LMFDB - Embedded newform 2912.2.k.i.1793.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1793,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2912.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.k (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:	$23.2524370686$
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} + 37x^{12} + 544x^{10} + 4060x^{8} + 16288x^{6} + 34160x^{4} + 33216x^{2} + 10816 $ x^14 + 37x^12 + 544x^10 + 4060x^8 + 16288x^6 + 34160x^4 + 33216x^2 + 10816
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.14
Root			$-2.99280i$ of defining polynomial
Character	$\chi$	$=$	2912.1793
Dual form			2912.2.k.i.1793.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99280 q^{3} +3.69704i q^{5} -1.00000i q^{7} +5.95684 q^{9} +5.69839i q^{11} +(1.34425 - 3.34559i) q^{13} +11.0645i q^{15} +4.74155 q^{17} -3.62948i q^{19} -2.99280i q^{21} +1.63311 q^{23} -8.66813 q^{25} +8.84922 q^{27} -1.25616 q^{29} -0.621770i q^{31} +17.0541i q^{33} +3.69704 q^{35} +10.6273i q^{37} +(4.02306 - 10.0127i) q^{39} -5.36611i q^{41} +11.3093 q^{43} +22.0227i q^{45} -4.39611i q^{47} -1.00000 q^{49} +14.1905 q^{51} -9.91227 q^{53} -21.0672 q^{55} -10.8623i q^{57} +0.279220i q^{59} -14.0573 q^{61} -5.95684i q^{63} +(12.3688 + 4.96974i) q^{65} +15.2364i q^{67} +4.88756 q^{69} +2.83477i q^{71} -3.94829i q^{73} -25.9420 q^{75} +5.69839 q^{77} -9.20943 q^{79} +8.61340 q^{81} -3.29212i q^{83} +17.5297i q^{85} -3.75943 q^{87} -11.8438i q^{89} +(-3.34559 - 1.34425i) q^{91} -1.86083i q^{93} +13.4183 q^{95} -0.554202i q^{97} +33.9444i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) $ 14 * q - 2 * q^3 + 32 * q^9 - 6 * q^13 + 36 * q^17 - 18 * q^23 - 18 * q^25 - 14 * q^27 - 16 * q^29 + 8 * q^35 + 34 * q^39 + 60 * q^43 - 14 * q^49 + 12 * q^51 + 4 * q^53 - 60 * q^55 + 6 * q^61 + 20 * q^65 - 2 * q^69 - 114 * q^75 - 2 * q^77 - 26 * q^79 + 22 * q^81 - 8 * q^87 + 16 * q^91 + 68 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.99280			1.72789			0.863946	−	0.503584i	$-0.167986\pi$
$3$	2.99280			1.72789			0.863946	+	0.503584i	$0.167986\pi$
$4$		0			0
$5$			3.69704i			1.65337i	0.562666	+	0.826684i	$0.309776\pi$
$5$			3.69704i			1.65337i	−0.562666	+	0.826684i	$0.690224\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	5.95684			1.98561
$10$		0			0
$11$			5.69839i			1.71813i	0.511867	+	0.859065i	$0.328954\pi$
$11$			5.69839i			1.71813i	−0.511867	+	0.859065i	$0.671046\pi$
$12$		0			0
$13$	1.34425	−	3.34559i	0.372827	−	0.927901i
$13$	1.34425	−	3.34559i	0.372827	−	0.927901i
$14$		0			0
$15$			11.0645i			2.85684i
$16$		0			0
$17$	4.74155			1.15000			0.574998	−	0.818155i	$-0.305003\pi$
$17$	4.74155			1.15000			0.574998	+	0.818155i	$0.305003\pi$
$18$		0			0
$19$		−	3.62948i		−	0.832659i	−0.909214	−	0.416329i	$-0.863316\pi$
$19$		−	3.62948i		−	0.832659i	0.909214	−	0.416329i	$-0.136684\pi$
$20$		0			0
$21$		−	2.99280i		−	0.653082i
$22$		0			0
$23$	1.63311			0.340527			0.170263	−	0.985399i	$-0.445538\pi$
$23$	1.63311			0.340527			0.170263	+	0.985399i	$0.445538\pi$
$24$		0			0
$25$	−8.66813			−1.73363
$26$		0			0
$27$	8.84922			1.70303
$28$		0			0
$29$	−1.25616			−0.233263			−0.116632	−	0.993175i	$-0.537210\pi$
$29$	−1.25616			−0.233263			−0.116632	+	0.993175i	$0.537210\pi$
$30$		0			0
$31$		−	0.621770i		−	0.111673i	−0.998440	−	0.0558366i	$-0.982217\pi$
$31$		−	0.621770i		−	0.111673i	0.998440	−	0.0558366i	$-0.0177826\pi$
$32$		0			0
$33$			17.0541i			2.96874i
$34$		0			0
$35$	3.69704			0.624914
$36$		0			0
$37$			10.6273i			1.74711i	0.486726	+	0.873555i	$0.338191\pi$
$37$			10.6273i			1.74711i	−0.486726	+	0.873555i	$0.661809\pi$
$38$		0			0
$39$	4.02306	−	10.0127i	0.644205	−	1.60331i
$40$		0			0
$41$		−	5.36611i		−	0.838046i	−0.907976	−	0.419023i	$-0.862373\pi$
$41$		−	5.36611i		−	0.838046i	0.907976	−	0.419023i	$-0.137627\pi$
$42$		0			0
$43$	11.3093			1.72466			0.862328	−	0.506350i	$-0.169006\pi$
$43$	11.3093			1.72466			0.862328	+	0.506350i	$0.169006\pi$
$44$		0			0
$45$			22.0227i			3.28295i
$46$		0			0
$47$		−	4.39611i		−	0.641239i	−0.947208	−	0.320619i	$-0.896109\pi$
$47$		−	4.39611i		−	0.641239i	0.947208	−	0.320619i	$-0.103891\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	14.1905			1.98707
$52$		0			0
$53$	−9.91227			−1.36155			−0.680777	−	0.732490i	$-0.738358\pi$
$53$	−9.91227			−1.36155			−0.680777	+	0.732490i	$0.738358\pi$
$54$		0			0
$55$	−21.0672			−2.84070
$56$		0			0
$57$		−	10.8623i		−	1.43874i
$58$		0			0
$59$			0.279220i			0.0363513i	0.999835	+	0.0181756i	$0.00578581\pi$
$59$			0.279220i			0.0363513i	−0.999835	+	0.0181756i	$0.994214\pi$
$60$		0			0
$61$	−14.0573			−1.79985			−0.899926	−	0.436042i	$-0.856380\pi$
$61$	−14.0573			−1.79985			−0.899926	+	0.436042i	$0.856380\pi$
$62$		0			0
$63$		−	5.95684i		−	0.750491i
$64$		0			0
$65$	12.3688	+	4.96974i	1.53416	+	0.616420i
$66$		0			0
$67$			15.2364i			1.86142i	0.365760	+	0.930709i	$0.380809\pi$
$67$			15.2364i			1.86142i	−0.365760	+	0.930709i	$0.619191\pi$
$68$		0			0
$69$	4.88756			0.588394
$70$		0			0
$71$			2.83477i			0.336425i	0.985751	+	0.168213i	$0.0537995\pi$
$71$			2.83477i			0.336425i	−0.985751	+	0.168213i	$0.946201\pi$
$72$		0			0
$73$		−	3.94829i		−	0.462112i	−0.972940	−	0.231056i	$-0.925782\pi$
$73$		−	3.94829i		−	0.462112i	0.972940	−	0.231056i	$-0.0742181\pi$
$74$		0			0
$75$	−25.9420			−2.99552
$76$		0			0
$77$	5.69839			0.649392
$78$		0			0
$79$	−9.20943			−1.03614			−0.518071	−	0.855338i	$-0.673350\pi$
$79$	−9.20943			−1.03614			−0.518071	+	0.855338i	$0.673350\pi$
$80$		0			0
$81$	8.61340			0.957045
$82$		0			0
$83$		−	3.29212i		−	0.361357i	−0.983542	−	0.180679i	$-0.942171\pi$
$83$		−	3.29212i		−	0.361357i	0.983542	−	0.180679i	$-0.0578294\pi$
$84$		0			0
$85$			17.5297i			1.90137i
$86$		0			0
$87$	−3.75943			−0.403054
$88$		0			0
$89$		−	11.8438i		−	1.25544i	−0.778438	−	0.627721i	$-0.783988\pi$
$89$		−	11.8438i		−	1.25544i	0.778438	−	0.627721i	$-0.216012\pi$
$90$		0			0
$91$	−3.34559	−	1.34425i	−0.350714	−	0.140915i
$92$		0			0
$93$		−	1.86083i		−	0.192959i
$94$		0			0
$95$	13.4183			1.37669
$96$		0			0
$97$		−	0.554202i		−	0.0562706i	−0.999604	−	0.0281353i	$-0.991043\pi$
$97$		−	0.554202i		−	0.0562706i	0.999604	−	0.0281353i	$-0.00895693\pi$
$98$		0			0
$99$			33.9444i			3.41154i
$100$		0			0
$101$	2.63834			0.262525			0.131263	−	0.991348i	$-0.458097\pi$
$101$	2.63834			0.262525			0.131263	+	0.991348i	$0.458097\pi$
$102$		0			0
$103$	6.67648			0.657853			0.328926	−	0.944356i	$-0.393313\pi$
$103$	6.67648			0.657853			0.328926	+	0.944356i	$0.393313\pi$
$104$		0			0
$105$	11.0645			1.07978
$106$		0			0
$107$	14.5625			1.40781			0.703906	−	0.710293i	$-0.251438\pi$
$107$	14.5625			1.40781			0.703906	+	0.710293i	$0.251438\pi$
$108$		0			0
$109$		−	4.66963i		−	0.447270i	−0.974673	−	0.223635i	$-0.928208\pi$
$109$		−	4.66963i		−	0.447270i	0.974673	−	0.223635i	$-0.0717923\pi$
$110$		0			0
$111$			31.8052i			3.01882i
$112$		0			0
$113$	0.471542			0.0443590			0.0221795	−	0.999754i	$-0.492939\pi$
$113$	0.471542			0.0443590			0.0221795	+	0.999754i	$0.492939\pi$
$114$		0			0
$115$			6.03767i			0.563016i
$116$		0			0
$117$	8.00746	−	19.9292i	0.740290	−	1.84245i
$118$		0			0
$119$		−	4.74155i		−	0.434657i
$120$		0			0
$121$	−21.4717			−1.95197
$122$		0			0
$123$		−	16.0597i		−	1.44805i
$124$		0			0
$125$		−	13.5612i		−	1.21295i
$126$		0			0
$127$	−4.97394			−0.441366			−0.220683	−	0.975346i	$-0.570829\pi$
$127$	−4.97394			−0.441366			−0.220683	+	0.975346i	$0.570829\pi$
$128$		0			0
$129$	33.8465			2.98002
$130$		0			0
$131$	−13.4693			−1.17682			−0.588410	−	0.808563i	$-0.700246\pi$
$131$	−13.4693			−1.17682			−0.588410	+	0.808563i	$0.700246\pi$
$132$		0			0
$133$	−3.62948			−0.314715
$134$		0			0
$135$			32.7159i			2.81574i
$136$		0			0
$137$			11.2758i			0.963353i	0.876349	+	0.481676i	$0.159972\pi$
$137$			11.2758i			0.963353i	−0.876349	+	0.481676i	$0.840028\pi$
$138$		0			0
$139$	−1.95984			−0.166232			−0.0831159	−	0.996540i	$-0.526487\pi$
$139$	−1.95984			−0.166232			−0.0831159	+	0.996540i	$0.526487\pi$
$140$		0			0
$141$		−	13.1567i		−	1.10799i
$142$		0			0
$143$	19.0645	+	7.66004i	1.59425	+	0.640565i
$144$		0			0
$145$		−	4.64408i		−	0.385670i
$146$		0			0
$147$	−2.99280			−0.246842
$148$		0			0
$149$		−	0.0759518i		−	0.00622222i	−0.999995	−	0.00311111i	$-0.999010\pi$
$149$		−	0.0759518i		−	0.00622222i	0.999995	−	0.00311111i	$-0.000990298\pi$
$150$		0			0
$151$		−	0.810323i		−	0.0659432i	−0.999456	−	0.0329716i	$-0.989503\pi$
$151$		−	0.810323i		−	0.0659432i	0.999456	−	0.0329716i	$-0.0104971\pi$
$152$		0			0
$153$	28.2447			2.28345
$154$		0			0
$155$	2.29871			0.184637
$156$		0			0
$157$	8.76714			0.699694			0.349847	−	0.936807i	$-0.386234\pi$
$157$	8.76714			0.699694			0.349847	+	0.936807i	$0.386234\pi$
$158$		0			0
$159$	−29.6654			−2.35262
$160$		0			0
$161$		−	1.63311i		−	0.128707i
$162$		0			0
$163$			16.2180i			1.27029i	0.772391	+	0.635147i	$0.219060\pi$
$163$			16.2180i			1.27029i	−0.772391	+	0.635147i	$0.780940\pi$
$164$		0			0
$165$	−63.0499			−4.90842
$166$		0			0
$167$		−	18.8403i		−	1.45790i	−0.684566	−	0.728951i	$-0.740008\pi$
$167$		−	18.8403i		−	1.45790i	0.684566	−	0.728951i	$-0.259992\pi$
$168$		0			0
$169$	−9.38600	−	8.99461i	−0.722000	−	0.691893i
$170$		0			0
$171$		−	21.6202i		−	1.65334i
$172$		0			0
$173$	14.5809			1.10857			0.554283	−	0.832328i	$-0.312992\pi$
$173$	14.5809			1.10857			0.554283	+	0.832328i	$0.312992\pi$
$174$		0			0
$175$			8.66813i			0.655249i
$176$		0			0
$177$			0.835648i			0.0628111i
$178$		0			0
$179$	7.23740			0.540949			0.270475	−	0.962727i	$-0.412819\pi$
$179$	7.23740			0.540949			0.270475	+	0.962727i	$0.412819\pi$
$180$		0			0
$181$	−23.4509			−1.74309			−0.871545	−	0.490315i	$-0.836882\pi$
$181$	−23.4509			−1.74309			−0.871545	+	0.490315i	$0.836882\pi$
$182$		0			0
$183$	−42.0707			−3.10995
$184$		0			0
$185$	−39.2894			−2.88862
$186$		0			0
$187$			27.0192i			1.97584i
$188$		0			0
$189$		−	8.84922i		−	0.643686i
$190$		0			0
$191$	23.1613			1.67589			0.837946	−	0.545753i	$-0.183756\pi$
$191$	23.1613			1.67589			0.837946	+	0.545753i	$0.183756\pi$
$192$		0			0
$193$		−	14.8845i		−	1.07141i	−0.844406	−	0.535703i	$-0.820047\pi$
$193$		−	14.8845i		−	1.07141i	0.844406	−	0.535703i	$-0.179953\pi$
$194$		0			0
$195$	37.0173	+	14.8734i	2.65087	+	1.06511i
$196$		0			0
$197$		−	12.6033i		−	0.897949i	−0.893545	−	0.448974i	$-0.851789\pi$
$197$		−	12.6033i		−	0.897949i	0.893545	−	0.448974i	$-0.148211\pi$
$198$		0			0
$199$	8.55932			0.606754			0.303377	−	0.952871i	$-0.401886\pi$
$199$	8.55932			0.606754			0.303377	+	0.952871i	$0.401886\pi$
$200$		0			0
$201$			45.5994i			3.21633i
$202$		0			0
$203$			1.25616i			0.0881652i
$204$		0			0
$205$	19.8387			1.38560
$206$		0			0
$207$	9.72817			0.676154
$208$		0			0
$209$	20.6822			1.43062
$210$		0			0
$211$	−14.1372			−0.973248			−0.486624	−	0.873611i	$-0.661772\pi$
$211$	−14.1372			−0.973248			−0.486624	+	0.873611i	$0.661772\pi$
$212$		0			0
$213$			8.48389i			0.581306i
$214$		0			0
$215$			41.8111i			2.85149i
$216$		0			0
$217$	−0.621770			−0.0422085
$218$		0			0
$219$		−	11.8164i		−	0.798480i
$220$		0			0
$221$	6.37382	−	15.8633i	0.428749	−	1.06708i
$222$		0			0
$223$		−	0.302897i		−	0.0202835i	−0.999949	−	0.0101417i	$-0.996772\pi$
$223$		−	0.302897i		−	0.0202835i	0.999949	−	0.0101417i	$-0.00322827\pi$
$224$		0			0
$225$	−51.6346			−3.44231
$226$		0			0
$227$		−	27.3442i		−	1.81490i	−0.420160	−	0.907450i	$-0.638026\pi$
$227$		−	27.3442i		−	1.81490i	0.420160	−	0.907450i	$-0.361974\pi$
$228$		0			0
$229$		−	13.6499i		−	0.902009i	−0.892522	−	0.451004i	$-0.851066\pi$
$229$		−	13.6499i		−	0.902009i	0.892522	−	0.451004i	$-0.148934\pi$
$230$		0			0
$231$	17.0541			1.12208
$232$		0			0
$233$	−21.2997			−1.39539			−0.697694	−	0.716396i	$-0.745791\pi$
$233$	−21.2997			−1.39539			−0.697694	+	0.716396i	$0.745791\pi$
$234$		0			0
$235$	16.2526			1.06020
$236$		0			0
$237$	−27.5620			−1.79034
$238$		0			0
$239$		−	21.9819i		−	1.42189i	−0.703246	−	0.710947i	$-0.748267\pi$
$239$		−	21.9819i		−	1.42189i	0.703246	−	0.710947i	$-0.251733\pi$
$240$		0			0
$241$		−	16.2846i		−	1.04898i	−0.851416	−	0.524492i	$-0.824255\pi$
$241$		−	16.2846i		−	1.04898i	0.851416	−	0.524492i	$-0.175745\pi$
$242$		0			0
$243$	−0.769477			−0.0493620
$244$		0			0
$245$		−	3.69704i		−	0.236195i
$246$		0			0
$247$	−12.1428	−	4.87891i	−0.772625	−	0.310438i
$248$		0			0
$249$		−	9.85265i		−	0.624386i
$250$		0			0
$251$	12.5122			0.789762			0.394881	−	0.918732i	$-0.370786\pi$
$251$	12.5122			0.789762			0.394881	+	0.918732i	$0.370786\pi$
$252$		0			0
$253$			9.30609i			0.585069i
$254$		0			0
$255$			52.4629i			3.28536i
$256$		0			0
$257$	−1.65213			−0.103057			−0.0515284	−	0.998672i	$-0.516409\pi$
$257$	−1.65213			−0.103057			−0.0515284	+	0.998672i	$0.516409\pi$
$258$		0			0
$259$	10.6273			0.660345
$260$		0			0
$261$	−7.48275			−0.463170
$262$		0			0
$263$	13.4712			0.830669			0.415335	−	0.909669i	$-0.363665\pi$
$263$	13.4712			0.830669			0.415335	+	0.909669i	$0.363665\pi$
$264$		0			0
$265$		−	36.6461i		−	2.25115i
$266$		0			0
$267$		−	35.4462i		−	2.16927i
$268$		0			0
$269$	−1.22799			−0.0748720			−0.0374360	−	0.999299i	$-0.511919\pi$
$269$	−1.22799			−0.0748720			−0.0374360	+	0.999299i	$0.511919\pi$
$270$		0			0
$271$			19.4724i			1.18287i	0.806354	+	0.591433i	$0.201438\pi$
$271$			19.4724i			1.18287i	−0.806354	+	0.591433i	$0.798562\pi$
$272$		0			0
$273$	−10.0127	−	4.02306i	−0.605995	−	0.243487i
$274$		0			0
$275$		−	49.3944i		−	2.97859i
$276$		0			0
$277$	27.4751			1.65082			0.825410	−	0.564534i	$-0.190944\pi$
$277$	27.4751			1.65082			0.825410	+	0.564534i	$0.190944\pi$
$278$		0			0
$279$		−	3.70378i		−	0.221740i
$280$		0			0
$281$		−	22.3246i		−	1.33178i	−0.746051	−	0.665888i	$-0.768053\pi$
$281$		−	22.3246i		−	1.33178i	0.746051	−	0.665888i	$-0.231947\pi$
$282$		0			0
$283$	14.6496			0.870829			0.435414	−	0.900230i	$-0.356602\pi$
$283$	14.6496			0.870829			0.435414	+	0.900230i	$0.356602\pi$
$284$		0			0
$285$	40.1583			2.37877
$286$		0			0
$287$	−5.36611			−0.316752
$288$		0			0
$289$	5.48232			0.322490
$290$		0			0
$291$		−	1.65861i		−	0.0972296i
$292$		0			0
$293$			9.49575i			0.554748i	0.960762	+	0.277374i	$0.0894641\pi$
$293$			9.49575i			0.554748i	−0.960762	+	0.277374i	$0.910536\pi$
$294$		0			0
$295$	−1.03229			−0.0601021
$296$		0			0
$297$			50.4263i			2.92603i
$298$		0			0
$299$	2.19530	−	5.46372i	0.126958	−	0.315975i
$300$		0			0
$301$		−	11.3093i		−	0.651859i
$302$		0			0
$303$	7.89603			0.453615
$304$		0			0
$305$		−	51.9704i		−	2.97582i
$306$		0			0
$307$			6.03718i			0.344560i	0.985048	+	0.172280i	$0.0551134\pi$
$307$			6.03718i			0.344560i	−0.985048	+	0.172280i	$0.944887\pi$
$308$		0			0
$309$	19.9813			1.13670
$310$		0			0
$311$	0.856441			0.0485643			0.0242821	−	0.999705i	$-0.492270\pi$
$311$	0.856441			0.0485643			0.0242821	+	0.999705i	$0.492270\pi$
$312$		0			0
$313$	25.3748			1.43427			0.717135	−	0.696935i	$-0.245453\pi$
$313$	25.3748			1.43427			0.717135	+	0.696935i	$0.245453\pi$
$314$		0			0
$315$	22.0227			1.24084
$316$		0			0
$317$		−	7.01047i		−	0.393747i	−0.980429	−	0.196874i	$-0.936921\pi$
$317$		−	7.01047i		−	0.393747i	0.980429	−	0.196874i	$-0.0630789\pi$
$318$		0			0
$319$		−	7.15809i		−	0.400776i
$320$		0			0
$321$	43.5827			2.43255
$322$		0			0
$323$		−	17.2093i		−	0.957554i
$324$		0			0
$325$	−11.6521	+	29.0000i	−0.646343	+	1.60863i
$326$		0			0
$327$		−	13.9753i		−	0.772834i
$328$		0			0
$329$	−4.39611			−0.242366
$330$		0			0
$331$		−	12.7027i		−	0.698203i	−0.937085	−	0.349101i	$-0.886487\pi$
$331$		−	12.7027i		−	0.698203i	0.937085	−	0.349101i	$-0.113513\pi$
$332$		0			0
$333$			63.3048i			3.46908i
$334$		0			0
$335$	−56.3295			−3.07761
$336$		0			0
$337$	−30.1177			−1.64061			−0.820307	−	0.571924i	$-0.806197\pi$
$337$	−30.1177			−1.64061			−0.820307	+	0.571924i	$0.806197\pi$
$338$		0			0
$339$	1.41123			0.0766475
$340$		0			0
$341$	3.54309			0.191869
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			18.0695i			0.972831i
$346$		0			0
$347$	2.81659			0.151203			0.0756013	−	0.997138i	$-0.475912\pi$
$347$	2.81659			0.151203			0.0756013	+	0.997138i	$0.475912\pi$
$348$		0			0
$349$		−	28.1480i		−	1.50673i	−0.657604	−	0.753364i	$-0.728430\pi$
$349$		−	28.1480i		−	1.50673i	0.657604	−	0.753364i	$-0.271570\pi$
$350$		0			0
$351$	11.8955	−	29.6059i	0.634937	−	1.58025i
$352$		0			0
$353$			26.4799i			1.40938i	0.709513	+	0.704692i	$0.248915\pi$
$353$			26.4799i			1.40938i	−0.709513	+	0.704692i	$0.751085\pi$
$354$		0			0
$355$	−10.4803			−0.556235
$356$		0			0
$357$		−	14.1905i		−	0.751041i
$358$		0			0
$359$		−	7.63333i		−	0.402872i	−0.979502	−	0.201436i	$-0.935439\pi$
$359$		−	7.63333i		−	0.402872i	0.979502	−	0.201436i	$-0.0645608\pi$
$360$		0			0
$361$	5.82691			0.306680
$362$		0			0
$363$	−64.2603			−3.37279
$364$		0			0
$365$	14.5970			0.764042
$366$		0			0
$367$	−3.64874			−0.190463			−0.0952314	−	0.995455i	$-0.530359\pi$
$367$	−3.64874			−0.190463			−0.0952314	+	0.995455i	$0.530359\pi$
$368$		0			0
$369$		−	31.9651i		−	1.66403i
$370$		0			0
$371$			9.91227i			0.514619i
$372$		0			0
$373$	24.4979			1.26845			0.634226	−	0.773148i	$-0.281319\pi$
$373$	24.4979			1.26845			0.634226	+	0.773148i	$0.281319\pi$
$374$		0			0
$375$		−	40.5860i		−	2.09585i
$376$		0			0
$377$	−1.68859	+	4.20260i	−0.0869668	+	0.216445i
$378$		0			0
$379$			25.3988i			1.30465i	0.757940	+	0.652324i	$0.226206\pi$
$379$			25.3988i			1.30465i	−0.757940	+	0.652324i	$0.773794\pi$
$380$		0			0
$381$	−14.8860			−0.762632
$382$		0			0
$383$			16.0037i			0.817753i	0.912590	+	0.408876i	$0.134079\pi$
$383$			16.0037i			0.817753i	−0.912590	+	0.408876i	$0.865921\pi$
$384$		0			0
$385$			21.0672i			1.07368i
$386$		0			0
$387$	67.3678			3.42450
$388$		0			0
$389$	9.18460			0.465678			0.232839	−	0.972515i	$-0.425199\pi$
$389$	9.18460			0.465678			0.232839	+	0.972515i	$0.425199\pi$
$390$		0			0
$391$	7.74347			0.391604
$392$		0			0
$393$	−40.3110			−2.03342
$394$		0			0
$395$		−	34.0477i		−	1.71312i
$396$		0			0
$397$		−	35.2963i		−	1.77147i	−0.464193	−	0.885734i	$-0.653656\pi$
$397$		−	35.2963i		−	1.77147i	0.464193	−	0.885734i	$-0.346344\pi$
$398$		0			0
$399$	−10.8623			−0.543794
$400$		0			0
$401$		−	24.0256i		−	1.19978i	−0.800082	−	0.599890i	$-0.795211\pi$
$401$		−	24.0256i		−	1.19978i	0.800082	−	0.599890i	$-0.204789\pi$
$402$		0			0
$403$	−2.08019	−	0.835812i	−0.103622	−	0.0416348i
$404$		0			0
$405$			31.8441i			1.58235i
$406$		0			0
$407$	−60.5582			−3.00176
$408$		0			0
$409$			6.55368i			0.324059i	0.986786	+	0.162029i	$0.0518039\pi$
$409$			6.55368i			0.324059i	−0.986786	+	0.162029i	$0.948196\pi$
$410$		0			0
$411$			33.7461i			1.66457i
$412$		0			0
$413$	0.279220			0.0137395
$414$		0			0
$415$	12.1711			0.597456
$416$		0			0
$417$	−5.86541			−0.287231
$418$		0			0
$419$	2.37134			0.115848			0.0579238	−	0.998321i	$-0.481552\pi$
$419$	2.37134			0.115848			0.0579238	+	0.998321i	$0.481552\pi$
$420$		0			0
$421$			10.5690i			0.515103i	0.966265	+	0.257551i	$0.0829156\pi$
$421$			10.5690i			0.515103i	−0.966265	+	0.257551i	$0.917084\pi$
$422$		0			0
$423$		−	26.1869i		−	1.27325i
$424$		0			0
$425$	−41.1004			−1.99366
$426$		0			0
$427$			14.0573i			0.680280i
$428$		0			0
$429$	57.0562	+	22.9250i	2.75470	+	1.10683i
$430$		0			0
$431$			31.4671i			1.51572i	0.652418	+	0.757859i	$0.273755\pi$
$431$			31.4671i			1.51572i	−0.652418	+	0.757859i	$0.726245\pi$
$432$		0			0
$433$	−27.4309			−1.31824			−0.659121	−	0.752037i	$-0.729072\pi$
$433$	−27.4309			−1.31824			−0.659121	+	0.752037i	$0.729072\pi$
$434$		0			0
$435$		−	13.8988i		−	0.666396i
$436$		0			0
$437$		−	5.92733i		−	0.283543i
$438$		0			0
$439$	20.1063			0.959623			0.479811	−	0.877372i	$-0.340705\pi$
$439$	20.1063			0.959623			0.479811	+	0.877372i	$0.340705\pi$
$440$		0			0
$441$	−5.95684			−0.283659
$442$		0			0
$443$	−22.1527			−1.05251			−0.526253	−	0.850328i	$-0.676404\pi$
$443$	−22.1527			−1.05251			−0.526253	+	0.850328i	$0.676404\pi$
$444$		0			0
$445$	43.7871			2.07571
$446$		0			0
$447$		−	0.227308i		−	0.0107513i
$448$		0			0
$449$			11.0867i			0.523215i	0.965174	+	0.261608i	$0.0842526\pi$
$449$			11.0867i			0.523215i	−0.965174	+	0.261608i	$0.915747\pi$
$450$		0			0
$451$	30.5782			1.43987
$452$		0			0
$453$		−	2.42513i		−	0.113943i
$454$		0			0
$455$	4.96974	−	12.3688i	0.232985	−	0.579859i
$456$		0			0
$457$			18.1977i			0.851254i	0.904899	+	0.425627i	$0.139946\pi$
$457$			18.1977i			0.851254i	−0.904899	+	0.425627i	$0.860054\pi$
$458$		0			0
$459$	41.9590			1.95848
$460$		0			0
$461$			15.2150i			0.708635i	0.935125	+	0.354317i	$0.115287\pi$
$461$			15.2150i			0.708635i	−0.935125	+	0.354317i	$0.884713\pi$
$462$		0			0
$463$			5.95248i			0.276635i	0.990388	+	0.138318i	$0.0441695\pi$
$463$			5.95248i			0.276635i	−0.990388	+	0.138318i	$0.955831\pi$
$464$		0			0
$465$	6.87957			0.319033
$466$		0			0
$467$	18.9812			0.878344			0.439172	−	0.898403i	$-0.355272\pi$
$467$	18.9812			0.878344			0.439172	+	0.898403i	$0.355272\pi$
$468$		0			0
$469$	15.2364			0.703550
$470$		0			0
$471$	26.2383			1.20900
$472$		0			0
$473$			64.4449i			2.96318i
$474$		0			0
$475$			31.4608i			1.44352i
$476$		0			0
$477$	−59.0458			−2.70352
$478$		0			0
$479$			30.0014i			1.37080i	0.728167	+	0.685400i	$0.240373\pi$
$479$			30.0014i			1.37080i	−0.728167	+	0.685400i	$0.759627\pi$
$480$		0			0
$481$	35.5545	+	14.2857i	1.62114	+	0.651370i
$482$		0			0
$483$		−	4.88756i		−	0.222392i
$484$		0			0
$485$	2.04891			0.0930361
$486$		0			0
$487$		−	12.3387i		−	0.559122i	−0.960128	−	0.279561i	$-0.909811\pi$
$487$		−	12.3387i		−	0.559122i	0.960128	−	0.279561i	$-0.0901888\pi$
$488$		0			0
$489$			48.5373i			2.19493i
$490$		0			0
$491$	32.6905			1.47530			0.737651	−	0.675182i	$-0.235935\pi$
$491$	32.6905			1.47530			0.737651	+	0.675182i	$0.235935\pi$
$492$		0			0
$493$	−5.95615			−0.268252
$494$		0			0
$495$	−125.494			−5.64053
$496$		0			0
$497$	2.83477			0.127157
$498$		0			0
$499$		−	4.59993i		−	0.205921i	−0.994685	−	0.102961i	$-0.967168\pi$
$499$		−	4.59993i		−	0.205921i	0.994685	−	0.102961i	$-0.0328316\pi$
$500$		0			0
$501$		−	56.3851i		−	2.51910i
$502$		0			0
$503$	19.1629			0.854430			0.427215	−	0.904150i	$-0.359495\pi$
$503$	19.1629			0.854430			0.427215	+	0.904150i	$0.359495\pi$
$504$		0			0
$505$			9.75407i			0.434050i
$506$		0			0
$507$	−28.0904	−	26.9190i	−1.24754	−	1.19552i
$508$		0			0
$509$			14.0308i			0.621902i	0.950426	+	0.310951i	$0.100648\pi$
$509$			14.0308i			0.621902i	−0.950426	+	0.310951i	$0.899352\pi$
$510$		0			0
$511$	−3.94829			−0.174662
$512$		0			0
$513$		−	32.1180i		−	1.41805i
$514$		0			0
$515$			24.6832i			1.08767i
$516$		0			0
$517$	25.0508			1.10173
$518$		0			0
$519$	43.6377			1.91548
$520$		0			0
$521$	2.99771			0.131332			0.0656661	−	0.997842i	$-0.479083\pi$
$521$	2.99771			0.131332			0.0656661	+	0.997842i	$0.479083\pi$
$522$		0			0
$523$	−21.4666			−0.938669			−0.469335	−	0.883020i	$-0.655506\pi$
$523$	−21.4666			−0.938669			−0.469335	+	0.883020i	$0.655506\pi$
$524$		0			0
$525$			25.9420i			1.13220i
$526$		0			0
$527$		−	2.94815i		−	0.128424i
$528$		0			0
$529$	−20.3330			−0.884042
$530$		0			0
$531$			1.66327i			0.0721796i
$532$		0			0
$533$	−17.9528	−	7.21338i	−0.777624	−	0.312446i
$534$		0			0
$535$			53.8383i			2.32763i
$536$		0			0
$537$	21.6601			0.934702
$538$		0			0
$539$		−	5.69839i		−	0.245447i
$540$		0			0
$541$		−	35.3946i		−	1.52173i	−0.648909	−	0.760866i	$-0.724774\pi$
$541$		−	35.3946i		−	1.52173i	0.648909	−	0.760866i	$-0.275226\pi$
$542$		0			0
$543$	−70.1838			−3.01187
$544$		0			0
$545$	17.2638			0.739502
$546$		0			0
$547$	−22.5940			−0.966051			−0.483025	−	0.875606i	$-0.660462\pi$
$547$	−22.5940			−0.966051			−0.483025	+	0.875606i	$0.660462\pi$
$548$		0			0
$549$	−83.7371			−3.57381
$550$		0			0
$551$			4.55920i			0.194229i
$552$		0			0
$553$			9.20943i			0.391625i
$554$		0			0
$555$	−117.585			−4.99122
$556$		0			0
$557$			12.6895i			0.537672i	0.963186	+	0.268836i	$0.0866390\pi$
$557$			12.6895i			0.537672i	−0.963186	+	0.268836i	$0.913361\pi$
$558$		0			0
$559$	15.2025	−	37.8364i	0.642998	−	1.60031i
$560$		0			0
$561$			80.8631i			3.41404i
$562$		0			0
$563$	11.0819			0.467045			0.233522	−	0.972351i	$-0.424975\pi$
$563$	11.0819			0.467045			0.233522	+	0.972351i	$0.424975\pi$
$564$		0			0
$565$			1.74331i			0.0733417i
$566$		0			0
$567$		−	8.61340i		−	0.361729i
$568$		0			0
$569$	24.7801			1.03884			0.519418	−	0.854520i	$-0.326149\pi$
$569$	24.7801			1.03884			0.519418	+	0.854520i	$0.326149\pi$
$570$		0			0
$571$	−26.6916			−1.11701			−0.558505	−	0.829501i	$-0.688625\pi$
$571$	−26.6916			−1.11701			−0.558505	+	0.829501i	$0.688625\pi$
$572$		0			0
$573$	69.3171			2.89576
$574$		0			0
$575$	−14.1560			−0.590346
$576$		0			0
$577$			0.572724i			0.0238428i	0.999929	+	0.0119214i	$0.00379479\pi$
$577$			0.572724i			0.0238428i	−0.999929	+	0.0119214i	$0.996205\pi$
$578$		0			0
$579$		−	44.5462i		−	1.85128i
$580$		0			0
$581$	−3.29212			−0.136580
$582$		0			0
$583$		−	56.4840i		−	2.33933i
$584$		0			0
$585$	73.6790	+	29.6039i	3.04625	+	1.22397i
$586$		0			0
$587$		−	18.9809i		−	0.783427i	−0.920087	−	0.391713i	$-0.871883\pi$
$587$		−	18.9809i		−	0.783427i	0.920087	−	0.391713i	$-0.128117\pi$
$588$		0			0
$589$	−2.25670			−0.0929856
$590$		0			0
$591$		−	37.7192i		−	1.55156i
$592$		0			0
$593$		−	0.244025i		−	0.0100209i	−0.999987	−	0.00501045i	$-0.998405\pi$
$593$		−	0.244025i		−	0.0100209i	0.999987	−	0.00501045i	$-0.00159488\pi$
$594$		0			0
$595$	17.5297			0.718649
$596$		0			0
$597$	25.6163			1.04841
$598$		0			0
$599$	35.0043			1.43024			0.715119	−	0.699003i	$-0.246372\pi$
$599$	35.0043			1.43024			0.715119	+	0.699003i	$0.246372\pi$
$600$		0			0
$601$	−39.3378			−1.60462			−0.802311	−	0.596906i	$-0.796396\pi$
$601$	−39.3378			−1.60462			−0.802311	+	0.596906i	$0.796396\pi$
$602$		0			0
$603$			90.7606i			3.69606i
$604$		0			0
$605$		−	79.3816i		−	3.22732i
$606$		0			0
$607$	−33.2454			−1.34939			−0.674695	−	0.738097i	$-0.735725\pi$
$607$	−33.2454			−1.34939			−0.674695	+	0.738097i	$0.735725\pi$
$608$		0			0
$609$			3.75943i			0.152340i
$610$		0			0
$611$	−14.7076	−	5.90946i	−0.595006	−	0.239071i
$612$		0			0
$613$		−	20.8329i		−	0.841434i	−0.907192	−	0.420717i	$-0.861778\pi$
$613$		−	20.8329i		−	0.841434i	0.907192	−	0.420717i	$-0.138222\pi$
$614$		0			0
$615$	59.3734			2.39417
$616$		0			0
$617$		−	21.5407i		−	0.867197i	−0.901106	−	0.433598i	$-0.857244\pi$
$617$		−	21.5407i		−	0.867197i	0.901106	−	0.433598i	$-0.142756\pi$
$618$		0			0
$619$		−	10.6047i		−	0.426237i	−0.977026	−	0.213119i	$-0.931638\pi$
$619$		−	10.6047i		−	0.426237i	0.977026	−	0.213119i	$-0.0683621\pi$
$620$		0			0
$621$	14.4517			0.579928
$622$		0			0
$623$	−11.8438			−0.474513
$624$		0			0
$625$	6.79582			0.271833
$626$		0			0
$627$	61.8975			2.47195
$628$		0			0
$629$			50.3897i			2.00917i
$630$		0			0
$631$			17.6795i			0.703810i	0.936036	+	0.351905i	$0.114466\pi$
$631$			17.6795i			0.703810i	−0.936036	+	0.351905i	$0.885534\pi$
$632$		0			0
$633$	−42.3099			−1.68167
$634$		0			0
$635$		−	18.3889i		−	0.729740i
$636$		0			0
$637$	−1.34425	+	3.34559i	−0.0532610	+	0.132557i
$638$		0			0
$639$			16.8863i			0.668010i
$640$		0			0
$641$	−28.2221			−1.11470			−0.557352	−	0.830276i	$-0.688183\pi$
$641$	−28.2221			−1.11470			−0.557352	+	0.830276i	$0.688183\pi$
$642$		0			0
$643$		−	36.6547i		−	1.44552i	−0.691099	−	0.722760i	$-0.742873\pi$
$643$		−	36.6547i		−	1.44552i	0.691099	−	0.722760i	$-0.257127\pi$
$644$		0			0
$645$			125.132i			4.92707i
$646$		0			0
$647$	−3.52087			−0.138420			−0.0692099	−	0.997602i	$-0.522048\pi$
$647$	−3.52087			−0.138420			−0.0692099	+	0.997602i	$0.522048\pi$
$648$		0			0
$649$	−1.59110			−0.0624562
$650$		0			0
$651$	−1.86083			−0.0729317
$652$		0			0
$653$	−10.8670			−0.425257			−0.212628	−	0.977133i	$-0.568202\pi$
$653$	−10.8670			−0.425257			−0.212628	+	0.977133i	$0.568202\pi$
$654$		0			0
$655$		−	49.7967i		−	1.94572i
$656$		0			0
$657$		−	23.5193i		−	0.917576i
$658$		0			0
$659$	−19.1972			−0.747818			−0.373909	−	0.927465i	$-0.621983\pi$
$659$	−19.1972			−0.747818			−0.373909	+	0.927465i	$0.621983\pi$
$660$		0			0
$661$		−	26.0169i		−	1.01194i	−0.862551	−	0.505971i	$-0.831134\pi$
$661$		−	26.0169i		−	1.01194i	0.862551	−	0.505971i	$-0.168866\pi$
$662$		0			0
$663$	19.0755	−	47.4757i	0.740833	−	1.84380i
$664$		0			0
$665$		−	13.4183i		−	0.520340i
$666$		0			0
$667$	−2.05145			−0.0794324
$668$		0			0
$669$		−	0.906508i		−	0.0350476i
$670$		0			0
$671$		−	80.1040i		−	3.09238i
$672$		0			0
$673$	25.4897			0.982555			0.491278	−	0.871003i	$-0.336530\pi$
$673$	25.4897			0.982555			0.491278	+	0.871003i	$0.336530\pi$
$674$		0			0
$675$	−76.7062			−2.95242
$676$		0			0
$677$	−20.0074			−0.768948			−0.384474	−	0.923136i	$-0.625617\pi$
$677$	−20.0074			−0.768948			−0.384474	+	0.923136i	$0.625617\pi$
$678$		0			0
$679$	−0.554202			−0.0212683
$680$		0			0
$681$		−	81.8357i		−	3.13595i
$682$		0			0
$683$			41.4205i			1.58491i	0.609929	+	0.792456i	$0.291198\pi$
$683$			41.4205i			1.58491i	−0.609929	+	0.792456i	$0.708802\pi$
$684$		0			0
$685$	−41.6870			−1.59278
$686$		0			0
$687$		−	40.8513i		−	1.55857i
$688$		0			0
$689$	−13.3245	+	33.1624i	−0.507624	+	1.26339i
$690$		0			0
$691$		−	2.22077i		−	0.0844823i	−0.999107	−	0.0422411i	$-0.986550\pi$
$691$		−	2.22077i		−	0.0844823i	0.999107	−	0.0422411i	$-0.0134498\pi$
$692$		0			0
$693$	33.9444			1.28944
$694$		0			0
$695$		−	7.24563i		−	0.274842i
$696$		0			0
$697$		−	25.4437i		−	0.963749i
$698$		0			0
$699$	−63.7456			−2.41108
$700$		0			0
$701$	−20.5299			−0.775405			−0.387703	−	0.921785i	$-0.626731\pi$
$701$	−20.5299			−0.775405			−0.387703	+	0.921785i	$0.626731\pi$
$702$		0			0
$703$	38.5714			1.45475
$704$		0			0
$705$	48.6408			1.83192
$706$		0			0
$707$		−	2.63834i		−	0.0992251i
$708$		0			0
$709$		−	35.4308i		−	1.33063i	−0.746562	−	0.665315i	$-0.768297\pi$
$709$		−	35.4308i		−	1.33063i	0.746562	−	0.665315i	$-0.231703\pi$
$710$		0			0
$711$	−54.8591			−2.05738
$712$		0			0
$713$		−	1.01542i		−	0.0380277i
$714$		0			0
$715$	−28.3195	+	70.4823i	−1.05909	+	2.63589i
$716$		0			0
$717$		−	65.7875i		−	2.45688i
$718$		0			0
$719$	7.91900			0.295329			0.147664	−	0.989038i	$-0.452824\pi$
$719$	7.91900			0.295329			0.147664	+	0.989038i	$0.452824\pi$
$720$		0			0
$721$		−	6.67648i		−	0.248645i
$722$		0			0
$723$		−	48.7365i		−	1.81253i
$724$		0			0
$725$	10.8886			0.404391
$726$		0			0
$727$	−27.7392			−1.02879			−0.514394	−	0.857554i	$-0.671983\pi$
$727$	−27.7392			−1.02879			−0.514394	+	0.857554i	$0.671983\pi$
$728$		0			0
$729$	−28.1431			−1.04234
$730$		0			0
$731$	53.6238			1.98335
$732$		0			0
$733$			13.8074i			0.509989i	0.966943	+	0.254994i	$0.0820736\pi$
$733$			13.8074i			0.509989i	−0.966943	+	0.254994i	$0.917926\pi$
$734$		0			0
$735$		−	11.0645i		−	0.408120i
$736$		0			0
$737$	−86.8228			−3.19816
$738$		0			0
$739$			35.4197i			1.30294i	0.758676	+	0.651468i	$0.225846\pi$
$739$			35.4197i			1.30294i	−0.758676	+	0.651468i	$0.774154\pi$
$740$		0			0
$741$	−36.3408	−	14.6016i	−1.33501	−	0.536403i
$742$		0			0
$743$			3.58383i			0.131478i	0.997837	+	0.0657391i	$0.0209405\pi$
$743$			3.58383i			0.131478i	−0.997837	+	0.0657391i	$0.979060\pi$
$744$		0			0
$745$	0.280797			0.0102876
$746$		0			0
$747$		−	19.6106i		−	0.717515i
$748$		0			0
$749$		−	14.5625i		−	0.532103i
$750$		0			0
$751$	28.6407			1.04512			0.522558	−	0.852604i	$-0.324978\pi$
$751$	28.6407			1.04512			0.522558	+	0.852604i	$0.324978\pi$
$752$		0			0
$753$	37.4465			1.36462
$754$		0			0
$755$	2.99580			0.109028
$756$		0			0
$757$	−42.8398			−1.55704			−0.778519	−	0.627621i	$-0.784029\pi$
$757$	−42.8398			−1.55704			−0.778519	+	0.627621i	$0.784029\pi$
$758$		0			0
$759$			27.8513i			1.01094i
$760$		0			0
$761$			5.07675i			0.184032i	0.995758	+	0.0920160i	$0.0293311\pi$
$761$			5.07675i			0.184032i	−0.995758	+	0.0920160i	$0.970669\pi$
$762$		0			0
$763$	−4.66963			−0.169052
$764$		0			0
$765$			104.422i			3.77538i
$766$		0			0
$767$	0.934155	+	0.375340i	0.0337304	+	0.0135527i
$768$		0			0
$769$		−	2.71257i		−	0.0978177i	−0.998803	−	0.0489088i	$-0.984426\pi$
$769$		−	2.71257i		−	0.0978177i	0.998803	−	0.0489088i	$-0.0155744\pi$
$770$		0			0
$771$	−4.94448			−0.178071
$772$		0			0
$773$			34.3409i			1.23516i	0.786509	+	0.617579i	$0.211886\pi$
$773$			34.3409i			1.23516i	−0.786509	+	0.617579i	$0.788114\pi$
$774$		0			0
$775$			5.38958i			0.193599i
$776$		0			0
$777$	31.8052			1.14101
$778$		0			0
$779$	−19.4762			−0.697806
$780$		0			0
$781$	−16.1536			−0.578022
$782$		0			0
$783$	−11.1160			−0.397255
$784$		0			0
$785$			32.4125i			1.15685i
$786$		0			0
$787$		−	6.58178i		−	0.234615i	−0.993096	−	0.117308i	$-0.962574\pi$
$787$		−	6.58178i		−	0.234615i	0.993096	−	0.117308i	$-0.0374263\pi$
$788$		0			0
$789$	40.3165			1.43531
$790$		0			0
$791$		−	0.471542i		−	0.0167661i
$792$		0			0
$793$	−18.8965	+	47.0300i	−0.671034	+	1.67009i
$794$		0			0
$795$		−	109.674i		−	3.88975i
$796$		0			0
$797$	10.6858			0.378509			0.189255	−	0.981928i	$-0.439393\pi$
$797$	10.6858			0.378509			0.189255	+	0.981928i	$0.439393\pi$
$798$		0			0
$799$		−	20.8444i		−	0.737422i
$800$		0			0
$801$		−	70.5517i		−	2.49282i
$802$		0			0
$803$	22.4989			0.793969
$804$		0			0
$805$	6.03767			0.212800
$806$		0			0
$807$	−3.67513			−0.129371
$808$		0			0
$809$	−6.25918			−0.220061			−0.110031	−	0.993928i	$-0.535095\pi$
$809$	−6.25918			−0.220061			−0.110031	+	0.993928i	$0.535095\pi$
$810$		0			0
$811$			47.7514i			1.67678i	0.545073	+	0.838389i	$0.316502\pi$
$811$			47.7514i			1.67678i	−0.545073	+	0.838389i	$0.683498\pi$
$812$		0			0
$813$			58.2771i			2.04387i
$814$		0			0
$815$	−59.9587			−2.10026
$816$		0			0
$817$		−	41.0469i		−	1.43605i
$818$		0			0
$819$	−19.9292	−	8.00746i	−0.696381	−	0.279803i
$820$		0			0
$821$		−	16.4692i		−	0.574780i	−0.957814	−	0.287390i	$-0.907212\pi$
$821$		−	16.4692i		−	0.574780i	0.957814	−	0.287390i	$-0.0927875\pi$
$822$		0			0
$823$	53.5016			1.86495			0.932474	−	0.361238i	$-0.117646\pi$
$823$	53.5016			1.86495			0.932474	+	0.361238i	$0.117646\pi$
$824$		0			0
$825$		−	147.827i		−	5.14669i
$826$		0			0
$827$		−	36.9341i		−	1.28432i	−0.766569	−	0.642162i	$-0.778038\pi$
$827$		−	36.9341i		−	1.28432i	0.766569	−	0.642162i	$-0.221962\pi$
$828$		0			0
$829$	14.5279			0.504574			0.252287	−	0.967652i	$-0.418817\pi$
$829$	14.5279			0.504574			0.252287	+	0.967652i	$0.418817\pi$
$830$		0			0
$831$	82.2275			2.85244
$832$		0			0
$833$	−4.74155			−0.164285
$834$		0			0
$835$	69.6532			2.41045
$836$		0			0
$837$		−	5.50218i		−	0.190183i
$838$		0			0
$839$		−	7.57108i		−	0.261383i	−0.991423	−	0.130691i	$-0.958280\pi$
$839$		−	7.57108i		−	0.261383i	0.991423	−	0.130691i	$-0.0417197\pi$
$840$		0			0
$841$	−27.4221			−0.945588
$842$		0			0
$843$		−	66.8131i		−	2.30117i
$844$		0			0
$845$	33.2535	−	34.7005i	1.14395	−	1.19373i
$846$		0			0
$847$			21.4717i			0.737775i
$848$		0			0
$849$	43.8433			1.50470
$850$		0			0
$851$			17.3555i			0.594938i
$852$		0			0
$853$			19.4050i			0.664416i	0.943206	+	0.332208i	$0.107794\pi$
$853$			19.4050i			0.664416i	−0.943206	+	0.332208i	$0.892206\pi$
$854$		0			0
$855$	79.9308			2.73358
$856$		0			0
$857$	−30.4706			−1.04085			−0.520427	−	0.853906i	$-0.674227\pi$
$857$	−30.4706			−1.04085			−0.520427	+	0.853906i	$0.674227\pi$
$858$		0			0
$859$	1.02714			0.0350455			0.0175227	−	0.999846i	$-0.494422\pi$
$859$	1.02714			0.0350455			0.0175227	+	0.999846i	$0.494422\pi$
$860$		0			0
$861$	−16.0597			−0.547313
$862$		0			0
$863$			6.55731i			0.223213i	0.993752	+	0.111607i	$0.0355997\pi$
$863$			6.55731i			0.223213i	−0.993752	+	0.111607i	$0.964400\pi$
$864$		0			0
$865$			53.9063i			1.83287i
$866$		0			0
$867$	16.4075			0.557227
$868$		0			0
$869$		−	52.4789i		−	1.78023i
$870$		0			0
$871$	50.9747	+	20.4814i	1.72721	+	0.693987i
$872$		0			0
$873$		−	3.30129i		−	0.111732i
$874$		0			0
$875$	−13.5612			−0.458453
$876$		0			0
$877$			1.19761i			0.0404405i	0.999796	+	0.0202202i	$0.00643674\pi$
$877$			1.19761i			0.0404405i	−0.999796	+	0.0202202i	$0.993563\pi$
$878$		0			0
$879$			28.4189i			0.958545i
$880$		0			0
$881$	37.9071			1.27712			0.638561	−	0.769571i	$-0.279530\pi$
$881$	37.9071			1.27712			0.638561	+	0.769571i	$0.279530\pi$
$882$		0			0
$883$	13.2224			0.444970			0.222485	−	0.974936i	$-0.428583\pi$
$883$	13.2224			0.444970			0.222485	+	0.974936i	$0.428583\pi$
$884$		0			0
$885$	−3.08943			−0.103850
$886$		0			0
$887$	14.1449			0.474939			0.237469	−	0.971395i	$-0.423682\pi$
$887$	14.1449			0.474939			0.237469	+	0.971395i	$0.423682\pi$
$888$		0			0
$889$			4.97394i			0.166821i
$890$		0			0
$891$			49.0825i			1.64433i
$892$		0			0
$893$	−15.9556			−0.533933
$894$		0			0
$895$			26.7570i			0.894388i
$896$		0			0
$897$	6.57009	−	16.3518i	0.219369	−	0.545971i
$898$		0			0
$899$			0.781043i			0.0260492i
$900$		0			0
$901$	−46.9995			−1.56578
$902$		0			0
$903$		−	33.8465i		−	1.12634i
$904$		0			0
$905$		−	86.6989i		−	2.88197i
$906$		0			0
$907$	5.66341			0.188050			0.0940252	−	0.995570i	$-0.470027\pi$
$907$	5.66341			0.188050			0.0940252	+	0.995570i	$0.470027\pi$
$908$		0			0
$909$	15.7162			0.521273
$910$		0			0
$911$	−47.3493			−1.56875			−0.784376	−	0.620285i	$-0.787017\pi$
$911$	−47.3493			−1.56875			−0.784376	+	0.620285i	$0.787017\pi$
$912$		0			0
$913$	18.7598			0.620858
$914$		0			0
$915$		−	155.537i		−	5.14190i
$916$		0			0
$917$			13.4693i			0.444796i
$918$		0			0
$919$	34.8875			1.15083			0.575417	−	0.817860i	$-0.304840\pi$
$919$	34.8875			1.15083			0.575417	+	0.817860i	$0.304840\pi$
$920$		0			0
$921$			18.0681i			0.595363i
$922$		0			0
$923$	9.48398	+	3.81063i	0.312169	+	0.125428i
$924$		0			0
$925$		−	92.1184i		−	3.02883i
$926$		0			0
$927$	39.7707			1.30624
$928$		0			0
$929$		−	22.7956i		−	0.747898i	−0.927449	−	0.373949i	$-0.878003\pi$
$929$		−	22.7956i		−	0.747898i	0.927449	−	0.373949i	$-0.121997\pi$
$930$		0			0
$931$			3.62948i			0.118951i
$932$		0			0
$933$	2.56315			0.0839139
$934$		0			0
$935$	−99.8912			−3.26679
$936$		0			0
$937$	−18.3968			−0.600997			−0.300499	−	0.953782i	$-0.597153\pi$
$937$	−18.3968			−0.600997			−0.300499	+	0.953782i	$0.597153\pi$
$938$		0			0
$939$	75.9417			2.47826
$940$		0			0
$941$			1.85031i			0.0603183i	0.999545	+	0.0301592i	$0.00960142\pi$
$941$			1.85031i			0.0603183i	−0.999545	+	0.0301592i	$0.990399\pi$
$942$		0			0
$943$		−	8.76345i		−	0.285377i
$944$		0			0
$945$	32.7159			1.06425
$946$		0			0
$947$			3.22237i			0.104713i	0.998628	+	0.0523565i	$0.0166732\pi$
$947$			3.22237i			0.104713i	−0.998628	+	0.0523565i	$0.983327\pi$
$948$		0			0
$949$	−13.2094	−	5.30747i	−0.428794	−	0.172288i
$950$		0			0
$951$		−	20.9809i		−	0.680353i
$952$		0			0
$953$	41.2811			1.33723			0.668614	−	0.743610i	$-0.266888\pi$
$953$	41.2811			1.33723			0.668614	+	0.743610i	$0.266888\pi$
$954$		0			0
$955$			85.6283i			2.77087i
$956$		0			0
$957$		−	21.4227i		−	0.692498i
$958$		0			0
$959$	11.2758			0.364113
$960$		0			0
$961$	30.6134			0.987529
$962$		0			0
$963$	86.7465			2.79537
$964$		0			0
$965$	55.0285			1.77143
$966$		0			0
$967$			15.0129i			0.482784i	0.970428	+	0.241392i	$0.0776039\pi$
$967$			15.0129i			0.482784i	−0.970428	+	0.241392i	$0.922396\pi$
$968$		0			0
$969$		−	51.5041i		−	1.65455i
$970$		0			0
$971$	−16.3347			−0.524206			−0.262103	−	0.965040i	$-0.584416\pi$
$971$	−16.3347			−0.524206			−0.262103	+	0.965040i	$0.584416\pi$
$972$		0			0
$973$			1.95984i			0.0628297i
$974$		0			0
$975$	−34.8724	+	86.7913i	−1.11681	+	2.77954i
$976$		0			0
$977$			24.0507i			0.769450i	0.923031	+	0.384725i	$0.125704\pi$
$977$			24.0507i			0.769450i	−0.923031	+	0.384725i	$0.874296\pi$
$978$		0			0
$979$	67.4907			2.15701
$980$		0			0
$981$		−	27.8162i		−	0.888104i
$982$		0			0
$983$		−	21.2826i		−	0.678810i	−0.940640	−	0.339405i	$-0.889774\pi$
$983$		−	21.2826i		−	0.678810i	0.940640	−	0.339405i	$-0.110226\pi$
$984$		0			0
$985$	46.5950			1.48464
$986$		0			0
$987$	−13.1567			−0.418782
$988$		0			0
$989$	18.4694			0.587291
$990$		0			0
$991$	−13.5680			−0.431002			−0.215501	−	0.976504i	$-0.569138\pi$
$991$	−13.5680			−0.431002			−0.215501	+	0.976504i	$0.569138\pi$
$992$		0			0
$993$		−	38.0166i		−	1.20642i
$994$		0			0
$995$			31.6442i			1.00319i
$996$		0			0
$997$	29.8893			0.946604			0.473302	−	0.880900i	$-0.343062\pi$
$997$	29.8893			0.946604			0.473302	+	0.880900i	$0.343062\pi$
$998$		0			0
$999$			94.0429i			2.97539i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.k.i.1793.14	yes	14
4.3	odd	2		2912.2.k.j.1793.2	yes	14
13.12	even	2	inner	2912.2.k.i.1793.13	✓	14
52.51	odd	2		2912.2.k.j.1793.1	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.k.i.1793.13	✓	14	13.12	even	2	inner
2912.2.k.i.1793.14	yes	14	1.1	even	1	trivial
2912.2.k.j.1793.1	yes	14	52.51	odd	2
2912.2.k.j.1793.2	yes	14	4.3	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.99280 q^{3} +3.69704i q^{5} -1.00000i q^{7} +5.95684 q^{9} +5.69839i q^{11} +(1.34425 - 3.34559i) q^{13} +11.0645i q^{15} +4.74155 q^{17} -3.62948i q^{19} -2.99280i q^{21} +1.63311 q^{23} -8.66813 q^{25} +8.84922 q^{27} -1.25616 q^{29} -0.621770i q^{31} +17.0541i q^{33} +3.69704 q^{35} +10.6273i q^{37} +(4.02306 - 10.0127i) q^{39} -5.36611i q^{41} +11.3093 q^{43} +22.0227i q^{45} -4.39611i q^{47} -1.00000 q^{49} +14.1905 q^{51} -9.91227 q^{53} -21.0672 q^{55} -10.8623i q^{57} +0.279220i q^{59} -14.0573 q^{61} -5.95684i q^{63} +(12.3688 + 4.96974i) q^{65} +15.2364i q^{67} +4.88756 q^{69} +2.83477i q^{71} -3.94829i q^{73} -25.9420 q^{75} +5.69839 q^{77} -9.20943 q^{79} +8.61340 q^{81} -3.29212i q^{83} +17.5297i q^{85} -3.75943 q^{87} -11.8438i q^{89} +(-3.34559 - 1.34425i) q^{91} -1.86083i q^{93} +13.4183 q^{95} -0.554202i q^{97} +33.9444i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) \) 14 * q - 2 * q^3 + 32 * q^9 - 6 * q^13 + 36 * q^17 - 18 * q^23 - 18 * q^25 - 14 * q^27 - 16 * q^29 + 8 * q^35 + 34 * q^39 + 60 * q^43 - 14 * q^49 + 12 * q^51 + 4 * q^53 - 60 * q^55 + 6 * q^61 + 20 * q^65 - 2 * q^69 - 114 * q^75 - 2 * q^77 - 26 * q^79 + 22 * q^81 - 8 * q^87 + 16 * q^91 + 68 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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