.76844
$590$	0	0
$591$	13.6048	0.559627
$592$	0	0
$593$	−18.8711	−0.774944	−0.387472	−	0.921881i	$-0.626652\pi$
$593$	−18.8711	−0.774944	−0.387472	+	0.921881i	$0.626652\pi$
$594$	0	0
$595$	0.144448	0.00592181
$596$	0	0
$597$	−14.2310	−0.582437
$598$	0	0
$599$	4.56833	0.186657	0.0933285	−	0.995635i	$-0.470249\pi$
$599$	4.56833	0.186657	0.0933285	+	0.995635i	$0.470249\pi$
$600$	0	0
$601$	8.16113	0.332899	0.166450	−	0.986050i	$-0.446770\pi$
$601$	8.16113	0.332899	0.166450	+	0.986050i	$0.446770\pi$
$602$	0	0
$603$	−3.80273	−0.154859
$604$	0	0
$605$	2.41844	0.0983235
$606$	0	0
$607$	−17.3331	−0.703529	−0.351764	−	0.936089i	$-0.614418\pi$
$607$	−17.3331	−0.703529	−0.351764	+	0.936089i	$0.614418\pi$
$608$	0	0
$609$	0.0213220	0.000864012 0
$610$	0	0
$611$	−23.5507	−0.952758
$612$	0	0
$613$	11.5952	0.468325	0.234162	−	0.972198i	$-0.424765\pi$
$613$	11.5952	0.468325	0.234162	+	0.972198i	$0.424765\pi$
$614$	0	0
$615$	6.47356	0.261039
$616$	0	0
$617$	−12.6206	−0.508087	−0.254044	−	0.967193i	$-0.581761\pi$
$617$	−12.6206	−0.508087	−0.254044	+	0.967193i	$0.581761\pi$
$618$	0	0
$619$	35.3244	1.41981	0.709904	−	0.704298i	$-0.248738\pi$
$619$	35.3244	1.41981	0.709904	+	0.704298i	$0.248738\pi$
$620$	0	0
$621$	5.58112	0.223963
$622$	0	0
$623$	7.59085	0.304121
$624$	0	0
$625$	21.6361	0.865446
$626$	0	0
$627$	−17.4441	−0.696649
$628$	0	0
$629$	−2.70592	−0.107892
$630$	0	0
$631$	10.5356	0.419417	0.209708	−	0.977764i	$-0.432749\pi$
$631$	10.5356	0.419417	0.209708	+	0.977764i	$0.432749\pi$
$632$	0	0
$633$	−32.4064	−1.28804
$634$	0	0
$635$	−0.0333570	−0.00132373
$636$	0	0
$637$	−7.18356	−0.284623
$638$	0	0
$639$	−0.876689	−0.0346813
$640$	0	0
$641$	22.8906	0.904124	0.452062	−	0.891987i	$-0.350689\pi$
$641$	22.8906	0.904124	0.452062	+	0.891987i	$0.350689\pi$
$642$	0	0
$643$	22.8893	0.902664	0.451332	−	0.892356i	$-0.350949\pi$
$643$	22.8893	0.902664	0.451332	+	0.892356i	$0.350949\pi$
$644$	0	0
$645$	−3.06879	−0.120834
$646$	0	0
$647$	36.5209	1.43578	0.717892	−	0.696154i	$-0.245107\pi$
$647$	36.5209	1.43578	0.717892	+	0.696154i	$0.245107\pi$
$648$	0	0
$649$	19.2535	0.755764
$650$	0	0
$651$	−9.79361	−0.383842
$652$	0	0
$653$	−15.4493	−0.604577	−0.302288	−	0.953217i	$-0.597751\pi$
$653$	−15.4493	−0.604577	−0.302288	+	0.953217i	$0.597751\pi$
$654$	0	0
$655$	−3.88891	−0.151952
$656$	0	0
$657$	−14.0833	−0.549441
$658$	0	0
$659$	46.1664	1.79839	0.899194	−	0.437550i	$-0.144154\pi$
$659$	46.1664	1.79839	0.899194	+	0.437550i	$0.144154\pi$
$660$	0	0
$661$	38.6047	1.50155	0.750774	−	0.660559i	$-0.229681\pi$
$661$	38.6047	1.50155	0.750774	+	0.660559i	$0.229681\pi$
$662$	0	0
$663$	2.77990	0.107962
$664$	0	0
$665$	2.67345	0.103672
$666$	0	0
$667$	0.0166785	0.000645794 0
$668$	0	0
$669$	4.42204	0.170966
$670$	0	0
$671$	25.4808	0.983674
$672$	0	0
$673$	−16.7906	−0.647232	−0.323616	−	0.946189i	$-0.604899\pi$
$673$	−16.7906	−0.647232	−0.323616	+	0.946189i	$0.604899\pi$
$674$	0	0
$675$	−26.6347	−1.02517
$676$	0	0
$677$	28.7560	1.10518	0.552591	−	0.833452i	$-0.313639\pi$
$677$	28.7560	1.10518	0.552591	+	0.833452i	$0.313639\pi$
$678$	0	0
$679$	1.12525	0.0431830
$680$	0	0
$681$	−5.42371	−0.207837
$682$	0	0
$683$	−10.1501	−0.388384	−0.194192	−	0.980964i	$-0.562209\pi$
$683$	−10.1501	−0.388384	−0.194192	+	0.980964i	$0.562209\pi$
$684$	0	0
$685$	4.49896	0.171896
$686$	0	0
$687$	18.0228	0.687612
$688$	0	0
$689$	−22.3671	−0.852119
$690$	0	0
$691$	3.33903	0.127023	0.0635113	−	0.997981i	$-0.479770\pi$
$691$	3.33903	0.127023	0.0635113	+	0.997981i	$0.479770\pi$
$692$	0	0
$693$	3.32615	0.126350
$694$	0	0
$695$	−6.45211	−0.244743
$696$	0	0
$697$	3.21214	0.121669
$698$	0	0
$699$	37.1045	1.40342
$700$	0	0
$701$	−35.3756	−1.33612	−0.668060	−	0.744107i	$-0.732875\pi$
$701$	−35.3756	−1.33612	−0.668060	+	0.744107i	$0.732875\pi$
$702$	0	0
$703$	−50.0811	−1.88885
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	9.46131	0.355829
$708$	0	0
$709$	41.7761	1.56894	0.784468	−	0.620170i	$-0.212936\pi$
$709$	41.7761	1.56894	0.784468	+	0.620170i	$0.212936\pi$
$710$	0	0
$711$	3.32615	0.124740
$712$	0	0
$713$	−7.66075	−0.286897
$714$	0	0
$715$	8.34898	0.312234
$716$	0	0
$717$	−24.9938	−0.933412
$718$	0	0
$719$	−3.24913	−0.121172	−0.0605861	−	0.998163i	$-0.519297\pi$
$719$	−3.24913	−0.121172	−0.0605861	+	0.998163i	$0.519297\pi$
$720$	0	0
$721$	0.348980	0.0129967
$722$	0	0
$723$	−13.5552	−0.504122
$724$	0	0
$725$	−0.0795946	−0.00295607
$726$	0	0
$727$	9.63289	0.357264	0.178632	−	0.983916i	$-0.442833\pi$
$727$	9.63289	0.357264	0.178632	+	0.983916i	$0.442833\pi$
$728$	0	0
$729$	25.5538	0.946437
$730$	0	0
$731$	−1.52272	−0.0563198
$732$	0	0
$733$	33.8899	1.25175	0.625876	−	0.779922i	$-0.284741\pi$
$733$	33.8899	1.25175	0.625876	+	0.779922i	$0.284741\pi$
$734$	0	0
$735$	0.610051	0.0225021
$736$	0	0
$737$	6.78192	0.249815
$738$	0	0
$739$	34.3805	1.26471	0.632353	−	0.774680i	$-0.282089\pi$
$739$	34.3805	1.26471	0.632353	+	0.774680i	$0.282089\pi$
$740$	0	0
$741$	51.4503	1.89007
$742$	0	0
$743$	24.2805	0.890766	0.445383	−	0.895340i	$-0.353067\pi$
$743$	24.2805	0.890766	0.445383	+	0.895340i	$0.353067\pi$
$744$	0	0
$745$	−6.87908	−0.252030
$746$	0	0
$747$	−16.0250	−0.586325
$748$	0	0
$749$	−0.223070	−0.00815082
$750$	0	0
$751$	−22.6514	−0.826560	−0.413280	−	0.910604i	$-0.635617\pi$
$751$	−22.6514	−0.826560	−0.413280	+	0.910604i	$0.635617\pi$
$752$	0	0
$753$	38.9615	1.41984
$754$	0	0
$755$	0.199888	0.00727468
$756$	0	0
$757$	−22.0528	−0.801523	−0.400762	−	0.916182i	$-0.631254\pi$
$757$	−22.0528	−0.801523	−0.400762	+	0.916182i	$0.631254\pi$
$758$	0	0
$759$	−3.11365	−0.113018
$760$	0	0
$761$	15.9416	0.577883	0.288941	−	0.957347i	$-0.406697\pi$
$761$	15.9416	0.577883	0.288941	+	0.957347i	$0.406697\pi$
$762$	0	0
$763$	−16.5264	−0.598295
$764$	0	0
$765$	0.197267	0.00713221
$766$	0	0
$767$	−56.7870	−2.05046
$768$	0	0
$769$	39.4438	1.42238	0.711190	−	0.703000i	$-0.248157\pi$
$769$	39.4438	1.42238	0.711190	+	0.703000i	$0.248157\pi$
$770$	0	0
$771$	−17.4298	−0.627719
$772$	0	0
$773$	−42.0375	−1.51198	−0.755992	−	0.654581i	$-0.772845\pi$
$773$	−42.0375	−1.51198	−0.755992	+	0.654581i	$0.772845\pi$
$774$	0	0
$775$	36.5593	1.31325
$776$	0	0
$777$	−11.4279	−0.409975
$778$	0	0
$779$	59.4503	2.13003
$780$	0	0
$781$	1.56352	0.0559470
$782$	0	0
$783$	0.0930847	0.00332658
$784$	0	0
$785$	−11.6223	−0.414818
$786$	0	0
$787$	6.98472	0.248978	0.124489	−	0.992221i	$-0.460271\pi$
$787$	6.98472	0.248978	0.124489	+	0.992221i	$0.460271\pi$
$788$	0	0
$789$	−12.3245	−0.438763
$790$	0	0
$791$	16.0181	0.569539
$792$	0	0
$793$	−75.1542	−2.66880
$794$	0	0
$795$	1.89949	0.0673679
$796$	0	0
$797$	26.2541	0.929969	0.464984	−	0.885319i	$-0.346060\pi$
$797$	26.2541	0.929969	0.464984	+	0.885319i	$0.346060\pi$
$798$	0	0
$799$	0.992388	0.0351082
$800$	0	0
$801$	10.3665	0.366283
$802$	0	0
$803$	25.1166	0.886344
$804$	0	0
$805$	0.477194	0.0168189
$806$	0	0
$807$	−32.8253	−1.15550
$808$	0	0
$809$	3.30337	0.116140	0.0580701	−	0.998313i	$-0.481505\pi$
$809$	3.30337	0.116140	0.0580701	+	0.998313i	$0.481505\pi$
$810$	0	0
$811$	29.5817	1.03875	0.519377	−	0.854545i	$-0.326164\pi$
$811$	29.5817	1.03875	0.519377	+	0.854545i	$0.326164\pi$
$812$	0	0
$813$	26.1820	0.918244
$814$	0	0
$815$	1.99778	0.0699793
$816$	0	0
$817$	−28.1824	−0.985979
$818$	0	0
$819$	−9.81029	−0.342799
$820$	0	0
$821$	−54.4549	−1.90049	−0.950245	−	0.311504i	$-0.899167\pi$
$821$	−54.4549	−1.90049	−0.950245	+	0.311504i	$0.899167\pi$
$822$	0	0
$823$	12.7484	0.444381	0.222191	−	0.975003i	$-0.428679\pi$
$823$	12.7484	0.444381	0.222191	+	0.975003i	$0.428679\pi$
$824$	0	0
$825$	14.8592	0.517333
$826$	0	0
$827$	−22.2940	−0.775239	−0.387620	−	0.921819i	$-0.626703\pi$
$827$	−22.2940	−0.775239	−0.387620	+	0.921819i	$0.626703\pi$
$828$	0	0
$829$	−2.25845	−0.0784392	−0.0392196	−	0.999231i	$-0.512487\pi$
$829$	−2.25845	−0.0784392	−0.0392196	+	0.999231i	$0.512487\pi$
$830$	0	0
$831$	−2.11255	−0.0732834
$832$	0	0
$833$	0.302704	0.0104881
$834$	0	0
$835$	−6.47821	−0.224188
$836$	0	0
$837$	−42.7555	−1.47785
$838$	0	0
$839$	−38.8393	−1.34088	−0.670442	−	0.741962i	$-0.733895\pi$
$839$	−38.8393	−1.34088	−0.670442	+	0.741962i	$0.733895\pi$
$840$	0	0
$841$	−28.9997	−0.999990
$842$	0	0
$843$	6.84200	0.235651
$844$	0	0
$845$	−18.4213	−0.633714
$846$	0	0
$847$	5.06804	0.174140
$848$	0	0
$849$	−0.934926	−0.0320866
$850$	0	0
$851$	−8.93916	−0.306431
$852$	0	0
$853$	3.34045	0.114375	0.0571873	−	0.998363i	$-0.481787\pi$
$853$	3.34045	0.114375	0.0571873	+	0.998363i	$0.481787\pi$
$854$	0	0
$855$	3.65102	0.124862
$856$	0	0
$857$	35.3153	1.20635	0.603174	−	0.797610i	$-0.293903\pi$
$857$	35.3153	1.20635	0.603174	+	0.797610i	$0.293903\pi$
$858$	0	0
$859$	−37.5433	−1.28096	−0.640481	−	0.767974i	$-0.721265\pi$
$859$	−37.5433	−1.28096	−0.640481	+	0.767974i	$0.721265\pi$
$860$	0	0
$861$	13.5659	0.462324
$862$	0	0
$863$	32.1921	1.09583	0.547916	−	0.836534i	$-0.315421\pi$
$863$	32.1921	1.09583	0.547916	+	0.836534i	$0.315421\pi$
$864$	0	0
$865$	−7.41051	−0.251965
$866$	0	0
$867$	21.6159	0.734114
$868$	0	0
$869$	−5.93196	−0.201228
$870$	0	0
$871$	−20.0029	−0.677773
$872$	0	0
$873$	1.53670	0.0520095
$874$	0	0
$875$	−4.66328	−0.157647
$876$	0	0
$877$	−9.78157	−0.330300	−0.165150	−	0.986268i	$-0.552811\pi$
$877$	−9.78157	−0.330300	−0.165150	+	0.986268i	$0.552811\pi$
$878$	0	0
$879$	13.1073	0.442099
$880$	0	0
$881$	43.3634	1.46095	0.730475	−	0.682939i	$-0.239299\pi$
$881$	43.3634	1.46095	0.730475	+	0.682939i	$0.239299\pi$
$882$	0	0
$883$	−46.7422	−1.57300	−0.786500	−	0.617590i	$-0.788109\pi$
$883$	−46.7422	−1.57300	−0.786500	+	0.617590i	$0.788109\pi$
$884$	0	0
$885$	4.82254	0.162108
$886$	0	0
$887$	38.5913	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$887$	38.5913	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$888$	0	0
$889$	−0.0699024	−0.00234445
$890$	0	0
$891$	−7.39924	−0.247884
$892$	0	0
$893$	18.3671	0.614632
$894$	0	0
$895$	−9.80870	−0.327869
$896$	0	0
$897$	9.18356	0.306630
$898$	0	0
$899$	−0.127770	−0.00426136
$900$	0	0
$901$	0.942515	0.0313997
$902$	0	0
$903$	−6.43092	−0.214008
$904$	0	0
$905$	2.89390	0.0961965
$906$	0	0
$907$	18.5641	0.616411	0.308205	−	0.951320i	$-0.400272\pi$
$907$	18.5641	0.616411	0.308205	+	0.951320i	$0.400272\pi$
$908$	0	0
$909$	12.9209	0.428560
$910$	0	0
$911$	4.69995	0.155716	0.0778581	−	0.996964i	$-0.475192\pi$
$911$	4.69995	0.155716	0.0778581	+	0.996964i	$0.475192\pi$
$912$	0	0
$913$	28.5796	0.945846
$914$	0	0
$915$	6.38234	0.210994
$916$	0	0
$917$	−8.14954	−0.269121
$918$	0	0
$919$	43.7947	1.44465	0.722326	−	0.691552i	$-0.243073\pi$
$919$	43.7947	1.44465	0.722326	+	0.691552i	$0.243073\pi$
$920$	0	0
$921$	−21.7127	−0.715458
$922$	0	0
$923$	−4.61151	−0.151790
$924$	0	0
$925$	42.6602	1.40266
$926$	0	0
$927$	0.476588	0.0156532
$928$	0	0
$929$	27.4615	0.900982	0.450491	−	0.892781i	$-0.351249\pi$
$929$	27.4615	0.900982	0.450491	+	0.892781i	$0.351249\pi$
$930$	0	0
$931$	5.60244	0.183613
$932$	0	0
$933$	10.3019	0.337268
$934$	0	0
$935$	−0.351813	−0.0115055
$936$	0	0
$937$	−26.5337	−0.866819	−0.433410	−	0.901197i	$-0.642690\pi$
$937$	−26.5337	−0.866819	−0.433410	+	0.901197i	$0.642690\pi$
$938$	0	0
$939$	4.71480	0.153862
$940$	0	0
$941$	54.5366	1.77784	0.888922	−	0.458059i	$-0.151455\pi$
$941$	54.5366	1.77784	0.888922	+	0.458059i	$0.151455\pi$
$942$	0	0
$943$	10.6115	0.345558
$944$	0	0
$945$	2.66328	0.0866363
$946$	0	0
$947$	16.3392	0.530951	0.265476	−	0.964118i	$-0.414471\pi$
$947$	16.3392	0.530951	0.265476	+	0.964118i	$0.414471\pi$
$948$	0	0
$949$	−74.0800	−2.40474
$950$	0	0
$951$	5.50952	0.178658
$952$	0	0
$953$	−5.02906	−0.162907	−0.0814536	−	0.996677i	$-0.525956\pi$
$953$	−5.02906	−0.162907	−0.0814536	+	0.996677i	$0.525956\pi$
$954$	0	0
$955$	−6.81616	−0.220566
$956$	0	0
$957$	−0.0519311	−0.00167869
$958$	0	0
$959$	9.42795	0.304444
$960$	0	0
$961$	27.6871	0.893132
$962$	0	0
$963$	−0.304638	−0.00981683
$964$	0	0
$965$	−5.91895	−0.190538
$966$	0	0
$967$	−55.4970	−1.78466	−0.892332	−	0.451379i	$-0.850932\pi$
$967$	−55.4970	−1.78466	−0.892332	+	0.451379i	$0.850932\pi$
$968$	0	0
$969$	−2.16804	−0.0696473
$970$	0	0
$971$	−53.3518	−1.71214	−0.856071	−	0.516858i	$-0.827101\pi$
$971$	−53.3518	−1.71214	−0.856071	+	0.516858i	$0.827101\pi$
$972$	0	0
$973$	−13.5209	−0.433462
$974$	0	0
$975$	−43.8266	−1.40357
$976$	0	0
$977$	10.0073	0.320161	0.160080	−	0.987104i	$-0.448825\pi$
$977$	10.0073	0.320161	0.160080	+	0.987104i	$0.448825\pi$
$978$	0	0
$979$	−18.4880	−0.590878
$980$	0	0
$981$	−22.5694	−0.720585
$982$	0	0
$983$	16.7241	0.533414	0.266707	−	0.963778i	$-0.414064\pi$
$983$	16.7241	0.533414	0.266707	+	0.963778i	$0.414064\pi$
$984$	0	0
$985$	5.07827	0.161807
$986$	0	0
$987$	4.19117	0.133406
$988$	0	0
$989$	−5.03039	−0.159957
$990$	0	0
$991$	−8.96165	−0.284676	−0.142338	−	0.989818i	$-0.545462\pi$
$991$	−8.96165	−0.284676	−0.142338	+	0.989818i	$0.545462\pi$
$992$	0	0
$993$	34.5536	1.09653
$994$	0	0
$995$	−5.31202	−0.168402
$996$	0	0
$997$	11.7263	0.371376	0.185688	−	0.982609i	$-0.440549\pi$
$997$	11.7263	0.371376	0.185688	+	0.982609i	$0.440549\pi$
$998$	0	0
$999$	−49.8905	−1.57847

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.ba.1.1		4
4.3	odd	2		1288.2.a.n.1.4	✓	4
28.27	even	2		9016.2.a.bf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.n.1.4	✓	4	4.3	odd	2
2576.2.a.ba.1.1		4	1.1	even	1	trivial
9016.2.a.bf.1.1		4	28.27	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27841 q^{3} -0.477194 q^{5} -1.00000 q^{7} -1.36566 q^{9} +O(q^{10})\)	\(q-1.27841 q^{3} -0.477194 q^{5} -1.00000 q^{7} -1.36566 q^{9} +2.43556 q^{11} -7.18356 q^{13} +0.610051 q^{15} +0.302704 q^{17} +5.60244 q^{19} +1.27841 q^{21} +1.00000 q^{23} -4.77229 q^{25} +5.58112 q^{27} +0.0166785 q^{29} -7.66075 q^{31} -3.11365 q^{33} +0.477194 q^{35} -8.93916 q^{37} +9.18356 q^{39} +10.6115 q^{41} -5.03039 q^{43} +0.651684 q^{45} +3.27841 q^{47} +1.00000 q^{49} -0.386981 q^{51} +3.11365 q^{53} -1.16223 q^{55} -7.16223 q^{57} +7.90514 q^{59} +10.4620 q^{61} +1.36566 q^{63} +3.42795 q^{65} +2.78454 q^{67} -1.27841 q^{69} +0.641953 q^{71} +10.3124 q^{73} +6.10096 q^{75} -2.43556 q^{77} -2.43556 q^{79} -3.03800 q^{81} +11.7343 q^{83} -0.144448 q^{85} -0.0213220 q^{87} -7.59085 q^{89} +7.18356 q^{91} +9.79361 q^{93} -2.67345 q^{95} -1.12525 q^{97} -3.32615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 4 * q^7 + q^9	\( 4 q + 3 q^{3} - 4 q^{7} + q^{9} + 10 q^{11} - 3 q^{13} + 6 q^{15} - 4 q^{17} + 10 q^{19} - 3 q^{21} + 4 q^{23} + 4 q^{25} + 9 q^{27} - 13 q^{29} - 3 q^{31} + 20 q^{33} + 11 q^{39} + q^{41} + 8 q^{43} + 4 q^{45} + 5 q^{47} + 4 q^{49} + 4 q^{51} - 20 q^{53} + 22 q^{55} - 2 q^{57} + 14 q^{59} + 8 q^{61} - q^{63} - 2 q^{65} + 18 q^{67} + 3 q^{69} + 25 q^{71} + 15 q^{73} + 11 q^{75} - 10 q^{77} - 10 q^{79} + 36 q^{83} - 28 q^{85} - q^{87} + 4 q^{89} + 3 q^{91} + 17 q^{93} + 36 q^{95} + 6 q^{97} + 28 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 4 * q^7 + q^9 + 10 * q^11 - 3 * q^13 + 6 * q^15 - 4 * q^17 + 10 * q^19 - 3 * q^21 + 4 * q^23 + 4 * q^25 + 9 * q^27 - 13 * q^29 - 3 * q^31 + 20 * q^33 + 11 * q^39 + q^41 + 8 * q^43 + 4 * q^45 + 5 * q^47 + 4 * q^49 + 4 * q^51 - 20 * q^53 + 22 * q^55 - 2 * q^57 + 14 * q^59 + 8 * q^61 - q^63 - 2 * q^65 + 18 * q^67 + 3 * q^69 + 25 * q^71 + 15 * q^73 + 11 * q^75 - 10 * q^77 - 10 * q^79 + 36 * q^83 - 28 * q^85 - q^87 + 4 * q^89 + 3 * q^91 + 17 * q^93 + 36 * q^95 + 6 * q^97 + 28 * q^99

Introduction

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