LMFDB - Embedded newform 1344.4.c.f.673.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.3
Root			$-3.79295 - 1.01632i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -2.27324i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -2.27324i q^{5} -7.00000 q^{7} -9.00000 q^{9} +1.28915i q^{11} +84.9431i q^{13} -6.81971 q^{15} +49.4316 q^{17} +2.27106i q^{19} +21.0000i q^{21} -96.7192 q^{23} +119.832 q^{25} +27.0000i q^{27} -216.670i q^{29} +108.361 q^{31} +3.86744 q^{33} +15.9126i q^{35} -92.2319i q^{37} +254.829 q^{39} -356.642 q^{41} -500.887i q^{43} +20.4591i q^{45} -104.336 q^{47} +49.0000 q^{49} -148.295i q^{51} +53.9429i q^{53} +2.93054 q^{55} +6.81318 q^{57} -395.641i q^{59} -76.1378i q^{61} +63.0000 q^{63} +193.096 q^{65} -55.9352i q^{67} +290.158i q^{69} -1005.21 q^{71} -149.692 q^{73} -359.497i q^{75} -9.02403i q^{77} -331.984 q^{79} +81.0000 q^{81} +729.693i q^{83} -112.370i q^{85} -650.009 q^{87} -7.31192 q^{89} -594.602i q^{91} -325.082i q^{93} +5.16265 q^{95} -1765.54 q^{97} -11.6023i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 2.27324i	− 0.203324i	−0.994819	−	0.101662i	$-0.967584\pi$
$5$	− 2.27324i	− 0.203324i	0.994819	−	0.101662i	$-0.0324161\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	1.28915i	0.0353357i	0.999844	+	0.0176678i	$0.00562414\pi$
$11$	1.28915i	0.0353357i	−0.999844	+	0.0176678i	$0.994376\pi$
$12$	0	0
$13$	84.9431i	1.81223i	0.423032	+	0.906115i	$0.360966\pi$
$13$	84.9431i	1.81223i	−0.423032	+	0.906115i	$0.639034\pi$
$14$	0	0
$15$	−6.81971	−0.117389
$16$	0	0
$17$	49.4316	0.705230	0.352615	−	0.935768i	$-0.385293\pi$
$17$	49.4316	0.705230	0.352615	+	0.935768i	$0.385293\pi$
$18$	0	0
$19$	2.27106i	0.0274219i	0.999906	+	0.0137110i	$0.00436447\pi$
$19$	2.27106i	0.0274219i	−0.999906	+	0.0137110i	$0.995636\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−96.7192	−0.876842	−0.438421	−	0.898770i	$-0.644462\pi$
$23$	−96.7192	−0.876842	−0.438421	+	0.898770i	$0.644462\pi$
$24$	0	0
$25$	119.832	0.958659
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 216.670i	− 1.38740i	−0.720265	−	0.693699i	$-0.755980\pi$
$29$	− 216.670i	− 1.38740i	0.720265	−	0.693699i	$-0.244020\pi$
$30$	0	0
$31$	108.361	0.627812	0.313906	−	0.949454i	$-0.398362\pi$
$31$	108.361	0.627812	0.313906	+	0.949454i	$0.398362\pi$
$32$	0	0
$33$	3.86744	0.0204011
$34$	0	0
$35$	15.9126i	0.0768494i
$36$	0	0
$37$	− 92.2319i	− 0.409806i	−0.978782	−	0.204903i	$-0.934312\pi$
$37$	− 92.2319i	− 0.409806i	0.978782	−	0.204903i	$-0.0656879\pi$
$38$	0	0
$39$	254.829	1.04629
$40$	0	0
$41$	−356.642	−1.35849	−0.679245	−	0.733912i	$-0.737693\pi$
$41$	−356.642	−1.35849	−0.679245	+	0.733912i	$0.737693\pi$
$42$	0	0
$43$	− 500.887i	− 1.77638i	−0.459473	−	0.888192i	$-0.651962\pi$
$43$	− 500.887i	− 1.77638i	0.459473	−	0.888192i	$-0.348038\pi$
$44$	0	0
$45$	20.4591i	0.0677748i
$46$	0	0
$47$	−104.336	−0.323807	−0.161903	−	0.986807i	$-0.551763\pi$
$47$	−104.336	−0.323807	−0.161903	+	0.986807i	$0.551763\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 148.295i	− 0.407165i
$52$	0	0
$53$	53.9429i	0.139804i	0.997554	+	0.0699021i	$0.0222687\pi$
$53$	53.9429i	0.139804i	−0.997554	+	0.0699021i	$0.977731\pi$
$54$	0	0
$55$	2.93054	0.00718461
$56$	0	0
$57$	6.81318	0.0158321
$58$	0	0
$59$	− 395.641i	− 0.873019i	−0.899700	−	0.436510i	$-0.856215\pi$
$59$	− 395.641i	− 0.873019i	0.899700	−	0.436510i	$-0.143785\pi$
$60$	0	0
$61$	− 76.1378i	− 0.159811i	−0.996802	−	0.0799053i	$-0.974538\pi$
$61$	− 76.1378i	− 0.159811i	0.996802	−	0.0799053i	$-0.0254618\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	193.096	0.368470
$66$	0	0
$67$	− 55.9352i	− 0.101994i	−0.998699	−	0.0509968i	$-0.983760\pi$
$67$	− 55.9352i	− 0.101994i	0.998699	−	0.0509968i	$-0.0162398\pi$
$68$	0	0
$69$	290.158i	0.506245i
$70$	0	0
$71$	−1005.21	−1.68023	−0.840113	−	0.542411i	$-0.817512\pi$
$71$	−1005.21	−1.68023	−0.840113	+	0.542411i	$0.817512\pi$
$72$	0	0
$73$	−149.692	−0.240002	−0.120001	−	0.992774i	$-0.538290\pi$
$73$	−149.692	−0.240002	−0.120001	+	0.992774i	$0.538290\pi$
$74$	0	0
$75$	− 359.497i	− 0.553482i
$76$	0	0
$77$	− 9.02403i	− 0.0133556i
$78$	0	0
$79$	−331.984	−0.472800	−0.236400	−	0.971656i	$-0.575968\pi$
$79$	−331.984	−0.472800	−0.236400	+	0.971656i	$0.575968\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	729.693i	0.964991i	0.875899	+	0.482495i	$0.160269\pi$
$83$	729.693i	0.964991i	−0.875899	+	0.482495i	$0.839731\pi$
$84$	0	0
$85$	− 112.370i	− 0.143390i
$86$	0	0
$87$	−650.009	−0.801014
$88$	0	0
$89$	−7.31192	−0.00870856	−0.00435428	−	0.999991i	$-0.501386\pi$
$89$	−7.31192	−0.00870856	−0.00435428	+	0.999991i	$0.501386\pi$
$90$	0	0
$91$	− 594.602i	− 0.684958i
$92$	0	0
$93$	− 325.082i	− 0.362468i
$94$	0	0
$95$	5.16265	0.00557554
$96$	0	0
$97$	−1765.54	−1.84808	−0.924039	−	0.382297i	$-0.875133\pi$
$97$	−1765.54	−1.84808	−0.924039	+	0.382297i	$0.875133\pi$
$98$	0	0
$99$	− 11.6023i	− 0.0117786i
$100$	0	0
$101$	976.161i	0.961699i	0.876803	+	0.480850i	$0.159672\pi$
$101$	976.161i	0.961699i	−0.876803	+	0.480850i	$0.840328\pi$
$102$	0	0
$103$	−392.656	−0.375627	−0.187813	−	0.982205i	$-0.560140\pi$
$103$	−392.656	−0.375627	−0.187813	+	0.982205i	$0.560140\pi$
$104$	0	0
$105$	47.7379	0.0443690
$106$	0	0
$107$	− 1609.16i	− 1.45386i	−0.686710	−	0.726932i	$-0.740946\pi$
$107$	− 1609.16i	− 1.45386i	0.686710	−	0.726932i	$-0.259054\pi$
$108$	0	0
$109$	1703.72i	1.49712i	0.663064	+	0.748562i	$0.269256\pi$
$109$	1703.72i	1.49712i	−0.663064	+	0.748562i	$0.730744\pi$
$110$	0	0
$111$	−276.696	−0.236602
$112$	0	0
$113$	1248.50	1.03937	0.519687	−	0.854357i	$-0.326049\pi$
$113$	1248.50	1.03937	0.519687	+	0.854357i	$0.326049\pi$
$114$	0	0
$115$	219.866i	0.178283i
$116$	0	0
$117$	− 764.488i	− 0.604077i
$118$	0	0
$119$	−346.021	−0.266552
$120$	0	0
$121$	1329.34	0.998751
$122$	0	0
$123$	1069.92i	0.784324i
$124$	0	0
$125$	− 556.562i	− 0.398243i
$126$	0	0
$127$	594.003	0.415033	0.207517	−	0.978231i	$-0.433462\pi$
$127$	594.003	0.415033	0.207517	+	0.978231i	$0.433462\pi$
$128$	0	0
$129$	−1502.66	−1.02560
$130$	0	0
$131$	− 2329.92i	− 1.55394i	−0.629537	−	0.776971i	$-0.716755\pi$
$131$	− 2329.92i	− 1.55394i	0.629537	−	0.776971i	$-0.283245\pi$
$132$	0	0
$133$	− 15.8974i	− 0.0103645i
$134$	0	0
$135$	61.3774	0.0391298
$136$	0	0
$137$	−2895.60	−1.80575	−0.902874	−	0.429906i	$-0.858547\pi$
$137$	−2895.60	−1.80575	−0.902874	+	0.429906i	$0.858547\pi$
$138$	0	0
$139$	37.9323i	0.0231466i	0.999933	+	0.0115733i	$0.00368398\pi$
$139$	37.9323i	0.0231466i	−0.999933	+	0.0115733i	$0.996316\pi$
$140$	0	0
$141$	313.007i	0.186950i
$142$	0	0
$143$	−109.504	−0.0640364
$144$	0	0
$145$	−492.541	−0.282092
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	2184.00i	1.20081i	0.799698	+	0.600403i	$0.204993\pi$
$149$	2184.00i	1.20081i	−0.799698	+	0.600403i	$0.795007\pi$
$150$	0	0
$151$	−453.710	−0.244519	−0.122259	−	0.992498i	$-0.539014\pi$
$151$	−453.710	−0.244519	−0.122259	+	0.992498i	$0.539014\pi$
$152$	0	0
$153$	−444.884	−0.235077
$154$	0	0
$155$	− 246.330i	− 0.127649i
$156$	0	0
$157$	− 1627.64i	− 0.827386i	−0.910416	−	0.413693i	$-0.864239\pi$
$157$	− 1627.64i	− 0.827386i	0.910416	−	0.413693i	$-0.135761\pi$
$158$	0	0
$159$	161.829	0.0807160
$160$	0	0
$161$	677.035	0.331415
$162$	0	0
$163$	− 1559.24i	− 0.749257i	−0.927175	−	0.374628i	$-0.877770\pi$
$163$	− 1559.24i	− 0.749257i	0.927175	−	0.374628i	$-0.122230\pi$
$164$	0	0
$165$	− 8.79161i	− 0.00414803i
$166$	0	0
$167$	2167.37	1.00429	0.502145	−	0.864784i	$-0.332544\pi$
$167$	2167.37	1.00429	0.502145	+	0.864784i	$0.332544\pi$
$168$	0	0
$169$	−5018.33	−2.28418
$170$	0	0
$171$	− 20.4395i	− 0.00914064i
$172$	0	0
$173$	200.341i	0.0880440i	0.999031	+	0.0440220i	$0.0140172\pi$
$173$	200.341i	0.0880440i	−0.999031	+	0.0440220i	$0.985983\pi$
$174$	0	0
$175$	−838.827	−0.362339
$176$	0	0
$177$	−1186.92	−0.504038
$178$	0	0
$179$	− 3769.55i	− 1.57402i	−0.616941	−	0.787009i	$-0.711628\pi$
$179$	− 3769.55i	− 1.57402i	0.616941	−	0.787009i	$-0.288372\pi$
$180$	0	0
$181$	3636.66i	1.49343i	0.665144	+	0.746715i	$0.268370\pi$
$181$	3636.66i	1.49343i	−0.665144	+	0.746715i	$0.731630\pi$
$182$	0	0
$183$	−228.413	−0.0922667
$184$	0	0
$185$	−209.665	−0.0833236
$186$	0	0
$187$	63.7246i	0.0249198i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	−2620.07	−0.992574	−0.496287	−	0.868158i	$-0.665304\pi$
$191$	−2620.07	−0.992574	−0.496287	+	0.868158i	$0.665304\pi$
$192$	0	0
$193$	−1220.05	−0.455030	−0.227515	−	0.973775i	$-0.573060\pi$
$193$	−1220.05	−0.455030	−0.227515	+	0.973775i	$0.573060\pi$
$194$	0	0
$195$	− 579.287i	− 0.212736i
$196$	0	0
$197$	− 744.290i	− 0.269180i	−0.990901	−	0.134590i	$-0.957028\pi$
$197$	− 744.290i	− 0.269180i	0.990901	−	0.134590i	$-0.0429717\pi$
$198$	0	0
$199$	−1537.26	−0.547607	−0.273803	−	0.961786i	$-0.588282\pi$
$199$	−1537.26	−0.547607	−0.273803	+	0.961786i	$0.588282\pi$
$200$	0	0
$201$	−167.806	−0.0588861
$202$	0	0
$203$	1516.69i	0.524387i
$204$	0	0
$205$	810.730i	0.276214i
$206$	0	0
$207$	870.473	0.292281
$208$	0	0
$209$	−2.92773	−0.000968973 0
$210$	0	0
$211$	− 1381.22i	− 0.450650i	−0.974284	−	0.225325i	$-0.927656\pi$
$211$	− 1381.22i	− 0.450650i	0.974284	−	0.225325i	$-0.0723444\pi$
$212$	0	0
$213$	3015.62i	0.970079i
$214$	0	0
$215$	−1138.63	−0.361182
$216$	0	0
$217$	−758.526	−0.237291
$218$	0	0
$219$	449.077i	0.138565i
$220$	0	0
$221$	4198.87i	1.27804i
$222$	0	0
$223$	−1873.11	−0.562477	−0.281239	−	0.959638i	$-0.590745\pi$
$223$	−1873.11	−0.562477	−0.281239	+	0.959638i	$0.590745\pi$
$224$	0	0
$225$	−1078.49	−0.319553
$226$	0	0
$227$	381.004i	0.111401i	0.998448	+	0.0557007i	$0.0177393\pi$
$227$	381.004i	0.111401i	−0.998448	+	0.0557007i	$0.982261\pi$
$228$	0	0
$229$	3386.71i	0.977293i	0.872482	+	0.488647i	$0.162509\pi$
$229$	3386.71i	0.977293i	−0.872482	+	0.488647i	$0.837491\pi$
$230$	0	0
$231$	−27.0721	−0.00771088
$232$	0	0
$233$	−3352.26	−0.942548	−0.471274	−	0.881987i	$-0.656206\pi$
$233$	−3352.26	−0.942548	−0.471274	+	0.881987i	$0.656206\pi$
$234$	0	0
$235$	237.179i	0.0658378i
$236$	0	0
$237$	995.953i	0.272971i
$238$	0	0
$239$	−193.704	−0.0524255	−0.0262127	−	0.999656i	$-0.508345\pi$
$239$	−193.704	−0.0524255	−0.0262127	+	0.999656i	$0.508345\pi$
$240$	0	0
$241$	−975.351	−0.260697	−0.130348	−	0.991468i	$-0.541610\pi$
$241$	−975.351	−0.260697	−0.130348	+	0.991468i	$0.541610\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 111.389i	− 0.0290463i
$246$	0	0
$247$	−192.911	−0.0496948
$248$	0	0
$249$	2189.08	0.557138
$250$	0	0
$251$	− 3566.26i	− 0.896814i	−0.893829	−	0.448407i	$-0.851991\pi$
$251$	− 3566.26i	− 0.896814i	0.893829	−	0.448407i	$-0.148009\pi$
$252$	0	0
$253$	− 124.685i	− 0.0309838i
$254$	0	0
$255$	−337.109	−0.0827865
$256$	0	0
$257$	−4455.77	−1.08149	−0.540746	−	0.841186i	$-0.681858\pi$
$257$	−4455.77	−1.08149	−0.540746	+	0.841186i	$0.681858\pi$
$258$	0	0
$259$	645.623i	0.154892i
$260$	0	0
$261$	1950.03i	0.462466i
$262$	0	0
$263$	−7380.88	−1.73051	−0.865256	−	0.501331i	$-0.832844\pi$
$263$	−7380.88	−1.73051	−0.865256	+	0.501331i	$0.832844\pi$
$264$	0	0
$265$	122.625	0.0284256
$266$	0	0
$267$	21.9358i	0.00502789i
$268$	0	0
$269$	− 7834.98i	− 1.77586i	−0.459976	−	0.887931i	$-0.652142\pi$
$269$	− 7834.98i	− 1.77586i	0.459976	−	0.887931i	$-0.347858\pi$
$270$	0	0
$271$	7758.06	1.73900	0.869500	−	0.493934i	$-0.164441\pi$
$271$	7758.06	1.73900	0.869500	+	0.493934i	$0.164441\pi$
$272$	0	0
$273$	−1783.81	−0.395461
$274$	0	0
$275$	154.482i	0.0338749i
$276$	0	0
$277$	− 5448.96i	− 1.18194i	−0.806695	−	0.590968i	$-0.798746\pi$
$277$	− 5448.96i	− 1.18194i	0.806695	−	0.590968i	$-0.201254\pi$
$278$	0	0
$279$	−975.247	−0.209271
$280$	0	0
$281$	230.340	0.0489001	0.0244501	−	0.999701i	$-0.492217\pi$
$281$	230.340	0.0489001	0.0244501	+	0.999701i	$0.492217\pi$
$282$	0	0
$283$	− 4384.96i	− 0.921055i	−0.887645	−	0.460528i	$-0.847660\pi$
$283$	− 4384.96i	− 0.921055i	0.887645	−	0.460528i	$-0.152340\pi$
$284$	0	0
$285$	− 15.4880i	− 0.00321904i
$286$	0	0
$287$	2496.49	0.513461
$288$	0	0
$289$	−2469.52	−0.502650
$290$	0	0
$291$	5296.63i	1.06699i
$292$	0	0
$293$	− 6777.54i	− 1.35136i	−0.737196	−	0.675679i	$-0.763850\pi$
$293$	− 6777.54i	− 1.35136i	0.737196	−	0.675679i	$-0.236150\pi$
$294$	0	0
$295$	−899.386	−0.177506
$296$	0	0
$297$	−34.8070	−0.00680036
$298$	0	0
$299$	− 8215.63i	− 1.58904i
$300$	0	0
$301$	3506.21i	0.671410i
$302$	0	0
$303$	2928.48	0.555237
$304$	0	0
$305$	−173.079	−0.0324934
$306$	0	0
$307$	− 4196.76i	− 0.780201i	−0.920772	−	0.390100i	$-0.872440\pi$
$307$	− 4196.76i	− 0.780201i	0.920772	−	0.390100i	$-0.127560\pi$
$308$	0	0
$309$	1177.97i	0.216868i
$310$	0	0
$311$	−6811.77	−1.24199	−0.620996	−	0.783813i	$-0.713272\pi$
$311$	−6811.77	−1.24199	−0.620996	+	0.783813i	$0.713272\pi$
$312$	0	0
$313$	−4822.52	−0.870879	−0.435440	−	0.900218i	$-0.643407\pi$
$313$	−4822.52	−0.870879	−0.435440	+	0.900218i	$0.643407\pi$
$314$	0	0
$315$	− 143.214i	− 0.0256165i
$316$	0	0
$317$	− 2655.85i	− 0.470560i	−0.971928	−	0.235280i	$-0.924399\pi$
$317$	− 2655.85i	− 0.470560i	0.971928	−	0.235280i	$-0.0756006\pi$
$318$	0	0
$319$	279.319	0.0490247
$320$	0	0
$321$	−4827.48	−0.839388
$322$	0	0
$323$	112.262i	0.0193388i
$324$	0	0
$325$	10178.9i	1.73731i
$326$	0	0
$327$	5111.15	0.864365
$328$	0	0
$329$	730.349	0.122387
$330$	0	0
$331$	7732.56i	1.28405i	0.766684	+	0.642024i	$0.221905\pi$
$331$	7732.56i	1.28405i	−0.766684	+	0.642024i	$0.778095\pi$
$332$	0	0
$333$	830.087i	0.136602i
$334$	0	0
$335$	−127.154	−0.0207378
$336$	0	0
$337$	−3505.13	−0.566578	−0.283289	−	0.959035i	$-0.591426\pi$
$337$	−3505.13	−0.566578	−0.283289	+	0.959035i	$0.591426\pi$
$338$	0	0
$339$	− 3745.51i	− 0.600083i
$340$	0	0
$341$	139.693i	0.0221842i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	659.597	0.102932
$346$	0	0
$347$	− 9604.66i	− 1.48589i	−0.669350	−	0.742947i	$-0.733427\pi$
$347$	− 9604.66i	− 1.48589i	0.669350	−	0.742947i	$-0.266573\pi$
$348$	0	0
$349$	− 1999.55i	− 0.306686i	−0.988173	−	0.153343i	$-0.950996\pi$
$349$	− 1999.55i	− 0.306686i	0.988173	−	0.153343i	$-0.0490039\pi$
$350$	0	0
$351$	−2293.46	−0.348764
$352$	0	0
$353$	10784.2	1.62602	0.813012	−	0.582247i	$-0.197826\pi$
$353$	10784.2	1.62602	0.813012	+	0.582247i	$0.197826\pi$
$354$	0	0
$355$	2285.07i	0.341631i
$356$	0	0
$357$	1038.06i	0.153894i
$358$	0	0
$359$	−2583.70	−0.379839	−0.189920	−	0.981800i	$-0.560823\pi$
$359$	−2583.70	−0.379839	−0.189920	+	0.981800i	$0.560823\pi$
$360$	0	0
$361$	6853.84	0.999248
$362$	0	0
$363$	− 3988.01i	− 0.576629i
$364$	0	0
$365$	340.286i	0.0487983i
$366$	0	0
$367$	−4724.95	−0.672045	−0.336023	−	0.941854i	$-0.609082\pi$
$367$	−4724.95	−0.672045	−0.336023	+	0.941854i	$0.609082\pi$
$368$	0	0
$369$	3209.77	0.452830
$370$	0	0
$371$	− 377.600i	− 0.0528411i
$372$	0	0
$373$	6604.60i	0.916818i	0.888741	+	0.458409i	$0.151580\pi$
$373$	6604.60i	0.916818i	−0.888741	+	0.458409i	$0.848420\pi$
$374$	0	0
$375$	−1669.68	−0.229926
$376$	0	0
$377$	18404.6	2.51428
$378$	0	0
$379$	4269.71i	0.578681i	0.957226	+	0.289341i	$0.0934360\pi$
$379$	4269.71i	0.578681i	−0.957226	+	0.289341i	$0.906564\pi$
$380$	0	0
$381$	− 1782.01i	− 0.239620i
$382$	0	0
$383$	5048.29	0.673514	0.336757	−	0.941592i	$-0.390670\pi$
$383$	5048.29	0.673514	0.336757	+	0.941592i	$0.390670\pi$
$384$	0	0
$385$	−20.5138	−0.00271553
$386$	0	0
$387$	4507.98i	0.592128i
$388$	0	0
$389$	3202.20i	0.417372i	0.977983	+	0.208686i	$0.0669187\pi$
$389$	3202.20i	0.417372i	−0.977983	+	0.208686i	$0.933081\pi$
$390$	0	0
$391$	−4780.98	−0.618375
$392$	0	0
$393$	−6989.77	−0.897169
$394$	0	0
$395$	754.679i	0.0961317i
$396$	0	0
$397$	− 7316.98i	− 0.925009i	−0.886617	−	0.462505i	$-0.846951\pi$
$397$	− 7316.98i	− 0.925009i	0.886617	−	0.462505i	$-0.153049\pi$
$398$	0	0
$399$	−47.6922	−0.00598395
$400$	0	0
$401$	7344.30	0.914605	0.457303	−	0.889311i	$-0.348816\pi$
$401$	7344.30	0.914605	0.457303	+	0.889311i	$0.348816\pi$
$402$	0	0
$403$	9204.51i	1.13774i
$404$	0	0
$405$	− 184.132i	− 0.0225916i
$406$	0	0
$407$	118.901	0.0144808
$408$	0	0
$409$	−8857.93	−1.07090	−0.535448	−	0.844568i	$-0.679857\pi$
$409$	−8857.93	−1.07090	−0.535448	+	0.844568i	$0.679857\pi$
$410$	0	0
$411$	8686.79i	1.04255i
$412$	0	0
$413$	2769.49i	0.329970i
$414$	0	0
$415$	1658.76	0.196206
$416$	0	0
$417$	113.797	0.0133637
$418$	0	0
$419$	− 5501.41i	− 0.641435i	−0.947175	−	0.320718i	$-0.896076\pi$
$419$	− 5501.41i	− 0.641435i	0.947175	−	0.320718i	$-0.103924\pi$
$420$	0	0
$421$	− 11163.2i	− 1.29231i	−0.763208	−	0.646153i	$-0.776377\pi$
$421$	− 11163.2i	− 1.29231i	0.763208	−	0.646153i	$-0.223623\pi$
$422$	0	0
$423$	939.021	0.107936
$424$	0	0
$425$	5923.50	0.676075
$426$	0	0
$427$	532.965i	0.0604027i
$428$	0	0
$429$	328.513i	0.0369714i
$430$	0	0
$431$	−6628.22	−0.740766	−0.370383	−	0.928879i	$-0.620774\pi$
$431$	−6628.22	−0.740766	−0.370383	+	0.928879i	$0.620774\pi$
$432$	0	0
$433$	1994.30	0.221339	0.110670	−	0.993857i	$-0.464700\pi$
$433$	1994.30	0.221339	0.110670	+	0.993857i	$0.464700\pi$
$434$	0	0
$435$	1477.62i	0.162866i
$436$	0	0
$437$	− 219.655i	− 0.0240447i
$438$	0	0
$439$	−3688.02	−0.400956	−0.200478	−	0.979698i	$-0.564249\pi$
$439$	−3688.02	−0.400956	−0.200478	+	0.979698i	$0.564249\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 10838.6i	− 1.16243i	−0.813749	−	0.581217i	$-0.802577\pi$
$443$	− 10838.6i	− 1.16243i	0.813749	−	0.581217i	$-0.197423\pi$
$444$	0	0
$445$	16.6217i	0.00177066i
$446$	0	0
$447$	6551.99	0.693286
$448$	0	0
$449$	−4813.96	−0.505980	−0.252990	−	0.967469i	$-0.581414\pi$
$449$	−4813.96	−0.505980	−0.252990	+	0.967469i	$0.581414\pi$
$450$	0	0
$451$	− 459.764i	− 0.0480032i
$452$	0	0
$453$	1361.13i	0.141173i
$454$	0	0
$455$	−1351.67	−0.139269
$456$	0	0
$457$	1434.31	0.146814	0.0734071	−	0.997302i	$-0.476613\pi$
$457$	1434.31	0.146814	0.0734071	+	0.997302i	$0.476613\pi$
$458$	0	0
$459$	1334.65i	0.135722i
$460$	0	0
$461$	9808.42i	0.990940i	0.868625	+	0.495470i	$0.165004\pi$
$461$	9808.42i	0.990940i	−0.868625	+	0.495470i	$0.834996\pi$
$462$	0	0
$463$	6737.06	0.676237	0.338119	−	0.941103i	$-0.390209\pi$
$463$	6737.06	0.676237	0.338119	+	0.941103i	$0.390209\pi$
$464$	0	0
$465$	−738.989	−0.0736985
$466$	0	0
$467$	− 7967.41i	− 0.789481i	−0.918793	−	0.394741i	$-0.870834\pi$
$467$	− 7967.41i	− 0.789481i	0.918793	−	0.394741i	$-0.129166\pi$
$468$	0	0
$469$	391.547i	0.0385500i
$470$	0	0
$471$	−4882.91	−0.477691
$472$	0	0
$473$	645.717	0.0627697
$474$	0	0
$475$	272.146i	0.0262883i
$476$	0	0
$477$	− 485.486i	− 0.0466014i
$478$	0	0
$479$	11687.0	1.11481	0.557405	−	0.830241i	$-0.311797\pi$
$479$	11687.0	1.11481	0.557405	+	0.830241i	$0.311797\pi$
$480$	0	0
$481$	7834.47	0.742663
$482$	0	0
$483$	− 2031.10i	− 0.191343i
$484$	0	0
$485$	4013.49i	0.375759i
$486$	0	0
$487$	4882.25	0.454283	0.227141	−	0.973862i	$-0.427062\pi$
$487$	4882.25	0.454283	0.227141	+	0.973862i	$0.427062\pi$
$488$	0	0
$489$	−4677.71	−0.432584
$490$	0	0
$491$	7670.20i	0.704992i	0.935813	+	0.352496i	$0.114667\pi$
$491$	7670.20i	0.704992i	−0.935813	+	0.352496i	$0.885333\pi$
$492$	0	0
$493$	− 10710.3i	− 0.978435i
$494$	0	0
$495$	−26.3748	−0.00239487
$496$	0	0
$497$	7036.45	0.635066
$498$	0	0
$499$	− 1140.60i	− 0.102326i	−0.998690	−	0.0511628i	$-0.983707\pi$
$499$	− 1140.60i	− 0.102326i	0.998690	−	0.0511628i	$-0.0162927\pi$
$500$	0	0
$501$	− 6502.12i	− 0.579827i
$502$	0	0
$503$	35.6993	0.00316452	0.00158226	−	0.999999i	$-0.499496\pi$
$503$	35.6993	0.00316452	0.00158226	+	0.999999i	$0.499496\pi$
$504$	0	0
$505$	2219.04	0.195537
$506$	0	0
$507$	15055.0i	1.31877i
$508$	0	0
$509$	− 15462.0i	− 1.34644i	−0.739441	−	0.673222i	$-0.764910\pi$
$509$	− 15462.0i	− 1.34644i	0.739441	−	0.673222i	$-0.235090\pi$
$510$	0	0
$511$	1047.85	0.0907123
$512$	0	0
$513$	−61.3186	−0.00527735
$514$	0	0
$515$	892.600i	0.0763741i
$516$	0	0
$517$	− 134.504i	− 0.0114419i
$518$	0	0
$519$	601.022	0.0508322
$520$	0	0
$521$	13515.3	1.13650	0.568248	−	0.822857i	$-0.307621\pi$
$521$	13515.3	1.13650	0.568248	+	0.822857i	$0.307621\pi$
$522$	0	0
$523$	− 4507.14i	− 0.376833i	−0.982089	−	0.188416i	$-0.939665\pi$
$523$	− 4507.14i	− 0.376833i	0.982089	−	0.188416i	$-0.0603354\pi$
$524$	0	0
$525$	2516.48i	0.209197i
$526$	0	0
$527$	5356.44	0.442752
$528$	0	0
$529$	−2812.39	−0.231149
$530$	0	0
$531$	3560.77i	0.291006i
$532$	0	0
$533$	− 30294.3i	− 2.46189i
$534$	0	0
$535$	−3658.00	−0.295606
$536$	0	0
$537$	−11308.6	−0.908760
$538$	0	0
$539$	63.1682i	0.00504796i
$540$	0	0
$541$	3529.31i	0.280475i	0.990118	+	0.140237i	$0.0447866\pi$
$541$	3529.31i	0.280475i	−0.990118	+	0.140237i	$0.955213\pi$
$542$	0	0
$543$	10910.0	0.862233
$544$	0	0
$545$	3872.95	0.304402
$546$	0	0
$547$	12222.7i	0.955404i	0.878522	+	0.477702i	$0.158530\pi$
$547$	12222.7i	0.955404i	−0.878522	+	0.477702i	$0.841470\pi$
$548$	0	0
$549$	685.240i	0.0532702i
$550$	0	0
$551$	492.069	0.0380451
$552$	0	0
$553$	2323.89	0.178701
$554$	0	0
$555$	628.994i	0.0481069i
$556$	0	0
$557$	− 3581.23i	− 0.272426i	−0.990680	−	0.136213i	$-0.956507\pi$
$557$	− 3581.23i	− 0.272426i	0.990680	−	0.136213i	$-0.0434932\pi$
$558$	0	0
$559$	42546.9	3.21921
$560$	0	0
$561$	191.174	0.0143875
$562$	0	0
$563$	12349.8i	0.924483i	0.886754	+	0.462241i	$0.152955\pi$
$563$	12349.8i	0.924483i	−0.886754	+	0.462241i	$0.847045\pi$
$564$	0	0
$565$	− 2838.14i	− 0.211330i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	20492.4	1.50981	0.754907	−	0.655832i	$-0.227682\pi$
$569$	20492.4	1.50981	0.754907	+	0.655832i	$0.227682\pi$
$570$	0	0
$571$	21122.8i	1.54809i	0.633128	+	0.774047i	$0.281771\pi$
$571$	21122.8i	1.54809i	−0.633128	+	0.774047i	$0.718229\pi$
$572$	0	0
$573$	7860.21i	0.573063i
$574$	0	0
$575$	−11590.1	−0.840592
$576$	0	0
$577$	−3049.70	−0.220036	−0.110018	−	0.993930i	$-0.535091\pi$
$577$	−3049.70	−0.220036	−0.110018	+	0.993930i	$0.535091\pi$
$578$	0	0
$579$	3660.14i	0.262712i
$580$	0	0
$581$	− 5107.85i	− 0.364732i
$582$	0	0
$583$	−69.5404	−0.00494008
$584$	0	0
$585$	−1737.86	−0.122823
$586$	0	0
$587$	14706.6i	1.03408i	0.855961	+	0.517041i	$0.172967\pi$
$587$	14706.6i	1.03408i	−0.855961	+	0.517041i	$0.827033\pi$
$588$	0	0
$589$	246.094i	0.0172158i
$590$	0	0
$591$	−2232.87	−0.155411
$592$	0	0
$593$	4886.30	0.338375	0.169187	−	0.985584i	$-0.445886\pi$
$593$	4886.30	0.338375	0.169187	+	0.985584i	$0.445886\pi$
$594$	0	0
$595$	786.587i	0.0541965i
$596$	0	0
$597$	4611.79i	0.316161i
$598$	0	0
$599$	−25238.6	−1.72157	−0.860785	−	0.508968i	$-0.830027\pi$
$599$	−25238.6	−1.72157	−0.860785	+	0.508968i	$0.830027\pi$
$600$	0	0
$601$	−11342.2	−0.769813	−0.384907	−	0.922956i	$-0.625766\pi$
$601$	−11342.2	−0.769813	−0.384907	+	0.922956i	$0.625766\pi$
$602$	0	0
$603$	503.417i	0.0339979i
$604$	0	0
$605$	− 3021.90i	− 0.203070i
$606$	0	0
$607$	17821.8	1.19171	0.595853	−	0.803094i	$-0.296814\pi$
$607$	17821.8	1.19171	0.595853	+	0.803094i	$0.296814\pi$
$608$	0	0
$609$	4550.06	0.302755
$610$	0	0
$611$	− 8862.59i	− 0.586812i
$612$	0	0
$613$	14938.7i	0.984290i	0.870513	+	0.492145i	$0.163787\pi$
$613$	14938.7i	0.984290i	−0.870513	+	0.492145i	$0.836213\pi$
$614$	0	0
$615$	2432.19	0.159472
$616$	0	0
$617$	19255.2	1.25638	0.628188	−	0.778061i	$-0.283797\pi$
$617$	19255.2	1.25638	0.628188	+	0.778061i	$0.283797\pi$
$618$	0	0
$619$	18907.1i	1.22769i	0.789426	+	0.613846i	$0.210378\pi$
$619$	18907.1i	1.22769i	−0.789426	+	0.613846i	$0.789622\pi$
$620$	0	0
$621$	− 2611.42i	− 0.168748i
$622$	0	0
$623$	51.1834	0.00329153
$624$	0	0
$625$	13713.9	0.877687
$626$	0	0
$627$	8.78319i	0 0.000559437i
$628$	0	0
$629$	− 4559.17i	− 0.289008i
$630$	0	0
$631$	−21736.8	−1.37136	−0.685681	−	0.727903i	$-0.740495\pi$
$631$	−21736.8	−1.37136	−0.685681	+	0.727903i	$0.740495\pi$
$632$	0	0
$633$	−4143.66	−0.260183
$634$	0	0
$635$	− 1350.31i	− 0.0843863i
$636$	0	0
$637$	4162.21i	0.258890i
$638$	0	0
$639$	9046.86	0.560076
$640$	0	0
$641$	8925.34	0.549969	0.274984	−	0.961449i	$-0.411327\pi$
$641$	8925.34	0.549969	0.274984	+	0.961449i	$0.411327\pi$
$642$	0	0
$643$	25304.1i	1.55194i	0.630772	+	0.775968i	$0.282738\pi$
$643$	25304.1i	1.55194i	−0.630772	+	0.775968i	$0.717262\pi$
$644$	0	0
$645$	3415.90i	0.208529i
$646$	0	0
$647$	−13834.3	−0.840625	−0.420312	−	0.907380i	$-0.638079\pi$
$647$	−13834.3	−0.840625	−0.420312	+	0.907380i	$0.638079\pi$
$648$	0	0
$649$	510.040	0.0308487
$650$	0	0
$651$	2275.58i	0.137000i
$652$	0	0
$653$	− 5409.70i	− 0.324193i	−0.986775	−	0.162096i	$-0.948174\pi$
$653$	− 5409.70i	− 0.324193i	0.986775	−	0.162096i	$-0.0518255\pi$
$654$	0	0
$655$	−5296.46	−0.315954
$656$	0	0
$657$	1347.23	0.0800007
$658$	0	0
$659$	− 14200.5i	− 0.839410i	−0.907661	−	0.419705i	$-0.862134\pi$
$659$	− 14200.5i	− 0.839410i	0.907661	−	0.419705i	$-0.137866\pi$
$660$	0	0
$661$	4777.95i	0.281151i	0.990070	+	0.140576i	$0.0448953\pi$
$661$	4777.95i	0.281151i	−0.990070	+	0.140576i	$0.955105\pi$
$662$	0	0
$663$	12596.6	0.737876
$664$	0	0
$665$	−36.1386	−0.00210736
$666$	0	0
$667$	20956.1i	1.21653i
$668$	0	0
$669$	5619.32i	0.324747i
$670$	0	0
$671$	98.1529	0.00564702
$672$	0	0
$673$	5088.95	0.291478	0.145739	−	0.989323i	$-0.453444\pi$
$673$	5088.95	0.291478	0.145739	+	0.989323i	$0.453444\pi$
$674$	0	0
$675$	3235.47i	0.184494i
$676$	0	0
$677$	5917.03i	0.335908i	0.985795	+	0.167954i	$0.0537160\pi$
$677$	5917.03i	0.335908i	−0.985795	+	0.167954i	$0.946284\pi$
$678$	0	0
$679$	12358.8	0.698508
$680$	0	0
$681$	1143.01	0.0643177
$682$	0	0
$683$	16751.3i	0.938462i	0.883076	+	0.469231i	$0.155469\pi$
$683$	16751.3i	0.938462i	−0.883076	+	0.469231i	$0.844531\pi$
$684$	0	0
$685$	6582.37i	0.367152i
$686$	0	0
$687$	10160.1	0.564241
$688$	0	0
$689$	−4582.08	−0.253357
$690$	0	0
$691$	− 18357.4i	− 1.01064i	−0.862933	−	0.505319i	$-0.831375\pi$
$691$	− 18357.4i	− 1.01064i	0.862933	−	0.505319i	$-0.168625\pi$
$692$	0	0
$693$	81.2163i	0.00445188i
$694$	0	0
$695$	86.2291	0.00470627
$696$	0	0
$697$	−17629.4	−0.958048
$698$	0	0
$699$	10056.8i	0.544180i
$700$	0	0
$701$	− 19642.8i	− 1.05834i	−0.848515	−	0.529171i	$-0.822503\pi$
$701$	− 19642.8i	− 1.05834i	0.848515	−	0.529171i	$-0.177497\pi$
$702$	0	0
$703$	209.464	0.0112377
$704$	0	0
$705$	711.538	0.0380115
$706$	0	0
$707$	− 6833.13i	− 0.363488i
$708$	0	0
$709$	609.731i	0.0322975i	0.999870	+	0.0161488i	$0.00514053\pi$
$709$	609.731i	0.0322975i	−0.999870	+	0.0161488i	$0.994859\pi$
$710$	0	0
$711$	2987.86	0.157600
$712$	0	0
$713$	−10480.6	−0.550492
$714$	0	0
$715$	248.929i	0.0130202i
$716$	0	0
$717$	581.113i	0.0302679i
$718$	0	0
$719$	33133.0	1.71857	0.859285	−	0.511497i	$-0.170909\pi$
$719$	33133.0	1.71857	0.859285	+	0.511497i	$0.170909\pi$
$720$	0	0
$721$	2748.59	0.141974
$722$	0	0
$723$	2926.05i	0.150513i
$724$	0	0
$725$	− 25964.0i	− 1.33004i
$726$	0	0
$727$	3739.07	0.190749	0.0953745	−	0.995441i	$-0.469595\pi$
$727$	3739.07	0.190749	0.0953745	+	0.995441i	$0.469595\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 24759.6i	− 1.25276i
$732$	0	0
$733$	22305.5i	1.12397i	0.827147	+	0.561986i	$0.189963\pi$
$733$	22305.5i	1.12397i	−0.827147	+	0.561986i	$0.810037\pi$
$734$	0	0
$735$	−334.166	−0.0167699
$736$	0	0
$737$	72.1088	0.00360402
$738$	0	0
$739$	350.664i	0.0174552i	0.999962	+	0.00872759i	$0.00277811\pi$
$739$	350.664i	0.0174552i	−0.999962	+	0.00872759i	$0.997222\pi$
$740$	0	0
$741$	578.732i	0.0286913i
$742$	0	0
$743$	−29648.6	−1.46393	−0.731966	−	0.681341i	$-0.761397\pi$
$743$	−29648.6	−1.46393	−0.731966	+	0.681341i	$0.761397\pi$
$744$	0	0
$745$	4964.74	0.244153
$746$	0	0
$747$	− 6567.24i	− 0.321664i
$748$	0	0
$749$	11264.1i	0.549509i
$750$	0	0
$751$	28940.1	1.40618	0.703088	−	0.711103i	$-0.251804\pi$
$751$	28940.1	1.40618	0.703088	+	0.711103i	$0.251804\pi$
$752$	0	0
$753$	−10698.8	−0.517776
$754$	0	0
$755$	1031.39i	0.0497167i
$756$	0	0
$757$	− 30841.3i	− 1.48077i	−0.672181	−	0.740387i	$-0.734642\pi$
$757$	− 30841.3i	− 1.48077i	0.672181	−	0.740387i	$-0.265358\pi$
$758$	0	0
$759$	−374.056	−0.0178885
$760$	0	0
$761$	23222.2	1.10618	0.553090	−	0.833122i	$-0.313449\pi$
$761$	23222.2	1.10618	0.553090	+	0.833122i	$0.313449\pi$
$762$	0	0
$763$	− 11926.0i	− 0.565860i
$764$	0	0
$765$	1011.33i	0.0477968i
$766$	0	0
$767$	33607.0	1.58211
$768$	0	0
$769$	8146.67	0.382024	0.191012	−	0.981588i	$-0.438823\pi$
$769$	8146.67	0.382024	0.191012	+	0.981588i	$0.438823\pi$
$770$	0	0
$771$	13367.3i	0.624399i
$772$	0	0
$773$	− 30694.9i	− 1.42823i	−0.700030	−	0.714113i	$-0.746830\pi$
$773$	− 30694.9i	− 1.42823i	0.700030	−	0.714113i	$-0.253170\pi$
$774$	0	0
$775$	12985.1	0.601858
$776$	0	0
$777$	1936.87	0.0894271
$778$	0	0
$779$	− 809.954i	− 0.0372524i
$780$	0	0
$781$	− 1295.86i	− 0.0593720i
$782$	0	0
$783$	5850.08	0.267005
$784$	0	0
$785$	−3700.00	−0.168228
$786$	0	0
$787$	38701.3i	1.75292i	0.481470	+	0.876462i	$0.340103\pi$
$787$	38701.3i	1.75292i	−0.481470	+	0.876462i	$0.659897\pi$
$788$	0	0
$789$	22142.6i	0.999111i
$790$	0	0
$791$	−8739.52	−0.392846
$792$	0	0
$793$	6467.38	0.289613
$794$	0	0
$795$	− 367.875i	− 0.0164115i
$796$	0	0
$797$	25536.4i	1.13494i	0.823395	+	0.567469i	$0.192077\pi$
$797$	25536.4i	1.13494i	−0.823395	+	0.567469i	$0.807923\pi$
$798$	0	0
$799$	−5157.47	−0.228358
$800$	0	0
$801$	65.8073	0.00290285
$802$	0	0
$803$	− 192.975i	− 0.00848064i
$804$	0	0
$805$	− 1539.06i	− 0.0673847i
$806$	0	0
$807$	−23504.9	−1.02529
$808$	0	0
$809$	39310.4	1.70838	0.854189	−	0.519962i	$-0.174054\pi$
$809$	39310.4	1.70838	0.854189	+	0.519962i	$0.174054\pi$
$810$	0	0
$811$	− 30741.9i	− 1.33107i	−0.746368	−	0.665533i	$-0.768204\pi$
$811$	− 30741.9i	− 1.33107i	0.746368	−	0.665533i	$-0.231796\pi$
$812$	0	0
$813$	− 23274.2i	− 1.00401i
$814$	0	0
$815$	−3544.51	−0.152342
$816$	0	0
$817$	1137.54	0.0487119
$818$	0	0
$819$	5351.42i	0.228319i
$820$	0	0
$821$	− 14406.2i	− 0.612398i	−0.951968	−	0.306199i	$-0.900943\pi$
$821$	− 14406.2i	− 0.612398i	0.951968	−	0.306199i	$-0.0990573\pi$
$822$	0	0
$823$	−15841.4	−0.670956	−0.335478	−	0.942048i	$-0.608898\pi$
$823$	−15841.4	−0.670956	−0.335478	+	0.942048i	$0.608898\pi$
$824$	0	0
$825$	463.445	0.0195577
$826$	0	0
$827$	3204.56i	0.134744i	0.997728	+	0.0673722i	$0.0214615\pi$
$827$	3204.56i	0.134744i	−0.997728	+	0.0673722i	$0.978539\pi$
$828$	0	0
$829$	877.989i	0.0367839i	0.999831	+	0.0183919i	$0.00585466\pi$
$829$	877.989i	0.0367839i	−0.999831	+	0.0183919i	$0.994145\pi$
$830$	0	0
$831$	−16346.9	−0.682391
$832$	0	0
$833$	2422.15	0.100747
$834$	0	0
$835$	− 4926.95i	− 0.204197i
$836$	0	0
$837$	2925.74i	0.120823i
$838$	0	0
$839$	9218.29	0.379321	0.189661	−	0.981850i	$-0.439261\pi$
$839$	9218.29	0.379321	0.189661	+	0.981850i	$0.439261\pi$
$840$	0	0
$841$	−22556.7	−0.924873
$842$	0	0
$843$	− 691.020i	− 0.0282325i
$844$	0	0
$845$	11407.9i	0.464429i
$846$	0	0
$847$	−9305.37	−0.377493
$848$	0	0
$849$	−13154.9	−0.531771
$850$	0	0
$851$	8920.60i	0.359335i
$852$	0	0
$853$	15797.2i	0.634100i	0.948409	+	0.317050i	$0.102692\pi$
$853$	15797.2i	0.634100i	−0.948409	+	0.317050i	$0.897308\pi$
$854$	0	0
$855$	−46.4639	−0.00185851
$856$	0	0
$857$	−30795.4	−1.22748	−0.613740	−	0.789508i	$-0.710336\pi$
$857$	−30795.4	−1.22748	−0.613740	+	0.789508i	$0.710336\pi$
$858$	0	0
$859$	15082.0i	0.599058i	0.954087	+	0.299529i	$0.0968295\pi$
$859$	15082.0i	0.599058i	−0.954087	+	0.299529i	$0.903171\pi$
$860$	0	0
$861$	− 7489.47i	− 0.296447i
$862$	0	0
$863$	−20334.2	−0.802069	−0.401034	−	0.916063i	$-0.631349\pi$
$863$	−20334.2	−0.802069	−0.401034	+	0.916063i	$0.631349\pi$
$864$	0	0
$865$	455.421	0.0179015
$866$	0	0
$867$	7408.56i	0.290205i
$868$	0	0
$869$	− 427.977i	− 0.0167067i
$870$	0	0
$871$	4751.31	0.184836
$872$	0	0
$873$	15889.9	0.616026
$874$	0	0
$875$	3895.93i	0.150522i
$876$	0	0
$877$	5439.67i	0.209446i	0.994501	+	0.104723i	$0.0333957\pi$
$877$	5439.67i	0.209446i	−0.994501	+	0.104723i	$0.966604\pi$
$878$	0	0
$879$	−20332.6	−0.780207
$880$	0	0
$881$	30529.7	1.16750	0.583752	−	0.811932i	$-0.301584\pi$
$881$	30529.7	1.16750	0.583752	+	0.811932i	$0.301584\pi$
$882$	0	0
$883$	23545.5i	0.897360i	0.893693	+	0.448680i	$0.148106\pi$
$883$	23545.5i	0.897360i	−0.893693	+	0.448680i	$0.851894\pi$
$884$	0	0
$885$	2698.16i	0.102483i
$886$	0	0
$887$	3763.59	0.142468	0.0712339	−	0.997460i	$-0.477306\pi$
$887$	3763.59	0.142468	0.0712339	+	0.997460i	$0.477306\pi$
$888$	0	0
$889$	−4158.02	−0.156868
$890$	0	0
$891$	104.421i	0.00392619i
$892$	0	0
$893$	− 236.952i	− 0.00887940i
$894$	0	0
$895$	−8569.07	−0.320036
$896$	0	0
$897$	−24646.9	−0.917432
$898$	0	0
$899$	− 23478.5i	− 0.871025i
$900$	0	0
$901$	2666.48i	0.0985942i
$902$	0	0
$903$	10518.6	0.387639
$904$	0	0
$905$	8266.99	0.303651
$906$	0	0
$907$	− 36390.1i	− 1.33221i	−0.745859	−	0.666104i	$-0.767961\pi$
$907$	− 36390.1i	− 1.33221i	0.745859	−	0.666104i	$-0.232039\pi$
$908$	0	0
$909$	− 8785.45i	− 0.320566i
$910$	0	0
$911$	−9706.33	−0.353002	−0.176501	−	0.984300i	$-0.556478\pi$
$911$	−9706.33	−0.353002	−0.176501	+	0.984300i	$0.556478\pi$
$912$	0	0
$913$	−940.682	−0.0340986
$914$	0	0
$915$	519.237i	0.0187601i
$916$	0	0
$917$	16309.5i	0.587335i
$918$	0	0
$919$	−25416.2	−0.912300	−0.456150	−	0.889903i	$-0.650772\pi$
$919$	−25416.2	−0.912300	−0.456150	+	0.889903i	$0.650772\pi$
$920$	0	0
$921$	−12590.3	−0.450449
$922$	0	0
$923$	− 85385.4i	− 3.04496i
$924$	0	0
$925$	− 11052.4i	− 0.392865i
$926$	0	0
$927$	3533.91	0.125209
$928$	0	0
$929$	43936.5	1.55168	0.775840	−	0.630930i	$-0.217326\pi$
$929$	43936.5	1.55168	0.775840	+	0.630930i	$0.217326\pi$
$930$	0	0
$931$	111.282i	0.00391742i
$932$	0	0
$933$	20435.3i	0.717065i
$934$	0	0
$935$	144.861	0.00506680
$936$	0	0
$937$	35754.0	1.24657	0.623283	−	0.781997i	$-0.285799\pi$
$937$	35754.0	1.24657	0.623283	+	0.781997i	$0.285799\pi$
$938$	0	0
$939$	14467.6i	0.502802i
$940$	0	0
$941$	− 1656.97i	− 0.0574023i	−0.999588	−	0.0287012i	$-0.990863\pi$
$941$	− 1656.97i	− 0.0574023i	0.999588	−	0.0287012i	$-0.00913712\pi$
$942$	0	0
$943$	34494.1	1.19118
$944$	0	0
$945$	−429.641	−0.0147897
$946$	0	0
$947$	− 56.9997i	− 0.00195590i	−1.00000		0.000977952i	$-0.999689\pi$
$947$	− 56.9997i	− 0.00195590i	1.00000		0.000977952i	$-0.000311292\pi$
$948$	0	0
$949$	− 12715.3i	− 0.434939i
$950$	0	0
$951$	−7967.55	−0.271678
$952$	0	0
$953$	−36757.1	−1.24940	−0.624700	−	0.780865i	$-0.714779\pi$
$953$	−36757.1	−1.24940	−0.624700	+	0.780865i	$0.714779\pi$
$954$	0	0
$955$	5956.04i	0.201814i
$956$	0	0
$957$	− 837.957i	− 0.0283044i
$958$	0	0
$959$	20269.2	0.682508
$960$	0	0
$961$	−18048.9	−0.605852
$962$	0	0
$963$	14482.4i	0.484621i
$964$	0	0
$965$	2773.45i	0.0925188i
$966$	0	0
$967$	−53075.2	−1.76503	−0.882514	−	0.470286i	$-0.844151\pi$
$967$	−53075.2	−1.76503	−0.882514	+	0.470286i	$0.844151\pi$
$968$	0	0
$969$	336.786	0.0111652
$970$	0	0
$971$	− 10735.7i	− 0.354815i	−0.984138	−	0.177407i	$-0.943229\pi$
$971$	− 10735.7i	− 0.354815i	0.984138	−	0.177407i	$-0.0567710\pi$
$972$	0	0
$973$	− 265.526i	− 0.00874860i
$974$	0	0
$975$	30536.8	1.00304
$976$	0	0
$977$	31916.0	1.04512	0.522561	−	0.852602i	$-0.324977\pi$
$977$	31916.0	1.04512	0.522561	+	0.852602i	$0.324977\pi$
$978$	0	0
$979$	− 9.42614i	0 0.000307723i
$980$	0	0
$981$	− 15333.5i	− 0.499042i
$982$	0	0
$983$	−59182.4	−1.92027	−0.960135	−	0.279536i	$-0.909820\pi$
$983$	−59182.4	−1.92027	−0.960135	+	0.279536i	$0.909820\pi$
$984$	0	0
$985$	−1691.95	−0.0547308
$986$	0	0
$987$	− 2191.05i	− 0.0706604i
$988$	0	0
$989$	48445.4i	1.55761i
$990$	0	0
$991$	−12663.5	−0.405924	−0.202962	−	0.979187i	$-0.565057\pi$
$991$	−12663.5	−0.405924	−0.202962	+	0.979187i	$0.565057\pi$
$992$	0	0
$993$	23197.7	0.741346
$994$	0	0
$995$	3494.56i	0.111342i
$996$	0	0
$997$	− 15499.4i	− 0.492349i	−0.969225	−	0.246175i	$-0.920826\pi$
$997$	− 15499.4i	− 0.492349i	0.969225	−	0.246175i	$-0.0791737\pi$
$998$	0	0
$999$	2490.26	0.0788673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.3	✓	12
4.3	odd	2		1344.4.c.g.673.9	yes	12
8.3	odd	2		1344.4.c.g.673.4	yes	12
8.5	even	2	inner	1344.4.c.f.673.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.3	✓	12	1.1	even	1	trivial
1344.4.c.f.673.10	yes	12	8.5	even	2	inner
1344.4.c.g.673.4	yes	12	8.3	odd	2
1344.4.c.g.673.9	yes	12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -2.27324i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -2.27324i q^{5} -7.00000 q^{7} -9.00000 q^{9} +1.28915i q^{11} +84.9431i q^{13} -6.81971 q^{15} +49.4316 q^{17} +2.27106i q^{19} +21.0000i q^{21} -96.7192 q^{23} +119.832 q^{25} +27.0000i q^{27} -216.670i q^{29} +108.361 q^{31} +3.86744 q^{33} +15.9126i q^{35} -92.2319i q^{37} +254.829 q^{39} -356.642 q^{41} -500.887i q^{43} +20.4591i q^{45} -104.336 q^{47} +49.0000 q^{49} -148.295i q^{51} +53.9429i q^{53} +2.93054 q^{55} +6.81318 q^{57} -395.641i q^{59} -76.1378i q^{61} +63.0000 q^{63} +193.096 q^{65} -55.9352i q^{67} +290.158i q^{69} -1005.21 q^{71} -149.692 q^{73} -359.497i q^{75} -9.02403i q^{77} -331.984 q^{79} +81.0000 q^{81} +729.693i q^{83} -112.370i q^{85} -650.009 q^{87} -7.31192 q^{89} -594.602i q^{91} -325.082i q^{93} +5.16265 q^{95} -1765.54 q^{97} -11.6023i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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