LMFDB - Embedded newform 1225.4.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
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} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.67516$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.67516 q^{2} +2.49396 q^{3} -5.19383 q^{4} -4.17779 q^{6} +22.1018 q^{8} -20.7802 q^{9} -57.5880 q^{11} -12.9532 q^{12} +45.5159 q^{13} +4.52655 q^{16} -92.0051 q^{17} +34.8101 q^{18} -125.177 q^{19} +96.4692 q^{22} +158.496 q^{23} +55.1211 q^{24} -76.2466 q^{26} -119.162 q^{27} -40.1708 q^{29} -49.5590 q^{31} -184.397 q^{32} -143.622 q^{33} +154.123 q^{34} +107.929 q^{36} +231.307 q^{37} +209.692 q^{38} +113.515 q^{39} -169.556 q^{41} -147.428 q^{43} +299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +11.2890 q^{48} -229.457 q^{51} -236.402 q^{52} +268.647 q^{53} +199.615 q^{54} -312.187 q^{57} +67.2926 q^{58} +240.843 q^{59} -90.4579 q^{61} +83.0194 q^{62} +272.683 q^{64} +240.591 q^{66} -406.498 q^{67} +477.859 q^{68} +395.283 q^{69} +330.782 q^{71} -459.279 q^{72} -546.255 q^{73} -387.477 q^{74} +650.149 q^{76} -190.156 q^{78} -25.3087 q^{79} +263.879 q^{81} +284.034 q^{82} -376.255 q^{83} +246.965 q^{86} -100.184 q^{87} -1272.80 q^{88} -1026.44 q^{89} -823.203 q^{92} -123.598 q^{93} -112.291 q^{94} -459.879 q^{96} +942.660 q^{97} +1196.69 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} - 10 q^{3} + 18 q^{4} - 6 q^{6} + 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 74 q^{16} - 238 q^{17} - 2 q^{18} + 36 q^{19} + 358 q^{22} + 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 - 10 * q^3 + 18 * q^4 - 6 * q^6 + 42 * q^8 + 23 * q^9 + 42 * q^11 - 136 * q^12 - 34 * q^13 + 74 * q^16 - 238 * q^17 - 2 * q^18 + 36 * q^19 + 358 * q^22 + 152 * q^23 + 36 * q^24 + 310 * q^26 - 334 * q^27 - 44 * q^29 - 60 * q^31 + 710 * q^32 - 426 * q^33 + 482 * q^34 - 210 * q^36 + 312 * q^37 - 280 * q^38 - 106 * q^39 + 426 * q^41 + 304 * q^43 + 712 * q^44 + 88 * q^46 - 370 * q^47 + 696 * q^48 + 638 * q^51 + 1156 * q^52 + 976 * q^53 - 498 * q^54 - 588 * q^57 - 2722 * q^58 + 432 * q^59 + 442 * q^61 + 956 * q^62 + 1362 * q^64 - 574 * q^66 - 804 * q^67 + 420 * q^68 + 2404 * q^69 + 440 * q^71 - 3150 * q^72 - 564 * q^73 - 1512 * q^74 + 1336 * q^76 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 + 3480 * q^82 - 1656 * q^83 + 1216 * q^86 + 1674 * q^87 - 1092 * q^88 - 746 * q^89 + 572 * q^92 + 676 * q^93 + 826 * q^94 - 2040 * q^96 - 518 * q^97 + 2652 * q^99

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 $

$2$	−1.67516	−0.592259	−0.296130	−	0.955148i	$-0.595696\pi$
$2$	−1.67516	−0.592259	−0.296130	+	0.955148i	$0.595696\pi$
$3$	2.49396	0.479963	0.239982	−	0.970777i	$-0.422859\pi$
$3$	2.49396	0.479963	0.239982	+	0.970777i	$0.422859\pi$
$4$	−5.19383	−0.649229
$5$	0	0
$5$	0	0
$6$	−4.17779	−0.284263
$7$	0	0
$7$	0	0
$8$	22.1018	0.976771
$9$	−20.7802	−0.769635
$10$	0	0
$11$	−57.5880	−1.57849	−0.789247	−	0.614076i	$-0.789529\pi$
$11$	−57.5880	−1.57849	−0.789247	+	0.614076i	$0.789529\pi$
$12$	−12.9532	−0.311606
$13$	45.5159	0.971066	0.485533	−	0.874218i	$-0.338626\pi$
$13$	45.5159	0.971066	0.485533	+	0.874218i	$0.338626\pi$
$14$	0	0
$15$	0	0
$16$	4.52655	0.0707273
$17$	−92.0051	−1.31262	−0.656309	−	0.754492i	$-0.727883\pi$
$17$	−92.0051	−1.31262	−0.656309	+	0.754492i	$0.727883\pi$
$18$	34.8101	0.455824
$19$	−125.177	−1.51145	−0.755726	−	0.654888i	$-0.772716\pi$
$19$	−125.177	−1.51145	−0.755726	+	0.654888i	$0.772716\pi$
$20$	0	0
$21$	0	0
$22$	96.4692	0.934878
$23$	158.496	1.43690	0.718451	−	0.695578i	$-0.244851\pi$
$23$	158.496	1.43690	0.718451	+	0.695578i	$0.244851\pi$
$24$	55.1211	0.468814
$25$	0	0
$26$	−76.2466	−0.575123
$27$	−119.162	−0.849360
$28$	0	0
$29$	−40.1708	−0.257225	−0.128613	−	0.991695i	$-0.541052\pi$
$29$	−40.1708	−0.257225	−0.128613	+	0.991695i	$0.541052\pi$
$30$	0	0
$31$	−49.5590	−0.287131	−0.143566	−	0.989641i	$-0.545857\pi$
$31$	−49.5590	−0.287131	−0.143566	+	0.989641i	$0.545857\pi$
$32$	−184.397	−1.01866
$33$	−143.622	−0.757619
$34$	154.123	0.777410
$35$	0	0
$36$	107.929	0.499670
$37$	231.307	1.02775	0.513874	−	0.857866i	$-0.328210\pi$
$37$	231.307	1.02775	0.513874	+	0.857866i	$0.328210\pi$
$38$	209.692	0.895171
$39$	113.515	0.466076
$40$	0	0
$41$	−169.556	−0.645859	−0.322929	−	0.946423i	$-0.604668\pi$
$41$	−169.556	−0.645859	−0.322929	+	0.946423i	$0.604668\pi$
$42$	0	0
$43$	−147.428	−0.522849	−0.261425	−	0.965224i	$-0.584192\pi$
$43$	−147.428	−0.522849	−0.261425	+	0.965224i	$0.584192\pi$
$44$	299.102	1.02480
$45$	0	0
$46$	−265.507	−0.851018
$47$	67.0327	0.208037	0.104018	−	0.994575i	$-0.466830\pi$
$47$	67.0327	0.208037	0.104018	+	0.994575i	$0.466830\pi$
$48$	11.2890	0.0339465
$49$	0	0
$50$	0	0
$51$	−229.457	−0.630008
$52$	−236.402	−0.630444
$53$	268.647	0.696254	0.348127	−	0.937447i	$-0.386818\pi$
$53$	268.647	0.696254	0.348127	+	0.937447i	$0.386818\pi$
$54$	199.615	0.503041
$55$	0	0
$56$	0	0
$57$	−312.187	−0.725441
$58$	67.2926	0.152344
$59$	240.843	0.531442	0.265721	−	0.964050i	$-0.414390\pi$
$59$	240.843	0.531442	0.265721	+	0.964050i	$0.414390\pi$
$60$	0	0
$61$	−90.4579	−0.189868	−0.0949340	−	0.995484i	$-0.530264\pi$
$61$	−90.4579	−0.189868	−0.0949340	+	0.995484i	$0.530264\pi$
$62$	83.0194	0.170056
$63$	0	0
$64$	272.683	0.532583
$65$	0	0
$66$	240.591	0.448707
$67$	−406.498	−0.741218	−0.370609	−	0.928789i	$-0.620851\pi$
$67$	−406.498	−0.741218	−0.370609	+	0.928789i	$0.620851\pi$
$68$	477.859	0.852190
$69$	395.283	0.689660
$70$	0	0
$71$	330.782	0.552910	0.276455	−	0.961027i	$-0.410840\pi$
$71$	330.782	0.552910	0.276455	+	0.961027i	$0.410840\pi$
$72$	−459.279	−0.751758
$73$	−546.255	−0.875812	−0.437906	−	0.899021i	$-0.644280\pi$
$73$	−546.255	−0.875812	−0.437906	+	0.899021i	$0.644280\pi$
$74$	−387.477	−0.608693
$75$	0	0
$76$	650.149	0.981279
$77$	0	0
$78$	−190.156	−0.276038
$79$	−25.3087	−0.0360436	−0.0180218	−	0.999838i	$-0.505737\pi$
$79$	−25.3087	−0.0360436	−0.0180218	+	0.999838i	$0.505737\pi$
$80$	0	0
$81$	263.879	0.361974
$82$	284.034	0.382516
$83$	−376.255	−0.497582	−0.248791	−	0.968557i	$-0.580033\pi$
$83$	−376.255	−0.497582	−0.248791	+	0.968557i	$0.580033\pi$
$84$	0	0
$85$	0	0
$86$	246.965	0.309662
$87$	−100.184	−0.123459
$88$	−1272.80	−1.54183
$89$	−1026.44	−1.22250	−0.611248	−	0.791439i	$-0.709332\pi$
$89$	−1026.44	−1.22250	−0.611248	+	0.791439i	$0.709332\pi$
$90$	0	0
$91$	0	0
$92$	−823.203	−0.932878
$93$	−123.598	−0.137812
$94$	−112.291	−0.123212
$95$	0	0
$96$	−459.879	−0.488919
$97$	942.660	0.986728	0.493364	−	0.869823i	$-0.335767\pi$
$97$	942.660	0.986728	0.493364	+	0.869823i	$0.335767\pi$
$98$	0	0
$99$	1196.69	1.21487
$100$	0	0
$101$	604.617	0.595659	0.297830	−	0.954619i	$-0.403737\pi$
$101$	604.617	0.595659	0.297830	+	0.954619i	$0.403737\pi$
$102$	384.378	0.373128
$103$	300.967	0.287914	0.143957	−	0.989584i	$-0.454017\pi$
$103$	300.967	0.287914	0.143957	+	0.989584i	$0.454017\pi$
$104$	1005.98	0.948509
$105$	0	0
$106$	−450.027	−0.412363
$107$	1511.66	1.36577	0.682886	−	0.730525i	$-0.260724\pi$
$107$	1511.66	1.36577	0.682886	+	0.730525i	$0.260724\pi$
$108$	618.907	0.551429
$109$	−1767.09	−1.55281	−0.776406	−	0.630233i	$-0.782959\pi$
$109$	−1767.09	−1.55281	−0.776406	+	0.630233i	$0.782959\pi$
$110$	0	0
$111$	576.871	0.493281
$112$	0	0
$113$	1045.27	0.870182	0.435091	−	0.900387i	$-0.356716\pi$
$113$	1045.27	0.870182	0.435091	+	0.900387i	$0.356716\pi$
$114$	522.963	0.429649
$115$	0	0
$116$	208.640	0.166998
$117$	−945.828	−0.747366
$118$	−403.451	−0.314751
$119$	0	0
$120$	0	0
$121$	1985.38	1.49164
$122$	151.532	0.112451
$123$	−422.866	−0.309988
$124$	257.401	0.186414
$125$	0	0
$126$	0	0
$127$	260.727	0.182171	0.0910857	−	0.995843i	$-0.470966\pi$
$127$	260.727	0.182171	0.0910857	+	0.995843i	$0.470966\pi$
$128$	1018.39	0.703233
$129$	−367.679	−0.250948
$130$	0	0
$131$	723.522	0.482553	0.241276	−	0.970456i	$-0.422434\pi$
$131$	723.522	0.482553	0.241276	+	0.970456i	$0.422434\pi$
$132$	745.950	0.491868
$133$	0	0
$134$	680.950	0.438993
$135$	0	0
$136$	−2033.48	−1.28213
$137$	773.693	0.482490	0.241245	−	0.970464i	$-0.422444\pi$
$137$	773.693	0.482490	0.241245	+	0.970464i	$0.422444\pi$
$138$	−662.164	−0.408457
$139$	2952.97	1.80192	0.900961	−	0.433899i	$-0.142863\pi$
$139$	2952.97	1.80192	0.900961	+	0.433899i	$0.142863\pi$
$140$	0	0
$141$	167.177	0.0998499
$142$	−554.114	−0.327466
$143$	−2621.17	−1.53282
$144$	−94.0624	−0.0544343
$145$	0	0
$146$	915.066	0.518708
$147$	0	0
$148$	−1201.37	−0.667244
$149$	2514.00	1.38225	0.691124	−	0.722736i	$-0.257116\pi$
$149$	2514.00	1.38225	0.691124	+	0.722736i	$0.257116\pi$
$150$	0	0
$151$	101.052	0.0544605	0.0272302	−	0.999629i	$-0.491331\pi$
$151$	101.052	0.0544605	0.0272302	+	0.999629i	$0.491331\pi$
$152$	−2766.64	−1.47634
$153$	1911.88	1.01024
$154$	0	0
$155$	0	0
$156$	−589.578	−0.302590
$157$	−2338.35	−1.18867	−0.594333	−	0.804219i	$-0.702584\pi$
$157$	−2338.35	−1.18867	−0.594333	+	0.804219i	$0.702584\pi$
$158$	42.3961	0.0213472
$159$	669.995	0.334176
$160$	0	0
$161$	0	0
$162$	−442.040	−0.214383
$163$	−1325.20	−0.636798	−0.318399	−	0.947957i	$-0.603145\pi$
$163$	−1325.20	−0.636798	−0.318399	+	0.947957i	$0.603145\pi$
$164$	880.646	0.419310
$165$	0	0
$166$	630.288	0.294698
$167$	−2086.20	−0.966675	−0.483338	−	0.875434i	$-0.660576\pi$
$167$	−2086.20	−0.966675	−0.483338	+	0.875434i	$0.660576\pi$
$168$	0	0
$169$	−125.299	−0.0570317
$170$	0	0
$171$	2601.20	1.16327
$172$	765.715	0.339449
$173$	−1918.19	−0.842990	−0.421495	−	0.906831i	$-0.638495\pi$
$173$	−1918.19	−0.842990	−0.421495	+	0.906831i	$0.638495\pi$
$174$	167.825	0.0731195
$175$	0	0
$176$	−260.675	−0.111643
$177$	600.652	0.255072
$178$	1719.45	0.724035
$179$	629.046	0.262665	0.131333	−	0.991338i	$-0.458074\pi$
$179$	629.046	0.262665	0.131333	+	0.991338i	$0.458074\pi$
$180$	0	0
$181$	2800.85	1.15020	0.575099	−	0.818084i	$-0.304964\pi$
$181$	2800.85	1.15020	0.575099	+	0.818084i	$0.304964\pi$
$182$	0	0
$183$	−225.599	−0.0911296
$184$	3503.05	1.40352
$185$	0	0
$186$	207.047	0.0816207
$187$	5298.39	2.07196
$188$	−348.157	−0.135063
$189$	0	0
$190$	0	0
$191$	740.255	0.280434	0.140217	−	0.990121i	$-0.455220\pi$
$191$	740.255	0.280434	0.140217	+	0.990121i	$0.455220\pi$
$192$	680.060	0.255620
$193$	4082.57	1.52264	0.761321	−	0.648375i	$-0.224551\pi$
$193$	4082.57	1.52264	0.761321	+	0.648375i	$0.224551\pi$
$194$	−1579.11	−0.584399
$195$	0	0
$196$	0	0
$197$	3414.89	1.23503	0.617515	−	0.786559i	$-0.288140\pi$
$197$	3414.89	1.23503	0.617515	+	0.786559i	$0.288140\pi$
$198$	−2004.65	−0.719515
$199$	3392.44	1.20846	0.604231	−	0.796809i	$-0.293480\pi$
$199$	3392.44	1.20846	0.604231	+	0.796809i	$0.293480\pi$
$200$	0	0
$201$	−1013.79	−0.355757
$202$	−1012.83	−0.352785
$203$	0	0
$204$	1191.76	0.409020
$205$	0	0
$206$	−504.169	−0.170520
$207$	−3293.58	−1.10589
$208$	206.030	0.0686809
$209$	7208.70	2.38582
$210$	0	0
$211$	3398.04	1.10867	0.554337	−	0.832292i	$-0.312972\pi$
$211$	3398.04	1.10867	0.554337	+	0.832292i	$0.312972\pi$
$212$	−1395.31	−0.452029
$213$	824.958	0.265376
$214$	−2532.28	−0.808892
$215$	0	0
$216$	−2633.69	−0.829630
$217$	0	0
$218$	2960.16	0.919667
$219$	−1362.34	−0.420358
$220$	0	0
$221$	−4187.70	−1.27464
$222$	−966.353	−0.292150
$223$	−182.611	−0.0548365	−0.0274183	−	0.999624i	$-0.508729\pi$
$223$	−182.611	−0.0548365	−0.0274183	+	0.999624i	$0.508729\pi$
$224$	0	0
$225$	0	0
$226$	−1750.99	−0.515373
$227$	3152.33	0.921707	0.460854	−	0.887476i	$-0.347543\pi$
$227$	3152.33	0.921707	0.460854	+	0.887476i	$0.347543\pi$
$228$	1621.45	0.470977
$229$	−6012.35	−1.73497	−0.867483	−	0.497466i	$-0.834264\pi$
$229$	−6012.35	−1.73497	−0.867483	+	0.497466i	$0.834264\pi$
$230$	0	0
$231$	0	0
$232$	−887.848	−0.251250
$233$	−940.660	−0.264484	−0.132242	−	0.991217i	$-0.542218\pi$
$233$	−940.660	−0.264484	−0.132242	+	0.991217i	$0.542218\pi$
$234$	1584.42	0.442635
$235$	0	0
$236$	−1250.90	−0.345027
$237$	−63.1188	−0.0172996
$238$	0	0
$239$	5158.82	1.39622	0.698109	−	0.715991i	$-0.254025\pi$
$239$	5158.82	1.39622	0.698109	+	0.715991i	$0.254025\pi$
$240$	0	0
$241$	463.836	0.123976	0.0619882	−	0.998077i	$-0.480256\pi$
$241$	463.836	0.123976	0.0619882	+	0.998077i	$0.480256\pi$
$242$	−3325.83	−0.883440
$243$	3875.47	1.02309
$244$	469.823	0.123268
$245$	0	0
$246$	708.370	0.183594
$247$	−5697.55	−1.46772
$248$	−1095.34	−0.280461
$249$	−938.365	−0.238821
$250$	0	0
$251$	−2290.25	−0.575934	−0.287967	−	0.957640i	$-0.592979\pi$
$251$	−2290.25	−0.575934	−0.287967	+	0.957640i	$0.592979\pi$
$252$	0	0
$253$	−9127.48	−2.26814
$254$	−436.760	−0.107893
$255$	0	0
$256$	−3887.43	−0.949079
$257$	802.202	0.194708	0.0973541	−	0.995250i	$-0.468962\pi$
$257$	802.202	0.194708	0.0973541	+	0.995250i	$0.468962\pi$
$258$	615.922	0.148627
$259$	0	0
$260$	0	0
$261$	834.756	0.197970
$262$	−1212.02	−0.285796
$263$	−286.978	−0.0672845	−0.0336423	−	0.999434i	$-0.510711\pi$
$263$	−286.978	−0.0672845	−0.0336423	+	0.999434i	$0.510711\pi$
$264$	−3174.31	−0.740020
$265$	0	0
$266$	0	0
$267$	−2559.90	−0.586753
$268$	2111.28	0.481221
$269$	−3561.22	−0.807180	−0.403590	−	0.914940i	$-0.632238\pi$
$269$	−3561.22	−0.807180	−0.403590	+	0.914940i	$0.632238\pi$
$270$	0	0
$271$	−1928.81	−0.432349	−0.216175	−	0.976355i	$-0.569358\pi$
$271$	−1928.81	−0.432349	−0.216175	+	0.976355i	$0.569358\pi$
$272$	−416.466	−0.0928380
$273$	0	0
$274$	−1296.06	−0.285759
$275$	0	0
$276$	−2053.04	−0.447747
$277$	6588.69	1.42916	0.714578	−	0.699556i	$-0.246619\pi$
$277$	6588.69	1.42916	0.714578	+	0.699556i	$0.246619\pi$
$278$	−4946.70	−1.06721
$279$	1029.84	0.220986
$280$	0	0
$281$	815.552	0.173138	0.0865689	−	0.996246i	$-0.472410\pi$
$281$	815.552	0.173138	0.0865689	+	0.996246i	$0.472410\pi$
$282$	−280.049	−0.0591371
$283$	6513.49	1.36815	0.684076	−	0.729411i	$-0.260206\pi$
$283$	6513.49	1.36815	0.684076	+	0.729411i	$0.260206\pi$
$284$	−1718.03	−0.358965
$285$	0	0
$286$	4390.89	0.907828
$287$	0	0
$288$	3831.80	0.783997
$289$	3551.94	0.722967
$290$	0	0
$291$	2350.96	0.473593
$292$	2837.16	0.568603
$293$	−435.520	−0.0868373	−0.0434186	−	0.999057i	$-0.513825\pi$
$293$	−435.520	−0.0868373	−0.0434186	+	0.999057i	$0.513825\pi$
$294$	0	0
$295$	0	0
$296$	5112.31	1.00387
$297$	6862.29	1.34071
$298$	−4211.36	−0.818650
$299$	7214.10	1.39533
$300$	0	0
$301$	0	0
$302$	−169.279	−0.0322547
$303$	1507.89	0.285895
$304$	−566.620	−0.106901
$305$	0	0
$306$	−3202.71	−0.598323
$307$	4915.99	0.913910	0.456955	−	0.889490i	$-0.348940\pi$
$307$	4915.99	0.913910	0.456955	+	0.889490i	$0.348940\pi$
$308$	0	0
$309$	750.601	0.138188
$310$	0	0
$311$	1831.11	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$311$	1831.11	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$312$	2508.89	0.455249
$313$	2442.96	0.441163	0.220582	−	0.975369i	$-0.429204\pi$
$313$	2442.96	0.441163	0.220582	+	0.975369i	$0.429204\pi$
$314$	3917.11	0.703998
$315$	0	0
$316$	131.449	0.0234006
$317$	1666.19	0.295214	0.147607	−	0.989046i	$-0.452843\pi$
$317$	1666.19	0.295214	0.147607	+	0.989046i	$0.452843\pi$
$318$	−1122.35	−0.197919
$319$	2313.36	0.406029
$320$	0	0
$321$	3770.02	0.655521
$322$	0	0
$323$	11516.9	1.98396
$324$	−1370.54	−0.235004
$325$	0	0
$326$	2219.93	0.377149
$327$	−4407.05	−0.745292
$328$	−3747.50	−0.630856
$329$	0	0
$330$	0	0
$331$	−5466.38	−0.907732	−0.453866	−	0.891070i	$-0.649956\pi$
$331$	−5466.38	−0.907732	−0.453866	+	0.891070i	$0.649956\pi$
$332$	1954.20	0.323045
$333$	−4806.60	−0.790991
$334$	3494.72	0.572522
$335$	0	0
$336$	0	0
$337$	10650.5	1.72157	0.860784	−	0.508970i	$-0.169973\pi$
$337$	10650.5	1.72157	0.860784	+	0.508970i	$0.169973\pi$
$338$	209.896	0.0337776
$339$	2606.86	0.417655
$340$	0	0
$341$	2854.01	0.453235
$342$	−4357.43	−0.688956
$343$	0	0
$344$	−3258.42	−0.510704
$345$	0	0
$346$	3213.28	0.499269
$347$	4019.13	0.621782	0.310891	−	0.950446i	$-0.399373\pi$
$347$	4019.13	0.621782	0.310891	+	0.950446i	$0.399373\pi$
$348$	520.341	0.0801529
$349$	−10544.9	−1.61735	−0.808674	−	0.588256i	$-0.799815\pi$
$349$	−10544.9	−1.61735	−0.808674	+	0.588256i	$0.799815\pi$
$350$	0	0
$351$	−5423.76	−0.824784
$352$	10619.1	1.60795
$353$	−2959.98	−0.446300	−0.223150	−	0.974784i	$-0.571634\pi$
$353$	−2959.98	−0.446300	−0.223150	+	0.974784i	$0.571634\pi$
$354$	−1006.19	−0.151069
$355$	0	0
$356$	5331.14	0.793680
$357$	0	0
$358$	−1053.75	−0.155566
$359$	2170.17	0.319045	0.159523	−	0.987194i	$-0.449005\pi$
$359$	2170.17	0.319045	0.159523	+	0.987194i	$0.449005\pi$
$360$	0	0
$361$	8810.30	1.28449
$362$	−4691.88	−0.681215
$363$	4951.46	0.715934
$364$	0	0
$365$	0	0
$366$	377.914	0.0539724
$367$	1252.20	0.178105	0.0890523	−	0.996027i	$-0.471616\pi$
$367$	1252.20	0.178105	0.0890523	+	0.996027i	$0.471616\pi$
$368$	717.441	0.101628
$369$	3523.40	0.497076
$370$	0	0
$371$	0	0
$372$	641.949	0.0894718
$373$	4646.02	0.644938	0.322469	−	0.946580i	$-0.395487\pi$
$373$	4646.02	0.644938	0.322469	+	0.946580i	$0.395487\pi$
$374$	−8875.66	−1.22714
$375$	0	0
$376$	1481.54	0.203204
$377$	−1828.41	−0.249783
$378$	0	0
$379$	1434.84	0.194466	0.0972331	−	0.995262i	$-0.469001\pi$
$379$	1434.84	0.194466	0.0972331	+	0.995262i	$0.469001\pi$
$380$	0	0
$381$	650.242	0.0874355
$382$	−1240.05	−0.166090
$383$	13216.7	1.76330	0.881649	−	0.471905i	$-0.156434\pi$
$383$	13216.7	1.76330	0.881649	+	0.471905i	$0.156434\pi$
$384$	2539.82	0.337526
$385$	0	0
$386$	−6838.97	−0.901799
$387$	3063.57	0.402403
$388$	−4896.02	−0.640612
$389$	−7755.01	−1.01078	−0.505391	−	0.862890i	$-0.668652\pi$
$389$	−7755.01	−1.01078	−0.505391	+	0.862890i	$0.668652\pi$
$390$	0	0
$391$	−14582.5	−1.88610
$392$	0	0
$393$	1804.44	0.231607
$394$	−5720.49	−0.731457
$395$	0	0
$396$	−6215.40	−0.788726
$397$	3560.83	0.450158	0.225079	−	0.974341i	$-0.427736\pi$
$397$	3560.83	0.450158	0.225079	+	0.974341i	$0.427736\pi$
$398$	−5682.89	−0.715723
$399$	0	0
$400$	0	0
$401$	−5430.61	−0.676288	−0.338144	−	0.941094i	$-0.609799\pi$
$401$	−5430.61	−0.676288	−0.338144	+	0.941094i	$0.609799\pi$
$402$	1698.26	0.210701
$403$	−2255.73	−0.278823
$404$	−3140.28	−0.386719
$405$	0	0
$406$	0	0
$407$	−13320.5	−1.62229
$408$	−5071.42	−0.615374
$409$	9698.79	1.17255	0.586277	−	0.810111i	$-0.300593\pi$
$409$	9698.79	1.17255	0.586277	+	0.810111i	$0.300593\pi$
$410$	0	0
$411$	1929.56	0.231577
$412$	−1563.17	−0.186922
$413$	0	0
$414$	5517.27	0.654974
$415$	0	0
$416$	−8393.01	−0.989186
$417$	7364.58	0.864856
$418$	−12075.7	−1.41302
$419$	−13830.9	−1.61261	−0.806307	−	0.591498i	$-0.798537\pi$
$419$	−13830.9	−1.61261	−0.806307	+	0.591498i	$0.798537\pi$
$420$	0	0
$421$	16703.0	1.93362	0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0	1.93362	0.966810	+	0.255498i	$0.0822393\pi$
$422$	−5692.26	−0.656623
$423$	−1392.95	−0.160112
$424$	5937.58	0.680081
$425$	0	0
$426$	−1381.94	−0.157172
$427$	0	0
$428$	−7851.31	−0.886699
$429$	−6537.10	−0.735698
$430$	0	0
$431$	8174.07	0.913530	0.456765	−	0.889588i	$-0.349008\pi$
$431$	8174.07	0.913530	0.456765	+	0.889588i	$0.349008\pi$
$432$	−539.392	−0.0600729
$433$	−14222.8	−1.57853	−0.789267	−	0.614051i	$-0.789539\pi$
$433$	−14222.8	−1.57853	−0.789267	+	0.614051i	$0.789539\pi$
$434$	0	0
$435$	0	0
$436$	9177.96	1.00813
$437$	−19840.1	−2.17181
$438$	2282.14	0.248961
$439$	−5537.38	−0.602016	−0.301008	−	0.953622i	$-0.597323\pi$
$439$	−5537.38	−0.602016	−0.301008	+	0.953622i	$0.597323\pi$
$440$	0	0
$441$	0	0
$442$	7015.07	0.754916
$443$	3974.09	0.426218	0.213109	−	0.977028i	$-0.431641\pi$
$443$	3974.09	0.426218	0.213109	+	0.977028i	$0.431641\pi$
$444$	−2996.17	−0.320252
$445$	0	0
$446$	305.903	0.0324774
$447$	6269.82	0.663428
$448$	0	0
$449$	−15243.1	−1.60216	−0.801078	−	0.598559i	$-0.795740\pi$
$449$	−15243.1	−1.60216	−0.801078	+	0.598559i	$0.795740\pi$
$450$	0	0
$451$	9764.40	1.01948
$452$	−5428.95	−0.564947
$453$	252.021	0.0261390
$454$	−5280.67	−0.545890
$455$	0	0
$456$	−6899.89	−0.708590
$457$	−10768.9	−1.10229	−0.551145	−	0.834410i	$-0.685809\pi$
$457$	−10768.9	−1.10229	−0.551145	+	0.834410i	$0.685809\pi$
$458$	10071.7	1.02755
$459$	10963.5	1.11489
$460$	0	0
$461$	332.605	0.0336029	0.0168015	−	0.999859i	$-0.494652\pi$
$461$	332.605	0.0336029	0.0168015	+	0.999859i	$0.494652\pi$
$462$	0	0
$463$	−8205.35	−0.823618	−0.411809	−	0.911270i	$-0.635103\pi$
$463$	−8205.35	−0.823618	−0.411809	+	0.911270i	$0.635103\pi$
$464$	−181.835	−0.0181929
$465$	0	0
$466$	1575.76	0.156643
$467$	167.628	0.0166100	0.00830501	−	0.999966i	$-0.497356\pi$
$467$	167.628	0.0166100	0.00830501	+	0.999966i	$0.497356\pi$
$468$	4912.47	0.485212
$469$	0	0
$470$	0	0
$471$	−5831.75	−0.570515
$472$	5323.06	0.519097
$473$	8490.07	0.825315
$474$	105.734	0.0102459
$475$	0	0
$476$	0	0
$477$	−5582.52	−0.535862
$478$	−8641.86	−0.826924
$479$	−6628.58	−0.632292	−0.316146	−	0.948711i	$-0.602389\pi$
$479$	−6628.58	−0.632292	−0.316146	+	0.948711i	$0.602389\pi$
$480$	0	0
$481$	10528.2	0.998010
$482$	−777.001	−0.0734262
$483$	0	0
$484$	−10311.7	−0.968419
$485$	0	0
$486$	−6492.05	−0.605937
$487$	−20641.6	−1.92065	−0.960327	−	0.278875i	$-0.910039\pi$
$487$	−20641.6	−1.92065	−0.960327	+	0.278875i	$0.910039\pi$
$488$	−1999.28	−0.185458
$489$	−3305.01	−0.305640
$490$	0	0
$491$	−16710.8	−1.53594	−0.767972	−	0.640484i	$-0.778734\pi$
$491$	−16710.8	−1.53594	−0.767972	+	0.640484i	$0.778734\pi$
$492$	2196.30	0.201253
$493$	3695.92	0.337639
$494$	9544.32	0.869270
$495$	0	0
$496$	−224.331	−0.0203080
$497$	0	0
$498$	1571.91	0.141444
$499$	−13728.7	−1.23162	−0.615812	−	0.787893i	$-0.711172\pi$
$499$	−13728.7	−1.23162	−0.615812	+	0.787893i	$0.711172\pi$
$500$	0	0
$501$	−5202.90	−0.463969
$502$	3836.54	0.341102
$503$	19523.7	1.73065	0.865326	−	0.501209i	$-0.167111\pi$
$503$	19523.7	1.73065	0.865326	+	0.501209i	$0.167111\pi$
$504$	0	0
$505$	0	0
$506$	15290.0	1.34333
$507$	−312.490	−0.0273731
$508$	−1354.17	−0.118271
$509$	8688.17	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$509$	8688.17	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$510$	0	0
$511$	0	0
$512$	−1635.04	−0.141131
$513$	14916.3	1.28377
$514$	−1343.82	−0.115318
$515$	0	0
$516$	1909.66	0.162923
$517$	−3860.28	−0.328385
$518$	0	0
$519$	−4783.89	−0.404604
$520$	0	0
$521$	−6771.36	−0.569402	−0.284701	−	0.958616i	$-0.591894\pi$
$521$	−6771.36	−0.569402	−0.284701	+	0.958616i	$0.591894\pi$
$522$	−1398.35	−0.117249
$523$	1365.89	0.114200	0.0570998	−	0.998368i	$-0.481815\pi$
$523$	1365.89	0.114200	0.0570998	+	0.998368i	$0.481815\pi$
$524$	−3757.85	−0.313287
$525$	0	0
$526$	480.735	0.0398499
$527$	4559.68	0.376894
$528$	−650.113	−0.0535844
$529$	12954.0	1.06469
$530$	0	0
$531$	−5004.75	−0.409016
$532$	0	0
$533$	−7717.51	−0.627171
$534$	4288.24	0.347510
$535$	0	0
$536$	−8984.34	−0.724001
$537$	1568.82	0.126070
$538$	5965.62	0.478060
$539$	0	0
$540$	0	0
$541$	−23250.1	−1.84769	−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1	−1.84769	−0.923844	+	0.382770i	$0.874970\pi$
$542$	3231.06	0.256063
$543$	6985.22	0.552052
$544$	16965.5	1.33711
$545$	0	0
$546$	0	0
$547$	−11552.7	−0.903033	−0.451516	−	0.892263i	$-0.649117\pi$
$547$	−11552.7	−0.903033	−0.451516	+	0.892263i	$0.649117\pi$
$548$	−4018.43	−0.313246
$549$	1879.73	0.146129
$550$	0	0
$551$	5028.47	0.388784
$552$	8736.48	0.673640
$553$	0	0
$554$	−11037.1	−0.846430
$555$	0	0
$556$	−15337.2	−1.16986
$557$	−16406.2	−1.24803	−0.624014	−	0.781413i	$-0.714499\pi$
$557$	−16406.2	−1.24803	−0.624014	+	0.781413i	$0.714499\pi$
$558$	−1725.16	−0.130881
$559$	−6710.31	−0.507721
$560$	0	0
$561$	13214.0	0.994465
$562$	−1366.18	−0.102542
$563$	−13631.9	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$563$	−13631.9	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$564$	−868.289	−0.0648255
$565$	0	0
$566$	−10911.2	−0.810300
$567$	0	0
$568$	7310.88	0.540067
$569$	−3086.83	−0.227428	−0.113714	−	0.993514i	$-0.536275\pi$
$569$	−3086.83	−0.227428	−0.113714	+	0.993514i	$0.536275\pi$
$570$	0	0
$571$	−3258.06	−0.238784	−0.119392	−	0.992847i	$-0.538095\pi$
$571$	−3258.06	−0.238784	−0.119392	+	0.992847i	$0.538095\pi$
$572$	13613.9	0.995152
$573$	1846.17	0.134598
$574$	0	0
$575$	0	0
$576$	−5666.39	−0.409895
$577$	−23758.4	−1.71417	−0.857083	−	0.515178i	$-0.827726\pi$
$577$	−23758.4	−1.71417	−0.857083	+	0.515178i	$0.827726\pi$
$578$	−5950.07	−0.428184
$579$	10181.8	0.730812
$580$	0	0
$581$	0	0
$582$	−3938.23	−0.280490
$583$	−15470.8	−1.09903
$584$	−12073.2	−0.855468
$585$	0	0
$586$	729.566	0.0514302
$587$	−596.893	−0.0419701	−0.0209850	−	0.999780i	$-0.506680\pi$
$587$	−596.893	−0.0419701	−0.0209850	+	0.999780i	$0.506680\pi$
$588$	0	0
$589$	6203.66	0.433985
$590$	0	0
$591$	8516.60	0.592768
$592$	1047.02	0.0726898
$593$	−19496.3	−1.35012	−0.675058	−	0.737765i	$-0.735881\pi$
$593$	−19496.3	−1.35012	−0.675058	+	0.737765i	$0.735881\pi$
$594$	−11495.5	−0.794047
$595$	0	0
$596$	−13057.3	−0.897396
$597$	8460.62	0.580017
$598$	−12084.8	−0.826395
$599$	3797.02	0.259001	0.129501	−	0.991579i	$-0.458663\pi$
$599$	3797.02	0.259001	0.129501	+	0.991579i	$0.458663\pi$
$600$	0	0
$601$	−5789.33	−0.392931	−0.196466	−	0.980511i	$-0.562946\pi$
$601$	−5789.33	−0.392931	−0.196466	+	0.980511i	$0.562946\pi$
$602$	0	0
$603$	8447.09	0.570468
$604$	−524.850	−0.0353573
$605$	0	0
$606$	−2525.96	−0.169324
$607$	−18536.4	−1.23949	−0.619745	−	0.784803i	$-0.712764\pi$
$607$	−18536.4	−1.23949	−0.619745	+	0.784803i	$0.712764\pi$
$608$	23082.3	1.53966
$609$	0	0
$610$	0	0
$611$	3051.06	0.202017
$612$	−9929.98	−0.655876
$613$	2163.47	0.142548	0.0712738	−	0.997457i	$-0.477294\pi$
$613$	2163.47	0.142548	0.0712738	+	0.997457i	$0.477294\pi$
$614$	−8235.08	−0.541272
$615$	0	0
$616$	0	0
$617$	22964.9	1.49843	0.749215	−	0.662327i	$-0.230431\pi$
$617$	22964.9	1.49843	0.749215	+	0.662327i	$0.230431\pi$
$618$	−1257.38	−0.0818433
$619$	1386.67	0.0900401	0.0450200	−	0.998986i	$-0.485665\pi$
$619$	1386.67	0.0900401	0.0450200	+	0.998986i	$0.485665\pi$
$620$	0	0
$621$	−18886.7	−1.22045
$622$	−3067.41	−0.197736
$623$	0	0
$624$	513.831	0.0329643
$625$	0	0
$626$	−4092.35	−0.261283
$627$	17978.2	1.14510
$628$	12145.0	0.771716
$629$	−21281.4	−1.34904
$630$	0	0
$631$	5969.39	0.376605	0.188303	−	0.982111i	$-0.439701\pi$
$631$	5969.39	0.376605	0.188303	+	0.982111i	$0.439701\pi$
$632$	−559.367	−0.0352064
$633$	8474.57	0.532123
$634$	−2791.14	−0.174843
$635$	0	0
$636$	−3479.84	−0.216957
$637$	0	0
$638$	−3875.25	−0.240474
$639$	−6873.70	−0.425539
$640$	0	0
$641$	30367.1	1.87118	0.935592	−	0.353084i	$-0.114867\pi$
$641$	30367.1	1.87118	0.935592	+	0.353084i	$0.114867\pi$
$642$	−6315.40	−0.388238
$643$	−28592.2	−1.75360	−0.876802	−	0.480851i	$-0.840328\pi$
$643$	−28592.2	−1.75360	−0.876802	+	0.480851i	$0.840328\pi$
$644$	0	0
$645$	0	0
$646$	−19292.7	−1.17502
$647$	14507.9	0.881555	0.440778	−	0.897616i	$-0.354703\pi$
$647$	14507.9	0.881555	0.440778	+	0.897616i	$0.354703\pi$
$648$	5832.21	0.353566
$649$	−13869.7	−0.838877
$650$	0	0
$651$	0	0
$652$	6882.89	0.413428
$653$	6999.85	0.419488	0.209744	−	0.977756i	$-0.432737\pi$
$653$	6999.85	0.419488	0.209744	+	0.977756i	$0.432737\pi$
$654$	7382.53	0.441406
$655$	0	0
$656$	−767.504	−0.0456799
$657$	11351.3	0.674056
$658$	0	0
$659$	7308.92	0.432041	0.216020	−	0.976389i	$-0.430692\pi$
$659$	7308.92	0.432041	0.216020	+	0.976389i	$0.430692\pi$
$660$	0	0
$661$	30097.2	1.77102	0.885512	−	0.464617i	$-0.153808\pi$
$661$	30097.2	1.77102	0.885512	+	0.464617i	$0.153808\pi$
$662$	9157.07	0.537613
$663$	−10444.0	−0.611779
$664$	−8315.91	−0.486024
$665$	0	0
$666$	8051.83	0.468472
$667$	−6366.92	−0.369608
$668$	10835.4	0.627594
$669$	−455.425	−0.0263195
$670$	0	0
$671$	5209.29	0.299706
$672$	0	0
$673$	−5400.26	−0.309309	−0.154654	−	0.987969i	$-0.549426\pi$
$673$	−5400.26	−0.309309	−0.154654	+	0.987969i	$0.549426\pi$
$674$	−17841.3	−1.01962
$675$	0	0
$676$	650.781	0.0370267
$677$	−6431.09	−0.365091	−0.182546	−	0.983197i	$-0.558434\pi$
$677$	−6431.09	−0.365091	−0.182546	+	0.983197i	$0.558434\pi$
$678$	−4366.91	−0.247360
$679$	0	0
$680$	0	0
$681$	7861.79	0.442385
$682$	−4780.92	−0.268433
$683$	−20865.8	−1.16897	−0.584486	−	0.811404i	$-0.698704\pi$
$683$	−20865.8	−1.16897	−0.584486	+	0.811404i	$0.698704\pi$
$684$	−13510.2	−0.755227
$685$	0	0
$686$	0	0
$687$	−14994.6	−0.832720
$688$	−667.339	−0.0369797
$689$	12227.7	0.676109
$690$	0	0
$691$	−18450.3	−1.01575	−0.507873	−	0.861432i	$-0.669568\pi$
$691$	−18450.3	−1.01575	−0.507873	+	0.861432i	$0.669568\pi$
$692$	9962.76	0.547294
$693$	0	0
$694$	−6732.70	−0.368256
$695$	0	0
$696$	−2214.26	−0.120591
$697$	15600.0	0.847766
$698$	17664.4	0.957890
$699$	−2345.97	−0.126942
$700$	0	0
$701$	12639.3	0.680996	0.340498	−	0.940245i	$-0.389404\pi$
$701$	12639.3	0.680996	0.340498	+	0.940245i	$0.389404\pi$
$702$	9085.69	0.488486
$703$	−28954.4	−1.55339
$704$	−15703.3	−0.840680
$705$	0	0
$706$	4958.45	0.264325
$707$	0	0
$708$	−3119.69	−0.165600
$709$	−23126.8	−1.22503	−0.612514	−	0.790460i	$-0.709842\pi$
$709$	−23126.8	−1.22503	−0.612514	+	0.790460i	$0.709842\pi$
$710$	0	0
$711$	525.918	0.0277404
$712$	−22686.1	−1.19410
$713$	−7854.92	−0.412579
$714$	0	0
$715$	0	0
$716$	−3267.16	−0.170530
$717$	12865.9	0.670134
$718$	−3635.39	−0.188957
$719$	24093.1	1.24968	0.624841	−	0.780752i	$-0.285164\pi$
$719$	24093.1	1.24968	0.624841	+	0.780752i	$0.285164\pi$
$720$	0	0
$721$	0	0
$722$	−14758.7	−0.760750
$723$	1156.79	0.0595041
$724$	−14547.2	−0.746741
$725$	0	0
$726$	−8294.49	−0.424019
$727$	−35983.4	−1.83570	−0.917849	−	0.396931i	$-0.870075\pi$
$727$	−35983.4	−1.83570	−0.917849	+	0.396931i	$0.870075\pi$
$728$	0	0
$729$	2540.55	0.129073
$730$	0	0
$731$	13564.1	0.686302
$732$	1171.72	0.0591640
$733$	1451.50	0.0731413	0.0365706	−	0.999331i	$-0.488357\pi$
$733$	1451.50	0.0731413	0.0365706	+	0.999331i	$0.488357\pi$
$734$	−2097.64	−0.105484
$735$	0	0
$736$	−29226.3	−1.46371
$737$	23409.4	1.17001
$738$	−5902.27	−0.294398
$739$	−5891.67	−0.293273	−0.146636	−	0.989190i	$-0.546845\pi$
$739$	−5891.67	−0.293273	−0.146636	+	0.989190i	$0.546845\pi$
$740$	0	0
$741$	−14209.5	−0.704451
$742$	0	0
$743$	7438.65	0.367292	0.183646	−	0.982992i	$-0.441210\pi$
$743$	7438.65	0.367292	0.183646	+	0.982992i	$0.441210\pi$
$744$	−2731.75	−0.134611
$745$	0	0
$746$	−7782.84	−0.381970
$747$	7818.64	0.382957
$748$	−27518.9	−1.34518
$749$	0	0
$750$	0	0
$751$	20272.4	0.985018	0.492509	−	0.870307i	$-0.336080\pi$
$751$	20272.4	0.985018	0.492509	+	0.870307i	$0.336080\pi$
$752$	303.427	0.0147139
$753$	−5711.80	−0.276427
$754$	3062.89	0.147936
$755$	0	0
$756$	0	0
$757$	−10193.8	−0.489432	−0.244716	−	0.969595i	$-0.578695\pi$
$757$	−10193.8	−0.489432	−0.244716	+	0.969595i	$0.578695\pi$
$758$	−2403.59	−0.115174
$759$	−22763.6	−1.08862
$760$	0	0
$761$	41117.6	1.95862	0.979311	−	0.202362i	$-0.0648618\pi$
$761$	41117.6	1.95862	0.979311	+	0.202362i	$0.0648618\pi$
$762$	−1089.26	−0.0517845
$763$	0	0
$764$	−3844.76	−0.182066
$765$	0	0
$766$	−22140.2	−1.04433
$767$	10962.2	0.516065
$768$	−9695.10	−0.455523
$769$	−11486.6	−0.538642	−0.269321	−	0.963050i	$-0.586799\pi$
$769$	−11486.6	−0.538642	−0.269321	+	0.963050i	$0.586799\pi$
$770$	0	0
$771$	2000.66	0.0934527
$772$	−21204.2	−0.988544
$773$	21799.2	1.01431	0.507156	−	0.861854i	$-0.330697\pi$
$773$	21799.2	1.01431	0.507156	+	0.861854i	$0.330697\pi$
$774$	−5131.98	−0.238327
$775$	0	0
$776$	20834.5	0.963807
$777$	0	0
$778$	12990.9	0.598645
$779$	21224.5	0.976185
$780$	0	0
$781$	−19049.1	−0.872765
$782$	24428.0	1.11706
$783$	4786.83	0.218477
$784$	0	0
$785$	0	0
$786$	−3022.72	−0.137172
$787$	24080.3	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$787$	24080.3	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$788$	−17736.4	−0.801817
$789$	−715.712	−0.0322941
$790$	0	0
$791$	0	0
$792$	26449.0	1.18665
$793$	−4117.28	−0.184374
$794$	−5964.96	−0.266610
$795$	0	0
$796$	−17619.8	−0.784569
$797$	17194.3	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$797$	17194.3	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$798$	0	0
$799$	−6167.35	−0.273073
$800$	0	0
$801$	21329.5	0.940876
$802$	9097.15	0.400538
$803$	31457.7	1.38246
$804$	5265.46	0.230968
$805$	0	0
$806$	3778.71	0.165136
$807$	−8881.55	−0.387417
$808$	13363.1	0.581823
$809$	33349.1	1.44931	0.724656	−	0.689111i	$-0.241999\pi$
$809$	33349.1	1.44931	0.724656	+	0.689111i	$0.241999\pi$
$810$	0	0
$811$	−4577.87	−0.198213	−0.0991066	−	0.995077i	$-0.531598\pi$
$811$	−4577.87	−0.198213	−0.0991066	+	0.995077i	$0.531598\pi$
$812$	0	0
$813$	−4810.37	−0.207512
$814$	22314.0	0.960819
$815$	0	0
$816$	−1038.65	−0.0445588
$817$	18454.6	0.790262
$818$	−16247.0	−0.694455
$819$	0	0
$820$	0	0
$821$	−5832.52	−0.247937	−0.123969	−	0.992286i	$-0.539562\pi$
$821$	−5832.52	−0.247937	−0.123969	+	0.992286i	$0.539562\pi$
$822$	−3232.33	−0.137154
$823$	37974.3	1.60838	0.804192	−	0.594369i	$-0.202598\pi$
$823$	37974.3	1.60838	0.804192	+	0.594369i	$0.202598\pi$
$824$	6651.92	0.281226
$825$	0	0
$826$	0	0
$827$	15796.2	0.664193	0.332096	−	0.943245i	$-0.392244\pi$
$827$	15796.2	0.664193	0.332096	+	0.943245i	$0.392244\pi$
$828$	17106.3	0.717976
$829$	12714.1	0.532666	0.266333	−	0.963881i	$-0.414188\pi$
$829$	12714.1	0.532666	0.266333	+	0.963881i	$0.414188\pi$
$830$	0	0
$831$	16431.9	0.685942
$832$	12411.4	0.517173
$833$	0	0
$834$	−12336.9	−0.512219
$835$	0	0
$836$	−37440.8	−1.54894
$837$	5905.55	0.243878
$838$	23169.0	0.955085
$839$	42924.2	1.76628	0.883140	−	0.469109i	$-0.155425\pi$
$839$	42924.2	1.76628	0.883140	+	0.469109i	$0.155425\pi$
$840$	0	0
$841$	−22775.3	−0.933835
$842$	−27980.2	−1.14520
$843$	2033.95	0.0830998
$844$	−17648.8	−0.719784
$845$	0	0
$846$	2333.42	0.0948281
$847$	0	0
$848$	1216.04	0.0492442
$849$	16244.4	0.656662
$850$	0	0
$851$	36661.3	1.47677
$852$	−4284.69	−0.172290
$853$	36172.5	1.45196	0.725980	−	0.687716i	$-0.241387\pi$
$853$	36172.5	1.45196	0.725980	+	0.687716i	$0.241387\pi$
$854$	0	0
$855$	0	0
$856$	33410.4	1.33405
$857$	32039.9	1.27709	0.638544	−	0.769586i	$-0.279537\pi$
$857$	32039.9	1.27709	0.638544	+	0.769586i	$0.279537\pi$
$858$	10950.7	0.435724
$859$	6798.07	0.270020	0.135010	−	0.990844i	$-0.456893\pi$
$859$	6798.07	0.270020	0.135010	+	0.990844i	$0.456893\pi$
$860$	0	0
$861$	0	0
$862$	−13692.9	−0.541046
$863$	−30179.3	−1.19040	−0.595200	−	0.803577i	$-0.702927\pi$
$863$	−30179.3	−1.19040	−0.595200	+	0.803577i	$0.702927\pi$
$864$	21973.1	0.865209
$865$	0	0
$866$	23825.5	0.934901
$867$	8858.39	0.346997
$868$	0	0
$869$	1457.47	0.0568946
$870$	0	0
$871$	−18502.1	−0.719772
$872$	−39055.9	−1.51674
$873$	−19588.6	−0.759421
$874$	33235.4	1.28627
$875$	0	0
$876$	7075.76	0.272908
$877$	−1700.51	−0.0654758	−0.0327379	−	0.999464i	$-0.510423\pi$
$877$	−1700.51	−0.0654758	−0.0327379	+	0.999464i	$0.510423\pi$
$878$	9276.02	0.356549
$879$	−1086.17	−0.0416787
$880$	0	0
$881$	−1678.46	−0.0641869	−0.0320935	−	0.999485i	$-0.510217\pi$
$881$	−1678.46	−0.0641869	−0.0320935	+	0.999485i	$0.510217\pi$
$882$	0	0
$883$	10285.8	0.392009	0.196005	−	0.980603i	$-0.437203\pi$
$883$	10285.8	0.392009	0.196005	+	0.980603i	$0.437203\pi$
$884$	21750.2	0.827532
$885$	0	0
$886$	−6657.24	−0.252432
$887$	12167.5	0.460593	0.230296	−	0.973121i	$-0.426030\pi$
$887$	12167.5	0.460593	0.230296	+	0.973121i	$0.426030\pi$
$888$	12749.9	0.481823
$889$	0	0
$890$	0	0
$891$	−15196.3	−0.571374
$892$	948.452	0.0356015
$893$	−8390.96	−0.314438
$894$	−10503.0	−0.392922
$895$	0	0
$896$	0	0
$897$	17991.7	0.669705
$898$	25534.7	0.948892
$899$	1990.83	0.0738574
$900$	0	0
$901$	−24716.9	−0.913916
$902$	−16356.9	−0.603799
$903$	0	0
$904$	23102.3	0.849968
$905$	0	0
$906$	−422.176	−0.0154811
$907$	−50766.0	−1.85850	−0.929250	−	0.369452i	$-0.879545\pi$
$907$	−50766.0	−1.85850	−0.929250	+	0.369452i	$0.879545\pi$
$908$	−16372.7	−0.598399
$909$	−12564.0	−0.458441
$910$	0	0
$911$	18451.1	0.671033	0.335517	−	0.942034i	$-0.391089\pi$
$911$	18451.1	0.671033	0.335517	+	0.942034i	$0.391089\pi$
$912$	−1413.13	−0.0513085
$913$	21667.8	0.785431
$914$	18039.6	0.652841
$915$	0	0
$916$	31227.1	1.12639
$917$	0	0
$918$	−18365.6	−0.660301
$919$	393.861	0.0141374	0.00706870	−	0.999975i	$-0.497750\pi$
$919$	393.861	0.0141374	0.00706870	+	0.999975i	$0.497750\pi$
$920$	0	0
$921$	12260.3	0.438643
$922$	−557.167	−0.0199016
$923$	15055.9	0.536912
$924$	0	0
$925$	0	0
$926$	13745.3	0.487795
$927$	−6254.15	−0.221589
$928$	7407.39	0.262025
$929$	−27452.3	−0.969517	−0.484759	−	0.874648i	$-0.661093\pi$
$929$	−27452.3	−0.969517	−0.484759	+	0.874648i	$0.661093\pi$
$930$	0	0
$931$	0	0
$932$	4885.63	0.171710
$933$	4566.72	0.160244
$934$	−280.803	−0.00983744
$935$	0	0
$936$	−20904.5	−0.730006
$937$	−21608.3	−0.753375	−0.376688	−	0.926340i	$-0.622937\pi$
$937$	−21608.3	−0.753375	−0.376688	+	0.926340i	$0.622937\pi$
$938$	0	0
$939$	6092.64	0.211742
$940$	0	0
$941$	16708.5	0.578832	0.289416	−	0.957203i	$-0.406539\pi$
$941$	16708.5	0.578832	0.289416	+	0.957203i	$0.406539\pi$
$942$	9769.12	0.337893
$943$	−26874.0	−0.928036
$944$	1090.19	0.0375874
$945$	0	0
$946$	−14222.2	−0.488800
$947$	13972.8	0.479466	0.239733	−	0.970839i	$-0.422940\pi$
$947$	13972.8	0.479466	0.239733	+	0.970839i	$0.422940\pi$
$948$	327.828	0.0112314
$949$	−24863.3	−0.850471
$950$	0	0
$951$	4155.42	0.141692
$952$	0	0
$953$	−11155.9	−0.379197	−0.189598	−	0.981862i	$-0.560719\pi$
$953$	−11155.9	−0.379197	−0.189598	+	0.981862i	$0.560719\pi$
$954$	9351.63	0.317369
$955$	0	0
$956$	−26794.0	−0.906466
$957$	5769.42	0.194879
$958$	11104.0	0.374481
$959$	0	0
$960$	0	0
$961$	−27334.9	−0.917556
$962$	−17636.4	−0.591081
$963$	−31412.5	−1.05115
$964$	−2409.09	−0.0804891
$965$	0	0
$966$	0	0
$967$	−1212.55	−0.0403238	−0.0201619	−	0.999797i	$-0.506418\pi$
$967$	−1212.55	−0.0403238	−0.0201619	+	0.999797i	$0.506418\pi$
$968$	43880.4	1.45699
$969$	28722.8	0.952227
$970$	0	0
$971$	48832.2	1.61390	0.806952	−	0.590617i	$-0.201115\pi$
$971$	48832.2	1.61390	0.806952	+	0.590617i	$0.201115\pi$
$972$	−20128.6	−0.664222
$973$	0	0
$974$	34578.0	1.13753
$975$	0	0
$976$	−409.462	−0.0134289
$977$	5939.19	0.194485	0.0972424	−	0.995261i	$-0.468998\pi$
$977$	5939.19	0.194485	0.0972424	+	0.995261i	$0.468998\pi$
$978$	5536.43	0.181018
$979$	59110.5	1.92970
$980$	0	0
$981$	36720.4	1.19510
$982$	27993.3	0.909677
$983$	21863.5	0.709398	0.354699	−	0.934980i	$-0.384583\pi$
$983$	21863.5	0.709398	0.354699	+	0.934980i	$0.384583\pi$
$984$	−9346.11	−0.302788
$985$	0	0
$986$	−6191.26	−0.199970
$987$	0	0
$988$	29592.1	0.952886
$989$	−23366.7	−0.751283
$990$	0	0
$991$	−44618.5	−1.43023	−0.715113	−	0.699009i	$-0.753625\pi$
$991$	−44618.5	−1.43023	−0.715113	+	0.699009i	$0.753625\pi$
$992$	9138.55	0.292489
$993$	−13632.9	−0.435678
$994$	0	0
$995$	0	0
$996$	4873.71	0.155050
$997$	37923.9	1.20468	0.602338	−	0.798241i	$-0.294236\pi$
$997$	37923.9	1.20468	0.602338	+	0.798241i	$0.294236\pi$
$998$	22997.8	0.729441
$999$	−27563.0	−0.872928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bh.1.2		5
5.2	odd	4		245.4.b.d.99.4		10
5.3	odd	4		245.4.b.d.99.7		10
5.4	even	2		1225.4.a.be.1.4		5
7.6	odd	2		175.4.a.j.1.2		5
21.20	even	2		1575.4.a.bn.1.4		5
35.2	odd	12		245.4.j.f.214.4		20
35.3	even	12		245.4.j.e.79.4		20
35.12	even	12		245.4.j.e.214.4		20
35.13	even	4		35.4.b.a.29.7	yes	10
35.17	even	12		245.4.j.e.79.7		20
35.18	odd	12		245.4.j.f.79.4		20
35.23	odd	12		245.4.j.f.214.7		20
35.27	even	4		35.4.b.a.29.4	✓	10
35.32	odd	12		245.4.j.f.79.7		20
35.33	even	12		245.4.j.e.214.7		20
35.34	odd	2		175.4.a.i.1.4		5
105.62	odd	4		315.4.d.c.64.7		10
105.83	odd	4		315.4.d.c.64.4		10
105.104	even	2		1575.4.a.bq.1.2		5
140.27	odd	4		560.4.g.f.449.4		10
140.83	odd	4		560.4.g.f.449.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.4	✓	10	35.27	even	4
35.4.b.a.29.7	yes	10	35.13	even	4
175.4.a.i.1.4		5	35.34	odd	2
175.4.a.j.1.2		5	7.6	odd	2
245.4.b.d.99.4		10	5.2	odd	4
245.4.b.d.99.7		10	5.3	odd	4
245.4.j.e.79.4		20	35.3	even	12
245.4.j.e.79.7		20	35.17	even	12
245.4.j.e.214.4		20	35.12	even	12
245.4.j.e.214.7		20	35.33	even	12
245.4.j.f.79.4		20	35.18	odd	12
245.4.j.f.79.7		20	35.32	odd	12
245.4.j.f.214.4		20	35.2	odd	12
245.4.j.f.214.7		20	35.23	odd	12
315.4.d.c.64.4		10	105.83	odd	4
315.4.d.c.64.7		10	105.62	odd	4
560.4.g.f.449.4		10	140.27	odd	4
560.4.g.f.449.7		10	140.83	odd	4
1225.4.a.be.1.4		5	5.4	even	2
1225.4.a.bh.1.2		5	1.1	even	1	trivial
1575.4.a.bn.1.4		5	21.20	even	2
1575.4.a.bq.1.2		5	105.104	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67516 q^{2} +2.49396 q^{3} -5.19383 q^{4} -4.17779 q^{6} +22.1018 q^{8} -20.7802 q^{9} -57.5880 q^{11} -12.9532 q^{12} +45.5159 q^{13} +4.52655 q^{16} -92.0051 q^{17} +34.8101 q^{18} -125.177 q^{19} +96.4692 q^{22} +158.496 q^{23} +55.1211 q^{24} -76.2466 q^{26} -119.162 q^{27} -40.1708 q^{29} -49.5590 q^{31} -184.397 q^{32} -143.622 q^{33} +154.123 q^{34} +107.929 q^{36} +231.307 q^{37} +209.692 q^{38} +113.515 q^{39} -169.556 q^{41} -147.428 q^{43} +299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +11.2890 q^{48} -229.457 q^{51} -236.402 q^{52} +268.647 q^{53} +199.615 q^{54} -312.187 q^{57} +67.2926 q^{58} +240.843 q^{59} -90.4579 q^{61} +83.0194 q^{62} +272.683 q^{64} +240.591 q^{66} -406.498 q^{67} +477.859 q^{68} +395.283 q^{69} +330.782 q^{71} -459.279 q^{72} -546.255 q^{73} -387.477 q^{74} +650.149 q^{76} -190.156 q^{78} -25.3087 q^{79} +263.879 q^{81} +284.034 q^{82} -376.255 q^{83} +246.965 q^{86} -100.184 q^{87} -1272.80 q^{88} -1026.44 q^{89} -823.203 q^{92} -123.598 q^{93} -112.291 q^{94} -459.879 q^{96} +942.660 q^{97} +1196.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} - 10 q^{3} + 18 q^{4} - 6 q^{6} + 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 74 q^{16} - 238 q^{17} - 2 q^{18} + 36 q^{19} + 358 q^{22} + 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 - 10 * q^3 + 18 * q^4 - 6 * q^6 + 42 * q^8 + 23 * q^9 + 42 * q^11 - 136 * q^12 - 34 * q^13 + 74 * q^16 - 238 * q^17 - 2 * q^18 + 36 * q^19 + 358 * q^22 + 152 * q^23 + 36 * q^24 + 310 * q^26 - 334 * q^27 - 44 * q^29 - 60 * q^31 + 710 * q^32 - 426 * q^33 + 482 * q^34 - 210 * q^36 + 312 * q^37 - 280 * q^38 - 106 * q^39 + 426 * q^41 + 304 * q^43 + 712 * q^44 + 88 * q^46 - 370 * q^47 + 696 * q^48 + 638 * q^51 + 1156 * q^52 + 976 * q^53 - 498 * q^54 - 588 * q^57 - 2722 * q^58 + 432 * q^59 + 442 * q^61 + 956 * q^62 + 1362 * q^64 - 574 * q^66 - 804 * q^67 + 420 * q^68 + 2404 * q^69 + 440 * q^71 - 3150 * q^72 - 564 * q^73 - 1512 * q^74 + 1336 * q^76 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 + 3480 * q^82 - 1656 * q^83 + 1216 * q^86 + 1674 * q^87 - 1092 * q^88 - 746 * q^89 + 572 * q^92 + 676 * q^93 + 826 * q^94 - 2040 * q^96 - 518 * q^97 + 2652 * q^99

Introduction

L-functions

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