LMFDB - Embedded newform 225.12.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,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, 12, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$172.877215626$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{151}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 151 $ x^2 - 151
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-12.2882$ 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)

$f(q)$	$=$	$q-83.7292 q^{2} +4962.58 q^{4} -15973.7 q^{7} -244036. q^{8} +O(q^{10})$	$q-83.7292 q^{2} +4962.58 q^{4} -15973.7 q^{7} -244036. q^{8} -339729. q^{11} -2.02328e6 q^{13} +1.33746e6 q^{14} +1.02696e7 q^{16} -2.45063e6 q^{17} -4.08504e6 q^{19} +2.84453e7 q^{22} +2.86497e7 q^{23} +1.69408e8 q^{26} -7.92706e7 q^{28} +9.41230e6 q^{29} +2.99399e8 q^{31} -3.60078e8 q^{32} +2.05190e8 q^{34} +4.57279e8 q^{37} +3.42037e8 q^{38} -1.83814e8 q^{41} -6.56811e8 q^{43} -1.68594e9 q^{44} -2.39882e9 q^{46} -1.97090e8 q^{47} -1.72217e9 q^{49} -1.00407e10 q^{52} +5.15890e9 q^{53} +3.89815e9 q^{56} -7.88085e8 q^{58} +6.62200e8 q^{59} +5.58296e8 q^{61} -2.50684e10 q^{62} +9.11694e9 q^{64} -1.01206e10 q^{67} -1.21615e10 q^{68} -1.78161e10 q^{71} +2.33380e10 q^{73} -3.82876e10 q^{74} -2.02724e10 q^{76} +5.42672e9 q^{77} +1.24957e10 q^{79} +1.53906e10 q^{82} +3.37037e10 q^{83} +5.49943e10 q^{86} +8.29062e10 q^{88} -2.94282e10 q^{89} +3.23192e10 q^{91} +1.42177e11 q^{92} +1.65022e10 q^{94} +1.13262e11 q^{97} +1.44196e11 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{2} + 6976 q^{4} - 57900 q^{7} - 246240 q^{8}+O(q^{10}) $ 2 * q - 20 * q^2 + 6976 * q^4 - 57900 * q^7 - 246240 * q^8	$ 2 q - 20 q^{2} + 6976 q^{4} - 57900 q^{7} - 246240 q^{8} + 618176 q^{11} - 3414260 q^{13} - 1334472 q^{14} + 6005632 q^{16} + 1317940 q^{17} + 5325320 q^{19} + 89491840 q^{22} + 58943940 q^{23} + 80761736 q^{26} - 163685760 q^{28} - 94140380 q^{29} + 244543464 q^{31} - 627301120 q^{32} + 445358072 q^{34} - 21003220 q^{37} + 941752240 q^{38} + 745743316 q^{41} - 629950100 q^{43} + 242725888 q^{44} - 468194856 q^{46} - 1402061540 q^{47} - 1941677414 q^{49} - 12841321600 q^{52} + 1138320580 q^{53} + 3990553920 q^{56} - 7387417960 q^{58} - 7317515560 q^{59} - 1516425676 q^{61} - 28564327440 q^{62} + 819531776 q^{64} - 15734290140 q^{67} - 4573774720 q^{68} - 32938471544 q^{71} + 29982848860 q^{73} - 68768198072 q^{74} - 1325392640 q^{76} - 34734748800 q^{77} - 3302823120 q^{79} + 74630515640 q^{82} + 13299102420 q^{83} + 56706093896 q^{86} + 80794874880 q^{88} + 12674770860 q^{89} + 90637859064 q^{91} + 203171571840 q^{92} - 60289765528 q^{94} + 3080703740 q^{97} + 130206802940 q^{98}+O(q^{100}) $ 2 * q - 20 * q^2 + 6976 * q^4 - 57900 * q^7 - 246240 * q^8 + 618176 * q^11 - 3414260 * q^13 - 1334472 * q^14 + 6005632 * q^16 + 1317940 * q^17 + 5325320 * q^19 + 89491840 * q^22 + 58943940 * q^23 + 80761736 * q^26 - 163685760 * q^28 - 94140380 * q^29 + 244543464 * q^31 - 627301120 * q^32 + 445358072 * q^34 - 21003220 * q^37 + 941752240 * q^38 + 745743316 * q^41 - 629950100 * q^43 + 242725888 * q^44 - 468194856 * q^46 - 1402061540 * q^47 - 1941677414 * q^49 - 12841321600 * q^52 + 1138320580 * q^53 + 3990553920 * q^56 - 7387417960 * q^58 - 7317515560 * q^59 - 1516425676 * q^61 - 28564327440 * q^62 + 819531776 * q^64 - 15734290140 * q^67 - 4573774720 * q^68 - 32938471544 * q^71 + 29982848860 * q^73 - 68768198072 * q^74 - 1325392640 * q^76 - 34734748800 * q^77 - 3302823120 * q^79 + 74630515640 * q^82 + 13299102420 * q^83 + 56706093896 * q^86 + 80794874880 * q^88 + 12674770860 * q^89 + 90637859064 * q^91 + 203171571840 * q^92 - 60289765528 * q^94 + 3080703740 * q^97 + 130206802940 * 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 $

$2$	−83.7292	−1.85017	−0.925086	−	0.379757i	$-0.876007\pi$
$2$	−83.7292	−1.85017	−0.925086	+	0.379757i	$0.876007\pi$
$3$	0	0
$3$	0	0
$4$	4962.58	2.42314
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−15973.7	−0.359224	−0.179612	−	0.983738i	$-0.557484\pi$
$7$	−15973.7	−0.359224	−0.179612	+	0.983738i	$0.557484\pi$
$8$	−244036.	−2.63305
$9$	0	0
$10$	0	0
$11$	−339729.	−0.636024	−0.318012	−	0.948087i	$-0.603015\pi$
$11$	−339729.	−0.636024	−0.318012	+	0.948087i	$0.603015\pi$
$12$	0	0
$13$	−2.02328e6	−1.51136	−0.755680	−	0.654941i	$-0.772693\pi$
$13$	−2.02328e6	−1.51136	−0.755680	+	0.654941i	$0.772693\pi$
$14$	1.33746e6	0.664626
$15$	0	0
$16$	1.02696e7	2.44846
$17$	−2.45063e6	−0.418610	−0.209305	−	0.977850i	$-0.567120\pi$
$17$	−2.45063e6	−0.418610	−0.209305	+	0.977850i	$0.567120\pi$
$18$	0	0
$19$	−4.08504e6	−0.378487	−0.189244	−	0.981930i	$-0.560604\pi$
$19$	−4.08504e6	−0.378487	−0.189244	+	0.981930i	$0.560604\pi$
$20$	0	0
$21$	0	0
$22$	2.84453e7	1.17675
$23$	2.86497e7	0.928149	0.464074	−	0.885796i	$-0.346387\pi$
$23$	2.86497e7	0.928149	0.464074	+	0.885796i	$0.346387\pi$
$24$	0	0
$25$	0	0
$26$	1.69408e8	2.79628
$27$	0	0
$28$	−7.92706e7	−0.870448
$29$	9.41230e6	0.0852132	0.0426066	−	0.999092i	$-0.486434\pi$
$29$	9.41230e6	0.0852132	0.0426066	+	0.999092i	$0.486434\pi$
$30$	0	0
$31$	2.99399e8	1.87828	0.939141	−	0.343532i	$-0.111623\pi$
$31$	2.99399e8	1.87828	0.939141	+	0.343532i	$0.111623\pi$
$32$	−3.60078e8	−1.89702
$33$	0	0
$34$	2.05190e8	0.774500
$35$	0	0
$36$	0	0
$37$	4.57279e8	1.08411	0.542053	−	0.840344i	$-0.317647\pi$
$37$	4.57279e8	1.08411	0.542053	+	0.840344i	$0.317647\pi$
$38$	3.42037e8	0.700267
$39$	0	0
$40$	0	0
$41$	−1.83814e8	−0.247780	−0.123890	−	0.992296i	$-0.539537\pi$
$41$	−1.83814e8	−0.247780	−0.123890	+	0.992296i	$0.539537\pi$
$42$	0	0
$43$	−6.56811e8	−0.681340	−0.340670	−	0.940183i	$-0.610654\pi$
$43$	−6.56811e8	−0.681340	−0.340670	+	0.940183i	$0.610654\pi$
$44$	−1.68594e9	−1.54117
$45$	0	0
$46$	−2.39882e9	−1.71724
$47$	−1.97090e8	−0.125350	−0.0626752	−	0.998034i	$-0.519963\pi$
$47$	−1.97090e8	−0.125350	−0.0626752	+	0.998034i	$0.519963\pi$
$48$	0	0
$49$	−1.72217e9	−0.870958
$50$	0	0
$51$	0	0
$52$	−1.00407e10	−3.66223
$53$	5.15890e9	1.69449	0.847247	−	0.531199i	$-0.178258\pi$
$53$	5.15890e9	1.69449	0.847247	+	0.531199i	$0.178258\pi$
$54$	0	0
$55$	0	0
$56$	3.89815e9	0.945854
$57$	0	0
$58$	−7.88085e8	−0.157659
$59$	6.62200e8	0.120588	0.0602939	−	0.998181i	$-0.480796\pi$
$59$	6.62200e8	0.120588	0.0602939	+	0.998181i	$0.480796\pi$
$60$	0	0
$61$	5.58296e8	0.0846351	0.0423176	−	0.999104i	$-0.486526\pi$
$61$	5.58296e8	0.0846351	0.0423176	+	0.999104i	$0.486526\pi$
$62$	−2.50684e10	−3.47515
$63$	0	0
$64$	9.11694e9	1.06135
$65$	0	0
$66$	0	0
$67$	−1.01206e10	−0.915787	−0.457894	−	0.889007i	$-0.651396\pi$
$67$	−1.01206e10	−0.915787	−0.457894	+	0.889007i	$0.651396\pi$
$68$	−1.21615e10	−1.01435
$69$	0	0
$70$	0	0
$71$	−1.78161e10	−1.17190	−0.585952	−	0.810346i	$-0.699279\pi$
$71$	−1.78161e10	−1.17190	−0.585952	+	0.810346i	$0.699279\pi$
$72$	0	0
$73$	2.33380e10	1.31761	0.658807	−	0.752312i	$-0.271061\pi$
$73$	2.33380e10	1.31761	0.658807	+	0.752312i	$0.271061\pi$
$74$	−3.82876e10	−2.00578
$75$	0	0
$76$	−2.02724e10	−0.917127
$77$	5.42672e9	0.228475
$78$	0	0
$79$	1.24957e10	0.456888	0.228444	−	0.973557i	$-0.426636\pi$
$79$	1.24957e10	0.456888	0.228444	+	0.973557i	$0.426636\pi$
$80$	0	0
$81$	0	0
$82$	1.53906e10	0.458436
$83$	3.37037e10	0.939179	0.469589	−	0.882885i	$-0.344402\pi$
$83$	3.37037e10	0.939179	0.469589	+	0.882885i	$0.344402\pi$
$84$	0	0
$85$	0	0
$86$	5.49943e10	1.26060
$87$	0	0
$88$	8.29062e10	1.67468
$89$	−2.94282e10	−0.558623	−0.279311	−	0.960201i	$-0.590106\pi$
$89$	−2.94282e10	−0.558623	−0.279311	+	0.960201i	$0.590106\pi$
$90$	0	0
$91$	3.23192e10	0.542916
$92$	1.42177e11	2.24903
$93$	0	0
$94$	1.65022e10	0.231920
$95$	0	0
$96$	0	0
$97$	1.13262e11	1.33918	0.669588	−	0.742732i	$-0.266471\pi$
$97$	1.13262e11	1.33918	0.669588	+	0.742732i	$0.266471\pi$
$98$	1.44196e11	1.61142
$99$	0	0
$100$	0	0
$101$	−1.19564e11	−1.13196	−0.565982	−	0.824417i	$-0.691503\pi$
$101$	−1.19564e11	−1.13196	−0.565982	+	0.824417i	$0.691503\pi$
$102$	0	0
$103$	9.46874e10	0.804799	0.402399	−	0.915464i	$-0.368176\pi$
$103$	9.46874e10	0.804799	0.402399	+	0.915464i	$0.368176\pi$
$104$	4.93753e11	3.97948
$105$	0	0
$106$	−4.31951e11	−3.13511
$107$	1.52769e11	1.05299	0.526497	−	0.850177i	$-0.323505\pi$
$107$	1.52769e11	1.05299	0.526497	+	0.850177i	$0.323505\pi$
$108$	0	0
$109$	2.85078e11	1.77467	0.887337	−	0.461122i	$-0.152553\pi$
$109$	2.85078e11	1.77467	0.887337	+	0.461122i	$0.152553\pi$
$110$	0	0
$111$	0	0
$112$	−1.64043e11	−0.879544
$113$	−2.35563e11	−1.20275	−0.601374	−	0.798968i	$-0.705380\pi$
$113$	−2.35563e11	−1.20275	−0.601374	+	0.798968i	$0.705380\pi$
$114$	0	0
$115$	0	0
$116$	4.67093e10	0.206483
$117$	0	0
$118$	−5.54455e10	−0.223108
$119$	3.91456e10	0.150375
$120$	0	0
$121$	−1.69896e11	−0.595474
$122$	−4.67457e10	−0.156590
$123$	0	0
$124$	1.48579e12	4.55133
$125$	0	0
$126$	0	0
$127$	1.25786e11	0.337840	0.168920	−	0.985630i	$-0.445972\pi$
$127$	1.25786e11	0.337840	0.168920	+	0.985630i	$0.445972\pi$
$128$	−2.59156e10	−0.0666664
$129$	0	0
$130$	0	0
$131$	2.97347e11	0.673397	0.336698	−	0.941613i	$-0.390690\pi$
$131$	2.97347e11	0.673397	0.336698	+	0.941613i	$0.390690\pi$
$132$	0	0
$133$	6.52530e10	0.135962
$134$	8.47389e11	1.69436
$135$	0	0
$136$	5.98043e11	1.10222
$137$	−6.29203e10	−0.111385	−0.0556926	−	0.998448i	$-0.517737\pi$
$137$	−6.29203e10	−0.111385	−0.0556926	+	0.998448i	$0.517737\pi$
$138$	0	0
$139$	−5.11416e11	−0.835974	−0.417987	−	0.908453i	$-0.637264\pi$
$139$	−5.11416e11	−0.835974	−0.417987	+	0.908453i	$0.637264\pi$
$140$	0	0
$141$	0	0
$142$	1.49173e12	2.16823
$143$	6.87368e11	0.961260
$144$	0	0
$145$	0	0
$146$	−1.95407e12	−2.43781
$147$	0	0
$148$	2.26929e12	2.62694
$149$	−7.94154e11	−0.885891	−0.442945	−	0.896549i	$-0.646066\pi$
$149$	−7.94154e11	−0.885891	−0.442945	+	0.896549i	$0.646066\pi$
$150$	0	0
$151$	1.24629e12	1.29195	0.645974	−	0.763360i	$-0.276452\pi$
$151$	1.24629e12	1.29195	0.645974	+	0.763360i	$0.276452\pi$
$152$	9.96896e11	0.996576
$153$	0	0
$154$	−4.54375e11	−0.422718
$155$	0	0
$156$	0	0
$157$	−9.73519e11	−0.814510	−0.407255	−	0.913315i	$-0.633514\pi$
$157$	−9.73519e11	−0.814510	−0.407255	+	0.913315i	$0.633514\pi$
$158$	−1.04625e12	−0.845322
$159$	0	0
$160$	0	0
$161$	−4.57641e11	−0.333413
$162$	0	0
$163$	−1.26084e12	−0.858281	−0.429140	−	0.903238i	$-0.641183\pi$
$163$	−1.26084e12	−0.858281	−0.429140	+	0.903238i	$0.641183\pi$
$164$	−9.12190e11	−0.600405
$165$	0	0
$166$	−2.82199e12	−1.73764
$167$	−3.01398e12	−1.79556	−0.897780	−	0.440443i	$-0.854821\pi$
$167$	−3.01398e12	−1.79556	−0.897780	+	0.440443i	$0.854821\pi$
$168$	0	0
$169$	2.30151e12	1.28421
$170$	0	0
$171$	0	0
$172$	−3.25948e12	−1.65098
$173$	−3.07482e11	−0.150857	−0.0754285	−	0.997151i	$-0.524032\pi$
$173$	−3.07482e11	−0.150857	−0.0754285	+	0.997151i	$0.524032\pi$
$174$	0	0
$175$	0	0
$176$	−3.48887e12	−1.55728
$177$	0	0
$178$	2.46400e12	1.03355
$179$	1.91470e12	0.778769	0.389384	−	0.921075i	$-0.372688\pi$
$179$	1.91470e12	0.778769	0.389384	+	0.921075i	$0.372688\pi$
$180$	0	0
$181$	1.10300e11	0.0422032	0.0211016	−	0.999777i	$-0.493283\pi$
$181$	1.10300e11	0.0422032	0.0211016	+	0.999777i	$0.493283\pi$
$182$	−2.70606e12	−1.00449
$183$	0	0
$184$	−6.99157e12	−2.44386
$185$	0	0
$186$	0	0
$187$	8.32552e11	0.266246
$188$	−9.78074e11	−0.303741
$189$	0	0
$190$	0	0
$191$	4.43991e12	1.26384	0.631918	−	0.775035i	$-0.282268\pi$
$191$	4.43991e12	1.26384	0.631918	+	0.775035i	$0.282268\pi$
$192$	0	0
$193$	−3.15713e12	−0.848647	−0.424324	−	0.905511i	$-0.639488\pi$
$193$	−3.15713e12	−0.848647	−0.424324	+	0.905511i	$0.639488\pi$
$194$	−9.48330e12	−2.47771
$195$	0	0
$196$	−8.54641e12	−2.11045
$197$	−3.58626e12	−0.861147	−0.430574	−	0.902555i	$-0.641689\pi$
$197$	−3.58626e12	−0.861147	−0.430574	+	0.902555i	$0.641689\pi$
$198$	0	0
$199$	4.14823e12	0.942260	0.471130	−	0.882064i	$-0.343846\pi$
$199$	4.14823e12	0.942260	0.471130	+	0.882064i	$0.343846\pi$
$200$	0	0
$201$	0	0
$202$	1.00110e13	2.09433
$203$	−1.50349e11	−0.0306106
$204$	0	0
$205$	0	0
$206$	−7.92810e12	−1.48902
$207$	0	0
$208$	−2.07782e13	−3.70050
$209$	1.38781e12	0.240727
$210$	0	0
$211$	1.14275e12	0.188104	0.0940519	−	0.995567i	$-0.470018\pi$
$211$	1.14275e12	0.188104	0.0940519	+	0.995567i	$0.470018\pi$
$212$	2.56015e13	4.10599
$213$	0	0
$214$	−1.27913e13	−1.94822
$215$	0	0
$216$	0	0
$217$	−4.78249e12	−0.674724
$218$	−2.38694e13	−3.28345
$219$	0	0
$220$	0	0
$221$	4.95832e12	0.632670
$222$	0	0
$223$	−3.54889e12	−0.430939	−0.215469	−	0.976511i	$-0.569128\pi$
$223$	−3.54889e12	−0.430939	−0.215469	+	0.976511i	$0.569128\pi$
$224$	5.75175e12	0.681454
$225$	0	0
$226$	1.97235e13	2.22529
$227$	3.90543e12	0.430058	0.215029	−	0.976608i	$-0.431015\pi$
$227$	3.90543e12	0.430058	0.215029	+	0.976608i	$0.431015\pi$
$228$	0	0
$229$	−4.85126e12	−0.509048	−0.254524	−	0.967066i	$-0.581919\pi$
$229$	−4.85126e12	−0.509048	−0.254524	+	0.967066i	$0.581919\pi$
$230$	0	0
$231$	0	0
$232$	−2.29694e12	−0.224370
$233$	1.03473e13	0.987115	0.493557	−	0.869713i	$-0.335696\pi$
$233$	1.03473e13	0.987115	0.493557	+	0.869713i	$0.335696\pi$
$234$	0	0
$235$	0	0
$236$	3.28622e12	0.292201
$237$	0	0
$238$	−3.27763e12	−0.278219
$239$	−1.85140e12	−0.153572	−0.0767859	−	0.997048i	$-0.524466\pi$
$239$	−1.85140e12	−0.153572	−0.0767859	+	0.997048i	$0.524466\pi$
$240$	0	0
$241$	2.73018e12	0.216321	0.108160	−	0.994133i	$-0.465504\pi$
$241$	2.73018e12	0.216321	0.108160	+	0.994133i	$0.465504\pi$
$242$	1.42252e13	1.10173
$243$	0	0
$244$	2.77059e12	0.205083
$245$	0	0
$246$	0	0
$247$	8.26518e12	0.572031
$248$	−7.30641e13	−4.94561
$249$	0	0
$250$	0	0
$251$	−1.18976e13	−0.753796	−0.376898	−	0.926255i	$-0.623009\pi$
$251$	−1.18976e13	−0.753796	−0.376898	+	0.926255i	$0.623009\pi$
$252$	0	0
$253$	−9.73316e12	−0.590324
$254$	−1.05319e13	−0.625062
$255$	0	0
$256$	−1.65016e13	−0.938007
$257$	3.53878e10	0.00196889	0.000984443	−	1.00000i	$-0.499687\pi$
$257$	3.53878e10	0.00196889	0.000984443		1.00000i	$0.499687\pi$
$258$	0	0
$259$	−7.30442e12	−0.389437
$260$	0	0
$261$	0	0
$262$	−2.48966e13	−1.24590
$263$	−1.19904e13	−0.587592	−0.293796	−	0.955868i	$-0.594919\pi$
$263$	−1.19904e13	−0.587592	−0.293796	+	0.955868i	$0.594919\pi$
$264$	0	0
$265$	0	0
$266$	−5.46358e12	−0.251553
$267$	0	0
$268$	−5.02243e13	−2.21908
$269$	3.81149e13	1.64990	0.824948	−	0.565208i	$-0.191204\pi$
$269$	3.81149e13	1.64990	0.824948	+	0.565208i	$0.191204\pi$
$270$	0	0
$271$	−1.13510e13	−0.471740	−0.235870	−	0.971785i	$-0.575794\pi$
$271$	−1.13510e13	−0.471740	−0.235870	+	0.971785i	$0.575794\pi$
$272$	−2.51670e13	−1.02495
$273$	0	0
$274$	5.26827e12	0.206082
$275$	0	0
$276$	0	0
$277$	−3.02533e13	−1.11464	−0.557320	−	0.830298i	$-0.688170\pi$
$277$	−3.02533e13	−1.11464	−0.557320	+	0.830298i	$0.688170\pi$
$278$	4.28205e13	1.54670
$279$	0	0
$280$	0	0
$281$	−4.72047e13	−1.60731	−0.803657	−	0.595093i	$-0.797115\pi$
$281$	−4.72047e13	−1.60731	−0.803657	+	0.595093i	$0.797115\pi$
$282$	0	0
$283$	8.68805e12	0.284510	0.142255	−	0.989830i	$-0.454565\pi$
$283$	8.68805e12	0.284510	0.142255	+	0.989830i	$0.454565\pi$
$284$	−8.84140e13	−2.83969
$285$	0	0
$286$	−5.75528e13	−1.77850
$287$	2.93617e12	0.0890085
$288$	0	0
$289$	−2.82663e13	−0.824766
$290$	0	0
$291$	0	0
$292$	1.15817e14	3.19276
$293$	2.17869e13	0.589417	0.294709	−	0.955587i	$-0.404777\pi$
$293$	2.17869e13	0.589417	0.294709	+	0.955587i	$0.404777\pi$
$294$	0	0
$295$	0	0
$296$	−1.11593e14	−2.85450
$297$	0	0
$298$	6.64939e13	1.63905
$299$	−5.79665e13	−1.40277
$300$	0	0
$301$	1.04917e13	0.244754
$302$	−1.04351e14	−2.39033
$303$	0	0
$304$	−4.19516e13	−0.926710
$305$	0	0
$306$	0	0
$307$	−2.05380e13	−0.429829	−0.214915	−	0.976633i	$-0.568947\pi$
$307$	−2.05380e13	−0.429829	−0.214915	+	0.976633i	$0.568947\pi$
$308$	2.69305e13	0.553626
$309$	0	0
$310$	0	0
$311$	7.37156e13	1.43674	0.718369	−	0.695663i	$-0.244889\pi$
$311$	7.37156e13	1.43674	0.718369	+	0.695663i	$0.244889\pi$
$312$	0	0
$313$	2.82026e13	0.530634	0.265317	−	0.964161i	$-0.414523\pi$
$313$	2.82026e13	0.530634	0.265317	+	0.964161i	$0.414523\pi$
$314$	8.15120e13	1.50698
$315$	0	0
$316$	6.20108e13	1.10710
$317$	2.56938e13	0.450819	0.225410	−	0.974264i	$-0.427628\pi$
$317$	2.56938e13	0.450819	0.225410	+	0.974264i	$0.427628\pi$
$318$	0	0
$319$	−3.19763e12	−0.0541976
$320$	0	0
$321$	0	0
$322$	3.83179e13	0.616872
$323$	1.00109e13	0.158439
$324$	0	0
$325$	0	0
$326$	1.05569e14	1.58797
$327$	0	0
$328$	4.48571e13	0.652417
$329$	3.14824e12	0.0450288
$330$	0	0
$331$	−6.14309e13	−0.849831	−0.424916	−	0.905233i	$-0.639696\pi$
$331$	−6.14309e13	−0.849831	−0.424916	+	0.905233i	$0.639696\pi$
$332$	1.67258e14	2.27576
$333$	0	0
$334$	2.52358e14	3.32210
$335$	0	0
$336$	0	0
$337$	−1.35566e14	−1.69898	−0.849488	−	0.527608i	$-0.823089\pi$
$337$	−1.35566e14	−1.69898	−0.849488	+	0.527608i	$0.823089\pi$
$338$	−1.92703e14	−2.37600
$339$	0	0
$340$	0	0
$341$	−1.01715e14	−1.19463
$342$	0	0
$343$	5.90945e13	0.672093
$344$	1.60285e14	1.79400
$345$	0	0
$346$	2.57452e13	0.279111
$347$	−2.88020e13	−0.307334	−0.153667	−	0.988123i	$-0.549108\pi$
$347$	−2.88020e13	−0.307334	−0.153667	+	0.988123i	$0.549108\pi$
$348$	0	0
$349$	3.70742e13	0.383294	0.191647	−	0.981464i	$-0.438617\pi$
$349$	3.70742e13	0.383294	0.191647	+	0.981464i	$0.438617\pi$
$350$	0	0
$351$	0	0
$352$	1.22329e14	1.20655
$353$	−5.64627e13	−0.548278	−0.274139	−	0.961690i	$-0.588393\pi$
$353$	−5.64627e13	−0.548278	−0.274139	+	0.961690i	$0.588393\pi$
$354$	0	0
$355$	0	0
$356$	−1.46040e14	−1.35362
$357$	0	0
$358$	−1.60316e14	−1.44086
$359$	1.96868e14	1.74243	0.871217	−	0.490899i	$-0.163332\pi$
$359$	1.96868e14	1.74243	0.871217	+	0.490899i	$0.163332\pi$
$360$	0	0
$361$	−9.98027e13	−0.856747
$362$	−9.23537e12	−0.0780831
$363$	0	0
$364$	1.60387e14	1.31556
$365$	0	0
$366$	0	0
$367$	1.00444e14	0.787514	0.393757	−	0.919214i	$-0.371175\pi$
$367$	1.00444e14	0.787514	0.393757	+	0.919214i	$0.371175\pi$
$368$	2.94221e14	2.27253
$369$	0	0
$370$	0	0
$371$	−8.24065e13	−0.608703
$372$	0	0
$373$	7.09849e13	0.509058	0.254529	−	0.967065i	$-0.418079\pi$
$373$	7.09849e13	0.509058	0.254529	+	0.967065i	$0.418079\pi$
$374$	−6.97090e13	−0.492600
$375$	0	0
$376$	4.80970e13	0.330054
$377$	−1.90437e13	−0.128788
$378$	0	0
$379$	6.79976e13	0.446661	0.223330	−	0.974743i	$-0.428307\pi$
$379$	6.79976e13	0.446661	0.223330	+	0.974743i	$0.428307\pi$
$380$	0	0
$381$	0	0
$382$	−3.71750e14	−2.33831
$383$	1.89007e14	1.17188	0.585942	−	0.810353i	$-0.300725\pi$
$383$	1.89007e14	1.17188	0.585942	+	0.810353i	$0.300725\pi$
$384$	0	0
$385$	0	0
$386$	2.64344e14	1.57014
$387$	0	0
$388$	5.62070e14	3.24501
$389$	2.90163e14	1.65165	0.825826	−	0.563924i	$-0.190709\pi$
$389$	2.90163e14	1.65165	0.825826	+	0.563924i	$0.190709\pi$
$390$	0	0
$391$	−7.02100e13	−0.388532
$392$	4.20271e14	2.29328
$393$	0	0
$394$	3.00275e14	1.59327
$395$	0	0
$396$	0	0
$397$	5.15115e13	0.262154	0.131077	−	0.991372i	$-0.458157\pi$
$397$	5.15115e13	0.262154	0.131077	+	0.991372i	$0.458157\pi$
$398$	−3.47328e14	−1.74334
$399$	0	0
$400$	0	0
$401$	1.11012e14	0.534657	0.267329	−	0.963605i	$-0.413859\pi$
$401$	1.11012e14	0.534657	0.267329	+	0.963605i	$0.413859\pi$
$402$	0	0
$403$	−6.05768e14	−2.83876
$404$	−5.93346e14	−2.74291
$405$	0	0
$406$	1.25886e13	0.0566349
$407$	−1.55351e14	−0.689517
$408$	0	0
$409$	−1.92951e14	−0.833623	−0.416811	−	0.908993i	$-0.636852\pi$
$409$	−1.92951e14	−0.833623	−0.416811	+	0.908993i	$0.636852\pi$
$410$	0	0
$411$	0	0
$412$	4.69894e14	1.95014
$413$	−1.05778e13	−0.0433180
$414$	0	0
$415$	0	0
$416$	7.28538e14	2.86707
$417$	0	0
$418$	−1.16200e14	−0.445386
$419$	−2.28750e14	−0.865336	−0.432668	−	0.901553i	$-0.642428\pi$
$419$	−2.28750e14	−0.865336	−0.432668	+	0.901553i	$0.642428\pi$
$420$	0	0
$421$	−4.00332e14	−1.47526	−0.737630	−	0.675205i	$-0.764055\pi$
$421$	−4.00332e14	−1.47526	−0.737630	+	0.675205i	$0.764055\pi$
$422$	−9.56816e13	−0.348024
$423$	0	0
$424$	−1.25896e15	−4.46169
$425$	0	0
$426$	0	0
$427$	−8.91803e12	−0.0304030
$428$	7.58132e14	2.55155
$429$	0	0
$430$	0	0
$431$	−1.29757e14	−0.420250	−0.210125	−	0.977675i	$-0.567387\pi$
$431$	−1.29757e14	−0.420250	−0.210125	+	0.977675i	$0.567387\pi$
$432$	0	0
$433$	3.09354e14	0.976725	0.488363	−	0.872641i	$-0.337594\pi$
$433$	3.09354e14	0.976725	0.488363	+	0.872641i	$0.337594\pi$
$434$	4.00435e14	1.24835
$435$	0	0
$436$	1.41473e15	4.30028
$437$	−1.17035e14	−0.351293
$438$	0	0
$439$	−6.15652e13	−0.180211	−0.0901054	−	0.995932i	$-0.528720\pi$
$439$	−6.15652e13	−0.180211	−0.0901054	+	0.995932i	$0.528720\pi$
$440$	0	0
$441$	0	0
$442$	−4.15156e14	−1.17055
$443$	−2.42894e13	−0.0676389	−0.0338194	−	0.999428i	$-0.510767\pi$
$443$	−2.42894e13	−0.0676389	−0.0338194	+	0.999428i	$0.510767\pi$
$444$	0	0
$445$	0	0
$446$	2.97146e14	0.797311
$447$	0	0
$448$	−1.45631e14	−0.381263
$449$	−6.65728e14	−1.72164	−0.860820	−	0.508910i	$-0.830049\pi$
$449$	−6.65728e14	−1.72164	−0.860820	+	0.508910i	$0.830049\pi$
$450$	0	0
$451$	6.24468e13	0.157594
$452$	−1.16900e15	−2.91442
$453$	0	0
$454$	−3.26999e14	−0.795681
$455$	0	0
$456$	0	0
$457$	−6.99716e14	−1.64204	−0.821018	−	0.570902i	$-0.806594\pi$
$457$	−6.99716e14	−1.64204	−0.821018	+	0.570902i	$0.806594\pi$
$458$	4.06192e14	0.941827
$459$	0	0
$460$	0	0
$461$	−3.43320e14	−0.767971	−0.383985	−	0.923339i	$-0.625449\pi$
$461$	−3.43320e14	−0.767971	−0.383985	+	0.923339i	$0.625449\pi$
$462$	0	0
$463$	−3.77708e14	−0.825012	−0.412506	−	0.910955i	$-0.635346\pi$
$463$	−3.77708e14	−0.825012	−0.412506	+	0.910955i	$0.635346\pi$
$464$	9.66603e13	0.208641
$465$	0	0
$466$	−8.66368e14	−1.82633
$467$	−5.01733e14	−1.04527	−0.522637	−	0.852555i	$-0.675052\pi$
$467$	−5.01733e14	−1.04527	−0.522637	+	0.852555i	$0.675052\pi$
$468$	0	0
$469$	1.61663e14	0.328972
$470$	0	0
$471$	0	0
$472$	−1.61601e14	−0.317513
$473$	2.23138e14	0.433348
$474$	0	0
$475$	0	0
$476$	1.94263e14	0.364378
$477$	0	0
$478$	1.55016e14	0.284134
$479$	−9.56649e14	−1.73343	−0.866717	−	0.498800i	$-0.833774\pi$
$479$	−9.56649e14	−1.73343	−0.866717	+	0.498800i	$0.833774\pi$
$480$	0	0
$481$	−9.25204e14	−1.63847
$482$	−2.28596e14	−0.400231
$483$	0	0
$484$	−8.43122e14	−1.44292
$485$	0	0
$486$	0	0
$487$	−1.81856e14	−0.300828	−0.150414	−	0.988623i	$-0.548061\pi$
$487$	−1.81856e14	−0.300828	−0.150414	+	0.988623i	$0.548061\pi$
$488$	−1.36244e14	−0.222848
$489$	0	0
$490$	0	0
$491$	−9.28536e14	−1.46842	−0.734210	−	0.678922i	$-0.762447\pi$
$491$	−9.28536e14	−1.46842	−0.734210	+	0.678922i	$0.762447\pi$
$492$	0	0
$493$	−2.30661e13	−0.0356711
$494$	−6.92037e14	−1.05835
$495$	0	0
$496$	3.07470e15	4.59889
$497$	2.84589e14	0.420976
$498$	0	0
$499$	1.10604e15	1.60036	0.800178	−	0.599763i	$-0.204739\pi$
$499$	1.10604e15	1.60036	0.800178	+	0.599763i	$0.204739\pi$
$500$	0	0
$501$	0	0
$502$	9.96177e14	1.39465
$503$	−1.07467e15	−1.48816	−0.744081	−	0.668089i	$-0.767112\pi$
$503$	−1.07467e15	−1.48816	−0.744081	+	0.668089i	$0.767112\pi$
$504$	0	0
$505$	0	0
$506$	8.14950e14	1.09220
$507$	0	0
$508$	6.24222e14	0.818632
$509$	1.50116e15	1.94750	0.973752	−	0.227610i	$-0.0730910\pi$
$509$	1.50116e15	1.94750	0.973752	+	0.227610i	$0.0730910\pi$
$510$	0	0
$511$	−3.72793e14	−0.473318
$512$	1.43474e15	1.80214
$513$	0	0
$514$	−2.96299e12	−0.00364278
$515$	0	0
$516$	0	0
$517$	6.69571e13	0.0797258
$518$	6.11593e14	0.720525
$519$	0	0
$520$	0	0
$521$	8.31060e13	0.0948473	0.0474237	−	0.998875i	$-0.484899\pi$
$521$	8.31060e13	0.0948473	0.0474237	+	0.998875i	$0.484899\pi$
$522$	0	0
$523$	9.42223e14	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$523$	9.42223e14	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$524$	1.47561e15	1.63173
$525$	0	0
$526$	1.00394e15	1.08715
$527$	−7.33717e14	−0.786267
$528$	0	0
$529$	−1.32002e14	−0.138540
$530$	0	0
$531$	0	0
$532$	3.23824e14	0.329454
$533$	3.71906e14	0.374485
$534$	0	0
$535$	0	0
$536$	2.46979e15	2.41131
$537$	0	0
$538$	−3.19133e15	−3.05259
$539$	5.85071e14	0.553950
$540$	0	0
$541$	1.18229e15	1.09683	0.548416	−	0.836206i	$-0.315231\pi$
$541$	1.18229e15	1.09683	0.548416	+	0.836206i	$0.315231\pi$
$542$	9.50410e14	0.872800
$543$	0	0
$544$	8.82418e14	0.794110
$545$	0	0
$546$	0	0
$547$	−1.57701e15	−1.37691	−0.688453	−	0.725281i	$-0.741710\pi$
$547$	−1.57701e15	−1.37691	−0.688453	+	0.725281i	$0.741710\pi$
$548$	−3.12247e14	−0.269902
$549$	0	0
$550$	0	0
$551$	−3.84496e13	−0.0322521
$552$	0	0
$553$	−1.99601e14	−0.164125
$554$	2.53309e15	2.06228
$555$	0	0
$556$	−2.53795e15	−2.02568
$557$	4.53070e14	0.358065	0.179033	−	0.983843i	$-0.442703\pi$
$557$	4.53070e14	0.358065	0.179033	+	0.983843i	$0.442703\pi$
$558$	0	0
$559$	1.32891e15	1.02975
$560$	0	0
$561$	0	0
$562$	3.95241e15	2.97381
$563$	4.23450e13	0.0315505	0.0157752	−	0.999876i	$-0.494978\pi$
$563$	4.23450e13	0.0315505	0.0157752	+	0.999876i	$0.494978\pi$
$564$	0	0
$565$	0	0
$566$	−7.27444e14	−0.526392
$567$	0	0
$568$	4.34777e15	3.08568
$569$	9.72594e14	0.683619	0.341810	−	0.939769i	$-0.388960\pi$
$569$	9.72594e14	0.683619	0.341810	+	0.939769i	$0.388960\pi$
$570$	0	0
$571$	−2.07663e15	−1.43173	−0.715863	−	0.698241i	$-0.753966\pi$
$571$	−2.07663e15	−1.43173	−0.715863	+	0.698241i	$0.753966\pi$
$572$	3.41112e15	2.32927
$573$	0	0
$574$	−2.45844e14	−0.164681
$575$	0	0
$576$	0	0
$577$	−2.62999e15	−1.71194	−0.855968	−	0.517028i	$-0.827038\pi$
$577$	−2.62999e15	−1.71194	−0.855968	+	0.517028i	$0.827038\pi$
$578$	2.36671e15	1.52596
$579$	0	0
$580$	0	0
$581$	−5.38371e14	−0.337375
$582$	0	0
$583$	−1.75263e15	−1.07774
$584$	−5.69531e15	−3.46934
$585$	0	0
$586$	−1.82420e15	−1.09052
$587$	−2.89275e15	−1.71317	−0.856587	−	0.516003i	$-0.827419\pi$
$587$	−2.89275e15	−1.71317	−0.856587	+	0.516003i	$0.827419\pi$
$588$	0	0
$589$	−1.22306e15	−0.710906
$590$	0	0
$591$	0	0
$592$	4.69606e15	2.65439
$593$	−1.15734e15	−0.648125	−0.324063	−	0.946036i	$-0.605049\pi$
$593$	−1.15734e15	−0.648125	−0.324063	+	0.946036i	$0.605049\pi$
$594$	0	0
$595$	0	0
$596$	−3.94106e15	−2.14663
$597$	0	0
$598$	4.85349e15	2.59536
$599$	−2.42056e14	−0.128253	−0.0641265	−	0.997942i	$-0.520426\pi$
$599$	−2.42056e14	−0.128253	−0.0641265	+	0.997942i	$0.520426\pi$
$600$	0	0
$601$	−3.08465e15	−1.60471	−0.802354	−	0.596848i	$-0.796419\pi$
$601$	−3.08465e15	−1.60471	−0.802354	+	0.596848i	$0.796419\pi$
$602$	−8.78460e14	−0.452836
$603$	0	0
$604$	6.18480e15	3.13057
$605$	0	0
$606$	0	0
$607$	−2.71223e15	−1.33595	−0.667973	−	0.744186i	$-0.732838\pi$
$607$	−2.71223e15	−1.33595	−0.667973	+	0.744186i	$0.732838\pi$
$608$	1.47093e15	0.717997
$609$	0	0
$610$	0	0
$611$	3.98768e14	0.189449
$612$	0	0
$613$	1.80466e15	0.842097	0.421049	−	0.907038i	$-0.361662\pi$
$613$	1.80466e15	0.842097	0.421049	+	0.907038i	$0.361662\pi$
$614$	1.71963e15	0.795258
$615$	0	0
$616$	−1.32431e15	−0.601585
$617$	4.28183e14	0.192780	0.0963898	−	0.995344i	$-0.469270\pi$
$617$	4.28183e14	0.192780	0.0963898	+	0.995344i	$0.469270\pi$
$618$	0	0
$619$	2.39110e15	1.05754	0.528772	−	0.848764i	$-0.322653\pi$
$619$	2.39110e15	1.05754	0.528772	+	0.848764i	$0.322653\pi$
$620$	0	0
$621$	0	0
$622$	−6.17215e15	−2.65821
$623$	4.70076e14	0.200671
$624$	0	0
$625$	0	0
$626$	−2.36138e15	−0.981763
$627$	0	0
$628$	−4.83117e15	−1.97367
$629$	−1.12062e15	−0.453818
$630$	0	0
$631$	3.26028e15	1.29746	0.648730	−	0.761019i	$-0.275301\pi$
$631$	3.26028e15	1.29746	0.648730	+	0.761019i	$0.275301\pi$
$632$	−3.04939e15	−1.20301
$633$	0	0
$634$	−2.15132e15	−0.834093
$635$	0	0
$636$	0	0
$637$	3.48443e15	1.31633
$638$	2.67735e14	0.100275
$639$	0	0
$640$	0	0
$641$	2.32104e15	0.847157	0.423579	−	0.905859i	$-0.360774\pi$
$641$	2.32104e15	0.847157	0.423579	+	0.905859i	$0.360774\pi$
$642$	0	0
$643$	2.55397e15	0.916336	0.458168	−	0.888866i	$-0.348506\pi$
$643$	2.55397e15	0.916336	0.458168	+	0.888866i	$0.348506\pi$
$644$	−2.27108e15	−0.807906
$645$	0	0
$646$	−8.38208e14	−0.293139
$647$	−1.70408e15	−0.590902	−0.295451	−	0.955358i	$-0.595470\pi$
$647$	−1.70408e15	−0.590902	−0.295451	+	0.955358i	$0.595470\pi$
$648$	0	0
$649$	−2.24969e14	−0.0766966
$650$	0	0
$651$	0	0
$652$	−6.25704e15	−2.07973
$653$	−2.20439e15	−0.726552	−0.363276	−	0.931682i	$-0.618342\pi$
$653$	−2.20439e15	−0.726552	−0.363276	+	0.931682i	$0.618342\pi$
$654$	0	0
$655$	0	0
$656$	−1.88769e15	−0.606679
$657$	0	0
$658$	−2.63600e14	−0.0833111
$659$	−1.97759e14	−0.0619821	−0.0309910	−	0.999520i	$-0.509866\pi$
$659$	−1.97759e14	−0.0619821	−0.0309910	+	0.999520i	$0.509866\pi$
$660$	0	0
$661$	−3.43414e15	−1.05855	−0.529274	−	0.848451i	$-0.677536\pi$
$661$	−3.43414e15	−1.05855	−0.529274	+	0.848451i	$0.677536\pi$
$662$	5.14356e15	1.57233
$663$	0	0
$664$	−8.22492e15	−2.47290
$665$	0	0
$666$	0	0
$667$	2.69660e14	0.0790905
$668$	−1.49571e16	−4.35089
$669$	0	0
$670$	0	0
$671$	−1.89670e14	−0.0538299
$672$	0	0
$673$	−2.01250e15	−0.561892	−0.280946	−	0.959724i	$-0.590648\pi$
$673$	−2.01250e15	−0.561892	−0.280946	+	0.959724i	$0.590648\pi$
$674$	1.13509e16	3.14340
$675$	0	0
$676$	1.14214e16	3.11181
$677$	−1.01903e15	−0.275392	−0.137696	−	0.990475i	$-0.543970\pi$
$677$	−1.01903e15	−0.275392	−0.137696	+	0.990475i	$0.543970\pi$
$678$	0	0
$679$	−1.80920e15	−0.481064
$680$	0	0
$681$	0	0
$682$	8.51648e15	2.21027
$683$	2.86885e15	0.738575	0.369287	−	0.929315i	$-0.379602\pi$
$683$	2.86885e15	0.738575	0.369287	+	0.929315i	$0.379602\pi$
$684$	0	0
$685$	0	0
$686$	−4.94793e15	−1.24349
$687$	0	0
$688$	−6.74517e15	−1.66823
$689$	−1.04379e16	−2.56099
$690$	0	0
$691$	−1.79931e15	−0.434487	−0.217243	−	0.976117i	$-0.569707\pi$
$691$	−1.79931e15	−0.434487	−0.217243	+	0.976117i	$0.569707\pi$
$692$	−1.52590e15	−0.365547
$693$	0	0
$694$	2.41157e15	0.568621
$695$	0	0
$696$	0	0
$697$	4.50460e14	0.103723
$698$	−3.10420e15	−0.709160
$699$	0	0
$700$	0	0
$701$	−2.99337e15	−0.667899	−0.333949	−	0.942591i	$-0.608381\pi$
$701$	−2.99337e15	−0.667899	−0.333949	+	0.942591i	$0.608381\pi$
$702$	0	0
$703$	−1.86800e15	−0.410321
$704$	−3.09729e15	−0.675045
$705$	0	0
$706$	4.72758e15	1.01441
$707$	1.90987e15	0.406629
$708$	0	0
$709$	2.74187e15	0.574769	0.287384	−	0.957815i	$-0.407214\pi$
$709$	2.74187e15	0.574769	0.287384	+	0.957815i	$0.407214\pi$
$710$	0	0
$711$	0	0
$712$	7.18154e15	1.47088
$713$	8.57770e15	1.74333
$714$	0	0
$715$	0	0
$716$	9.50185e15	1.88706
$717$	0	0
$718$	−1.64836e16	−3.22380
$719$	4.38021e15	0.850131	0.425065	−	0.905163i	$-0.360251\pi$
$719$	4.38021e15	0.850131	0.425065	+	0.905163i	$0.360251\pi$
$720$	0	0
$721$	−1.51250e15	−0.289103
$722$	8.35640e15	1.58513
$723$	0	0
$724$	5.47375e14	0.102264
$725$	0	0
$726$	0	0
$727$	−6.60205e15	−1.20570	−0.602851	−	0.797854i	$-0.705969\pi$
$727$	−6.60205e15	−1.20570	−0.602851	+	0.797854i	$0.705969\pi$
$728$	−7.88704e15	−1.42952
$729$	0	0
$730$	0	0
$731$	1.60960e15	0.285216
$732$	0	0
$733$	8.22795e15	1.43622	0.718108	−	0.695932i	$-0.245008\pi$
$733$	8.22795e15	1.43622	0.718108	+	0.695932i	$0.245008\pi$
$734$	−8.41006e15	−1.45704
$735$	0	0
$736$	−1.03161e16	−1.76071
$737$	3.43826e15	0.582462
$738$	0	0
$739$	−1.10770e16	−1.84875	−0.924376	−	0.381483i	$-0.875414\pi$
$739$	−1.10770e16	−1.84875	−0.924376	+	0.381483i	$0.875414\pi$
$740$	0	0
$741$	0	0
$742$	6.89984e15	1.12620
$743$	−3.86160e15	−0.625646	−0.312823	−	0.949811i	$-0.601275\pi$
$743$	−3.86160e15	−0.625646	−0.312823	+	0.949811i	$0.601275\pi$
$744$	0	0
$745$	0	0
$746$	−5.94352e15	−0.941846
$747$	0	0
$748$	4.13161e15	0.645150
$749$	−2.44029e15	−0.378260
$750$	0	0
$751$	1.05947e15	0.161834	0.0809168	−	0.996721i	$-0.474215\pi$
$751$	1.05947e15	0.161834	0.0809168	+	0.996721i	$0.474215\pi$
$752$	−2.02403e15	−0.306915
$753$	0	0
$754$	1.59452e15	0.238280
$755$	0	0
$756$	0	0
$757$	7.04375e15	1.02986	0.514928	−	0.857233i	$-0.327818\pi$
$757$	7.04375e15	1.02986	0.514928	+	0.857233i	$0.327818\pi$
$758$	−5.69339e15	−0.826400
$759$	0	0
$760$	0	0
$761$	−3.63598e15	−0.516424	−0.258212	−	0.966088i	$-0.583133\pi$
$761$	−3.63598e15	−0.516424	−0.258212	+	0.966088i	$0.583133\pi$
$762$	0	0
$763$	−4.55374e15	−0.637505
$764$	2.20334e16	3.06245
$765$	0	0
$766$	−1.58254e16	−2.16819
$767$	−1.33982e15	−0.182251
$768$	0	0
$769$	−5.34829e15	−0.717167	−0.358583	−	0.933498i	$-0.616740\pi$
$769$	−5.34829e15	−0.717167	−0.358583	+	0.933498i	$0.616740\pi$
$770$	0	0
$771$	0	0
$772$	−1.56675e16	−2.05639
$773$	8.06669e15	1.05126	0.525628	−	0.850715i	$-0.323831\pi$
$773$	8.06669e15	1.05126	0.525628	+	0.850715i	$0.323831\pi$
$774$	0	0
$775$	0	0
$776$	−2.76399e16	−3.52612
$777$	0	0
$778$	−2.42951e16	−3.05584
$779$	7.50886e14	0.0937816
$780$	0	0
$781$	6.05266e15	0.745359
$782$	5.87863e15	0.718851
$783$	0	0
$784$	−1.76859e16	−2.13250
$785$	0	0
$786$	0	0
$787$	8.56068e15	1.01076	0.505379	−	0.862897i	$-0.331353\pi$
$787$	8.56068e15	1.01076	0.505379	+	0.862897i	$0.331353\pi$
$788$	−1.77971e16	−2.08668
$789$	0	0
$790$	0	0
$791$	3.76279e15	0.432056
$792$	0	0
$793$	−1.12959e15	−0.127914
$794$	−4.31302e15	−0.485030
$795$	0	0
$796$	2.05859e16	2.28323
$797$	−1.47001e16	−1.61919	−0.809596	−	0.586987i	$-0.800314\pi$
$797$	−1.47001e16	−1.61919	−0.809596	+	0.586987i	$0.800314\pi$
$798$	0	0
$799$	4.82995e14	0.0524729
$800$	0	0
$801$	0	0
$802$	−9.29494e15	−0.989208
$803$	−7.92861e15	−0.838033
$804$	0	0
$805$	0	0
$806$	5.07205e16	5.25219
$807$	0	0
$808$	2.91779e16	2.98052
$809$	−5.20792e15	−0.528382	−0.264191	−	0.964470i	$-0.585105\pi$
$809$	−5.20792e15	−0.528382	−0.264191	+	0.964470i	$0.585105\pi$
$810$	0	0
$811$	1.95571e16	1.95745	0.978724	−	0.205183i	$-0.0657789\pi$
$811$	1.95571e16	1.95745	0.978724	+	0.205183i	$0.0657789\pi$
$812$	−7.46119e14	−0.0741737
$813$	0	0
$814$	1.30074e16	1.27573
$815$	0	0
$816$	0	0
$817$	2.68310e15	0.257879
$818$	1.61557e16	1.54235
$819$	0	0
$820$	0	0
$821$	−1.02440e16	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$821$	−1.02440e16	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$822$	0	0
$823$	1.40372e16	1.29593	0.647964	−	0.761671i	$-0.275621\pi$
$823$	1.40372e16	1.29593	0.647964	+	0.761671i	$0.275621\pi$
$824$	−2.31071e16	−2.11907
$825$	0	0
$826$	8.85667e14	0.0801457
$827$	8.80160e14	0.0791191	0.0395596	−	0.999217i	$-0.487405\pi$
$827$	8.80160e14	0.0791191	0.0395596	+	0.999217i	$0.487405\pi$
$828$	0	0
$829$	4.54642e14	0.0403292	0.0201646	−	0.999797i	$-0.493581\pi$
$829$	4.54642e14	0.0403292	0.0201646	+	0.999797i	$0.493581\pi$
$830$	0	0
$831$	0	0
$832$	−1.84461e16	−1.60408
$833$	4.22041e15	0.364592
$834$	0	0
$835$	0	0
$836$	6.88711e15	0.583314
$837$	0	0
$838$	1.91531e16	1.60102
$839$	−1.32346e16	−1.09906	−0.549529	−	0.835474i	$-0.685193\pi$
$839$	−1.32346e16	−1.09906	−0.549529	+	0.835474i	$0.685193\pi$
$840$	0	0
$841$	−1.21119e16	−0.992739
$842$	3.35195e16	2.72949
$843$	0	0
$844$	5.67099e15	0.455801
$845$	0	0
$846$	0	0
$847$	2.71386e15	0.213908
$848$	5.29797e16	4.14889
$849$	0	0
$850$	0	0
$851$	1.31009e16	1.00621
$852$	0	0
$853$	2.60074e15	0.197186	0.0985932	−	0.995128i	$-0.468566\pi$
$853$	2.60074e15	0.197186	0.0985932	+	0.995128i	$0.468566\pi$
$854$	7.46700e14	0.0562507
$855$	0	0
$856$	−3.72812e16	−2.77258
$857$	−1.68895e16	−1.24802	−0.624011	−	0.781415i	$-0.714498\pi$
$857$	−1.68895e16	−1.24802	−0.624011	+	0.781415i	$0.714498\pi$
$858$	0	0
$859$	8.98241e15	0.655285	0.327643	−	0.944802i	$-0.393746\pi$
$859$	8.98241e15	0.655285	0.327643	+	0.944802i	$0.393746\pi$
$860$	0	0
$861$	0	0
$862$	1.08645e16	0.777534
$863$	−1.16178e16	−0.826164	−0.413082	−	0.910694i	$-0.635548\pi$
$863$	−1.16178e16	−0.826164	−0.413082	+	0.910694i	$0.635548\pi$
$864$	0	0
$865$	0	0
$866$	−2.59020e16	−1.80711
$867$	0	0
$868$	−2.37335e16	−1.63495
$869$	−4.24514e15	−0.290592
$870$	0	0
$871$	2.04768e16	1.38408
$872$	−6.95694e16	−4.67280
$873$	0	0
$874$	9.79928e15	0.649952
$875$	0	0
$876$	0	0
$877$	−4.73202e15	−0.307999	−0.153999	−	0.988071i	$-0.549215\pi$
$877$	−4.73202e15	−0.307999	−0.153999	+	0.988071i	$0.549215\pi$
$878$	5.15481e15	0.333421
$879$	0	0
$880$	0	0
$881$	−2.81982e15	−0.179000	−0.0895002	−	0.995987i	$-0.528527\pi$
$881$	−2.81982e15	−0.179000	−0.0895002	+	0.995987i	$0.528527\pi$
$882$	0	0
$883$	−1.91741e16	−1.20208	−0.601038	−	0.799221i	$-0.705246\pi$
$883$	−1.91741e16	−1.20208	−0.601038	+	0.799221i	$0.705246\pi$
$884$	2.46061e16	1.53305
$885$	0	0
$886$	2.03373e15	0.125144
$887$	2.08389e15	0.127437	0.0637183	−	0.997968i	$-0.479704\pi$
$887$	2.08389e15	0.127437	0.0637183	+	0.997968i	$0.479704\pi$
$888$	0	0
$889$	−2.00926e15	−0.121360
$890$	0	0
$891$	0	0
$892$	−1.76117e16	−1.04422
$893$	8.05119e14	0.0474435
$894$	0	0
$895$	0	0
$896$	4.13967e14	0.0239481
$897$	0	0
$898$	5.57409e16	3.18533
$899$	2.81803e15	0.160054
$900$	0	0
$901$	−1.26426e16	−0.709332
$902$	−5.22863e15	−0.291576
$903$	0	0
$904$	5.74857e16	3.16689
$905$	0	0
$906$	0	0
$907$	−3.20443e16	−1.73345	−0.866723	−	0.498791i	$-0.833778\pi$
$907$	−3.20443e16	−1.73345	−0.866723	+	0.498791i	$0.833778\pi$
$908$	1.93810e16	1.04209
$909$	0	0
$910$	0	0
$911$	8.44282e15	0.445797	0.222898	−	0.974842i	$-0.428448\pi$
$911$	8.44282e15	0.445797	0.222898	+	0.974842i	$0.428448\pi$
$912$	0	0
$913$	−1.14501e16	−0.597340
$914$	5.85867e16	3.03805
$915$	0	0
$916$	−2.40748e16	−1.23349
$917$	−4.74971e15	−0.241900
$918$	0	0
$919$	−1.08277e16	−0.544880	−0.272440	−	0.962173i	$-0.587831\pi$
$919$	−1.08277e16	−0.544880	−0.272440	+	0.962173i	$0.587831\pi$
$920$	0	0
$921$	0	0
$922$	2.87460e16	1.42088
$923$	3.60470e16	1.77117
$924$	0	0
$925$	0	0
$926$	3.16252e16	1.52641
$927$	0	0
$928$	−3.38916e15	−0.161651
$929$	1.30338e16	0.617994	0.308997	−	0.951063i	$-0.400007\pi$
$929$	1.30338e16	0.617994	0.308997	+	0.951063i	$0.400007\pi$
$930$	0	0
$931$	7.03513e15	0.329647
$932$	5.13491e16	2.39191
$933$	0	0
$934$	4.20098e16	1.93394
$935$	0	0
$936$	0	0
$937$	2.06679e16	0.934819	0.467410	−	0.884041i	$-0.345187\pi$
$937$	2.06679e16	0.934819	0.467410	+	0.884041i	$0.345187\pi$
$938$	−1.35359e16	−0.608656
$939$	0	0
$940$	0	0
$941$	3.48692e16	1.54063	0.770317	−	0.637661i	$-0.220098\pi$
$941$	3.48692e16	1.54063	0.770317	+	0.637661i	$0.220098\pi$
$942$	0	0
$943$	−5.26621e15	−0.229977
$944$	6.80051e15	0.295254
$945$	0	0
$946$	−1.86832e16	−0.801769
$947$	−2.53080e15	−0.107977	−0.0539887	−	0.998542i	$-0.517193\pi$
$947$	−2.53080e15	−0.107977	−0.0539887	+	0.998542i	$0.517193\pi$
$948$	0	0
$949$	−4.72194e16	−1.99139
$950$	0	0
$951$	0	0
$952$	−9.55293e15	−0.395944
$953$	−1.68411e16	−0.693999	−0.347000	−	0.937865i	$-0.612799\pi$
$953$	−1.68411e16	−0.693999	−0.347000	+	0.937865i	$0.612799\pi$
$954$	0	0
$955$	0	0
$956$	−9.18773e15	−0.372126
$957$	0	0
$958$	8.00995e16	3.20715
$959$	1.00507e15	0.0400122
$960$	0	0
$961$	6.42312e16	2.52794
$962$	7.74666e16	3.03146
$963$	0	0
$964$	1.35488e16	0.524175
$965$	0	0
$966$	0	0
$967$	2.82761e16	1.07541	0.537705	−	0.843133i	$-0.319292\pi$
$967$	2.82761e16	1.07541	0.537705	+	0.843133i	$0.319292\pi$
$968$	4.14607e16	1.56791
$969$	0	0
$970$	0	0
$971$	−2.55089e16	−0.948386	−0.474193	−	0.880421i	$-0.657260\pi$
$971$	−2.55089e16	−0.948386	−0.474193	+	0.880421i	$0.657260\pi$
$972$	0	0
$973$	8.16918e15	0.300302
$974$	1.52266e16	0.556583
$975$	0	0
$976$	5.73346e15	0.207225
$977$	5.16659e16	1.85688	0.928441	−	0.371480i	$-0.121150\pi$
$977$	5.16659e16	1.85688	0.928441	+	0.371480i	$0.121150\pi$
$978$	0	0
$979$	9.99762e15	0.355297
$980$	0	0
$981$	0	0
$982$	7.77456e16	2.71683
$983$	1.06975e15	0.0371740	0.0185870	−	0.999827i	$-0.494083\pi$
$983$	1.06975e15	0.0371740	0.0185870	+	0.999827i	$0.494083\pi$
$984$	0	0
$985$	0	0
$986$	1.93131e15	0.0659976
$987$	0	0
$988$	4.10167e16	1.38611
$989$	−1.88175e16	−0.632385
$990$	0	0
$991$	2.05333e16	0.682423	0.341211	−	0.939987i	$-0.389163\pi$
$991$	2.05333e16	0.682423	0.341211	+	0.939987i	$0.389163\pi$
$992$	−1.07807e17	−3.56313
$993$	0	0
$994$	−2.38284e16	−0.778878
$995$	0	0
$996$	0	0
$997$	1.61931e16	0.520603	0.260301	−	0.965527i	$-0.416178\pi$
$997$	1.61931e16	0.520603	0.260301	+	0.965527i	$0.416178\pi$
$998$	−9.26076e16	−2.96093
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.a.h.1.1		2
3.2	odd	2		25.12.a.c.1.2		2
5.2	odd	4		225.12.b.f.199.1		4
5.3	odd	4		225.12.b.f.199.4		4
5.4	even	2		45.12.a.d.1.2		2
15.2	even	4		25.12.b.c.24.4		4
15.8	even	4		25.12.b.c.24.1		4
15.14	odd	2		5.12.a.b.1.1	✓	2
60.59	even	2		80.12.a.j.1.2		2
105.104	even	2		245.12.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.12.a.b.1.1	✓	2	15.14	odd	2
25.12.a.c.1.2		2	3.2	odd	2
25.12.b.c.24.1		4	15.8	even	4
25.12.b.c.24.4		4	15.2	even	4
45.12.a.d.1.2		2	5.4	even	2
80.12.a.j.1.2		2	60.59	even	2
225.12.a.h.1.1		2	1.1	even	1	trivial
225.12.b.f.199.1		4	5.2	odd	4
225.12.b.f.199.4		4	5.3	odd	4
245.12.a.b.1.1		2	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-83.7292 q^{2} +4962.58 q^{4} -15973.7 q^{7} -244036. q^{8} +O(q^{10})\)	\(q-83.7292 q^{2} +4962.58 q^{4} -15973.7 q^{7} -244036. q^{8} -339729. q^{11} -2.02328e6 q^{13} +1.33746e6 q^{14} +1.02696e7 q^{16} -2.45063e6 q^{17} -4.08504e6 q^{19} +2.84453e7 q^{22} +2.86497e7 q^{23} +1.69408e8 q^{26} -7.92706e7 q^{28} +9.41230e6 q^{29} +2.99399e8 q^{31} -3.60078e8 q^{32} +2.05190e8 q^{34} +4.57279e8 q^{37} +3.42037e8 q^{38} -1.83814e8 q^{41} -6.56811e8 q^{43} -1.68594e9 q^{44} -2.39882e9 q^{46} -1.97090e8 q^{47} -1.72217e9 q^{49} -1.00407e10 q^{52} +5.15890e9 q^{53} +3.89815e9 q^{56} -7.88085e8 q^{58} +6.62200e8 q^{59} +5.58296e8 q^{61} -2.50684e10 q^{62} +9.11694e9 q^{64} -1.01206e10 q^{67} -1.21615e10 q^{68} -1.78161e10 q^{71} +2.33380e10 q^{73} -3.82876e10 q^{74} -2.02724e10 q^{76} +5.42672e9 q^{77} +1.24957e10 q^{79} +1.53906e10 q^{82} +3.37037e10 q^{83} +5.49943e10 q^{86} +8.29062e10 q^{88} -2.94282e10 q^{89} +3.23192e10 q^{91} +1.42177e11 q^{92} +1.65022e10 q^{94} +1.13262e11 q^{97} +1.44196e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{2} + 6976 q^{4} - 57900 q^{7} - 246240 q^{8}+O(q^{10}) \) 2 * q - 20 * q^2 + 6976 * q^4 - 57900 * q^7 - 246240 * q^8	\( 2 q - 20 q^{2} + 6976 q^{4} - 57900 q^{7} - 246240 q^{8} + 618176 q^{11} - 3414260 q^{13} - 1334472 q^{14} + 6005632 q^{16} + 1317940 q^{17} + 5325320 q^{19} + 89491840 q^{22} + 58943940 q^{23} + 80761736 q^{26} - 163685760 q^{28} - 94140380 q^{29} + 244543464 q^{31} - 627301120 q^{32} + 445358072 q^{34} - 21003220 q^{37} + 941752240 q^{38} + 745743316 q^{41} - 629950100 q^{43} + 242725888 q^{44} - 468194856 q^{46} - 1402061540 q^{47} - 1941677414 q^{49} - 12841321600 q^{52} + 1138320580 q^{53} + 3990553920 q^{56} - 7387417960 q^{58} - 7317515560 q^{59} - 1516425676 q^{61} - 28564327440 q^{62} + 819531776 q^{64} - 15734290140 q^{67} - 4573774720 q^{68} - 32938471544 q^{71} + 29982848860 q^{73} - 68768198072 q^{74} - 1325392640 q^{76} - 34734748800 q^{77} - 3302823120 q^{79} + 74630515640 q^{82} + 13299102420 q^{83} + 56706093896 q^{86} + 80794874880 q^{88} + 12674770860 q^{89} + 90637859064 q^{91} + 203171571840 q^{92} - 60289765528 q^{94} + 3080703740 q^{97} + 130206802940 q^{98}+O(q^{100}) \) 2 * q - 20 * q^2 + 6976 * q^4 - 57900 * q^7 - 246240 * q^8 + 618176 * q^11 - 3414260 * q^13 - 1334472 * q^14 + 6005632 * q^16 + 1317940 * q^17 + 5325320 * q^19 + 89491840 * q^22 + 58943940 * q^23 + 80761736 * q^26 - 163685760 * q^28 - 94140380 * q^29 + 244543464 * q^31 - 627301120 * q^32 + 445358072 * q^34 - 21003220 * q^37 + 941752240 * q^38 + 745743316 * q^41 - 629950100 * q^43 + 242725888 * q^44 - 468194856 * q^46 - 1402061540 * q^47 - 1941677414 * q^49 - 12841321600 * q^52 + 1138320580 * q^53 + 3990553920 * q^56 - 7387417960 * q^58 - 7317515560 * q^59 - 1516425676 * q^61 - 28564327440 * q^62 + 819531776 * q^64 - 15734290140 * q^67 - 4573774720 * q^68 - 32938471544 * q^71 + 29982848860 * q^73 - 68768198072 * q^74 - 1325392640 * q^76 - 34734748800 * q^77 - 3302823120 * q^79 + 74630515640 * q^82 + 13299102420 * q^83 + 56706093896 * q^86 + 80794874880 * q^88 + 12674770860 * q^89 + 90637859064 * q^91 + 203171571840 * q^92 - 60289765528 * q^94 + 3080703740 * q^97 + 130206802940 * q^98

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