LMFDB - Embedded newform 3174.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, 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("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3174 = 2 \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3174.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:	$25.3445176016$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	3174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.89898 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.89898 q^{14} +1.00000 q^{16} +4.89898 q^{17} -1.00000 q^{18} -4.89898 q^{21} -4.89898 q^{22} -1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -4.89898 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -4.89898 q^{34} +1.00000 q^{36} -4.89898 q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.89898 q^{42} +9.79796 q^{43} +4.89898 q^{44} +1.00000 q^{48} +17.0000 q^{49} +5.00000 q^{50} +4.89898 q^{51} -2.00000 q^{52} +9.79796 q^{53} -1.00000 q^{54} +4.89898 q^{56} +6.00000 q^{58} -12.0000 q^{59} +4.89898 q^{61} +8.00000 q^{62} -4.89898 q^{63} +1.00000 q^{64} -4.89898 q^{66} +9.79796 q^{67} +4.89898 q^{68} -1.00000 q^{72} -2.00000 q^{73} +4.89898 q^{74} -5.00000 q^{75} -24.0000 q^{77} +2.00000 q^{78} -14.6969 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.89898 q^{83} -4.89898 q^{84} -9.79796 q^{86} -6.00000 q^{87} -4.89898 q^{88} -14.6969 q^{89} +9.79796 q^{91} -8.00000 q^{93} -1.00000 q^{96} -17.0000 q^{98} +4.89898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 2 q^{24} - 10 q^{25} + 4 q^{26} + 2 q^{27} - 12 q^{29} - 16 q^{31} - 2 q^{32} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 2 q^{48} + 34 q^{49} + 10 q^{50} - 4 q^{52} - 2 q^{54} + 12 q^{58} - 24 q^{59} + 16 q^{62} + 2 q^{64} - 2 q^{72} - 4 q^{73} - 10 q^{75} - 48 q^{77} + 4 q^{78} + 2 q^{81} + 12 q^{82} - 12 q^{87} - 16 q^{93} - 2 q^{96} - 34 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^18 - 2 * q^24 - 10 * q^25 + 4 * q^26 + 2 * q^27 - 12 * q^29 - 16 * q^31 - 2 * q^32 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 2 * q^48 + 34 * q^49 + 10 * q^50 - 4 * q^52 - 2 * q^54 + 12 * q^58 - 24 * q^59 + 16 * q^62 + 2 * q^64 - 2 * q^72 - 4 * q^73 - 10 * q^75 - 48 * q^77 + 4 * q^78 + 2 * q^81 + 12 * q^82 - 12 * q^87 - 16 * q^93 - 2 * q^96 - 34 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	−4.89898	−1.85164	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	−4.89898	−1.85164	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.89898	1.30931
$15$	0	0
$16$	1.00000	0.250000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.89898	−1.06904
$22$	−4.89898	−1.04447
$23$	0	0
$23$	0	0
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−4.89898	−0.925820
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	4.89898	0.852803
$34$	−4.89898	−0.840168
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.89898	0.755929
$43$	9.79796	1.49417	0.747087	−	0.664726i	$-0.231452\pi$
$43$	9.79796	1.49417	0.747087	+	0.664726i	$0.231452\pi$
$44$	4.89898	0.738549
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	17.0000	2.42857
$50$	5.00000	0.707107
$51$	4.89898	0.685994
$52$	−2.00000	−0.277350
$53$	9.79796	1.34585	0.672927	−	0.739709i	$-0.265037\pi$
$53$	9.79796	1.34585	0.672927	+	0.739709i	$0.265037\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.89898	0.654654
$57$	0	0
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	4.89898	0.627250	0.313625	−	0.949547i	$-0.398457\pi$
$61$	4.89898	0.627250	0.313625	+	0.949547i	$0.398457\pi$
$62$	8.00000	1.01600
$63$	−4.89898	−0.617213
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.89898	−0.603023
$67$	9.79796	1.19701	0.598506	−	0.801119i	$-0.295761\pi$
$67$	9.79796	1.19701	0.598506	+	0.801119i	$0.295761\pi$
$68$	4.89898	0.594089
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	4.89898	0.569495
$75$	−5.00000	−0.577350
$76$	0	0
$77$	−24.0000	−2.73505
$78$	2.00000	0.226455
$79$	−14.6969	−1.65353	−0.826767	−	0.562544i	$-0.809823\pi$
$79$	−14.6969	−1.65353	−0.826767	+	0.562544i	$0.809823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.89898	−0.537733	−0.268866	−	0.963177i	$-0.586649\pi$
$83$	−4.89898	−0.537733	−0.268866	+	0.963177i	$0.586649\pi$
$84$	−4.89898	−0.534522
$85$	0	0
$86$	−9.79796	−1.05654
$87$	−6.00000	−0.643268
$88$	−4.89898	−0.522233
$89$	−14.6969	−1.55787	−0.778936	−	0.627103i	$-0.784240\pi$
$89$	−14.6969	−1.55787	−0.778936	+	0.627103i	$0.784240\pi$
$90$	0	0
$91$	9.79796	1.02711
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−17.0000	−1.71726
$99$	4.89898	0.492366
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−4.89898	−0.485071
$103$	−4.89898	−0.482711	−0.241355	−	0.970437i	$-0.577592\pi$
$103$	−4.89898	−0.482711	−0.241355	+	0.970437i	$0.577592\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−9.79796	−0.951662
$107$	−4.89898	−0.473602	−0.236801	−	0.971558i	$-0.576099\pi$
$107$	−4.89898	−0.473602	−0.236801	+	0.971558i	$0.576099\pi$
$108$	1.00000	0.0962250
$109$	−4.89898	−0.469237	−0.234619	−	0.972088i	$-0.575384\pi$
$109$	−4.89898	−0.469237	−0.234619	+	0.972088i	$0.575384\pi$
$110$	0	0
$111$	−4.89898	−0.464991
$112$	−4.89898	−0.462910
$113$	−14.6969	−1.38257	−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969	−1.38257	−0.691286	+	0.722581i	$0.742955\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	−24.0000	−2.20008
$120$	0	0
$121$	13.0000	1.18182
$122$	−4.89898	−0.443533
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.89898	0.436436
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	9.79796	0.862662
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.89898	0.426401
$133$	0	0
$134$	−9.79796	−0.846415
$135$	0	0
$136$	−4.89898	−0.420084
$137$	−4.89898	−0.418548	−0.209274	−	0.977857i	$-0.567110\pi$
$137$	−4.89898	−0.418548	−0.209274	+	0.977857i	$0.567110\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.79796	−0.819346
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	17.0000	1.40214
$148$	−4.89898	−0.402694
$149$	−19.5959	−1.60536	−0.802680	−	0.596410i	$-0.796593\pi$
$149$	−19.5959	−1.60536	−0.802680	+	0.596410i	$0.796593\pi$
$150$	5.00000	0.408248
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	4.89898	0.396059
$154$	24.0000	1.93398
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−4.89898	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$157$	−4.89898	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$158$	14.6969	1.16923
$159$	9.79796	0.777029
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.89898	0.380235
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	4.89898	0.377964
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	9.79796	0.747087
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	24.4949	1.85164
$176$	4.89898	0.369274
$177$	−12.0000	−0.901975
$178$	14.6969	1.10158
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−4.89898	−0.364138	−0.182069	−	0.983286i	$-0.558279\pi$
$181$	−4.89898	−0.364138	−0.182069	+	0.983286i	$0.558279\pi$
$182$	−9.79796	−0.726273
$183$	4.89898	0.362143
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	24.0000	1.75505
$188$	0	0
$189$	−4.89898	−0.356348
$190$	0	0
$191$	−19.5959	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	17.0000	1.21429
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.89898	−0.348155
$199$	−14.6969	−1.04184	−0.520919	−	0.853606i	$-0.674411\pi$
$199$	−14.6969	−1.04184	−0.520919	+	0.853606i	$0.674411\pi$
$200$	5.00000	0.353553
$201$	9.79796	0.691095
$202$	−6.00000	−0.422159
$203$	29.3939	2.06305
$204$	4.89898	0.342997
$205$	0	0
$206$	4.89898	0.341328
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	9.79796	0.672927
$213$	0	0
$214$	4.89898	0.334887
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	39.1918	2.66052
$218$	4.89898	0.331801
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−9.79796	−0.659082
$222$	4.89898	0.328798
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	4.89898	0.327327
$225$	−5.00000	−0.333333
$226$	14.6969	0.977626
$227$	14.6969	0.975470	0.487735	−	0.872992i	$-0.337823\pi$
$227$	14.6969	0.975470	0.487735	+	0.872992i	$0.337823\pi$
$228$	0	0
$229$	−24.4949	−1.61867	−0.809334	−	0.587348i	$-0.800172\pi$
$229$	−24.4949	−1.61867	−0.809334	+	0.587348i	$0.800172\pi$
$230$	0	0
$231$	−24.0000	−1.57908
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−14.6969	−0.954669
$238$	24.0000	1.55569
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	19.5959	1.26228	0.631142	−	0.775667i	$-0.282587\pi$
$241$	19.5959	1.26228	0.631142	+	0.775667i	$0.282587\pi$
$242$	−13.0000	−0.835672
$243$	1.00000	0.0641500
$244$	4.89898	0.313625
$245$	0	0
$246$	6.00000	0.382546
$247$	0	0
$248$	8.00000	0.508001
$249$	−4.89898	−0.310460
$250$	0	0
$251$	24.4949	1.54610	0.773052	−	0.634343i	$-0.218729\pi$
$251$	24.4949	1.54610	0.773052	+	0.634343i	$0.218729\pi$
$252$	−4.89898	−0.308607
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	−9.79796	−0.609994
$259$	24.0000	1.49129
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	−9.79796	−0.604168	−0.302084	−	0.953281i	$-0.597682\pi$
$263$	−9.79796	−0.604168	−0.302084	+	0.953281i	$0.597682\pi$
$264$	−4.89898	−0.301511
$265$	0	0
$266$	0	0
$267$	−14.6969	−0.899438
$268$	9.79796	0.598506
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	4.89898	0.297044
$273$	9.79796	0.592999
$274$	4.89898	0.295958
$275$	−24.4949	−1.47710
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	4.89898	0.292249	0.146124	−	0.989266i	$-0.453320\pi$
$281$	4.89898	0.292249	0.146124	+	0.989266i	$0.453320\pi$
$282$	0	0
$283$	19.5959	1.16486	0.582428	−	0.812882i	$-0.302103\pi$
$283$	19.5959	1.16486	0.582428	+	0.812882i	$0.302103\pi$
$284$	0	0
$285$	0	0
$286$	9.79796	0.579365
$287$	29.3939	1.73507
$288$	−1.00000	−0.0589256
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−9.79796	−0.572403	−0.286201	−	0.958169i	$-0.592393\pi$
$293$	−9.79796	−0.572403	−0.286201	+	0.958169i	$0.592393\pi$
$294$	−17.0000	−0.991460
$295$	0	0
$296$	4.89898	0.284747
$297$	4.89898	0.284268
$298$	19.5959	1.13516
$299$	0	0
$300$	−5.00000	−0.288675
$301$	−48.0000	−2.76667
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	0	0
$305$	0	0
$306$	−4.89898	−0.280056
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−24.0000	−1.36753
$309$	−4.89898	−0.278693
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	2.00000	0.113228
$313$	−9.79796	−0.553813	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	−9.79796	−0.553813	−0.276907	+	0.960897i	$0.589309\pi$
$314$	4.89898	0.276465
$315$	0	0
$316$	−14.6969	−0.826767
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−9.79796	−0.549442
$319$	−29.3939	−1.64574
$320$	0	0
$321$	−4.89898	−0.273434
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	10.0000	0.554700
$326$	20.0000	1.10770
$327$	−4.89898	−0.270914
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.89898	−0.268866
$333$	−4.89898	−0.268462
$334$	0	0
$335$	0	0
$336$	−4.89898	−0.267261
$337$	9.79796	0.533729	0.266864	−	0.963734i	$-0.414012\pi$
$337$	9.79796	0.533729	0.266864	+	0.963734i	$0.414012\pi$
$338$	9.00000	0.489535
$339$	−14.6969	−0.798228
$340$	0	0
$341$	−39.1918	−2.12236
$342$	0	0
$343$	−48.9898	−2.64520
$344$	−9.79796	−0.528271
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−6.00000	−0.321634
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−24.4949	−1.30931
$351$	−2.00000	−0.106752
$352$	−4.89898	−0.261116
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−14.6969	−0.778936
$357$	−24.0000	−1.27021
$358$	12.0000	0.634220
$359$	19.5959	1.03423	0.517116	−	0.855915i	$-0.327005\pi$
$359$	19.5959	1.03423	0.517116	+	0.855915i	$0.327005\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	4.89898	0.257485
$363$	13.0000	0.682323
$364$	9.79796	0.513553
$365$	0	0
$366$	−4.89898	−0.256074
$367$	24.4949	1.27862	0.639312	−	0.768948i	$-0.279219\pi$
$367$	24.4949	1.27862	0.639312	+	0.768948i	$0.279219\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−48.0000	−2.49204
$372$	−8.00000	−0.414781
$373$	14.6969	0.760979	0.380489	−	0.924785i	$-0.375756\pi$
$373$	14.6969	0.760979	0.380489	+	0.924785i	$0.375756\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	4.89898	0.251976
$379$	−19.5959	−1.00657	−0.503287	−	0.864119i	$-0.667876\pi$
$379$	−19.5959	−1.00657	−0.503287	+	0.864119i	$0.667876\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	19.5959	1.00261
$383$	9.79796	0.500652	0.250326	−	0.968162i	$-0.419462\pi$
$383$	9.79796	0.500652	0.250326	+	0.968162i	$0.419462\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	9.79796	0.498058
$388$	0	0
$389$	19.5959	0.993552	0.496776	−	0.867879i	$-0.334517\pi$
$389$	19.5959	0.993552	0.496776	+	0.867879i	$0.334517\pi$
$390$	0	0
$391$	0	0
$392$	−17.0000	−0.858630
$393$	12.0000	0.605320
$394$	6.00000	0.302276
$395$	0	0
$396$	4.89898	0.246183
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	14.6969	0.736691
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−34.2929	−1.71250	−0.856252	−	0.516559i	$-0.827213\pi$
$401$	−34.2929	−1.71250	−0.856252	+	0.516559i	$0.827213\pi$
$402$	−9.79796	−0.488678
$403$	16.0000	0.797017
$404$	6.00000	0.298511
$405$	0	0
$406$	−29.3939	−1.45879
$407$	−24.0000	−1.18964
$408$	−4.89898	−0.242536
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−4.89898	−0.241649
$412$	−4.89898	−0.241355
$413$	58.7878	2.89276
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	14.6969	0.717992	0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969	0.717992	0.358996	+	0.933339i	$0.383119\pi$
$420$	0	0
$421$	4.89898	0.238762	0.119381	−	0.992849i	$-0.461909\pi$
$421$	4.89898	0.238762	0.119381	+	0.992849i	$0.461909\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−9.79796	−0.475831
$425$	−24.4949	−1.18818
$426$	0	0
$427$	−24.0000	−1.16144
$428$	−4.89898	−0.236801
$429$	−9.79796	−0.473050
$430$	0	0
$431$	29.3939	1.41585	0.707927	−	0.706286i	$-0.249631\pi$
$431$	29.3939	1.41585	0.707927	+	0.706286i	$0.249631\pi$
$432$	1.00000	0.0481125
$433$	9.79796	0.470860	0.235430	−	0.971891i	$-0.424350\pi$
$433$	9.79796	0.470860	0.235430	+	0.971891i	$0.424350\pi$
$434$	−39.1918	−1.88127
$435$	0	0
$436$	−4.89898	−0.234619
$437$	0	0
$438$	2.00000	0.0955637
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	17.0000	0.809524
$442$	9.79796	0.466041
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−4.89898	−0.232495
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−19.5959	−0.926855
$448$	−4.89898	−0.231455
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	5.00000	0.235702
$451$	−29.3939	−1.38410
$452$	−14.6969	−0.691286
$453$	−16.0000	−0.751746
$454$	−14.6969	−0.689761
$455$	0	0
$456$	0	0
$457$	19.5959	0.916658	0.458329	−	0.888783i	$-0.348448\pi$
$457$	19.5959	0.916658	0.458329	+	0.888783i	$0.348448\pi$
$458$	24.4949	1.14457
$459$	4.89898	0.228665
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	24.0000	1.11658
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	24.4949	1.13349	0.566744	−	0.823894i	$-0.308203\pi$
$467$	24.4949	1.13349	0.566744	+	0.823894i	$0.308203\pi$
$468$	−2.00000	−0.0924500
$469$	−48.0000	−2.21643
$470$	0	0
$471$	−4.89898	−0.225733
$472$	12.0000	0.552345
$473$	48.0000	2.20704
$474$	14.6969	0.675053
$475$	0	0
$476$	−24.0000	−1.10004
$477$	9.79796	0.448618
$478$	−24.0000	−1.09773
$479$	−19.5959	−0.895360	−0.447680	−	0.894194i	$-0.647750\pi$
$479$	−19.5959	−0.895360	−0.447680	+	0.894194i	$0.647750\pi$
$480$	0	0
$481$	9.79796	0.446748
$482$	−19.5959	−0.892570
$483$	0	0
$484$	13.0000	0.590909
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−4.89898	−0.221766
$489$	−20.0000	−0.904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	−29.3939	−1.32383
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	4.89898	0.219529
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	−24.4949	−1.09326
$503$	9.79796	0.436869	0.218435	−	0.975852i	$-0.429905\pi$
$503$	9.79796	0.436869	0.218435	+	0.975852i	$0.429905\pi$
$504$	4.89898	0.218218
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	16.0000	0.709885
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	9.79796	0.433436
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	30.0000	1.32324
$515$	0	0
$516$	9.79796	0.431331
$517$	0	0
$518$	−24.0000	−1.05450
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−4.89898	−0.214628	−0.107314	−	0.994225i	$-0.534225\pi$
$521$	−4.89898	−0.214628	−0.107314	+	0.994225i	$0.534225\pi$
$522$	6.00000	0.262613
$523$	9.79796	0.428435	0.214217	−	0.976786i	$-0.431280\pi$
$523$	9.79796	0.428435	0.214217	+	0.976786i	$0.431280\pi$
$524$	12.0000	0.524222
$525$	24.4949	1.06904
$526$	9.79796	0.427211
$527$	−39.1918	−1.70722
$528$	4.89898	0.213201
$529$	0	0
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	14.6969	0.635999
$535$	0	0
$536$	−9.79796	−0.423207
$537$	−12.0000	−0.517838
$538$	−18.0000	−0.776035
$539$	83.2827	3.58724
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−16.0000	−0.687259
$543$	−4.89898	−0.210235
$544$	−4.89898	−0.210042
$545$	0	0
$546$	−9.79796	−0.419314
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−4.89898	−0.209274
$549$	4.89898	0.209083
$550$	24.4949	1.04447
$551$	0	0
$552$	0	0
$553$	72.0000	3.06175
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−19.5959	−0.830306	−0.415153	−	0.909752i	$-0.636272\pi$
$557$	−19.5959	−0.830306	−0.415153	+	0.909752i	$0.636272\pi$
$558$	8.00000	0.338667
$559$	−19.5959	−0.828819
$560$	0	0
$561$	24.0000	1.01328
$562$	−4.89898	−0.206651
$563$	14.6969	0.619402	0.309701	−	0.950834i	$-0.399771\pi$
$563$	14.6969	0.619402	0.309701	+	0.950834i	$0.399771\pi$
$564$	0	0
$565$	0	0
$566$	−19.5959	−0.823678
$567$	−4.89898	−0.205738
$568$	0	0
$569$	−34.2929	−1.43763	−0.718816	−	0.695201i	$-0.755315\pi$
$569$	−34.2929	−1.43763	−0.718816	+	0.695201i	$0.755315\pi$
$570$	0	0
$571$	−9.79796	−0.410032	−0.205016	−	0.978759i	$-0.565725\pi$
$571$	−9.79796	−0.410032	−0.205016	+	0.978759i	$0.565725\pi$
$572$	−9.79796	−0.409673
$573$	−19.5959	−0.818631
$574$	−29.3939	−1.22688
$575$	0	0
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−7.00000	−0.291162
$579$	−10.0000	−0.415586
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	48.0000	1.98796
$584$	2.00000	0.0827606
$585$	0	0
$586$	9.79796	0.404750
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	17.0000	0.701068
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−4.89898	−0.201347
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−4.89898	−0.201008
$595$	0	0
$596$	−19.5959	−0.802680
$597$	−14.6969	−0.601506
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	5.00000	0.204124
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	48.0000	1.95633
$603$	9.79796	0.399004
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−6.00000	−0.243733
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	29.3939	1.19110
$610$	0	0
$611$	0	0
$612$	4.89898	0.198030
$613$	24.4949	0.989340	0.494670	−	0.869081i	$-0.335289\pi$
$613$	24.4949	0.989340	0.494670	+	0.869081i	$0.335289\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	24.0000	0.966988
$617$	24.4949	0.986127	0.493064	−	0.869993i	$-0.335877\pi$
$617$	24.4949	0.986127	0.493064	+	0.869993i	$0.335877\pi$
$618$	4.89898	0.197066
$619$	39.1918	1.57525	0.787626	−	0.616153i	$-0.211310\pi$
$619$	39.1918	1.57525	0.787626	+	0.616153i	$0.211310\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	72.0000	2.88462
$624$	−2.00000	−0.0800641
$625$	25.0000	1.00000
$626$	9.79796	0.391605
$627$	0	0
$628$	−4.89898	−0.195491
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−4.89898	−0.195025	−0.0975126	−	0.995234i	$-0.531089\pi$
$631$	−4.89898	−0.195025	−0.0975126	+	0.995234i	$0.531089\pi$
$632$	14.6969	0.584613
$633$	−20.0000	−0.794929
$634$	6.00000	0.238290
$635$	0	0
$636$	9.79796	0.388514
$637$	−34.0000	−1.34713
$638$	29.3939	1.16371
$639$	0	0
$640$	0	0
$641$	24.4949	0.967490	0.483745	−	0.875209i	$-0.339276\pi$
$641$	24.4949	0.967490	0.483745	+	0.875209i	$0.339276\pi$
$642$	4.89898	0.193347
$643$	−29.3939	−1.15918	−0.579591	−	0.814908i	$-0.696788\pi$
$643$	−29.3939	−1.15918	−0.579591	+	0.814908i	$0.696788\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	−58.7878	−2.30762
$650$	−10.0000	−0.392232
$651$	39.1918	1.53605
$652$	−20.0000	−0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	4.89898	0.191565
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	24.4949	0.954186	0.477093	−	0.878853i	$-0.341691\pi$
$659$	24.4949	0.954186	0.477093	+	0.878853i	$0.341691\pi$
$660$	0	0
$661$	24.4949	0.952741	0.476371	−	0.879245i	$-0.341952\pi$
$661$	24.4949	0.952741	0.476371	+	0.879245i	$0.341952\pi$
$662$	−4.00000	−0.155464
$663$	−9.79796	−0.380521
$664$	4.89898	0.190117
$665$	0	0
$666$	4.89898	0.189832
$667$	0	0
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	24.0000	0.926510
$672$	4.89898	0.188982
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−9.79796	−0.377403
$675$	−5.00000	−0.192450
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	14.6969	0.564433
$679$	0	0
$680$	0	0
$681$	14.6969	0.563188
$682$	39.1918	1.50073
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	0	0
$686$	48.9898	1.87044
$687$	−24.4949	−0.934539
$688$	9.79796	0.373544
$689$	−19.5959	−0.746545
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−6.00000	−0.228086
$693$	−24.0000	−0.911685
$694$	−12.0000	−0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	−29.3939	−1.11337
$698$	−2.00000	−0.0757011
$699$	−6.00000	−0.226941
$700$	24.4949	0.925820
$701$	29.3939	1.11019	0.555096	−	0.831786i	$-0.312682\pi$
$701$	29.3939	1.11019	0.555096	+	0.831786i	$0.312682\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	4.89898	0.184637
$705$	0	0
$706$	18.0000	0.677439
$707$	−29.3939	−1.10547
$708$	−12.0000	−0.450988
$709$	44.0908	1.65587	0.827933	−	0.560828i	$-0.189517\pi$
$709$	44.0908	1.65587	0.827933	+	0.560828i	$0.189517\pi$
$710$	0	0
$711$	−14.6969	−0.551178
$712$	14.6969	0.550791
$713$	0	0
$714$	24.0000	0.898177
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	−19.5959	−0.731313
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	19.0000	0.707107
$723$	19.5959	0.728780
$724$	−4.89898	−0.182069
$725$	30.0000	1.11417
$726$	−13.0000	−0.482475
$727$	14.6969	0.545079	0.272540	−	0.962145i	$-0.412136\pi$
$727$	14.6969	0.545079	0.272540	+	0.962145i	$0.412136\pi$
$728$	−9.79796	−0.363137
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	4.89898	0.181071
$733$	24.4949	0.904740	0.452370	−	0.891830i	$-0.350579\pi$
$733$	24.4949	0.904740	0.452370	+	0.891830i	$0.350579\pi$
$734$	−24.4949	−0.904123
$735$	0	0
$736$	0	0
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	48.0000	1.76214
$743$	−48.9898	−1.79726	−0.898631	−	0.438706i	$-0.855437\pi$
$743$	−48.9898	−1.79726	−0.898631	+	0.438706i	$0.855437\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−14.6969	−0.538093
$747$	−4.89898	−0.179244
$748$	24.0000	0.877527
$749$	24.0000	0.876941
$750$	0	0
$751$	4.89898	0.178766	0.0893832	−	0.995997i	$-0.471510\pi$
$751$	4.89898	0.178766	0.0893832	+	0.995997i	$0.471510\pi$
$752$	0	0
$753$	24.4949	0.892644
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−4.89898	−0.178174
$757$	34.2929	1.24640	0.623198	−	0.782064i	$-0.285833\pi$
$757$	34.2929	1.24640	0.623198	+	0.782064i	$0.285833\pi$
$758$	19.5959	0.711756
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−16.0000	−0.579619
$763$	24.0000	0.868858
$764$	−19.5959	−0.708955
$765$	0	0
$766$	−9.79796	−0.354015
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	−29.3939	−1.05997	−0.529985	−	0.848007i	$-0.677803\pi$
$769$	−29.3939	−1.05997	−0.529985	+	0.848007i	$0.677803\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	−10.0000	−0.359908
$773$	29.3939	1.05722	0.528612	−	0.848863i	$-0.322713\pi$
$773$	29.3939	1.05722	0.528612	+	0.848863i	$0.322713\pi$
$774$	−9.79796	−0.352180
$775$	40.0000	1.43684
$776$	0	0
$777$	24.0000	0.860995
$778$	−19.5959	−0.702548
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	17.0000	0.607143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−9.79796	−0.349260	−0.174630	−	0.984634i	$-0.555873\pi$
$787$	−9.79796	−0.349260	−0.174630	+	0.984634i	$0.555873\pi$
$788$	−6.00000	−0.213741
$789$	−9.79796	−0.348817
$790$	0	0
$791$	72.0000	2.56003
$792$	−4.89898	−0.174078
$793$	−9.79796	−0.347936
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−14.6969	−0.520919
$797$	−39.1918	−1.38825	−0.694123	−	0.719856i	$-0.744208\pi$
$797$	−39.1918	−1.38825	−0.694123	+	0.719856i	$0.744208\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	−14.6969	−0.519291
$802$	34.2929	1.21092
$803$	−9.79796	−0.345762
$804$	9.79796	0.345547
$805$	0	0
$806$	−16.0000	−0.563576
$807$	18.0000	0.633630
$808$	−6.00000	−0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	29.3939	1.03152
$813$	16.0000	0.561144
$814$	24.0000	0.841200
$815$	0	0
$816$	4.89898	0.171499
$817$	0	0
$818$	−14.0000	−0.489499
$819$	9.79796	0.342368
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	4.89898	0.170872
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	4.89898	0.170664
$825$	−24.4949	−0.852803
$826$	−58.7878	−2.04549
$827$	−14.6969	−0.511063	−0.255531	−	0.966801i	$-0.582250\pi$
$827$	−14.6969	−0.511063	−0.255531	+	0.966801i	$0.582250\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−2.00000	−0.0693375
$833$	83.2827	2.88557
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−14.6969	−0.507697
$839$	−48.9898	−1.69132	−0.845658	−	0.533726i	$-0.820792\pi$
$839$	−48.9898	−1.69132	−0.845658	+	0.533726i	$0.820792\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−4.89898	−0.168830
$843$	4.89898	0.168730
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	−63.6867	−2.18830
$848$	9.79796	0.336463
$849$	19.5959	0.672530
$850$	24.4949	0.840168
$851$	0	0
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	4.89898	0.167444
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	9.79796	0.334497
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	29.3939	1.00174
$862$	−29.3939	−1.00116
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−9.79796	−0.332948
$867$	7.00000	0.237732
$868$	39.1918	1.33026
$869$	−72.0000	−2.44243
$870$	0	0
$871$	−19.5959	−0.663982
$872$	4.89898	0.165900
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	40.0000	1.34993
$879$	−9.79796	−0.330477
$880$	0	0
$881$	24.4949	0.825254	0.412627	−	0.910900i	$-0.364611\pi$
$881$	24.4949	0.825254	0.412627	+	0.910900i	$0.364611\pi$
$882$	−17.0000	−0.572420
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−9.79796	−0.329541
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	4.89898	0.164399
$889$	−78.3837	−2.62890
$890$	0	0
$891$	4.89898	0.164122
$892$	8.00000	0.267860
$893$	0	0
$894$	19.5959	0.655386
$895$	0	0
$896$	4.89898	0.163663
$897$	0	0
$898$	30.0000	1.00111
$899$	48.0000	1.60089
$900$	−5.00000	−0.166667
$901$	48.0000	1.59911
$902$	29.3939	0.978709
$903$	−48.0000	−1.59734
$904$	14.6969	0.488813
$905$	0	0
$906$	16.0000	0.531564
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	14.6969	0.487735
$909$	6.00000	0.199007
$910$	0	0
$911$	9.79796	0.324621	0.162310	−	0.986740i	$-0.448105\pi$
$911$	9.79796	0.324621	0.162310	+	0.986740i	$0.448105\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	−19.5959	−0.648175
$915$	0	0
$916$	−24.4949	−0.809334
$917$	−58.7878	−1.94134
$918$	−4.89898	−0.161690
$919$	−14.6969	−0.484807	−0.242404	−	0.970175i	$-0.577936\pi$
$919$	−14.6969	−0.484807	−0.242404	+	0.970175i	$0.577936\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	−18.0000	−0.592798
$923$	0	0
$924$	−24.0000	−0.789542
$925$	24.4949	0.805387
$926$	16.0000	0.525793
$927$	−4.89898	−0.160904
$928$	6.00000	0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−24.0000	−0.785725
$934$	−24.4949	−0.801498
$935$	0	0
$936$	2.00000	0.0653720
$937$	−19.5959	−0.640171	−0.320085	−	0.947389i	$-0.603712\pi$
$937$	−19.5959	−0.640171	−0.320085	+	0.947389i	$0.603712\pi$
$938$	48.0000	1.56726
$939$	−9.79796	−0.319744
$940$	0	0
$941$	19.5959	0.638809	0.319404	−	0.947619i	$-0.396517\pi$
$941$	19.5959	0.638809	0.319404	+	0.947619i	$0.396517\pi$
$942$	4.89898	0.159617
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−14.6969	−0.477334
$949$	4.00000	0.129845
$950$	0	0
$951$	−6.00000	−0.194563
$952$	24.0000	0.777844
$953$	−34.2929	−1.11085	−0.555427	−	0.831565i	$-0.687445\pi$
$953$	−34.2929	−1.11085	−0.555427	+	0.831565i	$0.687445\pi$
$954$	−9.79796	−0.317221
$955$	0	0
$956$	24.0000	0.776215
$957$	−29.3939	−0.950169
$958$	19.5959	0.633115
$959$	24.0000	0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	−9.79796	−0.315899
$963$	−4.89898	−0.157867
$964$	19.5959	0.631142
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−13.0000	−0.417836
$969$	0	0
$970$	0	0
$971$	−24.4949	−0.786079	−0.393039	−	0.919522i	$-0.628576\pi$
$971$	−24.4949	−0.786079	−0.393039	+	0.919522i	$0.628576\pi$
$972$	1.00000	0.0320750
$973$	19.5959	0.628216
$974$	−8.00000	−0.256337
$975$	10.0000	0.320256
$976$	4.89898	0.156813
$977$	−24.4949	−0.783661	−0.391831	−	0.920037i	$-0.628158\pi$
$977$	−24.4949	−0.783661	−0.391831	+	0.920037i	$0.628158\pi$
$978$	20.0000	0.639529
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−4.89898	−0.156412
$982$	−12.0000	−0.382935
$983$	9.79796	0.312506	0.156253	−	0.987717i	$-0.450058\pi$
$983$	9.79796	0.312506	0.156253	+	0.987717i	$0.450058\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	29.3939	0.936092
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	8.00000	0.254000
$993$	4.00000	0.126936
$994$	0	0
$995$	0	0
$996$	−4.89898	−0.155230
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	20.0000	0.633089
$999$	−4.89898	−0.154997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3174.2.a.k.1.1	✓	2
3.2	odd	2		9522.2.a.bf.1.1		2
23.22	odd	2	inner	3174.2.a.k.1.2	yes	2
69.68	even	2		9522.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.k.1.1	✓	2	1.1	even	1	trivial
3174.2.a.k.1.2	yes	2	23.22	odd	2	inner
9522.2.a.bf.1.1		2	3.2	odd	2
9522.2.a.bf.1.2		2	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.89898 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.89898 q^{14} +1.00000 q^{16} +4.89898 q^{17} -1.00000 q^{18} -4.89898 q^{21} -4.89898 q^{22} -1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -4.89898 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -4.89898 q^{34} +1.00000 q^{36} -4.89898 q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.89898 q^{42} +9.79796 q^{43} +4.89898 q^{44} +1.00000 q^{48} +17.0000 q^{49} +5.00000 q^{50} +4.89898 q^{51} -2.00000 q^{52} +9.79796 q^{53} -1.00000 q^{54} +4.89898 q^{56} +6.00000 q^{58} -12.0000 q^{59} +4.89898 q^{61} +8.00000 q^{62} -4.89898 q^{63} +1.00000 q^{64} -4.89898 q^{66} +9.79796 q^{67} +4.89898 q^{68} -1.00000 q^{72} -2.00000 q^{73} +4.89898 q^{74} -5.00000 q^{75} -24.0000 q^{77} +2.00000 q^{78} -14.6969 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.89898 q^{83} -4.89898 q^{84} -9.79796 q^{86} -6.00000 q^{87} -4.89898 q^{88} -14.6969 q^{89} +9.79796 q^{91} -8.00000 q^{93} -1.00000 q^{96} -17.0000 q^{98} +4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 2 q^{24} - 10 q^{25} + 4 q^{26} + 2 q^{27} - 12 q^{29} - 16 q^{31} - 2 q^{32} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 2 q^{48} + 34 q^{49} + 10 q^{50} - 4 q^{52} - 2 q^{54} + 12 q^{58} - 24 q^{59} + 16 q^{62} + 2 q^{64} - 2 q^{72} - 4 q^{73} - 10 q^{75} - 48 q^{77} + 4 q^{78} + 2 q^{81} + 12 q^{82} - 12 q^{87} - 16 q^{93} - 2 q^{96} - 34 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^18 - 2 * q^24 - 10 * q^25 + 4 * q^26 + 2 * q^27 - 12 * q^29 - 16 * q^31 - 2 * q^32 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 2 * q^48 + 34 * q^49 + 10 * q^50 - 4 * q^52 - 2 * q^54 + 12 * q^58 - 24 * q^59 + 16 * q^62 + 2 * q^64 - 2 * q^72 - 4 * q^73 - 10 * q^75 - 48 * q^77 + 4 * q^78 + 2 * q^81 + 12 * q^82 - 12 * q^87 - 16 * q^93 - 2 * q^96 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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