LMFDB - Embedded newform 1148.2.d.a.1065.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1065,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1148.1065");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.d (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:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + \cdots + 144 $ x^20 + 39x^18 + 531x^16 + 3488x^14 + 12661x^12 + 27027x^10 + 34540x^8 + 25909x^6 + 10677x^4 + 2092*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1065.17
Root			$1.40239i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +O(q^{10})$	$q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +0.655586i q^{11} -1.32178i q^{13} +9.55950i q^{15} +1.55031i q^{17} +3.21822i q^{19} +2.39349 q^{21} -1.09317 q^{23} +10.9517 q^{25} +0.649079i q^{27} +4.14410i q^{29} -3.62085 q^{31} -1.56914 q^{33} -3.99395i q^{35} +6.76585 q^{37} +3.16367 q^{39} +(6.37033 + 0.647203i) q^{41} +6.56003 q^{43} -10.8988 q^{45} +2.13764i q^{47} -1.00000 q^{49} -3.71066 q^{51} -11.6378i q^{53} +2.61838i q^{55} -7.70279 q^{57} -8.90983 q^{59} -4.81119 q^{61} +2.72882i q^{63} -5.27912i q^{65} +0.603316i q^{67} -2.61650i q^{69} -5.43199i q^{71} -10.0698 q^{73} +26.2127i q^{75} +0.655586 q^{77} +4.77184i q^{79} -9.74001 q^{81} +4.09012 q^{83} +6.19186i q^{85} -9.91888 q^{87} +2.88671i q^{89} -1.32178 q^{91} -8.66648i q^{93} +12.8534i q^{95} +15.8789i q^{97} -1.78897i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) $ 20 * q + 4 * q^5 - 20 * q^9	$ 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) $ 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$.

$n$	$493$	$575$	$785$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.39349i			1.38188i	0.722910	+	0.690942i	$0.242804\pi$
$3$			2.39349i			1.38188i	−0.722910	+	0.690942i	$0.757196\pi$
$4$		0			0
$5$	3.99395			1.78615			0.893075	−	0.449908i	$-0.148543\pi$
$5$	3.99395			1.78615			0.893075	+	0.449908i	$0.148543\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.72882			−0.909605
$10$		0			0
$11$			0.655586i			0.197667i	0.995104	+	0.0988333i	$0.0315111\pi$
$11$			0.655586i			0.197667i	−0.995104	+	0.0988333i	$0.968489\pi$
$12$		0			0
$13$		−	1.32178i		−	0.366596i	−0.983057	−	0.183298i	$-0.941323\pi$
$13$		−	1.32178i		−	0.366596i	0.983057	−	0.183298i	$-0.0586773\pi$
$14$		0			0
$15$			9.55950i			2.46825i
$16$		0			0
$17$			1.55031i			0.376005i	0.982169	+	0.188003i	$0.0602013\pi$
$17$			1.55031i			0.376005i	−0.982169	+	0.188003i	$0.939799\pi$
$18$		0			0
$19$			3.21822i			0.738311i	0.929368	+	0.369155i	$0.120353\pi$
$19$			3.21822i			0.738311i	−0.929368	+	0.369155i	$0.879647\pi$
$20$		0			0
$21$	2.39349			0.522303
$22$		0			0
$23$	−1.09317			−0.227942			−0.113971	−	0.993484i	$-0.536357\pi$
$23$	−1.09317			−0.227942			−0.113971	+	0.993484i	$0.536357\pi$
$24$		0			0
$25$	10.9517			2.19033
$26$		0			0
$27$			0.649079i			0.124915i
$28$		0			0
$29$			4.14410i			0.769540i	0.923012	+	0.384770i	$0.125719\pi$
$29$			4.14410i			0.769540i	−0.923012	+	0.384770i	$0.874281\pi$
$30$		0			0
$31$	−3.62085			−0.650323			−0.325162	−	0.945658i	$-0.605419\pi$
$31$	−3.62085			−0.650323			−0.325162	+	0.945658i	$0.605419\pi$
$32$		0			0
$33$	−1.56914			−0.273153
$34$		0			0
$35$		−	3.99395i		−	0.675101i
$36$		0			0
$37$	6.76585			1.11230			0.556149	−	0.831082i	$-0.312278\pi$
$37$	6.76585			1.11230			0.556149	+	0.831082i	$0.312278\pi$
$38$		0			0
$39$	3.16367			0.506593
$40$		0			0
$41$	6.37033	+	0.647203i	0.994879	+	0.101076i
$41$	6.37033	+	0.647203i	0.994879	+	0.101076i
$42$		0			0
$43$	6.56003			1.00040			0.500198	−	0.865911i	$-0.333261\pi$
$43$	6.56003			1.00040			0.500198	+	0.865911i	$0.333261\pi$
$44$		0			0
$45$	−10.8988			−1.62469
$46$		0			0
$47$			2.13764i			0.311806i	0.987772	+	0.155903i	$0.0498288\pi$
$47$			2.13764i			0.311806i	−0.987772	+	0.155903i	$0.950171\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.71066			−0.519596
$52$		0			0
$53$		−	11.6378i		−	1.59857i	−0.600950	−	0.799287i	$-0.705211\pi$
$53$		−	11.6378i		−	1.59857i	0.600950	−	0.799287i	$-0.294789\pi$
$54$		0			0
$55$			2.61838i			0.353062i
$56$		0			0
$57$	−7.70279			−1.02026
$58$		0			0
$59$	−8.90983			−1.15996			−0.579980	−	0.814630i	$-0.696940\pi$
$59$	−8.90983			−1.15996			−0.579980	+	0.814630i	$0.696940\pi$
$60$		0			0
$61$	−4.81119			−0.616010			−0.308005	−	0.951385i	$-0.599661\pi$
$61$	−4.81119			−0.616010			−0.308005	+	0.951385i	$0.599661\pi$
$62$		0			0
$63$			2.72882i			0.343798i
$64$		0			0
$65$		−	5.27912i		−	0.654795i
$66$		0			0
$67$			0.603316i			0.0737068i	0.999321	+	0.0368534i	$0.0117335\pi$
$67$			0.603316i			0.0737068i	−0.999321	+	0.0368534i	$0.988267\pi$
$68$		0			0
$69$		−	2.61650i		−	0.314990i
$70$		0			0
$71$		−	5.43199i		−	0.644658i	−0.946628	−	0.322329i	$-0.895534\pi$
$71$		−	5.43199i		−	0.644658i	0.946628	−	0.322329i	$-0.104466\pi$
$72$		0			0
$73$	−10.0698			−1.17858			−0.589292	−	0.807920i	$-0.700593\pi$
$73$	−10.0698			−1.17858			−0.589292	+	0.807920i	$0.700593\pi$
$74$		0			0
$75$			26.2127i			3.02678i
$76$		0			0
$77$	0.655586			0.0747110
$78$		0			0
$79$			4.77184i			0.536874i	0.963297	+	0.268437i	$0.0865071\pi$
$79$			4.77184i			0.536874i	−0.963297	+	0.268437i	$0.913493\pi$
$80$		0			0
$81$	−9.74001			−1.08222
$82$		0			0
$83$	4.09012			0.448949			0.224474	−	0.974480i	$-0.427934\pi$
$83$	4.09012			0.448949			0.224474	+	0.974480i	$0.427934\pi$
$84$		0			0
$85$			6.19186i			0.671601i
$86$		0			0
$87$	−9.91888			−1.06342
$88$		0			0
$89$			2.88671i			0.305990i	0.988227	+	0.152995i	$0.0488919\pi$
$89$			2.88671i			0.305990i	−0.988227	+	0.152995i	$0.951108\pi$
$90$		0			0
$91$	−1.32178			−0.138560
$92$		0			0
$93$		−	8.66648i		−	0.898672i
$94$		0			0
$95$			12.8534i			1.31873i
$96$		0			0
$97$			15.8789i			1.61226i	0.591739	+	0.806130i	$0.298442\pi$
$97$			15.8789i			1.61226i	−0.591739	+	0.806130i	$0.701558\pi$
$98$		0			0
$99$		−	1.78897i		−	0.179799i
$100$		0			0
$101$		−	17.8206i		−	1.77321i	−0.462525	−	0.886606i	$-0.653056\pi$
$101$		−	17.8206i		−	1.77321i	0.462525	−	0.886606i	$-0.346944\pi$
$102$		0			0
$103$	13.7897			1.35874			0.679368	−	0.733797i	$-0.262254\pi$
$103$	13.7897			1.35874			0.679368	+	0.733797i	$0.262254\pi$
$104$		0			0
$105$	9.55950			0.932912
$106$		0			0
$107$	−12.8832			−1.24546			−0.622731	−	0.782436i	$-0.713977\pi$
$107$	−12.8832			−1.24546			−0.622731	+	0.782436i	$0.713977\pi$
$108$		0			0
$109$		−	16.1047i		−	1.54255i	−0.636502	−	0.771275i	$-0.719619\pi$
$109$		−	16.1047i		−	1.54255i	0.636502	−	0.771275i	$-0.280381\pi$
$110$		0			0
$111$			16.1940i			1.53707i
$112$		0			0
$113$	0.712810			0.0670556			0.0335278	−	0.999438i	$-0.489326\pi$
$113$	0.712810			0.0670556			0.0335278	+	0.999438i	$0.489326\pi$
$114$		0			0
$115$	−4.36608			−0.407139
$116$		0			0
$117$			3.60689i			0.333457i
$118$		0			0
$119$	1.55031			0.142117
$120$		0			0
$121$	10.5702			0.960928
$122$		0			0
$123$	−1.54908	+	15.2474i	−0.139676	+	1.37481i
$124$		0			0
$125$	23.7706			2.12611
$126$		0			0
$127$	−7.30114			−0.647871			−0.323936	−	0.946079i	$-0.605006\pi$
$127$	−7.30114			−0.647871			−0.323936	+	0.946079i	$0.605006\pi$
$128$		0			0
$129$			15.7014i			1.38243i
$130$		0			0
$131$	−5.57734			−0.487294			−0.243647	−	0.969864i	$-0.578344\pi$
$131$	−5.57734			−0.487294			−0.243647	+	0.969864i	$0.578344\pi$
$132$		0			0
$133$	3.21822			0.279055
$134$		0			0
$135$			2.59239i			0.223117i
$136$		0			0
$137$		−	3.00928i		−	0.257100i	−0.991703	−	0.128550i	$-0.958968\pi$
$137$		−	3.00928i		−	0.257100i	0.991703	−	0.128550i	$-0.0410323\pi$
$138$		0			0
$139$	−4.75504			−0.403317			−0.201659	−	0.979456i	$-0.564633\pi$
$139$	−4.75504			−0.403317			−0.201659	+	0.979456i	$0.564633\pi$
$140$		0			0
$141$	−5.11642			−0.430880
$142$		0			0
$143$	0.866540			0.0724637
$144$		0			0
$145$			16.5513i			1.37451i
$146$		0			0
$147$		−	2.39349i		−	0.197412i
$148$		0			0
$149$			15.3223i			1.25525i	0.778517	+	0.627624i	$0.215972\pi$
$149$			15.3223i			1.25525i	−0.778517	+	0.627624i	$0.784028\pi$
$150$		0			0
$151$		−	17.9639i		−	1.46188i	−0.682442	−	0.730940i	$-0.739082\pi$
$151$		−	17.9639i		−	1.46188i	0.682442	−	0.730940i	$-0.260918\pi$
$152$		0			0
$153$		−	4.23051i		−	0.342016i
$154$		0			0
$155$	−14.4615			−1.16157
$156$		0			0
$157$		−	11.6104i		−	0.926608i	−0.886199	−	0.463304i	$-0.846664\pi$
$157$		−	11.6104i		−	0.926608i	0.886199	−	0.463304i	$-0.153336\pi$
$158$		0			0
$159$	27.8550			2.20904
$160$		0			0
$161$			1.09317i			0.0861541i
$162$		0			0
$163$	−15.1493			−1.18659			−0.593293	−	0.804987i	$-0.702172\pi$
$163$	−15.1493			−1.18659			−0.593293	+	0.804987i	$0.702172\pi$
$164$		0			0
$165$	−6.26708			−0.487891
$166$		0			0
$167$		−	4.26888i		−	0.330336i	−0.986265	−	0.165168i	$-0.947183\pi$
$167$		−	4.26888i		−	0.330336i	0.986265	−	0.165168i	$-0.0528166\pi$
$168$		0			0
$169$	11.2529			0.865608
$170$		0			0
$171$		−	8.78193i		−	0.671571i
$172$		0			0
$173$	−11.8219			−0.898800			−0.449400	−	0.893331i	$-0.648362\pi$
$173$	−11.8219			−0.898800			−0.449400	+	0.893331i	$0.648362\pi$
$174$		0			0
$175$		−	10.9517i		−	0.827867i
$176$		0			0
$177$		−	21.3256i		−	1.60293i
$178$		0			0
$179$		−	9.10793i		−	0.680758i	−0.940288	−	0.340379i	$-0.889445\pi$
$179$		−	9.10793i		−	0.680758i	0.940288	−	0.340379i	$-0.110555\pi$
$180$		0			0
$181$		−	17.0621i		−	1.26821i	−0.773245	−	0.634107i	$-0.781368\pi$
$181$		−	17.0621i		−	1.26821i	0.773245	−	0.634107i	$-0.218632\pi$
$182$		0			0
$183$		−	11.5156i		−	0.851255i
$184$		0			0
$185$	27.0225			1.98673
$186$		0			0
$187$	−1.01636			−0.0743237
$188$		0			0
$189$	0.649079			0.0472136
$190$		0			0
$191$		−	18.8163i		−	1.36150i	−0.732516	−	0.680750i	$-0.761654\pi$
$191$		−	18.8163i		−	1.36150i	0.732516	−	0.680750i	$-0.238346\pi$
$192$		0			0
$193$		−	19.1003i		−	1.37487i	−0.726245	−	0.687436i	$-0.758736\pi$
$193$		−	19.1003i		−	1.37487i	0.726245	−	0.687436i	$-0.241264\pi$
$194$		0			0
$195$	12.6356			0.904851
$196$		0			0
$197$	−7.18645			−0.512014			−0.256007	−	0.966675i	$-0.582407\pi$
$197$	−7.18645			−0.512014			−0.256007	+	0.966675i	$0.582407\pi$
$198$		0			0
$199$		−	6.49015i		−	0.460074i	−0.973182	−	0.230037i	$-0.926115\pi$
$199$		−	6.49015i		−	0.460074i	0.973182	−	0.230037i	$-0.0738848\pi$
$200$		0			0
$201$	−1.44403			−0.101854
$202$		0			0
$203$	4.14410			0.290859
$204$		0			0
$205$	25.4428	+	2.58490i	1.77700	+	0.180537i
$206$		0			0
$207$	2.98307			0.207338
$208$		0			0
$209$	−2.10982			−0.145939
$210$		0			0
$211$			2.84127i			0.195601i	0.995206	+	0.0978004i	$0.0311807\pi$
$211$			2.84127i			0.195601i	−0.995206	+	0.0978004i	$0.968819\pi$
$212$		0			0
$213$	13.0014			0.890844
$214$		0			0
$215$	26.2004			1.78686
$216$		0			0
$217$			3.62085i			0.245799i
$218$		0			0
$219$		−	24.1021i		−	1.62867i
$220$		0			0
$221$	2.04917			0.137842
$222$		0			0
$223$	2.15843			0.144539			0.0722695	−	0.997385i	$-0.476976\pi$
$223$	2.15843			0.144539			0.0722695	+	0.997385i	$0.476976\pi$
$224$		0			0
$225$	−29.8850			−1.99234
$226$		0			0
$227$			4.85412i			0.322179i	0.986940	+	0.161090i	$0.0515008\pi$
$227$			4.85412i			0.322179i	−0.986940	+	0.161090i	$0.948499\pi$
$228$		0			0
$229$		−	17.9050i		−	1.18319i	−0.806233	−	0.591597i	$-0.798497\pi$
$229$		−	17.9050i		−	1.18319i	0.806233	−	0.591597i	$-0.201503\pi$
$230$		0			0
$231$			1.56914i			0.103242i
$232$		0			0
$233$		−	1.77544i		−	0.116313i	−0.998307	−	0.0581563i	$-0.981478\pi$
$233$		−	1.77544i		−	0.116313i	0.998307	−	0.0581563i	$-0.0185222\pi$
$234$		0			0
$235$			8.53761i			0.556932i
$236$		0			0
$237$	−11.4214			−0.741898
$238$		0			0
$239$			12.9001i			0.834438i	0.908806	+	0.417219i	$0.136995\pi$
$239$			12.9001i			0.834438i	−0.908806	+	0.417219i	$0.863005\pi$
$240$		0			0
$241$	−23.1109			−1.48870			−0.744352	−	0.667787i	$-0.767242\pi$
$241$	−23.1109			−1.48870			−0.744352	+	0.667787i	$0.767242\pi$
$242$		0			0
$243$		−	21.3654i		−	1.37059i
$244$		0			0
$245$	−3.99395			−0.255164
$246$		0			0
$247$	4.25378			0.270661
$248$		0			0
$249$			9.78967i			0.620395i
$250$		0			0
$251$	21.5199			1.35832			0.679162	−	0.733988i	$-0.262343\pi$
$251$	21.5199			1.35832			0.679162	+	0.733988i	$0.262343\pi$
$252$		0			0
$253$		−	0.716669i		−	0.0450566i
$254$		0			0
$255$	−14.8202			−0.928076
$256$		0			0
$257$			3.90586i			0.243641i	0.992552	+	0.121821i	$0.0388733\pi$
$257$			3.90586i			0.243641i	−0.992552	+	0.121821i	$0.961127\pi$
$258$		0			0
$259$		−	6.76585i		−	0.420409i
$260$		0			0
$261$		−	11.3085i		−	0.699978i
$262$		0			0
$263$		−	8.92846i		−	0.550553i	−0.961365	−	0.275276i	$-0.911231\pi$
$263$		−	8.92846i		−	0.550553i	0.961365	−	0.275276i	$-0.0887693\pi$
$264$		0			0
$265$		−	46.4808i		−	2.85529i
$266$		0			0
$267$	−6.90931			−0.422843
$268$		0			0
$269$	4.60411			0.280718			0.140359	−	0.990101i	$-0.455174\pi$
$269$	4.60411			0.280718			0.140359	+	0.990101i	$0.455174\pi$
$270$		0			0
$271$	21.6257			1.31367			0.656834	−	0.754035i	$-0.271895\pi$
$271$	21.6257			1.31367			0.656834	+	0.754035i	$0.271895\pi$
$272$		0			0
$273$		−	3.16367i		−	0.191474i
$274$		0			0
$275$			7.17975i			0.432955i
$276$		0			0
$277$	−24.6943			−1.48374			−0.741868	−	0.670546i	$-0.766060\pi$
$277$	−24.6943			−1.48374			−0.741868	+	0.670546i	$0.766060\pi$
$278$		0			0
$279$	9.88062			0.591537
$280$		0			0
$281$			5.86304i			0.349760i	0.984590	+	0.174880i	$0.0559537\pi$
$281$			5.86304i			0.349760i	−0.984590	+	0.174880i	$0.944046\pi$
$282$		0			0
$283$	20.0268			1.19047			0.595235	−	0.803552i	$-0.297059\pi$
$283$	20.0268			1.19047			0.595235	+	0.803552i	$0.297059\pi$
$284$		0			0
$285$	−30.7646			−1.82234
$286$		0			0
$287$	0.647203	−	6.37033i	0.0382032	−	0.376029i
$288$		0			0
$289$	14.5965			0.858620
$290$		0			0
$291$	−38.0061			−2.22796
$292$		0			0
$293$			25.5700i			1.49382i	0.664927	+	0.746908i	$0.268463\pi$
$293$			25.5700i			1.49382i	−0.664927	+	0.746908i	$0.731537\pi$
$294$		0			0
$295$	−35.5854			−2.07186
$296$		0			0
$297$	−0.425527			−0.0246916
$298$		0			0
$299$			1.44493i			0.0835627i
$300$		0			0
$301$		−	6.56003i		−	0.378114i
$302$		0			0
$303$	42.6534			2.45038
$304$		0			0
$305$	−19.2157			−1.10029
$306$		0			0
$307$	−21.7097			−1.23904			−0.619519	−	0.784982i	$-0.712672\pi$
$307$	−21.7097			−1.23904			−0.619519	+	0.784982i	$0.712672\pi$
$308$		0			0
$309$			33.0055i			1.87762i
$310$		0			0
$311$			23.8936i			1.35488i	0.735577	+	0.677441i	$0.236911\pi$
$311$			23.8936i			1.35488i	−0.735577	+	0.677441i	$0.763089\pi$
$312$		0			0
$313$			9.27380i			0.524186i	0.965043	+	0.262093i	$0.0844127\pi$
$313$			9.27380i			0.524186i	−0.965043	+	0.262093i	$0.915587\pi$
$314$		0			0
$315$			10.8988i			0.614075i
$316$		0			0
$317$			32.8603i			1.84562i	0.385255	+	0.922810i	$0.374114\pi$
$317$			32.8603i			1.84562i	−0.385255	+	0.922810i	$0.625886\pi$
$318$		0			0
$319$	−2.71682			−0.152112
$320$		0			0
$321$		−	30.8358i		−	1.72108i
$322$		0			0
$323$	−4.98924			−0.277609
$324$		0			0
$325$		−	14.4757i		−	0.802966i
$326$		0			0
$327$	38.5465			2.13163
$328$		0			0
$329$	2.13764			0.117852
$330$		0			0
$331$			5.23593i			0.287793i	0.989593	+	0.143897i	$0.0459632\pi$
$331$			5.23593i			0.287793i	−0.989593	+	0.143897i	$0.954037\pi$
$332$		0			0
$333$	−18.4627			−1.01175
$334$		0			0
$335$			2.40962i			0.131651i
$336$		0			0
$337$	−15.6868			−0.854513			−0.427257	−	0.904130i	$-0.640520\pi$
$337$	−15.6868			−0.854513			−0.427257	+	0.904130i	$0.640520\pi$
$338$		0			0
$339$			1.70611i			0.0926631i
$340$		0			0
$341$		−	2.37378i		−	0.128547i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	10.4502i		−	0.562619i
$346$		0			0
$347$		−	11.2286i		−	0.602782i	−0.953501	−	0.301391i	$-0.902549\pi$
$347$		−	11.2286i		−	0.602782i	0.953501	−	0.301391i	$-0.0974510\pi$
$348$		0			0
$349$	15.3491			0.821617			0.410809	−	0.911722i	$-0.365246\pi$
$349$	15.3491			0.821617			0.410809	+	0.911722i	$0.365246\pi$
$350$		0			0
$351$	0.857940			0.0457934
$352$		0			0
$353$	2.16013			0.114972			0.0574861	−	0.998346i	$-0.481692\pi$
$353$	2.16013			0.114972			0.0574861	+	0.998346i	$0.481692\pi$
$354$		0			0
$355$		−	21.6951i		−	1.15146i
$356$		0			0
$357$			3.71066i			0.196389i
$358$		0			0
$359$	21.8268			1.15198			0.575988	−	0.817458i	$-0.304617\pi$
$359$	21.8268			1.15198			0.575988	+	0.817458i	$0.304617\pi$
$360$		0			0
$361$	8.64305			0.454898
$362$		0			0
$363$			25.2997i			1.32789i
$364$		0			0
$365$	−40.2184			−2.10513
$366$		0			0
$367$	5.62271			0.293503			0.146751	−	0.989173i	$-0.453118\pi$
$367$	5.62271			0.293503			0.146751	+	0.989173i	$0.453118\pi$
$368$		0			0
$369$	−17.3835	−	1.76610i	−0.904947	−	0.0919394i
$370$		0			0
$371$	−11.6378			−0.604204
$372$		0			0
$373$	−19.9786			−1.03445			−0.517227	−	0.855848i	$-0.673036\pi$
$373$	−19.9786			−1.03445			−0.517227	+	0.855848i	$0.673036\pi$
$374$		0			0
$375$			56.8948i			2.93804i
$376$		0			0
$377$	5.47759			0.282110
$378$		0			0
$379$	5.35101			0.274863			0.137432	−	0.990511i	$-0.456115\pi$
$379$	5.35101			0.274863			0.137432	+	0.990511i	$0.456115\pi$
$380$		0			0
$381$		−	17.4752i		−	0.895283i
$382$		0			0
$383$		−	35.8618i		−	1.83245i	−0.400663	−	0.916226i	$-0.631220\pi$
$383$		−	35.8618i		−	1.83245i	0.400663	−	0.916226i	$-0.368780\pi$
$384$		0			0
$385$	2.61838			0.133445
$386$		0			0
$387$	−17.9011			−0.909964
$388$		0			0
$389$	34.7920			1.76402			0.882011	−	0.471228i	$-0.156189\pi$
$389$	34.7920			1.76402			0.882011	+	0.471228i	$0.156189\pi$
$390$		0			0
$391$		−	1.69476i		−	0.0857075i
$392$		0			0
$393$		−	13.3493i		−	0.673384i
$394$		0			0
$395$			19.0585i			0.958937i
$396$		0			0
$397$		−	7.53806i		−	0.378324i	−0.981946	−	0.189162i	$-0.939423\pi$
$397$		−	7.53806i		−	0.378324i	0.981946	−	0.189162i	$-0.0605772\pi$
$398$		0			0
$399$			7.70279i			0.385622i
$400$		0			0
$401$	6.84450			0.341798			0.170899	−	0.985289i	$-0.445333\pi$
$401$	6.84450			0.341798			0.170899	+	0.985289i	$0.445333\pi$
$402$		0			0
$403$			4.78596i			0.238406i
$404$		0			0
$405$	−38.9011			−1.93301
$406$		0			0
$407$			4.43560i			0.219864i
$408$		0			0
$409$	17.2857			0.854725			0.427363	−	0.904080i	$-0.359443\pi$
$409$	17.2857			0.854725			0.427363	+	0.904080i	$0.359443\pi$
$410$		0			0
$411$	7.20269			0.355282
$412$		0			0
$413$			8.90983i			0.438424i
$414$		0			0
$415$	16.3357			0.801889
$416$		0			0
$417$		−	11.3812i		−	0.557338i
$418$		0			0
$419$	27.5236			1.34462			0.672308	−	0.740271i	$-0.265303\pi$
$419$	27.5236			1.34462			0.672308	+	0.740271i	$0.265303\pi$
$420$		0			0
$421$		−	20.0785i		−	0.978564i	−0.872126	−	0.489282i	$-0.837259\pi$
$421$		−	20.0785i		−	0.978564i	0.872126	−	0.489282i	$-0.162741\pi$
$422$		0			0
$423$		−	5.83321i		−	0.283620i
$424$		0			0
$425$			16.9784i			0.823575i
$426$		0			0
$427$			4.81119i			0.232830i
$428$		0			0
$429$			2.07406i			0.100137i
$430$		0			0
$431$	−4.46647			−0.215142			−0.107571	−	0.994197i	$-0.534307\pi$
$431$	−4.46647			−0.215142			−0.107571	+	0.994197i	$0.534307\pi$
$432$		0			0
$433$	−0.143515			−0.00689691			−0.00344845	−	0.999994i	$-0.501098\pi$
$433$	−0.143515			−0.00689691			−0.00344845	+	0.999994i	$0.501098\pi$
$434$		0			0
$435$	−39.6155			−1.89942
$436$		0			0
$437$		−	3.51807i		−	0.168292i
$438$		0			0
$439$		−	14.6831i		−	0.700787i	−0.936603	−	0.350393i	$-0.886048\pi$
$439$		−	14.6831i		−	0.700787i	0.936603	−	0.350393i	$-0.113952\pi$
$440$		0			0
$441$	2.72882			0.129944
$442$		0			0
$443$	−29.0704			−1.38117			−0.690587	−	0.723249i	$-0.742648\pi$
$443$	−29.0704			−1.38117			−0.690587	+	0.723249i	$0.742648\pi$
$444$		0			0
$445$			11.5294i			0.546544i
$446$		0			0
$447$	−36.6737			−1.73461
$448$		0			0
$449$	−27.1979			−1.28355			−0.641774	−	0.766894i	$-0.721801\pi$
$449$	−27.1979			−1.28355			−0.641774	+	0.766894i	$0.721801\pi$
$450$		0			0
$451$	−0.424298	+	4.17630i	−0.0199794	+	0.196654i
$452$		0			0
$453$	42.9965			2.02015
$454$		0			0
$455$	−5.27912			−0.247489
$456$		0			0
$457$		−	1.02827i		−	0.0481005i	−0.999711	−	0.0240503i	$-0.992344\pi$
$457$		−	1.02827i		−	0.0481005i	0.999711	−	0.0240503i	$-0.00765617\pi$
$458$		0			0
$459$	−1.00627			−0.0469688
$460$		0			0
$461$	−0.925752			−0.0431166			−0.0215583	−	0.999768i	$-0.506863\pi$
$461$	−0.925752			−0.0431166			−0.0215583	+	0.999768i	$0.506863\pi$
$462$		0			0
$463$		−	1.17805i		−	0.0547485i	−0.999625	−	0.0273743i	$-0.991285\pi$
$463$		−	1.17805i		−	0.0547485i	0.999625	−	0.0273743i	$-0.00871458\pi$
$464$		0			0
$465$		−	34.6135i		−	1.60516i
$466$		0			0
$467$	0.0671690			0.00310821			0.00155410	−	0.999999i	$-0.499505\pi$
$467$	0.0671690			0.00310821			0.00155410	+	0.999999i	$0.499505\pi$
$468$		0			0
$469$	0.603316			0.0278586
$470$		0			0
$471$	27.7893			1.28047
$472$		0			0
$473$			4.30066i			0.197745i
$474$		0			0
$475$			35.2448i			1.61714i
$476$		0			0
$477$			31.7574i			1.45407i
$478$		0			0
$479$			13.0763i			0.597470i	0.954336	+	0.298735i	$0.0965646\pi$
$479$			13.0763i			0.597470i	−0.954336	+	0.298735i	$0.903435\pi$
$480$		0			0
$481$		−	8.94296i		−	0.407764i
$482$		0			0
$483$	−2.61650			−0.119055
$484$		0			0
$485$			63.4196i			2.87974i
$486$		0			0
$487$	−26.5480			−1.20301			−0.601503	−	0.798870i	$-0.705431\pi$
$487$	−26.5480			−1.20301			−0.601503	+	0.798870i	$0.705431\pi$
$488$		0			0
$489$		−	36.2598i		−	1.63973i
$490$		0			0
$491$	−12.5104			−0.564586			−0.282293	−	0.959328i	$-0.591095\pi$
$491$	−12.5104			−0.564586			−0.282293	+	0.959328i	$0.591095\pi$
$492$		0			0
$493$	−6.42464			−0.289351
$494$		0			0
$495$		−	7.14507i		−	0.321147i
$496$		0			0
$497$	−5.43199			−0.243658
$498$		0			0
$499$		−	25.9760i		−	1.16284i	−0.813602	−	0.581422i	$-0.802496\pi$
$499$		−	25.9760i		−	1.16284i	0.813602	−	0.581422i	$-0.197504\pi$
$500$		0			0
$501$	10.2175			0.456486
$502$		0			0
$503$			42.3299i			1.88740i	0.330806	+	0.943699i	$0.392679\pi$
$503$			42.3299i			1.88740i	−0.330806	+	0.943699i	$0.607321\pi$
$504$		0			0
$505$		−	71.1745i		−	3.16722i
$506$		0			0
$507$			26.9337i			1.19617i
$508$		0			0
$509$			18.1242i			0.803342i	0.915784	+	0.401671i	$0.131571\pi$
$509$			18.1242i			0.803342i	−0.915784	+	0.401671i	$0.868429\pi$
$510$		0			0
$511$			10.0698i			0.445463i
$512$		0			0
$513$	−2.08888			−0.0922263
$514$		0			0
$515$	55.0753			2.42691
$516$		0			0
$517$	−1.40140			−0.0616337
$518$		0			0
$519$		−	28.2956i		−	1.24204i
$520$		0			0
$521$			17.2055i			0.753786i	0.926257	+	0.376893i	$0.123008\pi$
$521$			17.2055i			0.753786i	−0.926257	+	0.376893i	$0.876992\pi$
$522$		0			0
$523$	41.5379			1.81633			0.908163	−	0.418616i	$-0.137485\pi$
$523$	41.5379			1.81633			0.908163	+	0.418616i	$0.137485\pi$
$524$		0			0
$525$	26.2127			1.14402
$526$		0			0
$527$		−	5.61343i		−	0.244525i
$528$		0			0
$529$	−21.8050			−0.948042
$530$		0			0
$531$	24.3133			1.05511
$532$		0			0
$533$	0.855460	−	8.42017i	0.0370541	−	0.364718i
$534$		0			0
$535$	−51.4547			−2.22458
$536$		0			0
$537$	21.7998			0.940730
$538$		0			0
$539$		−	0.655586i		−	0.0282381i
$540$		0			0
$541$	34.2684			1.47332			0.736658	−	0.676266i	$-0.236403\pi$
$541$	34.2684			1.47332			0.736658	+	0.676266i	$0.236403\pi$
$542$		0			0
$543$	40.8380			1.75252
$544$		0			0
$545$		−	64.3214i		−	2.75523i
$546$		0			0
$547$		−	28.0006i		−	1.19722i	−0.801041	−	0.598610i	$-0.795720\pi$
$547$		−	28.0006i		−	1.19722i	0.801041	−	0.598610i	$-0.204280\pi$
$548$		0			0
$549$	13.1289			0.560326
$550$		0			0
$551$	−13.3366			−0.568160
$552$		0			0
$553$	4.77184			0.202919
$554$		0			0
$555$			64.6781i			2.74543i
$556$		0			0
$557$			37.4618i			1.58731i	0.608370	+	0.793654i	$0.291824\pi$
$557$			37.4618i			1.58731i	−0.608370	+	0.793654i	$0.708176\pi$
$558$		0			0
$559$		−	8.67091i		−	0.366741i
$560$		0			0
$561$		−	2.43265i		−	0.102707i
$562$		0			0
$563$			1.87581i			0.0790559i	0.999218	+	0.0395279i	$0.0125854\pi$
$563$			1.87581i			0.0790559i	−0.999218	+	0.0395279i	$0.987415\pi$
$564$		0			0
$565$	2.84693			0.119771
$566$		0			0
$567$			9.74001i			0.409042i
$568$		0			0
$569$	−40.1225			−1.68202			−0.841012	−	0.541017i	$-0.818040\pi$
$569$	−40.1225			−1.68202			−0.841012	+	0.541017i	$0.818040\pi$
$570$		0			0
$571$		−	20.3308i		−	0.850817i	−0.905001	−	0.425409i	$-0.860130\pi$
$571$		−	20.3308i		−	0.850817i	0.905001	−	0.425409i	$-0.139870\pi$
$572$		0			0
$573$	45.0367			1.88144
$574$		0			0
$575$	−11.9720			−0.499269
$576$		0			0
$577$			34.0647i			1.41813i	0.705143	+	0.709065i	$0.250883\pi$
$577$			34.0647i			1.41813i	−0.705143	+	0.709065i	$0.749117\pi$
$578$		0			0
$579$	45.7165			1.89991
$580$		0			0
$581$		−	4.09012i		−	0.169687i
$582$		0			0
$583$	7.62957			0.315985
$584$		0			0
$585$			14.4058i			0.595605i
$586$		0			0
$587$			44.2327i			1.82568i	0.408317	+	0.912840i	$0.366116\pi$
$587$			44.2327i			1.82568i	−0.408317	+	0.912840i	$0.633884\pi$
$588$		0			0
$589$		−	11.6527i		−	0.480141i
$590$		0			0
$591$		−	17.2007i		−	0.707544i
$592$		0			0
$593$		−	28.1565i		−	1.15625i	−0.815949	−	0.578124i	$-0.803785\pi$
$593$		−	28.1565i		−	1.15625i	0.815949	−	0.578124i	$-0.196215\pi$
$594$		0			0
$595$	6.19186			0.253841
$596$		0			0
$597$	15.5341			0.635770
$598$		0			0
$599$	21.4765			0.877505			0.438753	−	0.898608i	$-0.355420\pi$
$599$	21.4765			0.877505			0.438753	+	0.898608i	$0.355420\pi$
$600$		0			0
$601$			21.6559i			0.883363i	0.897172	+	0.441681i	$0.145618\pi$
$601$			21.6559i			0.883363i	−0.897172	+	0.441681i	$0.854382\pi$
$602$		0			0
$603$		−	1.64634i		−	0.0670441i
$604$		0			0
$605$	42.2169			1.71636
$606$		0			0
$607$	−42.1733			−1.71176			−0.855880	−	0.517174i	$-0.826984\pi$
$607$	−42.1733			−1.71176			−0.855880	+	0.517174i	$0.826984\pi$
$608$		0			0
$609$			9.91888i			0.401933i
$610$		0			0
$611$	2.82548			0.114307
$612$		0			0
$613$	−35.3343			−1.42714			−0.713570	−	0.700584i	$-0.752923\pi$
$613$	−35.3343			−1.42714			−0.713570	+	0.700584i	$0.752923\pi$
$614$		0			0
$615$	−6.18694	+	60.8972i	−0.249482	+	2.45561i
$616$		0			0
$617$	2.82448			0.113709			0.0568546	−	0.998382i	$-0.481893\pi$
$617$	2.82448			0.113709			0.0568546	+	0.998382i	$0.481893\pi$
$618$		0			0
$619$	12.2616			0.492834			0.246417	−	0.969164i	$-0.420747\pi$
$619$	12.2616			0.492834			0.246417	+	0.969164i	$0.420747\pi$
$620$		0			0
$621$		−	0.709556i		−	0.0284735i
$622$		0			0
$623$	2.88671			0.115653
$624$		0			0
$625$	40.1804			1.60722
$626$		0			0
$627$		−	5.04984i		−	0.201671i
$628$		0			0
$629$			10.4892i			0.418230i
$630$		0			0
$631$	14.9954			0.596956			0.298478	−	0.954416i	$-0.403521\pi$
$631$	14.9954			0.596956			0.298478	+	0.954416i	$0.403521\pi$
$632$		0			0
$633$	−6.80055			−0.270298
$634$		0			0
$635$	−29.1604			−1.15719
$636$		0			0
$637$			1.32178i			0.0523708i
$638$		0			0
$639$			14.8229i			0.586385i
$640$		0			0
$641$			0.00935095i			0.000369340i	1.00000		0.000184670i	$5.87824e-5\pi$
$641$			0.00935095i			0.000369340i	−1.00000		0.000184670i	$0.999941\pi$
$642$		0			0
$643$		−	38.9540i		−	1.53619i	−0.640333	−	0.768097i	$-0.721204\pi$
$643$		−	38.9540i		−	1.53619i	0.640333	−	0.768097i	$-0.278796\pi$
$644$		0			0
$645$			62.7106i			2.46923i
$646$		0			0
$647$	−37.9934			−1.49368			−0.746838	−	0.665006i	$-0.768429\pi$
$647$	−37.9934			−1.49368			−0.746838	+	0.665006i	$0.768429\pi$
$648$		0			0
$649$		−	5.84116i		−	0.229286i
$650$		0			0
$651$	−8.66648			−0.339666
$652$		0			0
$653$			28.5842i			1.11859i	0.828970	+	0.559293i	$0.188927\pi$
$653$			28.5842i			1.11859i	−0.828970	+	0.559293i	$0.811073\pi$
$654$		0			0
$655$	−22.2756			−0.870380
$656$		0			0
$657$	27.4787			1.07205
$658$		0			0
$659$			2.10799i			0.0821157i	0.999157	+	0.0410579i	$0.0130728\pi$
$659$			2.10799i			0.0821157i	−0.999157	+	0.0410579i	$0.986927\pi$
$660$		0			0
$661$	−17.6021			−0.684642			−0.342321	−	0.939583i	$-0.611213\pi$
$661$	−17.6021			−0.684642			−0.342321	+	0.939583i	$0.611213\pi$
$662$		0			0
$663$			4.90467i			0.190482i
$664$		0			0
$665$	12.8534			0.498434
$666$		0			0
$667$		−	4.53022i		−	0.175411i
$668$		0			0
$669$			5.16619i			0.199736i
$670$		0			0
$671$		−	3.15415i		−	0.121765i
$672$		0			0
$673$			26.9746i			1.03979i	0.854229	+	0.519897i	$0.174030\pi$
$673$			26.9746i			1.03979i	−0.854229	+	0.519897i	$0.825970\pi$
$674$		0			0
$675$			7.10849i			0.273606i
$676$		0			0
$677$	−15.8186			−0.607958			−0.303979	−	0.952679i	$-0.598315\pi$
$677$	−15.8186			−0.607958			−0.303979	+	0.952679i	$0.598315\pi$
$678$		0			0
$679$	15.8789			0.609377
$680$		0			0
$681$	−11.6183			−0.445214
$682$		0			0
$683$			26.0082i			0.995177i	0.867413	+	0.497588i	$0.165781\pi$
$683$			26.0082i			0.995177i	−0.867413	+	0.497588i	$0.834219\pi$
$684$		0			0
$685$		−	12.0189i		−	0.459219i
$686$		0			0
$687$	42.8555			1.63504
$688$		0			0
$689$	−15.3826			−0.586030
$690$		0			0
$691$		−	34.1744i		−	1.30006i	−0.759910	−	0.650028i	$-0.774757\pi$
$691$		−	34.1744i		−	1.30006i	0.759910	−	0.650028i	$-0.225243\pi$
$692$		0			0
$693$	−1.78897			−0.0679575
$694$		0			0
$695$	−18.9914			−0.720385
$696$		0			0
$697$	−1.00337	+	9.87598i	−0.0380052	+	0.374079i
$698$		0			0
$699$	4.24950			0.160731
$700$		0			0
$701$	12.5140			0.472648			0.236324	−	0.971674i	$-0.424057\pi$
$701$	12.5140			0.472648			0.236324	+	0.971674i	$0.424057\pi$
$702$		0			0
$703$			21.7740i			0.821222i
$704$		0			0
$705$	−20.4347			−0.769616
$706$		0			0
$707$	−17.8206			−0.670211
$708$		0			0
$709$			27.6217i			1.03735i	0.854970	+	0.518677i	$0.173575\pi$
$709$			27.6217i			1.03735i	−0.854970	+	0.518677i	$0.826425\pi$
$710$		0			0
$711$		−	13.0215i		−	0.488343i
$712$		0			0
$713$	3.95821			0.148236
$714$		0			0
$715$	3.46092			0.129431
$716$		0			0
$717$	−30.8763			−1.15310
$718$		0			0
$719$			34.2303i			1.27657i	0.769798	+	0.638287i	$0.220357\pi$
$719$			34.2303i			1.27657i	−0.769798	+	0.638287i	$0.779643\pi$
$720$		0			0
$721$		−	13.7897i		−	0.513554i
$722$		0			0
$723$		−	55.3158i		−	2.05722i
$724$		0			0
$725$			45.3848i			1.68555i
$726$		0			0
$727$		−	16.6439i		−	0.617288i	−0.951178	−	0.308644i	$-0.900125\pi$
$727$		−	16.6439i		−	0.617288i	0.951178	−	0.308644i	$-0.0998751\pi$
$728$		0			0
$729$	21.9180			0.811778
$730$		0			0
$731$			10.1701i			0.376154i
$732$		0			0
$733$	−27.4973			−1.01564			−0.507819	−	0.861464i	$-0.669548\pi$
$733$	−27.4973			−1.01564			−0.507819	+	0.861464i	$0.669548\pi$
$734$		0			0
$735$		−	9.55950i		−	0.352608i
$736$		0			0
$737$	−0.395526			−0.0145694
$738$		0			0
$739$	−34.7601			−1.27867			−0.639335	−	0.768929i	$-0.720790\pi$
$739$	−34.7601			−1.27867			−0.639335	+	0.768929i	$0.720790\pi$
$740$		0			0
$741$			10.1814i			0.374023i
$742$		0			0
$743$	3.27081			0.119994			0.0599971	−	0.998199i	$-0.480891\pi$
$743$	3.27081			0.119994			0.0599971	+	0.998199i	$0.480891\pi$
$744$		0			0
$745$			61.1964i			2.24206i
$746$		0			0
$747$	−11.1612			−0.408366
$748$		0			0
$749$			12.8832i			0.470740i
$750$		0			0
$751$		−	41.0267i		−	1.49709i	−0.663087	−	0.748543i	$-0.730754\pi$
$751$		−	41.0267i		−	1.49709i	0.663087	−	0.748543i	$-0.269246\pi$
$752$		0			0
$753$			51.5078i			1.87705i
$754$		0			0
$755$		−	71.7469i		−	2.61114i
$756$		0			0
$757$			24.3275i			0.884199i	0.896966	+	0.442100i	$0.145766\pi$
$757$			24.3275i			0.884199i	−0.896966	+	0.442100i	$0.854234\pi$
$758$		0			0
$759$	1.71534			0.0622630
$760$		0			0
$761$	2.77842			0.100718			0.0503588	−	0.998731i	$-0.483963\pi$
$761$	2.77842			0.100718			0.0503588	+	0.998731i	$0.483963\pi$
$762$		0			0
$763$	−16.1047			−0.583029
$764$		0			0
$765$		−	16.8964i		−	0.610892i
$766$		0			0
$767$			11.7768i			0.425237i
$768$		0			0
$769$	26.0885			0.940775			0.470387	−	0.882460i	$-0.344114\pi$
$769$	26.0885			0.940775			0.470387	+	0.882460i	$0.344114\pi$
$770$		0			0
$771$	−9.34867			−0.336684
$772$		0			0
$773$			47.8018i			1.71931i	0.510875	+	0.859655i	$0.329322\pi$
$773$			47.8018i			1.71931i	−0.510875	+	0.859655i	$0.670678\pi$
$774$		0			0
$775$	−39.6543			−1.42442
$776$		0			0
$777$	16.1940			0.580957
$778$		0			0
$779$	−2.08284	+	20.5011i	−0.0746256	+	0.734529i
$780$		0			0
$781$	3.56114			0.127427
$782$		0			0
$783$	−2.68985			−0.0961274
$784$		0			0
$785$		−	46.3712i		−	1.65506i
$786$		0			0
$787$	−10.0915			−0.359725			−0.179862	−	0.983692i	$-0.557565\pi$
$787$	−10.0915			−0.359725			−0.179862	+	0.983692i	$0.557565\pi$
$788$		0			0
$789$	21.3702			0.760800
$790$		0			0
$791$		−	0.712810i		−	0.0253446i
$792$		0			0
$793$			6.35934i			0.225827i
$794$		0			0
$795$	111.251			3.94568
$796$		0			0
$797$	42.0491			1.48946			0.744728	−	0.667368i	$-0.232579\pi$
$797$	42.0491			1.48946			0.744728	+	0.667368i	$0.232579\pi$
$798$		0			0
$799$	−3.31399			−0.117241
$800$		0			0
$801$		−	7.87729i		−	0.278330i
$802$		0			0
$803$		−	6.60164i		−	0.232967i
$804$		0			0
$805$			4.36608i			0.153884i
$806$		0			0
$807$			11.0199i			0.387919i
$808$		0			0
$809$		−	53.6990i		−	1.88796i	−0.330007	−	0.943978i	$-0.607051\pi$
$809$		−	53.6990i		−	1.88796i	0.330007	−	0.943978i	$-0.392949\pi$
$810$		0			0
$811$	−8.88256			−0.311909			−0.155954	−	0.987764i	$-0.549845\pi$
$811$	−8.88256			−0.311909			−0.155954	+	0.987764i	$0.549845\pi$
$812$		0			0
$813$			51.7610i			1.81534i
$814$		0			0
$815$	−60.5056			−2.11942
$816$		0			0
$817$			21.1116i			0.738602i
$818$		0			0
$819$	3.60689			0.126035
$820$		0			0
$821$	−46.1354			−1.61014			−0.805068	−	0.593182i	$-0.797871\pi$
$821$	−46.1354			−1.61014			−0.805068	+	0.593182i	$0.797871\pi$
$822$		0			0
$823$		−	18.0075i		−	0.627701i	−0.949472	−	0.313850i	$-0.898381\pi$
$823$		−	18.0075i		−	0.627701i	0.949472	−	0.313850i	$-0.101619\pi$
$824$		0			0
$825$	−17.1847			−0.598294
$826$		0			0
$827$			26.6032i			0.925085i	0.886597	+	0.462543i	$0.153063\pi$
$827$			26.6032i			0.925085i	−0.886597	+	0.462543i	$0.846937\pi$
$828$		0			0
$829$	−10.6362			−0.369412			−0.184706	−	0.982794i	$-0.559133\pi$
$829$	−10.6362			−0.369412			−0.184706	+	0.982794i	$0.559133\pi$
$830$		0			0
$831$		−	59.1056i		−	2.05035i
$832$		0			0
$833$		−	1.55031i		−	0.0537150i
$834$		0			0
$835$		−	17.0497i		−	0.590029i
$836$		0			0
$837$		−	2.35022i		−	0.0812354i
$838$		0			0
$839$		−	52.9654i		−	1.82857i	−0.405073	−	0.914284i	$-0.632754\pi$
$839$		−	52.9654i		−	1.82857i	0.405073	−	0.914284i	$-0.367246\pi$
$840$		0			0
$841$	11.8264			0.407808
$842$		0			0
$843$	−14.0331			−0.483327
$844$		0			0
$845$	44.9435			1.54610
$846$		0			0
$847$		−	10.5702i		−	0.363197i
$848$		0			0
$849$			47.9340i			1.64509i
$850$		0			0
$851$	−7.39624			−0.253540
$852$		0			0
$853$	−24.9200			−0.853244			−0.426622	−	0.904430i	$-0.640296\pi$
$853$	−24.9200			−0.853244			−0.426622	+	0.904430i	$0.640296\pi$
$854$		0			0
$855$		−	35.0746i		−	1.19953i
$856$		0			0
$857$	0.394073			0.0134613			0.00673064	−	0.999977i	$-0.497858\pi$
$857$	0.394073			0.0134613			0.00673064	+	0.999977i	$0.497858\pi$
$858$		0			0
$859$	−48.2946			−1.64779			−0.823895	−	0.566742i	$-0.808204\pi$
$859$	−48.2946			−1.64779			−0.823895	+	0.566742i	$0.808204\pi$
$860$		0			0
$861$	15.2474	+	1.54908i	0.519628	+	0.0527924i
$862$		0			0
$863$	−49.4598			−1.68363			−0.841815	−	0.539766i	$-0.818513\pi$
$863$	−49.4598			−1.68363			−0.841815	+	0.539766i	$0.818513\pi$
$864$		0			0
$865$	−47.2160			−1.60539
$866$		0			0
$867$			34.9367i			1.18651i
$868$		0			0
$869$	−3.12835			−0.106122
$870$		0			0
$871$	0.797451			0.0270206
$872$		0			0
$873$		−	43.3306i		−	1.46652i
$874$		0			0
$875$		−	23.7706i		−	0.803593i
$876$		0			0
$877$	−5.54567			−0.187264			−0.0936319	−	0.995607i	$-0.529848\pi$
$877$	−5.54567			−0.187264			−0.0936319	+	0.995607i	$0.529848\pi$
$878$		0			0
$879$	−61.2017			−2.06428
$880$		0			0
$881$	24.2454			0.816848			0.408424	−	0.912792i	$-0.366078\pi$
$881$	24.2454			0.816848			0.408424	+	0.912792i	$0.366078\pi$
$882$		0			0
$883$			37.9289i			1.27641i	0.769866	+	0.638205i	$0.220323\pi$
$883$			37.9289i			1.27641i	−0.769866	+	0.638205i	$0.779677\pi$
$884$		0			0
$885$		−	85.1735i		−	2.86308i
$886$		0			0
$887$		−	25.8804i		−	0.868978i	−0.900677	−	0.434489i	$-0.856929\pi$
$887$		−	25.8804i		−	0.868978i	0.900677	−	0.434489i	$-0.143071\pi$
$888$		0			0
$889$			7.30114i			0.244872i
$890$		0			0
$891$		−	6.38542i		−	0.213920i
$892$		0			0
$893$	−6.87938			−0.230210
$894$		0			0
$895$		−	36.3766i		−	1.21594i
$896$		0			0
$897$	−3.45844			−0.115474
$898$		0			0
$899$		−	15.0052i		−	0.500450i
$900$		0			0
$901$	18.0422			0.601072
$902$		0			0
$903$	15.7014			0.522510
$904$		0			0
$905$		−	68.1451i		−	2.26522i
$906$		0			0
$907$	28.4230			0.943770			0.471885	−	0.881660i	$-0.343574\pi$
$907$	28.4230			0.943770			0.471885	+	0.881660i	$0.343574\pi$
$908$		0			0
$909$			48.6290i			1.61292i
$910$		0			0
$911$	6.16971			0.204412			0.102206	−	0.994763i	$-0.467410\pi$
$911$	6.16971			0.204412			0.102206	+	0.994763i	$0.467410\pi$
$912$		0			0
$913$			2.68142i			0.0887422i
$914$		0			0
$915$		−	45.9926i		−	1.52047i
$916$		0			0
$917$			5.57734i			0.184180i
$918$		0			0
$919$			17.8704i			0.589489i	0.955576	+	0.294744i	$0.0952345\pi$
$919$			17.8704i			0.589489i	−0.955576	+	0.294744i	$0.904765\pi$
$920$		0			0
$921$		−	51.9620i		−	1.71221i
$922$		0			0
$923$	−7.17989			−0.236329
$924$		0			0
$925$	74.0972			2.43630
$926$		0			0
$927$	−37.6295			−1.23591
$928$		0			0
$929$		−	1.76785i		−	0.0580012i	−0.999579	−	0.0290006i	$-0.990768\pi$
$929$		−	1.76785i		−	0.0580012i	0.999579	−	0.0290006i	$-0.00923247\pi$
$930$		0			0
$931$		−	3.21822i		−	0.105473i
$932$		0			0
$933$	−57.1892			−1.87229
$934$		0			0
$935$	−4.05930			−0.132753
$936$		0			0
$937$			17.7535i			0.579982i	0.957029	+	0.289991i	$0.0936523\pi$
$937$			17.7535i			0.579982i	−0.957029	+	0.289991i	$0.906348\pi$
$938$		0			0
$939$	−22.1968			−0.724365
$940$		0			0
$941$	−1.53136			−0.0499210			−0.0249605	−	0.999688i	$-0.507946\pi$
$941$	−1.53136			−0.0499210			−0.0249605	+	0.999688i	$0.507946\pi$
$942$		0			0
$943$	−6.96388	−	0.707505i	−0.226775	−	0.0230395i
$944$		0			0
$945$	2.59239			0.0843305
$946$		0			0
$947$	−20.5986			−0.669364			−0.334682	−	0.942331i	$-0.608629\pi$
$947$	−20.5986			−0.669364			−0.334682	+	0.942331i	$0.608629\pi$
$948$		0			0
$949$			13.3101i			0.432064i
$950$		0			0
$951$	−78.6510			−2.55043
$952$		0			0
$953$	−31.4112			−1.01751			−0.508755	−	0.860911i	$-0.669894\pi$
$953$	−31.4112			−1.01751			−0.508755	+	0.860911i	$0.669894\pi$
$954$		0			0
$955$		−	75.1514i		−	2.43184i
$956$		0			0
$957$		−	6.50268i		−	0.210202i
$958$		0			0
$959$	−3.00928			−0.0971746
$960$		0			0
$961$	−17.8895			−0.577079
$962$		0			0
$963$	35.1558			1.13288
$964$		0			0
$965$		−	76.2858i		−	2.45573i
$966$		0			0
$967$		−	33.7425i		−	1.08508i	−0.840028	−	0.542542i	$-0.817462\pi$
$967$		−	33.7425i		−	1.08508i	0.840028	−	0.542542i	$-0.182538\pi$
$968$		0			0
$969$		−	11.9417i		−	0.383623i
$970$		0			0
$971$			60.0751i			1.92790i	0.266079	+	0.963951i	$0.414272\pi$
$971$			60.0751i			1.92790i	−0.266079	+	0.963951i	$0.585728\pi$
$972$		0			0
$973$			4.75504i			0.152440i
$974$		0			0
$975$	34.6474			1.10961
$976$		0			0
$977$		−	56.5435i		−	1.80899i	−0.426488	−	0.904493i	$-0.640249\pi$
$977$		−	56.5435i		−	1.80899i	0.426488	−	0.904493i	$-0.359751\pi$
$978$		0			0
$979$	−1.89248			−0.0604840
$980$		0			0
$981$			43.9467i			1.40311i
$982$		0			0
$983$	38.7356			1.23548			0.617738	−	0.786384i	$-0.288049\pi$
$983$	38.7356			1.23548			0.617738	+	0.786384i	$0.288049\pi$
$984$		0			0
$985$	−28.7023			−0.914533
$986$		0			0
$987$			5.11642i			0.162857i
$988$		0			0
$989$	−7.17125			−0.228032
$990$		0			0
$991$			50.7023i			1.61061i	0.592861	+	0.805305i	$0.297999\pi$
$991$			50.7023i			1.61061i	−0.592861	+	0.805305i	$0.702001\pi$
$992$		0			0
$993$	−12.5322			−0.397697
$994$		0			0
$995$		−	25.9213i		−	0.821762i
$996$		0			0
$997$			38.3387i			1.21420i	0.794626	+	0.607100i	$0.207667\pi$
$997$			38.3387i			1.21420i	−0.794626	+	0.607100i	$0.792333\pi$
$998$		0			0
$999$			4.39157i			0.138943i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.17	yes	20
41.40	even	2	inner	1148.2.d.a.1065.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.4	✓	20	41.40	even	2	inner
1148.2.d.a.1065.17	yes	20	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +O(q^{10})\)	\(q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +0.655586i q^{11} -1.32178i q^{13} +9.55950i q^{15} +1.55031i q^{17} +3.21822i q^{19} +2.39349 q^{21} -1.09317 q^{23} +10.9517 q^{25} +0.649079i q^{27} +4.14410i q^{29} -3.62085 q^{31} -1.56914 q^{33} -3.99395i q^{35} +6.76585 q^{37} +3.16367 q^{39} +(6.37033 + 0.647203i) q^{41} +6.56003 q^{43} -10.8988 q^{45} +2.13764i q^{47} -1.00000 q^{49} -3.71066 q^{51} -11.6378i q^{53} +2.61838i q^{55} -7.70279 q^{57} -8.90983 q^{59} -4.81119 q^{61} +2.72882i q^{63} -5.27912i q^{65} +0.603316i q^{67} -2.61650i q^{69} -5.43199i q^{71} -10.0698 q^{73} +26.2127i q^{75} +0.655586 q^{77} +4.77184i q^{79} -9.74001 q^{81} +4.09012 q^{83} +6.19186i q^{85} -9.91888 q^{87} +2.88671i q^{89} -1.32178 q^{91} -8.66648i q^{93} +12.8534i q^{95} +15.8789i q^{97} -1.78897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 20 * q + 4 * q^5 - 20 * q^9	\( 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) \) 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more