LMFDB - Embedded newform 1925.2.b.n.1849.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1849,1925)] 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("1925.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-10,0,8,0,0,2,0,-6] 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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 385)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1925.1849
Dual form			1925.2.b.n.1849.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.67513i q^{2} +2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} -1.00000i q^{7} +8.44358i q^{8} -3.15633 q^{9} -1.00000 q^{11} -12.7938i q^{12} -5.83146i q^{13} -2.67513 q^{14} +12.2750 q^{16} +5.44358i q^{17} +8.44358i q^{18} +1.35026 q^{19} +2.48119 q^{21} +2.67513i q^{22} +3.19394i q^{23} -20.9502 q^{24} -15.5999 q^{26} -0.387873i q^{27} +5.15633i q^{28} +3.61213 q^{29} -5.28726 q^{31} -15.9502i q^{32} -2.48119i q^{33} +14.5623 q^{34} +16.2750 q^{36} -8.54420i q^{37} -3.61213i q^{38} +14.4690 q^{39} -5.02539 q^{41} -6.63752i q^{42} -5.89446i q^{43} +5.15633 q^{44} +8.54420 q^{46} -11.8315i q^{47} +30.4568i q^{48} -1.00000 q^{49} -13.5066 q^{51} +30.0689i q^{52} +0.231548i q^{53} -1.03761 q^{54} +8.44358 q^{56} +3.35026i q^{57} -9.66291i q^{58} -13.5999 q^{59} -1.41327 q^{61} +14.1441i q^{62} +3.15633i q^{63} -18.1187 q^{64} -6.63752 q^{66} -10.8568i q^{67} -28.0689i q^{68} -7.92478 q^{69} -15.5369 q^{71} -26.6507i q^{72} +11.3684i q^{73} -22.8568 q^{74} -6.96239 q^{76} +1.00000i q^{77} -38.7064i q^{78} -1.96968 q^{79} -8.50659 q^{81} +13.4436i q^{82} -10.6253i q^{83} -12.7938 q^{84} -15.7685 q^{86} +8.96239i q^{87} -8.44358i q^{88} -7.22425 q^{89} -5.83146 q^{91} -16.4690i q^{92} -13.1187i q^{93} -31.6507 q^{94} +39.5755 q^{96} -0.836381i q^{97} +2.67513i q^{98} +3.15633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} - 6 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} + 4 q^{21} - 52 q^{24} - 40 q^{26} + 20 q^{29} - 20 q^{31} + 12 q^{34} + 34 q^{36} + 24 q^{39} + 10 q^{44} + 32 q^{46} - 6 q^{49}+ \cdots - 2 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 + 8 * q^6 + 2 * q^9 - 6 * q^11 - 6 * q^14 + 10 * q^16 - 12 * q^19 + 4 * q^21 - 52 * q^24 - 40 * q^26 + 20 * q^29 - 20 * q^31 + 12 * q^34 + 34 * q^36 + 24 * q^39 + 10 * q^44 + 32 * q^46 - 6 * q^49 - 40 * q^51 - 28 * q^54 + 18 * q^56 - 28 * q^59 + 20 * q^61 - 66 * q^64 - 8 * q^66 - 4 * q^69 - 48 * q^71 - 76 * q^74 - 20 * q^76 - 16 * q^79 - 10 * q^81 - 24 * q^84 - 72 * q^86 - 40 * q^89 - 4 * q^91 - 76 * q^94 + 80 * q^96 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 2.67513i	− 1.89160i	−0.324745	−	0.945802i	$-0.605279\pi$
$2$	− 2.67513i	− 1.89160i	0.324745	−	0.945802i	$-0.394721\pi$
$3$	2.48119i	1.43252i	0.697834	+	0.716259i	$0.254147\pi$
$3$	2.48119i	1.43252i	−0.697834	+	0.716259i	$0.745853\pi$
$4$	−5.15633	−2.57816
$5$	0	0
$5$	0	0
$6$	6.63752	2.70976
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	8.44358i	2.98526i
$9$	−3.15633	−1.05211
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 12.7938i	− 3.69326i
$13$	− 5.83146i	− 1.61735i	−0.588252	−	0.808677i	$-0.700184\pi$
$13$	− 5.83146i	− 1.61735i	0.588252	−	0.808677i	$-0.299816\pi$
$14$	−2.67513	−0.714959
$15$	0	0
$16$	12.2750	3.06876
$17$	5.44358i	1.32026i	0.751150	+	0.660131i	$0.229499\pi$
$17$	5.44358i	1.32026i	−0.751150	+	0.660131i	$0.770501\pi$
$18$	8.44358i	1.99017i
$19$	1.35026	0.309771	0.154886	−	0.987932i	$-0.450499\pi$
$19$	1.35026	0.309771	0.154886	+	0.987932i	$0.450499\pi$
$20$	0	0
$21$	2.48119	0.541441
$22$	2.67513i	0.570340i
$23$	3.19394i	0.665982i	0.942930	+	0.332991i	$0.108058\pi$
$23$	3.19394i	0.665982i	−0.942930	+	0.332991i	$0.891942\pi$
$24$	−20.9502	−4.27644
$25$	0	0
$26$	−15.5999	−3.05939
$27$	− 0.387873i	− 0.0746462i
$28$	5.15633i	0.974454i
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	0	0
$31$	−5.28726	−0.949620	−0.474810	−	0.880088i	$-0.657483\pi$
$31$	−5.28726	−0.949620	−0.474810	+	0.880088i	$0.657483\pi$
$32$	− 15.9502i	− 2.81962i
$33$	− 2.48119i	− 0.431920i
$34$	14.5623	2.49741
$35$	0	0
$36$	16.2750	2.71251
$37$	− 8.54420i	− 1.40466i	−0.711853	−	0.702329i	$-0.752144\pi$
$37$	− 8.54420i	− 1.40466i	0.711853	−	0.702329i	$-0.247856\pi$
$38$	− 3.61213i	− 0.585964i
$39$	14.4690	2.31689
$40$	0	0
$41$	−5.02539	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$41$	−5.02539	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$42$	− 6.63752i	− 1.02419i
$43$	− 5.89446i	− 0.898897i	−0.893306	−	0.449448i	$-0.851621\pi$
$43$	− 5.89446i	− 0.898897i	0.893306	−	0.449448i	$-0.148379\pi$
$44$	5.15633	0.777345
$45$	0	0
$46$	8.54420	1.25977
$47$	− 11.8315i	− 1.72580i	−0.505379	−	0.862898i	$-0.668647\pi$
$47$	− 11.8315i	− 1.72580i	0.505379	−	0.862898i	$-0.331353\pi$
$48$	30.4568i	4.39605i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−13.5066	−1.89130
$52$	30.0689i	4.16980i
$53$	0.231548i	0.0318056i	0.999874	+	0.0159028i	$0.00506223\pi$
$53$	0.231548i	0.0318056i	−0.999874	+	0.0159028i	$0.994938\pi$
$54$	−1.03761	−0.141201
$55$	0	0
$56$	8.44358	1.12832
$57$	3.35026i	0.443753i
$58$	− 9.66291i	− 1.26880i
$59$	−13.5999	−1.77056	−0.885279	−	0.465061i	$-0.846032\pi$
$59$	−13.5999	−1.77056	−0.885279	+	0.465061i	$0.846032\pi$
$60$	0	0
$61$	−1.41327	−0.180950	−0.0904751	−	0.995899i	$-0.528839\pi$
$61$	−1.41327	−0.180950	−0.0904751	+	0.995899i	$0.528839\pi$
$62$	14.1441i	1.79630i
$63$	3.15633i	0.397660i
$64$	−18.1187	−2.26484
$65$	0	0
$66$	−6.63752	−0.817022
$67$	− 10.8568i	− 1.32638i	−0.748453	−	0.663188i	$-0.769203\pi$
$67$	− 10.8568i	− 1.32638i	0.748453	−	0.663188i	$-0.230797\pi$
$68$	− 28.0689i	− 3.40385i
$69$	−7.92478	−0.954031
$70$	0	0
$71$	−15.5369	−1.84389	−0.921946	−	0.387319i	$-0.873401\pi$
$71$	−15.5369	−1.84389	−0.921946	+	0.387319i	$0.873401\pi$
$72$	− 26.6507i	− 3.14081i
$73$	11.3684i	1.33057i	0.746591	+	0.665283i	$0.231689\pi$
$73$	11.3684i	1.33057i	−0.746591	+	0.665283i	$0.768311\pi$
$74$	−22.8568	−2.65705
$75$	0	0
$76$	−6.96239	−0.798641
$77$	1.00000i	0.113961i
$78$	− 38.7064i	− 4.38264i
$79$	−1.96968	−0.221607	−0.110803	−	0.993842i	$-0.535342\pi$
$79$	−1.96968	−0.221607	−0.110803	+	0.993842i	$0.535342\pi$
$80$	0	0
$81$	−8.50659	−0.945176
$82$	13.4436i	1.48460i
$83$	− 10.6253i	− 1.16628i	−0.812372	−	0.583139i	$-0.801824\pi$
$83$	− 10.6253i	− 1.16628i	0.812372	−	0.583139i	$-0.198176\pi$
$84$	−12.7938	−1.39592
$85$	0	0
$86$	−15.7685	−1.70036
$87$	8.96239i	0.960869i
$88$	− 8.44358i	− 0.900089i
$89$	−7.22425	−0.765769	−0.382885	−	0.923796i	$-0.625069\pi$
$89$	−7.22425	−0.765769	−0.382885	+	0.923796i	$0.625069\pi$
$90$	0	0
$91$	−5.83146	−0.611303
$92$	− 16.4690i	− 1.71701i
$93$	− 13.1187i	− 1.36035i
$94$	−31.6507	−3.26452
$95$	0	0
$96$	39.5755	4.03915
$97$	− 0.836381i	− 0.0849216i	−0.999098	−	0.0424608i	$-0.986480\pi$
$97$	− 0.836381i	− 0.0849216i	0.999098	−	0.0424608i	$-0.0135198\pi$
$98$	2.67513i	0.270229i
$99$	3.15633	0.317223
$100$	0	0
$101$	7.41327	0.737648	0.368824	−	0.929499i	$-0.379761\pi$
$101$	7.41327	0.737648	0.368824	+	0.929499i	$0.379761\pi$
$102$	36.1319i	3.57759i
$103$	− 4.21933i	− 0.415743i	−0.978156	−	0.207871i	$-0.933346\pi$
$103$	− 4.21933i	− 0.415743i	0.978156	−	0.207871i	$-0.0666536\pi$
$104$	49.2384	4.82822
$105$	0	0
$106$	0.619421	0.0601635
$107$	− 11.5369i	− 1.11531i	−0.830071	−	0.557657i	$-0.811700\pi$
$107$	− 11.5369i	− 1.11531i	0.830071	−	0.557657i	$-0.188300\pi$
$108$	2.00000i	0.192450i
$109$	2.18664	0.209442	0.104721	−	0.994502i	$-0.466605\pi$
$109$	2.18664	0.209442	0.104721	+	0.994502i	$0.466605\pi$
$110$	0	0
$111$	21.1998	2.01220
$112$	− 12.2750i	− 1.15988i
$113$	9.35026i	0.879599i	0.898096	+	0.439799i	$0.144950\pi$
$113$	9.35026i	0.879599i	−0.898096	+	0.439799i	$0.855050\pi$
$114$	8.96239	0.839405
$115$	0	0
$116$	−18.6253	−1.72932
$117$	18.4060i	1.70163i
$118$	36.3815i	3.34919i
$119$	5.44358	0.499012
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.78067i	0.342286i
$123$	− 12.4690i	− 1.12429i
$124$	27.2628	2.44827
$125$	0	0
$126$	8.44358	0.752214
$127$	− 16.9624i	− 1.50517i	−0.658496	−	0.752584i	$-0.728807\pi$
$127$	− 16.9624i	− 1.50517i	0.658496	−	0.752584i	$-0.271193\pi$
$128$	16.5696i	1.46456i
$129$	14.6253	1.28769
$130$	0	0
$131$	−9.92478	−0.867132	−0.433566	−	0.901122i	$-0.642745\pi$
$131$	−9.92478	−0.867132	−0.433566	+	0.901122i	$0.642745\pi$
$132$	12.7938i	1.11356i
$133$	− 1.35026i	− 0.117083i
$134$	−29.0435	−2.50898
$135$	0	0
$136$	−45.9633	−3.94132
$137$	− 10.9927i	− 0.939170i	−0.882887	−	0.469585i	$-0.844404\pi$
$137$	− 10.9927i	− 0.939170i	0.882887	−	0.469585i	$-0.155596\pi$
$138$	21.1998i	1.80465i
$139$	−6.88717	−0.584162	−0.292081	−	0.956394i	$-0.594348\pi$
$139$	−6.88717	−0.584162	−0.292081	+	0.956394i	$0.594348\pi$
$140$	0	0
$141$	29.3561	2.47223
$142$	41.5633i	3.48791i
$143$	5.83146i	0.487651i
$144$	−38.7440	−3.22867
$145$	0	0
$146$	30.4119	2.51690
$147$	− 2.48119i	− 0.204645i
$148$	44.0567i	3.62144i
$149$	−22.8119	−1.86883	−0.934414	−	0.356190i	$-0.884076\pi$
$149$	−22.8119	−1.86883	−0.934414	+	0.356190i	$0.884076\pi$
$150$	0	0
$151$	−3.24472	−0.264052	−0.132026	−	0.991246i	$-0.542148\pi$
$151$	−3.24472	−0.264052	−0.132026	+	0.991246i	$0.542148\pi$
$152$	11.4010i	0.924747i
$153$	− 17.1817i	− 1.38906i
$154$	2.67513	0.215568
$155$	0	0
$156$	−74.6067	−5.97332
$157$	− 5.42548i	− 0.433001i	−0.976283	−	0.216500i	$-0.930536\pi$
$157$	− 5.42548i	− 0.433001i	0.976283	−	0.216500i	$-0.0694643\pi$
$158$	5.26916i	0.419192i
$159$	−0.574515	−0.0455620
$160$	0	0
$161$	3.19394	0.251717
$162$	22.7562i	1.78790i
$163$	− 3.38058i	− 0.264787i	−0.991197	−	0.132394i	$-0.957734\pi$
$163$	− 3.38058i	− 0.264787i	0.991197	−	0.132394i	$-0.0422663\pi$
$164$	25.9126	2.02343
$165$	0	0
$166$	−28.4241	−2.20614
$167$	11.2750i	0.872489i	0.899828	+	0.436244i	$0.143692\pi$
$167$	11.2750i	0.872489i	−0.899828	+	0.436244i	$0.856308\pi$
$168$	20.9502i	1.61634i
$169$	−21.0059	−1.61584
$170$	0	0
$171$	−4.26187	−0.325913
$172$	30.3938i	2.31750i
$173$	8.98049i	0.682774i	0.939923	+	0.341387i	$0.110897\pi$
$173$	8.98049i	0.682774i	−0.939923	+	0.341387i	$0.889103\pi$
$174$	23.9756	1.81758
$175$	0	0
$176$	−12.2750	−0.925266
$177$	− 33.7440i	− 2.53636i
$178$	19.3258i	1.44853i
$179$	26.2374	1.96108	0.980539	−	0.196326i	$-0.0629010\pi$
$179$	26.2374	1.96108	0.980539	+	0.196326i	$0.0629010\pi$
$180$	0	0
$181$	−11.1998	−0.832476	−0.416238	−	0.909256i	$-0.636652\pi$
$181$	−11.1998	−0.832476	−0.416238	+	0.909256i	$0.636652\pi$
$182$	15.5999i	1.15634i
$183$	− 3.50659i	− 0.259214i
$184$	−26.9683	−1.98813
$185$	0	0
$186$	−35.0943	−2.57324
$187$	− 5.44358i	− 0.398074i
$188$	61.0068i	4.44938i
$189$	−0.387873	−0.0282136
$190$	0	0
$191$	11.1998	0.810390	0.405195	−	0.914230i	$-0.367204\pi$
$191$	11.1998	0.810390	0.405195	+	0.914230i	$0.367204\pi$
$192$	− 44.9560i	− 3.24442i
$193$	− 0.604833i	− 0.0435368i	−0.999763	−	0.0217684i	$-0.993070\pi$
$193$	− 0.604833i	− 0.0435368i	0.999763	−	0.0217684i	$-0.00692965\pi$
$194$	−2.23743	−0.160638
$195$	0	0
$196$	5.15633	0.368309
$197$	15.3054i	1.09046i	0.838286	+	0.545231i	$0.183558\pi$
$197$	15.3054i	1.09046i	−0.838286	+	0.545231i	$0.816442\pi$
$198$	− 8.44358i	− 0.600059i
$199$	12.5623	0.890518	0.445259	−	0.895402i	$-0.353112\pi$
$199$	12.5623	0.890518	0.445259	+	0.895402i	$0.353112\pi$
$200$	0	0
$201$	26.9380	1.90006
$202$	− 19.8315i	− 1.39534i
$203$	− 3.61213i	− 0.253522i
$204$	69.6444	4.87608
$205$	0	0
$206$	−11.2873	−0.786421
$207$	− 10.0811i	− 0.700685i
$208$	− 71.5814i	− 4.96327i
$209$	−1.35026	−0.0933996
$210$	0	0
$211$	−4.43866	−0.305570	−0.152785	−	0.988259i	$-0.548824\pi$
$211$	−4.43866	−0.305570	−0.152785	+	0.988259i	$0.548824\pi$
$212$	− 1.19394i	− 0.0819999i
$213$	− 38.5501i	− 2.64141i
$214$	−30.8627	−2.10973
$215$	0	0
$216$	3.27504	0.222838
$217$	5.28726i	0.358922i
$218$	− 5.84955i	− 0.396182i
$219$	−28.2071	−1.90606
$220$	0	0
$221$	31.7440	2.13533
$222$	− 56.7123i	− 3.80628i
$223$	7.78067i	0.521032i	0.965469	+	0.260516i	$0.0838927\pi$
$223$	7.78067i	0.521032i	−0.965469	+	0.260516i	$0.916107\pi$
$224$	−15.9502	−1.06572
$225$	0	0
$226$	25.0132	1.66385
$227$	− 10.4485i	− 0.693492i	−0.937959	−	0.346746i	$-0.887287\pi$
$227$	− 10.4485i	− 0.693492i	0.937959	−	0.346746i	$-0.112713\pi$
$228$	− 17.2750i	− 1.14407i
$229$	29.4518	1.94623	0.973116	−	0.230316i	$-0.0739759\pi$
$229$	29.4518	1.94623	0.973116	+	0.230316i	$0.0739759\pi$
$230$	0	0
$231$	−2.48119	−0.163251
$232$	30.4993i	2.00238i
$233$	8.73084i	0.571976i	0.958233	+	0.285988i	$0.0923218\pi$
$233$	8.73084i	0.571976i	−0.958233	+	0.285988i	$0.907678\pi$
$234$	49.2384	3.21881
$235$	0	0
$236$	70.1255	4.56478
$237$	− 4.88717i	− 0.317456i
$238$	− 14.5623i	− 0.943933i
$239$	21.2144	1.37225	0.686123	−	0.727486i	$-0.259311\pi$
$239$	21.2144	1.37225	0.686123	+	0.727486i	$0.259311\pi$
$240$	0	0
$241$	−9.33804	−0.601516	−0.300758	−	0.953700i	$-0.597240\pi$
$241$	−9.33804	−0.601516	−0.300758	+	0.953700i	$0.597240\pi$
$242$	− 2.67513i	− 0.171964i
$243$	− 22.2701i	− 1.42863i
$244$	7.28726	0.466519
$245$	0	0
$246$	−33.3561	−2.12671
$247$	− 7.87399i	− 0.501010i
$248$	− 44.6434i	− 2.83486i
$249$	26.3634	1.67071
$250$	0	0
$251$	1.87636	0.118435	0.0592174	−	0.998245i	$-0.481139\pi$
$251$	1.87636	0.118435	0.0592174	+	0.998245i	$0.481139\pi$
$252$	− 16.2750i	− 1.02523i
$253$	− 3.19394i	− 0.200801i
$254$	−45.3766	−2.84718
$255$	0	0
$256$	8.08840	0.505525
$257$	27.1392i	1.69290i	0.532472	+	0.846448i	$0.321263\pi$
$257$	27.1392i	1.69290i	−0.532472	+	0.846448i	$0.678737\pi$
$258$	− 39.1246i	− 2.43579i
$259$	−8.54420	−0.530911
$260$	0	0
$261$	−11.4010	−0.705707
$262$	26.5501i	1.64027i
$263$	− 12.8119i	− 0.790018i	−0.918677	−	0.395009i	$-0.870741\pi$
$263$	− 12.8119i	− 0.790018i	0.918677	−	0.395009i	$-0.129259\pi$
$264$	20.9502	1.28939
$265$	0	0
$266$	−3.61213	−0.221474
$267$	− 17.9248i	− 1.09698i
$268$	55.9814i	3.41961i
$269$	6.26187	0.381793	0.190896	−	0.981610i	$-0.438861\pi$
$269$	6.26187	0.381793	0.190896	+	0.981610i	$0.438861\pi$
$270$	0	0
$271$	−5.73813	−0.348567	−0.174283	−	0.984696i	$-0.555761\pi$
$271$	−5.73813	−0.348567	−0.174283	+	0.984696i	$0.555761\pi$
$272$	66.8202i	4.05157i
$273$	− 14.4690i	− 0.875702i
$274$	−29.4069	−1.77654
$275$	0	0
$276$	40.8627	2.45965
$277$	− 8.35756i	− 0.502157i	−0.967967	−	0.251078i	$-0.919215\pi$
$277$	− 8.35756i	− 0.502157i	0.967967	−	0.251078i	$-0.0807852\pi$
$278$	18.4241i	1.10500i
$279$	16.6883	0.999103
$280$	0	0
$281$	−8.44851	−0.503996	−0.251998	−	0.967728i	$-0.581088\pi$
$281$	−8.44851	−0.503996	−0.251998	+	0.967728i	$0.581088\pi$
$282$	− 78.5315i	− 4.67648i
$283$	− 0.836381i	− 0.0497177i	−0.999691	−	0.0248588i	$-0.992086\pi$
$283$	− 0.836381i	− 0.0497177i	0.999691	−	0.0248588i	$-0.00791363\pi$
$284$	80.1133	4.75385
$285$	0	0
$286$	15.5999	0.922442
$287$	5.02539i	0.296640i
$288$	50.3439i	2.96654i
$289$	−12.6326	−0.743094
$290$	0	0
$291$	2.07522	0.121652
$292$	− 58.6190i	− 3.43042i
$293$	2.71862i	0.158824i	0.996842	+	0.0794118i	$0.0253042\pi$
$293$	2.71862i	0.158824i	−0.996842	+	0.0794118i	$0.974696\pi$
$294$	−6.63752	−0.387108
$295$	0	0
$296$	72.1436	4.19326
$297$	0.387873i	0.0225067i
$298$	61.0249i	3.53508i
$299$	18.6253	1.07713
$300$	0	0
$301$	−5.89446	−0.339751
$302$	8.68006i	0.499481i
$303$	18.3938i	1.05669i
$304$	16.5745	0.950614
$305$	0	0
$306$	−45.9633	−2.62755
$307$	− 8.36344i	− 0.477326i	−0.971102	−	0.238663i	$-0.923291\pi$
$307$	− 8.36344i	− 0.477326i	0.971102	−	0.238663i	$-0.0767092\pi$
$308$	− 5.15633i	− 0.293809i
$309$	10.4690	0.595559
$310$	0	0
$311$	4.43629	0.251559	0.125779	−	0.992058i	$-0.459857\pi$
$311$	4.43629	0.251559	0.125779	+	0.992058i	$0.459857\pi$
$312$	122.170i	6.91651i
$313$	− 29.7889i	− 1.68377i	−0.539658	−	0.841885i	$-0.681446\pi$
$313$	− 29.7889i	− 1.68377i	0.539658	−	0.841885i	$-0.318554\pi$
$314$	−14.5139	−0.819066
$315$	0	0
$316$	10.1563	0.571338
$317$	15.4010i	0.865009i	0.901632	+	0.432504i	$0.142370\pi$
$317$	15.4010i	0.865009i	−0.901632	+	0.432504i	$0.857630\pi$
$318$	1.53690i	0.0861853i
$319$	−3.61213	−0.202240
$320$	0	0
$321$	28.6253	1.59771
$322$	− 8.54420i	− 0.476150i
$323$	7.35026i	0.408980i
$324$	43.8627	2.43682
$325$	0	0
$326$	−9.04349	−0.500873
$327$	5.42548i	0.300030i
$328$	− 42.4323i	− 2.34293i
$329$	−11.8315	−0.652289
$330$	0	0
$331$	6.26187	0.344183	0.172092	−	0.985081i	$-0.444947\pi$
$331$	6.26187	0.344183	0.172092	+	0.985081i	$0.444947\pi$
$332$	54.7875i	3.00685i
$333$	26.9683i	1.47785i
$334$	30.1622	1.65040
$335$	0	0
$336$	30.4568	1.66155
$337$	− 15.8700i	− 0.864495i	−0.901755	−	0.432248i	$-0.857721\pi$
$337$	− 15.8700i	− 0.864495i	0.901755	−	0.432248i	$-0.142279\pi$
$338$	56.1935i	3.05652i
$339$	−23.1998	−1.26004
$340$	0	0
$341$	5.28726	0.286321
$342$	11.4010i	0.616498i
$343$	1.00000i	0.0539949i
$344$	49.7704	2.68344
$345$	0	0
$346$	24.0240	1.29154
$347$	6.79147i	0.364585i	0.983244	+	0.182293i	$0.0583519\pi$
$347$	6.79147i	0.364585i	−0.983244	+	0.182293i	$0.941648\pi$
$348$	− 46.2130i	− 2.47728i
$349$	−26.7489	−1.43184	−0.715919	−	0.698183i	$-0.753992\pi$
$349$	−26.7489	−1.43184	−0.715919	+	0.698183i	$0.753992\pi$
$350$	0	0
$351$	−2.26187	−0.120729
$352$	15.9502i	0.850147i
$353$	16.8627i	0.897512i	0.893654	+	0.448756i	$0.148133\pi$
$353$	16.8627i	0.897512i	−0.893654	+	0.448756i	$0.851867\pi$
$354$	−90.2697	−4.79778
$355$	0	0
$356$	37.2506	1.97428
$357$	13.5066i	0.714844i
$358$	− 70.1886i	− 3.70958i
$359$	3.79289	0.200181	0.100091	−	0.994978i	$-0.468087\pi$
$359$	3.79289	0.200181	0.100091	+	0.994978i	$0.468087\pi$
$360$	0	0
$361$	−17.1768	−0.904042
$362$	29.9610i	1.57471i
$363$	2.48119i	0.130229i
$364$	30.0689	1.57604
$365$	0	0
$366$	−9.38058	−0.490331
$367$	− 6.36977i	− 0.332500i	−0.986084	−	0.166250i	$-0.946834\pi$
$367$	− 6.36977i	− 0.332500i	0.986084	−	0.166250i	$-0.0531658\pi$
$368$	39.2057i	2.04374i
$369$	15.8618	0.825731
$370$	0	0
$371$	0.231548	0.0120214
$372$	67.6444i	3.50720i
$373$	21.3317i	1.10451i	0.833674	+	0.552257i	$0.186233\pi$
$373$	21.3317i	1.10451i	−0.833674	+	0.552257i	$0.813767\pi$
$374$	−14.5623	−0.752998
$375$	0	0
$376$	99.8999	5.15194
$377$	− 21.0640i	− 1.08485i
$378$	1.03761i	0.0533690i
$379$	−24.7875	−1.27325	−0.636624	−	0.771174i	$-0.719670\pi$
$379$	−24.7875	−1.27325	−0.636624	+	0.771174i	$0.719670\pi$
$380$	0	0
$381$	42.0870	2.15618
$382$	− 29.9610i	− 1.53294i
$383$	5.45817i	0.278900i	0.990229	+	0.139450i	$0.0445334\pi$
$383$	5.45817i	0.278900i	−0.990229	+	0.139450i	$0.955467\pi$
$384$	−41.1124	−2.09801
$385$	0	0
$386$	−1.61801	−0.0823544
$387$	18.6048i	0.945737i
$388$	4.31265i	0.218942i
$389$	13.7235	0.695811	0.347906	−	0.937530i	$-0.386893\pi$
$389$	13.7235	0.695811	0.347906	+	0.937530i	$0.386893\pi$
$390$	0	0
$391$	−17.3865	−0.879271
$392$	− 8.44358i	− 0.426465i
$393$	− 24.6253i	− 1.24218i
$394$	40.9438	2.06272
$395$	0	0
$396$	−16.2750	−0.817851
$397$	2.11142i	0.105969i	0.998595	+	0.0529846i	$0.0168734\pi$
$397$	2.11142i	0.105969i	−0.998595	+	0.0529846i	$0.983127\pi$
$398$	− 33.6058i	− 1.68451i
$399$	3.35026	0.167723
$400$	0	0
$401$	19.1490	0.956257	0.478128	−	0.878290i	$-0.341315\pi$
$401$	19.1490	0.956257	0.478128	+	0.878290i	$0.341315\pi$
$402$	− 72.0625i	− 3.59415i
$403$	30.8324i	1.53587i
$404$	−38.2252	−1.90178
$405$	0	0
$406$	−9.66291	−0.479562
$407$	8.54420i	0.423520i
$408$	− 114.044i	− 5.64602i
$409$	18.6883	0.924077	0.462039	−	0.886860i	$-0.347118\pi$
$409$	18.6883	0.924077	0.462039	+	0.886860i	$0.347118\pi$
$410$	0	0
$411$	27.2750	1.34538
$412$	21.7562i	1.07185i
$413$	13.5999i	0.669208i
$414$	−26.9683	−1.32542
$415$	0	0
$416$	−93.0127	−4.56032
$417$	− 17.0884i	− 0.836822i
$418$	3.61213i	0.176675i
$419$	0.773377	0.0377819	0.0188910	−	0.999822i	$-0.493986\pi$
$419$	0.773377	0.0377819	0.0188910	+	0.999822i	$0.493986\pi$
$420$	0	0
$421$	−10.5198	−0.512702	−0.256351	−	0.966584i	$-0.582520\pi$
$421$	−10.5198	−0.512702	−0.256351	+	0.966584i	$0.582520\pi$
$422$	11.8740i	0.578017i
$423$	37.3439i	1.81572i
$424$	−1.95509	−0.0949478
$425$	0	0
$426$	−103.127	−4.99650
$427$	1.41327i	0.0683927i
$428$	59.4880i	2.87546i
$429$	−14.4690	−0.698569
$430$	0	0
$431$	−24.7308	−1.19124	−0.595621	−	0.803265i	$-0.703094\pi$
$431$	−24.7308	−1.19124	−0.595621	+	0.803265i	$0.703094\pi$
$432$	− 4.76116i	− 0.229071i
$433$	18.5599i	0.891933i	0.895050	+	0.445967i	$0.147140\pi$
$433$	18.5599i	0.891933i	−0.895050	+	0.445967i	$0.852860\pi$
$434$	14.1441	0.678939
$435$	0	0
$436$	−11.2750	−0.539976
$437$	4.31265i	0.206302i
$438$	75.4577i	3.60551i
$439$	1.42548	0.0680347	0.0340173	−	0.999421i	$-0.489170\pi$
$439$	1.42548	0.0680347	0.0340173	+	0.999421i	$0.489170\pi$
$440$	0	0
$441$	3.15633	0.150301
$442$	− 84.9194i	− 4.03920i
$443$	− 40.1925i	− 1.90960i	−0.297242	−	0.954802i	$-0.596067\pi$
$443$	− 40.1925i	− 1.90960i	0.297242	−	0.954802i	$-0.403933\pi$
$444$	−109.313	−5.18777
$445$	0	0
$446$	20.8143	0.985586
$447$	− 56.6009i	− 2.67713i
$448$	18.1187i	0.856029i
$449$	−12.6556	−0.597256	−0.298628	−	0.954370i	$-0.596529\pi$
$449$	−12.6556	−0.597256	−0.298628	+	0.954370i	$0.596529\pi$
$450$	0	0
$451$	5.02539	0.236636
$452$	− 48.2130i	− 2.26775i
$453$	− 8.05079i	− 0.378259i
$454$	−27.9511	−1.31181
$455$	0	0
$456$	−28.2882	−1.32472
$457$	− 0.544198i	− 0.0254565i	−0.999919	−	0.0127283i	$-0.995948\pi$
$457$	− 0.544198i	− 0.0254565i	0.999919	−	0.0127283i	$-0.00405164\pi$
$458$	− 78.7875i	− 3.68150i
$459$	2.11142	0.0985526
$460$	0	0
$461$	−11.5755	−0.539123	−0.269562	−	0.962983i	$-0.586879\pi$
$461$	−11.5755	−0.539123	−0.269562	+	0.962983i	$0.586879\pi$
$462$	6.63752i	0.308805i
$463$	− 23.7948i	− 1.10584i	−0.833235	−	0.552919i	$-0.813514\pi$
$463$	− 23.7948i	− 1.10584i	0.833235	−	0.552919i	$-0.186486\pi$
$464$	44.3390	2.05839
$465$	0	0
$466$	23.3561	1.08195
$467$	− 2.66784i	− 0.123453i	−0.998093	−	0.0617264i	$-0.980339\pi$
$467$	− 2.66784i	− 0.123453i	0.998093	−	0.0617264i	$-0.0196606\pi$
$468$	− 94.9072i	− 4.38709i
$469$	−10.8568	−0.501323
$470$	0	0
$471$	13.4617	0.620282
$472$	− 114.832i	− 5.28557i
$473$	5.89446i	0.271028i
$474$	−13.0738	−0.600500
$475$	0	0
$476$	−28.0689	−1.28654
$477$	− 0.730841i	− 0.0334629i
$478$	− 56.7513i	− 2.59574i
$479$	10.7104	0.489369	0.244685	−	0.969603i	$-0.421316\pi$
$479$	10.7104	0.489369	0.244685	+	0.969603i	$0.421316\pi$
$480$	0	0
$481$	−49.8251	−2.27183
$482$	24.9805i	1.13783i
$483$	7.92478i	0.360590i
$484$	−5.15633	−0.234378
$485$	0	0
$486$	−59.5755	−2.70240
$487$	17.4314i	0.789891i	0.918705	+	0.394945i	$0.129236\pi$
$487$	17.4314i	0.789891i	−0.918705	+	0.394945i	$0.870764\pi$
$488$	− 11.9330i	− 0.540183i
$489$	8.38787	0.379313
$490$	0	0
$491$	−28.3693	−1.28029	−0.640145	−	0.768254i	$-0.721126\pi$
$491$	−28.3693	−1.28029	−0.640145	+	0.768254i	$0.721126\pi$
$492$	64.2941i	2.89860i
$493$	19.6629i	0.885573i
$494$	−21.0640	−0.947712
$495$	0	0
$496$	−64.9013	−2.91415
$497$	15.5369i	0.696925i
$498$	− 70.5256i	− 3.16033i
$499$	−27.4763	−1.23001	−0.615003	−	0.788524i	$-0.710845\pi$
$499$	−27.4763	−1.23001	−0.615003	+	0.788524i	$0.710845\pi$
$500$	0	0
$501$	−27.9756	−1.24986
$502$	− 5.01951i	− 0.224032i
$503$	− 20.2981i	− 0.905046i	−0.891753	−	0.452523i	$-0.850524\pi$
$503$	− 20.2981i	− 0.905046i	0.891753	−	0.452523i	$-0.149476\pi$
$504$	−26.6507	−1.18712
$505$	0	0
$506$	−8.54420	−0.379836
$507$	− 52.1197i	− 2.31472i
$508$	87.4636i	3.88057i
$509$	24.2619	1.07539	0.537694	−	0.843140i	$-0.319296\pi$
$509$	24.2619	1.07539	0.537694	+	0.843140i	$0.319296\pi$
$510$	0	0
$511$	11.3684	0.502907
$512$	11.5017i	0.508306i
$513$	− 0.523730i	− 0.0231233i
$514$	72.6009	3.20229
$515$	0	0
$516$	−75.4128	−3.31986
$517$	11.8315i	0.520347i
$518$	22.8568i	1.00427i
$519$	−22.2823	−0.978086
$520$	0	0
$521$	2.20123	0.0964377	0.0482188	−	0.998837i	$-0.484646\pi$
$521$	2.20123	0.0964377	0.0482188	+	0.998837i	$0.484646\pi$
$522$	30.4993i	1.33492i
$523$	22.1378i	0.968017i	0.875063	+	0.484008i	$0.160820\pi$
$523$	22.1378i	0.968017i	−0.875063	+	0.484008i	$0.839180\pi$
$524$	51.1754	2.23561
$525$	0	0
$526$	−34.2736	−1.49440
$527$	− 28.7816i	− 1.25375i
$528$	− 30.4568i	− 1.32546i
$529$	12.7988	0.556468
$530$	0	0
$531$	42.9257	1.86282
$532$	6.96239i	0.301858i
$533$	29.3054i	1.26936i
$534$	−47.9511	−2.07505
$535$	0	0
$536$	91.6707	3.95957
$537$	65.1002i	2.80928i
$538$	− 16.7513i	− 0.722200i
$539$	1.00000	0.0430730
$540$	0	0
$541$	23.0640	0.991597	0.495799	−	0.868438i	$-0.334875\pi$
$541$	23.0640	0.991597	0.495799	+	0.868438i	$0.334875\pi$
$542$	15.3503i	0.659350i
$543$	− 27.7889i	− 1.19254i
$544$	86.8261	3.72264
$545$	0	0
$546$	−38.7064	−1.65648
$547$	21.3766i	0.913998i	0.889467	+	0.456999i	$0.151076\pi$
$547$	21.3766i	0.913998i	−0.889467	+	0.456999i	$0.848924\pi$
$548$	56.6820i	2.42133i
$549$	4.46073	0.190379
$550$	0	0
$551$	4.87732	0.207781
$552$	− 66.9135i	− 2.84803i
$553$	1.96968i	0.0837594i
$554$	−22.3576	−0.949882
$555$	0	0
$556$	35.5125	1.50606
$557$	− 9.19394i	− 0.389560i	−0.980847	−	0.194780i	$-0.937601\pi$
$557$	− 9.19394i	− 0.389560i	0.980847	−	0.194780i	$-0.0623992\pi$
$558$	− 44.6434i	− 1.88991i
$559$	−34.3733	−1.45384
$560$	0	0
$561$	13.5066	0.570249
$562$	22.6009i	0.953360i
$563$	9.79877i	0.412969i	0.978450	+	0.206484i	$0.0662023\pi$
$563$	9.79877i	0.412969i	−0.978450	+	0.206484i	$0.933798\pi$
$564$	−151.370	−6.37382
$565$	0	0
$566$	−2.23743	−0.0940461
$567$	8.50659i	0.357243i
$568$	− 131.187i	− 5.50449i
$569$	33.5125	1.40492	0.702458	−	0.711725i	$-0.252086\pi$
$569$	33.5125	1.40492	0.702458	+	0.711725i	$0.252086\pi$
$570$	0	0
$571$	43.1392	1.80532	0.902659	−	0.430356i	$-0.141612\pi$
$571$	43.1392	1.80532	0.902659	+	0.430356i	$0.141612\pi$
$572$	− 30.0689i	− 1.25724i
$573$	27.7889i	1.16090i
$574$	13.4436	0.561124
$575$	0	0
$576$	57.1886	2.38286
$577$	− 14.8510i	− 0.618254i	−0.951021	−	0.309127i	$-0.899963\pi$
$577$	− 14.8510i	− 0.618254i	0.951021	−	0.309127i	$-0.100037\pi$
$578$	33.7938i	1.40564i
$579$	1.50071	0.0623673
$580$	0	0
$581$	−10.6253	−0.440812
$582$	− 5.55149i	− 0.230117i
$583$	− 0.231548i	− 0.00958974i
$584$	−95.9897	−3.97208
$585$	0	0
$586$	7.27267	0.300431
$587$	14.7938i	0.610607i	0.952255	+	0.305304i	$0.0987580\pi$
$587$	14.7938i	0.610607i	−0.952255	+	0.305304i	$0.901242\pi$
$588$	12.7938i	0.527609i
$589$	−7.13918	−0.294165
$590$	0	0
$591$	−37.9756	−1.56211
$592$	− 104.880i	− 4.31056i
$593$	27.4191i	1.12597i	0.826467	+	0.562985i	$0.190347\pi$
$593$	27.4191i	1.12597i	−0.826467	+	0.562985i	$0.809653\pi$
$594$	1.03761	0.0425737
$595$	0	0
$596$	117.626	4.81814
$597$	31.1695i	1.27568i
$598$	− 49.8251i	− 2.03750i
$599$	−11.3258	−0.462761	−0.231380	−	0.972863i	$-0.574324\pi$
$599$	−11.3258	−0.462761	−0.231380	+	0.972863i	$0.574324\pi$
$600$	0	0
$601$	15.5393	0.633860	0.316930	−	0.948449i	$-0.397348\pi$
$601$	15.5393	0.633860	0.316930	+	0.948449i	$0.397348\pi$
$602$	15.7685i	0.642674i
$603$	34.2677i	1.39549i
$604$	16.7308	0.680768
$605$	0	0
$606$	49.2057	1.99884
$607$	17.7235i	0.719377i	0.933073	+	0.359688i	$0.117117\pi$
$607$	17.7235i	0.719377i	−0.933073	+	0.359688i	$0.882883\pi$
$608$	− 21.5369i	− 0.873437i
$609$	8.96239	0.363174
$610$	0	0
$611$	−68.9946	−2.79122
$612$	88.5945i	3.58122i
$613$	− 22.2941i	− 0.900450i	−0.892915	−	0.450225i	$-0.851344\pi$
$613$	− 22.2941i	− 0.900450i	0.892915	−	0.450225i	$-0.148656\pi$
$614$	−22.3733	−0.902912
$615$	0	0
$616$	−8.44358	−0.340202
$617$	− 30.9438i	− 1.24575i	−0.782321	−	0.622876i	$-0.785964\pi$
$617$	− 30.9438i	− 1.24575i	0.782321	−	0.622876i	$-0.214036\pi$
$618$	− 28.0059i	− 1.12656i
$619$	−32.4119	−1.30274	−0.651371	−	0.758759i	$-0.725806\pi$
$619$	−32.4119	−1.30274	−0.651371	+	0.758759i	$0.725806\pi$
$620$	0	0
$621$	1.23884	0.0497130
$622$	− 11.8677i	− 0.475850i
$623$	7.22425i	0.289434i
$624$	177.607	7.10998
$625$	0	0
$626$	−79.6893	−3.18502
$627$	− 3.35026i	− 0.133797i
$628$	27.9756i	1.11635i
$629$	46.5111	1.85452
$630$	0	0
$631$	−27.3258	−1.08782	−0.543912	−	0.839142i	$-0.683057\pi$
$631$	−27.3258	−1.08782	−0.543912	+	0.839142i	$0.683057\pi$
$632$	− 16.6312i	− 0.661553i
$633$	− 11.0132i	− 0.437734i
$634$	41.1998	1.63625
$635$	0	0
$636$	2.96239	0.117466
$637$	5.83146i	0.231051i
$638$	9.66291i	0.382558i
$639$	49.0395	1.93997
$640$	0	0
$641$	19.4460	0.768069	0.384034	−	0.923319i	$-0.374534\pi$
$641$	19.4460	0.768069	0.384034	+	0.923319i	$0.374534\pi$
$642$	− 76.5764i	− 3.02223i
$643$	− 5.29314i	− 0.208741i	−0.994538	−	0.104370i	$-0.966717\pi$
$643$	− 5.29314i	− 0.208741i	0.994538	−	0.104370i	$-0.0332828\pi$
$644$	−16.4690	−0.648969
$645$	0	0
$646$	19.6629	0.773627
$647$	35.0966i	1.37979i	0.723909	+	0.689896i	$0.242344\pi$
$647$	35.0966i	1.37979i	−0.723909	+	0.689896i	$0.757656\pi$
$648$	− 71.8261i	− 2.82159i
$649$	13.5999	0.533843
$650$	0	0
$651$	−13.1187	−0.514163
$652$	17.4314i	0.682665i
$653$	− 27.7988i	− 1.08785i	−0.839134	−	0.543925i	$-0.816938\pi$
$653$	− 27.7988i	− 1.08785i	0.839134	−	0.543925i	$-0.183062\pi$
$654$	14.5139	0.567538
$655$	0	0
$656$	−61.6869	−2.40847
$657$	− 35.8822i	− 1.39990i
$658$	31.6507i	1.23387i
$659$	−19.6180	−0.764209	−0.382105	−	0.924119i	$-0.624801\pi$
$659$	−19.6180	−0.764209	−0.382105	+	0.924119i	$0.624801\pi$
$660$	0	0
$661$	21.5633	0.838713	0.419357	−	0.907822i	$-0.362256\pi$
$661$	21.5633	0.838713	0.419357	+	0.907822i	$0.362256\pi$
$662$	− 16.7513i	− 0.651058i
$663$	78.7631i	3.05890i
$664$	89.7156	3.48164
$665$	0	0
$666$	72.1436	2.79551
$667$	11.5369i	0.446711i
$668$	− 58.1378i	− 2.24942i
$669$	−19.3054	−0.746388
$670$	0	0
$671$	1.41327	0.0545585
$672$	− 39.5755i	− 1.52666i
$673$	21.0679i	0.812109i	0.913849	+	0.406054i	$0.133096\pi$
$673$	21.0679i	0.812109i	−0.913849	+	0.406054i	$0.866904\pi$
$674$	−42.4544	−1.63528
$675$	0	0
$676$	108.313	4.16589
$677$	34.5174i	1.32661i	0.748349	+	0.663306i	$0.230847\pi$
$677$	34.5174i	1.32661i	−0.748349	+	0.663306i	$0.769153\pi$
$678$	62.0625i	2.38350i
$679$	−0.836381	−0.0320973
$680$	0	0
$681$	25.9248	0.993440
$682$	− 14.1441i	− 0.541606i
$683$	− 33.7802i	− 1.29256i	−0.763099	−	0.646282i	$-0.776323\pi$
$683$	− 33.7802i	− 1.29256i	0.763099	−	0.646282i	$-0.223677\pi$
$684$	21.9756	0.840257
$685$	0	0
$686$	2.67513	0.102137
$687$	73.0757i	2.78801i
$688$	− 72.3547i	− 2.75850i
$689$	1.35026	0.0514409
$690$	0	0
$691$	−13.8618	−0.527327	−0.263663	−	0.964615i	$-0.584931\pi$
$691$	−13.8618	−0.527327	−0.263663	+	0.964615i	$0.584931\pi$
$692$	− 46.3063i	− 1.76030i
$693$	− 3.15633i	− 0.119899i
$694$	18.1681	0.689651
$695$	0	0
$696$	−75.6747	−2.86844
$697$	− 27.3561i	− 1.03619i
$698$	71.5569i	2.70847i
$699$	−21.6629	−0.819367
$700$	0	0
$701$	40.5256	1.53063	0.765316	−	0.643655i	$-0.222583\pi$
$701$	40.5256	1.53063	0.765316	+	0.643655i	$0.222583\pi$
$702$	6.05079i	0.228372i
$703$	− 11.5369i	− 0.435123i
$704$	18.1187	0.682875
$705$	0	0
$706$	45.1100	1.69774
$707$	− 7.41327i	− 0.278805i
$708$	173.995i	6.53914i
$709$	0.850969	0.0319588	0.0159794	−	0.999872i	$-0.494913\pi$
$709$	0.850969	0.0319588	0.0159794	+	0.999872i	$0.494913\pi$
$710$	0	0
$711$	6.21696	0.233154
$712$	− 60.9986i	− 2.28602i
$713$	− 16.8872i	− 0.632429i
$714$	36.1319	1.35220
$715$	0	0
$716$	−135.289	−5.05598
$717$	52.6371i	1.96577i
$718$	− 10.1465i	− 0.378663i
$719$	−22.5769	−0.841976	−0.420988	−	0.907066i	$-0.638317\pi$
$719$	−22.5769	−0.841976	−0.420988	+	0.907066i	$0.638317\pi$
$720$	0	0
$721$	−4.21933	−0.157136
$722$	45.9502i	1.71009i
$723$	− 23.1695i	− 0.861683i
$724$	57.7499	2.14626
$725$	0	0
$726$	6.63752	0.246341
$727$	12.5174i	0.464244i	0.972687	+	0.232122i	$0.0745669\pi$
$727$	12.5174i	0.464244i	−0.972687	+	0.232122i	$0.925433\pi$
$728$	− 49.2384i	− 1.82490i
$729$	29.7367	1.10136
$730$	0	0
$731$	32.0870	1.18678
$732$	18.0811i	0.668297i
$733$	− 16.6678i	− 0.615641i	−0.951445	−	0.307820i	$-0.900400\pi$
$733$	− 16.6678i	− 0.615641i	0.951445	−	0.307820i	$-0.0995996\pi$
$734$	−17.0400	−0.628957
$735$	0	0
$736$	50.9438	1.87781
$737$	10.8568i	0.399917i
$738$	− 42.4323i	− 1.56196i
$739$	−42.7005	−1.57076	−0.785382	−	0.619011i	$-0.787533\pi$
$739$	−42.7005	−1.57076	−0.785382	+	0.619011i	$0.787533\pi$
$740$	0	0
$741$	19.5369	0.717706
$742$	− 0.619421i	− 0.0227397i
$743$	19.6873i	0.722259i	0.932516	+	0.361129i	$0.117609\pi$
$743$	19.6873i	0.722259i	−0.932516	+	0.361129i	$0.882391\pi$
$744$	110.769	4.06099
$745$	0	0
$746$	57.0651	2.08930
$747$	33.5369i	1.22705i
$748$	28.0689i	1.02630i
$749$	−11.5369	−0.421549
$750$	0	0
$751$	−5.85940	−0.213813	−0.106906	−	0.994269i	$-0.534094\pi$
$751$	−5.85940	−0.213813	−0.106906	+	0.994269i	$0.534094\pi$
$752$	− 145.232i	− 5.29605i
$753$	4.65562i	0.169660i
$754$	−56.3488	−2.05210
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 40.5863i	− 1.47513i	−0.675274	−	0.737567i	$-0.735975\pi$
$757$	− 40.5863i	− 1.47513i	0.675274	−	0.737567i	$-0.264025\pi$
$758$	66.3098i	2.40848i
$759$	7.92478	0.287651
$760$	0	0
$761$	21.8472	0.791960	0.395980	−	0.918259i	$-0.370405\pi$
$761$	21.8472	0.791960	0.395980	+	0.918259i	$0.370405\pi$
$762$	− 112.588i	− 4.07864i
$763$	− 2.18664i	− 0.0791618i
$764$	−57.7499	−2.08932
$765$	0	0
$766$	14.6013	0.527567
$767$	79.3073i	2.86362i
$768$	20.0689i	0.724173i
$769$	−45.2892	−1.63317	−0.816585	−	0.577226i	$-0.804135\pi$
$769$	−45.2892	−1.63317	−0.816585	+	0.577226i	$0.804135\pi$
$770$	0	0
$771$	−67.3376	−2.42510
$772$	3.11871i	0.112245i
$773$	33.8153i	1.21625i	0.793841	+	0.608125i	$0.208078\pi$
$773$	33.8153i	1.21625i	−0.793841	+	0.608125i	$0.791922\pi$
$774$	49.7704	1.78896
$775$	0	0
$776$	7.06205	0.253513
$777$	− 21.1998i	− 0.760539i
$778$	− 36.7123i	− 1.31620i
$779$	−6.78560	−0.243119
$780$	0	0
$781$	15.5369	0.555954
$782$	46.5111i	1.66323i
$783$	− 1.40105i	− 0.0500693i
$784$	−12.2750	−0.438394
$785$	0	0
$786$	−65.8759	−2.34972
$787$	1.27504i	0.0454502i	0.999742	+	0.0227251i	$0.00723425\pi$
$787$	1.27504i	0.0454502i	−0.999742	+	0.0227251i	$0.992766\pi$
$788$	− 78.9194i	− 2.81139i
$789$	31.7889	1.13172
$790$	0	0
$791$	9.35026	0.332457
$792$	26.6507i	0.946991i
$793$	8.24140i	0.292661i
$794$	5.64832	0.200452
$795$	0	0
$796$	−64.7753	−2.29590
$797$	− 42.5256i	− 1.50634i	−0.657829	−	0.753168i	$-0.728525\pi$
$797$	− 42.5256i	− 1.50634i	0.657829	−	0.753168i	$-0.271475\pi$
$798$	− 8.96239i	− 0.317265i
$799$	64.4055	2.27850
$800$	0	0
$801$	22.8021	0.805672
$802$	− 51.2262i	− 1.80886i
$803$	− 11.3684i	− 0.401181i
$804$	−138.901	−4.89865
$805$	0	0
$806$	82.4807	2.90526
$807$	15.5369i	0.546925i
$808$	62.5945i	2.20207i
$809$	14.7151	0.517356	0.258678	−	0.965964i	$-0.416713\pi$
$809$	14.7151	0.517356	0.258678	+	0.965964i	$0.416713\pi$
$810$	0	0
$811$	51.7743	1.81804	0.909021	−	0.416750i	$-0.136831\pi$
$811$	51.7743	1.81804	0.909021	+	0.416750i	$0.136831\pi$
$812$	18.6253i	0.653620i
$813$	− 14.2374i	− 0.499328i
$814$	22.8568	0.801132
$815$	0	0
$816$	−165.794	−5.80395
$817$	− 7.95906i	− 0.278452i
$818$	− 49.9937i	− 1.74799i
$819$	18.4060	0.643157
$820$	0	0
$821$	2.64974	0.0924765	0.0462383	−	0.998930i	$-0.485277\pi$
$821$	2.64974	0.0924765	0.0462383	+	0.998930i	$0.485277\pi$
$822$	− 72.9643i	− 2.54492i
$823$	− 5.76845i	− 0.201076i	−0.994933	−	0.100538i	$-0.967944\pi$
$823$	− 5.76845i	− 0.201076i	0.994933	−	0.100538i	$-0.0320563\pi$
$824$	35.6263	1.24110
$825$	0	0
$826$	36.3815	1.26588
$827$	− 13.4920i	− 0.469163i	−0.972096	−	0.234581i	$-0.924628\pi$
$827$	− 13.4920i	− 0.469163i	0.972096	−	0.234581i	$-0.0753719\pi$
$828$	51.9814i	1.80648i
$829$	4.70052	0.163256	0.0816280	−	0.996663i	$-0.473988\pi$
$829$	4.70052	0.163256	0.0816280	+	0.996663i	$0.473988\pi$
$830$	0	0
$831$	20.7367	0.719349
$832$	105.658i	3.66305i
$833$	− 5.44358i	− 0.188609i
$834$	−45.7137	−1.58294
$835$	0	0
$836$	6.96239	0.240799
$837$	2.05079i	0.0708855i
$838$	− 2.06888i	− 0.0714684i
$839$	38.8045	1.33968	0.669839	−	0.742506i	$-0.266363\pi$
$839$	38.8045	1.33968	0.669839	+	0.742506i	$0.266363\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	28.1417i	0.969828i
$843$	− 20.9624i	− 0.721983i
$844$	22.8872	0.787809
$845$	0	0
$846$	99.8999	3.43463
$847$	− 1.00000i	− 0.0343604i
$848$	2.84226i	0.0976036i
$849$	2.07522	0.0712215
$850$	0	0
$851$	27.2896	0.935476
$852$	198.777i	6.80998i
$853$	− 20.6824i	− 0.708153i	−0.935217	−	0.354076i	$-0.884795\pi$
$853$	− 20.6824i	− 0.708153i	0.935217	−	0.354076i	$-0.115205\pi$
$854$	3.78067	0.129372
$855$	0	0
$856$	97.4128	3.32950
$857$	26.3453i	0.899940i	0.893044	+	0.449970i	$0.148565\pi$
$857$	26.3453i	0.899940i	−0.893044	+	0.449970i	$0.851435\pi$
$858$	38.7064i	1.32141i
$859$	8.51151	0.290409	0.145205	−	0.989402i	$-0.453616\pi$
$859$	8.51151	0.290409	0.145205	+	0.989402i	$0.453616\pi$
$860$	0	0
$861$	−12.4690	−0.424942
$862$	66.1582i	2.25336i
$863$	− 7.56722i	− 0.257591i	−0.991671	−	0.128796i	$-0.958889\pi$
$863$	− 7.56722i	− 0.257591i	0.991671	−	0.128796i	$-0.0411111\pi$
$864$	−6.18664	−0.210474
$865$	0	0
$866$	49.6502	1.68718
$867$	− 31.3439i	− 1.06450i
$868$	− 27.2628i	− 0.925360i
$869$	1.96968	0.0668169
$870$	0	0
$871$	−63.3112	−2.14522
$872$	18.4631i	0.625239i
$873$	2.63989i	0.0893467i
$874$	11.5369	0.390242
$875$	0	0
$876$	145.445	4.91413
$877$	17.2955i	0.584028i	0.956414	+	0.292014i	$0.0943254\pi$
$877$	17.2955i	0.584028i	−0.956414	+	0.292014i	$0.905675\pi$
$878$	− 3.81336i	− 0.128695i
$879$	−6.74543	−0.227518
$880$	0	0
$881$	20.4504	0.688992	0.344496	−	0.938788i	$-0.388050\pi$
$881$	20.4504	0.688992	0.344496	+	0.938788i	$0.388050\pi$
$882$	− 8.44358i	− 0.284310i
$883$	49.6589i	1.67116i	0.549371	+	0.835578i	$0.314867\pi$
$883$	49.6589i	1.67116i	−0.549371	+	0.835578i	$0.685133\pi$
$884$	−163.682	−5.50524
$885$	0	0
$886$	−107.520	−3.61221
$887$	− 47.1100i	− 1.58180i	−0.611946	−	0.790900i	$-0.709613\pi$
$887$	− 47.1100i	− 1.58180i	0.611946	−	0.790900i	$-0.290387\pi$
$888$	179.002i	6.00693i
$889$	−16.9624	−0.568900
$890$	0	0
$891$	8.50659	0.284981
$892$	− 40.1197i	− 1.34331i
$893$	− 15.9756i	− 0.534602i
$894$	−151.415	−5.06407
$895$	0	0
$896$	16.5696	0.553551
$897$	46.2130i	1.54301i
$898$	33.8554i	1.12977i
$899$	−19.0982	−0.636962
$900$	0	0
$901$	−1.26045	−0.0419917
$902$	− 13.4436i	− 0.447622i
$903$	− 14.6253i	− 0.486700i
$904$	−78.9497	−2.62583
$905$	0	0
$906$	−21.5369	−0.715516
$907$	14.4591i	0.480107i	0.970760	+	0.240054i	$0.0771651\pi$
$907$	14.4591i	0.480107i	−0.970760	+	0.240054i	$0.922835\pi$
$908$	53.8759i	1.78793i
$909$	−23.3987	−0.776085
$910$	0	0
$911$	−31.5369	−1.04486	−0.522432	−	0.852681i	$-0.674975\pi$
$911$	−31.5369	−1.04486	−0.522432	+	0.852681i	$0.674975\pi$
$912$	41.1246i	1.36177i
$913$	10.6253i	0.351646i
$914$	−1.45580	−0.0481536
$915$	0	0
$916$	−151.863	−5.01770
$917$	9.92478i	0.327745i
$918$	− 5.64832i	− 0.186422i
$919$	−5.26328	−0.173620	−0.0868098	−	0.996225i	$-0.527667\pi$
$919$	−5.26328	−0.173620	−0.0868098	+	0.996225i	$0.527667\pi$
$920$	0	0
$921$	20.7513	0.683779
$922$	30.9659i	1.01981i
$923$	90.6028i	2.98223i
$924$	12.7938	0.420887
$925$	0	0
$926$	−63.6542	−2.09181
$927$	13.3176i	0.437407i
$928$	− 57.6140i	− 1.89127i
$929$	−26.0508	−0.854699	−0.427349	−	0.904087i	$-0.640553\pi$
$929$	−26.0508	−0.854699	−0.427349	+	0.904087i	$0.640553\pi$
$930$	0	0
$931$	−1.35026	−0.0442530
$932$	− 45.0191i	− 1.47465i
$933$	11.0073i	0.360363i
$934$	−7.13681	−0.233524
$935$	0	0
$936$	−155.412	−5.07981
$937$	− 29.3439i	− 0.958624i	−0.877645	−	0.479312i	$-0.840886\pi$
$937$	− 29.3439i	− 0.958624i	0.877645	−	0.479312i	$-0.159114\pi$
$938$	29.0435i	0.948304i
$939$	73.9121	2.41203
$940$	0	0
$941$	−28.6375	−0.933556	−0.466778	−	0.884374i	$-0.654585\pi$
$941$	−28.6375	−0.933556	−0.466778	+	0.884374i	$0.654585\pi$
$942$	− 36.0118i	− 1.17333i
$943$	− 16.0508i	− 0.522685i
$944$	−166.939	−5.43341
$945$	0	0
$946$	15.7685	0.512677
$947$	− 52.8178i	− 1.71635i	−0.513358	−	0.858174i	$-0.671599\pi$
$947$	− 52.8178i	− 1.71635i	0.513358	−	0.858174i	$-0.328401\pi$
$948$	25.1998i	0.818452i
$949$	66.2941	2.15200
$950$	0	0
$951$	−38.2130	−1.23914
$952$	45.9633i	1.48968i
$953$	− 37.1939i	− 1.20483i	−0.798183	−	0.602415i	$-0.794205\pi$
$953$	− 37.1939i	− 1.20483i	0.798183	−	0.602415i	$-0.205795\pi$
$954$	−1.95509	−0.0632985
$955$	0	0
$956$	−109.388	−3.53787
$957$	− 8.96239i	− 0.289713i
$958$	− 28.6516i	− 0.925693i
$959$	−10.9927	−0.354973
$960$	0	0
$961$	−3.04491	−0.0982228
$962$	133.289i	4.29740i
$963$	36.4142i	1.17343i
$964$	48.1500	1.55081
$965$	0	0
$966$	21.1998	0.682093
$967$	4.07125i	0.130923i	0.997855	+	0.0654613i	$0.0208519\pi$
$967$	4.07125i	0.130923i	−0.997855	+	0.0654613i	$0.979148\pi$
$968$	8.44358i	0.271387i
$969$	−18.2374	−0.585871
$970$	0	0
$971$	0.773377	0.0248188	0.0124094	−	0.999923i	$-0.496050\pi$
$971$	0.773377	0.0248188	0.0124094	+	0.999923i	$0.496050\pi$
$972$	114.832i	3.68324i
$973$	6.88717i	0.220792i
$974$	46.6312	1.49416
$975$	0	0
$976$	−17.3479	−0.555292
$977$	− 37.8740i	− 1.21170i	−0.795580	−	0.605848i	$-0.792834\pi$
$977$	− 37.8740i	− 1.21170i	0.795580	−	0.605848i	$-0.207166\pi$
$978$	− 22.4387i	− 0.717509i
$979$	7.22425	0.230888
$980$	0	0
$981$	−6.90175	−0.220356
$982$	75.8916i	2.42180i
$983$	15.5794i	0.496907i	0.968644	+	0.248453i	$0.0799223\pi$
$983$	15.5794i	0.496907i	−0.968644	+	0.248453i	$0.920078\pi$
$984$	105.283	3.35629
$985$	0	0
$986$	52.6009	1.67515
$987$	− 29.3561i	− 0.934416i
$988$	40.6009i	1.29169i
$989$	18.8265	0.598649
$990$	0	0
$991$	27.0982	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$991$	27.0982	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$992$	84.3327i	2.67756i
$993$	15.5369i	0.493049i
$994$	41.5633	1.31831
$995$	0	0
$996$	−135.938	−4.30737
$997$	50.4060i	1.59637i	0.602410	+	0.798187i	$0.294207\pi$
$997$	50.4060i	1.59637i	−0.602410	+	0.798187i	$0.705793\pi$
$998$	73.5026i	2.32668i
$999$	−3.31406	−0.104852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.n.1849.1		6
5.2	odd	4		1925.2.a.v.1.3		3
5.3	odd	4		385.2.a.f.1.1	✓	3
5.4	even	2	inner	1925.2.b.n.1849.6		6
15.8	even	4		3465.2.a.bh.1.3		3
20.3	even	4		6160.2.a.bn.1.3		3
35.13	even	4		2695.2.a.g.1.1		3
55.43	even	4		4235.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.f.1.1	✓	3	5.3	odd	4
1925.2.a.v.1.3		3	5.2	odd	4
1925.2.b.n.1849.1		6	1.1	even	1	trivial
1925.2.b.n.1849.6		6	5.4	even	2	inner
2695.2.a.g.1.1		3	35.13	even	4
3465.2.a.bh.1.3		3	15.8	even	4
4235.2.a.q.1.3		3	55.43	even	4
6160.2.a.bn.1.3		3	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.67513i q^{2} +2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} -1.00000i q^{7} +8.44358i q^{8} -3.15633 q^{9} -1.00000 q^{11} -12.7938i q^{12} -5.83146i q^{13} -2.67513 q^{14} +12.2750 q^{16} +5.44358i q^{17} +8.44358i q^{18} +1.35026 q^{19} +2.48119 q^{21} +2.67513i q^{22} +3.19394i q^{23} -20.9502 q^{24} -15.5999 q^{26} -0.387873i q^{27} +5.15633i q^{28} +3.61213 q^{29} -5.28726 q^{31} -15.9502i q^{32} -2.48119i q^{33} +14.5623 q^{34} +16.2750 q^{36} -8.54420i q^{37} -3.61213i q^{38} +14.4690 q^{39} -5.02539 q^{41} -6.63752i q^{42} -5.89446i q^{43} +5.15633 q^{44} +8.54420 q^{46} -11.8315i q^{47} +30.4568i q^{48} -1.00000 q^{49} -13.5066 q^{51} +30.0689i q^{52} +0.231548i q^{53} -1.03761 q^{54} +8.44358 q^{56} +3.35026i q^{57} -9.66291i q^{58} -13.5999 q^{59} -1.41327 q^{61} +14.1441i q^{62} +3.15633i q^{63} -18.1187 q^{64} -6.63752 q^{66} -10.8568i q^{67} -28.0689i q^{68} -7.92478 q^{69} -15.5369 q^{71} -26.6507i q^{72} +11.3684i q^{73} -22.8568 q^{74} -6.96239 q^{76} +1.00000i q^{77} -38.7064i q^{78} -1.96968 q^{79} -8.50659 q^{81} +13.4436i q^{82} -10.6253i q^{83} -12.7938 q^{84} -15.7685 q^{86} +8.96239i q^{87} -8.44358i q^{88} -7.22425 q^{89} -5.83146 q^{91} -16.4690i q^{92} -13.1187i q^{93} -31.6507 q^{94} +39.5755 q^{96} -0.836381i q^{97} +2.67513i q^{98} +3.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} - 6 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} + 4 q^{21} - 52 q^{24} - 40 q^{26} + 20 q^{29} - 20 q^{31} + 12 q^{34} + 34 q^{36} + 24 q^{39} + 10 q^{44} + 32 q^{46} - 6 q^{49}+ \cdots - 2 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 + 8 * q^6 + 2 * q^9 - 6 * q^11 - 6 * q^14 + 10 * q^16 - 12 * q^19 + 4 * q^21 - 52 * q^24 - 40 * q^26 + 20 * q^29 - 20 * q^31 + 12 * q^34 + 34 * q^36 + 24 * q^39 + 10 * q^44 + 32 * q^46 - 6 * q^49 - 40 * q^51 - 28 * q^54 + 18 * q^56 - 28 * q^59 + 20 * q^61 - 66 * q^64 - 8 * q^66 - 4 * q^69 - 48 * q^71 - 76 * q^74 - 20 * q^76 - 16 * q^79 - 10 * q^81 - 24 * q^84 - 72 * q^86 - 40 * q^89 - 4 * q^91 - 76 * q^94 + 80 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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