LMFDB - Embedded newform 1925.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,2,5,0,4,3,9,-1,0,-3,12,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.67513 q^{2} +2.48119 q^{3} +5.15633 q^{4} +6.63752 q^{6} +1.00000 q^{7} +8.44358 q^{8} +3.15633 q^{9} -1.00000 q^{11} +12.7938 q^{12} -5.83146 q^{13} +2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} +8.44358 q^{18} -1.35026 q^{19} +2.48119 q^{21} -2.67513 q^{22} +3.19394 q^{23} +20.9502 q^{24} -15.5999 q^{26} +0.387873 q^{27} +5.15633 q^{28} -3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -2.48119 q^{33} -14.5623 q^{34} +16.2750 q^{36} +8.54420 q^{37} -3.61213 q^{38} -14.4690 q^{39} -5.02539 q^{41} +6.63752 q^{42} -5.89446 q^{43} -5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +30.4568 q^{48} +1.00000 q^{49} -13.5066 q^{51} -30.0689 q^{52} +0.231548 q^{53} +1.03761 q^{54} +8.44358 q^{56} -3.35026 q^{57} -9.66291 q^{58} +13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +3.15633 q^{63} +18.1187 q^{64} -6.63752 q^{66} +10.8568 q^{67} -28.0689 q^{68} +7.92478 q^{69} -15.5369 q^{71} +26.6507 q^{72} +11.3684 q^{73} +22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} -38.7064 q^{78} +1.96968 q^{79} -8.50659 q^{81} -13.4436 q^{82} -10.6253 q^{83} +12.7938 q^{84} -15.7685 q^{86} -8.96239 q^{87} -8.44358 q^{88} +7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} -13.1187 q^{93} +31.6507 q^{94} +39.5755 q^{96} +0.836381 q^{97} +2.67513 q^{98} -3.15633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} - 3 q^{11} + 12 q^{12} - 2 q^{13} + 3 q^{14} + 5 q^{16} + 9 q^{18} + 6 q^{19} + 2 q^{21} - 3 q^{22} + 10 q^{23} + 26 q^{24} - 20 q^{26}+ \cdots + q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 4 * q^6 + 3 * q^7 + 9 * q^8 - q^9 - 3 * q^11 + 12 * q^12 - 2 * q^13 + 3 * q^14 + 5 * q^16 + 9 * q^18 + 6 * q^19 + 2 * q^21 - 3 * q^22 + 10 * q^23 + 26 * q^24 - 20 * q^26 + 2 * q^27 + 5 * q^28 - 10 * q^29 - 10 * q^31 + 11 * q^32 - 2 * q^33 - 6 * q^34 + 17 * q^36 + 16 * q^37 - 10 * q^38 - 12 * q^39 + 4 * q^42 + 2 * q^43 - 5 * q^44 + 16 * q^46 + 20 * q^47 + 34 * q^48 + 3 * q^49 - 20 * q^51 - 32 * q^52 + 12 * q^53 + 14 * q^54 + 9 * q^56 + 2 * q^58 + 14 * q^59 + 10 * q^61 - 6 * q^62 - q^63 + 33 * q^64 - 4 * q^66 + 2 * q^67 - 26 * q^68 + 2 * q^69 - 24 * q^71 + 23 * q^72 - 4 * q^73 + 38 * q^74 - 10 * q^76 - 3 * q^77 - 42 * q^78 + 8 * q^79 - 5 * q^81 - 24 * q^82 + 10 * q^83 + 12 * q^84 - 36 * q^86 - 16 * q^87 - 9 * q^88 + 20 * q^89 - 2 * q^91 + 18 * q^92 - 18 * q^93 + 38 * q^94 + 40 * q^96 + 3 * q^98 + 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 $

$2$	2.67513	1.89160	0.945802	−	0.324745i	$-0.105279\pi$
$2$	2.67513	1.89160	0.945802	+	0.324745i	$0.105279\pi$
$3$	2.48119	1.43252	0.716259	−	0.697834i	$-0.245853\pi$
$3$	2.48119	1.43252	0.716259	+	0.697834i	$0.245853\pi$
$4$	5.15633	2.57816
$5$	0	0
$5$	0	0
$6$	6.63752	2.70976
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	8.44358	2.98526
$9$	3.15633	1.05211
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	12.7938	3.69326
$13$	−5.83146	−1.61735	−0.808677	−	0.588252i	$-0.799816\pi$
$13$	−5.83146	−1.61735	−0.808677	+	0.588252i	$0.799816\pi$
$14$	2.67513	0.714959
$15$	0	0
$16$	12.2750	3.06876
$17$	−5.44358	−1.32026	−0.660131	−	0.751150i	$-0.729499\pi$
$17$	−5.44358	−1.32026	−0.660131	+	0.751150i	$0.729499\pi$
$18$	8.44358	1.99017
$19$	−1.35026	−0.309771	−0.154886	−	0.987932i	$-0.549501\pi$
$19$	−1.35026	−0.309771	−0.154886	+	0.987932i	$0.549501\pi$
$20$	0	0
$21$	2.48119	0.541441
$22$	−2.67513	−0.570340
$23$	3.19394	0.665982	0.332991	−	0.942930i	$-0.391942\pi$
$23$	3.19394	0.665982	0.332991	+	0.942930i	$0.391942\pi$
$24$	20.9502	4.27644
$25$	0	0
$26$	−15.5999	−3.05939
$27$	0.387873	0.0746462
$28$	5.15633	0.974454
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	0	0
$31$	−5.28726	−0.949620	−0.474810	−	0.880088i	$-0.657483\pi$
$31$	−5.28726	−0.949620	−0.474810	+	0.880088i	$0.657483\pi$
$32$	15.9502	2.81962
$33$	−2.48119	−0.431920
$34$	−14.5623	−2.49741
$35$	0	0
$36$	16.2750	2.71251
$37$	8.54420	1.40466	0.702329	−	0.711853i	$-0.252144\pi$
$37$	8.54420	1.40466	0.702329	+	0.711853i	$0.252144\pi$
$38$	−3.61213	−0.585964
$39$	−14.4690	−2.31689
$40$	0	0
$41$	−5.02539	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$41$	−5.02539	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$42$	6.63752	1.02419
$43$	−5.89446	−0.898897	−0.449448	−	0.893306i	$-0.648379\pi$
$43$	−5.89446	−0.898897	−0.449448	+	0.893306i	$0.648379\pi$
$44$	−5.15633	−0.777345
$45$	0	0
$46$	8.54420	1.25977
$47$	11.8315	1.72580	0.862898	−	0.505379i	$-0.168647\pi$
$47$	11.8315	1.72580	0.862898	+	0.505379i	$0.168647\pi$
$48$	30.4568	4.39605
$49$	1.00000	0.142857
$50$	0	0
$51$	−13.5066	−1.89130
$52$	−30.0689	−4.16980
$53$	0.231548	0.0318056	0.0159028	−	0.999874i	$-0.494938\pi$
$53$	0.231548	0.0318056	0.0159028	+	0.999874i	$0.494938\pi$
$54$	1.03761	0.141201
$55$	0	0
$56$	8.44358	1.12832
$57$	−3.35026	−0.443753
$58$	−9.66291	−1.26880
$59$	13.5999	1.77056	0.885279	−	0.465061i	$-0.153968\pi$
$59$	13.5999	1.77056	0.885279	+	0.465061i	$0.153968\pi$
$60$	0	0
$61$	−1.41327	−0.180950	−0.0904751	−	0.995899i	$-0.528839\pi$
$61$	−1.41327	−0.180950	−0.0904751	+	0.995899i	$0.528839\pi$
$62$	−14.1441	−1.79630
$63$	3.15633	0.397660
$64$	18.1187	2.26484
$65$	0	0
$66$	−6.63752	−0.817022
$67$	10.8568	1.32638	0.663188	−	0.748453i	$-0.269203\pi$
$67$	10.8568	1.32638	0.663188	+	0.748453i	$0.269203\pi$
$68$	−28.0689	−3.40385
$69$	7.92478	0.954031
$70$	0	0
$71$	−15.5369	−1.84389	−0.921946	−	0.387319i	$-0.873401\pi$
$71$	−15.5369	−1.84389	−0.921946	+	0.387319i	$0.873401\pi$
$72$	26.6507	3.14081
$73$	11.3684	1.33057	0.665283	−	0.746591i	$-0.268311\pi$
$73$	11.3684	1.33057	0.665283	+	0.746591i	$0.268311\pi$
$74$	22.8568	2.65705
$75$	0	0
$76$	−6.96239	−0.798641
$77$	−1.00000	−0.113961
$78$	−38.7064	−4.38264
$79$	1.96968	0.221607	0.110803	−	0.993842i	$-0.464658\pi$
$79$	1.96968	0.221607	0.110803	+	0.993842i	$0.464658\pi$
$80$	0	0
$81$	−8.50659	−0.945176
$82$	−13.4436	−1.48460
$83$	−10.6253	−1.16628	−0.583139	−	0.812372i	$-0.698176\pi$
$83$	−10.6253	−1.16628	−0.583139	+	0.812372i	$0.698176\pi$
$84$	12.7938	1.39592
$85$	0	0
$86$	−15.7685	−1.70036
$87$	−8.96239	−0.960869
$88$	−8.44358	−0.900089
$89$	7.22425	0.765769	0.382885	−	0.923796i	$-0.374931\pi$
$89$	7.22425	0.765769	0.382885	+	0.923796i	$0.374931\pi$
$90$	0	0
$91$	−5.83146	−0.611303
$92$	16.4690	1.71701
$93$	−13.1187	−1.36035
$94$	31.6507	3.26452
$95$	0	0
$96$	39.5755	4.03915
$97$	0.836381	0.0849216	0.0424608	−	0.999098i	$-0.486480\pi$
$97$	0.836381	0.0849216	0.0424608	+	0.999098i	$0.486480\pi$
$98$	2.67513	0.270229
$99$	−3.15633	−0.317223
$100$	0	0
$101$	7.41327	0.737648	0.368824	−	0.929499i	$-0.379761\pi$
$101$	7.41327	0.737648	0.368824	+	0.929499i	$0.379761\pi$
$102$	−36.1319	−3.57759
$103$	−4.21933	−0.415743	−0.207871	−	0.978156i	$-0.566654\pi$
$103$	−4.21933	−0.415743	−0.207871	+	0.978156i	$0.566654\pi$
$104$	−49.2384	−4.82822
$105$	0	0
$106$	0.619421	0.0601635
$107$	11.5369	1.11531	0.557657	−	0.830071i	$-0.311700\pi$
$107$	11.5369	1.11531	0.557657	+	0.830071i	$0.311700\pi$
$108$	2.00000	0.192450
$109$	−2.18664	−0.209442	−0.104721	−	0.994502i	$-0.533395\pi$
$109$	−2.18664	−0.209442	−0.104721	+	0.994502i	$0.533395\pi$
$110$	0	0
$111$	21.1998	2.01220
$112$	12.2750	1.15988
$113$	9.35026	0.879599	0.439799	−	0.898096i	$-0.355050\pi$
$113$	9.35026	0.879599	0.439799	+	0.898096i	$0.355050\pi$
$114$	−8.96239	−0.839405
$115$	0	0
$116$	−18.6253	−1.72932
$117$	−18.4060	−1.70163
$118$	36.3815	3.34919
$119$	−5.44358	−0.499012
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.78067	−0.342286
$123$	−12.4690	−1.12429
$124$	−27.2628	−2.44827
$125$	0	0
$126$	8.44358	0.752214
$127$	16.9624	1.50517	0.752584	−	0.658496i	$-0.228807\pi$
$127$	16.9624	1.50517	0.752584	+	0.658496i	$0.228807\pi$
$128$	16.5696	1.46456
$129$	−14.6253	−1.28769
$130$	0	0
$131$	−9.92478	−0.867132	−0.433566	−	0.901122i	$-0.642745\pi$
$131$	−9.92478	−0.867132	−0.433566	+	0.901122i	$0.642745\pi$
$132$	−12.7938	−1.11356
$133$	−1.35026	−0.117083
$134$	29.0435	2.50898
$135$	0	0
$136$	−45.9633	−3.94132
$137$	10.9927	0.939170	0.469585	−	0.882887i	$-0.344404\pi$
$137$	10.9927	0.939170	0.469585	+	0.882887i	$0.344404\pi$
$138$	21.1998	1.80465
$139$	6.88717	0.584162	0.292081	−	0.956394i	$-0.405652\pi$
$139$	6.88717	0.584162	0.292081	+	0.956394i	$0.405652\pi$
$140$	0	0
$141$	29.3561	2.47223
$142$	−41.5633	−3.48791
$143$	5.83146	0.487651
$144$	38.7440	3.22867
$145$	0	0
$146$	30.4119	2.51690
$147$	2.48119	0.204645
$148$	44.0567	3.62144
$149$	22.8119	1.86883	0.934414	−	0.356190i	$-0.115924\pi$
$149$	22.8119	1.86883	0.934414	+	0.356190i	$0.115924\pi$
$150$	0	0
$151$	−3.24472	−0.264052	−0.132026	−	0.991246i	$-0.542148\pi$
$151$	−3.24472	−0.264052	−0.132026	+	0.991246i	$0.542148\pi$
$152$	−11.4010	−0.924747
$153$	−17.1817	−1.38906
$154$	−2.67513	−0.215568
$155$	0	0
$156$	−74.6067	−5.97332
$157$	5.42548	0.433001	0.216500	−	0.976283i	$-0.430536\pi$
$157$	5.42548	0.433001	0.216500	+	0.976283i	$0.430536\pi$
$158$	5.26916	0.419192
$159$	0.574515	0.0455620
$160$	0	0
$161$	3.19394	0.251717
$162$	−22.7562	−1.78790
$163$	−3.38058	−0.264787	−0.132394	−	0.991197i	$-0.542266\pi$
$163$	−3.38058	−0.264787	−0.132394	+	0.991197i	$0.542266\pi$
$164$	−25.9126	−2.02343
$165$	0	0
$166$	−28.4241	−2.20614
$167$	−11.2750	−0.872489	−0.436244	−	0.899828i	$-0.643692\pi$
$167$	−11.2750	−0.872489	−0.436244	+	0.899828i	$0.643692\pi$
$168$	20.9502	1.61634
$169$	21.0059	1.61584
$170$	0	0
$171$	−4.26187	−0.325913
$172$	−30.3938	−2.31750
$173$	8.98049	0.682774	0.341387	−	0.939923i	$-0.389103\pi$
$173$	8.98049	0.682774	0.341387	+	0.939923i	$0.389103\pi$
$174$	−23.9756	−1.81758
$175$	0	0
$176$	−12.2750	−0.925266
$177$	33.7440	2.53636
$178$	19.3258	1.44853
$179$	−26.2374	−1.96108	−0.980539	−	0.196326i	$-0.937099\pi$
$179$	−26.2374	−1.96108	−0.980539	+	0.196326i	$0.937099\pi$
$180$	0	0
$181$	−11.1998	−0.832476	−0.416238	−	0.909256i	$-0.636652\pi$
$181$	−11.1998	−0.832476	−0.416238	+	0.909256i	$0.636652\pi$
$182$	−15.5999	−1.15634
$183$	−3.50659	−0.259214
$184$	26.9683	1.98813
$185$	0	0
$186$	−35.0943	−2.57324
$187$	5.44358	0.398074
$188$	61.0068	4.44938
$189$	0.387873	0.0282136
$190$	0	0
$191$	11.1998	0.810390	0.405195	−	0.914230i	$-0.367204\pi$
$191$	11.1998	0.810390	0.405195	+	0.914230i	$0.367204\pi$
$192$	44.9560	3.24442
$193$	−0.604833	−0.0435368	−0.0217684	−	0.999763i	$-0.506930\pi$
$193$	−0.604833	−0.0435368	−0.0217684	+	0.999763i	$0.506930\pi$
$194$	2.23743	0.160638
$195$	0	0
$196$	5.15633	0.368309
$197$	−15.3054	−1.09046	−0.545231	−	0.838286i	$-0.683558\pi$
$197$	−15.3054	−1.09046	−0.545231	+	0.838286i	$0.683558\pi$
$198$	−8.44358	−0.600059
$199$	−12.5623	−0.890518	−0.445259	−	0.895402i	$-0.646888\pi$
$199$	−12.5623	−0.890518	−0.445259	+	0.895402i	$0.646888\pi$
$200$	0	0
$201$	26.9380	1.90006
$202$	19.8315	1.39534
$203$	−3.61213	−0.253522
$204$	−69.6444	−4.87608
$205$	0	0
$206$	−11.2873	−0.786421
$207$	10.0811	0.700685
$208$	−71.5814	−4.96327
$209$	1.35026	0.0933996
$210$	0	0
$211$	−4.43866	−0.305570	−0.152785	−	0.988259i	$-0.548824\pi$
$211$	−4.43866	−0.305570	−0.152785	+	0.988259i	$0.548824\pi$
$212$	1.19394	0.0819999
$213$	−38.5501	−2.64141
$214$	30.8627	2.10973
$215$	0	0
$216$	3.27504	0.222838
$217$	−5.28726	−0.358922
$218$	−5.84955	−0.396182
$219$	28.2071	1.90606
$220$	0	0
$221$	31.7440	2.13533
$222$	56.7123	3.80628
$223$	7.78067	0.521032	0.260516	−	0.965469i	$-0.416107\pi$
$223$	7.78067	0.521032	0.260516	+	0.965469i	$0.416107\pi$
$224$	15.9502	1.06572
$225$	0	0
$226$	25.0132	1.66385
$227$	10.4485	0.693492	0.346746	−	0.937959i	$-0.387287\pi$
$227$	10.4485	0.693492	0.346746	+	0.937959i	$0.387287\pi$
$228$	−17.2750	−1.14407
$229$	−29.4518	−1.94623	−0.973116	−	0.230316i	$-0.926024\pi$
$229$	−29.4518	−1.94623	−0.973116	+	0.230316i	$0.926024\pi$
$230$	0	0
$231$	−2.48119	−0.163251
$232$	−30.4993	−2.00238
$233$	8.73084	0.571976	0.285988	−	0.958233i	$-0.407678\pi$
$233$	8.73084	0.571976	0.285988	+	0.958233i	$0.407678\pi$
$234$	−49.2384	−3.21881
$235$	0	0
$236$	70.1255	4.56478
$237$	4.88717	0.317456
$238$	−14.5623	−0.943933
$239$	−21.2144	−1.37225	−0.686123	−	0.727486i	$-0.740689\pi$
$239$	−21.2144	−1.37225	−0.686123	+	0.727486i	$0.740689\pi$
$240$	0	0
$241$	−9.33804	−0.601516	−0.300758	−	0.953700i	$-0.597240\pi$
$241$	−9.33804	−0.601516	−0.300758	+	0.953700i	$0.597240\pi$
$242$	2.67513	0.171964
$243$	−22.2701	−1.42863
$244$	−7.28726	−0.466519
$245$	0	0
$246$	−33.3561	−2.12671
$247$	7.87399	0.501010
$248$	−44.6434	−2.83486
$249$	−26.3634	−1.67071
$250$	0	0
$251$	1.87636	0.118435	0.0592174	−	0.998245i	$-0.481139\pi$
$251$	1.87636	0.118435	0.0592174	+	0.998245i	$0.481139\pi$
$252$	16.2750	1.02523
$253$	−3.19394	−0.200801
$254$	45.3766	2.84718
$255$	0	0
$256$	8.08840	0.505525
$257$	−27.1392	−1.69290	−0.846448	−	0.532472i	$-0.821263\pi$
$257$	−27.1392	−1.69290	−0.846448	+	0.532472i	$0.821263\pi$
$258$	−39.1246	−2.43579
$259$	8.54420	0.530911
$260$	0	0
$261$	−11.4010	−0.705707
$262$	−26.5501	−1.64027
$263$	−12.8119	−0.790018	−0.395009	−	0.918677i	$-0.629259\pi$
$263$	−12.8119	−0.790018	−0.395009	+	0.918677i	$0.629259\pi$
$264$	−20.9502	−1.28939
$265$	0	0
$266$	−3.61213	−0.221474
$267$	17.9248	1.09698
$268$	55.9814	3.41961
$269$	−6.26187	−0.381793	−0.190896	−	0.981610i	$-0.561139\pi$
$269$	−6.26187	−0.381793	−0.190896	+	0.981610i	$0.561139\pi$
$270$	0	0
$271$	−5.73813	−0.348567	−0.174283	−	0.984696i	$-0.555761\pi$
$271$	−5.73813	−0.348567	−0.174283	+	0.984696i	$0.555761\pi$
$272$	−66.8202	−4.05157
$273$	−14.4690	−0.875702
$274$	29.4069	1.77654
$275$	0	0
$276$	40.8627	2.45965
$277$	8.35756	0.502157	0.251078	−	0.967967i	$-0.419215\pi$
$277$	8.35756	0.502157	0.251078	+	0.967967i	$0.419215\pi$
$278$	18.4241	1.10500
$279$	−16.6883	−0.999103
$280$	0	0
$281$	−8.44851	−0.503996	−0.251998	−	0.967728i	$-0.581088\pi$
$281$	−8.44851	−0.503996	−0.251998	+	0.967728i	$0.581088\pi$
$282$	78.5315	4.67648
$283$	−0.836381	−0.0497177	−0.0248588	−	0.999691i	$-0.507914\pi$
$283$	−0.836381	−0.0497177	−0.0248588	+	0.999691i	$0.507914\pi$
$284$	−80.1133	−4.75385
$285$	0	0
$286$	15.5999	0.922442
$287$	−5.02539	−0.296640
$288$	50.3439	2.96654
$289$	12.6326	0.743094
$290$	0	0
$291$	2.07522	0.121652
$292$	58.6190	3.43042
$293$	2.71862	0.158824	0.0794118	−	0.996842i	$-0.474696\pi$
$293$	2.71862	0.158824	0.0794118	+	0.996842i	$0.474696\pi$
$294$	6.63752	0.387108
$295$	0	0
$296$	72.1436	4.19326
$297$	−0.387873	−0.0225067
$298$	61.0249	3.53508
$299$	−18.6253	−1.07713
$300$	0	0
$301$	−5.89446	−0.339751
$302$	−8.68006	−0.499481
$303$	18.3938	1.05669
$304$	−16.5745	−0.950614
$305$	0	0
$306$	−45.9633	−2.62755
$307$	8.36344	0.477326	0.238663	−	0.971102i	$-0.423291\pi$
$307$	8.36344	0.477326	0.238663	+	0.971102i	$0.423291\pi$
$308$	−5.15633	−0.293809
$309$	−10.4690	−0.595559
$310$	0	0
$311$	4.43629	0.251559	0.125779	−	0.992058i	$-0.459857\pi$
$311$	4.43629	0.251559	0.125779	+	0.992058i	$0.459857\pi$
$312$	−122.170	−6.91651
$313$	−29.7889	−1.68377	−0.841885	−	0.539658i	$-0.818554\pi$
$313$	−29.7889	−1.68377	−0.841885	+	0.539658i	$0.818554\pi$
$314$	14.5139	0.819066
$315$	0	0
$316$	10.1563	0.571338
$317$	−15.4010	−0.865009	−0.432504	−	0.901632i	$-0.642370\pi$
$317$	−15.4010	−0.865009	−0.432504	+	0.901632i	$0.642370\pi$
$318$	1.53690	0.0861853
$319$	3.61213	0.202240
$320$	0	0
$321$	28.6253	1.59771
$322$	8.54420	0.476150
$323$	7.35026	0.408980
$324$	−43.8627	−2.43682
$325$	0	0
$326$	−9.04349	−0.500873
$327$	−5.42548	−0.300030
$328$	−42.4323	−2.34293
$329$	11.8315	0.652289
$330$	0	0
$331$	6.26187	0.344183	0.172092	−	0.985081i	$-0.444947\pi$
$331$	6.26187	0.344183	0.172092	+	0.985081i	$0.444947\pi$
$332$	−54.7875	−3.00685
$333$	26.9683	1.47785
$334$	−30.1622	−1.65040
$335$	0	0
$336$	30.4568	1.66155
$337$	15.8700	0.864495	0.432248	−	0.901755i	$-0.357721\pi$
$337$	15.8700	0.864495	0.432248	+	0.901755i	$0.357721\pi$
$338$	56.1935	3.05652
$339$	23.1998	1.26004
$340$	0	0
$341$	5.28726	0.286321
$342$	−11.4010	−0.616498
$343$	1.00000	0.0539949
$344$	−49.7704	−2.68344
$345$	0	0
$346$	24.0240	1.29154
$347$	−6.79147	−0.364585	−0.182293	−	0.983244i	$-0.558352\pi$
$347$	−6.79147	−0.364585	−0.182293	+	0.983244i	$0.558352\pi$
$348$	−46.2130	−2.47728
$349$	26.7489	1.43184	0.715919	−	0.698183i	$-0.246008\pi$
$349$	26.7489	1.43184	0.715919	+	0.698183i	$0.246008\pi$
$350$	0	0
$351$	−2.26187	−0.120729
$352$	−15.9502	−0.850147
$353$	16.8627	0.897512	0.448756	−	0.893654i	$-0.351867\pi$
$353$	16.8627	0.897512	0.448756	+	0.893654i	$0.351867\pi$
$354$	90.2697	4.79778
$355$	0	0
$356$	37.2506	1.97428
$357$	−13.5066	−0.714844
$358$	−70.1886	−3.70958
$359$	−3.79289	−0.200181	−0.100091	−	0.994978i	$-0.531913\pi$
$359$	−3.79289	−0.200181	−0.100091	+	0.994978i	$0.531913\pi$
$360$	0	0
$361$	−17.1768	−0.904042
$362$	−29.9610	−1.57471
$363$	2.48119	0.130229
$364$	−30.0689	−1.57604
$365$	0	0
$366$	−9.38058	−0.490331
$367$	6.36977	0.332500	0.166250	−	0.986084i	$-0.446834\pi$
$367$	6.36977	0.332500	0.166250	+	0.986084i	$0.446834\pi$
$368$	39.2057	2.04374
$369$	−15.8618	−0.825731
$370$	0	0
$371$	0.231548	0.0120214
$372$	−67.6444	−3.50720
$373$	21.3317	1.10451	0.552257	−	0.833674i	$-0.313767\pi$
$373$	21.3317	1.10451	0.552257	+	0.833674i	$0.313767\pi$
$374$	14.5623	0.752998
$375$	0	0
$376$	99.8999	5.15194
$377$	21.0640	1.08485
$378$	1.03761	0.0533690
$379$	24.7875	1.27325	0.636624	−	0.771174i	$-0.280330\pi$
$379$	24.7875	1.27325	0.636624	+	0.771174i	$0.280330\pi$
$380$	0	0
$381$	42.0870	2.15618
$382$	29.9610	1.53294
$383$	5.45817	0.278900	0.139450	−	0.990229i	$-0.455467\pi$
$383$	5.45817	0.278900	0.139450	+	0.990229i	$0.455467\pi$
$384$	41.1124	2.09801
$385$	0	0
$386$	−1.61801	−0.0823544
$387$	−18.6048	−0.945737
$388$	4.31265	0.218942
$389$	−13.7235	−0.695811	−0.347906	−	0.937530i	$-0.613107\pi$
$389$	−13.7235	−0.695811	−0.347906	+	0.937530i	$0.613107\pi$
$390$	0	0
$391$	−17.3865	−0.879271
$392$	8.44358	0.426465
$393$	−24.6253	−1.24218
$394$	−40.9438	−2.06272
$395$	0	0
$396$	−16.2750	−0.817851
$397$	−2.11142	−0.105969	−0.0529846	−	0.998595i	$-0.516873\pi$
$397$	−2.11142	−0.105969	−0.0529846	+	0.998595i	$0.516873\pi$
$398$	−33.6058	−1.68451
$399$	−3.35026	−0.167723
$400$	0	0
$401$	19.1490	0.956257	0.478128	−	0.878290i	$-0.341315\pi$
$401$	19.1490	0.956257	0.478128	+	0.878290i	$0.341315\pi$
$402$	72.0625	3.59415
$403$	30.8324	1.53587
$404$	38.2252	1.90178
$405$	0	0
$406$	−9.66291	−0.479562
$407$	−8.54420	−0.423520
$408$	−114.044	−5.64602
$409$	−18.6883	−0.924077	−0.462039	−	0.886860i	$-0.652882\pi$
$409$	−18.6883	−0.924077	−0.462039	+	0.886860i	$0.652882\pi$
$410$	0	0
$411$	27.2750	1.34538
$412$	−21.7562	−1.07185
$413$	13.5999	0.669208
$414$	26.9683	1.32542
$415$	0	0
$416$	−93.0127	−4.56032
$417$	17.0884	0.836822
$418$	3.61213	0.176675
$419$	−0.773377	−0.0377819	−0.0188910	−	0.999822i	$-0.506014\pi$
$419$	−0.773377	−0.0377819	−0.0188910	+	0.999822i	$0.506014\pi$
$420$	0	0
$421$	−10.5198	−0.512702	−0.256351	−	0.966584i	$-0.582520\pi$
$421$	−10.5198	−0.512702	−0.256351	+	0.966584i	$0.582520\pi$
$422$	−11.8740	−0.578017
$423$	37.3439	1.81572
$424$	1.95509	0.0949478
$425$	0	0
$426$	−103.127	−4.99650
$427$	−1.41327	−0.0683927
$428$	59.4880	2.87546
$429$	14.4690	0.698569
$430$	0	0
$431$	−24.7308	−1.19124	−0.595621	−	0.803265i	$-0.703094\pi$
$431$	−24.7308	−1.19124	−0.595621	+	0.803265i	$0.703094\pi$
$432$	4.76116	0.229071
$433$	18.5599	0.891933	0.445967	−	0.895050i	$-0.352860\pi$
$433$	18.5599	0.891933	0.445967	+	0.895050i	$0.352860\pi$
$434$	−14.1441	−0.678939
$435$	0	0
$436$	−11.2750	−0.539976
$437$	−4.31265	−0.206302
$438$	75.4577	3.60551
$439$	−1.42548	−0.0680347	−0.0340173	−	0.999421i	$-0.510830\pi$
$439$	−1.42548	−0.0680347	−0.0340173	+	0.999421i	$0.510830\pi$
$440$	0	0
$441$	3.15633	0.150301
$442$	84.9194	4.03920
$443$	−40.1925	−1.90960	−0.954802	−	0.297242i	$-0.903933\pi$
$443$	−40.1925	−1.90960	−0.954802	+	0.297242i	$0.903933\pi$
$444$	109.313	5.18777
$445$	0	0
$446$	20.8143	0.985586
$447$	56.6009	2.67713
$448$	18.1187	0.856029
$449$	12.6556	0.597256	0.298628	−	0.954370i	$-0.403471\pi$
$449$	12.6556	0.597256	0.298628	+	0.954370i	$0.403471\pi$
$450$	0	0
$451$	5.02539	0.236636
$452$	48.2130	2.26775
$453$	−8.05079	−0.378259
$454$	27.9511	1.31181
$455$	0	0
$456$	−28.2882	−1.32472
$457$	0.544198	0.0254565	0.0127283	−	0.999919i	$-0.495948\pi$
$457$	0.544198	0.0254565	0.0127283	+	0.999919i	$0.495948\pi$
$458$	−78.7875	−3.68150
$459$	−2.11142	−0.0985526
$460$	0	0
$461$	−11.5755	−0.539123	−0.269562	−	0.962983i	$-0.586879\pi$
$461$	−11.5755	−0.539123	−0.269562	+	0.962983i	$0.586879\pi$
$462$	−6.63752	−0.308805
$463$	−23.7948	−1.10584	−0.552919	−	0.833235i	$-0.686486\pi$
$463$	−23.7948	−1.10584	−0.552919	+	0.833235i	$0.686486\pi$
$464$	−44.3390	−2.05839
$465$	0	0
$466$	23.3561	1.08195
$467$	2.66784	0.123453	0.0617264	−	0.998093i	$-0.480339\pi$
$467$	2.66784	0.123453	0.0617264	+	0.998093i	$0.480339\pi$
$468$	−94.9072	−4.38709
$469$	10.8568	0.501323
$470$	0	0
$471$	13.4617	0.620282
$472$	114.832	5.28557
$473$	5.89446	0.271028
$474$	13.0738	0.600500
$475$	0	0
$476$	−28.0689	−1.28654
$477$	0.730841	0.0334629
$478$	−56.7513	−2.59574
$479$	−10.7104	−0.489369	−0.244685	−	0.969603i	$-0.578684\pi$
$479$	−10.7104	−0.489369	−0.244685	+	0.969603i	$0.578684\pi$
$480$	0	0
$481$	−49.8251	−2.27183
$482$	−24.9805	−1.13783
$483$	7.92478	0.360590
$484$	5.15633	0.234378
$485$	0	0
$486$	−59.5755	−2.70240
$487$	−17.4314	−0.789891	−0.394945	−	0.918705i	$-0.629236\pi$
$487$	−17.4314	−0.789891	−0.394945	+	0.918705i	$0.629236\pi$
$488$	−11.9330	−0.540183
$489$	−8.38787	−0.379313
$490$	0	0
$491$	−28.3693	−1.28029	−0.640145	−	0.768254i	$-0.721126\pi$
$491$	−28.3693	−1.28029	−0.640145	+	0.768254i	$0.721126\pi$
$492$	−64.2941	−2.89860
$493$	19.6629	0.885573
$494$	21.0640	0.947712
$495$	0	0
$496$	−64.9013	−2.91415
$497$	−15.5369	−0.696925
$498$	−70.5256	−3.16033
$499$	27.4763	1.23001	0.615003	−	0.788524i	$-0.289155\pi$
$499$	27.4763	1.23001	0.615003	+	0.788524i	$0.289155\pi$
$500$	0	0
$501$	−27.9756	−1.24986
$502$	5.01951	0.224032
$503$	−20.2981	−0.905046	−0.452523	−	0.891753i	$-0.649476\pi$
$503$	−20.2981	−0.905046	−0.452523	+	0.891753i	$0.649476\pi$
$504$	26.6507	1.18712
$505$	0	0
$506$	−8.54420	−0.379836
$507$	52.1197	2.31472
$508$	87.4636	3.88057
$509$	−24.2619	−1.07539	−0.537694	−	0.843140i	$-0.680704\pi$
$509$	−24.2619	−1.07539	−0.537694	+	0.843140i	$0.680704\pi$
$510$	0	0
$511$	11.3684	0.502907
$512$	−11.5017	−0.508306
$513$	−0.523730	−0.0231233
$514$	−72.6009	−3.20229
$515$	0	0
$516$	−75.4128	−3.31986
$517$	−11.8315	−0.520347
$518$	22.8568	1.00427
$519$	22.2823	0.978086
$520$	0	0
$521$	2.20123	0.0964377	0.0482188	−	0.998837i	$-0.484646\pi$
$521$	2.20123	0.0964377	0.0482188	+	0.998837i	$0.484646\pi$
$522$	−30.4993	−1.33492
$523$	22.1378	0.968017	0.484008	−	0.875063i	$-0.339180\pi$
$523$	22.1378	0.968017	0.484008	+	0.875063i	$0.339180\pi$
$524$	−51.1754	−2.23561
$525$	0	0
$526$	−34.2736	−1.49440
$527$	28.7816	1.25375
$528$	−30.4568	−1.32546
$529$	−12.7988	−0.556468
$530$	0	0
$531$	42.9257	1.86282
$532$	−6.96239	−0.301858
$533$	29.3054	1.26936
$534$	47.9511	2.07505
$535$	0	0
$536$	91.6707	3.95957
$537$	−65.1002	−2.80928
$538$	−16.7513	−0.722200
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	23.0640	0.991597	0.495799	−	0.868438i	$-0.334875\pi$
$541$	23.0640	0.991597	0.495799	+	0.868438i	$0.334875\pi$
$542$	−15.3503	−0.659350
$543$	−27.7889	−1.19254
$544$	−86.8261	−3.72264
$545$	0	0
$546$	−38.7064	−1.65648
$547$	−21.3766	−0.913998	−0.456999	−	0.889467i	$-0.651076\pi$
$547$	−21.3766	−0.913998	−0.456999	+	0.889467i	$0.651076\pi$
$548$	56.6820	2.42133
$549$	−4.46073	−0.190379
$550$	0	0
$551$	4.87732	0.207781
$552$	66.9135	2.84803
$553$	1.96968	0.0837594
$554$	22.3576	0.949882
$555$	0	0
$556$	35.5125	1.50606
$557$	9.19394	0.389560	0.194780	−	0.980847i	$-0.437601\pi$
$557$	9.19394	0.389560	0.194780	+	0.980847i	$0.437601\pi$
$558$	−44.6434	−1.88991
$559$	34.3733	1.45384
$560$	0	0
$561$	13.5066	0.570249
$562$	−22.6009	−0.953360
$563$	9.79877	0.412969	0.206484	−	0.978450i	$-0.433798\pi$
$563$	9.79877	0.412969	0.206484	+	0.978450i	$0.433798\pi$
$564$	151.370	6.37382
$565$	0	0
$566$	−2.23743	−0.0940461
$567$	−8.50659	−0.357243
$568$	−131.187	−5.50449
$569$	−33.5125	−1.40492	−0.702458	−	0.711725i	$-0.747914\pi$
$569$	−33.5125	−1.40492	−0.702458	+	0.711725i	$0.747914\pi$
$570$	0	0
$571$	43.1392	1.80532	0.902659	−	0.430356i	$-0.141612\pi$
$571$	43.1392	1.80532	0.902659	+	0.430356i	$0.141612\pi$
$572$	30.0689	1.25724
$573$	27.7889	1.16090
$574$	−13.4436	−0.561124
$575$	0	0
$576$	57.1886	2.38286
$577$	14.8510	0.618254	0.309127	−	0.951021i	$-0.399963\pi$
$577$	14.8510	0.618254	0.309127	+	0.951021i	$0.399963\pi$
$578$	33.7938	1.40564
$579$	−1.50071	−0.0623673
$580$	0	0
$581$	−10.6253	−0.440812
$582$	5.55149	0.230117
$583$	−0.231548	−0.00958974
$584$	95.9897	3.97208
$585$	0	0
$586$	7.27267	0.300431
$587$	−14.7938	−0.610607	−0.305304	−	0.952255i	$-0.598758\pi$
$587$	−14.7938	−0.610607	−0.305304	+	0.952255i	$0.598758\pi$
$588$	12.7938	0.527609
$589$	7.13918	0.294165
$590$	0	0
$591$	−37.9756	−1.56211
$592$	104.880	4.31056
$593$	27.4191	1.12597	0.562985	−	0.826467i	$-0.309653\pi$
$593$	27.4191	1.12597	0.562985	+	0.826467i	$0.309653\pi$
$594$	−1.03761	−0.0425737
$595$	0	0
$596$	117.626	4.81814
$597$	−31.1695	−1.27568
$598$	−49.8251	−2.03750
$599$	11.3258	0.462761	0.231380	−	0.972863i	$-0.425676\pi$
$599$	11.3258	0.462761	0.231380	+	0.972863i	$0.425676\pi$
$600$	0	0
$601$	15.5393	0.633860	0.316930	−	0.948449i	$-0.397348\pi$
$601$	15.5393	0.633860	0.316930	+	0.948449i	$0.397348\pi$
$602$	−15.7685	−0.642674
$603$	34.2677	1.39549
$604$	−16.7308	−0.680768
$605$	0	0
$606$	49.2057	1.99884
$607$	−17.7235	−0.719377	−0.359688	−	0.933073i	$-0.617117\pi$
$607$	−17.7235	−0.719377	−0.359688	+	0.933073i	$0.617117\pi$
$608$	−21.5369	−0.873437
$609$	−8.96239	−0.363174
$610$	0	0
$611$	−68.9946	−2.79122
$612$	−88.5945	−3.58122
$613$	−22.2941	−0.900450	−0.450225	−	0.892915i	$-0.648656\pi$
$613$	−22.2941	−0.900450	−0.450225	+	0.892915i	$0.648656\pi$
$614$	22.3733	0.902912
$615$	0	0
$616$	−8.44358	−0.340202
$617$	30.9438	1.24575	0.622876	−	0.782321i	$-0.285964\pi$
$617$	30.9438	1.24575	0.622876	+	0.782321i	$0.285964\pi$
$618$	−28.0059	−1.12656
$619$	32.4119	1.30274	0.651371	−	0.758759i	$-0.274194\pi$
$619$	32.4119	1.30274	0.651371	+	0.758759i	$0.274194\pi$
$620$	0	0
$621$	1.23884	0.0497130
$622$	11.8677	0.475850
$623$	7.22425	0.289434
$624$	−177.607	−7.10998
$625$	0	0
$626$	−79.6893	−3.18502
$627$	3.35026	0.133797
$628$	27.9756	1.11635
$629$	−46.5111	−1.85452
$630$	0	0
$631$	−27.3258	−1.08782	−0.543912	−	0.839142i	$-0.683057\pi$
$631$	−27.3258	−1.08782	−0.543912	+	0.839142i	$0.683057\pi$
$632$	16.6312	0.661553
$633$	−11.0132	−0.437734
$634$	−41.1998	−1.63625
$635$	0	0
$636$	2.96239	0.117466
$637$	−5.83146	−0.231051
$638$	9.66291	0.382558
$639$	−49.0395	−1.93997
$640$	0	0
$641$	19.4460	0.768069	0.384034	−	0.923319i	$-0.374534\pi$
$641$	19.4460	0.768069	0.384034	+	0.923319i	$0.374534\pi$
$642$	76.5764	3.02223
$643$	−5.29314	−0.208741	−0.104370	−	0.994538i	$-0.533283\pi$
$643$	−5.29314	−0.208741	−0.104370	+	0.994538i	$0.533283\pi$
$644$	16.4690	0.648969
$645$	0	0
$646$	19.6629	0.773627
$647$	−35.0966	−1.37979	−0.689896	−	0.723909i	$-0.742344\pi$
$647$	−35.0966	−1.37979	−0.689896	+	0.723909i	$0.742344\pi$
$648$	−71.8261	−2.82159
$649$	−13.5999	−0.533843
$650$	0	0
$651$	−13.1187	−0.514163
$652$	−17.4314	−0.682665
$653$	−27.7988	−1.08785	−0.543925	−	0.839134i	$-0.683062\pi$
$653$	−27.7988	−1.08785	−0.543925	+	0.839134i	$0.683062\pi$
$654$	−14.5139	−0.567538
$655$	0	0
$656$	−61.6869	−2.40847
$657$	35.8822	1.39990
$658$	31.6507	1.23387
$659$	19.6180	0.764209	0.382105	−	0.924119i	$-0.375199\pi$
$659$	19.6180	0.764209	0.382105	+	0.924119i	$0.375199\pi$
$660$	0	0
$661$	21.5633	0.838713	0.419357	−	0.907822i	$-0.362256\pi$
$661$	21.5633	0.838713	0.419357	+	0.907822i	$0.362256\pi$
$662$	16.7513	0.651058
$663$	78.7631	3.05890
$664$	−89.7156	−3.48164
$665$	0	0
$666$	72.1436	2.79551
$667$	−11.5369	−0.446711
$668$	−58.1378	−2.24942
$669$	19.3054	0.746388
$670$	0	0
$671$	1.41327	0.0545585
$672$	39.5755	1.52666
$673$	21.0679	0.812109	0.406054	−	0.913849i	$-0.366904\pi$
$673$	21.0679	0.812109	0.406054	+	0.913849i	$0.366904\pi$
$674$	42.4544	1.63528
$675$	0	0
$676$	108.313	4.16589
$677$	−34.5174	−1.32661	−0.663306	−	0.748349i	$-0.730847\pi$
$677$	−34.5174	−1.32661	−0.663306	+	0.748349i	$0.730847\pi$
$678$	62.0625	2.38350
$679$	0.836381	0.0320973
$680$	0	0
$681$	25.9248	0.993440
$682$	14.1441	0.541606
$683$	−33.7802	−1.29256	−0.646282	−	0.763099i	$-0.723677\pi$
$683$	−33.7802	−1.29256	−0.646282	+	0.763099i	$0.723677\pi$
$684$	−21.9756	−0.840257
$685$	0	0
$686$	2.67513	0.102137
$687$	−73.0757	−2.78801
$688$	−72.3547	−2.75850
$689$	−1.35026	−0.0514409
$690$	0	0
$691$	−13.8618	−0.527327	−0.263663	−	0.964615i	$-0.584931\pi$
$691$	−13.8618	−0.527327	−0.263663	+	0.964615i	$0.584931\pi$
$692$	46.3063	1.76030
$693$	−3.15633	−0.119899
$694$	−18.1681	−0.689651
$695$	0	0
$696$	−75.6747	−2.86844
$697$	27.3561	1.03619
$698$	71.5569	2.70847
$699$	21.6629	0.819367
$700$	0	0
$701$	40.5256	1.53063	0.765316	−	0.643655i	$-0.222583\pi$
$701$	40.5256	1.53063	0.765316	+	0.643655i	$0.222583\pi$
$702$	−6.05079	−0.228372
$703$	−11.5369	−0.435123
$704$	−18.1187	−0.682875
$705$	0	0
$706$	45.1100	1.69774
$707$	7.41327	0.278805
$708$	173.995	6.53914
$709$	−0.850969	−0.0319588	−0.0159794	−	0.999872i	$-0.505087\pi$
$709$	−0.850969	−0.0319588	−0.0159794	+	0.999872i	$0.505087\pi$
$710$	0	0
$711$	6.21696	0.233154
$712$	60.9986	2.28602
$713$	−16.8872	−0.632429
$714$	−36.1319	−1.35220
$715$	0	0
$716$	−135.289	−5.05598
$717$	−52.6371	−1.96577
$718$	−10.1465	−0.378663
$719$	22.5769	0.841976	0.420988	−	0.907066i	$-0.361683\pi$
$719$	22.5769	0.841976	0.420988	+	0.907066i	$0.361683\pi$
$720$	0	0
$721$	−4.21933	−0.157136
$722$	−45.9502	−1.71009
$723$	−23.1695	−0.861683
$724$	−57.7499	−2.14626
$725$	0	0
$726$	6.63752	0.246341
$727$	−12.5174	−0.464244	−0.232122	−	0.972687i	$-0.574567\pi$
$727$	−12.5174	−0.464244	−0.232122	+	0.972687i	$0.574567\pi$
$728$	−49.2384	−1.82490
$729$	−29.7367	−1.10136
$730$	0	0
$731$	32.0870	1.18678
$732$	−18.0811	−0.668297
$733$	−16.6678	−0.615641	−0.307820	−	0.951445i	$-0.599600\pi$
$733$	−16.6678	−0.615641	−0.307820	+	0.951445i	$0.599600\pi$
$734$	17.0400	0.628957
$735$	0	0
$736$	50.9438	1.87781
$737$	−10.8568	−0.399917
$738$	−42.4323	−1.56196
$739$	42.7005	1.57076	0.785382	−	0.619011i	$-0.212467\pi$
$739$	42.7005	1.57076	0.785382	+	0.619011i	$0.212467\pi$
$740$	0	0
$741$	19.5369	0.717706
$742$	0.619421	0.0227397
$743$	19.6873	0.722259	0.361129	−	0.932516i	$-0.382391\pi$
$743$	19.6873	0.722259	0.361129	+	0.932516i	$0.382391\pi$
$744$	−110.769	−4.06099
$745$	0	0
$746$	57.0651	2.08930
$747$	−33.5369	−1.22705
$748$	28.0689	1.02630
$749$	11.5369	0.421549
$750$	0	0
$751$	−5.85940	−0.213813	−0.106906	−	0.994269i	$-0.534094\pi$
$751$	−5.85940	−0.213813	−0.106906	+	0.994269i	$0.534094\pi$
$752$	145.232	5.29605
$753$	4.65562	0.169660
$754$	56.3488	2.05210
$755$	0	0
$756$	2.00000	0.0727393
$757$	40.5863	1.47513	0.737567	−	0.675274i	$-0.235975\pi$
$757$	40.5863	1.47513	0.737567	+	0.675274i	$0.235975\pi$
$758$	66.3098	2.40848
$759$	−7.92478	−0.287651
$760$	0	0
$761$	21.8472	0.791960	0.395980	−	0.918259i	$-0.370405\pi$
$761$	21.8472	0.791960	0.395980	+	0.918259i	$0.370405\pi$
$762$	112.588	4.07864
$763$	−2.18664	−0.0791618
$764$	57.7499	2.08932
$765$	0	0
$766$	14.6013	0.527567
$767$	−79.3073	−2.86362
$768$	20.0689	0.724173
$769$	45.2892	1.63317	0.816585	−	0.577226i	$-0.195865\pi$
$769$	45.2892	1.63317	0.816585	+	0.577226i	$0.195865\pi$
$770$	0	0
$771$	−67.3376	−2.42510
$772$	−3.11871	−0.112245
$773$	33.8153	1.21625	0.608125	−	0.793841i	$-0.291922\pi$
$773$	33.8153	1.21625	0.608125	+	0.793841i	$0.291922\pi$
$774$	−49.7704	−1.78896
$775$	0	0
$776$	7.06205	0.253513
$777$	21.1998	0.760539
$778$	−36.7123	−1.31620
$779$	6.78560	0.243119
$780$	0	0
$781$	15.5369	0.555954
$782$	−46.5111	−1.66323
$783$	−1.40105	−0.0500693
$784$	12.2750	0.438394
$785$	0	0
$786$	−65.8759	−2.34972
$787$	−1.27504	−0.0454502	−0.0227251	−	0.999742i	$-0.507234\pi$
$787$	−1.27504	−0.0454502	−0.0227251	+	0.999742i	$0.507234\pi$
$788$	−78.9194	−2.81139
$789$	−31.7889	−1.13172
$790$	0	0
$791$	9.35026	0.332457
$792$	−26.6507	−0.946991
$793$	8.24140	0.292661
$794$	−5.64832	−0.200452
$795$	0	0
$796$	−64.7753	−2.29590
$797$	42.5256	1.50634	0.753168	−	0.657829i	$-0.228525\pi$
$797$	42.5256	1.50634	0.753168	+	0.657829i	$0.228525\pi$
$798$	−8.96239	−0.317265
$799$	−64.4055	−2.27850
$800$	0	0
$801$	22.8021	0.805672
$802$	51.2262	1.80886
$803$	−11.3684	−0.401181
$804$	138.901	4.89865
$805$	0	0
$806$	82.4807	2.90526
$807$	−15.5369	−0.546925
$808$	62.5945	2.20207
$809$	−14.7151	−0.517356	−0.258678	−	0.965964i	$-0.583287\pi$
$809$	−14.7151	−0.517356	−0.258678	+	0.965964i	$0.583287\pi$
$810$	0	0
$811$	51.7743	1.81804	0.909021	−	0.416750i	$-0.136831\pi$
$811$	51.7743	1.81804	0.909021	+	0.416750i	$0.136831\pi$
$812$	−18.6253	−0.653620
$813$	−14.2374	−0.499328
$814$	−22.8568	−0.801132
$815$	0	0
$816$	−165.794	−5.80395
$817$	7.95906	0.278452
$818$	−49.9937	−1.74799
$819$	−18.4060	−0.643157
$820$	0	0
$821$	2.64974	0.0924765	0.0462383	−	0.998930i	$-0.485277\pi$
$821$	2.64974	0.0924765	0.0462383	+	0.998930i	$0.485277\pi$
$822$	72.9643	2.54492
$823$	−5.76845	−0.201076	−0.100538	−	0.994933i	$-0.532056\pi$
$823$	−5.76845	−0.201076	−0.100538	+	0.994933i	$0.532056\pi$
$824$	−35.6263	−1.24110
$825$	0	0
$826$	36.3815	1.26588
$827$	13.4920	0.469163	0.234581	−	0.972096i	$-0.424628\pi$
$827$	13.4920	0.469163	0.234581	+	0.972096i	$0.424628\pi$
$828$	51.9814	1.80648
$829$	−4.70052	−0.163256	−0.0816280	−	0.996663i	$-0.526012\pi$
$829$	−4.70052	−0.163256	−0.0816280	+	0.996663i	$0.526012\pi$
$830$	0	0
$831$	20.7367	0.719349
$832$	−105.658	−3.66305
$833$	−5.44358	−0.188609
$834$	45.7137	1.58294
$835$	0	0
$836$	6.96239	0.240799
$837$	−2.05079	−0.0708855
$838$	−2.06888	−0.0714684
$839$	−38.8045	−1.33968	−0.669839	−	0.742506i	$-0.733637\pi$
$839$	−38.8045	−1.33968	−0.669839	+	0.742506i	$0.733637\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	−28.1417	−0.969828
$843$	−20.9624	−0.721983
$844$	−22.8872	−0.787809
$845$	0	0
$846$	99.8999	3.43463
$847$	1.00000	0.0343604
$848$	2.84226	0.0976036
$849$	−2.07522	−0.0712215
$850$	0	0
$851$	27.2896	0.935476
$852$	−198.777	−6.80998
$853$	−20.6824	−0.708153	−0.354076	−	0.935217i	$-0.615205\pi$
$853$	−20.6824	−0.708153	−0.354076	+	0.935217i	$0.615205\pi$
$854$	−3.78067	−0.129372
$855$	0	0
$856$	97.4128	3.32950
$857$	−26.3453	−0.899940	−0.449970	−	0.893044i	$-0.648565\pi$
$857$	−26.3453	−0.899940	−0.449970	+	0.893044i	$0.648565\pi$
$858$	38.7064	1.32141
$859$	−8.51151	−0.290409	−0.145205	−	0.989402i	$-0.546384\pi$
$859$	−8.51151	−0.290409	−0.145205	+	0.989402i	$0.546384\pi$
$860$	0	0
$861$	−12.4690	−0.424942
$862$	−66.1582	−2.25336
$863$	−7.56722	−0.257591	−0.128796	−	0.991671i	$-0.541111\pi$
$863$	−7.56722	−0.257591	−0.128796	+	0.991671i	$0.541111\pi$
$864$	6.18664	0.210474
$865$	0	0
$866$	49.6502	1.68718
$867$	31.3439	1.06450
$868$	−27.2628	−0.925360
$869$	−1.96968	−0.0668169
$870$	0	0
$871$	−63.3112	−2.14522
$872$	−18.4631	−0.625239
$873$	2.63989	0.0893467
$874$	−11.5369	−0.390242
$875$	0	0
$876$	145.445	4.91413
$877$	−17.2955	−0.584028	−0.292014	−	0.956414i	$-0.594325\pi$
$877$	−17.2955	−0.584028	−0.292014	+	0.956414i	$0.594325\pi$
$878$	−3.81336	−0.128695
$879$	6.74543	0.227518
$880$	0	0
$881$	20.4504	0.688992	0.344496	−	0.938788i	$-0.388050\pi$
$881$	20.4504	0.688992	0.344496	+	0.938788i	$0.388050\pi$
$882$	8.44358	0.284310
$883$	49.6589	1.67116	0.835578	−	0.549371i	$-0.185133\pi$
$883$	49.6589	1.67116	0.835578	+	0.549371i	$0.185133\pi$
$884$	163.682	5.50524
$885$	0	0
$886$	−107.520	−3.61221
$887$	47.1100	1.58180	0.790900	−	0.611946i	$-0.209613\pi$
$887$	47.1100	1.58180	0.790900	+	0.611946i	$0.209613\pi$
$888$	179.002	6.00693
$889$	16.9624	0.568900
$890$	0	0
$891$	8.50659	0.284981
$892$	40.1197	1.34331
$893$	−15.9756	−0.534602
$894$	151.415	5.06407
$895$	0	0
$896$	16.5696	0.553551
$897$	−46.2130	−1.54301
$898$	33.8554	1.12977
$899$	19.0982	0.636962
$900$	0	0
$901$	−1.26045	−0.0419917
$902$	13.4436	0.447622
$903$	−14.6253	−0.486700
$904$	78.9497	2.62583
$905$	0	0
$906$	−21.5369	−0.715516
$907$	−14.4591	−0.480107	−0.240054	−	0.970760i	$-0.577165\pi$
$907$	−14.4591	−0.480107	−0.240054	+	0.970760i	$0.577165\pi$
$908$	53.8759	1.78793
$909$	23.3987	0.776085
$910$	0	0
$911$	−31.5369	−1.04486	−0.522432	−	0.852681i	$-0.674975\pi$
$911$	−31.5369	−1.04486	−0.522432	+	0.852681i	$0.674975\pi$
$912$	−41.1246	−1.36177
$913$	10.6253	0.351646
$914$	1.45580	0.0481536
$915$	0	0
$916$	−151.863	−5.01770
$917$	−9.92478	−0.327745
$918$	−5.64832	−0.186422
$919$	5.26328	0.173620	0.0868098	−	0.996225i	$-0.472333\pi$
$919$	5.26328	0.173620	0.0868098	+	0.996225i	$0.472333\pi$
$920$	0	0
$921$	20.7513	0.683779
$922$	−30.9659	−1.01981
$923$	90.6028	2.98223
$924$	−12.7938	−0.420887
$925$	0	0
$926$	−63.6542	−2.09181
$927$	−13.3176	−0.437407
$928$	−57.6140	−1.89127
$929$	26.0508	0.854699	0.427349	−	0.904087i	$-0.359447\pi$
$929$	26.0508	0.854699	0.427349	+	0.904087i	$0.359447\pi$
$930$	0	0
$931$	−1.35026	−0.0442530
$932$	45.0191	1.47465
$933$	11.0073	0.360363
$934$	7.13681	0.233524
$935$	0	0
$936$	−155.412	−5.07981
$937$	29.3439	0.958624	0.479312	−	0.877645i	$-0.340886\pi$
$937$	29.3439	0.958624	0.479312	+	0.877645i	$0.340886\pi$
$938$	29.0435	0.948304
$939$	−73.9121	−2.41203
$940$	0	0
$941$	−28.6375	−0.933556	−0.466778	−	0.884374i	$-0.654585\pi$
$941$	−28.6375	−0.933556	−0.466778	+	0.884374i	$0.654585\pi$
$942$	36.0118	1.17333
$943$	−16.0508	−0.522685
$944$	166.939	5.43341
$945$	0	0
$946$	15.7685	0.512677
$947$	52.8178	1.71635	0.858174	−	0.513358i	$-0.171599\pi$
$947$	52.8178	1.71635	0.858174	+	0.513358i	$0.171599\pi$
$948$	25.1998	0.818452
$949$	−66.2941	−2.15200
$950$	0	0
$951$	−38.2130	−1.23914
$952$	−45.9633	−1.48968
$953$	−37.1939	−1.20483	−0.602415	−	0.798183i	$-0.705795\pi$
$953$	−37.1939	−1.20483	−0.602415	+	0.798183i	$0.705795\pi$
$954$	1.95509	0.0632985
$955$	0	0
$956$	−109.388	−3.53787
$957$	8.96239	0.289713
$958$	−28.6516	−0.925693
$959$	10.9927	0.354973
$960$	0	0
$961$	−3.04491	−0.0982228
$962$	−133.289	−4.29740
$963$	36.4142	1.17343
$964$	−48.1500	−1.55081
$965$	0	0
$966$	21.1998	0.682093
$967$	−4.07125	−0.130923	−0.0654613	−	0.997855i	$-0.520852\pi$
$967$	−4.07125	−0.130923	−0.0654613	+	0.997855i	$0.520852\pi$
$968$	8.44358	0.271387
$969$	18.2374	0.585871
$970$	0	0
$971$	0.773377	0.0248188	0.0124094	−	0.999923i	$-0.496050\pi$
$971$	0.773377	0.0248188	0.0124094	+	0.999923i	$0.496050\pi$
$972$	−114.832	−3.68324
$973$	6.88717	0.220792
$974$	−46.6312	−1.49416
$975$	0	0
$976$	−17.3479	−0.555292
$977$	37.8740	1.21170	0.605848	−	0.795580i	$-0.292834\pi$
$977$	37.8740	1.21170	0.605848	+	0.795580i	$0.292834\pi$
$978$	−22.4387	−0.717509
$979$	−7.22425	−0.230888
$980$	0	0
$981$	−6.90175	−0.220356
$982$	−75.8916	−2.42180
$983$	15.5794	0.496907	0.248453	−	0.968644i	$-0.420078\pi$
$983$	15.5794	0.496907	0.248453	+	0.968644i	$0.420078\pi$
$984$	−105.283	−3.35629
$985$	0	0
$986$	52.6009	1.67515
$987$	29.3561	0.934416
$988$	40.6009	1.29169
$989$	−18.8265	−0.598649
$990$	0	0
$991$	27.0982	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$991$	27.0982	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$992$	−84.3327	−2.67756
$993$	15.5369	0.493049
$994$	−41.5633	−1.31831
$995$	0	0
$996$	−135.938	−4.30737
$997$	−50.4060	−1.59637	−0.798187	−	0.602410i	$-0.794207\pi$
$997$	−50.4060	−1.59637	−0.798187	+	0.602410i	$0.794207\pi$
$998$	73.5026	2.32668
$999$	3.31406	0.104852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.v.1.3		3
5.2	odd	4		1925.2.b.n.1849.6		6
5.3	odd	4		1925.2.b.n.1849.1		6
5.4	even	2		385.2.a.f.1.1	✓	3
15.14	odd	2		3465.2.a.bh.1.3		3
20.19	odd	2		6160.2.a.bn.1.3		3
35.34	odd	2		2695.2.a.g.1.1		3
55.54	odd	2		4235.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.f.1.1	✓	3	5.4	even	2
1925.2.a.v.1.3		3	1.1	even	1	trivial
1925.2.b.n.1849.1		6	5.3	odd	4
1925.2.b.n.1849.6		6	5.2	odd	4
2695.2.a.g.1.1		3	35.34	odd	2
3465.2.a.bh.1.3		3	15.14	odd	2
4235.2.a.q.1.3		3	55.54	odd	2
6160.2.a.bn.1.3		3	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67513 q^{2} +2.48119 q^{3} +5.15633 q^{4} +6.63752 q^{6} +1.00000 q^{7} +8.44358 q^{8} +3.15633 q^{9} -1.00000 q^{11} +12.7938 q^{12} -5.83146 q^{13} +2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} +8.44358 q^{18} -1.35026 q^{19} +2.48119 q^{21} -2.67513 q^{22} +3.19394 q^{23} +20.9502 q^{24} -15.5999 q^{26} +0.387873 q^{27} +5.15633 q^{28} -3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -2.48119 q^{33} -14.5623 q^{34} +16.2750 q^{36} +8.54420 q^{37} -3.61213 q^{38} -14.4690 q^{39} -5.02539 q^{41} +6.63752 q^{42} -5.89446 q^{43} -5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +30.4568 q^{48} +1.00000 q^{49} -13.5066 q^{51} -30.0689 q^{52} +0.231548 q^{53} +1.03761 q^{54} +8.44358 q^{56} -3.35026 q^{57} -9.66291 q^{58} +13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +3.15633 q^{63} +18.1187 q^{64} -6.63752 q^{66} +10.8568 q^{67} -28.0689 q^{68} +7.92478 q^{69} -15.5369 q^{71} +26.6507 q^{72} +11.3684 q^{73} +22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} -38.7064 q^{78} +1.96968 q^{79} -8.50659 q^{81} -13.4436 q^{82} -10.6253 q^{83} +12.7938 q^{84} -15.7685 q^{86} -8.96239 q^{87} -8.44358 q^{88} +7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} -13.1187 q^{93} +31.6507 q^{94} +39.5755 q^{96} +0.836381 q^{97} +2.67513 q^{98} -3.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} - 3 q^{11} + 12 q^{12} - 2 q^{13} + 3 q^{14} + 5 q^{16} + 9 q^{18} + 6 q^{19} + 2 q^{21} - 3 q^{22} + 10 q^{23} + 26 q^{24} - 20 q^{26}+ \cdots + q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 2 * q^3 + 5 * q^4 + 4 * q^6 + 3 * q^7 + 9 * q^8 - q^9 - 3 * q^11 + 12 * q^12 - 2 * q^13 + 3 * q^14 + 5 * q^16 + 9 * q^18 + 6 * q^19 + 2 * q^21 - 3 * q^22 + 10 * q^23 + 26 * q^24 - 20 * q^26 + 2 * q^27 + 5 * q^28 - 10 * q^29 - 10 * q^31 + 11 * q^32 - 2 * q^33 - 6 * q^34 + 17 * q^36 + 16 * q^37 - 10 * q^38 - 12 * q^39 + 4 * q^42 + 2 * q^43 - 5 * q^44 + 16 * q^46 + 20 * q^47 + 34 * q^48 + 3 * q^49 - 20 * q^51 - 32 * q^52 + 12 * q^53 + 14 * q^54 + 9 * q^56 + 2 * q^58 + 14 * q^59 + 10 * q^61 - 6 * q^62 - q^63 + 33 * q^64 - 4 * q^66 + 2 * q^67 - 26 * q^68 + 2 * q^69 - 24 * q^71 + 23 * q^72 - 4 * q^73 + 38 * q^74 - 10 * q^76 - 3 * q^77 - 42 * q^78 + 8 * q^79 - 5 * q^81 - 24 * q^82 + 10 * q^83 + 12 * q^84 - 36 * q^86 - 16 * q^87 - 9 * q^88 + 20 * q^89 - 2 * q^91 + 18 * q^92 - 18 * q^93 + 38 * q^94 + 40 * q^96 + 3 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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