LMFDB - Embedded newform 2106.2.b.c.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2106,2,Mod(649,2106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2106, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2106.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.b (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:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 $ x^14 + 34x^12 + 435x^10 + 2617x^8 + 7651x^6 + 10260x^4 + 5589x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.97451i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.c.649.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.93284i q^{5} -2.20354i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.93284i q^{5} -2.20354i q^{7} +1.00000i q^{8} -2.93284 q^{10} +5.16934i q^{11} +(-3.40677 - 1.18064i) q^{13} -2.20354 q^{14} +1.00000 q^{16} +2.31124 q^{17} +5.16934i q^{19} +2.93284i q^{20} +5.16934 q^{22} -8.39252 q^{23} -3.60158 q^{25} +(-1.18064 + 3.40677i) q^{26} +2.20354i q^{28} -9.45310 q^{29} +6.22318i q^{31} -1.00000i q^{32} -2.31124i q^{34} -6.46263 q^{35} -0.646136i q^{37} +5.16934 q^{38} +2.93284 q^{40} +0.779346i q^{41} +3.49158 q^{43} -5.16934i q^{44} +8.39252i q^{46} -5.53442i q^{47} +2.14443 q^{49} +3.60158i q^{50} +(3.40677 + 1.18064i) q^{52} -8.68475 q^{53} +15.1609 q^{55} +2.20354 q^{56} +9.45310i q^{58} -2.99070i q^{59} -0.864857 q^{61} +6.22318 q^{62} -1.00000 q^{64} +(-3.46263 + 9.99153i) q^{65} +11.1778i q^{67} -2.31124 q^{68} +6.46263i q^{70} +12.9489i q^{71} -4.27533i q^{73} -0.646136 q^{74} -5.16934i q^{76} +11.3908 q^{77} -3.04350 q^{79} -2.93284i q^{80} +0.779346 q^{82} -3.68705i q^{83} -6.77851i q^{85} -3.49158i q^{86} -5.16934 q^{88} -3.34650i q^{89} +(-2.60158 + 7.50694i) q^{91} +8.39252 q^{92} -5.53442 q^{94} +15.1609 q^{95} -1.49094i q^{97} -2.14443i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4}+O(q^{10}) $ 14 * q - 14 * q^4	$ 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64} + 8 q^{65} - 8 q^{68} - 16 q^{74} + 24 q^{77} - 28 q^{79} - 24 q^{82} - 8 q^{92}+O(q^{100}) $ 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	2.93284i		−	1.31161i	−0.754931	−	0.655804i	$-0.772330\pi$
$5$		−	2.93284i		−	1.31161i	0.754931	−	0.655804i	$-0.227670\pi$
$6$		0			0
$7$		−	2.20354i		−	0.832858i	−0.909168	−	0.416429i	$-0.863281\pi$
$7$		−	2.20354i		−	0.832858i	0.909168	−	0.416429i	$-0.136719\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−2.93284			−0.927447
$11$			5.16934i			1.55861i	0.626642	+	0.779307i	$0.284429\pi$
$11$			5.16934i			1.55861i	−0.626642	+	0.779307i	$0.715571\pi$
$12$		0			0
$13$	−3.40677	−	1.18064i	−0.944868	−	0.327450i
$13$	−3.40677	−	1.18064i	−0.944868	−	0.327450i
$14$	−2.20354			−0.588920
$15$		0			0
$16$	1.00000			0.250000
$17$	2.31124			0.560558			0.280279	−	0.959919i	$-0.409573\pi$
$17$	2.31124			0.560558			0.280279	+	0.959919i	$0.409573\pi$
$18$		0			0
$19$			5.16934i			1.18593i	0.805229	+	0.592964i	$0.202042\pi$
$19$			5.16934i			1.18593i	−0.805229	+	0.592964i	$0.797958\pi$
$20$			2.93284i			0.655804i
$21$		0			0
$22$	5.16934			1.10211
$23$	−8.39252			−1.74996			−0.874981	−	0.484157i	$-0.839126\pi$
$23$	−8.39252			−1.74996			−0.874981	+	0.484157i	$0.839126\pi$
$24$		0			0
$25$	−3.60158			−0.720316
$26$	−1.18064	+	3.40677i	−0.231542	+	0.668123i
$27$		0			0
$28$			2.20354i			0.416429i
$29$	−9.45310			−1.75540			−0.877698	−	0.479214i	$-0.840922\pi$
$29$	−9.45310			−1.75540			−0.877698	+	0.479214i	$0.840922\pi$
$30$		0			0
$31$			6.22318i			1.11772i	0.829263	+	0.558858i	$0.188760\pi$
$31$			6.22318i			1.11772i	−0.829263	+	0.558858i	$0.811240\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	2.31124i		−	0.396374i
$35$	−6.46263			−1.09238
$36$		0			0
$37$		−	0.646136i		−	0.106224i	−0.998589	−	0.0531121i	$-0.983086\pi$
$37$		−	0.646136i		−	0.106224i	0.998589	−	0.0531121i	$-0.0169141\pi$
$38$	5.16934			0.838578
$39$		0			0
$40$	2.93284			0.463723
$41$			0.779346i			0.121713i	0.998147	+	0.0608567i	$0.0193833\pi$
$41$			0.779346i			0.121713i	−0.998147	+	0.0608567i	$0.980617\pi$
$42$		0			0
$43$	3.49158			0.532461			0.266230	−	0.963909i	$-0.414222\pi$
$43$	3.49158			0.532461			0.266230	+	0.963909i	$0.414222\pi$
$44$		−	5.16934i		−	0.779307i
$45$		0			0
$46$			8.39252i			1.23741i
$47$		−	5.53442i		−	0.807279i	−0.914918	−	0.403639i	$-0.867745\pi$
$47$		−	5.53442i		−	0.807279i	0.914918	−	0.403639i	$-0.132255\pi$
$48$		0			0
$49$	2.14443			0.306347
$50$			3.60158i			0.509340i
$51$		0			0
$52$	3.40677	+	1.18064i	0.472434	+	0.163725i
$53$	−8.68475			−1.19294			−0.596471	−	0.802635i	$-0.703431\pi$
$53$	−8.68475			−1.19294			−0.596471	+	0.802635i	$0.703431\pi$
$54$		0			0
$55$	15.1609			2.04429
$56$	2.20354			0.294460
$57$		0			0
$58$			9.45310i			1.24125i
$59$		−	2.99070i		−	0.389356i	−0.980867	−	0.194678i	$-0.937634\pi$
$59$		−	2.99070i		−	0.389356i	0.980867	−	0.194678i	$-0.0623662\pi$
$60$		0			0
$61$	−0.864857			−0.110734			−0.0553668	−	0.998466i	$-0.517633\pi$
$61$	−0.864857			−0.110734			−0.0553668	+	0.998466i	$0.517633\pi$
$62$	6.22318			0.790345
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−3.46263	+	9.99153i	−0.429486	+	1.23930i
$66$		0			0
$67$			11.1778i			1.36558i	0.730614	+	0.682791i	$0.239234\pi$
$67$			11.1778i			1.36558i	−0.730614	+	0.682791i	$0.760766\pi$
$68$	−2.31124			−0.280279
$69$		0			0
$70$			6.46263i			0.772432i
$71$			12.9489i			1.53675i	0.639998	+	0.768376i	$0.278935\pi$
$71$			12.9489i			1.53675i	−0.639998	+	0.768376i	$0.721065\pi$
$72$		0			0
$73$		−	4.27533i		−	0.500390i	−0.968196	−	0.250195i	$-0.919505\pi$
$73$		−	4.27533i		−	0.500390i	0.968196	−	0.250195i	$-0.0804947\pi$
$74$	−0.646136			−0.0751118
$75$		0			0
$76$		−	5.16934i		−	0.592964i
$77$	11.3908			1.29811
$78$		0			0
$79$	−3.04350			−0.342420			−0.171210	−	0.985235i	$-0.554768\pi$
$79$	−3.04350			−0.342420			−0.171210	+	0.985235i	$0.554768\pi$
$80$		−	2.93284i		−	0.327902i
$81$		0			0
$82$	0.779346			0.0860644
$83$		−	3.68705i		−	0.404706i	−0.979313	−	0.202353i	$-0.935141\pi$
$83$		−	3.68705i		−	0.404706i	0.979313	−	0.202353i	$-0.0648589\pi$
$84$		0			0
$85$		−	6.77851i		−	0.735232i
$86$		−	3.49158i		−	0.376506i
$87$		0			0
$88$	−5.16934			−0.551054
$89$		−	3.34650i		−	0.354728i	−0.984145	−	0.177364i	$-0.943243\pi$
$89$		−	3.34650i		−	0.354728i	0.984145	−	0.177364i	$-0.0567570\pi$
$90$		0			0
$91$	−2.60158	+	7.50694i	−0.272720	+	0.786942i
$92$	8.39252			0.874981
$93$		0			0
$94$	−5.53442			−0.570832
$95$	15.1609			1.55547
$96$		0			0
$97$		−	1.49094i		−	0.151382i	−0.997131	−	0.0756910i	$-0.975884\pi$
$97$		−	1.49094i		−	0.151382i	0.997131	−	0.0756910i	$-0.0241163\pi$
$98$		−	2.14443i		−	0.216620i
$99$		0			0
$100$	3.60158			0.360158
$101$	0.406692			0.0404674			0.0202337	−	0.999795i	$-0.493559\pi$
$101$	0.406692			0.0404674			0.0202337	+	0.999795i	$0.493559\pi$
$102$		0			0
$103$	17.1609			1.69091			0.845456	−	0.534046i	$-0.179329\pi$
$103$	17.1609			1.69091			0.845456	+	0.534046i	$0.179329\pi$
$104$	1.18064	−	3.40677i	0.115771	−	0.334061i
$105$		0			0
$106$			8.68475i			0.843538i
$107$	−0.329555			−0.0318592			−0.0159296	−	0.999873i	$-0.505071\pi$
$107$	−0.329555			−0.0318592			−0.0159296	+	0.999873i	$0.505071\pi$
$108$		0			0
$109$		−	9.64927i		−	0.924232i	−0.886819	−	0.462116i	$-0.847090\pi$
$109$		−	9.64927i		−	0.924232i	0.886819	−	0.462116i	$-0.152910\pi$
$110$		−	15.1609i		−	1.44553i
$111$		0			0
$112$		−	2.20354i		−	0.208215i
$113$	−3.03210			−0.285236			−0.142618	−	0.989778i	$-0.545552\pi$
$113$	−3.03210			−0.285236			−0.142618	+	0.989778i	$0.545552\pi$
$114$		0			0
$115$			24.6140i			2.29526i
$116$	9.45310			0.877698
$117$		0			0
$118$	−2.99070			−0.275316
$119$		−	5.09290i		−	0.466865i
$120$		0			0
$121$	−15.7221			−1.42928
$122$			0.864857i			0.0783005i
$123$		0			0
$124$		−	6.22318i		−	0.558858i
$125$		−	4.10135i		−	0.366836i
$126$		0			0
$127$	−9.03355			−0.801598			−0.400799	−	0.916166i	$-0.631267\pi$
$127$	−9.03355			−0.801598			−0.400799	+	0.916166i	$0.631267\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	9.99153	+	3.46263i	0.876315	+	0.303693i
$131$	−17.4172			−1.52175			−0.760874	−	0.648900i	$-0.775229\pi$
$131$	−17.4172			−1.52175			−0.760874	+	0.648900i	$0.775229\pi$
$132$		0			0
$133$	11.3908			0.987710
$134$	11.1778			0.965612
$135$		0			0
$136$			2.31124i			0.198187i
$137$			4.85601i			0.414877i	0.978248	+	0.207438i	$0.0665127\pi$
$137$			4.85601i			0.414877i	−0.978248	+	0.207438i	$0.933487\pi$
$138$		0			0
$139$	−11.6453			−0.987738			−0.493869	−	0.869536i	$-0.664418\pi$
$139$	−11.6453			−0.987738			−0.493869	+	0.869536i	$0.664418\pi$
$140$	6.46263			0.546192
$141$		0			0
$142$	12.9489			1.08665
$143$	6.10312	−	17.6108i	0.510369	−	1.47269i
$144$		0			0
$145$			27.7245i			2.30239i
$146$	−4.27533			−0.353829
$147$		0			0
$148$			0.646136i			0.0531121i
$149$			21.8055i			1.78637i	0.449687	+	0.893186i	$0.351536\pi$
$149$			21.8055i			1.78637i	−0.449687	+	0.893186i	$0.648464\pi$
$150$		0			0
$151$			14.6632i			1.19328i	0.802511	+	0.596638i	$0.203497\pi$
$151$			14.6632i			1.19328i	−0.802511	+	0.596638i	$0.796503\pi$
$152$	−5.16934			−0.419289
$153$		0			0
$154$		−	11.3908i		−	0.917899i
$155$	18.2516			1.46601
$156$		0			0
$157$	7.94869			0.634374			0.317187	−	0.948363i	$-0.397262\pi$
$157$	7.94869			0.634374			0.317187	+	0.948363i	$0.397262\pi$
$158$			3.04350i			0.242128i
$159$		0			0
$160$	−2.93284			−0.231862
$161$			18.4932i			1.45747i
$162$		0			0
$163$			0.222533i			0.0174302i	0.999962	+	0.00871508i	$0.00277413\pi$
$163$			0.222533i			0.0174302i	−0.999962	+	0.00871508i	$0.997226\pi$
$164$		−	0.779346i		−	0.0608567i
$165$		0			0
$166$	−3.68705			−0.286171
$167$		−	3.77070i		−	0.291785i	−0.989300	−	0.145893i	$-0.953395\pi$
$167$		−	3.77070i		−	0.291785i	0.989300	−	0.145893i	$-0.0466054\pi$
$168$		0			0
$169$	10.2122	+	8.04433i	0.785553	+	0.618795i
$170$	−6.77851			−0.519888
$171$		0			0
$172$	−3.49158			−0.266230
$173$	−10.8534			−0.825172			−0.412586	−	0.910919i	$-0.635374\pi$
$173$	−10.8534			−0.825172			−0.412586	+	0.910919i	$0.635374\pi$
$174$		0			0
$175$			7.93621i			0.599921i
$176$			5.16934i			0.389654i
$177$		0			0
$178$	−3.34650			−0.250831
$179$	−14.6400			−1.09425			−0.547123	−	0.837052i	$-0.684277\pi$
$179$	−14.6400			−1.09425			−0.547123	+	0.837052i	$0.684277\pi$
$180$		0			0
$181$	−2.81354			−0.209129			−0.104565	−	0.994518i	$-0.533345\pi$
$181$	−2.81354			−0.209129			−0.104565	+	0.994518i	$0.533345\pi$
$182$	7.50694	+	2.60158i	0.556452	+	0.192842i
$183$		0			0
$184$		−	8.39252i		−	0.618705i
$185$	−1.89502			−0.139324
$186$		0			0
$187$			11.9476i			0.873694i
$188$			5.53442i			0.403639i
$189$		0			0
$190$		−	15.1609i		−	1.09989i
$191$	−8.77640			−0.635038			−0.317519	−	0.948252i	$-0.602850\pi$
$191$	−8.77640			−0.635038			−0.317519	+	0.948252i	$0.602850\pi$
$192$		0			0
$193$			15.8231i			1.13897i	0.822002	+	0.569485i	$0.192857\pi$
$193$			15.8231i			1.13897i	−0.822002	+	0.569485i	$0.807143\pi$
$194$	−1.49094			−0.107043
$195$		0			0
$196$	−2.14443			−0.153174
$197$			0.395717i			0.0281936i	0.999901	+	0.0140968i	$0.00448731\pi$
$197$			0.395717i			0.0281936i	−0.999901	+	0.0140968i	$0.995513\pi$
$198$		0			0
$199$	3.20316			0.227066			0.113533	−	0.993534i	$-0.463783\pi$
$199$	3.20316			0.227066			0.113533	+	0.993534i	$0.463783\pi$
$200$		−	3.60158i		−	0.254670i
$201$		0			0
$202$		−	0.406692i		−	0.0286147i
$203$			20.8302i			1.46200i
$204$		0			0
$205$	2.28570			0.159640
$206$		−	17.1609i		−	1.19565i
$207$		0			0
$208$	−3.40677	−	1.18064i	−0.236217	−	0.0818625i
$209$	−26.7221			−1.84840
$210$		0			0
$211$	8.27972			0.570000			0.285000	−	0.958528i	$-0.408006\pi$
$211$	8.27972			0.570000			0.285000	+	0.958528i	$0.408006\pi$
$212$	8.68475			0.596471
$213$		0			0
$214$			0.329555i			0.0225279i
$215$		−	10.2403i		−	0.698380i
$216$		0			0
$217$	13.7130			0.930900
$218$	−9.64927			−0.653531
$219$		0			0
$220$	−15.1609			−1.02215
$221$	−7.87386	−	2.72874i	−0.529653	−	0.183555i
$222$		0			0
$223$		−	14.3199i		−	0.958928i	−0.877561	−	0.479464i	$-0.840831\pi$
$223$		−	14.3199i		−	0.958928i	0.877561	−	0.479464i	$-0.159169\pi$
$224$	−2.20354			−0.147230
$225$		0			0
$226$			3.03210i			0.201692i
$227$		−	20.0169i		−	1.32857i	−0.747481	−	0.664284i	$-0.768737\pi$
$227$		−	20.0169i		−	1.32857i	0.747481	−	0.664284i	$-0.231263\pi$
$228$		0			0
$229$		−	2.57747i		−	0.170324i	−0.996367	−	0.0851619i	$-0.972859\pi$
$229$		−	2.57747i		−	0.170324i	0.996367	−	0.0851619i	$-0.0271407\pi$
$230$	24.6140			1.62300
$231$		0			0
$232$		−	9.45310i		−	0.620626i
$233$	14.0412			0.919866			0.459933	−	0.887954i	$-0.347873\pi$
$233$	14.0412			0.919866			0.459933	+	0.887954i	$0.347873\pi$
$234$		0			0
$235$	−16.2316			−1.05883
$236$			2.99070i			0.194678i
$237$		0			0
$238$	−5.09290			−0.330124
$239$			4.70266i			0.304190i	0.988366	+	0.152095i	$0.0486019\pi$
$239$			4.70266i			0.304190i	−0.988366	+	0.152095i	$0.951398\pi$
$240$		0			0
$241$		−	22.6013i		−	1.45588i	−0.685642	−	0.727939i	$-0.740478\pi$
$241$		−	22.6013i		−	1.45588i	0.685642	−	0.727939i	$-0.259522\pi$
$242$			15.7221i			1.01065i
$243$		0			0
$244$	0.864857			0.0553668
$245$		−	6.28928i		−	0.401807i
$246$		0			0
$247$	6.10312	−	17.6108i	0.388332	−	1.12055i
$248$	−6.22318			−0.395173
$249$		0			0
$250$	−4.10135			−0.259392
$251$	3.40669			0.215028			0.107514	−	0.994204i	$-0.465711\pi$
$251$	3.40669			0.215028			0.107514	+	0.994204i	$0.465711\pi$
$252$		0			0
$253$		−	43.3838i		−	2.72752i
$254$			9.03355i			0.566815i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−21.8295			−1.36169			−0.680844	−	0.732429i	$-0.738387\pi$
$257$	−21.8295			−1.36169			−0.680844	+	0.732429i	$0.738387\pi$
$258$		0			0
$259$	−1.42378			−0.0884697
$260$	3.46263	−	9.99153i	0.214743	−	0.619649i
$261$		0			0
$262$			17.4172i			1.07604i
$263$	5.91025			0.364442			0.182221	−	0.983258i	$-0.441671\pi$
$263$	5.91025			0.364442			0.182221	+	0.983258i	$0.441671\pi$
$264$		0			0
$265$			25.4710i			1.56467i
$266$		−	11.3908i		−	0.698416i
$267$		0			0
$268$		−	11.1778i		−	0.682791i
$269$	1.05911			0.0645751			0.0322875	−	0.999479i	$-0.489721\pi$
$269$	1.05911			0.0645751			0.0322875	+	0.999479i	$0.489721\pi$
$270$		0			0
$271$			21.8135i			1.32508i	0.749027	+	0.662539i	$0.230521\pi$
$271$			21.8135i			1.32508i	−0.749027	+	0.662539i	$0.769479\pi$
$272$	2.31124			0.140139
$273$		0			0
$274$	4.85601			0.293362
$275$		−	18.6178i		−	1.12269i
$276$		0			0
$277$	−17.6765			−1.06208			−0.531040	−	0.847347i	$-0.678199\pi$
$277$	−17.6765			−1.06208			−0.531040	+	0.847347i	$0.678199\pi$
$278$			11.6453i			0.698436i
$279$		0			0
$280$		−	6.46263i		−	0.386216i
$281$			13.3218i			0.794714i	0.917664	+	0.397357i	$0.130073\pi$
$281$			13.3218i			0.794714i	−0.917664	+	0.397357i	$0.869927\pi$
$282$		0			0
$283$	5.29815			0.314942			0.157471	−	0.987524i	$-0.449666\pi$
$283$	5.29815			0.314942			0.157471	+	0.987524i	$0.449666\pi$
$284$		−	12.9489i		−	0.768376i
$285$		0			0
$286$	−17.6108	−	6.10312i	−1.04135	−	0.360885i
$287$	1.71732			0.101370
$288$		0			0
$289$	−11.6582			−0.685775
$290$	27.7245			1.62804
$291$		0			0
$292$			4.27533i			0.250195i
$293$		−	14.6790i		−	0.857556i	−0.903410	−	0.428778i	$-0.858944\pi$
$293$		−	14.6790i		−	0.857556i	0.903410	−	0.428778i	$-0.141056\pi$
$294$		0			0
$295$	−8.77126			−0.510682
$296$	0.646136			0.0375559
$297$		0			0
$298$	21.8055			1.26316
$299$	28.5914	+	9.90854i	1.65348	+	0.573025i
$300$		0			0
$301$		−	7.69382i		−	0.443464i
$302$	14.6632			0.843773
$303$		0			0
$304$			5.16934i			0.296482i
$305$			2.53649i			0.145239i
$306$		0			0
$307$		−	3.41724i		−	0.195032i	−0.995234	−	0.0975160i	$-0.968910\pi$
$307$		−	3.41724i		−	0.195032i	0.995234	−	0.0975160i	$-0.0310897\pi$
$308$	−11.3908			−0.649053
$309$		0			0
$310$		−	18.2516i		−	1.03662i
$311$	10.8292			0.614065			0.307033	−	0.951699i	$-0.400664\pi$
$311$	10.8292			0.614065			0.307033	+	0.951699i	$0.400664\pi$
$312$		0			0
$313$	19.3519			1.09384			0.546918	−	0.837186i	$-0.315801\pi$
$313$	19.3519			1.09384			0.546918	+	0.837186i	$0.315801\pi$
$314$		−	7.94869i		−	0.448570i
$315$		0			0
$316$	3.04350			0.171210
$317$		−	15.8904i		−	0.892493i	−0.894910	−	0.446247i	$-0.852760\pi$
$317$		−	15.8904i		−	0.892493i	0.894910	−	0.446247i	$-0.147240\pi$
$318$		0			0
$319$		−	48.8663i		−	2.73599i
$320$			2.93284i			0.163951i
$321$		0			0
$322$	18.4932			1.03059
$323$			11.9476i			0.664781i
$324$		0			0
$325$	12.2698	+	4.25216i	0.680604	+	0.235867i
$326$	0.222533			0.0123250
$327$		0			0
$328$	−0.779346			−0.0430322
$329$	−12.1953			−0.672349
$330$		0			0
$331$		−	28.6118i		−	1.57265i	−0.617813	−	0.786325i	$-0.711981\pi$
$331$		−	28.6118i		−	1.57265i	0.617813	−	0.786325i	$-0.288019\pi$
$332$			3.68705i			0.202353i
$333$		0			0
$334$	−3.77070			−0.206323
$335$	32.7827			1.79111
$336$		0			0
$337$	−5.50659			−0.299963			−0.149981	−	0.988689i	$-0.547921\pi$
$337$	−5.50659			−0.299963			−0.149981	+	0.988689i	$0.547921\pi$
$338$	8.04433	−	10.2122i	0.437554	−	0.555470i
$339$		0			0
$340$			6.77851i			0.367616i
$341$	−32.1698			−1.74209
$342$		0			0
$343$		−	20.1501i		−	1.08800i
$344$			3.49158i			0.188253i
$345$		0			0
$346$			10.8534i			0.583485i
$347$	−2.64618			−0.142054			−0.0710271	−	0.997474i	$-0.522628\pi$
$347$	−2.64618			−0.142054			−0.0710271	+	0.997474i	$0.522628\pi$
$348$		0			0
$349$		−	5.58483i		−	0.298949i	−0.988766	−	0.149474i	$-0.952242\pi$
$349$		−	5.58483i		−	0.298949i	0.988766	−	0.149474i	$-0.0477582\pi$
$350$	7.93621			0.424208
$351$		0			0
$352$	5.16934			0.275527
$353$			25.6467i			1.36504i	0.730869	+	0.682518i	$0.239115\pi$
$353$			25.6467i			1.36504i	−0.730869	+	0.682518i	$0.760885\pi$
$354$		0			0
$355$	37.9771			2.01562
$356$			3.34650i			0.177364i
$357$		0			0
$358$			14.6400i			0.773749i
$359$		−	5.78875i		−	0.305519i	−0.988263	−	0.152759i	$-0.951184\pi$
$359$		−	5.78875i		−	0.305519i	0.988263	−	0.152759i	$-0.0488160\pi$
$360$		0			0
$361$	−7.72208			−0.406425
$362$			2.81354i			0.147877i
$363$		0			0
$364$	2.60158	−	7.50694i	0.136360	−	0.393471i
$365$	−12.5389			−0.656315
$366$		0			0
$367$	−30.7984			−1.60766			−0.803831	−	0.594858i	$-0.797208\pi$
$367$	−30.7984			−1.60766			−0.803831	+	0.594858i	$0.797208\pi$
$368$	−8.39252			−0.437491
$369$		0			0
$370$			1.89502i			0.0985173i
$371$			19.1372i			0.993552i
$372$		0			0
$373$	−31.2882			−1.62004			−0.810021	−	0.586401i	$-0.800544\pi$
$373$	−31.2882			−1.62004			−0.810021	+	0.586401i	$0.800544\pi$
$374$	11.9476			0.617795
$375$		0			0
$376$	5.53442			0.285416
$377$	32.2046	+	11.1607i	1.65862	+	0.574805i
$378$		0			0
$379$			5.29769i			0.272124i	0.990700	+	0.136062i	$0.0434447\pi$
$379$			5.29769i			0.272124i	−0.990700	+	0.136062i	$0.956555\pi$
$380$	−15.1609			−0.777736
$381$		0			0
$382$			8.77640i			0.449040i
$383$		−	22.2050i		−	1.13462i	−0.823504	−	0.567311i	$-0.807984\pi$
$383$		−	22.2050i		−	1.13462i	0.823504	−	0.567311i	$-0.192016\pi$
$384$		0			0
$385$		−	33.4075i		−	1.70261i
$386$	15.8231			0.805373
$387$		0			0
$388$			1.49094i			0.0756910i
$389$	−8.77177			−0.444747			−0.222373	−	0.974962i	$-0.571380\pi$
$389$	−8.77177			−0.444747			−0.222373	+	0.974962i	$0.571380\pi$
$390$		0			0
$391$	−19.3971			−0.980955
$392$			2.14443i			0.108310i
$393$		0			0
$394$	0.395717			0.0199359
$395$			8.92611i			0.449121i
$396$		0			0
$397$			33.5447i			1.68356i	0.539819	+	0.841781i	$0.318492\pi$
$397$			33.5447i			1.68356i	−0.539819	+	0.841781i	$0.681508\pi$
$398$		−	3.20316i		−	0.160560i
$399$		0			0
$400$	−3.60158			−0.180079
$401$			2.04015i			0.101880i	0.998702	+	0.0509401i	$0.0162218\pi$
$401$			2.04015i			0.101880i	−0.998702	+	0.0509401i	$0.983778\pi$
$402$		0			0
$403$	7.34733	−	21.2010i	0.365997	−	1.05610i
$404$	−0.406692			−0.0202337
$405$		0			0
$406$	20.8302			1.03379
$407$	3.34010			0.165563
$408$		0			0
$409$		−	32.5774i		−	1.61085i	−0.592698	−	0.805424i	$-0.701937\pi$
$409$		−	32.5774i		−	1.61085i	0.592698	−	0.805424i	$-0.298063\pi$
$410$		−	2.28570i		−	0.112883i
$411$		0			0
$412$	−17.1609			−0.845456
$413$	−6.59011			−0.324278
$414$		0			0
$415$	−10.8135			−0.530816
$416$	−1.18064	+	3.40677i	−0.0578856	+	0.167031i
$417$		0			0
$418$			26.7221i			1.30702i
$419$	11.6609			0.569673			0.284837	−	0.958576i	$-0.408061\pi$
$419$	11.6609			0.569673			0.284837	+	0.958576i	$0.408061\pi$
$420$		0			0
$421$			1.63447i			0.0796590i	0.999206	+	0.0398295i	$0.0126815\pi$
$421$			1.63447i			0.0796590i	−0.999206	+	0.0398295i	$0.987319\pi$
$422$		−	8.27972i		−	0.403051i
$423$		0			0
$424$		−	8.68475i		−	0.421769i
$425$	−8.32411			−0.403779
$426$		0			0
$427$			1.90574i			0.0922254i
$428$	0.329555			0.0159296
$429$		0			0
$430$	−10.2403			−0.493829
$431$			2.92645i			0.140962i	0.997513	+	0.0704810i	$0.0224534\pi$
$431$			2.92645i			0.140962i	−0.997513	+	0.0704810i	$0.977547\pi$
$432$		0			0
$433$	−33.9770			−1.63283			−0.816414	−	0.577467i	$-0.804041\pi$
$433$	−33.9770			−1.63283			−0.816414	+	0.577467i	$0.804041\pi$
$434$		−	13.7130i		−	0.658246i
$435$		0			0
$436$			9.64927i			0.462116i
$437$		−	43.3838i		−	2.07533i
$438$		0			0
$439$	−16.4932			−0.787179			−0.393589	−	0.919286i	$-0.628767\pi$
$439$	−16.4932			−0.787179			−0.393589	+	0.919286i	$0.628767\pi$
$440$			15.1609i			0.722766i
$441$		0			0
$442$	−2.72874	+	7.87386i	−0.129793	+	0.374522i
$443$	−28.9569			−1.37578			−0.687891	−	0.725814i	$-0.741463\pi$
$443$	−28.9569			−1.37578			−0.687891	+	0.725814i	$0.741463\pi$
$444$		0			0
$445$	−9.81476			−0.465264
$446$	−14.3199			−0.678065
$447$		0			0
$448$			2.20354i			0.104107i
$449$		−	38.1292i		−	1.79943i	−0.436478	−	0.899715i	$-0.643774\pi$
$449$		−	38.1292i		−	1.79943i	0.436478	−	0.899715i	$-0.356226\pi$
$450$		0			0
$451$	−4.02871			−0.189704
$452$	3.03210			0.142618
$453$		0			0
$454$	−20.0169			−0.939439
$455$	22.0167	+	7.63003i	1.03216	+	0.357701i
$456$		0			0
$457$			31.3542i			1.46669i	0.679859	+	0.733343i	$0.262041\pi$
$457$			31.3542i			1.46669i	−0.679859	+	0.733343i	$0.737959\pi$
$458$	−2.57747			−0.120437
$459$		0			0
$460$		−	24.6140i		−	1.14763i
$461$		−	23.6894i		−	1.10332i	−0.834068	−	0.551662i	$-0.813994\pi$
$461$		−	23.6894i		−	1.10332i	0.834068	−	0.551662i	$-0.186006\pi$
$462$		0			0
$463$		−	6.76587i		−	0.314436i	−0.987564	−	0.157218i	$-0.949747\pi$
$463$		−	6.76587i		−	0.314436i	0.987564	−	0.157218i	$-0.0502526\pi$
$464$	−9.45310			−0.438849
$465$		0			0
$466$		−	14.0412i		−	0.650444i
$467$	−28.8066			−1.33301			−0.666505	−	0.745500i	$-0.732211\pi$
$467$	−28.8066			−1.33301			−0.666505	+	0.745500i	$0.732211\pi$
$468$		0			0
$469$	24.6306			1.13734
$470$			16.2316i			0.748708i
$471$		0			0
$472$	2.99070			0.137658
$473$			18.0492i			0.829901i
$474$		0			0
$475$		−	18.6178i		−	0.854243i
$476$			5.09290i			0.233433i
$477$		0			0
$478$	4.70266			0.215095
$479$			40.3680i			1.84446i	0.386642	+	0.922230i	$0.373635\pi$
$479$			40.3680i			1.84446i	−0.386642	+	0.922230i	$0.626365\pi$
$480$		0			0
$481$	−0.762853	+	2.20124i	−0.0347831	+	0.100368i
$482$	−22.6013			−1.02946
$483$		0			0
$484$	15.7221			0.714640
$485$	−4.37270			−0.198554
$486$		0			0
$487$		−	40.9866i		−	1.85728i	−0.370982	−	0.928640i	$-0.620979\pi$
$487$		−	40.9866i		−	1.85728i	0.370982	−	0.928640i	$-0.379021\pi$
$488$		−	0.864857i		−	0.0391502i
$489$		0			0
$490$	−6.28928			−0.284121
$491$	−8.51938			−0.384474			−0.192237	−	0.981348i	$-0.561574\pi$
$491$	−8.51938			−0.384474			−0.192237	+	0.981348i	$0.561574\pi$
$492$		0			0
$493$	−21.8484			−0.984001
$494$	−17.6108	−	6.10312i	−0.792346	−	0.274592i
$495$		0			0
$496$			6.22318i			0.279429i
$497$	28.5334			1.27990
$498$		0			0
$499$			21.9355i			0.981966i	0.871169	+	0.490983i	$0.163362\pi$
$499$			21.9355i			0.981966i	−0.871169	+	0.490983i	$0.836638\pi$
$500$			4.10135i			0.183418i
$501$		0			0
$502$		−	3.40669i		−	0.152048i
$503$	31.5620			1.40728			0.703641	−	0.710556i	$-0.251556\pi$
$503$	31.5620			1.40728			0.703641	+	0.710556i	$0.251556\pi$
$504$		0			0
$505$		−	1.19276i		−	0.0530773i
$506$	−43.3838			−1.92865
$507$		0			0
$508$	9.03355			0.400799
$509$			37.3745i			1.65660i	0.560287	+	0.828299i	$0.310691\pi$
$509$			37.3745i			1.65660i	−0.560287	+	0.828299i	$0.689309\pi$
$510$		0			0
$511$	−9.42084			−0.416754
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			21.8295i			0.962858i
$515$		−	50.3302i		−	2.21781i
$516$		0			0
$517$	28.6093			1.25824
$518$			1.42378i			0.0625575i
$519$		0			0
$520$	−9.99153	−	3.46263i	−0.438158	−	0.151846i
$521$	−17.2949			−0.757705			−0.378853	−	0.925457i	$-0.623681\pi$
$521$	−17.2949			−0.757705			−0.378853	+	0.925457i	$0.623681\pi$
$522$		0			0
$523$	−10.5189			−0.459960			−0.229980	−	0.973195i	$-0.573866\pi$
$523$	−10.5189			−0.459960			−0.229980	+	0.973195i	$0.573866\pi$
$524$	17.4172			0.760874
$525$		0			0
$526$		−	5.91025i		−	0.257699i
$527$			14.3833i			0.626545i
$528$		0			0
$529$	47.4345			2.06237
$530$	25.4710			1.10639
$531$		0			0
$532$	−11.3908			−0.493855
$533$	0.920126	−	2.65505i	0.0398551	−	0.115003i
$534$		0			0
$535$			0.966532i			0.0417868i
$536$	−11.1778			−0.482806
$537$		0			0
$538$		−	1.05911i		−	0.0456615i
$539$			11.0853i			0.477477i
$540$		0			0
$541$		−	21.7436i		−	0.934832i	−0.884038	−	0.467416i	$-0.845185\pi$
$541$		−	21.7436i		−	0.934832i	0.884038	−	0.467416i	$-0.154815\pi$
$542$	21.8135			0.936972
$543$		0			0
$544$		−	2.31124i		−	0.0990936i
$545$	−28.2998			−1.21223
$546$		0			0
$547$	27.9928			1.19689			0.598444	−	0.801165i	$-0.295786\pi$
$547$	27.9928			1.19689			0.598444	+	0.801165i	$0.295786\pi$
$548$		−	4.85601i		−	0.207438i
$549$		0			0
$550$	−18.6178			−0.793865
$551$		−	48.8663i		−	2.08177i
$552$		0			0
$553$			6.70646i			0.285187i
$554$			17.6765i			0.751004i
$555$		0			0
$556$	11.6453			0.493869
$557$		−	8.27318i		−	0.350546i	−0.984520	−	0.175273i	$-0.943919\pi$
$557$		−	8.27318i		−	0.350546i	0.984520	−	0.175273i	$-0.0560808\pi$
$558$		0			0
$559$	−11.8950	−	4.12229i	−0.503105	−	0.174354i
$560$	−6.46263			−0.273096
$561$		0			0
$562$	13.3218			0.561948
$563$	10.1195			0.426487			0.213243	−	0.976999i	$-0.431597\pi$
$563$	10.1195			0.426487			0.213243	+	0.976999i	$0.431597\pi$
$564$		0			0
$565$			8.89268i			0.374118i
$566$		−	5.29815i		−	0.222698i
$567$		0			0
$568$	−12.9489			−0.543324
$569$	−21.3523			−0.895136			−0.447568	−	0.894250i	$-0.647710\pi$
$569$	−21.3523			−0.895136			−0.447568	+	0.894250i	$0.647710\pi$
$570$		0			0
$571$	11.2426			0.470490			0.235245	−	0.971936i	$-0.424411\pi$
$571$	11.2426			0.470490			0.235245	+	0.971936i	$0.424411\pi$
$572$	−6.10312	+	17.6108i	−0.255184	+	0.736343i
$573$		0			0
$574$		−	1.71732i		−	0.0716794i
$575$	30.2263			1.26053
$576$		0			0
$577$			13.4761i			0.561018i	0.959851	+	0.280509i	$0.0905032\pi$
$577$			13.4761i			0.561018i	−0.959851	+	0.280509i	$0.909497\pi$
$578$			11.6582i			0.484916i
$579$		0			0
$580$		−	27.7245i		−	1.15120i
$581$	−8.12455			−0.337063
$582$		0			0
$583$		−	44.8944i		−	1.85934i
$584$	4.27533			0.176914
$585$		0			0
$586$	−14.6790			−0.606384
$587$			28.2513i			1.16605i	0.812453	+	0.583027i	$0.198132\pi$
$587$			28.2513i			1.16605i	−0.812453	+	0.583027i	$0.801868\pi$
$588$		0			0
$589$	−32.1698			−1.32553
$590$			8.77126i			0.361107i
$591$		0			0
$592$		−	0.646136i		−	0.0265560i
$593$			33.3901i			1.37117i	0.727993	+	0.685585i	$0.240453\pi$
$593$			33.3901i			1.37117i	−0.727993	+	0.685585i	$0.759547\pi$
$594$		0			0
$595$	−14.9367			−0.612344
$596$		−	21.8055i		−	0.893186i
$597$		0			0
$598$	9.90854	−	28.5914i	0.405190	−	1.16919i
$599$	20.0810			0.820489			0.410245	−	0.911976i	$-0.365443\pi$
$599$	20.0810			0.820489			0.410245	+	0.911976i	$0.365443\pi$
$600$		0			0
$601$	11.5667			0.471818			0.235909	−	0.971775i	$-0.424193\pi$
$601$	11.5667			0.471818			0.235909	+	0.971775i	$0.424193\pi$
$602$	−7.69382			−0.313577
$603$		0			0
$604$		−	14.6632i		−	0.596638i
$605$			46.1104i			1.87466i
$606$		0			0
$607$	24.0973			0.978080			0.489040	−	0.872261i	$-0.337347\pi$
$607$	24.0973			0.978080			0.489040	+	0.872261i	$0.337347\pi$
$608$	5.16934			0.209644
$609$		0			0
$610$	2.53649			0.102700
$611$	−6.53415	+	18.8545i	−0.264344	+	0.762772i
$612$		0			0
$613$		−	3.75444i		−	0.151641i	−0.997122	−	0.0758203i	$-0.975842\pi$
$613$		−	3.75444i		−	0.151641i	0.997122	−	0.0758203i	$-0.0241575\pi$
$614$	−3.41724			−0.137908
$615$		0			0
$616$			11.3908i			0.458949i
$617$			14.4142i			0.580295i	0.956982	+	0.290148i	$0.0937044\pi$
$617$			14.4142i			0.580295i	−0.956982	+	0.290148i	$0.906296\pi$
$618$		0			0
$619$			9.17348i			0.368713i	0.982859	+	0.184357i	$0.0590202\pi$
$619$			9.17348i			0.368713i	−0.982859	+	0.184357i	$0.940980\pi$
$620$	−18.2516			−0.733003
$621$		0			0
$622$		−	10.8292i		−	0.434210i
$623$	−7.37413			−0.295438
$624$		0			0
$625$	−30.0365			−1.20146
$626$		−	19.3519i		−	0.773458i
$627$		0			0
$628$	−7.94869			−0.317187
$629$		−	1.49338i		−	0.0595448i
$630$		0			0
$631$		−	32.8572i		−	1.30803i	−0.756483	−	0.654013i	$-0.773084\pi$
$631$		−	32.8572i		−	1.30803i	0.756483	−	0.654013i	$-0.226916\pi$
$632$		−	3.04350i		−	0.121064i
$633$		0			0
$634$	−15.8904			−0.631088
$635$			26.4940i			1.05138i
$636$		0			0
$637$	−7.30558	−	2.53180i	−0.289458	−	0.100313i
$638$	−48.8663			−1.93463
$639$		0			0
$640$	2.93284			0.115931
$641$	−21.3885			−0.844794			−0.422397	−	0.906411i	$-0.638811\pi$
$641$	−21.3885			−0.844794			−0.422397	+	0.906411i	$0.638811\pi$
$642$		0			0
$643$		−	5.29176i		−	0.208687i	−0.994541	−	0.104343i	$-0.966726\pi$
$643$		−	5.29176i		−	0.208687i	0.994541	−	0.104343i	$-0.0332741\pi$
$644$		−	18.4932i		−	0.728735i
$645$		0			0
$646$	11.9476			0.470071
$647$	0.564603			0.0221968			0.0110984	−	0.999938i	$-0.496467\pi$
$647$	0.564603			0.0221968			0.0110984	+	0.999938i	$0.496467\pi$
$648$		0			0
$649$	15.4599			0.606856
$650$	4.25216	−	12.2698i	0.166784	−	0.481259i
$651$		0			0
$652$		−	0.222533i		−	0.00871508i
$653$	−16.0224			−0.627003			−0.313502	−	0.949588i	$-0.601502\pi$
$653$	−16.0224			−0.627003			−0.313502	+	0.949588i	$0.601502\pi$
$654$		0			0
$655$			51.0819i			1.99594i
$656$			0.779346i			0.0304284i
$657$		0			0
$658$			12.1953i			0.475422i
$659$	−7.43221			−0.289518			−0.144759	−	0.989467i	$-0.546241\pi$
$659$	−7.43221			−0.289518			−0.144759	+	0.989467i	$0.546241\pi$
$660$		0			0
$661$		−	15.7226i		−	0.611537i	−0.952106	−	0.305769i	$-0.901087\pi$
$661$		−	15.7226i		−	0.611537i	0.952106	−	0.305769i	$-0.0989134\pi$
$662$	−28.6118			−1.11203
$663$		0			0
$664$	3.68705			0.143085
$665$		−	33.4075i		−	1.29549i
$666$		0			0
$667$	79.3354			3.07188
$668$			3.77070i			0.145893i
$669$		0			0
$670$		−	32.7827i		−	1.26650i
$671$		−	4.47074i		−	0.172591i
$672$		0			0
$673$	−39.6582			−1.52871			−0.764356	−	0.644795i	$-0.776943\pi$
$673$	−39.6582			−1.52871			−0.764356	+	0.644795i	$0.776943\pi$
$674$			5.50659i			0.212106i
$675$		0			0
$676$	−10.2122	−	8.04433i	−0.392776	−	0.309397i
$677$	−36.5227			−1.40368			−0.701840	−	0.712335i	$-0.747638\pi$
$677$	−36.5227			−1.40368			−0.701840	+	0.712335i	$0.747638\pi$
$678$		0			0
$679$	−3.28534			−0.126080
$680$	6.77851			0.259944
$681$		0			0
$682$			32.1698i			1.23184i
$683$		−	8.46240i		−	0.323805i	−0.986807	−	0.161902i	$-0.948237\pi$
$683$		−	8.46240i		−	0.323805i	0.986807	−	0.161902i	$-0.0517630\pi$
$684$		0			0
$685$	14.2419			0.544156
$686$	−20.1501			−0.769334
$687$		0			0
$688$	3.49158			0.133115
$689$	29.5870	+	10.2535i	1.12717	+	0.390629i
$690$		0			0
$691$			48.8294i			1.85756i	0.370635	+	0.928779i	$0.379140\pi$
$691$			48.8294i			1.85756i	−0.370635	+	0.928779i	$0.620860\pi$
$692$	10.8534			0.412586
$693$		0			0
$694$			2.64618i			0.100447i
$695$			34.1537i			1.29552i
$696$		0			0
$697$			1.80126i			0.0682274i
$698$	−5.58483			−0.211389
$699$		0			0
$700$		−	7.93621i		−	0.299960i
$701$	−40.2942			−1.52189			−0.760946	−	0.648816i	$-0.775265\pi$
$701$	−40.2942			−1.52189			−0.760946	+	0.648816i	$0.775265\pi$
$702$		0			0
$703$	3.34010			0.125974
$704$		−	5.16934i		−	0.194827i
$705$		0			0
$706$	25.6467			0.965226
$707$		−	0.896160i		−	0.0337036i
$708$		0			0
$709$			38.5105i			1.44629i	0.690695	+	0.723146i	$0.257305\pi$
$709$			38.5105i			1.44629i	−0.690695	+	0.723146i	$0.742695\pi$
$710$		−	37.9771i		−	1.42526i
$711$		0			0
$712$	3.34650			0.125415
$713$		−	52.2282i		−	1.95596i
$714$		0			0
$715$	−51.6496	−	17.8995i	−1.93159	−	0.669404i
$716$	14.6400			0.547123
$717$		0			0
$718$	−5.78875			−0.216034
$719$	−0.951205			−0.0354740			−0.0177370	−	0.999843i	$-0.505646\pi$
$719$	−0.951205			−0.0354740			−0.0177370	+	0.999843i	$0.505646\pi$
$720$		0			0
$721$		−	37.8146i		−	1.40829i
$722$			7.72208i			0.287386i
$723$		0			0
$724$	2.81354			0.104565
$725$	34.0461			1.26444
$726$		0			0
$727$	1.99813			0.0741065			0.0370532	−	0.999313i	$-0.488203\pi$
$727$	1.99813			0.0741065			0.0370532	+	0.999313i	$0.488203\pi$
$728$	−7.50694	−	2.60158i	−0.278226	−	0.0964209i
$729$		0			0
$730$			12.5389i			0.464085i
$731$	8.06987			0.298475
$732$		0			0
$733$			15.6304i			0.577322i	0.957431	+	0.288661i	$0.0932100\pi$
$733$			15.6304i			0.577322i	−0.957431	+	0.288661i	$0.906790\pi$
$734$			30.7984i			1.13679i
$735$		0			0
$736$			8.39252i			0.309353i
$737$	−57.7817			−2.12842
$738$		0			0
$739$			20.0122i			0.736162i	0.929794	+	0.368081i	$0.119985\pi$
$739$			20.0122i			0.736162i	−0.929794	+	0.368081i	$0.880015\pi$
$740$	1.89502			0.0696622
$741$		0			0
$742$	19.1372			0.702547
$743$		−	38.7890i		−	1.42303i	−0.702671	−	0.711515i	$-0.748009\pi$
$743$		−	38.7890i		−	1.42303i	0.702671	−	0.711515i	$-0.251991\pi$
$744$		0			0
$745$	63.9520			2.34302
$746$			31.2882i			1.14554i
$747$		0			0
$748$		−	11.9476i		−	0.436847i
$749$			0.726185i			0.0265342i
$750$		0			0
$751$	−14.9390			−0.545132			−0.272566	−	0.962137i	$-0.587872\pi$
$751$	−14.9390			−0.545132			−0.272566	+	0.962137i	$0.587872\pi$
$752$		−	5.53442i		−	0.201820i
$753$		0			0
$754$	11.1607	−	32.2046i	0.406448	−	1.17282i
$755$	43.0049			1.56511
$756$		0			0
$757$	−26.1383			−0.950011			−0.475006	−	0.879983i	$-0.657554\pi$
$757$	−26.1383			−0.950011			−0.475006	+	0.879983i	$0.657554\pi$
$758$	5.29769			0.192421
$759$		0			0
$760$			15.1609i			0.549943i
$761$		−	3.04980i		−	0.110555i	−0.998471	−	0.0552776i	$-0.982396\pi$
$761$		−	3.04980i		−	0.110555i	0.998471	−	0.0552776i	$-0.0176044\pi$
$762$		0			0
$763$	−21.2625			−0.769755
$764$	8.77640			0.317519
$765$		0			0
$766$	−22.2050			−0.802299
$767$	−3.53093	+	10.1886i	−0.127495	+	0.367890i
$768$		0			0
$769$		−	6.87645i		−	0.247971i	−0.992284	−	0.123985i	$-0.960432\pi$
$769$		−	6.87645i		−	0.247971i	0.992284	−	0.123985i	$-0.0395676\pi$
$770$	−33.4075			−1.20392
$771$		0			0
$772$		−	15.8231i		−	0.569485i
$773$			18.4949i			0.665216i	0.943065	+	0.332608i	$0.107929\pi$
$773$			18.4949i			0.665216i	−0.943065	+	0.332608i	$0.892071\pi$
$774$		0			0
$775$		−	22.4133i		−	0.805109i
$776$	1.49094			0.0535216
$777$		0			0
$778$			8.77177i			0.314483i
$779$	−4.02871			−0.144343
$780$		0			0
$781$	−66.9373			−2.39521
$782$			19.3971i			0.693640i
$783$		0			0
$784$	2.14443			0.0765868
$785$		−	23.3123i		−	0.832050i
$786$		0			0
$787$		−	0.0882504i		−	0.00314579i	−0.999999	−	0.00157289i	$-0.999499\pi$
$787$		−	0.0882504i		−	0.00314579i	0.999999	−	0.00157289i	$-0.000500668\pi$
$788$		−	0.395717i		−	0.0140968i
$789$		0			0
$790$	8.92611			0.317577
$791$			6.68135i			0.237561i
$792$		0			0
$793$	2.94637	+	1.02108i	0.104629	+	0.0362597i
$794$	33.5447			1.19046
$795$		0			0
$796$	−3.20316			−0.113533
$797$	30.9087			1.09484			0.547421	−	0.836857i	$-0.315610\pi$
$797$	30.9087			1.09484			0.547421	+	0.836857i	$0.315610\pi$
$798$		0			0
$799$		−	12.7914i		−	0.452526i
$800$			3.60158i			0.127335i
$801$		0			0
$802$	2.04015			0.0720402
$803$	22.1006			0.779915
$804$		0			0
$805$	54.2378			1.91163
$806$	−21.2010	−	7.34733i	−0.746772	−	0.258799i
$807$		0			0
$808$			0.406692i			0.0143074i
$809$	45.6038			1.60335			0.801673	−	0.597763i	$-0.203944\pi$
$809$	45.6038			1.60335			0.801673	+	0.597763i	$0.203944\pi$
$810$		0			0
$811$			20.3497i			0.714575i	0.933994	+	0.357287i	$0.116298\pi$
$811$			20.3497i			0.714575i	−0.933994	+	0.357287i	$0.883702\pi$
$812$		−	20.8302i		−	0.730998i
$813$		0			0
$814$		−	3.34010i		−	0.117070i
$815$	0.652656			0.0228615
$816$		0			0
$817$			18.0492i			0.631460i
$818$	−32.5774			−1.13904
$819$		0			0
$820$	−2.28570			−0.0798202
$821$			11.6741i			0.407430i	0.979030	+	0.203715i	$0.0653017\pi$
$821$			11.6741i			0.407430i	−0.979030	+	0.203715i	$0.934698\pi$
$822$		0			0
$823$	−35.6000			−1.24094			−0.620469	−	0.784231i	$-0.713058\pi$
$823$	−35.6000			−1.24094			−0.620469	+	0.784231i	$0.713058\pi$
$824$			17.1609i			0.597827i
$825$		0			0
$826$			6.59011i			0.229299i
$827$		−	13.5478i		−	0.471103i	−0.971862	−	0.235551i	$-0.924310\pi$
$827$		−	13.5478i		−	0.471103i	0.971862	−	0.235551i	$-0.0756896\pi$
$828$		0			0
$829$	52.4752			1.82254			0.911270	−	0.411809i	$-0.135103\pi$
$829$	52.4752			1.82254			0.911270	+	0.411809i	$0.135103\pi$
$830$			10.8135i			0.375344i
$831$		0			0
$832$	3.40677	+	1.18064i	0.118109	+	0.0409313i
$833$	4.95629			0.171725
$834$		0			0
$835$	−11.0589			−0.382708
$836$	26.7221			0.924202
$837$		0			0
$838$		−	11.6609i		−	0.402820i
$839$		−	49.2328i		−	1.69970i	−0.527021	−	0.849852i	$-0.676691\pi$
$839$		−	49.2328i		−	1.69970i	0.527021	−	0.849852i	$-0.323309\pi$
$840$		0			0
$841$	60.3611			2.08142
$842$	1.63447			0.0563274
$843$		0			0
$844$	−8.27972			−0.285000
$845$	23.5928	−	29.9508i	0.811616	−	1.03034i
$846$		0			0
$847$			34.6442i			1.19039i
$848$	−8.68475			−0.298236
$849$		0			0
$850$			8.32411i			0.285515i
$851$			5.42272i			0.185888i
$852$		0			0
$853$		−	44.8428i		−	1.53539i	−0.640815	−	0.767695i	$-0.721404\pi$
$853$		−	44.8428i		−	1.53539i	0.640815	−	0.767695i	$-0.278596\pi$
$854$	1.90574			0.0652132
$855$		0			0
$856$		−	0.329555i		−	0.0112639i
$857$	37.7559			1.28972			0.644858	−	0.764302i	$-0.276916\pi$
$857$	37.7559			1.28972			0.644858	+	0.764302i	$0.276916\pi$
$858$		0			0
$859$	−30.7045			−1.04762			−0.523812	−	0.851834i	$-0.675491\pi$
$859$	−30.7045			−1.04762			−0.523812	+	0.851834i	$0.675491\pi$
$860$			10.2403i			0.349190i
$861$		0			0
$862$	2.92645			0.0996752
$863$			4.16398i			0.141744i	0.997485	+	0.0708718i	$0.0225781\pi$
$863$			4.16398i			0.141744i	−0.997485	+	0.0708718i	$0.977422\pi$
$864$		0			0
$865$			31.8315i			1.08230i
$866$			33.9770i			1.15458i
$867$		0			0
$868$	−13.7130			−0.465450
$869$		−	15.7329i		−	0.533701i
$870$		0			0
$871$	13.1969	−	38.0801i	0.447160	−	1.29030i
$872$	9.64927			0.326765
$873$		0			0
$874$	−43.3838			−1.46748
$875$	−9.03748			−0.305522
$876$		0			0
$877$			2.14935i			0.0725784i	0.999341	+	0.0362892i	$0.0115537\pi$
$877$			2.14935i			0.0725784i	−0.999341	+	0.0362892i	$0.988446\pi$
$878$			16.4932i			0.556620i
$879$		0			0
$880$	15.1609			0.511073
$881$	−5.95507			−0.200632			−0.100316	−	0.994956i	$-0.531985\pi$
$881$	−5.95507			−0.200632			−0.100316	+	0.994956i	$0.531985\pi$
$882$		0			0
$883$	−27.9174			−0.939495			−0.469747	−	0.882801i	$-0.655655\pi$
$883$	−27.9174			−0.939495			−0.469747	+	0.882801i	$0.655655\pi$
$884$	7.87386	+	2.72874i	0.264827	+	0.0917774i
$885$		0			0
$886$			28.9569i			0.972825i
$887$	49.4513			1.66041			0.830206	−	0.557457i	$-0.188223\pi$
$887$	49.4513			1.66041			0.830206	+	0.557457i	$0.188223\pi$
$888$		0			0
$889$			19.9057i			0.667617i
$890$			9.81476i			0.328991i
$891$		0			0
$892$			14.3199i			0.479464i
$893$	28.6093			0.957374
$894$		0			0
$895$			42.9369i			1.43522i
$896$	2.20354			0.0736150
$897$		0			0
$898$	−38.1292			−1.27239
$899$		−	58.8284i		−	1.96204i
$900$		0			0
$901$	−20.0725			−0.668713
$902$			4.02871i			0.134141i
$903$		0			0
$904$		−	3.03210i		−	0.100846i
$905$			8.25169i			0.274295i
$906$		0			0
$907$	29.6345			0.983996			0.491998	−	0.870596i	$-0.336267\pi$
$907$	29.6345			0.983996			0.491998	+	0.870596i	$0.336267\pi$
$908$			20.0169i			0.664284i
$909$		0			0
$910$	7.63003	−	22.0167i	0.252933	−	0.729847i
$911$	58.0629			1.92371			0.961855	−	0.273559i	$-0.0882010\pi$
$911$	58.0629			1.92371			0.961855	+	0.273559i	$0.0882010\pi$
$912$		0			0
$913$	19.0596			0.630781
$914$	31.3542			1.03710
$915$		0			0
$916$			2.57747i			0.0851619i
$917$			38.3794i			1.26740i
$918$		0			0
$919$	−27.6437			−0.911882			−0.455941	−	0.890010i	$-0.650697\pi$
$919$	−27.6437			−0.911882			−0.455941	+	0.890010i	$0.650697\pi$
$920$	−24.6140			−0.811499
$921$		0			0
$922$	−23.6894			−0.780168
$923$	15.2880	−	44.1140i	0.503210	−	1.45203i
$924$		0			0
$925$			2.32711i			0.0765150i
$926$	−6.76587			−0.222340
$927$		0			0
$928$			9.45310i			0.310313i
$929$			34.9266i			1.14591i	0.819588	+	0.572953i	$0.194202\pi$
$929$			34.9266i			1.14591i	−0.819588	+	0.572953i	$0.805798\pi$
$930$		0			0
$931$			11.0853i			0.363306i
$932$	−14.0412			−0.459933
$933$		0			0
$934$			28.8066i			0.942581i
$935$	35.0404			1.14594
$936$		0			0
$937$	−8.39681			−0.274312			−0.137156	−	0.990549i	$-0.543796\pi$
$937$	−8.39681			−0.274312			−0.137156	+	0.990549i	$0.543796\pi$
$938$		−	24.6306i		−	0.804218i
$939$		0			0
$940$	16.2316			0.529417
$941$		−	25.3975i		−	0.827935i	−0.910292	−	0.413967i	$-0.864143\pi$
$941$		−	25.3975i		−	0.827935i	0.910292	−	0.413967i	$-0.135857\pi$
$942$		0			0
$943$		−	6.54068i		−	0.212994i
$944$		−	2.99070i		−	0.0973390i
$945$		0			0
$946$	18.0492			0.586829
$947$		−	24.5696i		−	0.798406i	−0.916863	−	0.399203i	$-0.869287\pi$
$947$		−	24.5696i		−	0.798406i	0.916863	−	0.399203i	$-0.130713\pi$
$948$		0			0
$949$	−5.04762	+	14.5651i	−0.163853	+	0.472802i
$950$	−18.6178			−0.604041
$951$		0			0
$952$	5.09290			0.165062
$953$	47.5723			1.54102			0.770509	−	0.637429i	$-0.220002\pi$
$953$	47.5723			1.54102			0.770509	+	0.637429i	$0.220002\pi$
$954$		0			0
$955$			25.7398i			0.832921i
$956$		−	4.70266i		−	0.152095i
$957$		0			0
$958$	40.3680			1.30423
$959$	10.7004			0.345534
$960$		0			0
$961$	−7.72802			−0.249291
$962$	2.20124	+	0.762853i	0.0709708	+	0.0245954i
$963$		0			0
$964$			22.6013i			0.727939i
$965$	46.4066			1.49388
$966$		0			0
$967$			43.1076i			1.38625i	0.720818	+	0.693124i	$0.243766\pi$
$967$			43.1076i			1.38625i	−0.720818	+	0.693124i	$0.756234\pi$
$968$		−	15.7221i		−	0.505327i
$969$		0			0
$970$			4.37270i			0.140399i
$971$	14.7164			0.472272			0.236136	−	0.971720i	$-0.424119\pi$
$971$	14.7164			0.472272			0.236136	+	0.971720i	$0.424119\pi$
$972$		0			0
$973$			25.6607i			0.822646i
$974$	−40.9866			−1.31330
$975$		0			0
$976$	−0.864857			−0.0276834
$977$		−	5.91487i		−	0.189233i	−0.995514	−	0.0946167i	$-0.969837\pi$
$977$		−	5.91487i		−	0.189233i	0.995514	−	0.0946167i	$-0.0301626\pi$
$978$		0			0
$979$	17.2992			0.552884
$980$			6.28928i			0.200904i
$981$		0			0
$982$			8.51938i			0.271864i
$983$			14.6215i			0.466353i	0.972434	+	0.233176i	$0.0749119\pi$
$983$			14.6215i			0.466353i	−0.972434	+	0.233176i	$0.925088\pi$
$984$		0			0
$985$	1.16058			0.0369790
$986$			21.8484i			0.695794i
$987$		0			0
$988$	−6.10312	+	17.6108i	−0.194166	+	0.560273i
$989$	−29.3031			−0.931786
$990$		0			0
$991$	35.1127			1.11539			0.557696	−	0.830045i	$-0.311685\pi$
$991$	35.1127			1.11539			0.557696	+	0.830045i	$0.311685\pi$
$992$	6.22318			0.197586
$993$		0			0
$994$		−	28.5334i		−	0.905024i
$995$		−	9.39436i		−	0.297821i
$996$		0			0
$997$	−35.3302			−1.11892			−0.559459	−	0.828858i	$-0.688991\pi$
$997$	−35.3302			−1.11892			−0.559459	+	0.828858i	$0.688991\pi$
$998$	21.9355			0.694355
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.c.649.2		14
3.2	odd	2		2106.2.b.d.649.13		14
9.2	odd	6		702.2.t.a.415.9		28
9.4	even	3		234.2.t.a.25.9	yes	28
9.5	odd	6		702.2.t.a.181.6		28
9.7	even	3		234.2.t.a.103.2	yes	28
13.12	even	2	inner	2106.2.b.c.649.13		14
39.38	odd	2		2106.2.b.d.649.2		14
117.25	even	6		234.2.t.a.103.9	yes	28
117.38	odd	6		702.2.t.a.415.6		28
117.77	odd	6		702.2.t.a.181.9		28
117.103	even	6		234.2.t.a.25.2	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.2	✓	28	117.103	even	6
234.2.t.a.25.9	yes	28	9.4	even	3
234.2.t.a.103.2	yes	28	9.7	even	3
234.2.t.a.103.9	yes	28	117.25	even	6
702.2.t.a.181.6		28	9.5	odd	6
702.2.t.a.181.9		28	117.77	odd	6
702.2.t.a.415.6		28	117.38	odd	6
702.2.t.a.415.9		28	9.2	odd	6
2106.2.b.c.649.2		14	1.1	even	1	trivial
2106.2.b.c.649.13		14	13.12	even	2	inner
2106.2.b.d.649.2		14	39.38	odd	2
2106.2.b.d.649.13		14	3.2	odd	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:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.93284i q^{5} -2.20354i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.93284i q^{5} -2.20354i q^{7} +1.00000i q^{8} -2.93284 q^{10} +5.16934i q^{11} +(-3.40677 - 1.18064i) q^{13} -2.20354 q^{14} +1.00000 q^{16} +2.31124 q^{17} +5.16934i q^{19} +2.93284i q^{20} +5.16934 q^{22} -8.39252 q^{23} -3.60158 q^{25} +(-1.18064 + 3.40677i) q^{26} +2.20354i q^{28} -9.45310 q^{29} +6.22318i q^{31} -1.00000i q^{32} -2.31124i q^{34} -6.46263 q^{35} -0.646136i q^{37} +5.16934 q^{38} +2.93284 q^{40} +0.779346i q^{41} +3.49158 q^{43} -5.16934i q^{44} +8.39252i q^{46} -5.53442i q^{47} +2.14443 q^{49} +3.60158i q^{50} +(3.40677 + 1.18064i) q^{52} -8.68475 q^{53} +15.1609 q^{55} +2.20354 q^{56} +9.45310i q^{58} -2.99070i q^{59} -0.864857 q^{61} +6.22318 q^{62} -1.00000 q^{64} +(-3.46263 + 9.99153i) q^{65} +11.1778i q^{67} -2.31124 q^{68} +6.46263i q^{70} +12.9489i q^{71} -4.27533i q^{73} -0.646136 q^{74} -5.16934i q^{76} +11.3908 q^{77} -3.04350 q^{79} -2.93284i q^{80} +0.779346 q^{82} -3.68705i q^{83} -6.77851i q^{85} -3.49158i q^{86} -5.16934 q^{88} -3.34650i q^{89} +(-2.60158 + 7.50694i) q^{91} +8.39252 q^{92} -5.53442 q^{94} +15.1609 q^{95} -1.49094i q^{97} -2.14443i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4}+O(q^{10}) \) 14 * q - 14 * q^4	\( 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64} + 8 q^{65} - 8 q^{68} - 16 q^{74} + 24 q^{77} - 28 q^{79} - 24 q^{82} - 8 q^{92}+O(q^{100}) \) 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

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