LMFDB - Embedded newform 1875.2.b.e.1249.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1875,2,Mod(1249,1875)] 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("1875.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 $ x^12 + 23x^10 + 199x^8 + 794x^6 + 1399x^4 + 783*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-2.78712i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.e.1249.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.78712i q^{2} -1.00000i q^{3} -5.76803 q^{4} -2.78712 q^{6} +3.15000i q^{7} +10.5020i q^{8} -1.00000 q^{9} +2.94681 q^{11} +5.76803i q^{12} +0.188171i q^{13} +8.77943 q^{14} +17.7341 q^{16} -2.62743i q^{17} +2.78712i q^{18} -1.94100 q^{19} +3.15000 q^{21} -8.21310i q^{22} -1.30447i q^{23} +10.5020 q^{24} +0.524454 q^{26} +1.00000i q^{27} -18.1693i q^{28} -1.23197 q^{29} +2.65174 q^{31} -28.4233i q^{32} -2.94681i q^{33} -7.32297 q^{34} +5.76803 q^{36} -10.1161i q^{37} +5.40980i q^{38} +0.188171 q^{39} +3.85940 q^{41} -8.77943i q^{42} -9.63734i q^{43} -16.9973 q^{44} -3.63570 q^{46} +11.9846i q^{47} -17.7341i q^{48} -2.92250 q^{49} -2.62743 q^{51} -1.08537i q^{52} +0.369521i q^{53} +2.78712 q^{54} -33.0812 q^{56} +1.94100i q^{57} +3.43364i q^{58} +6.90864 q^{59} +5.68643 q^{61} -7.39071i q^{62} -3.15000i q^{63} -43.7507 q^{64} -8.21310 q^{66} -3.24188i q^{67} +15.1551i q^{68} -1.30447 q^{69} +6.69982 q^{71} -10.5020i q^{72} -9.46938i q^{73} -28.1948 q^{74} +11.1957 q^{76} +9.28244i q^{77} -0.524454i q^{78} -0.420137 q^{79} +1.00000 q^{81} -10.7566i q^{82} -12.1635i q^{83} -18.1693 q^{84} -26.8604 q^{86} +1.23197i q^{87} +30.9473i q^{88} +15.1828 q^{89} -0.592737 q^{91} +7.52421i q^{92} -2.65174i q^{93} +33.4025 q^{94} -28.4233 q^{96} -13.9035i q^{97} +8.14536i q^{98} -2.94681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51}+ \cdots - 66 q^{96}+O(q^{100}) $ 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9 + 8 * q^14 + 34 * q^16 + 4 * q^19 + 4 * q^21 + 12 * q^24 + 74 * q^26 - 62 * q^29 - 4 * q^31 - 74 * q^34 + 22 * q^36 + 66 * q^41 + 22 * q^44 - 24 * q^46 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 60 * q^56 + 16 * q^59 + 68 * q^61 - 24 * q^64 - 18 * q^66 - 2 * q^69 - 6 * q^71 - 72 * q^74 + 54 * q^76 - 50 * q^79 + 12 * q^81 - 88 * q^84 - 60 * q^86 - 36 * q^89 + 56 * q^91 + 100 * q^94 - 66 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.78712i	− 1.97079i	−0.170281	−	0.985395i	$-0.554468\pi$
$2$	− 2.78712i	− 1.97079i	0.170281	−	0.985395i	$-0.445532\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.76803	−2.88402
$5$	0	0
$5$	0	0
$6$	−2.78712	−1.13784
$7$	3.15000i	1.19059i	0.803508	+	0.595294i	$0.202964\pi$
$7$	3.15000i	1.19059i	−0.803508	+	0.595294i	$0.797036\pi$
$8$	10.5020i	3.71300i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.94681	0.888496	0.444248	−	0.895904i	$-0.353471\pi$
$11$	2.94681	0.888496	0.444248	+	0.895904i	$0.353471\pi$
$12$	5.76803i	1.66509i
$13$	0.188171i	0.0521891i	0.999659	+	0.0260946i	$0.00830710\pi$
$13$	0.188171i	0.0521891i	−0.999659	+	0.0260946i	$0.991693\pi$
$14$	8.77943	2.34640
$15$	0	0
$16$	17.7341	4.43354
$17$	− 2.62743i	− 0.637246i	−0.947882	−	0.318623i	$-0.896780\pi$
$17$	− 2.62743i	− 0.637246i	0.947882	−	0.318623i	$-0.103220\pi$
$18$	2.78712i	0.656930i
$19$	−1.94100	−0.445296	−0.222648	−	0.974899i	$-0.571470\pi$
$19$	−1.94100	−0.445296	−0.222648	+	0.974899i	$0.571470\pi$
$20$	0	0
$21$	3.15000	0.687386
$22$	− 8.21310i	− 1.75104i
$23$	− 1.30447i	− 0.272000i	−0.990709	−	0.136000i	$-0.956575\pi$
$23$	− 1.30447i	− 0.272000i	0.990709	−	0.136000i	$-0.0434247\pi$
$24$	10.5020	2.14370
$25$	0	0
$26$	0.524454	0.102854
$27$	1.00000i	0.192450i
$28$	− 18.1693i	− 3.43368i
$29$	−1.23197	−0.228770	−0.114385	−	0.993436i	$-0.536490\pi$
$29$	−1.23197	−0.228770	−0.114385	+	0.993436i	$0.536490\pi$
$30$	0	0
$31$	2.65174	0.476266	0.238133	−	0.971233i	$-0.423465\pi$
$31$	2.65174	0.476266	0.238133	+	0.971233i	$0.423465\pi$
$32$	− 28.4233i	− 5.02457i
$33$	− 2.94681i	− 0.512973i
$34$	−7.32297	−1.25588
$35$	0	0
$36$	5.76803	0.961339
$37$	− 10.1161i	− 1.66308i	−0.555466	−	0.831540i	$-0.687460\pi$
$37$	− 10.1161i	− 1.66308i	0.555466	−	0.831540i	$-0.312540\pi$
$38$	5.40980i	0.877585i
$39$	0.188171	0.0301314
$40$	0	0
$41$	3.85940	0.602737	0.301368	−	0.953508i	$-0.402557\pi$
$41$	3.85940	0.602737	0.301368	+	0.953508i	$0.402557\pi$
$42$	− 8.77943i	− 1.35469i
$43$	− 9.63734i	− 1.46968i	−0.678240	−	0.734840i	$-0.737257\pi$
$43$	− 9.63734i	− 1.46968i	0.678240	−	0.734840i	$-0.262743\pi$
$44$	−16.9973	−2.56244
$45$	0	0
$46$	−3.63570	−0.536055
$47$	11.9846i	1.74814i	0.485804	+	0.874068i	$0.338527\pi$
$47$	11.9846i	1.74814i	−0.485804	+	0.874068i	$0.661473\pi$
$48$	− 17.7341i	− 2.55970i
$49$	−2.92250	−0.417500
$50$	0	0
$51$	−2.62743	−0.367914
$52$	− 1.08537i	− 0.150514i
$53$	0.369521i	0.0507576i	0.999678	+	0.0253788i	$0.00807919\pi$
$53$	0.369521i	0.0507576i	−0.999678	+	0.0253788i	$0.991921\pi$
$54$	2.78712	0.379279
$55$	0	0
$56$	−33.0812	−4.42066
$57$	1.94100i	0.257092i
$58$	3.43364i	0.450859i
$59$	6.90864	0.899428	0.449714	−	0.893173i	$-0.351526\pi$
$59$	6.90864	0.899428	0.449714	+	0.893173i	$0.351526\pi$
$60$	0	0
$61$	5.68643	0.728073	0.364037	−	0.931385i	$-0.381398\pi$
$61$	5.68643	0.728073	0.364037	+	0.931385i	$0.381398\pi$
$62$	− 7.39071i	− 0.938621i
$63$	− 3.15000i	− 0.396863i
$64$	−43.7507	−5.46884
$65$	0	0
$66$	−8.21310	−1.01096
$67$	− 3.24188i	− 0.396058i	−0.980196	−	0.198029i	$-0.936546\pi$
$67$	− 3.24188i	− 0.396058i	0.980196	−	0.198029i	$-0.0634541\pi$
$68$	15.1551i	1.83783i
$69$	−1.30447	−0.157039
$70$	0	0
$71$	6.69982	0.795122	0.397561	−	0.917576i	$-0.369857\pi$
$71$	6.69982	0.795122	0.397561	+	0.917576i	$0.369857\pi$
$72$	− 10.5020i	− 1.23767i
$73$	− 9.46938i	− 1.10831i	−0.832415	−	0.554153i	$-0.813042\pi$
$73$	− 9.46938i	− 1.10831i	0.832415	−	0.554153i	$-0.186958\pi$
$74$	−28.1948	−3.27758
$75$	0	0
$76$	11.1957	1.28424
$77$	9.28244i	1.05783i
$78$	− 0.524454i	− 0.0593827i
$79$	−0.420137	−0.0472691	−0.0236345	−	0.999721i	$-0.507524\pi$
$79$	−0.420137	−0.0472691	−0.0236345	+	0.999721i	$0.507524\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.7566i	− 1.18787i
$83$	− 12.1635i	− 1.33512i	−0.744557	−	0.667559i	$-0.767339\pi$
$83$	− 12.1635i	− 1.33512i	0.744557	−	0.667559i	$-0.232661\pi$
$84$	−18.1693	−1.98243
$85$	0	0
$86$	−26.8604	−2.89643
$87$	1.23197i	0.132081i
$88$	30.9473i	3.29899i
$89$	15.1828	1.60937	0.804687	−	0.593699i	$-0.202333\pi$
$89$	15.1828	1.60937	0.804687	+	0.593699i	$0.202333\pi$
$90$	0	0
$91$	−0.592737	−0.0621358
$92$	7.52421i	0.784453i
$93$	− 2.65174i	− 0.274972i
$94$	33.4025	3.44521
$95$	0	0
$96$	−28.4233	−2.90094
$97$	− 13.9035i	− 1.41169i	−0.708367	−	0.705845i	$-0.750568\pi$
$97$	− 13.9035i	− 1.41169i	0.708367	−	0.705845i	$-0.249432\pi$
$98$	8.14536i	0.822805i
$99$	−2.94681	−0.296165
$100$	0	0
$101$	−9.67188	−0.962388	−0.481194	−	0.876614i	$-0.659797\pi$
$101$	−9.67188	−0.962388	−0.481194	+	0.876614i	$0.659797\pi$
$102$	7.32297i	0.725082i
$103$	8.69434i	0.856679i	0.903618	+	0.428339i	$0.140901\pi$
$103$	8.69434i	0.856679i	−0.903618	+	0.428339i	$0.859099\pi$
$104$	−1.97616	−0.193778
$105$	0	0
$106$	1.02990	0.100033
$107$	2.56585i	0.248050i	0.992279	+	0.124025i	$0.0395804\pi$
$107$	2.56585i	0.248050i	−0.992279	+	0.124025i	$0.960420\pi$
$108$	− 5.76803i	− 0.555029i
$109$	15.2352	1.45926	0.729632	−	0.683840i	$-0.239691\pi$
$109$	15.2352	1.45926	0.729632	+	0.683840i	$0.239691\pi$
$110$	0	0
$111$	−10.1161	−0.960179
$112$	55.8626i	5.27852i
$113$	2.20391i	0.207326i	0.994612	+	0.103663i	$0.0330564\pi$
$113$	2.20391i	0.207326i	−0.994612	+	0.103663i	$0.966944\pi$
$114$	5.40980	0.506674
$115$	0	0
$116$	7.10602	0.659778
$117$	− 0.188171i	− 0.0173964i
$118$	− 19.2552i	− 1.77258i
$119$	8.27641	0.758697
$120$	0	0
$121$	−2.31633	−0.210575
$122$	− 15.8488i	− 1.43488i
$123$	− 3.85940i	− 0.347990i
$124$	−15.2953	−1.37356
$125$	0	0
$126$	−8.77943	−0.782133
$127$	14.9689i	1.32827i	0.747611	+	0.664136i	$0.231201\pi$
$127$	14.9689i	1.32827i	−0.747611	+	0.664136i	$0.768799\pi$
$128$	65.0920i	5.75337i
$129$	−9.63734	−0.848521
$130$	0	0
$131$	11.2957	0.986911	0.493456	−	0.869771i	$-0.335734\pi$
$131$	11.2957	0.986911	0.493456	+	0.869771i	$0.335734\pi$
$132$	16.9973i	1.47942i
$133$	− 6.11415i	− 0.530164i
$134$	−9.03550	−0.780548
$135$	0	0
$136$	27.5932	2.36610
$137$	− 4.17877i	− 0.357017i	−0.983938	−	0.178508i	$-0.942873\pi$
$137$	− 4.17877i	− 0.357017i	0.983938	−	0.178508i	$-0.0571271\pi$
$138$	3.63570i	0.309492i
$139$	15.3552	1.30241	0.651207	−	0.758900i	$-0.274263\pi$
$139$	15.3552	1.30241	0.651207	+	0.758900i	$0.274263\pi$
$140$	0	0
$141$	11.9846	1.00929
$142$	− 18.6732i	− 1.56702i
$143$	0.554502i	0.0463698i
$144$	−17.7341	−1.47785
$145$	0	0
$146$	−26.3923	−2.18424
$147$	2.92250i	0.241044i
$148$	58.3501i	4.79635i
$149$	−3.43364	−0.281294	−0.140647	−	0.990060i	$-0.544918\pi$
$149$	−3.43364	−0.281294	−0.140647	+	0.990060i	$0.544918\pi$
$150$	0	0
$151$	10.7716	0.876581	0.438290	−	0.898833i	$-0.355584\pi$
$151$	10.7716	0.876581	0.438290	+	0.898833i	$0.355584\pi$
$152$	− 20.3843i	− 1.65338i
$153$	2.62743i	0.212415i
$154$	25.8713	2.08477
$155$	0	0
$156$	−1.08537	−0.0868995
$157$	13.4045i	1.06980i	0.844916	+	0.534900i	$0.179651\pi$
$157$	13.4045i	1.06980i	−0.844916	+	0.534900i	$0.820349\pi$
$158$	1.17097i	0.0931574i
$159$	0.369521	0.0293049
$160$	0	0
$161$	4.10907	0.323840
$162$	− 2.78712i	− 0.218977i
$163$	− 15.4178i	− 1.20762i	−0.797129	−	0.603808i	$-0.793649\pi$
$163$	− 15.4178i	− 1.20762i	0.797129	−	0.603808i	$-0.206351\pi$
$164$	−22.2611	−1.73830
$165$	0	0
$166$	−33.9011	−2.63124
$167$	− 17.4045i	− 1.34680i	−0.739276	−	0.673402i	$-0.764832\pi$
$167$	− 17.4045i	− 1.34680i	0.739276	−	0.673402i	$-0.235168\pi$
$168$	33.0812i	2.55227i
$169$	12.9646	0.997276
$170$	0	0
$171$	1.94100	0.148432
$172$	55.5885i	4.23858i
$173$	− 19.2737i	− 1.46535i	−0.680578	−	0.732676i	$-0.738271\pi$
$173$	− 19.2737i	− 1.46535i	0.680578	−	0.732676i	$-0.261729\pi$
$174$	3.43364	0.260303
$175$	0	0
$176$	52.2591	3.93918
$177$	− 6.90864i	− 0.519285i
$178$	− 42.3163i	− 3.17174i
$179$	−23.7940	−1.77845	−0.889223	−	0.457475i	$-0.848754\pi$
$179$	−23.7940	−1.77845	−0.889223	+	0.457475i	$0.848754\pi$
$180$	0	0
$181$	−10.6891	−0.794516	−0.397258	−	0.917707i	$-0.630038\pi$
$181$	−10.6891	−0.794516	−0.397258	+	0.917707i	$0.630038\pi$
$182$	1.65203i	0.122457i
$183$	− 5.68643i	− 0.420353i
$184$	13.6995	1.00994
$185$	0	0
$186$	−7.39071	−0.541913
$187$	− 7.74253i	− 0.566190i
$188$	− 69.1277i	− 5.04165i
$189$	−3.15000	−0.229129
$190$	0	0
$191$	−14.3581	−1.03892	−0.519458	−	0.854496i	$-0.673866\pi$
$191$	−14.3581	−1.03892	−0.519458	+	0.854496i	$0.673866\pi$
$192$	43.7507i	3.15744i
$193$	1.95508i	0.140730i	0.997521	+	0.0703648i	$0.0224163\pi$
$193$	1.95508i	0.140730i	−0.997521	+	0.0703648i	$0.977584\pi$
$194$	−38.7508	−2.78214
$195$	0	0
$196$	16.8571	1.20408
$197$	1.17958i	0.0840417i	0.999117	+	0.0420209i	$0.0133796\pi$
$197$	1.17958i	0.0840417i	−0.999117	+	0.0420209i	$0.986620\pi$
$198$	8.21310i	0.583680i
$199$	6.55820	0.464899	0.232449	−	0.972608i	$-0.425326\pi$
$199$	6.55820	0.464899	0.232449	+	0.972608i	$0.425326\pi$
$200$	0	0
$201$	−3.24188	−0.228664
$202$	26.9567i	1.89666i
$203$	− 3.88069i	− 0.272371i
$204$	15.1551	1.06107
$205$	0	0
$206$	24.2322	1.68833
$207$	1.30447i	0.0906667i
$208$	3.33705i	0.231382i
$209$	−5.71975	−0.395643
$210$	0	0
$211$	12.3573	0.850715	0.425357	−	0.905026i	$-0.360148\pi$
$211$	12.3573	0.850715	0.425357	+	0.905026i	$0.360148\pi$
$212$	− 2.13141i	− 0.146386i
$213$	− 6.69982i	− 0.459064i
$214$	7.15134	0.488856
$215$	0	0
$216$	−10.5020	−0.714568
$217$	8.35298i	0.567037i
$218$	− 42.4622i	− 2.87591i
$219$	−9.46938	−0.639881
$220$	0	0
$221$	0.494405	0.0332573
$222$	28.1948i	1.89231i
$223$	− 7.30693i	− 0.489308i	−0.969610	−	0.244654i	$-0.921326\pi$
$223$	− 7.30693i	− 0.489308i	0.969610	−	0.244654i	$-0.0786744\pi$
$224$	89.5333	5.98219
$225$	0	0
$226$	6.14256	0.408597
$227$	21.0202i	1.39516i	0.716508	+	0.697579i	$0.245739\pi$
$227$	21.0202i	1.39516i	−0.716508	+	0.697579i	$0.754261\pi$
$228$	− 11.1957i	− 0.741457i
$229$	−19.5544	−1.29219	−0.646094	−	0.763258i	$-0.723599\pi$
$229$	−19.5544	−1.29219	−0.646094	+	0.763258i	$0.723599\pi$
$230$	0	0
$231$	9.28244	0.610740
$232$	− 12.9381i	− 0.849425i
$233$	12.1472i	0.795792i	0.917430	+	0.397896i	$0.130260\pi$
$233$	12.1472i	0.795792i	−0.917430	+	0.397896i	$0.869740\pi$
$234$	−0.524454	−0.0342846
$235$	0	0
$236$	−39.8493	−2.59397
$237$	0.420137i	0.0272908i
$238$	− 23.0673i	− 1.49523i
$239$	−13.4517	−0.870119	−0.435059	−	0.900402i	$-0.643273\pi$
$239$	−13.4517	−0.870119	−0.435059	+	0.900402i	$0.643273\pi$
$240$	0	0
$241$	19.9370	1.28426	0.642129	−	0.766597i	$-0.278051\pi$
$241$	19.9370	1.28426	0.642129	+	0.766597i	$0.278051\pi$
$242$	6.45588i	0.415000i
$243$	− 1.00000i	− 0.0641500i
$244$	−32.7995	−2.09978
$245$	0	0
$246$	−10.7566	−0.685816
$247$	− 0.365239i	− 0.0232396i
$248$	27.8485i	1.76838i
$249$	−12.1635	−0.770830
$250$	0	0
$251$	−2.76013	−0.174218	−0.0871088	−	0.996199i	$-0.527763\pi$
$251$	−2.76013	−0.174218	−0.0871088	+	0.996199i	$0.527763\pi$
$252$	18.1693i	1.14456i
$253$	− 3.84401i	− 0.241671i
$254$	41.7200	2.61775
$255$	0	0
$256$	93.9177	5.86986
$257$	17.2848i	1.07820i	0.842243	+	0.539098i	$0.181235\pi$
$257$	17.2848i	1.07820i	−0.842243	+	0.539098i	$0.818765\pi$
$258$	26.8604i	1.67226i
$259$	31.8658	1.98004
$260$	0	0
$261$	1.23197	0.0762568
$262$	− 31.4825i	− 1.94500i
$263$	− 3.96578i	− 0.244541i	−0.992497	−	0.122270i	$-0.960983\pi$
$263$	− 3.96578i	− 0.244541i	0.992497	−	0.122270i	$-0.0390175\pi$
$264$	30.9473	1.90467
$265$	0	0
$266$	−17.0409	−1.04484
$267$	− 15.1828i	− 0.929173i
$268$	18.6993i	1.14224i
$269$	−1.19564	−0.0728993	−0.0364496	−	0.999335i	$-0.511605\pi$
$269$	−1.19564	−0.0728993	−0.0364496	+	0.999335i	$0.511605\pi$
$270$	0	0
$271$	−2.18295	−0.132605	−0.0663023	−	0.997800i	$-0.521120\pi$
$271$	−2.18295	−0.132605	−0.0663023	+	0.997800i	$0.521120\pi$
$272$	− 46.5953i	− 2.82525i
$273$	0.592737i	0.0358741i
$274$	−11.6467	−0.703605
$275$	0	0
$276$	7.52421	0.452904
$277$	− 1.95174i	− 0.117268i	−0.998280	−	0.0586342i	$-0.981325\pi$
$277$	− 1.95174i	− 0.117268i	0.998280	−	0.0586342i	$-0.0186746\pi$
$278$	− 42.7969i	− 2.56679i
$279$	−2.65174	−0.158755
$280$	0	0
$281$	2.30131	0.137284	0.0686422	−	0.997641i	$-0.478133\pi$
$281$	2.30131	0.137284	0.0686422	+	0.997641i	$0.478133\pi$
$282$	− 33.4025i	− 1.98909i
$283$	− 10.6459i	− 0.632835i	−0.948620	−	0.316417i	$-0.897520\pi$
$283$	− 10.6459i	− 0.632835i	0.948620	−	0.316417i	$-0.102480\pi$
$284$	−38.6448	−2.29315
$285$	0	0
$286$	1.54546	0.0913852
$287$	12.1571i	0.717611i
$288$	28.4233i	1.67486i
$289$	10.0966	0.593918
$290$	0	0
$291$	−13.9035	−0.815039
$292$	54.6197i	3.19638i
$293$	14.9591i	0.873921i	0.899481	+	0.436960i	$0.143945\pi$
$293$	14.9591i	0.873921i	−0.899481	+	0.436960i	$0.856055\pi$
$294$	8.14536	0.475047
$295$	0	0
$296$	106.239	6.17502
$297$	2.94681i	0.170991i
$298$	9.56995i	0.554373i
$299$	0.245462	0.0141954
$300$	0	0
$301$	30.3576	1.74978
$302$	− 30.0217i	− 1.72756i
$303$	9.67188i	0.555635i
$304$	−34.4220	−1.97424
$305$	0	0
$306$	7.32297	0.418626
$307$	3.07333i	0.175404i	0.996147	+	0.0877021i	$0.0279524\pi$
$307$	3.07333i	0.175404i	−0.996147	+	0.0877021i	$0.972048\pi$
$308$	− 53.5414i	− 3.05081i
$309$	8.69434	0.494604
$310$	0	0
$311$	24.9095	1.41249	0.706244	−	0.707968i	$-0.250388\pi$
$311$	24.9095	1.41249	0.706244	+	0.707968i	$0.250388\pi$
$312$	1.97616i	0.111878i
$313$	9.28562i	0.524854i	0.964952	+	0.262427i	$0.0845230\pi$
$313$	9.28562i	0.524854i	−0.964952	+	0.262427i	$0.915477\pi$
$314$	37.3601	2.10835
$315$	0	0
$316$	2.42336	0.136325
$317$	− 20.7685i	− 1.16647i	−0.812302	−	0.583237i	$-0.801786\pi$
$317$	− 20.7685i	− 1.16647i	0.812302	−	0.583237i	$-0.198214\pi$
$318$	− 1.02990i	− 0.0577538i
$319$	−3.63037	−0.203261
$320$	0	0
$321$	2.56585	0.143212
$322$	− 11.4525i	− 0.638221i
$323$	5.09984i	0.283763i
$324$	−5.76803	−0.320446
$325$	0	0
$326$	−42.9713	−2.37996
$327$	− 15.2352i	− 0.842507i
$328$	40.5312i	2.23796i
$329$	−37.7515	−2.08131
$330$	0	0
$331$	−33.9172	−1.86426	−0.932128	−	0.362129i	$-0.882050\pi$
$331$	−33.9172	−1.86426	−0.932128	+	0.362129i	$0.882050\pi$
$332$	70.1595i	3.85050i
$333$	10.1161i	0.554360i
$334$	−48.5085	−2.65427
$335$	0	0
$336$	55.8626	3.04755
$337$	− 17.7792i	− 0.968494i	−0.874931	−	0.484247i	$-0.839094\pi$
$337$	− 17.7792i	− 0.968494i	0.874931	−	0.484247i	$-0.160906\pi$
$338$	− 36.1339i	− 1.96542i
$339$	2.20391	0.119700
$340$	0	0
$341$	7.81416	0.423161
$342$	− 5.40980i	− 0.292528i
$343$	12.8441i	0.693518i
$344$	101.211	5.45693
$345$	0	0
$346$	−53.7181	−2.88790
$347$	12.2251i	0.656277i	0.944630	+	0.328138i	$0.106421\pi$
$347$	12.2251i	0.656277i	−0.944630	+	0.328138i	$0.893579\pi$
$348$	− 7.10602i	− 0.380923i
$349$	−12.8425	−0.687441	−0.343720	−	0.939072i	$-0.611687\pi$
$349$	−12.8425	−0.687441	−0.343720	+	0.939072i	$0.611687\pi$
$350$	0	0
$351$	−0.188171	−0.0100438
$352$	− 83.7579i	− 4.46431i
$353$	17.4163i	0.926976i	0.886103	+	0.463488i	$0.153402\pi$
$353$	17.4163i	0.926976i	−0.886103	+	0.463488i	$0.846598\pi$
$354$	−19.2552	−1.02340
$355$	0	0
$356$	−87.5749	−4.64146
$357$	− 8.27641i	− 0.438034i
$358$	66.3167i	3.50494i
$359$	3.82947	0.202111	0.101056	−	0.994881i	$-0.467778\pi$
$359$	3.82947	0.202111	0.101056	+	0.994881i	$0.467778\pi$
$360$	0	0
$361$	−15.2325	−0.801712
$362$	29.7918i	1.56582i
$363$	2.31633i	0.121576i
$364$	3.41893	0.179201
$365$	0	0
$366$	−15.8488	−0.828428
$367$	29.6282i	1.54658i	0.634054	+	0.773289i	$0.281390\pi$
$367$	29.6282i	1.54658i	−0.634054	+	0.773289i	$0.718610\pi$
$368$	− 23.1336i	− 1.20592i
$369$	−3.85940	−0.200912
$370$	0	0
$371$	−1.16399	−0.0604314
$372$	15.2953i	0.793025i
$373$	11.4595i	0.593350i	0.954978	+	0.296675i	$0.0958779\pi$
$373$	11.4595i	0.593350i	−0.954978	+	0.296675i	$0.904122\pi$
$374$	−21.5794	−1.11584
$375$	0	0
$376$	−125.862	−6.49083
$377$	− 0.231820i	− 0.0119393i
$378$	8.77943i	0.451565i
$379$	2.19307	0.112650	0.0563251	−	0.998412i	$-0.482062\pi$
$379$	2.19307	0.112650	0.0563251	+	0.998412i	$0.482062\pi$
$380$	0	0
$381$	14.9689	0.766879
$382$	40.0177i	2.04749i
$383$	− 10.1997i	− 0.521181i	−0.965449	−	0.260591i	$-0.916083\pi$
$383$	− 10.1997i	− 0.521181i	0.965449	−	0.260591i	$-0.0839173\pi$
$384$	65.0920	3.32171
$385$	0	0
$386$	5.44904	0.277349
$387$	9.63734i	0.489894i
$388$	80.1960i	4.07134i
$389$	−28.5482	−1.44745	−0.723726	−	0.690088i	$-0.757572\pi$
$389$	−28.5482	−1.44745	−0.723726	+	0.690088i	$0.757572\pi$
$390$	0	0
$391$	−3.42740	−0.173331
$392$	− 30.6920i	− 1.55018i
$393$	− 11.2957i	− 0.569794i
$394$	3.28764	0.165629
$395$	0	0
$396$	16.9973	0.854146
$397$	− 37.1827i	− 1.86614i	−0.359688	−	0.933072i	$-0.617117\pi$
$397$	− 37.1827i	− 1.86614i	0.359688	−	0.933072i	$-0.382883\pi$
$398$	− 18.2785i	− 0.916218i
$399$	−6.11415	−0.306090
$400$	0	0
$401$	7.38648	0.368863	0.184432	−	0.982845i	$-0.440956\pi$
$401$	7.38648	0.368863	0.184432	+	0.982845i	$0.440956\pi$
$402$	9.03550i	0.450650i
$403$	0.498979i	0.0248559i
$404$	55.7877	2.77554
$405$	0	0
$406$	−10.8160	−0.536787
$407$	− 29.8102i	− 1.47764i
$408$	− 27.5932i	− 1.36607i
$409$	5.19942	0.257095	0.128547	−	0.991703i	$-0.458969\pi$
$409$	5.19942	0.257095	0.128547	+	0.991703i	$0.458969\pi$
$410$	0	0
$411$	−4.17877	−0.206124
$412$	− 50.1492i	− 2.47068i
$413$	21.7622i	1.07085i
$414$	3.63570	0.178685
$415$	0	0
$416$	5.34842	0.262228
$417$	− 15.3552i	− 0.751949i
$418$	15.9416i	0.779730i
$419$	14.9205	0.728914	0.364457	−	0.931220i	$-0.381255\pi$
$419$	14.9205	0.728914	0.364457	+	0.931220i	$0.381255\pi$
$420$	0	0
$421$	−0.0666925	−0.00325039	−0.00162520	−	0.999999i	$-0.500517\pi$
$421$	−0.0666925	−0.00325039	−0.00162520	+	0.999999i	$0.500517\pi$
$422$	− 34.4414i	− 1.67658i
$423$	− 11.9846i	− 0.582712i
$424$	−3.88069	−0.188463
$425$	0	0
$426$	−18.6732	−0.904719
$427$	17.9123i	0.866835i
$428$	− 14.7999i	− 0.715382i
$429$	0.554502	0.0267716
$430$	0	0
$431$	27.6996	1.33424	0.667121	−	0.744950i	$-0.267527\pi$
$431$	27.6996	1.33424	0.667121	+	0.744950i	$0.267527\pi$
$432$	17.7341i	0.853235i
$433$	− 30.5626i	− 1.46874i	−0.678748	−	0.734372i	$-0.737477\pi$
$433$	− 30.5626i	− 1.46874i	0.678748	−	0.734372i	$-0.262523\pi$
$434$	23.2807	1.11751
$435$	0	0
$436$	−87.8770	−4.20854
$437$	2.53197i	0.121120i
$438$	26.3923i	1.26107i
$439$	−11.0968	−0.529623	−0.264812	−	0.964300i	$-0.585310\pi$
$439$	−11.0968	−0.529623	−0.264812	+	0.964300i	$0.585310\pi$
$440$	0	0
$441$	2.92250	0.139167
$442$	− 1.37797i	− 0.0655432i
$443$	− 9.82831i	− 0.466957i	−0.972362	−	0.233479i	$-0.924989\pi$
$443$	− 9.82831i	− 0.466957i	0.972362	−	0.233479i	$-0.0750109\pi$
$444$	58.3501	2.76917
$445$	0	0
$446$	−20.3653	−0.964324
$447$	3.43364i	0.162405i
$448$	− 137.815i	− 6.51114i
$449$	19.1301	0.902805	0.451402	−	0.892320i	$-0.350924\pi$
$449$	19.1301	0.902805	0.451402	+	0.892320i	$0.350924\pi$
$450$	0	0
$451$	11.3729	0.535529
$452$	− 12.7122i	− 0.597933i
$453$	− 10.7716i	− 0.506094i
$454$	58.5858	2.74957
$455$	0	0
$456$	−20.3843	−0.954582
$457$	− 18.7578i	− 0.877452i	−0.898621	−	0.438726i	$-0.855430\pi$
$457$	− 18.7578i	− 0.877452i	0.898621	−	0.438726i	$-0.144570\pi$
$458$	54.5003i	2.54663i
$459$	2.62743	0.122638
$460$	0	0
$461$	−37.6208	−1.75217	−0.876087	−	0.482153i	$-0.839855\pi$
$461$	−37.6208	−1.75217	−0.876087	+	0.482153i	$0.839855\pi$
$462$	− 25.8713i	− 1.20364i
$463$	− 19.9275i	− 0.926111i	−0.886329	−	0.463056i	$-0.846753\pi$
$463$	− 19.9275i	− 0.926111i	0.886329	−	0.463056i	$-0.153247\pi$
$464$	−21.8479	−1.01426
$465$	0	0
$466$	33.8558	1.56834
$467$	− 27.3667i	− 1.26638i	−0.773997	−	0.633190i	$-0.781745\pi$
$467$	− 27.3667i	− 1.26638i	0.773997	−	0.633190i	$-0.218255\pi$
$468$	1.08537i	0.0501714i
$469$	10.2119	0.471542
$470$	0	0
$471$	13.4045	0.617649
$472$	72.5542i	3.33958i
$473$	− 28.3994i	− 1.30581i
$474$	1.17097	0.0537845
$475$	0	0
$476$	−47.7386	−2.18810
$477$	− 0.369521i	− 0.0169192i
$478$	37.4915i	1.71482i
$479$	38.5742	1.76250	0.881250	−	0.472650i	$-0.156703\pi$
$479$	38.5742	1.76250	0.881250	+	0.472650i	$0.156703\pi$
$480$	0	0
$481$	1.90356	0.0867946
$482$	− 55.5669i	− 2.53100i
$483$	− 4.10907i	− 0.186969i
$484$	13.3607	0.607303
$485$	0	0
$486$	−2.78712	−0.126426
$487$	18.4995i	0.838294i	0.907918	+	0.419147i	$0.137671\pi$
$487$	18.4995i	0.838294i	−0.907918	+	0.419147i	$0.862329\pi$
$488$	59.7187i	2.70334i
$489$	−15.4178	−0.697218
$490$	0	0
$491$	19.7211	0.890000	0.445000	−	0.895531i	$-0.353204\pi$
$491$	19.7211	0.890000	0.445000	+	0.895531i	$0.353204\pi$
$492$	22.2611i	1.00361i
$493$	3.23691i	0.145783i
$494$	−1.01796	−0.0458004
$495$	0	0
$496$	47.0263	2.11154
$497$	21.1044i	0.946663i
$498$	33.9011i	1.51915i
$499$	36.2906	1.62459	0.812296	−	0.583245i	$-0.198217\pi$
$499$	36.2906	1.62459	0.812296	+	0.583245i	$0.198217\pi$
$500$	0	0
$501$	−17.4045	−0.777578
$502$	7.69280i	0.343347i
$503$	2.66163i	0.118676i	0.998238	+	0.0593382i	$0.0188990\pi$
$503$	2.66163i	0.118676i	−0.998238	+	0.0593382i	$0.981101\pi$
$504$	33.0812	1.47355
$505$	0	0
$506$	−10.7137	−0.476283
$507$	− 12.9646i	− 0.575778i
$508$	− 86.3410i	− 3.83076i
$509$	−22.8322	−1.01202	−0.506010	−	0.862528i	$-0.668880\pi$
$509$	−22.8322	−1.01202	−0.506010	+	0.862528i	$0.668880\pi$
$510$	0	0
$511$	29.8285	1.31954
$512$	− 131.576i	− 5.81488i
$513$	− 1.94100i	− 0.0856972i
$514$	48.1748	2.12490
$515$	0	0
$516$	55.5885	2.44715
$517$	35.3163i	1.55321i
$518$	− 88.8137i	− 3.90225i
$519$	−19.2737	−0.846021
$520$	0	0
$521$	−12.3029	−0.539001	−0.269501	−	0.963000i	$-0.586859\pi$
$521$	−12.3029	−0.539001	−0.269501	+	0.963000i	$0.586859\pi$
$522$	− 3.43364i	− 0.150286i
$523$	− 33.3902i	− 1.46005i	−0.683420	−	0.730025i	$-0.739508\pi$
$523$	− 33.3902i	− 1.46005i	0.683420	−	0.730025i	$-0.260492\pi$
$524$	−65.1541	−2.84627
$525$	0	0
$526$	−11.0531	−0.481939
$527$	− 6.96726i	− 0.303499i
$528$	− 52.2591i	− 2.27429i
$529$	21.2984	0.926016
$530$	0	0
$531$	−6.90864	−0.299809
$532$	35.2666i	1.52900i
$533$	0.726225i	0.0314563i
$534$	−42.3163	−1.83121
$535$	0	0
$536$	34.0461	1.47057
$537$	23.7940i	1.02679i
$538$	3.33238i	0.143669i
$539$	−8.61204	−0.370947
$540$	0	0
$541$	5.98673	0.257389	0.128695	−	0.991684i	$-0.458921\pi$
$541$	5.98673	0.257389	0.128695	+	0.991684i	$0.458921\pi$
$542$	6.08413i	0.261336i
$543$	10.6891i	0.458714i
$544$	−74.6802	−3.20189
$545$	0	0
$546$	1.65203	0.0707003
$547$	− 4.47889i	− 0.191503i	−0.995405	−	0.0957516i	$-0.969475\pi$
$547$	− 4.47889i	− 0.191503i	0.995405	−	0.0957516i	$-0.0305255\pi$
$548$	24.1033i	1.02964i
$549$	−5.68643	−0.242691
$550$	0	0
$551$	2.39124	0.101870
$552$	− 13.6995i	− 0.583088i
$553$	− 1.32343i	− 0.0562780i
$554$	−5.43972	−0.231112
$555$	0	0
$556$	−88.5695	−3.75618
$557$	14.4278i	0.611326i	0.952140	+	0.305663i	$0.0988781\pi$
$557$	14.4278i	0.611326i	−0.952140	+	0.305663i	$0.901122\pi$
$558$	7.39071i	0.312874i
$559$	1.81346	0.0767014
$560$	0	0
$561$	−7.74253	−0.326890
$562$	− 6.41401i	− 0.270559i
$563$	27.6973i	1.16730i	0.812004	+	0.583652i	$0.198377\pi$
$563$	27.6973i	1.16730i	−0.812004	+	0.583652i	$0.801623\pi$
$564$	−69.1277	−2.91080
$565$	0	0
$566$	−29.6715	−1.24719
$567$	3.15000i	0.132288i
$568$	70.3612i	2.95229i
$569$	−35.6179	−1.49318	−0.746590	−	0.665285i	$-0.768310\pi$
$569$	−35.6179	−1.49318	−0.746590	+	0.665285i	$0.768310\pi$
$570$	0	0
$571$	5.85044	0.244833	0.122417	−	0.992479i	$-0.460936\pi$
$571$	5.85044	0.244833	0.122417	+	0.992479i	$0.460936\pi$
$572$	− 3.19839i	− 0.133731i
$573$	14.3581i	0.599818i
$574$	33.8833	1.41426
$575$	0	0
$576$	43.7507	1.82295
$577$	− 12.0524i	− 0.501746i	−0.968020	−	0.250873i	$-0.919282\pi$
$577$	− 12.0524i	− 0.501746i	0.968020	−	0.250873i	$-0.0807177\pi$
$578$	− 28.1404i	− 1.17049i
$579$	1.95508	0.0812503
$580$	0	0
$581$	38.3150	1.58958
$582$	38.7508i	1.60627i
$583$	1.08891i	0.0450979i
$584$	99.4470	4.11515
$585$	0	0
$586$	41.6928	1.72232
$587$	− 40.2350i	− 1.66068i	−0.557259	−	0.830339i	$-0.688147\pi$
$587$	− 40.2350i	− 1.66068i	0.557259	−	0.830339i	$-0.311853\pi$
$588$	− 16.8571i	− 0.695174i
$589$	−5.14702	−0.212079
$590$	0	0
$591$	1.17958	0.0485215
$592$	− 179.401i	− 7.37332i
$593$	33.4086i	1.37193i	0.727636	+	0.685964i	$0.240619\pi$
$593$	33.4086i	1.37193i	−0.727636	+	0.685964i	$0.759381\pi$
$594$	8.21310	0.336988
$595$	0	0
$596$	19.8053	0.811258
$597$	− 6.55820i	− 0.268409i
$598$	− 0.684132i	− 0.0279763i
$599$	24.6421	1.00685	0.503424	−	0.864040i	$-0.332073\pi$
$599$	24.6421	1.00685	0.503424	+	0.864040i	$0.332073\pi$
$600$	0	0
$601$	16.6327	0.678461	0.339230	−	0.940703i	$-0.389833\pi$
$601$	16.6327	0.678461	0.339230	+	0.940703i	$0.389833\pi$
$602$	− 84.6103i	− 3.44846i
$603$	3.24188i	0.132019i
$604$	−62.1310	−2.52807
$605$	0	0
$606$	26.9567	1.09504
$607$	− 29.4057i	− 1.19354i	−0.802412	−	0.596770i	$-0.796450\pi$
$607$	− 29.4057i	− 1.19354i	0.802412	−	0.596770i	$-0.203550\pi$
$608$	55.1695i	2.23742i
$609$	−3.88069	−0.157254
$610$	0	0
$611$	−2.25515	−0.0912337
$612$	− 15.1551i	− 0.612609i
$613$	12.1702i	0.491549i	0.969327	+	0.245775i	$0.0790423\pi$
$613$	12.1702i	0.491549i	−0.969327	+	0.245775i	$0.920958\pi$
$614$	8.56574	0.345685
$615$	0	0
$616$	−97.4838	−3.92774
$617$	39.1024i	1.57420i	0.616823	+	0.787102i	$0.288419\pi$
$617$	39.1024i	1.57420i	−0.616823	+	0.787102i	$0.711581\pi$
$618$	− 24.2322i	− 0.974761i
$619$	4.72717	0.190001	0.0950006	−	0.995477i	$-0.469715\pi$
$619$	4.72717	0.190001	0.0950006	+	0.995477i	$0.469715\pi$
$620$	0	0
$621$	1.30447	0.0523464
$622$	− 69.4258i	− 2.78372i
$623$	47.8258i	1.91610i
$624$	3.33705	0.133589
$625$	0	0
$626$	25.8801	1.03438
$627$	5.71975i	0.228425i
$628$	− 77.3179i	− 3.08532i
$629$	−26.5794	−1.05979
$630$	0	0
$631$	−33.8710	−1.34838	−0.674191	−	0.738557i	$-0.735508\pi$
$631$	−33.8710	−1.34838	−0.674191	+	0.738557i	$0.735508\pi$
$632$	− 4.41226i	− 0.175510i
$633$	− 12.3573i	− 0.491160i
$634$	−57.8842	−2.29888
$635$	0	0
$636$	−2.13141	−0.0845158
$637$	− 0.549929i	− 0.0217890i
$638$	10.1183i	0.400586i
$639$	−6.69982	−0.265041
$640$	0	0
$641$	15.9571	0.630269	0.315135	−	0.949047i	$-0.397950\pi$
$641$	15.9571	0.630269	0.315135	+	0.949047i	$0.397950\pi$
$642$	− 7.15134i	− 0.282241i
$643$	− 10.2436i	− 0.403968i	−0.979389	−	0.201984i	$-0.935261\pi$
$643$	− 10.2436i	− 0.403968i	0.979389	−	0.201984i	$-0.0647389\pi$
$644$	−23.7013	−0.933960
$645$	0	0
$646$	14.2139	0.559237
$647$	6.00244i	0.235980i	0.993015	+	0.117990i	$0.0376451\pi$
$647$	6.00244i	0.235980i	−0.993015	+	0.117990i	$0.962355\pi$
$648$	10.5020i	0.412556i
$649$	20.3584	0.799138
$650$	0	0
$651$	8.35298	0.327379
$652$	88.9305i	3.48279i
$653$	33.1069i	1.29557i	0.761822	+	0.647787i	$0.224305\pi$
$653$	33.1069i	1.29557i	−0.761822	+	0.647787i	$0.775695\pi$
$654$	−42.4622	−1.66040
$655$	0	0
$656$	68.4431	2.67226
$657$	9.46938i	0.369436i
$658$	105.218i	4.10183i
$659$	−19.5593	−0.761922	−0.380961	−	0.924591i	$-0.624407\pi$
$659$	−19.5593	−0.761922	−0.380961	+	0.924591i	$0.624407\pi$
$660$	0	0
$661$	30.5290	1.18744	0.593721	−	0.804671i	$-0.297658\pi$
$661$	30.5290	1.18744	0.593721	+	0.804671i	$0.297658\pi$
$662$	94.5312i	3.67406i
$663$	− 0.494405i	− 0.0192011i
$664$	127.741	4.95730
$665$	0	0
$666$	28.1948	1.09253
$667$	1.60706i	0.0622255i
$668$	100.390i	3.88421i
$669$	−7.30693	−0.282502
$670$	0	0
$671$	16.7568	0.646890
$672$	− 89.5333i	− 3.45382i
$673$	4.19687i	0.161778i	0.996723	+	0.0808888i	$0.0257759\pi$
$673$	4.19687i	0.161778i	−0.996723	+	0.0808888i	$0.974224\pi$
$674$	−49.5527	−1.90870
$675$	0	0
$676$	−74.7802	−2.87616
$677$	− 9.66481i	− 0.371449i	−0.982602	−	0.185724i	$-0.940537\pi$
$677$	− 9.66481i	− 0.371449i	0.982602	−	0.185724i	$-0.0594632\pi$
$678$	− 6.14256i	− 0.235903i
$679$	43.7961	1.68074
$680$	0	0
$681$	21.0202	0.805495
$682$	− 21.7790i	− 0.833961i
$683$	9.03781i	0.345822i	0.984937	+	0.172911i	$0.0553173\pi$
$683$	9.03781i	0.345822i	−0.984937	+	0.172911i	$0.944683\pi$
$684$	−11.1957	−0.428080
$685$	0	0
$686$	35.7981	1.36678
$687$	19.5544i	0.746045i
$688$	− 170.910i	− 6.51588i
$689$	−0.0695329	−0.00264899
$690$	0	0
$691$	−32.1359	−1.22251	−0.611254	−	0.791435i	$-0.709334\pi$
$691$	−32.1359	−1.22251	−0.611254	+	0.791435i	$0.709334\pi$
$692$	111.171i	4.22610i
$693$	− 9.28244i	− 0.352611i
$694$	34.0728	1.29338
$695$	0	0
$696$	−12.9381	−0.490416
$697$	− 10.1403i	− 0.384091i
$698$	35.7935i	1.35480i
$699$	12.1472	0.459451
$700$	0	0
$701$	−8.54116	−0.322595	−0.161298	−	0.986906i	$-0.551568\pi$
$701$	−8.54116	−0.322595	−0.161298	+	0.986906i	$0.551568\pi$
$702$	0.524454i	0.0197942i
$703$	19.6354i	0.740562i
$704$	−128.925	−4.85904
$705$	0	0
$706$	48.5413	1.82688
$707$	− 30.4664i	− 1.14581i
$708$	39.8493i	1.49763i
$709$	−5.04613	−0.189511	−0.0947556	−	0.995501i	$-0.530207\pi$
$709$	−5.04613	−0.189511	−0.0947556	+	0.995501i	$0.530207\pi$
$710$	0	0
$711$	0.420137	0.0157564
$712$	159.449i	5.97561i
$713$	− 3.45910i	− 0.129544i
$714$	−23.0673	−0.863274
$715$	0	0
$716$	137.244	5.12907
$717$	13.4517i	0.502363i
$718$	− 10.6732i	− 0.398319i
$719$	19.5253	0.728170	0.364085	−	0.931366i	$-0.381382\pi$
$719$	19.5253	0.728170	0.364085	+	0.931366i	$0.381382\pi$
$720$	0	0
$721$	−27.3872	−1.01995
$722$	42.4549i	1.58001i
$723$	− 19.9370i	− 0.741467i
$724$	61.6552	2.29140
$725$	0	0
$726$	6.45588	0.239600
$727$	− 40.4464i	− 1.50007i	−0.661395	−	0.750037i	$-0.730035\pi$
$727$	− 40.4464i	− 1.50007i	0.661395	−	0.750037i	$-0.269965\pi$
$728$	− 6.22490i	− 0.230710i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−25.3215	−0.936548
$732$	32.7995i	1.21231i
$733$	29.3862i	1.08540i	0.839926	+	0.542702i	$0.182599\pi$
$733$	29.3862i	1.08540i	−0.839926	+	0.542702i	$0.817401\pi$
$734$	82.5772	3.04798
$735$	0	0
$736$	−37.0772	−1.36668
$737$	− 9.55318i	− 0.351896i
$738$	10.7566i	0.395956i
$739$	30.7254	1.13025	0.565126	−	0.825005i	$-0.308828\pi$
$739$	30.7254	1.13025	0.565126	+	0.825005i	$0.308828\pi$
$740$	0	0
$741$	−0.365239	−0.0134174
$742$	3.24418i	0.119098i
$743$	41.4841i	1.52190i	0.648808	+	0.760952i	$0.275268\pi$
$743$	41.4841i	1.52190i	−0.648808	+	0.760952i	$0.724732\pi$
$744$	27.8485	1.02097
$745$	0	0
$746$	31.9390	1.16937
$747$	12.1635i	0.445039i
$748$	44.6592i	1.63290i
$749$	−8.08244	−0.295326
$750$	0	0
$751$	38.6123	1.40898	0.704491	−	0.709713i	$-0.251175\pi$
$751$	38.6123	1.40898	0.704491	+	0.709713i	$0.251175\pi$
$752$	212.537i	7.75042i
$753$	2.76013i	0.100585i
$754$	−0.646109	−0.0235299
$755$	0	0
$756$	18.1693	0.660811
$757$	23.0860i	0.839074i	0.907738	+	0.419537i	$0.137807\pi$
$757$	23.0860i	0.839074i	−0.907738	+	0.419537i	$0.862193\pi$
$758$	− 6.11234i	− 0.222010i
$759$	−3.84401	−0.139529
$760$	0	0
$761$	−31.7092	−1.14946	−0.574728	−	0.818344i	$-0.694892\pi$
$761$	−31.7092	−1.14946	−0.574728	+	0.818344i	$0.694892\pi$
$762$	− 41.7200i	− 1.51136i
$763$	47.9908i	1.73738i
$764$	82.8180	2.99625
$765$	0	0
$766$	−28.4278	−1.02714
$767$	1.30000i	0.0469404i
$768$	− 93.9177i	− 3.38896i
$769$	−38.6475	−1.39366	−0.696832	−	0.717235i	$-0.745408\pi$
$769$	−38.6475	−1.39366	−0.696832	+	0.717235i	$0.745408\pi$
$770$	0	0
$771$	17.2848	0.622497
$772$	− 11.2770i	− 0.405867i
$773$	38.9958i	1.40258i	0.712875	+	0.701291i	$0.247393\pi$
$773$	38.9958i	1.40258i	−0.712875	+	0.701291i	$0.752607\pi$
$774$	26.8604	0.965478
$775$	0	0
$776$	146.014	5.24161
$777$	− 31.8658i	− 1.14318i
$778$	79.5672i	2.85262i
$779$	−7.49109	−0.268396
$780$	0	0
$781$	19.7431	0.706463
$782$	9.55256i	0.341599i
$783$	− 1.23197i	− 0.0440269i
$784$	−51.8281	−1.85100
$785$	0	0
$786$	−31.4825	−1.12294
$787$	8.11107i	0.289128i	0.989495	+	0.144564i	$0.0461780\pi$
$787$	8.11107i	0.289128i	−0.989495	+	0.144564i	$0.953822\pi$
$788$	− 6.80387i	− 0.242378i
$789$	−3.96578	−0.141186
$790$	0	0
$791$	−6.94231	−0.246840
$792$	− 30.9473i	− 1.09966i
$793$	1.07002i	0.0379975i
$794$	−103.633	−3.67778
$795$	0	0
$796$	−37.8279	−1.34078
$797$	19.0152i	0.673554i	0.941584	+	0.336777i	$0.109337\pi$
$797$	19.0152i	0.673554i	−0.941584	+	0.336777i	$0.890663\pi$
$798$	17.0409i	0.603240i
$799$	31.4888	1.11399
$800$	0	0
$801$	−15.1828	−0.536458
$802$	− 20.5870i	− 0.726952i
$803$	− 27.9044i	− 0.984726i
$804$	18.6993	0.659472
$805$	0	0
$806$	1.39071	0.0489858
$807$	1.19564i	0.0420884i
$808$	− 101.574i	− 3.57335i
$809$	−49.9078	−1.75467	−0.877333	−	0.479882i	$-0.840679\pi$
$809$	−49.9078	−1.75467	−0.877333	+	0.479882i	$0.840679\pi$
$810$	0	0
$811$	30.9727	1.08760	0.543799	−	0.839215i	$-0.316985\pi$
$811$	30.9727	1.08760	0.543799	+	0.839215i	$0.316985\pi$
$812$	22.3840i	0.785523i
$813$	2.18295i	0.0765593i
$814$	−83.0847	−2.91212
$815$	0	0
$816$	−46.5953	−1.63116
$817$	18.7061i	0.654443i
$818$	− 14.4914i	− 0.506680i
$819$	0.592737	0.0207119
$820$	0	0
$821$	18.9246	0.660473	0.330236	−	0.943898i	$-0.392872\pi$
$821$	18.9246	0.660473	0.330236	+	0.943898i	$0.392872\pi$
$822$	11.6467i	0.406227i
$823$	− 9.31122i	− 0.324569i	−0.986744	−	0.162284i	$-0.948114\pi$
$823$	− 9.31122i	− 0.324569i	0.986744	−	0.162284i	$-0.0518862\pi$
$824$	−91.3076	−3.18085
$825$	0	0
$826$	60.6539	2.11042
$827$	− 23.7383i	− 0.825461i	−0.910853	−	0.412731i	$-0.864575\pi$
$827$	− 23.7383i	− 0.825461i	0.910853	−	0.412731i	$-0.135425\pi$
$828$	− 7.52421i	− 0.261484i
$829$	−1.91351	−0.0664590	−0.0332295	−	0.999448i	$-0.510579\pi$
$829$	−1.91351	−0.0664590	−0.0332295	+	0.999448i	$0.510579\pi$
$830$	0	0
$831$	−1.95174	−0.0677050
$832$	− 8.23260i	− 0.285414i
$833$	7.67867i	0.266050i
$834$	−42.7969	−1.48193
$835$	0	0
$836$	32.9917	1.14104
$837$	2.65174i	0.0916575i
$838$	− 41.5852i	− 1.43654i
$839$	−40.5105	−1.39858	−0.699289	−	0.714839i	$-0.746500\pi$
$839$	−40.5105	−1.39858	−0.699289	+	0.714839i	$0.746500\pi$
$840$	0	0
$841$	−27.4823	−0.947664
$842$	0.185880i	0.00640584i
$843$	− 2.30131i	− 0.0792612i
$844$	−71.2776	−2.45348
$845$	0	0
$846$	−33.4025	−1.14840
$847$	− 7.29643i	− 0.250708i
$848$	6.55314i	0.225036i
$849$	−10.6459	−0.365367
$850$	0	0
$851$	−13.1961	−0.452358
$852$	38.6448i	1.32395i
$853$	18.7735i	0.642792i	0.946945	+	0.321396i	$0.104152\pi$
$853$	18.7735i	0.642792i	−0.946945	+	0.321396i	$0.895848\pi$
$854$	49.9236	1.70835
$855$	0	0
$856$	−26.9465	−0.921012
$857$	10.0276i	0.342536i	0.985225	+	0.171268i	$0.0547863\pi$
$857$	10.0276i	0.342536i	−0.985225	+	0.171268i	$0.945214\pi$
$858$	− 1.54546i	− 0.0527613i
$859$	28.0443	0.956858	0.478429	−	0.878126i	$-0.341206\pi$
$859$	28.0443	0.956858	0.478429	+	0.878126i	$0.341206\pi$
$860$	0	0
$861$	12.1571	0.414313
$862$	− 77.2020i	− 2.62951i
$863$	9.84666i	0.335184i	0.985856	+	0.167592i	$0.0535992\pi$
$863$	9.84666i	0.335184i	−0.985856	+	0.167592i	$0.946401\pi$
$864$	28.4233	0.966979
$865$	0	0
$866$	−85.1815	−2.89459
$867$	− 10.0966i	− 0.342899i
$868$	− 48.1802i	− 1.63534i
$869$	−1.23806	−0.0419984
$870$	0	0
$871$	0.610026	0.0206699
$872$	159.999i	5.41826i
$873$	13.9035i	0.470563i
$874$	7.05690	0.238703
$875$	0	0
$876$	54.6197	1.84543
$877$	− 36.9671i	− 1.24829i	−0.781308	−	0.624146i	$-0.785447\pi$
$877$	− 36.9671i	− 1.24829i	0.781308	−	0.624146i	$-0.214553\pi$
$878$	30.9282i	1.04378i
$879$	14.9591	0.504558
$880$	0	0
$881$	−20.7137	−0.697863	−0.348932	−	0.937148i	$-0.613455\pi$
$881$	−20.7137	−0.697863	−0.348932	+	0.937148i	$0.613455\pi$
$882$	− 8.14536i	− 0.274268i
$883$	12.7254i	0.428243i	0.976807	+	0.214121i	$0.0686888\pi$
$883$	12.7254i	0.428243i	−0.976807	+	0.214121i	$0.931311\pi$
$884$	−2.85175	−0.0959146
$885$	0	0
$886$	−27.3927	−0.920275
$887$	21.6903i	0.728287i	0.931343	+	0.364144i	$0.118638\pi$
$887$	21.6903i	0.728287i	−0.931343	+	0.364144i	$0.881362\pi$
$888$	− 106.239i	− 3.56515i
$889$	−47.1520	−1.58143
$890$	0	0
$891$	2.94681	0.0987218
$892$	42.1466i	1.41117i
$893$	− 23.2621i	− 0.778437i
$894$	9.56995	0.320067
$895$	0	0
$896$	−205.040	−6.84990
$897$	− 0.245462i	− 0.00819574i
$898$	− 53.3178i	− 1.77924i
$899$	−3.26685	−0.108956
$900$	0	0
$901$	0.970891	0.0323451
$902$	− 31.6976i	− 1.05542i
$903$	− 30.3576i	− 1.01024i
$904$	−23.1454	−0.769803
$905$	0	0
$906$	−30.0217	−0.997406
$907$	3.28462i	0.109064i	0.998512	+	0.0545320i	$0.0173667\pi$
$907$	3.28462i	0.109064i	−0.998512	+	0.0545320i	$0.982633\pi$
$908$	− 121.245i	− 4.02366i
$909$	9.67188	0.320796
$910$	0	0
$911$	28.8264	0.955060	0.477530	−	0.878615i	$-0.341532\pi$
$911$	28.8264	0.955060	0.477530	+	0.878615i	$0.341532\pi$
$912$	34.4220i	1.13983i
$913$	− 35.8435i	− 1.18625i
$914$	−52.2802	−1.72927
$915$	0	0
$916$	112.790	3.72669
$917$	35.5815i	1.17500i
$918$	− 7.32297i	− 0.241694i
$919$	17.3985	0.573925	0.286962	−	0.957942i	$-0.407355\pi$
$919$	17.3985	0.573925	0.286962	+	0.957942i	$0.407355\pi$
$920$	0	0
$921$	3.07333	0.101270
$922$	104.854i	3.45317i
$923$	1.26071i	0.0414967i
$924$	−53.5414	−1.76138
$925$	0	0
$926$	−55.5404	−1.82517
$927$	− 8.69434i	− 0.285560i
$928$	35.0165i	1.14947i
$929$	24.3715	0.799603	0.399801	−	0.916602i	$-0.369079\pi$
$929$	24.3715	0.799603	0.399801	+	0.916602i	$0.369079\pi$
$930$	0	0
$931$	5.67257	0.185911
$932$	− 70.0657i	− 2.29508i
$933$	− 24.9095i	− 0.815501i
$934$	−76.2742	−2.49577
$935$	0	0
$936$	1.97616	0.0645928
$937$	− 14.5784i	− 0.476257i	−0.971234	−	0.238129i	$-0.923466\pi$
$937$	− 14.5784i	− 0.476257i	0.971234	−	0.238129i	$-0.0765339\pi$
$938$	− 28.4618i	− 0.929311i
$939$	9.28562	0.303025
$940$	0	0
$941$	−4.19686	−0.136814	−0.0684068	−	0.997658i	$-0.521792\pi$
$941$	−4.19686	−0.136814	−0.0684068	+	0.997658i	$0.521792\pi$
$942$	− 37.3601i	− 1.21726i
$943$	− 5.03445i	− 0.163944i
$944$	122.519	3.98765
$945$	0	0
$946$	−79.1525	−2.57347
$947$	5.53251i	0.179782i	0.995952	+	0.0898912i	$0.0286519\pi$
$947$	5.53251i	0.179782i	−0.995952	+	0.0898912i	$0.971348\pi$
$948$	− 2.42336i	− 0.0787071i
$949$	1.78186	0.0578416
$950$	0	0
$951$	−20.7685	−0.673464
$952$	86.9185i	2.81705i
$953$	50.4779i	1.63514i	0.575830	+	0.817570i	$0.304679\pi$
$953$	50.4779i	1.63514i	−0.575830	+	0.817570i	$0.695321\pi$
$954$	−1.02990	−0.0333442
$955$	0	0
$956$	77.5899	2.50944
$957$	3.63037i	0.117353i
$958$	− 107.511i	− 3.47352i
$959$	13.1631	0.425060
$960$	0	0
$961$	−23.9683	−0.773170
$962$	− 5.30544i	− 0.171054i
$963$	− 2.56585i	− 0.0826835i
$964$	−114.998	−3.70382
$965$	0	0
$966$	−11.4525	−0.368477
$967$	17.0340i	0.547777i	0.961761	+	0.273889i	$0.0883100\pi$
$967$	17.0340i	0.547777i	−0.961761	+	0.273889i	$0.911690\pi$
$968$	− 24.3260i	− 0.781867i
$969$	5.09984	0.163831
$970$	0	0
$971$	23.4172	0.751494	0.375747	−	0.926722i	$-0.377386\pi$
$971$	23.4172	0.751494	0.375747	+	0.926722i	$0.377386\pi$
$972$	5.76803i	0.185010i
$973$	48.3690i	1.55064i
$974$	51.5604	1.65210
$975$	0	0
$976$	100.844	3.22794
$977$	20.7626i	0.664253i	0.943235	+	0.332127i	$0.107766\pi$
$977$	20.7626i	0.664253i	−0.943235	+	0.332127i	$0.892234\pi$
$978$	42.9713i	1.37407i
$979$	44.7408	1.42992
$980$	0	0
$981$	−15.2352	−0.486422
$982$	− 54.9650i	− 1.75400i
$983$	42.1977i	1.34590i	0.739690	+	0.672948i	$0.234972\pi$
$983$	42.1977i	1.34590i	−0.739690	+	0.672948i	$0.765028\pi$
$984$	40.5312	1.29209
$985$	0	0
$986$	9.02164	0.287308
$987$	37.7515i	1.20164i
$988$	2.10671i	0.0670234i
$989$	−12.5716	−0.399753
$990$	0	0
$991$	−3.83926	−0.121958	−0.0609791	−	0.998139i	$-0.519422\pi$
$991$	−3.83926	−0.121958	−0.0609791	+	0.998139i	$0.519422\pi$
$992$	− 75.3711i	− 2.39303i
$993$	33.9172i	1.07633i
$994$	58.8206	1.86567
$995$	0	0
$996$	70.1595	2.22309
$997$	− 42.2091i	− 1.33678i	−0.743812	−	0.668389i	$-0.766984\pi$
$997$	− 42.2091i	− 1.33678i	0.743812	−	0.668389i	$-0.233016\pi$
$998$	− 101.146i	− 3.20173i
$999$	10.1161	0.320060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.e.1249.1		12
5.2	odd	4		1875.2.a.l.1.6	yes	6
5.3	odd	4		1875.2.a.i.1.1	✓	6
5.4	even	2	inner	1875.2.b.e.1249.12		12
15.2	even	4		5625.2.a.o.1.1		6
15.8	even	4		5625.2.a.r.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.1	✓	6	5.3	odd	4
1875.2.a.l.1.6	yes	6	5.2	odd	4
1875.2.b.e.1249.1		12	1.1	even	1	trivial
1875.2.b.e.1249.12		12	5.4	even	2	inner
5625.2.a.o.1.1		6	15.2	even	4
5625.2.a.r.1.6		6	15.8	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.78712i q^{2} -1.00000i q^{3} -5.76803 q^{4} -2.78712 q^{6} +3.15000i q^{7} +10.5020i q^{8} -1.00000 q^{9} +2.94681 q^{11} +5.76803i q^{12} +0.188171i q^{13} +8.77943 q^{14} +17.7341 q^{16} -2.62743i q^{17} +2.78712i q^{18} -1.94100 q^{19} +3.15000 q^{21} -8.21310i q^{22} -1.30447i q^{23} +10.5020 q^{24} +0.524454 q^{26} +1.00000i q^{27} -18.1693i q^{28} -1.23197 q^{29} +2.65174 q^{31} -28.4233i q^{32} -2.94681i q^{33} -7.32297 q^{34} +5.76803 q^{36} -10.1161i q^{37} +5.40980i q^{38} +0.188171 q^{39} +3.85940 q^{41} -8.77943i q^{42} -9.63734i q^{43} -16.9973 q^{44} -3.63570 q^{46} +11.9846i q^{47} -17.7341i q^{48} -2.92250 q^{49} -2.62743 q^{51} -1.08537i q^{52} +0.369521i q^{53} +2.78712 q^{54} -33.0812 q^{56} +1.94100i q^{57} +3.43364i q^{58} +6.90864 q^{59} +5.68643 q^{61} -7.39071i q^{62} -3.15000i q^{63} -43.7507 q^{64} -8.21310 q^{66} -3.24188i q^{67} +15.1551i q^{68} -1.30447 q^{69} +6.69982 q^{71} -10.5020i q^{72} -9.46938i q^{73} -28.1948 q^{74} +11.1957 q^{76} +9.28244i q^{77} -0.524454i q^{78} -0.420137 q^{79} +1.00000 q^{81} -10.7566i q^{82} -12.1635i q^{83} -18.1693 q^{84} -26.8604 q^{86} +1.23197i q^{87} +30.9473i q^{88} +15.1828 q^{89} -0.592737 q^{91} +7.52421i q^{92} -2.65174i q^{93} +33.4025 q^{94} -28.4233 q^{96} -13.9035i q^{97} +8.14536i q^{98} -2.94681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51}+ \cdots - 66 q^{96}+O(q^{100}) \) 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9 + 8 * q^14 + 34 * q^16 + 4 * q^19 + 4 * q^21 + 12 * q^24 + 74 * q^26 - 62 * q^29 - 4 * q^31 - 74 * q^34 + 22 * q^36 + 66 * q^41 + 22 * q^44 - 24 * q^46 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 60 * q^56 + 16 * q^59 + 68 * q^61 - 24 * q^64 - 18 * q^66 - 2 * q^69 - 6 * q^71 - 72 * q^74 + 54 * q^76 - 50 * q^79 + 12 * q^81 - 88 * q^84 - 60 * q^86 - 36 * q^89 + 56 * q^91 + 100 * q^94 - 66 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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