LMFDB - Embedded newform 2325.2.c.j.1024.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1024,2325)] 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("2325.1024"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,-4,0,8,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 465)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.j.1024.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -4.73205i q^{7} +1.73205i q^{8} -1.00000 q^{9} -1.46410 q^{11} -1.00000i q^{12} +1.26795i q^{13} +8.19615 q^{14} -5.00000 q^{16} -1.46410i q^{17} -1.73205i q^{18} -5.46410 q^{19} +4.73205 q^{21} -2.53590i q^{22} +0.535898i q^{23} -1.73205 q^{24} -2.19615 q^{26} -1.00000i q^{27} +4.73205i q^{28} -0.732051 q^{29} -1.00000 q^{31} -5.19615i q^{32} -1.46410i q^{33} +2.53590 q^{34} +1.00000 q^{36} -6.73205i q^{37} -9.46410i q^{38} -1.26795 q^{39} +3.46410 q^{41} +8.19615i q^{42} +4.00000i q^{43} +1.46410 q^{44} -0.928203 q^{46} -6.00000i q^{47} -5.00000i q^{48} -15.3923 q^{49} +1.46410 q^{51} -1.26795i q^{52} -12.3923i q^{53} +1.73205 q^{54} +8.19615 q^{56} -5.46410i q^{57} -1.26795i q^{58} +12.1962 q^{59} -10.3923 q^{61} -1.73205i q^{62} +4.73205i q^{63} -1.00000 q^{64} +2.53590 q^{66} -10.1962i q^{67} +1.46410i q^{68} -0.535898 q^{69} -3.80385 q^{71} -1.73205i q^{72} -5.66025i q^{73} +11.6603 q^{74} +5.46410 q^{76} +6.92820i q^{77} -2.19615i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -15.4641i q^{83} -4.73205 q^{84} -6.92820 q^{86} -0.732051i q^{87} -2.53590i q^{88} -7.26795 q^{89} +6.00000 q^{91} -0.535898i q^{92} -1.00000i q^{93} +10.3923 q^{94} +5.19615 q^{96} -2.00000i q^{97} -26.6603i q^{98} +1.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9} + 8 q^{11} + 12 q^{14} - 20 q^{16} - 8 q^{19} + 12 q^{21} + 12 q^{26} + 4 q^{29} - 4 q^{31} + 24 q^{34} + 4 q^{36} - 12 q^{39} - 8 q^{44} + 24 q^{46} - 20 q^{49} - 8 q^{51} + 12 q^{56}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 + 8 * q^11 + 12 * q^14 - 20 * q^16 - 8 * q^19 + 12 * q^21 + 12 * q^26 + 4 * q^29 - 4 * q^31 + 24 * q^34 + 4 * q^36 - 12 * q^39 - 8 * q^44 + 24 * q^46 - 20 * q^49 - 8 * q^51 + 12 * q^56 + 28 * q^59 - 4 * q^64 + 24 * q^66 - 16 * q^69 - 36 * q^71 + 12 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 12 * q^84 - 36 * q^89 + 24 * q^91 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times$.

$n$	$652$	$776$	$1801$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.73205	−0.707107
$7$	− 4.73205i	− 1.78855i	−0.447521	−	0.894274i	$-0.647693\pi$
$7$	− 4.73205i	− 1.78855i	0.447521	−	0.894274i	$-0.352307\pi$
$8$	1.73205i	0.612372i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.26795i	0.351666i	0.984420	+	0.175833i	$0.0562618\pi$
$13$	1.26795i	0.351666i	−0.984420	+	0.175833i	$0.943738\pi$
$14$	8.19615	2.19051
$15$	0	0
$16$	−5.00000	−1.25000
$17$	− 1.46410i	− 0.355097i	−0.984112	−	0.177548i	$-0.943183\pi$
$17$	− 1.46410i	− 0.355097i	0.984112	−	0.177548i	$-0.0568166\pi$
$18$	− 1.73205i	− 0.408248i
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	4.73205	1.03262
$22$	− 2.53590i	− 0.540655i
$23$	0.535898i	0.111743i	0.998438	+	0.0558713i	$0.0177936\pi$
$23$	0.535898i	0.111743i	−0.998438	+	0.0558713i	$0.982206\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	−2.19615	−0.430701
$27$	− 1.00000i	− 0.192450i
$28$	4.73205i	0.894274i
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 5.19615i	− 0.918559i
$33$	− 1.46410i	− 0.254867i
$34$	2.53590	0.434903
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.73205i	− 1.10674i	−0.832935	−	0.553371i	$-0.813341\pi$
$37$	− 6.73205i	− 1.10674i	0.832935	−	0.553371i	$-0.186659\pi$
$38$	− 9.46410i	− 1.53528i
$39$	−1.26795	−0.203034
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	8.19615i	1.26469i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	1.46410	0.220722
$45$	0	0
$46$	−0.928203	−0.136856
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	− 5.00000i	− 0.721688i
$49$	−15.3923	−2.19890
$50$	0	0
$51$	1.46410	0.205015
$52$	− 1.26795i	− 0.175833i
$53$	− 12.3923i	− 1.70221i	−0.524992	−	0.851107i	$-0.675932\pi$
$53$	− 12.3923i	− 1.70221i	0.524992	−	0.851107i	$-0.324068\pi$
$54$	1.73205	0.235702
$55$	0	0
$56$	8.19615	1.09526
$57$	− 5.46410i	− 0.723738i
$58$	− 1.26795i	− 0.166490i
$59$	12.1962	1.58780	0.793902	−	0.608046i	$-0.208046\pi$
$59$	12.1962	1.58780	0.793902	+	0.608046i	$0.208046\pi$
$60$	0	0
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	− 1.73205i	− 0.219971i
$63$	4.73205i	0.596182i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.53590	0.312148
$67$	− 10.1962i	− 1.24566i	−0.782358	−	0.622829i	$-0.785983\pi$
$67$	− 10.1962i	− 1.24566i	0.782358	−	0.622829i	$-0.214017\pi$
$68$	1.46410i	0.177548i
$69$	−0.535898	−0.0645146
$70$	0	0
$71$	−3.80385	−0.451434	−0.225717	−	0.974193i	$-0.572472\pi$
$71$	−3.80385	−0.451434	−0.225717	+	0.974193i	$0.572472\pi$
$72$	− 1.73205i	− 0.204124i
$73$	− 5.66025i	− 0.662483i	−0.943546	−	0.331241i	$-0.892533\pi$
$73$	− 5.66025i	− 0.662483i	0.943546	−	0.331241i	$-0.107467\pi$
$74$	11.6603	1.35548
$75$	0	0
$76$	5.46410	0.626775
$77$	6.92820i	0.789542i
$78$	− 2.19615i	− 0.248665i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	− 15.4641i	− 1.69741i	−0.528870	−	0.848703i	$-0.677384\pi$
$83$	− 15.4641i	− 1.69741i	0.528870	−	0.848703i	$-0.322616\pi$
$84$	−4.73205	−0.516309
$85$	0	0
$86$	−6.92820	−0.747087
$87$	− 0.732051i	− 0.0784841i
$88$	− 2.53590i	− 0.270328i
$89$	−7.26795	−0.770401	−0.385201	−	0.922833i	$-0.625868\pi$
$89$	−7.26795	−0.770401	−0.385201	+	0.922833i	$0.625868\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	− 0.535898i	− 0.0558713i
$93$	− 1.00000i	− 0.103695i
$94$	10.3923	1.07188
$95$	0	0
$96$	5.19615	0.530330
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 26.6603i	− 2.69309i
$99$	1.46410	0.147148
$100$	0	0
$101$	4.53590	0.451339	0.225669	−	0.974204i	$-0.427543\pi$
$101$	4.53590	0.451339	0.225669	+	0.974204i	$0.427543\pi$
$102$	2.53590i	0.251091i
$103$	− 3.26795i	− 0.322001i	−0.986954	−	0.161000i	$-0.948528\pi$
$103$	− 3.26795i	− 0.322001i	0.986954	−	0.161000i	$-0.0514720\pi$
$104$	−2.19615	−0.215350
$105$	0	0
$106$	21.4641	2.08478
$107$	− 15.8564i	− 1.53290i	−0.642306	−	0.766448i	$-0.722022\pi$
$107$	− 15.8564i	− 1.53290i	0.642306	−	0.766448i	$-0.277978\pi$
$108$	1.00000i	0.0962250i
$109$	2.53590	0.242895	0.121448	−	0.992598i	$-0.461246\pi$
$109$	2.53590	0.242895	0.121448	+	0.992598i	$0.461246\pi$
$110$	0	0
$111$	6.73205	0.638978
$112$	23.6603i	2.23568i
$113$	4.92820i	0.463606i	0.972763	+	0.231803i	$0.0744625\pi$
$113$	4.92820i	0.463606i	−0.972763	+	0.231803i	$0.925537\pi$
$114$	9.46410	0.886394
$115$	0	0
$116$	0.732051	0.0679692
$117$	− 1.26795i	− 0.117222i
$118$	21.1244i	1.94465i
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−8.85641	−0.805128
$122$	− 18.0000i	− 1.62964i
$123$	3.46410i	0.312348i
$124$	1.00000	0.0898027
$125$	0	0
$126$	−8.19615	−0.730171
$127$	− 10.5359i	− 0.934910i	−0.884017	−	0.467455i	$-0.845171\pi$
$127$	− 10.5359i	− 0.934910i	0.884017	−	0.467455i	$-0.154829\pi$
$128$	− 12.1244i	− 1.07165i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	6.73205	0.588182	0.294091	−	0.955777i	$-0.404983\pi$
$131$	6.73205	0.588182	0.294091	+	0.955777i	$0.404983\pi$
$132$	1.46410i	0.127434i
$133$	25.8564i	2.24203i
$134$	17.6603	1.52561
$135$	0	0
$136$	2.53590	0.217451
$137$	18.9282i	1.61715i	0.588396	+	0.808573i	$0.299760\pi$
$137$	18.9282i	1.61715i	−0.588396	+	0.808573i	$0.700240\pi$
$138$	− 0.928203i	− 0.0790139i
$139$	−9.85641	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$139$	−9.85641	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	− 6.58846i	− 0.552891i
$143$	− 1.85641i	− 0.155241i
$144$	5.00000	0.416667
$145$	0	0
$146$	9.80385	0.811372
$147$	− 15.3923i	− 1.26954i
$148$	6.73205i	0.553371i
$149$	−21.3205	−1.74664	−0.873322	−	0.487143i	$-0.838039\pi$
$149$	−21.3205	−1.74664	−0.873322	+	0.487143i	$0.838039\pi$
$150$	0	0
$151$	−20.9282	−1.70311	−0.851557	−	0.524263i	$-0.824341\pi$
$151$	−20.9282	−1.70311	−0.851557	+	0.524263i	$0.824341\pi$
$152$	− 9.46410i	− 0.767640i
$153$	1.46410i	0.118366i
$154$	−12.0000	−0.966988
$155$	0	0
$156$	1.26795	0.101517
$157$	− 12.9282i	− 1.03178i	−0.856654	−	0.515891i	$-0.827461\pi$
$157$	− 12.9282i	− 1.03178i	0.856654	−	0.515891i	$-0.172539\pi$
$158$	17.3205i	1.37795i
$159$	12.3923	0.982774
$160$	0	0
$161$	2.53590	0.199857
$162$	1.73205i	0.136083i
$163$	14.1962i	1.11193i	0.831206	+	0.555964i	$0.187651\pi$
$163$	14.1962i	1.11193i	−0.831206	+	0.555964i	$0.812349\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	13.8564i	1.07224i	0.844141	+	0.536120i	$0.180111\pi$
$167$	13.8564i	1.07224i	−0.844141	+	0.536120i	$0.819889\pi$
$168$	8.19615i	0.632347i
$169$	11.3923	0.876331
$170$	0	0
$171$	5.46410	0.417850
$172$	− 4.00000i	− 0.304997i
$173$	22.7846i	1.73228i	0.499800	+	0.866141i	$0.333407\pi$
$173$	22.7846i	1.73228i	−0.499800	+	0.866141i	$0.666593\pi$
$174$	1.26795	0.0961230
$175$	0	0
$176$	7.32051	0.551804
$177$	12.1962i	0.916719i
$178$	− 12.5885i	− 0.943545i
$179$	9.07180	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$179$	9.07180	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$180$	0	0
$181$	−10.3923	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$181$	−10.3923	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$182$	10.3923i	0.770329i
$183$	− 10.3923i	− 0.768221i
$184$	−0.928203	−0.0684280
$185$	0	0
$186$	1.73205	0.127000
$187$	2.14359i	0.156755i
$188$	6.00000i	0.437595i
$189$	−4.73205	−0.344206
$190$	0	0
$191$	−11.8038	−0.854096	−0.427048	−	0.904229i	$-0.640447\pi$
$191$	−11.8038	−0.854096	−0.427048	+	0.904229i	$0.640447\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	23.8564i	1.71722i	0.512628	+	0.858611i	$0.328672\pi$
$193$	23.8564i	1.71722i	−0.512628	+	0.858611i	$0.671328\pi$
$194$	3.46410	0.248708
$195$	0	0
$196$	15.3923	1.09945
$197$	13.0718i	0.931327i	0.884962	+	0.465663i	$0.154184\pi$
$197$	13.0718i	0.931327i	−0.884962	+	0.465663i	$0.845816\pi$
$198$	2.53590i	0.180218i
$199$	−16.9282	−1.20001	−0.600004	−	0.799997i	$-0.704834\pi$
$199$	−16.9282	−1.20001	−0.600004	+	0.799997i	$0.704834\pi$
$200$	0	0
$201$	10.1962	0.719181
$202$	7.85641i	0.552775i
$203$	3.46410i	0.243132i
$204$	−1.46410	−0.102508
$205$	0	0
$206$	5.66025	0.394369
$207$	− 0.535898i	− 0.0372475i
$208$	− 6.33975i	− 0.439582i
$209$	8.00000	0.553372
$210$	0	0
$211$	7.32051	0.503965	0.251982	−	0.967732i	$-0.418918\pi$
$211$	7.32051	0.503965	0.251982	+	0.967732i	$0.418918\pi$
$212$	12.3923i	0.851107i
$213$	− 3.80385i	− 0.260635i
$214$	27.4641	1.87741
$215$	0	0
$216$	1.73205	0.117851
$217$	4.73205i	0.321233i
$218$	4.39230i	0.297484i
$219$	5.66025	0.382485
$220$	0	0
$221$	1.85641	0.124875
$222$	11.6603i	0.782585i
$223$	0.392305i	0.0262707i	0.999914	+	0.0131353i	$0.00418123\pi$
$223$	0.392305i	0.0262707i	−0.999914	+	0.0131353i	$0.995819\pi$
$224$	−24.5885	−1.64289
$225$	0	0
$226$	−8.53590	−0.567800
$227$	13.3205i	0.884113i	0.896987	+	0.442057i	$0.145751\pi$
$227$	13.3205i	0.884113i	−0.896987	+	0.442057i	$0.854249\pi$
$228$	5.46410i	0.361869i
$229$	−18.7846	−1.24132	−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846	−1.24132	−0.620661	+	0.784079i	$0.713136\pi$
$230$	0	0
$231$	−6.92820	−0.455842
$232$	− 1.26795i	− 0.0832449i
$233$	− 10.9282i	− 0.715930i	−0.933735	−	0.357965i	$-0.883471\pi$
$233$	− 10.9282i	− 0.715930i	0.933735	−	0.357965i	$-0.116529\pi$
$234$	2.19615	0.143567
$235$	0	0
$236$	−12.1962	−0.793902
$237$	10.0000i	0.649570i
$238$	− 12.0000i	− 0.777844i
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	23.8564	1.53673	0.768363	−	0.640014i	$-0.221072\pi$
$241$	23.8564	1.53673	0.768363	+	0.640014i	$0.221072\pi$
$242$	− 15.3397i	− 0.986076i
$243$	1.00000i	0.0641500i
$244$	10.3923	0.665299
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 6.92820i	− 0.440831i
$248$	− 1.73205i	− 0.109985i
$249$	15.4641	0.979998
$250$	0	0
$251$	2.92820	0.184827	0.0924133	−	0.995721i	$-0.470542\pi$
$251$	2.92820	0.184827	0.0924133	+	0.995721i	$0.470542\pi$
$252$	− 4.73205i	− 0.298091i
$253$	− 0.784610i	− 0.0493280i
$254$	18.2487	1.14503
$255$	0	0
$256$	19.0000	1.18750
$257$	− 9.85641i	− 0.614826i	−0.951576	−	0.307413i	$-0.900537\pi$
$257$	− 9.85641i	− 0.614826i	0.951576	−	0.307413i	$-0.0994633\pi$
$258$	− 6.92820i	− 0.431331i
$259$	−31.8564	−1.97946
$260$	0	0
$261$	0.732051	0.0453128
$262$	11.6603i	0.720373i
$263$	17.0718i	1.05269i	0.850270	+	0.526346i	$0.176438\pi$
$263$	17.0718i	1.05269i	−0.850270	+	0.526346i	$0.823562\pi$
$264$	2.53590	0.156074
$265$	0	0
$266$	−44.7846	−2.74592
$267$	− 7.26795i	− 0.444791i
$268$	10.1962i	0.622829i
$269$	−9.80385	−0.597751	−0.298876	−	0.954292i	$-0.596612\pi$
$269$	−9.80385	−0.597751	−0.298876	+	0.954292i	$0.596612\pi$
$270$	0	0
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	7.32051i	0.443871i
$273$	6.00000i	0.363137i
$274$	−32.7846	−1.98059
$275$	0	0
$276$	0.535898	0.0322573
$277$	− 15.1244i	− 0.908734i	−0.890814	−	0.454367i	$-0.849865\pi$
$277$	− 15.1244i	− 0.908734i	0.890814	−	0.454367i	$-0.150135\pi$
$278$	− 17.0718i	− 1.02390i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−16.5359	−0.986449	−0.493224	−	0.869902i	$-0.664182\pi$
$281$	−16.5359	−0.986449	−0.493224	+	0.869902i	$0.664182\pi$
$282$	10.3923i	0.618853i
$283$	− 7.26795i	− 0.432035i	−0.976390	−	0.216017i	$-0.930693\pi$
$283$	− 7.26795i	− 0.432035i	0.976390	−	0.216017i	$-0.0693068\pi$
$284$	3.80385	0.225717
$285$	0	0
$286$	3.21539	0.190130
$287$	− 16.3923i	− 0.967607i
$288$	5.19615i	0.306186i
$289$	14.8564	0.873906
$290$	0	0
$291$	2.00000	0.117242
$292$	5.66025i	0.331241i
$293$	− 5.07180i	− 0.296298i	−0.988965	−	0.148149i	$-0.952669\pi$
$293$	− 5.07180i	− 0.296298i	0.988965	−	0.148149i	$-0.0473314\pi$
$294$	26.6603	1.55486
$295$	0	0
$296$	11.6603	0.677738
$297$	1.46410i	0.0849558i
$298$	− 36.9282i	− 2.13919i
$299$	−0.679492	−0.0392960
$300$	0	0
$301$	18.9282	1.09100
$302$	− 36.2487i	− 2.08588i
$303$	4.53590i	0.260581i
$304$	27.3205	1.56694
$305$	0	0
$306$	−2.53590	−0.144968
$307$	26.5885i	1.51748i	0.651392	+	0.758742i	$0.274186\pi$
$307$	26.5885i	1.51748i	−0.651392	+	0.758742i	$0.725814\pi$
$308$	− 6.92820i	− 0.394771i
$309$	3.26795	0.185907
$310$	0	0
$311$	13.6603	0.774602	0.387301	−	0.921953i	$-0.373407\pi$
$311$	13.6603	0.774602	0.387301	+	0.921953i	$0.373407\pi$
$312$	− 2.19615i	− 0.124333i
$313$	21.6603i	1.22431i	0.790738	+	0.612155i	$0.209697\pi$
$313$	21.6603i	1.22431i	−0.790738	+	0.612155i	$0.790303\pi$
$314$	22.3923	1.26367
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 22.9282i	− 1.28778i	−0.765120	−	0.643888i	$-0.777320\pi$
$317$	− 22.9282i	− 1.28778i	0.765120	−	0.643888i	$-0.222680\pi$
$318$	21.4641i	1.20365i
$319$	1.07180	0.0600091
$320$	0	0
$321$	15.8564	0.885018
$322$	4.39230i	0.244774i
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−24.5885	−1.36183
$327$	2.53590i	0.140236i
$328$	6.00000i	0.331295i
$329$	−28.3923	−1.56532
$330$	0	0
$331$	24.9282	1.37018	0.685089	−	0.728459i	$-0.259763\pi$
$331$	24.9282	1.37018	0.685089	+	0.728459i	$0.259763\pi$
$332$	15.4641i	0.848703i
$333$	6.73205i	0.368914i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	−23.6603	−1.29077
$337$	18.0526i	0.983386i	0.870769	+	0.491693i	$0.163622\pi$
$337$	18.0526i	0.983386i	−0.870769	+	0.491693i	$0.836378\pi$
$338$	19.7321i	1.07328i
$339$	−4.92820	−0.267663
$340$	0	0
$341$	1.46410	0.0792855
$342$	9.46410i	0.511760i
$343$	39.7128i	2.14429i
$344$	−6.92820	−0.373544
$345$	0	0
$346$	−39.4641	−2.12160
$347$	− 12.7846i	− 0.686314i	−0.939278	−	0.343157i	$-0.888504\pi$
$347$	− 12.7846i	− 0.686314i	0.939278	−	0.343157i	$-0.111496\pi$
$348$	0.732051i	0.0392420i
$349$	−6.53590	−0.349859	−0.174929	−	0.984581i	$-0.555970\pi$
$349$	−6.53590	−0.349859	−0.174929	+	0.984581i	$0.555970\pi$
$350$	0	0
$351$	1.26795	0.0676781
$352$	7.60770i	0.405492i
$353$	4.39230i	0.233779i	0.993145	+	0.116889i	$0.0372923\pi$
$353$	4.39230i	0.233779i	−0.993145	+	0.116889i	$0.962708\pi$
$354$	−21.1244	−1.12275
$355$	0	0
$356$	7.26795	0.385201
$357$	− 6.92820i	− 0.366679i
$358$	15.7128i	0.830448i
$359$	1.66025	0.0876249	0.0438124	−	0.999040i	$-0.486050\pi$
$359$	1.66025	0.0876249	0.0438124	+	0.999040i	$0.486050\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	− 18.0000i	− 0.946059i
$363$	− 8.85641i	− 0.464841i
$364$	−6.00000	−0.314485
$365$	0	0
$366$	18.0000	0.940875
$367$	− 21.8564i	− 1.14090i	−0.821334	−	0.570448i	$-0.806770\pi$
$367$	− 21.8564i	− 1.14090i	0.821334	−	0.570448i	$-0.193230\pi$
$368$	− 2.67949i	− 0.139678i
$369$	−3.46410	−0.180334
$370$	0	0
$371$	−58.6410	−3.04449
$372$	1.00000i	0.0518476i
$373$	− 26.3923i	− 1.36654i	−0.730165	−	0.683271i	$-0.760557\pi$
$373$	− 26.3923i	− 1.36654i	0.730165	−	0.683271i	$-0.239443\pi$
$374$	−3.71281	−0.191985
$375$	0	0
$376$	10.3923	0.535942
$377$	− 0.928203i	− 0.0478049i
$378$	− 8.19615i	− 0.421565i
$379$	−32.3923	−1.66388	−0.831940	−	0.554865i	$-0.812770\pi$
$379$	−32.3923	−1.66388	−0.831940	+	0.554865i	$0.812770\pi$
$380$	0	0
$381$	10.5359	0.539770
$382$	− 20.4449i	− 1.04605i
$383$	13.3205i	0.680646i	0.940309	+	0.340323i	$0.110536\pi$
$383$	13.3205i	0.680646i	−0.940309	+	0.340323i	$0.889464\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	−41.3205	−2.10316
$387$	− 4.00000i	− 0.203331i
$388$	2.00000i	0.101535i
$389$	−12.3397	−0.625650	−0.312825	−	0.949811i	$-0.601275\pi$
$389$	−12.3397	−0.625650	−0.312825	+	0.949811i	$0.601275\pi$
$390$	0	0
$391$	0.784610	0.0396794
$392$	− 26.6603i	− 1.34655i
$393$	6.73205i	0.339587i
$394$	−22.6410	−1.14064
$395$	0	0
$396$	−1.46410	−0.0735739
$397$	23.8564i	1.19732i	0.801004	+	0.598659i	$0.204300\pi$
$397$	23.8564i	1.19732i	−0.801004	+	0.598659i	$0.795700\pi$
$398$	− 29.3205i	− 1.46970i
$399$	−25.8564	−1.29444
$400$	0	0
$401$	9.51666	0.475239	0.237620	−	0.971358i	$-0.423633\pi$
$401$	9.51666	0.475239	0.237620	+	0.971358i	$0.423633\pi$
$402$	17.6603i	0.880813i
$403$	− 1.26795i	− 0.0631610i
$404$	−4.53590	−0.225669
$405$	0	0
$406$	−6.00000	−0.297775
$407$	9.85641i	0.488564i
$408$	2.53590i	0.125546i
$409$	17.3205	0.856444	0.428222	−	0.903674i	$-0.359140\pi$
$409$	17.3205	0.856444	0.428222	+	0.903674i	$0.359140\pi$
$410$	0	0
$411$	−18.9282	−0.933659
$412$	3.26795i	0.161000i
$413$	− 57.7128i	− 2.83986i
$414$	0.928203	0.0456187
$415$	0	0
$416$	6.58846	0.323026
$417$	− 9.85641i	− 0.482670i
$418$	13.8564i	0.677739i
$419$	27.1244	1.32511	0.662556	−	0.749013i	$-0.269472\pi$
$419$	27.1244	1.32511	0.662556	+	0.749013i	$0.269472\pi$
$420$	0	0
$421$	−15.3205	−0.746676	−0.373338	−	0.927695i	$-0.621787\pi$
$421$	−15.3205	−0.746676	−0.373338	+	0.927695i	$0.621787\pi$
$422$	12.6795i	0.617228i
$423$	6.00000i	0.291730i
$424$	21.4641	1.04239
$425$	0	0
$426$	6.58846	0.319212
$427$	49.1769i	2.37984i
$428$	15.8564i	0.766448i
$429$	1.85641	0.0896281
$430$	0	0
$431$	8.87564	0.427525	0.213762	−	0.976886i	$-0.431428\pi$
$431$	8.87564	0.427525	0.213762	+	0.976886i	$0.431428\pi$
$432$	5.00000i	0.240563i
$433$	− 15.5167i	− 0.745683i	−0.927895	−	0.372842i	$-0.878383\pi$
$433$	− 15.5167i	− 0.745683i	0.927895	−	0.372842i	$-0.121617\pi$
$434$	−8.19615	−0.393428
$435$	0	0
$436$	−2.53590	−0.121448
$437$	− 2.92820i	− 0.140075i
$438$	9.80385i	0.468446i
$439$	35.7128	1.70448	0.852240	−	0.523151i	$-0.175244\pi$
$439$	35.7128	1.70448	0.852240	+	0.523151i	$0.175244\pi$
$440$	0	0
$441$	15.3923	0.732967
$442$	3.21539i	0.152941i
$443$	− 30.3923i	− 1.44398i	−0.691902	−	0.721991i	$-0.743227\pi$
$443$	− 30.3923i	− 1.44398i	0.691902	−	0.721991i	$-0.256773\pi$
$444$	−6.73205	−0.319489
$445$	0	0
$446$	−0.679492	−0.0321749
$447$	− 21.3205i	− 1.00843i
$448$	4.73205i	0.223568i
$449$	−40.7321	−1.92226	−0.961132	−	0.276089i	$-0.910962\pi$
$449$	−40.7321	−1.92226	−0.961132	+	0.276089i	$0.910962\pi$
$450$	0	0
$451$	−5.07180	−0.238822
$452$	− 4.92820i	− 0.231803i
$453$	− 20.9282i	− 0.983293i
$454$	−23.0718	−1.08281
$455$	0	0
$456$	9.46410	0.443197
$457$	32.5885i	1.52442i	0.647328	+	0.762212i	$0.275887\pi$
$457$	32.5885i	1.52442i	−0.647328	+	0.762212i	$0.724113\pi$
$458$	− 32.5359i	− 1.52030i
$459$	−1.46410	−0.0683384
$460$	0	0
$461$	−0.732051	−0.0340950	−0.0170475	−	0.999855i	$-0.505427\pi$
$461$	−0.732051	−0.0340950	−0.0170475	+	0.999855i	$0.505427\pi$
$462$	− 12.0000i	− 0.558291i
$463$	− 39.7128i	− 1.84561i	−0.385266	−	0.922805i	$-0.625890\pi$
$463$	− 39.7128i	− 1.84561i	0.385266	−	0.922805i	$-0.374110\pi$
$464$	3.66025	0.169923
$465$	0	0
$466$	18.9282	0.876832
$467$	2.00000i	0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$	2.00000i	0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$	1.26795i	0.0586110i
$469$	−48.2487	−2.22792
$470$	0	0
$471$	12.9282	0.595700
$472$	21.1244i	0.972327i
$473$	− 5.85641i	− 0.269278i
$474$	−17.3205	−0.795557
$475$	0	0
$476$	6.92820	0.317554
$477$	12.3923i	0.567405i
$478$	24.0000i	1.09773i
$479$	27.8038	1.27039	0.635195	−	0.772352i	$-0.280920\pi$
$479$	27.8038	1.27039	0.635195	+	0.772352i	$0.280920\pi$
$480$	0	0
$481$	8.53590	0.389203
$482$	41.3205i	1.88210i
$483$	2.53590i	0.115387i
$484$	8.85641	0.402564
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	10.5359i	0.477427i	0.971090	+	0.238714i	$0.0767257\pi$
$487$	10.5359i	0.477427i	−0.971090	+	0.238714i	$0.923274\pi$
$488$	− 18.0000i	− 0.814822i
$489$	−14.1962	−0.641972
$490$	0	0
$491$	24.3923	1.10081	0.550405	−	0.834898i	$-0.314473\pi$
$491$	24.3923	1.10081	0.550405	+	0.834898i	$0.314473\pi$
$492$	− 3.46410i	− 0.156174i
$493$	1.07180i	0.0482713i
$494$	12.0000	0.539906
$495$	0	0
$496$	5.00000	0.224507
$497$	18.0000i	0.807410i
$498$	26.7846i	1.20025i
$499$	14.9282	0.668278	0.334139	−	0.942524i	$-0.391554\pi$
$499$	14.9282	0.668278	0.334139	+	0.942524i	$0.391554\pi$
$500$	0	0
$501$	−13.8564	−0.619059
$502$	5.07180i	0.226365i
$503$	11.8564i	0.528651i	0.964434	+	0.264326i	$0.0851493\pi$
$503$	11.8564i	0.528651i	−0.964434	+	0.264326i	$0.914851\pi$
$504$	−8.19615	−0.365086
$505$	0	0
$506$	1.35898	0.0604142
$507$	11.3923i	0.505950i
$508$	10.5359i	0.467455i
$509$	−16.7321	−0.741635	−0.370818	−	0.928706i	$-0.620922\pi$
$509$	−16.7321	−0.741635	−0.370818	+	0.928706i	$0.620922\pi$
$510$	0	0
$511$	−26.7846	−1.18488
$512$	8.66025i	0.382733i
$513$	5.46410i	0.241246i
$514$	17.0718	0.753005
$515$	0	0
$516$	4.00000	0.176090
$517$	8.78461i	0.386347i
$518$	− 55.1769i	− 2.42433i
$519$	−22.7846	−1.00013
$520$	0	0
$521$	21.3205	0.934068	0.467034	−	0.884239i	$-0.345322\pi$
$521$	21.3205	0.934068	0.467034	+	0.884239i	$0.345322\pi$
$522$	1.26795i	0.0554966i
$523$	11.6077i	0.507569i	0.967261	+	0.253785i	$0.0816754\pi$
$523$	11.6077i	0.507569i	−0.967261	+	0.253785i	$0.918325\pi$
$524$	−6.73205	−0.294091
$525$	0	0
$526$	−29.5692	−1.28928
$527$	1.46410i	0.0637773i
$528$	7.32051i	0.318584i
$529$	22.7128	0.987514
$530$	0	0
$531$	−12.1962	−0.529268
$532$	− 25.8564i	− 1.12102i
$533$	4.39230i	0.190252i
$534$	12.5885	0.544756
$535$	0	0
$536$	17.6603	0.762807
$537$	9.07180i	0.391477i
$538$	− 16.9808i	− 0.732093i
$539$	22.5359	0.970690
$540$	0	0
$541$	−6.53590	−0.281000	−0.140500	−	0.990081i	$-0.544871\pi$
$541$	−6.53590	−0.281000	−0.140500	+	0.990081i	$0.544871\pi$
$542$	− 29.0718i	− 1.24874i
$543$	− 10.3923i	− 0.445976i
$544$	−7.60770	−0.326177
$545$	0	0
$546$	−10.3923	−0.444750
$547$	− 34.1962i	− 1.46212i	−0.682312	−	0.731061i	$-0.739025\pi$
$547$	− 34.1962i	− 1.46212i	0.682312	−	0.731061i	$-0.260975\pi$
$548$	− 18.9282i	− 0.808573i
$549$	10.3923	0.443533
$550$	0	0
$551$	4.00000	0.170406
$552$	− 0.928203i	− 0.0395070i
$553$	− 47.3205i	− 2.01227i
$554$	26.1962	1.11297
$555$	0	0
$556$	9.85641	0.418005
$557$	− 9.46410i	− 0.401007i	−0.979693	−	0.200503i	$-0.935742\pi$
$557$	− 9.46410i	− 0.401007i	0.979693	−	0.200503i	$-0.0642578\pi$
$558$	1.73205i	0.0733236i
$559$	−5.07180	−0.214514
$560$	0	0
$561$	−2.14359	−0.0905026
$562$	− 28.6410i	− 1.20815i
$563$	− 40.6410i	− 1.71281i	−0.516301	−	0.856407i	$-0.672691\pi$
$563$	− 40.6410i	− 1.71281i	0.516301	−	0.856407i	$-0.327309\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	12.5885	0.529132
$567$	− 4.73205i	− 0.198727i
$568$	− 6.58846i	− 0.276446i
$569$	28.7321	1.20451	0.602255	−	0.798304i	$-0.294269\pi$
$569$	28.7321	1.20451	0.602255	+	0.798304i	$0.294269\pi$
$570$	0	0
$571$	−23.7128	−0.992350	−0.496175	−	0.868222i	$-0.665263\pi$
$571$	−23.7128	−0.992350	−0.496175	+	0.868222i	$0.665263\pi$
$572$	1.85641i	0.0776203i
$573$	− 11.8038i	− 0.493113i
$574$	28.3923	1.18507
$575$	0	0
$576$	1.00000	0.0416667
$577$	5.32051i	0.221496i	0.993849	+	0.110748i	$0.0353246\pi$
$577$	5.32051i	0.221496i	−0.993849	+	0.110748i	$0.964675\pi$
$578$	25.7321i	1.07031i
$579$	−23.8564	−0.991438
$580$	0	0
$581$	−73.1769	−3.03589
$582$	3.46410i	0.143592i
$583$	18.1436i	0.751431i
$584$	9.80385	0.405686
$585$	0	0
$586$	8.78461	0.362889
$587$	10.9282i	0.451055i	0.974237	+	0.225528i	$0.0724106\pi$
$587$	10.9282i	0.451055i	−0.974237	+	0.225528i	$0.927589\pi$
$588$	15.3923i	0.634768i
$589$	5.46410	0.225144
$590$	0	0
$591$	−13.0718	−0.537702
$592$	33.6603i	1.38343i
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	−2.53590	−0.104049
$595$	0	0
$596$	21.3205	0.873322
$597$	− 16.9282i	− 0.692825i
$598$	− 1.17691i	− 0.0481276i
$599$	−41.3731	−1.69046	−0.845229	−	0.534405i	$-0.820536\pi$
$599$	−41.3731	−1.69046	−0.845229	+	0.534405i	$0.820536\pi$
$600$	0	0
$601$	−37.3205	−1.52234	−0.761168	−	0.648555i	$-0.775374\pi$
$601$	−37.3205	−1.52234	−0.761168	+	0.648555i	$0.775374\pi$
$602$	32.7846i	1.33620i
$603$	10.1962i	0.415219i
$604$	20.9282	0.851557
$605$	0	0
$606$	−7.85641	−0.319145
$607$	− 14.8756i	− 0.603784i	−0.953342	−	0.301892i	$-0.902382\pi$
$607$	− 14.8756i	− 0.603784i	0.953342	−	0.301892i	$-0.0976182\pi$
$608$	28.3923i	1.15146i
$609$	−3.46410	−0.140372
$610$	0	0
$611$	7.60770	0.307774
$612$	− 1.46410i	− 0.0591828i
$613$	10.4449i	0.421864i	0.977501	+	0.210932i	$0.0676499\pi$
$613$	10.4449i	0.421864i	−0.977501	+	0.210932i	$0.932350\pi$
$614$	−46.0526	−1.85853
$615$	0	0
$616$	−12.0000	−0.483494
$617$	− 34.7846i	− 1.40038i	−0.713959	−	0.700188i	$-0.753100\pi$
$617$	− 34.7846i	− 1.40038i	0.713959	−	0.700188i	$-0.246900\pi$
$618$	5.66025i	0.227689i
$619$	−15.8564	−0.637323	−0.318661	−	0.947869i	$-0.603233\pi$
$619$	−15.8564	−0.637323	−0.318661	+	0.947869i	$0.603233\pi$
$620$	0	0
$621$	0.535898	0.0215049
$622$	23.6603i	0.948690i
$623$	34.3923i	1.37790i
$624$	6.33975	0.253793
$625$	0	0
$626$	−37.5167	−1.49947
$627$	8.00000i	0.319489i
$628$	12.9282i	0.515891i
$629$	−9.85641	−0.393001
$630$	0	0
$631$	2.92820	0.116570	0.0582850	−	0.998300i	$-0.481437\pi$
$631$	2.92820	0.116570	0.0582850	+	0.998300i	$0.481437\pi$
$632$	17.3205i	0.688973i
$633$	7.32051i	0.290964i
$634$	39.7128	1.57720
$635$	0	0
$636$	−12.3923	−0.491387
$637$	− 19.5167i	− 0.773278i
$638$	1.85641i	0.0734958i
$639$	3.80385	0.150478
$640$	0	0
$641$	−49.5167	−1.95579	−0.977895	−	0.209095i	$-0.932948\pi$
$641$	−49.5167	−1.95579	−0.977895	+	0.209095i	$0.932948\pi$
$642$	27.4641i	1.08392i
$643$	− 46.2487i	− 1.82387i	−0.410333	−	0.911936i	$-0.634588\pi$
$643$	− 46.2487i	− 1.82387i	0.410333	−	0.911936i	$-0.365412\pi$
$644$	−2.53590	−0.0999284
$645$	0	0
$646$	−13.8564	−0.545173
$647$	3.21539i	0.126410i	0.998001	+	0.0632050i	$0.0201322\pi$
$647$	3.21539i	0.126410i	−0.998001	+	0.0632050i	$0.979868\pi$
$648$	1.73205i	0.0680414i
$649$	−17.8564	−0.700925
$650$	0	0
$651$	−4.73205	−0.185464
$652$	− 14.1962i	− 0.555964i
$653$	25.8564i	1.01184i	0.862581	+	0.505920i	$0.168847\pi$
$653$	25.8564i	1.01184i	−0.862581	+	0.505920i	$0.831153\pi$
$654$	−4.39230	−0.171753
$655$	0	0
$656$	−17.3205	−0.676252
$657$	5.66025i	0.220828i
$658$	− 49.1769i	− 1.91712i
$659$	39.1244	1.52407	0.762034	−	0.647537i	$-0.224201\pi$
$659$	39.1244	1.52407	0.762034	+	0.647537i	$0.224201\pi$
$660$	0	0
$661$	−44.6410	−1.73633	−0.868167	−	0.496272i	$-0.834702\pi$
$661$	−44.6410	−1.73633	−0.868167	+	0.496272i	$0.834702\pi$
$662$	43.1769i	1.67812i
$663$	1.85641i	0.0720969i
$664$	26.7846	1.03944
$665$	0	0
$666$	−11.6603	−0.451826
$667$	− 0.392305i	− 0.0151901i
$668$	− 13.8564i	− 0.536120i
$669$	−0.392305	−0.0151674
$670$	0	0
$671$	15.2154	0.587384
$672$	− 24.5885i	− 0.948520i
$673$	9.26795i	0.357253i	0.983917	+	0.178627i	$0.0571654\pi$
$673$	9.26795i	0.357253i	−0.983917	+	0.178627i	$0.942835\pi$
$674$	−31.2679	−1.20440
$675$	0	0
$676$	−11.3923	−0.438166
$677$	28.3923i	1.09120i	0.838044	+	0.545602i	$0.183699\pi$
$677$	28.3923i	1.09120i	−0.838044	+	0.545602i	$0.816301\pi$
$678$	− 8.53590i	− 0.327819i
$679$	−9.46410	−0.363199
$680$	0	0
$681$	−13.3205	−0.510443
$682$	2.53590i	0.0971046i
$683$	19.1769i	0.733784i	0.930263	+	0.366892i	$0.119578\pi$
$683$	19.1769i	0.733784i	−0.930263	+	0.366892i	$0.880422\pi$
$684$	−5.46410	−0.208925
$685$	0	0
$686$	−68.7846	−2.62621
$687$	− 18.7846i	− 0.716678i
$688$	− 20.0000i	− 0.762493i
$689$	15.7128	0.598610
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	− 22.7846i	− 0.866141i
$693$	− 6.92820i	− 0.263181i
$694$	22.1436	0.840559
$695$	0	0
$696$	1.26795	0.0480615
$697$	− 5.07180i	− 0.192108i
$698$	− 11.3205i	− 0.428488i
$699$	10.9282	0.413343
$700$	0	0
$701$	6.78461	0.256251	0.128126	−	0.991758i	$-0.459104\pi$
$701$	6.78461	0.256251	0.128126	+	0.991758i	$0.459104\pi$
$702$	2.19615i	0.0828884i
$703$	36.7846i	1.38736i
$704$	1.46410	0.0551804
$705$	0	0
$706$	−7.60770	−0.286319
$707$	− 21.4641i	− 0.807241i
$708$	− 12.1962i	− 0.458359i
$709$	−32.9282	−1.23664	−0.618322	−	0.785925i	$-0.712187\pi$
$709$	−32.9282	−1.23664	−0.618322	+	0.785925i	$0.712187\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 12.5885i	− 0.471772i
$713$	− 0.535898i	− 0.0200696i
$714$	12.0000	0.449089
$715$	0	0
$716$	−9.07180	−0.339029
$717$	13.8564i	0.517477i
$718$	2.87564i	0.107318i
$719$	9.46410	0.352951	0.176476	−	0.984305i	$-0.443530\pi$
$719$	9.46410	0.352951	0.176476	+	0.984305i	$0.443530\pi$
$720$	0	0
$721$	−15.4641	−0.575913
$722$	18.8038i	0.699807i
$723$	23.8564i	0.887229i
$724$	10.3923	0.386227
$725$	0	0
$726$	15.3397	0.569311
$727$	17.5167i	0.649657i	0.945773	+	0.324828i	$0.105307\pi$
$727$	17.5167i	0.649657i	−0.945773	+	0.324828i	$0.894693\pi$
$728$	10.3923i	0.385164i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	5.85641	0.216607
$732$	10.3923i	0.384111i
$733$	− 8.92820i	− 0.329771i	−0.986313	−	0.164885i	$-0.947275\pi$
$733$	− 8.92820i	− 0.329771i	0.986313	−	0.164885i	$-0.0527254\pi$
$734$	37.8564	1.39731
$735$	0	0
$736$	2.78461	0.102642
$737$	14.9282i	0.549887i
$738$	− 6.00000i	− 0.220863i
$739$	39.5692	1.45558	0.727789	−	0.685802i	$-0.240548\pi$
$739$	39.5692	1.45558	0.727789	+	0.685802i	$0.240548\pi$
$740$	0	0
$741$	6.92820	0.254514
$742$	− 101.569i	− 3.72872i
$743$	− 48.4974i	− 1.77920i	−0.456743	−	0.889599i	$-0.650984\pi$
$743$	− 48.4974i	− 1.77920i	0.456743	−	0.889599i	$-0.349016\pi$
$744$	1.73205	0.0635001
$745$	0	0
$746$	45.7128	1.67366
$747$	15.4641i	0.565802i
$748$	− 2.14359i	− 0.0783775i
$749$	−75.0333	−2.74166
$750$	0	0
$751$	−8.78461	−0.320555	−0.160277	−	0.987072i	$-0.551239\pi$
$751$	−8.78461	−0.320555	−0.160277	+	0.987072i	$0.551239\pi$
$752$	30.0000i	1.09399i
$753$	2.92820i	0.106710i
$754$	1.60770	0.0585488
$755$	0	0
$756$	4.73205	0.172103
$757$	22.7321i	0.826210i	0.910683	+	0.413105i	$0.135556\pi$
$757$	22.7321i	0.826210i	−0.910683	+	0.413105i	$0.864444\pi$
$758$	− 56.1051i	− 2.03783i
$759$	0.784610	0.0284795
$760$	0	0
$761$	−31.6603	−1.14768	−0.573842	−	0.818966i	$-0.694548\pi$
$761$	−31.6603	−1.14768	−0.573842	+	0.818966i	$0.694548\pi$
$762$	18.2487i	0.661081i
$763$	− 12.0000i	− 0.434429i
$764$	11.8038	0.427048
$765$	0	0
$766$	−23.0718	−0.833618
$767$	15.4641i	0.558376i
$768$	19.0000i	0.685603i
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	0	0
$771$	9.85641	0.354970
$772$	− 23.8564i	− 0.858611i
$773$	17.1769i	0.617811i	0.951093	+	0.308905i	$0.0999626\pi$
$773$	17.1769i	0.617811i	−0.951093	+	0.308905i	$0.900037\pi$
$774$	6.92820	0.249029
$775$	0	0
$776$	3.46410	0.124354
$777$	− 31.8564i	− 1.14284i
$778$	− 21.3731i	− 0.766262i
$779$	−18.9282	−0.678173
$780$	0	0
$781$	5.56922	0.199282
$782$	1.35898i	0.0485972i
$783$	0.732051i	0.0261614i
$784$	76.9615	2.74863
$785$	0	0
$786$	−11.6603	−0.415907
$787$	− 53.4641i	− 1.90579i	−0.303301	−	0.952895i	$-0.598089\pi$
$787$	− 53.4641i	− 1.90579i	0.303301	−	0.952895i	$-0.401911\pi$
$788$	− 13.0718i	− 0.465663i
$789$	−17.0718	−0.607772
$790$	0	0
$791$	23.3205	0.829182
$792$	2.53590i	0.0901092i
$793$	− 13.1769i	− 0.467926i
$794$	−41.3205	−1.46641
$795$	0	0
$796$	16.9282	0.600004
$797$	− 1.85641i	− 0.0657573i	−0.999459	−	0.0328786i	$-0.989533\pi$
$797$	− 1.85641i	− 0.0657573i	0.999459	−	0.0328786i	$-0.0104675\pi$
$798$	− 44.7846i	− 1.58536i
$799$	−8.78461	−0.310777
$800$	0	0
$801$	7.26795	0.256800
$802$	16.4833i	0.582047i
$803$	8.28719i	0.292448i
$804$	−10.1962	−0.359591
$805$	0	0
$806$	2.19615	0.0773562
$807$	− 9.80385i	− 0.345112i
$808$	7.85641i	0.276387i
$809$	28.8372	1.01386	0.506930	−	0.861987i	$-0.330780\pi$
$809$	28.8372	1.01386	0.506930	+	0.861987i	$0.330780\pi$
$810$	0	0
$811$	−30.2487	−1.06218	−0.531088	−	0.847317i	$-0.678217\pi$
$811$	−30.2487	−1.06218	−0.531088	+	0.847317i	$0.678217\pi$
$812$	− 3.46410i	− 0.121566i
$813$	− 16.7846i	− 0.588662i
$814$	−17.0718	−0.598366
$815$	0	0
$816$	−7.32051	−0.256269
$817$	− 21.8564i	− 0.764659i
$818$	30.0000i	1.04893i
$819$	−6.00000	−0.209657
$820$	0	0
$821$	17.1244	0.597644	0.298822	−	0.954309i	$-0.403406\pi$
$821$	17.1244	0.597644	0.298822	+	0.954309i	$0.403406\pi$
$822$	− 32.7846i	− 1.14349i
$823$	23.7128i	0.826577i	0.910600	+	0.413288i	$0.135620\pi$
$823$	23.7128i	0.826577i	−0.910600	+	0.413288i	$0.864380\pi$
$824$	5.66025	0.197184
$825$	0	0
$826$	99.9615	3.47811
$827$	− 34.1051i	− 1.18595i	−0.805220	−	0.592976i	$-0.797953\pi$
$827$	− 34.1051i	− 1.18595i	0.805220	−	0.592976i	$-0.202047\pi$
$828$	0.535898i	0.0186238i
$829$	54.1051	1.87915	0.939574	−	0.342345i	$-0.111221\pi$
$829$	54.1051	1.87915	0.939574	+	0.342345i	$0.111221\pi$
$830$	0	0
$831$	15.1244	0.524658
$832$	− 1.26795i	− 0.0439582i
$833$	22.5359i	0.780823i
$834$	17.0718	0.591148
$835$	0	0
$836$	−8.00000	−0.276686
$837$	1.00000i	0.0345651i
$838$	46.9808i	1.62292i
$839$	44.5885	1.53936	0.769682	−	0.638427i	$-0.220415\pi$
$839$	44.5885	1.53936	0.769682	+	0.638427i	$0.220415\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	− 26.5359i	− 0.914487i
$843$	− 16.5359i	− 0.569527i
$844$	−7.32051	−0.251982
$845$	0	0
$846$	−10.3923	−0.357295
$847$	41.9090i	1.44001i
$848$	61.9615i	2.12777i
$849$	7.26795	0.249435
$850$	0	0
$851$	3.60770	0.123670
$852$	3.80385i	0.130318i
$853$	− 52.2487i	− 1.78896i	−0.447106	−	0.894481i	$-0.647545\pi$
$853$	− 52.2487i	− 1.78896i	0.447106	−	0.894481i	$-0.352455\pi$
$854$	−85.1769	−2.91469
$855$	0	0
$856$	27.4641	0.938704
$857$	9.71281i	0.331783i	0.986144	+	0.165892i	$0.0530502\pi$
$857$	9.71281i	0.331783i	−0.986144	+	0.165892i	$0.946950\pi$
$858$	3.21539i	0.109772i
$859$	−39.7128	−1.35498	−0.677492	−	0.735530i	$-0.736933\pi$
$859$	−39.7128	−1.35498	−0.677492	+	0.735530i	$0.736933\pi$
$860$	0	0
$861$	16.3923	0.558648
$862$	15.3731i	0.523609i
$863$	− 27.7128i	− 0.943355i	−0.881771	−	0.471678i	$-0.843649\pi$
$863$	− 27.7128i	− 0.943355i	0.881771	−	0.471678i	$-0.156351\pi$
$864$	−5.19615	−0.176777
$865$	0	0
$866$	26.8756	0.913272
$867$	14.8564i	0.504550i
$868$	− 4.73205i	− 0.160616i
$869$	−14.6410	−0.496662
$870$	0	0
$871$	12.9282	0.438055
$872$	4.39230i	0.148742i
$873$	2.00000i	0.0676897i
$874$	5.07180	0.171556
$875$	0	0
$876$	−5.66025	−0.191242
$877$	− 34.1051i	− 1.15165i	−0.817574	−	0.575824i	$-0.804681\pi$
$877$	− 34.1051i	− 1.15165i	0.817574	−	0.575824i	$-0.195319\pi$
$878$	61.8564i	2.08755i
$879$	5.07180	0.171067
$880$	0	0
$881$	48.8372	1.64537	0.822683	−	0.568500i	$-0.192476\pi$
$881$	48.8372	1.64537	0.822683	+	0.568500i	$0.192476\pi$
$882$	26.6603i	0.897697i
$883$	19.3205i	0.650187i	0.945682	+	0.325093i	$0.105396\pi$
$883$	19.3205i	0.650187i	−0.945682	+	0.325093i	$0.894604\pi$
$884$	−1.85641	−0.0624377
$885$	0	0
$886$	52.6410	1.76851
$887$	− 15.4641i	− 0.519234i	−0.965712	−	0.259617i	$-0.916404\pi$
$887$	− 15.4641i	− 0.519234i	0.965712	−	0.259617i	$-0.0835963\pi$
$888$	11.6603i	0.391293i
$889$	−49.8564	−1.67213
$890$	0	0
$891$	−1.46410	−0.0490492
$892$	− 0.392305i	− 0.0131353i
$893$	32.7846i	1.09710i
$894$	36.9282	1.23506
$895$	0	0
$896$	−57.3731	−1.91670
$897$	− 0.679492i	− 0.0226876i
$898$	− 70.5500i	− 2.35428i
$899$	0.732051	0.0244153
$900$	0	0
$901$	−18.1436	−0.604451
$902$	− 8.78461i	− 0.292496i
$903$	18.9282i	0.629891i
$904$	−8.53590	−0.283900
$905$	0	0
$906$	36.2487	1.20428
$907$	20.7321i	0.688396i	0.938897	+	0.344198i	$0.111849\pi$
$907$	20.7321i	0.688396i	−0.938897	+	0.344198i	$0.888151\pi$
$908$	− 13.3205i	− 0.442057i
$909$	−4.53590	−0.150446
$910$	0	0
$911$	−9.46410	−0.313560	−0.156780	−	0.987634i	$-0.550111\pi$
$911$	−9.46410	−0.313560	−0.156780	+	0.987634i	$0.550111\pi$
$912$	27.3205i	0.904672i
$913$	22.6410i	0.749308i
$914$	−56.4449	−1.86703
$915$	0	0
$916$	18.7846	0.620661
$917$	− 31.8564i	− 1.05199i
$918$	− 2.53590i	− 0.0836971i
$919$	10.5359	0.347547	0.173774	−	0.984786i	$-0.444404\pi$
$919$	10.5359	0.347547	0.173774	+	0.984786i	$0.444404\pi$
$920$	0	0
$921$	−26.5885	−0.876119
$922$	− 1.26795i	− 0.0417577i
$923$	− 4.82309i	− 0.158754i
$924$	6.92820	0.227921
$925$	0	0
$926$	68.7846	2.26040
$927$	3.26795i	0.107334i
$928$	3.80385i	0.124867i
$929$	50.1962	1.64688	0.823441	−	0.567402i	$-0.192051\pi$
$929$	50.1962	1.64688	0.823441	+	0.567402i	$0.192051\pi$
$930$	0	0
$931$	84.1051	2.75643
$932$	10.9282i	0.357965i
$933$	13.6603i	0.447217i
$934$	−3.46410	−0.113349
$935$	0	0
$936$	2.19615	0.0717835
$937$	− 0.143594i	− 0.00469100i	−0.999997	−	0.00234550i	$-0.999253\pi$
$937$	− 0.143594i	− 0.00469100i	0.999997	−	0.00234550i	$-0.000746596\pi$
$938$	− 83.5692i	− 2.72863i
$939$	−21.6603	−0.706856
$940$	0	0
$941$	−6.58846	−0.214778	−0.107389	−	0.994217i	$-0.534249\pi$
$941$	−6.58846	−0.214778	−0.107389	+	0.994217i	$0.534249\pi$
$942$	22.3923i	0.729581i
$943$	1.85641i	0.0604529i
$944$	−60.9808	−1.98475
$945$	0	0
$946$	10.1436	0.329797
$947$	23.1769i	0.753149i	0.926386	+	0.376574i	$0.122898\pi$
$947$	23.1769i	0.753149i	−0.926386	+	0.376574i	$0.877102\pi$
$948$	− 10.0000i	− 0.324785i
$949$	7.17691	0.232973
$950$	0	0
$951$	22.9282	0.743498
$952$	− 12.0000i	− 0.388922i
$953$	44.7846i	1.45072i	0.688372	+	0.725358i	$0.258326\pi$
$953$	44.7846i	1.45072i	−0.688372	+	0.725358i	$0.741674\pi$
$954$	−21.4641	−0.694926
$955$	0	0
$956$	−13.8564	−0.448148
$957$	1.07180i	0.0346463i
$958$	48.1577i	1.55590i
$959$	89.5692	2.89234
$960$	0	0
$961$	1.00000	0.0322581
$962$	14.7846i	0.476675i
$963$	15.8564i	0.510966i
$964$	−23.8564	−0.768363
$965$	0	0
$966$	−4.39230	−0.141320
$967$	− 20.6795i	− 0.665008i	−0.943102	−	0.332504i	$-0.892107\pi$
$967$	− 20.6795i	− 0.665008i	0.943102	−	0.332504i	$-0.107893\pi$
$968$	− 15.3397i	− 0.493038i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	27.8038	0.892268	0.446134	−	0.894966i	$-0.352800\pi$
$971$	27.8038	0.892268	0.446134	+	0.894966i	$0.352800\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	46.6410i	1.49524i
$974$	−18.2487	−0.584726
$975$	0	0
$976$	51.9615	1.66325
$977$	− 20.9282i	− 0.669553i	−0.942298	−	0.334776i	$-0.891339\pi$
$977$	− 20.9282i	− 0.669553i	0.942298	−	0.334776i	$-0.108661\pi$
$978$	− 24.5885i	− 0.786252i
$979$	10.6410	0.340088
$980$	0	0
$981$	−2.53590	−0.0809650
$982$	42.2487i	1.34821i
$983$	45.5692i	1.45343i	0.686938	+	0.726716i	$0.258954\pi$
$983$	45.5692i	1.45343i	−0.686938	+	0.726716i	$0.741046\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−1.85641	−0.0591200
$987$	− 28.3923i	− 0.903737i
$988$	6.92820i	0.220416i
$989$	−2.14359	−0.0681623
$990$	0	0
$991$	18.1436	0.576350	0.288175	−	0.957578i	$-0.406951\pi$
$991$	18.1436	0.576350	0.288175	+	0.957578i	$0.406951\pi$
$992$	5.19615i	0.164978i
$993$	24.9282i	0.791073i
$994$	−31.1769	−0.988872
$995$	0	0
$996$	−15.4641	−0.489999
$997$	− 29.6077i	− 0.937685i	−0.883282	−	0.468843i	$-0.844671\pi$
$997$	− 29.6077i	− 0.937685i	0.883282	−	0.468843i	$-0.155329\pi$
$998$	25.8564i	0.818470i
$999$	−6.73205	−0.212993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.j.1024.4		4
5.2	odd	4		2325.2.a.m.1.1		2
5.3	odd	4		465.2.a.d.1.2	✓	2
5.4	even	2	inner	2325.2.c.j.1024.1		4
15.2	even	4		6975.2.a.v.1.2		2
15.8	even	4		1395.2.a.f.1.1		2
20.3	even	4		7440.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.d.1.2	✓	2	5.3	odd	4
1395.2.a.f.1.1		2	15.8	even	4
2325.2.a.m.1.1		2	5.2	odd	4
2325.2.c.j.1024.1		4	5.4	even	2	inner
2325.2.c.j.1024.4		4	1.1	even	1	trivial
6975.2.a.v.1.2		2	15.2	even	4
7440.2.a.bk.1.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -4.73205i q^{7} +1.73205i q^{8} -1.00000 q^{9} -1.46410 q^{11} -1.00000i q^{12} +1.26795i q^{13} +8.19615 q^{14} -5.00000 q^{16} -1.46410i q^{17} -1.73205i q^{18} -5.46410 q^{19} +4.73205 q^{21} -2.53590i q^{22} +0.535898i q^{23} -1.73205 q^{24} -2.19615 q^{26} -1.00000i q^{27} +4.73205i q^{28} -0.732051 q^{29} -1.00000 q^{31} -5.19615i q^{32} -1.46410i q^{33} +2.53590 q^{34} +1.00000 q^{36} -6.73205i q^{37} -9.46410i q^{38} -1.26795 q^{39} +3.46410 q^{41} +8.19615i q^{42} +4.00000i q^{43} +1.46410 q^{44} -0.928203 q^{46} -6.00000i q^{47} -5.00000i q^{48} -15.3923 q^{49} +1.46410 q^{51} -1.26795i q^{52} -12.3923i q^{53} +1.73205 q^{54} +8.19615 q^{56} -5.46410i q^{57} -1.26795i q^{58} +12.1962 q^{59} -10.3923 q^{61} -1.73205i q^{62} +4.73205i q^{63} -1.00000 q^{64} +2.53590 q^{66} -10.1962i q^{67} +1.46410i q^{68} -0.535898 q^{69} -3.80385 q^{71} -1.73205i q^{72} -5.66025i q^{73} +11.6603 q^{74} +5.46410 q^{76} +6.92820i q^{77} -2.19615i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -15.4641i q^{83} -4.73205 q^{84} -6.92820 q^{86} -0.732051i q^{87} -2.53590i q^{88} -7.26795 q^{89} +6.00000 q^{91} -0.535898i q^{92} -1.00000i q^{93} +10.3923 q^{94} +5.19615 q^{96} -2.00000i q^{97} -26.6603i q^{98} +1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9} + 8 q^{11} + 12 q^{14} - 20 q^{16} - 8 q^{19} + 12 q^{21} + 12 q^{26} + 4 q^{29} - 4 q^{31} + 24 q^{34} + 4 q^{36} - 12 q^{39} - 8 q^{44} + 24 q^{46} - 20 q^{49} - 8 q^{51} + 12 q^{56}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 + 8 * q^11 + 12 * q^14 - 20 * q^16 - 8 * q^19 + 12 * q^21 + 12 * q^26 + 4 * q^29 - 4 * q^31 + 24 * q^34 + 4 * q^36 - 12 * q^39 - 8 * q^44 + 24 * q^46 - 20 * q^49 - 8 * q^51 + 12 * q^56 + 28 * q^59 - 4 * q^64 + 24 * q^66 - 16 * q^69 - 36 * q^71 + 12 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 12 * q^84 - 36 * q^89 + 24 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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