LMFDB - Embedded newform 1900.3.e.f.1101.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 5 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.7
Root			$0.630185i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.f.1101.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.630185i q^{3} +3.98530 q^{7} +8.60287 q^{9} +O(q^{10})$	$q+0.630185i q^{3} +3.98530 q^{7} +8.60287 q^{9} -5.54431 q^{11} +23.1242i q^{13} -16.5013 q^{17} +(11.6484 + 15.0105i) q^{19} +2.51147i q^{21} -6.86785 q^{23} +11.0931i q^{27} -37.0068i q^{29} -11.1127i q^{31} -3.49394i q^{33} +66.6951i q^{37} -14.5725 q^{39} +14.8753i q^{41} -34.7815 q^{43} -63.0151 q^{47} -33.1174 q^{49} -10.3989i q^{51} -36.8706i q^{53} +(-9.45937 + 7.34066i) q^{57} +19.0546i q^{59} +34.3686 q^{61} +34.2850 q^{63} +27.2417i q^{67} -4.32801i q^{69} -90.8927i q^{71} +19.7746 q^{73} -22.0957 q^{77} +111.986i q^{79} +70.4351 q^{81} +129.958 q^{83} +23.3211 q^{87} +66.5710i q^{89} +92.1568i q^{91} +7.00305 q^{93} +19.0605i q^{97} -47.6970 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 16 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 16 * q^9	$ 12 q + 12 q^{7} - 16 q^{9} - 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 124 q^{39} + 176 q^{43} + 72 q^{47} - 24 q^{49} - 140 q^{57} + 152 q^{61} - 48 q^{63} + 148 q^{73} - 376 q^{77} - 468 q^{81} + 208 q^{83} + 84 q^{87} + 184 q^{93} + 392 q^{99}+O(q^{100}) $ 12 * q + 12 * q^7 - 16 * q^9 - 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 124 * q^39 + 176 * q^43 + 72 * q^47 - 24 * q^49 - 140 * q^57 + 152 * q^61 - 48 * q^63 + 148 * q^73 - 376 * q^77 - 468 * q^81 + 208 * q^83 + 84 * q^87 + 184 * q^93 + 392 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			0.630185i			0.210062i	0.994469	+	0.105031i	$0.0334941\pi$
$3$			0.630185i			0.210062i	−0.994469	+	0.105031i	$0.966506\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.98530			0.569328			0.284664	−	0.958627i	$-0.408118\pi$
$7$	3.98530			0.569328			0.284664	+	0.958627i	$0.408118\pi$
$8$		0			0
$9$	8.60287			0.955874
$10$		0			0
$11$	−5.54431			−0.504028			−0.252014	−	0.967724i	$-0.581093\pi$
$11$	−5.54431			−0.504028			−0.252014	+	0.967724i	$0.581093\pi$
$12$		0			0
$13$			23.1242i			1.77878i	0.457145	+	0.889392i	$0.348872\pi$
$13$			23.1242i			1.77878i	−0.457145	+	0.889392i	$0.651128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−16.5013			−0.970664			−0.485332	−	0.874330i	$-0.661301\pi$
$17$	−16.5013			−0.970664			−0.485332	+	0.874330i	$0.661301\pi$
$18$		0			0
$19$	11.6484	+	15.0105i	0.613075	+	0.790024i
$19$	11.6484	+	15.0105i	0.613075	+	0.790024i
$20$		0			0
$21$			2.51147i			0.119594i
$22$		0			0
$23$	−6.86785			−0.298602			−0.149301	−	0.988792i	$-0.547702\pi$
$23$	−6.86785			−0.298602			−0.149301	+	0.988792i	$0.547702\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			11.0931i			0.410854i
$28$		0			0
$29$		−	37.0068i		−	1.27610i	−0.769996	−	0.638049i	$-0.779742\pi$
$29$		−	37.0068i		−	1.27610i	0.769996	−	0.638049i	$-0.220258\pi$
$30$		0			0
$31$		−	11.1127i		−	0.358474i	−0.983806	−	0.179237i	$-0.942637\pi$
$31$		−	11.1127i		−	0.358474i	0.983806	−	0.179237i	$-0.0573629\pi$
$32$		0			0
$33$		−	3.49394i		−	0.105877i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			66.6951i			1.80257i	0.433227	+	0.901285i	$0.357375\pi$
$37$			66.6951i			1.80257i	−0.433227	+	0.901285i	$0.642625\pi$
$38$		0			0
$39$	−14.5725			−0.373654
$40$		0			0
$41$			14.8753i			0.362811i	0.983408	+	0.181406i	$0.0580647\pi$
$41$			14.8753i			0.362811i	−0.983408	+	0.181406i	$0.941935\pi$
$42$		0			0
$43$	−34.7815			−0.808872			−0.404436	−	0.914566i	$-0.632532\pi$
$43$	−34.7815			−0.808872			−0.404436	+	0.914566i	$0.632532\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−63.0151			−1.34075			−0.670374	−	0.742024i	$-0.733866\pi$
$47$	−63.0151			−1.34075			−0.670374	+	0.742024i	$0.733866\pi$
$48$		0			0
$49$	−33.1174			−0.675865
$50$		0			0
$51$		−	10.3989i		−	0.203899i
$52$		0			0
$53$		−	36.8706i		−	0.695672i	−0.937555	−	0.347836i	$-0.886917\pi$
$53$		−	36.8706i		−	0.695672i	0.937555	−	0.347836i	$-0.113083\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−9.45937	+	7.34066i	−0.165954	+	0.128784i
$58$		0			0
$59$			19.0546i			0.322960i	0.986876	+	0.161480i	$0.0516267\pi$
$59$			19.0546i			0.322960i	−0.986876	+	0.161480i	$0.948373\pi$
$60$		0			0
$61$	34.3686			0.563420			0.281710	−	0.959500i	$-0.409098\pi$
$61$	34.3686			0.563420			0.281710	+	0.959500i	$0.409098\pi$
$62$		0			0
$63$	34.2850			0.544206
$64$		0			0
$65$		0			0
$66$		0			0
$67$			27.2417i			0.406593i	0.979117	+	0.203297i	$0.0651655\pi$
$67$			27.2417i			0.406593i	−0.979117	+	0.203297i	$0.934834\pi$
$68$		0			0
$69$		−	4.32801i		−	0.0627248i
$70$		0			0
$71$		−	90.8927i		−	1.28018i	−0.768300	−	0.640090i	$-0.778897\pi$
$71$		−	90.8927i		−	1.28018i	0.768300	−	0.640090i	$-0.221103\pi$
$72$		0			0
$73$	19.7746			0.270885			0.135443	−	0.990785i	$-0.456754\pi$
$73$	19.7746			0.270885			0.135443	+	0.990785i	$0.456754\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.0957			−0.286958
$78$		0			0
$79$			111.986i			1.41754i	0.705439	+	0.708771i	$0.250750\pi$
$79$			111.986i			1.41754i	−0.705439	+	0.708771i	$0.749250\pi$
$80$		0			0
$81$	70.4351			0.869569
$82$		0			0
$83$	129.958			1.56576			0.782881	−	0.622171i	$-0.213749\pi$
$83$	129.958			1.56576			0.782881	+	0.622171i	$0.213749\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	23.3211			0.268059
$88$		0			0
$89$			66.5710i			0.747989i	0.927431	+	0.373994i	$0.122012\pi$
$89$			66.5710i			0.747989i	−0.927431	+	0.373994i	$0.877988\pi$
$90$		0			0
$91$			92.1568i			1.01271i
$92$		0			0
$93$	7.00305			0.0753016
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.0605i			0.196500i	0.995162	+	0.0982498i	$0.0313244\pi$
$97$			19.0605i			0.196500i	−0.995162	+	0.0982498i	$0.968676\pi$
$98$		0			0
$99$	−47.6970			−0.481788
$100$		0			0
$101$	−106.469			−1.05415			−0.527073	−	0.849820i	$-0.676711\pi$
$101$	−106.469			−1.05415			−0.527073	+	0.849820i	$0.676711\pi$
$102$		0			0
$103$		−	123.262i		−	1.19672i	−0.801228	−	0.598359i	$-0.795820\pi$
$103$		−	123.262i		−	1.19672i	0.801228	−	0.598359i	$-0.204180\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	102.138i		−	0.954560i	−0.878751	−	0.477280i	$-0.841623\pi$
$107$		−	102.138i		−	0.954560i	0.878751	−	0.477280i	$-0.158377\pi$
$108$		0			0
$109$			189.408i			1.73769i	0.495083	+	0.868846i	$0.335138\pi$
$109$			189.408i			1.73769i	−0.495083	+	0.868846i	$0.664862\pi$
$110$		0			0
$111$	−42.0302			−0.378651
$112$		0			0
$113$		−	1.96880i		−	0.0174230i	−0.999962	−	0.00871150i	$-0.997227\pi$
$113$		−	1.96880i		−	0.0174230i	0.999962	−	0.00871150i	$-0.00277299\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			198.934i			1.70029i
$118$		0			0
$119$	−65.7625			−0.552626
$120$		0			0
$121$	−90.2606			−0.745956
$122$		0			0
$123$	−9.37417			−0.0762127
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.18645i		−	0.0172162i	−0.999963	−	0.00860809i	$-0.997260\pi$
$127$		−	2.18645i		−	0.0172162i	0.999963	−	0.00860809i	$-0.00274007\pi$
$128$		0			0
$129$		−	21.9188i		−	0.169913i
$130$		0			0
$131$	−37.7731			−0.288344			−0.144172	−	0.989553i	$-0.546052\pi$
$131$	−37.7731			−0.288344			−0.144172	+	0.989553i	$0.546052\pi$
$132$		0			0
$133$	46.4225	+	59.8212i	0.349041	+	0.449783i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−117.952			−0.860964			−0.430482	−	0.902599i	$-0.641656\pi$
$137$	−117.952			−0.860964			−0.430482	+	0.902599i	$0.641656\pi$
$138$		0			0
$139$	−165.428			−1.19013			−0.595063	−	0.803679i	$-0.702873\pi$
$139$	−165.428			−1.19013			−0.595063	+	0.803679i	$0.702873\pi$
$140$		0			0
$141$		−	39.7112i		−	0.281640i
$142$		0			0
$143$		−	128.208i		−	0.896558i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	20.8701i		−	0.141973i
$148$		0			0
$149$	121.995			0.818757			0.409378	−	0.912365i	$-0.365746\pi$
$149$	121.995			0.818757			0.409378	+	0.912365i	$0.365746\pi$
$150$		0			0
$151$			261.820i			1.73391i	0.498387	+	0.866955i	$0.333926\pi$
$151$			261.820i			1.73391i	−0.498387	+	0.866955i	$0.666074\pi$
$152$		0			0
$153$	−141.958			−0.927833
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−73.0950			−0.465573			−0.232787	−	0.972528i	$-0.574784\pi$
$157$	−73.0950			−0.465573			−0.232787	+	0.972528i	$0.574784\pi$
$158$		0			0
$159$	23.2353			0.146134
$160$		0			0
$161$	−27.3704			−0.170003
$162$		0			0
$163$	92.3779			0.566735			0.283368	−	0.959011i	$-0.408548\pi$
$163$	92.3779			0.566735			0.283368	+	0.959011i	$0.408548\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			300.141i			1.79725i	0.438716	+	0.898626i	$0.355433\pi$
$167$			300.141i			1.79725i	−0.438716	+	0.898626i	$0.644567\pi$
$168$		0			0
$169$	−365.729			−2.16407
$170$		0			0
$171$	100.210	+	129.133i	0.586023	+	0.755164i
$172$		0			0
$173$		−	25.8449i		−	0.149392i	−0.997206	−	0.0746962i	$-0.976201\pi$
$173$		−	25.8449i		−	0.149392i	0.997206	−	0.0746962i	$-0.0237987\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0079			−0.0678415
$178$		0			0
$179$			198.054i			1.10644i	0.833034	+	0.553222i	$0.186602\pi$
$179$			198.054i			1.10644i	−0.833034	+	0.553222i	$0.813398\pi$
$180$		0			0
$181$			202.353i			1.11797i	0.829177	+	0.558986i	$0.188810\pi$
$181$			202.353i			1.11797i	−0.829177	+	0.558986i	$0.811190\pi$
$182$		0			0
$183$			21.6586i			0.118353i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	91.4883			0.489242
$188$		0			0
$189$			44.2091i			0.233911i
$190$		0			0
$191$	5.50260			0.0288094			0.0144047	−	0.999896i	$-0.495415\pi$
$191$	5.50260			0.0288094			0.0144047	+	0.999896i	$0.495415\pi$
$192$		0			0
$193$			7.71141i			0.0399555i	0.999800	+	0.0199777i	$0.00635954\pi$
$193$			7.71141i			0.0399555i	−0.999800	+	0.0199777i	$0.993640\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−366.507			−1.86044			−0.930222	−	0.366997i	$-0.880386\pi$
$197$	−366.507			−1.86044			−0.930222	+	0.366997i	$0.880386\pi$
$198$		0			0
$199$	−317.466			−1.59531			−0.797653	−	0.603117i	$-0.793925\pi$
$199$	−317.466			−1.59531			−0.797653	+	0.603117i	$0.793925\pi$
$200$		0			0
$201$	−17.1673			−0.0854096
$202$		0			0
$203$		−	147.483i		−	0.726518i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−59.0832			−0.285426
$208$		0			0
$209$	−64.5825	−	83.2227i	−0.309007	−	0.398195i
$210$		0			0
$211$		−	285.703i		−	1.35404i	−0.735964	−	0.677020i	$-0.763271\pi$
$211$		−	285.703i		−	1.35404i	0.735964	−	0.677020i	$-0.236729\pi$
$212$		0			0
$213$	57.2792			0.268916
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	44.2874i		−	0.204089i
$218$		0			0
$219$			12.4617i			0.0569026i
$220$		0			0
$221$		−	381.579i		−	1.72660i
$222$		0			0
$223$		−	152.576i		−	0.684198i	−0.939664	−	0.342099i	$-0.888862\pi$
$223$		−	152.576i		−	0.684198i	0.939664	−	0.342099i	$-0.111138\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			405.597i			1.78677i	0.449289	+	0.893386i	$0.351677\pi$
$227$			405.597i			1.78677i	−0.449289	+	0.893386i	$0.648323\pi$
$228$		0			0
$229$	339.392			1.48206			0.741030	−	0.671472i	$-0.234338\pi$
$229$	339.392			1.48206			0.741030	+	0.671472i	$0.234338\pi$
$230$		0			0
$231$		−	13.9244i		−	0.0602787i
$232$		0			0
$233$	−245.708			−1.05454			−0.527271	−	0.849697i	$-0.676785\pi$
$233$	−245.708			−1.05454			−0.527271	+	0.849697i	$0.676785\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−70.5717			−0.297771
$238$		0			0
$239$	273.105			1.14270			0.571349	−	0.820707i	$-0.306420\pi$
$239$	273.105			1.14270			0.571349	+	0.820707i	$0.306420\pi$
$240$		0			0
$241$			114.540i			0.475271i	0.971354	+	0.237635i	$0.0763723\pi$
$241$			114.540i			0.475271i	−0.971354	+	0.237635i	$0.923628\pi$
$242$		0			0
$243$			144.225i			0.593517i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−347.105	+	269.361i	−1.40528	+	1.09053i
$248$		0			0
$249$			81.8978i			0.328907i
$250$		0			0
$251$	287.738			1.14637			0.573183	−	0.819427i	$-0.305708\pi$
$251$	287.738			1.14637			0.573183	+	0.819427i	$0.305708\pi$
$252$		0			0
$253$	38.0775			0.150504
$254$		0			0
$255$		0			0
$256$		0			0
$257$			238.684i			0.928730i	0.885644	+	0.464365i	$0.153717\pi$
$257$			238.684i			0.928730i	−0.885644	+	0.464365i	$0.846283\pi$
$258$		0			0
$259$			265.800i			1.02625i
$260$		0			0
$261$		−	318.365i		−	1.21979i
$262$		0			0
$263$	450.262			1.71202			0.856011	−	0.516958i	$-0.172936\pi$
$263$	450.262			1.71202			0.856011	+	0.516958i	$0.172936\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−41.9520			−0.157124
$268$		0			0
$269$		−	202.954i		−	0.754474i	−0.926117	−	0.377237i	$-0.876874\pi$
$269$		−	202.954i		−	0.754474i	0.926117	−	0.377237i	$-0.123126\pi$
$270$		0			0
$271$	−105.551			−0.389485			−0.194743	−	0.980854i	$-0.562387\pi$
$271$	−105.551			−0.389485			−0.194743	+	0.980854i	$0.562387\pi$
$272$		0			0
$273$	−58.0758			−0.212732
$274$		0			0
$275$		0			0
$276$		0			0
$277$	253.935			0.916734			0.458367	−	0.888763i	$-0.348435\pi$
$277$	253.935			0.916734			0.458367	+	0.888763i	$0.348435\pi$
$278$		0			0
$279$		−	95.6010i		−	0.342656i
$280$		0			0
$281$			265.663i			0.945418i	0.881219	+	0.472709i	$0.156724\pi$
$281$			265.663i			0.945418i	−0.881219	+	0.472709i	$0.843276\pi$
$282$		0			0
$283$	62.3633			0.220365			0.110183	−	0.993911i	$-0.464856\pi$
$283$	62.3633			0.220365			0.110183	+	0.993911i	$0.464856\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			59.2824i			0.206559i
$288$		0			0
$289$	−16.7075			−0.0578114
$290$		0			0
$291$	−12.0116			−0.0412770
$292$		0			0
$293$			219.282i			0.748401i	0.927348	+	0.374201i	$0.122083\pi$
$293$			219.282i			0.748401i	−0.927348	+	0.374201i	$0.877917\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	61.5034i		−	0.207082i
$298$		0			0
$299$		−	158.813i		−	0.531149i
$300$		0			0
$301$	−138.615			−0.460514
$302$		0			0
$303$		−	67.0950i		−	0.221436i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	217.314i		−	0.707865i	−0.935271	−	0.353932i	$-0.884844\pi$
$307$		−	217.314i		−	0.707865i	0.935271	−	0.353932i	$-0.115156\pi$
$308$		0			0
$309$	77.6778			0.251384
$310$		0			0
$311$	334.681			1.07615			0.538073	−	0.842899i	$-0.319153\pi$
$311$	334.681			1.07615			0.538073	+	0.842899i	$0.319153\pi$
$312$		0			0
$313$	50.1493			0.160222			0.0801108	−	0.996786i	$-0.474473\pi$
$313$	50.1493			0.160222			0.0801108	+	0.996786i	$0.474473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			147.124i			0.464113i	0.972702	+	0.232056i	$0.0745454\pi$
$317$			147.124i			0.464113i	−0.972702	+	0.232056i	$0.925455\pi$
$318$		0			0
$319$			205.177i			0.643189i
$320$		0			0
$321$	64.3657			0.200516
$322$		0			0
$323$	−192.214	−	247.692i	−0.595090	−	0.766848i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−119.362			−0.365022
$328$		0			0
$329$	−251.134			−0.763325
$330$		0			0
$331$		−	371.342i		−	1.12188i	−0.827857	−	0.560940i	$-0.810440\pi$
$331$		−	371.342i		−	1.12188i	0.827857	−	0.560940i	$-0.189560\pi$
$332$		0			0
$333$			573.769i			1.72303i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	588.911i		−	1.74751i	−0.486367	−	0.873755i	$-0.661678\pi$
$337$		−	588.911i		−	1.74751i	0.486367	−	0.873755i	$-0.338322\pi$
$338$		0			0
$339$	1.24071			0.00365990
$340$		0			0
$341$			61.6122i			0.180681i
$342$		0			0
$343$	−327.262			−0.954117
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−116.937			−0.336993			−0.168496	−	0.985702i	$-0.553891\pi$
$347$	−116.937			−0.336993			−0.168496	+	0.985702i	$0.553891\pi$
$348$		0			0
$349$	236.153			0.676657			0.338329	−	0.941028i	$-0.390138\pi$
$349$	236.153			0.676657			0.338329	+	0.941028i	$0.390138\pi$
$350$		0			0
$351$	−256.518			−0.730821
$352$		0			0
$353$	−176.061			−0.498755			−0.249378	−	0.968406i	$-0.580226\pi$
$353$	−176.061			−0.498755			−0.249378	+	0.968406i	$0.580226\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	41.4426i		−	0.116086i
$358$		0			0
$359$	−277.373			−0.772626			−0.386313	−	0.922368i	$-0.626251\pi$
$359$	−277.373			−0.772626			−0.386313	+	0.922368i	$0.626251\pi$
$360$		0			0
$361$	−89.6281	+	349.697i	−0.248277	+	0.968689i
$362$		0			0
$363$		−	56.8809i		−	0.156697i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	214.948			0.585689			0.292845	−	0.956160i	$-0.405398\pi$
$367$	214.948			0.585689			0.292845	+	0.956160i	$0.405398\pi$
$368$		0			0
$369$			127.970i			0.346802i
$370$		0			0
$371$		−	146.940i		−	0.396066i
$372$		0			0
$373$		−	217.265i		−	0.582479i	−0.956650	−	0.291239i	$-0.905932\pi$
$373$		−	217.265i		−	0.582479i	0.956650	−	0.291239i	$-0.0940676\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	855.753			2.26990
$378$		0			0
$379$		−	648.851i		−	1.71201i	−0.516970	−	0.856003i	$-0.672940\pi$
$379$		−	648.851i		−	1.71201i	0.516970	−	0.856003i	$-0.327060\pi$
$380$		0			0
$381$	1.37787			0.00361646
$382$		0			0
$383$		−	74.9159i		−	0.195603i	−0.995206	−	0.0978014i	$-0.968819\pi$
$383$		−	74.9159i		−	0.195603i	0.995206	−	0.0978014i	$-0.0311810\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−299.221			−0.773180
$388$		0			0
$389$	−539.065			−1.38577			−0.692886	−	0.721048i	$-0.743661\pi$
$389$	−539.065			−1.38577			−0.692886	+	0.721048i	$0.743661\pi$
$390$		0			0
$391$	113.328			0.289842
$392$		0			0
$393$		−	23.8040i		−	0.0605700i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	71.2291			0.179418			0.0897092	−	0.995968i	$-0.471406\pi$
$397$	71.2291			0.179418			0.0897092	+	0.995968i	$0.471406\pi$
$398$		0			0
$399$	−37.6984	+	29.2547i	−0.0944822	+	0.0733201i
$400$		0			0
$401$			471.620i			1.17611i	0.808821	+	0.588054i	$0.200106\pi$
$401$			471.620i			1.17611i	−0.808821	+	0.588054i	$0.799894\pi$
$402$		0			0
$403$	256.972			0.637648
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	369.778i		−	0.908546i
$408$		0			0
$409$			456.208i			1.11542i	0.830035	+	0.557712i	$0.188320\pi$
$409$			456.208i			1.11542i	−0.830035	+	0.557712i	$0.811680\pi$
$410$		0			0
$411$		−	74.3316i		−	0.180855i
$412$		0			0
$413$			75.9384i			0.183870i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	104.250i		−	0.250000i
$418$		0			0
$419$	367.066			0.876052			0.438026	−	0.898962i	$-0.355678\pi$
$419$	367.066			0.876052			0.438026	+	0.898962i	$0.355678\pi$
$420$		0			0
$421$			484.369i			1.15052i	0.817971	+	0.575260i	$0.195099\pi$
$421$			484.369i			1.15052i	−0.817971	+	0.575260i	$0.804901\pi$
$422$		0			0
$423$	−542.111			−1.28159
$424$		0			0
$425$		0			0
$426$		0			0
$427$	136.969			0.320771
$428$		0			0
$429$	80.7946			0.188332
$430$		0			0
$431$		−	227.149i		−	0.527028i	−0.964656	−	0.263514i	$-0.915119\pi$
$431$		−	227.149i		−	0.527028i	0.964656	−	0.263514i	$-0.0848815\pi$
$432$		0			0
$433$		−	243.982i		−	0.563468i	−0.959493	−	0.281734i	$-0.909090\pi$
$433$		−	243.982i		−	0.563468i	0.959493	−	0.281734i	$-0.0909095\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−79.9996	−	103.090i	−0.183066	−	0.235903i
$438$		0			0
$439$			106.971i			0.243671i	0.992550	+	0.121835i	$0.0388780\pi$
$439$			106.971i			0.243671i	−0.992550	+	0.121835i	$0.961122\pi$
$440$		0			0
$441$	−284.905			−0.646042
$442$		0			0
$443$	508.879			1.14871			0.574356	−	0.818606i	$-0.305253\pi$
$443$	508.879			1.14871			0.574356	+	0.818606i	$0.305253\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			76.8793i			0.171989i
$448$		0			0
$449$			158.585i			0.353195i	0.984283	+	0.176598i	$0.0565091\pi$
$449$			158.585i			0.353195i	−0.984283	+	0.176598i	$0.943491\pi$
$450$		0			0
$451$		−	82.4731i		−	0.182867i
$452$		0			0
$453$	−164.995			−0.364228
$454$		0			0
$455$		0			0
$456$		0			0
$457$	204.194			0.446814			0.223407	−	0.974725i	$-0.428282\pi$
$457$	204.194			0.446814			0.223407	+	0.974725i	$0.428282\pi$
$458$		0			0
$459$		−	183.050i		−	0.398801i
$460$		0			0
$461$	842.357			1.82724			0.913619	−	0.406571i	$-0.133276\pi$
$461$	842.357			1.82724			0.913619	+	0.406571i	$0.133276\pi$
$462$		0			0
$463$	677.168			1.46257			0.731283	−	0.682075i	$-0.238922\pi$
$463$	677.168			1.46257			0.731283	+	0.682075i	$0.238922\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.6256			−0.0334596			−0.0167298	−	0.999860i	$-0.505326\pi$
$467$	−15.6256			−0.0334596			−0.0167298	+	0.999860i	$0.505326\pi$
$468$		0			0
$469$			108.566i			0.231485i
$470$		0			0
$471$		−	46.0634i		−	0.0977991i
$472$		0			0
$473$	192.840			0.407695
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	317.193i		−	0.664975i
$478$		0			0
$479$	835.461			1.74418			0.872088	−	0.489349i	$-0.162765\pi$
$479$	835.461			1.74418			0.872088	+	0.489349i	$0.162765\pi$
$480$		0			0
$481$	−1542.27			−3.20638
$482$		0			0
$483$		−	17.2484i		−	0.0357110i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			171.272i			0.351688i	0.984418	+	0.175844i	$0.0562655\pi$
$487$			171.272i			0.351688i	−0.984418	+	0.175844i	$0.943734\pi$
$488$		0			0
$489$			58.2151i			0.119049i
$490$		0			0
$491$	−401.468			−0.817653			−0.408826	−	0.912612i	$-0.634062\pi$
$491$	−401.468			−0.817653			−0.408826	+	0.912612i	$0.634062\pi$
$492$		0			0
$493$			610.660i			1.23866i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	362.235i		−	0.728842i
$498$		0			0
$499$	711.651			1.42615			0.713077	−	0.701086i	$-0.247301\pi$
$499$	711.651			1.42615			0.713077	+	0.701086i	$0.247301\pi$
$500$		0			0
$501$	−189.144			−0.377534
$502$		0			0
$503$	333.140			0.662306			0.331153	−	0.943577i	$-0.392562\pi$
$503$	333.140			0.662306			0.331153	+	0.943577i	$0.392562\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	230.477i		−	0.454589i
$508$		0			0
$509$		−	571.857i		−	1.12349i	−0.827310	−	0.561746i	$-0.810130\pi$
$509$		−	571.857i		−	1.12349i	0.827310	−	0.561746i	$-0.189870\pi$
$510$		0			0
$511$	78.8078			0.154223
$512$		0			0
$513$	−166.512	+	129.217i	−0.324585	+	0.251884i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	349.375			0.675775
$518$		0			0
$519$	16.2870			0.0313816
$520$		0			0
$521$			193.824i			0.372022i	0.982548	+	0.186011i	$0.0595561\pi$
$521$			193.824i			0.372022i	−0.982548	+	0.186011i	$0.940444\pi$
$522$		0			0
$523$		−	749.167i		−	1.43244i	−0.697873	−	0.716221i	$-0.745870\pi$
$523$		−	749.167i		−	1.43244i	0.697873	−	0.716221i	$-0.254130\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			183.374i			0.347958i
$528$		0			0
$529$	−481.833			−0.910837
$530$		0			0
$531$			163.925i			0.308709i
$532$		0			0
$533$	−343.979			−0.645363
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−124.810			−0.232421
$538$		0			0
$539$	183.613			0.340655
$540$		0			0
$541$	156.375			0.289048			0.144524	−	0.989501i	$-0.453835\pi$
$541$	156.375			0.289048			0.144524	+	0.989501i	$0.453835\pi$
$542$		0			0
$543$	−127.520			−0.234843
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	453.990i		−	0.829964i	−0.909830	−	0.414982i	$-0.863788\pi$
$547$		−	453.990i		−	0.829964i	0.909830	−	0.414982i	$-0.136212\pi$
$548$		0			0
$549$	295.669			0.538559
$550$		0			0
$551$	555.489	−	431.071i	1.00815	−	0.782344i
$552$		0			0
$553$			446.297i			0.807046i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	481.620			0.864668			0.432334	−	0.901714i	$-0.357690\pi$
$557$	481.620			0.864668			0.432334	+	0.901714i	$0.357690\pi$
$558$		0			0
$559$		−	804.295i		−	1.43881i
$560$		0			0
$561$			57.6545i			0.102771i
$562$		0			0
$563$			182.504i			0.324163i	0.986777	+	0.162081i	$0.0518207\pi$
$563$			182.504i			0.324163i	−0.986777	+	0.162081i	$0.948179\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	280.705			0.495070
$568$		0			0
$569$			675.222i			1.18668i	0.804951	+	0.593341i	$0.202191\pi$
$569$			675.222i			1.18668i	−0.804951	+	0.593341i	$0.797809\pi$
$570$		0			0
$571$	633.002			1.10858			0.554292	−	0.832322i	$-0.312989\pi$
$571$	633.002			1.10858			0.554292	+	0.832322i	$0.312989\pi$
$572$		0			0
$573$			3.46765i			0.00605175i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	66.1452			0.114636			0.0573182	−	0.998356i	$-0.481745\pi$
$577$	66.1452			0.114636			0.0573182	+	0.998356i	$0.481745\pi$
$578$		0			0
$579$	−4.85961			−0.00839311
$580$		0			0
$581$	517.923			0.891433
$582$		0			0
$583$			204.422i			0.350638i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	730.051			1.24370			0.621850	−	0.783137i	$-0.286382\pi$
$587$	730.051			1.24370			0.621850	+	0.783137i	$0.286382\pi$
$588$		0			0
$589$	166.807	−	129.445i	0.283203	−	0.219771i
$590$		0			0
$591$		−	230.967i		−	0.390808i
$592$		0			0
$593$	591.105			0.996805			0.498403	−	0.866946i	$-0.333920\pi$
$593$	591.105			0.996805			0.498403	+	0.866946i	$0.333920\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	200.062i		−	0.335112i
$598$		0			0
$599$		−	885.437i		−	1.47819i	−0.673600	−	0.739096i	$-0.735253\pi$
$599$		−	885.437i		−	1.47819i	0.673600	−	0.739096i	$-0.264747\pi$
$600$		0			0
$601$			591.878i			0.984823i	0.870363	+	0.492411i	$0.163884\pi$
$601$			591.878i			0.984823i	−0.870363	+	0.492411i	$0.836116\pi$
$602$		0			0
$603$			234.357i			0.388652i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	747.061i		−	1.23074i	−0.788237	−	0.615371i	$-0.789006\pi$
$607$		−	747.061i		−	1.23074i	0.788237	−	0.615371i	$-0.210994\pi$
$608$		0			0
$609$	92.9416			0.152614
$610$		0			0
$611$		−	1457.17i		−	2.38490i
$612$		0			0
$613$	−1164.45			−1.89960			−0.949800	−	0.312859i	$-0.898713\pi$
$613$	−1164.45			−1.89960			−0.949800	+	0.312859i	$0.898713\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−352.505			−0.571321			−0.285660	−	0.958331i	$-0.592213\pi$
$617$	−352.505			−0.571321			−0.285660	+	0.958331i	$0.592213\pi$
$618$		0			0
$619$	−240.601			−0.388693			−0.194347	−	0.980933i	$-0.562259\pi$
$619$	−240.601			−0.388693			−0.194347	+	0.980933i	$0.562259\pi$
$620$		0			0
$621$		−	76.1854i		−	0.122682i
$622$		0			0
$623$			265.305i			0.425851i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	52.4457	−	40.6989i	0.0836454	−	0.0649106i
$628$		0			0
$629$		−	1100.55i		−	1.74969i
$630$		0			0
$631$	582.735			0.923510			0.461755	−	0.887007i	$-0.347220\pi$
$631$	582.735			0.923510			0.461755	+	0.887007i	$0.347220\pi$
$632$		0			0
$633$	180.045			0.284432
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	765.813i		−	1.20222i
$638$		0			0
$639$		−	781.938i		−	1.22369i
$640$		0			0
$641$			314.886i			0.491242i	0.969366	+	0.245621i	$0.0789919\pi$
$641$			314.886i			0.491242i	−0.969366	+	0.245621i	$0.921008\pi$
$642$		0			0
$643$	−650.686			−1.01195			−0.505977	−	0.862547i	$-0.668868\pi$
$643$	−650.686			−1.01195			−0.505977	+	0.862547i	$0.668868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.19128			−0.00338683			−0.00169341	−	0.999999i	$-0.500539\pi$
$647$	−2.19128			−0.00338683			−0.00169341	+	0.999999i	$0.500539\pi$
$648$		0			0
$649$		−	105.645i		−	0.162781i
$650$		0			0
$651$	27.9092			0.0428713
$652$		0			0
$653$	−955.448			−1.46317			−0.731584	−	0.681752i	$-0.761218\pi$
$653$	−955.448			−1.46317			−0.731584	+	0.681752i	$0.761218\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	170.119			0.258932
$658$		0			0
$659$		−	76.0625i		−	0.115421i	−0.998333	−	0.0577106i	$-0.981620\pi$
$659$		−	76.0625i		−	0.115421i	0.998333	−	0.0577106i	$-0.0183801\pi$
$660$		0			0
$661$		−	662.480i		−	1.00224i	−0.865378	−	0.501119i	$-0.832922\pi$
$661$		−	662.480i		−	1.00224i	0.865378	−	0.501119i	$-0.167078\pi$
$662$		0			0
$663$	240.465			0.362693
$664$		0			0
$665$		0			0
$666$		0			0
$667$			254.157i			0.381045i
$668$		0			0
$669$	96.1512			0.143724
$670$		0			0
$671$	−190.550			−0.283980
$672$		0			0
$673$			150.670i			0.223879i	0.993715	+	0.111939i	$0.0357063\pi$
$673$			150.670i			0.223879i	−0.993715	+	0.111939i	$0.964294\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	392.546i		−	0.579832i	−0.957052	−	0.289916i	$-0.906373\pi$
$677$		−	392.546i		−	0.579832i	0.957052	−	0.289916i	$-0.0936274\pi$
$678$		0			0
$679$			75.9616i			0.111873i
$680$		0			0
$681$	−255.601			−0.375332
$682$		0			0
$683$		−	737.192i		−	1.07934i	−0.841875	−	0.539672i	$-0.818548\pi$
$683$		−	737.192i		−	1.07934i	0.841875	−	0.539672i	$-0.181452\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			213.879i			0.311324i
$688$		0			0
$689$	852.603			1.23745
$690$		0			0
$691$	828.415			1.19886			0.599432	−	0.800426i	$-0.295393\pi$
$691$	828.415			1.19886			0.599432	+	0.800426i	$0.295393\pi$
$692$		0			0
$693$	−190.087			−0.274295
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	245.461i		−	0.352168i
$698$		0			0
$699$		−	154.842i		−	0.221519i
$700$		0			0
$701$	−1040.93			−1.48493			−0.742463	−	0.669887i	$-0.766343\pi$
$701$	−1040.93			−1.48493			−0.742463	+	0.669887i	$0.766343\pi$
$702$		0			0
$703$	−1001.12	+	776.893i	−1.42407	+	1.10511i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−424.310			−0.600155
$708$		0			0
$709$	29.3704			0.0414251			0.0207126	−	0.999785i	$-0.493407\pi$
$709$	29.3704			0.0414251			0.0207126	+	0.999785i	$0.493407\pi$
$710$		0			0
$711$			963.399i			1.35499i
$712$		0			0
$713$			76.3202i			0.107041i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			172.106i			0.240037i
$718$		0			0
$719$	−241.324			−0.335638			−0.167819	−	0.985818i	$-0.553672\pi$
$719$	−241.324			−0.335638			−0.167819	+	0.985818i	$0.553672\pi$
$720$		0			0
$721$		−	491.235i		−	0.681325i
$722$		0			0
$723$	−72.1815			−0.0998361
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−142.920			−0.196589			−0.0982943	−	0.995157i	$-0.531339\pi$
$727$	−142.920			−0.196589			−0.0982943	+	0.995157i	$0.531339\pi$
$728$		0			0
$729$	543.028			0.744894
$730$		0			0
$731$	573.940			0.785143
$732$		0			0
$733$	316.452			0.431721			0.215861	−	0.976424i	$-0.430744\pi$
$733$	316.452			0.431721			0.215861	+	0.976424i	$0.430744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	151.037i		−	0.204934i
$738$		0			0
$739$	−497.784			−0.673592			−0.336796	−	0.941578i	$-0.609343\pi$
$739$	−497.784			−0.673592			−0.336796	+	0.941578i	$0.609343\pi$
$740$		0			0
$741$	−169.747	−	218.740i	−0.229078	−	0.295196i
$742$		0			0
$743$		−	554.732i		−	0.746611i	−0.927708	−	0.373306i	$-0.878224\pi$
$743$		−	554.732i		−	0.746611i	0.927708	−	0.373306i	$-0.121776\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1118.01			1.49667
$748$		0			0
$749$		−	407.050i		−	0.543458i
$750$		0			0
$751$		−	694.808i		−	0.925177i	−0.886573	−	0.462589i	$-0.846921\pi$
$751$		−	694.808i		−	0.925177i	0.886573	−	0.462589i	$-0.153079\pi$
$752$		0			0
$753$			181.328i			0.240808i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	927.589			1.22535			0.612675	−	0.790335i	$-0.290094\pi$
$757$	927.589			1.22535			0.612675	+	0.790335i	$0.290094\pi$
$758$		0			0
$759$			23.9958i			0.0316151i
$760$		0			0
$761$	83.5130			0.109741			0.0548706	−	0.998493i	$-0.482525\pi$
$761$	83.5130			0.109741			0.0548706	+	0.998493i	$0.482525\pi$
$762$		0			0
$763$			754.849i			0.989317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−440.623			−0.574476
$768$		0			0
$769$	91.0434			0.118392			0.0591960	−	0.998246i	$-0.481146\pi$
$769$	91.0434			0.118392			0.0591960	+	0.998246i	$0.481146\pi$
$770$		0			0
$771$	−150.415			−0.195091
$772$		0			0
$773$			920.649i			1.19101i	0.803352	+	0.595504i	$0.203048\pi$
$773$			920.649i			1.19101i	−0.803352	+	0.595504i	$0.796952\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−167.503			−0.215577
$778$		0			0
$779$	−223.285	+	173.274i	−0.286630	+	0.222431i
$780$		0			0
$781$			503.937i			0.645246i
$782$		0			0
$783$	410.519			0.524290
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1386.63i			1.76192i	0.473188	+	0.880962i	$0.343103\pi$
$787$			1386.63i			1.76192i	−0.473188	+	0.880962i	$0.656897\pi$
$788$		0			0
$789$			283.748i			0.359630i
$790$		0			0
$791$		−	7.84625i		−	0.00991941i
$792$		0			0
$793$			794.747i			1.00220i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	327.713i		−	0.411183i	−0.978638	−	0.205592i	$-0.934088\pi$
$797$		−	327.713i		−	0.411183i	0.978638	−	0.205592i	$-0.0659118\pi$
$798$		0			0
$799$	1039.83			1.30142
$800$		0			0
$801$			572.701i			0.714983i
$802$		0			0
$803$	−109.637			−0.136534
$804$		0			0
$805$		0			0
$806$		0			0
$807$	127.898			0.158486
$808$		0			0
$809$	130.207			0.160948			0.0804738	−	0.996757i	$-0.474357\pi$
$809$	130.207			0.160948			0.0804738	+	0.996757i	$0.474357\pi$
$810$		0			0
$811$			732.273i			0.902926i	0.892290	+	0.451463i	$0.149098\pi$
$811$			732.273i			0.902926i	−0.892290	+	0.451463i	$0.850902\pi$
$812$		0			0
$813$		−	66.5163i		−	0.0818159i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−405.150	−	522.087i	−0.495900	−	0.639029i
$818$		0			0
$819$			792.813i			0.968025i
$820$		0			0
$821$	163.217			0.198803			0.0994014	−	0.995047i	$-0.468307\pi$
$821$	163.217			0.198803			0.0994014	+	0.995047i	$0.468307\pi$
$822$		0			0
$823$	533.268			0.647957			0.323978	−	0.946065i	$-0.394980\pi$
$823$	533.268			0.647957			0.323978	+	0.946065i	$0.394980\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1276.45i		−	1.54347i	−0.635947	−	0.771733i	$-0.719390\pi$
$827$		−	1276.45i		−	1.54347i	0.635947	−	0.771733i	$-0.280610\pi$
$828$		0			0
$829$		−	1194.31i		−	1.44067i	−0.693628	−	0.720333i	$-0.743989\pi$
$829$		−	1194.31i		−	1.44067i	0.693628	−	0.720333i	$-0.256011\pi$
$830$		0			0
$831$			160.026i			0.192571i
$832$		0			0
$833$	546.480			0.656038
$834$		0			0
$835$		0			0
$836$		0			0
$837$	123.274			0.147280
$838$		0			0
$839$			1239.77i			1.47768i	0.673883	+	0.738839i	$0.264625\pi$
$839$			1239.77i			1.47768i	−0.673883	+	0.738839i	$0.735375\pi$
$840$		0			0
$841$	−528.504			−0.628424
$842$		0			0
$843$	−167.416			−0.198596
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−359.715			−0.424694
$848$		0			0
$849$			39.3004i			0.0462902i
$850$		0			0
$851$		−	458.052i		−	0.538251i
$852$		0			0
$853$	644.073			0.755068			0.377534	−	0.925996i	$-0.376772\pi$
$853$	644.073			0.755068			0.377534	+	0.925996i	$0.376772\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			263.355i			0.307299i	0.988125	+	0.153650i	$0.0491027\pi$
$857$			263.355i			0.307299i	−0.988125	+	0.153650i	$0.950897\pi$
$858$		0			0
$859$	931.348			1.08422			0.542112	−	0.840307i	$-0.317625\pi$
$859$	931.348			1.08422			0.542112	+	0.840307i	$0.317625\pi$
$860$		0			0
$861$	−37.3588			−0.0433901
$862$		0			0
$863$		−	1043.40i		−	1.20903i	−0.796592	−	0.604517i	$-0.793366\pi$
$863$		−	1043.40i		−	1.20903i	0.796592	−	0.604517i	$-0.206634\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	10.5288i		−	0.0121439i
$868$		0			0
$869$		−	620.884i		−	0.714481i
$870$		0			0
$871$	−629.943			−0.723241
$872$		0			0
$873$			163.975i			0.187829i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	366.106i		−	0.417453i	−0.977974	−	0.208727i	$-0.933068\pi$
$877$		−	366.106i		−	0.417453i	0.977974	−	0.208727i	$-0.0669319\pi$
$878$		0			0
$879$	−138.188			−0.157210
$880$		0			0
$881$	742.098			0.842336			0.421168	−	0.906983i	$-0.361620\pi$
$881$	742.098			0.842336			0.421168	+	0.906983i	$0.361620\pi$
$882$		0			0
$883$	197.913			0.224137			0.112068	−	0.993701i	$-0.464252\pi$
$883$	197.913			0.224137			0.112068	+	0.993701i	$0.464252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			356.570i			0.401995i	0.979592	+	0.200998i	$0.0644184\pi$
$887$			356.570i			0.401995i	−0.979592	+	0.200998i	$0.935582\pi$
$888$		0			0
$889$		−	8.71367i		−	0.00980165i
$890$		0			0
$891$	−390.514			−0.438288
$892$		0			0
$893$	−734.027	−	945.886i	−0.821979	−	1.05922i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	100.082			0.111574
$898$		0			0
$899$	−411.245			−0.457447
$900$		0			0
$901$			608.412i			0.675263i
$902$		0			0
$903$		−	87.3529i		−	0.0967363i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			562.349i			0.620010i	0.950735	+	0.310005i	$0.100331\pi$
$907$			562.349i			0.620010i	−0.950735	+	0.310005i	$0.899669\pi$
$908$		0			0
$909$	−915.937			−1.00763
$910$		0			0
$911$		−	1310.45i		−	1.43847i	−0.694765	−	0.719237i	$-0.744491\pi$
$911$		−	1310.45i		−	1.43847i	0.694765	−	0.719237i	$-0.255509\pi$
$912$		0			0
$913$	−720.529			−0.789189
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−150.537			−0.164162
$918$		0			0
$919$	−768.042			−0.835736			−0.417868	−	0.908508i	$-0.637223\pi$
$919$	−768.042			−0.835736			−0.417868	+	0.908508i	$0.637223\pi$
$920$		0			0
$921$	136.948			0.148695
$922$		0			0
$923$	2101.82			2.27716
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	1060.41i		−	1.14391i
$928$		0			0
$929$	−478.127			−0.514669			−0.257334	−	0.966322i	$-0.582844\pi$
$929$	−478.127			−0.514669			−0.257334	+	0.966322i	$0.582844\pi$
$930$		0			0
$931$	−385.766	−	497.108i	−0.414356	−	0.533950i
$932$		0			0
$933$			210.911i			0.226057i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	427.941			0.456714			0.228357	−	0.973577i	$-0.426665\pi$
$937$	427.941			0.456714			0.228357	+	0.973577i	$0.426665\pi$
$938$		0			0
$939$			31.6033i			0.0336564i
$940$		0			0
$941$		−	1089.92i		−	1.15826i	−0.815237	−	0.579128i	$-0.803393\pi$
$941$		−	1089.92i		−	1.15826i	0.815237	−	0.579128i	$-0.196607\pi$
$942$		0			0
$943$		−	102.161i		−	0.108336i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−950.576			−1.00378			−0.501888	−	0.864933i	$-0.667361\pi$
$947$	−950.576			−1.00378			−0.501888	+	0.864933i	$0.667361\pi$
$948$		0			0
$949$			457.272i			0.481847i
$950$		0			0
$951$	−92.7151			−0.0974922
$952$		0			0
$953$		−	1262.54i		−	1.32481i	−0.749146	−	0.662405i	$-0.769536\pi$
$953$		−	1262.54i		−	1.32481i	0.749146	−	0.662405i	$-0.230464\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−129.300			−0.135109
$958$		0			0
$959$	−470.074			−0.490171
$960$		0			0
$961$	837.508			0.871497
$962$		0			0
$963$		−	878.679i		−	0.912439i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1658.76			1.71536			0.857682	−	0.514180i	$-0.171904\pi$
$967$	1658.76			1.71536			0.857682	+	0.514180i	$0.171904\pi$
$968$		0			0
$969$	156.092	−	121.130i	0.161085	−	0.125006i
$970$		0			0
$971$			1083.30i			1.11565i	0.829957	+	0.557827i	$0.188365\pi$
$971$			1083.30i			1.11565i	−0.829957	+	0.557827i	$0.811635\pi$
$972$		0			0
$973$	−659.278			−0.677573
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1248.74i			1.27813i	0.769152	+	0.639066i	$0.220679\pi$
$977$			1248.74i			1.27813i	−0.769152	+	0.639066i	$0.779321\pi$
$978$		0			0
$979$		−	369.090i		−	0.377007i
$980$		0			0
$981$			1629.45i			1.66101i
$982$		0			0
$983$		−	1806.81i		−	1.83806i	−0.394192	−	0.919028i	$-0.628976\pi$
$983$		−	1806.81i		−	1.83806i	0.394192	−	0.919028i	$-0.371024\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	158.261i		−	0.160345i
$988$		0			0
$989$	238.874			0.241531
$990$		0			0
$991$			697.379i			0.703712i	0.936054	+	0.351856i	$0.114449\pi$
$991$			697.379i			0.703712i	−0.936054	+	0.351856i	$0.885551\pi$
$992$		0			0
$993$	234.014			0.235664
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1233.03			1.23674			0.618370	−	0.785887i	$-0.287793\pi$
$997$	1233.03			1.23674			0.618370	+	0.785887i	$0.287793\pi$
$998$		0			0
$999$	−739.853			−0.740593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.f.1101.7		12
5.2	odd	4		1900.3.g.c.949.13		24
5.3	odd	4		1900.3.g.c.949.12		24
5.4	even	2		380.3.e.a.341.6	✓	12
15.14	odd	2		3420.3.o.a.721.1		12
19.18	odd	2	inner	1900.3.e.f.1101.6		12
20.19	odd	2		1520.3.h.b.721.7		12
95.18	even	4		1900.3.g.c.949.14		24
95.37	even	4		1900.3.g.c.949.11		24
95.94	odd	2		380.3.e.a.341.7	yes	12
285.284	even	2		3420.3.o.a.721.2		12
380.379	even	2		1520.3.h.b.721.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.6	✓	12	5.4	even	2
380.3.e.a.341.7	yes	12	95.94	odd	2
1520.3.h.b.721.6		12	380.379	even	2
1520.3.h.b.721.7		12	20.19	odd	2
1900.3.e.f.1101.6		12	19.18	odd	2	inner
1900.3.e.f.1101.7		12	1.1	even	1	trivial
1900.3.g.c.949.11		24	95.37	even	4
1900.3.g.c.949.12		24	5.3	odd	4
1900.3.g.c.949.13		24	5.2	odd	4
1900.3.g.c.949.14		24	95.18	even	4
3420.3.o.a.721.1		12	15.14	odd	2
3420.3.o.a.721.2		12	285.284	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.630185i q^{3} +3.98530 q^{7} +8.60287 q^{9} +O(q^{10})\)	\(q+0.630185i q^{3} +3.98530 q^{7} +8.60287 q^{9} -5.54431 q^{11} +23.1242i q^{13} -16.5013 q^{17} +(11.6484 + 15.0105i) q^{19} +2.51147i q^{21} -6.86785 q^{23} +11.0931i q^{27} -37.0068i q^{29} -11.1127i q^{31} -3.49394i q^{33} +66.6951i q^{37} -14.5725 q^{39} +14.8753i q^{41} -34.7815 q^{43} -63.0151 q^{47} -33.1174 q^{49} -10.3989i q^{51} -36.8706i q^{53} +(-9.45937 + 7.34066i) q^{57} +19.0546i q^{59} +34.3686 q^{61} +34.2850 q^{63} +27.2417i q^{67} -4.32801i q^{69} -90.8927i q^{71} +19.7746 q^{73} -22.0957 q^{77} +111.986i q^{79} +70.4351 q^{81} +129.958 q^{83} +23.3211 q^{87} +66.5710i q^{89} +92.1568i q^{91} +7.00305 q^{93} +19.0605i q^{97} -47.6970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 16 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 16 * q^9	\( 12 q + 12 q^{7} - 16 q^{9} - 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 124 q^{39} + 176 q^{43} + 72 q^{47} - 24 q^{49} - 140 q^{57} + 152 q^{61} - 48 q^{63} + 148 q^{73} - 376 q^{77} - 468 q^{81} + 208 q^{83} + 84 q^{87} + 184 q^{93} + 392 q^{99}+O(q^{100}) \) 12 * q + 12 * q^7 - 16 * q^9 - 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 124 * q^39 + 176 * q^43 + 72 * q^47 - 24 * q^49 - 140 * q^57 + 152 * q^61 - 48 * q^63 + 148 * q^73 - 376 * q^77 - 468 * q^81 + 208 * q^83 + 84 * q^87 + 184 * q^93 + 392 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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