LMFDB - Embedded newform 2912.2.h.b.2575.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(2575,2912)]

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([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("2912.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.28
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.b.2575.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.610846i q^{3} -0.780706 q^{5} +(2.18555 + 1.49110i) q^{7} +2.62687 q^{9} -3.13678 q^{11} +1.00000 q^{13} -0.476891i q^{15} +4.47267i q^{17} +1.92835i q^{19} +(-0.910831 + 1.33503i) q^{21} +5.54704i q^{23} -4.39050 q^{25} +3.43715i q^{27} -2.44239i q^{29} -5.76655 q^{31} -1.91609i q^{33} +(-1.70627 - 1.16411i) q^{35} +0.0497032i q^{37} +0.610846i q^{39} -9.71318i q^{41} -9.34658 q^{43} -2.05081 q^{45} +12.2688 q^{47} +(2.55326 + 6.51774i) q^{49} -2.73211 q^{51} -7.60088i q^{53} +2.44890 q^{55} -1.17792 q^{57} +13.2723i q^{59} +8.35048 q^{61} +(5.74115 + 3.91692i) q^{63} -0.780706 q^{65} -0.937154 q^{67} -3.38838 q^{69} +13.0139i q^{71} -0.406509i q^{73} -2.68192i q^{75} +(-6.85559 - 4.67725i) q^{77} -4.03568i q^{79} +5.78103 q^{81} +11.9237i q^{83} -3.49184i q^{85} +1.49193 q^{87} +15.7622i q^{89} +(2.18555 + 1.49110i) q^{91} -3.52247i q^{93} -1.50547i q^{95} -7.10470i q^{97} -8.23991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) $ 48 * q - 48 * q^9 + 4 * q^11 + 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 - 24 * q^45 + 40 * q^51 + 20 * q^63 + 4 * q^67 + 20 * q^77 + 64 * q^81 - 40 * q^87 - 20 * q^99

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
$3$			0.610846i			0.352672i	0.984330	+	0.176336i	$0.0564245\pi$
$3$			0.610846i			0.352672i	−0.984330	+	0.176336i	$0.943575\pi$
$4$		0			0
$5$	−0.780706			−0.349142			−0.174571	−	0.984645i	$-0.555854\pi$
$5$	−0.780706			−0.349142			−0.174571	+	0.984645i	$0.555854\pi$
$6$		0			0
$7$	2.18555	+	1.49110i	0.826060	+	0.563582i
$7$	2.18555	+	1.49110i	0.826060	+	0.563582i
$8$		0			0
$9$	2.62687			0.875622
$10$		0			0
$11$	−3.13678			−0.945775			−0.472888	−	0.881123i	$-0.656788\pi$
$11$	−3.13678			−0.945775			−0.472888	+	0.881123i	$0.656788\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$		−	0.476891i		−	0.123133i
$16$		0			0
$17$			4.47267i			1.08478i	0.840126	+	0.542391i	$0.182481\pi$
$17$			4.47267i			1.08478i	−0.840126	+	0.542391i	$0.817519\pi$
$18$		0			0
$19$			1.92835i			0.442393i	0.975229	+	0.221197i	$0.0709962\pi$
$19$			1.92835i			0.442393i	−0.975229	+	0.221197i	$0.929004\pi$
$20$		0			0
$21$	−0.910831	+	1.33503i	−0.198760	+	0.291328i
$22$		0			0
$23$			5.54704i			1.15664i	0.815811	+	0.578319i	$0.196291\pi$
$23$			5.54704i			1.15664i	−0.815811	+	0.578319i	$0.803709\pi$
$24$		0			0
$25$	−4.39050			−0.878100
$26$		0			0
$27$			3.43715i			0.661480i
$28$		0			0
$29$		−	2.44239i		−	0.453541i	−0.973948	−	0.226771i	$-0.927183\pi$
$29$		−	2.44239i		−	0.453541i	0.973948	−	0.226771i	$-0.0728167\pi$
$30$		0			0
$31$	−5.76655			−1.03570			−0.517851	−	0.855471i	$-0.673268\pi$
$31$	−5.76655			−1.03570			−0.517851	+	0.855471i	$0.673268\pi$
$32$		0			0
$33$		−	1.91609i		−	0.333548i
$34$		0			0
$35$	−1.70627	−	1.16411i	−0.288413	−	0.196770i
$36$		0			0
$37$			0.0497032i			0.00817115i	0.999992	+	0.00408557i	$0.00130048\pi$
$37$			0.0497032i			0.00817115i	−0.999992	+	0.00408557i	$0.998700\pi$
$38$		0			0
$39$			0.610846i			0.0978136i
$40$		0			0
$41$		−	9.71318i		−	1.51694i	−0.651706	−	0.758472i	$-0.725946\pi$
$41$		−	9.71318i		−	1.51694i	0.651706	−	0.758472i	$-0.274054\pi$
$42$		0			0
$43$	−9.34658			−1.42534			−0.712670	−	0.701499i	$-0.752514\pi$
$43$	−9.34658			−1.42534			−0.712670	+	0.701499i	$0.752514\pi$
$44$		0			0
$45$	−2.05081			−0.305717
$46$		0			0
$47$	12.2688			1.78959			0.894794	−	0.446479i	$-0.147322\pi$
$47$	12.2688			1.78959			0.894794	+	0.446479i	$0.147322\pi$
$48$		0			0
$49$	2.55326	+	6.51774i	0.364751	+	0.931105i
$50$		0			0
$51$	−2.73211			−0.382572
$52$		0			0
$53$		−	7.60088i		−	1.04406i	−0.852927	−	0.522031i	$-0.825175\pi$
$53$		−	7.60088i		−	1.04406i	0.852927	−	0.522031i	$-0.174825\pi$
$54$		0			0
$55$	2.44890			0.330210
$56$		0			0
$57$	−1.17792			−0.156020
$58$		0			0
$59$			13.2723i			1.72791i	0.503568	+	0.863955i	$0.332020\pi$
$59$			13.2723i			1.72791i	−0.503568	+	0.863955i	$0.667980\pi$
$60$		0			0
$61$	8.35048			1.06917			0.534585	−	0.845115i	$-0.320468\pi$
$61$	8.35048			1.06917			0.534585	+	0.845115i	$0.320468\pi$
$62$		0			0
$63$	5.74115	+	3.91692i	0.723317	+	0.493485i
$64$		0			0
$65$	−0.780706			−0.0968346
$66$		0			0
$67$	−0.937154			−0.114492			−0.0572458	−	0.998360i	$-0.518232\pi$
$67$	−0.937154			−0.114492			−0.0572458	+	0.998360i	$0.518232\pi$
$68$		0			0
$69$	−3.38838			−0.407914
$70$		0			0
$71$			13.0139i			1.54446i	0.635341	+	0.772232i	$0.280859\pi$
$71$			13.0139i			1.54446i	−0.635341	+	0.772232i	$0.719141\pi$
$72$		0			0
$73$		−	0.406509i		−	0.0475783i	−0.999717	−	0.0237891i	$-0.992427\pi$
$73$		−	0.406509i		−	0.0475783i	0.999717	−	0.0237891i	$-0.00757304\pi$
$74$		0			0
$75$		−	2.68192i		−	0.309681i
$76$		0			0
$77$	−6.85559	−	4.67725i	−0.781267	−	0.533022i
$78$		0			0
$79$		−	4.03568i		−	0.454049i	−0.973889	−	0.227025i	$-0.927100\pi$
$79$		−	4.03568i		−	0.454049i	0.973889	−	0.227025i	$-0.0728998\pi$
$80$		0			0
$81$	5.78103			0.642337
$82$		0			0
$83$			11.9237i			1.30879i	0.756151	+	0.654397i	$0.227078\pi$
$83$			11.9237i			1.30879i	−0.756151	+	0.654397i	$0.772922\pi$
$84$		0			0
$85$		−	3.49184i		−	0.378743i
$86$		0			0
$87$	1.49193			0.159951
$88$		0			0
$89$			15.7622i			1.67079i	0.549646	+	0.835397i	$0.314762\pi$
$89$			15.7622i			1.67079i	−0.549646	+	0.835397i	$0.685238\pi$
$90$		0			0
$91$	2.18555	+	1.49110i	0.229108	+	0.156309i
$92$		0			0
$93$		−	3.52247i		−	0.365263i
$94$		0			0
$95$		−	1.50547i		−	0.154458i
$96$		0			0
$97$		−	7.10470i		−	0.721373i	−0.932687	−	0.360687i	$-0.882542\pi$
$97$		−	7.10470i		−	0.721373i	0.932687	−	0.360687i	$-0.117458\pi$
$98$		0			0
$99$	−8.23991			−0.828142
$100$		0			0
$101$	−7.67995			−0.764184			−0.382092	−	0.924124i	$-0.624796\pi$
$101$	−7.67995			−0.764184			−0.382092	+	0.924124i	$0.624796\pi$
$102$		0			0
$103$	4.33561			0.427201			0.213600	−	0.976921i	$-0.431481\pi$
$103$	4.33561			0.427201			0.213600	+	0.976921i	$0.431481\pi$
$104$		0			0
$105$	0.711091	−	1.04227i	0.0693954	−	0.101715i
$106$		0			0
$107$	−16.2564			−1.57157			−0.785784	−	0.618501i	$-0.787740\pi$
$107$	−16.2564			−1.57157			−0.785784	+	0.618501i	$0.787740\pi$
$108$		0			0
$109$			3.71293i			0.355634i	0.984064	+	0.177817i	$0.0569035\pi$
$109$			3.71293i			0.355634i	−0.984064	+	0.177817i	$0.943097\pi$
$110$		0			0
$111$	−0.0303610			−0.00288174
$112$		0			0
$113$	−15.6349			−1.47081			−0.735403	−	0.677630i	$-0.763007\pi$
$113$	−15.6349			−1.47081			−0.735403	+	0.677630i	$0.763007\pi$
$114$		0			0
$115$		−	4.33060i		−	0.403831i
$116$		0			0
$117$	2.62687			0.242854
$118$		0			0
$119$	−6.66919	+	9.77525i	−0.611364	+	0.896096i
$120$		0			0
$121$	−1.16060			−0.105509
$122$		0			0
$123$	5.93325			0.534983
$124$		0			0
$125$	7.33122			0.655724
$126$		0			0
$127$		−	15.3966i		−	1.36623i	−0.730313	−	0.683113i	$-0.760626\pi$
$127$		−	15.3966i		−	1.36623i	0.730313	−	0.683113i	$-0.239374\pi$
$128$		0			0
$129$		−	5.70932i		−	0.502678i
$130$		0			0
$131$		−	17.2379i		−	1.50608i	−0.657972	−	0.753042i	$-0.728585\pi$
$131$		−	17.2379i		−	1.50608i	0.657972	−	0.753042i	$-0.271415\pi$
$132$		0			0
$133$	−2.87535	+	4.21450i	−0.249325	+	0.365443i
$134$		0			0
$135$		−	2.68340i		−	0.230950i
$136$		0			0
$137$	−10.9864			−0.938628			−0.469314	−	0.883031i	$-0.655499\pi$
$137$	−10.9864			−0.938628			−0.469314	+	0.883031i	$0.655499\pi$
$138$		0			0
$139$			18.0704i			1.53271i	0.642419	+	0.766354i	$0.277931\pi$
$139$			18.0704i			1.53271i	−0.642419	+	0.766354i	$0.722069\pi$
$140$		0			0
$141$			7.49434i			0.631138i
$142$		0			0
$143$	−3.13678			−0.262311
$144$		0			0
$145$			1.90679i			0.158350i
$146$		0			0
$147$	−3.98133	+	1.55965i	−0.328375	+	0.128637i
$148$		0			0
$149$			20.7169i			1.69720i	0.529038	+	0.848598i	$0.322553\pi$
$149$			20.7169i			1.69720i	−0.529038	+	0.848598i	$0.677447\pi$
$150$		0			0
$151$		−	12.7907i		−	1.04089i	−0.853895	−	0.520446i	$-0.825766\pi$
$151$		−	12.7907i		−	1.04089i	0.853895	−	0.520446i	$-0.174234\pi$
$152$		0			0
$153$			11.7491i			0.949860i
$154$		0			0
$155$	4.50198			0.361608
$156$		0			0
$157$	5.52377			0.440845			0.220423	−	0.975404i	$-0.429256\pi$
$157$	5.52377			0.440845			0.220423	+	0.975404i	$0.429256\pi$
$158$		0			0
$159$	4.64297			0.368211
$160$		0			0
$161$	−8.27117	+	12.1233i	−0.651860	+	0.955452i
$162$		0			0
$163$	0.876767			0.0686737			0.0343368	−	0.999410i	$-0.489068\pi$
$163$	0.876767			0.0686737			0.0343368	+	0.999410i	$0.489068\pi$
$164$		0			0
$165$			1.49590i			0.116456i
$166$		0			0
$167$	−22.5988			−1.74875			−0.874373	−	0.485254i	$-0.838727\pi$
$167$	−22.5988			−1.74875			−0.874373	+	0.485254i	$0.838727\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			5.06551i			0.387369i
$172$		0			0
$173$	1.41995			0.107956			0.0539782	−	0.998542i	$-0.482810\pi$
$173$	1.41995			0.107956			0.0539782	+	0.998542i	$0.482810\pi$
$174$		0			0
$175$	−9.59565	−	6.54666i	−0.725363	−	0.494881i
$176$		0			0
$177$	−8.10735			−0.609386
$178$		0			0
$179$	−12.0505			−0.900698			−0.450349	−	0.892852i	$-0.648701\pi$
$179$	−12.0505			−0.900698			−0.450349	+	0.892852i	$0.648701\pi$
$180$		0			0
$181$	23.9696			1.78164			0.890822	−	0.454352i	$-0.150129\pi$
$181$	23.9696			1.78164			0.890822	+	0.454352i	$0.150129\pi$
$182$		0			0
$183$			5.10086i			0.377066i
$184$		0			0
$185$		−	0.0388035i		−	0.00285289i
$186$		0			0
$187$		−	14.0298i		−	1.02596i
$188$		0			0
$189$	−5.12512	+	7.51206i	−0.372798	+	0.546422i
$190$		0			0
$191$			12.0391i			0.871119i	0.900160	+	0.435559i	$0.143449\pi$
$191$			12.0391i			0.871119i	−0.900160	+	0.435559i	$0.856551\pi$
$192$		0			0
$193$	3.94730			0.284133			0.142066	−	0.989857i	$-0.454625\pi$
$193$	3.94730			0.284133			0.142066	+	0.989857i	$0.454625\pi$
$194$		0			0
$195$		−	0.476891i		−	0.0341509i
$196$		0			0
$197$			16.8982i			1.20394i	0.798517	+	0.601972i	$0.205618\pi$
$197$			16.8982i			1.20394i	−0.798517	+	0.601972i	$0.794382\pi$
$198$		0			0
$199$	11.8887			0.842767			0.421383	−	0.906883i	$-0.361545\pi$
$199$	11.8887			0.842767			0.421383	+	0.906883i	$0.361545\pi$
$200$		0			0
$201$		−	0.572456i		−	0.0403780i
$202$		0			0
$203$	3.64185	−	5.33797i	0.255608	−	0.374652i
$204$		0			0
$205$			7.58313i			0.529629i
$206$		0			0
$207$			14.5713i			1.01278i
$208$		0			0
$209$		−	6.04880i		−	0.418405i
$210$		0			0
$211$	11.2892			0.777182			0.388591	−	0.921410i	$-0.372962\pi$
$211$	11.2892			0.777182			0.388591	+	0.921410i	$0.372962\pi$
$212$		0			0
$213$	−7.94947			−0.544689
$214$		0			0
$215$	7.29693			0.497646
$216$		0			0
$217$	−12.6031	−	8.59849i	−0.855553	−	0.583703i
$218$		0			0
$219$	0.248314			0.0167795
$220$		0			0
$221$			4.47267i			0.300865i
$222$		0			0
$223$	−6.21212			−0.415994			−0.207997	−	0.978129i	$-0.566694\pi$
$223$	−6.21212			−0.415994			−0.207997	+	0.978129i	$0.566694\pi$
$224$		0			0
$225$	−11.5333			−0.768884
$226$		0			0
$227$			2.41724i			0.160438i	0.996777	+	0.0802188i	$0.0255619\pi$
$227$			2.41724i			0.160438i	−0.996777	+	0.0802188i	$0.974438\pi$
$228$		0			0
$229$	0.777226			0.0513606			0.0256803	−	0.999670i	$-0.491825\pi$
$229$	0.777226			0.0513606			0.0256803	+	0.999670i	$0.491825\pi$
$230$		0			0
$231$	2.85708	−	4.18771i	0.187982	−	0.275531i
$232$		0			0
$233$	0.935862			0.0613104			0.0306552	−	0.999530i	$-0.490241\pi$
$233$	0.935862			0.0613104			0.0306552	+	0.999530i	$0.490241\pi$
$234$		0			0
$235$	−9.57832			−0.624821
$236$		0			0
$237$	2.46518			0.160131
$238$		0			0
$239$			8.07542i			0.522356i	0.965291	+	0.261178i	$0.0841109\pi$
$239$			8.07542i			0.522356i	−0.965291	+	0.261178i	$0.915889\pi$
$240$		0			0
$241$			13.0922i			0.843344i	0.906748	+	0.421672i	$0.138557\pi$
$241$			13.0922i			0.843344i	−0.906748	+	0.421672i	$0.861443\pi$
$242$		0			0
$243$			13.8428i			0.888014i
$244$		0			0
$245$	−1.99334	−	5.08843i	−0.127350	−	0.325088i
$246$		0			0
$247$			1.92835i			0.122698i
$248$		0			0
$249$	−7.28353			−0.461575
$250$		0			0
$251$			11.6794i			0.737199i	0.929588	+	0.368599i	$0.120163\pi$
$251$			11.6794i			0.737199i	−0.929588	+	0.368599i	$0.879837\pi$
$252$		0			0
$253$		−	17.3998i		−	1.09392i
$254$		0			0
$255$	2.13298			0.133572
$256$		0			0
$257$			22.9812i			1.43353i	0.697315	+	0.716765i	$0.254378\pi$
$257$			22.9812i			1.43353i	−0.697315	+	0.716765i	$0.745622\pi$
$258$		0			0
$259$	−0.0741123	+	0.108629i	−0.00460511	+	0.00674986i
$260$		0			0
$261$		−	6.41584i		−	0.397131i
$262$		0			0
$263$		−	8.58073i		−	0.529111i	−0.964371	−	0.264555i	$-0.914775\pi$
$263$		−	8.58073i		−	0.529111i	0.964371	−	0.264555i	$-0.0852252\pi$
$264$		0			0
$265$			5.93405i			0.364526i
$266$		0			0
$267$	−9.62830			−0.589243
$268$		0			0
$269$	20.5726			1.25433			0.627167	−	0.778885i	$-0.284214\pi$
$269$	20.5726			1.25433			0.627167	+	0.778885i	$0.284214\pi$
$270$		0			0
$271$	−7.63381			−0.463721			−0.231861	−	0.972749i	$-0.574481\pi$
$271$	−7.63381			−0.463721			−0.231861	+	0.972749i	$0.574481\pi$
$272$		0			0
$273$	−0.910831	+	1.33503i	−0.0551260	+	0.0807999i
$274$		0			0
$275$	13.7720			0.830485
$276$		0			0
$277$			11.3502i			0.681965i	0.940070	+	0.340982i	$0.110760\pi$
$277$			11.3502i			0.681965i	−0.940070	+	0.340982i	$0.889240\pi$
$278$		0			0
$279$	−15.1480			−0.906885
$280$		0			0
$281$	6.35231			0.378947			0.189474	−	0.981886i	$-0.439322\pi$
$281$	6.35231			0.378947			0.189474	+	0.981886i	$0.439322\pi$
$282$		0			0
$283$		−	23.3762i		−	1.38957i	−0.719217	−	0.694785i	$-0.755499\pi$
$283$		−	23.3762i		−	1.38957i	0.719217	−	0.694785i	$-0.244501\pi$
$284$		0			0
$285$	0.919611			0.0544731
$286$		0			0
$287$	14.4833	−	21.2286i	0.854922	−	1.25309i
$288$		0			0
$289$	−3.00481			−0.176753
$290$		0			0
$291$	4.33988			0.254408
$292$		0			0
$293$	0.300712			0.0175678			0.00878389	−	0.999961i	$-0.497204\pi$
$293$	0.300712			0.0175678			0.00878389	+	0.999961i	$0.497204\pi$
$294$		0			0
$295$		−	10.3618i		−	0.603287i
$296$		0			0
$297$		−	10.7816i		−	0.625611i
$298$		0			0
$299$			5.54704i			0.320793i
$300$		0			0
$301$	−20.4274	−	13.9367i	−1.17742	−	0.803296i
$302$		0			0
$303$		−	4.69127i		−	0.269506i
$304$		0			0
$305$	−6.51927			−0.373292
$306$		0			0
$307$		−	10.2092i		−	0.582671i	−0.956621	−	0.291336i	$-0.905900\pi$
$307$		−	10.2092i		−	0.582671i	0.956621	−	0.291336i	$-0.0940997\pi$
$308$		0			0
$309$			2.64839i			0.150662i
$310$		0			0
$311$	3.99969			0.226802			0.113401	−	0.993549i	$-0.463826\pi$
$311$	3.99969			0.226802			0.113401	+	0.993549i	$0.463826\pi$
$312$		0			0
$313$		−	14.2771i		−	0.806988i	−0.914982	−	0.403494i	$-0.867796\pi$
$313$		−	14.2771i		−	0.806988i	0.914982	−	0.403494i	$-0.132204\pi$
$314$		0			0
$315$	−4.48215	−	3.05796i	−0.252540	−	0.172296i
$316$		0			0
$317$			4.92436i			0.276580i	0.990392	+	0.138290i	$0.0441605\pi$
$317$			4.92436i			0.276580i	−0.990392	+	0.138290i	$0.955839\pi$
$318$		0			0
$319$			7.66126i			0.428948i
$320$		0			0
$321$		−	9.93017i		−	0.554248i
$322$		0			0
$323$	−8.62487			−0.479900
$324$		0			0
$325$	−4.39050			−0.243541
$326$		0			0
$327$	−2.26802			−0.125422
$328$		0			0
$329$	26.8141	+	18.2940i	1.47831	+	1.00858i
$330$		0			0
$331$	10.2132			0.561368			0.280684	−	0.959800i	$-0.409439\pi$
$331$	10.2132			0.561368			0.280684	+	0.959800i	$0.409439\pi$
$332$		0			0
$333$			0.130564i			0.00715484i
$334$		0			0
$335$	0.731641			0.0399738
$336$		0			0
$337$	−5.87706			−0.320144			−0.160072	−	0.987105i	$-0.551173\pi$
$337$	−5.87706			−0.320144			−0.160072	+	0.987105i	$0.551173\pi$
$338$		0			0
$339$		−	9.55050i		−	0.518712i
$340$		0			0
$341$	18.0884			0.979542
$342$		0			0
$343$	−4.13831	+	18.0520i	−0.223448	+	0.974716i
$344$		0			0
$345$	2.64533			0.142420
$346$		0			0
$347$	26.0891			1.40053			0.700267	−	0.713881i	$-0.253064\pi$
$347$	26.0891			1.40053			0.700267	+	0.713881i	$0.253064\pi$
$348$		0			0
$349$	−24.8684			−1.33118			−0.665588	−	0.746319i	$-0.731819\pi$
$349$	−24.8684			−1.33118			−0.665588	+	0.746319i	$0.731819\pi$
$350$		0			0
$351$			3.43715i			0.183461i
$352$		0			0
$353$			1.35013i			0.0718601i	0.999354	+	0.0359300i	$0.0114393\pi$
$353$			1.35013i			0.0718601i	−0.999354	+	0.0359300i	$0.988561\pi$
$354$		0			0
$355$		−	10.1600i		−	0.539237i
$356$		0			0
$357$	−5.97117	−	4.07385i	−0.316028	−	0.215611i
$358$		0			0
$359$		−	4.82833i		−	0.254829i	−0.991850	−	0.127415i	$-0.959332\pi$
$359$		−	4.82833i		−	0.254829i	0.991850	−	0.127415i	$-0.0406679\pi$
$360$		0			0
$361$	15.2815			0.804288
$362$		0			0
$363$		−	0.708947i		−	0.0372101i
$364$		0			0
$365$			0.317364i			0.0166116i
$366$		0			0
$367$	8.89488			0.464309			0.232155	−	0.972679i	$-0.425422\pi$
$367$	8.89488			0.464309			0.232155	+	0.972679i	$0.425422\pi$
$368$		0			0
$369$		−	25.5152i		−	1.32827i
$370$		0			0
$371$	11.3337	−	16.6121i	0.588414	−	0.862458i
$372$		0			0
$373$		−	35.8127i		−	1.85431i	−0.374674	−	0.927157i	$-0.622245\pi$
$373$		−	35.8127i		−	1.85431i	0.374674	−	0.927157i	$-0.377755\pi$
$374$		0			0
$375$			4.47824i			0.231255i
$376$		0			0
$377$		−	2.44239i		−	0.125790i
$378$		0			0
$379$	12.9413			0.664748			0.332374	−	0.943148i	$-0.392150\pi$
$379$	12.9413			0.664748			0.332374	+	0.943148i	$0.392150\pi$
$380$		0			0
$381$	9.40493			0.481829
$382$		0			0
$383$	16.3984			0.837918			0.418959	−	0.908005i	$-0.362395\pi$
$383$	16.3984			0.837918			0.418959	+	0.908005i	$0.362395\pi$
$384$		0			0
$385$	5.35220	+	3.65155i	0.272773	+	0.186100i
$386$		0			0
$387$	−24.5522			−1.24806
$388$		0			0
$389$		−	19.3191i		−	0.979516i	−0.871858	−	0.489758i	$-0.837085\pi$
$389$		−	19.3191i		−	0.979516i	0.871858	−	0.489758i	$-0.162915\pi$
$390$		0			0
$391$	−24.8101			−1.25470
$392$		0			0
$393$	10.5297			0.531154
$394$		0			0
$395$			3.15068i			0.158528i
$396$		0			0
$397$	27.8169			1.39609			0.698044	−	0.716055i	$-0.254054\pi$
$397$	27.8169			1.39609			0.698044	+	0.716055i	$0.254054\pi$
$398$		0			0
$399$	−2.57441	−	1.75640i	−0.128882	−	0.0879299i
$400$		0			0
$401$	−32.8827			−1.64208			−0.821041	−	0.570869i	$-0.806606\pi$
$401$	−32.8827			−1.64208			−0.821041	+	0.570869i	$0.806606\pi$
$402$		0			0
$403$	−5.76655			−0.287252
$404$		0			0
$405$	−4.51329			−0.224267
$406$		0			0
$407$		−	0.155908i		−	0.00772807i
$408$		0			0
$409$		−	22.6861i		−	1.12176i	−0.827899	−	0.560878i	$-0.810464\pi$
$409$		−	22.6861i		−	1.12176i	0.827899	−	0.560878i	$-0.189536\pi$
$410$		0			0
$411$		−	6.71097i		−	0.331028i
$412$		0			0
$413$	−19.7903	+	29.0074i	−0.973819	+	1.42736i
$414$		0			0
$415$		−	9.30889i		−	0.456956i
$416$		0			0
$417$	−11.0382			−0.540543
$418$		0			0
$419$		−	7.03620i		−	0.343741i	−0.985120	−	0.171870i	$-0.945019\pi$
$419$		−	7.03620i		−	0.343741i	0.985120	−	0.171870i	$-0.0549810\pi$
$420$		0			0
$421$			13.2449i			0.645515i	0.946482	+	0.322757i	$0.104610\pi$
$421$			13.2449i			0.645515i	−0.946482	+	0.322757i	$0.895390\pi$
$422$		0			0
$423$	32.2285			1.56700
$424$		0			0
$425$		−	19.6373i		−	0.952547i
$426$		0			0
$427$	18.2504	+	12.4514i	0.883199	+	0.602565i
$428$		0			0
$429$		−	1.91609i		−	0.0925097i
$430$		0			0
$431$		−	28.2469i		−	1.36060i	−0.732932	−	0.680302i	$-0.761849\pi$
$431$		−	28.2469i		−	1.36060i	0.732932	−	0.680302i	$-0.238151\pi$
$432$		0			0
$433$			13.3814i			0.643071i	0.946898	+	0.321535i	$0.104199\pi$
$433$			13.3814i			0.643071i	−0.946898	+	0.321535i	$0.895801\pi$
$434$		0			0
$435$	−1.16476			−0.0558457
$436$		0			0
$437$	−10.6966			−0.511688
$438$		0			0
$439$	15.9433			0.760933			0.380467	−	0.924795i	$-0.375763\pi$
$439$	15.9433			0.760933			0.380467	+	0.924795i	$0.375763\pi$
$440$		0			0
$441$	6.70707	+	17.1212i	0.319384	+	0.815297i
$442$		0			0
$443$	15.4639			0.734714			0.367357	−	0.930080i	$-0.380263\pi$
$443$	15.4639			0.734714			0.367357	+	0.930080i	$0.380263\pi$
$444$		0			0
$445$		−	12.3057i		−	0.583345i
$446$		0			0
$447$	−12.6548			−0.598553
$448$		0			0
$449$	−6.57371			−0.310233			−0.155116	−	0.987896i	$-0.549575\pi$
$449$	−6.57371			−0.310233			−0.155116	+	0.987896i	$0.549575\pi$
$450$		0			0
$451$			30.4681i			1.43469i
$452$		0			0
$453$	7.81314			0.367093
$454$		0			0
$455$	−1.70627	−	1.16411i	−0.0799912	−	0.0545743i
$456$		0			0
$457$	34.8367			1.62959			0.814796	−	0.579748i	$-0.196849\pi$
$457$	34.8367			1.62959			0.814796	+	0.579748i	$0.196849\pi$
$458$		0			0
$459$	−15.3732			−0.717562
$460$		0			0
$461$	10.9767			0.511236			0.255618	−	0.966778i	$-0.417721\pi$
$461$	10.9767			0.511236			0.255618	+	0.966778i	$0.417721\pi$
$462$		0			0
$463$			18.8055i			0.873963i	0.899470	+	0.436982i	$0.143953\pi$
$463$			18.8055i			0.873963i	−0.899470	+	0.436982i	$0.856047\pi$
$464$		0			0
$465$			2.75002i			0.127529i
$466$		0			0
$467$			14.3383i			0.663497i	0.943368	+	0.331748i	$0.107638\pi$
$467$			14.3383i			0.663497i	−0.943368	+	0.331748i	$0.892362\pi$
$468$		0			0
$469$	−2.04820	−	1.39739i	−0.0945769	−	0.0645254i
$470$		0			0
$471$			3.37417i			0.155474i
$472$		0			0
$473$	29.3182			1.34805
$474$		0			0
$475$		−	8.46640i		−	0.388465i
$476$		0			0
$477$		−	19.9665i		−	0.914204i
$478$		0			0
$479$	−5.36064			−0.244934			−0.122467	−	0.992473i	$-0.539081\pi$
$479$	−5.36064			−0.244934			−0.122467	+	0.992473i	$0.539081\pi$
$480$		0			0
$481$			0.0497032i			0.00226627i
$482$		0			0
$483$	−7.40548	−	5.05241i	−0.336961	−	0.229893i
$484$		0			0
$485$			5.54668i			0.251862i
$486$		0			0
$487$		−	14.7792i		−	0.669708i	−0.942270	−	0.334854i	$-0.891313\pi$
$487$		−	14.7792i		−	0.669708i	0.942270	−	0.334854i	$-0.108687\pi$
$488$		0			0
$489$			0.535569i			0.0242193i
$490$		0			0
$491$	21.0065			0.948011			0.474006	−	0.880522i	$-0.342808\pi$
$491$	21.0065			0.948011			0.474006	+	0.880522i	$0.342808\pi$
$492$		0			0
$493$	10.9240			0.491993
$494$		0			0
$495$	6.43295			0.289139
$496$		0			0
$497$	−19.4050	+	28.4425i	−0.870431	+	1.27582i
$498$		0			0
$499$	14.2078			0.636030			0.318015	−	0.948086i	$-0.396984\pi$
$499$	14.2078			0.636030			0.318015	+	0.948086i	$0.396984\pi$
$500$		0			0
$501$		−	13.8044i		−	0.616734i
$502$		0			0
$503$	−38.2658			−1.70619			−0.853095	−	0.521756i	$-0.825277\pi$
$503$	−38.2658			−1.70619			−0.853095	+	0.521756i	$0.825277\pi$
$504$		0			0
$505$	5.99578			0.266809
$506$		0			0
$507$			0.610846i			0.0271286i
$508$		0			0
$509$	−9.93667			−0.440435			−0.220217	−	0.975451i	$-0.570677\pi$
$509$	−9.93667			−0.440435			−0.220217	+	0.975451i	$0.570677\pi$
$510$		0			0
$511$	0.606145	−	0.888446i	0.0268143	−	0.0393025i
$512$		0			0
$513$	−6.62802			−0.292634
$514$		0			0
$515$	−3.38484			−0.149154
$516$		0			0
$517$	−38.4845			−1.69255
$518$		0			0
$519$			0.867368i			0.0380732i
$520$		0			0
$521$		−	10.2606i		−	0.449524i	−0.974414	−	0.224762i	$-0.927840\pi$
$521$		−	10.2606i		−	0.449524i	0.974414	−	0.224762i	$-0.0721605\pi$
$522$		0			0
$523$		−	10.2265i		−	0.447173i	−0.974684	−	0.223587i	$-0.928223\pi$
$523$		−	10.2265i		−	0.447173i	0.974684	−	0.223587i	$-0.0717765\pi$
$524$		0			0
$525$	3.99900	−	5.86146i	0.174531	−	0.255815i
$526$		0			0
$527$		−	25.7919i		−	1.12351i
$528$		0			0
$529$	−7.76963			−0.337810
$530$		0			0
$531$			34.8647i			1.51300i
$532$		0			0
$533$		−	9.71318i		−	0.420724i
$534$		0			0
$535$	12.6915			0.548701
$536$		0			0
$537$		−	7.36101i		−	0.317651i
$538$		0			0
$539$	−8.00901	−	20.4447i	−0.344972	−	0.880616i
$540$		0			0
$541$			46.1245i			1.98305i	0.129934	+	0.991523i	$0.458523\pi$
$541$			46.1245i			1.98305i	−0.129934	+	0.991523i	$0.541477\pi$
$542$		0			0
$543$			14.6417i			0.628336i
$544$		0			0
$545$		−	2.89870i		−	0.124167i
$546$		0			0
$547$	13.4615			0.575573			0.287787	−	0.957695i	$-0.407081\pi$
$547$	13.4615			0.575573			0.287787	+	0.957695i	$0.407081\pi$
$548$		0			0
$549$	21.9356			0.936189
$550$		0			0
$551$	4.70978			0.200643
$552$		0			0
$553$	6.01759	−	8.82018i	0.255894	−	0.375072i
$554$		0			0
$555$	0.0237030			0.00100614
$556$		0			0
$557$		−	18.6109i		−	0.788568i	−0.918989	−	0.394284i	$-0.870993\pi$
$557$		−	18.6109i		−	0.788568i	0.918989	−	0.394284i	$-0.129007\pi$
$558$		0			0
$559$	−9.34658			−0.395318
$560$		0			0
$561$	8.57005			0.361828
$562$		0			0
$563$		−	22.1148i		−	0.932026i	−0.884778	−	0.466013i	$-0.845690\pi$
$563$		−	22.1148i		−	0.932026i	0.884778	−	0.466013i	$-0.154310\pi$
$564$		0			0
$565$	12.2062			0.513521
$566$		0			0
$567$	12.6347	+	8.62009i	0.530609	+	0.362010i
$568$		0			0
$569$	25.8418			1.08335			0.541673	−	0.840590i	$-0.317791\pi$
$569$	25.8418			1.08335			0.541673	+	0.840590i	$0.317791\pi$
$570$		0			0
$571$	25.7728			1.07856			0.539279	−	0.842127i	$-0.318697\pi$
$571$	25.7728			1.07856			0.539279	+	0.842127i	$0.318697\pi$
$572$		0			0
$573$	−7.35403			−0.307219
$574$		0			0
$575$		−	24.3543i		−	1.01564i
$576$		0			0
$577$			22.8537i			0.951410i	0.879605	+	0.475705i	$0.157807\pi$
$577$			22.8537i			0.951410i	−0.879605	+	0.475705i	$0.842193\pi$
$578$		0			0
$579$			2.41119i			0.100206i
$580$		0			0
$581$	−17.7794	+	26.0598i	−0.737613	+	1.08114i
$582$		0			0
$583$			23.8423i			0.987447i
$584$		0			0
$585$	−2.05081			−0.0847906
$586$		0			0
$587$			5.51657i			0.227693i	0.993498	+	0.113847i	$0.0363172\pi$
$587$			5.51657i			0.227693i	−0.993498	+	0.113847i	$0.963683\pi$
$588$		0			0
$589$		−	11.1199i		−	0.458188i
$590$		0			0
$591$	−10.3222			−0.424598
$592$		0			0
$593$			5.37304i			0.220644i	0.993896	+	0.110322i	$0.0351883\pi$
$593$			5.37304i			0.220644i	−0.993896	+	0.110322i	$0.964812\pi$
$594$		0			0
$595$	5.20668	−	7.63160i	0.213453	−	0.312865i
$596$		0			0
$597$			7.26216i			0.297220i
$598$		0			0
$599$		−	4.38259i		−	0.179068i	−0.995984	−	0.0895338i	$-0.971462\pi$
$599$		−	4.38259i		−	0.179068i	0.995984	−	0.0895338i	$-0.0285377\pi$
$600$		0			0
$601$		−	32.9212i		−	1.34288i	−0.741057	−	0.671442i	$-0.765675\pi$
$601$		−	32.9212i		−	1.34288i	0.741057	−	0.671442i	$-0.234325\pi$
$602$		0			0
$603$	−2.46178			−0.100251
$604$		0			0
$605$	0.906087			0.0368377
$606$		0			0
$607$	−16.1526			−0.655615			−0.327807	−	0.944745i	$-0.606310\pi$
$607$	−16.1526			−0.655615			−0.327807	+	0.944745i	$0.606310\pi$
$608$		0			0
$609$	3.26068	+	2.22461i	0.132129	+	0.0901456i
$610$		0			0
$611$	12.2688			0.496342
$612$		0			0
$613$			38.2321i			1.54418i	0.635514	+	0.772090i	$0.280788\pi$
$613$			38.2321i			1.54418i	−0.635514	+	0.772090i	$0.719212\pi$
$614$		0			0
$615$	−4.63212			−0.186785
$616$		0			0
$617$	13.4473			0.541370			0.270685	−	0.962668i	$-0.412750\pi$
$617$	13.4473			0.541370			0.270685	+	0.962668i	$0.412750\pi$
$618$		0			0
$619$		−	3.01278i		−	0.121094i	−0.998165	−	0.0605468i	$-0.980716\pi$
$619$		−	3.01278i		−	0.121094i	0.998165	−	0.0605468i	$-0.0192845\pi$
$620$		0			0
$621$	−19.0660			−0.765092
$622$		0			0
$623$	−23.5030	+	34.4492i	−0.941630	+	1.38018i
$624$		0			0
$625$	16.2290			0.649159
$626$		0			0
$627$	3.69489			0.147560
$628$		0			0
$629$	−0.222306			−0.00886392
$630$		0			0
$631$		−	45.0426i		−	1.79312i	−0.442923	−	0.896560i	$-0.646059\pi$
$631$		−	45.0426i		−	1.79312i	0.442923	−	0.896560i	$-0.353941\pi$
$632$		0			0
$633$			6.89598i			0.274090i
$634$		0			0
$635$			12.0202i			0.477007i
$636$		0			0
$637$	2.55326	+	6.51774i	0.101164	+	0.258242i
$638$		0			0
$639$			34.1857i			1.35237i
$640$		0			0
$641$	−13.9305			−0.550223			−0.275111	−	0.961412i	$-0.588715\pi$
$641$	−13.9305			−0.550223			−0.275111	+	0.961412i	$0.588715\pi$
$642$		0			0
$643$			12.4706i			0.491793i	0.969296	+	0.245896i	$0.0790823\pi$
$643$			12.4706i			0.491793i	−0.969296	+	0.245896i	$0.920918\pi$
$644$		0			0
$645$			4.45730i			0.175506i
$646$		0			0
$647$	15.0662			0.592315			0.296157	−	0.955139i	$-0.404295\pi$
$647$	15.0662			0.592315			0.296157	+	0.955139i	$0.404295\pi$
$648$		0			0
$649$		−	41.6324i		−	1.63422i
$650$		0			0
$651$	5.25235	−	7.69854i	0.205856	−	0.301730i
$652$		0			0
$653$		−	4.21959i		−	0.165125i	−0.996586	−	0.0825627i	$-0.973690\pi$
$653$		−	4.21959i		−	0.165125i	0.996586	−	0.0825627i	$-0.0263105\pi$
$654$		0			0
$655$			13.4578i			0.525838i
$656$		0			0
$657$		−	1.06785i		−	0.0416606i
$658$		0			0
$659$	−3.00756			−0.117158			−0.0585790	−	0.998283i	$-0.518657\pi$
$659$	−3.00756			−0.117158			−0.0585790	+	0.998283i	$0.518657\pi$
$660$		0			0
$661$	−18.4307			−0.716871			−0.358436	−	0.933554i	$-0.616690\pi$
$661$	−18.4307			−0.716871			−0.358436	+	0.933554i	$0.616690\pi$
$662$		0			0
$663$	−2.73211			−0.106107
$664$		0			0
$665$	2.24481	−	3.29028i	0.0870498	−	0.127592i
$666$		0			0
$667$	13.5480			0.524583
$668$		0			0
$669$		−	3.79465i		−	0.146710i
$670$		0			0
$671$	−26.1936			−1.01119
$672$		0			0
$673$	34.6624			1.33614			0.668069	−	0.744099i	$-0.267121\pi$
$673$	34.6624			1.33614			0.668069	+	0.744099i	$0.267121\pi$
$674$		0			0
$675$		−	15.0908i		−	0.580845i
$676$		0			0
$677$	46.5739			1.78998			0.894990	−	0.446087i	$-0.147183\pi$
$677$	46.5739			1.78998			0.894990	+	0.446087i	$0.147183\pi$
$678$		0			0
$679$	10.5938	−	15.5277i	0.406553	−	0.595898i
$680$		0			0
$681$	−1.47656			−0.0565819
$682$		0			0
$683$	−13.1703			−0.503947			−0.251974	−	0.967734i	$-0.581080\pi$
$683$	−13.1703			−0.503947			−0.251974	+	0.967734i	$0.581080\pi$
$684$		0			0
$685$	8.57711			0.327715
$686$		0			0
$687$			0.474765i			0.0181134i
$688$		0			0
$689$		−	7.60088i		−	0.289571i
$690$		0			0
$691$		−	37.3690i		−	1.42158i	−0.703402	−	0.710792i	$-0.748337\pi$
$691$		−	37.3690i		−	1.42158i	0.703402	−	0.710792i	$-0.251663\pi$
$692$		0			0
$693$	−18.0087	−	12.2865i	−0.684095	−	0.466726i
$694$		0			0
$695$		−	14.1076i		−	0.535133i
$696$		0			0
$697$	43.4439			1.64555
$698$		0			0
$699$			0.571668i			0.0216225i
$700$		0			0
$701$			19.9712i			0.754302i	0.926152	+	0.377151i	$0.123096\pi$
$701$			19.9712i			0.754302i	−0.926152	+	0.377151i	$0.876904\pi$
$702$		0			0
$703$	−0.0958449			−0.00361486
$704$		0			0
$705$		−	5.85088i		−	0.220357i
$706$		0			0
$707$	−16.7849	−	11.4516i	−0.631262	−	0.430680i
$708$		0			0
$709$		−	16.5434i		−	0.621302i	−0.950524	−	0.310651i	$-0.899453\pi$
$709$		−	16.5434i		−	0.621302i	0.950524	−	0.310651i	$-0.100547\pi$
$710$		0			0
$711$		−	10.6012i		−	0.397576i
$712$		0			0
$713$		−	31.9873i		−	1.19793i
$714$		0			0
$715$	2.44890			0.0915838
$716$		0			0
$717$	−4.93284			−0.184220
$718$		0			0
$719$	11.5055			0.429084			0.214542	−	0.976715i	$-0.431174\pi$
$719$	11.5055			0.429084			0.214542	+	0.976715i	$0.431174\pi$
$720$		0			0
$721$	9.47570	+	6.46482i	0.352894	+	0.240763i
$722$		0			0
$723$	−7.99733			−0.297424
$724$		0			0
$725$			10.7233i			0.398254i
$726$		0			0
$727$	−24.4779			−0.907835			−0.453918	−	0.891044i	$-0.649974\pi$
$727$	−24.4779			−0.907835			−0.453918	+	0.891044i	$0.649974\pi$
$728$		0			0
$729$	8.88731			0.329160
$730$		0			0
$731$		−	41.8042i		−	1.54618i
$732$		0			0
$733$	6.67143			0.246415			0.123207	−	0.992381i	$-0.460682\pi$
$733$	6.67143			0.246415			0.123207	+	0.992381i	$0.460682\pi$
$734$		0			0
$735$	3.10825	−	1.21762i	0.114649	−	0.0449128i
$736$		0			0
$737$	2.93965			0.108283
$738$		0			0
$739$	29.0139			1.06730			0.533648	−	0.845707i	$-0.320821\pi$
$739$	29.0139			1.06730			0.533648	+	0.845707i	$0.320821\pi$
$740$		0			0
$741$	−1.17792			−0.0432721
$742$		0			0
$743$		−	6.25450i		−	0.229455i	−0.993397	−	0.114728i	$-0.963400\pi$
$743$		−	6.25450i		−	0.229455i	0.993397	−	0.114728i	$-0.0365996\pi$
$744$		0			0
$745$		−	16.1738i		−	0.592563i
$746$		0			0
$747$			31.3219i			1.14601i
$748$		0			0
$749$	−35.5292	−	24.2399i	−1.29821	−	0.885707i
$750$		0			0
$751$		−	13.3309i		−	0.486450i	−0.969970	−	0.243225i	$-0.921795\pi$
$751$		−	13.3309i		−	0.486450i	0.969970	−	0.243225i	$-0.0782053\pi$
$752$		0			0
$753$	−7.13433			−0.259989
$754$		0			0
$755$			9.98577i			0.363419i
$756$		0			0
$757$		−	42.5601i		−	1.54687i	−0.633875	−	0.773436i	$-0.718537\pi$
$757$		−	42.5601i		−	1.54687i	0.633875	−	0.773436i	$-0.281463\pi$
$758$		0			0
$759$	10.6286			0.385795
$760$		0			0
$761$		−	16.7302i		−	0.606470i	−0.952916	−	0.303235i	$-0.901933\pi$
$761$		−	16.7302i		−	0.606470i	0.952916	−	0.303235i	$-0.0980668\pi$
$762$		0			0
$763$	−5.53633	+	8.11478i	−0.200429	+	0.293775i
$764$		0			0
$765$		−	9.17261i		−	0.331636i
$766$		0			0
$767$			13.2723i			0.479236i
$768$		0			0
$769$			20.9434i			0.755238i	0.925961	+	0.377619i	$0.123257\pi$
$769$			20.9434i			0.755238i	−0.925961	+	0.377619i	$0.876743\pi$
$770$		0			0
$771$	−14.0380			−0.505566
$772$		0			0
$773$	7.79043			0.280202			0.140101	−	0.990137i	$-0.455257\pi$
$773$	7.79043			0.280202			0.140101	+	0.990137i	$0.455257\pi$
$774$		0			0
$775$	25.3180			0.909450
$776$		0			0
$777$	−0.0663554	−	0.0452712i	−0.00238049	−	0.00162409i
$778$		0			0
$779$	18.7304			0.671085
$780$		0			0
$781$		−	40.8217i		−	1.46072i
$782$		0			0
$783$	8.39487			0.300008
$784$		0			0
$785$	−4.31244			−0.153918
$786$		0			0
$787$		−	46.2745i		−	1.64951i	−0.565491	−	0.824755i	$-0.691313\pi$
$787$		−	46.2745i		−	1.64951i	0.565491	−	0.824755i	$-0.308687\pi$
$788$		0			0
$789$	5.24150			0.186602
$790$		0			0
$791$	−34.1708	−	23.3131i	−1.21497	−	0.828920i
$792$		0			0
$793$	8.35048			0.296534
$794$		0			0
$795$	−3.62479			−0.128558
$796$		0			0
$797$	37.9333			1.34367			0.671833	−	0.740703i	$-0.265507\pi$
$797$	37.9333			1.34367			0.671833	+	0.740703i	$0.265507\pi$
$798$		0			0
$799$			54.8743i			1.94131i
$800$		0			0
$801$			41.4053i			1.46299i
$802$		0			0
$803$			1.27513i			0.0449984i
$804$		0			0
$805$	6.45735	−	9.46475i	0.227592	−	0.333589i
$806$		0			0
$807$			12.5667i			0.442369i
$808$		0			0
$809$	27.1364			0.954066			0.477033	−	0.878885i	$-0.341712\pi$
$809$	27.1364			0.954066			0.477033	+	0.878885i	$0.341712\pi$
$810$		0			0
$811$			13.4314i			0.471639i	0.971797	+	0.235820i	$0.0757774\pi$
$811$			13.4314i			0.471639i	−0.971797	+	0.235820i	$0.924223\pi$
$812$		0			0
$813$		−	4.66308i		−	0.163541i
$814$		0			0
$815$	−0.684497			−0.0239769
$816$		0			0
$817$		−	18.0234i		−	0.630561i
$818$		0			0
$819$	5.74115	+	3.91692i	0.200612	+	0.136868i
$820$		0			0
$821$		−	22.5402i		−	0.786659i	−0.919398	−	0.393330i	$-0.871323\pi$
$821$		−	22.5402i		−	0.786659i	0.919398	−	0.393330i	$-0.128677\pi$
$822$		0			0
$823$		−	7.78576i		−	0.271394i	−0.990750	−	0.135697i	$-0.956673\pi$
$823$		−	7.78576i		−	0.271394i	0.990750	−	0.135697i	$-0.0433274\pi$
$824$		0			0
$825$			8.41259i			0.292889i
$826$		0			0
$827$	−0.985448			−0.0342674			−0.0171337	−	0.999853i	$-0.505454\pi$
$827$	−0.985448			−0.0342674			−0.0171337	+	0.999853i	$0.505454\pi$
$828$		0			0
$829$	−54.4732			−1.89193			−0.945967	−	0.324264i	$-0.894883\pi$
$829$	−54.4732			−1.89193			−0.945967	+	0.324264i	$0.894883\pi$
$830$		0			0
$831$	−6.93319			−0.240510
$832$		0			0
$833$	−29.1517	+	11.4199i	−1.01005	+	0.395675i
$834$		0			0
$835$	17.6430			0.610561
$836$		0			0
$837$		−	19.8205i		−	0.685096i
$838$		0			0
$839$	18.5284			0.639672			0.319836	−	0.947473i	$-0.396372\pi$
$839$	18.5284			0.639672			0.319836	+	0.947473i	$0.396372\pi$
$840$		0			0
$841$	23.0347			0.794301
$842$		0			0
$843$			3.88028i			0.133644i
$844$		0			0
$845$	−0.780706			−0.0268571
$846$		0			0
$847$	−2.53655	−	1.73057i	−0.0871568	−	0.0594630i
$848$		0			0
$849$	14.2792			0.490063
$850$		0			0
$851$	−0.275705			−0.00945105
$852$		0			0
$853$	−54.5504			−1.86777			−0.933886	−	0.357571i	$-0.883605\pi$
$853$	−54.5504			−1.86777			−0.933886	+	0.357571i	$0.883605\pi$
$854$		0			0
$855$		−	3.95467i		−	0.135247i
$856$		0			0
$857$		−	17.4890i		−	0.597413i	−0.954345	−	0.298706i	$-0.903445\pi$
$857$		−	17.4890i		−	0.597413i	0.954345	−	0.298706i	$-0.0965551\pi$
$858$		0			0
$859$			32.4405i			1.10686i	0.832897	+	0.553428i	$0.186681\pi$
$859$			32.4405i			1.10686i	−0.832897	+	0.553428i	$0.813319\pi$
$860$		0			0
$861$	12.9674	+	8.84706i	0.441928	+	0.301507i
$862$		0			0
$863$			2.87423i			0.0978400i	0.998803	+	0.0489200i	$0.0155779\pi$
$863$			2.87423i			0.0978400i	−0.998803	+	0.0489200i	$0.984422\pi$
$864$		0			0
$865$	−1.10856			−0.0376922
$866$		0			0
$867$		−	1.83547i		−	0.0623360i
$868$		0			0
$869$			12.6590i			0.429429i
$870$		0			0
$871$	−0.937154			−0.0317542
$872$		0			0
$873$		−	18.6631i		−	0.631651i
$874$		0			0
$875$	16.0227	+	10.9316i	0.541667	+	0.369554i
$876$		0			0
$877$		−	11.8009i		−	0.398489i	−0.979950	−	0.199244i	$-0.936151\pi$
$877$		−	11.8009i		−	0.398489i	0.979950	−	0.199244i	$-0.0638487\pi$
$878$		0			0
$879$			0.183689i			0.00619567i
$880$		0			0
$881$			11.1084i			0.374252i	0.982336	+	0.187126i	$0.0599173\pi$
$881$			11.1084i			0.374252i	−0.982336	+	0.187126i	$0.940083\pi$
$882$		0			0
$883$	2.08414			0.0701370			0.0350685	−	0.999385i	$-0.488835\pi$
$883$	2.08414			0.0701370			0.0350685	+	0.999385i	$0.488835\pi$
$884$		0			0
$885$	6.32946			0.212762
$886$		0			0
$887$	40.7788			1.36922			0.684609	−	0.728911i	$-0.259973\pi$
$887$	40.7788			1.36922			0.684609	+	0.728911i	$0.259973\pi$
$888$		0			0
$889$	22.9578	−	33.6500i	0.769980	−	1.12858i
$890$		0			0
$891$	−18.1338			−0.607507
$892$		0			0
$893$			23.6585i			0.791702i
$894$		0			0
$895$	9.40791			0.314472
$896$		0			0
$897$	−3.38838			−0.113135
$898$		0			0
$899$			14.0842i			0.469734i
$900$		0			0
$901$	33.9963			1.13258
$902$		0			0
$903$	8.51315	−	12.4780i	0.283300	−	0.415242i
$904$		0			0
$905$	−18.7132			−0.622047
$906$		0			0
$907$	−28.4897			−0.945984			−0.472992	−	0.881067i	$-0.656826\pi$
$907$	−28.4897			−0.945984			−0.472992	+	0.881067i	$0.656826\pi$
$908$		0			0
$909$	−20.1742			−0.669137
$910$		0			0
$911$		−	25.7994i		−	0.854772i	−0.904069	−	0.427386i	$-0.859435\pi$
$911$		−	25.7994i		−	0.854772i	0.904069	−	0.427386i	$-0.140565\pi$
$912$		0			0
$913$		−	37.4020i		−	1.23783i
$914$		0			0
$915$		−	3.98227i		−	0.131650i
$916$		0			0
$917$	25.7034	−	37.6744i	0.848802	−	1.24412i
$918$		0			0
$919$			48.7196i			1.60711i	0.595230	+	0.803556i	$0.297061\pi$
$919$			48.7196i			1.60711i	−0.595230	+	0.803556i	$0.702939\pi$
$920$		0			0
$921$	6.23626			0.205492
$922$		0			0
$923$			13.0139i			0.428357i
$924$		0			0
$925$		−	0.218222i		−	0.00717508i
$926$		0			0
$927$	11.3891			0.374067
$928$		0			0
$929$		−	16.1335i		−	0.529322i	−0.964341	−	0.264661i	$-0.914740\pi$
$929$		−	16.1335i		−	0.529322i	0.964341	−	0.264661i	$-0.0852601\pi$
$930$		0			0
$931$	−12.5685	+	4.92356i	−0.411915	+	0.161363i
$932$		0			0
$933$			2.44320i			0.0799866i
$934$		0			0
$935$			10.9531i			0.358206i
$936$		0			0
$937$			11.6153i			0.379454i	0.981837	+	0.189727i	$0.0607604\pi$
$937$			11.6153i			0.379454i	−0.981837	+	0.189727i	$0.939240\pi$
$938$		0			0
$939$	8.72109			0.284602
$940$		0			0
$941$	16.5828			0.540584			0.270292	−	0.962778i	$-0.412880\pi$
$941$	16.5828			0.540584			0.270292	+	0.962778i	$0.412880\pi$
$942$		0			0
$943$	53.8794			1.75455
$944$		0			0
$945$	4.00121	−	5.86471i	0.130159	−	0.190779i
$946$		0			0
$947$	20.6272			0.670295			0.335147	−	0.942166i	$-0.391214\pi$
$947$	20.6272			0.670295			0.335147	+	0.942166i	$0.391214\pi$
$948$		0			0
$949$		−	0.406509i		−	0.0131958i
$950$		0			0
$951$	−3.00803			−0.0975419
$952$		0			0
$953$	−44.3425			−1.43640			−0.718198	−	0.695839i	$-0.755033\pi$
$953$	−44.3425			−1.43640			−0.718198	+	0.695839i	$0.755033\pi$
$954$		0			0
$955$		−	9.39899i		−	0.304144i
$956$		0			0
$957$	−4.67985			−0.151278
$958$		0			0
$959$	−24.0112	−	16.3817i	−0.775363	−	0.528994i
$960$		0			0
$961$	2.25310			0.0726807
$962$		0			0
$963$	−42.7035			−1.37610
$964$		0			0
$965$	−3.08168			−0.0992028
$966$		0			0
$967$		−	11.2213i		−	0.360852i	−0.983589	−	0.180426i	$-0.942252\pi$
$967$		−	11.2213i		−	0.360852i	0.983589	−	0.180426i	$-0.0577477\pi$
$968$		0			0
$969$		−	5.26846i		−	0.169247i
$970$		0			0
$971$		−	53.3628i		−	1.71249i	−0.516566	−	0.856247i	$-0.672790\pi$
$971$		−	53.3628i		−	1.71249i	0.516566	−	0.856247i	$-0.327210\pi$
$972$		0			0
$973$	−26.9447	+	39.4937i	−0.863806	+	1.26611i
$974$		0			0
$975$		−	2.68192i		−	0.0858901i
$976$		0			0
$977$	−61.6226			−1.97148			−0.985741	−	0.168270i	$-0.946182\pi$
$977$	−61.6226			−1.97148			−0.985741	+	0.168270i	$0.946182\pi$
$978$		0			0
$979$		−	49.4427i		−	1.58020i
$980$		0			0
$981$			9.75336i			0.311401i
$982$		0			0
$983$	19.3693			0.617786			0.308893	−	0.951097i	$-0.400042\pi$
$983$	19.3693			0.617786			0.308893	+	0.951097i	$0.400042\pi$
$984$		0			0
$985$		−	13.1925i		−	0.420348i
$986$		0			0
$987$	−11.1748	+	16.3793i	−0.355698	+	0.521358i
$988$		0			0
$989$		−	51.8458i		−	1.64860i
$990$		0			0
$991$		−	10.3242i		−	0.327958i	−0.986464	−	0.163979i	$-0.947567\pi$
$991$		−	10.3242i		−	0.327958i	0.986464	−	0.163979i	$-0.0524329\pi$
$992$		0			0
$993$			6.23869i			0.197979i
$994$		0			0
$995$	−9.28157			−0.294245
$996$		0			0
$997$	−44.2145			−1.40029			−0.700143	−	0.714002i	$-0.746881\pi$
$997$	−44.2145			−1.40029			−0.700143	+	0.714002i	$0.746881\pi$
$998$		0			0
$999$	−0.170837			−0.00540505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.b.2575.28		48
4.3	odd	2		728.2.h.b.27.5	yes	48
7.6	odd	2		2912.2.h.a.2575.21		48
8.3	odd	2		2912.2.h.a.2575.28		48
8.5	even	2		728.2.h.a.27.6	yes	48
28.27	even	2		728.2.h.a.27.5	✓	48
56.13	odd	2		728.2.h.b.27.6	yes	48
56.27	even	2	inner	2912.2.h.b.2575.21		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.5	✓	48	28.27	even	2
728.2.h.a.27.6	yes	48	8.5	even	2
728.2.h.b.27.5	yes	48	4.3	odd	2
728.2.h.b.27.6	yes	48	56.13	odd	2
2912.2.h.a.2575.21		48	7.6	odd	2
2912.2.h.a.2575.28		48	8.3	odd	2
2912.2.h.b.2575.21		48	56.27	even	2	inner
2912.2.h.b.2575.28		48	1.1	even	1	trivial

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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.610846i q^{3} -0.780706 q^{5} +(2.18555 + 1.49110i) q^{7} +2.62687 q^{9} -3.13678 q^{11} +1.00000 q^{13} -0.476891i q^{15} +4.47267i q^{17} +1.92835i q^{19} +(-0.910831 + 1.33503i) q^{21} +5.54704i q^{23} -4.39050 q^{25} +3.43715i q^{27} -2.44239i q^{29} -5.76655 q^{31} -1.91609i q^{33} +(-1.70627 - 1.16411i) q^{35} +0.0497032i q^{37} +0.610846i q^{39} -9.71318i q^{41} -9.34658 q^{43} -2.05081 q^{45} +12.2688 q^{47} +(2.55326 + 6.51774i) q^{49} -2.73211 q^{51} -7.60088i q^{53} +2.44890 q^{55} -1.17792 q^{57} +13.2723i q^{59} +8.35048 q^{61} +(5.74115 + 3.91692i) q^{63} -0.780706 q^{65} -0.937154 q^{67} -3.38838 q^{69} +13.0139i q^{71} -0.406509i q^{73} -2.68192i q^{75} +(-6.85559 - 4.67725i) q^{77} -4.03568i q^{79} +5.78103 q^{81} +11.9237i q^{83} -3.49184i q^{85} +1.49193 q^{87} +15.7622i q^{89} +(2.18555 + 1.49110i) q^{91} -3.52247i q^{93} -1.50547i q^{95} -7.10470i q^{97} -8.23991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) \) 48 * q - 48 * q^9 + 4 * q^11 + 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 - 24 * q^45 + 40 * q^51 + 20 * q^63 + 4 * q^67 + 20 * q^77 + 64 * q^81 - 40 * q^87 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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