LMFDB - Embedded newform 1216.4.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 $ x^5 - 2x^4 - 28x^3 - 8x^2 + 73x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.169415$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.75519 q^{3} -10.5762 q^{5} +30.4413 q^{7} -4.38817 q^{9} +O(q^{10})$	$q+4.75519 q^{3} -10.5762 q^{5} +30.4413 q^{7} -4.38817 q^{9} +35.9632 q^{11} +84.5460 q^{13} -50.2919 q^{15} +46.5738 q^{17} -19.0000 q^{19} +144.754 q^{21} +21.5552 q^{23} -13.1438 q^{25} -149.257 q^{27} +241.965 q^{29} -143.527 q^{31} +171.012 q^{33} -321.953 q^{35} -57.8566 q^{37} +402.032 q^{39} -305.486 q^{41} -314.191 q^{43} +46.4102 q^{45} +228.640 q^{47} +583.671 q^{49} +221.467 q^{51} +85.6653 q^{53} -380.355 q^{55} -90.3486 q^{57} +52.0998 q^{59} +133.686 q^{61} -133.582 q^{63} -894.176 q^{65} +1074.37 q^{67} +102.499 q^{69} +245.509 q^{71} -501.891 q^{73} -62.5011 q^{75} +1094.77 q^{77} -730.048 q^{79} -591.263 q^{81} +915.448 q^{83} -492.574 q^{85} +1150.59 q^{87} -374.727 q^{89} +2573.69 q^{91} -682.498 q^{93} +200.948 q^{95} +440.507 q^{97} -157.813 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} - 5 q^{5} + 7 q^{7} - 5 q^{9}+O(q^{10}) $ 5 * q + 6 * q^3 - 5 * q^5 + 7 * q^7 - 5 * q^9	$ 5 q + 6 q^{3} - 5 q^{5} + 7 q^{7} - 5 q^{9} + 13 q^{11} + 72 q^{13} + 72 q^{15} - 59 q^{17} - 95 q^{19} + 224 q^{21} + 52 q^{23} - 86 q^{25} + 54 q^{27} + 128 q^{29} - 110 q^{31} - 68 q^{33} - 45 q^{35} + 436 q^{37} + 356 q^{39} - 804 q^{41} - 143 q^{43} + 579 q^{45} + 661 q^{47} - 406 q^{49} + 570 q^{51} + 898 q^{53} + 17 q^{55} - 114 q^{57} + 196 q^{59} + 1079 q^{61} + 143 q^{63} - 1632 q^{65} + 832 q^{67} + 1644 q^{69} + 834 q^{71} - 1375 q^{73} + 1054 q^{75} + 1473 q^{77} + 154 q^{79} - 1859 q^{81} + 2224 q^{83} + 1807 q^{85} + 2004 q^{87} - 542 q^{89} + 1890 q^{91} + 2788 q^{93} + 95 q^{95} - 1528 q^{97} + 601 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 - 5 * q^5 + 7 * q^7 - 5 * q^9 + 13 * q^11 + 72 * q^13 + 72 * q^15 - 59 * q^17 - 95 * q^19 + 224 * q^21 + 52 * q^23 - 86 * q^25 + 54 * q^27 + 128 * q^29 - 110 * q^31 - 68 * q^33 - 45 * q^35 + 436 * q^37 + 356 * q^39 - 804 * q^41 - 143 * q^43 + 579 * q^45 + 661 * q^47 - 406 * q^49 + 570 * q^51 + 898 * q^53 + 17 * q^55 - 114 * q^57 + 196 * q^59 + 1079 * q^61 + 143 * q^63 - 1632 * q^65 + 832 * q^67 + 1644 * q^69 + 834 * q^71 - 1375 * q^73 + 1054 * q^75 + 1473 * q^77 + 154 * q^79 - 1859 * q^81 + 2224 * q^83 + 1807 * q^85 + 2004 * q^87 - 542 * q^89 + 1890 * q^91 + 2788 * q^93 + 95 * q^95 - 1528 * q^97 + 601 * q^99

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
$3$	4.75519	0.915137	0.457568	−	0.889174i	$-0.348721\pi$
$3$	4.75519	0.915137	0.457568	+	0.889174i	$0.348721\pi$
$4$	0	0
$5$	−10.5762	−0.945965	−0.472983	−	0.881072i	$-0.656823\pi$
$5$	−10.5762	−0.945965	−0.472983	+	0.881072i	$0.656823\pi$
$6$	0	0
$7$	30.4413	1.64367	0.821837	−	0.569722i	$-0.192949\pi$
$7$	30.4413	1.64367	0.821837	+	0.569722i	$0.192949\pi$
$8$	0	0
$9$	−4.38817	−0.162525
$10$	0	0
$11$	35.9632	0.985757	0.492878	−	0.870098i	$-0.335945\pi$
$11$	35.9632	0.985757	0.492878	+	0.870098i	$0.335945\pi$
$12$	0	0
$13$	84.5460	1.80376	0.901878	−	0.431990i	$-0.142188\pi$
$13$	84.5460	1.80376	0.901878	+	0.431990i	$0.142188\pi$
$14$	0	0
$15$	−50.2919	−0.865687
$16$	0	0
$17$	46.5738	0.664459	0.332230	−	0.943199i	$-0.392199\pi$
$17$	46.5738	0.664459	0.332230	+	0.943199i	$0.392199\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	144.754	1.50419
$22$	0	0
$23$	21.5552	0.195416	0.0977079	−	0.995215i	$-0.468849\pi$
$23$	21.5552	0.195416	0.0977079	+	0.995215i	$0.468849\pi$
$24$	0	0
$25$	−13.1438	−0.105150
$26$	0	0
$27$	−149.257	−1.06387
$28$	0	0
$29$	241.965	1.54937	0.774684	−	0.632348i	$-0.217909\pi$
$29$	241.965	1.54937	0.774684	+	0.632348i	$0.217909\pi$
$30$	0	0
$31$	−143.527	−0.831555	−0.415777	−	0.909466i	$-0.636490\pi$
$31$	−143.527	−0.831555	−0.415777	+	0.909466i	$0.636490\pi$
$32$	0	0
$33$	171.012	0.902102
$34$	0	0
$35$	−321.953	−1.55486
$36$	0	0
$37$	−57.8566	−0.257069	−0.128535	−	0.991705i	$-0.541027\pi$
$37$	−57.8566	−0.257069	−0.128535	+	0.991705i	$0.541027\pi$
$38$	0	0
$39$	402.032	1.65068
$40$	0	0
$41$	−305.486	−1.16363	−0.581816	−	0.813321i	$-0.697657\pi$
$41$	−305.486	−1.16363	−0.581816	+	0.813321i	$0.697657\pi$
$42$	0	0
$43$	−314.191	−1.11427	−0.557136	−	0.830421i	$-0.688100\pi$
$43$	−314.191	−1.11427	−0.557136	+	0.830421i	$0.688100\pi$
$44$	0	0
$45$	46.4102	0.153743
$46$	0	0
$47$	228.640	0.709588	0.354794	−	0.934945i	$-0.384551\pi$
$47$	228.640	0.709588	0.354794	+	0.934945i	$0.384551\pi$
$48$	0	0
$49$	583.671	1.70167
$50$	0	0
$51$	221.467	0.608071
$52$	0	0
$53$	85.6653	0.222019	0.111010	−	0.993819i	$-0.464592\pi$
$53$	85.6653	0.222019	0.111010	+	0.993819i	$0.464592\pi$
$54$	0	0
$55$	−380.355	−0.932492
$56$	0	0
$57$	−90.3486	−0.209947
$58$	0	0
$59$	52.0998	0.114963	0.0574815	−	0.998347i	$-0.481693\pi$
$59$	52.0998	0.114963	0.0574815	+	0.998347i	$0.481693\pi$
$60$	0	0
$61$	133.686	0.280601	0.140301	−	0.990109i	$-0.455193\pi$
$61$	133.686	0.280601	0.140301	+	0.990109i	$0.455193\pi$
$62$	0	0
$63$	−133.582	−0.267138
$64$	0	0
$65$	−894.176	−1.70629
$66$	0	0
$67$	1074.37	1.95904	0.979519	−	0.201353i	$-0.0645338\pi$
$67$	1074.37	1.95904	0.979519	+	0.201353i	$0.0645338\pi$
$68$	0	0
$69$	102.499	0.178832
$70$	0	0
$71$	245.509	0.410374	0.205187	−	0.978723i	$-0.434220\pi$
$71$	245.509	0.410374	0.205187	+	0.978723i	$0.434220\pi$
$72$	0	0
$73$	−501.891	−0.804684	−0.402342	−	0.915489i	$-0.631804\pi$
$73$	−501.891	−0.804684	−0.402342	+	0.915489i	$0.631804\pi$
$74$	0	0
$75$	−62.5011	−0.0962267
$76$	0	0
$77$	1094.77	1.62026
$78$	0	0
$79$	−730.048	−1.03971	−0.519853	−	0.854256i	$-0.674013\pi$
$79$	−730.048	−1.03971	−0.519853	+	0.854256i	$0.674013\pi$
$80$	0	0
$81$	−591.263	−0.811061
$82$	0	0
$83$	915.448	1.21064	0.605322	−	0.795980i	$-0.293044\pi$
$83$	915.448	1.21064	0.605322	+	0.795980i	$0.293044\pi$
$84$	0	0
$85$	−492.574	−0.628555
$86$	0	0
$87$	1150.59	1.41788
$88$	0	0
$89$	−374.727	−0.446304	−0.223152	−	0.974784i	$-0.571635\pi$
$89$	−374.727	−0.446304	−0.223152	+	0.974784i	$0.571635\pi$
$90$	0	0
$91$	2573.69	2.96479
$92$	0	0
$93$	−682.498	−0.760986
$94$	0	0
$95$	200.948	0.217019
$96$	0	0
$97$	440.507	0.461100	0.230550	−	0.973060i	$-0.425947\pi$
$97$	440.507	0.461100	0.230550	+	0.973060i	$0.425947\pi$
$98$	0	0
$99$	−157.813	−0.160210
$100$	0	0
$101$	1605.82	1.58203	0.791013	−	0.611799i	$-0.209554\pi$
$101$	1605.82	1.58203	0.791013	+	0.611799i	$0.209554\pi$
$102$	0	0
$103$	493.039	0.471656	0.235828	−	0.971795i	$-0.424220\pi$
$103$	493.039	0.471656	0.235828	+	0.971795i	$0.424220\pi$
$104$	0	0
$105$	−1530.95	−1.42291
$106$	0	0
$107$	−1027.24	−0.928104	−0.464052	−	0.885808i	$-0.653605\pi$
$107$	−1027.24	−0.928104	−0.464052	+	0.885808i	$0.653605\pi$
$108$	0	0
$109$	2102.13	1.84722	0.923612	−	0.383328i	$-0.125222\pi$
$109$	2102.13	1.84722	0.923612	+	0.383328i	$0.125222\pi$
$110$	0	0
$111$	−275.119	−0.235254
$112$	0	0
$113$	2049.21	1.70596	0.852981	−	0.521943i	$-0.174792\pi$
$113$	2049.21	1.70596	0.852981	+	0.521943i	$0.174792\pi$
$114$	0	0
$115$	−227.972	−0.184856
$116$	0	0
$117$	−371.002	−0.293155
$118$	0	0
$119$	1417.77	1.09215
$120$	0	0
$121$	−37.6450	−0.0282833
$122$	0	0
$123$	−1452.64	−1.06488
$124$	0	0
$125$	1461.04	1.04543
$126$	0	0
$127$	−591.347	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$127$	−591.347	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$128$	0	0
$129$	−1494.04	−1.01971
$130$	0	0
$131$	1790.10	1.19391	0.596953	−	0.802276i	$-0.296378\pi$
$131$	1790.10	1.19391	0.596953	+	0.802276i	$0.296378\pi$
$132$	0	0
$133$	−578.384	−0.377085
$134$	0	0
$135$	1578.57	1.00638
$136$	0	0
$137$	−860.195	−0.536433	−0.268217	−	0.963359i	$-0.586434\pi$
$137$	−860.195	−0.536433	−0.268217	+	0.963359i	$0.586434\pi$
$138$	0	0
$139$	−409.173	−0.249680	−0.124840	−	0.992177i	$-0.539842\pi$
$139$	−409.173	−0.249680	−0.124840	+	0.992177i	$0.539842\pi$
$140$	0	0
$141$	1087.23	0.649370
$142$	0	0
$143$	3040.55	1.77807
$144$	0	0
$145$	−2559.07	−1.46565
$146$	0	0
$147$	2775.47	1.55726
$148$	0	0
$149$	−1686.88	−0.927481	−0.463741	−	0.885971i	$-0.653493\pi$
$149$	−1686.88	−0.927481	−0.463741	+	0.885971i	$0.653493\pi$
$150$	0	0
$151$	−378.811	−0.204154	−0.102077	−	0.994777i	$-0.532549\pi$
$151$	−378.811	−0.204154	−0.102077	+	0.994777i	$0.532549\pi$
$152$	0	0
$153$	−204.374	−0.107991
$154$	0	0
$155$	1517.97	0.786622
$156$	0	0
$157$	2486.91	1.26418	0.632092	−	0.774893i	$-0.282196\pi$
$157$	2486.91	1.26418	0.632092	+	0.774893i	$0.282196\pi$
$158$	0	0
$159$	407.355	0.203178
$160$	0	0
$161$	656.167	0.321200
$162$	0	0
$163$	−3138.16	−1.50797	−0.753987	−	0.656889i	$-0.771872\pi$
$163$	−3138.16	−1.50797	−0.753987	+	0.656889i	$0.771872\pi$
$164$	0	0
$165$	−1808.66	−0.853357
$166$	0	0
$167$	−158.068	−0.0732437	−0.0366218	−	0.999329i	$-0.511660\pi$
$167$	−158.068	−0.0732437	−0.0366218	+	0.999329i	$0.511660\pi$
$168$	0	0
$169$	4951.02	2.25354
$170$	0	0
$171$	83.3753	0.0372858
$172$	0	0
$173$	−4024.17	−1.76851	−0.884255	−	0.467004i	$-0.845333\pi$
$173$	−4024.17	−1.76851	−0.884255	+	0.467004i	$0.845333\pi$
$174$	0	0
$175$	−400.113	−0.172833
$176$	0	0
$177$	247.744	0.105207
$178$	0	0
$179$	2070.61	0.864607	0.432303	−	0.901728i	$-0.357701\pi$
$179$	2070.61	0.864607	0.432303	+	0.901728i	$0.357701\pi$
$180$	0	0
$181$	−4167.01	−1.71122	−0.855612	−	0.517618i	$-0.826819\pi$
$181$	−4167.01	−1.71122	−0.855612	+	0.517618i	$0.826819\pi$
$182$	0	0
$183$	635.700	0.256788
$184$	0	0
$185$	611.904	0.243179
$186$	0	0
$187$	1674.95	0.654995
$188$	0	0
$189$	−4543.56	−1.74865
$190$	0	0
$191$	1731.24	0.655853	0.327926	−	0.944703i	$-0.393650\pi$
$191$	1731.24	0.655853	0.327926	+	0.944703i	$0.393650\pi$
$192$	0	0
$193$	−4177.76	−1.55814	−0.779072	−	0.626935i	$-0.784309\pi$
$193$	−4177.76	−1.55814	−0.779072	+	0.626935i	$0.784309\pi$
$194$	0	0
$195$	−4251.98	−1.56149
$196$	0	0
$197$	3216.57	1.16330	0.581652	−	0.813438i	$-0.302407\pi$
$197$	3216.57	1.16330	0.581652	+	0.813438i	$0.302407\pi$
$198$	0	0
$199$	−2960.99	−1.05477	−0.527385	−	0.849626i	$-0.676828\pi$
$199$	−2960.99	−1.05477	−0.527385	+	0.849626i	$0.676828\pi$
$200$	0	0
$201$	5108.85	1.79279
$202$	0	0
$203$	7365.71	2.54666
$204$	0	0
$205$	3230.88	1.10075
$206$	0	0
$207$	−94.5878	−0.0317599
$208$	0	0
$209$	−683.302	−0.226148
$210$	0	0
$211$	2063.83	0.673365	0.336683	−	0.941618i	$-0.390695\pi$
$211$	2063.83	0.673365	0.336683	+	0.941618i	$0.390695\pi$
$212$	0	0
$213$	1167.44	0.375549
$214$	0	0
$215$	3322.95	1.05406
$216$	0	0
$217$	−4369.14	−1.36681
$218$	0	0
$219$	−2386.59	−0.736396
$220$	0	0
$221$	3937.63	1.19852
$222$	0	0
$223$	4574.68	1.37374	0.686868	−	0.726783i	$-0.258985\pi$
$223$	4574.68	1.37374	0.686868	+	0.726783i	$0.258985\pi$
$224$	0	0
$225$	57.6771	0.0170895
$226$	0	0
$227$	−3374.05	−0.986535	−0.493267	−	0.869878i	$-0.664198\pi$
$227$	−3374.05	−0.986535	−0.493267	+	0.869878i	$0.664198\pi$
$228$	0	0
$229$	−4022.93	−1.16089	−0.580443	−	0.814301i	$-0.697120\pi$
$229$	−4022.93	−1.16089	−0.580443	+	0.814301i	$0.697120\pi$
$230$	0	0
$231$	5205.82	1.48276
$232$	0	0
$233$	935.297	0.262976	0.131488	−	0.991318i	$-0.458025\pi$
$233$	935.297	0.262976	0.131488	+	0.991318i	$0.458025\pi$
$234$	0	0
$235$	−2418.15	−0.671245
$236$	0	0
$237$	−3471.52	−0.951474
$238$	0	0
$239$	−74.4072	−0.0201381	−0.0100690	−	0.999949i	$-0.503205\pi$
$239$	−74.4072	−0.0201381	−0.0100690	+	0.999949i	$0.503205\pi$
$240$	0	0
$241$	−3073.37	−0.821467	−0.410733	−	0.911756i	$-0.634727\pi$
$241$	−3073.37	−0.821467	−0.410733	+	0.911756i	$0.634727\pi$
$242$	0	0
$243$	1218.36	0.321638
$244$	0	0
$245$	−6173.03	−1.60972
$246$	0	0
$247$	−1606.37	−0.413810
$248$	0	0
$249$	4353.13	1.10791
$250$	0	0
$251$	4423.04	1.11227	0.556136	−	0.831092i	$-0.312284\pi$
$251$	4423.04	1.11227	0.556136	+	0.831092i	$0.312284\pi$
$252$	0	0
$253$	775.194	0.192632
$254$	0	0
$255$	−2342.28	−0.575214
$256$	0	0
$257$	−3713.10	−0.901232	−0.450616	−	0.892718i	$-0.648796\pi$
$257$	−3713.10	−0.901232	−0.450616	+	0.892718i	$0.648796\pi$
$258$	0	0
$259$	−1761.23	−0.422538
$260$	0	0
$261$	−1061.78	−0.251811
$262$	0	0
$263$	2218.24	0.520085	0.260042	−	0.965597i	$-0.416263\pi$
$263$	2218.24	0.520085	0.260042	+	0.965597i	$0.416263\pi$
$264$	0	0
$265$	−906.014	−0.210023
$266$	0	0
$267$	−1781.90	−0.408429
$268$	0	0
$269$	−2157.92	−0.489111	−0.244556	−	0.969635i	$-0.578642\pi$
$269$	−2157.92	−0.489111	−0.244556	+	0.969635i	$0.578642\pi$
$270$	0	0
$271$	−8020.04	−1.79772	−0.898862	−	0.438233i	$-0.855605\pi$
$271$	−8020.04	−1.79772	−0.898862	+	0.438233i	$0.855605\pi$
$272$	0	0
$273$	12238.4	2.71319
$274$	0	0
$275$	−472.692	−0.103652
$276$	0	0
$277$	−4481.14	−0.972005	−0.486003	−	0.873957i	$-0.661545\pi$
$277$	−4481.14	−0.972005	−0.486003	+	0.873957i	$0.661545\pi$
$278$	0	0
$279$	629.821	0.135148
$280$	0	0
$281$	5271.20	1.11905	0.559526	−	0.828813i	$-0.310983\pi$
$281$	5271.20	1.11905	0.559526	+	0.828813i	$0.310983\pi$
$282$	0	0
$283$	478.090	0.100422	0.0502111	−	0.998739i	$-0.484011\pi$
$283$	478.090	0.100422	0.0502111	+	0.998739i	$0.484011\pi$
$284$	0	0
$285$	955.546	0.198602
$286$	0	0
$287$	−9299.38	−1.91263
$288$	0	0
$289$	−2743.88	−0.558494
$290$	0	0
$291$	2094.70	0.421970
$292$	0	0
$293$	8287.67	1.65246	0.826231	−	0.563332i	$-0.190481\pi$
$293$	8287.67	1.65246	0.826231	+	0.563332i	$0.190481\pi$
$294$	0	0
$295$	−551.019	−0.108751
$296$	0	0
$297$	−5367.76	−1.04872
$298$	0	0
$299$	1822.40	0.352482
$300$	0	0
$301$	−9564.38	−1.83150
$302$	0	0
$303$	7635.96	1.44777
$304$	0	0
$305$	−1413.89	−0.265439
$306$	0	0
$307$	6522.60	1.21259	0.606294	−	0.795240i	$-0.292655\pi$
$307$	6522.60	1.21259	0.606294	+	0.795240i	$0.292655\pi$
$308$	0	0
$309$	2344.49	0.431630
$310$	0	0
$311$	−8881.78	−1.61942	−0.809710	−	0.586831i	$-0.800375\pi$
$311$	−8881.78	−1.61942	−0.809710	+	0.586831i	$0.800375\pi$
$312$	0	0
$313$	3784.75	0.683471	0.341736	−	0.939796i	$-0.388985\pi$
$313$	3784.75	0.683471	0.341736	+	0.939796i	$0.388985\pi$
$314$	0	0
$315$	1412.79	0.252703
$316$	0	0
$317$	−2672.35	−0.473482	−0.236741	−	0.971573i	$-0.576079\pi$
$317$	−2672.35	−0.473482	−0.236741	+	0.971573i	$0.576079\pi$
$318$	0	0
$319$	8701.83	1.52730
$320$	0	0
$321$	−4884.73	−0.849342
$322$	0	0
$323$	−884.902	−0.152437
$324$	0	0
$325$	−1111.25	−0.189665
$326$	0	0
$327$	9996.02	1.69046
$328$	0	0
$329$	6960.11	1.16633
$330$	0	0
$331$	317.128	0.0526614	0.0263307	−	0.999653i	$-0.491618\pi$
$331$	317.128	0.0526614	0.0263307	+	0.999653i	$0.491618\pi$
$332$	0	0
$333$	253.885	0.0417802
$334$	0	0
$335$	−11362.8	−1.85318
$336$	0	0
$337$	11574.9	1.87100	0.935500	−	0.353326i	$-0.114949\pi$
$337$	11574.9	1.87100	0.935500	+	0.353326i	$0.114949\pi$
$338$	0	0
$339$	9744.39	1.56119
$340$	0	0
$341$	−5161.69	−0.819711
$342$	0	0
$343$	7326.34	1.15331
$344$	0	0
$345$	−1084.05	−0.169169
$346$	0	0
$347$	3458.09	0.534986	0.267493	−	0.963560i	$-0.413805\pi$
$347$	3458.09	0.534986	0.267493	+	0.963560i	$0.413805\pi$
$348$	0	0
$349$	3664.39	0.562035	0.281018	−	0.959703i	$-0.409328\pi$
$349$	3664.39	0.562035	0.281018	+	0.959703i	$0.409328\pi$
$350$	0	0
$351$	−12619.1	−1.91896
$352$	0	0
$353$	3772.67	0.568836	0.284418	−	0.958700i	$-0.408200\pi$
$353$	3772.67	0.568836	0.284418	+	0.958700i	$0.408200\pi$
$354$	0	0
$355$	−2596.56	−0.388200
$356$	0	0
$357$	6741.75	0.999471
$358$	0	0
$359$	8649.19	1.27155	0.635775	−	0.771874i	$-0.280680\pi$
$359$	8649.19	1.27155	0.635775	+	0.771874i	$0.280680\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−179.009	−0.0258831
$364$	0	0
$365$	5308.11	0.761203
$366$	0	0
$367$	2133.33	0.303430	0.151715	−	0.988424i	$-0.451520\pi$
$367$	2133.33	0.303430	0.151715	+	0.988424i	$0.451520\pi$
$368$	0	0
$369$	1340.53	0.189119
$370$	0	0
$371$	2607.76	0.364928
$372$	0	0
$373$	9027.22	1.25311	0.626557	−	0.779375i	$-0.284463\pi$
$373$	9027.22	1.25311	0.626557	+	0.779375i	$0.284463\pi$
$374$	0	0
$375$	6947.51	0.956714
$376$	0	0
$377$	20457.1	2.79468
$378$	0	0
$379$	−6180.24	−0.837619	−0.418809	−	0.908074i	$-0.637552\pi$
$379$	−6180.24	−0.837619	−0.418809	+	0.908074i	$0.637552\pi$
$380$	0	0
$381$	−2811.97	−0.378114
$382$	0	0
$383$	9872.54	1.31714	0.658568	−	0.752521i	$-0.271163\pi$
$383$	9872.54	1.31714	0.658568	+	0.752521i	$0.271163\pi$
$384$	0	0
$385$	−11578.5	−1.53271
$386$	0	0
$387$	1378.73	0.181097
$388$	0	0
$389$	1557.35	0.202985	0.101492	−	0.994836i	$-0.467638\pi$
$389$	1557.35	0.202985	0.101492	+	0.994836i	$0.467638\pi$
$390$	0	0
$391$	1003.91	0.129846
$392$	0	0
$393$	8512.27	1.09259
$394$	0	0
$395$	7721.14	0.983526
$396$	0	0
$397$	−263.568	−0.0333202	−0.0166601	−	0.999861i	$-0.505303\pi$
$397$	−263.568	−0.0333202	−0.0166601	+	0.999861i	$0.505303\pi$
$398$	0	0
$399$	−2750.33	−0.345084
$400$	0	0
$401$	648.837	0.0808014	0.0404007	−	0.999184i	$-0.487137\pi$
$401$	648.837	0.0808014	0.0404007	+	0.999184i	$0.487137\pi$
$402$	0	0
$403$	−12134.6	−1.49992
$404$	0	0
$405$	6253.32	0.767235
$406$	0	0
$407$	−2080.71	−0.253408
$408$	0	0
$409$	2953.83	0.357109	0.178555	−	0.983930i	$-0.442858\pi$
$409$	2953.83	0.357109	0.178555	+	0.983930i	$0.442858\pi$
$410$	0	0
$411$	−4090.39	−0.490910
$412$	0	0
$413$	1585.98	0.188962
$414$	0	0
$415$	−9681.98	−1.14523
$416$	0	0
$417$	−1945.69	−0.228492
$418$	0	0
$419$	−16810.5	−1.96001	−0.980007	−	0.198962i	$-0.936243\pi$
$419$	−16810.5	−1.96001	−0.980007	+	0.198962i	$0.936243\pi$
$420$	0	0
$421$	−14263.8	−1.65124	−0.825621	−	0.564225i	$-0.809175\pi$
$421$	−14263.8	−1.65124	−0.825621	+	0.564225i	$0.809175\pi$
$422$	0	0
$423$	−1003.31	−0.115326
$424$	0	0
$425$	−612.155	−0.0698680
$426$	0	0
$427$	4069.56	0.461217
$428$	0	0
$429$	14458.4	1.62717
$430$	0	0
$431$	11861.2	1.32560	0.662799	−	0.748798i	$-0.269369\pi$
$431$	11861.2	1.32560	0.662799	+	0.748798i	$0.269369\pi$
$432$	0	0
$433$	−2927.69	−0.324933	−0.162467	−	0.986714i	$-0.551945\pi$
$433$	−2927.69	−0.324933	−0.162467	+	0.986714i	$0.551945\pi$
$434$	0	0
$435$	−12168.9	−1.34127
$436$	0	0
$437$	−409.548	−0.0448314
$438$	0	0
$439$	−8942.54	−0.972219	−0.486110	−	0.873898i	$-0.661584\pi$
$439$	−8942.54	−0.972219	−0.486110	+	0.873898i	$0.661584\pi$
$440$	0	0
$441$	−2561.25	−0.276563
$442$	0	0
$443$	489.215	0.0524680	0.0262340	−	0.999656i	$-0.491649\pi$
$443$	489.215	0.0524680	0.0262340	+	0.999656i	$0.491649\pi$
$444$	0	0
$445$	3963.20	0.422188
$446$	0	0
$447$	−8021.44	−0.848772
$448$	0	0
$449$	−10492.1	−1.10279	−0.551397	−	0.834243i	$-0.685905\pi$
$449$	−10492.1	−1.10279	−0.551397	+	0.834243i	$0.685905\pi$
$450$	0	0
$451$	−10986.3	−1.14706
$452$	0	0
$453$	−1801.32	−0.186828
$454$	0	0
$455$	−27219.9	−2.80459
$456$	0	0
$457$	−19039.9	−1.94891	−0.974454	−	0.224586i	$-0.927897\pi$
$457$	−19039.9	−1.94891	−0.974454	+	0.224586i	$0.927897\pi$
$458$	0	0
$459$	−6951.45	−0.706898
$460$	0	0
$461$	−13377.2	−1.35149	−0.675745	−	0.737136i	$-0.736178\pi$
$461$	−13377.2	−1.35149	−0.675745	+	0.737136i	$0.736178\pi$
$462$	0	0
$463$	11521.3	1.15646	0.578231	−	0.815873i	$-0.303743\pi$
$463$	11521.3	1.15646	0.578231	+	0.815873i	$0.303743\pi$
$464$	0	0
$465$	7218.24	0.719866
$466$	0	0
$467$	−18643.4	−1.84736	−0.923678	−	0.383171i	$-0.874832\pi$
$467$	−18643.4	−1.84736	−0.923678	+	0.383171i	$0.874832\pi$
$468$	0	0
$469$	32705.3	3.22002
$470$	0	0
$471$	11825.7	1.15690
$472$	0	0
$473$	−11299.3	−1.09840
$474$	0	0
$475$	249.732	0.0241231
$476$	0	0
$477$	−375.914	−0.0360837
$478$	0	0
$479$	9504.39	0.906611	0.453305	−	0.891355i	$-0.350245\pi$
$479$	9504.39	0.906611	0.453305	+	0.891355i	$0.350245\pi$
$480$	0	0
$481$	−4891.54	−0.463691
$482$	0	0
$483$	3120.20	0.293942
$484$	0	0
$485$	−4658.90	−0.436185
$486$	0	0
$487$	−17635.6	−1.64096	−0.820480	−	0.571675i	$-0.806294\pi$
$487$	−17635.6	−1.64096	−0.820480	+	0.571675i	$0.806294\pi$
$488$	0	0
$489$	−14922.6	−1.38000
$490$	0	0
$491$	−5196.09	−0.477589	−0.238795	−	0.971070i	$-0.576752\pi$
$491$	−5196.09	−0.477589	−0.238795	+	0.971070i	$0.576752\pi$
$492$	0	0
$493$	11269.2	1.02949
$494$	0	0
$495$	1669.06	0.151553
$496$	0	0
$497$	7473.61	0.674522
$498$	0	0
$499$	−14454.8	−1.29676	−0.648380	−	0.761317i	$-0.724553\pi$
$499$	−14454.8	−1.29676	−0.648380	+	0.761317i	$0.724553\pi$
$500$	0	0
$501$	−751.645	−0.0670280
$502$	0	0
$503$	10268.1	0.910202	0.455101	−	0.890440i	$-0.349603\pi$
$503$	10268.1	0.910202	0.455101	+	0.890440i	$0.349603\pi$
$504$	0	0
$505$	−16983.5	−1.49654
$506$	0	0
$507$	23543.1	2.06230
$508$	0	0
$509$	−8238.44	−0.717411	−0.358706	−	0.933451i	$-0.616782\pi$
$509$	−8238.44	−0.717411	−0.358706	+	0.933451i	$0.616782\pi$
$510$	0	0
$511$	−15278.2	−1.32264
$512$	0	0
$513$	2835.88	0.244068
$514$	0	0
$515$	−5214.49	−0.446170
$516$	0	0
$517$	8222.65	0.699481
$518$	0	0
$519$	−19135.7	−1.61843
$520$	0	0
$521$	−16940.7	−1.42454	−0.712271	−	0.701905i	$-0.752333\pi$
$521$	−16940.7	−1.42454	−0.712271	+	0.701905i	$0.752333\pi$
$522$	0	0
$523$	2198.70	0.183829	0.0919143	−	0.995767i	$-0.470701\pi$
$523$	2198.70	0.183829	0.0919143	+	0.995767i	$0.470701\pi$
$524$	0	0
$525$	−1902.61	−0.158165
$526$	0	0
$527$	−6684.59	−0.552534
$528$	0	0
$529$	−11702.4	−0.961813
$530$	0	0
$531$	−228.623	−0.0186844
$532$	0	0
$533$	−25827.6	−2.09891
$534$	0	0
$535$	10864.3	0.877954
$536$	0	0
$537$	9846.14	0.791234
$538$	0	0
$539$	20990.7	1.67743
$540$	0	0
$541$	−4380.65	−0.348131	−0.174065	−	0.984734i	$-0.555690\pi$
$541$	−4380.65	−0.348131	−0.174065	+	0.984734i	$0.555690\pi$
$542$	0	0
$543$	−19814.9	−1.56600
$544$	0	0
$545$	−22232.6	−1.74741
$546$	0	0
$547$	3714.48	0.290347	0.145173	−	0.989406i	$-0.453626\pi$
$547$	3714.48	0.290347	0.145173	+	0.989406i	$0.453626\pi$
$548$	0	0
$549$	−586.635	−0.0456047
$550$	0	0
$551$	−4597.33	−0.355450
$552$	0	0
$553$	−22223.6	−1.70894
$554$	0	0
$555$	2909.72	0.222542
$556$	0	0
$557$	−18491.1	−1.40663	−0.703313	−	0.710880i	$-0.748297\pi$
$557$	−18491.1	−1.40663	−0.703313	+	0.710880i	$0.748297\pi$
$558$	0	0
$559$	−26563.6	−2.00988
$560$	0	0
$561$	7964.68	0.599410
$562$	0	0
$563$	−13815.8	−1.03422	−0.517109	−	0.855919i	$-0.672992\pi$
$563$	−13815.8	−1.03422	−0.517109	+	0.855919i	$0.672992\pi$
$564$	0	0
$565$	−21672.9	−1.61378
$566$	0	0
$567$	−17998.8	−1.33312
$568$	0	0
$569$	−101.065	−0.00744616	−0.00372308	−	0.999993i	$-0.501185\pi$
$569$	−101.065	−0.00744616	−0.00372308	+	0.999993i	$0.501185\pi$
$570$	0	0
$571$	13667.8	1.00171	0.500857	−	0.865530i	$-0.333018\pi$
$571$	13667.8	1.00171	0.500857	+	0.865530i	$0.333018\pi$
$572$	0	0
$573$	8232.36	0.600195
$574$	0	0
$575$	−283.316	−0.0205480
$576$	0	0
$577$	4227.49	0.305014	0.152507	−	0.988302i	$-0.451265\pi$
$577$	4227.49	0.305014	0.152507	+	0.988302i	$0.451265\pi$
$578$	0	0
$579$	−19866.0	−1.42591
$580$	0	0
$581$	27867.4	1.98991
$582$	0	0
$583$	3080.80	0.218857
$584$	0	0
$585$	3923.80	0.277315
$586$	0	0
$587$	3596.44	0.252881	0.126440	−	0.991974i	$-0.459645\pi$
$587$	3596.44	0.252881	0.126440	+	0.991974i	$0.459645\pi$
$588$	0	0
$589$	2727.01	0.190772
$590$	0	0
$591$	15295.4	1.06458
$592$	0	0
$593$	8122.05	0.562450	0.281225	−	0.959642i	$-0.409259\pi$
$593$	8122.05	0.562450	0.281225	+	0.959642i	$0.409259\pi$
$594$	0	0
$595$	−14994.6	−1.03314
$596$	0	0
$597$	−14080.1	−0.965259
$598$	0	0
$599$	4944.42	0.337268	0.168634	−	0.985679i	$-0.446064\pi$
$599$	4944.42	0.337268	0.168634	+	0.985679i	$0.446064\pi$
$600$	0	0
$601$	−9845.46	−0.668228	−0.334114	−	0.942533i	$-0.608437\pi$
$601$	−9845.46	−0.668228	−0.334114	+	0.942533i	$0.608437\pi$
$602$	0	0
$603$	−4714.53	−0.318392
$604$	0	0
$605$	398.142	0.0267550
$606$	0	0
$607$	−5652.72	−0.377985	−0.188993	−	0.981979i	$-0.560522\pi$
$607$	−5652.72	−0.377985	−0.188993	+	0.981979i	$0.560522\pi$
$608$	0	0
$609$	35025.4	2.33054
$610$	0	0
$611$	19330.6	1.27992
$612$	0	0
$613$	13588.8	0.895343	0.447672	−	0.894198i	$-0.352253\pi$
$613$	13588.8	0.895343	0.447672	+	0.894198i	$0.352253\pi$
$614$	0	0
$615$	15363.5	1.00734
$616$	0	0
$617$	7930.36	0.517446	0.258723	−	0.965952i	$-0.416698\pi$
$617$	7930.36	0.517446	0.258723	+	0.965952i	$0.416698\pi$
$618$	0	0
$619$	−7464.30	−0.484678	−0.242339	−	0.970192i	$-0.577915\pi$
$619$	−7464.30	−0.484678	−0.242339	+	0.970192i	$0.577915\pi$
$620$	0	0
$621$	−3217.25	−0.207897
$622$	0	0
$623$	−11407.2	−0.733578
$624$	0	0
$625$	−13809.3	−0.883793
$626$	0	0
$627$	−3249.23	−0.206956
$628$	0	0
$629$	−2694.60	−0.170812
$630$	0	0
$631$	−6441.10	−0.406365	−0.203182	−	0.979141i	$-0.565128\pi$
$631$	−6441.10	−0.406365	−0.203182	+	0.979141i	$0.565128\pi$
$632$	0	0
$633$	9813.92	0.616221
$634$	0	0
$635$	6254.21	0.390852
$636$	0	0
$637$	49347.1	3.06939
$638$	0	0
$639$	−1077.34	−0.0666961
$640$	0	0
$641$	6811.72	0.419730	0.209865	−	0.977730i	$-0.432698\pi$
$641$	6811.72	0.419730	0.209865	+	0.977730i	$0.432698\pi$
$642$	0	0
$643$	−6151.76	−0.377297	−0.188648	−	0.982045i	$-0.560411\pi$
$643$	−6151.76	−0.377297	−0.188648	+	0.982045i	$0.560411\pi$
$644$	0	0
$645$	15801.3	0.964611
$646$	0	0
$647$	27024.4	1.64210	0.821049	−	0.570857i	$-0.193389\pi$
$647$	27024.4	1.64210	0.821049	+	0.570857i	$0.193389\pi$
$648$	0	0
$649$	1873.68	0.113326
$650$	0	0
$651$	−20776.1	−1.25081
$652$	0	0
$653$	−918.614	−0.0550508	−0.0275254	−	0.999621i	$-0.508763\pi$
$653$	−918.614	−0.0550508	−0.0275254	+	0.999621i	$0.508763\pi$
$654$	0	0
$655$	−18932.5	−1.12939
$656$	0	0
$657$	2202.39	0.130781
$658$	0	0
$659$	−14718.0	−0.870000	−0.435000	−	0.900430i	$-0.643252\pi$
$659$	−14718.0	−0.870000	−0.435000	+	0.900430i	$0.643252\pi$
$660$	0	0
$661$	−5027.57	−0.295839	−0.147920	−	0.988999i	$-0.547258\pi$
$661$	−5027.57	−0.295839	−0.147920	+	0.988999i	$0.547258\pi$
$662$	0	0
$663$	18724.2	1.09681
$664$	0	0
$665$	6117.11	0.356709
$666$	0	0
$667$	5215.59	0.302771
$668$	0	0
$669$	21753.5	1.25716
$670$	0	0
$671$	4807.77	0.276605
$672$	0	0
$673$	−26061.3	−1.49271	−0.746353	−	0.665550i	$-0.768197\pi$
$673$	−26061.3	−1.49271	−0.746353	+	0.665550i	$0.768197\pi$
$674$	0	0
$675$	1961.79	0.111866
$676$	0	0
$677$	−18689.9	−1.06102	−0.530510	−	0.847679i	$-0.678000\pi$
$677$	−18689.9	−1.06102	−0.530510	+	0.847679i	$0.678000\pi$
$678$	0	0
$679$	13409.6	0.757899
$680$	0	0
$681$	−16044.2	−0.902814
$682$	0	0
$683$	−12567.8	−0.704092	−0.352046	−	0.935983i	$-0.614514\pi$
$683$	−12567.8	−0.704092	−0.352046	+	0.935983i	$0.614514\pi$
$684$	0	0
$685$	9097.60	0.507447
$686$	0	0
$687$	−19129.8	−1.06237
$688$	0	0
$689$	7242.65	0.400469
$690$	0	0
$691$	5197.32	0.286129	0.143065	−	0.989713i	$-0.454304\pi$
$691$	5197.32	0.286129	0.143065	+	0.989713i	$0.454304\pi$
$692$	0	0
$693$	−4804.03	−0.263333
$694$	0	0
$695$	4327.50	0.236189
$696$	0	0
$697$	−14227.6	−0.773186
$698$	0	0
$699$	4447.51	0.240659
$700$	0	0
$701$	−3430.19	−0.184817	−0.0924084	−	0.995721i	$-0.529457\pi$
$701$	−3430.19	−0.184817	−0.0924084	+	0.995721i	$0.529457\pi$
$702$	0	0
$703$	1099.28	0.0589758
$704$	0	0
$705$	−11498.8	−0.614281
$706$	0	0
$707$	48883.1	2.60034
$708$	0	0
$709$	−32713.0	−1.73281	−0.866405	−	0.499342i	$-0.833575\pi$
$709$	−32713.0	−1.73281	−0.866405	+	0.499342i	$0.833575\pi$
$710$	0	0
$711$	3203.58	0.168978
$712$	0	0
$713$	−3093.75	−0.162499
$714$	0	0
$715$	−32157.5	−1.68199
$716$	0	0
$717$	−353.820	−0.0184291
$718$	0	0
$719$	1645.14	0.0853316	0.0426658	−	0.999089i	$-0.486415\pi$
$719$	1645.14	0.0853316	0.0426658	+	0.999089i	$0.486415\pi$
$720$	0	0
$721$	15008.7	0.775249
$722$	0	0
$723$	−14614.5	−0.751754
$724$	0	0
$725$	−3180.33	−0.162916
$726$	0	0
$727$	15943.8	0.813374	0.406687	−	0.913568i	$-0.366684\pi$
$727$	15943.8	0.813374	0.406687	+	0.913568i	$0.366684\pi$
$728$	0	0
$729$	21757.7	1.10540
$730$	0	0
$731$	−14633.1	−0.740389
$732$	0	0
$733$	5667.02	0.285561	0.142780	−	0.989754i	$-0.454396\pi$
$733$	5667.02	0.285561	0.142780	+	0.989754i	$0.454396\pi$
$734$	0	0
$735$	−29353.9	−1.47311
$736$	0	0
$737$	38637.9	1.93113
$738$	0	0
$739$	6727.80	0.334893	0.167447	−	0.985881i	$-0.446448\pi$
$739$	6727.80	0.334893	0.167447	+	0.985881i	$0.446448\pi$
$740$	0	0
$741$	−7638.61	−0.378693
$742$	0	0
$743$	−21244.5	−1.04897	−0.524485	−	0.851420i	$-0.675742\pi$
$743$	−21244.5	−1.04897	−0.524485	+	0.851420i	$0.675742\pi$
$744$	0	0
$745$	17840.8	0.877365
$746$	0	0
$747$	−4017.15	−0.196760
$748$	0	0
$749$	−31270.5	−1.52550
$750$	0	0
$751$	−10091.6	−0.490342	−0.245171	−	0.969480i	$-0.578844\pi$
$751$	−10091.6	−0.490342	−0.245171	+	0.969480i	$0.578844\pi$
$752$	0	0
$753$	21032.4	1.01788
$754$	0	0
$755$	4006.38	0.193122
$756$	0	0
$757$	28391.1	1.36314	0.681568	−	0.731755i	$-0.261299\pi$
$757$	28391.1	1.36314	0.681568	+	0.731755i	$0.261299\pi$
$758$	0	0
$759$	3686.19	0.176285
$760$	0	0
$761$	−22548.8	−1.07411	−0.537053	−	0.843549i	$-0.680462\pi$
$761$	−22548.8	−1.07411	−0.537053	+	0.843549i	$0.680462\pi$
$762$	0	0
$763$	63991.5	3.03624
$764$	0	0
$765$	2161.50	0.102156
$766$	0	0
$767$	4404.83	0.207365
$768$	0	0
$769$	6716.01	0.314936	0.157468	−	0.987524i	$-0.449667\pi$
$769$	6716.01	0.314936	0.157468	+	0.987524i	$0.449667\pi$
$770$	0	0
$771$	−17656.5	−0.824750
$772$	0	0
$773$	19385.3	0.901993	0.450996	−	0.892526i	$-0.351069\pi$
$773$	19385.3	0.901993	0.450996	+	0.892526i	$0.351069\pi$
$774$	0	0
$775$	1886.48	0.0874381
$776$	0	0
$777$	−8374.98	−0.386680
$778$	0	0
$779$	5804.23	0.266955
$780$	0	0
$781$	8829.31	0.404529
$782$	0	0
$783$	−36114.8	−1.64833
$784$	0	0
$785$	−26302.1	−1.19587
$786$	0	0
$787$	15943.7	0.722148	0.361074	−	0.932537i	$-0.382410\pi$
$787$	15943.7	0.722148	0.361074	+	0.932537i	$0.382410\pi$
$788$	0	0
$789$	10548.1	0.475949
$790$	0	0
$791$	62380.6	2.80404
$792$	0	0
$793$	11302.6	0.506136
$794$	0	0
$795$	−4308.27	−0.192199
$796$	0	0
$797$	−8261.93	−0.367193	−0.183596	−	0.983002i	$-0.558774\pi$
$797$	−8261.93	−0.367193	−0.183596	+	0.983002i	$0.558774\pi$
$798$	0	0
$799$	10648.7	0.471492
$800$	0	0
$801$	1644.37	0.0725355
$802$	0	0
$803$	−18049.6	−0.793223
$804$	0	0
$805$	−6939.76	−0.303844
$806$	0	0
$807$	−10261.3	−0.447604
$808$	0	0
$809$	−31673.1	−1.37647	−0.688236	−	0.725487i	$-0.741615\pi$
$809$	−31673.1	−1.37647	−0.688236	+	0.725487i	$0.741615\pi$
$810$	0	0
$811$	−32187.8	−1.39367	−0.696835	−	0.717231i	$-0.745409\pi$
$811$	−32187.8	−1.39367	−0.696835	+	0.717231i	$0.745409\pi$
$812$	0	0
$813$	−38136.8	−1.64516
$814$	0	0
$815$	33189.9	1.42649
$816$	0	0
$817$	5969.63	0.255632
$818$	0	0
$819$	−11293.8	−0.481852
$820$	0	0
$821$	−17264.8	−0.733915	−0.366958	−	0.930238i	$-0.619601\pi$
$821$	−17264.8	−0.733915	−0.366958	+	0.930238i	$0.619601\pi$
$822$	0	0
$823$	−22960.1	−0.972466	−0.486233	−	0.873829i	$-0.661629\pi$
$823$	−22960.1	−0.972466	−0.486233	+	0.873829i	$0.661629\pi$
$824$	0	0
$825$	−2247.74	−0.0948561
$826$	0	0
$827$	−1893.43	−0.0796142	−0.0398071	−	0.999207i	$-0.512674\pi$
$827$	−1893.43	−0.0796142	−0.0398071	+	0.999207i	$0.512674\pi$
$828$	0	0
$829$	−1263.35	−0.0529288	−0.0264644	−	0.999650i	$-0.508425\pi$
$829$	−1263.35	−0.0529288	−0.0264644	+	0.999650i	$0.508425\pi$
$830$	0	0
$831$	−21308.7	−0.889518
$832$	0	0
$833$	27183.8	1.13069
$834$	0	0
$835$	1671.76	0.0692859
$836$	0	0
$837$	21422.4	0.884665
$838$	0	0
$839$	19921.4	0.819743	0.409871	−	0.912143i	$-0.365574\pi$
$839$	19921.4	0.819743	0.409871	+	0.912143i	$0.365574\pi$
$840$	0	0
$841$	34157.9	1.40054
$842$	0	0
$843$	25065.6	1.02409
$844$	0	0
$845$	−52363.1	−2.13177
$846$	0	0
$847$	−1145.96	−0.0464885
$848$	0	0
$849$	2273.41	0.0919000
$850$	0	0
$851$	−1247.11	−0.0502354
$852$	0	0
$853$	42171.2	1.69275	0.846375	−	0.532587i	$-0.178780\pi$
$853$	42171.2	1.69275	0.846375	+	0.532587i	$0.178780\pi$
$854$	0	0
$855$	−881.795	−0.0352710
$856$	0	0
$857$	−9759.77	−0.389017	−0.194508	−	0.980901i	$-0.562311\pi$
$857$	−9759.77	−0.389017	−0.194508	+	0.980901i	$0.562311\pi$
$858$	0	0
$859$	11518.1	0.457501	0.228750	−	0.973485i	$-0.426536\pi$
$859$	11518.1	0.457501	0.228750	+	0.973485i	$0.426536\pi$
$860$	0	0
$861$	−44220.3	−1.75032
$862$	0	0
$863$	44074.7	1.73849	0.869247	−	0.494378i	$-0.164604\pi$
$863$	44074.7	1.73849	0.869247	+	0.494378i	$0.164604\pi$
$864$	0	0
$865$	42560.5	1.67295
$866$	0	0
$867$	−13047.7	−0.511098
$868$	0	0
$869$	−26254.9	−1.02490
$870$	0	0
$871$	90833.9	3.53363
$872$	0	0
$873$	−1933.02	−0.0749403
$874$	0	0
$875$	44475.8	1.71835
$876$	0	0
$877$	−20591.3	−0.792836	−0.396418	−	0.918070i	$-0.629747\pi$
$877$	−20591.3	−0.792836	−0.396418	+	0.918070i	$0.629747\pi$
$878$	0	0
$879$	39409.5	1.51223
$880$	0	0
$881$	−33758.4	−1.29098	−0.645488	−	0.763770i	$-0.723346\pi$
$881$	−33758.4	−1.29098	−0.645488	+	0.763770i	$0.723346\pi$
$882$	0	0
$883$	36310.7	1.38387	0.691933	−	0.721962i	$-0.256759\pi$
$883$	36310.7	1.38387	0.691933	+	0.721962i	$0.256759\pi$
$884$	0	0
$885$	−2620.20	−0.0995220
$886$	0	0
$887$	−10388.6	−0.393253	−0.196627	−	0.980478i	$-0.562999\pi$
$887$	−10388.6	−0.393253	−0.196627	+	0.980478i	$0.562999\pi$
$888$	0	0
$889$	−18001.4	−0.679130
$890$	0	0
$891$	−21263.7	−0.799509
$892$	0	0
$893$	−4344.17	−0.162791
$894$	0	0
$895$	−21899.2	−0.817888
$896$	0	0
$897$	8665.87	0.322570
$898$	0	0
$899$	−34728.4	−1.28838
$900$	0	0
$901$	3989.76	0.147523
$902$	0	0
$903$	−45480.4	−1.67607
$904$	0	0
$905$	44071.2	1.61876
$906$	0	0
$907$	−51435.2	−1.88299	−0.941497	−	0.337020i	$-0.890581\pi$
$907$	−51435.2	−1.88299	−0.941497	+	0.337020i	$0.890581\pi$
$908$	0	0
$909$	−7046.60	−0.257119
$910$	0	0
$911$	−50067.2	−1.82086	−0.910428	−	0.413668i	$-0.864247\pi$
$911$	−50067.2	−1.82086	−0.910428	+	0.413668i	$0.864247\pi$
$912$	0	0
$913$	32922.5	1.19340
$914$	0	0
$915$	−6723.30	−0.242913
$916$	0	0
$917$	54492.9	1.96239
$918$	0	0
$919$	33977.6	1.21961	0.609803	−	0.792553i	$-0.291249\pi$
$919$	33977.6	1.21961	0.609803	+	0.792553i	$0.291249\pi$
$920$	0	0
$921$	31016.2	1.10968
$922$	0	0
$923$	20756.8	0.740216
$924$	0	0
$925$	760.454	0.0270309
$926$	0	0
$927$	−2163.54	−0.0766559
$928$	0	0
$929$	−30854.4	−1.08967	−0.544833	−	0.838544i	$-0.683407\pi$
$929$	−30854.4	−1.08967	−0.544833	+	0.838544i	$0.683407\pi$
$930$	0	0
$931$	−11089.8	−0.390389
$932$	0	0
$933$	−42234.5	−1.48199
$934$	0	0
$935$	−17714.6	−0.619603
$936$	0	0
$937$	25083.7	0.874544	0.437272	−	0.899329i	$-0.355945\pi$
$937$	25083.7	0.874544	0.437272	+	0.899329i	$0.355945\pi$
$938$	0	0
$939$	17997.2	0.625470
$940$	0	0
$941$	−20650.9	−0.715408	−0.357704	−	0.933835i	$-0.616440\pi$
$941$	−20650.9	−0.715408	−0.357704	+	0.933835i	$0.616440\pi$
$942$	0	0
$943$	−6584.80	−0.227392
$944$	0	0
$945$	48053.7	1.65417
$946$	0	0
$947$	30301.4	1.03977	0.519885	−	0.854236i	$-0.325975\pi$
$947$	30301.4	1.03977	0.519885	+	0.854236i	$0.325975\pi$
$948$	0	0
$949$	−42432.9	−1.45145
$950$	0	0
$951$	−12707.5	−0.433301
$952$	0	0
$953$	30346.5	1.03150	0.515749	−	0.856740i	$-0.327514\pi$
$953$	30346.5	1.03150	0.515749	+	0.856740i	$0.327514\pi$
$954$	0	0
$955$	−18309.9	−0.620414
$956$	0	0
$957$	41378.9	1.39769
$958$	0	0
$959$	−26185.4	−0.881722
$960$	0	0
$961$	−9191.03	−0.308517
$962$	0	0
$963$	4507.71	0.150840
$964$	0	0
$965$	44184.9	1.47395
$966$	0	0
$967$	−7494.19	−0.249221	−0.124611	−	0.992206i	$-0.539768\pi$
$967$	−7494.19	−0.249221	−0.124611	+	0.992206i	$0.539768\pi$
$968$	0	0
$969$	−4207.88	−0.139501
$970$	0	0
$971$	2787.64	0.0921315	0.0460658	−	0.998938i	$-0.485332\pi$
$971$	2787.64	0.0921315	0.0460658	+	0.998938i	$0.485332\pi$
$972$	0	0
$973$	−12455.7	−0.410393
$974$	0	0
$975$	−5284.22	−0.173570
$976$	0	0
$977$	−49041.8	−1.60592	−0.802962	−	0.596030i	$-0.796744\pi$
$977$	−49041.8	−1.60592	−0.802962	+	0.596030i	$0.796744\pi$
$978$	0	0
$979$	−13476.4	−0.439947
$980$	0	0
$981$	−9224.51	−0.300220
$982$	0	0
$983$	23448.8	0.760835	0.380418	−	0.924815i	$-0.375780\pi$
$983$	23448.8	0.760835	0.380418	+	0.924815i	$0.375780\pi$
$984$	0	0
$985$	−34019.1	−1.10044
$986$	0	0
$987$	33096.6	1.06735
$988$	0	0
$989$	−6772.44	−0.217746
$990$	0	0
$991$	−10803.2	−0.346293	−0.173146	−	0.984896i	$-0.555393\pi$
$991$	−10803.2	−0.346293	−0.173146	+	0.984896i	$0.555393\pi$
$992$	0	0
$993$	1508.00	0.0481924
$994$	0	0
$995$	31316.1	0.997776
$996$	0	0
$997$	23890.7	0.758902	0.379451	−	0.925212i	$-0.376113\pi$
$997$	23890.7	0.758902	0.379451	+	0.925212i	$0.376113\pi$
$998$	0	0
$999$	8635.49	0.273488

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bd.1.4		5
4.3	odd	2		1216.4.a.y.1.2		5
8.3	odd	2		608.4.a.h.1.4	yes	5
8.5	even	2		608.4.a.e.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.e.1.2	✓	5	8.5	even	2
608.4.a.h.1.4	yes	5	8.3	odd	2
1216.4.a.y.1.2		5	4.3	odd	2
1216.4.a.bd.1.4		5	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+4.75519 q^{3} -10.5762 q^{5} +30.4413 q^{7} -4.38817 q^{9} +O(q^{10})\)	\(q+4.75519 q^{3} -10.5762 q^{5} +30.4413 q^{7} -4.38817 q^{9} +35.9632 q^{11} +84.5460 q^{13} -50.2919 q^{15} +46.5738 q^{17} -19.0000 q^{19} +144.754 q^{21} +21.5552 q^{23} -13.1438 q^{25} -149.257 q^{27} +241.965 q^{29} -143.527 q^{31} +171.012 q^{33} -321.953 q^{35} -57.8566 q^{37} +402.032 q^{39} -305.486 q^{41} -314.191 q^{43} +46.4102 q^{45} +228.640 q^{47} +583.671 q^{49} +221.467 q^{51} +85.6653 q^{53} -380.355 q^{55} -90.3486 q^{57} +52.0998 q^{59} +133.686 q^{61} -133.582 q^{63} -894.176 q^{65} +1074.37 q^{67} +102.499 q^{69} +245.509 q^{71} -501.891 q^{73} -62.5011 q^{75} +1094.77 q^{77} -730.048 q^{79} -591.263 q^{81} +915.448 q^{83} -492.574 q^{85} +1150.59 q^{87} -374.727 q^{89} +2573.69 q^{91} -682.498 q^{93} +200.948 q^{95} +440.507 q^{97} -157.813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 5 q^{5} + 7 q^{7} - 5 q^{9}+O(q^{10}) \) 5 * q + 6 * q^3 - 5 * q^5 + 7 * q^7 - 5 * q^9	\( 5 q + 6 q^{3} - 5 q^{5} + 7 q^{7} - 5 q^{9} + 13 q^{11} + 72 q^{13} + 72 q^{15} - 59 q^{17} - 95 q^{19} + 224 q^{21} + 52 q^{23} - 86 q^{25} + 54 q^{27} + 128 q^{29} - 110 q^{31} - 68 q^{33} - 45 q^{35} + 436 q^{37} + 356 q^{39} - 804 q^{41} - 143 q^{43} + 579 q^{45} + 661 q^{47} - 406 q^{49} + 570 q^{51} + 898 q^{53} + 17 q^{55} - 114 q^{57} + 196 q^{59} + 1079 q^{61} + 143 q^{63} - 1632 q^{65} + 832 q^{67} + 1644 q^{69} + 834 q^{71} - 1375 q^{73} + 1054 q^{75} + 1473 q^{77} + 154 q^{79} - 1859 q^{81} + 2224 q^{83} + 1807 q^{85} + 2004 q^{87} - 542 q^{89} + 1890 q^{91} + 2788 q^{93} + 95 q^{95} - 1528 q^{97} + 601 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 - 5 * q^5 + 7 * q^7 - 5 * q^9 + 13 * q^11 + 72 * q^13 + 72 * q^15 - 59 * q^17 - 95 * q^19 + 224 * q^21 + 52 * q^23 - 86 * q^25 + 54 * q^27 + 128 * q^29 - 110 * q^31 - 68 * q^33 - 45 * q^35 + 436 * q^37 + 356 * q^39 - 804 * q^41 - 143 * q^43 + 579 * q^45 + 661 * q^47 - 406 * q^49 + 570 * q^51 + 898 * q^53 + 17 * q^55 - 114 * q^57 + 196 * q^59 + 1079 * q^61 + 143 * q^63 - 1632 * q^65 + 832 * q^67 + 1644 * q^69 + 834 * q^71 - 1375 * q^73 + 1054 * q^75 + 1473 * q^77 + 154 * q^79 - 1859 * q^81 + 2224 * q^83 + 1807 * q^85 + 2004 * q^87 - 542 * q^89 + 1890 * q^91 + 2788 * q^93 + 95 * q^95 - 1528 * q^97 + 601 * q^99

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