LMFDB - Embedded newform 225.10.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.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:	$115.883063137$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4729}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1182 $ x^2 - x - 1182
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-33.8839$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+43.8839 q^{2} +1413.79 q^{4} +7861.50 q^{7} +39574.2 q^{8} +O(q^{10})$	$q+43.8839 q^{2} +1413.79 q^{4} +7861.50 q^{7} +39574.2 q^{8} +49373.3 q^{11} -24250.7 q^{13} +344993. q^{14} +1.01281e6 q^{16} +268222. q^{17} -168364. q^{19} +2.16669e6 q^{22} -2.12200e6 q^{23} -1.06422e6 q^{26} +1.11145e7 q^{28} -389624. q^{29} +90532.2 q^{31} +2.41838e7 q^{32} +1.17706e7 q^{34} +3.31991e6 q^{37} -7.38848e6 q^{38} -2.32694e7 q^{41} -1.91140e7 q^{43} +6.98036e7 q^{44} -9.31215e7 q^{46} +6.28153e7 q^{47} +2.14495e7 q^{49} -3.42855e7 q^{52} -180207. q^{53} +3.11112e8 q^{56} -1.70982e7 q^{58} -3.84564e7 q^{59} -553620. q^{61} +3.97290e6 q^{62} +5.42724e8 q^{64} +2.39163e8 q^{67} +3.79211e8 q^{68} -1.28653e8 q^{71} +2.39376e8 q^{73} +1.45690e8 q^{74} -2.38033e8 q^{76} +3.88148e8 q^{77} -5.28027e8 q^{79} -1.02115e9 q^{82} +2.12210e8 q^{83} -8.38797e8 q^{86} +1.95391e9 q^{88} +2.07724e8 q^{89} -1.90647e8 q^{91} -3.00007e9 q^{92} +2.75658e9 q^{94} -1.70780e9 q^{97} +9.41288e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 19 q^{2} + 1521 q^{4} + 11872 q^{7} + 49647 q^{8}+O(q^{10}) $ 2 * q + 19 * q^2 + 1521 * q^4 + 11872 * q^7 + 49647 * q^8	$ 2 q + 19 q^{2} + 1521 q^{4} + 11872 q^{7} + 49647 q^{8} - 35488 q^{11} - 143676 q^{13} + 245196 q^{14} + 707265 q^{16} + 385156 q^{17} - 403296 q^{19} + 4278368 q^{22} + 223704 q^{23} + 1907546 q^{26} + 11544484 q^{28} + 74572 q^{29} - 5027128 q^{31} + 26629583 q^{32} + 8860882 q^{34} - 5373628 q^{37} - 1542476 q^{38} - 14211332 q^{41} - 27748920 q^{43} + 60705952 q^{44} - 151491648 q^{46} + 95966440 q^{47} - 2819950 q^{49} - 47088706 q^{52} - 64305596 q^{53} + 351509340 q^{56} - 28649198 q^{58} - 187863136 q^{59} + 154080060 q^{61} + 131320056 q^{62} + 638301089 q^{64} - 33592376 q^{67} + 391747238 q^{68} + 228270976 q^{71} + 33122316 q^{73} + 362019226 q^{74} - 263218724 q^{76} + 47811456 q^{77} - 932406760 q^{79} - 1246549646 q^{82} + 207040152 q^{83} - 623926708 q^{86} + 1099114848 q^{88} - 224518164 q^{89} - 669602528 q^{91} - 2748592992 q^{92} + 1931650816 q^{94} - 387134596 q^{97} + 1545205739 q^{98}+O(q^{100}) $ 2 * q + 19 * q^2 + 1521 * q^4 + 11872 * q^7 + 49647 * q^8 - 35488 * q^11 - 143676 * q^13 + 245196 * q^14 + 707265 * q^16 + 385156 * q^17 - 403296 * q^19 + 4278368 * q^22 + 223704 * q^23 + 1907546 * q^26 + 11544484 * q^28 + 74572 * q^29 - 5027128 * q^31 + 26629583 * q^32 + 8860882 * q^34 - 5373628 * q^37 - 1542476 * q^38 - 14211332 * q^41 - 27748920 * q^43 + 60705952 * q^44 - 151491648 * q^46 + 95966440 * q^47 - 2819950 * q^49 - 47088706 * q^52 - 64305596 * q^53 + 351509340 * q^56 - 28649198 * q^58 - 187863136 * q^59 + 154080060 * q^61 + 131320056 * q^62 + 638301089 * q^64 - 33592376 * q^67 + 391747238 * q^68 + 228270976 * q^71 + 33122316 * q^73 + 362019226 * q^74 - 263218724 * q^76 + 47811456 * q^77 - 932406760 * q^79 - 1246549646 * q^82 + 207040152 * q^83 - 623926708 * q^86 + 1099114848 * q^88 - 224518164 * q^89 - 669602528 * q^91 - 2748592992 * q^92 + 1931650816 * q^94 - 387134596 * q^97 + 1545205739 * q^98

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 $

$2$	43.8839	1.93941	0.969706	−	0.244277i	$-0.0785506\pi$
$2$	43.8839	1.93941	0.969706	+	0.244277i	$0.0785506\pi$
$3$	0	0
$3$	0	0
$4$	1413.79	2.76132
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7861.50	1.23755	0.618777	−	0.785567i	$-0.287629\pi$
$7$	7861.50	1.23755	0.618777	+	0.785567i	$0.287629\pi$
$8$	39574.2	3.41591
$9$	0	0
$10$	0	0
$11$	49373.3	1.01678	0.508388	−	0.861128i	$-0.330242\pi$
$11$	49373.3	1.01678	0.508388	+	0.861128i	$0.330242\pi$
$12$	0	0
$13$	−24250.7	−0.235494	−0.117747	−	0.993044i	$-0.537567\pi$
$13$	−24250.7	−0.235494	−0.117747	+	0.993044i	$0.537567\pi$
$14$	344993.	2.40013
$15$	0	0
$16$	1.01281e6	3.86355
$17$	268222.	0.778888	0.389444	−	0.921050i	$-0.372667\pi$
$17$	268222.	0.778888	0.389444	+	0.921050i	$0.372667\pi$
$18$	0	0
$19$	−168364.	−0.296387	−0.148193	−	0.988958i	$-0.547346\pi$
$19$	−168364.	−0.296387	−0.148193	+	0.988958i	$0.547346\pi$
$20$	0	0
$21$	0	0
$22$	2.16669e6	1.97195
$23$	−2.12200e6	−1.58114	−0.790569	−	0.612373i	$-0.790215\pi$
$23$	−2.12200e6	−1.58114	−0.790569	+	0.612373i	$0.790215\pi$
$24$	0	0
$25$	0	0
$26$	−1.06422e6	−0.456720
$27$	0	0
$28$	1.11145e7	3.41728
$29$	−389624.	−0.102295	−0.0511475	−	0.998691i	$-0.516288\pi$
$29$	−389624.	−0.102295	−0.0511475	+	0.998691i	$0.516288\pi$
$30$	0	0
$31$	90532.2	0.0176066	0.00880330	−	0.999961i	$-0.497198\pi$
$31$	90532.2	0.0176066	0.00880330	+	0.999961i	$0.497198\pi$
$32$	2.41838e7	4.07709
$33$	0	0
$34$	1.17706e7	1.51058
$35$	0	0
$36$	0	0
$37$	3.31991e6	0.291218	0.145609	−	0.989342i	$-0.453486\pi$
$37$	3.31991e6	0.291218	0.145609	+	0.989342i	$0.453486\pi$
$38$	−7.38848e6	−0.574816
$39$	0	0
$40$	0	0
$41$	−2.32694e7	−1.28605	−0.643024	−	0.765846i	$-0.722321\pi$
$41$	−2.32694e7	−1.28605	−0.643024	+	0.765846i	$0.722321\pi$
$42$	0	0
$43$	−1.91140e7	−0.852597	−0.426298	−	0.904583i	$-0.640183\pi$
$43$	−1.91140e7	−0.852597	−0.426298	+	0.904583i	$0.640183\pi$
$44$	6.98036e7	2.80764
$45$	0	0
$46$	−9.31215e7	−3.06648
$47$	6.28153e7	1.87770	0.938848	−	0.344332i	$-0.111895\pi$
$47$	6.28153e7	1.87770	0.938848	+	0.344332i	$0.111895\pi$
$48$	0	0
$49$	2.14495e7	0.531539
$50$	0	0
$51$	0	0
$52$	−3.42855e7	−0.650273
$53$	−180207.	−0.00313711	−0.00156856	−	0.999999i	$-0.500499\pi$
$53$	−180207.	−0.00313711	−0.00156856	+	0.999999i	$0.500499\pi$
$54$	0	0
$55$	0	0
$56$	3.11112e8	4.22738
$57$	0	0
$58$	−1.70982e7	−0.198392
$59$	−3.84564e7	−0.413175	−0.206588	−	0.978428i	$-0.566236\pi$
$59$	−3.84564e7	−0.413175	−0.206588	+	0.978428i	$0.566236\pi$
$60$	0	0
$61$	−553620.	−0.00511950	−0.00255975	−	0.999997i	$-0.500815\pi$
$61$	−553620.	−0.00511950	−0.00255975	+	0.999997i	$0.500815\pi$
$62$	3.97290e6	0.0341464
$63$	0	0
$64$	5.42724e8	4.04361
$65$	0	0
$66$	0	0
$67$	2.39163e8	1.44996	0.724982	−	0.688768i	$-0.241848\pi$
$67$	2.39163e8	1.44996	0.724982	+	0.688768i	$0.241848\pi$
$68$	3.79211e8	2.15076
$69$	0	0
$70$	0	0
$71$	−1.28653e8	−0.600838	−0.300419	−	0.953807i	$-0.597127\pi$
$71$	−1.28653e8	−0.600838	−0.300419	+	0.953807i	$0.597127\pi$
$72$	0	0
$73$	2.39376e8	0.986569	0.493284	−	0.869868i	$-0.335796\pi$
$73$	2.39376e8	0.986569	0.493284	+	0.869868i	$0.335796\pi$
$74$	1.45690e8	0.564792
$75$	0	0
$76$	−2.38033e8	−0.818418
$77$	3.88148e8	1.25831
$78$	0	0
$79$	−5.28027e8	−1.52523	−0.762613	−	0.646855i	$-0.776084\pi$
$79$	−5.28027e8	−1.52523	−0.762613	+	0.646855i	$0.776084\pi$
$80$	0	0
$81$	0	0
$82$	−1.02115e9	−2.49418
$83$	2.12210e8	0.490812	0.245406	−	0.969420i	$-0.421079\pi$
$83$	2.12210e8	0.490812	0.245406	+	0.969420i	$0.421079\pi$
$84$	0	0
$85$	0	0
$86$	−8.38797e8	−1.65354
$87$	0	0
$88$	1.95391e9	3.47322
$89$	2.07724e8	0.350939	0.175469	−	0.984485i	$-0.443856\pi$
$89$	2.07724e8	0.350939	0.175469	+	0.984485i	$0.443856\pi$
$90$	0	0
$91$	−1.90647e8	−0.291436
$92$	−3.00007e9	−4.36602
$93$	0	0
$94$	2.75658e9	3.64162
$95$	0	0
$96$	0	0
$97$	−1.70780e9	−1.95868	−0.979341	−	0.202214i	$-0.935186\pi$
$97$	−1.70780e9	−1.95868	−0.979341	+	0.202214i	$0.935186\pi$
$98$	9.41288e8	1.03087
$99$	0	0
$100$	0	0
$101$	1.81384e8	0.173441	0.0867206	−	0.996233i	$-0.472361\pi$
$101$	1.81384e8	0.173441	0.0867206	+	0.996233i	$0.472361\pi$
$102$	0	0
$103$	1.70453e9	1.49223	0.746116	−	0.665816i	$-0.231917\pi$
$103$	1.70453e9	1.49223	0.746116	+	0.665816i	$0.231917\pi$
$104$	−9.59703e8	−0.804427
$105$	0	0
$106$	−7.90818e6	−0.00608415
$107$	1.73298e9	1.27811	0.639053	−	0.769163i	$-0.279326\pi$
$107$	1.73298e9	1.27811	0.639053	+	0.769163i	$0.279326\pi$
$108$	0	0
$109$	−1.17238e9	−0.795515	−0.397757	−	0.917491i	$-0.630211\pi$
$109$	−1.17238e9	−0.795515	−0.397757	+	0.917491i	$0.630211\pi$
$110$	0	0
$111$	0	0
$112$	7.96217e9	4.78135
$113$	−1.22180e9	−0.704935	−0.352467	−	0.935824i	$-0.614657\pi$
$113$	−1.22180e9	−0.704935	−0.352467	+	0.935824i	$0.614657\pi$
$114$	0	0
$115$	0	0
$116$	−5.50848e8	−0.282469
$117$	0	0
$118$	−1.68761e9	−0.801317
$119$	2.10863e9	0.963916
$120$	0	0
$121$	7.97750e7	0.0338324
$122$	−2.42950e7	−0.00992881
$123$	0	0
$124$	1.27994e8	0.0486174
$125$	0	0
$126$	0	0
$127$	9.50324e8	0.324157	0.162078	−	0.986778i	$-0.448180\pi$
$127$	9.50324e8	0.324157	0.162078	+	0.986778i	$0.448180\pi$
$128$	1.14347e10	3.76513
$129$	0	0
$130$	0	0
$131$	4.03050e9	1.19574	0.597872	−	0.801592i	$-0.296013\pi$
$131$	4.03050e9	1.19574	0.597872	+	0.801592i	$0.296013\pi$
$132$	0	0
$133$	−1.32360e9	−0.366795
$134$	1.04954e10	2.81207
$135$	0	0
$136$	1.06147e10	2.66061
$137$	5.51871e9	1.33843	0.669214	−	0.743070i	$-0.266631\pi$
$137$	5.51871e9	1.33843	0.669214	+	0.743070i	$0.266631\pi$
$138$	0	0
$139$	−3.97776e9	−0.903798	−0.451899	−	0.892069i	$-0.649253\pi$
$139$	−3.97776e9	−0.903798	−0.451899	+	0.892069i	$0.649253\pi$
$140$	0	0
$141$	0	0
$142$	−5.64579e9	−1.16527
$143$	−1.19734e9	−0.239445
$144$	0	0
$145$	0	0
$146$	1.05047e10	1.91336
$147$	0	0
$148$	4.69367e9	0.804145
$149$	2.26538e9	0.376533	0.188266	−	0.982118i	$-0.439713\pi$
$149$	2.26538e9	0.376533	0.188266	+	0.982118i	$0.439713\pi$
$150$	0	0
$151$	−8.49158e9	−1.32921	−0.664603	−	0.747197i	$-0.731399\pi$
$151$	−8.49158e9	−1.32921	−0.664603	+	0.747197i	$0.731399\pi$
$152$	−6.66288e9	−1.01243
$153$	0	0
$154$	1.70334e10	2.44039
$155$	0	0
$156$	0	0
$157$	−7.28379e9	−0.956774	−0.478387	−	0.878149i	$-0.658778\pi$
$157$	−7.28379e9	−0.956774	−0.478387	+	0.878149i	$0.658778\pi$
$158$	−2.31719e10	−2.95804
$159$	0	0
$160$	0	0
$161$	−1.66821e10	−1.95674
$162$	0	0
$163$	−1.38419e10	−1.53586	−0.767929	−	0.640535i	$-0.778712\pi$
$163$	−1.38419e10	−1.53586	−0.767929	+	0.640535i	$0.778712\pi$
$164$	−3.28981e10	−3.55119
$165$	0	0
$166$	9.31262e9	0.951887
$167$	−6.92038e9	−0.688503	−0.344252	−	0.938877i	$-0.611867\pi$
$167$	−6.92038e9	−0.688503	−0.344252	+	0.938877i	$0.611867\pi$
$168$	0	0
$169$	−1.00164e10	−0.944543
$170$	0	0
$171$	0	0
$172$	−2.70233e10	−2.35429
$173$	3.26987e9	0.277539	0.138769	−	0.990325i	$-0.455685\pi$
$173$	3.26987e9	0.277539	0.138769	+	0.990325i	$0.455685\pi$
$174$	0	0
$175$	0	0
$176$	5.00056e10	3.92836
$177$	0	0
$178$	9.11572e9	0.680614
$179$	1.12431e10	0.818556	0.409278	−	0.912410i	$-0.365781\pi$
$179$	1.12431e10	0.818556	0.409278	+	0.912410i	$0.365781\pi$
$180$	0	0
$181$	3.75039e9	0.259731	0.129865	−	0.991532i	$-0.458546\pi$
$181$	3.75039e9	0.259731	0.129865	+	0.991532i	$0.458546\pi$
$182$	−8.36633e9	−0.565215
$183$	0	0
$184$	−8.39764e10	−5.40103
$185$	0	0
$186$	0	0
$187$	1.32430e10	0.791954
$188$	8.88079e10	5.18491
$189$	0	0
$190$	0	0
$191$	−1.72426e10	−0.937460	−0.468730	−	0.883342i	$-0.655288\pi$
$191$	−1.72426e10	−0.937460	−0.468730	+	0.883342i	$0.655288\pi$
$192$	0	0
$193$	6.01060e9	0.311824	0.155912	−	0.987771i	$-0.450168\pi$
$193$	6.01060e9	0.311824	0.155912	+	0.987771i	$0.450168\pi$
$194$	−7.49448e10	−3.79869
$195$	0	0
$196$	3.03252e10	1.46775
$197$	−1.19077e10	−0.563289	−0.281645	−	0.959519i	$-0.590880\pi$
$197$	−1.19077e10	−0.563289	−0.281645	+	0.959519i	$0.590880\pi$
$198$	0	0
$199$	−6.10946e9	−0.276162	−0.138081	−	0.990421i	$-0.544093\pi$
$199$	−6.10946e9	−0.276162	−0.138081	+	0.990421i	$0.544093\pi$
$200$	0	0
$201$	0	0
$202$	7.95982e9	0.336374
$203$	−3.06303e9	−0.126596
$204$	0	0
$205$	0	0
$206$	7.48012e10	2.89405
$207$	0	0
$208$	−2.45613e10	−0.909842
$209$	−8.31271e9	−0.301359
$210$	0	0
$211$	−3.67892e10	−1.27776	−0.638881	−	0.769306i	$-0.720602\pi$
$211$	−3.67892e10	−1.27776	−0.638881	+	0.769306i	$0.720602\pi$
$212$	−2.54775e8	−0.00866256
$213$	0	0
$214$	7.60499e10	2.47877
$215$	0	0
$216$	0	0
$217$	7.11719e8	0.0217891
$218$	−5.14485e10	−1.54283
$219$	0	0
$220$	0	0
$221$	−6.50459e9	−0.183423
$222$	0	0
$223$	5.01307e9	0.135747	0.0678737	−	0.997694i	$-0.478379\pi$
$223$	5.01307e9	0.135747	0.0678737	+	0.997694i	$0.478379\pi$
$224$	1.90121e11	5.04562
$225$	0	0
$226$	−5.36175e10	−1.36716
$227$	−2.73858e10	−0.684555	−0.342278	−	0.939599i	$-0.611198\pi$
$227$	−2.73858e10	−0.684555	−0.342278	+	0.939599i	$0.611198\pi$
$228$	0	0
$229$	5.36868e10	1.29005	0.645027	−	0.764159i	$-0.276846\pi$
$229$	5.36868e10	1.29005	0.645027	+	0.764159i	$0.276846\pi$
$230$	0	0
$231$	0	0
$232$	−1.54191e10	−0.349431
$233$	7.25874e10	1.61347	0.806733	−	0.590916i	$-0.201234\pi$
$233$	7.25874e10	1.61347	0.806733	+	0.590916i	$0.201234\pi$
$234$	0	0
$235$	0	0
$236$	−5.43694e10	−1.14091
$237$	0	0
$238$	9.25348e10	1.86943
$239$	−1.73313e10	−0.343589	−0.171795	−	0.985133i	$-0.554957\pi$
$239$	−1.73313e10	−0.343589	−0.171795	+	0.985133i	$0.554957\pi$
$240$	0	0
$241$	4.06448e10	0.776119	0.388059	−	0.921634i	$-0.373146\pi$
$241$	4.06448e10	0.776119	0.388059	+	0.921634i	$0.373146\pi$
$242$	3.50084e9	0.0656149
$243$	0	0
$244$	−7.82704e8	−0.0141365
$245$	0	0
$246$	0	0
$247$	4.08296e9	0.0697973
$248$	3.58274e9	0.0601426
$249$	0	0
$250$	0	0
$251$	−1.00920e11	−1.60490	−0.802448	−	0.596723i	$-0.796469\pi$
$251$	−1.00920e11	−1.60490	−0.802448	+	0.596723i	$0.796469\pi$
$252$	0	0
$253$	−1.04770e11	−1.60766
$254$	4.17039e10	0.628673
$255$	0	0
$256$	2.23924e11	3.25852
$257$	5.03660e10	0.720176	0.360088	−	0.932918i	$-0.382747\pi$
$257$	5.03660e10	0.720176	0.360088	+	0.932918i	$0.382747\pi$
$258$	0	0
$259$	2.60995e10	0.360398
$260$	0	0
$261$	0	0
$262$	1.76874e11	2.31904
$263$	−5.58677e10	−0.720046	−0.360023	−	0.932943i	$-0.617231\pi$
$263$	−5.58677e10	−0.720046	−0.360023	+	0.932943i	$0.617231\pi$
$264$	0	0
$265$	0	0
$266$	−5.80845e10	−0.711366
$267$	0	0
$268$	3.38127e11	4.00381
$269$	1.05175e11	1.22469	0.612345	−	0.790590i	$-0.290226\pi$
$269$	1.05175e11	1.22469	0.612345	+	0.790590i	$0.290226\pi$
$270$	0	0
$271$	−5.44180e10	−0.612887	−0.306444	−	0.951889i	$-0.599139\pi$
$271$	−5.44180e10	−0.612887	−0.306444	+	0.951889i	$0.599139\pi$
$272$	2.71657e11	3.00927
$273$	0	0
$274$	2.42182e11	2.59576
$275$	0	0
$276$	0	0
$277$	−1.64528e11	−1.67912	−0.839560	−	0.543268i	$-0.817187\pi$
$277$	−1.64528e11	−1.67912	−0.839560	+	0.543268i	$0.817187\pi$
$278$	−1.74559e11	−1.75284
$279$	0	0
$280$	0	0
$281$	6.74255e10	0.645128	0.322564	−	0.946548i	$-0.395455\pi$
$281$	6.74255e10	0.645128	0.322564	+	0.946548i	$0.395455\pi$
$282$	0	0
$283$	−1.41917e11	−1.31521	−0.657603	−	0.753364i	$-0.728430\pi$
$283$	−1.41917e11	−1.31521	−0.657603	+	0.753364i	$0.728430\pi$
$284$	−1.81889e11	−1.65910
$285$	0	0
$286$	−5.25438e10	−0.464381
$287$	−1.82932e11	−1.59155
$288$	0	0
$289$	−4.66446e10	−0.393333
$290$	0	0
$291$	0	0
$292$	3.38428e11	2.72423
$293$	2.79097e10	0.221233	0.110617	−	0.993863i	$-0.464717\pi$
$293$	2.79097e10	0.221233	0.110617	+	0.993863i	$0.464717\pi$
$294$	0	0
$295$	0	0
$296$	1.31383e11	0.994776
$297$	0	0
$298$	9.94136e10	0.730252
$299$	5.14600e10	0.372349
$300$	0	0
$301$	−1.50265e11	−1.05513
$302$	−3.72643e11	−2.57788
$303$	0	0
$304$	−1.70520e11	−1.14510
$305$	0	0
$306$	0	0
$307$	−1.59767e11	−1.02651	−0.513257	−	0.858235i	$-0.671561\pi$
$307$	−1.59767e11	−1.02651	−0.513257	+	0.858235i	$0.671561\pi$
$308$	5.48761e11	3.47460
$309$	0	0
$310$	0	0
$311$	7.84292e10	0.475396	0.237698	−	0.971339i	$-0.423607\pi$
$311$	7.84292e10	0.475396	0.237698	+	0.971339i	$0.423607\pi$
$312$	0	0
$313$	1.26229e11	0.743378	0.371689	−	0.928357i	$-0.378779\pi$
$313$	1.26229e11	0.743378	0.371689	+	0.928357i	$0.378779\pi$
$314$	−3.19641e11	−1.85558
$315$	0	0
$316$	−7.46521e11	−4.21163
$317$	−9.59499e10	−0.533676	−0.266838	−	0.963741i	$-0.585979\pi$
$317$	−9.59499e10	−0.533676	−0.266838	+	0.963741i	$0.585979\pi$
$318$	0	0
$319$	−1.92370e10	−0.104011
$320$	0	0
$321$	0	0
$322$	−7.32074e11	−3.79493
$323$	−4.51591e10	−0.230852
$324$	0	0
$325$	0	0
$326$	−6.07435e11	−2.97866
$327$	0	0
$328$	−9.20867e11	−4.39303
$329$	4.93822e11	2.32375
$330$	0	0
$331$	7.73728e10	0.354293	0.177146	−	0.984185i	$-0.443313\pi$
$331$	7.73728e10	0.354293	0.177146	+	0.984185i	$0.443313\pi$
$332$	3.00022e11	1.35529
$333$	0	0
$334$	−3.03693e11	−1.33529
$335$	0	0
$336$	0	0
$337$	−1.73809e11	−0.734071	−0.367035	−	0.930207i	$-0.619627\pi$
$337$	−1.73809e11	−0.734071	−0.367035	+	0.930207i	$0.619627\pi$
$338$	−4.39558e11	−1.83186
$339$	0	0
$340$	0	0
$341$	4.46987e9	0.0179020
$342$	0	0
$343$	−1.48614e11	−0.579746
$344$	−7.56421e11	−2.91240
$345$	0	0
$346$	1.43495e11	0.538262
$347$	4.11335e11	1.52305	0.761523	−	0.648138i	$-0.224452\pi$
$347$	4.11335e11	1.52305	0.761523	+	0.648138i	$0.224452\pi$
$348$	0	0
$349$	2.97923e11	1.07495	0.537477	−	0.843278i	$-0.319377\pi$
$349$	2.97923e11	1.07495	0.537477	+	0.843278i	$0.319377\pi$
$350$	0	0
$351$	0	0
$352$	1.19404e12	4.14549
$353$	5.69499e9	0.0195212	0.00976061	−	0.999952i	$-0.496893\pi$
$353$	5.69499e9	0.0195212	0.00976061	+	0.999952i	$0.496893\pi$
$354$	0	0
$355$	0	0
$356$	2.93678e11	0.969052
$357$	0	0
$358$	4.93392e11	1.58752
$359$	−6.19565e11	−1.96862	−0.984310	−	0.176446i	$-0.943540\pi$
$359$	−6.19565e11	−1.96862	−0.984310	+	0.176446i	$0.943540\pi$
$360$	0	0
$361$	−2.94341e11	−0.912155
$362$	1.64582e11	0.503724
$363$	0	0
$364$	−2.69536e11	−0.804748
$365$	0	0
$366$	0	0
$367$	4.06833e9	0.0117063	0.00585314	−	0.999983i	$-0.498137\pi$
$367$	4.06833e9	0.0117063	0.00585314	+	0.999983i	$0.498137\pi$
$368$	−2.14917e12	−6.10880
$369$	0	0
$370$	0	0
$371$	−1.41670e9	−0.00388235
$372$	0	0
$373$	6.75397e11	1.80663	0.903315	−	0.428978i	$-0.141126\pi$
$373$	6.75397e11	1.80663	0.903315	+	0.428978i	$0.141126\pi$
$374$	5.81155e11	1.53592
$375$	0	0
$376$	2.48586e12	6.41405
$377$	9.44867e9	0.0240899
$378$	0	0
$379$	−3.67146e11	−0.914033	−0.457017	−	0.889458i	$-0.651082\pi$
$379$	−3.67146e11	−0.914033	−0.457017	+	0.889458i	$0.651082\pi$
$380$	0	0
$381$	0	0
$382$	−7.56672e11	−1.81812
$383$	2.22057e11	0.527314	0.263657	−	0.964616i	$-0.415071\pi$
$383$	2.22057e11	0.527314	0.263657	+	0.964616i	$0.415071\pi$
$384$	0	0
$385$	0	0
$386$	2.63768e11	0.604756
$387$	0	0
$388$	−2.41448e12	−5.40854
$389$	7.23043e11	1.60100	0.800499	−	0.599334i	$-0.204568\pi$
$389$	7.23043e11	1.60100	0.800499	+	0.599334i	$0.204568\pi$
$390$	0	0
$391$	−5.69168e11	−1.23153
$392$	8.48847e11	1.81569
$393$	0	0
$394$	−5.22558e11	−1.09245
$395$	0	0
$396$	0	0
$397$	−6.06161e11	−1.22470	−0.612352	−	0.790585i	$-0.709776\pi$
$397$	−6.06161e11	−1.22470	−0.612352	+	0.790585i	$0.709776\pi$
$398$	−2.68107e11	−0.535592
$399$	0	0
$400$	0	0
$401$	−6.72510e11	−1.29882	−0.649411	−	0.760438i	$-0.724984\pi$
$401$	−6.72510e11	−1.29882	−0.649411	+	0.760438i	$0.724984\pi$
$402$	0	0
$403$	−2.19547e9	−0.00414625
$404$	2.56439e11	0.478926
$405$	0	0
$406$	−1.34417e11	−0.245521
$407$	1.63915e11	0.296103
$408$	0	0
$409$	−3.02482e10	−0.0534497	−0.0267248	−	0.999643i	$-0.508508\pi$
$409$	−3.02482e10	−0.0534497	−0.0267248	+	0.999643i	$0.508508\pi$
$410$	0	0
$411$	0	0
$412$	2.40985e12	4.12052
$413$	−3.02325e11	−0.511326
$414$	0	0
$415$	0	0
$416$	−5.86476e11	−0.960130
$417$	0	0
$418$	−3.64794e11	−0.584459
$419$	1.61938e11	0.256676	0.128338	−	0.991731i	$-0.459036\pi$
$419$	1.61938e11	0.256676	0.128338	+	0.991731i	$0.459036\pi$
$420$	0	0
$421$	−8.06547e11	−1.25130	−0.625648	−	0.780105i	$-0.715165\pi$
$421$	−8.06547e11	−1.25130	−0.625648	+	0.780105i	$0.715165\pi$
$422$	−1.61445e12	−2.47810
$423$	0	0
$424$	−7.13154e9	−0.0107161
$425$	0	0
$426$	0	0
$427$	−4.35228e9	−0.00633565
$428$	2.45008e12	3.52925
$429$	0	0
$430$	0	0
$431$	−8.54653e11	−1.19300	−0.596502	−	0.802611i	$-0.703443\pi$
$431$	−8.54653e11	−1.19300	−0.596502	+	0.802611i	$0.703443\pi$
$432$	0	0
$433$	−8.21884e11	−1.12361	−0.561804	−	0.827270i	$-0.689893\pi$
$433$	−8.21884e11	−1.12361	−0.561804	+	0.827270i	$0.689893\pi$
$434$	3.12330e10	0.0422580
$435$	0	0
$436$	−1.65750e12	−2.19667
$437$	3.57269e11	0.468629
$438$	0	0
$439$	9.97807e11	1.28220	0.641100	−	0.767457i	$-0.278478\pi$
$439$	9.97807e11	1.28220	0.641100	+	0.767457i	$0.278478\pi$
$440$	0	0
$441$	0	0
$442$	−2.85447e11	−0.355733
$443$	−2.22586e11	−0.274588	−0.137294	−	0.990530i	$-0.543840\pi$
$443$	−2.22586e11	−0.274588	−0.137294	+	0.990530i	$0.543840\pi$
$444$	0	0
$445$	0	0
$446$	2.19993e11	0.263270
$447$	0	0
$448$	4.26662e12	5.00418
$449$	−1.28886e12	−1.49657	−0.748287	−	0.663375i	$-0.769123\pi$
$449$	−1.28886e12	−1.49657	−0.748287	+	0.663375i	$0.769123\pi$
$450$	0	0
$451$	−1.14889e12	−1.30762
$452$	−1.72738e12	−1.94655
$453$	0	0
$454$	−1.20179e12	−1.32763
$455$	0	0
$456$	0	0
$457$	−8.77644e11	−0.941230	−0.470615	−	0.882339i	$-0.655968\pi$
$457$	−8.77644e11	−0.941230	−0.470615	+	0.882339i	$0.655968\pi$
$458$	2.35599e12	2.50195
$459$	0	0
$460$	0	0
$461$	3.31142e11	0.341476	0.170738	−	0.985316i	$-0.445385\pi$
$461$	3.31142e11	0.341476	0.170738	+	0.985316i	$0.445385\pi$
$462$	0	0
$463$	1.05840e12	1.07037	0.535186	−	0.844735i	$-0.320242\pi$
$463$	1.05840e12	1.07037	0.535186	+	0.844735i	$0.320242\pi$
$464$	−3.94613e11	−0.395222
$465$	0	0
$466$	3.18542e12	3.12917
$467$	−1.49420e12	−1.45373	−0.726864	−	0.686782i	$-0.759023\pi$
$467$	−1.49420e12	−1.45373	−0.726864	+	0.686782i	$0.759023\pi$
$468$	0	0
$469$	1.88018e12	1.79441
$470$	0	0
$471$	0	0
$472$	−1.52188e12	−1.41137
$473$	−9.43722e11	−0.866900
$474$	0	0
$475$	0	0
$476$	2.98117e12	2.66168
$477$	0	0
$478$	−7.60563e11	−0.666361
$479$	−1.04497e12	−0.906970	−0.453485	−	0.891264i	$-0.649819\pi$
$479$	−1.04497e12	−0.906970	−0.453485	+	0.891264i	$0.649819\pi$
$480$	0	0
$481$	−8.05102e10	−0.0685801
$482$	1.78365e12	1.50521
$483$	0	0
$484$	1.12785e11	0.0934219
$485$	0	0
$486$	0	0
$487$	2.30343e12	1.85564	0.927820	−	0.373027i	$-0.121680\pi$
$487$	2.30343e12	1.85564	0.927820	+	0.373027i	$0.121680\pi$
$488$	−2.19091e10	−0.0174878
$489$	0	0
$490$	0	0
$491$	5.37950e11	0.417711	0.208855	−	0.977947i	$-0.433026\pi$
$491$	5.37950e11	0.417711	0.208855	+	0.977947i	$0.433026\pi$
$492$	0	0
$493$	−1.04506e11	−0.0796764
$494$	1.79176e11	0.135366
$495$	0	0
$496$	9.16915e10	0.0680239
$497$	−1.01141e12	−0.743570
$498$	0	0
$499$	−1.01572e12	−0.733364	−0.366682	−	0.930346i	$-0.619506\pi$
$499$	−1.01572e12	−0.733364	−0.366682	+	0.930346i	$0.619506\pi$
$500$	0	0
$501$	0	0
$502$	−4.42877e12	−3.11255
$503$	−1.07349e12	−0.747723	−0.373861	−	0.927485i	$-0.621966\pi$
$503$	−1.07349e12	−0.747723	−0.373861	+	0.927485i	$0.621966\pi$
$504$	0	0
$505$	0	0
$506$	−4.59772e12	−3.11792
$507$	0	0
$508$	1.34356e12	0.895099
$509$	1.48831e12	0.982799	0.491400	−	0.870934i	$-0.336485\pi$
$509$	1.48831e12	0.982799	0.491400	+	0.870934i	$0.336485\pi$
$510$	0	0
$511$	1.88185e12	1.22093
$512$	3.97208e12	2.55449
$513$	0	0
$514$	2.21026e12	1.39672
$515$	0	0
$516$	0	0
$517$	3.10140e12	1.90919
$518$	1.14534e12	0.698960
$519$	0	0
$520$	0	0
$521$	9.98544e11	0.593742	0.296871	−	0.954918i	$-0.404057\pi$
$521$	9.98544e11	0.593742	0.296871	+	0.954918i	$0.404057\pi$
$522$	0	0
$523$	1.59973e12	0.934954	0.467477	−	0.884005i	$-0.345163\pi$
$523$	1.59973e12	0.934954	0.467477	+	0.884005i	$0.345163\pi$
$524$	5.69829e12	3.30182
$525$	0	0
$526$	−2.45169e12	−1.39646
$527$	2.42828e10	0.0137136
$528$	0	0
$529$	2.70173e12	1.50000
$530$	0	0
$531$	0	0
$532$	−1.87129e12	−1.01284
$533$	5.64300e11	0.302857
$534$	0	0
$535$	0	0
$536$	9.46467e12	4.95295
$537$	0	0
$538$	4.61548e12	2.37518
$539$	1.05903e12	0.540456
$540$	0	0
$541$	5.11810e11	0.256875	0.128437	−	0.991718i	$-0.459004\pi$
$541$	5.11810e11	0.256875	0.128437	+	0.991718i	$0.459004\pi$
$542$	−2.38807e12	−1.18864
$543$	0	0
$544$	6.48665e12	3.17560
$545$	0	0
$546$	0	0
$547$	−1.40910e12	−0.672977	−0.336489	−	0.941688i	$-0.609239\pi$
$547$	−1.40910e12	−0.672977	−0.336489	+	0.941688i	$0.609239\pi$
$548$	7.80231e12	3.69582
$549$	0	0
$550$	0	0
$551$	6.55988e10	0.0303189
$552$	0	0
$553$	−4.15108e12	−1.88755
$554$	−7.22013e12	−3.25650
$555$	0	0
$556$	−5.62372e12	−2.49567
$557$	2.36835e12	1.04255	0.521276	−	0.853388i	$-0.325456\pi$
$557$	2.36835e12	1.04255	0.521276	+	0.853388i	$0.325456\pi$
$558$	0	0
$559$	4.63529e11	0.200781
$560$	0	0
$561$	0	0
$562$	2.95889e12	1.25117
$563$	−2.36245e12	−0.991005	−0.495502	−	0.868607i	$-0.665016\pi$
$563$	−2.36245e12	−0.991005	−0.495502	+	0.868607i	$0.665016\pi$
$564$	0	0
$565$	0	0
$566$	−6.22784e12	−2.55073
$567$	0	0
$568$	−5.09134e12	−2.05241
$569$	−2.25679e11	−0.0902582	−0.0451291	−	0.998981i	$-0.514370\pi$
$569$	−2.25679e11	−0.0902582	−0.0451291	+	0.998981i	$0.514370\pi$
$570$	0	0
$571$	−9.15389e11	−0.360366	−0.180183	−	0.983633i	$-0.557669\pi$
$571$	−9.15389e11	−0.360366	−0.180183	+	0.983633i	$0.557669\pi$
$572$	−1.69279e12	−0.661182
$573$	0	0
$574$	−8.02777e12	−3.08668
$575$	0	0
$576$	0	0
$577$	−2.22404e12	−0.835318	−0.417659	−	0.908604i	$-0.637149\pi$
$577$	−2.22404e12	−0.835318	−0.417659	+	0.908604i	$0.637149\pi$
$578$	−2.04694e12	−0.762835
$579$	0	0
$580$	0	0
$581$	1.66829e12	0.607407
$582$	0	0
$583$	−8.89741e9	−0.00318974
$584$	9.47310e12	3.37003
$585$	0	0
$586$	1.22478e12	0.429062
$587$	5.60186e12	1.94742	0.973712	−	0.227783i	$-0.0731477\pi$
$587$	5.60186e12	1.94742	0.973712	+	0.227783i	$0.0731477\pi$
$588$	0	0
$589$	−1.52424e10	−0.00521837
$590$	0	0
$591$	0	0
$592$	3.36242e12	1.12513
$593$	2.78326e12	0.924289	0.462145	−	0.886805i	$-0.347080\pi$
$593$	2.78326e12	0.924289	0.462145	+	0.886805i	$0.347080\pi$
$594$	0	0
$595$	0	0
$596$	3.20278e12	1.03973
$597$	0	0
$598$	2.25826e12	0.722137
$599$	1.36720e12	0.433922	0.216961	−	0.976180i	$-0.430386\pi$
$599$	1.36720e12	0.433922	0.216961	+	0.976180i	$0.430386\pi$
$600$	0	0
$601$	4.74171e12	1.48252	0.741259	−	0.671219i	$-0.234229\pi$
$601$	4.74171e12	1.48252	0.741259	+	0.671219i	$0.234229\pi$
$602$	−6.59420e12	−2.04634
$603$	0	0
$604$	−1.20053e13	−3.67036
$605$	0	0
$606$	0	0
$607$	5.74836e12	1.71868	0.859339	−	0.511406i	$-0.170875\pi$
$607$	5.74836e12	1.71868	0.859339	+	0.511406i	$0.170875\pi$
$608$	−4.07170e12	−1.20840
$609$	0	0
$610$	0	0
$611$	−1.52332e12	−0.442186
$612$	0	0
$613$	−2.99054e12	−0.855416	−0.427708	−	0.903917i	$-0.640679\pi$
$613$	−2.99054e12	−0.855416	−0.427708	+	0.903917i	$0.640679\pi$
$614$	−7.01120e12	−1.99083
$615$	0	0
$616$	1.53606e13	4.29829
$617$	6.27174e12	1.74223	0.871114	−	0.491081i	$-0.163398\pi$
$617$	6.27174e12	1.74223	0.871114	+	0.491081i	$0.163398\pi$
$618$	0	0
$619$	−2.67240e12	−0.731632	−0.365816	−	0.930687i	$-0.619210\pi$
$619$	−2.67240e12	−0.731632	−0.365816	+	0.930687i	$0.619210\pi$
$620$	0	0
$621$	0	0
$622$	3.44177e12	0.921989
$623$	1.63302e12	0.434305
$624$	0	0
$625$	0	0
$626$	5.53942e12	1.44172
$627$	0	0
$628$	−1.02978e13	−2.64195
$629$	8.90474e11	0.226826
$630$	0	0
$631$	2.73608e12	0.687064	0.343532	−	0.939141i	$-0.388377\pi$
$631$	2.73608e12	0.687064	0.343532	+	0.939141i	$0.388377\pi$
$632$	−2.08962e13	−5.21004
$633$	0	0
$634$	−4.21065e12	−1.03502
$635$	0	0
$636$	0	0
$637$	−5.20167e11	−0.125174
$638$	−8.44195e11	−0.201720
$639$	0	0
$640$	0	0
$641$	2.58542e12	0.604882	0.302441	−	0.953168i	$-0.402198\pi$
$641$	2.58542e12	0.604882	0.302441	+	0.953168i	$0.402198\pi$
$642$	0	0
$643$	4.53865e12	1.04707	0.523537	−	0.852003i	$-0.324612\pi$
$643$	4.53865e12	1.04707	0.523537	+	0.852003i	$0.324612\pi$
$644$	−2.35850e13	−5.40319
$645$	0	0
$646$	−1.98176e12	−0.447717
$647$	2.68597e12	0.602604	0.301302	−	0.953529i	$-0.402579\pi$
$647$	2.68597e12	0.602604	0.301302	+	0.953529i	$0.402579\pi$
$648$	0	0
$649$	−1.89872e12	−0.420106
$650$	0	0
$651$	0	0
$652$	−1.95696e13	−4.24099
$653$	−6.39120e12	−1.37554	−0.687770	−	0.725929i	$-0.741410\pi$
$653$	−6.39120e12	−1.37554	−0.687770	+	0.725929i	$0.741410\pi$
$654$	0	0
$655$	0	0
$656$	−2.35674e13	−4.96871
$657$	0	0
$658$	2.16708e13	4.50670
$659$	4.23105e12	0.873904	0.436952	−	0.899485i	$-0.356058\pi$
$659$	4.23105e12	0.873904	0.436952	+	0.899485i	$0.356058\pi$
$660$	0	0
$661$	−3.77872e12	−0.769907	−0.384953	−	0.922936i	$-0.625782\pi$
$661$	−3.77872e12	−0.769907	−0.384953	+	0.922936i	$0.625782\pi$
$662$	3.39542e12	0.687120
$663$	0	0
$664$	8.39806e12	1.67657
$665$	0	0
$666$	0	0
$667$	8.26782e11	0.161743
$668$	−9.78399e12	−1.90117
$669$	0	0
$670$	0	0
$671$	−2.73340e10	−0.00520538
$672$	0	0
$673$	−2.75736e12	−0.518114	−0.259057	−	0.965862i	$-0.583412\pi$
$673$	−2.75736e12	−0.518114	−0.259057	+	0.965862i	$0.583412\pi$
$674$	−7.62742e12	−1.42367
$675$	0	0
$676$	−1.41611e13	−2.60818
$677$	−2.40567e12	−0.440136	−0.220068	−	0.975485i	$-0.570628\pi$
$677$	−2.40567e12	−0.440136	−0.220068	+	0.975485i	$0.570628\pi$
$678$	0	0
$679$	−1.34259e13	−2.42397
$680$	0	0
$681$	0	0
$682$	1.96155e11	0.0347193
$683$	−7.88192e11	−0.138592	−0.0692961	−	0.997596i	$-0.522075\pi$
$683$	−7.88192e11	−0.138592	−0.0692961	+	0.997596i	$0.522075\pi$
$684$	0	0
$685$	0	0
$686$	−6.52178e12	−1.12436
$687$	0	0
$688$	−1.93588e13	−3.29405
$689$	4.37015e9	0.000738771 0
$690$	0	0
$691$	3.54288e12	0.591161	0.295580	−	0.955318i	$-0.404487\pi$
$691$	3.54288e12	0.591161	0.295580	+	0.955318i	$0.404487\pi$
$692$	4.62293e12	0.766372
$693$	0	0
$694$	1.80510e13	2.95381
$695$	0	0
$696$	0	0
$697$	−6.24137e12	−1.00169
$698$	1.30740e13	2.08478
$699$	0	0
$700$	0	0
$701$	3.66998e12	0.574027	0.287014	−	0.957926i	$-0.407337\pi$
$701$	3.66998e12	0.574027	0.287014	+	0.957926i	$0.407337\pi$
$702$	0	0
$703$	−5.58955e11	−0.0863133
$704$	2.67961e13	4.11144
$705$	0	0
$706$	2.49918e11	0.0378597
$707$	1.42595e12	0.214643
$708$	0	0
$709$	6.68373e12	0.993369	0.496685	−	0.867931i	$-0.334551\pi$
$709$	6.68373e12	0.993369	0.496685	+	0.867931i	$0.334551\pi$
$710$	0	0
$711$	0	0
$712$	8.22050e12	1.19878
$713$	−1.92109e11	−0.0278385
$714$	0	0
$715$	0	0
$716$	1.58954e13	2.26029
$717$	0	0
$718$	−2.71889e13	−3.81796
$719$	1.23056e13	1.71721	0.858604	−	0.512640i	$-0.171332\pi$
$719$	1.23056e13	1.71721	0.858604	+	0.512640i	$0.171332\pi$
$720$	0	0
$721$	1.34001e13	1.84672
$722$	−1.29168e13	−1.76904
$723$	0	0
$724$	5.30228e12	0.717198
$725$	0	0
$726$	0	0
$727$	1.07672e13	1.42954	0.714770	−	0.699360i	$-0.246531\pi$
$727$	1.07672e13	1.42954	0.714770	+	0.699360i	$0.246531\pi$
$728$	−7.54470e12	−0.995522
$729$	0	0
$730$	0	0
$731$	−5.12681e12	−0.664078
$732$	0	0
$733$	1.06410e13	1.36149	0.680744	−	0.732521i	$-0.261657\pi$
$733$	1.06410e13	1.36149	0.680744	+	0.732521i	$0.261657\pi$
$734$	1.78534e11	0.0227033
$735$	0	0
$736$	−5.13181e13	−6.44644
$737$	1.18083e13	1.47429
$738$	0	0
$739$	−4.24704e12	−0.523825	−0.261912	−	0.965092i	$-0.584353\pi$
$739$	−4.24704e12	−0.523825	−0.261912	+	0.965092i	$0.584353\pi$
$740$	0	0
$741$	0	0
$742$	−6.21701e10	−0.00752947
$743$	1.10732e13	1.33298	0.666488	−	0.745516i	$-0.267797\pi$
$743$	1.10732e13	1.33298	0.666488	+	0.745516i	$0.267797\pi$
$744$	0	0
$745$	0	0
$746$	2.96390e13	3.50380
$747$	0	0
$748$	1.87229e13	2.18684
$749$	1.36238e13	1.58172
$750$	0	0
$751$	1.45365e13	1.66755	0.833777	−	0.552102i	$-0.186174\pi$
$751$	1.45365e13	1.66755	0.833777	+	0.552102i	$0.186174\pi$
$752$	6.36197e13	7.25456
$753$	0	0
$754$	4.14644e11	0.0467202
$755$	0	0
$756$	0	0
$757$	5.75556e12	0.637025	0.318512	−	0.947919i	$-0.396817\pi$
$757$	5.75556e12	0.637025	0.318512	+	0.947919i	$0.396817\pi$
$758$	−1.61118e13	−1.77269
$759$	0	0
$760$	0	0
$761$	1.44303e13	1.55971	0.779854	−	0.625961i	$-0.215293\pi$
$761$	1.44303e13	1.55971	0.779854	+	0.625961i	$0.215293\pi$
$762$	0	0
$763$	−9.21664e12	−0.984492
$764$	−2.43775e13	−2.58862
$765$	0	0
$766$	9.74471e12	1.02268
$767$	9.32595e11	0.0973003
$768$	0	0
$769$	2.65690e12	0.273973	0.136986	−	0.990573i	$-0.456258\pi$
$769$	2.65690e12	0.273973	0.136986	+	0.990573i	$0.456258\pi$
$770$	0	0
$771$	0	0
$772$	8.49775e12	0.861046
$773$	−4.35569e12	−0.438783	−0.219391	−	0.975637i	$-0.570407\pi$
$773$	−4.35569e12	−0.438783	−0.219391	+	0.975637i	$0.570407\pi$
$774$	0	0
$775$	0	0
$776$	−6.75848e13	−6.69069
$777$	0	0
$778$	3.17299e13	3.10499
$779$	3.91774e12	0.381168
$780$	0	0
$781$	−6.35203e12	−0.610918
$782$	−2.49773e13	−2.38844
$783$	0	0
$784$	2.17242e13	2.05363
$785$	0	0
$786$	0	0
$787$	2.02013e13	1.87713	0.938564	−	0.345106i	$-0.112157\pi$
$787$	2.02013e13	1.87713	0.938564	+	0.345106i	$0.112157\pi$
$788$	−1.68351e13	−1.55542
$789$	0	0
$790$	0	0
$791$	−9.60521e12	−0.872394
$792$	0	0
$793$	1.34257e10	0.00120561
$794$	−2.66007e13	−2.37520
$795$	0	0
$796$	−8.63752e12	−0.762571
$797$	−2.07284e12	−0.181971	−0.0909857	−	0.995852i	$-0.529002\pi$
$797$	−2.07284e12	−0.181971	−0.0909857	+	0.995852i	$0.529002\pi$
$798$	0	0
$799$	1.68485e13	1.46251
$800$	0	0
$801$	0	0
$802$	−2.95124e13	−2.51895
$803$	1.18188e13	1.00312
$804$	0	0
$805$	0	0
$806$	−9.63458e10	−0.00804128
$807$	0	0
$808$	7.17812e12	0.592460
$809$	−1.41673e13	−1.16284	−0.581418	−	0.813605i	$-0.697502\pi$
$809$	−1.41673e13	−1.16284	−0.581418	+	0.813605i	$0.697502\pi$
$810$	0	0
$811$	2.04580e12	0.166062	0.0830310	−	0.996547i	$-0.473540\pi$
$811$	2.04580e12	0.166062	0.0830310	+	0.996547i	$0.473540\pi$
$812$	−4.33049e12	−0.349570
$813$	0	0
$814$	7.19322e12	0.574266
$815$	0	0
$816$	0	0
$817$	3.21812e12	0.252699
$818$	−1.32741e12	−0.103661
$819$	0	0
$820$	0	0
$821$	1.27265e13	0.977607	0.488804	−	0.872394i	$-0.337433\pi$
$821$	1.27265e13	0.977607	0.488804	+	0.872394i	$0.337433\pi$
$822$	0	0
$823$	−6.94698e12	−0.527833	−0.263917	−	0.964546i	$-0.585014\pi$
$823$	−6.94698e12	−0.527833	−0.263917	+	0.964546i	$0.585014\pi$
$824$	6.74552e13	5.09733
$825$	0	0
$826$	−1.32672e13	−0.991672
$827$	5.09129e11	0.0378489	0.0189244	−	0.999821i	$-0.493976\pi$
$827$	5.09129e11	0.0378489	0.0189244	+	0.999821i	$0.493976\pi$
$828$	0	0
$829$	−1.87663e13	−1.38001	−0.690006	−	0.723804i	$-0.742392\pi$
$829$	−1.87663e13	−1.38001	−0.690006	+	0.723804i	$0.742392\pi$
$830$	0	0
$831$	0	0
$832$	−1.31615e13	−0.952245
$833$	5.75324e12	0.414009
$834$	0	0
$835$	0	0
$836$	−1.17525e13	−0.832147
$837$	0	0
$838$	7.10645e12	0.497800
$839$	−6.68721e12	−0.465925	−0.232962	−	0.972486i	$-0.574842\pi$
$839$	−6.68721e12	−0.465925	−0.232962	+	0.972486i	$0.574842\pi$
$840$	0	0
$841$	−1.43553e13	−0.989536
$842$	−3.53944e13	−2.42678
$843$	0	0
$844$	−5.20124e13	−3.52830
$845$	0	0
$846$	0	0
$847$	6.27151e11	0.0418694
$848$	−1.82515e11	−0.0121204
$849$	0	0
$850$	0	0
$851$	−7.04484e12	−0.460456
$852$	0	0
$853$	−9.37029e12	−0.606014	−0.303007	−	0.952988i	$-0.597991\pi$
$853$	−9.37029e12	−0.606014	−0.303007	+	0.952988i	$0.597991\pi$
$854$	−1.90995e11	−0.0122874
$855$	0	0
$856$	6.85813e13	4.36590
$857$	8.81996e12	0.558538	0.279269	−	0.960213i	$-0.409908\pi$
$857$	8.81996e12	0.558538	0.279269	+	0.960213i	$0.409908\pi$
$858$	0	0
$859$	1.98932e13	1.24663	0.623313	−	0.781973i	$-0.285786\pi$
$859$	1.98932e13	1.24663	0.623313	+	0.781973i	$0.285786\pi$
$860$	0	0
$861$	0	0
$862$	−3.75055e13	−2.31373
$863$	−1.20953e13	−0.742278	−0.371139	−	0.928577i	$-0.621033\pi$
$863$	−1.20953e13	−0.742278	−0.371139	+	0.928577i	$0.621033\pi$
$864$	0	0
$865$	0	0
$866$	−3.60674e13	−2.17914
$867$	0	0
$868$	1.00622e12	0.0601666
$869$	−2.60704e13	−1.55081
$870$	0	0
$871$	−5.79987e12	−0.341458
$872$	−4.63959e13	−2.71741
$873$	0	0
$874$	1.56783e13	0.908864
$875$	0	0
$876$	0	0
$877$	2.52693e13	1.44243	0.721216	−	0.692710i	$-0.243584\pi$
$877$	2.52693e13	1.44243	0.721216	+	0.692710i	$0.243584\pi$
$878$	4.37876e13	2.48671
$879$	0	0
$880$	0	0
$881$	−4.23113e12	−0.236628	−0.118314	−	0.992976i	$-0.537749\pi$
$881$	−4.23113e12	−0.236628	−0.118314	+	0.992976i	$0.537749\pi$
$882$	0	0
$883$	−5.41348e12	−0.299677	−0.149839	−	0.988710i	$-0.547875\pi$
$883$	−5.41348e12	−0.299677	−0.149839	+	0.988710i	$0.547875\pi$
$884$	−9.19615e12	−0.506490
$885$	0	0
$886$	−9.76794e12	−0.532539
$887$	−6.28957e12	−0.341165	−0.170583	−	0.985343i	$-0.554565\pi$
$887$	−6.28957e12	−0.341165	−0.170583	+	0.985343i	$0.554565\pi$
$888$	0	0
$889$	7.47097e12	0.401161
$890$	0	0
$891$	0	0
$892$	7.08744e12	0.374841
$893$	−1.05759e13	−0.556525
$894$	0	0
$895$	0	0
$896$	8.98938e13	4.65955
$897$	0	0
$898$	−5.65603e13	−2.90247
$899$	−3.52735e10	−0.00180107
$900$	0	0
$901$	−4.83356e10	−0.00244346
$902$	−5.04176e13	−2.53602
$903$	0	0
$904$	−4.83519e13	−2.40800
$905$	0	0
$906$	0	0
$907$	1.27094e13	0.623580	0.311790	−	0.950151i	$-0.399072\pi$
$907$	1.27094e13	0.623580	0.311790	+	0.950151i	$0.399072\pi$
$908$	−3.87178e13	−1.89027
$909$	0	0
$910$	0	0
$911$	−9.10912e12	−0.438171	−0.219086	−	0.975706i	$-0.570307\pi$
$911$	−9.10912e12	−0.438171	−0.219086	+	0.975706i	$0.570307\pi$
$912$	0	0
$913$	1.04775e13	0.499046
$914$	−3.85144e13	−1.82543
$915$	0	0
$916$	7.59021e13	3.56225
$917$	3.16858e13	1.47980
$918$	0	0
$919$	−6.78526e12	−0.313795	−0.156898	−	0.987615i	$-0.550149\pi$
$919$	−6.78526e12	−0.313795	−0.156898	+	0.987615i	$0.550149\pi$
$920$	0	0
$921$	0	0
$922$	1.45318e13	0.662263
$923$	3.11993e12	0.141494
$924$	0	0
$925$	0	0
$926$	4.64466e13	2.07589
$927$	0	0
$928$	−9.42261e12	−0.417066
$929$	−3.16131e12	−0.139250	−0.0696252	−	0.997573i	$-0.522180\pi$
$929$	−3.16131e12	−0.139250	−0.0696252	+	0.997573i	$0.522180\pi$
$930$	0	0
$931$	−3.61134e12	−0.157541
$932$	1.02624e14	4.45529
$933$	0	0
$934$	−6.55713e13	−2.81938
$935$	0	0
$936$	0	0
$937$	−3.75670e13	−1.59213	−0.796065	−	0.605211i	$-0.793089\pi$
$937$	−3.75670e13	−1.59213	−0.796065	+	0.605211i	$0.793089\pi$
$938$	8.25094e13	3.48009
$939$	0	0
$940$	0	0
$941$	3.13713e13	1.30431	0.652153	−	0.758087i	$-0.273866\pi$
$941$	3.13713e13	1.30431	0.652153	+	0.758087i	$0.273866\pi$
$942$	0	0
$943$	4.93776e13	2.03342
$944$	−3.89488e13	−1.59632
$945$	0	0
$946$	−4.14142e13	−1.68127
$947$	−1.78574e13	−0.721513	−0.360757	−	0.932660i	$-0.617481\pi$
$947$	−1.78574e13	−0.721513	−0.360757	+	0.932660i	$0.617481\pi$
$948$	0	0
$949$	−5.80504e12	−0.232331
$950$	0	0
$951$	0	0
$952$	8.34473e13	3.29265
$953$	8.76602e12	0.344258	0.172129	−	0.985074i	$-0.444935\pi$
$953$	8.76602e12	0.344258	0.172129	+	0.985074i	$0.444935\pi$
$954$	0	0
$955$	0	0
$956$	−2.45028e13	−0.948758
$957$	0	0
$958$	−4.58572e13	−1.75899
$959$	4.33853e13	1.65638
$960$	0	0
$961$	−2.64314e13	−0.999690
$962$	−3.53310e12	−0.133005
$963$	0	0
$964$	5.74633e13	2.14311
$965$	0	0
$966$	0	0
$967$	−2.92145e13	−1.07443	−0.537217	−	0.843444i	$-0.680524\pi$
$967$	−2.92145e13	−1.07443	−0.537217	+	0.843444i	$0.680524\pi$
$968$	3.15703e12	0.115569
$969$	0	0
$970$	0	0
$971$	−1.10119e13	−0.397536	−0.198768	−	0.980047i	$-0.563694\pi$
$971$	−1.10119e13	−0.397536	−0.198768	+	0.980047i	$0.563694\pi$
$972$	0	0
$973$	−3.12711e13	−1.11850
$974$	1.01083e14	3.59885
$975$	0	0
$976$	−5.60709e11	−0.0197794
$977$	8.77989e12	0.308293	0.154146	−	0.988048i	$-0.450737\pi$
$977$	8.77989e12	0.308293	0.154146	+	0.988048i	$0.450737\pi$
$978$	0	0
$979$	1.02560e13	0.356826
$980$	0	0
$981$	0	0
$982$	2.36073e13	0.810113
$983$	−5.68473e13	−1.94186	−0.970932	−	0.239355i	$-0.923064\pi$
$983$	−5.68473e13	−1.94186	−0.970932	+	0.239355i	$0.923064\pi$
$984$	0	0
$985$	0	0
$986$	−4.58612e12	−0.154525
$987$	0	0
$988$	5.77246e12	0.192732
$989$	4.05599e13	1.34807
$990$	0	0
$991$	4.32479e13	1.42440	0.712202	−	0.701974i	$-0.247698\pi$
$991$	4.32479e13	1.42440	0.712202	+	0.701974i	$0.247698\pi$
$992$	2.18942e12	0.0717837
$993$	0	0
$994$	−4.43844e13	−1.44209
$995$	0	0
$996$	0	0
$997$	−5.30139e13	−1.69927	−0.849633	−	0.527374i	$-0.823177\pi$
$997$	−5.30139e13	−1.69927	−0.849633	+	0.527374i	$0.823177\pi$
$998$	−4.45735e13	−1.42229
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.j.1.2		2
3.2	odd	2		75.10.a.g.1.1		2
5.2	odd	4		225.10.b.g.199.4		4
5.3	odd	4		225.10.b.g.199.1		4
5.4	even	2		45.10.a.e.1.1		2
15.2	even	4		75.10.b.e.49.1		4
15.8	even	4		75.10.b.e.49.4		4
15.14	odd	2		15.10.a.c.1.2	✓	2
60.59	even	2		240.10.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.c.1.2	✓	2	15.14	odd	2
45.10.a.e.1.1		2	5.4	even	2
75.10.a.g.1.1		2	3.2	odd	2
75.10.b.e.49.1		4	15.2	even	4
75.10.b.e.49.4		4	15.8	even	4
225.10.a.j.1.2		2	1.1	even	1	trivial
225.10.b.g.199.1		4	5.3	odd	4
225.10.b.g.199.4		4	5.2	odd	4
240.10.a.m.1.2		2	60.59	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+43.8839 q^{2} +1413.79 q^{4} +7861.50 q^{7} +39574.2 q^{8} +O(q^{10})\)	\(q+43.8839 q^{2} +1413.79 q^{4} +7861.50 q^{7} +39574.2 q^{8} +49373.3 q^{11} -24250.7 q^{13} +344993. q^{14} +1.01281e6 q^{16} +268222. q^{17} -168364. q^{19} +2.16669e6 q^{22} -2.12200e6 q^{23} -1.06422e6 q^{26} +1.11145e7 q^{28} -389624. q^{29} +90532.2 q^{31} +2.41838e7 q^{32} +1.17706e7 q^{34} +3.31991e6 q^{37} -7.38848e6 q^{38} -2.32694e7 q^{41} -1.91140e7 q^{43} +6.98036e7 q^{44} -9.31215e7 q^{46} +6.28153e7 q^{47} +2.14495e7 q^{49} -3.42855e7 q^{52} -180207. q^{53} +3.11112e8 q^{56} -1.70982e7 q^{58} -3.84564e7 q^{59} -553620. q^{61} +3.97290e6 q^{62} +5.42724e8 q^{64} +2.39163e8 q^{67} +3.79211e8 q^{68} -1.28653e8 q^{71} +2.39376e8 q^{73} +1.45690e8 q^{74} -2.38033e8 q^{76} +3.88148e8 q^{77} -5.28027e8 q^{79} -1.02115e9 q^{82} +2.12210e8 q^{83} -8.38797e8 q^{86} +1.95391e9 q^{88} +2.07724e8 q^{89} -1.90647e8 q^{91} -3.00007e9 q^{92} +2.75658e9 q^{94} -1.70780e9 q^{97} +9.41288e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 19 q^{2} + 1521 q^{4} + 11872 q^{7} + 49647 q^{8}+O(q^{10}) \) 2 * q + 19 * q^2 + 1521 * q^4 + 11872 * q^7 + 49647 * q^8	\( 2 q + 19 q^{2} + 1521 q^{4} + 11872 q^{7} + 49647 q^{8} - 35488 q^{11} - 143676 q^{13} + 245196 q^{14} + 707265 q^{16} + 385156 q^{17} - 403296 q^{19} + 4278368 q^{22} + 223704 q^{23} + 1907546 q^{26} + 11544484 q^{28} + 74572 q^{29} - 5027128 q^{31} + 26629583 q^{32} + 8860882 q^{34} - 5373628 q^{37} - 1542476 q^{38} - 14211332 q^{41} - 27748920 q^{43} + 60705952 q^{44} - 151491648 q^{46} + 95966440 q^{47} - 2819950 q^{49} - 47088706 q^{52} - 64305596 q^{53} + 351509340 q^{56} - 28649198 q^{58} - 187863136 q^{59} + 154080060 q^{61} + 131320056 q^{62} + 638301089 q^{64} - 33592376 q^{67} + 391747238 q^{68} + 228270976 q^{71} + 33122316 q^{73} + 362019226 q^{74} - 263218724 q^{76} + 47811456 q^{77} - 932406760 q^{79} - 1246549646 q^{82} + 207040152 q^{83} - 623926708 q^{86} + 1099114848 q^{88} - 224518164 q^{89} - 669602528 q^{91} - 2748592992 q^{92} + 1931650816 q^{94} - 387134596 q^{97} + 1545205739 q^{98}+O(q^{100}) \) 2 * q + 19 * q^2 + 1521 * q^4 + 11872 * q^7 + 49647 * q^8 - 35488 * q^11 - 143676 * q^13 + 245196 * q^14 + 707265 * q^16 + 385156 * q^17 - 403296 * q^19 + 4278368 * q^22 + 223704 * q^23 + 1907546 * q^26 + 11544484 * q^28 + 74572 * q^29 - 5027128 * q^31 + 26629583 * q^32 + 8860882 * q^34 - 5373628 * q^37 - 1542476 * q^38 - 14211332 * q^41 - 27748920 * q^43 + 60705952 * q^44 - 151491648 * q^46 + 95966440 * q^47 - 2819950 * q^49 - 47088706 * q^52 - 64305596 * q^53 + 351509340 * q^56 - 28649198 * q^58 - 187863136 * q^59 + 154080060 * q^61 + 131320056 * q^62 + 638301089 * q^64 - 33592376 * q^67 + 391747238 * q^68 + 228270976 * q^71 + 33122316 * q^73 + 362019226 * q^74 - 263218724 * q^76 + 47811456 * q^77 - 932406760 * q^79 - 1246549646 * q^82 + 207040152 * q^83 - 623926708 * q^86 + 1099114848 * q^88 - 224518164 * q^89 - 669602528 * q^91 - 2748592992 * q^92 + 1931650816 * q^94 - 387134596 * q^97 + 1545205739 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more