LMFDB - Embedded newform 1197.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1197,2,Mod(1,1197)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1197, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1197.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1197 = 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1197.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:	$9.55809312195$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.09027$ of defining polynomial
Character	$\chi$	$=$	1197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.09027 q^{2} +2.36924 q^{4} -0.388134 q^{5} +1.00000 q^{7} -0.771813 q^{8} +0.811305 q^{10} -6.41859 q^{11} +3.88714 q^{13} -2.09027 q^{14} -3.12518 q^{16} +4.98727 q^{17} -1.00000 q^{19} -0.919582 q^{20} +13.4166 q^{22} -3.44207 q^{23} -4.84935 q^{25} -8.12518 q^{26} +2.36924 q^{28} -0.169801 q^{29} -8.62562 q^{31} +8.07611 q^{32} -10.4247 q^{34} -0.388134 q^{35} +7.37540 q^{37} +2.09027 q^{38} +0.299567 q^{40} +8.77968 q^{41} -9.11087 q^{43} -15.2072 q^{44} +7.19485 q^{46} +4.80672 q^{47} +1.00000 q^{49} +10.1365 q^{50} +9.20957 q^{52} -8.42933 q^{53} +2.49127 q^{55} -0.771813 q^{56} +0.354931 q^{58} -2.97652 q^{59} +5.82287 q^{61} +18.0299 q^{62} -10.6309 q^{64} -1.50873 q^{65} -14.9980 q^{67} +11.8160 q^{68} +0.811305 q^{70} -4.24879 q^{71} +13.5757 q^{73} -15.4166 q^{74} -2.36924 q^{76} -6.41859 q^{77} -1.01431 q^{79} +1.21299 q^{80} -18.3519 q^{82} +4.32147 q^{83} -1.93573 q^{85} +19.0442 q^{86} +4.95395 q^{88} -13.8955 q^{89} +3.88714 q^{91} -8.15508 q^{92} -10.0474 q^{94} +0.388134 q^{95} -13.7743 q^{97} -2.09027 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} + 7 q^{4} - 4 q^{5} + 5 q^{7} - 9 q^{8} - 6 q^{10} - 8 q^{11} - 6 q^{13} - 3 q^{14} + 19 q^{16} - 12 q^{17} - 5 q^{19} - 8 q^{20} - 12 q^{22} - 12 q^{23} + 15 q^{25} - 6 q^{26} + 7 q^{28}+ \cdots - 3 q^{98}+O(q^{100}) $ 5 * q - 3 * q^2 + 7 * q^4 - 4 * q^5 + 5 * q^7 - 9 * q^8 - 6 * q^10 - 8 * q^11 - 6 * q^13 - 3 * q^14 + 19 * q^16 - 12 * q^17 - 5 * q^19 - 8 * q^20 - 12 * q^22 - 12 * q^23 + 15 * q^25 - 6 * q^26 + 7 * q^28 - 4 * q^29 - 8 * q^31 - 31 * q^32 + 2 * q^34 - 4 * q^35 + 2 * q^37 + 3 * q^38 - 18 * q^40 - 10 * q^41 - 16 * q^43 + 4 * q^44 + 16 * q^46 + 2 * q^47 + 5 * q^49 + 39 * q^50 - 22 * q^52 - 12 * q^55 - 9 * q^56 + 2 * q^58 + 4 * q^59 - 14 * q^61 + 36 * q^62 + 11 * q^64 - 32 * q^65 - 20 * q^67 - 8 * q^68 - 6 * q^70 + 6 * q^71 + 10 * q^73 + 2 * q^74 - 7 * q^76 - 8 * q^77 + 20 * q^80 - 38 * q^82 - 6 * q^83 + 8 * q^85 + 36 * q^86 - 40 * q^88 - 26 * q^89 - 6 * q^91 + 48 * q^92 + 24 * q^94 + 4 * q^95 - 18 * q^97 - 3 * q^98

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 $

$2$	−2.09027	−1.47805	−0.739023	−	0.673680i	$-0.764713\pi$
$2$	−2.09027	−1.47805	−0.739023	+	0.673680i	$0.764713\pi$
$3$	0	0
$3$	0	0
$4$	2.36924	1.18462
$5$	−0.388134	−0.173579	−0.0867893	−	0.996227i	$-0.527661\pi$
$5$	−0.388134	−0.173579	−0.0867893	+	0.996227i	$0.527661\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.771813	−0.272877
$9$	0	0
$10$	0.811305	0.256557
$11$	−6.41859	−1.93528	−0.967638	−	0.252341i	$-0.918800\pi$
$11$	−6.41859	−1.93528	−0.967638	+	0.252341i	$0.918800\pi$
$12$	0	0
$13$	3.88714	1.07810	0.539049	−	0.842274i	$-0.318784\pi$
$13$	3.88714	1.07810	0.539049	+	0.842274i	$0.318784\pi$
$14$	−2.09027	−0.558649
$15$	0	0
$16$	−3.12518	−0.781295
$17$	4.98727	1.20959	0.604795	−	0.796381i	$-0.293255\pi$
$17$	4.98727	1.20959	0.604795	+	0.796381i	$0.293255\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.919582	−0.205625
$21$	0	0
$22$	13.4166	2.86043
$23$	−3.44207	−0.717720	−0.358860	−	0.933391i	$-0.616834\pi$
$23$	−3.44207	−0.717720	−0.358860	+	0.933391i	$0.616834\pi$
$24$	0	0
$25$	−4.84935	−0.969870
$26$	−8.12518	−1.59348
$27$	0	0
$28$	2.36924	0.447744
$29$	−0.169801	−0.0315313	−0.0157656	−	0.999876i	$-0.505019\pi$
$29$	−0.169801	−0.0315313	−0.0157656	+	0.999876i	$0.505019\pi$
$30$	0	0
$31$	−8.62562	−1.54921	−0.774604	−	0.632447i	$-0.782051\pi$
$31$	−8.62562	−1.54921	−0.774604	+	0.632447i	$0.782051\pi$
$32$	8.07611	1.42767
$33$	0	0
$34$	−10.4247	−1.78783
$35$	−0.388134	−0.0656066
$36$	0	0
$37$	7.37540	1.21251	0.606254	−	0.795271i	$-0.292671\pi$
$37$	7.37540	1.21251	0.606254	+	0.795271i	$0.292671\pi$
$38$	2.09027	0.339087
$39$	0	0
$40$	0.299567	0.0473656
$41$	8.77968	1.37116	0.685578	−	0.727999i	$-0.259550\pi$
$41$	8.77968	1.37116	0.685578	+	0.727999i	$0.259550\pi$
$42$	0	0
$43$	−9.11087	−1.38939	−0.694697	−	0.719302i	$-0.744462\pi$
$43$	−9.11087	−1.38939	−0.694697	+	0.719302i	$0.744462\pi$
$44$	−15.2072	−2.29257
$45$	0	0
$46$	7.19485	1.06082
$47$	4.80672	0.701132	0.350566	−	0.936538i	$-0.385989\pi$
$47$	4.80672	0.701132	0.350566	+	0.936538i	$0.385989\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.1365	1.43351
$51$	0	0
$52$	9.20957	1.27714
$53$	−8.42933	−1.15786	−0.578929	−	0.815378i	$-0.696529\pi$
$53$	−8.42933	−1.15786	−0.578929	+	0.815378i	$0.696529\pi$
$54$	0	0
$55$	2.49127	0.335923
$56$	−0.771813	−0.103138
$57$	0	0
$58$	0.354931	0.0466047
$59$	−2.97652	−0.387510	−0.193755	−	0.981050i	$-0.562067\pi$
$59$	−2.97652	−0.387510	−0.193755	+	0.981050i	$0.562067\pi$
$60$	0	0
$61$	5.82287	0.745542	0.372771	−	0.927923i	$-0.378408\pi$
$61$	5.82287	0.745542	0.372771	+	0.927923i	$0.378408\pi$
$62$	18.0299	2.28980
$63$	0	0
$64$	−10.6309	−1.32886
$65$	−1.50873	−0.187135
$66$	0	0
$67$	−14.9980	−1.83230	−0.916149	−	0.400837i	$-0.868719\pi$
$67$	−14.9980	−1.83230	−0.916149	+	0.400837i	$0.868719\pi$
$68$	11.8160	1.43290
$69$	0	0
$70$	0.811305	0.0969695
$71$	−4.24879	−0.504238	−0.252119	−	0.967696i	$-0.581127\pi$
$71$	−4.24879	−0.504238	−0.252119	+	0.967696i	$0.581127\pi$
$72$	0	0
$73$	13.5757	1.58891	0.794455	−	0.607323i	$-0.207757\pi$
$73$	13.5757	1.58891	0.794455	+	0.607323i	$0.207757\pi$
$74$	−15.4166	−1.79214
$75$	0	0
$76$	−2.36924	−0.271770
$77$	−6.41859	−0.731466
$78$	0	0
$79$	−1.01431	−0.114119	−0.0570594	−	0.998371i	$-0.518172\pi$
$79$	−1.01431	−0.114119	−0.0570594	+	0.998371i	$0.518172\pi$
$80$	1.21299	0.135616
$81$	0	0
$82$	−18.3519	−2.02663
$83$	4.32147	0.474343	0.237171	−	0.971468i	$-0.423780\pi$
$83$	4.32147	0.474343	0.237171	+	0.971468i	$0.423780\pi$
$84$	0	0
$85$	−1.93573	−0.209959
$86$	19.0442	2.05359
$87$	0	0
$88$	4.95395	0.528093
$89$	−13.8955	−1.47293	−0.736463	−	0.676478i	$-0.763505\pi$
$89$	−13.8955	−1.47293	−0.736463	+	0.676478i	$0.763505\pi$
$90$	0	0
$91$	3.88714	0.407483
$92$	−8.15508	−0.850226
$93$	0	0
$94$	−10.0474	−1.03631
$95$	0.388134	0.0398217
$96$	0	0
$97$	−13.7743	−1.39857	−0.699283	−	0.714845i	$-0.746497\pi$
$97$	−13.7743	−1.39857	−0.699283	+	0.714845i	$0.746497\pi$
$98$	−2.09027	−0.211149
$99$	0	0
$100$	−11.4893	−1.14893
$101$	−8.15009	−0.810964	−0.405482	−	0.914103i	$-0.632896\pi$
$101$	−8.15009	−0.810964	−0.405482	+	0.914103i	$0.632896\pi$
$102$	0	0
$103$	−12.8372	−1.26488	−0.632442	−	0.774608i	$-0.717947\pi$
$103$	−12.8372	−1.26488	−0.632442	+	0.774608i	$0.717947\pi$
$104$	−3.00014	−0.294188
$105$	0	0
$106$	17.6196	1.71137
$107$	8.12463	0.785437	0.392719	−	0.919659i	$-0.371535\pi$
$107$	8.12463	0.785437	0.392719	+	0.919659i	$0.371535\pi$
$108$	0	0
$109$	−20.3356	−1.94780	−0.973900	−	0.226978i	$-0.927115\pi$
$109$	−20.3356	−1.94780	−0.973900	+	0.226978i	$0.927115\pi$
$110$	−5.20743	−0.496509
$111$	0	0
$112$	−3.12518	−0.295302
$113$	−4.16980	−0.392262	−0.196131	−	0.980578i	$-0.562838\pi$
$113$	−4.16980	−0.392262	−0.196131	+	0.980578i	$0.562838\pi$
$114$	0	0
$115$	1.33598	0.124581
$116$	−0.402300	−0.0373526
$117$	0	0
$118$	6.22174	0.572758
$119$	4.98727	0.457182
$120$	0	0
$121$	30.1983	2.74530
$122$	−12.1714	−1.10194
$123$	0	0
$124$	−20.4362	−1.83522
$125$	3.82287	0.341927
$126$	0	0
$127$	−6.20025	−0.550184	−0.275092	−	0.961418i	$-0.588708\pi$
$127$	−6.20025	−0.550184	−0.275092	+	0.961418i	$0.588708\pi$
$128$	6.06927	0.536453
$129$	0	0
$130$	3.15366	0.276594
$131$	−15.5667	−1.36007	−0.680034	−	0.733181i	$-0.738035\pi$
$131$	−15.5667	−1.36007	−0.680034	+	0.733181i	$0.738035\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	31.3499	2.70822
$135$	0	0
$136$	−3.84924	−0.330069
$137$	−10.7222	−0.916058	−0.458029	−	0.888937i	$-0.651444\pi$
$137$	−10.7222	−0.916058	−0.458029	+	0.888937i	$0.651444\pi$
$138$	0	0
$139$	10.0987	0.856560	0.428280	−	0.903646i	$-0.359120\pi$
$139$	10.0987	0.856560	0.428280	+	0.903646i	$0.359120\pi$
$140$	−0.919582	−0.0777189
$141$	0	0
$142$	8.88112	0.745287
$143$	−24.9499	−2.08642
$144$	0	0
$145$	0.0659056	0.00547316
$146$	−28.3768	−2.34848
$147$	0	0
$148$	17.4741	1.43636
$149$	−9.75279	−0.798980	−0.399490	−	0.916738i	$-0.630813\pi$
$149$	−9.75279	−0.798980	−0.399490	+	0.916738i	$0.630813\pi$
$150$	0	0
$151$	−2.20942	−0.179800	−0.0899002	−	0.995951i	$-0.528655\pi$
$151$	−2.20942	−0.179800	−0.0899002	+	0.995951i	$0.528655\pi$
$152$	0.771813	0.0626023
$153$	0	0
$154$	13.4166	1.08114
$155$	3.34789	0.268909
$156$	0	0
$157$	−10.3519	−0.826173	−0.413087	−	0.910692i	$-0.635549\pi$
$157$	−10.3519	−0.826173	−0.413087	+	0.910692i	$0.635549\pi$
$158$	2.12018	0.168673
$159$	0	0
$160$	−3.13461	−0.247813
$161$	−3.44207	−0.271273
$162$	0	0
$163$	−8.52406	−0.667656	−0.333828	−	0.942634i	$-0.608340\pi$
$163$	−8.52406	−0.667656	−0.333828	+	0.942634i	$0.608340\pi$
$164$	20.8012	1.62430
$165$	0	0
$166$	−9.03305	−0.701101
$167$	−6.99801	−0.541522	−0.270761	−	0.962647i	$-0.587275\pi$
$167$	−6.99801	−0.541522	−0.270761	+	0.962647i	$0.587275\pi$
$168$	0	0
$169$	2.10985	0.162296
$170$	4.04620	0.310329
$171$	0	0
$172$	−21.5858	−1.64591
$173$	−14.6403	−1.11308	−0.556542	−	0.830820i	$-0.687872\pi$
$173$	−14.6403	−1.11308	−0.556542	+	0.830820i	$0.687872\pi$
$174$	0	0
$175$	−4.84935	−0.366577
$176$	20.0592	1.51202
$177$	0	0
$178$	29.0455	2.17705
$179$	12.9009	0.964258	0.482129	−	0.876100i	$-0.339864\pi$
$179$	12.9009	0.964258	0.482129	+	0.876100i	$0.339864\pi$
$180$	0	0
$181$	−12.3897	−0.920920	−0.460460	−	0.887680i	$-0.652316\pi$
$181$	−12.3897	−0.920920	−0.460460	+	0.887680i	$0.652316\pi$
$182$	−8.12518	−0.602279
$183$	0	0
$184$	2.65663	0.195849
$185$	−2.86264	−0.210466
$186$	0	0
$187$	−32.0112	−2.34089
$188$	11.3883	0.830575
$189$	0	0
$190$	−0.811305	−0.0588583
$191$	−25.0544	−1.81287	−0.906436	−	0.422343i	$-0.861208\pi$
$191$	−25.0544	−1.81287	−0.906436	+	0.422343i	$0.861208\pi$
$192$	0	0
$193$	8.79939	0.633394	0.316697	−	0.948527i	$-0.397426\pi$
$193$	8.79939	0.633394	0.316697	+	0.948527i	$0.397426\pi$
$194$	28.7920	2.06715
$195$	0	0
$196$	2.36924	0.169231
$197$	4.80856	0.342595	0.171298	−	0.985219i	$-0.445204\pi$
$197$	4.80856	0.342595	0.171298	+	0.985219i	$0.445204\pi$
$198$	0	0
$199$	9.61344	0.681479	0.340739	−	0.940158i	$-0.389323\pi$
$199$	9.61344	0.681479	0.340739	+	0.940158i	$0.389323\pi$
$200$	3.74279	0.264655
$201$	0	0
$202$	17.0359	1.19864
$203$	−0.169801	−0.0119177
$204$	0	0
$205$	−3.40769	−0.238003
$206$	26.8332	1.86956
$207$	0	0
$208$	−12.1480	−0.842313
$209$	6.41859	0.443983
$210$	0	0
$211$	2.22174	0.152951	0.0764756	−	0.997071i	$-0.475633\pi$
$211$	2.22174	0.152951	0.0764756	+	0.997071i	$0.475633\pi$
$212$	−19.9711	−1.37162
$213$	0	0
$214$	−16.9827	−1.16091
$215$	3.53624	0.241169
$216$	0	0
$217$	−8.62562	−0.585545
$218$	42.5070	2.87894
$219$	0	0
$220$	5.90242	0.397941
$221$	19.3862	1.30406
$222$	0	0
$223$	18.2606	1.22282	0.611408	−	0.791315i	$-0.290603\pi$
$223$	18.2606	1.22282	0.611408	+	0.791315i	$0.290603\pi$
$224$	8.07611	0.539608
$225$	0	0
$226$	8.71602	0.579781
$227$	5.13736	0.340978	0.170489	−	0.985360i	$-0.445465\pi$
$227$	5.13736	0.340978	0.170489	+	0.985360i	$0.445465\pi$
$228$	0	0
$229$	10.4750	0.692206	0.346103	−	0.938197i	$-0.387505\pi$
$229$	10.4750	0.692206	0.346103	+	0.938197i	$0.387505\pi$
$230$	−2.79257	−0.184136
$231$	0	0
$232$	0.131055	0.00860416
$233$	−17.5271	−1.14824	−0.574118	−	0.818772i	$-0.694655\pi$
$233$	−17.5271	−1.14824	−0.574118	+	0.818772i	$0.694655\pi$
$234$	0	0
$235$	−1.86565	−0.121702
$236$	−7.05210	−0.459052
$237$	0	0
$238$	−10.4247	−0.675736
$239$	21.7777	1.40868	0.704341	−	0.709862i	$-0.251243\pi$
$239$	21.7777	1.40868	0.704341	+	0.709862i	$0.251243\pi$
$240$	0	0
$241$	10.3611	0.667417	0.333708	−	0.942676i	$-0.391700\pi$
$241$	10.3611	0.667417	0.333708	+	0.942676i	$0.391700\pi$
$242$	−63.1226	−4.05768
$243$	0	0
$244$	13.7958	0.883183
$245$	−0.388134	−0.0247970
$246$	0	0
$247$	−3.88714	−0.247333
$248$	6.65736	0.422743
$249$	0	0
$250$	−7.99083	−0.505385
$251$	−10.9146	−0.688922	−0.344461	−	0.938801i	$-0.611938\pi$
$251$	−10.9146	−0.688922	−0.344461	+	0.938801i	$0.611938\pi$
$252$	0	0
$253$	22.0932	1.38899
$254$	12.9602	0.813197
$255$	0	0
$256$	8.57537	0.535961
$257$	3.28123	0.204677	0.102339	−	0.994750i	$-0.467367\pi$
$257$	3.28123	0.204677	0.102339	+	0.994750i	$0.467367\pi$
$258$	0	0
$259$	7.37540	0.458285
$260$	−3.57454	−0.221684
$261$	0	0
$262$	32.5386	2.01024
$263$	18.0034	1.11014	0.555069	−	0.831804i	$-0.312692\pi$
$263$	18.0034	1.11014	0.555069	+	0.831804i	$0.312692\pi$
$264$	0	0
$265$	3.27171	0.200979
$266$	2.09027	0.128163
$267$	0	0
$268$	−35.5339	−2.17058
$269$	0.886737	0.0540653	0.0270327	−	0.999635i	$-0.491394\pi$
$269$	0.886737	0.0540653	0.0270327	+	0.999635i	$0.491394\pi$
$270$	0	0
$271$	7.43917	0.451898	0.225949	−	0.974139i	$-0.427452\pi$
$271$	7.43917	0.451898	0.225949	+	0.974139i	$0.427452\pi$
$272$	−15.5861	−0.945047
$273$	0	0
$274$	22.4123	1.35398
$275$	31.1260	1.87697
$276$	0	0
$277$	−7.40189	−0.444736	−0.222368	−	0.974963i	$-0.571379\pi$
$277$	−7.40189	−0.444736	−0.222368	+	0.974963i	$0.571379\pi$
$278$	−21.1090	−1.26604
$279$	0	0
$280$	0.299567	0.0179025
$281$	1.98187	0.118228	0.0591141	−	0.998251i	$-0.481172\pi$
$281$	1.98187	0.118228	0.0591141	+	0.998251i	$0.481172\pi$
$282$	0	0
$283$	7.83519	0.465753	0.232877	−	0.972506i	$-0.425186\pi$
$283$	7.83519	0.465753	0.232877	+	0.972506i	$0.425186\pi$
$284$	−10.0664	−0.597331
$285$	0	0
$286$	52.1522	3.08382
$287$	8.77968	0.518248
$288$	0	0
$289$	7.87283	0.463108
$290$	−0.137761	−0.00808958
$291$	0	0
$292$	32.1640	1.88226
$293$	−3.83178	−0.223855	−0.111927	−	0.993716i	$-0.535702\pi$
$293$	−3.83178	−0.223855	−0.111927	+	0.993716i	$0.535702\pi$
$294$	0	0
$295$	1.15529	0.0672635
$296$	−5.69243	−0.330866
$297$	0	0
$298$	20.3860	1.18093
$299$	−13.3798	−0.773773
$300$	0	0
$301$	−9.11087	−0.525142
$302$	4.61830	0.265753
$303$	0	0
$304$	3.12518	0.179241
$305$	−2.26005	−0.129410
$306$	0	0
$307$	1.46279	0.0834861	0.0417431	−	0.999128i	$-0.486709\pi$
$307$	1.46279	0.0834861	0.0417431	+	0.999128i	$0.486709\pi$
$308$	−15.2072	−0.866509
$309$	0	0
$310$	−6.99801	−0.397460
$311$	12.0213	0.681665	0.340832	−	0.940124i	$-0.389291\pi$
$311$	12.0213	0.681665	0.340832	+	0.940124i	$0.389291\pi$
$312$	0	0
$313$	5.36938	0.303495	0.151748	−	0.988419i	$-0.451510\pi$
$313$	5.36938	0.303495	0.151748	+	0.988419i	$0.451510\pi$
$314$	21.6383	1.22112
$315$	0	0
$316$	−2.40314	−0.135187
$317$	12.6674	0.711471	0.355735	−	0.934587i	$-0.384230\pi$
$317$	12.6674	0.711471	0.355735	+	0.934587i	$0.384230\pi$
$318$	0	0
$319$	1.08988	0.0610218
$320$	4.12621	0.230662
$321$	0	0
$322$	7.19485	0.400954
$323$	−4.98727	−0.277499
$324$	0	0
$325$	−18.8501	−1.04562
$326$	17.8176	0.986826
$327$	0	0
$328$	−6.77627	−0.374157
$329$	4.80672	0.265003
$330$	0	0
$331$	15.2575	0.838630	0.419315	−	0.907841i	$-0.362270\pi$
$331$	15.2575	0.838630	0.419315	+	0.907841i	$0.362270\pi$
$332$	10.2386	0.561916
$333$	0	0
$334$	14.6278	0.800395
$335$	5.82123	0.318048
$336$	0	0
$337$	−19.5379	−1.06430	−0.532148	−	0.846652i	$-0.678615\pi$
$337$	−19.5379	−1.06430	−0.532148	+	0.846652i	$0.678615\pi$
$338$	−4.41017	−0.239881
$339$	0	0
$340$	−4.58620	−0.248722
$341$	55.3643	2.99814
$342$	0	0
$343$	1.00000	0.0539949
$344$	7.03189	0.379134
$345$	0	0
$346$	30.6023	1.64519
$347$	17.7777	0.954356	0.477178	−	0.878807i	$-0.341660\pi$
$347$	17.7777	0.954356	0.477178	+	0.878807i	$0.341660\pi$
$348$	0	0
$349$	13.6987	0.733275	0.366637	−	0.930364i	$-0.380509\pi$
$349$	13.6987	0.733275	0.366637	+	0.930364i	$0.380509\pi$
$350$	10.1365	0.541817
$351$	0	0
$352$	−51.8372	−2.76293
$353$	−8.81013	−0.468916	−0.234458	−	0.972126i	$-0.575332\pi$
$353$	−8.81013	−0.468916	−0.234458	+	0.972126i	$0.575332\pi$
$354$	0	0
$355$	1.64910	0.0875250
$356$	−32.9219	−1.74486
$357$	0	0
$358$	−26.9664	−1.42522
$359$	9.44808	0.498651	0.249325	−	0.968420i	$-0.419791\pi$
$359$	9.44808	0.498651	0.249325	+	0.968420i	$0.419791\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.8979	1.36116
$363$	0	0
$364$	9.20957	0.482712
$365$	−5.26917	−0.275801
$366$	0	0
$367$	−1.08439	−0.0566044	−0.0283022	−	0.999599i	$-0.509010\pi$
$367$	−1.08439	−0.0566044	−0.0283022	+	0.999599i	$0.509010\pi$
$368$	10.7571	0.560751
$369$	0	0
$370$	5.98370	0.311078
$371$	−8.42933	−0.437629
$372$	0	0
$373$	−3.64971	−0.188975	−0.0944874	−	0.995526i	$-0.530121\pi$
$373$	−3.64971	−0.188975	−0.0944874	+	0.995526i	$0.530121\pi$
$374$	66.9122	3.45995
$375$	0	0
$376$	−3.70989	−0.191323
$377$	−0.660041	−0.0339938
$378$	0	0
$379$	12.8846	0.661840	0.330920	−	0.943659i	$-0.392641\pi$
$379$	12.8846	0.661840	0.330920	+	0.943659i	$0.392641\pi$
$380$	0.919582	0.0471736
$381$	0	0
$382$	52.3705	2.67951
$383$	17.7242	0.905663	0.452831	−	0.891596i	$-0.350414\pi$
$383$	17.7242	0.905663	0.452831	+	0.891596i	$0.350414\pi$
$384$	0	0
$385$	2.49127	0.126967
$386$	−18.3931	−0.936185
$387$	0	0
$388$	−32.6346	−1.65677
$389$	−10.4750	−0.531102	−0.265551	−	0.964097i	$-0.585554\pi$
$389$	−10.4750	−0.531102	−0.265551	+	0.964097i	$0.585554\pi$
$390$	0	0
$391$	−17.1665	−0.868147
$392$	−0.771813	−0.0389824
$393$	0	0
$394$	−10.0512	−0.506372
$395$	0.393688	0.0198086
$396$	0	0
$397$	−34.3679	−1.72488	−0.862438	−	0.506163i	$-0.831064\pi$
$397$	−34.3679	−1.72488	−0.862438	+	0.506163i	$0.831064\pi$
$398$	−20.0947	−1.00726
$399$	0	0
$400$	15.1551	0.757755
$401$	10.7403	0.536346	0.268173	−	0.963371i	$-0.413580\pi$
$401$	10.7403	0.536346	0.268173	+	0.963371i	$0.413580\pi$
$402$	0	0
$403$	−33.5290	−1.67020
$404$	−19.3095	−0.960685
$405$	0	0
$406$	0.354931	0.0176149
$407$	−47.3397	−2.34654
$408$	0	0
$409$	26.0559	1.28838	0.644191	−	0.764865i	$-0.277194\pi$
$409$	26.0559	1.28838	0.644191	+	0.764865i	$0.277194\pi$
$410$	7.12300	0.351780
$411$	0	0
$412$	−30.4144	−1.49841
$413$	−2.97652	−0.146465
$414$	0	0
$415$	−1.67731	−0.0823358
$416$	31.3930	1.53917
$417$	0	0
$418$	−13.4166	−0.656227
$419$	1.36205	0.0665405	0.0332702	−	0.999446i	$-0.489408\pi$
$419$	1.36205	0.0665405	0.0332702	+	0.999446i	$0.489408\pi$
$420$	0	0
$421$	−1.93675	−0.0943912	−0.0471956	−	0.998886i	$-0.515028\pi$
$421$	−1.93675	−0.0943912	−0.0471956	+	0.998886i	$0.515028\pi$
$422$	−4.64405	−0.226069
$423$	0	0
$424$	6.50587	0.315953
$425$	−24.1850	−1.17315
$426$	0	0
$427$	5.82287	0.281788
$428$	19.2492	0.930445
$429$	0	0
$430$	−7.39170	−0.356459
$431$	−27.0267	−1.30183	−0.650915	−	0.759151i	$-0.725615\pi$
$431$	−27.0267	−1.30183	−0.650915	+	0.759151i	$0.725615\pi$
$432$	0	0
$433$	−38.2404	−1.83771	−0.918857	−	0.394590i	$-0.870887\pi$
$433$	−38.2404	−1.83771	−0.918857	+	0.394590i	$0.870887\pi$
$434$	18.0299	0.865463
$435$	0	0
$436$	−48.1800	−2.30740
$437$	3.44207	0.164656
$438$	0	0
$439$	−2.76497	−0.131965	−0.0659823	−	0.997821i	$-0.521018\pi$
$439$	−2.76497	−0.131965	−0.0659823	+	0.997821i	$0.521018\pi$
$440$	−1.92279	−0.0916656
$441$	0	0
$442$	−40.5224	−1.92746
$443$	−14.0564	−0.667839	−0.333919	−	0.942602i	$-0.608371\pi$
$443$	−14.0564	−0.667839	−0.333919	+	0.942602i	$0.608371\pi$
$444$	0	0
$445$	5.39333	0.255668
$446$	−38.1695	−1.80738
$447$	0	0
$448$	−10.6309	−0.502263
$449$	14.3792	0.678598	0.339299	−	0.940679i	$-0.389810\pi$
$449$	14.3792	0.678598	0.339299	+	0.940679i	$0.389810\pi$
$450$	0	0
$451$	−56.3531	−2.65357
$452$	−9.87926	−0.464681
$453$	0	0
$454$	−10.7385	−0.503982
$455$	−1.50873	−0.0707303
$456$	0	0
$457$	27.8024	1.30054	0.650271	−	0.759703i	$-0.274655\pi$
$457$	27.8024	1.30054	0.650271	+	0.759703i	$0.274655\pi$
$458$	−21.8955	−1.02311
$459$	0	0
$460$	3.16526	0.147581
$461$	−21.0697	−0.981312	−0.490656	−	0.871353i	$-0.663243\pi$
$461$	−21.0697	−0.981312	−0.490656	+	0.871353i	$0.663243\pi$
$462$	0	0
$463$	2.52304	0.117256	0.0586278	−	0.998280i	$-0.481327\pi$
$463$	2.52304	0.117256	0.0586278	+	0.998280i	$0.481327\pi$
$464$	0.530659	0.0246352
$465$	0	0
$466$	36.6364	1.69715
$467$	−21.2342	−0.982602	−0.491301	−	0.870990i	$-0.663479\pi$
$467$	−21.2342	−0.982602	−0.491301	+	0.870990i	$0.663479\pi$
$468$	0	0
$469$	−14.9980	−0.692544
$470$	3.89972	0.179881
$471$	0	0
$472$	2.29732	0.105743
$473$	58.4789	2.68886
$474$	0	0
$475$	4.84935	0.222504
$476$	11.8160	0.541587
$477$	0	0
$478$	−45.5213	−2.08210
$479$	9.86851	0.450904	0.225452	−	0.974254i	$-0.427614\pi$
$479$	9.86851	0.450904	0.225452	+	0.974254i	$0.427614\pi$
$480$	0	0
$481$	28.6692	1.30720
$482$	−21.6575	−0.986472
$483$	0	0
$484$	71.5469	3.25213
$485$	5.34626	0.242761
$486$	0	0
$487$	20.7038	0.938181	0.469090	−	0.883150i	$-0.344582\pi$
$487$	20.7038	0.938181	0.469090	+	0.883150i	$0.344582\pi$
$488$	−4.49416	−0.203441
$489$	0	0
$490$	0.811305	0.0366510
$491$	32.6041	1.47140	0.735700	−	0.677307i	$-0.236853\pi$
$491$	32.6041	1.47140	0.735700	+	0.677307i	$0.236853\pi$
$492$	0	0
$493$	−0.846844	−0.0381399
$494$	8.12518	0.365569
$495$	0	0
$496$	26.9566	1.21039
$497$	−4.24879	−0.190584
$498$	0	0
$499$	0.812534	0.0363740	0.0181870	−	0.999835i	$-0.494211\pi$
$499$	0.812534	0.0363740	0.0181870	+	0.999835i	$0.494211\pi$
$500$	9.05729	0.405054
$501$	0	0
$502$	22.8145	1.01826
$503$	16.6285	0.741427	0.370714	−	0.928747i	$-0.379113\pi$
$503$	16.6285	0.741427	0.370714	+	0.928747i	$0.379113\pi$
$504$	0	0
$505$	3.16333	0.140766
$506$	−46.1808	−2.05299
$507$	0	0
$508$	−14.6899	−0.651759
$509$	−33.8641	−1.50100	−0.750499	−	0.660871i	$-0.770187\pi$
$509$	−33.8641	−1.50100	−0.750499	+	0.660871i	$0.770187\pi$
$510$	0	0
$511$	13.5757	0.600552
$512$	−30.0634	−1.32863
$513$	0	0
$514$	−6.85866	−0.302523
$515$	4.98254	0.219557
$516$	0	0
$517$	−30.8524	−1.35689
$518$	−15.4166	−0.677366
$519$	0	0
$520$	1.16446	0.0510648
$521$	−28.3645	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$521$	−28.3645	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$522$	0	0
$523$	−1.32743	−0.0580443	−0.0290221	−	0.999579i	$-0.509239\pi$
$523$	−1.32743	−0.0580443	−0.0290221	+	0.999579i	$0.509239\pi$
$524$	−36.8812	−1.61116
$525$	0	0
$526$	−37.6320	−1.64083
$527$	−43.0183	−1.87391
$528$	0	0
$529$	−11.1522	−0.484878
$530$	−6.83876	−0.297057
$531$	0	0
$532$	−2.36924	−0.102720
$533$	34.1278	1.47824
$534$	0	0
$535$	−3.15344	−0.136335
$536$	11.5757	0.499992
$537$	0	0
$538$	−1.85352	−0.0799110
$539$	−6.41859	−0.276468
$540$	0	0
$541$	−19.0875	−0.820637	−0.410319	−	0.911942i	$-0.634583\pi$
$541$	−19.0875	−0.820637	−0.410319	+	0.911942i	$0.634583\pi$
$542$	−15.5499	−0.667925
$543$	0	0
$544$	40.2777	1.72689
$545$	7.89294	0.338096
$546$	0	0
$547$	−12.7660	−0.545834	−0.272917	−	0.962038i	$-0.587988\pi$
$547$	−12.7660	−0.545834	−0.272917	+	0.962038i	$0.587988\pi$
$548$	−25.4034	−1.08518
$549$	0	0
$550$	−65.0618	−2.77425
$551$	0.169801	0.00723377
$552$	0	0
$553$	−1.01431	−0.0431328
$554$	15.4720	0.657341
$555$	0	0
$556$	23.9262	1.01470
$557$	−23.1620	−0.981405	−0.490703	−	0.871327i	$-0.663260\pi$
$557$	−23.1620	−0.981405	−0.490703	+	0.871327i	$0.663260\pi$
$558$	0	0
$559$	−35.4152	−1.49790
$560$	1.21299	0.0512581
$561$	0	0
$562$	−4.14264	−0.174747
$563$	9.19938	0.387708	0.193854	−	0.981030i	$-0.437901\pi$
$563$	9.19938	0.387708	0.193854	+	0.981030i	$0.437901\pi$
$564$	0	0
$565$	1.61844	0.0680883
$566$	−16.3777	−0.688405
$567$	0	0
$568$	3.27927	0.137595
$569$	−42.8405	−1.79597	−0.897984	−	0.440028i	$-0.854969\pi$
$569$	−42.8405	−1.79597	−0.897984	+	0.440028i	$0.854969\pi$
$570$	0	0
$571$	10.6358	0.445095	0.222547	−	0.974922i	$-0.428563\pi$
$571$	10.6358	0.445095	0.222547	+	0.974922i	$0.428563\pi$
$572$	−59.1124	−2.47161
$573$	0	0
$574$	−18.3519	−0.765994
$575$	16.6918	0.696096
$576$	0	0
$577$	45.7190	1.90331	0.951653	−	0.307176i	$-0.0993842\pi$
$577$	45.7190	1.90331	0.951653	+	0.307176i	$0.0993842\pi$
$578$	−16.4564	−0.684494
$579$	0	0
$580$	0.156146	0.00648361
$581$	4.32147	0.179285
$582$	0	0
$583$	54.1044	2.24078
$584$	−10.4779	−0.433577
$585$	0	0
$586$	8.00946	0.330868
$587$	−13.8685	−0.572414	−0.286207	−	0.958168i	$-0.592395\pi$
$587$	−13.8685	−0.572414	−0.286207	+	0.958168i	$0.592395\pi$
$588$	0	0
$589$	8.62562	0.355412
$590$	−2.41487	−0.0994186
$591$	0	0
$592$	−23.0495	−0.947327
$593$	31.7063	1.30202	0.651011	−	0.759068i	$-0.274345\pi$
$593$	31.7063	1.30202	0.651011	+	0.759068i	$0.274345\pi$
$594$	0	0
$595$	−1.93573	−0.0793570
$596$	−23.1067	−0.946487
$597$	0	0
$598$	27.9674	1.14367
$599$	3.91479	0.159954	0.0799771	−	0.996797i	$-0.474515\pi$
$599$	3.91479	0.159954	0.0799771	+	0.996797i	$0.474515\pi$
$600$	0	0
$601$	7.68790	0.313596	0.156798	−	0.987631i	$-0.449883\pi$
$601$	7.68790	0.313596	0.156798	+	0.987631i	$0.449883\pi$
$602$	19.0442	0.776184
$603$	0	0
$604$	−5.23465	−0.212995
$605$	−11.7210	−0.476525
$606$	0	0
$607$	4.11499	0.167022	0.0835112	−	0.996507i	$-0.473387\pi$
$607$	4.11499	0.167022	0.0835112	+	0.996507i	$0.473387\pi$
$608$	−8.07611	−0.327529
$609$	0	0
$610$	4.72412	0.191274
$611$	18.6844	0.755890
$612$	0	0
$613$	14.0997	0.569482	0.284741	−	0.958604i	$-0.408092\pi$
$613$	14.0997	0.569482	0.284741	+	0.958604i	$0.408092\pi$
$614$	−3.05764	−0.123396
$615$	0	0
$616$	4.95395	0.199600
$617$	−26.1748	−1.05376	−0.526879	−	0.849941i	$-0.676638\pi$
$617$	−26.1748	−1.05376	−0.526879	+	0.849941i	$0.676638\pi$
$618$	0	0
$619$	13.8200	0.555473	0.277736	−	0.960657i	$-0.410416\pi$
$619$	13.8200	0.555473	0.277736	+	0.960657i	$0.410416\pi$
$620$	7.93196	0.318555
$621$	0	0
$622$	−25.1278	−1.00753
$623$	−13.8955	−0.556713
$624$	0	0
$625$	22.7630	0.910519
$626$	−11.2235	−0.448580
$627$	0	0
$628$	−24.5262	−0.978702
$629$	36.7831	1.46664
$630$	0	0
$631$	26.3799	1.05017	0.525084	−	0.851050i	$-0.324034\pi$
$631$	26.3799	1.05017	0.525084	+	0.851050i	$0.324034\pi$
$632$	0.782857	0.0311404
$633$	0	0
$634$	−26.4783	−1.05159
$635$	2.40653	0.0955001
$636$	0	0
$637$	3.88714	0.154014
$638$	−2.27815	−0.0901930
$639$	0	0
$640$	−2.35569	−0.0931168
$641$	43.5850	1.72151	0.860753	−	0.509024i	$-0.169993\pi$
$641$	43.5850	1.72151	0.860753	+	0.509024i	$0.169993\pi$
$642$	0	0
$643$	−37.6911	−1.48639	−0.743195	−	0.669075i	$-0.766691\pi$
$643$	−37.6911	−1.48639	−0.743195	+	0.669075i	$0.766691\pi$
$644$	−8.15508	−0.321355
$645$	0	0
$646$	10.4247	0.410156
$647$	41.8752	1.64628	0.823142	−	0.567835i	$-0.192219\pi$
$647$	41.8752	1.64628	0.823142	+	0.567835i	$0.192219\pi$
$648$	0	0
$649$	19.1051	0.749940
$650$	39.4019	1.54547
$651$	0	0
$652$	−20.1955	−0.790919
$653$	−20.9665	−0.820484	−0.410242	−	0.911977i	$-0.634556\pi$
$653$	−20.9665	−0.820484	−0.410242	+	0.911977i	$0.634556\pi$
$654$	0	0
$655$	6.04196	0.236079
$656$	−27.4381	−1.07128
$657$	0	0
$658$	−10.0474	−0.391687
$659$	−19.0359	−0.741532	−0.370766	−	0.928726i	$-0.620905\pi$
$659$	−19.0359	−0.741532	−0.370766	+	0.928726i	$0.620905\pi$
$660$	0	0
$661$	−18.6957	−0.727178	−0.363589	−	0.931559i	$-0.618449\pi$
$661$	−18.6957	−0.727178	−0.363589	+	0.931559i	$0.618449\pi$
$662$	−31.8924	−1.23953
$663$	0	0
$664$	−3.33536	−0.129437
$665$	0.388134	0.0150512
$666$	0	0
$667$	0.584467	0.0226306
$668$	−16.5800	−0.641498
$669$	0	0
$670$	−12.1680	−0.470090
$671$	−37.3746	−1.44283
$672$	0	0
$673$	42.1036	1.62297	0.811487	−	0.584370i	$-0.198658\pi$
$673$	42.1036	1.62297	0.811487	+	0.584370i	$0.198658\pi$
$674$	40.8395	1.57308
$675$	0	0
$676$	4.99875	0.192259
$677$	41.1045	1.57977	0.789887	−	0.613252i	$-0.210139\pi$
$677$	41.1045	1.57977	0.789887	+	0.613252i	$0.210139\pi$
$678$	0	0
$679$	−13.7743	−0.528608
$680$	1.49402	0.0572930
$681$	0	0
$682$	−115.726	−4.43140
$683$	2.64451	0.101189	0.0505947	−	0.998719i	$-0.483888\pi$
$683$	2.64451	0.101189	0.0505947	+	0.998719i	$0.483888\pi$
$684$	0	0
$685$	4.16164	0.159008
$686$	−2.09027	−0.0798070
$687$	0	0
$688$	28.4731	1.08553
$689$	−32.7660	−1.24828
$690$	0	0
$691$	8.59120	0.326825	0.163412	−	0.986558i	$-0.447750\pi$
$691$	8.59120	0.326825	0.163412	+	0.986558i	$0.447750\pi$
$692$	−34.6865	−1.31858
$693$	0	0
$694$	−37.1602	−1.41058
$695$	−3.91964	−0.148681
$696$	0	0
$697$	43.7866	1.65854
$698$	−28.6340	−1.08381
$699$	0	0
$700$	−11.4893	−0.434254
$701$	−24.5073	−0.925626	−0.462813	−	0.886456i	$-0.653160\pi$
$701$	−24.5073	−0.925626	−0.462813	+	0.886456i	$0.653160\pi$
$702$	0	0
$703$	−7.37540	−0.278168
$704$	68.2354	2.57172
$705$	0	0
$706$	18.4156	0.693079
$707$	−8.15009	−0.306516
$708$	0	0
$709$	30.1710	1.13310	0.566548	−	0.824029i	$-0.308278\pi$
$709$	30.1710	1.13310	0.566548	+	0.824029i	$0.308278\pi$
$710$	−3.44706	−0.129366
$711$	0	0
$712$	10.7248	0.401927
$713$	29.6899	1.11190
$714$	0	0
$715$	9.68391	0.362158
$716$	30.5653	1.14228
$717$	0	0
$718$	−19.7491	−0.737029
$719$	−22.5815	−0.842149	−0.421074	−	0.907026i	$-0.638347\pi$
$719$	−22.5815	−0.842149	−0.421074	+	0.907026i	$0.638347\pi$
$720$	0	0
$721$	−12.8372	−0.478081
$722$	−2.09027	−0.0777919
$723$	0	0
$724$	−29.3542	−1.09094
$725$	0.823426	0.0305813
$726$	0	0
$727$	16.6904	0.619013	0.309507	−	0.950897i	$-0.399836\pi$
$727$	16.6904	0.619013	0.309507	+	0.950897i	$0.399836\pi$
$728$	−3.00014	−0.111193
$729$	0	0
$730$	11.0140	0.407646
$731$	−45.4383	−1.68060
$732$	0	0
$733$	23.0956	0.853055	0.426528	−	0.904475i	$-0.359737\pi$
$733$	23.0956	0.853055	0.426528	+	0.904475i	$0.359737\pi$
$734$	2.26666	0.0836640
$735$	0	0
$736$	−27.7985	−1.02467
$737$	96.2660	3.54601
$738$	0	0
$739$	−23.6754	−0.870913	−0.435456	−	0.900210i	$-0.643413\pi$
$739$	−23.6754	−0.870913	−0.435456	+	0.900210i	$0.643413\pi$
$740$	−6.78229	−0.249322
$741$	0	0
$742$	17.6196	0.646836
$743$	8.20020	0.300836	0.150418	−	0.988622i	$-0.451938\pi$
$743$	8.20020	0.300836	0.150418	+	0.988622i	$0.451938\pi$
$744$	0	0
$745$	3.78539	0.138686
$746$	7.62889	0.279313
$747$	0	0
$748$	−75.8422	−2.77307
$749$	8.12463	0.296867
$750$	0	0
$751$	−53.7070	−1.95980	−0.979899	−	0.199494i	$-0.936070\pi$
$751$	−53.7070	−1.95980	−0.979899	+	0.199494i	$0.936070\pi$
$752$	−15.0219	−0.547791
$753$	0	0
$754$	1.37967	0.0502444
$755$	0.857552	0.0312095
$756$	0	0
$757$	48.5136	1.76326	0.881628	−	0.471945i	$-0.156448\pi$
$757$	48.5136	1.76326	0.881628	+	0.471945i	$0.156448\pi$
$758$	−26.9324	−0.978230
$759$	0	0
$760$	−0.299567	−0.0108664
$761$	−0.788644	−0.0285883	−0.0142942	−	0.999898i	$-0.504550\pi$
$761$	−0.788644	−0.0285883	−0.0142942	+	0.999898i	$0.504550\pi$
$762$	0	0
$763$	−20.3356	−0.736199
$764$	−59.3599	−2.14757
$765$	0	0
$766$	−37.0484	−1.33861
$767$	−11.5702	−0.417774
$768$	0	0
$769$	9.62974	0.347257	0.173629	−	0.984811i	$-0.444451\pi$
$769$	9.62974	0.347257	0.173629	+	0.984811i	$0.444451\pi$
$770$	−5.20743	−0.187663
$771$	0	0
$772$	20.8479	0.750331
$773$	35.2618	1.26828	0.634139	−	0.773219i	$-0.281355\pi$
$773$	35.2618	1.26828	0.634139	+	0.773219i	$0.281355\pi$
$774$	0	0
$775$	41.8287	1.50253
$776$	10.6312	0.381637
$777$	0	0
$778$	21.8955	0.784993
$779$	−8.77968	−0.314565
$780$	0	0
$781$	27.2712	0.975841
$782$	35.8827	1.28316
$783$	0	0
$784$	−3.12518	−0.111614
$785$	4.01793	0.143406
$786$	0	0
$787$	−40.3469	−1.43821	−0.719106	−	0.694900i	$-0.755449\pi$
$787$	−40.3469	−1.43821	−0.719106	+	0.694900i	$0.755449\pi$
$788$	11.3926	0.405845
$789$	0	0
$790$	−0.822915	−0.0292780
$791$	−4.16980	−0.148261
$792$	0	0
$793$	22.6343	0.803767
$794$	71.8383	2.54945
$795$	0	0
$796$	22.7766	0.807294
$797$	−5.29192	−0.187449	−0.0937247	−	0.995598i	$-0.529877\pi$
$797$	−5.29192	−0.187449	−0.0937247	+	0.995598i	$0.529877\pi$
$798$	0	0
$799$	23.9724	0.848083
$800$	−39.1639	−1.38465
$801$	0	0
$802$	−22.4502	−0.792744
$803$	−87.1365	−3.07498
$804$	0	0
$805$	1.33598	0.0470872
$806$	70.0847	2.46863
$807$	0	0
$808$	6.29034	0.221294
$809$	54.6544	1.92155	0.960774	−	0.277333i	$-0.0894507\pi$
$809$	54.6544	1.92155	0.960774	+	0.277333i	$0.0894507\pi$
$810$	0	0
$811$	34.1414	1.19887	0.599433	−	0.800425i	$-0.295393\pi$
$811$	34.1414	1.19887	0.599433	+	0.800425i	$0.295393\pi$
$812$	−0.402300	−0.0141180
$813$	0	0
$814$	98.9528	3.46829
$815$	3.30847	0.115891
$816$	0	0
$817$	9.11087	0.318749
$818$	−54.4640	−1.90429
$819$	0	0
$820$	−8.07363	−0.281944
$821$	47.8638	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$821$	47.8638	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$822$	0	0
$823$	−8.87160	−0.309244	−0.154622	−	0.987974i	$-0.549416\pi$
$823$	−8.87160	−0.309244	−0.154622	+	0.987974i	$0.549416\pi$
$824$	9.90789	0.345158
$825$	0	0
$826$	6.22174	0.216482
$827$	−22.8022	−0.792910	−0.396455	−	0.918054i	$-0.629760\pi$
$827$	−22.8022	−0.792910	−0.396455	+	0.918054i	$0.629760\pi$
$828$	0	0
$829$	−17.9774	−0.624381	−0.312190	−	0.950020i	$-0.601063\pi$
$829$	−17.9774	−0.624381	−0.312190	+	0.950020i	$0.601063\pi$
$830$	3.50603	0.121696
$831$	0	0
$832$	−41.3238	−1.43265
$833$	4.98727	0.172799
$834$	0	0
$835$	2.71616	0.0939967
$836$	15.2072	0.525951
$837$	0	0
$838$	−2.84705	−0.0983499
$839$	41.0481	1.41714	0.708569	−	0.705641i	$-0.249341\pi$
$839$	41.0481	1.41714	0.708569	+	0.705641i	$0.249341\pi$
$840$	0	0
$841$	−28.9712	−0.999006
$842$	4.04833	0.139515
$843$	0	0
$844$	5.26384	0.181189
$845$	−0.818905	−0.0281712
$846$	0	0
$847$	30.1983	1.03762
$848$	26.3432	0.904629
$849$	0	0
$850$	50.5533	1.73396
$851$	−25.3866	−0.870242
$852$	0	0
$853$	10.2504	0.350966	0.175483	−	0.984482i	$-0.443851\pi$
$853$	10.2504	0.350966	0.175483	+	0.984482i	$0.443851\pi$
$854$	−12.1714	−0.416496
$855$	0	0
$856$	−6.27069	−0.214328
$857$	−35.3195	−1.20649	−0.603246	−	0.797555i	$-0.706126\pi$
$857$	−35.3195	−1.20649	−0.603246	+	0.797555i	$0.706126\pi$
$858$	0	0
$859$	−29.0481	−0.991109	−0.495555	−	0.868577i	$-0.665035\pi$
$859$	−29.0481	−0.991109	−0.495555	+	0.868577i	$0.665035\pi$
$860$	8.37819	0.285694
$861$	0	0
$862$	56.4931	1.92416
$863$	−37.0768	−1.26211	−0.631054	−	0.775739i	$-0.717378\pi$
$863$	−37.0768	−1.26211	−0.631054	+	0.775739i	$0.717378\pi$
$864$	0	0
$865$	5.68241	0.193208
$866$	79.9328	2.71623
$867$	0	0
$868$	−20.4362	−0.693649
$869$	6.51043	0.220851
$870$	0	0
$871$	−58.2994	−1.97540
$872$	15.6953	0.531510
$873$	0	0
$874$	−7.19485	−0.243370
$875$	3.82287	0.129236
$876$	0	0
$877$	−40.8400	−1.37907	−0.689535	−	0.724253i	$-0.742185\pi$
$877$	−40.8400	−1.37907	−0.689535	+	0.724253i	$0.742185\pi$
$878$	5.77954	0.195050
$879$	0	0
$880$	−7.78567	−0.262455
$881$	20.8316	0.701835	0.350918	−	0.936406i	$-0.385870\pi$
$881$	20.8316	0.701835	0.350918	+	0.936406i	$0.385870\pi$
$882$	0	0
$883$	17.8077	0.599276	0.299638	−	0.954053i	$-0.403134\pi$
$883$	17.8077	0.599276	0.299638	+	0.954053i	$0.403134\pi$
$884$	45.9306	1.54481
$885$	0	0
$886$	29.3817	0.987097
$887$	12.0913	0.405985	0.202993	−	0.979180i	$-0.434933\pi$
$887$	12.0913	0.405985	0.202993	+	0.979180i	$0.434933\pi$
$888$	0	0
$889$	−6.20025	−0.207950
$890$	−11.2735	−0.377890
$891$	0	0
$892$	43.2636	1.44857
$893$	−4.80672	−0.160851
$894$	0	0
$895$	−5.00727	−0.167375
$896$	6.06927	0.202760
$897$	0	0
$898$	−30.0565	−1.00300
$899$	1.46464	0.0488485
$900$	0	0
$901$	−42.0393	−1.40053
$902$	117.793	3.92209
$903$	0	0
$904$	3.21831	0.107039
$905$	4.80886	0.159852
$906$	0	0
$907$	−48.9499	−1.62536	−0.812678	−	0.582713i	$-0.801991\pi$
$907$	−48.9499	−1.62536	−0.812678	+	0.582713i	$0.801991\pi$
$908$	12.1716	0.403930
$909$	0	0
$910$	3.15366	0.104543
$911$	−27.4903	−0.910796	−0.455398	−	0.890288i	$-0.650503\pi$
$911$	−27.4903	−0.910796	−0.455398	+	0.890288i	$0.650503\pi$
$912$	0	0
$913$	−27.7377	−0.917985
$914$	−58.1146	−1.92226
$915$	0	0
$916$	24.8177	0.820001
$917$	−15.5667	−0.514057
$918$	0	0
$919$	26.3177	0.868141	0.434070	−	0.900879i	$-0.357077\pi$
$919$	26.3177	0.868141	0.434070	+	0.900879i	$0.357077\pi$
$920$	−1.03113	−0.0339953
$921$	0	0
$922$	44.0413	1.45042
$923$	−16.5156	−0.543618
$924$	0	0
$925$	−35.7659	−1.17598
$926$	−5.27384	−0.173309
$927$	0	0
$928$	−1.37133	−0.0450162
$929$	−27.5260	−0.903099	−0.451550	−	0.892246i	$-0.649129\pi$
$929$	−27.5260	−0.903099	−0.451550	+	0.892246i	$0.649129\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−41.5258	−1.36022
$933$	0	0
$934$	44.3853	1.45233
$935$	12.4246	0.406329
$936$	0	0
$937$	−18.9196	−0.618075	−0.309038	−	0.951050i	$-0.600007\pi$
$937$	−18.9196	−0.618075	−0.309038	+	0.951050i	$0.600007\pi$
$938$	31.3499	1.02361
$939$	0	0
$940$	−4.42017	−0.144170
$941$	−33.7139	−1.09904	−0.549521	−	0.835480i	$-0.685190\pi$
$941$	−33.7139	−1.09904	−0.549521	+	0.835480i	$0.685190\pi$
$942$	0	0
$943$	−30.2202	−0.984106
$944$	9.30217	0.302760
$945$	0	0
$946$	−122.237	−3.97426
$947$	0.655109	0.0212882	0.0106441	−	0.999943i	$-0.496612\pi$
$947$	0.655109	0.0212882	0.0106441	+	0.999943i	$0.496612\pi$
$948$	0	0
$949$	52.7705	1.71300
$950$	−10.1365	−0.328870
$951$	0	0
$952$	−3.84924	−0.124754
$953$	−33.6974	−1.09157	−0.545783	−	0.837927i	$-0.683768\pi$
$953$	−33.6974	−1.09157	−0.545783	+	0.837927i	$0.683768\pi$
$954$	0	0
$955$	9.72446	0.314676
$956$	51.5966	1.66875
$957$	0	0
$958$	−20.6279	−0.666456
$959$	−10.7222	−0.346237
$960$	0	0
$961$	43.4013	1.40004
$962$	−59.9265	−1.93211
$963$	0	0
$964$	24.5479	0.790635
$965$	−3.41534	−0.109944
$966$	0	0
$967$	−31.1237	−1.00087	−0.500436	−	0.865774i	$-0.666827\pi$
$967$	−31.1237	−1.00087	−0.500436	+	0.865774i	$0.666827\pi$
$968$	−23.3074	−0.749128
$969$	0	0
$970$	−11.1751	−0.358812
$971$	−0.890148	−0.0285662	−0.0142831	−	0.999898i	$-0.504547\pi$
$971$	−0.890148	−0.0285662	−0.0142831	+	0.999898i	$0.504547\pi$
$972$	0	0
$973$	10.0987	0.323749
$974$	−43.2767	−1.38667
$975$	0	0
$976$	−18.1975	−0.582488
$977$	28.1686	0.901192	0.450596	−	0.892728i	$-0.351211\pi$
$977$	28.1686	0.901192	0.450596	+	0.892728i	$0.351211\pi$
$978$	0	0
$979$	89.1898	2.85052
$980$	−0.919582	−0.0293750
$981$	0	0
$982$	−68.1514	−2.17480
$983$	32.9894	1.05220	0.526099	−	0.850424i	$-0.323654\pi$
$983$	32.9894	1.05220	0.526099	+	0.850424i	$0.323654\pi$
$984$	0	0
$985$	−1.86636	−0.0594673
$986$	1.77013	0.0563726
$987$	0	0
$988$	−9.20957	−0.292995
$989$	31.3602	0.997197
$990$	0	0
$991$	32.9379	1.04631	0.523154	−	0.852238i	$-0.324755\pi$
$991$	32.9379	1.04631	0.523154	+	0.852238i	$0.324755\pi$
$992$	−69.6614	−2.21175
$993$	0	0
$994$	8.88112	0.281692
$995$	−3.73130	−0.118290
$996$	0	0
$997$	−36.5637	−1.15798	−0.578992	−	0.815334i	$-0.696554\pi$
$997$	−36.5637	−1.15798	−0.578992	+	0.815334i	$0.696554\pi$
$998$	−1.69842	−0.0537625
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1197.2.a.o.1.2		5
3.2	odd	2		399.2.a.g.1.4	✓	5
7.6	odd	2		8379.2.a.cb.1.2		5
12.11	even	2		6384.2.a.cf.1.3		5
15.14	odd	2		9975.2.a.bp.1.2		5
21.20	even	2		2793.2.a.bg.1.4		5
57.56	even	2		7581.2.a.w.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.4	✓	5	3.2	odd	2
1197.2.a.o.1.2		5	1.1	even	1	trivial
2793.2.a.bg.1.4		5	21.20	even	2
6384.2.a.cf.1.3		5	12.11	even	2
7581.2.a.w.1.2		5	57.56	even	2
8379.2.a.cb.1.2		5	7.6	odd	2
9975.2.a.bp.1.2		5	15.14	odd	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:	\( N \)	\(=\)	\( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1197.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09027 q^{2} +2.36924 q^{4} -0.388134 q^{5} +1.00000 q^{7} -0.771813 q^{8} +0.811305 q^{10} -6.41859 q^{11} +3.88714 q^{13} -2.09027 q^{14} -3.12518 q^{16} +4.98727 q^{17} -1.00000 q^{19} -0.919582 q^{20} +13.4166 q^{22} -3.44207 q^{23} -4.84935 q^{25} -8.12518 q^{26} +2.36924 q^{28} -0.169801 q^{29} -8.62562 q^{31} +8.07611 q^{32} -10.4247 q^{34} -0.388134 q^{35} +7.37540 q^{37} +2.09027 q^{38} +0.299567 q^{40} +8.77968 q^{41} -9.11087 q^{43} -15.2072 q^{44} +7.19485 q^{46} +4.80672 q^{47} +1.00000 q^{49} +10.1365 q^{50} +9.20957 q^{52} -8.42933 q^{53} +2.49127 q^{55} -0.771813 q^{56} +0.354931 q^{58} -2.97652 q^{59} +5.82287 q^{61} +18.0299 q^{62} -10.6309 q^{64} -1.50873 q^{65} -14.9980 q^{67} +11.8160 q^{68} +0.811305 q^{70} -4.24879 q^{71} +13.5757 q^{73} -15.4166 q^{74} -2.36924 q^{76} -6.41859 q^{77} -1.01431 q^{79} +1.21299 q^{80} -18.3519 q^{82} +4.32147 q^{83} -1.93573 q^{85} +19.0442 q^{86} +4.95395 q^{88} -13.8955 q^{89} +3.88714 q^{91} -8.15508 q^{92} -10.0474 q^{94} +0.388134 q^{95} -13.7743 q^{97} -2.09027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} + 7 q^{4} - 4 q^{5} + 5 q^{7} - 9 q^{8} - 6 q^{10} - 8 q^{11} - 6 q^{13} - 3 q^{14} + 19 q^{16} - 12 q^{17} - 5 q^{19} - 8 q^{20} - 12 q^{22} - 12 q^{23} + 15 q^{25} - 6 q^{26} + 7 q^{28}+ \cdots - 3 q^{98}+O(q^{100}) \) 5 * q - 3 * q^2 + 7 * q^4 - 4 * q^5 + 5 * q^7 - 9 * q^8 - 6 * q^10 - 8 * q^11 - 6 * q^13 - 3 * q^14 + 19 * q^16 - 12 * q^17 - 5 * q^19 - 8 * q^20 - 12 * q^22 - 12 * q^23 + 15 * q^25 - 6 * q^26 + 7 * q^28 - 4 * q^29 - 8 * q^31 - 31 * q^32 + 2 * q^34 - 4 * q^35 + 2 * q^37 + 3 * q^38 - 18 * q^40 - 10 * q^41 - 16 * q^43 + 4 * q^44 + 16 * q^46 + 2 * q^47 + 5 * q^49 + 39 * q^50 - 22 * q^52 - 12 * q^55 - 9 * q^56 + 2 * q^58 + 4 * q^59 - 14 * q^61 + 36 * q^62 + 11 * q^64 - 32 * q^65 - 20 * q^67 - 8 * q^68 - 6 * q^70 + 6 * q^71 + 10 * q^73 + 2 * q^74 - 7 * q^76 - 8 * q^77 + 20 * q^80 - 38 * q^82 - 6 * q^83 + 8 * q^85 + 36 * q^86 - 40 * q^88 - 26 * q^89 - 6 * q^91 + 48 * q^92 + 24 * q^94 + 4 * q^95 - 18 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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