LMFDB - Embedded newform 3025.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.672032$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67203 q^{2} +3.10994 q^{3} +0.795692 q^{4} -5.19992 q^{6} -3.08998 q^{7} +2.01364 q^{8} +6.67173 q^{9} +O(q^{10})$	\(q-1.67203 q^{2} +3.10994 q^{3} +0.795692 q^{4} -5.19992 q^{6} -3.08998 q^{7} +2.01364 q^{8} +6.67173 q^{9} +2.47456 q^{12} -3.37905 q^{13} +5.16655 q^{14} -4.95826 q^{16} -0.103701 q^{17} -11.1553 q^{18} +2.03005 q^{19} -9.60965 q^{21} -2.23435 q^{23} +6.26230 q^{24} +5.64988 q^{26} +11.4188 q^{27} -2.45867 q^{28} -6.49038 q^{29} -9.07559 q^{31} +4.26309 q^{32} +0.173391 q^{34} +5.30864 q^{36} +0.333342 q^{37} -3.39430 q^{38} -10.5086 q^{39} +3.93946 q^{41} +16.0677 q^{42} -7.39472 q^{43} +3.73591 q^{46} -6.62954 q^{47} -15.4199 q^{48} +2.54798 q^{49} -0.322502 q^{51} -2.68868 q^{52} -0.310989 q^{53} -19.0927 q^{54} -6.22211 q^{56} +6.31332 q^{57} +10.8521 q^{58} +11.2231 q^{59} -3.39136 q^{61} +15.1747 q^{62} -20.6155 q^{63} +2.78850 q^{64} -12.8133 q^{67} -0.0825137 q^{68} -6.94870 q^{69} -2.66814 q^{71} +13.4345 q^{72} +7.48626 q^{73} -0.557358 q^{74} +1.61529 q^{76} +17.5708 q^{78} +2.41593 q^{79} +15.4967 q^{81} -6.58691 q^{82} -0.143148 q^{83} -7.64633 q^{84} +12.3642 q^{86} -20.1847 q^{87} +13.9190 q^{89} +10.4412 q^{91} -1.77786 q^{92} -28.2246 q^{93} +11.0848 q^{94} +13.2579 q^{96} -5.92242 q^{97} -4.26030 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * 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 $

$2$	−1.67203	−1.18231	−0.591153	−	0.806560i	$-0.701327\pi$
$2$	−1.67203	−1.18231	−0.591153	+	0.806560i	$0.701327\pi$
$3$	3.10994	1.79552	0.897762	−	0.440480i	$-0.145192\pi$
$3$	3.10994	1.79552	0.897762	+	0.440480i	$0.145192\pi$
$4$	0.795692	0.397846
$5$	0	0
$5$	0	0
$6$	−5.19992	−2.12286
$7$	−3.08998	−1.16790	−0.583951	−	0.811789i	$-0.698494\pi$
$7$	−3.08998	−1.16790	−0.583951	+	0.811789i	$0.698494\pi$
$8$	2.01364	0.711930
$9$	6.67173	2.22391
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.47456	0.714343
$13$	−3.37905	−0.937180	−0.468590	−	0.883416i	$-0.655238\pi$
$13$	−3.37905	−0.937180	−0.468590	+	0.883416i	$0.655238\pi$
$14$	5.16655	1.38082
$15$	0	0
$16$	−4.95826	−1.23956
$17$	−0.103701	−0.0251511	−0.0125755	−	0.999921i	$-0.504003\pi$
$17$	−0.103701	−0.0251511	−0.0125755	+	0.999921i	$0.504003\pi$
$18$	−11.1553	−2.62934
$19$	2.03005	0.465724	0.232862	−	0.972510i	$-0.425191\pi$
$19$	2.03005	0.465724	0.232862	+	0.972510i	$0.425191\pi$
$20$	0	0
$21$	−9.60965	−2.09700
$22$	0	0
$23$	−2.23435	−0.465895	−0.232947	−	0.972489i	$-0.574837\pi$
$23$	−2.23435	−0.465895	−0.232947	+	0.972489i	$0.574837\pi$
$24$	6.26230	1.27829
$25$	0	0
$26$	5.64988	1.10803
$27$	11.4188	2.19756
$28$	−2.45867	−0.464646
$29$	−6.49038	−1.20523	−0.602617	−	0.798031i	$-0.705875\pi$
$29$	−6.49038	−1.20523	−0.602617	+	0.798031i	$0.705875\pi$
$30$	0	0
$31$	−9.07559	−1.63002	−0.815012	−	0.579444i	$-0.803270\pi$
$31$	−9.07559	−1.63002	−0.815012	+	0.579444i	$0.803270\pi$
$32$	4.26309	0.753614
$33$	0	0
$34$	0.173391	0.0297362
$35$	0	0
$36$	5.30864	0.884774
$37$	0.333342	0.0548010	0.0274005	−	0.999625i	$-0.491277\pi$
$37$	0.333342	0.0548010	0.0274005	+	0.999625i	$0.491277\pi$
$38$	−3.39430	−0.550628
$39$	−10.5086	−1.68273
$40$	0	0
$41$	3.93946	0.615241	0.307620	−	0.951509i	$-0.400467\pi$
$41$	3.93946	0.615241	0.307620	+	0.951509i	$0.400467\pi$
$42$	16.0677	2.47929
$43$	−7.39472	−1.12768	−0.563842	−	0.825883i	$-0.690677\pi$
$43$	−7.39472	−1.12768	−0.563842	+	0.825883i	$0.690677\pi$
$44$	0	0
$45$	0	0
$46$	3.73591	0.550830
$47$	−6.62954	−0.967017	−0.483509	−	0.875340i	$-0.660638\pi$
$47$	−6.62954	−0.967017	−0.483509	+	0.875340i	$0.660638\pi$
$48$	−15.4199	−2.22567
$49$	2.54798	0.363997
$50$	0	0
$51$	−0.322502	−0.0451594
$52$	−2.68868	−0.372853
$53$	−0.310989	−0.0427176	−0.0213588	−	0.999772i	$-0.506799\pi$
$53$	−0.310989	−0.0427176	−0.0213588	+	0.999772i	$0.506799\pi$
$54$	−19.0927	−2.59818
$55$	0	0
$56$	−6.22211	−0.831465
$57$	6.31332	0.836220
$58$	10.8521	1.42495
$59$	11.2231	1.46112	0.730558	−	0.682850i	$-0.239260\pi$
$59$	11.2231	1.46112	0.730558	+	0.682850i	$0.239260\pi$
$60$	0	0
$61$	−3.39136	−0.434219	−0.217109	−	0.976147i	$-0.569663\pi$
$61$	−3.39136	−0.434219	−0.217109	+	0.976147i	$0.569663\pi$
$62$	15.1747	1.92719
$63$	−20.6155	−2.59731
$64$	2.78850	0.348562
$65$	0	0
$66$	0	0
$67$	−12.8133	−1.56540	−0.782699	−	0.622400i	$-0.786158\pi$
$67$	−12.8133	−1.56540	−0.782699	+	0.622400i	$0.786158\pi$
$68$	−0.0825137	−0.0100063
$69$	−6.94870	−0.836526
$70$	0	0
$71$	−2.66814	−0.316650	−0.158325	−	0.987387i	$-0.550609\pi$
$71$	−2.66814	−0.316650	−0.158325	+	0.987387i	$0.550609\pi$
$72$	13.4345	1.58327
$73$	7.48626	0.876200	0.438100	−	0.898926i	$-0.355652\pi$
$73$	7.48626	0.876200	0.438100	+	0.898926i	$0.355652\pi$
$74$	−0.557358	−0.0647916
$75$	0	0
$76$	1.61529	0.185287
$77$	0	0
$78$	17.5708	1.98950
$79$	2.41593	0.271814	0.135907	−	0.990722i	$-0.456605\pi$
$79$	2.41593	0.271814	0.135907	+	0.990722i	$0.456605\pi$
$80$	0	0
$81$	15.4967	1.72186
$82$	−6.58691	−0.727402
$83$	−0.143148	−0.0157125	−0.00785627	−	0.999969i	$-0.502501\pi$
$83$	−0.143148	−0.0157125	−0.00785627	+	0.999969i	$0.502501\pi$
$84$	−7.64633	−0.834283
$85$	0	0
$86$	12.3642	1.33327
$87$	−20.1847	−2.16403
$88$	0	0
$89$	13.9190	1.47541	0.737704	−	0.675124i	$-0.235910\pi$
$89$	13.9190	1.47541	0.737704	+	0.675124i	$0.235910\pi$
$90$	0	0
$91$	10.4412	1.09454
$92$	−1.77786	−0.185354
$93$	−28.2246	−2.92675
$94$	11.0848	1.14331
$95$	0	0
$96$	13.2579	1.35313
$97$	−5.92242	−0.601330	−0.300665	−	0.953730i	$-0.597209\pi$
$97$	−5.92242	−0.601330	−0.300665	+	0.953730i	$0.597209\pi$
$98$	−4.26030	−0.430356
$99$	0	0
$100$	0	0
$101$	6.14926	0.611874	0.305937	−	0.952052i	$-0.401030\pi$
$101$	6.14926	0.611874	0.305937	+	0.952052i	$0.401030\pi$
$102$	0.539234	0.0533922
$103$	−6.41587	−0.632174	−0.316087	−	0.948730i	$-0.602369\pi$
$103$	−6.41587	−0.632174	−0.316087	+	0.948730i	$0.602369\pi$
$104$	−6.80420	−0.667206
$105$	0	0
$106$	0.519983	0.0505053
$107$	2.92433	0.282706	0.141353	−	0.989959i	$-0.454855\pi$
$107$	2.92433	0.282706	0.141353	+	0.989959i	$0.454855\pi$
$108$	9.08589	0.874290
$109$	−10.9549	−1.04929	−0.524644	−	0.851322i	$-0.675801\pi$
$109$	−10.9549	−1.04929	−0.524644	+	0.851322i	$0.675801\pi$
$110$	0	0
$111$	1.03667	0.0983966
$112$	15.3209	1.44769
$113$	−13.4734	−1.26748	−0.633738	−	0.773548i	$-0.718480\pi$
$113$	−13.4734	−1.26748	−0.633738	+	0.773548i	$0.718480\pi$
$114$	−10.5561	−0.988667
$115$	0	0
$116$	−5.16435	−0.479498
$117$	−22.5441	−2.08420
$118$	−18.7653	−1.72749
$119$	0.320433	0.0293740
$120$	0	0
$121$	0	0
$122$	5.67046	0.513379
$123$	12.2515	1.10468
$124$	−7.22138	−0.648499
$125$	0	0
$126$	34.4698	3.07081
$127$	−12.9324	−1.14757	−0.573784	−	0.819006i	$-0.694525\pi$
$127$	−12.9324	−1.14757	−0.573784	+	0.819006i	$0.694525\pi$
$128$	−13.1886	−1.16572
$129$	−22.9971	−2.02478
$130$	0	0
$131$	−3.73795	−0.326587	−0.163293	−	0.986578i	$-0.552212\pi$
$131$	−3.73795	−0.326587	−0.163293	+	0.986578i	$0.552212\pi$
$132$	0	0
$133$	−6.27280	−0.543921
$134$	21.4243	1.85078
$135$	0	0
$136$	−0.208816	−0.0179058
$137$	−17.4724	−1.49277	−0.746383	−	0.665517i	$-0.768211\pi$
$137$	−17.4724	−1.49277	−0.746383	+	0.665517i	$0.768211\pi$
$138$	11.6185	0.989029
$139$	−0.118114	−0.0100183	−0.00500917	−	0.999987i	$-0.501594\pi$
$139$	−0.118114	−0.0100183	−0.00500917	+	0.999987i	$0.501594\pi$
$140$	0	0
$141$	−20.6175	−1.73630
$142$	4.46122	0.374377
$143$	0	0
$144$	−33.0801	−2.75668
$145$	0	0
$146$	−12.5173	−1.03594
$147$	7.92406	0.653565
$148$	0.265237	0.0218024
$149$	5.53761	0.453659	0.226829	−	0.973935i	$-0.427164\pi$
$149$	5.53761	0.453659	0.226829	+	0.973935i	$0.427164\pi$
$150$	0	0
$151$	11.0776	0.901485	0.450742	−	0.892654i	$-0.351159\pi$
$151$	11.0776	0.901485	0.450742	+	0.892654i	$0.351159\pi$
$152$	4.08778	0.331563
$153$	−0.691861	−0.0559337
$154$	0	0
$155$	0	0
$156$	−8.36165	−0.669468
$157$	2.16524	0.172805	0.0864026	−	0.996260i	$-0.472463\pi$
$157$	2.16524	0.172805	0.0864026	+	0.996260i	$0.472463\pi$
$158$	−4.03952	−0.321367
$159$	−0.967156	−0.0767005
$160$	0	0
$161$	6.90411	0.544120
$162$	−25.9111	−2.03577
$163$	1.62467	0.127254	0.0636272	−	0.997974i	$-0.479733\pi$
$163$	1.62467	0.127254	0.0636272	+	0.997974i	$0.479733\pi$
$164$	3.13460	0.244771
$165$	0	0
$166$	0.239348	0.0185770
$167$	−22.7344	−1.75924	−0.879620	−	0.475676i	$-0.842203\pi$
$167$	−22.7344	−1.75924	−0.879620	+	0.475676i	$0.842203\pi$
$168$	−19.3504	−1.49292
$169$	−1.58202	−0.121694
$170$	0	0
$171$	13.5439	1.03573
$172$	−5.88392	−0.448645
$173$	−3.24107	−0.246414	−0.123207	−	0.992381i	$-0.539318\pi$
$173$	−3.24107	−0.246414	−0.123207	+	0.992381i	$0.539318\pi$
$174$	33.7495	2.55854
$175$	0	0
$176$	0	0
$177$	34.9030	2.62347
$178$	−23.2730	−1.74438
$179$	−3.11850	−0.233088	−0.116544	−	0.993186i	$-0.537182\pi$
$179$	−3.11850	−0.233088	−0.116544	+	0.993186i	$0.537182\pi$
$180$	0	0
$181$	−11.1248	−0.826899	−0.413450	−	0.910527i	$-0.635676\pi$
$181$	−11.1248	−0.826899	−0.413450	+	0.910527i	$0.635676\pi$
$182$	−17.4580	−1.29407
$183$	−10.5469	−0.779650
$184$	−4.49919	−0.331684
$185$	0	0
$186$	47.1924	3.46031
$187$	0	0
$188$	−5.27507	−0.384724
$189$	−35.2840	−2.56653
$190$	0	0
$191$	−7.71683	−0.558370	−0.279185	−	0.960237i	$-0.590064\pi$
$191$	−7.71683	−0.558370	−0.279185	+	0.960237i	$0.590064\pi$
$192$	8.67206	0.625852
$193$	13.5646	0.976403	0.488202	−	0.872731i	$-0.337653\pi$
$193$	13.5646	0.976403	0.488202	+	0.872731i	$0.337653\pi$
$194$	9.90247	0.710956
$195$	0	0
$196$	2.02741	0.144815
$197$	5.19874	0.370395	0.185197	−	0.982701i	$-0.440708\pi$
$197$	5.19874	0.370395	0.185197	+	0.982701i	$0.440708\pi$
$198$	0	0
$199$	23.1433	1.64059	0.820293	−	0.571943i	$-0.193810\pi$
$199$	23.1433	1.64059	0.820293	+	0.571943i	$0.193810\pi$
$200$	0	0
$201$	−39.8487	−2.81071
$202$	−10.2818	−0.723422
$203$	20.0551	1.40760
$204$	−0.256613	−0.0179665
$205$	0	0
$206$	10.7275	0.747423
$207$	−14.9070	−1.03611
$208$	16.7542	1.16170
$209$	0	0
$210$	0	0
$211$	−19.7696	−1.36100	−0.680498	−	0.732750i	$-0.738237\pi$
$211$	−19.7696	−1.36100	−0.680498	+	0.732750i	$0.738237\pi$
$212$	−0.247451	−0.0169950
$213$	−8.29776	−0.568553
$214$	−4.88958	−0.334245
$215$	0	0
$216$	22.9935	1.56451
$217$	28.0434	1.90371
$218$	18.3169	1.24058
$219$	23.2818	1.57324
$220$	0	0
$221$	0.350409	0.0235711
$222$	−1.73335	−0.116335
$223$	19.7365	1.32166	0.660828	−	0.750538i	$-0.270205\pi$
$223$	19.7365	1.32166	0.660828	+	0.750538i	$0.270205\pi$
$224$	−13.1729	−0.880148
$225$	0	0
$226$	22.5280	1.49854
$227$	−25.4084	−1.68641	−0.843207	−	0.537589i	$-0.819335\pi$
$227$	−25.4084	−1.68641	−0.843207	+	0.537589i	$0.819335\pi$
$228$	5.02346	0.332687
$229$	−13.1441	−0.868585	−0.434292	−	0.900772i	$-0.643002\pi$
$229$	−13.1441	−0.868585	−0.434292	+	0.900772i	$0.643002\pi$
$230$	0	0
$231$	0	0
$232$	−13.0693	−0.858041
$233$	4.83935	0.317036	0.158518	−	0.987356i	$-0.449328\pi$
$233$	4.83935	0.317036	0.158518	+	0.987356i	$0.449328\pi$
$234$	37.6945	2.46416
$235$	0	0
$236$	8.93010	0.581300
$237$	7.51341	0.488048
$238$	−0.535774	−0.0347290
$239$	14.5318	0.939981	0.469991	−	0.882671i	$-0.344257\pi$
$239$	14.5318	0.939981	0.469991	+	0.882671i	$0.344257\pi$
$240$	0	0
$241$	23.0851	1.48704	0.743520	−	0.668714i	$-0.233155\pi$
$241$	23.0851	1.48704	0.743520	+	0.668714i	$0.233155\pi$
$242$	0	0
$243$	13.9374	0.894085
$244$	−2.69848	−0.172752
$245$	0	0
$246$	−20.4849	−1.30607
$247$	−6.85963	−0.436467
$248$	−18.2750	−1.16046
$249$	−0.445182	−0.0282123
$250$	0	0
$251$	10.4272	0.658156	0.329078	−	0.944303i	$-0.393262\pi$
$251$	10.4272	0.658156	0.329078	+	0.944303i	$0.393262\pi$
$252$	−16.4036	−1.03333
$253$	0	0
$254$	21.6235	1.35678
$255$	0	0
$256$	16.4748	1.02968
$257$	2.20198	0.137356	0.0686779	−	0.997639i	$-0.478122\pi$
$257$	2.20198	0.137356	0.0686779	+	0.997639i	$0.478122\pi$
$258$	38.4519	2.39391
$259$	−1.03002	−0.0640023
$260$	0	0
$261$	−43.3020	−2.68033
$262$	6.24998	0.386125
$263$	−14.6341	−0.902374	−0.451187	−	0.892429i	$-0.648999\pi$
$263$	−14.6341	−0.902374	−0.451187	+	0.892429i	$0.648999\pi$
$264$	0	0
$265$	0	0
$266$	10.4883	0.643080
$267$	43.2872	2.64913
$268$	−10.1955	−0.622788
$269$	21.3004	1.29871	0.649353	−	0.760487i	$-0.275040\pi$
$269$	21.3004	1.29871	0.649353	+	0.760487i	$0.275040\pi$
$270$	0	0
$271$	29.8770	1.81490	0.907449	−	0.420162i	$-0.138027\pi$
$271$	29.8770	1.81490	0.907449	+	0.420162i	$0.138027\pi$
$272$	0.514174	0.0311764
$273$	32.4715	1.96526
$274$	29.2144	1.76491
$275$	0	0
$276$	−5.52903	−0.332809
$277$	18.8381	1.13187	0.565936	−	0.824449i	$-0.308515\pi$
$277$	18.8381	1.13187	0.565936	+	0.824449i	$0.308515\pi$
$278$	0.197491	0.0118447
$279$	−60.5499	−3.62503
$280$	0	0
$281$	−25.6017	−1.52727	−0.763634	−	0.645650i	$-0.776587\pi$
$281$	−25.6017	−1.52727	−0.763634	+	0.645650i	$0.776587\pi$
$282$	34.4731	2.05284
$283$	28.5624	1.69786	0.848928	−	0.528508i	$-0.177248\pi$
$283$	28.5624	1.69786	0.848928	+	0.528508i	$0.177248\pi$
$284$	−2.12302	−0.125978
$285$	0	0
$286$	0	0
$287$	−12.1729	−0.718541
$288$	28.4421	1.67597
$289$	−16.9892	−0.999367
$290$	0	0
$291$	−18.4184	−1.07970
$292$	5.95676	0.348593
$293$	−0.106639	−0.00622990	−0.00311495	−	0.999995i	$-0.500992\pi$
$293$	−0.106639	−0.00622990	−0.00311495	+	0.999995i	$0.500992\pi$
$294$	−13.2493	−0.772714
$295$	0	0
$296$	0.671231	0.0390145
$297$	0	0
$298$	−9.25906	−0.536363
$299$	7.54999	0.436627
$300$	0	0
$301$	22.8495	1.31703
$302$	−18.5222	−1.06583
$303$	19.1238	1.09863
$304$	−10.0655	−0.577295
$305$	0	0
$306$	1.15681	0.0661307
$307$	4.74173	0.270625	0.135312	−	0.990803i	$-0.456796\pi$
$307$	4.74173	0.270625	0.135312	+	0.990803i	$0.456796\pi$
$308$	0	0
$309$	−19.9530	−1.13508
$310$	0	0
$311$	−17.0889	−0.969021	−0.484510	−	0.874786i	$-0.661002\pi$
$311$	−17.0889	−0.969021	−0.484510	+	0.874786i	$0.661002\pi$
$312$	−21.1606	−1.19799
$313$	0.769404	0.0434893	0.0217446	−	0.999764i	$-0.493078\pi$
$313$	0.769404	0.0434893	0.0217446	+	0.999764i	$0.493078\pi$
$314$	−3.62036	−0.204309
$315$	0	0
$316$	1.92234	0.108140
$317$	6.92681	0.389048	0.194524	−	0.980898i	$-0.437684\pi$
$317$	6.92681	0.389048	0.194524	+	0.980898i	$0.437684\pi$
$318$	1.61712	0.0906834
$319$	0	0
$320$	0	0
$321$	9.09449	0.507605
$322$	−11.5439	−0.643316
$323$	−0.210517	−0.0117135
$324$	12.3306	0.685036
$325$	0	0
$326$	−2.71651	−0.150454
$327$	−34.0690	−1.88402
$328$	7.93266	0.438008
$329$	20.4851	1.12938
$330$	0	0
$331$	4.85136	0.266655	0.133327	−	0.991072i	$-0.457434\pi$
$331$	4.85136	0.266655	0.133327	+	0.991072i	$0.457434\pi$
$332$	−0.113902	−0.00625118
$333$	2.22396	0.121872
$334$	38.0127	2.07996
$335$	0	0
$336$	47.6471	2.59936
$337$	−3.88444	−0.211599	−0.105799	−	0.994387i	$-0.533740\pi$
$337$	−3.88444	−0.211599	−0.105799	+	0.994387i	$0.533740\pi$
$338$	2.64519	0.143879
$339$	−41.9016	−2.27578
$340$	0	0
$341$	0	0
$342$	−22.6458	−1.22455
$343$	13.7567	0.742790
$344$	−14.8903	−0.802832
$345$	0	0
$346$	5.41917	0.291337
$347$	−31.6400	−1.69853	−0.849263	−	0.527970i	$-0.822953\pi$
$347$	−31.6400	−1.69853	−0.849263	+	0.527970i	$0.822953\pi$
$348$	−16.0608	−0.860950
$349$	−2.68940	−0.143960	−0.0719802	−	0.997406i	$-0.522932\pi$
$349$	−2.68940	−0.143960	−0.0719802	+	0.997406i	$0.522932\pi$
$350$	0	0
$351$	−38.5849	−2.05951
$352$	0	0
$353$	−19.5791	−1.04209	−0.521046	−	0.853528i	$-0.674458\pi$
$353$	−19.5791	−1.04209	−0.521046	+	0.853528i	$0.674458\pi$
$354$	−58.3590	−3.10174
$355$	0	0
$356$	11.0752	0.586986
$357$	0.996526	0.0527417
$358$	5.21424	0.275581
$359$	−32.4908	−1.71480	−0.857399	−	0.514651i	$-0.827921\pi$
$359$	−32.4908	−1.71480	−0.857399	+	0.514651i	$0.827921\pi$
$360$	0	0
$361$	−14.8789	−0.783101
$362$	18.6010	0.977647
$363$	0	0
$364$	8.30798	0.435457
$365$	0	0
$366$	17.6348	0.921785
$367$	−29.6117	−1.54572	−0.772858	−	0.634579i	$-0.781174\pi$
$367$	−29.6117	−1.54572	−0.772858	+	0.634579i	$0.781174\pi$
$368$	11.0785	0.577507
$369$	26.2830	1.36824
$370$	0	0
$371$	0.960949	0.0498900
$372$	−22.4581	−1.16440
$373$	−20.3541	−1.05390	−0.526949	−	0.849897i	$-0.676664\pi$
$373$	−20.3541	−1.05390	−0.526949	+	0.849897i	$0.676664\pi$
$374$	0	0
$375$	0	0
$376$	−13.3495	−0.688448
$377$	21.9313	1.12952
$378$	58.9960	3.03443
$379$	−37.7782	−1.94054	−0.970269	−	0.242030i	$-0.922187\pi$
$379$	−37.7782	−1.94054	−0.970269	+	0.242030i	$0.922187\pi$
$380$	0	0
$381$	−40.2191	−2.06049
$382$	12.9028	0.660164
$383$	−17.9623	−0.917833	−0.458916	−	0.888479i	$-0.651762\pi$
$383$	−17.9623	−0.917833	−0.458916	+	0.888479i	$0.651762\pi$
$384$	−41.0159	−2.09308
$385$	0	0
$386$	−22.6805	−1.15441
$387$	−49.3355	−2.50787
$388$	−4.71242	−0.239237
$389$	−1.42640	−0.0723216	−0.0361608	−	0.999346i	$-0.511513\pi$
$389$	−1.42640	−0.0723216	−0.0361608	+	0.999346i	$0.511513\pi$
$390$	0	0
$391$	0.231704	0.0117178
$392$	5.13072	0.259140
$393$	−11.6248	−0.586394
$394$	−8.69245	−0.437920
$395$	0	0
$396$	0	0
$397$	14.0083	0.703058	0.351529	−	0.936177i	$-0.385662\pi$
$397$	14.0083	0.703058	0.351529	+	0.936177i	$0.385662\pi$
$398$	−38.6964	−1.93967
$399$	−19.5080	−0.976623
$400$	0	0
$401$	−1.03884	−0.0518772	−0.0259386	−	0.999664i	$-0.508257\pi$
$401$	−1.03884	−0.0518772	−0.0259386	+	0.999664i	$0.508257\pi$
$402$	66.6284	3.32312
$403$	30.6669	1.52763
$404$	4.89292	0.243432
$405$	0	0
$406$	−33.5329	−1.66421
$407$	0	0
$408$	−0.649404	−0.0321503
$409$	23.9062	1.18209	0.591043	−	0.806640i	$-0.298716\pi$
$409$	23.9062	1.18209	0.591043	+	0.806640i	$0.298716\pi$
$410$	0	0
$411$	−54.3380	−2.68030
$412$	−5.10506	−0.251508
$413$	−34.6790	−1.70644
$414$	24.9250	1.22500
$415$	0	0
$416$	−14.4052	−0.706272
$417$	−0.367329	−0.0179882
$418$	0	0
$419$	−3.98654	−0.194755	−0.0973776	−	0.995248i	$-0.531045\pi$
$419$	−3.98654	−0.194755	−0.0973776	+	0.995248i	$0.531045\pi$
$420$	0	0
$421$	21.5491	1.05024	0.525120	−	0.851028i	$-0.324021\pi$
$421$	21.5491	1.05024	0.525120	+	0.851028i	$0.324021\pi$
$422$	33.0554	1.60911
$423$	−44.2305	−2.15056
$424$	−0.626220	−0.0304119
$425$	0	0
$426$	13.8741	0.672203
$427$	10.4792	0.507125
$428$	2.32687	0.112473
$429$	0	0
$430$	0	0
$431$	21.6674	1.04368	0.521841	−	0.853043i	$-0.325245\pi$
$431$	21.6674	1.04368	0.521841	+	0.853043i	$0.325245\pi$
$432$	−56.6176	−2.72401
$433$	40.2792	1.93569	0.967846	−	0.251542i	$-0.0809376\pi$
$433$	40.2792	1.93569	0.967846	+	0.251542i	$0.0809376\pi$
$434$	−46.8895	−2.25077
$435$	0	0
$436$	−8.71672	−0.417455
$437$	−4.53584	−0.216979
$438$	−38.9279	−1.86005
$439$	6.95367	0.331881	0.165940	−	0.986136i	$-0.446934\pi$
$439$	6.95367	0.331881	0.165940	+	0.986136i	$0.446934\pi$
$440$	0	0
$441$	16.9994	0.809496
$442$	−0.585896	−0.0278682
$443$	−4.91087	−0.233322	−0.116661	−	0.993172i	$-0.537219\pi$
$443$	−4.91087	−0.233322	−0.116661	+	0.993172i	$0.537219\pi$
$444$	0.824873	0.0391467
$445$	0	0
$446$	−33.0001	−1.56260
$447$	17.2216	0.814555
$448$	−8.61641	−0.407087
$449$	−9.55932	−0.451132	−0.225566	−	0.974228i	$-0.572423\pi$
$449$	−9.55932	−0.451132	−0.225566	+	0.974228i	$0.572423\pi$
$450$	0	0
$451$	0	0
$452$	−10.7207	−0.504260
$453$	34.4508	1.61864
$454$	42.4837	1.99386
$455$	0	0
$456$	12.7128	0.595330
$457$	6.23193	0.291517	0.145759	−	0.989320i	$-0.453438\pi$
$457$	6.23193	0.291517	0.145759	+	0.989320i	$0.453438\pi$
$458$	21.9773	1.02693
$459$	−1.18414	−0.0552709
$460$	0	0
$461$	25.0983	1.16894	0.584472	−	0.811414i	$-0.301302\pi$
$461$	25.0983	1.16894	0.584472	+	0.811414i	$0.301302\pi$
$462$	0	0
$463$	31.3429	1.45663	0.728314	−	0.685244i	$-0.240304\pi$
$463$	31.3429	1.45663	0.728314	+	0.685244i	$0.240304\pi$
$464$	32.1810	1.49396
$465$	0	0
$466$	−8.09155	−0.374834
$467$	9.25098	0.428084	0.214042	−	0.976824i	$-0.431337\pi$
$467$	9.25098	0.428084	0.214042	+	0.976824i	$0.431337\pi$
$468$	−17.9382	−0.829192
$469$	39.5930	1.82823
$470$	0	0
$471$	6.73378	0.310276
$472$	22.5992	1.04021
$473$	0	0
$474$	−12.5627	−0.577022
$475$	0	0
$476$	0.254966	0.0116863
$477$	−2.07483	−0.0950000
$478$	−24.2976	−1.11135
$479$	27.3421	1.24929	0.624645	−	0.780909i	$-0.285244\pi$
$479$	27.3421	1.24929	0.624645	+	0.780909i	$0.285244\pi$
$480$	0	0
$481$	−1.12638	−0.0513584
$482$	−38.5990	−1.75814
$483$	21.4714	0.976981
$484$	0	0
$485$	0	0
$486$	−23.3038	−1.05708
$487$	−28.3705	−1.28559	−0.642795	−	0.766038i	$-0.722225\pi$
$487$	−28.3705	−1.28559	−0.642795	+	0.766038i	$0.722225\pi$
$488$	−6.82898	−0.309133
$489$	5.05264	0.228488
$490$	0	0
$491$	−35.0521	−1.58188	−0.790940	−	0.611894i	$-0.790408\pi$
$491$	−35.0521	−1.58188	−0.790940	+	0.611894i	$0.790408\pi$
$492$	9.74842	0.439493
$493$	0.673056	0.0303129
$494$	11.4695	0.516038
$495$	0	0
$496$	44.9991	2.02052
$497$	8.24450	0.369817
$498$	0.744359	0.0333555
$499$	29.9509	1.34079	0.670394	−	0.742006i	$-0.266125\pi$
$499$	29.9509	1.34079	0.670394	+	0.742006i	$0.266125\pi$
$500$	0	0
$501$	−70.7026	−3.15876
$502$	−17.4345	−0.778142
$503$	−24.7780	−1.10480	−0.552399	−	0.833580i	$-0.686287\pi$
$503$	−24.7780	−1.10480	−0.552399	+	0.833580i	$0.686287\pi$
$504$	−41.5122	−1.84910
$505$	0	0
$506$	0	0
$507$	−4.91999	−0.218504
$508$	−10.2902	−0.456556
$509$	−5.96034	−0.264188	−0.132094	−	0.991237i	$-0.542170\pi$
$509$	−5.96034	−0.264188	−0.132094	+	0.991237i	$0.542170\pi$
$510$	0	0
$511$	−23.1324	−1.02332
$512$	−1.16917	−0.0516707
$513$	23.1808	1.02346
$514$	−3.68178	−0.162396
$515$	0	0
$516$	−18.2986	−0.805553
$517$	0	0
$518$	1.72223	0.0756703
$519$	−10.0795	−0.442442
$520$	0	0
$521$	18.6673	0.817830	0.408915	−	0.912573i	$-0.365907\pi$
$521$	18.6673	0.817830	0.408915	+	0.912573i	$0.365907\pi$
$522$	72.4024	3.16897
$523$	−15.4090	−0.673787	−0.336894	−	0.941543i	$-0.609376\pi$
$523$	−15.4090	−0.673787	−0.336894	+	0.941543i	$0.609376\pi$
$524$	−2.97426	−0.129931
$525$	0	0
$526$	24.4686	1.06688
$527$	0.941144	0.0409969
$528$	0	0
$529$	−18.0077	−0.782942
$530$	0	0
$531$	74.8771	3.24939
$532$	−4.99122	−0.216397
$533$	−13.3116	−0.576591
$534$	−72.3775	−3.13208
$535$	0	0
$536$	−25.8015	−1.11445
$537$	−9.69835	−0.418515
$538$	−35.6149	−1.53547
$539$	0	0
$540$	0	0
$541$	−44.6943	−1.92156	−0.960780	−	0.277313i	$-0.910556\pi$
$541$	−44.6943	−1.92156	−0.960780	+	0.277313i	$0.910556\pi$
$542$	−49.9553	−2.14576
$543$	−34.5974	−1.48472
$544$	−0.442084	−0.0189542
$545$	0	0
$546$	−54.2934	−2.32354
$547$	21.0165	0.898601	0.449301	−	0.893381i	$-0.351673\pi$
$547$	21.0165	0.898601	0.449301	+	0.893381i	$0.351673\pi$
$548$	−13.9026	−0.593891
$549$	−22.6262	−0.965663
$550$	0	0
$551$	−13.1758	−0.561307
$552$	−13.9922	−0.595547
$553$	−7.46519	−0.317452
$554$	−31.4979	−1.33822
$555$	0	0
$556$	−0.0939828	−0.00398576
$557$	30.2560	1.28199	0.640995	−	0.767545i	$-0.278522\pi$
$557$	30.2560	1.28199	0.640995	+	0.767545i	$0.278522\pi$
$558$	101.241	4.28589
$559$	24.9871	1.05684
$560$	0	0
$561$	0	0
$562$	42.8068	1.80570
$563$	−17.8437	−0.752021	−0.376010	−	0.926616i	$-0.622704\pi$
$563$	−17.8437	−0.752021	−0.376010	+	0.926616i	$0.622704\pi$
$564$	−16.4052	−0.690782
$565$	0	0
$566$	−47.7572	−2.00738
$567$	−47.8846	−2.01097
$568$	−5.37268	−0.225433
$569$	9.36704	0.392687	0.196343	−	0.980535i	$-0.437093\pi$
$569$	9.36704	0.392687	0.196343	+	0.980535i	$0.437093\pi$
$570$	0	0
$571$	−25.9021	−1.08397	−0.541985	−	0.840388i	$-0.682327\pi$
$571$	−25.9021	−1.08397	−0.541985	+	0.840388i	$0.682327\pi$
$572$	0	0
$573$	−23.9989	−1.00257
$574$	20.3534	0.849535
$575$	0	0
$576$	18.6041	0.775171
$577$	−8.56279	−0.356474	−0.178237	−	0.983988i	$-0.557039\pi$
$577$	−8.56279	−0.356474	−0.178237	+	0.983988i	$0.557039\pi$
$578$	28.4066	1.18156
$579$	42.1852	1.75316
$580$	0	0
$581$	0.442325	0.0183507
$582$	30.7961	1.27654
$583$	0	0
$584$	15.0746	0.623793
$585$	0	0
$586$	0.178303	0.00736564
$587$	37.6184	1.55268	0.776339	−	0.630315i	$-0.217074\pi$
$587$	37.6184	1.55268	0.776339	+	0.630315i	$0.217074\pi$
$588$	6.30511	0.260019
$589$	−18.4239	−0.759142
$590$	0	0
$591$	16.1678	0.665053
$592$	−1.65279	−0.0679294
$593$	−15.6291	−0.641810	−0.320905	−	0.947111i	$-0.603987\pi$
$593$	−15.6291	−0.641810	−0.320905	+	0.947111i	$0.603987\pi$
$594$	0	0
$595$	0	0
$596$	4.40623	0.180486
$597$	71.9743	2.94571
$598$	−12.6238	−0.516227
$599$	18.4138	0.752367	0.376183	−	0.926545i	$-0.377236\pi$
$599$	18.4138	0.752367	0.376183	+	0.926545i	$0.377236\pi$
$600$	0	0
$601$	21.7297	0.886374	0.443187	−	0.896429i	$-0.353848\pi$
$601$	21.7297	0.886374	0.443187	+	0.896429i	$0.353848\pi$
$602$	−38.2052	−1.55713
$603$	−85.4871	−3.48130
$604$	8.81439	0.358652
$605$	0	0
$606$	−31.9757	−1.29892
$607$	31.6346	1.28401	0.642004	−	0.766702i	$-0.278103\pi$
$607$	31.6346	1.28401	0.642004	+	0.766702i	$0.278103\pi$
$608$	8.65426	0.350977
$609$	62.3703	2.52737
$610$	0	0
$611$	22.4015	0.906269
$612$	−0.550509	−0.0222530
$613$	1.53807	0.0621220	0.0310610	−	0.999517i	$-0.490111\pi$
$613$	1.53807	0.0621220	0.0310610	+	0.999517i	$0.490111\pi$
$614$	−7.92832	−0.319961
$615$	0	0
$616$	0	0
$617$	13.3595	0.537835	0.268917	−	0.963163i	$-0.413334\pi$
$617$	13.3595	0.537835	0.268917	+	0.963163i	$0.413334\pi$
$618$	33.3620	1.34202
$619$	19.5273	0.784868	0.392434	−	0.919780i	$-0.371633\pi$
$619$	19.5273	0.784868	0.392434	+	0.919780i	$0.371633\pi$
$620$	0	0
$621$	−25.5137	−1.02383
$622$	28.5731	1.14568
$623$	−43.0094	−1.72313
$624$	52.1046	2.08585
$625$	0	0
$626$	−1.28647	−0.0514176
$627$	0	0
$628$	1.72287	0.0687499
$629$	−0.0345677	−0.00137830
$630$	0	0
$631$	13.2500	0.527473	0.263736	−	0.964595i	$-0.415045\pi$
$631$	13.2500	0.527473	0.263736	+	0.964595i	$0.415045\pi$
$632$	4.86482	0.193512
$633$	−61.4823	−2.44370
$634$	−11.5818	−0.459974
$635$	0	0
$636$	−0.769559	−0.0305150
$637$	−8.60975	−0.341131
$638$	0	0
$639$	−17.8011	−0.704201
$640$	0	0
$641$	−26.6454	−1.05243	−0.526216	−	0.850351i	$-0.676390\pi$
$641$	−26.6454	−1.05243	−0.526216	+	0.850351i	$0.676390\pi$
$642$	−15.2063	−0.600144
$643$	38.4632	1.51684	0.758420	−	0.651766i	$-0.225972\pi$
$643$	38.4632	1.51684	0.758420	+	0.651766i	$0.225972\pi$
$644$	5.49355	0.216476
$645$	0	0
$646$	0.351991	0.0138489
$647$	35.4150	1.39231	0.696153	−	0.717893i	$-0.254893\pi$
$647$	35.4150	1.39231	0.696153	+	0.717893i	$0.254893\pi$
$648$	31.2049	1.22584
$649$	0	0
$650$	0	0
$651$	87.2133	3.41816
$652$	1.29274	0.0506277
$653$	43.3179	1.69516	0.847581	−	0.530667i	$-0.178058\pi$
$653$	43.3179	1.69516	0.847581	+	0.530667i	$0.178058\pi$
$654$	56.9645	2.22749
$655$	0	0
$656$	−19.5329	−0.762630
$657$	49.9463	1.94859
$658$	−34.2518	−1.33527
$659$	−13.2470	−0.516028	−0.258014	−	0.966141i	$-0.583068\pi$
$659$	−13.2470	−0.516028	−0.258014	+	0.966141i	$0.583068\pi$
$660$	0	0
$661$	9.94944	0.386988	0.193494	−	0.981101i	$-0.438018\pi$
$661$	9.94944	0.386988	0.193494	+	0.981101i	$0.438018\pi$
$662$	−8.11163	−0.315267
$663$	1.08975	0.0423224
$664$	−0.288249	−0.0111862
$665$	0	0
$666$	−3.71854	−0.144091
$667$	14.5018	0.561512
$668$	−18.0896	−0.699907
$669$	61.3794	2.37307
$670$	0	0
$671$	0	0
$672$	−40.9668	−1.58033
$673$	−35.8658	−1.38253	−0.691263	−	0.722603i	$-0.742945\pi$
$673$	−35.8658	−1.38253	−0.691263	+	0.722603i	$0.742945\pi$
$674$	6.49491	0.250175
$675$	0	0
$676$	−1.25880	−0.0484154
$677$	4.10288	0.157686	0.0788432	−	0.996887i	$-0.474877\pi$
$677$	4.10288	0.157686	0.0788432	+	0.996887i	$0.474877\pi$
$678$	70.0608	2.69067
$679$	18.3002	0.702295
$680$	0	0
$681$	−79.0186	−3.02800
$682$	0	0
$683$	−19.1256	−0.731821	−0.365910	−	0.930650i	$-0.619242\pi$
$683$	−19.1256	−0.731821	−0.365910	+	0.930650i	$0.619242\pi$
$684$	10.7768	0.412061
$685$	0	0
$686$	−23.0016	−0.878204
$687$	−40.8773	−1.55957
$688$	36.6649	1.39784
$689$	1.05085	0.0400341
$690$	0	0
$691$	−10.2002	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$691$	−10.2002	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$692$	−2.57889	−0.0980349
$693$	0	0
$694$	52.9032	2.00818
$695$	0	0
$696$	−40.6447	−1.54063
$697$	−0.408524	−0.0154740
$698$	4.49677	0.170205
$699$	15.0501	0.569247
$700$	0	0
$701$	−35.1829	−1.32884	−0.664420	−	0.747360i	$-0.731321\pi$
$701$	−35.1829	−1.32884	−0.664420	+	0.747360i	$0.731321\pi$
$702$	64.5151	2.43497
$703$	0.676699	0.0255222
$704$	0	0
$705$	0	0
$706$	32.7370	1.23207
$707$	−19.0011	−0.714609
$708$	27.7721	1.04374
$709$	19.8131	0.744095	0.372047	−	0.928214i	$-0.378656\pi$
$709$	19.8131	0.744095	0.372047	+	0.928214i	$0.378656\pi$
$710$	0	0
$711$	16.1184	0.604489
$712$	28.0278	1.05039
$713$	20.2781	0.759420
$714$	−1.66622	−0.0623569
$715$	0	0
$716$	−2.48137	−0.0927331
$717$	45.1929	1.68776
$718$	54.3257	2.02742
$719$	5.44231	0.202964	0.101482	−	0.994837i	$-0.467642\pi$
$719$	5.44231	0.202964	0.101482	+	0.994837i	$0.467642\pi$
$720$	0	0
$721$	19.8249	0.738318
$722$	24.8780	0.925864
$723$	71.7932	2.67002
$724$	−8.85191	−0.328979
$725$	0	0
$726$	0	0
$727$	12.7445	0.472669	0.236334	−	0.971672i	$-0.424054\pi$
$727$	12.7445	0.472669	0.236334	+	0.971672i	$0.424054\pi$
$728$	21.0248	0.779232
$729$	−3.14573	−0.116509
$730$	0	0
$731$	0.766836	0.0283625
$732$	−8.39210	−0.310181
$733$	15.2083	0.561730	0.280865	−	0.959747i	$-0.409379\pi$
$733$	15.2083	0.561730	0.280865	+	0.959747i	$0.409379\pi$
$734$	49.5117	1.82751
$735$	0	0
$736$	−9.52524	−0.351105
$737$	0	0
$738$	−43.9460	−1.61768
$739$	−39.9731	−1.47044	−0.735218	−	0.677831i	$-0.762920\pi$
$739$	−39.9731	−1.47044	−0.735218	+	0.677831i	$0.762920\pi$
$740$	0	0
$741$	−21.3330	−0.783688
$742$	−1.60674	−0.0589852
$743$	8.40886	0.308491	0.154246	−	0.988033i	$-0.450705\pi$
$743$	8.40886	0.308491	0.154246	+	0.988033i	$0.450705\pi$
$744$	−56.8341	−2.08364
$745$	0	0
$746$	34.0328	1.24603
$747$	−0.955045	−0.0349433
$748$	0	0
$749$	−9.03613	−0.330173
$750$	0	0
$751$	43.5786	1.59021	0.795104	−	0.606474i	$-0.207416\pi$
$751$	43.5786	1.59021	0.795104	+	0.606474i	$0.207416\pi$
$752$	32.8710	1.19868
$753$	32.4278	1.18174
$754$	−36.6699	−1.33544
$755$	0	0
$756$	−28.0752	−1.02109
$757$	−0.664899	−0.0241662	−0.0120831	−	0.999927i	$-0.503846\pi$
$757$	−0.664899	−0.0241662	−0.0120831	+	0.999927i	$0.503846\pi$
$758$	63.1664	2.29431
$759$	0	0
$760$	0	0
$761$	−20.4351	−0.740771	−0.370386	−	0.928878i	$-0.620774\pi$
$761$	−20.4351	−0.740771	−0.370386	+	0.928878i	$0.620774\pi$
$762$	67.2477	2.43613
$763$	33.8504	1.22547
$764$	−6.14022	−0.222145
$765$	0	0
$766$	30.0336	1.08516
$767$	−37.9232	−1.36933
$768$	51.2357	1.84881
$769$	18.5393	0.668543	0.334271	−	0.942477i	$-0.391510\pi$
$769$	18.5393	0.668543	0.334271	+	0.942477i	$0.391510\pi$
$770$	0	0
$771$	6.84803	0.246626
$772$	10.7933	0.388458
$773$	−3.48512	−0.125351	−0.0626755	−	0.998034i	$-0.519963\pi$
$773$	−3.48512	−0.125351	−0.0626755	+	0.998034i	$0.519963\pi$
$774$	82.4906	2.96506
$775$	0	0
$776$	−11.9256	−0.428105
$777$	−3.20330	−0.114918
$778$	2.38499	0.0855062
$779$	7.99729	0.286532
$780$	0	0
$781$	0	0
$782$	−0.387416	−0.0138540
$783$	−74.1127	−2.64857
$784$	−12.6335	−0.451198
$785$	0	0
$786$	19.4371	0.693297
$787$	−9.67351	−0.344823	−0.172412	−	0.985025i	$-0.555156\pi$
$787$	−9.67351	−0.344823	−0.172412	+	0.985025i	$0.555156\pi$
$788$	4.13659	0.147360
$789$	−45.5110	−1.62024
$790$	0	0
$791$	41.6327	1.48029
$792$	0	0
$793$	11.4596	0.406941
$794$	−23.4224	−0.831229
$795$	0	0
$796$	18.4150	0.652701
$797$	47.9880	1.69982	0.849912	−	0.526925i	$-0.176655\pi$
$797$	47.9880	1.69982	0.849912	+	0.526925i	$0.176655\pi$
$798$	32.6181	1.15467
$799$	0.687486	0.0243215
$800$	0	0
$801$	92.8636	3.28117
$802$	1.73697	0.0613347
$803$	0	0
$804$	−31.7073	−1.11823
$805$	0	0
$806$	−51.2760	−1.80612
$807$	66.2428	2.33186
$808$	12.3824	0.435611
$809$	−23.2707	−0.818153	−0.409076	−	0.912500i	$-0.634149\pi$
$809$	−23.2707	−0.818153	−0.409076	+	0.912500i	$0.634149\pi$
$810$	0	0
$811$	16.2259	0.569769	0.284884	−	0.958562i	$-0.408045\pi$
$811$	16.2259	0.569769	0.284884	+	0.958562i	$0.408045\pi$
$812$	15.9577	0.560006
$813$	92.9156	3.25869
$814$	0	0
$815$	0	0
$816$	1.59905	0.0559779
$817$	−15.0116	−0.525190
$818$	−39.9720	−1.39759
$819$	69.6608	2.43415
$820$	0	0
$821$	30.9788	1.08117	0.540584	−	0.841290i	$-0.318203\pi$
$821$	30.9788	1.08117	0.540584	+	0.841290i	$0.318203\pi$
$822$	90.8550	3.16893
$823$	17.0785	0.595321	0.297660	−	0.954672i	$-0.403794\pi$
$823$	17.0785	0.595321	0.297660	+	0.954672i	$0.403794\pi$
$824$	−12.9193	−0.450064
$825$	0	0
$826$	57.9844	2.01754
$827$	27.7250	0.964092	0.482046	−	0.876146i	$-0.339894\pi$
$827$	27.7250	0.964092	0.482046	+	0.876146i	$0.339894\pi$
$828$	−11.8614	−0.412211
$829$	−1.40780	−0.0488950	−0.0244475	−	0.999701i	$-0.507783\pi$
$829$	−1.40780	−0.0488950	−0.0244475	+	0.999701i	$0.507783\pi$
$830$	0	0
$831$	58.5854	2.03230
$832$	−9.42248	−0.326666
$833$	−0.264227	−0.00915491
$834$	0.614186	0.0212675
$835$	0	0
$836$	0	0
$837$	−103.633	−3.58207
$838$	6.66562	0.230260
$839$	24.0926	0.831767	0.415884	−	0.909418i	$-0.363472\pi$
$839$	24.0926	0.831767	0.415884	+	0.909418i	$0.363472\pi$
$840$	0	0
$841$	13.1250	0.452588
$842$	−36.0308	−1.24170
$843$	−79.6196	−2.74225
$844$	−15.7305	−0.541467
$845$	0	0
$846$	73.9548	2.54262
$847$	0	0
$848$	1.54196	0.0529512
$849$	88.8272	3.04854
$850$	0	0
$851$	−0.744803	−0.0255315
$852$	−6.60246	−0.226197
$853$	24.0161	0.822295	0.411148	−	0.911569i	$-0.365128\pi$
$853$	24.0161	0.822295	0.411148	+	0.911569i	$0.365128\pi$
$854$	−17.5216	−0.599577
$855$	0	0
$856$	5.88855	0.201267
$857$	−44.1725	−1.50890	−0.754452	−	0.656355i	$-0.772097\pi$
$857$	−44.1725	−1.50890	−0.754452	+	0.656355i	$0.772097\pi$
$858$	0	0
$859$	−24.9728	−0.852061	−0.426030	−	0.904709i	$-0.640088\pi$
$859$	−24.9728	−0.852061	−0.426030	+	0.904709i	$0.640088\pi$
$860$	0	0
$861$	−37.8569	−1.29016
$862$	−36.2286	−1.23395
$863$	6.81539	0.231999	0.115999	−	0.993249i	$-0.462993\pi$
$863$	6.81539	0.231999	0.115999	+	0.993249i	$0.462993\pi$
$864$	48.6795	1.65611
$865$	0	0
$866$	−67.3481	−2.28858
$867$	−52.8355	−1.79439
$868$	22.3139	0.757384
$869$	0	0
$870$	0	0
$871$	43.2969	1.46706
$872$	−22.0592	−0.747019
$873$	−39.5127	−1.33730
$874$	7.58407	0.256535
$875$	0	0
$876$	18.5252	0.625907
$877$	−48.5427	−1.63917	−0.819585	−	0.572958i	$-0.805796\pi$
$877$	−48.5427	−1.63917	−0.819585	+	0.572958i	$0.805796\pi$
$878$	−11.6268	−0.392384
$879$	−0.331640	−0.0111859
$880$	0	0
$881$	4.83974	0.163055	0.0815276	−	0.996671i	$-0.474020\pi$
$881$	4.83974	0.163055	0.0815276	+	0.996671i	$0.474020\pi$
$882$	−28.4236	−0.957071
$883$	−2.50364	−0.0842543	−0.0421271	−	0.999112i	$-0.513413\pi$
$883$	−2.50364	−0.0842543	−0.0421271	+	0.999112i	$0.513413\pi$
$884$	0.278818	0.00937766
$885$	0	0
$886$	8.21113	0.275858
$887$	−49.8268	−1.67302	−0.836509	−	0.547953i	$-0.815407\pi$
$887$	−49.8268	−1.67302	−0.836509	+	0.547953i	$0.815407\pi$
$888$	2.08749	0.0700515
$889$	39.9610	1.34025
$890$	0	0
$891$	0	0
$892$	15.7042	0.525816
$893$	−13.4583	−0.450364
$894$	−28.7951	−0.963053
$895$	0	0
$896$	40.7526	1.36145
$897$	23.4800	0.783975
$898$	15.9835	0.533376
$899$	58.9041	1.96456
$900$	0	0
$901$	0.0322497	0.00107439
$902$	0	0
$903$	71.0607	2.36475
$904$	−27.1307	−0.902353
$905$	0	0
$906$	−57.6028	−1.91372
$907$	−5.48535	−0.182138	−0.0910690	−	0.995845i	$-0.529028\pi$
$907$	−5.48535	−0.182138	−0.0910690	+	0.995845i	$0.529028\pi$
$908$	−20.2173	−0.670934
$909$	41.0262	1.36075
$910$	0	0
$911$	−59.1364	−1.95928	−0.979638	−	0.200770i	$-0.935656\pi$
$911$	−59.1364	−1.95928	−0.979638	+	0.200770i	$0.935656\pi$
$912$	−31.3031	−1.03655
$913$	0	0
$914$	−10.4200	−0.344662
$915$	0	0
$916$	−10.4586	−0.345563
$917$	11.5502	0.381421
$918$	1.97992	0.0653471
$919$	−4.61591	−0.152265	−0.0761324	−	0.997098i	$-0.524257\pi$
$919$	−4.61591	−0.152265	−0.0761324	+	0.997098i	$0.524257\pi$
$920$	0	0
$921$	14.7465	0.485913
$922$	−41.9652	−1.38205
$923$	9.01578	0.296758
$924$	0	0
$925$	0	0
$926$	−52.4063	−1.72218
$927$	−42.8049	−1.40590
$928$	−27.6691	−0.908281
$929$	−30.0805	−0.986908	−0.493454	−	0.869772i	$-0.664266\pi$
$929$	−30.0805	−0.986908	−0.493454	+	0.869772i	$0.664266\pi$
$930$	0	0
$931$	5.17251	0.169522
$932$	3.85063	0.126132
$933$	−53.1453	−1.73990
$934$	−15.4679	−0.506126
$935$	0	0
$936$	−45.3957	−1.48381
$937$	11.1260	0.363469	0.181735	−	0.983348i	$-0.441829\pi$
$937$	11.1260	0.363469	0.181735	+	0.983348i	$0.441829\pi$
$938$	−66.2007	−2.16153
$939$	2.39280	0.0780861
$940$	0	0
$941$	6.52918	0.212845	0.106423	−	0.994321i	$-0.466060\pi$
$941$	6.52918	0.212845	0.106423	+	0.994321i	$0.466060\pi$
$942$	−11.2591	−0.366841
$943$	−8.80215	−0.286637
$944$	−55.6468	−1.81115
$945$	0	0
$946$	0	0
$947$	32.0205	1.04053	0.520263	−	0.854006i	$-0.325834\pi$
$947$	32.0205	1.04053	0.520263	+	0.854006i	$0.325834\pi$
$948$	5.97836	0.194168
$949$	−25.2964	−0.821157
$950$	0	0
$951$	21.5420	0.698546
$952$	0.645236	0.0209122
$953$	47.5200	1.53932	0.769662	−	0.638451i	$-0.220425\pi$
$953$	47.5200	1.53932	0.769662	+	0.638451i	$0.220425\pi$
$954$	3.46919	0.112319
$955$	0	0
$956$	11.5628	0.373968
$957$	0	0
$958$	−45.7168	−1.47704
$959$	53.9893	1.74341
$960$	0	0
$961$	51.3664	1.65698
$962$	1.88334	0.0607214
$963$	19.5103	0.628712
$964$	18.3686	0.591613
$965$	0	0
$966$	−35.9008	−1.15509
$967$	−29.8462	−0.959790	−0.479895	−	0.877326i	$-0.659325\pi$
$967$	−29.8462	−0.959790	−0.479895	+	0.877326i	$0.659325\pi$
$968$	0	0
$969$	−0.654694	−0.0210318
$970$	0	0
$971$	−15.0784	−0.483890	−0.241945	−	0.970290i	$-0.577785\pi$
$971$	−15.0784	−0.483890	−0.241945	+	0.970290i	$0.577785\pi$
$972$	11.0899	0.355708
$973$	0.364971	0.0117004
$974$	47.4364	1.51996
$975$	0	0
$976$	16.8152	0.538242
$977$	0.719068	0.0230050	0.0115025	−	0.999934i	$-0.496339\pi$
$977$	0.719068	0.0230050	0.0115025	+	0.999934i	$0.496339\pi$
$978$	−8.44818	−0.270143
$979$	0	0
$980$	0	0
$981$	−73.0880	−2.33352
$982$	58.6083	1.87026
$983$	−5.71947	−0.182423	−0.0912114	−	0.995832i	$-0.529074\pi$
$983$	−5.71947	−0.182423	−0.0912114	+	0.995832i	$0.529074\pi$
$984$	24.6701	0.786454
$985$	0	0
$986$	−1.12537	−0.0358391
$987$	63.7076	2.02783
$988$	−5.45815	−0.173647
$989$	16.5224	0.525382
$990$	0	0
$991$	−8.53244	−0.271042	−0.135521	−	0.990774i	$-0.543271\pi$
$991$	−8.53244	−0.271042	−0.135521	+	0.990774i	$0.543271\pi$
$992$	−38.6900	−1.22841
$993$	15.0874	0.478785
$994$	−13.7851	−0.437236
$995$	0	0
$996$	−0.354228	−0.0112241
$997$	−36.6946	−1.16213	−0.581064	−	0.813858i	$-0.697363\pi$
$997$	−36.6946	−1.16213	−0.581064	+	0.813858i	$0.697363\pi$
$998$	−50.0789	−1.58522
$999$	3.80638	0.120428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bj.1.3		8
5.4	even	2		3025.2.a.bm.1.6		8
11.7	odd	10		275.2.h.e.126.3	yes	16
11.8	odd	10		275.2.h.e.251.3	yes	16
11.10	odd	2		3025.2.a.bn.1.6		8
55.7	even	20		275.2.z.c.49.2		32
55.8	even	20		275.2.z.c.174.2		32
55.18	even	20		275.2.z.c.49.7		32
55.19	odd	10		275.2.h.c.251.2	yes	16
55.29	odd	10		275.2.h.c.126.2	✓	16
55.52	even	20		275.2.z.c.174.7		32
55.54	odd	2		3025.2.a.bi.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.126.2	✓	16	55.29	odd	10
275.2.h.c.251.2	yes	16	55.19	odd	10
275.2.h.e.126.3	yes	16	11.7	odd	10
275.2.h.e.251.3	yes	16	11.8	odd	10
275.2.z.c.49.2		32	55.7	even	20
275.2.z.c.49.7		32	55.18	even	20
275.2.z.c.174.2		32	55.8	even	20
275.2.z.c.174.7		32	55.52	even	20
3025.2.a.bi.1.3		8	55.54	odd	2
3025.2.a.bj.1.3		8	1.1	even	1	trivial
3025.2.a.bm.1.6		8	5.4	even	2
3025.2.a.bn.1.6		8	11.10	odd	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67203 q^{2} +3.10994 q^{3} +0.795692 q^{4} -5.19992 q^{6} -3.08998 q^{7} +2.01364 q^{8} +6.67173 q^{9} +O(q^{10})\)	\(q-1.67203 q^{2} +3.10994 q^{3} +0.795692 q^{4} -5.19992 q^{6} -3.08998 q^{7} +2.01364 q^{8} +6.67173 q^{9} +2.47456 q^{12} -3.37905 q^{13} +5.16655 q^{14} -4.95826 q^{16} -0.103701 q^{17} -11.1553 q^{18} +2.03005 q^{19} -9.60965 q^{21} -2.23435 q^{23} +6.26230 q^{24} +5.64988 q^{26} +11.4188 q^{27} -2.45867 q^{28} -6.49038 q^{29} -9.07559 q^{31} +4.26309 q^{32} +0.173391 q^{34} +5.30864 q^{36} +0.333342 q^{37} -3.39430 q^{38} -10.5086 q^{39} +3.93946 q^{41} +16.0677 q^{42} -7.39472 q^{43} +3.73591 q^{46} -6.62954 q^{47} -15.4199 q^{48} +2.54798 q^{49} -0.322502 q^{51} -2.68868 q^{52} -0.310989 q^{53} -19.0927 q^{54} -6.22211 q^{56} +6.31332 q^{57} +10.8521 q^{58} +11.2231 q^{59} -3.39136 q^{61} +15.1747 q^{62} -20.6155 q^{63} +2.78850 q^{64} -12.8133 q^{67} -0.0825137 q^{68} -6.94870 q^{69} -2.66814 q^{71} +13.4345 q^{72} +7.48626 q^{73} -0.557358 q^{74} +1.61529 q^{76} +17.5708 q^{78} +2.41593 q^{79} +15.4967 q^{81} -6.58691 q^{82} -0.143148 q^{83} -7.64633 q^{84} +12.3642 q^{86} -20.1847 q^{87} +13.9190 q^{89} +10.4412 q^{91} -1.77786 q^{92} -28.2246 q^{93} +11.0848 q^{94} +13.2579 q^{96} -5.92242 q^{97} -4.26030 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * q^98

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