LMFDB - Embedded newform 3150.2.d.c.3149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(3149,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3150.3149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.d (of order $2$, degree $1$, not 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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3149.1
Root			$2.73923i$ of defining polynomial
Character	$\chi$	$=$	3150.3149
Dual form			3150.2.d.c.3149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} -1.00000 q^{8} -3.87386i q^{11} -1.60461 q^{13} +(1.93693 + 1.80230i) q^{14} +1.00000 q^{16} -8.11650i q^{17} -2.63803i q^{19} +3.87386i q^{22} -5.47847 q^{23} +1.60461 q^{26} +(-1.93693 - 1.80230i) q^{28} +5.47847i q^{29} -3.73074i q^{31} -1.00000 q^{32} +8.11650i q^{34} +4.51190i q^{37} +2.63803i q^{38} +1.60461 q^{41} +10.1165i q^{43} -3.87386i q^{44} +5.47847 q^{46} -11.1097i q^{47} +(0.503406 + 6.98188i) q^{49} -1.60461 q^{52} +2.26926 q^{53} +(1.93693 + 1.80230i) q^{56} -5.47847i q^{58} +4.61142 q^{59} +11.8472i q^{61} +3.73074i q^{62} +1.00000 q^{64} +6.90729i q^{67} -8.11650i q^{68} -2.63803i q^{71} -13.7477 q^{73} -4.51190i q^{74} -2.63803i q^{76} +(-6.98188 + 7.50341i) q^{77} +8.01698 q^{79} -1.60461 q^{82} +3.20921i q^{83} -10.1165i q^{86} +3.87386i q^{88} -17.8376 q^{89} +(3.10801 + 2.89199i) q^{91} -5.47847 q^{92} +11.1097i q^{94} -8.68768 q^{97} +(-0.503406 - 6.98188i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{13} + 8 q^{16} + 8 q^{23} - 8 q^{26} - 8 q^{32} - 8 q^{41} - 8 q^{46} - 4 q^{49} + 8 q^{52} + 8 q^{53} + 8 q^{64} - 48 q^{73} + 4 q^{77} - 8 q^{79} + 8 q^{82} + 8 q^{89} - 4 q^{91} + 8 q^{92} + 24 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^13 + 8 * q^16 + 8 * q^23 - 8 * q^26 - 8 * q^32 - 8 * q^41 - 8 * q^46 - 4 * q^49 + 8 * q^52 + 8 * q^53 + 8 * q^64 - 48 * q^73 + 4 * q^77 - 8 * q^79 + 8 * q^82 + 8 * q^89 - 4 * q^91 + 8 * q^92 + 24 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.93693	−	1.80230i	−0.732091	−	0.681207i
$7$	−1.93693	−	1.80230i	−0.732091	−	0.681207i
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$		−	3.87386i		−	1.16801i	−0.811749	−	0.584007i	$-0.801484\pi$
$11$		−	3.87386i		−	1.16801i	0.811749	−	0.584007i	$-0.198516\pi$
$12$		0			0
$13$	−1.60461			−0.445038			−0.222519	−	0.974928i	$-0.571428\pi$
$13$	−1.60461			−0.445038			−0.222519	+	0.974928i	$0.571428\pi$
$14$	1.93693	+	1.80230i	0.517667	+	0.481686i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	8.11650i		−	1.96854i	−0.176667	−	0.984271i	$-0.556532\pi$
$17$		−	8.11650i		−	1.96854i	0.176667	−	0.984271i	$-0.443468\pi$
$18$		0			0
$19$		−	2.63803i		−	0.605207i	−0.953117	−	0.302603i	$-0.902144\pi$
$19$		−	2.63803i		−	0.605207i	0.953117	−	0.302603i	$-0.0978557\pi$
$20$		0			0
$21$		0			0
$22$			3.87386i			0.825910i
$23$	−5.47847			−1.14234			−0.571170	−	0.820832i	$-0.693510\pi$
$23$	−5.47847			−1.14234			−0.571170	+	0.820832i	$0.693510\pi$
$24$		0			0
$25$		0			0
$26$	1.60461			0.314689
$27$		0			0
$28$	−1.93693	−	1.80230i	−0.366046	−	0.340603i
$29$			5.47847i			1.01733i	0.860966	+	0.508663i	$0.169860\pi$
$29$			5.47847i			1.01733i	−0.860966	+	0.508663i	$0.830140\pi$
$30$		0			0
$31$		−	3.73074i		−	0.670061i	−0.942207	−	0.335031i	$-0.891253\pi$
$31$		−	3.73074i		−	0.670061i	0.942207	−	0.335031i	$-0.108747\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			8.11650i			1.39197i
$35$		0			0
$36$		0			0
$37$			4.51190i			0.741751i	0.928683	+	0.370876i	$0.120942\pi$
$37$			4.51190i			0.741751i	−0.928683	+	0.370876i	$0.879058\pi$
$38$			2.63803i			0.427946i
$39$		0			0
$40$		0			0
$41$	1.60461			0.250597			0.125299	−	0.992119i	$-0.460011\pi$
$41$	1.60461			0.250597			0.125299	+	0.992119i	$0.460011\pi$
$42$		0			0
$43$			10.1165i			1.54275i	0.636379	+	0.771376i	$0.280431\pi$
$43$			10.1165i			1.54275i	−0.636379	+	0.771376i	$0.719569\pi$
$44$		−	3.87386i		−	0.584007i
$45$		0			0
$46$	5.47847			0.807756
$47$		−	11.1097i		−	1.62052i	−0.586074	−	0.810258i	$-0.699327\pi$
$47$		−	11.1097i		−	1.62052i	0.586074	−	0.810258i	$-0.300673\pi$
$48$		0			0
$49$	0.503406	+	6.98188i	0.0719152	+	0.997411i
$50$		0			0
$51$		0			0
$52$	−1.60461			−0.222519
$53$	2.26926			0.311706			0.155853	−	0.987780i	$-0.450187\pi$
$53$	2.26926			0.311706			0.155853	+	0.987780i	$0.450187\pi$
$54$		0			0
$55$		0			0
$56$	1.93693	+	1.80230i	0.258833	+	0.240843i
$57$		0			0
$58$		−	5.47847i		−	0.719358i
$59$	4.61142			0.600356			0.300178	−	0.953883i	$-0.402954\pi$
$59$	4.61142			0.600356			0.300178	+	0.953883i	$0.402954\pi$
$60$		0			0
$61$			11.8472i			1.51688i	0.651740	+	0.758442i	$0.274039\pi$
$61$			11.8472i			1.51688i	−0.651740	+	0.758442i	$0.725961\pi$
$62$			3.73074i			0.473805i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$			6.90729i			0.843860i	0.906628	+	0.421930i	$0.138647\pi$
$67$			6.90729i			0.843860i	−0.906628	+	0.421930i	$0.861353\pi$
$68$		−	8.11650i		−	0.984271i
$69$		0			0
$70$		0			0
$71$		−	2.63803i		−	0.313077i	−0.987672	−	0.156539i	$-0.949966\pi$
$71$		−	2.63803i		−	0.313077i	0.987672	−	0.156539i	$-0.0500335\pi$
$72$		0			0
$73$	−13.7477			−1.60905			−0.804525	−	0.593919i	$-0.797580\pi$
$73$	−13.7477			−1.60905			−0.804525	+	0.593919i	$0.797580\pi$
$74$		−	4.51190i		−	0.524497i
$75$		0			0
$76$		−	2.63803i		−	0.302603i
$77$	−6.98188	+	7.50341i	−0.795659	+	0.855092i
$78$		0			0
$79$	8.01698			0.901981			0.450990	−	0.892529i	$-0.351071\pi$
$79$	8.01698			0.901981			0.450990	+	0.892529i	$0.351071\pi$
$80$		0			0
$81$		0			0
$82$	−1.60461			−0.177199
$83$			3.20921i			0.352257i	0.984367	+	0.176128i	$0.0563574\pi$
$83$			3.20921i			0.352257i	−0.984367	+	0.176128i	$0.943643\pi$
$84$		0			0
$85$		0			0
$86$		−	10.1165i		−	1.09089i
$87$		0			0
$88$			3.87386i			0.412955i
$89$	−17.8376			−1.89078			−0.945392	−	0.325937i	$-0.894320\pi$
$89$	−17.8376			−1.89078			−0.945392	+	0.325937i	$0.894320\pi$
$90$		0			0
$91$	3.10801	+	2.89199i	0.325808	+	0.303163i
$92$	−5.47847			−0.571170
$93$		0			0
$94$			11.1097i			1.14588i
$95$		0			0
$96$		0			0
$97$	−8.68768			−0.882100			−0.441050	−	0.897482i	$-0.645394\pi$
$97$	−8.68768			−0.882100			−0.441050	+	0.897482i	$0.645394\pi$
$98$	−0.503406	−	6.98188i	−0.0508517	−	0.705276i
$99$		0			0
$100$		0			0
$101$	−6.21603			−0.618518			−0.309259	−	0.950978i	$-0.600081\pi$
$101$	−6.21603			−0.618518			−0.309259	+	0.950978i	$0.600081\pi$
$102$		0			0
$103$	12.8807			1.26917			0.634585	−	0.772853i	$-0.281171\pi$
$103$	12.8807			1.26917			0.634585	+	0.772853i	$0.281171\pi$
$104$	1.60461			0.157345
$105$		0			0
$106$	−2.26926			−0.220410
$107$	−19.9638			−1.92997			−0.964984	−	0.262308i	$-0.915516\pi$
$107$	−19.9638			−1.92997			−0.964984	+	0.262308i	$0.915516\pi$
$108$		0			0
$109$	15.0238			1.43902			0.719509	−	0.694483i	$-0.244367\pi$
$109$	15.0238			1.43902			0.719509	+	0.694483i	$0.244367\pi$
$110$		0			0
$111$		0			0
$112$	−1.93693	−	1.80230i	−0.183023	−	0.170302i
$113$	10.2330			0.962640			0.481320	−	0.876545i	$-0.340157\pi$
$113$	10.2330			0.962640			0.481320	+	0.876545i	$0.340157\pi$
$114$		0			0
$115$		0			0
$116$			5.47847i			0.508663i
$117$		0			0
$118$	−4.61142			−0.424515
$119$	−14.6284	+	15.7211i	−1.34098	+	1.44115i
$120$		0			0
$121$	−4.00681			−0.364256
$122$		−	11.8472i		−	1.07260i
$123$		0			0
$124$		−	3.73074i		−	0.335031i
$125$		0			0
$126$		0			0
$127$		−	2.81382i		−	0.249686i	−0.992177	−	0.124843i	$-0.960157\pi$
$127$		−	2.81382i		−	0.249686i	0.992177	−	0.124843i	$-0.0398427\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		0			0
$131$	8.62840			0.753867			0.376933	−	0.926240i	$-0.376979\pi$
$131$	8.62840			0.753867			0.376933	+	0.926240i	$0.376979\pi$
$132$		0			0
$133$	−4.75454	+	5.10969i	−0.412271	+	0.443066i
$134$		−	6.90729i		−	0.596699i
$135$		0			0
$136$			8.11650i			0.695984i
$137$	−0.723932			−0.0618496			−0.0309248	−	0.999522i	$-0.509845\pi$
$137$	−0.723932			−0.0618496			−0.0309248	+	0.999522i	$0.509845\pi$
$138$		0			0
$139$			2.84043i			0.240923i	0.992718	+	0.120461i	$0.0384374\pi$
$139$			2.84043i			0.240923i	−0.992718	+	0.120461i	$0.961563\pi$
$140$		0			0
$141$		0			0
$142$			2.63803i			0.221379i
$143$			6.21603i			0.519810i
$144$		0			0
$145$		0			0
$146$	13.7477			1.13777
$147$		0			0
$148$			4.51190i			0.370876i
$149$			3.00681i			0.246328i	0.992386	+	0.123164i	$0.0393041\pi$
$149$			3.00681i			0.246328i	−0.992386	+	0.123164i	$0.960696\pi$
$150$		0			0
$151$	−15.2262			−1.23909			−0.619545	−	0.784961i	$-0.712683\pi$
$151$	−15.2262			−1.23909			−0.619545	+	0.784961i	$0.712683\pi$
$152$			2.63803i			0.213973i
$153$		0			0
$154$	6.98188	−	7.50341i	0.562616	−	0.604642i
$155$		0			0
$156$		0			0
$157$	2.64767			0.211307			0.105653	−	0.994403i	$-0.466307\pi$
$157$	2.64767			0.211307			0.105653	+	0.994403i	$0.466307\pi$
$158$	−8.01698			−0.637797
$159$		0			0
$160$		0			0
$161$	10.6114	+	9.87386i	0.836297	+	0.778169i
$162$		0			0
$163$			9.32572i			0.730446i	0.930920	+	0.365223i	$0.119007\pi$
$163$			9.32572i			0.730446i	−0.930920	+	0.365223i	$0.880993\pi$
$164$	1.60461			0.125299
$165$		0			0
$166$		−	3.20921i		−	0.249083i
$167$		−	17.3257i		−	1.34070i	−0.742043	−	0.670352i	$-0.766143\pi$
$167$		−	17.3257i		−	1.34070i	0.742043	−	0.670352i	$-0.233857\pi$
$168$		0			0
$169$	−10.4252			−0.801941
$170$		0			0
$171$		0			0
$172$			10.1165i			0.771376i
$173$			2.99319i			0.227568i	0.993506	+	0.113784i	$0.0362972\pi$
$173$			2.99319i			0.227568i	−0.993506	+	0.113784i	$0.963703\pi$
$174$		0			0
$175$		0			0
$176$		−	3.87386i		−	0.292003i
$177$		0			0
$178$	17.8376			1.33699
$179$			7.38858i			0.552248i	0.961122	+	0.276124i	$0.0890501\pi$
$179$			7.38858i			0.552248i	−0.961122	+	0.276124i	$0.910950\pi$
$180$		0			0
$181$		−	11.1097i		−	0.825777i	−0.910782	−	0.412888i	$-0.864520\pi$
$181$		−	11.1097i		−	0.825777i	0.910782	−	0.412888i	$-0.135480\pi$
$182$	−3.10801	−	2.89199i	−0.230381	−	0.214368i
$183$		0			0
$184$	5.47847			0.403878
$185$		0			0
$186$		0			0
$187$	−31.4422			−2.29928
$188$		−	11.1097i		−	0.810258i
$189$		0			0
$190$		0			0
$191$		−	9.36197i		−	0.677408i	−0.940893	−	0.338704i	$-0.890011\pi$
$191$		−	9.36197i		−	0.677408i	0.940893	−	0.338704i	$-0.109989\pi$
$192$		0			0
$193$		−	17.4422i		−	1.25552i	−0.778408	−	0.627759i	$-0.783972\pi$
$193$		−	17.4422i		−	1.25552i	0.778408	−	0.627759i	$-0.216028\pi$
$194$	8.68768			0.623739
$195$		0			0
$196$	0.503406	+	6.98188i	0.0359576	+	0.498705i
$197$	5.27607			0.375904			0.187952	−	0.982178i	$-0.439815\pi$
$197$	5.27607			0.375904			0.187952	+	0.982178i	$0.439815\pi$
$198$		0			0
$199$			12.2160i			0.865971i	0.901401	+	0.432986i	$0.142540\pi$
$199$			12.2160i			0.865971i	−0.901401	+	0.432986i	$0.857460\pi$
$200$		0			0
$201$		0			0
$202$	6.21603			0.437358
$203$	9.87386	−	10.6114i	0.693009	−	0.744776i
$204$		0			0
$205$		0			0
$206$	−12.8807			−0.897439
$207$		0			0
$208$	−1.60461			−0.111259
$209$	−10.2194			−0.706889
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	2.26926			0.155853
$213$		0			0
$214$	19.9638			1.36469
$215$		0			0
$216$		0			0
$217$	−6.72393	+	7.22619i	−0.456450	+	0.490546i
$218$	−15.0238			−1.01754
$219$		0			0
$220$		0			0
$221$			13.0238i			0.876075i
$222$		0			0
$223$	−3.65784			−0.244947			−0.122473	−	0.992472i	$-0.539083\pi$
$223$	−3.65784			−0.244947			−0.122473	+	0.992472i	$0.539083\pi$
$224$	1.93693	+	1.80230i	0.129417	+	0.120421i
$225$		0			0
$226$	−10.2330			−0.680689
$227$		−	4.23301i		−	0.280955i	−0.990084	−	0.140477i	$-0.955136\pi$
$227$		−	4.23301i		−	0.280955i	0.990084	−	0.140477i	$-0.0448637\pi$
$228$		0			0
$229$			7.59497i			0.501890i	0.968001	+	0.250945i	$0.0807413\pi$
$229$			7.59497i			0.501890i	−0.968001	+	0.250945i	$0.919259\pi$
$230$		0			0
$231$		0			0
$232$		−	5.47847i		−	0.359679i
$233$	−9.94677			−0.651634			−0.325817	−	0.945433i	$-0.605639\pi$
$233$	−9.94677			−0.651634			−0.325817	+	0.945433i	$0.605639\pi$
$234$		0			0
$235$		0			0
$236$	4.61142			0.300178
$237$		0			0
$238$	14.6284	−	15.7211i	0.948218	−	1.01905i
$239$		−	9.36197i		−	0.605575i	−0.953058	−	0.302788i	$-0.902083\pi$
$239$		−	9.36197i		−	0.605575i	0.953058	−	0.302788i	$-0.0979173\pi$
$240$		0			0
$241$		−	23.0408i		−	1.48419i	−0.670296	−	0.742093i	$-0.733833\pi$
$241$		−	23.0408i		−	1.48419i	0.670296	−	0.742093i	$-0.266167\pi$
$242$	4.00681			0.257568
$243$		0			0
$244$			11.8472i			0.758442i
$245$		0			0
$246$		0			0
$247$			4.23301i			0.269340i
$248$			3.73074i			0.236902i
$249$		0			0
$250$		0			0
$251$	−23.8376			−1.50462			−0.752308	−	0.658811i	$-0.771060\pi$
$251$	−23.8376			−1.50462			−0.752308	+	0.658811i	$0.771060\pi$
$252$		0			0
$253$			21.2228i			1.33427i
$254$			2.81382i			0.176555i
$255$		0			0
$256$	1.00000			0.0625000
$257$			29.7441i			1.85539i	0.373341	+	0.927694i	$0.378212\pi$
$257$			29.7441i			1.85539i	−0.373341	+	0.927694i	$0.621788\pi$
$258$		0			0
$259$	8.13181	−	8.73923i	0.505286	−	0.543030i
$260$		0			0
$261$		0			0
$262$	−8.62840			−0.533064
$263$	22.7545			1.40310			0.701552	−	0.712618i	$-0.252491\pi$
$263$	22.7545			1.40310			0.701552	+	0.712618i	$0.252491\pi$
$264$		0			0
$265$		0			0
$266$	4.75454	−	5.10969i	0.291519	−	0.313295i
$267$		0			0
$268$			6.90729i			0.421930i
$269$	−0.202401			−0.0123406			−0.00617030	−	0.999981i	$-0.501964\pi$
$269$	−0.202401			−0.0123406			−0.00617030	+	0.999981i	$0.501964\pi$
$270$		0			0
$271$		−	11.4785i		−	0.697267i	−0.937259	−	0.348634i	$-0.886646\pi$
$271$		−	11.4785i		−	0.697267i	0.937259	−	0.348634i	$-0.113354\pi$
$272$		−	8.11650i		−	0.492135i
$273$		0			0
$274$	0.723932			0.0437343
$275$		0			0
$276$		0			0
$277$			23.7211i			1.42526i	0.701538	+	0.712632i	$0.252497\pi$
$277$			23.7211i			1.42526i	−0.701538	+	0.712632i	$0.747503\pi$
$278$		−	2.84043i		−	0.170358i
$279$		0			0
$280$		0			0
$281$			5.57194i			0.332394i	0.986093	+	0.166197i	$0.0531488\pi$
$281$			5.57194i			0.332394i	−0.986093	+	0.166197i	$0.946851\pi$
$282$		0			0
$283$	−32.1798			−1.91289			−0.956445	−	0.291914i	$-0.905708\pi$
$283$	−32.1798			−1.91289			−0.956445	+	0.291914i	$0.905708\pi$
$284$		−	2.63803i		−	0.156539i
$285$		0			0
$286$		−	6.21603i		−	0.367561i
$287$	−3.10801	−	2.89199i	−0.183460	−	0.170709i
$288$		0			0
$289$	−48.8776			−2.87515
$290$		0			0
$291$		0			0
$292$	−13.7477			−0.804525
$293$			15.8146i			0.923898i	0.886907	+	0.461949i	$0.152850\pi$
$293$			15.8146i			0.923898i	−0.886907	+	0.461949i	$0.847150\pi$
$294$		0			0
$295$		0			0
$296$		−	4.51190i		−	0.262249i
$297$		0			0
$298$		−	3.00681i		−	0.174180i
$299$	8.79079			0.508384
$300$		0			0
$301$	18.2330	−	19.5950i	1.05093	−	1.12944i
$302$	15.2262			0.876169
$303$		0			0
$304$		−	2.63803i		−	0.151302i
$305$		0			0
$306$		0			0
$307$	6.72393			0.383755			0.191878	−	0.981419i	$-0.438542\pi$
$307$	6.72393			0.383755			0.191878	+	0.981419i	$0.438542\pi$
$308$	−6.98188	+	7.50341i	−0.397829	+	0.427546i
$309$		0			0
$310$		0			0
$311$	−1.02379			−0.0580540			−0.0290270	−	0.999579i	$-0.509241\pi$
$311$	−1.02379			−0.0580540			−0.0290270	+	0.999579i	$0.509241\pi$
$312$		0			0
$313$	−23.2568			−1.31455			−0.657276	−	0.753650i	$-0.728291\pi$
$313$	−23.2568			−1.31455			−0.657276	+	0.753650i	$0.728291\pi$
$314$	−2.64767			−0.149417
$315$		0			0
$316$	8.01698			0.450990
$317$	−4.46830			−0.250965			−0.125482	−	0.992096i	$-0.540048\pi$
$317$	−4.46830			−0.250965			−0.125482	+	0.992096i	$0.540048\pi$
$318$		0			0
$319$	21.2228			1.18825
$320$		0			0
$321$		0			0
$322$	−10.6114	−	9.87386i	−0.591351	−	0.550249i
$323$	−21.4116			−1.19137
$324$		0			0
$325$		0			0
$326$		−	9.32572i		−	0.516504i
$327$		0			0
$328$	−1.60461			−0.0885996
$329$	−20.0230	+	21.5187i	−1.10391	+	1.18636i
$330$		0			0
$331$	−14.2160			−0.781383			−0.390692	−	0.920522i	$-0.627764\pi$
$331$	−14.2160			−0.781383			−0.390692	+	0.920522i	$0.627764\pi$
$332$			3.20921i			0.176128i
$333$		0			0
$334$			17.3257i			0.948021i
$335$		0			0
$336$		0			0
$337$			23.4286i			1.27624i	0.769938	+	0.638118i	$0.220287\pi$
$337$			23.4286i			1.27624i	−0.769938	+	0.638118i	$0.779713\pi$
$338$	10.4252			0.567058
$339$		0			0
$340$		0			0
$341$	−14.4524			−0.782641
$342$		0			0
$343$	11.6084	−	14.4307i	0.626794	−	0.779185i
$344$		−	10.1165i		−	0.545445i
$345$		0			0
$346$		−	2.99319i		−	0.160915i
$347$	20.1798			1.08331			0.541654	−	0.840602i	$-0.317798\pi$
$347$	20.1798			1.08331			0.541654	+	0.840602i	$0.317798\pi$
$348$		0			0
$349$		−	27.0372i		−	1.44727i	−0.690184	−	0.723634i	$-0.742470\pi$
$349$		−	27.0372i		−	1.44727i	0.690184	−	0.723634i	$-0.257530\pi$
$350$		0			0
$351$		0			0
$352$			3.87386i			0.206478i
$353$			12.3495i			0.657298i	0.944452	+	0.328649i	$0.106593\pi$
$353$			12.3495i			0.657298i	−0.944452	+	0.328649i	$0.893407\pi$
$354$		0			0
$355$		0			0
$356$	−17.8376			−0.945392
$357$		0			0
$358$		−	7.38858i		−	0.390499i
$359$			25.5757i			1.34983i	0.737894	+	0.674917i	$0.235821\pi$
$359$			25.5757i			1.34983i	−0.737894	+	0.674917i	$0.764179\pi$
$360$		0			0
$361$	12.0408			0.633725
$362$			11.1097i			0.583912i
$363$		0			0
$364$	3.10801	+	2.89199i	0.162904	+	0.151581i
$365$		0			0
$366$		0			0
$367$	20.8444			1.08807			0.544035	−	0.839062i	$-0.316896\pi$
$367$	20.8444			1.08807			0.544035	+	0.839062i	$0.316896\pi$
$368$	−5.47847			−0.285585
$369$		0			0
$370$		0			0
$371$	−4.39539	−	4.08989i	−0.228197	−	0.212336i
$372$		0			0
$373$		−	7.48810i		−	0.387719i	−0.981029	−	0.193860i	$-0.937899\pi$
$373$		−	7.48810i		−	0.387719i	0.981029	−	0.193860i	$-0.0621006\pi$
$374$	31.4422			1.62584
$375$		0			0
$376$			11.1097i			0.572939i
$377$		−	8.79079i		−	0.452749i
$378$		0			0
$379$	−32.6651			−1.67789			−0.838946	−	0.544215i	$-0.816827\pi$
$379$	−32.6651			−1.67789			−0.838946	+	0.544215i	$0.816827\pi$
$380$		0			0
$381$		0			0
$382$			9.36197i			0.479000i
$383$		−	0.890309i		−	0.0454927i	−0.999741	−	0.0227463i	$-0.992759\pi$
$383$		−	0.890309i		−	0.0454927i	0.999741	−	0.0227463i	$-0.00724101\pi$
$384$		0			0
$385$		0			0
$386$			17.4422i			0.887786i
$387$		0			0
$388$	−8.68768			−0.441050
$389$			13.5317i			0.686084i	0.939320	+	0.343042i	$0.111457\pi$
$389$			13.5317i			0.686084i	−0.939320	+	0.343042i	$0.888543\pi$
$390$		0			0
$391$			44.4660i			2.24874i
$392$	−0.503406	−	6.98188i	−0.0254258	−	0.352638i
$393$		0			0
$394$	−5.27607			−0.265804
$395$		0			0
$396$		0			0
$397$	25.2991			1.26973			0.634863	−	0.772625i	$-0.281057\pi$
$397$	25.2991			1.26973			0.634863	+	0.772625i	$0.281057\pi$
$398$		−	12.2160i		−	0.612334i
$399$		0			0
$400$		0			0
$401$			32.7619i			1.63605i	0.575182	+	0.818025i	$0.304931\pi$
$401$			32.7619i			1.63605i	−0.575182	+	0.818025i	$0.695069\pi$
$402$		0			0
$403$			5.98638i			0.298203i
$404$	−6.21603			−0.309259
$405$		0			0
$406$	−9.87386	+	10.6114i	−0.490032	+	0.526636i
$407$	17.4785			0.866376
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$		0			0
$412$	12.8807			0.634585
$413$	−8.93200	−	8.31117i	−0.439515	−	0.408966i
$414$		0			0
$415$		0			0
$416$	1.60461			0.0786723
$417$		0			0
$418$	10.2194			0.499846
$419$	4.17937			0.204176			0.102088	−	0.994775i	$-0.467448\pi$
$419$	4.17937			0.204176			0.102088	+	0.994775i	$0.467448\pi$
$420$		0			0
$421$	−32.6514			−1.59133			−0.795667	−	0.605735i	$-0.792879\pi$
$421$	−32.6514			−1.59133			−0.795667	+	0.605735i	$0.792879\pi$
$422$	−4.00000			−0.194717
$423$		0			0
$424$	−2.26926			−0.110205
$425$		0			0
$426$		0			0
$427$	21.3523	−	22.9473i	1.03331	−	1.11050i
$428$	−19.9638			−0.964984
$429$		0			0
$430$		0			0
$431$			13.3087i			0.641059i	0.947239	+	0.320530i	$0.103861\pi$
$431$			13.3087i			0.641059i	−0.947239	+	0.320530i	$0.896139\pi$
$432$		0			0
$433$	6.95358			0.334168			0.167084	−	0.985943i	$-0.446565\pi$
$433$	6.95358			0.334168			0.167084	+	0.985943i	$0.446565\pi$
$434$	6.72393	−	7.22619i	0.322759	−	0.346868i
$435$		0			0
$436$	15.0238			0.719509
$437$			14.4524i			0.691352i
$438$		0			0
$439$		−	26.9739i		−	1.28739i	−0.765280	−	0.643697i	$-0.777400\pi$
$439$		−	26.9739i		−	1.28739i	0.765280	−	0.643697i	$-0.222600\pi$
$440$		0			0
$441$		0			0
$442$		−	13.0238i		−	0.619479i
$443$	−2.80441			−0.133242			−0.0666208	−	0.997778i	$-0.521222\pi$
$443$	−2.80441			−0.133242			−0.0666208	+	0.997778i	$0.521222\pi$
$444$		0			0
$445$		0			0
$446$	3.65784			0.173204
$447$		0			0
$448$	−1.93693	−	1.80230i	−0.0915114	−	0.0851508i
$449$		−	30.1226i		−	1.42157i	−0.703409	−	0.710786i	$-0.748340\pi$
$449$		−	30.1226i		−	1.42157i	0.703409	−	0.710786i	$-0.251660\pi$
$450$		0			0
$451$		−	6.21603i		−	0.292701i
$452$	10.2330			0.481320
$453$		0			0
$454$			4.23301i			0.198665i
$455$		0			0
$456$		0			0
$457$		−	22.0340i		−	1.03071i	−0.856978	−	0.515353i	$-0.827661\pi$
$457$		−	22.0340i		−	1.03071i	0.856978	−	0.515353i	$-0.172339\pi$
$458$		−	7.59497i		−	0.354890i
$459$		0			0
$460$		0			0
$461$	−15.4116			−0.717790			−0.358895	−	0.933378i	$-0.616846\pi$
$461$	−15.4116			−0.717790			−0.358895	+	0.933378i	$0.616846\pi$
$462$		0			0
$463$			25.0605i			1.16466i	0.812953	+	0.582329i	$0.197858\pi$
$463$			25.0605i			1.16466i	−0.812953	+	0.582329i	$0.802142\pi$
$464$			5.47847i			0.254332i
$465$		0			0
$466$	9.94677			0.460775
$467$		−	9.62764i		−	0.445514i	−0.974874	−	0.222757i	$-0.928494\pi$
$467$		−	9.62764i		−	0.445514i	0.974874	−	0.222757i	$-0.0715056\pi$
$468$		0			0
$469$	12.4490	−	13.3789i	0.574843	−	0.617782i
$470$		0			0
$471$		0			0
$472$	−4.61142			−0.212258
$473$	39.1899			1.80196
$474$		0			0
$475$		0			0
$476$	−14.6284	+	15.7211i	−0.670492	+	0.720576i
$477$		0			0
$478$			9.36197i			0.428206i
$479$	−37.0238			−1.69166			−0.845830	−	0.533452i	$-0.820894\pi$
$479$	−37.0238			−1.69166			−0.845830	+	0.533452i	$0.820894\pi$
$480$		0			0
$481$		−	7.23982i		−	0.330107i
$482$			23.0408i			1.04948i
$483$		0			0
$484$	−4.00681			−0.182128
$485$		0			0
$486$		0			0
$487$		−	11.6386i		−	0.527394i	−0.964606	−	0.263697i	$-0.915058\pi$
$487$		−	11.6386i		−	0.527394i	0.964606	−	0.263697i	$-0.0849419\pi$
$488$		−	11.8472i		−	0.536300i
$489$		0			0
$490$		0			0
$491$		−	32.1069i		−	1.44896i	−0.689294	−	0.724481i	$-0.742079\pi$
$491$		−	32.1069i		−	1.44896i	0.689294	−	0.724481i	$-0.257921\pi$
$492$		0			0
$493$	44.4660			2.00265
$494$		−	4.23301i		−	0.190452i
$495$		0			0
$496$		−	3.73074i		−	0.167515i
$497$	−4.75454	+	5.10969i	−0.213270	+	0.229201i
$498$		0			0
$499$	−13.6122			−0.609365			−0.304682	−	0.952454i	$-0.598550\pi$
$499$	−13.6122			−0.609365			−0.304682	+	0.952454i	$0.598550\pi$
$500$		0			0
$501$		0			0
$502$	23.8376			1.06392
$503$			3.07573i			0.137140i	0.997646	+	0.0685700i	$0.0218436\pi$
$503$			3.07573i			0.137140i	−0.997646	+	0.0685700i	$0.978156\pi$
$504$		0			0
$505$		0			0
$506$		−	21.2228i		−	0.943470i
$507$		0			0
$508$		−	2.81382i		−	0.124843i
$509$	−14.7908			−0.655590			−0.327795	−	0.944749i	$-0.606306\pi$
$509$	−14.7908			−0.655590			−0.327795	+	0.944749i	$0.606306\pi$
$510$		0			0
$511$	26.6284	+	24.7776i	1.17797	+	1.09610i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	29.7441i		−	1.31196i
$515$		0			0
$516$		0			0
$517$	−43.0374			−1.89278
$518$	−8.13181	+	8.73923i	−0.357291	+	0.383980i
$519$		0			0
$520$		0			0
$521$	−27.0332			−1.18435			−0.592173	−	0.805811i	$-0.701730\pi$
$521$	−27.0332			−1.18435			−0.592173	+	0.805811i	$0.701730\pi$
$522$		0			0
$523$	−3.51472			−0.153688			−0.0768440	−	0.997043i	$-0.524484\pi$
$523$	−3.51472			−0.153688			−0.0768440	+	0.997043i	$0.524484\pi$
$524$	8.62840			0.376933
$525$		0			0
$526$	−22.7545			−0.992145
$527$	−30.2806			−1.31904
$528$		0			0
$529$	7.01362			0.304940
$530$		0			0
$531$		0			0
$532$	−4.75454	+	5.10969i	−0.206135	+	0.221533i
$533$	−2.57476			−0.111525
$534$		0			0
$535$		0			0
$536$		−	6.90729i		−	0.298350i
$537$		0			0
$538$	0.202401			0.00872612
$539$	27.0468	−	1.95013i	1.16499	−	0.0839979i
$540$		0			0
$541$	−39.6616			−1.70519			−0.852593	−	0.522576i	$-0.824971\pi$
$541$	−39.6616			−1.70519			−0.852593	+	0.522576i	$0.824971\pi$
$542$			11.4785i			0.493042i
$543$		0			0
$544$			8.11650i			0.347992i
$545$		0			0
$546$		0			0
$547$		−	34.3495i		−	1.46868i	−0.678782	−	0.734339i	$-0.737492\pi$
$547$		−	34.3495i		−	1.46868i	0.678782	−	0.734339i	$-0.262508\pi$
$548$	−0.723932			−0.0309248
$549$		0			0
$550$		0			0
$551$	14.4524			0.615692
$552$		0			0
$553$	−15.5283	−	14.4490i	−0.660332	−	0.614435i
$554$		−	23.7211i		−	1.00781i
$555$		0			0
$556$			2.84043i			0.120461i
$557$	38.4660			1.62986			0.814929	−	0.579561i	$-0.196776\pi$
$557$	38.4660			1.62986			0.814929	+	0.579561i	$0.196776\pi$
$558$		0			0
$559$		−	16.2330i		−	0.686583i
$560$		0			0
$561$		0			0
$562$		−	5.57194i		−	0.235038i
$563$			10.2194i			0.430696i	0.976537	+	0.215348i	$0.0690885\pi$
$563$			10.2194i			0.430696i	−0.976537	+	0.215348i	$0.930911\pi$
$564$		0			0
$565$		0			0
$566$	32.1798			1.35262
$567$		0			0
$568$			2.63803i			0.110689i
$569$		−	2.48129i		−	0.104021i	−0.998647	−	0.0520106i	$-0.983437\pi$
$569$		−	2.48129i		−	0.104021i	0.998647	−	0.0520106i	$-0.0165629\pi$
$570$		0			0
$571$	20.6345			0.863525			0.431762	−	0.901987i	$-0.357892\pi$
$571$	20.6345			0.863525			0.431762	+	0.901987i	$0.357892\pi$
$572$			6.21603i			0.259905i
$573$		0			0
$574$	3.10801	+	2.89199i	0.129726	+	0.120709i
$575$		0			0
$576$		0			0
$577$	−2.50455			−0.104266			−0.0521329	−	0.998640i	$-0.516602\pi$
$577$	−2.50455			−0.104266			−0.0521329	+	0.998640i	$0.516602\pi$
$578$	48.8776			2.03304
$579$		0			0
$580$		0			0
$581$	5.78397	−	6.21603i	0.239960	−	0.257884i
$582$		0			0
$583$		−	8.79079i		−	0.364077i
$584$	13.7477			0.568885
$585$		0			0
$586$		−	15.8146i		−	0.653294i
$587$			31.4422i			1.29776i	0.760891	+	0.648880i	$0.224762\pi$
$587$			31.4422i			1.29776i	−0.760891	+	0.648880i	$0.775238\pi$
$588$		0			0
$589$	−9.84183			−0.405526
$590$		0			0
$591$		0			0
$592$			4.51190i			0.185438i
$593$		−	27.1267i		−	1.11396i	−0.830526	−	0.556979i	$-0.811960\pi$
$593$		−	27.1267i		−	1.11396i	0.830526	−	0.556979i	$-0.188040\pi$
$594$		0			0
$595$		0			0
$596$			3.00681i			0.123164i
$597$		0			0
$598$	−8.79079			−0.359482
$599$		−	33.7747i		−	1.38000i	−0.723810	−	0.689999i	$-0.757611\pi$
$599$		−	33.7747i		−	1.38000i	0.723810	−	0.689999i	$-0.242389\pi$
$600$		0			0
$601$		−	24.4886i		−	0.998912i	−0.866339	−	0.499456i	$-0.833533\pi$
$601$		−	24.4886i		−	0.998912i	0.866339	−	0.499456i	$-0.166467\pi$
$602$	−18.2330	+	19.5950i	−0.743122	+	0.798631i
$603$		0			0
$604$	−15.2262			−0.619545
$605$		0			0
$606$		0			0
$607$	27.3331			1.10941			0.554707	−	0.832045i	$-0.312830\pi$
$607$	27.3331			1.10941			0.554707	+	0.832045i	$0.312830\pi$
$608$			2.63803i			0.106986i
$609$		0			0
$610$		0			0
$611$			17.8267i			0.721190i
$612$		0			0
$613$		−	11.3163i		−	0.457061i	−0.973537	−	0.228531i	$-0.926608\pi$
$613$		−	11.3163i		−	0.457061i	0.973537	−	0.228531i	$-0.0733921\pi$
$614$	−6.72393			−0.271356
$615$		0			0
$616$	6.98188	−	7.50341i	0.281308	−	0.302321i
$617$	10.6843			0.430135			0.215067	−	0.976599i	$-0.431003\pi$
$617$	10.6843			0.430135			0.215067	+	0.976599i	$0.431003\pi$
$618$		0			0
$619$		−	9.56437i		−	0.384424i	−0.981353	−	0.192212i	$-0.938434\pi$
$619$		−	9.56437i		−	0.384424i	0.981353	−	0.192212i	$-0.0615662\pi$
$620$		0			0
$621$		0			0
$622$	1.02379			0.0410504
$623$	34.5502	+	32.1488i	1.38423	+	1.28801i
$624$		0			0
$625$		0			0
$626$	23.2568			0.929529
$627$		0			0
$628$	2.64767			0.105653
$629$	36.6208			1.46017
$630$		0			0
$631$	17.1956			0.684546			0.342273	−	0.939601i	$-0.388803\pi$
$631$	17.1956			0.684546			0.342273	+	0.939601i	$0.388803\pi$
$632$	−8.01698			−0.318898
$633$		0			0
$634$	4.46830			0.177459
$635$		0			0
$636$		0			0
$637$	−0.807769	−	11.2032i	−0.0320050	−	0.443885i
$638$	−21.2228			−0.840220
$639$		0			0
$640$		0			0
$641$			16.5289i			0.652851i	0.945223	+	0.326426i	$0.105844\pi$
$641$			16.5289i			0.652851i	−0.945223	+	0.326426i	$0.894156\pi$
$642$		0			0
$643$	37.4286			1.47604			0.738020	−	0.674779i	$-0.235761\pi$
$643$	37.4286			1.47604			0.738020	+	0.674779i	$0.235761\pi$
$644$	10.6114	+	9.87386i	0.418148	+	0.389085i
$645$		0			0
$646$	21.4116			0.842429
$647$		−	5.52812i		−	0.217333i	−0.994078	−	0.108666i	$-0.965342\pi$
$647$		−	5.52812i		−	0.217333i	0.994078	−	0.108666i	$-0.0346580\pi$
$648$		0			0
$649$		−	17.8640i		−	0.701223i
$650$		0			0
$651$		0			0
$652$			9.32572i			0.365223i
$653$	−18.5159			−0.724583			−0.362291	−	0.932065i	$-0.618005\pi$
$653$	−18.5159			−0.724583			−0.362291	+	0.932065i	$0.618005\pi$
$654$		0			0
$655$		0			0
$656$	1.60461			0.0626494
$657$		0			0
$658$	20.0230	−	21.5187i	0.780579	−	0.838887i
$659$		−	43.3693i		−	1.68943i	−0.535217	−	0.844714i	$-0.679770\pi$
$659$		−	43.3693i		−	1.68943i	0.535217	−	0.844714i	$-0.320230\pi$
$660$		0			0
$661$			36.1142i			1.40468i	0.711842	+	0.702340i	$0.247861\pi$
$661$			36.1142i			1.40468i	−0.711842	+	0.702340i	$0.752139\pi$
$662$	14.2160			0.552522
$663$		0			0
$664$		−	3.20921i		−	0.124542i
$665$		0			0
$666$		0			0
$667$		−	30.0136i		−	1.16213i
$668$		−	17.3257i		−	0.670352i
$669$		0			0
$670$		0			0
$671$	45.8946			1.77174
$672$		0			0
$673$			33.8743i			1.30576i	0.757463	+	0.652879i	$0.226439\pi$
$673$			33.8743i			1.30576i	−0.757463	+	0.652879i	$0.773561\pi$
$674$		−	23.4286i		−	0.902436i
$675$		0			0
$676$	−10.4252			−0.400971
$677$		−	4.38446i		−	0.168509i	−0.996444	−	0.0842543i	$-0.973149\pi$
$677$		−	4.38446i		−	0.168509i	0.996444	−	0.0842543i	$-0.0268508\pi$
$678$		0			0
$679$	16.8274	+	15.6578i	0.645778	+	0.600893i
$680$		0			0
$681$		0			0
$682$	14.4524			0.553411
$683$	−1.95013			−0.0746195			−0.0373098	−	0.999304i	$-0.511879\pi$
$683$	−1.95013			−0.0746195			−0.0373098	+	0.999304i	$0.511879\pi$
$684$		0			0
$685$		0			0
$686$	−11.6084	+	14.4307i	−0.443211	+	0.550967i
$687$		0			0
$688$			10.1165i			0.385688i
$689$	−3.64126			−0.138721
$690$		0			0
$691$			12.1471i			0.462098i	0.972942	+	0.231049i	$0.0742157\pi$
$691$			12.1471i			0.462098i	−0.972942	+	0.231049i	$0.925784\pi$
$692$			2.99319i			0.113784i
$693$		0			0
$694$	−20.1798			−0.766014
$695$		0			0
$696$		0			0
$697$		−	13.0238i		−	0.493311i
$698$			27.0372i			1.02337i
$699$		0			0
$700$		0			0
$701$		−	11.0600i		−	0.417732i	−0.977944	−	0.208866i	$-0.933023\pi$
$701$		−	11.0600i		−	0.417732i	0.977944	−	0.208866i	$-0.0669773\pi$
$702$		0			0
$703$	11.9025			0.448913
$704$		−	3.87386i		−	0.146002i
$705$		0			0
$706$		−	12.3495i		−	0.464780i
$707$	12.0400	+	11.2032i	0.452811	+	0.421338i
$708$		0			0
$709$	−14.2330			−0.534532			−0.267266	−	0.963623i	$-0.586120\pi$
$709$	−14.2330			−0.534532			−0.267266	+	0.963623i	$0.586120\pi$
$710$		0			0
$711$		0			0
$712$	17.8376			0.668493
$713$			20.4388i			0.765438i
$714$		0			0
$715$		0			0
$716$			7.38858i			0.276124i
$717$		0			0
$718$		−	25.5757i		−	0.954477i
$719$	20.4660			0.763254			0.381627	−	0.924317i	$-0.375364\pi$
$719$	20.4660			0.763254			0.381627	+	0.924317i	$0.375364\pi$
$720$		0			0
$721$	−24.9490	−	23.2149i	−0.929149	−	0.864567i
$722$	−12.0408			−0.448111
$723$		0			0
$724$		−	11.1097i		−	0.412888i
$725$		0			0
$726$		0			0
$727$	25.5717			0.948402			0.474201	−	0.880417i	$-0.342737\pi$
$727$	25.5717			0.948402			0.474201	+	0.880417i	$0.342737\pi$
$728$	−3.10801	−	2.89199i	−0.115191	−	0.107184i
$729$		0			0
$730$		0			0
$731$	82.1106			3.03697
$732$		0			0
$733$	22.6624			0.837053			0.418527	−	0.908204i	$-0.362547\pi$
$733$	22.6624			0.837053			0.418527	+	0.908204i	$0.362547\pi$
$734$	−20.8444			−0.769382
$735$		0			0
$736$	5.47847			0.201939
$737$	26.7579			0.985640
$738$		0			0
$739$	44.8675			1.65048			0.825238	−	0.564785i	$-0.191041\pi$
$739$	44.8675			1.65048			0.825238	+	0.564785i	$0.191041\pi$
$740$		0			0
$741$		0			0
$742$	4.39539	+	4.08989i	0.161360	+	0.150145i
$743$	5.37536			0.197203			0.0986015	−	0.995127i	$-0.468563\pi$
$743$	5.37536			0.197203			0.0986015	+	0.995127i	$0.468563\pi$
$744$		0			0
$745$		0			0
$746$			7.48810i			0.274159i
$747$		0			0
$748$	−31.4422			−1.14964
$749$	38.6684	+	35.9807i	1.41291	+	1.31471i
$750$		0			0
$751$	13.6276			0.497280			0.248640	−	0.968596i	$-0.420016\pi$
$751$	13.6276			0.497280			0.248640	+	0.968596i	$0.420016\pi$
$752$		−	11.1097i		−	0.405129i
$753$		0			0
$754$			8.79079i			0.320142i
$755$		0			0
$756$		0			0
$757$		−	4.54586i		−	0.165222i	−0.996582	−	0.0826110i	$-0.973674\pi$
$757$		−	4.54586i		−	0.165222i	0.996582	−	0.0826110i	$-0.0263259\pi$
$758$	32.6651			1.18645
$759$		0			0
$760$		0			0
$761$	−15.6522			−0.567392			−0.283696	−	0.958914i	$-0.591561\pi$
$761$	−15.6522			−0.567392			−0.283696	+	0.958914i	$0.591561\pi$
$762$		0			0
$763$	−29.1001	−	27.0774i	−1.05349	−	0.980269i
$764$		−	9.36197i		−	0.338704i
$765$		0			0
$766$			0.890309i			0.0321682i
$767$	−7.39951			−0.267181
$768$		0			0
$769$		−	17.1922i		−	0.619968i	−0.950742	−	0.309984i	$-0.899676\pi$
$769$		−	17.1922i		−	0.619968i	0.950742	−	0.309984i	$-0.100324\pi$
$770$		0			0
$771$		0			0
$772$		−	17.4422i		−	0.627759i
$773$		−	42.6990i		−	1.53578i	−0.640584	−	0.767889i	$-0.721307\pi$
$773$		−	42.6990i		−	1.53578i	0.640584	−	0.767889i	$-0.278693\pi$
$774$		0			0
$775$		0			0
$776$	8.68768			0.311870
$777$		0			0
$778$		−	13.5317i		−	0.485135i
$779$		−	4.23301i		−	0.151663i
$780$		0			0
$781$	−10.2194			−0.365678
$782$		−	44.4660i		−	1.59010i
$783$		0			0
$784$	0.503406	+	6.98188i	0.0179788	+	0.249353i
$785$		0			0
$786$		0			0
$787$	30.3992			1.08361			0.541806	−	0.840503i	$-0.317741\pi$
$787$	30.3992			1.08361			0.541806	+	0.840503i	$0.317741\pi$
$788$	5.27607			0.187952
$789$		0			0
$790$		0			0
$791$	−19.8206	−	18.4430i	−0.704741	−	0.655757i
$792$		0			0
$793$		−	19.0102i		−	0.675071i
$794$	−25.2991			−0.897831
$795$		0			0
$796$			12.2160i			0.432986i
$797$			2.99319i			0.106024i	0.998594	+	0.0530121i	$0.0168822\pi$
$797$			2.99319i			0.106024i	−0.998594	+	0.0530121i	$0.983118\pi$
$798$		0			0
$799$	−90.1718			−3.19005
$800$		0			0
$801$		0			0
$802$		−	32.7619i		−	1.15686i
$803$			53.2568i			1.87939i
$804$		0			0
$805$		0			0
$806$		−	5.98638i		−	0.210861i
$807$		0			0
$808$	6.21603			0.218679
$809$			25.0334i			0.880128i	0.897966	+	0.440064i	$0.145044\pi$
$809$			25.0334i			0.880128i	−0.897966	+	0.440064i	$0.854956\pi$
$810$		0			0
$811$			19.4782i			0.683974i	0.939705	+	0.341987i	$0.111100\pi$
$811$			19.4782i			0.683974i	−0.939705	+	0.341987i	$0.888900\pi$
$812$	9.87386	−	10.6114i	0.346505	−	0.372388i
$813$		0			0
$814$	−17.4785			−0.612620
$815$		0			0
$816$		0			0
$817$	26.6877			0.933684
$818$		−	12.0000i		−	0.419570i
$819$		0			0
$820$		0			0
$821$		−	45.3787i		−	1.58373i	−0.610697	−	0.791864i	$-0.709111\pi$
$821$		−	45.3787i		−	1.58373i	0.610697	−	0.791864i	$-0.290889\pi$
$822$		0			0
$823$		−	25.8512i		−	0.901117i	−0.892747	−	0.450559i	$-0.851225\pi$
$823$		−	25.8512i		−	0.901117i	0.892747	−	0.450559i	$-0.148775\pi$
$824$	−12.8807			−0.448720
$825$		0			0
$826$	8.93200	+	8.31117i	0.310784	+	0.289183i
$827$	−19.3429			−0.672619			−0.336310	−	0.941751i	$-0.609179\pi$
$827$	−19.3429			−0.672619			−0.336310	+	0.941751i	$0.609179\pi$
$828$		0			0
$829$		−	43.4764i		−	1.51000i	−0.655726	−	0.754999i	$-0.727637\pi$
$829$		−	43.4764i		−	1.51000i	0.655726	−	0.754999i	$-0.272363\pi$
$830$		0			0
$831$		0			0
$832$	−1.60461			−0.0556297
$833$	56.6684	−	4.08590i	1.96344	−	0.141568i
$834$		0			0
$835$		0			0
$836$	−10.2194			−0.353445
$837$		0			0
$838$	−4.17937			−0.144374
$839$	24.2670			0.837789			0.418894	−	0.908035i	$-0.362418\pi$
$839$	24.2670			0.837789			0.418894	+	0.908035i	$0.362418\pi$
$840$		0			0
$841$	−1.01362			−0.0349526
$842$	32.6514			1.12524
$843$		0			0
$844$	4.00000			0.137686
$845$		0			0
$846$		0			0
$847$	7.76092	+	7.22149i	0.266668	+	0.248133i
$848$	2.26926			0.0779266
$849$		0			0
$850$		0			0
$851$		−	24.7183i		−	0.847332i
$852$		0			0
$853$	16.2439			0.556182			0.278091	−	0.960555i	$-0.410298\pi$
$853$	16.2439			0.556182			0.278091	+	0.960555i	$0.410298\pi$
$854$	−21.3523	+	22.9473i	−0.730662	+	0.785241i
$855$		0			0
$856$	19.9638			0.682347
$857$			40.5825i			1.38627i	0.720807	+	0.693136i	$0.243772\pi$
$857$			40.5825i			1.38627i	−0.720807	+	0.693136i	$0.756228\pi$
$858$		0			0
$859$		−	5.05310i		−	0.172410i	−0.996277	−	0.0862048i	$-0.972526\pi$
$859$		−	5.05310i		−	0.172410i	0.996277	−	0.0862048i	$-0.0274739\pi$
$860$		0			0
$861$		0			0
$862$		−	13.3087i		−	0.453297i
$863$	−48.3289			−1.64514			−0.822568	−	0.568666i	$-0.807460\pi$
$863$	−48.3289			−1.64514			−0.822568	+	0.568666i	$0.807460\pi$
$864$		0			0
$865$		0			0
$866$	−6.95358			−0.236292
$867$		0			0
$868$	−6.72393	+	7.22619i	−0.228225	+	0.245273i
$869$		−	31.0567i		−	1.05353i
$870$		0			0
$871$		−	11.0835i		−	0.375549i
$872$	−15.0238			−0.508770
$873$		0			0
$874$		−	14.4524i		−	0.488859i
$875$		0			0
$876$		0			0
$877$		−	55.3963i		−	1.87060i	−0.353854	−	0.935301i	$-0.615129\pi$
$877$		−	55.3963i		−	1.87060i	0.353854	−	0.935301i	$-0.384871\pi$
$878$			26.9739i			0.910326i
$879$		0			0
$880$		0			0
$881$	−49.8716			−1.68022			−0.840108	−	0.542419i	$-0.817509\pi$
$881$	−49.8716			−1.68022			−0.840108	+	0.542419i	$0.817509\pi$
$882$		0			0
$883$			43.3393i			1.45848i	0.684255	+	0.729242i	$0.260127\pi$
$883$			43.3393i			1.45848i	−0.684255	+	0.729242i	$0.739873\pi$
$884$			13.0238i			0.438038i
$885$		0			0
$886$	2.80441			0.0942160
$887$		−	33.1539i		−	1.11320i	−0.830781	−	0.556600i	$-0.812106\pi$
$887$		−	33.1539i		−	1.11320i	0.830781	−	0.556600i	$-0.187894\pi$
$888$		0			0
$889$	−5.07136	+	5.45018i	−0.170088	+	0.182793i
$890$		0			0
$891$		0			0
$892$	−3.65784			−0.122473
$893$	−29.3077			−0.980746
$894$		0			0
$895$		0			0
$896$	1.93693	+	1.80230i	0.0647083	+	0.0602107i
$897$		0			0
$898$			30.1226i			1.00520i
$899$	20.4388			0.681671
$900$		0			0
$901$		−	18.4184i		−	0.613607i
$902$			6.21603i			0.206971i
$903$		0			0
$904$	−10.2330			−0.340345
$905$		0			0
$906$		0			0
$907$			50.5553i			1.67866i	0.543622	+	0.839330i	$0.317052\pi$
$907$			50.5553i			1.67866i	−0.543622	+	0.839330i	$0.682948\pi$
$908$		−	4.23301i		−	0.140477i
$909$		0			0
$910$		0			0
$911$		−	21.6755i		−	0.718140i	−0.933311	−	0.359070i	$-0.883094\pi$
$911$		−	21.6755i		−	0.718140i	0.933311	−	0.359070i	$-0.116906\pi$
$912$		0			0
$913$	12.4321			0.411441
$914$			22.0340i			0.728819i
$915$		0			0
$916$			7.59497i			0.250945i
$917$	−16.7126	−	15.5510i	−0.551899	−	0.513539i
$918$		0			0
$919$	25.6616			0.846498			0.423249	−	0.906013i	$-0.360890\pi$
$919$	25.6616			0.846498			0.423249	+	0.906013i	$0.360890\pi$
$920$		0			0
$921$		0			0
$922$	15.4116			0.507554
$923$			4.23301i			0.139331i
$924$		0			0
$925$		0			0
$926$		−	25.0605i		−	0.823538i
$927$		0			0
$928$		−	5.47847i		−	0.179840i
$929$	31.6182			1.03736			0.518680	−	0.854968i	$-0.326424\pi$
$929$	31.6182			1.03736			0.518680	+	0.854968i	$0.326424\pi$
$930$		0			0
$931$	18.4184	−	1.32800i	0.603640	−	0.0435235i
$932$	−9.94677			−0.325817
$933$		0			0
$934$			9.62764i			0.315026i
$935$		0			0
$936$		0			0
$937$	38.7013			1.26432			0.632158	−	0.774839i	$-0.282169\pi$
$937$	38.7013			1.26432			0.632158	+	0.774839i	$0.282169\pi$
$938$	−12.4490	+	13.3789i	−0.406475	+	0.436838i
$939$		0			0
$940$		0			0
$941$	−32.9932			−1.07555			−0.537774	−	0.843089i	$-0.680734\pi$
$941$	−32.9932			−1.07555			−0.537774	+	0.843089i	$0.680734\pi$
$942$		0			0
$943$	−8.79079			−0.286267
$944$	4.61142			0.150089
$945$		0			0
$946$	−39.1899			−1.27418
$947$	−31.8452			−1.03483			−0.517415	−	0.855735i	$-0.673106\pi$
$947$	−31.8452			−1.03483			−0.517415	+	0.855735i	$0.673106\pi$
$948$		0			0
$949$	22.0597			0.716088
$950$		0			0
$951$		0			0
$952$	14.6284	−	15.7211i	0.474109	−	0.509524i
$953$	−45.9003			−1.48686			−0.743428	−	0.668817i	$-0.766801\pi$
$953$	−45.9003			−1.48686			−0.743428	+	0.668817i	$0.766801\pi$
$954$		0			0
$955$		0			0
$956$		−	9.36197i		−	0.302788i
$957$		0			0
$958$	37.0238			1.19618
$959$	1.40221	+	1.30474i	0.0452796	+	0.0421324i
$960$		0			0
$961$	17.0816			0.551018
$962$			7.23982i			0.233421i
$963$		0			0
$964$		−	23.0408i		−	0.742093i
$965$		0			0
$966$		0			0
$967$		−	41.2662i		−	1.32703i	−0.748162	−	0.663516i	$-0.769064\pi$
$967$		−	41.2662i		−	1.32703i	0.748162	−	0.663516i	$-0.230936\pi$
$968$	4.00681			0.128784
$969$		0			0
$970$		0			0
$971$	−11.0008			−0.353031			−0.176516	−	0.984298i	$-0.556483\pi$
$971$	−11.0008			−0.353031			−0.176516	+	0.984298i	$0.556483\pi$
$972$		0			0
$973$	5.11933	−	5.50173i	0.164118	−	0.176377i
$974$			11.6386i			0.372924i
$975$		0			0
$976$			11.8472i			0.379221i
$977$	−2.48528			−0.0795112			−0.0397556	−	0.999209i	$-0.512658\pi$
$977$	−2.48528			−0.0795112			−0.0397556	+	0.999209i	$0.512658\pi$
$978$		0			0
$979$			69.1005i			2.20846i
$980$		0			0
$981$		0			0
$982$			32.1069i			1.02457i
$983$		−	31.7408i		−	1.01237i	−0.862424	−	0.506187i	$-0.831055\pi$
$983$		−	31.7408i		−	1.01237i	0.862424	−	0.506187i	$-0.168945\pi$
$984$		0			0
$985$		0			0
$986$	−44.4660			−1.41609
$987$		0			0
$988$			4.23301i			0.134670i
$989$		−	55.4230i		−	1.76235i
$990$		0			0
$991$	57.3368			1.82136			0.910682	−	0.413108i	$-0.135557\pi$
$991$	57.3368			1.82136			0.910682	+	0.413108i	$0.135557\pi$
$992$			3.73074i			0.118451i
$993$		0			0
$994$	4.75454	−	5.10969i	0.150805	−	0.162070i
$995$		0			0
$996$		0			0
$997$	−52.6760			−1.66827			−0.834133	−	0.551564i	$-0.814031\pi$
$997$	−52.6760			−1.66827			−0.834133	+	0.551564i	$0.814031\pi$
$998$	13.6122			0.430886
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.d.c.3149.1		8
3.2	odd	2		3150.2.d.f.3149.1		8
5.2	odd	4		3150.2.b.e.251.3		8
5.3	odd	4		630.2.b.a.251.6	yes	8
5.4	even	2		3150.2.d.d.3149.8		8
7.6	odd	2		3150.2.d.a.3149.7		8
15.2	even	4		3150.2.b.f.251.7		8
15.8	even	4		630.2.b.b.251.2	yes	8
15.14	odd	2		3150.2.d.a.3149.8		8
20.3	even	4		5040.2.f.f.881.5		8
21.20	even	2		3150.2.d.d.3149.7		8
35.13	even	4		630.2.b.b.251.6	yes	8
35.27	even	4		3150.2.b.f.251.3		8
35.34	odd	2		3150.2.d.f.3149.2		8
60.23	odd	4		5040.2.f.i.881.5		8
105.62	odd	4		3150.2.b.e.251.7		8
105.83	odd	4		630.2.b.a.251.2	✓	8
105.104	even	2	inner	3150.2.d.c.3149.2		8
140.83	odd	4		5040.2.f.i.881.6		8
420.83	even	4		5040.2.f.f.881.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.2	✓	8	105.83	odd	4
630.2.b.a.251.6	yes	8	5.3	odd	4
630.2.b.b.251.2	yes	8	15.8	even	4
630.2.b.b.251.6	yes	8	35.13	even	4
3150.2.b.e.251.3		8	5.2	odd	4
3150.2.b.e.251.7		8	105.62	odd	4
3150.2.b.f.251.3		8	35.27	even	4
3150.2.b.f.251.7		8	15.2	even	4
3150.2.d.a.3149.7		8	7.6	odd	2
3150.2.d.a.3149.8		8	15.14	odd	2
3150.2.d.c.3149.1		8	1.1	even	1	trivial
3150.2.d.c.3149.2		8	105.104	even	2	inner
3150.2.d.d.3149.7		8	21.20	even	2
3150.2.d.d.3149.8		8	5.4	even	2
3150.2.d.f.3149.1		8	3.2	odd	2
3150.2.d.f.3149.2		8	35.34	odd	2
5040.2.f.f.881.5		8	20.3	even	4
5040.2.f.f.881.6		8	420.83	even	4
5040.2.f.i.881.5		8	60.23	odd	4
5040.2.f.i.881.6		8	140.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} -1.00000 q^{8} -3.87386i q^{11} -1.60461 q^{13} +(1.93693 + 1.80230i) q^{14} +1.00000 q^{16} -8.11650i q^{17} -2.63803i q^{19} +3.87386i q^{22} -5.47847 q^{23} +1.60461 q^{26} +(-1.93693 - 1.80230i) q^{28} +5.47847i q^{29} -3.73074i q^{31} -1.00000 q^{32} +8.11650i q^{34} +4.51190i q^{37} +2.63803i q^{38} +1.60461 q^{41} +10.1165i q^{43} -3.87386i q^{44} +5.47847 q^{46} -11.1097i q^{47} +(0.503406 + 6.98188i) q^{49} -1.60461 q^{52} +2.26926 q^{53} +(1.93693 + 1.80230i) q^{56} -5.47847i q^{58} +4.61142 q^{59} +11.8472i q^{61} +3.73074i q^{62} +1.00000 q^{64} +6.90729i q^{67} -8.11650i q^{68} -2.63803i q^{71} -13.7477 q^{73} -4.51190i q^{74} -2.63803i q^{76} +(-6.98188 + 7.50341i) q^{77} +8.01698 q^{79} -1.60461 q^{82} +3.20921i q^{83} -10.1165i q^{86} +3.87386i q^{88} -17.8376 q^{89} +(3.10801 + 2.89199i) q^{91} -5.47847 q^{92} +11.1097i q^{94} -8.68768 q^{97} +(-0.503406 - 6.98188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{13} + 8 q^{16} + 8 q^{23} - 8 q^{26} - 8 q^{32} - 8 q^{41} - 8 q^{46} - 4 q^{49} + 8 q^{52} + 8 q^{53} + 8 q^{64} - 48 q^{73} + 4 q^{77} - 8 q^{79} + 8 q^{82} + 8 q^{89} - 4 q^{91} + 8 q^{92} + 24 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^13 + 8 * q^16 + 8 * q^23 - 8 * q^26 - 8 * q^32 - 8 * q^41 - 8 * q^46 - 4 * q^49 + 8 * q^52 + 8 * q^53 + 8 * q^64 - 48 * q^73 + 4 * q^77 - 8 * q^79 + 8 * q^82 + 8 * q^89 - 4 * q^91 + 8 * q^92 + 24 * q^97 + 4 * q^98

Introduction

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