LMFDB - Embedded newform 925.2.b.i.149.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-14,0,12,0,0,-20,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 21x^{12} + 170x^{10} + 665x^{8} + 1280x^{6} + 1087x^{4} + 311x^{2} + 25 $ x^14 + 21x^12 + 170x^10 + 665x^8 + 1280x^6 + 1087x^4 + 311x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.9
Root			$0.548343i$ of defining polynomial
Character	$\chi$	$=$	925.149
Dual form			925.2.b.i.149.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.548343i q^{2} -2.30011i q^{3} +1.69932 q^{4} +1.26125 q^{6} +1.23269i q^{7} +2.02850i q^{8} -2.29051 q^{9} -0.976048 q^{11} -3.90863i q^{12} -6.96588i q^{13} -0.675937 q^{14} +2.28633 q^{16} +2.51901i q^{17} -1.25599i q^{18} +2.92164 q^{19} +2.83532 q^{21} -0.535210i q^{22} -8.99438i q^{23} +4.66577 q^{24} +3.81970 q^{26} -1.63190i q^{27} +2.09473i q^{28} +0.687767 q^{29} +6.60794 q^{31} +5.31069i q^{32} +2.24502i q^{33} -1.38128 q^{34} -3.89232 q^{36} -1.00000i q^{37} +1.60206i q^{38} -16.0223 q^{39} -0.193259 q^{41} +1.55473i q^{42} +9.21747i q^{43} -1.65862 q^{44} +4.93201 q^{46} -3.18723i q^{47} -5.25881i q^{48} +5.48048 q^{49} +5.79402 q^{51} -11.8373i q^{52} +1.28828i q^{53} +0.894839 q^{54} -2.50051 q^{56} -6.72011i q^{57} +0.377133i q^{58} -12.2588 q^{59} -3.67468 q^{61} +3.62342i q^{62} -2.82349i q^{63} +1.66057 q^{64} -1.23104 q^{66} -8.16762i q^{67} +4.28061i q^{68} -20.6881 q^{69} -2.82331 q^{71} -4.64630i q^{72} +0.837276i q^{73} +0.548343 q^{74} +4.96481 q^{76} -1.20316i q^{77} -8.78573i q^{78} -3.44333 q^{79} -10.6251 q^{81} -0.105972i q^{82} +4.66960i q^{83} +4.81812 q^{84} -5.05434 q^{86} -1.58194i q^{87} -1.97991i q^{88} +6.74431 q^{89} +8.58676 q^{91} -15.2843i q^{92} -15.1990i q^{93} +1.74770 q^{94} +12.2152 q^{96} -2.21791i q^{97} +3.00518i q^{98} +2.23565 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4} + 12 q^{6} - 20 q^{9} + 32 q^{11} + 6 q^{14} + 6 q^{16} - 18 q^{19} + 4 q^{21} + 48 q^{26} - 6 q^{29} + 22 q^{31} + 46 q^{34} + 38 q^{36} - 4 q^{39} + 58 q^{41} - 36 q^{44} + 14 q^{46} - 10 q^{49}+ \cdots - 78 q^{99}+O(q^{100}) $ 14 * q - 14 * q^4 + 12 * q^6 - 20 * q^9 + 32 * q^11 + 6 * q^14 + 6 * q^16 - 18 * q^19 + 4 * q^21 + 48 * q^26 - 6 * q^29 + 22 * q^31 + 46 * q^34 + 38 * q^36 - 4 * q^39 + 58 * q^41 - 36 * q^44 + 14 * q^46 - 10 * q^49 + 10 * q^51 - 16 * q^54 + 26 * q^56 - 84 * q^59 - 22 * q^61 + 32 * q^64 + 58 * q^66 - 4 * q^69 + 20 * q^71 - 2 * q^74 + 22 * q^76 + 18 * q^79 - 26 * q^81 + 82 * q^84 - 44 * q^86 - 24 * q^89 - 4 * q^91 - 18 * q^94 - 10 * q^96 - 78 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/925\mathbb{Z}\right)^\times$.

$n$	$76$	$852$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.548343i	0.387737i	0.981027	+	0.193869i	$0.0621036\pi$
$2$	0.548343i	0.387737i	−0.981027	+	0.193869i	$0.937896\pi$
$3$	− 2.30011i	− 1.32797i	−0.747746	−	0.663985i	$-0.768864\pi$
$3$	− 2.30011i	− 1.32797i	0.747746	−	0.663985i	$-0.231136\pi$
$4$	1.69932	0.849660
$5$	0	0
$5$	0	0
$6$	1.26125	0.514904
$7$	1.23269i	0.465912i	0.972487	+	0.232956i	$0.0748399\pi$
$7$	1.23269i	0.465912i	−0.972487	+	0.232956i	$0.925160\pi$
$8$	2.02850i	0.717182i
$9$	−2.29051	−0.763505
$10$	0	0
$11$	−0.976048	−0.294290	−0.147145	−	0.989115i	$-0.547008\pi$
$11$	−0.976048	−0.294290	−0.147145	+	0.989115i	$0.547008\pi$
$12$	− 3.90863i	− 1.12832i
$13$	− 6.96588i	− 1.93199i	−0.258564	−	0.965994i	$-0.583249\pi$
$13$	− 6.96588i	− 1.93199i	0.258564	−	0.965994i	$-0.416751\pi$
$14$	−0.675937	−0.180652
$15$	0	0
$16$	2.28633	0.571582
$17$	2.51901i	0.610951i	0.952200	+	0.305475i	$0.0988154\pi$
$17$	2.51901i	0.610951i	−0.952200	+	0.305475i	$0.901185\pi$
$18$	− 1.25599i	− 0.296039i
$19$	2.92164	0.670271	0.335136	−	0.942170i	$-0.391218\pi$
$19$	2.92164	0.670271	0.335136	+	0.942170i	$0.391218\pi$
$20$	0	0
$21$	2.83532	0.618718
$22$	− 0.535210i	− 0.114107i
$23$	− 8.99438i	− 1.87546i	−0.347368	−	0.937729i	$-0.612925\pi$
$23$	− 8.99438i	− 1.87546i	0.347368	−	0.937729i	$-0.387075\pi$
$24$	4.66577	0.952396
$25$	0	0
$26$	3.81970	0.749104
$27$	− 1.63190i	− 0.314059i
$28$	2.09473i	0.395867i
$29$	0.687767	0.127715	0.0638576	−	0.997959i	$-0.479660\pi$
$29$	0.687767	0.127715	0.0638576	+	0.997959i	$0.479660\pi$
$30$	0	0
$31$	6.60794	1.18682	0.593411	−	0.804900i	$-0.297781\pi$
$31$	6.60794	1.18682	0.593411	+	0.804900i	$0.297781\pi$
$32$	5.31069i	0.938806i
$33$	2.24502i	0.390808i
$34$	−1.38128	−0.236888
$35$	0	0
$36$	−3.89232	−0.648719
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	1.60206i	0.259889i
$39$	−16.0223	−2.56562
$40$	0	0
$41$	−0.193259	−0.0301820	−0.0150910	−	0.999886i	$-0.504804\pi$
$41$	−0.193259	−0.0301820	−0.0150910	+	0.999886i	$0.504804\pi$
$42$	1.55473i	0.239900i
$43$	9.21747i	1.40565i	0.711362	+	0.702826i	$0.248079\pi$
$43$	9.21747i	1.40565i	−0.711362	+	0.702826i	$0.751921\pi$
$44$	−1.65862	−0.250046
$45$	0	0
$46$	4.93201	0.727185
$47$	− 3.18723i	− 0.464905i	−0.972608	−	0.232453i	$-0.925325\pi$
$47$	− 3.18723i	− 0.464905i	0.972608	−	0.232453i	$-0.0746751\pi$
$48$	− 5.25881i	− 0.759043i
$49$	5.48048	0.782926
$50$	0	0
$51$	5.79402	0.811324
$52$	− 11.8373i	− 1.64153i
$53$	1.28828i	0.176959i	0.996078	+	0.0884795i	$0.0282008\pi$
$53$	1.28828i	0.176959i	−0.996078	+	0.0884795i	$0.971799\pi$
$54$	0.894839	0.121772
$55$	0	0
$56$	−2.50051	−0.334144
$57$	− 6.72011i	− 0.890100i
$58$	0.377133i	0.0495199i
$59$	−12.2588	−1.59596	−0.797978	−	0.602686i	$-0.794097\pi$
$59$	−12.2588	−1.59596	−0.797978	+	0.602686i	$0.794097\pi$
$60$	0	0
$61$	−3.67468	−0.470495	−0.235247	−	0.971936i	$-0.575590\pi$
$61$	−3.67468	−0.470495	−0.235247	+	0.971936i	$0.575590\pi$
$62$	3.62342i	0.460175i
$63$	− 2.82349i	− 0.355726i
$64$	1.66057	0.207572
$65$	0	0
$66$	−1.23104	−0.151531
$67$	− 8.16762i	− 0.997833i	−0.866650	−	0.498917i	$-0.833731\pi$
$67$	− 8.16762i	− 0.997833i	0.866650	−	0.498917i	$-0.166269\pi$
$68$	4.28061i	0.519100i
$69$	−20.6881	−2.49055
$70$	0	0
$71$	−2.82331	−0.335066	−0.167533	−	0.985866i	$-0.553580\pi$
$71$	−2.82331	−0.335066	−0.167533	+	0.985866i	$0.553580\pi$
$72$	− 4.64630i	− 0.547572i
$73$	0.837276i	0.0979958i	0.998799	+	0.0489979i	$0.0156028\pi$
$73$	0.837276i	0.0979958i	−0.998799	+	0.0489979i	$0.984397\pi$
$74$	0.548343	0.0637436
$75$	0	0
$76$	4.96481	0.569503
$77$	− 1.20316i	− 0.137113i
$78$	− 8.78573i	− 0.994788i
$79$	−3.44333	−0.387405	−0.193703	−	0.981060i	$-0.562050\pi$
$79$	−3.44333	−0.387405	−0.193703	+	0.981060i	$0.562050\pi$
$80$	0	0
$81$	−10.6251	−1.18057
$82$	− 0.105972i	− 0.0117027i
$83$	4.66960i	0.512555i	0.966603	+	0.256278i	$0.0824962\pi$
$83$	4.66960i	0.512555i	−0.966603	+	0.256278i	$0.917504\pi$
$84$	4.81812	0.525700
$85$	0	0
$86$	−5.05434	−0.545023
$87$	− 1.58194i	− 0.169602i
$88$	− 1.97991i	− 0.211059i
$89$	6.74431	0.714895	0.357448	−	0.933933i	$-0.383647\pi$
$89$	6.74431	0.714895	0.357448	+	0.933933i	$0.383647\pi$
$90$	0	0
$91$	8.58676	0.900137
$92$	− 15.2843i	− 1.59350i
$93$	− 15.1990i	− 1.57606i
$94$	1.74770	0.180261
$95$	0	0
$96$	12.2152	1.24671
$97$	− 2.21791i	− 0.225195i	−0.993641	−	0.112597i	$-0.964083\pi$
$97$	− 2.21791i	− 0.225195i	0.993641	−	0.112597i	$-0.0359170\pi$
$98$	3.00518i	0.303569i
$99$	2.23565	0.224692
$100$	0	0
$101$	4.83925	0.481523	0.240762	−	0.970584i	$-0.422603\pi$
$101$	4.83925	0.481523	0.240762	+	0.970584i	$0.422603\pi$
$102$	3.17711i	0.314581i
$103$	8.95741i	0.882600i	0.897360	+	0.441300i	$0.145483\pi$
$103$	8.95741i	0.882600i	−0.897360	+	0.441300i	$0.854517\pi$
$104$	14.1303	1.38559
$105$	0	0
$106$	−0.706420	−0.0686136
$107$	15.6463i	1.51259i	0.654233	+	0.756293i	$0.272992\pi$
$107$	15.6463i	1.51259i	−0.654233	+	0.756293i	$0.727008\pi$
$108$	− 2.77311i	− 0.266843i
$109$	12.4852	1.19586	0.597931	−	0.801548i	$-0.295990\pi$
$109$	12.4852	1.19586	0.597931	+	0.801548i	$0.295990\pi$
$110$	0	0
$111$	−2.30011	−0.218317
$112$	2.81833i	0.266307i
$113$	19.4522i	1.82991i	0.403561	+	0.914953i	$0.367772\pi$
$113$	19.4522i	1.82991i	−0.403561	+	0.914953i	$0.632228\pi$
$114$	3.68493	0.345125
$115$	0	0
$116$	1.16874	0.108514
$117$	15.9555i	1.47508i
$118$	− 6.72202i	− 0.618812i
$119$	−3.10516	−0.284650
$120$	0	0
$121$	−10.0473	−0.913394
$122$	− 2.01499i	− 0.182428i
$123$	0.444518i	0.0400808i
$124$	11.2290	1.00839
$125$	0	0
$126$	1.54824	0.137928
$127$	5.25778i	0.466553i	0.972410	+	0.233276i	$0.0749447\pi$
$127$	5.25778i	0.466553i	−0.972410	+	0.233276i	$0.925055\pi$
$128$	11.5319i	1.01929i
$129$	21.2012	1.86666
$130$	0	0
$131$	15.2972	1.33653	0.668263	−	0.743926i	$-0.267038\pi$
$131$	15.2972	1.33653	0.668263	+	0.743926i	$0.267038\pi$
$132$	3.81501i	0.332054i
$133$	3.60148i	0.312288i
$134$	4.47866	0.386897
$135$	0	0
$136$	−5.10981	−0.438163
$137$	3.28024i	0.280249i	0.990134	+	0.140125i	$0.0447503\pi$
$137$	3.28024i	0.280249i	−0.990134	+	0.140125i	$0.955250\pi$
$138$	− 11.3442i	− 0.965680i
$139$	−0.561230	−0.0476029	−0.0238015	−	0.999717i	$-0.507577\pi$
$139$	−0.561230	−0.0476029	−0.0238015	+	0.999717i	$0.507577\pi$
$140$	0	0
$141$	−7.33099	−0.617380
$142$	− 1.54815i	− 0.129918i
$143$	6.79904i	0.568564i
$144$	−5.23686	−0.436405
$145$	0	0
$146$	−0.459115	−0.0379966
$147$	− 12.6057i	− 1.03970i
$148$	− 1.69932i	− 0.139683i
$149$	−12.7327	−1.04311	−0.521553	−	0.853219i	$-0.674647\pi$
$149$	−12.7327	−1.04311	−0.521553	+	0.853219i	$0.674647\pi$
$150$	0	0
$151$	−18.1594	−1.47779	−0.738894	−	0.673822i	$-0.764651\pi$
$151$	−18.1594	−1.47779	−0.738894	+	0.673822i	$0.764651\pi$
$152$	5.92655i	0.480707i
$153$	− 5.76984i	− 0.466464i
$154$	0.659747	0.0531639
$155$	0	0
$156$	−27.2270	−2.17991
$157$	12.5146i	0.998777i	0.866378	+	0.499388i	$0.166442\pi$
$157$	12.5146i	0.998777i	−0.866378	+	0.499388i	$0.833558\pi$
$158$	− 1.88813i	− 0.150211i
$159$	2.96319	0.234996
$160$	0	0
$161$	11.0873	0.873799
$162$	− 5.82620i	− 0.457749i
$163$	− 8.40802i	− 0.658567i	−0.944231	−	0.329284i	$-0.893193\pi$
$163$	− 8.40802i	− 0.658567i	0.944231	−	0.329284i	$-0.106807\pi$
$164$	−0.328409	−0.0256444
$165$	0	0
$166$	−2.56055	−0.198737
$167$	− 11.9553i	− 0.925130i	−0.886585	−	0.462565i	$-0.846929\pi$
$167$	− 11.9553i	− 0.925130i	0.886585	−	0.462565i	$-0.153071\pi$
$168$	5.75144i	0.443733i
$169$	−35.5235	−2.73258
$170$	0	0
$171$	−6.69207	−0.511755
$172$	15.6634i	1.19433i
$173$	21.6980i	1.64967i	0.565375	+	0.824834i	$0.308732\pi$
$173$	21.6980i	1.64967i	−0.565375	+	0.824834i	$0.691268\pi$
$174$	0.867447	0.0657610
$175$	0	0
$176$	−2.23156	−0.168211
$177$	28.1966i	2.11938i
$178$	3.69820i	0.277192i
$179$	3.99480	0.298586	0.149293	−	0.988793i	$-0.452300\pi$
$179$	3.99480	0.298586	0.149293	+	0.988793i	$0.452300\pi$
$180$	0	0
$181$	17.2016	1.27859	0.639293	−	0.768964i	$-0.279227\pi$
$181$	17.2016	1.27859	0.639293	+	0.768964i	$0.279227\pi$
$182$	4.70849i	0.349017i
$183$	8.45218i	0.624803i
$184$	18.2451	1.34504
$185$	0	0
$186$	8.33428	0.611099
$187$	− 2.45868i	− 0.179797i
$188$	− 5.41612i	− 0.395011i
$189$	2.01162	0.146324
$190$	0	0
$191$	−20.4730	−1.48137	−0.740686	−	0.671852i	$-0.765499\pi$
$191$	−20.4730	−1.48137	−0.740686	+	0.671852i	$0.765499\pi$
$192$	− 3.81950i	− 0.275649i
$193$	− 20.4667i	− 1.47322i	−0.676317	−	0.736611i	$-0.736425\pi$
$193$	− 20.4667i	− 1.47322i	0.676317	−	0.736611i	$-0.263575\pi$
$194$	1.21618	0.0873164
$195$	0	0
$196$	9.31309	0.665220
$197$	17.1879i	1.22459i	0.790630	+	0.612294i	$0.209753\pi$
$197$	17.1879i	1.22459i	−0.790630	+	0.612294i	$0.790247\pi$
$198$	1.22591i	0.0871213i
$199$	12.8818	0.913164	0.456582	−	0.889681i	$-0.349074\pi$
$199$	12.8818	0.913164	0.456582	+	0.889681i	$0.349074\pi$
$200$	0	0
$201$	−18.7864	−1.32509
$202$	2.65357i	0.186704i
$203$	0.847803i	0.0595041i
$204$	9.84588	0.689350
$205$	0	0
$206$	−4.91174	−0.342217
$207$	20.6018i	1.43192i
$208$	− 15.9263i	− 1.10429i
$209$	−2.85167	−0.197254
$210$	0	0
$211$	14.6270	1.00697	0.503484	−	0.864005i	$-0.332051\pi$
$211$	14.6270	1.00697	0.503484	+	0.864005i	$0.332051\pi$
$212$	2.18920i	0.150355i
$213$	6.49394i	0.444957i
$214$	−8.57955	−0.586486
$215$	0	0
$216$	3.31030	0.225237
$217$	8.14554i	0.552955i
$218$	6.84615i	0.463680i
$219$	1.92583	0.130135
$220$	0	0
$221$	17.5472	1.18035
$222$	− 1.26125i	− 0.0846496i
$223$	3.59602i	0.240807i	0.992725	+	0.120404i	$0.0384189\pi$
$223$	3.59602i	0.240807i	−0.992725	+	0.120404i	$0.961581\pi$
$224$	−6.54642	−0.437401
$225$	0	0
$226$	−10.6665	−0.709523
$227$	− 21.7284i	− 1.44217i	−0.692848	−	0.721084i	$-0.743644\pi$
$227$	− 21.7284i	− 1.44217i	0.692848	−	0.721084i	$-0.256356\pi$
$228$	− 11.4196i	− 0.756282i
$229$	14.6078	0.965314	0.482657	−	0.875809i	$-0.339672\pi$
$229$	14.6078	0.965314	0.482657	+	0.875809i	$0.339672\pi$
$230$	0	0
$231$	−2.76741	−0.182082
$232$	1.39513i	0.0915950i
$233$	18.5161i	1.21303i	0.795071	+	0.606517i	$0.207434\pi$
$233$	18.5161i	1.21303i	−0.795071	+	0.606517i	$0.792566\pi$
$234$	−8.74907	−0.571944
$235$	0	0
$236$	−20.8316	−1.35602
$237$	7.92005i	0.514462i
$238$	− 1.70269i	− 0.110369i
$239$	−18.0845	−1.16979	−0.584896	−	0.811109i	$-0.698864\pi$
$239$	−18.0845	−1.16979	−0.584896	+	0.811109i	$0.698864\pi$
$240$	0	0
$241$	−12.4643	−0.802899	−0.401449	−	0.915881i	$-0.631493\pi$
$241$	−12.4643	−0.802899	−0.401449	+	0.915881i	$0.631493\pi$
$242$	− 5.50939i	− 0.354157i
$243$	19.5432i	1.25370i
$244$	−6.24446	−0.399761
$245$	0	0
$246$	−0.243748	−0.0155408
$247$	− 20.3518i	− 1.29496i
$248$	13.4042i	0.851167i
$249$	10.7406	0.680658
$250$	0	0
$251$	4.32919	0.273256	0.136628	−	0.990622i	$-0.456373\pi$
$251$	4.32919	0.273256	0.136628	+	0.990622i	$0.456373\pi$
$252$	− 4.79801i	− 0.302246i
$253$	8.77895i	0.551928i
$254$	−2.88307	−0.180900
$255$	0	0
$256$	−3.00231	−0.187645
$257$	16.4633i	1.02695i	0.858105	+	0.513475i	$0.171642\pi$
$257$	16.4633i	1.02695i	−0.858105	+	0.513475i	$0.828358\pi$
$258$	11.6255i	0.723775i
$259$	1.23269	0.0765955
$260$	0	0
$261$	−1.57534	−0.0975111
$262$	8.38813i	0.518221i
$263$	− 19.3023i	− 1.19023i	−0.803640	−	0.595116i	$-0.797106\pi$
$263$	− 19.3023i	− 1.19023i	0.803640	−	0.595116i	$-0.202894\pi$
$264$	−4.55402	−0.280280
$265$	0	0
$266$	−1.97485	−0.121086
$267$	− 15.5127i	− 0.949359i
$268$	− 13.8794i	− 0.847819i
$269$	9.62970	0.587133	0.293567	−	0.955939i	$-0.405158\pi$
$269$	9.62970	0.587133	0.293567	+	0.955939i	$0.405158\pi$
$270$	0	0
$271$	−3.65132	−0.221802	−0.110901	−	0.993831i	$-0.535374\pi$
$271$	−3.65132	−0.221802	−0.110901	+	0.993831i	$0.535374\pi$
$272$	5.75929i	0.349208i
$273$	− 19.7505i	− 1.19536i
$274$	−1.79870	−0.108663
$275$	0	0
$276$	−35.1557	−2.11612
$277$	− 25.0654i	− 1.50603i	−0.658002	−	0.753017i	$-0.728598\pi$
$277$	− 25.0654i	− 1.50603i	0.658002	−	0.753017i	$-0.271402\pi$
$278$	− 0.307747i	− 0.0184574i
$279$	−15.1356	−0.906144
$280$	0	0
$281$	11.5359	0.688177	0.344089	−	0.938937i	$-0.388188\pi$
$281$	11.5359	0.688177	0.344089	+	0.938937i	$0.388188\pi$
$282$	− 4.01990i	− 0.239381i
$283$	30.7318i	1.82682i	0.407045	+	0.913408i	$0.366559\pi$
$283$	30.7318i	1.82682i	−0.407045	+	0.913408i	$0.633441\pi$
$284$	−4.79771	−0.284692
$285$	0	0
$286$	−3.72821	−0.220454
$287$	− 0.238228i	− 0.0140622i
$288$	− 12.1642i	− 0.716783i
$289$	10.6546	0.626739
$290$	0	0
$291$	−5.10144	−0.299052
$292$	1.42280i	0.0832631i
$293$	16.2157i	0.947332i	0.880705	+	0.473666i	$0.157070\pi$
$293$	16.2157i	0.947332i	−0.880705	+	0.473666i	$0.842930\pi$
$294$	6.91226	0.403131
$295$	0	0
$296$	2.02850	0.117904
$297$	1.59281i	0.0924242i
$298$	− 6.98190i	− 0.404451i
$299$	−62.6538	−3.62336
$300$	0	0
$301$	−11.3623	−0.654910
$302$	− 9.95756i	− 0.572993i
$303$	− 11.1308i	− 0.639448i
$304$	6.67983	0.383115
$305$	0	0
$306$	3.16385	0.180865
$307$	− 0.434062i	− 0.0247732i	−0.999923	−	0.0123866i	$-0.996057\pi$
$307$	− 0.434062i	− 0.0247732i	0.999923	−	0.0123866i	$-0.00394288\pi$
$308$	− 2.04456i	− 0.116500i
$309$	20.6030	1.17207
$310$	0	0
$311$	−21.0126	−1.19151	−0.595756	−	0.803165i	$-0.703148\pi$
$311$	−21.0126	−1.19151	−0.595756	+	0.803165i	$0.703148\pi$
$312$	− 32.5012i	− 1.84002i
$313$	2.16272i	0.122244i	0.998130	+	0.0611222i	$0.0194679\pi$
$313$	2.16272i	0.122244i	−0.998130	+	0.0611222i	$0.980532\pi$
$314$	−6.86232	−0.387263
$315$	0	0
$316$	−5.85132	−0.329162
$317$	28.3991i	1.59505i	0.603284	+	0.797526i	$0.293858\pi$
$317$	28.3991i	1.59505i	−0.603284	+	0.797526i	$0.706142\pi$
$318$	1.62485i	0.0911168i
$319$	−0.671294	−0.0375853
$320$	0	0
$321$	35.9883	2.00867
$322$	6.07963i	0.338805i
$323$	7.35967i	0.409503i
$324$	−18.0554	−1.00308
$325$	0	0
$326$	4.61048	0.255351
$327$	− 28.7173i	− 1.58807i
$328$	− 0.392026i	− 0.0216460i
$329$	3.92886	0.216605
$330$	0	0
$331$	16.2418	0.892728	0.446364	−	0.894851i	$-0.352719\pi$
$331$	16.2418	0.892728	0.446364	+	0.894851i	$0.352719\pi$
$332$	7.93515i	0.435498i
$333$	2.29051i	0.125519i
$334$	6.55561	0.358707
$335$	0	0
$336$	6.48247	0.353648
$337$	11.0000i	0.599207i	0.954064	+	0.299604i	$0.0968544\pi$
$337$	11.0000i	0.599207i	−0.954064	+	0.299604i	$0.903146\pi$
$338$	− 19.4791i	− 1.05952i
$339$	44.7422	2.43006
$340$	0	0
$341$	−6.44967	−0.349269
$342$	− 3.66955i	− 0.198427i
$343$	15.3845i	0.830687i
$344$	−18.6976	−1.00811
$345$	0	0
$346$	−11.8980	−0.639638
$347$	− 8.15833i	− 0.437962i	−0.975729	−	0.218981i	$-0.929727\pi$
$347$	− 8.15833i	− 0.437962i	0.975729	−	0.218981i	$-0.0702733\pi$
$348$	− 2.68822i	− 0.144104i
$349$	28.2422	1.51177	0.755886	−	0.654704i	$-0.227207\pi$
$349$	28.2422	1.51177	0.755886	+	0.654704i	$0.227207\pi$
$350$	0	0
$351$	−11.3676	−0.606757
$352$	− 5.18349i	− 0.276281i
$353$	1.95167i	0.103877i	0.998650	+	0.0519385i	$0.0165400\pi$
$353$	1.95167i	0.103877i	−0.998650	+	0.0519385i	$0.983460\pi$
$354$	−15.4614	−0.821764
$355$	0	0
$356$	11.4607	0.607418
$357$	7.14222i	0.378006i
$358$	2.19052i	0.115773i
$359$	−25.2874	−1.33462	−0.667310	−	0.744780i	$-0.732554\pi$
$359$	−25.2874	−1.33462	−0.667310	+	0.744780i	$0.732554\pi$
$360$	0	0
$361$	−10.4640	−0.550736
$362$	9.43239i	0.495755i
$363$	23.1100i	1.21296i
$364$	14.5917	0.764811
$365$	0	0
$366$	−4.63470	−0.242259
$367$	− 7.03397i	− 0.367170i	−0.983004	−	0.183585i	$-0.941230\pi$
$367$	− 7.03397i	− 0.367170i	0.983004	−	0.183585i	$-0.0587703\pi$
$368$	− 20.5641i	− 1.07198i
$369$	0.442663	0.0230441
$370$	0	0
$371$	−1.58805	−0.0824474
$372$	− 25.8280i	− 1.33912i
$373$	25.0922i	1.29923i	0.760265	+	0.649613i	$0.225069\pi$
$373$	25.0922i	1.29923i	−0.760265	+	0.649613i	$0.774931\pi$
$374$	1.34820	0.0697138
$375$	0	0
$376$	6.46529	0.333422
$377$	− 4.79091i	− 0.246744i
$378$	1.10306i	0.0567352i
$379$	18.9929	0.975601	0.487801	−	0.872955i	$-0.337799\pi$
$379$	18.9929	0.975601	0.487801	+	0.872955i	$0.337799\pi$
$380$	0	0
$381$	12.0935	0.619568
$382$	− 11.2262i	− 0.574383i
$383$	− 30.2394i	− 1.54516i	−0.634918	−	0.772580i	$-0.718966\pi$
$383$	− 30.2394i	− 1.54516i	0.634918	−	0.772580i	$-0.281034\pi$
$384$	26.5247	1.35359
$385$	0	0
$386$	11.2228	0.571223
$387$	− 21.1127i	− 1.07322i
$388$	− 3.76894i	− 0.191339i
$389$	20.8628	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$389$	20.8628	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$390$	0	0
$391$	22.6570	1.14581
$392$	11.1171i	0.561500i
$393$	− 35.1853i	− 1.77487i
$394$	−9.42488	−0.474819
$395$	0	0
$396$	3.79909	0.190911
$397$	28.4743i	1.42908i	0.699593	+	0.714541i	$0.253365\pi$
$397$	28.4743i	1.42908i	−0.699593	+	0.714541i	$0.746635\pi$
$398$	7.06363i	0.354068i
$399$	8.28380	0.414709
$400$	0	0
$401$	−26.5071	−1.32370	−0.661851	−	0.749636i	$-0.730229\pi$
$401$	−26.5071	−1.32370	−0.661851	+	0.749636i	$0.730229\pi$
$402$	− 10.3014i	− 0.513788i
$403$	− 46.0302i	− 2.29293i
$404$	8.22343	0.409131
$405$	0	0
$406$	−0.464887	−0.0230720
$407$	0.976048i	0.0483809i
$408$	11.7531i	0.581867i
$409$	−7.94050	−0.392632	−0.196316	−	0.980541i	$-0.562898\pi$
$409$	−7.94050	−0.392632	−0.196316	+	0.980541i	$0.562898\pi$
$410$	0	0
$411$	7.54491	0.372163
$412$	15.2215i	0.749910i
$413$	− 15.1113i	− 0.743576i
$414$	−11.2968	−0.555209
$415$	0	0
$416$	36.9936	1.81376
$417$	1.29089i	0.0632153i
$418$	− 1.56369i	− 0.0764827i
$419$	−31.9064	−1.55873	−0.779366	−	0.626569i	$-0.784459\pi$
$419$	−31.9064	−1.55873	−0.779366	+	0.626569i	$0.784459\pi$
$420$	0	0
$421$	11.0436	0.538232	0.269116	−	0.963108i	$-0.413268\pi$
$421$	11.0436	0.538232	0.269116	+	0.963108i	$0.413268\pi$
$422$	8.02064i	0.390439i
$423$	7.30040i	0.354957i
$424$	−2.61327	−0.126912
$425$	0	0
$426$	−3.56091	−0.172527
$427$	− 4.52974i	− 0.219209i
$428$	26.5881i	1.28518i
$429$	15.6385	0.755036
$430$	0	0
$431$	40.0005	1.92676	0.963379	−	0.268143i	$-0.0864100\pi$
$431$	40.0005	1.92676	0.963379	+	0.268143i	$0.0864100\pi$
$432$	− 3.73105i	− 0.179510i
$433$	− 3.42091i	− 0.164398i	−0.996616	−	0.0821992i	$-0.973806\pi$
$433$	− 3.42091i	− 0.164398i	0.996616	−	0.0821992i	$-0.0261944\pi$
$434$	−4.46655	−0.214401
$435$	0	0
$436$	21.2163	1.01608
$437$	− 26.2784i	− 1.25707i
$438$	1.05602i	0.0504584i
$439$	−1.87199	−0.0893451	−0.0446725	−	0.999002i	$-0.514224\pi$
$439$	−1.87199	−0.0893451	−0.0446725	+	0.999002i	$0.514224\pi$
$440$	0	0
$441$	−12.5531	−0.597767
$442$	9.62187i	0.457666i
$443$	− 39.0684i	− 1.85620i	−0.372335	−	0.928098i	$-0.621443\pi$
$443$	− 39.0684i	− 1.85620i	0.372335	−	0.928098i	$-0.378557\pi$
$444$	−3.90863	−0.185495
$445$	0	0
$446$	−1.97185	−0.0933700
$447$	29.2867i	1.38521i
$448$	2.04697i	0.0967102i
$449$	−27.7382	−1.30905	−0.654523	−	0.756042i	$-0.727130\pi$
$449$	−27.7382	−1.30905	−0.654523	+	0.756042i	$0.727130\pi$
$450$	0	0
$451$	0.188630	0.00888225
$452$	33.0554i	1.55480i
$453$	41.7685i	1.96246i
$454$	11.9146	0.559182
$455$	0	0
$456$	13.6317	0.638364
$457$	− 3.94335i	− 0.184462i	−0.995738	−	0.0922310i	$-0.970600\pi$
$457$	− 3.94335i	− 0.184462i	0.995738	−	0.0922310i	$-0.0293998\pi$
$458$	8.01012i	0.374288i
$459$	4.11077	0.191874
$460$	0	0
$461$	31.1870	1.45252	0.726261	−	0.687419i	$-0.241256\pi$
$461$	31.1870	1.45252	0.726261	+	0.687419i	$0.241256\pi$
$462$	− 1.51749i	− 0.0706001i
$463$	− 29.8007i	− 1.38496i	−0.721439	−	0.692478i	$-0.756519\pi$
$463$	− 29.8007i	− 1.38496i	0.721439	−	0.692478i	$-0.243481\pi$
$464$	1.57246	0.0729996
$465$	0	0
$466$	−10.1532	−0.470338
$467$	− 17.4785i	− 0.808807i	−0.914581	−	0.404404i	$-0.867479\pi$
$467$	− 17.4785i	− 0.808807i	0.914581	−	0.404404i	$-0.132521\pi$
$468$	27.1134i	1.25332i
$469$	10.0681	0.464903
$470$	0	0
$471$	28.7851	1.32635
$472$	− 24.8669i	− 1.14459i
$473$	− 8.99670i	− 0.413669i
$474$	−4.34291	−0.199476
$475$	0	0
$476$	−5.27666	−0.241855
$477$	− 2.95083i	− 0.135109i
$478$	− 9.91653i	− 0.453572i
$479$	4.30979	0.196919	0.0984597	−	0.995141i	$-0.468608\pi$
$479$	4.30979	0.196919	0.0984597	+	0.995141i	$0.468608\pi$
$480$	0	0
$481$	−6.96588	−0.317617
$482$	− 6.83474i	− 0.311314i
$483$	− 25.5020i	− 1.16038i
$484$	−17.0736	−0.776074
$485$	0	0
$486$	−10.7164	−0.486105
$487$	− 38.1250i	− 1.72761i	−0.503829	−	0.863804i	$-0.668076\pi$
$487$	− 38.1250i	− 1.72761i	0.503829	−	0.863804i	$-0.331924\pi$
$488$	− 7.45408i	− 0.337430i
$489$	−19.3394	−0.874558
$490$	0	0
$491$	36.0217	1.62564	0.812818	−	0.582517i	$-0.197932\pi$
$491$	36.0217	1.62564	0.812818	+	0.582517i	$0.197932\pi$
$492$	0.755377i	0.0340550i
$493$	1.73250i	0.0780277i
$494$	11.1598	0.502103
$495$	0	0
$496$	15.1079	0.678366
$497$	− 3.48027i	− 0.156111i
$498$	5.88954i	0.263917i
$499$	−20.1879	−0.903733	−0.451867	−	0.892085i	$-0.649242\pi$
$499$	−20.1879	−0.903733	−0.451867	+	0.892085i	$0.649242\pi$
$500$	0	0
$501$	−27.4985	−1.22854
$502$	2.37388i	0.105952i
$503$	3.01511i	0.134437i	0.997738	+	0.0672185i	$0.0214124\pi$
$503$	3.01511i	0.134437i	−0.997738	+	0.0672185i	$0.978588\pi$
$504$	5.72744	0.255121
$505$	0	0
$506$	−4.81388	−0.214003
$507$	81.7081i	3.62878i
$508$	8.93465i	0.396411i
$509$	−24.8656	−1.10215	−0.551073	−	0.834457i	$-0.685782\pi$
$509$	−24.8656	−1.10215	−0.551073	+	0.834457i	$0.685782\pi$
$510$	0	0
$511$	−1.03210	−0.0456575
$512$	21.4176i	0.946532i
$513$	− 4.76782i	− 0.210504i
$514$	−9.02752	−0.398187
$515$	0	0
$516$	36.0276	1.58603
$517$	3.11089i	0.136817i
$518$	0.675937i	0.0296989i
$519$	49.9078	2.19071
$520$	0	0
$521$	38.3050	1.67817	0.839085	−	0.544000i	$-0.183091\pi$
$521$	38.3050	1.67817	0.839085	+	0.544000i	$0.183091\pi$
$522$	− 0.863828i	− 0.0378087i
$523$	3.02688i	0.132356i	0.997808	+	0.0661781i	$0.0210805\pi$
$523$	3.02688i	0.132356i	−0.997808	+	0.0661781i	$0.978919\pi$
$524$	25.9949	1.13559
$525$	0	0
$526$	10.5843	0.461497
$527$	16.6455i	0.725090i
$528$	5.13285i	0.223379i
$529$	−57.8989	−2.51734
$530$	0	0
$531$	28.0789	1.21852
$532$	6.12006i	0.265338i
$533$	1.34622i	0.0583113i
$534$	8.50626	0.368102
$535$	0	0
$536$	16.5680	0.715628
$537$	− 9.18850i	− 0.396513i
$538$	5.28038i	0.227653i
$539$	−5.34921	−0.230407
$540$	0	0
$541$	−26.8271	−1.15339	−0.576693	−	0.816961i	$-0.695657\pi$
$541$	−26.8271	−1.15339	−0.576693	+	0.816961i	$0.695657\pi$
$542$	− 2.00218i	− 0.0860008i
$543$	− 39.5656i	− 1.69792i
$544$	−13.3777	−0.573564
$545$	0	0
$546$	10.8301	0.463484
$547$	28.1420i	1.20326i	0.798773	+	0.601632i	$0.205483\pi$
$547$	28.1420i	1.20326i	−0.798773	+	0.601632i	$0.794517\pi$
$548$	5.57417i	0.238117i
$549$	8.41691	0.359225
$550$	0	0
$551$	2.00941	0.0856038
$552$	− 41.9657i	− 1.78618i
$553$	− 4.24455i	− 0.180497i
$554$	13.7444	0.583945
$555$	0	0
$556$	−0.953710	−0.0404463
$557$	− 16.3022i	− 0.690748i	−0.938465	−	0.345374i	$-0.887752\pi$
$557$	− 16.3022i	− 0.690748i	0.938465	−	0.345374i	$-0.112248\pi$
$558$	− 8.29950i	− 0.351346i
$559$	64.2078	2.71570
$560$	0	0
$561$	−5.65524	−0.238764
$562$	6.32566i	0.266832i
$563$	10.0364i	0.422985i	0.977380	+	0.211492i	$0.0678323\pi$
$563$	10.0364i	0.422985i	−0.977380	+	0.211492i	$0.932168\pi$
$564$	−12.4577	−0.524563
$565$	0	0
$566$	−16.8516	−0.708325
$567$	− 13.0974i	− 0.550040i
$568$	− 5.72709i	− 0.240303i
$569$	4.06296	0.170328	0.0851640	−	0.996367i	$-0.472859\pi$
$569$	4.06296	0.170328	0.0851640	+	0.996367i	$0.472859\pi$
$570$	0	0
$571$	−30.9715	−1.29612	−0.648058	−	0.761591i	$-0.724418\pi$
$571$	−30.9715	−1.29612	−0.648058	+	0.761591i	$0.724418\pi$
$572$	11.5537i	0.483086i
$573$	47.0901i	1.96722i
$574$	0.130631	0.00545243
$575$	0	0
$576$	−3.80357	−0.158482
$577$	1.38260i	0.0575583i	0.999586	+	0.0287792i	$0.00916196\pi$
$577$	1.38260i	0.0575583i	−0.999586	+	0.0287792i	$0.990838\pi$
$578$	5.84236i	0.243010i
$579$	−47.0756	−1.95639
$580$	0	0
$581$	−5.75616	−0.238806
$582$	− 2.79734i	− 0.115954i
$583$	− 1.25742i	− 0.0520772i
$584$	−1.69841	−0.0702808
$585$	0	0
$586$	−8.89178	−0.367316
$587$	− 20.3736i	− 0.840907i	−0.907314	−	0.420453i	$-0.861871\pi$
$587$	− 20.3736i	− 0.840907i	0.907314	−	0.420453i	$-0.138129\pi$
$588$	− 21.4211i	− 0.883393i
$589$	19.3061	0.795493
$590$	0	0
$591$	39.5341	1.62622
$592$	− 2.28633i	− 0.0939674i
$593$	28.4271i	1.16736i	0.811983	+	0.583681i	$0.198388\pi$
$593$	28.4271i	1.16736i	−0.811983	+	0.583681i	$0.801612\pi$
$594$	−0.873407	−0.0358363
$595$	0	0
$596$	−21.6370	−0.886284
$597$	− 29.6295i	− 1.21265i
$598$	− 34.3558i	− 1.40491i
$599$	−4.99991	−0.204291	−0.102145	−	0.994769i	$-0.532571\pi$
$599$	−4.99991	−0.204291	−0.102145	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−26.3404	−1.07445	−0.537223	−	0.843440i	$-0.680527\pi$
$601$	−26.3404	−1.07445	−0.537223	+	0.843440i	$0.680527\pi$
$602$	− 6.23043i	− 0.253933i
$603$	18.7080i	0.761850i
$604$	−30.8585	−1.25562
$605$	0	0
$606$	6.10350	0.247938
$607$	− 21.0686i	− 0.855147i	−0.903981	−	0.427573i	$-0.859369\pi$
$607$	− 21.0686i	− 0.855147i	0.903981	−	0.427573i	$-0.140631\pi$
$608$	15.5159i	0.629254i
$609$	1.95004	0.0790197
$610$	0	0
$611$	−22.2019	−0.898192
$612$	− 9.80480i	− 0.396336i
$613$	− 2.19848i	− 0.0887959i	−0.999014	−	0.0443980i	$-0.985863\pi$
$613$	− 2.19848i	− 0.0887959i	0.999014	−	0.0443980i	$-0.0141370\pi$
$614$	0.238015	0.00960551
$615$	0	0
$616$	2.44061	0.0983351
$617$	− 19.0587i	− 0.767275i	−0.923484	−	0.383638i	$-0.874671\pi$
$617$	− 19.0587i	− 0.767275i	0.923484	−	0.383638i	$-0.125329\pi$
$618$	11.2975i	0.454454i
$619$	−26.9627	−1.08372	−0.541862	−	0.840467i	$-0.682280\pi$
$619$	−26.9627	−1.08372	−0.541862	+	0.840467i	$0.682280\pi$
$620$	0	0
$621$	−14.6779	−0.589004
$622$	− 11.5221i	− 0.461994i
$623$	8.31363i	0.333079i
$624$	−36.6322	−1.46646
$625$	0	0
$626$	−1.18592	−0.0473987
$627$	6.55915i	0.261947i
$628$	21.2664i	0.848620i
$629$	2.51901	0.100440
$630$	0	0
$631$	−32.6629	−1.30029	−0.650145	−	0.759810i	$-0.725292\pi$
$631$	−32.6629	−1.30029	−0.650145	+	0.759810i	$0.725292\pi$
$632$	− 6.98479i	− 0.277840i
$633$	− 33.6438i	− 1.33722i
$634$	−15.5725	−0.618461
$635$	0	0
$636$	5.03541	0.199667
$637$	− 38.1764i	− 1.51260i
$638$	− 0.368100i	− 0.0145732i
$639$	6.46684	0.255824
$640$	0	0
$641$	−33.9051	−1.33917	−0.669586	−	0.742735i	$-0.733528\pi$
$641$	−33.9051	−1.33917	−0.669586	+	0.742735i	$0.733528\pi$
$642$	19.7339i	0.778836i
$643$	− 23.2105i	− 0.915331i	−0.889124	−	0.457666i	$-0.848686\pi$
$643$	− 23.2105i	− 0.915331i	0.889124	−	0.457666i	$-0.151314\pi$
$644$	18.8408	0.742432
$645$	0	0
$646$	−4.03562	−0.158779
$647$	− 23.4822i	− 0.923180i	−0.887093	−	0.461590i	$-0.847279\pi$
$647$	− 23.4822i	− 0.923180i	0.887093	−	0.461590i	$-0.152721\pi$
$648$	− 21.5530i	− 0.846680i
$649$	11.9652	0.469674
$650$	0	0
$651$	18.7356	0.734308
$652$	− 14.2879i	− 0.559558i
$653$	6.98761i	0.273446i	0.990609	+	0.136723i	$0.0436571\pi$
$653$	6.98761i	0.273446i	−0.990609	+	0.136723i	$0.956343\pi$
$654$	15.7469	0.615753
$655$	0	0
$656$	−0.441853	−0.0172515
$657$	− 1.91779i	− 0.0748203i
$658$	2.15437i	0.0839859i
$659$	−13.4171	−0.522656	−0.261328	−	0.965250i	$-0.584160\pi$
$659$	−13.4171	−0.522656	−0.261328	+	0.965250i	$0.584160\pi$
$660$	0	0
$661$	4.09587	0.159311	0.0796555	−	0.996822i	$-0.474618\pi$
$661$	4.09587	0.159311	0.0796555	+	0.996822i	$0.474618\pi$
$662$	8.90606i	0.346144i
$663$	− 40.3604i	− 1.56747i
$664$	−9.47227	−0.367596
$665$	0	0
$666$	−1.25599	−0.0486686
$667$	− 6.18604i	− 0.239524i
$668$	− 20.3159i	− 0.786045i
$669$	8.27125	0.319785
$670$	0	0
$671$	3.58667	0.138462
$672$	15.0575i	0.580856i
$673$	15.0709i	0.580939i	0.956884	+	0.290469i	$0.0938115\pi$
$673$	15.0709i	0.580939i	−0.956884	+	0.290469i	$0.906189\pi$
$674$	−6.03177	−0.232335
$675$	0	0
$676$	−60.3658	−2.32176
$677$	7.36551i	0.283080i	0.989933	+	0.141540i	$0.0452053\pi$
$677$	7.36551i	0.283080i	−0.989933	+	0.141540i	$0.954795\pi$
$678$	24.5341i	0.942225i
$679$	2.73399	0.104921
$680$	0	0
$681$	−49.9778	−1.91515
$682$	− 3.53664i	− 0.135425i
$683$	− 46.2794i	− 1.77083i	−0.464801	−	0.885415i	$-0.653874\pi$
$683$	− 46.2794i	− 1.77083i	0.464801	−	0.885415i	$-0.346126\pi$
$684$	−11.3720	−0.434818
$685$	0	0
$686$	−8.43601	−0.322088
$687$	− 33.5997i	− 1.28191i
$688$	21.0741i	0.803444i
$689$	8.97401	0.341883
$690$	0	0
$691$	5.06739	0.192773	0.0963863	−	0.995344i	$-0.469272\pi$
$691$	5.06739	0.192773	0.0963863	+	0.995344i	$0.469272\pi$
$692$	36.8718i	1.40166i
$693$	2.75586i	0.104687i
$694$	4.47357	0.169814
$695$	0	0
$696$	3.20896	0.121635
$697$	− 0.486823i	− 0.0184397i
$698$	15.4864i	0.586170i
$699$	42.5892	1.61087
$700$	0	0
$701$	20.9220	0.790213	0.395107	−	0.918635i	$-0.370708\pi$
$701$	20.9220	0.790213	0.395107	+	0.918635i	$0.370708\pi$
$702$	− 6.23335i	− 0.235262i
$703$	− 2.92164i	− 0.110192i
$704$	−1.62080	−0.0610862
$705$	0	0
$706$	−1.07019	−0.0402770
$707$	5.96528i	0.224348i
$708$	47.9150i	1.80075i
$709$	11.5698	0.434515	0.217257	−	0.976114i	$-0.430289\pi$
$709$	11.5698	0.434515	0.217257	+	0.976114i	$0.430289\pi$
$710$	0	0
$711$	7.88700	0.295786
$712$	13.6808i	0.512710i
$713$	− 59.4344i	− 2.22583i
$714$	−3.91639	−0.146567
$715$	0	0
$716$	6.78845	0.253696
$717$	41.5964i	1.55345i
$718$	− 13.8662i	− 0.517482i
$719$	−20.2226	−0.754176	−0.377088	−	0.926177i	$-0.623075\pi$
$719$	−20.2226	−0.754176	−0.377088	+	0.926177i	$0.623075\pi$
$720$	0	0
$721$	−11.0417	−0.411214
$722$	− 5.73786i	− 0.213541i
$723$	28.6694i	1.06623i
$724$	29.2310	1.08636
$725$	0	0
$726$	−12.6722	−0.470310
$727$	33.7740i	1.25261i	0.779579	+	0.626304i	$0.215433\pi$
$727$	33.7740i	1.25261i	−0.779579	+	0.626304i	$0.784567\pi$
$728$	17.4182i	0.645562i
$729$	13.0763	0.484307
$730$	0	0
$731$	−23.2189	−0.858784
$732$	14.3630i	0.530870i
$733$	− 49.8177i	− 1.84006i	−0.391849	−	0.920030i	$-0.628164\pi$
$733$	− 49.8177i	− 1.84006i	0.391849	−	0.920030i	$-0.371836\pi$
$734$	3.85703	0.142366
$735$	0	0
$736$	47.7663	1.76069
$737$	7.97199i	0.293652i
$738$	0.242731i	0.00893506i
$739$	6.25259	0.230005	0.115003	−	0.993365i	$-0.463312\pi$
$739$	6.25259	0.230005	0.115003	+	0.993365i	$0.463312\pi$
$740$	0	0
$741$	−46.8115	−1.71966
$742$	− 0.870796i	− 0.0319679i
$743$	− 0.401960i	− 0.0147465i	−0.999973	−	0.00737324i	$-0.997653\pi$
$743$	− 0.401960i	− 0.0147465i	0.999973	−	0.00737324i	$-0.00234700\pi$
$744$	30.8312	1.13032
$745$	0	0
$746$	−13.7592	−0.503758
$747$	− 10.6958i	− 0.391338i
$748$	− 4.17808i	− 0.152766i
$749$	−19.2870	−0.704733
$750$	0	0
$751$	48.4218	1.76694	0.883469	−	0.468490i	$-0.155202\pi$
$751$	48.4218	1.76694	0.883469	+	0.468490i	$0.155202\pi$
$752$	− 7.28705i	− 0.265731i
$753$	− 9.95763i	− 0.362876i
$754$	2.62706	0.0956719
$755$	0	0
$756$	3.41838	0.124325
$757$	− 16.3058i	− 0.592643i	−0.955088	−	0.296322i	$-0.904240\pi$
$757$	− 16.3058i	− 0.592643i	0.955088	−	0.296322i	$-0.0957601\pi$
$758$	10.4146i	0.378277i
$759$	20.1926	0.732944
$760$	0	0
$761$	−37.6100	−1.36336	−0.681681	−	0.731650i	$-0.738751\pi$
$761$	−37.6100	−1.36336	−0.681681	+	0.731650i	$0.738751\pi$
$762$	6.63138i	0.240230i
$763$	15.3903i	0.557167i
$764$	−34.7901	−1.25866
$765$	0	0
$766$	16.5816	0.599116
$767$	85.3932i	3.08337i
$768$	6.90566i	0.249187i
$769$	−35.9898	−1.29783	−0.648913	−	0.760863i	$-0.724776\pi$
$769$	−35.9898	−1.29783	−0.648913	+	0.760863i	$0.724776\pi$
$770$	0	0
$771$	37.8673	1.36376
$772$	− 34.7794i	− 1.25174i
$773$	11.1521i	0.401114i	0.979682	+	0.200557i	$0.0642752\pi$
$773$	11.1521i	0.401114i	−0.979682	+	0.200557i	$0.935725\pi$
$774$	11.5770	0.416128
$775$	0	0
$776$	4.49903	0.161506
$777$	− 2.83532i	− 0.101717i
$778$	11.4400i	0.410144i
$779$	−0.564634	−0.0202301
$780$	0	0
$781$	2.75569	0.0986064
$782$	12.4238i	0.444274i
$783$	− 1.12236i	− 0.0401100i
$784$	12.5302	0.447506
$785$	0	0
$786$	19.2936	0.688181
$787$	28.9665i	1.03254i	0.856425	+	0.516271i	$0.172680\pi$
$787$	28.9665i	1.03254i	−0.856425	+	0.516271i	$0.827320\pi$
$788$	29.2078i	1.04048i
$789$	−44.3975	−1.58059
$790$	0	0
$791$	−23.9785	−0.852576
$792$	4.53502i	0.161145i
$793$	25.5974i	0.908991i
$794$	−15.6137	−0.554109
$795$	0	0
$796$	21.8902	0.775879
$797$	− 54.2642i	− 1.92214i	−0.276312	−	0.961068i	$-0.589112\pi$
$797$	− 54.2642i	− 1.92214i	0.276312	−	0.961068i	$-0.410888\pi$
$798$	4.54237i	0.160798i
$799$	8.02868	0.284034
$800$	0	0
$801$	−15.4479	−0.545826
$802$	− 14.5350i	− 0.513249i
$803$	− 0.817222i	− 0.0288391i
$804$	−31.9241	−1.12588
$805$	0	0
$806$	25.2403	0.889053
$807$	− 22.1494i	− 0.779695i
$808$	9.81640i	0.345340i
$809$	−16.1657	−0.568355	−0.284177	−	0.958772i	$-0.591720\pi$
$809$	−16.1657	−0.568355	−0.284177	+	0.958772i	$0.591720\pi$
$810$	0	0
$811$	−29.0693	−1.02076	−0.510381	−	0.859948i	$-0.670496\pi$
$811$	−29.0693	−1.02076	−0.510381	+	0.859948i	$0.670496\pi$
$812$	1.44069i	0.0505582i
$813$	8.39844i	0.294546i
$814$	−0.535210	−0.0187591
$815$	0	0
$816$	13.2470	0.463738
$817$	26.9302i	0.942167i
$818$	− 4.35412i	− 0.152238i
$819$	−19.6681	−0.687259
$820$	0	0
$821$	30.7078	1.07171	0.535855	−	0.844310i	$-0.319990\pi$
$821$	30.7078	1.07171	0.535855	+	0.844310i	$0.319990\pi$
$822$	4.13720i	0.144301i
$823$	− 3.08648i	− 0.107588i	−0.998552	−	0.0537940i	$-0.982869\pi$
$823$	− 3.08648i	− 0.107588i	0.998552	−	0.0537940i	$-0.0171314\pi$
$824$	−18.1701	−0.632985
$825$	0	0
$826$	8.28615	0.288312
$827$	− 37.0441i	− 1.28815i	−0.764963	−	0.644074i	$-0.777243\pi$
$827$	− 37.0441i	− 1.28815i	0.764963	−	0.644074i	$-0.222757\pi$
$828$	35.0090i	1.21665i
$829$	45.3315	1.57443	0.787215	−	0.616679i	$-0.211522\pi$
$829$	45.3315	1.57443	0.787215	+	0.616679i	$0.211522\pi$
$830$	0	0
$831$	−57.6532	−1.99997
$832$	− 11.5674i	− 0.401026i
$833$	13.8054i	0.478329i
$834$	−0.707853	−0.0245109
$835$	0	0
$836$	−4.84589	−0.167599
$837$	− 10.7835i	− 0.372732i
$838$	− 17.4957i	− 0.604379i
$839$	45.1775	1.55970	0.779850	−	0.625967i	$-0.215296\pi$
$839$	45.1775	1.55970	0.779850	+	0.625967i	$0.215296\pi$
$840$	0	0
$841$	−28.5270	−0.983689
$842$	6.05569i	0.208693i
$843$	− 26.5340i	− 0.913879i
$844$	24.8560	0.855579
$845$	0	0
$846$	−4.00312	−0.137630
$847$	− 12.3852i	− 0.425561i
$848$	2.94543i	0.101147i
$849$	70.6866	2.42596
$850$	0	0
$851$	−8.99438	−0.308323
$852$	11.0353i	0.378062i
$853$	21.6482i	0.741220i	0.928789	+	0.370610i	$0.120851\pi$
$853$	21.6482i	0.741220i	−0.928789	+	0.370610i	$0.879149\pi$
$854$	2.48385	0.0849957
$855$	0	0
$856$	−31.7385	−1.08480
$857$	− 1.48050i	− 0.0505729i	−0.999680	−	0.0252865i	$-0.991950\pi$
$857$	− 1.48050i	− 0.0505729i	0.999680	−	0.0252865i	$-0.00804979\pi$
$858$	8.57529i	0.292756i
$859$	−20.1546	−0.687666	−0.343833	−	0.939031i	$-0.611725\pi$
$859$	−20.1546	−0.687666	−0.343833	+	0.939031i	$0.611725\pi$
$860$	0	0
$861$	−0.547952	−0.0186741
$862$	21.9340i	0.747076i
$863$	9.07168i	0.308803i	0.988008	+	0.154402i	$0.0493450\pi$
$863$	9.07168i	0.308803i	−0.988008	+	0.154402i	$0.950655\pi$
$864$	8.66649	0.294840
$865$	0	0
$866$	1.87583	0.0637434
$867$	− 24.5067i	− 0.832291i
$868$	13.8419i	0.469824i
$869$	3.36086	0.114009
$870$	0	0
$871$	−56.8946	−1.92780
$872$	25.3261i	0.857650i
$873$	5.08016i	0.171937i
$874$	14.4096	0.487411
$875$	0	0
$876$	3.27260	0.110571
$877$	3.52847i	0.119148i	0.998224	+	0.0595739i	$0.0189742\pi$
$877$	3.52847i	0.119148i	−0.998224	+	0.0595739i	$0.981026\pi$
$878$	− 1.02649i	− 0.0346424i
$879$	37.2980	1.25803
$880$	0	0
$881$	−32.2604	−1.08688	−0.543441	−	0.839448i	$-0.682879\pi$
$881$	−32.2604	−1.08688	−0.543441	+	0.839448i	$0.682879\pi$
$882$	− 6.88342i	− 0.231777i
$883$	31.5621i	1.06215i	0.847326	+	0.531074i	$0.178211\pi$
$883$	31.5621i	1.06215i	−0.847326	+	0.531074i	$0.821789\pi$
$884$	29.8182	1.00290
$885$	0	0
$886$	21.4229	0.719717
$887$	34.8507i	1.17017i	0.810971	+	0.585087i	$0.198939\pi$
$887$	34.8507i	1.17017i	−0.810971	+	0.585087i	$0.801061\pi$
$888$	− 4.66577i	− 0.156573i
$889$	−6.48121	−0.217373
$890$	0	0
$891$	10.3706	0.347428
$892$	6.11079i	0.204604i
$893$	− 9.31195i	− 0.311613i
$894$	−16.0592	−0.537098
$895$	0	0
$896$	−14.2153	−0.474899
$897$	144.111i	4.81172i
$898$	− 15.2100i	− 0.507566i
$899$	4.54473	0.151575
$900$	0	0
$901$	−3.24520	−0.108113
$902$	0.103434i	0.00344398i
$903$	26.1345i	0.869701i
$904$	−39.4587	−1.31238
$905$	0	0
$906$	−22.9035	−0.760918
$907$	36.6888i	1.21823i	0.793081	+	0.609116i	$0.208476\pi$
$907$	36.6888i	1.21823i	−0.793081	+	0.609116i	$0.791524\pi$
$908$	− 36.9236i	− 1.22535i
$909$	−11.0844	−0.367645
$910$	0	0
$911$	27.6843	0.917221	0.458610	−	0.888637i	$-0.348347\pi$
$911$	27.6843	0.917221	0.458610	+	0.888637i	$0.348347\pi$
$912$	− 15.3644i	− 0.508765i
$913$	− 4.55776i	− 0.150840i
$914$	2.16231	0.0715228
$915$	0	0
$916$	24.8234	0.820188
$917$	18.8567i	0.622704i
$918$	2.25411i	0.0743968i
$919$	−34.3200	−1.13211	−0.566057	−	0.824366i	$-0.691532\pi$
$919$	−34.3200	−1.13211	−0.566057	+	0.824366i	$0.691532\pi$
$920$	0	0
$921$	−0.998391	−0.0328981
$922$	17.1012i	0.563197i
$923$	19.6669i	0.647343i
$924$	−4.70272	−0.154708
$925$	0	0
$926$	16.3410	0.536999
$927$	− 20.5171i	− 0.673869i
$928$	3.65252i	0.119900i
$929$	22.9783	0.753894	0.376947	−	0.926235i	$-0.376974\pi$
$929$	22.9783	0.753894	0.376947	+	0.926235i	$0.376974\pi$
$930$	0	0
$931$	16.0120	0.524772
$932$	31.4649i	1.03067i
$933$	48.3312i	1.58229i
$934$	9.58421	0.313605
$935$	0	0
$936$	−32.3656	−1.05790
$937$	− 45.7823i	− 1.49564i	−0.663899	−	0.747822i	$-0.731100\pi$
$937$	− 45.7823i	− 1.49564i	0.663899	−	0.747822i	$-0.268900\pi$
$938$	5.52079i	0.180260i
$939$	4.97451	0.162337
$940$	0	0
$941$	−16.9929	−0.553952	−0.276976	−	0.960877i	$-0.589332\pi$
$941$	−16.9929	−0.553952	−0.276976	+	0.960877i	$0.589332\pi$
$942$	15.7841i	0.514274i
$943$	1.73825i	0.0566051i
$944$	−28.0276	−0.912219
$945$	0	0
$946$	4.93328	0.160395
$947$	23.4730i	0.762769i	0.924417	+	0.381384i	$0.124553\pi$
$947$	23.4730i	0.762769i	−0.924417	+	0.381384i	$0.875447\pi$
$948$	13.4587i	0.437118i
$949$	5.83237	0.189327
$950$	0	0
$951$	65.3211	2.11818
$952$	− 6.29881i	− 0.204146i
$953$	− 24.6541i	− 0.798623i	−0.916815	−	0.399312i	$-0.869249\pi$
$953$	− 24.6541i	− 0.798623i	0.916815	−	0.399312i	$-0.130751\pi$
$954$	1.61807	0.0523868
$955$	0	0
$956$	−30.7314	−0.993924
$957$	1.54405i	0.0499121i
$958$	2.36325i	0.0763530i
$959$	−4.04351	−0.130572
$960$	0	0
$961$	12.6649	0.408546
$962$	− 3.81970i	− 0.123152i
$963$	− 35.8381i	− 1.15487i
$964$	−21.1809	−0.682191
$965$	0	0
$966$	13.9838	0.449922
$967$	− 11.3662i	− 0.365512i	−0.983158	−	0.182756i	$-0.941498\pi$
$967$	− 11.3662i	− 0.365512i	0.983158	−	0.182756i	$-0.0585019\pi$
$968$	− 20.3810i	− 0.655070i
$969$	16.9281	0.543807
$970$	0	0
$971$	−1.93826	−0.0622017	−0.0311008	−	0.999516i	$-0.509901\pi$
$971$	−1.93826	−0.0622017	−0.0311008	+	0.999516i	$0.509901\pi$
$972$	33.2101i	1.06522i
$973$	− 0.691822i	− 0.0221788i
$974$	20.9056	0.669858
$975$	0	0
$976$	−8.40152	−0.268926
$977$	12.9186i	0.413303i	0.978415	+	0.206651i	$0.0662566\pi$
$977$	12.9186i	0.413303i	−0.978415	+	0.206651i	$0.933743\pi$
$978$	− 10.6046i	− 0.339099i
$979$	−6.58277	−0.210386
$980$	0	0
$981$	−28.5974	−0.913046
$982$	19.7523i	0.630320i
$983$	42.8843i	1.36780i	0.729578	+	0.683898i	$0.239717\pi$
$983$	42.8843i	1.36780i	−0.729578	+	0.683898i	$0.760283\pi$
$984$	−0.901703	−0.0287452
$985$	0	0
$986$	−0.950002	−0.0302542
$987$	− 9.03682i	− 0.287645i
$988$	− 34.5843i	− 1.10027i
$989$	82.9054	2.63624
$990$	0	0
$991$	−30.3017	−0.962564	−0.481282	−	0.876566i	$-0.659829\pi$
$991$	−30.3017	−0.962564	−0.481282	+	0.876566i	$0.659829\pi$
$992$	35.0927i	1.11419i
$993$	− 37.3579i	− 1.18552i
$994$	1.90838	0.0605302
$995$	0	0
$996$	18.2517	0.578328
$997$	9.76058i	0.309121i	0.987983	+	0.154560i	$0.0493961\pi$
$997$	9.76058i	0.309121i	−0.987983	+	0.154560i	$0.950604\pi$
$998$	− 11.0699i	− 0.350411i
$999$	−1.63190	−0.0516309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.b.i.149.9		14
5.2	odd	4		925.2.a.k.1.3	yes	7
5.3	odd	4		925.2.a.j.1.5	✓	7
5.4	even	2	inner	925.2.b.i.149.6		14
15.2	even	4		8325.2.a.cm.1.5		7
15.8	even	4		8325.2.a.cn.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
925.2.a.j.1.5	✓	7	5.3	odd	4
925.2.a.k.1.3	yes	7	5.2	odd	4
925.2.b.i.149.6		14	5.4	even	2	inner
925.2.b.i.149.9		14	1.1	even	1	trivial
8325.2.a.cm.1.5		7	15.2	even	4
8325.2.a.cn.1.3		7	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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.548343i q^{2} -2.30011i q^{3} +1.69932 q^{4} +1.26125 q^{6} +1.23269i q^{7} +2.02850i q^{8} -2.29051 q^{9} -0.976048 q^{11} -3.90863i q^{12} -6.96588i q^{13} -0.675937 q^{14} +2.28633 q^{16} +2.51901i q^{17} -1.25599i q^{18} +2.92164 q^{19} +2.83532 q^{21} -0.535210i q^{22} -8.99438i q^{23} +4.66577 q^{24} +3.81970 q^{26} -1.63190i q^{27} +2.09473i q^{28} +0.687767 q^{29} +6.60794 q^{31} +5.31069i q^{32} +2.24502i q^{33} -1.38128 q^{34} -3.89232 q^{36} -1.00000i q^{37} +1.60206i q^{38} -16.0223 q^{39} -0.193259 q^{41} +1.55473i q^{42} +9.21747i q^{43} -1.65862 q^{44} +4.93201 q^{46} -3.18723i q^{47} -5.25881i q^{48} +5.48048 q^{49} +5.79402 q^{51} -11.8373i q^{52} +1.28828i q^{53} +0.894839 q^{54} -2.50051 q^{56} -6.72011i q^{57} +0.377133i q^{58} -12.2588 q^{59} -3.67468 q^{61} +3.62342i q^{62} -2.82349i q^{63} +1.66057 q^{64} -1.23104 q^{66} -8.16762i q^{67} +4.28061i q^{68} -20.6881 q^{69} -2.82331 q^{71} -4.64630i q^{72} +0.837276i q^{73} +0.548343 q^{74} +4.96481 q^{76} -1.20316i q^{77} -8.78573i q^{78} -3.44333 q^{79} -10.6251 q^{81} -0.105972i q^{82} +4.66960i q^{83} +4.81812 q^{84} -5.05434 q^{86} -1.58194i q^{87} -1.97991i q^{88} +6.74431 q^{89} +8.58676 q^{91} -15.2843i q^{92} -15.1990i q^{93} +1.74770 q^{94} +12.2152 q^{96} -2.21791i q^{97} +3.00518i q^{98} +2.23565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4} + 12 q^{6} - 20 q^{9} + 32 q^{11} + 6 q^{14} + 6 q^{16} - 18 q^{19} + 4 q^{21} + 48 q^{26} - 6 q^{29} + 22 q^{31} + 46 q^{34} + 38 q^{36} - 4 q^{39} + 58 q^{41} - 36 q^{44} + 14 q^{46} - 10 q^{49}+ \cdots - 78 q^{99}+O(q^{100}) \) 14 * q - 14 * q^4 + 12 * q^6 - 20 * q^9 + 32 * q^11 + 6 * q^14 + 6 * q^16 - 18 * q^19 + 4 * q^21 + 48 * q^26 - 6 * q^29 + 22 * q^31 + 46 * q^34 + 38 * q^36 - 4 * q^39 + 58 * q^41 - 36 * q^44 + 14 * q^46 - 10 * q^49 + 10 * q^51 - 16 * q^54 + 26 * q^56 - 84 * q^59 - 22 * q^61 + 32 * q^64 + 58 * q^66 - 4 * q^69 + 20 * q^71 - 2 * q^74 + 22 * q^76 + 18 * q^79 - 26 * q^81 + 82 * q^84 - 44 * q^86 - 24 * q^89 - 4 * q^91 - 18 * q^94 - 10 * q^96 - 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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