LMFDB - Embedded newform 2475.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} +0.414214 q^{7} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} +0.414214 q^{7} -4.41421 q^{8} +1.00000 q^{11} -2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} -2.41421 q^{17} +6.41421 q^{19} -2.41421 q^{22} -1.00000 q^{23} +6.82843 q^{26} +1.58579 q^{28} -1.17157 q^{29} -8.48528 q^{31} +1.58579 q^{32} +5.82843 q^{34} -0.171573 q^{37} -15.4853 q^{38} +10.8995 q^{41} -11.6569 q^{43} +3.82843 q^{44} +2.41421 q^{46} +7.48528 q^{47} -6.82843 q^{49} -10.8284 q^{52} -7.65685 q^{53} -1.82843 q^{56} +2.82843 q^{58} -11.0000 q^{59} +8.82843 q^{61} +20.4853 q^{62} -9.82843 q^{64} -0.343146 q^{67} -9.24264 q^{68} -7.82843 q^{71} -8.82843 q^{73} +0.414214 q^{74} +24.5563 q^{76} +0.414214 q^{77} +13.2426 q^{79} -26.3137 q^{82} +4.48528 q^{83} +28.1421 q^{86} -4.41421 q^{88} -3.65685 q^{89} -1.17157 q^{91} -3.82843 q^{92} -18.0711 q^{94} +5.82843 q^{97} +16.4853 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{22} - 2 q^{23} + 8 q^{26} + 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} - 6 q^{37} - 14 q^{38} + 2 q^{41} - 12 q^{43} + 2 q^{44} + 2 q^{46} - 2 q^{47} - 8 q^{49} - 16 q^{52} - 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} - 12 q^{67} - 10 q^{68} - 10 q^{71} - 12 q^{73} - 2 q^{74} + 18 q^{76} - 2 q^{77} + 18 q^{79} - 30 q^{82} - 8 q^{83} + 28 q^{86} - 6 q^{88} + 4 q^{89} - 8 q^{91} - 2 q^{92} - 22 q^{94} + 6 q^{97} + 16 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^22 - 2 * q^23 + 8 * q^26 + 6 * q^28 - 8 * q^29 + 6 * q^32 + 6 * q^34 - 6 * q^37 - 14 * q^38 + 2 * q^41 - 12 * q^43 + 2 * q^44 + 2 * q^46 - 2 * q^47 - 8 * q^49 - 16 * q^52 - 4 * q^53 + 2 * q^56 - 22 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 - 12 * q^67 - 10 * q^68 - 10 * q^71 - 12 * q^73 - 2 * q^74 + 18 * q^76 - 2 * q^77 + 18 * q^79 - 30 * q^82 - 8 * q^83 + 28 * q^86 - 6 * q^88 + 4 * q^89 - 8 * q^91 - 2 * q^92 - 22 * q^94 + 6 * q^97 + 16 * q^98

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.414214	0.156558	0.0782790	−	0.996931i	$-0.475058\pi$
$7$	0.414214	0.156558	0.0782790	+	0.996931i	$0.475058\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	3.00000	0.750000
$17$	−2.41421	−0.585533	−0.292766	−	0.956184i	$-0.594576\pi$
$17$	−2.41421	−0.585533	−0.292766	+	0.956184i	$0.594576\pi$
$18$	0	0
$19$	6.41421	1.47152	0.735761	−	0.677242i	$-0.236825\pi$
$19$	6.41421	1.47152	0.735761	+	0.677242i	$0.236825\pi$
$20$	0	0
$21$	0	0
$22$	−2.41421	−0.514712
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	6.82843	1.33916
$27$	0	0
$28$	1.58579	0.299685
$29$	−1.17157	−0.217556	−0.108778	−	0.994066i	$-0.534694\pi$
$29$	−1.17157	−0.217556	−0.108778	+	0.994066i	$0.534694\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	5.82843	0.999567
$35$	0	0
$36$	0	0
$37$	−0.171573	−0.0282064	−0.0141032	−	0.999901i	$-0.504489\pi$
$37$	−0.171573	−0.0282064	−0.0141032	+	0.999901i	$0.504489\pi$
$38$	−15.4853	−2.51204
$39$	0	0
$40$	0	0
$41$	10.8995	1.70222	0.851108	−	0.524991i	$-0.175931\pi$
$41$	10.8995	1.70222	0.851108	+	0.524991i	$0.175931\pi$
$42$	0	0
$43$	−11.6569	−1.77765	−0.888827	−	0.458243i	$-0.848479\pi$
$43$	−11.6569	−1.77765	−0.888827	+	0.458243i	$0.848479\pi$
$44$	3.82843	0.577157
$45$	0	0
$46$	2.41421	0.355956
$47$	7.48528	1.09184	0.545920	−	0.837837i	$-0.316180\pi$
$47$	7.48528	1.09184	0.545920	+	0.837837i	$0.316180\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	0	0
$51$	0	0
$52$	−10.8284	−1.50163
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	0	0
$56$	−1.82843	−0.244334
$57$	0	0
$58$	2.82843	0.371391
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	20.4853	2.60163
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	−0.343146	−0.0419219	−0.0209610	−	0.999780i	$-0.506673\pi$
$67$	−0.343146	−0.0419219	−0.0209610	+	0.999780i	$0.506673\pi$
$68$	−9.24264	−1.12083
$69$	0	0
$70$	0	0
$71$	−7.82843	−0.929063	−0.464532	−	0.885556i	$-0.653777\pi$
$71$	−7.82843	−0.929063	−0.464532	+	0.885556i	$0.653777\pi$
$72$	0	0
$73$	−8.82843	−1.03329	−0.516645	−	0.856200i	$-0.672819\pi$
$73$	−8.82843	−1.03329	−0.516645	+	0.856200i	$0.672819\pi$
$74$	0.414214	0.0481513
$75$	0	0
$76$	24.5563	2.81681
$77$	0.414214	0.0472040
$78$	0	0
$79$	13.2426	1.48991	0.744957	−	0.667113i	$-0.232470\pi$
$79$	13.2426	1.48991	0.744957	+	0.667113i	$0.232470\pi$
$80$	0	0
$81$	0	0
$82$	−26.3137	−2.90586
$83$	4.48528	0.492324	0.246162	−	0.969229i	$-0.420831\pi$
$83$	4.48528	0.492324	0.246162	+	0.969229i	$0.420831\pi$
$84$	0	0
$85$	0	0
$86$	28.1421	3.03464
$87$	0	0
$88$	−4.41421	−0.470557
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−1.17157	−0.122814
$92$	−3.82843	−0.399141
$93$	0	0
$94$	−18.0711	−1.86389
$95$	0	0
$96$	0	0
$97$	5.82843	0.591787	0.295894	−	0.955221i	$-0.404383\pi$
$97$	5.82843	0.591787	0.295894	+	0.955221i	$0.404383\pi$
$98$	16.4853	1.66526
$99$	0	0
$100$	0	0
$101$	−14.8995	−1.48256	−0.741278	−	0.671199i	$-0.765780\pi$
$101$	−14.8995	−1.48256	−0.741278	+	0.671199i	$0.765780\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	12.4853	1.22428
$105$	0	0
$106$	18.4853	1.79545
$107$	5.31371	0.513696	0.256848	−	0.966452i	$-0.417316\pi$
$107$	5.31371	0.513696	0.256848	+	0.966452i	$0.417316\pi$
$108$	0	0
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	0	0
$111$	0	0
$112$	1.24264	0.117419
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	−4.48528	−0.416448
$117$	0	0
$118$	26.5563	2.44471
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	−21.3137	−1.92965
$123$	0	0
$124$	−32.4853	−2.91726
$125$	0	0
$126$	0	0
$127$	7.24264	0.642680	0.321340	−	0.946964i	$-0.395867\pi$
$127$	7.24264	0.642680	0.321340	+	0.946964i	$0.395867\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	0	0
$131$	−6.82843	−0.596602	−0.298301	−	0.954472i	$-0.596420\pi$
$131$	−6.82843	−0.596602	−0.298301	+	0.954472i	$0.596420\pi$
$132$	0	0
$133$	2.65685	0.230378
$134$	0.828427	0.0715652
$135$	0	0
$136$	10.6569	0.913818
$137$	12.1421	1.03737	0.518686	−	0.854965i	$-0.326421\pi$
$137$	12.1421	1.03737	0.518686	+	0.854965i	$0.326421\pi$
$138$	0	0
$139$	−18.9706	−1.60906	−0.804531	−	0.593911i	$-0.797583\pi$
$139$	−18.9706	−1.60906	−0.804531	+	0.593911i	$0.797583\pi$
$140$	0	0
$141$	0	0
$142$	18.8995	1.58601
$143$	−2.82843	−0.236525
$144$	0	0
$145$	0	0
$146$	21.3137	1.76394
$147$	0	0
$148$	−0.656854	−0.0539931
$149$	−7.72792	−0.633096	−0.316548	−	0.948576i	$-0.602524\pi$
$149$	−7.72792	−0.633096	−0.316548	+	0.948576i	$0.602524\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−28.3137	−2.29655
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−31.9706	−2.54344
$159$	0	0
$160$	0	0
$161$	−0.414214	−0.0326446
$162$	0	0
$163$	15.7990	1.23747	0.618736	−	0.785599i	$-0.287645\pi$
$163$	15.7990	1.23747	0.618736	+	0.785599i	$0.287645\pi$
$164$	41.7279	3.25840
$165$	0	0
$166$	−10.8284	−0.840449
$167$	−21.7990	−1.68686	−0.843428	−	0.537242i	$-0.819466\pi$
$167$	−21.7990	−1.68686	−0.843428	+	0.537242i	$0.819466\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	−44.6274	−3.40281
$173$	12.5563	0.954642	0.477321	−	0.878729i	$-0.341608\pi$
$173$	12.5563	0.954642	0.477321	+	0.878729i	$0.341608\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	8.82843	0.661719
$179$	−16.7990	−1.25562	−0.627808	−	0.778368i	$-0.716048\pi$
$179$	−16.7990	−1.25562	−0.627808	+	0.778368i	$0.716048\pi$
$180$	0	0
$181$	−21.9706	−1.63306	−0.816530	−	0.577304i	$-0.804105\pi$
$181$	−21.9706	−1.63306	−0.816530	+	0.577304i	$0.804105\pi$
$182$	2.82843	0.209657
$183$	0	0
$184$	4.41421	0.325420
$185$	0	0
$186$	0	0
$187$	−2.41421	−0.176545
$188$	28.6569	2.09002
$189$	0	0
$190$	0	0
$191$	−6.17157	−0.446559	−0.223280	−	0.974754i	$-0.571676\pi$
$191$	−6.17157	−0.446559	−0.223280	+	0.974754i	$0.571676\pi$
$192$	0	0
$193$	3.31371	0.238526	0.119263	−	0.992863i	$-0.461947\pi$
$193$	3.31371	0.238526	0.119263	+	0.992863i	$0.461947\pi$
$194$	−14.0711	−1.01024
$195$	0	0
$196$	−26.1421	−1.86730
$197$	−4.75736	−0.338948	−0.169474	−	0.985535i	$-0.554207\pi$
$197$	−4.75736	−0.338948	−0.169474	+	0.985535i	$0.554207\pi$
$198$	0	0
$199$	10.8284	0.767607	0.383803	−	0.923415i	$-0.374614\pi$
$199$	10.8284	0.767607	0.383803	+	0.923415i	$0.374614\pi$
$200$	0	0
$201$	0	0
$202$	35.9706	2.53088
$203$	−0.485281	−0.0340601
$204$	0	0
$205$	0	0
$206$	32.9706	2.29717
$207$	0	0
$208$	−8.48528	−0.588348
$209$	6.41421	0.443680
$210$	0	0
$211$	−13.3137	−0.916553	−0.458277	−	0.888810i	$-0.651533\pi$
$211$	−13.3137	−0.916553	−0.458277	+	0.888810i	$0.651533\pi$
$212$	−29.3137	−2.01327
$213$	0	0
$214$	−12.8284	−0.876933
$215$	0	0
$216$	0	0
$217$	−3.51472	−0.238595
$218$	−12.8284	−0.868851
$219$	0	0
$220$	0	0
$221$	6.82843	0.459330
$222$	0	0
$223$	−21.1716	−1.41775	−0.708877	−	0.705332i	$-0.750798\pi$
$223$	−21.1716	−1.41775	−0.708877	+	0.705332i	$0.750798\pi$
$224$	0.656854	0.0438879
$225$	0	0
$226$	−24.1421	−1.60591
$227$	18.4853	1.22691	0.613456	−	0.789729i	$-0.289779\pi$
$227$	18.4853	1.22691	0.613456	+	0.789729i	$0.289779\pi$
$228$	0	0
$229$	−2.51472	−0.166177	−0.0830886	−	0.996542i	$-0.526478\pi$
$229$	−2.51472	−0.166177	−0.0830886	+	0.996542i	$0.526478\pi$
$230$	0	0
$231$	0	0
$232$	5.17157	0.339530
$233$	−16.5563	−1.08464	−0.542321	−	0.840171i	$-0.682454\pi$
$233$	−16.5563	−1.08464	−0.542321	+	0.840171i	$0.682454\pi$
$234$	0	0
$235$	0	0
$236$	−42.1127	−2.74130
$237$	0	0
$238$	2.41421	0.156490
$239$	−23.6569	−1.53023	−0.765117	−	0.643891i	$-0.777319\pi$
$239$	−23.6569	−1.53023	−0.765117	+	0.643891i	$0.777319\pi$
$240$	0	0
$241$	14.1421	0.910975	0.455488	−	0.890242i	$-0.349465\pi$
$241$	14.1421	0.910975	0.455488	+	0.890242i	$0.349465\pi$
$242$	−2.41421	−0.155192
$243$	0	0
$244$	33.7990	2.16376
$245$	0	0
$246$	0	0
$247$	−18.1421	−1.15436
$248$	37.4558	2.37845
$249$	0	0
$250$	0	0
$251$	−8.97056	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$251$	−8.97056	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−17.4853	−1.09712
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−9.31371	−0.580973	−0.290487	−	0.956879i	$-0.593817\pi$
$257$	−9.31371	−0.580973	−0.290487	+	0.956879i	$0.593817\pi$
$258$	0	0
$259$	−0.0710678	−0.00441594
$260$	0	0
$261$	0	0
$262$	16.4853	1.01846
$263$	−22.9706	−1.41643	−0.708213	−	0.705999i	$-0.750498\pi$
$263$	−22.9706	−1.41643	−0.708213	+	0.705999i	$0.750498\pi$
$264$	0	0
$265$	0	0
$266$	−6.41421	−0.393281
$267$	0	0
$268$	−1.31371	−0.0802475
$269$	15.7990	0.963281	0.481641	−	0.876369i	$-0.340041\pi$
$269$	15.7990	0.963281	0.481641	+	0.876369i	$0.340041\pi$
$270$	0	0
$271$	8.89949	0.540606	0.270303	−	0.962775i	$-0.412876\pi$
$271$	8.89949	0.540606	0.270303	+	0.962775i	$0.412876\pi$
$272$	−7.24264	−0.439150
$273$	0	0
$274$	−29.3137	−1.77091
$275$	0	0
$276$	0	0
$277$	4.82843	0.290112	0.145056	−	0.989423i	$-0.453664\pi$
$277$	4.82843	0.290112	0.145056	+	0.989423i	$0.453664\pi$
$278$	45.7990	2.74684
$279$	0	0
$280$	0	0
$281$	−32.0711	−1.91320	−0.956600	−	0.291405i	$-0.905877\pi$
$281$	−32.0711	−1.91320	−0.956600	+	0.291405i	$0.905877\pi$
$282$	0	0
$283$	−0.899495	−0.0534694	−0.0267347	−	0.999643i	$-0.508511\pi$
$283$	−0.899495	−0.0534694	−0.0267347	+	0.999643i	$0.508511\pi$
$284$	−29.9706	−1.77843
$285$	0	0
$286$	6.82843	0.403773
$287$	4.51472	0.266495
$288$	0	0
$289$	−11.1716	−0.657151
$290$	0	0
$291$	0	0
$292$	−33.7990	−1.97794
$293$	−17.5858	−1.02737	−0.513686	−	0.857978i	$-0.671720\pi$
$293$	−17.5858	−1.02737	−0.513686	+	0.857978i	$0.671720\pi$
$294$	0	0
$295$	0	0
$296$	0.757359	0.0440206
$297$	0	0
$298$	18.6569	1.08076
$299$	2.82843	0.163572
$300$	0	0
$301$	−4.82843	−0.278306
$302$	−33.7990	−1.94491
$303$	0	0
$304$	19.2426	1.10364
$305$	0	0
$306$	0	0
$307$	6.68629	0.381607	0.190803	−	0.981628i	$-0.438891\pi$
$307$	6.68629	0.381607	0.190803	+	0.981628i	$0.438891\pi$
$308$	1.58579	0.0903586
$309$	0	0
$310$	0	0
$311$	−13.6569	−0.774409	−0.387205	−	0.921994i	$-0.626559\pi$
$311$	−13.6569	−0.774409	−0.387205	+	0.921994i	$0.626559\pi$
$312$	0	0
$313$	−27.1421	−1.53416	−0.767082	−	0.641549i	$-0.778292\pi$
$313$	−27.1421	−1.53416	−0.767082	+	0.641549i	$0.778292\pi$
$314$	−14.4853	−0.817452
$315$	0	0
$316$	50.6985	2.85201
$317$	−30.8284	−1.73150	−0.865748	−	0.500479i	$-0.833157\pi$
$317$	−30.8284	−1.73150	−0.865748	+	0.500479i	$0.833157\pi$
$318$	0	0
$319$	−1.17157	−0.0655955
$320$	0	0
$321$	0	0
$322$	1.00000	0.0557278
$323$	−15.4853	−0.861624
$324$	0	0
$325$	0	0
$326$	−38.1421	−2.11250
$327$	0	0
$328$	−48.1127	−2.65658
$329$	3.10051	0.170936
$330$	0	0
$331$	−32.1421	−1.76669	−0.883346	−	0.468722i	$-0.844715\pi$
$331$	−32.1421	−1.76669	−0.883346	+	0.468722i	$0.844715\pi$
$332$	17.1716	0.942412
$333$	0	0
$334$	52.6274	2.87964
$335$	0	0
$336$	0	0
$337$	4.14214	0.225637	0.112818	−	0.993616i	$-0.464012\pi$
$337$	4.14214	0.225637	0.112818	+	0.993616i	$0.464012\pi$
$338$	12.0711	0.656580
$339$	0	0
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	−5.72792	−0.309279
$344$	51.4558	2.77431
$345$	0	0
$346$	−30.3137	−1.62968
$347$	21.1716	1.13655	0.568275	−	0.822839i	$-0.307611\pi$
$347$	21.1716	1.13655	0.568275	+	0.822839i	$0.307611\pi$
$348$	0	0
$349$	−2.48528	−0.133034	−0.0665170	−	0.997785i	$-0.521189\pi$
$349$	−2.48528	−0.133034	−0.0665170	+	0.997785i	$0.521189\pi$
$350$	0	0
$351$	0	0
$352$	1.58579	0.0845227
$353$	4.48528	0.238727	0.119364	−	0.992851i	$-0.461915\pi$
$353$	4.48528	0.238727	0.119364	+	0.992851i	$0.461915\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	40.5563	2.14347
$359$	−15.5147	−0.818836	−0.409418	−	0.912347i	$-0.634268\pi$
$359$	−15.5147	−0.818836	−0.409418	+	0.912347i	$0.634268\pi$
$360$	0	0
$361$	22.1421	1.16538
$362$	53.0416	2.78781
$363$	0	0
$364$	−4.48528	−0.235093
$365$	0	0
$366$	0	0
$367$	1.31371	0.0685750	0.0342875	−	0.999412i	$-0.489084\pi$
$367$	1.31371	0.0685750	0.0342875	+	0.999412i	$0.489084\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	0	0
$371$	−3.17157	−0.164660
$372$	0	0
$373$	−23.6569	−1.22491	−0.612453	−	0.790507i	$-0.709817\pi$
$373$	−23.6569	−1.22491	−0.612453	+	0.790507i	$0.709817\pi$
$374$	5.82843	0.301381
$375$	0	0
$376$	−33.0416	−1.70399
$377$	3.31371	0.170665
$378$	0	0
$379$	9.17157	0.471112	0.235556	−	0.971861i	$-0.424309\pi$
$379$	9.17157	0.471112	0.235556	+	0.971861i	$0.424309\pi$
$380$	0	0
$381$	0	0
$382$	14.8995	0.762324
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	22.3137	1.13281
$389$	17.6569	0.895238	0.447619	−	0.894224i	$-0.352272\pi$
$389$	17.6569	0.895238	0.447619	+	0.894224i	$0.352272\pi$
$390$	0	0
$391$	2.41421	0.122092
$392$	30.1421	1.52241
$393$	0	0
$394$	11.4853	0.578620
$395$	0	0
$396$	0	0
$397$	−35.9411	−1.80383	−0.901917	−	0.431910i	$-0.857840\pi$
$397$	−35.9411	−1.80383	−0.901917	+	0.431910i	$0.857840\pi$
$398$	−26.1421	−1.31039
$399$	0	0
$400$	0	0
$401$	−31.7990	−1.58797	−0.793983	−	0.607940i	$-0.791996\pi$
$401$	−31.7990	−1.58797	−0.793983	+	0.607940i	$0.791996\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	−57.0416	−2.83793
$405$	0	0
$406$	1.17157	0.0581442
$407$	−0.171573	−0.00850455
$408$	0	0
$409$	−4.14214	−0.204815	−0.102408	−	0.994743i	$-0.532655\pi$
$409$	−4.14214	−0.204815	−0.102408	+	0.994743i	$0.532655\pi$
$410$	0	0
$411$	0	0
$412$	−52.2843	−2.57586
$413$	−4.55635	−0.224203
$414$	0	0
$415$	0	0
$416$	−4.48528	−0.219909
$417$	0	0
$418$	−15.4853	−0.757410
$419$	25.4853	1.24504	0.622519	−	0.782605i	$-0.286109\pi$
$419$	25.4853	1.24504	0.622519	+	0.782605i	$0.286109\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	32.1421	1.56465
$423$	0	0
$424$	33.7990	1.64142
$425$	0	0
$426$	0	0
$427$	3.65685	0.176968
$428$	20.3431	0.983323
$429$	0	0
$430$	0	0
$431$	−1.17157	−0.0564327	−0.0282163	−	0.999602i	$-0.508983\pi$
$431$	−1.17157	−0.0564327	−0.0282163	+	0.999602i	$0.508983\pi$
$432$	0	0
$433$	−13.3137	−0.639816	−0.319908	−	0.947449i	$-0.603652\pi$
$433$	−13.3137	−0.639816	−0.319908	+	0.947449i	$0.603652\pi$
$434$	8.48528	0.407307
$435$	0	0
$436$	20.3431	0.974260
$437$	−6.41421	−0.306833
$438$	0	0
$439$	2.27208	0.108440	0.0542202	−	0.998529i	$-0.482733\pi$
$439$	2.27208	0.108440	0.0542202	+	0.998529i	$0.482733\pi$
$440$	0	0
$441$	0	0
$442$	−16.4853	−0.784125
$443$	7.97056	0.378693	0.189346	−	0.981910i	$-0.439363\pi$
$443$	7.97056	0.378693	0.189346	+	0.981910i	$0.439363\pi$
$444$	0	0
$445$	0	0
$446$	51.1127	2.42026
$447$	0	0
$448$	−4.07107	−0.192340
$449$	6.48528	0.306059	0.153030	−	0.988222i	$-0.451097\pi$
$449$	6.48528	0.306059	0.153030	+	0.988222i	$0.451097\pi$
$450$	0	0
$451$	10.8995	0.513237
$452$	38.2843	1.80074
$453$	0	0
$454$	−44.6274	−2.09447
$455$	0	0
$456$	0	0
$457$	3.85786	0.180463	0.0902316	−	0.995921i	$-0.471239\pi$
$457$	3.85786	0.180463	0.0902316	+	0.995921i	$0.471239\pi$
$458$	6.07107	0.283682
$459$	0	0
$460$	0	0
$461$	32.7696	1.52623	0.763115	−	0.646263i	$-0.223669\pi$
$461$	32.7696	1.52623	0.763115	+	0.646263i	$0.223669\pi$
$462$	0	0
$463$	34.9706	1.62522	0.812610	−	0.582808i	$-0.198046\pi$
$463$	34.9706	1.62522	0.812610	+	0.582808i	$0.198046\pi$
$464$	−3.51472	−0.163167
$465$	0	0
$466$	39.9706	1.85160
$467$	−10.6274	−0.491778	−0.245889	−	0.969298i	$-0.579080\pi$
$467$	−10.6274	−0.491778	−0.245889	+	0.969298i	$0.579080\pi$
$468$	0	0
$469$	−0.142136	−0.00656321
$470$	0	0
$471$	0	0
$472$	48.5563	2.23499
$473$	−11.6569	−0.535983
$474$	0	0
$475$	0	0
$476$	−3.82843	−0.175476
$477$	0	0
$478$	57.1127	2.61227
$479$	24.4853	1.11876	0.559381	−	0.828911i	$-0.311039\pi$
$479$	24.4853	1.11876	0.559381	+	0.828911i	$0.311039\pi$
$480$	0	0
$481$	0.485281	0.0221269
$482$	−34.1421	−1.55513
$483$	0	0
$484$	3.82843	0.174019
$485$	0	0
$486$	0	0
$487$	−6.48528	−0.293876	−0.146938	−	0.989146i	$-0.546942\pi$
$487$	−6.48528	−0.293876	−0.146938	+	0.989146i	$0.546942\pi$
$488$	−38.9706	−1.76411
$489$	0	0
$490$	0	0
$491$	24.1421	1.08952	0.544760	−	0.838592i	$-0.316621\pi$
$491$	24.1421	1.08952	0.544760	+	0.838592i	$0.316621\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	43.7990	1.97061
$495$	0	0
$496$	−25.4558	−1.14300
$497$	−3.24264	−0.145452
$498$	0	0
$499$	35.1716	1.57450	0.787248	−	0.616637i	$-0.211505\pi$
$499$	35.1716	1.57450	0.787248	+	0.616637i	$0.211505\pi$
$500$	0	0
$501$	0	0
$502$	21.6569	0.966593
$503$	34.2843	1.52866	0.764330	−	0.644825i	$-0.223070\pi$
$503$	34.2843	1.52866	0.764330	+	0.644825i	$0.223070\pi$
$504$	0	0
$505$	0	0
$506$	2.41421	0.107325
$507$	0	0
$508$	27.7279	1.23023
$509$	−4.62742	−0.205107	−0.102553	−	0.994728i	$-0.532701\pi$
$509$	−4.62742	−0.205107	−0.102553	+	0.994728i	$0.532701\pi$
$510$	0	0
$511$	−3.65685	−0.161770
$512$	31.2426	1.38074
$513$	0	0
$514$	22.4853	0.991783
$515$	0	0
$516$	0	0
$517$	7.48528	0.329202
$518$	0.171573	0.00753848
$519$	0	0
$520$	0	0
$521$	36.1421	1.58342	0.791708	−	0.610900i	$-0.209192\pi$
$521$	36.1421	1.58342	0.791708	+	0.610900i	$0.209192\pi$
$522$	0	0
$523$	42.2132	1.84585	0.922927	−	0.384974i	$-0.125790\pi$
$523$	42.2132	1.84585	0.922927	+	0.384974i	$0.125790\pi$
$524$	−26.1421	−1.14202
$525$	0	0
$526$	55.4558	2.41799
$527$	20.4853	0.892353
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	10.1716	0.440994
$533$	−30.8284	−1.33533
$534$	0	0
$535$	0	0
$536$	1.51472	0.0654259
$537$	0	0
$538$	−38.1421	−1.64442
$539$	−6.82843	−0.294121
$540$	0	0
$541$	11.3137	0.486414	0.243207	−	0.969974i	$-0.421801\pi$
$541$	11.3137	0.486414	0.243207	+	0.969974i	$0.421801\pi$
$542$	−21.4853	−0.922872
$543$	0	0
$544$	−3.82843	−0.164142
$545$	0	0
$546$	0	0
$547$	35.8701	1.53369	0.766846	−	0.641831i	$-0.221825\pi$
$547$	35.8701	1.53369	0.766846	+	0.641831i	$0.221825\pi$
$548$	46.4853	1.98575
$549$	0	0
$550$	0	0
$551$	−7.51472	−0.320138
$552$	0	0
$553$	5.48528	0.233258
$554$	−11.6569	−0.495252
$555$	0	0
$556$	−72.6274	−3.08009
$557$	5.17157	0.219127	0.109563	−	0.993980i	$-0.465055\pi$
$557$	5.17157	0.219127	0.109563	+	0.993980i	$0.465055\pi$
$558$	0	0
$559$	32.9706	1.39451
$560$	0	0
$561$	0	0
$562$	77.4264	3.26604
$563$	−15.3137	−0.645396	−0.322698	−	0.946502i	$-0.604590\pi$
$563$	−15.3137	−0.645396	−0.322698	+	0.946502i	$0.604590\pi$
$564$	0	0
$565$	0	0
$566$	2.17157	0.0912780
$567$	0	0
$568$	34.5563	1.44995
$569$	−15.2426	−0.639005	−0.319502	−	0.947585i	$-0.603516\pi$
$569$	−15.2426	−0.639005	−0.319502	+	0.947585i	$0.603516\pi$
$570$	0	0
$571$	−9.02944	−0.377870	−0.188935	−	0.981990i	$-0.560504\pi$
$571$	−9.02944	−0.377870	−0.188935	+	0.981990i	$0.560504\pi$
$572$	−10.8284	−0.452759
$573$	0	0
$574$	−10.8995	−0.454936
$575$	0	0
$576$	0	0
$577$	23.9706	0.997908	0.498954	−	0.866629i	$-0.333718\pi$
$577$	23.9706	0.997908	0.498954	+	0.866629i	$0.333718\pi$
$578$	26.9706	1.12183
$579$	0	0
$580$	0	0
$581$	1.85786	0.0770772
$582$	0	0
$583$	−7.65685	−0.317115
$584$	38.9706	1.61261
$585$	0	0
$586$	42.4558	1.75383
$587$	36.6569	1.51299	0.756495	−	0.653999i	$-0.226910\pi$
$587$	36.6569	1.51299	0.756495	+	0.653999i	$0.226910\pi$
$588$	0	0
$589$	−54.4264	−2.24260
$590$	0	0
$591$	0	0
$592$	−0.514719	−0.0211548
$593$	−3.79899	−0.156006	−0.0780029	−	0.996953i	$-0.524854\pi$
$593$	−3.79899	−0.156006	−0.0780029	+	0.996953i	$0.524854\pi$
$594$	0	0
$595$	0	0
$596$	−29.5858	−1.21188
$597$	0	0
$598$	−6.82843	−0.279235
$599$	−36.3137	−1.48374	−0.741869	−	0.670545i	$-0.766060\pi$
$599$	−36.3137	−1.48374	−0.741869	+	0.670545i	$0.766060\pi$
$600$	0	0
$601$	−14.8284	−0.604864	−0.302432	−	0.953171i	$-0.597799\pi$
$601$	−14.8284	−0.604864	−0.302432	+	0.953171i	$0.597799\pi$
$602$	11.6569	0.475098
$603$	0	0
$604$	53.5980	2.18087
$605$	0	0
$606$	0	0
$607$	−2.97056	−0.120571	−0.0602857	−	0.998181i	$-0.519201\pi$
$607$	−2.97056	−0.120571	−0.0602857	+	0.998181i	$0.519201\pi$
$608$	10.1716	0.412512
$609$	0	0
$610$	0	0
$611$	−21.1716	−0.856510
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−16.1421	−0.651444
$615$	0	0
$616$	−1.82843	−0.0736694
$617$	9.85786	0.396863	0.198431	−	0.980115i	$-0.436415\pi$
$617$	9.85786	0.396863	0.198431	+	0.980115i	$0.436415\pi$
$618$	0	0
$619$	38.6274	1.55257	0.776283	−	0.630384i	$-0.217103\pi$
$619$	38.6274	1.55257	0.776283	+	0.630384i	$0.217103\pi$
$620$	0	0
$621$	0	0
$622$	32.9706	1.32200
$623$	−1.51472	−0.0606859
$624$	0	0
$625$	0	0
$626$	65.5269	2.61898
$627$	0	0
$628$	22.9706	0.916625
$629$	0.414214	0.0165158
$630$	0	0
$631$	26.6274	1.06002	0.530010	−	0.847991i	$-0.322188\pi$
$631$	26.6274	1.06002	0.530010	+	0.847991i	$0.322188\pi$
$632$	−58.4558	−2.32525
$633$	0	0
$634$	74.4264	2.95585
$635$	0	0
$636$	0	0
$637$	19.3137	0.765237
$638$	2.82843	0.111979
$639$	0	0
$640$	0	0
$641$	42.4853	1.67807	0.839034	−	0.544079i	$-0.183121\pi$
$641$	42.4853	1.67807	0.839034	+	0.544079i	$0.183121\pi$
$642$	0	0
$643$	−32.9706	−1.30023	−0.650116	−	0.759835i	$-0.725280\pi$
$643$	−32.9706	−1.30023	−0.650116	+	0.759835i	$0.725280\pi$
$644$	−1.58579	−0.0624887
$645$	0	0
$646$	37.3848	1.47088
$647$	−17.3431	−0.681829	−0.340915	−	0.940094i	$-0.610737\pi$
$647$	−17.3431	−0.681829	−0.340915	+	0.940094i	$0.610737\pi$
$648$	0	0
$649$	−11.0000	−0.431788
$650$	0	0
$651$	0	0
$652$	60.4853	2.36879
$653$	40.4853	1.58431	0.792156	−	0.610319i	$-0.208959\pi$
$653$	40.4853	1.58431	0.792156	+	0.610319i	$0.208959\pi$
$654$	0	0
$655$	0	0
$656$	32.6985	1.27666
$657$	0	0
$658$	−7.48528	−0.291807
$659$	−15.1127	−0.588707	−0.294354	−	0.955697i	$-0.595104\pi$
$659$	−15.1127	−0.588707	−0.294354	+	0.955697i	$0.595104\pi$
$660$	0	0
$661$	−34.6569	−1.34800	−0.673998	−	0.738733i	$-0.735424\pi$
$661$	−34.6569	−1.34800	−0.673998	+	0.738733i	$0.735424\pi$
$662$	77.5980	3.01593
$663$	0	0
$664$	−19.7990	−0.768350
$665$	0	0
$666$	0	0
$667$	1.17157	0.0453635
$668$	−83.4558	−3.22900
$669$	0	0
$670$	0	0
$671$	8.82843	0.340818
$672$	0	0
$673$	11.6569	0.449339	0.224669	−	0.974435i	$-0.427870\pi$
$673$	11.6569	0.449339	0.224669	+	0.974435i	$0.427870\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−19.1421	−0.736236
$677$	−40.4853	−1.55598	−0.777988	−	0.628279i	$-0.783760\pi$
$677$	−40.4853	−1.55598	−0.777988	+	0.628279i	$0.783760\pi$
$678$	0	0
$679$	2.41421	0.0926490
$680$	0	0
$681$	0	0
$682$	20.4853	0.784422
$683$	−29.4853	−1.12822	−0.564111	−	0.825699i	$-0.690781\pi$
$683$	−29.4853	−1.12822	−0.564111	+	0.825699i	$0.690781\pi$
$684$	0	0
$685$	0	0
$686$	13.8284	0.527972
$687$	0	0
$688$	−34.9706	−1.33324
$689$	21.6569	0.825060
$690$	0	0
$691$	15.4558	0.587968	0.293984	−	0.955810i	$-0.405019\pi$
$691$	15.4558	0.587968	0.293984	+	0.955810i	$0.405019\pi$
$692$	48.0711	1.82739
$693$	0	0
$694$	−51.1127	−1.94021
$695$	0	0
$696$	0	0
$697$	−26.3137	−0.996703
$698$	6.00000	0.227103
$699$	0	0
$700$	0	0
$701$	4.61522	0.174315	0.0871573	−	0.996195i	$-0.472222\pi$
$701$	4.61522	0.174315	0.0871573	+	0.996195i	$0.472222\pi$
$702$	0	0
$703$	−1.10051	−0.0415063
$704$	−9.82843	−0.370423
$705$	0	0
$706$	−10.8284	−0.407533
$707$	−6.17157	−0.232106
$708$	0	0
$709$	−0.857864	−0.0322178	−0.0161089	−	0.999870i	$-0.505128\pi$
$709$	−0.857864	−0.0322178	−0.0161089	+	0.999870i	$0.505128\pi$
$710$	0	0
$711$	0	0
$712$	16.1421	0.604952
$713$	8.48528	0.317776
$714$	0	0
$715$	0	0
$716$	−64.3137	−2.40352
$717$	0	0
$718$	37.4558	1.39784
$719$	1.65685	0.0617902	0.0308951	−	0.999523i	$-0.490164\pi$
$719$	1.65685	0.0617902	0.0308951	+	0.999523i	$0.490164\pi$
$720$	0	0
$721$	−5.65685	−0.210672
$722$	−53.4558	−1.98942
$723$	0	0
$724$	−84.1127	−3.12602
$725$	0	0
$726$	0	0
$727$	−16.9706	−0.629403	−0.314702	−	0.949191i	$-0.601904\pi$
$727$	−16.9706	−0.629403	−0.314702	+	0.949191i	$0.601904\pi$
$728$	5.17157	0.191671
$729$	0	0
$730$	0	0
$731$	28.1421	1.04087
$732$	0	0
$733$	3.85786	0.142493	0.0712467	−	0.997459i	$-0.477302\pi$
$733$	3.85786	0.142493	0.0712467	+	0.997459i	$0.477302\pi$
$734$	−3.17157	−0.117065
$735$	0	0
$736$	−1.58579	−0.0584529
$737$	−0.343146	−0.0126399
$738$	0	0
$739$	17.5858	0.646904	0.323452	−	0.946245i	$-0.395157\pi$
$739$	17.5858	0.646904	0.323452	+	0.946245i	$0.395157\pi$
$740$	0	0
$741$	0	0
$742$	7.65685	0.281092
$743$	−31.1127	−1.14141	−0.570707	−	0.821154i	$-0.693331\pi$
$743$	−31.1127	−1.14141	−0.570707	+	0.821154i	$0.693331\pi$
$744$	0	0
$745$	0	0
$746$	57.1127	2.09104
$747$	0	0
$748$	−9.24264	−0.337944
$749$	2.20101	0.0804232
$750$	0	0
$751$	−47.5980	−1.73687	−0.868437	−	0.495799i	$-0.834875\pi$
$751$	−47.5980	−1.73687	−0.868437	+	0.495799i	$0.834875\pi$
$752$	22.4558	0.818880
$753$	0	0
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	−33.3137	−1.21081	−0.605404	−	0.795919i	$-0.706988\pi$
$757$	−33.3137	−1.21081	−0.605404	+	0.795919i	$0.706988\pi$
$758$	−22.1421	−0.804239
$759$	0	0
$760$	0	0
$761$	10.8284	0.392530	0.196265	−	0.980551i	$-0.437119\pi$
$761$	10.8284	0.392530	0.196265	+	0.980551i	$0.437119\pi$
$762$	0	0
$763$	2.20101	0.0796819
$764$	−23.6274	−0.854810
$765$	0	0
$766$	48.2843	1.74458
$767$	31.1127	1.12341
$768$	0	0
$769$	3.65685	0.131870	0.0659348	−	0.997824i	$-0.478997\pi$
$769$	3.65685	0.131870	0.0659348	+	0.997824i	$0.478997\pi$
$770$	0	0
$771$	0	0
$772$	12.6863	0.456590
$773$	−4.82843	−0.173666	−0.0868332	−	0.996223i	$-0.527675\pi$
$773$	−4.82843	−0.173666	−0.0868332	+	0.996223i	$0.527675\pi$
$774$	0	0
$775$	0	0
$776$	−25.7279	−0.923579
$777$	0	0
$778$	−42.6274	−1.52827
$779$	69.9117	2.50485
$780$	0	0
$781$	−7.82843	−0.280123
$782$	−5.82843	−0.208424
$783$	0	0
$784$	−20.4853	−0.731617
$785$	0	0
$786$	0	0
$787$	−22.0711	−0.786749	−0.393374	−	0.919378i	$-0.628692\pi$
$787$	−22.0711	−0.786749	−0.393374	+	0.919378i	$0.628692\pi$
$788$	−18.2132	−0.648819
$789$	0	0
$790$	0	0
$791$	4.14214	0.147277
$792$	0	0
$793$	−24.9706	−0.886731
$794$	86.7696	3.07934
$795$	0	0
$796$	41.4558	1.46936
$797$	−7.02944	−0.248995	−0.124498	−	0.992220i	$-0.539732\pi$
$797$	−7.02944	−0.248995	−0.124498	+	0.992220i	$0.539732\pi$
$798$	0	0
$799$	−18.0711	−0.639308
$800$	0	0
$801$	0	0
$802$	76.7696	2.71083
$803$	−8.82843	−0.311548
$804$	0	0
$805$	0	0
$806$	−57.9411	−2.04089
$807$	0	0
$808$	65.7696	2.31376
$809$	17.7279	0.623281	0.311640	−	0.950200i	$-0.399122\pi$
$809$	17.7279	0.623281	0.311640	+	0.950200i	$0.399122\pi$
$810$	0	0
$811$	−32.2132	−1.13116	−0.565579	−	0.824694i	$-0.691347\pi$
$811$	−32.2132	−1.13116	−0.565579	+	0.824694i	$0.691347\pi$
$812$	−1.85786	−0.0651983
$813$	0	0
$814$	0.414214	0.0145182
$815$	0	0
$816$	0	0
$817$	−74.7696	−2.61586
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	23.7990	0.830590	0.415295	−	0.909687i	$-0.363678\pi$
$821$	23.7990	0.830590	0.415295	+	0.909687i	$0.363678\pi$
$822$	0	0
$823$	18.9706	0.661272	0.330636	−	0.943758i	$-0.392737\pi$
$823$	18.9706	0.661272	0.330636	+	0.943758i	$0.392737\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	11.0000	0.382739
$827$	35.3137	1.22798	0.613989	−	0.789315i	$-0.289564\pi$
$827$	35.3137	1.22798	0.613989	+	0.789315i	$0.289564\pi$
$828$	0	0
$829$	19.9411	0.692584	0.346292	−	0.938127i	$-0.387441\pi$
$829$	19.9411	0.692584	0.346292	+	0.938127i	$0.387441\pi$
$830$	0	0
$831$	0	0
$832$	27.7990	0.963757
$833$	16.4853	0.571181
$834$	0	0
$835$	0	0
$836$	24.5563	0.849299
$837$	0	0
$838$	−61.5269	−2.12541
$839$	24.6863	0.852265	0.426133	−	0.904661i	$-0.359876\pi$
$839$	24.6863	0.852265	0.426133	+	0.904661i	$0.359876\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	65.1838	2.24638
$843$	0	0
$844$	−50.9706	−1.75448
$845$	0	0
$846$	0	0
$847$	0.414214	0.0142325
$848$	−22.9706	−0.788812
$849$	0	0
$850$	0	0
$851$	0.171573	0.00588144
$852$	0	0
$853$	−24.8284	−0.850109	−0.425055	−	0.905168i	$-0.639745\pi$
$853$	−24.8284	−0.850109	−0.425055	+	0.905168i	$0.639745\pi$
$854$	−8.82843	−0.302103
$855$	0	0
$856$	−23.4558	−0.801704
$857$	−22.6985	−0.775365	−0.387683	−	0.921793i	$-0.626724\pi$
$857$	−22.6985	−0.775365	−0.387683	+	0.921793i	$0.626724\pi$
$858$	0	0
$859$	−3.51472	−0.119921	−0.0599603	−	0.998201i	$-0.519097\pi$
$859$	−3.51472	−0.119921	−0.0599603	+	0.998201i	$0.519097\pi$
$860$	0	0
$861$	0	0
$862$	2.82843	0.0963366
$863$	51.3137	1.74674	0.873369	−	0.487058i	$-0.161930\pi$
$863$	51.3137	1.74674	0.873369	+	0.487058i	$0.161930\pi$
$864$	0	0
$865$	0	0
$866$	32.1421	1.09223
$867$	0	0
$868$	−13.4558	−0.456721
$869$	13.2426	0.449226
$870$	0	0
$871$	0.970563	0.0328863
$872$	−23.4558	−0.794315
$873$	0	0
$874$	15.4853	0.523797
$875$	0	0
$876$	0	0
$877$	−47.1127	−1.59088	−0.795441	−	0.606031i	$-0.792761\pi$
$877$	−47.1127	−1.59088	−0.795441	+	0.606031i	$0.792761\pi$
$878$	−5.48528	−0.185119
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−6.62742	−0.223030	−0.111515	−	0.993763i	$-0.535570\pi$
$883$	−6.62742	−0.223030	−0.111515	+	0.993763i	$0.535570\pi$
$884$	26.1421	0.879255
$885$	0	0
$886$	−19.2426	−0.646469
$887$	−6.14214	−0.206233	−0.103116	−	0.994669i	$-0.532881\pi$
$887$	−6.14214	−0.206233	−0.103116	+	0.994669i	$0.532881\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	−81.0538	−2.71388
$893$	48.0122	1.60667
$894$	0	0
$895$	0	0
$896$	8.51472	0.284457
$897$	0	0
$898$	−15.6569	−0.522476
$899$	9.94113	0.331555
$900$	0	0
$901$	18.4853	0.615834
$902$	−26.3137	−0.876151
$903$	0	0
$904$	−44.1421	−1.46815
$905$	0	0
$906$	0	0
$907$	14.4853	0.480976	0.240488	−	0.970652i	$-0.422693\pi$
$907$	14.4853	0.480976	0.240488	+	0.970652i	$0.422693\pi$
$908$	70.7696	2.34857
$909$	0	0
$910$	0	0
$911$	−45.4853	−1.50699	−0.753497	−	0.657451i	$-0.771635\pi$
$911$	−45.4853	−1.50699	−0.753497	+	0.657451i	$0.771635\pi$
$912$	0	0
$913$	4.48528	0.148441
$914$	−9.31371	−0.308070
$915$	0	0
$916$	−9.62742	−0.318099
$917$	−2.82843	−0.0934029
$918$	0	0
$919$	44.2132	1.45846	0.729230	−	0.684269i	$-0.239879\pi$
$919$	44.2132	1.45846	0.729230	+	0.684269i	$0.239879\pi$
$920$	0	0
$921$	0	0
$922$	−79.1127	−2.60544
$923$	22.1421	0.728817
$924$	0	0
$925$	0	0
$926$	−84.4264	−2.77442
$927$	0	0
$928$	−1.85786	−0.0609874
$929$	−25.7990	−0.846437	−0.423219	−	0.906028i	$-0.639100\pi$
$929$	−25.7990	−0.846437	−0.423219	+	0.906028i	$0.639100\pi$
$930$	0	0
$931$	−43.7990	−1.43545
$932$	−63.3848	−2.07624
$933$	0	0
$934$	25.6569	0.839518
$935$	0	0
$936$	0	0
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	0.343146	0.0112041
$939$	0	0
$940$	0	0
$941$	3.10051	0.101074	0.0505368	−	0.998722i	$-0.483907\pi$
$941$	3.10051	0.101074	0.0505368	+	0.998722i	$0.483907\pi$
$942$	0	0
$943$	−10.8995	−0.354936
$944$	−33.0000	−1.07406
$945$	0	0
$946$	28.1421	0.914980
$947$	−2.79899	−0.0909549	−0.0454775	−	0.998965i	$-0.514481\pi$
$947$	−2.79899	−0.0909549	−0.0454775	+	0.998965i	$0.514481\pi$
$948$	0	0
$949$	24.9706	0.810579
$950$	0	0
$951$	0	0
$952$	4.41421	0.143065
$953$	−19.0416	−0.616819	−0.308409	−	0.951254i	$-0.599797\pi$
$953$	−19.0416	−0.616819	−0.308409	+	0.951254i	$0.599797\pi$
$954$	0	0
$955$	0	0
$956$	−90.5685	−2.92920
$957$	0	0
$958$	−59.1127	−1.90984
$959$	5.02944	0.162409
$960$	0	0
$961$	41.0000	1.32258
$962$	−1.17157	−0.0377730
$963$	0	0
$964$	54.1421	1.74380
$965$	0	0
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	−4.41421	−0.141878
$969$	0	0
$970$	0	0
$971$	27.6863	0.888495	0.444248	−	0.895904i	$-0.353471\pi$
$971$	27.6863	0.888495	0.444248	+	0.895904i	$0.353471\pi$
$972$	0	0
$973$	−7.85786	−0.251912
$974$	15.6569	0.501678
$975$	0	0
$976$	26.4853	0.847773
$977$	52.5685	1.68182	0.840908	−	0.541178i	$-0.182021\pi$
$977$	52.5685	1.68182	0.840908	+	0.541178i	$0.182021\pi$
$978$	0	0
$979$	−3.65685	−0.116874
$980$	0	0
$981$	0	0
$982$	−58.2843	−1.85993
$983$	5.28427	0.168542	0.0842710	−	0.996443i	$-0.473144\pi$
$983$	5.28427	0.168542	0.0842710	+	0.996443i	$0.473144\pi$
$984$	0	0
$985$	0	0
$986$	−6.82843	−0.217461
$987$	0	0
$988$	−69.4558	−2.20968
$989$	11.6569	0.370666
$990$	0	0
$991$	14.2843	0.453755	0.226877	−	0.973923i	$-0.427148\pi$
$991$	14.2843	0.453755	0.226877	+	0.973923i	$0.427148\pi$
$992$	−13.4558	−0.427223
$993$	0	0
$994$	7.82843	0.248303
$995$	0	0
$996$	0	0
$997$	43.2548	1.36989	0.684947	−	0.728593i	$-0.259825\pi$
$997$	43.2548	1.36989	0.684947	+	0.728593i	$0.259825\pi$
$998$	−84.9117	−2.68783
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.l.1.1		2
3.2	odd	2		825.2.a.f.1.2	yes	2
5.2	odd	4		2475.2.c.o.199.1		4
5.3	odd	4		2475.2.c.o.199.4		4
5.4	even	2		2475.2.a.w.1.2		2
15.2	even	4		825.2.c.d.199.4		4
15.8	even	4		825.2.c.d.199.1		4
15.14	odd	2		825.2.a.d.1.1	✓	2
33.32	even	2		9075.2.a.w.1.1		2
165.164	even	2		9075.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.d.1.1	✓	2	15.14	odd	2
825.2.a.f.1.2	yes	2	3.2	odd	2
825.2.c.d.199.1		4	15.8	even	4
825.2.c.d.199.4		4	15.2	even	4
2475.2.a.l.1.1		2	1.1	even	1	trivial
2475.2.a.w.1.2		2	5.4	even	2
2475.2.c.o.199.1		4	5.2	odd	4
2475.2.c.o.199.4		4	5.3	odd	4
9075.2.a.w.1.1		2	33.32	even	2
9075.2.a.ca.1.2		2	165.164	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} +0.414214 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} +0.414214 q^{7} -4.41421 q^{8} +1.00000 q^{11} -2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} -2.41421 q^{17} +6.41421 q^{19} -2.41421 q^{22} -1.00000 q^{23} +6.82843 q^{26} +1.58579 q^{28} -1.17157 q^{29} -8.48528 q^{31} +1.58579 q^{32} +5.82843 q^{34} -0.171573 q^{37} -15.4853 q^{38} +10.8995 q^{41} -11.6569 q^{43} +3.82843 q^{44} +2.41421 q^{46} +7.48528 q^{47} -6.82843 q^{49} -10.8284 q^{52} -7.65685 q^{53} -1.82843 q^{56} +2.82843 q^{58} -11.0000 q^{59} +8.82843 q^{61} +20.4853 q^{62} -9.82843 q^{64} -0.343146 q^{67} -9.24264 q^{68} -7.82843 q^{71} -8.82843 q^{73} +0.414214 q^{74} +24.5563 q^{76} +0.414214 q^{77} +13.2426 q^{79} -26.3137 q^{82} +4.48528 q^{83} +28.1421 q^{86} -4.41421 q^{88} -3.65685 q^{89} -1.17157 q^{91} -3.82843 q^{92} -18.0711 q^{94} +5.82843 q^{97} +16.4853 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{22} - 2 q^{23} + 8 q^{26} + 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} - 6 q^{37} - 14 q^{38} + 2 q^{41} - 12 q^{43} + 2 q^{44} + 2 q^{46} - 2 q^{47} - 8 q^{49} - 16 q^{52} - 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} - 12 q^{67} - 10 q^{68} - 10 q^{71} - 12 q^{73} - 2 q^{74} + 18 q^{76} - 2 q^{77} + 18 q^{79} - 30 q^{82} - 8 q^{83} + 28 q^{86} - 6 q^{88} + 4 q^{89} - 8 q^{91} - 2 q^{92} - 22 q^{94} + 6 q^{97} + 16 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^22 - 2 * q^23 + 8 * q^26 + 6 * q^28 - 8 * q^29 + 6 * q^32 + 6 * q^34 - 6 * q^37 - 14 * q^38 + 2 * q^41 - 12 * q^43 + 2 * q^44 + 2 * q^46 - 2 * q^47 - 8 * q^49 - 16 * q^52 - 4 * q^53 + 2 * q^56 - 22 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 - 12 * q^67 - 10 * q^68 - 10 * q^71 - 12 * q^73 - 2 * q^74 + 18 * q^76 - 2 * q^77 + 18 * q^79 - 30 * q^82 - 8 * q^83 + 28 * q^86 - 6 * q^88 + 4 * q^89 - 8 * q^91 - 2 * q^92 - 22 * q^94 + 6 * q^97 + 16 * q^98

Introduction

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