LMFDB - Embedded newform 1148.2.d.a.1065.11

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			1065.11
Root			$2.80197i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0281596i q^{3} -2.87346 q^{5} +1.00000i q^{7} +2.99921 q^{9} +O(q^{10})$	$q+0.0281596i q^{3} -2.87346 q^{5} +1.00000i q^{7} +2.99921 q^{9} +1.41531i q^{11} +0.163688i q^{13} -0.0809155i q^{15} -1.52886i q^{17} -0.870919i q^{19} -0.0281596 q^{21} -5.91912 q^{23} +3.25679 q^{25} +0.168935i q^{27} +6.54053i q^{29} -5.19635 q^{31} -0.0398546 q^{33} -2.87346i q^{35} -4.08685 q^{37} -0.00460937 q^{39} +(-4.64956 + 4.40246i) q^{41} +2.45069 q^{43} -8.61811 q^{45} +9.70449i q^{47} -1.00000 q^{49} +0.0430520 q^{51} +13.3388i q^{53} -4.06684i q^{55} +0.0245247 q^{57} -4.35387 q^{59} -5.06748 q^{61} +2.99921i q^{63} -0.470350i q^{65} +10.4414i q^{67} -0.166680i q^{69} +11.0791i q^{71} -1.62445 q^{73} +0.0917097i q^{75} -1.41531 q^{77} -8.29407i q^{79} +8.99286 q^{81} -7.40680 q^{83} +4.39311i q^{85} -0.184179 q^{87} -16.6621i q^{89} -0.163688 q^{91} -0.146327i q^{93} +2.50255i q^{95} -5.23253i q^{97} +4.24481i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			0.0281596i			0.0162579i	0.999967	+	0.00812897i	$0.00258756\pi$
$3$			0.0281596i			0.0162579i	−0.999967	+	0.00812897i	$0.997412\pi$
$4$		0			0
$5$	−2.87346			−1.28505			−0.642526	−	0.766264i	$-0.722113\pi$
$5$	−2.87346			−1.28505			−0.642526	+	0.766264i	$0.722113\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.99921			0.999736
$10$		0			0
$11$			1.41531i			0.426732i	0.976972	+	0.213366i	$0.0684427\pi$
$11$			1.41531i			0.426732i	−0.976972	+	0.213366i	$0.931557\pi$
$12$		0			0
$13$			0.163688i			0.0453988i	0.999742	+	0.0226994i	$0.00722606\pi$
$13$			0.163688i			0.0453988i	−0.999742	+	0.0226994i	$0.992774\pi$
$14$		0			0
$15$		−	0.0809155i		−	0.0208923i
$16$		0			0
$17$		−	1.52886i		−	0.370802i	−0.982663	−	0.185401i	$-0.940642\pi$
$17$		−	1.52886i		−	0.370802i	0.982663	−	0.185401i	$-0.0593585\pi$
$18$		0			0
$19$		−	0.870919i		−	0.199803i	−0.994997	−	0.0999013i	$-0.968147\pi$
$19$		−	0.870919i		−	0.199803i	0.994997	−	0.0999013i	$-0.0318527\pi$
$20$		0			0
$21$	−0.0281596			−0.00614492
$22$		0			0
$23$	−5.91912			−1.23422			−0.617111	−	0.786876i	$-0.711697\pi$
$23$	−5.91912			−1.23422			−0.617111	+	0.786876i	$0.711697\pi$
$24$		0			0
$25$	3.25679			0.651357
$26$		0			0
$27$			0.168935i			0.0325116i
$28$		0			0
$29$			6.54053i			1.21455i	0.794493	+	0.607273i	$0.207737\pi$
$29$			6.54053i			1.21455i	−0.794493	+	0.607273i	$0.792263\pi$
$30$		0			0
$31$	−5.19635			−0.933292			−0.466646	−	0.884444i	$-0.654538\pi$
$31$	−5.19635			−0.933292			−0.466646	+	0.884444i	$0.654538\pi$
$32$		0			0
$33$	−0.0398546			−0.00693779
$34$		0			0
$35$		−	2.87346i		−	0.485704i
$36$		0			0
$37$	−4.08685			−0.671874			−0.335937	−	0.941885i	$-0.609053\pi$
$37$	−4.08685			−0.671874			−0.335937	+	0.941885i	$0.609053\pi$
$38$		0			0
$39$	−0.00460937			−0.000738090
$40$		0			0
$41$	−4.64956	+	4.40246i	−0.726139	+	0.687548i
$41$	−4.64956	+	4.40246i	−0.726139	+	0.687548i
$42$		0			0
$43$	2.45069			0.373726			0.186863	−	0.982386i	$-0.440168\pi$
$43$	2.45069			0.373726			0.186863	+	0.982386i	$0.440168\pi$
$44$		0			0
$45$	−8.61811			−1.28471
$46$		0			0
$47$			9.70449i			1.41555i	0.706440	+	0.707773i	$0.250300\pi$
$47$			9.70449i			1.41555i	−0.706440	+	0.707773i	$0.749700\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.0430520			0.00602848
$52$		0			0
$53$			13.3388i			1.83222i	0.400922	+	0.916112i	$0.368690\pi$
$53$			13.3388i			1.83222i	−0.400922	+	0.916112i	$0.631310\pi$
$54$		0			0
$55$		−	4.06684i		−	0.548373i
$56$		0			0
$57$	0.0245247			0.00324838
$58$		0			0
$59$	−4.35387			−0.566825			−0.283413	−	0.958998i	$-0.591467\pi$
$59$	−4.35387			−0.566825			−0.283413	+	0.958998i	$0.591467\pi$
$60$		0			0
$61$	−5.06748			−0.648824			−0.324412	−	0.945916i	$-0.605166\pi$
$61$	−5.06748			−0.648824			−0.324412	+	0.945916i	$0.605166\pi$
$62$		0			0
$63$			2.99921i			0.377865i
$64$		0			0
$65$		−	0.470350i		−	0.0583397i
$66$		0			0
$67$			10.4414i			1.27562i	0.770195	+	0.637809i	$0.220159\pi$
$67$			10.4414i			1.27562i	−0.770195	+	0.637809i	$0.779841\pi$
$68$		0			0
$69$		−	0.166680i		−	0.0200659i
$70$		0			0
$71$			11.0791i			1.31485i	0.753519	+	0.657426i	$0.228355\pi$
$71$			11.0791i			1.31485i	−0.753519	+	0.657426i	$0.771645\pi$
$72$		0			0
$73$	−1.62445			−0.190128			−0.0950639	−	0.995471i	$-0.530306\pi$
$73$	−1.62445			−0.190128			−0.0950639	+	0.995471i	$0.530306\pi$
$74$		0			0
$75$			0.0917097i			0.0105897i
$76$		0			0
$77$	−1.41531			−0.161290
$78$		0			0
$79$		−	8.29407i		−	0.933156i	−0.884480	−	0.466578i	$-0.845487\pi$
$79$		−	8.29407i		−	0.933156i	0.884480	−	0.466578i	$-0.154513\pi$
$80$		0			0
$81$	8.99286			0.999207
$82$		0			0
$83$	−7.40680			−0.813002			−0.406501	−	0.913650i	$-0.633251\pi$
$83$	−7.40680			−0.813002			−0.406501	+	0.913650i	$0.633251\pi$
$84$		0			0
$85$			4.39311i			0.476500i
$86$		0			0
$87$	−0.184179			−0.0197460
$88$		0			0
$89$		−	16.6621i		−	1.76618i	−0.469202	−	0.883091i	$-0.655458\pi$
$89$		−	16.6621i		−	1.76618i	0.469202	−	0.883091i	$-0.344542\pi$
$90$		0			0
$91$	−0.163688			−0.0171591
$92$		0			0
$93$		−	0.146327i		−	0.0151734i
$94$		0			0
$95$			2.50255i			0.256757i
$96$		0			0
$97$		−	5.23253i		−	0.531283i	−0.964072	−	0.265641i	$-0.914416\pi$
$97$		−	5.23253i		−	0.531283i	0.964072	−	0.265641i	$-0.0855837\pi$
$98$		0			0
$99$			4.24481i			0.426620i
$100$		0			0
$101$		−	7.31740i		−	0.728109i	−0.931378	−	0.364054i	$-0.881392\pi$
$101$		−	7.31740i		−	0.728109i	0.931378	−	0.364054i	$-0.118608\pi$
$102$		0			0
$103$	0.742115			0.0731228			0.0365614	−	0.999331i	$-0.488360\pi$
$103$	0.742115			0.0731228			0.0365614	+	0.999331i	$0.488360\pi$
$104$		0			0
$105$	0.0809155			0.00789654
$106$		0			0
$107$	−14.5577			−1.40734			−0.703671	−	0.710526i	$-0.748457\pi$
$107$	−14.5577			−1.40734			−0.703671	+	0.710526i	$0.748457\pi$
$108$		0			0
$109$		−	4.21020i		−	0.403264i	−0.979461	−	0.201632i	$-0.935375\pi$
$109$		−	4.21020i		−	0.403264i	0.979461	−	0.201632i	$-0.0646245\pi$
$110$		0			0
$111$		−	0.115084i		−	0.0109233i
$112$		0			0
$113$	11.9578			1.12489			0.562445	−	0.826834i	$-0.309861\pi$
$113$	11.9578			1.12489			0.562445	+	0.826834i	$0.309861\pi$
$114$		0			0
$115$	17.0084			1.58604
$116$		0			0
$117$			0.490933i			0.0453868i
$118$		0			0
$119$	1.52886			0.140150
$120$		0			0
$121$	8.99689			0.817900
$122$		0			0
$123$	−0.123971	−	0.130930i	−0.0111781	−	0.0118055i
$124$		0			0
$125$	5.00906			0.448024
$126$		0			0
$127$	9.52108			0.844859			0.422430	−	0.906396i	$-0.361177\pi$
$127$	9.52108			0.844859			0.422430	+	0.906396i	$0.361177\pi$
$128$		0			0
$129$			0.0690103i			0.00607602i
$130$		0			0
$131$	−11.1689			−0.975830			−0.487915	−	0.872891i	$-0.662242\pi$
$131$	−11.1689			−0.975830			−0.487915	+	0.872891i	$0.662242\pi$
$132$		0			0
$133$	0.870919			0.0755183
$134$		0			0
$135$		−	0.485429i		−	0.0417791i
$136$		0			0
$137$		−	7.55058i		−	0.645090i	−0.946554	−	0.322545i	$-0.895462\pi$
$137$		−	7.55058i		−	0.645090i	0.946554	−	0.322545i	$-0.104538\pi$
$138$		0			0
$139$	0.156735			0.0132941			0.00664706	−	0.999978i	$-0.497884\pi$
$139$	0.156735			0.0132941			0.00664706	+	0.999978i	$0.497884\pi$
$140$		0			0
$141$	−0.273274			−0.0230138
$142$		0			0
$143$	−0.231669			−0.0193731
$144$		0			0
$145$		−	18.7940i		−	1.56075i
$146$		0			0
$147$		−	0.0281596i		−	0.00232256i
$148$		0			0
$149$			13.4649i			1.10309i	0.834146	+	0.551544i	$0.185961\pi$
$149$			13.4649i			1.10309i	−0.834146	+	0.551544i	$0.814039\pi$
$150$		0			0
$151$		−	21.7242i		−	1.76789i	−0.467592	−	0.883944i	$-0.654878\pi$
$151$		−	21.7242i		−	1.76789i	0.467592	−	0.883944i	$-0.345122\pi$
$152$		0			0
$153$		−	4.58536i		−	0.370704i
$154$		0			0
$155$	14.9315			1.19933
$156$		0			0
$157$			8.09661i			0.646180i	0.946368	+	0.323090i	$0.104722\pi$
$157$			8.09661i			0.646180i	−0.946368	+	0.323090i	$0.895278\pi$
$158$		0			0
$159$	−0.375615			−0.0297882
$160$		0			0
$161$		−	5.91912i		−	0.466492i
$162$		0			0
$163$	−18.7874			−1.47154			−0.735772	−	0.677229i	$-0.763181\pi$
$163$	−18.7874			−1.47154			−0.735772	+	0.677229i	$0.763181\pi$
$164$		0			0
$165$	0.114521			0.00891542
$166$		0			0
$167$			17.4354i			1.34919i	0.738187	+	0.674596i	$0.235682\pi$
$167$			17.4354i			1.34919i	−0.738187	+	0.674596i	$0.764318\pi$
$168$		0			0
$169$	12.9732			0.997939
$170$		0			0
$171$		−	2.61207i		−	0.199750i
$172$		0			0
$173$	−10.5109			−0.799126			−0.399563	−	0.916706i	$-0.630838\pi$
$173$	−10.5109			−0.799126			−0.399563	+	0.916706i	$0.630838\pi$
$174$		0			0
$175$			3.25679i			0.246190i
$176$		0			0
$177$		−	0.122603i		−	0.00921541i
$178$		0			0
$179$			9.34414i			0.698414i	0.937046	+	0.349207i	$0.113549\pi$
$179$			9.34414i			0.698414i	−0.937046	+	0.349207i	$0.886451\pi$
$180$		0			0
$181$		−	7.10252i		−	0.527926i	−0.964533	−	0.263963i	$-0.914970\pi$
$181$		−	7.10252i		−	0.527926i	0.964533	−	0.263963i	$-0.0850297\pi$
$182$		0			0
$183$		−	0.142698i		−	0.0105485i
$184$		0			0
$185$	11.7434			0.863392
$186$		0			0
$187$	2.16381			0.158233
$188$		0			0
$189$	−0.168935			−0.0122882
$190$		0			0
$191$		−	13.8790i		−	1.00425i	−0.864795	−	0.502125i	$-0.832552\pi$
$191$		−	13.8790i		−	1.00425i	0.864795	−	0.502125i	$-0.167448\pi$
$192$		0			0
$193$			7.48254i			0.538605i	0.963056	+	0.269303i	$0.0867932\pi$
$193$			7.48254i			0.538605i	−0.963056	+	0.269303i	$0.913207\pi$
$194$		0			0
$195$	0.0132449			0.000948484
$196$		0			0
$197$	−7.03739			−0.501393			−0.250697	−	0.968066i	$-0.580660\pi$
$197$	−7.03739			−0.501393			−0.250697	+	0.968066i	$0.580660\pi$
$198$		0			0
$199$			4.50679i			0.319478i	0.987159	+	0.159739i	$0.0510652\pi$
$199$			4.50679i			0.319478i	−0.987159	+	0.159739i	$0.948935\pi$
$200$		0			0
$201$	−0.294025			−0.0207389
$202$		0			0
$203$	−6.54053			−0.459055
$204$		0			0
$205$	13.3603	−	12.6503i	0.933125	−	0.883535i
$206$		0			0
$207$	−17.7527			−1.23389
$208$		0			0
$209$	1.23262			0.0852622
$210$		0			0
$211$			22.3372i			1.53776i	0.639394	+	0.768879i	$0.279185\pi$
$211$			22.3372i			1.53776i	−0.639394	+	0.768879i	$0.720815\pi$
$212$		0			0
$213$	−0.311984			−0.0213768
$214$		0			0
$215$	−7.04196			−0.480258
$216$		0			0
$217$		−	5.19635i		−	0.352751i
$218$		0			0
$219$		−	0.0457439i		−	0.00309109i
$220$		0			0
$221$	0.250255			0.0168340
$222$		0			0
$223$	−4.68504			−0.313734			−0.156867	−	0.987620i	$-0.550139\pi$
$223$	−4.68504			−0.313734			−0.156867	+	0.987620i	$0.550139\pi$
$224$		0			0
$225$	9.76777			0.651185
$226$		0			0
$227$			27.0940i			1.79829i	0.437646	+	0.899147i	$0.355812\pi$
$227$			27.0940i			1.79829i	−0.437646	+	0.899147i	$0.644188\pi$
$228$		0			0
$229$		−	1.42966i		−	0.0944746i	−0.998884	−	0.0472373i	$-0.984958\pi$
$229$		−	1.42966i		−	0.0944746i	0.998884	−	0.0472373i	$-0.0150417\pi$
$230$		0			0
$231$		−	0.0398546i		−	0.00262224i
$232$		0			0
$233$		−	22.3102i		−	1.46159i	−0.682597	−	0.730795i	$-0.739150\pi$
$233$		−	22.3102i		−	1.46159i	0.682597	−	0.730795i	$-0.260850\pi$
$234$		0			0
$235$		−	27.8855i		−	1.81905i
$236$		0			0
$237$	0.233557			0.0151712
$238$		0			0
$239$			16.4604i			1.06474i	0.846513	+	0.532368i	$0.178698\pi$
$239$			16.4604i			1.06474i	−0.846513	+	0.532368i	$0.821302\pi$
$240$		0			0
$241$	20.7587			1.33718			0.668592	−	0.743629i	$-0.266897\pi$
$241$	20.7587			1.33718			0.668592	+	0.743629i	$0.266897\pi$
$242$		0			0
$243$			0.760041i			0.0487566i
$244$		0			0
$245$	2.87346			0.183579
$246$		0			0
$247$	0.142559			0.00907079
$248$		0			0
$249$		−	0.208572i		−	0.0132177i
$250$		0			0
$251$	3.39645			0.214382			0.107191	−	0.994238i	$-0.465814\pi$
$251$	3.39645			0.214382			0.107191	+	0.994238i	$0.465814\pi$
$252$		0			0
$253$		−	8.37739i		−	0.526682i
$254$		0			0
$255$	−0.123708			−0.00774691
$256$		0			0
$257$		−	29.1711i		−	1.81964i	−0.415000	−	0.909821i	$-0.636219\pi$
$257$		−	29.1711i		−	1.81964i	0.415000	−	0.909821i	$-0.363781\pi$
$258$		0			0
$259$		−	4.08685i		−	0.253944i
$260$		0			0
$261$			19.6164i			1.21422i
$262$		0			0
$263$			6.88082i			0.424290i	0.977238	+	0.212145i	$0.0680449\pi$
$263$			6.88082i			0.424290i	−0.977238	+	0.212145i	$0.931955\pi$
$264$		0			0
$265$		−	38.3285i		−	2.35450i
$266$		0			0
$267$	0.469199			0.0287145
$268$		0			0
$269$	17.6439			1.07577			0.537884	−	0.843019i	$-0.319224\pi$
$269$	17.6439			1.07577			0.537884	+	0.843019i	$0.319224\pi$
$270$		0			0
$271$	12.2386			0.743443			0.371722	−	0.928344i	$-0.378768\pi$
$271$	12.2386			0.743443			0.371722	+	0.928344i	$0.378768\pi$
$272$		0			0
$273$		−	0.00460937i		−	0.000278972i
$274$		0			0
$275$			4.60936i			0.277955i
$276$		0			0
$277$	19.2464			1.15640			0.578201	−	0.815895i	$-0.303755\pi$
$277$	19.2464			1.15640			0.578201	+	0.815895i	$0.303755\pi$
$278$		0			0
$279$	−15.5849			−0.933045
$280$		0			0
$281$		−	1.34247i		−	0.0800849i	−0.999198	−	0.0400424i	$-0.987251\pi$
$281$		−	1.34247i		−	0.0800849i	0.999198	−	0.0400424i	$-0.0127493\pi$
$282$		0			0
$283$	12.1372			0.721481			0.360740	−	0.932666i	$-0.382524\pi$
$283$	12.1372			0.721481			0.360740	+	0.932666i	$0.382524\pi$
$284$		0			0
$285$	−0.0704709			−0.00417433
$286$		0			0
$287$	−4.40246	−	4.64956i	−0.259869	−	0.274455i
$288$		0			0
$289$	14.6626			0.862506
$290$		0			0
$291$	0.147346			0.00863756
$292$		0			0
$293$		−	11.1101i		−	0.649060i	−0.945875	−	0.324530i	$-0.894794\pi$
$293$		−	11.1101i		−	0.649060i	0.945875	−	0.324530i	$-0.105206\pi$
$294$		0			0
$295$	12.5107			0.728399
$296$		0			0
$297$	−0.239096			−0.0138737
$298$		0			0
$299$		−	0.968886i		−	0.0560321i
$300$		0			0
$301$			2.45069i			0.141255i
$302$		0			0
$303$	0.206055			0.0118375
$304$		0			0
$305$	14.5612			0.833773
$306$		0			0
$307$	1.43382			0.0818325			0.0409162	−	0.999163i	$-0.486972\pi$
$307$	1.43382			0.0818325			0.0409162	+	0.999163i	$0.486972\pi$
$308$		0			0
$309$			0.0208976i			0.00118883i
$310$		0			0
$311$			4.25405i			0.241225i	0.992700	+	0.120612i	$0.0384858\pi$
$311$			4.25405i			0.241225i	−0.992700	+	0.120612i	$0.961514\pi$
$312$		0			0
$313$			23.8526i			1.34823i	0.738626	+	0.674115i	$0.235475\pi$
$313$			23.8526i			1.34823i	−0.738626	+	0.674115i	$0.764525\pi$
$314$		0			0
$315$		−	8.61811i		−	0.485575i
$316$		0			0
$317$		−	4.21382i		−	0.236672i	−0.992974	−	0.118336i	$-0.962244\pi$
$317$		−	4.21382i		−	0.236672i	0.992974	−	0.118336i	$-0.0377560\pi$
$318$		0			0
$319$	−9.25688			−0.518286
$320$		0			0
$321$		−	0.409938i		−	0.0228805i
$322$		0			0
$323$	−1.33151			−0.0740873
$324$		0			0
$325$			0.533095i			0.0295708i
$326$		0			0
$327$	0.118558			0.00655625
$328$		0			0
$329$	−9.70449			−0.535026
$330$		0			0
$331$		−	12.6740i		−	0.696628i	−0.937378	−	0.348314i	$-0.886754\pi$
$331$		−	12.6740i		−	0.696628i	0.937378	−	0.348314i	$-0.113246\pi$
$332$		0			0
$333$	−12.2573			−0.671696
$334$		0			0
$335$		−	30.0029i		−	1.63923i
$336$		0			0
$337$	4.97103			0.270789			0.135395	−	0.990792i	$-0.456770\pi$
$337$	4.97103			0.270789			0.135395	+	0.990792i	$0.456770\pi$
$338$		0			0
$339$			0.336725i			0.0182884i
$340$		0			0
$341$		−	7.35445i		−	0.398266i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			0.478948i			0.0257857i
$346$		0			0
$347$			0.0300952i			0.00161559i	1.00000		0.000807796i	$0.000257130\pi$
$347$			0.0300952i			0.00161559i	−1.00000		0.000807796i	$0.999743\pi$
$348$		0			0
$349$	−16.3668			−0.876095			−0.438047	−	0.898952i	$-0.644330\pi$
$349$	−16.3668			−0.876095			−0.438047	+	0.898952i	$0.644330\pi$
$350$		0			0
$351$	−0.0276526			−0.00147599
$352$		0			0
$353$	28.4982			1.51681			0.758403	−	0.651786i	$-0.225980\pi$
$353$	28.4982			1.51681			0.758403	+	0.651786i	$0.225980\pi$
$354$		0			0
$355$		−	31.8355i		−	1.68965i
$356$		0			0
$357$			0.0430520i			0.00227855i
$358$		0			0
$359$	−10.6590			−0.562561			−0.281280	−	0.959626i	$-0.590759\pi$
$359$	−10.6590			−0.562561			−0.281280	+	0.959626i	$0.590759\pi$
$360$		0			0
$361$	18.2415			0.960079
$362$		0			0
$363$			0.253349i			0.0132974i
$364$		0			0
$365$	4.66781			0.244324
$366$		0			0
$367$	12.0181			0.627342			0.313671	−	0.949532i	$-0.398441\pi$
$367$	12.0181			0.627342			0.313671	+	0.949532i	$0.398441\pi$
$368$		0			0
$369$	−13.9450	+	13.2039i	−0.725947	+	0.687367i
$370$		0			0
$371$	−13.3388			−0.692516
$372$		0			0
$373$	−16.9296			−0.876580			−0.438290	−	0.898834i	$-0.644416\pi$
$373$	−16.9296			−0.876580			−0.438290	+	0.898834i	$0.644416\pi$
$374$		0			0
$375$			0.141053i			0.00728395i
$376$		0			0
$377$	−1.07060			−0.0551389
$378$		0			0
$379$	14.3469			0.736950			0.368475	−	0.929638i	$-0.379880\pi$
$379$	14.3469			0.736950			0.368475	+	0.929638i	$0.379880\pi$
$380$		0			0
$381$			0.268110i			0.0137357i
$382$		0			0
$383$			6.28154i			0.320972i	0.987038	+	0.160486i	$0.0513061\pi$
$383$			6.28154i			0.320972i	−0.987038	+	0.160486i	$0.948694\pi$
$384$		0			0
$385$	4.06684			0.207265
$386$		0			0
$387$	7.35012			0.373628
$388$		0			0
$389$	6.45966			0.327518			0.163759	−	0.986500i	$-0.447638\pi$
$389$	6.45966			0.327518			0.163759	+	0.986500i	$0.447638\pi$
$390$		0			0
$391$			9.04948i			0.457652i
$392$		0			0
$393$		−	0.314511i		−	0.0158650i
$394$		0			0
$395$			23.8327i			1.19915i
$396$		0			0
$397$			1.46970i			0.0737620i	0.999320	+	0.0368810i	$0.0117422\pi$
$397$			1.46970i			0.0737620i	−0.999320	+	0.0368810i	$0.988258\pi$
$398$		0			0
$399$			0.0245247i			0.00122777i
$400$		0			0
$401$	10.3420			0.516456			0.258228	−	0.966084i	$-0.416861\pi$
$401$	10.3420			0.516456			0.258228	+	0.966084i	$0.416861\pi$
$402$		0			0
$403$		−	0.850578i		−	0.0423703i
$404$		0			0
$405$	−25.8407			−1.28403
$406$		0			0
$407$		−	5.78416i		−	0.286710i
$408$		0			0
$409$	−30.8876			−1.52729			−0.763646	−	0.645635i	$-0.776593\pi$
$409$	−30.8876			−1.52729			−0.763646	+	0.645635i	$0.776593\pi$
$410$		0			0
$411$	0.212621			0.0104878
$412$		0			0
$413$		−	4.35387i		−	0.214240i
$414$		0			0
$415$	21.2832			1.04475
$416$		0			0
$417$			0.00441360i			0.000216135i
$418$		0			0
$419$	17.0674			0.833798			0.416899	−	0.908953i	$-0.363117\pi$
$419$	17.0674			0.833798			0.416899	+	0.908953i	$0.363117\pi$
$420$		0			0
$421$		−	25.5444i		−	1.24496i	−0.782637	−	0.622478i	$-0.786126\pi$
$421$		−	25.5444i		−	1.24496i	0.782637	−	0.622478i	$-0.213874\pi$
$422$		0			0
$423$			29.1058i			1.41517i
$424$		0			0
$425$		−	4.97916i		−	0.241525i
$426$		0			0
$427$		−	5.06748i		−	0.245233i
$428$		0			0
$429$		−	0.00652370i		−	0.000314967i
$430$		0			0
$431$	5.36700			0.258519			0.129260	−	0.991611i	$-0.458740\pi$
$431$	5.36700			0.258519			0.129260	+	0.991611i	$0.458740\pi$
$432$		0			0
$433$	−31.4504			−1.51141			−0.755706	−	0.654911i	$-0.772706\pi$
$433$	−31.4504			−1.51141			−0.755706	+	0.654911i	$0.772706\pi$
$434$		0			0
$435$	0.529230			0.0253746
$436$		0			0
$437$			5.15507i			0.246601i
$438$		0			0
$439$			15.6492i			0.746894i	0.927652	+	0.373447i	$0.121824\pi$
$439$			15.6492i			0.746894i	−0.927652	+	0.373447i	$0.878176\pi$
$440$		0			0
$441$	−2.99921			−0.142819
$442$		0			0
$443$	−9.19376			−0.436809			−0.218404	−	0.975858i	$-0.570085\pi$
$443$	−9.19376			−0.436809			−0.218404	+	0.975858i	$0.570085\pi$
$444$		0			0
$445$			47.8780i			2.26963i
$446$		0			0
$447$	−0.379166			−0.0179340
$448$		0			0
$449$	−15.9282			−0.751700			−0.375850	−	0.926680i	$-0.622649\pi$
$449$	−15.9282			−0.751700			−0.375850	+	0.926680i	$0.622649\pi$
$450$		0			0
$451$	−6.23085	−	6.58057i	−0.293399	−	0.309867i
$452$		0			0
$453$	0.611744			0.0287422
$454$		0			0
$455$	0.470350			0.0220503
$456$		0			0
$457$			6.12934i			0.286718i	0.989671	+	0.143359i	$0.0457904\pi$
$457$			6.12934i			0.286718i	−0.989671	+	0.143359i	$0.954210\pi$
$458$		0			0
$459$	0.258278			0.0120554
$460$		0			0
$461$	7.80317			0.363430			0.181715	−	0.983351i	$-0.441835\pi$
$461$	7.80317			0.363430			0.181715	+	0.983351i	$0.441835\pi$
$462$		0			0
$463$			4.70387i			0.218607i	0.994008	+	0.109304i	$0.0348621\pi$
$463$			4.70387i			0.218607i	−0.994008	+	0.109304i	$0.965138\pi$
$464$		0			0
$465$			0.420465i			0.0194986i
$466$		0			0
$467$	−33.8528			−1.56652			−0.783259	−	0.621695i	$-0.786444\pi$
$467$	−33.8528			−1.56652			−0.783259	+	0.621695i	$0.786444\pi$
$468$		0			0
$469$	−10.4414			−0.482138
$470$		0			0
$471$	−0.227997			−0.0105056
$472$		0			0
$473$			3.46849i			0.159481i
$474$		0			0
$475$		−	2.83640i		−	0.130143i
$476$		0			0
$477$			40.0058i			1.83174i
$478$		0			0
$479$		−	24.3707i		−	1.11352i	−0.830672	−	0.556762i	$-0.812044\pi$
$479$		−	24.3707i		−	1.11352i	0.830672	−	0.556762i	$-0.187956\pi$
$480$		0			0
$481$		−	0.668966i		−	0.0305022i
$482$		0			0
$483$	0.166680			0.00758419
$484$		0			0
$485$			15.0355i			0.682725i
$486$		0			0
$487$	−35.3561			−1.60214			−0.801069	−	0.598571i	$-0.795735\pi$
$487$	−35.3561			−1.60214			−0.801069	+	0.598571i	$0.795735\pi$
$488$		0			0
$489$		−	0.529046i		−	0.0239243i
$490$		0			0
$491$	27.3148			1.23270			0.616350	−	0.787472i	$-0.288611\pi$
$491$	27.3148			1.23270			0.616350	+	0.787472i	$0.288611\pi$
$492$		0			0
$493$	9.99953			0.450356
$494$		0			0
$495$		−	12.1973i		−	0.548228i
$496$		0			0
$497$	−11.0791			−0.496967
$498$		0			0
$499$			0.655972i			0.0293654i	0.999892	+	0.0146827i	$0.00467381\pi$
$499$			0.655972i			0.0293654i	−0.999892	+	0.0146827i	$0.995326\pi$
$500$		0			0
$501$	−0.490974			−0.0219351
$502$		0			0
$503$		−	12.0976i		−	0.539407i	−0.962943	−	0.269703i	$-0.913074\pi$
$503$		−	12.0976i		−	0.539407i	0.962943	−	0.269703i	$-0.0869257\pi$
$504$		0			0
$505$			21.0263i			0.935657i
$506$		0			0
$507$			0.365320i			0.0162244i
$508$		0			0
$509$		−	0.950920i		−	0.0421488i	−0.999778	−	0.0210744i	$-0.993291\pi$
$509$		−	0.950920i		−	0.0421488i	0.999778	−	0.0210744i	$-0.00670868\pi$
$510$		0			0
$511$		−	1.62445i		−	0.0718616i
$512$		0			0
$513$	0.147129			0.00649590
$514$		0			0
$515$	−2.13244			−0.0939665
$516$		0			0
$517$	−13.7349			−0.604059
$518$		0			0
$519$		−	0.295982i		−	0.0129922i
$520$		0			0
$521$			27.4516i			1.20268i	0.798995	+	0.601338i	$0.205366\pi$
$521$			27.4516i			1.20268i	−0.798995	+	0.601338i	$0.794634\pi$
$522$		0			0
$523$	17.1546			0.750117			0.375058	−	0.927001i	$-0.377623\pi$
$523$	17.1546			0.750117			0.375058	+	0.927001i	$0.377623\pi$
$524$		0			0
$525$	−0.0917097			−0.00400254
$526$		0			0
$527$			7.94448i			0.346067i
$528$		0			0
$529$	12.0359			0.523302
$530$		0			0
$531$	−13.0581			−0.566675
$532$		0			0
$533$	−0.720627	−	0.761074i	−0.0312138	−	0.0329658i
$534$		0			0
$535$	41.8309			1.80851
$536$		0			0
$537$	−0.263127			−0.0113548
$538$		0			0
$539$		−	1.41531i		−	0.0609618i
$540$		0			0
$541$	−15.8931			−0.683296			−0.341648	−	0.939828i	$-0.610985\pi$
$541$	−15.8931			−0.683296			−0.341648	+	0.939828i	$0.610985\pi$
$542$		0			0
$543$	0.200004			0.00858299
$544$		0			0
$545$			12.0979i			0.518215i
$546$		0			0
$547$			25.6595i			1.09712i	0.836111	+	0.548560i	$0.184824\pi$
$547$			25.6595i			1.09712i	−0.836111	+	0.548560i	$0.815176\pi$
$548$		0			0
$549$	−15.1984			−0.648653
$550$		0			0
$551$	5.69627			0.242669
$552$		0			0
$553$	8.29407			0.352700
$554$		0			0
$555$			0.330689i			0.0140370i
$556$		0			0
$557$			1.94337i			0.0823430i	0.999152	+	0.0411715i	$0.0131090\pi$
$557$			1.94337i			0.0823430i	−0.999152	+	0.0411715i	$0.986891\pi$
$558$		0			0
$559$			0.401147i			0.0169667i
$560$		0			0
$561$			0.0609319i			0.00257255i
$562$		0			0
$563$		−	18.8033i		−	0.792463i	−0.918151	−	0.396232i	$-0.870318\pi$
$563$		−	18.8033i		−	0.792463i	0.918151	−	0.396232i	$-0.129682\pi$
$564$		0			0
$565$	−34.3601			−1.44554
$566$		0			0
$567$			8.99286i			0.377665i
$568$		0			0
$569$	−6.54883			−0.274541			−0.137271	−	0.990534i	$-0.543833\pi$
$569$	−6.54883			−0.274541			−0.137271	+	0.990534i	$0.543833\pi$
$570$		0			0
$571$		−	10.1601i		−	0.425189i	−0.977141	−	0.212594i	$-0.931809\pi$
$571$		−	10.1601i		−	0.425189i	0.977141	−	0.212594i	$-0.0681913\pi$
$572$		0			0
$573$	0.390827			0.0163270
$574$		0			0
$575$	−19.2773			−0.803919
$576$		0			0
$577$			22.1349i			0.921489i	0.887533	+	0.460745i	$0.152418\pi$
$577$			22.1349i			0.921489i	−0.887533	+	0.460745i	$0.847582\pi$
$578$		0			0
$579$	−0.210705			−0.00875661
$580$		0			0
$581$		−	7.40680i		−	0.307286i
$582$		0			0
$583$	−18.8785			−0.781869
$584$		0			0
$585$		−	1.41068i		−	0.0583243i
$586$		0			0
$587$			44.1511i			1.82231i	0.412061	+	0.911156i	$0.364809\pi$
$587$			44.1511i			1.82231i	−0.412061	+	0.911156i	$0.635191\pi$
$588$		0			0
$589$			4.52560i			0.186474i
$590$		0			0
$591$		−	0.198170i		−	0.00815162i
$592$		0			0
$593$			9.55204i			0.392255i	0.980578	+	0.196128i	$0.0628367\pi$
$593$			9.55204i			0.392255i	−0.980578	+	0.196128i	$0.937163\pi$
$594$		0			0
$595$	−4.39311			−0.180100
$596$		0			0
$597$	−0.126909			−0.00519405
$598$		0			0
$599$	−40.3517			−1.64872			−0.824362	−	0.566063i	$-0.808466\pi$
$599$	−40.3517			−1.64872			−0.824362	+	0.566063i	$0.808466\pi$
$600$		0			0
$601$		−	34.9144i		−	1.42419i	−0.702084	−	0.712094i	$-0.747747\pi$
$601$		−	34.9144i		−	1.42419i	0.702084	−	0.712094i	$-0.252253\pi$
$602$		0			0
$603$			31.3159i			1.27528i
$604$		0			0
$605$	−25.8522			−1.05104
$606$		0			0
$607$	11.5578			0.469116			0.234558	−	0.972102i	$-0.424636\pi$
$607$	11.5578			0.469116			0.234558	+	0.972102i	$0.424636\pi$
$608$		0			0
$609$		−	0.184179i		−	0.00746329i
$610$		0			0
$611$	−1.58850			−0.0642640
$612$		0			0
$613$	28.2440			1.14077			0.570383	−	0.821379i	$-0.306795\pi$
$613$	28.2440			1.14077			0.570383	+	0.821379i	$0.306795\pi$
$614$		0			0
$615$	0.356227	+	0.376221i	0.0143645	+	0.0151707i
$616$		0			0
$617$	−20.0786			−0.808333			−0.404167	−	0.914685i	$-0.632438\pi$
$617$	−20.0786			−0.808333			−0.404167	+	0.914685i	$0.632438\pi$
$618$		0			0
$619$	−8.58459			−0.345044			−0.172522	−	0.985006i	$-0.555192\pi$
$619$	−8.58459			−0.345044			−0.172522	+	0.985006i	$0.555192\pi$
$620$		0			0
$621$		−	0.999947i		−	0.0401265i
$622$		0			0
$623$	16.6621			0.667554
$624$		0			0
$625$	−30.6773			−1.22709
$626$		0			0
$627$			0.0347101i			0.00138619i
$628$		0			0
$629$			6.24821i			0.249132i
$630$		0			0
$631$	37.9247			1.50976			0.754880	−	0.655863i	$-0.227695\pi$
$631$	37.9247			1.50976			0.754880	+	0.655863i	$0.227695\pi$
$632$		0			0
$633$	−0.629007			−0.0250008
$634$		0			0
$635$	−27.3585			−1.08569
$636$		0			0
$637$		−	0.163688i		−	0.00648554i
$638$		0			0
$639$			33.2286i			1.31450i
$640$		0			0
$641$			13.7949i			0.544866i	0.962175	+	0.272433i	$0.0878283\pi$
$641$			13.7949i			0.544866i	−0.962175	+	0.272433i	$0.912172\pi$
$642$		0			0
$643$		−	19.7443i		−	0.778641i	−0.921102	−	0.389320i	$-0.872710\pi$
$643$		−	19.7443i		−	0.778641i	0.921102	−	0.389320i	$-0.127290\pi$
$644$		0			0
$645$		−	0.198299i		−	0.00780800i
$646$		0			0
$647$	−37.4804			−1.47350			−0.736752	−	0.676163i	$-0.763642\pi$
$647$	−37.4804			−1.47350			−0.736752	+	0.676163i	$0.763642\pi$
$648$		0			0
$649$		−	6.16207i		−	0.241883i
$650$		0			0
$651$	0.146327			0.00573501
$652$		0			0
$653$		−	34.9038i		−	1.36589i	−0.730470	−	0.682945i	$-0.760699\pi$
$653$		−	34.9038i		−	1.36589i	0.730470	−	0.682945i	$-0.239301\pi$
$654$		0			0
$655$	32.0934			1.25399
$656$		0			0
$657$	−4.87207			−0.190078
$658$		0			0
$659$			1.40775i			0.0548383i	0.999624	+	0.0274191i	$0.00872888\pi$
$659$			1.40775i			0.0548383i	−0.999624	+	0.0274191i	$0.991271\pi$
$660$		0			0
$661$	2.74099			0.106612			0.0533061	−	0.998578i	$-0.483024\pi$
$661$	2.74099			0.106612			0.0533061	+	0.998578i	$0.483024\pi$
$662$		0			0
$663$			0.00704707i			0.000273686i
$664$		0			0
$665$	−2.50255			−0.0970449
$666$		0			0
$667$		−	38.7141i		−	1.49902i
$668$		0			0
$669$		−	0.131929i		−	0.00510066i
$670$		0			0
$671$		−	7.17206i		−	0.276874i
$672$		0			0
$673$			3.34620i			0.128987i	0.997918	+	0.0644934i	$0.0205431\pi$
$673$			3.34620i			0.128987i	−0.997918	+	0.0644934i	$0.979457\pi$
$674$		0			0
$675$			0.550186i			0.0211766i
$676$		0			0
$677$	7.55449			0.290343			0.145171	−	0.989407i	$-0.453627\pi$
$677$	7.55449			0.290343			0.145171	+	0.989407i	$0.453627\pi$
$678$		0			0
$679$	5.23253			0.200806
$680$		0			0
$681$	−0.762957			−0.0292366
$682$		0			0
$683$		−	1.76605i		−	0.0675760i	−0.999429	−	0.0337880i	$-0.989243\pi$
$683$		−	1.76605i		−	0.0675760i	0.999429	−	0.0337880i	$-0.0107571\pi$
$684$		0			0
$685$			21.6963i			0.828974i
$686$		0			0
$687$	0.0402586			0.00153596
$688$		0			0
$689$	−2.18339			−0.0831807
$690$		0			0
$691$		−	13.5746i		−	0.516401i	−0.966091	−	0.258200i	$-0.916870\pi$
$691$		−	13.5746i		−	0.516401i	0.966091	−	0.258200i	$-0.0831295\pi$
$692$		0			0
$693$	−4.24481			−0.161247
$694$		0			0
$695$	−0.450373			−0.0170836
$696$		0			0
$697$	6.73073	+	7.10851i	0.254945	+	0.269254i
$698$		0			0
$699$	0.628246			0.0237624
$700$		0			0
$701$	15.4909			0.585084			0.292542	−	0.956253i	$-0.405499\pi$
$701$	15.4909			0.585084			0.292542	+	0.956253i	$0.405499\pi$
$702$		0			0
$703$			3.55932i			0.134242i
$704$		0			0
$705$	0.785243			0.0295740
$706$		0			0
$707$	7.31740			0.275199
$708$		0			0
$709$			12.9458i			0.486191i	0.970002	+	0.243095i	$0.0781628\pi$
$709$			12.9458i			0.486191i	−0.970002	+	0.243095i	$0.921837\pi$
$710$		0			0
$711$		−	24.8756i		−	0.932909i
$712$		0			0
$713$	30.7578			1.15189
$714$		0			0
$715$	0.665691			0.0248955
$716$		0			0
$717$	−0.463519			−0.0173104
$718$		0			0
$719$			5.99870i			0.223714i	0.993724	+	0.111857i	$0.0356798\pi$
$719$			5.99870i			0.223714i	−0.993724	+	0.111857i	$0.964320\pi$
$720$		0			0
$721$			0.742115i			0.0276378i
$722$		0			0
$723$			0.584556i			0.0217399i
$724$		0			0
$725$			21.3011i			0.791103i
$726$		0			0
$727$		−	35.1664i		−	1.30425i	−0.758111	−	0.652125i	$-0.773878\pi$
$727$		−	35.1664i		−	1.30425i	0.758111	−	0.652125i	$-0.226122\pi$
$728$		0			0
$729$	26.9572			0.998414
$730$		0			0
$731$		−	3.74675i		−	0.138579i
$732$		0			0
$733$	33.7253			1.24567			0.622837	−	0.782352i	$-0.285980\pi$
$733$	33.7253			1.24567			0.622837	+	0.782352i	$0.285980\pi$
$734$		0			0
$735$			0.0809155i			0.00298461i
$736$		0			0
$737$	−14.7778			−0.544347
$738$		0			0
$739$	4.54479			0.167183			0.0835914	−	0.996500i	$-0.473361\pi$
$739$	4.54479			0.167183			0.0835914	+	0.996500i	$0.473361\pi$
$740$		0			0
$741$			0.00401439i			0.000147472i
$742$		0			0
$743$	−13.0762			−0.479719			−0.239860	−	0.970808i	$-0.577101\pi$
$743$	−13.0762			−0.479719			−0.239860	+	0.970808i	$0.577101\pi$
$744$		0			0
$745$		−	38.6909i		−	1.41753i
$746$		0			0
$747$	−22.2145			−0.812787
$748$		0			0
$749$		−	14.5577i		−	0.531925i
$750$		0			0
$751$		−	10.8923i		−	0.397465i	−0.980054	−	0.198733i	$-0.936317\pi$
$751$		−	10.8923i		−	0.397465i	0.980054	−	0.198733i	$-0.0636826\pi$
$752$		0			0
$753$			0.0956427i			0.00348541i
$754$		0			0
$755$			62.4236i			2.27183i
$756$		0			0
$757$			9.21144i			0.334795i	0.985889	+	0.167398i	$0.0535364\pi$
$757$			9.21144i			0.334795i	−0.985889	+	0.167398i	$0.946464\pi$
$758$		0			0
$759$	0.235904			0.00856277
$760$		0			0
$761$	2.50415			0.0907755			0.0453877	−	0.998969i	$-0.485548\pi$
$761$	2.50415			0.0907755			0.0453877	+	0.998969i	$0.485548\pi$
$762$		0			0
$763$	4.21020			0.152420
$764$		0			0
$765$			13.1759i			0.476374i
$766$		0			0
$767$		−	0.712674i		−	0.0257332i
$768$		0			0
$769$	−37.1466			−1.33954			−0.669770	−	0.742569i	$-0.733607\pi$
$769$	−37.1466			−1.33954			−0.669770	+	0.742569i	$0.733607\pi$
$770$		0			0
$771$	0.821446			0.0295836
$772$		0			0
$773$			21.4908i			0.772969i	0.922296	+	0.386484i	$0.126311\pi$
$773$			21.4908i			0.772969i	−0.922296	+	0.386484i	$0.873689\pi$
$774$		0			0
$775$	−16.9234			−0.607906
$776$		0			0
$777$	0.115084			0.00412861
$778$		0			0
$779$	3.83419	+	4.04939i	0.137374	+	0.145084i
$780$		0			0
$781$	−15.6804			−0.561090
$782$		0			0
$783$	−1.10493			−0.0394868
$784$		0			0
$785$		−	23.2653i		−	0.830375i
$786$		0			0
$787$	−13.2317			−0.471658			−0.235829	−	0.971795i	$-0.575781\pi$
$787$	−13.2317			−0.471658			−0.235829	+	0.971795i	$0.575781\pi$
$788$		0			0
$789$	−0.193761			−0.00689808
$790$		0			0
$791$			11.9578i			0.425169i
$792$		0			0
$793$		−	0.829484i		−	0.0294558i
$794$		0			0
$795$	1.07932			0.0382794
$796$		0			0
$797$	−6.32933			−0.224196			−0.112098	−	0.993697i	$-0.535757\pi$
$797$	−6.32933			−0.224196			−0.112098	+	0.993697i	$0.535757\pi$
$798$		0			0
$799$	14.8368			0.524887
$800$		0			0
$801$		−	49.9732i		−	1.76572i
$802$		0			0
$803$		−	2.29911i		−	0.0811337i
$804$		0			0
$805$			17.0084i			0.599466i
$806$		0			0
$807$			0.496845i			0.0174898i
$808$		0			0
$809$			31.5197i			1.10817i	0.832459	+	0.554087i	$0.186932\pi$
$809$			31.5197i			1.10817i	−0.832459	+	0.554087i	$0.813068\pi$
$810$		0			0
$811$	43.3399			1.52187			0.760934	−	0.648829i	$-0.224741\pi$
$811$	43.3399			1.52187			0.760934	+	0.648829i	$0.224741\pi$
$812$		0			0
$813$			0.344634i			0.0120869i
$814$		0			0
$815$	53.9849			1.89101
$816$		0			0
$817$		−	2.13435i		−	0.0746715i
$818$		0			0
$819$	−0.490933			−0.0171546
$820$		0			0
$821$	40.6234			1.41777			0.708884	−	0.705325i	$-0.249199\pi$
$821$	40.6234			1.41777			0.708884	+	0.705325i	$0.249199\pi$
$822$		0			0
$823$			2.70459i			0.0942762i	0.998888	+	0.0471381i	$0.0150101\pi$
$823$			2.70459i			0.0942762i	−0.998888	+	0.0471381i	$0.984990\pi$
$824$		0			0
$825$	−0.129798			−0.00451898
$826$		0			0
$827$			6.22340i			0.216409i	0.994129	+	0.108204i	$0.0345101\pi$
$827$			6.22340i			0.216409i	−0.994129	+	0.108204i	$0.965490\pi$
$828$		0			0
$829$	−8.46677			−0.294063			−0.147031	−	0.989132i	$-0.546972\pi$
$829$	−8.46677			−0.294063			−0.147031	+	0.989132i	$0.546972\pi$
$830$		0			0
$831$			0.541969i			0.0188007i
$832$		0			0
$833$			1.52886i			0.0529718i
$834$		0			0
$835$		−	50.1000i		−	1.73378i
$836$		0			0
$837$		−	0.877846i		−	0.0303428i
$838$		0			0
$839$			44.0081i			1.51933i	0.650315	+	0.759665i	$0.274637\pi$
$839$			44.0081i			1.51933i	−0.650315	+	0.759665i	$0.725363\pi$
$840$		0			0
$841$	−13.7785			−0.475121
$842$		0			0
$843$	0.0378033			0.00130202
$844$		0			0
$845$	−37.2780			−1.28240
$846$		0			0
$847$			8.99689i			0.309137i
$848$		0			0
$849$			0.341778i			0.0117298i
$850$		0			0
$851$	24.1905			0.829241
$852$		0			0
$853$	−22.6783			−0.776491			−0.388246	−	0.921556i	$-0.626919\pi$
$853$	−22.6783			−0.776491			−0.388246	+	0.921556i	$0.626919\pi$
$854$		0			0
$855$			7.50568i			0.256689i
$856$		0			0
$857$	−20.5351			−0.701466			−0.350733	−	0.936475i	$-0.614068\pi$
$857$	−20.5351			−0.701466			−0.350733	+	0.936475i	$0.614068\pi$
$858$		0			0
$859$	−30.5082			−1.04092			−0.520462	−	0.853885i	$-0.674240\pi$
$859$	−30.5082			−1.04092			−0.520462	+	0.853885i	$0.674240\pi$
$860$		0			0
$861$	0.130930	−	0.123971i	0.00446207	−	0.00422493i
$862$		0			0
$863$	−15.4387			−0.525541			−0.262770	−	0.964858i	$-0.584636\pi$
$863$	−15.4387			−0.525541			−0.262770	+	0.964858i	$0.584636\pi$
$864$		0			0
$865$	30.2026			1.02692
$866$		0			0
$867$			0.412893i			0.0140226i
$868$		0			0
$869$	11.7387			0.398208
$870$		0			0
$871$	−1.70912			−0.0579115
$872$		0			0
$873$		−	15.6934i		−	0.531142i
$874$		0			0
$875$			5.00906i			0.169337i
$876$		0			0
$877$	−29.1635			−0.984782			−0.492391	−	0.870374i	$-0.663877\pi$
$877$	−29.1635			−0.984782			−0.492391	+	0.870374i	$0.663877\pi$
$878$		0			0
$879$	0.312856			0.0105524
$880$		0			0
$881$	27.6228			0.930636			0.465318	−	0.885144i	$-0.345940\pi$
$881$	27.6228			0.930636			0.465318	+	0.885144i	$0.345940\pi$
$882$		0			0
$883$		−	8.80486i		−	0.296307i	−0.988964	−	0.148154i	$-0.952667\pi$
$883$		−	8.80486i		−	0.296307i	0.988964	−	0.148154i	$-0.0473330\pi$
$884$		0			0
$885$			0.352295i			0.0118423i
$886$		0			0
$887$			1.15960i			0.0389357i	0.999810	+	0.0194679i	$0.00619720\pi$
$887$			1.15960i			0.0389357i	−0.999810	+	0.0194679i	$0.993803\pi$
$888$		0			0
$889$			9.52108i			0.319327i
$890$		0			0
$891$			12.7277i			0.426394i
$892$		0			0
$893$	8.45183			0.282830
$894$		0			0
$895$		−	26.8500i		−	0.897497i
$896$		0			0
$897$	0.0272834			0.000910967
$898$		0			0
$899$		−	33.9869i		−	1.13353i
$900$		0			0
$901$	20.3931			0.679393
$902$		0			0
$903$	−0.0690103			−0.00229652
$904$		0			0
$905$			20.4088i			0.678412i
$906$		0			0
$907$	−28.5246			−0.947144			−0.473572	−	0.880755i	$-0.657036\pi$
$907$	−28.5246			−0.947144			−0.473572	+	0.880755i	$0.657036\pi$
$908$		0			0
$909$		−	21.9464i		−	0.727916i
$910$		0			0
$911$	−0.293188			−0.00971377			−0.00485688	−	0.999988i	$-0.501546\pi$
$911$	−0.293188			−0.00971377			−0.00485688	+	0.999988i	$0.501546\pi$
$912$		0			0
$913$		−	10.4829i		−	0.346934i
$914$		0			0
$915$			0.410038i			0.0135554i
$916$		0			0
$917$		−	11.1689i		−	0.368829i
$918$		0			0
$919$			12.4178i			0.409626i	0.978801	+	0.204813i	$0.0656586\pi$
$919$			12.4178i			0.409626i	−0.978801	+	0.204813i	$0.934341\pi$
$920$		0			0
$921$			0.0403758i			0.00133043i
$922$		0			0
$923$	−1.81352			−0.0596926
$924$		0			0
$925$	−13.3100			−0.437630
$926$		0			0
$927$	2.22576			0.0731034
$928$		0			0
$929$			36.6737i			1.20323i	0.798788	+	0.601613i	$0.205475\pi$
$929$			36.6737i			1.20323i	−0.798788	+	0.601613i	$0.794525\pi$
$930$		0			0
$931$			0.870919i			0.0285432i
$932$		0			0
$933$	−0.119792			−0.00392182
$934$		0			0
$935$	−6.21762			−0.203338
$936$		0			0
$937$			45.4981i			1.48636i	0.669092	+	0.743179i	$0.266683\pi$
$937$			45.4981i			1.48636i	−0.669092	+	0.743179i	$0.733317\pi$
$938$		0			0
$939$	−0.671680			−0.0219194
$940$		0			0
$941$	−26.8622			−0.875681			−0.437841	−	0.899053i	$-0.644257\pi$
$941$	−26.8622			−0.875681			−0.437841	+	0.899053i	$0.644257\pi$
$942$		0			0
$943$	27.5213	−	26.0587i	0.896216	−	0.848587i
$944$		0			0
$945$	0.485429			0.0157910
$946$		0			0
$947$	−7.66606			−0.249113			−0.124557	−	0.992212i	$-0.539751\pi$
$947$	−7.66606			−0.249113			−0.124557	+	0.992212i	$0.539751\pi$
$948$		0			0
$949$		−	0.265903i		−	0.00863157i
$950$		0			0
$951$	0.118660			0.00384780
$952$		0			0
$953$	30.5548			0.989768			0.494884	−	0.868959i	$-0.335211\pi$
$953$	30.5548			0.989768			0.494884	+	0.868959i	$0.335211\pi$
$954$		0			0
$955$			39.8808i			1.29051i
$956$		0			0
$957$		−	0.260670i		−	0.00842626i
$958$		0			0
$959$	7.55058			0.243821
$960$		0			0
$961$	−3.99795			−0.128966
$962$		0			0
$963$	−43.6614			−1.40697
$964$		0			0
$965$		−	21.5008i		−	0.692135i
$966$		0			0
$967$			57.7366i			1.85668i	0.371729	+	0.928341i	$0.378765\pi$
$967$			57.7366i			1.85668i	−0.371729	+	0.928341i	$0.621235\pi$
$968$		0			0
$969$		−	0.0374948i		−	0.00120451i
$970$		0			0
$971$		−	56.5156i		−	1.81367i	−0.421482	−	0.906837i	$-0.638490\pi$
$971$		−	56.5156i		−	1.81367i	0.421482	−	0.906837i	$-0.361510\pi$
$972$		0			0
$973$			0.156735i			0.00502470i
$974$		0			0
$975$	−0.0150117			−0.000480760
$976$		0			0
$977$		−	33.1335i		−	1.06004i	−0.847986	−	0.530018i	$-0.822185\pi$
$977$		−	33.1335i		−	1.06004i	0.847986	−	0.530018i	$-0.177815\pi$
$978$		0			0
$979$	23.5821			0.753687
$980$		0			0
$981$		−	12.6273i		−	0.403158i
$982$		0			0
$983$	−39.9797			−1.27515			−0.637577	−	0.770387i	$-0.720063\pi$
$983$	−39.9797			−1.27515			−0.637577	+	0.770387i	$0.720063\pi$
$984$		0			0
$985$	20.2217			0.644316
$986$		0			0
$987$		−	0.273274i		−	0.00869842i
$988$		0			0
$989$	−14.5059			−0.461261
$990$		0			0
$991$		−	39.6245i		−	1.25871i	−0.777116	−	0.629357i	$-0.783318\pi$
$991$		−	39.6245i		−	1.25871i	0.777116	−	0.629357i	$-0.216682\pi$
$992$		0			0
$993$	0.356895			0.0113257
$994$		0			0
$995$		−	12.9501i		−	0.410545i
$996$		0			0
$997$		−	45.0355i		−	1.42629i	−0.701016	−	0.713145i	$-0.747270\pi$
$997$		−	45.0355i		−	1.42629i	0.701016	−	0.713145i	$-0.252730\pi$
$998$		0			0
$999$		−	0.690412i		−	0.0218437i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.11	yes	20
41.40	even	2	inner	1148.2.d.a.1065.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.10	✓	20	41.40	even	2	inner
1148.2.d.a.1065.11	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)

\(f(q)\)	\(=\)	\(q+0.0281596i q^{3} -2.87346 q^{5} +1.00000i q^{7} +2.99921 q^{9} +O(q^{10})\)	\(q+0.0281596i q^{3} -2.87346 q^{5} +1.00000i q^{7} +2.99921 q^{9} +1.41531i q^{11} +0.163688i q^{13} -0.0809155i q^{15} -1.52886i q^{17} -0.870919i q^{19} -0.0281596 q^{21} -5.91912 q^{23} +3.25679 q^{25} +0.168935i q^{27} +6.54053i q^{29} -5.19635 q^{31} -0.0398546 q^{33} -2.87346i q^{35} -4.08685 q^{37} -0.00460937 q^{39} +(-4.64956 + 4.40246i) q^{41} +2.45069 q^{43} -8.61811 q^{45} +9.70449i q^{47} -1.00000 q^{49} +0.0430520 q^{51} +13.3388i q^{53} -4.06684i q^{55} +0.0245247 q^{57} -4.35387 q^{59} -5.06748 q^{61} +2.99921i q^{63} -0.470350i q^{65} +10.4414i q^{67} -0.166680i q^{69} +11.0791i q^{71} -1.62445 q^{73} +0.0917097i q^{75} -1.41531 q^{77} -8.29407i q^{79} +8.99286 q^{81} -7.40680 q^{83} +4.39311i q^{85} -0.184179 q^{87} -16.6621i q^{89} -0.163688 q^{91} -0.146327i q^{93} +2.50255i q^{95} -5.23253i q^{97} +4.24481i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 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