LMFDB - Embedded newform 425.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(1,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.66068$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.24214 q^{2} +1.66068 q^{3} -0.457096 q^{4} +2.06279 q^{6} +4.35698 q^{7} -3.05205 q^{8} -0.242137 q^{9} +0.760798 q^{11} -0.759092 q^{12} +3.53632 q^{13} +5.41196 q^{14} -2.87687 q^{16} +1.00000 q^{17} -0.300767 q^{18} +0.972823 q^{19} +7.23556 q^{21} +0.945015 q^{22} -7.47476 q^{23} -5.06848 q^{24} +4.39260 q^{26} -5.38416 q^{27} -1.99156 q^{28} -5.25686 q^{29} +8.62336 q^{31} +2.53063 q^{32} +1.26344 q^{33} +1.24214 q^{34} +0.110680 q^{36} -5.94137 q^{37} +1.20838 q^{38} +5.87271 q^{39} -4.29419 q^{41} +8.98755 q^{42} -3.98985 q^{43} -0.347758 q^{44} -9.28467 q^{46} -6.28404 q^{47} -4.77756 q^{48} +11.9833 q^{49} +1.66068 q^{51} -1.61644 q^{52} +1.54290 q^{53} -6.68786 q^{54} -13.2977 q^{56} +1.61555 q^{57} -6.52974 q^{58} +2.66849 q^{59} -3.32136 q^{61} +10.7114 q^{62} -1.05498 q^{63} +8.89713 q^{64} +1.56937 q^{66} -15.9868 q^{67} -0.457096 q^{68} -12.4132 q^{69} -11.0768 q^{71} +0.739013 q^{72} +15.3340 q^{73} -7.37999 q^{74} -0.444674 q^{76} +3.31478 q^{77} +7.29470 q^{78} -4.45680 q^{79} -8.21496 q^{81} -5.33397 q^{82} +6.71396 q^{83} -3.30735 q^{84} -4.95594 q^{86} -8.72998 q^{87} -2.32199 q^{88} +12.3839 q^{89} +15.4077 q^{91} +3.41669 q^{92} +14.3207 q^{93} -7.80564 q^{94} +4.20258 q^{96} +7.19823 q^{97} +14.8849 q^{98} -0.184217 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) $ 5 * q - q^2 + q^3 + 11 * q^4 + 3 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^11 + 17 * q^12 - 3 * q^13 - 7 * q^14 + 27 * q^16 + 5 * q^17 - 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 - 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 + 2 * q^29 + 21 * q^31 - 9 * q^32 + 12 * q^33 - q^34 + 22 * q^36 - 2 * q^37 - 34 * q^38 + 23 * q^39 - 8 * q^41 + 25 * q^42 - 4 * q^43 - 4 * q^44 - 6 * q^46 - 2 * q^47 + 51 * q^48 + 10 * q^49 + q^51 - 13 * q^52 + 21 * q^53 - 25 * q^54 - 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 - 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 - 12 * q^67 + 11 * q^68 + 21 * q^71 - 58 * q^72 + 22 * q^73 - 50 * q^74 - 14 * q^76 - 32 * q^77 - 53 * q^78 + 41 * q^79 - 35 * q^81 + 28 * q^82 - 8 * q^83 - 71 * q^84 - 36 * q^86 - 4 * q^87 + 14 * q^88 - 10 * q^89 + 13 * q^91 + 56 * q^92 + 9 * q^93 - 10 * q^94 - 85 * q^96 - 20 * q^97 + 38 * q^98 - 14 * 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 $

$2$	1.24214	0.878323	0.439162	−	0.898408i	$-0.355276\pi$
$2$	1.24214	0.878323	0.439162	+	0.898408i	$0.355276\pi$
$3$	1.66068	0.958795	0.479397	−	0.877598i	$-0.340855\pi$
$3$	1.66068	0.958795	0.479397	+	0.877598i	$0.340855\pi$
$4$	−0.457096	−0.228548
$5$	0	0
$5$	0	0
$6$	2.06279	0.842132
$7$	4.35698	1.64678	0.823392	−	0.567473i	$-0.192079\pi$
$7$	4.35698	1.64678	0.823392	+	0.567473i	$0.192079\pi$
$8$	−3.05205	−1.07906
$9$	−0.242137	−0.0807122
$10$	0	0
$11$	0.760798	0.229389	0.114695	−	0.993401i	$-0.463411\pi$
$11$	0.760798	0.229389	0.114695	+	0.993401i	$0.463411\pi$
$12$	−0.759092	−0.219131
$13$	3.53632	0.980800	0.490400	−	0.871498i	$-0.336851\pi$
$13$	3.53632	0.980800	0.490400	+	0.871498i	$0.336851\pi$
$14$	5.41196	1.44641
$15$	0	0
$16$	−2.87687	−0.719217
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.300767	−0.0708914
$19$	0.972823	0.223181	0.111590	−	0.993754i	$-0.464406\pi$
$19$	0.972823	0.223181	0.111590	+	0.993754i	$0.464406\pi$
$20$	0	0
$21$	7.23556	1.57893
$22$	0.945015	0.201478
$23$	−7.47476	−1.55859	−0.779297	−	0.626654i	$-0.784424\pi$
$23$	−7.47476	−1.55859	−0.779297	+	0.626654i	$0.784424\pi$
$24$	−5.06848	−1.03460
$25$	0	0
$26$	4.39260	0.861459
$27$	−5.38416	−1.03618
$28$	−1.99156	−0.376370
$29$	−5.25686	−0.976175	−0.488087	−	0.872795i	$-0.662305\pi$
$29$	−5.25686	−0.976175	−0.488087	+	0.872795i	$0.662305\pi$
$30$	0	0
$31$	8.62336	1.54880	0.774400	−	0.632696i	$-0.218052\pi$
$31$	8.62336	1.54880	0.774400	+	0.632696i	$0.218052\pi$
$32$	2.53063	0.447357
$33$	1.26344	0.219937
$34$	1.24214	0.213025
$35$	0	0
$36$	0.110680	0.0184466
$37$	−5.94137	−0.976755	−0.488378	−	0.872632i	$-0.662411\pi$
$37$	−5.94137	−0.976755	−0.488378	+	0.872632i	$0.662411\pi$
$38$	1.20838	0.196025
$39$	5.87271	0.940386
$40$	0	0
$41$	−4.29419	−0.670639	−0.335320	−	0.942104i	$-0.608844\pi$
$41$	−4.29419	−0.670639	−0.335320	+	0.942104i	$0.608844\pi$
$42$	8.98755	1.38681
$43$	−3.98985	−0.608447	−0.304223	−	0.952601i	$-0.598397\pi$
$43$	−3.98985	−0.608447	−0.304223	+	0.952601i	$0.598397\pi$
$44$	−0.347758	−0.0524265
$45$	0	0
$46$	−9.28467	−1.36895
$47$	−6.28404	−0.916621	−0.458311	−	0.888792i	$-0.651545\pi$
$47$	−6.28404	−0.916621	−0.458311	+	0.888792i	$0.651545\pi$
$48$	−4.77756	−0.689582
$49$	11.9833	1.71190
$50$	0	0
$51$	1.66068	0.232542
$52$	−1.61644	−0.224160
$53$	1.54290	0.211934	0.105967	−	0.994370i	$-0.466206\pi$
$53$	1.54290	0.211934	0.105967	+	0.994370i	$0.466206\pi$
$54$	−6.68786	−0.910102
$55$	0	0
$56$	−13.2977	−1.77698
$57$	1.61555	0.213985
$58$	−6.52974	−0.857397
$59$	2.66849	0.347408	0.173704	−	0.984798i	$-0.444426\pi$
$59$	2.66849	0.347408	0.173704	+	0.984798i	$0.444426\pi$
$60$	0	0
$61$	−3.32136	−0.425257	−0.212628	−	0.977133i	$-0.568202\pi$
$61$	−3.32136	−0.425257	−0.212628	+	0.977133i	$0.568202\pi$
$62$	10.7114	1.36035
$63$	−1.05498	−0.132916
$64$	8.89713	1.11214
$65$	0	0
$66$	1.56937	0.193176
$67$	−15.9868	−1.95310	−0.976552	−	0.215283i	$-0.930932\pi$
$67$	−15.9868	−1.95310	−0.976552	+	0.215283i	$0.930932\pi$
$68$	−0.457096	−0.0554311
$69$	−12.4132	−1.49437
$70$	0	0
$71$	−11.0768	−1.31458	−0.657288	−	0.753640i	$-0.728296\pi$
$71$	−11.0768	−1.31458	−0.657288	+	0.753640i	$0.728296\pi$
$72$	0.739013	0.0870935
$73$	15.3340	1.79471	0.897353	−	0.441314i	$-0.145488\pi$
$73$	15.3340	1.79471	0.897353	+	0.441314i	$0.145488\pi$
$74$	−7.37999	−0.857907
$75$	0	0
$76$	−0.444674	−0.0510076
$77$	3.31478	0.377755
$78$	7.29470	0.825963
$79$	−4.45680	−0.501429	−0.250715	−	0.968061i	$-0.580666\pi$
$79$	−4.45680	−0.501429	−0.250715	+	0.968061i	$0.580666\pi$
$80$	0	0
$81$	−8.21496	−0.912773
$82$	−5.33397	−0.589038
$83$	6.71396	0.736953	0.368476	−	0.929637i	$-0.379880\pi$
$83$	6.71396	0.736953	0.368476	+	0.929637i	$0.379880\pi$
$84$	−3.30735	−0.360861
$85$	0	0
$86$	−4.95594	−0.534413
$87$	−8.72998	−0.935952
$88$	−2.32199	−0.247525
$89$	12.3839	1.31269	0.656343	−	0.754462i	$-0.272102\pi$
$89$	12.3839	1.31269	0.656343	+	0.754462i	$0.272102\pi$
$90$	0	0
$91$	15.4077	1.61516
$92$	3.41669	0.356214
$93$	14.3207	1.48498
$94$	−7.80564	−0.805090
$95$	0	0
$96$	4.20258	0.428924
$97$	7.19823	0.730870	0.365435	−	0.930837i	$-0.380920\pi$
$97$	7.19823	0.730870	0.365435	+	0.930837i	$0.380920\pi$
$98$	14.8849	1.50360
$99$	−0.184217	−0.0185145
$100$	0	0
$101$	4.01473	0.399480	0.199740	−	0.979849i	$-0.435990\pi$
$101$	4.01473	0.399480	0.199740	+	0.979849i	$0.435990\pi$
$102$	2.06279	0.204247
$103$	−2.63686	−0.259817	−0.129909	−	0.991526i	$-0.541468\pi$
$103$	−2.63686	−0.259817	−0.129909	+	0.991526i	$0.541468\pi$
$104$	−10.7930	−1.05834
$105$	0	0
$106$	1.91650	0.186147
$107$	16.7283	1.61718	0.808591	−	0.588371i	$-0.200230\pi$
$107$	16.7283	1.61718	0.808591	+	0.588371i	$0.200230\pi$
$108$	2.46108	0.236817
$109$	5.76143	0.551845	0.275922	−	0.961180i	$-0.411017\pi$
$109$	5.76143	0.551845	0.275922	+	0.961180i	$0.411017\pi$
$110$	0	0
$111$	−9.86672	−0.936508
$112$	−12.5345	−1.18440
$113$	−12.2153	−1.14912	−0.574558	−	0.818464i	$-0.694826\pi$
$113$	−12.2153	−1.14912	−0.574558	+	0.818464i	$0.694826\pi$
$114$	2.00673	0.187948
$115$	0	0
$116$	2.40289	0.223103
$117$	−0.856273	−0.0791625
$118$	3.31463	0.305136
$119$	4.35698	0.399404
$120$	0	0
$121$	−10.4212	−0.947381
$122$	−4.12559	−0.373513
$123$	−7.13128	−0.643006
$124$	−3.94171	−0.353976
$125$	0	0
$126$	−1.31044	−0.116743
$127$	−14.3839	−1.27636	−0.638181	−	0.769887i	$-0.720313\pi$
$127$	−14.3839	−1.27636	−0.638181	+	0.769887i	$0.720313\pi$
$128$	5.99019	0.529463
$129$	−6.62588	−0.583376
$130$	0	0
$131$	4.65117	0.406374	0.203187	−	0.979140i	$-0.434870\pi$
$131$	4.65117	0.406374	0.203187	+	0.979140i	$0.434870\pi$
$132$	−0.577516	−0.0502663
$133$	4.23857	0.367531
$134$	−19.8578	−1.71546
$135$	0	0
$136$	−3.05205	−0.261711
$137$	−7.48784	−0.639729	−0.319865	−	0.947463i	$-0.603637\pi$
$137$	−7.48784	−0.639729	−0.319865	+	0.947463i	$0.603637\pi$
$138$	−15.4189	−1.31254
$139$	−6.43327	−0.545663	−0.272831	−	0.962062i	$-0.587960\pi$
$139$	−6.43327	−0.545663	−0.272831	+	0.962062i	$0.587960\pi$
$140$	0	0
$141$	−10.4358	−0.878852
$142$	−13.7589	−1.15462
$143$	2.69043	0.224985
$144$	0.696596	0.0580496
$145$	0	0
$146$	19.0469	1.57633
$147$	19.9004	1.64136
$148$	2.71578	0.223236
$149$	22.4881	1.84230	0.921150	−	0.389207i	$-0.127251\pi$
$149$	22.4881	1.84230	0.921150	+	0.389207i	$0.127251\pi$
$150$	0	0
$151$	24.0412	1.95644	0.978222	−	0.207560i	$-0.0665523\pi$
$151$	24.0412	1.95644	0.978222	+	0.207560i	$0.0665523\pi$
$152$	−2.96910	−0.240826
$153$	−0.242137	−0.0195756
$154$	4.11741	0.331791
$155$	0	0
$156$	−2.68439	−0.214923
$157$	−16.0081	−1.27759	−0.638795	−	0.769377i	$-0.720567\pi$
$157$	−16.0081	−1.27759	−0.638795	+	0.769377i	$0.720567\pi$
$158$	−5.53596	−0.440417
$159$	2.56227	0.203201
$160$	0	0
$161$	−32.5674	−2.56667
$162$	−10.2041	−0.801710
$163$	5.75463	0.450738	0.225369	−	0.974274i	$-0.427641\pi$
$163$	5.75463	0.450738	0.225369	+	0.974274i	$0.427641\pi$
$164$	1.96286	0.153273
$165$	0	0
$166$	8.33966	0.647283
$167$	22.0361	1.70521	0.852604	−	0.522558i	$-0.175022\pi$
$167$	22.0361	1.70521	0.852604	+	0.522558i	$0.175022\pi$
$168$	−22.0833	−1.70376
$169$	−0.494420	−0.0380323
$170$	0	0
$171$	−0.235556	−0.0180134
$172$	1.82375	0.139059
$173$	2.90159	0.220604	0.110302	−	0.993898i	$-0.464818\pi$
$173$	2.90159	0.220604	0.110302	+	0.993898i	$0.464818\pi$
$174$	−10.8438	−0.822068
$175$	0	0
$176$	−2.18872	−0.164981
$177$	4.43151	0.333093
$178$	15.3825	1.15296
$179$	−11.8511	−0.885793	−0.442897	−	0.896573i	$-0.646049\pi$
$179$	−11.8511	−0.885793	−0.442897	+	0.896573i	$0.646049\pi$
$180$	0	0
$181$	8.52974	0.634011	0.317005	−	0.948424i	$-0.397323\pi$
$181$	8.52974	0.634011	0.317005	+	0.948424i	$0.397323\pi$
$182$	19.1385	1.41864
$183$	−5.51573	−0.407734
$184$	22.8133	1.68182
$185$	0	0
$186$	17.7882	1.30429
$187$	0.760798	0.0556351
$188$	2.87241	0.209492
$189$	−23.4587	−1.70637
$190$	0	0
$191$	19.8182	1.43400	0.716999	−	0.697074i	$-0.245515\pi$
$191$	19.8182	1.43400	0.716999	+	0.697074i	$0.245515\pi$
$192$	14.7753	1.06632
$193$	−6.70723	−0.482797	−0.241398	−	0.970426i	$-0.577606\pi$
$193$	−6.70723	−0.482797	−0.241398	+	0.970426i	$0.577606\pi$
$194$	8.94119	0.641940
$195$	0	0
$196$	−5.47751	−0.391251
$197$	−10.5399	−0.750936	−0.375468	−	0.926835i	$-0.622518\pi$
$197$	−10.5399	−0.750936	−0.375468	+	0.926835i	$0.622518\pi$
$198$	−0.228823	−0.0162617
$199$	21.9649	1.55705	0.778525	−	0.627613i	$-0.215968\pi$
$199$	21.9649	1.55705	0.778525	+	0.627613i	$0.215968\pi$
$200$	0	0
$201$	−26.5490	−1.87263
$202$	4.98684	0.350873
$203$	−22.9040	−1.60755
$204$	−0.759092	−0.0531471
$205$	0	0
$206$	−3.27534	−0.228203
$207$	1.80991	0.125798
$208$	−10.1735	−0.705408
$209$	0.740122	0.0511953
$210$	0	0
$211$	11.8082	0.812910	0.406455	−	0.913671i	$-0.366765\pi$
$211$	11.8082	0.812910	0.406455	+	0.913671i	$0.366765\pi$
$212$	−0.705256	−0.0484372
$213$	−18.3951	−1.26041
$214$	20.7788	1.42041
$215$	0	0
$216$	16.4327	1.11810
$217$	37.5718	2.55054
$218$	7.15648	0.484698
$219$	25.4648	1.72075
$220$	0	0
$221$	3.53632	0.237879
$222$	−12.2558	−0.822557
$223$	11.4881	0.769303	0.384651	−	0.923062i	$-0.374322\pi$
$223$	11.4881	0.769303	0.384651	+	0.923062i	$0.374322\pi$
$224$	11.0259	0.736700
$225$	0	0
$226$	−15.1730	−1.00929
$227$	2.25207	0.149475	0.0747375	−	0.997203i	$-0.476188\pi$
$227$	2.25207	0.149475	0.0747375	+	0.997203i	$0.476188\pi$
$228$	−0.738462	−0.0489058
$229$	13.9788	0.923748	0.461874	−	0.886946i	$-0.347177\pi$
$229$	13.9788	0.923748	0.461874	+	0.886946i	$0.347177\pi$
$230$	0	0
$231$	5.50480	0.362189
$232$	16.0442	1.05335
$233$	−13.8452	−0.907032	−0.453516	−	0.891248i	$-0.649830\pi$
$233$	−13.8452	−0.907032	−0.453516	+	0.891248i	$0.649830\pi$
$234$	−1.06361	−0.0695303
$235$	0	0
$236$	−1.21976	−0.0793995
$237$	−7.40133	−0.480768
$238$	5.41196	0.350806
$239$	1.80564	0.116797	0.0583985	−	0.998293i	$-0.481401\pi$
$239$	1.80564	0.116797	0.0583985	+	0.998293i	$0.481401\pi$
$240$	0	0
$241$	19.8637	1.27953	0.639767	−	0.768569i	$-0.279031\pi$
$241$	19.8637	1.27953	0.639767	+	0.768569i	$0.279031\pi$
$242$	−12.9445	−0.832106
$243$	2.51004	0.161019
$244$	1.51818	0.0971917
$245$	0	0
$246$	−8.85802	−0.564767
$247$	3.44022	0.218896
$248$	−26.3189	−1.67125
$249$	11.1497	0.706587
$250$	0	0
$251$	−12.5070	−0.789436	−0.394718	−	0.918802i	$-0.629158\pi$
$251$	−12.5070	−0.789436	−0.394718	+	0.918802i	$0.629158\pi$
$252$	0.482230	0.0303776
$253$	−5.68678	−0.357525
$254$	−17.8667	−1.12106
$255$	0	0
$256$	−10.3536	−0.647102
$257$	−29.2572	−1.82501	−0.912506	−	0.409063i	$-0.865855\pi$
$257$	−29.2572	−1.82501	−0.912506	+	0.409063i	$0.865855\pi$
$258$	−8.23024	−0.512393
$259$	−25.8864	−1.60850
$260$	0	0
$261$	1.27288	0.0787893
$262$	5.77738	0.356928
$263$	7.23110	0.445889	0.222944	−	0.974831i	$-0.428433\pi$
$263$	7.23110	0.445889	0.222944	+	0.974831i	$0.428433\pi$
$264$	−3.85609	−0.237326
$265$	0	0
$266$	5.26488	0.322811
$267$	20.5657	1.25860
$268$	7.30753	0.446378
$269$	−24.6296	−1.50169	−0.750846	−	0.660478i	$-0.770354\pi$
$269$	−24.6296	−1.50169	−0.750846	+	0.660478i	$0.770354\pi$
$270$	0	0
$271$	20.8399	1.26594	0.632968	−	0.774178i	$-0.281836\pi$
$271$	20.8399	1.26594	0.632968	+	0.774178i	$0.281836\pi$
$272$	−2.87687	−0.174436
$273$	25.5873	1.54861
$274$	−9.30092	−0.561889
$275$	0	0
$276$	5.67403	0.341536
$277$	−2.32364	−0.139614	−0.0698070	−	0.997561i	$-0.522238\pi$
$277$	−2.32364	−0.139614	−0.0698070	+	0.997561i	$0.522238\pi$
$278$	−7.99100	−0.479268
$279$	−2.08803	−0.125007
$280$	0	0
$281$	6.22523	0.371366	0.185683	−	0.982610i	$-0.440550\pi$
$281$	6.22523	0.371366	0.185683	+	0.982610i	$0.440550\pi$
$282$	−12.9627	−0.771916
$283$	−14.5892	−0.867237	−0.433618	−	0.901097i	$-0.642763\pi$
$283$	−14.5892	−0.867237	−0.433618	+	0.901097i	$0.642763\pi$
$284$	5.06317	0.300444
$285$	0	0
$286$	3.34188	0.197609
$287$	−18.7097	−1.10440
$288$	−0.612759	−0.0361072
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.9540	0.700754
$292$	−7.00910	−0.410177
$293$	27.0877	1.58248	0.791239	−	0.611506i	$-0.209436\pi$
$293$	27.0877	1.58248	0.791239	+	0.611506i	$0.209436\pi$
$294$	24.7190	1.44164
$295$	0	0
$296$	18.1334	1.05398
$297$	−4.09626	−0.237689
$298$	27.9333	1.61814
$299$	−26.4332	−1.52867
$300$	0	0
$301$	−17.3837	−1.00198
$302$	29.8624	1.71839
$303$	6.66718	0.383020
$304$	−2.79869	−0.160516
$305$	0	0
$306$	−0.300767	−0.0171937
$307$	27.9509	1.59524	0.797622	−	0.603158i	$-0.206091\pi$
$307$	27.9509	1.59524	0.797622	+	0.603158i	$0.206091\pi$
$308$	−1.51518	−0.0863351
$309$	−4.37898	−0.249111
$310$	0	0
$311$	−5.02081	−0.284704	−0.142352	−	0.989816i	$-0.545467\pi$
$311$	−5.02081	−0.284704	−0.142352	+	0.989816i	$0.545467\pi$
$312$	−17.9238	−1.01473
$313$	0.909917	0.0514315	0.0257158	−	0.999669i	$-0.491814\pi$
$313$	0.909917	0.0514315	0.0257158	+	0.999669i	$0.491814\pi$
$314$	−19.8843	−1.12214
$315$	0	0
$316$	2.03719	0.114601
$317$	13.8870	0.779973	0.389986	−	0.920821i	$-0.372480\pi$
$317$	13.8870	0.779973	0.389986	+	0.920821i	$0.372480\pi$
$318$	3.18269	0.178476
$319$	−3.99941	−0.223924
$320$	0	0
$321$	27.7803	1.55055
$322$	−40.4531	−2.25436
$323$	0.972823	0.0541293
$324$	3.75503	0.208613
$325$	0	0
$326$	7.14804	0.395893
$327$	9.56790	0.529106
$328$	13.1061	0.723662
$329$	−27.3794	−1.50948
$330$	0	0
$331$	−21.1851	−1.16444	−0.582218	−	0.813032i	$-0.697815\pi$
$331$	−21.1851	−1.16444	−0.582218	+	0.813032i	$0.697815\pi$
$332$	−3.06893	−0.168429
$333$	1.43862	0.0788361
$334$	27.3719	1.49772
$335$	0	0
$336$	−20.8158	−1.13559
$337$	4.11972	0.224415	0.112208	−	0.993685i	$-0.464208\pi$
$337$	4.11972	0.224415	0.112208	+	0.993685i	$0.464208\pi$
$338$	−0.614137	−0.0334046
$339$	−20.2857	−1.10177
$340$	0	0
$341$	6.56064	0.355278
$342$	−0.292593	−0.0158216
$343$	21.7120	1.17234
$344$	12.1772	0.656552
$345$	0	0
$346$	3.60417	0.193761
$347$	−9.86542	−0.529603	−0.264802	−	0.964303i	$-0.585306\pi$
$347$	−9.86542	−0.529603	−0.264802	+	0.964303i	$0.585306\pi$
$348$	3.99044	0.213910
$349$	0.00227611	0.000121837 0	6.09187e−5	−	1.00000i	$-0.499981\pi$
$349$	0.00227611	0.000121837 0	6.09187e−5		1.00000i	$0.499981\pi$
$350$	0	0
$351$	−19.0401	−1.01629
$352$	1.92530	0.102619
$353$	−13.5739	−0.722468	−0.361234	−	0.932475i	$-0.617645\pi$
$353$	−13.5739	−0.722468	−0.361234	+	0.932475i	$0.617645\pi$
$354$	5.50454	0.292563
$355$	0	0
$356$	−5.66062	−0.300012
$357$	7.23556	0.382946
$358$	−14.7207	−0.778013
$359$	15.2378	0.804222	0.402111	−	0.915591i	$-0.368277\pi$
$359$	15.2378	0.804222	0.402111	+	0.915591i	$0.368277\pi$
$360$	0	0
$361$	−18.0536	−0.950190
$362$	10.5951	0.556866
$363$	−17.3063	−0.908344
$364$	−7.04280	−0.369143
$365$	0	0
$366$	−6.85129	−0.358122
$367$	7.78199	0.406217	0.203108	−	0.979156i	$-0.434896\pi$
$367$	7.78199	0.406217	0.203108	+	0.979156i	$0.434896\pi$
$368$	21.5039	1.12097
$369$	1.03978	0.0541288
$370$	0	0
$371$	6.72240	0.349010
$372$	−6.54592	−0.339390
$373$	−13.8234	−0.715747	−0.357874	−	0.933770i	$-0.616498\pi$
$373$	−13.8234	−0.715747	−0.357874	+	0.933770i	$0.616498\pi$
$374$	0.945015	0.0488656
$375$	0	0
$376$	19.1792	0.989092
$377$	−18.5900	−0.957432
$378$	−29.1389	−1.49874
$379$	11.2337	0.577035	0.288517	−	0.957475i	$-0.406838\pi$
$379$	11.2337	0.577035	0.288517	+	0.957475i	$0.406838\pi$
$380$	0	0
$381$	−23.8870	−1.22377
$382$	24.6170	1.25951
$383$	−6.55464	−0.334927	−0.167463	−	0.985878i	$-0.553558\pi$
$383$	−6.55464	−0.334927	−0.167463	+	0.985878i	$0.553558\pi$
$384$	9.94779	0.507646
$385$	0	0
$386$	−8.33129	−0.424052
$387$	0.966090	0.0491091
$388$	−3.29029	−0.167039
$389$	−29.9290	−1.51746	−0.758731	−	0.651404i	$-0.774180\pi$
$389$	−29.9290	−1.51746	−0.758731	+	0.651404i	$0.774180\pi$
$390$	0	0
$391$	−7.47476	−0.378015
$392$	−36.5735	−1.84724
$393$	7.72411	0.389630
$394$	−13.0920	−0.659565
$395$	0	0
$396$	0.0842050	0.00423146
$397$	−15.5987	−0.782876	−0.391438	−	0.920204i	$-0.628022\pi$
$397$	−15.5987	−0.782876	−0.391438	+	0.920204i	$0.628022\pi$
$398$	27.2834	1.36759
$399$	7.03892	0.352387
$400$	0	0
$401$	−16.0024	−0.799123	−0.399562	−	0.916706i	$-0.630838\pi$
$401$	−16.0024	−0.799123	−0.399562	+	0.916706i	$0.630838\pi$
$402$	−32.9775	−1.64477
$403$	30.4950	1.51906
$404$	−1.83512	−0.0913005
$405$	0	0
$406$	−28.4500	−1.41195
$407$	−4.52018	−0.224057
$408$	−5.06848	−0.250927
$409$	−8.18867	−0.404904	−0.202452	−	0.979292i	$-0.564891\pi$
$409$	−8.18867	−0.404904	−0.202452	+	0.979292i	$0.564891\pi$
$410$	0	0
$411$	−12.4349	−0.613369
$412$	1.20530	0.0593808
$413$	11.6266	0.572106
$414$	2.24816	0.110491
$415$	0	0
$416$	8.94914	0.438768
$417$	−10.6836	−0.523179
$418$	0.919333	0.0449660
$419$	0.547404	0.0267424	0.0133712	−	0.999911i	$-0.495744\pi$
$419$	0.547404	0.0267424	0.0133712	+	0.999911i	$0.495744\pi$
$420$	0	0
$421$	−11.1506	−0.543449	−0.271724	−	0.962375i	$-0.587594\pi$
$421$	−11.1506	−0.543449	−0.271724	+	0.962375i	$0.587594\pi$
$422$	14.6674	0.713998
$423$	1.52160	0.0739826
$424$	−4.70902	−0.228690
$425$	0	0
$426$	−22.8492	−1.10705
$427$	−14.4711	−0.700306
$428$	−7.64643	−0.369604
$429$	4.46794	0.215714
$430$	0	0
$431$	22.0900	1.06404	0.532019	−	0.846732i	$-0.321433\pi$
$431$	22.0900	1.06404	0.532019	+	0.846732i	$0.321433\pi$
$432$	15.4895	0.745240
$433$	−18.3983	−0.884165	−0.442083	−	0.896974i	$-0.645760\pi$
$433$	−18.3983	−0.884165	−0.442083	+	0.896974i	$0.645760\pi$
$434$	46.6693	2.24020
$435$	0	0
$436$	−2.63353	−0.126123
$437$	−7.27162	−0.347849
$438$	31.6308	1.51138
$439$	20.6254	0.984397	0.492198	−	0.870483i	$-0.336193\pi$
$439$	20.6254	0.984397	0.492198	+	0.870483i	$0.336193\pi$
$440$	0	0
$441$	−2.90159	−0.138171
$442$	4.39260	0.208935
$443$	−24.8065	−1.17859	−0.589296	−	0.807917i	$-0.700595\pi$
$443$	−24.8065	−1.17859	−0.589296	+	0.807917i	$0.700595\pi$
$444$	4.51004	0.214037
$445$	0	0
$446$	14.2698	0.675697
$447$	37.3456	1.76639
$448$	38.7646	1.83146
$449$	−1.43136	−0.0675502	−0.0337751	−	0.999429i	$-0.510753\pi$
$449$	−1.43136	−0.0675502	−0.0337751	+	0.999429i	$0.510753\pi$
$450$	0	0
$451$	−3.26701	−0.153837
$452$	5.58355	0.262628
$453$	39.9248	1.87583
$454$	2.79738	0.131287
$455$	0	0
$456$	−4.93074	−0.230903
$457$	3.35940	0.157146	0.0785730	−	0.996908i	$-0.474964\pi$
$457$	3.35940	0.157146	0.0785730	+	0.996908i	$0.474964\pi$
$458$	17.3636	0.811349
$459$	−5.38416	−0.251311
$460$	0	0
$461$	−10.9685	−0.510856	−0.255428	−	0.966828i	$-0.582216\pi$
$461$	−10.9685	−0.510856	−0.255428	+	0.966828i	$0.582216\pi$
$462$	6.83771	0.318119
$463$	−38.5578	−1.79193	−0.895966	−	0.444123i	$-0.853515\pi$
$463$	−38.5578	−1.79193	−0.895966	+	0.444123i	$0.853515\pi$
$464$	15.1233	0.702082
$465$	0	0
$466$	−17.1977	−0.796667
$467$	21.7749	1.00762	0.503810	−	0.863814i	$-0.331931\pi$
$467$	21.7749	1.00762	0.503810	+	0.863814i	$0.331931\pi$
$468$	0.391400	0.0180925
$469$	−69.6543	−3.21634
$470$	0	0
$471$	−26.5844	−1.22495
$472$	−8.14437	−0.374875
$473$	−3.03547	−0.139571
$474$	−9.19346	−0.422270
$475$	0	0
$476$	−1.99156	−0.0912830
$477$	−0.373594	−0.0171057
$478$	2.24285	0.102585
$479$	−4.62934	−0.211520	−0.105760	−	0.994392i	$-0.533728\pi$
$479$	−4.62934	−0.211520	−0.105760	+	0.994392i	$0.533728\pi$
$480$	0	0
$481$	−21.0106	−0.958001
$482$	24.6734	1.12384
$483$	−54.0840	−2.46091
$484$	4.76349	0.216522
$485$	0	0
$486$	3.11781	0.141427
$487$	−14.1331	−0.640432	−0.320216	−	0.947345i	$-0.603755\pi$
$487$	−14.1331	−0.640432	−0.320216	+	0.947345i	$0.603755\pi$
$488$	10.1370	0.458879
$489$	9.55662	0.432165
$490$	0	0
$491$	−19.1530	−0.864365	−0.432182	−	0.901786i	$-0.642256\pi$
$491$	−19.1530	−0.864365	−0.432182	+	0.901786i	$0.642256\pi$
$492$	3.25968	0.146958
$493$	−5.25686	−0.236757
$494$	4.27322	0.192261
$495$	0	0
$496$	−24.8083	−1.11392
$497$	−48.2614	−2.16482
$498$	13.8495	0.620611
$499$	29.8400	1.33582	0.667912	−	0.744241i	$-0.267188\pi$
$499$	29.8400	1.33582	0.667912	+	0.744241i	$0.267188\pi$
$500$	0	0
$501$	36.5950	1.63494
$502$	−15.5354	−0.693380
$503$	−16.5014	−0.735760	−0.367880	−	0.929873i	$-0.619916\pi$
$503$	−16.5014	−0.735760	−0.367880	+	0.929873i	$0.619916\pi$
$504$	3.21987	0.143424
$505$	0	0
$506$	−7.06376	−0.314022
$507$	−0.821074	−0.0364652
$508$	6.57481	0.291710
$509$	−11.5798	−0.513266	−0.256633	−	0.966509i	$-0.582613\pi$
$509$	−11.5798	−0.513266	−0.256633	+	0.966509i	$0.582613\pi$
$510$	0	0
$511$	66.8098	2.95549
$512$	−24.8410	−1.09783
$513$	−5.23783	−0.231256
$514$	−36.3414	−1.60295
$515$	0	0
$516$	3.02866	0.133330
$517$	−4.78089	−0.210263
$518$	−32.1545	−1.41279
$519$	4.81862	0.211514
$520$	0	0
$521$	−6.29332	−0.275715	−0.137858	−	0.990452i	$-0.544022\pi$
$521$	−6.29332	−0.275715	−0.137858	+	0.990452i	$0.544022\pi$
$522$	1.58109	0.0692024
$523$	−29.5093	−1.29035	−0.645175	−	0.764035i	$-0.723216\pi$
$523$	−29.5093	−1.29035	−0.645175	+	0.764035i	$0.723216\pi$
$524$	−2.12603	−0.0928761
$525$	0	0
$526$	8.98201	0.391634
$527$	8.62336	0.375639
$528$	−3.63476	−0.158183
$529$	32.8720	1.42922
$530$	0	0
$531$	−0.646139	−0.0280401
$532$	−1.93744	−0.0839985
$533$	−15.1856	−0.657763
$534$	25.5454	1.10546
$535$	0	0
$536$	48.7926	2.10752
$537$	−19.6809	−0.849294
$538$	−30.5933	−1.31897
$539$	9.11685	0.392691
$540$	0	0
$541$	14.6495	0.629829	0.314915	−	0.949120i	$-0.398024\pi$
$541$	14.6495	0.629829	0.314915	+	0.949120i	$0.398024\pi$
$542$	25.8861	1.11190
$543$	14.1652	0.607886
$544$	2.53063	0.108500
$545$	0	0
$546$	31.7829	1.36018
$547$	−2.44752	−0.104648	−0.0523242	−	0.998630i	$-0.516663\pi$
$547$	−2.44752	−0.104648	−0.0523242	+	0.998630i	$0.516663\pi$
$548$	3.42267	0.146209
$549$	0.804224	0.0343234
$550$	0	0
$551$	−5.11400	−0.217864
$552$	37.8857	1.61252
$553$	−19.4182	−0.825746
$554$	−2.88628	−0.122626
$555$	0	0
$556$	2.94063	0.124710
$557$	−21.6889	−0.918988	−0.459494	−	0.888181i	$-0.651969\pi$
$557$	−21.6889	−0.918988	−0.459494	+	0.888181i	$0.651969\pi$
$558$	−2.59362	−0.109797
$559$	−14.1094	−0.596765
$560$	0	0
$561$	1.26344	0.0533426
$562$	7.73259	0.326179
$563$	−21.2443	−0.895340	−0.447670	−	0.894199i	$-0.647746\pi$
$563$	−21.2443	−0.895340	−0.447670	+	0.894199i	$0.647746\pi$
$564$	4.77016	0.200860
$565$	0	0
$566$	−18.1218	−0.761714
$567$	−35.7924	−1.50314
$568$	33.8070	1.41851
$569$	13.7433	0.576150	0.288075	−	0.957608i	$-0.406985\pi$
$569$	13.7433	0.576150	0.288075	+	0.957608i	$0.406985\pi$
$570$	0	0
$571$	−5.50164	−0.230237	−0.115118	−	0.993352i	$-0.536725\pi$
$571$	−5.50164	−0.230237	−0.115118	+	0.993352i	$0.536725\pi$
$572$	−1.22979	−0.0514199
$573$	32.9118	1.37491
$574$	−23.2400	−0.970018
$575$	0	0
$576$	−2.15432	−0.0897634
$577$	17.2437	0.717865	0.358932	−	0.933364i	$-0.383141\pi$
$577$	17.2437	0.717865	0.358932	+	0.933364i	$0.383141\pi$
$578$	1.24214	0.0516661
$579$	−11.1386	−0.462903
$580$	0	0
$581$	29.2526	1.21360
$582$	14.8485	0.615489
$583$	1.17384	0.0486154
$584$	−46.8000	−1.93660
$585$	0	0
$586$	33.6466	1.38993
$587$	−10.0319	−0.414061	−0.207031	−	0.978334i	$-0.566380\pi$
$587$	−10.0319	−0.414061	−0.207031	+	0.978334i	$0.566380\pi$
$588$	−9.09640	−0.375129
$589$	8.38900	0.345663
$590$	0	0
$591$	−17.5034	−0.719994
$592$	17.0925	0.702499
$593$	30.0714	1.23488	0.617442	−	0.786616i	$-0.288169\pi$
$593$	30.0714	1.23488	0.617442	+	0.786616i	$0.288169\pi$
$594$	−5.08811	−0.208768
$595$	0	0
$596$	−10.2793	−0.421055
$597$	36.4767	1.49289
$598$	−32.8336	−1.34267
$599$	17.0655	0.697279	0.348640	−	0.937257i	$-0.386644\pi$
$599$	17.0655	0.697279	0.348640	+	0.937257i	$0.386644\pi$
$600$	0	0
$601$	5.26575	0.214794	0.107397	−	0.994216i	$-0.465748\pi$
$601$	5.26575	0.214794	0.107397	+	0.994216i	$0.465748\pi$
$602$	−21.5929	−0.880063
$603$	3.87100	0.157639
$604$	−10.9891	−0.447142
$605$	0	0
$606$	8.28155	0.336415
$607$	−30.6311	−1.24328	−0.621639	−	0.783304i	$-0.713533\pi$
$607$	−30.6311	−1.24328	−0.621639	+	0.783304i	$0.713533\pi$
$608$	2.46186	0.0998416
$609$	−38.0363	−1.54131
$610$	0	0
$611$	−22.2224	−0.899022
$612$	0.110680	0.00447397
$613$	−10.6054	−0.428348	−0.214174	−	0.976796i	$-0.568706\pi$
$613$	−10.6054	−0.428348	−0.214174	+	0.976796i	$0.568706\pi$
$614$	34.7189	1.40114
$615$	0	0
$616$	−10.1169	−0.407621
$617$	25.7600	1.03706	0.518529	−	0.855060i	$-0.326480\pi$
$617$	25.7600	1.03706	0.518529	+	0.855060i	$0.326480\pi$
$618$	−5.43929	−0.218800
$619$	0.534831	0.0214967	0.0107483	−	0.999942i	$-0.496579\pi$
$619$	0.534831	0.0214967	0.0107483	+	0.999942i	$0.496579\pi$
$620$	0	0
$621$	40.2453	1.61499
$622$	−6.23653	−0.250062
$623$	53.9562	2.16171
$624$	−16.8950	−0.676342
$625$	0	0
$626$	1.13024	0.0451735
$627$	1.22911	0.0490858
$628$	7.31727	0.291991
$629$	−5.94137	−0.236898
$630$	0	0
$631$	24.8614	0.989718	0.494859	−	0.868973i	$-0.335220\pi$
$631$	24.8614	0.989718	0.494859	+	0.868973i	$0.335220\pi$
$632$	13.6024	0.541074
$633$	19.6097	0.779414
$634$	17.2496	0.685068
$635$	0	0
$636$	−1.17121	−0.0464413
$637$	42.3767	1.67903
$638$	−4.96782	−0.196678
$639$	2.68210	0.106102
$640$	0	0
$641$	5.12703	0.202505	0.101253	−	0.994861i	$-0.467715\pi$
$641$	5.12703	0.202505	0.101253	+	0.994861i	$0.467715\pi$
$642$	34.5070	1.36188
$643$	0.0459139	0.00181067	0.000905334	−	1.00000i	$-0.499712\pi$
$643$	0.0459139	0.00181067	0.000905334		1.00000i	$0.499712\pi$
$644$	14.8864	0.586608
$645$	0	0
$646$	1.20838	0.0475430
$647$	13.6150	0.535259	0.267630	−	0.963522i	$-0.413760\pi$
$647$	13.6150	0.535259	0.267630	+	0.963522i	$0.413760\pi$
$648$	25.0725	0.984939
$649$	2.03018	0.0796917
$650$	0	0
$651$	62.3948	2.44544
$652$	−2.63042	−0.103015
$653$	43.7700	1.71285	0.856426	−	0.516269i	$-0.172680\pi$
$653$	43.7700	1.71285	0.856426	+	0.516269i	$0.172680\pi$
$654$	11.8846	0.464726
$655$	0	0
$656$	12.3538	0.482335
$657$	−3.71292	−0.144855
$658$	−34.0090	−1.32581
$659$	2.09312	0.0815364	0.0407682	−	0.999169i	$-0.487019\pi$
$659$	2.09312	0.0815364	0.0407682	+	0.999169i	$0.487019\pi$
$660$	0	0
$661$	−30.7363	−1.19550	−0.597751	−	0.801682i	$-0.703939\pi$
$661$	−30.7363	−1.19550	−0.597751	+	0.801682i	$0.703939\pi$
$662$	−26.3148	−1.02275
$663$	5.87271	0.228077
$664$	−20.4913	−0.795218
$665$	0	0
$666$	1.78697	0.0692436
$667$	39.2938	1.52146
$668$	−10.0726	−0.389722
$669$	19.0781	0.737604
$670$	0	0
$671$	−2.52689	−0.0975494
$672$	18.3105	0.706345
$673$	−17.7912	−0.685801	−0.342900	−	0.939372i	$-0.611409\pi$
$673$	−17.7912	−0.685801	−0.342900	+	0.939372i	$0.611409\pi$
$674$	5.11725	0.197109
$675$	0	0
$676$	0.225997	0.00869221
$677$	−1.89003	−0.0726398	−0.0363199	−	0.999340i	$-0.511564\pi$
$677$	−1.89003	−0.0726398	−0.0363199	+	0.999340i	$0.511564\pi$
$678$	−25.1976	−0.967707
$679$	31.3626	1.20358
$680$	0	0
$681$	3.73997	0.143316
$682$	8.14921	0.312049
$683$	27.6707	1.05879	0.529394	−	0.848376i	$-0.322419\pi$
$683$	27.6707	1.05879	0.529394	+	0.848376i	$0.322419\pi$
$684$	0.107672	0.00411694
$685$	0	0
$686$	26.9693	1.02969
$687$	23.2144	0.885685
$688$	11.4783	0.437606
$689$	5.45621	0.207865
$690$	0	0
$691$	−35.1592	−1.33752	−0.668759	−	0.743479i	$-0.733174\pi$
$691$	−35.1592	−1.33752	−0.668759	+	0.743479i	$0.733174\pi$
$692$	−1.32631	−0.0504186
$693$	−0.802630	−0.0304894
$694$	−12.2542	−0.465163
$695$	0	0
$696$	26.6443	1.00995
$697$	−4.29419	−0.162654
$698$	0.00282724	0.000107013 0
$699$	−22.9925	−0.869657
$700$	0	0
$701$	0.521416	0.0196936	0.00984680	−	0.999952i	$-0.496866\pi$
$701$	0.521416	0.0196936	0.00984680	+	0.999952i	$0.496866\pi$
$702$	−23.6504	−0.892628
$703$	−5.77990	−0.217993
$704$	6.76892	0.255113
$705$	0	0
$706$	−16.8607	−0.634561
$707$	17.4921	0.657857
$708$	−2.02563	−0.0761278
$709$	4.17407	0.156761	0.0783803	−	0.996924i	$-0.475025\pi$
$709$	4.17407	0.156761	0.0783803	+	0.996924i	$0.475025\pi$
$710$	0	0
$711$	1.07916	0.0404715
$712$	−37.7962	−1.41647
$713$	−64.4575	−2.41395
$714$	8.98755	0.336351
$715$	0	0
$716$	5.41710	0.202446
$717$	2.99859	0.111984
$718$	18.9275	0.706367
$719$	20.8193	0.776429	0.388215	−	0.921569i	$-0.373092\pi$
$719$	20.8193	0.776429	0.388215	+	0.921569i	$0.373092\pi$
$720$	0	0
$721$	−11.4887	−0.427863
$722$	−22.4251	−0.834574
$723$	32.9873	1.22681
$724$	−3.89892	−0.144902
$725$	0	0
$726$	−21.4968	−0.797819
$727$	31.0761	1.15255	0.576275	−	0.817256i	$-0.304506\pi$
$727$	31.0761	1.15255	0.576275	+	0.817256i	$0.304506\pi$
$728$	−47.0250	−1.74286
$729$	28.8133	1.06716
$730$	0	0
$731$	−3.98985	−0.147570
$732$	2.52122	0.0931869
$733$	−14.7053	−0.543151	−0.271576	−	0.962417i	$-0.587545\pi$
$733$	−14.7053	−0.543151	−0.271576	+	0.962417i	$0.587545\pi$
$734$	9.66630	0.356790
$735$	0	0
$736$	−18.9159	−0.697248
$737$	−12.1628	−0.448021
$738$	1.29155	0.0475426
$739$	16.0741	0.591294	0.295647	−	0.955297i	$-0.404465\pi$
$739$	16.0741	0.591294	0.295647	+	0.955297i	$0.404465\pi$
$740$	0	0
$741$	5.71310	0.209876
$742$	8.35014	0.306543
$743$	−18.5224	−0.679521	−0.339760	−	0.940512i	$-0.610346\pi$
$743$	−18.5224	−0.679521	−0.339760	+	0.940512i	$0.610346\pi$
$744$	−43.7073	−1.60239
$745$	0	0
$746$	−17.1705	−0.628658
$747$	−1.62570	−0.0594811
$748$	−0.347758	−0.0127153
$749$	72.8847	2.66315
$750$	0	0
$751$	36.7405	1.34068	0.670340	−	0.742054i	$-0.266148\pi$
$751$	36.7405	1.34068	0.670340	+	0.742054i	$0.266148\pi$
$752$	18.0784	0.659250
$753$	−20.7702	−0.756908
$754$	−23.0913	−0.840935
$755$	0	0
$756$	10.7229	0.389987
$757$	22.5807	0.820709	0.410354	−	0.911926i	$-0.365405\pi$
$757$	22.5807	0.820709	0.410354	+	0.911926i	$0.365405\pi$
$758$	13.9538	0.506823
$759$	−9.44394	−0.342793
$760$	0	0
$761$	30.7466	1.11456	0.557281	−	0.830324i	$-0.311845\pi$
$761$	30.7466	1.11456	0.557281	+	0.830324i	$0.311845\pi$
$762$	−29.6709	−1.07486
$763$	25.1024	0.908769
$764$	−9.05885	−0.327738
$765$	0	0
$766$	−8.14176	−0.294174
$767$	9.43664	0.340737
$768$	−17.1941	−0.620438
$769$	−22.2798	−0.803429	−0.401714	−	0.915765i	$-0.631586\pi$
$769$	−22.2798	−0.803429	−0.401714	+	0.915765i	$0.631586\pi$
$770$	0	0
$771$	−48.5868	−1.74981
$772$	3.06585	0.110342
$773$	−41.5644	−1.49497	−0.747483	−	0.664281i	$-0.768738\pi$
$773$	−41.5644	−1.49497	−0.747483	+	0.664281i	$0.768738\pi$
$774$	1.20002	0.0431337
$775$	0	0
$776$	−21.9694	−0.788654
$777$	−42.9891	−1.54223
$778$	−37.1760	−1.33282
$779$	−4.17748	−0.149674
$780$	0	0
$781$	−8.42722	−0.301550
$782$	−9.28467	−0.332019
$783$	28.3038	1.01149
$784$	−34.4743	−1.23123
$785$	0	0
$786$	9.59440	0.342221
$787$	44.0006	1.56845	0.784226	−	0.620475i	$-0.213060\pi$
$787$	44.0006	1.56845	0.784226	+	0.620475i	$0.213060\pi$
$788$	4.81775	0.171625
$789$	12.0086	0.427516
$790$	0	0
$791$	−53.2217	−1.89234
$792$	0.562240	0.0199783
$793$	−11.7454	−0.417092
$794$	−19.3757	−0.687618
$795$	0	0
$796$	−10.0401	−0.355861
$797$	20.5026	0.726238	0.363119	−	0.931743i	$-0.381712\pi$
$797$	20.5026	0.726238	0.363119	+	0.931743i	$0.381712\pi$
$798$	8.74330	0.309509
$799$	−6.28404	−0.222313
$800$	0	0
$801$	−2.99859	−0.105950
$802$	−19.8772	−0.701888
$803$	11.6661	0.411686
$804$	12.1355	0.427985
$805$	0	0
$806$	37.8789	1.33423
$807$	−40.9019	−1.43981
$808$	−12.2531	−0.431064
$809$	36.9474	1.29900	0.649500	−	0.760362i	$-0.274978\pi$
$809$	36.9474	1.29900	0.649500	+	0.760362i	$0.274978\pi$
$810$	0	0
$811$	31.3337	1.10028	0.550138	−	0.835074i	$-0.314575\pi$
$811$	31.3337	1.10028	0.550138	+	0.835074i	$0.314575\pi$
$812$	10.4694	0.367402
$813$	34.6085	1.21377
$814$	−5.61469	−0.196795
$815$	0	0
$816$	−4.77756	−0.167248
$817$	−3.88142	−0.135794
$818$	−10.1715	−0.355637
$819$	−3.73077	−0.130364
$820$	0	0
$821$	−19.1104	−0.666958	−0.333479	−	0.942757i	$-0.608223\pi$
$821$	−19.1104	−0.666958	−0.333479	+	0.942757i	$0.608223\pi$
$822$	−15.4459	−0.538736
$823$	−3.72559	−0.129866	−0.0649330	−	0.997890i	$-0.520683\pi$
$823$	−3.72559	−0.129866	−0.0649330	+	0.997890i	$0.520683\pi$
$824$	8.04782	0.280359
$825$	0	0
$826$	14.4418	0.502494
$827$	−52.6665	−1.83139	−0.915697	−	0.401868i	$-0.868361\pi$
$827$	−52.6665	−1.83139	−0.915697	+	0.401868i	$0.868361\pi$
$828$	−0.827305	−0.0287508
$829$	2.65690	0.0922780	0.0461390	−	0.998935i	$-0.485308\pi$
$829$	2.65690	0.0922780	0.0461390	+	0.998935i	$0.485308\pi$
$830$	0	0
$831$	−3.85882	−0.133861
$832$	31.4631	1.09079
$833$	11.9833	0.415196
$834$	−13.2705	−0.459520
$835$	0	0
$836$	−0.338307	−0.0117006
$837$	−46.4295	−1.60484
$838$	0.679950	0.0234885
$839$	34.0348	1.17501	0.587506	−	0.809220i	$-0.300110\pi$
$839$	34.0348	1.17501	0.587506	+	0.809220i	$0.300110\pi$
$840$	0	0
$841$	−1.36539	−0.0470824
$842$	−13.8506	−0.477324
$843$	10.3381	0.356064
$844$	−5.39749	−0.185789
$845$	0	0
$846$	1.89003	0.0649806
$847$	−45.4049	−1.56013
$848$	−4.43873	−0.152427
$849$	−24.2280	−0.831502
$850$	0	0
$851$	44.4103	1.52237
$852$	8.40831	0.288064
$853$	57.3607	1.96399	0.981997	−	0.188896i	$-0.0604909\pi$
$853$	57.3607	1.96399	0.981997	+	0.188896i	$0.0604909\pi$
$854$	−17.9751	−0.615095
$855$	0	0
$856$	−51.0555	−1.74504
$857$	−13.8195	−0.472065	−0.236033	−	0.971745i	$-0.575847\pi$
$857$	−13.8195	−0.472065	−0.236033	+	0.971745i	$0.575847\pi$
$858$	5.54980	0.189467
$859$	−44.4717	−1.51736	−0.758678	−	0.651466i	$-0.774154\pi$
$859$	−44.4717	−1.51736	−0.758678	+	0.651466i	$0.774154\pi$
$860$	0	0
$861$	−31.0708	−1.05889
$862$	27.4388	0.934570
$863$	24.4487	0.832242	0.416121	−	0.909309i	$-0.363389\pi$
$863$	24.4487	0.832242	0.416121	+	0.909309i	$0.363389\pi$
$864$	−13.6253	−0.463543
$865$	0	0
$866$	−22.8532	−0.776583
$867$	1.66068	0.0563997
$868$	−17.1739	−0.582921
$869$	−3.39073	−0.115023
$870$	0	0
$871$	−56.5346	−1.91560
$872$	−17.5842	−0.595475
$873$	−1.74296	−0.0589901
$874$	−9.03234	−0.305524
$875$	0	0
$876$	−11.6399	−0.393275
$877$	41.8463	1.41305	0.706524	−	0.707689i	$-0.250262\pi$
$877$	41.8463	1.41305	0.706524	+	0.707689i	$0.250262\pi$
$878$	25.6196	0.864619
$879$	44.9840	1.51727
$880$	0	0
$881$	−34.7002	−1.16908	−0.584540	−	0.811365i	$-0.698725\pi$
$881$	−34.7002	−1.16908	−0.584540	+	0.811365i	$0.698725\pi$
$882$	−3.60417	−0.121359
$883$	3.43890	0.115728	0.0578641	−	0.998324i	$-0.481571\pi$
$883$	3.43890	0.115728	0.0578641	+	0.998324i	$0.481571\pi$
$884$	−1.61644	−0.0543668
$885$	0	0
$886$	−30.8131	−1.03518
$887$	5.18772	0.174187	0.0870933	−	0.996200i	$-0.472242\pi$
$887$	5.18772	0.174187	0.0870933	+	0.996200i	$0.472242\pi$
$888$	30.1137	1.01055
$889$	−62.6702	−2.10189
$890$	0	0
$891$	−6.24993	−0.209380
$892$	−5.25119	−0.175823
$893$	−6.11326	−0.204572
$894$	46.3884	1.55146
$895$	0	0
$896$	26.0991	0.871911
$897$	−43.8971	−1.46568
$898$	−1.77795	−0.0593309
$899$	−45.3318	−1.51190
$900$	0	0
$901$	1.54290	0.0514016
$902$	−4.05807	−0.135119
$903$	−28.8688	−0.960694
$904$	37.2816	1.23997
$905$	0	0
$906$	49.5920	1.64758
$907$	−1.70414	−0.0565850	−0.0282925	−	0.999600i	$-0.509007\pi$
$907$	−1.70414	−0.0565850	−0.0282925	+	0.999600i	$0.509007\pi$
$908$	−1.02941	−0.0341623
$909$	−0.972112	−0.0322429
$910$	0	0
$911$	−5.77752	−0.191418	−0.0957089	−	0.995409i	$-0.530512\pi$
$911$	−5.77752	−0.191418	−0.0957089	+	0.995409i	$0.530512\pi$
$912$	−4.64773	−0.153902
$913$	5.10797	0.169049
$914$	4.17283	0.138025
$915$	0	0
$916$	−6.38968	−0.211121
$917$	20.2650	0.669210
$918$	−6.68786	−0.220732
$919$	−9.48267	−0.312804	−0.156402	−	0.987693i	$-0.549990\pi$
$919$	−9.48267	−0.312804	−0.156402	+	0.987693i	$0.549990\pi$
$920$	0	0
$921$	46.4176	1.52951
$922$	−13.6244	−0.448697
$923$	−39.1712	−1.28933
$924$	−2.51622	−0.0827777
$925$	0	0
$926$	−47.8940	−1.57390
$927$	0.638480	0.0209704
$928$	−13.3032	−0.436699
$929$	−42.7899	−1.40389	−0.701946	−	0.712230i	$-0.747685\pi$
$929$	−42.7899	−1.40389	−0.701946	+	0.712230i	$0.747685\pi$
$930$	0	0
$931$	11.6576	0.382063
$932$	6.32861	0.207300
$933$	−8.33797	−0.272973
$934$	27.0474	0.885017
$935$	0	0
$936$	2.61339	0.0854213
$937$	−33.6154	−1.09817	−0.549084	−	0.835767i	$-0.685023\pi$
$937$	−33.6154	−1.09817	−0.549084	+	0.835767i	$0.685023\pi$
$938$	−86.5202	−2.82498
$939$	1.51108	0.0493123
$940$	0	0
$941$	−24.8395	−0.809745	−0.404873	−	0.914373i	$-0.632684\pi$
$941$	−24.8395	−0.809745	−0.404873	+	0.914373i	$0.632684\pi$
$942$	−33.0215	−1.07590
$943$	32.0980	1.04525
$944$	−7.67690	−0.249862
$945$	0	0
$946$	−3.77047	−0.122589
$947$	46.0481	1.49636	0.748181	−	0.663494i	$-0.230927\pi$
$947$	46.0481	1.49636	0.748181	+	0.663494i	$0.230927\pi$
$948$	3.38312	0.109879
$949$	54.2259	1.76025
$950$	0	0
$951$	23.0619	0.747834
$952$	−13.2977	−0.430982
$953$	38.5949	1.25021	0.625106	−	0.780540i	$-0.285056\pi$
$953$	38.5949	1.25021	0.625106	+	0.780540i	$0.285056\pi$
$954$	−0.464054	−0.0150243
$955$	0	0
$956$	−0.825350	−0.0266937
$957$	−6.64175	−0.214697
$958$	−5.75027	−0.185783
$959$	−32.6244	−1.05350
$960$	0	0
$961$	43.3623	1.39878
$962$	−26.0980	−0.841435
$963$	−4.05053	−0.130526
$964$	−9.07963	−0.292435
$965$	0	0
$966$	−67.1798	−2.16147
$967$	−27.7170	−0.891318	−0.445659	−	0.895203i	$-0.647031\pi$
$967$	−27.7170	−0.891318	−0.445659	+	0.895203i	$0.647031\pi$
$968$	31.8060	1.02228
$969$	1.61555	0.0518989
$970$	0	0
$971$	−35.8211	−1.14955	−0.574777	−	0.818310i	$-0.694911\pi$
$971$	−35.8211	−1.14955	−0.574777	+	0.818310i	$0.694911\pi$
$972$	−1.14733	−0.0368006
$973$	−28.0296	−0.898589
$974$	−17.5552	−0.562506
$975$	0	0
$976$	9.55513	0.305852
$977$	−21.1597	−0.676960	−0.338480	−	0.940974i	$-0.609913\pi$
$977$	−21.1597	−0.676960	−0.338480	+	0.940974i	$0.609913\pi$
$978$	11.8706	0.379581
$979$	9.42162	0.301116
$980$	0	0
$981$	−1.39505	−0.0445406
$982$	−23.7907	−0.759192
$983$	53.4481	1.70473	0.852364	−	0.522948i	$-0.175168\pi$
$983$	53.4481	1.70473	0.852364	+	0.522948i	$0.175168\pi$
$984$	21.7650	0.693843
$985$	0	0
$986$	−6.52974	−0.207949
$987$	−45.4685	−1.44728
$988$	−1.57251	−0.0500282
$989$	29.8232	0.948322
$990$	0	0
$991$	23.3619	0.742115	0.371057	−	0.928610i	$-0.378995\pi$
$991$	23.3619	0.742115	0.371057	+	0.928610i	$0.378995\pi$
$992$	21.8226	0.692867
$993$	−35.1817	−1.11646
$994$	−59.9473	−1.90141
$995$	0	0
$996$	−5.09651	−0.161489
$997$	19.9898	0.633084	0.316542	−	0.948578i	$-0.397478\pi$
$997$	19.9898	0.633084	0.316542	+	0.948578i	$0.397478\pi$
$998$	37.0654	1.17328
$999$	31.9893	1.01210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.i.1.4	✓	5
3.2	odd	2		3825.2.a.bq.1.2		5
4.3	odd	2		6800.2.a.bz.1.2		5
5.2	odd	4		425.2.b.f.324.7		10
5.3	odd	4		425.2.b.f.324.4		10
5.4	even	2		425.2.a.j.1.2	yes	5
15.14	odd	2		3825.2.a.bl.1.4		5
17.16	even	2		7225.2.a.x.1.4		5
20.19	odd	2		6800.2.a.cd.1.4		5
85.84	even	2		7225.2.a.y.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.4	✓	5	1.1	even	1	trivial
425.2.a.j.1.2	yes	5	5.4	even	2
425.2.b.f.324.4		10	5.3	odd	4
425.2.b.f.324.7		10	5.2	odd	4
3825.2.a.bl.1.4		5	15.14	odd	2
3825.2.a.bq.1.2		5	3.2	odd	2
6800.2.a.bz.1.2		5	4.3	odd	2
6800.2.a.cd.1.4		5	20.19	odd	2
7225.2.a.x.1.4		5	17.16	even	2
7225.2.a.y.1.2		5	85.84	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24214 q^{2} +1.66068 q^{3} -0.457096 q^{4} +2.06279 q^{6} +4.35698 q^{7} -3.05205 q^{8} -0.242137 q^{9} +0.760798 q^{11} -0.759092 q^{12} +3.53632 q^{13} +5.41196 q^{14} -2.87687 q^{16} +1.00000 q^{17} -0.300767 q^{18} +0.972823 q^{19} +7.23556 q^{21} +0.945015 q^{22} -7.47476 q^{23} -5.06848 q^{24} +4.39260 q^{26} -5.38416 q^{27} -1.99156 q^{28} -5.25686 q^{29} +8.62336 q^{31} +2.53063 q^{32} +1.26344 q^{33} +1.24214 q^{34} +0.110680 q^{36} -5.94137 q^{37} +1.20838 q^{38} +5.87271 q^{39} -4.29419 q^{41} +8.98755 q^{42} -3.98985 q^{43} -0.347758 q^{44} -9.28467 q^{46} -6.28404 q^{47} -4.77756 q^{48} +11.9833 q^{49} +1.66068 q^{51} -1.61644 q^{52} +1.54290 q^{53} -6.68786 q^{54} -13.2977 q^{56} +1.61555 q^{57} -6.52974 q^{58} +2.66849 q^{59} -3.32136 q^{61} +10.7114 q^{62} -1.05498 q^{63} +8.89713 q^{64} +1.56937 q^{66} -15.9868 q^{67} -0.457096 q^{68} -12.4132 q^{69} -11.0768 q^{71} +0.739013 q^{72} +15.3340 q^{73} -7.37999 q^{74} -0.444674 q^{76} +3.31478 q^{77} +7.29470 q^{78} -4.45680 q^{79} -8.21496 q^{81} -5.33397 q^{82} +6.71396 q^{83} -3.30735 q^{84} -4.95594 q^{86} -8.72998 q^{87} -2.32199 q^{88} +12.3839 q^{89} +15.4077 q^{91} +3.41669 q^{92} +14.3207 q^{93} -7.80564 q^{94} +4.20258 q^{96} +7.19823 q^{97} +14.8849 q^{98} -0.184217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 5 * q - q^2 + q^3 + 11 * q^4 + 3 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^11 + 17 * q^12 - 3 * q^13 - 7 * q^14 + 27 * q^16 + 5 * q^17 - 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 - 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 + 2 * q^29 + 21 * q^31 - 9 * q^32 + 12 * q^33 - q^34 + 22 * q^36 - 2 * q^37 - 34 * q^38 + 23 * q^39 - 8 * q^41 + 25 * q^42 - 4 * q^43 - 4 * q^44 - 6 * q^46 - 2 * q^47 + 51 * q^48 + 10 * q^49 + q^51 - 13 * q^52 + 21 * q^53 - 25 * q^54 - 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 - 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 - 12 * q^67 + 11 * q^68 + 21 * q^71 - 58 * q^72 + 22 * q^73 - 50 * q^74 - 14 * q^76 - 32 * q^77 - 53 * q^78 + 41 * q^79 - 35 * q^81 + 28 * q^82 - 8 * q^83 - 71 * q^84 - 36 * q^86 - 4 * q^87 + 14 * q^88 - 10 * q^89 + 13 * q^91 + 56 * q^92 + 9 * q^93 - 10 * q^94 - 85 * q^96 - 20 * q^97 + 38 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

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Database

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