LMFDB - Embedded newform 3136.2.f.j.3135.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(3135,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3136.3135");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.f (of order $2$, degree $1$, not 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:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3135.8
Root			$1.50047 + 0.288947i$ of defining polynomial
Character	$\chi$	$=$	3136.3135
Dual form			3136.2.f.j.3135.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.182734 q^{3} +3.55765i q^{5} -2.96661 q^{9} +O(q^{10})$	$q-0.182734 q^{3} +3.55765i q^{5} -2.96661 q^{9} -5.47853i q^{11} -2.70791i q^{13} -0.650103i q^{15} +0.975864i q^{17} +5.47243 q^{19} +4.51192i q^{23} -7.65685 q^{25} +1.09030 q^{27} +2.96661 q^{29} +6.54184 q^{31} +1.00111i q^{33} +1.19542 q^{37} +0.494828i q^{39} +2.70791i q^{41} -1.02384i q^{43} -10.5541i q^{45} +4.03755 q^{47} -0.178324i q^{51} +6.85227 q^{53} +19.4907 q^{55} -1.00000 q^{57} -5.59856 q^{59} -8.69596i q^{61} +9.63381 q^{65} +3.95986i q^{67} -0.824482i q^{69} +13.3002i q^{71} +9.57828i q^{73} +1.39917 q^{75} -3.87891i q^{79} +8.70059 q^{81} +2.24331 q^{83} -3.47178 q^{85} -0.542101 q^{87} -14.2632i q^{89} -1.19542 q^{93} +19.4690i q^{95} +10.7412i q^{97} +16.2527i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} - 32 q^{25} - 16 q^{29} - 16 q^{37} - 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} + 16 q^{85} + 16 q^{93}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^25 - 16 * q^29 - 16 * q^37 - 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 + 16 * q^85 + 16 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.182734	−0.105502	−0.0527508	−	0.998608i	$-0.516799\pi$
$3$	−0.182734	−0.105502	−0.0527508	+	0.998608i	$0.516799\pi$
$4$	0	0
$5$	3.55765i	1.59103i	0.605935	+	0.795514i	$0.292799\pi$
$5$	3.55765i	1.59103i	−0.605935	+	0.795514i	$0.707201\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.96661	−0.988869
$10$	0	0
$11$	− 5.47853i	− 1.65184i	−0.563788	−	0.825920i	$-0.690656\pi$
$11$	− 5.47853i	− 1.65184i	0.563788	−	0.825920i	$-0.309344\pi$
$12$	0	0
$13$	− 2.70791i	− 0.751040i	−0.926814	−	0.375520i	$-0.877464\pi$
$13$	− 2.70791i	− 0.751040i	0.926814	−	0.375520i	$-0.122536\pi$
$14$	0	0
$15$	− 0.650103i	− 0.167856i
$16$	0	0
$17$	0.975864i	0.236682i	0.992973	+	0.118341i	$0.0377576\pi$
$17$	0.975864i	0.236682i	−0.992973	+	0.118341i	$0.962242\pi$
$18$	0	0
$19$	5.47243	1.25546	0.627731	−	0.778430i	$-0.283984\pi$
$19$	5.47243	1.25546	0.627731	+	0.778430i	$0.283984\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.51192i	0.940801i	0.882453	+	0.470400i	$0.155891\pi$
$23$	4.51192i	0.940801i	−0.882453	+	0.470400i	$0.844109\pi$
$24$	0	0
$25$	−7.65685	−1.53137
$26$	0	0
$27$	1.09030	0.209829
$28$	0	0
$29$	2.96661	0.550885	0.275443	−	0.961318i	$-0.411176\pi$
$29$	2.96661	0.550885	0.275443	+	0.961318i	$0.411176\pi$
$30$	0	0
$31$	6.54184	1.17495	0.587475	−	0.809243i	$-0.300122\pi$
$31$	6.54184	1.17495	0.587475	+	0.809243i	$0.300122\pi$
$32$	0	0
$33$	1.00111i	0.174272i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.19542	0.196525	0.0982627	−	0.995161i	$-0.468671\pi$
$37$	1.19542	0.196525	0.0982627	+	0.995161i	$0.468671\pi$
$38$	0	0
$39$	0.494828i	0.0792360i
$40$	0	0
$41$	2.70791i	0.422905i	0.977388	+	0.211453i	$0.0678194\pi$
$41$	2.70791i	0.422905i	−0.977388	+	0.211453i	$0.932181\pi$
$42$	0	0
$43$	− 1.02384i	− 0.156135i	−0.996948	−	0.0780674i	$-0.975125\pi$
$43$	− 1.02384i	− 0.156135i	0.996948	−	0.0780674i	$-0.0248749\pi$
$44$	0	0
$45$	− 10.5541i	− 1.57332i
$46$	0	0
$47$	4.03755	0.588938	0.294469	−	0.955661i	$-0.404857\pi$
$47$	4.03755	0.588938	0.294469	+	0.955661i	$0.404857\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 0.178324i	− 0.0249703i
$52$	0	0
$53$	6.85227	0.941232	0.470616	−	0.882338i	$-0.344032\pi$
$53$	6.85227	0.941232	0.470616	+	0.882338i	$0.344032\pi$
$54$	0	0
$55$	19.4907	2.62812
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−5.59856	−0.728871	−0.364435	−	0.931229i	$-0.618738\pi$
$59$	−5.59856	−0.728871	−0.364435	+	0.931229i	$0.618738\pi$
$60$	0	0
$61$	− 8.69596i	− 1.11340i	−0.830712	−	0.556702i	$-0.812067\pi$
$61$	− 8.69596i	− 1.11340i	0.830712	−	0.556702i	$-0.187933\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.63381	1.19493
$66$	0	0
$67$	3.95986i	0.483774i	0.970304	+	0.241887i	$0.0777663\pi$
$67$	3.95986i	0.483774i	−0.970304	+	0.241887i	$0.922234\pi$
$68$	0	0
$69$	− 0.824482i	− 0.0992560i
$70$	0	0
$71$	13.3002i	1.57844i	0.614108	+	0.789222i	$0.289516\pi$
$71$	13.3002i	1.57844i	−0.614108	+	0.789222i	$0.710484\pi$
$72$	0	0
$73$	9.57828i	1.12105i	0.828136	+	0.560527i	$0.189401\pi$
$73$	9.57828i	1.12105i	−0.828136	+	0.560527i	$0.810599\pi$
$74$	0	0
$75$	1.39917	0.161562
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 3.87891i	− 0.436412i	−0.975903	−	0.218206i	$-0.929980\pi$
$79$	− 3.87891i	− 0.436412i	0.975903	−	0.218206i	$-0.0700204\pi$
$80$	0	0
$81$	8.70059	0.966732
$82$	0	0
$83$	2.24331	0.246235	0.123118	−	0.992392i	$-0.460711\pi$
$83$	2.24331	0.246235	0.123118	+	0.992392i	$0.460711\pi$
$84$	0	0
$85$	−3.47178	−0.376567
$86$	0	0
$87$	−0.542101	−0.0581193
$88$	0	0
$89$	− 14.2632i	− 1.51189i	−0.654633	−	0.755947i	$-0.727177\pi$
$89$	− 14.2632i	− 1.51189i	0.654633	−	0.755947i	$-0.272823\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.19542	−0.123959
$94$	0	0
$95$	19.4690i	1.99748i
$96$	0	0
$97$	10.7412i	1.09061i	0.838239	+	0.545304i	$0.183586\pi$
$97$	10.7412i	1.09061i	−0.838239	+	0.545304i	$0.816414\pi$
$98$	0	0
$99$	16.2527i	1.63345i
$100$	0	0
$101$	8.53412i	0.849177i	0.905386	+	0.424588i	$0.139581\pi$
$101$	8.53412i	0.849177i	−0.905386	+	0.424588i	$0.860419\pi$
$102$	0	0
$103$	4.16369	0.410260	0.205130	−	0.978735i	$-0.434238\pi$
$103$	4.16369	0.410260	0.205130	+	0.978735i	$0.434238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.17832i	0.790628i	0.918546	+	0.395314i	$0.129364\pi$
$107$	8.17832i	0.790628i	−0.918546	+	0.395314i	$0.870636\pi$
$108$	0	0
$109$	5.49483	0.526309	0.263155	−	0.964754i	$-0.415237\pi$
$109$	5.49483	0.526309	0.263155	+	0.964754i	$0.415237\pi$
$110$	0	0
$111$	−0.218444	−0.0207337
$112$	0	0
$113$	−11.3574	−1.06842	−0.534209	−	0.845352i	$-0.679391\pi$
$113$	−11.3574	−1.06842	−0.534209	+	0.845352i	$0.679391\pi$
$114$	0	0
$115$	−16.0518	−1.49684
$116$	0	0
$117$	8.03332i	0.742681i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−19.0143	−1.72857
$122$	0	0
$123$	− 0.494828i	− 0.0446172i
$124$	0	0
$125$	− 9.45215i	− 0.845426i
$126$	0	0
$127$	14.2899i	1.26802i	0.773325	+	0.634010i	$0.218592\pi$
$127$	14.2899i	1.26802i	−0.773325	+	0.634010i	$0.781408\pi$
$128$	0	0
$129$	0.187091i	0.0164725i
$130$	0	0
$131$	13.0054	1.13629	0.568145	−	0.822928i	$-0.307661\pi$
$131$	13.0054	1.13629	0.568145	+	0.822928i	$0.307661\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.87891i	0.333844i
$136$	0	0
$137$	−14.6338	−1.25025	−0.625125	−	0.780524i	$-0.714952\pi$
$137$	−14.6338	−1.25025	−0.625125	+	0.780524i	$0.714952\pi$
$138$	0	0
$139$	11.3608	0.963613	0.481807	−	0.876278i	$-0.339981\pi$
$139$	11.3608	0.963613	0.481807	+	0.876278i	$0.339981\pi$
$140$	0	0
$141$	−0.737799	−0.0621339
$142$	0	0
$143$	−14.8354	−1.24060
$144$	0	0
$145$	10.5541i	0.876474i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.1620	0.832506	0.416253	−	0.909249i	$-0.363343\pi$
$149$	10.1620	0.832506	0.416253	+	0.909249i	$0.363343\pi$
$150$	0	0
$151$	9.67395i	0.787255i	0.919270	+	0.393627i	$0.128780\pi$
$151$	9.67395i	0.787255i	−0.919270	+	0.393627i	$0.871220\pi$
$152$	0	0
$153$	− 2.89501i	− 0.234047i
$154$	0	0
$155$	23.2736i	1.86938i
$156$	0	0
$157$	− 3.39581i	− 0.271015i	−0.990776	−	0.135507i	$-0.956734\pi$
$157$	− 3.39581i	− 0.271015i	0.990776	−	0.135507i	$-0.0432665\pi$
$158$	0	0
$159$	−1.25214	−0.0993015
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.5738i	− 1.61146i	−0.592281	−	0.805731i	$-0.701773\pi$
$163$	− 20.5738i	− 1.61146i	0.592281	−	0.805731i	$-0.298227\pi$
$164$	0	0
$165$	−3.56161	−0.277271
$166$	0	0
$167$	−2.01269	−0.155746	−0.0778732	−	0.996963i	$-0.524813\pi$
$167$	−2.01269	−0.155746	−0.0778732	+	0.996963i	$0.524813\pi$
$168$	0	0
$169$	5.66720	0.435938
$170$	0	0
$171$	−16.2346	−1.24149
$172$	0	0
$173$	9.23981i	0.702490i	0.936284	+	0.351245i	$0.114242\pi$
$173$	9.23981i	0.702490i	−0.936284	+	0.351245i	$0.885758\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.02305	0.0768970
$178$	0	0
$179$	3.90196i	0.291646i	0.989311	+	0.145823i	$0.0465830\pi$
$179$	3.90196i	0.291646i	−0.989311	+	0.145823i	$0.953417\pi$
$180$	0	0
$181$	12.8227i	0.953104i	0.879146	+	0.476552i	$0.158114\pi$
$181$	12.8227i	0.953104i	−0.879146	+	0.476552i	$0.841886\pi$
$182$	0	0
$183$	1.58905i	0.117466i
$184$	0	0
$185$	4.25287i	0.312678i
$186$	0	0
$187$	5.34630	0.390960
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.28311i	0.671702i	0.941915	+	0.335851i	$0.109024\pi$
$191$	9.28311i	0.671702i	−0.941915	+	0.335851i	$0.890976\pi$
$192$	0	0
$193$	3.57577	0.257390	0.128695	−	0.991684i	$-0.458921\pi$
$193$	3.57577	0.257390	0.128695	+	0.991684i	$0.458921\pi$
$194$	0	0
$195$	−1.76042	−0.126067
$196$	0	0
$197$	−6.41388	−0.456970	−0.228485	−	0.973547i	$-0.573377\pi$
$197$	−6.41388	−0.456970	−0.228485	+	0.973547i	$0.573377\pi$
$198$	0	0
$199$	−14.5664	−1.03259	−0.516294	−	0.856411i	$-0.672689\pi$
$199$	−14.5664	−1.03259	−0.516294	+	0.856411i	$0.672689\pi$
$200$	0	0
$201$	− 0.723601i	− 0.0510389i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.63381	−0.672854
$206$	0	0
$207$	− 13.3851i	− 0.930329i
$208$	0	0
$209$	− 29.9809i	− 2.07382i
$210$	0	0
$211$	− 16.3376i	− 1.12472i	−0.826891	−	0.562362i	$-0.809893\pi$
$211$	− 16.3376i	− 1.12472i	0.826891	−	0.562362i	$-0.190107\pi$
$212$	0	0
$213$	− 2.43040i	− 0.166528i
$214$	0	0
$215$	3.64248	0.248415
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 1.75028i	− 0.118273i
$220$	0	0
$221$	2.64256	0.177758
$222$	0	0
$223$	24.8100	1.66140	0.830700	−	0.556721i	$-0.187941\pi$
$223$	24.8100	1.66140	0.830700	+	0.556721i	$0.187941\pi$
$224$	0	0
$225$	22.7149	1.51433
$226$	0	0
$227$	21.8331	1.44912	0.724558	−	0.689214i	$-0.242044\pi$
$227$	21.8331	1.44912	0.724558	+	0.689214i	$0.242044\pi$
$228$	0	0
$229$	− 4.47568i	− 0.295761i	−0.989005	−	0.147880i	$-0.952755\pi$
$229$	− 4.47568i	− 0.295761i	0.989005	−	0.147880i	$-0.0472451\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.2803	1.00105	0.500523	−	0.865723i	$-0.333141\pi$
$233$	15.2803	1.00105	0.500523	+	0.865723i	$0.333141\pi$
$234$	0	0
$235$	14.3642i	0.937017i
$236$	0	0
$237$	0.708810i	0.0460421i
$238$	0	0
$239$	8.13818i	0.526415i	0.964739	+	0.263208i	$0.0847804\pi$
$239$	8.13818i	0.526415i	−0.964739	+	0.263208i	$0.915220\pi$
$240$	0	0
$241$	− 28.3067i	− 1.82339i	−0.410864	−	0.911697i	$-0.634773\pi$
$241$	− 28.3067i	− 1.82339i	0.410864	−	0.911697i	$-0.365227\pi$
$242$	0	0
$243$	−4.86080	−0.311821
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 14.8189i	− 0.942903i
$248$	0	0
$249$	−0.409929	−0.0259782
$250$	0	0
$251$	9.19315	0.580267	0.290133	−	0.956986i	$-0.406300\pi$
$251$	9.19315	0.580267	0.290133	+	0.956986i	$0.406300\pi$
$252$	0	0
$253$	24.7187	1.55405
$254$	0	0
$255$	0.634413	0.0397285
$256$	0	0
$257$	16.2290i	1.01234i	0.862435	+	0.506168i	$0.168938\pi$
$257$	16.2290i	1.01234i	−0.862435	+	0.506168i	$0.831062\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.80076	−0.544754
$262$	0	0
$263$	− 19.5835i	− 1.20757i	−0.797148	−	0.603784i	$-0.793659\pi$
$263$	− 19.5835i	− 1.20757i	0.797148	−	0.603784i	$-0.206341\pi$
$264$	0	0
$265$	24.3780i	1.49753i
$266$	0	0
$267$	2.60637i	0.159507i
$268$	0	0
$269$	2.02002i	0.123163i	0.998102	+	0.0615815i	$0.0196144\pi$
$269$	2.02002i	0.123163i	−0.998102	+	0.0615815i	$0.980386\pi$
$270$	0	0
$271$	25.6462	1.55789	0.778947	−	0.627090i	$-0.215754\pi$
$271$	25.6462	1.55789	0.778947	+	0.627090i	$0.215754\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	41.9483i	2.52958i
$276$	0	0
$277$	−31.1763	−1.87320	−0.936602	−	0.350395i	$-0.886047\pi$
$277$	−31.1763	−1.87320	−0.936602	+	0.350395i	$0.886047\pi$
$278$	0	0
$279$	−19.4071	−1.16187
$280$	0	0
$281$	−1.97695	−0.117935	−0.0589675	−	0.998260i	$-0.518781\pi$
$281$	−1.97695	−0.117935	−0.0589675	+	0.998260i	$0.518781\pi$
$282$	0	0
$283$	27.6231	1.64203	0.821013	−	0.570910i	$-0.193410\pi$
$283$	27.6231	1.64203	0.821013	+	0.570910i	$0.193410\pi$
$284$	0	0
$285$	− 3.55765i	− 0.210737i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.0477	0.943982
$290$	0	0
$291$	− 1.96279i	− 0.115061i
$292$	0	0
$293$	10.5289i	0.615105i	0.951531	+	0.307552i	$0.0995099\pi$
$293$	10.5289i	0.615105i	−0.951531	+	0.307552i	$0.900490\pi$
$294$	0	0
$295$	− 19.9177i	− 1.15965i
$296$	0	0
$297$	− 5.97326i	− 0.346604i
$298$	0	0
$299$	12.2179	0.706579
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 1.55948i	− 0.0895895i
$304$	0	0
$305$	30.9372	1.77146
$306$	0	0
$307$	27.0533	1.54401	0.772007	−	0.635615i	$-0.219253\pi$
$307$	27.0533	1.54401	0.772007	+	0.635615i	$0.219253\pi$
$308$	0	0
$309$	−0.760847	−0.0432831
$310$	0	0
$311$	25.3852	1.43946	0.719731	−	0.694253i	$-0.244265\pi$
$311$	25.3852	1.43946	0.719731	+	0.694253i	$0.244265\pi$
$312$	0	0
$313$	4.60181i	0.260109i	0.991507	+	0.130055i	$0.0415153\pi$
$313$	4.60181i	0.260109i	−0.991507	+	0.130055i	$0.958485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.53856	0.423408	0.211704	−	0.977334i	$-0.432099\pi$
$317$	7.53856	0.423408	0.211704	+	0.977334i	$0.432099\pi$
$318$	0	0
$319$	− 16.2527i	− 0.909974i
$320$	0	0
$321$	− 1.49446i	− 0.0834125i
$322$	0	0
$323$	5.34035i	0.297145i
$324$	0	0
$325$	20.7341i	1.15012i
$326$	0	0
$327$	−1.00409	−0.0555264
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 2.77412i	− 0.152480i	−0.997090	−	0.0762398i	$-0.975709\pi$
$331$	− 2.77412i	− 0.152480i	0.997090	−	0.0762398i	$-0.0242915\pi$
$332$	0	0
$333$	−3.54634	−0.194338
$334$	0	0
$335$	−14.0878	−0.769697
$336$	0	0
$337$	−25.2716	−1.37663	−0.688315	−	0.725412i	$-0.741649\pi$
$337$	−25.2716	−1.37663	−0.688315	+	0.725412i	$0.741649\pi$
$338$	0	0
$339$	2.07539	0.112720
$340$	0	0
$341$	− 35.8397i	− 1.94083i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.93322	0.157919
$346$	0	0
$347$	10.8827i	0.584216i	0.956385	+	0.292108i	$0.0943566\pi$
$347$	10.8827i	0.584216i	−0.956385	+	0.292108i	$0.905643\pi$
$348$	0	0
$349$	− 20.6437i	− 1.10503i	−0.833503	−	0.552516i	$-0.813668\pi$
$349$	− 20.6437i	− 1.10503i	0.833503	−	0.552516i	$-0.186332\pi$
$350$	0	0
$351$	− 2.95245i	− 0.157590i
$352$	0	0
$353$	− 1.82247i	− 0.0970004i	−0.998823	−	0.0485002i	$-0.984556\pi$
$353$	− 1.82247i	− 0.0970004i	0.998823	−	0.0485002i	$-0.0154442\pi$
$354$	0	0
$355$	−47.3174	−2.51135
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.36024i	0.124569i	0.998058	+	0.0622843i	$0.0198386\pi$
$359$	2.36024i	0.124569i	−0.998058	+	0.0622843i	$0.980161\pi$
$360$	0	0
$361$	10.9475	0.576185
$362$	0	0
$363$	3.47456	0.182367
$364$	0	0
$365$	−34.0761	−1.78363
$366$	0	0
$367$	−6.55477	−0.342156	−0.171078	−	0.985257i	$-0.554725\pi$
$367$	−6.55477	−0.342156	−0.171078	+	0.985257i	$0.554725\pi$
$368$	0	0
$369$	− 8.03332i	− 0.418198i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.852272	−0.0441290	−0.0220645	−	0.999757i	$-0.507024\pi$
$373$	−0.852272	−0.0441290	−0.0220645	+	0.999757i	$0.507024\pi$
$374$	0	0
$375$	1.72723i	0.0891938i
$376$	0	0
$377$	− 8.03332i	− 0.413737i
$378$	0	0
$379$	− 24.3954i	− 1.25311i	−0.779377	−	0.626555i	$-0.784464\pi$
$379$	− 24.3954i	− 1.25311i	0.779377	−	0.626555i	$-0.215536\pi$
$380$	0	0
$381$	− 2.61125i	− 0.133778i
$382$	0	0
$383$	−0.823249	−0.0420660	−0.0210330	−	0.999779i	$-0.506696\pi$
$383$	−0.823249	−0.0420660	−0.0210330	+	0.999779i	$0.506696\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.03735i	0.154397i
$388$	0	0
$389$	−22.0952	−1.12027	−0.560137	−	0.828400i	$-0.689251\pi$
$389$	−22.0952	−1.12027	−0.560137	+	0.828400i	$0.689251\pi$
$390$	0	0
$391$	−4.40302	−0.222670
$392$	0	0
$393$	−2.37654	−0.119880
$394$	0	0
$395$	13.7998	0.694343
$396$	0	0
$397$	25.1951i	1.26451i	0.774762	+	0.632253i	$0.217870\pi$
$397$	25.1951i	1.26451i	−0.774762	+	0.632253i	$0.782130\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−37.6521	−1.88025	−0.940127	−	0.340824i	$-0.889294\pi$
$401$	−37.6521	−1.88025	−0.940127	+	0.340824i	$0.889294\pi$
$402$	0	0
$403$	− 17.7148i	− 0.882435i
$404$	0	0
$405$	30.9536i	1.53810i
$406$	0	0
$407$	− 6.54913i	− 0.324628i
$408$	0	0
$409$	9.16418i	0.453139i	0.973995	+	0.226570i	$0.0727511\pi$
$409$	9.16418i	0.453139i	−0.973995	+	0.226570i	$0.927249\pi$
$410$	0	0
$411$	2.67410	0.131903
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.98091i	0.391767i
$416$	0	0
$417$	−2.07601	−0.101663
$418$	0	0
$419$	−10.9867	−0.536733	−0.268367	−	0.963317i	$-0.586484\pi$
$419$	−10.9867	−0.536733	−0.268367	+	0.963317i	$0.586484\pi$
$420$	0	0
$421$	6.99361	0.340848	0.170424	−	0.985371i	$-0.445486\pi$
$421$	6.99361	0.340848	0.170424	+	0.985371i	$0.445486\pi$
$422$	0	0
$423$	−11.9778	−0.582383
$424$	0	0
$425$	− 7.47205i	− 0.362448i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.71093	0.130885
$430$	0	0
$431$	− 25.8970i	− 1.24742i	−0.781657	−	0.623708i	$-0.785625\pi$
$431$	− 25.8970i	− 1.24742i	0.781657	−	0.623708i	$-0.214375\pi$
$432$	0	0
$433$	− 33.4629i	− 1.60812i	−0.594545	−	0.804062i	$-0.702668\pi$
$433$	− 33.4629i	− 1.60812i	0.594545	−	0.804062i	$-0.297332\pi$
$434$	0	0
$435$	− 1.92860i	− 0.0924694i
$436$	0	0
$437$	24.6912i	1.18114i
$438$	0	0
$439$	0.710043	0.0338885	0.0169443	−	0.999856i	$-0.494606\pi$
$439$	0.710043	0.0338885	0.0169443	+	0.999856i	$0.494606\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.0122i	− 0.855784i	−0.903830	−	0.427892i	$-0.859256\pi$
$443$	− 18.0122i	− 0.855784i	0.903830	−	0.427892i	$-0.140744\pi$
$444$	0	0
$445$	50.7433	2.40547
$446$	0	0
$447$	−1.85695	−0.0878307
$448$	0	0
$449$	31.2048	1.47264	0.736322	−	0.676631i	$-0.236561\pi$
$449$	31.2048	1.47264	0.736322	+	0.676631i	$0.236561\pi$
$450$	0	0
$451$	14.8354	0.698571
$452$	0	0
$453$	− 1.76776i	− 0.0830566i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.99125	0.467371	0.233685	−	0.972312i	$-0.424921\pi$
$457$	9.99125	0.467371	0.233685	+	0.972312i	$0.424921\pi$
$458$	0	0
$459$	1.06399i	0.0496627i
$460$	0	0
$461$	21.5729i	1.00475i	0.864650	+	0.502375i	$0.167540\pi$
$461$	21.5729i	1.00475i	−0.864650	+	0.502375i	$0.832460\pi$
$462$	0	0
$463$	− 14.6998i	− 0.683157i	−0.939853	−	0.341579i	$-0.889038\pi$
$463$	− 14.6998i	− 0.683157i	0.939853	−	0.341579i	$-0.110962\pi$
$464$	0	0
$465$	− 4.25287i	− 0.197222i
$466$	0	0
$467$	35.0429	1.62159	0.810797	−	0.585327i	$-0.199034\pi$
$467$	35.0429	1.62159	0.810797	+	0.585327i	$0.199034\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.620530i	0.0285925i
$472$	0	0
$473$	−5.60916	−0.257910
$474$	0	0
$475$	−41.9016	−1.92258
$476$	0	0
$477$	−20.3280	−0.930755
$478$	0	0
$479$	−0.583912	−0.0266796	−0.0133398	−	0.999911i	$-0.504246\pi$
$479$	−0.583912	−0.0266796	−0.0133398	+	0.999911i	$0.504246\pi$
$480$	0	0
$481$	− 3.23709i	− 0.147599i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−38.2135	−1.73519
$486$	0	0
$487$	27.8256i	1.26090i	0.776230	+	0.630450i	$0.217130\pi$
$487$	27.8256i	1.26090i	−0.776230	+	0.630450i	$0.782870\pi$
$488$	0	0
$489$	3.75953i	0.170012i
$490$	0	0
$491$	4.25265i	0.191920i	0.995385	+	0.0959598i	$0.0305920\pi$
$491$	4.25265i	0.191920i	−0.995385	+	0.0959598i	$0.969408\pi$
$492$	0	0
$493$	2.89501i	0.130385i
$494$	0	0
$495$	−57.8212	−2.59887
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 20.5738i	− 0.921009i	−0.887657	−	0.460504i	$-0.847669\pi$
$499$	− 20.5738i	− 0.921009i	0.887657	−	0.460504i	$-0.152331\pi$
$500$	0	0
$501$	0.367787	0.0164315
$502$	0	0
$503$	−29.2966	−1.30627	−0.653136	−	0.757241i	$-0.726547\pi$
$503$	−29.2966	−1.30627	−0.653136	+	0.757241i	$0.726547\pi$
$504$	0	0
$505$	−30.3614	−1.35106
$506$	0	0
$507$	−1.03559	−0.0459922
$508$	0	0
$509$	18.0910i	0.801869i	0.916107	+	0.400935i	$0.131315\pi$
$509$	18.0910i	0.801869i	−0.916107	+	0.400935i	$0.868685\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.96661	0.263432
$514$	0	0
$515$	14.8129i	0.652735i
$516$	0	0
$517$	− 22.1199i	− 0.972831i
$518$	0	0
$519$	− 1.68843i	− 0.0741138i
$520$	0	0
$521$	27.5141i	1.20541i	0.797963	+	0.602706i	$0.205911\pi$
$521$	27.5141i	1.20541i	−0.797963	+	0.602706i	$0.794089\pi$
$522$	0	0
$523$	−15.2617	−0.667347	−0.333673	−	0.942689i	$-0.608288\pi$
$523$	−15.2617	−0.667347	−0.333673	+	0.942689i	$0.608288\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.38395i	0.278089i
$528$	0	0
$529$	2.64256	0.114894
$530$	0	0
$531$	16.6087	0.720758
$532$	0	0
$533$	7.33280	0.317619
$534$	0	0
$535$	−29.0956	−1.25791
$536$	0	0
$537$	− 0.713021i	− 0.0307691i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	39.7998	1.71113	0.855563	−	0.517698i	$-0.173211\pi$
$541$	39.7998	1.71113	0.855563	+	0.517698i	$0.173211\pi$
$542$	0	0
$543$	− 2.34315i	− 0.100554i
$544$	0	0
$545$	19.5487i	0.837373i
$546$	0	0
$547$	− 3.19462i	− 0.136592i	−0.997665	−	0.0682961i	$-0.978244\pi$
$547$	− 3.19462i	− 0.136592i	0.997665	−	0.0682961i	$-0.0217563\pi$
$548$	0	0
$549$	25.7975i	1.10101i
$550$	0	0
$551$	16.2346	0.691616
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 0.777145i	− 0.0329880i
$556$	0	0
$557$	−3.88566	−0.164641	−0.0823204	−	0.996606i	$-0.526233\pi$
$557$	−3.88566	−0.164641	−0.0823204	+	0.996606i	$0.526233\pi$
$558$	0	0
$559$	−2.77248	−0.117264
$560$	0	0
$561$	−0.976951	−0.0412469
$562$	0	0
$563$	−41.0594	−1.73045	−0.865223	−	0.501387i	$-0.832823\pi$
$563$	−41.0594	−1.73045	−0.865223	+	0.501387i	$0.832823\pi$
$564$	0	0
$565$	− 40.4058i	− 1.69988i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.50531	0.314639	0.157319	−	0.987548i	$-0.449715\pi$
$569$	7.50531	0.314639	0.157319	+	0.987548i	$0.449715\pi$
$570$	0	0
$571$	− 6.46819i	− 0.270685i	−0.990799	−	0.135343i	$-0.956786\pi$
$571$	− 6.46819i	− 0.270685i	0.990799	−	0.135343i	$-0.0432135\pi$
$572$	0	0
$573$	− 1.69634i	− 0.0708657i
$574$	0	0
$575$	− 34.5471i	− 1.44071i
$576$	0	0
$577$	− 38.0283i	− 1.58314i	−0.611080	−	0.791569i	$-0.709265\pi$
$577$	− 38.0283i	− 1.58314i	0.611080	−	0.791569i	$-0.290735\pi$
$578$	0	0
$579$	−0.653416	−0.0271550
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 37.5404i	− 1.55476i
$584$	0	0
$585$	−28.5797	−1.18163
$586$	0	0
$587$	−27.0533	−1.11661	−0.558305	−	0.829636i	$-0.688548\pi$
$587$	−27.0533	−1.11661	−0.558305	+	0.829636i	$0.688548\pi$
$588$	0	0
$589$	35.7998	1.47510
$590$	0	0
$591$	1.17204	0.0482111
$592$	0	0
$593$	− 17.7734i	− 0.729868i	−0.931033	−	0.364934i	$-0.881092\pi$
$593$	− 17.7734i	− 0.729868i	0.931033	−	0.364934i	$-0.118908\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.66179	0.108940
$598$	0	0
$599$	− 14.0201i	− 0.572846i	−0.958103	−	0.286423i	$-0.907534\pi$
$599$	− 14.0201i	− 0.572846i	0.958103	−	0.286423i	$-0.0924663\pi$
$600$	0	0
$601$	26.0595i	1.06299i	0.847061	+	0.531495i	$0.178370\pi$
$601$	26.0595i	1.06299i	−0.847061	+	0.531495i	$0.821630\pi$
$602$	0	0
$603$	− 11.7473i	− 0.478389i
$604$	0	0
$605$	− 67.6462i	− 2.75021i
$606$	0	0
$607$	−20.0241	−0.812752	−0.406376	−	0.913706i	$-0.633208\pi$
$607$	−20.0241	−0.812752	−0.406376	+	0.913706i	$0.633208\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 10.9334i	− 0.442316i
$612$	0	0
$613$	−27.4487	−1.10864	−0.554322	−	0.832302i	$-0.687022\pi$
$613$	−27.4487	−1.10864	−0.554322	+	0.832302i	$0.687022\pi$
$614$	0	0
$615$	1.76042	0.0709872
$616$	0	0
$617$	24.6235	0.991303	0.495652	−	0.868521i	$-0.334929\pi$
$617$	24.6235	0.991303	0.495652	+	0.868521i	$0.334929\pi$
$618$	0	0
$619$	−14.3419	−0.576450	−0.288225	−	0.957563i	$-0.593065\pi$
$619$	−14.3419	−0.576450	−0.288225	+	0.957563i	$0.593065\pi$
$620$	0	0
$621$	4.91936i	0.197407i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.65685	−0.186274
$626$	0	0
$627$	5.47853i	0.218791i
$628$	0	0
$629$	1.16656i	0.0465140i
$630$	0	0
$631$	30.2517i	1.20430i	0.798383	+	0.602150i	$0.205689\pi$
$631$	30.2517i	1.20430i	−0.798383	+	0.602150i	$0.794311\pi$
$632$	0	0
$633$	2.98543i	0.118660i
$634$	0	0
$635$	−50.8383	−2.01746
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 39.4565i	− 1.56088i
$640$	0	0
$641$	6.17223	0.243789	0.121894	−	0.992543i	$-0.461103\pi$
$641$	6.17223	0.243789	0.121894	+	0.992543i	$0.461103\pi$
$642$	0	0
$643$	−28.7674	−1.13448	−0.567238	−	0.823554i	$-0.691988\pi$
$643$	−28.7674	−1.13448	−0.567238	+	0.823554i	$0.691988\pi$
$644$	0	0
$645$	−0.665605	−0.0262082
$646$	0	0
$647$	48.1068	1.89128	0.945638	−	0.325222i	$-0.105439\pi$
$647$	48.1068	1.89128	0.945638	+	0.325222i	$0.105439\pi$
$648$	0	0
$649$	30.6719i	1.20398i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0992	0.630010	0.315005	−	0.949090i	$-0.397994\pi$
$653$	16.0992	0.630010	0.315005	+	0.949090i	$0.397994\pi$
$654$	0	0
$655$	46.2688i	1.80787i
$656$	0	0
$657$	− 28.4150i	− 1.10858i
$658$	0	0
$659$	1.60357i	0.0624663i	0.999512	+	0.0312332i	$0.00994344\pi$
$659$	1.60357i	0.0624663i	−0.999512	+	0.0312332i	$0.990057\pi$
$660$	0	0
$661$	− 44.8275i	− 1.74359i	−0.489874	−	0.871793i	$-0.662957\pi$
$661$	− 44.8275i	− 1.74359i	0.489874	−	0.871793i	$-0.337043\pi$
$662$	0	0
$663$	−0.482885	−0.0187537
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.3851i	0.518273i
$668$	0	0
$669$	−4.53363	−0.175280
$670$	0	0
$671$	−47.6411	−1.83916
$672$	0	0
$673$	15.2430	0.587573	0.293787	−	0.955871i	$-0.405084\pi$
$673$	15.2430	0.587573	0.293787	+	0.955871i	$0.405084\pi$
$674$	0	0
$675$	−8.34829	−0.321326
$676$	0	0
$677$	37.7409i	1.45050i	0.688485	+	0.725250i	$0.258276\pi$
$677$	37.7409i	1.45050i	−0.688485	+	0.725250i	$0.741724\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.98966	−0.152884
$682$	0	0
$683$	− 8.92581i	− 0.341536i	−0.985311	−	0.170768i	$-0.945375\pi$
$683$	− 8.92581i	− 0.341536i	0.985311	−	0.170768i	$-0.0546249\pi$
$684$	0	0
$685$	− 52.0619i	− 1.98918i
$686$	0	0
$687$	0.817858i	0.0312032i
$688$	0	0
$689$	− 18.5554i	− 0.706903i
$690$	0	0
$691$	−42.8153	−1.62877	−0.814385	−	0.580325i	$-0.802926\pi$
$691$	−42.8153	−1.62877	−0.814385	+	0.580325i	$0.802926\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	40.4178i	1.53314i
$696$	0	0
$697$	−2.64256	−0.100094
$698$	0	0
$699$	−2.79224	−0.105612
$700$	0	0
$701$	13.3137	0.502852	0.251426	−	0.967877i	$-0.419101\pi$
$701$	13.3137	0.502852	0.251426	+	0.967877i	$0.419101\pi$
$702$	0	0
$703$	6.54184	0.246730
$704$	0	0
$705$	− 2.62483i	− 0.0988567i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−32.7378	−1.22949	−0.614747	−	0.788724i	$-0.710742\pi$
$709$	−32.7378	−1.22949	−0.614747	+	0.788724i	$0.710742\pi$
$710$	0	0
$711$	11.5072i	0.431554i
$712$	0	0
$713$	29.5163i	1.10539i
$714$	0	0
$715$	− 52.7791i	− 1.97383i
$716$	0	0
$717$	− 1.48712i	− 0.0555376i
$718$	0	0
$719$	21.4695	0.800678	0.400339	−	0.916367i	$-0.368892\pi$
$719$	21.4695	0.800678	0.400339	+	0.916367i	$0.368892\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.17259i	0.192371i
$724$	0	0
$725$	−22.7149	−0.843610
$726$	0	0
$727$	31.7591	1.17788	0.588940	−	0.808177i	$-0.299545\pi$
$727$	31.7591	1.17788	0.588940	+	0.808177i	$0.299545\pi$
$728$	0	0
$729$	−25.2135	−0.933835
$730$	0	0
$731$	0.999133	0.0369543
$732$	0	0
$733$	− 6.07847i	− 0.224513i	−0.993679	−	0.112257i	$-0.964192\pi$
$733$	− 6.07847i	− 0.224513i	0.993679	−	0.112257i	$-0.0358079\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.6942	0.799116
$738$	0	0
$739$	− 30.1727i	− 1.10992i	−0.831876	−	0.554961i	$-0.812733\pi$
$739$	− 30.1727i	− 1.10992i	0.831876	−	0.554961i	$-0.187267\pi$
$740$	0	0
$741$	2.70791i	0.0994777i
$742$	0	0
$743$	− 2.49181i	− 0.0914155i	−0.998955	−	0.0457078i	$-0.985446\pi$
$743$	− 2.49181i	− 0.0914155i	0.998955	−	0.0457078i	$-0.0145543\pi$
$744$	0	0
$745$	36.1529i	1.32454i
$746$	0	0
$747$	−6.65502	−0.243495
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15.3578i	0.560414i	0.959940	+	0.280207i	$0.0904032\pi$
$751$	15.3578i	0.560414i	−0.959940	+	0.280207i	$0.909597\pi$
$752$	0	0
$753$	−1.67990	−0.0612191
$754$	0	0
$755$	−34.4165	−1.25254
$756$	0	0
$757$	5.45255	0.198176	0.0990881	−	0.995079i	$-0.468407\pi$
$757$	5.45255	0.198176	0.0990881	+	0.995079i	$0.468407\pi$
$758$	0	0
$759$	−4.51695	−0.163955
$760$	0	0
$761$	18.8723i	0.684121i	0.939678	+	0.342060i	$0.111125\pi$
$761$	18.8723i	0.684121i	−0.939678	+	0.342060i	$0.888875\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.2994	0.372376
$766$	0	0
$767$	15.1604i	0.547412i
$768$	0	0
$769$	9.05295i	0.326458i	0.986588	+	0.163229i	$0.0521909\pi$
$769$	9.05295i	0.326458i	−0.986588	+	0.163229i	$0.947809\pi$
$770$	0	0
$771$	− 2.96559i	− 0.106803i
$772$	0	0
$773$	18.4366i	0.663117i	0.943435	+	0.331559i	$0.107574\pi$
$773$	18.4366i	0.663117i	−0.943435	+	0.331559i	$0.892426\pi$
$774$	0	0
$775$	−50.0899	−1.79928
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.8189i	0.530941i
$780$	0	0
$781$	72.8656	2.60734
$782$	0	0
$783$	3.23450	0.115592
$784$	0	0
$785$	12.0811	0.431192
$786$	0	0
$787$	35.0847	1.25064	0.625318	−	0.780370i	$-0.284969\pi$
$787$	35.0847	1.25064	0.625318	+	0.780370i	$0.284969\pi$
$788$	0	0
$789$	3.57857i	0.127400i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−23.5479	−0.836211
$794$	0	0
$795$	− 4.45469i	− 0.157991i
$796$	0	0
$797$	9.81203i	0.347560i	0.984785	+	0.173780i	$0.0555981\pi$
$797$	9.81203i	0.347560i	−0.984785	+	0.173780i	$0.944402\pi$
$798$	0	0
$799$	3.94010i	0.139391i
$800$	0	0
$801$	42.3133i	1.49507i
$802$	0	0
$803$	52.4749	1.85180
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.369127i	− 0.0129939i
$808$	0	0
$809$	−28.8190	−1.01322	−0.506611	−	0.862175i	$-0.669102\pi$
$809$	−28.8190	−1.01322	−0.506611	+	0.862175i	$0.669102\pi$
$810$	0	0
$811$	−13.4492	−0.472264	−0.236132	−	0.971721i	$-0.575880\pi$
$811$	−13.4492	−0.472264	−0.236132	+	0.971721i	$0.575880\pi$
$812$	0	0
$813$	−4.68643	−0.164360
$814$	0	0
$815$	73.1942	2.56388
$816$	0	0
$817$	− 5.60292i	− 0.196021i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.713158	0.0248894	0.0124447	−	0.999923i	$-0.496039\pi$
$821$	0.713158	0.0248894	0.0124447	+	0.999923i	$0.496039\pi$
$822$	0	0
$823$	− 42.6682i	− 1.48732i	−0.668558	−	0.743660i	$-0.733088\pi$
$823$	− 42.6682i	− 1.48732i	0.668558	−	0.743660i	$-0.266912\pi$
$824$	0	0
$825$	− 7.66539i	− 0.266875i
$826$	0	0
$827$	9.85293i	0.342620i	0.985217	+	0.171310i	$0.0548000\pi$
$827$	9.85293i	0.342620i	−0.985217	+	0.171310i	$0.945200\pi$
$828$	0	0
$829$	26.6142i	0.924349i	0.886789	+	0.462174i	$0.152931\pi$
$829$	26.6142i	0.924349i	−0.886789	+	0.462174i	$0.847069\pi$
$830$	0	0
$831$	5.69698	0.197626
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 7.16043i	− 0.247797i
$836$	0	0
$837$	7.13259	0.246538
$838$	0	0
$839$	20.6975	0.714559	0.357279	−	0.933998i	$-0.383704\pi$
$839$	20.6975	0.714559	0.357279	+	0.933998i	$0.383704\pi$
$840$	0	0
$841$	−20.1992	−0.696525
$842$	0	0
$843$	0.361256	0.0124423
$844$	0	0
$845$	20.1619i	0.693590i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.04769	−0.173236
$850$	0	0
$851$	5.39363i	0.184891i
$852$	0	0
$853$	− 3.45628i	− 0.118341i	−0.998248	−	0.0591704i	$-0.981154\pi$
$853$	− 3.45628i	− 0.118341i	0.998248	−	0.0591704i	$-0.0188455\pi$
$854$	0	0
$855$	− 57.7568i	− 1.97524i
$856$	0	0
$857$	30.3124i	1.03545i	0.855547	+	0.517726i	$0.173221\pi$
$857$	30.3124i	1.03545i	−0.855547	+	0.517726i	$0.826779\pi$
$858$	0	0
$859$	31.2116	1.06493	0.532463	−	0.846453i	$-0.321266\pi$
$859$	31.2116	1.06493	0.532463	+	0.846453i	$0.321266\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.81772i	0.266118i	0.991108	+	0.133059i	$0.0424800\pi$
$863$	7.81772i	0.266118i	−0.991108	+	0.133059i	$0.957520\pi$
$864$	0	0
$865$	−32.8720	−1.11768
$866$	0	0
$867$	−2.93246	−0.0995916
$868$	0	0
$869$	−21.2507	−0.720882
$870$	0	0
$871$	10.7230	0.363334
$872$	0	0
$873$	− 31.8650i	− 1.07847i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.9898	−0.573705	−0.286852	−	0.957975i	$-0.592609\pi$
$877$	−16.9898	−0.573705	−0.286852	+	0.957975i	$0.592609\pi$
$878$	0	0
$879$	− 1.92399i	− 0.0648945i
$880$	0	0
$881$	5.82308i	0.196185i	0.995177	+	0.0980923i	$0.0312741\pi$
$881$	5.82308i	0.196185i	−0.995177	+	0.0980923i	$0.968726\pi$
$882$	0	0
$883$	− 27.2143i	− 0.915835i	−0.888995	−	0.457918i	$-0.848595\pi$
$883$	− 27.2143i	− 0.915835i	0.888995	−	0.457918i	$-0.151405\pi$
$884$	0	0
$885$	3.63965i	0.122345i
$886$	0	0
$887$	−50.9982	−1.71235	−0.856177	−	0.516683i	$-0.827167\pi$
$887$	−50.9982	−1.71235	−0.856177	+	0.516683i	$0.827167\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 47.6664i	− 1.59689i
$892$	0	0
$893$	22.0952	0.739389
$894$	0	0
$895$	−13.8818	−0.464017
$896$	0	0
$897$	−2.23263	−0.0745453
$898$	0	0
$899$	19.4071	0.647262
$900$	0	0
$901$	6.68689i	0.222772i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45.6187	−1.51642
$906$	0	0
$907$	− 47.4031i	− 1.57400i	−0.616956	−	0.786998i	$-0.711634\pi$
$907$	− 47.4031i	− 1.57400i	0.616956	−	0.786998i	$-0.288366\pi$
$908$	0	0
$909$	− 25.3174i	− 0.839725i
$910$	0	0
$911$	− 37.4430i	− 1.24054i	−0.784388	−	0.620271i	$-0.787023\pi$
$911$	− 37.4430i	− 1.24054i	0.784388	−	0.620271i	$-0.212977\pi$
$912$	0	0
$913$	− 12.2900i	− 0.406741i
$914$	0	0
$915$	−5.65328	−0.186892
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.2267i	0.469295i	0.972081	+	0.234647i	$0.0753935\pi$
$919$	14.2267i	0.469295i	−0.972081	+	0.234647i	$0.924606\pi$
$920$	0	0
$921$	−4.94356	−0.162896
$922$	0	0
$923$	36.0158	1.18548
$924$	0	0
$925$	−9.15314	−0.300953
$926$	0	0
$927$	−12.3520	−0.405694
$928$	0	0
$929$	− 2.34731i	− 0.0770129i	−0.999258	−	0.0385065i	$-0.987740\pi$
$929$	− 2.34731i	− 0.0770129i	0.999258	−	0.0385065i	$-0.0122600\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4.63874	−0.151865
$934$	0	0
$935$	19.0203i	0.622029i
$936$	0	0
$937$	7.80691i	0.255041i	0.991836	+	0.127520i	$0.0407018\pi$
$937$	7.80691i	0.255041i	−0.991836	+	0.127520i	$0.959298\pi$
$938$	0	0
$939$	− 0.840907i	− 0.0274420i
$940$	0	0
$941$	− 49.1591i	− 1.60254i	−0.598303	−	0.801270i	$-0.704158\pi$
$941$	− 49.1591i	− 1.60254i	0.598303	−	0.801270i	$-0.295842\pi$
$942$	0	0
$943$	−12.2179	−0.397870
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.35544i	− 0.304011i	−0.988380	−	0.152006i	$-0.951427\pi$
$947$	− 9.35544i	− 0.304011i	0.988380	−	0.152006i	$-0.0485732\pi$
$948$	0	0
$949$	25.9372	0.841956
$950$	0	0
$951$	−1.37755	−0.0446702
$952$	0	0
$953$	19.2700	0.624216	0.312108	−	0.950047i	$-0.398965\pi$
$953$	19.2700	0.624216	0.312108	+	0.950047i	$0.398965\pi$
$954$	0	0
$955$	−33.0260	−1.06870
$956$	0	0
$957$	2.96991i	0.0960037i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11.7957	0.380506
$962$	0	0
$963$	− 24.2619i	− 0.781828i
$964$	0	0
$965$	12.7213i	0.409514i
$966$	0	0
$967$	− 3.48146i	− 0.111956i	−0.998432	−	0.0559782i	$-0.982172\pi$
$967$	− 3.48146i	− 0.111956i	0.998432	−	0.0559782i	$-0.0178277\pi$
$968$	0	0
$969$	− 0.975864i	− 0.0313493i
$970$	0	0
$971$	16.7071	0.536157	0.268079	−	0.963397i	$-0.413611\pi$
$971$	16.7071	0.536157	0.268079	+	0.963397i	$0.413611\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 3.78883i	− 0.121340i
$976$	0	0
$977$	20.3574	0.651292	0.325646	−	0.945492i	$-0.394418\pi$
$977$	20.3574	0.651292	0.325646	+	0.945492i	$0.394418\pi$
$978$	0	0
$979$	−78.1412	−2.49740
$980$	0	0
$981$	−16.3010	−0.520451
$982$	0	0
$983$	16.4443	0.524491	0.262246	−	0.965001i	$-0.415537\pi$
$983$	16.4443	0.524491	0.262246	+	0.965001i	$0.415537\pi$
$984$	0	0
$985$	− 22.8183i	− 0.727053i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.61951	0.146892
$990$	0	0
$991$	− 36.3934i	− 1.15608i	−0.816010	−	0.578038i	$-0.803819\pi$
$991$	− 36.3934i	− 1.15608i	0.816010	−	0.578038i	$-0.196181\pi$
$992$	0	0
$993$	0.506927i	0.0160868i
$994$	0	0
$995$	− 51.8223i	− 1.64288i
$996$	0	0
$997$	36.3583i	1.15148i	0.817633	+	0.575739i	$0.195286\pi$
$997$	36.3583i	1.15148i	−0.817633	+	0.575739i	$0.804714\pi$
$998$	0	0
$999$	1.30337	0.0412367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.f.j.3135.8		16
4.3	odd	2	inner	3136.2.f.j.3135.10		16
7.2	even	3		448.2.p.e.255.5		16
7.3	odd	6		448.2.p.e.383.4		16
7.6	odd	2	inner	3136.2.f.j.3135.9		16
8.3	odd	2		1568.2.f.b.1567.7		16
8.5	even	2		1568.2.f.b.1567.9		16
28.3	even	6		448.2.p.e.383.5		16
28.23	odd	6		448.2.p.e.255.4		16
28.27	even	2	inner	3136.2.f.j.3135.7		16
56.3	even	6		224.2.p.a.159.4	yes	16
56.5	odd	6		1568.2.p.b.31.5		16
56.11	odd	6		1568.2.p.b.607.5		16
56.13	odd	2		1568.2.f.b.1567.8		16
56.19	even	6		1568.2.p.b.31.4		16
56.27	even	2		1568.2.f.b.1567.10		16
56.37	even	6		224.2.p.a.31.4	✓	16
56.45	odd	6		224.2.p.a.159.5	yes	16
56.51	odd	6		224.2.p.a.31.5	yes	16
56.53	even	6		1568.2.p.b.607.4		16
168.59	odd	6		2016.2.cs.b.1279.7		16
168.101	even	6		2016.2.cs.b.1279.8		16
168.107	even	6		2016.2.cs.b.703.8		16
168.149	odd	6		2016.2.cs.b.703.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.p.a.31.4	✓	16	56.37	even	6
224.2.p.a.31.5	yes	16	56.51	odd	6
224.2.p.a.159.4	yes	16	56.3	even	6
224.2.p.a.159.5	yes	16	56.45	odd	6
448.2.p.e.255.4		16	28.23	odd	6
448.2.p.e.255.5		16	7.2	even	3
448.2.p.e.383.4		16	7.3	odd	6
448.2.p.e.383.5		16	28.3	even	6
1568.2.f.b.1567.7		16	8.3	odd	2
1568.2.f.b.1567.8		16	56.13	odd	2
1568.2.f.b.1567.9		16	8.5	even	2
1568.2.f.b.1567.10		16	56.27	even	2
1568.2.p.b.31.4		16	56.19	even	6
1568.2.p.b.31.5		16	56.5	odd	6
1568.2.p.b.607.4		16	56.53	even	6
1568.2.p.b.607.5		16	56.11	odd	6
2016.2.cs.b.703.7		16	168.149	odd	6
2016.2.cs.b.703.8		16	168.107	even	6
2016.2.cs.b.1279.7		16	168.59	odd	6
2016.2.cs.b.1279.8		16	168.101	even	6
3136.2.f.j.3135.7		16	28.27	even	2	inner
3136.2.f.j.3135.8		16	1.1	even	1	trivial
3136.2.f.j.3135.9		16	7.6	odd	2	inner
3136.2.f.j.3135.10		16	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.182734 q^{3} +3.55765i q^{5} -2.96661 q^{9} +O(q^{10})\)	\(q-0.182734 q^{3} +3.55765i q^{5} -2.96661 q^{9} -5.47853i q^{11} -2.70791i q^{13} -0.650103i q^{15} +0.975864i q^{17} +5.47243 q^{19} +4.51192i q^{23} -7.65685 q^{25} +1.09030 q^{27} +2.96661 q^{29} +6.54184 q^{31} +1.00111i q^{33} +1.19542 q^{37} +0.494828i q^{39} +2.70791i q^{41} -1.02384i q^{43} -10.5541i q^{45} +4.03755 q^{47} -0.178324i q^{51} +6.85227 q^{53} +19.4907 q^{55} -1.00000 q^{57} -5.59856 q^{59} -8.69596i q^{61} +9.63381 q^{65} +3.95986i q^{67} -0.824482i q^{69} +13.3002i q^{71} +9.57828i q^{73} +1.39917 q^{75} -3.87891i q^{79} +8.70059 q^{81} +2.24331 q^{83} -3.47178 q^{85} -0.542101 q^{87} -14.2632i q^{89} -1.19542 q^{93} +19.4690i q^{95} +10.7412i q^{97} +16.2527i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} - 32 q^{25} - 16 q^{29} - 16 q^{37} - 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} + 16 q^{85} + 16 q^{93}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^25 - 16 * q^29 - 16 * q^37 - 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 + 16 * q^85 + 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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