LMFDB - Embedded newform 2912.2.k.i.1793.9

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			1793.9
Root			$-1.80938i$ of defining polynomial
Character	$\chi$	$=$	2912.1793
Dual form			2912.2.k.i.1793.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80938 q^{3} -4.01229i q^{5} -1.00000i q^{7} +0.273853 q^{9} -1.12517i q^{11} +(-3.54502 + 0.657893i) q^{13} -7.25976i q^{15} +3.60098 q^{17} +0.211281i q^{19} -1.80938i q^{21} -6.65783 q^{23} -11.0985 q^{25} -4.93263 q^{27} -7.73269 q^{29} -4.41968i q^{31} -2.03585i q^{33} -4.01229 q^{35} +2.03868i q^{37} +(-6.41429 + 1.19038i) q^{39} +6.13460i q^{41} +11.5504 q^{43} -1.09878i q^{45} -11.9707i q^{47} -1.00000 q^{49} +6.51554 q^{51} +8.20624 q^{53} -4.51450 q^{55} +0.382287i q^{57} -3.68159i q^{59} +5.45038 q^{61} -0.273853i q^{63} +(2.63966 + 14.2237i) q^{65} +15.0159i q^{67} -12.0465 q^{69} +7.42623i q^{71} +3.80390i q^{73} -20.0814 q^{75} -1.12517 q^{77} -4.36938 q^{79} -9.74656 q^{81} -14.2682i q^{83} -14.4482i q^{85} -13.9914 q^{87} -9.68239i q^{89} +(0.657893 + 3.54502i) q^{91} -7.99688i q^{93} +0.847720 q^{95} -8.22069i q^{97} -0.308130i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.80938			1.04465			0.522323	−	0.852748i	$-0.325066\pi$
$3$	1.80938			1.04465			0.522323	+	0.852748i	$0.325066\pi$
$4$		0			0
$5$		−	4.01229i		−	1.79435i	−0.441673	−	0.897176i	$-0.645615\pi$
$5$		−	4.01229i		−	1.79435i	0.441673	−	0.897176i	$-0.354385\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	0.273853			0.0912844
$10$		0			0
$11$		−	1.12517i		−	0.339250i	−0.985509	−	0.169625i	$-0.945744\pi$
$11$		−	1.12517i		−	0.339250i	0.985509	−	0.169625i	$-0.0542557\pi$
$12$		0			0
$13$	−3.54502	+	0.657893i	−0.983212	+	0.182467i
$13$	−3.54502	+	0.657893i	−0.983212	+	0.182467i
$14$		0			0
$15$		−	7.25976i		−	1.87446i
$16$		0			0
$17$	3.60098			0.873366			0.436683	−	0.899615i	$-0.356153\pi$
$17$	3.60098			0.873366			0.436683	+	0.899615i	$0.356153\pi$
$18$		0			0
$19$			0.211281i			0.0484711i	0.999706	+	0.0242356i	$0.00771517\pi$
$19$			0.211281i			0.0484711i	−0.999706	+	0.0242356i	$0.992285\pi$
$20$		0			0
$21$		−	1.80938i		−	0.394839i
$22$		0			0
$23$	−6.65783			−1.38825			−0.694127	−	0.719853i	$-0.744209\pi$
$23$	−6.65783			−1.38825			−0.694127	+	0.719853i	$0.744209\pi$
$24$		0			0
$25$	−11.0985			−2.21970
$26$		0			0
$27$	−4.93263			−0.949286
$28$		0			0
$29$	−7.73269			−1.43593			−0.717963	−	0.696082i	$-0.754925\pi$
$29$	−7.73269			−1.43593			−0.717963	+	0.696082i	$0.754925\pi$
$30$		0			0
$31$		−	4.41968i		−	0.793798i	−0.917862	−	0.396899i	$-0.870086\pi$
$31$		−	4.41968i		−	0.793798i	0.917862	−	0.396899i	$-0.129914\pi$
$32$		0			0
$33$		−	2.03585i		−	0.354396i
$34$		0			0
$35$	−4.01229			−0.678201
$36$		0			0
$37$			2.03868i			0.335157i	0.985859	+	0.167578i	$0.0535947\pi$
$37$			2.03868i			0.335157i	−0.985859	+	0.167578i	$0.946405\pi$
$38$		0			0
$39$	−6.41429	+	1.19038i	−1.02711	+	0.190613i
$40$		0			0
$41$			6.13460i			0.958063i	0.877798	+	0.479031i	$0.159012\pi$
$41$			6.13460i			0.958063i	−0.877798	+	0.479031i	$0.840988\pi$
$42$		0			0
$43$	11.5504			1.76143			0.880713	−	0.473651i	$-0.157064\pi$
$43$	11.5504			1.76143			0.880713	+	0.473651i	$0.157064\pi$
$44$		0			0
$45$		−	1.09878i		−	0.163796i
$46$		0			0
$47$		−	11.9707i		−	1.74611i	−0.487623	−	0.873054i	$-0.662136\pi$
$47$		−	11.9707i		−	1.74611i	0.487623	−	0.873054i	$-0.337864\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	6.51554			0.912358
$52$		0			0
$53$	8.20624			1.12721			0.563607	−	0.826043i	$-0.309413\pi$
$53$	8.20624			1.12721			0.563607	+	0.826043i	$0.309413\pi$
$54$		0			0
$55$	−4.51450			−0.608734
$56$		0			0
$57$			0.382287i			0.0506351i
$58$		0			0
$59$		−	3.68159i		−	0.479303i	−0.970859	−	0.239651i	$-0.922967\pi$
$59$		−	3.68159i		−	0.479303i	0.970859	−	0.239651i	$-0.0770331\pi$
$60$		0			0
$61$	5.45038			0.697850			0.348925	−	0.937151i	$-0.386547\pi$
$61$	5.45038			0.697850			0.348925	+	0.937151i	$0.386547\pi$
$62$		0			0
$63$		−	0.273853i		−	0.0345023i
$64$		0			0
$65$	2.63966	+	14.2237i	0.327410	+	1.76423i
$66$		0			0
$67$			15.0159i			1.83449i	0.398324	+	0.917245i	$0.369592\pi$
$67$			15.0159i			1.83449i	−0.398324	+	0.917245i	$0.630408\pi$
$68$		0			0
$69$	−12.0465			−1.45023
$70$		0			0
$71$			7.42623i			0.881331i	0.897671	+	0.440665i	$0.145257\pi$
$71$			7.42623i			0.881331i	−0.897671	+	0.440665i	$0.854743\pi$
$72$		0			0
$73$			3.80390i			0.445212i	0.974908	+	0.222606i	$0.0714564\pi$
$73$			3.80390i			0.445212i	−0.974908	+	0.222606i	$0.928544\pi$
$74$		0			0
$75$	−20.0814			−2.31880
$76$		0			0
$77$	−1.12517			−0.128225
$78$		0			0
$79$	−4.36938			−0.491594			−0.245797	−	0.969321i	$-0.579050\pi$
$79$	−4.36938			−0.491594			−0.245797	+	0.969321i	$0.579050\pi$
$80$		0			0
$81$	−9.74656			−1.08295
$82$		0			0
$83$		−	14.2682i		−	1.56614i	−0.621933	−	0.783070i	$-0.713652\pi$
$83$		−	14.2682i		−	1.56614i	0.621933	−	0.783070i	$-0.286348\pi$
$84$		0			0
$85$		−	14.4482i		−	1.56713i
$86$		0			0
$87$	−13.9914			−1.50003
$88$		0			0
$89$		−	9.68239i		−	1.02633i	−0.858289	−	0.513166i	$-0.828473\pi$
$89$		−	9.68239i		−	1.02633i	0.858289	−	0.513166i	$-0.171527\pi$
$90$		0			0
$91$	0.657893	+	3.54502i	0.0689659	+	0.371619i
$92$		0			0
$93$		−	7.99688i		−	0.829237i
$94$		0			0
$95$	0.847720			0.0869742
$96$		0			0
$97$		−	8.22069i		−	0.834685i	−0.908749	−	0.417342i	$-0.862962\pi$
$97$		−	8.22069i		−	0.834685i	0.908749	−	0.417342i	$-0.137038\pi$
$98$		0			0
$99$		−	0.308130i		−	0.0309683i
$100$		0			0
$101$	12.1334			1.20732			0.603658	−	0.797243i	$-0.293709\pi$
$101$	12.1334			1.20732			0.603658	+	0.797243i	$0.293709\pi$
$102$		0			0
$103$	−13.1207			−1.29282			−0.646411	−	0.762989i	$-0.723731\pi$
$103$	−13.1207			−1.29282			−0.646411	+	0.762989i	$0.723731\pi$
$104$		0			0
$105$	−7.25976			−0.708480
$106$		0			0
$107$	−12.4981			−1.20824			−0.604120	−	0.796893i	$-0.706475\pi$
$107$	−12.4981			−1.20824			−0.604120	+	0.796893i	$0.706475\pi$
$108$		0			0
$109$			5.47007i			0.523938i	0.965076	+	0.261969i	$0.0843718\pi$
$109$			5.47007i			0.523938i	−0.965076	+	0.261969i	$0.915628\pi$
$110$		0			0
$111$			3.68874i			0.350120i
$112$		0			0
$113$	−16.9170			−1.59141			−0.795707	−	0.605682i	$-0.792900\pi$
$113$	−16.9170			−1.59141			−0.795707	+	0.605682i	$0.792900\pi$
$114$		0			0
$115$			26.7132i			2.49102i
$116$		0			0
$117$	−0.970816	+	0.180166i	−0.0897519	+	0.0166564i
$118$		0			0
$119$		−	3.60098i		−	0.330101i
$120$		0			0
$121$	9.73400			0.884909
$122$		0			0
$123$			11.0998i			1.00084i
$124$		0			0
$125$			24.4690i			2.18857i
$126$		0			0
$127$	−3.42935			−0.304306			−0.152153	−	0.988357i	$-0.548621\pi$
$127$	−3.42935			−0.304306			−0.152153	+	0.988357i	$0.548621\pi$
$128$		0			0
$129$	20.8991			1.84007
$130$		0			0
$131$	8.66700			0.757239			0.378620	−	0.925552i	$-0.376399\pi$
$131$	8.66700			0.757239			0.378620	+	0.925552i	$0.376399\pi$
$132$		0			0
$133$	0.211281			0.0183204
$134$		0			0
$135$			19.7912i			1.70335i
$136$		0			0
$137$		−	13.1920i		−	1.12707i	−0.826091	−	0.563536i	$-0.809441\pi$
$137$		−	13.1920i		−	1.12707i	0.826091	−	0.563536i	$-0.190559\pi$
$138$		0			0
$139$	−4.86578			−0.412710			−0.206355	−	0.978477i	$-0.566160\pi$
$139$	−4.86578			−0.412710			−0.206355	+	0.978477i	$0.566160\pi$
$140$		0			0
$141$		−	21.6596i		−	1.82406i
$142$		0			0
$143$	0.740239	+	3.98874i	0.0619019	+	0.333555i
$144$		0			0
$145$			31.0258i			2.57656i
$146$		0			0
$147$	−1.80938			−0.149235
$148$		0			0
$149$		−	10.6762i		−	0.874629i	−0.899309	−	0.437314i	$-0.855930\pi$
$149$		−	10.6762i		−	0.874629i	0.899309	−	0.437314i	$-0.144070\pi$
$150$		0			0
$151$		−	19.7429i		−	1.60666i	−0.595535	−	0.803329i	$-0.703060\pi$
$151$		−	19.7429i		−	1.60666i	0.595535	−	0.803329i	$-0.296940\pi$
$152$		0			0
$153$	0.986140			0.0797247
$154$		0			0
$155$	−17.7330			−1.42435
$156$		0			0
$157$	11.3604			0.906660			0.453330	−	0.891343i	$-0.350236\pi$
$157$	11.3604			0.906660			0.453330	+	0.891343i	$0.350236\pi$
$158$		0			0
$159$	14.8482			1.17754
$160$		0			0
$161$			6.65783i			0.524710i
$162$		0			0
$163$		−	7.29297i		−	0.571230i	−0.958344	−	0.285615i	$-0.907802\pi$
$163$		−	7.29297i		−	0.571230i	0.958344	−	0.285615i	$-0.0921978\pi$
$164$		0			0
$165$	−8.16843			−0.635912
$166$		0			0
$167$		−	11.0452i		−	0.854705i	−0.904085	−	0.427353i	$-0.859446\pi$
$167$		−	11.0452i		−	0.854705i	0.904085	−	0.427353i	$-0.140554\pi$
$168$		0			0
$169$	12.1344	−	4.66449i	0.933412	−	0.358807i
$170$		0			0
$171$			0.0578599i			0.00442466i
$172$		0			0
$173$	0.727064			0.0552776			0.0276388	−	0.999618i	$-0.491201\pi$
$173$	0.727064			0.0552776			0.0276388	+	0.999618i	$0.491201\pi$
$174$		0			0
$175$			11.0985i			0.838968i
$176$		0			0
$177$		−	6.66140i		−	0.500701i
$178$		0			0
$179$	−1.52061			−0.113656			−0.0568280	−	0.998384i	$-0.518099\pi$
$179$	−1.52061			−0.113656			−0.0568280	+	0.998384i	$0.518099\pi$
$180$		0			0
$181$	2.65339			0.197225			0.0986124	−	0.995126i	$-0.468560\pi$
$181$	2.65339			0.197225			0.0986124	+	0.995126i	$0.468560\pi$
$182$		0			0
$183$	9.86181			0.729006
$184$		0			0
$185$	8.17978			0.601390
$186$		0			0
$187$		−	4.05170i		−	0.296290i
$188$		0			0
$189$			4.93263i			0.358796i
$190$		0			0
$191$	0.252114			0.0182424			0.00912118	−	0.999958i	$-0.497097\pi$
$191$	0.252114			0.0182424			0.00912118	+	0.999958i	$0.497097\pi$
$192$		0			0
$193$			11.7157i			0.843316i	0.906755	+	0.421658i	$0.138552\pi$
$193$			11.7157i			0.843316i	−0.906755	+	0.421658i	$0.861448\pi$
$194$		0			0
$195$	4.77615	+	25.7360i	0.342027	+	1.84299i
$196$		0			0
$197$		−	9.53382i		−	0.679257i	−0.940560	−	0.339628i	$-0.889699\pi$
$197$		−	9.53382i		−	0.679257i	0.940560	−	0.339628i	$-0.110301\pi$
$198$		0			0
$199$	−11.4508			−0.811727			−0.405863	−	0.913934i	$-0.633029\pi$
$199$	−11.4508			−0.811727			−0.405863	+	0.913934i	$0.633029\pi$
$200$		0			0
$201$			27.1695i			1.91639i
$202$		0			0
$203$			7.73269i			0.542729i
$204$		0			0
$205$	24.6138			1.71910
$206$		0			0
$207$	−1.82327			−0.126726
$208$		0			0
$209$	0.237726			0.0164438
$210$		0			0
$211$	−28.6273			−1.97079			−0.985393	−	0.170298i	$-0.945527\pi$
$211$	−28.6273			−1.97079			−0.985393	+	0.170298i	$0.945527\pi$
$212$		0			0
$213$			13.4369i			0.920678i
$214$		0			0
$215$		−	46.3438i		−	3.16062i
$216$		0			0
$217$	−4.41968			−0.300027
$218$		0			0
$219$			6.88269i			0.465089i
$220$		0			0
$221$	−12.7656	+	2.36906i	−0.858704	+	0.159360i
$222$		0			0
$223$		−	2.14547i		−	0.143671i	−0.997416	−	0.0718357i	$-0.977114\pi$
$223$		−	2.14547i		−	0.143671i	0.997416	−	0.0718357i	$-0.0228857\pi$
$224$		0			0
$225$	−3.03936			−0.202624
$226$		0			0
$227$			3.76294i			0.249755i	0.992172	+	0.124877i	$0.0398538\pi$
$227$			3.76294i			0.249755i	−0.992172	+	0.124877i	$0.960146\pi$
$228$		0			0
$229$			9.03262i			0.596892i	0.954426	+	0.298446i	$0.0964683\pi$
$229$			9.03262i			0.596892i	−0.954426	+	0.298446i	$0.903532\pi$
$230$		0			0
$231$	−2.03585			−0.133949
$232$		0			0
$233$	11.5866			0.759066			0.379533	−	0.925178i	$-0.376085\pi$
$233$	11.5866			0.759066			0.379533	+	0.925178i	$0.376085\pi$
$234$		0			0
$235$	−48.0300			−3.13313
$236$		0			0
$237$	−7.90586			−0.513541
$238$		0			0
$239$		−	22.2221i		−	1.43743i	−0.695307	−	0.718713i	$-0.744731\pi$
$239$		−	22.2221i		−	1.43743i	0.695307	−	0.718713i	$-0.255269\pi$
$240$		0			0
$241$		−	14.8107i		−	0.954040i	−0.878892	−	0.477020i	$-0.841717\pi$
$241$		−	14.8107i		−	0.954040i	0.878892	−	0.477020i	$-0.158283\pi$
$242$		0			0
$243$	−2.83733			−0.182015
$244$		0			0
$245$			4.01229i			0.256336i
$246$		0			0
$247$	−0.139000	−	0.748994i	−0.00884436	−	0.0476574i
$248$		0			0
$249$		−	25.8166i		−	1.63606i
$250$		0			0
$251$	−24.8658			−1.56952			−0.784759	−	0.619802i	$-0.787213\pi$
$251$	−24.8658			−1.56952			−0.784759	+	0.619802i	$0.787213\pi$
$252$		0			0
$253$			7.49116i			0.470965i
$254$		0			0
$255$		−	26.1423i		−	1.63709i
$256$		0			0
$257$	23.1265			1.44259			0.721296	−	0.692627i	$-0.243547\pi$
$257$	23.1265			1.44259			0.721296	+	0.692627i	$0.243547\pi$
$258$		0			0
$259$	2.03868			0.126677
$260$		0			0
$261$	−2.11762			−0.131078
$262$		0			0
$263$	27.7840			1.71324			0.856618	−	0.515952i	$-0.172562\pi$
$263$	27.7840			1.71324			0.856618	+	0.515952i	$0.172562\pi$
$264$		0			0
$265$		−	32.9259i		−	2.02262i
$266$		0			0
$267$		−	17.5191i		−	1.07215i
$268$		0			0
$269$	11.6180			0.708365			0.354182	−	0.935176i	$-0.384759\pi$
$269$	11.6180			0.708365			0.354182	+	0.935176i	$0.384759\pi$
$270$		0			0
$271$		−	4.49910i		−	0.273301i	−0.990619	−	0.136651i	$-0.956366\pi$
$271$		−	4.49910i		−	0.273301i	0.990619	−	0.136651i	$-0.0436338\pi$
$272$		0			0
$273$	1.19038	+	6.41429i	0.0720450	+	0.388210i
$274$		0			0
$275$			12.4877i			0.753034i
$276$		0			0
$277$	2.37569			0.142741			0.0713706	−	0.997450i	$-0.477263\pi$
$277$	2.37569			0.142741			0.0713706	+	0.997450i	$0.477263\pi$
$278$		0			0
$279$		−	1.21034i		−	0.0724614i
$280$		0			0
$281$		−	7.26749i		−	0.433542i	−0.976222	−	0.216771i	$-0.930447\pi$
$281$		−	7.26749i		−	0.433542i	0.976222	−	0.216771i	$-0.0695526\pi$
$282$		0			0
$283$	20.2964			1.20649			0.603247	−	0.797555i	$-0.293873\pi$
$283$	20.2964			1.20649			0.603247	+	0.797555i	$0.293873\pi$
$284$		0			0
$285$	1.53385			0.0908573
$286$		0			0
$287$	6.13460			0.362114
$288$		0			0
$289$	−4.03294			−0.237231
$290$		0			0
$291$		−	14.8743i		−	0.871950i
$292$		0			0
$293$			0.801148i			0.0468035i	0.999726	+	0.0234018i	$0.00744969\pi$
$293$			0.801148i			0.0468035i	−0.999726	+	0.0234018i	$0.992550\pi$
$294$		0			0
$295$	−14.7716			−0.860038
$296$		0			0
$297$			5.55003i			0.322045i
$298$		0			0
$299$	23.6021	−	4.38014i	1.36495	−	0.253310i
$300$		0			0
$301$		−	11.5504i		−	0.665756i
$302$		0			0
$303$	21.9539			1.26122
$304$		0			0
$305$		−	21.8685i		−	1.25219i
$306$		0			0
$307$		−	18.0895i		−	1.03242i	−0.856461	−	0.516212i	$-0.827342\pi$
$307$		−	18.0895i		−	1.03242i	0.856461	−	0.516212i	$-0.172658\pi$
$308$		0			0
$309$	−23.7403			−1.35054
$310$		0			0
$311$	−1.08563			−0.0615602			−0.0307801	−	0.999526i	$-0.509799\pi$
$311$	−1.08563			−0.0615602			−0.0307801	+	0.999526i	$0.509799\pi$
$312$		0			0
$313$	−2.66346			−0.150547			−0.0752737	−	0.997163i	$-0.523983\pi$
$313$	−2.66346			−0.150547			−0.0752737	+	0.997163i	$0.523983\pi$
$314$		0			0
$315$	−1.09878			−0.0619092
$316$		0			0
$317$			11.5464i			0.648512i	0.945969	+	0.324256i	$0.105114\pi$
$317$			11.5464i			0.648512i	−0.945969	+	0.324256i	$0.894886\pi$
$318$		0			0
$319$			8.70056i			0.487138i
$320$		0			0
$321$	−22.6139			−1.26218
$322$		0			0
$323$			0.760818i			0.0423330i
$324$		0			0
$325$	39.3444	−	7.30163i	2.18244	−	0.405021i
$326$		0			0
$327$			9.89743i			0.547329i
$328$		0			0
$329$	−11.9707			−0.659967
$330$		0			0
$331$			20.8490i			1.14596i	0.819568	+	0.572982i	$0.194214\pi$
$331$			20.8490i			1.14596i	−0.819568	+	0.572982i	$0.805786\pi$
$332$		0			0
$333$			0.558299i			0.0305946i
$334$		0			0
$335$	60.2484			3.29172
$336$		0			0
$337$	−20.7215			−1.12877			−0.564387	−	0.825511i	$-0.690887\pi$
$337$	−20.7215			−1.12877			−0.564387	+	0.825511i	$0.690887\pi$
$338$		0			0
$339$	−30.6092			−1.66246
$340$		0			0
$341$	−4.97287			−0.269296
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			48.3342i			2.60223i
$346$		0			0
$347$	13.2038			0.708819			0.354410	−	0.935090i	$-0.384682\pi$
$347$	13.2038			0.708819			0.354410	+	0.935090i	$0.384682\pi$
$348$		0			0
$349$			15.7494i			0.843046i	0.906818	+	0.421523i	$0.138504\pi$
$349$			15.7494i			0.843046i	−0.906818	+	0.421523i	$0.861496\pi$
$350$		0			0
$351$	17.4863	−	3.24515i	0.933349	−	0.173213i
$352$		0			0
$353$		−	31.2703i		−	1.66435i	−0.554515	−	0.832174i	$-0.687096\pi$
$353$		−	31.2703i		−	1.66435i	0.554515	−	0.832174i	$-0.312904\pi$
$354$		0			0
$355$	29.7962			1.58142
$356$		0			0
$357$		−	6.51554i		−	0.344839i
$358$		0			0
$359$		−	26.5386i		−	1.40065i	−0.713822	−	0.700327i	$-0.753038\pi$
$359$		−	26.5386i		−	1.40065i	0.713822	−	0.700327i	$-0.246962\pi$
$360$		0			0
$361$	18.9554			0.997651
$362$		0			0
$363$	17.6125			0.924417
$364$		0			0
$365$	15.2623			0.798868
$366$		0			0
$367$	−7.27668			−0.379840			−0.189920	−	0.981800i	$-0.560823\pi$
$367$	−7.27668			−0.379840			−0.189920	+	0.981800i	$0.560823\pi$
$368$		0			0
$369$			1.67998i			0.0874562i
$370$		0			0
$371$		−	8.20624i		−	0.426047i
$372$		0			0
$373$	−20.4623			−1.05950			−0.529748	−	0.848155i	$-0.677714\pi$
$373$	−20.4623			−1.05950			−0.529748	+	0.848155i	$0.677714\pi$
$374$		0			0
$375$			44.2737i			2.28628i
$376$		0			0
$377$	27.4126	−	5.08729i	1.41182	−	0.262009i
$378$		0			0
$379$		−	38.1485i		−	1.95956i	−0.200080	−	0.979780i	$-0.564120\pi$
$379$		−	38.1485i		−	1.95956i	0.200080	−	0.979780i	$-0.435880\pi$
$380$		0			0
$381$	−6.20500			−0.317892
$382$		0			0
$383$		−	3.31983i		−	0.169635i	−0.996396	−	0.0848176i	$-0.972969\pi$
$383$		−	3.31983i		−	0.169635i	0.996396	−	0.0848176i	$-0.0270308\pi$
$384$		0			0
$385$			4.51450i			0.230080i
$386$		0			0
$387$	3.16313			0.160791
$388$		0			0
$389$	9.74598			0.494141			0.247070	−	0.968998i	$-0.420532\pi$
$389$	9.74598			0.494141			0.247070	+	0.968998i	$0.420532\pi$
$390$		0			0
$391$	−23.9747			−1.21245
$392$		0			0
$393$	15.6819			0.791046
$394$		0			0
$395$			17.5312i			0.882092i
$396$		0			0
$397$			22.8867i			1.14865i	0.818628	+	0.574324i	$0.194735\pi$
$397$			22.8867i			1.14865i	−0.818628	+	0.574324i	$0.805265\pi$
$398$		0			0
$399$	0.382287			0.0191383
$400$		0			0
$401$			28.1273i			1.40461i	0.711876	+	0.702306i	$0.247846\pi$
$401$			28.1273i			1.40461i	−0.711876	+	0.702306i	$0.752154\pi$
$402$		0			0
$403$	2.90768	+	15.6679i	0.144842	+	0.780471i
$404$		0			0
$405$			39.1061i			1.94320i
$406$		0			0
$407$	2.29385			0.113702
$408$		0			0
$409$		−	21.3432i		−	1.05535i	−0.849446	−	0.527676i	$-0.823063\pi$
$409$		−	21.3432i		−	1.05535i	0.849446	−	0.527676i	$-0.176937\pi$
$410$		0			0
$411$		−	23.8694i		−	1.17739i
$412$		0			0
$413$	−3.68159			−0.181159
$414$		0			0
$415$	−57.2483			−2.81021
$416$		0			0
$417$	−8.80404			−0.431136
$418$		0			0
$419$	21.2141			1.03638			0.518189	−	0.855266i	$-0.326606\pi$
$419$	21.2141			1.03638			0.518189	+	0.855266i	$0.326606\pi$
$420$		0			0
$421$		−	26.3337i		−	1.28342i	−0.766946	−	0.641712i	$-0.778224\pi$
$421$		−	26.3337i		−	1.28342i	0.766946	−	0.641712i	$-0.221776\pi$
$422$		0			0
$423$		−	3.27822i		−	0.159392i
$424$		0			0
$425$	−39.9655			−1.93861
$426$		0			0
$427$		−	5.45038i		−	0.263762i
$428$		0			0
$429$	1.33937	+	7.21714i	0.0646655	+	0.348447i
$430$		0			0
$431$		−	12.7641i		−	0.614826i	−0.951576	−	0.307413i	$-0.900537\pi$
$431$		−	12.7641i		−	0.614826i	0.951576	−	0.307413i	$-0.0994634\pi$
$432$		0			0
$433$	−17.1130			−0.822400			−0.411200	−	0.911545i	$-0.634890\pi$
$433$	−17.1130			−0.822400			−0.411200	+	0.911545i	$0.634890\pi$
$434$		0			0
$435$			56.1375i			2.69159i
$436$		0			0
$437$		−	1.40667i		−	0.0672902i
$438$		0			0
$439$	7.63549			0.364422			0.182211	−	0.983259i	$-0.441675\pi$
$439$	7.63549			0.364422			0.182211	+	0.983259i	$0.441675\pi$
$440$		0			0
$441$	−0.273853			−0.0130406
$442$		0			0
$443$	6.84923			0.325417			0.162708	−	0.986674i	$-0.447977\pi$
$443$	6.84923			0.325417			0.162708	+	0.986674i	$0.447977\pi$
$444$		0			0
$445$	−38.8486			−1.84160
$446$		0			0
$447$		−	19.3173i		−	0.913677i
$448$		0			0
$449$			32.9532i			1.55516i	0.628784	+	0.777580i	$0.283553\pi$
$449$			32.9532i			1.55516i	−0.628784	+	0.777580i	$0.716447\pi$
$450$		0			0
$451$	6.90244			0.325023
$452$		0			0
$453$		−	35.7225i		−	1.67839i
$454$		0			0
$455$	14.2237	−	2.63966i	0.666816	−	0.123749i
$456$		0			0
$457$		−	2.40115i		−	0.112321i	−0.998422	−	0.0561604i	$-0.982114\pi$
$457$		−	2.40115i		−	0.112321i	0.998422	−	0.0561604i	$-0.0178858\pi$
$458$		0			0
$459$	−17.7623			−0.829074
$460$		0			0
$461$			27.5006i			1.28083i	0.768030	+	0.640414i	$0.221237\pi$
$461$			27.5006i			1.28083i	−0.768030	+	0.640414i	$0.778763\pi$
$462$		0			0
$463$		−	21.4114i		−	0.995071i	−0.867444	−	0.497535i	$-0.834238\pi$
$463$		−	21.4114i		−	0.995071i	0.867444	−	0.497535i	$-0.165762\pi$
$464$		0			0
$465$	−32.0858			−1.48794
$466$		0			0
$467$	0.485886			0.0224841			0.0112421	−	0.999937i	$-0.496421\pi$
$467$	0.485886			0.0224841			0.0112421	+	0.999937i	$0.496421\pi$
$468$		0			0
$469$	15.0159			0.693372
$470$		0			0
$471$	20.5553			0.947138
$472$		0			0
$473$		−	12.9962i		−	0.597564i
$474$		0			0
$475$		−	2.34490i		−	0.107591i
$476$		0			0
$477$	2.24731			0.102897
$478$		0			0
$479$			19.4648i			0.889368i	0.895687	+	0.444684i	$0.146684\pi$
$479$			19.4648i			0.889368i	−0.895687	+	0.444684i	$0.853316\pi$
$480$		0			0
$481$	−1.34123	−	7.22716i	−0.0611550	−	0.329530i
$482$		0			0
$483$			12.0465i			0.548136i
$484$		0			0
$485$	−32.9838			−1.49772
$486$		0			0
$487$			23.3887i			1.05984i	0.848047	+	0.529922i	$0.177779\pi$
$487$			23.3887i			1.05984i	−0.848047	+	0.529922i	$0.822221\pi$
$488$		0			0
$489$		−	13.1958i		−	0.596733i
$490$		0			0
$491$	−13.3745			−0.603583			−0.301791	−	0.953374i	$-0.597585\pi$
$491$	−13.3745			−0.603583			−0.301791	+	0.953374i	$0.597585\pi$
$492$		0			0
$493$	−27.8453			−1.25409
$494$		0			0
$495$	−1.23631			−0.0555680
$496$		0			0
$497$	7.42623			0.333112
$498$		0			0
$499$			26.7979i			1.19964i	0.800136	+	0.599818i	$0.204760\pi$
$499$			26.7979i			1.19964i	−0.800136	+	0.599818i	$0.795240\pi$
$500$		0			0
$501$		−	19.9850i		−	0.892864i
$502$		0			0
$503$	−29.2886			−1.30591			−0.652957	−	0.757395i	$-0.726472\pi$
$503$	−29.2886			−1.30591			−0.652957	+	0.757395i	$0.726472\pi$
$504$		0			0
$505$		−	48.6827i		−	2.16635i
$506$		0			0
$507$	21.9556	−	8.43983i	0.975084	−	0.374826i
$508$		0			0
$509$		−	5.88848i		−	0.261002i	−0.991448	−	0.130501i	$-0.958341\pi$
$509$		−	5.88848i		−	0.261002i	0.991448	−	0.130501i	$-0.0416586\pi$
$510$		0			0
$511$	3.80390			0.168274
$512$		0			0
$513$		−	1.04217i		−	0.0460129i
$514$		0			0
$515$			52.6442i			2.31978i
$516$		0			0
$517$	−13.4690			−0.592368
$518$		0			0
$519$	1.31553			0.0577455
$520$		0			0
$521$	11.7267			0.513755			0.256878	−	0.966444i	$-0.417306\pi$
$521$	11.7267			0.513755			0.256878	+	0.966444i	$0.417306\pi$
$522$		0			0
$523$	−3.12214			−0.136521			−0.0682607	−	0.997668i	$-0.521745\pi$
$523$	−3.12214			−0.136521			−0.0682607	+	0.997668i	$0.521745\pi$
$524$		0			0
$525$			20.0814i			0.876424i
$526$		0			0
$527$		−	15.9152i		−	0.693276i
$528$		0			0
$529$	21.3267			0.927247
$530$		0			0
$531$		−	1.00822i		−	0.0437529i
$532$		0			0
$533$	−4.03591	−	21.7473i	−0.174815	−	0.941979i
$534$		0			0
$535$			50.1462i			2.16801i
$536$		0			0
$537$	−2.75136			−0.118730
$538$		0			0
$539$			1.12517i			0.0484643i
$540$		0			0
$541$		−	11.1538i		−	0.479541i	−0.970830	−	0.239770i	$-0.922928\pi$
$541$		−	11.1538i		−	0.479541i	0.970830	−	0.239770i	$-0.0770721\pi$
$542$		0			0
$543$	4.80099			0.206030
$544$		0			0
$545$	21.9475			0.940129
$546$		0			0
$547$	10.5189			0.449757			0.224878	−	0.974387i	$-0.427802\pi$
$547$	10.5189			0.449757			0.224878	+	0.974387i	$0.427802\pi$
$548$		0			0
$549$	1.49260			0.0637028
$550$		0			0
$551$		−	1.63377i		−	0.0696009i
$552$		0			0
$553$			4.36938i			0.185805i
$554$		0			0
$555$	14.8003			0.628239
$556$		0			0
$557$			43.2660i			1.83324i	0.399760	+	0.916620i	$0.369094\pi$
$557$			43.2660i			1.83324i	−0.399760	+	0.916620i	$0.630906\pi$
$558$		0			0
$559$	−40.9466	+	7.59896i	−1.73185	+	0.321402i
$560$		0			0
$561$		−	7.33106i		−	0.309518i
$562$		0			0
$563$	27.0595			1.14042			0.570211	−	0.821498i	$-0.306861\pi$
$563$	27.0595			1.14042			0.570211	+	0.821498i	$0.306861\pi$
$564$		0			0
$565$			67.8758i			2.85556i
$566$		0			0
$567$			9.74656i			0.409317i
$568$		0			0
$569$	−10.0465			−0.421170			−0.210585	−	0.977576i	$-0.567537\pi$
$569$	−10.0465			−0.421170			−0.210585	+	0.977576i	$0.567537\pi$
$570$		0			0
$571$	44.6276			1.86761			0.933803	−	0.357788i	$-0.116469\pi$
$571$	44.6276			1.86761			0.933803	+	0.357788i	$0.116469\pi$
$572$		0			0
$573$	0.456171			0.0190568
$574$		0			0
$575$	73.8919			3.08151
$576$		0			0
$577$		−	16.7156i		−	0.695879i	−0.937517	−	0.347940i	$-0.886881\pi$
$577$		−	16.7156i		−	0.695879i	0.937517	−	0.347940i	$-0.113119\pi$
$578$		0			0
$579$			21.1982i			0.880966i
$580$		0			0
$581$	−14.2682			−0.591946
$582$		0			0
$583$		−	9.23338i		−	0.382408i
$584$		0			0
$585$	0.722880	+	3.89520i	0.0298874	+	0.161047i
$586$		0			0
$587$			15.2771i			0.630555i	0.948999	+	0.315278i	$0.102098\pi$
$587$			15.2771i			0.630555i	−0.948999	+	0.315278i	$0.897902\pi$
$588$		0			0
$589$	0.933793			0.0384763
$590$		0			0
$591$		−	17.2503i		−	0.709583i
$592$		0			0
$593$		−	6.70680i		−	0.275415i	−0.990473	−	0.137708i	$-0.956027\pi$
$593$		−	6.70680i		−	0.275415i	0.990473	−	0.137708i	$-0.0439735\pi$
$594$		0			0
$595$	−14.4482			−0.592318
$596$		0			0
$597$	−20.7189			−0.847967
$598$		0			0
$599$	5.30563			0.216782			0.108391	−	0.994108i	$-0.465430\pi$
$599$	5.30563			0.216782			0.108391	+	0.994108i	$0.465430\pi$
$600$		0			0
$601$	30.1041			1.22797			0.613986	−	0.789317i	$-0.289565\pi$
$601$	30.1041			1.22797			0.613986	+	0.789317i	$0.289565\pi$
$602$		0			0
$603$			4.11217i			0.167460i
$604$		0			0
$605$		−	39.0557i		−	1.58784i
$606$		0			0
$607$	−11.1086			−0.450883			−0.225442	−	0.974257i	$-0.572382\pi$
$607$	−11.1086			−0.450883			−0.225442	+	0.974257i	$0.572382\pi$
$608$		0			0
$609$			13.9914i			0.566959i
$610$		0			0
$611$	7.87545	+	42.4364i	0.318607	+	1.71679i
$612$		0			0
$613$		−	17.1277i		−	0.691783i	−0.938275	−	0.345891i	$-0.887577\pi$
$613$		−	17.1277i		−	0.691783i	0.938275	−	0.345891i	$-0.112423\pi$
$614$		0			0
$615$	44.5357			1.79585
$616$		0			0
$617$		−	33.9390i		−	1.36633i	−0.730264	−	0.683166i	$-0.760603\pi$
$617$		−	33.9390i		−	1.36633i	0.730264	−	0.683166i	$-0.239397\pi$
$618$		0			0
$619$		−	20.9785i		−	0.843196i	−0.906783	−	0.421598i	$-0.861469\pi$
$619$		−	20.9785i		−	0.843196i	0.906783	−	0.421598i	$-0.138531\pi$
$620$		0			0
$621$	32.8406			1.31785
$622$		0			0
$623$	−9.68239			−0.387917
$624$		0			0
$625$	42.6842			1.70737
$626$		0			0
$627$	0.430136			0.0171780
$628$		0			0
$629$			7.34125i			0.292715i
$630$		0			0
$631$			19.7336i			0.785583i	0.919628	+	0.392791i	$0.128491\pi$
$631$			19.7336i			0.785583i	−0.919628	+	0.392791i	$0.871509\pi$
$632$		0			0
$633$	−51.7977			−2.05877
$634$		0			0
$635$			13.7596i			0.546032i
$636$		0			0
$637$	3.54502	−	0.657893i	0.140459	−	0.0260667i
$638$		0			0
$639$			2.03370i			0.0804518i
$640$		0			0
$641$	35.1738			1.38928			0.694641	−	0.719356i	$-0.255563\pi$
$641$	35.1738			1.38928			0.694641	+	0.719356i	$0.255563\pi$
$642$		0			0
$643$			19.4656i			0.767647i	0.923407	+	0.383823i	$0.125393\pi$
$643$			19.4656i			0.767647i	−0.923407	+	0.383823i	$0.874607\pi$
$644$		0			0
$645$		−	83.8534i		−	3.30173i
$646$		0			0
$647$	−5.98561			−0.235319			−0.117659	−	0.993054i	$-0.537539\pi$
$647$	−5.98561			−0.235319			−0.117659	+	0.993054i	$0.537539\pi$
$648$		0			0
$649$	−4.14240			−0.162604
$650$		0			0
$651$	−7.99688			−0.313422
$652$		0			0
$653$	−48.5204			−1.89875			−0.949375	−	0.314146i	$-0.898282\pi$
$653$	−48.5204			−1.89875			−0.949375	+	0.314146i	$0.898282\pi$
$654$		0			0
$655$		−	34.7745i		−	1.35875i
$656$		0			0
$657$			1.04171i			0.0406409i
$658$		0			0
$659$	0.268593			0.0104629			0.00523145	−	0.999986i	$-0.498335\pi$
$659$	0.268593			0.0104629			0.00523145	+	0.999986i	$0.498335\pi$
$660$		0			0
$661$			29.2823i			1.13895i	0.822009	+	0.569475i	$0.192854\pi$
$661$			29.2823i			1.13895i	−0.822009	+	0.569475i	$0.807146\pi$
$662$		0			0
$663$	−23.0977	+	4.28653i	−0.897042	+	0.166475i
$664$		0			0
$665$		−	0.847720i		−	0.0328732i
$666$		0			0
$667$	51.4829			1.99343
$668$		0			0
$669$		−	3.88197i		−	0.150086i
$670$		0			0
$671$		−	6.13258i		−	0.236746i
$672$		0			0
$673$	45.2733			1.74516			0.872579	−	0.488473i	$-0.162446\pi$
$673$	45.2733			1.74516			0.872579	+	0.488473i	$0.162446\pi$
$674$		0			0
$675$	54.7448			2.10713
$676$		0			0
$677$	3.68744			0.141720			0.0708599	−	0.997486i	$-0.477426\pi$
$677$	3.68744			0.141720			0.0708599	+	0.997486i	$0.477426\pi$
$678$		0			0
$679$	−8.22069			−0.315481
$680$		0			0
$681$			6.80858i			0.260905i
$682$		0			0
$683$		−	17.4994i		−	0.669597i	−0.942290	−	0.334798i	$-0.891332\pi$
$683$		−	17.4994i		−	0.669597i	0.942290	−	0.334798i	$-0.108668\pi$
$684$		0			0
$685$	−52.9303			−2.02236
$686$		0			0
$687$			16.3434i			0.623541i
$688$		0			0
$689$	−29.0913	+	5.39883i	−1.10829	+	0.205679i
$690$		0			0
$691$		−	41.1786i		−	1.56651i	−0.621701	−	0.783254i	$-0.713558\pi$
$691$		−	41.1786i		−	1.56651i	0.621701	−	0.783254i	$-0.286442\pi$
$692$		0			0
$693$	−0.308130			−0.0117049
$694$		0			0
$695$			19.5229i			0.740547i
$696$		0			0
$697$			22.0906i			0.836740i
$698$		0			0
$699$	20.9646			0.792955
$700$		0			0
$701$	44.5878			1.68406			0.842028	−	0.539433i	$-0.181362\pi$
$701$	44.5878			1.68406			0.842028	+	0.539433i	$0.181362\pi$
$702$		0			0
$703$	−0.430734			−0.0162454
$704$		0			0
$705$	−86.9045			−3.27301
$706$		0			0
$707$		−	12.1334i		−	0.456323i
$708$		0			0
$709$			0.794798i			0.0298493i	0.999889	+	0.0149246i	$0.00475083\pi$
$709$			0.794798i			0.0298493i	−0.999889	+	0.0149246i	$0.995249\pi$
$710$		0			0
$711$	−1.19657			−0.0448748
$712$		0			0
$713$			29.4255i			1.10199i
$714$		0			0
$715$	16.0040	−	2.97006i	0.598515	−	0.111074i
$716$		0			0
$717$		−	40.2082i		−	1.50160i
$718$		0			0
$719$	−6.03984			−0.225248			−0.112624	−	0.993638i	$-0.535926\pi$
$719$	−6.03984			−0.225248			−0.112624	+	0.993638i	$0.535926\pi$
$720$		0			0
$721$			13.1207i			0.488641i
$722$		0			0
$723$		−	26.7981i		−	0.996634i
$724$		0			0
$725$	85.8213			3.18732
$726$		0			0
$727$	24.2424			0.899099			0.449550	−	0.893255i	$-0.351584\pi$
$727$	24.2424			0.899099			0.449550	+	0.893255i	$0.351584\pi$
$728$		0			0
$729$	24.1059			0.892811
$730$		0			0
$731$	41.5929			1.53837
$732$		0			0
$733$			39.6138i			1.46317i	0.681751	+	0.731585i	$0.261219\pi$
$733$			39.6138i			1.46317i	−0.681751	+	0.731585i	$0.738781\pi$
$734$		0			0
$735$			7.25976i			0.267780i
$736$		0			0
$737$	16.8954			0.622351
$738$		0			0
$739$		−	28.1283i		−	1.03472i	−0.855769	−	0.517358i	$-0.826915\pi$
$739$		−	28.1283i		−	1.03472i	0.855769	−	0.517358i	$-0.173085\pi$
$740$		0			0
$741$	−0.251504	−	1.35521i	−0.00923923	−	0.0497851i
$742$		0			0
$743$			15.0355i			0.551599i	0.961215	+	0.275799i	$0.0889426\pi$
$743$			15.0355i			0.551599i	−0.961215	+	0.275799i	$0.911057\pi$
$744$		0			0
$745$	−42.8361			−1.56939
$746$		0			0
$747$		−	3.90740i		−	0.142964i
$748$		0			0
$749$			12.4981i			0.456672i
$750$		0			0
$751$	−33.5964			−1.22595			−0.612975	−	0.790102i	$-0.710027\pi$
$751$	−33.5964			−1.22595			−0.612975	+	0.790102i	$0.710027\pi$
$752$		0			0
$753$	−44.9917			−1.63959
$754$		0			0
$755$	−79.2145			−2.88291
$756$		0			0
$757$	19.1463			0.695884			0.347942	−	0.937516i	$-0.386881\pi$
$757$	19.1463			0.695884			0.347942	+	0.937516i	$0.386881\pi$
$758$		0			0
$759$			13.5543i			0.491992i
$760$		0			0
$761$			44.3671i			1.60831i	0.594422	+	0.804153i	$0.297381\pi$
$761$			44.3671i			1.60831i	−0.594422	+	0.804153i	$0.702619\pi$
$762$		0			0
$763$	5.47007			0.198030
$764$		0			0
$765$		−	3.95669i		−	0.143054i
$766$		0			0
$767$	2.42210	+	13.0513i	0.0874568	+	0.471256i
$768$		0			0
$769$			26.3343i			0.949640i	0.880083	+	0.474820i	$0.157487\pi$
$769$			26.3343i			0.949640i	−0.880083	+	0.474820i	$0.842513\pi$
$770$		0			0
$771$	41.8446			1.50700
$772$		0			0
$773$			29.6069i			1.06489i	0.846466	+	0.532443i	$0.178726\pi$
$773$			29.6069i			1.06489i	−0.846466	+	0.532443i	$0.821274\pi$
$774$		0			0
$775$			49.0518i			1.76199i
$776$		0			0
$777$	3.68874			0.132333
$778$		0			0
$779$	−1.29612			−0.0464384
$780$		0			0
$781$	8.35573			0.298992
$782$		0			0
$783$	38.1425			1.36310
$784$		0			0
$785$		−	45.5813i		−	1.62687i
$786$		0			0
$787$		−	4.27900i		−	0.152530i	−0.997088	−	0.0762650i	$-0.975701\pi$
$787$		−	4.27900i		−	0.152530i	0.997088	−	0.0762650i	$-0.0242995\pi$
$788$		0			0
$789$	50.2718			1.78972
$790$		0			0
$791$			16.9170i			0.601498i
$792$		0			0
$793$	−19.3217	+	3.58577i	−0.686134	+	0.127334i
$794$		0			0
$795$		−	59.5754i		−	2.11292i
$796$		0			0
$797$	2.58662			0.0916229			0.0458114	−	0.998950i	$-0.485413\pi$
$797$	2.58662			0.0916229			0.0458114	+	0.998950i	$0.485413\pi$
$798$		0			0
$799$		−	43.1063i		−	1.52499i
$800$		0			0
$801$		−	2.65156i		−	0.0936881i
$802$		0			0
$803$	4.28001			0.151038
$804$		0			0
$805$	26.7132			0.941515
$806$		0			0
$807$	21.0214			0.739990
$808$		0			0
$809$	−34.5870			−1.21601			−0.608007	−	0.793931i	$-0.708031\pi$
$809$	−34.5870			−1.21601			−0.608007	+	0.793931i	$0.708031\pi$
$810$		0			0
$811$			6.36884i			0.223640i	0.993728	+	0.111820i	$0.0356680\pi$
$811$			6.36884i			0.223640i	−0.993728	+	0.111820i	$0.964332\pi$
$812$		0			0
$813$		−	8.14059i		−	0.285503i
$814$		0			0
$815$	−29.2615			−1.02499
$816$		0			0
$817$			2.44038i			0.0853782i
$818$		0			0
$819$	0.180166	+	0.970816i	0.00629552	+	0.0339230i
$820$		0			0
$821$		−	0.0417676i		−	0.00145770i	−1.00000		0.000728849i	$-0.999768\pi$
$821$		−	0.0417676i		−	0.00145770i	1.00000		0.000728849i	$-0.000232000\pi$
$822$		0			0
$823$	46.7977			1.63126			0.815632	−	0.578571i	$-0.196389\pi$
$823$	46.7977			1.63126			0.815632	+	0.578571i	$0.196389\pi$
$824$		0			0
$825$			22.5949i			0.786653i
$826$		0			0
$827$		−	12.2318i		−	0.425343i	−0.977124	−	0.212671i	$-0.931784\pi$
$827$		−	12.2318i		−	0.425343i	0.977124	−	0.212671i	$-0.0682164\pi$
$828$		0			0
$829$	5.64980			0.196226			0.0981128	−	0.995175i	$-0.468719\pi$
$829$	5.64980			0.196226			0.0981128	+	0.995175i	$0.468719\pi$
$830$		0			0
$831$	4.29852			0.149114
$832$		0			0
$833$	−3.60098			−0.124767
$834$		0			0
$835$	−44.3167			−1.53364
$836$		0			0
$837$			21.8007i			0.753541i
$838$		0			0
$839$		−	39.5226i		−	1.36447i	−0.731132	−	0.682236i	$-0.761008\pi$
$839$		−	39.5226i		−	1.36447i	0.731132	−	0.682236i	$-0.238992\pi$
$840$		0			0
$841$	30.7946			1.06188
$842$		0			0
$843$		−	13.1497i		−	0.452898i
$844$		0			0
$845$	−18.7153	−	48.6866i	−0.643826	−	1.67487i
$846$		0			0
$847$		−	9.73400i		−	0.334464i
$848$		0			0
$849$	36.7238			1.26036
$850$		0			0
$851$		−	13.5732i		−	0.465283i
$852$		0			0
$853$			45.3952i			1.55430i	0.629314	+	0.777151i	$0.283336\pi$
$853$			45.3952i			1.55430i	−0.629314	+	0.777151i	$0.716664\pi$
$854$		0			0
$855$	0.232151			0.00793939
$856$		0			0
$857$	28.4003			0.970137			0.485068	−	0.874476i	$-0.338795\pi$
$857$	28.4003			0.970137			0.485068	+	0.874476i	$0.338795\pi$
$858$		0			0
$859$	−27.7622			−0.947235			−0.473618	−	0.880731i	$-0.657052\pi$
$859$	−27.7622			−0.947235			−0.473618	+	0.880731i	$0.657052\pi$
$860$		0			0
$861$	11.0998			0.378281
$862$		0			0
$863$			11.3324i			0.385758i	0.981223	+	0.192879i	$0.0617824\pi$
$863$			11.3324i			0.385758i	−0.981223	+	0.192879i	$0.938218\pi$
$864$		0			0
$865$		−	2.91719i		−	0.0991875i
$866$		0			0
$867$	−7.29711			−0.247823
$868$		0			0
$869$			4.91628i			0.166773i
$870$		0			0
$871$	−9.87889	−	53.2318i	−0.334733	−	1.80369i
$872$		0			0
$873$		−	2.25126i		−	0.0761937i
$874$		0			0
$875$	24.4690			0.827203
$876$		0			0
$877$		−	44.4641i		−	1.50145i	−0.660618	−	0.750723i	$-0.729705\pi$
$877$		−	44.4641i		−	1.50145i	0.660618	−	0.750723i	$-0.270295\pi$
$878$		0			0
$879$			1.44958i			0.0488931i
$880$		0			0
$881$	44.7090			1.50629			0.753143	−	0.657857i	$-0.228537\pi$
$881$	44.7090			1.50629			0.753143	+	0.657857i	$0.228537\pi$
$882$		0			0
$883$	−9.34785			−0.314580			−0.157290	−	0.987552i	$-0.550276\pi$
$883$	−9.34785			−0.314580			−0.157290	+	0.987552i	$0.550276\pi$
$884$		0			0
$885$	−26.7275			−0.898435
$886$		0			0
$887$	35.8117			1.20244			0.601220	−	0.799084i	$-0.294682\pi$
$887$	35.8117			1.20244			0.601220	+	0.799084i	$0.294682\pi$
$888$		0			0
$889$			3.42935i			0.115017i
$890$		0			0
$891$			10.9665i			0.367392i
$892$		0			0
$893$	2.52918			0.0846358
$894$		0			0
$895$			6.10114i			0.203939i
$896$		0			0
$897$	42.7052	−	7.92533i	1.42589	−	0.264619i
$898$		0			0
$899$			34.1760i			1.13983i
$900$		0			0
$901$	29.5505			0.984471
$902$		0			0
$903$		−	20.8991i		−	0.695479i
$904$		0			0
$905$		−	10.6462i		−	0.353891i
$906$		0			0
$907$	35.4034			1.17555			0.587776	−	0.809024i	$-0.300004\pi$
$907$	35.4034			1.17555			0.587776	+	0.809024i	$0.300004\pi$
$908$		0			0
$909$	3.32277			0.110209
$910$		0			0
$911$	2.19563			0.0727446			0.0363723	−	0.999338i	$-0.488420\pi$
$911$	2.19563			0.0727446			0.0363723	+	0.999338i	$0.488420\pi$
$912$		0			0
$913$	−16.0541			−0.531314
$914$		0			0
$915$		−	39.5685i		−	1.30809i
$916$		0			0
$917$		−	8.66700i		−	0.286209i
$918$		0			0
$919$	−45.9837			−1.51686			−0.758430	−	0.651754i	$-0.774033\pi$
$919$	−45.9837			−1.51686			−0.758430	+	0.651754i	$0.774033\pi$
$920$		0			0
$921$		−	32.7308i		−	1.07852i
$922$		0			0
$923$	−4.88566	−	26.3261i	−0.160814	−	0.866535i
$924$		0			0
$925$		−	22.6263i		−	0.743948i
$926$		0			0
$927$	−3.59315			−0.118015
$928$		0			0
$929$			34.7040i			1.13860i	0.822130	+	0.569300i	$0.192786\pi$
$929$			34.7040i			1.13860i	−0.822130	+	0.569300i	$0.807214\pi$
$930$		0			0
$931$		−	0.211281i		−	0.00692444i
$932$		0			0
$933$	−1.96431			−0.0643086
$934$		0			0
$935$	−16.2566			−0.531648
$936$		0			0
$937$	−1.79230			−0.0585518			−0.0292759	−	0.999571i	$-0.509320\pi$
$937$	−1.79230			−0.0585518			−0.0292759	+	0.999571i	$0.509320\pi$
$938$		0			0
$939$	−4.81920			−0.157269
$940$		0			0
$941$		−	23.4155i		−	0.763324i	−0.924302	−	0.381662i	$-0.875352\pi$
$941$		−	23.4155i		−	0.763324i	0.924302	−	0.381662i	$-0.124648\pi$
$942$		0			0
$943$		−	40.8431i		−	1.33003i
$944$		0			0
$945$	19.7912			0.643807
$946$		0			0
$947$		−	48.2052i		−	1.56646i	−0.621732	−	0.783230i	$-0.713571\pi$
$947$		−	48.2052i		−	1.56646i	0.621732	−	0.783230i	$-0.286429\pi$
$948$		0			0
$949$	−2.50256	−	13.4849i	−0.0812364	−	0.437738i
$950$		0			0
$951$			20.8919i			0.677466i
$952$		0			0
$953$	−55.1431			−1.78626			−0.893130	−	0.449800i	$-0.851495\pi$
$953$	−55.1431			−1.78626			−0.893130	+	0.449800i	$0.851495\pi$
$954$		0			0
$955$		−	1.01156i		−	0.0327332i
$956$		0			0
$957$			15.7426i			0.508887i
$958$		0			0
$959$	−13.1920			−0.425993
$960$		0			0
$961$	11.4664			0.369885
$962$		0			0
$963$	−3.42266			−0.110294
$964$		0			0
$965$	47.0069			1.51321
$966$		0			0
$967$			25.7119i			0.826839i	0.910541	+	0.413419i	$0.135666\pi$
$967$			25.7119i			0.826839i	−0.910541	+	0.413419i	$0.864334\pi$
$968$		0			0
$969$			1.37661i			0.0442230i
$970$		0			0
$971$	0.594076			0.0190648			0.00953240	−	0.999955i	$-0.496966\pi$
$971$	0.594076			0.0190648			0.00953240	+	0.999955i	$0.496966\pi$
$972$		0			0
$973$			4.86578i			0.155990i
$974$		0			0
$975$	71.1890	−	13.2114i	2.27987	−	0.423104i
$976$		0			0
$977$			49.2389i			1.57529i	0.616127	+	0.787647i	$0.288701\pi$
$977$			49.2389i			1.57529i	−0.616127	+	0.787647i	$0.711299\pi$
$978$		0			0
$979$	−10.8943			−0.348183
$980$		0			0
$981$			1.49800i			0.0478274i
$982$		0			0
$983$			11.3672i			0.362557i	0.983432	+	0.181279i	$0.0580236\pi$
$983$			11.3672i			0.362557i	−0.983432	+	0.181279i	$0.941976\pi$
$984$		0			0
$985$	−38.2525			−1.21883
$986$		0			0
$987$	−21.6596			−0.689432
$988$		0			0
$989$	−76.9008			−2.44530
$990$		0			0
$991$	−13.0008			−0.412985			−0.206493	−	0.978448i	$-0.566205\pi$
$991$	−13.0008			−0.412985			−0.206493	+	0.978448i	$0.566205\pi$
$992$		0			0
$993$			37.7237i			1.19713i
$994$		0			0
$995$			45.9440i			1.45652i
$996$		0			0
$997$	−39.3875			−1.24741			−0.623707	−	0.781658i	$-0.714374\pi$
$997$	−39.3875			−1.24741			−0.623707	+	0.781658i	$0.714374\pi$
$998$		0			0
$999$		−	10.0561i		−	0.318160i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.k.i.1793.9	✓	14
4.3	odd	2		2912.2.k.j.1793.5	yes	14
13.12	even	2	inner	2912.2.k.i.1793.10	yes	14
52.51	odd	2		2912.2.k.j.1793.6	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.k.i.1793.9	✓	14	1.1	even	1	trivial
2912.2.k.i.1793.10	yes	14	13.12	even	2	inner
2912.2.k.j.1793.5	yes	14	4.3	odd	2
2912.2.k.j.1793.6	yes	14	52.51	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)

\(f(q)\)	\(=\)	\(q+1.80938 q^{3} -4.01229i q^{5} -1.00000i q^{7} +0.273853 q^{9} -1.12517i q^{11} +(-3.54502 + 0.657893i) q^{13} -7.25976i q^{15} +3.60098 q^{17} +0.211281i q^{19} -1.80938i q^{21} -6.65783 q^{23} -11.0985 q^{25} -4.93263 q^{27} -7.73269 q^{29} -4.41968i q^{31} -2.03585i q^{33} -4.01229 q^{35} +2.03868i q^{37} +(-6.41429 + 1.19038i) q^{39} +6.13460i q^{41} +11.5504 q^{43} -1.09878i q^{45} -11.9707i q^{47} -1.00000 q^{49} +6.51554 q^{51} +8.20624 q^{53} -4.51450 q^{55} +0.382287i q^{57} -3.68159i q^{59} +5.45038 q^{61} -0.273853i q^{63} +(2.63966 + 14.2237i) q^{65} +15.0159i q^{67} -12.0465 q^{69} +7.42623i q^{71} +3.80390i q^{73} -20.0814 q^{75} -1.12517 q^{77} -4.36938 q^{79} -9.74656 q^{81} -14.2682i q^{83} -14.4482i q^{85} -13.9914 q^{87} -9.68239i q^{89} +(0.657893 + 3.54502i) q^{91} -7.99688i q^{93} +0.847720 q^{95} -8.22069i q^{97} -0.308130i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) \) 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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