LMFDB - Embedded newform 3575.2.a.t.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3575,2,Mod(1,3575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3575.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3575 = 5^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3575.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-1,-2,15,0,-1,4,-3,23,0,-9,-6,9,16,0,15,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.5465187226$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 $ x^9 - x^8 - 16x^7 + 14x^6 + 86x^5 - 57x^4 - 179x^3 + 64x^2 + 118*x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 715)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.06779$ of defining polynomial
Character	$\chi$	$=$	3575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.06779 q^{2} -0.343328 q^{3} -0.859816 q^{4} -0.366604 q^{6} -1.31980 q^{7} -3.05369 q^{8} -2.88213 q^{9} -1.00000 q^{11} +0.295199 q^{12} +1.00000 q^{13} -1.40927 q^{14} -1.54108 q^{16} -7.11691 q^{17} -3.07752 q^{18} +4.66833 q^{19} +0.453123 q^{21} -1.06779 q^{22} +3.09689 q^{23} +1.04842 q^{24} +1.06779 q^{26} +2.01950 q^{27} +1.13478 q^{28} -4.83303 q^{29} +5.60339 q^{31} +4.46183 q^{32} +0.343328 q^{33} -7.59940 q^{34} +2.47810 q^{36} -3.95696 q^{37} +4.98481 q^{38} -0.343328 q^{39} +6.94860 q^{41} +0.483842 q^{42} -0.0908316 q^{43} +0.859816 q^{44} +3.30684 q^{46} -5.18536 q^{47} +0.529097 q^{48} -5.25814 q^{49} +2.44344 q^{51} -0.859816 q^{52} -12.6253 q^{53} +2.15641 q^{54} +4.03025 q^{56} -1.60277 q^{57} -5.16068 q^{58} +10.3805 q^{59} +14.4830 q^{61} +5.98326 q^{62} +3.80382 q^{63} +7.84648 q^{64} +0.366604 q^{66} -0.452157 q^{67} +6.11924 q^{68} -1.06325 q^{69} +10.7371 q^{71} +8.80113 q^{72} +10.7422 q^{73} -4.22522 q^{74} -4.01390 q^{76} +1.31980 q^{77} -0.366604 q^{78} +5.81132 q^{79} +7.95303 q^{81} +7.41967 q^{82} +14.5762 q^{83} -0.389602 q^{84} -0.0969895 q^{86} +1.65931 q^{87} +3.05369 q^{88} +1.82629 q^{89} -1.31980 q^{91} -2.66276 q^{92} -1.92380 q^{93} -5.53690 q^{94} -1.53187 q^{96} +5.98946 q^{97} -5.61461 q^{98} +2.88213 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} - 2 q^{3} + 15 q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - 9 q^{11} - 6 q^{12} + 9 q^{13} + 16 q^{14} + 15 q^{16} - 13 q^{17} - 3 q^{18} - 3 q^{19} + 14 q^{21} + q^{22} + 2 q^{24} - q^{26}+ \cdots - 23 q^{99}+O(q^{100}) $ 9 * q - q^2 - 2 * q^3 + 15 * q^4 - q^6 + 4 * q^7 - 3 * q^8 + 23 * q^9 - 9 * q^11 - 6 * q^12 + 9 * q^13 + 16 * q^14 + 15 * q^16 - 13 * q^17 - 3 * q^18 - 3 * q^19 + 14 * q^21 + q^22 + 2 * q^24 - q^26 - 8 * q^27 + 27 * q^28 + 16 * q^29 + 12 * q^31 + 18 * q^32 + 2 * q^33 + 14 * q^34 - 4 * q^36 - q^37 + 9 * q^38 - 2 * q^39 + 5 * q^41 + 11 * q^42 + 3 * q^43 - 15 * q^44 + 4 * q^46 - 27 * q^47 - q^48 + 31 * q^49 + 15 * q^52 - 34 * q^53 - 49 * q^54 + 73 * q^56 + 16 * q^57 + 38 * q^58 - 4 * q^59 + 30 * q^61 + 8 * q^62 + 30 * q^63 + 3 * q^64 + q^66 + 11 * q^67 - 52 * q^68 + 52 * q^69 - 4 * q^71 + 8 * q^72 + 12 * q^73 - 16 * q^74 - 16 * q^76 - 4 * q^77 - q^78 - 16 * q^79 + 61 * q^81 + 42 * q^82 - 34 * q^83 - 79 * q^84 + 52 * q^86 + 64 * q^87 + 3 * q^88 - 10 * q^89 + 4 * q^91 - 7 * q^92 + 22 * q^93 - 18 * q^94 + 15 * q^97 - 24 * q^98 - 23 * q^99

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 $

$2$	1.06779	0.755044	0.377522	−	0.926001i	$-0.376776\pi$
$2$	1.06779	0.755044	0.377522	+	0.926001i	$0.376776\pi$
$3$	−0.343328	−0.198221	−0.0991103	−	0.995076i	$-0.531600\pi$
$3$	−0.343328	−0.198221	−0.0991103	+	0.995076i	$0.531600\pi$
$4$	−0.859816	−0.429908
$5$	0	0
$5$	0	0
$6$	−0.366604	−0.149665
$7$	−1.31980	−0.498836	−0.249418	−	0.968396i	$-0.580239\pi$
$7$	−1.31980	−0.498836	−0.249418	+	0.968396i	$0.580239\pi$
$8$	−3.05369	−1.07964
$9$	−2.88213	−0.960709
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.295199	0.0852166
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.40927	−0.376643
$15$	0	0
$16$	−1.54108	−0.385271
$17$	−7.11691	−1.72611	−0.863053	−	0.505114i	$-0.831450\pi$
$17$	−7.11691	−1.72611	−0.863053	+	0.505114i	$0.831450\pi$
$18$	−3.07752	−0.725378
$19$	4.66833	1.07099	0.535494	−	0.844539i	$-0.320126\pi$
$19$	4.66833	1.07099	0.535494	+	0.844539i	$0.320126\pi$
$20$	0	0
$21$	0.453123	0.0988795
$22$	−1.06779	−0.227654
$23$	3.09689	0.645747	0.322873	−	0.946442i	$-0.395351\pi$
$23$	3.09689	0.645747	0.322873	+	0.946442i	$0.395351\pi$
$24$	1.04842	0.214008
$25$	0	0
$26$	1.06779	0.209412
$27$	2.01950	0.388653
$28$	1.13478	0.214453
$29$	−4.83303	−0.897471	−0.448736	−	0.893665i	$-0.648126\pi$
$29$	−4.83303	−0.897471	−0.448736	+	0.893665i	$0.648126\pi$
$30$	0	0
$31$	5.60339	1.00640	0.503199	−	0.864171i	$-0.332156\pi$
$31$	5.60339	1.00640	0.503199	+	0.864171i	$0.332156\pi$
$32$	4.46183	0.788747
$33$	0.343328	0.0597657
$34$	−7.59940	−1.30329
$35$	0	0
$36$	2.47810	0.413016
$37$	−3.95696	−0.650521	−0.325261	−	0.945624i	$-0.605452\pi$
$37$	−3.95696	−0.650521	−0.325261	+	0.945624i	$0.605452\pi$
$38$	4.98481	0.808643
$39$	−0.343328	−0.0549765
$40$	0	0
$41$	6.94860	1.08519	0.542594	−	0.839995i	$-0.317442\pi$
$41$	6.94860	1.08519	0.542594	+	0.839995i	$0.317442\pi$
$42$	0.483842	0.0746584
$43$	−0.0908316	−0.0138517	−0.00692585	−	0.999976i	$-0.502205\pi$
$43$	−0.0908316	−0.0138517	−0.00692585	+	0.999976i	$0.502205\pi$
$44$	0.859816	0.129622
$45$	0	0
$46$	3.30684	0.487567
$47$	−5.18536	−0.756363	−0.378182	−	0.925731i	$-0.623450\pi$
$47$	−5.18536	−0.756363	−0.378182	+	0.925731i	$0.623450\pi$
$48$	0.529097	0.0763686
$49$	−5.25814	−0.751163
$50$	0	0
$51$	2.44344	0.342150
$52$	−0.859816	−0.119235
$53$	−12.6253	−1.73422	−0.867109	−	0.498119i	$-0.834024\pi$
$53$	−12.6253	−1.73422	−0.867109	+	0.498119i	$0.834024\pi$
$54$	2.15641	0.293450
$55$	0	0
$56$	4.03025	0.538565
$57$	−1.60277	−0.212292
$58$	−5.16068	−0.677631
$59$	10.3805	1.35143	0.675716	−	0.737162i	$-0.263835\pi$
$59$	10.3805	1.35143	0.675716	+	0.737162i	$0.263835\pi$
$60$	0	0
$61$	14.4830	1.85435	0.927177	−	0.374624i	$-0.122228\pi$
$61$	14.4830	1.85435	0.927177	+	0.374624i	$0.122228\pi$
$62$	5.98326	0.759875
$63$	3.80382	0.479236
$64$	7.84648	0.980810
$65$	0	0
$66$	0.366604	0.0451258
$67$	−0.452157	−0.0552398	−0.0276199	−	0.999618i	$-0.508793\pi$
$67$	−0.452157	−0.0552398	−0.0276199	+	0.999618i	$0.508793\pi$
$68$	6.11924	0.742066
$69$	−1.06325	−0.128000
$70$	0	0
$71$	10.7371	1.27425	0.637127	−	0.770759i	$-0.280123\pi$
$71$	10.7371	1.27425	0.637127	+	0.770759i	$0.280123\pi$
$72$	8.80113	1.03722
$73$	10.7422	1.25728	0.628640	−	0.777696i	$-0.283612\pi$
$73$	10.7422	1.25728	0.628640	+	0.777696i	$0.283612\pi$
$74$	−4.22522	−0.491172
$75$	0	0
$76$	−4.01390	−0.460426
$77$	1.31980	0.150405
$78$	−0.366604	−0.0415097
$79$	5.81132	0.653824	0.326912	−	0.945055i	$-0.393992\pi$
$79$	5.81132	0.653824	0.326912	+	0.945055i	$0.393992\pi$
$80$	0	0
$81$	7.95303	0.883670
$82$	7.41967	0.819365
$83$	14.5762	1.59995	0.799973	−	0.600036i	$-0.204847\pi$
$83$	14.5762	1.59995	0.799973	+	0.600036i	$0.204847\pi$
$84$	−0.389602	−0.0425091
$85$	0	0
$86$	−0.0969895	−0.0104586
$87$	1.65931	0.177897
$88$	3.05369	0.325525
$89$	1.82629	0.193586	0.0967931	−	0.995305i	$-0.469141\pi$
$89$	1.82629	0.193586	0.0967931	+	0.995305i	$0.469141\pi$
$90$	0	0
$91$	−1.31980	−0.138352
$92$	−2.66276	−0.277612
$93$	−1.92380	−0.199489
$94$	−5.53690	−0.571088
$95$	0	0
$96$	−1.53187	−0.156346
$97$	5.98946	0.608138	0.304069	−	0.952650i	$-0.401655\pi$
$97$	5.98946	0.608138	0.304069	+	0.952650i	$0.401655\pi$
$98$	−5.61461	−0.567161
$99$	2.88213	0.289665
$100$	0	0
$101$	−18.1302	−1.80402	−0.902009	−	0.431717i	$-0.857908\pi$
$101$	−18.1302	−1.80402	−0.902009	+	0.431717i	$0.857908\pi$
$102$	2.60909	0.258338
$103$	18.7381	1.84632	0.923159	−	0.384419i	$-0.125598\pi$
$103$	18.7381	1.84632	0.923159	+	0.384419i	$0.125598\pi$
$104$	−3.05369	−0.299439
$105$	0	0
$106$	−13.4812	−1.30941
$107$	−14.3491	−1.38718	−0.693590	−	0.720370i	$-0.743972\pi$
$107$	−14.3491	−1.38718	−0.693590	+	0.720370i	$0.743972\pi$
$108$	−1.73640	−0.167085
$109$	3.65773	0.350347	0.175174	−	0.984538i	$-0.443951\pi$
$109$	3.65773	0.350347	0.175174	+	0.984538i	$0.443951\pi$
$110$	0	0
$111$	1.35854	0.128947
$112$	2.03392	0.192187
$113$	−18.7717	−1.76589	−0.882945	−	0.469477i	$-0.844443\pi$
$113$	−18.7717	−1.76589	−0.882945	+	0.469477i	$0.844443\pi$
$114$	−1.71142	−0.160290
$115$	0	0
$116$	4.15552	0.385830
$117$	−2.88213	−0.266453
$118$	11.0843	1.02039
$119$	9.39287	0.861043
$120$	0	0
$121$	1.00000	0.0909091
$122$	15.4648	1.40012
$123$	−2.38565	−0.215107
$124$	−4.81788	−0.432658
$125$	0	0
$126$	4.06169	0.361844
$127$	8.65119	0.767669	0.383834	−	0.923402i	$-0.374603\pi$
$127$	8.65119	0.767669	0.383834	+	0.923402i	$0.374603\pi$
$128$	−0.545230	−0.0481919
$129$	0.0311850	0.00274569
$130$	0	0
$131$	13.0268	1.13816	0.569079	−	0.822283i	$-0.307300\pi$
$131$	13.0268	1.13816	0.569079	+	0.822283i	$0.307300\pi$
$132$	−0.295199	−0.0256938
$133$	−6.16123	−0.534247
$134$	−0.482811	−0.0417085
$135$	0	0
$136$	21.7329	1.86358
$137$	10.7958	0.922349	0.461175	−	0.887309i	$-0.347428\pi$
$137$	10.7958	0.922349	0.461175	+	0.887309i	$0.347428\pi$
$138$	−1.13533	−0.0966459
$139$	−18.8935	−1.60252	−0.801262	−	0.598313i	$-0.795838\pi$
$139$	−18.8935	−1.60252	−0.801262	+	0.598313i	$0.795838\pi$
$140$	0	0
$141$	1.78028	0.149927
$142$	11.4650	0.962118
$143$	−1.00000	−0.0836242
$144$	4.44160	0.370133
$145$	0	0
$146$	11.4705	0.949303
$147$	1.80527	0.148896
$148$	3.40226	0.279664
$149$	−6.97104	−0.571090	−0.285545	−	0.958365i	$-0.592175\pi$
$149$	−6.97104	−0.571090	−0.285545	+	0.958365i	$0.592175\pi$
$150$	0	0
$151$	−19.9988	−1.62748	−0.813740	−	0.581229i	$-0.802572\pi$
$151$	−19.9988	−1.62748	−0.813740	+	0.581229i	$0.802572\pi$
$152$	−14.2556	−1.15629
$153$	20.5118	1.65828
$154$	1.40927	0.113562
$155$	0	0
$156$	0.295199	0.0236348
$157$	14.2815	1.13979	0.569895	−	0.821717i	$-0.306984\pi$
$157$	14.2815	1.13979	0.569895	+	0.821717i	$0.306984\pi$
$158$	6.20529	0.493666
$159$	4.33462	0.343757
$160$	0	0
$161$	−4.08726	−0.322121
$162$	8.49219	0.667210
$163$	3.29640	0.258194	0.129097	−	0.991632i	$-0.458792\pi$
$163$	3.29640	0.258194	0.129097	+	0.991632i	$0.458792\pi$
$164$	−5.97451	−0.466531
$165$	0	0
$166$	15.5644	1.20803
$167$	−4.87554	−0.377281	−0.188640	−	0.982046i	$-0.560408\pi$
$167$	−4.87554	−0.377281	−0.188640	+	0.982046i	$0.560408\pi$
$168$	−1.38370	−0.106755
$169$	1.00000	0.0769231
$170$	0	0
$171$	−13.4547	−1.02891
$172$	0.0780985	0.00595495
$173$	−13.2332	−1.00610	−0.503049	−	0.864258i	$-0.667789\pi$
$173$	−13.2332	−1.00610	−0.503049	+	0.864258i	$0.667789\pi$
$174$	1.77181	0.134320
$175$	0	0
$176$	1.54108	0.116164
$177$	−3.56393	−0.267881
$178$	1.95010	0.146166
$179$	−18.9190	−1.41408	−0.707038	−	0.707176i	$-0.749969\pi$
$179$	−18.9190	−1.41408	−0.707038	+	0.707176i	$0.749969\pi$
$180$	0	0
$181$	24.5749	1.82663	0.913317	−	0.407248i	$-0.133512\pi$
$181$	24.5749	1.82663	0.913317	+	0.407248i	$0.133512\pi$
$182$	−1.40927	−0.104462
$183$	−4.97241	−0.367571
$184$	−9.45696	−0.697176
$185$	0	0
$186$	−2.05422	−0.150623
$187$	7.11691	0.520440
$188$	4.45846	0.325167
$189$	−2.66532	−0.193874
$190$	0	0
$191$	−8.28475	−0.599464	−0.299732	−	0.954023i	$-0.596897\pi$
$191$	−8.28475	−0.599464	−0.299732	+	0.954023i	$0.596897\pi$
$192$	−2.69392	−0.194417
$193$	20.0813	1.44549	0.722743	−	0.691116i	$-0.242881\pi$
$193$	20.0813	1.44549	0.722743	+	0.691116i	$0.242881\pi$
$194$	6.39551	0.459171
$195$	0	0
$196$	4.52103	0.322931
$197$	15.8972	1.13263	0.566314	−	0.824189i	$-0.308369\pi$
$197$	15.8972	1.13263	0.566314	+	0.824189i	$0.308369\pi$
$198$	3.07752	0.218710
$199$	−11.2488	−0.797403	−0.398701	−	0.917081i	$-0.630539\pi$
$199$	−11.2488	−0.797403	−0.398701	+	0.917081i	$0.630539\pi$
$200$	0	0
$201$	0.155238	0.0109497
$202$	−19.3593	−1.36211
$203$	6.37861	0.447691
$204$	−2.10091	−0.147093
$205$	0	0
$206$	20.0084	1.39405
$207$	−8.92563	−0.620374
$208$	−1.54108	−0.106855
$209$	−4.66833	−0.322915
$210$	0	0
$211$	12.0831	0.831834	0.415917	−	0.909403i	$-0.363461\pi$
$211$	12.0831	0.831834	0.415917	+	0.909403i	$0.363461\pi$
$212$	10.8554	0.745554
$213$	−3.68633	−0.252583
$214$	−15.3219	−1.04738
$215$	0	0
$216$	−6.16693	−0.419607
$217$	−7.39532	−0.502027
$218$	3.90570	0.264528
$219$	−3.68810	−0.249219
$220$	0	0
$221$	−7.11691	−0.478735
$222$	1.45064	0.0973604
$223$	11.8085	0.790755	0.395377	−	0.918519i	$-0.370614\pi$
$223$	11.8085	0.790755	0.395377	+	0.918519i	$0.370614\pi$
$224$	−5.88870	−0.393455
$225$	0	0
$226$	−20.0443	−1.33333
$227$	16.8935	1.12126	0.560632	−	0.828065i	$-0.310558\pi$
$227$	16.8935	1.12126	0.560632	+	0.828065i	$0.310558\pi$
$228$	1.37808	0.0912659
$229$	16.5925	1.09647	0.548233	−	0.836326i	$-0.315301\pi$
$229$	16.5925	1.09647	0.548233	+	0.836326i	$0.315301\pi$
$230$	0	0
$231$	−0.453123	−0.0298133
$232$	14.7586	0.968949
$233$	−19.2647	−1.26207	−0.631036	−	0.775753i	$-0.717370\pi$
$233$	−19.2647	−1.26207	−0.631036	+	0.775753i	$0.717370\pi$
$234$	−3.07752	−0.201184
$235$	0	0
$236$	−8.92535	−0.580991
$237$	−1.99519	−0.129601
$238$	10.0297	0.650126
$239$	28.3128	1.83140	0.915702	−	0.401857i	$-0.131635\pi$
$239$	28.3128	1.83140	0.915702	+	0.401857i	$0.131635\pi$
$240$	0	0
$241$	−17.7918	−1.14607	−0.573036	−	0.819530i	$-0.694235\pi$
$241$	−17.7918	−1.14607	−0.573036	+	0.819530i	$0.694235\pi$
$242$	1.06779	0.0686404
$243$	−8.78899	−0.563814
$244$	−12.4527	−0.797202
$245$	0	0
$246$	−2.54738	−0.162415
$247$	4.66833	0.297038
$248$	−17.1110	−1.08655
$249$	−5.00442	−0.317142
$250$	0	0
$251$	18.5434	1.17045	0.585225	−	0.810871i	$-0.301006\pi$
$251$	18.5434	1.17045	0.585225	+	0.810871i	$0.301006\pi$
$252$	−3.27058	−0.206027
$253$	−3.09689	−0.194700
$254$	9.23768	0.579624
$255$	0	0
$256$	−16.2752	−1.01720
$257$	3.31960	0.207071	0.103535	−	0.994626i	$-0.466984\pi$
$257$	3.31960	0.207071	0.103535	+	0.994626i	$0.466984\pi$
$258$	0.0332992	0.00207312
$259$	5.22238	0.324503
$260$	0	0
$261$	13.9294	0.862208
$262$	13.9100	0.859360
$263$	11.0823	0.683364	0.341682	−	0.939816i	$-0.389003\pi$
$263$	11.0823	0.683364	0.341682	+	0.939816i	$0.389003\pi$
$264$	−1.04842	−0.0645257
$265$	0	0
$266$	−6.57893	−0.403380
$267$	−0.627016	−0.0383728
$268$	0.388772	0.0237480
$269$	−0.765738	−0.0466879	−0.0233439	−	0.999727i	$-0.507431\pi$
$269$	−0.765738	−0.0466879	−0.0233439	+	0.999727i	$0.507431\pi$
$270$	0	0
$271$	−12.5639	−0.763205	−0.381602	−	0.924327i	$-0.624628\pi$
$271$	−12.5639	−0.763205	−0.381602	+	0.924327i	$0.624628\pi$
$272$	10.9678	0.665019
$273$	0.453123	0.0274242
$274$	11.5277	0.696415
$275$	0	0
$276$	0.914199	0.0550283
$277$	−24.3104	−1.46067	−0.730334	−	0.683090i	$-0.760636\pi$
$277$	−24.3104	−1.46067	−0.730334	+	0.683090i	$0.760636\pi$
$278$	−20.1744	−1.20998
$279$	−16.1497	−0.966855
$280$	0	0
$281$	15.3784	0.917400	0.458700	−	0.888591i	$-0.348315\pi$
$281$	15.3784	0.917400	0.458700	+	0.888591i	$0.348315\pi$
$282$	1.90097	0.113201
$283$	3.73537	0.222045	0.111022	−	0.993818i	$-0.464587\pi$
$283$	3.73537	0.222045	0.111022	+	0.993818i	$0.464587\pi$
$284$	−9.23189	−0.547812
$285$	0	0
$286$	−1.06779	−0.0631400
$287$	−9.17072	−0.541331
$288$	−12.8595	−0.757756
$289$	33.6505	1.97944
$290$	0	0
$291$	−2.05635	−0.120545
$292$	−9.23632	−0.540515
$293$	−2.21953	−0.129666	−0.0648332	−	0.997896i	$-0.520652\pi$
$293$	−2.21953	−0.129666	−0.0648332	+	0.997896i	$0.520652\pi$
$294$	1.92765	0.112423
$295$	0	0
$296$	12.0834	0.702331
$297$	−2.01950	−0.117183
$298$	−7.44363	−0.431198
$299$	3.09689	0.179098
$300$	0	0
$301$	0.119879	0.00690972
$302$	−21.3546	−1.22882
$303$	6.22459	0.357594
$304$	−7.19428	−0.412620
$305$	0	0
$306$	21.9024	1.25208
$307$	26.8988	1.53520	0.767598	−	0.640932i	$-0.221452\pi$
$307$	26.8988	1.53520	0.767598	+	0.640932i	$0.221452\pi$
$308$	−1.13478	−0.0646602
$309$	−6.43331	−0.365978
$310$	0	0
$311$	−15.9839	−0.906365	−0.453183	−	0.891418i	$-0.649711\pi$
$311$	−15.9839	−0.906365	−0.453183	+	0.891418i	$0.649711\pi$
$312$	1.04842	0.0593550
$313$	−2.48431	−0.140422	−0.0702108	−	0.997532i	$-0.522367\pi$
$313$	−2.48431	−0.140422	−0.0702108	+	0.997532i	$0.522367\pi$
$314$	15.2497	0.860592
$315$	0	0
$316$	−4.99666	−0.281084
$317$	7.91673	0.444648	0.222324	−	0.974973i	$-0.428636\pi$
$317$	7.91673	0.444648	0.222324	+	0.974973i	$0.428636\pi$
$318$	4.62848	0.259552
$319$	4.83303	0.270598
$320$	0	0
$321$	4.92645	0.274968
$322$	−4.36436	−0.243216
$323$	−33.2241	−1.84864
$324$	−6.83814	−0.379897
$325$	0	0
$326$	3.51987	0.194948
$327$	−1.25580	−0.0694460
$328$	−21.2189	−1.17162
$329$	6.84362	0.377301
$330$	0	0
$331$	−21.2305	−1.16693	−0.583467	−	0.812137i	$-0.698304\pi$
$331$	−21.2305	−1.16693	−0.583467	+	0.812137i	$0.698304\pi$
$332$	−12.5329	−0.687830
$333$	11.4045	0.624961
$334$	−5.20607	−0.284864
$335$	0	0
$336$	−0.698300	−0.0380954
$337$	−21.6075	−1.17704	−0.588518	−	0.808484i	$-0.700288\pi$
$337$	−21.6075	−1.17704	−0.588518	+	0.808484i	$0.700288\pi$
$338$	1.06779	0.0580803
$339$	6.44484	0.350036
$340$	0	0
$341$	−5.60339	−0.303440
$342$	−14.3668	−0.776870
$343$	16.1782	0.873543
$344$	0.277372	0.0149549
$345$	0	0
$346$	−14.1303	−0.759649
$347$	25.8757	1.38908	0.694540	−	0.719455i	$-0.255608\pi$
$347$	25.8757	1.38908	0.694540	+	0.719455i	$0.255608\pi$
$348$	−1.42671	−0.0764794
$349$	−17.8972	−0.958014	−0.479007	−	0.877811i	$-0.659003\pi$
$349$	−17.8972	−0.958014	−0.479007	+	0.877811i	$0.659003\pi$
$350$	0	0
$351$	2.01950	0.107793
$352$	−4.46183	−0.237816
$353$	−30.2693	−1.61107	−0.805535	−	0.592548i	$-0.798122\pi$
$353$	−30.2693	−1.61107	−0.805535	+	0.592548i	$0.798122\pi$
$354$	−3.80554	−0.202262
$355$	0	0
$356$	−1.57027	−0.0832243
$357$	−3.22484	−0.170676
$358$	−20.2016	−1.06769
$359$	−5.68638	−0.300116	−0.150058	−	0.988677i	$-0.547946\pi$
$359$	−5.68638	−0.300116	−0.150058	+	0.988677i	$0.547946\pi$
$360$	0	0
$361$	2.79326	0.147014
$362$	26.2409	1.37919
$363$	−0.343328	−0.0180200
$364$	1.13478	0.0594787
$365$	0	0
$366$	−5.30951	−0.277532
$367$	−16.1935	−0.845291	−0.422646	−	0.906295i	$-0.638899\pi$
$367$	−16.1935	−0.845291	−0.422646	+	0.906295i	$0.638899\pi$
$368$	−4.77257	−0.248788
$369$	−20.0267	−1.04255
$370$	0	0
$371$	16.6628	0.865089
$372$	1.65411	0.0857618
$373$	9.81582	0.508244	0.254122	−	0.967172i	$-0.418214\pi$
$373$	9.81582	0.508244	0.254122	+	0.967172i	$0.418214\pi$
$374$	7.59940	0.392956
$375$	0	0
$376$	15.8345	0.816603
$377$	−4.83303	−0.248914
$378$	−2.84602	−0.146383
$379$	−15.3268	−0.787283	−0.393642	−	0.919264i	$-0.628785\pi$
$379$	−15.3268	−0.787283	−0.393642	+	0.919264i	$0.628785\pi$
$380$	0	0
$381$	−2.97019	−0.152168
$382$	−8.84641	−0.452622
$383$	5.98657	0.305899	0.152950	−	0.988234i	$-0.451123\pi$
$383$	5.98657	0.305899	0.152950	+	0.988234i	$0.451123\pi$
$384$	0.187193	0.00955263
$385$	0	0
$386$	21.4427	1.09141
$387$	0.261788	0.0133074
$388$	−5.14984	−0.261443
$389$	28.5530	1.44769	0.723847	−	0.689960i	$-0.242372\pi$
$389$	28.5530	1.44769	0.723847	+	0.689960i	$0.242372\pi$
$390$	0	0
$391$	−22.0403	−1.11463
$392$	16.0568	0.810988
$393$	−4.47247	−0.225606
$394$	16.9749	0.855185
$395$	0	0
$396$	−2.47810	−0.124529
$397$	15.3127	0.768522	0.384261	−	0.923225i	$-0.374456\pi$
$397$	15.3127	0.768522	0.384261	+	0.923225i	$0.374456\pi$
$398$	−12.0114	−0.602075
$399$	2.11532	0.105899
$400$	0	0
$401$	19.1032	0.953968	0.476984	−	0.878912i	$-0.341730\pi$
$401$	19.1032	0.953968	0.476984	+	0.878912i	$0.341730\pi$
$402$	0.165762	0.00826748
$403$	5.60339	0.279125
$404$	15.5886	0.775562
$405$	0	0
$406$	6.81104	0.338026
$407$	3.95696	0.196139
$408$	−7.46151	−0.369400
$409$	23.2991	1.15206	0.576032	−	0.817427i	$-0.304600\pi$
$409$	23.2991	1.15206	0.576032	+	0.817427i	$0.304600\pi$
$410$	0	0
$411$	−3.70651	−0.182829
$412$	−16.1113	−0.793747
$413$	−13.7002	−0.674142
$414$	−9.53074	−0.468410
$415$	0	0
$416$	4.46183	0.218759
$417$	6.48666	0.317653
$418$	−4.98481	−0.243815
$419$	30.6438	1.49705	0.748525	−	0.663107i	$-0.230762\pi$
$419$	30.6438	1.49705	0.748525	+	0.663107i	$0.230762\pi$
$420$	0	0
$421$	5.40310	0.263331	0.131666	−	0.991294i	$-0.457967\pi$
$421$	5.40310	0.263331	0.131666	+	0.991294i	$0.457967\pi$
$422$	12.9022	0.628071
$423$	14.9449	0.726645
$424$	38.5538	1.87234
$425$	0	0
$426$	−3.93624	−0.190712
$427$	−19.1146	−0.925018
$428$	12.3376	0.596360
$429$	0.343328	0.0165760
$430$	0	0
$431$	−13.9555	−0.672212	−0.336106	−	0.941824i	$-0.609110\pi$
$431$	−13.9555	−0.672212	−0.336106	+	0.941824i	$0.609110\pi$
$432$	−3.11222	−0.149737
$433$	9.59238	0.460980	0.230490	−	0.973075i	$-0.425967\pi$
$433$	9.59238	0.460980	0.230490	+	0.973075i	$0.425967\pi$
$434$	−7.89668	−0.379053
$435$	0	0
$436$	−3.14498	−0.150617
$437$	14.4573	0.691586
$438$	−3.93813	−0.188171
$439$	−1.99577	−0.0952529	−0.0476264	−	0.998865i	$-0.515166\pi$
$439$	−1.99577	−0.0952529	−0.0476264	+	0.998865i	$0.515166\pi$
$440$	0	0
$441$	15.1546	0.721649
$442$	−7.59940	−0.361467
$443$	−27.1573	−1.29028	−0.645141	−	0.764063i	$-0.723202\pi$
$443$	−27.1573	−1.29028	−0.645141	+	0.764063i	$0.723202\pi$
$444$	−1.16809	−0.0554352
$445$	0	0
$446$	12.6090	0.597055
$447$	2.39335	0.113202
$448$	−10.3558	−0.489263
$449$	−13.6479	−0.644085	−0.322043	−	0.946725i	$-0.604369\pi$
$449$	−13.6479	−0.644085	−0.322043	+	0.946725i	$0.604369\pi$
$450$	0	0
$451$	−6.94860	−0.327197
$452$	16.1402	0.759170
$453$	6.86615	0.322600
$454$	18.0388	0.846604
$455$	0	0
$456$	4.89436	0.229199
$457$	−12.1666	−0.569132	−0.284566	−	0.958656i	$-0.591849\pi$
$457$	−12.1666	−0.569132	−0.284566	+	0.958656i	$0.591849\pi$
$458$	17.7174	0.827880
$459$	−14.3726	−0.670855
$460$	0	0
$461$	24.8253	1.15623	0.578116	−	0.815955i	$-0.303788\pi$
$461$	24.8253	1.15623	0.578116	+	0.815955i	$0.303788\pi$
$462$	−0.483842	−0.0225104
$463$	−14.9250	−0.693625	−0.346812	−	0.937935i	$-0.612736\pi$
$463$	−14.9250	−0.693625	−0.346812	+	0.937935i	$0.612736\pi$
$464$	7.44811	0.345770
$465$	0	0
$466$	−20.5707	−0.952921
$467$	21.1572	0.979040	0.489520	−	0.871992i	$-0.337172\pi$
$467$	21.1572	0.979040	0.489520	+	0.871992i	$0.337172\pi$
$468$	2.47810	0.114550
$469$	0.596755	0.0275556
$470$	0	0
$471$	−4.90325	−0.225930
$472$	−31.6990	−1.45906
$473$	0.0908316	0.00417644
$474$	−2.13045	−0.0978548
$475$	0	0
$476$	−8.07614	−0.370169
$477$	36.3877	1.66608
$478$	30.2323	1.38279
$479$	−14.9078	−0.681156	−0.340578	−	0.940216i	$-0.610623\pi$
$479$	−14.9078	−0.681156	−0.340578	+	0.940216i	$0.610623\pi$
$480$	0	0
$481$	−3.95696	−0.180422
$482$	−18.9980	−0.865336
$483$	1.40327	0.0638511
$484$	−0.859816	−0.0390825
$485$	0	0
$486$	−9.38483	−0.425705
$487$	−22.2067	−1.00628	−0.503140	−	0.864205i	$-0.667822\pi$
$487$	−22.2067	−1.00628	−0.503140	+	0.864205i	$0.667822\pi$
$488$	−44.2266	−2.00204
$489$	−1.13175	−0.0511793
$490$	0	0
$491$	−21.0059	−0.947984	−0.473992	−	0.880529i	$-0.657187\pi$
$491$	−21.0059	−0.947984	−0.473992	+	0.880529i	$0.657187\pi$
$492$	2.05122	0.0924760
$493$	34.3963	1.54913
$494$	4.98481	0.224277
$495$	0	0
$496$	−8.63529	−0.387736
$497$	−14.1707	−0.635644
$498$	−5.34369	−0.239456
$499$	−5.96703	−0.267121	−0.133560	−	0.991041i	$-0.542641\pi$
$499$	−5.96703	−0.267121	−0.133560	+	0.991041i	$0.542641\pi$
$500$	0	0
$501$	1.67391	0.0747848
$502$	19.8006	0.883742
$503$	−6.17193	−0.275193	−0.137596	−	0.990488i	$-0.543938\pi$
$503$	−6.17193	−0.275193	−0.137596	+	0.990488i	$0.543938\pi$
$504$	−11.6157	−0.517404
$505$	0	0
$506$	−3.30684	−0.147007
$507$	−0.343328	−0.0152477
$508$	−7.43843	−0.330027
$509$	14.3534	0.636205	0.318103	−	0.948056i	$-0.396954\pi$
$509$	14.3534	0.636205	0.318103	+	0.948056i	$0.396954\pi$
$510$	0	0
$511$	−14.1775	−0.627176
$512$	−16.2881	−0.719837
$513$	9.42768	0.416242
$514$	3.54465	0.156348
$515$	0	0
$516$	−0.0268134	−0.00118039
$517$	5.18536	0.228052
$518$	5.57643	0.245014
$519$	4.54331	0.199429
$520$	0	0
$521$	10.8081	0.473511	0.236755	−	0.971569i	$-0.423916\pi$
$521$	10.8081	0.473511	0.236755	+	0.971569i	$0.423916\pi$
$522$	14.8737	0.651006
$523$	17.0609	0.746019	0.373009	−	0.927828i	$-0.378326\pi$
$523$	17.0609	0.746019	0.373009	+	0.927828i	$0.378326\pi$
$524$	−11.2007	−0.489303
$525$	0	0
$526$	11.8336	0.515970
$527$	−39.8788	−1.73715
$528$	−0.529097	−0.0230260
$529$	−13.4093	−0.583011
$530$	0	0
$531$	−29.9180	−1.29833
$532$	5.29753	0.229677
$533$	6.94860	0.300977
$534$	−0.669524	−0.0289731
$535$	0	0
$536$	1.38075	0.0596393
$537$	6.49544	0.280299
$538$	−0.817650	−0.0352514
$539$	5.25814	0.226484
$540$	0	0
$541$	−4.72643	−0.203205	−0.101602	−	0.994825i	$-0.532397\pi$
$541$	−4.72643	−0.203205	−0.101602	+	0.994825i	$0.532397\pi$
$542$	−13.4157	−0.576253
$543$	−8.43724	−0.362077
$544$	−31.7544	−1.36146
$545$	0	0
$546$	0.483842	0.0207065
$547$	−4.77101	−0.203994	−0.101997	−	0.994785i	$-0.532523\pi$
$547$	−4.77101	−0.203994	−0.101997	+	0.994785i	$0.532523\pi$
$548$	−9.28242	−0.396525
$549$	−41.7417	−1.78149
$550$	0	0
$551$	−22.5622	−0.961180
$552$	3.24684	0.138195
$553$	−7.66975	−0.326151
$554$	−25.9585	−1.10287
$555$	0	0
$556$	16.2449	0.688938
$557$	16.5718	0.702169	0.351084	−	0.936344i	$-0.385813\pi$
$557$	16.5718	0.702169	0.351084	+	0.936344i	$0.385813\pi$
$558$	−17.2445	−0.730018
$559$	−0.0908316	−0.00384177
$560$	0	0
$561$	−2.44344	−0.103162
$562$	16.4210	0.692678
$563$	25.9594	1.09406	0.547030	−	0.837113i	$-0.315758\pi$
$563$	25.9594	1.09406	0.547030	+	0.837113i	$0.315758\pi$
$564$	−1.53071	−0.0644547
$565$	0	0
$566$	3.98861	0.167654
$567$	−10.4964	−0.440806
$568$	−32.7877	−1.37574
$569$	−6.86052	−0.287608	−0.143804	−	0.989606i	$-0.545933\pi$
$569$	−6.86052	−0.287608	−0.143804	+	0.989606i	$0.545933\pi$
$570$	0	0
$571$	−28.0004	−1.17178	−0.585890	−	0.810390i	$-0.699255\pi$
$571$	−28.0004	−1.17178	−0.585890	+	0.810390i	$0.699255\pi$
$572$	0.859816	0.0359507
$573$	2.84439	0.118826
$574$	−9.79244	−0.408729
$575$	0	0
$576$	−22.6145	−0.942273
$577$	21.7112	0.903850	0.451925	−	0.892056i	$-0.350737\pi$
$577$	21.7112	0.903850	0.451925	+	0.892056i	$0.350737\pi$
$578$	35.9318	1.49456
$579$	−6.89449	−0.286525
$580$	0	0
$581$	−19.2376	−0.798110
$582$	−2.19576	−0.0910171
$583$	12.6253	0.522886
$584$	−32.8034	−1.35742
$585$	0	0
$586$	−2.37000	−0.0979039
$587$	−21.5111	−0.887857	−0.443929	−	0.896062i	$-0.646416\pi$
$587$	−21.5111	−0.887857	−0.443929	+	0.896062i	$0.646416\pi$
$588$	−1.55220	−0.0640115
$589$	26.1584	1.07784
$590$	0	0
$591$	−5.45795	−0.224510
$592$	6.09802	0.250627
$593$	43.7709	1.79745	0.898727	−	0.438509i	$-0.144493\pi$
$593$	43.7709	1.79745	0.898727	+	0.438509i	$0.144493\pi$
$594$	−2.15641	−0.0884785
$595$	0	0
$596$	5.99381	0.245516
$597$	3.86201	0.158062
$598$	3.30684	0.135227
$599$	−19.5116	−0.797221	−0.398611	−	0.917120i	$-0.630508\pi$
$599$	−19.5116	−0.797221	−0.398611	+	0.917120i	$0.630508\pi$
$600$	0	0
$601$	−0.382954	−0.0156210	−0.00781051	−	0.999969i	$-0.502486\pi$
$601$	−0.382954	−0.0156210	−0.00781051	+	0.999969i	$0.502486\pi$
$602$	0.128006	0.00521714
$603$	1.30317	0.0530693
$604$	17.1953	0.699667
$605$	0	0
$606$	6.64658	0.269999
$607$	−15.5471	−0.631039	−0.315519	−	0.948919i	$-0.602179\pi$
$607$	−15.5471	−0.631039	−0.315519	+	0.948919i	$0.602179\pi$
$608$	20.8293	0.844738
$609$	−2.18996	−0.0887415
$610$	0	0
$611$	−5.18536	−0.209777
$612$	−17.6364	−0.712910
$613$	−20.8664	−0.842786	−0.421393	−	0.906878i	$-0.638459\pi$
$613$	−20.8664	−0.842786	−0.421393	+	0.906878i	$0.638459\pi$
$614$	28.7224	1.15914
$615$	0	0
$616$	−4.03025	−0.162383
$617$	20.8763	0.840450	0.420225	−	0.907420i	$-0.361951\pi$
$617$	20.8763	0.840450	0.420225	+	0.907420i	$0.361951\pi$
$618$	−6.86945	−0.276330
$619$	14.5455	0.584633	0.292316	−	0.956322i	$-0.405574\pi$
$619$	14.5455	0.584633	0.292316	+	0.956322i	$0.405574\pi$
$620$	0	0
$621$	6.25417	0.250971
$622$	−17.0675	−0.684346
$623$	−2.41033	−0.0965678
$624$	0.529097	0.0211809
$625$	0	0
$626$	−2.65273	−0.106025
$627$	1.60277	0.0640083
$628$	−12.2795	−0.490005
$629$	28.1614	1.12287
$630$	0	0
$631$	−2.96489	−0.118030	−0.0590152	−	0.998257i	$-0.518796\pi$
$631$	−2.96489	−0.118030	−0.0590152	+	0.998257i	$0.518796\pi$
$632$	−17.7460	−0.705897
$633$	−4.14846	−0.164887
$634$	8.45344	0.335729
$635$	0	0
$636$	−3.72697	−0.147784
$637$	−5.25814	−0.208335
$638$	5.16068	0.204313
$639$	−30.9455	−1.22419
$640$	0	0
$641$	−34.5769	−1.36570	−0.682852	−	0.730557i	$-0.739261\pi$
$641$	−34.5769	−1.36570	−0.682852	+	0.730557i	$0.739261\pi$
$642$	5.26043	0.207613
$643$	−3.43687	−0.135537	−0.0677684	−	0.997701i	$-0.521588\pi$
$643$	−3.43687	−0.135537	−0.0677684	+	0.997701i	$0.521588\pi$
$644$	3.51429	0.138483
$645$	0	0
$646$	−35.4765	−1.39580
$647$	35.0062	1.37624	0.688119	−	0.725598i	$-0.258437\pi$
$647$	35.0062	1.37624	0.688119	+	0.725598i	$0.258437\pi$
$648$	−24.2861	−0.954049
$649$	−10.3805	−0.407472
$650$	0	0
$651$	2.53902	0.0995121
$652$	−2.83429	−0.111000
$653$	17.3441	0.678727	0.339364	−	0.940655i	$-0.389788\pi$
$653$	17.3441	0.678727	0.339364	+	0.940655i	$0.389788\pi$
$654$	−1.34094	−0.0524348
$655$	0	0
$656$	−10.7084	−0.418092
$657$	−30.9604	−1.20788
$658$	7.30758	0.284879
$659$	1.99929	0.0778812	0.0389406	−	0.999242i	$-0.487602\pi$
$659$	1.99929	0.0778812	0.0389406	+	0.999242i	$0.487602\pi$
$660$	0	0
$661$	23.6273	0.918994	0.459497	−	0.888179i	$-0.348030\pi$
$661$	23.6273	0.918994	0.459497	+	0.888179i	$0.348030\pi$
$662$	−22.6698	−0.881088
$663$	2.44344	0.0948952
$664$	−44.5113	−1.72737
$665$	0	0
$666$	12.1776	0.471873
$667$	−14.9674	−0.579539
$668$	4.19207	0.162196
$669$	−4.05418	−0.156744
$670$	0	0
$671$	−14.4830	−0.559109
$672$	2.02176	0.0779909
$673$	21.2862	0.820523	0.410261	−	0.911968i	$-0.365437\pi$
$673$	21.2862	0.820523	0.410261	+	0.911968i	$0.365437\pi$
$674$	−23.0723	−0.888714
$675$	0	0
$676$	−0.859816	−0.0330698
$677$	−44.5518	−1.71227	−0.856133	−	0.516756i	$-0.827139\pi$
$677$	−44.5518	−1.71227	−0.856133	+	0.516756i	$0.827139\pi$
$678$	6.88176	0.264292
$679$	−7.90487	−0.303361
$680$	0	0
$681$	−5.80003	−0.222258
$682$	−5.98326	−0.229111
$683$	−36.0211	−1.37831	−0.689154	−	0.724615i	$-0.742018\pi$
$683$	−36.0211	−1.37831	−0.689154	+	0.724615i	$0.742018\pi$
$684$	11.5686	0.442335
$685$	0	0
$686$	17.2750	0.659563
$687$	−5.69668	−0.217342
$688$	0.139979	0.00533666
$689$	−12.6253	−0.480985
$690$	0	0
$691$	6.72904	0.255985	0.127992	−	0.991775i	$-0.459147\pi$
$691$	6.72904	0.255985	0.127992	+	0.991775i	$0.459147\pi$
$692$	11.3781	0.432530
$693$	−3.80382	−0.144495
$694$	27.6299	1.04882
$695$	0	0
$696$	−5.06704	−0.192066
$697$	−49.4526	−1.87315
$698$	−19.1105	−0.723343
$699$	6.61411	0.250169
$700$	0	0
$701$	15.7973	0.596657	0.298328	−	0.954463i	$-0.403571\pi$
$701$	15.7973	0.596657	0.298328	+	0.954463i	$0.403571\pi$
$702$	2.15641	0.0813884
$703$	−18.4724	−0.696700
$704$	−7.84648	−0.295725
$705$	0	0
$706$	−32.3213	−1.21643
$707$	23.9281	0.899909
$708$	3.06432	0.115164
$709$	31.3162	1.17610	0.588052	−	0.808823i	$-0.299895\pi$
$709$	31.3162	1.17610	0.588052	+	0.808823i	$0.299895\pi$
$710$	0	0
$711$	−16.7489	−0.628135
$712$	−5.57693	−0.209004
$713$	17.3531	0.649878
$714$	−3.44346	−0.128868
$715$	0	0
$716$	16.2669	0.607922
$717$	−9.72059	−0.363022
$718$	−6.07188	−0.226601
$719$	48.9739	1.82642	0.913210	−	0.407490i	$-0.133596\pi$
$719$	48.9739	1.82642	0.913210	+	0.407490i	$0.133596\pi$
$720$	0	0
$721$	−24.7304	−0.921009
$722$	2.98263	0.111002
$723$	6.10843	0.227175
$724$	−21.1299	−0.785285
$725$	0	0
$726$	−0.366604	−0.0136059
$727$	29.3947	1.09019	0.545094	−	0.838375i	$-0.316494\pi$
$727$	29.3947	1.09019	0.545094	+	0.838375i	$0.316494\pi$
$728$	4.03025	0.149371
$729$	−20.8416	−0.771910
$730$	0	0
$731$	0.646441	0.0239095
$732$	4.27536	0.158022
$733$	−21.8190	−0.805904	−0.402952	−	0.915221i	$-0.632016\pi$
$733$	−21.8190	−0.805904	−0.402952	+	0.915221i	$0.632016\pi$
$734$	−17.2913	−0.638233
$735$	0	0
$736$	13.8178	0.509331
$737$	0.452157	0.0166554
$738$	−21.3844	−0.787171
$739$	4.31975	0.158905	0.0794523	−	0.996839i	$-0.474683\pi$
$739$	4.31975	0.158905	0.0794523	+	0.996839i	$0.474683\pi$
$740$	0	0
$741$	−1.60277	−0.0588791
$742$	17.7924	0.653181
$743$	7.58424	0.278239	0.139119	−	0.990276i	$-0.455573\pi$
$743$	7.58424	0.278239	0.139119	+	0.990276i	$0.455573\pi$
$744$	5.87470	0.215377
$745$	0	0
$746$	10.4813	0.383747
$747$	−42.0104	−1.53708
$748$	−6.11924	−0.223741
$749$	18.9379	0.691975
$750$	0	0
$751$	4.01713	0.146587	0.0732936	−	0.997310i	$-0.476649\pi$
$751$	4.01713	0.146587	0.0732936	+	0.997310i	$0.476649\pi$
$752$	7.99108	0.291405
$753$	−6.36648	−0.232007
$754$	−5.16068	−0.187941
$755$	0	0
$756$	2.29169	0.0833479
$757$	−25.5698	−0.929348	−0.464674	−	0.885482i	$-0.653829\pi$
$757$	−25.5698	−0.929348	−0.464674	+	0.885482i	$0.653829\pi$
$758$	−16.3658	−0.594434
$759$	1.06325	0.0385935
$760$	0	0
$761$	−27.1804	−0.985287	−0.492644	−	0.870231i	$-0.663969\pi$
$761$	−27.1804	−0.985287	−0.492644	+	0.870231i	$0.663969\pi$
$762$	−3.17156	−0.114893
$763$	−4.82746	−0.174766
$764$	7.12336	0.257714
$765$	0	0
$766$	6.39242	0.230968
$767$	10.3805	0.374820
$768$	5.58772	0.201629
$769$	8.02400	0.289353	0.144676	−	0.989479i	$-0.453786\pi$
$769$	8.02400	0.289353	0.144676	+	0.989479i	$0.453786\pi$
$770$	0	0
$771$	−1.13971	−0.0410457
$772$	−17.2663	−0.621426
$773$	3.39505	0.122111	0.0610557	−	0.998134i	$-0.480553\pi$
$773$	3.39505	0.122111	0.0610557	+	0.998134i	$0.480553\pi$
$774$	0.279536	0.0100477
$775$	0	0
$776$	−18.2900	−0.656572
$777$	−1.79299	−0.0643232
$778$	30.4887	1.09307
$779$	32.4383	1.16222
$780$	0	0
$781$	−10.7371	−0.384202
$782$	−23.5345	−0.841593
$783$	−9.76030	−0.348805
$784$	8.10324	0.289401
$785$	0	0
$786$	−4.77568	−0.170343
$787$	44.7193	1.59407	0.797036	−	0.603932i	$-0.206400\pi$
$787$	44.7193	1.59407	0.797036	+	0.603932i	$0.206400\pi$
$788$	−13.6687	−0.486926
$789$	−3.80487	−0.135457
$790$	0	0
$791$	24.7748	0.880889
$792$	−8.80113	−0.312735
$793$	14.4830	0.514305
$794$	16.3508	0.580268
$795$	0	0
$796$	9.67186	0.342810
$797$	22.6005	0.800549	0.400275	−	0.916395i	$-0.368915\pi$
$797$	22.6005	0.800549	0.400275	+	0.916395i	$0.368915\pi$
$798$	2.25873	0.0799582
$799$	36.9038	1.30556
$800$	0	0
$801$	−5.26360	−0.185980
$802$	20.3983	0.720288
$803$	−10.7422	−0.379084
$804$	−0.133476	−0.00470735
$805$	0	0
$806$	5.98326	0.210751
$807$	0.262899	0.00925449
$808$	55.3640	1.94770
$809$	34.0475	1.19705	0.598523	−	0.801105i	$-0.295754\pi$
$809$	34.0475	1.19705	0.598523	+	0.801105i	$0.295754\pi$
$810$	0	0
$811$	−9.83446	−0.345335	−0.172667	−	0.984980i	$-0.555239\pi$
$811$	−9.83446	−0.345335	−0.172667	+	0.984980i	$0.555239\pi$
$812$	−5.48443	−0.192466
$813$	4.31355	0.151283
$814$	4.22522	0.148094
$815$	0	0
$816$	−3.76554	−0.131820
$817$	−0.424032	−0.0148350
$818$	24.8786	0.869859
$819$	3.80382	0.132916
$820$	0	0
$821$	−13.8180	−0.482253	−0.241126	−	0.970494i	$-0.577517\pi$
$821$	−13.8180	−0.482253	−0.241126	+	0.970494i	$0.577517\pi$
$822$	−3.95779	−0.138044
$823$	11.6302	0.405403	0.202702	−	0.979241i	$-0.435028\pi$
$823$	11.6302	0.405403	0.202702	+	0.979241i	$0.435028\pi$
$824$	−57.2204	−1.99337
$825$	0	0
$826$	−14.6290	−0.509007
$827$	−20.6783	−0.719054	−0.359527	−	0.933135i	$-0.617062\pi$
$827$	−20.6783	−0.719054	−0.359527	+	0.933135i	$0.617062\pi$
$828$	7.67440	0.266704
$829$	22.8936	0.795127	0.397563	−	0.917575i	$-0.369856\pi$
$829$	22.8936	0.795127	0.397563	+	0.917575i	$0.369856\pi$
$830$	0	0
$831$	8.34643	0.289535
$832$	7.84648	0.272028
$833$	37.4217	1.29659
$834$	6.92642	0.239842
$835$	0	0
$836$	4.01390	0.138824
$837$	11.3160	0.391139
$838$	32.7213	1.13034
$839$	−39.2466	−1.35494	−0.677471	−	0.735550i	$-0.736924\pi$
$839$	−39.2466	−1.35494	−0.677471	+	0.735550i	$0.736924\pi$
$840$	0	0
$841$	−5.64182	−0.194545
$842$	5.76940	0.198827
$843$	−5.27985	−0.181848
$844$	−10.3892	−0.357612
$845$	0	0
$846$	15.9580	0.548649
$847$	−1.31980	−0.0453487
$848$	19.4566	0.668144
$849$	−1.28246	−0.0440138
$850$	0	0
$851$	−12.2543	−0.420072
$852$	3.16957	0.108588
$853$	24.4193	0.836101	0.418051	−	0.908424i	$-0.362713\pi$
$853$	24.4193	0.836101	0.418051	+	0.908424i	$0.362713\pi$
$854$	−20.4104	−0.698430
$855$	0	0
$856$	43.8178	1.49766
$857$	−34.2627	−1.17039	−0.585196	−	0.810892i	$-0.698982\pi$
$857$	−34.2627	−1.17039	−0.585196	+	0.810892i	$0.698982\pi$
$858$	0.366604	0.0125156
$859$	22.5047	0.767850	0.383925	−	0.923364i	$-0.374572\pi$
$859$	22.5047	0.767850	0.383925	+	0.923364i	$0.374572\pi$
$860$	0	0
$861$	3.14857	0.107303
$862$	−14.9016	−0.507550
$863$	5.81216	0.197848	0.0989240	−	0.995095i	$-0.468460\pi$
$863$	5.81216	0.197848	0.0989240	+	0.995095i	$0.468460\pi$
$864$	9.01066	0.306549
$865$	0	0
$866$	10.2427	0.348061
$867$	−11.5532	−0.392366
$868$	6.35862	0.215826
$869$	−5.81132	−0.197135
$870$	0	0
$871$	−0.452157	−0.0153208
$872$	−11.1696	−0.378250
$873$	−17.2624	−0.584243
$874$	15.4374	0.522178
$875$	0	0
$876$	3.17109	0.107141
$877$	23.6694	0.799260	0.399630	−	0.916677i	$-0.369139\pi$
$877$	23.6694	0.799260	0.399630	+	0.916677i	$0.369139\pi$
$878$	−2.13107	−0.0719202
$879$	0.762027	0.0257025
$880$	0	0
$881$	−18.6126	−0.627074	−0.313537	−	0.949576i	$-0.601514\pi$
$881$	−18.6126	−0.627074	−0.313537	+	0.949576i	$0.601514\pi$
$882$	16.1820	0.544877
$883$	35.6983	1.20134	0.600671	−	0.799496i	$-0.294900\pi$
$883$	35.6983	1.20134	0.600671	+	0.799496i	$0.294900\pi$
$884$	6.11924	0.205812
$885$	0	0
$886$	−28.9984	−0.974221
$887$	29.9209	1.00465	0.502323	−	0.864680i	$-0.332479\pi$
$887$	29.9209	1.00465	0.502323	+	0.864680i	$0.332479\pi$
$888$	−4.14856	−0.139216
$889$	−11.4178	−0.382941
$890$	0	0
$891$	−7.95303	−0.266436
$892$	−10.1531	−0.339952
$893$	−24.2070	−0.810055
$894$	2.55561	0.0854723
$895$	0	0
$896$	0.719592	0.0240399
$897$	−1.06325	−0.0355009
$898$	−14.5732	−0.486313
$899$	−27.0813	−0.903213
$900$	0	0
$901$	89.8531	2.99344
$902$	−7.41967	−0.247048
$903$	−0.0411579	−0.00136965
$904$	57.3229	1.90653
$905$	0	0
$906$	7.33164	0.243577
$907$	−39.8957	−1.32472	−0.662358	−	0.749188i	$-0.730444\pi$
$907$	−39.8957	−1.32472	−0.662358	+	0.749188i	$0.730444\pi$
$908$	−14.5253	−0.482040
$909$	52.2534	1.73314
$910$	0	0
$911$	−34.8570	−1.15486	−0.577431	−	0.816439i	$-0.695945\pi$
$911$	−34.8570	−1.15486	−0.577431	+	0.816439i	$0.695945\pi$
$912$	2.47000	0.0817898
$913$	−14.5762	−0.482402
$914$	−12.9915	−0.429720
$915$	0	0
$916$	−14.2665	−0.471379
$917$	−17.1927	−0.567754
$918$	−15.3470	−0.506526
$919$	6.14603	0.202739	0.101369	−	0.994849i	$-0.467678\pi$
$919$	6.14603	0.202739	0.101369	+	0.994849i	$0.467678\pi$
$920$	0	0
$921$	−9.23511	−0.304307
$922$	26.5083	0.873006
$923$	10.7371	0.353415
$924$	0.389602	0.0128170
$925$	0	0
$926$	−15.9368	−0.523717
$927$	−54.0055	−1.77377
$928$	−21.5642	−0.707878
$929$	54.1710	1.77729	0.888647	−	0.458592i	$-0.151646\pi$
$929$	54.1710	1.77729	0.888647	+	0.458592i	$0.151646\pi$
$930$	0	0
$931$	−24.5467	−0.804486
$932$	16.5641	0.542575
$933$	5.48773	0.179660
$934$	22.5916	0.739219
$935$	0	0
$936$	8.80113	0.287674
$937$	0.682282	0.0222892	0.0111446	−	0.999938i	$-0.496452\pi$
$937$	0.682282	0.0222892	0.0111446	+	0.999938i	$0.496452\pi$
$938$	0.637211	0.0208057
$939$	0.852934	0.0278345
$940$	0	0
$941$	11.4586	0.373539	0.186770	−	0.982404i	$-0.440198\pi$
$941$	11.4586	0.373539	0.186770	+	0.982404i	$0.440198\pi$
$942$	−5.23566	−0.170587
$943$	21.5190	0.700757
$944$	−15.9973	−0.520667
$945$	0	0
$946$	0.0969895	0.00315340
$947$	3.11253	0.101143	0.0505717	−	0.998720i	$-0.483896\pi$
$947$	3.11253	0.101143	0.0505717	+	0.998720i	$0.483896\pi$
$948$	1.71549	0.0557167
$949$	10.7422	0.348707
$950$	0	0
$951$	−2.71804	−0.0881384
$952$	−28.6830	−0.929620
$953$	26.1414	0.846802	0.423401	−	0.905942i	$-0.360836\pi$
$953$	26.1414	0.846802	0.423401	+	0.905942i	$0.360836\pi$
$954$	38.8545	1.25796
$955$	0	0
$956$	−24.3438	−0.787335
$957$	−1.65931	−0.0536380
$958$	−15.9185	−0.514303
$959$	−14.2483	−0.460101
$960$	0	0
$961$	0.397933	0.0128366
$962$	−4.22522	−0.136227
$963$	41.3559	1.33268
$964$	15.2977	0.492706
$965$	0	0
$966$	1.49841	0.0482104
$967$	38.3726	1.23398	0.616989	−	0.786972i	$-0.288352\pi$
$967$	38.3726	1.23398	0.616989	+	0.786972i	$0.288352\pi$
$968$	−3.05369	−0.0981495
$969$	11.4068	0.366438
$970$	0	0
$971$	−48.2005	−1.54683	−0.773415	−	0.633900i	$-0.781453\pi$
$971$	−48.2005	−1.54683	−0.773415	+	0.633900i	$0.781453\pi$
$972$	7.55692	0.242388
$973$	24.9355	0.799397
$974$	−23.7122	−0.759787
$975$	0	0
$976$	−22.3195	−0.714429
$977$	11.1779	0.357611	0.178806	−	0.983884i	$-0.442777\pi$
$977$	11.1779	0.357611	0.178806	+	0.983884i	$0.442777\pi$
$978$	−1.20847	−0.0386426
$979$	−1.82629	−0.0583685
$980$	0	0
$981$	−10.5420	−0.336582
$982$	−22.4300	−0.715770
$983$	1.54272	0.0492050	0.0246025	−	0.999697i	$-0.492168\pi$
$983$	1.54272	0.0492050	0.0246025	+	0.999697i	$0.492168\pi$
$984$	7.28504	0.232239
$985$	0	0
$986$	36.7281	1.16966
$987$	−2.34961	−0.0747888
$988$	−4.01390	−0.127699
$989$	−0.281296	−0.00894468
$990$	0	0
$991$	24.4899	0.777948	0.388974	−	0.921249i	$-0.372830\pi$
$991$	24.4899	0.777948	0.388974	+	0.921249i	$0.372830\pi$
$992$	25.0013	0.793793
$993$	7.28903	0.231310
$994$	−15.1314	−0.479939
$995$	0	0
$996$	4.30288	0.136342
$997$	16.3909	0.519104	0.259552	−	0.965729i	$-0.416425\pi$
$997$	16.3909	0.519104	0.259552	+	0.965729i	$0.416425\pi$
$998$	−6.37156	−0.201688
$999$	−7.99109	−0.252827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3575.2.a.t.1.6		9
5.4	even	2		715.2.a.i.1.4	✓	9
15.14	odd	2		6435.2.a.bq.1.6		9
55.54	odd	2		7865.2.a.w.1.6		9
65.64	even	2		9295.2.a.t.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
715.2.a.i.1.4	✓	9	5.4	even	2
3575.2.a.t.1.6		9	1.1	even	1	trivial
6435.2.a.bq.1.6		9	15.14	odd	2
7865.2.a.w.1.6		9	55.54	odd	2
9295.2.a.t.1.6		9	65.64	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3575 = 5^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3575.a (trivial)

\(f(q)\)	\(=\)	\(q+1.06779 q^{2} -0.343328 q^{3} -0.859816 q^{4} -0.366604 q^{6} -1.31980 q^{7} -3.05369 q^{8} -2.88213 q^{9} -1.00000 q^{11} +0.295199 q^{12} +1.00000 q^{13} -1.40927 q^{14} -1.54108 q^{16} -7.11691 q^{17} -3.07752 q^{18} +4.66833 q^{19} +0.453123 q^{21} -1.06779 q^{22} +3.09689 q^{23} +1.04842 q^{24} +1.06779 q^{26} +2.01950 q^{27} +1.13478 q^{28} -4.83303 q^{29} +5.60339 q^{31} +4.46183 q^{32} +0.343328 q^{33} -7.59940 q^{34} +2.47810 q^{36} -3.95696 q^{37} +4.98481 q^{38} -0.343328 q^{39} +6.94860 q^{41} +0.483842 q^{42} -0.0908316 q^{43} +0.859816 q^{44} +3.30684 q^{46} -5.18536 q^{47} +0.529097 q^{48} -5.25814 q^{49} +2.44344 q^{51} -0.859816 q^{52} -12.6253 q^{53} +2.15641 q^{54} +4.03025 q^{56} -1.60277 q^{57} -5.16068 q^{58} +10.3805 q^{59} +14.4830 q^{61} +5.98326 q^{62} +3.80382 q^{63} +7.84648 q^{64} +0.366604 q^{66} -0.452157 q^{67} +6.11924 q^{68} -1.06325 q^{69} +10.7371 q^{71} +8.80113 q^{72} +10.7422 q^{73} -4.22522 q^{74} -4.01390 q^{76} +1.31980 q^{77} -0.366604 q^{78} +5.81132 q^{79} +7.95303 q^{81} +7.41967 q^{82} +14.5762 q^{83} -0.389602 q^{84} -0.0969895 q^{86} +1.65931 q^{87} +3.05369 q^{88} +1.82629 q^{89} -1.31980 q^{91} -2.66276 q^{92} -1.92380 q^{93} -5.53690 q^{94} -1.53187 q^{96} +5.98946 q^{97} -5.61461 q^{98} +2.88213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} - 2 q^{3} + 15 q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - 9 q^{11} - 6 q^{12} + 9 q^{13} + 16 q^{14} + 15 q^{16} - 13 q^{17} - 3 q^{18} - 3 q^{19} + 14 q^{21} + q^{22} + 2 q^{24} - q^{26}+ \cdots - 23 q^{99}+O(q^{100}) \) 9 * q - q^2 - 2 * q^3 + 15 * q^4 - q^6 + 4 * q^7 - 3 * q^8 + 23 * q^9 - 9 * q^11 - 6 * q^12 + 9 * q^13 + 16 * q^14 + 15 * q^16 - 13 * q^17 - 3 * q^18 - 3 * q^19 + 14 * q^21 + q^22 + 2 * q^24 - q^26 - 8 * q^27 + 27 * q^28 + 16 * q^29 + 12 * q^31 + 18 * q^32 + 2 * q^33 + 14 * q^34 - 4 * q^36 - q^37 + 9 * q^38 - 2 * q^39 + 5 * q^41 + 11 * q^42 + 3 * q^43 - 15 * q^44 + 4 * q^46 - 27 * q^47 - q^48 + 31 * q^49 + 15 * q^52 - 34 * q^53 - 49 * q^54 + 73 * q^56 + 16 * q^57 + 38 * q^58 - 4 * q^59 + 30 * q^61 + 8 * q^62 + 30 * q^63 + 3 * q^64 + q^66 + 11 * q^67 - 52 * q^68 + 52 * q^69 - 4 * q^71 + 8 * q^72 + 12 * q^73 - 16 * q^74 - 16 * q^76 - 4 * q^77 - q^78 - 16 * q^79 + 61 * q^81 + 42 * q^82 - 34 * q^83 - 79 * q^84 + 52 * q^86 + 64 * q^87 + 3 * q^88 - 10 * q^89 + 4 * q^91 - 7 * q^92 + 22 * q^93 - 18 * q^94 + 15 * q^97 - 24 * q^98 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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