LMFDB - Embedded newform 1275.2.b.k.1174.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1275,2,Mod(1174,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1275.1174"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-18,0,2,0,0,-8,0,4] 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:	$10.1809262577$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12131700736.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 33x^{4} + 24x^{2} + 4 $ x^8 + 11x^6 + 33x^4 + 24*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.8
Root			$-0.489088i$ of defining polynomial
Character	$\chi$	$=$	1275.1174
Dual form			1275.2.b.k.1174.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.60015i q^{2} +1.00000i q^{3} -4.76079 q^{4} -2.60015 q^{6} -2.81754i q^{7} -7.17848i q^{8} -1.00000 q^{9} +4.38277 q^{11} -4.76079i q^{12} -5.52159i q^{13} +7.32602 q^{14} +9.14356 q^{16} +1.00000i q^{17} -2.60015i q^{18} -1.18246 q^{19} +2.81754 q^{21} +11.3959i q^{22} +9.15666i q^{23} +7.17848 q^{24} +14.3570 q^{26} -1.00000i q^{27} +13.4137i q^{28} +0.817537 q^{29} +7.20030 q^{31} +9.41769i q^{32} +4.38277i q^{33} -2.60015 q^{34} +4.76079 q^{36} -2.70405i q^{37} -3.07459i q^{38} +5.52159 q^{39} +9.58307 q^{41} +7.32602i q^{42} -1.95635i q^{43} -20.8655 q^{44} -23.8087 q^{46} +4.38277i q^{47} +9.14356i q^{48} -0.938512 q^{49} -1.00000 q^{51} +26.2871i q^{52} -9.58307i q^{53} +2.60015 q^{54} -20.2256 q^{56} -1.18246i q^{57} +2.12572i q^{58} +7.52159 q^{59} -5.52159 q^{61} +18.7219i q^{62} +2.81754i q^{63} -6.20030 q^{64} -11.3959 q^{66} -13.4779i q^{67} -4.76079i q^{68} -9.15666 q^{69} -2.04365 q^{71} +7.17848i q^{72} +4.33912i q^{73} +7.03094 q^{74} +5.62946 q^{76} -12.3486i q^{77} +14.3570i q^{78} +10.7219 q^{79} +1.00000 q^{81} +24.9175i q^{82} -2.04365i q^{83} -13.4137 q^{84} +5.08682 q^{86} +0.817537i q^{87} -31.4616i q^{88} +10.2871 q^{89} -15.5573 q^{91} -43.5930i q^{92} +7.20030i q^{93} -11.3959 q^{94} -9.41769 q^{96} +8.40061i q^{97} -2.44027i q^{98} -4.38277 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 18 q^{4} + 2 q^{6} - 8 q^{9} + 4 q^{11} + 22 q^{14} + 22 q^{16} - 24 q^{19} + 8 q^{21} - 8 q^{29} + 12 q^{31} + 2 q^{34} + 18 q^{36} + 4 q^{39} - 30 q^{44} - 48 q^{46} - 44 q^{49} - 8 q^{51} - 2 q^{54}+ \cdots - 4 q^{99}+O(q^{100}) $ 8 * q - 18 * q^4 + 2 * q^6 - 8 * q^9 + 4 * q^11 + 22 * q^14 + 22 * q^16 - 24 * q^19 + 8 * q^21 - 8 * q^29 + 12 * q^31 + 2 * q^34 + 18 * q^36 + 4 * q^39 - 30 * q^44 - 48 * q^46 - 44 * q^49 - 8 * q^51 - 2 * q^54 - 96 * q^56 + 20 * q^59 - 4 * q^61 - 4 * q^64 - 42 * q^66 - 4 * q^69 - 40 * q^71 - 22 * q^74 + 74 * q^76 + 8 * q^81 + 2 * q^84 - 16 * q^86 - 20 * q^89 + 36 * q^91 - 42 * q^94 - 38 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times$.

$n$	$52$	$751$	$851$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	2.60015i	1.83859i	0.393575	+	0.919293i	$0.371238\pi$
$2$	2.60015i	1.83859i	−0.393575	+	0.919293i	$0.628762\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.76079	−2.38040
$5$	0	0
$5$	0	0
$6$	−2.60015	−1.06151
$7$	− 2.81754i	− 1.06493i	−0.846452	−	0.532464i	$-0.821266\pi$
$7$	− 2.81754i	− 1.06493i	0.846452	−	0.532464i	$-0.178734\pi$
$8$	− 7.17848i	− 2.53798i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.38277	1.32145	0.660727	−	0.750626i	$-0.270248\pi$
$11$	4.38277	1.32145	0.660727	+	0.750626i	$0.270248\pi$
$12$	− 4.76079i	− 1.37432i
$13$	− 5.52159i	− 1.53141i	−0.643191	−	0.765706i	$-0.722390\pi$
$13$	− 5.52159i	− 1.53141i	0.643191	−	0.765706i	$-0.277610\pi$
$14$	7.32602	1.95796
$15$	0	0
$16$	9.14356	2.28589
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 2.60015i	− 0.612862i
$19$	−1.18246	−0.271276	−0.135638	−	0.990758i	$-0.543308\pi$
$19$	−1.18246	−0.271276	−0.135638	+	0.990758i	$0.543308\pi$
$20$	0	0
$21$	2.81754	0.614837
$22$	11.3959i	2.42961i
$23$	9.15666i	1.90930i	0.297738	+	0.954648i	$0.403768\pi$
$23$	9.15666i	1.90930i	−0.297738	+	0.954648i	$0.596232\pi$
$24$	7.17848	1.46530
$25$	0	0
$26$	14.3570	2.81563
$27$	− 1.00000i	− 0.192450i
$28$	13.4137i	2.53495i
$29$	0.817537	0.151813	0.0759064	−	0.997115i	$-0.475815\pi$
$29$	0.817537	0.151813	0.0759064	+	0.997115i	$0.475815\pi$
$30$	0	0
$31$	7.20030	1.29321	0.646606	−	0.762824i	$-0.276188\pi$
$31$	7.20030	1.29321	0.646606	+	0.762824i	$0.276188\pi$
$32$	9.41769i	1.66483i
$33$	4.38277i	0.762942i
$34$	−2.60015	−0.445922
$35$	0	0
$36$	4.76079	0.793465
$37$	− 2.70405i	− 0.444543i	−0.974985	−	0.222271i	$-0.928653\pi$
$37$	− 2.70405i	− 0.444543i	0.974985	−	0.222271i	$-0.0713471\pi$
$38$	− 3.07459i	− 0.498764i
$39$	5.52159	0.884161
$40$	0	0
$41$	9.58307	1.49662	0.748312	−	0.663347i	$-0.230864\pi$
$41$	9.58307	1.49662	0.748312	+	0.663347i	$0.230864\pi$
$42$	7.32602i	1.13043i
$43$	− 1.95635i	− 0.298341i	−0.988811	−	0.149171i	$-0.952340\pi$
$43$	− 1.95635i	− 0.298341i	0.988811	−	0.149171i	$-0.0476603\pi$
$44$	−20.8655	−3.14559
$45$	0	0
$46$	−23.8087	−3.51040
$47$	4.38277i	0.639292i	0.947537	+	0.319646i	$0.103564\pi$
$47$	4.38277i	0.639292i	−0.947537	+	0.319646i	$0.896436\pi$
$48$	9.14356i	1.31976i
$49$	−0.938512	−0.134073
$50$	0	0
$51$	−1.00000	−0.140028
$52$	26.2871i	3.64537i
$53$	− 9.58307i	− 1.31634i	−0.752871	−	0.658168i	$-0.771332\pi$
$53$	− 9.58307i	− 1.31634i	0.752871	−	0.658168i	$-0.228668\pi$
$54$	2.60015	0.353836
$55$	0	0
$56$	−20.2256	−2.70276
$57$	− 1.18246i	− 0.156621i
$58$	2.12572i	0.279121i
$59$	7.52159	0.979227	0.489613	−	0.871940i	$-0.337138\pi$
$59$	7.52159	0.979227	0.489613	+	0.871940i	$0.337138\pi$
$60$	0	0
$61$	−5.52159	−0.706967	−0.353483	−	0.935441i	$-0.615003\pi$
$61$	−5.52159	−0.706967	−0.353483	+	0.935441i	$0.615003\pi$
$62$	18.7219i	2.37768i
$63$	2.81754i	0.354976i
$64$	−6.20030	−0.775038
$65$	0	0
$66$	−11.3959	−1.40273
$67$	− 13.4779i	− 1.64659i	−0.567612	−	0.823296i	$-0.692133\pi$
$67$	− 13.4779i	− 1.64659i	0.567612	−	0.823296i	$-0.307867\pi$
$68$	− 4.76079i	− 0.577331i
$69$	−9.15666	−1.10233
$70$	0	0
$71$	−2.04365	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$71$	−2.04365	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$72$	7.17848i	0.845992i
$73$	4.33912i	0.507856i	0.967223	+	0.253928i	$0.0817226\pi$
$73$	4.33912i	0.507856i	−0.967223	+	0.253928i	$0.918277\pi$
$74$	7.03094	0.817330
$75$	0	0
$76$	5.62946	0.645744
$77$	− 12.3486i	− 1.40725i
$78$	14.3570i	1.62561i
$79$	10.7219	1.20631	0.603153	−	0.797625i	$-0.293911\pi$
$79$	10.7219	1.20631	0.603153	+	0.797625i	$0.293911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	24.9175i	2.75167i
$83$	− 2.04365i	− 0.224319i	−0.993690	−	0.112160i	$-0.964223\pi$
$83$	− 2.04365i	− 0.224319i	0.993690	−	0.112160i	$-0.0357768\pi$
$84$	−13.4137	−1.46356
$85$	0	0
$86$	5.08682	0.548526
$87$	0.817537i	0.0876491i
$88$	− 31.4616i	− 3.35382i
$89$	10.2871	1.09043	0.545216	−	0.838295i	$-0.316447\pi$
$89$	10.2871	1.09043	0.545216	+	0.838295i	$0.316447\pi$
$90$	0	0
$91$	−15.5573	−1.63084
$92$	− 43.5930i	− 4.54488i
$93$	7.20030i	0.746637i
$94$	−11.3959	−1.17539
$95$	0	0
$96$	−9.41769	−0.961189
$97$	8.40061i	0.852953i	0.904499	+	0.426476i	$0.140245\pi$
$97$	8.40061i	0.852953i	−0.904499	+	0.426476i	$0.859755\pi$
$98$	− 2.44027i	− 0.246505i
$99$	−4.38277	−0.440485
$100$	0	0
$101$	−9.04317	−0.899829	−0.449915	−	0.893072i	$-0.648546\pi$
$101$	−9.04317	−0.899829	−0.449915	+	0.893072i	$0.648546\pi$
$102$	− 2.60015i	− 0.257453i
$103$	1.08682i	0.107087i	0.998566	+	0.0535436i	$0.0170516\pi$
$103$	1.08682i	0.107087i	−0.998566	+	0.0535436i	$0.982948\pi$
$104$	−39.6366	−3.88669
$105$	0	0
$106$	24.9175	2.42020
$107$	− 10.4006i	− 1.00546i	−0.864442	−	0.502732i	$-0.832328\pi$
$107$	− 10.4006i	− 1.00546i	0.864442	−	0.502732i	$-0.167672\pi$
$108$	4.76079i	0.458107i
$109$	8.40061	0.804632	0.402316	−	0.915501i	$-0.368205\pi$
$109$	8.40061	0.804632	0.402316	+	0.915501i	$0.368205\pi$
$110$	0	0
$111$	2.70405	0.256657
$112$	− 25.7623i	− 2.43431i
$113$	5.60888i	0.527639i	0.964572	+	0.263819i	$0.0849823\pi$
$113$	5.60888i	0.527639i	−0.964572	+	0.263819i	$0.915018\pi$
$114$	3.07459	0.287961
$115$	0	0
$116$	−3.89212	−0.361374
$117$	5.52159i	0.510471i
$118$	19.5573i	1.80039i
$119$	2.81754	0.258283
$120$	0	0
$121$	8.20866	0.746242
$122$	− 14.3570i	− 1.29982i
$123$	9.58307i	0.864077i
$124$	−34.2792	−3.07836
$125$	0	0
$126$	−7.32602	−0.652654
$127$	18.2435i	1.61885i	0.587226	+	0.809423i	$0.300220\pi$
$127$	18.2435i	1.61885i	−0.587226	+	0.809423i	$0.699780\pi$
$128$	2.71364i	0.239854i
$129$	1.95635	0.172247
$130$	0	0
$131$	−9.95635	−0.869891	−0.434945	−	0.900457i	$-0.643232\pi$
$131$	−9.95635	−0.869891	−0.434945	+	0.900457i	$0.643232\pi$
$132$	− 20.8655i	− 1.81610i
$133$	3.33163i	0.288889i
$134$	35.0447	3.02740
$135$	0	0
$136$	7.17848	0.615550
$137$	− 14.4526i	− 1.23477i	−0.786661	−	0.617385i	$-0.788192\pi$
$137$	− 14.4526i	− 1.23477i	0.786661	−	0.617385i	$-0.211808\pi$
$138$	− 23.8087i	− 2.02673i
$139$	7.67872	0.651301	0.325650	−	0.945490i	$-0.394417\pi$
$139$	7.67872	0.651301	0.325650	+	0.945490i	$0.394417\pi$
$140$	0	0
$141$	−4.38277	−0.369096
$142$	− 5.31379i	− 0.445924i
$143$	− 24.1998i	− 2.02369i
$144$	−9.14356	−0.761963
$145$	0	0
$146$	−11.2824	−0.933736
$147$	− 0.938512i	− 0.0774072i
$148$	12.8734i	1.05819i
$149$	−3.12098	−0.255680	−0.127840	−	0.991795i	$-0.540804\pi$
$149$	−3.12098	−0.255680	−0.127840	+	0.991795i	$0.540804\pi$
$150$	0	0
$151$	−9.21815	−0.750162	−0.375081	−	0.926992i	$-0.622385\pi$
$151$	−9.21815	−0.750162	−0.375081	+	0.926992i	$0.622385\pi$
$152$	8.48829i	0.688491i
$153$	− 1.00000i	− 0.0808452i
$154$	32.1083	2.58736
$155$	0	0
$156$	−26.2871	−2.10465
$157$	1.23446i	0.0985209i	0.998786	+	0.0492605i	$0.0156864\pi$
$157$	1.23446i	0.0985209i	−0.998786	+	0.0492605i	$0.984314\pi$
$158$	27.8785i	2.21790i
$159$	9.58307	0.759987
$160$	0	0
$161$	25.7992	2.03326
$162$	2.60015i	0.204287i
$163$	14.6262i	1.14562i	0.819690	+	0.572808i	$0.194146\pi$
$163$	14.6262i	1.14562i	−0.819690	+	0.572808i	$0.805854\pi$
$164$	−45.6230	−3.56256
$165$	0	0
$166$	5.31379	0.412430
$167$	− 8.76554i	− 0.678298i	−0.940733	−	0.339149i	$-0.889861\pi$
$167$	− 8.76554i	− 0.678298i	0.940733	−	0.339149i	$-0.110139\pi$
$168$	− 20.2256i	− 1.56044i
$169$	−17.4879	−1.34522
$170$	0	0
$171$	1.18246	0.0904252
$172$	9.31379i	0.710170i
$173$	− 22.5141i	− 1.71172i	−0.517212	−	0.855858i	$-0.673030\pi$
$173$	− 22.5141i	− 1.71172i	0.517212	−	0.855858i	$-0.326970\pi$
$174$	−2.12572	−0.161150
$175$	0	0
$176$	40.0741	3.02070
$177$	7.52159i	0.565357i
$178$	26.7481i	2.00485i
$179$	20.7655	1.55209	0.776045	−	0.630678i	$-0.217223\pi$
$179$	20.7655	1.55209	0.776045	+	0.630678i	$0.217223\pi$
$180$	0	0
$181$	9.52159	0.707734	0.353867	−	0.935296i	$-0.384867\pi$
$181$	9.52159	0.707734	0.353867	+	0.935296i	$0.384867\pi$
$182$	− 40.4513i	− 2.99845i
$183$	− 5.52159i	− 0.408167i
$184$	65.7309	4.84575
$185$	0	0
$186$	−18.7219	−1.37276
$187$	4.38277i	0.320500i
$188$	− 20.8655i	− 1.52177i
$189$	−2.81754	−0.204946
$190$	0	0
$191$	−4.47841	−0.324047	−0.162023	−	0.986787i	$-0.551802\pi$
$191$	−4.47841	−0.324047	−0.162023	+	0.986787i	$0.551802\pi$
$192$	− 6.20030i	− 0.447468i
$193$	− 23.4438i	− 1.68752i	−0.536720	−	0.843760i	$-0.680337\pi$
$193$	− 23.4438i	− 1.68752i	0.536720	−	0.843760i	$-0.319663\pi$
$194$	−21.8429	−1.56823
$195$	0	0
$196$	4.46806	0.319147
$197$	13.1661i	0.938049i	0.883185	+	0.469025i	$0.155394\pi$
$197$	13.1661i	0.938049i	−0.883185	+	0.469025i	$0.844606\pi$
$198$	− 11.3959i	− 0.809869i
$199$	−21.0352	−1.49115	−0.745573	−	0.666424i	$-0.767824\pi$
$199$	−21.0352	−1.49115	−0.745573	+	0.666424i	$0.767824\pi$
$200$	0	0
$201$	13.4779	0.950660
$202$	− 23.5136i	− 1.65441i
$203$	− 2.30344i	− 0.161670i
$204$	4.76079	0.333322
$205$	0	0
$206$	−2.82589	−0.196889
$207$	− 9.15666i	− 0.636432i
$208$	− 50.4870i	− 3.50064i
$209$	−5.18246	−0.358478
$210$	0	0
$211$	26.7576	1.84207	0.921034	−	0.389483i	$-0.127346\pi$
$211$	26.7576	1.84207	0.921034	+	0.389483i	$0.127346\pi$
$212$	45.6230i	3.13340i
$213$	− 2.04365i	− 0.140028i
$214$	27.0432	1.84863
$215$	0	0
$216$	−7.17848	−0.488434
$217$	− 20.2871i	− 1.37718i
$218$	21.8429i	1.47939i
$219$	−4.33912	−0.293211
$220$	0	0
$221$	5.52159	0.371422
$222$	7.03094i	0.471886i
$223$	− 10.9489i	− 0.733190i	−0.930381	−	0.366595i	$-0.880523\pi$
$223$	− 10.9489i	− 0.733190i	0.930381	−	0.366595i	$-0.119477\pi$
$224$	26.5347	1.77292
$225$	0	0
$226$	−14.5839	−0.970109
$227$	13.7992i	0.915886i	0.888981	+	0.457943i	$0.151414\pi$
$227$	13.7992i	0.915886i	−0.888981	+	0.457943i	$0.848586\pi$
$228$	5.62946i	0.372820i
$229$	−8.42641	−0.556833	−0.278417	−	0.960460i	$-0.589810\pi$
$229$	−8.42641	−0.556833	−0.278417	+	0.960460i	$0.589810\pi$
$230$	0	0
$231$	12.3486	0.812479
$232$	− 5.86867i	− 0.385297i
$233$	27.1923i	1.78143i	0.454563	+	0.890715i	$0.349796\pi$
$233$	27.1923i	1.78143i	−0.454563	+	0.890715i	$0.650204\pi$
$234$	−14.3570	−0.938544
$235$	0	0
$236$	−35.8087	−2.33095
$237$	10.7219i	0.696462i
$238$	7.32602i	0.474876i
$239$	−7.55727	−0.488839	−0.244420	−	0.969670i	$-0.578597\pi$
$239$	−7.55727	−0.488839	−0.244420	+	0.969670i	$0.578597\pi$
$240$	0	0
$241$	−13.4343	−0.865379	−0.432689	−	0.901543i	$-0.642435\pi$
$241$	−13.4343	−0.865379	−0.432689	+	0.901543i	$0.642435\pi$
$242$	21.3438i	1.37203i
$243$	1.00000i	0.0641500i
$244$	26.2871	1.68286
$245$	0	0
$246$	−24.9175	−1.58868
$247$	6.52907i	0.415435i
$248$	− 51.6873i	− 3.28214i
$249$	2.04365	0.129511
$250$	0	0
$251$	−28.3228	−1.78772	−0.893860	−	0.448347i	$-0.852013\pi$
$251$	−28.3228	−1.78772	−0.893860	+	0.448347i	$0.852013\pi$
$252$	− 13.4137i	− 0.844984i
$253$	40.1315i	2.52305i
$254$	−47.4358	−2.97639
$255$	0	0
$256$	−19.4565	−1.21603
$257$	24.0863i	1.50246i	0.660038	+	0.751232i	$0.270540\pi$
$257$	24.0863i	1.50246i	−0.660038	+	0.751232i	$0.729460\pi$
$258$	5.08682i	0.316691i
$259$	−7.61876	−0.473406
$260$	0	0
$261$	−0.817537	−0.0506042
$262$	− 25.8880i	− 1.59937i
$263$	− 18.0178i	− 1.11103i	−0.831507	−	0.555514i	$-0.812522\pi$
$263$	− 18.0178i	− 1.11103i	0.831507	−	0.555514i	$-0.187478\pi$
$264$	31.4616	1.93633
$265$	0	0
$266$	−8.66276	−0.531148
$267$	10.2871i	0.629562i
$268$	64.1657i	3.91954i
$269$	−0.590561	−0.0360071	−0.0180036	−	0.999838i	$-0.505731\pi$
$269$	−0.590561	−0.0360071	−0.0180036	+	0.999838i	$0.505731\pi$
$270$	0	0
$271$	13.7580	0.835742	0.417871	−	0.908506i	$-0.362776\pi$
$271$	13.7580	0.835742	0.417871	+	0.908506i	$0.362776\pi$
$272$	9.14356i	0.554410i
$273$	− 15.5573i	− 0.941569i
$274$	37.5790	2.27023
$275$	0	0
$276$	43.5930	2.62399
$277$	− 9.68573i	− 0.581959i	−0.956729	−	0.290980i	$-0.906019\pi$
$277$	− 9.68573i	− 0.581959i	0.956729	−	0.290980i	$-0.0939812\pi$
$278$	19.9658i	1.19747i
$279$	−7.20030	−0.431071
$280$	0	0
$281$	10.1230	0.603886	0.301943	−	0.953326i	$-0.402365\pi$
$281$	10.1230	0.603886	0.301943	+	0.953326i	$0.402365\pi$
$282$	− 11.3959i	− 0.678614i
$283$	− 4.31293i	− 0.256377i	−0.991750	−	0.128188i	$-0.959084\pi$
$283$	− 4.31293i	− 0.256377i	0.991750	−	0.128188i	$-0.0409162\pi$
$284$	9.72938	0.577332
$285$	0	0
$286$	62.9232	3.72073
$287$	− 27.0007i	− 1.59380i
$288$	− 9.41769i	− 0.554943i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.40061	−0.492452
$292$	− 20.6577i	− 1.20890i
$293$	11.2181i	0.655371i	0.944787	+	0.327686i	$0.106269\pi$
$293$	11.2181i	0.655371i	−0.944787	+	0.327686i	$0.893731\pi$
$294$	2.44027	0.142320
$295$	0	0
$296$	−19.4110	−1.12824
$297$	− 4.38277i	− 0.254314i
$298$	− 8.11501i	− 0.470090i
$299$	50.5593	2.92392
$300$	0	0
$301$	−5.51210	−0.317712
$302$	− 23.9686i	− 1.37924i
$303$	− 9.04317i	− 0.519517i
$304$	−10.8119	−0.620107
$305$	0	0
$306$	2.60015	0.148641
$307$	4.44426i	0.253647i	0.991925	+	0.126824i	$0.0404782\pi$
$307$	4.44426i	0.253647i	−0.991925	+	0.126824i	$0.959522\pi$
$308$	58.7892i	3.34982i
$309$	−1.08682	−0.0618269
$310$	0	0
$311$	−7.61723	−0.431934	−0.215967	−	0.976401i	$-0.569290\pi$
$311$	−7.61723	−0.431934	−0.215967	+	0.976401i	$0.569290\pi$
$312$	− 39.6366i	− 2.24398i
$313$	1.85122i	0.104637i	0.998630	+	0.0523185i	$0.0166611\pi$
$313$	1.85122i	0.104637i	−0.998630	+	0.0523185i	$0.983339\pi$
$314$	−3.20979	−0.181139
$315$	0	0
$316$	−51.0447	−2.87149
$317$	− 6.16415i	− 0.346213i	−0.984903	−	0.173106i	$-0.944620\pi$
$317$	− 6.16415i	− 0.346213i	0.984903	−	0.173106i	$-0.0553805\pi$
$318$	24.9175i	1.39730i
$319$	3.58307	0.200614
$320$	0	0
$321$	10.4006	0.580505
$322$	67.0819i	3.73833i
$323$	− 1.18246i	− 0.0657940i
$324$	−4.76079	−0.264488
$325$	0	0
$326$	−38.0305	−2.10631
$327$	8.40061i	0.464555i
$328$	− 68.7919i	− 3.79840i
$329$	12.3486	0.680801
$330$	0	0
$331$	26.8532	1.47599	0.737993	−	0.674808i	$-0.235774\pi$
$331$	26.8532	1.47599	0.737993	+	0.674808i	$0.235774\pi$
$332$	9.72938i	0.533969i
$333$	2.70405i	0.148181i
$334$	22.7917	1.24711
$335$	0	0
$336$	25.7623	1.40545
$337$	10.7040i	0.583087i	0.956558	+	0.291543i	$0.0941688\pi$
$337$	10.7040i	0.583087i	−0.956558	+	0.291543i	$0.905831\pi$
$338$	− 45.4712i	− 2.47331i
$339$	−5.60888	−0.304632
$340$	0	0
$341$	31.5573	1.70892
$342$	3.07459i	0.166255i
$343$	− 17.0785i	− 0.922150i
$344$	−14.0436	−0.757183
$345$	0	0
$346$	58.5401	3.14713
$347$	− 29.9222i	− 1.60631i	−0.595772	−	0.803154i	$-0.703154\pi$
$347$	− 29.9222i	− 1.60631i	0.595772	−	0.803154i	$-0.296846\pi$
$348$	− 3.89212i	− 0.208640i
$349$	−11.6966	−0.626103	−0.313051	−	0.949736i	$-0.601351\pi$
$349$	−11.6966	−0.626103	−0.313051	+	0.949736i	$0.601351\pi$
$350$	0	0
$351$	−5.52159	−0.294720
$352$	41.2755i	2.19999i
$353$	26.2256i	1.39585i	0.716171	+	0.697925i	$0.245893\pi$
$353$	26.2256i	1.39585i	−0.716171	+	0.697925i	$0.754107\pi$
$354$	−19.5573	−1.03946
$355$	0	0
$356$	−48.9749	−2.59566
$357$	2.81754i	0.149120i
$358$	53.9936i	2.85365i
$359$	22.7917	1.20290	0.601451	−	0.798910i	$-0.294590\pi$
$359$	22.7917	1.20290	0.601451	+	0.798910i	$0.294590\pi$
$360$	0	0
$361$	−17.6018	−0.926409
$362$	24.7576i	1.30123i
$363$	8.20866i	0.430843i
$364$	74.0649	3.88206
$365$	0	0
$366$	14.3570	0.750450
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	83.7245i	4.36444i
$369$	−9.58307	−0.498875
$370$	0	0
$371$	−27.0007	−1.40180
$372$	− 34.2792i	− 1.77729i
$373$	− 18.7655i	− 0.971643i	−0.874058	−	0.485821i	$-0.838521\pi$
$373$	− 18.7655i	− 0.971643i	0.874058	−	0.485821i	$-0.161479\pi$
$374$	−11.3959	−0.589266
$375$	0	0
$376$	31.4616	1.62251
$377$	− 4.51410i	− 0.232488i
$378$	− 7.32602i	− 0.376810i
$379$	9.08682	0.466758	0.233379	−	0.972386i	$-0.425022\pi$
$379$	9.08682	0.466758	0.233379	+	0.972386i	$0.425022\pi$
$380$	0	0
$381$	−18.2435	−0.934642
$382$	− 11.6446i	− 0.595788i
$383$	12.0685i	0.616672i	0.951278	+	0.308336i	$0.0997720\pi$
$383$	12.0685i	0.616672i	−0.951278	+	0.308336i	$0.900228\pi$
$384$	−2.71364	−0.138480
$385$	0	0
$386$	60.9574	3.10265
$387$	1.95635i	0.0994470i
$388$	− 39.9936i	− 2.03037i
$389$	−4.59190	−0.232819	−0.116409	−	0.993201i	$-0.537138\pi$
$389$	−4.59190	−0.232819	−0.116409	+	0.993201i	$0.537138\pi$
$390$	0	0
$391$	−9.15666	−0.463072
$392$	6.73709i	0.340274i
$393$	− 9.95635i	− 0.502232i
$394$	−34.2340	−1.72468
$395$	0	0
$396$	20.8655	1.04853
$397$	29.8702i	1.49914i	0.661924	+	0.749571i	$0.269740\pi$
$397$	29.8702i	1.49914i	−0.661924	+	0.749571i	$0.730260\pi$
$398$	− 54.6947i	− 2.74160i
$399$	−3.33163	−0.166790
$400$	0	0
$401$	17.4601	0.871916	0.435958	−	0.899967i	$-0.356410\pi$
$401$	17.4601	0.871916	0.435958	+	0.899967i	$0.356410\pi$
$402$	35.0447i	1.74787i
$403$	− 39.7571i	− 1.98044i
$404$	43.0527	2.14195
$405$	0	0
$406$	5.98929	0.297244
$407$	− 11.8512i	− 0.587443i
$408$	7.17848i	0.355388i
$409$	−6.79134	−0.335810	−0.167905	−	0.985803i	$-0.553700\pi$
$409$	−6.79134	−0.335810	−0.167905	+	0.985803i	$0.553700\pi$
$410$	0	0
$411$	14.4526	0.712895
$412$	− 5.17411i	− 0.254910i
$413$	− 21.1923i	− 1.04281i
$414$	23.8087	1.17013
$415$	0	0
$416$	52.0006	2.54954
$417$	7.67872i	0.376029i
$418$	− 13.4752i	− 0.659093i
$419$	1.89487	0.0925702	0.0462851	−	0.998928i	$-0.485262\pi$
$419$	1.89487	0.0925702	0.0462851	+	0.998928i	$0.485262\pi$
$420$	0	0
$421$	−10.5128	−0.512360	−0.256180	−	0.966629i	$-0.582464\pi$
$421$	−10.5128	−0.512360	−0.256180	+	0.966629i	$0.582464\pi$
$422$	69.5738i	3.38680i
$423$	− 4.38277i	− 0.213097i
$424$	−68.7919	−3.34083
$425$	0	0
$426$	5.31379	0.257454
$427$	15.5573i	0.752869i
$428$	49.5151i	2.39340i
$429$	24.1998	1.16838
$430$	0	0
$431$	4.34709	0.209392	0.104696	−	0.994504i	$-0.466613\pi$
$431$	4.34709	0.209392	0.104696	+	0.994504i	$0.466613\pi$
$432$	− 9.14356i	− 0.439920i
$433$	1.21776i	0.0585217i	0.999572	+	0.0292608i	$0.00931534\pi$
$433$	1.21776i	0.0585217i	−0.999572	+	0.0292608i	$0.990685\pi$
$434$	52.7496	2.53206
$435$	0	0
$436$	−39.9936	−1.91534
$437$	− 10.8274i	− 0.517945i
$438$	− 11.2824i	− 0.539093i
$439$	−32.4171	−1.54718	−0.773592	−	0.633684i	$-0.781542\pi$
$439$	−32.4171	−1.54718	−0.773592	+	0.633684i	$0.781542\pi$
$440$	0	0
$441$	0.938512	0.0446910
$442$	14.3570i	0.682891i
$443$	4.53155i	0.215300i	0.994189	+	0.107650i	$0.0343327\pi$
$443$	4.53155i	0.215300i	−0.994189	+	0.107650i	$0.965667\pi$
$444$	−12.8734	−0.610945
$445$	0	0
$446$	28.4687	1.34803
$447$	− 3.12098i	− 0.147617i
$448$	17.4696i	0.825360i
$449$	−10.8695	−0.512965	−0.256483	−	0.966549i	$-0.582564\pi$
$449$	−10.8695	−0.512965	−0.256483	+	0.966549i	$0.582564\pi$
$450$	0	0
$451$	42.0004	1.97772
$452$	− 26.7027i	− 1.25599i
$453$	− 9.21815i	− 0.433106i
$454$	−35.8801	−1.68394
$455$	0	0
$456$	−8.48829	−0.397501
$457$	− 21.0020i	− 0.982432i	−0.871038	−	0.491216i	$-0.836553\pi$
$457$	− 21.0020i	− 0.982432i	0.871038	−	0.491216i	$-0.163447\pi$
$458$	− 21.9100i	− 1.02379i
$459$	1.00000	0.0466760
$460$	0	0
$461$	−7.24395	−0.337384	−0.168692	−	0.985669i	$-0.553954\pi$
$461$	−7.24395	−0.337384	−0.168692	+	0.985669i	$0.553954\pi$
$462$	32.1083i	1.49381i
$463$	− 9.27811i	− 0.431190i	−0.976483	−	0.215595i	$-0.930831\pi$
$463$	− 9.27811i	− 0.431190i	0.976483	−	0.215595i	$-0.0691691\pi$
$464$	7.47519	0.347027
$465$	0	0
$466$	−70.7042	−3.27531
$467$	− 12.2955i	− 0.568967i	−0.958681	−	0.284483i	$-0.908178\pi$
$467$	− 12.2955i	− 0.568967i	0.958681	−	0.284483i	$-0.0918221\pi$
$468$	− 26.2871i	− 1.21512i
$469$	−37.9746	−1.75350
$470$	0	0
$471$	−1.23446	−0.0568811
$472$	− 53.9936i	− 2.48526i
$473$	− 8.57424i	− 0.394244i
$474$	−27.8785	−1.28050
$475$	0	0
$476$	−13.4137	−0.614816
$477$	9.58307i	0.438779i
$478$	− 19.6500i	− 0.898772i
$479$	−2.82589	−0.129118	−0.0645591	−	0.997914i	$-0.520564\pi$
$479$	−2.82589	−0.129118	−0.0645591	+	0.997914i	$0.520564\pi$
$480$	0	0
$481$	−14.9306	−0.680778
$482$	− 34.9312i	− 1.59107i
$483$	25.7992i	1.17391i
$484$	−39.0797	−1.77635
$485$	0	0
$486$	−2.60015	−0.117945
$487$	− 4.72985i	− 0.214330i	−0.994241	−	0.107165i	$-0.965823\pi$
$487$	− 4.72985i	− 0.214330i	0.994241	−	0.107165i	$-0.0341773\pi$
$488$	39.6366i	1.79426i
$489$	−14.6262	−0.661422
$490$	0	0
$491$	16.6782	0.752679	0.376339	−	0.926482i	$-0.377183\pi$
$491$	16.6782	0.752679	0.376339	+	0.926482i	$0.377183\pi$
$492$	− 45.6230i	− 2.05684i
$493$	0.817537i	0.0368200i
$494$	−16.9766	−0.763813
$495$	0	0
$496$	65.8364	2.95614
$497$	5.75805i	0.258284i
$498$	5.31379i	0.238117i
$499$	−13.2510	−0.593195	−0.296597	−	0.955003i	$-0.595852\pi$
$499$	−13.2510	−0.593195	−0.296597	+	0.955003i	$0.595852\pi$
$500$	0	0
$501$	8.76554	0.391615
$502$	− 73.6436i	− 3.28688i
$503$	− 27.0789i	− 1.20739i	−0.797217	−	0.603693i	$-0.793695\pi$
$503$	− 27.0789i	− 1.20739i	0.797217	−	0.603693i	$-0.206305\pi$
$504$	20.2256	0.900921
$505$	0	0
$506$	−104.348	−4.63884
$507$	− 17.4879i	− 0.776665i
$508$	− 86.8534i	− 3.85350i
$509$	−26.4101	−1.17061	−0.585304	−	0.810814i	$-0.699025\pi$
$509$	−26.4101	−1.17061	−0.585304	+	0.810814i	$0.699025\pi$
$510$	0	0
$511$	12.2256	0.540830
$512$	− 45.1626i	− 1.99592i
$513$	1.18246i	0.0522070i
$514$	−62.6282	−2.76241
$515$	0	0
$516$	−9.31379	−0.410017
$517$	19.2087i	0.844796i
$518$	− 19.8099i	− 0.870398i
$519$	22.5141	0.988259
$520$	0	0
$521$	1.63468	0.0716168	0.0358084	−	0.999359i	$-0.488599\pi$
$521$	1.63468	0.0716168	0.0358084	+	0.999359i	$0.488599\pi$
$522$	− 2.12572i	− 0.0930402i
$523$	− 4.96384i	− 0.217054i	−0.994094	−	0.108527i	$-0.965387\pi$
$523$	− 4.96384i	− 0.217054i	0.994094	−	0.108527i	$-0.0346133\pi$
$524$	47.4001	2.07068
$525$	0	0
$526$	46.8491	2.04272
$527$	7.20030i	0.313650i
$528$	40.0741i	1.74400i
$529$	−60.8444	−2.64541
$530$	0	0
$531$	−7.52159	−0.326409
$532$	− 15.8612i	− 0.687671i
$533$	− 52.9138i	− 2.29195i
$534$	−26.7481	−1.15750
$535$	0	0
$536$	−96.7511	−4.17901
$537$	20.7655i	0.896099i
$538$	− 1.53555i	− 0.0662022i
$539$	−4.11328	−0.177172
$540$	0	0
$541$	−21.0187	−0.903665	−0.451832	−	0.892103i	$-0.649229\pi$
$541$	−21.0187	−0.903665	−0.451832	+	0.892103i	$0.649229\pi$
$542$	35.7730i	1.53658i
$543$	9.52159i	0.408610i
$544$	−9.41769	−0.403780
$545$	0	0
$546$	40.4513	1.73115
$547$	− 6.67785i	− 0.285524i	−0.989757	−	0.142762i	$-0.954402\pi$
$547$	− 6.67785i	− 0.285524i	0.989757	−	0.142762i	$-0.0455984\pi$
$548$	68.8059i	2.93924i
$549$	5.52159	0.235656
$550$	0	0
$551$	−0.966707	−0.0411831
$552$	65.7309i	2.79769i
$553$	− 30.2093i	− 1.28463i
$554$	25.1844	1.06998
$555$	0	0
$556$	−36.5568	−1.55035
$557$	31.2168i	1.32270i	0.750078	+	0.661349i	$0.230016\pi$
$557$	31.2168i	1.32270i	−0.750078	+	0.661349i	$0.769984\pi$
$558$	− 18.7219i	− 0.792561i
$559$	−10.8022	−0.456883
$560$	0	0
$561$	−4.38277	−0.185041
$562$	26.3213i	1.11030i
$563$	33.6163i	1.41676i	0.705833	+	0.708379i	$0.250573\pi$
$563$	33.6163i	1.41676i	−0.705833	+	0.708379i	$0.749427\pi$
$564$	20.8655	0.878594
$565$	0	0
$566$	11.2143	0.471371
$567$	− 2.81754i	− 0.118325i
$568$	14.6703i	0.615551i
$569$	−7.63507	−0.320079	−0.160039	−	0.987111i	$-0.551162\pi$
$569$	−7.63507	−0.320079	−0.160039	+	0.987111i	$0.551162\pi$
$570$	0	0
$571$	−25.6009	−1.07136	−0.535682	−	0.844420i	$-0.679946\pi$
$571$	−25.6009	−1.07136	−0.535682	+	0.844420i	$0.679946\pi$
$572$	115.210i	4.81719i
$573$	− 4.47841i	− 0.187088i
$574$	70.2058	2.93033
$575$	0	0
$576$	6.20030	0.258346
$577$	34.9242i	1.45391i	0.686683	+	0.726957i	$0.259066\pi$
$577$	34.9242i	1.45391i	−0.686683	+	0.726957i	$0.740934\pi$
$578$	− 2.60015i	− 0.108152i
$579$	23.4438	0.974290
$580$	0	0
$581$	−5.75805	−0.238884
$582$	− 21.8429i	− 0.905416i
$583$	− 42.0004i	− 1.73948i
$584$	31.1483	1.28893
$585$	0	0
$586$	−29.1689	−1.20496
$587$	− 18.0535i	− 0.745149i	−0.928002	−	0.372574i	$-0.878475\pi$
$587$	− 18.0535i	− 0.745149i	0.928002	−	0.372574i	$-0.121525\pi$
$588$	4.46806i	0.184260i
$589$	−8.51410	−0.350817
$590$	0	0
$591$	−13.1661	−0.541583
$592$	− 24.7246i	− 1.01618i
$593$	29.4958i	1.21125i	0.795752	+	0.605623i	$0.207076\pi$
$593$	29.4958i	1.21125i	−0.795752	+	0.605623i	$0.792924\pi$
$594$	11.3959	0.467578
$595$	0	0
$596$	14.8583	0.608620
$597$	− 21.0352i	− 0.860914i
$598$	131.462i	5.37587i
$599$	19.7214	0.805795	0.402898	−	0.915245i	$-0.368003\pi$
$599$	19.7214	0.805795	0.402898	+	0.915245i	$0.368003\pi$
$600$	0	0
$601$	−3.29556	−0.134429	−0.0672144	−	0.997739i	$-0.521411\pi$
$601$	−3.29556	−0.134429	−0.0672144	+	0.997739i	$0.521411\pi$
$602$	− 14.3323i	− 0.584141i
$603$	13.4779i	0.548864i
$604$	43.8857	1.78568
$605$	0	0
$606$	23.5136	0.955176
$607$	− 30.5222i	− 1.23886i	−0.785052	−	0.619430i	$-0.787364\pi$
$607$	− 30.5222i	− 1.23886i	0.785052	−	0.619430i	$-0.212636\pi$
$608$	− 11.1361i	− 0.451627i
$609$	2.30344	0.0933401
$610$	0	0
$611$	24.1998	0.979020
$612$	4.76079i	0.192444i
$613$	23.5478i	0.951086i	0.879693	+	0.475543i	$0.157748\pi$
$613$	23.5478i	0.951086i	−0.879693	+	0.475543i	$0.842252\pi$
$614$	−11.5557	−0.466352
$615$	0	0
$616$	−88.6443	−3.57158
$617$	9.43429i	0.379810i	0.981802	+	0.189905i	$0.0608180\pi$
$617$	9.43429i	0.379810i	−0.981802	+	0.189905i	$0.939182\pi$
$618$	− 2.82589i	− 0.113674i
$619$	12.1657	0.488980	0.244490	−	0.969652i	$-0.421380\pi$
$619$	12.1657	0.488980	0.244490	+	0.969652i	$0.421380\pi$
$620$	0	0
$621$	9.15666	0.367444
$622$	− 19.8060i	− 0.794147i
$623$	− 28.9843i	− 1.16123i
$624$	50.4870	2.02110
$625$	0	0
$626$	−4.81345	−0.192384
$627$	− 5.18246i	− 0.206968i
$628$	− 5.87702i	− 0.234519i
$629$	2.70405	0.107817
$630$	0	0
$631$	14.0710	0.560157	0.280078	−	0.959977i	$-0.409640\pi$
$631$	14.0710	0.560157	0.280078	+	0.959977i	$0.409640\pi$
$632$	− 76.9669i	− 3.06158i
$633$	26.7576i	1.06352i
$634$	16.0277	0.636542
$635$	0	0
$636$	−45.6230	−1.80907
$637$	5.18207i	0.205321i
$638$	9.31654i	0.368845i
$639$	2.04365	0.0808454
$640$	0	0
$641$	−9.35610	−0.369544	−0.184772	−	0.982781i	$-0.559155\pi$
$641$	−9.35610	−0.369544	−0.184772	+	0.982781i	$0.559155\pi$
$642$	27.0432i	1.06731i
$643$	50.6619i	1.99791i	0.0456899	+	0.998956i	$0.485451\pi$
$643$	50.6619i	1.99791i	−0.0456899	+	0.998956i	$0.514549\pi$
$644$	−122.825	−4.83997
$645$	0	0
$646$	3.07459	0.120968
$647$	− 14.0436i	− 0.552113i	−0.961141	−	0.276056i	$-0.910972\pi$
$647$	− 14.0436i	− 0.552113i	0.961141	−	0.276056i	$-0.0890276\pi$
$648$	− 7.17848i	− 0.281997i
$649$	32.9654	1.29400
$650$	0	0
$651$	20.2871	0.795115
$652$	− 69.6325i	− 2.72702i
$653$	9.04317i	0.353887i	0.984221	+	0.176943i	$0.0566209\pi$
$653$	9.04317i	0.353887i	−0.984221	+	0.176943i	$0.943379\pi$
$654$	−21.8429	−0.854124
$655$	0	0
$656$	87.6234	3.42112
$657$	− 4.33912i	− 0.169285i
$658$	32.1083i	1.25171i
$659$	7.32176	0.285215	0.142608	−	0.989779i	$-0.454451\pi$
$659$	7.32176	0.285215	0.142608	+	0.989779i	$0.454451\pi$
$660$	0	0
$661$	−42.1329	−1.63878	−0.819389	−	0.573238i	$-0.805687\pi$
$661$	−42.1329	−1.63878	−0.819389	+	0.573238i	$0.805687\pi$
$662$	69.8225i	2.71373i
$663$	5.52159i	0.214441i
$664$	−14.6703	−0.569317
$665$	0	0
$666$	−7.03094	−0.272443
$667$	7.48590i	0.289855i
$668$	41.7309i	1.61462i
$669$	10.9489	0.423308
$670$	0	0
$671$	−24.1998	−0.934224
$672$	26.5347i	1.02360i
$673$	3.82637i	0.147496i	0.997277	+	0.0737478i	$0.0234960\pi$
$673$	3.82637i	0.147496i	−0.997277	+	0.0737478i	$0.976504\pi$
$674$	−27.8322	−1.07205
$675$	0	0
$676$	83.2563	3.20216
$677$	− 11.5311i	− 0.443175i	−0.975140	−	0.221588i	$-0.928876\pi$
$677$	− 11.5311i	− 0.443175i	0.975140	−	0.221588i	$-0.0711239\pi$
$678$	− 14.5839i	− 0.560093i
$679$	23.6690	0.908334
$680$	0	0
$681$	−13.7992	−0.528787
$682$	82.0537i	3.14200i
$683$	− 3.82541i	− 0.146375i	−0.997318	−	0.0731877i	$-0.976683\pi$
$683$	− 3.82541i	− 0.146375i	0.997318	−	0.0731877i	$-0.0233172\pi$
$684$	−5.62946	−0.215248
$685$	0	0
$686$	44.4066	1.69545
$687$	− 8.42641i	− 0.321488i
$688$	− 17.8880i	− 0.681975i
$689$	−52.9138	−2.01585
$690$	0	0
$691$	5.95635	0.226590	0.113295	−	0.993561i	$-0.463859\pi$
$691$	5.95635	0.226590	0.113295	+	0.993561i	$0.463859\pi$
$692$	107.185i	4.07456i
$693$	12.3486i	0.469085i
$694$	77.8023	2.95333
$695$	0	0
$696$	5.86867	0.222451
$697$	9.58307i	0.362985i
$698$	− 30.4128i	− 1.15114i
$699$	−27.1923	−1.02851
$700$	0	0
$701$	−31.6436	−1.19516	−0.597581	−	0.801808i	$-0.703871\pi$
$701$	−31.6436	−1.19516	−0.597581	+	0.801808i	$0.703871\pi$
$702$	− 14.3570i	− 0.541869i
$703$	3.19744i	0.120594i
$704$	−27.1745	−1.02418
$705$	0	0
$706$	−68.1906	−2.56639
$707$	25.4795i	0.958254i
$708$	− 35.8087i	− 1.34577i
$709$	−15.6708	−0.588528	−0.294264	−	0.955724i	$-0.595074\pi$
$709$	−15.6708	−0.588528	−0.294264	+	0.955724i	$0.595074\pi$
$710$	0	0
$711$	−10.7219	−0.402102
$712$	− 73.8459i	− 2.76749i
$713$	65.9307i	2.46913i
$714$	−7.32602	−0.274170
$715$	0	0
$716$	−98.8604	−3.69459
$717$	− 7.55727i	− 0.282231i
$718$	59.2620i	2.21164i
$719$	4.15579	0.154985	0.0774925	−	0.996993i	$-0.475309\pi$
$719$	4.15579	0.154985	0.0774925	+	0.996993i	$0.475309\pi$
$720$	0	0
$721$	3.06215	0.114040
$722$	− 45.7673i	− 1.70328i
$723$	− 13.4343i	− 0.499627i
$724$	−45.3303	−1.68469
$725$	0	0
$726$	−21.3438	−0.792141
$727$	− 7.36445i	− 0.273132i	−0.990631	−	0.136566i	$-0.956393\pi$
$727$	− 7.36445i	− 0.273132i	0.990631	−	0.136566i	$-0.0436066\pi$
$728$	111.678i	4.13905i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.95635	0.0723583
$732$	26.2871i	0.971600i
$733$	− 50.3228i	− 1.85872i	−0.369181	−	0.929358i	$-0.620362\pi$
$733$	− 50.3228i	− 1.85872i	0.369181	−	0.929358i	$-0.379638\pi$
$734$	20.8012	0.767787
$735$	0	0
$736$	−86.2346	−3.17865
$737$	− 59.0707i	− 2.17590i
$738$	− 24.9175i	− 0.917224i
$739$	10.0863	0.371032	0.185516	−	0.982641i	$-0.440604\pi$
$739$	10.0863	0.371032	0.185516	+	0.982641i	$0.440604\pi$
$740$	0	0
$741$	−6.52907	−0.239851
$742$	− 70.2058i	− 2.57734i
$743$	− 34.7646i	− 1.27539i	−0.770289	−	0.637694i	$-0.779888\pi$
$743$	− 34.7646i	− 1.27539i	0.770289	−	0.637694i	$-0.220112\pi$
$744$	51.6873	1.89495
$745$	0	0
$746$	48.7933	1.78645
$747$	2.04365i	0.0747731i
$748$	− 20.8655i	− 0.762916i
$749$	−29.3041	−1.07075
$750$	0	0
$751$	−16.4171	−0.599069	−0.299534	−	0.954085i	$-0.596831\pi$
$751$	−16.4171	−0.599069	−0.299534	+	0.954085i	$0.596831\pi$
$752$	40.0741i	1.46135i
$753$	− 28.3228i	− 1.03214i
$754$	11.7373	0.427449
$755$	0	0
$756$	13.4137	0.487852
$757$	37.8611i	1.37608i	0.725670	+	0.688042i	$0.241530\pi$
$757$	37.8611i	1.37608i	−0.725670	+	0.688042i	$0.758470\pi$
$758$	23.6271i	0.858175i
$759$	−40.1315	−1.45668
$760$	0	0
$761$	6.16415	0.223450	0.111725	−	0.993739i	$-0.464362\pi$
$761$	6.16415	0.223450	0.111725	+	0.993739i	$0.464362\pi$
$762$	− 47.4358i	− 1.71842i
$763$	− 23.6690i	− 0.856876i
$764$	21.3208	0.771360
$765$	0	0
$766$	−31.3799	−1.13380
$767$	− 41.5311i	− 1.49960i
$768$	− 19.4565i	− 0.702076i
$769$	−29.0007	−1.04579	−0.522895	−	0.852397i	$-0.675148\pi$
$769$	−29.0007	−1.04579	−0.522895	+	0.852397i	$0.675148\pi$
$770$	0	0
$771$	−24.0863	−0.867448
$772$	111.611i	4.01697i
$773$	− 29.3045i	− 1.05401i	−0.849862	−	0.527005i	$-0.823315\pi$
$773$	− 29.3045i	− 1.05401i	0.849862	−	0.527005i	$-0.176685\pi$
$774$	−5.08682	−0.182842
$775$	0	0
$776$	60.3036	2.16477
$777$	− 7.61876i	− 0.273321i
$778$	− 11.9396i	− 0.428057i
$779$	−11.3316	−0.405998
$780$	0	0
$781$	−8.95683	−0.320501
$782$	− 23.8087i	− 0.851398i
$783$	− 0.817537i	− 0.0292164i
$784$	−8.58134	−0.306476
$785$	0	0
$786$	25.8880	0.923396
$787$	5.09517i	0.181623i	0.995868	+	0.0908116i	$0.0289461\pi$
$787$	5.09517i	0.181623i	−0.995868	+	0.0908116i	$0.971054\pi$
$788$	− 62.6813i	− 2.23293i
$789$	18.0178	0.641452
$790$	0	0
$791$	15.8032	0.561898
$792$	31.4616i	1.11794i
$793$	30.4879i	1.08266i
$794$	−77.6671	−2.75630
$795$	0	0
$796$	100.144	3.54952
$797$	− 16.6786i	− 0.590787i	−0.955376	−	0.295394i	$-0.904549\pi$
$797$	− 16.6786i	− 0.590787i	0.955376	−	0.295394i	$-0.0954508\pi$
$798$	− 8.66276i	− 0.306658i
$799$	−4.38277	−0.155051
$800$	0	0
$801$	−10.2871	−0.363478
$802$	45.3989i	1.60309i
$803$	19.0174i	0.671108i
$804$	−64.1657	−2.26295
$805$	0	0
$806$	103.375	3.64121
$807$	− 0.590561i	− 0.0207887i
$808$	64.9162i	2.28375i
$809$	−13.3574	−0.469623	−0.234811	−	0.972041i	$-0.575447\pi$
$809$	−13.3574	−0.469623	−0.234811	+	0.972041i	$0.575447\pi$
$810$	0	0
$811$	34.8344	1.22320	0.611601	−	0.791166i	$-0.290526\pi$
$811$	34.8344	1.22320	0.611601	+	0.791166i	$0.290526\pi$
$812$	10.9662i	0.384838i
$813$	13.7580i	0.482516i
$814$	30.8150	1.08006
$815$	0	0
$816$	−9.14356	−0.320089
$817$	2.31332i	0.0809327i
$818$	− 17.6585i	− 0.617416i
$819$	15.5573	0.543615
$820$	0	0
$821$	27.0282	0.943291	0.471645	−	0.881788i	$-0.343660\pi$
$821$	27.0282	0.943291	0.471645	+	0.881788i	$0.343660\pi$
$822$	37.5790i	1.31072i
$823$	6.60727i	0.230315i	0.993347	+	0.115157i	$0.0367373\pi$
$823$	6.60727i	0.230315i	−0.993347	+	0.115157i	$0.963263\pi$
$824$	7.80170	0.271785
$825$	0	0
$826$	55.1033	1.91729
$827$	29.1400i	1.01330i	0.862153	+	0.506648i	$0.169116\pi$
$827$	29.1400i	1.01330i	−0.862153	+	0.506648i	$0.830884\pi$
$828$	43.5930i	1.51496i
$829$	1.22763	0.0426375	0.0213188	−	0.999773i	$-0.493214\pi$
$829$	1.22763	0.0426375	0.0213188	+	0.999773i	$0.493214\pi$
$830$	0	0
$831$	9.68573	0.335994
$832$	34.2355i	1.18690i
$833$	− 0.938512i	− 0.0325175i
$834$	−19.9658	−0.691361
$835$	0	0
$836$	24.6726	0.853321
$837$	− 7.20030i	− 0.248879i
$838$	4.92694i	0.170198i
$839$	−11.7759	−0.406549	−0.203274	−	0.979122i	$-0.565158\pi$
$839$	−11.7759	−0.406549	−0.203274	+	0.979122i	$0.565158\pi$
$840$	0	0
$841$	−28.3316	−0.976953
$842$	− 27.3348i	− 0.942018i
$843$	10.1230i	0.348654i
$844$	−127.387	−4.38485
$845$	0	0
$846$	11.3959	0.391798
$847$	− 23.1282i	− 0.794694i
$848$	− 87.6234i	− 3.00900i
$849$	4.31293	0.148019
$850$	0	0
$851$	24.7600	0.848764
$852$	9.72938i	0.333323i
$853$	− 38.7139i	− 1.32554i	−0.748823	−	0.662770i	$-0.769381\pi$
$853$	− 38.7139i	− 1.32554i	0.748823	−	0.662770i	$-0.230619\pi$
$854$	−40.4513	−1.38421
$855$	0	0
$856$	−74.6606	−2.55185
$857$	31.6351i	1.08063i	0.841462	+	0.540317i	$0.181696\pi$
$857$	31.6351i	1.08063i	−0.841462	+	0.540317i	$0.818304\pi$
$858$	62.9232i	2.14816i
$859$	−17.6337	−0.601655	−0.300828	−	0.953679i	$-0.597263\pi$
$859$	−17.6337	−0.601655	−0.300828	+	0.953679i	$0.597263\pi$
$860$	0	0
$861$	27.0007	0.920180
$862$	11.3031i	0.384985i
$863$	52.4541i	1.78556i	0.450493	+	0.892780i	$0.351248\pi$
$863$	52.4541i	1.78556i	−0.450493	+	0.892780i	$0.648752\pi$
$864$	9.41769	0.320396
$865$	0	0
$866$	−3.16635	−0.107597
$867$	− 1.00000i	− 0.0339618i
$868$	96.5828i	3.27823i
$869$	46.9916	1.59408
$870$	0	0
$871$	−74.4196	−2.52161
$872$	− 60.3036i	− 2.04214i
$873$	− 8.40061i	− 0.284318i
$874$	28.1529	0.952287
$875$	0	0
$876$	20.6577	0.697958
$877$	− 5.19195i	− 0.175320i	−0.996150	−	0.0876599i	$-0.972061\pi$
$877$	− 5.19195i	− 0.175320i	0.996150	−	0.0876599i	$-0.0279389\pi$
$878$	− 84.2894i	− 2.84463i
$879$	−11.2181	−0.378379
$880$	0	0
$881$	−42.0867	−1.41794	−0.708969	−	0.705240i	$-0.750839\pi$
$881$	−42.0867	−1.41794	−0.708969	+	0.705240i	$0.750839\pi$
$882$	2.44027i	0.0821683i
$883$	− 22.2602i	− 0.749115i	−0.927204	−	0.374557i	$-0.877795\pi$
$883$	− 22.2602i	− 0.749115i	0.927204	−	0.374557i	$-0.122205\pi$
$884$	−26.2871	−0.884132
$885$	0	0
$886$	−11.7827	−0.395848
$887$	− 31.3574i	− 1.05288i	−0.850212	−	0.526440i	$-0.823527\pi$
$887$	− 31.3574i	− 1.05288i	0.850212	−	0.526440i	$-0.176473\pi$
$888$	− 19.4110i	− 0.651389i
$889$	51.4017	1.72396
$890$	0	0
$891$	4.38277	0.146828
$892$	52.1253i	1.74528i
$893$	− 5.18246i	− 0.173425i
$894$	8.11501	0.271407
$895$	0	0
$896$	7.64578	0.255428
$897$	50.5593i	1.68812i
$898$	− 28.2625i	− 0.943130i
$899$	5.88651	0.196326
$900$	0	0
$901$	9.58307	0.319258
$902$	109.207i	3.63621i
$903$	− 5.51210i	− 0.183431i
$904$	40.2632	1.33914
$905$	0	0
$906$	23.9686	0.796303
$907$	29.9837i	0.995592i	0.867294	+	0.497796i	$0.165857\pi$
$907$	29.9837i	0.995592i	−0.867294	+	0.497796i	$0.834143\pi$
$908$	− 65.6952i	− 2.18017i
$909$	9.04317	0.299943
$910$	0	0
$911$	−2.74770	−0.0910352	−0.0455176	−	0.998964i	$-0.514494\pi$
$911$	−2.74770	−0.0910352	−0.0455176	+	0.998964i	$0.514494\pi$
$912$	− 10.8119i	− 0.358019i
$913$	− 8.95683i	− 0.296428i
$914$	54.6084	1.80629
$915$	0	0
$916$	40.1164	1.32548
$917$	28.0524i	0.926372i
$918$	2.60015i	0.0858178i
$919$	−12.3653	−0.407894	−0.203947	−	0.978982i	$-0.565377\pi$
$919$	−12.3653	−0.407894	−0.203947	+	0.978982i	$0.565377\pi$
$920$	0	0
$921$	−4.44426	−0.146443
$922$	− 18.8354i	− 0.620310i
$923$	11.2842i	0.371423i
$924$	−58.7892	−1.93402
$925$	0	0
$926$	24.1245	0.792780
$927$	− 1.08682i	− 0.0356958i
$928$	7.69930i	0.252742i
$929$	23.0625	0.756657	0.378329	−	0.925671i	$-0.376499\pi$
$929$	23.0625	0.756657	0.378329	+	0.925671i	$0.376499\pi$
$930$	0	0
$931$	1.10976	0.0363708
$932$	− 129.457i	− 4.24051i
$933$	− 7.61723i	− 0.249377i
$934$	31.9701	1.04609
$935$	0	0
$936$	39.6366	1.29556
$937$	− 54.0609i	− 1.76609i	−0.469285	−	0.883047i	$-0.655488\pi$
$937$	− 54.0609i	− 1.76609i	0.469285	−	0.883047i	$-0.344512\pi$
$938$	− 98.7397i	− 3.22397i
$939$	−1.85122	−0.0604122
$940$	0	0
$941$	39.2009	1.27791	0.638956	−	0.769243i	$-0.279367\pi$
$941$	39.2009	1.27791	0.638956	+	0.769243i	$0.279367\pi$
$942$	− 3.20979i	− 0.104581i
$943$	87.7489i	2.85750i
$944$	68.7741	2.23841
$945$	0	0
$946$	22.2943	0.724852
$947$	21.8453i	0.709878i	0.934889	+	0.354939i	$0.115498\pi$
$947$	21.8453i	0.709878i	−0.934889	+	0.354939i	$0.884502\pi$
$948$	− 51.0447i	− 1.65785i
$949$	23.9588	0.777737
$950$	0	0
$951$	6.16415	0.199886
$952$	− 20.2256i	− 0.655517i
$953$	11.6364i	0.376940i	0.982079	+	0.188470i	$0.0603529\pi$
$953$	11.6364i	0.376940i	−0.982079	+	0.188470i	$0.939647\pi$
$954$	−24.9175	−0.806732
$955$	0	0
$956$	35.9786	1.16363
$957$	3.58307i	0.115824i
$958$	− 7.34774i	− 0.237395i
$959$	−40.7208	−1.31494
$960$	0	0
$961$	20.8444	0.672400
$962$	− 38.8219i	− 1.25167i
$963$	10.4006i	0.335155i
$964$	63.9579	2.05994
$965$	0	0
$966$	−67.0819	−2.15832
$967$	25.4096i	0.817118i	0.912732	+	0.408559i	$0.133969\pi$
$967$	25.4096i	0.817118i	−0.912732	+	0.408559i	$0.866031\pi$
$968$	− 58.9257i	− 1.89394i
$969$	1.18246	0.0379862
$970$	0	0
$971$	−44.9654	−1.44301	−0.721504	−	0.692411i	$-0.756549\pi$
$971$	−44.9654	−1.44301	−0.721504	+	0.692411i	$0.756549\pi$
$972$	− 4.76079i	− 0.152702i
$973$	− 21.6351i	− 0.693589i
$974$	12.2983	0.394064
$975$	0	0
$976$	−50.4870	−1.61605
$977$	− 3.51210i	− 0.112362i	−0.998421	−	0.0561810i	$-0.982108\pi$
$977$	− 3.51210i	− 0.112362i	0.998421	−	0.0561810i	$-0.0178924\pi$
$978$	− 38.0305i	− 1.21608i
$979$	45.0861	1.44096
$980$	0	0
$981$	−8.40061	−0.268211
$982$	43.3660i	1.38386i
$983$	9.97381i	0.318115i	0.987269	+	0.159057i	$0.0508455\pi$
$983$	9.97381i	0.318115i	−0.987269	+	0.159057i	$0.949154\pi$
$984$	68.7919	2.19301
$985$	0	0
$986$	−2.12572	−0.0676967
$987$	12.3486i	0.393061i
$988$	− 31.0836i	− 0.988900i
$989$	17.9137	0.569621
$990$	0	0
$991$	−4.24443	−0.134829	−0.0674143	−	0.997725i	$-0.521475\pi$
$991$	−4.24443	−0.134829	−0.0674143	+	0.997725i	$0.521475\pi$
$992$	67.8102i	2.15298i
$993$	26.8532i	0.852161i
$994$	−14.9718	−0.474877
$995$	0	0
$996$	−9.72938	−0.308287
$997$	43.8336i	1.38822i	0.719868	+	0.694111i	$0.244202\pi$
$997$	43.8336i	1.38822i	−0.719868	+	0.694111i	$0.755798\pi$
$998$	− 34.4545i	− 1.09064i
$999$	−2.70405	−0.0855523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.b.k.1174.8		8
5.2	odd	4		255.2.a.d.1.1	✓	4
5.3	odd	4		1275.2.a.t.1.4		4
5.4	even	2	inner	1275.2.b.k.1174.1		8
15.2	even	4		765.2.a.m.1.4		4
15.8	even	4		3825.2.a.bi.1.1		4
20.7	even	4		4080.2.a.bt.1.2		4
85.67	odd	4		4335.2.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.d.1.1	✓	4	5.2	odd	4
765.2.a.m.1.4		4	15.2	even	4
1275.2.a.t.1.4		4	5.3	odd	4
1275.2.b.k.1174.1		8	5.4	even	2	inner
1275.2.b.k.1174.8		8	1.1	even	1	trivial
3825.2.a.bi.1.1		4	15.8	even	4
4080.2.a.bt.1.2		4	20.7	even	4
4335.2.a.z.1.1		4	85.67	odd	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:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.60015i q^{2} +1.00000i q^{3} -4.76079 q^{4} -2.60015 q^{6} -2.81754i q^{7} -7.17848i q^{8} -1.00000 q^{9} +4.38277 q^{11} -4.76079i q^{12} -5.52159i q^{13} +7.32602 q^{14} +9.14356 q^{16} +1.00000i q^{17} -2.60015i q^{18} -1.18246 q^{19} +2.81754 q^{21} +11.3959i q^{22} +9.15666i q^{23} +7.17848 q^{24} +14.3570 q^{26} -1.00000i q^{27} +13.4137i q^{28} +0.817537 q^{29} +7.20030 q^{31} +9.41769i q^{32} +4.38277i q^{33} -2.60015 q^{34} +4.76079 q^{36} -2.70405i q^{37} -3.07459i q^{38} +5.52159 q^{39} +9.58307 q^{41} +7.32602i q^{42} -1.95635i q^{43} -20.8655 q^{44} -23.8087 q^{46} +4.38277i q^{47} +9.14356i q^{48} -0.938512 q^{49} -1.00000 q^{51} +26.2871i q^{52} -9.58307i q^{53} +2.60015 q^{54} -20.2256 q^{56} -1.18246i q^{57} +2.12572i q^{58} +7.52159 q^{59} -5.52159 q^{61} +18.7219i q^{62} +2.81754i q^{63} -6.20030 q^{64} -11.3959 q^{66} -13.4779i q^{67} -4.76079i q^{68} -9.15666 q^{69} -2.04365 q^{71} +7.17848i q^{72} +4.33912i q^{73} +7.03094 q^{74} +5.62946 q^{76} -12.3486i q^{77} +14.3570i q^{78} +10.7219 q^{79} +1.00000 q^{81} +24.9175i q^{82} -2.04365i q^{83} -13.4137 q^{84} +5.08682 q^{86} +0.817537i q^{87} -31.4616i q^{88} +10.2871 q^{89} -15.5573 q^{91} -43.5930i q^{92} +7.20030i q^{93} -11.3959 q^{94} -9.41769 q^{96} +8.40061i q^{97} -2.44027i q^{98} -4.38277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 18 q^{4} + 2 q^{6} - 8 q^{9} + 4 q^{11} + 22 q^{14} + 22 q^{16} - 24 q^{19} + 8 q^{21} - 8 q^{29} + 12 q^{31} + 2 q^{34} + 18 q^{36} + 4 q^{39} - 30 q^{44} - 48 q^{46} - 44 q^{49} - 8 q^{51} - 2 q^{54}+ \cdots - 4 q^{99}+O(q^{100}) \) 8 * q - 18 * q^4 + 2 * q^6 - 8 * q^9 + 4 * q^11 + 22 * q^14 + 22 * q^16 - 24 * q^19 + 8 * q^21 - 8 * q^29 + 12 * q^31 + 2 * q^34 + 18 * q^36 + 4 * q^39 - 30 * q^44 - 48 * q^46 - 44 * q^49 - 8 * q^51 - 2 * q^54 - 96 * q^56 + 20 * q^59 - 4 * q^61 - 4 * q^64 - 42 * q^66 - 4 * q^69 - 40 * q^71 - 22 * q^74 + 74 * q^76 + 8 * q^81 + 2 * q^84 - 16 * q^86 - 20 * q^89 + 36 * q^91 - 42 * q^94 - 38 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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