LMFDB - Embedded newform 968.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,4,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("968.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	968.a (trivial)

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:	yes
Analytic conductor:	$57.1138488856$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 $ x^8 - 2x^7 - 87x^6 + 12x^5 + 2157x^4 + 2939x^3 - 5906x^2 - 3030*x + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 11 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.40099$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.71590 q^{3} -19.9654 q^{5} -4.43442 q^{7} +5.67153 q^{9} +O(q^{10})$	$q-5.71590 q^{3} -19.9654 q^{5} -4.43442 q^{7} +5.67153 q^{9} +65.8393 q^{13} +114.120 q^{15} -79.1744 q^{17} -91.2003 q^{19} +25.3467 q^{21} +90.4612 q^{23} +273.618 q^{25} +121.911 q^{27} +157.770 q^{29} -44.8791 q^{31} +88.5351 q^{35} +94.0637 q^{37} -376.331 q^{39} +104.777 q^{41} +251.870 q^{43} -113.235 q^{45} +538.141 q^{47} -323.336 q^{49} +452.553 q^{51} +588.409 q^{53} +521.292 q^{57} -466.276 q^{59} -506.686 q^{61} -25.1499 q^{63} -1314.51 q^{65} -228.386 q^{67} -517.067 q^{69} +117.053 q^{71} -632.669 q^{73} -1563.98 q^{75} -857.361 q^{79} -849.965 q^{81} +261.765 q^{83} +1580.75 q^{85} -901.800 q^{87} -441.266 q^{89} -291.959 q^{91} +256.525 q^{93} +1820.85 q^{95} -1567.18 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9	$ 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39} + 74 q^{41} - 35 q^{43} - 566 q^{45} + 197 q^{47} + 221 q^{49} + 608 q^{51} - 247 q^{53} + 979 q^{57} - 1208 q^{59} - 2061 q^{61} + 1617 q^{63} - 837 q^{65} - 829 q^{67} - 1860 q^{69} - 2005 q^{71} - 320 q^{73} - 2323 q^{75} + 1581 q^{79} - 3752 q^{81} - 692 q^{83} - 21 q^{85} + 131 q^{87} - 3205 q^{89} - 2724 q^{91} - 722 q^{93} + 432 q^{95} - 4702 q^{97}+O(q^{100}) $ 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−5.71590	−1.10003	−0.550013	−	0.835156i	$-0.685377\pi$
$3$	−5.71590	−1.10003	−0.550013	+	0.835156i	$0.685377\pi$
$4$	0	0
$5$	−19.9654	−1.78576	−0.892881	−	0.450292i	$-0.851320\pi$
$5$	−19.9654	−1.78576	−0.892881	+	0.450292i	$0.851320\pi$
$6$	0	0
$7$	−4.43442	−0.239436	−0.119718	−	0.992808i	$-0.538199\pi$
$7$	−4.43442	−0.239436	−0.119718	+	0.992808i	$0.538199\pi$
$8$	0	0
$9$	5.67153	0.210057
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	65.8393	1.40466	0.702328	−	0.711853i	$-0.252144\pi$
$13$	65.8393	1.40466	0.702328	+	0.711853i	$0.252144\pi$
$14$	0	0
$15$	114.120	1.96438
$16$	0	0
$17$	−79.1744	−1.12957	−0.564783	−	0.825240i	$-0.691040\pi$
$17$	−79.1744	−1.12957	−0.564783	+	0.825240i	$0.691040\pi$
$18$	0	0
$19$	−91.2003	−1.10120	−0.550599	−	0.834770i	$-0.685601\pi$
$19$	−91.2003	−1.10120	−0.550599	+	0.834770i	$0.685601\pi$
$20$	0	0
$21$	25.3467	0.263386
$22$	0	0
$23$	90.4612	0.820107	0.410054	−	0.912061i	$-0.365510\pi$
$23$	90.4612	0.820107	0.410054	+	0.912061i	$0.365510\pi$
$24$	0	0
$25$	273.618	2.18895
$26$	0	0
$27$	121.911	0.868958
$28$	0	0
$29$	157.770	1.01025	0.505124	−	0.863047i	$-0.331447\pi$
$29$	157.770	1.01025	0.505124	+	0.863047i	$0.331447\pi$
$30$	0	0
$31$	−44.8791	−0.260017	−0.130009	−	0.991513i	$-0.541500\pi$
$31$	−44.8791	−0.260017	−0.130009	+	0.991513i	$0.541500\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	88.5351	0.427576
$36$	0	0
$37$	94.0637	0.417945	0.208973	−	0.977921i	$-0.432988\pi$
$37$	94.0637	0.417945	0.208973	+	0.977921i	$0.432988\pi$
$38$	0	0
$39$	−376.331	−1.54516
$40$	0	0
$41$	104.777	0.399108	0.199554	−	0.979887i	$-0.436051\pi$
$41$	104.777	0.399108	0.199554	+	0.979887i	$0.436051\pi$
$42$	0	0
$43$	251.870	0.893251	0.446625	−	0.894721i	$-0.352626\pi$
$43$	251.870	0.893251	0.446625	+	0.894721i	$0.352626\pi$
$44$	0	0
$45$	−113.235	−0.375111
$46$	0	0
$47$	538.141	1.67013	0.835063	−	0.550155i	$-0.185431\pi$
$47$	538.141	1.67013	0.835063	+	0.550155i	$0.185431\pi$
$48$	0	0
$49$	−323.336	−0.942670
$50$	0	0
$51$	452.553	1.24255
$52$	0	0
$53$	588.409	1.52499	0.762493	−	0.646996i	$-0.223975\pi$
$53$	588.409	1.52499	0.762493	+	0.646996i	$0.223975\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	521.292	1.21135
$58$	0	0
$59$	−466.276	−1.02888	−0.514441	−	0.857526i	$-0.672000\pi$
$59$	−466.276	−1.02888	−0.514441	+	0.857526i	$0.672000\pi$
$60$	0	0
$61$	−506.686	−1.06352	−0.531759	−	0.846896i	$-0.678469\pi$
$61$	−506.686	−1.06352	−0.531759	+	0.846896i	$0.678469\pi$
$62$	0	0
$63$	−25.1499	−0.0502952
$64$	0	0
$65$	−1314.51	−2.50838
$66$	0	0
$67$	−228.386	−0.416444	−0.208222	−	0.978082i	$-0.566768\pi$
$67$	−228.386	−0.416444	−0.208222	+	0.978082i	$0.566768\pi$
$68$	0	0
$69$	−517.067	−0.902139
$70$	0	0
$71$	117.053	0.195657	0.0978283	−	0.995203i	$-0.468810\pi$
$71$	117.053	0.195657	0.0978283	+	0.995203i	$0.468810\pi$
$72$	0	0
$73$	−632.669	−1.01436	−0.507180	−	0.861840i	$-0.669312\pi$
$73$	−632.669	−1.01436	−0.507180	+	0.861840i	$0.669312\pi$
$74$	0	0
$75$	−1563.98	−2.40790
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−857.361	−1.22102	−0.610510	−	0.792008i	$-0.709036\pi$
$79$	−857.361	−1.22102	−0.610510	+	0.792008i	$0.709036\pi$
$80$	0	0
$81$	−849.965	−1.16593
$82$	0	0
$83$	261.765	0.346174	0.173087	−	0.984907i	$-0.444626\pi$
$83$	261.765	0.346174	0.173087	+	0.984907i	$0.444626\pi$
$84$	0	0
$85$	1580.75	2.01714
$86$	0	0
$87$	−901.800	−1.11130
$88$	0	0
$89$	−441.266	−0.525552	−0.262776	−	0.964857i	$-0.584638\pi$
$89$	−441.266	−0.525552	−0.262776	+	0.964857i	$0.584638\pi$
$90$	0	0
$91$	−291.959	−0.336326
$92$	0	0
$93$	256.525	0.286026
$94$	0	0
$95$	1820.85	1.96648
$96$	0	0
$97$	−1567.18	−1.64044	−0.820221	−	0.572046i	$-0.806150\pi$
$97$	−1567.18	−1.64044	−0.820221	+	0.572046i	$0.806150\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	150.666	0.148434	0.0742171	−	0.997242i	$-0.476354\pi$
$101$	150.666	0.148434	0.0742171	+	0.997242i	$0.476354\pi$
$102$	0	0
$103$	−485.986	−0.464909	−0.232455	−	0.972607i	$-0.574676\pi$
$103$	−485.986	−0.464909	−0.232455	+	0.972607i	$0.574676\pi$
$104$	0	0
$105$	−506.058	−0.470345
$106$	0	0
$107$	1351.37	1.22096	0.610478	−	0.792033i	$-0.290977\pi$
$107$	1351.37	1.22096	0.610478	+	0.792033i	$0.290977\pi$
$108$	0	0
$109$	44.7283	0.0393046	0.0196523	−	0.999807i	$-0.493744\pi$
$109$	44.7283	0.0393046	0.0196523	+	0.999807i	$0.493744\pi$
$110$	0	0
$111$	−537.659	−0.459751
$112$	0	0
$113$	1355.72	1.12863	0.564317	−	0.825558i	$-0.309140\pi$
$113$	1355.72	1.12863	0.564317	+	0.825558i	$0.309140\pi$
$114$	0	0
$115$	−1806.10	−1.46452
$116$	0	0
$117$	373.409	0.295057
$118$	0	0
$119$	351.093	0.270459
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−598.895	−0.439029
$124$	0	0
$125$	−2967.23	−2.12318
$126$	0	0
$127$	138.447	0.0967336	0.0483668	−	0.998830i	$-0.484598\pi$
$127$	138.447	0.0967336	0.0483668	+	0.998830i	$0.484598\pi$
$128$	0	0
$129$	−1439.66	−0.982599
$130$	0	0
$131$	2642.31	1.76229	0.881144	−	0.472848i	$-0.156774\pi$
$131$	2642.31	1.76229	0.881144	+	0.472848i	$0.156774\pi$
$132$	0	0
$133$	404.420	0.263667
$134$	0	0
$135$	−2434.01	−1.55175
$136$	0	0
$137$	−2038.21	−1.27107	−0.635533	−	0.772074i	$-0.719219\pi$
$137$	−2038.21	−1.27107	−0.635533	+	0.772074i	$0.719219\pi$
$138$	0	0
$139$	620.989	0.378932	0.189466	−	0.981887i	$-0.439324\pi$
$139$	620.989	0.378932	0.189466	+	0.981887i	$0.439324\pi$
$140$	0	0
$141$	−3075.96	−1.83718
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3149.95	−1.80406
$146$	0	0
$147$	1848.16	1.03696
$148$	0	0
$149$	−2221.86	−1.22162	−0.610810	−	0.791777i	$-0.709156\pi$
$149$	−2221.86	−1.22162	−0.610810	+	0.791777i	$0.709156\pi$
$150$	0	0
$151$	476.794	0.256960	0.128480	−	0.991712i	$-0.458990\pi$
$151$	476.794	0.256960	0.128480	+	0.991712i	$0.458990\pi$
$152$	0	0
$153$	−449.040	−0.237273
$154$	0	0
$155$	896.032	0.464329
$156$	0	0
$157$	−508.043	−0.258256	−0.129128	−	0.991628i	$-0.541218\pi$
$157$	−508.043	−0.258256	−0.129128	+	0.991628i	$0.541218\pi$
$158$	0	0
$159$	−3363.29	−1.67752
$160$	0	0
$161$	−401.143	−0.196363
$162$	0	0
$163$	−2709.91	−1.30219	−0.651093	−	0.758998i	$-0.725689\pi$
$163$	−2709.91	−1.30219	−0.651093	+	0.758998i	$0.725689\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2389.61	1.10727	0.553635	−	0.832760i	$-0.313240\pi$
$167$	2389.61	1.10727	0.553635	+	0.832760i	$0.313240\pi$
$168$	0	0
$169$	2137.81	0.973059
$170$	0	0
$171$	−517.245	−0.231314
$172$	0	0
$173$	−4203.47	−1.84731	−0.923653	−	0.383231i	$-0.874811\pi$
$173$	−4203.47	−1.84731	−0.923653	+	0.383231i	$0.874811\pi$
$174$	0	0
$175$	−1213.34	−0.524113
$176$	0	0
$177$	2665.19	1.13180
$178$	0	0
$179$	3654.09	1.52581	0.762904	−	0.646512i	$-0.223773\pi$
$179$	3654.09	1.52581	0.762904	+	0.646512i	$0.223773\pi$
$180$	0	0
$181$	1275.06	0.523617	0.261809	−	0.965120i	$-0.415681\pi$
$181$	1275.06	0.523617	0.261809	+	0.965120i	$0.415681\pi$
$182$	0	0
$183$	2896.17	1.16990
$184$	0	0
$185$	−1878.02	−0.746351
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−540.607	−0.208060
$190$	0	0
$191$	2538.57	0.961698	0.480849	−	0.876803i	$-0.340329\pi$
$191$	2538.57	0.961698	0.480849	+	0.876803i	$0.340329\pi$
$192$	0	0
$193$	−1157.08	−0.431547	−0.215773	−	0.976443i	$-0.569227\pi$
$193$	−1157.08	−0.431547	−0.215773	+	0.976443i	$0.569227\pi$
$194$	0	0
$195$	7513.61	2.75929
$196$	0	0
$197$	−1796.63	−0.649769	−0.324885	−	0.945754i	$-0.605325\pi$
$197$	−1796.63	−0.649769	−0.324885	+	0.945754i	$0.605325\pi$
$198$	0	0
$199$	2834.34	1.00965	0.504827	−	0.863220i	$-0.331556\pi$
$199$	2834.34	1.00965	0.504827	+	0.863220i	$0.331556\pi$
$200$	0	0
$201$	1305.43	0.458099
$202$	0	0
$203$	−699.620	−0.241890
$204$	0	0
$205$	−2091.92	−0.712712
$206$	0	0
$207$	513.053	0.172269
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3634.77	1.18591	0.592957	−	0.805234i	$-0.297960\pi$
$211$	3634.77	1.18591	0.592957	+	0.805234i	$0.297960\pi$
$212$	0	0
$213$	−669.063	−0.215227
$214$	0	0
$215$	−5028.69	−1.59513
$216$	0	0
$217$	199.013	0.0622575
$218$	0	0
$219$	3616.27	1.11582
$220$	0	0
$221$	−5212.79	−1.58665
$222$	0	0
$223$	−5981.82	−1.79629	−0.898145	−	0.439700i	$-0.855085\pi$
$223$	−5981.82	−1.79629	−0.898145	+	0.439700i	$0.855085\pi$
$224$	0	0
$225$	1551.83	0.459803
$226$	0	0
$227$	4286.98	1.25347	0.626733	−	0.779234i	$-0.284392\pi$
$227$	4286.98	1.25347	0.626733	+	0.779234i	$0.284392\pi$
$228$	0	0
$229$	−3472.03	−1.00191	−0.500957	−	0.865472i	$-0.667018\pi$
$229$	−3472.03	−1.00191	−0.500957	+	0.865472i	$0.667018\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1003.86	0.282254	0.141127	−	0.989992i	$-0.454927\pi$
$233$	1003.86	0.282254	0.141127	+	0.989992i	$0.454927\pi$
$234$	0	0
$235$	−10744.2	−2.98245
$236$	0	0
$237$	4900.59	1.34315
$238$	0	0
$239$	−2780.96	−0.752660	−0.376330	−	0.926486i	$-0.622814\pi$
$239$	−2780.96	−0.752660	−0.376330	+	0.926486i	$0.622814\pi$
$240$	0	0
$241$	2394.67	0.640060	0.320030	−	0.947407i	$-0.396307\pi$
$241$	2394.67	0.640060	0.320030	+	0.947407i	$0.396307\pi$
$242$	0	0
$243$	1566.71	0.413598
$244$	0	0
$245$	6455.54	1.68339
$246$	0	0
$247$	−6004.56	−1.54681
$248$	0	0
$249$	−1496.22	−0.380801
$250$	0	0
$251$	4834.70	1.21579	0.607896	−	0.794017i	$-0.292014\pi$
$251$	4834.70	1.21579	0.607896	+	0.794017i	$0.292014\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9035.42	−2.21890
$256$	0	0
$257$	−401.466	−0.0974426	−0.0487213	−	0.998812i	$-0.515515\pi$
$257$	−401.466	−0.0974426	−0.0487213	+	0.998812i	$0.515515\pi$
$258$	0	0
$259$	−417.118	−0.100071
$260$	0	0
$261$	894.799	0.212209
$262$	0	0
$263$	−7790.07	−1.82645	−0.913225	−	0.407457i	$-0.866416\pi$
$263$	−7790.07	−1.82645	−0.913225	+	0.407457i	$0.866416\pi$
$264$	0	0
$265$	−11747.8	−2.72326
$266$	0	0
$267$	2522.23	0.578121
$268$	0	0
$269$	6576.46	1.49061	0.745304	−	0.666724i	$-0.232304\pi$
$269$	6576.46	1.49061	0.745304	+	0.666724i	$0.232304\pi$
$270$	0	0
$271$	1367.10	0.306441	0.153221	−	0.988192i	$-0.451035\pi$
$271$	1367.10	0.306441	0.153221	+	0.988192i	$0.451035\pi$
$272$	0	0
$273$	1668.81	0.369967
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5470.68	1.18665	0.593323	−	0.804964i	$-0.297816\pi$
$277$	5470.68	1.18665	0.593323	+	0.804964i	$0.297816\pi$
$278$	0	0
$279$	−254.533	−0.0546183
$280$	0	0
$281$	−4239.13	−0.899948	−0.449974	−	0.893042i	$-0.648567\pi$
$281$	−4239.13	−0.899948	−0.449974	+	0.893042i	$0.648567\pi$
$282$	0	0
$283$	−1377.20	−0.289279	−0.144640	−	0.989484i	$-0.546202\pi$
$283$	−1377.20	−0.289279	−0.144640	+	0.989484i	$0.546202\pi$
$284$	0	0
$285$	−10407.8	−2.16318
$286$	0	0
$287$	−464.626	−0.0955609
$288$	0	0
$289$	1355.59	0.275918
$290$	0	0
$291$	8957.84	1.80453
$292$	0	0
$293$	−25.2459	−0.00503372	−0.00251686	−	0.999997i	$-0.500801\pi$
$293$	−25.2459	−0.00503372	−0.00251686	+	0.999997i	$0.500801\pi$
$294$	0	0
$295$	9309.41	1.83734
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5955.90	1.15197
$300$	0	0
$301$	−1116.90	−0.213877
$302$	0	0
$303$	−861.193	−0.163281
$304$	0	0
$305$	10116.2	1.89919
$306$	0	0
$307$	1098.15	0.204153	0.102076	−	0.994777i	$-0.467451\pi$
$307$	1098.15	0.204153	0.102076	+	0.994777i	$0.467451\pi$
$308$	0	0
$309$	2777.85	0.511412
$310$	0	0
$311$	−5983.38	−1.09095	−0.545476	−	0.838126i	$-0.683651\pi$
$311$	−5983.38	−1.09095	−0.545476	+	0.838126i	$0.683651\pi$
$312$	0	0
$313$	−721.406	−0.130276	−0.0651378	−	0.997876i	$-0.520749\pi$
$313$	−721.406	−0.130276	−0.0651378	+	0.997876i	$0.520749\pi$
$314$	0	0
$315$	502.130	0.0898152
$316$	0	0
$317$	2154.81	0.381787	0.190893	−	0.981611i	$-0.438862\pi$
$317$	2154.81	0.381787	0.190893	+	0.981611i	$0.438862\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7724.32	−1.34308
$322$	0	0
$323$	7220.73	1.24388
$324$	0	0
$325$	18014.8	3.07472
$326$	0	0
$327$	−255.663	−0.0432360
$328$	0	0
$329$	−2386.34	−0.399889
$330$	0	0
$331$	6178.03	1.02591	0.512954	−	0.858416i	$-0.328551\pi$
$331$	6178.03	1.02591	0.512954	+	0.858416i	$0.328551\pi$
$332$	0	0
$333$	533.485	0.0877922
$334$	0	0
$335$	4559.82	0.743670
$336$	0	0
$337$	−3087.24	−0.499029	−0.249514	−	0.968371i	$-0.580271\pi$
$337$	−3087.24	−0.499029	−0.249514	+	0.968371i	$0.580271\pi$
$338$	0	0
$339$	−7749.17	−1.24153
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2954.81	0.465146
$344$	0	0
$345$	10323.5	1.61101
$346$	0	0
$347$	−6879.45	−1.06429	−0.532145	−	0.846654i	$-0.678614\pi$
$347$	−6879.45	−1.06429	−0.532145	+	0.846654i	$0.678614\pi$
$348$	0	0
$349$	−6725.05	−1.03147	−0.515736	−	0.856748i	$-0.672481\pi$
$349$	−6725.05	−1.03147	−0.515736	+	0.856748i	$0.672481\pi$
$350$	0	0
$351$	8026.56	1.22059
$352$	0	0
$353$	−4363.90	−0.657980	−0.328990	−	0.944333i	$-0.606708\pi$
$353$	−4363.90	−0.657980	−0.328990	+	0.944333i	$0.606708\pi$
$354$	0	0
$355$	−2337.01	−0.349396
$356$	0	0
$357$	−2006.81	−0.297512
$358$	0	0
$359$	−8274.48	−1.21646	−0.608232	−	0.793760i	$-0.708121\pi$
$359$	−8274.48	−1.21646	−0.608232	+	0.793760i	$0.708121\pi$
$360$	0	0
$361$	1458.49	0.212639
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12631.5	1.81141
$366$	0	0
$367$	−5690.38	−0.809360	−0.404680	−	0.914458i	$-0.632617\pi$
$367$	−5690.38	−0.809360	−0.404680	+	0.914458i	$0.632617\pi$
$368$	0	0
$369$	594.246	0.0838353
$370$	0	0
$371$	−2609.26	−0.365137
$372$	0	0
$373$	12046.8	1.67227	0.836137	−	0.548521i	$-0.184809\pi$
$373$	12046.8	1.67227	0.836137	+	0.548521i	$0.184809\pi$
$374$	0	0
$375$	16960.4	2.33555
$376$	0	0
$377$	10387.5	1.41905
$378$	0	0
$379$	−614.208	−0.0832447	−0.0416223	−	0.999133i	$-0.513253\pi$
$379$	−614.208	−0.0832447	−0.0416223	+	0.999133i	$0.513253\pi$
$380$	0	0
$381$	−791.349	−0.106409
$382$	0	0
$383$	5525.70	0.737207	0.368603	−	0.929587i	$-0.379836\pi$
$383$	5525.70	0.737207	0.368603	+	0.929587i	$0.379836\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1428.49	0.187633
$388$	0	0
$389$	2194.43	0.286020	0.143010	−	0.989721i	$-0.454322\pi$
$389$	2194.43	0.286020	0.143010	+	0.989721i	$0.454322\pi$
$390$	0	0
$391$	−7162.21	−0.926365
$392$	0	0
$393$	−15103.2	−1.93856
$394$	0	0
$395$	17117.6	2.18045
$396$	0	0
$397$	−4636.14	−0.586099	−0.293049	−	0.956097i	$-0.594670\pi$
$397$	−4636.14	−0.586099	−0.293049	+	0.956097i	$0.594670\pi$
$398$	0	0
$399$	−2311.63	−0.290040
$400$	0	0
$401$	−3408.81	−0.424508	−0.212254	−	0.977215i	$-0.568080\pi$
$401$	−3408.81	−0.424508	−0.212254	+	0.977215i	$0.568080\pi$
$402$	0	0
$403$	−2954.81	−0.365235
$404$	0	0
$405$	16969.9	2.08208
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9164.08	−1.10791	−0.553954	−	0.832547i	$-0.686882\pi$
$409$	−9164.08	−1.10791	−0.553954	+	0.832547i	$0.686882\pi$
$410$	0	0
$411$	11650.2	1.39820
$412$	0	0
$413$	2067.67	0.246352
$414$	0	0
$415$	−5226.26	−0.618185
$416$	0	0
$417$	−3549.51	−0.416835
$418$	0	0
$419$	−5733.34	−0.668478	−0.334239	−	0.942488i	$-0.608479\pi$
$419$	−5733.34	−0.668478	−0.334239	+	0.942488i	$0.608479\pi$
$420$	0	0
$421$	−7274.74	−0.842160	−0.421080	−	0.907024i	$-0.638349\pi$
$421$	−7274.74	−0.842160	−0.421080	+	0.907024i	$0.638349\pi$
$422$	0	0
$423$	3052.08	0.350821
$424$	0	0
$425$	−21663.6	−2.47256
$426$	0	0
$427$	2246.86	0.254645
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−462.548	−0.0516941	−0.0258470	−	0.999666i	$-0.508228\pi$
$431$	−462.548	−0.0516941	−0.0258470	+	0.999666i	$0.508228\pi$
$432$	0	0
$433$	5839.35	0.648086	0.324043	−	0.946042i	$-0.394958\pi$
$433$	5839.35	0.648086	0.324043	+	0.946042i	$0.394958\pi$
$434$	0	0
$435$	18004.8	1.98452
$436$	0	0
$437$	−8250.09	−0.903101
$438$	0	0
$439$	−3730.11	−0.405532	−0.202766	−	0.979227i	$-0.564993\pi$
$439$	−3730.11	−0.405532	−0.202766	+	0.979227i	$0.564993\pi$
$440$	0	0
$441$	−1833.81	−0.198014
$442$	0	0
$443$	−7913.68	−0.848736	−0.424368	−	0.905490i	$-0.639504\pi$
$443$	−7913.68	−0.848736	−0.424368	+	0.905490i	$0.639504\pi$
$444$	0	0
$445$	8810.07	0.938511
$446$	0	0
$447$	12699.9	1.34381
$448$	0	0
$449$	7672.44	0.806425	0.403213	−	0.915106i	$-0.367894\pi$
$449$	7672.44	0.806425	0.403213	+	0.915106i	$0.367894\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2725.31	−0.282663
$454$	0	0
$455$	5829.09	0.600598
$456$	0	0
$457$	10308.6	1.05518	0.527591	−	0.849499i	$-0.323095\pi$
$457$	10308.6	1.05518	0.527591	+	0.849499i	$0.323095\pi$
$458$	0	0
$459$	−9652.26	−0.981545
$460$	0	0
$461$	8440.80	0.852770	0.426385	−	0.904542i	$-0.359787\pi$
$461$	8440.80	0.852770	0.426385	+	0.904542i	$0.359787\pi$
$462$	0	0
$463$	2466.79	0.247606	0.123803	−	0.992307i	$-0.460491\pi$
$463$	2466.79	0.247606	0.123803	+	0.992307i	$0.460491\pi$
$464$	0	0
$465$	−5121.63	−0.510774
$466$	0	0
$467$	−16299.0	−1.61505	−0.807524	−	0.589834i	$-0.799193\pi$
$467$	−16299.0	−1.61505	−0.807524	+	0.589834i	$0.799193\pi$
$468$	0	0
$469$	1012.76	0.0997118
$470$	0	0
$471$	2903.92	0.284089
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−24954.1	−2.41047
$476$	0	0
$477$	3337.18	0.320333
$478$	0	0
$479$	15208.2	1.45069	0.725343	−	0.688387i	$-0.241681\pi$
$479$	15208.2	1.45069	0.725343	+	0.688387i	$0.241681\pi$
$480$	0	0
$481$	6193.09	0.587069
$482$	0	0
$483$	2292.90	0.216005
$484$	0	0
$485$	31289.4	2.92944
$486$	0	0
$487$	−7335.48	−0.682551	−0.341275	−	0.939963i	$-0.610859\pi$
$487$	−7335.48	−0.682551	−0.341275	+	0.939963i	$0.610859\pi$
$488$	0	0
$489$	15489.6	1.43244
$490$	0	0
$491$	7374.59	0.677822	0.338911	−	0.940818i	$-0.389941\pi$
$491$	7374.59	0.677822	0.338911	+	0.940818i	$0.389941\pi$
$492$	0	0
$493$	−12491.4	−1.14114
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−519.062	−0.0468473
$498$	0	0
$499$	6513.25	0.584314	0.292157	−	0.956370i	$-0.405627\pi$
$499$	6513.25	0.584314	0.292157	+	0.956370i	$0.405627\pi$
$500$	0	0
$501$	−13658.8	−1.21802
$502$	0	0
$503$	9956.94	0.882620	0.441310	−	0.897355i	$-0.354514\pi$
$503$	9956.94	0.882620	0.441310	+	0.897355i	$0.354514\pi$
$504$	0	0
$505$	−3008.12	−0.265068
$506$	0	0
$507$	−12219.5	−1.07039
$508$	0	0
$509$	−15514.3	−1.35100	−0.675498	−	0.737362i	$-0.736071\pi$
$509$	−15514.3	−1.35100	−0.675498	+	0.737362i	$0.736071\pi$
$510$	0	0
$511$	2805.52	0.242875
$512$	0	0
$513$	−11118.4	−0.956896
$514$	0	0
$515$	9702.92	0.830217
$516$	0	0
$517$	0	0
$518$	0	0
$519$	24026.6	2.03208
$520$	0	0
$521$	−9885.26	−0.831250	−0.415625	−	0.909536i	$-0.636437\pi$
$521$	−9885.26	−0.831250	−0.415625	+	0.909536i	$0.636437\pi$
$522$	0	0
$523$	15675.5	1.31059	0.655297	−	0.755371i	$-0.272543\pi$
$523$	15675.5	1.31059	0.655297	+	0.755371i	$0.272543\pi$
$524$	0	0
$525$	6935.33	0.576538
$526$	0	0
$527$	3553.28	0.293706
$528$	0	0
$529$	−3983.77	−0.327424
$530$	0	0
$531$	−2644.50	−0.216123
$532$	0	0
$533$	6898.45	0.560610
$534$	0	0
$535$	−26980.8	−2.18034
$536$	0	0
$537$	−20886.4	−1.67843
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1697.56	0.134905	0.0674527	−	0.997722i	$-0.478513\pi$
$541$	1697.56	0.134905	0.0674527	+	0.997722i	$0.478513\pi$
$542$	0	0
$543$	−7288.14	−0.575992
$544$	0	0
$545$	−893.021	−0.0701886
$546$	0	0
$547$	13188.6	1.03090	0.515450	−	0.856919i	$-0.327625\pi$
$547$	13188.6	1.03090	0.515450	+	0.856919i	$0.327625\pi$
$548$	0	0
$549$	−2873.69	−0.223399
$550$	0	0
$551$	−14388.7	−1.11248
$552$	0	0
$553$	3801.90	0.292357
$554$	0	0
$555$	10734.6	0.821005
$556$	0	0
$557$	−9639.49	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$557$	−9639.49	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$558$	0	0
$559$	16582.9	1.25471
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2922.28	−0.218756	−0.109378	−	0.994000i	$-0.534886\pi$
$563$	−2922.28	−0.218756	−0.109378	+	0.994000i	$0.534886\pi$
$564$	0	0
$565$	−27067.6	−2.01547
$566$	0	0
$567$	3769.10	0.279167
$568$	0	0
$569$	−17432.5	−1.28437	−0.642186	−	0.766549i	$-0.721972\pi$
$569$	−17432.5	−1.28437	−0.642186	+	0.766549i	$0.721972\pi$
$570$	0	0
$571$	−19497.7	−1.42899	−0.714493	−	0.699642i	$-0.753343\pi$
$571$	−19497.7	−1.42899	−0.714493	+	0.699642i	$0.753343\pi$
$572$	0	0
$573$	−14510.2	−1.05789
$574$	0	0
$575$	24751.9	1.79517
$576$	0	0
$577$	−16422.5	−1.18489	−0.592443	−	0.805612i	$-0.701837\pi$
$577$	−16422.5	−1.18489	−0.592443	+	0.805612i	$0.701837\pi$
$578$	0	0
$579$	6613.76	0.474712
$580$	0	0
$581$	−1160.78	−0.0828867
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−7455.28	−0.526902
$586$	0	0
$587$	2279.52	0.160283	0.0801413	−	0.996784i	$-0.474463\pi$
$587$	2279.52	0.160283	0.0801413	+	0.996784i	$0.474463\pi$
$588$	0	0
$589$	4092.99	0.286331
$590$	0	0
$591$	10269.4	0.714763
$592$	0	0
$593$	4992.56	0.345734	0.172867	−	0.984945i	$-0.444697\pi$
$593$	4992.56	0.345734	0.172867	+	0.984945i	$0.444697\pi$
$594$	0	0
$595$	−7009.72	−0.482975
$596$	0	0
$597$	−16200.8	−1.11065
$598$	0	0
$599$	−4039.61	−0.275550	−0.137775	−	0.990464i	$-0.543995\pi$
$599$	−4039.61	−0.275550	−0.137775	+	0.990464i	$0.543995\pi$
$600$	0	0
$601$	2520.28	0.171055	0.0855276	−	0.996336i	$-0.472742\pi$
$601$	2520.28	0.171055	0.0855276	+	0.996336i	$0.472742\pi$
$602$	0	0
$603$	−1295.30	−0.0874768
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6608.30	0.441882	0.220941	−	0.975287i	$-0.429087\pi$
$607$	6608.30	0.441882	0.220941	+	0.975287i	$0.429087\pi$
$608$	0	0
$609$	3998.96	0.266085
$610$	0	0
$611$	35430.8	2.34595
$612$	0	0
$613$	6209.23	0.409117	0.204558	−	0.978854i	$-0.434424\pi$
$613$	6209.23	0.409117	0.204558	+	0.978854i	$0.434424\pi$
$614$	0	0
$615$	11957.2	0.784002
$616$	0	0
$617$	7237.79	0.472257	0.236128	−	0.971722i	$-0.424121\pi$
$617$	7237.79	0.472257	0.236128	+	0.971722i	$0.424121\pi$
$618$	0	0
$619$	−7189.81	−0.466855	−0.233427	−	0.972374i	$-0.574994\pi$
$619$	−7189.81	−0.466855	−0.233427	+	0.972374i	$0.574994\pi$
$620$	0	0
$621$	11028.3	0.712639
$622$	0	0
$623$	1956.76	0.125836
$624$	0	0
$625$	25039.8	1.60254
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7447.44	−0.472097
$630$	0	0
$631$	−6066.32	−0.382720	−0.191360	−	0.981520i	$-0.561290\pi$
$631$	−6066.32	−0.382720	−0.191360	+	0.981520i	$0.561290\pi$
$632$	0	0
$633$	−20776.0	−1.30454
$634$	0	0
$635$	−2764.15	−0.172743
$636$	0	0
$637$	−21288.2	−1.32413
$638$	0	0
$639$	663.869	0.0410990
$640$	0	0
$641$	−7205.54	−0.443996	−0.221998	−	0.975047i	$-0.571258\pi$
$641$	−7205.54	−0.443996	−0.221998	+	0.975047i	$0.571258\pi$
$642$	0	0
$643$	8330.27	0.510908	0.255454	−	0.966821i	$-0.417775\pi$
$643$	8330.27	0.510908	0.255454	+	0.966821i	$0.417775\pi$
$644$	0	0
$645$	28743.5	1.75469
$646$	0	0
$647$	−21335.6	−1.29643	−0.648215	−	0.761458i	$-0.724484\pi$
$647$	−21335.6	−1.29643	−0.648215	+	0.761458i	$0.724484\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1137.54	−0.0684849
$652$	0	0
$653$	−2788.08	−0.167084	−0.0835420	−	0.996504i	$-0.526623\pi$
$653$	−2788.08	−0.167084	−0.0835420	+	0.996504i	$0.526623\pi$
$654$	0	0
$655$	−52754.9	−3.14703
$656$	0	0
$657$	−3588.20	−0.213073
$658$	0	0
$659$	−7469.63	−0.441541	−0.220770	−	0.975326i	$-0.570857\pi$
$659$	−7469.63	−0.441541	−0.220770	+	0.975326i	$0.570857\pi$
$660$	0	0
$661$	−12187.9	−0.717176	−0.358588	−	0.933496i	$-0.616742\pi$
$661$	−12187.9	−0.717176	−0.358588	+	0.933496i	$0.616742\pi$
$662$	0	0
$663$	29795.8	1.74536
$664$	0	0
$665$	−8074.43	−0.470846
$666$	0	0
$667$	14272.1	0.828513
$668$	0	0
$669$	34191.5	1.97596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16457.1	0.942605	0.471302	−	0.881972i	$-0.343784\pi$
$673$	16457.1	0.942605	0.471302	+	0.881972i	$0.343784\pi$
$674$	0	0
$675$	33357.2	1.90210
$676$	0	0
$677$	−34499.1	−1.95851	−0.979253	−	0.202643i	$-0.935047\pi$
$677$	−34499.1	−1.95851	−0.979253	+	0.202643i	$0.935047\pi$
$678$	0	0
$679$	6949.53	0.392781
$680$	0	0
$681$	−24503.9	−1.37884
$682$	0	0
$683$	−9503.98	−0.532444	−0.266222	−	0.963912i	$-0.585775\pi$
$683$	−9503.98	−0.532444	−0.266222	+	0.963912i	$0.585775\pi$
$684$	0	0
$685$	40693.7	2.26982
$686$	0	0
$687$	19845.8	1.10213
$688$	0	0
$689$	38740.5	2.14208
$690$	0	0
$691$	−3214.40	−0.176963	−0.0884816	−	0.996078i	$-0.528201\pi$
$691$	−3214.40	−0.176963	−0.0884816	+	0.996078i	$0.528201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12398.3	−0.676683
$696$	0	0
$697$	−8295.66	−0.450819
$698$	0	0
$699$	−5737.97	−0.310487
$700$	0	0
$701$	−24993.3	−1.34663	−0.673313	−	0.739358i	$-0.735129\pi$
$701$	−24993.3	−1.34663	−0.673313	+	0.739358i	$0.735129\pi$
$702$	0	0
$703$	−8578.63	−0.460241
$704$	0	0
$705$	61412.9	3.28077
$706$	0	0
$707$	−668.118	−0.0355405
$708$	0	0
$709$	−101.995	−0.00540269	−0.00270135	−	0.999996i	$-0.500860\pi$
$709$	−101.995	−0.00540269	−0.00270135	+	0.999996i	$0.500860\pi$
$710$	0	0
$711$	−4862.54	−0.256483
$712$	0	0
$713$	−4059.82	−0.213242
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15895.7	0.827945
$718$	0	0
$719$	−16282.9	−0.844576	−0.422288	−	0.906462i	$-0.638773\pi$
$719$	−16282.9	−0.844576	−0.422288	+	0.906462i	$0.638773\pi$
$720$	0	0
$721$	2155.07	0.111316
$722$	0	0
$723$	−13687.7	−0.704083
$724$	0	0
$725$	43168.9	2.21138
$726$	0	0
$727$	−10002.9	−0.510299	−0.255149	−	0.966902i	$-0.582125\pi$
$727$	−10002.9	−0.510299	−0.255149	+	0.966902i	$0.582125\pi$
$728$	0	0
$729$	13993.9	0.710964
$730$	0	0
$731$	−19941.6	−1.00899
$732$	0	0
$733$	−11752.8	−0.592224	−0.296112	−	0.955153i	$-0.595690\pi$
$733$	−11752.8	−0.592224	−0.296112	+	0.955153i	$0.595690\pi$
$734$	0	0
$735$	−36899.2	−1.85177
$736$	0	0
$737$	0	0
$738$	0	0
$739$	21112.0	1.05090	0.525452	−	0.850823i	$-0.323896\pi$
$739$	21112.0	1.05090	0.525452	+	0.850823i	$0.323896\pi$
$740$	0	0
$741$	34321.5	1.70153
$742$	0	0
$743$	−3845.40	−0.189871	−0.0949354	−	0.995483i	$-0.530264\pi$
$743$	−3845.40	−0.189871	−0.0949354	+	0.995483i	$0.530264\pi$
$744$	0	0
$745$	44360.3	2.18152
$746$	0	0
$747$	1484.61	0.0727162
$748$	0	0
$749$	−5992.56	−0.292341
$750$	0	0
$751$	−9365.45	−0.455060	−0.227530	−	0.973771i	$-0.573065\pi$
$751$	−9365.45	−0.455060	−0.227530	+	0.973771i	$0.573065\pi$
$752$	0	0
$753$	−27634.7	−1.33740
$754$	0	0
$755$	−9519.40	−0.458870
$756$	0	0
$757$	−27359.6	−1.31361	−0.656805	−	0.754061i	$-0.728092\pi$
$757$	−27359.6	−1.31361	−0.656805	+	0.754061i	$0.728092\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12749.8	0.607331	0.303665	−	0.952779i	$-0.401790\pi$
$761$	12749.8	0.607331	0.303665	+	0.952779i	$0.401790\pi$
$762$	0	0
$763$	−198.344	−0.00941094
$764$	0	0
$765$	8965.27	0.423713
$766$	0	0
$767$	−30699.3	−1.44522
$768$	0	0
$769$	35071.1	1.64460	0.822300	−	0.569054i	$-0.192691\pi$
$769$	35071.1	1.64460	0.822300	+	0.569054i	$0.192691\pi$
$770$	0	0
$771$	2294.74	0.107189
$772$	0	0
$773$	−6908.44	−0.321448	−0.160724	−	0.986999i	$-0.551383\pi$
$773$	−6908.44	−0.321448	−0.160724	+	0.986999i	$0.551383\pi$
$774$	0	0
$775$	−12279.8	−0.569164
$776$	0	0
$777$	2384.21	0.110081
$778$	0	0
$779$	−9555.70	−0.439497
$780$	0	0
$781$	0	0
$782$	0	0
$783$	19234.0	0.877864
$784$	0	0
$785$	10143.3	0.461184
$786$	0	0
$787$	−37429.7	−1.69533	−0.847665	−	0.530532i	$-0.821992\pi$
$787$	−37429.7	−1.69533	−0.847665	+	0.530532i	$0.821992\pi$
$788$	0	0
$789$	44527.3	2.00914
$790$	0	0
$791$	−6011.84	−0.270236
$792$	0	0
$793$	−33359.9	−1.49388
$794$	0	0
$795$	67149.5	2.99566
$796$	0	0
$797$	−23307.5	−1.03588	−0.517938	−	0.855418i	$-0.673300\pi$
$797$	−23307.5	−1.03588	−0.517938	+	0.855418i	$0.673300\pi$
$798$	0	0
$799$	−42607.0	−1.88652
$800$	0	0
$801$	−2502.65	−0.110396
$802$	0	0
$803$	0	0
$804$	0	0
$805$	8009.00	0.350659
$806$	0	0
$807$	−37590.4	−1.63971
$808$	0	0
$809$	5184.45	0.225310	0.112655	−	0.993634i	$-0.464065\pi$
$809$	5184.45	0.225310	0.112655	+	0.993634i	$0.464065\pi$
$810$	0	0
$811$	−32812.7	−1.42073	−0.710363	−	0.703836i	$-0.751469\pi$
$811$	−32812.7	−1.42073	−0.710363	+	0.703836i	$0.751469\pi$
$812$	0	0
$813$	−7814.23	−0.337093
$814$	0	0
$815$	54104.5	2.32540
$816$	0	0
$817$	−22970.6	−0.983647
$818$	0	0
$819$	−1655.85	−0.0706474
$820$	0	0
$821$	8438.18	0.358702	0.179351	−	0.983785i	$-0.442600\pi$
$821$	8438.18	0.358702	0.179351	+	0.983785i	$0.442600\pi$
$822$	0	0
$823$	−5702.44	−0.241525	−0.120762	−	0.992681i	$-0.538534\pi$
$823$	−5702.44	−0.241525	−0.120762	+	0.992681i	$0.538534\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38070.8	−1.60079	−0.800395	−	0.599473i	$-0.795377\pi$
$827$	−38070.8	−1.60079	−0.800395	+	0.599473i	$0.795377\pi$
$828$	0	0
$829$	4210.59	0.176405	0.0882025	−	0.996103i	$-0.471888\pi$
$829$	4210.59	0.176405	0.0882025	+	0.996103i	$0.471888\pi$
$830$	0	0
$831$	−31269.9	−1.30534
$832$	0	0
$833$	25599.9	1.06481
$834$	0	0
$835$	−47709.7	−1.97732
$836$	0	0
$837$	−5471.28	−0.225944
$838$	0	0
$839$	−26556.3	−1.09276	−0.546380	−	0.837537i	$-0.683995\pi$
$839$	−26556.3	−1.09276	−0.546380	+	0.837537i	$0.683995\pi$
$840$	0	0
$841$	502.480	0.0206027
$842$	0	0
$843$	24230.5	0.989966
$844$	0	0
$845$	−42682.3	−1.73765
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7871.94	0.318215
$850$	0	0
$851$	8509.12	0.342760
$852$	0	0
$853$	26639.1	1.06929	0.534646	−	0.845076i	$-0.320445\pi$
$853$	26639.1	1.06929	0.534646	+	0.845076i	$0.320445\pi$
$854$	0	0
$855$	10327.0	0.413072
$856$	0	0
$857$	8010.37	0.319287	0.159643	−	0.987175i	$-0.448966\pi$
$857$	8010.37	0.319287	0.159643	+	0.987175i	$0.448966\pi$
$858$	0	0
$859$	−44103.2	−1.75178	−0.875892	−	0.482508i	$-0.839726\pi$
$859$	−44103.2	−1.75178	−0.875892	+	0.482508i	$0.839726\pi$
$860$	0	0
$861$	2655.75	0.105119
$862$	0	0
$863$	16819.3	0.663426	0.331713	−	0.943380i	$-0.392374\pi$
$863$	16819.3	0.663426	0.331713	+	0.943380i	$0.392374\pi$
$864$	0	0
$865$	83924.1	3.29885
$866$	0	0
$867$	−7748.39	−0.303517
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−15036.8	−0.584961
$872$	0	0
$873$	−8888.30	−0.344586
$874$	0	0
$875$	13158.0	0.508366
$876$	0	0
$877$	25370.0	0.976835	0.488418	−	0.872610i	$-0.337574\pi$
$877$	25370.0	0.976835	0.488418	+	0.872610i	$0.337574\pi$
$878$	0	0
$879$	144.303	0.00553722
$880$	0	0
$881$	5973.34	0.228430	0.114215	−	0.993456i	$-0.463565\pi$
$881$	5973.34	0.228430	0.114215	+	0.993456i	$0.463565\pi$
$882$	0	0
$883$	−22225.1	−0.847038	−0.423519	−	0.905887i	$-0.639205\pi$
$883$	−22225.1	−0.847038	−0.423519	+	0.905887i	$0.639205\pi$
$884$	0	0
$885$	−53211.7	−2.02112
$886$	0	0
$887$	18065.0	0.683838	0.341919	−	0.939730i	$-0.388923\pi$
$887$	18065.0	0.683838	0.341919	+	0.939730i	$0.388923\pi$
$888$	0	0
$889$	−613.932	−0.0231615
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−49078.6	−1.83914
$894$	0	0
$895$	−72955.5	−2.72473
$896$	0	0
$897$	−34043.3	−1.26720
$898$	0	0
$899$	−7080.60	−0.262682
$900$	0	0
$901$	−46587.0	−1.72257
$902$	0	0
$903$	6384.07	0.235270
$904$	0	0
$905$	−25457.2	−0.935056
$906$	0	0
$907$	42154.2	1.54323	0.771613	−	0.636092i	$-0.219450\pi$
$907$	42154.2	1.54323	0.771613	+	0.636092i	$0.219450\pi$
$908$	0	0
$909$	854.508	0.0311796
$910$	0	0
$911$	−16926.0	−0.615568	−0.307784	−	0.951456i	$-0.599587\pi$
$911$	−16926.0	−0.615568	−0.307784	+	0.951456i	$0.599587\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−57823.3	−2.08916
$916$	0	0
$917$	−11717.1	−0.421956
$918$	0	0
$919$	42794.9	1.53610	0.768048	−	0.640392i	$-0.221228\pi$
$919$	42794.9	1.53610	0.768048	+	0.640392i	$0.221228\pi$
$920$	0	0
$921$	−6276.93	−0.224573
$922$	0	0
$923$	7706.68	0.274830
$924$	0	0
$925$	25737.6	0.914860
$926$	0	0
$927$	−2756.28	−0.0976572
$928$	0	0
$929$	−10916.0	−0.385515	−0.192757	−	0.981246i	$-0.561743\pi$
$929$	−10916.0	−0.385515	−0.192757	+	0.981246i	$0.561743\pi$
$930$	0	0
$931$	29488.3	1.03807
$932$	0	0
$933$	34200.4	1.20008
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31597.7	−1.10166	−0.550829	−	0.834618i	$-0.685688\pi$
$937$	−31597.7	−1.10166	−0.550829	+	0.834618i	$0.685688\pi$
$938$	0	0
$939$	4123.49	0.143307
$940$	0	0
$941$	3420.29	0.118489	0.0592446	−	0.998243i	$-0.481131\pi$
$941$	3420.29	0.118489	0.0592446	+	0.998243i	$0.481131\pi$
$942$	0	0
$943$	9478.26	0.327311
$944$	0	0
$945$	10793.4	0.371546
$946$	0	0
$947$	−47614.1	−1.63384	−0.816921	−	0.576749i	$-0.804321\pi$
$947$	−47614.1	−1.63384	−0.816921	+	0.576749i	$0.804321\pi$
$948$	0	0
$949$	−41654.5	−1.42483
$950$	0	0
$951$	−12316.7	−0.419975
$952$	0	0
$953$	−30793.5	−1.04669	−0.523347	−	0.852119i	$-0.675317\pi$
$953$	−30793.5	−1.04669	−0.523347	+	0.852119i	$0.675317\pi$
$954$	0	0
$955$	−50683.6	−1.71736
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9038.28	0.304339
$960$	0	0
$961$	−27776.9	−0.932391
$962$	0	0
$963$	7664.36	0.256470
$964$	0	0
$965$	23101.6	0.770640
$966$	0	0
$967$	−14753.3	−0.490626	−0.245313	−	0.969444i	$-0.578891\pi$
$967$	−14753.3	−0.490626	−0.245313	+	0.969444i	$0.578891\pi$
$968$	0	0
$969$	−41273.0	−1.36830
$970$	0	0
$971$	−12541.3	−0.414491	−0.207245	−	0.978289i	$-0.566450\pi$
$971$	−12541.3	−0.414491	−0.207245	+	0.978289i	$0.566450\pi$
$972$	0	0
$973$	−2753.73	−0.0907302
$974$	0	0
$975$	−102971.	−3.38227
$976$	0	0
$977$	−56562.7	−1.85220	−0.926101	−	0.377276i	$-0.876861\pi$
$977$	−56562.7	−1.85220	−0.926101	+	0.377276i	$0.876861\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	253.678	0.00825618
$982$	0	0
$983$	−143.116	−0.00464363	−0.00232182	−	0.999997i	$-0.500739\pi$
$983$	−143.116	−0.00464363	−0.00232182	+	0.999997i	$0.500739\pi$
$984$	0	0
$985$	35870.5	1.16033
$986$	0	0
$987$	13640.1	0.439888
$988$	0	0
$989$	22784.5	0.732562
$990$	0	0
$991$	18914.6	0.606299	0.303150	−	0.952943i	$-0.401962\pi$
$991$	18914.6	0.606299	0.303150	+	0.952943i	$0.401962\pi$
$992$	0	0
$993$	−35313.0	−1.12852
$994$	0	0
$995$	−56588.9	−1.80300
$996$	0	0
$997$	52584.3	1.67037	0.835187	−	0.549966i	$-0.185359\pi$
$997$	52584.3	1.67037	0.835187	+	0.549966i	$0.185359\pi$
$998$	0	0
$999$	11467.4	0.363177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.o.1.3		8
4.3	odd	2		1936.4.a.bv.1.6		8
11.2	odd	10		88.4.i.a.81.2	yes	16
11.6	odd	10		88.4.i.a.25.2	✓	16
11.10	odd	2		968.4.a.n.1.3		8
44.35	even	10		176.4.m.e.81.3		16
44.39	even	10		176.4.m.e.113.3		16
44.43	even	2		1936.4.a.bw.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.a.25.2	✓	16	11.6	odd	10
88.4.i.a.81.2	yes	16	11.2	odd	10
176.4.m.e.81.3		16	44.35	even	10
176.4.m.e.113.3		16	44.39	even	10
968.4.a.n.1.3		8	11.10	odd	2
968.4.a.o.1.3		8	1.1	even	1	trivial
1936.4.a.bv.1.6		8	4.3	odd	2
1936.4.a.bw.1.6		8	44.43	even	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q-5.71590 q^{3} -19.9654 q^{5} -4.43442 q^{7} +5.67153 q^{9} +O(q^{10})\)	\(q-5.71590 q^{3} -19.9654 q^{5} -4.43442 q^{7} +5.67153 q^{9} +65.8393 q^{13} +114.120 q^{15} -79.1744 q^{17} -91.2003 q^{19} +25.3467 q^{21} +90.4612 q^{23} +273.618 q^{25} +121.911 q^{27} +157.770 q^{29} -44.8791 q^{31} +88.5351 q^{35} +94.0637 q^{37} -376.331 q^{39} +104.777 q^{41} +251.870 q^{43} -113.235 q^{45} +538.141 q^{47} -323.336 q^{49} +452.553 q^{51} +588.409 q^{53} +521.292 q^{57} -466.276 q^{59} -506.686 q^{61} -25.1499 q^{63} -1314.51 q^{65} -228.386 q^{67} -517.067 q^{69} +117.053 q^{71} -632.669 q^{73} -1563.98 q^{75} -857.361 q^{79} -849.965 q^{81} +261.765 q^{83} +1580.75 q^{85} -901.800 q^{87} -441.266 q^{89} -291.959 q^{91} +256.525 q^{93} +1820.85 q^{95} -1567.18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9	\( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39} + 74 q^{41} - 35 q^{43} - 566 q^{45} + 197 q^{47} + 221 q^{49} + 608 q^{51} - 247 q^{53} + 979 q^{57} - 1208 q^{59} - 2061 q^{61} + 1617 q^{63} - 837 q^{65} - 829 q^{67} - 1860 q^{69} - 2005 q^{71} - 320 q^{73} - 2323 q^{75} + 1581 q^{79} - 3752 q^{81} - 692 q^{83} - 21 q^{85} + 131 q^{87} - 3205 q^{89} - 2724 q^{91} - 722 q^{93} + 432 q^{95} - 4702 q^{97}+O(q^{100}) \) 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * q^97

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